Let Δ E designate the energy gap between the valence band and the conduction band. The population of conduction electrons (and of the holes) is roughly proportional to e^ -Δ E/2kT . Determine the approximate ratio of

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Quantrex Academy · Accepted Answer

Correct answer: C. Ratio 7.6 × 10^ -42 , almost zero

The concentration of conduction electrons is given by

n \propto e^{-\Delta E / 2kT}

.

The ratio of the concentration of conduction electrons in diamond to that in silicon is:

\dfrac{n_{\text{diamond}}}{n_{\text{silicon}}} = \dfrac{e^{-\Delta E_{\text{diamond}} / 2kT}}{e^{-\Delta E_{\text{silicon}} / 2kT}} = e^{-(\Delta E_{\text{diamond}} - \Delta E_{\text{silicon}}) / 2kT}

Given

\Delta E_{\text{diamond}} = 6.0 \text{ eV}

and

\Delta E_{\text{silicon}} = 1.1 \text{ eV}

.

\Delta E_{\text{diamond}} - \Delta E_{\text{silicon}} = 6.0 - 1.1 = 4.9 \text{ eV}

At

T = 300 \text{ K}

, the thermal energy

kT

in electron-volts is:

kT = \dfrac{1.38 \times 10^{-23} \times 300}{1.6 \times 10^{-19}} \text{ eV} \approx 0.025875 \text{ eV}

2kT = 2 \times 0.025875 = 0.05175 \text{ eV}

Substituting these values into the ratio expression:

\text{Ratio} = e^{-4.9 / 0.05175} = e^{-94.686}

Converting to base

10

using

\log_{10}(e) \approx 0.4343

:

\text{Ratio} = 10^{-94.686 \times 0.4343} = 10^{-41.12}

\text{Ratio} = 10^{0.88} \times 10^{-42} \approx 7.6 \times 10^{-42}

Since this ratio is extremely small, the number of conduction electrons in one cubic metre of diamond is practically zero.

Let Δ E designate the energy gap between the valence band and the...

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