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Frozen Phonon

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by

Andrew Scullion

on 5 October 2015

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Transcript of Frozen Phonon

Thansk!
Thanks!
+
...
TEM contrast is complicated!
by Andrew Scullion
Classically:
N: number of electrons
z: material thickness
k: absorption coefficient
Quantum Mechanically:
: electron wavefunction
e: electron charge
V: atomic potential
E: kinetic energy
f
Plane Wave
Fast Electron Approximation
Too slow!
Integrated potential
Propagator
In 2D:
Euler's Solution:
What does this mean?
Phase Shift:
Phase shift depends on how close the electron passes to the atom
What is the propagator?
modeled based on Kirkland [1]
For elastic interactions:
Phase of k :
2
phase shift based on position (real space)
small angle approximation:
Phase shift based on frequency (angle)
Top view of InAs(110)
Side view
CTEM
A 5x7 superlattice is chosen so that the height and width are about the same
The the brightest spot is not the largest atom!
CBED
Separation between spots depends on number of units cells in the chosen supercell.
Diffraction Patterns
5x7
3x4
12x17
Multislice &
The Frozen Phonon Approximation

nm
The highest available frequency depends on the size of the supercell
Atoms are always moving!
acceptance angle: 10 mrad
STEM
collection angle: 86-143 mrad
nm
Phonons: Quantized lattice vibrations
How can we model electron-phonon interactions?
In 1D:
5x10 s
1x10 s
-16
-13
Electrons are super fast!
v = 0.7 c
given 100 nm sample
A realistic phonon model:
[4] http://www.ioffe.ru/SVA/NSM/Semicond/
All phonons are active!
"... the effort required to determine these quantities is tremendous ..."
[3] Loane et. al. Acta Crystal. 47(3) 267-279
Uncorrelated Gaussian displacements are used instead as per Einstein's model of atomic vibrations.
Multislice Algorithm
Choose initial wavefunction
Calculate integrated atomic potentials for next slice
Apply phase change to wavefunction
Apply the propagator
# of slices
Look at wavefunction or diffraction pattern
Frozen Phonon Algorithm
Choose initial wavefunction
Randomly displace atoms for next slice
Calculate integrated atomic potentials for next slice
Apply phase change to wavefunction
Apply the propagator
Save resulting wavefunction or diffraction pattern
Incoherently add wavefunctions for different atomic configurations
Reset initial wavefunction
# of slices
# of configurations
1
2
4
8
16
32
64
Channelling
Atomic contrast can be shifted.
Does not consider atom vibrations!
displacement:
+
=
...
In Practice
[1] E. J. Kirkland, Advanced Computing in Electron Microscopy (Springer US, Boston, MA, 2010).
[2] G. D. Reid, Multislice Simulation of TEM Images, Tech. Rep. May (2012).
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512x512 pixels
columns!
Atoms appear as being stationary or
"frozen"
Full transcript