Lecture #12 Program 1. Bloch solutions 2. Reciprocal space 3. Alternate derivation of Bloch’s theorem 4. Transforming the search for egenfunctions and eigenvalues from solving a PDE to finding the e-vectors and e-values of a matrix: The central equation. 5. Graphical description of eigenfunctions and eigenvalues - band diagrams. 6. Band gaps what happens near the edge of the BZ?

Eigenfunctions of infinite crystal - The effect of a periodic potential The solutions of the single electron Hamiltonian were shown to be Bloch wave of the form:

( ) ( ), ,ikr

n k n ku r e f r=r rr

Following from the Bloch Theorem which says: The eigenfunctions of H can be chosen such that with each eigenfunction ( ),n k

u rr ris associated a

wave vector k such that:

( ) ( ), ,ikR

n k n ku r R e u r+ =

rrr r

rr r

which is an equivalent form of Bloch’s Theorem, when the eigenfunction is observed at a distance R it differs by a phase factor. Reciprocal space Suppose we found a set of vectors G that satisfy the following relation:

1iG Re ⋅ =r r

that would lead to

( ) ( ), ,n k G n ku r u r

+=r rr r r

Eigenfunctions which differ by G are identical! These G’s are called the reciprocal lattice vectors, since they occur at regular intervals. The lattice vectors R are defined as follows:

1 1 2 2 3 3R n a n a n a= + +r r r r

1 1 2 2 3 3G m b m b m b= + +r r rr

Let a’s be the the set of primitive vectors for the direct lattice and the b’s are the primitive vectors for the reciprocal lattice given by:

( ) ( ) ( )2 3 3 1 1 2

1 2 31 2 3 1 2 3 1 2 3

2 , 2 , 2a a a a a a

b b ba a a a a a a a a

π π π× × ×

= = =⋅ × ⋅ × ⋅ ×

r r r r r rr r rr r r r r r r r r

Example simple cubic lattice: 1 2 3ˆ ˆ ˆ, , a ax a ay a az= = =

r r r

1 2 3

2 2 2ˆ ˆ ˆ, , b x b y b za a aπ π π

= = =r r r

For a 1D lattice we would just have for our direct lattice the number a and for the reciprocal

lattice “vector” the number 2

n= 1, 2, 3....naπ

± ± ± .

Plot 1D direct and reciprocal lattice. Any point k in the reciprocal space (not necessarily a lattice point) can be expressed as:

:k k Gwhere

ka aπ π

′ = +

− ≤ ≤

r r r

r

This range is called the first Brilllouin zone, it is the Wigner Seitz cell of the reciprocal lattice.

Alternate derivation of Bloch’s theorem We are considering a periodic potential,

( ) ( )V x a V x+ = Since the function is periodic with period a it can be expanded in a Fourier series where G’s are the set of reciprocal lattice vectors

( ) ( ) iGx

G

V x V G e= ∑

in a more familiar form,

( ){ {

2 2cos sinn n

nG G

n nV x A x B x

a aπ π

= +∑

We want the potential to be real, ( ) ( ) ( ) ( )

0 0

2 cosiGx iGx

G G

V x V G e e V G Gx−

> >

= + =∑ ∑

Where we have assumed that the crystal is symmetric about x=0. We are searching for eigenfunctions of the Hamiltonian:

( ) ( ) ( )2 2

22iGx

G

dV G e u x Eu x

m dx

− + =

∑h

let’s assume that the eigenfunctions can be written as a Fourier series since there is no reason to assume that the eigenfunctions is periodic in lattice period we expand it in k,

( ) ( ) ikx

k

u x C k e= ∑

(Born Von-Karman boundary conditions) Substituting in the Hamiltonian,

( ) ( ) ( )2 2

22iGx ikx ikx

G k k

dV G e C k e E C k e

m dx

− + =

∑ ∑ ∑h

( ) ( ) ( ) ( ) ( )2

2

2i G k xikx ikx

k G k k

C k k e V G C k e E C k em

++ =∑ ∑∑ ∑h

Equating the coefficients of the exponentials with the same arguments,

( ) ( ) ( )2

2 02 G

k E C k V G C k Gm

− + − =

∑h

Consequences: 1. A set of coupled linear algebraic equations (sometimes called the central equation) where

the unknowns are the ( )C k G− ’s. 2. The solution of the central equation for a given k involves in principle an infinite number

of coefficients that all differ by a reciprocal lattice vector G. 3. The number of nonzero matrix elements in the equations are determined by the number of

Fourier components that are needed to express the potential function.

Thus the eigenfunctions should have the following form:

( ) ( ) ( )i k G xk

G

u x C k G e −= −∑

which can be re-written: ( ) ( )ikx iGx

kG

u x e C k G e−= −∑

which is just a restatement of Bloch’s theorem, since ( ) ( )iGx

kG

C k G e f x−− =∑

where f is a periodic function with the periodicity of the lattice. Example of an explicit solution for a simple cos potential The solution of the central equation for a particular potential

( ) ( )2 cos igx igxV x V gx V e e−= = +

This defines an infinite set of equations (which we truncated after 5 terms), the equations can have non-trivial solutions for the C’s if the determinant vanishes:

( )

( )

( )

( )

22

22

22

22

22

2 0 0 02

0 02

00 02

0 02

0 0 0 22

k g E Vm

V k g E Vm

V k E Vm

V k g E Vm

V k g Em

− −

− −

=−

+ −

+ −

h

h

h

h

h

For a given k there will be 5 energy eigenvalues labeled ( )nE k which basically defines the band diagram and corresponding eigenvectors C’s which in turn define the Bloch wave eigenfunctions. As we mentioned in principle the matrix is infinite and one needs to calculate all of the C(k-ng)’s, in practice just a few coefficients are usually sufficient. In the following example we calculate the coefficients and corresponding ground state eigenvalue (note: the Mathematica program involves the generation of a tridiagonal matrix and will be emailed to you). C(k-4g) C(k-3g) C(k-2g) C(k-g) C(k) C(k+g) C(k+2g) C(k+3g) C(k+4g) E0

0.3765 0.8828 0.2807 -0.3622 0.0479 0.3828 0.8668 0.3014 0.0945 -0.3847 0.0027 0.0482 0.3827 0.86651 0.3017 0.0961 0.0048 -0.3849

0.0000 0.0027 0.0482 0.3827 0.86651 0.3017 0.0961 0.0048 0.0001 -0.3849

Observations:

1. The ground state energy is dominated by the contributions from C(k) where k is the smallest vector i.e it lies in the first BZ.

2. If we were to choose a reciprocal vector that is out of the first BZ k k G′ = + the form of the matrix above would remain unchanged and the eigenvectors would be identical. i.e. the ground state calculated for k’ and the one calculated for k’+g would be the same.

3. This means that the eigenvectors and energy eigenvalues are periodic in G ( ) ( )k G ku x u x+ =

and ( ) ( )n nE k G E k+ =

and we therefore can choose k’s that are restricted to the first BZ - reduced zone scheme.

This leads to a description of the energy levels of an electron in a periodic potential in terms of a family of continuous functions ( ) ( )n nE k G E k+ =

r rrwith the periodicity of the reciprocal lattice

vector. The information contained in these functions is referred to as the band structure. ( ) ( ), ,n k G n k

u r u r+

=r rr r r

, ,n k G n kε ε

+=r rr

For each n the set of electronic levels specified by ( )n kεr

is called an energy band.

Band gaps: what happens near the edge of the BZ?

The edge of the BZ is defined by:

2g

k = ±

review the master equation:

( ) ( ) ( )2

2 02 G

k E C k V G C k Gm

− + − =

∑h

And consider only the first terms in the sum:

22

22

2 20

2 2

gE V

m

gV E

m

− =

− −

h

h

The energy eigenvalues are 22

2 2g

E Vm±

= ±

h

Thus the energy has two roots which are separated by 2V, keep in mind that this result is valid for the particular form of the potential that we used:

( ) ( )2 cos igx igxV x V gx V e e−= = +

So we are able to predict the existence of a band gap which to first order is proportional to the magnitude of the periodic potential. The form of the eigenfunctions near the band edge is:

( ) 2 2

2

igx igx

gu x e e−

= ±

One solution gives the eigenfunctions near the bottom of the gap the other gives it near the top of the gap. The differences in energies can be explained by looking at the charge distribution relative to the “ion” positions.

( )

( )

2cos

2 sin

x xi ia a

x xi i

a a

xu x e e

ax

u x e e ia

π π

π π

π

π

+

−

= + =

= − =

The two standing waves concentrate their electrons in different regions – the probability density for a particle is:

( ) ( ) 2cosx

u x u xa

π∗+ + =

This function “piles” electrons (with negative charge) on the positive ions centered at x=0, a, 2a,… where the potential energy is lowest. For the other standing wave,

( ) ( ) 2sinx

u x u xa

π∗− − =

It can be shown that an electron is a level specified by band n has a non-vanishing mean velocity

( ) ( )1n k nv k kε= ∇

r rrrh

This is quite remarkable since it states that there are time independent eigenfunctions which in spite of the interaction of the electron with the fixed lattice move forever without any degradation of the mean velocity

Generating band diagrams (displaying eigenfunctions graphically) What happen as we transition from a free electron system to one which has a periodic potential?

.