Intrinsic Spin Hall Conductivity of MoTe2 and WTe2 Semimetals

TL;DR

This study uses ab initio calculations to analyze the intrinsic spin Hall conductivity of MoTe2 and WTe2 semimetals, revealing large SHC, anisotropic behavior, and methods for optimization through doping, advancing 2D spintronics research.

Contribution

It provides the first comprehensive ab initio analysis of SHC in MoTe2 and WTe2, highlighting anisotropy, doping effects, and mechanisms behind SHC variation.

Findings

Large intrinsic SHC and spin Hall angles found

Anisotropic SHC due to crystal symmetry

SHC can be optimized by electron doping

Abstract

We report a comprehensive study on the intrinsic spin Hall conductivity (SHC) of semimetals MoTe2 and WTe2 by ab initio calculation. Large SHC and desirable spin Hall angles have been discovered, due to the strong spin orbit coupling effect and low charge conductivity in semimetals. Diverse anisotropic SHC values, attributed to the unusual reduced-symmetry crystalline structure, have been revealed. We report an effective method on SHC optimization by electron doping, and exhibit the mechanism of SHC variation respect to the energy shifting by the spin Berry curvature. Our work provides insights into the realization of strong spin Hall effects in 2D systems.

Tables

Table 1. Table 1: The calculated non-zero SHC tensor elements in MoTe 2 and WTe 2 at Fermi energy, in unit of ( ℏ / e Planck-constant-over-2-pi 𝑒 \hbar/e ) ( Ω Ω \Omega cm) -1 The maximum SHA values are listed in the last column.

Semimetals	${\sigma }_{yz}^x$	${\sigma }_{zy}^x$	${\sigma }_{xz}^y$	${\sigma }_{zx}^y$	${\sigma }_{xy}^z$	${\sigma }_{yx}^z$	SHAMAX
MoTe2	$-18$	$45$	$-88$	$286$	$-176$	$-361$	-0.72
WTe2	$-44$	$-44$	$-61$	$103$	$-204$	$-15$	-0.54

Equations

$$\begin{array}{rl}{\sigma }_{xy}(\omega )=& \frac{\hslash }{V{N}_k^3}\sum _{\mathbf{k}}\sum _n{f}_{n\mathbf{k}}\\ & \times \sum _{m\ne n}\frac{2\mathrm{Im}[⟨n\mathbf{k}|{\hat{j}}_x|m\mathbf{k}⟩⟨m\mathbf{k}|-e{\hat{v}}_y|n\mathbf{k}⟩]}{({ϵ}_{n\mathbf{k}}-{ϵ}_{m\mathbf{k}}{)}^2-(\hslash \omega +i\eta {)}^2},\end{array}$$

$$\begin{array}{r}{\Omega }_{n,xy}^{\text{spin}z}(\mathbf{k})={\hslash }^2\sum _{m\ne n}\frac{-2\mathrm{Im}[⟨n\mathbf{k}|\frac{1}{2}\{{\hat{\sigma }}_z,{\hat{v}}_x\}|m\mathbf{k}⟩⟨m\mathbf{k}|{\hat{v}}_y|n\mathbf{k}⟩]}{({ϵ}_{n\mathbf{k}}-{ϵ}_{m\mathbf{k}}{)}^2-(\hslash \omega +i\eta {)}^2}\end{array}$$

$${\Omega }_{xy}^{\text{spin}z}(\mathbf{k})=\sum _n{f}_{n\mathbf{k}}{\Omega }_{n,xy}^{\text{spin}z}(\mathbf{k}).$$

$${\sigma }_{xy}^{\text{spin}z}(\omega )=-\frac{e^2}{\hslash }\frac{1}{V{N}_k^3}\sum _{\mathbf{k}}{\Omega }_{xy}^{\text{spin}z}(\mathbf{k}).$$

Full text

I introduction

Spin-orbit torque (SOT) generated by materials with strong spin-orbit coupling (SOC) is a promising approach for energy-efficient manipulation of nonvolatile magnetic memory and spin logic devicesBaek et al. (2018); Zhang et al. (2018); Shi et al. (2018); Liu et al. (2012a); Manchon and Zhang (2009). The form of the in-plane SOT alone does not allow for deterministic switching of perpendicular magnetic anisotropy (PMA) devices, and magnetization switching requires an additional external in-plane magnetic fieldLiu et al. (2012b); Han et al. (2017). Efforts are devoted to find efficient methods on PMA devices switching by field-free SOT. It is found that lateral structural asymmetry, the wedge film, makes it possible to switch the PMA device by in-plane-current SOT without external magnetic fieldsYu et al. (2014). Besides, the out-of-plane SOT can be introduced by WTe2, a layered orthorhombic transition-metal dichalcogenide (TMD) with strong SOC and reduced crystalline symmetryMacNeill et al. (2016, 2017). WTe2/Py device produces an out-of-plane antidamping torque when current is applied along a low-symmetry axis, but not when current is applied along a high-symmetry axis. This is due to the absence of two-fold rotational symmetry in WTe2/Py bilayer.The anisotropic enhancement of spin-orbit torques in WTe2/Py devices has been observedLi et al. (2018).

On the other hand, one SOT mechanism is the bulk spin Hall effect (SHE). A charge current flowing in the spin Hall layer can generate a pure spin current that exerts a spin torque on the recording layerKhang et al. (2018). The SHE can be separated into intrinsic and extrinsic parts. The intrinsic SHE, significantly contributing to the total SHE in materials with strongly spin-orbit-coupled bands, can be calculated accurately based on ab initio theoriesQiao et al. (2018); Guo et al. (2008); Sui et al. (2017). As the materials with broken crystal symmetry provide anisotropic SOTMacNeill et al. (2016); Li et al. (2018), it triggers an intriguing question on the intrinsic SHE of the low-symmetry crystals. Exploring the intrinsic SHE in low-symmetry crystal would greatly contribute to research on novel materials with diverse anisotropy and applications. Besides, it is expected to discover high spin-charge conversion efficiency in semimetals considering their low conductivity.

In this work, we report the SHC in semimetals MoTe2 and WTe2 using ab initio calculations. These studies reveal large SHC and high spin Hall angle in MoTe2 and WTe2, as well as diverse anisotropic SHC values. We analyze the mechanism on SHC by spin Berry curvature, and illustrate that energy shifting is an effective method to enhance SHC.

II METHOD

The general form of Kubo formula SHC is given by Yao et al. (2004); Guo et al. (2005); Matthes et al. (2016)

[TABLE]

where $\hat{j}_{x}=\frac{1}{2}\{\hat{s}_{z},\hat{v}_{x}\}$, $V$ is the primitive cell volume, and $N_{k}^{3}$ is the number of $k$-points in the Brillouin zone (BZ). To facilitate further analysis, we separate the Equ.(1) into the band-projected Berry curvature-like term

[TABLE]

and $k$-resolved term

[TABLE]

The SHC is the sum over occupied bands

[TABLE]

All the ab initio calculations were performed using Quantum ESPRESSO package based on projector-augmented wave (PAW) method and a plane wave basis set Giannozzi et al. (2009, 2017). The exchange and correlation terms were described using generalized gradient approximation (GGA) in the scheme of Perdew-Burke-Ernzerhof (PBE) parameterization, as implemented in the pslibrary Corso (2014). The plane-wave cutoff energy is 600 eV and a $k$-point grid with $12\times 10\times 6$ was used in self-consistent calculation. We employed the atomic structure in previous reportsWang et al. (2016); Soluyanov et al. (2015). Then, density functional theory (DFT) wave functions were projected to maximally localized Wannier functions using the WANNIER90 package Marzari and Vanderbilt (1997); Souza et al. (2001); Marzari et al. (2012) and the Kubo formula was employed to calculate the SHC. A dense $k$ mesh of 500$\times$500$\times$500 was employed to perform the BZ integration for the intrinsic SHC, and adapt $k$ mesh was used to deal with the dramatic variation in the spin Berry curvature. More details can be found in our previous workQiao et al. (2018).

III RESULTS AND DISCUSSIONS

Figure 1(a) shows the atomic structure of MTe2 (M = Mo or W) semimetal. Both MoTe2 and WTe2 are layered orthorhombic transition-metal dichalcogenides (TMD) with strong SOCSun et al. (2015); Feng et al. (2016). The space group is identified to be Pnm21 (No. 31). The crystal structure possesses one mirror plane $x$ = 0 and one glide mirror plane parallel to $y$ = 0, which transform to $k_{x}$ = 0 and $k_{y}$ = 0 plane, respectively, in the momentum space.

Table 1 shows the calculated SHC for MoTe2 and WTe2 at Fermi energy. Common metals exhibiting $C_{4v}$ symmetry such as Pt, W, Ta, have the limitation on $\sigma_{xy}^{z}$ = $-\sigma_{yx}^{z}$, $\sigma_{yz}^{x}$ = $-\sigma_{xz}^{y}$, and $\sigma_{zx}^{y}$ = $-\sigma_{zy}^{x}$. $\beta-$Ta and $\beta-$W have more strict restrictions as $\sigma_{xy}^{z}$ = $-\sigma_{yx}^{z}$ = $\sigma_{yz}^{x}$ = $-\sigma_{zy}^{x}$ = $\sigma_{zx}^{y}$ = $-\sigma_{xz}^{y}$. But in MoTe2 and WTe2, we found more diverse anisotropies than heavy metals. Both MoTe2 and WTe2 have strong anisotropies with six non-zero SHC tensors with different absolute values, which vary a lot from positive to negative scale. For MoTe2, the SHC max is $\sigma_{yx}^{z}$ = -360($\hbar/e$)($\Omega$cm)-1, and $\sigma_{zx}^{y}$ = 286($\hbar/e$)($\Omega$cm)-1is also a considerable one. For WTe2, the maximum of SHC is $\sigma_{xy}^{z}$ =-204($\hbar/e$)($\Omega$cm)-1. These results will provide helpful information for the experimental detection of the SHE. We also analyzed the spin Hall angle (SHA). SHA is the SHE efficiency and the critical current density for magnetization switching, which measures the efficiency of the charge current to spin current conversionZhang et al. (2016). The SHAMAX of MoTe2 is -0.72 and the SHAMAX of WTe2 is -0.54. We take Pt to make a comparison. Pt is a 5$d$ heavy metal with the SHC in the order of 103 and the conductivity in the range of 104–106 ($\Omega$cm)-1Guo et al. (2008), consequently the spin Hall angle is relatively smallWang et al. (2014); Zhang et al. (2015). On the other hand, the conductivities of these two semimetals are much lower. The conductivity of WTe2 is 7.4$\times$102($\Omega$cm)-1Jana et al. (2015) while the conductivity of MoTe2 is 1$\times$103($\Omega$cm)-1 Qi et al. (2016). As a result, high SHA can be expected in MoTe2 and WTe2 as shown in Tab. 1.

According to the Kubo formula, SHC varies quickly with the Fermi energy. Figure 2 presents the variation of SHC with respect to the position of Fermi energy. In the left panel of Fig.2, MoTe2 shows peaks around Fermi energy for $\sigma_{yx}^{z}$, $\sigma_{zx}^{y}$ and $\sigma_{xy}^{z}$, as high as -390, 306 and -178($\hbar/e$)($\Omega$cm)-1, respectively. The relatively high SHC, over 200($\hbar/e$)($\Omega$cm)-1, remains from -0.5 eV to 0 eV for $\sigma_{yx}^{z}$. In the right panel of Fig. 2, WTe2 exhibits SHC as high as $\sigma_{yx}^{z}$ = -528($\hbar/e$)($\Omega$cm)-1when the Fermi energy lies at -0.094 eV, the high SHC around -400($\hbar/e$)($\Omega$cm)-1, arises at 0.47 for $\sigma_{xy}^{z}$, and the $\sigma_{zx}^{y}$ value reaches up to 370 when the Fermi energy lies at -0.048 eV. The energy-dependent analysis indicates the general route to optimizing the SHE in these materials. Shifting Fermi energy by weak electron doping is an effective way to enhance SHC in MoTe2 and WTe2.

To elucidate the underlying mechanism of the enhancement of SHC induced by energy shift, we performed band-projected and $k$-resolved spin Berry curvature in Figure 3. We take $\sigma_{xy}^{z}$ in WTe2 as an example, as it varies a lot and reverses in a small energy range, from -204($\hbar/e$)($\Omega$cm)-1at $E=E_{F}$ to 127($\hbar/e$)($\Omega$cm)-1at $E=E_{F}-0.09$, marked by green horizontal dashed line crossing Fig. 3(a) and (b). Fig. 3(a) shows the $k$-resolved band projected by spin Berry curvature $\Omega_{n,xy}^{\text{spin}z}(\bm{k})$ with log scale, where red (blue) color denotes a positive (negative) contribution of spin Berry curvature. The bands close to Fermi energy mainly contribute to SHC, especially at spin-orbit splitting points. As the SHC is the sum over the occupied bands, the Fermi energy plays an important role in SHC scale. SHC at Fermi energy presents a minus sign due to negative contribution along Z-U and X-$\Gamma$ point. When the energy is shifted by -0.09, positive spin Berry curvature rises and overwhelms the negative one, resulting in a 127($\hbar/e$)($\Omega$cm)-1SHC. To make it more visualized, Fig. 3(c) and (d) show the $k$-resolved spin Berry curvature with log scale. At Fermi energy, the negative spin Berry curvature dominants along the Z-U and X-$\Gamma$ path. Some sharp peaks are caused by the bands crossing with Fermi energy. When the energy is shifted by -0.09eV, the negative spin Berry curvature along the X-$\Gamma$ path is weakened while the positive one strengthens, especially at the $\Gamma$ point. As a result the SHC turns to be positive. It is remarkable that the spin Berry curvature varies dramatically along the $k$-path. In such cases, the method of adaptive $k$ mesh refinement can be effective for refined SHC calculation.

We show the $k$-resolved spin Berry curvature in the two-dimensional BZ at $k_{z}$ = 0 at the $E=E_{F}$ and $E=E_{F}-0.09$ points in Figure4(a) and (b), respectively. The spin Berry curvature is also in the log scale and red (blue) color denotes a positive (negative) contribution. It can be seen that the spin Berry curvature depends sensitively on energy shift including a sign reversal throughout a large fraction of the BZ, especially around the $\Gamma$ point. Fig.4(a) shows that at $E=E_{F}$ point, the spin Berry curvature at $\Gamma$ point is close to zero, and the negative region surrounded by black line is very conspicuous considering the log scale. At $E=E_{F}-0.09$ point in Fig.4(b), the spin Berry curvature around $\Gamma$ point becomes positive shown by red color. Besides, the positive region in red color expands widely while the negative region is weakened. As a result of the spin Berry curvature integration, the SHC inverts its sign from the negative one at $E=E_{F}$ to the positive one at $E=E_{F}-0.09$. The analysis above clarifies the mechanism on SHC variation with energy, and sheds light on the SHC optimization.

IV CONCLUSION

In summary, by ab initio calculations, we present the anisotropic spin Hall conductivity in MoTe2 and WTe2 semimetals. The SHC is desirable in both semimetals, and the spin Hall angle is conspicuously higher compared with heavy metals. We find the reduced symmetry results in the diverse SHC anisotropies, which provides potential for various application. We report the possibility to improve SHC by energy shift, which can induce a large enhancement or reversal of the spin Berry curvature integration by doping. Our investigations are conducive to the study on the spin Hall effect and spin-orbit torque in 2D systems.

Acknowledgements.

The authors gratefully acknowledge the National Natural Science Foundation of China (Grant No. 61627813, 61571023), the International Collaboration Project B16001, and the National Key Technology Program of China 2017ZX01032101 for their financial support of this work. This work is supported by the Academic Excellence Foundation of BUAA for PhD Students.