2.3 Internal interfaces

The internal interfaces between the different layers modify the drift-diffusion equations. In order to account for differences in the conduction/valence band levels and the effective densities of states, one can replace the potential \(V\) in Eqs (??) and (??) with the generalised potentials \(V_{gn}\) and \(V_{gp}\) for electrons and holes, respectively.

If the conduction/valence band edge (\(E_{c/v}(x)\)) differs across the layers in the simulation volume, then the generalised potentials are defined as [ 4 , 5 ]

\begin{equation} V_{gn}(x)=V(x) - \frac{\Delta E_c}{q}, \end{equation}

and

\begin{equation} V_{gp}(x)=V(x) - \frac{\Delta E_v}{q}, \end{equation}

where \(\Delta E_{c/v}\) is the change in conduction/valence band offset.

To account for changes in the effective densities of states, ¹ these generalised potentials are modified by the introduction of \(\Gamma \) as in Eqs (14) and (15) in Ref. [ 7 ] . By taking the effective density of states in a layer (\(N_c\)) as a reference, [ 11 ] we have

\begin{equation} \Gamma =\frac{kT}{q} \ln \frac{N_c(x)}{N_c}. \end{equation}

This changes to generalised potentials to

\begin{equation} V_{gn}(x)= V(x) - \frac{\Delta E_c}{q} + \Gamma , \end{equation}

and

\begin{equation} V_{gp}(x)= V(x) - \frac{\Delta E_v}{q} - \Gamma . \end{equation}

The current density across interfaces requires a little care. In the thermionic emission model, the current across an interface between grid points \(i\) and \(i+1\)—shown for electrons—is given by [ 6 ]

\begin{equation} \label{eq_thermionic_emission} J_{ni} = q S_T \left[ n_i - n_{i+1}\exp \left( -\frac{\Delta E_c}{kT} \right)\right], \end{equation}

where \(S_T\) is the transfer velocity. Note, this expression is only valid if the effective densities of states at either side of the interface are equal. [ 6 ] Moreover, it does not take into account any electric field across the interface.

\(\mathtt{\require{color}{\color[rgb]{0.000000000000000,0.500000000000000,0.500000000000000} SIMsalabim}}\) uses a slightly different expression across the interface that does take into account any difference in effective density of states and electric field: [ 7 ]

\begin{equation} \label{eq_interface} J_{ni}=q \nu _{int}\left[ n_i B\left(\frac{V_{gni}-V_{gni+1}}{V_t}\right) - n_{i+1} B\left(\frac{V_{gni+1}-V_{gni}}{V_t} \right) \right], \end{equation}

where \(\nu _{int}\) (parameter \(\mathtt{\require{color}{\color[rgb]{0.000000000000000,0.500000000000000,0.500000000000000} nu\_ int\_ n}}\)) is the interface transfer velocity. In the absence of an electric field across the interface and assuming equal densities of states at either side of the interface, Eq. (??) reduces to

\begin{equation} \label{eq_simple_interface} J_{ni} = q \nu _{int} B\left(- \frac{\Delta E_c}{kT}\right) \left[ n_i - n_{i+1}\exp \left(-\frac{\Delta E_c}{kT}\right) \right]. \end{equation}

By comparing Eqs (??) and (??), we can convert \(S_T\) into \(\nu _{int}\):

\begin{equation} S_T = \nu _{int} B\left(-\frac{\Delta E_c}{kT} \right). \end{equation}