Ab initio study of Z 2 topological phases in perovskite (111) ( SrTiO 3 ) 7 /( SrIrO 3 ) 2 and ( KTaO 3 ) 7 /( KPtO 3 ) 2 multilayers
AAb initio study of Z topological phases in perovskite (111) ( SrTiO ) / ( SrIrO ) and ( KTaO ) / ( KPtO ) multilayers J. L. Lado,
1, 2, ∗ V. Pardo,
1, 2, † and D. Baldomir
1, 2 Departamento de F´ısica Aplicada, Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Spain Instituto de Investigaci´ons Tecnol´oxicas, Universidade deSantiago de Compostela, E-15782 Santiago de Compostela, Spain (Dated: October 31, 2018)Honeycomb structures formed by the growth of perovskite 5d transition metal oxide heterostruc-tures along the (111) direction in t g configuration can give rise to topological ground statescharacterized by a topological index ν =1, as found in Nature Commun. 2, 596 (2011). Us-ing a combination of a tight binding model and ab initio calculations we study the multilayers(SrTiO ) / (SrIrO ) and (KTaO ) / (KPtO ) as a function of parity asymmetry, on-site interactionand uniaxial strain and determine the nature and evolution of the gap. According to our DFT cal-culations, (SrTiO ) / (SrIrO ) is found to be a topological semimetal whereas (KTaO ) / (KPtO ) is found to present a topological insulating phase that can be understood as the high U limit of theprevious one, that can be driven to a trivial insulating phase by a perpendicular external electricfield. I. INTRODUCTION
Topological insulators (TI) are a type of mate-rials which show a gapped bulk spectrum but gap-less surface states. The topological nature of thesurface states protects them against perturbationsand backscattering.
In addition, the surface of ad-dimensional TI is such that the effective Hamil-tonian defined on its surface cannot be representedby the Hamiltonian of a d-1 dimensional materialwith the same symmetries, so the physics of a (d-1)-surface of a topological insulator may show com-pletely different behavior from that of a conventional(d-1)-dimensional material. Surface states can beunderstood in terms of solitonic states which inter-polate between two topologically different vacuums,the topological vacuum of the TI and the trivial vac-uum of a conventional insulator or empty space.The ground state of a system can be classi-fied by a certain topological number depend-ing on its dimensionality and symmetries presentwhich define its topological classification. Two-dimensional single-particle Hamiltonians with timereversal (TR) invariance are classified in a Z ( ν =0 ,
1) topological class. Two-dimensional TR sys-tems with nontrivial topological index ( ν = 1)show the so-called quantum spin Hall effect (QSHE)which is characterized by a non-vanishing spin Chernnumber and a helical edge current. This state hasbeen theoretically predicted and experimentally con- firmed in HgTe quantum wells, as well as pre-dicted in several materials such as two-dimensionalSi and Ge and transition metal oxide (TMO)heterostructures. All these systems have in com-mon that they present a honeycomb lattice structurewith two atoms (A,B) as atomic basis. In that situa-tion, it is tempting to think that the effective Hamil-tonian in certain k-points could have the form of aDirac Hamiltonian. The components of the spinorwould be some combination of localized orbitals inthe A or B atoms, whereas the coupling would takeplace via non-diagonal elements due to the bipar-tite geometry of the lattice. The simplest and bestknown example is graphene, where the Hamiltonianis a Dirac equation in two nonequivalent points Kand K’. At half filling, graphene with TR and inver-sion symmetry (IS) has ν = 1, so a term whichdoes not break those symmetries and opens a gap inthe whole Brillouin zone would give rise to the QSHEstate. The way in which an IS term can arise in agraphene Hamiltonian is due to spin-orbit coupling(SOC), however it is known that the gap opened thisway is too small. In contrast, a sublattice asym-metry, which breaks IS, will open a trivial gap, as inBN.
How a honeycomb structure can be constructedfrom a perovskite unit cell can be seen in Fig. 1aand 1b, a perovskite bilayer grown along the (111)direction made of an open-shell oxide is sandwichedby an isostructural band insulating oxide. The metalatoms of the bilayer form a buckled honeycomb a r X i v : . [ c ond - m a t . s t r- e l ] S e p attice. It has been shown that perovskite (andalso pyrochlore) (111) multilayers can develop topo-logical phases, as well as spin-liquid phasesand non-trivial superconducting states. Topolog-ical insulating phases have been predicted for var-ious fillings of the d shell, here we will focuson the large SOC limit (5d electrons) and formal d filling. We will study two different multilayers,(SrTiO ) / (SrIrO ) and (KTaO ) / (KPtO ) andwe will focus on the realization of a nontrivial ν = 1ground state. SrIrO ( a SrIrO = 3 .
94 ˚A) is a cor-related metal whose lattice match with SrTiO (STO) would be close enough ( a SrTiO = 3 .
905 ˚A) for them to grow epitaxially with standard growthtechniques. KPtO has not been synthesized (tothe best of our knowledge) but our calculations( a KPtO = 4 .
02 ˚A) show a reasonable lattice matchwith KTaO would be possible ( a KTaO = 3 .
98 ˚A). The first multilayer is an iridate very similar to thewell known Na IrO . This system presents alayered honeycomb lattice of Ir atoms at t g filling,whereas the present bilayers show a buckled honey-comb lattice, and is predicted to develop the QSHE,however electron correlation would lead to an anti-ferromagnetic order in the edges. We will study thedependence of the topological ground state on theapplied uniaxial strain and the electron-electron in-teraction and we will determine a transition betweentwo topological phases in both materials. The workis organized as follows. In Section II we introducea simple tight binding (TB) model as in Ref. 20 fo-cusing on the t g case. In Section III we use densityfunctional theory (DFT) calculations to study theevolution of both multilayers with uniaxial strainand on-site Coulomb repulsion and we determinethe ground state of each material. In Section IV westudy the stability of the topological phase againstTR and IS breaking using both TB and DFT cal-culations. Finally in Section V we summarize theresults obtained. II. TIGHT BINDING MODEL
The qualitative behavior of this system can be un-derstood using a simple TB model for the 5d elec-trons in the TM atoms, as shown in Ref. 20. InSection IIA we will give the qualitative behavior ofthe effective Hamiltonian. In Section IIB we willshow numerical calculations of the full model.
FIG. 1: (Color online) (a) Scheme of the cubic perovskitestructure
XY O . (b) Construction of the bilayer, theTM atoms are arranged in a triangular A and B latticein the (111) direction in such a way the two atoms willform a honeycomb lattice. The Z atom corresponds tothe insulating layer (in our case SrTiO or KTaO ) anddoes not participate in the honeycomb. (c) Scheme ofthe multilayer considered in the DFT calculations. A. Full Hamiltonian
The octahedral environment of oxygen atoms sur-rounding the transition metal atoms decouples the dlevels in a t g sextuplet and an e g quadruplet. Giventhat the crystal field gap is higher than the otherparameters considered we will retain only the t g or-bitals. The Hamiltonian considered for the t g levelstakes the form H = H SO + H t + H tri + H m (1) H SO is the SOC term, which gives rise to an effec-tive angular momentum J eff = S − L which decou-ples the t g levels into a filled j=3/2 quadruplet anda half filled j = 1 / H t is the hopping be-tween neighboring atoms via oxygen that couples thelocal orbitals. H tri is a local trigonal term which isresponsible for opening a gap (as we will see below)without breaking TR and IS. H m is a term whichbreaks IS making the two sublattices nonequivalenttending to open a trivial gap by decoupling them.In the following discussion we will suppose that this2ast term is zero, but we will analyze its role in Sec-tion IV.We are interested in two different regimes as afunction of SOC strength: strong and intermediate.We call strong SOC to the regime where the j=3/2and j=1/2 are completely decoupled so that thereis a trivial gap between them. We will refer to anintermediate regime if the two subsets are coupledby the hopping. The key point to understand thetopological character of the calculations is that a t g configuration can be adiabatically connected fromthe strong to the intermediate regime without clos-ing the gap. The argument is the following, begin-ning in the strong SOC regime it is expected thata four-band effective model will be a good approxi-mation. In this regime the mathematical structureof the effective Hamiltonian turns out to be equiv-alent to graphene. The trigonal term is responsiblefor opening a gap ∆ via a third order process inperturbation theory∆ ∼ λ tri t α (2)where λ tri is the trigonal coupling, t is the hop-ping parameter and α the SOC strength. It canbe checked by symmetry considerations that the re-striction of the matrix representations leads to thisterm as the first non-vanishing contribution in per-turbation theory. Eq. (2) has been checked by alogarithmic fitting of numerical calculations of thefull model. This term will open a gap in the K pointconserving TR and IS and thus realizing a ν = 1ground state. As SOC decreases, the gap becomes larger whilethe system evolves from the strong to the inter-mediate regime, so the intermediate regime is ex-pected to be a topological configuration with anon-vanishing gap. Note that even though perturba-tion theory will only hold in the strong SOC regime,the increase in the gap as the system goes to theintermediate regime suggests that the t g configura-tion will always remain gapped. This argument ischecked by the numerical calculations shown below. B. Results from tight binding calculations
In Fig. 2 we show the results of a calculation usingthe TB Hamiltonian proposed above. Figure 2a is
FIG. 2: (Color online) (a) Band structure of the TBmodel with intermediate α = t (left) and strong α = 2 t (right) SOC strength and λ tri = − . t . The differencebetween the two cases relies on the ν b invariant of thelast filled band. The red lines are the band structurewith λ tri = 0. (b) Band structure zoomed for the j=1/2bands near the K point for negative ( λ tri = − . t ), zeroand positive ( λ tri = 0 . t ) trigonal coupling. In the threecases the topological invariant gives a topological groundstate. (c) Evolution of the gap in the K point with λ tri for the intermediate SOC regime. The two topologicalphases found will be identified as HUTI and LUTI in theDFT calculations. the bulk band structure for strong and intermediateSOC strength α . If a non-vanishing trigonal term isincluded, it opens a gap in the Dirac points of theband structure generating topologically non-trivialconfigurations. We can see this clearly in Fig. 2b,where the band structure close to the Fermi level inthe vicinity of the K point is shown.The topological character of each configuration isdefined by the ν topological invariant which for a3and in an IS Hamiltonian can be calculated as ( − ν b = (cid:89) TRIM (cid:104) Ψ b | P | Ψ b (cid:105) (3)where the product runs over the four time reversalinvariant momenta (TRIM). The full invariant of aconfiguration will be the product of the last equationover all the occupied bands ν = (cid:88) occ.bands ν b (mod 2) (4)For a t g filling the first unfilled band has always ν b = 1 but the difference between strong and in-termediate SOC is the ν b invariant of the last filledband. For strong SOC the j=1/2 and j=3/2 arecompletely decoupled so a t g configuration wouldbe topologically trivial, being the invariant of thefifth band ν b = 1. However, when SOC is not sizablethe bandwidths are large enough to couple the j=1/2and j=3/2 levels so that the t g filling is a topologicalconfiguration. In both cases the t g filling is topo-logically non-trivial. We will see below using DFTcalculations that the systems under study (TMO’swith 5d electrons in a perovskite bilayer structure)are in this intermediate SOC regime.Figure 2b shows the bulk band structure, focus-ing now on the j=1/2 bands, for negative, zero andpositive trigonal terms. The left numbers are the ν b invariant of the band while the right numbers arethe sum of the invariants of that band and the bandsbelow it. No matter what the sign of λ tri is, the con-figuration becomes non-trivial, being its role to opena gap in the K point around the Fermi level. In theDFT calculations below, it will be seen that a changeof sign of the trigonal term can be understood as atopological transition between a low U topologicalinsulating phase (LUTI) and a high U topologicalinsulator (HUTI), across a boundary where the sys-tem behaves as a topological semimetal (TSM). III. DFT CALCULATIONSA. Computational procedures
Ab initio electronic structure calculations havebeen performed using the all-electron full potentialcode wien2k The unit cell chosen is shown in Fig.1c. It consists of 9 perovskite layers grown along the (111) direction, 2 layers of SrIrO (KPtO ) whichconform the honeycomb and 7 layers of SrTiO (KTaO ) which isolate one honeycomb from theother.For the different off-plane lattice parameters alongthe (111) direction of the perovskite considered, thestructure was relaxed using the full symmetry ofthe original cell. The exchange-correlation termis parametrized depending on the case using thegeneralized gradient approximation (GGA) in thePerdew-Burke-Ernzerhof scheme, local density ap-proximation+U (LDA+U) in the so-called ”fully lo-cated limit” and the Tran-Blaha modified Becke-Jonsson (TB-mBJ) potential. The calculations were performed with a k-meshof 7 × ×
1, a value of R mt K max = 7.0. SOC wasintroduced in a second variational manner using thescalar relativistic approximation. The R mt valuesused were in a.u. 1.89 for Ti, 1.91 for Ir, 2.5 for Srand 1.67 for O in the (SrTiO ) / (SrIrO ) multilayerand 1.93 for Ta, 1.92 for Pt, 2.5 for K, 1.7 for O inthe (KTaO ) / (KPtO ) multilayer. B. Band structure of the non-magnetic groundstate
We have already discussed that the systemschosen to study a d filling in a honeycomblattice with substantial SOC are the multilay-ers (SrTiO ) / (SrIrO ) and (KTaO ) / (KPtO ) formed by perovskites grown along the (111) direc-tion.First the structure is optimized for different c lat-tice parameters respecting IS using GGA and with-out SOC. This means that mainly only the inter-planar distances in the multilayers are relaxed. Forthe energy minimum the band structure is calculatedturning on SOC.The band structure using three exchange-correlation schemes (GGA, LDA+U and TB-mBJ)develops the same structure. The ν b topologicalinvariant of each band is calculated as in the TBmodel, the topological invariant being the sum ofthe ν invariants over all the occupied bands. Figure3 shows the band structure calculated with TB-mBJas well as the ν b invariants also obtained ab initio.The difference in the curvature of the bands withrespect to the result obtained with the TB modelis due to the existence of bands near the bottom of4he j=3/2 t g quadruplet which are not consideredin the TB Hamiltonian. Each band has double de-generacy due to the combination of TR and IS. Atthe optimized c, the gap between the last filled andthe first unfilled band is located at the corner of theBrillouin zone (K-point). At low c (unstable ener-getically but attainable via uniaxial compression),GGA predicts that the system can become a metalby closing an indirect gap between the K and Mpoints, however TB-mBJ calculations predict that adirect gap is localized at the K point. For all thecalculations the ground state has ν = 1 and thus itdevelops a topological phase. The last filled bandhas ν b = 0 so by comparison with the TB results thesystem corresponds to the intermediate SOC limit inwhich the J eff = 1 / J eff = 3 / If a 5d electron system like thisis in the intermediate SOC limit, it is hard to imag-ine how one can build a TMO heterostructe closer tothe strong SOC limit (the only simple solution wouldbe to weaken the hopping between the TM somehowto increase the α/t ratio). The first unfilled band has ν b = 1 as expected since a t g configuration will be atrivial insulator with a gap opened by the octahedralcrystal field.The way a trigonal field is present in the DFT cal-culations is mainly in two ways. On one hand, strainalong the z direction varies the distance to the firstneighbors in that direction, so that the electronic re-pulsion varies as well. We define this deformation as (cid:15) zz = c − c c where c is the off-plane lattice constantwith lowest energy. On the other hand, an on-siteCoulomb repulsion defined on the TM by using theLDA+U method has precisely the symmetry of thebilayer, i.e. trigonal symmetry, so varying in someway the on-site potential (always preserving paritysymmetry) will have the effect of a trigonal term inthe Hamiltonian (see A.3 for further details).According to this, it is expected that in a certainregime, variations in (cid:15) zz can be compensated by tun-ing U. In this regime, similar to what we discussedabove, the system will develop a transition betweentwo topological phases: a LUTI and a HUTI. At evenhigher U the system will show magnetic order. Wewill address this point later and by now we will focusfirst on the non-magnetic (NM) phase. In order tostudy the phase diagram defined by the parameters (cid:15) zz and U, we will perform calculations keeping oneof them constant and determine how the gap closesas the other parameter varies, keeping track of the FIG. 3: (Color online) Band structure obtained in theDFT calculations for the optimized lattice parameter c.The calculations were performed with TB-mBJ for both(SrTiO ) / (SrIrO ) (a) and (KTaO ) / (KPtO ) (b).The right panels are the band structure zoomed in nearthe Fermi level with the topological invariants displayed. ν b is the invariant of the band considered calculated byEq. 3 whereas ν is the sum over all the bands up to theone considered, in Eq. 4. parities at both sides of the transition. C. Evolution of the gap with uniaxial strain
Here we will discuss the behavior of the gap inthe K point with (cid:15) zz for various U and J values.First we analyze the behavior of both materials inparameter space showing their similarities finishingcharacterizing the actual position of the ground stateof the system in the general phase diagram.First we focus on the (SrTiO ) / (SrIrO ) case.As shown in Figure 4a, the gap closes as a functionof c, so uniaxial strain can drive the system betweentwo insulating phases just as λ tri does in the TBmodel. However, for high U (see below the discus-sion on the plausible U values) the transition pointdisappears (two such cases are plot in Fig. 4a). Forlow U (see Figs. 4 c,d), (cid:15) zz can drive the systemfrom a positive trigonal term to a negative one. Thismeans that uniaxial strain can change the sign of the5 IG. 4: (Color online) Evolution of the gap as a function of the uniaxial strain for the Ir-based multilayer (a) andPt-based multilayer (b). Evolution of the gap with the on-site interaction for the Ir-based multilayer (c,d) and Pt-based multilayer (e,f) for small (c,e) and large (d,f) J. It is seen that both systems show a similar behavior althoughin a different U regime. In (a) the values of U,J are U = 2 . J = 0 .
95 eV for the circles and U = 2 . J = 0 . U = − . J = 0 .
95 eV for the circles and U = 2 . J = 0 .
27 eV for the stars. effective trigonal term of the Hamiltonian. However,for high U (Fig. 4a), strain is not capable of chang-ing the trigonal field, so the system remains in thesame topological phase for every (cid:15) zz . For the calcu-lations using the TB-mBJ scheme (we will see belowto what effective U this situation would correspond),the transition point takes place almost at (cid:15) zz = 0, sobased on this scheme, the Ir-based multilayer wouldbe classified rather as a TSM than as a TI.Now we will focus on the (KTaO ) / (KPtO ) system (see Fig. 4 b,e,f). For the GGA calculationsthe transition with (cid:15) zz disappears, being the systemalways in the HUTI phase. If the system is calcu-lated using LDA+U with U negative (circles in Fig.4b), the behavior is similar to the previous systemand the transition point across a TSM reappears.For more realistic values of U and J (stars in Fig.4b) the transition point disappears again. Thus,the present system (Pt-based), though isoelectronicand isostructural, can be understood as the strong-U limit of the previous system (Ir-based). The band gap is larger, which is a sought-after feature of theseTI, but not large enough to make it suitable for roomtemperature applications. We observe that changingJ does not vary the overall picture, just displacesslightly the phase diagram in U-space. D. Evolution of the gap with U at constant (cid:15) zz The Hamiltonian felt by the electrons dependsalso on the on-site Coulomb interaction betweenthem. If the variation in the term of the Hamiltonianthat controls it takes place only in the TM atoms,the symmetry of the varying term will have the samelocal symmetry as the TM atoms, i.e. trigonal sym-metry. Thus, it is expected that a variation in U willhave a similar effect as λ tri in the TB model, so thegap can also be tuned by the on-site interaction.Figure 4 c,d,e,f shows the behavior of the gap forboth systems in an LDA+U scheme with J=0.27 (re-alistic) and 0.95 (large) eV as the parameter U is6aried.The slow increase of the gap with U suggests thatthe gap opened is not that of a usual Mott insulatorand reminds rather to the slow increase obtainedin the TB model. In fact, the calculation of the ν invariant shows again a topological phase on bothsides of the transition.Both systems develop a transition between theLUTI to the HUTI by increasing U. However, thetransition point of (SrTiO ) / (SrIrO ) is at posi-tive U’s, for the (KTaO ) / (KPtO ) the transitionappears at negative U’s, so for all possible reasonableU values the system will be in the HUTI phase.TB-mBJ calculations have proven to give accu-rate results of band gaps in various systems, including s-p semiconductors, correlated insulatorsand d systems, however it might give an inaccurateposition of semicore d orbitals and overestimatemagnetic moments for ferromagnetic metals. For(SrTiO ) / (SrIrO ) it is possible to use the transi-tion between the LUTI and the HUTI with the TB-mBJ scheme to estimate which value of U shouldbe used in an LDA+U calculation for these 5d sys-tems to reproduce the result of the TB-mBJ calcu-lation. The actual value of U needed for a correctprediction of the properties under study is often amatter of contention when dealing with insulatingoxides containing 5d TM’s. From Fig. 4 c,d itcan be checked that the gap closes at U=1.4 eV forJ=0.95 eV and at U=1.0 eV for the more realisticJ=0.27 eV, so this suggests that the values whichmight be used in an LDA+U calculation to mimicthe TB-mBJ result (a zero gap at (cid:15) zz = 0) are onthe order of U = 1 . however due to the well known propertydependence of the value of U, it is still unclearwhich is the correct value to study these topolog-ical phases. Therefore, our result can serve as areference for other ab initio based phase diagramsfor iridates where topological phases have been pre-dicted as a function of U. Moreover, we study thissystem in a broad range of U values and using differ-ent exchange-correlation schemes to provide a broadpicture of the system, rather than using a fixed Uvalue that would yield a more restricted view of theproblem.For the Pt-based multilayer we could also considerthe hypothetical effective value for which gap wouldclose at negative U ( U eff = U − J ) for both values of J , we obtain the values U eff = − .
42 eV for J = 0 .
27 eV and U eff = − . J = 0 .
95. So,it is clearly seen that the gap of these systems is notonly dependent on the parameter U − J , but alsohas an strong dependence on J , both in the realisticpicture of the Ir-based multilayer as well as in thenegative U regime of the Pt-based multilayer.Recently topological phases dominated by inter-actions, called topological Mott insulating phases,have been theoretically found, being this termemployed for physically different phenomena. Inthe HUTI phase, the topological gap of the sys-tems is enhanced by increasing the U parameter sothat the system seems to be robust against electron-electron interactions. In the same fashion a usualband insulator can be connected to a Mott insulat-ing phase, the previous robustness suggests thatthe HUTI phase might be adiabatically connected toa Z non-trivial interacting topological phase. Whether this is an artifact of the DFT method oran acceptable mean field approach of a many bodyproblem is something that can only be clarified withexperiments.
IV. STABILITY OF THE TOPOLOGICALPHASE
So far we have considered a system with both TRand IS. However, given that the Z classification isvalid only for TR invariant systems it is necessary todetermine if the ground state possesses this symme-try. IS breaking could destroy the topological phaseopening a trivial gap by decoupling both sublattices,as would happen if the honeycomb is sandwichedby two different materials. TR symmetry breakingwill be fulfilled by a magnetic ground state whereasIS breaking will be realized by a structural instabil-ity. In this Section we will study the two possibilitiesand conclude that both systems are structurally sta-ble and remain NM according to TB-mBJ in theirground states.
A. Stability of time reversal symmetry
Increasing electronic interactions will drive theNM ground state to a magnetic trivial Mott in-sulating phase at very high U. For a magnetic d S=1/2 localized-electron system, from Goode-7
IG. 5: (Color online) Difference of the sublatticemagnetization for the (SrTiO ) / (SrIrO ) (a) and(KTaO ) / (KPtO ) (b) as a function of U for two differ-ent J. (c) Phase diagram in (cid:15) zz -U space, the approximateposition of the ground state of both systems according toTB-mBJ is indicated. At high U the systems develop anAF order, however at realistic U’s both systems remainnon-magnetic. nough’s rules an antiferromagnetic (AF) exchangebetween the two sublattices is expected which wouldcreate an AF ground state breaking both TR and IS.To check this result, we have performed DFT calcu-lations within an LDA+U scheme taking J I = 0 . J II = 0 .
27 eV and varying U. For both sys-tems the calculations have been carried out at dif-ferent U’s for the NM and AF configurations at thetwo J’s. In both cases the ground state is AF athigh U. Also, the sublattice magnetization increaseswith U. Figures 5a and 5b show the evolution of the sublattice magnetic moment for both compounds.Also, a ferromagnetic (FM) phase has been ana-lyzed, being the least preferred one. In the Ir com-pound a FM phase can be stabilized but has alwayshigher energy than the NM or AF. In the Pt com-pound a FM phase could not be stabilized for any ofthe U values considered. The true ground state ofthe system would be a TI phase in the Pt-based mul-tilayer (or TSM in the Ir-based multilayer accordingto TBmBJ) depending on the value of U employed,so if the correct value to be used is larger than thecritical value (of about 3 eV, which is large accord-ing to our previous discussions), the topological Z phase will break down and the Kramers protection ofthe gapless edge states will disappear; whether theedge states would become gapped or not requiresfurther study. Thus the experimental measurementof the magnetic moment of the ground state of thesebilayers would shed light into the correct value ofU which should be used in these and other simi-lar compounds. In the Pt-based multilayer the sub-lattice magnetization is almost only dependent on U eff as can be checked by the shifting of the curves(Fig. 5b), however in the Ir-based multilayer thereis a stronger dependence on J (Fig. 5a). Again,simplifying the evolution in terms of the effective U eff = U − J is discouraged for these systems ac-cording to our results. The non trivial effect of J has been also observed in several compounds suchas multi-band materials. To summarize, we have obtained the magnetiza-tion of both compounds as a function of U, showingthe system shows a NM ground state for both com-pounds until a certain U (larger than the value of Uthat would be equivalent to TB-mBJ calculations)where the system becomes AF (see Fig. 5c).
B. Stability of inversion symmetry
The topological properties of this system rely onboth TR and IS. Tight-binding calculations pre-dicted that non-invariant parity terms with energyassociated of the order of magnitude of the topolog-ical gap could eventually drive the topological phaseto a trivial one.First we will discuss the (SrTiO ) / (SrIrO ) sys-tem. The simplest IS breaking could be driven bya structural instability. To study the structural sta-bility, we have displaced one of the TM atoms from8 IG. 6: (Color online) (a) Evolution of the gap in the TB model as a function of m , the closing point marksthe transition between topological and trivial phase. (b) Bands of zig-zag ribbons at different m , for low m thereare gapless edge states while for large m the edge states become gapped. The color marks the position of thewavefunction, the left edge (blue), the right edge (red) or the bulk (green). (c) Evolution of the eigenvalues obtainedin the DFT calculation (using the GGA scheme) of (KTaO ) / (KPtO ) , the dashed lines correspond to the fullyrelaxed structure. (d) Evolution of the four eigenvalues closest to the Fermi level for the TB model. its symmetric position and then relaxed the struc-ture. As a result, the structure returned to the sym-metric configuration. However, due to being in thetransition point between the two topological phases,any external perturbation (such as a perpendicularelectric field) could break inversion symmetry. Thissystem should be classified more as a topologicalsemimetal rather than a topological insulator due tothe (almost) vanishing gap of the relaxed structure.Now we will proceed to (KTaO ) / (KPtO ) . Thefirst difference between this and the previous systemis that in the present case the structure is well im-mersed in the HUTI phase. A large IS breaking isexpected to drive the system to a trivial phase wherethe sublattices A and B would be decoupled. How-ever, to change its topological class, the system hasto cross a critical point where the gap vanishes insome point of the Brillouin zone. According to that,the expected behavior is that the gap closes and re-opens as the sublattice asymmetry grows, going from the original topological phase to a trivial insulatingphase.To check this, we will compare the results obtainedfrom the TB model and DFT calculations. In the TBcase, we introduce a new parameter which is a diag-onal on-site energy whose value is + m for A atomsand − m for B atoms. This new parameter will breakIS and its value will take into account the amountof breaking. When IS is broken the index ν cannotbe calculated with the parities at the TRIM’s. How-ever, we can study the topological character search-ing for gapless edge states. For that sake we cal-culate the band structure of a zig-zag ribbon of 40dimers width with α = t and λ tri = − t . The calcula-tions were also carried out in an armchair ribbon andthe same behavior is found (not shown), however themixing of valleys makes the band structure harderto understand. The color of the bands indicates theexpectation value of the position along the widthof the ribbon of the eigenfunction corresponding to9hat eigenvalue, it is checked that the edge states arelocated in the two edges (red and blue) whereas therest of the states are bulk states (green). Accordingto the result of the TB model shown in Figs. 6a and6b, for small m the system remains a topological in-sulator although the introduction of m weakens thegap. If m keeps increasing the system reaches a crit-ical point where the gap vanishes and if m increaseseven more the gap reopens but now the edge statesbecome gapped so the system is in a trivial insulat-ing phase.To model the symmetry breaking in the DFT cal-culations we move one of the Pt atoms in the z direc-tion. As the distance to the original point increasesIS gets more broken. We show the four closest eigen-values to the Fermi level in the K point for the DFTcalculations (Fig. 6c) and TB model (Fig. 6d). Forthe symmetric structure (∆ z = 0), the combinationof TR and IS guarantees that each eigenvalue is two-fold degenerate. Once the atom is moved the degen-eracy is broken and the eigenvalues evolve with theIS breaking parameter. The analogy between thetwo calculations suggests that in the DFT calcula-tion once the gap reopens the new state is also atrivial insulator. The dashed lines in Fig. 6c cor-respond to the eigenvalues at the K point for thefully relaxed structure allowing IS breaking. Com-paring with the curves obtained for the evolution ofthe eigenvalues it is observed that the relaxed struc-ture is in a slightly asymmetric configuration but itremains in the HUTI phase.Due to the dependence of the topological state onIS, tuning this behavior would allow to make a de-vice formed by a perovskite heterostructure whichcan be driven from a topological phase to a trivialphase applying a perpendicular electric field. Thedevice will be formed by the TM honeycomb latticesandwiched between layers of the same insulating(111) perovskite from above and below. In this con-figuration the system will be in a HUTI phase. How-ever, a perpendicular electric field will break morethe sublattice symmetry inducing a much greatermass term in the Hamiltonian proportional to theapplied field. Modifying the value of the electricfield it would be possible to drive the system fromthe topological phase ( (cid:126)E = 0), to the trivial phase(high (cid:126)E ). This can be exploited as an application ofthis TI in nanoelectronic and spintronic devices. The sublattice asymmetry needed to make the tran-sition is of the same order of magnitude as the topo- logical gap, as can be checked in Fig. 6a.
V. SUMMARY
We have studied the gap evolution in the t g perovskite multilayers (SrTiO ) / (SrIrO ) and(KTaO ) / (KPtO ) as a function of the on-siteCoulomb interactions and uniaxial perpendicularstrain conserving time reversal and inversion sym-metry. The behavior of the system has been un-derstood with a simple TB model where SOC gaverise to an effective j=1/2 four-band Hamiltonian.Uniaxial strain and on-site interactions have beenidentified as a trigonal term in the TB model whosestrength controls the magnitude of a topological gap.The topological invariant ν has been calculated us-ing the parities at the TRIM’s both in TB and DFTcalculations. Comparisons between the invariants ofthe bands determines that both of these 5d electronsystems stay in the intermediate SOC regime. Thesmall value of the gap in the K-point comes from be-ing a contribution of third order in perturbation the-ory. In contrast, sublattice asymmetry contributesas a first order term, so the topological phase canbe easily destroyed by an external perturbation thatgives rise to an IS-breaking term in the Hamiltonian.(SrTiO ) / (SrIrO ) has been found to be a topo-logical semimetal at equilibrium (cid:15) zz . ComparingTB-mBJ and LDA+U calculations reasonable re-sults can be obtained for U in the range 1 - 2eV. In (KTaO ) / (KPtO ) a HUTI phase at equi-librium has been found. This last system can bedriven from topological insulating state to a trivialone by switching on a perpendicular electric fieldwhich would break inversion symmetry. Also, wehave verified that the properties of these systemsare dependent on both U and J instead of only in U eff = U − J .Although the smallness of the gap (less than 10meV according to TB-mBJ) makes the t g config-uration not particularly attractive for technologicalapplications, the simple understanding of the sys-tem turns it physically very interesting. The presentsystem can be thought of as an adiabatic deforma-tion of a mathematical realization of the four bandgraphene with SOC, with an experimentally accessi-ble gap, the roles of (cid:126)S and H SO being played now by (cid:126)J eff and H tri , but with a different physical natureof the topological state.10 cknowledgments The authors thank financial support fromthe Spanish Government via the project MAT-200908165, and V. P. through the Ram´on y CajalProgram. We also thank W. E. Pickett for fruitfuldiscussions.
Appendix A: Tight binding model
In this Section we explain the form of the differentterms of the TB Hamiltonian H = H SO + H t + H tri + H m (A1)
1. Spin-orbit term
We want to obtain the form of the SOC restrictedto the t g subspace. Taking the basis | yz (cid:105) = | xz (cid:105) = | xy (cid:105) = (A2)can be easily seen that the representation L i = ¯ hl i takes the form l x = i − i l y = i − i (A3) l z = i − i The SOC term has the usual form H SO = 2 α ¯ h (cid:126)L · (cid:126)S = α(cid:126)l · (cid:126)σ (A4)where σ i are the usual Pauli matrices acting onspin space. The representation follows the commu-tation relation [ l i , l j ] = − i(cid:15) ijk l k so defining J i = S i − L i the usual commutation relations hold. Thischange of sign introduces an overall minus sign onthe eigenvalues so the spectrum results in a j = 3 / E j =3 / = − α and j = 1 / E j =1 / = 2 α .
2. Hopping term
Each site has three bonds directed along the edgesof the perovskite structure. The bonds link the Aand B sublattices so there will be three differentoverlaps t i = (cid:104) A | H t | B (cid:105) depending on the direction( t x , t y , t z ) = t ( τ x , τ y , τ z ) (A5)The hopping takes place through the overlap ofthe t g orbitals of the TM and the oxygen p orbitals,It can be easily checked that the main contributiongives a matrix form that can be casted in the form( τ x , τ y , τ z ) = ( l x , l y , l z ) (A6)
3. Trigonal term
Due to the geometry of the system, a possible termin the Hamiltonian that does not break the spatialsymmetries of the system is a trigonal term. Thisterm will differentiate the perpendicular directionfrom the in-plane directions. This term will behaveas an on-site term which mixes the t g states withoutbreaking the symmetry between them so the generalform in the t g basis is H tri = λ tri (A7)given that the previous matrix is diagonal in the(111) direction, being the perpendicular eigenvaluesdegenerated, thus preserving the trigonal symmetry.The way (cid:15) zz enters in this term is on one hand byanisotropy of charge density due to the lack of localspatial inversion and on the other hand by distor-tion of the cubic perovskite edges (and thus of theoctahedral environment) by expansion/contractionof the (111) direction. The absence of local octahe-dral rotational symmetry is also responsible for thedependence of λ tri on U. Since varying the onsiteinteraction will modify the local electron density,provided the local trigonal symmetry is conservedbut not the local inversion (as it is broken explic-itly by the multilayer), this modification in the elec-tronic density will influence the electrons across theHartree and exchange-correlation terms, by a term11ith those symmetries. In summary, local symme-try forces that a spin-independent perturbation, thatincludes (cid:15) zz and spin-independent U-terms, can berecasted on the previous form.Whether spin-mixing trigonal terms are relevantfor the effective model of this system or not should beclarified in the future. Either way, the agreement ofthe predictions both of the TB model and the DFTcalculations, in addition with the explicitly checkedtopological phase at high U suggests that this modelis a well behaved effective model to study the NMphase of this type of systems.
4. Mass term
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