Shockley–Read–Hall (SRH) Recombination Model

Shockley–Read–Hall (SRH) recombination describes carrier loss through localized defect states (traps) within the band gap. It is a trap-assisted process: an electron (or hole) is first captured into a defect state, and then may either recombine with an opposite carrier or re-emerge by thermal excitation. ?? illustrates the four main pathways for a single mid-gap trap state, with transitions labelled (a–f):

• 1. Electron trapping: capture of an electron (a), followed by recombination with a trapped hole (b).
• 2. Hole trapping: capture of a hole (d), followed by recombination with a trapped electron (c).
• 3. Hole trapping with escape: capture of a hole, followed by thermal re-emission (f).
• 4. Electron trapping with escape: capture of an electron, followed by thermal re-emission (e).

These processes show that the same trap can mediate both recombination and carrier release. SRH recombination is inherently a two-stage mechanism: first, a carrier is captured (a, d), and only later does recombination occur when the opposite carrier is captured (b, c). If no opposite carrier arrives, the trapped carrier may thermally escape (e, f). The overall efficiency of SRH recombination depends on trap density, energy level within the band gap, carrier capture cross-sections, and the relative lifetimes of trapped carriers.

Standard SRH Recombination Rate

The net trap-assisted recombination rate under the steady-state Shockley–Read–Hall (SRH) model is given by

\[ R_{\mathrm{SRH}} = \frac{np - n_{\mathrm{eq}} p_{\mathrm{eq}}} {\tau_{p}(n + n_{1}) + \tau_{n}(p + p_{1})} \]

where \(n\) and \(p\) are the local electron and hole densities, and \(n_{\mathrm{eq}}\) and \(p_{\mathrm{eq}}\) denote their equilibrium values. Writing the numerator in this form ensures that the net recombination rate vanishes exactly at equilibrium.

The effective electron and hole lifetimes associated with the trap are

\[ \tau_n = \frac{1}{\sigma_n v_{\mathrm{th}} N_t}, \qquad \tau_p = \frac{1}{\sigma_p v_{\mathrm{th}} N_t}, \]

where \(N_t\) is the trap density, \(\sigma_n\) and \(\sigma_p\) are the electron and hole capture cross-sections, and \(v_{\mathrm{th}}\) is the thermal velocity.

The auxiliary SRH quantities \(n_1\) and \(p_1\) represent the carrier densities for which the trap is in equilibrium with the conduction and valence bands, respectively. They are defined as

\[ n_1 = n_i \exp\!\left(\frac{E_t - E_{\mathrm{ref}}}{k_B T}\right), \qquad p_1 = n_i \exp\!\left(\frac{E_{\mathrm{ref}} - E_t}{k_B T}\right), \]

where \(E_t\) is the trap energy level and \(E_{\mathrm{ref}} = E_g/2\) is the mid-gap reference energy. The intrinsic carrier concentration \(n_i\) is defined through the equilibrium condition

\[ n_i^2 = n_{\mathrm{eq}} p_{\mathrm{eq}}. \]

A trap energy \(E_t = E_{\mathrm{ref}}\) therefore corresponds to a mid-gap defect, while positive or negative values shift the trap towards the conduction or valence band, respectively.

In this formulation, recombination is mediated by a single defect level that can capture both electrons and holes. Despite its simplicity, the SRH model captures the dominant role of defect-assisted recombination in many semiconductor devices and provides a computationally efficient description suitable for steady-state device simulations.

Limitations of the Standard SRH Model

While powerful, the standard SRH equation comes with several important limitations:

• Single-level approximation — it accounts for only one discrete trap energy, whereas real semiconductors (particularly organics and disordered systems) typically contain a broad distribution of trap states.
• No explicit electrostatics — the model treats traps as purely recombination centers. Charges temporarily captured in traps are not included in the electrostatic potential of the device, so their effect on space charge and internal fields is ignored.
• Purely recombination-focused — the expression does not describe trapping and de-trapping dynamics explicitly, only their net effect as recombination.

To overcome these limitations, one must explicitly solve the SRH formalism across a distribution of trap states, allowing both the recombination rate and the trap occupation (and thus their electrostatic contribution) to be correctly represented. This more general treatment is described here.

In OghmaNano, the standard SRH recombination term can be enabled or disabled in the Electrical parameter editor, and the lifetimes \(\tau_{n}\) and \(\tau_{p}\) can be specified by the user.

Using SRH in OghmaNano