### Expressions for the scattering cross section

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

2–

4]:

$\frac{{d}^{2}\sigma}{d\omega d\Omega}=\frac{{\omega}_{2}}{{\omega}_{1}}{\left(\frac{e}{c}\right)}^{4}\frac{{n}_{\omega}+1}{\Pi}\text{Im}\left({L}_{2}-\frac{2\Pi {e}^{2}}{q\kappa}\frac{{L}_{1}{\stackrel{~}{L}}_{1}}{\epsilon}\right),$

(1)

where

*L*_{2},

*L*_{1}, and

${\stackrel{~}{L}}_{1}$ are respectively given by the expressions

$\begin{array}{ll}{L}_{2}=\frac{1}{S}\sum _{{\beta}^{\prime}\beta}|{\gamma}_{{\beta}^{\prime}\beta}{|}^{2}{F}_{{\beta}^{\prime}\beta},& {L}_{1}=\frac{1}{S}\sum _{{\beta}^{\prime}\beta}{\gamma}_{{\beta}^{\prime}\beta}^{\ast}{J}_{{\beta}^{\prime}\beta}\left(\mathbf{q}\right){F}_{{\beta}^{\prime}\beta},\phantom{\rule{2em}{0ex}}\\ {\stackrel{~}{L}}_{1}=\frac{1}{S}\sum _{{\beta}^{\prime}\beta}{\gamma}_{{\beta}^{\prime}\beta}{J}_{{\beta}^{\prime}\beta}^{\ast}\left(\mathbf{q}\right){F}_{{\beta}^{\prime}\beta}.\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2em}{0ex}}\end{array}$

(2)

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*^{
i
q
r
},

*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;

$\widehat{\gamma}$ 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

$\epsilon (\omega ,q)=1+\frac{2\Pi {e}^{2}}{q\kappa}\frac{1}{S}\sum _{{\beta}^{\prime}\beta}\left|{J}_{{\beta}^{\prime}\beta}\right(\mathbf{q}){|}^{2}{F}_{{\beta}^{\prime}\beta},$

(3)

${F}_{{\beta}^{\prime}\beta}=\frac{{f}_{{\beta}^{\prime}}-{f}_{\beta}}{(\omega +{\epsilon}_{\beta {\beta}^{\prime}}+i\delta )},\phantom{\rule{1em}{0ex}}(\delta =+0),$

(4)

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${\epsilon}_{\beta {\beta}^{\prime}}={\epsilon}_{\beta}-{\epsilon}_{{\beta}^{\prime}}$.

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

_{III}B

_{V} semiconductor). In this case, the operator of scattering

$\widehat{\gamma}$ reads

$\widehat{\gamma}=\widehat{{\gamma}_{1}}+\widehat{{\gamma}_{2}}=A\left(\right({\mathbf{e}}_{\mathbf{1}}{\mathbf{e}}_{\mathbf{2}}^{\ast})+i(\mathbf{\sigma}\mathbf{a}\left)\right)J\left(\mathbf{q}\right),\phantom{\rule{1em}{0ex}}A=\frac{1}{3}\frac{{P}^{2}}{{E}_{g}-{\omega}_{1}},$

(5)

where *E*_{
g
} is the effective bandgap width,$\mathbf{a}=[{\mathbf{e}}_{1},{\mathbf{e}}_{2}^{\ast}]$, *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:

$\frac{{d}^{2}\sigma}{d\omega d\Omega}=\frac{{\omega}_{2}}{{\omega}_{1}}{\left(\frac{e}{c}\right)}^{4}\frac{{n}_{\omega}+1}{\Pi}R\left(\omega \right),R\left(\omega \right)=\sum _{j=1}^{4}{R}_{j}\left(\omega \right),$

(6)

where

${R}_{1}\left(\omega \right)=-\frac{{A}^{2}\kappa q}{2\Pi {e}^{2}}|{\mathbf{e}}_{1}\xb7{\mathbf{e}}_{2}^{\ast}{|}^{2}\text{Im}\left(\frac{1}{\epsilon}\right),$

(7)

${R}_{2}\left(\omega \right)=\frac{1}{S}\sum _{{\beta}^{\prime}\beta}\left|{\left({\gamma}_{2}\right)}_{{\beta}^{\prime}\beta}{|}^{2}\text{Im}\right({F}_{{\beta}^{\prime}\beta}),$

(8)

${R}_{3}\left(\omega \right)=-\frac{2\Pi {e}^{2}}{q\kappa}\text{Im}\left(\frac{Z\stackrel{~}{Z}}{\epsilon}\right),$

(9)

${R}_{4}\left(\omega \right)=A\phantom{\rule{1em}{0ex}}\text{Im}\left(\frac{1}{\epsilon}\left(\left({\mathbf{e}}_{1}{\mathbf{e}}_{2}^{\ast}\right)Z+\left({\mathbf{e}}_{1}^{\ast}{\mathbf{e}}_{2}\right)\stackrel{~}{Z}\right)\right).$

(10)

The values of *Z* and$\stackrel{~}{Z}$ are given by expressions for *L*_{1} and${\stackrel{~}{L}}_{1}$ 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$\stackrel{~}{Z}$ 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]

${\mathcal{\mathscr{H}}}_{0}=\frac{{\mathbf{p}}^{2}}{2m}+\alpha (\mathit{\sigma}\xb7[\mathbf{p},\mathbf{n}\left]\right).$

(11)

Here,

**p** is the 2D momentum of the electron,

*m* is the effective mass,

*α* is the Rashba parameter, and

**n** is the unit vector normal to the plane of the system. The spectrum of this Hamiltonian has the form

${\epsilon}_{\beta}={p}^{2}/2m+\mu \alpha p,$

(12)

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

${\psi}_{\mathbf{p},\mu}=\frac{{e}^{i\mathbf{pr}}}{\sqrt{2S}}\left(\begin{array}{c}i\mu {e}^{-i{\phi}_{\mathbf{p}}}\\ \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}1\end{array}\right).$

(13)