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Question #43843

Difficulty: 8/10 | 🟢 Free

The Question

An n-type semiconductor occupies the region $x \ge 0$ . In thermal equilibrium at temperature $T$ , the electron concentration profile is maintained as $n(x) = n_0 (1 + \alpha x)^2$ , where $n_0$ and $\alpha$ are positive constants. The hole concentration is negligible. Assuming the permittivity of the semiconductor is $\epsilon$ and using the Einstein relation $\frac{D_n}{\mu_n} = \frac{kT}{q}$ (where $k$ is the Boltzmann constant, $q$ is the elementary charge, $D_n$ is the diffusion coefficient, and $\mu_n$ is the mobility), determine the net space charge density $\rho(x)$ required in this region to maintain this profile.

Options

A.
$\frac{2\epsilon k T \alpha^2}{q(1+\alpha x)^2}$
B.
$-\frac{2\epsilon k T \alpha^2}{q(1+\alpha x)^2}$
C.
$-\frac{2\epsilon k T \alpha^2}{q}$
D.
$\frac{2\epsilon k T \alpha^2}{q}$

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

Drift vs Diffusion Current|Conservative Force Field|Potential vs Field Relationship

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental thermodynamic equilibrium relationship between carrier concentration, electric field, and space charge in a semiconductor.

A common lethal mistake is assuming a linear relationship between potential $V(x)$ and carrier concentration $n(x)$ (i.e., incorrectly guessing $V(x) \propto n(x)$ ).

If a student falls into this trap, they will calculate a linear electric field $E(x) \propto (1+\alpha x)$ and subsequently a constant space charge density $\rho(x)$ (Option C).

This fundamentally ignores the Boltzmann distribution $n(x) \propto \exp(qV/kT)$ , which dictates a logarithmic relationship $V(x) \propto \ln(n(x))$ .

By forcing a linear potential assumption, students violate the Poisson equation for the given quadratic concentration profile.

The rigorous approach requires balancing drift and diffusion to find the true non-linear electric field before applying Gauss's law.

Quick Facts About This Question

Question Information

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