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Equation 1.11 shows that for a given semiconductor and T the product of np is a constant, free of the concentrations of the dopant. It is a significant relation and is generally shown in the following form:
(1.14)
The equation 1.13 is called law of mass action and it is true for any semiconductor at equilibrium. The intrinsic carrier concentration in a semiconductor is dependant only on the bandgap Eg. It is a material property (at a given temperature).
When in the intrinsic material we represent the Fermi level position Ei one can obtain from equations 1.5 and 1.9:
From equation 1.15 we can deduce easily the position of Ei, which is given by:
(1.16)
The Fermi level Ei is nearby to the midgap; a minor alteration is due to the variance in the effective densities of conduction and valance and band.
Also using Equation 1.15 and reference energy as Ei, we have for N-type materials the alternate equations given as:
(1.17)
And for P-type materials:
(1.18)
Donors and acceptors
The number of holes and electrons in their bands respectively and therefore the conductivity of the semiconductor, may be regulated by adding different dopants or different types of impurities called acceptors and donors. The doped semiconductor, known as extrinsic material.
A semiconductor has four kinds of charged species: holes, electrons, donor ions which are positive Nd, and acceptor ions which are negative Na. Generally, there is no net charge in all samples.
The densities of the negative and positive particles should be equal in order to maintain the charge neutrality:
For n and p we can solve equations 1.19 and 1.13:
The equations 1.20a and 1.20b are seldom used. Alternatively, either of the following cases is mostly valid:
1 Nd – Na ≫ ni (i.e., N-type material),(1.21)(1.22)If additionally, Nd ≫ Na, then(1.23)The Fermi energy can then be written as:(1.24)
2 Na – Nd ≫ ni (i.e., P-type material)(1.25)(1.26)If furthermore, Na ≫ Nd, then(1.27)The Fermi energy can then be written as:(1.28)
B. Nonequilibrium carrier concentration
Many semiconductor devices work by producing charge carriers that exceed the thermal equilibrium values, such excess carriers can be generated either by optical excitation or by adding an electrical field.
Through producing excess carriers (electrons and holes), thermal equilibrium in the semiconductor is no longer maintained, and the Fermi energy is no longer strictly specified. For the nonequilibrium semiconductor, we can outline two novel parameters: the quasi-Fermi energy for electrons and holes respectively.
The electron and hole concentrations of for the nonequilibrium are then given by:
(1.29)
(1.30)
EFn and EFp are the quasi-Fermi levels for electron and hole.
When electrons and holes are at equilibrium, EFn and EFp coincide and this is known as EF.
(II) Basic equations for semiconductor modeling
The so-called equations of semiconductor devices could explain the activity of most of the devices made up of semiconductor materials, including solar cells.
The elementary equations for the operation of semiconductor-based devices describe the static and dynamic carrier behavior in semiconductors under external factors, like optical excitation or applied field, both causing deviations from conditions of thermal equilibrium.
Poisson’s Equation
The electric potential is related to the density of the space-charge by Poisson equation:
(1.31)
where ψ represents the electrostatic potential, the local permittivity is given by ε, and ρ represents the density of the local space charge.
The total contribution from all charges which may be mobile or fixed, including holes, electrons and ionized impurities is expressed as local space charge density.
(1.32)
Where the elementary charge is given by q, n gives the electron concentration, p represents the concentration of holes, Na is the concentration of ionized acceptors and Nd is the concentration of ionized donor.
The electric field is obtained by the gradient of the potential as:
(1.33)
Continuity