k - Dipartimento di Fisica

Metodi diffrattivi per studi cristallografici di volume e di superficie
Le informazioni cristallografiche possono essere ottenute con esperimenti di
scattering effettuati con opportune particelle sonda (elettroni, neutroni, fotoni X).
L’onda diffusa (o scatterata) è descritta da:
Ψj = Ψ0 eik·R /R f j (k0, k) ei(k-ko)·Rj
dove
fj (k0, k)
fattore di scattering atomico
ei(k-ko)·Rj
sfasamento tra centri scatteratori
Se Rj=(n1a1,n2a2)+rj la funzione dell’onda scatterata totale si partiziona:
Ψ= ∑jΨj ∝ ∑j fj (k0,k) e
i(k-ko)·Rj
Fattore di forma
fj
=F⋅G
Fattore di struttura
dipendente dalla posizione delle
singole celle unitarie
Scattering factor atomico dipendente dall’interazione sonda-atomo
r r
F∆k = ∑ f j exp(i∆k ⋅ rj )
j
fattore di forma dipendente dalla posizione
degli atomi all’interno della cella unitaria
Fattore di
scattering fj
Il fattore di scattering dipende dall’energia degli elettroni. Ad alta
energia (1100 eV) la sezione d’urto è tutta in avanti (0°). Ad energie
minori (300-500 eV) diventa importante il picco di backscattering
(180°), mentre a bassa energia (<100 eV) si hanno fenomeni di
interferenza che portano intensità anche a grandi angoli dalla
traiettoria incidente (picchi a 60° e 120° in figura).
Fattore di Struttura G in 2D
Esempio: caso 2 Dim.
G = ∑n = 0 e
M1
1
posto Rj=(n1a1,n2a2) e supponendo M1 x M2 celle si ottiene:
r r
in1a1 ( k − k 0 )
∑
M2
n2 = 0
e
r r
in2 a2 ( k − k 0 )
L’intensità nei singoli canali di diffrazione vale quindi
r r 2
r r 2
1
1
[sin( M 1a1 (k − k0 ))] [sin( M 2 a2 (k − k0 ))]
2
2
2
2
2
G =
⇒
M
M
1
2
r r 2
r r 2
1
1
[sin( a1 (k − k0 )]
[sin( a2 (k − k0 ))]
2
2
che ha massimi per a·(k-ko)=nπ
π, ossia in corrispondenza dei vettori del
reticolo reciproco G=(2π
πn/a1, 2π
πm/a2).
Nella terza dimensione i picchi si estendono indefinitamente (barre o rods).
La forma angoalre dei picchi di diffrazione è Lorenziana. Sono stretti se M1 e M2 >>1.
M1 ed M2 sono limitati da:
lunghezza di coerenza L, che dipende da come si prepara la superficie. Essa può
eccedere 1000 Å per monocristalli o essere limitata dalla presenza di dislocazioni.
Lunghezza di trasferimento W, dipendente dall’apparato sperimentale
Definizione: spazio reciproco e zona di Brillouin
Caso 3D: una volta definiti i vettori a1, a2, a3 nello spazio diretto
si possono definire i vettori b1, b2, b3
a 3 × a1
a1 × a 2
a2 × a23
b2 = 2π
b 3 = 2π
b1 = 2π
a1 ⋅ a 2 × a 3
a1 ⋅ a 2 × a 3
a1 ⋅a2 × a3
b i ⋅ a j = 2πδ ij
gν 1ν 2ν 3 = ν 1b1 +ν 2b 2 +ν 3b 3
con v1, v2, v3 numeri interi
Dimensione di R [L]
Dimensione di g [1/L]
La zona di Brillouin è definita come la cella di Wigner-Seitz cell
nello spazio reciproco
Zone di Brillouin in 3D
Costruzione di Ewald
Le particelle sonda rimbalzano elasticamente conservando energia
(ħk)2 e quantità di moto (ħk)
k = (k )
r' r r
k = k + ghkl
2
' 2
Conservazione dell’energia
Conservazione della quantità di moto
Dove ghkl è un vettore del reticolo reciproco, (2 indici per problemi
in 2 Dim, 3 per problemi in 3 Dim). Per avere diffrazione k’- k deve
essere confrontabile o maggiore di g.
Ewald Sphere Construction in 3D Reciprocal Space
2π
2mE 

k=
=
2 

λ
h
12
1) A vector k is drawn terminating at the origin of the reciprocal space
2) A sphere of radius k is constructed about the beginning of k
3) For any point at which the sphere passes through a reciprocal
lattice point, a line to this point from the center of the sphere
represents a diffracted beam k’
4) Notice the reciprocal lattice vector ghkl
Diffraction Techniques in 3D Space
There is a variety of techniques based on the diffraction
process suitable for volume structural determination:
Electrons IN/Electrons OUT
TED (Transmission High Energy Electron Diffraction)
Photons IN/Photons OUT
XRD (X-Ray Diffraction)
Neutrons IN/Neutrons OUT
…. (Elastic Neutron Scattering)
……Photon IN/Electron OUT : EXAFS (see later …)
The best probe particles to study 3D crystallography by diffraction are
neutrons and X-rays, for which the single scattering approximation
holds. Inverting the scattered intensities allows to reconstruct the
position of the scatterers in direct space.
Electrons are suited for microscopes since they can be easily focused.
Diffraction in 2D
k = (k )
r 2 r 2 r' 2 r' 2
k|| + k⊥ = k || + k ⊥
2
' 2
r
r r
k '|| = k|| + g hk
Conservation of Energy
Conservation of Momentum
r
k⊥ is
not conserved since the translational
symmetry normal to the surface is now broken
The indexing of the diffracted beams is, by convention, referenced to
the substrate real and reciprocal net.
If the selvedge or adsorbate structures have larger periodicities, the
surface reciprocal net is smaller than that of the substrate alone. The
extra reciprocal net points and associated diffracted beams are
denoted by fractional rather than integral indices.
Ewald Sphere Construction in 2D Reciprocal Space
Notice that the reciprocal lattice
is now replaced by infinite
reciprocal lattice rods
perpendicular to the surface and
passing through the reciprocal
net points
1) At surfaces 2D translational symmetry holds thereby only the
wave vector parallel to the surface is conserved with the addition of
a reciprocal net vector
2) The dashed scattered wave vectors propagate into the solid and
are not observable
2D Crystallography: Surface Reciprocal Lattice
The reciprocal net vectors
c1* and c2* of the surface
mesh are defined as
c1 • c2* = c2 • c1* = 0
c1 • c1* = c2 • c2* = 2π (or 1)
The reciprocal net points of
a diperiodic net may be
thought of (in 3D space) as rods.
The rods are infinite in extent
and normal to the surface plane
where they pass through the
3D reciprocal net points.
Zone di Brillouin 2D
bcc
fcc
Diffraction Techniques in 2D Space
Techniques based on the diffraction process suitable
for surface structural determination:
Electrons IN/Electrons OUT
LEED (Low Energy Electron Diffraction)
RHEED (Reflected High Energy Electron Diffraction)
Photons IN/Photons OUT
GIXD (Grazing Incidence X-Ray Diffraction)
Atoms IN/Atoms OUT
ABS (Atom Beam Scattering)
Electrons are the least expensive tool for diffraction
experiments, but multiple scattering is important which
makes the inversion of the diffraction intensities complicated
2D-Superstructures
and nomenclature
Superstructures
correspond to the
formation of larger
lattices due to surface
reconstruction or to
the presence of an
adsorbate layer at the
surface.
Examples of LEED Patterns
(a) Si(111)
(b) GaAs(110)
(c) Sr2CuO2Cl2
20 eV<Ei<500 eV
Notice that the LEED spots
span a variety of relative
intensities (see below)
LEED Process
1) The incident wave is exponentially attenuated while the electron
propagates in the solid. Inelastic mean free path depends on electron
energy and is smallest around 100 eV.
2) Since atom-electron scattering cross sections can be very large
(≈ 1 Å2, i.e. 1010 times larger than in X-ray diffraction), multiple
scattering must be included. This means that each incident electron is
treated as a superposition of the primary wave and the scattered
waves.
3) The atomic scattering cross section (fj) involves a phase shift (also
dependent on k) and is thus complex.
4) The diffraction intensity is modulated by the Bragg conditions in
vertically to the surface.
LEED
Since the penetration depth of the primary electrons
is finite, one has thickness-modulated vertical
reciprocal rods instead of infinite uniform-thickness
reciprocal rods within the Ewald sphere construction
Ewald sphere construction
in 2D space
LEED dinamico: l’intensità nei singoli canali di
diffrazione dipende dall’interferenza. Invertendo tali
intensità si può risalire alla posizione dei nuclei
LEED
Esempio di curve intensità energia (I-V)
La modulazione dell’intensità e
la posizione dei picchi sono
date grossolanamente dalle
condizioni di interferenza
positiva tra i piani
cristallografici. I picchi sono
spostati dalla posizione attesa
a causa del potenziale interno
al cristallo (che modifica
l’energia degli elettroni) e
dallo scattering multiplo.
Quest’ultimo può anche
indurre ulteriori strutture.
LEED
Examples of spot profiles as a function of the primary
beam energy
Spot profile
analysis reveals
the inadequacy
of the simple
scattering
approach.
Multiple
scattering plays
a major role and
it must be
included in the
theory.
Esempio: film sottile Fe3O4 cresciuto epitassialmente su Pt(111)
Basic Instrumental Components of a LEED Experiment
•UHV Apparatus
•Sample
•Sample Preparation Facilities
•LEED Optics
•Electron Gun
•Grids & Fluorescent Screen
•CCD Camera/Computer
Schematic of a Three-Grid
LEED Optics
Usually the electron beam
at a selected kinetic
energy hits the surface
along the normal direction
Surface sensitivity in LEED
is achieved by combining
the low IMFP (≈ 5 Å) with
the high backscattering
probability (ion core cross
section as high as 1 Å2)
LEED Optics
In this case one
UHV flange is just
needed
Recording LEED with a CCD Camera
Effetto energia incidente
90 eV
Effetto energia incidente
140 eV
Effetto sovrastrutture su G
Effetto sovrastrutture su G
Effetto sovrastrutture su G
RHEED: Reflection high energy electron diffraction
The primary beam hits the surface and the diffracted beam emerges at
very grazing incidence (<5°). This, combined with the high forward
scattering probability and notwithstanding the high primary energy (30÷100
keV), ensures a high surface sensitivity.
This primary-beam/sample/screen geometry is typically preferred during
MBE growth, i.e. when the sample must face the Knudsen cells.
The Ewald sphere is now very large relative to the spacing of the reciprocal
net rods (see LEED) and cuts these rods at grazing angles.
The combination of the very grazing angle of intersection of the Ewald
sphere with the broadened reciprocal net rods (due to the real quality of
surface order and the instrumental transfer width) leads to streaking of
the diffraction spots.
Diffracted Electron Beams
Edge of Ewald sphere
Primary Electron Beam
k
Sample
φ
Grid
Fluorescent Screen
RHEED
Ewald
sphere
Reciprocal
Lattice
Rods
Reciprocal
Lattice
Rods
Allowed
Reciprocal
Lattice Vectors
First
Order
Second
Order
Perfectly flat
surface Reciprocal
rods have no width
Ewald
sphere
Reciprocal
lattice
points
First
Order
Second
Order
Surface with
monolayer roughness.
Broadened rods.
First
Order
Second
Order
Surface with large
roughness.
Transmission features.
RHEED: Ideal Pattern
Fractional-order spots in some
azimuths indicate surface
reconstruction
The spacing gives the surface
lattice constant
Splitting of the beams indicates
a faceted surface. The
separation gives the surface tilt.
This distance gives the
angle of incidence
Image of the incident
electron beam
RHEED Apparatus
RHEED and Ewald Sphere
RHEED
Scattering and Diffraction Geometry
(01 )
(00)
L
s
θ
(01)
RHEED Patterns
RHEED oscillations
How RHEED oscillations are
applied to monitor MBE growth
processes.
The (0,0) beam is recorded for
destructive interference
conditions for scattering from the
first layer and from the added
layer. The scattered intensity is
lowest when the added layer has
half a monolayer coverage.
Monitoring the beam intensity vs
layer growth time allows therefore
to determine when the layer is
compete.
RHEED Oscillations
The intensity of the spots oscillates as layers are deposited if the
surface becomes alternately rough and smooth.
The period of the oscillation gives the
growth rate and can be used to
determine composition
in some cases
Intensity
Envelope shows the
surface roughness
evolution.
Surface smoothness recovers
if growth is stopped
one monolayer
Time
High Resolution in k space
High k-resolution is needed to observe large objects since scattering
must be coherent over them. Large objects give rise to small
momentum transfers --- small angle scattering
Easily achieved with neutrons and TEAS for which the sources are
far away from the target, more difficult with electrons which may
easily be deflected by stray fields .
Limiti nella risoluzione: Lunghezza di trasferimento
La Lunghezza di trasferimento definisce la zona su cui l’onda incidente è
colpisce il campione con la stessa fase:
Definendo il momento trasferito
q||=kf||-ki ||
q||= k |sinθf-sin θi|
posto w=2π/∆q|| con ∆q|| incertezza su q||
(larghezza macchie di diffrazione)
∆q||=(∂q|| /∂E) ∆E+(∂q||/∂θ) ∆θ
si ottiene:
wθ=2π/(kcosθf ∆θf)
wE=2π 2 k/(∆E |sinθf -sinθi|)
Per la diffrazione di elettroni lenti (LEED)
k≅ 0.51 √E con E ≅ 100 eV e ∆θf ≅10-2 rad wθ ≅ 100 Å<< wE
la nitidezza dell’immagine di diffrazione è limitata dalla divergenza angolare del
fascio di elettroni (l’onda non è piana, ma sferica)
Diffraction Intensities for particular configurations
Spot Separation
Halfwidth
(relative to 2π/a)
One atom
Two atoms
(distance a)
∞
∞
1
1/2
1
1/N
N atoms in a row
(regular distance a)
Several (M) groups of N atoms
each (regularly spaced)
[distance of group centers
(N+1/2)a]
1/[N+(1/2)]
Several groups of varying
size (arranged as in (d))
1
N atoms randomly
1
distributed over 2N regular
sites
1/{M[N+(1/2)]}
dependening on
spot size and mixture
1/(2N)
LEED
Stepped Surfaces: Ewald Sphere Construction
LEED
How surface defects
affect the reciprocal
lattice rods and the
resultant LEED spot
profile
High Resolution LEED: SPA-LEED
Instrumental Apparatus for Spot profile Analysis
LEED vs SPA LEED
Reciprocal space
E gun
crystal
shield
window Ewald sphere
The deflection is achieved
via an octupole field.
Since the reciprocal space is
scanned changing also the
angle of incidence the
Ewald construction implies
a twice as large “modified
Ewals sphere
Reciprocal space
modified Ewald sphere
Standard
sphere
SPA-LEED