Electrons and Holes

a) Derive a formula for the concentration of electrons in the conduction band of a semiconductor with a fixed chemical potential (Fermi level) μ, assuming that in the conduction band ε – μ >> τ (nondegenerate electrons).

b) What is the relationship between hole and electron concentrations in a semiconductor with arbitrary impurity concentration and band gap E_g?

c) Find the concentration of electrons and holes for an intrinsic semiconductor (no impurities), and calculate the chemical potential if the electron mass is equal to the mass of the hole: m_e = m_h.

SOLUTION

a) Let the zero of energy be the bottom of the conduction band, so μ ≤ 0 (see Figure 1.1). The number of electrons may be found from

(1)

Figure 1.1

where 2 = 2s + 1 for electrons, and the Fermi distribution formula has been approximated by

(2)

The concentration of electrons is then

(3)

where 1/α ≡ 2mτ

b) In an intrinsic semiconductor

(4)

since a hole is defined as the absence of an electron. We may then write

(5)

where ε^* is the energy of a hole and we have used the non-degeneracy condition for holes μ – ε^* >> τ. The number of holes is

(6)

The energy of a hole (from the bottom of the conduction band) is

(7)

Therefore, similar to (a):

(8)

The product of the concentrations of electrons and holes does not depend on the chemical potential μ, as we see by multiplying (3) and (8):

(9)

We did not use the fact that there are no impurities. The only important assumption is that μ – ε >> τ, which implies that the chemical potential μ is not too close to either the conduction or valence bands.

c) Since, in the case of an intrinsic semiconductor n_i = p_i (every electron in the conduction band leaves behind a hole in the valence band), we can write, using (9),

(10)

Therefore,

(11)

Equating (3) and (11), we can find the chemical potential for an intrinsic semiconductor:

(12)

If m_e = m_h, then the chemical potential is in the middle of the band gap: