Section 1.2-3Homework from this

section: 1.5(We will do a similar problem in

class today)

Bloch’s Theorem and Krönig-Penney

Model

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

Learning Objectives for TodayAfter today’s class you should be able to:Apply Bloch’s theorem to the Kronig-

Penney model or any other periodic potential

Explain the meaning and origin of “forbidden band gaps”

Begin to understand the Brillouin zone For another source on today’s topics, see

Ch. 7 of Kittel’s Intro to Solid State Physics. Crystal basics to prepare us for next class

“Realistic” Potential in Solids

Multi-electron atomic potentials are complex

Even for hydrogen atom with a “simple” Coulomb potential solutions are quite complex

So we use a model infinite one-dimensional periodic potential to get insight into the problem (last time, looked at 1-6 atoms)

Periodic Potential

For one dimensional case where atoms (ions) are separated by distance a, we can write condition of periodicity as

)()( naxVxV

a

Section 1.3: Bloch’s Theorem

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

We will prove 1-D version, AKA Floquet’s theorem.(3D proof in the book)

When using this theorem, we still use the time-indep. Schrodinger equation for an electron in a periodic potential

ErVm

)(

22

2

)()( rVTrV

where the potential energy is invariant under a lattice translation of a

In 3D (vector): cwbvauT

)()( naxVxV

Bloch Wavefunctions

Bloch’s Theorem states that for a particle moving in the periodic potential, the wavefunctions ψ(x) are of the form

uk(x) has the periodicity of the atomic potential The exact form of u(x) depends on the potential

associated with atoms (ions) that form the solid

)()(

)(,)()(

axuxu

xuwhereexux

kk

kikx

k

function periodic a is

a

Main points in the proof of Bloch’s Theorem in 1-D

1. First notice that Bloch’s theorem implies:

Can show that this formally implies Bloch’s theorem, so if we can prove it we will have proven Bloch’s theorem.

Tkirkikk

eeTruTr

)()( Tkirki

keeru

)( Tki

ker

)(

Or just:

Tkikk

erTr

)()(

2. To prove the statement shown above in 1-D:Consider N identical lattice points around a circular ring, each separated by a distance a. Our task is to prove: ikaexax )()( 1

2 N

3

)()( xNax

Built into the ring model is the periodic boundary

condition:

)()(

,)()(

axuxu

exux

kk

ikxk

Proof of Bloch’s Theorem in 1-D: Conclusion

If we apply this translation N times we will return to the initial atom position:

Now that we know C we can rewrite

)()()( xxCNax N

This requires 1NC

And has the most general solution:

,...2,1,02 neC niN

ikaNni eeC /2Or:

Where we define the Bloch wavevector: Na

nk

2

...)()()( DEQxexCax ika

The symmetry of the ring (and last lecture) implies that we can find a solution to the wave equation:

)()( xCax

12 N

3

Consequence of Bloch’s Theorem

Probability of finding the electron

Each electron in a crystalline solid “belongs” to each and every atom forming the solid

Very accurate for metals where electrons are free to move around the

crystal! Makes sense to talk about a specific x

(± n a)

)()( axPxP

Using Bloch’s Theorem: The Krönig-Penney Model

Bloch’s theorem allows us to calculate the energy bands of electrons in a crystal if we know the potential energy function.

First done for a chain of finite square well potentials model by Krönig and Penney in 1931 with E<V0

Each atom is represented by a finite square well of width a and depth V0. The atomic spacing is a+b.

We can solve the SE in each region of space: ExV

dx

d

m

)(

2 2

22

0 < x < aiKxiKx

I BeAex )( m

KE

2

22

-b < x < 0

V

x0 a a+b

2a+b 2(a+b)

V0

-b

xxII DeCex )(

mEV

2

22

0

I wish the book had selected different letters than K and ,

but staying consistent

Boundary Conditions and Bloch’s Theorem

x = 0

The solutions of the SE require that the wavefunction and its derivative be continuous across the potential boundaries. Thus, at the two boundaries (which are infinitely repeated):

iKxiKxI BeAex )(

xxII DeCex

)(

Now using Bloch’s theorem for a periodic potential with period a+b:

x = a )(aBeAe IIiKaiKa

DCBA (1) )()( DCBAiK (2)

)()()( baikIIII eba k = Bloch

wavevector

Now we can write the boundary conditions at x = a:

)()( baikbbiKaiKa eDeCeBeAe (3)

)())()(()()( baikbbiKaiKa eDeikCeikBeikiKAeikiK (4, deriv.)

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

ikxk exux )()(

Results of the Krönig-Penney Model

Since the values of a and b are inputs to the model, and depends on V0 and the energy E, we can solve this system of equations 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

eeikeeikeikiKeikiK

eeeeee

iKiK

baikbbaikbiKaiKa

baikbbaikbiKaiKa

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

)(coscoshcossinhsin2

22

bakbKabKaK

K

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

Graphical Approach

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

)cos(cossin2

2

kaKaKaK

b

Ka

Plotting left side of equation

Gap occurs at Ka=N or

K=N/a

)(coscoshcossinhsin2

22

bakbKabKaK

K

small b

m

KE

2

22

Not really much different

Single Atom

Multiple Atoms

Greek Theater Analogy: Energy Gaps

What Else Can We Learn From

This Model?

Exercise 1.4  )cos(cossin

2

2

kaKaKaK

b

Different Ways to Plot ItExtended Zone Scheme

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

Ka

The range -<ka< is called the first Brillouin zone.

Atoms

Different x axis

k = Bloch wavevector

y=cos ka

)cos(cossin2

2

kaKaKaK

b

Different Representations of E(k)

Reduced zone scheme

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

Frequently only one side is shown as they are degenerate.

In 3D, often show one side along with dispersion along two other directions (e.g. 100, 110, 111)

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

momentum space k

Real space examples

Momentum space example

21

Compare to the free-electron model

Free electron dispersion E

k

Let’s slowly turn on the periodic potential

...with first Brillouin zone:–/a ® /a

(a the lattice constant)

–/a /a2

2 2 2( )2 x y zE k k km

Let’s draw it in 3D!

Electron Wavefunctions in a Periodic Potential

(Another way to understand the energy gap)

Consider the following cases:

Electrons wavelengths much larger than atomic spacing a, so wavefunctions and energy bands are nearly the same as above

01 V)( tkxiAe

m

kE

2

22

ak

V

01

Wavefunctions are plane waves and energy bands are parabolic:

E

k–/a /a

V

x0 a a+b

2a+b 2(a+b)

V1

-b

How do X-rays Work?

The soft tissue in your body is composed of smaller atoms, and so does not absorb X-ray photons particularly well. The calcium atoms that make up your bones are much larger, so they are better at absorbing X-ray photons.

Consequence of Bloch’s Theorem

Similar to how radio waves pass through us without affecting

Electron Wavefunctions in a Periodic PotentialU=barrier potential

Consider the following cases:

Electrons wavelengths much larger than a, so wavefunctions and energy bands are nearly the same as above

01 V)( tkxiAe

m

kE

2

22

ak

V

01

Wavefunctions are plane waves and energy bands are parabolic:

ak

V01 Electrons waves are strongly back-scattered (Bragg

scattering) so standing waves are formed:

tiikxikxtkxitkxi eeeAeeC

21)()(

ak

V01 Electrons wavelengths approach a, so waves begin to

be strongly back-scattered by the potential:)()( tkxitkxi BeAe

AB

E

k–/a /a

The nearly-free-electron model (Standing Waves)

Either: Nodes at ions

Or: Nodes midway between ions

a

Due to the ±, there are two such standing waves possible:

titiikxikx ekxAeeeA )cos(2

21

21

titiikxikx ekxiAeeeA )sin(2

21

21

These two approximate solutions to the S. E. at have very different potential energies. has its peaks at x = a, 2a, 3a, …at the positions of the atoms, where V is at its minimum (low energy wavefunction). The other solution, has its peaks at x = a/2, 3a/2, 5a/2,… at positions in between atoms, where V is at its maximum (high energy wavefunction).

ak

tiikxikx eeeA

21

The nearly-free-electron model

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

a

~ 2

2

2

titiikxikx ekxAeeeA )cos(2

21

21

titiikxikx ekxiAeeeA )sin(2

21

21

)(cos2 22*axA

)(sin2 22*axA

Different energies for electron standing waves

Symmetric and

Antisymmetric Solutions

28

E

k

Summary: The nearly-free-electron model

BAND GAPS APPEAR AT EACH BRILLOUIN ZONE EDGE

-2π/a –π/a π/a 2π/a

In between the two energies there are no allowed energies; i.e., wavelike solutions of the Schrodinger equation do not exist.

Forbidden energy bands form called band gaps.

The periodic potential V(x) splits the free-

electron E(k) into “energy bands” separated by

gaps at each BZ boundary.

E-

E+

Eg

E

k

Approximating the Band Gap

BAND GAPS APPEAR AT EACH BRILLOUIN ZONE EDGE

-2π/a –π/a π/a 2π/a

a

xax

a dxxVEE0

22 )(cos)(

E-

E+

Eg

a

x

g dxxVEEE0

22])[(

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

)(cos2 22*axA

)(sin2 22*axA

Crystal Directions

Figure shows [111] direction

Choose one lattice point on the line as an origin (point O). Choice of origin is completely arbitrary, since every lattice point is identical.

Then choose the lattice vector joining O to any point on the line, say point T. This vector can be written as;

R = n1 a + n2 b + n3 c

To distinguish a lattice direction from a lattice point, the triplet is enclosed in square brackets

[ ...]. Example: [n1n2n3]

[n1n2n3] is the smallest integer of the same

relative ratios. Example: [222] would not be used instead of [111].

Negative directions can be written as

][ 321 nnnAlso sometimes

[-1-1-1]