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Question #43836

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The Question

The energy band gap $E_g$ of a semiconductor decreases with absolute temperature $T$ due to non-linear thermal expansion of the lattice. This dependence is modeled as $E_g(T) = E_0(1 - \beta T^2)$ , where $E_0$ is the band gap at $0\text{ K}$ and $\beta$ is a positive constant with units of $\text{K}^{-2}$ . A spherical sample of this semiconductor is heated. At a specific temperature $T_0$ , the peak wavelength of its thermal blackbody emission spectrum exactly matches the threshold wavelength required to excite an electron across its band gap. Assuming Wien's displacement constant is $b$ , Planck's constant is $h$ , and the speed of light in vacuum is $c$ , which of the following expressions correctly gives the temperature $T_0$ ?

Options

A.
$\frac{\sqrt{h^2 c^2 + 4 b^2 E_0^2 \beta} - hc}{2 b E_0 \beta}$
B.
$\frac{b E_0}{hc}$
C.
$\frac{b E_0}{hc + b E_0 \sqrt{\beta}}$
D.
$\frac{\sqrt{h^2 c^2 + 4 b^2 E_0^2 \beta} + hc}{2 b E_0 \beta}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Energy Band Gap Temperature Dependency|Wien’s Displacement Law|Thermal Properties of Matter

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to couple quantum solid-state phenomena (temperature-dependent band gap) with classical thermodynamics (Wien's Displacement Law).

The primary trap lies in the assumption of a constant band gap, which leads to the incorrect simple inverse relation (Option B).

A secondary trap is assuming a standard linear thermal expansion model $E_g = E_0(1 - \sqrt{\beta} T)$ to maintain dimensional consistency instead of the given non-linear model, leading to another incorrect algebraic form (Option C).

The student must correctly set up the non-linear dependency, form the quadratic equation, and carefully extract the positive root, avoiding sign errors in the quadratic formula (Option D).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Semiconductor Electronics: Materials, Devices, and Simple Circuits - Class 12
Concept	Energy Band Gap Temperature Dependency
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free