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Question #42719

Difficulty: 8/10 | 🟢 Free

The Question

An n-type semiconductor bar is positioned along the $x$ -axis. The donor impurity concentration is non-uniform, resulting in a steady-state electron concentration profile $n(x) = n_0 \left(1 + \frac{x}{L}\right)^{-2}$ for $x \ge 0$ , where $n_0$ and $L$ are positive constants. The semiconductor is in thermal equilibrium at an absolute temperature $T$ . Assuming the Einstein relation $D_e = \frac{\mu_e k_B T}{e}$ holds (where $D_e$ is the diffusion coefficient, $\mu_e$ is the electron mobility, $k_B$ is the Boltzmann constant, and $e$ is the elementary charge), what is the built-in potential difference $\Delta V = V(L) - V(0)$ between $x = L$ and $x = 0$ ?

Options

A.
$-\frac{2 k_B T}{e} \ln 2$
B.
$\frac{2 k_B T}{e} \ln 2$
C.
$-\frac{k_B T}{e} \ln 2$
D.
0

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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

Ohm's Law Validity|Drift vs Diffusion Current|Integration (Poisson/Maxwell)

Why This Question Is Tricky

Common Mistakes

Students often incorrectly assume that Ohm's Law ( $J = \sigma E$ ) is the only source of current, leading them to conclude that zero net current in thermal equilibrium implies zero electric field (and thus zero potential difference, Option D).

This ignores the diffusion current driven by the spatial gradient of charge carriers.

Furthermore, students who do account for diffusion often make sign errors by forgetting that electrons carry a negative charge, or they fail to correctly apply the Einstein relation to link mobility and the diffusion coefficient, leading to an inverted potential sign (Option B) or missing the exponent multiplier (Option C).

This question forces the reconciliation of non-ohmic diffusion with the requirement for zero net current, testing the fundamental origin of the built-in potential.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Current Electricity - Class 12
Concept	Ohm's Law Validity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free