Spin torques and charge transport on the surface of topological insulator

We study various aspects of interplay between two-dimensional helical electrons, realized on the surface of a three-dimensional topological insulator, and the magnetization of a ferromagnet coupled to them. The magnetization is assumed to be perpendicular to the surface, with small transverse fluctuations $\mathit{u}$. In the first part of this paper, we calculate spin torques that the helical electrons exert on the magnetization. Up to first orders with respect to $\mathit{u}$, space/time derivative and electric current, we have determined all torques, which include Gilbert damping, spin renormalization, current-induced spin-orbit torques, and gradient corrections to them. Thanks to the identity between the velocity and spin in this model, these torques have exact interpretation in terms of transport phenomena, namely, diagonal conductivity, (anomalous) Hall conductivity, and corrections to them due to ordinary Hall effect on top of the anomalous one. These torque (and transport) coefficients are studied in detail with particular attention to the effects of vertex corrections and type of impurities (normal and magnetic). It is shown rigorously that the conventional current-induced torques, namely, spin-transfer torque and the so-called $\beta$ term, are absent. An electromotive force generated by spin dynamics, which is the inverse to the current-induced spin-orbit torque, is also studied. In the second part, we study the feedback effects arising as combinations of current-induced spin-orbit torques and spin-dynamics-induced electromotive force. It is demonstrated that the Gilbert damping process in this system is completely understood as a feedback effect. Another feedback effect, which may be called “magnon-drag electrical conductivity,” is shown to violate the exact correspondence between spin-torque and transport phenomena demonstrated in the first part.

Figure 1

Self-energy in the Born approximation.

Figure 2

Feynman diagram for transverse spin susceptibility (upper diagram). ${\Lambda }^{\beta }$ represents the renormalized vertex defined by the lower diagrams.

Figure 3

Normalized Gilbert damping constant $\tilde{\alpha }$ [Eq. (35)] as a function of chemical potential $\mu$ for ${\gamma }_0=0.1$, with (1) and without (2) vertex corrections. Partial contributions ${\tilde{\alpha }}^{\prime }$ and ${\tilde{\alpha }}^{\prime \prime }$ are also plotted as 3 and 4, respectively.

Figure 4

Normalized spin renormalization $\delta \tilde{S}$ [Eq. (35)] as a function of chemical potential $\mu$ for ${\gamma }_0=0.1$, with (1) and without (2) vertex corrections. Partial contributions $\delta {\tilde{S}}^{\prime }$ and $\delta {\tilde{S}}^{\prime \prime \prime }$ are also plotted as 3 and 4, respectively.

Figure 5

Parameters $c_0,c_1,\rho ,\theta$ [see Eqs. (A59, A60, A61)], $\xi$ and $C_{\zeta }$ [see Eq. (B12)] as functions of $\mu$ for ${\gamma }_0=0.1$. $c_0$ and $\rho$ are almost overlapped.

Figure 6

Normalized Gilbert damping constant $\tilde{\alpha }$ as a function of chemical potential $\mu$ in the presence of magnetic impurities (${\gamma }_z$) in addition to normal impurities (${\gamma }_0$). Their relative magnitudes are changed by keeping ${\gamma }_0+{\gamma }_z=0.1$ and ${\gamma }_{\perp }=0$. See Eq. (A76) for the notations.

Figure 7

Normalized spin renormalization $\delta \tilde{S}$ as a function of chemical potential $\mu$ in the presence of magnetic impurities (${\gamma }_z$) in addition to normal impurities (${\gamma }_0$). Their relative magnitudes are changed by keeping ${\gamma }_0+{\gamma }_z=0.1$ and ${\gamma }_{\perp }=0$.

Figure 8

Diagrammatic expression for $K_{ij}^{\alpha \beta }$ without (a) and with (b), (c) vertex corrections. The shaded triangles in (b) and (c) represent renormalized vertices: the one with ${\Lambda }^{\alpha }$ represents a renormalized spin (${\sigma }^{\alpha }$) vertex, and that with ${\tilde{\Lambda }}_i$ represents a renormalized velocity ($v_i$) vertex. The former has been defined in Fig. 2, and the latter is defined by ${\tilde{\Lambda }}_i=-v_{\mathrm{F}}{\varepsilon }_{i\alpha }{\Lambda }^{\alpha }$. The double line with a cross in (c) represents a ladder defined in (d).

Figure 9

The coefficients $A$ and $B$ of current-induced torques with magnetization gradient as functions of chemical potential $\mu$ for ${\gamma }_0=0.1$, with (1) and without (2) vertex corrections. For $B$, partial contributions $B^{\prime }$ and $B^{\prime \prime \prime }$ are also plotted as 3 and 4, respectively.

Figure 10

The coefficients $A$ and $B$ of current-induced torques with magnetization gradient as functions of chemical potential $\mu$ in the presence of magnetic impurities (${\gamma }_z$) in addition to normal impurities (${\gamma }_0$). Their relative magnitudes are changed by keeping ${\gamma }_0+{\gamma }_z=0.1$ and ${\gamma }_{\perp }=0$.

Figure 11

Diagrammatic expressions for the two feedback effects (a), (b) and the renormalization of magnon propagator (c). (a) The feedback torque due to induced current. This is actually identical to the one calculated in Sec. 3. The dashed rectangle indicates the intermediate state discussed in the main text. (b) The feedback correction to conductivity due to induced magnetization dynamics. The wavy line represents the propagator ($\widehat{D}$) of induced magnons, and the electron bubble is essentially the conductivity tensor $i\omega \widehat{\sigma }$, without feedback effects. (c) Dyson equation for the magnon propagator. The thick (thin) wavy line represents the renormalized (bare) propagator $\widehat{D}$ (${\widehat{D}}_0$). The electron bubble contributes to the self-energy $\widehat{\Pi }=M^2{\widehat{\chi }}_{\perp }=-M^2\widehat{\varepsilon }\widehat{\sigma }\widehat{\varepsilon }/{\left(e{v}_{\mathrm{F}}\right)}^2$, which is essentially the transverse spin susceptibility ${\widehat{\chi }}_{\perp }$, or the (irreducible) conductivity tensor, given by (a).

Figure 12

Contributions to $K_{ij}^{\alpha \beta }$. For simplicity, those without vertex corrections are shown. The original contribution is shown in Fig. 8 in which each line represents $G=(g_{\parallel }+g_{\perp })D$. Expansion with respect to $g_{\parallel }$ and $g_{\perp }$ generates processes shown in (a)–(c) in this figure. Only (a) contributes to the final result since each diagram of (b) vanishes and the four diagrams of (c) sum to zero. In the figure, a line with a letter $m$ attached represents $g_m{\sigma }^m$, $m=x,y$, and the one without such a letter represents $g_{\parallel }$. In (c), $\overline{m}=y$ for $m=x$, and $\overline{m}=x$ for $m=y$. Sum over $m$ is understood. These features remain valid even if the vertex corrections (${\sigma }^{\alpha }\to {\Lambda }^{\alpha }$, $v_i\to {\tilde{\Lambda }}_i$) are included.