2.6.2 Bulk SRH Recombination

For simplicity and notational convenience, we show the recombination expressions per energy level, i.e. this is what is used if there is but a single trap level. If there are multiple trap levels, then the total SRH recombination rate is simply the sum over the individual trap levels.

In steady state, bulk SRH recombination is calculated based on the rate

\begin{equation} R_{\rm SRH, bulk} = \frac{\mathtt{\require{color}{\color[rgb]{0.000000000000000,0.500000000000000,0.500000000000000} C\_ n\_ bulk}}~ \mathtt{\require{color}{\color[rgb]{0.000000000000000,0.500000000000000,0.500000000000000} C\_ p\_ bulk}}~ \mathtt{\require{color}{\color[rgb]{0.000000000000000,0.500000000000000,0.500000000000000} N\_ t\_ bulk}}~ }{\mathtt{\require{color}{\color[rgb]{0.000000000000000,0.500000000000000,0.500000000000000} C\_ n\_ bulk}}~ (n+n_1) + \mathtt{\require{color}{\color[rgb]{0.000000000000000,0.500000000000000,0.500000000000000} C\_ p\_ bulk}}~ (p+p_1)} (np -n_i^2), \end{equation}
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where \(\mathtt{\require{color}{\color[rgb]{0.000000000000000,0.500000000000000,0.500000000000000} C\_ n,p\_ bulk}}\) are the capture coefficients, \(\mathtt{\require{color}{\color[rgb]{0.000000000000000,0.500000000000000,0.500000000000000} N\_ t\_ bulk}}\) is the density of traps, and \(n(p)_1\) is the electron (hole) density when the quasi-Fermi level matches the trap energy (\(\mathtt{\require{color}{\color[rgb]{0.000000000000000,0.500000000000000,0.500000000000000} E\_ t\_ bulk}}\)).

In transient simulations, we use SRH trapping and detrapping rates to form net rates for electron and hole trapping, viz.

\begin{equation} R_n = C_n n N_{tb} (1-f_{tb}) - C_n n_1 N_{tb} f_{tb} \end{equation}
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and

\begin{equation} R_p = C_p p N_{tb} f_{tb} - C_p p_1 N_{tb} (1-f_{tb}), \end{equation}
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where \(f_{tb}\) is calculated using the value from the previous time step and calculating the emission and absorption rates based on the new carrier densities found in the current time step. This is done by solving the ordinary differential equation,

\begin{equation} \frac{\partial f_{tb}(t)}{\partial t} = C_n n (1-f_{tb}) - C_n n_1 f_{tb} + C_p p_1 (1-f_{tb}) - C_p p f_{tb}, \end{equation}
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where \(C_n\) and \(C_p\) are the capture coefficients for electrons and holes and \(n(p)_1\) is the electron (hole) density when the quasi-Fermi level matches the trap energy (\(\mathtt{\require{color}{\color[rgb]{0.000000000000000,0.500000000000000,0.500000000000000} E\_ t\_ bulk}}\)). We get a solution

\begin{align} f_{tb}(t) = & \frac{C_n n + C_p p_1}{C_n n + C_n n_1 + C_p p_1 + C_p p} \\ \nonumber & + c_1 \exp \bigl(-(C_n n + C_n n_1 + C_p p_1 + C_p p)t\bigr), \end{align}

where we can express \(c_1\) in terms of the trap filling at time zero (\(f_{tb}(0)\)) as

\begin{equation} c_1 = f_{tb}(0) - \frac{C_n n + C_p p_1}{C_n n + C_n n_1 + C_p p_1 + C_p p}. \end{equation}
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Now we can calculate a trap filling at a time \(t\) with the new electron and hole densities \(n\), \(p\) and \(f_{tb}(0)\). This can be done analogously for interface traps. [ 3 ]