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Question #42297

Difficulty: 8/10 | 🟢 Free

The Question

A cubic semiconductor crystal is subjected to a uniform tensile stress $\sigma$ along the $[100]$ crystallographic axis. The elastic stiffness constants of the crystal are given by $C_{11}$ and $C_{12}$ in Voigt notation. According to the deformation potential theory, the shift in the energy band gap is given by $\Delta E_g = a \cdot \text{Tr}(\epsilon)$ , where $a$ is the hydrostatic deformation potential and $\epsilon$ is the strain tensor. Which of the following expressions correctly represents the shift in the energy band gap $\Delta E_g$ ?

Options

A.
$\frac{a \sigma}{C_{11} + 2C_{12}}$
B.
$\frac{a \sigma}{C_{11}}$
C.
$\frac{3 a \sigma}{C_{11} + 2C_{12}}$
D.
$\frac{a \sigma (C_{11} - C_{12})}{C_{11} + C_{12}}$

Check Your Answer

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

Young's Modulus Sensitivity|Energy Band Gap Temperature Dependency|Vector Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition from scalar 1D Hooke's Law to the 3D tensorial nature of elasticity in anisotropic (cubic) crystals.

A common lethal trap is assuming that a uniaxial stress only produces a uniaxial strain ( $\epsilon_{xx} = \sigma/C_{11}$ ), completely ignoring the Poisson effect (transverse strains $\epsilon_{yy}$ and $\epsilon_{zz}$ ) which directly contribute to the volumetric strain $\text{Tr}(\epsilon)$ .

Another trap is confusing uniaxial stress with hydrostatic stress, leading to an incorrect factor of 3.

By linking continuum mechanics (stiffness tensor) to solid-state physics (Deformation Potential Theory), it rigorously filters out rote-learners who memorize $\Delta L/L = F/AY$ without understanding the underlying tensor mechanics.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Solids - Class 11 Part 2
Concept	Young's Modulus Sensitivity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free