Chiral response of spin-spiral states as the origin of chiral transport fingerprints of spin textures

The transport properties of nontrivial spin textures are coming under closer scrutiny as the amount of experimental data and theoretical simulations is increasing. To extend the commonly accepted yet simplifying and approximate picture of transport effects taking place in systems with spatially varying magnetization, it is important to understand the transport properties of building blocks for spin textures—the homochiral spin-spiral states. In this work, by referring to phenomenological symmetry arguments based on the gradient expansion, and explicit calculations within the Kubo framework, we study the transport properties of various types of spin-spirals in a two-dimensional model with strong spin-orbit interaction. In particular, we focus on the contributions to the magnetoconductivity, the planar Hall effect, and the anomalous Hall effect, which are sensitive to the sense of chirality of the spiral states. We analyze the emergence, symmetry, and microscopic properties of the resulting chiral magnetoconductivity, chiral planar Hall effect, and chiral Hall effect in terms of spin-spiral propagation direction, cone angle, spiral pitch, and disorder strength. Our findings suggest that the presence of spin-spiral states in magnets can be readily detected in various types of magnetotransport setups. Moreover, the sizable magnitude of chiral contributions to the conductivity of skyrmions estimated from homochiral spirals implies that chiral, as opposed to topological, magnetotransport can play a prominent role for the detection of nontrivial spin textures.

Figure 1

Parametrization of Néel, Bloch, and cone-type spirals. Parts (a)–(c) show the parametrization of Néel (a), Bloch (b), and cone-type (c) spin-spirals on a sphere spanned by magnetic moments. The spiral propagates along a direction $\mathbf{q}$, and the magnetization rotates around an axis ${\mathbf{e}}_r$, which encloses a cone angle $\alpha$ with the initial magnetic moment ${\widehat{\mathit{n}}}_0$. For the Néel spiral, ${\mathbf{e}}_r\perp \mathbf{q}$, while for the Bloch spiral, ${\mathbf{e}}_r\parallel \mathbf{q}$. In this work, the plane of the honeycomb lattice is the $xy$-plane (axis $\widehat{\mathit{z}}$ is perpendicular to it), which contains $\mathbf{q}$. Plots (d)–(f) show the real-space distribution (as seen from the top) of corresponding magnetization textures for $\mathbf{q}\parallel \widehat{\mathit{x}}$.

Figure 2

Electronic structure of Néel, Bloch, and cone-type spirals. Band structures of Néel (a) and Bloch (b) spin-spirals in the energy interval $[-1,1]$ eV for $\mathbf{q}=\pm 0.31a_0^{-1}·\widehat{\mathit{x}}$ (blue and red). (c),(d) Band structures of a cone-type spiral with the cone angle $\alpha =0.6\text{rad}\approx 34.38$ in the energy range of $[-1,1]$ eV for opposite orientations of the wave vector $\mathbf{q}=\pm 0.31a_0^{-1}·\widehat{\mathit{x}}$, reflecting the changes in the electronic structure due to an opposite sense of chirality. The color of the bands in (c),(d) indicates the value of the band-resolved Berry curvature ${\mathrm{\Omega }}_{xy}\left(\mathbf{k}\right)$. (e)–(h) The real-space magnetic texture corresponding to band structures in the respective columns. For more details, see the main text.

Figure 3

Chiral magnetoconductivity (CMC), chiral planar Hall effect (CPHE), and chiral Hall effect (CHE). (a,d) Chiral magnetoconductivity (CMC), ${\sigma }_{xx}^c$, calculated for a Néel-type spiral with $\mathbf{q}=\pm 0.47a_0^{-1}·\widehat{\mathit{x}}$ as a function of Fermi energy $E_f$ and broadening $\mathrm{\Gamma }$, (a), and as a function of $E_f$ and $q$ (along $\widehat{\mathit{x}}$) at $\mathrm{\Gamma }$ of 100 meV, (d). The insets depict the corresponding behavior of the nonchiral part of the conductivity tensor element, ${\sigma }_{xx}^{\text{nc}}$. (b),(e) Chiral planar Hall effect (CPHE), ${\sigma }_{\left(xy\right)}^c$, calculated for a Bloch spiral with $\mathbf{q}=\pm 0.47a_0^{-1}·\widehat{\mathit{x}}$ as a function of $E_f$ and $\mathrm{\Gamma }$, (b), and as a function of $E_f$ and $q$ (along $\widehat{\mathit{x}}$) at $\mathrm{\Gamma }$ of 100 meV, (e). The corresponding nonchiral part of the conductivity tensor element, ${\sigma }_{\left(xx\right)}^{\text{nc}}$, vanishes in this case; see Table 4. (c),(f) Chiral Hall effect (CHE), ${\sigma }_{\left[xy\right]}^c$, calculated for a cone-type spiral with $\mathbf{q}=\pm 0.47a_0^{-1}·\widehat{\mathit{x}}$ and a cone angle of $\alpha =0.6\text{rad}\approx 34.38$ as a function of $E_f$ and $\mathrm{\Gamma }$, (c), and as a function of $E_f$ and $q$ (along $\widehat{\mathit{x}}$) at $\mathrm{\Gamma }$ of 100 meV, (f). The insets depict the corresponding behavior of the nonchiral part of the conductivity tensor element, ${\sigma }_{\left[xy\right]}^{\text{nc}}$.

Figure 4

Scaling with the cone angle and spin-spiral pitch. (a) Chiral magnetoconductivity (CMC) for a cone-type spiral (blue) calculated for cone angles in the range $\alpha \in [0,\pi /2]$ and CMC for a Néel-type spiral (red) calculated for cone angle $\alpha =0$ and extrapolated to larger cone angles by multiplying with $sin{\left(\alpha \right)}^3$; see (43). The relationship between CMC for cone-type and Néel spirals obtained from the gradient expansion is reproduced with explicit tight-binding calculations. The calculation was done for spin-orbit coupling ${\alpha }_R=0.04\text{eV}$ at a spiral pitch $q=0.031a_0^{-1}$. (b) Ferromagnetic limit of the chiral Hall effect (CHE) for a cone-type spiral. Surface term (orange), sea term (red), and the sum of the two (blue) yielding the CHE for nonzero broadening $\mathrm{\Gamma }$. (c) The CHE signal near the FM limit of $q=0$. The linear slope $\gamma \approx -10.752e^2{a}_0/h$ in the limit of $q\to 0$, participating in a gradient expansion prediction (50), can be estimated from calculations. For all plots, the data were calculated for the values of $E_f=-0.625\text{eV}$ and $\mathrm{\Gamma }=0.2\text{eV}$. A value of ${\alpha }_R=0.4\text{eV}$ was used in (b) and (c).