Persistent Skyrmion Lattice of Noninteracting Electrons with Spin-Orbit Coupling

A persistent spin helix (PSH) is a robust helical spin-density pattern arising in disordered 2D electron gases with Rashba $\alpha$ and Dresselhaus $\beta$ spin-orbit (SO) tuned couplings, i.e., $\alpha =\pm \beta$. Here, we investigate the emergence of a persistent Skyrmion lattice (PSL) resulting from the coherent superposition of PSHs along orthogonal directions—crossed PSHs—in wells with two occupied subbands $\nu =1$, 2. For realistic GaAs wells, we show that the Rashba ${\alpha }_{\nu }$ and Dresselhaus ${\beta }_{\nu }$ couplings can be simultaneously tuned to equal strengths but opposite signs, e.g., ${\alpha }_1={\beta }_1$ and ${\alpha }_2=-{\beta }_2$. In this regime, and away from band anticrossings, our noninteracting electron gas sustains a topologically nontrivial Skyrmion-lattice spin-density excitation, which inherits the robustness against spin-independent disorder and interactions from its underlying crossed PSHs. We find that the spin relaxation rate due to the interband SO coupling is comparable to that of the cubic Dresselhaus term as a mechanism of the PSL decay. Near anticrossings, the interband-induced spin mixing leads to unusual spin textures along the energy contours beyond those of the Rahsba-Dresselhaus bands. Our PSL opens up the unique possibility of observing topological phenomena, e.g., topological and Skyrmion Hall effects, in ordinary GaAs wells with noninteracting electrons.

Figure 1

(a) Energy dispersion for a GaAs double well with two subbands (no disorder) and (b) its potential profile and wave functions. (c) Calculated SO couplings vs $V_g$: intraband (interband) Rashba ${\alpha }_{\nu }$ ($\eta$) and Dresselhaus ${\beta }_{\nu }$ ($\mathrm{\Gamma }$). The dotted-dashed vertical line (orange) indicates the crossed PSH symmetry point ${\alpha }_1={\beta }_1$ and ${\alpha }_2=-{\beta }_2$. (d) Energy contours: the arrows pointing along the orthogonal axes $x$ (pink) and $y$ (green) define the subband SO fields within subbands 1 and 2, respectively. (e) PSL pattern in the 2DEG. The size of the circles and arrows denote $⟨s_z⟩$ and $⟨s_{x,y}⟩$, respectively. Blue (dark gray) circles stand for spins up and red (light gray) for spins down.

Figure 2

(a) Energy dispersions $E_{\mathbf{k},{\lambda }_1,{\lambda }_2}$ (scaled by a factor of 10 for visibility) along $k_{\overline{x}}\parallel [100]$ (or $k_x=k_y$) for an InSb double well. The black solid lines correspond to the uncoupled ($\eta =\mathrm{\Gamma }=0$) bands $E_{\nu ,\mathbf{k}}^{\pm }$ and cross at $k_c$. For $\eta$, $\mathrm{\Gamma }\ne 0$, these bands anticross (dashed lines). Away from $k_c$, the coupled and uncoupled cases coincide. The label sets (1, 2, 3, 4) and (5, 6, 7, 8) denote Fermi points along $k_{\overline{x}}$ at $E_F^{\prime }$ and $E_F$, respectively. Panels (b) and (c) show $⟨{\sigma }_{\overline{x}}⟩$ and $⟨{\sigma }_{\overline{y}}⟩$, respectively, along $k_{\overline{x}}$. The solid and dashed lines correspond to the respective energy branches in (a).