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Question #43717

Difficulty: 8/10 | 🟢 Free

The Question

Consider a non-degenerate n-type semiconductor at thermal equilibrium at an absolute temperature $T$ . The donor impurity concentration varies spatially along the x-axis ( $x > 0$ ) as $N_D(x) = N_0 e^{-\alpha x^2}$ , where $N_0$ and $\alpha$ are positive constants. Assume complete ionization such that the electron concentration $n(x) \approx N_D(x)$ , and $N_D(x) \gg n_i$ everywhere, where $n_i$ is the intrinsic carrier concentration. Let $k_B$ be the Boltzmann constant and $q$ be the elementary charge. Which of the following expressions correctly represents the built-in electric field $\vec{E}(x)$ and the minority hole concentration $p(x)$ as a function of position?

Options

A.
$\vec{E}(x) = \frac{2\alpha k_B T}{q} x \hat{i}, \quad p(x) = \frac{n_i^2}{N_0} e^{\alpha x^2}$
B.
$\vec{E}(x) = -\frac{2\alpha k_B T}{q} x \hat{i}, \quad p(x) = \frac{n_i^2}{N_0} e^{\alpha x^2}$
C.
$\vec{E}(x) = \frac{k_B T}{q} \alpha x \hat{i}, \quad p(x) = \frac{n_i^2}{N_0} e^{-\alpha x^2}$
D.
$\vec{E}(x) = -\frac{2\alpha k_B T N_0}{q} x e^{-\alpha x^2} \hat{i}, \quad p(x) = \frac{n_i^2}{N_0} e^{\alpha x^2}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Law of Mass Action|Fermi Level Position|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to link a spatially varying doping profile to the induced built-in electric field and minority carrier distribution.

A common trap is assuming that the electric field is directly proportional to the concentration gradient $dn/dx$ (leading to Option D) rather than the gradient of the logarithm of the concentration $d(\ln n)/dx$ .

Furthermore, students often misapply the sign convention for the electric field, forgetting that electrons diffuse down the concentration gradient (towards $+x$ ), so the built-in field must point in the $+x$ direction to provide an opposing drift current for equilibrium.

Finally, the Law of Mass Action ( $np = n_i^2$ ) must be applied locally at every position $x$ , which counter-intuitively means the minority carrier concentration grows exponentially where the majority carrier concentration decays.

Quick Facts About This Question

Question Information

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