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Question #43449

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The Question

A cylindrical semiconductor rod containing a p-n junction is rigidly clamped between two immovable walls along its length (the $x$ -axis). The lateral surface of the rod is completely free to expand. The rod has a coefficient of linear thermal expansion $\alpha$ , Poisson's ratio $\nu$ , and Young's modulus $Y$ . The temperature of the rod is uniformly increased by $\Delta T$ .

According to Deformation Potential Theory, the bandgap energy $E_g$ of the semiconductor changes with the volumetric strain $\frac{\Delta V}{V}$ as $\Delta E_g = \Xi \frac{\Delta V}{V}$ , where $\Xi$ is the deformation potential constant (in Joules). Assume the built-in potential $V_{bi}$ of the junction changes solely due to this mechanically-induced bandgap shift, given by $\Delta V_{bi} = \frac{\Delta E_g}{e}$ (where $e$ is the elementary charge in Coulombs).

The depletion width of the junction is given by $W = \sqrt{\frac{2\varepsilon V_{bi}}{e N_{eff}}}$ . If the initial built-in potential is $V_{bi0}$ and the initial depletion width is $W_0$ , which of the following expressions correctly represents the fractional change in the square of the depletion width, $\frac{\Delta (W^2)}{W_0^2}$ ?

Options

A.
$\frac{2 \Xi \alpha \Delta T (1+\nu)}{e V_{bi0}}$
B.
$\frac{3 \Xi \alpha \Delta T}{e V_{bi0}}$
C.
$\frac{\Xi \alpha \Delta T (1-2\nu)}{e V_{bi0}}$
D.
$\frac{2 \Xi \alpha \Delta T}{e V_{bi0}}$

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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

Depletion Region Dynamics|Temperature and Kinetic Energy|Thermal Stress in Constrained Rods|Approximations

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of solid-state mechanics and semiconductor physics.

The primary trap is the 'Linearity Assumption' regarding thermal expansion.

Students commonly assume either a free 3D thermal expansion (yielding a volumetric strain of $3\alpha \Delta T$ ) or they incorrectly assume that a 1D constraint simply removes one dimension of expansion (yielding $2\alpha \Delta T$ ).

They fail to account for the Poisson effect: the mechanical compressive stress required to keep the rod constrained in the longitudinal direction induces an additional outward mechanical strain in the transverse directions.

This mechanical-electrical coupling demonstrates how macroscopic boundary conditions directly modulate microscopic electronic properties like the depletion width.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Semiconductor Electronics: Materials, Devices, and Simple Circuits - Class 12
Concept	Depletion Region Dynamics
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free