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Question #43846

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The Question

Consider a 1D space charge region (depletion region) in a semiconductor device spanning $0 \le x \le W(t)$ . The electric potential at $x=0$ is maintained at $0$ , and at the moving boundary $x=W(t)$ it is maintained at a constant value $V_0 > 0$ . The electric field at the boundary $x=W(t)$ is always zero. Due to a transient recombination process, the uniform positive space charge density in this region decays exponentially with time as $\rho(t) = \rho_0 e^{-\lambda t}$ , where $\rho_0$ and $\lambda$ are positive constants. Assuming the permittivity of the medium is $\epsilon$ , determine the magnitude of the displacement current density $J_d$ at $x=0$ as a function of time $t$ .

Options

A.
$\frac{\lambda}{2} \sqrt{2\epsilon V_0 \rho_0} e^{-\lambda t / 2}$
B.
$\lambda \sqrt{2\epsilon V_0 \rho_0} e^{-\lambda t}$
C.
$\frac{\lambda}{2} \sqrt{\epsilon V_0 \rho_0} e^{-\lambda t / 2}$
D.
$\lambda \sqrt{\epsilon V_0 \rho_0} e^{-\lambda t}$

Check Your Answer

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

Depletion Region Dynamics|Drift Velocity Mechanism|Radioactive Decay Law Dynamics|Differential Equations

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to couple time-varying electromagnetic fields with semiconductor electrostatics, specifically targeting the misconception of treating a depletion region as a static capacitor.

A common trap (Option B) is to assume the depletion width $W$ remains constant, leading students to directly differentiate $\rho(t) \propto e^{-\lambda t}$ and obtain a displacement current proportional to $e^{-\lambda t}$ .

However, to maintain a constant built-in potential $V_0$ as the charge density decays, the depletion boundary $W(t)$ must dynamically expand (drift).

By forcing the use of Poisson's equation, students must realize that $W(t) \propto \rho(t)^{-1/2}$ , which alters the time dependence of the electric field at the origin to $e^{-\lambda t / 2}$ , fundamentally changing the displacement current derivative.

Quick Facts About This Question

Question Information

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