Carrier transport

Drift-Diffusion Model

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

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

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

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

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

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

which we call current equation.

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

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 \(\big(n(\mathbf{x}, \phi, E_{\text{F},n}),p(\mathbf{x}, \phi, E_{\text{F},p})\big)\) 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: 04/12/2024