A comparative study of the electronic and magnetic properties of BaFe_2As_2 and BaMn_2As_2 using the Gutzwiller approximation
aa r X i v : . [ c ond - m a t . s t r- e l ] S e p A comparative study of the electronic and magnetic properties ofBaFe As and BaMn As using the Gutzwiller approximation Y. X. Yao
Ames Laboratory-U.S. DOE. and Department of Physics and Astronomy,Iowa State University, Ames, Iowa 50011, USA
J. Schmalian
Ames Laboratory-U.S. DOE. and Department of Physics and Astronomy,Iowa State University, Ames, Iowa 50011, USA andKarlsruhe Institute of Technology, Institute for Theoryof Condensed Matter, 76131 Karlsruhe, Germany
C. Z. Wang, K. M. Ho
Ames Laboratory-U.S. DOE. and Department of Physics and Astronomy,Iowa State University, Ames, Iowa 50011, USA
G. Kotliar
Department of Physics and Astronomy,Rutgers University, Piscataway, New Jersey 08854, USA bstract To elucidate the role played by the transition metal ion in the pnictide materials, we comparethe electronic and magnetic properties of BaFe As with BaMn As . To this end we employ theLDA+Gutzwiller method to analyze the mass renormalizations and the size of the ordered magneticmoment of the two systems. We study a model that contains all five transition metal 3d orbitalstogether with the Ba-5d and As-4p states (ddp-model) and compare these results with a downfoldedmodel that consists of Fe/Mn d-states only (d-model). Electronic correlations are treated using themultiband Gutzwiller approximation. The paramagnetic phase has also been investigated usingLDA+Gutzwiller method with electron density self-consistency. The renormalization factors forthe correlated Mn 3d orbitals in the paramagnetic phase of BaMn As are shown to be generallysmaller than those of BaFe As , which indicates that BaMn As has stronger electron correlationeffect than BaFe As . The screening effect of the main As 4p electrons to the correlated Fe/Mn 3delectrons is evident by the systematic shift of the results to larger Hund’s rule coupling J side fromthe ddp-model compared with those from the d-model. A gradual transition from paramagneticstate to the antiferromagnetic ground state with increasing J is obtained for the models of BaFe As which has a small experimental magnetic moment; while a rather sharp jump occurs for the modelsof BaMn As , which has a large experimental magnetic moment. The key difference between thetwo systems is shown to be the d-level occupation. BaMn As , with approximately five d-electronsper Mn atom, is for same values of the electron correlations closer to the transition to a Mottinsulating state than BaFe As . Here an orbitally selective Mott transition, required for a systemwith close to six electrons only occurs at significantly larger values for the Coulomb interactions. . INTRODUCTION The discovery of high T c superconductivity in LaO − x F x FeAs has initiated a detailedinvestigation of these and related iron based compounds with the ultimate aim to identifynew high temperature superconductors[1]. Many binary, ternary and quaternary compoundshave been synthesized and investigated experimentally, some of which were quickly foundto exhibit high T c superconductivity with electron or hole doping or under pressure orintrinsically [2–7]. Meanwhile, several closely related and physically interesting systems havealso been studied. An interesting example is the BaMn As compound, which has the samelayered tetragonal ThCr Si -type crystal structure as the BaFe As at room temperature[8–11]. BaMn As is an antiferromagnetic (AFM) insulator with a small gap; while BaFe As ,a parent compound of the Fe-based high T c superconductors, is an AFM metal at lowtemperatures. The more localized behavior of the ordered magnetic state in the Mn-basedsystem suggests that electronic correlations in this system are stronger or more efficient inchanging the electronic properties of the material. A closer, comparative investigation ofdistinct transition metal ions therefore promises to reveal further whether iron is indeedspecial among its neighbors in the periodic table. This is also expected on qualitativegrounds on the basis of the Hund’s rule picture[13].Theoretically, the description of the normal states for iron pnictide superconductors iscomplicated by the multi-orbital nature and the presence of electron correlation effects.Interesting results can be obtained from low energy theories that are primarily concernedwith states in the immediate vicinity of the Fermi surface[14–19]. However, material specificinsight often requires a careful analysis of states over a larger energy window. In the ironpinctide and chalcogenides the strength of the correlations is controlled by the Hund’s rulecoupling J , rather than by the Hubbard U or the p-d charge transfer energy[13]. Thismakes these materials a new class of correlated metals, different from doped Mott or chargetransfer insulators. Calculations based on density functional theory (DFT) usually yield toowide Fe 3d bands and too large magnetic moment in the AFM ground state[12]. Recentlythe AFM phase of the iron pnictide parent compounds has been successfully described byusing a combination of DFT and dynamical mean field theory (DMFT) with full chargeself-consistency[20]. These investigations clearly demonstrated the important role played bythe Hund’s rule coupling J for the magnetic and electronic properties of the iron pnictides.3 systematic analysis of a large class of Fe-based materials further demonstrated that thenature and strength of local correlations are rather universal, while material specifics resultfrom changes in the kinetic energy, amplified by the frustrated nature of electron hoppingpath due to Fe-As-Fe and direct Fe-Fe overlaps[21]. This universality in the value of the localCoulomb interactions U , U ′ , J only exists if one includes in addition to the Fe-3d states atleast the As 4p orbitals. Downfolding to models with iron states only is sufficient to lead tomaterial specific variations in the local correlations. In view of these insights is it importantto compare the behavior of Fe-based systems with systems based on other transition metals,also in view of the ongoing debate whether these systems should be considered weakly orstrongly correlated.On a technical level, quite a few efforts have been devoted to study the electronic andmagnetic properties of the systems using the Gutzwiller variational method[22] and simpli-fied tight-binding models[23, 24] with the particular appeal that the Gutzwiller approachis computationally cheaper if compared to more sophisticated DMFT approaches such asthe usage of Quantum Monte Carlo algorithms. For example. the experimental magneticmoment was obtained in a three band Hubbard model of LaOFeAs which contains only Fe3d t g orbitals[23]. Calculations based on the five-band model show that the e g orbitals arealso important and the system exhibits a sharp transition from paramagnetic (PM) state toAFM state with large magnetic moment when J increases, which is different from the resultsof three-band model[24]. Another important question is whether, in addition to the Fe 3d e g orbitals, the inclusion of As 4p elections is required for a more quantitative descriptionof the AFM and paramagnetic phases. Finally, recent calculations for LaOFeAs based onthe LDA+Gutzwiller approach that includes a Fe3d–As4p Wannier-orbital basis does indeedreproduce the experimentally observed small ordered magnetic moment over a large regionof ( U, J ) parameter space[25].In this paper, we report a comparative study of the BaFe As and BaMn As on theelectronic and magnetic properties by investigating the Fe/Mn d-band model and the ddp-model that includes Fe/Mn 3d, Ba 5d, and As 4p orbitals with Gutzwiller approximation[26–28] which has been shown to be equivalent to a slave-boson mean field theory[29]. Theparamagnetic phase for the two systems has also been investigated using LDA+Gutzwillermethod with electron density self-consistency[30, 31]. BaMn As is shown to be a morecorrelated material than BaFe As . The screening effect of the As 4p electrons is evident4y the comparison between the results of the Fe/Mn d-model and those of the ddp-modelor LDA+Gutzwiller method. The key difference in the behavior of these systems is shownto be the distinct electron count of the 3d states, a result that is robust against changesin the detailed description of these systems. While the even number of electrons for anFe site requires that the system undergoes an orbitally selective Mott transition, Mn withfive electrons can more easily localize and therefore has a much stronger tendency towardslarge moment magnetism and Mott localization, even for the nominally same values of theCoulomb interactions. The prospect to tune the tendency towards localization by varyingthe transition metal ions may therefore play an important role in optimizing the strength ofelectronic correlations for high temperature superconductivity. II. FORMALISM
For the sake of being self-contained, we outline the main formalisms for the multibandmodel with the Gutzwiller approximation[26–28] and the LDA+Gutzwiller approach withelectron density self-consistency[30, 31].
A. Multiband model with the Gutzwiller approximation
The multi-band Hamiltonian with local many-body interactions is written as H = H + H (1)where the bare paramagnetic band Hamiltonian ( H ) is H = X ( iα ) =( jβ ) ,σ t iαjβ c † iασ c jβσ + X i,α,σ ε iα c † iασ c iασ (2) t iαjβ is the electron hopping element between orbital ϕ α at site i and orbital ϕ β at site j . ε iα is the orbital level. For the Fe/Mn d-model, α and β run over the Fe/Mn 3d orbitals. Forthe ddp-model, α and β run over the Fe/Mn 3d orbitals, Ba 5d orbitals, and As 4p orbitals. c † ( c ) is the electron creation (annihilation) operator. σ is the spin index.5he typical Hubbard density-density type interaction term ( H ) for Fe 3d electrons is H = U X i,γ n iγ ↑ n iγ ↓ + U ′ X i,γ<γ ′ ,σσ ′ n iγσ n iγ ′ σ ′ − J X i,γ<γ ′ ,σ n iγσ n iγ ′ σ (3)which is usually sufficient for describing collinear magnetic order[23]. Here γ is the correlatedFe/Mn 3d orbital index, U ′ = U − J , and n iγσ = c † iγσ c iγσ .We take a variational wave function of the Gutzwiller form, | Ψ G i = ˆ G | Ψ i rD Ψ (cid:12)(cid:12)(cid:12) ˆ G (cid:12)(cid:12)(cid:12) Ψ E (4)with the Gutzwiller approximation to calculate the expectation values of Eq. 1 [26–28]. Ψ is the uncorrelated wave function. ˆ G is the Gutzwiller projection operator, which adoptsthe following form ˆ G = e − P i F g i F |F i ihF i | (5)where |F i i is the Fock state generated by a set of n c † iγσ o |F i i = Y γσ (cid:16) c † iγσ (cid:17) n F iγσ | i (6)with n F iγσ = 0 or 1, which identifies whether there is an electron with spin σ occupied inorbital γ for Fock state F at the i th site. g i F is a variational parameter which controls theweight of the local Fock state |F i i in the Gutzwiller wave function. g i F = 1 for the empty andsingly occupied configurations because in these cases there are no electron-electron repulsioninvolved. According to Ref. [28], the expectation value of the electron Hamiltonian H canbe expressed as hHi G = X iα,jβ,σ ( z iασ z jβσ t iαjβ + ε iα δ αβ δ ij ) D c † iασ c jβσ E + X i F U i F p i F (7)where D c † iασ c jβσ E = D Ψ (cid:12)(cid:12)(cid:12) c † iασ c jβσ (cid:12)(cid:12)(cid:12) Ψ E and U i F = hF i |H | F i i . We define the Gutzwillerorbital renormalization factor z iασ ≡ z iγσ = 1 q n iγσ (cid:0) − n iγσ (cid:1) X F , F ′ √ p i F p i F ′ (cid:12)(cid:12)(cid:12)D F i (cid:12)(cid:12)(cid:12) c † iγσ (cid:12)(cid:12)(cid:12) F ′ i E(cid:12)(cid:12)(cid:12) (8)with n iγσ = h n iγσ i . p i F is the occupation probability of configuration |F i i , which absorbs g i F and serves as the additional variational parameter due to the Gutzwiller wave function.Since H in the Hamiltonian of Eq. 1 is obtained by downfolding the DFT band structure,some contributions from H have already been taken into account by H in a mean-field way.Such contributions are commonly referred to as the double counting term. Consequently,the total energy of the system is given by E tot = hHi G + E d.c. (9)with E d.c. = − X i (cid:0) ¯ U N id ( N id − / − ¯ J N id ( N id / − / (cid:1) (10)following the treatment in DFT+U method[32, 33]. Here ¯ U ( ¯ J ) is the averaged Coulomb U(Hund’s rule coupling J) parameter, which for d orbitals is given as[31, 34]¯ U = U + 4 U ′ J = ¯ U − U ′ + J (12) N id = P γσ n iγσ is the expectation value of the total Fe/Mn d-orbital occupancy at site i .The double counting term E d.c. introduces some spin-independent chemical potential shiftfor the Fe/Mn 3d orbitals,˜ ε iγ = − ¯ U (cid:18) N id − (cid:19) + ¯ J (cid:18) N id / − (cid:19) (13)In the Fe/Mn d-model that contains only 3d states, ˜ ε iγ has no effect since it merely shiftsall the Fe/Mn 3d levels by a constant. However, in the ddp-models ˜ ε iγ becomes importantbecause it changes the position of the Fe/Mn 3d levels with respect to Ba 4d and As 4porbitals and hence affects the electron flow between these orbitals.The minimization of E tot with the constraints of P F p i F = 1 and P F p i F n F iγσ = n iγσ yields an effective single-particle Hamiltonian H eff = X iα,jβ,σ ( z iασ z jβσ t iαjβ + µ iασ δ αβ δ ij ) c † iασ c jβσ (14)7here µ iασ = ε iα + ˜ ε iα + ∂z iασ ∂n iασ e iασ + η iασ if α belongs to the correlated Fe/Mn 3d orbitalsand µ iασ = ε iα otherwise. It has been shown that H eff describes the Landau-Gutzwillerquasiparticle bands[37], and the square of the orbital renormalization factor z correspondsto the quasiparticle weight[31]. { z iγσ } and { η iγσ } are obtained by solving the followingequations. X F ′ M i FF ′ √ p i F ′ = η i √ p i F (15)where M i F , F ′ = X γσ e iγσ q n iγσ (cid:0) − n iγσ (cid:1) (cid:12)(cid:12)(cid:12)D F i (cid:12)(cid:12)(cid:12) c † iγσ + c iγσ (cid:12)(cid:12)(cid:12) F ′ i E(cid:12)(cid:12)(cid:12) + δ FF ′ U i F − X γσ η iγσ n F iγσ ! (16)with e iγσ = P jβ (cid:16) z jβσ t iαjβ D c † iγσ c jβσ E + c.c. (cid:17) . Ψ from Eq. 14 and { p i F } from Eq. 15 needto be solved self-consistently to obtain the final solution[35]. B. LDA+Gutzwiller
The above formalism based on the Gutzwiller approximation on the multiband modelcan be naturally combined with DFT, in which full electron density convergence can beachieved[30, 31, 36]. The total energy density functional can be written as E [ ρ ] = D Ψ G (cid:12)(cid:12)(cid:12) ˆ T (cid:12)(cid:12)(cid:12) Ψ G E + E H [ ρ ] + E xc + Z ρ ( r ) v ion ( r ) d r + E ion − ion (17)where ˆ T is the kinetic energy operator and E H is the Hartree potential energy. The electron-ion and ion-ion potential energies are given by R ρ ( r ) v ion ( r ) d r and E ion − ion , respectively.In the LDA+Gutzwiller method[30, 31], the approximate exchange-correlation energy E xc takes the following form E xc = E LDAxc [ ρ ] + X i F U i F p i F + E d.c. (18)The electron density ρ under the Gutzwiller approximation can be written as ρ ( r ) = ′ X iα,jβ,σ z iασ z jβσ ϕ ∗ iα ( r ) ϕ jβ ( r ) D c † iασ c jβσ E + X iα,σ | ϕ iα ( r ) | h n iασ i (19)where { ϕ iα } is a general complete basis set including local correlated orbitals (e.g., Fe/Mn3d orbitals) as well as nonlocal orbitals. P ′ indicates that the summation excludes the term8 d-model ddp-model LDA -5-4-3-2-1012 BaFe As E ( e V ) X FIG. 1: (Color online) Band structure of BaFe As in PM phase based on LDA (solid line), Fed-model (circle) and ddp-model calculations (cross). Fermi level is shifted to zero. with iα = jβ . The kinetic energy is also renormalized D Ψ G (cid:12)(cid:12)(cid:12) ˆ T (cid:12)(cid:12)(cid:12) Ψ G E = ′ X iα,jβ,σ z iασ z jβσ D ϕ iα (cid:12)(cid:12)(cid:12) ˆ T (cid:12)(cid:12)(cid:12) ϕ jβ E D c † iασ c jβσ E + X iα,σ D ϕ iα (cid:12)(cid:12)(cid:12) ˆ T (cid:12)(cid:12)(cid:12) ϕ iα E h n iασ i (20)The total energy E [ ρ ] of the closed form can be similarly minimized as in the above multi-band model with respect to Ψ and { p i F } . The resultant effective single-particle Hamiltonianhas the same form as Eq. 14 with the hopping element t iαjβ = D ϕ iα (cid:12)(cid:12)(cid:12) ˆ T (cid:12)(cid:12)(cid:12) ϕ jβ E + Z (cid:0) v LDAscr ( r ) + v ion ( r ) (cid:1) ϕ ∗ iα ( r ) ϕ jβ ( r ) d r (21)The LDA screening potential v LDAscr ( r ) is the sum of the Hartree potential v H ( r ) = δE H [ ρ ] δρ and the exchange-correlation potential v LDAxc ( r ) = δE LDAxc [ ρ ] δρ . The renormalized orbital level µ iασ = ε iα + ˜ ε iα + ∂z iασ ∂n iασ ˜ e iασ + η iασ if α belongs to the correlated Fe/Mn 3d orbitals and µ iασ = ε iα otherwise. Here ε iα = t iαiα and e iγσ = P jβ (cid:16) z jβσ t iαjβ D c † iγσ c jβσ E + c.c. (cid:17) . { z iγσ } and { η iγσ } are obtained by solving the same equation as Eq.15. One can see that the majordifference is that the hopping element t iαjβ and the orbital level ε iα will be updated self-consistently by the LDA screening potential v LDAscr ( r ) in LDA+Gutzwiller method, whilethey are fixed in the multiband model. III. EFFECTS CAUSED BY DOWNFOLDING
The Fe/Mn d-model and ddp-model for BaFe As and BaMn As have been obtained bydownfolding the first-principles LDA electron bands of the experimentally determined crystal9 d-model ddp-model LDA -4-3-2-1012 E ( e V ) X BaMn As FIG. 2: (Color online) Band structure of BaMn As with settings same as Fig. 1. xyz Fe2 As Fe3Fe1
FIG. 3: (Color online) The atomic configuration of the Fe-As layer with the coordinate system.Note that the center As atom is above the Fe-plane, while the corner As atoms are below theFe-plane. structures[9, 38] using the quasi-atomic minimal basis-set orbitals (QUAMBOs)[39–42]. Themain idea of the QUAMBO approach is to recover a set of local quasi-atomic orbitals byperforming an inverse unitary transformation for the “bonding” and “anti-bonding” states.In practice, the “bonding states” are some occupied bands or low energy bands which areintended to be preserved in the new local orbital representation. The “anti-bonding” statesare constructed from the orthogonal subspace of the “bonding” subspace and are optimizedsuch that resultant QUAMBOs have maximal similarity with the free atomic orbitals. The10 e /Mn xy yz z xz x − y Fe /Mn : xy -0.34 /-0.32 ( -0.35 /-0.33) /0.20 ( /0.17) /0.28 ( /0.24) -0.25 /-0.20 ( -0.21 /-0.17) 0 yz /0.20 ( /0.17) -0.22 /-0.16 ( -0.14 /-0.10) -0.11 /-0.10 ( -0.09 /-0.08) /0.11 ( /0.08) /0.16 ( /0.15) z /0.28 ( /0.24) -0.11 /-0.10 ( -0.09 /-0.08) /0.03 ( /-0.01) /0.10 ( /0.08) 0 xz -0.25 /-0.20 ( -0.21 /-0.17) /0.11 ( / 0.08) /0.10 ( /0.08) -0.22 /-0.16 ( -0.14 /-0.10) /0.16 ( /0.14) x − y /0.16 ( /0.15) 0 /0.16 ( /0.15) -0.13 /-0.07 ( -0.09 /-0.03)Fe /Mn : xy -0.07 /-0.05 ( -0.06 /-0.05) /0.11 ( /0.08) 0 0 0 yz -0.14 /-0.11 ( -0.10 /-0.08) /0.11 ( /0.07) 0 0 0 z -0.01 /-0.02 ( /-0.01) /0.15 ( /0.12) -0.18 /-0.13 ( -0.12 /-0.08) xz -0.16 /-0.15 ( -0.13 /-0.12) /0.27 ( /0.17) /0.06 ( /0.05) x − y -0.17 /-0.13 ( -0.12 /-0.08) -0.04 /-0.06 ( -0.04 /-0.05) /0.09 ( /0.07)As: y ( -0.50 /-0.42) ( /0.33) 0 0 0 z /0.29) ( /0.12) ( -0.49 /-0.41) x /0.13) ( -0.69 /-0.58) ( /0.25) TABLE I: The hopping parameters between the orbitals of Fe1/Mn1 and those of its nearestneighbour Fe2/Mn2, next nearest neighbour Fe3/Mn3 and the center As atom. The entries of A1 /B1( A2 /B2) mean that A1 is the hopping parameter in the Fe d-model and B1 in the Mnd-model. A2 and B2 are the corresponding values in the ddp-models. orthogonal subspace is spanned by the wave functions in some energy window. It becomescomplete in the infinite energy or band limit, where succinct closed form for QUAMBOs canstill be obtained[41]. In principle, different choices of the “bonding” states and the energywindow for the “anti-bonding” states will yield different set of QUAMBOs and tight-bindingparameters, which is reasonable and acceptable since the downfolding of the band structureshould not be unique. We find that the ddp model will give too many Fe/Mn d-electrons if itis obtained by treating all the occupied states and virtual states up to 2eV above Fermi levelas the “bonding” states in the infinite band limit: 7 . As (close to the analysis of Ref.[43]) and 6 . As , while the nominal value would be6 and 5, respectively. Such large deviation of the correlated electron number clearly do notproperly reflect the correct physics of these materials. This problem does not exist for theFe/Mn d-model where the electron filling is fixed by assumption. In order to have a generaltight-binding model suitable for the correlated local orbital-based approaches, we choose afinite energy window around the Fermi level for the generation of the “anti-bonding” statesand another smaller energy window to control the Bloch bands which will contribute to theFe/Mn d orbitals. We can then construct tight-binding models with reasonable number ofcorrelated d-electrons. As a result, we have 6 . . .00.20.40.60.81.00.0 0.2 0.4 0.6 0.80.00.20.40.60.81.0 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8U=2eV Mn-d modelFe-d model U=2eV z xy xz/yz z x -y U=4eVU=3eV U=4eV
U=3eV
BaFe As BaMn As z J (eV)PM
J (eV)
J (eV)
FIG. 4: (Color online) The square of the orbital renormalization factor z iγ as a function of Hund’srule coupling J parameter at Hubbard U=2eV, 3eV and 4eV in the Fe d-model (upper panels) forBaFe As and Mn d-model for BaMn As (lower pannels) in the paramagnetic state. the ddp-model. Those are the initial occupancies that are further reduced due to the self-consistent determination of the occupancies within the Gutzwiller approach and the doublecounting corrections, reaching values close to 6 Fe 3d-electrons and 5 Mn 3d-electrons.Figure 1 and 2 show the bare band structures ( U = J = 0) of the d-model and theddp-model for BaFe As and BaMn As , respectively. The LDA band structures have alsobeen plotted for comparison. One can see that the band structure from the ddp-modelagrees very well with the LDA result in the low energy window for both systems. TheFe/Mn d-model, however, fails to reproduce one low energy unoccupied band near Γ-pointwhich is mainly contributed by the Ba 5d orbitals. The total number of electrons is 6(5)per Fe(Mn) atom for the d-model and 12(11) per Fe(Mn) atom for the ddp-model. Thetight-binding parameters between one Fe/Mn atom and its nearest Fe/Mn atom, secondthe nearest Fe/Mn atom and the nearest As atom are listed in Table I, with the atomicconfiguration and the coordinate system shown in Fig.3. The hopping parameters of theBaFe As system are overall larger in magnitude than those of BaMn As system, which ismainly owing to the fact that BaMn As has a much larger volume than BaFe As , althoughthe ionic radii of Fe and Mn are very close. The direct Fe-Fe/Mn-Mn hopping parametersbecome smaller in the ddp-model, as expected. The Fe d-model has been compared withthe one downfolded using maximally localized Wannier function (MLWF)[44, 45]. Althoughthe MLWF approach can achieve better fitting after some fine tuning[46], we find that theresults reported here are not sensitive to this detail.12 .00.20.40.60.81.00.0 0.2 0.4 0.6 0.80.00.20.40.60.81.0 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8U=2eV U=2eV z xy xz/yz z x -y U=4eVU=3eV U=4eV
U=3eV ddp-model
BaFe As BaMn As z J (eV)PM
J (eV)
J (eV)
FIG. 5: (Color online) Same as Fig. 4 but based on ddp-model.
IV. COMPARATIVE STUDY OF THE D AND DDP MODEL FOR PM PHASE
Figure 4 and 5 show the variation of the z -factors for the correlated d-orbitals withincreasing Hund’s rule coupling J and Hubbard U fixed at 2eV, 3eV and 4eV in the param-agnetic state of the Fe/Mn d-model and the ddp-model, respectively. On the small J side,the z -factors for both models of the compounds stay almost the same. With increasing J,the z -factors start to drop rapidly but remain finite for the two models of BaFe As . Incontrast, the z -factors exhibit a sharp decrease to zero for both models of BaMn As , i.e.,Mott localization for all the orbitals sets in beyond a threshold value of J . This indicatesthat BaMn As is a more correlated system than BaFe As . By increasing U from 2eV to4eV, the transition of the z -curves occurs at somewhat smaller J -values for all the modelcalculations. The results show a much stronger dependence on the Hund’s rule coupling J rather than Hubbard U , which is typically the case in correlated multiband systems. Com-paring the Fe/Mn d-model and the ddp-model, we observe a systematic shift of the z -curvesto larger J value with the inclusion of the Ba 5d and As 4p orbitals to the Fe/Mn d-model.Thus, a larger value of J is needed in a ddp-model to obtain the similar z -factors as in theFe/Mn d-model. This can be attributed to the screening effect of additional electrons in theddp-model as caused the hybridization of the Fe/Mn 3d orbitals with other, less correlateddegrees of freedom.The general feature of the local d-orbital occupations in the model calculations as shown inFig. 6 and 7 is consistent with that of the z -factors described above. While the Hubbard Uparameter tends to polarize the orbitals, the Hund’s rule coupling J favors the equalization13 .500.550.600.650.700.750.0 0.2 0.4 0.6 0.80.40.50.60.7 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8U=2eV Mn-d modelFe-d model U=2eV n xy xz/yz z x -y U=4eVU=3eV U=4eV
U=3eV
BaFe As BaMn As n J (eV)PM
J (eV)
J (eV)
FIG. 6: (Color online) Orbital occupations for the Mn/Fe d-model with settings same as Fig. 4. ddp-model
U=2eV n xy xz/yz z x -y U=4eVU=3eV U=4eV
U=3eV
BaFe As BaMn As n J (eV)PM
J (eV)
J (eV)
FIG. 7: (Color online) Local Fe/Mn 3d rbital occupations for the ddp-model with settings sameas Fig. 4. of the local orbital occupations. All local d-orbitals become half occupied as the systemapproaches the Mott transition for the models of BaMn Fe which has 5 electrons in each5 Mn 3d orbitals. For the BaFe As system, there is 6 electrons in each 5 Fe 3d orbitals.Hence we expect an orbital selective Mott transition would occur with large U and J [47, 48].We did perform model calculations at even larger U and find an orbitally selective transitionwhere the z factors of all orbitals except the d xy -orbital vanish, while z xy = 1 as the orbitalbecomes completely filled. This behavior requires however unphysically large values of U .Indeed some signature can be identified as one can see that n x − y becomes closest to 0.5 and z x − y drops most rapidly with increasing U and J for the Fe d-model. When comparing theresults of Mn/Fe d-model and the ddp-model, a similar systematic shift of the local orbitaloccupation curves to the large J side is observed due to the screening and hybridization effect.14 .0 0.2 0.4 0.6 0.80.00.20.40.60.81.0 BaFe As dxy dxz/yz dz2 dx2 z J (eV)
BaMn As U=3eVddp-modelNe=11 per Fe/Mn atom
FIG. 8: (Color online) The square of the orbital renormalization factor z iγ as a function of Hund’srule coupling J parameter for the ddp-model of BaMn As (solid symbol) and that of BaFe2As2(open symbol) with total number of electrons fixed to 11 per Fe/Mn atom and U=3eV. We conclude that BaMn Fe is approaching a Mott insulator with large localized moment.Combining our calculations in the magnetically ordered state and in the paramagnetic statewe obtain J ≃ . − . As is significantly away from Mott localization, yet there are clearly visible polarizations ofthe orbital populations that demonstrate that the system is in an intermediate regime. Todemonstrate that this is indeed the case we determined the quasiparticle weight of a systemwith the tight binding parameters (ddp-model) of BaFe As , yet with a total electron countof 11 per Fe atom as appropriate for BaMn As (see Fig. 8). Now a Mott transition similarto the actual BaMn As calculation occurs, albeit at a somewhat larger value of J . V. EFFECTS OF CHARGE SELF-CONSISTENCY
The variation of the z -factors and occupations of the correlated Fe/Mn 3d orbitals withU and J has further been investigated by the LDA+Gutzwiller method[30, 31], which is builtupon the LDA+U approach[49, 50]. We use the same correlated orbitals obtained via theQUAMBO procedure as in the Fe/Mn d-models for the calculations and achieve the electrondensity self-consistency. As shown in Fig. 9, the z -curves exhibit a similar trend but vary ina way much slower than that in the Fe/Mn d-models, which can again be attributed to the15 .00.20.40.60.81.0 0.1 0.3 0.5 0.7 0.90.00.20.40.60.81.0 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9U=2eV U=2eV z xy xz/yz z x -y U=4eVU=3eV U=4eV
U=3eVLDA+Gutzwiller
BaFe As BaMn As z J (eV)PM
J (eV)
J (eV)
FIG. 9: (Color online) Same as Fig. 4 but based on LDA+G calculations.
LDA+Gutzwiller
U=2eV n xy xz/yz z x -y U=4eVU=3eV U=4eV
U=3eV
BaFe As BaMn As n J (eV)PM
J (eV)
J (eV)
FIG. 10: (Color online) Local Fe/Mn 3d orbital occupations based on LDA+G calculations withsettings same as Fig. 4. screening effect of the other electrons in the systems. Our results on the BaFe As system arein good agreement with those reported in Ref. [22]. Overall, the z -factors for BaMn As are systematically smaller than those for BaFe As . Note that the calculated screenedU(J) is 3.6(0.76)eV for Fe and 3.2(0.70)eV for Mn based on the constrained random phaseapproximation and the maximally localized Wannier function[51], it is fairly reasonable toassume that the screened interaction parameters in the models investigated here are alsovery close. Therefore BaMn As is a more correlated material than BaFe As . VI. COMPARATIVE STUDY OF THE D AND DDP MODEL FOR AFM PHASE
The ground state AFM phase has also been studied for the models of the two systems.Figure 11 shows the behavior of the magnetic moment as a function of J with U fixed at16
BaFe As ddp-modelFe d-model U=2eV
U=3eV
U=4eV M () S-AFM M () J (eV)
FIG. 11: (Color online) Magnetic moment vs J at U=2eV (squares), 3eV (circles) and 4eV (trian-gles) in the Fe d-model (upper panel) and the ddp-model (lower panel) for BaFe As . The solidsymbols indicate the results using Gutzwiller approximation, and open symbols using Hartree-Fockapproximation. As . In contrast to theFe d-model including complete local interactions for LaOFeAs in which the system exhibitsa sharp transition from PM state to AFM state with large magnetic moment using theGutzwiller approximation[24], the models with only density-density type interactions forBaFe As describe the PM-AFM transition fairly well as the magnetic moment increasesrapidly yet smoothly when J rises as indicated by the solid symbols in the upper panel.The magnetic moment varies from 0 . µ B to 1 . µ B in the range of J = 0 . − . U = 3eV. For reference the experimental magnetic moment is 0 . µ B for BaFe As [52].With the inclusion of the Ba 5d and As 4p orbitals in the ddp-model, the magnetic momentincreases slower as seen in the lower panel. The high sensitivity of the magnetic momentto Hund’s rule coupling J is consistent with the DFT+DMFT calculation results[20]. Forcomparison, Fig. 11 also shows the magnetic moment of the AFM phase in the Fe d-model with Hartree-Fock approximation by the open symbols in the upper panel. WithinHartree-Fock approximation, the PM-AFM transition takes place at much smaller J. Themagnetic moment exhibits quite smooth variation with increasing J at U=2eV. However, anabrupt transition from PM to AFM state with large magnetic moment occurs at U=3eV. In17 ddp-modelMn d-model U=2eV
U=3eV
U=4eV M () G-AFM M () J (eV)
BaMn As FIG. 12: (Color online) Magnetic moment vs J curves for BaMn As with setting as as Fig. 11. comparison, the models of BaMn As yield a very sharp transition from PM to the G-typeAFM ground state with magnetic moment M ≈ µ B as shown in Fig. 12. However, it isstill reasonable since the experimental magnetic moment is 3.88 µ B [9]. The z -factors for themodels of BaFe As in the AFM ground state are always or order unity ( > . VII. CONCLUSION
In summary, we have studied the Fe/Mn d-model and the ddp-model on the electronicand magnetic properties of the BaFe As and BaMn As systems. The renormalizationfactors of the correlated Mn 3d orbitals are found to be systematically smaller than those ofFe 3d orbitals, implying that the electron correlation for BaMn As are much more efficientto cause Mott localization physics compared to BaFe As . Ultimately this is due to the factthat Fe, with its close to six electrons, must undergo an orbital selective Mott transition whilethe odd number of electrons in case of Mn allow for a more ordinary Mott transition. Whilethe strength of the interactions are not sufficient for Mott localization in either system theMn-based material is significantly closer to localization. The LDA+Gutzwiller results on theparamagnetic phase with electron density self-consistency also confirm the conclusion. Thevariation of the magnetic moment in the AFM ground state of the two compounds seems inaccordance with the experimental results: a smooth increase of the magnetic moment with18ising J for the stripe-AFM ground state of BaFe As with small experimental magneticmoment and an abrupt jump of the magnetic moment from 0 to about 3 µ B for the G-AFMground state of BaFe As with large experimental magnetic moment. We also checked thatthe two different ordered states are indeed energetically lower in the respective systems.The Gutzwiller approximation, however, is not able to provide a quantitatively consistentdescription of the AFM ground state for the iron pnictide systems with both correct magneticmoment and band renormalization factor for the same parameters. Nevertheless it is acomparatively easy to implement and powerful tool that allows for a first analysis of the roleof magnetic correlations, the nature of magnetically ordered states in strongly correlatedmulti-orbital materials. Acknowledgments
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