Giant Collective Spin-Orbit Field in a Quantum Well: Fine Structure of Spin Plasmons

We employ inelastic light scattering with magnetic fields to study intersubband spin plasmons in a quantum well. We demonstrate the existence of a giant collective spin-orbit (SO) field that splits the spin-plasmon spectrum into a triplet. The effect is remarkable as each individual electron would be expected to precess in its own momentum-dependent SO field, leading to D’yakonov-Perel’ dephasing. Instead, many-body effects lead to a striking organization of the SO fields at the collective level. The macroscopic spin moment is quantized by a uniform collective SO field, five times higher than the individual SO field. We provide a momentum-space cartography of this field.

Figure 1

Fine structure model of ISB spin plasmons. (a) Threefold splitting, induced by SO coupling, of the ISB spin-plasmon modes. $\Delta E$ denotes the difference of the mode energies with and without SO coupling, calculated for the studied GaAs quantum well, and $q$ is the magnitude of the plasmon momentum (here $\mathbf{q}\parallel [110]$). The splitting $\delta$ between the transverse $m_{\pm }$ modes is almost linear in $q$. (b) For a fixed $q=8.0\mu {\mathrm{m}}^{-1}$, calculated modulation of the splitting $\delta$ with the in-plane orientation of $\mathbf{q}$, labeled by the angle $\phi$ to [110]. (c) Sketch of the proposed interpretation of the transverse ISB spin plasmons $m_{\pm }$, as the precession of antiparallel $\mu$ collective magnetic moments about ${\mathbf{B}}_{\mathrm{SO}}^{\mathrm{coll}}$ at zero external field (left), and about the superposition ${\mathbf{B}}_{\mathrm{SO}}^{\mathrm{coll}}+{\mathbf{B}}_{\mathrm{ext}}$ when an external magnetic field ${\mathbf{B}}_{\mathrm{ext}}$ is applied (right).

Figure 2

Anisotropic splitting of the ISB spin-plasmon modes. (a) Top panel: Inelastic light-scattering spectrum of the ISB excitations, in polarized (dashed line) and depolarized (solid line) geometry. Bottom panel: Depolarized spectra obtained at fixed $q=8.0\mu {\mathrm{m}}^{-1}$, by varying the in-plane angle $\phi$ measured from $[110]$ (vertical offset for clarity). The single, quasi-Lorentzian peak observed is the sum of two Lorentzians (red dashed lines for the $\phi =90°$ spectrum) of same amplitude and linewidth, corresponding to the transverse spin-plasmon modes $m_{+}$ and $m_{-}$ split by an amount $\delta$. Inset: Scattering geometry showing angle definitions; ${\mathbf{k}}_{\mathbf{i}}$ and ${\mathbf{k}}_{\mathbf{s}}$ are the incoming and scattered light wave vectors. (b–d) Variation of the linewidth $w$ with $\phi$ for $q=10.2$, 8.0, and $5.4\mu {\mathrm{m}}^{-1}$ respectively. Lines: Theory (see text).

Figure 3

Variation of the composite linewidth $w$ with the external magnetic field ${\mathbf{B}}_{\mathrm{ext}}$. (a) $w(B_{\mathrm{ext}})$ plots obtained in the configuration ${\mathbf{B}}_{\mathrm{ext}}\parallel \mathbf{q}$ for a fixed $q=8.0\mu {\mathrm{m}}^{-1}$ and various in-plane angles $\phi$ (measured from [110]). (b) Corresponding $w(B_{\mathrm{ext}})$ plots obtained for ${\mathbf{B}}_{\mathrm{ext}}\perp \mathbf{q}$. Lines are guides for the eyes. Each $w(B_{\mathrm{ext}})$ plot is symmetric about a certain value of the external field (marked by a dotted circle) which cancels the corresponding component of the collective SO field ${\mathbf{B}}_{\mathrm{SO}}^{\mathrm{coll}}(\mathbf{q})$.

Figure 4

SO collective field and magnetic moment. (a) Components of the collective SO field parallel ($B_{\mathrm{SO},\parallel }^{\mathrm{coll}}$, filled circles) and perpendicular ($B_{\mathrm{SO},\perp }^{\mathrm{coll}}$, open circles) to $\mathbf{q}$ for $q=8.0\mu {\mathrm{m}}^{-1}$, as extracted from the data of Fig. 3, and compared with theory (lines). (b) Spin-plasmon magnetic moment $\mu$, experimental (squares) and theoretical (line). (c) Minimum (open diamonds) and maximum (filled diamonds) SO field $|{\mathbf{B}}_{\mathrm{SO}}^{\mathrm{coll}}|$ versus $q$, compared to theoretical values (lines). (d) Spin-plasmon magnetic moment averaged over $\phi$, experimental (squares) and theoretical (line), as a function of $q$. (e) Variation of the SO splitting $\delta$ with external magnetic field ${\mathbf{B}}_{\mathrm{ext}}\parallel \mathbf{q}$, for $q=8.0\mu {\mathrm{m}}^{-1}$ (symbols, same as in Fig. 3). The experimental data are very well reproduced by Eq. (3) (lines).