Coulomb repulsion and correlation strength in LaFeAsO from Density Functional and Dynamical Mean-Field Theories
V. I. Anisimov, Dm. M. Korotin, M. A. Korotin, A. V. Kozhevnikov, J. Kuneš, A. O. Shorikov, S. L. Skornyakov, S. V. Streltsov
aa r X i v : . [ c ond - m a t . s t r- e l ] O c t Coulomb repulsion and correlation strength inLaFeAsO from Density Functional and DynamicalMean-Field Theories
V. I. Anisimov , Dm. M. Korotin , M. A. Korotin ,A. V. Kozhevnikov , , J. Kuneˇs , A. O. Shorikov ,S.L. Skornyakov and S. V. Streltsov Institute of Metal Physics, Russian Academy of Sciences, 620041 YekaterinburgGSP-170, Russia Joint Institute for Computational Sciences, Oak Ridge National Laboratory P.O.Box 2008 Oak Ridge, TN 37831-6173, USA Theoretical Physics III, Center for Electronic Correlations and Magnetism, Instituteof Physics, University of Augsburg, Augsburg 86135, Germany
Abstract.
LDA+DMFT (Local Density Approximation combined with Dynamical Mean-Field Theory) computation scheme has been used to calculate spectral propertiesof LaFeAsO – the parent compound for new high-T c iron oxypnictides. Coulombrepulsion U and Hund’s exchange J parameters for iron 3 d electrons were calculatedusing first principles constrained density functional theory scheme in Wannier functionsformalism. Resulting values strongly depend on the number of states taken into accountin calculations: when full set of O-2 p , As-4 p , and Fe-3 d orbitals with correspondingbands are included, computation results in U =3 ÷ d orbitals and bands only, computation givesmuch smaller parameter values F =0.8 eV, J =0.5 eV. However, DMFT calculationswith both parameter sets and corresponding to them choice of basis functions resultin weakly correlated electronic structure that is in agreement with experimental X-rayand photoemission spectra.PACS numbers: 74.25.Jb, 71.45.Gm oulomb repulsion and correlation strength in LaFeAsO F =0 02 d -Fe D en s i t y o f s t a t e s , ( e V . a t o m ) - p -As 02 p -O Figure 1.
Total and partial densities of states for LaFeAsO obtained in DFTcalculation in frame of LMTO method.
Recent discovery of high- T c superconductivity in iron oxypnictides LaO − x F x FeAs [1]has stimulated an intense experimental and theoretical activity. In striking similaritywith high- T c cuprates, undoped material LaFeAsO is not superconducting with antifer-romagnetic commensurate spin density wave developing below 150 K [2]. Only whenelectrons (or holes) are added to the system via doping, antiferromagnetism is suppressedand superconductivity appears. As it is generally accepted that Coulomb correlationsbetween copper 3 d electrons are responsible for cuprates anomalous properties, it istempting to suggest that the same is true for iron 3 d electrons in LaFeAsO.The ratio of Coulomb interaction parameter U and band width W determinescorrelation strength. If U/W < U value is comparable with W oreven larger then the system is in intermediate or strongly correlated regime and Coulombinteractions must be explicitly treated in electronic structure calculations. The partiallyfilled bands formed by Fe-3 d states in LaFeAsO have the width ≈ U should be compared with this value.In practical calculations, U is often considered as a free parameter to achievethe best agreement of calculated and measured properties of investigated system.Alternatively, U value could be estimated from the experimental spectra. However,the most attractive approach is to determine Coulomb interaction parameter U valuein first principles non-empirical way. There are two such methods: constrainedDFT scheme [3, 4], where d -orbital occupancies in DFT calculations are fixed to thecertain values and U is numerically determined as a derivative of d -orbital energyover its occupancy, and Random Phase Approximation (RPA) method [5], wherescreened Coulomb interaction between d -electrons is calculated via perturbation theory. oulomb repulsion and correlation strength in LaFeAsO Figure 2. (Colour online) Module square of d x − y -like Wannier function computedfor Fe-3 d bands only (left panel) and for full set of O-2 p , As-4 p and Fe-3 d bands (rightpanel). Big sphere in the center marks Fe ion position and four small spheres aroundit correspond to As neighbors. Recently, the RPA calculations for Coulomb interaction parameter U in LaFeAsO werereported [6], where U value was estimated as 1.8 ÷ U =4 eV obtained in RPA calculations for metallic iron [8].This value for Coulomb parameter (with Hund’s exchange parameter J =0.7 eV) wasused in Dynamical Mean-Field Theory (DMFT) [9] calculations for LaFeAsO [7, 11, 10].Results of these works show iron 3 d electrons being in intermediate or strongly correlatedregime, as it is natural to be expected for Coulomb parameter value U =4 eV and Fe-3 d band width ≈ K -edge) spectroscopy [13], and photoemission spectroscopy [14].In all these works the conclusion was that DOS obtained in DFT calculationsgave good agreement with the experimental spectra and the estimations [13] forCoulomb parameter value are U < U =4 eV shows that first principles calculation of Coulombinteraction parameter U value for LaFeAsO is needed to determine the correlation effectsstrength in this material. Results of such calculations by constrained DFT calculationsare reported in the present work. We have obtained the value U < d bands in agreement with the estimates from spectroscopy.It is important to note that Coulomb interaction parameter U value depends onthe choice of the model and, more specifically, on the choice of the orbital set that isused in the model. For example, in constrained DFT calculations for high-T c cupratesthe resulting U value for Cu d -shell was found [15] between 8 and 10 eV. The U valuein this range was used in cluster calculations where all Cu d -orbitals and p -orbitals of oulomb repulsion and correlation strength in LaFeAsO x − y orbital percopper atom is explicitly included in the calculations [17], the U value giving goodagreement with experimental data falls down to 2.5 ÷ U value fromconstrained DFT calculations is ≈ d -orbitalsand p -orbitals of neighboring oxygens were taken into account with U close to thisvalue gave good agreement between calculated and experimental spectra [18]. However,in the model where only partially filled t g orbitals are included, much smaller U value (corresponding to Slater integral F =3.5 eV) gives the results in agreement withexperimental data [19].It is interesting that such a small U value can be obtained in constrained DFTcalculations for titanates and vanadates where only t g -orbital occupancies are fixedwhile all other states ( e g -orbitals of vanadium and p -orbitals of oxygens) are allowed torelax in self-consistent iterations [20, 19]. So the calculation scheme used in constrainedDFT (the set of the orbitals with fixed occupancies) should be consistent with basis setof the model where the calculated U value will be used.Another source of uncertainty in constrained DFT calculation scheme is a definitionof atomic orbitals whose occupancies are fixed and energies are calculated. In some DFTmethods, like Linearized Muffin-Tin Orbitals (LMTO), these orbitals could be identifiedwith LMTO. However, in other DFT calculation schemes, where plane waves are usedas a basis, like in pseudopotential method one should use more general definition forlocalized atomic like orbitals such as Wannier functions [23] (WFs). The practical wayto calculate WFs for specific materials using projection of atomic orbitals on Blochfunctions was developed in Ref. [24].In Fig. 1 the total and partial DOS for LaFeAsO obtained in LMTO calculationsare shown. Crystal field splitting for Fe-3 d orbitals in this material is rather weak(∆ cf =0.25 eV) and all five d orbitals of iron form common band in the energy region( −
2, +2) eV relative to the Fermi level (see grey region on the bottom panel in Fig. 1).There is a strong hybridization of iron t g orbitals with p orbitals of arsenic atoms whichform nearest neighbors tetrahedron around iron ion. This effect becomes apparent inthe energy interval ( − −
2) eV (white region on the bottom panel in Fig. 1) whereband formed by p orbitals of arsenic is situated. More week hybridization with oxygen Table 1.
The constrained DFT calculated values of average Coulomb interaction ¯ U and Hund’s exchange J (eV) parameters for d -symmetry Wannier functions computedwith two different sets of bands and orbitals. DFT method restricted to Fe-3 d bands full bands setTB-LMTO-ASA ¯ U =0.49, J =0.51 ¯ U =3.10, J =0.81PWSCF ¯ U =0.59, J =0.53 ¯ U =4.00, J =1.02 oulomb repulsion and correlation strength in LaFeAsO S pe c t r a l f un c t i on , e V - xy yz, zx -r -3 -2 -1 0 1 2 3Energy, eV, E F =000,5 x -y Figure 3.
Partial densities of states for Fe-3 d orbitals obtained within the DFT (filledareas) and LDA+DMFT orbitally resolved spectral functions for “restricted basis” and F =0.8 eV, J =0.5 eV (bold lines). p states reveals in ( − −
3) eV energy window (black region on the bottom panel inFig. 1).We have calculated Coulomb interaction U and Hund’s exchange J parametersfor WFs basis set via constrained DFT procedure with fixed occupancies for WFs of d symmetry. For this purpose we have used two calculation schemes based on twodifferent DFT methods. One of them involves linearized muffin-tin orbitals produced bythe TB-LMTO-ASA code [21]; corresponding WFs calculation procedure is described indetails in Ref. [25]. The second one is based on the pseudopotential plane-wave methodPWSCF, as implemented in the Quantum ESPRESSO package [22] and is describedin Ref. [26]. The difference between the results of these two schemes could give anestimation for the error of U and J determination.The WFs are defined by the choice of Bloch functions Hilbert space and by a setof trial localized orbitals that will be projected on these Bloch functions [25]. Weperformed calculations for two different choices of Bloch functions and atomic orbitals.One of them includes only bands predominantly formed by Fe-3 d orbitals in the energywindow ( −
2, +2) eV and equal number of Fe-3 d orbitals to be projected on the Blochfunctions for these bands. That choice corresponds to the model where only five d -orbitalper Fe site are included but all arsenic and oxygen p -orbitals are omitted. Second choiceincludes all bands in energy window ( − p , As-4 p andFe-3 d states and correspondingly full set of O-2 p , As-4 p and Fe-3 d atomic orbitals to beprojected on Bloch functions for these bands. That would correspond to the extendedmodel where in addition to d -orbitals all p -orbitals are included too.In both cases we obtained Hamiltonian in WF basis that reproduces exactly bandsformed by Fe-3 d states in the energy window ( −
2, +2) eV (Fig. 1), but in the secondcase in addition to that bands formed by p -orbitals in the energy window ( − −
2) eV oulomb repulsion and correlation strength in LaFeAsO -6 -5 -4 -3 -2 -1 0 1 2 30.3 Fe d xy S pe c t r a l f un c t i on , e V - Fe d yz, zx
Fe d -r Fe d x -y As p -6 -5 -4 -3 -2 -1 0 1 2 3Energy, eV012
O p
Figure 4.
Partial densities of states for Fe-3 d , As-4 p and O-2 p states obtained withinthe DFT (filled areas) and LDA+DMFT orbitally resolved spectral functions for “fullbasis” calculations with F =3.5 eV, J =0.81 eV (bold lines). will be reproduced too. However, WFs with d -orbital symmetry computed in those twocases have very different spatial distribution. In Fig. 2 the module square of d x − y -like WF is plotted. While for the case when full set of bands and atomic orbitals wasused (right panel) WF is nearly pure atomic d-orbital (iron states contribute 99%), WFcomputed using Fe-3 d bands only is much more extended in space (left panel). It hassignificant weight on neighboring As ions with only 67% contribution from central ironatom.The physical reason for such effect is p - d hybridization that is treated explicitly inthe case where both p - and d -orbitals are included. In the case where only Fe-3 d bandsare included in calculation p - d hybridization reveals itself in the shape of WF. Fe-3 d bands in the energy window ( −
2, +2) eV correspond to antibonding combination ofFe-3 d and As-4 p states and that is clearly seen on the left panel of Fig. 2.The different spatial distribution for two WFs calculated with full and restrictedorbital bases can be expected to lead to different effective Coulomb interaction forelectrons occupying these states. The results of constrained DFT calculations of theaverage Coulomb interaction ¯ U and Hund’s exchange J parameters for electrons onWFs computed with two different set of bands and orbitals (and using two differentDFT methods: LMTO and pseudopotential) are presented in Tab. 1.One can see that very different Coulomb interaction strength is obtained for oulomb repulsion and correlation strength in LaFeAsO R e ( Σ ( ω )) -5 0 5 ω , eV-0,5-0,4-0,3-0,2-0,10 I m ( Σ ( ω )) d(xy)d(yz)d(3z -r)d(zx)d(x -y ) Figure 5. (Colour online) Real (upper panel) and imaginary (lower panel) parts ofLDA+DMFT self energy interpolated on real axis with the use of Pad´e approximantfor “restricted basis” and F =0.8 eV, J =0.5 eV. separate Fe-3 d band and full bands set calculations. While the latter gives value 3 ÷ d band restricted calculation results in0.5 ÷ d band set”WF with 33% of WF on neighboring As atoms. Another reason for strong reduction ofcalculated ¯ U value in going from “full bands set” to “Fe-3 d band set” WF is additionalscreening via p - d hybridization with As-4 p band that is situated just below Fe-3 d band(see Fig. 1). The effect of decreasing of the effective ¯ U value in several times goingfrom full orbital model to restricted basis was found previously for high-T c cuprates( U =8 ÷
10 eV for full p - d -orbitals basis [15] and 2.5 ÷ U that can be estimated[4] as ¯ U = F − J/ F can be calculatedas F = ¯ U + J/
2. For “Fe-3 d band set” WF that gives F =0.8 eV at J =0.5 eV. Coulombparameters for ”full basis” were calculated in the same way using data from the first rowin Tab. 1, and were taken as F =3.5 eV and J =0.81 eV. With this set of parameterswe performed the LDA+DMFT [27] calculations (for detailed description of the presentcomputation scheme see Ref. [25]). The DFT band structure was calculated within theTB-LMTO-ASA method [21]. Crystal structure parameters were taken from Ref. [1].The LDA+DMFT calculations were performed for both models: with restricted toFe-3 d states basis and with full basis including also As- p and O- p states. For the later oulomb repulsion and correlation strength in LaFeAsO R e ( Σ ( ω )) -5 0 5 ω , eV-2-10 I m ( Σ ( ω )) d(xy)d(yz)d(3z -r)d(zx)d(x -y ) Figure 6. (Colour online) Real (upper panel) and imaginary (lower panel) parts ofLDA+DMFT self energy interpolated on real axis with the use of Pad´e approximantfor “full basis” and F =3.5 eV, J =0.81 eV. case double counting term ¯ U ( n DMF T − ) was used to obtain noninteracting Hamiltonian[26]. Here n DMF T is the total number of d -electrons obtained selfconsistently within theLDA+DMFT scheme. The effective impurity model for the DMFT was solved by theQMC method in Hirsh-Fye algorithm [28]. Calculations for both cases were performedfor the value of inverse temperature β =10 eV − . Inverse temperature interval 0 < τ < β was divided into 100 slices. 6 · QMC sweeps were used in self-consistency loop withinthe LDA+DMFT scheme and 12 · of QMC sweeps were used to calculate the spectralfunctions.The iron 3 d orbitally resolved spectral functions obtained within DFT andLDA+DMFT calculations for “restricted basis” with F =0.8 eV, J =0.5 eV are presentedin Fig. 3. The influence of correlation effects on the electronic structure of LaFeAsOis minimal: there are relatively small changes of peak positions for 3 z − r , xy and x − y orbitals (the shift toward the Fermi energy) and practically unchanged pictureof spectral function distribution for yz, zx bands. There is no appearance of eitherKondo resonance peak on the Fermi level or Hubbard bands in the energy spectrumwith such small values of U and J .Results for “full basis” LDA+DMFT calculations are presented in Fig. 4. Note,that though much larger Coulomb U and J parameters ( F =3.5 eV, J =0.81 eV) wereused in this calculation the spectra around the Fermi energy are very similar to thoseobtained in “restricted basis” calculations (Fig. 3). The general shape of spectra doesnot show either Kondo resonance peak on the Fermi level or Hubbard bands and thefeatures in Fe- d spectral functions below -2 eV correspond to hybridization with As- p oulomb repulsion and correlation strength in LaFeAsO -0,4-0,200,20,4 E n e r gy , e V d(xy)d(3z -r )d(yz),d(zx)d(x -y )LDA+DMFTLDAFe-d Figure 7. (Colour online) Splitting of Fe- d orbitals obtained in LDA and LDA+DMFTfor “full basis” and F =3.5 eV, J =0.81 eV. -6 -4 -2 0 2Energy, eV, E F =0 eV S p ec t r a l f un c ti on ( A r b . un it s ) Fe L XESLDA+DMFT Fe-d
Figure 8. (Colour online) Calculated Fe- d LDA+DMFT (“full basis” and F =3.5 eV, J =0.81 eV) spectral function (green) and experimental Fe L XES spectrum (blackcircles) from Ref. [12]. and O- p bands. The reason for such weak correlation effects in spite of relatively largeCoulomb repulsion parameters is very strong hybridization of Fe- d orbitals with As- p states (see peaks in Fe- d spectral function in the region -2 ÷ -3 eV corresponding toadmixture to As- p bands). This hybridization results in additional very effective channelof screening for Coulomb interaction between Fe- d electrons.This agrees with the results of soft X-ray absorption and emission spectroscopystudy [12]. It was concluded there that LaFeAsO does not represent strongly correlatedsystem since Fe L X-ray emission spectra do not show any features that would indicatethe presence of the low Hubbard band or the quasiparticle peak that were predicted bythe LDA+DMFT analysis [7, 11, 10] with the large U =4 eV. A comparison of the X-rayabsorption spectra (O K -edge) with the LDA calculations gave [13] an upper limit of theon-site Hubbard U ≈ p core-level spectra correspond to an itinerant character ofFe 3 d electrons. It was demonstrated there that the valence-band spectra are generallyconsistent with band-structure calculations except for the shifts of Fe 3 d -derived peakstoward the Fermi level. Such a shift is indeed observed in our LDA+DMFT results(Fig. 3). oulomb repulsion and correlation strength in LaFeAsO -6 -5 -4 -3 -2 -1 0Energy, eV, E F =0 eV S p ec t r a l f un c ti on ( A r b . un it s ) NdFeAsO x F PES single srystal 77 eVLDA+DMFT
Figure 9. (Colour online) Calculated total LDA+DMFT (“full basis” and F =3.5 eV, J =0.81 eV) spectral function (green) and experimental NdFeAsO x F − x PES spectrum(black circles) from Ref. [30].
The behavior of real part of self energy near zero frequency Σ( ω ) | ω → could give animportant information about band narrowing and renormalization of electron mass inthe system under consideration. Pad´e approximant [29] was used to obtain self energyon real frequencies. Results for calculation with both bases are presented in Figs. 5, 6.The calculated value of quasiparticle renormalization amplitude Z = (1 − ∂ Σ( ω ) ∂ω | ω =0) ) − was found to be 0.77, 0.63, 0.71, 0.46 for “restricted basis” and 0.56, 0.54, 0.45,0.56 for “full basis” for d xy , d yz (or d zx ), d z − r , d x − y orbitals, respectively. Thevalues for “restricted basis” agree well with the effective narrowing of the LDA+DMFTspectral functions comparing with LDA DOS curves (Fig. 3). Note, that quasiparticlerenormalization amplitude Z = (1 − ∂ Σ( ω ) ∂ω | ω =0) ) − from full basis calculations can notbe directly used to estimate effective narrowing of the bands. Due to the explicithybridization with As- p and O- p states actual narrowing should be weaker. Keepingin mind the above mentioned, we can conclude that both models agree well with eachother. The effective masses m ∗ = Z − are 1.3, 1.59, 1.41, 2.17 (“restricted basis”) for d xy , d yz (or d zx ), d z − r , d x − y orbitals, respectively. The d x − y orbital has the largesteffective mass and exhibits most evident narrowing of LDA spectrum (see Fig. 3). Thisorbital has its lobes directed to the empty space between nearest iron neighbors in ironplane. Hence it has the weakest overlapping, the smallest band width, and the largest U/W ratio.Small effective mass enhancement value shows that LaOFeAs belongs to the weakly oulomb repulsion and correlation strength in LaFeAsO et al where a strongly renormalized low energy band with a fraction of the originalwidth ( Z ≈ ≈ U = 4.5 eV) opens the gap” so that the systemis in strongly correlated regime on the edge of metal-insulator transition.In LaFeAsO the iron ion is in tetrahedral coordination of four As ions with slighttetragonal distortion of the tetrahedron. In tetrahedral symmetry group T d five d -orbitals should be split by crystal field ‡ on low-energy doubly degenerate set 3 z − r , xy corresponding to irreducible representation e g and high-energy triply degenerate set x − y , xz , yz for representation t g . We have calculated Wannier functions energy andhave found that t g – e g crystal field splitting parameter is very small ∆ cf ≈ t g and e g levels with the following values for orbital energies (energy of the lowest 3 z − r orbitalis taken as a zero): ε z − r =0.00 eV, ε xy =0.03 eV, ε xz,yz =0.26 eV, ε x − y =0.41 eV.Correlation effects lead not only to narrowing of the bands but also to substantial shiftof Fe- d orbitals energies. For “full basis” calculation adding of Re (Σ(0)) to the LDAorbital energies results in ε z − r =0.00 eV, ε xy =-0.37 eV, ε xz,yz =0.10 eV, ε x − y =0.20 eV(see Fig. 7). Note that the actual splitting should be smaller due to p - d hybridization.Comparisons of spectral function calculated within LDA+DMFT for full basisset with various experimental spectra are presented in Figs. 8, 9. One can see agood agreement in comparison of calculated Fe- d spectral function with Fe L XESspectrum (black circles) from Ref. [12]. The shoulder in experimental curve near -2.5eV correspond to the low energy peak in calculated spectrum that appears due to stronghybridization between Fe- d and As- p states (see also Fig. 1). Note that this features incalculated curve is absent in the calculations with restricted basis (see orbital resolvedspectra in Fig. 3).The total spectrum (Fig. 9) is in reasonable agreement with experimentalphotoemission data. The sharp peak at the Fermi energy corresponds to partially filledFe- d band. Broad bands between -2 and 6 eV represent oxygen and arsenicum p bands.In conclusion, we have calculated the values of Coulomb interaction parameters U and J via constrained DFT procedure in the basis of Wannier functions. Forminimal model including only Fe-3 d orbitals and bands we have obtained Coulombparameters F =0.8 eV, J =0.5 eV. For “full basis” calculations Coulomb parameters are F =3.5 eV, J =0.8 eV. The LDA+DMFT calculation for both models with calculatedparameters results in weakly correlated nature of iron d bands in this compound. Thisconclusion is supported by spectroscopic investigations of this material. As the choiceof the orbitals to construct the effective Hubbard model from LDA results is to someextent arbitrary, showing that different choices leading to largely different interactionparameters yield similar physical observables provides an important consistency test of ‡ The coordinate axes x, y in LaOFeAs crystal structure are rotated on 45 degrees from the standardtetrahedral notation so that xy and x − y orbitals are interchanged. oulomb repulsion and correlation strength in LaFeAsO References [1] Kamihara Y, Watanabe T, Hirano M and Hosono H 2008
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