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These are semiconducting materials which are chemically pure, structurally perfect and stoichiometric in composition if compound. They have properties characteristic of the material alone. The carriers contributing to electrical conduction are generated due to the breaking of covalent bonds by thermal activation process. Pure germanium or silicon are examples of intrinsic semiconductors. These materials have four valence electrons and require four more to complete the subshell. Hence, they have a tendency to form four covalent bonds with neighbouring atoms. This is shown schematically in two dimensions in Fig.5.4.

It may be mentioned that the four covalent bonds are equally inclined to each other in three dimensions and are referred to as tetrahedral bonding. The two dimensional representation is used only for ease of explaining the generation and motion of carriers in the semiconductor lattice.

5.2.1 Carrier generation in intrinsic semiconductors

Carrier generation in an intrinsic semiconductor is due to the breaking of covalent

Figure 5.4 Formation of covalent bonds in silicon and germanium.

bonds. Number of covalent bonds breaking is a strong function of temperature. Break- ing of a covalent bond result in the release of an electron into the conduction band. Consequently, an electron energy state becomes vacant in the valence band. This va- cant site provides an opportunity for the valence electrons to move under the influence of an applied electric field. The movement of electrons in the valence band will thus equivalent to the movement of vacant site in the opposite direction (Fig.5.5). The va- cant sites thus created by the transfer of an electron to the conduction band are called ‘holes’.

E

Figure 5.5 Breaking of covalent bond and generation of electron - hole pairs. The movement of electrons in the valence band is conveniently explained as equiv- alent to the movement of holes. Since the direction of movement of these holes is opposite to that of electrons, holes behave as if they are positively charged. Thus, breaking of each covalent bond results in the formation of an electron-hole pair, and they contribute to the electrical conductivity of the material. Since the electron and the hole are created together in pair, the number of electrons in the conduction band will be equal to the number of holes in the valence band at any given temperature.

5.2.2 Fermi factor and Fermi energy

In order to understand the electrical conductivity of semiconductors, it is necessary to know the distribution of electrons over the available energy states. Electrons in solids obey Fermi-Dirac statistics. Fermi-Dirac distribution function or Fermi factor which gives the probability that an available energy state at E will be occupied by an electron at a temperature T is given by

F(E) = 1

1 + exp {(E − EF)/kT }

(5.1)

The quantity EF is called Fermi energy and it represents a reference level in a semiconductor. Fermi energy may be defined as the energy corresponding to a level whose probability of occupation by an electron is half. The variation of Fermi factor with energy is shown in Fig. 5.6. It is easy to show that

At T = 0, for E < EF, F(E) = 1 At T = 0, for E > EF, F(E) = 0 At T , 0, for E = EF, F(E) = 0.5 E_F E T=0 K T>0 K 1 0.5 0 F(E)

Figure 5.6 Fermi distribution factor as a function of energy.

At absolute zero temperature, a semiconductor behaves like an insulator since all the covalent bonds are intact. The valence band is full and the conduction band is

empty. It can be shown that for an intrinsic semiconductor, the Fermi level lies at the centre of the band gap and remains invariant with temperature. We notice that the Fermi factor has a non-vanishing value even for the energy values corresponding to the forbidden energy gap. However, no carriers are present in the forbidden gap since there are no sites available for occupation by the charge carriers.

5.2.3 Conductivity of an intrinsic semiconductor

When an electric field is applied to an intrinsic semiconductor, there will be movement of electrons in the conduction band and of holes in the valence band. Hence, there will be two currents, an electron current in the conduction band and a hole current in the valence band. As these currents are constituted by oppositely charged carriers and are in opposite directions, the net current will be sum of the two currents.

If ve represents the drift velocity of electrons due to the applied field E, the current density due to electrons is given by

Je = neve (5.2)

where n is the number of conduction electrons available per unit volume and e is the charge associated with these electrons (compare this with equation (3.9) for the case of a metal). Similarly, the current density due to holes is given by

Jh = pevh (5.3)

The total current density is given by

J = (Je+ Jh) = neve+ pevh = ne(ve+ vh) (5.4) since the number of conduction electrons will be equal to the number of holes in an intrinsic semiconductor. Further,it is observed that the drift velocity is directly propor- tional to the applied field E.

i.e. v α E or v =µE (5.5)

where µ is called the mobility of carriers. Hence the above equation becomes

J = neE(µe+µh) (5.6) where µeand µhare the the electron and hole mobilities respectively. The conductivity, which is defined as the current density per applied field, is given by

This equation gives the conductivity of an intrinsic semiconductor. The mobility of holes will be usually less than that of the electrons. This is due to the fact that the movement of holes in the valence band is nothing but the movement of electrons in the opposite direction. Being tightly bound, they move with a smaller drift velocity.

5.2.4 Effect of temperature on conductivity

The conductivity of an intrinsic semiconductor depends strongly on temperature. The higher the temperature, more will be the number of electrons excited to the conduc- tion band. Thermal energy much lower than the forbidden gap energy is sufficient to excite a large number of electrons from valence band to the conduction band. Hence, the electrical conductivity of a semiconductor increases with increase in temperature. This accounts for the negative value of the temperature coefficient of resistivity of semiconductors. The carrier concentration may be expressed as

n = k1T3/2 exp (−Eg/2kT ) (5.8) where Egis the energy gap of the semiconductor and k1is a constant. The mobility of electrons and holes generally depend on temperature as

µ = k2T−3/2 (5.9)

where k2is a constant. It may be mentioned that equation (5.9) is valid for a case where the scattering of charge carriers is predominantly by phonons rather than ionised im- purities. Substituting from equations (5.8) and (5.9) in equation (5.7), the conductivity can be expressed as

σ = k1k2e exp (−Eg/2kT )

σ = K exp (−Eg/2kT ) (5.10)

where K = k1k2e

A plot of lnσ versus (1/T ) will be a straight line (Fig.5.7) with a slope equal to (Eg/2k). Hence, the band gap energy of the semiconductor can be calculated.