4.12 Sspd_chapter_2.2.6. theoretical formulation of thermal

2.2.6. Theoretical Formulation of thermal equilibrium values of majority and minority carriers in Semi-conductors.

The broad view of Silicon(intrinsic or doped) is that there is thermal generation of EHP(electron-hole pair), EHP are recombining and there is the ionization of the dopent.At any given temperature we have a thermal equilibrium established. At this thermal equilibrium, the concentrations achieved are time invariant and are referred to as thermal equilibrium concentrations.

In intrinsic Silicon: electron thermal equilibrium concentration = hole thermal equilibrium concentration = intrinsic concentration.

The bar indicates thermal equilibrium value.

In N-Type Silicon we have:

In P-Type Silicon we have:

By Law of Mass Action in each case:

2.2.6.1. Majority and Minority Carrrier thermal equilibrium concentrations as a function of Temperature in Kelvin.

In intrinsic Silicon, thermal equilibrium concentrations of electrons and holes are equal and are called intrinsic concentration. In Figure 2.2.26. the Intrinsic Concentration of Silicon as a function of Temperature(K) is given. It should be noted that conducting electron and holes in intrinsic Si is only due to thermal generation of EHP. Hence below 155K(liquid Nitrogen temperature), thermal generation stops and intrinsic Si becomes a perfect insulator.

In Figure 2.2.29. Fermi-Dirac distribution is given at T=155K and at T=300K. At T= 300K, Fermi-Dirac distribution is more skewed hence the tail end of the distribution has a greater distribution as compared to that at 155K.

At 300K, n _i = 4×10 ⁹ and P(E _C ) = 4×10 ^-10 ;

At 155K, n _i = 6.55 and P(E _C ) = 6.55×10 ^-19 ;

For all practical purposes, thermal generation of EHP has altogether stopped and the sample freezes out. It becomes an insulator.

In Extrinsic Silicon, below 155K, there is no thermal generation of EHP as well as there is no ionization of the dopent. Hence there are no charge carriers whatsoever in Si below 155K.

From 155K to 200K, there is thermal generation of EHP but there is no ionization because the thermal lattice vibration does not provide enough energy for ionization. Above 200K the thermal energy is kT=17meV whereas ionization energy is only 10meV hence practically all dopent atoms get ionized and majority and minority carriers remain constant inspite of the fact that EHP thermal generation is exponentially increasing. This increase is felt only above 473K when intrinsic carrier concentration starts dominating the dopent concentration.

In extrinsic Si, dopent concentration is in the range 10 ¹² /cc to 10 ¹⁵ /cc whereas the intrinsic concentration at 473K and above is above 10 ¹³ /cc.

Well above 473K depending on the dopent concentration, the sample becomes intrinsic type.

The thermal equilibrium values of majority and minority carriers in N-Type Silicon with a donor doping density of 10 ¹³ /cc as a function of temperature is given in Figure 2.2.30.