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Solid State TheorySolid State TheoryPhysics 545Physics 545

Kronni -Penne Model

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Periodic Potential ApproachPeriodic Potential Approach

can give rise to energy bands. But to calculate the

can be a difficult task. Here we will consider a

, ,

represent the actual potential in a real crystal. This

- .

realistic, it allows us to calculate exactly the energy

theorem.

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Weak vs. Stron Cr stal Potentials

Previously we used the NEARLY-FREE ELECTRON model to discuss the propagation ofelectrons in the presence of a PERIODIC crystal potential

* In this model we assume that the onl effect of the cr stal otential is to DIFFRACT

electrons whenever the Bragg condition is satisfied

* The effect of this diffraction was found to be to open forbidden GAPS in the energys ectrum of the electrons The SIZE of these gaps was also shown to be proportional to the STRENGTH of

the crystal potential

pen ng o an energy gap n e near y- ree e ec ronmodel

wavelength is commensurate with the inter-atomic

spacing (l = 2a in the case shown here)

ENERGY GAP

ENERGY GAP

in this model the gap is understood to result frombragg diffraction of electrons by the crystal

structurekk

/a/a

/a/a

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Energy Levels and BandsIsolated atoms have precise allowed energy levels.

In the presence of the periodic lattice potential bandsof allowedstates are separated by energy gaps for which there are no allowed

energy states.e a owe s a es n con uc ors can e cons ruc e rom

combinations of free electron states (the nearly free electron model)or from linear combinations of the states of the isolated atoms (the

tight binding model).

E

+ + + + +position

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Bound States in atoms

Electrons in isolated00

allowed energy levels

E0, E1, E2 etc. .

-1 V(r)E2Increasin

The potential energy of

-2 E1BindingEnergy

an electron a distance rfrom a positively chargenucleus of char e is

-3

r4

qe=)r(V

o

2

-4

-8 -6 -4 -2 0 2 4 6 8

-5

rr

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Bound and free states in solids

000

The 1D potential energyof an electron due to anarray of nuclei of charge

0

-1-1-1

V(r)E2

q separated by a distance

a is 2e

-2-2-2 E1n naro4

=r

-3-3-30

Where n = 0, +/-1, +/-2 etc.This is shown as the

rSolid

-4-4-4.

V(r) lower in solid (work

-8 -6 -4 -2 0 2 4 6 8

-5

r

-8 -6 -4 -2 0 2 4 6 8

-5

r

-8 -6 -4 -2 0 2 4 6 8

-5

rr0

+ + + + +

Naive picture: lowestbinding energy states can

aNuclear positionsbecome free to movethroughout crystal

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Influence of the lattice eriodicit

In the free electron model, the allowed energy states are2

where for periodic boundary conditions

)(2

zyx kkkm

E ++=

L

nk

L

nk

L

nk zz

y

yx

x

;; === E

k0

0

-2

-1

Exact form of potential is complicated

-4

-3

R = m1a + m2b + m3c

-5

r

w ere m1, m2, m3 are n egers an a , ,c

are the primitive lattice vectors.

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The formation of forbidden gaps is NOT unique to the nearly-free electron problem

however

* We demonstrate this here by considering the formation of energy gaps in the so-calledKRONIG-PENNEY MODEL

* Unlike the nearly-free electron model in this problem we consider the case where thee ectrons move in a STRONG ut PERIODICALLY-VARYING crysta potentia

The model potential considered in the kronig-enne model

the periodic potential of the crystal is crudelymodeled as a s uare-well su erlattice

Vod

each potential well can be considered to

represent the potential associated with aV= 0particularatom in the crystal

in this model the amplitude of the potentiala

b

modulation vo is taken to be large compared to

the electron energy

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The Kroni -Penne Model

The problem that we consider here is a ONE-DIMENSIONAL one in which we assume thatthe potential variation occurs only in thex-direction

* We start from the SCHRDINGER EQUATION for motion in this direction

)(22 xd=

* Since the electron moves in a PERIODIC potential its wavefunction must satisfy

.2 2dxm

=

BLOCHS THEOREM which for the superlattice of interest here may be expressed as

)21.4()()( xedxdikx =+

In the above equation dis the PERIOD of the superlattice potential while kxis the value

of the wavenumber associated with the motion in the direction

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To solve this problem we must RECALL the solutions for electron motion in thepresence of a CONSTANT potential barrier

or a constant arr er o e g t t e wave unct on an energy so ut ons aregiven as

ikxikx .

)23.4()(22

VEm

k =

These solutions correspond to PLANE-WAVE propagation with wavenumber k

When the barrier height EXCEEDS the energy the electron wavenumber is

IMAGINARY inside the barrier and this ives rise to TUNNELING

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Taking our ORIGIN as indicated below we can use Blochs theorem to write thefollowing WAVEFUNCTIONS

= xixi .,

)25.4(0,)( axFeDex xixi

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Next we apply CONTINUITY CONDITIONS that require the wavefunction and itsderivative to be SINGLE-VALUED functions at any one of the potential steps

pp cat on o t e oun ary con t ons atx= y e s t e o ow ng re at ons

)30.4(FDBA +=+

* =

)31.4()()(FDB =

)32.4()(

aiaibibidik

FeDeBeAeex

+=+

)33.4()()( aiaibibidik

FeDeBeAee x =

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The only way in which solutions for the wavefunction variablesA,B, C, andD can beobtained is if the DETERMINANT of their coefficients vanishes

* This requirement leads in turn to the following conditions on the ALLOWED electron

energies and wavenumbers

When the electron energy is LESS than the height of the barriers (E< Vo) thiscondition may be written as shown in Equation 4.34

)34.4()sinh()sin(2

)cosh()cos()cos(22

bBaB

BbBadkx

=

35.42 iEVmB == 2 o=

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In terms of E, Equation 4.34 becomes

34.4)(2

sinh2

sin2)(2

cosh2

coscos000

EVmbmEaEVEVmbmEadk

=

This is a DISPERSION RELATION for electrons since it provides a connection

)(2 0 ==== EVE

e ween e r wavenum er x on e an e r energy on e

The importance of this equation is that it provides a CONDITION on the

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In the figure below we plot the variation of the RHS of Equation 4.34 as a function of the

parameter a

* The shaded regions correspond to ranges ofa (E) for which solution of the LHS ofqua on . poss e

The UNSHADED regions allow NO solution of Equation 4.34 and so correspond

2

In this figure the rhs of equation 4.34 is plottedas a function of the parametera which is related

to the energy of the electron

1

ON4.3

8

.

solution we require that1 rhs 1

this condition is only satisfied for the ranges of

a (energy) that are shown shaded

0

Q

UATI

all values ofa outside of these ranges do notallow solution of the lhs of equation 4.34 and

so correspond toforbidden energies

- 1

RHSOF

- 2

- 1 0 -5 0 5 1 0

aoa

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2

Allowed bands.38

0

1

ATION

1)cos( =dkx

- 1

OF

EQU

1)cos( =dkx

- 2

- 1 0 - 5 0 5 1 0

aoa

E1 E2 E3 E4 E5

E6RHS

The allowed and forbidden

bands are plotted in the E vs. k