Downloaded by guest on September 25, 2021 rniinmtldichalcogenides transition-metal polarization spin current-induced out-of-plane quantum valley the for number. of way control the electric paves CISP and rotation The magnetization space. momentum the with out-of-plane in anisotropic together splitting an spin magnetization inducing in-plane coupling an spin–orbit to intrinsic due is CISP out- the of-plane that find and ferromagnetic monolayers easy-plane dichalcogenide for transition-metal model low-energy We general methods. a first-principles deduce and theory linear-response on based fpaemgei reigealsacs oanme of effect number Hall a anomalous to quantum access e.g., enables phenomena, out- ordering intriguing controlling Besides, magnetic torque. perpendicular of-plane spin–orbit on by switching based perpendicular-magneti- zation facilitates concepts it out-of-plane device where highly anisotropy, in The magnetic are important CISP. is mechanisms out-of-plane CISP and generating systems. for materials Rashba of desirable prototypical number large new a in Both the occurs with that compared fewer CISP relatively in-plane still are materials a out-of-plane real the in in of CISP grown examples the well However, quantum (15). direction and certain the 11–14) in (7, proposed further heterostructure been bilayer from recently arising has breaking CISP symmetry of crystalline component redistribu- carrier space. out-of-plane the The momentum after in-plane tion. the always chiral in is CISP gives components resulting two-dimensional The which spin (8–10), in-plane by SOC of captured break- textures Rashba-type minimally symmetry with is inversion models The interface (2D) (5–7). the used CISP at widely the is ing of layer study nonmagnetic fer- the heavy a in a of composed and structure layer bilayer by romagnetic A driven (5). torque dynamics motion, spin–orbit Landau–Lifshitz–Gilbert the spintronic domain-wall to many switching, according etc., seen magnetization have in the applications phenomena on spin-galvanic torque, These coupling. spin–orbit inverse exchange the i.e., through torque, parameter magnetic order a a magnetic in applies (CISP) further polarization (1– system currents spin electric current-induced by The induced 4). be can polarization carrier-spin I freedom of degree valley we monolayer ferromagnetic Taking Here, in CISP. VSe emerge monolayers. of can dichalcogenide study CISP for transition-metal out-of-plane the an search promote that and to propose out- spin the desirable generate carrier highly to of-plane mechanisms is and materials it prototypical new elec- evolution, but and CISP, structure in-plane switching the tronic of perpendicular-magnetization that that for than fewer Given important polar- relatively spin are polarization. current-induced (CISP) spin ization out-of-plane of in-plane realizations to material leading Rashba the model, using two- analyzed 2019) in conventionally 20, which, is currents, July nonequilibrium systems, electric review dimensional by for a induced (received allows be 2020 to 1, symmetry polarization June spin approved inversion and spatial NJ, Piscataway, of Jersey, New Absence of University State The Rutgers, Vanderbilt, David by Edited d b a ioL ( Li Xiao current electric under dichalcogenides transition-metal in spin carrier Out-of-plane www.pnas.org/cgi/doi/10.1073/pnas.1912472117 etrfrQatmTasotadTemlEeg cec,Sho fPyisadTcnlg,NnigNra nvriy ajn 103 China; 210023, Nanjing University, Normal Nanjing Technology, and Physics of School Science, Energy Thermal and Transport Quantum for Center colo dacdMtrasDsoey ooaoSaeUiest,Fr oln,C 80523 CO Collins, Fort University, State 78712; Colorado TX Discovery, Austin, Materials Austin, Advanced at of Texas School of University Physics, of Department odcigsse ihboe neso symmetry inversion broken with nonequilibrium (SOC), system coupling spin–orbit conducting nonnegligible and a n 2 n VTe and ) 2 a,b,1 seape,w aclt h u-fpaeCISP out-of-plane the calculate we examples, as u Chen Hua , | nrni pnobtcoupling spin–orbit intrinsic c,d,1 n inNiu Qian and , | b | c eateto hsc,Clrd tt nvriy otClis O853 and 80523; CO Collins, Fort University, State Colorado Physics, of Department rprisofrptnilapiain neeg-fcetelec- 26). optoelectronic energy-efficient (22, valley-contrasting in spintronics dimen- and applications that a tronics potential add given offer may CISP, freedom, properties the of 2H-MX to of degree bands electronic sion low-energy an More- the paper. as in this valleys Valley, out-of- in two an addressed are fully to there have over, leads we splitting which ques- spin CISP, attractive plane out-of-plane an thus the is whether It tion 22). in (21, results splitting and spin out-of-plane Zeeman-type therefore M- is field momentum, the 2D magnetic the through being effective gradient, plane SOC-induced potential interfacial mirror the The the that a ensures from of layer structure SOC atom existence crystal Rashba The bulk to asymmetry. the contrast 2H-MX from in monolayer results asymmetries, of theory effective SOC non-Rashba energy the intrinsic for 2H-MX The systems in 2H-MX SOC. 2D that CISP representative emphasizing worth possible are is monolayers of It lacking. study still is to the monolayers paid (6), been structures recently bilayer has in attention CISP break- special symmetry the inversion While and (21–24). SOC strong ing both have spin-related phase 2H exploring dichalco- for transition-metal monolayer genides, platforms particular, In powerful (20). physics and simple of splitting 19). valley (18, and (17), transition semiconductor-to-metal (16), doi:10.1073/pnas.1912472117/-/DCSupplemental at online information supporting contains article This 1 the under Published Submission.y Direct PNAS a is article This interest.y paper. X.L., y competing the no research; wrote declare H.C. performed authors and The X.L. X.L. and research; data; analyzed designed Q.N. Q.N. and H.C., and X.L. contributions: Author owo orsodnemyb drse.Eal iionn.d.no huachen@ or [email protected] Email: addressed. be colostate.edu.y may correspondence whom To ep oilmnt hs hsc n rae h cp of scope the broaden and physics CISP. these the CISP. illuminate of also to study model the helps effective non-Rashba to and freedom, argument symmetry of The degree adds valley and dimension, current electric a associated to is responses polarization valley-dependent spin with the Moreover, opportunities of splitting. control provides valley electric which and rotation space, magnetization out-of-plane spin for symme- the the in breaking by try dichalcogenides transition-metal of is com- in help CISP out-of-plane designed the fewer intriguing With an systems. coupling, relatively Rashba spin–orbit intrinsic in are symmetry CISP crystalline in-plane examples the with pared material by realized its been breaking, has polarization reorien- it perpendicular-magnetization and spin for tation, important current-induced is (CISP) out-of-plane Although Significance w-iesoa a e al aeil rvd plethora a provide materials Waals der van Two-dimensional MX MX 2 NSlicense.y PNAS M=V o ;X=S e e t.,i the in etc.), Te, Se, S, = X W; Mo, V, = (M 2 gahn iae 2)and (25) bilayer /graphene . y https://www.pnas.org/lookup/suppl/ NSLts Articles Latest PNAS ∇V ∝ MX ∇ V 2 2 /ferromagnet sin-plane. is , × ntelow- the in with p, | f7 of 1 2 p 2 2 .
APPLIED PHYSICAL SCIENCES 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. δs is therefore absent δs = τχE, [1] in a perpendicularly magnetized or nonmagnetic monolayer 2H-MX2 that has the C3 symmetry. Similarly, a x component where δs is the carrier spin induced by the electric field E, of E is required for nonzero δsz due to σv -related symme- and τ is the relaxation time. χ is the part of the intraband tries for a magnetization M along the high-symmetry x or y response tensor that does not depend on τ in the good metal directions. limit and is invariant under any unitary symmetry operations of the system in equilibrium. More details can be found in SI First-Principles Calculations of CISP. Armed with the symmetry Appendix. analysis, we perform first-principles calculations of the CISP in We now consider symmetries of the 2H-VX2 (X = Se or Te) monolayer VSe2 and VTe2 by density functional theory and monolayer that can help us identify possible nonzero compo- linear-response theory. The calculated lattice constants are 3.33 nents of χ. Fig. 1 A and B shows the atomic structure of the and 3.60 A˚ for VSe2 and VTe2, respectively. For both monolay- VX2 monolayer with the D3h point group. Two sublattices of a ers, the stable ferromagnetic order has a magnetic moment of honeycomb lattice are respectively occupied by vanadium and 1.0 µB per unit cell, agreeing with previous results (23). chalcogen atoms. Each vanadium atom lies at the center of a Fig. 1 C and D shows band structures of the ferromagnetic trigonal prismatic cage of six chalcogen atoms. There are there- VSe2 monolayer, and similar band structures are also obtained fore three atomic layers stacked as X-V-X. Without considering for VTe2. When the magnetization is in-plane, there are two the magnetization, there are two types of mirror planes. The inequivalent but degenerate valleys at K± points of the Brillouin vanadium layer is a mirror, σh , lying in the monolayer (Fig. 1B). zone. For an out-of-plane magnetization, the valley degeneracy The other type of mirrors consists of three planes perpendicular is lifted. There are valley splittings of 78 and 92 meV for VSe2 to the monolayer and through different V-X bonds. One of them and VTe2, respectively, by comparing valence band maxima at lies in the y–z plane, as denoted by σv in Fig. 1A. The others two valleys, similar to magnetized MoX2 (18, 19). The valley are obtained by rotating σv through 2π/3 and 4π/3 around the z splitting leads to rich valley-spin physics, e.g., valley polariza- axis due to the threefold rotational symmetry, C3. Since the VX2 tion induced by carrier doping and nonpolarized optical pumping monolayer has an easy-plane magnetic anisotropy (23), the joint besides the well-known chiral optical field (18, 19, 24). The val- transformation of σh and time reversal T becomes a new sym- ley structure and optoelectronic responses are therefore tunable metry, and the C3 symmetry is broken, when the ferromagnetic by tilting the magnetization. The out-of-plane CISP, if realized, order is further taken into account. provides opportunities for electric control of the magnetization Applying the symmetry operation σh T on different quanti- and valleys. ties in Eq. 1, we find the right-hand side is invariant up to a We compute the CISP in the VSe2 and VTe2 monolayers by sign change under this operation (SI Appendix), if E is in-plane. the Kubo linear-response formula. For the Fermi surface contri-
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Fig. of thickness monolayer of order maxi- the The 0.01 sign. about is its VSe CISP out-of-plane changes the of CISP size mum the which beyond that directions magnetization the of the component in-plane breaks an is for there as values long extreme opposite has the in- along it for emerges and indeed magnetization, CISP plane out-of-plane zero, is CISP in-plane in found of CISP angle, lated polar the by respectively. energy, band electronic the η and energy Fermi the (4): as element, the bution, pnigcagso h em e contribution. to sea Fermi the of changes sponding and rotation the magnetization within in-plane rotation with out-of-plane contribution surface Fermi the of ie al. et Li C AB = k x i.2 Fig. G x ˆ and ~/2τ 2 k n hnnraie yalniuia odciiy ti on is it conductivity, longitudinal a by normalized When . R and β ae ad n aevcos respectively. vectors, wave and bands label k χ e h vlto fcluae urn-nue pni ntcl with cell unit a in spin current-induced calculated of evolution The z (G = αβ orsod oteeeti-eddirection. electric-field the to corresponds η steeeto charge, electron the is A ietos respectively. directions, epciey hl the While respectively. S2, and S1 Figs. Appendix, SI E x stebn raeigdet h iodrpotential. disorder the to due broadening band the is 0 = = C and F and n 3 10 A = .01 π e k VTe ymty xetfrsm pca magnetization special some for except symmetry, S ) η −7 . V h antzto ieto sdenoted is direction magnetization The eV. −0.2 B ∗ y Re Vbsdo rvosrslsof results previous on based eV 1 = ietos h u-fpaeCS tl xssas exists still CISP out-of-plane The directions. µ hw acltditaadCS nthe in CISP intraband calculated shows 2 B X θ χ n h eedneo IPon CISP of dependence the and n /(E e /m( A/cm per n h zmta angle, azimuthal the and , ,k αβ hs ∼6 F ftersos esri Eq. in tensor response the of , α − i x osdrd(4,ti corresponds this (34), considered A n ˚ –z ε k E n hv k ln,rsetvl.(C respectively. plane, k α, S x β + .Wt the With S2). Fig. Appendix, SI and i 2 β stemnlyrsae,and area, monolayer’s the is n D i ooae.(A monolayer. k η eoeCreindirections. Cartesian denote (G where ), E F n = A k ϕ . eV. −0.2 G = n R k and 0 − E and µ G F B hs and θ n h calcu- The ϕ. R α e V/ 1 per k = h changes The B) and G MX G i correspond 0 n h corre- The D) n R E n R k k ( 1 k F 2 A) (hv ), ε sgiven is n a be can mono- sthe is k VSe β for A ˚ i are [2] n M k n 2 ) iemgei edatn namgei oeto 1 of relationship moment the magnetic by a on estimated acting effec- field the magnetic dynamics, tive magnetization the in field magnetic effective of that than larger of tude size maximum the particular, In VTe in CISP staggered of order an to ln IPae8 and 89 out-of- are CISP maximum plane to corresponding spin fields the magnetic to effective according The eV 0.5 mag- be the of to splitting and estimated spin and carrier moment, the netic between interaction exchange the E)det ni-ln antzto ihteaiuhlangle azimuthal the with respectively. interactions magnetization in-plane exchange ϕ, an and to Eq. SOC, due Hamiltonian, intrinsic (EI) the states, of Dirac parts massive three the Here, space fer- low-energy the of the projecting composed to and mainly term element, interaction X the exchange of contributions romagnetic SOC second-order atomic the considering from critically By Appendix. out-of-plane generic K the a of 2H-MX of Hamiltonian ferromagnetic effective origin easy-plane an derive physical we the CISP, intraband Model. of Low-Energy understanding a from itive CISP the of Origin intraband the than found smaller is relatively out- it is contribution. the CISP Besides, interband 2D. for the Fig. found that in also rotation is magnetization cosine dependence of-plane magnetization. orientation or the similar in sine contributions The standard high-order to the due is from orientation function first-order derivation the The demonstrating dependence. con- respectively, in 2C, in-plane, contribution. Fig. nearly surface Fermi be the to from it δ CISP found out-of-plane and to 31) trast (30, CISP the to analysis. symmetry the with consistent are results for found when also For is CISP rotation. perpendicular intraband out-of-plane the the of for dependence orientation vanishing is Besides, it magnetization. per- that magnetization the except tilting pendicularly, for the opportunity an tune provides the which within to rotated exploited is (ϕ magnetization be the may When CISP. dependence orientation order (θ (7). resonance ferromagnetic spin–torque the of technique the were (37) VX doping considerable and in 36) achieved (35, fields electric for erable especially field, magnetic effective smaller V/ much 1 is than needed field electric the Therefore, respectively. eobtain we s ± x eas acltdteitrad(.. em e)contribution sea) Fermi (i.e., interband the calculated also We rotation magnetization in-plane considering further When = 0 = 2 and aly 3,3) h eiaindtiscnb on in found be can details derivation The 39). (38, valleys 2 nFg 2A, Fig. in 2) π/ r xetdt eraiyosre xeietlyb,e.g., by, experimentally observed readily be to expected are M ooae smc agrta hto the of that than larger much is monolayer nFg 2B, Fig. in ) δ saogtehgl symmetric highly the along is ,eg,odr f10 of orders e.g., A, ˚ s H y i.S3 ). Fig. , Appendix (SI magnetization the from eV ∼1 SOC H MX H r prxmtl rprinlt sinϕ to proportional approximately are H EI 0 10 2 = (M = = = σ −8 ooaes h IPadascae oqein torque associated and CISP the monolayers, α Mn γ H v + /s F (λ µ 0 M (γσ d α δ 0 B δ + 2 0 0 s ↑ σ 1.5 s N u(0 1.Teoto-ln IPo the of CISP out-of-plane The 31). (30, Au σ z VSe (α , z per H + D (σ x + D δ d laspit otesm direction, same the to points always spootoa to proportional is p SOC s × + 0 = ↓ + + N x x E , 2 10 10 M + s x ie htteCS ly oeof role a plays CISP the that Given . d λ per swl.Tesecond-order The well. as appears −4 k + N ±2 , + ↑ B 2 7 2 4 y σ ˆ y e σ σ eff o10 to h corresponding the , y A/cm e V/ 1 per T H , and , i − D − D p ϕ z 2 EI = y )s )(s r al arcsfrthe for matrices Pauli are ) VTe + )s ooae ntevcnt of vicinity the in monolayer , z δ −3 σ + , 0 N s 2 d x VTe z oprbet,eg,the e.g., to, comparable , − N + e 2 ±2 J ↓ V/ NSLts Articles Latest PNAS and s −i ex stoodr fmagni- of orders two is − ∆ N 2 for A ,t nueasizable a induce to A, ˚ ttso h atom, X the of states ˚ ϕ /(µ 2 e h second- The 2ϕ. cos ogtamr intu- more a get To σ ie htconsid- that Given . + −i y z s B ietos These directions. s ϕ VSe 0 VSe − N ) ). describe b–d, 3 , 2 e where , i 2 ϕ 2 δ monolayer. ) rcosϕ or and s x µ –z B vanishes | salso is VTe J plane f7 of 3 [3d] [3b] [3a] ex [3c] SI in 2 is ,
APPLIED PHYSICAL SCIENCES orbital pseudospin/real spin, while σ0 and s0 are 2 × 2 iden- AB tity matrices. The orbital parts of the basis functions have the K K 1 - + relations d0 = d 2 and d±2 = √ (d 2 2 ± idxy ) for K± valley, K K z 2 x −y - + E with subscripts 0, ±2 denoting the magnetic quantum numbers. E Up-spin and down-spin are denoted by ↑ and ↓, respectively. D 1 D 1 γ = ±1 labels two valleys. σ± ≡ (σ0 ± σz ), s± ≡ (s0 ± sz ) and -1 -1 2 2 -1 -1 N 1 N 1 CDkx (Å ) kx (Å ) kx (Å ) kx (Å ) σ± ≡ (σx ± iγσy ), s± ≡ (sx ± isy ) are defined to separate the 2 2 -0.12 0 0.12 -0.12 0 0.12 -0.12 0 0.12 -0.12 0 0.12 diagonal (D) and nondiagonal (N ) elements of the orbital and 0.12 0.12
spin Pauli matrices. vF and ∆ are the Fermi velocity and the ) )
-1 -1 crystal-field splitting between d0 and d±2 orbitals, respectively. 0 (Å 0 (Å y y
The two orbitals have different SOC strengths denoted by λ0 k k and λ2, respectively, which are dependent on the atomic SOC -0.12 -0.12 strength of the X atom, λ. We note the SOC terms in HSOC lead
such a low-energy model is applicable. Note the first term in -1 -1
H (Å 0 (Å 0
EI is diagonal in orbital space and leads to a conventional spin y y splitting, with orbital-dependent exchange-coupling strengths k k -0.12 -0.12 denoted by M0 and M2. The second term is, however, nondiag- df df onal in both spin and orbital basis and arises from a combined (per V/Å)-1500 0 1500 (per V/Å) -1500 0 1500 effect of the magnetization and atomic SOC. Specifically, MN is given by -1 -1 k (Å-1 ) k (Å-1 ) k (Å ) k (Å ) √ GHx x x x 2 -0.12 0 0.12 -0.12 0 0.12 -0.12 0 0.12 -0.12 0 0.12 3λ M1 K+ : MN = √ , 0.12 0.12
2(+1 − 0)(+1 − +2) ) )
-1 √ [4] -1
2 (Å 0 (Å 0 3λ M1 y y K− : MN = √ , k k 0 0 0 0 2(−1 − 0)(−1 − −2) -0.12 -0.12
-13 0 13 0 11 where M1 stands for the strength of exchange coupling for d±1 (µB per V/Å) (µ per V/Å) -11 0 B orbitals, and m /m (m = 0, ±1, ±2) is the on-site energy of the dm orbital at K+/K− when the SOC and the magnetization are Fig. 3. Momentum-resolved electronic structures near K± valleys. (A and 0 B) Schematics of the carrier redistribution when E k x and E k y, respectively. absent, with m = −m owing to time-reversal symmetry. As a 2 (C) hszi texture. (D) Band-energy contours with respect to the band-edge result, MN is valley-independent. The λ factor in MN suggests energy. (E and F) δfk for two electric-field directions. (G and H) δsz,k for two this term is due to second-order processes of the atomic SOC. electric-field directions. In C–H, Left and Right correspond to the K and ↑ ↓ ↓ ↑ ↓ − That is, d0 and d+2 (d0 and d−2) states are coupled with d+1 K+ valley, with the gray dot denoting the K− and K+ point. The highest ↑ ↑ ↓ energy band is chosen with more eccentric distributions of hszi and band and d+1 (d−1 and d−1) states at K+ (K−) valley, respectively, by energy. All model parameters can be found in SI Appendix, and a larger MN the ladder operations of the SOC (38, 39). The d+1 (d−1) states with opposite spins are further coupled by the in-plane exchange is adopted to amplify its role compared with the fitted value. interaction. The process is summarized as
↑ SOC ↓ EI ↑ SOC ↓ K+ : d ←−−→ d ←−−→ d ←−−→ d , which is schematically illustrated in Fig. 3 A and B. To under- 0 +1 +1 +2 [5] ↓ SOC ↑ EI ↓ SOC ↑ stand why δsz is nonzero, we plot hsz ik, εk, δfk, and δsz,k of the K− : d0 ←−−→ d−1 ←−−→ d−1 ←−−→ d−2. highest band in Fig. 3 C–H. Due to the intrinsic HSOC ∝ sz , hsz ik appears despite the dominant in-plane components induced by Since the MN term mixes different orbitals with opposite spins, the in-plane magnetization, and it is opposite at the two valleys it converts the anisotropy in the spin space arising from the in- (Fig. 3C). Moreover, due to the MN term in HEI (Eqs. 3d and plane magnetization to the orbital space, which is reflected by an 5), hsz ik becomes eccentric with respect to the Kγ point at a anisotropic low-energy band structure with the rotational sym- given valley but in a way different from that of the band energy metry breaking at each valley (see Fig. 3D). In contrast, all bands (Fig. 3D). It also exhibits the rotational symmetry breaking. As are isotropic in the minimal model with respect to corresponding a result, sampling hsz ik by the δfk in the neighborhood of the Brillouin zone corners if MN = 0, irrespective of the first term Fermi surface determined by εk gives a nonzero result at each in HEI. valley when E k x. Given that two valleys are related by the ver- Based on our analysis above, the rotational symmetry breaking tical mirror σv , both hsz ik and δfk are mirror-antisymmetric by in the orbital space is the ultimate reason for the out-of-plane comparing their values in two valleys, which ensures that their CISP. We nevertheless give a more intuitive explanation by the product δsz,k is mirror-symmetric and two valleys equally con- semiclassical Boltzmann formalism. In a semiclassical Boltzmann tribute to a nonzero total δsz (Fig. 3 C, E, and G). When E k y, 1 P formalism, the CISP is obtained as δs = S nkhsinkδfnk. δfnk is the total δsz is vanishing by combining antisymmetric hsz ik and the nonequilibrium part of the single-particle distribution func- symmetric δfk under the σv symmetry (Fig. 3 C, F, and H). tion and is related to the Kubo formula result through δfnk = Moreover, the vertical CISP induced by low-symmetric out-of- eE~ A R R R 2π hvinkRe(GnkGnk − GnkGnk). It can be roughly understood plane spin splitting applies to other regions of the momentum as the change to the equilibrium Fermi–Dirac distribution func- space, e.g., Γ valley that participates in transport with further tion under a momentum shift proportional to the electric field, hole doping (SI Appendix, Figs. S4 and S5). Besides, the degree
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Landau–Lifshitz–Gilbert the tion to according CISPs dynamics, magnetization–orientation-dependent the in above the the study CISP. of to performed roles the further by are calculations Induced zation-dynamics Dynamics Magnetization The etre,rsetvl.Tesed-tt rcsin rudtoplsare poles two around in precessions given steady-state The respectively. jectories, reduced the of function a as angle elevation The strengths, (A) field CISPs. from arising 4. Fig. CISP. out-of-plane of size the struc- band determines VTe the texture in spin symmetry and rotational ture the from deviation the of respectively. magnetiza- in realized the are representing B magnetizations sphere resting The unit direction. on tion dynamics magnetization the with S6 compared Fig. CISP, larger a to leading ture, inter 0 A B h antzto yaiscnla ovrosfia rest- final various to lead can dynamics magnetization The ieto,udrteato of action the under direction, /B 2 u 0 a oecnieal eiaini t lcrncstruc- electronic its in derivation considerable more has ). h antzto yaisudrteato fefciefields effective of action the under dynamics magnetization The en 16 ) 0 .) n 16 .)frbu,bak n e tra- red and black, blue, for 0.6) (1.6, and 0.6), (0, 0), (1.6, being ) C and B ihrdcdsrntso (1.6, of strengths reduced with D, intra 0 /B u 0 CD and B B x intra 0 inter 0 ieto,and direction, /B /B u 0 u 0 ersnaietaetre of trajectories Representative (B–D) . and E k x B h fetv magnetic effective The . inter 0 B /B u 0 .)ad(0.6, and −0.6) VSe u 0 stesrnt of strength the is Here, . ih(B with B, 2 h magneti- The IAppendix, (SI B intra/inter 0 intra 0 −0.8), /B u 0 , swrhntn htalrefml f2 ermgeswt the with ferromagnets 2D of family large it a systems, that experimental noting promising worth is indeed are that monolayers 11–14). in (7, components symmetry crystalline spin lower nonequilibrium only with more the of is with presence compared CISP symmetry, the out-of-plane crystalline higher the par- to plane, owing is component monolayer magnetization the the the to When needs allel SOC. CISP intrinsic magnetization in-plane and out-of-plane nonvanishing component a direc- The both of the magnetization. role on combination the dependence intraband of second-order The tion a characteristics. demonstrates time-reversal distinct the CISP by has induced breaking pre- CISP in The symmetry breaking 11–15). symmetry (7, crystalline studies the vious fur- only can from out-of-plane arises the strain contrast, CISP In CISP. uniaxial out-of-plane the the enhance creation ther while transition-metal the ferromagnetic monolayer, and pristine dichalcogenides reduction in CISP symmetry out-of-plane overall of the enables that and T, 20 For dichalcogenides. transition-metal ferromagnetic in dynamics tion magnetization out-of-plane rotation. considerable realizing for preferred nonvanishing have poles angle, two magnetization-orientation magnetiza- elevation the (average) resting around the precession out-of-plane the to The and tion CISPs. due the complicated of dependency very the 4 be is Fig. can ics in axis 4B. shown procession Fig. as the in poles procession, shown steady-state as magneti- motion, the the damping For and representing the can semisphere sphere after magnetization unit upper direction the zation the equator, states, of the semisphere on resting lower rest final to For come 4A. eventually Fig. in ary IPi ovnsigfralmgeiaindrcin including directions vertical magnetization the to all the perpendicular for Besides, nonvanishing rotation. is CISP CISP magnetization help out-of-plane can out-of-plane which the Appendix), the (SI enhances case unstrained space 1 the with orbital (Fig. compared the preserved to in well reduced breaking is still group uniaxial are the point leys For crystal reduction. the symmetry strain, crystalline the strained and uniaxially zation of the CISP where the layer, study further We Discussion applied e.g., by, in increased (SI demonstrated be as resonance can strain, CISPs ferromagnetic the and spin–torque Besides, ferromagnet the Appendix). adjacent of the technique of the rotation magnetization the of be by orders to the enough of 10 field to large anisotropy 10 is weak relatively T a 0.1 of with romagnet of heterostructure order A the experimentally. observed of avail- an field out-of-plane, effective itself able magnetization the in tilt cannot realized CISPs are be splitting valley to of likely control electric corresponding and degrees V/ to up increase can with of order the of to likely very are the Since rotation. above magnetization proposed out-of-plane out-of-plane CISPs induce considerable the are large Therefore, there space. with that rotations found is magnetization It plane. tor lhuhw aebe ouigo the on focusing been have we Although space spin the in breaking symmetry the that noted is It magnetiza- the of realization material the discuss further We .Teoto-ln antzto oainwith rotation magnetization out-of-plane The A. ˚ B VSe ξ inter 0 0 = 2 2 /B Tmyb osrce oehbtterl fteCISP the of role the exhibit to constructed be may mT h adai oto-ln)aiorp edi about is field anisotropy (out-of-plane) axis hard the , oee,for However, 4A. Fig. to according B u 0 intra 0 xs eaieylrertoof ratio large relatively a axis, /B 10 C −3 u 0 3 o neeti edo h re of order the of field electric an for ∼1 si h re f00 o neeti field electric an for 0.01 of order the in is VSe ymtyi rknb ohtemagneti- the both by broken is symmetry V/ .A npaertto cusin occurs rotation in-plane An A. ˚ VTe 2 C ξ Discussion. sheet. = and 2 2 π/ .Ee huhthe though Even Appendix). (SI D − rteeutr h dynam- The equator. the or θ ξ C esrdfo h equa- the from measured , naszbeparameter sizable a in and NSLts Articles Latest PNAS VTe .Tesymmetry The D). C VSe B VSe 2v 2 intra 0 u w val- two but , the , ξ 2 VSe 2 szr near zero is ξ to and n fer- a and ftn of tens of B 2 B intra 0 | mono- inter VTe 0 VSe MX 10 f7 of 5 /B −3 is u 2 2 0 2
APPLIED PHYSICAL SCIENCES symmetry breaking in the spin space can also generate out-of- the Vienna Ab initio Simulation Package (45, 46). Here, the projector aug- plane CISP, e.g., VS2 (23), NbX2 (40), TaX2 (41), group-III mented wave potentials (46) and the Perdew–Burke–Ernzerhof exchange- monochalcogenides (42), transition metal-doped MX2 (19), etc. correlation functional (47) are used. A plane-wave cutoff of 400 eV and On the other hand, the low-energy Hamiltonian proposed here a k-mesh of 12×12×1 are adopted. Vacuum slabs more than 16 A˚ thick is also expected to become an effective model like the Rashba are inserted to minimize the interaction between the monolayer and its periodic images. Structure optimizations are performed with a convergence model when it comes to discussing out-of-plane CISP and other threshold of 0.01 eV/A˚ on the interatomic forces. The SOC is included in the spin-related physics. calculation of electronic structure, unless otherwise specified. Finally, we did not consider higher-order scattering processes The ab initio tight-binding model is then constructed based on such as the vertex correction (4, 25, 43), with the expectation that Wannier functions with first-principles input (48, 49) to calculate the CISP. they will not change the main conclusions of this work quali- The Wannier basis set consists of 11 orbitals, i.e., five d-orbitals of one tatively. As for VSe2 and VTe2, the band splittings induced by vanadium atom and six p-orbitals of two chalcogen atoms in a unit cell. the SOC have magnitudes of 0.08 and 0.1 eV in the low-energy We perform a real-space truncation of the spinless Hamiltonian up to fifth- nearest neighbors and symmetrize corresponding hopping and on-site ener- bands at K± valleys (SI Appendix, Fig. S3), respectively, while the band broadening due to the disorder potential, η = 0.01 eV, gies (38, 39). The symmetrized Hamiltonian reproduces the first-principles bands very well (SI Appendix, Fig. S3), and the symmetrization ensures is used in our calculation, and it represents the disorder strength. that symmetry-disallowed components of CISP are vanishing. The SOC is Since the ratio of the disorder strength to the SOC is ∼0.1 and further incorporated by operating on-site λ0ˆL · ˆs term on atomic orbitals the SOC induced band splitting is well defined under the action (38, 39), with ˆL (ˆs) being the orbital (spin) angular momentum opera- of disorders, our calculation of the CISP in the ferromagnetic tor. The magnetization is taken into account by the exchange interaction, 0 transition-metal dichalcogenides are done in the weak disorder m(sx sinθ cos φ + sy sinθ sin φ + sz cos θ). The above strength parameters, λ regime for which the finite lifetime approximation is appropri- and m, are element-dependent and orbital-dependent, respectively. They ate for giving a good qualitative description of nonequilibrium are obtained by comparing spinless and spinful first-principles calculations. responses. In contrast, the vertex correction is indispensable in the diffusive transport regime with a large strength ratio of the Data Availability disorder to SOC (44). Nevertheless, quantitative dependence of The data supporting the findings and the parameters for repro- the out-of-plane CISP on different impurity types, concentra- ducing the calculated results are included in the main text and SI tions, and distributions is interesting and worth further study, by, Appendix. e.g., using Keldysh nonequilibrium Green’s function formalism in a two-terminal device (10). ACKNOWLEDGMENTS. X.L. is grateful to Shiang Fang, Jakub Zeleznˇ y,´ Huawei Gao, Hua Jiang, and Cong Xiao for valuable discussions. X.L. was supported by National Natural Science Foundation of China Grant 11904173 Methods and the Jiangsu Specially-Appointed Professor Program; Q.N. was supported We study geometric and electronic structures of the VSe2 and VTe2 mono- by the US Department of Energy Grant DE-FG03-02ER45958, Division of layers by density functional theory calculations, which are implemented in Materials Science and Engineering, for initial theoretical formulation.
1. E. L. Ivchenko, G. E. Pikus, New photogalvanic effect in gyrotropic crystals. JETP Lett. 20. J. Park, Opportunities and challenges of 2D magnetic van der waals materials: 27, 604–608 (1978). Magnetic graphene?J. Phys.: Condens. Matter 28, 301001 (2016). 2. E. L. Ivchenko, Y. B. Lyanda-Geller, G. E. Pikus, Photocurrent in structures with quan- 21. Z. Zhu, Y. Cheng, U. Schwingenschlogl,¨ Giant spin-orbit-induced spin splitting in two- tum wells with an optical orientation of free carriers. JETP Lett. 50, 175–177 dimensional transition-metal dichalcogenide semiconductors. Phys. Rev. B 84, 153402 (1989). (2011). 3. A. G. Aronov, Y. B. Lyanda-Geller, Nuclear electric resonance and orientation of 22. D. Xiao, G. B. Liu, W. Feng, X. Xu, W. Yao, Coupled spin and valley physics in mono- carrier spins by an electric field. JETP Lett. 50, 431–434 (1989). layers of MoS2 and other group-VI dichalcogenides. Phys. Rev. Lett. 108, 196802 4. V. M. Edelstein, Spin polarization of conduction electrons induced by electric current (2012). in two-dimensional asymmetric electron systems. Solid State Commun. 73, 233–235 23. H. R. Fuh et al., Newtype single-layer magnetic semiconductor in transition-metal (1990). dichalcogenides VX2 (X = S, Se and Te). Sci. Rep. 6, 32625 (2016). 5. P. Gambardella, I. M. Miron, Current-induced spin-orbit torques. Phil. Trans. R. Soc. A 24. W. Y. Tong, S. J. Gong, X. Wan, C. G. Duan, Concepts of ferrovalley material and 369, 3175–3197 (2011). anomalous valley hall effect. Nat. Commun. 7, 13612 (2016). 6. Q. Shao et al., Strong Rashba-Edelstein effect-induced spin–orbit torques in mono- 25. M. Offidani, M. Milletar`ı, R. Raimondi, A. Ferreira, Optimal charge-to-spin conver- layer transition metal dichalcogenide/ferromagnet bilayers. Nano Lett. 16, 7514–7520 sion in graphene on transition-metal dichalcogenides. Phys. Rev. Lett. 119, 196801 (2016). (2017). 7. D. MacNeill et al., Control of spin-orbit torques through crystal symmetry in 26. T. Cao et al., Valley-selective circular dichroism of monolayer molybdenum disulphide. WTe2/ferromagnet bilayers. Nat. Phys. 13, 300–305 (2017). Nat. Commun. 3, 887 (2012). 8. K. S. Lee et al., Angular dependence of spin-orbit spin-transfer torques. Phys. Rev. B 27. W. Zhao et al., Colloidal synthesis of VSe2 single-layer nanosheets as novel elec- 91, 144401 (2015). trocatalysts for the hydrogen evolution reaction. Chem. Commun. 52, 9228–9231 9. J. Lee, J. Fabian, Magnetotransport signatures of the proximity exchange and spin- (2016). orbit couplings in graphene. Phys. Rev. B 94, 195401 (2016). 28. M. Bonilla et al., Strong room-temperature ferromagnetism in VSe2 monolayers on 10. A. Kalitsov, S. A. Nikolaev, J. Velev, M. Chshiev, O. Mryasov, Intrinsic spin-orbit torque van der Waals substrates. Nat. Nano. 13, 289–293 (2018). in a single-domain nanomagnet. Phys. Rev. B 96, 214430 (2017). 29. H. Kurebayashi et al., An antidamping spin-orbit torque originating from the Berry 11. D. MacNeill et al., Thickness dependence of spin-orbit torques generated by WTe2. curvature. Nat. Nano. 9, 211–217 (2014). Phys. Rev. B 96, 054450 (2017). 30. J. Zeleznˇ y` et al., Relativistic Neel-order´ fields induced by electrical current in 12. M. H. Guimaraes, G. M. Stiehl, D. MacNeill, N. D. Reynolds, D. C. Ralph, Spin–orbit antiferromagnets. Phys. Rev. Lett. 113, 157201 (2014). torques in NbSe2/permalloy bilayers. Nano Lett. 18, 1311–1316 (2018). 31. J. Zeleznˇ y´ et al., Spin-orbit torques in locally and globally noncentrosymmetric 13. G. M. Stiehl et al., Current-induced torques with Dresselhaus symmetry due to crystals: Antiferromagnets and ferromagnets. Phys. Rev. B 95, 014403 (2017). resistance anisotropy in 2D materials. ACS Nano 13, 2599–2605 (2019). 32. B. Radisavljevic, A. Radenovic, J. Brivio, V. Giacometti, A. Kis, Single-layer MoS2 14. G. M. Stiehl et al., Layer-dependent spin-orbit torques generated by the cen- transistors. Nat. Nano. 6, 147–150 (2011). trosymmetric transition metal dichalcogenide β-MoTe2. Phys. Rev. B 100, 184402 33. Y. Zhang, T. Oka, R. Suzuki, J. Ye, Y. Iwasa, Electrically switchable chiral light-emitting (2019). transistor. Science 344, 725–728 (2014). 15. S. D. Ganichev, L. E. Golub, Interplay of Rashba/Dresselhaus spin splittings probed by 34. F. Li, K. Tu, Z. Chen, Versatile electronic properties of VSe2 bulk, few-layers, mono- photogalvanic spectroscopy–A review. Phys. Status Solidi B 251, 1801–1823 (2014). layer, nanoribbons, and nanotubes: A computational exploration. J. Phys. Chem. C 16. T. Cai et al., Single-spin Dirac fermion and Chern insulator based on simple oxides. 118, 21264–21274 (2014). Nano Lett. 15, 6434–6439 (2015). 35. D. Lembke, A. Kis, Breakdown of high-performance monolayer MoS2 transistors. ACS 17. J. L. Lado, J. Fernandez-Rossier,´ Magnetic edge anisotropy in graphenelike honey- Nano 6, 10070–10075 (2012). comb crystals. Phys. Rev. Lett. 113, 027203 (2014). 36. M. Si et al., Steep-slope hysteresis-free negative capacitance MoS2 transistors. Nature 18. J. Qi, X. Li, Q. Niu, J. Feng, Giant and tunable valley degeneracy splitting in MoTe2. Nano. 13, 24–28 (2018). Phys. Rev. B 92, 121403 (2015). 37. T. Brumme, M. Calandra, F. Mauri, First-principles theory of field-effect doping 19. X. Chen, L. Zhong, X. Li, J. Qi, Valley splitting in the transition-metal dichalcogenides in transition-metal dichalcogenides: Structural properties, electronic structure, Hall monolayer via atom adsorption. Nanoscale 9, 2188–2194 (2017). coefficient, and electrical conductivity. Phys. Rev. B 91, 155436 (2015).
6 of 7 | www.pnas.org/cgi/doi/10.1073/pnas.1912472117 Li et al. Downloaded by guest on September 25, 2021 Downloaded by guest on September 25, 2021 2 .Co .L,S .Lue ual ants n afmtliiyi hole-doped in half-metallicity and magnetism Tunable Louie, G. S. Li, Z. Cao, T. 42. dichalco- Ta of Magnetism NbSe Skomski, R. in Sellmyer, states J. D. magnetic Yu, of H. Sharma, switching V. Manchanda, induced P. strain 41. Tensile Guo, W. Liu, X. Xu, Y. 40. 3 .A envg .Vfk iz-anteeti fet npdpdsemiconductors. p-doped in effects Piezo-magnetoelectric Vafek, O. Bernevig, A. B. 43. for model Fang tight-binding Three-band S. Xiao, 39. D. Yao, W. Yao, Y. Shan, Y. W. Liu, B. G. 38. ie al. et Li ooae GaSe. monolayer hydrogenation. and (2015). strain by tuned monolayers genide NbS and genides. hs e.B Rev. Phys. dichalcogenides. metal (2013). transition group-VIB of monolayers 2 biii ih-idn aitna o rniinmtldichalco- metal transition for Hamiltonian tight-binding initio Ab al., et hs e.B Rev. Phys. igelayers. single 323(2005). 033203 72, hs e.Lett. Rev. Phys. 018(2015). 205108 92, Nanoscale 362(2015). 236602 114, 22–23 (2014). 12929–12933 6, pl hs Lett. Phys. Appl. hs e.B Rev. Phys. 032402 107, 085433 88, 2 8 .Mrai .Vnebl,MxmlylclzdgnrlzdWannier generalized localized Maximally Mostofi A. Vanderbilt, A. 49. D. made approximation Marzari, gradient Generalized Ernzerhof, N. M. 48. Burke, K. Perdew, augmented- projector P. the J. to 47. pseudopotentials ultrasoft From Joubert, D. Kresse, G. 46. Furthm J. Kresse, G. regime. 45. Boltzmann the beyond torque Rashba Niu, Q. Xiao, C. 44. ucin o opst nrybands. functions. energy composite (1997). for functions simple. set. method. wave basis plane-wave a (1996). using calculations (2017). hs e.Lett. Rev. Phys. opt hs Commun. Phys. Comput. ane9:Ato o bann aial-oaie Wannier maximally-localised obtaining for tool A wannier90: al., et hs e.B Rev. Phys. le,Efiin trtv cee for schemes iterative Efficient uller, ¨ 8536 (1996). 3865–3868 77, 7817 (1999). 1758–1775 59, 8–9 (2008). 685–699 178, hs e.B Rev. Phys. hs e.B Rev. Phys. NSLts Articles Latest PNAS binitio ab hs e.B Rev. Phys. 12847–12865 56, 11169–11186 54, total-energy 035423 96, | f7 of 7
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