Nuclear spin relaxation in Rashba nanowires

We study the nuclear spin relaxation in a ballistic nanowire with hyperfine and Rashba spin-orbit interactions (SOI) and in the presence of magnetic field and electron interactions. The relaxation rate shows pronounced peaks as a function of magnetic field and chemical potential due to van Hove singularities in the Rashba bands. As a result, the regimes of weak and strong SOIs can be distinguished by the number of peaks in the rate. The relaxation rate increases with increasing magnetic field if both Rashba subbands are occupied, whereas it decreases if only the lowest one is occupied.

Figure 1

Schematics of a nanowire of cross-sectional area, $d_x\times d_y$, confining a one-dimensional electron gas with Rashba electric field ${\mathbf{E}}_R$ directed along the $y$ axis and with magnetic field $\mathbf{B}$ along the $z$ axis. The itinerant electrons are coupled by hyperfine interaction to localized nuclear spins (arrows). The Fermi wave-length ${\lambda }_F\approx d_x,d_y$ is much larger than the lattice spacing between nuclear spins.

Figure 2

The nuclear spin relaxation rate $1/{\tilde{T}}_1=T_1\left(0\right)/\left(2T_1(\mu /h)\right)$ as a function of $\mu /h$ for small (a) and large (b) SOI strengths $m{\lambda }^2/h=0.2$ and $m{\lambda }^2/h=10$, respectively, as numerically obtained from Eq. (5). The rate is normalized by its value at $\mu =0$ (we set $\mu =0$ at the middle of the Zeeman gap between the Rashba subbands at $p=0$). The pronounced peaks are due to van Hove singularities at the edges of the Rashba subbands. (a) Curves from top to bottom are for $T/h=0.07,0.1,0.2,0.3$, respectively. (b) Curves from top to bottom are for $T/h=0.05,0.07,0.1$, respectively. Inset (a): The Rashba spectrum with possible Fermi levels (lines). Strong SOI gives rise to regions in the spectrum (dashed-dotted line) with negative group velocity and nonmonotonic relaxation rate as shown in (b). The strong increase of the rate for $\mu /h$ approaching the band bottom $E_0/h=-5$ [see also inset (b)] signals the breakdown of perturbation theory.

Figure 3

Nuclear spin relaxation rate $1/{\tilde{T}}_1=T_1{(h=0)}_{\mu =0}/(2T_1(h/m{\alpha }^2))$ as a function of Zeeman energy $h$ normalized by the SOI energy $h/m{\alpha }^2$ for two cases: (a) $\mu /m{\alpha }^2=2$, $T/m{\alpha }^2=0.01,0.02,0.025$. The corresponding Fermi level is shown by the solid line in Fig. 2. (b) $\mu /m{\alpha }^2=-0.38$, $T/m{\alpha }^2=0.005,0.0075,0.01$. The corresponding Fermi level is shown by the dashed-dotted line in Fig. 2. Note that the singularities are smoothened into finite peaks by temperature effects.