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.

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