PROPERTIES OF MOTT-HUBBARD BANDS IN THE ATOMIC LIMIT

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PROPERTIES OF MOTT-HUBBARD BANDS IN THE ATOMIC LIMIT

W. Brinkman, T. Rice

To cite this version:

W. Brinkman, T. Rice. PROPERTIES OF MOTT-HUBBARD BANDS IN THE ATOMIC LIMIT.

Journal de Physique Colloques, 1971, 32 (C1), pp.C1-1102-C1-1104. �10.1051/jphyscol:19711392�.

�jpa-00214432�

TRANSITION MÉTAL-ISOLANT ;

SEMI-CONDUCTEURS MAGNÉTIQUES-

PROPERTIES OF MOTT-HUBBARD BANDS IN THE ATOMIC LIMIT

by W. F. BRINKMAN and T. M. RICE

Bell Telephone Laboratories, Incorporated Murray Hill, New Jersey

Résumé. — Nous étudions la forme de la conductibilité en fonction de la fréquence, dans le modèle de Hubbard pour un électron par atome. La conductibilité s'écrit en fonction des chemins possibles d'un électron et d'un trou supplé- mentaires, la paire particule-trou évitant de se trouver simultanément sur le même site du réseau. La part d'absorption de la conductibilité peut se décrire en gros comme une raie gaussienne, centrée sur l'énergie U de largeur 2(zt}/3, ainsi que l'indique un calcul des moments. La raie est quelque peu plus étroite pour un système de spins aléatoires que pour l'état de Néel antiferromagnétique. Nous envisageons aussi le problème de l'anisotropie de la conductibilité dans les structures de spin en couches et montrons qu'on ne la prévoit pas grande.

Abstract. — We consider the form of the frequency dependent conductivity function for the Hubbard Model when there is one electron per atom. The conductivity is written in terms of the possible paths of an extra electron and hole in which the particle-hole pair avoid being simultaneously on the same lattice site. The absorptive part of the conductivity can be crudely described as a Gaussian line centered at energy U of width 2(zt)^y3 as inferred from a calculation of the moments. The line is somewhat more narrow for a random spin system than for the antiferromagnetic Neel state. We also consider the question of the anisotropy of the conductivity in layered spin structures and show that it is not expected to be large.

I. Introduction. — In this paper we would like to consider two properties of the Hubbard model that can be calculated in the atomic limit [1, 2]. The first is the absorptive part of the conductivity at energies of the order of the intra-atomic Coulomb energy. We consider there to be one electron per atom. The cal- culation is presented in Chapter II and the result is shown to be somewhat different than one would expect from the one-electron density of states [3].

The second quantity we calculate is the anisotropy in the resistivity above and below the ordering tem- perature when the magnetic ordering occurs in fer- romagnetic layers ordered anti-parallel in the third direction such as in V203 [4, 5]. In Chapter III we calculate the band averaged resistivity for a hexagonal lattice in order to estimate this effect.

II. High frequency conductivity. — In this section we study the absorptive part of the conductivity at frequencies of the order of the intra-atomic Coulomb interaction U. We will work in the atomic limit assu- ming the hopping matrix element t is small compared to U and that it is non-zero only for nearest neighbors.

It is assumed that there is one electron per atom and we consider the two cases, one when the local spins are arranged in an antiferromagnetic N6el state (AF) and the other when the spins are randomly arranged on the lattice (R). U is assumed sufficiently large so that double occupancy of a site is unimportant in the ground state. The absorption we calculate is thus that due to the transfer of an electron from one atom to the next. The interest here is to see if there is any rela- tion between the shape of the absorption band at this energy and the one particle densities of states calcu- lated in a previous paper by the authors [3].

We start with the Kubo formula [6] and introduce resolvant operators. Then by carrying out the time integrals we obtain

(2.1) Here

(2.2) H is the full Hubbard Hamiltonian, e is electronic charge and a is the lattice constant, at creates an electron of spin a on the site /. The contour c encloses the poles of the resolvant operators (co — H )- 1. The expectation value is taken with respect to all the states of the system. If Q is assumed to be positive and the temperature is small compared to U only states with no double occupancy are allowed. The states omitted are weighted by a factor exp(— piT) which is assumed small. The energies of the remaining states in the ensemble are of order t2/U and in the limit we are working can be taken as zero [3]. The (ci^ — H )- 1

can, therefore, be replaced by mi and the (O integrals performed.

(2.3) The subscript 0 on the trace indicates that only states with all sites singly occupied are to be kept. At temperatures large compared to t2/U the average includes all possible spin configurations while for temperatures equal to zero, we will keep only the Neel state. Let us separate the Hamiltonian as

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711392

PROPERTIES OF MOTT-HUBBARD BANDS I N THE ATOMIC LIMIT C 1 - 1 1 0 3

and expand the resolvant in powers of H,.

The nth order term in this expansion is described as follows. The first j, operator creates a particle-hole pair at two near neighbor sites on the lattice (the hole being an empty site and the particle a doubly occu- pied site). The particle or hole then moves n steps on the lattice and they recombine when the second j, operator acts. The condition on the paths is that the spins along it all be returned to their original configu- ration. Paths from which the particle and hole come together at an intermediate step can easily be shown to cancel. This means that the (Q - H,') in the denomi- nators of the expansion (2.5) are all equal to (Q - U).

Because the maximum change in kinetic energy is 2:zt the Re o(o) is non-zero only for

Let us define moments

Now oo defines the total intensity which is equal to 2 n e2 a2 t2 p per unit volume where p is the probabi- lity that one can find a nearest neighbor pair with opposite spin. In the simple cubic lattice p =

+

the random configuration and p = 1 for AF configu- ration. We have calculated the 2nd and 4th moments.

by counting the possible paths and giving them their proper weight. We find for the simple cubic lattice that

and - =

These moments are quite close to being those of a Gaussian line of width 2(zt)/3. This is considerably more narrow than one would guess from twice the width of the one electron density of states

-

It should also be noted that the R configuration gives a somewhat narrower peak than the AF configuration.

This is in contrast to the narrower one electron density of states for the AF configuration. The reason for this effect is that while regions which are ferromagnetic tended to broaden the one particle density of states they tend to narrow the absorption band because in these regions a quasi-momentum conservation is established. This is easily seen in that if you create a particle-hole excitation in an otherwise completely ferromagnetic region the resulting line is a delta function. The quasi-momentum conservation is intro- duced into the calculation of the moments by partial phase cancellations between various walks for the R configuration.

In conclusion it appears that the interaction effects

in the Hubbard Model have strongly modified the overall shape of the band. In contrast to the one- particle density of states, the absorption band broa- dens slightly as the temperature is lowered below the ordering temperature and the calculated moments suggest a single peak of a Gaussian form. In this model there is no sharp edge to the absorption band.

111. Anisotropy in the resistivity. - In crystals where there is an ordered phase with ferromagnetic planes one might expect a difference in the conducti- vity along and perpendicular to these planes. In this section we make an attempt at estimating this difference. Consider a simple hexagonal lattice in which the hopping matrix element along the c axis is t, and that in the hexagonal plane t,. We will assume that the ordered state is one in which the hexagonal planes are ferromagnetic and that these planes line up anti-parallel along the c axis. To obtain a crude estimate of the effect of the order we compare the ratio of the band average conductivity along and perpendicular to the c axis calculated both above and below the Ndel temperature. By band average conduc- tivity we mean that we assume the carriers are distri- buted evenly over the states of the band. (Not accor- ding to thermal equilibrium.) This is easily done since in the atomic limit the d. c. conductivity can be written as a function of energy in th'e band which is averaged over the thermal distribution of carriers. Our calcula- tion is equivalent to taking the high temperature limit with respect to the distribution of carriers. Under these circumstances the procedure used by Ohata and Kubo [7] can be used to calculate the averaged d. c.

conductivity. In this procedure one simply calculates the first three non-zero moments of a(o) and then depending on the ratio of the second and fourth moments deduces a value for the d. c. conductivity by fitting to a Lorentzian or Gaussian. We have done this for both the AF and R configurations and find the moments to be best fit by a Gaussian function. In the AF phase we find that the ratio of the conductivity along the c axis o,, to that in the hexagonal plane ox, is

Here c is the lattice constant along the hexagonal axis and a that in the plane. If we calculate the same quan- tities for a random spin distribution we find that

The main effect of the random configuration is to introduce resistance in the hexagonal plane and this gives rise to the square root form in (3.2). If t, = ta we see that the increase in the anisotropy is

- ^1/43

the Ndel temperature. This number becomes closer to one if to > ta and becomes larger if t, > t,. The net result of this calculation is that if the anisotropy in the resistivity is not large in the disordered phase one expects that the AF ordering will increase the aniso- tropy by approximately a factor of two.

W. F. BRINKMAN AND T. M. RICE

References

[I] HUBBARD (J.), Proc. Roy. Soc., 1963, A276, 238. [7] OHATA (N.) and KUBO (R.), J. Phys. Soc. (Japan), [2] HUBBARD (J.), Proc. Roy. Soc., 1964, A281, 401. 1970, 28, (to be published).

[3] BRINKMAN (W. F.) and RICE (T. M.), Phys. Rev. [8] GUTZWILLER (M. C.), Phys. Rev., 1965, 137, A 1726.

(in press). [9] SOKOLOFF (J. B.), preprint.

[4] MOON (R. M.), J. Appl. Phys., 1970,41, 883. [lo] RICE (T. M.) and BIUNKMAN (W. F.), Proc. of the [5] MCWH~N (D. B.), RICE (T. M.) and REMEIKA (J. P.), Battelle Colloquium on Critical Phenomena,

Phys. Rev. Letters, 1969, 23, 1384. Gstaad, 1970 (to be published).

[6] KUBO (R.), Can. J. of Physics, 1956, 34, 1274. [ l l ] NAGAOKA (Y.), Solid State Comm., 1965, 3, 409, and Phys. Rev., 1966, 147, 392.