2.2 Boundary conditions

The boundary condition on the potential \(V\) is given by

\begin{equation} \label{Eq_BC_potential} qV_R - qV_L = \mathtt{\require{color}{\color[rgb]{0.000000000000000,0.500000000000000,0.500000000000000} W\_ L}}~ -\mathtt{\require{color}{\color[rgb]{0.000000000000000,0.500000000000000,0.500000000000000} W\_ R}}~ + q \mathtt{\require{color}{\color[rgb]{0.000000000000000,0.500000000000000,0.500000000000000} Vint}}~ , \end{equation}
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where \(V_{R(L)}\) is the potential at the right (left) electrode, \(\mathtt{\require{color}{\color[rgb]{0.000000000000000,0.500000000000000,0.500000000000000} W\_ R(L)}}\) is the work function of the right (left) electrode, and \(\mathtt{\require{color}{\color[rgb]{0.000000000000000,0.500000000000000,0.500000000000000} Vint}}\) is the internally applied voltage (see section 2.8).

The boundary conditions on the carrier densities can take different forms, depending on the surface recombination velocities: If the surface recombination velocities (see \(\mathtt{\require{color}{\color[rgb]{0.000000000000000,0.500000000000000,0.500000000000000} S\_ n/p\_ L/R}}\)) are infinitely large, we simply assume that the carrier densities at the contacts follow from the difference between the local conduction or valence band edge and the work function of the electrodes. At the right electrode (anode), we have for electrons

\begin{equation} \label{eq_elec_BC} n_R = n_{eq} = \mathtt{\require{color}{\color[rgb]{0.000000000000000,0.500000000000000,0.500000000000000} N\_ c}}~ \exp \left( \frac{\mathtt{\require{color}{\color[rgb]{0.000000000000000,0.500000000000000,0.500000000000000} E\_ c}}~ -\mathtt{\require{color}{\color[rgb]{0.000000000000000,0.500000000000000,0.500000000000000} W\_ R}}~ }{kT} \right), \end{equation}
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where \(n_{eq}\) is the equilibrium electron density at this contact. If the surface recombination velocity is finite, then we use

\begin{equation} J_n = q \mathtt{\require{color}{\color[rgb]{0.000000000000000,0.500000000000000,0.500000000000000} S\_ n\_ R}}~ (n_R - n_{eq}), \end{equation}
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but only if the electrons are extracted at this electrode (\(J_n{\lt}0\)). If electrons are injected, we use Eq. (??). For holes at the right electrode, one has

\begin{equation} \label{eq_hole_BC} p_R = p_{eq} = \mathtt{\require{color}{\color[rgb]{0.000000000000000,0.500000000000000,0.500000000000000} N\_ c}}~ \exp \left(-\frac{\mathtt{\require{color}{\color[rgb]{0.000000000000000,0.500000000000000,0.500000000000000} E\_ v}}~ -\mathtt{\require{color}{\color[rgb]{0.000000000000000,0.500000000000000,0.500000000000000} W\_ R}}~ }{kT} \right), \end{equation}
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where \(p_{eq}\) is the equilibrium hole density at this contact. Again, if the surface recombination velocity is finite, then we use

\begin{equation} J_p = q \mathtt{\require{color}{\color[rgb]{0.000000000000000,0.500000000000000,0.500000000000000} S\_ p\_ R}}~ (p_R - p_{eq}), \end{equation}
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but only if the holes are extracted at this electrode (\(J_p{\lt}0\)). If holes are injected, we use Eq. (??). The densities at the left electrode are analogous.