Distribution of Energy

Abstract

The conducting solids are studied from the point of view of quantum statistics. The concept of Fermi energy is introduced and its physical meaning is explained.

Questions

• 1. By using the expression (3.5) for the occupation density, verify Eq. (3.3) for the case of an ideal gas. Hint: Set E₁ = 0, E₂ = ∞, and use the integral formula

$$ {\int}_0^{\infty }{E}^{1/2}{e}^{-E/ kT} dE=\kern0.34em \frac{1}{2}\kern0.32em {\left(\pi {(kT)}^3\right)}^{1/2} $$

• 2. What is the fundamental difference between the classical Maxwell-Boltzmann theory and Quantum Statistics? In your opinion, which theory is the most general of the two?

• 3. Imagine a bizarre world in which the electrons would be bosons while the photons would be fermions. (a) What would be your grade in a Chemistry class? (b) What would be the cost of a laser pointer? [Hint: (a) Bosons do not obey the Pauli exclusion principle. All atomic electrons would therefore tend to occupy the lowest energy level, that is, the very first subshell. What would then be the structure of an atom? Would there be any chemical reactions? (b) A laser beam is a huge system of identical photons, i.e., photons in (almost) the same quantum state. Would such a beam exist if the photons obeyed the Pauli exclusion principle?]

• 4. Can the occupation density exceed the density of states in a system of fermions? How about a system of bosons?

• 5. What is the physical significance of the Fermi-Dirac distribution function? What is the physical significance of the Fermi energy? Suggest a method for deriving the probability function for holes in a semiconductor. What is the physical significance of that function?

• 6. Derive an expression for the Fermi energy E_F of a metal. What is the physical significance of E_F in this case? How does the situation differ in comparison to the classical theory of ideal gases?

• 7. Justify physically the presence of the Fermi level inside the forbidden band of a pure semiconductor. (Examine the cases T = 0 and T > 0 separately.)

• 8. Consider an n-doped semiconductor crystal. Describe the modification of the Fermi level of the system if (a) we add more donor atoms; (b) we add acceptor atoms; (c) we increase the temperature.

• 9. Consider an n-doped semiconductor crystal. We recall that the donor introduces a new energy level E_D in the forbidden band, very close to the conduction band. At absolute temperature T → 0, the level E_D is occupied by the fifth valence electron of the donor atom (at very low temperatures the donor atoms are not ionized). Show that, in the limit T → 0, the Fermi level E_F of the system passes above E_D. [Hint: Remember the physical significance of E_F for T = 0.]

• 10. The Fermi energy of a metal is known, equal to E_F. The mobility of the electrons in this metal is μ. Show that the resistivity of the metal is equal to

$$ \rho =\frac{3}{2\kern0.2em q\kern0.2em \gamma \kern0.1em \mu}\kern0.34em {E_F}^{-3/2} $$

where q is the absolute value of the charge of the electron and γ is the constant defined in Eq. (3.9). [Hint: Find the conductivity σ = 1/ρ of the metal (cf. Sect. 2.4).]

• 11. Consider two metals M₁ and M₂ . For the electron mobilities and the Fermi energies of these metals we are given that μ₁ = 4 μ₂ and E_F,2 = 4 E_F,1 . The resistivity of M₂ is ρ₂ = 1.5 × 10⁻⁸Ω. m. Find the resistivity ρ₁ of M₁ . [Answer: ρ₁ = 3 × 10⁻⁸Ω. m].

• 12. The electronic density of a metal is known, equal to n. The external conditions are such that, according to the classical theory, the average kinetic energy of the air molecules is very close to zero. Determine the maximum kinetic energy of the free electrons in the metal according to the quantum theory.

• 13. Consider a crystal of an intrinsic semiconductor. In the energy-band diagram the Fermi level lies 0.4 eV above the valence band. Determine the maximum wavelength of radiation absorbed by the crystal. Given: h = 6.63 × 10^–34J. s ; c = 3 × 10⁸m/s ; 1eV = 1.6 × 10⁻¹⁹J . [Answer: λ_max = 15.54 × 10^–7m].