Electron density magnification of the collective spin-orbit field in quantum wells

The spin-orbit field acting on the spin waves of a spin-polarized electron gas is studied by inelastic light scattering on a CdMnTe quantum well. Above-barrier illumination allows us to vary the electronic density and control the collective Rashba and Dresselhaus coupling constants. We demonstrate that the enhancement between the single-particle and the collective spin-orbit field increases with increasing electronic density. This result is reproduced by a first-principles calculation. This behavior, which is opposite to usual Coulombic spin enhancements, reveals a novel aspect of the interplay of spin-orbit and Coulomb interactions in collective spin modes.

Figure 1

Determination of the collective spin-orbit field. (a) Inelastic light scattering (ILS) geometry: ${\mathit{k}}_i$ and ${\mathit{k}}_s$ are the incoming and scattered light wave vectors, respectively; $\mathbf{q}$ is the transferred momentum of in-plane orientation measured by the angle $\phi$ from $\left[100\right]$. An external magnetic field ${\mathbf{B}}_{\mathrm{ext}}$ is applied perpendicularly to $\mathbf{q}$. (b) ILS spectra of the spin-flip wave, obtained at $B_{\mathrm{ext}}=2\mathrm{T}$ and $\phi =\pi /4$, for a series of transferred momenta $q$. (c) Wave-vector dispersion of the SFW for $\phi =\pi /4$ and $\phi =3\pi /4$. (d) Variation of the linear term $E_1$ of the SFW dispersion (see text) as a function of the in-plane angle $\phi$.

Figure 2

Tuning of the 2DEG density. (a) Photoluminescence (PL) spectra, at $B_{\mathrm{ext}}=0$, for a series of power densities $F_{\mathrm{green}}$ of secondary illumination. (b) For the same values of $F_{\mathrm{green}}$, cross-polarized ILS spectra of the single-particle excitations at $B_{\mathrm{ext}}=0$ and $q=10.4\mu {\mathrm{m}}^{-1}$. (c) 2DEG density as a function of $F_{\mathrm{green}}$, as extracted from the PL (solid circles) and ILS (empty circles) data. The line is a fit to the theory of Chaves et al. [33].

Figure 3

Optical control of the collective spin-orbit field. (a) Collective Rashba ($\tilde{\alpha }$) and Dresselhaus ($\tilde{\beta }$) coupling constants as a function of the secondary illumination power density. (b) Same quantities as a function of the 2DEG density, as deduced from Fig. 2.

Figure 4

Enhancement factor $\mathcal{C}$ of the collective spin-orbit field, obtained from $\mathcal{C}=\tilde{\alpha }/\alpha$ (black squares) and $\mathcal{C}=\tilde{\beta }/\beta$ (red circles) as a function of the 2DEG density $n_{2\mathrm{D}}$. The black line corresponds to the theory of Eq. (4). Inset: Calculated single-particle Rashba ($\alpha$, black) and Dresselhaus ($\beta$, red) coefficients versus $n_{2\mathrm{D}}$.