ON THE METAL-INSULATOR TRANSITION IN V2O3

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ON THE METAL-INSULATOR TRANSITION IN

V2O3

C. Castellani, C. Natoli, J. Ranninger

To cite this version:

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JOURNAL DE PHYSIQUE Colloque C4, supplkment au no 10, Tome 37, Octobre 1976, page C4-199

ON THE

'

METAL-INSULATOR TRANSITION

I N

V, 0,

C . CASTELLANI (*), C. R. NATOLI (**) an& J. RANNINGER Groupe des Transitions de Phases, Centre National de la Recherche Scientifique,

B. P. 166, 38042 Grenoble Cedex, France

Rksumk.

-

Nous avons Btudi6 la structure magnktique pour V203 dans la limite atomique et

Bgalement pour le cas W/U

-

1 en utilisant une approximation Hartree Fock unrestricted. Avec

la m&me mBthode nous avons examine la phase m6tallique et avons Btudie les possibilit6s d'ouvrir dans la densite d'Btat un gap dQ aux corr6lations Blectroniques.

Abstract. - The magnetic structure of V 2 0 3 inthe insulating phase is studied in the atomic limit

and by means of an unrestricted Hartree Fock calculation for the case W / U

-

1. Within the same

approximation the metallic phase is investigated and the possibilities of opening a correlation gap are discussed.

Any attempt to understand the metal-insulator transition in V203 must start with a correct description of the two magnetic 3d electrons in the t2, subband. A way to do that, which at the same time takes into proper account the role of the diamagnetic ligands, has been indicated by Anderson [l]. It consists in describing the 3d antibonding electrons of the cations as being in Wannier states of the form.

where

cpfi

is the 3d normalized wave function for the

electron at a site i and Xc)(r) is the corresponding

normalized wave function of the cluster of six oxygen

atoms surrounding the cation site.

NF1

is the proper

normalization factor given by

where S(3d, 2pn) is the overlap integral of n type

between the 3d electrons of symmetry t2, and the

corresponding cluster wave function

X,

transforming

according to the same symmetry. In relation (1) L, is

the covalent mixture parameter obtainable from NMR measurement at the ligand sites. Actually a more correct form of Wannier function relative to a site i, is one which is normalized to the ones centred on the neighbouring sites and is given by

to first order in the overlap integral Sij = ($i/$j).

Using these Wannier type of functions the effective

(*) Instituto di Fisica, University of Rome, Italy.

(**) Laboratori Nazionali I. N. F. N. Frascati, Italy.

transfer integral between two 3d electrons sitting on site

i and j is

where E,,

-

EZpn is the average energy difference

between the energy of the 3d electrons and that of the 2p, electrons in the cluster. The quantity ki/xj) is a numerical coefficient easely calculated from the knowledge of the cluster of the ligand wave functions

centred on neighbouring cation sites i and j. Equa-

tion (3) shows how covalency effects can drastically modify the direct transfer integrals between 3d electrons. Expression (3) allows a reasonable estimation of the

effective transfer integrals since ;1, is known from

NMR measurement at ligand sites, the energy diffe-

rence AE = E,,

-

E,,- is obtainable from optical or

photoemission spectra and the quantity (cpfd/~o/cpi) can be calculated with a fair degree of accuracy. In

the case of V 2 0 3 1: turns out to be 0.4 with 50

%

of

accuracy [2] and AE = 4 - 6 eV as obtained from

photoemission spectra [3] (C&, H. 40id) having been

calculated by Ashkenazi and Chuchem [4] in the metallic phase, and thus a sound basis is provided for a complete investigation of the problem.

The most relevant transfer integrals from a site up to

its four neighbours are shown in figure 1. Table I

gives for the same transfer integrals the magnitude and the sign of the quantity (xi/xj) as well as the

direct transfer integral H. (P:d) for V2O3. The

relative signs of these last two quantities apply to the corundum structure in general, since they depend only on geometrical factors, in particular they apply to Ti203 and Cr203 which allows us to treat their magnetic structures together with V203.

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ON THE METAL-INSULATOR TRANSITION IN V203 C4-201

FIG. 1.

-

Characteristic cluster of V-atoms representing the corundum structure of V203 and the first neighbours (up to the

fourth) of atom 1 (a).

In the case of Ti203 there is only one magnetic electron for which the effective transfer integral for

pairs along the c axis of the corundum lattice is big,

since the interference between the direct and the covalent contribution is additive. On the contrary transfer to nearest neighbours in the basal plane is small since the above interference is destructive. For most distant neighbours the direct transfer integral is negligible and the covalent contribution predominate, being somewhat smaller than for nearest neighours along

the c axis. All this is consistent with the ground state

structure of Ti203 at low temperature. The system gains energy by forming diamagnetic pairs along the c axis, since interaction of the two electrons in the bond with neighbouring pairs is either small or hindered by the Coulomb repulsion. The transition observed at

around 500 K is then explained on the basis that the

excitation of these bonding electrons to the upper conducting band and the consequent creation of broken pairs reduces by screening the Coulomb repulsion and favours a conducting state through

modification of the c/a ratio and consequent increase

of the jumping possibilities.

Cr203 on the other hand is an insulator. Its magnetic structure consists of ferromagnetic planes perpen-

dicular to the c axis stacked antiferromagnetically.

One can apply here perturbation theory and study the.

most stable magnetic order. Again the peculiarity of the interference of direct and covalent contribution to the effective transfer integrals explains this structure. There are three 3d electrons in the t2, subband which

are coupled to a spin 312 state by intratomic exchange

(Hund's rule coupling) and the band is half full. Neglecting transfer to the higher empty Eg band (which would favor weak ferromagnetic coupling along all directions) the kinetic exchange mechanism would lead to an antiferromagnetic coupling of a site to all its nearest neighbours. However only transfer

integrals along the c axis are strong enough to stabi-

lize the antiferromagnetic coupling. The transfer integrals along the basal plane toward the three neighbours are relatively small due to the covalence destructive interference and an antiferromagnetic coupling with the much more numerous neighbours (nine) in the adjacent planes through a moderatly strong purely covalent exchange is preferred. Hence the actually observed structure is obtained.

V203 is intermediate between the two previous situations. We have found it useful to takle the problem in two successive steps. Since there are two electrons in a triply degenerate t,, subband which is split by the trigonal symmetry in a non degenerate a,, band and a doubly degenerate e, band, we have chosen to study the orbital and magnetic instability of an electron in a doubly degenerate e, band in the corundum structure. Instead of studying suitable response function in the R. P. A. (quite laborious to calculate due to the complexity of the crystal structure) we have preferred

to study the atomic limit ( W / U + 0) with the hope

and the intuitive conviction that the instabilities

hereby found would persist into the region W/U

-

1

which is of physical interest. A realistic unrestricted Hartree-Fock calculation has born this out. We start from a generalized Hubbard Hamiltonian for two degenerate e, bands having the symmetry of the

metallic state of V,03 (D;, symmetry group) and

study the symmetry braking solutions in the atomic limit. The phase diagram obtained is shown in figure 2. (We have limited ourselves to first and second neighbour interaction for sake of simplicity although the results are similar if third nearest neighbours are taken into account.) There are two free parameters

FIG. 2. - Phase diagram of stable orbitally and spin ordered symmetry braking configurations of a corundum structure with

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C4-202 C. CASTELLANI, C. R. NATOLI AND J. RANNINGER

(apart from an overall constant factor) in our problem,

namely J/U12 and P/a, where J is the intratomic

exchange integral between the two e, states, U,, is the intratomic Coulomb repulsion for electrons in diffe-

rent orbital states (in our case U,, = U,,

+

2 J )

and a,

p

are the transfer integral defined in table I.

Figure 2 shows that there is a quite w i g range of

physical parameters for which a symmetry braking solution exists, namely an ordered magnetic and orbital solution not having the trigonal symmetry. Figure 3

FIG. 3. - Orbital band RS spin ordered solutions of phase diagram in figure 2 depicted in form of clusters representing the corundum lattice. The left and right half of the circles represent orbital 1 and 2 respectively. 3a) AORS structure, 3b) RORS, the f signs underneath each circle indicate the wave function

fi 4 f2 for that site with

f i = Jhdtp

+

J x d ~ and fi = ,/Zdp2-tz

-

J $ d p C

.

illustrates the various solutions found in the phase

diagram in figure 2. In particular the RS spin structure

(RS stands for real that is to say actually observed, spin structure) is of interest. According to the value of the parameters, it is associated with either a totally antiferromagnetic orbital structure (AO-RS) or a real orbital structure(R0-RS) where the orbitals of adjacent atoms are occupied according to the symmetric and antisymmetric combination of the two e, states. The

reason for such instability is clear. If there in enough correlation, the vertical pair constitutes a diatomic molecule with one electron per site in a doubly degene-

rate Wannier (- atomic) state. The ground state of this

molecule is known to be ferromagnetic, since the two electrons prefer to occupy different orbitals on the two sites in a triplet spin state in order to pay the minimum

repulsive energies (U,, - J ) when they find themselves

on the same site (in the trigonal symmetry ta,b,, the cross transfer integral between different orbitals is zero). The same thing can happen along one of three basal plane bonds, since if the chosen Wannier functions are not already such that the cross transfer integral tYz equals zero, one can always choose a Wannier basis in which this is the case. The trigonal symmetry then determines the transfer integrals along the other two directions d

and c (Fig. 1). A simple calculation shows that the t,,,

t,,, t,, along these directions are comparable in magnitude. Hence an antiferromagnetic coupling both in spin and orbital occupancy is more stable along these two directions since it allows the maximum lowering of kinetic energy compatible with the minimum playing of Coulomb repulsive energy when two electrons happen to be on the same site. As already stated, the results of the atomic calculation hold true if an unrestricted Hartree-Fock calculation is performed. The boundaries of the phase diagrams are only slightly changed.

In a restricted Hartree-Fock calculation (that is

to say all the occupation numbers

<

ni,,

>

equal) a

band structure of the type shown in figure 4a is found

(the degeneration of the two e, band being maintained under the trigonal symmetry). A gap 3s found if one

s t a t e s e t =

-

e v l s p 8 n N a t o m e v / s p l n N o t o m s t a t e s e v / s p l n / V o t o m

""T

l e g eLect./s~te

FIG. 4. - Density of states for corundum structure with one elec- tron per site in a doubly degenerate eg band. 4a) without electro- nic correlation ; 4b) with electronic correlation ; 4c) including an extra electron per site diamagnetically bonded in alp without mutual polarization of the alp and eg electron ; 4d) same as in figure 4c but allowing for mutual polarization of the alg and eg

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ON THE METAL-INSULATOR TRANSITION IN V203 C4-203

allows a symmetry braking suggested by the atomic limit calculation (Fig. 4b). (The occupation numbers

<

n,,,

>

are determined self-consistently.) Typical values of the physical parameters are (refer to Table I)

p = 0 . 4 0 e V 7 a = 0 . 3 0 e V , P - 0 , U , , = 1 . 5 eV,

UI2 = 1.3 eV, J = 0.1 eV, E,,, = 0.2 eV. This is

interesting in two respects. It shows firstly the possi- bility of opening a correlation gap and secondly that this occurs far away from the critical values of the band width and Coulomb repulsion which one would

expect from a Gutzwiller-Brinkman-Rice type of

description of the metallic phase in V203.

At this point it would be tempting to say that the addition of an extra a,, electron per site does not modify this picture. Since the transfer integral along the c axis for a,, electrons is quite big one would be led to say that the added a,, electron goes into the bonding a,, band (which is full with one electron per site) well below the Fermi level. (Fig. 4c). In this way the correlation gap in the e, band would remain and the transition to the metallic state would be viewed as a pure Mott transition.

Unfortunately this interpretation is not supported by

the experimental evidence [5] that there are 1.2 p, per

site, pointing to a strong and net polarization of the a,, electrons. (Itinerancy and a considerable covalency

reduction factor (-- 1 -

A:)

would reduce - as born

out by actual calculation - the total magnetic moment

of 2 p, to the observed value).

Moreover the increase of the cation-cation distances with parallel magnetic spins which one observes when going from the metallic phase to the antiferromagnetic insulating phase would have no immediate

explanation in the above interpretation. _,

Including an a,, electron in our self-consistent unrestricted Hartree-Fock calculation and trying to pola- rize it by interatomic exchange interaction results in

the following situation. For JIU,,

<

1.5 the RO-RS,

structure for the e, electron and a diamagnetic a,, band is found to be the most stable (lowest ground state energy) among various possible self-consistent structures which we tried, as for example the RS structure obtained by putting the two electrons in the e, band in the real spin ordering and leaving the a,, band empty.

If one increases the ratio J / U (- 0.2) it happens that

whatever the input structure is, the most stable self- consistent solution is the one in which the e, band

contains 1.5 electrons per site and is ordered in the RS

structure, whereas the a,, band is half full with 0.5 electron per site and polarized in the same structure. The

the AI insulating phase of V203 and the fact that now the crystal symmetry, not only the magnetic symmetry, is lower we are forced to introduce new transfer integrals (necessarly proportionnal to the

distortion for small displacements of the atoms) : for

example along the vertical pairs a non zero interband transfer integral may appear which would be rigo- rously zero if the trigonal symmetry were valid.

Introducing these new quantities as parameters in our calculation and modifying according to the distortion the preexisting parameters (for example we

introduce a small reduction factor (-- 0.8) for the

direction ab and ac (Fig. l ) corresponding to an increase

of the distances of the cation pairs) we are able to open a gap for reasonable values of these parameters.

Accepting this last point of view, we are led to the conclusion that the distortion (although magnetostric- tive in origine as a consequence of the tendency of the system to a certain magnetic instability), nonetheless plays a somewhat independant role in localizing the electrons and is determinant for the opening of the gap. It should be clearly stated that the distortion alone without the kind of magnetic instability we have been discussing would be insufficient to open the gap and make V203 a semi-conductor at low temperature.

Obviously a more fundamental study of the electronic level of the distorded cluster is needed before we accept this conclusion. Our aim is to show that the consideration of the experimental evidence available up to now supports, or at least does not contradict this view and leads to a consistent interpretation of the metal-insulator transition.

Let us first try to understand the nature of the metal-

lic phase at about 150 K immediatly above the transi-

tion. That the electronic system is one of high entropy content is a speculation that has already been put forward by several authors (Rice, MC Whan [4]). The argument is that this entropy content can be evaluated as the integral

along a path which goes from the metallic state at zero temperature and high pressure to the final state at

atmospheric pressure and

T,

= 150 K. The electronic

specific heat y T has been measured at low temperature

and high pressure [6] giving a density of states at the Fermi level of the order of 6 states/eV/Vatomj spin if we use

band scheme is shown in figure 4d with the relative _n2

occupation of the electrons. Unfortunately this

Y

= - 3 N k 2 P,E(&F)

-

structure is metallic, since now we must put 0.5 elec-

tron in the e, band above the point where it would be This is one order of magnitude more than what is

easy to open a correlation g. usually encountered in transition metals and indicates a

All this holds if we leave for the transfer integrals the strong mass enhancement of the electrons at the Fermi

trigonal symmetry, although braking it with a spin and level. On the other hand we have a measurement of the

orbital structure which is not invariant under C,. But bare density of the states for the system at atmospheric

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C4-204 C. CASTELLANI, C. R. NATOLI AND J. RANNINGER

measurements of JCrome and coworkers [7], if correctly interpreted by taking into account the dipolar relaxa- tion which is not negligible in strongly exchange enhanced systems lead to a bare band density of the states at the Fermi level pb(eF) of the order of 1 state/eV/V atom/Spin and to a Stoner enhancement factor of the order of 10. In this interpretation the system is described as having an exchange enhanced susceptibi-

lity proportional to

-

P~(")

.

The a value derived from

l - a

experiment is of the order of 0.9 which together with the measured susceptibility

of the order of 1.6 X 10-3 emu/mole (the total measured

value of the susceptibility is 2 X 10-3 emulmole from

which the Van Vleck susceptibility estimated to be

0.4 X 10-3 emu/mole is substracted) leads to pb(cF)

of about 1 state/eV/Vatom/spin (in this scheme the temperature dependence of the susceptibility in the temperature range of 150-306 K is due substantially to the temperature dependance of the enhancement

factor a, 8).

Furthermore because of

where m is the number of bands at the Fermi surface

(in our case m = 2, since in the metallic phase the den-

sity of the a,, band at the Fermi level is negligible)

we derive U,,

+

J

-

2 eV, which is in fact the value

we have used initially in our calculations. In our interpretation

is also the order of magnitude of the electronic specific

heat at 7' = T,, since there is practically no variation

(0

fi

0.5 cal/mole/K) of the total specific heat as

measured before and after the transition [9] (this assumes that tfie lattice specific heat does not vary very much and indeed we shall see that, if at all, it should increase in going from the AI phase to the metallic state, confirming our assumption concerning the electronic specific heat).

Moreover the resistivity data at low temperature under pressure [l01 show a T' behaviour indicating a strong electron hole correlation which fades away at about 100 K where there is a turnover to a more

metallic behaviour, proportionality to T. This would

indicate a correlation energy of the electronic system of the order of 0.01 eV. Correspondingly we expect a

decrease of the y value of the specific heat with a

marked drop around T = 100 K (Fig. 5). In this

picture the area underneath the y(T) curve gives an estimate of the electronic entropy content of the metallic phase. One finds that it roughly represents half

states

FIG. 5. - Temperature dependence of the value of the specific heat as conjectured from specific heat, NMR and conductivity

measurement.

the total entropy change observed at the transition. The rest of the entropy must then be provided by the lattice.

That this latter conclusion should hold is seen from the following considerations. If one observes figure 1 representing a central atom a surrounded by four immediate neighbours b, c, d, e, we see that with 2 electrons and three nearly degenerate levels per site one can have an orbital ground state which is degenerate, this degeneracy being lifted by the distortion of the molecule according to the instability already discussed

Thus our picture-of the transition at T = T, = 150 K

is as follows. There are two states of the system, the M state and the AI which have nearly equal internal

energies, the M state having a big entropy content,

the ordered AI state having low entropy content. The metallic state is spin and orbit correlated at least at short range and this correlation becomes stronger as one approaches it from above. The transfer integrals are big enough not to support a long range ordered

state. At T = Tt some clusters begin to distort, since

locally the system gains energy by localizing the electrons and taking advantage of internal exchange energy. The arbitrary choice of a distortion axis of a molecule influence the next one in a sort of cooperative Jahn-Teller effect, all three possible axis of distortion being possible (the crystal is in fact twinned in the AI state). On the other hand, increasing temperature in the

AI phase creates spin deviations which at T = T, are

sufficient enough to perturb some distorted molecules (the distortion being a consequence of the spin order-

ing !) which serve as nucleation centers to make

the electronic solid melt. The bigger entropy content of the M phase makes the corresponding free energy

vary much faster as a function of T as compared to

the AI phase, so that a first order transition is expected. This view explains also the fact the system is sensi- tive to applied axial pressure with basal plane and not

along the c axis. This is because the ferromagnetic

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ON THE METAL-INSULATOR TRANSITION IN V203

Finally we want to point out that in this interpretation of the transition the variation in the relative population of the a,, band plays little role. Actually in both phases the a,, band is to a certain extent depopu- lated because of the intraatomic exchange correlation. This explain the variation of the c/a ratio in the corun-

dum compounds (Fig. 6). For Ti,03 there is no choice ;

for the reasons given above it has to form stable bonds along the c-axis pairs. For Cr203 with three electrons per site coupled to spin 312 the at, is occupied and the maximum of bonding is achieved by an anti- ferromagnetic coupling along the c axis pairs. For V203 where c/a is a maximum, there is competitions between bonding of the a,, pairs along the c axis (due

to diamagnetic spin coupling) and intratomic exdhange

2

Ti V Cr Fe Ga

correlation with the eg electrons tending to introduce _FIG. _6._-_c,a_ratio_{for transition}_{metal oxydes of}_corundum

a triplet component in those pairs and thereby loose- _{structure (after D. 13}_MC_Whan,_T._{M. Rice}_and_J._{P. Rerneika,}

ning the strenght of the bonds. Phys. Rev. Lett. 23 (1969) 1384.

References

[l] ANDERSON, P. W., Phys. Rev. 115 (1959) 2. [7] KERLIN, A. L., NAGASAWA, H. and JEROME, D., Solid State

GOSSARD, A. C. and REMEIKA, J . P,, Solid State Commun.

15 (1974) 609.

HONIG, J. M., VAN ZANDT, L. L., BOARD, R. D. and

WEAVER, H. E., Phys. Rev. 6 (1972) 1323.

ASHKENAZI, J. and CHUCHEM, T., Phil. Mag. 32 (1975) 763.

MOON, R. M., Phys. Rev. Lett. 25 (1970) 527.

Commun. 13 (1973) 1125.

[S] DE BOER, F. R., SCHLENKBR, C. J., BRESTERBOS, J. and

PROOST, S., J. Appl. Phys. 40 (1969) 1049.

[g] COOK, 0 . A., J. Mer. Cham. Soc. 69 (1947) 331 and

COEY, 3. M. D., ROUX-BUISSON, H., SCHLENKER, C.,

LAKKIS, S. and DUMAS, J., Conference Proceedings

(1975).

[6] MC WHAN, D. B., REMEIKA, J. P., BADER, S. D., TRIPLET, B. [l01 MC WHAN, D. B. and RICE, T. M., Phys. Rev. Lett. 22 (1969)