# Transport Properties¶

## Boltzmann transport equation¶

The Boltzmann transport equation for electrons under a small electric field, $\mathbf{F}$, can be written

$\frac{e \mathbf{F}}{\hbar} \nabla_\mathbf{k} f_\mathrm{T} (\mathbf{k}, T) = \left ( \frac{\delta f_\mathrm{T} (\mathbf{k}, T)}{\delta t} \right ).$

Under low field conditions, the total distribution function, $f_\mathrm{T}$, is

$f_\mathrm{T}(\mathbf{k}) = f(\mathbf{k}) + xg(\mathbf{k}),$

where $x$ is the cosine of the angle between $\mathbf{F}$ and $\mathbf{k}$, $g$ is the perturbation part of the distribution. and $f$ is the distribution at equilibrium (Fermi–Dirac distribution), given as

$f = \frac{1}{e^{(E-E_\mathrm{F})/k_\mathrm{B}T} + 1}$

## Elastic versus inelastic scattering¶

The perturbation is dependent on the scattering rate $s$. The differential scattering rate from state $\mathbf{k}$ to $\mathbf{k}^\prime$ is given as

$s(\mathbf{k}, \mathbf{k}^\prime ) = s_\mathrm{inel} (\mathbf{k}, \mathbf{k}^\prime ) + s_\mathrm{el} (\mathbf{k}, \mathbf{k}^\prime )$

where the subscripts $\mathrm{el}$ and $\mathrm{inel}$ refer to elastic and inelastic scattering processes, respectively.

Elastic scattering is completely randomizing and has no effect on $f$, i.e.,

$s(\mathbf{k}, \mathbf{k}^\prime) \equiv s(\mathbf{k}^\prime, \mathbf{k}),$

whereas ineleastic scattering is not randomizing and does affect $f$, i.e.,

$s(\mathbf{k}, \mathbf{k}^\prime) \neq s(\mathbf{k}^\prime, \mathbf{k}).$

The non-randomizing behaviour follows from the different rates for emission and absorption processes in inelastic scattering. Accordingly, we define the scattering out rate from k-point, $\mathbf{k}$, as

$s_\mathrm{out}(\mathbf{k}) = \int \left [ s( \mathbf{k}, \mathbf{k}^\prime ) (1 - f^\prime) + s(\mathbf{k}^\prime, \mathbf{k}) f^\prime \right ] \mathrm{d}\mathbf{k}^\prime,$

and scattering in rate to k-point, $\mathbf{k}$, as

$s_\mathrm{in}(\mathbf{k}) = \int X g^\prime \left [ s(\mathbf{k}^\prime, \mathbf{k}) (1 - f) + s(\mathbf{k}, \mathbf{k}^\prime) f \right ] \mathrm{d}\mathbf{k}^\prime,$

where $f=f(\mathbf{k})$, $f^\prime = f(\mathbf{k}^\prime)$, $g^\prime = g(\mathbf{k}^\prime)$ and $X$ is the angle between $\mathbf{k}$ and $\mathbf{k}^\prime$.

## Iterative Boltzmann transport¶

As detailed in Rode1, the perturbation to the total distribution function can be written,

$g (\mathbf{k}) = \frac{s_\mathrm{in}(g, \mathbf{k})- \frac{e \mathbf{F}}{\hbar} \nabla_\mathbf{k} f_\mathrm{T} (\mathbf{k}, T)} {s_\mathrm{out}(\mathbf{k}) + s_\mathrm{el}(\mathbf{k})}$

As $g$ depends upon itself, an iterative procedure is needed to solve the perturbation to the distribution function. Fortunately, the functional form of $g$ ensures it converges exponentially, in most cases requiring fewer than 5 iterations to achieve accurate results.

In the absence of inelastic scattering, $s_\mathrm{in}(g) = 0$ and $g_1$, is the exact solution. In addition, the $g_1$ solution is often termed the relaxation time approximation (RTA). Currently, AMSET only support the RTA but the full iterative solution will be added in a future release.

## Calculating transport properties¶

Transport properties are calculated using the transport density of states (DOS), defined as:

$\sigma(E, T) = \int \sum_n \mathbf{v}_{n\mathbf{k}} \otimes \mathbf{v}_{n\mathbf{k}} \tau_{n\mathbf{k},T} \delta (E - E_{n\mathbf{k}}) \frac{\mathrm{d}\mathbf{k}}{8 \pi^3},$

where $\mathbf{v}_{n\mathbf{k}}$ is the group velocity of k-point $\mathbf{k}$ and band $n$, $\tau_{n\mathbf{k}}$ is the lifetime defined as $1/s_n(\mathbf{k})$ and $E$ is energy.

The transport DOS is used to calculate the moments of the generalized transport coefficients

$\mathcal{L}^{(\alpha)}(E_\mathrm{F}, T) = q^2 \int \sigma(E, T)(E - E_\mathrm{F})^\alpha \left ( - \frac{ \delta f_\mathrm{T}(E, E_\mathrm{F}, T)}{\delta E} \right ) \mathrm{d}E,$

where $E_\mathrm{F}$ is the Fermi level at doping concentration, $c$.

Finally, the electrical conductivity, $\sigma$, Seebeck coefficient, $S$, and electronic contribution to the thermal conductivity, $\kappa_\mathrm{e}$, are calculated according to

\begin{aligned} \sigma = {}& \mathcal{L}^{(0)},\\ S = {}& \frac{1}{qT} \frac{\mathcal{L}^{(1)}}{\mathcal{L}^{(0)}},\\ \kappa_\mathrm{e} = {}& \frac{1}{q^2T} \left [ \frac{(\mathcal{L}^{(1)})^2}{\mathcal{L}^{(0)}} - \mathcal{L}^{(2)} \right ] . \end{aligned}

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