Resonant tunneling through Gaussian superlattices

REVISTA MEXICANA DE FÍSICA 48 SUPLEMENTO 3, 40–42
DICIEMBRE 2002
Resonant tunneling through Gaussian superlattices
G.A. Lara
Dpto. de Fı́sica, Universidad de Antofagasta
Casilla 170 Antofagasta
e-mail: [email protected]
P. Orellana
Dpto. de Fı́sica, Universidad Católica del Norte
Casilla 180 Antofagasta
e-mail: [email protected]
Recibido el 18 de enero del 2001; aceptado el 8 de junio del 2001
We study the tunneling of electrons in semiconductor superlattices (SL) where the width of the barrier is modulated by a Gaussian function.
The system is modelated using the effective mass approximation for the electrons and it is solved with the transfer-matrix method. We found,
as in the previous models of Gaussian SL, that the probability of transmission is almost equal to unity in the miniband. In comparison with
previous designs of SLs, our system shows experimental advantages.
Keywords: Quantum wells, tunel effect, superlattice.
Estudiamos el tunelamiento de electrones en super-redes semiconductoras donde el ancho de las barreras es modulado por una función
gaussiana. Modelamos el sistema usando la aproximación de masa efectiva y los resolvemos usando el método de matriz de transferencia.
Como en los diseños anteriores de super-redes gaussianas encontramos que la probabilidad de transmisión es muy próxima a la unidad en
toda la minibanda. En comparación con los modelos anteriores de super-redes gaussianas, nuestro sistema presenta ventajas experimentales.
Descriptores: Pozos cuánticos, efecto túnel, super-redes.
PACS: 68.65.+g;73.20.x;73.61.-r
Introduction
Model
The analysis of resonant tunneling through semiconductor
superlattices (SLs) has a lot of interest whether from fundamental point of view as for its applications in microelectronic
devices. Since the pioneering work of Esaki and Tsu [1], the
transport properties of those structures have been subjected
to intense experimental and theoretical investigations [2–5].
Due to advances in manufacturing techniques of semiconductor heterostructures, it is possible to tailor the band structure
of SLs to the particular needs of every experiment. Recently,
Tung and Lee [6] proposed a novel SL where the heights of
the barrier and the bottom of the quantum wells are modulated by Gaussian functions. These authors found some
plateaus in the transmission characteristic where electrons are
almost unscattered. This is quite different from uniform SLs,
where the transmission probability have great oscillations in
each miniband. More recently, Gomez et al. [7] have reported
similar effects in GaAs-AlGaAs SLs where only the heights
of the barriers are modulated, whereas the width of the wells
remain constant. They found that J - V characteristic present
negative resistance with peak-to-valley ratios much greater
than for uniform SLs. This design requires minor Al concentration values in the barrier. In this context and considering
that experimentally the control on the width is better than on
the height of the barrier, we present a model where the width
of the barrier is modulated by a Gaussian function.
We consider a three-dimensional GaAs − Alx Ga1−x As SL
with z the growth direction. The width of barriers is modulated by a Gaussian function centered in the middle of the
superlattice (z = L/2). The height of the barriers are fixed
as well as the width and bottom of the wells. The number
of monolayers (ML) of the barrier centred in zb is the integer
closest to
(
µ
¶2 )
1 zb − L/2
Nb = No exp −
.
(1)
2
σ
We calculate the envelope function with the effectivemass parabolic-band aproximation
EF (z) = −
·
¸
1 dF
~2 d
+ V (z)F (z),
2 dz m(z) dz
(2)
where V (z) include the SL potential profile and bias voltage. This model is reasonable for low applied bias, below
0.3 ∼ 0.4 eV, where the intervalley tunneling can be neglected.
In the left side of the SL (z < 0) the solution of this
equation is the superposition of a incident plane wave with
amplitude I and a reflect wave of amplitude r. In the right
RESONANT TUNNELING THROUGH GAUSSIAN SUPERLATTICES
41
side of the SL (z > L) only a transmitted wave is expected,
therefore the wave function is assumed to be
½ ikz
Ie + re−ikz , for z < 0,
0
F (z) =
(3)
te−ik x ,
for z > L;
where k, k 0 are the wave vectors in the emitter region and the
collector region respectively.
We calculate the tunneling current by the usual approximation [8]. The current density is
Z
em∗ kB T
J(V ) =
N (E, V )T (E, V )dE,
(4)
2π 2 ~3
where V is the bias voltage and kB is the Boltzmann constant. The transmission probability
F IGURE 1. Shows the potential profile of Gaussian SL with 16 ML
in the wells and 11 barriers with No =10, Vo =0.15 eV, σ=18 nm, at
zero bias.
T (E, V ) = |t/I|2
is obtained using the continuity conditions to the wave function, continuity of the probability flux and the transfer-matrix
method [7]. N (E, V ) consider the states of occupation on
both sides of the SL in acordance with the Fermi distribution
function and is given by
·
¸
1 + eβ(E−µ)
N (E, V ) = ln
,
(5)
1 + eβ(E+eV −µ)
F IGURE 2.
where µ is the chemical potential.
Results
The tunneling probability through such potential profile is illustrated in Fig. 2 for a effective-mass m∗ = 0.067 me in the
wells and m∗ = 0.082 me in the barriers. For certain range
of energies below the barrier height the particle can tunnel
almost unattenuated (solid line), in contrast with the uniform
SL with the same parameters and 6 ML width, shows a transmission probability that oscillates with the energy (dashed
line). The shape of plateau in the transmission probability
does not depend of the difference in the effective mass between wells and barriers. It is obtained similar results if the
effective mass is constant in all SL. These results are for a
SL with 70 nm approximately where the electron transport is
coherent for these materials [9].
Figure 3 display the current density for the same parameters of Fig. 2 at 77 K. Note that the J-V characteristics shows
negative differential conductivities (NDC) for both Gaussian
and uniform SLs. The details of NDC depends of the kind of
SL. Moreover, as in the previous design [6, 7], the Gaussian
SL shows a peak-to-valley ratio much greater that the uniform SL, nevertheless to difference them, in our design, the
mole concentration remains constant in barriers and wells.
This is a great advantage from the experimental point of view
at the time of growing the SL.
F IGURE 3.
In conclusion, we have studied the resonant tunneling
through a Gaussian SLs where the width of the barrier is
modulated by a gaussian function. Our design has an advantage: when growing the samples it is easier to control the
barriers widths than their heights. We hope that future experiments will confirm our results.
Acknowledgments
This work was sopported in part by Dirección de Investigación U. de Antofagasta, Proyecto Milenio ”Fı́sica de la Materia Condensada” ICM P99-135F and Cátedra Presidencial
en Ciencias.
Rev. Mex. Fı́s. 48 S3 (2002) 40–42
42
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Rev. Mex. Fı́s. 48 S3 (2002) 40–42