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

u

• 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?

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Dynamical Matrix Method

W. L. Park et al, Nano Letters, 19 August 2008

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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 ….

Daryoush Shiri, IQC, Waterloo 24

• 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