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 4^th 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 8^th 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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