REVISTA MEXICANA DE FISICA S53 (7) 29–33 DICIEMBRE 2007

Pressure induced insulator-metal transition in LaMnO3

J.D. Fuhr and B. AlascioInstituto Balseiro and Centro Atomico Bariloche, CNEA,

8400 San Carlos de Bariloche, Argentina.

M. AvignonInstitut Neel, CNRS and Universite Joseph Fourier,

BP 166, 38042 Grenoble Cedex 9, France.

Recibido el 30 de noviembre de 2006; aceptado el 8 de octubre de 2007

The recent observation of an insulator to metal transition (IMT) [1] in pure LaMnO3 at 32 GPa and at room temperature, well above the Neeltemperature (145 K) and below the Jahn-Teller transition temperature (780 K), opens the way to a study of the role of the orbital degrees offreedom on the electronic structure in a stoichiometric material. In this paper we focus our attention on the orbital aspects of the insulator tometal transition. We use a Hamiltonian model for theeg orbitals of Mn that includes the on site Coulomb repulsionU , the hoppingt, and itspressure dependence. In order to include in an appropriate way the strong correlations induced by the dominant electron-electron interactions,we introduce auxiliary fields (Slave Bosons,SB) to the description of the low energy states. We use the O-Mn distance (d) dependence oft according to [2] and the pressure-d relation from the experimental data to describe the evolution of the electronic structure with pressure.Our results confirm and make transparent the conclusion reached in previous ab-initio calculations: the inclusion of the Coulomb energy isnecessary and constitutes an important factor enhancing the orbital polarization in these compounds.

Keywords: Manganites; Jahn-Teller effect; orbital order; strongly correlated electrons; insulator-metal transition.

La observacion de la transicion metal aislante (IMT) [1] del LaMnO3 a 32 GPa y temperatura ambiente, por encima de la temperatura deNeel (145 K) y por debajo de la temperatura de transicion Jahn-Teller (780 K), da lugar al estudio del rol de los grados de libertad orbitalesen la estructura electronica de un cristal puro. En este artıculo nos focalizamos en los aspectos orbitales de la transicion aislante metal. Paralo cual usamos un Hamiltoniano modelo para los orbitaleseg del Mn que incluye la repulsion coulombianaU , el elemento de matriz desaltot y su dependencia con la presion. Para describir de forma apropiada la correlacion inducida por la fuerte interaccion electron-electron,introducimos campos auxiliares (bosones esclavos) para la descripcion de los estados de baja energıa. Usamos la dependencia det con ladistancia O-Mn (d) y la relacion presion-d de los datos experimentales para describir la evolucion de la estructura electronica con la presion.Nuestros resultados confirman y hacen transparentes las conclusiones obtenidas en calculos ab-initio anteriores: la inclusion de la energıacoulombiana es necesaria y constituye un factor importante en incrementar la polarizacion orbital en estos compuestos.

Descriptores: Manganitas; efecto Jahn-Teller; orden orbital, electrones fuertemente correlacionados; transicion aislante-metal.

PACS: 71.30.+h; 71.27.+a; 71.70.Ej; 75.47.Lx

1. Introduction

The discovery of colossal magneto-resistance in manganeseperovskytes, of relevance to spintronics and other techno-logical applications, has opened a new field to the study ofhighly correlated systems (HCS). In the three dimensionalcompounds like La1−xSrxMnO3, or in the two dimensionalones, such as La2−2xSr2+2xMn2O7, the interplay of severaldegrees of freedom, charge, spin, orbit and lattice displace-ments determines the physical properties of the system. Theinteractions between the different degrees of freedom cannotbe reduced to a perturbation theory, as in other materials, andthe disorder produced by alloying to obtain the metallic statedoes not contribute to the clarification of theory nor to theinterpretation of experiments.

The recent observation of an insulator to metal transition(IMT) [1] in pure LaMnO3 at 32 GPa and at room tempera-ture, well above the Neel temperature (145 K) and below theJahn-Teller transition temperature (780K), opens the way toa study of the role of the orbital degrees of freedom on theelectronic structure in a stoechiometric material.

Two theoretical studies relevant to this matter have ap-peared in the literature almost simultaneously after the pub-lication of the experimental result: both papers are based onlocal density approximation+Hubbard U (LDA+U) approxi-mation.

In the first one by Wei-go Yin et al. [3], the electronicstructure of the ground state of LaMnO3 (antiferromagneticA phase) is analysed to determine the relative importance ofthe electron-electron (e-e) against electron-lattice (e-l) Jahn-Teller interactions. It concludes that the e-l interaction byitself is not sufficient to stabilize the orbital ordered state andemphasizes the importance of the e-e interaction to facilitatethe Jahn-Teller distortion.

In the second paper by Yamasaki et al. [4], LDA+U andLDA+ dynamical mean field theories are used to analyze themetal insulator transition taking place at 32 GPa in the para-magnetic phase of the same material. The authors concludethat the transition at 32 GPa is caused by the orbital split-ting of theeg bands and that both e-e and e-l interactions areneeded to explain the insulating character of the substance atlower pressures.

30 J.D. FUHR, B. ALASCIO, AND M. AVIGNON

In this paper we focus our attention on the orbital aspectsof the insulator to metal transition. In order to include in anappropriate way the strong correlations induced by the dom-inant e-e interactions, we follow Feiner and Oles in introduc-ing auxiliary fields (Slave Bosons,SB) to the description ofthe low energy states. [5]

We start by defining a simplified Hamiltonian, two pa-rameters:U on site Coulomb repulsion, and∆ε a measureof the Jahn Teller splitting, to describe theeg states of thesystem and use SB to calculate the lowest energy varying theparameters to obtain a phase diagram. This procedure allowsus to identify the values of the parameters that are appropriateto describe the transition, as well as the thermodynamics andtransport properties of LaMnO3 in the paramagnetic phase.

2. Methods

2.1. Hamiltonian

To describe the active electrons in LaMnO3, we use thedouble-exchange model that contains the essential physics ofmanganites [6]. The four 3d electrons in each Mn3+ site arepolarized in the same direction due to the large Hund cou-pling. Three of them occupy thet2g orbitals and are con-sidered localized forming a spinS = 3/2, while the fourthoccupying theeg state is itinerant. We treat the localized spinclassically, and since the fourth has to be parallel to the localspin, we can consider theeg electron as spinless with Hamil-tonian

H = Ht + HU + HJT (1)

where the first term,Ht, is the kinetic energy given by

Ht =∑

〈ij〉αβ

tαβij c†iαcjβ (2)

with α, β = 1, 2 corresponding to the (possibly site depen-dent) orthogonal basis for the twoeg orbitals. The valuesof the hopping integralstαβ

ij depend both on the type of or-bitals involved and on the direction between sitesi, j. InLaMnO3, there is a staggered order in thex-y plane, and theorbitals are stacked ferromagnetically along thez axis. Thedominantly occupied orbitals alternating in thex-y plane are|x〉 =

∣∣3x2 − r2⟩

and |y〉 =∣∣3y2 − r2

⟩, which define then

the low energy orbitals|1〉 in each of the sublattices, respec-tively A andB. The corresponding higher energy orbitals|2〉 are therefore, respectively

∣∣y2 − z2⟩

and∣∣x2 − z2

⟩. The

hopping parameters between these orbitals are the following:

tABx = t

(1/2 0

−√3/2 0

)

tABy = t

(1/2 −√3/20 0

)

tAAz = tBB

z = t

( −1/4 −√3/4−√3/4 −3/4

)(3)

t being the hopping between|x〉 (|y〉) orbitals along thex (y)direction.

The magnetic order is introduced by modulating the hop-ping integrals by the factorexp(iAij) cos(θij/2), [7], withθij being the angle between thet2g localized spins in theneighboring sitesi, j andAij a hopping phase. As we areinterested in the paramagnetic phase that appears at roomtemperatureT À TN , we assume that the localized spinsare completely random and consider a mean field approx-imation in which this factor is averaged, giving a value〈exp(iAij) cos(θij/2)〉 = 2/3. Thetαβ

ij are the same as for aferromagnetic phase with a factor2/3. Henceforth, we takethis renormalized hopping as the referencet.

The on-site Coulomb interaction betweeneg electrons oc-cuping both orbitals on the same site is given by

HU = U∑

i

ni1ni2 (4)

with niα = c†iαciα the number operators.Finally, to model the effect of the Jahn-Teller (JT) defor-

mation, we add a term that shifts the on-site energies of theeg orbitals1, 2 in opposite directions

HJT = ∆ε∑

i

(ni2 − ni1) (5)

which corresponds to a JT splitting of2∆ε.

2.2. Slave bosons method

In order to treat the Hamiltonian Eq. (1), we used the slaveboson theory of Kotliar and Ruckenstein [8] adapted to ourcase of two orbitals instead of the two spin projection. There-fore, we introduce new boson (ei, di, biα) and pseudofermion(fiα) operators. The boson numberse†iei, b†iαbiα andd†idi

represent the projectors onto the possible states|0i〉 〈0i|,|αi〉 〈αi| and|di〉 〈di| so that

e†iei + b†i1bi1 + b†i2bi2 + d†idi = 1 (6)

andb†iαbiα + d†idi = c†iαciα

The original fermion operators are replaced by

c†i1 =(b†i1ei + d†i bi2

)f†i1

c†i2 =(b†i2ei + d†i bi1

)f†i2 (7)

which correspond to a representation of the empty (|0i〉), sin-gle occupied (|1i〉, |2i〉) and doubled occupied (|di〉) localstates by

|0i〉 = e†i |vac〉|1i〉 = b†i1f

†i1 |vac〉

|2i〉 = b†i2f†i2 |vac〉

|di〉 = d†if†i2f

†i1 |vac〉 (8)

Rev. Mex. Fıs. S53 (7) (2007) 29–33

PRESSURE INDUCED INSULATOR-METAL TRANSITION IN LAMNO3 31

with |vac〉 correspondig to the vacuum state.The anticommutation rules for the original fermions are

guaranteed, provided the following constraints are satisfied

b†iαbiα + d†idi = f†iαfiα α = 1, 2 (9)

which are implemented by means of the corresponding La-grange multipliers{λi, µi1, µi2}. To recover the correct re-sult in the uncorrelated (U = 0) limit, a renormalization ofthe bosonic factor in Eq. (7) is necessary. In analogy with thespin case, the renormalized bosons factors take the form

z†i1 =b†i1ei + d†i bi2√(

1− e†iei − b†i2bi2

)(1− d†idi − b†i1bi1

)

z†i2 =b†i2ei + d†i bi1√(

1− e†iei − b†i1bi1

)(1− d†idi − b†i2bi2

) (10)

With this slave boson representation, the Hamiltonian (1)reduces to

H =∑

〈ij〉αβ

tαβij z†iαzjβf†iαfjβ

+ U∑

i

d†idi + ∆ε∑

i

(ni2 − ni1)

+∑

iα

µiα

(f†iαfiα − b†iαbiα − d†idi

)

+∑

i

λi

(e†iei + b†i1bi1 + b†i2bi2 + d†idi − 1

)(11)

where the pseudo-fermions number operators areniα = f†iαfiα = c†iαciα. We have studied the solutions of thisHamiltonian in the mean-field approximation, in which wereplace the boson operators by their averages obtained fromthe minimization of the band energy. In pure LaMnO3, thenumber of conduction electrons isn = 1, and in the absenceof charge orderingni1 + ni2 = ni = 1. In this case, the bandrenormalization factor is independent of the type of orbitalsinvolvedq = z†iαzjβ .

3. Results

To characterize the solutions of Eq. (11) for different valuesof the two parametersU and ∆ε, we considered both themagnitude of the pseudofermion band gap and the orbital po-larization. In Fig. 1 we show the dependence of the band gapwith the value ofU . We can see that for∆ε = 0 there is ajump in the band gap at a critical valueUc, corresponding toa first order metal-insulating transition (in the parameterU ).When we begin to increase the value of the JT splitting,Uc

shifts to lower values and also the initial band gap in the in-sulating phase also diminishes. Beyond a certain value of∆ε(between0.225t and0.250t), the metal-insulating transitionis no mlonger of first order, and the band gap opens smoothly.

FIGURE 1. Dependence of the band gap as a function ofU .

FIGURE 2. Dependence of the orbital polarization as a functionof U .

FIGURE 3. Phase diagramU -∆ε. Solid line: metal-insulating tran-sition. Gray tone: occupancy of the low energy orbitaln1.

Rev. Mex. Fıs. S53 (7) (2007) 29–33

32 J.D. FUHR, B. ALASCIO, AND M. AVIGNON

FIGURE 4. DOS for the different phases: (a) metallic phase withvery small orbital polarization, (b) metallic phase with orbital po-larization, and (c) insulating phase.

Similar features can be observed in the orbital polariza-tion shown in Fig. 2, where for small values of∆ε there is ajump in the polarization at the same critical valueUc. Fromthese results, we can see that an insulating phase with or-bital polarization is possible even without a JT deformation.However, an orbital polarization will induce a JT deforma-tion due to the electron-lattice interaction. We also note thatfor ∆ε = 0, the orbital polarization is symmetric,i.e. the low

energy orbital can be any combination ofeg orbitals, and theactual one will depend on the electron-lattice interaction.

In Fig. 3 we show the phase diagramU −∆ε where theline corresponds to the value ofUc at the metal-insulatingtransition, and the gray tone represents the orbital polariza-tion given by the occupancy of the low energy orbitaln1.While in the insulating phase there is always a high orbitalpolarization, in the metallic phase the amount of orbital po-larization depends on∆ε: the higher the value of∆ε, thehigher the polarization. The critical value is close toUc = 7tfor the case without JT splitting, and descends to almost2tfor a splitting∆ε = t.

To better characterize the different phases, we have calcu-lated the pseudofermion density of states (DOS). In Fig. 4 weshow this DOS for three different representative parametersin the phase diagram:

• (U = 3.5t, ∆ε = 0.0t): metallic phase with very smallorbital polarization, which we have named orbital liq-uid (MOL). In this case, there is almost no orbital po-larization (n1 = 0.502) and the renormalization factoris q = 0.917.

• (U = 3.5t, ∆ε = 0.4t): metallic phase with orbitalorder (MOO). The orbital polarization is now higher(n1 = 0.764), and the renormalization factor is closerto one (q = 0.941).

• (U = 3.5t,∆ε = 0.8t): insulating phase, also withorbital order (IOO). In this case, the system is almostfully polarizad withn1 = 0.920 ,and as a consequence,q = 0.986 is also close to one.

FIGURE 5. Phase diagramU -P . Solid line: metal-insulating tran-sition. Dot line: U = 11.5t0. Gray tone: occupancy of the lowenergy orbitaln1.

Rev. Mex. Fıs. S53 (7) (2007) 29–33

PRESSURE INDUCED INSULATOR-METAL TRANSITION IN LAMNO3 33

To simulate the effect of pressure on the electronic prop-erties, we take the experimental data on the IM transition andmodel the dependence of the parameters with the pressure.All magnitudes are given in terms of the effective hopping atP = 0, which is of the order oft0 ' 0.4 eV [3]. The JT split-ting is taken to vary linearly from a value of∆ε = 1.85t0 [4]atP = 0 to ∆ε = 0 at the experimental JT suppression pres-sureP = 18 GPa. Thee-einteraction constant is consideredconstant. Finally, the effective hopping has a dependence ofthe form [2]

t(P ) = t0

(d0

d(P )

)7

(12)

whered(P ) is the mean Mn-O distance as a function of thepressure, taken from the experimental data, andd0 = d(0).

From this dependence of the parameters with the pres-sure, in Fig. 5 we show the phase diagramU − P wherethe solid line corresponds to the value ofUc at the metal-insulating transition, the gray tone represents the orbital po-larization given by the occupancy of the low energy orbitaln1. Note that the values ofU in the vertical axis are inunits of t0, and we have added a dot line showing that forthe estimated value ofU = 11.5t0, we obtain an insulatorto metal transition pressure close to the experimental one ofP ' 32 GPa.

4. Conclusions

We use a minimum parameter Hamiltonian model to studythe evolution of orbital polarization in LaMnO3. In order to

appropriately include the effects of correlation, we resort tothe Slave Bosons technique, which was previously used byFeiner and Oles in the context of manganites.

We calculate the electronic structure and from it, the elec-tronic energy to obtain a phase diagram in terms of two inde-pendent parametersU/t and∆ε/t. The results can be trans-lated to the effect of pressure on the material by modellingthe variation of the parameters with volume and connectingto pressure through the compressibility.

The same Hamiltonian could be used to represent the or-bital state of other compounds where trivalent Rare Earthspartially or totally substitute La, again though modelling ofthe variation of the hopping ,U or ∆ε as an effect of substi-tution and pressure, as for example in equation 12.

Our results confirm and make transparent the conclusionreached in previous ab-initio calculations: The inclusion ofthe Coulomb energy is necessary and constitute an importantfactor enhancing the orbital polarization in these compounds.

From the density of the states, it is possible to calculatethe gap in the insulating phases and the number of carriers asa function of temperature and pressure in order to compare itwith the results of Loaet al. [1]

A natural continuation of these results is the evaluation ofthe elastic energy involved in the Jahn-Teller distortion thatwill appear at the polarized phases; these results will be pub-lished elsewhere.

Acknowledgements

We acknowledge support by CONICET.

1. I. Loaet al, Phys. Rev. Lett.87 (2001) 125501.

2. W.A. Harrison, “Electronic structure and the properties ofsolids : the physics of the chemical bond”, San Francisco, W.H.Freeman ,1980.

3. Wei-Guo Yinet al., Phys. Rev. Lett.96 (2006) 116405.

4. A. Yamasakiet al., Phys. Rev. Lett.96, 166401 (2006).

5. L.F. Feiner and Andrzej M. Oles, Phys. Rev B71 (2005)144422.

6. F. Popescuet al., Phys. Rev. B73 (2006) R180404.

7. E. Muller-Hartmann and E. Dagotto,Phys. Rev. B54 (1996)R6819.

8. G. Kotliar and A.E. Ruckenstein,Phys. Rev. Lett.57 (1986)1362.

Rev. Mex. Fıs. S53 (7) (2007) 29–33