Determine the temperature at which there is 1 percent probability that an energy state is empty.

Assume that the Fermi energy level for a particular material is $6.25 \mathrm{eV}$ and that the electrons in this material follow the Fermi-Dirac distribution function. Calculate the temperature at which there is a 1 percent probability that a state $0.30 \mathrm{eV}$ below the Fermi energy level will not contain an electron.

Question

Determine the temperature at which there is 1 percent probability that an energy state is empty.

Assume that the Fermi energy level for a particular material is [latex]6.25 \mathrm{eV}[/latex] and that the electrons in this material follow the Fermi-Dirac distribution function. Calculate the temperature at which there is a 1 percent probability that a state [latex]0.30 \mathrm{eV}[/latex] below the Fermi energy level will not contain an electron.

Accepted Answer

The probability that a state is empty is

[latex] 1-f_{F}(E)=1-\frac{1}{1+\exp \left(\frac{E-E_{F}}{k T}\right)}[/latex]

Then

[latex]0.01=1-\frac{1}{1+\exp \left(\frac{5.95-6.25}{k T}\right)}[/latex]

Solving for [latex]k T[/latex], we find [latex]k T=0.06529 \mathrm{eV}[/latex], so that the temperature is [latex]T=756 \mathrm{~K}[/latex].

Comment

The Fermi probability function is a strong function of temperature.

Question 3.7: Determine the temperature at which there is 1 percent probab...

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