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BAND THEORY OF SOLIDS

11.6.1 Energy Bands in Solids

When a large number of atoms are brought together to form a solid, a single energy level of an isolated atom is split into an energy band consisting of very closely spaced levels of slightly differing energy. It is known from quantum theory that the energy of the bound electron Inan atom is quantisedi.e., the electrons can take on only discrete values of energy and we knowthat the electron is not localized at a given radius. Rather the electron location is obtained from a probability density function, which is the probability of finding the electron at a particular distance from the nucleus. Application of quantum mechanics and Schrodinger’s wave equation is needed to find out the electron energy bands in a solid for assumed potential functions.

The probability density function for lowest energy state of a single isolated atom
is shown in Figure 11.9 (a). When two atoms are brought close together as shown in Figure 11.9 (b),the wave functions of the electrons in the two atoms overlap and in the overlapping region the discrete single energy level splits into two levels [Figure 11.9 (c)] due to interacting potential so that each electron can occupy a distinct quantum state according to Pauli exclusion principle In an assembly of N atoms, each of the energy level splits into N energy levels and forms an energy bond of very closely spaced levels differing slightly in energy. For example, a solid containing 1019atoms will have 1019possible energy states. The maximum number of electronsthat can be accommodated in these energy state is 2 × 1019since only two electrons with opposite spin can occupy the same state in accordance with Pauli exclusion principle. If the width of the energy band be 2 eV, then the energy levels are separated by % × 10-19i.e., 10-19eV. Since this spacing of energy level is extremely small, a quasi continuum level is formed.

FIGURE 11.9 (a) Probability density function for lowest electron energy state of a single isolated atom

FIGURE 11.9 (b) Probability density function of two atoms brought in close proximity to each other

FIGURE 11.9 (c) Splitting of energy level

Figure 11.10 shows the splitting of n=1, n=2, n=3 i.e., K, L, M energy levels as the distance between different atoms is reduced. At first n=3 (M-shell) energy level is affected and as the separation is reduced inner shells i.e., n = 2 and n = 1 are gradually affected. Thus bans of allowed energies that the electrons may occupy are separated by bands of forbidden energy when the interatomic distance reaches the equilibrium value r0. This is essence of energy-band theory of single crystal material.

FIGURE 11.10 Splitting of energy levels

11.6.2 Vaence and Conduction Bands

The energy band occupied by outermost electrons of an atom i.e., valence electron is known as valence band, which is the highest occupied energy band. It may be completely filled or partially filled by electrons.

The next higher allowed band is called the conduction band, which is also may be completely empty or partially filled by electrons. In this band electrons are free to move.

The energy gap (Eg) between valence and conduction band is known as the band gap.

The splitting of energy band is a crystal like silicon is visualized in the Figure 11.11.

Silicon has fourteen electrons in 1s2, 2s2, 2p6, 3s2, 3p6,configuration.

FIGURE 11.11. Energy band of silicon

Ten  out of fourteen electrons occupy the innermost shells (K and L shells). The four remaining electrons in the outermost shell (M shell) which are called valence electrons. The 3 s states corresponds to n=3 and l=0 which contain 2 quantum states per atoms i.e., 2 N states for N atoms. This state will contain 2 N electrons at T = 0 K. The 3 p state corresponds to n = 3 and l = 1, which contain 6 quantum states per atom i.e., 6 N states and 2 N remaining valence electrons will occupy these states. As the atoms are brought closer, the 3s and 3p states interact and overlap, which causes the splitting of energy levels into bands. A equilibrium interatomic distance, the single band splits, into two bands separated by the energy gap Eg. The lower band contains 4 N states. At 0 K, all the states in the lower band (valence band) will be occupied by valence electrons but those in the upper band will be empty (conduction band). The energy band gap Eg, between the top of the valence band and the bottom of the conduction band is the width of the forbidden gap, which contains no available states for the electrons to occupy. Excitation of electrons from the valence band through the forbidden gap into the conduction band is mainly responsible for electrical conduction.

11.6.3 Insulators, Semiconductors and Conductors

The distinction among insulators, semiconductors and conductors can be understood for their large variations in band structures and wide range of electrical conductivity.

A material having either completely full or empty energy band is an insulator. The conductivity of an insulator is very small as no free electron is available in the upper empty band for flow of current in this material. Between the full and empty band is a forbidden region whose width is so large that at physically realizable temperature electrons from the top of the highest filled band (valence band) can not excited across the wide forbidden region to the bottom of the lowest empty band (conduction band). The width of the forbidden region is called the band gap energy Eg whose value is of the order of 2.5 V to 6 V or larger for insulators. This situation is depicted in Figure 11.12 (a).

FIGURE 11.12. Energy band diagram

In semiconductor, the band gap energy (Eg) between full and empty band is small fo the order of 1 eV. Electrons from states near the top of the lower full band will have an appreciable probability to be excited termally across the small gap to states near the bottom of upper empty band. Thus a limited number of electrons will be available near the bottom of the upper empty band for conduction of electric current when an electric field is applied. The vacant states left behind by the thermally excited electrons will create positively charged empty states near the top of the lower full band (valence band).

At T > 0K, valence electrons gains thermal energy to top into empty states alternatively filling one empty state and creating a new empty state. The movement of these positive charges are called holes in the valence band will also give rise to an electric current in a semiconductor. The electrical conductivity is much smaller than that of a metal owing to few number of free electrons and holes available of the energy bands. The conductivity is strongly temperature dependent and also a function of Eg. The semiconductor possesses a negative temperature coefficient of resistance, since the resistance decreases sharply with increase in temperature. Further the resistivity can be controlled over a wide range by means of doping i.e., introducing proper impurity in the semiconductor.

The resistivity of semiconductor ranges for 10-2 to 106 cm, that of metal is of the order of 10-6cm while insulators have resistivity greater than 1010 Ω cm.

The distinction between semiconductor and insulator is of degree only as all semiconductors are insulators at T = 0 K. Similarly all insulators are semiconductor at sufficiently high temperature, which is not realizable experimentally.

Figure 11.12 (c) shows the energy band diagram of metals in which a part of the upper region of valence band overlaps into a part of lower region of conduction band. Thus a large number of electrons and holes will be available for conduction. So these materials exhibits high thermal conductivity.

11.6.4 Intrinsic Semiconductor

An intrinsic semiconductor is a pure semiconductor. Silicon and germanium are the semiconductor material. To form a stable covalent bond, 8 electrons are required. The silicon atom at the centre has 4 valence electrons as shown in Figure 11.13 (a). It shares 4 electrons from the neighbour atoms  to form the covalent bond. At absolute zero temperature, no energy is supplied to the crystal. All the electrons are engaged in forming the covalent bond and no free electrons are available. Hence there is no conduction.

Thus the semiconductors acts as insulator at 0 K. the conduction band is empty, as no conduction electrons are available. When thermal energy is supplied to the semiconductor, some of the covalent bonds are broken due to the energy supplied. These electrons jump from valence band (VB) to the conduction band (CB).

FIGURE 11.13 (a) Covalent bond

FIGURE 11.13 (b) Symbol

FIGURE 11.13 (c) At 0K

FIGURE 11.13 (d) At room temperature

Carrier Concentration and Fermi Level

We have assumed here that the width of the allowed energy band is comparable to the forbidden gap of width Eg. Conduction band electrons have energy lying between Ec and ∞, and, valence band electrons have energy lying between – ∞ to EV. This is shown in Figure 11.14.

FIGURE 11.14 Energy band diagram of an intrinsic semiconductor

(a)             Electron Concentration in conduction band: We know the total numbers of electrons per unit volume in conduction band is given by in terms of Fermi-Dirac distribution function f(E) and density of state function g(E) dE as

Here f(E) decreases as we rise through the CB because f(E) = 0 for E >> Eg.

So we have set the upper limit for energy as infinity for CB.

Hence, equation (1) becomes

11.6.5 Extrinsic Semiconductor

A doped semiconductor is called an extrinsic semiconductor. The semiconductors are classified as N-type and P-type.

(a)             Negative type or N-type: Covalent bond and energy band diagram are shown in Figure 11.15. When the intrinsic semiconductor is doped with pentavalent impurity, negative type semiconductor is formed. The pentavalent impurities are antimony, arsenic and bismuth. The pentavalent atom at the centre has 5 valence electrons. This atom shares four electrons from the neighbor atoms. For the formation of stable covalent bond, only 8 electron need to rotate in the valence orbit. Thus one excess electron is produced by each impurity atom.

FIGURE 11.15 (a) Covalent bond

FIGURE 11.15 (b) Energy band diagram

Several impurity atoms donate several electrons. Since the impurity atoms donate electrons, they are known as donors. A few covalent bonds are broken at room temperature due to the thermal energy supplied. The vacancies are shown as holes in the valence band. The majority carriers are electrons and minority carriers are holes.

Fermi level corresponds to the centre of gravity of the electrons and holes. In the case of intrinsic semiconductor, the number of electrons are equal to the number of holes. The Fermi level lies midway between the valence band and conduction band. In N-type semiconductor the Fermi level is lifted towards the conduction band as the conduction electrons are the majority carriers.

For N-type semiconductor a donor level Ed is formed below the Fermi level. At a temperature T, the density of conduction electron from equation (4) is given by

From equation (15), we say that electron concentration is proportional to the square root of the donor concentration of the semiconductor. With more increase EF reaches up to the middle of the conduction band and valence band to make the material intrinsic.

(a)             Positive Type or P-type: When the intrinsic semiconductor is doped with trivalent atoms, positive type semiconductor is formed. The trivalent atoms are indium, gallium, boron and aluminum. The trivalent atom at the centre has 3 valence electrons. This atom shares 4 electrons from the neighbor atoms, 8 electrons are required to form the valence orbit. In other words the trivalent atom can accept one electron. This vacancy is known as hole. The holes have positive charge. Millions of impurity atoms can accept million of electrons. Hence they are called as acceptors. The majority carriers are holes and the minority carriers are electrons. The covalent bond and energy band diagram are shown in Figure 11.16.

FIGURE 11.16 (a) Covalent bond

FIGURE 11.16 (b) Energy band diagram

The electrons jumped from the valence band to the conduction band due to the thermal energy are represented in the conduction band. The valence orbit of each impurity atom has one hole. Thus the holes in the valence orbits of the impurity atoms are represented in the valence band. The Fermi level is shifted down as the majority carriers are the holes in the valence orbits.

For p-type semiconductors an acceptor level Ea is formed above the Fermi level and the valence band. At a temperature T, the density of holes is similar to equation (4) and is given by