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Out-Of-Plane Carrier Spin in Transition-Metal Dichalcogenides

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Out-Of-Plane Carrier Spin in Transition-Metal Dichalcogenides Out-of-plane carrier spin in transition-metal dichalcogenides under electric current Xiao Li ( )a,b,1, Hua Chenc,d,1 , and Qian Niub aCenter for Quantum Transport and Thermal Energy Science, School of Physics and Technology, Nanjing Normal University, Nanjing 210023, China; bDepartment of Physics, University of Texas at Austin, Austin, TX 78712; cDepartment of Physics, Colorado State University, Fort Collins, CO 80523; and dSchool of Advanced Materials Discovery, Colorado State University, Fort Collins, CO 80523 Edited by David Vanderbilt, Rutgers, The State University of New Jersey, Piscataway, NJ, and approved June 1, 2020 (received for review July 20, 2019) Absence of spatial inversion symmetry allows a nonequilibrium (16), semiconductor-to-metal transition (17), and valley splitting spin polarization to be induced by electric currents, which, in two- (18, 19). dimensional systems, is conventionally analyzed using the Rashba Two-dimensional van der Waals materials provide a plethora model, leading to in-plane spin polarization. Given that the of simple and powerful platforms for exploring spin-related material realizations of out-of-plane current-induced spin polar- physics (20). In particular, monolayer transition-metal dichalco- ization (CISP) are relatively fewer than that of in-plane CISP, but genides, MX2 (M = V, Mo, W; X = S, Se, Te, etc.), in the important for perpendicular-magnetization switching and elec- 2H phase have both strong SOC and inversion symmetry break- tronic structure evolution, it is highly desirable to search for ing (21–24). While special attention has recently been paid to new prototypical materials and mechanisms to generate the out- the CISP in MX2/graphene bilayer (25) and MX2/ferromagnet of-plane carrier spin and promote the study of CISP. Here, we bilayer structures (6), the study of possible CISP in 2H-MX2 propose that an out-of-plane CISP can emerge in ferromagnetic monolayers is still lacking. It is worth emphasizing that 2H-MX2 transition-metal dichalcogenide monolayers. Taking monolayer monolayers are representative 2D systems for the non-Rashba VSe2 and VTe2 as examples, we calculate the out-of-plane CISP SOC. The intrinsic SOC of monolayer 2H-MX2 in the low- based on linear-response theory and first-principles methods. We energy effective theory results from the bulk crystal structure deduce a general low-energy model for easy-plane ferromagnetic asymmetries, in contrast to Rashba SOC from the interfacial transition-metal dichalcogenide monolayers and find that the out- asymmetry. The existence of a mirror plane through the M- of-plane CISP is due to an in-plane magnetization together with atom layer ensures that the potential gradient, rV , is in-plane. APPLIED PHYSICAL SCIENCES intrinsic spin–orbit coupling inducing an anisotropic out-of-plane The SOC-induced effective magnetic field / rV × p, with p spin splitting in the momentum space. The CISP paves the way for being the 2D momentum, is therefore out-of-plane and results in magnetization rotation and electric control of the valley quantum Zeeman-type spin splitting (21, 22). It is thus an attractive ques- number. tion whether the out-of-plane spin splitting leads to an out-of- plane CISP, which we have fully addressed in this paper. More- out-of-plane current-induced spin polarization j over, there are two valleys in the low-energy bands of 2H-MX2. transition-metal dichalcogenides j intrinsic spin–orbit coupling j Valley, as an electronic degree of freedom, may add a dimen- valley degree of freedom sion to the CISP, given that valley-contrasting optoelectronic properties offer potential applications in energy-efficient elec- tronics and spintronics (22, 26). n a conducting system with broken inversion symmetry Iand nonnegligible spin–orbit coupling (SOC), nonequilibrium Significance carrier-spin polarization can be induced by electric currents (1– 4). The current-induced spin polarization (CISP) in a magnetic Although out-of-plane current-induced spin polarization system further applies a torque, i.e., spin–orbit torque, on the (CISP) is important for perpendicular-magnetization reorien- magnetic order parameter through the exchange coupling. These tation, and it has been realized by the crystalline symmetry inverse spin-galvanic phenomena have seen many spintronic breaking, its material examples are relatively fewer com- applications in magnetization switching, domain-wall motion, pared with in-plane CISP in Rashba systems. With the help of etc., according to Landau–Lifshitz–Gilbert dynamics driven by intrinsic spin–orbit coupling, an intriguing out-of-plane CISP is the spin–orbit torque (5). A bilayer structure composed of a fer- designed in transition-metal dichalcogenides by the symme- romagnetic layer and a heavy nonmagnetic layer is widely used try breaking in the spin space, which provides opportunities in the study of the CISP (5–7). The inversion symmetry break- for out-of-plane magnetization rotation and electric control of ing at the interface is minimally captured by two-dimensional valley splitting. Moreover, the spin polarization is associated (2D) models with Rashba-type SOC (8–10), which gives chiral with valley-dependent responses to electric current and adds textures of in-plane spin components in the momentum space. a dimension, valley degree of freedom, to the study of CISP. The resulting CISP is always in-plane after the carrier redistribu- The symmetry argument and non-Rashba effective model also tion. The out-of-plane component of CISP arising from further helps to illuminate these physics and broaden the scope of crystalline symmetry breaking has recently been proposed in the the CISP. bilayer heterostructure (7, 11–14) and quantum well grown in a certain direction (15). However, the examples of the out-of-plane Author contributions: X.L. and Q.N. designed research; X.L. performed research; X.L., CISP in real materials are still relatively fewer compared with the H.C., and Q.N. analyzed data; and X.L. and H.C. wrote the paper.y in-plane CISP that occurs in a large number of Rashba systems. The authors declare no competing interest.y Both new prototypical materials and mechanisms are highly This article is a PNAS Direct Submission.y desirable for generating out-of-plane CISP. The out-of-plane Published under the PNAS license.y CISP is important in device concepts based on perpendicular 1 To whom correspondence may be addressed. Email: [email protected] or huachen@ magnetic anisotropy, where it facilitates perpendicular-magneti- colostate.edu.y zation switching by spin–orbit torque. Besides, controlling out- This article contains supporting information online at https://www.pnas.org/lookup/suppl/ of-plane magnetic ordering enables access to a number of doi:10.1073/pnas.1912472117/-/DCSupplemental.y intriguing phenomena, e.g., quantum anomalous Hall effect www.pnas.org/cgi/doi/10.1073/pnas.1912472117 PNAS Latest Articles j 1 of 7 Downloaded by guest on September 25, 2021 In this paper, we use symmetry analysis and first-principles A v B calculations to point out that there is out-of-plane CISP in the prototypical ferromagnet transition-metal dichalcogenides VSe2 and VTe2 monolayers (23). Especially for the VSe2 mono- z y layer, it has been synthesized recently (27, 28). We derive a x low-energy model suitable for generic easy-plane ferromagnetic h y 2H-MX2 monolayers, and use it to elucidate the physical ori- gin of out-of-plane CISP, which is the anisotropic spin splitting in the momentum space due to both non-Rashba SOC and in- x plane magnetization. The symmetry breaking in the spin space enables the out-of-plane CISP in ferromagnet transition-metal C 0.75 D dichalcogenides, in contrast to the crystalline symmetry breaking in previous works (7, 11–15). Moreover, two valleys exhibit dis- 0.50 tinct responses depending on the electric-field direction, which M -2% is a convenient experimental knob to tune CISP. This CISP pro- K K 0.25 - + 0% vides opportunities for magnetization reorientation and electric 2% control of valleys. The proposed non-Rashba system and the low- 0 energy model will help to broaden the scope of the study of the Energy (eV) CISP and other spin-related physics. -0.25 Results -0.50 Symmetry Analysis. Symmetry arguments provide a powerful tool M K K M - + M K- K+ M for finding possible nonvanishing tensor components of any response function. For the CISP, the corresponding response Fig. 1. Atomic and band structures of the VX2 monolayer. Top view (A) function is of spin density responding to electric fields. In the and side view (B) of the atomic structure. Blue and yellow spheres represent good metal limit, we can separate this response function into vanadium and chalcogen atoms, respectively. (C and D) Band structures of two contributions: one arises from the nonequilibrium redistri- VSe2 with the magnetization along the x and z directions, respectively. Black bution of carriers in the neighborhood of the Fermi level, under bands correspond to the pristine monolayer without strain, while blue and red bands correspond to the monolayers under −2 and 2% uniaxial strain the combined effect of electric field and disorder scattering; the along the y direction, respectively. The valence band maximum is set to zero other is from the interband transitions of Fermi sea electrons energy. The Brillouin zone is given in C (Inset). induced by electric field (29–31). We first study the first contri- bution that results in out-of-plane CISP and discuss the second contribution at the end of First-Principles Calculations of CISP. For the convenience of discussion, we adopt the relaxation time Therefore, in order for Eq. 1 to hold, δs must change sign, which approximation, under which the Fermi surface contribution can means only the out-of-plane component, δsz , is allowed. The be described by C3 symmetry breaking by the in-plane magnetization is also cru- cial for generating δsz , since both χ and sz are invariant under the C3 rotation, but an in-plane E is not.
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