Electronic Conduction in Solids

1. Introduction

Electronic conduction refers to the movement of electrons through a solid material when an external electric field or voltage is applied. The ability of a material to conduct electricity depends mainly on its electronic band structure and the availability of empty energy states for electrons to move into.

In crystalline solids, electrons occupy specific energy bands that arise from the interaction of atomic orbitals in the periodic crystal lattice.

2. Allowed Energy States

In a crystal lattice, electrons are described using wave vectors (k-values). The allowed k-values extend up to the edges of the Brillouin zone.

• Each allowed k-state can contain two electrons.
• The two electrons must have opposite spin (spin-up and spin-down).
• If a crystal contains N unit cells, each energy band contains 2N states.

In three-dimensional crystals, the total number of states in a band becomes 2N_u, where N_u is the number of unit cells.

3. Energy Bands in Solids

The two most important bands in electronic conduction are:

• Valence Band (VB) – highest occupied band.
• Conduction Band (CB) – next higher empty band.

The energy difference between these bands is called the band gap.

4. Role of Valence Electrons

The total number of electrons present in the crystal depends on the number of valence electrons per unit cell (z).

Total electrons in crystal:

Total electrons = z × N_u

Depending on whether the number of valence electrons is even or odd, the electronic structure and conductivity of the material changes. If z is even, then one energy band is completely filled, with the next band being completely empty. The highest filled band is the valence band, and the next, empty band, is the conduction band. The electrons in the valence band cannot participate in electrical conduction, because there are no available states for them to move into consistent with the small increase in energy required by motion in response to an externally applied voltage: hence this configuration results in an insulator or, if the band gap is sufficiently small, a semiconductor. Alternatively, if z is odd, then the highest occupied energy band is only half full. In such a material, there are many vacant states immediately adjacent in energy to the highest occupied states, therefore electrical conduction occurs very efficiently and the material is a metal. Figure shows schematic energy diagrams for insulators, metals and semiconductors respectively. There is one further, special case which gives rise to metallic behaviour: namely, when the valence band is completely full (z is even), but the valence and conduction bands overlap in energy, such that there are vacant states immediately adjacent to the top of the valence band, just as in the case of a half-filled band. Such a material is called a semi-metal.

In the same way as defined in the molecular orbital theory, the uppermost occupied energy level in a solid is the Fermi level \(E_F\), and the corresponding Fermi wavevector is given by:

\[ E_F = \frac{\hbar^2 k_F^2}{2m_e} \]

The volume of k-space per state is \(8\pi^3/V\). Therefore the volume of k-space filled by N electrons is:

\[ \frac{4N\pi^3}{V} \]

Accounting for the fact that two electrons of opposite spins can occupy each wavevector state, we equate this volume to the volume of a sphere in k-space (the Fermi sphere) with radius \(k_F\). This gives:

\[ k_F = (3\pi^2 n_e)^{1/3} \]

where \(n_e = N/V\) is the electron density. Hence the Fermi energy becomes:

\[ E_F = \frac{\hbar^2 (3\pi^2 n_e)^{2/3}}{2m_e} \]

If the Fermi sphere extends beyond the first Brillouin zone, as occurs in many metals, the appropriate mapping back into the zone results in a Fermi surface of complex topology.

The density of states \(N(E)dE\) is defined such that

\[ N_s = \int N(E)dE \]

gives the total number of states per unit crystal volume in an energy band.

The number of wavevector states per unit volume of k-space is:

\[ \frac{V}{8\pi^3} \]

Thus the total number of states per band is:

\[ N_s = 2 \times \frac{V}{8\pi^3} \int dk \]

The factor of 2 accounts for the two spin states per k value.

In three dimensions:

\[ dk = 4\pi k^2 dk \]

Thus we obtain:

\[ N_s = \frac{V}{4\pi^3} \int 4\pi k^2 \frac{dk}{dE} dE \]

For parabolic bands:

\[ E = \frac{\hbar^2 k^2}{2m_e} \]

and hence

\[ \frac{dk}{dE} = \frac{m_e}{\hbar^2 k} \]

From which the density of states becomes:

\[ N(E) = \frac{4\pi (2m_e)^{3/2} E^{1/2}}{h^3} \]

The dependence of the density of states on \(E^{1/2}\) arises from the increasing volume of phase space available at larger energies.

The actual population of electrons as a function of energy is given by the product of the density of states and the occupation probability \(f(E)\), described by the Fermi–Dirac distribution:

\[ f(E) = \frac{1}{\exp((E - E_F)/k_B T) + 1} \]

from which we may observe that the Fermi energy corresponds to the energy at which the occupation probability is exactly one half.

5. Electrical Behaviour of Different Materials

1. Insulators

• Valence band completely filled.
• Conduction band completely empty.
• Large band gap between VB and CB.
• Electrons cannot easily move to higher energy levels.

Result: Very poor electrical conduction.

2. Semiconductors

• Valence band filled.
• Conduction band empty at absolute zero.
• Small band gap between VB and CB.

At higher temperatures, some electrons gain enough energy to move from the valence band to the conduction band.

Result: Moderate electrical conductivity.

3. Metals (Conductors)

• The highest occupied band is only partially filled.
• Many empty energy states exist near the Fermi level.
• Electrons can move easily under an applied electric field.

Result: Very high electrical conductivity.

6. Special Case: Band Overlap

Another situation that produces metallic conduction occurs when the valence band overlaps with the conduction band.

Even if the valence band is completely filled, overlapping bands create vacant energy states that allow electrons to move freely.

This overlapping of bands leads to excellent electrical conduction.

7. Summary

• Electrical conduction depends on the availability of empty energy states.
• Materials with partially filled bands conduct electricity well.
• Large band gaps prevent electron movement, producing insulators.
• Small band gaps allow limited electron excitation, producing semiconductors.
• Band overlap or partially filled bands produce metals.

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