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Question #42818

Difficulty: 8/10 | 🟢 Free

The Question

An n-type semiconductor at a constant temperature $T$ has a non-uniform donor impurity profile. Assuming complete ionization, the electron concentration is given by $n(x) = n_0 e^{-x/L}$ for $x \ge 0$ , where $n_0$ and $L$ are positive constants. The electron mobility is dominated by ionized impurity scattering, such that the momentum relaxation time $\tau(x)$ is inversely proportional to the local impurity concentration. At $x = 0$ , the electron diffusion coefficient is $D_0$ . Which of the following expressions gives the magnitude of the electron diffusion current density $J_{\text{diff}}$ at $x = 2L$ ? (Let $e$ be the elementary charge).

Options

A.
$\frac{e D_0 n_0}{L}$
B.
$\frac{e D_0 n_0}{L} e^{-2}$
C.
$\frac{e D_0 n_0}{L} e^{-4}$
D.
$\frac{2 e D_0 n_0}{L} e^{-2}$

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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

Relaxation Time Dependency|Drift vs Diffusion Current|Transistor Biasing Stability

Why This Question Is Tricky

Common Mistakes

Students often memorize the diffusion current equation $J_{\text{diff}} = e D \frac{dn}{dx}$ and treat the diffusion coefficient $D$ as a universal constant for a given material and temperature.

In a graded junction where ionized impurity scattering dominates, the local relaxation time $\tau(x)$ (and thus mobility $\mu(x)$ ) depends strongly on the local doping concentration.

By the Einstein relation ( $D/\mu = k_B T/e$ ), $D(x)$ must also vary spatially.

The trap is calculating the concentration gradient at $x=2L$ and multiplying it by the constant $D_0$ , leading to the incorrect exponentially decaying current (Option B).

The correct approach recognizes that the exponential increase in $D(x)$ exactly cancels the exponential decrease in the concentration gradient, resulting in a spatially uniform diffusion current.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Current Electricity - Class 12
Concept	Relaxation Time Dependency
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free