By S. L. DUDAREV, D. NGUYEN MANH and A. P. SUTTON
Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, England
Abstract:
The influence of Mott-Hubbard electron-ectron correlations on the electronic structure and structural stability of uranium dioxide (U02) has been analysed using the local spin-density approximation (LSDA) + U approach. We have found that the inclusion of a term describing the Hubbard on-site repulsion between Sf electrons results in a dramatic improvement in the description of the equilibrium electronic and magnetic structure of U02 for which conventional LSDA calculations incorrectly predict a non-magnetic metallic ground state. We have found that the presence of electron-electron correlations in the 5f band modifies the character of chemical bonding in the material, leading to a Heitler-London type of hybridization between the Sf orbitals and giving rise to a larger value of the equilibrium lattice constant in better agreement with experimental observations.
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1. INTRODUCTION
The problem of metallic ground states predicted by density functional theory (DFT) within the local-density approximation (LDA) for a number of insulating materials (e.g. for transition-metal monoxides) has attracted considerable attention in recent years and remains a debatable subject. It has been established that qualitatively the origin of the failure of conventional DFT LDA to predict a bandgap in these materials originates from the presence of strong (larger than or comparable with the effective bandwidth) on-site Coulomb repulsion between electrons localized on the same metallic ion. This idea was first proposed by Mott (1974) and Hubbard (1963, 1964a,b) who discovered the mechanism responsible for the appearance of band gaps in the density of electronic states of compounds where a conventional DFT LDA calculation would lead to a metallic ground state. Uranium dioxide (U02) is a typical representative of this class of materials and it is predicted to be metallic in the DFT LDA approach (Arko et al. 1986, Kelly and Brooks 1987, Goodman 1992, Petit el al. 1996). However, pure U02 is known to be a good insulator (Castell, Muggelberg, Briggs and Goddard 1996, Muggelberg, Castell, Briggs and Goddard 1997) and it is also known that, at low temperatures, the magnetic moments on uranium ions order antiferromagnetically (Faber and Lander 1976). This points to the presence of strong electron4ectron correlations in the ground state of this compound (Anderson 1963, Spalek 1990).
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2. COMPUTATIONAL APPROACH
Actinide elements have 5f shells which in many compounds are only partly filled, for example a U4+ ion in U02 has two electrons in its outer 5f shell. In a solid, 5f electrons become itinerant while retaining some of their localized character. Competition between localization and itineracy of 5f electrons in actinide compounds leads to unusual electronic properties and in the recent past there has appeared an extensive literature in which two alternative approximate treatments have been developed. The first treats 5f electrons as localized and leads to a cluster-type (configuration interaction or Anderson impurity model) approach (Gunnarsson, Sharma, Hillebrecht and Schonhammer 1988). This approach has been successfully applied to the interpretation of X-ray photoelectron and optical spectra of UOz (Kotani and Yamazaki 1992, Kotani and Ogasawara 1993, Krupa and Gajek 1991). However, the cluster approach becomes invalid if we are interested in the description of spatially delocalized electronic states leading, for example, to the onset of antiferromagnetism, or to the appearance of spatially delocalized electronic states around surface defects which have been recently discovered by Castell et al. (1997) on the (001) surface of NiO (which is also a material characterized by the presence of strong correlations between valence electrons (Zaanen, Sawatzky and Allen 1985, Zaanen and Sawatzky 1987)).
To understand the electronic properties of U02 on a macroscopic scale, an approximate approach has been developed which involves periodic boundary conditions and treats 5f electrons as itinerant and effectively independent particles moving in a self-consistent field calculated using the LDA (Arko et al. 1986, Kelly and Brooks 1987). This approximation was shown to lead to the incorrect result that the ground state of UOz is metallic (Goodman 1992, Petit et al. 1996), a result which follows from the fact that the 5f shell of uranium ions is only partly filled. This fact, in addition to the existence of antiferromagnetic ordering in the ground state of U02, points to the presence of strong electron-electron correlations in the ground state of this material (Fulde 1984).
In this paper, we report the results of numerical studies of the electronic structure of U02 which have been performed using three different methods: firstly the spinindependent LDA, secondly the LSDA and thirdly the LSDA + U approach. All the methods have been realized using the same linear muffin-tin orbitals (LMTO) atomic spheres approximation (ASA) program by van Schilfgaarde, Paxton and Methfessel (1994), although the third method required substantial modifications to be made to the program. The first two methods (LDA and LSDA) are known to give reasonably accurate descriptions of the electronic structure of weakly correlated electronic systems such as simple or transition metals and semiconductors (Fulde 1984). The third method has been recently shown to open the possibility of calculating the mean field electronic structure of materials with strongly correlated electrons, for example transition-metal oxides (Anisimov et al. 1991, 1993) or rare-earth compounds (Liechtenstein and Mazin 1995). In what follows, we present the results obtained using all three techniques and discuss the origin of differences between the results obtained using the different approaches.
U02 adopts the fluorite crystal structure shown schematically in fig. 1. In the non-magnetic state, a primitive unit cell contains two oxygen atoms located at (a0/4, ao/4, ao/4) and at (3a0/4, ao/4, 44) and a uranium ion is positioned at the origini. Numerical implementation of the LMTO ASA method requires introducing empty spheres in the space separating uranium ions (one empty sphere per uranium site). The coordinates of the centre of an empty sphere corresponding to the above choice of positions of uranium and oxygen sites are (ao/2, 0,O). All the calculations discussed below were performed using muffin-tin sphere radii RU : R, : Rempty = 1.15 : 0.9 : 1.007 chosen in such a way as to reduce the contribution of the electrostatic interionic interaction to the total energy of the system. 7s, 6p, 6d and 5f orbitals of uranium have been taken into account in the expansion of one-electron Bloch states. In choosing this set of orbitals, we followed Petit et al. (1996) who found that the results obtained using this set were in good agreement with full-potential LMTO calculations. Sets of Is, 2p, 3d and 4f orbitals and 2s, 2p, 3d and 4f orbitals were used to describe states located on empty sphere and oxygen sites respectively. All the calculations used the scalar-relativistic version of the LMTO program and the von Barth-Hedin (1972) exchangexorrelation potential. We did not include gradient corrections in our calculations since it has been shown that they result in no improvement in the calculated properties of UOz (Petit et al. 1996).
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