1. Phonon dispersion calculation
spin-relaxation rates depends on τm
Momentum relaxation time (τm )  electron-phonon scattering
electron-phonon scattering  phonon spectrum (dispersion)

2. Lattice vibrations in mono-atomic crystals
Daryoush Shiri, IQC

3. Lattice vibrations in …
Daryoush Shiri, IQC

4. WHAT if we have a more complex solid?
Lattice vibrations in Diatomic lattice…
It is instructive to consider the
boundaries and limiting cases e.g. k=0 and k=π/a.
WHAT if we have a more complex solid?
e.g.
Bulk Si , ge, GaAs crystals
Nanowires
Amorphous Si, oxide, atomic clusters u
Daryoush Shiri, IQC

5. Dynamical Matrix Method
The generalization of the previous method to large solids
Challenges: Computationally intensive for large number of atoms
In the last example (1 and 2 atoms in each unit cell of a periodic 1D solid)  we found 1 and 2 modes, respectively.
What about bulk Si and Ge?
What about a nanowire?
W. L. Park et al, Nano Letters, 19 August 2008
Daryoush Shiri, IQC

6. Dynamical Matrix Method
Daryoush Shiri, IQC

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21. Daryoush Shiri, IQC

22. SIESTA calculation of phonon spectrum
bulkGe_phonon.fdf
Dynamic Matrix Equation is solved in SIESTA using Vibra package (see: )
STEP 1: Building a super-cell from the unit cell of a given structure e.g. bulk Si or Ge (2 atoms per unit cell)
Daryoush Shiri, IQC, Waterloo

23. Daryoush Shiri, IQC, Waterloo
SIESTA ….
LOCAL MACHINE:
After downloading, unzipping & installing SIESTA you
can go directly to Vibra/Src and copy your .fdf files there.
Make sure you have a Fortran compiler to compile fcbuild.f, Vibra.f etc
SUPERCOMPUTING FACILITY:
If you have access to e.g. SHARCNET (
just copy /Src from your local machine to your work directory.
Daryoush Shiri, IQC, Waterloo

24. Daryoush Shiri, IQC, Waterloo
SIESTA ….
bulkGe_phonon_2014.ifc.fdf
STEP 2: Displace the atoms and calculate the IFC
$siesta path/./siesta < example.ifc.fdf > example.ifc.out
OR in my example I used MPI version of SIESTA on Sharcnet
Daryoush Shiri, IQC, Waterloo

25. Daryoush Shiri, IQC, Waterloo
SIESTA ….
BandLinesScale pi/a
%block BandLines
\Gamma X K
\Gamma L
%endblock BandLines
STEP 3: Computing Dynamical matrix and Diagonalize
A Fourier transform carries Force matrix from position to momentum(k-space)
K grid points are defined in example.fdf file.
$path/Utils/Vibra/Src/./vibrator <bulkGe_phonon.fdf
 OUTPUT: Ge_bulk_2014.bands
$path/ ./bandline.x < Ge_bulk_2014.bands > Ge_bulk_2014.gnubands.dat
Daryoush Shiri, IQC, Waterloo

26. SIESTA calculated phonon spectrum of Bulk Ge
1/cm = x Hz
BandLinesScale pi/a
%block BandLines
\Gamma X K
\Gamma L
%endblock BandLines
Daryoush Shiri, IQC, Waterloo

27. LO TO

28. Electronic dispersion
[110] Si nanowire d = 1.7nm
Phonon dispersion
Electronic dispersion
From: J. Appl. Phys. 104,

29. Daryoush Shiri, IQC

30. Quantization of phonon modes