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Question #42515

Difficulty: 8/10 | 🟢 Free

The Question

Consider an intrinsic semiconductor with a band gap energy $E_g$ . The charge carriers behave as a classical ideal gas, and their mean free path $\lambda$ is independent of temperature. The carrier number density varies with temperature $T$ as $n(T) = C T^{3/2} e^{-E_g / 2k_B T}$ , where $C$ is a constant and $k_B$ is the Boltzmann constant. When a constant uniform electric field $E$ is applied, the current density is $J$ . Assuming the effective mass of the carriers is constant and the drift velocity is governed by the Drude model, what is the ratio of the current density at temperature $T$ to the current density at temperature $2T$ ?

Options

A.
$\frac{1}{2} e^{-\frac{E_g}{4k_B T}}$
B.
$\sqrt{2}$
C.
$\frac{1}{2\sqrt{2}} e^{-\frac{E_g}{4k_B T}}$
D.
$\frac{1}{2} e^{-\frac{3E_g}{4k_B T}}$

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

Mean Free Path|Ohm's Law Validity|Energy Band Gap Temperature Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the integration of the Drude model with semiconductor physics, specifically targeting the misconception that resistivity in all materials follows a linear temperature dependence or that carrier density is constant (as in metals).

Students often fall into the trap of treating the semiconductor like a metal, ignoring the exponential $n(T)$ dependence and only considering the collision time $\tau \propto T^{-1/2}$ , which leads to the incorrect ratio $\sqrt{2}$ .

The correct approach requires synthesizing kinetic theory ( $v_{rms} \propto \sqrt{T}$ ) to find the temperature dependence of mobility, and combining it with the intrinsic carrier concentration to find the true temperature dependence of conductivity.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Kinetic Theory - Class 11 Part 2
Concept	Mean Free Path
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free