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MCQEasyJEE 2023Semiconductors Basics

JEE Physics 2023 Question with Solution

The resistivity (ρ\rho) of a semiconductor varies with temperature. Which of the following curves represents the correct behavior?

  • A
    Graph of resistivity $$\rho$$ versus temperature $$T$$ showing a straight line increasing upward from left to right after a small positive temperature intercept.
  • B
    Graph of resistivity $$\rho$$ versus temperature $$T$$ showing a curved increase upward from left to right, becoming steeper at higher temperature.
  • C
    Graph of resistivity $$\rho$$ versus temperature $$T$$ showing a decreasing curve that falls rapidly first and then gradually flattens as temperature increases.
  • D
    Graph of resistivity $$\rho$$ versus temperature $$T$$ showing a straight line decreasing downward from left to right toward zero as temperature increases.

Answer

Correct answer:C

Step-by-step solution

Standard Method

Given: The variation of resistivity ρ\rho of a semiconductor with temperature TT is asked through graphs.

Find: Which graph correctly represents how ρ\rho changes with increasing TT.

For a semiconductor,

ρ=mne2τ\rho = \frac{m}{n e^2 \tau}

As temperature increases, the relaxation time τ\tau decreases, but the carrier concentration nn increases significantly. The effect of increase in nn dominates over the decrease in τ\tau.

Therefore, the resistivity ρ\rho decreases as temperature increases.

So, the correct graph is the one in which ρ\rho decreases with increasing TT. The correct option is C.

Common mistakes

  • Confusing a semiconductor with a metallic conductor. In metals, resistivity usually increases with temperature, but in semiconductors the carrier concentration increases strongly with temperature. Use semiconductor carrier behavior, not metallic behavior.

  • Focusing only on the decrease of τ\tau with temperature. Although τ\tau decreases, the increase in carrier concentration nn is the dominant effect. Consider both factors in ρ=mne2τ\rho = \frac{m}{n e^2 \tau}.

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