ch2theory
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Chapter 2: Theory of Size-Dependent Optical Properties
Summary
Experimental observations of spherical CdSe nanocrystals have shown that as the
average radius,R, increases from 0.6 nm to 4.15 nm, the photoluminescence emission
color changes from blue to red, and the photon energy of the first absorbance peak,E12,
decreases from 3.02 eV to 1.88 eV24
and eventually approaches the bandgap energy,Eg=
1.7 eV, of bulk CdSe.25 Various theoretical approaches have been developed to explain
the relationship betweenRandE12. The effective mass approximation predicts that a
positive, size-dependent energy shift will be proportional to 1/R2, based on quantum
confinement,26
while classical physics predicts a positive energy shift will be proportional
to1/R, based on the dielectric properties of spheres.27
Complex simulations of the band
structure in semiconductor nanocrystals attempt to provide more accurate predictions of
the allowed photon transitions for quantum dots of a particular size,28
but these results do
not produce analytical expressions forEversusR. Empirically, a positive energy shift
proportional to 1/Rclosely follows experimental observations and is useful for converting
observed absorbance spectra into estimates of particle size in order to track quantum dot
growth kinetics.
Quantum Confinement Theory
Quantum dots are so named because their tunable luminescent emission may
originate from quantum confinement. The number of degrees of freedom equals the
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Chapter 2: Theory of Size-Dependent Optical Properties
dimensionality. Reducing the effective dimensionality of the region that constrains the
electrons wave function, from a 3D bulk solid, to a 2D nano-thin plane (in quantum well
lasers and high mobility field effect transistors), to 1D quantum wires, and finally to 0D
quantum dots, can produce technologically useful properties. As the electron is more
confined, its allowed energy states are also more restricted and more size-dependent.
Unconfined conduction electrons in a 3D bulk semiconductor experience the
seemingly boundless periodic electric potential of the crystal lattice. When electrons in
the conduction band recombine radiatively with holes (i.e.missing electrons) in the
valence band and essentially return to their ground state, photons are emitted with
energies near or just belowEg, which is a material property independent of sample size.
However, size-dependent quantum confinement effects develop when the
thickness of an electronic layer approaches the de Broglie wavelength of the electron in a
quantum well structure,25
and when the radius of a semiconductor sphere is smaller than
the bulk-exciton Bohr radius, in a nanocrystal.26
In a quantum dot, the electron and hole
wave functions are confined on all sides by the crystal boundaries, where the electric
potential is higher. Such constrained electron-hole pairs can only have discrete energies
and the transition energy between the first two energy states is higher thanEgby a size-
dependent energy shift, En,l. Various theories are used to explain and estimate the
magnitude of this energy shift.
The effective mass approximation comes from solving the Schrdinger equation
for an isolated electron and then for an isolated hole in a sphere, and assuming that the
effective masses of carriers in the quantum dot are the same as in a bulk semiconductor.
In this case, the discrete energy levels,Enl, are given by Equation (2-1), where his
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Chapter 2: Theory of Size-Dependent Optical Properties
Plank's constant, mois the rest mass of an electron, meand mhare the effective masses of
electrons in the conduction band and holes in the valence band, respectively, and n,l is a
dimensionless series that takes discrete values. The value of 1,0 is equal to for the
lowest allowed interband transition between the 1s electron and the 1s hole states. The
next values in this sequence, 11=4.4934, 12=5.7635, 20=2 ,26
can be used to
predict the structure of an absorbance or excitation spectra for a given sample (with a
fixed averageR, me, and mh) by noting that En,l/E10=(n,l/)2. This will be examined
later in the results section.
2
2,
2
2., 8
111
ln
he
glngln
h
mmREEEE
++=+= (2-1)
To calculate the first photon transition energy,E12, (in eV) it is useful to express
Equation (2-1) in the form shown by Equation (2-2). Here is the wavelength of the first
absorbance peak (in nm), and c is the speed of light. We note that the product hchas a
value of 1239.77 eV nm,25
and that the term h2/(8mo) equals 0.376036 eV nm2. For bulk
CdSe,Egis 1.7 eV, me/mois 0.13, and mh/mois 0.45, according to Sze.25
Then the photon
energy in eV can be estimated from the radius in nm.
)(2
)(
)(
2
211 273.3
17.1
8
1nmeV
nm
eV
oh
o
e
ogss
Rm
h
m
m
m
m
RE
hcE
+=
++==
(2-2)
To test this theory, numerous authors have measured the average radius of
colloidal CdSe nanocrystals in each sample using transmission electron microscopy
(TEM), and they have also measured the photon energy of the first absorbance
peak.24,27,29,28
This experimental data is plotted as symbols in Figure 2-1 and in
Figure 2-2.
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Chapter 2: Theory of Size-Dependent Optical Properties
CdSe Quantum Dots
1.7
1.8
1.9
2
2.1
2.22.3
2.4
2.5
2.6
2.7
2.8
2.9
3
3.1
0 1 2 3 4 5 6
Nanocrystal Radius (nm)
AbsorbancePea
kPhoton
Energy(eV)
Data: Murray 2000
Data: Murray 1993
Data: Peng 1998
a) E=1.7+3.73/R^2
b) E=1.7+1.5/R^2
c) E=1.7+3.73/R^2 -0.26/R
d) E=1.7+0.82/R
Figure 2-1. Size dependent photon transition energy in CdSe quantum dots. Symbolsindicate experimental estimates of the radius,R, from TEM images, and the photon
energy,E, of each samples primary absorbance peak according to several authors.24,27,29
Curves show the following models forEversusR: a) effective mass approximation, b)parabolic 2-band model, c) effective mass with Coulomb interaction,
30and d) size-
dependent capacitance.
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Chapter 2: Theory of Size-Dependent Optical Properties
The effective mass approximation significantly overestimates the first absorbance
peak energy for quantum dots with a radius smaller than 4 nm, as shown in Figure 2-1
curve a). Noting this discrepancy, Murray et. alproposed that the effective masses of
carriers in quantum dots may be different than those in the bulk semiconductor.24
However, if the effective masses are adjusted to improve the energy estimate in this size
range, as shown by curve b) in Figure 2-1, the assumption of two parabolic bands inE-k
space (one valence band and one conduction band) produces a 1/R2dependence that just
does not follow observations very well.
Since the effective mass approximation treats the electron and the hole
independently, it ignores any interaction between them. The 1/R2behavior in Equations
(2-1) and (2-2) approximates only the kinetic energy contribution to the electron-hole
(e-h) pair energy.26
Since the oppositely charged electron and hole attract each other,
confining them closer together lowers their net potential energy. Therefore, this
Coulomb interaction energy decreases proportional to 1/R.26
Because it is difficult to
solve the Schrdinger equation with an added 1/Rpotential energy term defined into the
problem, some authors use perturbation theory26
to justify tacking it onto the solution
energy,30,31
as seen in Equation (2-3), where eis the charge of the electron, ois the
permittivity of free space, and ris the relative dielectric constant of the semiconductor.
For CdSe, ris 10.25
When evaluated, the negative 1/Rterm only reduces the energy by
about 0.1 eV, as shown in Figure 2-1 curve c). Even including Coulomb interaction, the
effective mass approximation still diverges from observations for smaller quantum dots.
++=
roo
ln
h
o
e
o
gln
e
Rm
h
m
m
m
m
REE
4
8.11
8
1 2
2
2
,
2
2, (2-3)
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Chapter 2: Theory of Size-Dependent Optical Properties
Another possible energy correction factor also has 1/Rdependence. Holding
some number, z, of like elemental charges on the surface of a small spherical capacitor
would increase the electric potential by Vcapacitanceas shown in Equation (2-5).27,32
( )()(
2
tan 072.08
nmeV
nmro
cecapaciR
ze
R
zV +=
+=
) (2-5)
Empirically, the photon energy of the first absorbance peak,E, seems to increase
as 1/Rabove the bandgap energy of CdSe, as described in Equation (2-6). By
comparison with Equation (2-5), this behavior could be interpreted as arising from the
size-dependent capacitance of a sphere with about eleven or twelve elemental units of
surface charge.
++=
ro
g
nm
nmeV
eVss
e
R
zE
R
hcE
8~
82.07.1
2
)(
)(
)(11 (2-6)
This capacitive interpretation presents a dilemma. Can quantum confinement still
yield discrete energy states without a significant size-dependent contribution to the first
photon transition energy?
Band simulation models are providing increasingly more rigorous and accurate
estimates of allowed transitions between electronic energy states. When valence band
interaction and degeneracy is numerically simulated, the effective mass approximation is
improved enough to closely predict the first excited state energy of CdSe quantum dots as
a function ofR, as shown by Figure 2-2. The differences between data sets from different
sources may illustrate the need for more consistent calibrated methods of estimating
nanocrystal size using TEM. The details of the band structure yield slight deviations
from 1/R2behavior, so that most features of the absorbance spectra for CdSe quantum
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Chapter 2: Theory of Size-Dependent Optical Properties
dots have been reasonably estimated.28,31
The main difficulty with this approach is that
the finalEversusRrelationships are not analytical.
CdSe Quantum Dots
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
3.1
0 1 2 3 4 5 6
Nanocrystal Radius (nm)
Absorbance
Peak
Photon
Energ
y
(eV)
Data: Norris 1996
Data: Murray 2000
Data: Murray 1993
Data: Peng 1998
e) Band degeneracy model
Figure 2-2. Comparison of band degeneracy model predictions with experimental data.
The primary photon transition energy as a function of CdSe quantum dot radius is
predicted by the band degeneracy model of Norris and Bawendi,28
as shown by the greyline e). Their model follows their data, labeled Data: Norris 1996. However there are
significant differences between the data sets from different sources.24,27,29
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Chapter 2: Theory of Size-Dependent Optical Properties
Although quantum confinement predicts discrete energy levels, continuous
Gaussian energy peaks are observed. Individual CdSe quantum dots show extremely
photoluminescence emission. Phonon and Coulomb scattering of an electron in a
quantum dot may produce a small degree of homogeneous line broadening around the
allowed energy states.26
In the laboratory, quantum dot suspensions are characterized by
a mean nanocrystal size and a standard distribution rather than by a single radius. Thus
inhomogeneous line broadening is also expected, due to the superposition of billions of
discrete quantum dot spectra.26
Of course, the slit width of the spectrophotometer can
also contribute to the observed line width, but this contribution can be controlled, and it
can usually be made small enough to be ignored, compared to the dominant
inhomogeneous broadening. As in standard error analysis, the net squared standard
deviation in the PL emission wavelength is the sum of the squared contributions. Often
the observed emission width is determined primarily by the quantum dot size distribution.
When the absorbance peak wavelength, ,is monitored as a function of reaction
time and synthesis temperature, solving Equation (2-6) forRis a convenient way to
estimate changes in the average nanocrystal radius, thereby providing a tool to study
synthesis kinetics.
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