THEORY OF THE MOTT INSULATORS NiO AND CoO

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THEORY OF THE MOTT INSULATORS NiO AND

CoO

B. Brandow

To cite this version:

JOURNAL DE PHYSIQUE Colloque

C4, supplkment

1976,

THEORY OF THE MOTT INSULATORS NiO AND

COO

(")

B. H. BRANDOW

Theoretical Division, Los Alamos Scientific Laboratory Los Alamos, New Mexico 87545 U. S. A.

Rkum6. - Le sujet de cet article doit paraStre prochainement [ l ] et sera expose de f a ~ o n plus detaillee qu'ici ; en outre, un rapport complet parattra sous peu 121. Ce qui suit n'est qu'un aperw de notre travail.

Nous prhentons un point de vue theorique qui tente de resoudre les difficultes concernant les isolants de Mott, de fawn simple et intuitive et qui fournit Bgalement quelques resultats semi- quantitatifs en accord avec l'experience. Nous assimilons les isolants de Mott aux isolants magne- tiques habituels, c'est-h-dire des materiaux (periodiques et stcechiometriques) dans lesquels des moments locaux et un gap persiste au-dessus de la temperature de destruction de l'ordre h longue distance (TN ou Tc). Bien que la grande majorite de ces materiaux soit antiferromagnktique, quel- ques ferromagnetiques sont connus. Nous trouvons que les caracteristiques principales des isolants de Mott peut btre comprise dans le formalisme Hartree-Fock (HF), en considerant une C ~ ? S S ~

convenable de solutions H F differentes, Comme exemples specifiques, nous traitons NiO, en ralson de sa simplicite relative et de son histoire. C'est en effet un des exemples le plus 6tudie d'isolant de Mott. COO est egalement un trks bon exemple car il attire notre attention sur un problkme supplb mentaire, celui de la d6gentrescence due au champ cristallin.

Avant ce prhent travail, aucune explication satisfaisante n'existait pour les gaps optiques Blevks dans ces materiaux.

Afin #interpreter les gaps a T = OK nous utilisons la theorie de Slater des bandes pour les antiferromagnetiques 131. L'approximation de l'kchange local p 113 ne peut rendre compte de l'impor-

tance du gap dans NiO, mais un traitement plus detail16 des interactions intra-atomiques 3 d, bask sur les paramktres U, U', J de Kanamori [4], donne de bons resultats [ S ] . Nous utilisons aussi les integrales de hopping, en liaisons fortes, obtenues par Mattheiss dans son calcul de bandes [61.

Ainsi, on dkrit les electrons 3 d par une approche faisant intervenir la degenkrescence orbitale dans l'hamiltonien de Hubbard.

La question de la degenerescence de champ cristallin dans COO est risolue en reduisant la fonc- tion d'onde dans COO a un simple determinant, par des transformations canoniques sur les orbitales tzg de chaque Co2+. Avec les parametres de Kanamori et dans l'approximation H F on trouve que tous les Btats inoccupes des centroldes de la sous-bande 3 d sont plus hauts que les Btats occupes d'une quantite de l'ordre de U. Ceci explique, pour la premiere fois [ S ] pourquoi COO est, du point de vue experimental, si proche de NiO.

Nous devons aussi expliquer pourquoi les gaps persistent au-dessus de TN. NOUS regardons pour cela un reseau d'atomes d'hydrogkne. Pour une repulsion coulombienne U a un site suffisamment forte, on peut supposer l'existence de 2N solutions autoconsistantes differentes des equations H F ; ces solutions representent des configurations isolantes qui correspondent de faqon biurivoque aux

2 N configurations differentes de spin d'un modele dYIsing pour un systeme a N sites. Bien que cette

hypothkse ne soit pas nouvelle [7], elle n'a pas attire l'attention. Nous avons present6 la premikre preuve concrete [ 5 ] pour Btayer cette hypothkse. Cependant, I'objection majeure B cette description simple est que le systhme de fonctions d'onde de base (les 2 N determinants) n'est pas complbte-

ment orthogonal. Ceci a la consequence fticheuse que, si l'on essaie de calculer un hamiltonien de spin effectif, on rencontre un cas de catastrophe due & la non-orthogonulite'. Nous avons resolu ce problkme [8] par un developpement en perturbation de clusters [9], un analogue d6gkner6 du forma- lisme de Bruekner-Goldstone.

Notre etude contient aussi une reconciliation de plusieurs donnees sur NiO avec les integrales de hopping de Mattheiss [ 6 ] . Ceci entraine l'estimation de plusieurs effets de rearrangement des orbitales associes aux trous 3 d. De plus, une analyse dktaillke des effets de champ cristallin sur les etats finaux 3 dn-1 a conduit B un meilleur accord avec les donnees de XPS, ce que les analyses anterieures ne permettaient pas.

Un survol des stabilitks de phase des diverses solutions H F demontre que la ph6nomenologie des isolants de Mott (mis en kvidence dans les diagrammes de phase de V 2 0 3, NiS 2-NiSe2, et les perovs-

kites) se comprend qualitativement dans notre sch6ma Hartree-Fock, quand ceci est renforce par d'autres effets physiques connus tel qu'un effet d'kcran plus violent dans 1'6tat mktallique.

Abstract. - The present subject matter is currently being published [ l ] in more detail than possible here, and a thorough report will appear elsewhere [ 2 ] . What follows is merely an outline. We offer a theoretical picture which resolves the main conceptual difficulties of Mott insulators in a very simple and intuitive manner, and which has also yielded some semi-quantitative results in (*) Part of this work was performed under the auspices of the U. S. Energy Research and Development Administration.

B. H. BRANDOW

agreement with experiment. We identify Mott insulators with ordinary magnetic insulators, i. e. materials (nominally periodic and stoichiometric) in which local moments and an insulating gap persist above the temperature (TN or Tc) for the destruction of long-range spin order. (Although the vast majority of such materials is antiferromagnetic, a few ferromagnetic examples are known.) We find that the main features of Mott insulators can be understood within the general framework of the Hartree-Fock (HF) approximation, by considering a suitable variety of different HF solu- tions. As specific examples, we consider NiO, because of its relative simplicity as well as the fact that this has historically been the best-known and most extensively studied example of a Mott insulator, and also COO, because this focusses attention on an additional problem-that of crystal-field degene- racy. Previous to the present work, there has been no satisfactory explanation for the large insula- ting (optical) gaps in either of these materials.

To explain the gaps at zero temperature, we use essentially the antiferromagnetic band theory of Slater [3]. The local p 113 exchange approximation is unable to account for the large magnitude of the gap in NiO, but a more detailed treatment of the 3 d intra-atomic interactions, based on the U, U', J parameters of Kanamori [4] is found to be adequate [ 5 ] . We also use the tight-binding hopping integrals obtained by Mattheiss from his band calculations [6], thus the 3 d electrons are described by an orbitally degenerate version of the familiar Hubbard Hamiltonian. The crystal-field degene- racy problem for COO is resolved by formally reducing the wavefunction for COO to a single- determinant form, by means of suitable canonical transformations among, the minority t ~ , orbitals of each Co2+ cation. Use of the Kanamori parameters within the H F approximation then leads to the result that all of the unoccupied 3 d sub-band centroids lie higher than their occupied counter- parts, by amounts of order U. This explains, for the first time [5], why COO is experimentally so similar to NiO.

We must also explain why these insulating gaps persist for the disordered-spin configurations encountered above TN. We focus on the prototype system consisting of a lattice of hydrogen atoms. For a sufficiently large one-site Coulomb interaction U, it is plausible to assume the exis- tence of 2 N different self-consistent solutions of the HF equations, solutions which represent insulating configurations in an obvious one-to-one correspondence with the 2 N different Ising spin configurations for an N-site system. Although thi hypothesis is not at all new [7], it has not previously attracted much attention. We have presented the first concrete evidence [5] in support of this hypothesis. However, the most cogent objection to this elementary picture is that the resulting set of basis functions (the 2 N different N-electron determinants) is not fully orthogonal. This has the unfortunate consequence that, in attempting to derive an effective (Heisenberg-like) spin Hamilto- nian, one encounters a form of the old nonorthogonality catastrophe. We have resolved this pro- blem [S] by means of a suitable linked-cluster perturbation expansion [9], a degenerate (open-shell) analog of the Brueckner-Goldstone formalism.

Our study also includes~a semi-quantitative reconciliation of several kinds of NiO data with the effective 3 d hopping integrals obtained by Mattheiss [6]. This involves the estimation of several orbital rearrangement effects associated with physical 3 d holes. Furthermore, a detailed crystal- field analysis of the 3 dn-1 final states has led to far better agreement with the high-resolution XPS data than any previous analyses. A survey of the phase stabilities of various types of HF solutions demonstrates that much of the basic phenomenology of Mott insulators (as evidenced by the gene- ralized phase diagrams for V203, NiS2-NiSe2, and perovskites) can be qualitatively understood within the present H F framework, when this is supplemented by other known physical effects such as stronger screening in the metallic state.

References

[l] BRANDOW, B. H., Int. J. Quantum Chem. To be published. [7] ADLER, D., Solid State Phys. 21 (1968) l ; [2] BRANDOW, B. H., Adv. Phys. To be published. &ROT, M., J. Physique 33 (1972) 125 ; [3] SLATER, J. C., Phys. Rev. 82 (1951) 538. _{[8] BRANDOW,} CYROT, M., Phil. Mug. _B. 25 (1972) 1031.

H., Advances in Quantum Chemistry (Vol. 10),

[4] KANAMORI, J., Prog. Theor. Phys. 30 (1963) 275. _{Ed. Lowdin (Academic Press, New York) 1976. To be}

[5] BRANDOW, B. H., J. Solid State Chem. 12 (1975) 397. _bublished.