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Density of States and Fermi Dirac Distribution in Solid State Physics, Lecture notes of Solid State Physics

Heythrop College, University of London Solid State Physics

This document from the colorado school of mines explores the concept of density of states and its relationship with the fermi dirac distribution in solid state physics. The development of the density of states expression, its proportionality to energy in three dimensions, and the filling of states using statistical thermodynamics and the fermi-dirac distribution.

Typology: Lecture notes

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Uploaded on 09/27/2022

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Colorado School of Mines Solid State Physics in a Nutshell

solidstate.mines.edu

Topic 8-2: Density of States and Fermi Dirac Distribution

Kittel Pages: 136-140

Summary: In this video we develop the density of states for electrons using the Fermi Dirac

distribution. We then discuss how electrons fill states using the density of states expression and

look at the Fermi Dirac distribution as temperature is increased.

• Developed a dispersion relation for a one electron system in previous video, 𝐸=ħ2

2𝑚 �𝑘

�



�2

• Now develop a density of states expression, D(E)

• Recall the density of states is found by 𝐷(𝐸)=𝑑𝑁

𝑑𝐸

o The number of electronic states within some dE is given by dN

o N(E) is the number of states with energy less than or equal to E

• Already have an expression for N(E) of the traveling wave solution from phonons

𝑁(𝐸)=𝑉𝑠𝑝ℎ𝑒𝑟𝑒 ∙1 𝑘 𝑝𝑜𝑖𝑛𝑡

𝑉𝑜𝑙𝑢𝑚𝑒 𝑝𝑒𝑟 𝑘 𝑝𝑜𝑖𝑛𝑡 ∙2 [1]

 2 comes from 2 spin orientations

 𝑉𝑠𝑝ℎ𝑒𝑟𝑒 =4

3𝜋�𝑘

�



�3

 Volume per k point = (2𝜋

𝐿)3

• Need N(E) in terms of energy not k as it is above

• Can get this by solving for k in terms of E from the dispersion

𝑉𝑠𝑝ℎ𝑒𝑟𝑒 =4

3𝜋�𝑘

�



�3 [2]

𝐸=ħ2

2𝑚 �𝑘

�



�2 → �𝑘

�



�= (2𝑚

ħ𝐸)12

� [3]

𝑉=4

3𝜋(2𝑚𝐸

ħ2)32

� [4]

𝑁(𝐸)=4

3𝜋�2𝑚𝐸

ħ2�32

�(1 𝑘 𝑝𝑜𝑖𝑛𝑡

8𝜋3

𝐿3

)(2) [5]

• Taking the derivative:

𝑑𝑁

𝑑𝐸 =𝑉

2𝜋2(2𝑚

ħ)32

�𝐸12

�=𝐷(𝐸) [6]

• Normalizing for volume we get:

Partial preview of the text

Download Density of States and Fermi Dirac Distribution in Solid State Physics and more Lecture notes Solid State Physics in PDF only on Docsity!

solidstate.mines.edu

Topic 8-2: Density of States and Fermi Dirac Distribution

Kittel Pages: 136-

Summary: In this video we develop the density of states for electrons using the Fermi Dirac

distribution. We then discuss how electrons fill states using the density of states expression and

look at the Fermi Dirac distribution as temperature is increased.

Developed a dispersion relation for a one electron system in previous video, 𝐸 =

ħ

2

2𝑚

2

Now develop a density of states expression, D(E)
Recall the density of states is found by 𝐷(𝐸) =

𝑑𝑁

𝑑𝐸

o The number of electronic states within some dE is given by dN

o N(E) is the number of states with energy less than or equal to E

Already have an expression for N(E) of the traveling wave solution from phonons

𝑠𝑝ℎ𝑒𝑟𝑒

1 𝑘 𝑝𝑜𝑖𝑛𝑡

𝑉𝑜𝑙𝑢𝑚𝑒 𝑝𝑒𝑟 𝑘 𝑝𝑜𝑖𝑛𝑡

∙ 2 [1]

 2 comes from 2 spin orientations

𝑠𝑝ℎ𝑒𝑟𝑒

4

3

 Volume per k point = (

2𝜋

𝐿

3

Need N(E) in terms of energy not k as it is above
Can get this by solving for k in terms of E from the dispersion

𝑠𝑝ℎ𝑒𝑟𝑒

4

3

[2]

ħ

2

2𝑚

2

2𝑚

ħ

1

2

�

[3]

4

3

2𝑚𝐸

ħ

2

3

2

�

[4]

4

3

2𝑚𝐸

ħ

2

3

2

�

1 𝑘 𝑝𝑜𝑖𝑛𝑡

8𝜋

3

𝐿

3

)(2) [5]

Taking the derivative:

𝑑𝑁

𝑑𝐸

𝑉

2𝜋

2

2𝑚

ħ

3

2

�

1

2

�

= 𝐷(𝐸) [6]

Normalizing for volume we get:

solidstate.mines.edu

𝐷(𝐸)

𝑉

𝐸

1

2

�

2𝜋

2

2𝑚

ħ

3

2

�

[7]

o The right side of the expression is independent of sample volume

o

We see that D(E) is proportional to E

1/

in 3D; this is a good relation to keep in

mind.

Now use statistical thermodynamics to see how these states are filled as a function of

temperature

Imagine we have a Fermi gas at 0 K

o System will have N electrons

∞

0

[8]

o 𝑓(𝐸) is the occupation probability

Instead of using the Planck distribution which is for bosons we will use the Fermi-Dirac

distribution since electrons are fermions

1

𝑒

(𝐸−𝜇)/𝑘

𝐵

𝑇

[9]

Since we are dealing with fermions we can imagine our density of states as a massive

bucket

When we pour in electrons they will start by filling in the origin and will then move out

o No two electrons can occupy the same state

Since 𝐷 ∝ 𝐸

1

2

�

1

2

�

so we can graph as below

How high in energy the electrons fill this bucket is material dependent

o Depends on the number of electrons per volume

Define this maximum energy as the Fermi energy, E f
At 0 K 𝑓(𝐸) = 1 𝑖𝑓 𝐸 ≤ 𝐸

𝑓

solidstate.mines.edu

Questions to Ponder

Compare and contrast phonons in a harmonic lattice with the free electron model.

Density of States and Fermi Dirac Distribution in Solid State Physics, Lecture notes of Solid State Physics

Related documents

Partial preview of the text

Download Density of States and Fermi Dirac Distribution in Solid State Physics and more Lecture notes Solid State Physics in PDF only on Docsity!

∙ 2 [1]

[2]

[3]

[4]

)(2) [5]

= 𝐷(𝐸) [6]

[7]

[8]

[9]