Post on 21-Jul-2020
Lecture 1
Periodicity and Spatial ConfinementPeriodicity and Spatial ConfinementCrystal structureTranslational symmetryEnergy bandsk·p theory and effective mass Theory of invariantsHeterostructures
J. Planelles
SUMMARY (keywords)
Lattice → Wigner-Seitz unit cell
Periodicity → Translation group → wave-function in Block form
Reciprocal lattice → k-labels within the 1rst Brillouin zone
Schrodinger equation → BCs depending on k; bands E(k); gaps
Gaps→ metal, isolators and semiconductorsGaps→ metal, isolators and semiconductors
Machinery: kp Theory → effective mass J character table
Theory of invariants: ΓΓΓΓƒ ΓΓΓΓœA1; H = ΣΣΣΣ NiΓΓΓΓ k i
ΓΓΓΓ
Heterostructures: EFA
QWell QWire QDotoffsetbandVtconfinemen
ipk
c
ˆ
====→→→→∇∇∇∇−−−−====→→→→
Crystal structure
A crystal is solid material whose constituent atoms, molecules
or ions are arranged in an orderly repeating pattern
+ +
Crystal = lattice + basis
Bravais lattice: the lattice looks the same when seen from anypoint
a1
P’ = P + (m,n,p) * (a1,a2,a3)
P
a1
a2
integers
P
a1a2
Unit cell: a region of the space that fills the entire crystal by translation
Primitive: smallest unit cells (1 point)
Wigner-Seitz unit cell: primitive and captures the point symmetry
Centered in one point. It is the region which is closer to that point than to any other.
Body-centered cubic
How many types of lattices exist?
C ?4
How many types of lattices exist?
C ?6
How many types of lattices exist?
14 (in 3D)
C ?5
Bravais Lattices
We have a periodic system.
Is there a simple function to approximate its properties?
a
)(xρ
Translational symmetry
)()(|)(||)(|)()( 22 xfenaxfnaxfxfnaxx iφρρ =+⇒+=⇔+=
→+= )()( naxfxfTn GroupnTranslatioTn →GroupAbelianTT mn 0],[ →=
q
q
qpnai
n pq
naieT ˆ
!
) (ˆ ∑== )()(!
)()()( ˆ anxf
dx
fdi
q
ianxfexfT
q
q
qpian
n ++++====−−−−======== ∑∑∑∑
Eigenfunctions of the linear momentum
basis of translation group irreps
ikxiankikxpianikxn eeeeeT == ˆ
MLMLMM
LL nTE
q
q
qpnai
n pq
naieT ˆ
!
) (ˆ ∑∑∑∑========
MLMLMM
LL
MLMLMMikxikna eek 1
Range of k and 3D extension
irrep A symmetricfully thelabels 0k ; ],[ 1=−∈aa
kππ
3D extension → rx
Bloch functions as basis of irreps:
iknam]na
a
πi[k
ee =+ 2
same character:Zmma
kkk ∈+= ,2
'~π
3D extension
→ kkBloch functions as basis of irreps:
)()( );()( rarrr rkk uuuei ====++++==== ⋅⋅⋅⋅ΨΨΨΨ
)()()( ( rarr rkaka)rkka ueeueT i
character
ii ⋅⋅+⋅ =+=Ψ
Reciprocal Lattice
1 :2
''~ :1 ===−→ iKaea
πKkkkkD
vectors) (lattice eD i c b,a,aKkk'k'k iaK i ===−→ ⋅ ,1 :~ :3
Z∈××
=++= i321 p ,)(
)( 2 ,ppp ?
ikj
kji321 aaa
aakkkkKK π
ipπ 2=⋅ iaK ,, lattice l reciproca →kkk
First Brillouin zone: Wigner-Seitz cell of the reciprocal lattice
ipπ 2=⋅ iaK ,, lattice l reciproca →321 kkk
(-1/2,1/2) zy, x,,zy x ,0 : ∈++==Γ 321 kkkkk
Solving Schrödinger equation: Von-Karman BCs
Crystals are infinite... How are we supposed to deal with that?
We use periodic boundary conditions
Group of translations:k is a quantum number due to
translational symmetry
We solve the Schrödinger equation for each k value:
The plot En(k) represents an energy band
],[ ),2/()2/(
)( )(
ππππππππφφφφΨΨΨΨΨΨΨΨ
ΨΨΨΨΨΨΨΨφφφφ −−−−
⋅⋅⋅⋅
∈∈∈∈====
====
aea-
rerT
ki
k
kaki
ka
1rst Brillouin zone
How does the wave function look like?
0]ˆ,ˆ[ =HT Hamiltonian eigenfunctions are basis of the Tn group irreps
We require Ψ=Ψ tkieTrr
ˆ
)()( ruer krki
k
rr rr
=Ψ Bloch function
Periodic (unit cell) partEnvelope part
)()( rutru kk
rrr =+
Envelope part
Periodic part
Energy bands
How do electrons behave in crystals?
quasi-free electrons
ions
)(xVc
)()()(2
2
xxxVm
pkkkc Ψ=Ψ
+ ε
)(xVck >>ε Empty lattice
xkik eNx =Ψ )(
a
pk
π2= Plane wave
)(Ψ )(Ψ: xeaxBC k
ika
k ====++++
)(xVc
)()(ˆ xkxp kk Ψ=Ψ hm
kk 2
22h=ε
kε
FεT=0 K
Fermi energy
m
kk 2
22h=ε
k
Fε
FkFk−
Fermi energy
m
kk 2
22h=ε
)(Ψ )(Ψ: xeaxBC k
ika
k ====++++
Band folded into the first Brillouin zone
ε (k): single parabola
1 :2
''~ ============−−−−→→→→ iKaea
πKkkkk
folded parabola
)()()(2
2
xxxVm
pkkkc Ψ=Ψ
+ ε
Znnka ∈= ,2π
Bragg diffraction
ka/2π0
a
gap
a/4πa/2π−
)()()(2
2
xxxVm
pkkkc Ψ=Ψ
+ ε
kε
fraction eV
Types of crystals
kε
Fermi
levelgap
Conduction Band
kε
few eV
Valence
Band
k
Semiconductor
• Switches from conducting
to insulating at will
k
Metal
• Empty orbitals
available at low-energy:
high conductivity
k
Insulator
• Low conductivity
Band
k·p Theory
How do we calculate realistic band diagrams?Tight-binding
Pseudopotentials
k·p theory
)()( ruer krki
k
rr rr
=Ψ
+= )(
2ˆ
2
rVm
pH c
rr
)()(ˆ rerHe krki
kkrki rr rrrr
Ψ=Ψ −− ε
)()(2
)(2
222
rurum
pk
m
krV
m
pkkkc
rrrr
hhr
r
ε=
⋅+++
The k·p Hamiltonian
MgO
k=Γk=0
CB
HH
LH
Split-off
uCBk
uHHk
uLHk
uSOk
k
)()(2
)(2
222
rurum
pk
m
krV
m
pkkkc
rrrr
hhr
r
ε=
⋅+++
k=Γ
)()( 0 rucru n
nnk
nk
rr∑∞
=
'00',
22
0'
00 2
||ˆ nnnn
nnkp
n upum
k
m
kuHu
rr
h
rh +
+= δε
gapKane
parameter
CB
HH
LH
uCBk
uHHk
uLH
)()( 0 rucru n
nnk
nk
rr∑∞
=
4-band H
1-band H
'00',
22
0'
00 2
||ˆ nnnn
nnkp
n upum
k
m
kuHu
rr
h
rh +
+= δε
k=0
LH
Split-off
uLHk
uSOk
k
4-band H
↑s ↓s
↑sm
k
2
2
0
↓s 0
m
k
2
2
1-band H
k=0
CB
HH
LH
Split-off
uCBk
uHHk
uLHk
uSOk
k
kPkPkPkPkPk
s
ss
210
120
2
1,
2
1
2
1,
2
1
2
3,
2
3
2
1,
2
3
2
1,
2
3
2
3,
2
3
2
−−−↑
−−−↓↑
8-band H
m
kkPkP
m
kkPkP
m
kkP
m
kkPkP
m
kkPkP
m
kkP
kPkPkPkPkPm
ks
kPkPkPkPkPm
ks
z
z
z
z
zz
zz
200000
3
1
3
2
2
1,
2
1
02
00003
2
3
1
2
1,
2
1
002
00002
3,
2
3
0002
003
2
3
1
2
1,
2
3
00002
03
1
3
2
2
1,
2
3
000002
02
3,
2
33
1
3
2
3
2
3
10
20
3
2
3
10
3
1
3
20
2
2
0
2
0
2
0
2
0
2
0
2
0
2
+∆−−−−
+∆−−
+−−
+−−−−
+−−
+−
−↓
−−−↑
−
−
−
−
−
+
−−−
−−+
ε
ε
ε
ε
ε
ε
One-band Hamiltonian for the conduction band
'00',
22
0'
00 2
||ˆ nnnn
nnkp
n upum
k
m
kuHu
rr
h
rh +
+= δε
m
kcbcbk 2
|| 22
0
rh+= εε
This is a crude approximation... Let’s include remote bands perturbationally
∑∑≠= −
++=cbn
ncb
ncb
zyx
cbcbk
upuk
mm
k
00
2
0022
2
,,
22
0 ||2
||
εεεε
αα
α
α hh
1/m*Effective mass
*
22
02 αααα
ααααεεεεεεεεm
kcbcbk
h++++====
Free electron InAs
m=1 a.u. m*=0.025 a.u.
k
CB
kε
negative mass?
22
1
*
1
km
cbk
∂∂= ε
h
Theory of invariants
1. Perturbation theory becomes more complex for many-band models
2. Nobody calculate the huge amount of integrals involved
grup them and fit to experimentgrup them and fit to experiment⇒
Alternative (simpler and deeper) to perturbation theory:
Determine the Hamiltonian H by symmetry considerations
Theory of invariants (basic ideas)
1. Second order perturbation: H second order in k:j
jiiij kkMH ∑
≥=
3
2. H must be an invariant under point symmetry (Td ZnBl, D6h wurtzite)
A·B is invariant (A1 symmetry) if A and B are of the same symmetry
e.g. (x, y, z) basis of T2 of Td: x·x+y·y+z·z = r2 basis of A1 of Td
Theory of invariants (machinery)
1. k basis of T2 2. ki kj basis of ][ 12122 TTEATT ⊕⊕⊕=⊗
3. Character Table:
) (
,,
,2
1
2
22222
2221
tensorsymmetrickkNOT
kkkkkkT
kkkkkE
kkkA
ji
zyzxyx
yxyxz
zyx
→
→
−−−→
++→ notation: elements
of these basis: . Γik
) ( 1 tensorsymmetrickkNOT ji→
4.Invariant: sum of invariants:
fitting parameter
(not determined by symmetry)
ΓΓ
Γ
ΓΓ∑ ∑= i
ii kNaH
)dim(
irrep
basis element
Machinery (cont.)How can we determine the matrices?Γ
iN
→→→→ we can use symmetry-adapted JiJj products
11221 , ),,( TTTTandTofbasisJJJ zyx ⊗⊗⊗⊗====⊗⊗⊗⊗
Example: 4-th band model: >−>−>> 2/3,2/3|,2/1,2/3|,2/1,2/3|,2/3,2/3| >−>−>> 2/3,2/3|,2/1,2/3|,2/1,2/3|,2/3,2/3|
Machinery (cont.)We form the following invariants
Finally we build the Hamiltonian
Luttinger parameters: determined by fitting
Four band Hamiltonian:
Exercise: Show that the 2-bands |1/2,1/2>,|1/2,-1/2> conduction band k·p Hamiltonian reads H= a k2 I , where I is the 2x2 unit matrix, k the modulus of the linear momentum and a is a fitting parameter (that we cannot fix by symmetry considerations)
Hints: 1.2. Angular momentum components in the ±1/2 basis: Si=1/2 σi, with
][ 1211122 TTEATTTT ⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕⊕====⊗⊗⊗⊗====⊗⊗⊗⊗
3. Character tables and basis of irreps
yzzyxzzxxyyx
yzzyxzzxxyyx
yxyxz
zyx
kkkkkkkkkkkkT
kkkkkkkkkkkkT
kkkkkE
kkkA
'','',''
,,
,2
1
2
22222
2221
−−−−−−−−−−−−→→→→
++++++++++++→→→→
−−−−−−−−−−−−→→→→
++++++++→→→→
Heterostructures
How do we study this?
Brocken translational symmetry
B A B
z
B A B
z
z
A
quantum well quantum wirequantum dot
Heterostructures
How do we study this?
If A and B have:
...we use the “envelope function approach”
rr rr
• the same crystal structure
• similar lattice constants
• no interface defects
rr rr
B A B
)()( ruer krki
k
rr rr
=Ψ )()()( ruzer krki
k
rr rr
χ⊥⊥=Ψ
Project Hkp onto Ψnk, considering that:
Envelope part
Periodic part
∫∫∫ΩΩ
⋅⋅Ω
≈ drrfdrrudrrurfcellunit
nknk )()(1
)()(
z
Heterostructures
B A B
In a one-band model we finally obtain:
)()(2
)(2
22
2
22
zzm
kzV
dz
d
mχεχ =
++− ⊥hh
0, =kBεz V(z)
0, =kAε
0, =kBε
1D potential well: particle-in-the-box problem
Quantum well
Most prominent applications:
• Laser diodes
• LEDs
• Infrared photodetectors
Quantum well
Image: CNRS France
Image: C. Humphrey, Cambridge
B A B
zImage: U. Muenchen
Quantum wire
Most prominent applications:
• Transport
• Photovoltaic devices
),(),(2
),()(2
2222
2
zyzym
kzyV
mx
zy χεχ =
++∇+∇−hh
Image: U. Muenchen
),,(),,(),,(22
zyxzyxzyxV χεχ =
+∇− h
z
A
Quantum dot
Most prominent applications:
• Single electron transistor
• LEDs
• In-vivo imaging
• Cancer therapy
• Photovoltaics
• Memory devices
• Qubits?
),,(),,(),,(2
2 zyxzyxzyxVm
χεχ =
+∇− h
SUMMARY (keywords)
Lattice → Wigner-Seitz unit cell
Periodicity → Translation group → wave-function in Block form
Reciprocal lattice → k-labels within the 1rst Brillouin zone
Schrodinger equation → BCs depending on k; bands E(k); gaps
Gaps→ metal, isolators and semiconductorsGaps→ metal, isolators and semiconductors
Machinery: kp Theory → effective mass J character table
Theory of invariants: ΓΓΓΓƒ ΓΓΓΓœA1; H = ΣΣΣΣ NiΓΓΓΓ k i
ΓΓΓΓ
Heterostructures: EFA
QWell QWire QDotoffsetbandVtconfinemen
ipk
c
ˆ
====→→→→∇∇∇∇−−−−====→→→→