Effects of nuclear spins on the transport properties of the edge of two-dimensional topological insulators

The electrons in the edge channels of two-dimensional topological insulators can be described as a helical Tomonaga-Luttinger liquid. They couple to nuclear spins embedded in the host materials through the hyperfine interaction, and are therefore subject to elastic spin-flip backscattering on the nuclear spins. We investigate the nuclear-spin-induced edge resistance due to such backscattering by performing a renormalization-group analysis. Remarkably, the effect of this backscattering mechanism is stronger in a helical edge than in nonhelical channels, which are believed to be present in the trivial regime of InAs/GaSb quantum wells. In a system with sufficiently long edges, the disordered nuclear spins lead to an edge resistance which grows exponentially upon lowering the temperature. On the other hand, electrons from the edge states mediate an anisotropic Ruderman-Kittel-Kasuya-Yosida nuclear spin-spin interaction, which induces a spiral nuclear spin order below the transition temperature. We discuss the features of the spiral order, as well as its experimental signatures. In the ordered phase, we identify two backscattering mechanisms, due to charge impurities and magnons. The backscattering on charge impurities is allowed by the internally generated magnetic field, and leads to an Anderson-type localization of the edge states. The magnon-mediated backscattering results in a power-law resistance, which is suppressed at zero temperature. Overall, we find that in a sufficiently long edge the nuclear spins, whether ordered or not, suppress the edge conductance to zero as the temperature approaches zero.

Figure 1

In a 2DTI of rectangular shape, electrons propagate along the edges. In this work, without loss of generality, we focus on one of the four edges (labeled by the coordinate $r$), where the up-spin electron $L_{\uparrow }$ (blue) moves in the left direction, and the down-spin electron $R_{\downarrow }$ (red) moves in the right direction. The electron spin quantization axis is defined as the $z$ axis, which is perpendicular to the 2DTI plane. Nuclear spins (indicated by the green arrows) are randomly oriented (displayed at the top edge) above the transition temperature $T_0$, and form a spiral order below $T_0$. In the ordered phase, the nuclear spins align ferromagnetically within each cross section, and rotate along the edge, as demonstrated at the bottom edge. For clarity, the spins are drawn only for two opposite edges, and the spin up and spin down edge states are separated spatially.

Figure 2

Dependence of the localization temperature $T_{\mathrm{hf}}$ on the strength of electron-electron interactions for various materials. For the horizontal axis, we take the Luttinger liquid parameter $K$ for 2DTI helical edge states, and $K_c$ (while fixing $K_s=1$) for nonhelical channels in GaAs wires and in the trivial regime of InAs/GaSb. The other parameters are listed in Table 1. (Inset) The localization length ${\xi }_{\mathrm{hf}}$ as a function of the number of nuclei per cross section $N_{\perp }$.

Figure 3

Dependence of the resistance $R_{\mathrm{hf}}$ on different system parameters in the disordered phase for the parameters of InAs/GaSb. (a) $R_{\mathrm{hf}}$ vs length ($L$) at the temperature $T=0.3\mathrm{K}$. (b) $R_{\mathrm{hf}}$ vs $T$ for $L=10\mu \mathrm{m}$. (c) $R_{\mathrm{hf}}$ vs $T$ for $L=20\mu \mathrm{m}$. (d) Differential resistance as a function of the bias voltage ($V$) for $L=10\mu \mathrm{m}$ at $T=0.3\mathrm{K}$. In all the panels we take the Luttinger liquid parameter $K=0.2$. The other parameters are listed in Table 1.

Figure 4

The effect of nuclear helix ${\langle \tilde{\mathbf{I}}\rangle }_{\pm }$ on the electron spectrum. (a) For the spiral order ${\langle \tilde{\mathbf{I}}\rangle }_{+}$, a gap of the size ${\mathrm{\Delta }}_{\mathrm{m}}$ is opened at the Fermi surface (indicated by red line) by the corresponding Overhauser field. (b) For ${\langle \tilde{\mathbf{I}}\rangle }_{-}$, the gap is induced below the Fermi surface.

Figure 5

Magnon energy $E_{\mathrm{mag}}$ as a function of the momentum $q$ in an infinitely long edge. We take the order parameter $m_{2k_F}=1$ and an exaggerated temperature $T=10\mathrm{K}$ to make the dip at $q=0$ visible. For a realistic temperature, the dips are very narrow, and the magnon energy is approximately momentum independent $E_{\mathrm{mag}}\left(q\right)\approx \hslash {\omega }_{\mathrm{mag}}$. The $T$ dependence of $m_{2k_F}$ ($\hslash {\omega }_{\mathrm{mag}}$) given by Eq. (25) [by Eq. (28)] is shown in the left (right) inset.

Figure 6

Transition temperature $T_0$ as a function of the Luttinger liquid parameter for various materials. We take $K$ for 2DTIs and $K_c$ (with fixed $K_s=1$) for nonhelical channels. The other parameters are listed in Table 1.

Figure 7

The dependence of the real part of the nonlocal conductivity $\mathrm{Re}\left[{\sigma }_{\mathrm{nl}}(0,0,\omega )\right]$ on angular frequency ($\omega$), obtained from Eq. (32). Here we take the Luttinger liquid parameter $K=0.2$ ($K_{\mathrm{L}}=1$) for the edge (lead) electrons.

Figure 8

The dependence of the gap ${\mathrm{\Delta }}_{\mathrm{m}}$ opened in the electron spectrum on temperature $T$. On the left axis, the gap value is converted into the frequency ${\mathrm{\Delta }}_{\mathrm{m}}/h$. The Luttinger liquid parameter is taken to be $K=0.2$ ($K=0.25$) for the black solid (blue dashed) curve. The other parameters are listed in Table 1. The red shaded region marks the temperature region within which ${\mathrm{\Delta }}_{\mathrm{m}}/h\le 20\mathrm{GHz}$. (Inset) Zero-temperature gap ${\mathrm{\Delta }}_{\mathrm{m}}(T=0)$ as a function of $K$.

Figure 9

Schematics of Overhauser-field-assisted backscattering on impurities. The Overhauser field admixes $R_{\downarrow }\left(q\right)$ and $L_{\uparrow }(q+2k_F)$. The mixed $L_{\uparrow }$ component can then forward scatter to the Fermi surface due to the impurity random potential $V_{\mathrm{imp}}$, giving rise to the effective backscattering. For clarity, other scattering processes (e.g., electrons with the opposite velocities) are not shown.

Figure 10

Characteristic temperature $T_{\mathrm{hx}}$ for various materials as a function of the Luttinger liquid parameter $[K$ for 2DTI and $K_c$ (with fixed $K_s=1$) for nonhelical channels]. We take ${\lambda }_{\mathrm{mfp}}=1\mu \mathrm{m}$, and the other parameters are listed in Table 1. (Inset) Zero-temperature localization length ${\xi }_{\mathrm{hx}}(T=0)$ as a function of the same variable.

Figure 11

A summary of the resistance ($R$) as a function of the temperature $T$ in both the disordered and ordered phases for the parameters of InAs/GaSb. We take $K=0.2$, the mean free path ${\lambda }_{\mathrm{mfp}}=0.1\mu \mathrm{m}$, and $L=10\mu \mathrm{m}$, so that ${\xi }_{\mathrm{hx}}<L<{\xi }_{\mathrm{hf}}$. The other parameters are listed in Table 1. In the disordered phase ($T>T_0$), $R$ is given by Eqs. (18) and (17) for $T>\hslash u/\left(k_{B}L\right)$ and $T<\hslash u/\left(k_{B}L\right)$, respectively. In the ordered phase ($T<T_0$), three backscattering processes contribute to $R$, including the Overhauser-field-assisted backscattering on impurities $R_{\mathrm{hx}}$ [Eq. (40)], the magnon emission $R_{\mathrm{mag}}^{\mathrm{em}}$ [Eq. (44)], and the magnon absorption $R_{\mathrm{mag}}^{\mathrm{abs}}$ [Eq. (45)]. The gray curve gives the upper limit on the sum of the last two. Inset: $T$ dependence of the gap ${\mathrm{\Delta }}_{\mathrm{hx}}$.