Q5Electrical Materials

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Q.5. Calculate the intrinsic concentration of charge carriers at 300 K given that:

m_{e}^{*} = 0.12 m_{0}, m_{h}^{*} = 0.28 m_{0}, m_{0} = 9.1 \times 1 0^{- 31} kg, E_{g} = 0.67 eV

Answer 1

The intrinsic carrier concentration of a semiconductor is given by:

n_{i} = 2 (\frac{2 π k T}{h ^{2}})^{3/2} (m_{e}^{*} m_{h}^{*})^{3/4} exp (- \frac{E _{g}}{2 k T})

Step 1 — evaluate the exponential factor: at T = 300 K, kT = 1.38×10⁻²³ × 300 = 4.14×10⁻²¹ J, and Eg = 0.67 eV = 0.67×1.6×10⁻¹⁹ = 1.072×10⁻¹⁹ J:

\frac{E _{g}}{2 k T} = \frac{1.072 \times 1 0 ^{- 19}}{2 \times 4.14 \times 1 0 ^{- 21}} = 12.95 \Rightarrow e^{- 12.95} \approx 2.4 \times 1 0^{- 6}

Step 2 — evaluate the pre-exponential factor: with me = 0.12m₀, mh = 0.28m₀, m₀ = 9.1×10⁻³¹ kg:

(\frac{2 π k T}{h ^{2}})^{3/2} = (\frac{2 π \times 4.14 \times 1 0 ^{- 21}}{( 6.626 \times 1 0 ^{- 34} ) ^{2}})^{3/2} = (5.92 \times 1 0^{46})^{3/2} \approx 1.44 \times 1 0^{70}

(m_{e}^{*} m_{h}^{*})^{3/4} = (0.12 \times 0.28 \times m_{0}^{2})^{3/4} = (2.78 \times 1 0^{- 62})^{3/4} \approx 6.8 \times 1 0^{- 47}

Step 3 — combine:

n_{i} = 2 \times 1.44 \times 1 0^{70} \times 6.8 \times 1 0^{- 47} \times 2.4 \times 1 0^{- 6} \approx 4.7 \times 1 0^{18} m^{- 3}

The intrinsic carrier concentration is therefore approximately 4.7×10¹⁸ m⁻³ (equivalently ≈4.7×10¹² cm⁻³). The 0.67 eV gap corresponds to germanium, and the computed value is of the correct order for Ge at room temperature (the exact textbook figure ≈2.4×10¹⁹ m⁻³ uses slightly different effective masses), confirming the calculation's physical reasonableness with the given reduced effective-mass values.

Sensitivity of the result to the band gap: the exponential term dominates the temperature and material dependence of ni far more strongly than the polynomial prefactor, since ni scales as exp(−Eg/2kT); a change of only 0.1 eV in the assumed gap alters ni by roughly a factor of exp(0.1×1.6×10⁻¹⁹/(2×4.14×10⁻²¹)) ≈ exp(1.93) ≈ 6.9, nearly an order of magnitude. This extreme exponential sensitivity is precisely why silicon (Eg = 1.1 eV, ni ≈ 1.5×10¹⁶ m⁻³ at 300 K) has an intrinsic carrier concentration roughly 3000 times smaller than germanium despite the modest 0.43 eV difference in gap width, and is the fundamental physical reason silicon devices tolerate far higher operating temperatures before intrinsic conduction overwhelms the intentionally doped carrier population — the same reasoning applied qualitatively in the earlier comparison of silicon and germanium as device materials.

Q5Electrical Materials

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