Field-Tunable Topological Phase Transitions and Spin-Hall Effects in 2D Crystals
FField-Tunable Topological Phase Transitions and Spin-Hall Effects in 2D Crystals
Maxwell Fishman ∗ and Debdeep Jena Department of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14850 (Dated: December 18, 2020)As recent additions to the catalog of 2D crystals, silicene and other silicene-class crystals havenumerous unique properties currently being investigated and considered for use in novel device ap-plications. In this paper, we investigate electronic and transport properties of silicene in a field effecttransistor geometry. We find that the Berry curvature of silicene-class crystals can be continuouslytuned by a perpendicular electric field. By direct calculation of the Z invariant, we confirm that anelectronic phase transition from a topological insulator to a band insulator occurs when the electricfield passes a critical value. In a device setting with asymmetric gate voltages, this field-tunableBerry curvature generates a large spin current transverse to the charge current. When the electricfield strength surpasses the critical value, the bulk spin current is found to change direction andgreatly decrease in magnitude. This finding of a large magnitude, switchable spin current suggeststhat the silicene family of 2D crystals could be an attractive candidate for field-tunable charge-spin conversion. Such field-tunable phase transitions between topologically distinct phases could beuseful for robust qubits as well. In an effort to overcome the issues that arise as 3Ddevices approach near atomic scale lengths, a great dealof investigation is being done into the use of 2D mate-rials [1]. Graphene, the quintessential 2D material, hasmany unique properties that could make it useful in de-vices, but preventing current leakage is difficult due toits gapless band structure [2, 3]. Consequently, 2D crys-tals of the silicene family are receiving significant atten-tion. Silicene has the same honeycomb lattice structureas graphene, but this class of 2D crystals have largerband gaps due to larger atomic radii and buckled latticestructures. Until recently, silicene and related 2D crystalswere primarily considered theoretical since monolayers ofsilicon rapidly decompose in air. However, advances inmaterial growth processes show the potential to stabilizesuch 2D crystals and use them in devices [4, 5].Silicene can be considered a silicon analog of grapheneand has numerous unique properties [5–7]. The largersilicon atoms in silicene cause it to have a unique buckledstructure and stronger spin orbit coupling (SOC) thangraphene. DFT analysis has shown that the SOC opensup a band gap in silicene of 1 .
55 meV [8]. The SOCgap is small relative to compounds such as InSb (split-offband gap ∼ . ∼
50 K. The properties of silicene can also be generalizedto most honeycomb structured 2D materials with brokeninversion symmetry [5, 10–12]. Silicene has also beenresearched for use in ferromagnetic junction spin devices[13, 14] as well as bilayer silicene in a FET [15].In this work, we investigate how monolayer silicene isaffected by a perpendicular electric field and how theseaffects can be exploited in a independent double gate fieldeffect transistor (IDGFET) setting. In an IDGFET, sil-icene can be placed in band insulator (BI) and quantumspin hall (QSH) states through gate voltage control. Afield-tunable spin hall effect (SHE) causes a pure spin current transverse to the drain charge current. The mag-nitude and direction of this spin current can be switchedbetween two states, depending on whether the silicenechannel is in the BI or QSH state. This as well as themagnitude of the spin current are particularly interestingproperties of the silicene IDGFET and silicene-class 2Dmaterials may serve as a lightweight alternative materialfor spintronic devices.Due to the buckled structure of silicene, application ofan electric field perpendicular to the silicene plane createsa potential difference between the two sublattices. Thispotential difference exposes many of the interesting prop-erties of silicene and similar 2D crystals. A low energyapproximation of a four-band tight binding Hamiltonianis [8, 16] H η = (cid:126) v f ( k x τ x − ηk y τ y ) + ητ z h + q(cid:96)F z τ z , (1)where v f = √ at (cid:126) = 5 . × cm/s is the Fermi velocity, a = 3 .
86 ˚A is the lattice constant, τ i is the i th Paulisublattice matrix, η = ± K ± Dirac point, F z is the vertical electric field strength, (cid:96) = 0 .
23 ˚A ishalf the vertical distance between the sublattices, and h = − λ SO σ z − aλ R ( k y σ x − k x σ y ). σ i is the Pauli i thspin matrix and λ SO = 3 . λ R = 0 . E η,s z = ± (cid:118)(cid:117)(cid:117)(cid:116) (cid:126) v f k + (cid:32) q(cid:96)F z − ηs z (cid:113) λ SO + a λ R k (cid:33) , (2)where s z = ± k = k x + k y .The energy dispersion near the K + Dirac point is plot-ted in Fig. 1(a) and (c). At F z = 0 the bands arespin and valley degenerate and there is a band gap of a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec λ SO ∼ . | F z | is increased, spin degener-acy is broken due to the broken inversion symmetry. Nearthe K + Dirac point, as F z increases from zero the bandgap between the spin up states decreases while the gapbetween the spin down states increases. When F z reachesthe critical value F c = λ SO /q(cid:96) ≈
17 mV/˚A, spin up en-ergy dispersion becomes gapless. Increasing F z furtherre-opens the spin up band gap. When F z is decreasedbelow zero, the spin down band becomes gapless at − F c .Near the K − Dirac point, the modified spin states areopposite that of the K + Dirac point. Therefore, the twonotable regimes are | F z | < | F c | and | F z | > | F c | . Thetransition from one regime to the other occurs when theband gap closes and re-opens, signifying a non-adiabaticchange in the material’s Hamiltonian and a topologicalphase change in its eigenvalue spectrum [8, 17]. FIG. 1. Band structure and total density of states (DOS)near the K + Dirac point. Due to SOC, applying a verticalelectric field breaks spin degeneracy into spin up (blue) andspin down (red) bands. (a) Energy dispersion and (b) DOSwhen the 0 < F z < F c . (c) Energy dispersion and (d) DOSwhen F z = F c . Relative to effective SOC, k -space dependent RashbaSOC has a negligibly small effect on the properties of sil-icene being investigated in this paper. From this pointforward the Rashba spin orbit component will be ne-glected ( λ R = 0).The DOS is found by equating the number of states ina infinitesimal volume of k -space to that of energy space. Summing over all bands, the total low-energy DOS is g total ( E ) = | E | π (cid:126) v f (cid:18) Θ (cid:16) | E | − | eF z (cid:96) + λ SO | (cid:17) +Θ (cid:16) | E | − | eF z (cid:96) − λ SO | (cid:17)(cid:19) , (3)where Θ( ... ) is the Heaviside step function.The total DOS is shown in Fig. 1(b) and (d) for twodifferent electric fields. At F z = 0, the bands are spindegenerate, resulting in only one step and a linear DOS.When F z (cid:54) = 0 two steps occur in the DOS representingthe spin split upper and lower bands. At | F z | = | F c | ,the DOS of the lower bands become gapless. As | F z | increases beyond | F c | , the first step in the DOS re-opens,matching the behavior of the band structure.Non-trivial Berry curvature is expected in materialswith non-zero band gap and broken inversion or time re-versal symmetry, which can lead to anomalous hall cur-rents [18]. Rather than finding the Berry curvature di-rectly from the Berry potential, a more computationallyconvenient form is [18] Ω nµν ( R ) = i (cid:88) n (cid:54) = n (cid:48) (cid:104) n | ∂ R µ H| n (cid:48) (cid:105)(cid:104) n (cid:48) | ∂ R ν H| n (cid:105) ( ε n − ε n (cid:48) ) −(cid:104) n | ∂ R ν H| n (cid:48) (cid:105)(cid:104) n (cid:48) | ∂ R µ H| n (cid:105) ( ε n − ε n (cid:48) ) , (4)in which H is the Hamiltonian, R µ ( ν ) is the µ th ( ν th)component of R parameter space, which in this case is k -space, and ε n is the energy dispersion of the n th band.The only non-zero Berry curvature component will beΩ nxy . Therefore, the Berry curvature is locked in the z (out of plane) direction and is found to be Ω xy = ξη (cid:126) v f ( qF z (cid:96) − ηs z λ SO )2[ (cid:126) k v f + ( − ηs z qF z (cid:96) + λ SO ) ] / ˆ z, (5)where ξ = ± K + Dirac point can be seen in Fig. 2(a) and (b). At F z = 0,the Berry curvature for the spin up and spin down statesare equal in magnitude, but point in opposite directions.Maximum curvature occurs at the Dirac points. Depen-dent on the field direction, increasing the field strengthcauses the curvature of one spin state to increase whilethe other spin state decreases.As F z approaches F c , the spin up Berry curvature inFig. 2(a) diverges while the spin down curvature con-tinues to decrease. When F z becomes greater than F c (Fig. 2(b)), the direction of the spin up Berry cur-vature changes sign and decreases in magnitude as F z is increased beyond this point. F c is the critical fieldat which the spin up and spin down Berry curvaturestransition from pointing in opposite directions to point-ing in the same direction. The electron group veloc-ity in a material with a non-trivial Berry curvature is v n ( k ) = ∂ε n ( k ) (cid:126) ∂ k − q (cid:126) F × Ω n ( k ), where the additionallast term is the “anomalous velocity” [18]. In the pres-ence of a planar electric field, this field-dependent Berrycurvature will strongly modify the group velocity andconsequently the current density in a silicene-based de-vice. This Berry curvature relationship further indicatesa topological phase change, which is evaluated by calcu-lating the Z invariant. FIG. 2. Conduction band Berry curvature near the K + Dirac point when (a) 0 < F z < F c and (b) F z > F c . Sweeping F z through F c causes the spin up (blue) Berry curvature tochange direction, but the spin down (red) Berry curvaturedoes not change direction. It has been reported that silicene is a topological in-sulator (TI) intrinsically and can be transitioned to a BIby application of a vertical electric field [8, 19]. Since s z is conserved in this low energy regime, the Z invariantcan be evaluated using spin Chern numbers. Integratingthe Berry curvatures of each occupied band over the Bril-louin zone [20–22], the spin Chern numbers are found tobe n s z = (cid:88) η − ( ηqF z (cid:96) − s z λ SO )2 (cid:112) ( − ηs z qF z (cid:96) + λ SO ) , (6)where the sum is taken over the Dirac points.Using the spin Chern number difference, n σ = ( n ↑ − n ↓ ) /
2, the Z invariant is found from ν = n σ mod 2. TheChern invariant is always zero, indicating that silicene iseither be a BI or TI. When | F z | < | F c | , the Z invari-ant is that of a TI ( Z = 1). When | F z | > | F c | , the Z invariant changes to that of a BI ( Z = 0). As pre-dicted by the closing and re-opening of the band gap, sil-icene transitions from TI to BI when F z is swept through F c . This transition point greatly affects bulk transport,which dominates near room temperature.Using Eq. (2) and (5), the electron group velocitycomponents in the conduction band are found to be v i,η,s z = (cid:126) k i v f (cid:113) (cid:126) k v f + ( − ηs z qF z (cid:96) + λ SO ) ˆ i, (7) v yBerry,η,s z = − ηqF x (cid:126) v f ( qF z (cid:96) − ηs z λ SO )2( (cid:126) k v f + ( − ηs z qF z (cid:96) + λ SO ) ) / ˆ y, (8) where i = x or y and v yT otal = v y + v yBerry . It is as-sumed that an electric field in the negative x direction isbeing applied. The Berry curvature velocity component, v yBerry,η,s z , can be recast as v yBerry,η,s z = − µ B,η,s z F x ˆ y, (9)where µ B is µ B,η,s z = − ηq (cid:126) v f ( qF z (cid:96) − ηs z λ SO )2( (cid:126) k v f + ( − ηs z qF z (cid:96) + λ SO ) ) / . (10)In this form, µ B can be interpreted as a F z -dependenttransverse mobility, or “Berry mobility”.Due to this “Berry mobility”, when F x (cid:54) = 0 and F z (cid:54) = 0,there is always an extra non-zero y component in the ve-locity due to the Berry curvature. This creates asymme-try in the velocity profile and also shifts the zero-velocitypoint away from the Dirac points. The Berry curvaturevelocity term rapidly goes to zero as one moves away fromthe Dirac points. When | F z | < | F c | the y direction ve-locity of spin up and spin down states point in oppositedirections and when | F z | > | F c | the y direction veloci-ties transition to pointing in the same direction. Thispredicts a transitioning in the direction and magnitudeof a spin Hall effect when silicene changes phase fromTI to BI. It should be noted that relativistic effects onelectron-electric field dynamics are not included in thisanalysis.Using silicene’s band structure and velocity profile, aballistic FET model is built based on the method devel-oped by Natori [23]. This method is modified to supportthe use of silicene as the channel material and the use ofindependent double gates. A layout of the silicene FETis shown in Fig. 3(a).Assuming no short channel effects, application of adrain bias will not effect the total carrier density at thesource injection point. One can find the source injectionpoint Fermi level using the relationship between the gatevoltage and total carrier density and imposing carrierdensity conservation under a drain bias.By summing over the energies of the two barriers sep-arately, one finds qφ b,T G + qV b,T G − ∆ E CB + ( E f − E CB ) = qV gs,T G qφ b,BG + qV b,BG − ∆ E CB + ( E f − E CB ) = qV gs,BG , (11)where qφ b is the metal-insulator barrier height, V b is thevoltage drop across the insulator (as seen in Fig 3(b)),∆ E CB is the insulator-semiconductor conduction banddifference, E f is the semiconductor quasi-Fermi level, and E CB is the conduction band edge. ∆ E CB and E CB aremodified by the vertical electric field. Using Gauss’ law, V b is found to be equal to qn m t b /ε b ε . Grouping φ b and∆ E CB into a threshold voltage term, qV T = qφ b − ∆ E CB ,Eq. (11) are combined into q n s t b,T G ε b,T G ε + (1 + G )( E f − E CB ) = q ( V gs,T G + GV gs,BG ) , (12)where G = ε b,BG t b,TG t b,BG ε b,TG and the charge neutrality condi-tion, n s = n m,T G + n m,BG , has been used. n s is the totalcarrier density in the semiconductor, t b,T G and t b,BG arethe top and bottom gate oxide thickness, ε b,T G and ε b,BG are the top and bottom gate oxide relative permittivity,and V gs,T G and V gs,BG are the top and bottom gate volt-ages. It is assumed that the gate voltages are in referenceto the threshold voltages.By summing the occupation function over silicene’sconduction bands and applying carrier density conser-vation, a relationship can be found between n s and E f under any drain bias. This relationship is n η,s z = k b T πv f (cid:126) (cid:90) ∞ βE CB du u e u − η = k b T πv f (cid:126) (cid:90) ∞ βE CB du (cid:32) u e u − η → + u e u − η ← (cid:33) , (13)where E CB = | qF z (cid:96) − ηs z λ SO | after modification by thevertical electric field, β = 1 /k b T , and u = βE . The Fermilevel is given by η = βE f . η represents the Fermi levelwithout the application of a drain bias and η → and η ← = η → − qV ds represent the quasi-Fermi levels of the rightand left going carriers when a drain bias is applied. Thetotal carrier density n s = (cid:80) η (cid:80) s z n η,s z = 2( n + , ↑ + n + , ↓ )results from the symmetry of the bands. Solving Eq.(13) for E f ( n s ), one can find n s ( V gs,T G , V gs,BG ) using Eq.(12). In order to find F z , the electric field is assumed tolinearly change through the channel and the mid-point istaken as the vertical field to which the channel is exposed.Using η → ( V ds , V gs,T G , V gs,BG ), the total drain chargecurrent can be found using Eq. (7) and (8). The currentis found on a per-band basis and the total current isthe summation over all bands. Since the only differencebetween a right-going and left-going carrier is the Fermilevel, the total current per conduction band is found tobe J x,η,s z = J (cid:90) ∞ βE CB du (cid:112) u − β ( − ηs z qF z (cid:96) + λ SO ) (cid:32)
11 + e u − η → −
11 + e u − η ← (cid:33) , (14)where J = qk b T π (cid:126) v f , and the total current over all bandsis J T otalx = (cid:80) η (cid:80) s z J η,s z = 2( J x, + , ↑ + J x, + , ↓ ).Fig. 3(c) shows the relationship between current den-sity and symmetric gate voltages at V ds = 0 . J spin = J ↑ − J ↓ , is always zero. Further-more, it should also be noted that the model developedthus far is for conduction band transport. As experimen-tally shown [4], once the gate voltages are low enough va-lence band transport takes over and the current increases.If valence band transport is included, the ON/OFF ratiois reduced at room temperature. Using the symmetryof the conduction and valence bands, the valence bandand total current density is shown in Fig. 3(c). In or-der to increase ON/OFF ratio and reduce leakage, onecould operate at lower temperatures (Fig. 3(c) and (d)).Overall, the standard source-drain characteristics of thissilicene IDGFET are similar to that of a traditional FET. FIG. 3. Sample layout and drain current density of siliceneIDGFET. (a) Sample layout of silicene FET. (b) Band dia-gram of a IDGFET in the gate-channel-gate direction. (c)Drain current density at 300K, V ds = 0 . t b,TG = t b,BG =2 nm, and ε b,TG = ε b,BG = 10 when considering conduc-tion band transport (blue), valence band transport (red), andthe total current density (green, solid). Total transport at50K (green, dashed) is also shown. (d) I ON /I OFF ratio at V gs,TG = 0 . V (solid) and V gs,TG = 0 . V (dashed). All plotsuse symmetric gate voltages. Due to the non-trivial Berry curvature, silicene has acurrent component which is transverse to the longitudi-nal current. Since there is no voltage applied across thechannel in the y direction, the standard group velocityinduced current is zero. Therefore, the transverse currentis solely a result of silicene’s Berry curvature (Fig. 4(a)).Using the Berry curvature velocity expression, the bal-listic current density relation is found to be J Berry,η,s z = J Θ (cid:90) ∞ βE CB du u − e u − η , (15)where J Θ = − q ηF x ( qF z (cid:96) − ηs z λ SO )4 π (cid:126) k b T and F x = V ds /L ch isthe x direction electric field due to the drain bias. Thisassumes a linear voltage drop across the channel with L ch being the channel length. This approximation remainsvalid as long as the transistor is below saturation.As before, Eq. (15) applies to conduction band trans-port. Valence band transport can be found using bandand Berry curvature symmetries. Using the previouslyfound source injection Fermi level, Fig. 4(b) shows thetotal transverse spin current at various gate voltages.Due to symmetries of the Berry curvatures, the totalBerry charge current, J totaly = (cid:80) η (cid:80) s z J Berry,η,s z , is al-ways zero. The total transverse spin current, J spiny = (cid:80) η J Berry,η, ↑ − (cid:80) η J Berry,η, ↓ , is non-zero and displaysa unique switching effect. When | F z | < | F c | there is alarge total spin current. The total spin current switchesdirection and reduces magnitude when | F z | > | F c | . FIG. 4. Transverse transport diagram and spin current den-sity. (a) Due to the non-trivial berry curvature, the electricfield in the x direction causes carrier transport in the y direc-tion. This transport is spin polarized, creating a transversespin current. (b) Total spin current density when | F z | < | F c | (red) and | F z | > | F c | (blue). Inset: spin current ratio, | J s /J e | ,in the TI (red) and BI (blue) regimes. V gs,TG = V gs,BG forthe TI regime (red) and V gs,BG = V gs,TG − . V for the BIregime (blue). L ch = 10 nm and V ds = 0 . Since the spin current is relatively small when | F z | > | F c | , this can be considered as a means to switch the cur-rent between a HIGH and LOW state. When the trans-verse spin current is HIGH the spin current magnitudeis comparable to the drain current. An appropriate wayto examine this is the spin current ratio, θ SH = | J s /J e | ,where J e is the charge drain current density and J s isthe spin current density. Recent values that have beenexperimentally measured range from | θ SH | = 0 . . ∼ .
15 at L ch = 10 nm. The ratiocan be tuned higher if the channel length is reduced orif V ds is increased, but a ratio of 1 .
15 at 0 . V is quiteremarkable. However, one can see from the ∼ /E CB form of v yBerry that relativistic effects will be a limitingfactor in the maximum spin hall current density as theband gap is reduced. This large spin current magnitudeand the switching capability makes silicene an excitingcandidate for low power spintronic devices.In conclusion, when silicene is exposed to an electricfield perpendicular to the material plane, spin degeneracyis broken and the Berry curvature and electron group ve-locity are strongly modified. In an IDGFET setting, sil-icene’s Berry curvature induces a large pure spin currentwhich can be switched between HIGH and LOW states.This spin current is produced without the need for fer-romagetic junctions or external magnetic fields. This isparticularly surprising considering that heavy elementsare usually used to create such spin currents [24–26, 28].Sweeping | F z | through | F c | also causes silicene to transi-tion from TI to BI. This model could lead to a good plat-form for experimentally investigating relativistic effectson electron dynamics. There have also been theoreticalpredictions that quantum anomalous hall states occur inresponse to an exchange field [19]. Furthermore, thereis work on manipulating the electronic and thermal con-ductivity properties of silicene through mechanical strain[29, 30]. This, as well as looking at similar properties inother X-ene materials, is a sampling of the numerous ar-eas into which this work could be expanded. This workdemonstrates the potential for the use of silicene-classof 2D materials in novel nanoelectronic and spintronicsapplications.We thank Zexuan Zhang for the helpful discussions andproof reading. ∗ fi[email protected][1] D. Jena, Proceedings of the IEEE , 1585 (2013).[2] M. O. Goerbig, Reviews of Modern Physics (2011),10.1103/RevModPhys.83.1193, arXiv:1004.3396.[3] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S.Novoselov, and A. K. Geim, Reviews of Modern Physics , 109 (2009), arXiv:0709.1163.[4] L. Tao, E. Cinquanta, D. Chiappe, C. Grazianetti,M. Fanciulli, M. Dubey, A. Molle, and D. Akinwande,Nature nanotechnology , 1 (2015).[5] A. Molle, J. Goldberger, M. Houssa, Y. Xu, S.-C.Zhang, and D. Akinwande, Nature Materials (2017),10.1038/nmat4802.[6] A. Kara, H. Enriquez, A. P. Seitsonen, L. C. Lew YanVoon, S. Vizzini, B. Aufray, and H. Oughaddou, SurfaceScience Reports , 1 (2012). [7] Y. Yamada-Takamura and R. Friedlein, Science andTechnology of Advanced Materials , 064404 (2014).[8] M. Ezawa, New Journal of Physics (2012),10.1088/1367-2630/14/3/033003, arXiv:1201.3687.[9] R. Winkler, Spin Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems (2003) p. 228.[10] D. Akinwande, C. Huyghebaert, C. H. Wang, M. I. Serna,S. Goossens, L. J. Li, H. S. Wong, and F. H. Koppens,Nature , 507 (2019).[11] M. Ezawa, Journal of the Physical Society of Japan ,1 (2015), arXiv:1503.08914.[12] S. Chowdhury and D. Jana, Reports on Progress inPhysics (2016), 10.1088/0034-4885/79/12/126501.[13] M. A. Kharadi, G. F. A. Malik, F. A. KHANDAY,K. Shah, S. Mittal, and B. K. Kaushik, ECS Jour-nal of Solid State Science and Technology (2020),10.1149/2162-8777/abd09a.[14] V. Vargiamidis and P. Vasilopoulos, Applied Physics Let-ters , 1 (2014).[15] X. F. Ouyang, Z. Y. Song, and Y. Z. Zhang, PhysicalReview B , 1 (2018).[16] C. C. Liu, H. Jiang, and Y. Yao, Physical Review B -Condensed Matter and Materials Physics , 1 (2011),arXiv:1108.2933.[17] C. L. Kane and E. J. Mele, Physical Review Letters ,1 (2005), arXiv:0411737 [cond-mat].[18] D. Xiao, M. C. Chang, and Q. Niu, Reviews of ModernPhysics , 1959 (2010).[19] M. Ezawa, Physical Review Letters , 1 (2012),arXiv:1203.0705.[20] M. Z. Hasan and C. L. Kane, Reviews of Modern Physics , 3045 (2010), arXiv:1002.3895.[21] X. L. Qi and S. C. Zhang, Reviews of Modern Physics (2011), 10.1103/RevModPhys.83.1057, arXiv:1008.2026.[22] D. N. Sheng, Z. Y. Weng, L. Sheng, and F. D. M.Haldane, Physical Review Letters , 1 (2006),arXiv:0603054 [cond-mat].[23] K. Natori, Journal of Applied Physics , 4879 (1994).[24] Z. Xu, G. D. Hwee Wong, J. Tang, E. Liu, W. Gan, F. Xu,and W. S. Lew, ACS Applied Materials and Interfaces ,32898 (2020).[25] L. Zhu, L. Zhu, S. Shi, D. C. Ralph, and R. A. Buhrman,Advanced Electronic Materials , 1 (2020).[26] L. Zhu, L. Zhu, S. Shi, M. Sui, D. C. Ralph, andR. A. Buhrman, Physical Review Applied , 1 (2019),arXiv:1904.07800.[27] M. Dc, R. Grassi, J. Y. Chen, M. Jamali, D. ReifsnyderHickey, D. Zhang, Z. Zhao, H. Li, P. Quarterman, Y. Lv,M. Li, A. Manchon, K. A. Mkhoyan, T. Low, and J. P.Wang, Nature Materials , 800 (2018).[28] L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C.Ralph, and R. A. Buhrman, Science , 555 (2012),arXiv:1203.2875.[29] H. Xie, T. Ouyang, ´E. Germaneau, G. Qin, M. Hu, andH. Bao, Physical Review B , 4 (2016).[30] J. A. Yan, S. P. Gao, R. Stein, and G. Coard, PhysicalReview B - Condensed Matter and Materials Physics91