QMMS Lecture #5 (Venables/Heggie)

Notes for QMMS Lecture 5 (Venables/Heggie)

Lecture notes by John A. Venables and Edward HernŠndez. Lecture scheduled for 1 Nov 05. Latest version 28th October 2005.

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5. Metallic band structures and methods of calculation

5.1 Methods based on plane waves

There have been many developments since Lang and Kohn to extend the free electron approach, first to s-p bonded metals and then to the complications of transition metals involving d-electrons, and in the case of the rare earths, f-electrons as well. The d-electrons give an angular character to the bonding, often resulting in structures which are not close-packed, e.g. b.c.c (Fe, Mo, W, etc.) or complex structures like a-Mn. This is in contrast to s-p bonded metals which typically are either f.c.c or h.c.p. There are many challenges left for quantum mechanical models of metals.

To start we need a few names of the methods, to supplement the NFE methods discussed in Lecture 3 and Tight-binding, to be discussed in the next lecture. Some of these are pseudopotentials, orthogonalized plane waves (OPW), augmented plane waves (APW), Korringa-Kohn-Rostoker (KKR), etc. These long-standing methods are described by Ashcroft and Mermin, chapter 11, with experimentally determined band structures in chapter 15. There are descriptions of these methods with a more modern "feel" to them in the books by Kaxiras, chapter 4 and by Martin, chapter 16. A shorter version is given by Marder, chapter 10. These books, especially Richard Martin's, almost amount to "how-to" manuals, and are backed up with detailed computational examples. Such computations would be suitable for a detailed project, but are not to be undertaken lightly.

Typically tight-binding (where interatomic overlap integrals are thought of as small) is taken as the opposite extreme to the nearly-free electron model (where Fourier coefficients of the lattice potential are thought of as small). However, this is more apparent than real, in that both pictures can work for arbitrarily large overlap integrals or lattice potentials; the only requirement is that the basis sets are complete for the problem being studied. This of course can lead to some semantic problems: methods which sound different may not in fact be so different; in particular, when additional effects are included they are almost certainly not simply additive.

The basic feature caused by including the ions via any of these methods is that the electron density is now modulated in x, y and z with the periodicity of the lattice. So there are now two length scales in the problem which compete. When defects of any kind are present, there is no need for the two length scales to bear any relationship to each other. For example, when translational invariance in the z-direction is lost due to the presence of a surface, the resulting surface states have oscillation periods with no simple relation to the lattice period in the z-direction.

One of the first such calculations was to metallic Lithium, which, with only three electrons per atom, is amenable to full first principles calculations. An example of a surface-related calculation is shown in figure 5.1, taken from an early review article by Appelbaum & Hamann (1976). Large scale calculations of the band structures for many metals using similar methods have been performed by Moruzzi et al. (1978) and by Papaconstantopoulos (1986).

Figure 5.1: Valence electron density at several x// points for Li(001) in a pseudo-potential calculation (from Alldredge & Kleinman 1974, after Appelbaum & Hamann 1976).

5.2 Embedded atom and/or Effective medium theories

Contrary to the free electron starting point, it has more recently proved fruitful to consider models based on wave functions relatively localized in real, rather than reciprocal space, and to construct interatomic potential functions arising from atomic-like entities interacting with the electron gas in which the Ďatomí is embedded. The resulting methods are known as embedded atom models (EAM) or effective medium theories (EMT); in these models the embedding energy DE is expressed in terms of the cohesive function Ec(n), as DE = Ec(n) + DEc, where the correction energy DEc differs between the various schemes, but is relatively small. The terms EAM and EMT are often used as though the methods form a continuum, so that it is difficult for the non-specialist to be sure what each method represents, or whether it is unique. I will concentrate on the (largely Danish) EMT literature; other references can be found via Sutton chapter 9, Pettifor chapters 5 and 6, and/or Venables, chapter 6.

The cohesive function Ec(n) is a function of the homogeneous electron gas density n in which the atom is embedded (Jacobsen et al. 1987, Jacobsen 1988, NÝrskov et al. 1993). A major effect of these models is to show clearly that metallic binding is strongly non-linear with coordination number. The first Ďbondsí to form are strong, and get progressively weaker as extra metal atoms are added to the first coordination shell. Some examples will be shown and discussed in the lecture.

Increasingly what counts is the speed of the computer code; if this speed scales with a lower power of the number, N, of electrons in the system, then more complex/ larger problems can be tackled; O(N) methods are discussed later in Lecture 8. For example, because the interactions between atoms and the electron gas are parameterised initially, EMT calculations are fast enough that they can be used to simulate dynamic processes such as adsorption, nucleation or melting on metal surfaces; here an approximate electronic structure calculation is being done for each set of positions of the nuclei, i.e. at each time step (Jacobsen et al. 1987, Stolze 1994, 1997). This requires computer speeds that would have been inconceivable just a few years ago. It is now feasible to download EMT programs from Per Stoltze's website in Denmark in order to run them anywhere. There are real possibilities for experiment-theory collaborations here which were impractical just a few years ago. If this is of interest as a project, let me know.

For the general reader, we should bear in mind that that both EAM and EMT are approximations, which are aimed at increasing broad-brush understanding, i.e. getting most of the story, most of the time. There are examples where these approaches simply don't work, despite the best efforts of the specialists involved. Detailed studies of such cases, and comparison of methods have then been used to find out where they went wrong. An example is that both EAM and EMT generally get too low a surface energy, and an extreme case is that of Pd. This metal has become a test case in that it can have two chemical configurations. The ground state is 4d10, but the 4d95s state is quite low lying (Balasubramanian 1988). Thus, at a surface or other special configuration, we can have changes in the amount, or even the type, of hybridisation; such effects disturb the continuous variation of energies with coordination number which are a feature of these approximate methods.

A chemist might wonder what the fuss is all about, since ideas of hybridisation are central to their world view. We explore these ideas in the context of silicon in Lecture 7. But first we survey the effects of d-electrons in relation to magnetism in the next section, and then we explore the tight-binding and related methods from a more chemical viewpoint in the next lecture.

The 3d, 4d and 5d series have a major contribution to cohesion from both s-d and d-d interactions. In the case of 3d, the overall cohesion peaks before and after the middle of the transition series, unlike the 4d and 5d series, where cohesion from the d-bands peaks in the middle. An example of a comparison of EMT with KKR methods (Morruzzi et al. 1978, Jacobsen et al. 1987, Jacobsen 1988) is given in figure 5.2. Note that these particular calculations do not include spin-correlation effects, which are discussed later.

Figure 5.2: Calculated cohesive energies and the equilibrium radius rs for the 3d transition series, comparing EMT (open circles) with KKR methods (Morruzzi et al. 1978, closed circles). The modified EMT corresponds to EMT applied at the density given by the KKR method (after Jacobsen et al. 1987 and Jacobsen 1988).

5.3 Magnetism and spin-dependent calculations

Magnetic interactions in 3-d metals are dominated by the d-electrons and perturbed by s-d hybridization. The 3d-electrons, responsible for the magnetism of Fe, Ni and Co form a relatively narrow band which overlaps with the wide 4s band. The question of why these members of the 3d series are ferromagnetic, while others are antiferromagnetic, and why the 4d and 5d series are not magnetic, is a typically subtle problem in cohesive energy, in which several terms of differing sign are closely balanced.

The magnetism of the parent atoms is a result of Hundís rule, which asserts that the first 5 d-electrons are populated with parallel spins, and the remaining 5 then fill up the band with antiparallel alignment. This is due to the reduced electron-electron Coulomb interaction between pairs with parallel spins, because the exchange-correlation hole which accompanies each electron keeps these electrons further apart on average. The rare earth elements are an important class of magnetic materials based on 4f electrons, but are not discussed here.

When these atoms are assembled into solids, several effects occur which we should not try to oversimplify. The d-band is very important for cohesion, and the simplest model is that due to Friedel (1969), which predicts a parabolic dependence of the bond energy as the number of d-electrons Nd is increased across the series. This model leads to the contribution of d-d bonding to the pair-bond energy, Eb

-Eb = 2ÚEF (E - ed)(5/W)dE = -(W/20)Nd(10 - Nd),         (5.1)

where ed is the unperturbed atomic d-level energy and W is the d-band width in the solid. In terms of the second moment of the energy distribution m2, the overlap integrals between d-orbitals of strength b, and the band width are related by W = (12z)1/2|b|, with z nearest neighbors; this can be derived for a rectangular d-band, where the second moment m2 = W2/12, as explained by Sutton on pages 174-175. This parabolic behaviour with Nd is quite closely obeyed by the 4d and 5d series, leading to surface energies displaying similar trends (Skriver and Rosengaard 1992).

However, when magnetic effects are considered, the shape of the d-band is also very important, and ferromagnetism only results when both the d-d nearest neighbor overlap is strong and the density of states near the Fermi energy is large. These conditions are fulfilled towards the end of the 3d series, aided by the two-peaked character of the density of states, sketched in figure 5.3; this energy distribution has a large fourth moment m4, which is also implicated in the discussion of why Fe has the b.c.c structure, points which can be explored further via project 5.4.2.

Figure 5.3: Schematic distribution of s-d band overlap with the d-band having a double-peaked density of states, and hence a large fourth moment m4.

When detailed band structure calculations are done including magnetic interactions, we have to account separately for the majority spin-up (r≠) and minority spin-down () densities. By analogy to LDA, there is a corresponding local spin density (LSD) approximation, which can be explored further via project 5.4.3.

5.4 Problems and projects relating to this topic

5.4.1: Plane wave pseudopotential calculation for aluminium

The nearly free electron (NFE) model applies to aluminium, but there is a subtlety pointed out by Pettifor on page 122, where he explains how the Fourier coefficient of the lattice potential can have the opposite sign to what one would expect due to orthogonalisation of valence and core electron wavefunctions. Follow through his section 5.5 (pages 121-127), and explain the steps needed to set up a calculation based on (a few) orthogonalised plane waves (OPW's). Note: problem 25 in Sutton, coupled with the discussion on his page 151-153, covers the same material in general terms.

5.4.2: Influence of d-bands on structure and reactivity

You will have noticed that some of the 3d transition metals (e.g. Fe) have the b.c.c structure and that many catalysts are formed from d-band metals (e.g. Fe, Ni, Pt, Pd, Ag etc). Somehow these facts have to be related to the band structure, as expressed in terms of moments, m2, m4, etc. Explore one of these topics by delving into references supplied, and see how far Effective Medium Theory (EMT) has been able to come up with general explanations.

5.4.3: Magnetism in the 3d, 4d and 5d series

The end of the 3d series contains the magnetic elements (Fe, Co and Ni), but the 4d and 5d elements are not magnetic. This means that magnetism is a subtle phenomenon, and results from competing interactions which determine whether or not the magnetic state results in the lowest (free) energy. See what you can find out about these interactions, starting from the relevant sections of Sutton and Pettifor and references quoted.

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