[The Chemical Physics of Solid Surfaces] Oxide Surfaces Volume 9 || Clean Oxide Surfaces: a...

Oxide Surfaces D.P. Woodruff, editor © 2001 Elsevier Science B. V. All rights reserved. 35

Chapter 2

Clean Oxide Surfaces: a theoretical review

Claudine Noguera

Laboratoire de Physique des Solides, UMR CNRS 8502, Universite Paris-Sud, 91405 Orsay, France

1. INTRODUCTION

The last five years have witnessed a tremendous effort to better produce, characterise and study insulating oxide surfaces. Several reasons stem for the rapid development of this field. They are related to experimental considerations — a better control of the fabrication of surfaces, a more thorough use of advanced spectroscopic or structural tools — but also to a more accurate recognition of the technological importance of high quality oxide surfaces in catalysis, magnetic recording, as sensors or as constituents of artificial nano-materials.

A simultaneous effort to simulate these surfaces has been performed. While most theoretical works quoted in the Henrich's and Noguera's books [1,2] five years ago, referred to semi-empirical methods, first principles approaches are now routinely used to calculate the ground state properties of surfaces. The delay, with respect to the simulation of semi-conductor surfaces, may be assigned to the larger computational resources necessary to quantum-mechanically treat oxygen and transition metal atoms. However, ab initio methods find their limitations as soon as defects break the symmetry of the surface or when large cell reconstructions take place. In addition, most of them are based on a variational principle which limits their use to the prediction of ground state properties. Accounting quantitatively for quasi-particle spectra, optical excitations or d -^ d excitations on oxide surfaces remains a challenge. The same is true as regards analytical theories describing the basic microscopic processes at work on these surfaces.

These considerations will be examplified here on three aspects of oxide

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surface physics, namely (i) the non-polar stoichiometric surfaces, (ii) the oxygen deficient non-polar surfaces and finally (iii) the polar surfaces. The specific properties of the first family are induced by the low coordination of the surface atoms and are shared by other low-dimensional oxide systems, such as non-supported clusters or ultra-thin films [3]. In the second family, the presence of oxygen vacancies is responsible for strong electron redistributions in the surface layers which can result in a higher surface reactivity and/or surface reconstructions. A similar situation is met in the third family due to the strong electrostatic polar instability [4]. After a presentation of the available numerical methods in Section 2 and a review of simulation works in Section 3, we will consider successively the three families of surfaces in Sections 4, 5 and 6, from the point of view of their generic properties and the theoretical arguments to understand them. Finally, we will conclude on open questions for future investigations.

2. NUMERICAL METHODS

The first historical step in the simulation of insulating oxides relied on the idea that most of these materials are highly ionic. It has led to the development of classical models of cohesion in which the leading forces result from Coulomb interactions between ionic charges. These approaches usually give a quite reasonable description of atomic positions and vibrations, but cannot yield information on electronic degrees of freedom, which require quantum treatments. As a function of time, the latters have evolved from semi-empirical — tight-binding or semi-empirical Hartree-Fock (HF) — to first principles methods, in which there are no adjustable parameters. In parallel, quantum-chemistry all-electron approaches allow a treatment of electron correlations beyond the HF approximation, but only for small finite size systems. We will stress the strengths and limitations of these methods when applied to oxide surfaces, and we will also place special emphasis on some methods which can describe excited state properties, or ground state properties of highly correlated oxides.

2.1. Classical approaches In classical approaches, the lattice energy E, which is the diff'erence

between the total energy of the system and that of its component ions at infinity, is expanded into a sum of pair, triplet, etc terms:

E = lEEij + l Z Eijk + ... (1)

Keeping only the Eij terms yields the so-called pair-potential approximation. The pair interactions include Coulomb (l/r^j dependence), van der

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Waals {1/rfj dependence) and short range terms. The latters result from PauU's principle, which impedes the overlap of closed electronic shells. It may be taken under a Lennard-Jones ( l/rjj variation), or a under a Born Mayer form {Bij exp{—rij/p)). The parameters entering Eij are fitted, so that the properties of the perfect lattice are reproduced. In the applications to surfaces, they are kept equal to their bulk values, in particular the ionic charges Qi.

Many body effects are included in the triplet and higher order terms in Eq. (1). The triplet terms, for example, give energy variations with bond angles. In addition, low-coordinated atoms, which are submitted to non-vanishing electric fields, may be strongly polarized. The shell model [5] was designed to describe these effects, in an approximate way. Rather than being written under empirical forms, potentials can also be derived from the functional expression of the free ion energies as a function of the electron density [6]. Corrections may also be included to account for the instantaneous environment in which an ion finds itself, through modifications of the ionic radii, shapes and charge distributions [7,8]. A good review of the state-of-the-art interatomic potentials in solid state chemistry can be found in Ref. 9.

In order to predict the ground state geometry, energy minimization with respect to the ionic positions can be performed by various techniques: static ones — steepest descent, conjugate gradient methods — or dynamical ones — damped dynamics, or molecular dynamics [10]. The latters allow an exploration of the configurational space, through the technique of simulated annealing.

Classical simulations have long been performed to predict surface phenomena. Due to the increasing availability of first principles calculations for medium size systems, they are nowadays essentially restricted to large size systems, like zeolites, grain boundaries, complex interfaces, etc.

2.2. Semi-empirical quantumi methods At the lowest level of sophistication of quantum treatments, the tight-

binding method and the semi-empirical HF method reduce the complexity of the interacting electron system to the diagonalization of an effective one-electron Hamiltonian matrix, whose elements contain empirical parameters. The electronic wave functions are expanded on a minimal basis set of atomic or Slater orbitals centered on the atoms and usually restricted to valence orbitals. The matrix elements are self-consistently determined or not, depending upon the method.

In the tight-binding method, the elements of the Hamiltonian matrix are treated as adjustable parameters to be fitted to experimental or first-

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principles calculation results. To estimate the total energy, an additional short-range repulsion term is added to the electronic contribution. The dependence of the parameters upon interatomic distances are fitted in order to account for the bulk equilibrium structure and elastic constants. It is assumed that the parameters are transferable from bulk to surfaces [11]. This method has been widely used in the field of semi-conductors, and has yielded a good qualitative description of their surface properties. However, in more ionic systems, it completely disregards the variations of the electrostatic potential with the local environment of the atoms (see below), which is the reason for some wrong predictions made in the past.

The semi-empirical HF approaches can be viewed as self-consistent tight-binding methods, since they incorporate the self-consistent relationship between charges and electrostatic potentials in the expression of the Hamil-tonian matrix elements. The empirical parameters either fix the spatial dependence of the basis set wave functions, or are incorporated directly into some Hamiltonian matrix elements [12-14]. In the application to oxide surfaces, these methods can account for the shifts of the atomic orbital effective levels, i.e. for the surface modification of electronegativity, for example (Section 4.4). This is a very important point, both for obtaining reliable eigen-energy spectra, and to properly describe the reactivity of certain types of atoms as a function of their environment. It can also account for the surface charge redistribution, and for its influence on the structural degrees of freedom, contrary to classical approaches.

These methods can be combined with geometry optimization as well as with molecular dynamics algorithms, with forces obtained from the gradients of the total quantum energy [10]. This equally applies to all quantum methods, quoted in the following.

2.3. The ab initio H F method and methods beyond The HF method treats electron-electron interactions at a mean field

level, with the Hartree and exchange interactions exactly written. The method can be implemented either in its spin restricted form (RHF), for closed shell systems, or in the unrestricted form (UHF) for open-shell or strongly correlated systems. In the first case, the one-electron orbitals are identical for electrons of both spin directions, while UHF can account for a non-uniform spin density. The one electron orbitals, which are determined in the course of the self-consistent resolution of the HF equations, are expanded on an over-complete basis set of optimized variational functions.

The HF method has been implemented on periodic systems [15], including bulk and surface crystalline materials. It has proved very useful in the description of magnetic insulators, but it has also successfully been used for describing surface properties of a large number of simple oxide surfaces.

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One of the very serious problems with the HF approach is the unscreened nature of the Coulomb and exchange interactions. The bare value of a typical intra-atomic Coulomb integral U is in the range 15-20 eV, while screening in solids weakens it by more than a factor of 3. A large over-estimation of the HOMO-LUMO (HOMO= Highest Occupied Molecular Orbital; LUMO=Lowest Unoccupied Molecular Orbital) gap in insulators results. However, there have been quite accurate predictions of gap values or d ^ d excitation energies, based on HF total energy differences (Sections 4.4 and 4.5).

Quantum chemical approaches have long tried to solve N-electron Hamil-tonians of finite systems, beyond HF, at levels of increasing sophistication. The essential ingredient is the expansion of the ground state and excited state N-electron wave functions on a basis set of Slater determinants, built from one-electron HF eigen-functions. This so-called configuration interaction method (CI) includes explicitely electron correlation effects. Depending upon the basis set size and the number and nature of configurations kept, ground state and some types of excited state properties may be calculated [16].

By essence, these methods can only treat finite size systems containing a limited number of electrons, although some hybrid DFT-CI schemes (DFT== Density Functional Theory, see Section 2.4) try to deal with extended systems [17]. Bulks or surfaces are represented by a "quantum part" containing a small number of atoms, which is embedded in an array of point charges. Varying the cluster size gives hints on the validity of the representation. However, due to the heavy demand in computational resources, these methods, usually, can only treat few atoms.

2.4. The ab initio Density Functional Theory and methods beyond The DFT provides another route to include correlation effects. It gives a

prescription for calculating the ground state energy of an assembly of atoms at fixed positions, as a function of the electron density n{r) [18]. Assuming that there exists a system of non-interacting electrons with the same n{r)j the ground state energy of the interacting system may be expressed as a function of the eigen-functions and eigen-values of the non-interacting system. The latters are solutions of one-electron self-consistent equations, named Kohn Sham equations, which include a Hartree potential, an exchange term, due to the Pauli's principle and a correlation term. No exact expression exists for the correlation energy functional.

The Local Density Approximation (LDA) replaces the exchange and correlation potential Vxc, which is by essence non-local, by a local potential built from the properties of the homogeneous electron gas (HEG). At each point f of the inhomogeneous system, the exchange correlation energy is

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equal to that of the HEG: £'^^^(n), with the uniform electron density n functionally replaced by the actual density n{r). In open-shell or magnetic systems, it is possible to build an exchange and correlation functional which depends on both the electron density and the spin density. This is the Local Spin Density Approximation (LSDA). Excellent reviews of DFT-LDA or LSDA may be found in Ref. 19-21 One step beyond the LDA, the GGA (generalized gradient approximation) represents a semi-local approximation, in which not only the density but its gradient are locally taken into account [22-24]. The GGA corrects a large part of the systematic overestimation of the LDA cohesive energies. In particular, at surfaces, the GGA yields a much better description of adsorption phenomena than the LDA.

In order to solve the Kohn Sham equations, an expansion of the one-electron wave functions on a basis set is performed. Both localized basis sets and plane wave ones are currently used. Localized basis sets have the advantage of their small size. However they are attached to the atomic positions, which yields non-zero Pulay forces in geometry optimization and molecular dynamics. Plane waves, on the other hand, provide a uniform sampling of space, whatever the specific conformation of the system; they are independent of the atomic positions, but they require the use of pseudo-potentials to mimick core electrons and a very large number of vectors is necessary in standard surface calculations.

The density functional theory is a theory for the ground state. It cannot predict excitation properties. In particular, there is no theoretical justification to identify the Kohn-Sham eigen-values with quasi-particle energies, although this is currently done. In addition, DFT-LDA fails in localizing electrons in highly correlated systems. Several schemes have been proposed, in order to overcome these drawbacks. However, their computational cost is usually very high and, at present, they cannot be used routinely in surface physics. Despite this fact, we will now present them, because there is no doubt that an improved understanding of oxide surface properties will be gained thanks to them, in the coming years.

2.4-1' The Self-Interaction Correction (SIC) method In the mean-field approximation for electron-electron interactions, the

Hartree U[n] and exchange Ex energies read:

rrr ^ f i'^ A ,nir)n{r') U\n] = d^rd^r'^-^—^

^ ^ J 2\r-r'\ Ex = - Y: /dVdV^-^^^-^^^^'^^^^^^^^^^^"!j^^^^^^^^^ (2)

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Ex involves a sum over orbital (a, a') and spin (cr) quantum numbers, of a product of one-particle orbitals "ipao- with occupation numbers faa obeying Fermi statistics. U[n] includes a spurious Coulomb self-interaction, as can be realized if Eq. (2) is applied to a single electron system. This Coulomb self-interaction is exactly cancelled out by the a = a^ terms in Ex (self-exchange term).

In the DFT-LSDA approach, the local approximation to the exchange functional achieves only partial cancellation of the self-Coulomb term. Only for orbitals which are delocalized over the whole system does this self-interaction vanish. In the general case, for finite systems and localized states in extended systems, it leads to systematic errors, which have been summarized in Ref. 25. This work proposes a method for SIC, in which the LSDA exchange-correlation energy functional £'^^^^[n>|^,nj, which depends upon the electron densities for t and I spins, is replaced by:

Ell^ = E ^ f A ^ , „ j _ Y: U[n,,,] + El!^^[n,,,, 0] (3) a,cr

Modified Kohn-Sham equations result which contain an orbital-dependent local potential. The correction usually leads to a greater localization of the electronic wave functions. It was shown that SIC heals most of the LSDA drawbacks, including a large part of the gap problem in insulators and it corrects the bad shape of the exchange hole around electrons. In addition, in atoms, the negative of the SIC eigen-energies closely approximates the relaxed excitation energies for electron removal.

24.2. The GW method The calculation of a quasi-particle (QP) spectrum in a crystal requires

the resolution of an equation of the type:

- ^ + 14xt(r) + VH(r-)

(4) in which E^j^ and (t>^%{'f^ are the quasi-particle energy and wave function

in a band n at a given k point in the Brillouin zone. Kxt(r) i ^^^ ionic potential, V]i{r) is the Hartree potential and E is the self-energy operator, which is in general non local, non-Hermitian and energy-dependent. It contains the exchange and correlation effects and is state [nk) dependent. By comparison, in the Kohn Sham equations, E is replaced by T4c(r)^(^"" f^), and the resulting eigen-values fail in reproducing quasiparticle energies in solids, as has been stressed by many authors [26-28].

The estimation of E(r, r';£^^^) is a very difficult task. A possible approximation is the GW one, [29,30] in which a perturbation expansion for

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the self-energy is constructed and stopped at the first order:

E(f, P; Lj) = i /_"^J ^e+i^^' G{r, ?• u + J) W{r, P; a;') (5)

In Eq. (5), G is the one-particle Green's function, W is the screened Coulomb interaction and (5 = 0" . The real part of the self-energy contains a screened exchange contribution, which requires an explicit calculation of the dielectric matrix of the system, and a Coulomb-hole term which takes into account the actual presence of the quasi-particle (excess electron or hole) in the system and its screening by the surrounding electrons.

Although many improvements have been proposed for the calculation of the dielectric response function from first principles [31,32], still this stage is both computer time and memory very demanding when a large basis set is used for the description of the electronic structure. Efficient GW methods have thus been developed, in which a model dielectric function is used to mimick the screening properties of the system under study [28,33,34].

It should be noted that the resolution of Eq. (4) may be performed self-consistently or in a perturbative way with respect to E(r, r';£J^^) — 14c(^)- In the second case, HF as well as DFT-LDA eigen-states and eigen-energies may be used as starting points for the implementation of the GW approximation.

24.3. The LDA-hU method The LDA+U method assumes an approximate non-local and frequency

independent form for T,{r^r^; ^nk)^ which makes it a HF-type theory with a screened exchange. The idea consists in describing delocalized s — p electrons by an LDA-type approach, and in adding a Hubbard term ^C/ E [ui the occupancy operator of the di orbital) to the d- or /-electron Hamil-tonian [35-37]. This procedure stabilizes the occupied LDA bands, and destabilizes the unoccupied ones. This jump in energy between occupied and unoccupied states, which exists in an exact DFT theory, as well as the correlated jump of potential when the number of electrons goes through an integer value [38], are absent in the LDA, which thus misses an important contribution to the gap value.

Taking into account exchange interactions (J) and the non-sphericity of electron-electron interactions (dependence of U and J upon the orbitals i), the orbital-dependent potential reads (C/eff = U — J/2):

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Fig. 1: Domains of the projected atomic structure of the Q;-Al203-(\/3l x \ /3l)R±9° reconstruction, where the unit cells as well as domain walls are drawn. The two constituting Al planes are shown separately, with evidence of one being much better ordered than the other. Numerical relaxation has shown that the ordered layer could be associated to the outermost layer and the more disordered one to the layer adjacent to the substrate (from Ref. 43).

Via{r) = VLDA{r) + E{UiJ-Ues)n^ J-a 3

+ YliUij - Jij - Ues)nja + C/effCo ~ ^*>) ~~ 7*^ (^)

The LDA+U theory may be regarded as an approximate GW method [37]. The screened Coulomb and exchange parameters U and J are usually estimated in a supercell approximation [39]. However, there is some arbitrariness in the choice of the localized orbitals when performing the partitioning of the Hamiltonian. A further step in the improvement of LDA+U consists in adding dynamical effects — frequency dependence in S(r, r';a;). This may be performed using a DMFT-type approach (DMFT= Dynamical Mean Field Theory) [40] as part of the so-called LDA++ approaches [41].

2.5. General features for simulating surfaces Whatever the choice of the quantum mechanical method, two repre

sentations are currently used to simulate semi-infinite surfaces: slabs or clusters. Only the CI method is restricted to cluster geometries.

In slab calculations, a finite number of layers mimicks the semi-infinite system, with a two-dimensional (2D) translational periodicity. A minimal thickness dmin is required, so that the layers in the slab centre display bulk characteristics. Practically speaking, dmin should be at least equal to twice the damping length of surface relaxation effects, which depend upon the surface orientation. In plane wave codes, the slab is periodically repeated

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in the direction perpendicular to the surface, in order to make finite the required number of plane waves. The procedure generates spurious electric fields and energy contributions: in order to minimize them, one has to introduce thick enough vacuum regions between the repeated slabs and has to construct symmetric slabs with a zero total dipole moment. This question is particularly crucial for polar orientations.

Cluster calculations seek to determine surface properties by simulating only the local active sites [42]. They are usually implemented when it is believed that local bonding prevails in the electronic structure, when the property under study is local by essence, or when one wishes to model excited states through CI methods. It has been realized that a correct modelling requires the effect of the electrostatic potential created by the ions outside the cluster: the clusters are thus embedded in an array of point charges and/or in effective core potentials to describe atoms at the fringe of the clusters. A good embedding is more and more necessary as the cluster size decreases.

Treating non-stoichiometric surfaces requires special care in several respects. First, the long range elastic deformations (surface stress) induced by the defects are not easy to account for in a realistic way, neither in periodic calculations, nor in cluster models. When isolated vacancies are simulated in slab calculations, the size of the 2D unit cell, which is, most of the time, fixed by computational rather than physical constraints, corresponds to vacancy densities by far larger than realistic concentrations. At present, large size reconstructions, such as the (\/3T x \/31)R9^ on the a-Al2O3(0001) surface (Fig. 1) are still beyond the possibilities of quantum simulations. The presence of oxygen vacancies is usually associated to a strong redistribution of electrons, some of them being trapped at the vacancy site. When using localized basis sets to expand the electronic wave functions, it is mandatory either to place virtual orbitals at the location of the missing oxygen or to use strongly polarized orbitals on the neighbouring cations, in order to span the whole space for the electron density (Section 5.2). Finally, in order to account for strong electron redistributions, self-consistent approaches which properly solve the Poisson's equation have to be used. The same is true for polar surfaces, whose stability relies on a well-defined electrostatic condition.

3. REVIEW OF LITTERATURE

This section summarizes the present stage of our knowledge on clean oxide surfaces, obtained from numerical works. It is presented according to the oxide bulk crystal structures, with special emphasis on those oxides which have been more thoroughly studied.

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3.1. Fluorite MO2 and anti-fluorite M2O structures The anti-fluorite structure consists in an fee oxygen lattice, in which

every 0 - 0 pair is bridged by two metal ions, of charge 4-1. It is found in highly ionic oxides [44], like alkali-oxides Li20, Na20 and CS2O. The fluorite structure is similar, with a mere exchange of oxygens and cations, as in Zr02, Ce02, UO2, etc. The non-polar surfaces of lowest Miller indices are the (111) and (110) surfaces. The former is more dense than the latter, with a single broken bond per formula unit, to be compared to 2 on the (110). The (100) surface is polar.

Various works have shown that the (111) surface has always a lower energy than the (110), whether Li20 [45-47], Ce02 [48,49], UO2 [50], or Zr02 [45] are concerned. Relaxation eflFects depend upon the oxide: on 1102(111), inward atomic displacements take place, while on the (110) surface, there are small vertical atomic displacements associated to large in-plane displacements [50]; on tetragonal ZrO2(001), oxygens move inwards and cations outwards [51], while the (110) and (101) surfaces display some rumpling, with no preferred outward displacements of the surface oxygens [52]. The relaxation strength is an increasing function of the unrelaxed surface energy.

The modifications of electronic structure at the Li20 surfaces include a shift of the oxygen Is core levels, and a narrowing of the oxygen-derived valence band (VB) width, especially on the (110) surface. No surface states are found in the bulk band gap [46,47]. In an effort to interpret STM images (STM= Scanning Tunneling Microscopy) of U02( l l l ) , electronic charge density maps were obtained, using a DFT-LDA-i-U approach, and confirm the high degree of ionicity of the oxide [53]. Noticeable charge redistributions take place at the tetragonal (001) Zr02 surface [51].

3.2. Rocksalt structure The rocksalt structure consists in two interpenetrating fee lattices of

anions and cations, in which all atoms are in an octahedral environment. It is met in alkaline-earth oxides (MgO, CaO, SrO, BaO) and in some transition metal oxides like TiO, VO, MnO, FeO, CoO, NiO, etc, with cations in a +2 oxidation state. The non-polar surfaces of lowest Miller indices are the (100) and (110) surfaces: they have neutral layers, with a many cations as oxygen ions, and their outermost atoms are 5- and 4-fold coordinated, respectively. Actually, planar surfaces can only be produced along the (100) orientation. The polar direction of lowest indices is (111): it has an hexagonal 2D unit cell, three-fold coordinated surface atoms and equidistant layers of either metal or oxygen composition.

As regards the (100) surface, all theoretical works predict very small atomic displacements, on MgO [45,54-69], as well as on CaO, SrO, BaO

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- M -4.0 ^ . 0 0.0 2.0 ENERGY (eV)

0.10 0.08 0.06 0.04 0.(»

4.0

" ^ 2 . 0

1 0.0 I 4.0 » 2.0 UJ H 0.0

a 2.0 CM

O 0.0

1«llay«rofMnpty| •bovt 0 sltet

ml ^iiiwHwibw a«^ [t I' turfac* layer

• " I »W|W"I

1 St •ubturtac* lay«r |

"I »»l»i

N I ¥*<»' 3rd MibturtM* layar

•• I » ^ m

'•'U.O -*.0 -AJIt -2.0 0.0 2.0 ENERGY (eV)

Fig. 2: Projected DOS of Ni 3d states (left pannel) and O 2p states (right pannel) for ions belonging to the first four layers of NiO(lOO). The left and right top pannels show the density of Is states corresponding to empty spheres in the vacuum regions above the Ni and O sites, respectively. The chemical potential is in the middle of the band gap (from Ref. 74)

[45,58-60,70] NiO [71,72] and FeO [73]. A rumpling with the oxygen atoms outwards is found on the alkaline earth oxide surface, in agreement with older and more recent structural determinations. On NiO(100) however, the sign of the rumpling is reversed, the Ni being located more outwards than the oxygens. The (110) surface energy is found to be about twice that of (100); the relaxation is much larger and yields a larger energy stabilization.

Authors do not really agree on the charge redistribution which takes place at the MgO(lOO) surface, because the ionicity of the oxide appears to be nearly total in HF approaches [61,63], while in DFT and semi-empirical methods, the Mg-0 bonds present a small but non-negligible covalent character [60,66]. The surface projected Density of States (DOS) displays surface states just at the top of the VB and bottom of the conduction band (CB), originating from a reduction of the Madelung potential on the surface atoms. These states also exist on NiO(100) [74] and CoO(100)[75] and were shown to play a role in the formation of STM images (Fig. 2). A small reduction of the HOMO-LUMO gap results in MgO(lOO) as well as CaO(lOO) [60,64,67,68], and NiO(lOO) [74], which is qualitatively consistent with experimental observation [76-79].

Electronic d ^ d excitations at the NiO (100) surface have been calculated either with the CI method on small clusters [80-82], or with a periodic ah initio HF method by means of total energy differences [83]. Both ap-

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Fig. 3: Iso-density surfaces (about 15% of maximum density) for the two gap states per (2\/2 X 2\/2) 2D unit cellassociated with a 25% concentration of vacancies in a close packed configuration, (a) on the left: bonding combination of the two localized states trapped at the vacancy sites; (b) on the right, antibonding combination. A top view of the surface is presented with oxygen atoms as black dots and magnesiums as light spheres. The location of the vacancies is indicated by black boxes, (from Ref. 69)

proaches reproduce satisfactorily surface-specific transitions, which are due to the modification of the crystal field splitting of the surface d orbitals. It was further shown that monotonic variations of the d —> d excitation energies take place, whose sign depends upon the nature of the orbitals involved, when the Ni coordination number Z varies in thin film geometries [84,85]. This result agrees with suggestions by Freund et al. [86] that d -^ d excitation spectroscopy could be used as a tool for characterizing under-coordinated atoms on NiO surfaces.

Finally, the characteristics of the NiO surface antiferromagnetism have been looked at and in particular the value of the direct Jd and superex-change Jse coupling constants at the surface. A reduction of Jge was found on NiO(lOO) and in thin films, which is a function of the Ni coordination [84,85,87].

Oxygen vacancies on MgO(lOO) have been described by numerous authors [69,88-97]. The formation energy Eyf of an isolated vacancy in bulk MgO is large of the order of 10 eV. It decreases monotonically when the vacancy is located closer and closer to the surface [91], or in surface sites of

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lower and lower coordination Z (step edges, corners, etc) [91,95]. Outward atomic displacements around the vacancy take place, whose strength also increases as Z decreases. In the case of neutral vacancies, each missing oxygen leaves two electrons behind itself, which are redistributed in the surface layers. They are strongly trapped at the site of the vacancy by the electrostatic potential of the surrounding Mg. However, the degree of localisation found in the calculation is dependent upon the basis set used [92,95,98]. These electrons occupy an energy level in the band gap region and can be excited into empty states. Optical transitions as well as electron addition or substraction energies have been calculated using CI [96] or total energy difference [97] approaches. The EPR spectrum (EPR= Electron Paramagnetic Resonance) of charged vacancies or divacancies has also been simulated [95,99]. In an effort to understand the strength and nature of vacancy-vacancy interactions, finite vacancy concentrations and various vacancy arrangements have been considered [69,92,93]. In the most advanced simulation, effective repulsions between vacancies have been found, presenting a non-monotonic behaviour as a function of vacancy separation. The nature of the gap states changes as a function of vacancy concentration: there is a competition between localized states trapped at the vacancy sites (Fig. 3) and delocalized states on surface magnesiums, the latter starting to be occupied above a critical concentration.

In an ionic picture, the two-dimensional unit cell of the polar (111) orientation bears a charge ±2, so that a reduction of charge by a factor of 2 is required in the outer layers to heal polarity (Section 6). When charge compensation is provided by changes in stoichiometry, simple electrostatic arguments suggest two stable surface configurations. One is obtained by removing every other atom in the outermost layer, which yields a missing row surface structure with a (2 x 1) reconstructed unit cell. The second stable surface configuration, named the octo-polar (2 x 2) reconstruction [100], is obtained by removing 75% of the atoms in the outermost layer and 25% in the layer beneath, in a way which produces nano {100} facets.

All quantum mechanical studies of the unreconstructed MgO(lll) surface [101-103] find a metallicity of the outer layers, which arises from large shifts and overlap of the surface VB and CB. This produces the reduction of charge in the surface layers which fulfills the electrostatic criterion. At variance with ionic models, which predict an infinite surface energy, the calculated one is large {E^ > 5 J.m"^), but not diverging. This however suggests that the unreconstructed surface is not the lowest energy configuration.

Simulations of reconstructed surface [102,104-107] have considered the (2x1), the octopolar (2x2), the micro-facetted (\/3x \/3)R30° and (2\/3x

49

2\/3)R30° surfaces. The (2x2) surface energy is found to be lower than the (2x1) one, which supports the observation of the octo-polar (2x2) configuration on the parent oxide NiO(ll l ) . In FeO(ll l) however, experiments show that a two-layer film grown on Pt(lOO) and (111) substrates does not reconstruct. Its outer layer is made of oxygen atoms and the interlayer spacing is strongly reduced with respect to bulk FeO (0.68 A versus 1.25 A). The subsequent reduction of the dipole moment is consistent with a quantitative interpretation of STM images [108].

3.3. Corundum structure Aluminium oxide as well as transition metal sesquoxides (Cr203, Ti203,

V2O3, Fe203) crystallize in the corundum structure. The bulk unit cell contains six formula units with metal atoms in octahedral environment and four-fold coordinated oxygen atoms. The surface orientation which has been most thoroughly studied is the basal (0001) surface. Along (0001), the layer sequence is M/O3/M, with a very short inter-unit distance (e.g. 0.485 A in AI2O3). As a result, three chemically distinct terminations may be produced, which expose a single metal layer (M/O3/M/...), an oxygen layer (O3/M/M/...), or two metal layers (M/M/O3/...). Only the first one is non-polar and stable, on which the surface metal atoms are three-fold coordinated. The (lOTO) (the prismatic face), the (0lT2), the (1120) and the(lOll) orientations have also been considered.

The structure of the unreconstructed non-polar (0001) surfaces has been the suject of all types of numerical studies, in AI2O3 [45,109-116], Cr203 [117-119] and Fe203 [120-122]. All approaches predict large surface relaxations. However, the predicted amount of relaxation depends upon the symmetry constraints which are assumed in the modelization. Surprisingly, in Al2O3(0001), even when all atomic positions are relaxed, there remains a sizeable discrepancy between ab initio and exprimental [123-125] results, which is larger than the standard error bars of the theoretical methods.

As regards other surface orientations, early classical simulations had pointed out that their relative stability was strongly dependent upon relaxation effects [45]. The prismatic face turns out to have a surface energy very close to that of the basal face. An interesting point raised in Ref. 45 is the different behaviour of AI2O3 and Fe203 in this respect. While in AI2O3, the (0001) surface is found to be the most stable, in Fe203, it is the (0112), with an energy just below the (0001) surface. In addition, the prismatic Fe203 face has a higher surface energy than the (1120) face. Such differences manifest themselves in the crystal morphologies of the two oxides.

50

355 Oxygen-poor Oxygen-rich

-2 .0 -1 .0 Ho-Ho(gas) (eV)

Fig. 4: Surface energies of different Fe2O3(0001) surface terminations, /io(gas) is the chemical potential per oxygen atom of molecular O2. The allowed range of / - /io(gas) is indicated by the vertical dotted lines. Solid lines show results for relaxed geometries, and dashed lines give, for comparison, results for unrelaxed surfaces (from Ref. 122).

The non-polar AI2O3 (0001) surface is insulating and retains a large part of the ionic character of the bulk [115]. The surface DOS displays empty surface states derived from under-coordinated aluminium atoms. The position of these surface states is very sensitive to the degree of relaxation [110,113,126] and differs according to the calculations. The authors thus do not agree on the existence of a gap narrowing at the surface, which is however found to be of the order of 1 eV by EELS experiments [126] (EELS= Electron Energy loss Spectroscopy). The Gibbs free energy of all other terminations of AI2O3 (0001) compatible with a (1 x 1) unit cell turns out to be higher than that of the non-polar surface, in the whole range of accessible oxygen chemical potential [127,128], and a nearly similar situation is found for Fe203 (0001), except at high oxygen partial pressure where the oxygen termination is stabilized [122] (Fig. 4). Energies of surface d -^ d and charge transfer excitations have been calculated on Cr2O3(0001). A reduction of ionicity at the surface was found [129,130].

The structure of ultra-thin AI2O3 films, grown on metal substrates, was shown experimentally to be more closely related to 7-AI2O3 than to a-AI2O3, with part of the aluminium atoms located in tetrahedral sites [131-136]. Based on ab initio DFT-LDA calculations, a model for a 5 A-thick layer on-top an Al( l l l ) substrate has been proposed, in which the first oxygen layer in contact with the substrate is hexagonally packed as in the 0(1 X 1)/A1(111) system. Aluminium atoms bind on-top those interfacial oxygens and to three oxygens in the layer above, thus adopting a tetrahedral environment. The outer oxygen layer is strongly relaxed inwards so as to bring its atoms nearly coplanar to the aluminiums [137].

51

© #^i| i

Fig. 5: Iso-density map of an excess electron state on TiO2(110). Titanium and oxygen atoms are marked by small and large circles, respectively, and the semi-infinite system is represented by a slab with vacuum above and below and lateral periodic boundary conditions (from Ref. 157).

3.4. Rutile structure Sn02 as well as several transition metal oxides of 1:2 composition, such

as Ti02 and VO2, crystallize in the rutile structure. Their cation-oxygen bonding usually presents a large part of covalent character. The cations and oxygens are six- and three-fold coordinated, respectively. The three most stable surfaces in the rutile structure are the (110), (100) and (001) surfaces. They display arrays of two-fold-coordinated oxygen atoms — named bridging oxygens — and under-coordinated titaniums in the outer layers, with either 5-fold coordination — in the case of (110) and (100) surfaces — or 4-fold coordination —for the (001) surface. The TiO2(110) face has emerged as one of the most important oxide surfaces, which makes it a model system to understand the surface physics and chemistry of transition metal oxides.

52

Many numerical studies have been devoted to the TiO2(110) surface structure [58-60,138-147]. All approaches predict large surface relaxations, with inward displacement of the undercoordinated atoms and outward displacement of the fully coordinated atoms in the close vicinity of the surface. However, despite important efforts to increase the numerical accuracy of the calculations, by a careful check of the slab and vacuum thicknesses, etc [147], the calculated displacement of the bridging oxygens remains significantly underestimated with respect to the experimental value derived from GIXD experiments [148] (GIXD= Grazing Incidence X-Ray Diffraction). More recently, the surface structure of TiO2(110) has been re-examined by Li-impact collision ion scattering spectroscopy (ICISS) [149], which finds a relaxation of the bridging oxygens smaller than in Ref. 148, and even smaller than the calculated values.

The TiO2(110) surface is found to be insulating, with no deep surface states in the bulk band gap [1]. There are oxygen derived surface states located just above the top of the bulk VB and the bottom of the CB is made of 3d orbitals located on the 5-fold coordinated titaniums [138,150-156]. These local DOS features were used to interpret STM image contrast [154,156]. The analysis of an excess electron state on TiO2(110) [157] shows that the 5-fold coordinated titaniums experience the lowest Madelung potential on the unrelaxed surface and are thus the electron-acceptor sites. However, in the presence of surface relaxation, the Madelung potential is the lowest on the titaniums located just below them in the sub-surface layer, and the excess charge is thus shared between these two types of titaniums (Fig. 5). The calculated HOMO-LUMO gap is narrower at the surface than in the bulk [58-60,139,140,152,155,156]. The difference is largely reduced on relaxed surfaces because the contraction of the 0-Ti bondlength partly balances the bond breaking effect in the surface Madelung potential.

The two other surfaces of low Miller indices, namely the (100) and (001) surfaces, are less stable than the (110) [58-60,138,140,141,143,152,158-160]. This is especially true for the (001) surface. Their atomic and electronic characteristics are qualitatively similar to those of the (110) surface.

Aside from a first attempt to model isolated oxygen vacancies on TiO2(110) by a cluster model [153], more recent self-consistent works have considered large densities of vacancies, associated with (1 x 1) [142,143,146,156,161], (1 X 2) [143,146] or (2 x 1) [143,146,156,162] unit cells. The complete removal of bridging oxygens leads to a (1 x 1) configuration, while half removal leads to (1 x 2) or (2 x 1) reconstructions, depending upon whether one oxygen upon two is missing in each bridging oxygen row, or whether one row upon two is missing. In addition, a (2 x 1) structure may also be produced by adding rows according to a model first proposed by On-

53

ishi et al. [163]. Removing half of the bridging oxygens in each row costs less energy than removing half of the rows [143], and the (2 x 1) added row model is found to be more stable than the missing row [162]. In addition, calculated densities of states of the former [156,162] are in better agreement with STM images. DFT-LDA approaches usually predict that non-stoichiometric surfaces are metallic. However, insulating solutions are found for the ( 1 x 1 ) configuration using UHF approach [142] and for the (2 X 1) using DFT-LSDA [146]. In the latter case, a symmetry breaking takes place between the two titaniums close to a vacancy and the excess electrons localize preferentially on one of them. The presence of vacancies is thus associated with a spin polarisation of the surface layer; the ground state of the (1 x 1) surface is anti-ferromagnetic.

Finally, we mention studies of the rutile SnO2(110) surface [145,164-166]. Despite the absence of partially filled d orbitals on Sn cations, the surface characteristics of Sn02 are qualitatively similar to those of Ti02. The same is true for the non-stoichiometric (1x1) and (1x2) surfaces, which present a distribution of defect states in the gap [166]. The authors, however, argue that some quantitative diflFerences with respect to Ti02 take place, which are due to the much larger polarisability of the Tin atoms, compared to the Titaniums.

3.5. Wurtzite structure There exist three polymorphs of ZnO, with zinc-blende, wurtzite and

rocksalt structures. Atoms are tetra-coordinated in the two first forms, while in the latter, which is stabilized at higher pressure, the local environment is octahedral. The most thoroughly studied phase is the wurtzite one. Non polar (lOTO) and (1120) as well as polar (0001) or (OOOT) planar surfaces can be prepared.

Relaxation effects at the non-polar surfaces are important [56,167-172]. In particular, on the (1010) surface, there is a sizeable contraction of the intra-plane as well as interplane Zn-0 bondlengths in the vicinity of the surface, with a very small rotation {6 ^ 3°) of the surface ZnO dimers which pushes the oxygens outwards. The short Zn-0 bond can be viewed as a double bond. At variance, based on earlier tight binding models [56], a bond-length conserving mechanism of relaxation had been proposed in which a much larger rotation of the ZnO dimers was taking place {9 ^ 18°). More recently [173], it was shown that the dimer bondlength is very sensitive to the choice of the exchange correlation functional, and that the rotation angle 0 had been previously underestimated because of an unsuflScient slab thickness. The most recent calculated value 9^5° falls close to the lower edge of the experimental error bar (11.5° =b 5°) [174].

The theoretical prediction of the surface gap width is especially diflScult

54

in ZnO because it nearly vanishes in DFT-LDA calculations. The work done in Ref. 173, which makes use of another exchange and correlation functional, yields a correct bulk band gap of 3.2 eV and predicts a tiny reduction at surface (3.12 eV).

Along the polar (0001) or (OOOT) directions, ZnO exhibits a Zn/0/ . . . stacking of the hexagonal type with interplane distances equal to i?i=0.61 A and i?2 =1.99 A inside and in between the double layers, respectively. The surface is Zn-terminated or 0-terminated, respectively, with three-fold coordinated atoms. The condition for stabilization of the polar orientation requires surface charge densities equal to a' ^ .75a (Section 6).

Experimentally, unreconstructed or reconstructed terminations can be obtained, depending upon the surface preparation conditions. In particular, both cleaved and polished surfaces submitted to ion bombardment below 600° exhibit a p(l x 1) LEED diagram (LEED= Low Energy Electron Diffraction). Annealing at higher temperature may lead to ( 2 x 2 ) , {VS X V3), (3 X 3) or (4^3 x 4^3) reconstructions, some of them being possibly due to surface contamination [175-180].

Using electrostatic considerations, Nosker et al. [181] have predicted that the polar surface may be stabilized either by removing 1/4 of the outer atoms on each termination, leading to a reconstructed configuration, or by facetting, the latter process yielding a lower surface energy. Quantum mechanical approaches [182-186], on the other hand, have shown that a spontaneous charge redistribution takes place on the stoichiometric terminations, associated to a partial filling of states at the Fermi level. With this in mind, early observations of unreconstructed stoichiometric ZnO(OOOl) or (0001) (1 X 1) surfaces, with no anomalous electronic structure, seemed puzzling. A recent GIXD study [187] which finds partial occupancy of surface sites, without long range order of the surface vacancies, partly clarifies the problem and should be confronted with numerical simulations in the future.

3.6, Perovskite s t ruc ture The perovskite ABO3 structure possesses a cubic unit cell with the A

cations at the cube vertices, the B cations at the cube centre and the oxygens at the face centers. The lowest Miller index face is the (100), which can exhibit either an SrO or a Ti02 termination in SrTiOa. Using formal charges yields an estimation of zero for the dipole moment in the repeat unit. However, we discussed elsewhere the "weakly polar" character of this surface [4,188]. The (110) and (111) orientations, on the other hand, are polar with equidistant layers (i?i = R2) in their repeat units, of the B/AO3-, and 02/ABO-type, respectively. The layers formally bear charge densities equal to ±4 per 2D unit cell, and require compensating charges

55

(a)

(b)

(c)

(d)

IA. .^-A

hK /vH

^ ii llld

h~ '\ J -2.0 0.0

ENERGY(eV)

Fig. 6: Density of states of ideal and non-stoichiometric SrTiO3(100) slabs, (a) 1 x 1 cell; (b) \/2 X y/2 cell; (c) >/5 x v^ cell; (d) ideal slab. The energy scale of each graph is adjusted to Fermi level (from Ref. 210).

equal to ±2 (Section 6). The surface relaxation on both SrTiOa (001) terminations has been cal

culated by various numerical approaches [45,189-193]. The surface rumpling is usually reasonably accounted for, but all calculations predict an inward relaxation for the Ti02 termination, in contradiction to experiments [194-196]. A particular attention has been focused on the energy of surface states, since the first study based on non-self-consistent calculations predicted that they were located deep in the gap [197,198]. All subsequent self-consistent calculations have contradicted this prediction [191,192,199,200], in agreement with photoemission and EELS results [201,202]. When calculated [191,192], the surface energy is rather low, an indication that no surface instability takes place, and there is no evidence of anomalous filling of electronic states.

On defective surfaces, for example after Ar bombardment, surface states appear in the band gap region [203]. They were assigned to Ti^"*'-0-vacancy complexes [202]. Various reconstructions have been observed on vacuum annealed surfaces: (2 x 2) [204], (x75 x v/5)R26.6° [205-208], c(2 x 2)[209]. They were assigned to an ordering of oxygen vacancies, the latter reconstruction being possibly due to Ca segregation on the surface. Simula-

56

tions of oxygen vacancies agree on the existence of defect states in the gap [199,200,210]. They predict that the (\/2 x V2) and (\/5 x \/5)R26.6° reconstructed surfaces with one oxygen vacancy per unit cell have a metallic character [200,210] (Fig. 6). The excess electrons are shared between the vacancy and its two neighbouring titaniums, in agreement with STM images. The calculated interaction energies between vacancies depends not only upon the termination, but also upon the inequivalent positions of the vacancy pair on first neighbour sites, on the Ti02 face. On the SrO termination, the vacancies are predicted to repell each other [94]. Due to the lack of knowledge of the precise surface stoichiometry in the experiments, it is presently diflScult to propose a structure for the reconstructed surfaces.

The SrTiOa (111) [211-214] and (110) polar surfaces [215-217] and the BaTi03(ll l) surface [218] have been produced and studied. At variance with rocksalt polar surfaces, many of these investigations suggest that one can obtain non-reconstructed quasi-planar polar surfaces. It should be realized that the perovskite structure is such that there exist ordered configurations of vacancies in the surface layers compatible with (1x1) diffraction patterns. In addition, SrTiO3(110) displays a variety of reconstructions, such as c(2 x 6) [215,217], under reducing conditions. No precise determination of the layer stoichiometry has been performed.

The atomic and electronic structures of SrTiOa (111) and (110) (1 x 1) surfaces have been investigated using a total energy, semi-empirical HF method [219,220]. Terminations of various stoichiometrics, whether compensated or not, exhibit strong electron redistributions, which suppress the macroscopic component of the dipole moment. On stoichiometric surfaces (e.g. SrOa or T i ( l l l ) layers), this is an expected result, but redistributions are also present on compensated surfaces (e.g. Sr02 or TiO( l l l ) layers), leading to an anomalous filling of surfaces states and anomalous charge states for some surface atoms. All terminations were found to be insulating, a result assigned to the specific value ±2 of the compensating charges, which allows the complete filling of one surface state per 2D unit cell. The average surface energies were found to be rather low, a necessary condition for the obtention of planar surfaces. The relative stability of terminations of different stoichiometrics was calculated as a function of oxygen chemical potential and it was anticipated that non-stoichiometric reconstructions are not necessarily efficient mechanisms to stabilize the surface. It would be interesting, in the future, to assess whether surface stability is due to the presence of disordered surface vacancies, ordered vacancies, or whether the stoichiometric surface, with a complete filling of cation-derived surface states, has indeed a low surface energy, as predicted theoretically.

57

3.7. Other oxides Other oxide surfaces have been the subject of less numerical investiga

tions. We will quote here some studies performed on Fe304, Si02, V2O5 and M0O3.

Magnetite Fe304 crystallizes in the inverse spinel structure. It is based on a slightly distorted face centered cubic lattice of oxygen atoms, with the iron ions located in the tetrahedral A and octahedral B interstitial sites. Formally, A sites are occupied by ferric ions Fe^" , while B sites contain an equal amount of ferrous Fe "*" and ferric Fe "*" ions. Magnetite possesses two polar surfaces of low index, the (100) and (111) faces. Along the (111) direction, the stacking sequence contains two formula units and is described by: Fe^/40/3FeB /40/Fe^/Fe5. Six different terminations may be produced. STM images [221] show surface protrusions which are assigned to the topmost Fe atoms. An ab initio HF study in a slab geometry [222] predicts that the termination by an iron bilayer (Fe^/Fe^/40... sequence) is energetically favored, thanks to an exchange of iron and oxygen layers inside the slab, which reduces the total dipole moment. It is suggested that the stability of an Fe monolayer-terminated slab (Fe^/40... sequence) may be enhanced by adsorption of hydrogen: the calculated relaxations then agree with the LEED structure refinement performed in Ref. 223.

Crystalline Si02 surfaces have not been the subject of intensive structural or electronic determinations, because their large gap induces severe charging problems in spectroscopic experiments. There are several al-lotropes, such as a-quartz, /?-tridymite, and /3-cristobalite. The energetics of the lowest index a-quartz surfaces has been modelled by classical potential approaches [224] and the electronic structure of unrelaxed (0001) and (1010) surfaces by a semi-empirical HF method [58,60]. According to the latters, the (1010) face has a higher surface energy, and displays larger ionic charges and smaller band gaps than the bulk. On the /?-tridymite (1010) and the /3-cristobalite (110) surfaces, a tight-binding total energy calculation predicts that relaxation effects involve bond rotations, shifting the oxygens outwards and the silicon atoms inwards. This bondlength conservation mechanism allows a rehybridization of non-bonding oxygen orbitals and thus a lowering of electronic energy [56]. No deep surface states are found in the bulk band gap.

Vanadium oxides of various stoichiometrics exist, related to the different oxidation states of the cation. V2O5 has the vanadium atoms with the highest valency and a dP configuration. It is insulating at low temperature, with an orthorhombic layered structure made of VO5 pyramids sharing corners and edges. A transition towards a metallic phase takes

58

0(1) 0(2) 0(3)

Fig. 7: Crystal structure of orthorhombic V2O5 with netplane stacking along (010). Vanadium (oxygen) centers are shown by large (small) balls. Inequivalent oxygen centers, 0(1), 0(2), 0(3), with 1, 2 and 3 coordination number, respectively, are labelled accordingly. (From Ref. 229).

place at high temperature. DFT calculations appUed to an orthorhombic ¥205(010) surface, either in a cluster [225,226] or in a periodic slab [227] geometry have evidenced a spectroscopic signature of the three types of surface oxygen in the Local DOS (LDOS) [228]: the terminal oxygens belonging to vanadyl groups, the two-fold bridging oxygens and the threefold oxygens (Fig. 7). These oxygens are differently charged and present a distinct reactivity towards electrophilic attack by adsorbates. The energy to remove them is also different, increasing from 6.68 eV to 7.26 eV as the oxygen coordination number grows [229].

Molybdenum oxide crystallizes in an orthorhombic structure built from a piling of bilayers along the (010) direction. The bilayers are made of MoOe octahedra sharing oxygen atoms both in and between sub-layers. They are weakly coupled to each others by van der Waals forces. Ab initio DFT cluster calculations have examined the electronic structure of the ideal (010) and (100) surfaces [230]. The first one is obtained by cleavage of the crystal and involves three types of oxygens: terminal oxygens bound to a single Molybdenum atom, asymmetrically bridging oxygen with one short and one long Mo-0 bond-lengths, and symmetrically bridging oxygens with two short and one long Mo-0 bonds. The (100) surface on the other hand is less dense and only the two first oxygen types are found. Mul-liken population analysis shows large charge variations between bulk and surface atoms, and an especially large reduction of charge on the terminal and asymmetrically bridging oxygens. This inhomogeneous charge distribution manifests itself in the spatial variations of the surface electrostatic potential, an important quantity to understand the binding of ad-particles on surfaces.

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4. DISCUSSION OF NON-POLAR STOICHIOMETRIC SURFACES

This section summarises some properties common to non-polar stoichiometric oxide surfaces and presents theoretical arguments to explain them. Their specificities come from the local environment of the surface sites, which have a lower coordination number than in the bulk. From this point of view, a close parallel with unsupported clusters or ultra-thin films can be established [3]. We will not explicitely consider here the properties associated to structural defects, such as steps or kinks, for the reason of space limitation. However, most of the time, the same concepts as those akin to terrace sites apply, but with an even larger strength since the local environment is more reduced. We will successively analyse structural characteristics, energetics, electron distribution, one-particle and two-particle excitations.

4 .1 . Atomic structure There are numerous examples of bond-length modifications when local

environments of the atoms involved in the bond change. The usual trend is a contraction when the atom coordination Z decreases. The table of ionic radii given in Ref. 231 illustrates this effect. An increase in Z is always associated with an increase of the ionic radius, and since first neighbour interatomic distances are assumed to be the sum of two ionic radii, both quantities follow the same trends.

The same is true at oxide surfaces. Usually, inward relaxation is observed, whose strength is larger on more open surfaces. Systematic calculations on low index faces of rocksalt oxides examplify this tendency: the (001) surface, on which atoms are 5-fold coordinated, is hardly relaxed, while the (110) and (211) surfaces, with respectively 4-fold and 3-fold coordinated atoms, display increasing inward relaxation, associated to increasing bond-length contraction. In-plane bond-lengths are also affected, which yields a surface stress, and the same is true for bond-lengths close to surface defects like steps or kinks. It can thus be concluded that bond-length contraction around under-coordinated atoms is a general phenomenon and simple analytical models of cohesion, whether applied to ionic or covalent bonding, can account for this effect [2].

However, very often, relaxation effects involve more complex atomic displacements, yielding bond-angle distortions and/or surface rumpling. For example, a relaxation mechanism via bond rotation, has been invoked for Si02 and AI2O3 surfaces [56,110]. In the latter case, the huge contraction of the first interplane distance (shortening by «.16 A with respect to its mean value in the bulk) is not reflected strictly into the Al-0 bond-lengths

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[0001]

® - > [loToi [OiTo]

Fig. 8: Illustration of the sapphire (0001) surface: truncated bulk and best fit model obtained from crystal truncation rods measurements. The two top planes are strongly shifted with respect to their bulk positions in a nearly bond-length conservative displacement (from Ref. 124).

because the oxygens in the second layer move radially away from the surface Al (Fig. 8). The Al-0 bond-lengths are only shortened by 4.5%, according to GIXD studies [124], and the relaxation is said to nearly obey a bond-length conservation principle.

Surface layers which contain both oxygens and cations may also rumple, when forces of different strength act on the different atoms. Rumphng and relaxation effects often occur simultaneously. The rumpling amplitude is weak on dense rocksalt oxide surfaces - of the order of 0.05 A on MgO(lOO) - and much stronger in more covalent systems - about 0.4-0.5 A on the non-polar ZnO surfaces. Two mechanisms have been proposed in the literature to account for it. The first one relies on the existence of different electrostatic forces on the anions and cations, originating from polarization effects [232]. The most polarizable atom experiences the largest force [233,234]. On MgO(lOO), due to the small polarizability of magnesiums, Verwey's model predicts the oxygens to be located more outwards than the magnesiums [235], in agreement with exprimental refinements of the surface structure. On NiO(lOO), on the other hand, classical pair potential calculations [71,72] predict a rumpling of opposite sign, due to the larger polarizability of Ni ions. To our knowledge, no systematic extension of Verwey's model to other oxide surfaces has been performed. For example, rumpling has been found on the outer TiO layer of the (110) face of rutile Ti02, as well as on zirconia [52] and on Ce02( l l l ) [48] surfaces, but the

61

electrostatic arguments should be re-examined for each specific geometry, especially when cations and oxygens do not occupy symmetry-equivalent positions.

On compound semi-conductor surfaces, the rumpling is generally much stronger than on rocksalt oxide surfaces, although the anion and cation polarizabilities are comparable [11,236]. In these materials, it was argued that it is easier to bend a bond than to change its length, because the radial elastic constant is higher than the angular ones. The structural distortions thus obey a bond-length conservation principle, whenever possible. In addition, according to an argument first proposed by Chadi [237], a surface rumpling with the anions outwards favors the hybridization of the anion dangling bonds with the atomic orbitals in the underlying layer. This lowers their energy and pushes them into the valence band. Other covalent forces, such as those due to 0 - 0 orbital overlap can also lead to a surface rumpling [60].

4.2. Energetics In insulating oxides, the surface energies Eg range from 0.5 to 1 J/m^,

for the most stable orientations, but quantitatively, very few measurements have been performed [238,239]. Surface energies have several contributions, among which an electrostatic and a covalent one, but it is difficult to quantify them, because no general consensus upon energy partitioning exists among theorists. When several surface orientations of a given compound are simulated with the same numerical method, surface energies display a monotonic increase with the number of broken bonds at the surface. Qualitatively, the models of cohesion developed for relaxation effects account for the gross dependence of the surface energy upon Z [2]. But for the sake of comparing two surfaces with a nearly equivalent number of broken bonds, the value of the Madelung potential acting on surface atoms is a better indicator of stability than the coordination. Smaller Madelung potentials are associated to less stable surfaces.

Surfaces are always stabilised by relaxation effects. Since the latters are larger on more open surfaces, the energy lowering AEg is also larger. In their study of zirconia surfaces, Christensen et al. [52] note a linear relation between unrelaxed and relaxed surface energies {AEs/Eg constant) for both tetragonal and monoclinic faces, but this does not seem to be a general feature, as proved by the values of AEg/Eg for MgO found in Ref. 45. When two surface orientations of a given oxide correspond to numbers of broken bonds which are close to each other, the energy lowering depends upon the details of the structure. Inversions of stability between unrelaxed and relaxed surfaces may result. Within a given crystal structure, the na-

62

ture of the oxygen-metal bonding also contributes in a subtle way, yielding different surface orderings and thus different crystal morphologies. This was examphfied in Ref. 45 for a-Al203 and a-Fe203 surfaces.

4.3. Electron distribution The electron distribution in a compound reflects the topology of the

system and the characteristics of the bonding. It was proved by Hohenberg and Kohn [18] that the ground state energy is univoquely specified by the electron density n{f)^ which is the quantity self-consistently determined in DFT-type numerical approaches. n{r) sensitively changes when the local environment of the atoms is modified and thus the electron distribution at surfaces may differ noticeably from the bulk distribution. Discussions of surface effects usually rely on integrated quantities of n(r), such as the ionic charges, although the latters are not measurable quantities.

There are several ways to define ionic charges. When the eigen-states of the effective one-electron Hamiltonian are expanded on a localised basis set, the probability of presence of the electrons contains a 'site' contribution (square modulus of the projection of the wave function on the sites) and a 'bond' term, related to the overlap of the basis functions. In standard Mulliken population analysis [16], each bond contribution is equally shared between the atom pair. The charges then depend sensitively upon the choice of the basis set and it is meaningless to compare absolute values obtained with different methods. Only charge variations within a given method bear significance.

Another way to define ionic charges consists in partitioning space into elementary volumes associated to each atom. One method has been proposed by Bader [240,241]. Bader noted that, although the concept of atoms seems to lose significance when one considers the total electron density in a molecule or in a condensed phase, chemical intuition still relies on the notion that a molecule or a solid is a collection of atoms linked by a network of bonds. Consequently, Bader proposes to define an "atom in molecule" as a closed system, which can be described by a Schrodinger equation, and whose volume is defined in such a way that no electron flux passes through its surface. The mathematical condition which defines the partitioning of space into atomic bassins is thus:

Vn(r) • N{r) = 0 (7)

in which N{f) is the unit vector normal to the surface at f. Usually, and more specifically in ionic systems, each atomic bassin includes one nucleus. The advantage of Bader's approach is that the partitioning of space is uniquely defined, and in particular is independent of the basis set used in the electronic structure calculation. A Bader analysis was for example

63

Fig. 9: Electron transfer per bond A as a function of the difference SVMad in the Madelung potentials (in Hartree per electron) acting on the Ti and O in neutral stoichiometric clusters (filled circles), in bulk rutile Ti02 (plus), and on the TiO2(110) surface (diamonds) (from Ref. 243).

performed on bulk MgO and AI2O3 [242], and on free clusters and surfaces ofTi02 [243].

In most of the surfaces which have been considered numerically, it appears that surface charges display only minor variations with respect to their bulk values. This comes from the fact that charges are a complex result of oxygen-cation orbital hybridization and local environment factors which compete with one another. The Bond Transfer Method that we have recently worked out [244] makes this distinction more explicit. It can be shown, starting from the ionic limit in which oxygens and cations bear formal charges (Qo and Q^, respectively), that the sharing of electrons due to orbital hybridization is fully accounted for by positive quantities AoiCj^ defined for each bond. For simple insulating oxides with a completely filled oxygen-derived VB, the charges on an oxygen Oj or a cation Cj are expressed as:

Cj Oi (8)

The summation runs over the first neighbour atoms. Each transfer A, being related to a bond, concerns two atoms and appears in the expression of their charges. This description presents some analogy with the Bond Orbital Model [14].

As expected AoiCj goes to zero in the fully ionic limit (zero resonance integrals /3oiA,Cj//) and increases when the degree of covalency /3o.x,Cjfi/{^Cj —

64

Fig. 10: Electron transfer per bond A as a function of the Ti-0 bond length d (atomic units) in neutral (filled circles), anionic (down triangles) and cationic (up triangles) stoichiometric clusters, in non-stoichiometric clusters (stars), in bulk rutile Ti02 (plus) and at the (110) surface (diamonds) (from Ref. 243).

CQ.) of the bond increases (ecj and CQ. the relevant effective atomic orbital energies for the bond).

In the framework of a Hartree theory, the effective atomic orbital energies Ci defined as the diagonal matrix elements of the self consistent Hartree operator H on the atomic orbital basis set, read:

ei = {i\H\i) e f - UiQi - Vi (9)

In this expression, e\ is the atomic orbital energy in the isolated neutral atoms i] Ui is the intra-atomic electron-electron repulsion integral, Qi is the ionic charge, and K the Madelung potential exerted by the surrounding charges.

Since in most compounds, oxygens are surrounded by cations and vice-versa, the Madelung potential is positive on O and negative on C, which shifts their effective levels towards lower and higher energies, respectively [2]. The Madelung potential acting on surface atoms is usually smaller than its bulk counterpart. A decrease of the eQ. — CQ. energy difference results. Surface relaxation also modifies the values of the resonance integrals /?OiA,Cj/i with respect to the bulk. The contraction of the anion-cation bonds, usually observed at surfaces, corresponds to a strengthening of the |3oiX,Cj^J,• The AoiCj at the surface — and the same is true around any kind of under-coordinated atom — are thus enhanced with respect to the bulk, which means that the anion-cation bonding at surfaces is more covalent

65

than in the bulk [243] (Fig. 9). On the other hand, the number of atoms on which the summation in Eq. (8) runs is smaller at the surface than in the bulk. The topological factors thus compete with the electronic ones, leading usually to a rough cancellation of the effects. It is likely that having surface charges close to bulk charges also avoids to create huge electric fields and is favorable from an energetic point of view.

It was shown in Ref. 243, devoted to a study oiTinOm clusters, TiO2(110) surface and bulk rutile Ti02, that, although the AoiCj are not measurable quantities, their variations are strongly and systematically correlated to variations of bond-lengths (Fig. 10). They are thus pertinent quantities to discuss modifications of the iono-covalent character at surfaces.

4.4. One particle excitations We considere in this section one particle excitations, i.e. electron re

moval (hole creation) and electron addition processes. We will restrict our considerations to charge transfer insulators [245].

We first recall the relationship between the first ionization potential / , the electron affinity A and the MuUiken electronegativity x ^^ ^^ atom. According to MuUiken, x = —dE/dN is equal to the derivative of the total energy E of an atom with respect to its electron number N. It is the opposite of the chemical potential, which, itself, is equal to either minus / or minus A^ depending upon whether substraction {dN < 0) or addition (dN > 0) of electrons is considered. According to Koopman's and Janak's theorems [246] — the latter may be considered as a generalization of the former in the framework of DFT — / and A are equal to minus the HOMO or LUMO levels enoMO ^^^ ^LUMOI respectively [19], provided that no electronic relaxation takes place. In stoichiometric charge transfer insulating oxides, and more specifically when under-coordinated sites are involved, the HOMO level, at the top of the VB, is usually of non-bonding character and localized on the oxygens: CHOMO ^o- The LUMO level, at the bottom of the CB, has a prevalent cation character CLUMO ~ ^c- Using the expressions Eq. (9) for the effective atomic energies, one can write the MuUiken electronegativity under the following form:

Xi = x'^ + UiQi + Vi (10)

which ascertains that the electronegativity Xi oi the atom in the solid [2] is different from that of the isolated neutral atom x?- Since reactivity, acid-base strengths, etc, are often discussed in terms of the electronegativity of the active sites, Eq. (9) suggests how they vary as a function of the local environment of the surface atoms involved in the reaction: the effective levels of the surface cations (oxygen) being lower (higher) than in the bulk, electron removal is easier to achieve on the oxygens experiencing

66

the smallest Madelung potential. Similarly, electron addition preferentially takes place on the non-bonding cationic orbital of lowest energy, i.e. subject to the lowest Madelung potential (in absolute value).

However, electron affinities and ionization potentials are not accurately given by the opposite of the HOMO or LUMO energies, because of the existence of electronic relaxation effects. To obtain quantitative estimates, a simulation of both the neutral and the ionized systems is required. Noting E{N) the ground state energy of the neutral system with N electrons, and E{N — 1) and E{N + 1) the ground state energies of the ionized systems with an excess hole and electron, respectively, / and A are defined by the following total energy differences:

/ = E{N - 1) - E{N) A = E{N) - E{N + 1) (11)

Vertical and adiabatic quantities are obtained if the atomic structure of the neutral system is frozen upon ionization or not. Total energy differences yield much more reliable results for / and A than HOMO and LUMO energies derived from Fock or Kohn-Sham eigenvalues.

One way of measuring the "gap" G, in insulators, consists in superposing direct and inverse photoemission spectra (quasi-particle spectra of the compound), and recording the smallest energy difference between them. G is thus related to the ionization potential and electron affinity by:

G = I-A = E{N + 1) + E{N - 1) - 2E{N) (12)

The excess electron, relevant for the estimation of the electron affinity, and the hole — relevant for the ionization potential — are never simultaneously present in the system and do not interact. If one neglects electron relaxation effects, G ^ ^LUMO — ^HOMO^ but this energy difference does not give an accurate estimate of G: there is a systematic under-estimation in the DFT-LSDA, and a systematic over-estimation in UHF. However, part of the variation of G with the conformation, size and dimensionality of the system results from one-electron processes, and is reasonably described by the variations of eiuMO — ^HOMO- AS discussed above, the ionic contribution €c — eo is smaller at the surface than in the bulk. This acts towards a narrowing of the gap, and the effect is stronger on more open surfaces.

It should be noted that gap reduction at surfaces is always overestimated if relaxation effects are neglected, because the latters, by inducing bond-length contractions, strengthen the surface Madelung potential. The prediction of metallicity on NiO(llO) [247], based on unrelaxed geometry, should thus be reconsidered at the light of the present remark.

The qualitative behaviour of G as a function of the local surface environment has been confirmed by systematic studies of planar [60] and

67

A / ionised state

conduction bai

-10.0 -5.0 0.0 5.0 10.0 Energy (eV)

15.0 20.0

Fig. 11: Oxygen-projected unoccupied DOS of an NiO(lOO) monolayer. Plain and dashed lines refer to the positively ionized and neutral layers, respectively (from Ref. 85).

stepped surfaces [102,248]. Even in the most advanced calculations of surface gap width, such as the LDA+U approach of Ref. 74, the small gap reduction on NiO(lOO) was assigned to a Madelung potential effect. Incidentally, this explains that calculations of the surface electronic structure, using non-self-consistent methods — in which the effective atomic orbital energies are assumed to be the same throughout the material —, generally conclude that the surface gap width is of the same order as the bulk gap, while it is found to be reduced when self-consistent methods are used. Actually, for the two oxide surfaces on which precise gap measurements have been performed, namely MgO(lOO) [76,77] and Cr2O3(0001) [133], G is smaller than in the bulk in agreement with Madelung potential arguments.

It has been argued that, while the HOMO-LUMO energy difference suffers from systematic errors, a more reliable estimate of G can be obtained from the quasi-particle spectrum of the ionized system. For example, in NiO, the presence of an excess hole leads to the formation of a narrow band of unoccupied states in the VB region [249] (Fig. 11). It has been stressed [250] that the gap between this band and the CB edge approximates the optical/conductivity band gap. In LiiNiO [251] and bulk NiO [249], this method yields a value of 4 eV, in agreement with optical absorption measurements [252]. It was also used to estimate G in an NiO(lOO) monolayer [85].

A first approximation to the total quasi-particle spectrum, is given by the DOS. As was the case for / , A and G, a large part of the LDOS modifications on under-coordinated atoms is well accounted for by an effective one electron approach, because the shape of the LDOS reveals the pecu-

68

liarities of the hybridization between oxygen and cation orbitals. The DOS shapes at surfaces usually present two distinctive features: first, there is a systematic shift of the surface VB and CB, leading to a narrowing of the gap, as discussed above. In addition a systematic narrowing of the bands takes place, which is mainly due to the reduction of the surface coordination numbers [2]. To our knowledge, there has been no GW calculation of quasi particle spectra on oxide surfaces.

4.5. Two-particle excitations Electron-hole excitations can be produced in photon absorption or elec

tron energy loss experiments: one distinguishes d -^ d excitations, and charge transfer excitations.

d -^ d excitations show up in EELS spectroscopy as peaks in the region of the optical gap, for oxides having a sufficiently large band-gap and involving cations with a partially filled d shell. They correspond to optically forbidden transitions within the d manifold. In CI approaches, in an embedded cluster geometry, the ground state as well as some types of excited state are determined, which allows a calculation of the d -> d transition energies. Surface and bulk d -> d excitations in NiO(lOO) [80,81] and Cr2O3(0001) [129,130] have been obtained in that way. Another method consists in applying a variational procedure with some symmetry breaking to obtain specific types of excited states. The self-consistent loop then converges towards the state of lowest energy within the assumed symmetry class. This was done, in an ab initio periodic UHF approach in NiO [83,85], and the d -^ d transition energies were obtained by total energy differences. To our knowledge, there has been no similar attempt based on the DFT method, because the very localised nature of the d electrons in NiO is badly accounted for by DFT.

The nature and number of d —> d transitions in an oxide are determined by the place of the cation in the transition metal series (filling of the d levels) and the crystal field splitting of the d levels. At a surface, an additional lifting of degeneracy usually takes place, induced by the lower symmetry of the local environment of the surface atoms, d -^ d transitions may thus take place with an energy lower than in the bulk, which was indeed observed experimentally. Moreover, the dependence of given di —)-dj transition energies in NiO as a function of environment was studied [84]. It is nearly linear with respect to the coordination Z of the Ni " ion, with a positive of negative slope according to whether the final or initial state of the transition is d^^^ the orbital pointing perpendicular to the surface (Fig. 12).

Charge transfer excitations simulated by quasi-particle techniques lack the electron-hole interaction (excitonic effect) because the two excess charges

69

fe 5

4i

3

2

1

0

(z2—x2-y2)

(x2-y2 —z2)

(xy—z2)

(xz/yz-^z2/x2-y2)

- (xy^x2-y2)

coordination

Fig. 12: Evolution of calculated d -> d excitation energies in NiO (001) monolayer {Z = 4), NiO (001)/MgO (001) bilayers {Z = 5) and MgO (001)/NiO (001)/MgO (001) trilayers (Z = 6), as a function of the Ni coordination. Lines are drawn to guide the eye (From Ref. 84).

are never simultaneously present in the system. In order to interpret optical absorption experiments or EELS measurements, excitonic effects, have to be explicitely taken into account. The gap value obtained in this way will be denoted G*. Embedded-cluster CI techniques may be used to simulate charge transfer excitations. This was done in a study of planar [253] or defective [254,255] MgO surfaces. The results support the argument of decreasing G* as Z decreases. They are consistent with the arguments of Madelung potentials developed above. Total energy difference techniques also yields values of G*. This procedure was followed in order to determine charge transfer gaps in various geometries of the NiO(100) monolayer, thanks to symmetry breaking tricks [84]. An increase of G* with the coordination number of the Ni ions was found in a series including the unsupported NiO monolayer, a NiO/MgO bilayer and a MgO/NiO/MgO trilayer, which again may be rationalized with Madelung potential arguments. On the other hand, a variation of G* with an opposite slope is found between the monolayer and bulk NiO {G* =5.1 eV and 4 eV, respectively), whose origin remains at present unclear.

5. NON-STOICHIOMETRIC SURFACES

Non-stoichiometry in oxides is not a rare phenomenon, and especially at surfaces, where the process of annealing in ultra high vacuum after surface cleaning, induces a desorption of oxygen. Depending upon the annealing temperature and time, surfaces with various oxygen contents may be produced, whose exact stoichiometry is not easily quantified. However, some bench marks can be found when the vacancies order and give rise

70

to reconstructed structures, which allows to prepare surfaces in a roughly reproducible manner. There is a peculiarity of oxide surfaces, which is not yet understood, that no intrinsic reconstruction has ever been observed, at variance with metals or semi-conductors. In this section, we will discuss energetic, structural and electronic signatures of isolated or ordered oxygen vacancies at surfaces. We will only consider the case of neutral oxygen vacancies, in which neutral atoms are removed from the surface and leave two electrons behind them.

5.1. Atomic relaxation Structural relaxations are induced around vacancies by the breaking of

oxygen-cation bonds. The importance and sign of the atomic displacements depends upon the crystal structure, the surface orientation, and more generally upon the coordination number of the missing oxygen. The general trend is a displacement of the neighbouring cations away from the vacancy, leading to a contraction of the remaining cation-oxygen bonds. This effect is qualitatively similar to relaxations on stoichiometric surfaces, as discussed in Section 4.1. It relies on the property of interatomic distances to decrease when the coordination number Z of the partner atoms is reduced. When a vacancy is created on an MgO(lOO) surface, for example, the neighbouring cations, which are five-fold coordinated on the perfect surface, become four-fold coordinated, and their displacement away from the vacancy shortens some surface Mg-0 bonds [69,91,95]. This effect is also found on SrTiO3(100) [210] and on TiO2(110) [143,249]. Its strength increases close to surface defect sites — step edges, kinks —, whose Z is smaller.

However, there are exceptions to this rule, when a cation-cation bond across the vacancy site may be formed. This is the case in bulk Si02, where the creation of an oxygen vacancy breaks two 0-Si bonds. The atomic displacements of the two neighbouring silicon atoms are directed inwards in such a way that the Si-Si distance across the vacancy site is reduced by about 0.5 A with respect to its value in the perfect oxide [256].

5.2. Charge and spin distribution The spatial distribution of the two electrons associated to a neutral oxy

gen vacancy depends upon the degree of covalency of the oxygen-cation bond, and on topological factors: the oxide crystal structure and the coordination of the vacancy site. In some cases, as in MgO, the electrons are trapped in the vacancy site by the strong electrostatic potential exerted by the neighbouring cations, thus forming a surface F centre (denoted Fg centre). In more covalent oxides, Ti02 and SrTiOs for example [89,210], the electron redistribution is more diffuse and reaches the neighbouring

71

d2 43

Fig. 13: Characteristics of the saddle point configuration in the diflFusion path of an oxygen vacancy on MgO(lOO) (top view), (a) atomic configuration; (b) iso-density surface for the vacancy gap state, plotted at about 15% of the maximum state density. The magnesiums and oxygens are shown as large and small circles, respectively (from Ref. 69).

cations and sometimes the second neighbour oxygens. In all cases, the electron density is asymmetric with a larger contribution above the surface plane than below. Similar distortions are also found when the vacancies are located at surface defect sites.

However, the precise localization of the excess electrons is not straightforwardly found in numerical simulations, and contradictory predictions have been made as a result of different choices in the internal parameters of the computation. One of them concerns the basis set on which the wave functions are expanded. According to whether it consists of plane waves or localized functions, and, in this last case, according to whether or not virtual orbitals are introduced at the site of the vacancy, space is thoroughly spanned or not and electrons can or cannot redistribute freely in order to minimize energy. The absence of orbitals at the vacancy site, for example, prevents a correct description of an Fg centre et constrains the excess electrons to redistribute on the neighbouring cations [92]. To examplify the difficulty, we show on Fig. 13 the excess electron distribution during the migration of an oxygen vacancy in the MgO(lOO) surface plane: when the oxygen atom reaches the saddle point between two lattice positions, the excess electrons are equally shared between these two sites [69]. It is quite clear that describing this distribution with a localized basis set requires an extreme care. In addition, when localized basis sets are used, it is important to introduce polarization orbitals on the neighbouring cations, to account for their response to the strong electrostatic fields created by their highly asymmetric environment. Part of the differences found between non-stoichiometric Sn02 and Ti02 surfaces was for example

72

assigned to the larger polarisability of the former [166]. The MuUiken charges are also extremely sensitive to the choice of the ba

sis set. This is due to the fact that bond charges, which are equally shared between the two partner atoms, result from the overlap between the neighbouring orbitals [98]. To my knowledge, no Bader analysis — which is basis set independent — ha.s been performed on non-stoichiometric surfaces. In any Ccuse, a consensus now exists that the direct view of charge density maps give more reliable — although not quantitative — information than a MuUiken charge analysis on these surfaces.

The electron distribution also depends sensitively upon the treatment of exchange and correlations. For example, a comparison of DFT-LSDA and UHF simulations of non-stoichiometric bulk Ti02 shows that the electron density is larger in the vacancy site in UHF than in LSDA [161]. Similarly, on the reconstructed (2 x 1) TiO2(110) surface, a symmetry breaking induced by spin degrees of freedom — and consequently absent in DFT-LDA calculations — takes place in which the excess charge is gathered on a single titanium, rather than equally shared between two titaniums. It was argued that the complete filling of spin-unpaired orbitals is more effective in reducing on-site repulsion — which is a dominant factor in narrow d band compounds — than the partial occupancy of spin-paired orbitals [161,249].

5.3. Spectroscopic signature Spectroscopic experiments have revealed several specific signatures of the

presence of oxygen vacancies, in the electronic spectra. The two electrons left by the missing oxygen fill one additional electronic level, usually located in the the gap of the oxide and which pins the Fermi level. The density of states is distorted, with a transfer of weight from non-bonding states at the top of the VB to bonding states at the bottom of the VB. Finally, core level shifts on the neighbouring cation(s) take place. The numerical studies, quoted in Section 3, support these observations.

Due to the difficulties in reproducing correctly gap widths by standard ab initio methods, the position of the vacancy level e in the band gap is only given reliably through its distance in energy to the top of the VB. One should note that several factors affect the value of Cy. First, a good prediction requires a correct account of self-consistency effects, because the electron redistribution induces non-negligible electrostatic potentials. All the effects mentioned above — basis set limitations, choice of the exchange-correlation functional, etc —, which may change the electron localization, can also shift Cy, since energies and wave functions are simultaneously determined in the self-consistent procedure. In addition, by modifying the Madelung potential at the vacancy site, relaxation effects around a vacancy

73

5r 0.0

Fig. 14: Band structure of a fully oxygen defective (1x1) MgO(lOO) surface along the three symmetry lines J-F-M of the 2D Brillouin Zone, as obtained through the FP-LMTO calculation (Full Potential- Linear Muffin-Tin Orbital method). The dashed horizontal line represents the Fermi level, black dots (star) indicate the energy positions of the filled (empty) Bloch states at F calculated in a (2v^ x 2\/2) supercell. The dashed line in the gap of the projected bulk bandstructure gives the dispersion of the F^ centre band. The dashed-dotted line is used for the surface conduction band of lowest energy (from Ref. 69).

can move Cy, This is especially important when the vacancy is located at step edges or kinks, where large atomic displacements take place. It was shown that optical transitions associated to F centres on sites of low coordination of MgO are red shifted with respect to their bulk value, due to a reduced Pauh repulsion in the excited state [96].

The nature of the gap states may change as a function of the vacancy concentration c. On MgO (100), at low values of c, the Fg levels broaden into a band, which is completely filled. The band width was interpreted within a simple tight-binding model, involving effective hopping processes between F^ orbitals on neighbouring vacancies [69]. Above a critical concentration, this band overlaps the surface conduction band, and the surface becomes metallic (Fig. 14). Two types of quantum states, available for the excess electrons, thus coexist on the surface: the Fg orbitals close to the vacancy sites, and surface CB states localized on the surface magnesiums. On rutile Ti02 (HO) surfaces with large vacancy concentrations, it was shown that the metaUic state is less stable than an insulating spin polarized state [249].

5.4. Energetics The energetics of non-stoichiometric surfaces can be characterized in

two different ways. One consists in defining the vacancy formation energy Eyf. It is the energy required to extract one neutral oxygen atom from the

74

surface and to send it at infinite distance into vacuum, either as an isolated atom or as part of an oxygen molecule. In the first case, for example, Eyf is equal to:

Est - Enan-st " NE{0) ^vf = -j^ (13)

with Est^ Enon-st ctud £"(0) the energies of the stoichiometric surface, the non-stoichiometric surface with N vacancies, and the neutral oxygen atom, respectively. Most of the simulations quoted in Section 3 give values of Eyf^ which allows to discuss its variations as a function of the oxide, the surface orientation and the coordination Z of the vacancy site at the surface. For a given oxide, E^f systematically decreases as Z gets smaller, i.e. as the number of broken oxygen-cation bonds gets smaller. This is checked, for example, for various vacancy sites on the MgO(lOO) surface [69,91,95] , and on inequivalent oxygen sites on the ¥205(010) surface [229].

Another approach consists in comparing the energetics of surfaces with different oxygen contents, to understand which configuration is the most stable under given experimental conditions. The surface is then assumed to be in contact with an external reservoir. The relevant thermodynamical potential is no longer the internal energy E of the surface, but rather its grand potential Vt\

n = E + PV-TS-Y:f^iNi (14) i

For typical pressures P and temperatures T, the PV and —TS terms can usually be neglected, f] is a function of the chemical potentials //j of the different species i, which are not independent variables. For a binary system XnOm, for example, the surface is in equilibrium both with the bulk oxide and with the outer oxygen atmosphere. The first condition yields the relationship niix + mjio = I^XnOm between the cation and oxygen chemical potentials and the bulk energy per formula unit /J^XnOm • Knowing the internal energy for a given surface termination, Q can be calculated, according to Eq. (14), for the whole range of accessible values of /JLO-Stability of non-stoichiometric Al2O3(0001) [127,128], Fe2O3(0001) [122] and SrTi03(lll) and (110) [220] surfaces has been discussed in this context, and a typical Q — f{fio) graph is shown in Fig. 4.

The nature and strength of vacancy-vacancy interactions have scarcely been investigated, although they drive the thermodynamics of vacancy ordering on defective surfaces, and are thus key quantities to understand the numerous reconstructions observed on surfaces annealed in vacuum [1,2]. However, the ordering process may be inhibited by kinetic effects, relying on parameters, such as the activation energy for vacancy diffusion and the temperature. On MgO(lOO), vacancies have been found to weakly repell

75

each other, in a non-monotonic way as a function of their distance [69]. The overall repulsion is typical of an hybridization process — between two Fg orbitals — in which both bonding and anti-bonding states are filled. The non-monotonic behaviour is a consequence of the surface topology, the second and third neighbour interactions being differently mediated, by magnesiums and oxygens, respectively. On this surface, no strong tendency towards order could be found, in agreement with experimental observation. On SrTiO3(100), the same problem was addressed [94], but the interactions are of a different nature due to the more diffuse character of the electron redistribution. An interesting point concerns the comparison between the Ti02 and SrO terminations. On SrO, the excess electrons are shared between the vacancy site in the outermost layer and the titanium located in the sub-surface layer, thus producing a vertical electric dipole. On the Ti02 termination, the excess electrons are spread over the vacancy site and its two titanium neighbours in the surface layer [199], resulting in an horizontal quadrupolar distribution. This difference manifests itself in the sign of vacancy-vacancy interactions on the two terminations, which are found to be repulsive on SrO, and either repulsive or attractive on Ti02, depending upon the relative orientation of the two quadrupoles under consideration [94].

Finally, activation barriers have been determined for the diffusion of a vacancy in the MgO(lOO) surface, between sub-surface and surface layers, and close to surface defects [69,95]. As expected, the barrier is nearly twice lower in the first case than in the second one. Moreover, it is of the order of .20 to .25 times the formation energy, a quite reasonable ratio, in view of similar results on metal surfaces [257].

6. POLAR SURFACES

Polarity represents a peculiarity of some surface orientations in compounds involving simultaneously atoms of different electronegativity. It has been thoroughly studied in semi-conductor compounds [236,258]. A polar orientation is such that each repeat unit in the direction perpendicular to the surface bears a non-zero dipole moment. An electrostatic instability results from the presence of this macroscopic dipole, which can only be cancelled by the presence of compensating charges in the outer planes. This can be achieved either by a deep modification of the surface electronic structure — total or partial filling of surface states, sometimes leading to surface metallization— or by strong changes in the surface stoichiometry — spontaneous desorption of atoms, faceting, large cell reconstructions due to the ordering of surface vacancies, etc. Polar oxide surfaces present a large diversity, compared to sp^ semi-conductors. They exhibit a vast

76

number of crystallographic structures — rocksalt, corundum, spinel, inverse spinel, wurtzite, perovskite, for the simplest ones — which reflect the subtle mixing of ionicity and covalency in the metal-oxygen bonding and the specificities of the d electrons in transition metal oxides. In addition, mixed valence compounds, such as magnetite Fe304, can form when metal atoms with several oxidation states are involved, and, playing with experimental parameters, such as temperature, partial oxygen pressure, etc., allows to stabilize oxides of different stoichiometrics. It is clear that these peculiarities demand a generalization, if not a total reconsideration, of some theoretical concepts, a work which is still in its first stages [4]. In this section, we will discuss successively classical electrostatic arguments, some models of electronic structure which are useful in this context and finally the various processes which can stabilise a polar surface, including non-stoichiometric reconstructions.

6.1. Criterion for surface polarity According to classical electrostatic criteria, the stability of a compound

surface depends on the characteristics of the charge distribution in the structural unit which repeats itself in the direction perpendicular to the surface [259]. Type 1 or 2 surfaces — with neutral or charged layers, respectively — have a zero dipole moment fl in their repeat unit and are thus potentially stable. At variance, polar type 3 surfaces have a diverging electrostatic surface energy [181] due to the presence of a non-zero dipole moment not only on the outer layers (which would not distinguish them from non-polar rumpled or reconstructed surfaces), but on all the repeat units throughout the material. The total dipole moment is thus proportional to the number of repeat units and so is the electrostatic contribution to the surface energy per unit area (Fig. 15). This is the origin of the surface instability.

The estimation of the dipole moment in the repeat unit is not always an easy matter. In binary compounds, the difference in electronegativity of the constituents readily points out the existence and sign of the charge transfer between the ions. In simple crystal structures and for orientations such that layers containing cations only and anions only alternate, the presence of a dipole moment in the repeat unit is thus unquestionable, whatever the charge values — provided they are non-zero. These surfaces are undoubtly polar surfaces. The same is also true for some surfaces of ternary compounds, such as the (110) and (111) faces of ABO3 perovskites. For example, the (110) and (111) repeat units contain alternating ABO and O2 layers in the first case, and alternating AO3 and B layers in the second one, which are undoubtly charged, because, in each case, one of the layers contains a single constituent.

77

(a)

-a 0 -a a -CG

4iio

l.4n(5R.

(b)

-GO-GO -GG

4710 ^7

-47ia ^

/?j+/?2 J_4-

4jto_^2?j

Fig. 15: Spatial variations of the electrostatic field E and potential F in a slab cut along a polar direction.

There are less obvious cases, among which is the (100) perovskite sur~ face. SrTiO3(100) , for example, presents alternating layers of SrO and Ti02 composition. If formal charges are assigned to the ions (Sr " , Ti'*" and 0^~), each layer is charge neutral, the repeat unit bears no dipole moment, and the orientation is considered as non-polar. This is the statement most often encountered in the literature. However, SrTiOs is not fully ionic. Its gap width, equal to 3 eV, places it on the border line between semiconductors and insulators and the Ti-0 bond presents a non-negligible part of covalent character. The actual charges are thus likely not equal to the formal ones and Qsr + Qo and Qxi + 2Qo are not likely to vanish. SrTiO3(100) should thus be considered as a polar surface and this examplifies how careful one should be in the classification of surfaces.

In addition, the surface orientation h is not always sufficient to fully characterize a semi-infinite system, especially when various terminations may be produced. In the rutile structure, for example, the bulk repeat unit in the (110) direction is made of three layers of O and (M0)2 composition, and there exist three chemically inequivalent terminations, which expose a single oxygen layer (0 / (M0)2 /0 sequence), two oxygen layers (0 /0 / (M0)2 sequence) or one mixed cation-oxygen layer ( (M0)2 /0 /0 sequence). Only in the first case, the repeat unit bears no dipole mo-

78

merit. Similarly, on the basal (0001) face of the corundum structure, three chemically distinct terminations may be produced (Section 3.3). Only the M/O3/M termination is non-polar.

According to classical electrostatics, polar surfaces are thus unstable. However, it can be shown [2,94] that the macroscopic dipole moment can be cancelled out by modifications of the charge densities on the outer layers. Assuming that m outer layers have a charge density CTJ different from the bulk {\aj\ 7« cr for 1 < j < m) and that the layer spacing is alternatively equal to Ri and i?27 the electrostatic condition for surface stability reads:

E ^ m + l ^j = — 5 -

(_!)"» _ -R2 - -Ri i?2 + -Rl

(15)

Several scenarios, in which either the charge or the stoichiometry of the layers is changed can thus lead to polarity healing:

• one or several surface layers have their composition which differs from the bulk stoichiometry. Reconstruction or terracing will result, depending upon how the vacancies or adatoms order. If no order takes place, no information on the surface stoichiometry can be extracted from surface diffraction patterns, unless a quantitative analysis is performed.

• foreign atoms or ions, coming from the residual atmosphere in the experimental set-up, provide the charge compensation.

• on stoichiometric surfaces, charge compensation may be provided by the electron redistribution which takes place in response to the polar electrostatic field. This is well examplified in self-consistent electronic structure calculations.

Which process actually takes place depends upon energetic as well as kinetic considerations. As will become clear in the following, if stoichiometric ideal polar surfaces are not observed, this is never because their surface energy diverges. There always exist enough electronic degrees of freedom in a material to reach charge compensation through the third mechanism. However, in most cases, the resulting surface energy is high and other processes may be more efficient.

6.2. Models of electronic structure in iono-covalent materials: application to polar surfaces

The characteristics of the charge distribution in the bulk and in the outer layers are thus key factors to understand the physics of polar surfaces. A zeroth-order description of the electronic structure of oxides is the fully

79

ionic model, in which formal charges are assigned to bulk as well as to surface atoms. This model successfully assesses whether, in a binary oxide, a bulk repeat unit bears a non-zero dipole moment, and can predict the stable surface terminations in some complex structures. It applies well to highly ionic materials, but cannot be justified in more covalent compounds or when surface charge redistributions take place. For example, according to it, the surface energy of stoichiometric SrTiOs and BaTi03(l l l ) and (110) surfaces should be infinite, as well as those of the oxygen or iron bilayer terminations of a-Fe203 (0001), in contradiction with both experiment or ab initio calculations. In addition, it may even give wrong predictions on the nature (polar or not) of a surface, in ternary compounds, as in the case ofSrTiO3(100)

In tetrahedral semi-conductor compounds, on the other hand, the most widely used criterion for the stability of polar surfaces relies on specific fillings of dangling bonds — sps atomic orbitals which remain unpaired due to surface bond breaking. The criterion is referred to as the auto-compensating model [260,261]. It states that a surface is stable if anion-derived and cation-derived dangling bonds are occupied and empty, respectively, because such a filling induces a net lowering of the surface energy. According to electron counting rules, it has been realized that, at polar surfaces, the condition of auto-compensation implies that Eq. (15) is fulfilled, and the surfaces are then said to be "charge neutral" [262]. Later, the auto-compensation principle was generalized to describe mineral surfaces [263]. assuming that the electron number per bond is the same at the surface and in the bulk (but not necessarily equal to 2 as in tetraedral semiconductors). This principle has been applied for example to a-Al2O3(0001) [110], Fe3O4(100) [264-267] and a-Fe2O3(0001) [268,269].

This generalization may be useful in a number of cases. However, the vocabulary taken from the physics of semiconductors is highly unsuited to oxides, because, in most cases, dangling bonds have no physical significance. The conservation of the number of electrons per bond between bulk and surface is also not a well founded assumption.

The description of the charge distribution in terms of electron transfer per bond [244] given in Section 4.3, allows to make a bridge between the two limits and can account for the metal-oxygen bonding in the whole range of ionicities. We will use it in the following to understand the mechanisms at work on polar surfaces. It has to be noted that, for this purpose, it will never be necessary to know quantitatively the values of the parameters AoiCj, neither in the bulk nor at surfaces. It is the principle of electron sharing per bond which is the key ingredient.

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6.3. Surface processes relevant for polarity healing We will successively discuss surface relaxation effects, changes of cova-

lency in the outer layers, partial fillings of surface states and stoichiometry changes.

It is often thought that surface relaxation is one of the most efficient processes to stabilize polar surfaces, especially open surfaces, which experience large atomic displacements. Actually, two aspects of the problem should be distinguished. First, as far as the electrostatic criterion Eq. (15) is concerned, the macroscopic component of the dipole moment is entirely determined by the charges and layer spacings in the bulk repeat unit. Consequently, in the vicinity of the surface, a mere contraction or expansion of the interlayer spacings, not accompanied by a change in the charge densities, can never heal polarity However, once charge compensation is achieved — by whatever mechanism —, surface relaxation induces a lowering of surface energy, as it does on non-polar surfaces, and the strength of the effect increases as the coordination of the surface atoms gets lower [60] (c.f Section 4.1), as exemplified on the a-Al203 (0001) surface (Section 3.3), and on the reconstructed (2 x 1) NiO(ll l) surface [270].

Since charge compensation requires a modification of the charge density, changes of covalency at the surface are often assumed to heal the polarity. With the help of the bond transfer model, one can show that this statement is incorrect, as far as semi-infinite polar surfaces are concerned. It is useful to make a distinction between weakly polar surfaces, in which the dipole moment in the repeat unit is entirely due to covalent effects, and truely polar surfaces whose dipole moment contains an integer contribution. As already said, in the fully ionic limit, the first ones are considered as non-polar, while the second ones are recognized as polar.

A prototype of the first family is SrTiO3(100). Applying the Bond Transfer Model [244], it is found that the charge densities per two-dimensional unit cell, on the Ti02 and SrO bulk layers, read (JB = IQxi + 2(5o| = QST + Qo — 2ATi-o — 8Asr-o, in terms of the electron transfers Axi-o and Asr-o per Ti-0 and SrO bonds. The dipole moment in the bulk repeat unit is thus non-zero. However, it has no integer contribution and depends only on covalent effects. On the (100) surface, due to the change in local environment, due to possible structural distortions and due to shifts of atomic levels, redistributions of charge take place (Fig. 16). On the Ti02 termination, for instance, assuming modified Afpi_o and Agj._o values yields atomic charges which differ from the bulk in two layers. However, it is found that a\ + a2 = Axi-o — 4Asr-o, which fulfills the condition ^1 + cr2 — —crB/2, Eq. (15), whatever the specific values of the A and A' parameters. On SrTiO3(100), the bond breaking mechanism, by itself.

81

Ti02 -f^o

« - ^ - »

SrTiO3(001)

000000 0

MgO(lll)

0 =oxygen • =titanium # =strontium or magnesium

Fig. 16: Electron transfers per bonds introduced in the modelization of an SrTiO3(001) surface (left pannel) and of an MgO(l l l ) surface (right pannel). On SrTiO3(001), specific transfers A^i-o ind Asi._o (represented by thick arrows) are introduced inside the surface layer and in between the surface and sub-surface layers for Ti-0 bonds and Sr-0 bonds involving surface atoms. On MgO(l l l ) , only the first inter-plane transfer is assumed to be modified, (from Ref. 4).

thus yields the charge redistribution which suppresses the divergent part of the electrostatic potential. The surface charges are different from the bulk, both because the bond covalency in the surface layers is different from the bulk (A' ^ A) and because the coordination of surface atoms is reduced with respect to the bulk. However, the model shows that only the second factor is effective for polarity healing, while the change of covalency plays no role in the charge compensation process.

On the other hand, on truely polar surfaces, in which the dipole moment borne by the bulk repeat unit contains an integer contribution a change in covalency in the surface layers cannot either heal polarity, MgO(ll l) is representative of this family. The charge density per ( 1 x 1 ) unit cell on bulk (111) layers is equal to GB — ±(2 — 6A). On a stoichiometric (111) magnesium termination, surface atoms are three-fold coordinated to atoms located in the underlying layers. Their charges QMgi are different from those in the bulk. But the same is also true for oxygen atoms in the sub-surface layer (n = 2) because A' 7^ A (Fig. 16). ( Mgi and QQ^ are such that (Ji -h (J2 = 3A. This does not fulfill the electrostatic criterion, which, instead, requires that ai + a2 = 3A — 1. The surface layers have a deficit of one electron, which is independent of the values of A' and A. It should be noted that the same result is obtained in a fully ionic picture (A' = A = 0) or when neglecting the change of covalency (A' = A). Neither in this family, thus, can charge compensation be achieved by change of covalency at the surface.

What we have demonstrated here, is very general: a mere change of

82

I

(b)

v t y ^

E p

-

o.o Energy (eV)

-4 .0 O.O Energy (eV)

Fig. 17: Calculated DOS for five-layer symmetric MgO(l l l ) slabs, with oxygen (a) and magnesium (b) terminations. Plain and dashed lines represent the DOS in the slab and in the bulk, respectively. The top and lower panels refer to projections on majority and minority spin states, respectively. All DOS have been convoluted by a 0.1 eV wide Gaussian function (from Ref. 103).

covalency in the outer layer can never provide the compensating charges, because it concerns several layers whose contributions cancel out in the expression of the electrostatic criterion.

On these stoichiometric polar surfaces, charge compensation can only be achieved by a partial/total filling of CB surface states or depletion of VB surface state. The Bond Transfer Model helps understanding that this filling / can be determined by ionic arguments, even in very covalent materials, and that it is independent of the specificities of hybridization in the outer layers [244]. In the case of MgO(lll) , for example, follovi ing the charge analysis just given above, charge compensation is only achieved if / = 1/2. The analysis of the oxygen termination leads to the symmetric conclusion that a surface valence band has to be half-filled. It should be noted that the condition / = 1/2 is independent of the precise values of the electron transfers per bond. It would have been similar using a fully ionic picture. The result that we have obtianed is not restricted to ionic oxides such as MgO. On ZnO(OOOl) or SrTi03(l l l) or (110), the occupancy of surface states is also dictated by the values of the formal charges and of the interlayer spacings Ri and i?2, despite the fact that these oxides are rather covalent. The ionic limit can thus be currently used to estimate the value of / required for charge compensation. Two cases occur. In the first case, / is non-integer. The surface states are only partly filled or empty and the surface layers have a "metallic" character ^

^We use here the term "metallic" for the sake of simplicity, meaning that electronic excitations of zero energy can be

83

(Fig. 17). This occurs on the rocksalt oxide (111) surfaces ( / = 1/2), the ZnO(OOOl) surface ( / = 1/4), the oxygen termination of corundum oxide (0001) surface ( / = 1.5), etc. These partial fillings were indeed found in the quantum calculations. For some other polar surfaces, / is integer, and the surface can remain insulating. This takes place for example, on the (111) or (110) polar surfaces of SrTiOa (/ = 1).

Charge compensation can also be achieved by changing the atomic density in the outer layers. Again, according to the bond transfer model, the required modifications of stoichiometry are found to be entirely determined by ionic arguments, and not by the specificities of hybridization in the outer layers. For example, in the case of MgO(ll l) , in a two-dimensional supercell containing M surface Mg, from which one atom is removed, the electrostatic criterion requires that / = (M — 2)/(2M — 2). This relationship is independent of the degree of covalency of the Mg-0 bond Its simplest achievement, with no filled conduction band ( / = 0), is obtained for M = 2. This amounts to removing half of the surface atoms from the outermost layer. A similar reasoning, assuming modifications of stoichiometry in two surface layers instead of one, would lead to the octopolar surface configuration. It is thus possible to predict which stoichiometry, in the surface layers, compatible with an insulating band structure, yields charge compensation, only on the basis of ionic considerations, even in very covalent compounds. This does not mean that covalency effects are absent, or that they are the same as in the bulk, but rather that they cancel out in the condition for charge compensation. Charge compensated non-stoichiometric surfaces, such as the (2 x 2) octopolar reconstructed rocksalt (111) surfaces, have electronic properties which present no anomaly. However, due to the presence of surface atoms with low coordination numbers, interesting phenomena for applications, such as a reduction of the gap, an increase in basicity of surface oxygens, or an increase in acidity of surface metal atoms can be expected, as on open non-polar surfaces [58,59,271,272].

6.4. Summary Polar oxide surfaces present a wide variety of electronic and atomic char

acteristics, which are dependent upon the crystal structure, the ionicity of the metal-oxygen bonding, the surface orientation and its stoichiometry. The nature of the microscopic processes responsible for the cancellation of polarity provides a means to introduce a classification among these surfaces.

Weakly polar surfaces are met each time that the dipole moment in produced. However, when the Fermi level crosses a narrow band, as met on the oxygen termination of MgO(lll) for example, and when the surface metallization would survive in the limit of zero bond width (no electron delocalization, zero value of the resonance integrals), the system should rather be named an "open-shell" system

84

the repeat unit is a function of the electron transfers per bond only and does not involve integer contributions, as in the case of SrTiO3(100). On stoichiometric surfaces, the charge compensation may be assigned to the effect of bond breaking. The surface LDOS differs from the bulk one because of the reduced local environment of the surface atoms (different effective atomic orbital energies, bond-length modifications, change in the coordination numbers). However, the terminations are insulating, with filled oxygen-derived surface states and empty metal-derived surface states. These surfaces have low surface energies and can be produced stoichiometric and planar. We have called them previously weakly polar surfaces^ because their polar instability is weak.

For fully polar surfaces, the surface dipole in the repeat unit contains an integer contribution, independently of the values of the electron transfer per bond. For semi-infinite surfaces, it was shown that a mere change of covalency in the outer layers can never provide the compensating charges, because it modifies the charges of several types of atoms whose contributions cancel out in the expression of the electrostatic criterion. Similarly, and for the same reason, surface relaxation cannot provide the compensating charges. Only a modification in the filling of surface states or a change in the stoichiometry of the surface layers may yield charge compensation.

It turns out that if stoichiometric polar surfaces are unstable, this is never because they present a diverging electrostatic surface energy. The compounds have enough degrees of freedom, and in particular enough flexibility of their electronic structure in response to the surface potential, to heal the polarity while remaining stoichiometric. If a partial filling of surface states is required — and, as shown above, this can be estimated assuming a fully ionic limit —, as in the rocksalt (111), wurtzite (0001) or (0001), etc surfaces, the terminations present a "metallic" character. This is usually not favourable, from an energetic point of view, as recognized in the expression of the auto-compensation principle.

On other stoichiometric polar surfaces, an integer filling of surface states is required. This is the case on SrTiOa or BaTiOs (110) and (111) surfaces. The surfaces thus may remain insulating but a conduction state is located below the Fermi Energy Ep- Self-consistent calculations give a hint that this electronic structure is not associated with a high surface energy. They support preliminary experimental observations of non-reconstructed perovskite (111) and (110) surfaces. This suggests that the autocompensa-tion principle (filled anion-derived surface states and empty cation-derived surface states) should be extended to include all insulating surface configurations, whatever the nature of the filled and empty states. However, in perovskite compounds, the absence of reconstruction is not necessarily

85

synonymous with stoichiometry, and further work should assess this point in the future, by determining quantitatively surface compositions.

Polarity may also be healed by the removal of a certain percentage of atoms in the outer layers. When the vacancies order, most of the time, this leads to surface reconstructions. The surface concentrations compatible with a vanishing macroscopic dipole moment can be correctly estimated within a fully ionic picture. Low energy configurations are expected to be insulating, and indeed, on non-polar oxide surfaces, a correlation has been found between the stability of a surface orientation and the surface gap value [58,59].

It is clear that much work remains to be done to extend our understanding to polar surfaces of transition metal oxides in which the cations have partially filled d orbitals. An especially challenging issue is related to mixed valence metal oxides, such as Fe304, in which the cations exist under two oxidation states. In addition, considering the rapid development of ultra-thin film synthesis and characterization, a simultaneous effort should be performed on the theoretical side to settle the conditions of stability of polar films. More generally, on the experimental side, it seems that one of the present bottlenecks is in a quantitative determination of the surface stoichiometry, an information of prominent interest to interpret the presence or absence of reconstruction.

7. CONCLUSION

We end this review on clean oxide surfaces by stressing some directions in which theoretical improvements can be expected in a near future

One deals with the ah initio description of electronic excited states. These include the attachment or removal of electrons, the account of direct or inverse photo-emission spectra, and the electron-hole excitations of the d -> d or charge transfer type. Advanced methods are presently under development to account for them: the GW method, the SIC method, the LDA-hU method, etc. However, they imply an increased computation cost, which is not routinely accessible for complex systems, such as most oxide surfaces. These methods are also expected to open the field of strongly correlated materials, among which transition metal oxides, which have important technological applications: high-Tc superconductivity, giant magneto-resistance, etc.

A second challenge is in the understanding of surface magnetism: elementary spin-spin interactions, atomic magnetic moments on surface sites, collective properties such as surface magnetic ordering, spin polarised transport across oxide ultra-thin barriers, etc. We have not developed this problematics here, because, until now, there have been very few related the-

86

oretical studies. However, new beautiful experimental results have been recently published which should encourage theorists to join this field.

Finally, understanding the reconstructions of oxide surfaces from the point of view of non-stoichiometry and/or polarity represents a necessary step for the production and use of high quality nano-structured surfaces. On the experimental side, it seems that one of the present bottlenecks is in a quantitative determination and control of the surface stoichiometry. On the theoretical side, there are strong limitations to apply ab initio methods to reconstructed surfaces because of their computational cost. In addition, it is not yet clear why intrinsic reconstructions do not exist and what are the driving forces for vacancy ordering. No doubt that these points will be seriously considered in a near future if one is to use reconstructed oxide surfaces as substrates to grow artificial structures.

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