Download Terahertz driven intraband dynamics of excitons in nanorods Fredrik Sy

Terahertz driven intraband dynamics of
excitons in nanorods
by
Fredrik Sy
A thesis submitted to the
graduate program in Physics, Engineering Physics & Astronomy
in conformity with the requirements for
the degree of Master of Science
Queen’s University
Kingston, Ontario, Canada
May 2014
Copyright © Fredrik Sy, 2014
Abstract
Quantum dots and nanorods are becoming increasingly important structures due to
their potential applications that range from photovoltaic devices to medicine. The
majority of the research on carrier dynamics in these structures has been in the optical
regime, with little work performed at Terahertz frequencies where excitonic dynamics
can be more directly probed. In this work, we examine theoretically the interaction of
Terahertz radiation with colloidal CdSe nanorods to determine the dynamics of excitons generated via a short optical pulse. We calculate the energies and wavefunctions
for the excitons within the envelope function approximation in the low density limit
where there is at most one exciton per nanorod. The linear Terahertz transmittance
and absorbance is found for nanorods that are approximately 70 nm in length and
7 nm in diameter and are compared with experimental results that have shown the
first observation of intra-excitonic transitions in nanorods. We find absorbance peaks
at 8.5 THz and 11 THz that result from polarizations in the longitudinal (rod axis)
and transverse directions respectively. Our theoretical results show that the 8.5 THz
and 11 Thz peaks are due to 1s − 2pz and 1s − 2px transitions respectively. The
theoretical absorbance spectra is in good agreement with the experimental one and
show that only the ground state is significantly populated 1 ps after optical excitation. This provides strong evidence of rapid trapping of excited holes into the ligand
i
used to passivate the nanorods. A full set of dynamical equations were then constructed from Heisenberg’s equation of motion, and were used to model the excitonic
correlations as a function of time. Transmittance and absorbance were calculated for
different nanorod orientations and electric field strengths in both the linear and nonlinear regime. These results were then averaged over nanorod orientation in order to
more accurately reflect experimental conditions. Nonlinearity was found to become
significant at peak pulse field strengths of 7 kV/cm and greater. Due to two-photon
processes, we predict the 2pz − 3dz transition that is not observed in the linear regime
will be clearly seen in the nonlinear absorbance spectrum.
ii
Acknowledgments
I thank Marc Dignam for his patience and guidance as my supervisor during my
studies at Queen’s University. I thank my parents for their love and support as they
raised me. I thank my friends for all the times in between.
iii
Contents
Abstract
i
Acknowledgments
iii
Contents
iv
List of Tables
vii
List of Figures
viii
Chapter 1:
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Theoretical models . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3
Excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.4
Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.5
Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
Chapter 2:
2.1
Excitonic Basis . . . . . . . . . . . . . . . . . . . . . . . . .
11
Solving the Time Independent Schrödinger Equation . . . . . . . . .
12
iv
2.2
Non-interacting energies for electrons and holes . . . . . . . . . . . .
20
2.3
Light and Heavy Holes . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.4
Calculating excitonic states . . . . . . . . . . . . . . . . . . . . . . .
23
Chapter 3:
Terahertz driven intraband transitions: linear response .
34
3.1
Intraband Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . .
34
3.2
Linear Response
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
3.3
Theoretical results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
3.4
Intraband transition frequencies, dipole moments and binding energies
3.5
as a function of radius . . . . . . . . . . . . . . . . . . . . . . . . . .
48
Comparison to Experimental Results . . . . . . . . . . . . . . . . . .
53
Chapter 4:
Terahertz driven nonlinear response . . . . . . . . . . . .
61
4.1
Nanorods suspended in air . . . . . . . . . . . . . . . . . . . . . . . .
61
4.2
Heisenberg equation of motion: modeling exciton dynamics . . . . . .
65
4.3
Calculating the transmitted THz electric field . . . . . . . . . . . . .
68
4.4
Terahertz driven nonlinear response . . . . . . . . . . . . . . . . . . .
72
4.5
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
Chapter 5:
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
5.1
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
5.2
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
v
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
Appendix A:
Numerical Methods . . . . . . . . . . . . . . . . . . . . .
vi
95
List of Tables
1
Table of Abbreviations and Notations . . . . . . . . . . . . . . . . . .
1.1
Dielectric constant, exciton binding energy, and exciton radius in various materials [16, 30, 29] . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
1
9
Effective masses of electrons, light holes and heavy holes for CdSe
nanorod taken from Ref. [30]. Only masses inside the rod are needed
for this thesis as we will always assume hard boundary conditions. . .
2.2
23
Exciton energies for the first 31 excitonic states and their respective
labels. The label 1sx denotes a 1s like state but with an x number of
nodes in the zt (center of mass) direction. Some states are not labeled
as they exhibit complicated properties than cannot be adequately explained with the introduced notation. There are two 2px states for
angular quantum numbers -1 and 1 . . . . . . . . . . . . . . . . . . .
vii
32
List of Figures
1.1
THz radiation in electromagnetic spectrum [12] . . . . . . . . . . . .
2.1
Nanorod diagram showing the longitudinal direction (z) along the rod
axis and the transverse direction in the plane of the radius . . . . . .
2.2
3
12
Energy for the light hole states as a function of kh (quantum number
associated with longitudinal direction for holes) for the lowest energy
light hole radial states for a CdSe nanorod 70nm in length, 3.5 nm in
radius and with hard boundary conditions . . . . . . . . . . . . . . .
2.3
21
Energy for the heavy hole states as a function of kh (quantum number
associated with longitudinal direction for holes) for the lowest energy
heavy hole radial for a CdSe nanorod 70nm in length, 3.5 nm in radius
and with hard boundary conditions . . . . . . . . . . . . . . . . . . .
2.4
22
Energy for the electron states as a function of ke (quantum number associated with longitudinal direction for electrons) for the lowest energy
electron radial for a CdSe nanorod 70nm in length, 3.5 nm in radius
and with hard boundary conditions . . . . . . . . . . . . . . . . . . .
viii
22
2.5
Energy of hole states for both light and heavy holes. Energy is shown
as function of longitudinal quantum number kh for a CdSe nanorod
with radius 3.5 nm and 70 nm in length. Only states with quantum
numbers nh = 0, νh = 0 are shown. . . . . . . . . . . . . . . . . . . .
2.6
Plot of the exciton wavefunction as a function of the relative and center
of mass coordinates along the rod axis for the excitonic ground state .
2.7
24
29
Plot of the exciton wavefunction as a function of the relative and center
of mass coordinates along the rod axis for the higher order center of
mass motion excited states . . . . . . . . . . . . . . . . . . . . . . . .
2.8
Plot of the exciton wavefunction as a function of the relative and center
of mass coordinates along the rod axis for the 2pz state . . . . . . . .
2.9
29
30
Plot of the exciton wavefunction as a function of the relative and center of mass coordinates along the rod axis for the two 2px states corresponding to angular quantum numbers -1 and 1 . . . . . . . . . . .
30
2.10 Plot of the exciton wavefunction as a function of the relative and center
of mass coordinates along the rod axis for the 3dz state . . . . . . . .
30
2.11 Plot of the exciton wavefunction as a function of the relative and center
of mass coordinates along the rod axis for the higher order center of
mass motion pz -like states . . . . . . . . . . . . . . . . . . . . . . . .
31
2.12 Schematic diagram presenting the energies of the 1s, 2pz , 2px and 3dz
excitonic states in meV. The distance between the energies are not
presented to scale and the dots denote there are states in between the
labeled excitonic states. The band gap of 1.74 eV is for a temperature
of 0 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
33
3.1
Experimental setup with randomly oriented nanorods. The diagram
shows four layers for illustration purposes and there could me more.
The THz electric field is polarized in the x − z plane. The nanorods
also lie in the x − z plane . . . . . . . . . . . . . . . . . . . . . . . .
3.2
44
Reduction in the transverse electric field due to different dielectric constants outside (0 ) and inside (r ) the nanorod. The subscripts T and
L denote the transverse and longitudinal directions. The electric field
in the longitudinal direction is the same both outside and inside the
nanorod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3
45
Linear absorbance for longitudinal and transverse polarizations as well
as for the total angularly-averaged response at a temperature of 0 K
with a linewidth of 5.5 THz using light holes
3.4
. . . . . . . . . . . . .
Linear absorbance of CdSe nanorod for different temperatures with a
linewidth of 5.5 THz using light holes . . . . . . . . . . . . . . . . . .
3.5
50
2pz − 3dz light-hole exciton transition frequency as a function of radius
for a 70 nm long CdSe nanorod . . . . . . . . . . . . . . . . . . . . .
3.9
50
1s − 2px light-hole exciton transition frequency as a function of radius
for a 70 nm long CdSe nanorod . . . . . . . . . . . . . . . . . . . . .
3.8
49
1s − 2pz light-hole exciton transition frequency as a function of radius
for a 70 nm long CdSe nanorod . . . . . . . . . . . . . . . . . . . . .
3.7
48
Linear absorbance of CdSe nanorod for different temperatures with a
linewidth of 5.5 THz using heavy holes . . . . . . . . . . . . . . . . .
3.6
47
51
1s − 2pz light-hole exciton dipole moment as a function of radius for a
70 nm long CdSe nanorod . . . . . . . . . . . . . . . . . . . . . . . .
x
51
3.10 1s − 2px light-hole exciton dipole moment as a function of radius for a
70 nm long CdSe nanorod . . . . . . . . . . . . . . . . . . . . . . . .
52
3.11 2pz − 3dz light-hole exciton dipole moment as a function of nanorod
radius for a 70 nm long CdSe nanorod . . . . . . . . . . . . . . . . .
52
3.12 Binding energy as a function of radius for a 70 nm long CdSe nanorod
using light holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
3.13 Schematic of the THz pump-probe experimental setup. Excitons are
generated in a CdSe nanorod layer (approximately 1 µm in thickness)
with a 400 nm optical pump. The CdSe nanorods are deposited on an
Indium-Tin-Oxide (ITO) substrate on a glass slide. A THz pulse is
then applied to the nanorods after a delay time and is reflected by the
ITO. This pump-probe setup is used to measure the absorbance of the
CdSe nanorods in the THz frequency range. . . . . . . . . . . . . . .
55
3.14 Experimental and theoretical absorbance as a function of linear frequency for colloidal CdSe nanorods roughly 70 nm in length and 3.5 nm
in radius. The absorbance was measured using a pump-probe teachnique with a delay time of 1 ps between the pump and probe. The
two theoretical fits are the Lorentzian and Gaussian fits at 0 K. The
Gaussian fit is done by averaging Lorentzian fits of different nanorod
radii in increments of 0.05 nm. [14] . . . . . . . . . . . . . . . . . . .
xi
57
3.15 Excitonic States Energy Diagram depicting how the optical pulse first
creates the exciton bast the band gap. The ligand states are hypothesized to be in between the 1s and 2pz state. Higher energy excited
stated then relax down to the ligand states rather than the 1s state
and get trapped . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
3.16 Absorbance as a function of linear frequency for various of pump delay
time for a colloidal CdSe nanorods of length 70 nm and radius 3.5 nm
(courtesty of David Cooke [14]) . . . . . . . . . . . . . . . . . . . . .
4.1
60
Boundary Condition Diagram for Terahertz radiation incident on a
sheet of nanorods. The nanorod sheet is thin enough so that the current
can be described a surface current. . . . . . . . . . . . . . . . . . . .
4.2
Incident and transmitted field amplitude vs time for a weak field of
0.07 kV/cm peak . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3
72
Exciton population in the 2pz and 3dz as a function of time for a peak
field strength of 70 kV/cm . . . . . . . . . . . . . . . . . . . . . . . .
4.6
71
Exciton population in the 2pz and 3dz as a function of time for a peak
field strength of 7 kV/cm . . . . . . . . . . . . . . . . . . . . . . . . .
4.5
71
Incident and transmitted field amplitude in frequency space for a weak
field of 0.07 kV/cm peak . . . . . . . . . . . . . . . . . . . . . . . . .
4.4
63
73
Transmittance for field intensities from 0.7 kV/cm to 70 kV/cm with
THz field polarized along the longitudinal direction. Dephasing time
was 1 ps, population decay time was also 1 ps but with the 1s state
not decaying over time. A linear response is seen up to 0.7 kV cm peak
field strength and with strong nonlinearity at 70 kV/cm . . . . . . . .
xii
75
4.7
Absorbance for field intensities from 0.7 kV/cm to 70 kV/cm with THz
field polarized along the longitudinal direction. Dephasing time was
1 ps, population decay time was also 1 ps but with the 1s state not
decaying over time. A linear response is seen up to 0.7 kV cm peak
field strength and with strong nonlinearity at 70 kV/cm . . . . . . . .
4.8
THz field orientation with respect to the nanorod. θ is the THz field
polarization angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.9
77
78
Populations of 2pz and 2px states over time for a THz electric field
polarization angle of θ = 20◦
. . . . . . . . . . . . . . . . . . . . . .
79
4.10 Populations of 2pz and 2px states over time for a THz electric field
polarization angle of θ = 45◦ . . . . . . . . . . . . . . . . . . . . . . .
79
4.11 Transmittance for a THz electric field polarization angle of θ = 20◦ .
The transmittance due to the field components in the z and x direction
are shown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
4.12 4θ,x and 4θ,z as a function of theta for different THz field intensities.
4θ,x and 4θ,z reach zero for θ values of 0 and 90 degrees respectively.
Strong nonlinearity begins at approximately 7 kV/cm . . . . . . . . .
82
4.13 Transmittance nanorods of uniformly distributed random orientations
with respect to the THz field electric field . . . . . . . . . . . . . . .
xiii
83
Symbol
THz
CdSe
H
L0
R0
Jv (x)
Kv (x)
Eρ , Ez
n
v
k
µ
T
t
B̂µ†
Γµ,ν
E1s
R̂e , R̂h
E
EI ,ER ,ET
D
B
H
K
σ
ω1s−2pz
n0
c
n2d
Table 1: Table of Abbreviations and Notations
Name
Terahertz
Cadmium-Selenide
Hamiltonian
nanorod length
nanorod radius
Bessel function of the first kind
Modified Bessel function of the second kind
Energies associated with the radial and longitudinal wavefunctions
radial excitation quantum number
angular momentum quantum number
longitudinal excitation quantum number
exciton quantum number
temperature
time
exciton creation operator for state µ
dephasing time constant between excitonic states µ and ν
energy of 1s excitonic state
position operators for the electron and the hole
Terahertz electric field
Incident, reflected and transmitted Terahertz electric field
D field
magnetic field
applied magnetic field
surface current
surface charge density
transition frequency of 1s to the 2pz state
index of refraction
speed of light
exciton density per unit area
Chapter 1
Introduction
1.1
Motivation
Quantum dots and nanorods have been studied extensively due to their size dependent electronic properties. The size dependence comes from the strong quantum
confinement in three dimensions, in the case of quantum dots, and effectively in two
dimensions, in the case of nanorods. The potential applications range from nanodevices, to photovoltaics [1] to biological markers [2]. Nanorods differ from dots in
their electronic and optical properties; one difference is that nanorods demonstrate
linearly polarized emission [3]. The majority of the research into carrier dynamics
in these structures has been in the optical regime, with little work performed at
Terahertz frequencies where excitonic dynamics can be directly probed.
Quantum dots have exhibited many interesting properties based on their confinement such as emitting light at a frequency that can be tuned by choosing the
quantum dot diameter [4]. They have also been fabricated from a wide variety of
1
CHAPTER 1. INTRODUCTION
2
materials such as Silicon, Lead selenide (PbSe), Indium phosphide (InP) and Cadmium selenide (CdSe) to name a few [4,5,6]. Due to their wide array of potential
applications, numerous theoretical studies have also been done on the optical and
electronic properties of quantum dots with the focus being on quantities such as the
exciton binding energy, carrier lifetime and band gap of these quantum dots[7]. The
exciton Bohr radius is the characteristic length scale of these quantum dots and gives
a measure of the average separation of the electron and the hole. In the case where
the quantum dot radius is smaller than exciton Bohr radius, we say that we are in the
strong confinement regime. The exciton Bohr radius varies significantly depending
on the material. Depending how strong one wishes the confinement to be, quantum
dot sizes can typically range from 1 to 10 nm which corresponds to as few as several hundred atoms to well over ten thousand atoms [5,9]. One important feature of
nanorods, is that when the length is sufficiently large, the energy difference between
the excitonic states can be considerably smaller than in quantum dots. This means
that intraexcitonic transitions can be excited using THz fields, something that is not
possible in smaller quantum dots.
Nanorods are cylindrically shaped nanostructures with lengths larger than their
diameter. Nanorods exhibit similar applications to quantum dots but with the possibility of the previously mentioned linearly polarized emission and have seen recent
work in both photovoltaics and biosensors. Nanorods exhibit huge variation in size
with diameters ranging from 2 to 50 nm and lengths from 50 nm to 1µm. The polarized emission, specifically in ZnO nanorods, has been suggested to have possible
application to devices such as flat panel displays and power devices [10,2,11].
Terahertz (THz) radiation, which lies between microwave and infrared, is the last
CHAPTER 1. INTRODUCTION
3
Figure 1.1: THz radiation in electromagnetic spectrum [12]
region of the electromagnetic spectrum that has yet to be rigorously studied. Figure
1.1 shows THz radiation spectrum that is typically between 0.1 THz and 10 THz.
This absence of development for this range has been due to a lack of inexpensive
and convenient sources and detectors. In recent years there have been significant
improvements in the field, not the least of which is due to Terahertz time-domain
spectroscopy [13]. THz applications have ranged from imaging and spectroscopy to
high speed wireless communication. There is still a lot of work that needs to be done
in finding better sources and detectors to further the technology. This thesis explores
how Terahertz waves can be used to induce intraband transitions in nanorods to
better understand the underlying dynamics. To the author’s best knowledge, there
has been no other work on intraband transitions in CdSe nanorods aside from the
work presented by the author and co-workers in Ref [14].
CHAPTER 1. INTRODUCTION
1.2
4
Theoretical models
Solving many-body carrier dynamics in a nanostructures is complicated and calculations can quickly become too computationally intensive. To combat this many
approximation methods have been applied to calculate the electronic structure and
to describe the carrier dynamics in these nanostructures. In this work we calculate
the energies and wavefunctions of the electron and hole both in the non-interacting
picture and with the electron-hole Coulomb interaction included. This section details
two commonly used methods to find these energies and wavefunctions.
1.2.1
Tight-binding method
The tight binding approximation is a common method of computing the electronic
states in bulk semiconductors and in semiconductor nanostructures [15]. Assuming
that the electrons are localized strongly to each atomic site in a lattice, we can
assign an electron wave function to each lattice site. Since the electrons are confined
mostly to the lattice site, each electron wavefunction has very little overlap with other
wavefunctions on atoms that are farther away than the nearest neighbor. This allows
us to ignore interactions between atomic sites that are deemed too far, and thus too
weak. The method has been widely used to model a wide variety of structures among
them are quantum dots [20-23] and small nanorods [24].
In the bulk case, for a lattice of atoms, the eigenvalue problem is given by
#
"
2 2
X
~∇
+
V0 (r − Rl ) − Eλ (k) Φλ (k, r) = 0,
(1.1)
2m0
l
where Rl is the position of the lth atom, V0 (r − Rl ) is the potential due to the lth
atom, k is the Bloch wave vector, m0 is the electron mass and λ is the quantum
5
CHAPTER 1. INTRODUCTION
number of the energy. The standard ansatz, given by a sum of atomic wavefunctions
that fulfills Bloch’s theorem, is given by
Φλ (k, r) =
X eik·Rl
l
L3/2
φ (r − Rl ) ,
(1.2)
where L corresponds to the system length and the subscript l corresponds to lattice
sites and φ (r − Rl ) corresponds to the atomic wavefunction. This method can be
generalized for nonperiodic structures such as quantum dots and nanorods where the
electron wavefunction is similarly constructed based on atomic wavefunctions. The
overlap of the nearby wavefunctions give the interactions terms and can be done for
nearest neighbor interactions or beyond nearest neighbor if more accuracy is needed.
The determination of the wavefunctions can be based on other ab initio calculations such as density functional theory and pseudo potential calculations [8,18]. Alternatively, pseudo-empirical or empirical tight-binding approaches also exist, where
the tight-binding atomic potential overlap constants are empirically determined as to
provide a good fit to the band structure [19,21-23,25,26].
Although tight-binding methods are widely used in bulk semiconductors and small
quantum dots, this method becomes too computationally expensive when there is a
large number of lattice sites. In the case of this thesis upwards of 40,000 lattice sites
would be needed and tight-binding would be too computationally expensive.
1.2.2
Envelope function method
In the case of nanostructures where the system is small enough so that quantum confinement is strong but large enough so that the wavefunction is significantly larger
than the atomic lattice, we can use the envelope function method. The envelope
function approximation assumes that the electron wave function is made of a slowly
6
CHAPTER 1. INTRODUCTION
varying envelope function and a Bloch function. In the case of the a bulk semiconductor, the Schrödinger equation for an electron in a crystal lattice is given by
2
p
Hψλ (k, r) =
+ V0 (r) Φλ (k, r) = Eλ (k) Φλ (k, r) ,
(1.3)
2m0
where H is the Hamiltonian for the system, V0 (r) is the potential due to the lattice
,
p2
2m0
is the kinetic energy term, λ is the energy eigenvalue and ~k is the crystal
momentum. The Bloch wave functions can be written in the form
ψλ (k, r) =
eik·r
uλ (k, r) ,
L3/2
(1.4)
where uλ (k, r) is the atomic function that has the lattice periodicity and is dependent
on the details of the atomic lattice [15]. In the case of nanorods and other quantum
confined structures, the periodicity is broken and in the approximation that multiband coupling can be neglected, we can replace the plane wave function with an
envelope function given by
Φλ (r) = C (r) uλ (k = 0, r) ,
(1.5)
where C (r) is the envelope function and the assumption that we can use only the
k = 0 part of the Bloch function is expected to be valid as long as the Bloch states
needed to construct our nanorod state are close to the Γ point [15].
In the effective mass approximation, the effective envelope function Hamiltonian
is given by
H=−
~2 ∇2e ~2 ∇2h
−
+ Vee + Vhh + Veh + Vsys ,
2me
2mh
(1.6)
where the subscript e and h refer to the electron and hole respectively. Vee , Vhh and
Veh refer to the electron-electron, hole-hole and electron-hole Coulomb interactions.
Vsys refers to the potential specific to the system, which would be the confinement
CHAPTER 1. INTRODUCTION
7
barrier in the case of nanorods and quantum dots. The effective masses, me and mh ,
are obtained from the diagonal elements of the Luttinger-Kohn Hamiltonian based
on k · p theory [30]. This new Hamiltonian can now be analyzed to study the system
of interest. The Hamiltonian in Eq. (1.6) can be generalized to include coupling to
higher bands using higher order k · p theory that also takes into account light and
heavy hole mixing [30].
The envelope function method has been used to model the band structure of
quantum dots and quantum wires [30,31,32,33]. The method does not, however,
take into account surface effects (ligand or surface chemistry) that might be more
accurately modeled by tight-binding or pseudo-potential methods. In the case of this
study, however, where we are modeling a large quantum rod with many atoms, the
envelope function is expected to be sufficiently accurate for the calculation of interexcitonic transitions. In this thesis we create an excitonic basis based on the effective
mass envelope function approximation. A basis is first formed by solving Eq.(1.6) but
without the Coulomb terms. These eigenstates form a non-interacting electron-hole
basis that can then be diagonalized to form an excitonic basis.
There are variety of advantages to this approach. Most notable is the ability to
capture the carrier dynamics with a relatively small basis size in comparison to a noninteracting basis. A second advantage is that the excitonic basis properly accounts
for coherence between the electron and hole within an exciton [39].
1.3
Excitons
Excitons first arose from questions involving pure semiconducting crystals absorbing
light. The energy from the light was later found to be creating these electron and
CHAPTER 1. INTRODUCTION
8
hole pairs. Research on excitons was pioneered by Frenkel and Wannier’s work in the
1930’s and became sporadic during the 1950’s. Since then excitons have been studied
extensively and used to explain charge carrier dynamics in many solids. There are
also biexcitons which are pairs of excitons but will not be the topic of this report due
to the low density of excitons considered in this work [27].
1.3.1
Type of Excitons
Excitons are electron-hole pairs that are attracted and bound together due to Coulomb
interaction. Exciton lifetimes are finite and excitons are destroyed when the electron
and the hole recombine. Lifetimes are typically on the order of nanoseconds but can
vary significantly depending on the material. There are two main kinds of excitons,
Frenkel excitons and Wannier excitons. Frenkel excitons are localized to a single
lattice site and are commonly found in molecular crystals, polymers and biological
molecules. Wannier excitons span many lattices and are the excitons considered in
this work. Table 1.1 gives the exciton parameters for a variety of materials. We see
that the radius can vary immensely over a variety of materials and that the radius
is on the order of a nanometer, which while small relative to the macroscopic scale
spans over several lattice sites [27].
1.4
Thesis overview
In this thesis, the Terahertz driven intraband transitions are modeled in a CdSe
nanorod in an excitonic basis using hard boundary conditions. The calculations are
done using the effective mass, envelope function and two band approximation. It is
9
CHAPTER 1. INTRODUCTION
Table 1.1: Dielectric constant, exciton binding energy, and exciton radius in various
materials [16, 30, 29]
Material
Binding Energy (meV)
KCl
4.6
580
CuCl
5.6
190
Cu2 O
7.1
150
CdSe
9.5
15
Si
11.4
12
GaAs
13.1
4.9
◦
Approximate radius(A)
3
7
7
56
50
150
assumed that there is less than one exciton per nanorod avoiding possible excitonexciton interactions. The Terahertz driven linear response is found for both excitons
made of light and heavy holes. Calculated theoretical linear response is found to
be in good agreement with experimental results of David Cooke [14]. Several key
transitions are identified in the THz absorption spectrum namely the 1s−2pz , 1s−2px
and 2pz − 3dz transitions. Finally, the nonlinear absorption spectrum of the nanorods
is performed and we show that due to two-photon processes, the nonlinear spectrum
exhibits absorption due to the 2pz − 3dz transition that is not observed in the linear
response.
1.5
Thesis outline
The organization of this thesis is as follows. Chapter 2 discusses the method by
which we calculate an excitonic basis. The Schrodinger equation is first solved for
the non-interacting electron and hole. The Coulomb potential is then evaluated in
this non-interacting electron-hole pair basis and, together with the kinetic energy,
diagonalized to form the excitonic basis.
Chapter 3 presents an examination of intraband transitions in the linear regime.
CHAPTER 1. INTRODUCTION
10
The dipole matrix elements for the excitonic basis is defined, calculated and related to
the Terahertz absorbance. Important states such as the 1s, 2pz , 2px and 3dz states are
discussed and give a deeper understand of the intraband transitions. The absorbance
for CdSe nanorods, assuming less than one exciton per nanorod, is calculated and is
compared to experimental data provided by David Cooke of McGill University with
good agreement. Some critical differences are discussed between the theoretical and
experimental data mostly due to lack of information about the initial population of
the excitonic states as well as lack of detailed information on the effect of the ligand
covering the CdSe nanorods.
In chapter 4, Heisenberg’s equation of motion is used with the constructed excitonic basis to describe nonlinear carrier dynamics. The nonlinear absorbance and
transmittance are calculated and are analyzed in relation to the shifting populations
and correlations of the excitonic states. Nonlinear effects start to be seen for THz field
intensities of approximately 7 kV/cm most notably characterized by higher order transitions such as the 2pz − 3dz transition. Finally we find the total nonlinear response
for a randomly oriented film of nanorods by averaging over nanorod orientation.
Chapter 2
Excitonic Basis
In this work, we employ an excitonic basis to study carrier dynamics in nanorods.
Figure 2.1 shows a diagram of the nanorod, where the longitudinal direction is along
the rod axis and the transverse direction is perpendicular to it. The time independent
Schrödinger equation (TISE) is first solved for both electrons and holes in cylindrical coordinates to obtain the non-interacting electron and hole eigenstates. These
eigenstates form our non-interacting electron hole basis. The full Hamiltonian which
includes the kinetic and electron-hole Coulomb energy is then evaluated in this noninteracting basis. The Hamiltonian is then diagonalized to find the eigenstates for
the interacting electron and holes in the nanorods, which forms our excitonic basis.
11
12
CHAPTER 2. EXCITONIC BASIS
Longitudinal, z
R0
Transverse, x
y
Figure 2.1: Nanorod diagram showing the longitudinal direction (z) along the rod
axis and the transverse direction in the plane of the radius
2.1
Solving the Time Independent Schrödinger Equation
The TISE is given by
~2 2
∇ Φ + V (r) = EΦ,
−
2m
(2.1)
V (r) ≈ Vρ (ρ) + Vz (z) ,
(2.2)
where V (r) is given by
with the radial and longitudinal potentials given by
Vρ (ρ) =
and



V
ρ > R0


0
ρ < R0
0
(2.3)
13
CHAPTER 2. EXCITONIC BASIS
Vz (z) =




V0




0






V 0
z > L0
0 < z < L0
(2.4)
z<0
respectively. The constant V0 is the confinement potential for a rod of length L0 and
radius R0 . This approximation to the true potential clearly has issues when both
ρ > R0 and z > L0 or z < 0 where (using the above prescription), we would have
V (ρ, z) = 2V0 while the potential should actually be V0 . However, in the limit that V0
is very large, which is the case in this study, the approximate expression is essentially
exact. The advantage of the above approximation for the potential function is that
separation of variables can now be used to obtain an analytical solution to the TISE.
Section 2.1.1 shows the details of solving Eq. (2.1) with separation of variables as
well as the various boundary conditions used.
2.1.1
Separation of variables
The non-interacting solutions of the nanorod are found via separation of variable for
both the electrons and the holes. The TISE, Eq. (2.1), is solved for an electron or a
hole in cylindrical coordinates. The cylindrical wave function
Φ (ρ, z, φ) = R (ρ) Z (z) Θ(φ),
(2.5)
is substituted into Eq. (2.1). Writing the TISE in cylindrical coordinates and generalizing for anisotropic masses we obtain
~2
−
2mρ
∂ 2 Φ 1 ∂Φ
1 ∂ 2Φ
+
+
∂ρ2
ρ ∂ρ
ρ2 ∂φ2
−
~2 ∂ 2 Φ
= (E − V ) Φ.
2mz ∂z 2
(2.6)
14
CHAPTER 2. EXCITONIC BASIS
The constants, mρ and mz are the effective masses from k·p theory (first order approximation) for either electron or the hole in the transverse and longitudinal directions.
Substitution of Eq. (2.5) into Eq. (2.6) allows us to separate the variables. We then
obtain
~2
−
2mρ
1 0
1
00
R (ρ) Z (z) Θ(φ) + R (ρ) Z (z) Θ(φ) + 2 R (ρ) Z (z) Θ (φ)
ρ
ρ
~2
(R (ρ) Z 00 (z) Θ(φ)) = (E − V (ρ, z)) · R (ρ) Z (z) Θ(φ)
−
2mz
00
(2.7)
with primes denoting a derivative with respect to the argument of the given function.
Dividing both sides by R (ρ) Z (z) Θ(φ) yields
~2
−
2mρ
R00 (ρ) 1 R0 (ρ)
1 Θ00 (φ)
+
+ 2
R (ρ)
ρ R (ρ)
ρ Θ(φ)
~2
−
2mz
Z 00 (z)
Z (z)
= (E − V (ρ, z))
(2.8)
In the following subsection we will proceed to separate the θ, ρ and z variables to
find solutions for Θ(φ), R (ρ) and Z (z). In the process we will also find the allowed
energies and assign quantum numbers associated with the transverse and longitudinal
directions.
2.1.2
Angular Solutions
Isolating for the φ dependent terms gives
~2
−
2mρ
R00 (ρ) 1 R0 (ρ)
+
R (ρ)
ρ R (ρ)
~2 Z 00 (z)
−
+ (E − V (ρ, z)) ρ2
2mz Z (z)
~2 Θ00 (φ)
~2 2
=
≡
ν ,
2mρ Θ(φ)
2mρ
(2.9)
CHAPTER 2. EXCITONIC BASIS
15
where we have defined the constant ν for convenience. Eq. (2.9) gives the single
ordinary differential equation
Θ00 (φ) = −ν 2 Θ(φ),
(2.10)
Θν (φ) = eiνφ .
(2.11)
with solutions given by
The variable ν in Eq. (2.11) needs to be an integer for Θν (φ) to be single valued
and is interpreted as the quantum number for the component of the orbital angular
momentum along the rod axis.
2.1.3
Zed Solutions
Isolating for the z variable in Eq. (2.9) yields
~2
−
2mρ
1
R00 (ρ) 1 R0 (ρ)
2
+
+ 2 (−ν ) + (E − Vρ (ρ))
R (ρ)
ρ R (ρ)
ρ
~2 Z 00 (z)
= Vz (z) −
≡ Ez ,
2mz Z (z)
(2.12)
where Ez corresponds to the energy in the longitudinal direction. The above equation
yields the eigenvalue equation
Z 00 (z)
2mz
= 2 (Vz (z) − Ez ) .
Z (z)
~
(2.13)
Inside the nanorod, this has the solution
Zk (z) = A sin (pk z) + B cos (pk z) ,
(2.14)
16
CHAPTER 2. EXCITONIC BASIS
where pk is defined by
pk ≡
2mz,r Ez,k
,
~2
(2.15)
where k is the quantum number associated with the motion in the longitudinal direction. k is an integer with range {0, , 1 , 2 , 3 ...} with higher integers implying a higher
energy state. Ez,k is the energy for a particular solution and mz,r is the effective mass
of the electron or hole inside the nanorod in the radial direction. In the limit of hard
boundary conditions pk reduces to the wavenumber for a particle in the box given by
pk =
π (k + 1)
.
L0
(2.16)
Similarly, the solutions outside the nanorod are
Zk (z) = Aeqk z + Be−qk z ,
(2.17)
where qk is defined by
r
qk ≡
2mz,b (V0 − Ez,k )
,
~2
(2.18)
and mz,b is the effective mass of the electron or hole in the barrier. The boundary
conditions that the wavefunction is continuous gives
Zr (zB ) = Zb (zB ) ,
(2.19)
and the equation for the relation of the derivatives is given by
1 ∂
1 ∂
Zr (zB ) =
Zb (zB ) ,
mz,r ∂z
mz,b ∂z
(2.20)
17
CHAPTER 2. EXCITONIC BASIS
where zB is either 0 or L and the subscripts r and b denote inside and outside the
nanorod respectively. Taking into account that the wavefunction must go to zero far
from the nanorod, the full solutions to Eq. (2.13) are given by
Zk (z) =




Ak eqk z




z<0
Ck sin (pk z) + D cos (pk z)





F e−qk z

k
(2.21)
0 < z < L0
z > L0
whose constants are determined based on the normalization condition
ˆ∞
(2.22)
dk (Zk (z))2 = 1,
0
and previously stated boundary conditions. In order to find the energy Ek,z that
satisfies the boundary conditions we first construct the matrix equation based on
Eqs. (2.19) and (2.20) given by

0
−1
0
 1

 qk
−m? pk
0
0


 0
sin (pk L0 )
cos (pk L0 )
−e−qk L0


0 m? pk cos (pk L) −m? k sin (kk L0 ) qe−qk L0




  Ak   0 

  
  Ck   0 

  

= 
 D   0 
 k   

  
Fk
0
(2.23)
where m? ≡ mz,r /mz,b . The determinant of the matrix must be equal to zero to
obtain non-trivial solutions to Ak , Ck , Dk , Fk . Setting the determinant to zero and
solving for the roots yields the allowed Ek,z values.
2.1.4
Radial Solutions
Finally, proceeding with the radial portion of the Eq. (2.12), we obtain
18
CHAPTER 2. EXCITONIC BASIS
ρ2
R00 (ρ)
R0 (ρ)
2mρ
+ρ
− ( 2 (Vρ (ρ) − Eρ ))ρ2 − ν 2 = 0.
R (ρ)
R (ρ)
~
(2.24)
where we introduce the variable Eρ defined by
Eρ ≡ E − Ez .
(2.25)
Using the substitution
r
x=ρ
(
2mρ
Eρ )
~2
(2.26)
in Eq. (2.24) we obtain
x2
∂ 2R
∂R
+x
+ (x2 − ν 2 )R = 0,
∂x
∂x
(2.27)
whose solutions are known to be Bessel functions of the first kind for the region ρ < R0
given by
Rν (ρ) = Jv (x).
(2.28)
Similarly using the substitution
r
x=ρ
(
2mρ
(Vρ − Eρ ))
~2
(2.29)
in Eq. (2.24) gives
x2
∂ 2R
∂R
+x
− (x2 + ν 2 )R = 0
∂x
∂x
(2.30)
whose solutions for ρ > R0 are known to be modified Bessel functions of the second
kind given by
19
CHAPTER 2. EXCITONIC BASIS
Rν (ρ) = Kv (x).
(2.31)
The Radial portion of the wavefunction is therefore

q


A
2m
n,v
ρ,r

En,v,ρ
ρ < R0
 Nn,v Jv ρ
~2
Rn,v (ρ) =
,
q

2mρ,b
B

n,v

(Vρ − En,v,ρ )
ρ > R0
 Nn,v Kv ρ
~2
(2.32)
where n is a quantum number associated with the allowed radial energy. n is an
integer with range {0, , 1 , 2 , 3 ...} with higher integers implying a higher energy state.
The coefficients An,v and Bn,v are constants for a particular radial energy En,v,ρ and
Nn,v is the normalization factor which is defined by
ˆ∞
dρ ρ (R (ρ))2 = 1.
(2.33)
0
The boundary conditions are given by
An,v Jv (fn,v R0 ) = Bn,v Kv (gn,v R0 )
(2.34)
An,v ∂
Bn,v ∂
Jv (fn,v R0 ) =
Kv (gn,v R0 ) ,
mρ,r ∂ρ
mρ,b ∂ρ
(2.35)
and
where the masses mρ,r and mρ,b correspond to the effective masses inside and outside
the nanorod, fn,v is defined by
r
fn,v ≡
and gn,v is defined by
2mρ,r
En,v,ρ
~2
(2.36)
20
CHAPTER 2. EXCITONIC BASIS
r
gn,v ≡
2mρ,b
(Vρ − En,v,ρ ).
~2
(2.37)
To find Eρ the transcendental Eq. based on Eqs. (2.34) and (2.35). The matrix
representation of the boundary conditions is given by







  An,v   0 

 =  ,
1 ∂
1 ∂
J (fn,v R0 ) mρ,b ∂ρ Kv (gn,v R0 )
Bn,v
0
mρ,r ∂ρ v
Jv (fn,v R0 )
Kv (gn,v R0 )
(2.38)
where the determinant of the matrix must be equal to zero to obtain non-trivial
solutions to An,v and Bn,v . Setting the determinant to zero and solving for the roots
yields the allowed En,v,ρ values. Once the allowed En,v,ρ can all the other constants
can be found to find the radial solutions.
2.2
Non-interacting energies for electrons and holes
We are now in a position to find the wavefunctions and energies of the electron and
holes in the non-interacting basis. Figures 2.2, 2.3 and 2.4 show the energy of light
holes, heavy holes and electron states as a function of the longitudinal quantum
number for some of the states with the lowest radial and angular quantum numbers
for the hole and electron states. The plots shown are for a CdSe nanorod 70 nm in
length, 3.5 nm in radius and with hard boundary conditions. The effective masses
used are given in table 2.1 taken from Ref. [30]. Only the effective masses inside the
rod are needed in this thesis as we will always assume hard boundary conditions. As
shown in the previous sections each electron-hole pair state is characterized by six
quantum numbers (ne , nh , νe , νh , ke , kh ) but for clarity we discuss the electron and
hole states separately. We observe that for low kh values the heavy holes are higher in
CHAPTER 2. EXCITONIC BASIS
21
energy while for high kh values, light holes are more energetic with corresponding nh ,
vh radial quantum numbers. The spacing between between the electron radial states is
approximately 200 meV which is significantly larger than any of the energy differences
between radial states of either light or heavy holes shown. This implies that the lowest
energy electron-hole pair states are composed of only the lowest electron radial state
and the first few hole radial states.
Figure 2.2: Energy for the light hole states as a function of kh (quantum number
associated with longitudinal direction for holes) for the lowest energy light hole radial
states for a CdSe nanorod 70nm in length, 3.5 nm in radius and with hard boundary
conditions
2.3
Light and Heavy Holes
Before we can continue we must first deal with the issue of both light and heavy holes
being present. So far we have not specified which type of hole is being considered in
our excitons, and we have not included any light and heavy hole mixing. Figure 2.5
shows the energy of light and heavy holes in the non-interacting basis as a function
CHAPTER 2. EXCITONIC BASIS
22
Figure 2.3: Energy for the heavy hole states as a function of kh (quantum number
associated with longitudinal direction for holes) for the lowest energy heavy hole
radial for a CdSe nanorod 70nm in length, 3.5 nm in radius and with hard boundary
conditions
Figure 2.4: Energy for the electron states as a function of ke (quantum number
associated with longitudinal direction for electrons) for the lowest energy electron
radial for a CdSe nanorod 70nm in length, 3.5 nm in radius and with hard boundary
conditions
CHAPTER 2. EXCITONIC BASIS
23
Table 2.1: Effective masses of electrons, light holes and heavy holes for CdSe nanorod
taken from Ref. [30]. Only masses inside the rod are needed for this thesis as we will
always assume hard boundary conditions.
Name
effective mass (units of free electron mass)
electrons mρ,r
0.11
electrons mz,r
0.11
light holes mρ,r
0.645
light holes mz,r
0.313
heavy holes mρ,r
0.377
heavy holes mz,r
1.0
of kh for a CdSe nanorod (L0 = 70 nm, R0 = 3.5 nm as in the previous section).
Only the states with quantum numbers nh = 0, νh = 0, which corresponds to the
lowest energy hole radial states are shown. For low k values, the light holes have a
lower energy than the heavy holes due to a difference in effective masses in the radial
direction. This gap in radial energy (18 meV) between the light and heavy holes for
the low energy longitudinal states makes the light hole states more important. The
important excitonic states that will be discussed in later sections (1s, 2pz , 2px , 3dz )
are mostly comprised of hole states that have longitudinal quantum numbers less
than kh = 10. The light holes being lower energy than the heavy holes allows us,
to first order, ignore the heavy hole contribution. Later sections will show that this
approximation of ignoring the heavy hole contribution is quite good.
2.4
Calculating excitonic states
In section 2.1.1, the solution to the Time independent Schrödinger equation was presented without the electron-hole Coulomb interaction. In order to obtain an excitonic
basis we need to evaluate the Hamiltonian that includes not just barrier potentials and
24
CHAPTER 2. EXCITONIC BASIS
Figure 2.5: Energy of hole states for both light and heavy holes. Energy is shown as
function of longitudinal quantum number kh for a CdSe nanorod with radius 3.5 nm
and 70 nm in length. Only states with quantum numbers nh = 0, νh = 0 are shown.
the kinetic energy but the electron-hole Coulomb interaction as well. The Coulomb
Hamiltonian for an electron-pair is given by
ĤCoulomb
e2
=
4π R̂
1
,
electron − R̂hole (2.39)
where R̂electron and R̂hole are the position operators for the electron and the hole, respectively. We write a particular electron hole pair state as |ne , ke , ve > ⊗|nh , kh , vh >.
The pair state wavefunction is given by
Rne ,νe (ρe ) Zke (ze ) Θνe (φe )Rnh ,νh (ρh ) Zkh (zh ) Θνh (φh )
= < ρe , φe , ze |⊗ < ρh , φh , zh |ne , ke , νe > ⊗|nh , kh , νh >
(2.40)
where e and h subscript correspond to the electron and the hole respectively. is the
dielectric constant of the nanorod.We first need to diagonalize the full Hamiltonian,
which is given by
Ĥ = ĤKinetic + ĤCoulomb
(2.41)
25
CHAPTER 2. EXCITONIC BASIS
and make a new excitonic basis with the eigenstates . The eigenstate of the Hamiltonian can be written as
|ψµ >=
X
ζµ,ne ,ke ,ve ,nh ,kh ,vh |ne , ke , ve > ⊗|nh , kh , vh >
(2.42)
ne ,ke ,ve ,nh ,kh ,vh
which we define as the excitonic states, where the quantum number µ denotes the
energy of the exciton and the ζµ,ne ,ke ,ve ,nh ,kh ,vh are the appropriate expansion coefficients.
2.4.1
Evaluation of Coulomb matrix elements
The Coulomb Hamiltonian in the electron-hole pair state basis is given by
< ne1 , ke1 , ve1 |⊗ < nh1 , kh1 , vh1 |ĤCoulomb |ne2 , ke2 , ve2 > ⊗|nh2 , kh2 , vh2 >
1
e2
|ne2 , ke2 , ve2 > ⊗|nh2 , kh2 , vh2 >
4π R̂
electron − R̂hole ˆ
ˆ
e2
3
=
d relectron d3 rhole Φne1 ,ke1 ,ve1 (ρe , φe , ze )Φnh1 ,kh1 ,vh1 (ρh , φh , zh )
4π
1
Φn ,k ,v (ρe , φe , ze )Φnh2 ,kh2 ,vh2 (ρh , φh , zh(2.43)
) .
|relectron − rhole | e2 e2 e2
=< ne1 , ke1 , ve1 |⊗ < nh1 , kh1 , vh1 |
A direct approach to evaluating this Hamiltonian in the non-interacting electron-hole
pair basis would require the calculation of real space integrals via a six dimensional
Monte Carlo routine. This method, however, is far too computationally expensive
and so spatial function expansions are first done analytically that take advantage of
the cylindrical symmetry of the system. The
1
|relectron −rhole |
term can be written using
the identity
ˆ
∞
1
2 X
dk eim(φe −φh ) cos [k (ze − zh )] Im (kρ< ) Km (kρ> ) ,
=
|relectron − rhole |
π m=−∞
∞
0
(2.44)
26
CHAPTER 2. EXCITONIC BASIS
where ρ> is the larger of ρe or ρh , ρ< is the smaller of ρe or ρh , Im and Km are the
modified Bessel functions of the first and second kind respectively [34]. The elements
of the Coulomb Hamiltonian can now be written as
< ne1 , ke1 , ve1 |⊗ < nh1 , kh1 , vh1 |ĤCoulomb |ne2 , ke2 , ve2 > ⊗|nh2 , kh2 , vh2 >
ˆ
ˆ
∞
X
e2 2
3
3
=
d relectron d rhole
4π π
m=−∞
ˆ∞
dk eim(φe −φh ) cos [k (ze − zh )] Im (kρ< ) Km (kρ> )
×
0
× Φ?e1 (ρe , φe , ze )Φ?h1 (ρh , φh , zh )Φe2 (ρe , φe , ze )Φh2 (ρh , φh , zh ).
(2.45)
The advantage of the above expression is that the angular terms can be changed
into Dirac delta functions and the z dependence can be integrated separately and
analytically from the ρ terms reducing computation time. Thus we obtain
ˆ
∞
e2 2 X
=
dk
4π π m=−∞
∞
ĤCoulomb
0
ˆR0
×
×
ˆR0
dρe
dρh ρe ρh Im (kρ< ) Km (kρ> )
0
0
Rn? e1 ,ve1
(ρe ) Rn? h1 ,vh1 (ρh ) Rne2 ,ve2 (ρe ) Rnh2 ,vh2 (ρh )
× δve2 −ve1 +m,0 δvh2 −vh1 −m,0
ˆ
ˆ
×
dze dzh Zk?e1 Zk?h1 Zke2 Zkh2 cos [k (ze − zh )] .
(2.46)
The Coulomb Hamiltonian can now be evaluated with a 3D integral using Monte
Carlo routines since the z integral can be done analytically and the angular portion
of the integral has been been reduced to delta functions. The z integrals can be done
27
CHAPTER 2. EXCITONIC BASIS
analytically since they are piecewise defined exponential and sinusoidal functions.
This reduction from an initially 6D integral to a 3D integral makes the problem
computationally feasible.
2.4.2
Exciton wavefunctions
The excitonic wave functions based on the Coulomb matrix elements calculated in
section 2.4.1 are strongly confined in the radial direction. This confinement causes
the radial function of the excitonic states to be mostly composed of J0 (kρ) for both
the electron and hole states. The ground state and the 2pz state (to be described in a
later section), for example, have a radial wavefunction that is 99.9% comprised of the
n = 0, v = 0 state of the non-interacting basis. The 2px state is similarly comprised
almost exclusively of n = 0, v = −1, 1 states of the non-interacting basis.
Figure 2.6 shows the ground state exciton (1s) wavefunction of a CdSe rod 70 nm
in length and 7 nm diameter. The 1s state is plotted as a function of the relative
and center of mass electron-hole coordinates in the z-direction, where the relative
electron-hole coordinate (zr ) given by
zr = ze − zh
(2.47)
and the center of mass coordinate is given by
zt =
me ze + mh zh
.
me + mh
(2.48)
= 9.5 0
(2.49)
A dielectric constant of
is used for CdSe where 0 is the vacuum permittivity. An infinite potential (V0 = ∞)
was used to mimic hard boundary conditions. The radial and angular portion of the
CHAPTER 2. EXCITONIC BASIS
28
wavefunction have been plotted along the rod axis. As can be seen, the ground state
wavefunction is more concentrated along the zr = 0 line where the distance between
the electron and hole is small; this implies a large binding energy. Figure 2.7 shows
the µ = 1, 2, 3, 4 states that have increasing number of nodes along the zr = 0 line
representing the higher center of mass states analogous to excited states a particle in
a box. The asymmetry along the zt is due to the difference in effective masses between
the electron and hole; if the masses for the two carriers were the same then we would
expect a perfectly symmetric plot. Figure 2.8 shows that the 2pz state is zero along
the zr = 0 and large near zr = −5 , 5 signifying a large separation between the
electron and the hole (we will see later sections that this large dipole moment along
the longitudinal direction makes the 1s − 2pz the dominant transition). Figure 2.9
shows the two 2px states, corresponding to the v = −1, 1 angular momentum quantum
numbers; this state as it is plotted looks identical to the excitonic ground state because
the excitation is in the radial/angular direction rather than the longitudinal direction.
Figure 2.11 shows the pz -like states that the states in Figure 2.7 would transition to,
analogous to the 1s − 2pz transition. Another important longitudinal transition is
the 2pz − 3dz transition with 3dz state wavefunction shown in figure 2.10. The states
shown here comprise a small subset of all the states but they are the most significant
states involved with Terahertz intraband dynamics to be discussed in later sections.
Calling these states 1s, 2pz , 2px and 3dz is meant to draw analogies to the hydrogen
atom but since this is a pseudo one dimensional system rather than one with spherical
symmetry, the number prefix (2 in 2pz ) is somewhat arbitrary.
Figure 2.12 shows the energies of the 1s, 2pz , 2px and 3dz excitonic states without
the band gap. We will see in later sections that the 1s − 2pz transitions at 34 meV
CHAPTER 2. EXCITONIC BASIS
29
Figure 2.6: Plot of the exciton wavefunction as a function of the relative and center
of mass coordinates along the rod axis for the excitonic ground state
Figure 2.7: Plot of the exciton wavefunction as a function of the relative and center
of mass coordinates along the rod axis for the higher order center of mass motion
excited states
CHAPTER 2. EXCITONIC BASIS
30
Figure 2.8: Plot of the exciton wavefunction as a function of the relative and center
of mass coordinates along the rod axis for the 2pz state
Figure 2.9: Plot of the exciton wavefunction as a function of the relative and center of
mass coordinates along the rod axis for the two 2px states corresponding to angular
quantum numbers -1 and 1
Figure 2.10: Plot of the exciton wavefunction as a function of the relative and center
of mass coordinates along the rod axis for the 3dz state
CHAPTER 2. EXCITONIC BASIS
31
Figure 2.11: Plot of the exciton wavefunction as a function of the relative and center
of mass coordinates along the rod axis for the higher order center of mass motion
pz -like states
(8.5 THz), 1s − 2px transition at 46 meV (11 THz) and 2pz − 3dz 5 meV (1 THz) are
the dominant transitions. Table 2.2 gives a list of the energies of the excitons for µ
values 0 to 30 with the important states labeled. The 1s and 2p like states are also
labeled with the superscript denoting the number of nodes in the zt axis.
CHAPTER 2. EXCITONIC BASIS
32
Table 2.2: Exciton energies for the first 31 excitonic states and their respective labels.
The label 1sx denotes a 1s like state but with an x number of nodes in the zt (center
of mass) direction. Some states are not labeled as they exhibit complicated properties
than cannot be adequately explained with the introduced notation. There are two
2px states for angular quantum numbers -1 and 1
µ Name Energy
µ
Name
Energy
3
0
1s
129.3
16
2pz
167.1
1
1
1s
130.0
17
3dz
168.0
2
13
2
1s
131.0
18
1s
168.2
3
1s3
132.5
19
169.0
4
4
1s
134.5
20
169.5
5
5
1s
136.8
21
170.9
6
6
1s
139.4
22
171.6
7
1s7
142.4
23
172.0
8
8
1s
145.7
24
172.9
9
9
1s
149.5
25
173.0
10 1s10
153.6
26
1s14
174.0
11
11 1s
158.2
27
174.3
12
12 1s
162.7
28
174.8
13
2pz
163.1
29 2px ,v = 1
175.1
14
2p1z
164.0
30 2px ,v = −1 175.1
15
2p2z
165.4
33
CHAPTER 2. EXCITONIC BASIS
2px 175 meV
Energy
3dz 168 meV
2pz 163 meV
1s 129 meV
Band Gap 1.74 eV
Figure 2.12: Schematic diagram presenting the energies of the 1s, 2pz , 2px and 3dz
excitonic states in meV. The distance between the energies are not presented to scale
and the dots denote there are states in between the labeled excitonic states. The
band gap of 1.74 eV is for a temperature of 0 K.
Chapter 3
Terahertz driven intraband
transitions: linear response
This chapter details the theory and experimental methods used to examine the interaction of Terahertz radiation with the nanorod excitonic states described in Chapter
2. The intraband transitions are calculated in the rotating wave approximation assuming that there is at most one exciton per nanorod. The predicted absorbance
is then compared and found to be in good agreement with experimental results by
David Cooke et. al. [14].
3.1
Intraband Hamiltonian
In order to calculate the linear response of the excitons in the presence of THz radiation, we first need the part of the Hamiltonian that accounts for the intraband
transitions in the excitonic basis. We call this Hamiltonian Ĥintraband . In the dipole
approximation, the part of the Hamiltonian that accounts for the interaction of the
34
CHAPTER 3. TERAHERTZ DRIVEN INTRABAND TRANSITIONS: LINEAR RESPONSE35
THz field with the excitons is given by a dot product between the electric field and
the electron hole separation R̂:
Ĥintraband = eÊ · R̂,
(3.1)
R̂ = R̂electron − R̂hole .
(3.2)
where R̂ is given by
The matrix elements of this intraband Hamiltonian can now be evaluated in position
space. Since the wavelength of a Terahertz wave is large in comparison to the dimensions of a nanorod, the electric field is assumed to be constant in space. In the
non-interacting electron-hole basis, the intraband Hamiltonian matrix elements are
given by
< ne1 , ke1 , ve1 |⊗ < nh1 , kh1 , vh1 |Ĥintraband |ne2 , ke2 , ve2 > ⊗|nh2 , kh2 , vh2 >
= eE· < ne1 , ke1 , ve1 |⊗ < nh1 , kh1 , vh1 |R̂|ne2 , ke2 , ve2 > ⊗|nh2 , kh2 , vh2 >
ˆ
= eE · [δh1,h2
d3 re Φ?ne1 ,ke1 ,ve1 (ρe , φe , ze )re Φne2 ,ke2 ,ve2 (ρe , φe , ze )
cylinder
ˆ
−δe1,e2
d3 rh Φ?nh1 ,kh1 ,vh1 (ρh , φh , zh )rh Φnh2 ,kh2 ,vh2 (ρh , φh , zh )] , (3.3)
cylinder
where δe1,e2 requires that the electron states are identical with a similar condition for
holes. The two integrals in Eq. (3.3) both take the form
CHAPTER 3. TERAHERTZ DRIVEN INTRABAND TRANSITIONS: LINEAR RESPONSE36
ˆ
d3 rΦ?n1 ,k1 ,v1 (ρ, φ, z)rΦn2 ,k2 ,v2 (ρ, φ, z)
cylinder
ˆ∞
1
=
2π
ˆ
ˆ∞
ˆ
ˆ
2π
dφ e
dz Zk1 (z) Zk2 (z) ρ sin (φ) ŷ
−∞
ˆ
2π
dρ ρRn1 ,v1 (ρ) Rn2 ,v2 (ρ)
dφ e
0
0
∞
i(ν2 −ν1 )φ
0
ˆ
dz Zk1 (z) Zk2 (z) ρ cos (φ) x̂
−∞
dρ ρRn1 ,v1 (ρ) Rn2 ,v2 (ρ)
ˆ∞
∞
i(ν2 −ν1 )φ
0
0
1
+
2π
dφ e
dρ ρRn1 ,v1 (ρ) Rn2 ,v2 (ρ)
0
1
+
2π
ˆ
2π
∞
i(ν2 −ν1 )φ
dz Zk1 (z) Zk2 (z) z ẑ.
(3.4)
−∞
Evaluating the angular integrals gives
ˆ
d3 rΦ?n1 ,k1 ,v1 (ρ, φ, z)rΦn2 ,k2 ,v2 (ρ, φ, z)
cylinder
ˆ∞
=
1
2
dρ ρ2 Rn1 ,v1 (ρ) Rn2 ,v2 (ρ) (δν2 −ν1 ,−1 + δν2 −ν1 ,1 ) δk1 ,k2 x̂
0
−
i
2
ˆ∞
dρ ρ2 Rn1 ,v1 (ρ) Rn2 ,v2 (ρ) (δν2 −ν1 ,−1 − δν2 −ν1 ,1 ) δk1 ,k2 ŷ
0
ˆ
∞
+ δv1 ,v2 δn1 ,n2
dz Zk1 (z) Zk2 (z) z ẑ.,
−∞
and grouping terms finally gives
(3.5)
CHAPTER 3. TERAHERTZ DRIVEN INTRABAND TRANSITIONS: LINEAR RESPONSE37
ˆ
d3 rΦ?n1 ,k1 ,v1 (ρ, φ, z)rΦn2 ,k2 ,v2 (ρ, φ, z)
cylinder
1
(δν −ν ,−1 (x̂ − iŷ) + δν2 −ν1 ,1 (x̂ + iŷ))
2 2 1
ˆ∞
× δk1 ,k2 dρ ρ2 Rn1 ,v1 (ρ) Rn2 ,v2 (ρ)
=
0
ˆ
∞
+ δv1 ,v2 δn1 ,n2
dz Zk1 (z) Zk2 (z) z ẑ.
(3.6)
−∞
Eq. (3.6) shows that there are two types of transitions. The first type requires
identical zed wavefunction between the first and second states while requiring a difference between the angular momentum quantum numbers of +1,-1 corresponding
to left and right circularly polarized THz radiation in the transverse direction. The
second term arises from transitions along the zed directions and requires the radial
and angular portions of the wavefunctions to be identical. Eq. (3.6) allows the evaluation of Eq. (3.3). Transitions along the longitudinal (transverse) direction occur for
THz radiation polarized in the longitudinal (transverse) direction. With the intraband Hamiltonian in the electron-hole pair basis, we can perform a change of basis
transformation to convert it to the excitonic basis.
Now that we have calculated Ĥintraband in the electron-hole pair basis we wish to
express it in the excitonic basis. The previously evaluated Hamiltonian, Ĥ with the
kinetic energy and Coulomb energy given by Eq. (2.41) evaluated in electron-hole
pair basis can now be diagonalized. This diagonalization is given by
Ĥ = P DP −1 ,
(3.7)
where P is the matrix whose column vectors are eigenvectors of Ĥ and D is the
diagonal matrix with corresponding eigenvalues. These eigenvectors form excitonic
CHAPTER 3. TERAHERTZ DRIVEN INTRABAND TRANSITIONS: LINEAR RESPONSE38
states and the eigenvalues give the corresponding energies. We can thus perform a
change of basis operation on Ĥintraband given by
Ĥintra,ex = P −1 Ĥintraband P,
(3.8)
where Ĥintra,ex is now the intraband Hamiltonian in the excitonic basis. A discussion of the numerical techniques and computational times required to calculate the
excitonic states and to perform the nonlinear response presented in Chapter 4 are
discussed in Appendix A.
3.2
Linear Response
We consider now the situation where the excitons in a nanorod are first generated by
optical pulse with energy above the band gap. The excitons are then given enough
time to relax down to a quasi-thermal equilibrium state at some temperature T . The
nanorods considered are made of CdSe with a length of 70 nm and radius of 3.5
nm. The nanorods are covered with passivating organic ligand dodecylphosphonic
acid (DDPA) approximately 2 nm long. The nanorod axes all lie in a plane with
uniformly randomly distributed orientations. If we assume that the absorbance of
the Terahertz wave, with electric field in the the plane of nanorods, is in the linear
regime, the excitonic intraband transitions are proportional to the matrix elements
of Ĥintra,ex . We define the linear electric susceptibility tensor, χ̄ij (ω) given by
P̄ii (ω) =
X
χ̄ij (ω) Ej (ω) ,
(3.9)
j
where P̄ii (ω) is the polarization, Ej (ω) is the THz electric field and the indices i, j
denote the direction (x, y, z) [38]. The linear electric susceptibility tensor is thus
given by
CHAPTER 3. TERAHERTZ DRIVEN INTRABAND TRANSITIONS: LINEAR RESPONSE39
D ED E
ψn R̂i ψm ψm R̂j ψn e−~ωm /kB T − e−~ωn /kB T
X
X
eN
, (3.10)
χ̄ij (ω) =
0 m n>m
~ (ωnm − ω − iΓnm ) Z
2
where N is the density of excitons per unit volume and the energies levels are given
by
En = ~ωn ,
(3.11)
ωnm ≡ ωn − ωm .
(3.12)
where
The term ψm denotes an excitonic state with quantum number µ = m. The partition
function is given by
Z −1 =
X
e−~ωn /kB T ,
(3.13)
n
and the linear frequency full width half maximum of the absorbance line ∆f i of each
transition is given by
∆f i =
Γf i
.
π
(3.14)
Γf i is a phenomenological constant, R̂i refers to the operator in the i direction, R̂z
would, for example, equal to Ẑ. 0 is the vacuum permittivity. For our choice of
D ED E
coordinate, The product ψn R̂i ψm ψm R̂j ψn is nonzero only if i 6= j since
transitions from n to m correspond only to a particular polarization and requires
that the indices i and j are the same. We can thus remove the j index and only take
CHAPTER 3. TERAHERTZ DRIVEN INTRABAND TRANSITIONS: LINEAR RESPONSE40
the diagonal components of the tensor rewriting the linear electric susceptibility in
the i direction as
D E2 ψn R̂i ψm e−~ωm /kB T − e−~ωn /kB T
X
X
eN
.
χ̄ii (ω) =
0 m n>m
~ (ωnm − ω − iΓnm ) Z
2
(3.15)
For a THz field polarized in the ith direction, the real part of the susceptibility is
given by
D E2
ψn R̂i ψn (ωnm − ω) e−~ωm /kB T − e−~ωn /kB T
X
X
e
N
χ̄0ii (ω) ≡ Re {χ̄ii (ω)} =
~0 m n>m
(ωnm − ω)2 + (Γnm )2 Z
(3.16)
2
and the imaginary part is given by
D E2
X X ψn R̂i ψn (Γnm ) e−~ωm /kB T − e−~ωn /kB T
e
N
χ̄00ii (ω) ≡ Im {χ̄ii (ω)} =
.
~0 m n>m
(ωnm − ω)2 + (Γnm )2 Z
(3.17)
2
The complex wavenumber for the plane wave propagating through the medium containing the nanorods with polarization in the ith direction is given by
k = k0
p
n0 + χii (ω),
(3.18)
where n0 is the average index of refraction of the medium (nanorod and the ligand
layer) when the excitonic resonances are not included, χii (ω) is the linear electric
susceptibility in the i direction and
k0 ≡
Now, we can write the wavenumber as
ω
.
c
(3.19)
CHAPTER 3. TERAHERTZ DRIVEN INTRABAND TRANSITIONS: LINEAR RESPONSE41
i
k = n (ω) k0 + αi (ω) ,
2
(3.20)
where n (ω) is the index of refraction and αi (ω) is the absorbance coefficient in the i
direction. Substitution yields
1
k02 n20 + χ0ii (ω) + iχ00ii (ω) = n2 (ω) k02 − αi2 (ω) + in (ω) k0 αi (ω) .
4
(3.21)
Solving, we obtain
αi (ω) =
k0 χ00ii (ω)
n (ω)
(3.22)
and
n2 (ω) = n20 + χ0ii (ω) +
1 2
α (ω)
4k0 i
(3.23)
This second equation becomes
1
n4 (ω) − n20 + χ0ii (ω) n2 (ω) − χ00ii (ω) ,
4
(3.24)
and solving this we obtain
r
n (ω) =
1 2
1
[n0 + χ0ii (ω)] +
2
2
q
2
[n20 + χ0ii (ω)] + χ00ii (ω).
(3.25)
Now, in our case, the background index of refraction will be an average between the
index of ligand DPPA (nD = 1.55) and the nanorods (nr ' 2.3). If we average the
two index of refraction weighted by the volume of the ligand and the nanorod we
obtain n ' 2.2.
CHAPTER 3. TERAHERTZ DRIVEN INTRABAND TRANSITIONS: LINEAR RESPONSE42
Now, for a frequency near the main resonance for polarization along the longitudinal direction of 8.5 THz, we would expect χ0ii (ω) to be much smaller than χ00ii (ω)
on resonance. Assuming ω = ω1s−2pz and the excitons are initially in the 1s state we
can rewrite Eq. (3.17) as
D
E2
R̂
ψ
ψ
i
2pz 1s
eN
00
.
(3.26)
χ̄ii (ω) ≡ Im {χ̄ii (ω)} =
~0
Γnm
D
E
Now using the calculated value of ψ1s Ẑ ψ2pz = 3.0 nm and an estimated exciton
2
volume density of N =
1
(based
6000 nm3
on a CdSe nanorod of 70 nm in length, 3.5
nm radius, 2 nm thick ligands covering and 1 exciton per nanorod), we find then
for the longitudinal polarization χ00zz (ω) ' 5. Substituting this to Eq. (3.25) yields
n (ω) = 2.25 for n0 = 2.2. Thus, it is a reasonably good approximation to neglect
the effect of the excitonic resonances on the index of refraction when calculating the
absorbance. We thus obtain
αi (ω) =
ωχ00ii (ω)
.
cn0
(3.27)
Our final expression for the absorbance coefficient is then
D E2
ψn R̂ ψm (Γnm ) e−~ωm /kB T − e−~ωn /kB T
X
X
e Nω
.
ᾱ (ω) =
~cn0 0 m n>m
(ωnm − ω)2 + (Γnm )2 Z
2
3.2.1
(3.28)
Orientation averaging
Figure 3.1 illustrates the experimental setup where colloidal nanorods are randomly
oriented but with the rod axis and the THz radiation always in the the x-z plane.
There are also multiple layers of nanorods that must be taken into account when
CHAPTER 3. TERAHERTZ DRIVEN INTRABAND TRANSITIONS: LINEAR RESPONSE43
calculating the absorbance. Figure 3.2 shows a diagram of the reduction of the THz
electric field in the transverse direction but with no change in the longitudinal direction. This reduction in electric field must be taken into account when calculating the
absorbance. The electric field inside a long cylinder is given by
EL,r = EL,0
(3.29)
and
ET,r =
2
ET,0
1 + r /0
(3.30)
where the subscripts T and L refer to the transverse and longitudinal directions
respectively. The subscripts r and 0 denote inside and outside the rod respectively
[34]. In a later sections we take the value of r = 6.25 for CdSe and 0 = 2.40 for the
organic ligand to be described when comparing experimental and theoretical results.
For colloidal nanorods, we want to average over all possible orientations in relation
to the THz radiation polarization. Since both the THz electric field and nanorods
lie in the x − z plane, we can ignore the y component of the transition dipole and
averaging the nanorod orientations gives the absorbance of
e2 N ω X X (Γnm ) e−~ωm /kB T − e−~ωn /kB T
hα (ω)i =
~cn0 0 m n>m
(ωnm − ω)2 + (Γnm )2 Z
ˆ2π
D E2
2
2
ψn X̂ ψm sin (θ)
1 + r /0
0
2
e N ω X X (Γnm ) e−~ωm /kB T − e−~ωn /kB T
=
~cn0 0 m n>m
(ωnm − ω)2 + (Γnm )2 Z
D E2 2
1 D E2
×
(3.31)
ψn Ẑ ψm +
ψn X̂ ψm .
2
1 + r /0
1
×
2π
D E
2
dθ ψn Ẑ ψm cos2 (θ) +
CHAPTER 3. TERAHERTZ DRIVEN INTRABAND TRANSITIONS: LINEAR RESPONSE44
Ez
Ex
Figure 3.1: Experimental setup with randomly oriented nanorods. The diagram shows
four layers for illustration purposes and there could me more. The THz electric field
is polarized in the x − z plane. The nanorods also lie in the x − z plane
CHAPTER 3. TERAHERTZ DRIVEN INTRABAND TRANSITIONS: LINEAR RESPONSE45
Figure 3.2: Reduction in the transverse electric field due to different dielectric constants outside (0 ) and inside (r ) the nanorod. The subscripts T and L denote the
transverse and longitudinal directions. The electric field in the longitudinal direction
is the same both outside and inside the nanorod.
CHAPTER 3. TERAHERTZ DRIVEN INTRABAND TRANSITIONS: LINEAR RESPONSE46
Eq. (3.31) gives the absorbance with longitudinal (z) and transverse components (x)
projected out. The components are then averaged over all nanorod orientations and
then added together after taking into account the reduction in electric field in the
transverse direction. The linear absorbance is therefore given simply by the average
of the absorbance for polarization in the longitudinal and transverse directions.
3.3
Theoretical results
In this section and all subsequent sections for the chapter we consider CdSe nanorod
of length 70nm, using light holes and hard boundary conditions. Figure 3.3 shows
the calculated linear response absorbance coefficient of the CdSe nanorods (70 nm
length, 7 nm diameter) using light holes for longitudinal polarization, transverse
polarizations and orientation-averaged result at 0 K (all states initially in the 1s
state) and Γnm = 2.75 THz (linewidth of 5.5 THz). The choice of Γnm = 2.75
THz is chosen to make plots easily understandable. We will consider what value is
experimentally appropriate in a later section where we compare to experiment. The
longitudinal intraband excitation has a dominant transition at 8.5 THz (35 meV)
caused by a transition from the 1s, as shown in figure 2.7 , (lowest energy excitonic
state) to the 2pz state, as shown in figure 2.8. This 1s − 2pz transition causes a large
peak due to the large dipole moment, d¯zn,m given by
D E
z
¯
dn,m = ψn Ẑ ψn ,
(3.32)
along the longitudinal direction. The zed component (longitudinal) of d¯n,m between
the 1s and 2pz state has a dipole moment of dz1s,2pz = 3 nm. The absorbance in the
transverse direction is dominated by the 1s − 2px transition with a dipole moment of
CHAPTER 3. TERAHERTZ DRIVEN INTRABAND TRANSITIONS: LINEAR RESPONSE47
dx1s−2px =1.1 nm, with the 2px shown in figure 2.9 .
Figure 3.3: Linear absorbance for longitudinal and transverse polarizations as well as
for the total angularly-averaged response at a temperature of 0 K with a linewidth of
5.5 THz using light holes
In the following discussion we ignore the transitions between light and heavy holes
as the intraband matrix elements between them are very small. Figures 3.4 and 3.5
show the linear absorbance for both light and heavy holes at different temperatures.
We note that heavy hole absorbance is significantly less than the light holes due to
their lower occupation probability (heavy holes having overall higher energy). The
light holes having a greater contribution to the absorbance gives us assurance that
we can ignore the heavy hole contributions. Another thing to note is that the linear
absorbance for both light and heavy holes are very similar with regards to where the
peaks occur. This similarity gives further assurance that even if the heavy holes had
a larger contribution it would not significantly change the qualitative aspects of the
result.
The most distinct feature in the linear absorbance for both light and heavy holes
CHAPTER 3. TERAHERTZ DRIVEN INTRABAND TRANSITIONS: LINEAR RESPONSE48
is that for increasing temperature there is a growing peak at approximately 2 THz.
This peak is attributed to the 2pz − 3dz , with wavefunctions shown in figures 2.8
and 2.10, and other higher order zed transitions. At room temperature the 2 THz
peak begins to dominate the linear absorbance. It’s important to note that while
the 8.5 THz and 11 THz peaks are formed almost exclusively from the 1s − 2pz and
1s − 2px transitions, the broad 2 THz peak comes from dozens of different higher
order longitudinal transitions of which the 2pz − 3dz , with a very large dipole moment
of 11.9 nm, is the most significant.
Figure 3.4: Linear absorbance of CdSe nanorod for different temperatures with a
linewidth of 5.5 THz using light holes
3.4
Intraband transition frequencies, dipole moments
and binding energies as a function of radius
Calculations have shown that due to the weak quantum confinement in the longitudinal direction, transition frequencies and dipole moments have weak dependence
CHAPTER 3. TERAHERTZ DRIVEN INTRABAND TRANSITIONS: LINEAR RESPONSE49
Figure 3.5: Linear absorbance of CdSe nanorod for different temperatures with a
linewidth of 5.5 THz using heavy holes
on the nanorod length. Increasing or decreasing the length of the nanorod has little
effect on the features of interest. Thus in this section we study the effect of nanorod
radius on the 1s − 2pz , 1s − 2px , 2pz − 3dz transitions frequencies and dipole moments
as shown in figures 3.6-3.11. The transition frequencies in figures 3.6-3.8 show a decreasing trend with increasing nanorod radius. This decrease in transitions frequency
is due to the energy spacing between the states decreasing with the radius. Figure 3.8
for the 2pz − 3dz transition shows the same downward trend but exhibits some noise
that is due to numerical errors in the Monte Carlo routines. Figures 3.9-3.11 show the
dipole moments increasing with increasing radius. The increase in the dipole moment
is attributed to increasing radius reducing the quantum confinement and allowing the
greater electron and hole separation yielding a larger dipole moment. The noisy result
for the 2pz − 3dz dipole moment in figure 3.11 are as before attributed to numerical
errors. Note that while the result for the 2pz − 3dz transitions appears noisy due to
the scale of the graph, the variation are on the order of 0.01 meV and are quite small.
CHAPTER 3. TERAHERTZ DRIVEN INTRABAND TRANSITIONS: LINEAR RESPONSE50
Figure 3.6: 1s − 2pz light-hole exciton transition frequency as a function of radius for
a 70 nm long CdSe nanorod
Figure 3.7: 1s − 2px light-hole exciton transition frequency as a function of radius for
a 70 nm long CdSe nanorod
CHAPTER 3. TERAHERTZ DRIVEN INTRABAND TRANSITIONS: LINEAR RESPONSE51
Figure 3.8: 2pz − 3dz light-hole exciton transition frequency as a function of radius
for a 70 nm long CdSe nanorod
Figure 3.9: 1s − 2pz light-hole exciton dipole moment as a function of radius for a 70
nm long CdSe nanorod
CHAPTER 3. TERAHERTZ DRIVEN INTRABAND TRANSITIONS: LINEAR RESPONSE52
Figure 3.10: 1s − 2px light-hole exciton dipole moment as a function of radius for a
70 nm long CdSe nanorod
Figure 3.11: 2pz − 3dz light-hole exciton dipole moment as a function of nanorod
radius for a 70 nm long CdSe nanorod
CHAPTER 3. TERAHERTZ DRIVEN INTRABAND TRANSITIONS: LINEAR RESPONSE53
We introduce the exciton binding energy given by
Ebinding = E1s − Ef ree ,
(3.33)
where E1s is the energy of the excitonic 1s state and Ef ree is the energy of the lowest
energy state of the non-interacting electron hole pair basis described in Chapter 2.
The exciton binding energy gives a rough measure of electron hole Coulomb energy.
The exciton binding energy is shown in figure 3.12 as a function of nanorod radius with
the same parameters as listed above. This decrease in binding energy is expected as
the nanorod increases in size since the electron and hole will be further apart resulting
in a lower Coulomb energy and thus a lower exciton binding energy. The binding
energies shown are larger than the bulk case of 15 meV giving further confirmation to
our calculations [29]. To the author’s best knowledge, no experimental work has been
done to determine the binding energy of CdSe nanorod with comparable dimensions
to those studied in this thesis.
A study by Efros et. al. have shown the exciton binding energy to be substantially
changed (a factor of 2) if one takes into account the dielectric mismatch inside and
outside a quantum dot [36]. Work on the effect of this dielectric mismatch on optically
induced transitions have been done on nanorods but with no work THz driven transitions [37]. This dielectric mismatch could be significant in determining intraband
transition energies but further work needs to be done to quantify this effect.
3.5
Comparison to Experimental Results
This section described a THz pump-probe experiment performed by David Cooke
of McGill university on colloidal CdSe nanorods. His experiment give absorbance
CHAPTER 3. TERAHERTZ DRIVEN INTRABAND TRANSITIONS: LINEAR RESPONSE54
Figure 3.12: Binding energy as a function of radius for a 70 nm long CdSe nanorod
using light holes
spectra that can be compared with the theoretical calculations in this thesis. Colloidal
CdSe nanorods made by Jun Yan Lek of Nanyang Technological University using the
hot coordinating solvents method described in Ref. [35], were passivated with organic
ligand dodecylphosphonic acid (DDPA) roughly 2 nm in length. The DDPA’s purpose
is to electronically isolate the nanorods from each other. The nanorods were then
dropcast onto an Indium-Tin-Oxide (ITO) coated glass slide which reflects 98% of
the THz pulse used in the experiment. Subsequent thickness measurements indicate
that the nanorod layer is roughly 30 nanorods thick. Excitons were optically injected
into the nanorods via 400 nm pulses and were then probed by a THz pulse. Note
that since the optical pump is creating excitons at an energy well past the band gap,
it is not resonant with any of the low energy excitonic states described in Chapter
2. Time-resolved multi-THz spectroscopy was applied in the 1-13 THz (1-50 meV)
range to probe exciton relaxation dynamics in the nanorods at room temperature.
The optically generated carriers were allowed to relax down to lower excited states
CHAPTER 3. TERAHERTZ DRIVEN INTRABAND TRANSITIONS: LINEAR RESPONSE55
before being probed via THz pulse at different delay times ranging from 1 ps to 10 ps.
Figure 3.13 shows the experimental setup where the nanorods form a ∼1 micron film
on a SiO2 substrate that was first coated with an ITO film. The sample is probed in
reflection geometry, with the ITO film reflecting the THz field. The nanorods were
on average 7 nm in diameter and 70 nm in length. Visual inspection of TEM images
of the nanorods suggest a distribution of radii of 3.5 ± 0.5 nm. Optical pump-THz
probe spectroscopy performed by David Cooke was used to characterize the THz
absorbance of the CdSe nanorods. All experimental results discussed in this section
were performed by David Cooke unless otherwise specified [14].
Figure 3.13: Schematic of the THz pump-probe experimental setup. Excitons are
generated in a CdSe nanorod layer (approximately 1 µm in thickness) with a 400
nm optical pump. The CdSe nanorods are deposited on an Indium-Tin-Oxide (ITO)
substrate on a glass slide. A THz pulse is then applied to the nanorods after a delay
time and is reflected by the ITO. This pump-probe setup is used to measure the
absorbance of the CdSe nanorods in the THz frequency range.
Figure 3.14 shows experimentally found absorbance spectrum for CdSe nanorods
after a 1 ps delay as well as theoretical Lorentzian and Gaussian fits to the data at
CHAPTER 3. TERAHERTZ DRIVEN INTRABAND TRANSITIONS: LINEAR RESPONSE56
T = 0 K. The choice for T = 0 K will be explained in a later section. A reduction in
the experimental absorption spectrum was observed at approximately 6.2 THz and
has been attributed to longitudinal optical phonon (6.15 THz in CdSe). This phonon
induced reduction has been found to have a linewidth of 1 THz. It should be noted
that the possibility that the absorbance peak observed was due to plasmon-Fröhlich
phonon mode was also ruled out due to the 8.5 THz peak not shifting with exciton
density [14]. The Lorentzian fit, as described in section 3.2, used a fitted linewidth of
5.5 THz. The Lorentzian and the experimental absorbance are in agreement at the
8.5 THz (1s − 2pz ) peak but differ for the 11 THz (1s − 2px ) transition. There is
another issue in that the 5.5 THz linewidth is quite broad when compared with the
experimentally determined phonon-mediated linewidth of 1 THz. The Lorentzian fit
also does not take into account the variance of the radius or the length. To improve the
fit to the data, we use a linewidth value of 1 THz based on the previously mentioned
phonon linewidth and then take into account of the homogeneous broadening arising
from differences in the radii of the nanorods that form the ensemble.
The distribution of nanorod radii is modelled using a Gaussian fit and the final
absorbance is calculated by averaging over the calculated absorbance spectrum for
different radii. Varying the CdSe nanorod length (70 ± 10 nm) had little effect on
the calculated absorbance spectrum. Assuming a normal distribution for the radii,
a fit to the experimental data yielded a radius of 3.25 ± 0.7 nm which is within the
experimentally measured radius range of 3.5 ± 0.5 nm based on image analysis of
the nanorods [14]. In order to obtain a reasonable computation time, the absorbance
was calculated for different radii in steps of 0.05 nm. This discretization resulted
in the small fluctuations seen in the Gaussian fit and are expected to disappear for
CHAPTER 3. TERAHERTZ DRIVEN INTRABAND TRANSITIONS: LINEAR RESPONSE57
sufficiently small radii steps.
Figure 3.14: Experimental and theoretical absorbance as a function of linear frequency
for colloidal CdSe nanorods roughly 70 nm in length and 3.5 nm in radius. The
absorbance was measured using a pump-probe teachnique with a delay time of 1
ps between the pump and probe. The two theoretical fits are the Lorentzian and
Gaussian fits at 0 K. The Gaussian fit is done by averaging Lorentzian fits of different
nanorod radii in increments of 0.05 nm. [14]
It is observed that the experimental results, show in figure 3.14 fit the theoretical
calculations in figure 3.4 at low temperatures. Thus, we chose a temperature of T = 0
K for the Lorentzian and Gaussian fit in figure 3.14. The theoretical peak at 2 THz
is not seen experimentally despite the fact that the nanorods were at room temperature during the experiment. One has to be careful when determining an appropriate
temperature since we are discussing the temperature of the excitons and not the
nanorods. The excitons, once excited optically and allowed to relax down to reach
a quasi equilibrium state, may or may not be at the temperature of the electronic
lattice/nanorods. One would expect, however, that the excitons have a higher temperature than the nanorods which would imply an even more prevalent 2 THz peak
CHAPTER 3. TERAHERTZ DRIVEN INTRABAND TRANSITIONS: LINEAR RESPONSE58
than that shown in figure 3.4. Since the 2 THz peak is from higher order longitudinal
transitions, such as 2pz −3dz , this suggests that only the 1s state is being significantly
populated by the optical pulse. Figure 3.15 shows a schematic representation of the
excitonic state energies based on our interpretation of the suppression of the 2pz −3dz
and higher order longitudinal transitions. The excitons are first generated via optical pump (~ωoptical ) and allowed relax before being probed via THz pulse (~ωT Hz ).
We postulate that the holes for the higher energy excitons are being trapped into
ligand states and therefore have negligible contribution to intraband transitions. It
was also shown that changing the ligand from DDPA to pyridine reduced the absorbance, further affirming the importance of the ligand [14]. Figure 3.16, provided
by David Cooke, shows experimentally found absorbance spectra for varying probe
delays. The 8.5 THz peak (1s − 2pz transition), is found to decrease for increasing
delay times. This change in absorbance suggests that the excitonic population distribution is significantly changing in the first 10 ps after excitons are generated. Further
experimental is needed to understand both the decrease in the 1s − 2pz absorbance
peak with increasing pump delay and how the excitonic population behaves in the
early times.
One of the critical assumptions that we have made is that there is less than
one exciton per nanorod. Eq. (3.31) allows us to estimate the number of excitons
per nanorod via comparison to experimental results. Using a scratch test, David
Cooke found that there were approximately 30 nanorod layers. Using the nanorod
dimensions of 70 nm for the length and 3.25 nm for the radius and a ligand length
of 2 nm, a nanorod density of approximately 1.92x10−4 nm3 is found. The measured
absorbance in figure 3.14 has a peak of 7.3x10−5 nm−3 at 8.5 THz (provided by David
CHAPTER 3. TERAHERTZ DRIVEN INTRABAND TRANSITIONS: LINEAR RESPONSE59
Figure 3.15: Excitonic States Energy Diagram depicting how the optical pulse first
creates the exciton bast the band gap. The ligand states are hypothesized to be in
between the 1s and 2pz state. Higher energy excited stated then relax down to the
ligand states rather than the 1s state and get trapped
Cooke and not shown in the figure). These values, in conjunction with our expression
for the absorbance in Eq. (3.31), and using a Γ = 2.75 THz to match the full width
half maximum of the experimental results, we obtain an estimate of 0.45 excitons per
nanorod. This estimated exciton density per nanorod is in agreement with our initial
assumption of not more than one exciton per nanorod.
CHAPTER 3. TERAHERTZ DRIVEN INTRABAND TRANSITIONS: LINEAR RESPONSE60
Figure 3.16: Absorbance as a function of linear frequency for various of pump delay
time for a colloidal CdSe nanorods of length 70 nm and radius 3.5 nm (courtesty of
David Cooke [14])
Chapter 4
Terahertz driven nonlinear response
All discussions thus far have concerned the linear response that occurs under low
intensity Terahertz radiation. The next step is to model the nonlinear carrier dynamics that are expected for higher field intensities. Heisenberg’s equation of motion
is used to model the populations and correlations of the excitons assuming less than
one exciton per nanorod. The transmittance and absorbance are first calculated with
all nanorods uniformly aligned. The calculations are then repeated for a randomly
oriented nanorod system. In this chapter, for simplicity, we assume that all nanorods
are made of CdSe with a radius of 3.5 nm, length of 70 nm and with hard boundary
conditions. We also only consider light-hole excitons.
4.1
Nanorods suspended in air
We first describe a hypothetical case where the nanorods are suspended in air. Since
there are currently no experimental studies done on the THz driven nonlinear response
of CdSe nanorods, we restrict our study on the simple case of nanorods in air to focus
61
CHAPTER 4. TERAHERTZ DRIVEN NONLINEAR RESPONSE
62
on the nonlinear response. This subsection discusses how the Terahertz radiation
relates to the current due to the excitons in the nanorod. Our nanorods are suspended
in a medium all lying along the z direction with the field incident on the plane of
nanorods. Due to the nanorods being much much smaller than the wavelength of the
Terahertz waves we can treat the nanorod layer as an effective medium (as we did in
the previous chapter). For simplicity the field is modeled as a plane wave. We model
the response of the excitons in the collection of nanorods to the THz field as a sheet
current.
4.1.1
Calculating the transmitted and reflected fields
This section details the calculating the transmitted and reflected fields based on an
incident field. Maxwell’s Equations gives the boundary conditions as
n̂ × (H2 − H1 ) = K
(4.1)
n̂ × (E2 − E1 ) = 0
n̂ · (D2 − D1 ) = σ
n̂ · (B2 − B1 ) = 0
where the subscripts 1 and 2 correspond to the different regions divided by a thin
nanorod sheet as shown in figure 4.1, σ is the surface charge density, n̂ is the unit
vector with direction normal to the surface from region 1 to region 2, K is the surface
current, E is the electric field, D is the displacement field, H is the applied magnetic
field, B is the magnetic field. Note that n̂ is also the direction of the incident THz
field.
For simplicity we assume that both regions 1 and 2 are the same and have an
63
CHAPTER 4. TERAHERTZ DRIVEN NONLINEAR RESPONSE
Region 1
Region 2
EIncident
ETransmitted
BIncident
BTransmitted
K
BReflected
EReflected
Nanorod sheet
Figure 4.1: Boundary Condition Diagram for Terahertz radiation incident on a sheet
of nanorods. The nanorod sheet is thin enough so that the current can be described
a surface current.
incident of refraction n and we choose n such that it matches the n of the nanorod
sheet avoiding possible impedance issues. Since we assume that the incident field is
normal to the plane (E and B lie along the surface plane) and that that the nanorods
are undoped we can ignore the last two boundary conditions. The first two equations
require that the parallel (to the surface) component of the electric field is continuous
and that the parallel component of H has a discontinuity determined by the surface
current. Assuming no perpendicular component of the field and that the the incident
and reflected field are in region 1 as described by Eq. (4.1) we can write them as
E1 = EI + ER
(4.2)
E2 = ET ,
(4.3)
and
which gives the constraint
CHAPTER 4. TERAHERTZ DRIVEN NONLINEAR RESPONSE
EI + ER = ET .
64
(4.4)
We can relate the incident electric field to the magnetic field by
n̂ × EI = cBI ,
where c ≡
√1
µ
(4.5)
is the speed of light in region 1 and 2. The magnetic field of the
reflected field is given by
−n̂ × ER = cBR ,
(4.6)
where the negative sign denotes that the reflected field travels in the opposite direction. Adding the reflected field to the equation gives
n̂ × (EI − ER ) = c (BI + BR ) .
(4.7)
Now rewriting Eq. (4.1)
1
B
µ
(4.8)
1
(BT − BI − BR ) = K
µ
(4.9)
H=
yields
n̂ ×
and applying −n̂× to both side gives
−n̂ × n̂ ×
1
(BT − BI − BR ) = −n̂ × K
µ
1
(BT − BI − BR ) = −n̂ × K.
µ
(4.10)
(4.11)
CHAPTER 4. TERAHERTZ DRIVEN NONLINEAR RESPONSE
65
Inserting Eq. (4.11) into Eq. (4.7) gives
n̂ × (EI − ER ) = c (BT − n̂ × µ0 K)
(4.12)
and again relating the magnetic and electric fields using
n̂ × ET = cB̂T
(4.13)
n̂ × (EI − ER ) = n̂ × (ET − µK) .
(4.14)
we obtain
Substituting Eq. (4.4) into Eq. (4.14) yields
1
ER = −
2
r
1
ET = Ei −
2
µ
K
r
µ
K
(4.15)
(4.16)
which gives the transmitted fields and reflected field for a given incident field and
surface current.
4.2
Heisenberg equation of motion: modeling exciton dynamics
Now that we have described how the Terahertz radiation behaves in the system we can
describe how it interacts with the excitons. We begin with the Heisenberg equation
of motion for a general operator  given by
i
∂ Â h
−i~
= Ĥ, Â .
∂t
(4.17)
CHAPTER 4. TERAHERTZ DRIVEN NONLINEAR RESPONSE
66
In our case the Hamiltonian Ĥ is given by
Ĥ =
X
Eµ B̂µ† B̂µ + ET Hz ·
µ
X
Gµ,ν B̂µ† B̂ν ,
(4.18)
µ,ν
taken from Ref. [39] where B̂µ† is the bosonic creation operator for an exciton in state
µ and Eµ contains the kinetic and Coulomb energy of the excitonic state. B̂µ† can be
taken to obey the Boson commutation relations when the exciton density is sufficient
low exciton density [40]. In our case, we assume no more than one exciton per nanorod
and so this assumption is exact. Gµ,ν is the intraband polarization transition matrix
element between two excitonic states given by
E
D Gµ,ν = e ψµ R̂e − R̂h ψν ,
(4.19)
where e is the elementary charge. The intraband polarization is given by Ref. [39]
P=
X
Gµ,ν
D
E
† ψµ B̂µ B̂ν ψν .
(4.20)
µ,ν
If we assume that the nanorods make a thin sheet then we can describe the current
from these nanorods as a surface current. We can relate the polarization to the surface
current K via
d
P.
dt
X
d D † E
= n2d
Gµ,ν
B̂µ B̂v
dt
µ,ν
K = n2d
(4.21)
where n2d is the exciton per unit area as seen by the incident field. n2d is be given by
n2d =
ρex
,
dex
(4.22)
where ρex is the exciton density per unit volume and dex is the nanorod sheet thickness.
Thus, calculating the time evolution of the intraband correlations allows us to find
CHAPTER 4. TERAHERTZ DRIVEN NONLINEAR RESPONSE
67
the intraband current density which can then be used to determine the current and
related it back to the THz absorbance, reflection and transmission from the thin
sheets of nanorods.
Using the Heisenberg equation of motion, we obtain
Eν − Eµ D † E
B̂µ B̂v
i~
D
E
D
E
X
†
†
+ ET Hz ·
Gµ,γ B̂γ B̂v − Gν,γ B̂µ B̂γ .
d D † E
B̂µ B̂v
= −
dt
(4.23)
γ
D
E
The diagonal element B̂µ† B̂µ describes the population of the µ excitonic state while
E
D
the off diagonal element B̂µ† B̂ν describe the correlations between the µ and ν excitonic states. To take into account dephasing we introduce phenomenological time
constants Γµ,ν into Eq. (4.23) giving
D
E
1
Eν − Eµ
+
B̂µ† B̂v
i~
Γµ,ν
D
E
D
E
X
†
†
Gµ,γ B̂γ B̂v − Gν,γ B̂µ B̂γ .
+ ET Hz ·
d D † E
B̂µ B̂v
= −
dt
(4.24)
γ
Γµ,ν describes the population decay when µ = ν and describes dephasing between
states when µ 6= ν. The Terahertz field ET Hz , refers to the transmitted field and not
the incident field since the field the nanorods see will be either the incident field plus
the reflected field or the transmitted field. The subsequent sections describe how to
find the transmitted field based on the applied incident field.
68
CHAPTER 4. TERAHERTZ DRIVEN NONLINEAR RESPONSE
4.3
Calculating the transmitted THz electric field
To find the transmitted field, ET , for the dynamical equation we first need to find
the surface current density previously described as
K=
X
Gµ,ν
µ,ν
d D † E
B̂µ B̂v n2d .
dt
(4.25)
The above Eq. (4.25) expresses the surface current as the rate of change of the exciton
dipole moment. We use the substitution
D
E
ρµ,ν = B̂µ† B̂v
(4.26)
for brevity. Substituting Eq. (4.24) and Eq. (4.25) into (4.16) gives
ET
1
=
2
(
2EI − n2d Z0
X
−
Gµ,ν
µ,ν
1
ωv,u
+
i
Tµ,ν
ρµ,ν + ET ·
X
Gµ,γ ργ,ν − G?ν,γ ρµ,ν
!)
γ
(4.27)
where
r
Z0 =
µ0
.
0
(4.28)
Using the substitutions
ξµ,ν = −
1
ωv,u
+
i
Tµ,ν
(4.29)
and
Cµ,ν =
X
Gµ,γ ργ,ν − G?ν,γ ρµ,ν
(4.30)
γ
simplifies the equation to
(
)
X
1
ET =
2EI − n2d Z0
Gµ,ν (ξµ,ν ρµ,ν + ET · Cµ,ν ) .
2
µ,ν
(4.31)
69
CHAPTER 4. TERAHERTZ DRIVEN NONLINEAR RESPONSE
The issue with this expression for ET is that it is dependent on ET . If we assume that
polarization of the Terahertz field is along either the z (longitudinal) or x (transverse)
direction with all rods lying on the x-z plane then ETy = 0 and we obtain
ETz
ETx
1
=
2
(
1
=
2
(
)
2EIz
− n2d Z0
X
Gzµ,ν
ξµ,ν ρµ,ν +
x
ETx Cµ,ν
+
z
ETz Cµ,ν
(4.32)
µ,ν
2EIx − n2d Z0
X
x
z
Gxµ,ν ξµ,ν ρµ,ν + ETx Cµ,ν
+ ETz Cµ,ν
)
.
(4.33)
µ,ν
Rewriting the above equations and taking into account the reduction in the transverse
electric field described in Chapter 3 we obtain
=
=
P
2
z
x
z
x
z
z
+
ζZ
(4.34)
E
C
C
G
G
0
µ,ν µ,ν
µ,ν µ,ν ET
T
µ,ν
µ,ν
1+r /0
P
Eiz − ζZ0 µ,ν Gzµ,ν ξµ,ν ρµ,ν
2 P
P
2
z
z
x
x
x
x
µ,ν Gµ,ν Cµ,ν ET + 1 + ζZ0 1+r /0
µ,ν Gµ,ν Cµ,ν ET(4.35)
P
Eix − ζZ0 1+2r /0 µ,ν Gxµ,ν ξµ,ν ρµ,ν
1 + ζZ0
ζZ0 1+2r /0
P
with the substitution
1
ζ = n2d
2
(4.36)
gives a 2x2 set of equations that can be solved for the transmitted electric field. The
transmitted electric field ET can now be calculated at any given time if the incident
field and initial values for ρµ,ν are known. We can now model the carrier dynamics
of the nanorods given the applied THz radiation and the initial state of the system.
Figure 4.2 shows the transmitted field as compared with the incident field over
time for a weak THz field. Before considering the previously described system of a
sheet of randomly oriented nanorods we first consider the case of all nanorods rod axis
CHAPTER 4. TERAHERTZ DRIVEN NONLINEAR RESPONSE
70
oriented along the z direction. After we calculate the response for a single nanorod
we can then calculate the response for multiple nanorod orientations and average over
them. We have an incident THz field polarized in the z direction given by
1
1
EI = √ exp −
2
σ 2π
t − tmean
σ
2 !
sin (ω1s−2p (t − tmean )) ẑ,
(4.37)
which is a sin wave pulse with a Gaussian envelope. Note that the field polarization
is along the longitudinal direction and that transverse excitation are forbidden. The
field is chosen to be resonant with the 1s − 2p transition. Figure 4.3 shows the
incident and transmitted field amplitudes in frequency domain. The noticeable dip in
the transmitted field around 8 THz corresponding to the 1s − 2pz transition. A basis
size of 60 excitonic states was used for the simulations and found to have minimal
improvement (<1%) for higher basis sizes. A dephasing time of 1 ps was used to be
consistent with the Lorentzian line width of 1 THz, for m 6= n. The population decay
times were also chosen to be 1 ps but with Γ00 = ∞ so that the 1s state population
did not decay over time. It can be argued that the dephasing and population decay
times are not the same but due to a lack of experimental data this is a reasonable
assumption to make. There is a ringing effect caused by the excitons absorbing some
of the energy from the incident field and then re-radiating it over time. The net effect
is that the transmitted field is lower in amplitude than the incident field but exhibits
a longer tail on the order of the dephasing time (in this case we choose a dephasing
time of 1 ps that will be justified in later sections).
CHAPTER 4. TERAHERTZ DRIVEN NONLINEAR RESPONSE
71
Figure 4.2: Incident and transmitted field amplitude vs time for a weak field of 0.07
kV/cm peak
Figure 4.3: Incident and transmitted field amplitude in frequency space for a weak
field of 0.07 kV/cm peak
CHAPTER 4. TERAHERTZ DRIVEN NONLINEAR RESPONSE
4.4
4.4.1
72
Terahertz driven nonlinear response
Calculations of populations, transmittance and absorbance
This section discusses how exciton populations behave as a function of time. We also
calculate the transmittance and absorbance for different THz peak intensities. All the
calculations done for this subsection are with a THz pulse polarized in the longitudinal
(rod axis) direction and assumes all nanorods have the same orientation. Figure 4.4
shows the population of the 2pz and 3dz excitonic states as a function of time for low
field strengths, where the 1s state starts with a population of 1 corresponding to 1
exciton per nanorod. The incident field is a near single cycle pulse with a peak of
7 kV/cm as described in the previous section and the medium chosen is air making
n = 1. We can see the 2pz population rises and then decays over time on the order
of the dephasing time of 1 picosecond.
Figure 4.4: Exciton population in the 2pz and 3dz as a function of time for a peak
field strength of 7 kV/cm
Figure 4.5 shows a similar graph but now with a field of peak intensity of 70
CHAPTER 4. TERAHERTZ DRIVEN NONLINEAR RESPONSE
73
kV/cm. The large population of the 2pz state now allows for the higher order 2pz −3dz
transitions. The response is clearly complicated with the population sharply rising
and falling as compared to the peak field of 7 kV/cm indicating a nonlinear response.
Figure 4.5: Exciton population in the 2pz and 3dz as a function of time for a peak
field strength of 70 kV/cm
Figure 4.6 shows the transmittance for varying Terahertz field intensities where
the transmittance in the longitudinal direction (T z (ω)) is given by
ITz (ω)
T (ω) = z
II (ω)
z
(4.38)
where ITz (ω) and IIz (ω) are the field intensities of the transmitted and incident field
respectively. The subscript z denotes the longitudinal direction. There are clear dips
in the transmittance especially at 8.5 THz corresponding to the 1s − 2pz transition.
The smaller peaks in the 9-12 THz region corresponds to other higher energy longitudinal transitions. At higher fields a dip is observed in the 1 THz region that
corresponds to the 2pz − 3dz like transitions that can only occur once higher excited
states such as 2pz is significantly populated. At a peak field strength of 70 kV/cm,
CHAPTER 4. TERAHERTZ DRIVEN NONLINEAR RESPONSE
74
shoulders appears on either side of the 1s − 2pz frequency with a dip on the right side
but a peak on the left. This effect similar to off peak resonances related to the Rabi
frequency. The effect is complicated but has been verified to occur even in a strict
two states system. Calculations of absorbance in pump probe studies for a two level
system show resonances at ω, ω + Ω0 and ω − Ω0 , where Ω0 is the generalized Rabi
frequency as given in Ref [38]. These resonances exhibit peaks and dips similar in
feature to those in the 70 kV/cm peak field transmittance. Note that since dressed
states occur under continuous wave radiation, we cannot strictly call these transitions to dressed states. Though the calculations in such a pump probe experiment is
different from our system, it gives some insight on the key features of the 70 kV/cm
transmittance graph. Experimental measurements at high field strengths observing
both these shoulders and the response near 1 THz can then be used as confirmation
for the model. Another possibility is that these shoulder features are an artifact of
the phase of the incident field and may disappear if averaged over different phases
[41].
Figure 4.7 show the Absorbance for a variety of intensities and showing similar
picture as the transmittance, with strong absorbance in the 1 THz regime that occur
from multiple 2pz − 3dz like higher order transitions. Absorbance due field polarized
in the longitudinal direction, Az (ω), in this case is defined as
Az (ω) =
IIz (ω) − IRz (ω) − ITz (ω)
,
II (ω)
(4.39)
where IIz (ω), IRz (ω) and ITz (ω) are the intensities of incident, reflected and transmitted field respectively with superscript z denoting the longitudinal direction. Overall
CHAPTER 4. TERAHERTZ DRIVEN NONLINEAR RESPONSE
75
Figure 4.6: Transmittance for field intensities from 0.7 kV/cm to 70 kV/cm with THz
field polarized along the longitudinal direction. Dephasing time was 1 ps, population
decay time was also 1 ps but with the 1s state not decaying over time. A linear
response is seen up to 0.7 kV cm peak field strength and with strong nonlinearity at
70 kV/cm
CHAPTER 4. TERAHERTZ DRIVEN NONLINEAR RESPONSE
76
the nonlinearity is observed when fields are so high that the the majority of the excitons are in the excited states. The splitting that occurs near the ω1s−2pz transition
is, as mentioned previously, similar to transitions to dressed states.
4.4.2
Transmittance and absorbance as a function of nanorod
orientation to THz polarization
In the previous sections only polarization along the longitudinal direction was discussed. In this section we discuss the populations and transmittance for various
orientations of nanorods with respect to the THz electric field. In a more realistic
scenario the nanorods will be randomly oriented and will yield an average response.
It is thus important to model the response for each different nanorod orientation and
average over them, which is what we will finally do. Figure 4.8 shows how the incident
THz field can oriented at an angle θ with respect to a nanorod. The incident field is
the same as in earlier sections where there is a Gaussian pulse whose intensity peaks
at ω1s−2pz in frequency space but with the THz polarization angle of 20 degrees. The
longitudinal and transverse parts of the field are described by
ETx = ET cos (θ)
(4.40)
ETz = ET sin (θ) ,
(4.41)
where θ is the field polarization angle that the field makes with the x-axis where z is
along the rod axis and ET is the THz electric field amplitude.
Figure 4.9 shows the populations of the 2pz and 2px state over time for a peak
field strength of 0.07 kV/cm for θ = 20◦ . A sharp rise is seen in both populations as
the THz field peaks at around 1.5 ps (when the incident field peaks) proceeded by
CHAPTER 4. TERAHERTZ DRIVEN NONLINEAR RESPONSE
77
Figure 4.7: Absorbance for field intensities from 0.7 kV/cm to 70 kV/cm with THz
field polarized along the longitudinal direction. Dephasing time was 1 ps, population
decay time was also 1 ps but with the 1s state not decaying over time. A linear
response is seen up to 0.7 kV cm peak field strength and with strong nonlinearity at
70 kV/cm
CHAPTER 4. TERAHERTZ DRIVEN NONLINEAR RESPONSE
78
z
x
ET
θ
ETz
ETx
Figure 4.8: THz field orientation with respect to the nanorod. θ is the THz field
polarization angle
a steady decay characterized by a dephasing time of 1 ps. In contrast Figure 4.10
shows a similar graph but with a polarization of θ = 45◦ , which yields smaller 2px
population since less of the field is polarized along the transverse direction.
Figure 4.11 shows the transmittance of a THz electric field with polarization angle
of θ = 20◦ . The transmittance due to the field polarization in both x and z direction
are shown. We see the dips corresponding to the 1s − 2pz transitions due to the field
component in the longitudinal direction and a similar 1s − 2px transition due field
component in the transverse direction. We see that while the 1s − 2pz and 1s − 2px
transitions are dominant there are other smaller but still significant longitudinal transitions. It can then be come difficult, in the context of an experiment, to confidently
identify the 1s − 2px transition unless the nanorods are oriented perpendicular to the
electric field.
CHAPTER 4. TERAHERTZ DRIVEN NONLINEAR RESPONSE
79
Figure 4.9: Populations of 2pz and 2px states over time for a THz electric field
polarization angle of θ = 20◦
Figure 4.10: Populations of 2pz and 2px states over time for a THz electric field
polarization angle of θ = 45◦
80
CHAPTER 4. TERAHERTZ DRIVEN NONLINEAR RESPONSE
Figure 4.11: Transmittance for a THz electric field polarization angle of θ = 20◦ . The
transmittance due to the field components in the z and x direction are shown
4.4.3
Nonlinearity as a function of Terahertz polarization
We now examine the nonlinearity of the field as function of the THz polarization.
Γµν values are the same as before with populations described by Γµ=ν,µ6=0 = 1 ps,
Γµ=0,ν=0 = ∞ and dephasing times described by Γµ6=ν = 1 ps. If a nanorod is subjected
to an incident field of Eix (ω) in the transverse direction it yields a transmitted field of
ETx (ω). The same is true for fields polarized in the longitudinal direction with fields
Eiz (ω) and ETz (ω). In the linear regime we expect a field of AEix (ω) + BEiz (ω) to
give a response of AETx (ω) + BETz (ω). In the nonlinear regime however, this is not
the case. To quantify the deviation from the simple linear result we introduce the
quantity
´∞
4θ,x
x
E (ω) − E x
dθ
(ω)
cos
(θ)
T,pure
´ ∞T
.
= −∞
dθ ETx (ω) cos (θ)
−∞
(4.42)
The difference is calculated between the transmitted field, ETx (ω), and scaled down
x
x
field ET,pure
(ω) cos (θ). ET,pure
(ω) is the resulting transmitted field that would arise
CHAPTER 4. TERAHERTZ DRIVEN NONLINEAR RESPONSE
81
had the incident field been only been along the x direction. This difference between
what we expect to obtain in the linear case and the calculated field is then summed
over all frequencies. A similar definition exists for 4θ,z given by
z
´∞
E (ω) − E z
dθ
(ω)
sin
(θ)
T,pure
´ ∞T
.
4θ,z = −∞
z
dθ
E
(ω)
sin
(θ)
T
−∞
(4.43)
Figure 4.12 shows 4θ,x and 4θ,z as a function of the THz field polarization angle
θ for varying field intensities. For low intensity fields there is little difference between
the nonlinear calculations and the simple linear result. This nonlinearity increases by
7 kV/cm and has 4θ tapering out after fields strength of 60 kV/cm. The transverse
and longitudinal errors are zero at θ equal to 0 and
π
2
respectively. What this shows
is that at higher THz field intensities one has to calculate the nonlinear response for
all nanorod orientations to obtain an accurate average response for a colloidal set of
nanorods with this effect becoming significant (>1%) for peak intensities around 7
kV/cm. This is what we will do in the next section.
4.4.4
Field averaging and the lab reference frame
The results presented thus far have all been for a particular nanorod orientation
relative to the field. In experiments, however, we want to see the transmittance
and absorbance for a uniform distribution of nanorod orientations. As discussed
in the previous section, it is important to calculate the response for all nanorod
orientations and then average over them. Figure 4.13 shows the transmittance of
colloidal nanorods for varying field intensities, where we assume equal distribution
in all direction for the nanorods and they all lie along the plane of incidence of the
incident Terahertz field. The peaks seen at low field strength (0.07 kV/cm) are from
longitudinal and transverse excitations at 8.5 THz (1s − 2pz transition) and 11 THz
82
Δθ,z
Δθ,x
CHAPTER 4. TERAHERTZ DRIVEN NONLINEAR RESPONSE
Figure 4.12: 4θ,x and 4θ,z as a function of theta for different THz field intensities. 4θ,x and 4θ,z reach zero for θ values of 0 and 90 degrees respectively. Strong
nonlinearity begins at approximately 7 kV/cm
CHAPTER 4. TERAHERTZ DRIVEN NONLINEAR RESPONSE
83
(1s − 2px transition) respectively. We again see that for high field strengths a strong
transmittance around 1 THz corresponding to 2pz − 3dz like transitions appears.
We also see some amplification near 8 THz which we interpret as energy is being
transferred from some frequencies to others.
Figure 4.13: Transmittance nanorods of uniformly distributed random orientations
with respect to the THz field electric field
CHAPTER 4. TERAHERTZ DRIVEN NONLINEAR RESPONSE
4.5
84
Summary
In this chapter, we have constructed a dynamical set of equations to model the THz
driven nonlinear response of excitons in CdSe nanorods. Calculations were done for
nanorods in air using light-hole excitons and hard boundary conditions. The transmittance and absorbance were calculated for different THz field polarization angles
and related to transitions described in Chapter 3. The nonlinearity was measured as
a function of field polarization angle and was found to become significant at field intensities higher than 7 kV/cm. Finally the transmittance for a uniform distribution of
nanorod orientations was calculated for possible comparison with future experiments.
Chapter 5
Conclusions
5.1
Summary
In this thesis, colloidal CdSe nanorods were modeled using the two band envelope
function approximation. The excitonic basis was defined, constructed and showed
the important states (1s, 2pz ,2px ,3dz ) with regard to THz driven intraband transitions. The dipole matrix elements were then defined between states and used to
calculate absorbance in the linear response (low field amplitude) regime. The key
transitions, 1s − 2pz and 1s − 2px were found to dominate the absorbance graphs with
the 2pz − 3dz like transitions becoming more significant at higher field intensities.
The appearance of the peak due to the 2pz − 3dz transition resulted from a two photo
transition from the 1s state. The absorbance was then found to be good agreement
to experimental data provided by David Cooke from McGill University giving confirmation to the model. Similar results were found for both light and heavy holes
but with a significantly weaker contribution for the heavy holes. As such, the light
holes were used when using the full set of dynamical equation and the heavy holes
85
CHAPTER 5. CONCLUSIONS
86
ignored as an approximation. Heisenberg equation of motion was used to obtain a full
dynamical set of equations to describe the system in the linear and nonlinear regime.
The nonlinearity was defined and measured as a function of polarization angle of the
THz field in relation to the nanorod. Transmittance and absorbance were calculated
with strong nonlinearity starting at approximately 60 kV/cm. To relate to a more
realistic experimental setting, the transmittance was found for a uniform distribution
of rod orientation in relation to applied THz field and averaged over to give a net
transmittance.
5.2
Future Work
Future experiments that delve into high THz field intensities could be used to compare to results in chapter 4 and probe more deeply into the 2pz − 3dz and other higher
order transitions that have yet to be observed. Studies into the relation of different
dephasing times of the populations as well as the effect of different THz probe delays
could give more insight on the fundamental nature that these excitons are behaving in the very early (1 ps) time scales before being probed. Including the ligand
in the model might also be interesting and more accurately model the hypothesized
hole trapping. Higher order corrections to the excitonic states such as dipole-dipole
interactions could be included to improve the accuracy of the model and perhaps
allow higher excitonic densities. Taking into account light and heavy hole mixing
would allow analysis of structures where the length of the nanorod is comparable to
the radius which would invalidate the approximation that only light holes are significant. Applying the calculations/simulation done in chapter 4 but with a different
experimental setup such as reflective geometry would possibly allow for comparison
CHAPTER 5. CONCLUSIONS
87
to more experimental work. Future work can also involve looking into THz radiation
with coupled nanorods/quantum dots that could lead to interesting resonances and
dynamics.
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Appendix A
Numerical Methods
The majority of the code for the project was written in C++ while a combination
of Perl, awk and bash were employed to manage the file system, data analysis and
systematic study of CdSe nanorod parameter space (such as on the nanorod radii).
The code is segmented into separate pieces for calculating the excitonic basis, linear
response and modeling the dynamical equations described in Chapter 2, 3 and 4
respectively. The underlying concept was to write the code such that each piece
was modular. As an example, code that diagonalized that Hamiltonian should not be
rewritten if the code for calculating the excitonic basis changed. Figures and diagrams
were constructed using an combination of Libre Draw and Gnuplot [43,44].
The excitonic basis used root solving and adaptive Monte-Carlo integration routines of the GNU Scientific library [42]. Containers for the data made use of the Boost
library containers [45]. The calculations for the Coulomb matrix elements in Chapter
3 were substantial and so the code was multi-threaded using a library written by the
author based on the Boost thread library [45]. For a CdSe nanorod 70 nm in length
and 3.5 nm in radius, a basis size of the lowest 600 non-interacting pair basis states
95
APPENDIX A. NUMERICAL METHODS
96
yielded a Coulomb matrix whose elements had error less than 1%. Computation time
using 16 cores each with a processor speed of 2 GHz required 6 hours.
Calculation the nonlinear response required an excitonic basis size of 60 at a time
step of 0.001 ps for 18 ps to reach convergence. The partial differential equations was
solved using the odeint library [46]. Frequency space was calculated based on Fourier
transform functions written with algorithms of the GNU scientific library [42]. A
computation run for 18 ps with the dynamical equations takes 2 hours.