GATE EC 2016 Set 3 — Question 65
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Electronic Devices → Carrier Transport → Junction Formation & Depletion
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Question
Consider the charge profile shown in the figure. The resultant potential distribution is best described by
Correct answer
(D) [figure]
Solution
The relationship between charge density and electric potential is governed by Poisson's equation in one dimension:Also, the electric field is related to the potential by:And from Gauss's Law:Step 1: Analyze the Electric Field
The potential distribution starts flat at , increases with a concave-up shape until , then continues increasing with a concave-down shape until it levels off at . This behavior is correctly represented in option (D).
- For , .
- For , (a negative constant). Thus, . Since and , is negative and decreases linearly from at to its minimum value at .
- For , (a positive constant). Thus, . Since , increases linearly from its minimum at back to at (assuming total charge neutrality).
- Since is negative throughout the region , the slope of the potential is positive. Therefore, is strictly increasing from to .
- Curvature: The second derivative determines the concavity.
- In : , so the curve is concave up (parabolic).
- In : , so the curve is concave down (parabolic).
- Boundary Slopes: At and , , so . The potential curve must be horizontal (zero slope) at these points.
The potential distribution starts flat at , increases with a concave-up shape until , then continues increasing with a concave-down shape until it levels off at . This behavior is correctly represented in option (D).
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