GATE 2023 EC Q.61: solution and answer

GATE 2023 · ECQ.61 · 2 marks · NAT

In a semiconductor device, the Fermi-energy level is 0.35 eV above the valence band energy. The effective density of states in the valence band at $T = 300 K$ is $1 \times 1 0^{19} cm^{- 3}$ . The thermal equilibrium hole concentration in silicon at 400 K is _______ $\times 1 0^{13} cm^{- 3}$ (rounded off to two decimal places). Given kT at 300 K is 0.026 eV.

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Answer and solution

ANSWER

The thermal equilibrium hole concentration in silicon at 400 K is 63.49 × 10¹³ cm⁻³.

Official answer: 60.00 to 70.00Source: Official answer key, GATE 2023 EC (IIT Kanpur)

Given: Fermi level above valence band edge, $E_{F} - E_{V} = 0.35 eV$ (taken as unchanged with temperature, since no doping concentration is given to compute its shift). Effective density of states in the valence band, $N_{V} (300 K) = 1 \times 1 0^{19} cm^{- 3}$ . Thermal voltage, $k T ∣_{300 K} = 0.026 eV$ .
Hole concentration: $p = N_{V} e^{- (E_{F} - E_{V}) / k T}$ .
$N_{V}$ scales with temperature as $T^{3/2}$ : $N_{V} (400 K) = N_{V} (300 K) (\frac{400}{300})^{3/2} = 1 \times 1 0^{19} \times 1.5396 = 1.5396 \times 1 0^{19} cm^{- 3}$
$k T$ scales linearly with temperature: $k T ∣_{400 K} = 0.026 \times \frac{400}{300} = 0.034667 eV$
Compute the exponent: $\frac{E _{F} - E _{V}}{k T ∣ _{400 K}} = \frac{0.35}{0.034667} = 10.096$
Compute $p$ : $p = 1.5396 \times 1 0^{19} \times e^{- 10.096} = 1.5396 \times 1 0^{19} \times 4.124 \times 1 0^{- 5}$
$p \approx 6.349 \times 1 0^{14} cm^{- 3} = 63.49 \times 1 0^{13} cm^{- 3}$

KEY IDEA

Hole concentration is $p = N_{V} e^{- (E_{F} - E_{V}) / k T}$ . Both $N_{V}$ (through the $T^{3/2}$ density-of-states scaling) and $k T$ change with temperature, while the doping-fixed energy gap $(E_{F} - E_{V})$ is taken as unchanged between 300 K and 400 K.

WATCH OUT FOR

Scaling only $N_{V}$ with temperature but forgetting to also scale $k T$ (i.e. plugging $400 K$ carrier density with the $300 K$ value of $k T = 0.026 eV$ in the exponential), which changes the exponent from about $10.10$ to about $13.46$ and gives an answer far outside the accepted range.

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GATE 2023 EC Q.61: solution and answer

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