Electron and Hole Concentrations at Equilibrium

• The F-D distribution function can be used to calculate the electron and hole concentrations in semiconductors, if the densities of available states in the conduction and valence bands are known.
• In equilibrium, concentration of the electrons in conduction band can be given by

        (2.8)

here N(E)dE is density of available states/cm³ in energy range dE.

• Note that the upper limit of theoretically not proper, as the conduction band does not extend to the infinite energies; but, f(E) decreases rapidly with the increasing E, the contribution to this integral for the higher energies is negligible.
• Using the solution of the schrodinger's wave equation under the periodic boundary conditions, it can be shown that

(2.9)

• Thus, N(E) increases with the increase in E, however, f(E) decreases rapidly with E, therefore, the product f(E)N(E) decreases rapidly with E, and a few electrons occupy states far above conduction band edge, that is most electrons occupy the narrow energy band near conduction band edge.
• Likewise, the probability of finding the empty state in valence band [1 - f(E)] decreases rapidly below , and most holes occupy the states near the top of valence band.
• Therefore, a mathematical simplification can be made supposing that all available states in conduction band can be represented by the effective density of states NC located at conduction band edge and making use of Boltzmann approximation.

Thus,   (2.10)

where . 

• Note: as ( E_c -E_F) decreases, that is the Fermi level moves closer to the conduction band, the concentration of electron increases.

• By similar arguments,

              (2.11)

here  N_V is the effective density of states located at the valence band edge E_V.

• Note that the only terms separating the expressions for  N_C and  N_V are effective masses of electrons (   ) and holes (    ) respectively, hence  N_C N_V ,
• Thus, as ( E_F - E_V) decreases, that is the Fermi level moves closer to the valence band edge, and concentration of the hole increases.
• These equations for n₀ and P₀are valid in the equilibrium, irrespective of material being intrinsic or doped.
• For intrinsic material E_F lies at an intrinsic level E_i (very near the middle of the band gap), and the intrinsic electron and hole concentrations are given by

  (2.12)

• Note: At equilibrium, the product is a constant for a particular material and temperature, even though the doping is varied,

That is   (2.13)

• This equation gives the expression for intrinsic carrier concentration ni as the function of N_C, N_V, and temperature cab be given by  (2.14)

• These relations are very significant, and are frequently used for calculations.
• Note - If N_C were to be equal to  N_V, then  E_i would have been exactly at the mid gap (E_c- E_i =  E_i - E_v = E_g/2). 
• However, since  N_C, N_V, E_i is displaced a bit from mid gap (more for GaAs than that for Si).
• Alternate expressions for n₀ and p₀:

                 (2.15)

• Note: the concentration of electron is equal to ni when E_F is at E_i, and n₀ increases exponentially as  E_F moves away from  E_i towards conduction band.
• Likewise, the hole concentration P₀ varies from n_i to larger values as E_F moves from E_i to the valence band.

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