Strain-Induced Room-Temperature Ferromagnetic Semiconductors With

arXiv:1901.09306v2 [cond-mat.mtrl-sci] 16 Jul 2019 oia nuao (Bi,Sb) insulator logical neprmn with experiment in neprmn nte6n l fCr of film 6-nm the reported in recently experiment was in anisotropy magnetic out-of-plane [ experiment a eetydsusdi em fKte neato [ interaction Kitaev of terms in discussed recently was a been (Ga,Mn)As has semiconductor magnetic studied extensively endsusd[ discussed also been have effects magneto-optical and Photoluminescence CrI monolayer in structure magnetic layered a as along known spin was with bulk ferromagnet corresponding the Cr where 2D in obtained was state ferromagnetic berg mono- of in study observed CrI was layer the magnetization for out-of-plane platform with [ new semiconductors a magnetic provided ma- have Waals der terials van (2D) two-dimensional advances Recent in magnetism applications. in by semiconduc- required highly ferromagnetic are temperature tors room The ture. lscluain a rvd eiberfrnefrexper- for princi- [ reference first reliable iments provide by can candidates calculations exper- ples promising in predicting realized materials iment, ferromagnetic monolayer of a lorpre neprmn ntemnlyrVSe monolayer the in experiment [ in reported also was a oraiesitoi plctosbsdo s of de- electronic use in on freedom [ of based vices degrees spin applications and charge spintronic promising both realize a to semiconductors, way magnetic of development ∗ † 12 [email protected] [email protected] obnn ants n eiodco nbe the enables semiconductor and magnetism Combining n MnSe and ] tanIdcdRo-eprtr ermgei Semicond Ferromagnetic Room-Temperature Strain-Induced 2 1 , al nttt o hoeia cecsadCSCne fo Center CAS Sciences,and Theoretical for Institute Kavli 14 3 2 T 3 .TehgetCretmeaueo h most the of temperature Curie highest The ]. neprmn with experiment in , c hsclSineLbrtr,HaruNtoa Comprehens National Huairou Laboratory, Science Physical 15 1 eiodco Cr semiconductor omtmeaueb pligafwpretsri.I additi In strain. percent Cr few 2D a in applying by temperature room fF n i n sa re fmgiuelre hnta ndi in that than larger magnitude of enhan order the an interactions, superexchange is on and Based Ni, Ga(Mn,As). and Fe of eudrto ytedcesdeeg ieec ewe 3 between difference energy decreased the by understood be u nighglgt h irsoi ehns ooti t obtain to mechanism strain. microscopic by the semiconductors highlights finding Our 7 0 [ K 200 = .Alrermnn antzto with magnetization remanent large A ]. 2 colo hsclSine,Uiest fCieeAcademy Chinese of University Sciences, Physical of School .Rsaceshv losuidpossible studied also have Researchers ]. ydniyfntoa hoycluain,w rdc sta a predict we calculations, theory functional density By 10 [ .INTRODUCTION I. 13 nmlu alCnutvt nToDmninlCr Two-Dimensional in Conductivity Hall Anomalous , .Dsieterltvl ml number small relatively the Despite ]. 11 T 2 .Tefroants ihhigh with ferromagnetism The ]. c Ge 8Ki iae Cr bilayer in K 28 = 3 2 ,silfrblwro tempera- room below far still ], u-unDong Xue-Juan 2 Te Se nvriyo hns cdm fSine,Biig100190, Beijing Sciences, of Academy Chinese of University 4 .TeIigferromagnetism Ising The ]. 3 6 2 T n Cr and with Ge c c xsand axis 2 5K[ K 45 = Se T 6 2 hr h ui temperature Curie the where , 2 c Ge Ge 3 0K[ K 80 = 2 n Cr and 2 1 Te 5 Te igYn You Jing-Yang , T .TeHeisen- The ]. 6 c 2 6 speitdt ecmaal ota nfroantcmetal ferromagnetic in that to comparable be to predicted is Ge 1Kin K 61 = natopo- a on Dtd uy1,2019) 17, July (Dated: 2 2 2 8 Te Ge Ge .The ]. 6 2 2 Te Te [ T 9 6 ]. ], 6 6 2 c 1 oGu Bo , nmgei utlyrCr multilayer magnetic in eie,wihaennmgei,tesri sue to [ properties used electronics is and strain optical the the non-magnetic, modify are which genides, Fe 2D in ferromagnetism of xeiet[ experiment n ermgei ttsi iae CrI bilayer in states ferromagnetic antiferromagnetic tun- and between By switching the materials. voltage, 2D gate ing discovered recently these in states tan and strain, T ndltdmgei eiodco aM,s [ Ga(Mn,As) semiconductor magnetic diluted in xeiet[ experiment hc a eehne to enhanced be can which in epeitasal Dfroantcsemiconduc- ferromagnetic 2D stable Cr tor a predict we tions aclto,coet h au of value the to close calculation, h tani on odces h nrydffrnebe- difference energy the decrease 3 to tween found is strain The stud- been also have [ materials ied 2D these of erostructures ferromagnetism the affect to studied Cr monolayer is in strain of effect [ learning machine by materials ferromagnetic 2D iiyi DCr 2D in tivity [ calculations high-throughput ecmaal ota nfroantcmtl fF and Fe of metals ferromagnetic [ in Ni that to comparable be eteprmn [ experiment cent cpcmcaimt bantero eprtr mag- strain. temperature by room semiconductors the netic obtain to micro- the mechanism highlights scopic the finding on Our based picture. coupling superexchange ferromagnetic enhanced the duces [ 22 c eea ehd r sdt oto h magnetic the control to used are methods Several nti ae,b est ucinlter calcula- theory functional density by paper, this In xelnei oooia unu Computation, Quantum Topological in Excellence r .Teeeti edeetwsdsusdi experiment in discussed was effect field electric The ]. n2 eiodco Cr semiconductor 2D in 27 24 T – 2 – v cec etr ejn 040 China 101400, Beijing Center, Science ive c 29 Ge l w-iesoa 2)ferromagnetic (2D) two-dimensional ble 2 d 26 d a edaaial nacdbeyond enhanced dramatically be can ced , 3 ,adi nodro antd agrta that than larger magnitude of order an is and ], riaso rad4 and Cr of orbitals n h nmlu alconductivity Hall anomalous the on, ]. 2 , riaso rad4 and Cr of orbitals ero eprtr ferromagnetic temperature room he ∗ fSine,Biig104,China 100049, Beijing Sciences, of Se T T 6 18 n agSu Gang and c 6 .I diin h nmlu alconduc- Hall anomalous the addition, In ]. c efidthat find We . 2 y5 tan nteohrhand, other the On strain. 5% by K 421 = .Uiggt otg,teenhancement the voltage, gate Using ]. n2 Cr 2D in ue antcsemiconductor magnetic luted 2 2 Ge Ge 19 2 .I Dtasto ea dichalco- metal transition 2D In ]. 2 Se Te 6 2 6 Ge China n Cr and [ 21 2 1 coswt Large with uctors 2 2 T , Ge Se 2 Ge 3 n CrX and ] p c , GeTe 2 3 T p 6 17 Ge 2 † riaso Se. of orbitals 2 yapyn 3% applying by K 326 = 2 c Te ysri can strain by Te riaso e hc in- which Se, of orbitals 2 ]. 4 nCr in K 144 = Ge 6 2 6 2 T Se 2 [ saot3 nour in K 30 about is c a bevdi re- in observed was 23 Te 8Ki recent in K 28 = 6 3 3 .Rcnl,het- Recently, ]. 6 a bandin obtained was X=C,B,I) Br, Cl, = (X speitdto predicted is s 20 2 Ge 30 16 .The ]. , and ] 2 Se 31 6 ]. , 2

(a)

4 Frequency (THz)

FIG. 1. Crystal structure of Cr2Ge2Se6 of (a) top view in M K a-b plane and (b) side view in a-c plane. The two-dimensional Brillouin zone is shown in (c). (b) 1.5

1.0 II. CALCULATION METHODS

0.5 The density functional theory (DFT) calculations are 0.0 done by the Vienna ab initio simulation package (VASP)

-0.5 [32]. The spin-polarized calculation with projector Energy (eV) augmented wave (PAW) method, and general gradient approximations (GGA) in the Perdew-Burke-Ernzerhof -1.0 (PBE) exchange correlation functional are used. The spin-orbit coupling (SOC) is included in the calculations. -1.5 Total energies are obtained with k-grids 9 × 9 × 1 by M K Mohnkhorst-Pack approach. The lattice constants and FIG. 2. (a) Phonon spectrum and (b) electron band structure atom coordinates are optimized with the energy conver- −6 of two-dimensional Cr2Ge2Se6, obtained by the spin-polarized gence less than 10 eV and the force less than 0.01 GGA + SOC + U calculations. eV/A˚, where a large vacuum of 20 A˚ is used to model a 2D system. The phonon calculations are performed with density functional perturbation theory (DFPT) by Cr2Ge2Se6 is shown in Fig. 1 (a) for top view and the PHONOPY code [33]. Size of the supercell is 3 × 3 1 (b) for side view, where the space group number is × 1, where the displacement is taken by 0.01 A˚. 162 (P-31m). The structure is obtained by the fully re- For the on-site Coulomb interaction U of Cr ion, the laxed calculation. The optimized lattice constant of 2D values in range of 3 ∼ 5 eV are usually reasonable for Cr2Ge2Se6 is calculated as 6.413 A˚, which is smaller than 3d transition metal insulators [34], while a small value the lattice constant 6.8275 A˚ of Cr2Ge2Te6 in experi- of U < 2 eV was adopted in the DFT calculation of 2D ment. It is reasonable because Se has a smaller radius Cr2Ge2Te6 [6]. We fixed the parameter U = 4 eV in than Te. We examine the stability in terms of phonon most of our following calculations for 2D Cr2Ge2Te6 and spectrum, where the Brillouin zone is shown in Fig. 1(c). Cr2Ge2Se6, and will discuss the effect of different U pa- As shown in Fig. 2(a), there is no imaginary frequency rameters later. in the phonon dispersion near Γ point. It predicts that Based on the DFT results, the Curie temperature is the crystal structure of monolayer Cr2Ge2Se6 is stable. calculated by using the Monte Carlo simulations[21, 22] To further check the crystal stability, we study based on the 2D Ising model, where a 60 × 60 supercell 5 the formation energy of monolayer Cr2Ge2Se6 and is adopted, and 10 steps are performed for every tem- Cr2Ge2Te6 by DFT calculations. The formation energy perature to acquire the equilibrium. The anomalous Hall of Cr2Ge2Se6 is given by Eform = Etot − 2ECr − 2EGe − conductivity is calculated with Wannier90 code [35] and 6ESe, where Etot is the total energy of the 2D Cr2Ge2Se6, WannierTools code [36]. and ECr, EGe and ESe are the total energy per atom for the bulk chromium, germanium and selenium, respec- tively. We found that the formation energy of monolayer III. CRYSTAL STABILITY Cr2Ge2Se6 is -1.13 eV, lower than the 0.92 eV of monolayer Cr2Ge2Te6. It suggests that the 2D Cr2Ge2Se6 We study the stability of the 2D new material should be more stable than 2D Cr2Ge2Te6, the latter Cr2Ge2Se6. We propose this material guided by the re- was realized in recent experiment [6]. cently reported 2D material Cr2Ge2Te6 in experiment, The electronic band structure of Cr2Ge2Se6 is shown in and we replace Te by Se. The crystal structure of Fig. 2(b). An indirect band gap of 0.748 eV is observed, 3

(a)

1.0

Cr Ge Te

2 2 6

Expt.

0.8

Calc

Cr Ge Se (strain)

2 2 6

0.6

-1%

0.4

3% Magnetization

0.2

0.0

FIG. 3. (a) Four possible sates for two-dimensional 0 200 400 600 800

2 2 6 T (K) Cr Ge Se : ferromagnetic (FM) state, antiferromagnetic (AFM) N´eel state, AFM stripe state, and AFM zigzag state. (b) Balls denote Cr atoms, and arrows denote spins. (b) Total 600 energy as a function of angle θ for the FM state, obtained by the spin-polarized GGA + SOC + U calculations. 300 Cr Ge Te )

2 2 6 -1 cm and monolayer Cr2Ge2Se6 is a semiconductor. Recently, 0 (

it has been found that the change of the magnetization xy Cr Ge Se (strain)

2 2 6 direction can alter the band structures in 2D CrI3 [37]. -300 0% 2 2 6 In our case, we calculate the band structure of Cr Ge Se 3%

with the in-plane magnetization, and ﬁnd that the band 7% -600 structure does not change dramatically. In addition, we uncover that a larger band gap of 1.38 eV is obtained by -0.4 -0.2 0.0 0.2 0.4

E-E (eV) using the HSE06 hybrid functional, whereas the proﬁles F of band structure do not change.

FIG. 4. For two-dimensional Cr2Ge2Te6 and Cr2Ge2Se6 with different strains, (a) the normalized magnetization as a func- IV. MAGNETIC PROPERTIES tion of temperature, and (b) the anomalous Hall conductivity as a function of energy. The experimental result of Cr2Ge2Te6 To study the magnetic ground state of 2D Cr2Ge2Se6, is taken from Ref. [6]. The calculation results are obtained we examine the possible states with paramagnetic, ferro- by the DFT calculations and Monte Carlo simulations. magnetic and antiferromagnetic configurations. The calculations reveal that the energy of the paramagnetic state is 6 eV higher than the ferromagnetic and antiferromagnetic sates. Fig. 3(a) shows four possible magnetic sates of 2D Cr2Ge2Se6: ferromagnetic (FM) state, antiferro- TABLE I. For 2D semiconductor Cr2Ge2Te6, Cr2Ge2Se6, and magnetic (AFM) Néel state, AFM stripe state, and AFM Cr2Ge2Se6 with 5% strain, DFT results of total energy of ferromagnetic state EFM, total energy of antiferromagnetic state zigzag state. The calculations show that the FM state is AFM E , Curie temperature T c, 3d orbital occupation number the most stable state, and the energy difference between n , spin (S) and orbital (L) momentum of Cr atom, and bond the FM and the AFM configuration is more than 30 meV d length dCr−T e/Se . as listed in Table I. Furthermore, as shown in Fig. 3(b), the calculations show that the lowest energy of ferromag- Cr2Ge2Se6 DFT Cr2Ge2Te6 netic state is obtained when the magnetization direction no strain 5% strain is perpendicular to the two-dimensional materials, with EFM(eV) -89.1376 -97.3460 -96.5658 magnetic anisotropy energy of 0.32 meV per unit cell of EAFM(eV) -89.1309 -97.3135 -96.4710 Cr2Ge2Se6. As listed in Table I, the calculation shows AFM FM E -E (meV) 6.7 32.5 98.8 that for 2D Cr2Ge2Se6, the spin momentum of Cr atom T c(K) 30 144 421 is 3.4 µB, and the occupation number of Cr 3d orbitals is 3.976. nd(Cr) 4.043 3.976 3.951 The Curie temperature T c can be estimated by the S(µB )(Cr) 3.586 3.404 3.487 Monte Carlo simulation based on a 2D Ising model. The L(µB )(Cr) -0.018 -0.076 -0.105 exchange coupling parameter is estimated as the total dCr−T e/Se (A˚) 2.827 2.64 2.697 energy difference of ferromagnetic and antiferromagnetic states EAFM-EFM as listed in Table I, which was ob- 4

tained by the DFT calculations. The obtained normalized magnetization as function of temperature is shown 500 in Fig.4 (a). The experimental result of magnetiza- Cr Ge Te

2 2 6 tion for Cr2Ge2Te6 is taken from temperature-dependent 400 Exp 2 2 6 T Kerr rotation of bilayer Cr Ge Te with c = 28 K [6]. Cal

300 The calculated Curie temperature is T c = 30 K for the Cr Ge Se (strain)

2 2 6 monolayer Cr2Ge2Te6, which is close to the experimental

200 value. It is noted that for simplicity the Ising model is ap- Tc (K) 5% plied in our Monte Carlo simulation, while the Heisenberg model with magnetic anisotropy was suggested in exper- 100 iment [6]. In addition, the estimated Curie temperature

T c could be even larger by using the mean field approxi- 0 mation [17]. The calculated Curie temperature for mono- 1 2 3 4 layer Cr2Ge2Se6 is T c = 144 K, which is about 5 times U (eV) higher than the T c = 30 K for monolayer Cr2Ge2Te6 by the Monte Carlo simulation. More interestingly, the FIG. 5. The Curie temperature Tc as a function of parame- Curie temperature can be enhanced to T c = 326K by ap- ter U for 2D Cr2Ge2Te6, Cr2Ge2Se6 without strain and with plying tensile 3% strain, and T c = 421 K with 5% strain. 5% strain. The experimental result of Cr2Ge2Te6 and the We find that T c is decreased to 67 K with 1% compres- calculations are the same in Fig. 4(a), except with different sion strain, as shown in Fig. 4(a), and the system be- values of U. comes antiferromagnetic when applying 2% compression strain. Our result predicts that monolayer Cr2Ge2Se6 by applying a few percent tensile strain can be a promising TABLE II. For 2D semiconductor Cr2Ge2Se6 witout strain and with 5% strain, DFT results of hopping matrix element candidate for room-temperature ferromagnetic semicon- |V | and energy difference |Ep − Ed| between 4p orbitals of Se ductor. and 3d orbitals of Cr. As the magnetization direction is out-of-plane with strain hopping matrix element |V | (eV) an easy axis along c direction in 2D Cr2Ge2Te6 and Cr2Ge2Se6, it is interesting to study the anomalous Hall pz-dz2 pz-dxz pz-dyz pz-dx2−y2 pz-dxy conductivity due to the Berry curvature of band struc- 0% 0.257983 0.29527 0.380283 0.190041 0.445906 ture. By the DFT calculations, the results are shown in 5% 0.258289 0.22747 0.306872 0.201524 0.416315 Fig. 4(b). The magnitude of anomalous Hall conductiv- energy difference |Ep − Ed| (eV) ity σxy for the p-type Cr2Ge2Se6 and n-type Cr2Ge2Te6 can be as large as 4 × 102 (Ω cm)−1. This value is com- pz-dz2 pz-dxz pz-dyz pz-dx2−y2 pz-dxy 0% 1.415092 0.4458 0.45512 0.581666 0.581166 parable to the σxy in some ferromagnetic metals, such 2 as σxy = 7.5 × 10 in bcc Fe [27, 28], and σxy = 4.8 × 5% 1.552237 0.09143 0.09775 0.695055 0.693946 102 (Ω cm)−1 in fcc Ni [29] due to the Berry curvature of band structures. More importantly, the estimated σxy in 2D magnetic semiconductors Cr2Ge2Te6 and Cr2Ge2Se6 For 2D Cr2Ge2Se6 with 5% strain, the calculated Tc is an order of magnitude larger than the σxy of classic is always higher than room temperature with parame- diluted magnetic semiconductor Ga(Mn,As) [30, 31]. ter U in the range of 1 ∼ 4 eV, as shown in Fig. 5. Thus, our prediction about the room temperature ferromagnetic semiconductor Cr2Ge2Se6 with a few percent V. DISCUSSION strain is robust with the values of parameter U. How to understand the enhancement of T c in 2D To study the effect of the on-site Coulomb interaction Cr2Ge2Se6 by applying strain? According to the superex- U of 3d orbitals of Cr, we calculate the Curie tempera- change interaction [38–40], the FM coupling is expected since the Cr-Se-Cr bond angle is close to 90 degree. The ture Tc as a function of parameter U for 2D Cr2Ge2Te6, indirect FM coupling between Cr atoms is proportional Cr2Ge2Se6 without strain and with 5% strain, as shown to the direct AFM coupling between neighboring Cr and in Fig. 5. The experimental Tc of 2D Cr2Ge2Te6 is taken Se atom. The magnitude of this direct AFM coupling from Ref. [6]. For 2D Cr2Ge2Te6, the calculated Tc is de- 2 can be roughly estimated as J = |V | , where |V | is creased with increasing U, and close to the experimental |Ep−Ed| Tc with U = 4 eV. In addition, we found that the mag- the hopping matrix element between 4p orbitals of Se netization direction of Cr2Ge2Te6 becomes in-plane with and 3d orbitals of Cr, and |Ep − Ed| is the energy dif- U = 1 eV, which disagrees with experiment. The antifer- ference between 4p orbitals of Se and 3d orbitals of Cr. romagnetic ground state is obtained for Cr2Ge2Te6 with By DFT calculations, we can obtain these parameters U = 5 eV, which is also against with the experiment. for 2D Cr2Ge2Se6 without strain and with 5% strain. Our calculations suggest that the reasonable U for 2D The results of |V | and |Ep − Ed| are listed in Table II. Cr2Ge2Te6 may be in the range of 2 ∼ 4 eV. The results suggest that the strain has little effect on the 5 hopping |V |, while the strain can make energy difference on the superexchange interaction, the decreased energy between pz orbital of Se and dxz and dyz orbitals of Cr difference between 4p orbitals of Se and 3d orbitals of decreased to one-fifth of the value without strain, causing Cr is found to be important to enhance the Tc in 2D a dramatical enhancement of the AFM coupling between Cr2Ge2Se6 by applying strain. Our finding highlights the Cr and Se atom. This leads to the enhanced FM cou- microscopic mechanism to obtain the room temperature pling between Cr atoms, and makes Tc beyond the room ferromagnetic semiconductors by strain. temperature.

ACKNOWLEDGMENTS VI. SUMMARY The authors acknowledge Q. B. Yan, Z. G. Zhu, and By the DFT calculations we predict a stable 2D ferro- Z. C. Wang for many valuable discussions. BG is sup- magnetic semiconductor Cr2Ge2Se6. We ﬁnd the Curie ported by NSFC(Grant No. Y81Z01A1A9), CAS (Grant temperature T c of Cr2Ge2Se6 can be dramatically en- No.Y929013EA2) and UCAS (Grant No.110200M208). hanced beyond room temperature by applying 3% strain, GS is supported in part by the the National Key R&D which is much higher than the T c = 28 K in 2D Program of China (Grant No. 2018FYA0305800), the Cr2Ge2Te6 in recent experiment. In addition, the anoma- Strategic Priority Research Program of CAS (Grant lous Hall conductivity in 2D Cr2Ge2Se6 and Cr2Ge2Te6 Nos.XDB28000000, XBD07010100), the NSFC (Grant is predicted to be an order of magnitude larger than that No.11834014), and Beijing Municipal Science and Tech- in diluted magnetic semiconductor Ga(Mn,As). Based nology Commission (Grant No. Z118100004218001).

[1] H. Ohno, Making nonmagnetic semiconductors ferromag- X. Xu, Ligand-field helical luminescence in a 2D ferro- netic, Science 281, 951 (1998). magnetic insulator, Nat. Phys. 14, 277 (2017). [2] T. Dietl, A ten-year perspective on dilute magnetic semi- [11] Y. Fang, S. Wu, Z.-Z. Zhu, and G.-Y. Guo, conductors and oxides, Nat. Mater. 9, 965 (2010). Large magneto-optical effects and magnetic [3] L. Chen, X. Yang, F. Yang, J. Zhao, J. Mis- anisotropy energy in two-dimensional Cr2Ge2Te6, uraca, P. Xiong, and S. von Molnár, Enhancing Phys. Rev. B 98, 125416 (2018). the curie temperature of ferromagnetic semiconductor [12] M. Bonilla, S. Kolekar, Y. Ma, H. C. Diaz, V. Kalap- (Ga,Mn)As to 200 K via nanostructure engineering, pattil, R. Das, T. Eggers, H. R. Gutierrez, M.-H. Nano Lett. 11, 2584 (2011). Phan, and M. Batzill, Strong room-temperature ferro- [4] K. S. Burch, D. Mandrus, and J.-G. Park, Mag- magnetism in VSe2 monolayers on van der Waals sub- netism in two-dimensional van der Waals materials, strates, Nat. Nanotech. 13, 289 (2018). Nature 563, 47 (2018). [13] D. J. O’Hara, T. Zhu, A. H. Trout, A. S. Ahmed, [5] B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, Y. K. Luo, C. H. Lee, M. R. Brenner, S. Rajan, R. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A. J. A. Gupta, D. W. McComb, and R. K. Kawakami, McGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo- Room temperature intrinsic ferromagnetism in epitax- Herrero, and X. Xu, Layer-dependent ferromagnetism in ial manganese selenide films in the monolayer limit, a van der Waals crystal down to the monolayer limit, Nano Lett. 18, 3125 (2018). Nature 546, 270 (2017). [14] B. Shabbir, M. Nadeem, Z. Dai, M. S. Fuhrer, Q.-K. Xue, [6] C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao, X. Wang, and Q. Bao, Long range intrinsic ferromag- W. Bao, C. Wang, Y. Wang, Z. Q. Qiu, R. J. Cava, S. G. netism in two dimensional materials and dissipationless Louie, J. Xia, and X. Zhang, Discovery of intrinsic fer- future technologies, Appl. Phys. Rev. 5, 041105 (2018). romagnetism in two-dimensional van der Waals crystals, [15] N. Sivadas, M. W. Daniels, R. H. Swendsen, S. Okamoto, Nature 546, 265 (2017). and D. Xiao, Magnetic ground state of semicon- [7] V. Carteaux, D. Brunet, G. Ouvrard, and G. An- ducting transition-metal trichalcogenide monolayers, dre, Crystallographic, magnetic and electronic structures Phys. Rev. B 91, 235425 (2015). of a new layered ferromagnetic compound Cr2Ge2Te6, [16] T. D. Rhone, W. Chen, S. Desai, A. Yacoby, and E. Kaxi- J. Phys.: Condens. Matter 7, 69 (1995). ras, Data-driven studies of magnetic two-dimensional ma- [8] M. Mogi, A. Tsukazaki, Y. Kaneko, R. Yoshimi, K. S. terials, arXiv preprint arXiv:1806.07989 (2018). Takahashi, M. Kawasaki, and Y. Tokura, Ferromagnetic [17] H. Liu, J.-T. Sun, M. Liu, and S. Meng, insulator Cr2Ge2Te6 thin films with perpendicular rema- Screening magnetic two-dimensional atomic nence, APL Mater. 6, 091104 (2018). crystals with nontrivial electronic topology, [9] C. Xu, J. Feng, H. Xiang, and L. Bellaiche, In- J. Phys. Chem. Lett. 9, 6709 (2018). terplay between kitaev interaction and single ion [18] B. Huang, G. Clark, D. R. Klein, D. MacNeill, anisotropy in ferromagnetic CrI3 and CrGeTe3 monolay- E. Navarro-Moratalla, K. L. Seyler, N. Wilson, M. A. ers, npj Comput. Mater. 4, 57 (2018). McGuire, D. H. Cobden, D. Xiao, W. Yao, P. Jarillo- [10] K. L. Seyler, D. Zhong, D. R. Klein, S. Gao, X. Zhang, Herrero, and X. Xu, Electrical control of 2D magnetism B. Huang, E. Navarro-Moratalla, L. Yang, D. H. Cobden, in bilayer CrI3, Nat. Nanotech. 13, 544 (2018). M. A. McGuire, W. Yao, D. Xiao, P. Jarillo-Herrero, and 6

[19] Y. Deng, Y. Yu, Y. Song, J. Zhang, N. Z. wannier interpolation, Phys. Rev. B 74, 195118 (2006). Wang, Z. Sun, Y. Yi, Y. Z. Wu, S. Wu, J. Zhu, [29] X. Wang, D. Vanderbilt, J. R. Yates, and I. Souza, Fermi- J. Wang, X. H. Chen, and Y. Zhang, Gate-tunable surface calculation of the anomalous Hall conductivity, room-temperature ferromagnetism in two-dimensional Phys. Rev. B 76, 195109 (2007). Fe3GeTe2, Nature 563, 94 (2018). [30] T. Jungwirth, J. Sinova, K. Y. Wang, K. W. Ed- [20] R. Roldán, A. Castellanos-Gomez, E. Cap- monds, R. P. Campion, B. L. Gallagher, C. T. pelluti, and F. Guinea, Strain engineering Foxon, Q. Niu, and A. H. MacDonald, Dc-transport in semiconducting two-dimensional crystals, properties of ferromagnetic (Ga,Mn)As semiconductors, J. Phys.: Condens. Matter 27, 313201 (2015). Appl. Phys. Lett. 83, 320 (2003). [21] X. Li and J. Yang, CrXTe3(X = Si, Ge) nanosheets: [31] J. Sinova, T. Jungwirth, S.-R. E. Yang, two dimensional intrinsic ferromagnetic semiconductors, J. Kuˇcera, and A. H. MacDonald, Infrared con- J. Mater. Chem. C 2, 7071 (2014). ductivity of metallic (III,Mn)V ferromagnets, [22] J. Liu, Q. Sun, Y. Kawazoe, and P. Jena, Exfoliat- Phys. Rev. B 66, 041202 (2002). ing biocompatible ferromagnetic Cr-trihalide monolayers, [32] G. Kresse and J. Furthmüller, Efficient iterative schemes Phys. Chem. Chem. Phys. 18, 8777 (2016). for ab initio total-energy calculations using a plane-wave [23] W. Xing, Y. Chen, P. M. Odenthal, X. Zhang, W. Yuan, basis set, Phys. Rev. B 54, 11169 (1996). T. Su, Q. Song, T. Wang, J. Zhong, S. Jia, X. C. [33] A. Togo and I. Tanaka, First principles phonon calcula- Xie, Y. Li, and W. Han, Electric field effect in tions in materials science, Scr. Mater. 108, 1 (2015). multilayer Cr2Ge2Te6: a ferromagnetic 2D material, [34] S. Maekawa, T. Tohyama, S. E. Barnes, S. Ishihara, 2D Mater. 4, 024009 (2017). W. Koshibae, and G. Khaliullin, Physics of Transition [24] D. Zhong, K. L. Seyler, X. Linpeng, R. Cheng, Metal Oxides (Springer Berlin Heidelberg, 2004), Series N. Sivadas, B. Huang, E. Schmidgall, T. Taniguchi, in Solid State Sciences, Vol. 144. K. Watanabe, M. A. McGuire, W. Yao, D. Xiao, K.- [35] A. A. Mostofi, J. R. Yates, G. Pizzi, Y.-S. Lee, M. C. Fu, and X. Xu, Van der Waals engineering of fer- I. Souza, D. Vanderbilt, and N. Marzari, An romagnetic semiconductor heterostructures for spin and updated version of wannier90: A tool for ob- valleytronics, Sci. Adv. 3, e1603113 (2017). taining maximally-localised wannier functions, [25] Z. Wang, T. Zhang, M. Ding, B. Dong, Y. Li, Comput. Phys. Commun. 185, 2309 (2014). M. Chen, X. Li, J. Huang, H. Wang, X. Zhao, Y. Li, [36] Q. Wu, S. Zhang, H.-F. Song, M. Troyer, and D. Li, C. Jia, L. Sun, H. Guo, Y. Ye, D. Sun, A. A. Soluyanov, WannierTools: An open-source Y. Chen, T. Yang, J. Zhang, S. Ono, Z. Han, and software package for novel topological materials, Z. Zhang, Electric-field control of magnetism in a few- Comput. Phys. Commun. 224, 405 (2018). layered van der Waals ferromagnetic semiconductor, [37] P. Jiang, L. Li, Z. Liao, Y. X. Zhao, and Nat. Nanotech. 13, 554 (2018). Z. Zhong, Spin direction-controlled electronic band [26] M. Lohmann, T. Su, B. Niu, Y. Hou, M. Alghamdi, structure in two-dimensional ferromagnetic CrI3, M. Aldosary, W. Xing, J. Zhong, S. Jia, W. Han, R. Wu, Nano Lett. 18, 3844 (2018). Y.-T. Cui, and J. Shi, Probing magnetism in insulat- [38] J. B. Goodenough, Theory of the role of covalence ing Cr2Ge2Te6 by induced anomalous Hall effect in Pt, in the perovskite-type manganites[La, M(II)]MnO3, Nano Lett. 19, 2397 (2019). Phys. Rev. 100, 564 (1955). [27] Y. Yao, L. Kleinman, A. H. MacDonald, J. Sinova, [39] J. Kanamori, Crystal distortion in magnetic compounds, T. Jungwirth, D. sheng Wang, E. Wang, and J. Appl. Phys. 31, S14 (1960). Q. Niu, First principles calculation of anoma- [40] P. W. Anderson, New approach to the theory of superex- lous Hall conductivity in ferromagnetic bcc Fe, change interactions, Phys. Rev. 115, 2 (1959). Phys. Rev. Lett. 92, 037204 (2004). [28] X. Wang, J. R. Yates, I. Souza, and D. Vanderbilt, Ab initio calculation of the anomalous Hall conductivity by