Using group theory and Kane-like k · p model together with the Löwdining partition method, we derive the expressions of spin-orbit coupling of electrons and holes, including the linear-k Rashba term due to the intrinsic structure inversion asymmetry and the cubic-k Dresselhaus term due to the bulk inversion asymmetry in wurtzite semiconductors. The coefficients of the electron and hole Dresselhaus terms of ZnO and GaN in wurtzite structure and GaN in zinc-blende structure are calculated using the nearest-neighbor sp 3 and sp 3 s * tight-binding models separately.

The Rashba and Dresselhaus spin-orbit (SO) interactions in 2D electron gases act as effective magnetic fields with momentum-dependent directions, which cause spin decay as the spins undergo arbitrary precessions about these randomly oriented SO fields due to momentum scattering. Theoretically and experimentally, it has been established that by fine-tuning the Rashba α and renormalized Dresselhaus β couplings to equal fixed strengths α ¼ β, the total SO field becomes unidirectional, thus rendering the electron spins immune to decay due to momentum scattering. A robust persistent spin helix (PSH), i.e., a helical spin-density wave excitation with constant pitch P ¼ 2π=Q, Q ¼ 4mα=ℏ 2 , has already been experimentally realized at this singular point α ¼ β, enhancing the spin lifetime by up to 2 orders of magnitude. Here, we employ the suppression of weak antilocalization as a sensitive detector for matched SO fields together with independent electrical control over the SO couplings via top gate voltage V T and back gate voltage V B to extract all SO couplings when combined with detailed numerical simulations. We demonstrate for the first time the gate control of the renormalized β and the continuous locking of the SO fields at α ¼ β; i.e., we are able to vary both α and β controllably and continuously with V T and V B , while keeping them locked at equal strengths. This makes possible a new concept: "stretchable PSHs," i.e., helical spin patterns with continuously variable pitches P over a wide parameter range. Stretching the PSH, i.e., gate controlling P while staying locked in the PSH regime, provides protection from spin decay at the symmetry point α ¼ β, thus offering an important advantage over other methods. This protection is limited mainly by the cubic Dresselhaus term, which breaks the unidirectionality of the total SO field and causes spin decay at higher electron densities. We quantify the cubic term, and find it to be sufficiently weak so that the extracted spin-diffusion lengths and decay times show a significant enhancement near α ¼ β. Since within the continuous-locking regime quantum transport is diffusive (2D) for charge while ballistic (1D) for spin and thus amenable to coherent spin control, stretchable PSHs could provide the platform for the much heralded long-distance communication ∼ 8-25 μm between solid-state spin qubits, where the spin diffusion length for α ≠ β is an order of magnitude smaller. DOI: 10.1103/PhysRevX.7.031010 Subject Areas: Condensed Matter Physics, Quantum Information, SpintronicsThe inextricable coupling between the electron spatial and spin degrees of freedom-the spin-orbit (SO) interactionunderlies many fundamental phenomena such as the spin Hall effects-quantum and anomalous [1]-and plays a crucial role in newly discovered quantum materials hosting FIG. 1. Stretchable PSHs. Illustration of spin helices at different values of α¼β accessible in the measurements. The position x þ for one 2π rotation (dashed curve) is changing for the gate-locked regime α¼β. The gray box h...

A persistent spin helix (PSH) is a robust helical spin-density pattern arising in disordered 2D electron gases with Rashba α and Dresselhaus β spin-orbit (SO) tuned couplings, i.e., α = ±β. 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 ν = 1, 2. For realistic GaAs wells we show that the Rashba αν and Dresselhaus βν couplings can be simultaneously tuned to equal strengths but opposite signs, e.g., α1 = β1 and α2 = −β2. In this regime and away from band anticrossings, our non-interacting electron gas sustains a topologically non-trivial 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 RahsbaDresselhaus bands. Our PSL opens up the unique possibility of observing topological phenomena, e.g., topological and skyrmion Hall effects, in ordinary GaAs wells with non-interacting electrons.PACS numbers: 71.70. Ej, 75.70.Tj, 72.25.Rb Topological spin textures in crystals arise in connection with the electron-electron interaction. Skyrmions in the fractional quantum Hall regime [1, 2], magnetic and multiferroic systems [3] exemplify spin patterns characterized by topological invariants associated with the nontrivial winding of the spins. Non-topological helical spin patterns, e.g., spin-density waves in metals [4] can also occur. When coupled to conduction electrons, the emergent electrodynamics of the non-trivial spin textures gives rise to fundamental phenomena, e.g., the topological and skyrmion Hall effects in chiral magnets [5].Here we show that non-interacting 2D electrons in twosubband quantum wells [6,7] with matched SO couplings of opposite signs α 1 = β 1 > 0, α 2 = −β 2 < 0, can sustain a Persistent Skyrmion Lattice (PSL), Fig. 1. This should allow the observation of fundamental topological phenomena in ordinary (non-magnetic) GaAs wells.The formation of this skyrmion lattice can be easily understood for ballistic electrons (later on we include disorder). For a single-subband well with α 1 = β 1 , the Rashba-Dresselhaus Hamiltoninan is effectively 1D: H RD1 = 2α 1 σ y p x / = gµ B σ y B y /2, i.e., an electronmomentum (p x )-dependent Zeeman interaction with a unidirectional effective magnetic field B y (y [110], x [110]). Here σ y is the Pauli matrix and µ B the Bohr magneton. The corresponding quantum evolution operator is: U RD1 (t) = e −igµB σy Byt/2 = e −iσyQ1x/2 , where, and m * the electron mass. Hence a spin-up electron injected at x = y = 0 precesses around this B y field such that