Berry Curvature Engineering by Gating Two-Dimensional Antiferromagnets
Shiqiao Du, Peizhe Tang, Jiaheng Li, Zuzhang Lin, Yong Xu, Wenhui Duan, Angel Rubio
BBerry Curvature Engineering by Gating Two-Dimensional Antiferromagnets
Shiqiao Du, Peizhe Tang, ∗ Jiaheng Li, Zuzhang Lin, Yong Xu,
1, 4, 5, † Wenhui Duan,
1, 4, 3, ‡ and Angel Rubio
6, 7, 8, § State Key Laboratory of Low-Dimensional Quantum Physics,Department of Physics, Tsinghua University, Beijing 100084, China, Max Planck Institute for the Structure and Dynamics of Matter,Center for Free-Electron Laser Science, Luruper Chaussee 149, 22761 Hamburg, Germany. Institute for Advanced Study, Tsinghua University, Beijing 100084, China, Collaborative Innovation Center of Quantum Matter, Beijing 100084, China, RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan, Max Planck Institute for the Structure and Dynamics of Matter,Center for Free-Electron Laser Science, Luruper Chaussee 149, 22761 Hamburg, Germany, Nano-Bio Spectroscopy Group and ETSF, Dpto. Fisica de Materiales,Universidad del Pa´ıs Vasco UPV/EHU, 20018 San Sebasti´an, Spain Center for Computational Quantum Physics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA. (Dated: September 4, 2019)Recent advances in tuning electronic, magnetic, and topological properties of two-dimensional(2D) magnets have opened a new frontier in the study of quantum physics and promised excitingpossibilities for future quantum technologies. In this study, we find that the dual gate technologycan well tune the electronic and topological properties of antiferromagnetic (AFM) even septuple-layer (SL) MnBi Te thin films. Under an out-of-plane electric field that breaks PT symmetry,the Berry curvature of the thin film could be engineered efficiently, resulting in a huge change ofanomalous Hall (AH) signal. Beyond the critical electric field, the double-SL MnBi Te thin filmbecomes a Chern insulator with a high Chern number of 3. We further demonstrate that such 2Dmaterial can be used as an AFM switch via electric-field control of the AH signal. These discoveriesinspire the design of low-power memory prototype for future AFM spintronic applications. AFM spintronics, aiming to use antiferromagnets tocomplement or replace ferromagnets as active compo-nents of spintronic devices, opens a new era in the field ofspintronics owing to its numerous advantages, includingthe robustness against perturbation of external magneticfield, the absence of stray field, and the ultrafast dynam-ics [1–4]. The key issue to design AFM spintronic devicesis to find an efficient approach to manipulate and detectthe magnetic or electronic quantum states of an antifer-romagent. By using spin-transfer torque and spin-orbittorque [1], external electric currents are able to write in-formation by controlling the AFM order inside spintronicdevices [5–9] or by generating spin currents at interfacesbetween antiferromagnets and non-magnets [10]. For ex-ample, via applying local current in the tetragonal CuM-nAs thin film, AFM spin orientations can be manipulatedfor information storage, which can be readout via detect-ing the anisotropic magnetoresistance [11, 12]. Basedon such proposal, room-temperature AFM memory cellshave been fabricated experimentally [13].The electric detection of anomalous Hall effect (AHE)provides an alternative powerful scenario to design spin-tronic devices based on antiferromagnets. The physicalorigin of the AHE has been under debate for decades[14]. While, in recent years, people realize that Berrycurvature Ω ( k ) in the momentum space plays an impor-tant role in generating the intrinsic AHE. Based on thesymmetry argument, the Berry curvature could be non-zero in a system without the combination of time rever-sal symmetry ( T ) and inversion symmetry ( P ), namely PT symmetry. Such a constrain indicate the antifer-romagnets with non-collinear magnetism and antifer-romagnet heterostructures as potential material candi-dates, including Mn Ge [15–17], Mn Sn [18], Mn Ir[19],Mn Pt/BaTiO [20] and CrSb/Cr-doped (Bi,Sb) Te su-perlattices [21]. However, the external control of theirAHE is challenging in practice. For AFM structureswith collinear magnetism, the possible existence of PT symmetry guarantees the vanishing of Berry curvature,although non-trivial topological fermions may survivein some three-dimensional (3D) antiferromagnets [22].Therefore, to find an effective way to manipulate andengineer Berry curvature is essential to use AHE as thedetection signal in AFM spintronic devices.The recently discovered 3D AFM topological insula-tor (TI) MnBi Te provides us a new chance to designtopological quantum devices controlled by external fields.Similar to the prototype of 3D TI ( e.g. , Bi Se family),MnBi Te crystal is a layered material [23–26]. There ex-ist strong chemical bonding within each SL (consisting ofTe-Bi-Te-Mn-Te-Bi-Te) and weak van der Waals bondingbetween SLs. Each Mn atom has a magnetic moment of ∼ µ B ( µ B is the Bohr magneton) and its spin orienta-tion is along the out-of-plane z direction [26, 27]. In theground state, a long-range ferromagnetic order is formedin each SL, and adjacent SLs couple with each other an-tiferromagnetically, displaying an N´eel temperature T N ∼
25 K [28, 29]. In contrast to AFM transition-metaloxides [30], ultrathin films of layered MnBi Te can beeasily cleaved by mechanical exfoliation [31–33] or grown a r X i v : . [ c ond - m a t . m t r l - s c i ] S e p via molecular beam epitaxy [23]. The quantized Halleffect without Landau levels has been theoretically pre-dicted and experimentally observed in these thin films[26, 27, 31–33].Herein, we focus on 2D AFM MnBi Te thin film withdouble-SL whose intrinsic electronic property is topolog-ically trivial. By using ab initio calculations, we demon-strate that the out-of-plane electric field induced by dual-gate technology breaks PT symmetry in AFM double-SL, engineers its band structures significantly, and gen-erates huge AH signals. Beyond the critical value, theelectric field induces a topological phase transition, driv-ing the double-SL MnBi Te to be a quantized anoma-lous Hall (QAH) insulator with a high Chern numberof 3. Based on these useful properties, we propose aprototype device as the electric field controlled topologi-cal AFM memory, whose performance is expected to bemuch higher than the current AFM random-access mem-ory. Thanks to AFM ground state, its electric switchshould be stable and robust under external magnetic fieldand the switching time is very fast.The first-principles DFT calculations are performedby using the projector-augmented wave method imple-mented in Vienna ab initio Simulation Package (VASP)with the GGA-PBE exchange-correlation functional [34].The energy cut-off for the plane wave basis is set to be350 eV and the energy convergence criterion is 10 − eVfor self-consistent electronic-structure calculations. A 20˚A vacuum layer is chosen to eliminate the periodic effectalong the z-axis direction. The van der Waals interactionis included through the DFT-D3 method [35], which hasbeen tested to have the best performance on describingthe lattice parameters compared with experiments. Thecrystal structures are relaxed until the force on each atomis less than 0.01 eV/˚A. A 24 × × k -pointmesh is sampled uniformly over the BZ. The GGA+ U method with U =4 eV is applied to describe the localized3 d orbitals of Mn atom in the electronic structure calcula-tions. The electric field is applied along the non-periodicdirection with the dipole corrections. The HSE06 hybridfunctional is also applied to check the band gap value [36].The maximally-localized Wannier functions are obtainedby the Wannier90 packages [37]. The DFT calculatedBloch functions are projected onto Mn d , Te p and Bi s and p orbitals to construct the corresponding Wannierfunctions. The edge-state and AH conductance is calcu-lated based on the tight binding model from maximallylocalized Wannier functions [38]. To include the tem-perature effect, the Fermi-Dirac distribution function isapplied in the Kubo-Greenwood formula for the calcula-tion of AH conductance.For the bilayer AFM insulating thin film with the PT symmetry, such as double-SL MnBi Te , every Blochstate at any k point is doubly degenerated. Figure 1demonstrates the schematics of AFM thin-film structuresin real space and the corresponding electronic bands in FIG. 1.
Schematics of PT symmetry in AFM doublelayer. ( A ) Schematic of AFM double layer with PT symme-try. The purple balls in each layer represent magnetic atomswhose magnetic orientations (out-of-plane) are depicted byarrows (e.g. Mn atoms in double-SLs MnBi Te ). With PT symmetry, the Berry curvature Ω ( k ) at each k point is zero.( B ) Schematic of AFM double layer with an out-of-plane elec-tric field in which PT symmetry is broken. Correspondingly, Ω ( k ) becomes nonzero. ( C - E ) Schematic band structures ofAFM double-SL MnBi Te thin film without and with electricfield. In C , Each band is doublely degenerate and the inter-band coupling is forbidden due to PT symmetry. In D and E , Under the electric field, PT symmetry is broken and theinter-band coupling will induce a Zeeman-like splitting in thevalence band and a Rashba-like splitting in the conductionband. Because the valence band maximum (VBM) and theconduction band minimum (CBM) are mainly contributed byout-of-plane and in-plane orbitals. momentum space including the influence of out-of-planeelectric field. In the intrinsic film (see Figs. 1A and1C), the Berry curvature of the n th Bloch band satisfies Ω n ( k ) = − Ω n ( − k ) with T symmetry, and P symme-try enforces Ω n ( k ) = Ω n ( − k ). Thus, Ω n ( k ) is zero inthe presence of PT symmetry. Due to the 2D geometricproperty, the double-SL MnBi Te can be applied withan out-of-plane electric field easily by using the dual-gatetechnology. As shown in Fig. 1B, the electrostatic po-tential is different in each SL, which breaks the PT sym-metry. Therefore, the double degeneracy of each bandgets broken and Ω n ( k ) becomes non-zero, which enablesthe possible observation of AHE via changing the carrierdensity in this system. In Figs. 1D and 1E, we showthe schematics of the change of electronic structure un-der electric field. On the edge of the top valance band,the degenerated states are mainly contributed by out-of-plane orbitals with opposite spins localized at differentSLs. The spacial variance of electrostatic potential cansplit the band edge, giving rise to a Zeeman-like bandsplitting. On the other hand, in-plane orbitals are ma-jor components on the edge of the bottom conductionband, and the applied electric field results in a Rashba-like spin splitting. When increasing the magnitude ofelectric field, the band gap decreases gradually. At thecritical value, the band gap closes, which possibly leadsto a topological phase transition.By using density functional theory (DFT) (see calcu-lation details in Methods), we calculate electronic struc-tures of the double-SL MnBi Te thin film with and with-out electric field to verify the above physical picture. Fig-ure 2A displays the lattice structure and the first Bril-louin zone (BZ) of double-SL MnBi Te . The calculatedband structures and Berry curvature Ω ( k ) for occupiedbands are shown in Figs. 2B-2K. In the absence of exter-nal field, the AFM ground state has neither P symmetrynor T symmetry, but has the PT symmetry and C z ro-tational symmetry along the z direction. The intrinsicdouble-SL MnBi Te is a trivial semiconductor with adirect band gap at the Γ point (the calculated gap sizeis 76 meV) and Ω ( k ) is zero. We find that each SL inMnBi Te thin film is antiferromagnetically coupled withthe other in the ground state, consistent with previouscalculations [26, 27].When applying an out-of-plane electric field to the thinfilm, its magnetism keeps the AFM order as the groundstate in a finite field range (see Fig. S2 in SupplementaryMaterials (SM)). But the electronic structure varies con-siderably. Even under a small electric field (see Fig. 2C),a sizable Zeeman-like splitting is observed at the valancebands, whose magnitude at the Γ point is comparableto the electrostatic potential difference between adjacentSLs. And these states possess opposite spin textures.On the edge of conduction bands, the Rashba-like split-ting can be observed but the splitting is much smallercompared with that of the valance bands. To under-stand such phenomenon, we calculate the charge densitydistribution in real space for split bands on the conduc-tion and valance bands (see Fig. S3 in SM). We foundthat the valance band edge is mainly contributed by the p z orbital of Te atoms localized on different SLs. Theyare partner states under PT symmetry when the electricfield is zero, whose spin-polarizations are locked with theSL index. The out-of-plane external field can shift thesespin-polarized bands easily, and the energy splitting hasthe same order of magnitude as the electrostatic potentialdifference between different SLs. Correspondingly, thesebands carry non-zero Berry curvatures. Similar argumenthas been used to understand the tuning of Berry curva-ture and valley magnetic moments in bilayer MoS [39].The band edges of conduction bands are contributed by p orbitals of Bi atoms and Te atoms in the same SL. Theout-of-plane electric field will enhance the Rashba-likesplitting, which is consistent with our DFT calculations.When increasing the electric field, band splittings ofvalence and conduction bands become larger and larger.We then find topological phase transitions in the double-SL MnBi Te thin film. The first critical value E c is 0.021 V/˚A. Figure 2D shows its band structure alonghigh symmetric lines and the 2D electronic structure fortwo low-energy bands around the Γ point is displayedin Fig. 2H. The band crossing points are along lines ofΓ-K’. Because the applied electric field is along the z di-rection, it does not break C z symmetry, then three gap-less points are observed in the first BZ. Figure 2J showsthe momentum resolved distribution of the Berry curva-ture Ω ( k ). Different from conventional topological phasetransition that just hosts singularities of Berry curva-ture at the crossing points [40], we find a triangle regionaround the Γ point with small gap size, in which all stateshave large Ω ( k ). Beyond E c , double-SL MnBi Te be-comes a AFM QAH system with a high Chern number of3, as shown in Fig. 2I. To the best of our knowledge, thisresult is the first proposal to realize the QAH effect withhigh Chern numbers based on realistic AFM materials.If we further increase the electric field beyond the secondcritical point ( E c = 0.027 V/˚A), the gap closing processoccurs along lines of Γ-K. Then the double-SL MnBi Te becomes a trivial AFM metal. Due to the breaking of PT symmetry, this metallic AFM thin film has the non-zero Berry curvature distribution (Fig. 2K), contributingto an intrinsic AH signal. But its value is not quantizedanymore. When we flip the direction of electric field andkeep the magnetic structures, the AH signals change signcorrespondingly.In order to confirm the predicted non-trivial bandtopology of the double-SL MnBi Te thin film under cer-tain electric fields, we build the 2D tight-binding modelwith semi-infinite boundary conditions to calculate theedge states, whose effective hopping terms are obtainedfrom ab initio calculations. In Figs. 3A and 3C, weplot the band structure of edge states and AH conduc-tance as a function of chemical potential in double-SLMnBi Te thin film under the electric field of 0.023 V/˚A.Inside the 2D bulk gap, we can observe three chiral edgestates clearly. They connect the conduction and valancebands, contributing to the quantized AH conductance of σ xy = 3 e /h (see Fig. 3C). Beside the topologically non-trivial edge states, some other edge states are also foundin Fig. 3A along the line of Γ-K. These edge states aretopologically trivial without the intrinsic contribution tothe AH conductance.Figure 3B demonstrates the AH signal as a functionof chemical potential and external perpendicular electricfield that can be well controlled via dual-gate technology.In the trivial semiconducting phase (marked as Phase Iin Fig. 3B), we observe the negative AH effect when tun-ing the Fermi level to the conduction band with minorelectron doping. Its band structure, the Fermi surface,and related spin textures are shown in Fig. S5. Wefind the Rashba-like splitting on the conduction band,but different from the traditional Rashba splitting of thenon-magnetic semiconductor interface, a finite band gapexists at the Γ point without the Kramers’ degeneracy. FIG. 2.
Electronic band structures and Berry curvature of the double-SL MnBi Te thin film under electricfields. ( A ) The lattice structure of double-SL MnBi Te thin film and the first BZ. The magnetic orientations on Mn atomsare marked by red arrows. ( B-G ) Band structures of double-SL MnBi Te thin film under different electric fields, whosemagnitudes are 0, 0 . . . . .
029 V/˚A, respectively. The Fermi levels marked by the black dash lines are setto be zero. It notes that in ( E ) the black dash line is overlapped with the red one. ( H ) 2D electronic structures around theFermi level and the Γ point when AFM thin film under the critical electric field. The band crossing points are marked as redstars. ( I ) A schematic drawing depicting the QAH edge states with a Chern number of 3 in the double-SL MnBi Te thin filmunder electric field. The magnetic Mn atoms is indicated by the purple balls with arrows and the the electric field direction isindicated by black arrow. ( J - K ) Distributions of the Berry curvature in the first BZ for double-SLs MnBi Te thin film under E = 0 .
023 V/˚A and E = 0 .
029 V/˚A. Corresponding energy levels are marked as the red dash lines in ( E ) and ( G ). Such anti-crossing point originates from the breaking of T symmetry and contributes a large value to the Berrycurvature, whose sign is determined by the local spin tex-ture. Away from the anti-crossing point, the out-of-planecomponent in spin polarization will change the sign, re-sulting in the positive Berry curvature whose value issmaller compared with that at the anti-crossing point(see Fig. S5E in SM). Therefore, the total AH signalis negative. With increasing the electric field, the anti-crossing gap becomes larger and the value of correspond-ing negative Berry curvature becomes smaller, thus theAH signal decreases and even changes the sign. Fur-ther increasing the electric field, we obtain the quantizedAH signal inside the band gap that is guaranteed by thenon-trivial band topology of double-SL MnBi Te thin film (see Phase II in Fig. 3B). The thin film becomesa trivial AFM metal in Phase III when the electric fieldis beyond E c . Its AH signal has finite value when thechemical potential is zero but increases rapidly when weshift down the Fermi level to lower energies. The corre-sponding band structures and Fermi surfaces are shownin Fig. S6. These states are spin-polarized and host largeBerry curvature.The drastic change of AH signal caused by varyingchemical potential and electric field offers an opportu-nity to design AFM spintronic devices via dual-gate tech-nology. Figures 4A and 4B show the proposed device-prototype of the AFM memory. We use the standarddual-gate to simultaneously control the perpendicularelectric field and Fermi level to encode the information as FIG. 3.
Nontrivial topological properties of thedouble-SL MnBi Te thin film under electric fields. ( A ) Edge states in double-SL MnBi Te thin film under theelectric field of E = 0 .
023 V/˚A. ( B ) The AH signal as a func-tion of chemical potential and external perpendicular electricfields. The chemical potentials are aligned to Fermi levels.The phase regions I, II and III represent trivial semiconduct-ing phase, QAH phase, and trivial AFM metal phase, re-spectively. The black dashed lines mark the energy positionsof conduction and valence band edges. ( C ) AH conductancewith varying chemical potential in double-SLs MnBi Te thinfilm under the electric field of E = 0 .
023 V/˚A. The chemicalpotential scale is marked as the red solid line in ( B ). the “write-in” process and use the Hall bar to detect theAH conductance as the “read-out” process. In principle,the AH conductance in “Off” state is zero and its value in“On” state could be as large as 3 e /h , so its ideal on/offratio is infinite. In the realistic experimental conditionswith dissipation and at finite temperatures, the double-SL MnBi Te thin film could be supported on substrateswith large dielectric constant, such as BN, silicon, andSrTiO . The contacting interface might induce an effec-tive electric field that slightly breaks the PT symmetry.The AH conductance in “Off” state thus gains a finitevalue. In order to simulate such effect, we put double-SLMnBi Te thin film on BN substrate and estimate theeffective electric field strength (see Fig. S7 in SM). Thenwe evaluate its performance at different working temper-atures. The calculated results are shown in Figs. 4C and4D. With increasing the temperature, more states arethermally excited and the on/off ratio becomes smaller,but its value (10 -10 ) still much larger than the cur-rently realized AFM random-access memory [5–7, 10–13]. In our current calculations, we mainly consider theintrinsic Berry-phase-related AHE. Some other extrinsicmechanisms (e.g. skew scattering) may also contributeto AH signals [14], but their contributions should be in-significant. Because, in contrast to the magnetic alloys,MnBi Te thin films are cleaved from high-quality singlecrystals with lower density of impurities and disorders[31–33]. The intrinsic mechanism should dominate thecontribution of AHE.Beyond the double-SL, the discussed physics and pro- FIG. 4.
Memory devices based on AFM double-SLMnBi Te thin film. ( A ) Top view of schematics for aHall bar device. The blue arrows indicate the edge currentdirection. The red areas show the double-SL MnBi Te thinfilm. The gray and yellow areas represent the substrate andattached electrodes, respectively. ( B ) The schematic oper-ation for the AFM memory device. “On” and “Off” statesare for double-SL MnBi Te thin film in the trivial semicon-ductor phase and QAH phase. ( C - D ) AH conductance withvarying temperature in double-SL MnBi Te thin film underthe electric field of E = 0 .
010 V/˚A and E = 0 .
023 V/˚A, thetemperature is normalized to N´eel temperature T N =25K. posed spintronic device-prototype in this work could begeneralized to other MnBi Te thin films with even-SLthickness as long as PT symmetry is not intrinsicallybroken. As the confirmation, we apply the external elec-tric field to a four-SL MnBi Te thin film that was re-garded as an axion insulator [25]. The calculated elec-tronic structures are shown in Fig. S8. We can stillobserve the spin-polarized band splitting and topologi-cal phase transitions driven by the electric field. While,its critical values are smaller than those in double-SLMnBi Te and the topologically non-trivial phase regionbecomes narrower. To conclude, we believe that the elec-tric field is an efficient method to manipulate the Berrycurvature effects in even-SL MnBi Te thin films, result-ing in a large change of AH signal. Thus the even-SLMnBi Te thin film is a promising material platform tobuild the low-power AFM memory bit based on the AHsignal with electric write-in and read-out. We expect itsperformance to be stable and robust under the externalmagnetic field. This work paves the way for using even-SL MnBi Te thin films, and perhaps AFM topologicalthin films more generally, in a new generation of electri-cally switchable AFM spintronic devices.S.D., J.L., Z.L., Y.X., and W.D. acknowledge finan-cial supports from the Basic Science Center Project ofNSFC (Grant No. 51788104), the Ministry of Scienceand Technology of China (Grants No. 2016YFA0301001,No. 2018YFA0307100, and No. 2018YFA0305603), theNational Natural Science Foundation of China (GrantsNo. 11674188 and No. 11874035), and the Beijing Ad-vanced Innovation Center for Future Chip (ICFC). A.R.and P.T. acknowledge financial supports from the Euro-pean Research Council (ERC-2015-AdG-694097). P.T.acknowledges the received funding from the EuropeanUnion Horizon 2020 research and innovation programmeunder the Marie Sklodowska-Curie grant agreement No793609. The Flatiron Institute is a division of the SimonsFoundation. ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] § E-mail: [email protected][1] V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono,and Y. Tserkovnyak, Rev. Mod. Phys. , 015005 (2018).[2] O. Gomonay, T. Jungwirth, and J. Sinova, Phys. StatusSolidi RRL , 1700022 (2017).[3] T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich,Nat. Nanotechnol. , 231 (2016).[4] A. H. MacDonald and M. Tsoi, Phil. Trans. R. Soc. A , 3098 (2011).[5] J. ˇZelezn´y, H. Gao, K. V´yborn´y, J. Zemen, J. Maˇsek,A. Manchon, J. Wunderlich, J. Sinova, and T. Jung-wirth, Phys. Rev. Lett. , 157201 (2014).[6] X. F. Zhou, J. Zhang, F. Li, X. Z. Chen, G. Y. Shi, Y. Z.Tan, Y. D. Gu, M. S. Saleem, H. Q. Wu, F. Pan, andC. Song, Phys. Rev. Appl. , 054028 (2018).[7] X. Marti, I. Fina, C. Frontera, J. Liu, P. Wadley, Q. He,R. J. Paull, J. D. Clarkson, J. Kudrnovsk´y, I. Turek,J. Kuneˇs, D. Yi, J.-H. Chu, C. T. Nelson, L. You,E. Arenholz, S. Salahuddin, J. Fontcuberta, T. Jung-wirth, and R. Ramesh, Nat. Mater. , 367 (2014).[8] X. Z. Chen, R. Zarzuela, J. Zhang, C. Song, X. F. Zhou,G. Y. Shi, F. Li, H. A. Zhou, W. J. Jiang, F. Pan, andY. Tserkovnyak, Phys. Rev. Lett. , 207204 (2018).[9] M. Wang, W. Cai, D. Zhu, Z. Wang, J. Kan, Z. Zhao,K. Cao, Z. Wang, Y. Zhang, T. Zhang, C. Park, J.-P.Wang, A. Fert, and W. Zhao, Nat. Electron. , 582(2018).[10] T. Moriyama, W. Zhou, T. Seki, K. Takanashi, andO. Teruo, Phys. Rev. Lett. , 167202 (2018).[11] P. Wadley, B. Howells, J. ˇZelezn´y, C. Andrews, V. Hills,R. P. Campion, V. Nov´ak, K. Olejn´ık, F. Maccherozzi,S. S. Dhesi, S. Y. Martin, T. Wagner, J. Wunderlich,F. Freimuth, Y. Mokrousov, J. Kunˇes, J. S. Chauhan,M. J. Grzybowski, A. W. Rushforth, K. W. Edmonds,B. L. Gallagher, and T. Jungwirth, Science , 587(2016).[12] S. Y. Bodnar, L. ˇSmejkal, I. Turek, T. Jungwirth,O. Gomonay, J. Sinova, A. A. Sapozhnik, H.-J. Elmers,M. Kl¨aui, and M. Jourdan, Nat. Comm. , 348 (2018).[13] K. Olejn´ık, V. Schuler, X. Marti, V. Nov´ak, Z. Kaˇspar,P. Wadley, R. P. Campion, K. W. Edmonds, B. L. Gal-lagher, J. Garc´es, M. Baumgartner, P. Gambardella, andT. Jungwirth, Nat. Comm. , 15434 (2017).[14] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, andN. P. Ong, Rev. Mod. Phys. , 1539 (2010).[15] N. Kiyohara, T. Tomita, and S. Nakatsuji, Phys. Rev. Appl. , 064009 (2016).[16] J. Liu and L. Balents, Phys. Rev. Lett. , 087202(2017).[17] A. K. Nayak, J. E. Fischer, Y. Sun, B. Yan, J. Karel,A. C. Komarek, C. Shekhar, N. Kumar, W. Schnelle,J. K¨ubler, C. Felser, and S. S. P. Parkin, Sci. Adv. ,e1501870 (2016).[18] S. Nakatsuji, N. Kiyohara, and T. Higo, Nature ,212 (2015).[19] H. Chen, Q. Niu, and A. H. MacDonald, Phys. Rev.Lett. , 017205 (2014).[20] Z. Liu, H. Chen, J. Wang, J. Liu, K. Wang, Z. Feng,H. Yan, X. Wang, C. Jiang, J. Coey, and A. H. Mac-Donald, Nat. Electro. , 172 (2018).[21] Q. L. He, X. Kou, A. J. Grutter, G. Yin, L. Pan, X. Che,Y. Liu, T. Nie, B. Zhang, S. M. Disseler, B. J. Kirby,W. Ratcliff II, Q. Shao, K. Murata, X. Zhu, G. Yu,Y. Fan, M. Montazeri, X. Han, J. A. Borchers, and K. L.Wang, Nat. Mater. , 94 (2017).[22] P. Tang, Q. Zhou, G. Xu, and S.-C. Zhang, Nat. Phys. , 1100 (2016).[23] Y. Gong, J. Guo, J. Li, K. Zhu, M. Liao, X. Liu,Q. Zhang, L. Gu, L. Tang, X. Feng, D. Zhang, W. Li,C. Song, L. Wang, P. Yu, X. Chen, Y. Wang, H. Yao,W. Duan, Y. Xu, S.-C. Zhang, X. Ma, Q.-K. Xue, andK. He, Chin. Phys. Lett. , 076801 (2019).[24] M. M. Otrokov, I. I. Klimovskikh, H. Bentmann,A. Zeugner, Z. S. Aliev, S. Gass, A. U. Wolter, A. V.Koroleva, D. Estyunin, A. M. Shikin, M. Blanco-Rey,M. Hoffmann, A. Y. Vyazovskaya, S. V. Eremeev, Y. M.Koroteev, I. R. Amiraslanov, M. B. Babanly, N. T.Mamedov, N. A. Abdullayev, V. N. Zverev, B. B¨uchner,E. F. Schwier, S. Kumar, A. Kimura, L. Petaccia, G. D.Santo, R. C. Vidal, S. Schatz, K. Kißner, C. H. Min,S. K. Moser, T. R. F. Peixoto, F. Reinert, A. Ernst,P. M. Echenique, A. Isaeva, and E. V. Chulkov,arXiv:1809.07389 (2018).[25] D. Zhang, M. Shi, T. Zhu, D. Xing, H. Zhang, andJ. Wang, Phys. Rev. Lett. , 206401 (2019).[26] J. Li, Y. Li, S. Du, Z. Wang, B.-L. Gu, S.-C. Zhang,K. He, W. Duan, and Y. Xu, Sci. Adv. , eaaw5685(2019).[27] M. M. Otrokov, I. P. Rusinov, M. Blanco-Rey, M. Hoff-mann, A. Y. Vyazovskaya, S. V. Eremeev, A. Ernst, P. M.Echenique, A. Arnau, and E. V. Chulkov, Phys. Rev.Lett. , 107202 (2019).[28] A. Zeugner, F. Nietschke, A. U. B. Wolter, S. Gaß,R. C. Vidal, T. R. F. Peixoto, D. Pohl, C. Damm,A. Lubk, R. Hentrich, S. K. Moser, C. Fornari, C. H. Min,S. Schatz, K. Kißner, M. ¨Unzelmann, M. Kaiser, F. Scar-avaggi, B. Rellinghaus, K. Nielsch, C. Hess, B. B¨uchner,F. Reinert, H. Bentmann, O. Oeckler, T. Doert, M. Ruck,and A. Isaeva, Chem. Mater. , 2795 (2019).[29] J. Cui, M. Shi, H. Wang, F. Yu, T. Wu, X. Luo, J. Ying,and X. Chen, Phys. Rev. B , 155125 (2019).[30] X.-Y. Dong, S. Kanungo, B. Yan, and C.-X. Liu, Phys.Rev. B , 245135 (2016).[31] Y. Deng, Y. Yu, M. Z. Shi, J. Wang, X. H. Chen, andY. Zhang, arXiv:1904.11468 (2019).[32] C. Liu, Y. Wang, H. Li, Y. Wu, Y. Li, J. Li, K. He, Y. Xu,J. Zhang, and Y. Wang, arXiv:1905.00715 (2019).[33] J. Ge, Y. Liu, J. Li, H. Li, T. Luo, Y. Wu, Y. Xu, andJ. Wang, arXiv:1907.09947 (2019).[34] G. Kresse and J. Furthmller, Comput. Mater. Sci. , 15 (1996).[35] S. Grimme, J. Comput. Chem. , 1787 (2006),.[36] J. Heyd, G. E. Scuseria, and M. Ernzerhof, J. Chem.Phys. , 8207 (2003),.[37] G. Pizzi, V. Vitale, R. Arita, S. Bl¨ugel, F. Freimuth,G. G´eranton, M. Gibertini, D. Gresch, C. Johnson,T. Koretsune, J. Iba˜nez − Azpiroz, H. Lee, J.-M. Lihm,D. Marchand, A. Marrazzo, Y. Mokrousov, J. I. Mustafa,Y. Nohara, Y. Nomura, L. Paulatto, S. Ponc´e, T. Pon-weiser, J. Qiao, F. Th¨ole, S. S. Tsirkin, M. Wierzbowska, N. Marzari, D. Vanderbilt, I. Souza, A. A. Mostofi, andJ. R. Yates, arXiv:1907.09788 (2019).[38] Q. S. Wu, S. N. Zhang, H.-F. Song, M. Troyer, and A. A.Soluyanov, Comput. Phys. Commu. , 405 (2018).[39] S. Wu, J. S. Ross, G.-B. Liu, G. Aivazian, A. Jones,Z. Fei, W. Zhu, D. Xiao, W. Yao, D. Cobden, and X. Xu,Nat. Phys. , 149 (2013).[40] S. Murakami, New J. Phys.9