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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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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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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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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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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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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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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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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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