Boltzmann Transport Theory

The Boltzmann Transport Equation (BTE) is given by

$$\partial_t n_i + \boldsymbol{v}_i\cdot\nabla_{\boldsymbol{r}} + \boldsymbol{F}\cdot\nabla_{\boldsymbol{k}} n_i + \Gamma_i = 0$$

where $n_i(\boldsymbol{r})$ is the distribution function of carriers in combined state index $i$ (encompassing the wavevector $\boldsymbol{k}$ and band $\nu$), $\boldsymbol{v}_i$ is the group velocity of state $i$, $\boldsymbol{F}$ is an external driving force, and $\Gamma_i[\{n_j(\boldsymbol{r})\}]$ is the collision operator, which specifies the rate at which carriers scatter into and out of state $i$ as a function of the full carrier distribution $n_j$ at position $\boldsymbol{r}$.

SpaRTaNS solves the BTE with a few simplifying assumptions:

1. Steady state

   i.e. $\partial_t n_i = 0$.

2. Linearized collision operator

   i.e. write $\delta n_i = n_i - n_i^0$, where $n_i^0$ is an equilibrium distribution, then expand $\Gamma_i$ to linear order in $\delta n_i$.

3. Temperature and material properties are spatially uniform

   i.e. $n_i^0$ is not a function of position and $\boldsymbol{F}\cdot\nabla_{\boldsymbol{k}} n_i \approx \boldsymbol{v}_i \cdot \frac{\partial n_i^0}{\partial \epsilon_i}\boldsymbol{F} \equiv - S_i$

These allow us to write the BTE as

$$\left(\frac{\partial \Gamma_i}{\partial n_j} + \delta_{ij} \boldsymbol{v}_i\cdot\nabla_{\boldsymbol{r}}\right)\delta n_j = S_i,$$

where summation is implied over repeated indices and we have written the forcing term as a source of carriers $S_i$.

Next, we separate the collision operator into diagonal terms, representing decay with lifetime $\tau_i$, and off-diagonal ‘mixing’ terms:

$$\frac{\partial \Gamma_i}{\partial n_j} = \tau_i^{-1} \delta_{ij} - M_{ij}.$$

Using this decomposition, we write the BTE as

$$\left(1 + \tau_i\boldsymbol{v}_i\cdot\nabla_{\boldsymbol{r}}\right)\delta n_j = \tau_i S_i + G_{ij} \delta n_j,$$

where summation is again implied over repeated indices. Here

$$G_{ij} \equiv \tau_i M_{ij}$$

with no summation implied.

SpaRTaNS solves this equation iteratively, by expressing $\delta n_j$ as a power series in $G$:

$$\delta n_j = \delta n_j^0 + \delta n_j^1 + ...$$

where

$$\begin{aligned} (1+\tau_i \boldsymbol{v}_i\cdot\nabla_{\boldsymbol{r}})\delta n_i^0 &= \tau_i S_i \\ (1+\tau_i \boldsymbol{v}_i\cdot\nabla_{\boldsymbol{r}})\delta n_i^k &= G_{ij} \delta n_j^{k-1} \end{aligned}$$

This approach converges so long as the spectral radius of $G_{ij}$ is less than unity. Otherwise more sophisticated approaches like Jacobi weighting or alternate decompositions (e.g. choosing artificially smaller $\tau_{ij}^{-1}$, so that $M_{ij}$ has non-zero diagonal entries) must be used.

References

Spatially-Resolved Transport

For more details on the formalism behind SpaRTaNS implementation, please consult the following papers:

• Georgios Varnavides, Adam S. Jermyn, Polina Anikeeva, and Prineha Narang (2019), Phys. Rev. B 100, 115402. [publisher's copy], [pre-print copy]

• Adam S. Jermyn, Giulia Tagliabue, Harry A. Atwater, William A. Goddard, III, Prineha Narang, and Ravishankar Sundararaman (2019), Phys. Rev. Materials 3, 075201. [publisher's copy], [pre-print copy]

Generating Collision Operators

For more details on the formalism behind generating physically-plausible linearized collision operators, please consult the following paper:

• Georgios Varnavides, Adam S. Jermyn, Polina Anikeeva, and Prineha Narang (2022), ArXiv:2204.06004. [pre-print copy]