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

qq

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

====→→→→∇∇∇∇−−−−====→→→→