Electrical conductivity

Theory behind electron simulations

Essentially, we want to solve the Boltzmann transport equation:

We use the relaxation time approximation to compute the collision term:

And we can compute the average time of flight $\langle ToF \rangle$ i.e. the average time it takes for an electron to travel from one side of the structure to another. Once this time is known, we can compute the flux per simulated electron as $F(E) = \langle ToF \rangle^{-1}$, from which, we can compute the transport distribution function:

$\Xi (E) = C \times F(E) \times g(E)$

where $g(E)$ is the density of states, and $C$ is the mapping constant. Then, the electrical conductivity is given by the:

$\sigma = q^2 \int \Xi(E) \left(-\frac{\partial f}{\partial E} \right) dE$

and Seeback coefficient it given by:

$S = \frac {qk_b}{\sigma} \int \Xi(E) \left(-\frac{\partial f}{\partial E} \right) \left(\frac{E - E_f}{k_bT} \right) dE$

Also, we want our prediction to match the bulk electron conductivity before we proceed to the calculations on nanostructures, so we need to compute the mapping constant as:

See additional details in the Ref. [1].

References

1. Priyadarshia and Neophytou, J. Appl. Phys. 133, 054301 (2023)