The Kronig-Penney model describes electron behavior in periodic solid structures by modeling the potential as a series of barriers and wells. Solving the time-independent Schrodinger equation under these conditions yields allowed energy bands separated by gaps. As the energy increases, the bands get wider and the gaps get narrower. The model also introduces the concept of effective mass to describe how electrons behave as if they have a different mass inside solid materials. Based on how the energy bands are filled, materials can be classified as metals, insulators, or semiconductors.

2. INTRODUCTION
NAME: K V SURYA VINEETH
BRANCH: ECE
ROLL NO: 229519A04P3
SUBJECT: PHYSICS
TOPICS: KORNIG PENNY MODEL

3. The Kronig-Penney Model
* Solving for tunneling through the potentials
between the atoms
* Introducing periodicity into the wave solutions
⇒ Electron bands
⇒ Energy gaps
Effective Mass

4. V
+
+ +
ION
ION
ION
POSITION
POTENTIAL
ENERGY
V = 0
We simplify the potential,
in order to be able to
solve the problem in any
simple manner.
Potential core around the
atom.
We will eventually let
V→∞ and d →0 in the
X=0 X=a
X=−d
Potential barrier
between the atoms.

5. We now solve the time-independent Schrödinger equation.

7. This simple b.c. enforces the
periodicity onto the solution.

8. Since the RHS is 0, there must be an intrinsic solution that
arises without any forcing functions.
This requires the determinant of the large square matrix to vanish:

9. To simplify this, we take the limit V →∞, d →0, in such a manner that Vd = Q.
Function of the energy E Depends only upon the
Wavevector k
The wavevector k is real only for certain allowed ranges of E,
which we illustrate by a graphical solution.
Applying Bloch theorem and solving Schrödinger equation yields

10. In general, as the energy increases (αa increases), each
successive band gets wider, and each successive gap gets
narrower.
No solution
exists, k2 < 0
Regions where the equation is satisfied, hence where
the solution exists.
1
-1
Boundaries are for ka =
nπ.
αa

11. αa
1
-1
The Kronig-Penney model
gives us DETAILED solutions
for the bands, which are
almost, but not, cosinusoidal
in nature.

12. Extended zone scheme Reduced zone scheme
As energy increases, the bands get WIDER
and the gaps get NARROWER

13. The Electron’s Effective Mass
J The energy bands are closer to cosines than to a
free electron parabola.
J Hence, we will define an effective mass, which will
vary with energy!
J As a result, we must return to our basic
connection for momentum:
We introduce our effective mass through
this defining equation, which relates the
crystal momentum to the real momentum.

14. For a wave packet the group velocity is given by:
=
In presence of an electric field E, the energy change is:
Now we can say:
where p is the electron's momentum.

15. Substitute the expression for the group velocity into this last
equation and we get:
From this follows the definition of effective mass:
effective mass is the mass it seems to carry in the semiclassical model of
transport in a crystal. It can be shown that, under most conditions,
electrons and holes in a crystal respond to electric and magnetic fields
almost as if they were free particles in a vacuum, but with a different mass.
This mass is usually stated in units of the ordinary mass of an electron me
(9.11×10-31 kg).

16. The Electron’s Effective Mass
Some values of electron effective mass:
GaAs 0.067
InAs 0.22
InSb 0.13
Si 0.19,0.91*
* Minima are not at center of zone, but are ellipsoids.

17. O.K. We have energy bands and we have gaps. How do we know
whether the material is an insulator, a metal, or a semiconductor?
Well, let us reconsider some of the things we have learned so far:
1. The crystal potential and the wave functions are periodic functions.
If the crystal has length L, then we require
1. Hence, we have that the exponential part of the wave function must
satisfy

18. 3. This means that each band can hold 2N electrons (the factor of 2 is
for spin).
3. Thus, a material with only 1 (outer shell) electron per atom, such as
Li, K, Cu, Au, Ag, etc., will be a metal, since only one-half of the
available states are filled. The highest band (which we will call the
conduction band) is one-half filled. (We assume that, in 3D, the
material has a comparable band structure to the simple cubic.)
3. In Si, however, there are 8 atoms per FCC cell: 8 corner atoms,
shared between 8 cubes, gives 1; 6 face atoms, shared between 2
cubes each, give 3; and 4 internal atoms, which are not shared with
any other cube, gives 4. But, this is considering the basis. The basic
FCC cell has only 4 atoms, and each can contribute 8 states, so that
there are 32 states per unit cell in the band. Now, we have 8 atoms,
each with 4 electrons, and this means 32 electrons. Hence, all the
states in the band are filled, and Si should be an insulator!
3. An insulator has all the states in the topmost occupied band FULL.

19. 8. In a metal, the number of electrons does not change with
temperature. The scattering does increase with temperature, so that
the conductance goes down with T.
8. In a semiconductor, the number of electrons increases exponentially
with temperature, so that even though the scattering increases, the
conductance increases with T.