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 mono-atomic crystals Daryoush Shiri, IQC
Lattice vibrations in … Daryoush Shiri, IQC
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
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
Dynamical Matrix Method Daryoush Shiri, IQC
Daryoush Shiri, IQC
Daryoush Shiri, IQC
Daryoush Shiri, IQC
Daryoush Shiri, IQC
Daryoush Shiri, IQC
Daryoush Shiri, IQC
Daryoush Shiri, IQC
Daryoush Shiri, IQC
Daryoush Shiri, IQC
Daryoush Shiri, IQC
Daryoush Shiri, IQC
Daryoush Shiri, IQC
Daryoush Shiri, IQC
Daryoush Shiri, IQC
Daryoush Shiri, IQC
SIESTA calculation of phonon spectrum bulkGe_phonon.fdf Dynamic Matrix Equation is solved in SIESTA using Vibra package (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
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 (www.sharcnet.ca), just copy /Src from your local machine to your work directory. Daryoush Shiri, IQC, Waterloo
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
Daryoush Shiri, IQC, Waterloo 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 %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
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 %endblock BandLines Daryoush Shiri, IQC, Waterloo
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Electronic dispersion [110] Si nanowire d = 1.7nm Phonon dispersion Electronic dispersion From: J. Appl. Phys. 104, 053716 2008
Daryoush Shiri, IQC
Quantization of phonon modes