Density of states in 1D, 2D, and 3D - Engineering physics



Density of states in 1D, 2D, and 3D

In 1-dimension

The density of state for 1-D is defined as the number of electronic or quantum states per unit energy range per unit length and is usually denoted by

 ...(1)

Where dN is the number of quantum states present in the energy range between E and E+dE

 ...(2)

Here factor 2 comes because each quantum state contains two electronic states, one for spin up and other for spin down.

Equation (1) can be written as 

 ...(3)
From eq. (2), we have 

 ...(4)
As we know 

                                                                 

So 

                                                                
And 

 ...(5)

By using Eqs. (4)and (5), eq. (3) becomes 

     


 ...(6)


This result is shown plotted in the figure. 

It shows that all the states up to Fermi-level  are filled at 0K.

In 2-Dimensional

The density of state for 2D is defined as the number of electronic or quantum states per unit energy range per unit area and is usually defined as 

 ...(7)

Area (A) 


Area of the 4th part of the circle in K-space  

 ...(8)

Here factor 2 comes because each quantum state contains two electronic states, one for spin up and other for spin down.

Eq. (7) can be written as 
 ...(9)

From eq. (8) we have 
 ...(10)
As we know 

So 
And 
 ...(11)

By using eqns. (10)and (11), eq. (9) becomes 


   

This result is shown in figure. 

It is significant that the 2D density of states does not depend on energy. Immediately as the top of the energy-gap is reached, there is a significant number of available states.

In 3-Dimension

The density of state for 3D is defined as the number of electronic or quantum states per unit energy range per unit volume and is usually defined as  

 ...(12)
Volume 
Volume of the 8th part of the sphere in K-space  

 ...(13)

Here factor 2 comes because each quantum state contains two electronic states, one for spin up and other for spin down.

Eq. (12) can be written as 

 ...(14)
From eq. (13) we have 
 ...(15)
As we know 

So 
And 
 ...(16)

By using Eqs. (15)and (16), eq. (14) becomes 

    


This result is shown in figure. 

The fig. shows that the density of the state is a step function with steps occurring at the energy of each quantized level.


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3 Comments

  1. Hi, I am a year 3 Physics engineering student from Hong Kong. I cannot understand, in the 3D part, why is that only 1/8 of the sphere has to be calculated, instead of the whole sphere.
    Hope someone can explain this to me. Many thanks.

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    Replies
    1. I think this is because in reciprocal space the dimension of reciprocal length is ratio of 1/2Pi and for a volume it should be (1/2Pi)^3. It has written 1/8 th here since it already has somewhere included the contribution of Pi. i hope this helps.

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