The central idea underlying the description of the electronic structure of solids is that the valence electrons supplied by the atoms spread through the entire structure. This concept is expressed more formally by making a simple extension of MO theory in which the solid is treated like an indefinitely large molecule. In solid-state physics, this approach is called the tight-binding approximation. The description in terms of delocalized electrons can also be used to describe non- metallic solids. We therefore begin by showing how metals are described in terms of molecular orbitals. Then we go on to show that the same principles can be applied, but with a different outcome, to ionic and molecular solids.
(a) Band formation by orbital overlap
KEY POINT The overlap of atomic orbitals in solids gives rise to bands of energy levels separated by energy gaps.
The overlap of a large number of atomic orbitals in a solid leads to a large number of molecular orbitals that are closely spaced in energy and so form an almost continuous band of energy levels (Fig. 4.61). Bands are separated by band gaps,
1 10 100 1000
T / K 1
Conductivity / (S cm–1)
Semi- conductor Superconductor
Metal
10–8 10–4 104 108
FIGURE 4.60 The variation of the electrical conductivity of a substance with temperature is the basis of the classification of the substance as a metallic conductor, a semiconductor, or a superconductor.
Band Band Band Band gap
Band gap
Energy
FIGURE 4.61 The electronic structure of a solid is characterized by a series of bands of orbitals separated by gaps at energies where orbitals do not occur.
which are values of the energy for which there is no molecu- lar orbital.
The formation of bands can be understood by consider- ing a line of atoms, and supposing that each atom has an s orbital that overlaps the s orbitals on its immediate neigh- bours (Fig. 4.62). When the line consists of only two atoms, there is a bonding and an antibonding molecular orbital.
When a third atom joins them, there are three molecular orbitals. The central orbital of the set is nonbonding and the outer two are at low energy and high energy, respectively.
As more atoms are added, each one contributes an atomic orbital, and hence one more molecular orbital is formed.
When there are N atoms in the line, there are N molecular orbitals. The orbital of lowest energy has no nodes between neighbouring atoms and the orbital of highest energy has a node between every pair of neighbours. The remaining orbitals have successively 1, 2, . . . internuclear nodes and a corresponding range of energies between the two extremes.
The total width of the band, which remains finite even as N approaches infinity (as shown in Fig. 4.63), depends on the strength of the interaction between neighbouring atoms.
The greater the strength of interaction (in broad terms, the greater the degree of overlap between neighbours), the greater the energy separation of the non-node orbital and the all-node orbital. However, whatever the number of atomic orbitals used to form the molecular orbitals, there is only a finite spread of orbital energies (as depicted in Fig. 4.63). It follows that the separation in energy between neighbouring orbitals must approach zero as N approaches infinity, otherwise the range of orbital energies could not be finite. That is, a band consists of a countable number but near-continuum of energy levels.
The band just described is built from s orbitals and is called an s band. If there are p orbitals available, a p band can be constructed from their overlap, as shown in Fig. 4.64.
Because p orbitals lie higher in energy than s orbitals of the
same valence shell, there is often an energy gap between the s band and the p band (Fig. 4.65). However, if the bands span a wide range of energy and the atomic s and p energies are similar (as is often the case), then the two bands overlap.
The d band is similarly constructed from the overlap of d orbitals. The formation of bands is not restricted to one type of atomic orbital and bands may be formed in compounds by combinations of different orbital types; for example, the d orbitals of a metal atom may overlap the p orbitals of neighbouring O atoms.
In general a band structure diagram can be produced for any solid and it is constructed using the frontier orbitals on all the atoms present. The energies of these bands, and whether they overlap, will depend on the energies of the contributing atomic orbitals, and bands may be empty, full, or partially filled depending on the total number of elec- trons in the system.
Most bonding Most antibonding
Intermediate bonding
Energy
FIGURE 4.62 A band can be thought of as formed by bringing up atoms successively to form a line of atoms. N atomic orbitals give rise to N molecular orbitals.
1 2 3 4 5 6 7 8 9101112 ∞
Energy
0
Most antibonding
Most bonding
Number of atoms, N
FIGURE 4.63 The energies of the orbitals that are formed when N atoms are brought up to form a one-dimensional array. This produces a density-of-states diagram similar to that shown in Fig. 4.68.
Most bonding Most antibonding
Intermediate orbitals
FIGURE 4.64 An example of a p band in a one-dimensional solid.
EXAMPLE 4.21 Identifying orbital overlap
Decide whether any d orbitals on titanium in TiO (with the rock- salt structure) can overlap to form a band.
Answer We need to decide whether there are d orbitals on neighbouring metal atoms that can overlap with one another.
Figure 4.66 shows one face of the rock-salt structure with the dxy orbital drawn in on each of the Ti atoms. The lobes of these orbitals point directly towards each other and will overlap to give a band. In a similar fashion the dzx and dyz orbitals overlap in the directions perpendicular to the xz and yz faces.
Self-test 4.21 Which d orbitals can overlap in a metal having a primitive structure?
(b) The Fermi level
KEY POINT The Fermi level is the highest occupied energy level in a solid at T = 0.
At T = 0, electrons occupy the individual molecular orbitals of the bands in accordance with the building-up principle.
If each atom supplies one s electron, then at T = 0 the low- est N orbitals are occupied. The highest occupied orbital at T = 0 is called the Fermi level; it lies near the centre of the band (Fig. 4.67). When the band is not completely full, the electrons close to the Fermi level can easily be promoted to nearby empty levels. As a result, they are mobile and can move relatively freely through the solid, and the substance is an electrical conductor.
The solid is in fact a metallic conductor. We have seen that the criterion for metallic conduction is the decrease of electrical conductivity with increasing temperature. This behaviour is the opposite of what we might expect if the conductivity were governed by thermal promotion of elec- trons above the Fermi level. The competing effect can be identified once we recognize that the ability of an electron to travel smoothly through the solid in a conduction band depends on the uniformity of the arrangement of the atoms.
An atom vibrating vigorously at a site is equivalent to an impurity that disrupts the orderliness of the orbitals. This decrease in uniformity reduces the ability of the electron to travel from one edge of the solid to the other, so the con- ductivity of the solid is less than at T = 0. If we think of the electron as moving through the solid, then we would say that it was ‘scattered’ by the atomic vibration. This carrier scattering increases with increasing temperature as the
Band gap
Energy
p Band
s Band s Band
p Band
(a) (b)
FIGURE 4.65 (a) The s and p bands of a solid and the gap between them. Whether or not there is in fact a gap depends on the separation of the s and p orbitals of the atoms and the strength of the interaction between them in the solid. (b) If the interaction is strong, the bands are wide and may overlap.
O Ti
FIGURE 4.66 One face of the TiO rock-salt structure showing how orbital overlap can occur for the dxy, dyz, and dzx orbitals.
Energy
Empty band
Occupied levels
Fermi level
Empty band
Occupied levels
Metal Insulator
(a) (b)
FIGURE 4.67 (a) Typical band structure for a metal showing the Fermi level; if each of the N atoms supplies one s electron, then at T = 0 the lower ẵN orbitals are occupied and the Fermi level lies near the centre of the band. (b) Typical band structure for an insulator with the Fermi level midway in the band gap.
lattice vibrations increase, and the increase accounts for the observed inverse temperature dependence of the conductiv- ity of metals.
(c) Densities of states and width of bands
KEY POINT The density of states is not uniform across a band: in most cases, the states are densest close to the centre of the band.
The number of energy levels in an energy range divided by the width of the range is called the density of states, ρ (Fig.
4.68a). The density of states is not uniform across a band because the energy levels are packed together more closely at some energies than at others. In three dimensions, the variation of density of states is like that shown in Fig. 4.69, with the greatest density of states near the centre of the band and the lowest density at the edges. The reason for this behaviour can be traced to the number of ways of producing a particular linear combination of atomic orbitals. There is only one way of forming a fully bonding molecular orbital (the lower edge of the band) and only one way of forming a fully antibonding orbital (the upper edge). However, there are many ways (in a three-dimensional array of atoms) of forming a molecular orbital with an energy corresponding to the interior of a band.
The number of orbitals contributing to a band deter- mines the total number of states within it—that is, the area enclosed by the density of states curve. Large numbers of contributing atomic orbitals which have strong overlap
produce broad (in energy terms) bands with a high den- sity of states. If only a relatively few atoms contribute to a band and these are well separated in the solid, as is the case for a dopant species, then the band associated with this dopant atom type is narrow and contains only a few states (Figure 4.68b).
The density of states is zero in the band gap itself—there is no energy level in the gap. In certain special cases, however, a full band and an empty band might coincide in energy but with a zero density of states at their conjunction (Fig. 4.70).
Solids with this band structure are called semimetals. One important example is graphite, which is a semimetal in directions parallel to the sheets of carbon atoms.
E E + dE
Energy
Density of states (a)
Density of states (b) Energy
FIGURE 4.68 (a) The density of states in a metal is the number of energy levels in an infinitesimal range of energies between E and E + dE. (b) The density of states associated with a low concentration of dopant.
Energy
Fermi level Empty
Full Part filled
FIGURE 4.69 Typical density of states diagram for a three- dimensional metal.
Energy Fermi
level Empty
Full
FIGURE 4.70 The density of states in a semimetal.
A NOTE ON GOOD PRACTICE
This use of the term ‘semimetal’ should be distinguished from its other use as a synonym for metalloid. In this text we avoid the latter usage.
(d) Insulators
KEY POINT An insulator is a solid with a large band gap.
A solid is an insulator if enough electrons are present to fill a band completely and there is a considerable energy
gap before an empty orbital and its associated band become available (Fig. 4.71). In a sodium chloride crystal, for instance, the N Cl− ions are nearly in contact and their 3s and three 3p valence orbitals overlap to form a narrow band consisting of 4N levels. The Na+ ions are also nearly in contact and also form a band. The electronegativity of chlo- rine is so much greater than that of sodium that the chlorine band lies well below the sodium band, and the band gap is about 7 eV. A total of 8N electrons are to be accommo- dated (seven from each Cl atom, one from each Na atom).
These 8N electrons enter the lower chlorine band, fill it, and leave the sodium band empty. Because the energy of ther- mal motion available at room temperature is kT ≈ 0.03 eV (k is Boltzmann’s constant), very few electrons have enough energy to occupy the orbitals of the sodium band.
In an insulator the band of highest energy that contains electrons (at T = 0) is normally termed the valence band. The next-higher band (which is empty at T = 0) is called the con- duction band. In NaCl the band derived from the Cl orbitals is the valence band and the band derived from the Na orbit- als is the conduction band.
We normally think of an ionic or molecular solid as consisting of discrete ions or molecules. According to the picture just described, however, they can be regarded as hav- ing a band structure. The two pictures can be reconciled because it is possible to show that a full band is equivalent to a sum of localized electron densities. In sodium chloride, for example, a full band built from Cl orbitals is equivalent to a collection of discrete Cl− ions, and an empty band built from Na orbitals is equivalent to the array of Na+ ions.