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PHY583 – Note 1d – Band Theory of Solids

BAND THEORY OF SOLIDS

In general, there are 2 approaches to determine the electronic energies in solids:

1. Follow the behaviour of energy levels of isolated atoms as they are brought closer and closer to form a solid.

2. Show that the energy bands arise when the Schrodinger equation is solved for electrons subject to a periodic potential representing the lattice ions.

Isolated-Atom Approach to Band Theory

If 2 identical atoms are very far apart, they do not interact and their electronic energy levels can be considered to be those of isolated atoms.

E.g. 2 sodium atoms, each having an outermost 3s electron with a specific energy.

As the two sodium atoms are brought closer together, their wavefunction overlap, and the two degenerate, isolated 3s energy levels are split into 2 different levels, as shown in Fig. 12.16a.

The width of an energy band ( depend only on the number of atoms close enough

to interact strongly, which is always a small number.

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PHY583 – Note 1d – Band Theory of SolidsIf the total number of atoms in solids (N 1023 atoms/cm3), we find a very large number of levels (determined by N) spaced within the width E, so the levels may be regarded as a continuous band of energy levels (Fig. 12.16c).

*Two or more different quantum states are said to be degenerate if they are all at the same energy level. An energy level is said to be degenerate if it contains two or more different states. The number of different states at a particular energy level is called the level’s degeneracy and this phenomenon is generally known as the quantum degeneracy.

Degenerate state: a quantum state of a system, having the same energy level as another state of the system, but different wavefunction.

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For sodium, it is common to refer to the continuous distribution of allowed energy levels as the 3s band, because it originate from the 3s levels of individual sodium atoms.

In general, a crystalline solid has numerous allowed energy bands, one band arising from each atomic energy level.

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Electrical Conduction in Metals, Insulators & Semiconductors

The variation in the electrical conductivity of metals, insulator & semiconductors can be explained qualitatively in terms of energy bands.

The position & the electronic occupation of the highest band, or at most, of the highest two bands determine the conductivity of solid.

Metals

Consider the half-filled 3s band of sodium.

Fig. 12.19 shows this typical half-filled metallic band at T = 0 K, where the shaded region represents levels that are filled with electrons.

Since electrons obey Fermi-Dirac statistics, all levels below the Fermi energy, EF, are filled with electrons, while all levels above EF are empty.

In the case of sodium, the Fermi energy lied in the middle of the band.

At temperature greater than 0 K, some electrons are thermally excited to levels above EF (as shown by the Fermi-Dirac distribution to the left of Fig. 12.19), but overall there is little change from the 0-K case.

If an electric field is applied to the metal, electrons with energies near the Fermi energy can gain a small amount of additional energy from the field and reach nearby empty energy state.

Thus, electrons are free to move with only a small applied field in a metal because there are many unoccupied energy states very close to occupied energy state.

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Insulators

Consider: two highest-energy bands of a material having the lower band completely filled with electrons & the higher completely empty at 0 K (Fig.12-20).

The separation between the outermost filled and empty bands is commonly referred to as the energy gap, Eg, of the material.

The lower band filled with electrons is called the valence band, and the upper empty band is the conduction band.

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The Fermi energy is at the midpoint of the energy gap, as shown in Fig. 12.20.

Since the energy gap for an insulator is large ( 10 eV) compared to kBT at room temperature

(kBT = 0.025 eV at 300 K), the Fermi-Dirac distribution predicts that very few electrons will be thermally excited into the upper band at normal temperatures, as can be seen by the smaller value of fFD at the bottom of the conduction band in Fig. 12.20.

Although an insulator has many vacant states in the conduction band that can except electrons, there are so few electrons actually occupying conduction-band states at room temperature that the overall contribution to electrical conductivity is very small, resulting in a higher resistivity for insulators.

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Semiconductors

Semiconductor: A material with much smaller energy gap 1 eV.

Table 12.8 shows the energy gap for some semiconductors.

At T = 0 K, all electrons are in the valence band & no electrons in the conduction band.

Hence, semiconductors are poor conductors at low temperatures.

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Fig.12.20 An insulator at T = 0 K has a filled valance band and empty conduction band. The Fermi level lies midway between these bands. The Fermi-Dirac probability that an energy state E is occupied at T> 0 K is shown to the left.

However, at ordinary temperature, the population of the valence & conduction bands are altered as shown in Fig. 12.21.

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Fig. 12.21 The band structure of a semiconductor at ordinary temperatures (T 300 K). Note that the energy gap is much smaller than in an insulator & that many electrons occupy states in the conduction band.

Because EF, is located the middle of the gap & Eg is small, appreciable number of electrons are thermally excited from the valance band to the conduction band.

Since there are many empty nearby states in the conduction band, a small applied potential can easily raise the energy of the electrons in the conduction band, resulting in a moderate current.

Because thermal excitation across the narrow gap is more probable at higher temperatures, the conductivity of semiconductors depend strongly on temperature & increases rapidly with temperature.

This contrasts sharply with a metal, which decreases slowly with temperature.

There are both positive & negative charge carriers in a semicondcutor.

When an electron moves from the valence band into the conduction band, it leaves behind a vacant valence electron site, called hole, in the otherwise filled valence band.

This hole (electron-deficient site) appears as a positive charge, +e.

The hole acts as a charge carrier in the sense that a valence electron from the nearby site can transfer into the hole, thereby filling it & leaving a hole behind in the electron’s original place.

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Thus the hole migrates through the valence band.

In a pure crystal, containing only one element or compound,

number of conduction electrons = number of holes.

These combinations of charges are electron-hole pairs, & a pure semiconductor that contains such pairs is called an intrinsic semiconductor as illustrated in Fig. 12.22.

In the presence of an electric field, the holes move in the direction of the field & the conduction electrons move opposite to the field.

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Energy Bands from Electron Wave Reflection

This is a different approach to understand the origin of energy bands in solids.

This approach involves modifying the free electron wavefunctions to take into account the scattering of electron waves by the periodic crystal lattice.

A completely free electron moving in the +x direction, is represented by a travelling wave with

wavenumber , described by:

According to de Broglie, the wave carries momentum and energy .

The energy of a free electron as a function of k is

A plot of E versus k is given in Fig.12.23.

Fig. 12.23 Energy versus wavenumber k for a free electron, where .

Fig. 12.23 shows that allowed energy values are distributed continuously from zero to infinity, & there are no breaks or gaps in energy at particular k values.

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Consider what happen to a travelling electron wave when it passes through a one-dimensional crystal lattice with atomic spacing a.

When the wavelength of the incident electron waves is very long (or small k) & low energy travelling to the right as in Fig. 12.24a, the waves reflected from successive atoms & travelling to the left are all slightly out of phase & on average cancel out.

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PHY583 – Note 1d – Band Theory of SolidsThus the electron does not get reflected, & the electron moves through the lattice like a free particle.

However, if we make the electron wavelength shorter & shorter, when we reach the condition

= 2a, waves reflected from adjacent atoms will be in phase as shown in Fig. 12.24b & there will be a strong reflected wave.

This occurs because the path length difference is 2a for a wave reflected to the left directly from atom 1 compared to a wave that travels from 1 to 2 & then is reflected back to 1, as shown in Fig. 12.24c.

The longest wavelength at which constructive interference of reflected waves occurs is: 2a = .

In general, constructive interference also occurs at other, shorter wavelengths:

2a = n n = 1,2,3,... ..............12.29

The negative sign arises from reflection from the atom to the left of atom 1.

In terms of wavenumber k, eqn. 12.29 predicts strong reflected electron waves when,

..................12.30

Thus for the electron wavefunctions are not just wave travelling to the right but are

composed of equal parts of wave travelling to the right (incident) and to the left (reflected).

Since the wave travelling to the right and left can be added or subtracted, we have two different

possible standing wave types, denoted by and :

..................12.31

....................12.32

Where describes a wave travelling to the right and a wave travelling to the

left.

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PHY583 – Note 1d – Band Theory of SolidsImportant point:

has a slightly higher energy than & has a slightly lower energy than at .

This leads to a discontinuity in energy or the band gap, at as shown in Fig. 12.25.

Fig. 12.25 Energy versus wavenumber for a one dimensional lattice with atomic separation a. The E versus k curve for the free electron case is shown dashed. Note the forbidden bands corresponding to impossible energy states for the electron.

state has higher energy than the state

kinetic energy of electron in state = kinetic energy of electron in state;

potential energy of electron in state > potential energy of electron in state;

total energy of electron in state > total energy of electron in state; (at )

i.e. total energy of electron in state is slightly above the free electron energy;

total energy of electron in state is slightly below the free electron energy.

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PHY583 – Note 1d – Band Theory of SolidsFig 12.25 also shows that the overall effect of the lattice is to introduce forbidden gaps in the free

electron E versus k plot at values of .

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