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Phonon dispersion calculation. spin-relaxation rates depends on τ m Momentum relaxation time ( τ m )  electron-phonon scattering electron-phonon scattering  phonon spectrum (dispersion). Lattice vibrations in mono-atomic crystals. Lattice vibrations in ….

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phonon dispersion calculation
Phonon dispersion calculation
  • spin-relaxation rates depends on τm
  • Momentum relaxation time (τm ) electron-phonon scattering
  • electron-phonon scattering phonon spectrum (dispersion)
lattice vibrations in
Lattice vibrations in …

Daryoush Shiri, IQC

slide4

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

slide5

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

slide6

Dynamical Matrix Method

Daryoush Shiri, IQC

siesta calculation of phonon spectrum
SIESTA calculation of phonon spectrum

bulkGe_phonon.fdf

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)

Dynamic Matrix Equation is solved in SIESTA using Vibra package

(see: http://departments.icmab.es/leem/siesta/Documentation/Tutorials/index.html )

Daryoush Shiri, IQC, Waterloo

siesta
SIESTA ….
  • LOCAL MACHINE:
    • After downloading, unzipping & installing SIESTA you

can go directly to Vibra/Src and copy your .fdffiles there.

Make sure you have a Fortran compiler to compile fcbuild.f, Vibra.fetc

  • SUPERCOMPUTING FACILITY:
    • If you have access to e.g. SHARCNET (www.sharcnet.ca),

just copy /Srcfrom your local machine to your work directory.

Daryoush Shiri, IQC, Waterloo

siesta1

bulkGe_phonon_2014.ifc.fdf

SIESTA ….
  • 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

siesta2
SIESTA ….
  • 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
  • 1.000 1.000 1.000 L
  • %endblockBandLines
  • 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

siesta calculated phonon spectrum of bulk ge
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
  • 1.000 1.000 1.000 L
  • %endblockBandLines

Daryoush Shiri, IQC, Waterloo

slide27

LO

TO

slide28

[110] Si nanowire d = 1.7nm

Phonon dispersion

Electronic dispersion

From: J. Appl. Phys. 104, 053716 2008