l04 band structuresangalet/stato solido/l04_band...fisica dello stato solido - i modulo bande...

From weak periodic potentialsto band structure

Bande elettronicheFisica dello Stato solido - I Modulo

Università Cattolica del Sacro Cuore A.A. 2005-06

One can gain substantial insight into the structure imposed on the electronic energylevels by a periodic potential, if that potential is very weak.

When the periodic potential is zero, the solution to the Schroedinger Equation are planewaves (unperturbed states). A reasonable starting place for the treatment of weakperiodic potential is therefore the expansion of the exact solution in plane waves.

We now provide a second proof of the Bloch Theorem, that will allow us to look forsolutions of the Schroedinger equation with periodic U in terms of plane waves. Thisformulation of the Bloch theorem will be useful to find a solution when U(r) can betreated as a perturbation.

real) is )r U((since and electrons) (free 0

)exp()(1 )exp()(

)exp()(

r

rrrrrrr

rrr

rr

rr

r

rr

*GG-

cellG

GG

qq

UUU

rdrGirUv

UwithrGiUrU

rqicr

==

⋅−=⋅=

⋅=Ψ

∫∑

∑



Lets us substitute the first two equations in the Schrödinger equation

The original problem is split up in N indipendent problems (one for each k). ck coupleswith all ck’ with k-k’=G. This is the second Proof of the Block Theorem.

( ) 02

''..1,

02

)exp(by multipling

0'2

)'(exp

)'exp())(exp()()(

)exp(22

)(2

'''

22

22

'2

2

','

,

22

222

'

=+

−−

−→∈−→

=+

−

−

=

+

−⋅

=⋅=⋅+=

⋅=Ψ∇−=

∑

∑

∫

∑∑

∑∑

∑

−−−

−

−

−

GGkGGGk

st

GGqGq

GGqGq

qGGqG

qGqG

qq

cUcGkm

GGGZBkGkq

cUcqm

riqrd

cUcqm

rqi

rqicUrGqicUrrU

rqicqmm

rmp

q

rrrrrrr

rrvrv

rrrv

rrrvr

rrvr

vr

rrh

r

rh

rr

rhrr

rrrrrrr

rrrhhr

r

ε

ε

ε

ψ

ψ



G

qk

G

The original problem is split up in N indipendent problems (one for each k). ck coupleswith all ck’ with k-k’=G. For a given k in the 1st BZ, there are infinite solutions to the system of infinite equations we have obtained. These solutions are labelled by n, the band index.

If we write this as

then this is the Block form with the periodic function u(r) given by:

∑ ⋅−=Ψ −G

Gkk rGkicr r rrrrrrr )(exp)(

))exp((exp)( ∑ ⋅−⋅=Ψ −G

Gkk rGicrkir r rrrrrrrr

)exp()( ∑ ⋅−= −G

Gk rGicru r rrrrr

Case U=0( )

( ) ( )( ) ( )

( ) ( )( ) ( ) )'exp( ofn combinatiolinear any is ,

2

' ofset finite afor ' If

solutions space free )exp(,2

'any for ' If

02

22

22

22

22

22

rGkiGkm

E

GGkGk

rGkiGkm

E

GGkGk

cEGkm Gk

rrrrrh

rrrrr

rrrrrh

rrrrr

rrhrr

⋅−Ψ−=

−=−

⋅−∝Ψ−=

−≠−

=

−−

−





What happens when we consider U as a weak perturbation to the free electron behavior?

We can treat the Schroedinger equation solution with stationay perturbation theory.Two cases can be considered, i.e. the non-degenerate and the degenerate cases(see. Ashcroft and Mermin, Chapt.9). We now discuss the degenerate case.

When the value of k is such that there are reciprocal lattice vectors K1,..,Km with all within order U of each other, but far apart from the other

energies on the scale of U,

the approximate solution, to leading corrections in U, can be shown to be:

00 ,,1 mKkKk

rrrr L−−

εε

mKkKk KKKmiUirrr

L rrrr ,...,,,...,1,,, 100

1≠=>>

−−εε

( ) micUc mj

KkKKKkKk jijii,...,1

1

0 ==− ∑=

−−−−rrrrrrrrεε

The simplest and most importantexample is when two free electron levelsare within order U of each other, but far compared with U form all other levels.

At ½ G, we have: k2=(k-G)2. We keep only the terms with ck and ck-G1:

We solve the secular equation with U1= UG1= U-G1

( )( ) 0

0

11

11

0

0

=−+

=+−

−−

−

GkkkkG

GkGkkk

cEEcU

cUcEE

0

0

1

10

=−

−

kk

kk

EEUUEE

( ) ( )2/1

21

20011 4

121

+−±+=⇒ −− UEEEEE GkkGkkk





The energy correction due to the weak perturbation are easily calculated at the Bragg plane.At the ½ G1 point in k-space, we have: k2=(k-G1)2. and therefore

Once the eigenvalues have been extracted, we can calculate the correspondingeigenvectors, through the ck. We have:

UEEandEE kkGkk ±== −000 ,

( )

( )

( )

( )

( )1

1

1

1

11

02

12

02

12

02

12

02

12

1

,21cos

,21sin

0

,21sin

,21cos

0

sgn

Gk

Gk

G

Gk

Gk

G

GkGk

UEErGr

UEErGr

UWhen

UEErGr

UEErGr

UWhen

cUc

rr

rr

r

rr

rr

r

rrrr

vrr

vrr

vrr

vrr

−=

⋅∝

+=

⋅∝

<

−=

⋅∝

+=

⋅∝

>

±=−

ψ

ψ

ψ

ψ

sin(…) -> p-like waves, cos(…) -> s-like waves

It is easy to verify that when k is on the Bragg plane, from the equation

one obtains:

i.e., when k is on a Bragg plane the gradient of E(k) is parallel to the plane.

Since the gradient is perpendicular to the surfaces on which a function is constant, the contant-energy surfaces at the Braggplane are perpendicular to the plane.

−=

∂∂ Gk

mkE rrhr

212



( ) ( )2/1

21

20011 4

121

+−±+= −− UEEEEE GkkGkkk

k

k-G

G/2G/2

G

0

k-G/2

Nearly Free Electron ModelThe presence of a periodic potential leads to the existence of a forbidden energy region, or band gap, near k=G/2

G

G/2



2|U|

Energy bands

The two quantum numbers n and k have quite different meanings. The band index is aninteger. The allowed values of wave vector k are fixed by the boundary conditions in the solid.



1D Energy BandFree electron, parabolic, energydispersion curve

Distortion of the free parabola due to a weakperiodic potential

Extended-zone scheme

Reduced-zone scheme

Repeated-zone scheme



Energy bands in 3 DimensionsFree electronenergy levels foran fcc Bravaislattice



l04 band structuresangalet/stato solido/l04_band...fisica dello stato solido - i modulo bande...

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