Theory of non-equilibrium 'hot' carriers in direct band-gap semiconductors under continuous illumination

The first term in equation (1), ${\left(\frac{\partial {f}_{\mathrm{c}}}{\partial t}\right)}_{\mathrm{e}\mathrm{x}\mathrm{c}}$, denotes the change in the population of carriers in the conduction and valence bands due to absorption of a photon of energy ℏω, see figure 1. In semiconductors, following the absorption of a photon with energy $\hslash \omega > {\mathcal{E}}_{\mathrm{g}}$, the band-gap, an electron–hole pair is created either across the band-gap or inside a specific band. We adapt the formalism introduced in [66] to incorporates the band-gap corresponding to semiconductors. The formalism of [66] is simple enough, and correctly incorporate the quantum-like term (∼|E|², incoherent) in the photo-excitation term of the BE, as has been argued in a previous study in connection to the metal nanoparticles [54]. It further bypasses the need for a priori exact determination of the electron–photon interaction matrix element by using physical conditions, such as the normalization of the total probability of all the electronic processes and the particle number conservation, with an input of the absorption lineshape [66]. The change in carrier distribution at energy $\mathcal{E}$, due to photon absorption is given by

$$\begin{equation}{\left(\frac{\partial {f}_{\mathrm{c}}(\mathcal{E})}{\partial t}\right)}_{\mathrm{e}\mathrm{x}\mathrm{c}}=\frac{{N}_{\mathrm{e}\mathrm{x}\mathrm{c}}{\phi }_{\mathrm{e}\mathrm{x}\mathrm{c}}^{(\mathrm{c})}(\mathcal{E})}{{\rho }_{\mathrm{c}}(\mathcal{E})},\end{equation} \tag{ 2 }$$

where ${\phi }_{\mathrm{e}\mathrm{x}\mathrm{c}}^{(\mathrm{c})}(\mathcal{E})$ is the net change in the carrier population at $\mathcal{E}$ in the conduction band (for c = e) and valence band (for c = v). See appendices A.1 and A.2 for detailed derivation of the explicit formulae. The rate of number of electrons excited due to the absorption of photon of energy ℏω is then given by $\frac{\mathrm{d}{n}_{\mathrm{c}}}{\mathrm{d}t}=\int {\rho }_{\mathrm{c}}(\mathcal{E}){\left(\frac{\partial {f}_{\mathrm{e}}(\mathcal{E})}{\partial t}\right)}_{\mathrm{e}\mathrm{x}\mathrm{c}}$, which is equal to N_exc electrons per unit time per unit volume. A positive value of N_exc signifies an increase in the particle number, i.e., an increase in the number of electrons in the conduction band and holes in the valence band. When the semi-conductor is continuously illuminated by light, an ever increasing particle number indicates that the system shall never reach a steady-state until all the electrons (holes) are moved to the conduction (valence) band, thereby losing the semiconducting property. A balancing process that comes to rescue this catastrophe is the spontaneous recombination which we formulate in section 2.2.

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In our formalism of the photo-excitation we do not explicitly incorporate momentum conservation, because in the presence of interactions of electrons and holes with phonons, defects and impurities present in the system, as well as for systems with finite size, the momentum is no longer a good quantum number. In this regard, for direct band-gap SCs, our formulation of the photo-excitation process provides an upper bound for the number of excitation events (i.e., the maximum number of photo-excited carriers).

For indirect band-gap SCs, the interband transition due to photon absorption may require intraband carrier–phonon interaction to provide the necessary momentum transfer. This is because the conduction band minimum and the valence band maximum do not appear at the same k-point in the Brillouin zone while an optical transition is associated with near-zero momentum transfer. Moreover, any inter-valley (intraband optical) transition [for example, from the Γ-valley to L (or X)-valley] requires a direct intraband optical transition accompanied by a carrier–phonon and/or a carrier–carrier Umklapp scattering. Therefore, due to the absence of explicit momentum conservation our formulation of photo-excitation process can incorporate both indirect interband transitions and inter-valley (intraband) optical transitions only in the sense that it provides the upper bound for the number of excitation events. Nevertheless, for the intensities considered in this study, we explicitly show in appendix A.1 (equations (A9) and (A10)) that intraband optical transitions are negligible compared to their interband counterpart. We show this to be true irrespective of whether the transition is direct or indirect. However, for even higher intensities and $\hslash \omega \sim {\mathcal{E}}_{\mathrm{g}}$ intra-band optical transitions should be considered explicitly. Moreover, how much the explicit incorporation of inter-valley scattering would change the NESS quantitatively at the steady state in our CW illumination case is beyond the scope of present study.

The second term in equation (1), ${\left(\frac{\partial {f}_{\mathrm{c}}}{\partial t}\right)}_{\mathrm{r}\mathrm{e}\mathrm{c}}$, denotes the interband recombination of excited carriers, by which excited electrons and holes lose energy by spontaneously emitting photons leading to PL, see figure 1. We adopt the standard formulation, see appendices B.1 and B.2, for PL [36, 67, 68], which is also consistent with the formulation of the photo-excitation (section 2.1) considered above. The rate of change of electron population due to the recombination of carriers with energy $\mathcal{E}$ is given by

$$\begin{equation}{\left(\frac{\partial {f}_{\mathrm{c}}}{\partial t}\right)}_{\mathrm{r}\mathrm{e}\mathrm{c}}=-\frac{{N}_{\mathrm{r}\mathrm{e}\mathrm{c}}{\phi }_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{c})}(\mathcal{E})}{{\rho }_{\mathrm{c}}(\mathcal{E})},\end{equation} \tag{ 3 }$$

and the rate of change of particles due to recombination is given by $\frac{\mathrm{d}{n}_{\mathrm{r}\mathrm{e}\mathrm{c}}}{\mathrm{d}t}=\int \mathrm{d}\mathcal{E}\,{\rho }_{\mathrm{c}}(\mathcal{E}){\left(\frac{\partial {f}_{\mathrm{e}}}{\partial t}\right)}_{\mathrm{r}\mathrm{e}\mathrm{c}}\ =-{N}_{\mathrm{r}\mathrm{e}\mathrm{c}}$, a negative sign signifying a loss of particles. Here, ${\phi }_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{c})}(\mathcal{E})$ is the net change in the carrier population at energy $\mathcal{E}$ due to the recombination. See appendix B for details of the formulation. At the steady-state, the rate of photo-excitation of electrons would become the same as that of recombination of electrons, and this fixes ${N}_{\mathrm{r}\mathrm{e}\mathrm{c}}=\frac{\mathrm{d}{n}_{\mathrm{e}}}{\mathrm{d}t}$ so that $\frac{\mathrm{d}{n}_{\mathrm{e}}}{\mathrm{d}t}+\frac{\mathrm{d}{n}_{\mathrm{r}\mathrm{e}\mathrm{c}}}{\mathrm{d}t}=0$. It is worthwhile to point out the recombination rate, therefore, indirectly depends on the photo-excitation rate via $\frac{\mathrm{d}{n}_{\mathrm{e}}}{\mathrm{d}t}$.

To this end, we relax the requirement of the momentum conservation for the same set of reasons corresponding to the photo-excitation process. Therefore, our formulation of recombination provides only an upper limit of the number of recombination events for indirect band-gap SCs.

Furthermore, we assume absorption of spontaneously emitted photons is a weak effect. Such an assumption is valid for systems with size smaller than the optical skin depth, such as nanoparticles and thin films. Most semiconductor applications, such as solar cells, lighting applications [3, 5, 6, 29, 30, 36] etc meet this assumption. At this system size, the effective local electric field corresponding to the illumination is homogeneous throughout the system, and almost all the spontaneously emitted photons from inside the bulk of the system are radiated out of the system with a negligible fraction getting reabsorbed thus, justifying our assumption.

The electron–phonon (e–ph) and hole–phonon (h–ph) interactions (third term in equation (1)) are responsible for transferring energy from the electrons and holes to the lattice. These are therefore, responsible for the heating of the lattice and cooling of the carriers and shown schematically shown in figure 1. They occur within an energy window comparable to the Debye energy near the band edge (thus, it is typically narrow with respect to the photon energy). For simplicity, we consider the deformation potential approximation for the interaction between the carriers (both electrons and holes) and the phonons [18], where only the longitudinal acoustic phonons are taken into account; nevertheless, our formulation can be extended to incorporate optical phonons, both polar and longitudinal [65]. In general, a more accurate quantitative estimation of the energy transfer from carriers to phonon in semiconductors would require explicit inclusion of other phonon modes, such as, transverse acoustic, polar optical phonons [65] (arguably, known to be the dominant one in III–V semiconductors [13, 19]), inter-valley carrier–phonon scatterings [65]. However, such a detailed analysis is not the primary motivation of the present study. Our calculations can be considered as an order of magnitude estimation of the carrier–phonon energy transfer and in section 4.3 we show that our formalism agrees well with the experimental predictions in the appropriate regime.

The energy of an electron in the conduction band is measured from the conduction band edge, and $\mathcal{E}={\mathcal{E}}_{\mathrm{c}}+\frac{{\hslash }^{2}{k}^{2}}{2{m}_{\mathrm{e}}^{\ast }}$ where ${m}_{\mathrm{e}}^{\ast }$ is the conduction band effective mass of the electron and ${\mathcal{E}}_{\mathrm{c}}$ is the energy of the conduction band edge. Following references [18, 65], the Fermi golden rule expression for the scattering between the conduction band electrons and phonons is obtained to be

$$\begin{align}\hfill {\left(\frac{\partial {f}_{\mathrm{e}}(\mathcal{E})}{\partial t}\right)}_{\mathrm{e}-\mathrm{p}\mathrm{h}}& =\frac{{{\Xi}}_{\mathrm{c}}^{2}}{4\pi {\rho }_{\mathrm{c}}}\frac{\sqrt{{m}_{\mathrm{e}}^{\ast }}}{\sqrt{2(\mathcal{E}-{\mathcal{E}}_{\mathrm{c}})}}{\int }_{0}^{\hslash {\omega }_{\mathrm{D}}}\frac{\mathrm{d}{\mathcal{E}}_{q}\,{\mathcal{E}}_{q}^{2}}{{(\hslash {v}_{\mathrm{p}\mathrm{h}})}^{4}}\left[({n}_{\mathrm{B}}({\mathcal{E}}_{q},{T}_{\mathrm{p}\mathrm{h}})+1)\left({\,f}_{\mathrm{e}}(\mathcal{E}+{\mathcal{E}}_{q})[1-{f}_{\mathrm{e}}(\mathcal{E})]\right.\right.\hfill \\ \hfill & \quad -\left.{f}_{\mathrm{e}}(\mathcal{E})[1-{f}_{\mathrm{e}}(\mathcal{E}-{\mathcal{E}}_{q})]\right)+{n}_{\mathrm{B}}({\mathcal{E}}_{q},{T}_{\mathrm{p}\mathrm{h}})\left({\,f}_{\mathrm{e}}(\mathcal{E}-{\mathcal{E}}_{q})[1-{f}_{\mathrm{e}}(\mathcal{E})]\right.\hfill \\ \hfill & \quad -\left.\left.{f}_{\mathrm{e}}(\mathcal{E})[1-{f}_{\mathrm{e}}(\mathcal{E}+{\mathcal{E}}_{q})]\right)\right],\hfill \end{align} \tag{ 4 }$$

where we have assumed that a phonons are in thermal equilibrium characterized by a Bose–Einstein distribution ${n}_{\mathrm{B}}({\mathcal{E}}_{q},{T}_{\mathrm{p}\mathrm{h}})$ at the lattice temperature T_ph, with ${n}_{\mathrm{B}}({\mathcal{E}}_{q},{T}_{\mathrm{p}\mathrm{h}})={({\mathrm{e}}^{{\mathcal{E}}_{q}/{k}_{\mathrm{B}}{T}_{\mathrm{p}\mathrm{h}}}-1)}^{-1}$, k_B being the Boltzmann constant expressed in eV and ℏω_D is the 'Debye' energy. Within the deformation potential approximation the acoustic phonons exhibit a linear dispersion, viz, ω_q = v_ph|q|, v_ph being the speed of sound (see table 1 for the numerical values used) [69]. In (4), Ξ_c is the deformation potential constant, ρ_c is the material density, V is the volume of the system [69].

Table 1. Values of parameters corresponding to intrinsic (undoped) GaAs. The chemical potential μ is completely arbitrary and is used in fixing the centre of the band-gap only.

Parameter	Parameter's symbol	Value
Effective mass of electron	${m}_{\mathrm{e}}^{\ast }/{m}_{\mathrm{e}}$	0.063 [60]
Effective mass of hole	${m}_{\mathrm{h}}^{\ast }/{m}_{\mathrm{e}}$	0.49 [60]
Band-gap	${\mathcal{E}}_{\mathrm{g}}$	1.42 eV [60]
Chemical potential	μ	4.2 eV
Bottom of valence band	${\mathcal{E}}_{\mathrm{b}}$	−μ/10
Top of conduction band	${\mathcal{E}}_{\mathrm{t}}$	2μ + μ/10
Imaginary part of the dielectric function	''(ω)	3.30 [83]
Debye frequency	ωD	0.0243 eV
Deformation potential in conduction band	Ξc	7.04 [78]
Deformation potential in valence band	Ξv	3.6 [86]
Sound velocity	vph	3650 m s−1
Ambient temperature	Tamb	297 K
Mass density	ρm	5320 kg m−3
Free space permittivity	0	8.854 2 × 10−12 F m−1
Lattice–environment coupling	Gph–env	5 × 1014 W m−3 K−1

Analogously, the energy of a hole in the valence band is measured from the valence band edge, and $\mathcal{E}={\mathcal{E}}_{\mathrm{v}}-\frac{{\hslash }^{2}{k}^{2}}{2{m}_{\mathrm{h}}^{\ast }}$, where ${m}_{\mathrm{h}}^{\ast }$ is the valence band effective mass of the hole and ${\mathcal{E}}_{\mathrm{v}}$ is the energy of the valence band edge. Therefore, energy of a hole with momentum k is ${\mathcal{E}}_{\mathrm{h}}={\mathcal{E}}_{\mathrm{v}}-\mathcal{E}$, i.e., the lower the value of $\mathcal{E}$ the more energy the hole exhibits. A Fermi golden rule expression corresponding to the hole–phonon interaction, analogous to the electron–phonon one, is given by,

$$\begin{align}\hfill {\left(\frac{\partial {f}_{\mathrm{h}}(\mathcal{E})}{\partial t}\right)}_{\mathrm{h}-\mathrm{p}\mathrm{h}}& =\frac{{{\Xi}}_{\mathrm{v}}^{2}}{4\pi {\rho }_{\mathrm{m}}}\frac{\sqrt{{m}_{\mathrm{h}}^{\ast }}}{\sqrt{2({\mathcal{E}}_{\mathrm{v}}-\mathcal{E})}}{\int }_{0}^{\hslash {\omega }_{\mathrm{D}}}\frac{\mathrm{d}{\mathcal{E}}_{q}\,{\mathcal{E}}_{q}^{2}}{{(\hslash {v}_{\mathrm{p}\mathrm{h}})}^{4}}\left[({n}_{\mathrm{B}}({\mathcal{E}}_{q},{T}_{\mathrm{p}\mathrm{h}})+1)\left({\,f\enspace }_{\mathrm{h}}(\mathcal{E}-{\mathcal{E}}_{q})[1-{f}_{\enspace \mathrm{h}}(\mathcal{E})]\right.\right.\hfill \\ \hfill & \quad -\left.{f }_{\mathrm{h}}(\mathcal{E})[1-{f\enspace }_{\mathrm{h}}(\mathcal{E}+{\mathcal{E}}_{q})]\right)+{n}_{\mathrm{B}}({\mathcal{E}}_{q},{T}_{\mathrm{p}\mathrm{h}})\left({\,f\enspace }_{\mathrm{h}}(\mathcal{E}+{\mathcal{E}}_{q})[1-{f\enspace }_{\mathrm{h}}(\mathcal{E})]\right.\hfill \\ \hfill & \quad -\left.\left.{f}_{\enspace \mathrm{h}}(\mathcal{E})[1-{f}_{\mathrm{e}}(\mathcal{E}-{\mathcal{E}}_{q})]\right)\right],\hfill \end{align} \tag{ 5 }$$

where Ξ_v is the deformation potential for the h–ph interaction corresponding to the holes in the valence band (see table 1).

The electron–electron interaction in the conduction band and hole–hole interaction in the valence band (fourth term in equation (1), altogether carrier–carrier scattering) are responsible for thermalization of the electrons and holes in their respective bands. This enables carriers to reach a quasi-equilibrium where a Fermi–Dirac distribution with a carrier temperature [T_e(h) for electrons (holes)] can be associated to the carriers.

Following the standard notations from the [18], we obtain the net scattering rate corresponding to the electrons with energy within $\mathcal{E}$ and $\mathcal{E}+\mathrm{d}\mathcal{E}$ is given by

$$\begin{align}\hfill {\left(\frac{\partial {f}_{\mathrm{e}}(\mathcal{E})}{\partial t}\right)}_{\mathrm{e}-\mathrm{e}}& =\frac{4\pi }{\hslash }\frac{{V}^{2}}{16{\pi }^{4}}\int \mathrm{d}{\mathcal{E}}_{1}\,\mathrm{d}{\mathcal{E}}_{2}\,\mathrm{d}{\mathcal{E}}_{3}\left[{\int }_{{k}_{\mathrm{l}\mathrm{o}\mathrm{w}}}^{{k}_{\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}}}\mathrm{d}\kappa \vert M(\kappa ){\vert }^{2}\right]\delta ({\mathcal{E}}_{1}-{\mathcal{E}}_{2}+\mathcal{E}-{\mathcal{E}}_{3})\hfill \\ \hfill & \quad \times \frac{\hslash }{\sqrt{(\mathcal{E}-{\mathcal{E}}_{\mathrm{c}})}}\left([1-{f}_{\mathrm{e}}(\mathcal{E})][1-{f}_{\mathrm{e}}({\mathcal{E}}_{1})]{f}_{\mathrm{e}}({\mathcal{E}}_{2}){f}_{\mathrm{e}}({\mathcal{E}}_{3})-{f}_{\mathrm{e}}(\mathcal{E}){f}_{\mathrm{e}}({\mathcal{E}}_{1})[1-{f}_{\mathrm{e}}({\mathcal{E}}_{2})][1-{f}_{\mathrm{e}}({\mathcal{E}}_{3})]\right),\hfill \end{align} \tag{ 6 }$$

where V is the volume of the system under consideration and M(κ) is the matrix element of the electron–electron interaction Hamiltonian within the first-order perturbation theory; it depends on the momentum exchange, κ = k − k₂ = k₁ − k₃. The limits of the integration over the matrix element are k_low = max(|k − k₂|, |k₁ − k₃|) and k_high = min(k + k₂, k₁ + k₃) where the electron wave-number k_j corresponds to ${\mathcal{E}}_{j}$.

We consider the short-ranged, hard sphere interaction between the electrons for which ${\int }_{{k}_{\mathrm{l}\mathrm{o}\mathrm{w}}}^{{k}_{high}}\mathrm{d}\kappa \vert M(\kappa ){\vert }^{2}=\frac{{\sigma }_{\mathrm{t}}{({m}_{\mathrm{e}}^{\ast })}^{2}}{4{\pi }^{2}{\hslash }^{3}{V}^{2}}({k}_{\text{high}}-{k}_{\text{low}})$, where σ_t = 6 × 10⁻¹⁶ m² is the total scattering cross-section corresponding to electrons in GaAs [18]. Notice the fact that the matrix element M²(κ) contains a V⁻² factor which cancels the V² appearing in equation (6), thereby making the expression for electron–electron scattering volume normalized. We point out in passing that our choice of the hard-sphere scattering is a trade-off between the computation time and a more realistic model of screened Coulomb interaction, the Debye screening [59, 65]. In [18, 70, 71] it has been pointed out that at low densities (n_e < 10²² m⁻³) the 'Debye' screening formula [59, 65] breaks down and electrons act as unscreened particles. Therefore, following [18] we take hard-sphere interaction with the total scattering cross-section σ_t representing the value of the scattering cross-section of unscreened electrons.

An expression for the hole–hole interaction in the valence band, analogous to the electron–electron interaction term, is given by

$$\begin{align}\hfill {\left(\frac{\partial {f}_{\mathrm{h}}(\mathcal{E})}{\partial t}\right)}_{\mathrm{h}-\mathrm{h}}& =\frac{4\pi }{\hslash }\frac{{V}^{2}}{16{\pi }^{4}}\int \mathrm{d}{\mathcal{E}}_{1}\,\mathrm{d}{\mathcal{E}}_{2}\,\mathrm{d}{\mathcal{E}}_{3}\left[{\int }_{{k}_{\mathrm{l}\mathrm{o}\mathrm{w}}}^{{k}_{\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}}}\mathrm{d}\kappa \vert M(\kappa ){\vert }^{2}\right]\delta ({\mathcal{E}}_{1}-{\mathcal{E}}_{2}+\mathcal{E}-{\mathcal{E}}_{3})\hfill \\ \hfill & \quad \times \frac{\hslash }{\sqrt{({\mathcal{E}}_{\mathrm{v}}-\mathcal{E})}}\left([1-{f}_{\mathrm{h}}(\mathcal{E})][1-{f}_{\mathrm{h}}({\mathcal{E}}_{1})]{f}_{\mathrm{h}}({\mathcal{E}}_{2}){f}_{\mathrm{h}}({\mathcal{E}}_{3})-{f}_{\mathrm{h}}(\mathcal{E}){f}_{\mathrm{h}}({\mathcal{E}}_{1})[1-{f}_{\mathrm{h}}({\mathcal{E}}_{2})][1-{f}_{\mathrm{h}}({\mathcal{E}}_{3})]\right).\hfill \end{align} \tag{ 7 }$$

The hard sphere interaction between the holes is given by ${\int }_{{k}_{\mathrm{l}\mathrm{o}\mathrm{w}}}^{{k}_{\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}}}\mathrm{d}\kappa \vert M(\kappa ){\vert }^{2}=\frac{{\sigma }_{\mathrm{t}}{({m}_{\mathrm{h}}^{\ast })}^{2}}{4{\pi }^{2}{\hslash }^{3}{V}^{2}}({k}_{\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}}-{k}_{\mathrm{l}\mathrm{o}\mathrm{w}})$, where σ_t = 6 × 10⁻¹⁶ m², being equal to that of the electrons, is the total scattering cross-section corresponding to holes in GaAs.

The quantum nature of the carrier–carrier scattering comes from the Pauli exclusion principle incorporated into the distribution-dependent term. The carrier–carrier scattering rate depends on the density of (excited) carriers, and therefore, on the intensity of the incident light and the carrier temperature [70, 72]. For carrier densities n_e < 10²⁴ m⁻³, we replace the screened Coulomb interaction by the hard sphere interaction involving a total carrier–carrier scattering cross-section that determines the strength of carrier–carrier scattering [70, 71, 73]. In this carrier density regime screening energy, ${\mathcal{E}}_{\mathrm{T}\mathrm{F}}=\frac{{\hslash }^{2}{\kappa }_{\mathrm{T}\mathrm{F}}^{2}}{2{m}_{\mathrm{e}}^{\ast }}$ with κ_TF being the Thomas–Fermi wave-vector [59], always remain much smaller than the average energy of the electrons which validates the use of hard-sphere interaction (this applies to the holes too). However, in this situation carrier–carrier exchange interaction also become important [74, 75] but within hard-sphere interaction it can be incorporated within an adjustable parameter in the total carrier–carrier scattering cross-section [76]. In most of the undoped III–V binary SCs with wider band-gaps [with exceptions being InSb $({\mathcal{E}}_{\mathrm{g}}=0.235\enspace \mathrm{e}\mathrm{V})$, InAs $({\mathcal{E}}_{\mathrm{g}}=0.417\enspace \mathrm{e}\mathrm{V})$ at room temperature] carrier density does not reach such high values for the illumination intensities considered in our study [70, 71, 77]. Therefore, our formulation of carrier–carrier interaction applies to a large class of III–V binary SCs. For ternary and quaternary alloys energy band-gaps and room temperature carrier density strongly depend on the alloy composition [78], and therefore, these are needed to be studied case by case. Importantly, we discard Auger (non-radiative) recombination here, as it is negligible for the intensities we consider [45, 79, 80], and become appreciable only under extremely high carrier densities with spatial confinement (such as small nano-structures with system size even much less than that we already assumed in connection to absorption of the spontaneously emitted photons in section 2.2) where spatial confinement can lead to a large overlap between electron and hole wave functions [81]. Moreover, we consider only pristine SCs, such that carrier–impurity scattering is neglected [60].

Figure 2(a) shows the steady-state distributions of holes f_h and electrons f_e as a function of the energy $\mathcal{E}$ in the valence and conduction bands, respectively, for electric field levels ranging from |E|² = 4 (V m⁻¹)² (1.976 × 10⁻⁶ W cm⁻²⁾ to |E|² = 4 × 10⁹ (V m⁻¹)² (1.976 × 10³ W cm⁻²⁾. For comparison, we plot the thermal (Fermi–Dirac) distributions for electrons and holes ${f}_{\mathrm{e}}^{\mathrm{T}}$ and ${f}_{\mathrm{h}}^{\mathrm{T}}$ at ambient temperature.

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The population of the carriers increases with the illumination intensity while the (logarithmic) slopes of the steady-state distribution are nearly the same as that of the ambient thermal distribution at lower intensities. The common textbook interpretation is that the carrier temperatures do not change at all, and only the effective chemical potential for each of the carriers are shifted to their respective band edges, viz, for electrons towards the conduction band edges and for holes towards the valence band edges [59, 60]. However, in the CW illuminated SC the distribution can only be thermalized via carrier–carrier scattering. Due to the negligible carrier–carrier scattering the distributions are indeed not thermalized. Therefore, one can interpret the distributions in the following alternative way. Since for a thermalized (i.e., Fermi–Dirac) distribution the slope is inversely proportional to the carrier temperature, an increase of the distribution without a change in slope is indicative of a non-thermal distribution. It is only at higher intensities, when e–e and h–h interactions become comparable to the e(h)–ph coupling, that the slopes of f_e(h) start to deviate from ${f}_{\mathrm{e}(\mathrm{h})}^{\mathrm{T}}(\mathcal{E},{T}_{\mathrm{a}\mathrm{m}\mathrm{b}})$. To demonstrate this, in figure 2(b) we plot the slopes of ${f}_{\mathrm{e}}(\mathcal{E})$ at $\mathcal{E}=4.914$ eV (near the conduction band edge, where the strength of the e–e interaction is the maximum and therefore, can affect ${f}_{\mathrm{e}}(\mathcal{E})$ the most), as a function of the local field |E|². The slope of ${f}_{\mathrm{e}}(\mathcal{E})$ remains unchanged, until the intensity reaches a critical value such that the e–e interaction becomes comparable to the e–ph interaction. This will also be reflected in the carrier temperature (see discussion below and figure 3(b)). Thus, at low intensities due to inefficient carrier–carrier interaction, the distributions are truly non-thermal, and only start to thermalize at higher illumination intensities. This conclusion is independent of incident photon energies, as demonstrated in appendix C.

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We now use the NESS distributions to extract macroscopic characteristics of interest. First, we extract ('effective') electron and hole temperatures from the carrier–phonon power transfers. See appendix E for the details of the formalism. We denote ΔT_e and ΔT_h as the deviation of electron and hole temperatures, respectively, from the ambient temperature. These are plotted as a function of the local field in figure 3(a), showing a nonlinear increase with the illumination intensity by several hundreds of degrees. Such a nonlinear dependence stems from the fact that W_e(h)–ph exhibits a nonlinear dependence on T_e(h) − T_ph. The difference between the values of ΔT_e and ΔT_h is due to the different effective band masses of electrons and holes, and the different deformation potentials in the conduction and valence bands, see table 1.

Figure 3(b) shows the rise in the carrier temperature above ambient, denoted by ΔT_c, extracted from the PL (see appendix F), as a function of the local field. It is worth emphasizing that the rise in electron and hole temperatures, obtained from the PL spectra is the same, because the recombination process that gives rise to PL is symmetric with respect to electrons and holes. ΔT_c exhibits a nonlinear increase with the intensity, and differs considerably from ΔT_e(h) both qualitatively and quantitatively throughout the entire range of the local fields. In particular, T_c obtained from the PL spectra deviate 5–15 K from T_amb which is much smaller compared to the deviation of T_e(h) from T_amb obtained from carrier–phonon power transfers. However, as explained above, with increasing intensity, thermalization becomes more important, especially above a critical intensity (corresponding to a carrier density n_e ≈ 5.2 × 10¹⁹ m⁻³ here). The value of ΔT_c starts increasing above this critical intensity indicating the role of thermalization in a PL-based temperature measurement. This is also evident from the similarity between figures 2(b) and 3(b).

In figure 3(c) we plot the increase in phonon (lattice) temperature from the ambient, ΔT_ph, as a function of the local field, showing a linear dependence on the square of the local electric field. The total power transferred to the lattice from both electron and hole sub-systems exhibits the same linear dependence on the square of the local field (not shown). At the steady-state we find that ΔT_ph = (W_e–ph + W_h–ph)/G_ph–env, explaining the linear dependence of ΔT_ph on |E|², see equation (9). The tiny increase in T_ph with respect to the ambient temperature implies that the lattice hardly heats up at all. However, since ${\Delta}{T}_{\mathrm{p}\mathrm{h}}\propto {G}_{\mathrm{p}\mathrm{h}-\mathrm{e}\mathrm{n}\mathrm{v}}^{-1}$, weaker lattice–environment coupling would lead to more heating of the lattice and vice versa. Interestingly, in the case of metals, electron–phonon coupling G_e–ph was orders of magnitude higher due to the larger number of electrons, and even exceeded G_ph–env [54]. In that sense, while the bottleneck of the energy flow in metals was the heat transfer from the phonons to the environment, for semiconductors, the bottleneck is the e(h)–ph coupling.

Our formulation provides a unique prediction for the partition between the two channels by which the electron/hole sub-system dissipates its energy, viz, interband recombination and energy transfer to the lattice via phonons. This is crucial for the correct prediction of semiconductor heating and associated photo-thermal nonlinearity in SC nanostructures (for example, recently studied in silicon nanostructures; notably an indirect band-gap SC) [87, 88], quantification of the strength of PL, applications relying on these quantities such as thermometry and imaging, and most importantly, lighting applications of semiconductors [29, 89, 90].

Thus, we define ${\eta }_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{c})}={W}_{\mathrm{c}-\mathrm{r}\mathrm{e}\mathrm{c}}/{W}_{\mathrm{c}-\mathrm{e}\mathrm{x}\mathrm{c}}$ as the ratio of the power dissipated from the electron/hole sub-systems through recombination W_c–rec to the power absorbed by the electron/hole sub-systems, W_c–exc; similarly, ${\eta }_{\mathrm{p}\mathrm{h}}^{(\mathrm{c})}={W}_{\mathrm{c}-\mathrm{p}\mathrm{h}}/{W}_{\mathrm{c}-\mathrm{e}\mathrm{x}\mathrm{c}}[\approx (1-{\eta }_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{c})})]$ as the ratio between the power transferred from the electron/hole sub-systems to the phonons, W_c–ph, and the power absorbed by the electron/hole sub-systems. Figure 4 shows ${\eta }_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{e})}$ and ${\eta }_{\mathrm{p}\mathrm{h}}^{(\mathrm{e})}$ for two different photon energies 1.65 and 2.05 eV (${\eta }_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{h})}$ and ${\eta }_{\mathrm{p}\mathrm{h}}^{(\mathrm{h})}$ are quantitatively the same as that of the electrons). We note that the energy partition is inversely proportional to the photon energy, i.e., ${\eta }_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{c})}\propto {(\hslash \omega -{\mathcal{E}}_{\mathrm{g}})}^{-1}$, but does not depend on intensity, see figure 4. To further clarify, we plot ${\eta }_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{e})}$ for four different values photon energy in figures 8(a) and (b) in appendix G, which corroborate these conclusions. The reason is rather straightforward; for higher photon energies, electrons cross the energy gap to higher energies in the conduction band. They, thus, have more energy to lose to phonons before reaching the band edge and recombining.

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Most importantly, the energy partitions can also be used to evaluate the efficiency of the PL process. The total power absorbed W_abs by the system is ℏωN_exc, where N_exc is the rate of photon absorption per unit volume, and the total power lost due to PL is ${W}_{\mathrm{P}\mathrm{L}}={W}_{\mathrm{e}-\mathrm{r}\mathrm{e}\mathrm{c}}+{W}_{\mathrm{h}-\mathrm{r}\mathrm{e}\mathrm{c}}+{\mathcal{E}}_{\mathrm{g}}{N}_{\mathrm{e}\mathrm{x}\mathrm{c}}$ (note that N_exc = N_rec, the rate of recombination being equal to the rate of excitation at the steady-state, as required by the particle number conservation). From equations (D2) and (D4) we find ${W}_{\mathrm{e}-\mathrm{e}\mathrm{x}\mathrm{c}}=\frac{\hslash \omega -{\mathcal{E}}_{\mathrm{g}}}{2}{N}_{\mathrm{e}\mathrm{x}\mathrm{c}}$ and ${W}_{\mathrm{h}-\mathrm{e}\mathrm{x}\mathrm{c}}=\frac{\hslash \omega -{\mathcal{E}}_{\mathrm{g}}}{2}{N}_{\mathrm{e}\mathrm{x}\mathrm{c}}$; these are the powers absorbed by the electron and hole subsystems measured from the conduction and valence band edges, respectively. This signifies that out of the total power absorbed by the semiconductor, viz ℏωN_exc, a power of $(\hslash \omega -{\mathcal{E}}_{\mathrm{g}}){N}_{\mathrm{e}\mathrm{x}\mathrm{c}}$ is absorbed together by electron and hole subsystems and is equally distributed between them; it also means that a power of ${\mathcal{E}}_{\mathrm{g}}{N}_{\mathrm{e}\mathrm{x}\mathrm{c}}$ is lost in overcoming the band-gap. W_c–rec and W_h–rec defined in equations (D2) and (D4), respectively, measure the power lost by the electrons and holes, respectively. Therefore, the total power lost due to PL is ${W}_{\mathrm{P}\mathrm{L}}={W}_{\mathrm{e}-\mathrm{r}\mathrm{e}\mathrm{c}}+{W}_{\mathrm{h}-\mathrm{r}\mathrm{e}\mathrm{c}}+{\mathcal{E}}_{\mathrm{g}}{N}_{\mathrm{e}\mathrm{x}\mathrm{c}}$ which incorporates all possible recombination events from the electrons and holes away from the band edges. Then, the efficiency of PL is defined by ${\eta }_{\mathrm{P}\mathrm{L}}^{\mathrm{Q}\mathrm{E}}=\frac{{W}_{\mathrm{P}\mathrm{L}}}{\hslash \omega {N}_{\mathrm{e}\mathrm{x}\mathrm{c}}}$, which leads to equation (4). In deriving the final form of equation (4) we define ${\eta }_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{c})}=\frac{{W}_{\mathrm{c}-\mathrm{r}\mathrm{e}\mathrm{c}}}{{W}_{\mathrm{c}-\mathrm{e}\mathrm{x}\mathrm{c}}}$ (c = e for electrons and c = h for holes) as the ratio of the power dissipated from the electron/hole sub-systems through the recombination W_c–rec to the power absorbed by the electron/hole sub-systems, W_c–exc.

$$\begin{equation}{\eta }_{\mathrm{P}\mathrm{L}}^{\mathrm{Q}\mathrm{E}}=\frac{{W}_{\mathrm{P}\mathrm{L}}}{{W}_{\mathrm{a}\mathrm{b}\mathrm{s}}}=\frac{{\eta }_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{e})}+{\eta }_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{h})}}{2}+\frac{{\mathcal{E}}_{\mathrm{g}}}{\hslash \omega }\left[1-\frac{{\eta }_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{e})}+{\eta }_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{h})}}{2}\right].\end{equation} \tag{ 11 }$$

For the example we study here, we find the efficiency ${\eta }_{\mathrm{P}\mathrm{L}}^{\mathrm{Q}\mathrm{E}}$ to be 90.6% and 73% at photon energies 1.65 and 2.05 eV, respectively. We compare this to experimental observations [45], where the PL efficiency was measured as a function of illumination intensity (under laser light of 633 nm). In these experiments, the low-intensity regime is dominated by trap-assisted recombination (e.g., Shockley–Read–Hall recombination), an effect which we do not account for. At high intensities interband recombination is dominant, and the PL efficiency saturates at around 71% (obtained by interpolating the experimental data), very close to our theoretical result, showing that the saturation of efficiency comes due to the phonon-mediated dissipation of carrier energy.

It is worth emphasizing again that although existing formulations for evaluation of PL efficiency account for trap-assisted and Auger recombination [30, 43–46], these are macroscopic formulations based on (rate equations of) carrier density instead of carrier distribution and ignore the role of phonons. This prevents these formulations from being able to explain the high intensity PL efficiency. Put simply, without considering carrier–phonon interactions, PL efficiency should reach 100% at high intensities, which is in contradiction to experimental observations. Our theory thus provides a much improved theoretical prediction for PL efficiency. Moreover, our theory may also serve as a basis for a more rigorous microscopic formulation (in terms of carrier distribution f_c) of laser cooling of semiconductors [52], and can allow the quantification of the possible role of carrier–phonon scattering as a heating path-way that hinders cooling [53].

A.1.1. Interband transition

Upon absorption of a photon of energy ℏω, an electron–hole pair is created across the band-gap, i.e., if the electron is created in the conduction band at energy $\mathcal{E}$, then the corresponding hole is created in the valence band at energy $\mathcal{E}-\hslash \omega$. The joint probability of absorption of a photon of frequency between ω and ω + dω and the excitation of an electron in the conduction band at energy $\mathcal{E}$ is given by

$$\begin{equation}{A}_{\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{v}\to \mathrm{c}}({\mathcal{E}}_{\mathrm{f}}=\mathcal{E},\hslash \omega )={K}_{\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{v}\to \mathrm{c}}{D}_{\mathrm{J}}(\mathcal{E},\mathcal{E}-\hslash \omega ){\rho }_{\mathrm{J},\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{v}\to \mathrm{c}}(\mathcal{E},\mathcal{E}-\hslash \omega ),\end{equation} \tag{ A1 }$$

where ${D}_{\mathrm{J}}({\mathcal{E}}_{\mathrm{fi}\mathrm{n}\mathrm{a}\mathrm{l}},{\mathcal{E}}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}})$ is the square of the matrix element corresponding to the electron–photon interaction leading to the electronic process ${\mathcal{E}}_{\mathrm{fi}\mathrm{n}\mathrm{a}\mathrm{l}}\to {\mathcal{E}}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}}$, ${K}_{\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{v}\to \mathrm{c}}$ is the proportionality constant to be defined later, and the superscript 'v → c' denotes the excitation (denoted by the subscript 'exc') process from the valence band to the conduction band. In equation (A1), ${\rho }_{\mathrm{J},\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{v}\to \mathrm{c}}(\mathcal{E},\mathcal{E}-\hslash \omega )$ is the joint density of states of the excitation of an electron at $\mathcal{E}$ in the conduction band leaving a hole at $\mathcal{E}-\hslash \omega$ in the valence band, and is given by

$$\begin{equation}{\rho }_{\mathrm{J},\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{v}\to \mathrm{c}}(\mathcal{E},\mathcal{E}-\hslash \omega )=[1-{f}_{\mathrm{h}}(\mathcal{E}-\hslash \omega )]{\rho }_{\mathrm{h}}(\mathcal{E}-\hslash \omega )\left[1-{f}_{\mathrm{e}}(\mathcal{E})\right]{\rho }_{\mathrm{e}}(\mathcal{E}),\quad \mathcal{E} > {\mathcal{E}}_{\mathrm{c}},\end{equation} \tag{ A2 }$$

where $[1-{f}_{\mathrm{h}}(\mathcal{E}-\hslash \omega )]{\rho }_{\mathrm{h}}(\mathcal{E}-\hslash \omega )$ is the electron distribution in the valence band and $[1-{f}_{\mathrm{e}}(\mathcal{E})]{\rho }_{\mathrm{e}}(\mathcal{E})$ is the hole distribution in the conduction band, and the density of electron and hole states ${\rho }_{\mathrm{e}}(\mathcal{E})$ and ${\rho }_{\mathrm{h}}(\mathcal{E})$, respectively, are plotted in figure 5 (considering the parameters of GaAs). Equation (A2) is schematically shown by the black arrow in figure 5.

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A.1.2. Intraband transition

The intraband transition, due to the absorption of a photon, can lead to the excitation of an electron–hole pair both within the conduction band and within the valence band with the corresponding probability to be defined below. The joint probability of absorption of a photon of frequency between ω and ω + dω and the excitation of an electron–hole pair within the conduction band (denoted by superscript c → c), with the electron at energy $\mathcal{E}+\hslash \omega$ and hole at energy $\mathcal{E}$, is given by

$$\begin{equation}{A}_{\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{c}\to \mathrm{c};\mathrm{o}\mathrm{u}\mathrm{t}}({\mathcal{E}}_{\mathrm{f}}=\mathcal{E}+\hslash \omega ,\hslash \omega )={K}_{\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{c}\to \mathrm{c};\mathrm{o}\mathrm{u}\mathrm{t}}{D}_{\mathrm{J}}(\mathcal{E}+\hslash \omega ,\mathcal{E}){\rho }_{\mathrm{J},\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{c}\to \mathrm{c};\mathrm{o}\mathrm{u}\mathrm{t}}(\mathcal{E}+\hslash \omega ,\mathcal{E}),\quad \mathcal{E} > {\mathcal{E}}_{\mathrm{c}},\end{equation} \tag{ A3 }$$

where ${K}_{\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{c}\to \mathrm{c};\mathrm{o}\mathrm{u}\mathrm{t}}$ is the proportionality constant, and the corresponding joint density of state is given by

$$\begin{equation}{\rho }_{\mathrm{J},\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{c}\to \mathrm{c};\mathrm{o}\mathrm{u}\mathrm{t}}(\mathcal{E}+\hslash \omega ,\mathcal{E})={f}_{\mathrm{e}}(\mathcal{E}){\rho }_{\mathrm{e}}(\mathcal{E})[1-{f}_{\mathrm{e}}(\mathcal{E}+\hslash \omega )]{\rho }_{\mathrm{e}}(\mathcal{E}+\hslash \omega ),\end{equation} \tag{ A4 }$$

and the superscript 'out' represents the fact that the electron is going out of the state with energy $\mathcal{E}$. The equation (A4) is schematically shown by the red arrow in figure 5.

Similarly, there exists a finite joint probability for absorption of a photon of frequency between ω and ω + dω, and the excitation of an electron–hole pair within the valence band (denoted by superscript v → v), with the electron at energy $\mathcal{E}-\hslash \omega$ and hole at energy $\mathcal{E}-2\hslash \omega$, is given by

$$\begin{equation}{A}_{\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{v}\to \mathrm{v};\mathrm{i}\mathrm{n}}({\mathcal{E}}_{\mathrm{f}}=\mathcal{E}-\hslash \omega ,\hslash \omega )={K}_{\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{v}\to \mathrm{v};\mathrm{i}\mathrm{n}}{D}_{\mathrm{J}}(\mathcal{E}-\hslash \omega ,\mathcal{E}-2\hslash \omega ){\rho }_{\mathrm{J},\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{v}\to \mathrm{v};\mathrm{i}\mathrm{n}}(\mathcal{E}-\hslash \omega ,\mathcal{E}-2\hslash \omega ),\quad \mathcal{E} > {\mathcal{E}}_{\mathrm{c}},\end{equation} \tag{ A5 }$$

where ${K}_{\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{v}\to \mathrm{v};\mathrm{i}\mathrm{n}}$ is the proportionality constant, and the corresponding joint density of state is given by

$$\begin{equation}{\rho }_{\mathrm{J},\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{v}\to \mathrm{v};\mathrm{i}\mathrm{n}}(\mathcal{E}-\hslash \omega ,\mathcal{E}-2\hslash \omega )=[1-{f}_{\mathrm{h}}(\mathcal{E}-2\hslash \omega )]{\rho }_{\mathrm{h}}(\mathcal{E}-2\hslash \omega ){f}_{\mathrm{h}}(\mathcal{E}-\hslash \omega ){\rho }_{\mathrm{h}}(\mathcal{E}-\hslash \omega ),\end{equation} \tag{ A6 }$$

and the superscript 'in' represents the fact that the electron is going in to the state with energy $\mathcal{E}-\hslash \omega$ by creating a hole at energy $\mathcal{E}-2\hslash \omega$ in the valence band. Equation (A6) is schematically shown by the blue arrow in figure 5. The above-mentioned joint probability densities corresponding to equations (A1), (A3), and (A5) satisfy the following property,

$$\begin{equation}{\int }_{-\infty }^{\infty }\mathrm{d}\mathcal{E}[{A}_{\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{v}\to \mathrm{c}}(\mathcal{E},\hslash \omega )+{A}_{\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{c}\to \mathrm{c};\mathrm{o}\mathrm{u}\mathrm{t}}(\mathcal{E}+\hslash \omega ,\hslash \omega )+{A}_{\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{v}\to \mathrm{v};\mathrm{i}\mathrm{n}}(\mathcal{E}-\hslash \omega ,\hslash \omega )]=\frac{{n}_{\mathrm{e}\mathrm{x}\mathrm{c}}(\omega )}{{N}_{\mathrm{e}\mathrm{x}\mathrm{c}}},\end{equation} \tag{ A7 }$$

where n_exc(ω) is the number density of absorbed ℏω photons per unit time between ω and ω + dω, and

$$\begin{equation}{N}_{\mathrm{e}\mathrm{x}\mathrm{c}}=\int \mathrm{d}\omega \,{n}_{\mathrm{e}\mathrm{x}\mathrm{c}}(\omega )={\hslash }^{-1}{{\epsilon}}^{\prime \prime }(\omega ,{T}_{\mathrm{e}},{T}_{\mathrm{h}},{T}_{\mathrm{p}\mathrm{h}}){\langle \mathbf{E}(t)\cdot \mathbf{E}(t)\rangle }_{\mathrm{t}},\end{equation} \tag{ A8 }$$

⟨⋅⟩_t being the temporal average over a single optical cycle. In the calculation we use ${N}_{\mathrm{e}\mathrm{x}\mathrm{c}}=\frac{2{{\epsilon}}_{0}{{\epsilon}}^{{\prime\prime}}(\omega )}{\hslash }\vert E{\vert }^{2}$, ℏ being in the units of (J s), which is the total number density of absorbed photons per unit time (in units of m⁻³ s⁻¹). Its value is known from electromagnetic simulations. For simplicity, we consider the effective local electric field corresponding to the illumination to be homogeneous throughout the sample. In that sense, we implicitly assume the system size to be smaller than that of the optical skin-depth of the illumination.

To simplify equation (A7) we approximate ${D}_{\mathrm{J}}({\mathcal{E}}_{\mathrm{fi}\mathrm{n}\mathrm{a}\mathrm{l}},{\mathcal{E}}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}})$ by a constant, and consider all the processes corresponding to equations (A1), (A3), and (A5) to be equally probable [54, 66] such that ${K}_{\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{v}\to \mathrm{v};\mathrm{i}\mathrm{n}}{D}_{\mathrm{J}}(\mathcal{E}-\hslash \omega ,\mathcal{E}-2\hslash \omega )={K}_{\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{c}\to \mathrm{c};\mathrm{o}\mathrm{u}\mathrm{t}}{D}_{\mathrm{J}}(\mathcal{E}+\hslash \omega ,\mathcal{E})={K}_{\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{v}\to \mathrm{c}}{D}_{\mathrm{J}}(\mathcal{E},\mathcal{E}-\hslash \omega )={K}_{\mathrm{e}\mathrm{x}\mathrm{c}}^{(\mathrm{e})}$; the normalization constant (with the superscript '(e)' denoting that this normalization corresponds to the excitation of electrons), is determined via the following condition,

$$\begin{equation}{K}_{\mathrm{e}\mathrm{x}\mathrm{c}}^{(\mathrm{e})}{\int }_{-\infty }^{\infty }\mathrm{d}\mathcal{E}[{\rho }_{\mathrm{J},\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{v}\to \mathrm{c}}(\mathcal{E},\mathcal{E}-\hslash \omega )+{\rho }_{\mathrm{J},\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{c}\to \mathrm{c};\mathrm{o}\mathrm{u}\mathrm{t}}(\mathcal{E}+\hslash \omega ,\mathcal{E})+{\rho }_{\mathrm{J},\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{v}\to \mathrm{v};\mathrm{i}\mathrm{n}}(\mathcal{E}-\hslash \omega ,\mathcal{E}-2\hslash \omega )]=\frac{{n}_{\mathrm{e}\mathrm{x}\mathrm{c}}(\omega )}{{N}_{\mathrm{e}\mathrm{x}\mathrm{c}}}.\end{equation} \tag{ A9 }$$

To simplify further, we can consider all the intraband photo-excitations probabilities to be negligible, and we show that $\frac{{\rho }_{\mathrm{J},\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{c}\to \mathrm{c};\mathrm{o}\mathrm{u}\mathrm{t}}(\mathcal{E}+\hslash \omega ,\mathcal{E})}{{\rho }_{\mathrm{J},\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{v}\to \mathrm{c}}(\mathcal{E},\mathcal{E}-\hslash \omega )}\ll 1$ and $\frac{{\rho }_{\mathrm{J},\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{v}\to \mathrm{v};\mathrm{i}\mathrm{n}}(\mathcal{E}-\hslash \omega ,\mathcal{E}-2\hslash \omega )}{{\rho }_{\mathrm{J},\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{v}\to \mathrm{c}}(\mathcal{E},\mathcal{E}-\hslash \omega )}\ll 1$.

Firstly, from equations (A2) and (A4), it is easy to see,

$$\begin{equation}\frac{{\rho }_{\mathrm{J},\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{c}\to \mathrm{c};\mathrm{o}\mathrm{u}\mathrm{t}}(\mathcal{E}+\hslash \omega ,\mathcal{E})}{{\rho }_{\mathrm{J},\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{v}\to \mathrm{c}}(\mathcal{E},\mathcal{E}-\hslash \omega )}=\frac{{f}_{\mathrm{e}}(\mathcal{E})[1-{f}_{\mathrm{e}}(\mathcal{E}+\hslash \omega )]{\rho }_{\mathrm{e}}(\mathcal{E}+\hslash \omega )}{[1-{f}_{\mathrm{h}}(\mathcal{E}-\hslash \omega )]{\rho }_{\mathrm{h}}(\mathcal{E}-\hslash \omega )[1-{f}_{\mathrm{e}}(\mathcal{E})]}\ll 1,\end{equation} \tag{ A10 }$$

given the fact that for an electron at energy $\mathcal{E}$ in the conduction band, $[1-{f}_{\mathrm{e}}(\mathcal{E})]\approx 1$, $[1-{f}_{\mathrm{e}}(\mathcal{E}+\hslash \omega )]\approx 1$, ${f}_{\mathrm{e}}(\mathcal{E})\ll 1$, and $[1-{f}_{\mathrm{h}}(\mathcal{E}-\hslash \omega )]\approx 1$, these are ensured by the presence of the gap because ${f}_{\mathrm{e}(\mathrm{h})}(\mathcal{E})\ll 1$ far away from the chemical potential, a condition that is mostly satisfied for intensities considered in the present study. Moreover, the ratio of the density of states in the above equation brings in a factor ${({m}_{\mathrm{e}}^{\ast }/{m}_{\mathrm{h}}^{\ast })}^{3/2}\ll 1$ (0.046 for GaAs, for example). Similar arguments lead to ${\rho }_{\mathrm{J},\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{v}\to \mathrm{v};\mathrm{i}\mathrm{n}}(\mathcal{E}-\hslash \omega ,\mathcal{E}-2\hslash \omega )/{\rho }_{\mathrm{J},\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{v}\to \mathrm{c}}(\mathcal{E},\mathcal{E}-\hslash \omega )\ll 1$. Therefore, we can rewrite equation (A9) as,

$$\begin{equation}{K}_{\mathrm{e}\mathrm{x}\mathrm{c}}^{(\mathrm{e})}{\int }_{-\infty }^{\infty }\mathrm{d}\mathcal{E}\,{\rho }_{\mathrm{J},\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{v}\to \mathrm{c}}(\mathcal{E},\mathcal{E}-\hslash \omega )=\frac{{n}_{\mathrm{e}\mathrm{x}\mathrm{c}}(\omega )}{{N}_{\mathrm{e}\mathrm{x}\mathrm{c}}}.\end{equation} \tag{ A11 }$$

We have explicitly verified these arguments in the numerical calculations for intensities we have considered and found the interband transition to be dominant so that using either equation (A9), or (A11) does not change the final result. Finally, the net change in the electronic population at $\mathcal{E}$ in the conduction band is, therefore,

$$\begin{equation}{\phi }_{\mathrm{e}\mathrm{x}\mathrm{c}}^{(\mathrm{e})}(\mathcal{E})={\int }_{0}^{\infty }\mathrm{d}\omega \frac{{n}_{\mathrm{e}\mathrm{x}\mathrm{c}}(\omega )}{{N}_{\mathrm{e}\mathrm{x}\mathrm{c}}}\left[\frac{{\rho }_{\mathrm{J},\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{v}\to \mathrm{c}}(\mathcal{E},\mathcal{E}-\hslash \omega )}{{\int }_{-\infty }^{\infty }\mathrm{d}\mathcal{E}\,{\rho }_{\mathrm{J},\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{v}\to \mathrm{c}}(\mathcal{E},\mathcal{E}-\hslash \omega )}\right],\quad \mathcal{E}\geqslant {\mathcal{E}}_{\mathrm{c}},\end{equation} \tag{ A12 }$$

where the explicit expression for the normalization constant ${K}_{\mathrm{e}\mathrm{x}\mathrm{c}}^{(\mathrm{e})}$, given by equation (A11), has been used. Therefore, the change in electron distribution at energy $\mathcal{E}$, in the conduction band due to photon absorption is given by

$$\begin{equation}{\left(\frac{\partial {f}_{\mathrm{e}}(\mathcal{E})}{\partial t}\right)}_{\mathrm{e}\mathrm{x}\mathrm{c}}=\frac{{N}_{\mathrm{e}\mathrm{x}\mathrm{c}}{\phi }_{\mathrm{e}\mathrm{x}\mathrm{c}}^{(\mathrm{e})}(\mathcal{E})}{{\rho }_{\mathrm{e}}(\mathcal{E})},\end{equation} \tag{ A13 }$$

which is (2) for c = e in the conduction band.

We aim to formulate the rate of change of hole population ${f}_{\mathrm{h}}(\mathcal{E})$ in the valence band due to photon absorption by adapting the formalism introduced in [66]. The population probability of holes in the valence band changes both due to the interband and intra-band transitions. However, following the explanations given in the appendix A.1 (see equations (A9)–(A11), and nearby discussions), we neglect any intra-band transition due to the absorption of a photon. Therefore, the joint probability of absorption of a photon of frequency between ω and ω + dω and excitation of a hole in the valence band is given by

$$\begin{equation}{A}_{\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{c}\to v}(\hslash \omega ,{\mathcal{E}}_{\mathrm{f}}=\mathcal{E})={K}_{\mathrm{e}\mathrm{x}\mathrm{c}}^{(\mathrm{h})}{D}_{\mathrm{J}}(\mathcal{E},\mathcal{E}+\hslash \omega ){\rho }_{\mathrm{J},\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{c}\to v}(\mathcal{E},\mathcal{E}+\hslash \omega ),\end{equation} \tag{ A14 }$$

where ${K}_{\mathrm{e}\mathrm{x}\mathrm{c}}^{(\mathrm{h})}$ is the normalization to be defined later and the superscript '(h)' denotes that the normalization constant corresponds to the excitation of hole. The joint density states corresponding to equation (A14) is

$$\begin{equation}{\rho }_{\mathrm{J},\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{c}\to v}(\mathcal{E},\mathcal{E}+\hslash \omega )=[1-{f}_{\mathrm{h}}(\mathcal{E})]{\rho }_{\mathrm{h}}(\mathcal{E})[1-{f}_{\mathrm{e}}(\mathcal{E}+\hslash \omega )]{\rho }_{\mathrm{e}}(\mathcal{E}+\hslash \omega ),\end{equation} \tag{ A15 }$$

where a hole at $\mathcal{E}+\hslash \omega$ in the conduction band moves to the valence band at $\mathcal{E}$ upon absorption of a photon of energy ℏω, $[1-{f}_{\mathrm{h}}(\mathcal{E})]{\rho }_{\mathrm{h}}(\mathcal{E})$ being the electron distribution in the valence band and $[1-{f}_{\mathrm{e}}(\mathcal{E}+\hslash \omega )]{\rho }_{\mathrm{e}}(\mathcal{E}+\hslash \omega )$ being the hole distribution in the conduction band. Assuming ${D}_{\mathrm{J}}(\mathcal{E},\mathcal{E}+\hslash \omega )$ to be a constant, i.e., all hole transitions are equally probable, and absorbing them into the normalization constant ${K}_{\mathrm{e}\mathrm{x}\mathrm{c}}^{(\mathrm{h})}$, we find the joint density of states in equation (A2) satisfies the following condition,

$$\begin{equation}{K}_{\mathrm{e}\mathrm{x}\mathrm{c}}^{(\mathrm{h})}{\int }_{-\infty }^{\infty }\mathrm{d}\mathcal{E}\,{\rho }_{\mathrm{J},\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{v}\to \mathrm{c}}(\mathcal{E},\mathcal{E}+\hslash \omega )=\frac{{n}_{\mathrm{e}\mathrm{x}\mathrm{c}}(\omega )}{{N}_{\mathrm{e}\mathrm{x}\mathrm{c}}}.\end{equation} \tag{ A16 }$$

Then the net change in the hole population at $\mathcal{E}$ in the valence band is given by

$$\begin{equation}{\phi }_{\mathrm{e}\mathrm{x}\mathrm{c}}^{(\mathrm{h})}(\mathcal{E})={\int }_{0}^{\infty }\mathrm{d}\omega \frac{{n}_{\mathrm{e}\mathrm{x}\mathrm{c}}(\omega )}{{N}_{\mathrm{e}\mathrm{x}\mathrm{c}}}\left[\frac{{\rho }_{\mathrm{J},\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{c}\to \mathrm{v}}(\mathcal{E},\mathcal{E}+\hslash \omega )}{{\int }_{-\infty }^{\infty }\mathrm{d}\mathcal{E}\,{\rho }_{\mathrm{J},\mathrm{e}\mathrm{x}\mathrm{c}}^{\mathrm{v}\to \mathrm{c}}(\mathcal{E},\mathcal{E}+\hslash \omega )}\right],\quad \mathcal{E}\leqslant {\mathcal{E}}_{\mathrm{v}}.\end{equation} \tag{ A17 }$$

Therefore, the rate of change in the hole population in the valence band due to the absorption of a photon is given by

$$\begin{equation}{\left(\frac{\mathrm{\partial }{f}_{\mathrm{h}}(\mathcal{E})}{\mathrm{\partial }t}\right)}_{\mathrm{e}\mathrm{x}\mathrm{c}}=\frac{{N}_{\mathrm{e}\mathrm{x}\mathrm{c}}{\phi }_{\mathrm{e}\mathrm{x}\mathrm{c}}^{(\mathrm{h})}(\mathcal{E})}{{\rho }_{\mathrm{h}}(\mathcal{E})},\end{equation} \tag{ A18 }$$

where ${\rho }_{\mathrm{h}}(\mathcal{E})$ the eDOS in the valence band. The number of holes excited due to photon absorption is $\frac{\mathrm{d}{n}_{\mathrm{h}}}{\mathrm{d}t}=\int \mathrm{d}\mathcal{E}\,{\rho }_{\mathrm{h}}(\mathcal{E}){\left(\frac{\partial {f}_{\mathrm{h}}}{\partial t}\right)}_{\mathrm{e}\mathrm{x}\mathrm{c}}$ which is equal to N_exc holes per unit time. Our formulation thus ensures that the number of photo-excited electrons and holes are the same.

We adapt the theoretical formalism for the spontaneous emission obtained using the usual Fermi golden rule, as explained in [67], to make it well-matched with the formulation of the photo-absorption. The rate of change of electron population due to the recombination of electrons with energy $\mathcal{E}$ in the conduction band is given by

$$\begin{equation}{\left(\frac{\partial {f}_{\mathrm{e}}}{\partial t}\right)}_{\mathrm{r}\mathrm{e}\mathrm{c}}=-\frac{{N}_{\mathrm{r}\mathrm{e}\mathrm{c}}{\phi }_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{e})}(\mathcal{E})}{{\rho }_{\mathrm{e}}(\mathcal{E})},\end{equation} \tag{ B1 }$$

and the rate of change of particles due to recombination is given by $\frac{\mathrm{d}{n}_{\mathrm{r}\mathrm{e}\mathrm{c}}}{\mathrm{d}t}=\int \mathrm{d}\mathcal{E}\,{\rho }_{\mathrm{c}}(\mathcal{E}){\left(\frac{\partial {f}_{\mathrm{e}}}{\partial t}\right)}_{\mathrm{r}\mathrm{e}\mathrm{c}}\ =-{N}_{\mathrm{r}\mathrm{e}\mathrm{c}}$, a negative sign signifying a loss of particles. Here, ${\phi }_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{e})}(\mathcal{E})$ is the net change in the electronic population at energy $\mathcal{E}$ due to the recombination in the conduction band which is given by

$$\begin{equation}{\phi }_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{e})}(\mathcal{E})={\int }_{0}^{\infty }\mathrm{d}{\omega }^{\prime }\frac{{n}_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{e})}(\hslash {\omega }^{\prime })}{{N}_{\mathrm{r}\mathrm{e}\mathrm{c}}}\left[\frac{{\rho }_{\mathrm{J},\mathrm{r}\mathrm{e}\mathrm{c}}^{\mathrm{c}\to v}(\mathcal{E},\mathcal{E}-\hslash {\omega }^{\prime })}{{\int }_{-\infty }^{\infty }\mathrm{d}\mathcal{E}\,{\rho }_{\mathrm{J},\mathrm{r}\mathrm{e}\mathrm{c}}^{\mathrm{c}\to v}(\mathcal{E}-\hslash {\omega }^{\prime },\mathcal{E})}\right],\quad \mathcal{E}\geqslant {\mathcal{E}}_{\mathrm{c}},\end{equation} \tag{ B2 }$$

where the joint density of states are given by

$$\begin{equation}{\rho }_{\mathrm{J},\mathrm{r}\mathrm{e}\mathrm{c}}^{\mathrm{c}\to v}(\mathcal{E}-\hslash {\omega }^{\prime },\mathcal{E})={f}_{\mathrm{e}}(\mathcal{E}){\rho }_{\mathrm{e}}(\mathcal{E}){f}_{\mathrm{h}}(\mathcal{E}-\hslash {\omega }^{\prime }){\rho }_{\mathrm{h}}(\mathcal{E}-\hslash {\omega }^{\prime }),\quad \mathcal{E}\geqslant {\mathcal{E}}_{\mathrm{c}},\end{equation} \tag{ B3 }$$

and is normalized such that

$$\begin{equation}{\int }_{-\infty }^{\infty }\mathrm{d}\mathcal{E}\,{K}_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{e})}{\rho }_{\mathrm{J},\mathrm{r}\mathrm{e}\mathrm{c}}^{\mathrm{c}\to v}(\mathcal{E}-\hslash {\omega }^{\prime },\mathcal{E})=\frac{{n}_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{e})}({\omega }^{\prime })}{{N}_{\mathrm{r}\mathrm{e}\mathrm{c}}},\end{equation} \tag{ B4 }$$

where ${n}_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{e})}({\omega }^{\prime })$ is the number density of emitted photons (per unit time, unit volume, and per unit frequency) of energy ℏω' within the range ω' and ω' + dω'. The total number of emitted photons is given by ${N}_{\mathrm{r}\mathrm{e}\mathrm{c}}=\int \mathrm{d}{\omega }^{\prime }\,{n}_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{e})}(\hslash {\omega }^{\prime })$ which equals the number of electrons recombining per unit volume. The intraband downward transition induced by photon emission, i.e., the intraband recombination is assumed to be negligible [66].

We initially choose ${n}_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{e})}({\omega }^{\prime })$ by the analytic result of the lhs of equation (B4) for thermal distributions [67, 68], namely,

$$\begin{equation}{n}_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{e})}({\omega }^{\prime })=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.\,{\rho }_{\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{t}}(\hslash {\omega }^{\prime })\sqrt{\hslash {\omega }^{\prime }-{\mathcal{E}}_{\mathrm{g}}}\,{\text{e}}^{-{\beta }_{\mathrm{e}}^{\ast }(\hslash {\omega }^{\prime }-{\mathcal{E}}_{\mathrm{g}})},\end{equation} \tag{ B5 }$$

where ρ_phot(ℏω') is the photonic density of states. In vacuum ${\rho }_{\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{t}}(\hslash {\omega }^{\prime })=\frac{\sqrt{{\epsilon}({\omega }^{\prime })}{\hslash }^{2}{\omega }^{\prime 2}}{{\pi }^{2}{(\hslash c)}^{3}}$ (in units of ${(\mathrm{e}\mathrm{V}\enspace {\mathrm{m}}^{3})}^{-1}$) where c is the speed of light and $\sqrt{{\epsilon}({\omega }^{\prime })}$ is the frequency dependent refractive index corresponding to the bulk SC. Note the ${\beta }_{\mathrm{e}}^{\ast }$ appearing in the above equation represents (the inverse of) an energy scale coming from the energy conservation and aids the convergence in the self-consistent calculation, thereby serving as a parameter for convergence.

It is worthwhile to point out that an alternative choice of ${n}_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{e})}({\omega }^{\prime })={\Theta}(\hslash {\omega }^{\prime }-{{\epsilon}}_{\mathrm{g}}){\Theta}(2\hslash \omega -{{\epsilon}}_{\mathrm{g}}-\hslash {\omega }^{\prime })$, ℏω being the energy of the incident light, which does not use any convergence parameter like ${\beta }_{\mathrm{e}}^{\ast }$, also brings in the energy conservation. This indicates that the matrix element ${K}_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{e})}$ is quite insensitive to the choice of ${n}_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{e})}({\omega }^{\prime })$ which can be attributed to the condition of the particle number conservation, viz, at the steady-state the number of electrons excited must be equal to the number of electrons recombine. The same holds for the holes too.

Analogous to the recombination of electrons, the joint probability of recombination of a hole from the valence band with an electron in the conduction band and emission of a photon of energy in the range ω' and ω' + dω', similar to that of the photo-excitation, is given by

$$\begin{equation}{A}_{\mathrm{r}\mathrm{e}\mathrm{c}}^{v\to \mathrm{c}}({\mathcal{E}}_{\mathrm{f}}=\mathcal{E}+\hslash {\omega }^{\prime },\hslash {\omega }^{\prime })={K}_{\mathrm{r}\mathrm{e}\mathrm{c}}^{\mathrm{c}\to v}{D}_{\mathrm{J}}(\mathcal{E}+\hslash {\omega }^{\prime },\mathcal{E}){\rho }_{\mathrm{J},\mathrm{r}\mathrm{e}\mathrm{c}}^{v\to \mathrm{c}}(\mathcal{E}+\hslash {\omega }^{\prime },\mathcal{E}),\end{equation} \tag{ B6 }$$

with the corresponding joint density of states,

$$\begin{equation}{\rho }_{\mathrm{J},\mathrm{r}\mathrm{e}\mathrm{c}}^{v\to \mathrm{c}}(\mathcal{E}+\hslash {\omega }^{\prime },\mathcal{E})={f}_{\mathrm{h}}(\mathcal{E}){\rho }_{\mathrm{h}}(\mathcal{E}){f}_{\mathrm{e}}(\mathcal{E}+\hslash {\omega }^{\prime }){\rho }_{\mathrm{e}}(\mathcal{E}+\hslash {\omega }^{\prime }),\quad \mathcal{E}\leqslant {\mathcal{E}}_{v},\end{equation} \tag{ B7 }$$

where ${K}_{\mathrm{r}\mathrm{e}\mathrm{c}}^{\mathrm{c}\to \mathrm{v}}$ is the normalization. Assuming all the electronic (in this case hole) transitions corresponding to the recombination to be equally probable, i.e., ${D}_{\mathrm{J}}(\mathcal{E},\mathcal{E}+\hslash {\omega }^{\prime }){K}_{\mathrm{r}\mathrm{e}\mathrm{c}}^{v\to \mathrm{c}}={K}_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{h})}$, (the superscript '(h)' denote that this normalization corresponds to the recombination of a hole), equation (B6) satisfy the following condition,

$$\begin{equation}{\int }_{-\infty }^{\infty }\mathrm{d}\mathcal{E}\,{A}_{\mathrm{r}\mathrm{e}\mathrm{c}}^{v\to \mathrm{c}}(\mathcal{E}+\hslash {\omega }^{\prime },\hslash {\omega }^{\prime })={\int }_{-\infty }^{\infty }\mathrm{d}\mathcal{E}\,{K}_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{h})}{\rho }_{\mathrm{J},\mathrm{r}\mathrm{e}\mathrm{c}}^{v\to \mathrm{c}}(\mathcal{E}+\hslash {\omega }^{\prime },\mathcal{E})=\frac{{n}_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{h})}({\omega }^{\prime })}{{N}_{\mathrm{r}\mathrm{e}\mathrm{c}}},\end{equation} \tag{ B8 }$$

which define the normalization constant ${K}_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{h})}$. The total number of emitted photons is given by ${N}_{\mathrm{r}\mathrm{e}\mathrm{c}}=\int \mathrm{d}{\omega }^{\prime }\,{n}_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{h})}({\omega }^{\prime })$ which is equal to the number of re-combinations happening, where ${n}_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{h})}({\omega }^{\prime })$ is the number density of emitted photons of energy ℏω' per unit time per unit frequency of emitted photons. We take the expression for ${n}_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{h})}({\omega }^{\prime })$ to be the same as that of the electrons but with ${\beta }_{\mathrm{h}}^{\ast }$ in place of ${\beta }_{\mathrm{e}}^{\ast }$.

Then, the rate of change of electron population due to the recombination of a hole at energy $\mathcal{E}$ in the valence band is given by

$$\begin{equation}{\left(\frac{\partial {f}_{\mathrm{h}}(\mathcal{E})}{\partial t}\right)}_{\mathrm{r}\mathrm{e}\mathrm{c}}=-\frac{{N}_{\mathrm{r}\mathrm{e}\mathrm{c}}{\phi }_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{h})}(\mathcal{E})}{{\rho }_{\mathrm{h}}(\mathcal{E})},\end{equation} \tag{ B9 }$$

where the net change in the electronic population at energy $\mathcal{E}$ in the valence band due to the recombination of holes with the electrons in the conduction band is given by

$$\begin{equation}{\phi }_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{h})}(\mathcal{E})={\int }_{0}^{\infty }\mathrm{d}{\omega }^{\prime }\frac{{n}_{\mathrm{r}\mathrm{e}\mathrm{c}}^{(\mathrm{h})}(\hslash {\omega }^{\prime })}{{N}_{\mathrm{r}\mathrm{e}\mathrm{c}}}\left[\frac{{\rho }_{\mathrm{J},\mathrm{r}\mathrm{e}\mathrm{c}}^{\mathrm{v}\to \mathrm{c}}(\mathcal{E}+\hslash {\omega }^{\prime },\mathcal{E})}{{\int }_{-\infty }^{\infty }\mathrm{d}\mathcal{E}\,{\rho }_{\mathrm{J},\mathrm{r}\mathrm{e}\mathrm{c}}^{\mathrm{v}\to \mathrm{c}}(\mathcal{E}+\hslash {\omega }^{\prime },\mathcal{E})}\right],\quad \mathcal{E}\leqslant {\mathcal{E}}_{v}.\end{equation} \tag{ B10 }$$