Calculating scattering rates¶

AMSET calculates mode dependent scattering rates within the Born approximation using common materials parameters. The differential scattering rate from state $\mathinner{|n\mathbf{k}\rangle}$ to state $\mathinner{|m\mathbf{k} + \mathbf{q}\rangle}$ is calculated using Fermi's golden rule as

$\tilde{\tau}_{n\mathbf{k}\rightarrow m\mathbf{k}+\mathbf{q}}^{-1} = \frac{2\pi}{\hbar} \lvert g_{nm}(\mathbf{k}, \mathbf{q}) \rvert^2 \delta(\varepsilon_{n\mathbf{k}} - \varepsilon_{m\mathbf{k}+\mathbf{q}}),$

where $\varepsilon_{n\mathbf{k}}$ is the energy of state $\mathinner{|n\mathbf{k}\rangle}$, and $g_{nm}(\mathbf{k}, \mathbf{q})$ is the matrix element for scattering from state $\mathinner{|n\mathbf{k}\rangle}$ into state $\mathinner{|m\mathbf{k} + \mathbf{q}\rangle}$.

Info

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¶

Mechanism Code Requires Type
Acoustic deformation potential scattering ADP n- and p-type deformation potential, elastic constant Elastic
Piezoelectric scattering PIE high-frequency dielectric constant, elastic constant, piezoelectric coefficient ($\mathbf{e}$) Elastic
Polar optical phonon scattering POP polar optical phonon frequency, static and high-frequency dielectric constants Inelastic
Ionized impurity scattering IMP static dielectric constant Elastic

Acoustic deformation potential scattering¶

The acoustic deformation potential matrix element is given by

$g_{nm}^\mathrm{ad}(\mathbf{k}, \mathbf{q}) = \sqrt{k_\mathrm{B} T} \sum_{\mathbf{G} \neq -\mathbf{q}} \left[ \frac{\mathbf{\tilde{D}}_{n\mathbf{k}} \mathbin{:} \mathbf{\hat{S}}_l}{c_l\sqrt{\rho}} + \frac{\mathbf{\tilde{D}}_{n\mathbf{k}} \mathbin{:} \mathbf{\hat{S}}_{t_1}}{c_{t_1}\sqrt{\rho}} + \frac{\mathbf{\tilde{D}}_{n\mathbf{k}} \mathbin{:} \mathbf{\hat{S}}_{t_2}}{c_{t_2}\sqrt{\rho}} \right] \mathinner{\langle m\mathbf{k}+\mathbf{q} \left | e^{i(\mathbf{q} + \mathbf{G})\cdot\mathbf{r}} \right | n\mathbf{k} \rangle},$

where $\mathbf{\tilde{D}}_{n\mathbf{k}} = \mathbf{D}_{n\mathbf{k}} + \mathbf{v}_{n\mathbf{k}} \otimes \mathbf{v}_{n\mathbf{k}}$ in which $\mathbf{D}_{n\mathbf{k}}$ is the rank 2 deformation potential tensor, $\mathbf{\hat{S}} = \mathbf{\hat{q}}\otimes\mathbf{\hat{u}}$ is the unit strain associated with an acoustic mode, $\mathbf{\hat{u}}$ is the unit vector of phonon polarization, and the subscripts $l$, $t_1$, and $t_2$ indicate properties belonging to the longitudinal and transverse modes.

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

Piezoelectric scattering¶

The piezoelectric differential scattering rate is given by

$g_{nm}^\mathrm{pi}(\mathbf{k}, \mathbf{q}) = \sqrt{k_\mathrm{B} T} \sum_{\mathbf{G} \neq -\mathbf{q}} \left[ \frac{\mathbf{\hat{n}} \mathbf{h} \mathbin{:} \mathbf{\hat{S}}_l}{c_l\sqrt{\rho}} + \frac{\mathbf{\hat{n}} \mathbf{h} \mathbin{:} \mathbf{\hat{S}}_{t_1}}{c_{t_1}\sqrt{\rho}} + \frac{\mathbf{\hat{n}} \mathbf{h} \mathbin{:} \mathbf{\hat{S}}_{t_2}}{c_{t_2}\sqrt{\rho}} \right ] \frac{\mathinner{\langle m\mathbf{k}+\mathbf{q} \left | e^{i(\mathbf{q} + \mathbf{G})\cdot\mathbf{r}} \right | n\mathbf{k} \rangle}} {\left | \mathbf{q} + \mathbf{G} \right |},$

where $\mathbf{h}$ is the full piezoelectric stress tensor and $\mathbf{\hat{n}} = (\mathbf{q} + \mathbf{G}) / \left | \mathbf{q} + \mathbf{G} \right |$ is a unit vector in the direction of scattering.

• 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

$g_{nm}^\mathrm{po}(\mathbf{k}, \mathbf{q}) = \left [ \frac{\hbar \omega_\mathrm{po}}{2} \right ] ^ {1/2} \sum_{\mathbf{G} \neq -\mathbf{q}} \left (\frac{1}{\mathbf{\hat{n}}\cdot\boldsymbol{\epsilon}_\infty\cdot\mathbf{\hat{n}}} - \frac{1}{\mathbf{\hat{n}}\cdot\boldsymbol{\epsilon}_\mathrm{s}\cdot\mathbf{\hat{n}}}\right) ^\frac{1}{2} \frac{\mathinner{\langle m\mathbf{k}+\mathbf{q} \left | e^{i(\mathbf{q} + \mathbf{G})\cdot\mathbf{r}} \right | n\mathbf{k} \rangle}} {\left | \mathbf{q} + \mathbf{G} \right |},$

where $\boldsymbol{\epsilon}_\mathrm{s}$ and $\boldsymbol{\epsilon}_\infty$ are the static and high-frequency dielectric tensors and $\omega_\mathrm{po}$ is the polar optical phonon frequency. To capture scattering from the full phonon band structure in a single phonon frequency, each phonon mode is weighted by the dipole moment it produces.

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

Ionized impurity scattering¶

The ionized impurity matrix element is given by

$g_{nm}^\mathrm{ii}(\mathbf{k}, \mathbf{q}) = \sum_{\mathbf{G} \neq -\mathbf{q}} \frac{n_\mathrm{ii}^{1/2} Z e }{\mathbf{\hat{n}} \cdot \boldsymbol{\epsilon}_\mathrm{s} \cdot \mathbf{\hat{n}}} \frac{\mathinner{\langle m\mathbf{k}+\mathbf{q} \left | e^{i(\mathbf{q} + \mathbf{G})\cdot\mathbf{r}} \right | n\mathbf{k} \rangle}} {\left | \mathbf{q} + \mathbf{G} \right | ^2 + \beta^2},$

where $Z$ is the charge state of the impurity center, $n_\mathrm{ii}$ is the concentration of ionized impurities (i.e., $C \times (n_\mathrm{holes} - n_\mathrm{electrons}) / Z$, where $C$ is the amount of charge compensation), and $\beta$ is the inverse screening length, defined as

$\beta^2 = \frac{e^2}{\epsilon_\mathrm{s} k_\mathrm{B} T} \int \frac{\mathrm{d}\varepsilon}{V}\,D(\varepsilon) f(1-f),$

where $V$ is the unit cell volume, $D$ is the density of states, and $f$ is the Fermi–Dirac distribution given in the transport properties section.

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

Elastic vs inelastic scattering¶

AMSET treats elastic and inelastic scattering mechanisms separately.

Inelastic¶

The differential scattering rate for inelastic processes is calculated as

\begin{aligned} \tau_{n\mathbf{k}\rightarrow m\mathbf{k}+\mathbf{q}}^{-1} = \frac{2\pi}{\hbar} \lvert g_{nm}(\mathbf{k}, \mathbf{q}) \rvert^2 \times [ &{} (n_\mathrm{po} + 1 - f_{m\mathbf{k} + \mathbf{q}}) \delta(\varepsilon_{n\mathbf{k}} - \varepsilon_{m\mathbf{k}+\mathbf{q}} - \hbar\omega_\mathrm{po}) \\ &{} (n_\mathrm{po} + f_{m\mathbf{k} + \mathbf{q}}) \delta(\varepsilon_{n\mathbf{k}} - \varepsilon_{m\mathbf{k}+\mathbf{q}} + \hbar\omega_\mathrm{po})], \end{aligned}

where $\omega_\mathrm{po}$ is an effective phonon frequency, $n_\mathrm{po} = 1 / [\exp (\hbar \omega_\mathrm{po} / k_\mathrm{B} T) - 1]$ denotes the Bose–Einstein distribution of phonons, and the $-\hbar \omega_\mathrm{po}$ and $+\hbar \omega_\mathrm{po}$ terms correspond to scattering by phonon absorption and emission, respectively.

The overall inelastic scattering rate for state $\mathinner{|n\mathbf{k}\rangle}$ is calculated as

$\tau^{-1}_{n\mathbf{k}} = \sum_m \int \frac{\mathrm{d}^3q}{\Omega} \tau_{n\mathbf{k}\rightarrow m\mathbf{k}+\mathbf{q}}^{-1}$

where $\Omega$ is the volume of the Brillouin zone.

Elastic¶

Elastic rates are calculated using the momentum relaxation time approximation (MRTA), given by

$\tilde{\tau}^{-1}_{n\mathbf{k}} = \sum_m \int \frac{\mathrm{d}^3q}{\Omega} \left [ 1 - \frac{\mathbf{v}_{n\mathbf{k}} \cdot \mathbf{v}_{m\mathbf{k} + \mathbf{q}}}{\lvert \mathbf{v}_{n\mathbf{k}} \rvert^2} \right ] \tilde{\tau}_{n\mathbf{k}\rightarrow m\mathbf{k}+\mathbf{q}}^{-1}$

where $\tilde{\tau}_{n\mathbf{k}\rightarrow m\mathbf{k}+\mathbf{q}}^{-1}$ is the elastic differential scattering rate defined at the top of this page and $\mathbf{v}_{n\mathbf{k}}$ is the group velocity of state $\mathinner{|n\mathbf{k}\rangle}$.

Overlap integral¶

In the Born approximation, the scattering rate equations depend on the wavefunction overlap $\mathinner{\langle m\mathbf{k}+\mathbf{q} \left | e^{i(\mathbf{q} + \mathbf{G})\cdot\mathbf{r}} \right | n\mathbf{k} \rangle}$. AMSET uses pawpyseed to obtain wavefunction coefficients including PAW core regions from the pseudo wavefunction coefficients written by VASP. The wavefunctions coefficient are linearly interpolated onto the mesh used to calculate scattering rates.

Brillouin zone integration¶

All scattering rates depend on the Dirac delta function $\delta$, 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 $1 / \lvert\mathbf{q}\rvert ^2$ 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. 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.

1. Rode, D. L. Low-field electron transport. Semiconductors and semimetals 10, (Elsevier, 1975).

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).

5. Fröhlich, H. Electrons in lattice fields. Adv. Phys. 3, 325–361 (1954).

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