2.7. Carrier transport

2.7.1. Drift-Diffusion Model

The continuity equations in the presence of generation \(G\) recombination \(R\) of electron-hole pairs read

(2.7.1.1)\[\begin{split}\begin{aligned} -e\frac{\partial n}{\partial t} + \nabla\cdot \big[-e\mathbf{j}_n(\mathbf{r})\big] &= -e\big[G(\mathbf{r})-R(\mathbf{r})\big]\\ e\frac{\partial p}{\partial t} + \nabla\cdot e\mathbf{j}_p(\mathbf{r}) &= e\big[G(\mathbf{r})- R(\mathbf{r})\big] \end{aligned}\end{split}\]

where the current is proportional to the gradient of quasi Fermi levels \(E_{\text{F},n/p}(\mathbf{r})\).

(2.7.1.2)\[\begin{split}\begin{aligned} \mathbf{j}_n(\mathbf{r}) &= -\mu_n(\mathbf{r})n(\mathbf{r})\nabla E_{\text{F},n}(\mathbf{r})\\ \mathbf{j}_p(\mathbf{r}) &= \mu_p(\mathbf{r})p(\mathbf{r})\nabla E_{\text{F},p}(\mathbf{r}) \end{aligned}\end{split}\]

Here the charge current has the unit of (area)\(^{-1}\)(time)\(^{-1}\). \(\mu_{n/p}\) are the mobilities of each carrier. In nextnano++, \(\mu_{n/p}\) are determined using the mobility model specified in the input file under currents{ }.

Hereafter we consider stationary solutions and set \(\dot{n}=\dot{p}=0\). The governing equations then reduce to

(2.7.1.3)\[\begin{split}\begin{aligned} \nabla\cdot\big[\mu_n(\mathbf{r})n(\mathbf{r})\nabla E_{\text{F},n}(\mathbf{r})\big]&=-\big[G(\mathbf{r})-R(\mathbf{r})\big]\\ \nabla\cdot\big[\mu_p(\mathbf{r})p(\mathbf{r})\nabla E_{\text{F},p}(\mathbf{r})\big]&=G(\mathbf{r})-R(\mathbf{r}) \end{aligned}\end{split}\]

which we call current equation.

We can also say that the current equation governs the relationship between the carrier densities \(n(\mathbf{r})\), \(p(\mathbf{r})\) and quasi Fermi levels \(E_{\text{F},n/p}(\mathbf{r})\).

The nextnano++ tool solves this equation and Poisson equation (and also Schrödinger equation) self-consistently.

In their solution, the corresponding calculation of the carrier densities \(\big[n(\mathbf{r}, \phi, E_{\text{F},n}),p(\mathbf{r}, \phi, E_{\text{F},p})\big]\) and Poisson equation are firstly iterated for a given quasi-Fermi levels until the carreir densities converge. Then the resulting carrier densities are substituted into the current equation and the quasi-Fermi levels are updated. This whole cycle is iterated until the quasi-Fermi levels satisfies the convergence criteria, which can be tuned by the users from run{ current_poisson{ } } or run{ quantum_current_poisson{ } }.

Last update: 2025-09-18