Spin Hall effect emerging from a chiral magnetic lattice without spin-orbit coupling

TL;DR

This paper proposes a novel intrinsic spin Hall effect mechanism in chiral magnetic lattices that does not require spin-orbit coupling, expanding the understanding and potential design of spin Hall materials.

Contribution

It introduces an alternative spin Hall effect mechanism based on chiral magnetic structures, independent of spin-orbit coupling and scalar spin chirality.

Findings

Large spin Hall conductivities in Mn3X compounds from ab initio calculations

Demonstration of SHE without spin-orbit coupling or scalar spin chirality

Potential for designing spin Hall materials without heavy elements

Abstract

The spin Hall effect (SHE), which converts a charge current into a transverse spin current, has long been believed to be a phenomenon induced by the spin--orbit coupling. Here, we propose an alternative mechanism to realize the intrinsic SHE through a chiral magnetic structure that breaks the spin rotation symmetry. No spin--orbit coupling is needed even when the scalar spin chirality vanishes, different from the case of the topological Hall effect. In known chiral antiferromagnetic compounds Mn$_3X$ ($X=$ Ga, Ge, and Sn), for example, we indeed obtain large spin Hall conductivities based on \textit{ab initio} calculations. Apart further developing the conceptual understanding of the SHE, our work suggests an alternative strategy to design spin Hall materials without involving heavy elements, which may be advantageous for technological applications.

Figures

Equations

$$H=t\sum _{<ij>\alpha }{c}_{i\alpha }^{†}{c}_{j_{\alpha }}-J\sum _{i\alpha \beta }(\mathbf{\sigma }\cdot {\mathbf{n}}_i{)}_{\alpha \beta }{c}_{i\alpha }^{†}{c}_{i\beta },$$

$${\Omega }_{n,\alpha \beta }(\vec{k})=2i{\hslash }^2\sum _{m\ne n}\frac{⟨{\psi }_{n\mathbf{k}}|{\hat{v}}_{\alpha }|{\psi }_{m\mathbf{k}}⟩⟨{\psi }_{m\mathbf{k}}|{\hat{v}}_{\beta }|{\psi }_{n\mathbf{k}}⟩}{(E_n(\vec{k})-E_m(\vec{k}){)}^2},$$

$${\sigma }_{\alpha \beta }^{\gamma }$$

$${\Omega }_{n,\alpha \beta }^{\gamma }(\mathbf{k})$$

$$H_{\text{so}}=i{t}_2\sum _{<ij>\alpha \beta }{v}_{ij}(\mathbf{\sigma }\cdot {\mathbf{n}}_{ij}{)}_{\alpha \beta }{c}_{i\alpha }^{†}{c}_{j\beta },$$

Full text

I introduction

The spin Hall effect (SHE) Sinova et al. (2015) is one of the most important ways to create and detect spin currents in the field of spintronics, which aims to realize low-power-consumption and high-speed devices. Wolf et al. (2001) It converts electric currents into transverse spin currents and vice versa. The SHE in materials is generally believed to rely on spin–orbit coupling (SOC) Sinova et al. (2015); Maekawa et al. (2012); Hoffmann (2013); Gradhand et al. (2012). In typical SHE devices, the generated spin current is injected into a ferromagnet (FM) and consequently switches its magnetization via the spin-transfer torque Miron et al. (2011); Liu et al. (2012) or drives an efficient motion of magnetic domain walls. Parkin and Yang (2015); Yang et al. (2015)

The SHE is conceptually similar to the well established anomalous Hall effect (AHE). In recent decades, the understanding of the intrinsic AHE Nagaosa et al. (2010) and intrinsic SHE Murakami et al. (2003); Sinova et al. (2004) was significantly advanced based on the concept of the Berry phase, Xiao et al. (2010) which originates directly from the electronic band structure. Although the AHE requires the existence of SOC in a FM, it also appears in a non-coplanar magnetic lattice without SOC, where an electron acquires a Berry phase by hopping through sites with a noncoplanar magnetic structure (nonzero scalar spin chirality) Ohgushi et al. (2000); Taguchi et al. (2001), later referred as the topological Hall effect (THE) Bruno et al. (2004). Thus, in experiment the THE-induced Hall signal is considered as a signature of the skyrmion phase with chiral spin texture Neubauer et al. (2009); Kanazawa et al. (2011). Provoked by the THE, recent numerical simulations of the spin scattering by a single skyrmion indicated the presence of a finite SHE even without SOC Yin et al. (2015); Buhl et al. (2017); Ndiaye et al. (2017), which is termed as a topological SHE. Thus, the topological SHE has been presumed to stem from the Berry phase due to the nonzero spin chirality of the skyrmion. However, the origin of the spin current is illusive in the topological SHE, for it is hard to separate it from the spin-polarized charge current of the THE. Here, we pose new questions one step further. Is the skyrmion-like spin texture (nonzero scalar spin chirality) always necessary to generate a SHE without SOC? What is the generic condition for a SHE without SOC?

In this article, we propose a mechanism to realize the SHE with the noncollinear magnetic structure but without SOC. The crucial role of SOC is to break the spin rotational symmetry (SRS) in SHE. Alternatively, it is known that common noncollinear magnetic textures can also violate the SRS, thus resulting in the SHE. Different from the THE and topological SHE in symmetry, such an SHE appears universally for the noncollinear magnetic lattice, regardless of the scalar spin chirality. For example, it can even emerge in a coplanar magnetic structure where the scalar spin chirality is zero. Here, we first prove the principle from the symmetry analysis in a simple lattice model. Then, we demonstrate the existence of a strong SHE in several known materials Mn${}_{3}X$ ($X=$Ga, Ge, and Sn)Nayak et al. (2016); Nakatsuji et al. (2015) by ab initio calculations without including SOC.

II results

Double-exchange model and symmetry analysis – The existence of the SHE and AHE in metals is determined by symmetry (in insulators apart from symmetry also the topology of the electronic structure is important). The symmetry of magnetic systems is normally described in terms of magnetic space groups, which contain, apart from the spatial symmetry operations, also the time-reversal symmetry operation. In absence of SOC (or other terms in the Hamiltonian that couple the magnetic moments to the lattice such as the shape anisotropy), however, the symmetry of the magnetic systems is higher than that contained in the magnetic space groups since the spins can be rotated independently of the lattice. This can be illustrated by considering the following minimal Hamiltonian

[TABLE]

known as the double-exchange model ($s$-$d$ model) Zener (1951); Anderson and Hasegawa (1955); de Gennes (1960) which describes itinerant $s$ electrons interacting with local $d$ magnetic moments. We assume that magnetic moments are only contributed by the spin degrees of freedom. Here, $\alpha$ and $\beta$ stand for spin up and spin down, respectively. The first term is the nearest neighbor hopping term with $<ij>$ denoting the nearest neighbor lattice sites. In the second term, $J$ is the Hund’s coupling strength between the conduction electron and the on-site spin moment, $\bm{\sigma}$ is the vector of Pauli matrices, and $\bm{n}_{i}$ is the spin magnetic moment on site $i$. The magnetic texture is defined by the pattern of $\bm{n}_{i}$. In such a Hamiltonian, spin rotation only rotates the magnetic moments $\mathbf{n}_{i}$. The corresponding symmetry groups are thus formed by combining the magnetic space groups with spin rotations.Litvin and Opechowski (1974); Brinkman and Elliott (1966) Such symmetry groups are referred to as spin-space groups. They apply generally for non-interacting Hamiltonians in absence of spin-orbit coupling.

We focus here only on the intrinsic contribution to the AHE and SHE, however, the other (extrinsic) contributions have the same symmetry and thus the symmetry discussion in the following is general. The intrinsic AHE and intrinsic SHE are well characterized via the Berry curvature formalism. Xiao et al. (2010); Nagaosa et al. (2010); Sinova et al. (2015); Gradhand et al. (2012) The anomalous Hall conductivity(AHC) $\sigma_{\alpha\beta}$ can be evaluated by the integral of the Berry curvature $\Omega^{n}_{\alpha\beta}(\mathbf{k})$ over the Brillouin zone (BZ) for all the occupied bands, where $n$ is the band index. It should be noted that this method can also be applied to the THE, although it is commonly interpreted using the real space spin texture. Here, $\sigma_{\alpha\beta}$ corresponds to a $3\times 3$ matrix and indicates a transverse Hall current $j_{\alpha}$ generated by a longitudinal electric field $\mathbf{E}$, which satisfies $J_{\alpha}=\sigma_{\alpha\beta}E_{\beta}$. Within a linear response, Berry curvature can be expressed as

[TABLE]

where $n$ and $m$ are band indices, and $\psi_{n\mathbf{k}}$ and $E_{n\mathbf{k}}$ denote the Bloch wave functions and eigenvalues, respectively, and $\hat{\mathbf{v}}$ is the velocity operator. Replacing the velocity operator with the spin current operator $\hat{J}_{\alpha}^{\gamma}=\dfrac{1}{2}\{\hat{v}_{\alpha},\hat{s}_{\gamma}\}$, where $\hat{s}_{\gamma}$ is the spin operator, we obtain the spin Berry curvature and corresponding spin Hall conductivity (SHC),

[TABLE]

The SHC ($\sigma_{\alpha\beta}^{\gamma}$; $\alpha,\beta,\gamma=x,y,z$) is a third-order tensor ($3\times 3\times 3$) and represents the spin current ${J}_{s,\alpha}^{\gamma}$ generated by an electric field $\mathbf{E}$ via ${J}_{s,\alpha}^{\gamma}=\sigma_{\alpha\beta}^{\gamma}{E}_{\beta}$, where ${J}_{s,\alpha}^{\gamma}$ is a spin current flowing along the $\alpha$-direction with the spin-polarization along the $\gamma$-direction, and $f_{n}(\mathbf{k})$ is the temperature dependent Fermi-Diract distribution.

We know that AHE vanishes while SHE remains if the time-reversal symmetry (operator $T$) exists in the system. In Eq. 2, $T$ reverses the velocities $\hat{v}_{\alpha,\beta}$ and brings an additional “$-$” sign by the complex conjugate. Thus, $\sigma_{\alpha\beta}=0$ owing to $\Omega_{n,\alpha\beta}(\vec{k})=-\Omega_{n,\alpha\beta}(\vec{-k})$. In contrast, In Eq. 3, $T$ generates one more “$-$” sign by reversing the spin in ${J}_{\alpha}^{\gamma}$. Then, $\sigma_{\alpha\beta}^{\gamma}$ can be nonzero since $\Omega_{n,\alpha\beta}^{\gamma}(\vec{k})$ is even in $k$-space. In a magnetic system without SOC, $T$ is broken, but a combination of $T$ and a spin rotation (operator $S$) can still be a symmetry. For example, a coplanar magnetic system shows a $TS$ symmetry, in which $S$ rotates all spins by $180^{\circ}$ around the axis perpendicular to the plane. Since $S$ does not act on $\hat{v}_{\alpha,\beta}$, $TS$ causes vanishing $\sigma_{\alpha\beta}$ just as $T$ alone. In a general noncoplanar magnetic lattice, the $TS$ symmetry is naturally broken, because one cannot find a common axis about which all spins can be rotated $180^{\circ}$ at the same time, and thus the AHE can exist without SOC.

The situation is different for the SHE since ${J}_{\alpha}^{\gamma}$ in Eq. 3 contains an additional spin operator. As a consequence, (assuming that $S$ is a rotation around the $z$ axis) $TS$ forces $\Omega_{n,\alpha\beta}^{x/y}(\vec{k})$ to be odd where spin $\hat{s}_{x}$ or $\hat{s}_{y}$ is reversed by $TS$, but $\Omega_{n,\alpha\beta}^{z}(\vec{k})$ to be even where $\hat{s}_{z}$ is unchanged by $TS$. Then, one can obtain zero $\sigma_{\alpha\beta}^{x/y}$ but nonzero $\sigma_{\alpha\beta}^{z}$. In a collinear magnetic lattice there exists more than one spin rotation $S$ such that $TS$ is a symmetry of the system and thus all of the $\sigma_{\alpha\beta}^{\gamma}$ components have to vanish. Therefore, we can argue that SHE can exist without SOC in general noncollinear magnetic lattices, regardless of FM, AFM, or the scalar spin chirality (coplanar or noncoplanar). In contrast, the AHC is zero for a coplanar magnetic lattice (zero scalar spin chirality), since $TS$ acts as $T$ alone in Eq. 2.

When SOC is included, SHE is allowed by symmetry in any crystal,Seemann et al. (2015) while the AHE on the other hand can be present in magnetic systems that are not symmetric under time reversal combined with a translation or inversion (for example, a conventional collinear AFM). We summarize the necessary conditions for the existence of AHE and SHE in systems with and without SOC in Fig. 1.

To demonstrate that the SHE can indeed be nonzero without SOC, we consider the $s$–$d$ Hamiltonian (1) projected on a kagome lattice with the so-called $q=0$ magnetic order, shown in Fig. 2(a). Such a coplanar AFM order is well studied in theory Ohgushi et al. (2000); Chen et al. (2014) and appears in many realistic materials even at room temperature, for example Mn${}_{3}X$ ($X=$ Ir, Ga, Ge, and Sn) Zhang et al. (2013); Kübler and Felser (2014); Nakatsuji et al. (2015); Nayak et al. (2016); Zhang et al. (2016, 2017) as we discuss in the following. For comparison, the SOC effect is also considered by adding to $H$ in Eq. 1,

[TABLE]

where $v_{ij}$ is the antisymmetric Levi-Civita symbol and $n_{ij}$ are a set of coplanar vectors anticlockwise perpendicular to the lattice vector $R_{ij}$, as defined in Ref. Chen et al., 2014 and $t_{2}$ is the SOC strength.

We first analyze the symmetry of the SHC tensor for the $q=0$ magnetic order. Note that we use the Cartesian coordinate systems defined in Fig. 2. As discussed above, the existence of the $TS$ symmetry leaves only $\sigma_{\alpha\beta}^{z}$ terms in the absence of SOC. Further, the combined symmetry $TM_{x}$, in which $M_{x}$ is the mirror reflection along $x$ and flips $\hat{s}_{z}$ and $\hat{v}_{x}$ in Eq. 3, leads to $\sigma_{xx}^{z}=\sigma_{yy}^{z}=0$. We further obtain only two nonzero SHC tensor element $\sigma^{z}_{xy}=-\sigma^{z}_{yx}$ by considering the three-fold rotation around $z$. The magnetic order shown in Fig. 2(b) will also be relevant for the discussion in the following. This magnetic configuration differs from the $q=0$ case only by a two-fold spin rotation around the $y$-axis and thus, without SOC its symmetry is exactly the same as that of the $q=0$ case.

Setting the Hund coupling constant $J=1.7t$ and SOC strength $t_{2}=0$, we calculate the spin Berry curvature via Eq.  3. As expected, we find nonzero SHC $\sigma^{z}_{xy}$ fully in agreement with the symmetry considerations. Figures 3a and 3b show the band structures with $t_{2}=0$ and $t_{2}=0.2t$, respectively. One can see that SOC modifies slightly the band structure by gaping some band crossing points such as the BZ corners ($K$). Without SOC, we already observe nonzero $\sigma^{z}_{xy}$, while adding SOC reduces $\sigma^{z}_{xy}$ slightly at the Fermi energy that is set between the first and second bands at about –2.7 eV. We plot corresponding spin Berry curvature $\Omega^{z}_{xy}$ in the hexagonal BZ In Figs. 3d and 3e. Large $\Omega^{z}_{xy}$ appears in the BZ without SOC, leading to net $\sigma^{z}_{xy}$. The SOC simply brings an extra contribution to $\sigma^{z}_{xy}$ at the band anti-crossing region around the $K$ point.

Realistic materials – After proving the principle, we now identify materials that show strong SHE with negligible contribution from SOC. We naturally consider Mn3X (X=Ga, Ge, Sn, and Ir) compounds, since they exhibit non:w collinear AFM order at room temperature (the AFM Néel temperature is over 365 K). Our recent ab  initio calculations showed a sizable intrinsic SHE by including SOC Zhang et al. (2017). Here, we further point out that SHE still presents without SOC and SOC actually plays a negligible effect for SHE in these materials.

The primitive unit cell of Mn3Ga, Mn3Ge and Mn3Sn (space group $P6_{3}/mmc$, No. 194) includes two Mn${}_{3}X$ planes that are stacked along the $c$-axis according to a “–AB–AB–” sequence. Inside each plane, Mn atoms form a kagome-type lattice with Ga, Ge, or Sn lying at the center of the hexagon formed by Mn. Both the ab initio calculation Zhang et al. (2013) and neutron diffraction measurements Krén and Kádár (1970); Kádár and Krén (1971); Ohoyama et al. (1961) show that the Mn magnetic moments exhibit noncollinear AFM order, where the neighboring moments are aligned at an angle of 120∘, as in Fig. 2(b). Large AHE in room temperature has recently been reported in Mn3Ge and Mn3Sn. Nakatsuji et al. (2015); Nayak et al. (2016) These materials also exhibit other exotic phenomena including the Weyl semimetal phase, Yang et al. (2016) magneto-optical Kerr effect, Feng et al. (2015) anomalous Nernst effect, Li et al. (2016) and topological defects. Liu and Balents (2017) Distinct from hexagonal Mn${}_{3}X$ compounds, the Mn3Ir (space group $Pm\bar{3}m$, No. 221) crystallizes in a face-centered cubic structure with Mn atoms in the [111] planes forming a kagome lattice with the $q=0$ magnetic order.

The symmetry of the SHE without SOC in these materials can be understood using a similar approach as we used for the 2D kagome lattice. The hexagonal Ga, Ge, and Sn materials can be viewed as stacking versions of the kagome lattice and thus we find that the symmetry of SHE is the same as the 2D kagome lattice, i.e. only $\sigma_{xy}^{z}=-\sigma_{yx}^{z}$ is nonzero. However, we find that SHE must vanish in Mn3Ir without SOC, which is imposed by the higher symmetry of the cubic magnetic lattice. For completeness, we list the tensor matrices without and with SOC for all these compounds in the appendix.

Since the SHC tensor shape imposed by the symmetry has been systematically investigated for these materials in Ref. Zhang et al., 2017, we only discuss one of the largest SHC tensor elements $\sigma_{xy}^{z}$ based on the ab initio calculations Kresse and Furthmüller (1996) of the SHC. For comparison, we show the SHC without and together with SOC in Fig. 4. In the absence of SOC, Ga, Ge, and Sn compounds indeed exhibit nonzero SHC $\sigma_{xy}^{z}=-613,115,$ and $90~{}(\hbar/e)(\Omega\cdot cm)^{-1}$, respectively, at the Fermi energy. One can see that SOC induces very few changes in the band structure and thereafter modifies the SHC weakly, especially at the Fermi energy for Ga, Ge, and Sn compounds. It is intuitive to observe comparable $\sigma_{xy}^{z}$ values for Ge and Sn compounds, despite the fact that Sn exhibits much larger SOC than Ge. These facts further verifies that the noncollinear magnetic structure, rather than SOC, is dominant for the SHE. The Ga compound shows an opposite sign in SHC compared to the Ge/Sn compound, since Ga has one valence electron fewer than Ge/Sn and the Fermi energy is lower in Mn3Ga than in Mn3Ge/Mn3Sn.

III discussion

Understanding the role SOC plays in the SHE is important for the fundamental understanding of the SHE, but also for practical reasons. It can help with the search for materials with large SHE since in non-magnetic or collinear magnetic materials, SOC is necessary for SHE and thus the presence of heavy elements is generally required for large SHE. The SHE without SOC proposed in this work suggests a new strategy to design SHE materials without necessarily involving heavy elements. In noncollinear systems, the Rashba effect can also appear without SOC Liu et al. (2013). The spin texture in the band structure may depend sensitively on the real space spin texture. For example, we found that band structure spin texture is different between the Kagome lattice and the triangular lattice.

We propose the general, necessary symmetry-breaking requirements (Fig. 1) for SHE without SOC. It is worth noting that SHE can become zero without SOC in some noncollinear magnetic lattice where additional symmetries forces the SHE to vanish. For example, we have shown that in Mn3Ir the SHE vanishes in absence of SOC even though it has a noncollinear magnetic structure. This is a consequence of its high-symmetrical cubic structure. Similar situation could happen for AHE without SOC in a noncoplanar magnetic lattice, such as the AFM skyrmion system Ndiaye et al. (2017).

In conclusion, we have shown that the SHE can be realized by a non-chiral coplanar magnetic structure without involving the SOC. The noncollinearity of the magnetic lattice can break the spin rotation symmetry and consequently allow the existence of SHE. By ab initio calculations, we further predicted that such an SHE without SOC can be observed in noncollinear AFM compounds Mn${}_{3}X$ ($X=$ Ga, Ge, and Sn). From our symmetry considerations, an extrinsic SHE can appear when breaking the spin rotational symmetry. Thus, we expect the extrinsic effect to also exist in our systems. Its amplitude will depend on the details of the scattering, and cannot be estimated without microscopic calculations, though in general for SHE the intrinsic contribution tends to be the dominant contribution. By providing a general theoretical, symmetry based understanding of the SHE, our work motivates a comprehensive search for SHE materials among noncollinear magnetic systems, that not necessarily involve heavy elements. In addition, the close relation between the SHE and the magnetic order suggests that the SHE may be used vice versa, as a probe to establish and symmetry restrict the ground state magnetic structures of long-range ordered antiferromagnets.

Regarding the strong correlated system, we would discuss the spin liquid material as an example. RuCl3 has a rich magnetic phase diagram, with a complex zig-zag type AFM long-range order at low temperatures, and even a quantum spin liquid phase in applied magnetic field. According to our work, non-magnetic and collinearly ordered phases have vanishing SHE without SOC. For the quantum spin liquid phase, it remains unclear whether SHE appears without SOC. If yes, SHE would be a promising probe to the spin liquid phase. We will study this very interesting question in the future.

IV method

To calculate the SHE in these compounds we obtain the DFT Bloch wave functions from Vienna $ab-initio$ Simulation Package (vasp) Kresse and Furthmüller (1996) within the generalized gradient approximation (GGA). Perdew et al. (1996) By projecting the Bloch wave functions onto maximally localized Wannier functions (MLWFs), Mostofi et al. (2008) we get a tight-binding Hamiltonian which we use for efficient evaluation of the SHC. For the integrals of Eq. 3, the BZ was sampled by $k$-grids from $50\times 50\times 50$ to $200\times 200\times 200$. Satisfactory convergence was achieved for a $k$-grid of size $150\times 150\times 150$ for all three compounds. Increasing the grid size to $200\times 200\times 200$ only varied the SHC by no more than 5%. The results of the calculations agree well with the symmetry analysis.

V acknowledgments

We thank Prof. Claudia Felser at Max Planck Institute in Dresden and Prof. Carsten Timm at the Technical University Dresden, Prof. Yuval Oreg in the Weizmann Institute of Science for helpful discussions. Y.Z. and J.v.d.B. acknowledge the German Research Foundation (DFG) SFB 1143. Y.Z. and J.Z. acknowledge the the ASPIN project (EU FET Open RIA Grant No. 766566 grant (ASPIN)). B.Y. acknowledges the support by the Ruth and Herman Albert Scholars Program for New Scientists in Weizmann Institute of Science and by a Grant from the GIF, the German-Israeli Foundation for Scientific Research and Development.

VI author contributions

B.Y. conceived the project. Y.Z., J.Z. and Y.S. performed the DFT calculation and the data analysis. Y.Z., J.Z., and B.Y. performed the tight-binding modeling and symmetry analysis. Y.Z. wrote the manuscript with contributions from all co-authors. J.v.d.B. and B.Y. supervised the work. We acknowledge the support by ASPIN ( EU FET Open RIA Grant No. 766566).

VII competing interests

The authors declare no competing financial interests.