Insulator-metal-insulator transition and selective spectral-weight transfer in a disordered stronglycorrelated system

P. Lombardo and R. HaynLaboratoire Matériaux et Microélectronique de Provence associé au Centre National de la Recherche Scientifique, UMR 6137,

Université de Provence, Marseille, France

G. I. JaparidzeGeorgian Academy of Sciences, Tamarashvili 6, 0177 Tbilisi, Georgia

�Received 29 December 2005; revised manuscript received 9 March 2006; published 28 August 2006�

We investigate the metal-insulator transitions at finite temperature for the Hubbard model with diagonalalloy disorder. We solve the dynamical mean-field-theory equations with the noncrossing approximation andwe use the coherent potential approximation to handle disorder. The excitation spectrum is given for variouscorrelation strength U and disorder. Two successive metal-insulator transitions are observed at integer-fillingvalues as U is increased. An important selective transfer of spectral weight arises upon doping. The stronginfluence of the temperature on the low-energy dynamics is studied in detail.

DOI: 10.1103/PhysRevB.74.085116 PACS number�s�: 71.27.�a, 71.30.�h, 71.10.Fd

I. INTRODUCTION

Strongly correlated materials have been the subject of in-tensive research for the last forty years. The simplest latticemodel accounting for correlations is the one-band Hubbardmodel. Great progress has been achieved within the last tenyears with the development of the dynamical mean-fieldtheory �DMFT� �for a review, see Ref. 1�. One importantresult of the DMFT is the description of the metal-to-insulator transition driven by correlations �Mott transition2�for the half-filled Hubbard model. Today, the Mott metal-insulator transition �MIT� is the subject of intensive experi-mental and theoretical studies.3 Recently, a renewal of inter-est in the Mott transition has occurred in relation to the so-called orbital-selective Mott transition4–7 observed inmultiorbital compounds.

In most of the experimental studies, strong electron cor-relations are closely connected with disorder effects. So, tochange experimentally the band filling, the bandwidth, orother parameters, one usually replaces one atom of a givencompound by another one. That is true for strongly corre-lated transition-metal oxides or similar compounds. For ex-ample, replacing La by Sr in LaxSr1−xTiO3 changes the dop-ing, i.e., the band filling.8,9 In Ca1−xSrxVO3,10,11 Ca and Sratoms have the same valence but different atomic radii,which changes the angle of the V-O-V bond and, corre-spondingly, the bandwidth. More recently, a very interestingstudy12 on the metal-insulator transition in the disordered andcorrelated system SrTi1−xRuxO3 has been performed. Fromtheir transport and optical data, the authors propose a classi-fication scheme with six kinds of electronic states dependingon x. The disorder and the electron-correlation effects shouldbe considered together to understand the measuredelectronic-structure evolutions.

A generic model to study the common influence of disor-der and correlations is the Hubbard model including diagonaldisorder. As we will outline below, it shows many featuresand metal-insulator transitions not present in the standard

Hubbard model. A possible method to solve the disorderedHubbard model is the combination of DMFT with the coher-ent potential approximation �CPA�14 to treat disorder. It wasused in the seminal work of Laad et al.,13 where the localimpurity model of the DMFT was solved by the iterationperturbation theory �IPT�. Interesting results concerning thestability of the Fermi-liquid metal against small disorder andthe effect of the interplay between correlations and disorderon the optical and Raman response had been obtained.

For the disordered Hubbard model, metal-insulator transi-tions at noninteger fillings have been found for the first timeby Byczuk et al.15,16 They have shown that, at a particulardensity n=x, the interplay between disorder-induced bandsplitting and correlation-induced Mott transition gives rise toa new type of MIT. A first classification of the metal-insulator transitions at noninteger fillings �and T=0� wasgiven in Ref. 17. It used an analogy to the Zaanen-Sawatzky-Allen scheme of charge-transfer or Mott-Hubbard insulators,respectively �see also Ref. 18 for the relationship to Ander-son localization�. Our contribution concerning the disorderedHubbard model at noninteger fillings is to clarify the natureof the different MIT by computing site-specific densities ofstates and corresponding transfers of spectral weights.Besides, we will concentrate on the Hubbard model withdiagonal alloy disorder at integer filling. With increasingHubbard correlation U we find two metal-insulator transi-tions: the first from a noncorrelated band insulator to a metaland a second transition from a metal to a Mott-Hubbard in-sulator.

The transition from a band insulator to a Mott-Hubbardinsulator has been the subject of intensive current studieswithin the framework of the half-filled ionic Hubbard model�IHM�.19,20 The IHM corresponds to the spatially orderedlimit of the binary alloy system, which is considered in thepresent paper, AxB1−x, where the A atoms are separated fromeach other by an identical sequence of B atoms. In particular,intensive analytical and numerical efforts have been per-formed to analyze the band-insulator �BI� to Mott-insulator

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�MI� transition in the Hubbard chain with alternating on-siteenergies.21–26 These studies show that the half-filled ionicHubbard chain has two transitions as U is increased. The firstis from the BI to a bond-ordered, spontaneously dimerized,ferroelectric insulator �FI�. The second, when U is furtherincreased, involves a transition between the FI and the MI. Inthis phase diagram the very unconventional metallic state isrealized only at the BI-FI transition point.

Very recent studies of the generalized ABn−1 IH chain forn�3, where n−1 is the length of domain of B atoms, whichseparates A atoms, show that at commensurate band filling1/n the system shows, with increasing U, a similar sequenceof transitions BI-FI-MI.27 Therefore, in the case of one-dimensional �1D� ordered binary chains, the BI phase is al-ways separated from a MI by the insulating ferroelectricphase. Although the Mott transition in the disordered 1Dbinary chain has not been studied in detail, the Andersonlocalization28 excludes the presence of a metallic phase in afinite sector of the ground-state phase diagram of a 1D dis-ordered IH model.

Therefore, the approach presented in this paper allows usto investigate in detail the band-insulator to correlated-metaland metal to Mott-Hubbard insulator transitions in a disor-dered binary-alloy system for higher dimensions.29 Belowwe analyze these metal-insulator transitions in detail. Forthat purpose we consider the spectral-weight transfer withdoping, and we find that it occurs in a selective way, differ-ent for the two disorder sites in question.

In the present work, we propose a finite-temperature cal-culation of the dynamical properties by using the noncross-ing approximation �NCA�30,31 to solve the DMFT localAnderson problem in the self-consistent loop. Then we com-bine the DMFT with CPA which is a very natural way sinceboth methods use a local self-energy. Spectral densities andspectral weight transfers will be computed with the NCA.The role played by the temperature for the low-energy phys-ics will be discussed in detail.

In the following Sec. II the model Hamiltonian is pre-sented. This section also presents the application of theDMFT on this Hamiltonian and a solving method based onthe NCA. In Sec. III, the main results are discussed. Theseare the insulator-metal-insulator transition at integer fillingand a detailed analysis of possible transitions at integer andnoninteger fillings by using their spectral properties.

II. HAMILTONIAN AND SOLVING METHOD

We consider the following Hamiltonian, where correla-tions and diagonal disorder are present:

H = − t ��i,j�,�

�ci�+ cj� + H.c.� + �

i,��ini� + U�

i

ni↑ni↓. �1�

In expression �1� the sum �i , j� is the sum over nearest-neighbor sites of a Bethe lattice. ci�

+ �respectively, ci�� de-notes the creation �respectively, annihilation� operator of anelectron at the lattice site i with spin � and ni� is the occu-pation number per spin. U is the on-site Coulomb repulsionand t is the hopping term. A fraction 1−x of sites �sites A�

have a local on-site energy �i=�A and a fraction x of sites�sites B� have a local on-site energy �i=�B. We will note�=�A−�B the energy difference between the two types ofsites.

Following the DMFT procedure of Ref. 13, by integratingout all fermionic degrees of freedom except for a central sitei=o, the lattice model �1� can be mapped onto an effectiveimpurity model. The expression of the corresponding Hamil-tonian Heff

A,B depends on the nature of the central site i=o. Itcontains a local part Hloc

A,B and a part corresponding to thecoupling to the effective medium. This coupling is deter-mined self-consistently. We then write

HeffA,B = Hloc

A,B + Hmed, �2�

where

HlocA,B = Uno↑no↓ + �

�

��A,B − ��co�+ co�

is depending on the nature �A or B� of the central site i=o. Inthe expression of Hloc

A,B, we have written explicitly the chemi-cal potential �. The coupling Hamiltonian with the effectivemedium is

Hmed = �k��

�Wk�bk��+ co� + H.c.� + �

k��

�k�bk��+ bk��.

Wk� represents the hybridization between the site i=o and theeffective medium. �k� is the band energy of the effective me-dium. bk��

+ �respectively, bk��� is the creation �respectively, an-nihilation� operator of an electron in the effective medium.The effective medium can be characterized by the effectivedynamical hybridizations

J��� = �k�

�Wk��2

� + i0+ − �k�.

On a Bethe lattice, the self-consistent equations of theDMFT can be simply written

J��� = t2G���� ,

where G���� is obtained by averaging over disorder with theCPA procedure. We have G����=xG�

B���+ �1−x�G�A���

where G�A��� �respectively, G�

B���� is the Green’s-functionsolution of the effective local-impurity problem representedby the Hamiltonian Heff

A �respectively, HeffB �.

To solve the local-impurity problem Hloc� �where �=A ,B�

with the NCA, we introduce four local states �m�� and theirlocal energies Em

� . m=1 corresponds to an empty local site,m=2 �respectively, m=3� corresponds to a singly occupiedlocal site with an electron of spin ↑ �respectively, ↓�, andm=4 corresponds to a doubly occupied local site. Of course,energy E1

� is � independent and because of spin degeneracy,we have E2

�=E3�=��−�. For the doubly occupied site

E4�=2��−2�+U.

LOMBARDO, HAYN, AND JAPARIDZE PHYSICAL REVIEW B 74, 085116 �2006�

085116-2

The NCA equations30 read

1���� = 2 d����f���P2

��� + �� ,

2���� = d����f�− ��P1

��� − ��

+ d����f���P4��� + �� ,

3���� = 2

���� ,

4���� = 2 d����f�− ��P2

��� − �� ,

where the local propagators Pm� are defined by

Pm���� =

1

� + i0+ − Em� − m

����

and the effective medium-spectral density by���=− 1

� Im�J����. f is the Fermi-Dirac distribution at tem-perature T. We finally obtain the site-specific one-particleGreen’s function

G����� =

1

Z0� d�e−���p1

����P2��� + �� + p2

����P4��� + ��

− p2����P1

��� − ��� − p4����P2

��� − ���� ,

where pm����=− 1

� Im�Pm����� and the partition function is

Z0�=d�e−���p1

����+2p2����+ p4

�����.

III. RESULTS AND DISCUSSION

In this section, we present the calculated densities ofstates for the disordered Hubbard model. In Sec. III A, wefocus on the possible metal-insulator transitions and, in par-ticular, the nature of these transitions.

In the Sec. III B, we show how the site-selective spectral-weight transfer can help to understand the interplay of disor-der and correlations for dynamical properties.

A. Metal-insulator transitions

One has to distinguish the metal-insulator transitions atinteger filling �n=1� from those that occur at noninteger fill-ings �n=x or n=1+x for � 0�,15,17 since at �=0 the MItransition with increasing on-site repulsion U takes placeonly in the former case.

Let us first concentrate on those with n=1. In that case,the concentration of sites A has to be x=0.5. Then, for each� and sufficiently large values of U, we observe an insulat-ing state for n=1. Before discussing the commensuraten=1 case it is very instructive to consider the case of adoping slightly below half filling �see Fig. 1�. In this case wehave a strongly correlated metal with heavy quasiparticlesfor all U. That is illustrated in Fig. 1 for �=2 eV, U=6 eV,and n=0.9.

NCA applied to DMFT is known31,32 to supply a fairlygood solution even away from half-filling. However, thistechnique gives unphysical behavior at very low temperatureand is inaccurate for the weak-correlation regime �in particu-lar, for the low-energy excitations�. Consequently, the fol-lowing results for low values of U and for very low elec-tronic occupation numbers should be considered asqualitative results only. For this strongly correlated situation,we found �plain line� a metallic state with four incoherentbands and one coherent resonance at the Fermi level, whichis a signature of the presence of quasiparticles. The tempera-ture dependence of the quasiparticle peak is shown in part �b�of the figure. Although the density of states is almost con-stant at the Fermi level, the low-energy states near the Fermilevel are strongly temperature dependent. In part �a� of Fig.1, we plot as well the spectral densities obtained from thesolutions of the two impurity models with a site A or Bembedded in the same effective medium. This gives enlight-enment on the nature of the different bands forming the den-sity of states for the disordered system and then allows us toclarify the nature of the near-insulating state. Using theNCA, it is possible to reduce arbitrarily the doping with in-creasing numerical efforts. Band A1 �respectively, B1� ismainly composed of singly occupied A sites �respectively, Bsites� and spectral weights are proportional to the probabilityof removing one electron from a site A �respectively, B�.Correspondingly, the spectral weight of band A2 �respec-tively, B2� is proportional to the probability of adding oneelectron to a singly occupied site A�respectively, B�. Theenergy gap between band A1 and band A2 �or B1 and B2� isof the order of U=6 eV. We can identify A1 to a so-calledlower Hubbard band �LHB� and B2 to a so-called upperHubbard band �UHB�. We can therefore consider that thesystem is a strongly correlated metal, close to a Mott insula-tor, with an insulating gap between the LHB �A1� and the

-5 0 5energy-eF [eV]

0

0.2

0.4

d.o.

s. [

1/eV

] total d.o.s.A siteB site

-1 0 1energy-eF [eV]

0

0.2

0.4

d.o.

s. [

1/eV

]

T=1000KT=750KT=500K

(a)

(b)

B1 A1 B2 A2

A1

FIG. 1. �a� Density of states �plain line� obtained from DMFT-NCA and CPA for U=6 eV, �=2 eV, and T=1000 K. The dottedline �respectively, dashed line� shows the corresponding density ofstates for the impurity problem with a site A �respectively, B� sur-rounded by the effective medium. �b� Temperature dependence ofthe low energy part of the calculated spectrum.

INSULATOR-METAL-INSULATOR TRANSITION AND¼ PHYSICAL REVIEW B 74, 085116 �2006�

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UHB �B2� and a reduced effective value of the correlationstrength U−�. Note the asymmetric role played by the twotypes of site. For n=0.9, charge fluctuations are alreadyblocked by correlations on sites B which are half filled. TheFermi-level resonance peak is therefore only built up byelectrons evolving on sites A. This result is confirmed by therespective local occupations nA=0.4 and nB=0.5.

Part �a� of Fig. 2 shows the densities of states for a de-creasing U �U=7,5 ,4 ,3 ,2 eV� and a given �=4 eV atT=1000 K for x=0.5 and n=1. We observe two phase tran-sitions. As expected, starting from the strongly correlatedsituation U=7 eV, the system undergoes a transition to ametallic state for decreasing U. There is a second criticalvalue at which the system enters into a new insulating statewith further decreasing U. The nature of the insulator is com-pletely different here. This can be clarified by examining part�b� of Fig. 2. Sites B are almost filled �nB=0.915� whereassites A are almost empty �nA=0.045�. The two side bands ofthe gap correspond to singly occupied sites and cannot beinterpreted in terms of Hubbard bands. Therefore, this insu-lator can be understood as an ordinary band insulator. Themetallic state for intermediate values of U can be understoodsince in that case the energies for doubly occupied B sites�B+U and singly occupied A sites �A are nearly degenerate.For U=� we have a special situation �see Fig. 2�. In thatcase we have a single peak in the DOS at the Fermi energy,whereas in all other situations a double peak develops.

We believe that the series of insulator-metal-insulatorphase transitions is a rather generic situation and should oc-cur in a wide class of models with two different species atequal concentration in dimensions higher than one. Very re-

cently a similar insulator-metal-insulator transition was ob-tained for the ordered Hubbard model with two sites on abipartite lattice.33 Also, there is a quite large similarity to thetransition between a Mott-Hubbard insulator and a band in-sulator observed in the 1D ionic Hubbard model; however,due to the peculiarity of 1D systems the metallic state in thiscase is reduced to only one critical point corresponding tothe transition from a BI to FI.

To characterize the different phases in more detail wecomputed also the effective masses in a slightly dopedsituation �n=0.96, not shown�. We used the equationm�

� /m=1− �� Re������ /����=0. Crossing the upperinsulator-metal transition, we found a strong reduction ofmB

� /m from 13.5 �for U=7 eV� to 2.4 �for U=5 eV� whilemA

� /m evolves slightly from 1.5 to 1.9. This is in contrast tothe band-insulator case �U=2 eV� with the effective massesmA

� /m=1.004 and mB� /m=1.07 proving the absence of heavy

quasiparticles.Finally, let us discuss the metal-insulator transitions at

noninteger fillings. They have been found by Byczuk et al.15

Using a quantum Monte Carlo simulation, they have foundthat a new Mott-Hubbard-type MIT occurs because of theinterplay between band splitting by disorder and correlationeffects. Another study17 based on the numerical renormaliza-tion group method at zero temperature has shown that thesystem becomes a Mott insulator at strong interaction forn=x or n=1+x �for � 0�. Here we propose a visual inter-pretation of this result. Figure 3 displays the densities ofstates for U=2�=4 eV, n=0.9, and various disorder concen-trations x of site B between 0 and 1. For x=0, there areonly sites A whose spectral densities are distributed over

FIG. 2. �a� DMFT-NCA-CPA densities of states forU=7,5 ,4 ,3 ,2 eV, �=4 eV, T=1000 K, x=0.5, and n=1.0. �b�Corresponding results for site-selective densities of states.

-5 0 5energy [eV]

00.10.2

dens

ities

of

stat

es [

1/eV

]

A1 A2

B1 B2

x = 0

x = 0.1

x = 0.2

x = 0.3

x = 0.4

x = 0.5

x = 0.6

x = 0.7

x = 0.8

x = 0.9

x = 1

FIG. 3. Density of states obtained from DMFT-NCA and CPAfor U=2�=4 eV, n=0.9, and various disorder concentrations x ofsite B between 0 and 1.

LOMBARDO, HAYN, AND JAPARIDZE PHYSICAL REVIEW B 74, 085116 �2006�

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two Hubbard bands, and the lower one is partially filled.Increasing the concentration of B sites leads to an increasingstrength of the two Hubbard peaks corresponding toB �B1 and B2�. For x=0.9, the lower Hubbard peak of site Bis completely filled and sites A are nearly empty, leading toan insulating situation. There is another insulating state atx=1.9, which is not shown. If we exchange the role of A andB sites ���0� we would observe insulating states at x=0.1and x=1.1.

B. Selective spectral-weight transfer

In this section, we investigate a larger range of dopingvalues, from the quarter-filled situation where n=0.5 ton=1.5, but keeping x=0.5. We found an important transfer ofspectral weight and explain it by using the selective spectraldensities supplied by our approach. Two particular cases ofnoninteger filling n=0.5 and n=1.5 will be discussed in de-tail, with a new type of insulating state. In part �a� of Fig. 4the densities of states are displayed for U=�=4 eV,T=1000 K, and for various fillings from n=0.55 to n=0.95.Total densities of states for n 1 can be deduced from thepresented ones by particle-hole symmetry. For A and B spe-cific densities of states, the particle-hole symmetry has to beaccompanied by the exchange A↔B. The position of theFermi level is marked with an asterisk on each spectrum. Asthe filling is increased, an important transfer of spectralweight takes place between the different bands. The spectral

weights are reported in part �b� of the figure. Part �c� �respec-tively, part �d�� of the figure shows site-selective densities ofstates for sites A �respectively, B�. We found that the ob-served spectral-weight transfer is characteristic of stronglycorrelated bands and is site specific. For n going from 0.55 to0.95 the spectral-weight transfer occurs for sites A �from1.05 to 1.45 for sites B�. For n=0.55 and n=1.45 the systemis close to an insulating state. Nevertheless, the nature of theinsulating phase is strongly different from the half-filled situ-ation, where we have a Mott insulator and two half-filledspectral densities for sites A and B. For the almost quarter-filled situation �n=0.55� we found a hybrid type of insulatingphase, where sites B are half-filled and sites A are almostempty. For B sites, charge fluctuations are frozen by correla-tions like in the usual Mott insulator. For A sites, the blockedmicroscopic processes are of charge-transfer type like in aband insulator. For n=1.45 we have the symmetric situation.

IV. CONCLUSION

In this paper, we proposed an approach based on the dy-namical mean-field theory, which is able to handle diagonaldisorder in a strongly correlated Hubbard model at finitetemperature for any doping. We used the coherent potentialapproximation in the self-consistent equation of the DMFTand we applied the noncrossing approximation to the localimpurity problem on which the lattice model is mapped.

-5 0 5energy [eV]

0

0,1

dens

ities

of

stat

es [

1/eV

]

*

*

*

*

*

n=0.55

n=0.95

(a)

0.5 1 1.5n

0

0.1

0.2

0.3

0.4

0.5

spec

tral

wei

ghts

A2

A1

B1

B2(b)

-5 0 5energy [eV]

0

0,1

dens

ities

of

stat

es [

1/eV

]

n=0.55

n=0.95

*

*

*

*

A1A2

sites A

*

(c)

-5 0 5energy [eV]

0

0,1

dens

ities

of

stat

es [

1/eV

]

*

*

*

*

*B1 B2 n=0.55

n=0.95

sites B(d)

FIG. 4. �a� Densities of statesfor U=�=4 eV, T=1000 K, andfor various fillings from n=0.55 ton=0.95. �b� Spectral weights ofthe different bands. �c� Corre-sponding densities of states forsites A. �d� Corresponding densi-ties of states for sites B. The posi-tion of the Fermi level is markedwith an asterisk on each spectrum.For fillings n 1, densities ofstates can be found by particle-hole symmetry and exchanging Aand B.

INSULATOR-METAL-INSULATOR TRANSITION AND¼ PHYSICAL REVIEW B 74, 085116 �2006�

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For a decreasing U we showed that the disordered half-filled system undergoes two successive metal-insulator tran-sitions of different nature. The first transition is a Mott-typetransition with a reduced effective-correlation strength. Thesecond transition is a charge-transfer-like transition. Wepointed out that this situation is a very generic one. We alsodiscussed the differences and similarities between the con-sidered disordered binary-alloy model and the ionic Hubbardmodel in one and higher spatial dimensions.

In addition to the insulating phase at half filling whereboth types of sites are half filled and all charge fluctuationsare blocked by correlations effects, we studied the MITobserved15 for noninteger fillings n=x or n=1+x andshowed the hybrid character of the insulating state. We illus-trated it for the quarter-filled and the three-quarter-filled situ-

ations. Charge fluctuations are blocked by correlations forone type of sites and by charge-transfer excitations for theother type of sites.

It could be interesting to include in the present model twoorbitals with different bandwidth to investigate the role ofdisorder in the orbital-selective Mott transition.

ACKNOWLEDGMENTS

We thank M. Laad and L. Craco for useful discussionsand the NATO science division for financial support �GrantNo. CLG 98 1255�. The authors would like to thank thegenerous hospitality of the MPI-PKS Dresden, where theirjoint work on the given problem had started.

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