This is the accepted version of the journal article: Garcia, José H.; You, Jinxuan; García-Mota, Mónica (Simune Atomistics S.L.); [et al.]. «Electrical control of spin-polarized topological currents in monolayer WTe2». Physical review B, Vol. 106, issue 16 (Oct. 2022), art. L161410. DOI 10.1103/PhysRevB.106.L161410 This version is available at https://ddd.uab.cat/record/269762 under the terms of the license Electrical Control of Spin-polarized Topological Currents in Monolayer WTe2 Jose H. Garcia,1 Jinxuan You,1, 2, 3 Mónica Garcı́a-Mota,2 Peter Koval,2 Pablo Ordejón,1 Ramón Cuadrado,1 Matthieu J. Verstraete,4 Zeila Zanolli,5, 1 and Stephan Roche1, 6 1 Catalan Institute of Nanoscience and Nanotechnology - ICN2, (CSIC and BIST), Campus UAB, Bellaterra, 08193 Barcelona, Spain 2 Simune Atomistics S.L., Tolosa Hiribidea, 76, 20018 Donostia-San Sebastian, Spain 3 Department of Materials Science, Universitat Autònoma de Barcelona, Cerdanyola del Vallès, 08193 Bellaterra, Spain 4 nanomat/QMAT/CESAM and European Theoretical Spectroscopy Facility Universite de Liege, Allee du 6 Aout 19 (B5a), 4000 Liege, Belgium 5 Dept. of Chemistry, Debye Institute for Nanomaterials Science, and ETSF, Utrecht University, The Netherlands 6 ICREA–Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain (Dated: May 30, 2022) We evidence the possibility for coherent electrical manipulation of the spin orientation of topolog- ically protected edge states in a low-symmetry quantum spin Hall insulator. By using a combination of ab-initio simulations, symmetry-based modeling, and large-scale calculations of the spin Hall con- ductivity, it is shown that small electric fields can efficiently vary the spin textures of edge currents in monolayer 1T’-WTe2 by up to a 90-degree spin rotation, without jeopardizing their topological character. These findings suggest a new kind of gate-controllable spin-based device, topologically protected against disorder and of relevance for the development of topological spintronics. Introduction. The existence of the quantum spin Hall SOC is more generally a consequence of uncompensated (QSH) insulator [1, 2] has boosted opportunities for spin- electric fields in noncentrosymmetric systems either orig- tronics and quantum metrology, given the ability of topo- inated from the substrate, strain, or the crystal geom- logically protected states to convey spin information over etry [21]. Additionally, electrostatic gates larger than long distances at ultralow dissipation rate [3–5]. QSH is a 1 V/nm typically lead to appreciable modulation of the manifestation of strong spin-orbit coupling which funda- Rashba SOC strength [22–24]. Actually, transition metal mentally depends on the symmetries of the system [6–9]. dichalcogenides in the 1T’ phase have already shown a However, even in time-reversal symmetric systems, the variety of tunable properties under electric fields [23, 25– lack of a spin conservation axis in QSH insulators allows 28], resulting from spin splitting and modulation of the backscattering effects for edge states, limiting their ballis- SOC parameters [29]. However a possible control of the tic transport [10–12]. In some situations, the emergence canted QSH phase via electric fields remains unexplored of a phenomenon known as persistent spin texture (PST) to date, albeit it could enable the design of all-electrically enforces spin conservation and favors long spin lifetimes controlled spin devices, such as spin-dependent topologi- even in the presence chemical disorder and structural cal switches, that would enrich the prospects for dissipa- imperfections [13]. Such an effect is deeply rooted in tionaless spintronics and quantum metrology. the underlying symmetries of the system [14] and opens This Letter reports on the possibility of a fully control- promising prospects for spintronics when combined with lable variation of up to 90 degrees rotation of the spin po- the manifestation of dissipationless chiral edge states. larization of chiral edge-states, dictating the canted QSH As a matter of fact, the recent prediction [15] and ex- effect, while preserving spin conservation. By combin- perimental observations [16, 17] of a PST-driven canted ing density functional theory (DFT) with tight-binding quantum spin Hall effect in low-symmetry monolayer methods and quantum transport simulations, we show WTe2 provide new ingredient for the use of topologi- that the emerging PST can be continuously varied from cal materials in spintronic applications [18, 19]. Such in-plane to out-of-plane under electric fields below 0.1 phenomenon is rooted in the lack of multiple vertical V/nm, making this effect experimentally accessible. The mirror planes enabling a constant spin-texture in the experimental confirmation of such fully electrically tun- Fermi level vicinity [15]. The topologically protected able spin-polarized topological currents would establish edge-states inherit such canted spin polarization from the a new milestone towards replacing magnetic components bulk bands leading to a quantized spin Hall conductiv- in spintronic devices and all-electric spin circuit architec- ity (SHC) plateau of 2e2 /h along the canting axis. No- tures, as well as optimized resistance quantum standards. tably, such state is more robust against inversion sym- Model and electronic properties. The relevant spin metry breaking when compared with higher symmetry transport properties of monolayer WTe2 in its 1T’ and systems (such as graphene), in which broken inversion 1Td structural phases originate from the presence of symmetry usually generates a Rashba spin-orbit coupling two charge pockets, Q and −Q, symmetrically located (SOC) effects, inducing momentum-dependent spin tex- around the Γ point and related by time-reversal symme- tures and hindering spin conservation [20]. The Rashba try. Those pockets have fundamentally p − d hybridized 2 orbital symmetry that can be well captured by a 4-band the spin-splitting provides a method to determine the Hamiltonian [15, 27, 30, 31]. Ref.[15] demonstrated the strength of the SOC while the spin texture gives the cant- 1Td phase conserves spin along a direction given by a ing angle. canted angle ϕ prescribed by the SOC parameters. Since To determine the microscopic parameters, first- we are focused on the physics around the Q points, we principles calculations were performed with the Siesta shifted our Brillouin zone origin to these points, and [34, 35] implementation of DFT, using GGA-PBE through this process our model becomes a tilted massive exchange-correlation functional, optimized norm- Dirac Hamiltonian [32, 33] conserving pseudopotentials [36, 37], and a standard double-ζ polarized basis set [38]. SOC is included in the H = ~ vFx (τz σx qx + ζσy qy ) + ~vT · q + σ · B + Hq2 + Hsoc , calculation using the fully-relativistic pseudopotential (1) method proposed by Hemstreet et al. [39] and imple- where q = k − τz Q the moment measured from the Q mented in Siesta by Cuadrado et al. [35, 40, 41]. The points; σi with i = x, y, z the Pauli matrices acting on the electronic properties of monolayer WTe2 converged for a pseudospin space, vFx the Fermi velocity of electrons trav- 500 Ry real-space grid cutoff, a shifted 14×8×1 k-point eling along y axis close to the Q points, ζ an adimensional mesh, and a Fermi-Dirac smearing of the electronic tem- parameter that produces anisotropy in the Fermi veloc- perature of k B T = 8 meV. A vacuum thickness of more ity, B a Zeeman-like field acting on the pseudospin; vT than 80 Å was employed to avoid spurious interactions tilt velocity vector pointing to Q and responsible for the between the periodic replicae of the monolayer. WTe2 warping of the Fermi contour, Hq2 = q 2 (1/mp − σx /md ) was fully relaxed in the 1T’ phase using the conjugate a quadratic term that describes the behavior of the elec- gradient algorithm until the forces on atoms were smaller trons in the proximity of the Γ point. The SOC term is than 0.01 eV/Å and the pressure on the cell less than given by 0.037 kBar. The optimized cell parameters (3.53Å and 6.33Å) are within 1% of previous studies [42]. Hsoc = τz σz (λy sy + λz sz ) + (Υs × q) · σ, (2) Fig. 1 shows the band-structure and spin-texture com- where si with i = x, y, z the Pauli matrices acting on the puted using DFT (dashed lines with symbols) and our spin subspace, λx and λy the strength of a momentum- model (solid lines) for 1T’-WTe2 . Since our model was independent spin-orbit coupling, and Υ a tensor that derived to describe the Q points, we highlight in shaded characterized an anisotropic Rashba-like term. The mo- red the range between two poinwts Q+ and Q− that we mentum independent SOC can be rewritten as a Kane- use for the fits(technical details of the fits are given else- Mele Hamiltonian by performing a rotation by an an- where [43]). As seen in Fig. 1, the model reproduces well gle ϕ = arctan(λy /λz ) around the x-axis [15]. We de- the main electronic and spin properties in the energy win- rived the model by imposing the P21 /m symmetry of dow of interest. From the band structure, we determined 1T’-WTe2 by choosing a mirror axis along the x direc- the Dirac velocity vxF = 0.7 nm/fs, the anisotropy pa- tion Mx , and two-fold rotation symmetry along the z- rameter ζ = 0.2, the tilt velocity vxT = 0.4 nm/fs, and axis Cz2 . We consider also the possibility for a broken the pseudo Zeeman field Bx = 0.7411 eV. From the spin- horizontal mirror plane symmetries which will lift also texture, we identify the SOC strength λ = 0.188 eV and inversion symmetry. Such situation could arise due to a canting angle of ϕ ≈ 30◦ , in agreement with other theo- the action of an electric field applied perpendicularly to retical and experimental calculations [15–17, 44]. It is im- the layer, the substrates, geometrical distortions (1Td portant to highlight that the DFT results predict a per- phase) or electrostatic gates, and is characterized by the sistent spin texture, indicated by the constant plateaus Bz parameter. At the Q points, the energy is given by spotted in the shaded region, also captured by our sim- εQ = α p B 2 + (B + σλ)2 where α = ±1 for conduction plified model. α,τ x z and valence bands and σ = ±1 for spins up and down and Electric field effects. To unveil the effect of external λ = λ2x + λ2y . For small values of Bz , the spin-splitting is electric fields on the spin properties, we use DFT to com- given by pute the band structure and spin textures of 1T’-WTe2 (relaxed at zero field) in presence of different electric fields ranging from -10 mV/nm to 10 mV/nm and fit the p ∆ = 2λBz / Bx2 + λ2 . (3) results to our model. As expected, all parameters remain Therefore, spin-splitting vanishes for Bz = 0, in insensitive to the field except Bz that breaks the inver- agreement with prior statements using symmetry ar- sion symmetry and the SOC parameters that define the guments and microscopic calculations [15, 29]. The canting angle ϕ. Moreover, the canting angle saturates at spin texture is defined as the expectation value of the 30 degrees for electric field above 1 mV/nm, a value that spin operator Si (q) ≡ hε(q)| si |ε(q)i where |ε(q)i the matches previous experimental works. Fig.2a shows the eigenvectors of the system. At the Q points, the difference between the bands with opposite spins (orange spin texture is prescribed solely by the SOC parame- crosses), as well as the parameter Bz obtained from a fit ters S(0) = (0, sin(ϕ), cos(ϕ)). Therefore, measuring to the DFT at each electric field (blue dots). The shaded 3 E (meV/nm) (a) −10 −5 0 5 10 25 ∆s (meV) (a) 0 −25 90 (b) ϕ(◦) 60 30 (b) 0 (c) (d) (f ) 0.025 qy (2π/ay ) 0.000 −0.025 0.00 0.05 0.00 0.05 0.00 0.05 qx (2π/ax ) qx (2π/ax ) qx (2π/ax ) FIG. 1. (a) Band structure of 1T’-WTe2 computed using DFT (dashed lines with symbols) and our model (solid lines) fitted FIG. 2. (a) Comparison of the conduction band spin-splitting considering the points in the shaded areas. The Q points are obtained ab-initio calculations (blue dots) and the prediction defined by the minimum of the conduction band, highlighted of our fitted model (orange crosses). (b) The canting angle of in red. (b) Spin texture predicted by the DFT calculations the spin texture computed via DFT compared to the canting and our fitted model. the structure of 1T’-WTe2 is illustrated angle computed from the spin-orbit coupling parameters of in inset (a). The definitions of Q± is provided in [43] our model (orange crosses). To determine the canting angle and spin-splitting from DFT we average over a region of the Brillouin zone around the Q points. The shaded region in (a) and (b) represents the standard deviation of the splitting and region indicates the standard deviation of the splitting canting angle around the Q points, and is a measurement of and canting angle around the Q points, as a measure of the fluctuations away from Q. (c-e) Examples of the regions their variations. We found a band splitting proportional we used to compute the spin-texture for different electric felds to the applied electric field, while the spin splitting ∆ E =-0.8, 0.0 and 8 mV/nm. depends linearly with the electric field. This splitting is due to the Stark effect and is consistent with previous previous researches[30, 45]. More importantly, our fitted using the Kubo-Bastin formula [46, 47]: Bz evolves linearly with the electric field, and the ratio EF   +  α dG Z ∆/Bz ≈ 0.49 matches perfectly with Eq. 3 when using α σij = −2~Ω dE Im Tr δ(E − H)Js,i Jj , the fitted values for λ and Bx . −∞ dE (4) Fig. 2b shows how the canting angle (ϕ) (related to the PST) evolves with the external electric field. It where Ω the area of the sample, Js,i α ≡ {Ji , sα }/2 is the is clear that ϕ is varying substantially at low electric i-th component of the spin current density operator, with field, whereas it saturates to a critical angle ϕc = 30 α = x, y, z denoting the spin polarization direction and for high electric fields. These observations indicate two Jj ≡ (ie/Ω~)[H, Rj ] the j-th component of the current important things, (i) the PST is robust against electric density operator, with e the electron charge and Rj the field and its direction is electrically tunable, and (ii) our position operator [48]. The spectral operators δ(E − H) model captures this phenomenon through the renormal- and G+ ≡ 1/(E − H + i0+ ) are the Dirac delta and ization of the SOC parameters. This is one of the cen- the retarded Green’s function, respectively. We numeri- tral results of our article. The shaded regions displays cally computed the Kubo-Bastin formula using the kernel the standard deviation, indicating a small discrepancy polynomial method [47–50] with 2000 Chebyshev expan- from the PST landscape. This is clearer from Fig. 2(c-e) sion moments (equivalent to a 5 meV of broadening), on where variations of the spin texture above the Fermi con- a system containing 4 millions orbitals. tour (white line) are seen to increase for higher electric The results are presented in Fig. 3a, where the spin fields. Such variations are attributable to the anisotropic Hall conductivity is plotted along the z 0 direction (pre- Rashba field. scribed by different canting angles depending on the elec- Electrically controlled chiral spin currents. In WTe2 tric field strength) for three different electric fields E =- the persistent spin texture defines the spin polariza- 0.8, 0.0 and 8 mV/nm. Within the energy gap, the topo- tion direction in the QSH insulating regime [15–17]. To logical plateau remains pinned to its quantized value at demonstrate that the electrically tunable PST enables 2e2 /h (three curves are almost superimposed), demon- full control of the spin polarization of spin currents (or strating that in this range of fields the small variations canting angle), we determine the spin Hall conductivities due to the anisotropic Rashba term do not jeopardize spin 4 2 does not compromise their topological robustness. These (e2 /h) results promote 1T’-WTe2 as an alternative source of 1 tunable spin currents that could lead to all-electrical σxy 0 z spintronic devices and open a door to explore other mech- 0 anism to tune the persistent spin texture such as strain −0.2 −0.1 0.0 0.1 0.2 0.3 Energy (eV) or pressure. Supported by the EU H2020 Programme under Grants 0.2 No. 881603 (Graphene Flagship), No 824140 (TOCHA, H2020-FETPROACT-01-2018) and No. 824143 (MaX EF (eV) 0.0 Materials Design at the Exascale CoE) and by Spanish MICIU, AEI and EU FEDER (Grants No. PGC2018- −0.2 096955-B-C43 and PCI2018-093120). Z.Z. acknowledges −1.0 −0.5 0.0 0.5 1.0 E (V/nm) support by the Ramón y Cajal program RYC-2016-19344 (MINECO/AEI/FSE, UE), the Netherlands sector plan FIG. 3. (a) Spin Hall conductivity oriented along the z 0 direc- program 2019-2023, Spanish MINECO (FIS2015-64886- tion for E = −0.8 mV/nm (red), 0.0 mV/nm (black), and 8 C5-3-P). R.C. acknowledges the funding from the EU mV/nm. (blue). The curves fell on top of each other demon- H2020 Programme under the Marie Sklodoswka–Curie strating that these fields do not alter appreciable the topo- Grant No. 665919. JY, PK, MGM and ZZ acknowledge logical properties of the system. (b) Spin polarization of the spin currents as a function of the electric field and Fermi level. the computer resources at MareNostrum and the tech- The arrows indicate the direction of the spin polarization in nical support provided by the Barcelona Supercomput- the yz plane. The dashed lines indicates the energy range of ing Center through Red Española de Supercomputación the conductivity plateau. (Grants No. RES-FI-2020-1-0018, RES-FI-2020-1-0014 and RES-FI-2020-2-0039). ICN2 is funded by the Gen- eralitat de Catalunya (CERCA Programme and Grants conservation and topological protection. 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