Class 24: Density of States
The solution to the Schrödinger wave equation showed us that confinement leads to quantization.
The smaller the region within which the electron is confined, the more widely spaced are the
allowed values of as well as the corresponding values of . The larger the region within which
the electron is confined, the more closely spaced are the allowed values of as well as the
corresponding values of . Figure 24.1 below shows an illustrative schematic of the allowed
values of energy for a bound electron and for a nearly free electron.
Figure 24.1: An illustrative schematic showing the allowed values of energy for nearly free
electrons and for bound electrons.
The gap between adjacent allowed energy levels for the nearly free electrons is of the order of
10-10 eV, while that between allowed energy levels for a bound electron, is of the order of several
eV to several tens of eV. Therefore the energy levels of nearly free electrons, drawn to the scale
of the energy levels of bound electrons, appear continuous, even though they are actually
discrete. This appearance of continuity arises because there are 1010 nearly free electron energy
levels occupying the same magnitude in the energy scale that is occupied by two adjacent bound
electron energy levels.
Although, we will discuss semiconductors in detail later, it is of interest to note now that the
bands are considered to be almost continuous sets of allowed energy values, relative to the band
gaps which are several electron volts wide. This is for the same reason as indicated above.
While we have plotted the energy levels as a function of position in several of the plots so far, we
note that
Where is the wave vector and is the quantum number (since it quantizes the allowed values
of energy). These relationships have been obtained in a one dimensional sense. In three
dimensions we have
And
Therefore we can make plots of the system using the x, y, and z components of the vector, or
the components of the quantum number. In the language of Physics, this is referred to as plotting
in space or plotting in quantum number space respectively.
We can create plots describing the system using any of the variables that define the system, since
the other variables are related to it. The shape of the plot may look different based on the
variable chosen, but the information presented will be the same. Based on the information
desired, one or the other set of axis, and hence the corresponding „space‟, will be the more
convenient choice.
Since
( )
When energy is constant,
Therefore all points of constant energy lie on the surface of a sphere in space.
Given that we have now incorporated the detail of confinement and associated quantization, in
the model for the material, it is possible to extract further understanding of the workings of
materials.
While we made a plot of the Fermi-Dirac distribution, we noted that we did not know how many
states were available at a given energy level. The parameter we are looking for, ( ), is called
the density of allowed states, and is the number of states in the energy range and .
If we define ( ) as the density of occupied states,
( ) ( ) ( )
The factor „2‟ accounts for the fact that we can have an electron with spin up or with spin down
in each state.
Stated in words, the above equation means that density of occupied states = the density of
allowed states ( ), times the probability of occupancy of the states ( ).
It is of interest to us to obtain an expression for ( ). To obtain this expression, in the
discussion that follows, we will approach the problem from two different perspectives and obtain
two different expressions which we will be able to relate to each other. Through this relationship
we will get an expression for ( ).
Firstly, we note that all states of equal energy lie on the surface of a sphere in space. Since ,
, and are quantum numbers, they can only be integers with positive values. This implies
that only the positive octant of a sphere in space contains the allowed states. Therefore the total
number of states up to an energy value , which is given by:
Becomes 1/8 of the volume of a sphere in space, divided by 13, since the unit volume in
space is defined by a cube with unit dimensions, i.e.
Therefore the total number of states up to an energy value , is given by:
( )
Since
We have
( )
Therefore the total number of states up to an energy value , is given by:
( ) ( )
Secondly, if ( ), is the density of allowed states, or the number of states in the energy level
and , then another expression for the total number of states up to an energy value , is as
given below:
( ) ∫ ( )
Or,
( )
( )
Since we independently have arrived at an expression for ( ), in our discussion earlier, we can
differentiate the same and arrive at the expression for the density of allowed states ( ).
Therefore:
( ) [ ( ) ]
Simplifying,
( ) ( )
Therefore ( ) is related to through a parabolic relationship.
The plots of ( ), ( ), and hence ( ) ( ) ( ), as a function of energy, are shown in
Figure 24.2 below, as also their variation with temperature.
Figure 24.2: Plots of the density of allowed states ( ), the probability of occupancy ( ), and
the density of occupied states ( ), as a function of energy, and at different temperatures.
In a subsequent class we will look at the calculations associated with estimating , which has
only been addressed descriptively so far. Given our present knowledge of the density of occupied
states, we can calculate , consistent with the Drude-Sommerfeld model.
In the next class we will look more closely at aspects associated with the Fermi energy.
