Physics - Conductivity in Metals - Density of States

Lecture 4

Density of States and Fermi Energy Concepts

Reading: (Cont’d) Notes and Anderson2 sections 2.8-2.13

Georgia Tech

ECE 3080 - Dr. Alan Doolittle

How do electrons and holes populate the bands?
Density of States Concept In lower level courses, we state that “Quantum Mechanics” tells us that the number of available states in a cubic cm per unit of energy, the density of states, is given by:
* * mn 2mn ( E  Ec ) gc (E)  , E  Ec 2 3  

gv (E) 

m * 2 m * ( Ev  E ) p p

 
2

3

, E  Ev

 Number of States    3 cm  unit  
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eV
ECE 3080 - Dr. Alan Doolittle

How do electrons and holes populate the bands?
Density of States Concept

Thus, the number of states per cubic centimeter between energy E’ and E’+dE is g c (E’ )dE if E’  E c and , g v (E’ )dE if E’  E v 0 otherwise
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and ,

But where does it come from?

ECE 3080 - Dr. Alan Doolittle

How do electrons and holes populate the bands?
Where Does the Density of States Concept come from?

Approach: 1. Find the smallest volume of k-space that can hold an electron. This will turn out to be related to the largest volume of real space that can confine the electron. 2. Next assume that the average energy of the free electrons (free to move), the fermi energy Ef, corresponds to a wave number kf. This value of kf, defines a volume in k-space for which all the electrons must be within. 3. We then simply take the ratio of the total volume needed to account for the average energy of the system to the smallest volume able to hold an electron ant that tells us the number of electrons. 4. From this we can differentiate to get the density of states distribution.

Georgia Tech

ECE 3080 - Dr. Alan Doolittle

How do electrons and holes populate the bands?
Derivation of Density of States Concept
First a needed tool: Consider an electron trapped in an energy well with infinite potential barriers. Recall that the reflection