Micro-espectroscopía óptica en microcavidades de cristal fotónico

Micro-espectroscopía óptica en microcavidades
de cristal fotónico con nanoestructuras cúanticas
semiconductoras III-V.
Per
Josep Canet Ferrer
Per a optar al grau de Doctor en Física
Director:
Juan Pascual Martínez Pastor
Chapter 1. Motivation.
Chapter 2. Theoretical Background.
1.
Optical properties of semiconductors III-V.
a) Optical density of states.
b) Radiative transition rate of a single emitter.
c) Spectral linewidth.
2. Quantum confinement: from bulk material to 1D and 0d nano-structures.
a) Electronic density of states in confined sistems.
3. Photonic crystal microcavities.
4. Light-matter interaction: coupling degree.
a) Free-Space spontaneous emission.
b) Weak Coupling.
c) Strong Coupling.
Chapter 3. Set-up, experimental & Characterization techniques.
1. Introduction to confocal microscopy.
2. Devices.
a) Optical Excitation.
b) Cryogenics, optics and optomechanics.
c) Detectors.
3. Experimental techniques.
4. Data processing.
Chapter 4. Characterization of site controlled InAs Quantum dots grown on
GaAs(001) pre-patterned substrates.
1. Introduccion.
2. Experimental details.
3. Optical characterization.
a) Electronic structure.
b) Effects of the buffer layer on the optical properties of the QDs.
c) Temperature evolution of the micro-PL.
4. Conclusions.
Chapter 5. Optical properties of self-assembled Quantum Wires.
1. Introduction to the InAs/InP system: comparison with InAs/GaAs.
2. Experimental details and samples.
3. Optical Characterization.
a) PL dependence on the temperature.
b) PL dependence on the excitation power.
c) Time resolved measurements.
4. Conclusions.
Chapter 6. L7.type photonic crystal micro-cavities with embedded selfassembled InAs/InP quantum wires.
1. Photonic crystal microcavities with embedded semiconductor nanostructures.
2. Experimental details and samples.
a) QWR Epitaxy and basic properties
b) Fabrication details of L7 microcavities
Index
i
c) Set-up for optical micro-spectroscopy
3. Expression of the Purcell factor for an ensemble of quantum
emitters embedded into a photonic crystal microcavity.
4. Results and Discussion.
a) Figure of merit.
b) Polarization Mismatch.
c) Spectral and Spatial detuning factors.
5. Optimization possibilities and future work: Isolated QWRs.
6. Conclusions.
Chapter 7. Purcell factor dependence on the excitation power and
temperature.
1. General remarks
2. Purcell Factor as a function of the excitation power.
3. Purcell Factor as a function of the temperature.
4. Conclusions
Chapter 8. General Conclusions and outlook.
Index
ii
Capítulo 1. Motivación
The superstitions of today are the scientific facts of tomorrow.
Von Helsing (Jonh Balderston), Dracula; 1927.
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Desde hace ya más de dos décadas, las nanoestructuras semiconductoras se están
abriendo camino en el campo de la optoelectrónica y la fotónica. Sus propiedades
ópticas estrechamente relacionadas con sus bajas dimensionalidades confieren a estos
materiales gran interés a la vez que un enorme potencial tecnológico [1-3]. A través de
múltiples investigaciones se ha desarrollado una amplia gama de aplicaciones y
dispositivos aprovechando el gran número de posibles combinaciones de materiales,
sus distintas morfologías y las diferentes técnicas empleadas para la fabricación de
nanoestructuras [4]. Se podría decir que todo empezó en la década de los ochenta, con
la aparición de los pozos cuánticos, pues originarían una pequeña revolución debido a
su uso en láseres de alta potencia a 808 nm (bombeo de cristales dopados) y en láseres
a 780 nm para lectores de discos compactos [5]. Estos sistemas (conocidos como
heteroestructuras en el caso más general) estaban formados por una capa delgada
(GaAs, por ejemplo) de grosor que del orden de la longitud de onda de De Broglie,
encerrada por sendas barreras de un material de anchura de banda prohibida superior
(aleación AlGaAs, por ejemplo) al de dicha capa. En el interior de ésta los portadores
de carga (electrones y huecos) estaban sometidos a un potencial de confinamiento
determinado por la diferencia entre las bandas prohibidas de ambos materiales, según
una determinada dirección del espacio (la dirección del crecimiento de las capas),
viendo reducida su movilidad en esta dirección. Consecuentemente, la relación de
dispersión E(k) será sensible a “los efectos de borde”, típicamente despreciados en los
cálculos macroscópicos, generando en consecuencia la aparición de estados
adicionales a los del material masivo. Este efecto se denominó efecto de
confinamiento cuántico por tamaño [6]. Hoy en día es bien sabido que la
fotoluminiscencia de pozos cuánticos es considerablemente mayor que en un material
masivo (en éste la densidad de estados es nula en el borde de absorción), y, además, la
energía de emisión en materiales nanoestructurados se puede sintonizar levemente
variando el tamaño o la estequiometria de las diferentes fases solidas que componen
la heteroestructura. Siguiendo ese procedimiento y la evolución de las técnicas de
crecimiento epitaxial se han obtenido más recientemente nanostructuras
semiconductoras que confinan los portadores en dos dimensiones (hilos cuánticos, con
dimensionalidad 1D para el movimiento de los portadores) e incluso en las tres
dimensiones espaciales (puntos cuánticos, con dimensionalidad 0D pues los
portadores estarían atrapados en su interior). Estos últimos también se les suele llamar
“átomos artificiales”, ya que su espectro de emisión está caracterizado por un
conjunto de líneas discretas. En la práctica, debido a la interacción de los portadores
confinados con la red cristalina (fonones acústicos y ópticos) la imagen del punto
cuántico como átomo requiere de algunos matices, como se mostrará en el capítulo 4
de esta Tesis.
Recientemente, las nuevas técnicas de caracterización óptica de alta resolución
espacial abrieron el campo de la espectroscopia de nanoestructuras semiconductoras
aisladas y de sus aplicaciones [7]. Gracias a estas técnicas se puede obtener
información correspondiente a una sola nanoestructura [8]; como los fenómenos
asociados a la interacción coulombiana en complejos excitónicos formados por
combinaciones de varios portadores. Cada nivel electrónico en la banda de conducción
(valencia) de un punto cuántico sólo puede ser ocupado por dos electrones (huecos)
con distinto espín, debido al principio de exclusión de Pauli. Al ir añadiendo electrones
y huecos al punto cuántico se forman lo que se conoce como complejos excitónicos,
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como el caso del exciton neutro (un electron y un hueco), trión negativo (dos
electrones y un hueco), el trión positivo (dos huecos y un electrón), el biexciton (dos
electrones y dos huecos) [9]; así como complejos con mayor numero de portadores. Al
estudiar estos sistemas con técnicas ópticas de alta resolución espacial, se pueden
observar los efectos de estructura fina asociados a la interacción de intercambio, la
correlación coulombiana entre las distintas cargas confinadas en el punto cuántico, los
procesos de desfase y relajación, o incluso los efectos de interacción entre los núcleos
atómicos con electrones o huecos confinados [10]. Por otra parte, el excitón también
ha sido etiquetado como un átomo hidrogenoide, pues surge de la cuantización del
movimiento relativo electrón-hueco debido a la atracción Coulombiana entre ambos.
Esta imagen conceptual también tiene en cuenta que los portadores tienen una vida
finita. Esto es, al cabo de un determinado tiempo el estado fundamental del excitón se
desocupa: el electrón que se encuentra en la banda de conducción cae a la banda de
valencia, y el hueco al que estaba ligado desaparece generando un fotón. En la
naturaleza de este proceso intervienen los parámetros internos del material, como su
morfología o su tamaño, a través de su influencia sobre la estabilidad y dinámica de los
complejos excitónicos. Algunas especies excitonicas que no se pueden observar en
semiconductores con una densidad de estados continua (semiconductores masivos o
pozos cuánticos), como el biexcitón, trión, triexcitón, …, son estables en
nanoestructuras semiconductoras con una densidad de estados 0D (puntos cuánticos).
Los parámetros externos, tales como la temperatura, el bombeo óptico o las
perturbaciones en forma de campo eléctrico o magnético, también determinarán e
influirán sobre la configuración excitónica [11, 12]. Toda esta fenomenología explica
por que, aparte del interés tecnológico, este tipo de materiales también pueden ser
usados para llevar a cabo experimentos de gran importancia desde el punto de vista de
la investigación fundamental, pues relacionan diversas ramas de la ciencia como la
electrodinámica, la física del estado sólido, la termodinámica y física estadística, la
óptica o la mecánica cuántica. Para estos fines, la microscopía confocal es la técnica de
caracterización óptica más extendida pues resulta una manera eficiente de colectar la
luz procedente de un único emisor puntual y además permite la obtención de
información en condiciones muy variables con pequeños ligeros cambios del esquema
experimental (temperatura ambiente, bajas temperaturas, presión ambiental, alto
vacio, campo eléctrico o magnético externo). Frente a la microscopía convencional, las
ventajas del microscopio confocal derivan principalmente de la posibilidad de
seccionamiento óptico que permite aproximar la resolución espacial de confocal al
limite de Rayleigh. En la mayoría de sus aplicaciones, el microscopio confocal es
utilizado para colectar luz de manera extremadamente eficiente (en su plano focal)
evitando al mismo tiempo la influencia de la luz no deseada (procedente de los planos
desenfocados). De esta manera, se pueden estudiar por separado las diferentes
secciones de una muestra transparente y gracias a este hecho, es posible realizar
reconstrucciones tridimensionales de dicho tipo de muestras. En el capitulo 3 se van a
mostrar algunos ejemplos explicando estas aplicaciones de la microscopia confocal,
que pueden ser entendidos mediante sencillos principios de al óptica geométrica. La
aplicación en la caracterización de nanoestructuras semiconductoras es ligeramente
diferente ya que más que las propiedades de seccionamiento óptico se aprovechan la
resolución espacial del confocal, que estará por debajo de la micra en algunas
ocasiones. Esta resolución es suficiente para poder estudiar nano-estructuras aisladas
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en muestras de baja densidad de nanostructuras y por lo tanto también para aislar la
luz procedente de un dispositivo fotónico [7].
En el contexto de la actual sociedad de la información, basada en el manejo de una
cantidad cada vez más grande de datos, se requieren nuevas soluciones para el
procesado y almacenamiento de dicha información de una manera rápida, eficiente,
económica y a ser posible de bajo consumo energético. A corto plazo, una de las
propuestas más prometedoras para aumentar la velocidad de respuesta de nuestros
circuitos electrónicos (reduciendo a la vez el consumo energético) consiste en la
substitución parcial de las actuales pistas electrónicas por componentes ópticos
capaces de trabajar a frecuencias del orden del THz (con potencias de consumo en el
orden de los nanowatios). Entre todos los candidatos para implementar este tipo de
pistas opticas, las nanostructuras semiconductoras son una puerta de enlace entre
nuevas aplicaciones (que surgen de la manipulación en la nanoescala) y la tecnología
convencional. Uno de los campos de aplicación más interesantes es el procesado de la
información a través de un fenómeno cuántico: la recombinación de un electrón y un
hueco generando un fotón, por lo que también se podrían postular como dispositivos
puente para futuras tecnologías de computación y criptografía cuántica. Este concepto
se muestra en la Fig. 1.
Figura 1. El desarrollo de materiales nanoestructurados suele conducir a
aplicaciones o dispositivos fotónicos, ya que incluso en el caso de no poder aislar
una única nanoestructura los materiales resultanten siempre van a poder
emplearse como emisores o medios activos de alta eficiencia.
El paso del material nanoestructurado a los emisores de fotones individuales
supone dos retos importantes: en primer lugar se tienen que obtener nanoestructuras
de tamaño suficientemente pequeño para que el confinamiento cuántico permita la
aparición de estados discretos de energía. En segundo lugar, las nano-estructuras
asociadas a dichos estados tienen que estar suficientemente separadas para que los
sistemas de caracterización/manipulación óptica puedan colectar la luz que proceda
de una única nanoestructura. Puede que la tendencia en un futuro sea re-escalar la
tecnología del silicio, cuyos dispositivos operan en el rango de los voltios y los
miliamperios, para que puedan ser compatibles con dispositivos de emisores
individuales. Sin embargo, a corto plazo parece más probable que se imponga una
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adaptación de los nanodispositivos a la tecnología actual. Desde el punto de vista
práctico esto supone un problema, pues a pesar de la gran eficiencia de emisión de las
nanoestructuras cúanticas, la luz procedente de un único emisor no es lo
suficientemente intensa para pensar en la integración en un sistema de
telecomunicación actual. Una de las soluciones más prometedoras para resolver el
problema de esta baja señal consiste en el acoplamiento entre un único emisor
cuántico y el modo óptico de una cavidad resonante, Fig. 2. Este fenómeno se conoce
con el nombre de efecto Purcell y consiste en un aumento de la emisión espontanea
de la nanoesturtura debido al gran numero de estados ópticos accesibles en el interior
de la cavidad.
Figura 2. Efecto Purcell como solución al problema de la baja señal obtenida de un
emisor aislado.
De entre todos los tipos de cavidades ópticas existentes, las microcavidades de
cristal fotónico han sido las más estudiadas en los últimos años. El interés en este tipo
de cavidades radica en la posibilidad de fabricar dispositivos cuyos modos, con un gran
factor de calidad, presentan volúmenes modales muy reducidos, del orden de (/n)3,
siendo n el índice de refracción del material en el que se fabrica el cristal fotonico. Los
cristales fotonicos fueron propuestos por E. Yablonovich [13] and S. John [14] a finales
de los ochenta. En estos primeros trabajos se planteaba el uso de los cristales
fotónicos para controlar el flujo de luz (transmitancia y reflectancia) en interior del
material con una constante dieléctrica periódica. Por ejemplo un cristal fotónico 2D,
consiste en un substrato de un semiconductor en el que se graba un patrón periódico
de agujeros idénticos. Normalmente, este semiconductor queda suspendido en el aire
en forma de membrana, con el fin de aumentar el confinamiento la dirección vertical,
mientras que el patrón de agujeros se graba en el plano horizontal. Por tanto, la
periodicidad de la constante dieléctrica se consigue debido al contraste periódico de
índice de refracción entre el aire y el semiconductor. Esta periodicidad favorece la
aparición de unas soluciones muy particulares de las ecuaciones de Maxwell, que
incluso permiten el diseño de materiales con un gap óptico, es decir, materiales en el
interior de los cuales la luz no se puede propagar para cierto rango de frecuencias. Las
microcavidades de cristal fotónico se fabrican provocando un defecto en dicho patrón
periódico (no practicando uno o varios agujeros del patrón), a partir del cual aparece
una cierta densidad de estados ópticos dentro del gap fotónico. La luz generada o
existente en el interior del defecto no se va a poder propagar en el plano horizontal
debido al gap fotónico, mientras que en la dirección vertical va a sufrir el
confinamiento debido a las interfaces semiconductor/aire existentes.
En trabajos recientes de los grupos más activos en el campo de investigación
(Imamoglu, Finley, Forchel, Vucovik, …) se ha puesto de manifiesto que es posible
aumentar en más de un factor 100 el numero de fotones procedentes de un único
punto cuántico, si éste se acopla con éxito al modo óptico de una cavidad de cristal
fotónico [15-17]. Pero esta tecnología sigue siendo novedosa, y requiere de un mayor
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estudio antes de pensar en desarrollarla a nivel industrial. En la Fig. 3 se propone un
esquema realimentado para desarrollo tecnológico de nanodispositivos. Siguiendo
este modelo de trabajo se pretende mejorar sucesivamente las propiedades de los
dispositivos basados en cristales fotónicos con el fin de obtener prototipos
susceptibles de ser integrados en la tecnología actual. El trabajo experimental que se
ha llevado a cabo en esta Tesis ha consistido en seguir el esquema marcado para
diferentes tipos de heteroestructuras semiconductoras. En el proceso se han probado
y descartado varios materiales siendo los resultados que vamos a presentar los
obtenidos en dos de los más prometedores: puntos cuánticos de InAs crecidos en
substratos grabados de GaAs y los hilos cuánticos auto-organizados de InAs/InP.
Los puntos cuanticos de InAs/GaAs son las nanoestructuras semiconductoras más
estudiadas en la literatura. Cuando se empezó el trabajo de esta Tesis ya se había
demostrado el efecto Purcell en el modo fundamental de cavidades de tipo micro-pilar
conteniendo un único punto cuántico auto-organizado de InAs/GaAs [18]. Poco
después también se demostró la mejora del factor de Purcell al trabajar en cavidades
de cristal fotónico [19]. En esta Tesis se partió de cavidades ya existentes y se
contribuyó (a través de los estudios ópticos realizados) a abordar el problema de
conseguir fabricar la cavidad justo sobre el punto cuántico (localizando el emisor en el
centro del defecto de la cavidad). Notese que los puntos cuánticos auto-organizados
suelen crecerse por el método de Stransky-Krastanow tras la deposición de unas pocas
monocapas de InAs sobre un buffer de GaAs. Los puntos se forman por efecto de las
tensiones acumuladas debidas a la diferencia en el parámetro de red de ambos
materiales y es, por tanto, un fenómeno aleatorio. De esta forma, en muestras
conteniendo del orden de mil cavidades solo se conseguía acoplar de forma eficiente
un punto cuántico en unas pocas de ellas. Existían, por tanto, dos grandes retos, la
sintonización espacial o localización del punto cuántico respecto a la cavidad y la
sintonización espectral o control de la emisión.
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Figura 3. Esquema del método desarrollo tecnológico empleado.
Por este motivo se impulsó (en el IMM, dentro del marco del proyecto de
investigación coordinado con nuestro grupo) el desarrollo de la fabricación de puntos
cuánticos de InAs sobre substratos pregrabados de GaAs. Este tipo de subtratos se
preparan mediante litografía de oxidación local por microscopia de fuerzas atómicas
(AFM, del inglés Atomic Force Microscopy). Con esta técnica se ha demostrado
precisión en el crecimiento de untos cuánticos de InAs/GaAs del orden de la resolución
del AFM, unos 20 nm, resolviendo el problema de la dispersión espacial al azar
intrínseca al crecimiento de los puntos cuánticos auto-organizados [20]. Después de
demostrar que estos nuevos puntos cuánticos son emisores de fotones individuales
(realizando un experimento de interferometría cuántica Hanbury-Brown-Twiss [21]),
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también se han estudiado con detalle sus propiedades ópticas (capítulo 4 de esta
Tesis), requisito necesario para el futuro diseño de dispositivos. En cuanto a las
cavidades se consideraron varias aproximaciones para su fabricación atendiendo a
ligeras variaciones del método:
Figura 4. Metodo de fabricación de cavidades sobre un punto cuántico de InAs crecido sobre un
substrato pregrabado de GaAs. (a) En esta aproximación la cavidad se posiciona con la ayuda de unas
marcas (reglas) litografiadas previamente. (b) Imagen SEM de una de estas reglas. (c) AFM de un motivo
litografiado por oxidación local respecto a la regla.

Se prepara un substrato de GaAs y se graba con AFM.

Se fabrica mediante SEM un sistema referencia para el posicionamiento.

Se crecen puntos cuánticos en los motivos del patrón grabado.

Se recrece con GaAs hasta dar al material las dimensiones requeridas
por las cavidades de cristal fotónico.

Se miden los espectros de emisión de cada uno de los puntos cuanticos
mediante microespectrocopia óptica.

Se fabrican las cavidades atendiendo a las energías de emisión de cada
punto cuántico y dejando estos en la posición deseada.
Siguiendo este método ya se ha conseguido fabricar una serie de cavidades para las
cuales, tras las primeras medidas preliminares a 80 K, parece posible medir efecto
Purcell, cosa que permitiría plantearse la optimización de los diseños propuestos
obteniendo en breve un prototipo.
Otro de los materiales estudiados con éxito en esta Tesis son los hilos cuánticos de
InAs/InP. Estas nanoestructuras semiconductoras de tipo 1D poseen un gran interés
tecnológico, ya que presentan propiedades electrónicas y ópticas intermedias entre las
de los puntos y los pozos cuánticos. A pesar de ello los hilos cuánticos de InAs/InP no
han sido estudiados tan profundamente como los puntos cuánticos, entre otros
motivos porque apenas unos pocos grupos en el mundo disponen del conocimiento
base para su crecimiento (entre ellos el grupo de la Dra. Luisa González, del IMM, con
quienes venimos colaborando en su caracterización óptica desde 1999). Por este
motivo una de las primeras tareas en esta Tesis fue realizar una caracterización
completa de los hilos cuánticos usados para fabricar las cavidades fotónicas (capítulo
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5). Los datos experimentales de esta caracterización son necesarios para entender los
estudios ópticos posteriores cuando los hilos están enterrados en cavidades de cristal
fotónico (capítulos 6 y 7 de esta Tesis). Concretamente, se han estudiado los
parámetros más importantes para su optimización, como el efecto Purcell y el factor
de calidad de la cavidad (capítulo 6), así como el efecto en estos parámetros de la
potencia y la temperatura (capítulo 7). Si bien se han medido valores moderados para
el factor de Purcell (magnitud que representa el grado de acoplamiento entre el
emisor y la cavidad), los primeros prototipos han resultado ser unos emisores de luz
extremadamente eficientes en ambas ventanas de telecomunicaciones, 1310 y 1550
nm, permitiendo la emisión laser con umbrales muy bajos (bombeo óptico en la escala
de los nanovatios), incluso a temperatura ambiente.
En cuanto a las limitaciones de este tipo de heteroestructuras y dispositivos
basados en ellas, se ha identificado el motivo del bajo factor de Purcell obtenido, en
gran parte atribuible a la gran cantidad de hilos en el interior de la cavidad. Se ha
desarrollado un modelo bastante realista (capítulo 6) con el que se demuestra que
incluso en el caso más favorable siempre hay acoplados con el modo óptico de la
cavidad (L7, consistente en tapar 7 agujeros alineados de la red fotónica) al menos tres
o más hilos. Este hecho no solo impide la obtención de fuentes cuánticas de luz
(emisión de fotones uno a uno), sino que, además, reduce el efecto Purcell promedio.
La solución es obvia, la siguiente generación de cavidades se han de desarrollar
utilizando como base una distribución de hilos cuánticos con menor densidad. Sin
embargo, todavía no se ha conseguido demostrar la emisión de fotones individuales en
estas nanoestructuras, dado que no se dispone de los detectores adecuados para
realizar el experimento de Hanbury-Brown-Twiss [21] en el rango de longitudes de
onda de las telecomunicaciones. Dicho detectores están basados en InGaAs y
presentan un gran número de cuentas de ruido (la corriente de oscuridad es 105 veces
mayor que en los dispositivos de silicio), por lo que resulta muy difícil plantear
experimentos de auto-correlación de fotones.
En resumen, el espectro aislado de las estructuras semiconductoras como los hilos
y los puntos cuánticos está formando por transiciones ópticas discretas (como ocurre
con el caso de las líneas de emisión de un átomo). Uno de los retos experimentales de
la tecnología actual de semiconductores III-V es introducir una única de estas nanoestructuras en el interior de una cavidad de cristal fotónico, pues ello favorece
enormemente su eficiencia de emisión. Para poder conseguir superar este reto con
éxito se tiene que conocer al detalle la estructura de niveles de niveles electrónicos (y
excitónicos) de los emisores individuales. Esto es así pues hay que conseguir el
acoplamiento entre la emisión de la nano-estructura y alguno de los modos ópticos de
la cavidad [4], consistente en la sintonización perfecta de la energía de emisión de la
nano-estructura con la del modo óptico de la cavidad, además de la sintonización de la
posición de la nanoestructura respecto al centro de la distribución de campo
electromagnético del modo óptico. Por estos motivos, en cada material se puede
encontrar un límite para dicho acoplamiento que requiere de la optimización continua
de las propiedades ópticas de las cavidades resonantes, pero, sobretodo, de los
emisores cuánticos que se utilicen como medio activo.
Para facilitar el entendimiento de los resultados que se van exponer, en el capítulo
2 se reúnen algunos conceptos fundamentales, como son las propiedades ópticas y
electrónicas de los semiconductores III-V y de los cristales fotónicos (estructura de
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bandas, densidad de estados, …). También se justifica brevemente por qué cambian las
propiedades ópticas del material masivo cuando se reduce el tamaño del material
hasta una escala nanometrica (aparición de confinamiento cuántico), relacionando
dichas propiedades con los fenómenos de la interacción materia y radiación más
relevantes para el caso de emisores individuales.
Puesto que el trabajo que se expone es principalmente experimental, en el capítulo
3 se describirán las técnicas de caracterización óptica empleadas, describiendo varios
de los montajes utilizados. En este capítulo también se pueden encontrar detalles más
específicos relacionados con el instrumental utilizado, que varía notablemente para la
caracterización del sistema InAs/GaAs respecto al del InAs/InP.
Como ya se señaló más arriba, el capítulo 4 se centra en el estudio de los puntos
cuánticos de InAs/GaAs fabricados sobre substratos pregrabados mediante oxidación
local por AFM. Se describirá brevemente la técnica de fabricación y crecimiento
desarrollada para la obtención de estos puntos cuánticos. Los resultados
experimentales están orientados a poner de manifiesto la calidad óptica lograda en
estos puntos cuánticos, la cual supera ampliamente otras soluciones propuestas para
el control del crecimiento de puntos cuánticos. También demostraremos la
reproducibilidad de los mecanismos de crecimiento de las nanoestruturas, así como la
importancia en el control de la evolución de los motivos grabados en las diferentes
etapas de crecimiento. Para este fin se estudiaron mediante microespectroscopia de
fotoluminiscencia y microespectroscopia de fotoluminiscencia resuelta en tiempo dos
muestras de puntos cuánticos crecidas en condiciones ligueramente diferentes.
También se han identificado los complejos excitonicos presentes así como la estructura
electrónica de estos puntos cuánticos. Los resultados muestran líneas de emisión
homogéneamente ensanchadas debido a efectos de difusión espectral relacionados
con el entorno de carga del punto cuántico. Pese a este ensanchamiento, la intensidad
integrada de su emisión, así como los tiempos de vida excitónicos medidos en estos
puntos cuánticos, revelan que el proceso de fabricación protege las nanoestructuras
de efectos no radiativos relacionados con defectos en la intercara (del punto cuántico
con la capa tampón de GaAs), típicamente observados en otro tipo de substratos
pregrabados.
En el capítulo 5 se estudia la dinámica de recombinación en hilos cuanticos autoorganizados de InAs crecidos en el interior de una lamina /2 de InP. Esta estructura
fue depositada sobre una capa de 700 nm de InGaAs idéntica a la utilizada como capa
de sacrificio en la fabricación de microcavidades de cristal fotónico. Los resultados de
este capitulo se centran en determinar la propiedades ópticas de los materiales que
van a ser más relevantes para la caracterización de los dispositivos basados en ellos. En
ese sentido se ha obtenido información sobre la interacción excitónica con fonones
acústicos mediante un estudio de la evolución de la intensidad de emisión con la
temperatura de la muestra, y se han identificado los mecanismos de pérdidas no
radiativas que inducen una reducción de los tiempos de recombinación excitónica por
encima de 100 K. Por debajo de esta temperatura podemos distinguir dos dinámicas
de recombinación bien diferenciadas. Desde 4 hasta 50 K la recombinación está
dominada por la emisión a través de centros de localización atribuidos a defectos o
impurezas en los hilos. A temperaturas más altas, dichos centros de localización se
ionizan térmicamente y la dinámica pasa a estar dominada por los excitones libres
característicos de estructuras 1D. Todas estas medidas se realizaron en condiciones
1-9
parecidas a las utilizadas para la caracterización de los dispositivos fotónicos de los
capítulos posteriores.
En el capítulo 6 las medidas fueron destinadas a demostrar el efecto Purcell en
hilos cuánticos enterrados en cavidades L7 (realizadas en una red triangular de
agujeros cilíndricos) a 80 K, con el fin de evitar efectos de localización en los hilos. En
este capitulo se obtienen diferentes valores del factor de Purcell para cavidades con
parámetros de fabricación muy similares, que se atribuyen al diferente contenido de
hilos en cada cavidad. A parte de los altos valores del factor de calidad obtenidos en las
simulaciones de FDTD para estas cavidades, resulta interesante el hecho de que las
cavidades tipo Ln (donde n es un numero entero) permiten fijar la polarización de sus
modos en paralelo o en perpendicular a la polarización espontanea de los hilos,
mostrándose la influencia de la polarización del medio activo sobre el factor de Purcell.
Los resultados experimentales de este capitulo se discuten extensamente con el apoyo
de un modelo teórico el cual, partiendo de la formula de factor de Purcell para un
único emisor puntual, se deriva una expresión general representativa de un colectivo
de emisores cualquiera.
Finalmente, en el capítulo 7, haciendo uso de dichas expresiones, se discute la
evolución medida para el factor de Purcell con la potencia de excitación y con la
temperatura de la muestra. El primer estudio es necesario para confirmar la validez del
valor del factor de Purcell obtenido en el capitulo 6, pues al tratarse de dispositivos
con los que se puede obtener emisión laser, la medida de los tiempos de vida de estos
modos láser podrían estar influenciados por la emisión estimulada. Por otro lado, el
estudio con la temperatura es más bien un estudio de la influencia de los estados
localizados sobre el factor de Purcell. Finalmente, en este capítulo se demuestra y
explica como en el régimen en que la emisión de luz esta dominada por los centros de
localización, se produce una disminución del factor de Purcell atribuible a una menor
probabilidad de acoplamiento espacial respecto al caso de los excitones libres
(deslocalizados a lo largo del hilo).
References
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2.
3.
4.
5.
6.
7.
8.
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building blocks for nanoscale electronic and optoelectronic devices” Nature 409, 66
(2001).
K.J. Vahala, “Optical Microcavities” Nature 424, 839 (2003).
P. Michler, “A quantum dot single-photon turnstile device” Science 290, 2282 (2000).
D. Bimberg, M. Grundmann and N. N. Ledentsov, “Quantum dot heterostructures” Wiley;
1 ed. (1999).
Y. Arakawa and H. Sakaki, “Multidimensional quantum well laser and temperaturedependence of its threshold current” Appl. Phys. Lett. 40, 939 (1982).
P. Yu and M. Cardona, “Fundamental of semiconductors : Physics and materials propeties”
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1-10
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dots” Nanotechnology 19, 145711 (2008).
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and Exchange Energies in Semiconductor Quantum Dots” Phys. Rev. B 78, 915 (1997).
S.A. Empedocles and M.G. Bawendi, “Quantum-confined stark effect in single CdSe
nanocrystallite quantum dots” Science 278, 2114 (1997).
S. Tarucha, D.G. Austing, T. Honda, R.J. van der Hage, L.P. Kouwenhoven, “Shell filling and
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E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics”
Phys. Rev. Lett. 58, 2059–2062 (1987).
S. John, “Light control at will,” Nature 460, 337 (2009).
C. Reese, C. Becher, A. Imamoglu, B.D. Gerardot, P.M. Petroff, “Photonic crystal
microcavities with self-assembled InAs quantum dots as active emitters” Appl. Phys. Lett.
78, 2279 (2001).
M. Kaniber, A. Laucht, A. Neumann, J.M. Villas-Bôas, M. Bichler, M.-C. Amann, and J.J.
Finley, “Investigation of the nonresonant dot-cavity coupling in two-dimensional photonic
crystal nanocavities,” Phys. Rev. B 77, 161303 (2008).
J. Vuckovic and Y. Yamamoto, “Photonic crystal microcavities for cavity quantum
electrodynamics with a single quantum dot” Appl. Phys. Lett. 82, 2374 (2003).
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spontaneous emission by quantum boxes in a monolithic optical microcavity” Phys. Rev.
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Alonso-Gonzalez, D. Fuster, L. Gonzalez, J. Martínez-Pastor and F. Briones, “Single Photon
Emission from Site-Controlled InAs Quantum Dots Grown on GaAs(001) Patterned
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1-11
Chapter 2. Theoretical Background.
“La velocidad de grupo que aparece en esta expresión
es la velocidad de grupo, pues… de toda la vida”.
L. J. Martínez
2-0
1. Optical properties of semiconductors III-V.
Figure1. Unit Cell corresponding to the Zinc-blende structure of InAs, InP and GaAs.
Figure 2. First Brillouin zone of the Zinc-Blende structure. The main simmetry points are labeled.
Figure 3. Band structure of GaAs, InP and InAs obtained by means of the pseudopotential methord.
2-1
The nanostructures studied in the present work are based on III-V semiconductors. Concretely,
we have characterized a new kind of site-controlled InAs/GaAs quantum dots (QDs) and an
improved sample containing a single monolayer of InAs/InP self-assembled quantum wires
(QWRs). The InAs/GaAs QDs can be grown by the Stranski-Krastanow method due to the lattice
parameter mismatch between indium and gallium arsenide (around 7%). The optical emission of
these QDs can be tuned in the wavelength range of 800 to 1200 nm [1]. On the other hand, the
InAs/InP QWRs can be tuned into the range of 1200 to 1600 nm wavelength (1010-780 meV)
covering the two most important telecommunication windows [2, 3]. From the growth point of
view the InAs/InP system presents two main differences with respect to the InAs/GaAs one: firstly,
onto InP the InAs grown front is produced under lower strain conditions (than onto GaAs);
secondly, the growth requires the combination of two elements belonging to the V group (arsenic
and phosphorous).
As aforementioned InAs, InP and GaAs belongs to the group of the III-V semiconductors. These
binary compounds are formed by the covalent-ionic bonding of a cation from the group III (In or
Ga) with an anion belonging to the group V (As or P). The resulting molecules are organized
forming tetrahedric structures due to the hibridation of valence electron orbitals. The material can
be observed as a two superimposed fcc structures (one composed by the cations and one by the
anions), see Fig 1. As a result, the unit cell contains eight atoms (four molecules): 4 inside the cube
and 4 corresponding to the corners and face centres. Due the fact that the lattices are formed by
different atoms the structure is characterized by the lack of inversion symmetry. Setting one
corner of the cube as the origin of coordinates it can be shown that the structure is invariant to
the point group Td [4], also called Zinc-blende group. The Wigner-Seitz cell associated to the
reciprocal lattice of the Zinc-blende structure (first Brillouin cell) forms a truncated octahedron, as
shown in Fig. 2. In this figure we can distinguish several symmetry points: the center of this cell is
known as the  point, being X, L and K the larger symmetry points at the Brillouin zone borders.
The electronic structure of semiconductors can be described by considering the motion of an
electron in the presence of and effective potential:
( ⃗)
(1)
The Hamiltonian accounts for the interaction between the electron with other electrons or
atoms (V(r)) including the spin-orbit interaction, Hso. The problem is many times solved by means
of the pseudopotential method because of its accuracy [5]. The pseudopotential method simplifies
the electron structure calculation considering just the valence electrons which are considered in
the presence of an effective potential consisting of the superposition of the atomic interaction in
the entire crystal. In Fig. 3 it is shown the electronic structure for GaAs, InP and InAs published in
Ref 6. An important characteristic of the III-V semiconductors is that the band structure around
the  point can be approached to the scheme of Fig. 4. The conduction band (6 in Fig. 3), arises
from the S-type orbitals of the cations and it is drawn as a double degenerate concave band on the
top of Fig. 4. On the other hand, the valence band is formed by the P-type orbital of the anions,
resulting in three different subbands (convex bands in Fig. 4). At k=0, the spin-orbit interaction
splits such subbands driving to the four time degenerate band 8 and the double degenerate band
7. The energy difference between 7 and 8 (ESO in Fig. 4) is known as spin-orbit coupling. For
this reason, the lowest energy subband is called split-off band (soh) while the other subbands are
known as heavy (hh) and light hole (lh) bands due to their respective effective masses.
2-2
Figure 4. Parabolic approach of the band structure around the  point for the Zinc Blende family.
The optical properties of III-V semiconductors can be described by the electronic structure
around the  point. This electronic structure can be successfully described by means of the
method known as kp which consists of describing electron wavefunction by means of the Bloch’s
theorem as follows [7]:
⃗⃗ (
⃗⃗ )
⃗⃗ ⃗
⃗⃗ (
⃗⃗ ) (2)
In fact, we can apply in Eq. 1 this wavefunction, i.e. time independent Schrödinger’s equation,
for obtaining the energies around the point. With this method, once the energies are found the
conduction band terms depending on k can be treated as perturbations, for instance the second
order approach would drive to the quasi-free electron approach. In Ref 8, it is reported the
solutions after considering higher order terms):
⃗⃗
( ⃗⃗ )
(3)
In contrast, the description of P-type electrons of the conduction bands require a 6x6
Hamiltonian (for example the Kohn-Luttinger Hamiltonian) or even more complicated operators if
the semiconductor band gap is narrow. In spite of the larger difficulty of calculus, we can also
obtain effective masses for the heavy (mhh) and light hole (mlh) band.
2-3
Material Eg(eV)
1.519
GaAs
1.423
InP
0.418
InAs
ESO(eV)
me/m0 mlh/m0 mhh/m0
0.34
0.0665
0.082
0.45
0.108
0.079
0.12
0.65
0.38
0.024
0.025
0.41
Table 1. Parameter of the III-V semiconductor materials taken from Ref 7. The energy gap corresponds to the low
temperature values.
Sometimes, it is required a more detailed study in order to express phenomena as non-linear
effects, the presence of radiation or other external fields. In that case, the Hamiltonian must be
representative of processes involving a great amount of particles and quasi-particles (electrons,
holes, photons, phonons, etc). Furthermore, the Coulomb interaction between some of these
particles can play an important role. For these reasons, this kind of Hamiltonians are usually
proposed using the second quantization formalism into the Heisenberg representation in such a
way that the problem is reduced to a homogeneous differential expression called Wannier
equation [9]:
[
r 2
r 2
⃗]
( ⃗)
( ⃗) (4)
where the energy Eeh takes into account the Coulomb interaction between the electron and the
hole (also called exciton). The Wannier equation can be solved by means of a separable variable
technique if the wavefunction is expressed: on one hand, as a function of the coordinates of the
center of mass (of the electron and holes) and; on the other hand, as a function of the relative
coordinates, rrR which lead to the next expressions (with r = re+rh):
[
2
[
] ( ⃗)
R2
] ( ⃗⃗ )
( ⃗) (5)
( ⃗⃗ ) (6)
⃗⃗
where
(7)
⃗⃗
(8)
⃗⃗
⃗⃗⃗⃗
⃗⃗⃗⃗⃗
(9)
(10).
The equation 5 describes the relative motion of an electron-hole pair around their center of
mass. The solution of such expression is well known since it is similar to that of the hydrogen
atom, in this case modified by considering the exciton reduced mass () and the dielectric
2-4
constant of the semiconductor (b). Eq. 6 is the Schrödinger equation for a free particle of mass M.
Thanks to these analogies the exciton energy can be determined by Eq. 11, where Eg can be
determined from the dispersion relation obtained by the pseudopotential method,
( ⃗⃗ )
(11)
Obviously, the range of validity of this result is limited by the approaches done during the
deduction of the Wannier equation, where it is assumed that the Coulomb potential variation is
negligible into the unit cell. This assumption can be done if the excitonic Borh radius (aX) is
noticeably larger than the crystal lattice parameter:
(12)
where aH=0.0529 nm is the Borh radius for the hydrogen atom. In that case, the electron-hole
interactions are called Wannier exctions. For the nanostructures studied in this Thesis we can
consider the Borh radius for the bulk InAs which using Eq. 12 is around 34 nm at 12 K. For a better
estimation of the Borh radius in QWRs and QDs we must obtain the effective masses of each kind
of nanostructures (calculated by means of the kp method) having into account the size
confinement effects and the Coulomb interactions.
After this brief introduction of the theoretical tools available for describing the optical
transitions of semiconductors we are going to focus on the magnitudes more relevant for the
discussion of the experimental results of this work: the density of states, the spontaneous
emission rate and the homogeneous linewidth of an optical transition. A more detailed
development of such magnitudes can be found in Refs. 4 and 10. Such properties of the bulk
semiconductor are modified at the nanometer scale due to size quantum confinement
phenomena, as will be described in section 2. On the other hand, in section 3 it is going to be
introduced the concept of Photonic Crystal (PhC) together with the technological possibilities of
PhC based devices [11]. The description of all these concepts is necessary for understanding the
main topic in this Thesis, the Pucell effect, which is presented in section 4 in the context of the
light-matter interaction.
a) Optical density of states.
The concept of the density of states arises in many branches of physics. First, we focus on the
photon density of states, which is important for the discussion of the radiative rate of quantum
emitters in free space. After that, we also explain how the derivation for photon modes can be
adapted to electrons in semiconductors. On behalf the calculation, it is considered the
electromagnetic field within a finite volume V of free space as shown in Fig. 5. For simplicity, we
assume that the volume comprises a cube of edge length L, so that V = L 3. Notice that the
argument can be generalized to volumes of other shapes, without affecting the final result. The
volume is assumed to be large enough so that its dimensions have no significant effects on the
physical result. The general solution for the electromagnetic field within V can be written as a
superposition of travelling waves of the form:
(
)
(⃗⃗ ⃗
∑⃗⃗
2-5
)
(13)
Figure 5. Finite volume of the free space considered for calculating the electromagnetic density of states.
Eq. 13 gives us a general expression for the field within volume V, but attending to the
transverse condition of the Maxwell equations we could expect two independent wave
polarizations, one for each value of wavevector (remember that E x k = 0). The electromagnetic
field can be expanded as sine waves, and its expression can be written as Fourier series for a finite
volume:
(
)
∑
(14)
The values of kx, ky, and kz are determined by the dimensions of V accomplishing the relations:
(15)
where nx, ny, and nz are positive and negative integers. The possible values of the wave vector can
thus be written in the form:
⃗⃗
(
)
(
) (16)
Each set of integers (nx, ny, nz) represent two modes of the electromagnetic field (one for each
polarization). Figure 6 shows a plot of the allowed values of the wave vector in the k-space. The
phase diagram forms a 3D-grid with a spacing of 2/L between successive points, and hence each
allowed value of the k-vector occupies an effective volume of (2/L)3 of this 3D k-space. Therefore,
we could calculate the number of states in the region of the k-space limited by k+dk, lets say g(k)
dk, which in the three-dimensional case can be calculated by working out the area of k-space
enclosed by k-vectors with magnitudes between k and k+dk and then dividing by the effective area
per k-state, see blue shell in Fig. 6:
2-6
Figure 6. Sphere shell or radius k and thickness dk represented with the number of states in the k-space.
( )
(
(17)
⁄ )
Finally the result can be normalized by V to obtain:
( )
(18)
Note that this value does not depend on the volume and confirms that the subdivision of the
space is merely a computational tool. From this density of states we can deduce the number of
states per unit volume and unit angular frequency range, g(). To do this we map the values of k
and k+dk onto their corresponding angular frequencies, namely +d, and remember that there
are two photon polarizations for each k-state. We thus write:
( )
( )
(19)
and using the relationship  = ck we finally obtain:
( )
( )
(20)
Where we can conclude that, the photon density of states is proportional to the square of the
photon frequency. As aforementioned, the derivation of the density of states for photon modes
can be adapted to other branches of physics. In the case of electron waves in crystals, as for
example the III-V semiconductor, we usually express de density of electronic states per unit
volume as a function of the energy, g(E). Notice that, once the density of states in momentum
space is worked out, the derivation is identical to that given above, with g(k) given by Eq. 18, in
analogy with Eq. 20.
2-7
( )
( )
(21)
In Eq. 21 the factor 2 comes from the fact that there are two electron spin states for each
available k-state, namely spin up and spin down. In the case of free electrons we have
(22)
which then gives:
( )
(
)
⁄
(23)
For the case of electronic states in semiconductors within the parabolic approximation the
solution of the free electron is modified by using the effective mass (m*) instead of the free
electron mass (m0). It is worth noting that Eq. 23 arises for the density of electronic states in a bulk
material, in the next section we are going to show that working with nanostructures the density of
states must be revisited since in that cases it is expected a variation of the dispersion relation due
to the confinement potential.
b) Radiative transition rate of a single emitter.
Figure 7. Optical transitions between discrete electronic states (like for atoms or QDs) involving photon emission
into a continuum of states of optical states.
The calculation of radiative transition rates by quantum mechanics is based on time-dependent
perturbation theory. The light–matter interaction is described by transition probabilities, which
can be calculated for the case of spontaneous emission by using Fermi’s golden rule. According to
this method, the emission rate of an optical transition, relating the final state 1 and the initial state
2, is given by:
|
| ( ) (24)
where M12 is known as the matrix element of the optical transition and g(ħ) the density of
states. As an example we could consider the standard case of transitions between quantized levels
in a single emitter, where the initial and final electron states are discrete. In this case, the density
of final states of Eq. 24 corresponds to the density of photon states in the material. For describing
spontaneous emission, we are interested in the situation where the photons are emitted into the
2-8
free space. In this case, the photons are emitted into a continuum of states, as illustrated
schematically in Fig. 7. The density of photon modes in free space and therefore the
recombination probability is proportional to 2, see Eq. 20. In the next, it will be also shown that
the photon density of states can be modified by embedding the emitters into an optical cavity.
This modification of the photon density of states can produce deep effects on the radiative
emission rate, as we will be discussed in Section 4.
For the moment let us consider the matrix element that appears in the Fermi’s golden rule:
⟨ | | ⟩
∫
( )
( )
( )
(25)
where H’ is the perturbation caused by the light, r is the position vector of the electron, and
1(r) and 2(r) are the wave functions of the initial and final states of the electron. For this kind of
calculation it is convenient to adopt a semi-classical approach in which the atoms are treated as
quantum-mechanical objects but the light is treated classically. In this framework, there are
different types of interactions that can be considered between the light and the emitter, and this
gives rise to a wide classification of the radiative transitions. It is well known that the transition
rate decreases by several orders of magnitude each time the multipolarity increases (i.e. dipole →
quadrupole → octupole) and also that the magnetic interactions are weaker than the equivalent
electric ones by a similar factor [12]. For this reason, it is common to concentrate on the electric
dipole interaction, which is the strongest of the different possibilities [10]. Therefore the emitter
perturbation is given by the interaction between the electric field amplitude of the light and an
electric dipole p associated to the emitter, H’=-pE0. Assuming that only electrons can respond to
optical frequencies p=-er, being e the electron charge, and hence H’=e(xEx+yEy+zEz). Ex, Ey and Ez
are the component of the field amplitude along the x-axis, etc. Since radiant dipoles are usually
small compared to the wavelength of light, the amplitude of the electric field will not vary
significantly over the dimensions of a point-like emitter as an atom or as a QD. We can therefore
take Ex, Ey, and Ez to be constants in the evaluation of the integrals in Eq. 25 to obtain:
( )
∫
( )
( )
( )
∫
( )
( )
( )
∫
( )
( )
(26)
These matrix elements can be written in the more succinct form:
(27)
where
(⟨ | | ⟩ ⟨ | | ⟩ ⟨ | | ⟩) (28)
is the electric dipole moment of the transition. The dipole moment is thus the key parameter that
determines the transition rate for the electric-dipole process. The results given in Eq. 26 allow us
to evaluate the matrix elements for particular transitions if the wave-functions of the initial and
final states are known. We can then use Fermi’s golden rule to calculate the transition rate of a
point-like emitter as:
2-9
|
| (29)
To conclude, we define the oscillator strength f12 of the transition which is directly related to
the dipole moment according to [13]:
|⟨ | | ⟩|
|
| (30)
The oscillator strength was introduced before quantum theory to explain how some atomic
absorption and emission lines are stronger than others. After the development of the quantum
mechanics, it is easy to understand that this is simply caused by the different dipole moments for
the transitions.
c)
Spectral linewidth.
The radiation emitted by means of electronic transitions is not perfectly monochromatic. The
shape of the emission line is described by the spectral lineshape function g().This is a function
that peaks at the line centre defined by ħ12=(E2-E1), and is normalized such a way that:
∫
( )
(31)
The most important parameter of the lineshape function are the peak energy, ħ, and the full
width at half maximum (FWHM) , which quantifies the width of the spectral line. Before going
into the details, it is important to remember the general classification of the broadening
mechanisms as either homogeneous or inhomogeneous. In the first case, the optical transitions
are produced by a unique transition, while in the second case the spectrum are produced by
transitions that contribute to different parts of the total spectrum. From the experimental point of
view, this distinction is important since the homogeneous and inhomogenous broadening would
affect the emission rate in a different way. In general, light is emitted when an electron in an
excited state drops to a lower level by spontaneous emission. The rate at which this event occurs
is determined by the corresponding emission rate, which in turn determines the radiative lifetime
, as discussed below. The finite lifetime of the excited state leads to a broadening of the spectral
line in accordance with the energy–time uncertainty principle [14, 15], it is E t ≥ ħ. If we call
t= the broadening in angular-frequency units will be:
(32)
Since this broadening mechanism is intrinsic to the transition, it is alternatively called natural
broadening or simply radiative broadening. However, optical transitions are usually affected by
different scattering mechanisms (optic and acoustic photons, impurities, defects, …) giving rise to
shorter  and hence larger  values. The detailed form of the lifetime broadening can be
deduced by taking the Fourier transform of an optical signal that decays exponentially with time
constant . This gives the well known Lorentzian lineshape function, where the FWHM is given by
 = 1/:
2-10
( )
(
)
(
)
(33)
The spectrum described by Eq. 33 is called a Lorentzian lineshape, and is plotted in Fig. 8. Note
that the concrete  = 1/ agrees with the approximate one of Eq. 32 derived from the
uncertainty principle.
Figure 8. The lorentzian lineshape function peaks at the centre 0 and has a FWHM of 1/. The function is
normalized so that the total area is unity.
2. Quantum size confinement: from bulk material to 1D and 0D nanostructures
The electronic states of crystals have been obtained by the band theory of solids. The atoms in
a solid are packed very close to each other separated periodical distances approximately equal to
the size of the atoms. Hence, the outer orbitals of the atoms can overlap and interact strongly with
each other. This broadens the discrete levels of the free atoms into the characteristic electronic
bands of solids above described. Each band can contain 2N electrons per volume unit, where N is
the number of atoms in such volume unit [16]. The factor of two arises from the two possible
values of the spin for each available electron space wave function. The atoms of GaAs, InP and
InAs come from groups III and V of the periodic table, giving an average of four per atom when the
covalent bond is formed.
In Figure 4 it was shown the generic energy-level diagram for the III-V semiconductor as
determined by conduction and valence bands, respectively. At very low temperatures, the
electrons completely fill the valence band; there is then a gap in energy Eg (called the band gap) to
the next band (the conduction band) which is completely empty. As the temperature rises up from
absolute zero, an increasing number of electrons can be excited from the valence band to the
conduction band, leaving empty states (holes) in the valence band. The holes are equivalent to the
absence of an electron, and behave like particles with a positive charge (+e) against an external
force. The electrons in the conduction band and the holes in the valence band can be treated like
free particles but with a different mass from that of free electrons. This modified mass (effective
mass) is usually different for electrons and holes. In a semiconductor at room temperature, there
are typically a smaller number of these free carriers than in a metal, which explains why the
materials are called semiconductors. The semiconductors employed for growing the
heterostructres studied in this Thesis present a direct band gap which means that the maximum of
2-11
the valence band and the minimum of the conduction band occur for the same electron
wavevector, typically k=0, i.e. the  point. This means that an electron in the conduction band can
recombine directly with a hole in the valence band by emitting a photon conserving the
momentum without the interaction with additional particles (usually phonons).
Figure 9. (a) absorption of a photon. (b) and (c) relaxation processes of the electron and hole by phonon emission.
(d) radiative recombination of an electron hole pair.
Figure 10. Absorption of the bulk GaAs at different temperatures. Extracted from [4]
The optical characterization consists of studying the electronic structures of materials by
analyzing the optical transitions occurring at different frequencies. For example, the energy of the
ground state transition on a semiconductor coincides with the bulk material band gap. This
transition is generally called interband transitions and its measurement involves the different
phenomena shown in Fig. 9. This plot illustrates the mater radiation interaction between the
particles of the conduction and the valence bands of a semiconductor. We consider the case of
optical excitation the (photoinjection) which consist of illuminating the material typically with laser
light, see Fig. 9(a). If the photon energy of the incident light is larger than the semiconductor gap,
ħ> Eg, we can expect an important absorption of light which drives the promotion of electrons
to the conduction band, generating holes in the valence band. In that case, the electrons (holes)
are thermalized to the bottom (top) of the conduction (valence) band by phonon emission or
charge interaction (notice that the hole energies are measured downwards from the top of the
valence band), as shown in Fig. 9(b) and (c). Finally the electron-hole pair recombine emitting a
photon of energy ħ’≈ Eg. In the case of a bulk material the emission linewidth is determined by
the thermal spread of the Boltzmann tail (of charge carriers) or eventually due to inhomogeneous
effects.
2-12
As a result, in a semiconductor the band gap (Eg) represents the threshold for absorption to
occur, whereas it corresponds to the transition energy in the case of emission. In the absorption
spectrum, a threshold Eg= 1.52 eV is observed at the band gap at 21 K (Eg = 1.46 eV at 80 K) and a
continuous absorption band is observed for photon energies that exceed this threshold. Having
this fact into account, during study of InAs/GaAs QDs the sample is optically pumped using an
excitation wavelength of 780 nm (1.59 eV). In this way, the GaAs surrounding the nanostructures
absorbs the incident light in an efficient way in a wide range of temperatures (~104 cm-1 from 4300 K) producing a large amount of electron-hole pairs [4]. After that, the carriers are relaxed by
phonon emission to the lower energy levels, as described in Fig. 9. But in nanostructure materials
these levels do not correspond to the bottom (top) of the valence (conduction) band but they
consist of the electronic states of the QDs and eventually of wetting layer. The absorption spectra
of the InP presents a very similar behavior of GaAs, but with an absorption threshold of 1.36 eV at
80 K. At the QWRs and the photonic devices studied in Chapters 5-7 the carriers are injected by
means of resonant excitation (980 nm wavelength, close to 1.27 eV). Under these conditions the
light absorption is reduced allowing the optical characterization in conditions of very low carrier
population, even in pulsed operation. As the emission spectra of the bulk materials as the ones of
the heterostructures consist of discrete lines with peak energies coinciding with the corresponding
optical transition. In Fig. 11(a) we show an example of the typical PL spectrum of a sample
containing InAs/GaAs QDs. The emission of the QDs is observed at lower energies (SQD from small
QDs), accompanied of the emission of the wetting layer (WL) and an impurity band and the bulk
GaAs. The characterization of all these peaks can be performed thanks to the wavelength
employed (up to the GaAs emission). These transitions are not observed working in resonant
excitation, since the carriers can not be injected to the states with optical transitions higher than
the excitation wavelength. For example, in Fig. 11 (b) it is show the PL spectrum of sample
containing self-assembled QWR excited in resonant conditions. Due to the absence of WL in this
kind of nanostructures we just can observe the emission of QWRs, peaks P1-P4.
Figure 11. Optical transitions of InAs/GaAs and InAs/InP nanostructures. (a) Exciting above the gap of GaAs we can
find high energy trasitions. (b) In resonant excitation we just can study the emission of the nanostructures.
a) Electronic density of states in confined systems.
Throughout this work, we are going to refer to a number of quantum optical effects relating to
excitons in low-dimensional semiconductor structures which optical properties differs with respect
to the bulk material due to the quantum confinement effects on the electron and hole
wavefunctions. Low-dimensional semiconductor structures are of considerable importance in
modern optoelectronics. This has lead to the development of crystal growth techniques which
now routinely make semiconductor layers with atomic monolayer precision. In this framework, the
2-13
electron waves are characterized by the de Broglie wavelength (similar to the Borh radius for
excitons) B defined by [17]:
√
(34)
where kB is the Boltzmann constant. In bulk semiconductors, the electrons in the conduction
band are free to move in all three directions, and their de Broglie wavelength is governed by the
thermal kinetic energy at temperature T, where me*, is the effective mass. In normal
circumstances, the de Broglie wavelength is much smaller than the dimensions of the crystal, and
the motion is governed by the laws of Classical Physics. However, when one or more of the
dimensions of the crystal are comparable to B, then the motion in that direction is quantized.
Based in this discussion, there are three general classifications of quantum confinement effects. If
the motion is confined in one direction (e.g. the z-direction), the structure is called Quantum Well
(QW). The electrons in a QW are free to move in the other two directions of the space being
carrier motion just quantized in the first direction. If the motion is confined in two directions the
structure is called a QWR. The electrons in a QWR have free motion just in one dimension (e.g. the
y-axis) while it is quantized in the other two directions. Finally, if the motion is confined in all
directions the structure is called a QD. The general scheme of classifying quantum-confined
structures is illustrated schematically in Fig. 12.
Figure 12.Comparison the electronic density of states between the QWs, QWRs and QDs with respect to the bulk
material.
The main effect of quantum confinement consists of modifying the emission energy and the
density of states of the nanostructures under study [18]. We have demonstrated in section 1 that
the density of estates is proportional to (E −Eg)1/2 for a bulk material using geometric
considerations and neglecting border effects. But in the case of confined electrons the functional
form of the density of states is altered. In the case of a QW the calculus of the DOS can be done in
an analogue way, but notice that in such case the number of states would be contained in a 2D kspace. Hence, let consider an area or the k-space
2-14
( ) (35)
We can define a ring of radius k with thickness dk, considering the number of states lying into
the ring, with surface 2kdk, we obtain an expression equivalent to Eq. 18 for the case of the
electronic states of a QW.
( )
(36)
Notice we have also introduced a factor 2 and divided by L2 for accounting the spin
degeneration and showing the result per surface unit. The use of the relation between the energy
and the wave-number drives to the next expression:
( )
√
(
⁄
)
(37)
As a clear difference with the case of a bulk material, the DOS of a QW does not depend on the
energy, as shown in Fig. 12. This fact has got important consequences from the technological point
of view, since in these materials we can find an important number of available electronic
transitions occurring at the ground state.
Figure 13. Representation of the 2D density of states in the k-space, in analogy to the case of bulk material (Fig. 6).
Now we discuss about the case of 1D nanostructures, like QWRs. In 1D-materials the electrons
are confined in two directions, therefore the k-space become a length and the area the ring is
reduced to a short line (dk) into such length. Let consider then a region of the 1D k-space with
finite dimension:
L=
(38)
2-15
The density of states per unit length in this case is reduced to
( )
(39)
which as a function of the electron energy is given by
( )
(
)
⁄
√
(40)
As a difference with the case of the QW, there is observed an important dependence of the
density of states on the energy. The larger amount of available states coincides with the ground
state transition, see Fig. 12. From this distribution of states we can expect optical transitions
narrower than in the case of a QW. Finally, in 0D nanostructures, like a QD, the values of k are
quantized in all directions. As a result the dispersion relation becomes a group of Dirac’s deltas, in
such a way that we just can find states at discrete energies with a fix wavenumber. Due to this fact
the linewidth of the PL peak obtained from isolated QDs is attributed to broadening effects as
phonon scattering or spectral diffusion by impurities and defects.
3. Photonic crystal micro-cavities.
Nowadays, information society needs to handle a continuously increasing amount of
information. It is required new ways to manage a huge amount of data with high speed and lower
energy consumption [19, 20]. In this sense, a promising candidate for enhancing the quick
response of electronic circuits consists of the partial substitution of electronic tracks by photonic
gates working at optical frequencies. This fact is expected to reduce the electronic signal delays
and parasite impedances which are actually limiting the working frequency of the electronic
devices. On the other hand, it has been also considered the potential application of this
technology in quantum computing since such kind of optoelectronic circuits is expected to be
mainly based on semiconductor nanoheterostructures. In fact, the connection between the
standard silicon based electronics and the future quantum technology could be done thought
optoelectronic integrated circuits, based in single photon emitters (like quantum dot or quantum
wires). From the technological point of view, the main problem of the single photon emitter
devices is related with the low optical signal obtained from isolated emitters. A possible approach
to solve the low signal problem of single nanostructures consists of controlling the spontaneous
emission rate of the quantum emitter by means of photonic or plasmonic materials.
2-16
Figure 14. (a) Illustration of a 1D PhC made of alternating thin layers of two dielectric materials. (b) The photonic
band structure (normal incidence) calculated for a typical 1D PhC crystal where 1 and 2 are equal to 13 and 12,
respectively. The dashed region indicates the photonic bandgap. Any electromagnetic wave whose frequency falls
anywhere in this gap cannot propagate through this material. (c) The photonic band structure of a PhC crystal can be
probed experimentally by measuring its transmission spectra using electromagnetic waves with different wavelengths.
Figure Extracted from Xia et al. Adv. Matter. (2000).
In this context, one of the most promising platforms to manipulate light is the use of photonic
crystals (PhCs)[21-23]. Motivated by these facts, research in the field of PhCs has suffered an
exponential growth since the proposals of E. Yablonovich and S. John in the late eighties. [24]. We
can define a photonic crystal as a material with a periodic dielectric function. The periodicity of the
dielectric function affects the electromagnetic field behavior into the sample and therefore their
optical properties. In fact, the Maxwell equation solutions of such problem can be expressed in
form of a complete base of Bloch functions, as in the case of electrons into a crystal, establishing a
good analogy between the solid state physics and the photonics. In addition, the scalable nature of
the Maxwell equations allows us to define dimensionless variables describing the mentioned Bloch
functions. If the dimensionless parameters are adjusted to the material periodicity we obtain
solutions to the Maxwell equations on form of dimensionless variables maintaining therefore the
2-17
scaling provided by photonic crystals translational, rotational and time reversal symmetry [11].
For example, in Fig. 14 it is illustrated the typical case of the 1D photonic crystal. If we calculate
the dispersion relation by using the dimensionless parameter “a” the solution can be expressed in
terms of a dimensionless frequency and wavenumber (which depend on the parameter “a”). The
point marked with a red arrow in Fig. 14 (b) (called photonic band) consists of a small range of
frequencies without any possible propagation direction producing a minimum in the transmittance
at the corresponding normalized wavelength [25]. Since the result is calculated in terms of the
dimensionless frequencies, we could set the suitable lattice parameter in order to fix the bandgap
at the desired wavelength obtaining something similar to a filter while embedding an active
medium into a photonic device allows additional possibilities. It is well known, that the
spontaneous emission of isolated emitters embedded into a PhC is strongly suppressed if radiative
transitions overlap with the photonic bandgap due to the absence of available optical states, [24,
26]. On the other hand, when a defect is created in the PhC structure a strong confinement of the
light can be obtained in a small volume. In fact, if the dielectric medium shows suitable properties,
different effects and functionalities can be obtained like Purcell effect [27] and vacuum Rabi
splitting [28-30] both with potential applications in telecommunication technology: single photon
sources [31, 32], low-threshold lasers [33-35], high-speed modulation lasers [36], etc.
Figure 15. Photonic band structure of a 2D photonic crystal slab of InP (c=10.1) on air for the triangular lattice (a)
and the graphite lattice (b). The parameter of the structures are: (a) hole radius r/a=0.35 and slab thickness d/a=0.45.
(b) hole radius r/a=0.15 slab thickness d/a=0.32. Only even modes (σxy = +1) are plotted. Green lines are the light lines
of the cladding and average core materials.
The photonic cavities studied in this work are based in two dimensional photonic crystal slabs.
A 2D-PhC slab consists of a material which is periodically patterned in the plane presenting a null
variation of the dielectric constant in the vertical direction (translation symmetry). In that case,
the optical in-plane confinement is due to the 2D-PhC pattern and in the third direction is given by
total internal reflection. If the upper and lower sides of the slab have the same dielectric constant
the structure is also symmetric with respect to a plane (lets say XY) across the center of the slab
and the modes can be classified either as even (TE) or odd (TM). Fig. 15 shows two examples of a
2D photonic bandgap structure calculated by the Guided-Mode Expansion (GME) model [37]. An
important feature of this kind of structures is the light line (upper green lines in the plot) of the
surrounding material. This line is the limit between the Bloch modes: the ones that lie below the
2-18
light line are confined by index guiding (non radiative, with no intrinsic losses); and the ones lying
above the line are leaky modes (with intrinsic losses by diffraction out of the plane).
Figure 16. Dispersion relationship of a W1 photonic crystal waveguide on InP material. The parameters of the
stimulation are: hole radius r/a=0.27 and slab thickness d/a=0.539. Blue color: even modes (σ yz = +1). Red color: odd
modes (σyz = -1). Only even modes σxy = +1 (TE-like) are plotted. Black line is the light line of the cladding core material.
We can obtain a PhC microcavity by creating a defect in the periodic 2D-PhC slab, see Fig. 16. In
that case, a set of optical states appears inside the photonic band gap of the periodic structure.
Depending on the characteristics of such defect, the number of cavity modes is different. In 2DPhC slabs, the microcavity modes have intrinsic losses due to the incomplete band gap of the PhC
structures. For the characterization of a microcavity mode two main parameters are commonly
used: the quality factor (Q) and the modal volume (V). Q, a dimensionless parameter, accounts for
temporal confinement of energy in the cavity. It can be considered as the number of optical cycles
before the cavity energy decays by a factor e-2≈ 2x10-3, even if we will consider this parameter
from another point of view in next section. The parameter V is a measurement of the spatial
confinement of the microcavity mode and typically expressed in units of (/n)3 can be estimated
as follows:
∭
( )| ( )|
( ( )| ( )| )
(41)
One of the main properties of the PhC microcavities is that they confine light in volumes in the
order of V≈(/n)3, considered by some authors as close to the minimum limit for an optical cavity
[38]. This allows for the possibility of obtaining high ratios of Q/V with moderate values of Q that
can be used to improve the properties of single photon sources [31] and to get high rate
modulation lasers [36]. Another parameter related to the cavity mode is the effective refractive
index of the mode (neff). This gives a numerical value for the overlap between the cavity mode and
the dielectric material. For example, a low value of neff indicates a large overlap between the mode
and the air region. This effective refractive index is defined as:
∭ ( )| ( )|
∭| ( )|
2-19
(42)
4. Light-mater interaction: coupling degree.
We can start by discussing the interaction between an emitter and the light inside a cavity, as
shown schematically in Fig. 17. For simplicity, we assume that a point-like emitter is inserted in a
planar resonator in such a way that it can absorb photons from the cavity modes and therefore
emit photons into the cavity [39]. We are particularly interested in the case where the transition
frequency of the emitter coincides with one of the resonant modes of the cavity. In these
circumstances, we can expect that the interaction between the emitter and the light field will be
strongly affected, since the emitter and cavity can exchange photons in a resonant way [40]. The
transition frequencies of the emitter are determined by its internal structure and are taken as
fixed in this analysis. The resonance condition is then achieved by tuning the cavity so that the
frequency of one of the cavity modes coincides with that of the emitter transition. At resonance
we find that the relative strength of the emitter–cavity interaction is determined mainly by three
parameters:
• the photon decay rate of the cavity WP =1/p,
• the non-resonant decay rate ,
• the emitter–photon coupling parameter g0.
Each of these three parameters defines a characteristic time-scale for the dynamics of the
emitter–photon system [10]. The interaction is said to be in the strong coupling limit when g0 >>
(WP, ), where (WP, ) represents the larger of WP and . Conversely, we have weak coupling when
g0 << (WP, ). Those limits mean that in the strong coupling case, the emitter–photon interaction is
faster than the irreversible processes due to loss of photons out of the cavity mode. This makes
the emission of the photon a reversible process in which the photon can be re-absorbed by the
emitter before escaping from the cavity. By contrast, in the weak coupling the emission of the
photon is an irreversible process, as in normal free-space spontaneous emission, but the
magnitude of spontaneous emission rate (W0) varies if the emitter is embedded into the cavity.
Figure 17. A two-level atom in a resonant cavity with modal volume V 0. The cavity is described by three
parameters: g0, κ, and  which, respectively quantify the atom-cavity coupling, the photon decay rate.
Therefore, previous to study the spontaneous emission of an emitter coupled to a cavity mode
we need to consider the relative magnitudes of WP, , and g0. First, the photon decay rate (WP) is
governed by the properties of the cavity that determine its quality factor Q, WP = ω/Q. Hence high
Q values mean relatively small photon loss rates. In practice, very high Q factors are required
before any of the effects described in this section were observed. Secondly, the non-resonant
decay rate  is determined by several factors. In the example of Fig. 17 the emitter could emit a
photon of the resonant frequency in a direction that does not coincide with the cavity mode, for
2-20
example, sideways. Alternatively, the emitter could decay to other levels, emitting a photon of a
different frequency that is not in resonance with the cavity. Yet again, the emitter in the excited
state could be scattered to other states and perhaps decay without emission of a photon at all.
The first of these processes is a property of the cavity, in the case of 3D resonators with optical
loss are related with the leaky field which are negligible for PhC slabs. The second is determined by
the internal dynamics of the emitter, and represents a breakdown of the two-level atom
approximation which we are not going to consider for our quantum emitters. The final process is
connected with the same sort of scattering events that cause dephasing. Therefore, for the case of
radiative decay to non-resonant photon modes, we can set  equal to the transverse dephasing
rate:
(43)
where p is the longitudinal decay rate given by:
(
) (44)
being ΔΩ the solid angle subtended by the cavity mode [41]. The analysis of p is of great
importance in planar cavities or micropillars due to the contribution of leaky modes. In contrast, in
PhC slabs the optical loss is dominated by the photon decay rate of the cavity. This leaves us to the
comparison of Wp with the third parameter, g0, namely the emitter–cavity coupling rate.
Previously it has been illustrated how two-level emitters interact with resonant light fields
originating from external sources. The situation we are considering here is slightly more
complicated, because there is not an external source to determine the field strength. We
therefore have to consider the interaction between the emitter and the vacuum field that exists in
the cavity due to the zero-point fluctuations of the electromagnetic field.
The interaction energy ΔE between the emitter and the cavity vacuum field is set by the electric
dipole interaction
|
| (45)
where μ12 = −e<1|x|2> is the electric dipole matrix element of the transition, and Evac is the
magnitude of the vacuum field. On setting ΔE equal to ħg0 we then find:
(
)
(46)
It is therefore apparent that the emitter–photon coupling rate is determined by the dipole
moment μ12, the angular frequency ω and the modal volume V0. Eq. 46 allows us to compare the
emitter–photon coupling rate directly with the dissipative loss rate, and hence determine whether
we are in the strong or weak coupling regime, respectively. If we assume that the cavity loss rate
WP is the dominant loss mechanism, the strong coupling occurs when:
g0 >> ω/Q (47),
it is, when
(
)
2-21
(48)
It is worth noting that this condition is very strict, and it requires cavities with very high Q
values. In most cases, single emitter systems will be thus in the weak coupling regime, especially
when the loss rate to non-resonant modes is significant. The situation improves, however, if we
have N atoms in the cavity. The criterion for strong coupling is then given by the next expression
as demonstrated in Ref. 42.
(
√
) (49)
As a result we expect weak coupling when the emitter–cavity coupling constant g0 is smaller
than the loss rate due to either leakage of photons from the cavity (WP) or decay to non-resonant
modes (). This means that photons are lost from the emitter–cavity system faster than the
characteristic interaction time between the emitter and the cavity. Roughly speaking there is not
time enough to observe energy exchange between the electronic states of the emitter and the
cavity modes [10]. Since the effect of the cavity is relatively small in the weak coupling limit, it is
appropriate to treat the emitter–cavity interaction by perturbation theory. In order to illustrate
this interaction, we use Fermi’s golden rule to calculate the emission rate for the atom in free
space, and then we calculate the revised rate when the atom is coupled resonantly to a single
mode of a high-Q cavity. We will see that the main effect of the cavity is to enhance or suppress
the photon density of states (in comparison with the free-space value), depending on whether the
cavity mode is resonant with the optical transition or not. This either enhances or suppresses the
radiative emission rate through the density of states factor that appears in Fermi’s golden rule.
a) Free-space spontaneous emission.
Before considering the spontaneous emission of a single emitter into a resonant cavity mode, it
is helpful to remind ourselves of the theory of dipole emission in free space. To do this we
consider the properties of an optical transition in a large volume V0. The volume is large enough
for avoid border effects on the properties of the emitter, and is merely incorporated to simplify
the calculation. The transition rate for spontaneous emission is given by:
|
|
( ) (50)
where M12 is the transition matrix element and g(ω) is the optical density of states. For the
density of states we use the standard result for photon modes in free space described in section 1.
( )
(51)
while for the matrix element we can use the electric dipole interaction:
〈
〉 (52)
Assuming that there is not an external field source within the cavity, we must use the vacuum
field for Ε for describing the matrix element interaction. Averaging over all possible orientations of
the atomic dipole with respect to the field direction [10], we then obtain:
2-22
(53)
Hence from Eq. 50 we find the final result:
(54)
where τR is the radiative lifetime. We thus conclude that the emission rate is proportional to the
cube of the frequency and the square of the transition moment.
b) Weak coupling.
Figure 18. (a) A two-level atom in a single-mode cavity with volume V0. (b) Density of states function g(ω) for the
cavity. The angular frequency of the cavity mode is ωc, and Δωc is its linewidth.
We now come to the main task of this section: to calculate the spontaneous emission rate for a
two-level electronic transition of an emitter coupled to a single-mode resonant cavity in the weak
coupling limit. This problem was first considered by E. M. Purcell in 1946 [40], and the resulting
change to the emission properties of the emitter is now called the Purcell effect. For illustrating
such effect we consider a single-mode cavity of volume V0 as shown in Fig. 18. There will of course
be other modes in the cavity, but we neglect them in this analysis because they are assumed to be
far from optical transitions of the emitter. In the weak coupling limit it is possible to use a
perturbative approach similar to that for an atom in free space. Then, the emission rate is also
given by Fermi’s Golden rule. We assume that the cavity mode has an angular frequency c with a
half width Δωc determined by the quality factor Q. The density of optical states g(ω) for the cavity
will then take the form shown in Fig. 15 (b) [43]. Since there is only one resonant mode, we must
have:
∫
( )
(55)
which is satisfied if we use a normalized Lorentzian function,
2-23
( )
(
(56)
)
If the frequency of the optical transition is ω0, then we must evaluate Eq. 56 at ω0 to obtain:
(
)
(
(57)
)
From this expression we can conclude that the emitter would reach the maximum number of
available states at exact resonance with the cavity (i.e. ω0 = ωc):
(
)
(58)
As with the free space, we use the electric dipole matrix element given in Eq. 52 and write in
analogy to Eq. 53:
(59)
The factor is the normalized dipole orientation factor defined by:
|
|
| ‖ |
(60)
We can recall it is usually averaged to 1/3 for the case of the randomly orientated dipole in free
space. On substituting Eq. 57 and 59 into Fermi’s golden rule, we obtain:
(
)
(61)
This rate can be compared to the free-space value given in Eq. 54. We now introduce the
Purcell Factor (PF) defined by:
(62)
Substituting from Eqs. 54 and 61 we then find:
(
)
(
)
(63)
where we have replaced c/ω by (λ/n)/2π, being λ the free-space wavelength of the light and n
the refractive index of the medium inside the cavity. At exact resonance and with the dipoles
orientated along the field direction, Eq. 63 reduces the magnitude, we will call here as figure of
merit intrinsic value, only depending on Q and Vo of the Purcell effect:
(
2-24
)
(64)
The PF is a convenient parameter that characterizes the effects of the cavity. A PF greater than
unity imply that the spontaneous emission rate is enhanced by the cavity, while FP < 1 implies that
the cavity inhibits the emission. Eq. 64 shows that large PF require high Q cavities with small
modal volumes. Furthermore, Eq. 63 indicates that we need to have close matching between the
cavity mode and the atomic transition, and we also need to ensure that the dipole is orientated as
parallel with the mode field as possible. The enhancement of the emission rate on resonance is
related to the relatively large density of states function at the cavity mode frequency. By contrast,
the inhibition of emission when the emitter is off-resonance is caused by the absence of photon
modes.
c)
Strong Coupling regime.
For reaching the strong coupling conditions it is required the emitter–cavity coupling rate g0 to
be larger than the cavity decay rate. The emitter generates a photon into the resonant mode,
which in Fig. 18 “bounces between the mirrors” and can be re-absorbed faster than it is lost.
Roughly speaking, the reversible interaction between the emitter and the cavity field is thus faster
than the irreversible processes due to photon losses. The interaction between a resonant cavity
mode and the emitter was first analyzed in detail by Jaynes and Cummings in 1963 [44]. The
Jaynes–Cummings model describes the interaction of a two-level atom with a single quantized
mode of the radiation field. The description of this model is beyond the scope of this Thesis so the
reader is referred to the bibliography for further details. Here we just summarize the main
conclusions. We consider first the “bare” states of an uncoupled resonant system with just a single
atom of angular frequency ω in the cavity. As aforementioned, in the strong coupling regime the
emitter and the cavity mode can exchange energy in a reversible way. The reason is that the
system can oscillate some cycles before the photon leaves the cavity. This effect produces an
emission energy splitting called Vacuum Rabi Splitting whose magnitude can be deduced from the
Jaynes-Cummings model given by:
( )
√ (
)
(65)
The vacuum Rabi splitting can be understood as an AC Stark effect induced by the vacuum field
[45]. Such emitter-cavity interaction can be described by means of a quasi-classical explanation by
considering the properties of two coupled classical oscillators. If 1 and 2 are the frequencies of
the two oscillators (in our case the optical transition and the cavity mode) the frequencies of the
coupling modes will be given by:
(
)
(
(
) )
(66)
where is the coupling strength. As a result, the spectrum would consists of two emission lines
at very close frequencies, 1=2=

(67)

2-25
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2-27
Chapter 3. Set-up, Instrumental & Characterization
techniques.
“If it bleeds we can kill it”.
Alan Schaeffer (Arnold Schwarzenegger), Predator; 1987.
3-0
1. Introduction to the confocal microscopy.
The confocal microscope was proposed by a post-doctoral researcher at the
University of Harvard; M. Minsky, 1955 [1]. The technique could not been really
exploited until the late sixties with the appearance of high intensity light sources and
the development of numerical calculation capabilities (allowing to handle large
amount of data). During the seventies, the advances in experimental techniques were
accompanied by the development of scanning-image-formation theory; Shepard &
Choudhury, 1977 [2]. As a result, the confocal microscopy became applied to different
research fields driving to the design of the first commercial set-up in 1987. Nowadays,
the confocal microscopy is extended for different requests from which we could
highlight tomography and the examination of biological specimens, even the confocal
microscope contribution is also essential in different research topics: spectroscopy,
interferometry, optical profilometry, etc.
In front of the conventional microscopy, the confocal advantages are mainly
derived from possibility of optical sectioning. The confocal microscope is able to obtain
an extremely efficient light collection at the focal plane avoiding the influence of light
from unfocused planes. This way, the different sections of a transparent sample can be
separately studied without carrying the contribution of defocused light. Thanks to this
fact, it is allowed to perform tridimensional (3D) reconstructions, as in the example of
Fig. 1(a). At the top of this figure it is shown a confocal image of a porous silicon wafer
obtained by scanning the sample in plane of the surface (i.e. XY). The surface details
(porous of about 80 nm diameter) can not be distinguished since they are quite below
the optical resolution limit but we can distinguish the sample roughness. At the
bottom of Fig. 1(a) it is shown a transversal cut obtained by scanning the sample in the
plane XZ. The transversal cut is acquired without moving the sample, but changing the
slow scan axis (the Y direction is substituted by a step Z motion). As a result, the
sample-edge-like image is obtained by scanning a determined profile (marked with a
red line at the top image) at different work distances. In order to explain this
capabilities, in Fig. 1(b) it is shown how different lens systems work. At the case 1, it is
represented the optical ray diagram of a convergent lens. Employing words of the
geometrical optics, the image-lens distance is related with the object-lens distance,
such a way, that the image of different points of the sample is generated at different
points of the imaginary space. Having this fact into account, the light from each point
of the object can be filtered by locating a pinhole in determined points of the light
path. For example, in the top of case 2, a pinhole allows the propagation of the
depicted light rays (top of the image) while most of the light coming from other points
of the sample are reflected, bottom configuration. This is the fundamentals of the
confocal microscopy: selecting an observation plane by filtering the contribution of the
defocused planes. The typical confocal configuration is described in case 3 where, in
addition to the pinholes, the light source is focused in the observation point favoring
the pinhole filtering effect. This fact is really useful for the study of isolated
semiconductor nanostructures since at the same time that the collection area is
reduced by the presence of the pinhole, the excitation light (and therefore the carrier
photoinjection) is focused on such area. This fact is better described in Fig. 2, where it
is shown how using a confocal set-up the volumes under study are considerable
reduced allowing the study of a single emitter.
3-1
Figure 1. (a) At the top, scanning confocal microscopy image of a thin film of porous silicon;at the
bottom it is shown the transversal cut-edge corresponding to the profile marked with a red line. (b)
Description of several lens systems. (Case 1) the light coming from different focal planes are propagated
forming different light cones. (Case 2) The light coming from different focal planes can be filtered by
locating a pinhole at the apex of the light cone formed during the propagation. (Case 3) In confocal
microspopy while a pinhole is used to filter the corresponding focal plane the illumination is focused on in
such point (as in transmission as in reflection).
Figure 2. Effective illumination volumes for widefield (left) and confocal (right) microscopy.
The fundamental properties of confocal microscopy and the generation of scanning
images are perfectly explained by means of the scalar theory of the diffraction [3]. In
this formalism the propagating light is treated as a spatial dependent scalar field
distribution. In the free space, the field amplitude is diffracted according with the
corresponding transference function which describes the field propagation, as a linear
3-2
invariant system (LIS) [4]. In the same way, the light transmitted in optical systems
(through lenses, pinholes, mirrors…) can be determined by introducing the
transference function of each component. Thanks to this fact, the field amplitude can
be perfectly known at each plane of the space. For example, the optical intensity at the
focus of a lens can be described by the field distribution at its focal plane, usually
called PSF (point spread function). Working with circular apertures the optical intensity
of the PSF forms a characteristic distribution known as Airy’s Disk. The Airy’s Disk
consists of a circular diffraction pattern forming an intensity maximum at the center of
the distribution, surrounded by minor intensity maxima and separated by dark regions.
In Figure 3, it is shown the field distribution in different planes of the optical system
showing the Airy’s Disk at the focus point.
Figure 3. Diffraction (propagation) of the Airy’s Disk produced by a circular microscope objective.
Figure 4. Amplitude profile (top) and diffraction pattern (bottom) of two identical Airy’s Disks. The
pattern can be resolved while the intensity maxima do not illuminate the first minima.
3-3
The importance of the PSF is explained due to their relation with the system
resolution. In the optical microscopy framework, the resolution is defined as the
minimum distance at which two objects could be distinguished in the resulting image.
The resolution of an optical system is really determined by the PSF since such
parameter determines the focus size. But independently of the focus characteristics,
different criteria can be used to determine the resolution of an optical system. The
Rayleigh’s criterion is the generally accepted for estimating minimum resolvable
details. According to this criterion, the imaging formation is said to be diffractionlimited when the first minimum of SPF of a point-like object coincides with the
absolute maximum of another, see Figure 4.
Accordingly, the lateral resolution of a microscope objective whose focus is
described by means of an Airy’s Disk would be expressed as:
(1)
where  is the light wavelength, NA the numerical aperture and 0.61 a dimensionless
magnitude considering the half distance between the maximum and the first minimum
of the Airy’s Disk. But notice that in the case of the confocal microscope, the collection
point is not homogeneously excited since the illumination light is also focused and
therefore it also produces an excitation Airy’s Disk, see Fig. 5. In fact, experimental
results in fluorescence confocal microscopy have shown a reduction of the confocal
PSF of around a 30%, with respect to a system with a unique collection pinhole. The
resolution enhancement is consistent with the next definitions:
(2)
(3)
Figure 5. (a) PSF expected of a simple optical system like case 2 in Fig. 1. (b) PSF of a confocal
microscope.
3-4
It is worth noting that, the properties above described are observed just if the
collection (excitation) pinhole is smaller enough to generate an Airy’s Disk at the focus.
On the other hand, an excessive reduction of the apertures could drive to a noticeable
diminution of the optical signal with any resolution enhancement. In this sense, the
sample properties and the working wavelength usually determine the optimum
parameters of the microscope components.
2. Devices.
In this section we are going to describe the devices working in the different set-ups
employed for characterization techniques. The instrumental can be classified in three
different main types: a) light sources for optical excitation, b) cryogenics, optics and
optomechanics or c) detection systems.
a) Optical excitation.
The more extended excitation source in semiconductor spectroscopy is
the laser light due to its coherence, collimation and monochromatic
properties. For the study of III-V semiconductors it is required a laser source
working at wavelength range 700-1000 nm (at the semiconductor gap or
eventually close to). In our laboratory it is employed a solid state laser
(Ti:Sapphire) and several laser diodes in order to cover this range.

Ti:Sapphire (Mira 900D from Coherent)
The titanium sapphire laser (Ti:Al2O3) is a solid state laser extensively
used due to its wide tuning range, since the titanium emission into a
sapphire matrix offers a broad luminescence band around 700-1000
nm. In the Mira 900 model, the operation wavelength can be selected
by means of birefringence filters, as in continuous wave (CW) as in
pulsed operation (75 MHz repetition rate). It is provided with two
different cavity configurations allowing the generation of picosecond
or femtosencod pulses. The pulse generation is done by means of the
Kerr Lens Modelocking technique (KLM). For this purpose, it is located
a micrometric slit which is acting as a saturable absorber thanks to the
self-focusing properties related with the Kerr effect in the active
medium.
The titanium fluorescence is obtained by the optical pumping of the
doped crystal by means of the second harmonic of a Nd:YVO4 laser
(Verdi V5 from Coherent). The neodymium is excited by several laser
diodes coupled to a high power optical fiber producing emission at
1064 nm wavelength. The second harmonic (532 nm) is generated by
means of a non-linear crystal (LBO at 148 °C). Applying a current of 12
A to the laser diodes it is obtained an optical power of 5W at 532 nm
wavelength. In such pumping conditions the Ti:Sapphire offers a
maximum optical power of 1.2 W at 808 nm wavelength. On the other
hand, the Ti:Sapphire beam present a multimodal structure required
for the generation of ultra short pulses. But sometimes, this
3-5
characteristic is not convenient for characterization techniques. For
this reason, the use of the solid state laser is substituted by
complementary light sources.
Figure 6. Saturable absorber based on the Kerr effect.

Diode Lasers (Thorlabs).
The diode laser is a versatile tool for spectroscopy applications due to
its compact design and low price. A single chip with current and
temperature control can reach a really stable light output achieving
power close to 1W. Due to the different available active media, laser
diodes are able to cover an extended region of the spectra, from the
near ultraviolet to the middle infrared. The diodes are usually
composed by an optical monomode cavity responsible of the
stimulated emission of the active medium. In characterization
experiments we are going to describe in the next section, the laser
diodes have been used for optical excitation of the micro-PL
measurements, but sometimes they can be included in strategic
points of the set-up in order to help for the alignment and
optimization process. In addition, laser diodes can also operate under
pulsed conditions. For example, some of the time resolved
measurements made on InAs/InP quantum wires has been performed
by using a pulsed diode laser working at 980 nm wavelength.
b) Cryogenics, optics and optomechanics.

Cryogenics
In order to prevent thermal escape of carriers during the study of the
optical properties of quantum nanostructures it is necessary to
perform measurements at low temperatures (4-80K). In addition, the
emission broadening by phonon scattering is practically negligible in
this temperature range. Depending on the characterization set-up it
can be used different cryogenic techniques. For the optical
characterization of ensemble nano-structures, the samples are held in
the cold finger of a close-cycle He cryogenerator shown in figure 7(a),
where the sample can reach 12 K. The temperature can be fixed by
3-6
means of a temperature control system based on a silicon measuring
device controlling a resistance heater.
In order to prevent vibrations the low-temperature confocal microspectroscopy measurements are carried out using two different
adapted dewars: a 60L liquid helium dewar (4-140K) and a 100L liquid
nitrogen dewar (80-300K), see Fig. 7(b). For this purpose, it has been
necessary to design security mechanisms for control of the pressure
into the dewar during the immersion process. Thanks to this, the
confocal microscope can be maintained at very low temperatures for
several weeks. The use of the cryogenerator, as well as the immersion
dewars, requires high vacuum chambers in order to avoid water
condensation at the sample surface. In the case of our confocal
microscope, a vacuum system consisting of a rotary and a turbomolecular pump is used to reach a vacuum in the order of 10-5 Torr.
Figure 7. (a) Macro-PL set up and criogenerator. (b) Micro-PL set up and liquid gas immersion dewar.

Optics and Optomechanics.
The light coming from the excitation sources must be guided to the
different stages of the experimental set-up by means of lenses,
mirrors, beam splitters, optical fibres, filters and polarizers, whose
position with respect to the other components of the optical system
must be accurately controlled. For this reason, the components are
mounted using commercial holders and positioning systems
(Thorlabs). The characteristics of the optics depend on the
experimental conditions. In the case of macro-PL, the excitation and
collection light is managed by means of 50 m core multimode optical
fibres (NA=0.22). The monomode optical fibres (4 m core and
NA=0.12) are used for the micro-PL study, in this case the cut-off
wavelength must be selected according to the optical signal. The
3-7
optical fibre inputs (outputs) are focused (collimated) with the aid of
microscope objectives and lenses. Previously to the detection, the
optical signal is usually dispersed and analyzed using monochromator
or spectrometers (respectively for single channel or multichannel
detection). We have employed mainly two spectrometers: Spectra
Pro 2500i and DSP 300i, both from Acton Research. These devices are
controled by means of a Visual Basic interface.
c) Detectors.
In general, single semiconductor nano-structures offer a relative weak
photoluminescence signal, typically around hundred photons per second at
the exit of a monochromator. For this reason, optical characterization of
quantum emitters requires the use of detection devices with an optimum
signal-to-noise ratio (S/N) and high quantum efficiency (QE):
√ (12)
√
In Eq. 12 np represents the number of photons per second, nd is the number of
electrons generated by the dark current while t is the time. From this
expression it is directly deduced that weak optical signals (low values of n p)
requires low noise (nd) or larger integration times in order to obtain a good
signal-to-noise relation.

Silicon CCD
For the multichannel detection in the visible and near-infrared (4001050 nm) it has been employed a silicon back thinned CCD (DU-401
BR-DD from Andor Technology). This device is provided of a
thermoelectric system allowing a detector working temperature close
to -80°C. This fact drives to an important reduction of the dark current
(0.1 e-/pixel*s) which, combined with the high quantum efficiency at
this wavelengths, allows the measurement of weak optical signals in a
relative short time.

InGaAs Array
In order to complete the detection spectra from 900 to 1600 nm
wavelength, it is employed an InGaAs detector array (iDus 492-1.7
from Andor Technology). It is also provided of a cooling system but in
the case of InGaAs technology the dark current can not be reduced
more than 11700 e-/pixel*s at -75°C.

Single photon detectors
The time resolved measurements requires the use of ultra fast
detectors combined with photon counting electronics. We can use
silicon or a InGaAs avalanche diodes depending on the optical signal
wavelength. A SPCM-AQRH15 FC (from Perkin Elmer) is employed in
3-8
the silicon detection range; while an Id200 (idQuantique) is used in
the near infrared range (NIR). The data coming from the APDs are
managed by a photon correlation board (Edinburgh Instruments)
which monitors delay of the optical signal with respect to the
excitation pulse drawing the PL transients as a result.
3. Experimental techniques.
In this section we describe the main experimental techniques and set ups employed
during the experimental data acquisition. Let notice that the particular details of their
corresponding components have been described in section 2.3.

Macro-photoluminescence (macro-PL)
Figure 8. Basic scheme of the macro-PL set-up.
Figure 8 shows the experimental set-up employed for the macro-PL
study used in the optical characterization of nanostructure
ensembles. The excitation light is directed to the sample (hold at the
cold finger of the cryogenerator) by means of mirrors, filters, beam
splitters and lenses. The laser output is propagated a distance close to
3 meters producing a beam divergence around 8 mm previous to the
focusing objective, which leads to an excitation spot of around 6 m.
Using the same objective, the sample emision is coupled to a
3-9
multimode optical fiber in order to be guided to the detection system.
The excitation power and energy can be tuned allowing PL studies on
temperature, power and on excitation energy (PLE).

Micro-photoluminescence (micro-PL)
Figure 9. Basic scheme of the micro-PL set-up.
The single quantum emitter measurements are performed by means
of confocal microscopy Fig. 9. The main part of our confocal set-up
consists of a commercial cryogenic microscope stick, attoCFM I
(Attocube Systems). The excitation and collection light is guided by
monomode optical fibers whose cores act as pinholes but the
confocal conditions are achieved by a free optics alignment system in
order to allow polarization measurements. As afore mentioned, the
low temperatures are achieved by immersing the microscope stick in
liquid nitrogen or helium dewar. The sample holder is located at the
bottom of the microscope, which consists of a piezoelectric inertial
position system covering a range of 7 mm along the directions X and Y
(the plane of sample surface) and 6 mm along the Z direction
(propagation direction of the light). In general the Mira 900 is used as
excitation source but it can be substituted for several fiber coupled
laser diodes. In any case, the excitation light is split previously to
3-10
arrive to the microscope objective (BS1) allowing the measurement to
the excitation power. The second beam splitter (BS2) is employed to
produce optical images of the sample surface during the
characterization of the photonic devices. Finally, the optical signal of
the sample is coupled to the collection optical fiber and analyzed by
the same detection system used for macro-PL.
4. Data processing.
The confocal set-up shown in Fig. 9 can be used to obtain PL specta of isolated nanostructures in low density samples. Some of the different excitonic species confined at
the nanostructure can be identified by studying the power evolution of the relative
peak intensities. As an example in Fig. 10 it is represented the emission spectra of
three isolated InAs/GaAs quantum dots from three different samples fabricated by
MBE. The samples are grown in the same conditions being the unique difference the
buffer layer deposited between the GaAs and the InAs. The buffer layer is deposited in
order to reduce the non-radiative mechanisms produced by the defects in the
interface between the epitaxy material and the GaAs substrate. The buffer is also
deposited by MBE and it consists of a 3.5, 30 and 150 nm thin film of GaAs in samples
of Figs. 10(a), 10(b) and 10(c), respectively. From the corresponding spectrum we can
verify that the buffer layer of 3.5 nm is not thick enough to avoid non-radiative
recombination since the optical intensity is almost a factor five smaller than in the
other QDs. For the same reason, we have not observed noticeable differences in the
integrated optical intensity measured from QDs in Figs. 10(b) and (c); where it is
concluded that depositing a 30 nm buffer the recombination at the interface defects is
noticeable avoided. The PL characterization is completed by carrying out time resolved
photoluminescence (recorded after PL spectrum). In Fig. 10(d) it is shown the PL
transients corresponding to three QDs (green, red and magenta scatters) for samples
with 3.5, 30 and 150 nm buffers respectively. The decays are measured at the exciton
wavelength, which at the excitation power regime corresponds to the highest intensity
peak. As expected from the steady state PL results the green transient (3.5 nm buffer
layer) presents a shortest decay time due to the contribution of non-radiative
mechanisms, in contrast to the other QDs (with similar times). The radiative PL decay
times can be determined by the deconvolution of the system response, black scatters,
obtaining quantitative results: 0.64, 1.47 and 1.83 ns life time for green, red and
magenta decays.
3-11
Figure 10. PL spectra of self-assembled QDs grown onto different thickness spacer: 3.5, 30 and 150 nm
for (a), (b) and (c). The time resolved measurements are performed at the maximum intensity peak. From
the decays of TRPL we can extract the lifetime of the studied optical transitions (d).
References.
[1] M. Minsky, “Memoir on inventing the confocal scanning microscope” Scanning 10, 128 (1988).
[2] C.J.R. Sheppard and A. Choudhury, “Imaging formation in the scanning microscope” Optica Acta 24,
1051 (1977).
[3] J.W. Goodman, “Introduction to fourier optics” Roberts & Company Publishers, third edition (2004).
[4] J.D. Gaskill, “Linear systems, fourier transforms and optics” Willey-Interscience, First Edition (1978).
3-12
Chapter 4. Characterization of site controlled InAs Quantum
Dots grown on GaAs(001) pre-patterned substrates.
"The principles of physics, as far as I can see,
do not speak against the possibility of maneuvering things atom by atom."
R. Feynman
4-0
1. Introduccion.
From the technological point of view, the interest in QDs is related with their emission
properties. The photoluminescence (PL) spectra of the QDs are typically observed as narrow peaks
associated to discrete energy levels. In fact, the density of states in semiconductor QDs is usually
represented by a Dirac’s delta, and for this reason the QDs are usually called artificial atoms. The
emission properties of QDs together with the new developments in fabrication techniques have
allowed the improvement of opto-electronic devices and other new applications using QDs as
active medium. For instance, the epitaxial materials with high density of QDs are usually employed
for the design of heterojunction lasers [1], infrared (IR) detectors [2] or intermediate band solar
cells [3]. For these purposes, a narrow QD size distribution is required but maximizing the active
medium volume. In contrast, low QD density samples are the basis of single photon emitter
devices and therefore they are potential candidates to quantum cryptography and quantum
information processing applications. For these requests, it is mandatory the control of the QD
density, position into the device and emission energy. For example, one of the more extended
approaches consists of enhancing the single emitter signal by the coupling of the nano-structure
PL with the optical mode of a photonic crystal microcavity (PCM). As explained in Chapter 2, this
kind of devices can operate in two different regimes depending on the interaction between the
optical mode and the cavity emitter: weak or strong coupling. The emitter spontaneous emission
(SE) rate (and therefore the SE enhancement) depends on the relative position of the emitter and
the maximum electric field of optical mode (see spatial detuning in section 3 of Chapter 6). On the
other hand, quantum information applications sometimes require of the fabrication of quantum
dot molecules. In this sense, vertically aligned QDs have been successfully fabricated thanks to the
control of stress propagation effects by Stranski-Krastanow fabrication. However, lateral
molecules are considered better candidates since they offer more facilities for scaling-up to the
electronic coupling of multiple QDs (e.g. quantum systems with several qbits). The fabrication of
QDs based on semiconductor materials can be performed by means of three main approaches:
fabrication techniques based on lithography and chemical etching; self-assembled processes and
epitaxial growth onto pre-patterned substrates.
The first kind is referred to the processes used for the fabrication of nano-structures after the
epitaxial growth of the desired material (top-down technology). The main disadvantages of this
approach are related with the introduction of contaminants and surface defects produced during
the manipulation processes. In addition, the density of QDs is conditioned by the limitations of the
lithography spatial resolution.
The self-assembled QDs are spontaneously obtained during the epitaxial growth of a layer with
lattice mismatch with respect to the wafer material, i.e. Stranski-Krastanow method. As a result,
sample contamination and surface defects are reduced allowing in addition, the control of the
density and size distribution of the resulting nano-structures. The problem is that this kind of QDs
is obtained in random positions.
In the past the possibility of using epitaxial grown onto pre-patterned substrates has appeared
[4], allowing the positioning QDs with nanometer precision. This approach is in between the latter
ones, that is, the self-assembled processes are combined with the preferential QD nucleation onto
lithography patterns. In literature it can be found several approaches for obtaining preferential
nucleation of quantum dots. The most used techniques to pattern are based on e-beam, AFM,
nano-imprint, oxide/deoxide or alloying surfaces, etc [5-10]. Using these different methods it has
been successfully obtained several kinds of patterns, either for low or for high density samples,
demonstrating position capabilities in the range of ten nanometers [11-13]. Unfortunately, in most
4-1
cases the experimental results are still far away from the technological needs because the
quantum efficiency of the emitters is strongly affected by the chemical process during pattern
fabrication. Between the different nano-structures grown onto pre-patterned surfaces, we would
like to point out the following:
 Droplet Epitaxy. The nano-hole formation on a GaAs wafer is produced first by effect of
Ga droplets. After that the nano-hole is refilled by GaAs deposition driving to
elongated GaAs mounds along the [110] direction. As a result of the capillarity-induced
diffusion, InAs is influenced to grow inside such nano-holes, providing the possibility to
fabricate InAs/GaAs isolated QDs and even quantum molecules [14-16]. Other benefit
of the droplet epitaxy growth is related to the positioning capabilities. After an
appropriate capping of InAs QD, the resulting GaAs top surface show a characteristic
mounding morphology that allows a direct localization of the buried nano-structures.
Different experiments based on the nucleation of new nanostructures onto capping
layers have lead to establishing a one-to-one correspondence between the mounds
and the buried nanostructures [17]. This result is of high-technological interest for the
fabrication of single emitter photonic crystal micro-cavities.
 V-Groove QWRs. Metallorganic chemical vapor deposition MOCVD and MBE on
substrates patterned with V-groove arrays and piramidal features or masked with SiO2
stripes have been successfully employed as an alternative approach to the
spontaneous self-ordering method. This technique drives to better nanostructure
uniformity, since it overcomes the intrinsic randomness of the nucleation process. The
resulting nano-structures can be staked and easily embedded into a micro-cavity using
the pre-pattern as a position reference during the fabrication process [18].
 Ga-assisted deoxidation patterns. It has been recently demonstrated the sitecontrolled growth of isolated InAs QDs and QD pairs onto electron-beam patterned
(001) GaAs substrates using in situ Ga-assisted deoxidation prior to overgrowth. The
deposition of few layers of gallium (in the absence of arsenic) followed by a brief
anneal under low arsenic pressure is used to remove the surface oxide maintaining the
pattern. Good site-control of InAs QDs has been achieved using this technique,
allowing in addition the vertical staking of the QDs, as has been already demonstrated
in dense ordered arrays [19].
In this chapter we are going to describe the technological developments achieved to obtain
efficient single quantum emitters (and emitter pairs) grown by MBE presenting a location control
at nanometer scale. The fabrication is carried out by controlling the growth of InAs QDs onto
GaAs(001) wafers pre-patterned by means of atomic force microscopy (AFM) local oxidation nanolithography. As a difference with other patterning techniques, AFM lithography allows the
fabrication of patterns with nanometer resolution avoiding surface contamination and interface
damages. The samples are prepared by means of a three step process that we will briefly describe
in the following. In the PhD Thesis of J. Martín-Sanchez, from the IMM (Madrid), it has been
studied the evolution of the pattern motifs during the grown process as the influence of the motifs
in the resulting nano-structures. It was identified two reproducible types of motifs, namely “simple
oxide” and “double oxide” nano-holes. The “simple oxide” nano-holes were pre-patterned for the
fabrication of isolated QDs matrices with a pitch period around 2 m, a very suitable geometry for
4-2
single QD location. On the other hand, almost the 65% of nano-structures grown on the “double
oxide” nano-holes consist of QD pairs, which could be good candidates for the growth of QD
molecules. The main objective of this work would consist of determining the optical properties of
the nano-structures fabricated by using both kinds of motifs.
2. Experimental details.
In this work we present results on InAs/GaAs QDs and on QD paris obtained in a fabrication
process combining AFM local oxidation lithography and molecular beam epitaxy (MBE) growth
techniques. Using patterned substrates we found optically active InAs QDs with decay times
similar to those of self-assembled ones. We also compare the optical properties of the InAs QDs
grown in identical samples differing just in their buffer layer. The results demonstrate that the
control and fine tuning of pattern motifs during the buffer layer growth are key factors to control
the optical properties the resulting nano-structures.
All samples were fabricated by the Molecular Beam Epitaxy Group at the “Instituto de
Microelectrónica de Madrid (CSIC)”. The GaAs epitaxial substrates are patterned by AFM local
oxidation nanolithography using a commercial Nanotec AFM system in tapping mode under
ambient conditions. The pattern consists of 7x7 square arrays of oxide motifs about 10 nm heights
with a diameter close to 160 nm with a pitch period of 2μm. The MBE re-growth process on
patterned substrates starts with in situ oxide desorption by atomic H exposure of the GaAs surface
at Ts= 490 °C for 30 minutes while As4 is also supplied in order to avoid As losses. Atomic H is
obtained by thermal cracking of H2 with base pressure PH2=1.1x10-5 Torr. This process efficiently
removes the oxide material, leading to the formation of nanoholes at the predefined positions,
providing a clean surface for the growth of nanostructures with good optical properties [20]. The
reflection high energy electron diffraction (RHEED) shows a 2x2 periodicity after 5 minutes of
atomic H exposure, changing to a c(4x4)-GaAs(001) reconstruction diagram when the atomic H flux
is closed.
Prior to InAs deposition, a 15 nm-thick GaAs buffer layer was grown at low temperature (Ts =
490 °C) in order to preserve the pattern motifs. Two growth methods were employed for the
growth of the GaAs buffer layer: atomic layer molecular beam epitaxy (ALMBE) [21] and
conventional molecular beam epitaxy (MBE). The MBE growth is carried out at a sample
temperature Ts = 490 °C and growth rate of 0.5ML/s under As4 beam equivalent pressure
BEP(As4)=2x10-6 Torr. The ALMBE growth is carried out under identical temperature, with Ga
growth rate and BEP(As4), but pulsing the As4 flux with a time sequence of 1s (As4 OFF, Ga ON)/ 1s
(As4 ON, Ga ON). On the top of the GaAs buffer layer, 1.5ML InAs is deposited with a growth rate
of 0.01ML/s at TS= 490°. The InAs coverage is below the critical thickness for QD formation as
observed by 2D-3D RHEED transition on non-patterned substrates (1.7 ML) under these growth
conditions, but it is enough for obtaining nucleation of InAs inside the nanoholes obtaining InAs
QD [22]. After 2 minutes of growth interruption under As4 flux, samples are capped with 20 nmthick GaAs layer for optical characterization. The first 8 nm of the GaAs capping layer are grown at
Ts = 490 °C and the remainder 12 nm are grown while Ts is being increased to 580 °C.
4-3
3. Optical Characterization.
Figure 1. (a) AFM image in a region of the 7x7 double oxide matrix sample obtained previously to the HF etching.
(b) Zoom of the InAs QDs and dot pairs grown on the pre-patterned sample shown in (a) after etching. (c) -PL image
of InAs QDs.
Figure 1(a) shows an AFM image corresponding to a region of the oxide matrix fabricated in the
lithographic step. Concretely, in the double oxide motif patterns the InAs is expected to nucleate
forming QD pairs almost with a 65% of probability, see Fig. 1 (b). In fact, the AFM characterization
of the patterned area after InAs deposition shows the high selectivity obtaining only QDs inside
the pattern. On the other hand, the nucleation of self-assembled QDs in the non-patterned region
has been completely suppressed. From morphological characterization previously performed in
samples grown in similar conditions we can conclude that the resulting nanostructures are
shallow, with an average height h≈7 nm with a height to diameter ratio h/D = 0.08. The aim of our
present study is to evaluate the emission properties of the QDs formed inside the patterned nanoholes after the H flux etching. For this reason, a 20x20 m2 area enclosing the patterned matrix
was delimited by metallic markers fabricated after growth by electron beam lithography. In this
way, we are able to find the markers by using the confocal microscope described in Chapter 3 and
investigate the emission spectra of the site-controlled QDs at low temperatures. Figure 1(c) shows
the micro-photoluminescence (-PL) image of the area under study as obtained by integration of
the photoluminescence in a 50-nm wide spectral window around 970 nm (≈1.25-1.30 eV) within a
region of the double oxide matrix sample. The distribution of the nano-hole sites is reproduced by
4-4
the most of the emission of site controlled QDs, as can be seen by comparison between Fig. 1(c)
and 1(a). Most of the sites without PL signal are explained by the presence of a QD emitting at
different energy. This fact is corroborated by reproducing the experiment in other spectral
windows, not shown.
Analyzing the different spectra composing the -PL images the optical properties of the simple
and the double oxide matrix can be studied and compared. It has been studied the emission
energy and linewidth of characteristic single QDs and dot pairs in both samples. In the simple
oxide matrix we have restricted our study to 10 simple and 2 double QDs, while in the double
oxide matrix the analysis is performed on 3 isolated QDs, 6 dot pairs and 1 triplet (not shown). In
both cases the emitters are distributed in 12 nano-holes. Such number of QDs is small and hence
we just can consider qualitative differences or similarities. Having this fact into account we can
conclude that:




The PL intensity is similar to self-assembled QDs.
The PL peak energy is centered at ~1.28 eV.
The exciton linewidth is about 500 eV but it can even reach 1.5 meV.
The exciton lifetime oscillates around 1 ns (0.8-1.2 ns).
In Figure 2 we summarize the optical characterization of a typical QD from the simple oxide
matrix sample at 5K. The inset in Fig. 2(a) shows a typical -PL spectrum of the wafer recorded in a
region far away from the matrix. As expected, we do not find spectral features which could be
associated to self-assembled QDs. The broad peaks around 1.360 eV correspond to emission from
the substrate associated to CuGa and its two longitudinal optical phonon replicas at 1.324 and
1.288 eV, respectively. Meanwhile, the peaks at 1.42 and 1.46 eV must be associated to the
emission from a wetting layer (WL) of varying thickness resulting from the deposition of InAs
below the critical thickness. Exciton and biexciton recombination lines can be recognized in the
excitation power dependent spectra shown in Figure 5(b). Their relative splitting is typical of selfassembled QDs emitting at this energy. Weaker resonances, marked with down arrows in Figure
2(a) can be observed by increasing the excitation power. Some of them can be related to charge
complexes associated to the impurity background and/or the presence of defects in the interface,
as will be discussed below. Indeed, in our method the preservation of the nano-holes during
growth imposes the vicinity of a re-grown interface which could be the origin of undesired effects
in the QD emission. One of them would be related to the PL line broadening due to spectral
diffusion caused by the electrical carriers trapped at the interface [23-25].
In our case, the fitting of PL lines to Lorenzian functions reproduces quite well the observed
line shape with a full width at half maximum (FWHM) typically below 1 meV. Another effect of the
QD environment is a reduction of the exciton lifetime if non-radiative recombination channels are
created at the interface. It can be found several examples related with this fact in literature, for
instance Wang et al. reported a strong decrease of the PL intensity and lifetime of QDs when they
are close to the sample surface [26]. Decreasing the distance from 15 to 9 nm, they found a QD
lifetime reduction from 550 to 65 ps. In our samples, this effect has been tested by means of the
time-resolved PL of the exciton and biexciton resonances. The experimental decay curves from
one of the best quality QDs are shown in Figure 2(c) together with their best fittings to single
exponential decays. The biexciton decay curve is well described by a single decay constant XX ≈
740 ± 20 ps (red line). This is approximately half of the value found for the exciton line just
considering its initial decay (X ≈ 1210 ± 40 ps, blue line). This is a promising result since it suggests
that the optical quality of the interface is maintained in spite of the spectral diffusion observed,
4-5
i.e. 1.22 and 1.18 meV for the biexciton and excition transitions respectively. Thanks to a good
control of the nano-hole evolution during the ALMBE growth, in the present sample the QDs and
the patterned surface are separated by 15 nm without apparent influences of interface defects.
Moreover, we have also reported similar values for the exciton and biexciton lifetimes in samples
with buffer layer thickness of about 7 nm [27]. The minor dependence of the lifetimes on the
buffer layer thickness demonstrates the reduction of non-radiative effect due to the development
of the re-growth process above explained.
Figure 2. (a) PL spectrum of a typical InAs QD grown onto a pre-patterned substrate, in the inset it is shown the PL
spectra far a way of the QD matrix. (b) L-L plot of the integrated optical intensity dependence on the excitation power
for the exiton and biexciton transitions. (c) PL transients of the X and XX transitions identified in (b).
In the following, we are going to study more in detail the optical properties of these novel QDs
in order to prove their potential applications for quantum optics devices. For that purpose it is
required the identification of excitonic complexes, and the understanding of the nature of the
exciton peaks observed at 4K. Once the details of the present QDs are know they are compared to
those obtained from a twin sample grown under identical conditions, except that the buffer of the
latter was deposited by MBE instead of employing ALMBE. We will also describe the optical
emission dependence of both samples on the excitation power and temperature evolution. In this
way we prove the importance of controlling the nano-hole evolution during the growth process.
4-6
a) Electronic structure.
The excitonic complexes inside the QDs are attributed to multiplets coming up from different
numbers of electrons and holes and their relative spin configurations. [28, 29]. Particularly the
biexciton (XX0) and charged excitons (X+ or X-, i.e. the positively or negatively charged trions) are
observed in single QD emission spectra, together with the single electron-hole pair associated to
the neutral exciton (X0). For this reason, the correct attribution of the different lines in the PL
multiplets is of great interest in order to control and manipulate the charge and spin of the exciton
species confined in QDs. The optical transitions and their relative energy shift must be well known
to design single emitter based devices. For this purpose, different methods for assignment are
available, such as measurements of the fine structure splitting (FSS), [30] intensity correlation
methods, [31] and the most common that is related to the integrated optical intensity
dependence on the excitation power (photo-injection dependence analysis) [32].
Figure 3. (a) Evolution of the -PL as a function of the excitation power. (b) and (c) Integrated optical intensity
dependence on the excitation power of the different peaks.
In fact, the relative intensity between the different exciton complexes is clearly dependent on
the carrier population. At intermediate excitation power, the PL spectra of our QDs grown on prepatterned substrates are usually composed by two main transitions, usually ascribed to X0 and XX0
recombination. As afore mentioned the excitonic transitions are broadened by the homogeneous
scattering due to the presence of a certain charge environment (charged impurities of vacancies)
surrounding the nano-hole. Comparable broadening effects have been observed on other sitecontrolled QDs as for example in the case of InAs/GaAs QDs grown by means of droplet epitaxy
[24], where it has been reported a dominant role of negatively charged species ascribed to arsenic
vacancies. Despite these extrinsic broadening effects we are more interested on the optical
properties of the excitonic complexes appearing at low excitation powers. More difficulties are
found in order to identify excitonic species dominating at high excitation powers, due to the
presence of a vast number of optical transitions into an spectral region of about 10-20 meV.
As an example in Fig 3(a) it is shown the PL dependence on the optical pumping in a
representative isolated QD. The more intense peaks are labeled from P1 to P7 depending of their
emission energy. The biexciton line (P5) is clearly distinguished from the excitonic lines (P3, P6 and
P7) thanks to the fact that their contribution to the PL is notably enhanced with the increase of the
excitation power. Once XX0 is found, we can think of several possible assignments for the rest of
the peaks. Indeed, the number of possible assignations is considerable reduced having into
account the relative emission energies. For example, it has been reported for InAs/GaAs selfassembled QDs emitting at 1.23 eV that: i) the energy emission of the negatively charged trion (X -)
is observed typically at 4 meV below that of the exciton emission (bonding position); ii) the energy
4-7
emission of the positively charged trion (X+) is anti-bonding, i.e., typically 2 meV above the exciton
emission line [33-35]. Finally, the energy emission of the biexciton can be either bonding or antibonding, but typically around 1 meV bellow/above the exciton emission [36]. In Table 1 we fix the
more provable assignment for several of the PL lines shown in Fig. 3 according to the above
exposed assumptions:
P3
P5
P6
P7
Emission Energy
1.2242 eV
1.2251 eV
1.2264 eV
1.2287 eV
0
0
+
Assign
X
XX
X
X
Energy shift
-3.3 meV
-1.3 meV
2.2 meV
Table 1. Emission energies of the excitonic complexes and assign of the main peaks.
The energetic considerations are in agreement with the slopes of the double Log plot of the
integrated intensities in front of the excitation power, as shown in Fig 3(b). As expected, X0
presents the smaller slope m(X0)=0.7 while the slope of XX0 is a factor two larger m(XX0)=1.38.
Considering just the photoinjection mechanism it would be observed a linear behavior for the X0
transition. Therefore, the non-linear dependence of X0 suggests certain contribution of carrier
relaxation and capture processes into the studied QD [37]. The case of the charged excitons is
slightly different since these transitions are more often related with individual carrier capture [38].
In comparison with X0 we could expect two trends: i) a similar slope or ii) a larger slope. Both
behaviors would be interpreted as a consequence of the nature of the additional charge in the
trion recombination, since the extra carrier in the trion configuration can be obtained either by
residual doping of the heterostructure or by the optical injection. When extra carrier of the trion is
obtained from the photoinjected population the slope of this transition in the Log-log plot
becomes larger than the one of X0. On the contrary, if the extra carrier is captured from an
impurity the trion is formed without additional contribution of the photoinjected carriers so that,
it is expected a slope similar than in the case of X0. Since X+ and X- present slopes just slightly larger
than the exciton (like for P3 m3=0.75 and for P7 m7=0.8) we can conclude the charge environment
is composed by both donor and acceptor impurities.
On the other hand, the above discussed assignment is confirmed by measuring the polarization
resolved spectrum for the exciton and biexciton lines. For an ideal QD the optical states are double
degenerated for rotational symmetry. But in real QDs the rotational symmetry is broken, for
example due to a geometrical elongation of the nano-structure, strain fields, alloy randomness,
etc. As a result, the exciton states are mixed by the anisotropic e-h Coulomb exchange interaction
[39]. The spin degeneracy is lifted in two linearly polarized states and their energy separation is
denominated fine structure splitting (FSS). The FSS is usually present in III-V QDs as due to the
unavoidable potential asymmetry, which value for the InAs/GaAs QDs is in the range from 20 to
100 eV depending on the nano-structure geometry [30]. On the other hand, the positive and
negative charged exciton recombination does not exhibit FSS. For example, in the case of the
positive trion X+; the two holes are in a singlet spin state, so the electron in the conduction band
interacts with a zero spin in the valence band and the corresponding exchange interaction
vanishes; consequently, the trion recombination line is not split. The XX0 state is also a singlet spin
state and it does not split by exchange interaction. However, the XX0 recombination is cascade of
two possible exciton states, being the sum of the two photon energies the same in the two
polarization paths, see Fig. 4. Therefore, in a given polarization path and due to the FSS of the X0
line, the XX0 line should show an opposite FSS with respect to the X0 recombination.
4-8
Figure 4. Energy scheme of the exciton and biexciton tranitions.
These effects are observed in Fig. 5. As expected, the X0 and the XX0 peaks experience opposite
shifts upon 90° turn of the linear polarization analyzer aligned along the [1-10] crystal direction. In
fact, a splitting around 65 eV [XX0(+)-XX0(-)≈130eV] is measured due to the anisotropic
exchange interaction between electrons and holes. We must also discuss about the possible
nature of the rest of peaks appearing at the high power regime all of them presenting superlinear
behavior. The largest slope belongs to the peak P4 with m4=2.24 and could be explained by several
transitions. Since the slope of P4 is close to three times the exciton slope it could be ascribed to
the triexciton XXX0 but it could be also explained by charged biexciton transitions like the complex
XX-. The peaks P1 and P2 could be tentatively associated to the singlet and triplet transitions of the
X2-, since they present a slope larger than X± but smaller than XX0 and their energy separation (1.6
meV) is consistent with this interpretation [40].
Finally, by characterization of several QDs it has been studied the influence of the emission
energy on the splitting of different excitonic complexes. This fact is explained by the influence of
the QD size in the Coulomb forces (between electrons and holes) since the magnitude of such
interaction determines the binding energies. In Figure 6 we show the energy shift (with respect to
X0) of the excitonic complexes as a function of the X0 emission energy. Notice that this plot is
somehow representing the energy splitting as a function of the QD size. The bonding transitions,
XX0 (black symbols) and X- (green symbols), are described by a negative energy (the emission
energy of the complex is smaller than the X0 transition). In contrast, the positive trion (red dots) is
anti-bonding since its emission occurs at energies above X0 (positive energy shift). From the
comparison of the energy splitting with experiments performed in self-assembled QDs we can
conclude that there is a larger attraction between the electrons and holes forming charged species
on the present QDs. For example, in self-assembled QDs emitting at these energies the biexciton
can be as binding as anti-binding depending of the emitter size and the energy shift is usually close
to ±1 meV. The binding biexcitons are characteristic of QDs bigger than those anti-bonding XX0
which are mostly observed in QDs emitting above 1.3 eV. In the QDs studied here the biexciton is
always observed as a binding transition even for the smaller QDs. For instance, a QD emitting at
1.36 eV with an energy shift larger than -2 meV is shown in Fig. 6. This different behavior is
attributed to the fact that the energetic configuration does not just depend on the emitter size,
4-9
but also the shape and stechiometry could have certain influence [33]. Therefore, the present
results suggest differences in geometry and composition with respect to typical self-assembled
QDs.
Figure 5. Polarization resolved -PL of X , XX and X .
0
0
+
0
Figure 6. Energy shift between X and the other excitonic complexes.
Other evidence of the quality of these QDs is the existence of excited states. The excited states
are associated with the allowed optical transitions between different deep shells in the QD. The
energy of these transitions is shifted to higher energies with respect to the ground-state
(transition X0). For instance, Fig. 7 contains the PL spectra obtained at 9K in a characteristic QD.
The peak groups labeled as  and  are not observed at low excitation powers appearing just up to
20 W of irradiated power. As expected, the intensity of the ground state emission (around 1.229
eV) saturates whereas the intensity of the excited states increases. The transition is composed
by a single peak that occurs at 1.268 eV, while in  we can distinguish several peaks at 1.284,
1.286, 1.289 and 1.290 eV. Such electronic structure can be interpreted in the framework of the
most recent atomistic calculations. For example, in Ref. 31 A. Narvaez and A. Zunger calculated
optical transitions in two-electron-shell In(Ga)As lens-shaped QDs and reported: a single transition
2Se-1Shh and four resolved 1Pe-1Phh transitions with similar oscillator strength, both comparable
with the transitions  and  observed in Fig. 5. The other lines found between  and  could be
4-10
attributed to nominally forbidden (partially forbidden in a real case) transitions among unpaired
shells. Comparing the excited states in QDs emitting at different energies we can conclude that 
and  transitions usually occur around 40 and 52 meV from the ground state transition,
respectively. On the other hand, it is observed that the energy of the excited states depends on
the ground state emission in a different way. It is, the  (or ) transition energy rises up (or
decreases) with the ground state energy. This trend is illustrated in table 2 where it is shown the
energy of transitions corresponding to QDs emitting at different energies.
It is worth noting that excited states of the dot pairs present the same structure of excited
states even the identification of the peaks forming transitions results quite difficult due to the
great amount of spectral lines and their energetic proximity. Notice that the emission of QDs
forming a QD pair always occurs at slightly different energies and the energy emission at the
multipeaks of their excited states are usually overlapped.
Figure 7. Excited states of QDs from the simple oxide matrix emitting at different energies.
4-11
1Se-1Shh (ev)
1.229
1.315*
1.265
1.284
1.300
2Se-1Shh (eV)
S (meV)
1.268
39
1.336
21
1.308
43
1.325
41
-
1Pe-1Phh (eV)
P (meV)
1.286
57
1.369
54
1.321
56
-
1.341
41
1.321*
1.337
1.357
16
36
1.349
1.398
1.411
48
62
Table 2. Optical transitions of QDs from the simple oxide matrix emitting at different energies.
b) Effects of the buffer layer on the optical properties of the QDS.
The optical characterization is finished by studying the influence of the buffer layer growth on
the optical properties of this new kind of QDs. For this purpose we have compared the QDs of the
previous section with the ones of a new sample. These could be considered as twin samples since
the pre-patterning process has been developed in identical conditions. In both samples a buffer
layer 15 nm thick was grown previously to the InAs deposition, but while the buffer layer of the
first sample (SampleA) is deposited by means of ALMBE as described in the experimental section
at the second one (SampleB) the GaAs buffer is grown by MBE. The AFM characterization revealed
certain differences on the evolution of the nano-holes during the GaAs growth. Such differences
could lead to variations in the size and shape of the resulting InAs nano-structures. If these minor
variations exist we would expect consequences on the QD optical properties. Due to this fact, we
have compared the emission spectra of several QDs in SampleB with those in SampleA.
Firstly, the QDs of the ALMBE sample emit at higher energies as observed in Fig. 8(a). In this
plot we show the decay time as a function of the emission energy for both kinds of QDs. It is
clearly demonstrated that the emitters from SampleA are emitting around 1.29 eV (blue spheres)
while the dots of SampleB emit at energies clearly lower (around 1.24), see the red spheres. In
SampleA and in SampleB, the exciton life times are close to 1 ns even with an important dispersion
in the experimental data. It is also worth noting that the shorter times correspond to the QDs of
SampleB. Regarding the spectral diffusion this phenomenon is more important in SampleB; see
Fig. 8(b). In fact, the mean value for the FWHM of QDs in SampleB is 1.14 meV, while in the case of
SampleA the broadening oscillates around of 0.81 meV. Finally, in Fig. 8(c) the life time is
represented as a function of the QD broadening in order to prove that there is not any relationship
between these magnitudes. Having into account these results, we can conclude that in both
samples the process of fabrication prevents non-radiative effects, obtaining efficient quantum
emitters. This fact would be indicative that the optical quality of the QDs is reached thanks to the
H treatment performed previously to the buffer deposition. But due to the chemical processes of
the pre-patterning we could also expect for the presence of impurities surrounding the QDs which,
un-removable by means of the H treatment, would be the main responsible of the broadening
effects. On the one hand, the time resolved measurements suggest a minor influence of such
impurities in the emission efficiency, since the spectral diffusion might not be related with non4-12
radiative channels. On the other hand, residual impurities determine the PL linewidth and a larger
broadening in SampleB where we should conclude that the buffer layer grown by MBE can
introduce more charged defects/impurities.
Figure 8. (a) and (b) Exciton lifetime and linewidth of several QDs grown onto a ALMBE (in blue) and a MBE (in red)
GaAs buffer. (c) For complete the lifetime is represented as a function of the emitter linewidth for the same QDs.
Other than the above given properties a more intrinsic difference between both samples is
observed in the electronic structure. In the previous section a single peak transition (2Shh-1Se)
followed by a multipeak transition (1Phh-1Pe) around 40 and 52 meV from the ground state was
observed. In contrast, in SampleB all the excited states present multi peak transitions typically
around 60 and 110 meV (see Fig. 9). By comparison with the aforementioned calculus of Zunger
[35], in SampleB we can ascribe the first excited state to (1Phh-1Se) while the second excited state
would be ascribed to (S+P)lh+hh-1Se (five peak transitions). These structural and energetic variations
suggest not just size but also morphological differences with respect to SampleA. All these facts
reveal the importance of the nano-hole evolution during the buffer growth which is determining
the QD morphology and size and hence the emitter energy, linewidth and electronic state
configuration.
Figure 9. (a) Power evolution on the ground state and excited states (b) on a single QD grown onto the MBE GaAs
buffer.
4-13
c) Temperature evolution of the PL.
Figure 10. (a) and (b) PL evolution with temperature of QDs characteristic from dot A and dot B.
The dependence of the QD emission on the temperature is usually studied by macro-PL since
the data obtained in this way are representative of the emitter ensemble. But the samples in this
work have been designed for single QD characterization. As a result, the temperature evolution
must be studied by analyzing several isolated QDs. Obviously it has not been obtained data
enough for a statistically representative description of the temperature effects, but despite this
fact, we have obtained some valuable information about our QDs. Figure 10 shows the PL
evolution with temperature for two different QDs (namely “dotA” from SampleA and “dotB” from
SampleB). At a first sight, it can be observed a 10 meV red shift, as typically observed in
semiconductor nano-structures by increasing the sample temperature up to 80K. The evolution of
the emission energy of these QDs follows approximately the Varshni’s Law (not shown), which is
an empirical expression representing the energy gap variation of a bulk semiconductor as a
function of the temperature [41]. Thanks to this fact, the emission energy dependence on the
temperature can be attributed to a reduction of the confinement potential into the QD produced
by the decrease of the GaAs and InAs gap forming the nano-structures.
As expected, the red shift is accompanied by the broadening of the emission linewidths as
observed in Fig 11. The emission of the QD at low temperatures is expected to depend on the QD
quality, but also on the corresponding charge environment, which could differ at each nano-hole.
For the analysis of the linewidth evolution our attention is focused in the X0 and XX0 transitions.
The FWHM of X0 and XX0 present very close values for both dotA and dotB. The observed
evolution with temperature can be fitted according to the next expression:
(T)=0ACT+LO/(e(Elo/KT)-1)
(1)
where 0 represents the broadening mainly limited by spectral diffusion, AC represents the
broadening rate produced by the interaction of carriers with acoustic phonons while LO
represents a similar contribution associated to the longitudinal optical phonons. The best fittings
are shown Fig. 11 while the best fitting parameters are listed in Table 2. Limiting our discussion to
a qualitative level we can point out that, even if dot A is more affected by the spectral diffusion, it
suffers the scattering with phonons in a minor manner as can be deduced from the values
4-14
obtained for AC and LO. This fact would be consistent with an enhancement of the dielectric
screening in dotA [42]. On one hand, the emission broadening of our QDs is produced by
surrounding impurities which create a charge environment responsible of the wide spectral lines
obtained. On the other hand, such charge environment isolates the QD electronic states from the
lattice reducing phonon scattering.
Figure 11. Unhomogeneous broadening of the excitonic lines in dot A and dot B as a function temperature
Peak
 (meV)
0
 (eV/K)
 (meV)
AC
LO
ELO
XXA
1.528
3.6
80.1
32
XA
1.452
1.5
36.0
32
XXB
0.93
9.0
121
32
XB
0.86
8.2
35.2
32
Table 3. Best fitting parameters of the experimental data to Eq. 1
In self-assembled QDs, the photoluminescence is expected to decrease with the temperature,
producing an optical signal two orders of magnitude weaker at 80-100K. Such reduction is usually
produced by escape of carriers towards the wetting layer states. In order to clarify such
phenomenon it is represented the Arrhenius plot corresponding to each QD. In figure 12 it is
plotted the PL spectra and the temperature evolution of the integrated optical intensity of the
exciton and biexciton (and the whole QD band) for dotA and dotB. In dotA, it is observed how the
temperature increase is accompanied of a slight enhancement of the PL intensity previously to the
PL quenching. In contrast, in dotB we observe a minor reduction of the PL intensity even at very
low temperatures. The data of dotB is fitted to an expression containing two Boltzman-type
quenching mechanisms, like Eq. 2, while the data of dotA are fitted with just one quenching
mechanism (Eq. 3). Notice that the PL enhacement observed below 50K is not representative
enough to be determined in the fitting:
I (0)
I (T ) 
1   (1
I (T ) 
E
 1
KT
e
 2 e

E2
KT
)
(2)
I (0)
1   (1e
4-15

E1
KT
)
(3)
In these expressions I(0) is the optical intensity at 0K,  the decay time of the whole band and K
the Boltzman constant, while 1 and 2 are the scattering rate of the loss mechanisms with
activation energies E1 and E2. The value of I(0) for each family is extracted from the integrated
optical intensity at 9K which is the lowest temperature raised during the temperature sweep.
Notice that the possible dependence of  on the temperature of emission energy are not
considered (for simplicity it is treated as a parameter). The best fittings for dotA and dotB are
shown in Tables 4 and 5, this way, it can be estimated the activation energy of the quenching
mechanism.
Figure 12. Arrhenius plots of the dot A and dot B.
Whole Band
X
XX
I0 (a.u.)
108
43
68
tau1 (ns)
1.7
2.36
1.33
-1
G1 (ns )
3736
1152
11043
E1 (meV)
33.4
24.6
37
Table 4. Best fitting parameters to the Arrhenius plots for dot A
I0 (a.u.)
23
tau1 (ns)
1
-1
G1 (ns )
5418
E1 (meV)
64.9
G2
5418
E2 (meV)
Whole Band
64.9
X
51.1
XX
7
0.5
4.5
5.6
302E5
146
Table 5. Best fitting parameters to the Arrhenius plots for dot B.
Focusing on the whole band we found different activation energies for the quenching
mechanisms of each dot; E1≈30 for the dotA or E1≈60 meV and E2≈140 meV for the dot B. Notice
that the values of the activation energies obtained can not be quantitatively considered since due
to the low signal provided by the dots (at the excitation power employed) we could not obtain
points enough (mainly for dotA). Indeed, we should reach almost 140-160 K for obtaining a
reliable value for the activation energy which in Fig. 12 is clearly underestimated. In spite of the
great error of the fitting parameters, we could qualitatively discuss the result with the aim to
explain the nature the PL quenching mechanism. The quenching mechanism of InAs/GaAs nanostrucctures at these temperatures is typically explained by thermal escape of excitons or carriers
(electrons and holes separately) and by loss toward impurities and defects:
4-16
 Since the emission of the QDs occurs around 1.2-1.3 eV, the thermal escape could be done
to the wetting layer located at EWL1=1.42 and EWL2=1.46 or to the GaAs barrier (EGaAs=1.52 eV).
Therefore we could expect activation energies of 180-140 meV and 240 meV for the escape
toward the wetting layer and the GaAs barrier respectively. This value fits with the activation
energy (E2) obtained for dotB, but is noticeable larger than E1 in both cases.
 The activation energies around 30 meV (E1) observed for dot A should be more properly
related with optical loss to impurities and defects.
4. Conclusions.
We have briefly described the method for fabrication of site controlled InAs/GaAs quantum
dots with high optical quality using local oxidation nanolithography and molecular beam epitaxial
re-growth. The nucleation selectivity of the method was demonstrated in previous works by
fabricating a QD square matrix with 2 m pitch period. In the present work we paid more
attention to the optical properties of QDs fabricated in the matrices using two different kinks of
motifs. The simple and the double oxide motifs have been proved to reproduce isolated dots or
dot pairs, respectively, with great reproducibility. We have also studied the effects of the nanohole evolution during the growth process by comparison of both kind of matrices in samples
whose buffer layer has been deposited by ALMBE and MBE, SampleA and SampleB respectively.
Focusing on SampleA, it is shown that the PL emission of the simple and double oxide matrix nanostructures occurs at the same energy with typical linewidth of about 800eV, being the main
difference associated to the power evolution. Finally, we have found similar optical properties in
SampleB, the differences observed in the spectra are attributed to morphological differences (of
the resulting nano-structures), probably due to a different evolution of the corresponding nanoholes during the growth process.
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4-19
Chapter 5. Optical properties of self-assembled
Quantum Wires.
There might have been wires,
but I have this ability to make myself light.
Albert Finney.
5-0
1. Introduction to the InAs/InP system: comparison with InAs/GaAs.
Actually the InAs/GaAs quantum dots (QDs) are the more studied nano-structures. As
aforementioned, this kind of emitters is usually grown by solid source Molecular Beam Epitaxy
(MBE) using the Stranski-Krastanow method. Deposited onto a GaAs buffer, the InAs suffers a
noticeable biaxial strain due to the lattice parameter mismatch between both semiconductors
(around 7%), hence the mechanical energy accumulated after deposition of several InAs
monolayers (ML) is relaxed forming self-assembled nano-structures. The optical emission of such
nano-structures can be tuned into the wavelength range of 920 to 1200 nm (1.35-1 eV) by
controlling substrate temperature, As pressure and deposition rate during the growth [1].
However more complicated methods are required in order to reach the telecommunication
windows beyond 1310 nm (0.946 eV). For instance, we can mention the introduction of small
quantities of nitrogen (forming a ternary material) [2] or the modification of the confinement
barrier by capping the QDs with an InGaAs barrier [3].
In this sense, one of the benefits of the InAs/InP heteroepitaxial system is the fact that the
resulting nano-structures can be tuned in the wavelength range of 1200 to 1600 nm (1033-775
meV), a spectral region of difficult access through the InAs/GaAs system [4-7]. From the growth
point of view the InAs/InP nanostructures present two main differences with respect to the
InAs/GaAs ones: firstly, onto InP the InAs growth front is produced under lower strain conditions;
secondly, the growth requires the combination of two elements belonging to the V group (As and
P). Concretely, the lattice parameter mismatch is close to 3% in the InAs/InP system. Such
mismatch is lower enough to allow the formation of elongated self-assembled nano-structures,
instead of QDs [8,9]. Let notice that QDs are more efficient structures for relaxing elastic energy
since the accumulated strain is dissipated in all directions. In contrast, the elastic energy at the
InAs/InP system is relaxed just in one direction, the [110], forming, consequently, one dimensional
(1D) structures oriented parallel to the [1-10] direction [10]. The second important difference with
respect to the InAs/GaAs system lies in the As/P exchange ratio produced during the growth. Such
unavoidable exchange process results of great importance due to the fact that the InAs
nanostructures size depends on it, since such phenomenon determines the quantity of InAs
present at the surface. For more details about the relaxation mechanisms or about the As/P
exchange see [11, 12].
From the optical characterization point of view, another peculiarity of self-assembled InAs/InP
QWRs is the lack of wetting layer. For the formation of efficient emitters the nominal quantity of
InAs deposited is typically about 2.5 ML, while the height of the QWR could vary between 7 to 13
ML. Therefore, it is supposed that all the material deposited is involved in the formation of QWRs
which drives to the extinction of the previously formed wetting layer. The knowledge and fine
control of the growth processes offer the possibility to obtain self-assembled QWRs and also to
tune accurately their emission energy by controlling their size [Fig. 1(a)]. In addition, monitoring
the growth front at the first stages of the QWR formation by means of in situ techniques has
permitted the fabrication of low density samples (previously to the disappearance of the wetting
layer) and the study of isolated QWRs, as shown in Fig. 1(b).
In previous works of B. Alén et al. the main characteristics of self-assembled InAs/InP QWRs
has been reported for both arrays and single QWRs [6, 13]. In the most general case, the emission
spectra of QWRs are formed by multiple Gaussian peaks that are attributed to quantum wires of
heights differing by a discrete number of monolayers. There are some works studying theoretically
the charge confinement in InAs/InP self-assembled quantum wires using the effective-mass
approximation and taking into account the strain in the nanostructures [14-17]. In such works, it is
5-1
shown how for thin wires the electron wave function is significantly spilled out of the wire along
the growth direction (100), because the electron spill over is inversely proportional to the wire
height [14].
Figure 1. Atomic Force Microscope (AFM) images of uncapped samples containing self-assembled QWRs: (a) array
of QWRs and (b) isolated QWRs obtained by controlling the growth front at the fist stage of the InAs deposition.
Figure 2. Schematic drawing of the change in confinement with wire height.
5-2
The inverse relation of the exciton wavefuntion extent to the wire height is illustrated in Fig. 2.
If the potential well is wide enough (big QWRs), a size reduction leads to a reduction of the
wavefunction extent; as occurs passing from the case of Fig. 2(a) to Fig. 2(b). However, for thinner
nanostructures, the confinement energy is very close to the InP barrier and the carrier
wavefunction significantly spreads out of the barrier material and thus increases its extent with a
further decrease in the wire height [compare Figs. 2 (b) and (c)].
On the other hand, strong polarization anisotropy is observed for absorption and emission
spectra. Generally, the polarization anisotropy in nanostructures is also associated to their
confinement potentials. If the confinement potential is not isotropic, some optical transitions (as
merged states with angular momentum L=1) would be sensitive to the linear polarization of light
[18, 19]. Due to the QWR geometry, the linear polarization axis is parallel (perpendicular) to the to
the QWR axis [1-10] ([110]). On the other hand, the different peaks are explained by the existence
of nanostructures with different height. This conclusion is not just in agreement with PL and
absorption measurements, but it is also supported by theoretical predictions [20, 21]. Indeed, the
emission energy differences between consecutive Gaussian families forming the broad QWR PL
spectra correspond to fluctuations of one ML, being the inhomogeneous broadening mainly
attributed to QWR width fluctuations.
2. Experimental details and samples.
One of the objectives of this Thesis consists of proving the possibilities of InAs/InP QWRs as
active media for novel applications in nanophotonic devices. In order to achieve this purpose, the
samples will suffer some modifications with respect to previously fabricated epitaxies. For
example, the QWRs used as active media are fabricated in order to present their maximum
emission wavelength close to 1500 nm at room temperature, which means growing relatively large
QWRs and reducing their height distribution to a small number of families. In addition, QWRs
embedded into micro-cavities must be grown onto a 700 nm thick In53Ga47As film, which is
typically used as a sacrificial layer necessary for the membrane fabrication, see Fig. 3. The InAs
nanostructures are grown at the middle of a 237-nm-thick InP slab after the deposition of 1.7 ML
of InAs at a substrate temperature of 515 °C and a growth rate of 0.1 ML/s.
Figure 3. Scheme of the epitaxy.
The sample was held in the cold finger of a closed-cycle cryostat operating between 12 and 300
K, as described in Chapter 3. The PL experiments were carried out exciting the sample just below
the InP absorption band edge by means of a 980 nm pulsed laser diode with a repetition rate of 40
MHz and pulse width around 100 ps. The excitation light was focused using a 100 mm focal lens (1
inch diameter) arriving to the sample surface with an incidence angle i≈ 30°. The emitted light
was collected by means of a large focal length microscope objective, dispersed by a 0.5 m focal
length spectrograph and detected with an InGaAs Peltier cooled 512-photodiode array in order to
5-3
obtain the steady state PL spectra. For the PL transient acquisition the optical signal was dispersed
by the same spectrometer but detected by a Peltier cooled InGaAs avalanche photodiode. The PL
transients were recorded using standard time correlated single photon counting techniques with
an overall time resolution after the experimental response deconvolution of around 50-100 ps,
depending on the optical intensity and the integration time.
3. Optical characterization.
Figure 4. (a) PL spectra of samples H1, H3 and H6 normalized to the maximum intensity peak at 80K. (b) Polarization
resolved spectra and polarization degree of sample H1.
The three samples employed for the devices studied in this Thesis (labeled H1, H3 and H6) are
fabricated using identical growing conditions, and therefore they basically contains similar size
distribution of nano-structures. All of them have been used for the fabrication of 2D-photonic
crystal micro-cavities with embedded QWRs and, after the microcavity characterization, sample H1
was selected to perform an extensive study of the active medium regarding light emission
properties. We have selected H1 because of its emission is in between H6 (which presents a minor
intensity contribution at 1500 nm) and H3 (emitting at larger wavelengths), see Fig. 4(a). In this
figure, the spectra of samples H3 and H6 have been acquired by micro-PL, in contrast, the
spectrum corresponding to H1 has been studied using the ensemble-PL setup described in section
3. Thinking of future applications of these QWRs, the maximum emission of the three spectra is
located close to 0,85 eV (1450nm) and it is expected to move towards 0.83 eV (1500 nm) at room
temperature (RT). Firstly we will investigate the PL anisotropy found under linear polarization and
the PL dependence on temperature and excitation power, while the evolution of the radiative and
non-radiative PL decay times (mostly associated to exciton dynamics) with temperature is
discussed in section 4.
In Fig. 4(b), it is shown the linear polarization resolved PL spectra and the corresponding
polarization degree in sample H1. Consistently with the previous works the light emission of our
QWRs presents a larger component parallel to the QWRs ([1-10] direction). The polarization
degree,
P(%)=100*(IPL[1-10] – IPL[110])/(IPL[1-10] + IPL[110])
(1),
5-4
slightly depends on the height of the QWRs (from 40 to 30 % in sample H1 Fig. 4(b)). This smooth
variation can be attributed to a certain increment of the penetration of the exciton wavefunction
into the InP barrier, which is more important for the smaller QWRs of the distribution.
Figure 5. (a) The PL spectra of the QWRs at 30K is mainly composed by four Gausssian peaks accompanied by a
low energy tail. (b) Every peak can be associated to different family of QWRs with the same height as those previously
calcutlated in Ref. 6.
Figure 5(a) shows the PL spectrum of sample H1 at 30 and 12 K. The PL spectrum at 30 K is
acquired in order to compare with the results previously published by B. Alen et al. [6]. At this
temperature the PL consists of a broad band centered at 1460 nm (0.85 eV) and composed by
several peaks. The PL spectra (in red) shows the deconvolution of 5 Gaussian peaks which would
be ascribed to four QWRs families, superimposed to a certain low energy PL tail. This PL tail
becomes two well defined emission lines at ~1.79 and ~1.77 eV [black line in Fig. 5(a)] when
reducing the lattice temperature, which are attributed to excitonic recombination at the InGaAs
sacrificial layer [22]. Each Gaussian component is typically attributed to the emission of QWRs with
the same height, in fact, it can be demonstrated that the emission energy difference between
consecutive families is consistent with QWRs whose height differs just in one ML. Using the model
and data published by Alen et al. [6] we can determine the height of the QWRs contained in our
sample at 30 K. The calculation consists of an eigenfunction expansion method used to obtain the
ground states of electron and holes into an infinite array of rectangular QWRs. Fig. 5(b) shows
these theoretical results for the band-to-band optical transitions as a function of the wire height.
The entire number of monolayers is represented by the black solid line while the experimental
data of the PL peak at 30 K are represented by red symbols. A good matching between the
5-5
energies calculated for 11 to 8 ML and the PL peak energies of families P1-P4 is observed. These
heights are in agreement with the structural characterization made by AFM and XTEM [23].
Quantitatively, the experimental emission energies are slightly smaller than the theoretical
simulations. Such overestimation is similar to the previously published results and it is related to
the calculation method. On one hand the electron and hole ground states are calculated by
supposing a biaxial deformation and neglecting the excitonic energy. Several authors have been
taken into account the effects of a triaxial deformation showing that the consequence always
consists of an enhancement of the effective forbidden bandgap. On the other hand, taking into
account the exciton binding energy the optical transition energy must be corrected compensating
the deformation effects partially.
a) PL Dependence on temperature.
The dependence of the PL on temperature has been accurately studied, as shown in Fig. 6. In
the range of very low temperatures it is observed an almost constant intensity of the QWR peaks
which just present a smaller variation of their integrated PL intensity. In contrast to the QWRs, the
PL of the InGaAs sacrificial layer is quickly suppressed by increasing the temperature, as shown in
Fig. 6(a). The temperature sweep is completed in Fig. 6(b) where it can be identified three main
effects: redshift of the PL peak energy, homogeneous broadening and decrease of the PL peak
integrated intensity.
Figure 6. (a) PL evolution from 12 to 30 K. (b) PL evolution from 50 to 140 K.
Figure 7 (a) illustrates the redshift of the PL peak energy with increasing temperature for every
QWR family. The data are depicted together with the Varshni’s Law applied to InAs and InP (grey
lines). The Varshni’s Law is an empirical expression representing the energy gap of a given bulk
semiconductor as a function of the temperature:
Egap(T)= Egap(T=0)-T2/(T+)
5-6
(2).
Figure 7. (a) Evolution of the PL peak energy of the QWRs in comparison to the bulk InP, InAs and InGaAs. (b)
Temperature evolution of the full width at half maximum. (c) PL intensity of families P1-P4.
The Varshni parameters employed for the InAs are Egap(T=0) =0.415 eV,  2,76x10-4 eV/K, and
= 83K [24] [Fig. 7(a) bottom panel], while the InP parameters are Egap(T=0) =1.421 eV,  4,90x104
eV/K, and = 327K [Fig. 7(a) top panel]. We can see that families P1, P2 and P3 seem to follow the
Varshni’s Law, which means that the redshift can be interpreted as a reduction of the InP barrier
by rising up the temperature. This fact is not clearly observed in P4 probably due to its low optical
signal and the influence of the atmospheric water absorption. The PL peak energy shift is
accompanied by an enhancement of the homogeneous broadening, as shown in Fig. 7(b). At low
5-7
temperatures, all the families present a similar linewidth around 20 meV, which means a
comparable size fluctuations between families. The broadening evolution can be fitted according
to the expression:
(T)=0ACT+LO/(e(ELO/KT)-1) (3)
where 0 would be related to the nanostructure size fluctuations, AC represents the
homogeneous broadening produced by the interaction of carriers with acoustic phonons while LO
represents the contribution of the interaction with longitudinal optical phonons. The results of the
best fitting of the experimental data with Eq. 3 [26] are shown in Table 1. We will return to these
parameters during the discussion on the Purcell effect figure of merit.
P1
P2
P3
P4
0(meV) AC(eV/K) LO(meV) ELO(meV)
18.5
14
1
39
22
62
2
39
22
6
26
39
23
60
91
39
Table 1. Best fitting parameters to Eq. 3.
Finally in Fig. 7(c) it is represented the evolution of the integrated optical intensity with
temperature, where we can observe a different variation for the different families. The observed
behavior might be related to different contributions of the carrier loss mechanisms. For this
reason, the integrated intensities of each component have been separately represented like
Arrhenius plots in Fig. 8. In this figure it is shown how, while the PL intensity of P1 and P2 remain
constant or even increases slightly below 70 K, the PL intensity of families P3 and P4 exhibit a
continuous decrease from very low temperatures. In fact, the data of P3 and P4 fits with an
expression containing two Boltzman-type quenching mechanisms:
I (T ) 
I (0)

 E 
 E 
1   d G 1 exp  1   G 2 exp  2 
 kT 
 kT 

(4)
being I(0) the optical intensity at 0 K, d the decay time of the PL and k the Boltzmann constant,
while G1 and G2 are the times associated to two loss mechanisms with activation energies E1 and
E2. The value of I(0) for each family is extracted from the integrated optical intensity at 12K which
is the minor temperature reached at the cryogenerator. The value of d is set to the PL decay time
measured at 12K, but its possible dependence on the temperature is neglected to simplify the
discussion. The first loss mechanism (below 100 K) has an activation energy around 12 meV for
both families. This value is consistent with thermal emission of electrons to impurity centers (or
other kind of defects), but it could be also partially explained by the presence of the localized
states (associated to size fluctuations and disorder). The optical losses are more affected by the
second mechanism whose activation energy is larger than 200 meV above 100 K. This energy, far
from the estimable to the InP barrier, would consistent with a unipolar escape of electrons.
As aforementioned, the temperature evolution of the integrated optical intensity for families P1
and P2 is slightly different. In spite of this fact, the data for family P2 are also fitted to Eq. 1 but just
considering the unipolar escape with an activation energy E2= 220 meV. This fact is explained by a
negligible contribution of the lower activation energy mechanism (i.e. G1≈0) as deduced from the
near constant variation of the integrated intensity in the Arrhenius plot below 100 K. As a
5-8
difference with respect to the other components, in family P1 it is required a carrier feeding
mechanism to fit the experimental data below 100 K. For this reason, the family P1 is fitted by
means of the expression:
I (T ) 
I (0 )

1   dG 2 exp 





A
1 

E2  
1
 E3  
1

exp



kT  
a
 kT  
(5)
Figure 8. Arrhenius plots of the PL integrated intensity for families P1-P4.
The parameter A accounts for the carrier population ratio being transferred towards the
QWRs, whereas a is the ratio between the injection rate of such carriers and the radiative
recombination rate of excitons for family P1. The activation energy of this transfer mechanism is
represented by E3. The PL quenching mechanism at high temperatures is again very similar to the
other families, E2=240 meV while for the transfer process an activation energy of E3= 31 meV is
obtained. On one hand, this activation energy would not agree with the most common transfer
mechanisms from small to big nanostructures. On the other hand, the value of E3 is close to the
gap between the P1 component and the PL peaks of the InGaAs sacrificial layer (777 and 797 meV
with respect to the 832 meV of the P1 family). For this reason and having into account the fact that
such transfer mechanism has been not observed in samples without sacrificial layer, this gain has
been tentatively associated to a certain injection of carriers starting from the InGaAs layer
5-9
probably through deep impurity levels present in the InP and resonant with the P1 component.
Finally, it is worth noting that family P2 is also susceptible to be affected by the proposed carrier
transfer, even in a minor manner due to the larger energetic distance with respect to the InGaAs,
being therefore the optical losses partially compensated by carrier diffusion below 100 K. Such a
negligible contribution could affect the fitting values obtained for E1 and G1 in family P2.
Component I(0) d (ns) G1 (ns-1) E1 (meV) G2 (ns-1) E2 (meV)
4900
1.56
4x106
240
P1
6
11750 1.56
20
45
10
220
P2
9960
1.64
1.4
11.7
25x105
231
P3
6
1470
1.93
4.4
11.5
>10
>200
P4
Table 2. Experimental data and Best fitting parameters of the PL integrated intensity at the
different families (PL-Gaussian components) to (1) and (2) respectively.
b) PL dependence on the excitation power.
Figure 9. (a) and (b) PL dependence on the excitation power in CW and pulsed operation, in the inset we plot the
double log plot of the integrated intensity of family P2 under CW (black scatters) and pulsed operation (blue scatters).
The dependence of the PL spectrum of the active medium on the excitation power was studied
in sample H3 at 80 K. This sample is selected instead of H1 because the cavities fabricated in this
expitaxy are used in Chapter 7 for studying the Purcell factor as a function of the excitation power.
One of the parameters required for the device characterization consists of the emitter broadening.
The inhomogeneous linewidth associated to the QWRs families in the PL band spectrum of H3
sample is similar to that observed in H1. Due to the low signal obtained from P1 and P4 QWR
families we center our attention in families P2 and P3. In figure 9 we can see several PL spectra
acquired at 80K for different excitation powers. The study is performed by using two kinds of laser
diodes both emitting at 980nm, in Fig. 9(a) the measurement is done in continuous wave (CW),
while in Fig. 9(b) it is employed pulsed excitation (100ps pulse working at 40MHz of repetition
rate). The excitation power varies from 20W to 5mW (depending on the laser diode), but the
sample does not show evidences of heating by light absorption, due to the fact the excitation
wavelength is below the InP barrier. In fact, at the high power regime the emission broadening is
expected to increase due to a larger contribution of the carrier-phonon interaction or more
complex phenomena attributable to large carrier populations. However, in the power range used
5-10
here the broadening is negligible for both cases. The intensity emission dependence on the
excitation power presents a linear behavior and it does not differs between CW and pulsed
excitation, as can be seen at the double log plot of the inset in Fig. 9(b). Here, the integrated
optical intensity of family P2 is plotted with respect to the excitation power, where we measured a
slope close to the unit for each family. This result is important for the discussion presented in
Chapter 7 since it is shown that any remarkable power dependence is expected by working in
pulsed excitation conditions, as occurs under CW conditions.
c) Time resolved measurements.
Time resolved experiments are essential for the complete understanding of the light-matter
interactions occurring at the nano-structures since important information about the excitonic
dynamics will be obtained. On the other hand, an accurate quantitative determination of the
Purcell Factor precise the comparison between exciton the lifetime at bare QWRs and QWRs
coupled to optical modes of the microcavities. For these reasons, the TRPL measurements have
been focused on two main directions: i) the exciton lifetime evolution with temperature previous
to the analysis of the coupling to cavity resonant modes in order to evaluate the optical
confinement effects; ii) time resolved measurements on large QWRs emitting at 1550 nm could
complete previous carrier dynamics studies developed by D. Fuster limited to QWRs emitting at
wavelengths shorter than 1200 nm [7, 25]. The first part of the study is necessary to obtain
comparable results from measurements taken in different cavities as well as different QWRs. In
contrast, the second part offers a more complete understanding of the self-assembled QWR
exitonic dynamics, contributing with new information about the effect of localized excitons in
QWRs emitting at energies below 1 eV.
Due to the complex system merging localized and free states, the exciton dynamics in our
QWRs will be strongly dependent on the measurement conditions: excitation power, emission
energy and temperature. The effect of excitation power is typically observed as a reduction of the
PL decay time at very high powers. In order to avoid power effects the lifetime characterization is
performed under constant excitation power conditions (concretely 600 W at 980 nm
wavelength). As a signature of the low excitation power regime, the time resolved measurements
were obtained as mono-exponential PL transients, instead of the more complex decays typically
reported at the high power regime [25, 26]. In Figure 10(a) a typical PL transient is shown for
QWRs of family P1 (0.827 eV) at 12 K. The decay of the optical signal is accompanied by the system
response for the laser excitation. As shown in the figure, a mono-exponential decay function is
used to fit the experimental data (red continuous line) after convolution with the system response,
in black.
The PL decay time dependence on the emission energy is illustrated in Figs. 10(b)-(c) for two
different temperatures. At 12K the decay time increases from 1.4 up to 2 ns in the emission energy
range from 0.825 to 0.94 eV. Such tendency can be explained by taking into account the influence
of electron-wavefunction spill-over effect mentioned above: in smaller QWRs (corresponding to
higher emission energies) the electron wavefunction is extended out of the QWR (through the InP
barrier) while the hole-wavefunction is mainly confined at the QWR, see Ref. 17. As a result, it is
produced a noticeable reduction of the electron-hole wavefunction overlap that leads to an
increase of the exciton lifetime (as reported for QWs and QDs [27, 28]). Given that localized
excitons dominate over free ones at low temperatures, we can conclude that the spillover effect
applies for localized electrons too. At higher temperatures the decay times become a bit larger.
For instance at 80 K the lifetimes vary from 1.7 to 2.6 ns, see Figure 10(c). In this figure we can
5-11
distinguish two different behaviors: at the low energy region of the 80 K PL spectrum (coinciding
with families P1 and P2) the decay time is still increasing with the emission energy. On the other
hand, above 0.85 eV (families P3 and P4) the lifetime decreases with the emission energy.
Consistently with the Arrhenius plots, this lifetime reduction could be associated the non-radiative
mechanisms.
Figure 10. (a) System response (black continuous line), low temperature PL transient at 1500 nm for family P1 (red
dots) and best mono-exponential fit (red continuous line). The PL decay time as a function of the emission energy (red
circles) and the corresponding PL spectrum (black continuous line) are recorded using non-resonant excitation at 12 K
(b) and 80 K (c).
5-12
Figure 11. Temperature dependence of the PL decay time for families P1-P4: (a) - (d). Continuous lines stand for
direct fits of the experimental data to the Lomascolo’s Model. In the case of results for PL decay times for families P3P4 the fit is done on radiative lifetimes obtained from the experimental PL decay time values, as explain in the text. The
excitation power in the experiment was kept constant with temperature at 600 W.
In order to present a better description of the exciton dynamics, the lifetime evolution with
temperature has been studied from 12 to 200 K, as shown in Fig. 11. These results point out that
the different behavior of families P1 and P2 with respect P3 and P4 is mainly produced by the loss
mechanisms previously identified which mainly affects families P3 and P4. However the square
root dependence on temperature expected for ideal 1D QWRs is not observed for any family.
Indeed, the behavior shown by families P1-P3 contains qualitative coincidences with works
studying localization effects in nano-structures [29, 30]. Among these works we have selected the
model proposed by Lomascolo et al. since in this model, the lifetime evolution with temperature is
described by means parameters representative of our self-assembled QWRs [31]. In fact, fitting the
data in Figure 5 to the next expression we are going estimate the localization energy and density of
localization centers of the QWRs studied in the present sample:
rad(T ) 
 EL   2 Mk 
ND exp


 kT    2  2 
ND
L
1
2
 EL  1  2 ME0 
exp



 kT   0   2  
5-13
T
1
2
(6)
In this expression rad(T) is the radiative recombination time, 0 the intrinsic radiative lifetime
and L the lifetime of localized excitons. The localization energy and the effective linear density of
localization sites are represented by EL and ND, respectively. E0 is the energy related to the centerof-mass exciton wavenumber K0. For InP, K0 = 1.3x107 m-1 (considered emission wavelength of
1500 nm and InP refractive index equal to 3.1) and M the total exciton mass (we have used the
value given by Andreani et al. for InGaAs QWs, M = 0.18m0 [32]). Given that M can be slightly
different for excitons in InAs QWRs (due to a different valence band mixing) the results we obtain
should be taken for a realistic but semi-quantitative discussion. In the case of families P1 and P2
we observe an increase of the PL decay time by more than a factor 2, as shown in Figs. 11(a)-(b).
Above 140-160K the PL decay decreases quickly because of the unipolar escape of electrons to the
InP barrier. In contrast, we observe a PL decay time practically constant up to 50-60 K for family P4
and a weak increase up to 110 K for family P3, Figs. 11(c)-(d). From these results we can conclude
that for families P1 and P2 the PL decay time is mainly radiative. In contrast, for fitting the results
from families P3 and P4 to Eq. 6 the non-radiative effects must be taken into account:
1
d

1
rad
IPL(T )  I (0)

1
nr (7)
d(T )
rad(T )
(8)
where nr stands for the non-radiative recombination time associated to the electron loss at low
temperatures and IPL(T)/I0 accounts for the PL intensity decrease. After calculating the radiative
lifetimes, the best fitting parameters to Eq. 6 can be qualitatively compared (Table 3). As expected,
the values obtained for 0 are larger than the value reported for GaAs V-grooved QWRs (see for
example Ref. 33), which is indicative of the different wavefunction for confined excitons. The
lifetimes obtained for localized excitons (L) are practically the values of the PL decay times
measured at 12 K, consistently with their larger contribution at low temperatures. The value of L
increases from 1.5 to 1.9 ns for families from P1 to P4, respectively. As aforementioned this fact is
attributed to the spill-over effect of the electron wavefunction, demonstrating that it occurs for
both free and localized excitons. The exciton localization energies deduced for our QWRs, EL≈ 8-16
meV (Table 3), suggest that localization effects would be considerably larger than those reported
in the GaAs/AlGaAs system [30, 31]. This result can be compared with structural characterization
studies: many elongated QWR defects 500-1000 nm long (zones where the height is different than
the average) are typically measured by AFM characterization [23]. Along the QWRs width
fluctuations are also observed, in average extending 100-150 nm. By considering the emission
energy differences between the different families we can estimate localization energies in the
order of 30 meV for height fluctuations around 1 ML, while we expect localization energies close to
12 meV considering the inhomogeneous broadening produced by the wire width fluctuations. Let
notice that exciton localization is produced if the dimensions of the defect are comparable with
Bohr radius (smaller than 40 nm for bulk InAs), in such a way that the height or width fluctuations
determines the localization energy. In this sense, the elongated height fluctuations observed by
AFM can be considered as independent QWRs given the finite extension of the exciton along the
wire axis. In contrast, even the largest width fluctuations observed, i.e. 150 nm in length, could be
responsible of relative weak localization phenomena. The density of localization centers deduced
in the present work (listed in Table 3) varies from 7-11 m-1 for families P1-P3 (even with an
important relative error), which suggests a separation between defects around of 90-140 nm, in
5-14
agreement with the AFM data. From this result we can conclude that exciton localization effects
can be mainly ascribed to wire width fluctuations.
The optical properties and exciton recombination dynamics in InAs QWRs has been discussed
in the present sample bearing in mind its role as the active medium in future photonic devices. In
this sense, the presence of a sacrificial layer below the QWR layer does not affect the emission
properties of the QWRs, as compared to previous published results when they are grown onto a
thick InP buffer. Particularly, the observed PL quenching is characterized by a low loss behavior at
intermediate temperatures (below 140 K) for families P1 and P2, which is important for laser
design. In addition, the determination of the linear density of exciton localization centers suggests
a moderate density of defects being the recombination dynamics dominated by free excitons
above 40-50 K. In this way we could define an optimum temperature in the range 50-120 K for the
operation of devices based in QWRs belonging to families P1 and P2. Above 120 K, the
spontaneous emission rate would be affected by the main carrier loss mechanism of the QWRs,
associated to the thermal escape of electrons out of the QWRs. On the other hand, the mentioned
optimum operation temperature range would be more difficult to be identified in QWRs of families
P3 and P4 due to an additional non-radiative mechanism observed even at low temperatures.
Component L (ns) 0 (ns) EL (meV) ND (m-1)
P1
1.48
0.7
16.3
7±5
P2
1.59
0.7
12.3
11±8
P3
1.66
0.7
9.3
7±5
P4
1.85
0.6
8.5
2±5
Table 3. Best fitting parameters of TRPL measurements to Eq. 6.
4. Conclusions
In the present Chapter we have studied the exciton recombination dynamics of self-assembled
InAs QWRs embedded into a /2 thick InP layer (the central emission of the QWRs takes place at
around 1450 nm at 12 K) deposited onto an InGaAs sacrificial film, a promising epitaxy for photonic
crystal based devices. Two electron loss mechanisms have been identified as responsible of the PL
quenching at intermediate and high temperatures under both low excitation power and resonant
conditions. The TRPL results gives us further information about exciton dynamics, which is
determined by excitons localized in QWR size fluctuations at the lowest temperatures. When
increasing the lattice temperature the radiative exciton lifetime increases according to the
expected thermal equilibrium between free and localized exciton populations. The localization
energies obtained here suggest a major contribution of the wire width fluctuations to the exciton
localization dominating the exciton recombination at low temperatures. When increasing the
temperature free excitons become important due to the thermal ionization of localized excitons.
As a result, some parameters of the excitonic dynamics in InAs/InP QWRs have been obtained,
some of them are key for the application of this system as active media in photonic devices
working at the telecommunication windows.
References:
1. J.M. García et al. “Electronic states tuning of InAs self-organized InAs quantum dots” Appl. Phys. Lett. 72 3172
(1998).
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dots with an intense and narrow photoluminescence peak at 1.3 m” PhysicaE 17, 127 (2003).
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3. L. Seravalli, G. Trevisi, P. Frigeri, D. Rivas, G. Munoz-Matutano, I. Suarez, B. Alén, J. Canet-Ferrer, J.P. MartínezPastor, “Single quantum dot emission at telecom wavelengths from metamorphic InAs/InGaAs nanostructures
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5-17
Chapter 6. L7-type photonic crystal micro-cavities with
embedded self-assembled InAs/InP quantum wires.
“Three hundred lives of men I walked this earth
and now, at the end of my days, I have no time”
Gandalf, The Lord of the Rings.
6-0
1. Photonic crystal microcavities with embedded semiconductor nano-structures.
Photonic crystal microcavities (PCM) combine high quality factors with small effective modal
volumes, which lead to enhanced values of the spontaneous emission (SE). [1-3]. This
phenomenon is known as Purcell effect (PE) and is explained by the larger density of optical states
available into the cavity with respect to the vacuum [4]. In the case of a solid state emitter inside a
PCM, the enhancement of the SE rate is determined by the detuning of the emitter wavelength
with respect to that of the cavity mode (spectral detuning) [5], the position of the emitter in the
cavity (spatial detuning) [6], the cavity quality factor, the quantum emitter linewidth [7] and the
polarization mismatch [8]. It is worth noting that the quantum emitter characteristics are crucial
for the successful coupling with the cavity modes. In this sense, the quantum wires (QWRs)
present optical properties in between those of quantum wells (QW) and quantum dots (QDs).
Ideally, the density-of-states (DOS) in a QD is described by a serie of Dirac’s deltas while in the
case of QWs and QWRs it is a continuum of states above the fundamental transition, an important
fact for technological applications, as nano-laser design. In spite of their particular properties, the
QWRs are still less studied than other kind of nano-structures, and just a few works studying the
Purcell effect on QWRs embedded into PCM have been reported up to date. Among them, Atlasov
et al. demonstrated the integration of site-controled InGaAs/GaAs V-groove QWRs into a PCM
where the coupling of the quantum emitter and the cavity modes exhibit a SE enhancement by a
factor 2.5 [9, 10].
In this work we present a systematic study of L7-type PCMs embedding InAs/InP self-assembled
QWRs [11, 12]. One of the most interesting properties of the InAs/InP QWRs is that they can be
used as active medium in devices working at two optical telecom windows in 1.3 and 1.5 m [1315]. The fundamental optical modes of Ln cavities show a large linear polarization anisotropy
which is maximum in the direction perpendicular to the linear defect [16]. Given the linear
polarization anisotropy of QWRs themselves (maximum along the QWR axis or 1-10 crystal
direction), [14] several L7 cavities have been fabricated aligned either parallel or perpendicular to
the QWRs. We will show that the Purcell factor values measured for both types of cavities is
different. The average value and its dispersion will be described attending to the optical properties
of finite size QWRs and the electromagnetic field distribution of the different optical modes. In
particular, the spatial and spectral detuning will be estimated taking into account the statistical
deviations of an amount of QWRs embedded in the cavity. We show how the Purcell factor
exhibits fluctuating values in the range of 1.2-2.4, with is attributed to the characteristics of the
PCM and the finite number of QWRs embedded , even of the maximum value (the figure of merit)
can be as high as 15.5.
2. Experimental details.
a) QWR Epitaxy and basic properties
6-1
Figure 1. (a) Atomic force microscopy image of an uncapped sample of self-assembled InAs/InP QWRs. (b)
Photoluminescence spectrum of the active media measured at 80K out of the photonic crystal structure. (c) and (d)
scanning electron microscope images of the microcavities fabricated with the defect perpendicular and parallel to the
direction of the QWRs.
The active medium consists of a single layer of self-assembled QWRs embedded into a 237 nm
thick film of InP. The InP is grown by molecular beam epitaxy (MBE) onto a 700 nm thick layer of
In0.53Ga0.47As which is deposited onto InP(001). InAs QWRs are aligned along the [1-10] direction
and form a periodic array, being the average height and width 3.5 and 14 nm, respectively, as
observed in atomic force microscopy (AFM) images like the one in Fig. 1 (a). The PL emission at 80
K is composed by four Gaussian peaks, being consistent with the most repeated QWR heights
measured by AFM, as shown in Fig. 1(b). The emission band typically exhibits around 30% linear
polarization anisotropy along the QWR axis direction. This behavior is explained by the
confinement potential anisotropy related with the QWRs geometry [14]. Indeed, the
nanostructures are clearly oriented along the polarization direction forming an almost periodic
array of wires with a period close to 18 nm. Even from AFM measurements we can observe QWRs
larger than 1 m we estimate an averaged QWR length of 500 nm with width fluctuations
responsible of the localization effects observed during the optical characterization. The influence
of this kind of defects has been shown in Chapter 5 as a function of the lattice temperature [15],
giving parameters for the model presented below. At very low temperatures the exciton dynamics
is dominated by localization centers. Fitting the data to Lomascolo’s model we deduced a density
of localization defects ≈ 7 m-1 along the QWR, in agreement with the 150 nm width fluctuations
observed by AFM. It was also demonstrated that free exciton recombination is visible above 40-50
6-2
K, therefore, in order to neglect the contribution of localized states the present study of the
Purcell effect is carried out at 80 K.
b) Fabrication details of L7 microcavities
The L7 cavity defect is designed by removing seven holes of a triangular lattice with lattice
constant a = 410 nm. Several cavities of each kind were fabricated varying the r/a ratio, r being the
radius of the hole, to tune the cavity modes with the emission of the QWRs. The PCMs were
fabricated by electron beam lithography on a polymethylmetracrylate (PMMA). The PhC holes
were opened by reactive ion beam etching (RIBE) at the hard SiOx mask before being transferred
to the semiconductor epitaxy by reactive ion etching (RIE). The remaining material and the InGaAs
sacrificial layer were removed by a time controlled bath of the sample into a hydrofluoridric acid
water solution. For more details about the process of fabrication see [17]. The resulting devices
are shown in Figs. 1 (c)-(d). One kind of cavities is designed with the longitudinal defect direction
parallel to the QWR axis (labeled “type(-)”), while the other is fabricated perpendicular to this
direction (labeled “type(+)”).
c) Set-up for optical micro-spectroscopy
The optical characterization of the QWR/L7-PCM structures was performed by microphotoluminescence (PL) and time resolved micro-photoluminescence (TRPL). The sample was
held at 80 K by immersing the confocal microscope described in Chapter 3 in a liquid nitrogen
bath. The PL measurements were carried out by using as excitation source a 980 nm pulsed laser
diode (40 ps pulsewidth, 40 MHz repetition rate). This way, photogenerated excitons are created
below barrier directly at the QWRs continuum. The excitation and emitted light were coupled to
single mode fibers and focused by means of the same microscope objective (NA ≈ 0.6), which
determines a combined spatial resolution around 1.5 m. The collected light was dispersed by a
0.5 m focal length monochromator and detected with a cooled InGaAs photodiode array. The
TRPL measurements were performed using the same optical set-up, except for the detector
(InGaAs APD single photon detector). Time correlation electronics were used to record the
emission decays. To avoid saturation effects which might obscure the determination of the Purcell
factor, the excitation power was kept at the minimum possible value as established independently
through optical characterization of these QWRs [15].
3. Expression of the Purcell factor for an ensemble of quantum emitters embedded into a
photonic crystal microcavity.
To explain our experimental results, we first study the equations at play in a L7-PCM containing
a single QWR of finite size. This will be the basis to simulate statistically the case of a QWR
ensemble embedded in the cavity (section 4). We make use of the expression proposed by
Meldrum et al. for the Purcell factor of a point-like emitter exhibiting an emission linewidth
comparable to that of the optical mode [18]:
6-3
-
[
∫- |
∫ - |
|
|
]
(1)
where WCAV and W0 are the SE rate of the emitter into the cavity or in the vacuum, V ef is the
effective modal volume, and /n is the cavity mode bulk emission wavelength, respectively. The
polarization mismatch between the optical mode and the quantum emitter is represented by 2.
The cavity and emitter angular frequencies are c and e, being c and e the linewidths
defined by their full widths at half maximum (FWHM), respectively. Meldrum deduced this
expression to describe the Purcell factor when both linewidths can be treated as Lorentzian
profiles. We can factorize such expression introducing the effective quality factor, as defined by
Gerard & Gayral [19]:
(2)
This way, we recover the well known figure of merit for a perfectly matched point-like emitter, FP:
(3)
where
(4)
(5)
(6)
-
Eq. 6 accounts for the spectral detuning of the emitter for a point-like emitter spatially matched
to the mode. In general, this is not the case, and the spatial detuning needs to be introduced
through a new factor, 
(7)
In the most studied case of a point-like emitter (atom or QD) QD is given by [5, 20]:
|
|
|
|
(8)
where E(r) is the electric field amplitude of optical mode at the emitter position, r, while EMAX
represents the electric field amplitude at its maximum value. However, Eq. 8 does not describe
correctly situations where a large nano-structure (comparable with the size of electric field lobes)
has to be matched to particular electromagnetic mode. In that case, a possible approach would
consist of integrating Eq. 8 into the emitter volume (Ve) [21]:
6-4
|
∫
|
|
(9)
|
which would be equivalent to describe the extended quantum emitter as a distribution of equal
point-like emitters covering the volume Ve. This approach is useful for describing extended single
emitters but it is expected a loss of accuracy in cavities with narrow electric field distributions. For
instance, we can evaluate Eq. 9 for the particular case of a PCM using a QW as active medium. The
field distribution is approximately constant across the narrow thickness of the QW, and, thus, the
integral can be evaluated in two dimensions across the emitter area, Se. Furthermore, when the
optical mode is strongly confined within the cavity volume E(r)  0 far from the cavity defect area,
Scav. Introducing VeSeScav:
∫
|
|
|
|
(10)
This definition of  leads to values smaller than unity for a QW, except in cavities with a
homogeneous electric field distribution (E(x,y) = cte = EMAX), as approximately occurs at the
fundamental optical mode in a micropilar [22]. One would expect that the spatial detuning factor
in any cavity containing a QW should not affect the Purcell Factor (=1), since the QW fills
uniformly the available space. The origin of this paradox lies in the introduction of Eq. 8 for a
point-like emitter. Eq. 8 stands for the ratio between the optical density of states available for the
emitter allocated at an arbitrary position (proportional to |E(r)|2), and for the same emitter
centered at the cavity maximum (proportional to |EMAX|2). For an extended emitter, even if its
geometrical center is perfectly allocated, we must expect that the electric field of the mode varies
within the emitter volume. Consequently, we propose a new definition of  to account for such
variations in the perfect matching case.
∫ |
|
∫ |
|
(11)
where V stands for the optical mode volume which can be substituted by the cavity volume (V CAV)
for tightly confined modes, or by the cavity surface (SCAV) when emitters with negligible thickness
are considered as explained above. We introduce the shape function H(r-re), to account for the
spatial extension of the emitter centered at re, H(r-re) = 1 into the emitter volume and null in the
rest of the space. In the spirit of Eq. 8, Eq 11 gives the ratio between the available optical density
of states when the emitter is allocated at an arbitrary position or at the antinodal position of the
electric field, r0, where E(r0) = EMAX. It can be particularized to the QW case:
∫
|
|
∫
|
|
∫
|
|
∫
|
|
6-5
(12)
introducing HQW(x-xi, y-yi) = 1; or the QD case:
∫
|
|
|
∫
|
(13)
introducing HQD(x-xi,y-yi) = (x-xi)(y-yi).
In both cases, we recover the expected results. If we apply this formalism for a QWR of length
2L and negligible width oriented along the y direction, the spatial detuning factor can be
determined as:
∫
|
|
∫
|
|
∫
|
|
∫
|
|
(14)
where we have introduced the function HQWR(x-xe,y-ye) = (x-xe) if |y-ye| <= L and HQWR(x-xe,y-ye) =
0 elsewhere.
Let us discuss finally the simultaneous coupling of an amount of quantum emitters justifying
the computation developed in next section. From the density of states point of view, the emission
line of a single emitter can be expressed by a spectrally narrow Lorentzian line (≈ 400
eV)Nevertheless, the PL spectrum for an ensemble formed by hundreds of emitters will exhibit
a Gaussian lineshape representing its inhomogeneous broadening. Evidently, the behavior of an
ensemble formed by many narrow emitters embedded into an optical cavity differs from a single
emitter spectrally broadened by several scattering mechanisms (homogeneous broadening of the
PL sepctrum). In fact, the cavity SE rate depends on the homogeneous broadening of the emitter,
because this parameter is determining the number of available states for their related optical
transitions. For example, the Purcell factor for a QW is limited by the broad homogeneous
linewidth (low Qe) of its emission line. In contrast, for an ensemble of single emitters, the
inhomogeneous broadening is mainly related with the number of emitters coupled to each cavity
mode. In fact, in the same cavity we would find perfectly tuned emitters presenting a high SE rate
together with off-resonance emitters whose emission is inhibited [23]. The Purcell factor in active
media composed by an ensemble of single emitters has been extensively reported because this is
the most common case when one develop the fabrication of cavities (micropillars, microdisks,
photonic crystal defects, …) on a sample containing a QD layer, for example. Some authors have
estimated the Purcell factor of an ensemble of QDs by averaging the SE rate [18, 24-27]. In other
works the experimental results are discussed by assuming that the Purcell Factor of an ensemble is
smaller than in the case of isolated quantum emitters coupled to the cavity [10, 28-30]. We
propose here a simple method to estimate the Purcell factor of an ensemble of quantum emitters,
now valid for our QWRs embedded in the microcavity, by averaging their SE rate:
6-6
∑
〈
〉
∑
∑
∑
(15)
In this expression we consider the optical intensity of the cavity mode as the superposition of
the emission of individual optical transitions. Since the Figure of Merit and the polarization factor
can be assumed constant for all the emitters they are out of the sum. The magnitudes i and i
accounts for the spectral and the spatial detuning of each single emitter, labeled with the integer
“i”. The average is weighted having into account the fact that the different emitters coupled to an
optical mode contribute in a different way to the PL band depending on their coupling degree. This
fact is considered by means of the weighting factor P i that is proportional to the contribution of
the emitter “i” to the PL ensemble I(i)PL. As aforementioned this contribution depends on emission
frequency and spatial location being proportional to the SE rate WiCAV.
(16)
WiCAV can be estimated by simplifying Eq. 7 and assuming W0 as a constant in the frequency range
around the cavity mode. In this way it can be shown that the weighting factor is proportional to
the product of the spatial and the spectral detuning factors:
(17)
This fact means that the best tuned emitters present the largest contribution to the PL intensity
whuch drives to a higher influence in the SE. Finally, Eq. 15 can be rewritten as:
〈
〉
∑
〈
∑
〉 (18)
Since the theoretical approach developed above has been carried out in a general way, it can
be employed for every kind of active media. In fact, the spatial detuning factor in Eq. 7 can be
employed for describing the Purcell effect extended active media, while Eq. 18 drives to
appropriate estimation of the Purcell factor in cavities containing an ensemble active media. For
the successful use of Eq. 7, the size and shape for the single emitter must be estimated, as done in
section 2. On the other hand, for applying Eq. 18 it is required more accurate data consisting of the
emission frequency/wavelength and position of each of the single emitter embedded into the
cavity defect. Since these data cannot be experimentally obtained for a determined cavity, in the
next section the Purcell factor is determined by studying the SE rate of simulated cavities with an
active media presenting optical properties similar to our self-assembled QWRs. During every
simulation the emission of the QWRs is randomly distributed according to the ensemble PL
6-7
emission shown in Fig. 2(b) and their position into the cavity defect varies attending geometrical
considerations.
4. Results and Discussion.
Figure 2. (a) Photoluminescence spectra of a parallel-type(-) cavity at the polarization directions [110] (in red)
and [1-10] (in blue). (b) Idem with a perpependicular-type (+) cavity.
The spectra of L7-type PCM typically present three or four emission resonances, depending on
the matching between their optical modes and the PL of the QWRs. The optical modes can be
labeled as O1, O2, O3 and E1 according to their different symmetry (the O-labeled modes are odd
while the E-labeled ones are even with respect to the center of the center of the cavity defect) [11,
16]. In this work, we will focus our attention on the odd modes, mainly on O1 and O2, since their
intensity and Q-factor are higher. Fig. 2 shows the PL polarization spectra corresponding to a
type(-) and a type(+) cavities. In both cases the emission of the odd optical modes is strongly
polarized perpendicular to the linear defect elongated axis independently of the QWR orientation
within the cavity. Due to this fact, polarization the odd modes at the type(+) PCM coincides with
the spontaneous polarization of the active media while in the case of the type(-) PCM it is
observed a noticeable polarization mismatch. In order to illustrate this fact, in Fig. 2 it is
represented the polarization resolved spectra of both kind of cavities (type(-) Fig. 2(a) and type(+)
in Fig 2(b)). It is clearly shown that the emission of the odd modes is clearly polarized parallel to
the linear defect direction ([110] and [1-10] for the type(-) and type(+) PCM respectively).
Therefore, a higher SE rate is expected in perpendicular-type(+) L7-cavities since QWRs also exhibit
a preferred polarization emission along the [1-10] direction, as was shown in Fig. 1(b).
6-8
Figure 3. Photoluminescence spectrum of a parallel-type(-) cavity. The PL is accompanied by the decay times of the
modes O1, O2 and O3 (blue scatter) and by the decay time of the active media at the corresponding wavelength (in
red). b) Photoluminescence transients of the mode O1 and the QWRs emitting at the same wavelength. c) Mode O1
Decay time measured on twelve different cavities of both kinds parallel-type(-) (x scatter) and perpendicular-type(+) (+
scatter). The red circles correspond to the decay times of the active media at the same wavelength of the modes. d)
Idem for the modes O2 of the same cavities.
In a given L7-cavity we can register simultaneously the steady state and transient PL spectra
under the same excitation conditions, as shown in Figs. 3(a)-(b). In this way the optical mode
wavelength, quality factor and decay time can be determined for the same excitation conditions.
The corresponding transients for the QWRs are measured far away from the PCMs and at the
same wavelengths for comparison. An example is shown in Fig. 3(b) for the O1 mode of a paralleltype(-) cavity emitting at 1509 nm. The decay time of the QWRs stays approximately constant over
the entire emission band and it shows only a smooth decrease from 2.6 to 2.3 ns between 1460
nm and 1510 nm in Fig. 3(c)-(d). Meanwhile, the decay times measured at the optical modes of 12
different L7-cavities, although typically smaller than that of the QWRs, exhibit a noticeable
dispersion as shown in the same figures. We have summarized the Purcell factors determined as
0/m, where m is the cavity mode decay time (m = 1 and 2 for modes O1 and O2) and 0 is the
6-9
decay time of the corresponding active media in Tables 1 and 2. From these results, we point out
three main facts:
i) Despite their different Q-factor, the average Purcell factors for O1 and O2 optical modes
are practically the same, within the dispersion error.
ii) The Purcell factors of type(+) L7-cavities are larger than those of type(-) ones for both O1
and O2 modes.
iii) The Purcell factors measured in many cavities exhibit a great dispersion, even in adjacent
cavities with similar characteristics.
Average
0/1
1.78
Q1
7890
0/2
1.70
Q2
4720
1.35
1.41
1.72
2.03
1.32
7740
7620
7880
6820
7430
1.65
1.71
1.76
1.89
1.22
4550
4830
4330
4540
4390
1.6 ± 0.3
7560±170
1.65 ± 0.20
4560±140
Table 1. SE enhancement of the type(+) cavity modes obtained
from the experimental data represented in Fig. 3 (c)-(d).

Average
/1
1.12
Q1
7500
/2
1.07
Q2
4060
1.29
1.41
1.29
1.27
1.16
6880
4910
6540
5740
5420
1.34
1.28
1.18
1.33
1.22
3140
2450
4220
3250
3160
1.26 ± 0.10
6200±600
1.21 ± 0.12
3400±500
Table 2. SE enhancement of the type(-) cavity modes obtained
from the experimental data represented in Fig. 3 (c)-(d).
a) Figure of Merit
In the analysis of the figure for merit in L7-type PCM, the determinant parameters are the
linewidths of the cavity modes and the active medium, since the modal volume of the three first
optical modes are approximately equal V1ef ≈ V2ef ≈ V3ef ≈ 1.1(/n)3. The figure of merit becomes
thus directly proportional to the corresponding Qef for the three modes. First of all we need to
calculate Qe, which is inversely proportional to the PL emission linewidth that can be taken from
direct experimental results on high quality single QWRs at 4 K. In order to obtain the emission
6-10
linewidth at 80 K we can take into account the temperature broadening of the optical transition by
acoustic and optical phonon scattering:
(T) = 0ACT+LO/(e(ELO/KT)-1) (19)
Accordingly to the single QWR characterization is set to 0.5 meV [31], AC is the scattering
rate of excitons by acoustic phonons and LO the linewidth associated to the scattering of excitons
by longitudinal optical phonons of energy ELO. We consider here 0.035 meV/K and 25 meV as
approximate values for AC and LO (with ELO = 40 meV), as determined by PL measurements in
QWR ensembles [32]. In this way we can deduce Qe ≈ 235 for QWRs emitting at around 1550 nm
(0.8 eV).
Figure 4. Figure of Merit of the Purcell Factor as a funtion the cavity quality factor for the three case: narrow
emitters (doted black line), broad emitters (dashed black line) and self-assembled QWRs emitter at a lattice
temperature of 80 K (red continuous line).
The value Qe is one order of magnitude smaller than the cavity quality factors (see Fig. 2(a)),
which directly enables us to conclude that Qef (Eq. 2) will be mainly determined by the PL
linewidth of the single QWR emitter, as depicted in Fig. 4. An upper limit for the figure of merit is
given by the curve Qef = Qc (dotted black line in Fig. 4) that is representative of a narrow emitter
coupled to a wide optical mode: it is the case of QDs into a micropillar [20]. In contrast, the lower
limit is given by a curve with Qef = Qe = 40 (dashed black line in Fig. 4), representing a broad band
emitter coupled to a narrow optical mode: this is the case of an InGaAsP QW emitting at 0.8 eV
with a 20 meV linewidth [33]. The Figure of Merit in our cavities is about Fp=15.5 (solid red line in
Fig. 4) and it is expected to be practically constant with Qc, because the emitters are wide as
compared to the optical modes. This result is consistent with observation i) in section 3 where
minor differences in the Purcell factor of O1 and O2 optical modes were noted in both kinds of
cavities.
6-11
b) Polarization Mismatch.
It is well known than the non-polarized light coming from QDs reduces three times the Purcell
factor. Indeed, some authors introduce directly a factor 1/3 in Eq. 1 instead of the term  [25, 34].
In the case of our QWRs, the value of the polarization factor is calculated as the projection of the
spontaneous polarization vector of the emitter along the electromagnetic field direction which can
be expressed as the following scalar product:
⃗
⃗
⃗
⃗
(20)
As described in section 2, the L7-cavities of the present work were fabricated in such a way that
the polarization of the odd optical modes is defined either parallel (in type(+) cavities) or
perpendicular (in type(-) cavities) to the QWR axis direction. Using this criterion the normalized
polarization vectors of these modes at both types of cavities are simply the unit vectors defining
our X and Y axes:
(21)
(-)
(
)
(
)
( ) (22)
√
The PL intensity of the emitted light by our QWRs is mainly polarized along the Y direction (i.e.
[1-1 0]), obtaining a polarization degree of 32 % perpendicular to this direction, as was described
in Fig. 1(b). By comparing the polarization resolved spectra in different planes of the sample (not
shown) we can approximate spontaneous polarization vector to
√
√
(√
)
(√
)
(23)
√
√
The lower polarization degree in the axis perpendicular to the grow direction is comparable with
the proposed for InGaAsP/InP QWs in Ref. 33. The scalar product (Eq. 20) of vectors defined in Eqs.
21-23 gives us polarization factors (-) = 0.29 and (+) = 0.49, explaining the observed
smaller SE rates (and hence smaller Purcell Factor) in type(+) L7-cavities as compared to type(-)
ones.
6-12
c) Spectral and Spatial detuning factors
In this section we are going to develop a numerical simulation of the average values for the
spectral and detuning factor product, <>, in the case of a finite number of QWRs embedded in
a L7-cavity whose emission wavelength is randomly distributed within the observed PL band of the
ensemble [Fig. 1(b)]. The L7-cavities can support up to about 150-620 QWRs depending of the
considered QWR length and the cavity type (type(+) cavities have more QWRs because they are
orientated perpendicularly). However, the introduction of the weighting factor in Eq. 18 will
introduce a huge reduction of QWRs that really contribute to the average: less than a 10 % of the
emitters will be coupled to the optical modes in the most favorable case. We have developed an
algorithm to evaluate <*> in a system consisting 500 nm long QWRs embedded in type(-) [Fig.
7(a)] and type(+) [Fig. 7(b)] L7-cavities. The spatial distribution of the electromagnetic field
intensity for the mode O1 in a L7 micro-cavity is illustrated in Fig. 7(a). The represented cavity is
characterized by a lattice parameter a = 410 nm and filling factor r/a = 0.29. From the
experimental measurements we would expected the emission of the fundamental mode occurring
at 1522 nm with a quality factor Qc ≈ 6500. The active medium is constructed by considering each
QWR located at a random position into the cavity defect up to filling completely (execpt those at
borders cut by PhC holes which contribution is neglected) and presenting an arbitrary emission
frequency. Each frequency is selected using a random value function modulated by means of the
ensemble DOS function, proportional to the PL spectra of Fig. 1(b).
Two simulated QWR ensembles for the type(-) cavity are depicted in Figs 7(d) and (e) giving rise
to two different values of <> for the selected L7-cavity. The two simulated emission bands
[black lines in Figs. 7(d)-(e)] cannot be equal to the measured PL of a big ensemble given the finite
number of QWRs that can be embedded in the simulated cavity. In this sense we can see how the
peak energy of the simulated emission band is at around 1470 nm [Fig. 7(d)] and 1410 nm [Fig.
7(e)] instead of the 1450-1460 nm observed in the experimental PL [Fig. 1(b)]. At the same time,
we can also see how the number of QWRs emitting at 1522 nm [mode O1 depicted in blue in Figs.
7(d)-(e)] is different in the two simulations. This fact is related with different values found for
<> in Figs. 7(d)-(e). Notice that fluctuations of  between 0.1 and 0.5 are obtained in 100
simulations (corresponding to 100 different QWR finite ensembles), as shown in Fig. 7(f). It is
worth noting that we have obtained the same average value for both kind of cavities which after
the 30-40 simulations tends to a stationary value <*> ≈ 0.32 in both cases.
6-13
Figure 5. (a) Electromagnetic fiel distribution of mode O1 in a L7-type PCM the orientation of the QWRs
differences the type(-) (b) and type(+) (c) cavities. (d) and (e) represent two simulated distributions for the optical
mode O1of type(-) L7-cavity with similar QWRs content but leading to different values of ; (f) 100 simulated
values of  for the optical mode O1.
Applying this simulation of <> in 16 different type(-) and type(+) L7-cavities for O1 and O2
modes (to sweep the most important region of the QWR PL band) and calculating the effective
Purcell factor by taking into account the Purcell Figure of merit and the Polarization Factor
discussed above we obtain the results shown in Fig. 6. These calculated values are compared to
6-14
the experimental data (red scatters). From these results we can conclude that the dispersion of
the experimental Purcell factor is produced by the finite content and random distribution of QWRs
coupled in a different way in a given cavity. This effect also reduces the SE rate by a factor 0.32
with respect to the figure of merit, driving to an average Purcell factor as low as 1.2-1.3 for type(-)
L7-cavities for both O1 and O2 modes, in agreement with the experiment. In the case of the
type(+) L7-cavities the average Purcell Factor can increase up to 2.0-2.3, even if an important
reduction is predicted for optical modes above 1520 nm. This is due to the emission intensity
decrease at the low energy tail of the PL band. The highest dispersion in the calculated Purcell
factor is found at the mode O1 of the type(+) L7-cavity, where values in the range 1.5-3 can be
obtained, also in correspondence with the experimental case (values found in the range 1.3-2).
Figure 6. (a)-(b) Calculated average values of the Purcell factor by using simulated values of <>, as
explained in the text, for the case of type(-) and type(+) cavities for mode O1; (c)-(d) idem for mode O2. The error
bars stand for the dispersion in the simulated values of < > as was illustrated in Fig. 7(f).
5. Optimization possibilities and future work: Isolated QWRs
6-15
We have also studied cavities containing isolated InAs/InP QWRs and, even the Purcell effect is not
yet demonstrated, preliminary results show the different coupling degree between similar cavities.
In the left we show the typical PL spectra of a cavity containing just a few of QWRs. As a difference
with respect to the cavities embedding an ensemble of QWRs, we can point out the lower optical
signal observed in the fundamental modes, since such modes has been tuned with the low energy
tail of the active media PL, see grey shadowed spectra in Fig. 7(a). Due to the low signal generally
measured the fundamental modes must be distinguished from other peaks (like off-resonance
QWRs) by means of polarization resolved spectroscopy (in red or blue parallel to the [110] or [110] direction). But sometimes, we also observe cavities where the fundamental mode dominates
the PL Fig. 7(b). Even this kind of cavities is currently under investigation it seems there are two
probable explanations for this fact. On one hand, it can be produced by a larger amount of QWRs
emitting at the mode wavelength attributable to the random content of QWRs. The other
explanation would consist of the perfect tuning with the optical mode and a single emitter.
Figure 7. (a) typical PL spectra of a L7 type(-) cavity containing isolated QWRs, the fundamental modes are resolved in
the inset. (b) PL spectrum of a L7 type(-) dominated by the fundamental mode O1.
6. Conclusions.
In the present work the spontaneous emission rate of L7-type PCMs with embedded selfassembled QWRs is studied. With the aim to observe polarization effects, two cavity designs are
proposed: type(+) cavities with the polarization of the fundamental mode parallel to the
spontaneous polarization of the QWRs and the type(-) presenting the largest polarization
mismatch between the fundamental modes and the QWRs. In particular, our attention has been
focused on the modes O1 and O2, whose experimental Q factors are about 6500 and 3500,
respectively. The spontaneous emission rate has been measured by TRPL, observing a certain
lifetime reduction with respect to the case of bare QWRs outside the cavities. The ratio between
both decays give us the experimental Purcell factor that, in average, results around 1.6 and 1.2 for
type(+) and type(-) cavities, respectively, even if a value ≈ 15.5 can be expected theoretically.
6-16
These results have been extensively discussed after grouping the parameters contained into the
Purcell factor expression into four main factors: the Figure of Merit (F p), the polarization mismatch
(), the spectral detuning factor () and the spatial detuning factor (). The two first factors are
approximately constant for different cavity dimensions, except the polarization mismatch that
changes significantly from type(+) to type(-) cavities. In contrast, the two last factors depend on
the QWR content which would be different at each point of the sample. The possible effects of
such finite ensemble of QWRs filling the PhC micro-cavity have been determined by means of an
algorithm simulating the random content of QWRs embedded into both kinds of L7-type cavities.
Finally, we can conclude that the measured moderate value of the Purcell factor is mainly
attributed to the homogeneous broadening of the active medium, with a notable reduction due to
the spectral and spatial detuning factors, while the great oscillations of the experimental data are
attributed to the finite number of QWRs embedded into the cavity.
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6-18
Chapter 7. Purcell Factor dependence on the
excitation power and temperature.
“Oh, yeah, what are you gonna do?
Release the dogs?
Or the bees?
Or the dogs with bees in their mouth
and when they bark they shoot bees at you?”
Homer J. Simpson; The Simpsons, Seasson 5 Chapter 18th; 1994.
7-0
1. General remarks
Previously, we studied optical properties in QWR samples (mainly H1, but also H3 and H6 in
some cases). Sample H6 (employed for the SE measurements of Chapter 6) presents quality factors
around 6500, a value considerable lower than in samples H3 and H1. The lower Q vales of cavities
in sample H6 was favorable for the determination of the Purcell factor, since due to the ultra low
threshold laser operation observed in type(-) of H3 the determination of the SE rate in the cavities
of this sample become complicated. The dificulties are described in the next section, where we
propouse an extrapolation method in order to estimate the Purcell factor in lasing modes. It is
worth also noting that there is not observed laser operation in type(+) cavities. This result,
currently under investigation, seems to be related to the carrier loss at the fabricated hole edges
of the photonic crystal. This effect would be reduced in type(-) PMC due to the large confinement
of the QWRs normal to the linear defect direction.
Another difference between the three samples could be found in the content of QWRs emitting
at 1500 nm that is larger in the H3 sample, as shown Figure 1(a). However this does not seen to be
decisive for the experimental Purcell factor, as illustrated in Figs 1 (b) and (c). The mean values of
the Purcell factor of cavities contained in both samples are close to the values estimated
theoretically in Chapter 6. Nevertheless, a few cavities in sample H3 reach the highest measured
Purcell factor values (>2). At the same time, some of them also show inhibition of the SE, that is a
Purcell factor smaller than the unit. In section 4 of Chapter 6, the dispersion of the experimental
measurements was related to the finite number (and random content) of coupled emitters. In this
sense we can conclude that sample H3 seems to be more sensitive to such kind of effects. This
suggests a dependence of the experimental data dispersion on the cavity Qs. The reason is that
even the coupling of the best tuned QWRs is enhanced for higher Q modes; the narrower modal
linewidth might reduce the coupling with emitters spectrally located away from the optical mode,
reducing the effective number of QWRs coupled to the cavity. In section 3 we are going to deep
into this hypothesis by means of the study of the PF evolution on the temperature.
From the different factors determining the SE rate of our cavities, the figure of merit is the
most dependent on temperature since it is inversely proportional to the active medium
homogeneous broadening. At very low temperatures (in the absence of phonon-scattering) the
the linewidth of the single InAs/InP QWRs is expected to be around 0.5 meV, which would lead to
an enhancement of the figure of merit by a factor 4 with respect to the values obtained at 80K in
Chapter 6. In contrast to this prediction, the Purcell factor measurements at 6 K [Fig. 2(a)] do not
differ noticeably from data at 80 K, as shown in Figure 1 (a) and (b). Indeed, the SE rate measured
in a given cavity at 80 K is larger than that at lower temperatures and, even in the cases where it is
observed a larger Purcell factor at 6 K the temperature evolution is smooth. On the other hand,
the PL decay time associated to cavities with lager Purcell factors [marked with arrows in Fig. 2(a)]
could be influenced by to the contribution of stimulated emission, since these data correspond to
optical modes excited close to their lasing threshold [see Figure 2 (b)]. With the aim to give more
consistency to low temperature results and detect possible artifacts related with emission close to
the lasing threshold, in the following we proceed to develop an accurate study of the stimulated
emission influence on the PL decay time measured at the cavity modes. For this purpose it is
separately monitored the power and temperature evolution of the PL decay times in both lasing
and non-lasing cavity modes on some of the cavities belonging to sample H3. As a result, we are
going to deduce an appropriate way for determining the SE rate at both kinds of modes. This study
has been focused in type(-) cavities since they present more difficulties for the measurement of
the SE due to the low lasing threshold power.
7-1
Figure 1. (a) PL emission of the three samples studied in the present Chapter. (b) and (c) Purcell factor and PL
decay times at 80K measured at the modes O1 and O2 of the samples H3 and H6.
Mode O1
Mode O2
Sample
Q1
Purcell Factor Q2 Purcell Factor
H6 (80K) 6165
1.26
3380
1.21
H3 (80K) 14470
1.19
6315
1.19
H3 (6K)
14470
1.58
6315
1.23
Table I. Quality factor and Purcell factors of the modes at the samples H6 and H3.
7-2
Figure 2. (a) Purcell Factor at 6K and lasing threshold (b)for cavities of sample H3.
2. Purcell Factor as a Function of the Excitation Power.
In this chapter our attention is focused in H3 cavities, which as aforementioned present
noticeable higher Qs. In this sampe, it has been demonstrated ultra low threshold laser operation
at room temperature [1], when the fundamental mode (O1) is located between families P1 and P2.
Particularly, a power threshold (Pth) around 20 W is found by using an excitation laser working at
780 nm. If we decrease temperature down to 80 K and use pulsed operation (required for time
resolved measurements) a noticeable reduction of the lasing threshold is expected. Firstly because
the pulsed operation leads to a magnification of the instantaneous pumping power, and secondly
due to the minor contribution of non-radiative exciton recombination channels. Despite that low
threshold lasers is a major finding for technological applications, it makes difficult the
measurement of the SE rate enhancement (e.g., the Purcell Factor). For this reason the PL
transients were measured under near resonant excitation conditions, since the carrier
photoinjection is reduced by using a pumping laser diode operating at 980 nm, well below the InP
absorption band edge. Figure 3 shows the basic optical properties of a typical lasing PCM
measured in the mentioned conditions. These data corresponds to a L7 PCM with r/a ≈ 0.36,
whose fundamental optical mode takes place at (O1) = 1442 nm. The power threshold for lasing
operation is close to 8 W, as shown in Fig. 3(a), which is used to normalize the Q dependence on
pumping power in Fig. 3(b), in order to be considered as a characteristic curve in our L7 cavities (if
we do the same normalization for other cavities we observe the same dependence). The Q value is
7-3
nearly constant, ≈13500 below 0.7Pth, but increases and reaches a maximum value ≈21000 at
around 1.8Pth, as shown in Fig. 3 (b). This phenomenon, known as Schawlow-Towns effect, is
related to the optical gain dependence on the excitation power [2]. As expected, the optical
amplification will affect the PL transients measured at the wavelength of the O1 mode, as shown
in Fig. 3 (c)-(e). At low powers the PL transient is basically mono-exponential [Fig. 3(c)], whereas a
above this power a second long decay component is required to fit the experimental data [Fig.
3(d)-(e)]. At the same time, the main and fast decay component progressively approaches the
system response as the pumping power reaches Pth [Fig. 3(e)]. Quantitatively, the PL transients can
be fitted with PL decay times (fast component) of 0.6 ns for 0.2Pth [Fig. 2(c), 0.3 ns for 0.6Pth [Fig.
2(d)] and 0.1 ns for 0.8Pth [Fig. 2(e)], which is very small as compared to exciton lifetimes, Fig. 1(b).
On the other hand, the long decay time is found to be around 2 ns and it could be due to a
background of QWRs not coupled to the optical mode, but also to photon recycling processes, as
reported in H-type PCMs with an embedded InGaAsP quantum well [3].
Figure 3. Lasing properties of a typical L7 PCM: integrated optical intensity (a) and quality factor (b) as a function of
the excitation power. PL transients for three different values of incident power: 0.2P th (c), 0.6Pth (d) and 0.8Pth (e).
As highlighted above, a more detailed study is necessary to extract the true Purcell Factor, given
that their values are highly dependent on power and typically they are deduced below the lasing
threshold. The results of this study are summarized in Fig. 4. Figure 4(a) shows the PL spectra of
two different cavities, both with lattice parameter a = 410 nm but different filling factors, r/a =
0.24 and 0.28. The first cavity is designed to present the mode O1 at the lowest energy side of
family P1 (dark blue line), NL(O1) = 1516 nm, and it does not exhibit laser operation. The second
cavity is fabricated with L(O1) = 1483 nm (blue line) in order to allow laser operation at the
fundamental mode. This mode is very close to the mode O2 of the non-lasing cavity that is
measured at NL(O2) = 1490 nm, allowing its comparison with the lasing mode. The power
dependence of the optical intensity for those three optical modes is represented in Fig. 4(b). The
7-4
lasing threshold for mode O1 of the second cavity is clearly observed at Pth ≈ 13 W, whereas a
linear power dependence (any evidence of light amplification) is observed for optical modes O1
and O2 of the non lasing cavity. Figures 4(c) and 4(d) compare the power dependence of exciton
recombination times in QWRs emitting at 1516 and 1486 nm, respectively, together with that of
PL decay times in the corresponding optical modes at similar wavelengths.
The exciton recombination time in bare QWRs exhibit a slight decrease with the excitation power
of about 10 % (1.85 to 1.67 ns) at 1515 nm and 15 % (2.17 to 1.86 ns) at 1486 nm [red solid
squares in Figs. 4(c) and (d), respectively]. In QWRs at low temperatures this dependence can be
explained by the transition from a dynamics dominated by localized excitons (low power regime)
to a dynamics dominated by free excitons (high power regime) [4]. At 80 K the exciton
recombination dynamics in these QWRs would be mainly dominated by free excitons due to the
thermal ionization from localization centers [5], but localization effects cannot be completely
neglected, as evidenced by data in Figs. 4 (c) and (d). If we compare the exciton lifetime
dependence on the excitation power with that of the PL decay time for optical modes O1 [dark
blue solid circles in Fig. 3(c)] and O2 [dark blue solid stars in Fig. 3(d)] we observe a similar trend,
despite a reduction given by the Purcell Factor. Nevertheless, the deduced PF is not strictly
constant, but it changes from (CAV/0) = 1.24 (1.29) at the lowest excitation power to (CAV/0) =
1.44 (1.58) for optical mode O1 (O2). The values found at low excitation powers are consistent
with our previous findings in L7-cavities [6]. This result suggests that in non lasing modes the
observed small difference in PF between low and high excitation power can be ascribed to the
different nature of the emitters, localized and free excitons.
Figure 4. (a) and (b) PL spectra and integrated optical intensity plot (as a function of the incident power) of a lasing
(r/a=0.24 in blue) and non-lasing (r/a=0.28 in cyan) modes. (c) PL decay time of the mode O1 of the non-lasing cavity
and QWRs emitting at 1516 nm wavelength. (d) lifetime of the mode O2 of the non-lasing cavity and the mode O1 of
the lasing cavity. They are accompanied of the data corresponding to QWRs emitting at 1486 nm wavelength.
Light amplification induce drastic changes in the measured PL decay times, as observed in Fig.
4(d) for lasing modes (blue solid circles), where the PF would increase from less than 2 at very low
excitation powers to more than 4 close to 0.5Pth. On the other hand, excitation powers smaller
7-5
than 3 W are needed in order to avoid the influence of the optical amplification on the PL
transients and hence extract a decay time characteristic of the QWR spontaneous emission.
Nevertheless, the PL intensity of bare QWRs for such extremely low excitation power is several
orders of magnitude smaller than the PL signal produced by optical modes of a lasing cavity. The
measurement of a representative PL transient in for the active media would need high integration
times that would induce a noticeable experimental error in the extraction of the PL decay time.
The solution here proposed consists of studying the power evolution of both decay times, that of
the cavity mode and that of the bare QWRs emitting at the same wavelength. The PF of lasing
modes can be evaluated by extrapolating the experimental data towards zero power, where the
optical amplification will not exist. In the case of data presented in Fig. 4(d) a PF about 1.2 would
be deduced, very close to the ones experimentally measured in non-lasing modes at low excitation
powers (7.5 W).
Quantum Wires
Non-lasing Cavity
Lasing Cavity
Family P1 Family P2 Mode O1 Mode O2
Mode O1
Wavelength
Quality Factor
 (P=0) (ns)
 (P=7.5 W) (ns)
 (P=60 W) (ns)
CAV/0)P=0
CAV/0)P=7.5
CAV/0)P=70
1516
1.85
1.83
1.67
-
1486
2.17
2.09
1.86
-
1516
18500*
1.51
1.47
1.16
1.23
1.24
1.44
1490
7400*
1.80
1.62
1.17
1.20
1.29
1.58
1483
17200*
1.80
0.33
1.20
-
*Quality factor measured at 2.5 W, i.e Pirr = 0.2Pth in the lasing mode
3. Purcell Factor as a Function of Temperature.
From the possible effects related to temperature our discussion will be centered in:
i) Linewidth broadening.
ii) Emission energy shift.
iii) Appearance of localization.
As shown shown in Chapter 5 [5], all three effects are important in the temperature range from
10 to 80 K. Their evolution can influence either positively or negatively to the SE enhancement (i.e.
the PF) depending on the tuning of the active medium with the cavity mode. The emitter linewidth
at 10K is the narrowest due to a reduction of the phonon scattering rate by reducing temperature.
This is, a priory, a positive effect since it leads to an important increase of the figure of merit. On
the opposite, the linewidth would produce a reduction of the number of quantum emitters
coupled to the cavity modes. This would be translated into a decrease of the spectral detuning,
and therefore in the averaged factor <>, as discussed in Chapter 6. The emission energy shift
affects to both the cavity mode and the emitter. On the one hand, the red shift of the optical
mode is produced by the refractive index variation of the InP (material conforming the photonic
crystal membranes) [7]. On the other hand, the emitter suffers a bigger redshift associated to the
bandgap reduction with increasing the temperature of both the InAs (QWRs) and the InP (barrier)
materials [8]. That variation can be accounted for by means of the Varshni’s Law, as discussed in
7-6
Chapter 5. In this way, a relative approach (separation) of the cavity modes with respect to the
emitters spectrally located at larger (shorter) wavelengths is expected by reducing lattice
temperature since the shift of the QWRs emission is stronger. Obviously, this phenomenon can
influence the temperature evolution of the Purcell factor, but in this case, it would occur in a
rather random way depending on the relative spectral position of the cavity mode and the
emitter. Finally, since the exciton dynamics is dominated by localized excitons below 50 K, the SE
rate of the cavity modes at lowest temperatures is also expected to suffer from localization
effects, like the ones described in the last section of chapter 5. As discussed there, the localization
centres in our QWRs are mainly attributed to wire width fluctuations. That would lead to a
considerable reduction of the localization length, being the exciton wavefunction limited by the
dimensions of the QWR fluctuation [9]. This fact produces important consequences on the spatial
detuning, which for extended emitters (like QWRs and QWs) depends on the emitter dimensions.
Figure 5. Polarization resolved spectrum for different lattice temperatures of the PCM with its fundamental mode
located at long wavelenghts.
Firstly we analyze in parallel the evolution of the first two and most important effects (the
linewidth broadening and the energy shift), given that the effects of localization are more difficult
to be distingushed from them. For this purpose, Fig. 5 shows a particular example illustrating the
emitter broadening and energy shift by changing temperature. The spectra are obtained from a
type(-) PCM and therefore the odd modes O1 and O2 are polarized in the direction [110], but in
contrast, the spectra are polarized along the direction [1-10]. In these conditions the cavity modes
offer a weak optical signal, in such a way that the PL emission of a spectrally isolated QWR can be
distinguished among them. At 90 K, the broadening of the emission line makes difficult an
accurate determination of the PL peak. However, reducing the latice temperature to 60 or 40 K we
can find an isolated QWR emitting around 8 and 5 nm away from the mode O2, respectively. The
spectral distance between the mode O2 and the emitter is reduced to 2 nm at 10K. Unfortunately,
in spite of such proximity the narrowing of the QWRs avoids an important coupling of such QWR
to the O2. In this way it is illustrated how the coupling probabilities are reduced at very low
temperatures due to the linewidth narrowing. Another example is given in Fig. 6, where the
number of QWRs coupled to the fundamental mode O1 of a L7 type(-) PCM (Q1 = 14500 approx.)
is calculated at different wavelengths and temperatures. The temperature effects are considered
according to the energy shift of the band PL, the reduction of Qe=235 (≈1000 at 10K) and its
influence on Qeff, see Chapter 6. The number of emitters coupled to the cavity is estimated by
7-7
considering the probability to find QWRs into the spectral region comprised by c-eff and
c+eff, where eff is extracted from the effective quality factor (Qeff). Doing this simple statistic
estimate, it is observed a noticeable reduction (almost a factor 4) of the averaged number of
QWRs around the considered cavity mode, by reducing the sample temperature from 80 to 10 K.
Figure 6. statistical number of QWRs coupled to a cavity mode with qualitity factor Q=6500 at 80 and 10 K.
The data of Figs. 5 and 6 clearly illustrate the phenomena expected during the temperature
evolution, but it results quite difficult to conclude if a reduction in the number of coupled emitters
to the optical mode could compensate the SE enhancement produced by the increase of the
Figure of Merit. For this reason, in Fig. 7 we are going to simulate the temperature evolution
neglecting localization effects, for the moment. As done in Chapter 6, the predictions for the
Purcell factor will be deduced by averaging multiple cavities with a random content of QWRs. For
the simulations we will neglect possible (but small) dependences of the QWR polarization in the
considered temperature range [10]. It is also assumed a homogeneous linewidth for all the QWRs
in such a way that Qe does not depend on the emission wavelength, i. e. ≠ (T) = 0.23 and Fp=
Fp(T) ≠ Fp(). In contrast, the detuning factors are considered dependent on both the temperature
and the emission wavelength. The simulation is performed in the temperature range 10-200K by
using the homogeneous linewidth variation and Varshni parameters obtained for QWRs of sample
H1 in Chapter 5. The quality factors are taken Q1= 14500 and Q2=6500 based on the experimental
data shown in Table I. The modes selected for the simulation are located in three different spectral
points [see Fig. 7(a)]: at the low energy tail of family P1 (cavity 2), at the high energy side of family
P1 (Cavity 4) and half intensity of the family P2 (Cavity 8). Figures 7(b) and (c) show the simulations
of the detuning factor  of the thee cavities for the modes O1 and O2, respectively. The three
cavities present a similar behavior: an increase of by raising the lattice temperature.
Quantitatively, the mode O1 of the cavity 2 (blue scatters) presents detuning factors slightly lower
than in the other cases, which we attribute to the smaller number of QWRs emitting at the optical
mode wavelength, as deduced from Fig. 6. Consistenly, there is not observed noticeable
differences in the O2 modes of this cavities due to they are better tuned with the active medium.
7-8
Figure 7. (a) PL of the active medium and the fundamental modes (solid symbols) for cavities 2, 4, 6, 8 and 10. (b)
and (c) averaged spectral and spatial detuning factor simulated for modes O1 and O2 in cavities 2, 4 and 8.
Figure 8. (a) Figure of Merit of the Purcell effect as a function of the temperature. (b) simulated spatial and spectral
detuning factor simulated for cavity 2(-) at different temperaures. (c) and (d) Purcell Factor and PL decay times of the
modes O1 and O2 for the cavity 2(-).
The simulations are compared with the experimental data acquired in the cavity 2(-). Figures 7
(a) and (b) show the calculated temperature evolution of Fp* and the simulation for the factor
<> at different temperatures. Figure 7(c) stands for the experimental measurement of the PL
decay times as a function of the temperature for the optical modes O1 (in blue) and O2 (in red) to
be compared with the exciton recombination times for bare QWRs (family P1). From these results
the evolution of the PF can be determined [Fig. 7(d)]. The signature of exciton localization is
observed, consisting of an almost constant exciton lifetime below 50 K. Focusing on the Purcell
factor, we can identify two different behaviours labelled as Region I and Region II in Fig. 8(d).
Region I, from very low temperatures to more or less 100 K, is the temperature range where the
7-9
Purcell factor suffers minor variations, while it clearly decreases in Region II (above 150 K). In the
temperature range between both regions the Purcell effect can not be clearly determined. This is
attributed to the finite number of QWRs that can produce uncertainty in the lifetime
measurements due to local differences in the contribution of the non-radiative channels.
Neglecting the influence of such data, the results reproduce qualitatively at Region II the tendency
expected by Fp*, while in Region I the smooth variation of the Purcell factor is attributed to the
reduction of predicted in the simulations [Fig. 8(d)].
As aforementioned, we must estimate the effects of the localized excitons for a more accurate
determination of the Purcell factor. For this purpose, the localization centres can be considered as
QWRs whose exciton wavefunction is limited by the localization potential (i.e. the QWR fluctuation
size). Having into account the relatively high localization energies obtained in Chapter 5, the
localization centres could be considered like QDs, being the difference with respect to QWRs the
spatial detuning factor which is determined by means of Eq. 8 in Chapter 6. Figure 9 shows the
averaged value of the Purcell factor, at the cavity 2 simulated for three different active media. The
first case, black squares, represents the Purcell Factor considering 500 nm length QWRs. The result
for QWRs is compared with the simulation for active media composed just by localized excitons
(without QWRs) for two densities of localization centres, 4 m-1 (in green) and 1 m-1 (in blue).
Since localization centers are approached by point-like emitters (similar to QDs) their size is limited
to the pixel of the FDTD simulation of the optical mode. The number of emitters embedded in the
cavity defect is determined by the linear density of localization centers which therefore become a
critical parameter for obtaining a good average of the spectral and spatial detuning factor.
Considering the cavity defect geometry, in the active medium with a density of 1 m-1 we would
find approximately 80 emitters embedded into the cavity defect and close to 300 in the active
medium with a deffect density of 4 m-1. This difference leads to larger values of the PF for the
active medium with a larger density. The density of an active medium composed by 500 nm length
QWRs could be compared to the case of a defecte density of about 2 m-1, considering the QWR
dimensions. Indeed the corresponding cavity support an average of 156 emitters. In spite of this
fact, we obtain larger values of the Purcell factor for the QWRs than for the sample with 4 m-1 of
localization centres. This suggests a larger <> factor for more extended emitters, as can be
deduced through the theoretical expresions of section 3 in Chapter 6. Comparing with the
experimental data obtained in cavity 2, the results fits very well with the simulation performed for
500 nm length QWRs around 80 K, but in contrast the PF approaches to the values simulated for
localized excitons at lower temperatures. This means that the effective density of localization
centres increases when reducing the lattice temperature, as occurs in bare QWRs. In all the
studied cavities, the introduction of localized excitons is required to explain the experimental
Purcell factor at 10 K, since as we showed in Chapter 5, the recombination dynamics is dominated
by localized states at this temperature.
7-10
Figure 9. Temperature evolutuion of the experimental and simulated Purcell factor. The green and blue lines
corresponds to a simulation of an active media just composed by localization centers while the black lines accounts for
an active media composed with 500 nm length QWRs without localization effects.
From measurements performed in different cavities we should outline the observation of
nearly constant or even a soft reduction of the PF at lower temperatures. This fact is attributed to
the lower tuning probability between the active medium and the cavity resonances. As we wanted
to show, the expected increase of the Figure of Merit at very low temperatures is partially
compensated by the reduction of <> and the localization effects. Both contributions are
necessary to obtain a better quantitative agreement with the experimental PF measurements at
Region I. Above 150K (Region II), where it is not expected neither the contribution of localized
exactions nor important variations in the factor <>, the Purcell factor evolution simply follows
the behavior of Fp. In between, we avoid to explain the temperature range around 120-140K, since
we can not simulate properly† the contribution of non-radiative recombination mechanisms.
Looking for more experimental information in this temperature range we have studied the
lasing threshold evolution with temperature, because this magnitude is directly related to the
number of emitters coupled to the lasing mode. The study is carried out for cavities 4, 6 and 8
whose optical modes are located at different wavelengths. As aforementioned, the cavity 4 is
tuned with the maximum of P1 family (O1 emitting at 1494 nm and O2 emitting at 1476 nm). The
optical modes O1 and O2 in cavity 6 occur at 1479 and 1460 nm respectively, whereas the modes
of cavity 8 are located close to the maximum of P2, i.e. O1 at 1465 nm and O2 at 1448 nm. As
expected, the PF evolution of these cavities at Region II is characterized by a strong reduction with
increasing temperature (Figs. 10(a)-(c)). We also find the expected behavior in Region I, but in this
case we can observe certain differences between cavities. As in cavity 2, in cavity 8 the PF suffers a
slight decrease with increasing temperature, which was attributed to the fact that <> at 80 K
would not compensate the broadening of the QWRs emission lines. In contrast, in the case of
cavities 4 and 6 the PF is slightly increasing with temperature, in spite of the fact that simulations
provide the same for the fundamental modes of the three cavities. This different
phenomenology at Region I must be attributed to random effects, as local variations in the
number of coupled QWRs or in the
†
See Fig. 8(c) the bare QWR life time decreases above 130K while the cavity mode decay time is still rising up until 150K. This fact can be
explained by different reasons: a different contribution the carrier loss mechanisms, a different density of localized excitons with respect to the
ensemble average or by an enhancement of the spatial detuning factor due to delocalization of effects related with the unipolar escape of
electrons. Simulating each of them would require different considerations dificulting the simulations without offering valuable information.
7-11
Figure 10. (a), (b) and (c) Purcell factor versus temperature in cavities 4, 6, and 8 respectively. (d), (e), and (f) lasing
threshold of the same cavities.
density of localization centres. Let compare the evolution of the Purcell factor in region I with the
evolution of the lasing threshold at the three cavities, as don in Figs 9 (d), (e) and (f). Both
magnitudes depend on the number of QWRs coupled into the cavity modes. In cavity 8 the lasing
threshold is close to 250W at 10 K and reaches a value in the range of 400-500 W above 40K.
This fact could be explained by a worse coupling produced due to increase the temperature rising.
In contrast, cavity 6 present laser operation above 700 W at 10K and the lasing threshold of this
cavity is reduced until ≈250 W above 60 K. The cavity 4 presents a laser threshold considerably
larger (Pth≈ 800W at 80 K), due to the fact that its fundamental mode is located in a region with
7-12
a minor content of QWRs and it is practically double at 60 K (below this temperature laser
operation is not observed.
In summary, the a reduction of the lasing power threshold is clearly correlated with an
enhancement of the Purcell factor in Region I. In this way, there is a better coupling between the
active medium and the optical mode in cavities 4 and 6, whereas it occurs at 10 K in the case of
cavity 8. Given that the active medium is an ensemble of quantum emitters it is quite reasonable
to attribute the corresponding Purcell factor enhancement to the contribution of QWRs, which are
uncoupled at low temperatures, but become coupled when increasing temperature due to the
homogeneous broadening of the emission line, the peak energy shift and a reduction of
localization effects.
4. Conclusions
We have obtained PCMs with embedded self-assembled QWRs with high quality factors (in some
cases close to 20000). The study of the PF in these cavities presents some difficulties related to the
fact that it is observed laser operation of fundamental modes emitting at the wavelength range of
interest. The evolution of the integrated PL intensity (and other magnitudes) is scalable to the
lasing threshold value, typically around 8 W of irradiated power under pulsed resonant
excitation. Using the threshold power as a reference we can distinguish three different power
regimes (below 0.2Pth, around 0.5 Pth and above 0.8Pth) depending on the contribution of the
optical amplification. The contribution of the stimulated emission to the emission of the
fundamental modes leads to an overestimation of the PF. In this way, the most suitable estimation
of the PF is obtained by the extrapolation of the experimental data to zero excitation power. As a
result, it is recovered the characteristic value of the PF, i.e. 1.2 for L7cavities with the linear defect
oriented in the direction [1-10]. Once we have stablished the procedure for measuring the PF in
high Q samples, we have studied their temperature evolution. The cavities studied here do not
present a clear enhancement of the SE by reducing the sample temperature. This is explained by
the reduction of the spectral detuning and a reduction of the number of emitters coupled to the
optical mode due to the narrowing of the emitter linewidth at very low temperatures. This
hypothesis is supported by simultaneous determination of the lasing threshold (again as a function
of the temperature) that results is an additional evidence of the coupling between the active
medium and the lasing mode.
References.
L.J. Martinez, B. Alen, I. Prieto, D. Fuster, L. Gonzalez, Y. Gonzalez, M.L. Dotor, P.A. Postigo, “Room temperature continuous wave
operation in a photonic crystal microcavity laser with a single laser of InAs/InP self-assembled quantum wires,” Opt. Express 17, 14993
(2009).
2. A.L. Schawlow and C.H. Townes, “Infrared and optical masers”, Phys. Rev. 112, 1940 (1958).
3. K. Nozaki, S. Kita, T. Baba. “Room temperature and continuous wave operation and controlled spontaneous emission in ultra small
photonic crystal nanolaser” Opt. Exp. 15 7506-7514 (2007).
4. D. Fuster, J. Martinez-Pastor, L. Gonzalez, Y. Gonzalez, “Exciton recombination dynamics in InAs/InP self-assembled quantum wires,”
Phys. Rev. B, 71, 205329 (2005).
5. Submited to Journal of Applied Physics.
6. Submited to Optics Express.
7. L.J. Martínez, I. Prieto, B. Alen, P.A. Postigo, “Fabrication of high quality factor photonic crystal microcavities in InAsP/InP membranes
combining reactive ion beam etching and reactive etching” J. Vac. Sci and Tec. B 27, 1801-1804 (2009).
8. Z.M. Fang, K.Y. Ma, D.H. Jaw, R.M. Cohen and G.B. Stringfellow, “Potoluminescence of InAb, InAs, and InAsSb grown by
organometallic vapor phase epitaxy” J. Appl. Phys. 67, 7034 (1990).
P. Michler “Single Quantum Dots: fundamentals, applications and new concepts”, 1st ed, Springer, (2003).
9. M. Lomascolo, P. Ciccarese, R. Cingolani, R. Rinaldi and F.K. Reinhart, “Free versus localized exciton in GaAs V-shaped quantum wires”
J. Appl. Phys. 83 302 (1998).
10. Y.I. Mazur, V.G. Dorogan, O. Bierwagen, G.G. Tarasov, E.A. De Cuir, S. Noda, Z.Y. Zhuchenko, M.O. Manasrhe, W.T. Masselink, G. J.
Salamo, “Spectroscopy of shalow InAs/InP quantum wire nanostructures” Nanotechnology 20, 065401 (2009).
1.
7-13
Chapter 8. General conclusions and Outlook
We have combined continuous wave photoluminescence (PL) and time resolved
photoluminescence (TRPL) to the study two different kinds of samples: novel
InAs/GaAs site controlled quantum dots (SCQDs) and photonic crystal microcavities
(PCM) containing self-assembled quantum wires (QWRs).
Following the scheme shown in Chapter 1, we have performed a fundamental
analysis in order to determine the optical properties and capabilities of the SCQDs. In
Chapter 4 it is described the method for fabrication SCQDs with high optical quality
using local oxidation nanolithography and molecular beam epitaxial re-growth. The
nucleation selectivity of the method was demonstrated in previous works by
fabricating a QD square matrix with 2 m pitch period. The samples studied have been
proved to reproduce isolated QDs or QD pairs (depending on the oxide motif) with
great reproducibility. We have also studied the effects of the nano-hole evolution
during the growth process by comparison of samples whose buffer layer has been
deposited by ALMBE and MBE respectively. This way we showed the influence of the
buffer layer on the evolution of this kind of nanostructures during the growth. Finally,
we have performed preliminary measurements of the emission of photonic crystal
microcavities embedding SCQDs. In the future we expect to demonstrate Pucell effect
in this kind of devices.
Previously to the study of the PCM (in the Chapter 5) we have studied the decay
time of the active medium as a function of temperature which is presented and fitted
to a model accounting for the exciton localization effects. For this purpose we have
performed a macro-PL study in a sample containing self-assembled InAs QWRs
embedded into a /2 thick InP layer (the central emission of the QWRs takes place at
around 1450 nm at 12 K) deposited onto an InGaAs sacrificial film. Two electron loss
mechanisms have been identified as responsible of the PL quenching at intermediate
and high temperatures both under low excitation power and resonant conditions. The
TRPL results gives us further information about exciton dynamics, which is determined
by excitons localized in QWR size fluctuations at the lowest temperatures. When
increasing the lattice temperature the radiative exciton lifetime increases according to
the expected thermal equilibrium between free and localized exciton populations. The
localization energies obtained here suggest a major contribution of the wire width
fluctuations to the exciton localization dominating the exciton recombination at low
temperatures. When increasing temperature the free excitons become important due
to the thermal ionization of the localized excitons. Summarizing all this results, some
parameters of the excitonic dynamics in InAs/InP QWRs have been obtained, some of
1-1
them key for the application of this system as active medium in photonic devices
working at the telecommunication windows.
After that we have studied the spontaneous emission rate of the L7-type PCM with
embedded self-assembled QWRs. In particular, our attention has been focused on
modes O1 and O2 with experimental Q factors close to 6500 and 3500, respectively.
The spontaneous emission rate has been measured by TRPL observing a certain
lifetime reduction with respect to the case of bare QWRs outside the cavities. The ratio
between both decays give us the experimental Purcell factor that, in average, results
around 1-2 with a clear dependence on the polarization match between the emission
of the optical mode and the spontaneous polarization of the active medium. These
results have been extensively discussed after grouping the parameters contained into
the Purcell factor expression into four main factors: the Figure of Merit (Fp), the
polarization mismatch (), the spectral detuning factor () and the spatial detuning
factor (). As a result we can conclude that the two last factors depend on the QWR
content which would be different at each point of the sample. The possible effects of
such finite ensemble of QWRs filling the PhC micro-cavity has been determined by
means of an algorithm simulating random content of QWRs embedded into a both
kinds of L7-type cavities. Therefore the moderate Purcell factor measured is mainly
attributed to the homogeneous broadening of the active medium, with a notable
reduction due to the spectral and spatial detuning factors, while the great oscillation of
the experimental data is attributed to the finite number of QWRs embedded into the
cavity.
During the determination of the Purcell factor we realize that, the study of the PF in
cavities with higher Q than the shown in chapter 6, usually presents some difficulties
related to the fact that in such cavities it is allowed laser operation of fundamental
modes. Indeed, using the threshold power as a reference we can distinguish three
different power regimes (below 0.2Pth, around 0.5Pth and above 0.8Pth) depending
on the contribution of the optical amplification. The contribution of the stimulated
emission to the emission of the fundamental modes leads to an overestimation of the
PF. As a result, the more suitable estimation of the PF is obtained by extrapolation of
the experimental data to zero excitation power. In this conditions it is recovered the
characteristic value of the PF, i.e. 1.2 for L7cavities with the linear defect oriented in
the direction [1-10]. Once we have established the procedure for measuring the PF in
high Q samples, we have studied their temperature evolution. The cavities studied
here do not present a clear enhancement of the SE by reducing the sample
temperature. This is explained by the reduction of the spectral detuning and a
reduction of the number of coupled emitters due to the narrowing of the emitter
linewidth at very low temperatures. This hypothesis is supported the simultaneous
determination of the lasing threshold (again as a function of the temperature) that
results is an additional evidence of the coupling between the active media and the
lasing mode.
Nowadays, the author of this PhD is working at the University of Valencia, where
further experiments are proposed to directly measure PCM containing isolated QWRs.
En este trabajo se ha combinado la fotoluminiscencia de en excitación en onda
continua (PL del inglés photoluminescence) y fotoluminiscencia resuelta en el tiempo
(TRPL del inglés time resolved photoluminescence) para el estudio de dos tipos
1-2
diferentes de muestras: puntos cuánticos (QDs del inglés quantum dots) de InAs
crecidos sobre substratos pregrabados de GaAs y microcavidades de cristal fotónico
(PCM del inglés photonic cristal microcavities) con hilos cuánticos auto-ensamblados (
QWRs del inglés quantum wires).
Siguiendo el esquema mostrado en el capítulo 1, hemos realizado un estudio
básico de las propiedades ópticas de los SCQDs (del inglés site controlled quantum
dots). En el capítulo 4 se describe el método de fabricación de SCQDs mediante nanolitografía por oxidación local combinada con crecimiento MBE (del inglés molecular
beam epitaxy). En las muestras estudiadas se ha demostrado reproducibilidad en el
crecimiento tanto de los puntos cuánticos aislados como de pares de puntos cuánticos,
dependiendo la aparición uno u otro tipo de estructura de la forma del motivo inicial.
También hemos estudiado los efectos de la evolución del grabado (los nano-agujeros)
durante el proceso de crecimiento. Pare ellos se han comparado dos muestras cuya
capa tampón de GaAs ha sido depositada por ALMBE (del inglés atomic layer molecular
beam epitaxy) y MBE respectivamente. Por último, se han realizado medidas
preliminares de la emisión de microcavidades de cristal fotónico conteniendo
SCQDs. En el futuro esperamos poder demostrar el efecto Pucell en este tipo de
dispositivos.
Previamente al estudio de las PCM se ha determinado el tiempo de vida del medio
activo en función de la temperatura. Para esto se ha utilizado una muestra con QWRs
autoensamblados de InAs/InP crecidos en el centro de un film de InP con un grosor de
/2. El film de InP se depositó sobre una capa InGaAs de 700 nm que es usada como
capa de sacrificio en la fabricación de cavidades. Como resultado se identifican dos
mecanismos diferentes de pérdidas de portadores que explican la extinción PL a
temperaturas intermedias y altas. Los resultados de TRPL nos dan información sobre la
dinámica excitonica, que dependiendo de la temperatura puede estar determinada
por los efectos de localización debidos las fluctuaciones de anchura en los hilos. Esto
es, al aumentar la temperatura la contribución de los excitones libres se vuelve más
importante, debido a la ionización de los excitones localizados.
Después de conocer la emisión espontanea del medio activo nos centramos en el
estudio la las cavidades, concretamente en las de tipo L7. Nos centramos sobretodo en
los modos O1 y O2 cuyos factores Q experimentales están de 6500 y 3500,
respectivamente. El tiempo de vida de la emisión espontánea de dichos modos se ha
medido por TRPL y se ha comparado con los resultados del Capitulo 5 para el medio
activo fuera de la cavidad. De la relación entre ambos tiempos de vida podemos
obtener valores experimentales del factor de Purcell que, en promedio, varían
alrededor de 1-2 con una clara dependencia de la polarización la emisión del modo en
relación a la polarización espontánea del medio activo. Estos resultados han sido
ampliamente discutidos después de agrupar los parámetros contenidos en la expresión
de factor de Purcell en cuatro factores principales: la figura de mérito (FP), el factor de
polarización (), el factor espectral (), el factor espacial (). Como resultado se puede
concluir que los dos últimos factores dependen de los QWRs contenidos en el defecto,
que serían diferentes en cada punto de la muestra. Los posibles efectos del debidos al
tamaño finito de dichos QWRs dentro del defecto de la PCM se ha determinado por
medio de un algoritmo que simula el contenido aleatorio de QWRs adaptado a la
geometría del los dispositivos y promedia el PF resultante. Gracias a esta simulación
los podemos atribuir los valores moderados del factor de Purcell por una parte a la
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anchura homogénea del medio activo que por otro lado estaría acompañada de una
reducción de los factores espacial y espectral.
Durante la determinación del factor de Purcell nos dimos cuenta de que el estudio
del PF en las cavidades de alto Q suele presentar algunas dificultades relacionadas con
el hecho de que este tipo de cavidades presentan emisión laser en sus modos
fundamentales. De hecho, utilizando la umbral del modo laser como referencia,
podemos distinguir tres regímenes de potencia diferentes (por debajo de 0.2Pth,
alrededor de 0.5Pth y por encima 0.8Pth). La contribución de la emisión estimulada a
la emisión de los modos fundamentales produce a una sobreestimación del PF. En
efecto, la estimación más adecuada de la PF se obtiene mediante la extrapolación de
los datos experimentales a la potencia de excitación cero. Como resultado, se recupera
el valor característico de la PF, es decir, 1.2 para cavidades L7 con el defecto lineal
orientado en la dirección [1-10]. Una vez establecido el procedimiento para la
medición de la PF en muestras de alta Q, se ha estudiado su evolución de la
temperatura. Las cavidades estudiadas aquí no representan una mejora clara de la SE
mediante la reducción de la temperatura de la muestra. Esto se explica por la
reducción del factor espectral y del número de emisores acoplados cuando se estrecha
de la anchura de la línea emisión a muy bajas temperaturas. Esta hipótesis esta de
acuerdo con la evolución del umbral láser en función de la temperatura.
A día de hoy, el autor de esta tesis doctoral está trabajando en la Universidad de
Valencia, donde se están proponiendo nuevos experimentos para medir directamente
PCMs conteniendo QWRs aislados.
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