Calculate the probability that an energy state above $E_{F}$ is occupied by an electron.

Let $T=300 \mathrm{~K}$ . Determine the probability that an energy level $3 k T$ above the Fermi energy is occupied by an electron.

Question

Calculate the probability that an energy state above [latex]E_{F}[/latex] is occupied by an electron.

Let [latex]T=300 \mathrm{~K}[/latex]. Determine the probability that an energy level [latex]3 k T[/latex] above the Fermi energy is occupied by an electron.

Accepted Answer

From Equation (3.79), we can write

[latex]\begin{array}{c} \frac{N(E)}{g(E)}=f_{F}(E)=\frac{1}{1+\exp \left(\frac{E-E_{F}}{k T} ight)} \ \end{array}[/latex] (3.79)

[latex]\begin{array}{c}f_{F}(E)=\frac{1}{1+\exp \left(\frac{E-E_{F}}{k T} ight)}=\frac{1}{1+\exp \left(\frac{3 k T}{k T} ight)} \end{array}[/latex]

which becomes

[latex] f_{F}(E)=\frac{1}{1+20.09}=0.0474=4.74 \% [/latex]

Comment

At energies above [latex]E_{F}[/latex], the probability of a state being occupied by an electron can become significantly less than unity, or the ratio of electrons to available quantum states can be quite small.

Question 3.6: Calculate the probability that an energy state above E F is ...

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