1

5µm

Nanoscale Systems for Opto-Electronics

1.80 1.85 1.90 1.95 2.00 2.05

PL

inte

nsity

[a

rb. u

nits

]

Energy [eV]

700 675 650 625 600

Wavelength [nm]

2

Nanoscale Systems for Opto-ElectronicsLecture 8

Interaction of Light with Nanoscale Systems- general introdcution and motivation- nano-metals (Au, Ag, Cu, Al ...)

introduction to optical propertiesmie scatteringmie scattering in the near-fieldmie scattering with nano rodsresonant optical antennas

- artificial quantum structures (semiconductor quantum dots, ...)- quantum dot lasers

Optical Interactions between Nanoscale Systems- Förster energy transfer (dipole-dipole interaction)- super-emitter concept- SERS (surface enhanced Raman spectroscopy: bio-sensors)

Beating the diffraction limit with Nanoscale Systems- surface plasmon polariton (SPP) - light confinement at nanoscale- plasmonic chips- plasmonic nanolithography

3

QM Bulk Picture

QM: HΨ = E Ψ

Potential for carriers in crystal:→ translational symmetry Va(x) = Va(x+a)

Ψ-function modulated (Bloch ansatz):

→Ψk(x) = uk(x) exp(ikx)

Energy-Dispersion

“modulation of plane waves“

unit cell delocalization

V

x

E(k)

k

CB

VB

4

Introduction – Solid State

Semiconductor

light absorption

relaxation

light emissionE

nerg

y

VB

CB

ΔEgap

E(k)

k

CB

VB

V

x

5

Introduction – Solid State

Semiconductor

light absorption

relaxation

light emissionE

nerg

y

VB

CB

ΔEgap

CB

VB

V

xFree Excitons (Wannier-Mott)

6

Free Exciton SpectroscopyA

bsor

ptio

n, α

Photon Energy

(ħω – Eg)1/2

n=1

n=2

For T < RX/kB: hydrogenic line series observable

E(n) = Eg – RX / n2

Ene

rgy

light absorption

Val

ence

Ban

dC

ondu

ctio

n B

and

7

Excitons in CdSe Bulk - Energetic Aspect

• Binding energy: RX,CdSe = (µ/m0ε) RH 15 meV

with me* = 0.119 me

0 , mh* = 0.5 me

0

→ RX,CdSe / kB = 174 K

•Exciton Bohr radius: aX = (m0 ε / µ) aH 6 nm

→ N = V/V0 = (4/3 π aX3) / (a2c) ≈ 8*105 unit cells

8

Electronic DOS does matter !

Exciton Bohr radius >> crystal dimension

3 D 2 D 1 D 0 D

E E E E

DOS

DOS

bulk

se

mic

ondu

ctor

arti

fici

al a

tom

Early motivation for semiconductor nanostructures

9

Outlook: Squeeze the Exciton Bohr radius

Energy

Small sphere

1-10nm 'particle-in-a-spherical-box' problem

10

Outlook: Synthesis - Bottom-up Approach

20 nm

TEM image of core CdSe nanocrystals Eisler HJ, unpublished data

C.B. Murray, D.J. Norris, and M.G. Bawendi, J. Amer. Chem. Soc. 1993, 115, 8706

T=330ºC

N2

TOPO

Tri-octylphosphineoxide

TOPSe

CdO

11

Concept of Confinement

Quantum dots can be decribed in analogy to a particle in a 3D potential-box (sphere).

Since the actual physical length scale of the semiconductor system is smaller than the exciton Bohr radius or deBroglie wavlength.

In turn, the energies of the carriers appear to be discrete states, rather than being continious.

Look at analytical easy problem:

• particle in spherical potential to get a feeling for the wavefunction and the eigenvalues as a fucntion of sphere radius a.

• adjust the Hamiltonian to be a confined exciton.

• define confinement regimes depending on the energetic dependencies of Coulomb interatcion(1/r) vs. confinement energy (1/r2)

12

Particle in a Sphere

Ψ=Ψ+Ψ∇− εVm

22

2

¯

arrV

arrV

≥∞=<=

if )(

if 0)(

Ψ=Ψ∇− ε22

2m

¯

time-indep. SE

potential

for V=0 (inside the sphere):

with ( ) ( ) ( ) 2

2

222

22

sin

1sin

sin

11

φθθθ

θθ ∂∂+

∂∂

∂∂+

∂∂

∂∂=∇

rrrr

rr

radius of sphere

13

Particle in a Sphere

Ψ=Ψ∇− ε2222 2mrr¯

( ) ( ) ( ) 0sin

1

sin

1

sin

12

2

22 =Ψ

∂∂+−

∂∂

∂∂−

φθθθθθ↓Ψ−

∂∂

∂∂− ε222 2mr

rr

r¯

time-indep. SE(both sides 2mr2)

rearrange to give:

recall:

then

( ) ( ) ( )

∂∂+−

∂∂

∂∂−=

2

2

222

sin

1

sin

1

sin

1ˆφθθθθθ

L

Ψ−

∂∂

∂∂− ε222 2mr

rr

r¯ 0ˆ2 =Ψ+ L

14

Particle in a Sphere

( )( ) 012 2222 =+−Ψ−

∂Ψ∂

∂∂− llmr

rr

r¯ ε

( ) 012 =Ψ++ ll¯Ψ−

∂Ψ∂

∂∂− ε222 2mr

rr

r¯rearrange to give:

recall:

then simplify

( )Ψ+=Ψ 1ˆ 22 llL ¯

( ) 012 2

2

22 =

+−Ψ+

∂Ψ∂

∂∂

llmr

rr

r¯

ε

let 2

2 2

εm

k =

( )( ) 012222 =+−Ψ+

∂Ψ∂

∂∂

llrkr

rr

¯

15

Particle in a Sphere

( )( ) 012 2222 =+−+′+′′ llrkxxrxr ¯

( ) ( )( ) 012 2222 =+−+′+′′ llrkxxrxr ¯

( )( ) 012 222

22 =+−++ llzx

dz

dxz

dz

xdz ¯

replace to give:

solution with spherical bessel functions:

rearrange

( ) ( )φθ ,yrx=Ψ

let krz =

( ) ( )( ) 012 2222 =+−+′+′′ llrkxyxrxry ¯

general solution: spherical Bessel equation ( ) ( ) ( )zByzAjrx ll +=

physical solution: spherical Bessel function first kind ( ) ( )zAjrx l=

16

Particle in a Sphere

find eigenvalues of: ( ) 0=krjl

2

22

2malk

l

αε =Calling αl,k the kth zero of jl, we have

examples of spherical Bessel functions: ( )

( )

( )z

z

z

z

z

zzj

z

z

z

zzj

z

zzj

)sin()cos(3

)sin(3

)cos()sin(

)sin(

232

21

0

−−=

−=

=

αε

α

=

=

am

ka

2

2

¯

with

17

Exciton Schrödinger Equation

Hamiltonian for two independent particles (electron and hole) coupled by a Coulomb term

( ) hehehe

hh

ee rr

e

mmΨΨ=ΨΨ

−

−∇−∇− επεε0

22

22

2

422

¯

r

rh

reR

center of mass coordinate

he

hhee

mmM

withM

rm

M

rmR

+=

+=

he

he

hh

ee

mm

mm

with

m

rRr

m

rRr

+=

+=

+=

µ

µ

µ

reduced mass

18

Exciton Schrödinger Equation

Hamiltonian for two independent particles (electron and hole) coupled by a Coulomb term

( ) hehehe

hh

ee rr

e

mmΨΨ=ΨΨ

−

−∇−∇− επεε0

22

22

2

422

Re

re

ee

eeee

M

m

RM

m

r

r

R

Rr

r

rr

∇

+∇=∇

∂∂

+

∂∂=∇

∂∂

∂∂+

∂∂

∂∂=

∂∂=∇

Rh

rh

hh

hhhh

M

m

RM

m

r

r

R

Rr

r

rr

∇

+−∇=∇

∂∂

+

∂∂−=∇

∂∂

∂∂+

∂∂

∂∂=

∂∂=∇

19

Exciton Schrödinger Equation

Hamiltonian for two independent particles (electron and hole) coupled by a Coulomb term

)()()()(422 0

22

22

2

RrRrr

e

M Rr ΨΨ=ΨΨ

−∇−∇− ε

πεεµ

relcm

cmR

relr

with

RRM

rrr

e

εεε

ε

επεεµ

+=

Ψ=Ψ

∇−

Ψ=Ψ

−∇−

)()(2

)()(42

22

0

22

2

exciton relative motion and center of mass motion

20

Confinement Regimes

In bulk materials: Coulomb term is important for exciton (1/r)In nanocrystals: confinement energy for exciton scales like 1/r2

Strong confinement regime:aNC < aBohr,e , aBohr,h

Optical properties are dominated by quantum confinement effects of electrons and holes

Intermediate confinement regime:aBohr,h < aNC < aBohr,e

Usually the effective mass of the electron is smaller than the effective mass of the hole.

Weak confinement regime: aNC > aBohr,e , aBohr,h

21

Tunability at wish ?

Bulk band gap

bulkghe

exciton Eamam

E ,2

22

2

22

22++= ππ ↓

22

Optical Properties of Artificial Atoms

2 nm 8 nmCdSe

23

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4

no

rma

l. In

ten

sity

Energy [eV]

800 700 600 500 400

1.47 470

1.85 950

2.5 2350

3.05 4220

Reff

(nm) #atoms

Wavelength [nm]

Optical Properties of Artificial Atoms

VR(r) = VR(r+a) “spherical potential“ as boundary condition

Φik(r) = uk(r) Ylm J(κ nlr/R) / Jl+1(κ nl) spherical harmonics and bessel fct

for i = e- and h+

ε nli = (ħ / 2mi) κnl

2 / k2 → n=1, l=0 (S state) κ nl = π

24

Refinement

•Introduction of Coulomb interaction (e- and h+) as perturbation (only important for large nanocrystals with sizes near Bohr radius; → strong confinement regime)

Econf prop. 1/R2

Ecoulomb prop. 1/R

•Valence band mixing: interaction of h+ states due to confinement

•Crystal field splitting

•Ellipsoidal shape of nanocrystal: effective radius Reff = (b2 c)1/3

•Exchange interaction of e- and h+ spins

25

Wavefunctions of Excitons

radial probability functionJ. Phys. Chem B Vol. 101, No.46, pp. 9463, 1997

26

Optical Properties of Type-I Artificial Atoms

27

[CdSe]core{ZnS}shell Type-I Heterostructure

M. A. Hines, P. Guyot-Sionnest, J. Phys. Chem. 1996, 100, 468-471.B. O. Dabbousi et al., J. Phys. Chem. B 1997, 101, 9463-9475.

400 500 600 700 800

Abs

orb

anc

e, P

hoto

lum

ines

cenc

e

Wavelength [nm]

400 500 600 700 800

Abs

orb

anc

e, P

hoto

lum

ines

cenc

e

Wavelength [nm]

ZnEt2

(TMS)2S

~200oCTOP/TOPO

28

Wavefunctions of Excitons in Core-Shell Systems

radial probability functionJ. Phys. Chem B Vol. 101, No.46, pp. 9463, 1997

29

Wavefunction2 in STS and STM

Phys. Rev. Lett. Vol. 86, No. 24, pp. 5751 (2001)

InAs/ZnSe

30

Optical Properties of Type-II Artificial Atoms

S. Kim, B. Fisher, H.-J. Eisler, M. G. Bawendi, J. AM. CHEM. SOC. 125, 11466 (2003)

31

Optical Properties of Type-II Artificial Atoms

S. Kim, B. Fisher, H.-J. Eisler, M. G. Bawendi, J. AM. CHEM. SOC. 125, 11466 (2003)

CdSe

32

Optical Properties of Type-II Artificial Atoms

S. Kim, B. Fisher, H.-J. Eisler, M. G. Bawendi, J. AM. CHEM. SOC. 125, 11466 (2003)

CdSe

33

Absorption and Photoluminescence

34

Absorption

Lambert-Beer Law:

[cm]path optical

][cmt coefficien absorption

with1-

0

l

eII l

α

α−=

]/cm[#ion concentratn

][cmsection cross absorption 3

2σσα n=

35

Absorption

[cm]path optical

][cmt coefficien absorption

with1-

0

l

eII l

α

α−=

[cm]path optical

]cm[Mt coefficien extinctionmolar

absorbance 1-1-

l

A

clA

ε

ε=

36

Absorption

[cm]path optical

][cmt coefficien absorption

with1-

0

l

eII l

α

α−=

const. Avogadro

)(

)1000)(303.2(

)(log

)1000(

aN

N

Ne

a

a

εσ

εσ

=

=

37

Absorption and Photoluminescence