Spin-helix Larmor mode

Abstract

A two-dimensional electron gas (2DEG) with equal-strength Rashba and Dresselhaus spin-orbit coupling sustains persistent helical spin-wave states, which have remarkably long lifetimes. In the presence of an in-plane magnetic field, there exist single-particle excitations that have the character of propagating helical spin waves. For magnon-like collective excitations, the spin-helix texture reemerges as a robust feature, giving rise to a decoupling of spin-orbit and electronic many-body effects. We prove that the resulting spin-flip wave dispersion is the same as in a magnetized 2DEG without spin-orbit coupling, apart from a shift by the spin-helix wave vector. The precessional mode about the persistent spin-helix state is shown to have an energy given by the bare Zeeman splitting, in analogy with Larmor's theorem. We also discuss ways to observe the spin-helix Larmor mode experimentally.

Figure 1

Reference frames $ℛ\prime$ (black) and $ℛ$ (red) for the electronic states in a 2DEG with SOC and in-plane magnetic field B. The striped pattern along the φ = 45° direction indicates the persistent spin helix state of the 2DEG, with wave vector Q, which forms in the absence of B if the Rashba and Dresselhaus coupling strengths are equal.

Figure 2

Single-particle energies E_−,k and E_+,k for α = 0.05 and k along the [110] direction, see Eq. (13). (a) No magnetic field (Z^* = 0). Linear combinations of E_−,k and E_+,k+Q states have spin helix texture, but these cancel out if summed over all occupied states below E_F. A persistent spin helix appears if a quasiparticle is injected at the Fermi surface, as shown. (b) Finite magnetic field (Z^* = 0.0381). Single-particle excitations across the Fermi energy with momentum transfer Q (thick green arrow) give rise to propagating spin helices.

Figure 3

(a) Spin-wave dispersion ω_sw,0(q) (line) and single-particle spin-flip continuum (shaded area) without SOC. (b) Spin-wave dispersion ω_sw(q) and single-particle spin-flip continuum, plotted along [110], for α = β = 0.003 (Q = 0.012). The 2DEG parameters are r_s = 2, ζ = −0.0762, Z = 0.02, and Z^* = 0.0381 (all values are in atomic units). The inset shows the position of Larmor’s mode in the wave vector plane (q_x′, q_y′).

Figure 4

Proposed experimental design for the direct excitation of the spin-helix Larmor mode. Left: photo-conductive antenna on top of the sample, converting an infrared optical pulse into a short current pulse. Right: close-up view of the metal stripes on top of the spin-polarized 2DEG. The currents (blue arrows) are in alternating directions in neighboring stripes; the induced magnetic fields (pink circles) trigger a standing spin wave in the 2DEG.

Figure 5

Spin-wave stiffness of the 2DEG, obtained with the ALDA, for various values of the spin polarization ζ between 0.05 and 0.95.