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arXiv:cond-mat/0309102v3 [cond-mat.str-el] 19 Jan 2004

Mott transition and suppression of orbital fluctuations in orthorhombic 3d1perovskites

E. Pavarini,1S. Biermann,2A. Poteryaev,3A. I. Lichtenstein,3A. Georges,2and O.K. Andersen4

1INFM and Dipartimento di Fisica “A. Volta”, Universit` a di Pavia, Via Bassi 6, I-27100 Pavia, Italy

2Centre de Physique Th´ eorique, Ecole Polytechnique, 91128 Palaiseau Cedex, France

3NSRIM, University of Nijmegen, NL-6525 ED Nijmegen, The Netherlands

4Max-Planck-Institut f¨ ur Festk¨ orperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany

Using t2g Wannier-functions, a low-energy Hamiltonian is derived for orthorhombic 3d1transition-

metal oxides. Electronic correlations are treated with a new implementation of dynamical mean-field

theory for non-cubic systems. Good agreement with photoemission data is obtained. The interplay

of correlation effects and cation covalency (GdFeO3-type distortions) is found to suppress orbital

fluctuations in LaTiO3, and even more in YTiO3, and to favor the transition to the insulating state.

PACS numbers: 71.27.+a, 71.30.+h, 71.15.Ap

Transition-metal perovskites have attracted much in-

terest because of their unusual electronic and magnetic

properties arising from narrow 3d bands and strong

Coulomb correlations [1]. The 3d1perovskites are partic-

ularly interesting, since seemingly similar materials have

very different electronic properties: SrVO3 and CaVO3

are correlated metals with mass-enhancements of respec-

tively 2.7 and 3.6 [2] while LaTiO3and YTiO3are Mott

insulators with gaps of respectively 0.2 and 1eV [3].

In the Mott-Hubbard picture the metal-insulator tran-

sition occurs when the ratio of the on-site Coulomb re-

pulsion and the one-electron bandwidth exceeds a criti-

cal value Uc/W,which increases with orbital degeneracy

[4, 5]. In the ABO3perovskites the transition-metal ions

(B) are on a nearly cubic (orthorhombic) lattice and

at the centers of corner-sharing O6 octahedra. The 3d

band splits into pdπ-coupled t2gbands and pdσ-coupled

egbands, of which the former lie lower, have less O char-

acter, and couple less to the octahedra than the latter.

Simplest theories for the d1perovskites[1] are therefore

based on a Hubbard model with 3 degenerate,

t2gbands per B-ion, and the variation of the electronic

properties along the series is ascribed to a progressive

reduction of W due to the increased bending of the pdπ

hopping paths (BOB bonds).

This may not be the full explanation of the Mott tran-

sition however, because a splitting of the t2glevels can

effectively lower the degeneracy. In the correlated metal,

the relevant energy scale is the reduced bandwith asso-

ciated with quasiparticle excitations. Close to the tran-

sition, this scale is of order ∼ ZW, with Z ∼ 1 − U/Uc,

and hence much smaller than the original bandwith W.

A level splitting by merely ZW is sufficient to lower

the effective degeneracy all the way from three-fold to a

non-degenerate single band[6]. This makes the insulating

state more favorable by reducing Uc/W[5, 6]. Unlike in

eg-band perovskites, such as LaMnO3, where large (10%)

cooperative Jahn-Teller (JT) distortions of the octahedra

indicate that the orbitals are spatially ordered, in the t2g-

band perovskites the octahedra are almost perfect. The

t2gorbitals have therefore often been assumed to be de-

1

6-filled

generate. If that is true, it is conceivable that quantum

fluctuations lead to an orbital liquid [7] rather than or-

bital ordering. An important experimental constraint on

the nature of the orbital physics is the observation of an

isotropic, small-gap spin-wave spectrum in both insula-

tors[8]. This is remarkable because LaTiO3is a G-type

antiferromagnet with TN=140K, m=0.45µB, and a 3%

JT stretching along a [9], while YTiO3is a ferromagnet

with TC=30K, m0∼0.8µB, and a 3% stretching along y

on sites 1 and 3, and x on 2 and 4 [10] (see Fig.1).

We shall find that the t2g degeneracy is lifted at the

classical level. This is not due to the small JT distortions

via OB pdπ-coupling, but to the GdFeO3-type distortion

which tilts the corner-sharing octahedra around the b-

axis (by 0,9,12, and 20◦) and rotates them around the

c-axis (by 0,7,9, and 13◦), as we progress from cubic

SrVO3via CaVO3and LaTiO3to YTiO3[9, 10, 11, 12].

This distortion is driven by the increasing oxygen-cation

(OA) pdσ-covalency, and it primarily pulls closer 4 of the

12 oxygens neighboring a given cation[13]. Moreover, 2

to 4 of the 8 cations neighboring a given B ion are pulled

closer [14]. The t2g orbitals couple to the OA distor-

tion via oxygen (BOAdpπ-pdσ), and they couple directly

(AB ddσ) to the AB distortion. As seen in Fig.1, the

orthorhombic GdFeO3-type distortion also leads to qua-

drupling of the cell. These findings are consistent with

conclusions drawn in the most recent model Hartree-Fock

study[15]. The correct magnetic orders in LaTiO3 and

YTiO3were also obtained with the LDA+U method [16].

The predicted moment and orbital order in YTiO3were

confirmed by NMR [17] and neutron scattering [18], but

not in LaTiO3. These static mean-field methods are not

appropriate for the metallic systems, however.

In this letter, we shall (i) present a new implementa-

tion of a many-body method [19, 20], which allows for

a quantitative, material-specific description of both the

Mott transition and the orbital physics, and (ii) use it to

explain why some of the d1perovskites are metallic and

others are insulators, why the metals have different mass

enhancements and the insulators different gaps. Such

properties can be described by a low-energy, multi-band

Page 2

2

FIG. 1:

the occupied t2g orbitals for LaTiO3 (top) and YTiO3

(bottom) according to the LDA+DMFT calculation.

oxygens are violet, the octahedra yellow and the cations

orange. In the global, cubic xyz-system directed ap-

proximatelyalongthe Ti-O

translations are a=(1,−1,0)(1 + α),

and c=(0,0,2)(1 + γ), with α,β,γ small.

to 4 are: a/2, b/2, (a+c)/2, and (b+c)/2.

ab-plane is a mirror (z ↔ −z), and so is the Ti bc-plane

(x ↔ y) when combined with the translation (b-a)/2. See:

http://www.mpi-stuttgart.mpg.de/andersen/cm/0309102.html

Pbnm primitive cell (right), subcell1 (left), and

The

bonds, the

b=(1,1,0)(1 + β),

The Ti sites 1

The La(Y)

orthorhombic

Hubbard Hamiltonian,

H = HLDA+1

2

?

imm′σUmm′nimσnim′−σ

+1

2

?

im(?=m′)σ(Umm′ − Jmm′)nimσnim′σ,(1)

where nimσ = a+

tron with spin σ in a localized orbital m at site i.

This Hamiltonian depends on how the imσ-orbitals

are chosen.HLDAis the one-electron part given by

density-functional theory (LDA), which should provide

the proper material dependence. Recently it has become

feasible to solve (1) using the dynamical mean-field ap-

proximation (DMFT)[19] and to obtain realistic physical

properties[20]. In the original LDA+DMFT implemen-

tations it was assumed that the on-site block(s) of the

single-particle Green function is diagonal in the space of

the correlated orbitals, and these were taken as orthonor-

mal LMTOs approximated by truncated and renormal-

ized partial waves. Although these approximations are

good for cubic t2gsystems such as SrVO3[21], they dete-

riorate with the degree of distortion. Our new implemen-

tation of LDA+DMFT uses a set of localized Wannier

functions in order to construct a realistic Hamiltonian

(1), which is then solved by DMFT, including the non-

imσaimσ, and a+

imσcreates an elec-

000

0.5 0.5

-1-1-1000111

Nm,m(E)

E (eV) E (eV)E (eV)

Nm,m(E)

Nm,m(E)

000

0.50.5 0.5

Nm,m(E)

Nm,m(E)

Nm,m(E)

-1 0 1

-0.1 -0.1-0.1

0 0 0

0.1 0.1 0.1

Nm,n(E) Nm,n(E) Nm,n(E)

-101

E (eV)

1 0

1

CaVO3

SrVO3

LaTiO3YTiO3

E (eV)

FIG. 2: t2g LDA DOS matrix (states/eV/spin/band) in the

Wannier representation. On-site-1 elements: Nxz,xz (red)

Nyz,yz (green), and Nxy,xy (blue).

Nxz,xy (green), and Nxy,yz (blue). εF ≡ 0.

Insets:

Nyz,xz (red),

diagonal part of the on-site self-energy.

For an isolated set of bands, a set of Wannier functions

constitutes a complete, orthonormal set of orbitals with

one orbital per band. For the d1perovskites we take

the correlated orbitals to be three localized t2gWannier-

orbitals, and in HLDAwe neglect the degrees of free-

dom from all other bands. In order to be complete, such

a Wannier orbital must have a tail with e.g.Opπ and

Ad characters. Our Wannier orbitals are symmetrically

orthonormalized Nth-order muffin-tin orbitals (NMTOs)

[22], which have all partial waves other than Bxy, yz,

andzx downfolded. Such a t2g NMTO can have on-site

egcharacter, and that allows the orbital to orient itself

after the surroundings, although xy,yz, and zx refers to

the global cubic axes defined in Fig.1. Fourier transfor-

mation of the orthonormalized 12×12 NMTO Hamilto-

nian, HLDA(k), yields on-site blocks and hopping inte-

grals. For the on-site Coulomb interactions in Eq.(1),

we use the common assumption that, as in the isotropic

case, they can be expressed in terms of two parameters:

Umm=U, Umm′=U − 2J, and Jmm′(?=m)=J [23]. From

Ref.24, J=0.68eV for the vanadates and 0.64eV for the

titanates. Since our Hamiltonian involves only correlated

orbitals, so that the number of correlated electrons is

fixed, the double-counting correction amounts to an ir-

relevant shift of the chemical potential. H is now solved

within DMFT, i.e.under the assumption that the com-

ponents of the self-energy between different sites can be

neglected. As a result, the self-energy can be obtained

from the solution of an effective local impurity model

which involves only 3 correlated orbitals.

to all previous studies, we take all components of the

self-energy matrix Σmm′ between different Wannier func-

tions on a given B-site into account [25]. From this 3×3

matrix, by use of the Pbnm symmetry (Fig.1), we con-

In contrast

Page 3

3

struct a 12×12 block-diagonal self-energy matrix. The

latter is then used together with HLDA(k) to obtain the

Green function at a given k-point. Fourier transforma-

tion over the entire Brillouin zone yields the local Green

function associated with a primitive cell and its 3×3 on-

site block is used in the DMFT self-consistency condition

in the usual manner. The 3-orbital impurity problem

is solved by a numerically exact quantum Monte Carlo

scheme[26]. To access temperatures down to 770K, we

use up to 100 slices in imaginary time. 106QMC sweeps

and 15-20 DMFT iterations suffice to reach convergence.

Finally, the spectral function is obtained using the max-

imum entropy method [27].

We now present the LDA results for the four per-

ovskites. Fig. 2 displays the on-site DOS matrix Nmm′(ε)

in the representation of the xy,yz, and zx Wannier func-

tions. SrVO3 is cubic and its t2g band with a width

W=2.8 eV consists of 3 non-interacting subbands, each of

which is nearly 2D and gives rise to a nearly logarithmic

DOS peak. In CaVO3, W is reduced to 2.4eV because

the Wannier orbitals are misaligned by the GdFeO3-type

distortion and because some of their O2p character is

stolen by the increased OA covalency, which drives this

distortion. The energy of the xy Wannier orbital (the

center of gravity of Nxy,xy) is 80meV lower than that of

the degenerate xz and yz orbitals, and small off-diagonal

DOS elements appear. Going from the vanadates to

the titanates, the effects of OA and (Ad)(Bt2g) cova-

lency increase dramatically, because now A and B are

1st rather than 3rd-nearest neighbors in the periodic ta-

ble. As consequences, the increased misalignment and

loss of oxygen character reduces the bandwidths to 2.1

and 2.0eV in LaTiO3and YTiO3, and weak hybridiza-

tion with the Ad bands deforms the t2gband.A pseudo-

gap which starts out as a splitting of the van Hove peak

in CaVO3, deepens and moves to lower occupancy as

we progress to LaTiO3 and YTiO3. The deep pseudo-

gap in YTiO3is mainly caused by the hybridization with

the Yxy orbital. The xy,yz,andzx Wannier orbitals are

now strongly mixed and diagonalization of the on-site

blocks of HLDAyields three singly-degenerate levels with

the middle (highest) being 140 (200) meV above the

lowest in LaTiO3, and 200 (330) meV in YTiO3. This

splitting is not only an order of magnitude smaller than

the t2g bandwidth, but also smaller than the subband-

widths, in particular for LaTiO3. As a consequence, the

eigenfunction for the lowest level is occupied by merely

0.45electron in LaTiO3and 0.50 in YTiO3, while the re-

maining 0.55 (0.50)electron occupies the two other eigen-

functions. The eigenfunction on site 1 with the lowest en-

ergy is 0.604|xy? + 0.353|xz? + 0.714|yz? in LaTiO3and

0.619|xy?−0.073|xz?+0.782|yz?in YTiO3. The splittings

are large compared with the spin-orbit splitting (20meV)

and kT, and they are not caused by the JT distortions, as

we have verified by turning them off in the calculations.

Next, we turn to the LDA+DMFT results. Calcula-

-4 -4-4666 -3 -3-3 -2 -2-2 -1 -1-1000111222333444555666

E(eV) E(eV)E(eV)

000

0.20.2 0.2

0.40.4 0.4

0.60.60.6

0.80.80.8

111

-4 -4 -4 -3-3 -3 -2-2 -2-1 -1-1000111222333444555

DOS/states/eV/spin/band

E(eV)E(eV)E(eV)

DOS/states/eV/spin/bandDOS/states/eV/spin/band

000

0.2 0.2 0.2

0.40.4 0.4

0.60.6 0.6

0.8 0.80.8

11

DOS states/ev/spin/bandDOS states/ev/spin/band DOS states/ev/spin/band

J=0.68 eV

U=5 eV

J=0.68 eV

U=5 eV

J=0.64 eVJ=0.64 eV

U=5 eV

YTiO3

LaTiO3

SrVO3

CaVO3

U=5 eV

FIG. 3: DMFT spectral function at T = 770K (thick line)

and LDA DOS (thin line). µ ≡ 0.

tions were performed for several values of U between 3

and 6eV. We found that the critical ratio Uc/W decreases

when going along the series: SrVO3, CaVO3, LaTiO3,

and YTiO3. This is consistent with the increasing split-

ting of the t2glevels and indicates that the Mott transi-

tion in the d1series is driven as much by the decrease

of effective degeneracy as by the reduction of bandwidth.

The main features of the photoemission spectra for all

four materials, as well as the correct values of the Mott-

Hubbard gap for the insulators [3], are reproduced by

taking U constant ∼5eV. This is satisfying, because U

is expected to be similar for vanadates and titanates,

although slightly smaller for the latter [24].

we show the DMFT spectral functions together with the

LDA total DOS. For cubic SrVO3we reproduce the re-

sults of previous calculations [21, 28]: the lower Hubbard

band (LHB) is around −1.8eV and the upper Hubbard

band (UHB) around 3eV. Going to CaVO3, the quasi-

particle peak looses weight to the LHB, which remains at

−1.8eV, while the UHB moves down to 2.5eV. These re-

sults are in good agreement with photoemission data[29]

and show that SrVO3and CaVO3are rather similar, with

the latter slightly more correlated. Similar conclusions

were drawn in Ref.21. From the linear regime of the

self-energy at small Matsubara frequencies we estimate

the quasi-particle weight to be Z=0.45 for SrVO3 and

0.29 for CaVO3. For a k-independent self-energy, as as-

sumed in DMFT, this yields m∗/m = 1/Z = 2.2 for

SrVO3and 3.5 for CaVO3, in reasonable agreement with

the optical-conductivity values 2.7 and 3.6[2].

For LaTiO3 and YTiO3 the LHB is around −1.5eV,

in accord with photoemission[30].

ilar bandwidths, the gaps are very different, 0.3 and

1eV, and this also agrees with experiments[3].

is consistent with our findings that the t2g-level split-

tings are smaller and (U − 2J)c/W is larger in LaTiO3

than in YTiO3, where the orbital degeneracy is effec-

In Fig.3

Despite very sim-

This

Page 4

4

tively 1.

numbers reveals that for the titanates one orbital per site

is nearly full, in contrast to LDA. It contains 0.88electron

in LaTiO3and 0.96 in YTiO3. The orbital polarization

increases around the metal-insulator transition and be-

comes complete thereafter. Thus, for the vanadates, each

orbital is approximately 1/3 occupied for all U in the

range 0to 6eV. The nearly complete orbital polarization

found by LDA+DMFT for the two insulators indicates

that correlation effects in the paramagnetic Mott insu-

lating state considerably decrease orbital fluctuations,

and makes it unlikely that YTiO3is a realization of an

orbital liquid[7]. In LaTiO3 some orbital fluctuations

are still active, although quite weak. The occupied or-

bital in LDA+DMFT is 0.586|xy?+0.275|xz?+0.762|yz?

for LaTiO3 and 0.622|xy? − 0.029|xz? + 0.782|yz? for

YTiO3. Hence, it is nearly identical with the ones we

obtained from the LDA as having the lowest energy.

For YTiO3 our orbital is similar to the one obtained

in Ref.[16] and for LaTiO3 it is similar to the one ob-

tained in Refs.[9, 15]. Our accurate Wannier func-

tions show why these orbitals (Fig.1, left) have the

lowest energy: The positive (blue) lobes have bonding

3z2

nearest cations –those along [111]– and the negative (red)

lobes have bonding xy character on the next-nearest

cations –those along [1-11]– whose oxygen surrounding

is favorable for this type of bond, i.e.where (Op)(Yxy)

pdσ-hybridization is strong. The former type of AB co-

valency dominates in LaTiO3, while the latter dominates

in YTiO3, where the shortest YO bond is merely 10%

longer than the TiO bond. The difference seen (Fig.1,

right side) between the orbital orders in the two com-

pounds is therefore quantitative rather than qualitative;

it merely reflects the extent to which the orbital has

the bc-plane as mirror. The two different JT distortions

of the oxygen square is a reaction to, rather than the

cause of the difference in the orbital orders. This differ-

ence is reflected in the hopping integrals between nearest

neighbors: tx=ty=99 (38) meV and tz=105 (48) meV

for LaTiO3(YTiO3). These hoppings are fairly isotropic

and twice larger in LaTiO3 than in YTiO3. Moreover,

the hoppings to the two excited orbitals are stronger in

YTiO3 than in LaTiO3. All of this is consistent with

LaTiO3being G-type anti- and YTiO3ferromagnetic at

low temperature, and it warrants detailed future calcu-

lations of the spin-wave spectra.

In conclusion, we have extended the LDA+DMFT ap-

proach to the non-cubic case using ab-initio downfolding

in order to obtain a low-energy Wannier Hamiltonian.

Applying this method to the Mott transition in 3d1per-

ovskites, we have explained the photoemission spectra

and the values of the Mott gap without adjustable pa-

rameters, except a single value of U. The Mott transition

is driven by correlation effects and GdFeO3-type distor-

tion through reduction, not only of bandwidth, but also

Diagonalization of the matrix of occupation

111− 1 = (|xy? + |xz? + |yz?)/√3 character on the

of effective orbital degeneracy. Correlation effects and

cation covalency suppress orbital fluctuations in the high-

temperature paramagnetic insulating phase of LaTiO3

and YTiO3.

We thank J.Nuss, G.Khaliullin, E.Koch, J.Merino,

M.Rozenberg,A.Bringer,M.Imada for useful discussions

and the KITP Santa Barbara for hospitality and sup-

port (NSF Grant PHY99-07949). Calculations were per-

formed at MPI-FKF Stuttgart and IDRIS Orsay (project

No.021393). S.B. acknowledges support from the CNRS

and the EU (Contract No.HPMF-CT-2000-00658).

[1] M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys.

70, 1039 (1998).

[2] H. Makino et al., Phys. Rev. B 58, 4384 (1998).

[3] Y. Okimoto, et al., Phys. Rev. B 51, 9581 (1995).

[4] M.J. Rozenberg, Phys. Rev. B 55, R4855 (1997).

[5] E. Koch, O.Gunnarsson, R.M. Martin, Phys. Rev. B 60,

15714 (1999); S. Florens et al., ibid. 66, 205102 (2002).

[6] N. Manini, G.E. Santoro, A. Dal Corso, and E. Tosatti,

Phys. Rev. B 66, 115107 (2002).

[7] G.Khaliullin,S.Maekawa,Phys.Rev.Lett.85,3950(2000).

[8] B.Keimer et al., Phys. Rev. Lett. 85, 3946 (2000);

C.Ulrich et al., Phys.Rev.Lett.89,167202(2002).

[9] M. Cwik et al., cond-mat/0302087.

[10] M. Eitel et al., J. Less-Common Met. 116, 95 (1986).

[11] M.J. Rey et al., J. Solid State Chem. 86, 101 (1990).

[12] M.H. Jung, and H. Nakotte, unpublished.

[13] By 0, 10, 13, and 21% of the average OA distance.

[14] By 0, 3, 4, and 10% of the average AB distance.

[15] M.Mochizuki,andM.Imada, Phys.Rev.Lett.91, 167203

(2003); T. Mizokawa, D.I. Khomskii, and G.A. Sawatzky,

Phys. Rev. B 60, 7309 (1999).

[16] I. Solovyev, N. Hamada, and K. Terakura, Phys. Rev B

53, 7158 (1996); H. Sawada, and K. Terakura, ibid. 58,

6831 (1998).

[17] M.Itho et al. J.Phys.Soc.Jap. 68, 2783 (1999).

[18] J. Akimitsu et al. J. Phys. Soc. Japan 70, 3475 (2001).

[19] A. Georges, G. Kotliar, W. Kraut, M.J. Rozenberg, Rev.

Mod. Phys. 68, 13 (1996).

[20] V.Anisimov et al. J J.Phys:Cond.Matt.9,7359(1997);

A.I.Lichtenstein and M. I. Katsnelson, Phys.Rev.B 57,

6884 (1998).

[21] I.A. Nekrasov et al., cond-mat/0211508.

[22] O.K. Andersen and T. Saha-Dasgupta, Phys. Rev B 62,

16219 (2000); Bull. Mater. Sci. 26, 19 (2003).

[23] R.FresardandG.Kotliar, Phys.Rev.B56,12909 (1997).

[24] T.Mizokawa andA.Fujimori,Phys.Rev.B54,5368(1996).

[25] Σmm′(?=m)(ω) ?= 0 in non-cubic systems, also in the eigen-

representation of the density matrix. In Fig.2, a large

Nmm′(?=m)points to a large Σmm′(?=m).

[26] J.E. Hirsch and R.M.Fye,Phys.Rev.Lett.56,2521(1986).

[27] J.E. Gubernatis, M. Jarrell, R. N. Silver and D. S. Sivia,

Phys. Rev. B 44, 6011 (1991).

[28] A.Liebsch, Phys.Rev.Lett.90,096401(2003).

[29] K. Maiti et al., Europhys. Lett. 55, 246 (2001); A.

Sekiyama et al., cond-mat/0206471.

[30] A. Fujimori et al, Phys. Rev. B 46, 9841 (1992); K.

Morikawa et al., ibid. 54, 8446 (1996).