Phonon dispersion calculation
• spin-relaxation rates depends on τm
• Momentum relaxation time (τm )  electron-phonon scattering
• electron-phonon scattering  phonon spectrum (dispersion)
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Lattice vibrations in mono-atomic crystals
Daryoush Shiri, IQC
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Lattice vibrations in …
Daryoush Shiri, IQC
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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
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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
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Dynamical Matrix Method
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SIESTA calculation of phonon spectrum
Dynamic Matrix Equation is solved in SIESTA using Vibra package
bulkGe_phonon.fdf
(see: http://departments.icmab.es/leem/siesta/Documentation/Tutorials/index.html )
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
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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 (www.sharcnet.ca),
just copy /Src from your local machine to your work directory.
Daryoush Shiri, IQC, Waterloo
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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
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SIESTA ….
BandLinesScale
• 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
pi/a
%block BandLines
1 0.000 0.000 0.000 \Gamma
45 2.000 0.000 0.000 X
17 2.000 0.500 0.500 K
48 2.000 2.000 2.000 \Gamma
41 1.000 1.000 1.000 L
%endblock BandLines
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SIESTA calculated phonon spectrum of Bulk Ge
1/cm = 2.997 93 x 10+10 Hz
BandLinesScale
pi/a
%block BandLines
1 0.000 0.000 0.000 \Gamma
45 2.000 0.000 0.000 X
17 2.000 0.500 0.500 K
48 2.000 2.000 2.000 \Gamma
41 1.000 1.000 1.000 L
%endblock BandLines
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LO
TO
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[110] Si nanowire d = 1.7nm
Phonon dispersion
Electronic dispersion
From: J. Appl. Phys. 104, 053716 2008
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Quantization of phonon modes