bloch’s theorem and krönig-penney model

8/13/2019 Blochs Theorem and Krnig-Penney Model

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Section 1.2-3Homework from this section: 1.5

(We will do a similar problem in class today)

Blochs Theorem andKrnig-Penney Model

For review/introduction of Schrodinger equation:http://web.monroecc.edu/manila/webfiles/spiral/6schrodingereqn.pdf
http://web.monroecc.edu/manila/webfiles/spiral/6schrodingereqn.pdfhttp://web.monroecc.edu/manila/webfiles/spiral/6schrodingereqn.pdfhttp://web.monroecc.edu/manila/webfiles/spiral/6schrodingereqn.pdf


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Learning Objectives for TodayAfter todays class you should be able to:

Apply Blochs theorem to the Kronig-Penney model or any other periodicpotentialExplain the meaning and origin of

forbidden band gaps

Begin to understand the Brillouin zoneFor another source on todays topics, seeCh. 7 of Kittels Intro to Solid State Physics.

Crystal basics to prepare us for next class


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Realistic Potential in Solids

Multi-electron atomic potentials are complex

Even for hydrogen atom with a simpleCoulomb potential solutions are quitecomplex

So we use a model infinite one-dimensionalperiodic potential to get insight into theproblem (last time, looked at 1-6 atoms)


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Periodic Potential

For one dimensional case where atoms (ions) areseparated by distance a , we can write condition ofperiodicity as

)()( na xV xV

a


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Section 1.3: Blochs Theorem

This theorem gives the electron wavefunction in thepresence of a periodic potential energy.

We will prove 1- D version, AKA Floquets theorem.

(3D proof in the book)

When using this theorem, we still usethe time-indep. Schrodinger equationfor an electron in a periodic potential

E r V m

)(2

22

)()( r V T r V

where the potential energyis invariant under a latticetranslation of a

In 3D (vector): cwbvauT

)()( na xV xV


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Bloch Wavefunctions

Blochs Theorem states that for a particle movingin the periodic potential, the wavefunctions (x)are of the form

uk(x) has the periodicity of the atomic potentialThe exact form of u(x) depends on the potentialassociated with atoms (ions) that form the solid

)()(

)(,)()(

a xu xu

xuwheree xu x

k k

k ikx

k functionperiodicais

a


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Main points in the proof ofBlochs Theorem in 1 -D

1. First notice that Blochs theorem implies:

Can show that this formally impliesBlochs theorem, so if we can prove itwe will have proven Blochs theorem .

T k ir k ik k eeT r uT r

)()( T k ir k ik eer u )( T k ik er

)(

Or just:

T k ik k er T r

)()(

2. To prove the statement shown above in 1-D:Consider N identical lattice points around acircular ring, each separated by a distance a.

Our task is to prove: ikae xa x )()( 12 N

3

)()( x Na x

Built into the ringmodel is the periodicboundary condition:

)()(

,)()(

a xu xu

e xu x

k k

ikxk


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Consequence of Blochs Theorem Probability of finding the electron

Each electron in a crystalline solid belongsto each and every atom forming the solid

Very accurate for metals where electrons arefree to move around the crystal!

Makes sense to talk about a specific x ( n a)

)()( a x P x P


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Using Blochs Theorem: The Krnig -Penney Model

Blochs theorem allows us to calculate the energy bands of electronsin a crystal if we know the potential energy function.

First done for a chain of finite square well potentials model by Krnigand Penney in 1931 with E


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Boundary Conditions and Blochs Theorem

x = 0

The solutions of the SE require that thewavefunction and its derivative becontinuous across the potential boundaries.Thus, at the two boundaries (which areinfinitely repeated):

iKxiKx I Be Ae x

)(

x x II DeCe x

)(

Now using Blochs theorem for a

periodic potential with period a+b:

x = a )(a Be Ae II iKaiKa

DC B A (1) )()( DC B AiK (2)

)()()( baik II II eba

k = Bloch

wavevector

Now we can write the boundary conditions at x = a:)()( baik bbiKaiKa e DeCe Be Ae (3)

)())()(()()( baik bbiKaiKa e Deik Ceik Beik iK Aeik iK (4, deriv.)

The four simultaneous equations(1-4) can be written compactly inmatrix form

ikxk e xu x )()(


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Results of the Krnig-Penney Model

Since the values of a and b are inputs to the model, and depends on V 0 and the energy E, we can solve this system ofequations to find the energy E at any specified value of the

Bloch wavevector k. What is the easiest way to do this?

0

)()()()(

1111

)()(

)()(

D

C

B

A

eeik eeik eik iK eik iK

eeeeee

iK iK

baik bbaik biKaiKa

baik bbaik biKaiKa

Taking the determinant, setting it equal to zero and lots of algebra gives:

)(coscoshcossinhsin2

22

bak b Kab Ka K

K

By reducing the barrier width b ( small b) , this can be simplified to:


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Graphical Approach

Right hand side cannot exceed 1, so values exceeding willmean that there is no wavelike solutions of the Schrodinger eq.(forbidden band gap)

)cos(cossin2

2

ka Ka Ka K

b

Ka

Plotting left side of equation

Gap occursat Ka=N orK=N /a

)(coscoshcossinhsin2

22

bak b Kab Ka K

K

small b

m K

E 2

22


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Not really much different

Single Atom

Multiple Atoms


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Greek Theater Analogy: Energy Gaps


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What Else Can WeLearn From This

Model?


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Exercise 1.4

)cos(cossin2

2

ka Ka Ka K

b


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Different Ways to Plot ItExtended Zone Scheme

Note that the larger the energy, the larger the band/gap is (untilsome limit).

Ka

The range -


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Different Representations of E(k)

Reduced zone scheme

All states with |k| > /a aretranslated back into 1 st BZ

Frequently only one side isshown as they aredegenerate.

In 3D, often show one sidealong with dispersionalong two other directions(e.g. 100, 110, 111)


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Band diagrams can refer to either E vs.real space or E vs momentum space k

Real space examples

Momentum space example


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21

Compare to the free-electron model

Free electron dispersion E

k

Lets slowly turn on the periodic potential

...with first Brillouin zone: /a /a

(a the lattice constant)

/a /a2

2 2 2( )2

x y z E k k k m

Lets draw it in 3D!

E


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Electron Wavefunctions in aPeriodic Potential

(Another way to understand the energy gap)

Consider the following cases:

Electrons wavelengths much larger than atomic spacinga, so wavefunctions and energy bands are nearly thesame as above

01 V )( t kxi Ae

mk

E 2

22

ak V

01

Wavefunctions are planewaves and energy bandsare parabolic:

E

k /a /a

V

x0 a a+b

2a+b 2(a+b)

V1

-b


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How do X-rays Work?

The soft tissue in yourbody is composed ofsmaller atoms, and sodoes not absorb X-rayphotons particularlywell. The calciumatoms that make up

your bones are muchlarger, so they arebetter at absorbingX-ray photons .


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Consequence of Blochs Theorem

Similar to how radio waves pass through us without affecting

l f E


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Electron Wavefunctions in aPeriodic Potential

U=barrier potential

Consider the following cases:

Electrons wavelengths much larger than a, sowavefunctions and energy bands are nearly the sameas above

01 V )( t kxi Ae

mk

E 2

22

ak

V

01

Wavefunctions are planewaves and energy bandsare parabolic:

ak

V

01 Electrons waves are strongly back-scattered (Braggscattering) so standing waves are formed:

t iikxikxt kxit kxi eee AeeC

21)()(

ak

V

01 Electrons wavelengths approach a, so waves begin tobe strongly back-scattered by the potential:

)()( t kxit kxi

Be Ae

A B

E

k /a /a

Th l f l d l


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The nearly-free-electron model(Standing Waves)

Either: Nodes at ions

Or: Nodes midway

between ionsa

Due to the , there are two such standing waves possible:t it iikxikx ekx Aeee A )cos(22121t it iikxikx ekxiAeee A

)sin(22

12

1

These two approximate solutions to the S. E. at havevery different potential energies. has its peaks at x = a,

2a, 3a, at the positions of the atoms, where V is at itsminimum (low energy wavefunction). The other solution,has its peaks at x = a/2, 3a/2, 5a/2, at positions in betweenatoms, where V is at its maximum (high energy wavefunction).

ak

t iikxikx eee A

21


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The nearly-free-electron model

Strictly speaking we should have looked at the probabilitiesbefore coming to this conclusion:

a

~ 2

2

2

t it iikxikx ekx Aeee A )cos(2

21

21

t it iikxikx ekxiAeee A )sin(2

21

21

)(cos2 22* a x A

)(sin2 22* a x A

Different energies for electron standing waves


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28

E

k

Summary: The nearly-free-electron model

BAND GAPS APPEAR AT EACH BRILLOUIN ZONE EDGE

- 2/a /a /a 2/a

In between the two energies

there are no allowed energies;i.e., wavelike solutions of theSchrodinger equation do notexist.

Forbidden energy bands form

called band gaps.

The periodic potentialV(x) splits the free-

electron E(k) into energy

bands separated by gapsat each BZ boundary.

E-E+

Eg


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E

k

Approximating the Band Gap

BAND GAPS APPEAR AT EACH BRILLOUIN ZONE EDGE

- 2/a /a /a 2/a

a

xa x

a dx xV E E 0

22 )(cos)( E-

E+Eg

a

x g dx xV E E E

0

22])[(

For square potential: V(x) =V o for specific values of x (changes integration limits)

)(cos2 22* a x A

)(sin2 22* a x A


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Crystal Directions

Figure shows[111] direction

Choose one lattice point on the line as anorigin (point O). Choice of origin iscompletely arbitrary, since everylattice point is identical.Then choose the lattice vector joining O to

any point on the line, say point T. Thisvector can be written as;

R = n 1 a + n 2 b + n 3 c

To distinguish a lattice direction from alattice point, the triplet is enclosed insquare brackets [ ...]. Example: [n 1n 2n 3]

[n 1n 2n 3] is the smallest integer of thesame relative ratios . Example: [222]

would not be used instead of [111] ][ Also sometimes

bloch’s theorem and krönig-penney model

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