Calculate the intrinsic carrier concentration in silicon at $T=250 \mathrm{~K}$ and at $T=400 \mathrm{~K}$ .

The values of $N_{c}$ and $N_{v}$ for silicon at $T=300 \mathrm{~K}$ are $2.8 \times 10^{19} \mathrm{~cm}^{-3}$ and $1.04 \times 10^{19}$ $\mathrm{cm}^{-3}$ , respectively. Both $N_{c}$ and $N_{v}$ vary as $T^{3 / 2}$ . Assume the bandgap energy of silicon is $1.12 \mathrm{eV}$ and does not vary over this temperature range.

Question

Calculate the intrinsic carrier concentration in silicon at [latex]T=250 \mathrm{~K}[/latex] and at [latex]T=400 \mathrm{~K}[/latex].

The values of [latex]N_{c}[/latex] and [latex]N_{v}[/latex] for silicon at [latex]T=300 \mathrm{~K}[/latex] are [latex]2.8 imes 10^{19} \mathrm{~cm}^{-3}[/latex] and [latex]1.04 imes 10^{19}[/latex] [latex]\mathrm{cm}^{-3}[/latex], respectively. Both [latex]N_{c}[/latex] and [latex]N_{v}[/latex] vary as [latex]T^{3 / 2}[/latex]. Assume the bandgap energy of silicon is [latex]1.12 \mathrm{eV}[/latex] and does not vary over this temperature range.

Accepted Answer

Using Equation (4.23), we find, at [latex]T=250 \mathrm{~K}[/latex]

[latex]\begin{array}{c} n_{i}^{2}=N_{c} N_{v} \exp \left[\frac{-\left(E_{c}-E_{v} ight)}{k T} ight]=N_{c} N_{v} \exp \left[\frac{-E_{g}}{k T} ight] \ \end{array}[/latex] (4.23)

[latex]\begin{array}{c}n_{i}^{2} =\left(2.8 imes 10^{19} ight)\left(1.04 imes 10^{19} ight)\left(\frac{250}{300} ight)^{3} \exp \left[\frac{-1.12}{(0.0259)(250 / 300)} ight] \ =4.90 imes 10^{15} \end{array}[/latex]

or

[latex]n_{i}=7.0 imes 10^{7} \mathrm{~cm}^{-3}[/latex]

At [latex]T=400 \mathrm{~K}[/latex], we find

[latex] \begin{aligned} n_{i}^{2} &=\left(2.8 imes 10^{19} ight)\left(1.04 imes 10^{19} ight)\left(\frac{400}{300} ight)^{3} \exp \left[\frac{-1.12}{(0.0259)(400 / 300)} ight] \ &=5.67 imes 10^{24} \end{aligned}[/latex]

or

[latex]n_{i}=2.38 imes 10^{12} \mathrm{~cm}^{-3}[/latex]

Comment

We may note from this example that the intrinsic carrier concentration increased by over 4 orders of magnitude as the temperature increased by [latex]150^{\circ} \mathrm{C}[/latex].

Question 4.3: Calculate the intrinsic carrier concentration in silicon at ...

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