Colloquium: Persistent spin textures in semiconductor nanostructures

Device concepts in semiconductor spintronics make long spin lifetimes desirable, and the requirements put on spin control by proposals for quantum information processing are even more demanding. Unfortunately, due to spin-orbit coupling electron spins in semiconductors are generically subject to rather fast decoherence. In two-dimensional quantum wells made of zinc-blende semiconductors, however, the spin-orbit interaction can be engineered to produce persistent spin structures with extraordinarily long spin lifetimes even in the presence of disorder and imperfections. Experimental and theoretical developments on this subject for both $n$-doped and $p$-doped structures are reviewed and possible device applications are discussed.

Figure 1

Left: Dispersion relation of free electrons showing a gap of about 1 MeV between solutions of positive and negative energy. Right: Schematic band structure of III-V zinc-blende semiconductors with a band gap of typically 1 eV. The $p$-type valence band consists of the heavy and light hole branches and the split-off band.

Figure 2

Schematic of a spin field-effect transistor: An electron is emitted from a spin-polarized source and enters a semiconductor region with spin-orbit coupling being externally controllable via a perpendicular gate voltage. In the above example the spin-orbit interaction reverses the electron’s spin during its path, and since the drain electrode is polarized opposite to the source, the conductance of the device is high.

Figure 3

Fermi contours for an electron system with band mass $m=0.067m_0$ (corresponding to GaAs), a typical Fermi energy of ${ϵ}_f=10\mathrm{meV}$, and a Dresselhaus parameter of $\beta =10\mathrm{meV}\mathrm{nm}$. With a growing Rashba parameter the energy dispersion becomes increasingly anisotropic. For the case $\alpha =\beta$ (bottom right) the spin directions being independent of wave vector are indicated. Adapted from [141].

Figure 4

Schematic of the persistent spin helix occurring in a [001] quantum well for spin-orbit coupling tuned to $\alpha =\beta$ according to Eq. (38). The shift vector $\overrightarrow{Q}$ defines the pitch of the helix and points along the lateral direction. The spin density component in the longitudinal direction vanishes, ${\overline{s}}^2=0$. Adapted from [55].

Figure 5

Upper left panel: Decay curves of spin gratings obtained by [85] for different wave numbers $q$ of the initially modulated spin density. The data show decay on two distinct time scales. Upper right panel: Lifetimes of the spin helices with enhanced (${\tau }_E$) and reduced (${\tau }_R$) stability. The former one corresponds to the spin density configuration (38) with a maximum lifetime at $q=2Q$. Lower panels: Lifetime of the spin helix configuration as a function of the wave number for different doping asymmetry (varying the Rashba coupling, left) and well width (varying the Dresselhaus term, right). Adapted from [85].

Figure 6

Kerr rotation data obtained by [165]. The detection method is sensitive to the out-of-plane component $s^z$ of the spin density. Upper panels: Time evolution of the persistent spin helix from an initial local spin polarization along the growth direction (left) generated by a pump laser. Here the $x$ and $y$ directions point along $[1\overline{1}0]$ and [110], respectively, and the detection method is sensitive only to the out-of-plane component $s^z(\overrightarrow{r},t)$ of the spin density shown. Lower panels: Time evolution of the spin density in zero magnetic field (left) and in an in-plane field of $B=-1\mathrm{T}$ along $[1\overline{1}0]$ rotating the in-plane spin components into the growth direction. Adapted from [165].

Figure 7

Measured magnetoconductance for different gate voltages in two quantum wells differing in width. In the narrower well (left) a transition from weak antilocalization to weak localization and back occurs. From [83].

Figure 8

Upper panels (a) Time- and spatially resolved Kerr rotation data by [34]. The dynamics of an initial spin polarization along the [110] growth direction is followed along the $[1\overline{1}0]$ and [001] directions. To generate nontrivial dynamics a magnetic field $B_{\mathrm{ext}}$ of various strength is applied along $[1\overline{1}0]$. Lower panels (b) Corresponding Monte Carlo simulation results. From [34].

Figure 9

(a) Sketch of a curved two-dimensional electron system with Rashba spin-orbit coupling induced by asymmetric radial confinement. (b) General case: Spin-orbit interaction leads to anisotropic Fermi contours ($k_t=q_{\phi }$, spin direction given by arrows). (c) Isotropic Fermi contours of a flat system with Rashba coupling [cf. Eq. (8)]. (d) Fermi contours of the curved system with Rashba coupling tuned according to Eq. (41): Two circles displaced by a shift vector. From [161].