2004 Debye Lecture 3
C. B. Murray
Semiconductor Nanocrystals
Quantum Dots Part 1
Basic Physics of Semiconductor Quantum Dots
C. R. Kagan, IBM T. J. Watson Research Center,
Yorktown Heights, NY
Lowest
Unoccupied
Molecular
Orbital
Conduction
Band
Energy
Gap
Highest
Occupied
Molecular
Orbital
Valence
Band
Bulk Semiconductor
Quantum Dot
Like a
Molecule
Quantum Confinement
Low Dimensional Structures
ρ c (E ) ∝
(E − EC )
ρ c (E ) = cons tan t
ρ c (E ) ∝
1
(E − En )
ρ c (E ) ∝ δ (E − E n )
Particle-in-a-Sphere
j l (k n,l r )Yl m (θ ,φ )
Φ (r ,θ ,φ ) = C
r
a
Yl m (θ ,φ ) is a spherical harmonic
Potential V
∞
2s
1s
0
j l (k n,l r ) is the lth order spherical Bessel function
k n,l =
α n,l
a
r
solutions give
hydrogen-like orbitals with
quantum numbers
n (1, 2, 3 …)
l (s, p, d …)
m
E n,l
η2k n2,l
η2α n2,l
=
=
2mo
2mo a 2
Discrete energy levels
size-dependence
The Quantum Dot is a Semiconductor
The Effective Mass Approximation
parabolic conduction and valence bands
Bloch’s Theorem
ρ ρ
ρ
ρ
Ψnk (r ) = unk (r )exp ik ⋅ r
(
)
E
n=conduction band
with periodicity of crystal lattice
η2 k 2
E =
+ Eg
c
2meff
c
k
Eg
Free particles treated
by effective mass:
k
n=valence band
η2 k 2
E =−
v
2meff
v
k
Direct Bandgap Semiconductor
• describing graphically the
curvature of the bands
• representing the potential
presented by the lattice
Combining the Effective Mass Approximation with a
Spherical Boundary Condition
Single Particle (sp) Wavefunction
E
(1)
ρ ρ
ρ
ρ
Ψsp (r ) = ∑ Cnk unk (r )exp ik ⋅ r
(
)
k
linear combination of Bloch functions
Eg
Ehν
(2)
k
ρ ρ
ρ
ρ
ρ
ρ
(
)
(
)
(
)
(
Ψsp r = un 0 r ∑ Cnk exp ik ⋅ r = un 0 r fsp r )
(
)
k
assume unk has weak k-dependence
Envelope Function Approximation
valid for rQD > lattice constant
which for QDs is given by
the “Particle-in-a-Sphere”
(3)
ρ
ρ ρ
un 0 (r ) = ∑ Cni ϕ n (r − ri )
i
linear combination of atomic orbitals with
atomic wavefunctions ϕn (n= CB or VB)
i=lattice sites
Coulomb Attraction
e-
e-
h•
Bulk semiconductors,
Coulomb attraction
creates bound excitons
h•
Confinement Energy ∝ 1/a2
Coulomb Attraction ∝ 1/a
For small a:
• Confinement Energy>Coulomb Attraction
electron and hole are treated independently
• Coulomb interaction added as a correction
η2
Eehp (nh Lh ne Le ) = E g + 2
2a
⎧⎪ϕ n2h,Lh ϕ n2e,Le ⎫⎪
⎨ v + c ⎬ − Ecoulomb
meff ⎪⎭
⎪⎩ meff
For 1Se pairs of states
Ecoulomb=1.8e2/εa
Brus, J. Phys. Chem. 90, 2555 (1986).
Size Dependence of Electronic Structure
E
Eg
E
Ehν
Eg
k
Decreasing Dot Diameter
Ehν
k
Development of Electronic Structure
Similar to Length Dependence in 1D polyenes
p-orbitals form π-bonds
with associated energy levels
with an energy separation
in the visible
Lowest
Unoccupied
Molecular
Orbital
Highest
Occupied
Molecular
Orbital
Example of alternating double/single bond
π-bond extends over many C atoms
LUMO
π*
HOMO
π
Ground
Polyene’s with increasing
chain length
Size Dependent Absorption
Example: CdSe
150 Å
Absorbance (arbitrary units)
Absorbance (arbitrary units)
150Å
90 Å
72 Å
55 Å
45 Å
33 Å
29 Å
21 Å
17Å
17 Å
1.5 2.0 2.5 3.0 3.5
1.5 2.0 2.5 3.0 3.5
Energy (eV)
Energy (eV)
Semiconductor Materials
Range from 30 nm QDs to bulk crystal
Graph from H. Weller, Pure Appl. Chem. 72, 295 (2000).
Absorption Spectra of Semiconductor Nanocrystals
Changing the Core
InAs
Core
HgS
PbSe
A. P. Alivisatos, UC Berkeley
PbS
PbTe
InP
8.0 nm
7.0 nm
6.0 nm
5.0 nm
4.0 nm
5.2 nm
7.4 nm
4.0 nm
IR Absorption (Arb. Unit)
12 nm
IR Absorption (Arb. Unit)
IR Absorption (Arb. Unit)
NRL group
8.5 nm
8 nm
3.0 nm
3.5 nm
3.0 nm
800 1200 1600 2000 2400 2800
Wavelength (nm)
O. Micic, A. Nozik, NREL
800
1200 1600 2000
Wavelength (nm)
C. B. Murray, IBM
800 1200 1600 2000 2400
Wavelength (nm)
Real Band Structure
Example: CdSe
E
Cd 5s orbitals
2-fold degenerate at k=0
J=1/2
Eg
∆cf
crystal field splitting
k
hh
Se 4p orbitals
6-fold degenerate at k=0
Introduces splitting of bands
J=L+S
∆so
where
heavy hole
J=3/2
lh
light hole
so
spin-orbit splitoff
L=orbital angular momentum
S=spin angular momentum
J good quantum number due to strong spin-orbit coupling
J=1/2
Size Evolution of Electronic States
InAs
CdSe
1P3/21Pe
2S3/21Se
Low Band gap InAs
modeling must also
account for valenceconduction band
coupling
D. J. Norris, M. G. Bawendi, Phys. Rev. B 53,
16338 (1996).
where
2S3/21Se
1S3/21Se
1S3/21Se
F=J+L
1P3/21Pe
U. Banin et al., J. Chem. Phys. 109,
2306 (1998).
L=envelope angular momentum
J=Bloch-band edge angular momentum
Hole states labeled by nhLF [LF=L + (L+2)]
Electron states labeled neLe
Selection Rules
ρ
P = Ψe e ⋅ pˆ Ψh
momentum
operator
polarization
vector of light
ρ
P = uc e ⋅ pˆ uv
ρ
P = uc e ⋅ pˆ uv
2
2
2
acts only on unit cell
portion of wavefunction
2
fe fh
δ n ,n δ L ,L
e
h
e
h
Overlap of the electron and hole wavefunctions within the QDs
Luminescence (arbitrary units)
Absorbance (arbitrary units)
Towards the Homogeneous Distribution: Photoluminescence
and Photoluminescence Excitation
3.0
10 K
2.5
2.0
1.5
1.0
0.5
0.0
400 450 500 550 600 650 700
1.0
10 K
0.8
0.6
0.4
0.2
0.0
600
650
700
750
800
Wavelength (nm)
Absorption
Luminescence
Wavelength (nm)
Wavelength
Photoluminescence Excitation
“smallest” QDS
Wavelength
Photoluminescence
“largest” QDS
Distribution in ensemble from size, structure, and environmental inhomogeneities
Fluorescence Line Narrowing and
Photoluminescence Excitation
Band Edge Exciton Structure
Splitting due to crystal field, non-spherical shape, and exchange
interactions of quantum dots
D. J. Norris, Al. L. Efros, M. Rosen, M. G. Bawendi, Phys. Rev. B 53, 16347 (1996)
Single Molecule Spectroscopy
Diffraction Limited Spot
Fluorescence Intermittancy in CdSe QDs
On-period decreases with increasing
illumination intensity
Off-period intensity independent
Excitation every 10-5 sec
Relaxation every 10-8 sec
But occasionally
Two electron-hole pairs may
exist in a single QD
QDs “blink” like molecules
Auger ionization
Probability of photoionization/excitation 10-6
Neutralization time ~0.5 sec
M. Nirmal, L. E. Brus, Acc. Chem. Res. 32, 407 (1999).
Auger Ionization
Consistent with single molecule spectroscopy and photodarkening
observed in QD doped glasses
Al. L. Efros, M. Rosen, Phys. Rev. Lett., 78, 1110 (1997).
Single Dot Spectroscopy
Individual
quantum dots
Histogram of 513
43 Å QDs
Including all
phonon lines
Single Quantum Dot Emission
Spectral diffusion driven by environment
S. A. Empedocles, M. G. Bawendi, Acc. Chem. Res. 32, 389 (1999).
Metal Nanoparticles
- - - - --
-- Metal
- Particle
-
--
-- - - --
Surface Plasmon Resonance
• dipolar, collective excitation between
negatively charge free electrons and
positively charged core
Au nanoparticle
absorption
• energy depends on free electron density
and dielectric surroundings
• resonance sharpens with increasing
particle size as scattering distance to
surface increases
Electronic Properties of Semiconductor and Metal
Nanoparticles
ε
a
Charge not completely solvated
as in infinite solid
Nanoparticle capacitance C = 4πε oεa
Charging Energy
e2
Ec =
2C (a )
10 nm Al NC
Courtesy of C. T. Black, Thesis, Harvard U.
Coulomb blockade at
kBT<Ec
Structure from discrete electronic states of
metal NC
STM Measurements on Single QDs
InAs QDs
U. Banin et al. Nature 400, 542 (1999).
Synthesis of monodisperse CdSe nanocrystals
HDA −TOPO −TOP, 300°C
Cd(CH 3 )2 + (oct )3 PSe ⎯⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯⎯→ CdSe + ...
absorbance, PL intensity [a.u.]
TEM and HRTEM images of as-prepared CdSe
nanocrystals.
add. inj.
add. inj.
200 min
120 min
60 min
12 min
0.5 min
400
600
800
wavelength [nm]
UV-Vis and PL spectra of CdSe
nanocrystals in growth at 300°C
D. V. Talapin, A. L. Rogach, A. Kornowski, M. Haase, H. Weller. Nano Lett. 2001, 1, 207.
Wet Chemical Synthesis of PbSe
Nanocrystals and Superlattices
oleic acid,
Synthesis
Pb(OAc)2 + R3PSe
R3P, T=150 C
R= octyl
Size Selective
Processing
Size selective precipitation in
solvent/ non solvent pairs like
hexane-methanol
Self Assembly
Evaporation
of the solvent
PbSe
T. J. Watson Research Center
PbSe Nanocrystals and Nanowires
Nanocrystals:
Conduction band
LUMO
1.5 – 10 nm diameters
100 – 100 000 atoms
HOMO
Valence band
bulk
nanocrystals
molecular clusters
PbSe Nanocrystal
Small Bandgap (0.28 eV, cf CdSe : 1.70 eV) ⇒ IR detector, IR diode Laser Material
Larger Bohr Radius (PbSe 46 nm, CdSe 12nm) ⇒ Strong Confinement of Electron-Hole Pair
Larger Optical Nonlinearity, Thermoelectric Cooling (ZT = 1 : PbTe)
Semiconducting, Solar Cells, Thermoelectric, Biological Application
PbSe Nanowire
Solution Phase Synthesis using the Nanoparticles as a Building Block
Formation of the Nanowires from the Self Assembling the Particles
Controlling the wire Properties by Changing the Size and Shape of the Particles
Semiconducting device, Interconnect, Building Blocks for the Nanodevice
A
0
Growth
From Solution
Nucleation
Injection
(arbitrary units)
Concentration of Precursors
Monodisperse Colloid Growth (La Mer)
Nucleation Threshold
200
Ostwald Ripening
400
600
Seconds
Staturation
800
1000
Normalized Counts
\
\/\/
/\/
/\//\/\//\/\/\/\\/\//\/\
\/
\/\P\/\
\/\/\/ /\/
P/=\ =0
/\/\/\/\
\/\/\ \/\
P 0
P==0
\/\/
0
0/\=//\P/\/\/\
0=P
P=0
/\/\
0=/\/\/\/\
/
/\/ P \
\
/
/ \
\/\/ \/\/\/\/\
\/\/
/\/\/\/\/\/
\
\//\\//\/\/\
r
\/
\/\/
00==\P/P
Se=P \/\
0=P
/\/
/\//\\/\
/\
0=P
0=\/\/\/\/
/\/\/\/\
P
0=\/P
\/
0\=//\/\/\/\
P/\/\\//\/ \
\
/\/\/\//\\/
/\/\
/\/\
\/\/
\
/\/\ /\/
/\/\
/\
/\/
/\/\/\
\/\/
\
\/\/
\//\\//\/\/\
/\/\/\/\/\/
\
\//\\//\/\/\
\/\/
\
/\/\ /\/
/\/\
\
\/\/
/\/
/\//\/\//\/\/\/\\/\//\/\
\/\P\/\
P/\ =
P==00
P= 0
0
P=0
/\/\
/\//\\//\\/\
\/\/
\/\/
\
\//\\=//P\/\/
\/ \
Se \/\\/\ /
/ /\/\/
\/\
\/\/\/\/\
/\/\
/\/
/\/\/\/\
/\
/\/
/\/\/\
\/\/
\
\/\/
\//\\//\/\/\
\
\//\\=//P\/\/
\/ \
Se/\ /\\/ \/
\/\/\/\/
\/\/\/\/\
/\/\
/\/
/\/\/\/\
\/\/\/\/
\/\/\/\/
P
P= =0
O
\/\/P=0
\/\/
P=Se
P=0
/\/\/\/\ =0
P
/\/\
/\/\
r
\/
\/\/
00==\P/P
Se=P \/\
0=P
/\/
/\//\\/\
/\
0=P
0=\/\/\/\/
/\/\/\/\
P
0=\/P
\/
0\=//\/\/\/\
P \/\/ \
/\/\\//\\///\\\//
R
\/\/\/\/ =0
P
\/\/
\/\/
/\/\
/\/\
/\/\
/\/\
P
P= =0
O
\/\/P=0
\/\/
P=Se
P=0
/\/\/\/\ =0
P
/\/\
/\//\\//\\/\
\/\/\/\/ =0
P
\/
\/\/\/ /\/
/\/\ \/\
\/\/\/\/\
\/\/
0/\=//\P/\/\/\
0=P
/\/\
0=/\/\/\/\
/
/\/ P \
\
/ /\
\/\/ \/\/\/\/\
\/\/
Size selective processing:
10 0
G ro w th S o lutio n
80
37Å M ajor D ia.
σ = 12%
M eOH
40
20
0
Normalized Counts
Energy (arbitrary units)
-6
van der Waals (R )
Attraction
20
30
40
50
60
B
S ize S e lecte d
10 0
d
10
M ajor D iam e te r < 0 02 >
-12
Steric Repulsion (R )
c
C
60
0
a
b
A
39Å M ajor D ia.
σ = 4.5%
80
60
40
H e xa ne
B uO H
20
0
0
Center to Center Distance R
(arbitrary units)
10
20
30
40
50
M ajor D iam e te r < 002 >
60
Results of size selected Percipitation
B
(a) 37Å+12%
(b) 39Å+8%
(c) 40Å+5%
(d) 42Å+<4%
(e) 39Å+11%
(f) 41Å+6%
(g) 45Å+<4%
400 500 600 700 800 400 500 600 700 800
Wavlength (nm)
Wavelength (nm)
Absorbance (arbitrary units)
Absorbance (arbitrary units)
A
Absorption and Photoluminescence
of PbSe Nanocrystals
7 7 K
(g )
8.0 nm
7.0 nm ( f )
Relative intensity
IR Absorption
(Arb. Unit)
Absorbance (arbitrary units)
12 nm
6.0 nm
(e )
5.0 nm
4.0 nm
(d )
PbSe Nanocrystals
2 9 8 K
(c )
(b )
3.5 nm
size: 4.0 nm
(a )
3.0 nm
800
12001600200024002800
1 0 0 0
2 0 0 0
3 0 0
0
1 0 0 0
W a v e le n g t h ( n m )
Wavelength (nm)
1 5 0 0
Wavelength (nm)
PbSe nanowires
Confinement Energy (eV)
2.0
Based on IR & TEM
Calculated using Scherrer
Formula (XRD)
(1)
PbSe in Phosphate Glass
Effective Mass Approx.
1.5
1.0
0.5
0.0
0
10
20
30
40
50
Equivalent Radius (A)
60
70
T. J. Watson Research Center
Reflected Intensity (Arb. Uint)
Shape Change from Sphere to Cubic
and SAXS in Polymer Matrix
7
10
6
10
5
10
4
10
3
Experiment Sphere Particle
D = 9.8 nm, Rg = 3.8 nm
(σ = 8 %)
1
5 nm
Reflected Intensity (Arb. Unit)
50 nm
10
10
7
10
6
10
5
10
4
10
3
10 nm
3
4
2θ (Degree)
5
6
Experiment Cubic Particle
L = 10.5 nm, Rg = 5.25 nm
(σ = 10 %)
1
50 nm
2
2
3
4
2θ (Degree)
5
6
K.-S. Cho, W. Gaschler
PbSe Quantum Cubes
Reflected Intensity (Arb. Unit)
(220)
34
2θ (Degree)
intensity / a.u.
Reflected Intensity (Arb. Unit)
32
(200)
36
(200)
< 3 nm
4.3 nm
5.1 nm
20
Sphere
25
30
35
2Θ
(200)
40
45
intensity / a.u.
7.6 nm
8.3 nm
9.5 nm
(111)
20
30
(200)
(220)
40
50
2θ (Degree)
13.2 nm
> 16 nm
(311)(222)
60
(220)
Cube
20
70
25
30
35
2Θ
40
45
WAXS of 10 nm PbSe quantum cubes
slowly deposited from toluene (top)
and rapidly precipitated from
methanol (bottom)
Shape evolution of PbSe Nanocrystals
Highly symmetric rock salt structure
100
111
Modeling of x-ray diffraction:
The Debye equation which is valid in the kinematical approximation is shown in
equation 4.6 (8).
I(q ) = I 0 ∑ m ∑ n Fm Fn
sin(qrm , n )
qrm , n
(4.6)
Where I(q) is the scattered intensity , Io is the incident intensity, q is the scattering
parameter [q = 4πsin(θ)/l] for X-rays of wavelength l diffracted through angle θ.
The distance between atoms m and n is rmn . A discrete form of the Debye is
shown in eguation (4.7)(9).
(4.7)
ρ ( rk )
f 2 (q )
I(q ) = I o
sin ( qr )
∑
q
rk
k
where is the incident intensity, f(q) is the angle dependent scattering factor q is the
scattering parameter[4πsin(θ)/λ] for X-rays of wavelength λ diffracted through
angle θ. The sum is over all inter atomic distances, and ρ(rk) is the number of
times a given interatomic distance rk occurs. Since the number of discrete
interatomic distances in an ordered structure grows much more slowly than the
total number of distances, using the discrete form of the equation is significantly
more efficient in the simulation of large crystallites(9).
Modeling NP Shape
Modeling Stacking faults
Scattered Intensity (arbitrary units)
Equivalent Diameter ~63Å
Spherical
1:1
(b)
Prolate
1.22
(a)
10
20
30
40
2θ
50
60
Small angle X-ray Scattering SAXS
4
sin( qR ) − qR cos( qR ) 2
π R 3 [3
]]
3
( qR ) 3
Where ρ and ρo are the electron density of the particle and the dispersing medium respectively. Io is the incident intensity and N is the number of
particles. F(q) is the material form factor (the fourier transform of the shape of the scattering object) and is the origin of the oscillations observed.
Thus for a spherical particle of radius R
(4.8)
I ( q ) = I o N [( ρ − ρ o ) 2
(4.9)
I(q ) = I o N( ρ − ρ o ) 2 F 2 (q )
(4.10)
F( q ) =
4 3 sin(qR) − qR cos(qR )
πR [3
]
3
(qR ) 3
Combined SAXS and WAXS Modeling.
Qunatum cubes:
Cubic 12 nm PbSe nanocrystals Assembling into a superlattice.
(100) (112)
(111) (101)
Self-assembled CdSe nanorod solids
without
polarizers
with
crossed
polarizers
20 µm
20 µm
Optical micrograph of self-assembled CdSe
nanorods (between crossed polarizers).
III-V semiconductor nanocrystals : InP
TOPO −TOP, 180 − 260° C
InCl3 ⋅ (oct ) 3 P + [(CH ) 3 Si]3 P ⎯⎯
⎯ ⎯ ⎯ ⎯ ⎯→ InP + ...
InP
etching agent
hν
~8nm
absorbance, a.u.
HF , TOPO, hν
InP (as - prepared, weak PL) ⎯⎯ ⎯ ⎯ ⎯⎯→ InP ( strong PL)
<2nm
400
600
800
wavelength, nm
PL quantum efficiency ~25-40%
Size-dependent evolution of absorption
spectra of InP colloidal quantum dots
J. Phys. Chem. B, 2002, 106, 12659.
CdSe/CdS quantum dot - quantum rods
CdSe/CdS
CdSe/CdS
CdSe cores
40 nm
Cd:S=1:1
Cd:S=1:3
⊥
CdSe
||
CdS
||
⊥
0°
180°
Luminescent II-VI nanocrystals
Room temperature PL quantum efficiencies 50-70%
Colloidal solutions of CdSe/ZnS coreshell nanocrystals.
Single particle luminescence of CdSe/ZnS
nanocrystals
CdSe/CdS core-shell nanocrystals in a polymer
matrix