Q34P Calculate the Fermi energy for n... [FREE SOLUTION]

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Introduction to Quantum Mechanics

David J. Griffiths

Physics

2nd Edition 路 469 pages

ISBN: 9780131118928

Chapter 5: Q34P (page 246)

Calculate the Fermi energy for noninteracting electrons in a two-dimensional infinite square well. Let 蟽 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

$鈭� E_{F} = \frac{蟺 h^{2} 蟽}{m}$

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

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

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鈥�-this time a plane-and each state occupies an area $蟺^{2} / l_{x} l_{y} = 蟺^{2} / A$ ( whereA $鈮� l_{x} l_{y}$ is the area of the well). Two electrons per state means

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

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

where $蟽鈮� N q / A$ is the number of free electrons per unit area.

$E_{F} = \frac{h^{2} k_{F}^{2}}{2 m} = \frac{h^{2}}{2 m} 2 蟺蟽 = \frac{蟺 h^{2} 蟽}{m}$

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

$E_{F} = \frac{蟺 h^{2} 蟽}{m}$

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91影视

Calculate the Fermi energy for noninteracting electrons in a two-dimensional infinite square well. Let 蟽 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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