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

V

++ +

IONIONION

POSITIONP

OT

EN

TIA

L E

NE

RG

Y

V = 0

We simplify the potential, in order to be able to solve the problem in any simple manner.

Potential core around theatom.

We will eventually letV and d 0 in the problem.

X=0 X=a

X=d

Potential barrier between the atoms.

We now solve the time-independent Schrödinger equation.

ax 0 0 xd

0

2

12

21

2

121

22

dx

d

Edx

d

m

0

2

22

22

2

2222

22

dx

d

EVdx

d

m

22 )(2

EVm

( ) ( ),ikxi ix e u x )()( xudaxu ii

22 2

mE

Bloch Theorem: The eigenfunction of the wave equation for a

periodic potential are the product of a plane wave exp( )

times a function ( ) with the periodicity of the crystal lattice.k

ik r

u r

0)(2 1221

21

2

kdx

duik

dx

ud0)(2 2

22222

2

ukdx

duik

dx

ud

xi eu

0)(2 221

21 kik 0)(2 22

222 kik

iik 1 ik2

xiikxxiikx BeAeu 1

xikxxikx DeCeu 2

)0()0( 21 uu DCBA

0

2

0

1

xx dx

du

dx

du

DikCik

AkiAki

)()(

)()(

)()()( 221 duauau

dikdik

akiaki

DeCe

BeAe

)()(

)()(

dxax dx

du

dx

du

21

dikdik

akiaki

DeikCeik

BekiAeki

)()(

)()(

)()(

)()(

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

0

)()(

)()(

1111

)()()()(

)()()()(

D

C

B

A

eikeikekieki

eeee

ikikkiki

dikdikakiaki

dikdikakiaki

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:

0.det coeff

)](cos[)cos()cosh()sin()sinh(2

22

dakadad

To simplify this, we take the limit V , d 0, in such a manner that Vd = Q.

1)sinh(

, 1)cosh( d

dd

)cos()cos()sin(2

2

kaaad

Function of the energy E Depends only upon theWavevector 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

)cos()sin(

aa

aP

In general, as the energy increases (a increases), each successive band gets wider, and each successive gap gets narrower.

No solutionexists, k2 < 0

Regions where the equation is satisfied, hence wherethe solution exists.

)cos()cos()sin(2

2

kaaad

1

-1

Boundaries are for ka = n.

a

a

1

-1

d

d

2

d

3

d

4

d

d

2

d

3

d

4

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

d

d

2

d

3

d

4

d

d

2

d

3

d

4

Extended zone scheme

d

d

2

d

3

d

4

d

d

2

d

3

d

4

Reduced zone scheme

As energy increases, the bands get WIDER and the gaps get NARROWER

The Electron’s Effective Mass

The energy bands are closer to cosines than to a

free electron parabola.

Hence, we will define an effective mass, which will

vary with energy!

As a result, we must return to our basic

connection for momentum: v*mk

We introduce our effective mass through this defining equation, which relates the crystal momentum to the real momentum.

22Remember in free e Fermi gas,

2

k k

m

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.

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).

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.

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

2. Hence, we have that the exponential part of the wave function must satisfy

There are N, where N is the number of atoms, values of n.

dx

d

dx

Ld

L

)0()(

)0()(

NnL

nk

nkL

ee ikikL

,...,2,12

2

10

3. This means that each band can hold 2N electrons (the factor of 2 is for spin).

4. 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.)

5. 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!

6. An insulator has all the states in the topmost occupied band FULL. Si is therefore an insulator.

7. But, most insulators have a band gap of 4-10 eV. Silicon has a gap of only 1.0 eV, so electrons can be excited over the gap, and we call Si a semiconductor.

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.

9. In a semiconductor, the number of electrons increases exponentially with temperature, so that even though the scattering increases, the conductance increases with T.