Raman scattering in a two-dimensional Fermi liquid with spin-orbit coupling

We present a microscopic theory of Raman scattering in a two-dimensional Fermi liquid (FL) with Rashba and Dresselhaus types of spin-orbit coupling and subject to an in-plane magnetic field $\left(\vec{B}\right)$. In the long-wavelength limit, the Raman spectrum probes the collective modes of such a FL: the chiral spin waves. The characteristic features of these modes are a linear-in-$q$ term in the dispersion and the dependence of the mode frequency on the directions of both $\vec{q}$ and $\vec{B}$. All of these features have been observed in recent Raman experiments on ${\mathrm{Cd}}_{1-x}{\mathrm{Mn}}_x\mathrm{Te}$ quantum wells.

Figure 1

(a) Schematically: excitation spectrum (at $q=0)$ of a 2D Fermi liquid with spin-orbit coupling and subject to an in-plane magnetic field $\vec{B}$. Shaded region: continuum of spin-flip excitations; lines: chiral spin waves. (b) Continuum as a function of the angle ${\theta }_{\vec{B}}$ between $\vec{B}$ and the (100) direction. ${\mathrm{\Delta }}_R/{\mathrm{\Delta }}_D=0.5,\eta ={\mathrm{\Delta }}_Z/{\mathrm{\Delta }}_D$, where ${\mathrm{\Delta }}_{D/R/Z}$ is Dresselhaus/Rashba/Zeeman splitting. (c) Continuum (shaded) and collective-mode frequency at $q=0$ (line) as a function of $\varphi ={\theta }_{\vec{B}}-\pi /2$ for a ${\mathrm{Cd}}_{1-x}{\mathrm{Mn}}_x\mathrm{Te}$ quantum well.

Figure 2

Diagrams for the differential cross section of Raman scattering in cross-polarized geometry. Shaded bubbles denote vertex corrections due to the exchange part of eei, calculated in the ladder approximation. $V\left(q\right)$ is the bare Coulomb potential.

Figure 3

(a) Variation of the CSW velocity $w$, as defined in Eq. (1), with the angle $\varphi ={\theta }_{\vec{B}}-\pi /2$. Inset: variation of the frequency at $q=0$ with $\varphi$. (b) Dispersion of a CSW in the CdMnTe quantum well in the magnetic field of 2 T for $\varphi =\pi /4$. To be consistent with experimental geometry, we fixed $\vec{B}\perp \vec{q}$. The negative sign of the field implies flipping its direction.