**The Kronig-Penney Model **

Kronig and Penney
examined the behavior of electrons in a periodic potential by considering a relatively simple and one-dimensional model. It is assumed that the potential
energy of an electron has the shape of a square well as shown in fig. The period
of potential is (a+b)

This type of periodic
potential is an approximation of the actual picture.

Now Schrodinger wave
eq. for the two regions can be written as,

It is assumed that E the energy of an electron is smaller than

Whereandare two real quantities.

So eq. (1) and
(2) become

Since the potential is periodic, so the solution of eq. (4) and (5) must be of the form of Bloch functions i.e.

Where is the periodic function x with period *(a+b)*

Or

Putting eq. (6) and
(7) in eq. (4) and (5), we get

Where represents the value ofin the intervalandrepresents the value of

The solutions of these equations are

Where A, B, C and D are arbitrary constants which can be determined by following boundary conditions.

Applying these boundary conditions to eqns. (10) and (11), we get

These eqns can be
solved for non-zero value of A, B, C and D only if the determinant of the
coefficient of A, B, C and D becomes zero.

On solving, we get

In order to make the
situation more simple, Kronig and Penney considered the height of the potential
barrier is very large. i.eand simultaneously the width of the barrierin such a way that the productremain finite.

Under these
circumstances, eq. (14) becomes

This eq. can be written more simply as

Where ...(16)Which is a measure of
the areaof the potential barrier. Large value of P
means that given electron is more strongly bound to a particular potential well.

Now R.H.S of eq. (1.72)
can assume values between +1 and -1 and hence only those value ofare allowed which make L.H.S of this eq. lie
between +1

The conclusions exclude
from the above figure are given below.

**1.** The energy spectrum
of the electrons consists of energy bands allowed and forbidden.

**2.** The width of the allowed
energy region or band increases with increasing values of

**3.** In case,
the allowed region becomes infinitely narrow.

and from Eq. (3), we
obtain

for

This is the result we
obtain for a particle in a box of atomic dimensions with a constant potential
i.e. electron tightly bound and tunneling through the barriers becomes
improbable. This shows the case of an insulator.

This represents the energy
of a completely free electron for which any energy value is possible. This
shows the case of the conductor.

**E-K diagram**

Right-hand side ofof eq. (15) is an even periodic function and
its value does not change whether *Ka**K*

However, there are two
other schemes, the reduced zone scheme, and the extended zone scheme. These are shown in figures below-

We find that
discontinuities occur at

These K value define
the boundaries of the first, second and third Brillouin zones.

*For Energy Bands in Solids - Click Here*

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