Nanotechnology-Enabled Sensors

6.2 Density and Number of States 287
Fig. 6.2 The time-independent Schrödinger equation solutions for a twodimensional
infinite potential well.
6.2.4 Definition of Density of States
The electrical, mechanical, optical and magnetic properties of materials
depend strongly on how the energies are distributed. To assess these properties,
we need to consider the NOS and DOS.
We define the number of free particles having an energy of E as N(E).
The density of states is then defined as D(E) = dN(E)/dE. DOS is then expressed
as D(E)dE, which is the number of allowed energy levels per unit
volume of the material, within the energy range E to E + dE.
6.2.5 DOS in Three-Dimensional Materials
In three-dimensional k space, the points of vectors of equal magnitude in
all directions form a spherical shell. These vectors represent equal energy
states, proportional to k 2 = kx 2 +ky 2 +kz 2 , given by:
E
2
�
2
2 2 2 �
( kx
+ k y + k z ) =
2
k
= . (6.10)
2m
2m
If we consider the free particles being contained in a cube of length L,
the total volume will be equal to V = L 3 , while the difference between kspace
points will be equal to 2π/L.