Quantum Anomalous Hall Effect in Magnetic Topological Insulator GdBiTe3
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A ug Quantum Anomalous Hall Effect in Magnetic TopologicalInsulator GdBiTe Hai-Jun Zhang, Xiao Zhang & Shou-Cheng Zhang Department ofPhysics,McCulloughBuilding,Stanford University,Stanford,CA 94305-4045
The quantum anomalous Hall (QAH) state is a two-dimensional bulk insulator with a non-zero Chern number in absence of external magnetic fields. Protected gapless chiral edgestates enable dissipationless current transport in electronic devices. Doping topological in-sulators with random magnetic impurities could realize the QAH state, but magnetic orderis difficult to establish experimentally in the bulk insulating limit. Here we predict that thesingle quintuple layer of GdBiTe film could be a stoichiometric QAH insulator based on ab-initio calculations, which explicitly demonstrate ferromagnetic order and chiral edge statesinside the bulk gap. We further investigate the topological quantum phase transition by tun-ing the lattice constant and interactions. A simple low-energy effective model is presented tocapture the salient physical feature of this topological material. Recently time-reversal-invariant topological insulators(TIs) have attracted broad attention infields of the condensed matter physics, material science and electrical engineering . Quantumspin Hall(QSH) insulators
5, 6 are two dimensions(2D) TIs with a bulk energy gap and gapless he-lical edge states which are protected by the Z topological invariant . Both 2D and 3D TIs arenatural platforms to realize the QAH when the time-reversal symmetry is spontaneously brokenby the ferromagnetic order . The QAH state is the quantum version of the intrinsic anomalous1all effect , and is characterized by a bulk 2D Chern number and gapless chiral edge states with-out the external magnetic field
8, 13, 14 , and is sometimes also referred to as the 2D Chern insulator.The QSH state has two counter-propagating edge states; doping the 2D TIs with magnetic impu-rities could break the time reversal symmetry, annihilate one edge state and leave the other intact,thus realizing the QAH state with chiral edge state. Alternatively, doping the surface of 3D TIswith magnetic impurities breaks time reversal symmetry, and opens up a gap for the Dirac surfacestates
11, 15, 16 . Chiral edge states exist on the magnetic domain walls on the surface . In the QAHstate, electrons move like cars on a highway, where oppositely moving traffics are spatially sepa-rated into opposite lanes . Realizing such a dissipationless transport mechanism without extremecondition could greatly improve the performance of electronic devices.Normally, ferromagnetic order among magnetic impurities in randomly doped semiconduc-tors are mediated by the free carriers , it is difficult to establish magnetic order in the bulk insu-lating limit. For this reason, we search for stoichiometric ferromagnetic insulators with a nonzeroChern number for the 2D band structure. Rare earth elements have partially filled f electrons, inparticular, Gd has exact half filled f electrons. It is possible to realize a stoichiometric QAHinsulator by replacing Bi by Gd in the Bi Te family of TIs . GdBiTe has a stable structurephase, synthesized thirty years ago . In this work, we predict that stoichiometric QAH insulatorscould be realized in the single quintuple layer(SQL) of GdBiTe , based on ab-initio calculations. Crystal structure and band structure
GdBiTe has a rhombohedral crystal structure with the space group R m . Its crystal consists2f close-packed layers stacked along [111] direction with the A-B-C · · · order. The quintuple layer(QL) (Te-X-Te-X-Te) structure is the basic crystal unit, where X presents Bi and Gd with a certainpattern, similar to the case of LaBiTe . It has strong coupling within one QL and weak van derWaals coupling between neighbor QLs, so this material can be grown as two-dimensional (2D)thin film along [111] direction. In addition, though its lattice type and lattice constant have beenmeasured experimentally , positions of Bi and Gd have not been clearly resolved yet. Since Biand Gd have different electronegativity, it is possible to grow the non-mixing structure as Te-Gd-Te-Bi-Te with Molecular beam epitaxy (MBE) technique. This crystal structure of GdBiTe3 isshown in Fig. 1a. It holds the three-fold rotation symmetry C with z axis as the trigonal axisand the reflection symmetry with x axis as its normal axis. Compared with Bi Te , the inversionsymmetry is broken for this structure. Since SQL GdBiTe film is the simplest system to studythe QAH effect, in this work we focus on its SQL structure with Te1-Bi-Te2-Gd-Te1 ′ , where Te1and Te1 ′ are marked for the Te closest to Bi and Gd layer separately, and Te2 denotes the Te in thecenter of this SQL.All ab-initio calculations are carried out in the framework of density functional theory (DFT)with Perdew-Burke-Ernzerbof-type generalized-gradient approximation . Both BSTATE package with plane-wave pseudo-potential method and the Vienna ab initio simulation package (VASP)with the projected augmented wave method are employed. The k -point grid is taken as × × ,and the kinetic energy cutoff is fixed to 340eV in all self-consistent calculations. A free standingslab model is employed with SQL. Its lattice constant ( a = 4 . ˚ A ) is taken from experiments ,and the inner atomic positions are obtained through the ionic relaxation with the force cutoff3.001eV/ ˚A. The spin-orbit coupling (SOC) is taken into account, because of its importance torealize the QAH effect. GGA+U method is also employed to study the strong correlation effectin GdBiTe because of the existence of narrow occupied f bands of Gd. We find that the ferromag-netic phase is more stable than the non-magnetic and collinear anti-ferromagnetic phases for SQLGdBiTe . In addition, the ferromagnetic phase with magnetic moment along [111] direction haslower energy than that with the magnetic moment along [010] and [110] directions. All GdBiTe calculations are carried out here with the ferromagnetic moments along [111] direction.The ferromagnetic phase of SGL GdBiTe is an insulator state with exact magnetic moment S = 7 / . Because of the lattice similarity between GdBiTe and GdN(or EuO) with [111] direc-tion, the known magnetic exchange mechanisms in GdN or EuO could provide possible explana-tion for the ferromagnetic state in SQL GdBiTe . The first mechanism is a third-order perturbationprocess . A virtual excitation, which takes a 4 f to a 5 d state, leads to a f-f interaction through the d-f exchange due to the wave-function overlap between neighboring rare-earth atoms. Recently,Mitra and Lambrecht , based on ab-initio calculations, presented another magnetic mechanismfor the ferromagnetic ground state in GdN. The anti-ferromagnetic ordering, between N p and Gd d small magnetic moments, stabilizes the ferromagnetic structure between nearest neighbor Gdatoms due to the d-f exchange interaction. In addition, our calculations also indicate the similaranti-ferromagnetic ordering between Te p and Gd d small magnetic moments.Bi Te , LaBiTe and GdBiTe have very similar SQL structure. First of all, Bi Te have boththe inversion symmetry and the time-reversal symmetry. However, inversion symmetry is broken4n LaBiTe , and both inversion symmetry and time-reversal symmetry are broken in GdBiTe .The band evolution from Bi Te to LaBiTe , and finally to GdBiTe is shown in Fig. 1c-e. Thebands of Bi Te have double degeneracy because of both inversion and time-reversal symmetries.Its energy gap is calculated to be about . eV , consistent with the experiments . The bottomof the conduction bands at Γ point with a “V” shape originates from the p x,y orbitals of Bi, Te1and Te1 ′ . The top of the valence band at the Γ point, which lies below valence bands at othermomenta, originates from the p x,y orbitals of Te2. For LaBiTe , the double degeneracy of thebands is lifted except at the time-reversal points, due to the lack of the inversion symmetry. Thoughthe bottom of conduction bands still shows the “V” shape, the top of the valence bands moves to Γ , which mainly originates from the p x,y orbitals of Te2 and Te1 ′ . The schematic evolution of thetop of valence bands and the bottom of conduction bands from LaBiTe to GdBiTe is shown inFig. 2a-b. The Kramers double degeneracy at the Γ point in GdBiTe is broken due to the lackof time-reversal symmetry. The band inversion occurs between the | p x + i p y , Bi + Te1 , ↓i and | p x + i p y , Te1 ′ + Te2 , ↑i due to the strong SOC and the large magnetic moment. The existence ofthis band inversion is the key point to realize the QAH effect in this SQL GdBiTe system.Fig. 3a shows that the energy gap is quite sensitive with the lattice constant. Smaller latticeconstant a increases the band inversion. Conversely, increasing the lattice constant a would de-stroy the band inversion. The QAH effect exists for a < . a , where a is the experimentallydetermined bulk lattice constant. Because the SQL GdBiTe is very thin, its lattice constant couldbe controlled by the substrate, giving us a broad tunability range. In addition, due to the narrow f bands, the strong correlation needs to be checked by GGA+U calculations. First of all, the band5tructure calculations show that the occupied f bands are located at − eV below Fermi level (FL),and the unoccupied f bands are located at . eV above FL. Since both occupied and unoccupied f bands are quite far from FL, correlation effects should not influence bands very close to the Fermilevel. Our calculations indicate that this conclusion is true. The dependence of the energy gap on U(from eV to eV ) is shown in Fig. 3b. The correlation U lifts up the top spin-down valence band,and pulls down the bottom spin-up valence band. Therefore, the strong correlation U likes to pullGdBiTe from the topologically non-trivial QAH phase to the topologically trivial ferromagneticinsulator phase. But the energy gap is much more sensitive with the lattice constant than with thecorrelation U. We have shown that the SQL GdBiTe is close to the quantum critical point betweenthe QAH state and topologically trivial magnetic insulator state, and it is possible to realize theQAH state by tuning the lattice constant with proper substrates. Also the 2D Dirac-type band dis-persion shows up at the critical point, shown in Fig. 3c. This is very similar to the case of QSHstate at the critical point . Topological chiral edge states
The existence of topologically protected chiral edge states is the direct and intrinsic evidenceof the QAH phase. It is important to show the explicit features of these topologically protected chi-ral edge states. In order to calculate the edge states, we employ the tight-binding method based onmaximally localized Wannier functions (MLWF), developed by Vanderbilt and his co-workers ,in the framework of ab-initio calculations. The edge with Te1 ′ and Bi terminated along the [11]direction is chosen to show the edge states, shown in the inset of Fig. 4b. Because the topological6ature is completely determined by its bulk electronic structure, here we ignore the edge recon-struction of the atoms on the edge, and also make another approximation that the MLWF hoppingparameters close to the edge are the same to ones of the bulk. The bulk MLWF hopping param-eters are obtained from the ab-initio calculations of the SQL free-standing GdBiTe slab. Similarto the method described in our previous work , we use an iterative method to obtain the edgeGreen’s function of the semi-infinite system with an edge along [11] direction. The local densityof states (LDOS) can be calculated with the imaginary part of these edge Green’s functions, shownin Fig. 4a,b. The LDOS in Fig. 4a shows two edge states for topologically non-trivial GdBiTe with the experimental lattice constant a . One topologically trivial edge state only stays withinthe valence bands, while the topologically non-trivial edge state ties the conduction band withthe valence band. This topologically non-trivial edge state is chiral, and its Fermi velocity in theenergy gap is v F ≃ . × m/s (or . eV · Bohr ). The edge states of topologically trivialferromagnetic GdBiTe with the lattice constant a = 1 . a are shown in Fig. 4b. Both of thesetwo topologically trivial edge states only stay within the valence bands, and do not connect thevalence and conduction bands. Low-energy effective model
As the topological nature is determined by the physics near the Γ point for this material,it is possible to write down a four-band effective Hamiltonian to characterize the low-energylong-wavelength properties of the system. Starting from the four low-lying states | B − , − ↑i = c | p x − ip y , T e , ↑i + c | p x − ip y , Bi, ↑i , | B + , ↓i = c | p x + ip y , T e , ↓i + c | p x + ip y , Bi, ↓i ,7 T + , ↑i = d | p x + ip y , T e , ↑i + d | p x + ip y , T e ′ , ↑i and | T − , − ↓i = d | p x − ip y , T e , ↓i + d | p x + ip y , T e ′ , ↑i at the Γ point, where c , c , d and d are constants and ± ( ) are the an-gular momenta in z direction. In addition, the “ ± ” subscripts are used to present the z componentof orbital angular momentum l z = ± . Such a Hamiltonian can be constructed by the theory ofinvariants for the finite wave vector k . On the basis of the symmetries of the system, the genericform of the × effective Hamiltonian can be constructed up to the order of O ( k ) , and the pa-rameters of the Hamiltonian can be obtained by fitting to ab initio calculations. First we considerthe Te-La-Te-Bi-Te system without magnetic order. The important symmetries of this system are(1) time-reversal symmetry T , (2) reflection symmetry σ x with x axis as the normal axis, and (3)three-fold rotation symmetry C along the z axis. In the basis of | B + , ↓i , | T + , ↑i , | B − , − ↑i and | T − , − ↓i , the representation of the symmetry operations is given by T = K · iσ y ⊗ τ z , σ x = − iσ x ⊗ I and C = exp ( − i πJ z ) , where K is the complex conjugation operator, σ x,y,z and τ x,y,z denote the Pauli matrices in the spin and orbital space, respectively and J z is the angularmomentum in z direction. For low energy effective model, we ignore the in-plane anisotropy andimpose continuous rotational symmetry R ( θ ) = exp ( − iθJ z ) . By requiring these three symmetries,we obtain the following generic form of the effective Hamiltonian: H = M ( k ) Ak + − i ∆ e k − A ′ k − Ak − −M ( k ) A ′ k − i ∆ e k + A ′ k M ( k ) Ak − A ′ k Ak + −M ( k ) + ǫ ( k ) (1)with k ± = k x ± ik y , ǫ ( k ) = C + D ( k x + k y ) , M ( k ) = M + B ( k x + k y ) . The block diagonalpart is just the BHZ model , and the off-block diagonal ± i ∆ e k ± , A ′ k ± terms breaks the inversion8ymmetry. Next we add the ferromagnetic exchange term which breaks time reversal symmetry, H ex = − g B M g T M g B M − g T M (2)with g B and g T as Land ´ e - g factors for two different orbitals and M the exchange field introducedby the FM ordering of Gd with half-occupied f electrons. Then the full Hamiltonian can be writtenas H = H + H ex . By fitting the energy spectrum of the effective Hamiltonian with that of the ab initio calculations, the parameters in the effective model can be determined. For GdBiTe , ourfitting leads to M = 27 meV , A = 2 . eV ˚ A , B = 7 . eV ˚ A , C = − meV , D = 4 . eV ˚ A , A ′ ≈ , ∆ e = 1 . eV ˚ A , g B M = 18 meV and g T M = 59 meV , which agree well with the ab initio results. Such an effective model can be used for illustration of the formation of the GdBiTe QAHsystem. As shown in Fig. 2c-e, when we have no FM ordering, the fact that M , B > suggestsit is the topologically trivial insulator as SQL LaBiTe , and the lack of inversion symmetry splitsthe double degeneracy except at Γ point. As we adiabatically increase the exchange field M , the | B + , ↓i and | T + , ↑i states start to move towards each other. At the transition point, thosetwo bands form a 2D Dirac cone, and as we further increase the exchange field, those two bandsbecome inverted and the Ak ± term hybridizes them and creates a band gap again. In this process,a topological phase transition of SQL GdBiTe is shown clearly from a topologically trivial phaseto the QAH phase.Our theoretical calculations show that a SQL of GdBiTe is a 2D stoichiometric magnetic9opological insulator, realizing the long-sought-after QAH state. This topological material can begrown by MBE, or by exfoliating the bulk crystal. In addition, YBiTe with the same structure isfound to be a normal insulator , and could serve as the best substrate for MBE growth. Experi-mentally, the best way to see the QAH effect is to measure the four-terminal Hall conductance asa function of gate voltage. A quantized plateau in Hall conductance should be observed when thechemical potential is inside the gap. These intrinsic QAH materials could be used for applicationswith dissipationless electronic transport. 10 d eba Quintuplelayer d e layer
TeBiGd
Figure 1: | Crystal Structure, Brillouin zone and band structure. a , Bulk crystal structure ofGdBiTe . A quintuple layer(QL) with T e − Bi − T e − Gd − T e ′ is indicated by the red box. b , Brillouin zone for GdBiTe with space group R m , which has four inequivalent time-reversal-invariant points Γ(0 , , , L ( π, , , F ( π, π, and Z ( π, π, π ) . The projected(111) 2D Brillouinzone is marked by the blue hexagon with its high-symmetry k points Γ , K and M . c, d, e , Bandstructure with spin-orbit coupling(SOC) for (c) single QL Bi Te , (d) single QL LaBiTe and (e) single QL GdBiTe . The Fermi level is fixed at eV .11 (Px-iPy),(Bi+Te1) >|(Px+iPy),(Bi+Te1) >|(Px+iPy),(Te1’+Te2) >|(Px-iPy),(Te1’+Te2) > LaBiTe |(Px-iPy),(Bi+Te1) >|(Px+iPy),(Te1’+Te2) > GdBiTe |(Px+iPy),(Bi+Te1) >|(Px-iPy),(Te1’+Te2) > a b Fermi level c d e
LaBiTe GdBiTe c d e Figure 2: | Schematic representation of the topological phase transition. a,b , The phasetransition from (a) the topologically trivial insulator phase of LaBiTe to (b) the topologicallynon-trivial QAH phase of GdBiTe . Because of the Kramers degeneracy, | p x − ip y , Bi + T e , ↑i and | p x + ip y , Bi + T e , ↓i states at the bottom of the conduction band, as well as states | p x + ip y , T e ′ + T e , ↑i and | p x − ip y , T e ′ + T e , ↓i at the top of the valence band are dou-ble degenerated for LaBiTe . The time-reversal symmetry is broken for GdBiTe because ofthe ferromagnetism of Gd with the half filled f bands. The Kramers degeneracy at Γ is re-moved. Due to the large SOC and the ferromagnetic moment, the band inversion occurs between | p x + ip y , Bi + T e , ↓i and | p x + ip y , T e ′ + T e , ↑i . c,d,e , Phase transition based on the Low-energy effective model. The Kramers degeneracy at Γ for LaBiTe is shown in the band structure (c) with the time-reversal symmetry, but the Kramers degeneracy for the band structure of (e) GdBiTe is broken due to the lack of the time-reversal symmetry. The gapless Dirac-type disper-sion is shown in (d) the band structure at the phase transition point.12 b ac Figure 3: | Phase diagram. a , The topological phase diagram depending on the lattice constant. a is GdBiTe ’s experimental bulk lattice constant. The system is in topologically non-trivial QAHphase with the small lattice constant( a < . a ) marked by yellow, and it becomes topologicallytrivial ferromagnetic insulator with the large lattice constant( a > . a ) marked by blue. b , Thetopological phase diagram depending on the correlation U with fixed experimental lattice constant a . The GdBiTe is in QAH phase with small U( < . eV ), and it changes to the topologicallytrivial ferromagnetic insulator phase with large U( > . eV ). c , The 2D Dirac-type dispersion ofthe lowest conduction band and the highest valence band at the transition point.13igure 4: | Edge states a,b , Energy and momentum dependence of the LDOS for GdBiTe with (a) QAH phase, and with (b) topologically trivial ferromagnetic insulator phase. The red regionsindicate the 2D bulk bands and the dark blue regions indicate the bulk energy gap. The edge stateare clearly shown in the bulk energy gap. The edge states in (a) , which connect the bulk conductionbands and the bulk valence bands, are chiral, and the detailed dispersions around Γ are zoomed inin the inset. Comparing with the case of QAH, (b) the topologically trivial ferromagnetic insulatorphase has no chiral edge states. The edge of the single QL is taken to be along x direction, whichis the normal axis of the reflection symmetric plane, shown in the inset of (b) .14. Qi, X.-L. & Zhang, S.-C. The quantum spin Hall effect and topological insulators. PhysicsToday , 33 (2010).2. Hasan, M. Z. & Kane, C. L. Colloquium: Topological insulators. Rev. Mod. Phys. , 3045–3067 (2010).3. Moore, J. E. The birth of topological insulators. Nature , 194–198 (2010).4. Qi, X. L. & Zhang, S. C. Topological insulators and superconductors. arXiv: 1008.2026v1 (2010).5. Bernevig,B. A. , Hughes, T. L. & Zhang, S. C. Quantum spin Hall effect and topologicalphase transition in HgTe quantum wells.
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We are indebted to B.H. Yan at University of Bremen, X.L. Qi and Q.F. Zhang atStanford University for their great help. We would like to thank Y.L. Chen and D.S. Kong for their usefuldiscussion. This work is supported by the Army Research Office (No.W911NF-09-1-0508) and the KeckFoundation.
Competing Interests
The authors declare that they have no competing financial interests.
Correspondence