An Unusual Dresselhaus Spin-Orbit Contribution of Even Order in Momentum

Abstract

The spin-orbit (SO) coupling is conventionally known to manifest as \emph{odd} functions of momentum. Here, through both model calculations and symmetry analysis along with the method of invariants, we reveal that, in ordinary semiconductor heterostructures, a \emph{quadratic} Dresselhaus SO term -- inheriting from its bulk crystal form -- emerges via the interband effect, while complying with time-reversal and spatial symmetries. Furthermore, we observe that this unusual SO term gives rise to a range of striking quantum phenomena, including hybridized swirling texture, anisotropic energy dispersion, avoided band crossing, longitudinal \emph{Zitterbewegung}, and opposite spin evolution between different bands in quantum dynamics. These stand in stark contrast to those associated with the usual \emph{linear} SO terms. Our findings uncover a previously overlooked route for exploiting interband effects and open new avenues for spintronic functionalities that leverage unusual SO terms of \emph{even} orders in momentum.

Figures (4)

• Figure 1: (Color online) (a) Growth profile of an n-doped ${\rm Al_{0.48}In_{0.52}As/Ga_{0.47}In_{0.53}As}$quantum well subject to gate voltage $V_{\rm g}$, and the potential profile for 2DEGs with two-band energies $\varepsilon_1$ and $\varepsilon_2$. (b) Illustration of intra- and interband Dresselhaus SO couplings with spin and band degrees of freedom. Here $\beta_{\nu}$$(\nu=1,2)$ denotes the intraband SO connecting distinct spins within the same $\nu$-th band, and $\Gamma^{(1)}$ ($\Gamma^{(2)}$) represents the linear (quadratic) interband SO between spins of opposite (same) orientations. The red and blue arrows stand for the spin-up and spin-down states, respectively.
• Figure 2: (Color online) (a) The quadratic (interband $\Gamma^{(2)}$) and (b) the linear (intraband $\beta_\nu$ and interband $\Gamma^{(1)}$) Dresselhaus SO coefficients versus $V_{g}$. (c), (d) Spin-resolved two-band energy dispersion (scaled by a factor of 100 for visibility) along the $k_x$ direction, with interband coupling $\Gamma^{(1)}$(c) and $\Gamma^{(2)}$(d). The size of markers scales with the degree of spin polarization and the color represents spin orientations up (red) and down (blue). The gray (dashed) lines indicate the uncoupled ($\Gamma^{(1)}=\Gamma^{(2)}=0$) bands. (e), (f) Constant-energy contours and spin textures at ${E_{\rm F}=-0.098}$ eV [see horizontal (green) dashed line in (c) and (d)], for our system with $\Gamma^{(1)}$ (e) and $\Gamma^{(2)}$ (f).
• Figure 3: (Color online) (a), (b) Zitterbewegung induced by the linear ($\Gamma^{(1)}$) (a) and quadratic ($\Gamma^{(2)}$) (b) interband SO terms, for the initial (group) velocity $\mathbf{v}_{\rm g}=\hbar k_{0y}/m^*\hat{\mathbf{y}}$ (along the $y$ direction) with $k_{0y}=\{0.5, 1.0, 1.5\}\times 10^{6}$ cm$^{-1}$. In (b), the $x$-direction motion vanishes for all $k_{0y}$; the inset shows the $y$-direction trajectory over time. (c), (d) Time evolution of spin components $\sigma_{x,y,z}^\nu$ ($\nu=1,2$) for the two bands in the presence of $\Gamma^{(1)}$ (c) and $\Gamma^{(2)}$ (d), at $k_{0y}=1.0\times 10^6$ cm$^{-1}$.
• Figure 4: (Color online) (a), (b) Time evolution of the two-band spin components $\sigma_{x,y,z}^\nu$ (a) and Zitterbewegung (b), incorporating both linear ($\Gamma^{(1)}$) and quadratic ($\Gamma^{(2)}$) SO contributions. In (a), $k_{0y}=0.05\times 10^6$ cm$^{-1}$; in (b), $k_{0y}=\{0.5, 1.0, 1.5\}\times 10^{6}$ cm$^{-1}$.