Calculation of model Hamiltonian parameters for LaMnO_3 using maximally localized Wannier functions
CCalculation of model Hamiltonian parameters for LaMnO using maximally localizedWannier functions Roman Kov´aˇcik and Claude Ederer
School of Physics, Trinity College Dublin, Dublin 2, Ireland ∗ (Dated: October 27, 2018)Maximally localized Wannier functions (MLWFs) based on Kohn-Sham band-structures provide asystematic way to construct realistic, materials specific tight-binding models for further theoreticalanalysis. Here, we construct MLWFs for the Mn e g bands in LaMnO , and we monitor changes inthe MLWF matrix elements induced by different magnetic configurations and structural distortions.From this we obtain values for the local Jahn-Teller and Hund’s rule coupling strength, the hoppingamplitudes between all nearest and further neighbors, and the corresponding reduction due to theGdFeO -type distortion. By comparing our results with commonly used model Hamiltonians formanganites, where electrons can hop between two ” e g -like” orbitals located on each Mn site, wefind that the most crucial limitation of such models stems from neglecting changes in the underlyingMn( d )-O( p ) hybridization. I. INTRODUCTION
The theoretical description of complex transition metaloxides and similar materials is very often based on effec-tive tight-binding (TB) models, i.e. a representation ofthe electronic structure within a certain energy region interms of localized atomic-like orbitals. Simple TB mod-els with a small number of orbitals can be used to studythe essential mechanisms governing complex physical be-havior, such as for example that found in the colossalmagneto-resistive manganites.
The electronic properties of manganites R − x A x MnO ( R : trivalent rare earth cation, A : divalent alkalineearth cation) are often described within an effective “two-band” TB model, where electrons can hop between thetwo e g levels on each Mn site. The corresponding Hamil-tonian typically also contains several local terms describ-ing the coupling of the e g states to the t g core spin,to the Jahn-Teller (JT) distortion of the oxygen octa-hedra, and/or the electron-electron Coulomb repulsion.It has recently been shown, that such a model (with pa-rameters obtained partly from first principles calculationsand partly by fitting to experimental data) is able to re-produce the basic structure of the phase diagram as afunction of doping and temperature found in manganitesystems such as La − x (Ca,Sr) x MnO . An elegant and systematic way to obtain realistic(materials-specific) TB models is the construction ofmaximally localized Wannier functions (MLFWs) fromthe Kohn-Sham states calculated using density functionaltheory (DFT). DFT calculations are known to give arealistic description of electronic structure for systemswhere electronic correlation effects are not too strong.
Furthermore, for materials where correlation effects areimportant, a Wannier representation of the Kohn-Shamband structure can be used to define a subset of orbitals(the “correlated subspace”), which can then be used asbasis for a more elaborate treatment of correlation ef-fects beyond standard DFT. This is done for example inDFT+DMFT (DMFT = dynamical mean-field theory) calculations, which aim at an accurate quantitativedescription of materials where electronic correlation can-not be ignored.In this work we construct MLWFs corresponding to theMn e g states for LaMnO , the parent compound for manymanganite systems, based on DFT calculations withinthe generalized gradient approximation (GGA). We cal-culate the real space Hamiltonian matrix elements in theMLWF basis for different structural modifications andfor different magnetic configurations, and we comparethe obtained results with assumptions made in commonlyused two band TB models.Our analysis is closely related to earlier work presentedin Ref. 10, which examined the validity of the two bandpicture by fitting TB model parameters (including thehopping between nearest and next-nearest neighbors) tothe DFT band structure obtained within the local densityapproximation (LDA). The approach based on MLWFsused in the present work is less biased and more generallyapplicable, and thus allows for a more systematic anal-ysis than the manual fitting of TB parameters discussedin Ref. 10. It is also well suited for the constructionof the correlated orbital subspace used for DFT+DMFTcalculations. This paper is organized as follows. In the followingsection we describe the theoretical background of ourwork. Thereby, Sec. II A summarizes the effective twoband model that is often used for a theoretical treatmentof manganites, Sec. II B presents the definition of the ML-WFs, Sec. II C describes the various structural modifica-tions of LaMnO investigated throughout this work, andSec. II D lists some of the calculational details. The pre-sentation of results starts with the case of the ideal cubicperovskite structure in Sec. III A. The individual effectsof the staggered JT and the GdFeO -type distortions arethen presented in Secs. III B and III C, respectively. Thisis followed by the results for the combined distortion inSec. III D, and the construction of a refined TB modeland its application to the full experimental structure ofLaMnO in Sec. III E. Finally, the most important resultsand conclusions are summarized in Sec. IV. a r X i v : . [ c ond - m a t . s t r- e l ] D ec FIG. 1: (Color online) Different structural modifications ofLaMnO investigated in this work, viewed along the [001]direction: (i) ideal cubic perovskite, (ii) purely Jahn-Tellerdistorted, (iii) purely GdFeO -type distorted, and (v) exper-imental P bnm structure. Pictures have been generated usingVESTA. II. METHOD AND THEORETICALBACKGROUNDA. Effective two-band models for LaMnO LaMnO crystallizes in an orthorhombically dis-torted perovskite structure with P bnm space group (seeFig. 1v), and A-type antiferromagnetic (A-AFM) order ofthe magnetic moments of the Mn cations.
The devi-ation from the simple cubic perovskite structure (Fig. 1i)can be decomposed into a staggered JT distortion ofthe MnO octahedra within the x - y plane (Fig. 1ii), theso-called GdFeO -type (GFO) distortion, consisting ofcollective tiltings and rotations of the oxygen octahedra(Fig. 1iii), and “the rest”, i.e. displacements of the Lacations from their ideal positions plus a homogeneous or-thorhombic strain (Fig. 1v). The electronic structure of LaMnO close to the Fermienergy is dominated by Mn 3 d states, which are split bythe cubic component of the crystal field into the lower-lying three-fold degenerate t g and the higher-lying two-fold degenerate e g states. The formal electronicconfiguration Mn : [Ar] 3 d leads to a high spin state ofthe Mn cation with fully occupied local majority spin t g states and one electron per local majority spin e g state,while both t g and e g minority spin states are empty.Based on this electronic structure, the theoretical de-scription of manganites often employs an effective two-band TB picture, where electrons can hop between thetwo e g levels on each Mn site. This hopping is facilitatedby hybridization with the oxygen 2 p states, which, how-ever, are not explicitly included in the TB model. It is therefore understood, that the “atomic” e g states usedin the TB model are indeed somewhat extended Wan-nier orbitals that also include the hybridization with theO 2 p states. In contrast, the three t g electrons are as-sumed to be tightly bound to a specific Mn site wherethey give rise to a local “core spin” S = 3 /
2. This corespin then interacts with the valence e g electron spin viaHund’s rule coupling. In addition, a JT distortion of thesurrounding oxygen octahedron splits the two e g levelson the corresponding Mn site, whereas elastic couplingbetween neighboring oxygen octahedra gives rise to a co-operative effect. The GFO distortion in this picture isusually assumed to simply reduce the effective hoppingamplitudes between neighboring Mn sites due to the re-sulting non-ideal Mn-O-Mn bond angle. In addition, alocal electron-electron interaction between electrons oc-cupying the same Mn site can be included in the model. The electronic Hamiltonian for such a model can beexpressed as: ˆ H = ˆ H kin + ˆ H local , (1)whereˆ H kin = (cid:88) a,b, R , ∆ R ,σ t ab (∆ R ) ˆ c † b ( R +∆ R ) σ ˆ c a R σ + h.c. (2)describes the electron hopping between orbital | a (cid:105) (spin σ ) at site R and orbital | b (cid:105) at site R + ∆ R , and it isassumed that all sites are translationally equivalent, sothat the hopping amplitudes t ab (∆ R ) depend only on therelative position between the two sites.Representing the e g orbital subspace within the usualbasis | (cid:105) = | z − r (cid:105) and | (cid:105) = | x − y (cid:105) , and assumingcubic symmetry, the nearest neighbor hopping along thethree cartesian directions has the following form: t ( ± a c ˆ z ) = (cid:18) t t (cid:48) (cid:19) (3) t ( ± a c ˆ x ) = t (cid:32) − √ − √
34 34 (cid:33) + t (cid:48) (cid:32) √ √
34 14 (cid:33) (4) t ( ± a c ˆ y ) = t (cid:32) √ √
34 34 (cid:33) + t (cid:48) (cid:32) − √ − √
34 14 (cid:33) . (5)Here, a c is the lattice constant of the underlying cubicperovskite structure. The hopping t (cid:48) between two neigh-boring | x − y (cid:105) -type orbitals along ˆ z is small due tothe planar shape of this orbital, and it is therefore oftenneglected. In this case, the nearest neighbor hopping de-pends only on a single parameter t , the hopping along ˆ z between | z − r (cid:105) -type orbitals.ˆ H local contains all local interaction terms included inthe model, i.e. Hund’s rule coupling with the t g corespin, the JT coupling to the oxygen octahedra distortion,and eventually also the electron-electron interaction. Inthis work we will discuss only the Hund’s rule and JTcoupling, which are of the form:ˆ H Hund = − J (cid:88) R S R · s R , and (6)ˆ H JT = − λ (cid:88) R ,σ,a,b (cid:16) Q x R ˆ c † a R σ τ xab ˆ c b R σ + Q z R ˆ c † a R σ τ zab ˆ c b R σ (cid:17) . (7)Here, J is the Hund’s rule coupling strength and S R isthe t g core spin at site R , which in the following wewill consider as classical vector normalized to | S R | = 1. s R = (cid:80) a,σ,σ (cid:48) c † a R σ τ σσ (cid:48) c a R σ (cid:48) is the corresponding e g va-lence spin, λ describes the strength of the JT coupling,and τ σσ (cid:48) are the usual Pauli matrices. The quantities Q x R and Q z R describe the JT distortion of the oxygenoctahedron surrounding site R : Q x R = 12 √ d x R − d y R ) , (8) Q z R = 12 √ d z R − d x R − d y R ) , (9)where d x R , d y R , and d z R indicate the O-O distances alongthe x , y , and z directions, corresponding to the oxygenoctahedron located at site R . B. Maximally localized Wannier functions
As is well known from basic solid state physics, theeigenfunctions within a periodic crystal potential are ex-tended Bloch waves, classified by a wave-vector k anda band-index m . These Bloch waves can in turn be ex-pressed as a Bloch sum of “atomic-like” localized TB ba-sis functions or Wannier functions . For an isolated groupof N Bloch states | ψ m k (cid:105) , i.e. a group of bands that are en-ergetically separated from all lower- or higher-lying bandsthroughout the entire Brillouin zone (BZ), a set of N lo-calized Wannier functions | w n T (cid:105) , associated with latticevector T , is defined via the following transformation: | w n T (cid:105) = V (2 π ) (cid:90) BZ (cid:34) N (cid:88) m =1 U ( k ) mn | ψ m k (cid:105) (cid:35) e − i k · T d k . (10)Here, U ( k ) is a unitary matrix mixing Bloch states atwave-vector k . Different U ( k ) lead to different Wannierorbitals, which are not uniquely determined by Eq. (10).However, Marzari and Vanderbilt showed that a uniqueset of maximally localized Wannier functions (MLWFs)can be obtained by minimizing the total quadratic spreadof the Wannier orbitals, defined as: Ω = N (cid:88) n (cid:2) (cid:104) r (cid:105) n − (cid:104) r (cid:105) n (cid:3) , (11) where (cid:104) ˆ O (cid:105) n = (cid:104) w n | ˆ O | w n (cid:105) .For the case of entangled Bloch bands, i.e. bandsthat are not energetically separated from other groupsof higher- or lower-lying states, an energy window[ E min , E max ] can be defined such that there are N ( k )win > N Bloch bands within the energy window at each k vec-tor, and then an N -dimensional manifold of mixed Blochstates is obtained as: | ψ dis m k (cid:105) = (cid:88) l ∈ N ( k )win U dis( k ) lm | ψ l k (cid:105) . (12)The corresponding Wannier functions can then be ob-tained from the mixed Bloch states by replacing | ψ m k (cid:105) with | ψ dis m k (cid:105) in Eq. (10). The unitary rectangular N ( k )win × N matrix U dis( k ) is also uniquely determined by the condi-tion of maximal localization, i.e. it can be obtained byminimizing Ω (cid:0) U ( k ) , U dis( k ) (cid:1) . Once a set of MLWFs is obtained, the correspondingHamilton matrix, H (W) ( k ), is constructed by a unitarytransformation: H (W) ( k ) = (cid:0) U ( k ) (cid:1) † (cid:0) U dis( k ) (cid:1) † H (B) ( k ) U dis( k ) U ( k ) , (13)from the (diagonal) Hamilton matrix in the Bloch basis, H (B) nm ( k ) = ε n k δ nm , with eigenvalues ε n k . The MLWFHamiltonian in real space is then calculated as a Fouriertransform of H (W) ( k ), which in practice is replaced by asum over N k points in k -space: h T nm = 1 N k (cid:88) k e − i k · T H (W) nm ( k ) . (14)Thus, the real space representation of the Hamiltonianin the MLWF basis is equivalent to a TB descriptionof the full Hamiltonian within the corresponding orbitalsubspace: ˆ H = (cid:88) T , ∆ T h ∆ T nm ˆ c † n T +∆ T ˆ c m T + h.c. , (15)where c m T is the annihilation operator for an electron inorbital | w m T (cid:105) . The real space MLWF matrix elements h T nm can therefore be interpreted as hopping amplitudeswithin a TB picture of MLWFs [compare Eq. (15) withEq. (2)]. Note that ∆ T in Eq. (15) refers to lattice vec-tors, whereas ∆ R in Eq. (2) refers to Mn sites. Thesubscripts n and m in Eq. (15) can thus in general indi-cate both site and orbital/spin character (for cases withmore than one site per unit cell).For the case when MLWFs are constructed from anisolated set of bands, the TB model, Eq. (15), exactlyreproduces the band dispersion within the correspond-ing energy window. For the entangled case, the energybands calculated from Eq. (15) do not necessarily haveto coincide with the underlying Bloch bands. C. Structural decomposition
To analyze the effect of the various distinct structuraldistortions within the experimental
P bnm structure onthe electronic properties of LaMnO we investigate sev-eral different atomic configurations (similar to Ref. 10):(i) The ideal cubic perovskite structure (Fig. 1i).(ii) A purely JT distorted structure (Fig. 1ii), whichresults from alternating long and short O-O dis-tances within the x - y plane, i.e. a staggered JTdistortion Q x R = ± Q x and Q z R = 0. This distortiondoubles the unit cell within the x - y plane, leadingto new in-plane lattice vectors a ii = a c ( ˆy + ˆx ) and b ii = a c ( ˆy − ˆx ) and tetragonal symmetry.(iii) A purely GFO-distorted structure (Fig. 1iii), re-sulting from rotations of the oxygen octahedraaround the z direction and octahedral tilts awayfrom z , alternating along all three cartesian di-rections. This distortion quadruples the unit cellcompared to the undistorted structure (i), yield-ing orthorhombic P bnm symmetry. The result-ing in-plane lattice vectors are identical to thoseof structure (ii) and the new lattice vector along z is c iii = 2 a c ˆ z .(iv) A superposition of JT and GFO distortion, whichalso leads to orthorhombic P bnm symmetry andunit cell vectors unchanged with respect to struc-ture (iii).(v) The full experimental structure (Fig. 1v),with orthorhombically strained lattice vectors( | a v | (cid:54) = | b v | (cid:54) = | c v | , resulting in Q z R (cid:54) = 0) anddisplaced La cations compared to structure (iv).For each of these structural modifications we use thesame volume V = 60 .
91 ˚A per formula unit as in theexperimentally observed P bnm structure. This leadsto a cubic lattice parameter a c = 3 . z direction for both (i) and (ii)structures in order to accommodate the magnetic order,thus changing the symmetry to tetragonal in case (i).Starting from the ideal cubic perovskite structure, weanalyze the effect of a specific distortion by graduallyincreasing the amount of this distortion, i.e. we performseries of calculations using a linear superposition of theWyckoff positions in the cubic perovskite structure andin structure ( x ): R ( α x ) = (1 − α x ) R (i) + α x R ( x ) , (16) TABLE I: Wyckoff parameters of the O(4c), ( x , y , 0.25),O(8d), ( x , y , z ), and La(4c), ( x , y , 0.25), sites for the vari-ous structural configurations used in this work (compare withTable I in Ref. 10). Expt. (Ref. 17) (ii) (iii) (iv) (v)O(4c) x -0.0733 0.0 -0.0733 -0.0733 -0.0733 y -0.0107 0.0 -0.0107 -0.0107 -0.0107O(8d) x y z x y α x between 0 and 1. The following cases areconsidered: ( x =ii) (pure JT distortion), ( x =iii) (pureGFO distortion), ( x =iv) (combined JT and GFO dis-tortions). D. Computational details
We perform spin-polarized first principles DFT calcu-lations using the Quantum-ESPRESSO program pack-age, the GGA exchange-correlation functional ofPerdew, Burke, and Ernzerhof, and Vanderbilt ul-trasoft pseudopotentials in which the La (5 s ,5 p ) andMn (3 s ,3 p ) semicore states are included in the valence.Convergence has been tested for the total energyand total magnetization using the ideal cubic perovskitestructure and ferromagnetic (FM) order. We find the to-tal energy converged to an accuracy better than 1 mRyand the total magnetization converged to an accuracyof 0.05 µ B for a a plane-wave energy cut-off of 35 Ryand a Γ-centered 10 × ×
10 k-point grid using a Gaus-sian broadening of 0.01 Ry. These values for plane-wave cutoff and Gaussian broadening are used in allcalculations presented in this work. The 10 × ×
10 k-point grid is used in all calculations for the cubic struc-ture (i), whereas appropriately reduced k-point grids of10 × ×
5, 7 × ×
10, and 7 × × z direction, doubledin the x - y plane, and quadrupled, respectively.After obtaining the DFT Bloch bands within GGA,we construct MLWFs using the wannier90 program inte-grated into the Quantum-ESPRESSO package. Start-ing from an initial projection of the Bloch bands ontoatomic d basis functions | z − r (cid:105) and | x − y (cid:105) centeredat the different Mn sites within the unit cell, we obtaina set of two e g -like MLWFs per spin channel for eachsite. The spread functional (both gauge-invariant andnon-gauge-invariant parts) is considered to be convergedif the corresponding fractional change between two suc-cessive iterations is smaller than 10 − . For cases withentangled bands a suitable energy window is chosen asdescribed in the corresponding “Results” section. FIG. 2: (Color online) Projected DOS and band structurealong high symmetry lines within the BZ calculated for the cu-bic structure (i) and both FM and A-AFM order. Filled (red)areas and solid (green) lines represent the projected DOS cor-responding to Mn( e g ) and Mn( t g ) states, respectively, whiledashed (blue) lines represent the site and orbitally averagedprojected DOS corresponding to the O p states. For the A-AFM case the left (right) panel corresponds to local majority(minority) spin projection. In the band structure plots, thedispersion calculated from the e g -like MLWFs are representedby thick (red) lines. The Fermi level is indicated by the hori-zontal dashed lines. III. RESULTS AND DISCUSSIONA. Perfect cubic perovskite – structure (i)
The projected densities of states (DOS) and bandstructure calculated for LaMnO in the ideal cubic per-ovskite structure (i) for both FM and A-AFM order areshown in Fig. 2. A metallic state is obtained for both FM and A-AFM order, in agreement with previous band-structurecalculations.
The projected DOS show that the(local) majority spin bands around the Fermi energy havemainly Mn( e g ) character and are half-filled while the (lo-cal) minority spin bands with mainly Mn( e g ) characterare unoccupied, as expected from the formal electron con-figuration. Bands with Mn( t g ) character are lying justbelow the Mn( e g ) bands, and slightly overlap with the FIG. 3: (Color online) Real space representation of the ML-WFs for majority and minority spin projections in the cubicstructure (i) with FM order, projected on the x - z plane pass-ing through Mn (large/blue sphere) and O (small/red spheres)sites (in arbitrary units). latter. O( p ) bands are located below the Mn( t g ) bands(between ∼ d ) and O( p ) electrons can beseen from the substantial amount of Mn( d ) character inthe energy range around 8 eV, i.e. towards the bottom ofthe bands with predominant O( p ) character. The statesabove the Mn( e g ) bands have predominantly La( d ) char-acter.One can see from the band structures depicted in Fig. 2that for the FM majority spin channel the bands withpredominant e g character are nearly completely isolatedfrom both higher and lower-lying bands, while for the FMminority spin channel and in the A-AFM case, the “ e g bands” overlap strongly with other bands (with mostlyMn( t g ) minority and La( d ) character). As describedin section II D, in order to construct e g -like MLWFs forthe various cases, we define an energy window for thedisentanglement procedure [see Eq. (12)], and then ini-tialize the Wannier functions from a projection of theKohn-Sham states within that energy window on atomic e g wave-functions (see Ref. 16). A suitable energy win-dow is chosen based on the e g projected DOS and cal-culated band structure (see discussion below for moredetails). Two MLWFs per spin channel for the single Mnsite within the cubic unit cell are constructed for FM or-der, and two pairs of MLWFs, localized at the two Mnsites within the magnetic unit cell, are constructed forA-AFM order (for global spin up projection only).Figure 3 shows the real space representation of thetwo e g -like MLWFs for both majority and minorityspin and FM order, calculated for an energy windowof [12.0, 17.0] eV and [15.9, 20.0] eV, respectively. Theshape of the MLWFs resembles the antibonding σ ∗ char-acter of hybridization between Mn( e g ) and O( p ) states inthis energy range. The hybridization is notably strongerfor the majority spin MLWFs (individual spread per WF2 .
90 ˚A compared to 1 .
65 ˚A for the minority spin ML-WFs), which is due to the smaller energy separation be-tween the atomic Mn( e g ) and O( p ) levels for the majorityspin channel. The difference between the real space rep-resentation of the MLWFs for FM and A-AFM order (notshown here) is more subtle. A quantitative comparisonof the corresponding differences in the real space Hamil-ton matrix elements between MLWFs will be presentedbelow.The dispersion calculated from the obtained e g ML-WFs is also shown in Fig. 2. It can be seen that even inthe cases with strongly entangled e g bands (FM minor-ity spin and A-AFM) the MLWF bands follow certainDFT bands almost exactly, except around some bandcrossings with higher lying La d bands. This representsthe fact that within cubic symmetry the e g states can-not hybridize with the t g bands, and hybridize only veryweakly with the La d states.In order to reproduce the two majority spin bandsaround the Fermi energy for the FM case, the lowerbound of the energy window, E min , has to be abovethe lower peaks in the Mn( e g ) projected DOS at around10.5 eV and 8 eV, which correspond to the bonding com-bination of hybridized atomic O( p ) and Mn( e g ) states. Ifthese bands are included in the energy window, the bond-ing and antibonding combinations of atomic orbitals be-come disentangled and the e g Wannier functions becomeessentially “atomic-like” (compare also with the case ofSrVO described in Ref. 9). On the other hand, varying E min within 0.4 eV below the Γ point energy of the e g -like bands changes the MLWF bands by less than 1 meVfor any k . Similarly, varying the upper bound of theenergy window has only minor influence on the result-ing MLWF bands, due to the negligible hybridization ofthe e g states with higher-lying bands. Additional testcalculations for different k-point grids showed that theMLWF band structure is converged within 0.5 meV atany k-point for the 10 × ×
10 grid which was used forthe energy window test calculations.We now turn to the analysis of the hopping parame-ters, i.e. the real space matrix elements h ∆ R ab , Eq. (14),between MLWFs located at different Mn sites. The mag-nitudes of all calculated hopping parameters for the FMmajority spin case are shown in Fig. 4. It is noticeablethat the hopping amplitudes along the three cartesianaxes are most dominant and that their decay with dis-tance is rather slow, so that the terms corresponding tointer-site distances of 2 a c and 3 a c are of comparable mag-nitude as the hopping between next-nearest neighbors forwhich | ∆ R | = √ a c .The exact MLWF representation in terms of H (W) ( k )is well suited for further numerical calculations, e.g.within a DFT+DMFT approach. On the other hand,for the analysis of specific physical mechanisms withina semi-analytical TB model, one generally wants to use FIG. 4: (Color online) Magnitude of all calculated non-zerohopping parameters for FM order in the ideal cubic structureas a function of the inter-site distance | ∆ R | (open circles:hopping along the unit cell directions; open diamonds: hop-ping between next-nearest neighbors; filled circles: all otherhoppings). Inset: Comparison of the full MLWF band struc-ture (solid lines) and the one calculated from a simplifiedTB model (filled circles) which includes only the inter-sitehoppings for which the largest matrix element is larger than10 meV (see main text). only a very limited number of hopping parameters h ∆ R between closest neighbors. We therefore identify a min-imal subset of hopping parameters, corresponding to in-tersite distances | ∆ R | /a c ∈ { , √ , , } , i.e. where onlyhopping between sites, for which the leading term (i.e.the corresponding matrix element with largest magni-tude) is larger than 10 meV, are considered, while therest is set to zero. This model yields an overall verygood agreement with the full MLWF band structure (seeinset in Fig. 4), deviating not more than 0.11 eV for anyk-point on the 10 × ×
10 k-point grid used. On theother hand, a TB model where only the hopping am-plitudes between nearest and next-nearest neighbors aretaken into account leads to deviations of up to 0.29 eV forsome k-points, which might still be acceptable for certainpurposes. However, the overall bandwidth for the lattermodel is reduced by about 0.2 eV compared to the fullMLWF band structure.The calculated matrix elements of the real space ma-trix elements h ∆ R ab for nearest and next nearest neigh-bor hopping as well as the corresponding on-site terms(∆ R = 0) are summarized in Table II. Here and inthe following we use the abbreviated notation h z , cor-responding to ∆ R = ± a c ˆ z , and h xz , corresponding to∆ R = a c ( ± ˆ x ± ˆ z ) (and analogously for all other cartesiandirections). We note that in the A-AFM case the trans-lational equivalence between the two Mn sites within theunit cell is broken, and ∆ R = ± a c ˆ z is not a lattice vec-tor in this case. Nevertheless, in order to simplify thenotation, we stick to the site-based index and note thatfor A-AFM order a translation along ˆ z is equivalent toreversing the two spin projections. In the following we TABLE II: Calculated values of the on-site, nearest, and next-nearest neighbor matrix elements h ∆ R ab (in meV) for FM andA-AFM order within structure (i) for the two different spinprojections. As described in the text, in the A-AFM case allmatrix elements refer to the Mn site closest to the origin. (a) FM( ↑ ) FM( ↓ ) A-AFM( ↑ ) A-AFM( ↓ ) h . . . . h . . . . h z − . − . − . h z . − . − . h x − . − . − . − . h x . . . . h x − . − . − . − . h xz . . . h xz − . − . − . − . h xz − . . . h xy − . − . − . − . h xy . . . . | z − r (cid:105) -like MLWFs along the z direction, h z ( ≡ t inthe effective model description), is the leading term forthe nearest neighbor hopping, and that overall the nextnearest neighbor hopping is about an order of magni-tude smaller than the nearest neighbor hopping. Thehopping amplitude between two | x − y (cid:105) -like functionsalong the z direction, h z ( ≡ t (cid:48) in the model description),is indeed very small compared to h z . In the FM case,all nearest neighbor hopping amplitudes for the minorityspin orbitals (except h z ) are reduced (to about 75-85%)compared to the majority spin channel. This reflects theweaker hybridization between minority spin e g and O(2 p )states, leading to more localized minority spin MLWFswith reduced hopping amplitudes. For A-AFM order, h z corresponds to the hopping between a local majorityand a local minority spin orbital, and its value, (92 %of h z for FM ( ↑ )), is intermediate between the corre-sponding FM majority and minority values. The A-AFMhopping amplitudes within ferromagnetically ordered x - y planes for local majority/minority spin directions arevery similar to the corresponding FM hoppings (differ-ing by less than 5 meV), with the exception of the (local)majority spin h x value, which is larger than that. Simi-lar relations between the FM majority and minority spinand A-AFM values are also observed for the next-nearestneighbor hoppings.It can easily be verified, that the hopping parameters for FM majority and minority spin fulfill the relationsdescribed in Eqs. (3)-(5), as required for cubic symme-try. However, if the terms proportional to t (cid:48) ≡ h z areneglected, the corresponding equations are not exactlyfulfilled. Thus, simply neglecting h z while keeping allother nearest neighbor hopping amplitudes unchanged,leads to slight deviations from cubic symmetry. Further-more, Eqs. (3)-(5) are clearly not fulfilled for the A-AFMhopping amplitudes, which reflects the overall tetragonalsymmetry resulting from the magnetic order.This symmetry reduction for the A-AFM case is alsovisible in the on-site matrix elements h and h , whichdiffer by about 100 meV. On the other hand, the smallasymmetry ( ∼ J (treating S R as clas-sical unit vector). From the calculated on-site MLWFmatrix elements, we thus obtain a value of J = 1 .
499 eVfor the Hund’s rule coupling parameter in the FM case,and J = 1 . / .
451 eV from the A-AFM on-site terms.The differences between these values indicate the limitsof the assumption of a fixed t g core spin. We note thatall these values are slightly larger than the results ob-tained in previous LSDA calculations ( J = 1 .
34 eV), which reflects the fact that GGA in general leads to astronger magnetic splitting than LSDA. Overall, the results obtained via MLWFs are in a verygood qualitative agreement with the previous study us-ing TB fits to DFT band structures. However, thedirect comparison between the values calculated fromMLWF in this work and the values reported in Ref. 10 isslightly hampered by the different exchange correlationfunctionals and pseudopotentials used in the two studies.The same fitting method as described in Ref. 10 appliedto the GGA band structure calculated in the presentwork, leads to a nearest neighbor hopping parameter t = −
688 meV, i.e. slightly larger than the −
648 meVobtained from the MLWFs. This is due to the larger ma-jority spin e g bandwidth obtained here, W ↑ = 4 .
126 eV,compared to the value of 3.928 eV reported in Ref. 10.Thus, the difference in bandwidth compensates the ne-glect of further neighbor hopping in the simple TB fit,leading to the apparent very good agreement between h z = −
648 meV listed in Table II and the correspond-ing value ( t = −
655 meV) given in Ref. 10.In the following sections, we will analyze the influ-ence of the structural distortions only for the on-site andnearest-neighbor hopping terms. We have verified thatthe resulting changes in the further neighbor hopping am-plitudes do not lead to significant differences in the dis-persion characteristics of the e g bands, even though thecorresponding relative changes of the next-nearest neigh-bor hoppings are comparable with those of the nearest-neighbor hoppings. FIG. 5: (Color online) DFT band structure (thin lines) forthe JT-distorted structure (ii): a) majority spin FM, b) mi-nority spin FM, and c) A-AFM. MLWF bands are depictedas thick/red lines. The Fermi level is indicated by the dashedline.
B. Jahn-Teller distortion – structure (ii)
As described in Sec. II C, the staggered JT distortion, Q x R = ± Q x , leads to a unit cell doubling within the x - y plane. In the case of FM order, we therefore constructtwo pairs of e g MLWFs for each spin channel, localizedat the two different Mn sites within the unit cell, whilefor A-AFM order we construct four pairs of MLWFs, lo-calized at the four different Mn sites within the corre-sponding unit cell (for global spin up projection only).The same approach for choosing the energy window forthe disentanglement procedure was used as described inthe previous section.The calculated DFT band structure and e g -like MLWFdispersion for the JT distorted structure (ii) are shownin Fig. 5. As a result of the unit cell doubling, thereare now 4 and 8 bands with e g character per spin chan-nel for the FM and A-AFM order, respectively. As forthe cubic perovskite structure, the calculated MLWF dis-persion largely follows the DFT band structure, exceptwhere there is strong hybridization with states of a differ-ent orbital character. It can be seen that several degen-eracies and potential band crossings, which would resultfrom a simple “backfolding” of the cubic band-structureonto the smaller tetragonal BZ, are lifted due to the JTdistortion. This can be seen for example for the FMmajority spin bands, where the highest-lying band alongΓZ acquires some dispersion, leading to a splitting of thehigher energy e g states at Z. Similarly, the degeneracy ofthe two lowest-lying e g states at Γ is lifted, and a poten-tial crossing of e g bands is prevented between Γ and M.The latter splitting, together with the reduced dispersionalong ΓZ for A-AFM order, appears crucial for the open-ing of an energy gap in the JT-distorted A-AFM orderedstructure (Fig. 5c).To further analyze the influence of the JT distortion FIG. 6: (Color online) MLWF Hamiltonian matrix elements h ∆ R ab as function of the JT distortion. Large/black andsmall/red symbols correspond to FM and A-AFM order, re-spectively. Matrix elements associated with pure (local) ma-jority and minority spin character are shown as trianglespointing up and down, respectively. Closed circles in (b) rep-resent the A-AFM h z hopping. on the e g electronic structure, we perform a series of cal-culations where we gradually change the oxygen posi-tions from the ideal perovskite structure (i) to the fullyJT distorted structure (ii), according to Eq. (16), andmonitor the resulting changes in the MLWF Hamilto-nian matrix elements. In all these calculations, we usethe same energy windows of [12 . , .
5] eV, [15 . , .
0] eVand [12 . , .
0] eV for the disentanglement in the caseof FM majority, FM minority and A-AFM, respectively.The resulting MLWF matrix elements are depicted inFig. 6. As discussed in the previous section, we reportonly hopping from and to the Mn site at the origin. Thehopping amplitudes corresponding to other sites in theunit cell follow from symmetry. We find a strong lin-ear dependence on the JT distortion for both the off-diagonal on-site matrix elements h (Fig. 6a) as well asfor the off-diagonal in-plane hopping h x / (Fig. 6c/d).All other on-site and nearest neighbor hopping matrixelements show only a weak or moderate quadratic de-pendence on α ii .Within the model described in Sec. II A the sole effectof the JT distortion ( Q x R , Q z R ) is a linear coupling to theon-site terms at site R according to: t = (cid:18) e − λQ z R − λQ x R − λQ x R e + λQ z R (cid:19) . (17)In our case Q z R = 0 and Q x R = ± α ii Q x ; e is the on-siteenergy of the e g orbitals. It can be seen from Fig. 6a thatthe off-diagonal element h indeed shows a linear depen-dence on α , consistent with Eq. (17). The correspond-ing slope, − λQ x = 482 meV, is nearly identical for theFM majority and A-AFM local majority spin elements,whereas it is significantly smaller for the (local) minorityspin matrix elements ( − λQ x = 246 /
155 meV). This in-dicates that the JT splitting is also a ligand-field effect,i.e. it is mediated by hybridization with the surround-ing oxygen orbitals, which, as pointed out previously, isstronger for the energetically lower majority spin states.The values for the JT coupling constant λ obtained fromthe data shown in Fig. 6a are 3.19 eV/˚A, 1.63 eV/˚A, and1.02 eV/˚A, for majority, FM minority, and A-AFM localminority spin states, respectively. We note that the valueof λ obtained for majority spin is approximately a fac-tor of two larger than the value obtained from the fittingprocedure described in Ref. 10. As we will discuss inmore detail below, the source for this discrepancy is thestrong linear splitting observed for the off-diagonal in-plane nearest neighbor hoppings h x / , which is inducedby the JT distortion (see Fig. 6c/d).This splitting between h x / again results from theunderlying hopping between atomic Mn( e g ) and O( p )states, which (in leading order) depends linearly on theMn-O distance. Since this dependence will be differentfor the | z − r (cid:105) and | x − y (cid:105) orbitals, it can easilybe verified that the effective hopping across a combina-tion of one long and one short Mn-O bond within the x - y plane between two different e g orbitals will also de-pend linearly on the JT distortion, whereas the effectivehopping between the same type of e g orbitals will showonly a quadratic dependence. We have verified, by con-structing atomic-like Wannier functions for both Mn( e g )and O( p ) orbitals (corresponding to larger energy win-dows), that indeed the dependence on the Mn-O distanceis much stronger for the hopping amplitude between the | z − r (cid:105) -type orbital and a neighboring O( p ) orbitalthan for the corresponding | x − y (cid:105) -type hopping, con-sistent with the observed splitting in the effective hoppingamplitudes h x / shown in Fig. 6c/d.It can be verified within a TB model where the lin-ear splitting between h x and h x (and analogously forthe hopping along the y direction) is taken into accountvia one extra parameter derived from the MLWF data,that this splitting partially cancels the effect of the on- site JT term on the band dispersion. In particular, theJT-induced “gap” between the second and third e g bandbetween Γ and M is reduced by increasing the h x / h x splitting, whereas it is enhanced by increasing the JTcoupling strength λ . Thus, the band dispersion result-ing from reduced λ and no splitting between h x and h x looks very similar to the one obtained from the MLWFparameters (i.e. including the spitting between h x / ).This is the reason why the fitting of the DFT band struc-ture on a TB model that does not incorporate a h x / h x splitting (see Ref. 10) leads to a smaller value of λ thanthe one obtained from the MLWF parameters. An inter-esting question arising from this is whether, despite thevery similar band dispersion, the two different TB pa-rameterizations would lead to noticeable differences incalculated ordering temperatures for the collective JTdistortion.The differences between the off-diagonal in-plane hop-ping parameters induced by the JT distortion indicatechanges of the MLWFs themselves, i.e. the JT distortionalters the basis-set of a MLWF-based TB model. We notethat this is an unavoidable result of the effective “two-band” picture. The definition of a distortion-independentbasis-set is only possible within a full d - p TB model,based on truly atomic-like functions. On the other hand,a splitting between h x and h x can in principle also re-sult from a unitary mixing of the | z − r (cid:105) and | x − y (cid:105) basis functions. In order to check whether (at least partof) the observed splitting is due to such a mixing, we haveapplied a local unitary transformation between the twoMLWFs on each site, and studied the resulting changesin the MLWF matrix elements. In essence, we find thatit is impossible to retrieve the “cubic symmetry”, i.e. theform described in Eqs. (3)-(5) and (17), simultaneouslyfor h , h z , and h x , and that a transformation of one ofthese terms to the desired form in general increases thecorresponding deviations in the other two terms. It ap-pears that the basis functions resulting directly from themaximum localization procedure using initial projectionson atomic | z − r (cid:105) and | x − y (cid:105) functions represent thebest overall compromise.The leading hopping term in z direction, h z (Fig. 6b),exhibits only a weak quadratic change as a function of α ii . We also find a similar weak quadratic dependenceon the JT distortion in the hopping parameters h x / (not shown), and a moderately strong quadratic changein the on-site diagonal matrix elements (Fig. 6e/f), whichintroduces a splitting of about 150 meV between h and h for the fully JT distorted structure.Finally, we note that the Hund’s rule coupling pa-rameters derived from the local spin splitting betweenMLWFs obtained for the fully JT distorted structure( J = 1 . / .
484 eV for FM order, J = 1 . / .
465 eVfor A-AFM order) are not significantly changed comparedwith the ones obtained for structure (i).0
FIG. 7: (Color online) DFT band structure (thin lines) for thepurely GFO-distorted structure (iii): a) FM majority spin, b)FM minority spin, c) A-AFM. MLWF bands are depicted asthick/red lines. The Fermi level is indicated by dashed lines.
C. GdFeO -type distortion – structure (iii) The band dispersion calculated for the purely GFO-distorted structure (iii) is presented in Fig. 7. The ro-tation and tilting of oxygen octahedra in structure (iii)distorts the ideal 180 ◦ Mn-O-Mn bond angle, which is ex-pected to reduce the hopping amplitudes. Indeed, it canbe seen in Fig. 7 that the GFO distortion leads to sig-nificantly smaller bandwidth (2.951 eV and 2.139 eV forFM majority and minority spin, respectively, comparedto 4.126 eV and 3.156 eV in the undistorted structure(i)). As a result, the FM majority spin e g bands becomecompletely separated from the lower-lying t g bands andthe La( d ) bands at higher energy. Unlike in the JT dis-torted structure (ii), the system stays metallic for bothFM and A-AFM order.Since the unit cell for structure (iii) is quadrupled withrespect to the cubic perovskite structure, there are now8 bands with dominant e g character for each spin direc-tion. However, due to the tilt/rotation of the oxygen oc-tahedra, “ e g -like” orbitals at a certain site can hybridizewith “ t g -like” orbitals at a neighboring site, leading tobands with mixed e g / t g character. In the FM casethis does not represent a problem for the disentanglementprocedure, since the bands with predominant e g charac-ter are separated from the predominantly t g bands forboth spin direction. For FM order, we can therefore con-struct four pairs of MLWFs, localized at the four differentsites within the unit cell, by defining appropriate energywindows separately for each spin direction. This is notpossible in the A-AFM case, where the local minority t g bands overlap strongly with the local majority e g bandsin the energy region between 14 eV and 16 eV. In thiscase, the standard disentanglement procedure employedfor structures (i) and (ii), i.e. defining an energy window[12 . , .
0] eV and initializing 8 Wannier functions fromprojections on atomic e g orbitals at the various sites, re- FIG. 8: (Color online) Hamiltonian matrix elements inthe basis of MLWFs as a function of the GFO distortion.Large/black and small/red symbols correspond to FM andA-AFM order, respectively. Elements associated with purely(local) majority and minority spin character are representedby triangles pointing up and down, respectively. sults in MLWFs with mixed t g / e g orbital character. Inparticular, the resulting local minority spin MLWFs ex-hibit a rather strong t g character.One possible way to overcome this problem would beto construct all 20 d -like MLWFs (5 per Mn site), i.e.both e g and t g orbitals. However, the resulting ML-WFs still contain some amount of e g / t g mixing, andthe corresponding MLWF matrix elements exhibit sys-tematic deviations from the results obtained in the pre-vious sections, which are derived from a smaller set ofMLWFs. In the following, we therefore adopt a differ-ent strategy to obtain model parameters for the A-AFMcase, and construct the 4 local majority and 4 local mi-nority spin e g -like MLWFs separately, using two differentenergy windows. From this, we obtain the on-site matrixelements h as well as the hopping parameters h x withinthe x - y plane (and of course all further neighbor hoppingamplitudes within this plane). On the other hand we donot obtain the hopping amplitudes h z between adjacentplanes in the z direction, which would connect the twoseparate sets of MLWFs. Similar to the purely JT dis-torted case, we analyze the effect of the GFO distortionon the e g bands by performing calculations with varyingdegree of distortion, i.e. by changing the oxygen positionsaccording to Eq. (16). In this case we always adjust theenergy window for the construction of the MLWFs to theactual e g bandwidth corresponding to a particular α iii .We find that the main effect of the GFO distortion1 FIG. 9: (Color online) DFT band structure (thin lines) forstructure (iv): a) majority spin FM, b) minority spin FM,and c) A-AFM. MLWF are depicted as thick/red lines. Fermilevel is indicated by dashed line. is indeed a systematic reduction of all hopping ampli-tudes by ≈ −
30 %, consistent with what was re-ported in Ref. 10. Fig. 8a shows the overall reduc-tion for all obtained nearest neighbor hopping ampli-tudes for both FM and A-AFM order, while Fig. 8b re-solves the reduction factors of the various hopping am-plitudes for full GFO distortion ( α iii = 1). It can beseen, that even though there is a significant spread inthe reduction factors for the various hopping parameters,the overall reduction can approximately be described as h x/z ( α iii ) = h x/z (0) (cid:0) − ηα (cid:1) , with an average value of η = 0 . p and d orbitals. Since the effective e g bands correspond to the antibonding combination ofthese atomic orbitals, a reduction of the underlying p - d hopping amplitudes results in a decrease of the Γ-pointenergy of the e g states. The Hund’s rule coupling param-eter J = 1 .
502 eV obtained from the on-site splitting forFM order and α iii = 1 is very similar to the correspond-ing value for the cubic perovskite structure. D. Combined Jahn-Teller and GdFeO -typedistortion – structure (iv) So far we have analyzed the individual effects of theJT and GFO distortion. We now discuss whether the su-perposition of both distortions gives rise to any changesin the MLWF matrix elements that go beyond a simplesuperposition of the individual effects. The correspond-ing band structure and MLWF dispersion for structure
FIG. 10: (Color online) MLWF Hamiltonian matrix elementsas function of combined JT and GFO distortion. Large/blackand small/red symbols correspond to the FM and A-AFMorder, respectively. Elements associated with purely (local)majority and minority spin character are represented by tri-angles pointing up and down, respectively. (iv), i.e. the combined JT and GFO distortion, is pre-sented in Fig. 9. It can be seen that the band structurein this case closely resembles the one of the purely GFOdistorted structure (iii), Fig. 7, but with the additionalJT-induced effects (avoided band-crossings and lifted de-generacies) as described in Sec. III B. Note that, as in thepurely JT distorted structure, the FM case is metallic,whereas a band gap opens only for A-AFM order.As described in the previous section we construct 8MLWFs per spin direction for the FM case and two sep-arate sets of 4 local majority and 4 local minority e g -likeMLWFs for the A-AFM case. Fig. 10 shows the evolu-tion of selected MLWF matrix elements as a function ofdistortion. The atomic positions are changed accordingto Eq. (16) with x = iv. By comparing Fig. 10a withFig. 6a, it can be seen that the GFO distortion does alsosignificantly reduce the on-site matrix elements h (to ≈ −
80 %), which are otherwise proportional to the JTdistortion. This is further evidence for the ligand-fieldnature of the JT coupling, i.e. that it is mediated bythe Mn-O hybridization (which is reduced by the GFOdistortion). Furthermore, it can be seen that the leadinghopping along z , h z , follows very closely the trend ob-served for the purely GFO distorted structures. In thecase of the off-diagonal hopping amplitudes within the x - y plane, the superposition of GFO-distortion-inducedreduction and JT-induced splitting leads to an initial in-2crease of h x for small distortion, followed by a decreasefor larger α iv . Overall, the observed trends can indeedbe well understood as independent superposition of theindividual effects of JT and GFO distortions. We notethat the kinks observed in some of the minority spin hop-ping terms around α iv ≈ . e g -like and t g -like minority spin bandsfor this amount of distortion, which represents a certain“discontinuity” for the disentanglement procedure. E. Simplified TB models for LaMnO in the fullexperimental P bnm structure (v)
The analysis presented so far showed that the effect ofdifferent structural distortions on the e g bands can, toa good extent, be treated independently of each other.In this section, we attempt to incorporate the most sig-nificant effects described in the previous sections into arefined effective TB model. Then, in order to test theaccuracy of the resulting parameterization, we comparethe resulting band dispersion with the full GGA andMLWF band-structure, calculated for the full experimen-tal P bnm structure of LaMnO and A-AFM order.For the refined TB model we introduce different hop-ping amplitudes for local majority/minority spin pro-jections to describe the hopping between ferromagnet-ically aligned nearest neighbors within the x - y planes( t ↑↑ / t ↓↓ ), and an intermediate value for the nearest neigh-bor hopping between antiferromagnetically aligned near-est neighbors along the z direction ( t ↑↓ ), i.e. hoppingbetween two different local spin projections. This is inaccordance with our results presented in Sec. III A. Forthe corresponding hopping amplitudes we use the valuesof h z calculated for the ideal cubic perovskite structure(see Table II) for FM and A-AFM order, which are thenreduced by the same factor (1 − ηα ), where α iii describesthe amount of pure GFO distortion. Apart from thesemodifications we assume the usual cubic symmetry of thenearest neighbor hopping matrices, i.e.: t ss (cid:48) ( ± a c ˆ z ) = (1 − ηα ) t ss (cid:48) (cid:18) (cid:19) , (18) t ss (cid:48) ( ± a c ˆ x ) = (1 − ηα ) t ss (cid:48) (cid:32) − √ − √
34 34 (cid:33) , (19)(and analogously for t ss (cid:48) ( ± a c ˆ y )). Note that s and s (cid:48) inthese equations should be read as a local spin index, i.e. itdesignates the spin projection relative to the orientationof the local core spin. We use the average value η = 0 . x - y plane discussed inSec. III B is incorporated in the TB model as an addi- tional contribution to the in-plane hopping:∆ t ( ± a c ˆ x ) = ˜ λQ x α ii (1 − ηα iii ) (cid:18) − (cid:19) (20)(and analogously for ∆ t ( ± a c ˆ y )). Here, α ii describesthe amplitude of the staggered JT distortion, i.e. Q x R = ± Q x α ii , and the parameter ˜ λ is determined fromthe average splitting over all hopping amplitudes in thepurely JT-distorted structure (shown in Fig. 6c/d). Inaddition, we include the usual on-site JT effect in essen-tially the same form as described in Eq. (7), but witha spin-dependent JT coupling constant that is also re-duced by the GFO distortion (with the same factor asthe hopping amplitudes): λ → λ s (1 − ηα ) . (21)We note that the orthorhombic strain in the experimentalstructure of LaMnO gives rise to a homogeneous Q z component to the JT distortion, i.e. the same Q z R (cid:54) = 0on all sites, which we take into account within the modelby using the same coupling constant λ s as for the Q x component.We also include hopping between next nearest neigh-bors and between second nearest neighbors along thecartesian coordinate axes in the refined TB model, butwe do not consider any spin-dependence of the corre-sponding hopping amplitudes. We describe the hoppingbetween next-nearest neighbors by spin-independent pa-rameters t xy corresponding to the hopping between two | z − r (cid:105) -type orbitals along the ± a c ˆ x ± a c ˆ y directions.The parameter t xy is taken as spin average over the corre-sponding MLWF matrix elements h xy calculated for thecubic structure. All other hopping matrix elements be-tween next nearest neighbors are determined from thisvia the following relations, which are derived assumingcubic symmetry and indirect hopping only (see Ref. 10): t xz = t xy (cid:0) − ηα (cid:1) (cid:18) − √ √ (cid:19) (22) t xy = t xy (cid:0) − ηα (cid:1) (cid:18) − (cid:19) (23)The same GFO-distortion-induced reduction as for thenearest neighbor hopping matrices is applied. The hop-ping between second nearest neighbors along the coordi-nate axes [ t ( ± a c ˆ x ), t ( ± a c ˆ y ), t ( ± a c ˆ z )] is included ac-cording to the ideal cubic symmetry relations describedby Eqs. (3)-(5), with a c replaced by 2 a c , t (cid:48) = 0, and t = t z , where t z is estimated from the MLWF matrixelements for the purely GFO distorted structure. Wenote that the reduction of this parameter compared tothe undistorted case is significantly stronger than for thenearest (and next nearest) neighbor hopping amplitudes.Furthermore, the hopping between third nearest neigh-bors along the coordinate axes [ t ( ± a c ˆ x ), t ( ± a c ˆ y ), t ( ± a c ˆ z )], that was considered in Sec. III A, becomes3 TABLE III: Parameters used in the TB models. refined simple t ↑↑ (eV) -0.648 -0.492 t ↓↓ (eV) -0.512 -0.492 t ↑↓ (eV) -0.569 -0.492 η − ˜ λ (eV˚A − ) 0.53 0 λ ↑ (eV˚A − ) 3.19 1.64 λ ↓ (eV˚A − ) 1.33 1.64 t xy (eV) -0.018 0 t z (eV) -0.020 0 e (eV) 15.356 15.505 J (eV) 1.5 1.805negligible as result of the GFO distortion.Finally, we include the Hund’s rule coupling in the re-fined TB model using the standard form [Eq. (6)] withan average value of J obtained from the MLWF on-sitesplitting. In order to relate the obtained TB bands tothe full GGA and MLWF band-structures, we determinethe on-site energy e as the spin and orbital average ofthe corresponding h aa matrix elements for the A-AFMexperimental structure.The values of all parameters used in the refined TBmodel are summarized in Table III. The JT distortion inthe experimental P bnm structure corresponds to α ii =1, Q x = − .
161 ˚A, and Q z = − .
048 ˚A, and the corre-sponding amplitude of the GFO distortion is α iii = 1.We also compare with a very simple TB model thatincludes only nearest neighbor hopping according toEqs. (3)-(5) with t (cid:48) = 0, and the standard JT and Hund’srule coupling as described by Eqs. (7) and (6). The pa-rameters for this model are chosen via typical simpli-fied fitting procedures: the nearest neighbor hopping pa-rameter − t is obtained as one sixth of the majority spinbandwidth W for the fully GFO distorted structure (iii)and FM order; the JT coupling constant λ is taken fromRef. 10, where it was obtained by fitting a similar TBmodel (including also next nearest neighbor hopping) toa DFT band-structure; J is calculated from the spin split-ting between FM majority and minority bands at the Γ-point for the cubic structure (i); and e is fitted suchthat the Fermi energy is aligned with the DFT calcula-tion value.Figure 11a shows the band dispersion obtained fromthe GGA calculation for the full experimental struc-ture (v) and A-AFM order as well as the correspond-ing MLWF bands. Fig. 11b/c shows the comparison be-tween the MLWF bands and the two different simplifiedTB models. It can be seen that the orthorhombic latticestrain and La displacements do not lead to significantqualitative changes in the band-structure as compared tostructure (iv) (see Fig. 9c). The comparison between theMLWF dispersion and the refined TB model (Fig. 11b)shows that, despite the many simplifications made, this FIG. 11: (Color online) (a) DFT bands (thin/grey lines) andMLWF bands (thick/red lines) for the A-AFM experimental
P bnm structure (v). Comparison of the MLWF bands withthe refined TB model (b) and with the simple nearest neighborTB model (c) The Fermi level is indicated by dashed lines. model reproduces the MLWF bands to a remarkable ac-curacy. The only major discrepancy can be seen for thelowest-lying local minority band along Γ-Z at E ∼
16 eV,which is slightly lower than the corresponding MLWFband. This can be traced back to an overestimation ofthe h x hopping amplitude, which results from the factthat we use the same reduction factor η for all hoppings.As can be seen in Fig. 8b, h x is affected more stronglyby the GFO distortion than any other nearest neighborhopping (for A-AFM order). The very simple nearestneighbor TB model depicted in Fig. 11c deviates muchstronger from the MLWF band structure than the refinedmodel, but still captures the overall dispersion surpris-ingly well. Consistent with our analysis from the previ-ous sections, the deviations are more pronounced for theenergetically higher local minority spin bands, which isclearly due to the neglected spin dependence of the hop-ping. As discussed in Sec. III B, the smaller JT couplingconstant used in the simple model partially cancels themissing effect of the JT distortion on the inter-orbital in-plane hopping parameters, leading to the relative goodagreement of the simpler model with the MLWF bandsaround the Fermi level. IV. SUMMARY AND CONCLUSIONS
We have shown that the construction of maximally lo-calized Wannier functions together with the disentangle-ment procedure described in Ref. 16 can be used to ex-tract effective e g bands in LaMnO even for cases wherethese bands are strongly entangled with other states.This procedure thus provides a very robust way for ex-tracting the “correlated subspace” used for example inDFT+DMFT calculations.We have used this procedure to obtain a TB param-4eterization of the e g bands for different structural mod-ifications of LaMnO with both FM and A-AFM order.By monitoring the effect of the individual distortions onthe MLWF matrix elements, we can assess the qualityof the various approximations and simplifications thatare commonly used in model Hamiltonians for mangan-ite systems. In particular, we find the following: • While the nearest neighbor hopping is clearly domi-nant, the further neighbor hopping along the carte-sian axes decays rather slowly. Taking into accountnearest, next-nearest, as well as second and thirdnearest hopping along the cartesian axes leads todeviations of less than 0.11 eV from the (cubic FM)DFT band structure. • In addition to the linear on-site coupling to theJT distortion, we observe a strong effect on thein-plane hopping amplitudes between different or-bitals. The corresponding splitting, which is due tothe underlying Mn-O hopping, partially cancels theeffect of the on-site term on the band dispersion,which has a strong influence on the determinationof the local JT coupling strength. • The GFO distortion leads to an overall reductionof all hopping amplitudes by about 25-30 %, andalso reduces the local JT splitting. This reductionis due to the weaker hybridization between Mn( e g )and O( p ) states for non-180 ◦ bond angle. • The higher energy of the (local) minority spinstates reduces the hybridization between the cor-responding atomic e g and O( p ) states, leading toreduced hopping amplitudes and JT coupling com-pared to the majority spin states. • The splitting between (local) majority and minorityspin states is generally well described by the localHund’s rule coupling, even though small variationsin the corresponding J values indicate the limits ofthe core spin approximation.It is apparent that the most crucial deviations from thesimple two band description are a result of the underly-ing Mn-O hybridization. Nevertheless, we have shown that a refined TB model that incorporates the effects de-scribed above using the parameters listed in Table IIIreproduces the DFT band structure calculated for thefull experimental crystal structure of LaMnO with re-markable accuracy. Whether this accuracy, at the prizeof more parameters in the model, is desirable depends ofcourse on the specific application of the model descrip-tion.Furthermore, our analysis shows that the effects of thevarious distinct structural distortions present in LaMnO are (to a good approximation) independent from eachother and can therefore be assessed individually. How-ever, the GFO distortion has to be taken into account toobtain the correct magnitude of the Jahn-Teller coupling.In comparison with the manual TB fits presented inRef. 10, the construction of MLWFs is less biased andmore universally applicable. It allows to calculate pa-rameters of the model instead of fitting them to eitherexperimental or computational data. In particular, it ispossible to obtain accurate TB representations even forrather complex band structures. However, care has to beapplied when parameters corresponding to a more com-plex parameterization are used for simpler models. Forexample, using the MLWF nearest neighbor hopping am-plitudes within a simple model that neglects all furtherneighbor hoppings, can lead to a significant underesti-mation of the total bandwidth, so that in certain casesa parameterization with renormalized nearest neighborhoppings, leading to a more accurate total bandwidth,might be more desirable. The analysis presented in thiswork demonstrates that, depending on the specific ap-plication at hand, MLWFs can in principle be used toconstruct more and more refined TB parameterizationswhich lead to realistic, materials-specific band structureswith very high accuracy. Acknowledgments
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