Voltage-induced conversion of helical to uniform nuclear spin polarization in a quantum wire

We study the effect of bias voltage on the nuclear spin polarization of a ballistic wire, which contains electrons and nuclei interacting via hyperfine interaction. In equilibrium, the localized nuclear spins are helically polarized due to the electron-mediated Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. Focusing here on nonequilibrium, we find that an applied bias voltage induces a uniform polarization, from both helically polarized and unpolarized spins available for spin flips. Once a macroscopic uniform polarization in the nuclei is established, the nuclear spin helix rotates with frequency proportional to the uniform polarization. The uniform nuclear spin polarization monotonically increases as a function of both voltage and temperature, reflecting a thermal activation behavior. Our predictions offer specific ways to test experimentally the presence of a nuclear spin helix polarization in semiconducting quantum wires.

Figure 1

A sketch of a conducting wire (yellow cylinder) with itinerant electrons (not shown) that couple to localized nuclear spins (red arrows) via hyperfine interaction. As a result, a helical nuclear polarization emerges below a critical temperature. The blue spiral is a guide to the eye showing the direction of the helical polarization. The helical plane is chosen to be perpendicular to the wire axis (which need not be the case in general).

Figure 2

Sketch of the energy spectrum given in Eq. (18) and the direction of the electron spins in the presence of the helical Overhauser field ${\mathit{B}}_h$ and the uniform Overhauser field ${\mathit{B}}_u$ perpendicular to the plane of the helix. Red arrows denote the spin directions of the electrons in the lower subband, and the blue arrows label the spin directions for the upper subband. The coordinate system for the spins is formed by $\mathit{h}$ and $\mathit{u}$ shown in the right lower corner. The chemical potentials for left and right movers are denoted as ${\mu }_L$ and ${\mu }_R$, respectively. The voltage applied to the wire is $e{V}_{RL}={\mu }_R-{\mu }_L$.

Figure 3

(a) The voltage dependence of the polarization $p_h$ along helical direction $\mathit{h}$ (blue), polarization $p_u$ in the direction of $\mathit{u}$ perpendicular to the helix plane (red), and the overall polarization of nuclei $\sqrt{p_u^2+p_h^2}$ (black). (b) Enlarged from (a) the voltage dependence of $p_h$ and $\sqrt{p_u^2+p_h^2}$. We use $T=90\text{mK}$ and other parameters as given in the text. We recall that our parametrization is such that $p_u=1$ corresponds to $B_u=B_{\max }$, which is about 5 T in GaAs, and analogously for $B_h$.

Figure 4

Plot of the temperature dependence of the polarization $p_u$ (upper panel, red) and the polarization $p_h$ (lower panel, blue). For these plots the same parameters were used as in Fig. 3 and the applied voltage is $eV=0.5\mu \text{eV}$. We note that our calculation is valid for $eV,k_{B}T<2{\mu }_e{B}_h$, therefore the smallest value of $p_h$ we consider here is $p_h\simeq 0.2$.