Bulk and surface spin conductivity in topological insulators with hexagonal warping

We investigate the spin conductivity of topological insulators taking into account both the surface and quasi-two-dimensional bulk states. We apply a low-energy expansion of the Hamiltonian up to the third order in momentum and take into account the vertex corrections arising due to the short-range disorder. Hexagonal warping gives rise to the additional anisotropic components in the spin conductivity tensor. Typically, the isotropic part of the spin conductivity is larger than the anisotropic one. The helical regime for the bulk states, in which the electrons in the Fermi level have the same projection of the spin on the direction of momentum, have been studied in more detail. In this regime, a substantial increase of the spin conductivity contribution from the bulk states at the Fermi level is observed. We find that the bulk spin conductivity is insensitive to disorder if Rashba spin-orbit coupling is larger than disorder strength, otherwise, it is strongly suppressed. The contribution to the spin conductivity from the surface states is almost independent of the chemical potential, robust to disorder and its value is comparable to the spin conductivity contribution from the bulk states per layer. The obtained results are in agreement with experimental data.

Figure 1

Energy spectrum, Eq. (2), and corresponding Fermi surface for different values of parameters. Spin direction of the states in the Fermi level is shown by arrows. (a)–(d) illustrate the spectrum and the Fermi surface for the bulk states at $s=0$ and $\lambda {\alpha }_R/r^2=0.2$; $r\mu /{\alpha }_R^2=-2$ in (a) and (b), $r\mu /{\alpha }_R=0.2$ in (c) and (d). Bold lines in (a) and (c) indicate the helical regime. (e) and (f) illustrate the spectrum and the Fermi surface for the surface states at $s{\alpha }_R^2/r^2=10$, $\lambda /s{\alpha }_R=0.2$, and $r\mu /{\alpha }_R=0.2$. Orange dashed lines indicate zero of the chemical potential and green dot-dashed lines show the chemical potential.

Figure 2

Scattering rate $\mathrm{\Gamma }$ for the bulk states as a function of the dimensionless chemical potential $\mu /{\mathrm{\Gamma }}_0$. Black line corresponds to the case, when the spin-orbit interaction and the hexagonal warping are absent, ${\alpha }_R,\lambda \to 0$. Green line corresponds to $r{\mathrm{\Gamma }}_0/{\alpha }_R^2=0.001$, $\lambda =0$, and $s=0$; red line to $r{\mathrm{\Gamma }}_0/{\alpha }_R^2=0.001$, $\lambda {\alpha }_R/r^2=0.1$, and $s=0$; blue line to $r{\mathrm{\Gamma }}_0/{\alpha }_R^2=0.001$, $s{\alpha }_R^2/r^2=0.1$, and $\lambda =0$.

Figure 3

Scattering rate $\mathrm{\Gamma }$ for the surface states as a function of the chemical potential $\mu$ for ${\gamma }_b=0.001$. Black line corresponds to the case $r=s=\lambda =0$, green line to $\lambda {\varepsilon }_0^2/{\alpha }_R^3=0.5$, $s=r=0$, red line to $s{\varepsilon }_0^2/{\alpha }_R^2=0.1$, $\lambda =r=0$, blue line to $s{\varepsilon }_0^2/{\alpha }_R^2=0.1$, $\lambda =0$. Normalization parameter is chosen as ${\varepsilon }_0={\alpha }_R{k}_{\mathrm{cut}}/10$ where $k_{\mathrm{cut}}$ is the cutoff momentum.

Figure 4

Dependence of ${\alpha }_R^{VC}$ for the bulk states on the chemical potential, ${\gamma }_b=0.001$. Black line corresponds to $\lambda =s=0$, red line to $\lambda {\alpha }_R/r^2=0.1$ and $s=0$, blue line to $s{\alpha }_R/r^2=0.1$ and $\lambda =0$.

Figure 5

Dependence of the vertex correction ${\alpha }_R^{VC}$ on the chemical potential for the surface states, ${\gamma }_b=0.001$. Black line corresponds to the case $r=\lambda =s=0$, blue line to $s{\varepsilon }_0^2/{\alpha }_R^2=0.1$, $s{\alpha }_R^2/r^2=10$, and $\lambda =0$, green line to $\lambda {\varepsilon }_0^2/{\alpha }_R^3=0.5$ and $r=s=0$, red line to $s{\varepsilon }_0^2/{\alpha }_R^2=0.1$ and $r=\lambda =0$.

Figure 6

Isotropic spin conductivity ${\sigma }_{xy}^{Iz}$ (top) and anisotropic spin conductivity ${\sigma }_{xx}^{Ix}$ (bottom) for the bulk states as a function of the chemical potential $\mu$, ${\gamma }_b=0.01$. In the top panel black line corresponds to $s=\lambda =0$, red line to $\lambda {\alpha }_R/r^2=0.1$ and $s=0$, blue line to $s{\alpha }_R^2/r^2=0.1$ and $\lambda =0$, green line to $s{\alpha }_R^2/r^2=0.05$ and $\lambda =0$. In the bottom panel black line corresponds to $\lambda {\alpha }_R/r^2=0.1$ and $s=0$, red to $\lambda {\alpha }_R/r^2=0.05$ and $s=0$, blue line to $\lambda {\alpha }_R/r^2=0.05$ and $s=\lambda /{\alpha }_R$.

Figure 7

Isotropic spin conductivity ${\sigma }_{xy}^{Iz}$ (top) and anisotropic spin conductivity ${\sigma }_{xx}^{Ix}$ (bottom) for the surface states as a function of the chemical potential, ${\gamma }_b=0.01$. In the top panel black line corresponds to $s{\alpha }_R^2/r^2=1$ and $\lambda =0$, red line to $s{\alpha }_R^2/r^2=5$ and $\lambda =0$, blue line to $s{\alpha }_R^2/r^2=5$ and $\lambda =s{\alpha }_R$. In the bottom panel black line corresponds to $s{\alpha }_R^2/r^2=5$ and $\lambda =s{\alpha }_R$, red line to $s{\alpha }_R^2/r^2=1$ and $\lambda =5s{\alpha }_R$, blue line to $s{\alpha }_R^2/r^2=5$ and $\lambda =s{\alpha }_R/5$.

Figure 8

Contribution to the spin conductivity from the states at the Fermi level, ${\sigma }_{\alpha \beta }^{I\gamma }$, for the bulk and surface states as a function of disorder strength ${\gamma }_b$. For all curves $\mu =-{\alpha }_R^2/r$. Black line corresponds to the contribution from the bulk states in ${\sigma }_{xy}^{Iz}$ for $s$, and $\lambda =0$, red line to the contribution from the bulk states in ${\sigma }_{xx}^{Ix}$ for $\lambda {\alpha }_R/r^2=0.1$ and $s=0$, blue line is the contribution from the surface states in ${\sigma }_{xy}^{Iz}$ for $s{\alpha }_R^2/r^2=5$ and $\lambda =0$, green line is the contribution from the surface states in ${\sigma }_{xx}^{Ix}$ for $s{\alpha }_R^2/r^2=5$ and $\lambda =s{\alpha }_R$.

Figure 9

Isotropic spin conductivity ${\sigma }_{xy}^{IIIz}$ (top) and anisotropic spin conductivity ${\sigma }_{xx}^{IIIx}$ (bottom) for the bulk states as a function of the chemical potential in the clean limit, $\mathrm{\Gamma }=0$. In the top panel black line corresponds to $\lambda$ and $s=0$, red line to $\lambda {\alpha }_R/r^2=0.1$ and $s=0$, blue line to $s{\alpha }_R^2/r^2=0.1$ and $\lambda =0$. In the bottom panel black line corresponds to $\lambda {\alpha }_R/r^2=0.05$ and $s=0$, red line to $\lambda {\alpha }_R/r^2=0.1$ and $s=0$, blue line to $\lambda {\alpha }_R/r^2=0.1$, $s{\alpha }_R=2\lambda$.

Figure 10

Isotropic spin conductivity ${\sigma }_{xy}^{IIIz}$ (top) and anisotropic spin conductivity ${\sigma }_{xx}^{IIIx}$ (bottom) for the surface states as a function of the chemical potential for $\mathrm{\Gamma }=0$. Black line corresponds to $s{\alpha }_R^2/r^2=5$ and $\lambda =0$, red line to $s{\alpha }_R^2/r^2=1$ and $\lambda =0$, green line to $s{\alpha }_R^2/r^2=1$ and $2\lambda =s{\alpha }_R$.

Figure 11

Bulk and surface contributions to the spin conductivity from the filled states, ${\sigma }_{\alpha \beta }^{III\gamma }$, as a function of disorder parameter $\mathrm{\Gamma }$ at $\mu =-{\alpha }_R^2/r$. Black line corresponds to the isotropic component due the bulk states, ${\sigma }_{xy}^{IIIz}$, for $s$ and $\lambda =0$, red line presents the anisotropic component due to the bulk states, ${\sigma }_{xx}^{IIIx}$, for $\lambda {\alpha }_R/r^2=0.1$ and $s=0$, blue line is the isotropic component due to the surface states, ${\sigma }_{xy}^{IIIz}$, for $s{\alpha }_R^2/r^2=5$ and $\lambda =0$, green line is the anisotropic component due to the surface states, ${\sigma }_{xx}^{IIIx}$, for $s{\alpha }_R^2/r^2=5$ and $\lambda =s{\alpha }_R$.

Figure 12

Total bulk spin conductivity per layer (top) and total surface spin conductivity (bottom) calculated for parameters from Table 1 and ${\gamma }_b={10}^{-3}$. Black line shows isotropic spin conductivity ${\sigma }_{xy}^z$ and red line shows anisotropic component ${\sigma }_{xx}^x$.