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The thermal equilibrium value of the majority carrier n _{n} -bar and the thermal equilibrium value of the minority carrier p _{n} -bar are calculated from the following set of equations:
By re-arranging the above two equations we get the following equation:
In second last equation we see that majority carriers consist two terms: first term is electrons contributed by ionization of the N _{D} dopent atoms/cc and the second term is the electrons contributed through thermal generation of EHP. But since electrons contributed by EHP thermal generation is negligible compared to N _{D} dopent atoms/cc as seen below so without any loss of accuracy we can use the last equation above.
In case of Silicon, doping is in the range from 10 ^{12} Dopent Atoms/cc to 10 ^{18} Atoms/cc.
Hence the thermal equilibrium value of the majority carrier in N-Type Si will range from n _{n} -bar=10 ^{12} electrons ^{} /cc to 10 ^{18} electrons/cc.
The thermal equilibrium value of the minority carier in N-Type will range from p _{n} -bar=10 ^{8} holes ^{} /cc to 10 ^{2} holes/cc. assuming n _{i} =10 ^{10} /cc.
2.2.5.1. The Energy-Band Diagram of N-Type and P-Type Semiconductor.
Left hand side diagram in Figure 2.2.28 is that of N-Type Material. The dopent N _{D} donor atoms/cc introduce filled energy level E _{D} very close to the lower edge (E _{C} ) of the Conduction band.At room temperature the thermal energy is of the order of 25meV whereas 10meV is required to jump to the conduction band hence at Room Temperature all the donor atoms easily get positively ionized contributing their 5 ^{th} valence electrons to the conduction band. There are few and far between thermal generation of EHP also as indicated by EE’.
Right hand side diagram in Figure 2.2.28 is that of P-Type Material. The dopent N _{A} acceptor atoms/cc introduce empty energy level E _{A} very close to the upper edge (E _{V} ) of the Valence band.At room temperature the thermal energy is of the order of 25meV whereas 10meV is required to jump to the empty energy level E _{A} hence at Room Temperature all the acceptor atoms easily get negatively ionized by accepting the valence electron from the neighbouring Si-Atom thereby completing its octave and contributing equal number of vacancies in the Valence Band. These vacanicies wander about the Si-lattice as conducting positive carriers which we call holes. There are few and far between thermal generation of EHP also as indicated by EE’.
Comparing these two diagrams it is clear that holes will wander about the Si-lattice in much more sluggish manner as compared to conducting electrons. This has implications for its mobility.
2.2.5.2. Compensation in Semiconductors.
If both donor N _{d} dopent/cc and acceptor N _{a} dopent/cc are introduced simultaneously then the two compensate. The fifth electron from the donor atom goes to fulfil the octave of the acceptor atom. The Acceptor need not make its pick of the 4 ^{th} electron from the valence band. Thus equal number of donor and acceptor will compensate each other.Whatever is leftover decides the Type of the semiconductor.
If donor N _{d} dopent/cc<acceptor N _{a} dopent/cc then we have P-Type Si with net doping density = N _{a} - N _{d} =N _{A} .
If donor N _{d} dopent/cc = acceptor N _{a} dopent/cc then we have Intrinsic Si with net doping density = N _{a} - N _{d} = 0 .
If donor N _{d} dopent/cc>acceptor N _{a} dopent/cc then we have N-Type Si with net doping density = N _{d} - N _{a} =N _{D} .
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