Two-dimensional Weyl Half Semimetal and Tunable Quantum Anomalous Hall Effect in Monolayer PtCl 3
Jing-Yang You, Cong Chen, Zhen Zhang, Xian-Lei Sheng, Shengyuan A. Yang, Gang Su
TTwo-dimensional Weyl Half Semimetal and Tunable Quantum Anomalous Hall Effectin Monolayer PtCl Jing-Yang You, Cong Chen,
2, 3
Zhen Zhang, Xian-Lei Sheng,
2, 3, ∗ Shengyuan A. Yang, † and Gang Su
1, 4, ‡ School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China Department of Physics, Key Laboratory of Micro-nano Measurement-Manipulationand Physics (Ministry of Education), Beihang University, Beijing 100191, China Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore 487372, Singapore Kavli Institute for Theoretical Sciences, and CAS Center for Excellence in Topological Quantum Computation,University of Chinese Academy of Sciences, Beijing 100190, China
We propose a new topological quantum state of matter—the two-dimensional (2D) Weyl halfsemimetal (WHS), which features 2D Weyl points at Fermi level belonging to a single spin channel,such that the low-energy electrons are described by fully spin-polarized 2D Weyl fermions. We pre-dict its realization in the ground state of monolayer PtCl . We show that the material is a half metalwith an in-plane magnetization, and its Fermi surface consists of a pair of fully spin-polarized Weylpoints protected by a mirror symmetry, which are robust against spin-orbit coupling. Remarkably,we show that the WHS state is a critical state at the topological phase transition between two quan-tum anomalous Hall insulator phases with opposite Chern numbers, such that a switching betweenquantum anomalous Hall states can be readily achieved by rotating the magnetization direction.Our findings demonstrate that WHS offers new opportunity to control the chiral edge channels,which will be useful for designing new topological electronic devices. Introduction. —Weyl semimetals have been attractingextensive attention in recent research [1–9]. In a Weylsemimetal, the conduction and valence bands cross lin-early at isolated twofold degenerate nodal points in theBrillouin zone (BZ), such that the low-energy electronsresemble the relativistic Weyl fermions. Thus, many in-triguing phenomena in relativity and high-energy physicscan be explored in condensed matter experiments [10–12]. In order to achieve the Weyl point, it is necessary tobreak the inversion ( P ) or the time reversal ( T ) symme-try to remove the spin degeneracy of the bands. So far,most Weyl semimetals are realized in crystals with broken P , while the candidates with broken T , i.e., the magneticWeyl semimetals, are much less [3, 13–17]. Moreover,the studies are mainly for three dimensional (3D) sys-tems. A Weyl point in 3D has a topological protection,characterized by the Chern number defined on a surfaceenclosing the point. In comparison, a Weyl point in 2Dmust require additional symmetry protection [18]. Suchreduction of protection means that the Weyl semimetalphase in 2D is less robust than its 3D counterpart. Onthe other hand, however, it also leads to the opportunityto more easily manipulate the topological phase transi-tions in 2D, especially the interplay between magnetismand band topology if the Weyl phase is realized in a mag-netic state.In this work, we propose a new topological state in2D—the 2D Weyl half semimetal (WHS), which is botha half metal and a semimetal, with fully spin-polarizedWeyl points at Fermi level formed in a single spin chan-nel. Consequently, the low-energy electrons are fully spin-polarized 2D Weyl fermions. We predict the real-ization of this novel phase in monolayer PtCl . Basedon first-principles calculations, we show that the ground state of monolayer PtCl is a 2D WHS with an in-planemagnetization, which preserves a vertical mirror plane.A pair of 2D Weyl points are protected by the mirrorsymmetry and are robust even under spin-orbit coupling(SOC). Furthermore, we find that the 2D WHS state rep-resents a critical point between two quantum anomalousHall (QAH) insulator phases with opposite Chern num-bers ±
1. By breaking the mirror, e.g., by rotating themagnetization vector, one can readily control the real-ization of QAH phases and the propagating direction ofthe chiral edge channels. Our findings not only reveala new state of matter, but also offer promising materialplatforms for novel topological spintronics applications.
Computational method. —Our first-principles calcula-tions were based on the density-functional theory (DFT)as implemented in the Vienna ab initio simulation pack-age (VASP) [19, 20], using the projector augmentedwave method [21]. The generalized gradient approx-imation with Perdew-Burke-Ernzerhof [22] realizationwas adopted for the exchange-correlation functional.The plane-wave cutoff energy was set to 520 eV. TheMonkhorst-Pack k -point mesh [23] of size 11 × × U method [24, 25] was used for calculating the band struc-tures. The crystal structure was optimized until theforces on the ions were less than 0.01 eV/˚A. The surfacespectrum was calculated by using the Wannier functionsand the iterative Green’s function method [26–29]. Structure and magnetism. —Monolayer PtCl consistsof a Pt atomic layer sandwiched by two Cl atomic lay-ers, where the Pt atoms form a honeycomb lattice andeach Pt is surrounded by six Cl atoms forming an octa-hedral crystal field, as shown in Fig. 1(a). It takes the a r X i v : . [ c ond - m a t . m t r l - s c i ] M a r (a) PtCl (b) M y M + M – Γ KM (c) F r e qu e n cy ( T H z ) (d) FM NAFMSAFM ZAFMxy K ’ FIG. 1. (a) Top and side view of monolayer PtCl , with edgesharing PtCl octahedron forming a honeycomb lattice. (b)First Brillouin zone for monolayer PtCl with high symme-try points labeled. We also mark the orientation of the threevertical mirror planes for the lattice structure (red lines). (c)Phonon spectrum for monolayer PtCl . (d) Possible magneticconfigurations considered: ferromagnet (FM), N´eel antifer-romagnet (NAFM), stripe AFM (SAFM), and zigzag AFM(ZAFM). The magnetic moments are on the Pt sites forminga honeycomb lattice.TABLE I. The total energy E tot per unit cell (in meV, rela-tive to E tot of the FM y ground state) as well as spin (cid:104) S (cid:105) andorbital (cid:104) O (cid:105) moments (in µ B ) for several magnetic configura-tions calculated by GGA+SOC+ U method. The superscriptin each configuration indicates the magnetic polarization di-rection. FM y NAFM y SAFM y ZAFM y FM z FM x PM E tot (cid:104) S (cid:105) (cid:104) O (cid:105) same structure as monolayer CrI [30] and RuCl [31]that have been shown to be 2D magnetic materials. Thepoint group symmetry is D d , with generators of a rotore-flection S and a vertical mirror σ d . Combining these twooperations leads to another two vertical mirror planes, asillustrated in Fig. 1(b). The three vertical mirrors play animportant role in the discussion of the WHS state below.The optimized lattice constant from our first-principlescalculations is 6.428 ˚A. To confirm its stability, we cal-culate the phonon spectrum, which shows no imaginaryfrequency mode [see Fig. 1(c)], indicating that monolayerPtCl is dynamically stable.Pt is a transition metal element with partially filled d shell, which may give rise to magnetism. Indeed, ourfirst-principles calculations show that monolayer PtCl favors a ferromagnetic (FM) ground state than the anti- (a) E F e g t (b) (c) P DO S E - E F ( e V ) -0.4 E-E F (eV) Г M K Г (d) E - E F ( m e V ) -50500 M K Г (e) total with SOCtotal w/o SOCPt t Pt e g Cl p K ’ K m with SOC w/o SOC FIG. 2. (a) Schematic depiction of the orbital splitting inmonolayer PtCl . (b) Spin-resolved partial density of states(PDOS) for monolayer PtCl projected on different orbitals.(c) Band structure without spin-orbit coupling (SOC). Thered and blue bands are for spin majority (spin-up) and mi-nority (spin-down) channels, respectively. (d) Enlarged viewof the band structure around the Weyl point. The red solid(blue dashed) lines are for the bands with (without) SOC. (e)Two Weyl points are located at K/K (cid:48) points without SOC(blue points), and they are shifted along x direction on themirror-invariant line after considering SOC (red points). ferromagnetic (AFM) or the paramagnetic (PM) states[see Fig. 1(d) and Table I]. Furthermore, the FM state isfound to be a half metal, i.e., with a single spin channelpresent at the Fermi level, as can be observed from theprojected density of states (DOS) in Fig. 2(b) and theband structure in Fig. 2(c).To understand this, we note that under the octahe-dral crystal field, Pt-5 d orbitals are split into t g and e g groups, with the latter energetically higher. For Pt with seven valence electrons, Pt- t g orbitals will be fully-filled. Since the crystal field in PtCl is stronger than ex-change field, the fully-filled t g orbitals are away from theFermi level. On the other hand, Pt- e g orbitals are filledby one electron, hence are fully spin-polarized. Becausethere are two Pt atoms in the primitive cell, the bandsdominated by e g orbitals are half-filled for one spin chan-nel and empty for the other, as schematically depicted inFig. 2(a). The bands around Fermi level are completelyfrom the spin-up subband of e g orbitals, therefore makingit a half metal with 100% spin polarization.Next, we shall pin down the magnetization directionfor the FM ground state. We compare the energies byscanning the magnetization direction m (with SOC in-cluded), and find that: (i) in-plane directions are ener-getically preferred over the out-of-plane ones; (ii) amongthe in-plane ones, the directions perpendicular to the ver-tical mirrors (i.e., the armchair direction for the Pt hon-eycomb lattice) have the lowest energy (see Table I). Itfollows that the magnetic interaction around the groundstate configuration may be approximately described bythe following spin Hamiltonian H = − (cid:88) (cid:104) i,j (cid:105) J ( S xi S xj + S yi S yj ) − (cid:88) i D ( S yi ) , (1)where S x,y is the spin operator, (cid:104) i, j (cid:105) denotes the sum-mation over nearest neighboring sites, J and D denotethe strengths for exchange interaction and anisotropy,respectively. The values of J and D can be extractedfrom the first-principles calculations. Approximating themodel as an anisotropic 2D XY ferromagnet, the Curietemperature for the FM state can be estimated [32–34]as T C ≈
200 K.
2D Weyl half semimetal. —In the band structure plot inFig. 2(c), one notices a remarkable feature: the conduc-tion and valence bands form a linear crossing point at theFermi level. Since the two crossing bands are fully spinpolarized (spin-up), the crossing point is twofold degen-erate and represents a 2D Weyl point. Thus, the groundstate for PtCl is a 2D WHS, with the low-energy elec-trons being 100% spin-polarized 2D Weyl fermions.In the absence of SOC, a pair of Weyl points are lo-cated at the K and K (cid:48) points of the BZ, similar tographene, but they are formed by a single spin species.Without SOC, the spin and the orbital part of the elec-tronic wave function are decoupled, and hence all crys-talline symmetries are preserved for each spin channelseparately as for spinless particles. To characterize thelow-energy band structure, we construct a k · p effectivemodel expanded around the K/K (cid:48) point. It is subjectedto the C v little group at K ( K (cid:48) ), with two generators C z and M y . The effective Hamiltonian must satisfy C z H ( q + , q − ) C − z = H ( q + e i π/ , q − e − i π/ ) , (2) M y H ( q x , q y ) M − y = H ( q x , − q y ) , (3)where q is measured from K , and q ± = q x ± iq y . And thetwo Weyl points are related by inversion. In the basis ofthe 2D irreducible representation E for C v , we find thatto linear order in q , the effective model takes the form ofthe 2D Weyl model H ( q ) = v F ( τ q x σ x + q y σ y ) , (4)where v F is the Fermi velocity, τ = ± for the K/K (cid:48) point, and σ i ’s are the Pauli matrices acting in the spaceof the two basis states. Thus, the low-energy electronsindeed resemble 2D Weyl fermions. It is worth notingthat despite the similarity to the low-energy model forgraphene [35], the model basis and hence the describedfermions here are fully spin polarized.As we have mentioned, the inclusion of SOC pins theground state magnetization perpendicular to one of thevertical mirrors (taken to be M y here). It follows that the C z symmetry is broken but the M y symmetry is stillpreserved. The preserved M y dictates that the spin-upand spin-down bands are still fully spin polarized (along y ) without hybridization by SOC. Remarkably, one findsthat the two Weyl points are maintained, only their lo-cations slightly shifted from K and K (cid:48) to some nearbypoints on the path K - M and K (cid:48) - M which are invariantunder the remaining mirror [see Fig. 2(d) and 2(e)]. TheWeyl points are still protected, since the two crossingbands have opposite M y eigenvalues. On the level of theeffective model, to leading order in k , SOC introducesthe following term H SOC = ησ x (5)for both K/K (cid:48) points. As a result, the original Weylpoint at
K/K (cid:48) is shifted by ∓ η/v F along the x direction(i.e., on the mirror-invariant line) but does not open agap. This is consistent with the first-principles calcula-tion result in Fig. 2(d). We mention that it is quite rareto have robust 2D Weyl point under SOC [36]. To ourknowledge, this is the first time to find such Weyl pointin a magnetic state.The above discussion demonstrates that the groundstate of monolayer PtCl is indeed a WHS with a pairof fully spin-polarized 2D Weyl points, and this state isrobust under SOC. It is in contrast to the Dirac points ingraphene, which are unpolarized and are removed whenSOC is turned on. Below, we shall show that the WHSstate represents a critical point between two QAH insu-lator phases with Chern numbers C = ±
1. This in turnrequires that the WHS state must be gapless, since it islocated at a topological phase transition.
Tuning QAH phases. —Because the Weyl points areprotected by M y , breaking M y will generally remove theWeyl points and open an energy gap. For example, theresult for m along the x direction (zigzag direction forthe Pt honeycomb lattice) is shown in Fig. 3(a). Clearly,a finite band gap ∼ . H ∆ = ∆2 σ z with | ∆ | the gap size, such that the effectivemodel becomes H = H + H SOC + H ∆ . (6)It is known that the gap opening at a 2D Weylpoint would induce a finite Berry curvature Ω( q ) = − (cid:104) ∂ q x u v | ∂ q y u v (cid:105) , where | u v (cid:105) is the eigenstate of thevalence band. The integral of Berry curvature in a regionaround the Weyl point gives a valley topological charge of ± / , T is broken, and the two Weyl pointsare related by P . Because the Berry curvature is aneven function under P , the valley topological charge forthe two points after gap opening must be the same, andtherefore a finite Chern number C = (cid:82) BZ Ω( k ) d k = ± (a) E - E F ( m e V ) -50 E g =15.5 meV (b) M K Γ m C=+1C=+1 C=+1
C= 1 C= 1C= 1
FIG. 3. (a) A gap is opened at the original Weyl pointwhen the magnetization is along the zigzag direction. (b) Theflower-like curve (red line) shows the band gap as a function ofthe azimuthal angle φ for the magnetization direction, wherepolar radius indicates the gap value (in meV). The blue (or-ange) color indicates the regions with Chern number C = +1( − The analysis above is confirmed by the first-principlescalculations. In Fig. 3(b), we plot the band gap and theChern number as functions of angle φ , which is the az-imuthal angle for the magnetization vector m , assuming m is rotated in-plane. One observes that the gap van-ishes at φ = ± π , ± π , and ± π , at which one of themirror planes is preserved, and the state corresponds toa WHS. In regions between these values, the gap be-comes nonzero, and the Chern number takes values al-ternating between +1 and −
1. Since the finite bandgap and the (quantized) Hall conductivity σ xy = e h C are tied together in the current case, the gap closing inthe WHS state can also be understood as a general sym-metry requirement. It was shown by Liu et al. [39] thatto maintain the invariance of the Hall response equation j x = σ xy E y for a nonzero σ xy , all the vertical mirrorsmust be broken. Thus, at the special values of φ thatpreserve a mirror, σ xy hence the gap must be zero.The hallmark of the QAH phase is the existence of chi-ral edge states, i.e., gapless channels at the edge propa-gating unidirectionally. Figure 4(a) shows the edge spec-trum obtained from first-principles calculations, whichconfirms the existence of one chiral channel per edge.The chirality of the edge channel is determined by the x yzm (a) ϕ m m E - E F ( m e V ) -50500 (b) -π 0 π (c) V xy V xx I I
FIG. 4. (a) The edge spectrum corresponding to the case inFig. 3(a), showing the existence of gapless chiral edge states.(b) Schematic top views of a finite size sample. By tuning themagnetization direction to regions with opposite Chern num-bers, one can switch the propagation direction of the chiraledge channel. This can be probed by the standard transportmeasurement setup as in (c). sign of Chern number. Consequently, by tuning acrossthe topological phase transition at the WHS state, thepropagating direction of the edge channel will be reversed[see Fig. 4(b)]. This can be detected in electrical trans-port measurement as shown in Fig. 4(c).
Discussion. —We have revealed a new topologicalquantum state—the 2D WHS, and demonstrated its re-alization in monolayer PtCl . Since it is a critical stateat the topological phase transition between two QAHphases, it offers great advantage to control the QAHphases. In experiment, the switching can be readilyachieved by applying an in-plane magnetic field to rotatethe magnetization vector. By switching between C = +1and − d elements like Pt. Here, we test the effect ofcorrelation via the DFT+ U approach [24, 25]. We findthat the results are qualitatively unchanged for U valuesup to 2 eV, and only for very large U ( > . U value for 5 d elements is less than 1.5 eV, theresults presented here should be robust.Finally, we mention that since the WHS state here isprotected by the mirror symmetry, it is robust under bi-axial strain or uniaxial strains along the high symmetrydirections (zigzag or armchair), which preserve the mir-ror. For more general strains (like shear strain), the WHSwould transform into the QAH phase. Strains can furtherbe used to tune the gap of the QAH state. For example,the band gap for the case in Fig. 3(a) can be increasedto ∼
20 meV under a biaxial 5% compressive strain.We thank D. L. Deng for helpful discussion. Thiswork is supported in part by the National Key R&DProgram of China (Grant No. 2018FYA0305800), theStrategic Priority Research Program of CAS (GrantNos. XDB28000000, XBD07010100), the NSFC (GrantNo. 11834014, 14474279, 11504013), Beijing Munic-ipal Science and Technology Commission (Grant No.Z118100004218001), and the Singapore Ministry of Edu-cation AcRF Tier 2 (MOE2015-T2-2-144). ∗ [email protected] † shengyuan [email protected] ‡ [email protected][1] S. Murakami, New J. Phys. , 356 (2007).[2] A. A. Burkov and L. Balents, Phys. Rev. Lett. ,127205 (2011).[3] X. Wan, A. M. Turner, A. Vishwanath, and S. Y.Savrasov, Phys. Rev. B , 205101 (2011).[4] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu,Rev. Mod. Phys. , 035005 (2016).[5] A. A. Burkov, Nat. Mater. , 1145 (2016).[6] S. A. Yang, SPIN , 1640003 (2016).[7] X. Dai, Nat. Phys. , 727 (2016).[8] A. Bansil, H. Lin, and T. Das, Rev. Mod. Phys. ,021004 (2016).[9] N. P. Armitage, E. J. Mele, and A. Vishwanath, Rev.Mod. Phys. , 015001 (2018).[10] H. Nielsen and M. Ninomiya, Phys. Lett. B , 389(1983).[11] D. T. Son and B. Z. Spivak, Phys. Rev. B , 104412(2013).[12] S. Guan, Z.-M. Yu, Y. Liu, G.-B. Liu, L. Dong, Y. Lu,Y. Yao, and S. A. Yang, npj Quant. Mater. , 23 (2017).[13] G. Xu, H. Weng, Z. Wang, X. Dai, and Z. Fang, Phys.Rev. Lett. , 186806 (2011).[14] Z. Wang, M. G. Vergniory, S. Kushwaha,M. Hirschberger, E. V. Chulkov, A. Ernst, N. P.Ong, R. J. Cava, and B. A. Bernevig, Phys. Rev. Lett. , 236401 (2016).[15] J. Kbler and C. Felser, EPL (Europhysics Letters) ,47005 (2016).[16] Q. Xu, E. Liu, W. Shi, L. Muechler, J. Gayles, C. Felser,and Y. Sun, Phys. Rev. B , 235416 (2018).[17] N. Morali, R. Batabyal, P. K. Nag, E. Liu, Q. Xu, Y. Sun, B. Yan, C. Felser, N. Avraham, and H. Beidenkopf,arXiv:1903.00509.[18] Y. X. Zhao and Z. D. Wang, Phys. Rev. Lett. , 240404(2013).[19] G. Kresse and J. Hafner, Phys. Rev. B , 14251 (1994).[20] G. Kresse and J. Furthm¨uller, Phys. Rev. B , 11169(1996).[21] P. E. Bl¨ochl, Phys. Rev. B , 17953 (1994).[22] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.Lett. , 3865 (1996).[23] H. J. Monkhorst and J. D. Pack, Phys. Rev. B , 5188(1976).[24] V. I. Anisimov, J. Zaanen, and O. K. Andersen, Phys.Rev. B , 943 (1991).[25] S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J.Humphreys, and A. P. Sutton, Phys. Rev. B , 1505(1998).[26] N. Marzari and D. Vanderbilt, Phys. Rev. B , 12847(1997).[27] I. Souza, N. Marzari, and D. Vanderbilt, Phys. Rev. B , 035109 (2001).[28] Q. Wu, S. Zhang, H.-F. Song, M. Troyer, and A. A.Soluyanov, Comput. Phys. Commun. , 405 (2018).[29] M. P. L´opez Sancho, J. M. L´opez Sancho, and J. Rubio,J. Phys. F 14, 1205 (1984); 15, 851 (1985).[30] B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein,R. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A.McGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-Herrero, and X. Xu, Nature , 270 (2017).[31] A. Banerjee, C. A. Bridges, J. Q. Yan, A. A. Aczel, L. Li,M. B. Stone, G. E. Granroth, M. D. Lumsden, Y. Yiu,J. Knolle, S. Bhattacharjee, D. L. Kovrizhin, R. Moess-ner, D. A. Tennant, D. G. Mandrus, and S. E. Nagler,Nat. Mater. , 733 (2016).[32] D. Spirin and Y. Fridman, Physica B , 410 (2003).[33] B. V. Costa, A. R. Pereira, and A. S. T. Pires, Phys.Rev. B , 3019 (1996).[34] Y.-q. Ma and W. Figueiredo, Phys. Rev. B , 5604(1997).[35] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S.Novoselov, and A. K. Geim, Rev. Mod. Phys. , 109(2009).[36] W. Wu, Y. Jiao, S. Li, X.-L. Sheng, Z.-M. Yu, and S. A.Yang, arXiv:1902.09283.[37] W. Yao, S. A. Yang, and Q. Niu, Phys. Rev. Lett. ,096801 (2009).[38] H. Pan, X. Li, F. Zhang, and S. A. Yang, Phys. Rev. B , 041404 (2015).[39] X. Liu, H.-C. Hsu, and C.-X. Liu, Phys. Rev. Lett.111