10.1.1 Density of states
The DOS of a system,n(e), is defined as the number of states in an energy interval between eande+δe. Thusn(e)is non-zero in the allowed energy
Fig. 10.1
π-type electronic structure of the cyclobutadiene molecule according to the H¨uckel approach: (a) molecular orbital diagram and (b) density of states.
region of a band and vanishes in the forbidden energy region. The concept of the DOS can be easily illustrated by considering the example of the cyclobu- tadiene molecule (Fig. 2.1). As shown in Fig. 10.1a, theπ-type orbitals of this molecule are, according to a H¨uckel approach: a) a non-degenerate bonding orbital atα+2β, b) a set of two non-bonding degenerate orbitals atα, and c) a non-degenerate antibonding orbital atα−2β(see Fig. 2.10). Thus the number ofπ states at energyαis twice the number ofπ states atα+2β andα−2β and theπ-type DOS of cyclobutadiene can be represented as in Fig. 10.1b.
For the simple case of an infinite chain of hydrogen atoms, as discussed in Section 3.2, the dispersion relation of the band according to a H¨uckel approach is given by a cosine function ofk(see Fig. 10.2a and eqn. (3.20)). Since the k-space is a set of equally spaced k points, the DOS is proportional to the inverse of the slope of the energy versuskcurve.
n(e)∝
de(k) dk
−1
(10.1) Thus,n(e)will be minimum at the centre of the band and will tend to infinity at the borders, i.e. at k= 0 andk =π/a (see Fig. 10.2b). The last feature is known as a van Hove singularity and is physically meaningless because the integrated DOS of the whole band must be equal to the number of electrons per unit cell associated with the band, i.e. two if the band is completely filled.1
1Let us recall that, for simplicity, throughout this book we have assumed that the molecular orbitals for the α (or ‘spin-up’) and β (or ‘spin-down’) electrons are identical. In other words, we assume that a molecular orbital can accommodate two electrons and a band can accommodate two electrons per cell.
Since n(e)is zero outside the band, the typical shape of the DOS for a 1D band is that shown in Fig. 10.2c. Thus the wider the band the lower the DOS will be. In other words, strong interactions along the chain are associated with a large energy dispersion and thus with lower DOS values. Conversely, very sharp peaks in the DOS are associated with flat bands and thus with electrons which tend to be localised.
The shape of the DOS curves for 2D and 3D systems is quite different.
Although the detailed shape strongly depends upon the nature of the lattice, high DOS values generally occur around the centre of the band. In practice, to determine the DOS of a system thek-dependent Hamiltonian is diagonalised for a fine mesh ofk-points in the Brillouin zone. This leads to an histogram of energy values, which is then numerically smoothed using a set of appropriate functions, for example Gaussians, leading to the DOS of the material. The finer
(b)
(a) (c)
Fig. 10.2
(a) Band structure and (b) density of states for an infinite chain of hydrogen atoms according to a H¨uckel approach;
(c) DOS for a typical 1D band.
the mesh ofk points used, the more precise will be the DOS. The electronic structure of solids can be discussed solely on the basis of their DOS values.
The DOS of a real solid, i.e. a multi-band system, can have a quite complex shape. Thus it is important to develop a qualitative way to analyse this type of plot. In this chapter we illustrate some chemically appealing ways to do it.
10.1.2 Projected density of states
Then(e)values associated with a given band,i, should satisfy the normalisa- tion condition:
∞
−∞ni(e)de=1 (10.2)
Consequently, the DOS plot counts the number of levels available for a certain structure. Since a band can accommodate two electrons per unit cell, integra- tion of twice the DOS up to the Fermi level gives the total number of electrons of the system. Since the COs have been written as a linear combination of BOs associated with the different AOs of the unit cell, the DOS curves can be analysed in terms of contributions per AO of the unit cell. In other words, once the DOS plot has been calculated, we can project the contribution of certain AOs or fragment orbitals (FOs) of interest for the analysis.
The easiest way to illustrate this process is by considering a simple two- centre MO:
=c1χ1+c2χ2 (10.3)
The electron distribution in this MO is given by the normalisation condition:
1=c12+c22+2c1c2S12 (10.4) where S12is the overlap integral between the AOsχ1andχ2. Although it is clear that c12 andc22 represent the fraction of the electron described by this orbital, which on average is found near the centres 1 and 2, respectively, and consequently should be assigned toχ1 andχ2, it is not clear how to assign the term 2c1c2S12, which is anoverlap density. The simplest solution has been suggested by Mulliken. [3] In the Mulliken population analysis, this term is equally shared between centres 1 and 2. Thus centre 1 is assigned a total of c21+c1c2S12 electrons, which is thegross populationof centre 1, and centre 2 a total ofc22+c1c2S12 electrons, which is thegross populationof centre 2.
These contributions should now be multiplied by the occupation number of
this MO. This type of analysis can be performed for the different orbitals of a molecule or a solid. Consequently, a local DOS can be projected out from the total DOS. These projected DOS can be those of an AO, an FO, an atom, or a group of atoms that are convenient for the analysis. The projected DOS can be used in different ways. For instance, one can be interested in knowing with which atom or group of atoms of a complex structure the electrons in a certain energy range are associated. Typically, this question can be an important one when trying to correlate the structure and properties of complex materials. For instance, let us consider the 2D purple bronze KMo6O17. [4] This material has been the subject of many studies because it is a 2D metal and exhibits a resis- tivity anomaly at 120 K, which is the result of a metal-to-metal transition (see Chapter 11). The structure of KMo6O17is schematically shown in Fig. 10.3a.
This bronze contains Mo6O−17layers made up of condensed MoO6octahedra and MoO4 tetrahedra. Between these layers are found the K+ cations. The electrostatic interactions between these K+cations and the outer oxygen atoms of the covalently bonded Mo6O−17layers hold the solid together. The layers of Fig. 10.3a contain only three different types of Mo atoms: a) the Mo atoms of the MoO4tetrahedra (denoted MoIin Fig. 10.3a), b) the Mo atoms of the MoO6
octahedra in the two outer octahedral sublayers (denoted MoIIin Fig. 10.3a), and c) the Mo atoms of the MoO6 octahedra in the two inner octahedral sublayers (denoted MoIII in Fig. 10.3a). With the usual oxidation states K+ and O2−, it turns out that three electrons per formula unit remain to fill the lower d-block levels of the Mo6O−17 layers. Since there are three electrons to occupy these levels but six Mo atoms per formula unit, it is of interest to
(a) (b)
Fig. 10.3
(a) Perspective view of the crystal structure of KMo6O17, with the three different types of molybdenum atom labelled. (b) Projected DOS of the Mo atoms in KMo6O17as well as the individual contributions of MoI, MoIIand MoIII.