In the random-phase approximation, the differential cross section for the scattering by a 2D system can be written as follows 2 3 4 :

d 2 σ d ω d Ω = ω 2 ω 1 e c 4 n ω + 1 Π Im L 2 − 2 Π e 2 q κ L 1 L ~ 1 ε ,

where L ₂,L ₁, and L ~ 1 are respectively given by the expressions

L 2 = 1 S ∑ β ′ β | γ β ′ β | 2 F β ′ β , L 1 = 1 S ∑ β ′ β γ β ′ β ∗ J β ′ β ( q ) F β ′ β , L ~ 1 = 1 S ∑ β ′ β γ β ′ β J β ′ β ∗ ( q ) F β ′ β .

Here, ω _1,2 are the incident and scattered light frequencies, respectively; q = q ₁ − q ₂, q _1,2 are the in-plane components of the incident and scattered light wave vectors, respectively; ω = ω ₁−ω ₂ is the frequency shift in the inelastic light scattering; n _ω = 1/(e ^ω/T −1) is the Bose distribution function; J(q) = e ^iqr , β is the set of quantum numbers characterizing an electron state in the conduction band; S is the normalization area; κ is the background dielectric constant; γ ̂ is the scattering operator; and ℏ = 1 is assumed throughout this paper. The longitudinal dielectric function of electrons in the conduction band ε has the form

ε ( ω , q ) = 1 + 2 Π e 2 q κ 1 S ∑ β ′ β | J β ′ β ( q ) | 2 F β ′ β ,

F β ′ β = f β ′ − f β ( ω + ε β β ′ + i δ ) , ( δ = + 0 ) ,

where f _β ≡ f(ε _β ), f(ε) is the Fermi distribution function, ε _β is the energy of an electron in the conduction band, and ε β β ′ = ε β − ε β ′ .

The resonant situation is considered when the frequencies of incident (scattered) wave ω ₁(ω ₂) are close to E ₀ + Δ ₀, i.e., resonance with the spin-orbit split-off band takes place (E ₀ and Δ ₀ are the band parameters of the bulk A_IIIB_V semiconductor). In this case, the operator of scattering γ ̂ reads

γ ̂ = γ 1 ̂ + γ 2 ̂ = A ( ( e 1 e 2 ∗ ) + i ( σ a ) ) J ( q ) , A = 1 3 P 2 E g − ω 1 ,

where E _g is the effective bandgap width, a = [ e 1 , e 2 ∗ ] , P ≡ p _cv /m ₀ is the Kane parameter, e _1,2 are the polarizations of incident and scattered photons, and σ are the Pauli matrices. We treat here the enhanced resonant factor A in Equation 5 just as a constant that is true for not extremely resonant regime: the denominator in Equation 5 is much larger than the Fermi energy of electrons. We do this in order to simplify calculations because our main goal in this paper is to demonstrate the qualitatively new features of the scattering process due to spin-orbit interaction.

The substitution of Equation 5 into Equation 1 yields an expression comprising four characteristic contributions to the scattering:

d 2 σ d ω d Ω = ω 2 ω 1 e c 4 n ω + 1 Π R ( ω ) , R ( ω ) = ∑ j = 1 4 R j ( ω ) ,

where

R 1 ( ω ) = − A 2 κ q 2 Π e 2 | e 1 · e 2 ∗ | 2 Im 1 ε ,

R 2 ( ω ) = 1 S ∑ β ′ β | ( γ 2 ) β ′ β | 2 Im ( F β ′ β ) ,

R 3 ( ω ) = − 2 Π e 2 q κ Im Z Z ~ ε ,

R 4 ( ω ) = A Im 1 ε ( e 1 e 2 ∗ ) Z + ( e 1 ∗ e 2 ) Z ~ .

The values of Z and Z ~ are given by expressions for L ₁ and L ~ 1 in Equation 2 with γ replaced by γ ₂.

The contribution R ₁ determines the scattering of light by fluctuations of charge density. The value R ₂ determines unscreened mechanism of scattering and corresponds to single-particle excitations. It can be shown that in the absence of SOI in the conduction band, the values of Z and Z ~ and, respectively, R ₃ and R ₄ are equal to 0 identically.

Equations 7 to 10 are general. They are valid for any Hamiltonian, describing electron states in the conduction band. In this paper, we consider the light scattering for the so-called Rashba plane, namely 2D electron gas in the presence of SOI. Such a system is described by the Hamiltonian 5

ℋ 0 = p 2 2 m + α ( σ · [ p , n ] ) .

Here, p is the 2D momentum of the electron, m is the effective mass, α is the Rashba parameter, and nis the unit vector normal to the plane of the system. The spectrum of this Hamiltonian has the form

ε β = p 2 / 2 m + μ α p ,

where β = (p,μ) and the parameter μ = ± 1 labels two branches of the spin-split spectrum. The wave functions of the Hamiltonian (Equation 11) are

ψ p , μ = e i pr 2 S i μ e − i φ p 1 .

Equations 7 to 10, 12, and 13 were used for numerical calculations of scattering cross section as a function of frequency shift ω. They were carried out for 2D electron gas at temperature T = 0 in the scattering geometry when incident and scattered beams make a right angle and lie in the same plane (Figure 1). The structure InAs/GaSb with α = 1.44 × 10⁶ cm/s, m = 0.055 m ₀, and κ = 15.69 was considered at the areal concentration n _s = 10¹¹ cm⁻². The contributions R ₁,R ₃, and R ₄ contain plasmon poles (zeros of ε). To get finite results, it is necessary to introduce a finite damping. We replace δ in Equation 4 by the relaxation frequency ν = e/mμ (μ is the mobility). For q and ν, we have chosen the following values: ℏq = 0.004 p _F , ℏν = 0.001 ε _F .

If the polarizations of incident and scattered waves are strictly parallel (e ₁||e ₂), the cross section is determined by only R ₁. The spectrum of scattering has two peaks: the plasmon peak ω 0 ( q ) = v F q / a B and the SOI-induced peak at 2α p _F . The dependence of the cross section on the frequency shift for both peaks is similar to the plasmon absorption and is given by Figure 2.

Let the geometry of scattering in such a way that incident and scattered waves are linearly polarized and, moreover, e ₁ ⊥ e ₂ and a _y = 0. It can be realized, e.g., if incident and scattered beams are perpendicular and one of them is polarized in the incidence plane but the other is perpendicular to it. In this case, the spectrum demonstrates peculiarities due to only single-particle transitions (contribution R ₂): one peak near the frequency q v _F and another peak near the frequency 2α p _F . This case is demonstrated by Figure 3.

Due to SOI, the plasmon peak in the light scattering spectrum can occur even at strictly perpendicular polarizations of incident and scattered waves. It occurs when vector a has a nonzero projection onto axis y (axes z and x were chosen along vectors n and q, respectively). The nonvanishing contribution to the cross section is due to the sum R ₂ + R ₃. This case is presented by Figure 4.

For the existence of the contribution R ₄, polarization vectors e ₁ and e ₂ should be arbitrarily oriented with respect to each other (neither parallel nor perpendicular). Besides, at least one of the waves should not be linearly polarized. When these conditions are justified and R ₄ ≠ 0, all other contributions (R ₁,R ₂,R ₃) also exist. Herewith, due to the sensitivity of the contribution R ₄ to the sign of the effective Rashba SOI αand to polarization vector phases, the total cross section of scattering also depends on these parameters. Therefore, measurements of inelastic light scattering can be, in principle, used for the determination of the sign of the constant α. Figure 5 shows an example of the inelastic light scattering spectrum in the most interesting case when the incident wave has right or left circular polarization while the scattered one is linearly polarized at the angle Π/4 to the incidence plane. It is seen that at α > 0, the amplitude of the plasmon peak for left polarization is distinctly larger than that for right polarization. For α < 0, the curves should be interchanged.