Energy Bands in Solids | Conductivity | Electrical Engineering

In this article we will discuss about the classical picture of energy bands in solids.

For the transition of an electron from the inner orbit to the outer orbit of the atom, some energy is required. Thus, the different orbits have the different levels of potential energy.

In an atom, the electrons try to occupy the inner orbit (having minimum energy) but according to Pauli’s exclusion principle not more than two electrons can exist in the one energy state. The electrons start occupying the energy states having low energy. Therefore, in normal atom, the lower energy levels are completely filled, while the higher energy levels remain unoccupied.

When two similar atoms are brought close together, then there is an interaction between the orbits of their electrons. This interaction causes a splitting of each individual energy level into two slightly different levels. The atoms in almost every crystalline solid are so close together that the energy levels produced after splitting due to interaction between the various orbits of different electrons will be very large and so close together as to form a band.

If the energy band contains the number of electrons equal to the number of electrons allowed by Pauli Exclusion Principle, then the band is said to be completely filled. In a completely filled band, there is no free electron for the conduction of current. On the other hand, the conduction is possible in a partially filled band. The energy band as a function of inter-atomic spacing in shown in Fig. 5.17.

In a solid, many atoms are brought together, so that the split energy levels form essentially continuous bands of energies. As an example, Fig. 5.17 illustrates the imaginary formation of a diamond crystal from isolated carbon atoms. Each isolated carbon atom has an electronic structure 1s²2s²2p² in the ground state. Each atom has available two 1s states, two 2s states, six 2p states and higher states.

If we consider, N atoms, there will be 2N, 2N and 6N states of type 1s, 2s and 2p respectively. As the interatomic spacing decreases, their energy level split into bands, beginning with the outer shell, i.e., n = 2. As the 2s and 2p bands grow, they merge into a single band composed of a mixture of energy levels. This band of 2s-2p levels contains 8N available states.

As the distance between atoms approaches the equilibrium interatomic spacing of the diamond, this band splits into two bands separated by an energy gap E_g. The upper band (conduction band) contains 4N states, as does the lower band (valence band). Thus, apart from the low lying and tightly bound 1s levels, the diamond crystal has two bands of available energy levels separated by an energy gap E_g wide which contains no allowed energy levels for electrons to occupy.

The lower 1s band is filled with the 2N electrons which originally resided in the collective 1s states of the isolated atoms. However, there were 4N electrons in the original isolated (n = 2) shell (2N in 2s and 2N in 2p states). These 4N electrons must occupy states in the valence band or the conduction band in the crystal.

At 0 K the electrons will occupy the lowest energy states available to them. In the case of diamond crystal, there are exactly 4N states in the valence band available to the 4N electrons. Thus at 0 K, every state in the valence band will be filled, while the conduction band will be completely empty of electrons.

In an insulator and pure semiconductor, lower band is completely filled and the upper band is completely empty. The energy of the forbidden gap is denoted by E_g. The conduction takes place only when the electron in valence band jumps to the conduction band.

In other words, the electron in valence band requires energy equal to E_g to jump to the conduction band. When the electron jumps from the valence band to the conduction band, then a vacancy electron called a hole is created in the valence band. Since hole is a deficiency of an electron and hence is positively charged.

The forbidden energy gap in an insulator is of the order of 5 to 10 eV. The amount of energy cannot be imparted to the electrons in the valence band and hence the electron cannot jump from the valence to conduction band. Therefore, conduction is not possible in the insulators.

The forbidden energy gap in case of semiconductor is usually, of the order of 0.75 to 1 eV. This amount of energy can be easily imparted to the electrons in the valence band by thermal agitation of the crystal lattice.

Thus, with the increase in temperature, many electrons from the valence band acquire the required amount of energy to jump to the conduction band and these results in the increase of electron hole pairs. The forbidden energy gap E_g is the energy required to break the covalent bands so as to make the electron free for conduction. 

Band Gap in Energy Band Calculations:

In a typical energy band calculation, a single electron is assumed to travel through a perfectly periodic lattice. The wave function of the electron is assumed to be in the form of a plane wave, e.g., moving in the x-direction with propagation constant K, so called a wave vector. The space-dependent wave function for the electron is-

Ψ(x) = Vk(x) e^ikx

Where, the function Vk(x) modulates the wave function according to the periodicity of the lattice. In such calculation, allowed values of energy can be plotted versus the propagation constant K. Since the periodicity of most lattices is different in various directions, the E-K diagram must be plotted for the various crystal directions, and the full relationship between E and K is a complex surface which should be visualized in three-dimensions.

The band structure of GaAs has a minimum in the conduction band and a maximum in the valance band for the same K value (K = 0). On the other hand, Si has its valence band maximum at a different value of K than its conduction band minimum.

Thus an electron making a smallest energy transition from the conduction band to the valence band in GaAs can do so without a change in K value; on the other hand, a transition from the minimum point in the Si conduction band to the maximum point of the valence band requires some change in K.

Thus there are two classes of semiconductor energy bands; direct and indirect (Fig. 5.14). In case of a n indirect transition, a change in K requires a change in momentum for the electron. In a direct semiconductor, an electron in the conduction band can fall to an empty state in the valence band, giving off the energy difference E_g as a photon of light.

 

On the other hand, an electron in the conduction band minimum of an indirect semiconductor cannot fall directly to the valence band minimum but must undergo a momentum change as well as changing its energy. For example, it may go through some defect state E_t within the band gap. In an indirect transition which involves a change in K, the energy is generally given up as heat to the lattice rather than as an emitted photon.

This difference between direct and indirect band structures is very important for deciding which semiconductors can be used in devices requiring light output, e.g., semiconductor light emitters and laser generally must be made of materials capable of direct band-to-band transition or of indirect materials with vertical transitions between defect states.