Quantum Anomalous Hall Effect in a Perovskite and Inverse-Perovskite Sandwich Structure
aa r X i v : . [ c ond - m a t . m e s - h a ll ] D ec Quantum Anomalous Hall Effect in a Perovskite and Inverse-Perovskite SandwichStructure
Long-Hua Wu , ∗ and Qi-Feng Liang , , and Xiao Hu , † International Center for Materials Nanoarchitectonics (WPI-MANA),National Institute for Materials Science, Tsukuba 305-0044, Japan Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba 305-8571, Japan Department of Physics, Shaoxing University, Shaoxing 312000, China
Based on first-principles calculations, we propose a sandwich structure composed of a G-type anti-ferromagnetic (AFM) Mott insulator LaCrO grown along the [001] direction with one atomic layerreplaced by an inverse-perovskite material Sr PbO. We show that the system is in a topologicallynontrivial phase characterized by simultaneous nonzero charge and spin Chern numbers, which cansupport a spin-polarized and dissipationless edge current in a finite system. Since these two materialsare stable in bulk and match each other with only small lattice distortions, the composite materialis expected easy to synthesize.
I. INTRODUCTION
Discovery of the quantum Hall effect (QHE) by vonKlitzing has opened a new era in condensed matterphysics . It is revealed that the quantization of Hallconductance is a manifestation of topologically non-trivial Bloch wavefunctions . Topological matters havevarious promising applications in many fields, suchas fault-tolerant topological quantum computations ,spintronics and photonics .The quantum spin Hall effect (QSHE) was first pre-dicted theoretically in graphene and later studiedin a two-dimensional (2D) HgTe quantum well boththeoretically and experimentally . A 3D topolog-ical insulator Bi − x Sb x and its family members werealso reported . Breaking time-reversal symmetry candrive a topological insulator into the quantum anoma-lous Hall effect (QAHE) . There are two categoriesof the QAHE classified by the spin Chern number .One subclass of the QAHE is characterized by a van-ishing spin Chern number. The Cr-doped Bi Se thinfilm belongs to this class, where the topologicalband gap is opened by hybridizations between the spin-up and -down channels. The other subclass of the QAHEhas a nonzero spin Chern number. One representativematerial is the Mn-doped HgTe , where the s -type elec-trons of Hg and the p -type holes of Te experience opposite g -factors when they couple with the d electrons of Mn.The opposite exchange fields felt by the s and p orbitalsenlarge the energy gap in one spin channel, and closeand then reopen the energy gap in the other spin chan-nel, which induces a nontrivial topology in the latter spinchannel due to the band inversion mechanism , for alarge enough g -factor. However, its experimental real-ization turns out to be difficult due to the paramagneticstate of Mn spins. Two other materials are proposed torealize the QAHE with nonzero spin Chern numbers inhoneycomb lattice, a silicene sheet sandwiched by twoferromagnets with magnetization directions aligned anti-parallelly , and a perovskite material LaCrO grownalong the [111] direction with Cr atoms replaced by Ag or Au in one atomic layer . In both systems, in additionto the anti-ferromagnetic (AFM) exchange field and spin-orbit coupling (SOC), a strong electric field is requiredto break the inversion symmetry in order to realize theQAHE. For the former one, the weak SOC of silicene lim-its the novel QAHE to low temperatures, while for thelatter one, growth of the perovskite material along the[111] direction seems to be difficult.In the present work, we propose a new material torealize the second subclass of the QAHE without any ex-trinsic operation and easy to synthesize. It is based onLaCrO grown along the [001] direction, where we in-sert one atomic layer of an inverse-perovskite materialSr PbO such that the Pb atom feels the exchangefield established by the Cr atoms in the parent material.With first-principles calculations, we reveal that there isa band inversion at the Γ point between the d orbital ofCr and the p orbital of Pb in the spin-up channel inducedby the SOC, whereas the spin-down bands are pushed faraway from the Fermi level by the AFM exchange field.Constructing an effective low-energy Hamiltonian, we ex-plicitly show that the system is characterized by simul-taneous nonzero charge and spin Chern numbers. Pro-jecting the bands near the Fermi level onto the subspacecomposed of the spin-up d and p orbitals by maximallylocalized Wannier functions , we confirm that a spin-polarized and dissipationless current flows along the edgeof a finite sample. Since these two materials are stable inbulk and match each other with small lattice distortions,the composite material is expected easy to synthesize. II. FIRST-PRINCIPLES CALCULATIONS
The parent material LaCrO exhibits the perovskitestructure with formula ABO , where the oxygen atomsform an octahedron surrounding the B atom. It is awell-known Mott insulator with a large energy gap ∼ PbO shows the inverse-
FIG. 1. (Color online) (a) Crystal structure of bulk LaCrO grown along the [001] direction with one atomic layer re-placed by Sr PbO. (b) Enlarged interface between LaCrO and Sr PbO with grey and blue arrows representing spin mo-ments on Pb and Cr sites, respectively. ~a , ~b and ~c are latticevectors. perovskite structure with formula A BO, where the Aatoms form an octahedron surrounding the oxygen. Itwas revealed recently that there is a topological bandgap in bulk Sr PbO . We notice that the ~a - ~b planelattice constant is 3.88 ˚A for LaCrO , and 5.15 ˚A forSr PbO, different from each other by a factor close to √
2. Therefore, with a π/ ~c axis, these two materials match each other quite well[see Figs. 1(a) and (b)]. At the interface the oxygen ofSr PbO completes the CrO octahedron of the perovskitestructure [see Fig. 1(b)], which minimizes the distortionto the two materials when grown together. As shown inFig. 1(b) zoom-in at the interface, there are two types ofCr atoms in each CrO unit cell, where Cr1 sits at thecorners of the square and above the Pb atom in the ~c axis, whereas Cr2 sits at the center of square and abovethe oxygen.We have performed first-principles calculations by us-ing density functional theory (DFT) implemented in theVienna Ab-initio
Simulation Package (VASP) , whichuses the projected augmented wave (PAW) method .The exchange correlation potential is described by thegeneralized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) type . The cut-off energy of theplane waves is chosen to be 500 eV. The Brillouin zone ismeshed into a 10 × × d electrons with U = 5 . J = 0 . by usingthe Dudarev method. For the lattice structure, we take a = b = 5 .
48 ˚A (= √ × .
88 ˚A) and c LaCrO = 3 .
88 ˚A.The height of the inserted Sr PbO layer is determinedby a relaxation process to achieve the minimal energy: c SrPbO = 5 .
46 ˚A, the distance from the Pb atom to theCr just above it (that to the Cr below it is c LaCrO ). Af-
FIG. 2. (Color online) Band structure of the supercell shownin Fig. 1 without (a) and with (b) SOC. Solid cyan, solid redand dashed blue curves are for Cr1- d ↑ z , Pb- p ↑± and Pb- p ↓± orbitals, respectively. Other colors indicate their hybridiza-tions. (c) Schematic band evolution of Pb- p x,y and Cr- d z atthe Γ point caused by SOC. terwards, the positions of atoms are determined by a sec-ond relaxation process with all lattice constants fixed. Inboth processes, the criterion on forces between atoms isset to below 0.01 eV/˚A. The results shown below are fora superlattice structure with five layers of LaCrO andone atomic layer of Sr PbO. We confirm that the resultsremain unchanged as far as the number of LaCrO layersis above five and for U eff = U − J > . .
18 eV at the Γpoint. As shown in Fig. 2(a), the topmost valence bandis occupied by the spin-up p ± (= p x ± ip y ) orbitals ofPb, and the lowest conduction band is contributed bythe spin-up d z of Cr1. The reason for this band ar- FIG. 3. (Color online) Energy dispersion around the Γ pointfitted by the 2 × k · P Hamiltonian (4) on the basis [ d ↑ z , p ↑ + ].The fitted curves collapse with the DFT results within theregion | k x | ≤ . πa and | k y | ≤ . πb , where a and b arelattice constants given in text. rangement is that the Cr1 atom does not live in a closedoctahedron due to the absence of an oxygen in the cornerof Sr O layer as shown in Fig. 1(b), which weakens thecrystal field splitting and lowers the energy of the unoc-cupied Cr1- d z band, whereas the Cr2 shares one oxygenwith Sr PbO, thus is closed by a complete oxygen octahe-dron, which keeps its d z far away from the Fermi level.Therefore, only the spin-up Cr1- d z band appears justabove the Fermi level, in contrast to the original Mottinsulator. Meanwhile, the Pb acquires a magnetic mo-ment 0 . µ B polarized downwards [see Fig. 1(b)], whichmatches the overall AFM order of LaCrO and splits thespin-up and spin-down p orbitals of Pb [see Fig. 2(a)].In this way, both the topmost valence band and the bot-tommost conduction band are occupied by the spin-upchannel. We notice that the total magnetic moment inthe present system is compensated to zero, distinct fromthe Cr-doped Bi Se .The band structure of the material is then calculatedwith SOC turned on, which lifts the degeneracy of the p + and p − bands in both spin channels. Remarkably,the strong SOC of the heavy element Pb pushes the p + orbital with up spin even above the Fermi energy E F around the Γ point as displayed in Figs. 2(b) and (c).The Cr1- d z orbital with up spin then has to sink acrossthe Fermi level partially in order to maintain the chargeneutrality of the system, which causes a band inversionbetween the p and d orbitals around the Γ point, as shownin Fig. 2(b). An energy gap of 59 meV is observed ac-cording to the first-principles calculations. III. EFFECTIVE LOW-ENERGY MODEL
We now derive an effective low-energy k · P Hamil-tonian to describe topological properties of the system.Noticing that the topological band gap is opened by hy-bridizations between the spin-up p + orbital of Pb andthe spin-up d z orbital of Cr1, it is then reasonable totake these two orbitals as a basis to construct a 2 × = d ↑ z and Γ = p ↑ + . The effective k · P Hamiltonianaround the Γ point is H ( k ) = H + H ′ (1)on the basis [Γ , Γ ], where H = (cid:18) ǫ + γ k ǫ + γ k (cid:19) (2)and H ′ = k · P = ( k − P + + k + P − ) / k ± = k x ± ik y and P ± = P x ± iP y ( P x/y is the momentum operator in the x/y direction).Since the crystal is symmetric with respect to the C rotation around the ~c axis, H ′ must be invariant underthe C = e − i π J z transformation, where J z the is the z -component of the total angular momentum. The sym-metry constraint allows us to determine nonzero entriesof H ′ . It is easy to check that C Γ = e − i π Γ and C Γ = e − i π Γ because J z = 1 / / andΓ respectively. Since h Γ | P + | Γ i = h Γ | C † C P + C † C | Γ i = h Γ | e i π e − i π P + e − i π | Γ i = − h Γ | P + | Γ i , (3) h Γ | k − P + | Γ i must vanish. Performing similar calcula-tions for all other terms, we arrive at the Hamiltonianrespecting the crystal symmetry H ( k ) = (cid:0) ǫ + γ k (cid:1) I × + (cid:18) ǫ + γ k αk + α ∗ k − − ǫ − γ k (cid:19) (4)up to the lowest orders of k , with ǫ = ( ǫ + ǫ ) / ǫ = ( ǫ − ǫ ) / γ = ( γ + γ ) / γ = ( γ − γ ) /
2. Byfitting the energy dispersion of H ( k ) in Eq. (4) againstthe first-principles results given in Fig. 2(b), we obtainthe parameters as follows: ǫ = − .
007 eV, γ = − . · ˚A , ǫ = − .
031 eV, γ = 9 . · ˚A and α = 1 . · ˚A (see Fig. 3). Since ǫ and γ take opposite signs,the electronic wavefunction of the spin-up channel be-comes topologically nontrivial due to the band inversionmechanism with Chern number C ↑ = 1. Since the spin-down electronic bands are kept far away from the Fermilevel [see Figs. 2(b) and (c)], one clearly has C ↓ = 0.It is therefore confirmed that the system is character-ized by simultaneous charge and spin Chern numbers: C c = C ↑ + C ↓ = 1 and C s = C ↑ − C ↓ = 1. FIG. 4. (Color online) Band structure for a slab of the systemshown in Fig. 1 based on Wannierized wavefunctions down-folded from the results of the first-principles calculations inFig. 2(b). Red curves are for topological edge states in thespin-up channel, and grey ones are for bulk states.
IV. TOPOLOGICAL EDGE STATES
The nontrivial topology gives rise to gapless edge statesin a finite sample. To illustrate this feature, we calculatethe dispersion relation for a slab of the topological mate-rial with 100 a along the ~a axis and infinite along the ~b axis(see Fig. 1). Since the bulk bands close to the Fermi levelare mainly contributed by the Pb- p x , Pb- p y and Cr1- d z orbitals, it is reasonable to downfold the wavefunctionsobtained by the first-principles calculations in Fig. 2(b)onto these three orbitals. Employing the maximally-localized Wannier functions , we obtain the hopping in-tegrals within the six-dimensional subspace including thespin degree of freedom. It is then straightforward to cal-culate the band structure of the slab system. As shownin Fig. 4, a gapless edge state with up spin appears insidethe bulk gap, manifesting the nontrivial topology of thepresent system. V. DISCUSSION
Liu proposed a 3D spinless model for a layered squarelattice with A-type AFM (intra-plane ferromagnetic andinter-plane AFM orderings). At each lattice site, thereare three orbitals: s , p x and p y . Each layer can bedriven into a QAHE in a same way as that for the Mn-doped HgTe . Since every two adjacent layers have op-posite magnetic moments, their chiral edge states prop-agate counter to each other. Therefore, the system canbe viewed as a stack of quantum spin Hall insulators[see also ], where the combination of the time-reversaland the primitive-lattice translational symmetries is pre-served. In contrast, all symmetries are broken in oursystem, giving rise to a Chern insulator. VI. CONCLUSION
We propose a novel topological material composed ofthe LaCrO of perovskite structure grown along the [001]direction with one atomic layer replaced by an inverse-perovskite material Sr PbO. Based on first-principles cal-culations and an effective low-energy Hamiltonian, wedemonstrate that the topological state is characterizedby simultaneous nonzero charge and spin Chern num-bers, which can support a spin-polarized and dissipa-tionless edge current in a finite sample. Supported bythe anti-ferromagnetic exchange field and spin-orbit cou-pling inherent in the compounds, no extrinsic operationis required for achieving the novel topological state. Im-portantly, these two materials are stable in bulk andmatch each other with only small lattice distortions,which makes the composite material easy to synthesize.
ACKNOWLEDGMENTS
This work was supported by the WPI Initiative on Ma-terials Nanoarchitectonics, Ministry of Education, Cul-ture, Sports, Science and Technology of Japan. QFL ac-knowledges support from the National Natural ScienceFoundation of China (No.11574215) and the ScientificResearch Foundation for the Returned Overseas ChineseScholars, State Education Ministry. ∗ [email protected] † [email protected] K. v. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. , 494 (1980). D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M.den Nijs, Phys. Rev. Lett. , 405 (1982). C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. DasSarma, Rev. Mod. Phys. , 1083 (2008). T. D. Stanescu and S. Tewari, J. Phys.: Condens. Matter , 233201 (2013). C. W. J. Beenakker, Annu. Rev. Condens. Matter Phys. , 113 (2013). L.-H. Wu, Q.-F. Liang, and X. Hu, Sci. Technol. Adv.Mater. , 064402 (2014). T. Kawakami and X. Hu, Phys. Rev. Lett. , 177001(2015). O. V. Yazyev, J. E. Moore, and S. G. Louie, Phys. Rev.Lett. , 266806 (2010). D. Pesin and A. H. MacDonald, Nat. Mater. , 409(2012). F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. ,013904 (2008). A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian,A. H. MacDonald, and G. Shvets, Nat. Mater. , 233(2013). L.-H. Wu and X. Hu, Phys. Rev. Lett. , 223901 (2015). C. L. Kane and E. J. Mele, Phys. Rev. Lett. , 226801(2005). C. L. Kane and E. J. Mele, Phys. Rev. Lett. , 146802(2005). M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. , 3045(2010). B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science , 1757 (2006). X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. , 1057(2011). M. K¨onig, S. Wiedmann, C. Br¨une, A. Roth, H. Buhmann,L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Science ,766 (2007). D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava,and M. Z. Hasan, Nature , 970 (2008). H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C.Zhang, Nat. Phys. , 438 (2009). Y. Ando, J. Phys. Soc. Jpn. , 102001 (2013). F. D. M. Haldane, Phys. Rev. Lett. , 2015 (1988). H.-M. Weng, R. Yu, X. Hu, X. Dai, and Z. Fang, Adv.Phys. , 227 (2015). E. Prodan, Phys. Rev. B , 125327 (2009). Y. Yang, Z. Xu, L. Sheng, B. Wang, D. Y. Xing, and D.N. Sheng, Phys. Rev. Lett. , 066602 (2011). R. Yu, W. Zhang, H.-J. Zhang, S.-C. Zhang, X. Dai, andZ. Fang, Science , 61 (2010). C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M.Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, Z.-Q. Ji, Y. Feng,S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He,Y. Wang, L. Lu, X.-C. Ma, and Q.-K. Xue, Science ,167 (2013). C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C. Zhang,Phys. Rev. Lett. , 146802 (2008). M. Ezawa, Phys. Rev. B , 155415 (2013). Q.-F. Liang, L.-H. Wu, and X. Hu, New J. Phys. ,063031 (2013) A. Widera and H. Sch¨afer, Mater. Res. Bull. , 1805(1980). T. Kariyado and M. Ogata, J. Phys. Soc. Jpn. , 064701(2012). A. A. Mostofi, J. R. Yates, Y.-S. Lee, I. Souza, D. Vander-bilt, and N. Marzari, Comput. Phys. Commun. , 685(2008). M. Klintenberg, arXiv:1007.4838. T. H. Hsieh, J. Liu, and L. Fu, Phys. Rev. B , 081112(R)(2014). G. Kresse and J. Furthm¨uller, Phys. Rev. B , 11169(1996). P. E. Bl¨ochl, Phys. Rev. B , 17953 (1994). G. Kresse and D. Joubert, Phys. Rev. B , 1758 (1999). J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.Lett. , 3865 (1996). Z. Yang, Z. Huang, L. Ye, and X. Xie, Phys. Rev. B ,15674 (1999). R. Yu, H.-M. Weng, X. Hu, Z. Fang, and X. Dai, New J.Phys. , 023012 (2015). C.-X. Liu, arXiv:1304.6455. R. S. K. Mong, A. M. Essin, and J. E. Moore, Phys. Rev.B81