Carrier transport

Drift-Diffusion Model

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

(2.2.20)

\begin{array}{r} \begin{aligned} - e \frac{\partial n}{\partial t} + \nabla \cdot (- e j_{n} (x)) & = - e (G (x) - R (x)), \\ e \frac{\partial p}{\partial t} + \nabla \cdot e j_{p} (x) & = e (G (x) - R (x)), \end{aligned} \end{array}

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

(2.2.21)

\begin{array}{r} \begin{aligned} j_{n} (x) & = - μ_{n} (x) n (x) \nabla E_{F, n} (x), \\ j_{p} (x) & = μ_{p} (x) p (x) \nabla E_{F, p} (x) . \end{aligned} \end{array}

Here the charge current has the unit of (area) $^{- 1}$ (time) $^{- 1}$ . $μ_{n / p}$ are the mobilities of each carrier. In nextnano++, $μ_{n / p}$ 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

(2.2.22)

\begin{array}{r} \begin{aligned} \nabla \cdot μ_{n} (x) n (x) \nabla E_{F, n} (x) & = - (G (x) - R (x)), \\ \nabla \cdot μ_{p} (x) p (x) \nabla E_{F, p} (x) & = G (x) - R (x), \end{aligned} \end{array}

which we call current equation.

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

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 $(n (x, ϕ, E_{F, n}), p (x, ϕ, E_{F, p}))$ 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