the incoming photons, and is an integral, I = ∫ dωg ( ω ), of the spectral density g ( ω ) representing the probability for the incoming photon to scatter inelastically with energy ˜ Ω = Ω − ω , where ω is the Raman shift. The quantum efficiency expresses the total probabil- ity for single incoming photon to scatter inelastically in a proccess under consideration - that is, to scatter on an electron and excite an electron-hole pair in the low- energy bands. The probability for the incoming photon to scatter with the Raman shift ω in a particular direction (defined by the momentum ˜ q of the scattered photon), is, in turn, characterised by the angle-resolved probabil- ity of scattering w (˜ q ). Finally, the scattering probability w (˜ q ) that one photon is scattered with the excitation of an electron-hole (e-h) pair in the final state is related, as w ∝ |R| $^{2}$, to the scattering amplitude R of the Raman process. The amplitude R is the sum of the amplitudes corre- sponding to the one-step and two-step processes. Only terms quadratic in electron momentum p which appear in the addition δ ˆ H to the Hamiltonian in (1), contribute to the contact interaction. We show in Appendix that this contribution is much smaller than the leading con- tribution from the two-step processes, thus, we neglect it in further considerations. To find R , illustrated us- ing Feynman diagrams shown in Fig. 1(d), we describe a two-step transition which involves an intermediate vir- tual state | ν 〉 with energy ϵ$_{ν}$ , as structure in the vicinity of Brillouin zone corners, the electron initial and final energies ϵ$_{i}$ and ϵ$_{f}$ are related, ϵ$_{f}$ ≈ − ϵ$_{i}$ . To include the interaction of the electrons with photons, we construct the canonical momentum [ p − e ( A ( r , t $^{′}$)+ ˜ A ( r , t $^{′′}$))], where A ( r , t $^{′}$) and ˜ A ( r , t $^{′′}$) are the vector potentials of the incoming and outgoing light, re- spectively, and b$_{q}$$_{,q}$$_{z}$$_{,}$$_{l}$ is an annihilation operator for a photon with in-plane momentum q , out-of-plane momentum compo- nent q$_{z}$ and polarisation l . We expand the resulting Hamiltonian up to the second order in the vector po- tential and write down the interaction part, where j = − e ∂ ˆ H$_{0}$ ∂ p is the current vertex. We aim to calculate the spectral density g ( ω ) and the quantum efficiency (intensity) of the Raman scattering, I . The quantum efficiency describes the ratio of the flux of outgoing, inelastically scattered photons to the flux of The virtual state | ν 〉 may belong to any of the four bands, since an electron is excited from a state with momentum p to a state with momentum p + q or p − ˜ q depending on the accompanying photon process. At this step of the calculation we still work with the four-band Hamiltonian (1), to include the influence of the high-energy ”split” bands. In Eq. (4), the first (second) term corresponds to processes in which the photon is absorbed (emitted) in the first step and emitted (absorbed) in the second step of the process and is given by the first (second) diagram in the expression for R in Fig. 1(d). Integration in the time-dependent perturbation theory in Eq. (4) can be performed by changing variables to τ = t ′ − t $^{′′}$, which varies at the scale of ω − $^{1}$, ω = Ω − ˜ Ω, and ¯ t = ( t $^{′}$+ t $^{′′}$) / 2, which varies at the scale of ¯ Ω − $^{1}$, ¯ Ω = (Ω+Ω $^{′}$) / 2 ≫ ω . For incoming and outgoing photons, Ω , ˜ Ω ≫ γ$_{1}$ , and we also study the low energy excitations in the final states with Below, we analyse the contribution of electronic modes ω ≪ γ$_{1}$ . This allows us to expand factors 1 ± ¯ Ω − ϵ$_{ν}$ resulting from the integration over τ in powers of ( ϵ$_{ν}$/ Ω), keeping terms of the order of 1 and ( γ$_{1}$/ Ω) [the latter appear when the virtual state is taken to be in the high-energy bands] and to perform summation over the intermediate virtual states of the process. Consequently, the amplitude R takes the form of a matrix R = − 1 2 ϵ$_{0}$ √ Ω ˜ Ω ∑ ν ∫ ∞ −∞ ∫ t ′ −∞ e i $_{ℏ}$( ϵ$_{f}$ − ϵ$_{ν}$ ) t $^{′}$( j · ˜ l $^{∗}$) e − i $_{ℏ}$(˜ q · r − ˜ Ω t $^{$^{′}$)}$| ν 〉〈 ν | e i $_{ℏ}$( q · r − Ω t $^{$^{′′}$)}$( j · l ) e i $_{ℏ}$( ϵ$_{ν}$ − ϵ$_{i}$ ) t $^{′′}$dt $^{′}$dt$^{′′}$− − 1 2 ϵ$_{0}$ √ Ω ˜ Ω ∑ ν ∫ ∞ −∞ ∫ t ′ −∞ e i $_{ℏ}$( ϵ$_{f}$ − ϵ$_{ν}$ ) t $^{′}$( j · l ) e i $_{ℏ}$( q · r − Ω t $^{$^{′}$)}$| ν 〉〈 ν | e − i $_{ℏ}$(˜ q · r − ˜ Ω t $^{$^{′′}$)}$( j · ˜ l $^{∗}$) e i $_{ℏ}$( ϵ$_{ν}$ − ϵ$_{i}$ ) t $^{′′}$dt $^{′}$dt$^{′′}$. (4) R ≈ e 2 ℏ $^{2}$v2 ϵ$_{0}$ Ω 2 {$_{−}$ i ( σ$_{z}$ 0 0 σ$_{z}$ ) ( l × ˜ l ∗ ) $_{z}$+ M · d Ω } δ ( ϵ$_{f}$ − ϵ$_{i}$ − ω ); d = ( l$_{x}$ ˜ l ∗ y + l$_{y}$ ˜ l ∗ $_{x}$, l$_{x}$ ˜ l ∗ x − l$_{y}$ ˜ l ∗ $_{y}$); M = ( M $_{x}$, M $_{y}$); M $_{x}$= ( γ$_{1}$σ$_{y}$ ξv ( σ$_{y}$p$_{x}$ + σ$_{x}$p$_{y}$ ) ξv ( σ$_{y}$p$_{x}$ + σ$_{x}$p$_{y}$ ) 0 ) ; M $_{y}$= ( γ$_{1}$σ$_{x}$ ξv ( σ$_{x}$p$_{x}$ − σ$_{y}$p$_{y}$ ) ξv ( σ$_{x}$p$_{x}$ − σ$_{y}$p$_{y}$ ) 0 ) . ˆ H$_{int}$ = j · ( A ( r , t $^{′}$) + ˜ A ( r , t $^{′′}$) ) + e 2 2 ∑ i,j ∂ 2 ˆ H$_{0}$ ∂p$_{i}$∂p$_{j}$ A$_{i}$ ˜ A$_{j}$ , (3) A ( r , t $^{′}$) = 1 √$_{2}$$_{ϵ}$$_{0}$$_{Ω}$ ( l e i ( q · r − Ω t $^{′}$) / $^{ℏ}$b$_{q}$$_{,q}$$_{z}$$_{,}$$_{l}$ + h.c ) ; ˜ A ( r , t $^{′′}$) = 1 √ 2 ϵ$_{0}$ ˜ Ω ( ˜ l $^{∗}$e− i (˜ q · r − ˜ Ω t $^{′′}$) / $^{ℏ}$b † ˜ q , ˜ q$_{z}$, ˜ l + h.c. ) ; (2) 3