2.1.1 The Potential

First of all, the Poisson equation relates the potential \(V(x)\) to the charge density

\begin{equation} \label{eq_Poisson} \frac{\partial }{\partial x}\left( \varepsilon (x) \frac{\partial V(x)}{\partial x}\right) = q (n(x) - p(x) + C(x)), \end{equation}
1

where \(\varepsilon (x)\) is the dielectric constant, \(n\) and \(p\) are the density of free electrons and holes. \(C(x)\) denotes any other charges that may be present, such as doping, ions, trapped charge carriers. In detail, the term \(C(x)\) follows from

\begin{align} C(x) = & N_A(x) - N_D(x) + n_{\rm ion}(x) - p_{\rm ion}(x)) \\ \nonumber & + \sum _{j=1}^{M} (s^e_{tb,j}(x) - f_{tb,j}(x))N_{tb,j}(x) + (s^e_{ti,j}(x) - f_{ti,j}(x))N_{ti,j}(x), \end{align}

where \(N_D\) and \(N_A\) are n- and p-doping densities (see \(\mathtt{\require{color}{\color[rgb]{0.000000000000000,0.500000000000000,0.500000000000000} N\_ D}}\) and \(\mathtt{\require{color}{\color[rgb]{0.000000000000000,0.500000000000000,0.500000000000000} N\_ A}}\)), \(n_{\rm ion}\) and \(p_{\rm ion}\) are the negative and positive ion densities. In this expression, we sum over the different trap levels \(j=1 \ldots M\), where \(M\) is the number of trap levels. For each trap level, \(s^e_{ti,j}\) and \(s^e_{tb,j}\) is the charge-type of empty (no electron in the trap) interface / bulk traps and can have value 1 or 0, \(f_{tb,j}\) and \(f_{ti,j}\) are the fraction of filled bulk traps and interface traps in steady-state as defined in Refs [ 2 ] and [ 3 ] , and \(N_{tb,j}\) and \(N_{ti,j}\) are the densities of bulk traps and interface traps. For more details on how \(f_{tb,j}\) and \(f_{ti,j}\) are calculated, see section 2.6.

\includegraphics[width=0.8\linewidth ]{figs/basic_band_diagram.svg}
Figure 2.1 Schematic band diagram showing a device that consists of three layers (main absorber, electron and hole transport layers), cathode and anode, and interface traps.