Chapter 5: Q34P (page 246)

Calculate the Fermi energy for noninteracting electrons in a two-dimensional infinite square well. Let σ be the number of free electrons per unit area.

Short Answer

Expert verified

The Fermi energy for electrons in a two-dimensional infinite square well is

$∴ E_{F} = \frac{π h^{2} σ}{m}$

Step by step solution

Definition of Fermi energy of electron

The greatest energy that an electron may hold at 0K is known as the Fermi energy.

Equation 5.50

$E_{n_{x} n_{y}} = \frac{π^{2} h^{2}}{2 m} (\frac{n_{x}^{2}}{l_{x}^{2}} + \frac{n_{y}^{2}}{l_{y}^{2}}) = \frac{h^{2} k^{2}}{2 m}, with k = (\frac{π n_{x}}{l_{x}}, \frac{π n_{y}}{l_{y}})$

Calculating the Fermi energy for electrons in a two-dimensional infinite square well

Each state is represented by an intersection on a grid in k-space”-this time a plane-and each state occupies an area $π^{2} / l_{x} l_{y} = π^{2} / A$ ( whereA $\equiv l_{x} l_{y}$ is the area of the well). Two electrons per state means

$E_{n_{x} n_{y} n_{z}} = \frac{h^{2}}{2 m} (\frac{n_{x}^{2}}{l_{x}^{2}} + \frac{n_{y}^{2}}{l_{y}^{2}} + \frac{n_{z}^{2}}{l_{z}^{2}}) = \frac{h^{2} k^{2}}{2 m}$ …(5.50).

$\frac{1}{4} π k^{2} = \frac{N q}{2} (\frac{π^{2}}{A}), o r k_{F} = {(2 π \frac{N q}{A})}^{1 / 2} = {(2 π σ)}^{1 / 2}$

where $σ \equiv N q / A$ is the number of free electrons per unit area.

$E_{F} = \frac{h^{2} k_{F}^{2}}{2 m} = \frac{h^{2}}{2 m} 2 π σ = \frac{π h^{2} σ}{m}$

Thus the Fermi energy for electrons in a two-dimensional infinite square well is

$E_{F} = \frac{π h^{2} σ}{m}$

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Calculate the Fermi energy for noninteracting electrons in a two-dimensional infinite square well. Let σ be the number of free electrons per unit area.

Short Answer

Step by step solution

Definition of Fermi energy of electron

Calculating the Fermi energy for electrons in a two-dimensional infinite square well

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