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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TEM image of core CdSe nanocrystals Eisler HJ, unpublished data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