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Recombination is the opposite process of generation, and involves the annihilation of an electron-hole pair. Recombination is classified as either intrinsic or extrinsic, whereby intrinsic recombination processed in silicon are radiative and Auger recombination, and extrinsic recombination is recombination via defects — commonly referred to as Shockley Read Hall (SRH, also referred to as trap-assisted) recombination.

Radiative Recombination

Radiative recombination is the reverse process of electron-hole pair generation via the absorption of a photon. In this process, an electron in the conduction band relaxes to the valence band, recombining with an empty state, emitting all or some of its excess energy as a photon. For indirect bandgap semiconductors, where the minimum energy state in the conduction band and the maximum energy state in the valence band have a different momentum k-vector, this process involves conservation of momentum via phonon emission. Due to the involvement of a phonon, radiative recombination is suppressed in indirect semiconductors such as silicon. For a direct semiconductor the maximum in the valence band and the minimum in the conduction band are aligned in k-space, consequently, the transition does not require a phonon an its probability is higher. The net recombination rate for radiative recombination is given by [1]:

$U_{rad}=Bnp$ , (1)

where B is the spectral radiance of a body, or radiative recombination coefficient for the material, n and p are the electron and hole concentrations respectively. Values for B have been evaluated experimentally and in this work a value of B = 4.73×10^-15 cm³·s^-1at 300 K is used [2]. The corresponding lifetime component for radiative recombination is given by the expression [1]:

$\tau_{rad}=\frac{\triangle n}{U_{rad}}=\frac{1}{B(n_{0}+p_{0}+\triangle n})$ , (2)

At present, radiative recombination is not the dominating loss mechanism in silicon-based photovoltaic devices and is included here for completeness.

Auger Recombination

Band-to-band Auger recombination involves an electron in the conduction band transmitting excess energy to a third charge carrier (either a hole in the valence band or an electron in the conduction band) and relaxes to the valence band. This process does not involve the emission of a photon since the energy is transferred to a third charge carrier which absorbs both the energy and momentum and returns to its original state, for instance, by the emission of phonons [3-6]. In Auger recombination the excess energy and momentum can either be transferred to another electron (“eeh” process) or to another hole (“ehh” process) [7]. These transitions are modelled to be interactions between noninteracting quasi-free particles so that the rate of recombination is proportional to the product of the concentration of each participating particle [3, 4, 7, 8]. The corresponding recombination rates of “ehh” and “eeh” processes are given by the formulae U_eeh = C_nn²p and U_ehh = C_pnp², where C_n and C_p are the Auger coefficients of electrons and holes respectively. Commonly used values of C_n and C_p were found by Dziewior and Schmid to be C_n = 2.8×10^-32cm⁶s^-1 and C_p = 9.9×10^‑32cm⁶s^-1 at 300 K [9]. The net recombination rate from Auger processes is the sum of U_eeh and U_ehh given by:

$U_{Aug}=U_{eeh}+U_{ehh}=C_{n}n^{2}p+C_{p}np^2$ , (3)

In reality, the particles involved are not non-interacting quasi-free particles and, consequently, the Auger recombination rate is enhanced by Coulombic interactions of holes and electrons. To account for these effects, the Coulomb-enhanced Auger recombination rate enhancement factors g_eeh and g_ehh are multiplied by C_n and C_p respectively, giving enhanced Auger coefficients, C^*_n = C_n.g_eeh and C^*_p = C_p.g_ehh [10]. Kerr and Cuevas devised an empirical parameterisation for the intrinsic lifetime at 300 K taking into account Auger processes, radiative recombination according to Schlangenotto et al. [12], the dopant level and the excess carrier density. In that work the Auger recombination rate is given as [13]

$U_{Aug}=np\left(1.8\ \times\ {10}^{-24}n_0^{0.65}+6\ \times{\ 10}^{-25}p_0^{0.65}+3\ \times{\ 10}^{-27}\mathrm{\Delta}n^{0.8}\right)$ (4)

with the intrinsic lifetime (taking into account radiative and Auger processes) given as

$\tau_{intrinsic,\ Kerr}=\frac{\mathrm{\Delta n}}{np\left(1.8\ \times{10}^{-24}n_0^{0.65}+6\ \times{10}^{-25}p_0^{0.65}+3\times{10}^{-27}\mathrm{\Delta}n^{0.8}+9.5\ \times{10}^{-15}\right)}$ . (5)

As surface passivation techniques improved, measurements exceeding the intrinsic limit proposed by Kerr and Cuevas were observed, consequently Richter et al. further refined the parameterisation based off new measurements. In their work, the intrinsic lifetime is given as

$\tau_{intrinsic,\ Richter}=\frac{\mathrm{\Delta n}}{(np-ni_{eff}^2)\left(2.5\ \times{10}^{-31}g_{eeh}n_0+8.5\ \times{10}^{-32}g_{ehh}p_0+3\times{10}^{-20}\mathrm{\Delta}n^{0.92}+B_{rel}B_{low}\right)}$ . (6)

The enhancement factors are given by

$g_{eeh}\left(n_0\right)=1+13\left\{1-tanh\left[\left(\frac{n_0}{N_{0,eeh}}\right)^{0.66}\right]\right\}$ , (7)

$g_{eeh}\left(p_0\right)=1+7.5\left\{1-tanh\left[\left(\frac{p_0}{N_{0,eeh}}\right)^{0.63}\right]\right\}$ , (8)

where N_0,eeh = 3.3×10¹⁷ cm^-3and N_0ehh = 7.0×10¹⁷cm^-3, B_low is the relative radiative recombination coefficient = 4.73×10^-15 cm³·s^-1at 300 K [2], and B_rel is the relative radiative recombination coefficient according to Ref. [14]. Auger recombination is an intrinsic property of materials like silicon and together with radiative recombination determines the upper limit of photovoltaic device performance. Auger recombination processes is the dominant intrinsic recombination mechanism for a wide range of doping an injection levels—particularly in the high injection and highly doped case [15].

Recombination via Defects

Defects and impurities in the crystal lattice form discrete energy levels within the forbidden energy gap. These levels allow for electrons in the conduction band to be captured by these states before relaxing into the conduction band via recombination with a hole. The statistical model for this process, known as Shockley‑Read‑Hall (SRH) theory [16, 17], models the transitions of four processes, shown in Figure 1:

An electron in the conduction band is captured by an empty defect level.
A hole in the conduction band is captured by a filled defect level.
Electron emission by a filled defect level towards the valence band.
Hole emission by an empty defect level.

Recombination processes.png — Figure 1: Capture processes involved in defect-assisted recombination; a) electron capture, b) electron emission, c) hole capture and d) hole emission. Adapted from [16].

It is assumed that the states are non-interacting and the capture and relaxation time of carriers by the defect is non-limiting. For a single state located at some energy E_t, the recombination rate U_SRH is given by

$U_{SRH}=\ \frac{pn-n_i^2}{\tau_{p0}\left(n+n_1\right)+\tau_{n0}\left(p+p_1\right)},$ , (9)

where the numerator term represents the deviation from thermal equilibrium, which drives the net recombination rate. The terms n₁ and p₁ are the electron and hole concentrations the defect energy level, E_t (i.e. E_Fn = E_Fp = E_t), given by

$n_1\equiv N_cexp\left(\frac{E_t-E_c}{kT}\right)$ , (10a)

$p_1\equiv N_vexp\left(\frac{E_v-E_t}{kT}\right)$ . (10b)

These latter terms may be approximated by their non-degenerate forms, by replacing the exponent by the Fermi-Dirac operator, F_1/2, when located away from the band edges. The terms τ_p0 and τ_n0 are the capture time constants of holes and electrons respectively are defined as

$\tau_{p0}\equiv{\sigma_pv_{th,\ p}N}_t$ , (11a)

$\tau_{n0}\equiv{\sigma_nv_{th,n}N}_t$ . (11b)

Where v_th is the thermal velocity, N_t is density of defect sites and σ_n and σ_p are the capture cross section of electron and holes respectively. The capture cross sections represent the cross-section for capture, such that the multiple of the capture cross-section and the thermal velocity represents the probability per unit time that an electron or hole in the energy range is captured at a site. The thermal velocity of electrons and holes and its temperature dependence can be modelled independently according to Green [18], however more commonly, this value is considered equal for electrons and holes and is approximated as 10⁷ cm/s at 300 K in silicon.

The text in this section has been reproduced with permission from To, A., Improved carrier selectivity of diffused silicon wafer solar cells. 2017, University of New South Wales, Sydney , Green, M.a., Solar Cells: Operating Principle. 1982. p. 2-2.