space, Fermi energy, electron waves)27/Feb/1999
The Density of States
Essay
(Keywords: quantum-mechanical treatment of the free-electron model, space, Fermi energy, electron waves)
Density of states
[1]If we are studying a sample of metal we are not interested in the total number of electrons for instance. We need quantities that are independent of the size of the sample, e.g. the electron density 
(total number of electrons per unit volume).
At 
all electrons in a crystal occupy a state with their energy equal or lower than the Fermi energy 
, i.e. all the states up to the Fermi level are occupied, but the number of electrons in each energy state varies because these states are highly degenerate:
with 
(1)
Here the energy is a function in space. For 
we get a degeneracy of 16 (including spin).
If we know the degeneracy of each state, we automatically know the number of electrons with corresponding energy 
and thus the electron density 
of the electrons with any given energy .
This quantity is not very useful because the distribution of the energy is practically continuous (increments of order 
) and the probability of determining a precise value of 
is nil. Thus we define the very important quantity density of states:
The density of electrons with energy in the range 
around the stated energy 
, per unit range of energy.
The stated density of electrons is divided by the interval to let the density of states be independent of .
Therefore
the density of electrons in the range 
around 
. (2)
(1) gives for a chosen value of E the surface of a sphere of radius 
. The electrons in the range 
will thus form a spherical shell in 
space of thickness 
(spherical shell only for Fermi electron gas). Their number is given by
volume of shell / volume occupied by each state 
(area of sphere radius 
) 
/ 
. (3)
(3) divided by 
(crystal volume) gives the density of the electrons and thus
with 
because of (1). (4)
(1) in (4) and replacing with leads to
. (5)
Therefore we obtain for the density of states
, (6)
which is shown in Fig. 1. At 
all states are occupied below the Fermi energy 
(thick curve). Thermal excitation of the free-electron gas, which is 
, or about 
at normal temperature, will blur the Fermi edge somewhat (thin line).
Fig. 1 Density of states for a free-electron gas
Eigenfunctions of free electrons in a metal (one dimension)
[2]Assumptions:
and length 
because potential 
Solving the time-independent Schrödinger equation
, (7)
we get the eigenfunctions and energies of the free electrons in a metal.
(8)
(9)
With periodic boundary conditions (Born-von Kármán boundary conditions, i.e. one-dimensional box being twisted around back onto itself)
, (10)
in order to use exponential wave functions, we get
(11)
(12)
Eigenfunctions of free electrons in a metal (two dimensions)
[3]Assumptions:
with potential 
(13)
The solutions of Schrödinger’s equation are standing waves of the form
with 
, 
(14)
The wavefunction (14) corresponds to an energy
(15)
with crystal momentum 
.
The allowed values of the wavevector components can be represented as a lattice in space and lie on a square lattice of side 
in the positive quadrant.
(a) (b)
Fig. 2 (a) Standing wave pattern in a two-dimensional box of side L
(b) Diagram in k space of the allowed modes; the mode illustrated in (a) is indicated as point P
Density of states of the two-dimensional electron gas
[4]Assumptions:
This two-dimensional electron gas is like a thin film of thickness 
, with infinite potential barriers at 
and 
. The motion of the electrons is in the 
-plane.
with potential 
at and (16)
(17)
The resulting (unnormalized) wavefunction looks like a travelling wave for motion in the -plane and a standing wave for motion along z.
(18)
To satisfy the boundary conditions:
, 
, 
, 
(19)
The values for 
and 
are the same as the values for the three-dimensional electrons. The 
dependence of the wavefunction corresponds to the stationary states of a one-dimensional infinite square potential well and the allowed values of 
correspond to fitting an integer number of half-wavelengths into the well.
The energy corresponding to the wavefunction of (18) is
. (20)
For 
less than 
Å and 
this exceeds 
at room temperature. This means that all the electrons can be frozen into the 
state at low temperature. Therefore the motion of the electrons in direction is frozen and the electrons behave like free particles in a two-dimensional space.
For states associated with the 
th bound state of the potential well, the motion of the electrons in the -plane is described by the 
and 
values (19), which are components of a 
vector 
in 
space. The states lie on a simple square lattice of side 
and the area of space per is therefore 
. In the area 
of space between circles of radius 
and 
there are
(21)
allowed states.
Fig. 3 The allowed wavevectors for 2D motion of electrons in the xy plane.
The states form a simple square lattice of side 2p /L. To calculate the density of states the number of wavevectors in the shaded circular ring between k and k+dk must be determined.
The energies of the states (20) can be written with 
as
. (22)
To obtain the density of states per unit energy range 
in unit area of the film (
) we use 
. The factor 
is due to the two possible spin states for the electron.
Thus for fixed 
, the density of states of the 
electrons associated with each bound state is independent of energy (using (21) and (22)).
(23)
Adding together the densities associated with all bound states gives the total density shown in Fig. 4 (
). As the film thickness increases the bound state energies become more closely spaced and approache the 
curve.
Density of states for three dimensions
[5]Assumptions:
and volume
(24)
The solutions of Schrödinger’s equation are plane waves
. (25)
To satisfy the boundary conditions (24) the wavevector components are
. (26)
The wavefunction (25) corresponds to an energy
(27)
with crystal momentum .
(27) in (4) leads to the density of states in three dimensions (6).
Dimensionality effects
[6]Fig. 4 shows the functions of the density of states for zero, one, two and three dimensions.
Fig. 4 Density of states for 0D, 1D, 2D and 3D
(28)
(29)
(30)
(31)
From dimensional considerations we obtain very different energy dependent behaviour for the density of states. Therefore there are also related differences for the Fermi energy, Fermi wavevector, and so on. These dimensionality effects play important roles in various areas of science.
: surface science, inversion layers and superlattices
Heat capacity of the free electron gas
[7]The density of states is, for instance, used to obtain the heat capacity 
of the free electron gas.
(32)
One of the failures of the free electron theory was that the heat capacity of the electrons was 
, which did not correspond to experiment. Quantum theory solved this problem and lead to a 
-dependence of the heat capacity.
The total heat capacity at low temperatures is of the form
. (33)
Density of states also applies for the nearly free electron model.
Further information:
Kittel, Introduction to Solid State Physics, John Wiley & Sons
References
[1] Altmann, S. - Band Theory of Solids: An Introduction from the Point of View of Symmetry - Oxford University Press - New York - 1991 - p. 14ff
[2] Burns, G. - Solid State Physics - Academic Press - London - 1985 - p. 203f
[3] Hook, R. - Hall, H. - Solid State Physics - John Wiley & Sons - Chichester - 1991 - 2nd ed. - p. 54f
[4] [3] p. 400ff
[5] [3] p. 77ff
[6] [2] p. 212f
[7] [3] p. 82