Calculating scattering rates

AMSET calculates mode dependent scattering rates within the Born approximation using common materials parameters. The differential scattering rate, snm(k,k)s_{nm}(\mathbf{k}, \mathbf{k}^\prime), gives the rate from band nn, k-point k\mathbf{k} to band mm, k-point k\mathbf{k}^\prime .

The final scattering rate at each band and k-point is obtained by integrating the differential scattering rates over the full Brillouin zone. In this section report the differential scattering rate equations and references for each scattering mechanism. More information on calculating transport properties is given in the transport properties section.

Summary of scattering rates

Mechanism Code Requires Type
Acoustic deformation potential scattering ACD 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 differential scattering rate is given by

snm(k,k)=e2kBTEd24π2Celψmkψnk2δ(EE), s_{nm}(\mathbf{k}, \mathbf{k}^\prime) = \frac{e^2 k_\mathrm{B}T E_\mathrm{d}^2}{4 \pi^2 \hbar C_\mathrm{el}} \lvert \mathinner{\langle{\psi_{m\mathbf{k}^\prime}|\psi_{n\mathbf{k}}}\rangle}\rvert^2 \delta ( E - E^\prime ),

where EdE_\mathrm{d} is the acoustic-phonon deformation-potential, and CelC_\mathrm{el} is the elastic constant.

Acoustic deformation potential scattering information

  • Abbreviation: ACD
  • Type: Elastic
  • References: 3, 2, 1
  • Requires: deformation_potential, elastic_constant

Ionized impurity scattering

The ionized impurity differential scattering rate is given by

snm(k,k)=e4Nimp4π2ϵs2ψmkψnk2(kk2+β2)2δ(EE), s_{nm}(\mathbf{k}, \mathbf{k}^\prime) = \frac{e^4 N_\mathrm{imp}}{4 \pi^2 \hbar \epsilon_\mathrm{s}^2} \frac{\lvert \mathinner{\langle{\psi_{m\mathbf{k}^\prime}|\psi_{n\mathbf{k}}}\rangle}\rvert^2} {(\left | \mathbf{k} - \mathbf{k}^\prime \right | ^2 + \beta^2)^2} \delta ( E - E^\prime ),

where ϵs\epsilon_\mathrm{s} is the static dielectric constant, NimpN_\mathrm{imp} is the concentration of ionized impurities (i.e., Nholes+NelectronsN_\mathrm{holes} + N_\mathrm{electrons}), and β\beta is the inverse screening length, defined as

β2=e2ϵskBT(k/π)2f(1f)dk. \beta^2 = \frac{e^2}{\epsilon_\mathrm{s} k_\mathrm{B} T} \int (\mathbf{k} / \pi)^2 f(1-f) \,\mathrm{d}\mathbf{k}.

where $f$ is the Fermi dirac distribution given in the transport properties section.

Ionized impurity scattering information

  • Abbreviation: IMP
  • Type: Elastic
  • References: 4, 1
  • Requires: static_dielectric

Piezoelectric scattering

The piezoelectric differential scattering rate is given by

snm(k,k)=e2kBTPpie24πϵsψmkψnk2kk2δ(EE), s_{nm}(\mathbf{k}, \mathbf{k}^\prime) = \frac{e^2 k_\mathrm{B} T P_\mathrm{pie}^2}{4 \pi \hbar \epsilon_\mathrm{s}} \frac{\lvert \mathinner{\langle{\psi_{m\mathbf{k}^\prime}|\psi_{n\mathbf{k}}}\rangle}\rvert^2} {\left | \mathbf{k} - \mathbf{k}^\prime \right | ^2 } \delta ( E - E^\prime ),

where ϵs\epsilon_\mathrm{s} is the static dielectric constant and PpieP_\mathrm{pie} is the dimensionless piezoelectric coefficient.

Piezoelectric scattering information

  • Abbreviation: PIE
  • Type: Elastic
  • References: 1
  • Requires: piezoelectric_coefficient, static_dielectric

Polar optical phonon scattering

The polar optical phonon differential scattering rate is given by

snm(k,k)=e2ωpo8π2(1ϵ1ϵs)ψmkψnk2kk2×{δ(EE+ωpo)(Npo+1),emission,δ(EEωpo)(Npo),absorption, \begin{aligned} s_{nm}(\mathbf{k}, \mathbf{k}^\prime) = {}& \frac{e^2 \omega_\mathrm{po}}{8 \pi^2} \left (\frac{1}{\epsilon_\infty} - \frac{1}{\epsilon_\mathrm{s}}\right) \frac{\lvert \mathinner{\langle{\psi_{m\mathbf{k}^\prime}|\psi_{n\mathbf{k}}}\rangle}\rvert^2} {\lvert \mathbf{k} - \mathbf{k}^\prime \rvert ^2 } \\ {}& \times \begin{cases} \delta ( E - E^\prime + \hbar \omega_\mathrm{po})(N_\mathrm{po} + 1), & \text{emission},\\ \delta ( E - E^\prime - \hbar \omega_\mathrm{po})(N_\mathrm{po}), & \text{absorption},\\ \end{cases} \end{aligned}

where ωpo\omega_\mathrm{po} is the polar optical phonon frequency, ϵ\epsilon_\infty is the high-frequency dielectric constant, and NpoN_\mathrm{po} is the phonon density of states. The ωpo-\hbar \omega_\mathrm{po} and +ωpo+\hbar \omega_\mathrm{po} terms correspond to scattering by phonon absorption and emission, respectively.

The phonon density of states is given by the Bose-Einstein distribution, according to

Npo=1exp(ωpo/kBT)1. N_\mathrm{po} = \frac{1}{\exp (\hbar \omega_\mathrm{po} / k_\mathrm{B} T) - 1}.

Polar optical phonon scattering information

  • Abbreviation: POP
  • Type: Inelastic
  • References: 5, 6, 1
  • Requires: pop_frequency, static_dielectric, high_frequency_dielectric

Overlap integral

In the Born approximation, the scattering rate equations depend on the wavefunction overlap ψmkψnk\mathinner{\langle{\psi_{m\mathbf{k}^\prime}|\psi_{n\mathbf{k}}}\rangle}. AMSET uses pawpyseed to obtain wavefunction coefficients including PAW core regions from the pseudo wavefunction coefficients written by VASP. The wavefunctions coefficients are linearly interpolated onto the mesh used to calculate scattering rates.

Brillouin zone integration

All scattering rates depend on the Dirac delta function, δ(EE)\delta(E - E^\prime), which imposes conservation of energy. Due to finite k-point sampling and numerical noise, it is unlikely that two states will ever have exactly the same energy. Furthermore, many scattering rates have a 1/kk21 / {\lvert \mathbf{k} - \mathbf{k}^\prime \rvert ^2 } dependence which requires extremely dense k-point meshes to achieve convergence.

To account for this, AMSET employs a modified tetrahedron integration scheme. Similar to the traditional implementation of the tetrahedron method, tetrahedra cross sections representing regions of the constant energy surface are identified. If the area of these cross section is integrated using the analytical expressions detailed by Blöchl, this will not satisfactorily account for the strong k-point dependence at small kk2{\lvert \mathbf{k} - \mathbf{k}^\prime \rvert ^2 } values. In AMSET, we explicitly calculate the scattering rates on a ultra-fine mesh on the tetrahedron cross sections and integrate numerically. As the cross sections represent a constant energy surface, the band structure does not need to be interpolated onto the ultra-fine mesh resulting in significant speed-ups.

The methodology for combining scattering rates for multiple scattering mechanisms is given in the transport properties section.

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  2. Shockley, W. & others. Electrons and holes in semiconductors: with applications to transistor electronics. (van Nostrand New York, 1950). 

  3. Bardeen, J. & Shockley, W. Deformation potentials and mobilities in non-polar crystals. Phys. Rev. 80, 72-80 (1950). 

  4. 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). 

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  6. Conwell, E. M. High Field Transport in Semiconductors. Academic Press, New York (1967).