bloch’s theorem - zhejiang...
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BlochBloch’’s theorems theorem
The The eigenstateseigenstates of such a oneof such a one--electron Hamiltonian can be written aselectron Hamiltonian can be written as
lattice Bravais ain allfor )()( where),(2
22
RrURrUrUm
H
General properties for a singeGeneral properties for a singe--electron Hamiltonianelectron Hamiltonian
)()( ruer knrki
kn
wherewhere
)()( ruRru knkn
Another statement of BlochAnother statement of Bloch’’s theorems theorem
The The eigenstateseigenstates of the Hamiltonian can be chosen to satisfy thatof the Hamiltonian can be chosen to satisfy that
)()( reRr Rki
Proof of BlochProof of Bloch’’s theorem:s theorem:
Defining a lattice translation operatorDefining a lattice translation operator RT
)()( RrfrfTR
Lattice translational symmetryLattice translational symmetry
RR HTRrrHRrRrHHT )()()()(
RR HTHT
On the other handOn the other hand
''' RRRRRR TTTTT
Now we will find the eigenvaluesNow we will find the eigenvalues )(Rc
)'(
)'()()'(
''
'
RRcTTT
RcRcTRcTT
RRRR
RRR
)'()()'( RcRcRRc
332211 anananR LetLet
321 )()()()( 321nnn acacacRc
RT Each is associated with a good quantum number, Each is associated with a good quantum number, consequencelyconsequencely, the , the eigenstateseigenstates can be chosen as can be chosen as eigenstateseigenstates of simultaneously. of simultaneously. RT
)(RcTEH
R
where are three primitive vectors for the Bravais latticewhere are three primitive vectors for the Bravais lattice..ia
We can always write the asWe can always write the as)( iac
ixii eac 2)(
IntroducingIntroducing332211 bxbxbxk
where are the reciprocal lattice vectors satisfying where are the reciprocal lattice vectors satisfying ,,ib
ijji ab 2
we find thatwe find thatRkieRc)(
So thatSo that
)()( reRreT RkiRkiR
BornBorn--von Karman boundary conditionvon Karman boundary condition
)()( raNr ii
integral ,
11 2
ii
ii
xNiakiN
mNmx
ee iiii
The general form of allowed Bloch wave vectorThe general form of allowed Bloch wave vector
integral ,3
1i
ii
i
i mbNmk
Volume occupied by per allowed wave vector in kVolume occupied by per allowed wave vector in k--spacespace
V
bbbNN
bNb
Nbk
3
3213
3
2
2
1
1 )2(1
rqi
qqecr
)(
rKi
KKeUrU
)(
cell
rKiK rUerd
vU )(1
Another proof of BlochAnother proof of Bloch’’s theorems theorem
We can expand any function satisfying periodic boundary conditioWe can expand any function satisfying periodic boundary condition as follows, n as follows,
On the other hand, the periodic potential can be expanded asOn the other hand, the periodic potential can be expanded as
where the Fourier coefficients read where the Fourier coefficients read
Then we can study the SchrThen we can study the Schröödinger equation in dinger equation in kk-- space.space.
vector in reciprocal lattice
rqiq
q
ecmq
mmp
222
22222
rqiKq
qKK
rqKiq
qKK
rqi
rKi
KK
ecUecU
eceUU
''
',
)(
,
02 '
''
22
q KKqKq
rqi cUcEmqe
Kinetic partKinetic part
The term in potential energyThe term in potential energy
SchrSchröödinger equation in dinger equation in kk-- space.space.
02 '
''
22
KKqKq cUcE
mq
KKKKkq '',
02
)('
''
22
K
KkKKKk cUcEm
Kk
SchrSchröödinger equation in dinger equation in kk-- space.space.
For any fix in the first Brillouin zone, it only couples to ,Then the wave function will be of the form
k
,',, KkKkk
)()( )( rueeceecr rkirKi
KKk
rkirKki
KKkk
rKi
KKk ecru
)(where satisfies )()( ruRru
General remarks on BlochGeneral remarks on Bloch’’s theorems theorem
Wave vector , crystal momentum , and electron momentumWave vector , crystal momentum , and electron momentum
Equivalence between different wave vectorsEquivalence between different wave vectors
Band indexBand index
For a given , we expect to find infinite family of solutions For a given , we expect to find infinite family of solutions with discretely spaced with discretely spaced energy eigenvalues, which we label with the band index energy eigenvalues, which we label with the band index nn..
Band structure: Band structure: Periodic functions of in the reciprocal spacePeriodic functions of in the reciprocal space
CollisionlessCollisionless electron movement in a perfect crystalelectron movement in a perfect crystal
k
k
i
p
)())(( rui
ekrueii
p knrki
knknrki
knkn
not a momentum eigenstate,broken continuous symmetry
k
k
knKknknKkn EErr
,,,, ),()(
)(1)( kEkv nkn
It contradicts Drude’s ansatz
)()()()()(
)()(,'by Replace
')(
''
'
ruRrueRrurueru
rueerueKkkk
knknRrKi
knknrKi
kn
knrKirki
knrki
Electrons in a weak periodic Electrons in a weak periodic potentialpotential
•• Perturbation theory:Perturbation theory:•• Applicability: Applicability:
–– Metals in groups, I, II, III and IV, Metals in groups, I, II, III and IV, ss and and pp electronselectrons–– Pauli exclusion, conduction electronPauli exclusion, conduction electron--ion separation is not ion separation is not
very small, where core electrons have occupied the very small, where core electrons have occupied the immediate neighborhood of the ions.immediate neighborhood of the ions.
–– Screening effect will reduce the total effective potential.Screening effect will reduce the total effective potential.
Generic perturbation approachGeneric perturbation approach
,02
)('
''
22
K
KkKKKk cUcEm
Kk
.)( )( rKki
KKkk ecr
We begin with SchrWe begin with Schröödinger equation in periodic potential in kdinger equation in periodic potential in k--space space
with Bloch wave function with Bloch wave function
For real potential, we haveFor real potential, we have
,)( *KK UU
cell
rUrdv
U 0)(10
and can choose
by shifting the energy zero.
Free electronFree electron
,02
)(022
Kkk cE
mKkU
Introducing Introducing nonperturbativenonperturbative energyenergymqEq 2
220
rKkikKk eEE
)(0 ,
Issue: Issue: For a given there may be several reciprocal lattice vectorFor a given there may be several reciprocal lattice vectors s satisfyingsatisfying
k
,,,1 mKK
001 mKkKk EE
Case 1: Fix and consider a particular reciprocal lattice veCase 1: Fix and consider a particular reciprocal lattice vector such that ctor such that
the free electron energy is far away from the values the free electron energy is far away from the values of (for all of (for all other ) compared withother ) compared with
Case 2: Suppose the value of is such that there are recipCase 2: Suppose the value of is such that there are reciprocal lattice rocal lattice
vectors with all vectors with all within order of each other, but within order of each other, but far apart from the on the scale offar apart from the on the scale of
k
1K
01KkE
0
KkE
K
U
100 all and fixedfor ,||
1KKkUEE KkKk
k
,,,1 mKK
00 ,,
1 mKkKk EE U
0KkE
U
mKkKk KKKmiUEEi
,,,,,1,|| 1
00
Case 1: Case 1: nondegeneratenondegenerate perturbation theoryperturbation theory
'
''0 )(
KKkKKKkKk cUcEE
K
KkKKKkKk cUcEE
111
)( 0
|,||| on,perturbati ofspirit in the ,For 11 KkKk ccKK
)( 20
'0
''0
'0
''
11
1
11
UOEEcU
EEcU
EEcU
EEcU
c
Kk
KkKK
KK Kk
KkKK
Kk
KkKK
K Kk
KkKKKk
KKk
Kk
KKKKKkKk UOc
EEUU
cEE
)()( 30
01
11
11
K Kk
KKKk UO
EEU
EE
)(||
30
20 1
1
The leading correction is of the order of The leading correction is of the order of UU22
Case 2: degenerate perturbation theoryCase 2: degenerate perturbation theory
mijij
iii
KKKKkKK
m
jKkKK
KKkKKKkKk
cUcU
cUcEE
,1
0
1
)(
,,,For 1 mKKK
)(1
1
2
10
'',,'1
01
UOcUEE
cUcUEE
c
m
jKkKK
Kk
KkKKKKK
m
jKkKK
KkKk
jj
mjj
)(
)(
3
,0
1
1
0
1
UOcEE
UU
cUcEE
jm
ji
jijii
KkKKK Kk
KKKKm
j
m
jKkKKKkKk
)(1 2
10 UOcU
EEc
m
jKkKK
KkKk jj
The leading corrections in The leading corrections in UU is given byis given by
additional terms
micUcEEm
jKkKKKkKk jijii
,,1,)(1
0
The leading correction is of the order of The leading correction is of the order of UU but not but not UU22..
Energy levels near a single Bragg planeEnergy levels near a single Bragg plane
12122
21211
)(
)(0
0
KkKKKkKk
KkKKKkKk
cUcEE
cUcEE
Degenerate perturbation theory, only two reciprocal lattice vectors K1 and K2 are involved.
121, KKKKkq
KqKKqKq
KqKqq
cUcEE
cUcEE
)(
)(0
0
twotwo--level problemlevel problem
0,'for ||, 0'
000 KKUEEEE KqqKqq
KqKKqKq
KqKqq
cUcEE
cUcEE
)(
)(0
0
2/1
2200
00
2)(
21
KKqq
Kqq UEE
EEE
When When qq is on the Bragg planeis on the Bragg plane
Kq
Kqq
UEE
EE
0
00
Energy bands in one dimensionEnergy bands in one dimension
SummarySummary• Bloch’s theorem
– The concept of lattice momentum– The wave function is a superposition of plane-wave states with
momenta which are different by reciprocal lattice vectors– Periodic band structure in k-space– Short-range varying potential → extra degrees of freedom →
discrete energy bands– Coherent (non-dissipative) motion of electrons in a perfect crystal
• Applicability of weak potential• The leading order correction by weak potention
– Non-degenerate case: U2 correction– Degenerate case: U correction