1

Ch. 8: Electron Levels In Periodic Potential

What are the consequences of a periodic potential on the form of the wave functions, irrespective of the exact form of the potential and irrespective of whether the material is a conductor or an insulator? U is the effective one-

)()( rURrU rrr=+

electron potential.

ψεψψ =⎟⎟⎠

⎞⎜⎜⎝

⎛+∇−= )(

22

2

rUm

H rh

Bloch’s Theorem: The eigenstates of the one-electron Hamiltonian, with a periodic potential in a Bravais lattice, can be chosen to have the form of a plane wave times a function with the periodicity of the Bravais lattice.

Or, equivalently

)()( ruer knrki

kn

rrr

rr

r⋅=ψ )()( Rruru knkn

rrrrr +=

)()( reRr knRki

kn

rrrr

rr

r ψψ ⋅=+

1st Proof Of Bloch’s Theorem

Define translation operator TR:

)()( RrfrfTRrrr

r +=

Hamiltonian is periodic,

)()()()( RrRrHrrHTRrrrrrr

r ++= ψψ

RR HTHT rr =

RRRRRR TTTTT rrrrrr +′′′ ==But, also

Since H and all TR’s commute with one another, eigenstate of H can be chosen to be eigenstate of all TR’s.

ψεψ =H )()()( rRcrTRrrr

r ψψ =

a number

2

1st Proof Of Bloch’s Theorem

ψεψ =H )()()( rRcrTRrrr

r ψψ =

)()()( RcRcRRc ′′+rrrr

)()()( RcRcRRc =+

Let iixi eacπ2)( =r

332211 anananRrrrr

++=)()( reRr Rki r

rr rr ψψ ⋅+

primitive vectora (complex) number

332211 222)( nixnixnix eeeRc πππ=r

RkieRcrrr⋅=)(

332211 bxbxbxkrrrr

++=

)()( reRr knkn rr ψψ =+

To see the constraints on k, we need to introduce boundary conditions.

Born – von Karman Boundary Condition

3,2,1)()( ==+ iraNr iirrr ψψ PBC

i t

3,2,1)()( ==+ ⋅ ireaNr knakiN

iiknii

rrrr

rr

r ψψ

12 =ii xNie π332211 bxbxbxkrrrr

++=

Nmx / 33

22

11 b

mbmbmk

rrrr++=mi integer

volume per allowed state

iii Nmx /= 332

21

1

bN

bN

bN

k ++

( )3213

3

2

2

1

1 1 bbbNN

bNb

Nb

krrr

rrrr

×⋅=⎟⎟⎠

⎞⎜⎜⎝

⎛×⋅=Δ

cellVNk

3)2( π=Δr

same as free electron gas!

3

2nd Proof of Bloch’s Theorem (Plane Wave Expansion)

Working within the Born-von Karman periodic boundary condition, we expand both the one-electron wave function and the effective potential in plane waves

rKi

KK eUrU

rr

rr

r ⋅∑=)( k’s are all the wave vectors allowed by the PBC, but K’s are only reciprocal lattice vectors because the potential is periodic.)(1 rUerd

VU rKi

cellcell

K

rr rrr ⋅−∫=

Plugging into Schroedinger’s eq. to look for eigen states

rki

kk ecr

rr

rr

r ⋅∑=)(ψ

ψεψψ =⎟⎟⎠

⎞⎜⎜⎝

⎛+∇−= )(

22

2

rUm

H rh

∑∑ ∑ ⋅⋅+⋅ =⎬⎫

⎨⎧

+ rkirKkirki ceecUeckrrrrrrrh ε)(

22

*KK UU rr −=

∑∑ ∑ ⋅−⋅ =⎭⎬⎫

⎩⎨⎧

+k

krki

k KKkKk

rki cecUcmke

rr

rr

r rrrrr

rr h ε2

22

Since k’s form a complete orthogonal set, coefficient for a specific plane wave on LHS must equal coefficient for the same k on the RHS.

∑∑ ∑ =⎭⎬

⎩⎨ +

kk

k KkKk ceecUecm r

rr r

rrr ε)(2

k k’-K

Plane Wave Expansion (cont.)

02

22

=+⎟⎟⎠

⎞⎜⎜⎝

⎛− ∑ −

KKkKk cUcm

kr

rrrrh ε

The C coefficient for plane wave with a specific k is coupled to the coefficients for all wave vectors k’ related to k by k’=k+K, where K is a

i i i⎠⎝ K reciprocal lattice vector and UK is non-zero.

Note that U can always be chosen (shifted) so that UK=0=0.

Plane waves not differing by a reciprocal lattice vector are unrelated. Or, if a plane wave is to couple, it can only couple to other plane waves with wave vectors differing by K from its own. Grab all plane waves with wave vectors related to k and form a subset (Hilbert space) that is independent of other subsets. Now a plane wave can only couple to waves within its own subset. To distinguish subsets we can identify them by

For a specific ε, this equation has non-trivial solutions only for a few k’s (or for none).

What does it mean physically?

(1) If UK vanishes for large K’s, say, K>G ?

(2) If UK vanishes for a series of K’s (e.g. nG, where n is an odd integer)?

coup e o w ves w s ow subse . o d s gu s subse s we c de y e bythe one wave vector, q, in each subset that belongs to the first Brillouin zone.

4

Plane Wave Expansion (cont.)

02

22

=+⎟⎟⎠

⎞⎜⎜⎝

⎛− ∑ −

KKkKk cUcm

kr

rrrrh ε

qnqnqnH rrr ψεψ =rKqi

Kqnqn ecrrrr

rrrr ⋅+

+∑= )(,)(ψ

There is an infinite number of plane waves in each subset (because the number of reciprocal lattice points is infinite). However, the number of subsets is finite ( number of allowed q = number of unit cells in the Born von Karman box). Within each subset, the above matrix can be diagonalized to yield an infinite number of discreteeigenvalues εn(q), n=0, 1, 2,.... Each eigenstate is composed of a (sub)set of plane waves from one subset, with their proper C coefficients, which satisfies the Schroedinger equation

qnqnqn qKr ,

What can we say about the C’s, if the periodic potential is weak?

Eigenstates with same q but different n can be made to be orthogonal to each other. Perturbation theory tells us that for an incremental change in q, (i.e. neighboring subsets) there must be allowed eigenvalues that are incremental changed from each other. These states can be viewed as forming bands and n is then the band index.

Plane Wave Expansion (cont.)

∑∑ ∑ ⋅−⋅ =⎭⎬⎫

⎩⎨⎧

+k

krki

k KKkKk

rki cecUcmke

rr

rr

r rrrrr

rr h ε2

22rki

kk ecr

rr

rr

r ⋅∑=)(ψ

Looking at the Schroedinger equation and requesting that Ψ be an eigenstate with eigenvalue ε, we see that only a very small number of subsets could have ε as one of their eigenvalues. A specific Ψ can be expanded in plane waves from these few independent subsets with a specific, allowed, eigenvalue.

a’s are independent and arbitrary

)()(),( ,,,

, qnqnqn

qn rar rrr

rrr εεδψεψ −=∑

few terms

rKqiKqn

Kqn ecr

rrrrr

rrr ⋅+

+∑= )(,)(ψ )(),( ,,,

rar qnqnqn

rrrr

r

ψεψεε

∑=

=

We have as many choices as there are terms in the sum to define eigenstates. Certainly, any eigenfunction can be chosen to have only contribution(s) from one q.

5

2nd Proof of Bloch’s Theorem

rKqiKqn

Kqn ecr

rrrrr

rrr ⋅+

+∑= )(,)(ψ any eigen function of a periodic lattice

all terms periodic with lattice

K

rKiKqn

K

rqiqn ecer

rr

rrr

rrrr ⋅

+⋅ ∑= ,)(ψ

)()( Rruru knknrrr

rr +=)()( ruer knrki

kn

rrr

rr

r⋅=ψ

Velocity of Bloch Electrons

22 rrh ⎟

⎞⎜⎛

)()()(2

22

rrrUm knknkn

rrrrhrrr ψεψ =⎟⎟

⎠

⎞⎜⎜⎝

⎛+∇

−

222

2)( q

mkiq

mHH kqk

hrrrhrrr ++∇−⋅+=

+

...)()()(2

* ++∇−⋅+=+ ∫ nknknn ukiqmurdkqkrrrhrrrr εε

)()()()(2

)( 2 rururUkim

ruH kkkkkrrrrrhr

rrrrr ε=⎟⎟⎠

⎞⎜⎜⎝

⎛++∇−=

)()( ruer knrki

kn

rrr

rr

r⋅=ψ

1)()(2

+∂

+∇++ ∑ nkk εεεε rrrr

But, we also know that

)()( Rruru kkrrr

rr +=

...2

)()(,

+∂∂

+∇⋅+=+ ∑ji

jiji

nnknn qqkk

qkqk εεε r

nknknkn vri

mrrd

krrr

hrr

rh

=∇−=∂∂

∫ )()(1)(* ψψε

know that

6

About Bloch’s Theorem

1. Allowed wave vector k follow same PBC as free l t b t k i l t /h belectrons, but k is no longer momentum/h bar.

2. k can always be confined to the first Brilloun zone.

3. band index n

4 repeated zone representation

)()( Rruru kkrrr

rr +=

)()()()(2

)( 22

rururUkim

ruH kkkkkrrrrrhr

rrrrr ε=⎟⎟⎠

⎞⎜⎜⎝

⎛++∇−=

Kkk rr rrrrr

+= )()( ψψ4. repeated zone representation

5. velocity of electron

6. Filled bands, empty bands, partially filled bands and Fermi surface.

Kknkn

Kknkn

rrr+

+

= ,,,, )()(

εεψψ

Density of Levels

)(8

2)(2 3 kQkdVkQQ nn

rrr∫∑∑ ==

One is often interested in summing quantities from occupied individual electronic levels. In the limit of large crystals, the summation can be

l d b i t l

over Brillouin zone

8 3,n

nknnr ∫∑∑ πreplaced by an integral.

If the interested quantity only depends on k through energy, then the integral can be carried out

)()( εεε QgdVQq ∫== ))((4)()( 3 k

kdgg nn

nn

rr

εεδπ

εε −== ∫∑∑Consider equi-energy surfaces in reciprocal space. For small enough energy difference, we get

)()( kkdSdgr

δεε ∫= )(4)( )( 3 kkdg nSn δπεε ε∫=

)(|)(| kkkd nrr

δεε ∇=

|)(|1

4)(

)( 3 kdSg

nSn n

rεπ

εε ∇

= ∫

7

van Hove Singularities

van Hove singularities in 3D: gn is finite. However, slope divergesdiverges.

Ch. 9: Electrons In Weak Periodic Potential

Assuming weak periodic potential in some metals, a reasonable starting point for eigenstates of conduction electrons is plane waves.

Why do we even expect the periodic potential experienced by some conduction (outer-shell) electrons to be weak in some metals?

(1) Outer shell electrons are forbidden from coming close to the ion cores.

(2) Outside the cores, potential is effectively screened by mobile electrons.

Bloch’s theorem rKki

KKkk ecr

rrr

rrrr

r ⋅−−∑= )()(ψ

2h ⎞⎛

02

)( 22=+⎟⎟

⎠

⎞⎜⎜⎝

⎛−

− ∑′

′−−′−K

KkKKKk cUcmKk

rrrrrrr

rrh ε

Weak Potential: UK’s ~ U

8

Extreme Case: Free Electrons

In the case of ultimately weak potential (free electron gas),

02

22

=⎟⎟⎠

⎞⎜⎜⎝

⎛− qcm

qr

h ε

rqiq e

rrr

⋅∝ψ222

qmqh

r =ε

22

0

2q

mqh

r =ε

00 ....1 mKqKq

rrrr++ == εε

For free electrons or nearly free electrons, energy is related to the plane wave vector in an obvious way. Inconvenient to use band index. Convenient to use extend zone scheme, i.e. full wave vector.

degenerate case

rKim

jj

rqiq

jeaerrrr

r⋅

=

⋅ ∑=1

ψ

a’s are independent and arbitrary

Nearly Free Electrons: Non-degenerate

22

0

2q

mqh

r =ε∑∑ ∑ ⋅−⋅ =⎭⎬⎫

⎩⎨⎧

+k

krki

k KKkKk

ok

rki cecUcer

rrr

r rrrrrr

rr

εε

First, non-degenerate cases, by which we mean

( ) 0111

0 =+− ∑ −−−−K

KkKKKkKk cUc r rrrrrrrr εε

KKallforUKkKkrr

rrrr ≠′>>− ′−− ||00 εε

Concentrate on eigenstate “derived” from k-K1 plane wave

Expect energies and wave functions to be only slightly modified from FEG case.

factor of U/εsmaller than ck-K1

summing over all K (U0 = 0)

∑∑ ∑ −⋅−′

′−−′−−⋅− =

⎭⎬⎫

⎩⎨⎧

+k

KkrKki

k KKKkKKk

oKk

rKki cecUcer

rrrrr

r rrrrrrrrr

rrr )()( εεTo find

we rewrite the topmost eq.

eigenvalue “derived” from k-K1, not k-K

Kkc rr−

∑≠′

′−−′−−−−+=−

111

)(KK

KkKKKkKKKko

Kk cUcUc rr rrrrrrrrrrrrεε

9

Non-Degenerate NFE

∑′

′−−′−−

− −+

−= 11 00

KK

KkKKKkKKKk

cUcUc

rr r

rrrr

r

rrrr

rr

εεεε

To order of U2,

≠′ −− 1KK KkKkrrrr εεεε

)( 2UO

( ) 0111

0 =+− ∑ −−−−K

KkKKKkKk cUc r rrrrrrrr εε

...1 000 111

1+

−+= ∑ −−−−

K KkKk

KkKKKK

KkKk

cUU

c r rrrr

rrrrrr

rr

rr

εεεε

copied from last page

small

)(||

300

20

1

1

1UO

U

K KkKk

KKKk +−+= ∑

−−

−

− r rrrr

rr

rr

εεεε

Shift in energy is to order of U2. Repelled by other energy levels.

11 −−− K KkKkKk

Perturbation Theory

Equivalently, we could have used perturbation theory and would have gotten the answer quicker

22

0 2∇−=

mH h )(1 rUH

r=

21

20 )(

2Kk

mrrh

−=ε

rKkierrr⋅−∝ )(0 1ψ

...||||||2

111

0 +−

>−−<+>−−−−

+>−=>−−−

−∑ KkU

KkKkKkKk

KK

K

newrrrrrr

rrrr

rr

r εε

1/ KkKk cc rrrr −−=

1KKU rr−=

10

Nearly Free Electrons: m-Fold Near-Degeneracy

Near degeneracy means:

mjiU 1|| 00 =≤εε

mKkKk KKKmiUirrr

rrrr ,...,...,,1,|| 100 ≠=>>−−−

εε

mjiUij KkKk

...,,1,,|| =≤−−−rrrr εε

∑ ′−−′−− += 11 00 KkKKKkKK

Kk

cUcUc

rrrrrrrr

rrIn non-degenerate case the c’s for the plane wave component to the ∑

≠′ −−− −−

1

00KK KkKk

Kk rr rrrr εεεεthe plane wave component to the state derived from k-K1 is

In case of near-degeneracy, c’s are not necessarily small. Eigen states are not (necessarily) dominated by one plane wave component, even for very weak potential.

Degenerate NFE

( ) 0111

0 =+− ∑ −−−−K

KkKKKkKk cUc r rrrrrrrr εεfrom plane wave expansion

di ti i h t

( ) micUcUcm

ijijiiKKK

KkKK

m

jKkKKKkKk ,...,1,

...1

0

1

=+=− ∑∑≠

−−=

−−−− rrrrrrrrrrrrrrrεε

mKKK

KkKK

m

jKkKK

KkKk KKKcUcUc

mjj

rrr

rrrrrrrrrrr

rr

rr ,...,,1 1...1

01

≠⎟⎟⎠

⎞⎜⎜⎝

⎛+

−= ∑∑

≠′′−−′

=−−

−− εε

distinguish two groups

m

m

jKkKK

KkKk KKKUOcUc jj

rrrrrrr

rr

rr ,...,,)(1 12

10 ≠+−

= ∑=

−−−

− εε

11

Degenerate NFE

( ) miUOUUccUc m KKKKKkm

KkKKKkKkji

jjijii,...,1,)( 30

0 =+⎟⎟⎠

⎞⎜⎜⎝

⎛+=− ∑ ∑∑ −−−−−−−

rrrr

rrrrrrrrrr

εεεε

( ) miUOcUc mj

KkKKKkKk jijii,...,1,)( 2

1

0 =+=− ∑=

−−−−rrrrrrrrεε

( )j KKK Kk

Kkj

KkKkKkm

jjijii1 ...1 1

⎟⎠

⎜⎝ −

∑ ∑∑= ≠ −=

rrr rrεε

To first order in U only need to consider the nearly degenerate statesTo first order in U, only need to consider the nearly degenerate states.

Eigenvalue evolves with wave vector. When degeneracy occurs for certain k-K1, it occurs for a small volume around that point in the k-space.

Splitting Near Single Bragg Plane

Only two vectors K1 and K2 leads to near-degeneracy (within U) of the k-K level near a single Bragg plane. Namely, not 3-or more-fold degenerate.

Choose U such that Uq=0=0.

0,|| 00rrr

rrr KKforUKqq ≠′>>− ′−εεUqKq

12

Single Bragg Plane

For non-trivial solutions to be found, 00*

0

=−−−−

KqK

Kq

UU

rrr

rr

εεεε

−KqK

( ) 2200

00

221

KKqq

Kqq U rrrr

rrr +⎟⎟⎠

⎞⎜⎜⎝

⎛ −±+= −−

εεεεε

( ) 0|| 200002 =−++− −− KqKqqKq U rrrrrrr εεεεεε

⎠⎝

Kq U rr ±=0εεFor q lying in the Bragg plane

Additional Comments: Single Bragg Plane

( ) ( ) 22000041

21

KKqqKqq U rrrrrrr +−±+= −− εεεεε 42

⎟⎠⎞

⎜⎝⎛ −=

∂∂ Kq

mqrrh

r 212ε Gradient lies in the Bragg plane.

Constant energy surface perp. to plane.

stationary state (standing wave)

( ) KqKqq cUc rrrrr −=− 0εε

KqKq cUc rrrr −±= )sgn( Assume UK to be real (potential has inversion symmetry)

( ) KqKqqhigher band

lower band

13

Nature of States

rrrrrrrrr

KqKq cUc rrrr −±= )sgn(

0>KU r ⎪⎩

⎪⎨⎧

−=⋅∝

+=⋅∝

Kq

Kq

UrKrUrKr

rr

rr

rrr

rrr

02212

02212

)(sin)()(cos)(

εεψ

εεψIon cores at a/2.

p-like

2/)cos(~ 21)2/()()( rKeee rKqirKqirqi r

rrrrrrr⋅=+ ⋅−⋅−⋅+ψ

2/)sin(~ 21)2/()()( rKieee rKqirKqirqi r

rrrrrrrrr⋅=− ⋅−⋅−⋅−ψ

Direction of propagation?

0

14

3D Examples

FEG energy bands. NoteFEG energy bands. Note degeneracy.

Energy gaps at Bragg planes.

There are as many k-states in a Brillouin zone as there are cells in the Born von Karman box. Two electrons (spin up and down) to each k-state. At 1 electron per unit cell, the lowest band should be exactly half-filled or, in some extreme cases, about half-filled.

Brillouin Zones And Fermi Surfaces

||22

2||

22

Kinner UkK

mE rh −

⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟

⎠⎞

⎜⎝⎛=

||22

2||

22

Kouter UkKE rh +

⎥⎥⎤

⎢⎢⎡

+⎟⎠⎞

⎜⎝⎛=

(1) Draw the free electron Fermi surface

(2) Deform it slightly near Bragg plane

(3) Translate the portion within n-th Brillouin zone through all reciprocal lattice vectors. This constructs a repeated zone Fermi surface.

||22 || Kouter m ⎥

⎥⎦⎢

⎢⎣

⎟⎠

⎜⎝

15

Shape of N-th Brillouin Zones

Free Electron Fermi Surface

valence 4

16

FCC Valence 2 and 3

Lattices with Basis (Identical Atoms)

)()( jdRrrUrrrr

−−= ∑∑ φ

∑∫ −−= ⋅−jR

jrKi

cellKdRrerd

vU

,

)(1r

rr

r

rrrr φ

Geometrical

Structure Factor

)()( jjR

r ∑∑ φ

*)(1 KK SKvU rr

rφ=

r r

∫

∑∫ −= ⋅−j

jrKi

spacedrerd

v)(1

rrr rr φ

∑ ⋅−= dKiK jeSrr

r

Structure Factor)()( rerdK rKispace

rrr r φφ ⋅−∫= j

No splitting on Bragg planes with vanishing structure factors, e.g. basal planes of hcp metal. Convenient to use “Jones zones”: zero-gap planes ignored.

17

Other Contributions To Degeneracy Removal

22

∇H h 20 2∇−=

mH

)(1 rUHr

=

SLHrr⋅=ξ2

Spin-orbit coupling

Spin-orbit coupling is important for heavy elements and for degeneracy not removed by weak periodic potential.

Homework Chapter 8 - 10

Chapter 8: Problems 1

Chapter 9: Problems 1, 3

Chapter 10: Problem 1

Due 3/16/2012