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3.
Periodic Potential in a Crystalline Environment
The potential energy of electrons in a crystal is a result of the positively charged
atomic cores producing a columbic attraction. All the electron-electron interactions are
neglected at first by the reasoning that these secondary interactions average out to be
zero. The potential energy for the electron can be viewed as Figure 3.1 in Pierret. The
electronic consequences of such a potential shown in the above figure will be analyzed
starting with the very useful and illuminating Bloch’s Theorem.
3.1 Bloch’s Theorem
The solutions of the Schrodinger equation for a periodic potential must be of the form:

 k  uk r e ik r





 
where u k r   u k r  T is an amplitude function of the planewave e ik r and T is a
translation vector of the crystal. The eigenfunctions of the wave equation for a periodic


potential are the product of a planewave e ik r times a function u k r  with the periodicity
of the lattice. Wavefunctions of this form are called Bloch’s functions and they are very
useful in calculations because they allow us to concentrate on only one period of the
lattice to solve for the wavefunction of the electrons throughout the entire crystal.
Pierret has four general statements:
1.
It can be shown that, for a 1D system, two and only two distinct values of k exist
for each and every allowed value of energy E.

2.
For a given E, values of k differing by a reciprocal lattice vector K give rise to
one and the same wavefunction solution. Therefore, a complete set of distinct k
values will always be obtained if the allowed k-values are limited to 0 to K or


equivalently  K 2 to K 2 . This leads to the concept of Brillouin Zones.
3.
For an infinite crystal, k can assume a continuum of real values in the range
specified in statement 2.
4.
For a finite crystal, we adopt periodic boundary conditions. This means that we
construct a “ring” of N atoms such that:
 k x    k x  Na
 k x    k x  Na   e ikNa k x 
Hence, e ikNa  1 and therefore:
k
2m
Na
m  1, 2, 3, ...
It is extremely difficult to solve Schrödinger’s equation for the potential shown in
Pierret Figure 3.1. In introductory courses in semiconductors, the Kronig-Penny model is
used to solve a simple periodic potential as illustrated in Fig. 3.2. This model is very
useful and it describes many important aspects of energy levels in a real crystal. The
extremely important concept of energy bands and energy band gaps are developed when
analyzing the Kronig-Penny model. This will be done for a homework problem and it
will be seen that there is not a continuum of allowed states. In fact, it is shown that there
are bands of states with varying energy and crystal momentum and also energy ranges
where no states are allowed to exist. These gaps in states for these energy ranges are
called energy bandgaps.
The resulting energy bands for electron states are shown in the figure below:
3.2 Effective Mass and Particle Motion
The term m* is called the effective mass and is inversely proportional to the curvature
of the E-k curves and given by the equation:
m* 
1
1 2E
 2 k 2
Note that m* can be both positive or negative indicating that for some situations,
when an external force is applied to the system, electrons can move in a direction
opposite of the direction predicted by classical physics. Near the top or bottom of any
band, the E-k curve is, to a good approximation, parabolic in nature allowing us to write:
E  E edge 
const.
k k edge 2
2
and hence it is seen that:
const .  m *
It is seen that m* is positive near the bottoms of all bands and negative near the tops of
all bands.
One additional property of each energy band is that there are N distinct k-states in
each band for a finite crystal of N atoms. If the atoms contribute 2 electrons per atom to
these states, then the first two energy bands are totally filled with higher energy bands
vacant. Two important facts about energy bands are:
1. Totally empty bands will not contribute to conduction.
2. Totally filled bands will not contribute to conduction.
Energy bands are either mostly filled (as in the case of the valence band defined later),
or mostly empty (as in the case of the conduction band defined later). To calculate the
current contribution from electrons in the nearly empty energy bands, the following
summation needs to be evaluated:
q
I 3    vi
L filled
For the nearly filled band:
I2  
but since
v
i
all _ states
q
L
v
i
filled
 0 we have:
I3  
q
L
v
filled
i

q
q
vi   vi

L all _ states L empty
where vi is the velocity of the empty state.
Hence, the concept of the charge carrier denoted as a “hole” is born that is:
1. Positively charged vacancy near the top of a band
2. When you do the procedure –e e you have to let m* -m* and you have
completed the conversion from the negative mass and negatively charged electron
state to a postive mass and posively charged hole.
3.3 Brillouin Zones and E-k diagrams

When working with real crystals which are 3 dimensional, the crystal momentum k is

now a 3D vector k . Just as in the 1D case where we can describe all distinct states using
2


a the range 0  k 
or equivalently  k  , we can restrict k-space (i.e.,
d
d
d
reciprocal space) in 3D crystals to the Wigner-Seitz primative unit cell. This WignerSeitz primative unit cell is called the the First Brillouin Zone.
To try to plot out the energy for each k-value in the first Brillouin zone is extremely
difficult, so generally the energies of the states for important directions in k-space are
plotted. These important directions are from   L and   X .
The valence band is defined as the top-most band that is filled with electrons except
for a few vacancies or holes. Ev is the maximum energy of this valence band. Ev always
occurs at  . The conduction band is defined as the bottom-most band that is empty
except for a few thermally excited or otherwise present electrons. Ec is the minimum
energy of this band which does not have to occur at  . The bandgap is defined as
Egap=Ec-Ev and is one of the most important electronic properties of a semiconductor that
effects almost every aspect of its electrical behavior as will be seen throughout the
remainder of this course. If Ec occurs at  , then the bandgap is called a direct bandgap.
If Ec does not occur at  , then the bandgap is called an indirect bandgap.
One additional fact is that an applied electric field  causes a bending in the valence
and conduction band and hence the intrinsic Fermi level according to the following
equation:

1 dE c 1 dEV 1 dEi


q dx
q dx
q dx