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Question #42569

Difficulty: 8/10 | 🟢 Free

The Question

Consider an intrinsic semiconductor at temperature $T$ with a temperature-dependent energy band gap $E_g(T) = E_0 - \alpha T$ , where $E_0$ and $\alpha$ are positive constants. The intrinsic carrier density is given by $n_i(T) = A T^{3/2} \exp\left(-\frac{E_g(T)}{2k_B T}\right)$ , where $A$ is a temperature-independent constant and $k_B$ is the Boltzmann constant. Treating the thermally excited electrons and holes as a non-degenerate ideal gas, the Helmholtz free energy per unit volume of this electron-hole gas is $f = -2 n_i k_B T$ . Which of the following represents the electronic entropy per unit volume ( $S_e$ ) and the internal energy per unit volume ( $u_e$ ) of the electron-hole gas?

Options

A.
$S_e = n_i \left( 5 k_B + \frac{E_0}{T} \right), \quad u_e = n_i (E_0 + 3 k_B T)$
B.
$S_e = n_i \left( 5 k_B + \frac{E_g(T)}{T} \right), \quad u_e = n_i (E_g(T) + 3 k_B T)$
C.
$S_e = n_i \left( 3 k_B + \frac{E_0}{T} \right), \quad u_e = n_i \left(E_0 + \frac{3}{2} k_B T\right)$
D.
$S_e = n_i \left( 3 k_B + \frac{E_g(T)}{T} \right), \quad u_e = n_i \left(E_g(T) + \frac{3}{2} k_B T\right)$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Entropy and Reversibility|Equipartition Theorem|Energy Band Gap Temperature Dependency

Why This Question Is Tricky

Common Mistakes

This question targets a profound misconception in semiconductor thermodynamics: conflating the temperature-dependent band gap $E_g(T)$ with the internal energy of electron-hole pair formation.

Students often naively assume the internal energy is $u_e = n_i(E_g(T) + 3k_B T)$ by applying the equipartition theorem and adding the instantaneous gap energy.

However, $E_g(T)$ actually represents the change in Helmholtz free energy ( $\Delta F$ ) required to create a pair.

By forcing students to derive entropy from the free energy $f = -2n_i k_B T$ , the math reveals that the $\alpha T$ term in $E_g(T)$ cancels out during differentiation.

The true internal energy of formation is $E_0$ , demonstrating that the temperature dependence of the band gap is purely an entropic effect driven by electron-phonon interactions and lattice expansion.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	Entropy and Reversibility
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free