Coulomb-driven organization and enhancement of spin-orbit fields in collective spin excitations

Spin-orbit (SO) fields in a spin-polarized electron gas are studied by angle-resolved inelastic light scattering on a CdMnTe quantum well. We demonstrate a striking organization and enhancement of SO fields acting on the collective spin excitation (spin-flip wave). While individual electronic SO fields have a broadly distributed momentum dependence, giving rise to D'yakonov-Perel' dephasing, the collective spin dynamics is governed by a single collective SO field which is drastically enhanced due to many-body effects. The enhancement factor is experimentally determined. These results provide a powerful indication that these constructive phenomena are universal to collective spin excitations of conducting systems.

Figure 1

Intrasubband spin excitations of a CdMnTe quantum well. (a) Scattering geometry: ${\mathbf{k}}_{\mathbf{i}}$ and ${\mathbf{k}}_{\mathbf{s}}$ are the incoming and scattered light wave vectors; $\mathbf{q}$ is the transferred momentum, of in-plane orientation $\phi$ measured from $\left[100\right]$, and amplitude $\left|\mathbf{q}\right|\simeq \frac{4\pi }{\lambda }\sin \theta$, where $\lambda \simeq 771\mathrm{n}\mathrm{m}$ is the incoming wavelength. An external magnetic field ${\mathbf{B}}_{\mathrm{ext}}$ is applied perpendicularly to $\mathbf{q}$ (the arrow defines $B_{\mathrm{ext}}>0$). (b) Typical inelastic light scattering (ILS) spectrum obtained at $q\simeq 0$ in cross-polarized geometry. Two lines are observed, corresponding to the spin-flip wave (SFW, energy $Z$) and to the spin-flip single-particle excitations (SPE, energy $Z^{*}$). (c) Magnetic dispersion of $Z$ and $Z^{*}$. (d) ILS spectra obtained for $B_{\mathrm{ext}}=3\mathrm{T}$ and a series of transferred momenta $q$, at fixed in-plane angle $\phi =\pi /4$.

Figure 2

Spin-orbit induced modulation of the spin-flip wave energy, (a) for $B_{\mathrm{ext}}=3\mathrm{T}$ and (b) $B_{\mathrm{ext}}=-3\mathrm{T}$, and for three amplitudes of the transferred momentum $q$. The modulation shows a two-fold symmetry, and its amplitude increases with $q$. Furthermore, as compared with $B_{\mathrm{ext}}>0$, the modulation for $B_{\mathrm{ext}}<0$ is out of phase, and lies at higher energy. The lines reproduce Eq. (6).

Figure 3

Extraction of the collective spin-orbit coupling constants. (a) Schematics of the four experimental configurations $(\phi ,B_{\mathrm{ext}})$ used to extract $\tilde{\alpha }$ and $\tilde{\beta }$. ${\mathbf{B}}_{\mathrm{SO},\mathrm{R}}^{\mathrm{coll}}$ is the Rashba component of the collective SO field (blue), and ${\mathbf{B}}_{\mathrm{SO},\mathrm{D}}^{\mathrm{coll}}$ the Dresselhaus component (purple). (b) SFW dispersions obtained for the configurations depicted above (same color code). Lines correspond to Eq. (6). (c) For $\left|B_{\mathrm{ext}}\right|=2$, 3 and $4\mathrm{T}$, plots of $E_{\pm }\left(q,3\pi /4\right)-E_{\pm }\left(q,\pi /4\right)$. Both quantities are linear in $q$. (d) For the same values of $\left|B_{\mathrm{ext}}\right|$, plots of the quantity $〈E_{-}〉\left(q\right)-〈E_{+}〉\left(q\right)$.

Figure 4

Extraction of the individual spin-orbit coupling constants. (a) Experimental ILS spectra obtained at $B_{\mathrm{ext}}=3\mathrm{T}$ and $q=0.6\mu {\mathrm{m}}^{-1}$, for a series of equally spaced in-plane orientations between $\phi =\pi /4$ (bottom curve) and $\phi =3\pi /4$ (top curve). (b) With the same color code, calculated imaginary part of the Lindhard polarizability of Eq. (7) using $\alpha =3.3\mathrm{m}\mathrm{e}\mathrm{V}\mathring{\mathrm{A}}$ and $\beta =4.6\mathrm{m}\mathrm{e}\mathrm{V}\mathring{\mathrm{A}}$. The grey points are guides to the eye.