Introduction

As discussed in chapter 3, very recently attempts have been made to use magnetic tips in NC-AFM to try to image the magnetic structure of materials on the atomic scale. Due to the ease of preparation of an atomically clean surface [2], most of these attempts have tried to measure this phenomenon on the anti-ferromagnetic nickel oxide surface. As yet, no difference in contrast over opposite spin Ni ions has been observed, so it would be useful to study the interactions involved theoretically to establish whether it should be possible. In this chapter a simple model of a magnetic tip has been used to measure the difference in interaction over opposite spin Ni ions in an ab initio simulation. The contribution of the exchange force to the tip-surface interaction is calculated and used to predict whether the spin contrast over Ni ions would be observable in NC-AFM experiments.

Nickel oxide is a classic example of one of the class of materials which have excited and perplexed over the past 70 years, first-row transition metal monoxides. In a purely ionic picture of NiO the Ni $^{2+}$ ions have a partially filled $d$ shell in a $3d^{8}$ ground-state configuration. According to conventional band-theory this should result in metallic behaviour, yet NiO is an insulator with a bulk band gap of 4.3 eV [182]. It crystallizes in the rock-salt structure (as MgO) with a lattice constant of 0.417 nm and a high-spin anti-ferromagnetic spin structure at low temperatures (see fig. 9.1). Its Néel temperature ( $T_{N}$ ) is 523 K and it undergoes a magnetic phase transition above this point.

**Figure 9.1:** Structure of NiO, showing the AF $_{2}$ anti-ferromagnetic spin structure with adjacent (111) planes of similar spin Ni ions.
$\resizebox*{13.77cm}{10cm}{\includegraphics{figures/nio_schem.eps}}$

Understanding the electronic structure of NiO, and other transition metal monoxides, has been a topic of great interest for many theorists. Mott [183] proposed that the band gaps in these materials were due to strong on-site repulsion between the $d$ -electrons of the metal ions. Coulomb repulsion between the $d$ -electrons localized on the metal ions increases their effective band volume and produces a pseudo-filled band. Early band theory calculations of NiO based on the local spin density approximation (LSDA) [184] predicted a narrow gap spanned by unoccupied Ni $d$ -states, but underestimation of the repulsion inherent in LSDA methods meant that the antiferromagnetic ordering was required to produce the gap. In the absence of antiferromagnetic ordering, LSDA gives the same one-electron energies for the occupied and unoccupied $d$ -orbitals with the same spin, since they experience the same local effective potential. Unrestricted Hartree Fock (UHF) methods, although by definition including no controlled representation of correlation, have been successfully used to study the structural and magnetic properties of some of these transition metal monoxides [185]. In the HF approximation the effective potential is non-local and occupied and unoccupied $d$ -orbitals have different energies. However, HF methods overestimate the size of the band gap substantially and combinations of LSDA and UHF [186,187] are needed to provide a more complete picture of NiO's electronic structure. The treatment of correlation in these oxides remains difficult and the role of O(2 $p$ ) - Ni(3 $d$ ) hybridization in the valence band has proved especially difficult to predict reliably [187]. However, most studies [187,188,189] predict that the upper edge of the valence band is of O(2 $p$ ) character and that NiO should be classed as a 'charge-transfer' insulator, as opposed to a Mott-Hubbard insulator where the conduction and upper valence edge are of the same character. This consensus can only really be thoroughly confirmed by more calculations with an improved treatment of electron correlation.

In contrast, the magnetic properties of NiO are well known and various techniques have established the antiferromagnetic AF $_{2}$ structure as the most stable with each Ni ion having 2 unpaired electrons. The AF $_{2}$ structure is shown in fig. 9.1, with adjacent (111) sheets of similar spin Ni ions. This spin configuration has two components, both due to the non-local exchange interaction between Ni electrons. The direct exchange interaction between nearest neighbour Ni ions favours pairing of spins to lower the energy. However, a much stronger interaction comes from the super-exchange between the next-nearest neighbour Ni ions [184,185,188]. The hybridization of the O(2 $p$ ) and Ni(3 $d$ ) states, i.e. covalent bonding between the Ni and O, is stronger than the coupling of the $d-d$ states between nearest neighbour Ni ions and opposite spin next-nearest neighbour ions are energetically favoured. This makes the anti-ferromagnetic spin structure the ground state of NiO.

The (001) surface of NiO has been shown by LEED studies [190,191] to be an almost perfect bulk termination, with no rumpling and only a 2% relaxation of the outer layer. The surface has also been atomically resolved by NC-AFM [192] and elevated temperature STM [188], clearly demonstrating the stability of the ( $1\times 1$ ) reconstruction. Thorough UHF calculations [193] on a monolayer of NiO confirmed the stability of the anti-ferromagnetic phase and also demonstrated the usefulness of HF methods on these systems. Although it cannot model all the subtleties of the surface electronic structure (for discussion of the complexities of NiO surface electronic structure see refs. [194,195]) of NiO, UHF is an excellent technique for modelling of the structural and magnetic properties of the surface.