Phonon dispersion calculation
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• spin-relaxation rates depends on τm • Momentum relaxation time (τm ) electron-phonon scattering • electron-phonon scattering phonon spectrum (dispersion)
Lattice vibrations in mono-atomic crystals
Daryoush Shiri, IQC 2
Daryoush Shiri, IQC 3
Lattice vibrations in …
Daryoush Shiri, IQC 4
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 crystalsNanowiresAmorphous Si, oxide, atomic clusters
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• 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?
Daryoush Shiri, IQC 5
Dynamical Matrix Method
W. L. Park et al, Nano Letters, 19 August 2008
Daryoush Shiri, IQC 6
Dynamical Matrix Method
Daryoush Shiri, IQC 7
Daryoush Shiri, IQC 8
Daryoush Shiri, IQC 9
Daryoush Shiri, IQC 10
Daryoush Shiri, IQC 11
Daryoush Shiri, IQC 12
Daryoush Shiri, IQC 13
Daryoush Shiri, IQC 14
Daryoush Shiri, IQC 15
Daryoush Shiri, IQC 16
Daryoush Shiri, IQC 17
Daryoush Shiri, IQC 18
Daryoush Shiri, IQC 19
Daryoush Shiri, IQC 20
Daryoush Shiri, IQC 21
SIESTA calculation of phonon spectrumDynamic Matrix Equation is solved in SIESTA using Vibra package
(see: http://departments.icmab.es/leem/siesta/Documentation/Tutorials/index.html )
Daryoush Shiri, IQC, Waterloo 22
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)
bulkGe_phonon.fdf
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 23
• 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
bulkGe_phonon_2014.ifc.fdfSIESTA ….
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• 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
SIESTA …. BandLinesScale pi/a
%block BandLines
1 0.000 0.000 0.000 \Gamma45 2.000 0.000 0.000 X17 2.000 0.500 0.500 K48 2.000 2.000 2.000 \Gamma41 1.000 1.000 1.000 L
%endblock BandLines
Daryoush Shiri, IQC, Waterloo 25
SIESTA calculated phonon spectrum of Bulk Ge
BandLinesScale pi/a%block BandLines 1 0.000 0.000 0.000 \Gamma45 2.000 0.000 0.000 X17 2.000 0.500 0.500 K48 2.000 2.000 2.000 \Gamma41 1.000 1.000 1.000 L%endblock BandLines
Daryoush Shiri, IQC, Waterloo 26
1/cm = 2.997 93 x 10+10 Hz
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TO
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[110] Si nanowire d = 1.7nm
Phonon dispersion
Electronic dispersion
From: J. Appl. Phys. 104, 053716 2008
Daryoush Shiri, IQC 29
Quantization of phonon modes
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