Calculating scattering rates¶
AMSET calculates mode dependent scattering rates within the Born approximation using common materials parameters. The differential scattering rate from state to state is calculated using Fermi's golden rule as
where is the energy of state , and is the matrix element for scattering from state into state .
Note, this is the expression for elastic scattering. Inelastic scattering contains addition terms, as detailed in the elastic vs inelastic scattering section.
The overall mode-dependent scattering rate is obtained by integrating the scattering rates over the full Brillouin zone. In this section, we report the matrix elements for each scattering mechanism implemented in AMSET. Information on calculating transport properties is given in the transport properties section.
Summary of scattering rates¶
|Acoustic deformation potential scattering||ADP||n- and p-type deformation potential, elastic constant||Elastic|
|Ionized impurity scattering||IMP||static dielectric constant||Elastic|
|Piezoelectric scattering||PIE||static dielectric constant, piezoelectric coefficient||Elastic|
|Polar optical phonon scattering||POP||polar optical phonon frequency, static and high-frequency dielectric constants||Inelastic|
Acoustic deformation potential scattering¶
The acoustic deformation potential matrix element is given by
where is the rank 2 volume deformation potential tensor for state , is the acoustic tensor in the direction of , is the rank 4 elastic constant tensor, and is the unit scattering direction. The vectors and are unit vectors perpendicular to and to each other, such that the three form an orthogonal basis. The first term given in square brackets accounts for scattering due to longitudinal phonons, whereas the second two terms describe scattering from transverse modes.
Ionized impurity scattering¶
The ionized impurity matrix element is given by
where is the static dielectric constant, is the concentration of ionized impurities (i.e., ), and is the inverse screening length, defined as
where is the unit cell volume, is the density of states, and is the Fermi–Dirac distribution given in the transport properties section.
The piezoelectric differential scattering rate is given by
where is the dimensionless piezoelectric coefficient.
- Abbreviation: PIE
- Type: Elastic
- References: 1
Polar optical phonon scattering¶
The polar optical phonon differential scattering rate is given by
where is the polar optical phonon frequency, and is the high-frequency dielectric constant tensor.
Elastic vs inelastic scattering¶
AMSET treats elastic and inelastic scattering mechanisms separately.
The differential scattering rate for inelastic processes is calculated as
where is an effective phonon frequency, denotes the Bose–Einstein distribution of phonons, and the and terms correspond to scattering by phonon absorption and emission, respectively.
The overall inelastic scattering rate for state is calculated as
where is the volume of the Brillouin zone.
Elastic rates are calculated using the momentum relaxation time approximation (MRTA), given by
where is the elastic differential scattering rate defined at the top of this page and is the group velocity of state .
In the Born approximation, the scattering rate equations depend on the wavefunction overlap . AMSET uses pawpyseed to obtain wavefunction coefficients including PAW core regions from the pseudo wavefunction coefficients written by VASP. The wavefunctions coefficient are linearly interpolated onto the mesh used to calculate scattering rates.
Brillouin zone integration¶
All scattering rates depend on the Dirac delta function , which imposes conservation of energy. Due to finite k-point sampling and numerical noise, it is unlikely that this condition will ever be satisfied exactly. Furthermore, many scattering rates have a dependence which requires an extremely dense k-point mesh to achieve convergence.
To account for this, AMSET employs a modified tetrahedron integration scheme. AMSET first identifies a constant energy surface by computing tetrahedral cross sections using the tetrahedron method. Next, the constant energy surface is resampled using an ultra-fine mesh of k-points generated using the quadpy numerical integration package. The wavefunction coefficients and group velocities are reinterpolated into the ultra-fine mesh using linear interpolation and the matrix elements are calculated directly. This methodology allows for significantly faster convergence than the regular tetrahedron method.
The methodology for combining rates from multiple scattering mechanisms is given in the transport properties section.
Shockley, W. & others. Electrons and holes in semiconductors: with applications to transistor electronics. (van Nostrand New York, 1950). ↩
Bardeen, J. & Shockley, W. Deformation potentials and mobilities in non-polar crystals. Phys. Rev. 80, 72-80 (1950). ↩
Dingle, R. B. XCIV. Scattering of electrons and holes by charged donors and acceptors in semiconductors. London, Edinburgh, Dublin Philos. Mag. J. Sci. 46, 831-840 (1955). ↩
Fröhlich, H. Electrons in lattice fields. Adv. Phys. 3, 325–361 (1954). ↩
Conwell, E. M. High Field Transport in Semiconductors. Academic Press, New York (1967). ↩