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Optical studies in photonics: terahertz detection and propagation in slot waveguide
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Optical studies in photonics: terahertz detection and propagation in slot waveguide
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Content
OPTICAL STUDIES IN PHOTONICS: TERAHERTZ DETECTION
AND PROPAGATION IN SLOT WAVEGUIDE
by
Yinying Xiao Li
____________________
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(ELECTRICAL ENGINEERING)
December 2014
Copyright 2014 Yinying Xiao Li
ii
Dedication
to my parents
iii
Acknowledgments
I would like to thank my PhD supervisor, Dr. John D. O’Brien, for the
education, instruction, and inspiration I received from him throughout my PhD
studies. I could not imagine finishing the dissertation without his guidance,
knowledge, perceptiveness, and patience. It was a great honor for me to have him
as my PhD supervisor.
I would like to acknowledge my dissertation and qualifying exam committee
members for their patience and assistances, Dr. William H. Steier, Dr. Daniel
Dapkus, Dr. Jack Feinberg, and Dr. Wei Wu. It was my privilege and honor to be
mentored by such a group of first-class educators and scientists during my PhD
studies at USC. Especially, I would like to thank Dr. Steier for his continued
support and encouragement during my PhD at USC.
I feel indebted to a large group of people for their help and support during
my graduate study at USC. I would like to thank Dr. Alan Willner for recruiting
me to his research group where I learned a great deal of computational and
experimental skills in optical communications and published my first journal
paper. I would also like to thank my fellow colleagues Dr. Raymond Sarkissian,
Ashkan Seyedi, Aaron Friesz, Dr. Hari Mahalingam, Dr. Lin Zhang, Dr. Yang
Yue, and Dr. Jian Wang for their help. It was a great experience studying and
iv
living in USC because of a group of fantastic individuals whom I had the honor to
share friendship with.
I am also indebted to Dr. Haisheng Rong at Intel for offering me the
internship opportunity to work at Intel’s world-class R&D laboratories where I
became an expert in designing, fabricating, and testing silicon-based passive and
active photonic devices.
I would also like to mention my undergraduate education received at the
College of Optical Sciences at the University of Arizona. For the entire four years
I spent at the University of Arizona, I was supported with a full scholarship
without which I would not have come from Mexico to study in the United States.
Finally, it is my great honor to thank Viterbi School of Engineering at USC
for offering me an Annenberg fellowship for four years. I would also like to thank
my advisor Dr. O'Brien and Dr. Dapkus for supporting me in the rest of my PhD
with research and teaching assistantships.
v
Table of Contents
Dedication ii
Acknowledgments iii
List of Tables viii
List of Figures viii
Preface xv
PART I: TERAHERTZ DETECTION 1
Chapter 1: Introduction 2
1.1 Abstract for PART I ...................................................................................... 2
1.2 Challenges in Terahertz detection ................................................................. 4
1.3 Integrated photonics in Terahertz .................................................................. 7
1.4 Nanopore superlattice .................................................................................... 8
1.5 Theoretical treatment ................................................................................... 12
1.6 Proposed model device ................................................................................ 15
Chapter 2: Subbands in nanopore superlattice 18
2.1 Model for nanopore superlattice .............................................................. 18
2.2 Modified Schrödinger equation ............................................................... 20
2.3 The 3D finite difference method ................................................................. 21
2.4 Subband structure and density of states ....................................................... 25
2.5 Intersubband absorption .............................................................................. 28
vi
2.6 Selection rules .............................................................................................. 32
Chapter 3: Electrical transport 35
3.1. Boltzmann equation .................................................................................. 35
3.2. Intrasubband current ................................................................................. 36
3.3. Numerical solution with finite difference method ....................................... 38
3.4. Rigorous solution with a Taylor expansion ................................................. 39
3.5. Nonlinear dependence of the intrasubband current .................................... 42
Chapter 4: Photocurrent 45
4.1. Redistribution of carriers ......................................................................... 45
4.2. Light-induced current ................................................................................. 47
4.3. Self-consistent transitions .......................................................................... 51
Chapter 5: Device design 57
5.1. A model device for terahertz detection .................................................... 57
5.2. Noise analysis ............................................................................................ 59
5.3. Figure of merits ........................................................................................... 64
5.4. Wideband detection ..................................................................................... 67
5.5. Tailoring the nanopore superlattice ............................................................. 69
5.6. Conclusions of PART I ................................................................................. 75
PART II: PROPAGATION IN SLOT WAVEGUIDE 77
Chapter 6: Introduction 78
6.1 Abstract of PART II .................................................................................. 78
6.2 Slot waveguide ............................................................................................. 80
6.3 Dual slot waveguide ..................................................................................... 88
6.4 Engineering of slot waveguides ................................................................... 90
vii
Chapter 7: Dual slot waveguide 96
7.1 Model and simulation ................................................................................ 96
7.2 Index and size dependence ........................................................................ 100
Chapter 8: Engineering a slot waveguide 104
8.1 Properties of dual slot waveguides ......................................................... 104
8.2 Properties of slot waveguides ................................................................. 108
8.3 Conclusions of PART II ............................................................................ 124
References 127
viii
List of Tables
6.1 Dispersion and nonlinearity comparison between the published
and proposed waveguides. .................................................................. 94
List of Figures
1.1 Schematic diagram of the GaAs/InGaAs nanopore superlattice. ......... 10
1.2 Calculated electronic distribution in a GaAs/InGaAs nanopore
superlattice for subband n=2, and in-plane wavenumber k=0.
The dotted lines are for guide of eyes, indicating the positions
of GaAs cylinders..Picture to illuminate the idea for terahertz
detection... ........................................................................................... 11
1.3 Picture to illuminate the idea for terahertz detection... ....................... 16
2.1 (a) Schematic diagram of the nanopore superlattice. (b) Top
view of the 2D triangular cell lattice, where R is the nanopore
radius, a the lattice constant. (c) The parallelogram and
rectangular unit cells, and the 2 / a circularly shifted clone of
the rectangular unit cell. (d) The first Brillouin zone, the
irreducible Brillouin zone and the highly symmetric points. .............. 19
2.2 (a) Energy band structure as a function of the 2D wave number
//
k , along the boundary of the irreducible Brillouin zone (see
Figure 2.5) between the high symmetry points for filling factor
and cylinder radius from 2267 . 0 = f and nm R 20 = . (b) The
corresponding density of states, normalized to the volume and
the energy interval used in the calculations, and in units of
( ) ( )
1 3 − −
meV cm . .................................................................................... 25
2.3 Electron probability distribution ( )
2
, , z y x ψ normalized over a
unit cell, k at Γ point, (a) for 2267 . 0 = f and 1 = n , (b) for
ix
2267 . 0 = f and 2 = n , (c) for 6 . 0 = f
and 1 = n , (d) for
6 . 0 = f and 2 = n . The dotted circles are a guide for the eye
indicating the InGaAs cylinders. ......................................................... 27
2.4 Optical absorption coefficient for (a) 2267 . 0 = f ,
Γ
k ,
meV E
f
40 = , meV 1 = Γ , s-polarization, at K T 77 = , (b)
2267 . 0 = f ,
Γ
k , meV E
f
40 = , meV 1 = Γ , s-polarization, at
K T 300 = , (c) 2267 . 0 = f ,
Γ
k , meV E
f
60 = , meV 1 = Γ , s-
polarization, at K T 77 = , (d) 2267 . 0 = f ,
Γ
k , meV E
f
60 = ,
meV 1 = Γ , s-polarization, at
K T 300 = , (e)
2267 . 0 = f ,
Γ
k ,
meV E
f
60 = , meV 1 = Γ , p-polarization at K T 77 = , (f)
2267 . 0 = f ,
Γ
k , meV E
f
60 = , meV 1 = Γ , p-polarization at
K T 300 = . .......................................................................................... 32
2.5 Individual contributions to the absorption curve in Figure
2.4(a) for meV 25 < ω , corresponding to transitions from the
ground state
1 = n
E . The two numbers marked for each peak
indicate the two states involved in the specific transition, with
green background for transitions from
1 = n
E and pink
background for other transitions (a) K T 0 = , (b) K T 77 = . .............. 33
4.1 The intrasubband current as a function of bias field for various
light intensities. The photon energy is meV 15 . The operation
temperature is assumed K T 77 = . ..................................................... 47
4.2 The intrasubband current as a function of light intensity for
various bias fields. The photon energy is meV 15 . The
operation temperature is assumed K T 77 = . ..................................... 48
4.3 The intrasubband current as a function of photon energy for
different bias voltages. The operation temperature is assumed
K T 77 = . ............................................................................................ 50
4.4 The intrasubband current as a function of light intensity for
various bias fields. The operation temperature is assumed
K T 300 = . .......................................................................................... 51
4.5 Intrasubband current as a function of bias field for various light
x
intensities. The photon energy is 15meV. The operation
temperature is assumed K T 77 = . The redistribution of
carriers is calculated with the self-consistent solution ....................... 55
4.6 Intrasubband current as a function of light intensity for various
bias fields. The operation temperature is assumed K T 77 = .
The redistribution of carriers is calculated with the self-
consistent solution. ............................................................................. 56
5.1 Schematics showing a model device for terahertz detection. A
bias voltage is applied to the nanopore thin film and the current
is measured. Light to be detected comes in vertically of s-
polarization with photon energy ω and light intensity I. ................. 58
5.2 Spectra of generation-recombination noise, shot noise and
thermal noise at 77K and 300K. ......................................................... 63
5.3 Spectrums of SNR in dB for various light intensities. The other
parameters are assumed the same as those for Figure 4.1. ................. 65
5.4 Spectrums of BER in dB for various light intensities. The other
parameters are assumed the same as those for Figure 4.1. . .............. 66
5.5 Schematic of a probe integrated with 6 gradually rescaled
chips. The bias voltage at each chip can be calibrated
individually so that the spectral response to a scanning
Gaussian light beam of half width 10meV appears like the
simulated curves in the upper right inset. The photocurrent
from each chip is compared to select the highest, so that the
probe spectrum becomes wideband as indicated in the bottom-
right inset. ........................................................................................... 68
5.6 (a) Energy band structure as a function of the 2D wave number
//
k , along the boundary of the irreducible Brillouin zone (see
Figure 2.1) between the high symmetry points for filling factor
and cylinder radius from % 90 2267 . 0 × = f and nm R 19 = . (b)
The corresponding density of states, normalized to the volume
and the energy interval used in the calculations, and in units of
( ) ( )
1 3 − −
meV cm . ................................................................................... 70
xi
5.7 Energy band structure as a function of the 2D wave number
//
k , along the boundary of the irreducible Brillouin zone (see
Figure 2.1) between the high symmetry points for filling factor
and cylinder radius and the corresponding density of states,
normalized to the volume and the energy interval used in the
calculations, and in units of ( ) ( )
1 3 − −
meV cm , for (a)(b)
4 . 0 = f , nm R 6 . 26 = , (c)(d) 6 . 0 = f , nm R 5 . 32 = , and (e)(f)
8 . 0 = f , nm R 6 . 37 = ......................................................................... 72
5.8 Intrasubband current as a function of bias field for various light
intensities. The photon energy is 15meV. The operation
temperature is assumed K T 77 = . .................................................... 74
5.9 Intrasubband current as a function of light intensity for various
bias fields. The operation temperature is assumed K T 77 = . ........... 74
6.1 Schematic showing the difference between a strip waveguide
and a slot waveguide, and how slot waveguide can be
converted from a strip waveguide. The substrate material is
assumed of silica SiO2, and the strip material be of silicon Si. .......... 82
6.2 Schematic showing a cross section of a slot waveguide, where
the refractive indexes are respectively marked as in the slot
region, in the high-index slabs region, and in the cladding
region. The cross section is assumed in x direction and the
origin of the coordinate is located at the center. The width of
the slot is a 2 , and the width of the waveguide including the
two slabs and the slot is assumed b 2 . The refractive indexes in
the three regions are respectively, in the slot region
S
n , in the
strip regions
H
n , and in the cladding region
C
n ................................. 84
6.3 Normalized field profile along a cross section of a slot
waveguide. The width of the slot is a 2 , and the width of the
waveguide including the two stabs and the slot is assumed b 2 .. ........ 86
6.4 Normalized field distribution along a cross section of a slot
waveguide. The height of the slot waveguide is finite (The
vertical direction as indicated in the figure ). ..................................... 87
xii
7.1 Schematics of the slot waveguide (a) and the dual slot
waveguide (b). In the text, the geometric terms, width and
thickness, are respectively referred to the lateral and vertical
lengths of the layers described in these schematics.. .......................... 98
7.2 Intensity distribution of the slot waveguide (a) and the dual slot
waveguide (b); Intensity distribution in surf format of the slot
waveguide (c) and the dual slot waveguide (d). The sizes are
chosen waveguide width 375nm, slot thickness 21nm, and sub-
core width 50nm. ................................................................................ 99
7.3 Cross-sectional curve of the intensity distribution in Fig. 7.2
along the center axis of the slot of the slot waveguide (a) and
the dual slot waveguide (b); Cross-sectional curve of the
intensity distribution in Figure 7.2 across the slot of the slot
waveguide (c) and the dual slot waveguide (d). ............................... 100
7.4 (a) Average peak intensity and peak height normalized to the
peak of the slot waveguide as a function of the slot refractive
index; (b) FWHM of the slot waveguide (dot curve) and
FWHM of the dual slot waveguide (square curve) as a function
of the slot refractive index. The sizes chosen are waveguide
width 375nm, slot thickness 21nm, and sub-core width 50nm.. ....... 102
7.5 (a) Average intensity in the sub-core normalized to average
intensity in the slot as a function of sub-core width for fixed
sub-core thickness 21nm; (b) Power ratio between light in the
sub-core and light in the slot. ............................................................ 103
8.1 (a) Group velocity dispersion; (b) Nonlinear coefficient
dispersion; (c) Birefringence dispersion.. ......................................... 105
8.2 Dispersions of dual slot waveguides with (a) fixed slot
thickness 21nm, (b) fixed slot thickness 50nm, and (c) fixed
sub-core width 60nm......................................................................... 107
8.3 Slot and strip modes strongly interact with each other due to
index-matching at the crossing point, producing a sharp index
change of symmetric and anti-symmetric modes. Modal power
xiii
distributions of the symmetric mode at different wavelengths
(Media 1). .......................................................................................... 109
8.4 (a) Dispersion profiles of the symmetric and anti-symmetric
modes, and a negative dispersion of -181520 ps/nm/km can be
obtained from the symmetric mode. (b) The dispersion profile
red-shifts with a small peak value as the slot thickness
increases ............................................................................................ 110
8.5 Dispersion value and peak wavelength are examined as
functions of the slot thickness (a) and the silicon-layer
thickness (b), respectively. ................................................................ 112
8.6 (a) The dispersion profile is fixed at the same wavelength as
the dispersion’s peak value is changed from -181520 to -28473
ps/nm/km by varying the silica base thickness. (b) A trade-off
is found between the dispersion peak value and dispersion
bandwidth. ......................................................................................... 112
8.7 (a) Dispersion shifts with different slot thicknesses, exhibiting
almost unchanged dispersion value and bandwidth. (b)
Dispersion properties change with the waveguide width.. ............... 114
8.8 Dispersion compensation for very high-speed signals
transmitted over 11.4-km single mode fiber. Eye-opening
penalty increases with bit rate. Eye-diagrams are in the same
scale................................................................................................... 114
8.9 (a) Waveguides with variable waveguide thickness or width are
cascaded. (b). The dispersion shifts over wavelength by
changing waveguide thickness. (c). Flat dispersion of -31300
ps/nm/km over 147nm. 6.3 ns/m tunable delay can be obtained
by 230-nm wavelength conversion. .................................................. 116
8.10 (a) Slot waveguide with silicon layers surrounding a highly
nonlinear slot layer. (b) Dispersion profiles in 10 cm long
chalcogenide slot waveguides with different slot heights. ................ 117
8.11 Dispersion profiles in 10-cm-long chalcogenide slot
waveguides with different (a) waveguide widths and (b) upper
xiv
silicon heights. .................................................................................. 119
8.12 For chalcogenide slot waveguides, nonlinear coefficient γ and
figure of merit are examined over wavelength with different (a)
slot heights and (b) waveguide widths, respectively. ....................... 121
8.13 For 10-cm-long Si nano-crystal slot waveguides, (a) dispersion
profiles change with slot height. (b). Dispersion profile red-
shifts as lower silicon height increases. ............................................ 122
8.14 For 10-cm Si nano-crystal slot waveguides, dispersion
sensitivity changes with Hs. .............................................................. 123
xv
Preface
This PhD dissertation is divided in two parts. In the first part of the dissertation, a
proposal is presented for optical detection in terahertz based on subbands
supported by nanopore superlattices. In the second part of the dissertation, studies
on light propagation in slot waveguides are presented. The two parts are presented
with separate introductions and conclusions. The numbering of the chapters is
however unified, so are the figure list and the reference list. Some contents of the
dissertation have been published. The relevant publications are listed in the
reference list and cited respectively.
1
PART I
TERAHERTZ DETECTION
2
Chapter 1
Introduction
1.1 Abstract for PART I
In PART I of the present PhD dissertation, a design for terahertz detection is
proposed based on externally biased nanopore superlattices. In Chapter 1,
important aspects related to terahertz detection are introduced, including the
current status of available techniques and the challenges for terahertz detection,
the challenges with solid state integrated photonic devices for terahertz detection,
an introduction of nanopore superlattices and their possible usefulness for
terahertz detection, the methods to study the electronic subband structure as well
as the electrical transport in the nanopore superlattices, and the proposed model
device for terahertz detection.
After the introduction in Chapter 1, theoretical foundations, simulation
results, and device designs are presented in detail in Chapters 2 ~ 5. Firstly,
electronic subband structures of GaAs/InGaAs nanopore superlattices in the
triangular cell configuration are analyzed by solving the 3D Schrödinger equation
with the finite difference method. The results demonstrated that subband gaps
could be observed ranging from less than meV 1 up to meV 20 . The
corresponding density of states spectrum has been calculated, which also shows
promise for terahertz detection. Optical absorption due to intersubband transitions
is also studied within the dipole-dipole approximation. Strong and narrow
3
absorption peaks covering the entire terahertz and far-infrared ranges are observed
at liquid nitrogen temperature (77K) as well as at room temperature (300K).
Selection rules governing the intersubband transitions are discovered and
analyzed.
Secondly, external electric field induced intrasubband currents are analyzed
by solving the Boltzmann transport equation rigorously, and it is found that as the
bias voltage increases, after a linear conductivity region, the current increases
with higher orders of the external bias field, i.e. the nonlinear conductivity region.
It is further revealed that the light-induced intersubband absorption brings about
changes in the external-electric-field-induced nonlinear current. Therefore,
detection of terahertz light becomes apparently feasible. In addition, a self-
consistent approach is implemented in treating the light-induced intersubband
absorption while keeping the total carriers conserved, so that the intersubband
absorption process is described more precisely.
Finally, based on the above findings, a model device for terahertz detection
is proposed. A tunable bias voltage is applied to the nanopore structure and the
induced current is measured. The device is found suitable for high speed analog
and digital terahertz detection. The main properties and capabilities of the
proposed detector are analyzed. Noises, such as generation-recombination,
thermal, and shot noises, as well as figures of merit, such as signal to noise rate
and bit error rate, are estimated. A design is also proposed to construct a
wideband detector based on an array of gradually rescaled lattices. The wideband
design is also capable of analyzing the frequency of the incoming light. At the end
4
of Chapter 5, a number of numerical simulations that tailor the nanopore
superlattice with the key parameter, namely the filing factor, are presented, and
the results are discussed with the emphasis on the influences of the parameter
tailoring to the subband structures, the light-induced intersubband transitions, and
the intrasubband current.
1.2 Challenges in terahertz detection
Recently there has been a growing interest in optical terahertz detection. In
various fields in physics, chemistry and biology, the demand to detect light
sources in spectral ranges covering the terahertz and far-infrared frequencies has
been dramatically increased thanks to the more extensive research activities at
sub-cell and molecule levels in these fields. Light scattering and fluorescent
emission from chemical and biological species often fall in this frequency range
as the components of the species are of molecular sizes. For nondestructive
sample testing, a convenient way is to carry out optical spectroscopy. In order to
realize the spectroscopy, optical detection in terahertz becomes therefore required.
Another important application of terahertz detection is to image in terahertz,
including microscopy on chemical and biological samples and photographing in
special situations where the main light sources are in the terahertz range.
However, to realize terahertz detection is rather challenging, because the
wavelength seems too short to make use of common techniques that employ
traditional solid state electronics and antennas to pick up and amplify signals of
electromagnetic radiation in radio and microwave frequencies. On the other hand,
5
optical detection in the terahertz range is also challenging for the techniques based
on semiconductor photonics, because the wavelength now becomes too long for
most of the common semiconductor materials to have compatible band gaps that
are suitable for detecting the interband transitions in these frequency ranges. For
similar reasons, it is also equally difficult to construct light sources in terahertz
with compact semiconductor chips, The above situation has been referred in the
literature as the terahertz gap [SIR02, SIE02].
Despite the challenges, some techniques are presently available for terahertz
detection. These techniques, based on various distinguishable operation
principles, have mostly serious compromises in size, cost, operation temperature,
detection speed, and reliability, as tradeoffs for circumventing the terahertz gap.
In the following, a brief discussion on the current status of the available
techniques for terahertz detection is presented.
Up to the present times, commercially available terahertz detectors such as
bolometers, golay cells and pyroelectric sensors are in general very slow in
detection speed, large in size and expensive. On the other hand, bolometers must
operate at cryogenic temperatures (near absolute zero). Below, detailed
discussions of these classic equipment for terahertz detection are skipped, as the
emphasis in the present work is to find a satisfactory solution with on-chip
semiconductor devices.
Recently, a few solid state terahertz detectors have been proposed, which are
mostly based on field effects and/or quantum cascade systems. The main
challenges of the devices are in noise control, repetition rate, and spectrum
6
coverage. There are a few publications on semiconductor terahertz quantum well
photodetectors. In the following list, some background on the current status of
terahertz detectors with emphasis on the recent progress in solid state terahertz
detectors is presented.
a) Terahertz quantum well photodetectors [LLS07, VIT12]
Presently, most terahertz quantum well photodetectors are
photoconductive detectors, covering peak response frequency from 2 to 7
THz. The operation temperature is still not very high, about 30K or lower.
Scattering-assisted escape is the dominant process for a typical quantum
well photodetector, especially at low fields. Electrons associated with the
confined ground state in the well and distributed on the 2D in-plane
dispersion curve undergo a scattering event to get out of the well and then
become 3D mobile carriers in the barrier.
b) Semiconductor nanowire field effect transistors [CLF92]
The terahertz detection principle in field effect transistors was first
explained by Dyakonov −Shur plasma wave theory: the nonlinear
properties of plasma wave excitations in nanoscale field effect transistor
channels enable their response at frequencies appreciably higher than the
device cutoff frequency.
c) THz GaAs metal-semiconductor-metal photodetectors [PAJ08]
Metal-semiconductor-metal photodetectors with finger spacing and finger
width of 25 nm on molecular beam epitaxy grown GaAs were fabricated.
7
The full width at half maximum impulse response is as short as 0.25 ps
and the 3 dB bandwidth is 0.4 THz.
Having reviewed the current status of terahertz detectors, especially the solid state
terahertz detectors, one realizes that much fundamental work remains to be done
for terahertz detection. The intention of the present work is to work out a feasible
solution for terahertz detection. The efforts concentrate on the photonics side of
the terahertz gap. The purpose is to find a way to construct on-chip photon
detectors for optical terahertz detection.
1.3 Integrated photonics in terahertz
There has been a growing interest in recent years in making use of integrated
semiconductor devices to construct compact light sources and detectors ranging
from far infrared to terahertz [KTB02, WIL07].
In order to span the terahertz gap, among others, a hopeful research
direction for terahertz source and detection is to make use of systems that support
subbands, i.e. the conduction band in a semiconductor is split into subbands due
to quantum confinement to the carriers. The gaps between the subbands fall in the
frequency range of interest. Actually, creating and engineering subbands in the
conduction band of semiconductors have long been considered [BK89, HF91].
More recently, quantum cascade lasers, as a compact terahertz source, have
shown promise, mainly because bound states can be established and tuned at ease,
8
and electrons can be effectively and selectively injected into each barrier [KTB02,
WIL07].
On the other hand, optical detection in the terahertz range based on compact
semiconductor devices has yet to make any significant progress [EC08]. This is
especially due to the difficulty in establishing stable subbands in the terahertz
range and also to the lack of a proper channel for reading out optical intersubband
transitions. A successful device should contain stable subbands in the terahertz
range, and also include a reliable readout channel that directly responds to the
optical intersubband absorptions.
In the present work, a novel scheme for optical terahertz detection based on
an externally biased GaAs/InGaAs nanopore superlattice is proposed. This
detector scheme does not rely on the generation of free carriers upon absorption
of terahertz radiation, but rather on the fact that an excited electron population
within the conduction subbands experiences different transport properties than a
quasi-equilibrium population.
1.4 Nanopore superlattice
In the present work, the possibility of using nanopore superlattices for terahertz
detection is studied. The nanopore superlattices referred here are basically thin
active layers incorporated in a periodic lattice. The structure of a nanopore
superlattice looks similar to that of the widely studied photonic crystals in that
both nanopore superlattice and photonic crystal are composed of thin films with
two-dimensional periodic features, and thus support band structures that can be
9
obtained by solving an eigenvalue problem. However, the physics behind the two
devices are fundamentally different. In the case of photonic crystals, the device
supports the photonic band structures which is a result of solving Maxwell’s
equations under periodic boundary conditions. The eigenvalues are the
electromagnetic modes that are allowed to be in the photonic crystals, whereas the
corresponding eigenfunctions are the electromagnetic distributions. On the other
hand, in the case of nanopore superlattices, the device supports electronic band
structures which is a result of solving Schrödinger equation under periodic
boundary conditions. Here, the eigenvalues are the electronic modes that are
allowed to be in the nanopore superlattice, whereas the corresponding
eigenfunctions are the electronic distributions, or wave functions of the carriers.
The idea to make use of nanopore superlattices for terahertz detection comes
from a recently reported development with nanopore superlattices [VEC09]. In
the report, subband structures have been observed in GaAs/InGaAs active layers.
In other words, a GaAs/InGaAs/GaAs thin-film structure that is patterned by a
lithographically defined and etched periodic triangular lattice with GaAs barriers
supports subbands and can be referred as a nanopore superlattice [VEC09]. This
was experimentally confirmed in [VEC09] that the observed radiation spectra
contained peaks corresponding to small energy intervals (less than a few meV),
which showed that subbands were created in the conduction band and the
observed radiation spectra were due to the intersubband transitions.
Let us now have a close look at the nanopore superlattice. In Figure 1.1, a
schematic of the nanopore superlattice is drawn [VEC09].
10
Figure 1.1: Schematic diagram of the GaAs/InGaAs nanopore superlattice
One can see from the schematic diagram in Figure 1.1 that the nanopore active
layer is of InGaAs and sandwiched by GaAs layers, and the cylinders in the
InGaAs active layer are filled with GaAs too. The GaAs cylinders are arranged in
a periodic triangular lattice.
One should not confuse the nanopore superlattice with aligned quantum
dots. They look similar, but are completely different in terms of confinement to
the carriers. It is important to note as follows. In contrast to the situation of
quantum dot arrays where the electrons are confined locally inside the quantum
dot regions, in the nanopore superlattice, the electrons are mostly confined in the
plane of the InGaAs quantum well. This modifies the overall electron distribution
in the conduction band (and also in the valance band for the holes). A two-
dimensional electron distribution map is presented below in Figure 1.2 for one of
the subbands (n=2) for k=0, where k is the wave vector in the horizontal plane.
The detailed model and theory as well as the calculation details will be given in
Chapter 2.
11
Figure 1.2: Calculated electronic distribution in a GaAs/InGaAs nanopore
superlattice for subband n=2, and in-plane wavenumber k=0. The dotted lines are
for guide of eye, indicating the positions of GaAs cylinders.
One can see from Figure 1.2 that the carriers are confined in the area between the
GaAs barriers. This is completely different from the case of quantum dot arrays.
For this reason, the structure is referred as nanopore superlattice.
The starting point of our work is to theoretically model the nanopore
superlattices so as to find out the possible electronic subband structures. On the
basis of the subband structures, one is able to analyze the intersubband absorption.
It is then interesting to extend the range of the intersubband transitions in the
GaAs/InGaAs nanopore superlattice to cover the entire terahertz and far-infrared
ranges. This corresponds to transition energies ranging from a few to several
dozens of meV. It is highly desirable to tailor the subband structure by tailoring
parameters that are controllable during the fabrication process. Once the subband
structures are engineered, it is then interesting to further explore their potential
applications in light emission and detection in these ranges, especially at room or
at least liquid nitrogen temperatures.
12
Obtaining the electronic subband structure of the nanopore superlattice is
the first task of the work. As far as light detection is concerned, the next challenge
is to pick up the signal from the light-induced intersubband transitions. Unlike the
usual situation in semiconductor probes where the current caused by light-induced
transitions between the valence and conduction bands can be measured to reflect
the strength of the incoming light, in the case of subbands within the conduction
band, since the global carrier density has to be conserved, it is difficult to read out
the changes due to light-induced intersubband transitions. The electric transport
along the nanopore superlattice, which requires accurate solutions of the
Boltzmann equation for the electric distributions, is analyzed extensively. Based
on the simulation results for the electric distributions, the challenge is overcome
by introducing an external bias along the nanopore superlattice. In the following
section, the details of the theoretical treatment will be introduced.
1.5 Theoretical treatment
In order to explore the optical and optoelectronic properties of the GaAs/InGaAs
nanopore superlattice, theoretical simulations have been carried out to map the
detailed subband structures, i.e. the energy dispersion relations in the conduction
band, as well as the spatial electronic distributions [AM76]. On the basis of the
eigenenergy and wave function, further studies are performed on the density of
states and the optical absorption. In calculating the optical absorption, additional
aspects have been considered, such as the dipole-dipole matrix elements and the
selection rules for the possible subband transitions. The above studies aim
13
primarily at determining the necessary parameters for the structure to become
suitable for building sensitive terahertz probes.
The numerical results from the simulations based on the solutions of
Schrödinger equation enable us to demonstrate the following. The conduction
band of the active layer is split into subbands due to the extra-confinement to the
carriers caused by the incorporated periodic nanopore lattice, i.e. the GaAs
cylinders shown in Figure 6.1, and the density of states distribution contains
enhanced peaks due to the subbands. The shapes of the subbands and separations
between the subbands can be tailored to fit the need for terahertz detection.
Once the suitable subband structures are obtained, within the framework of
dipole-dipole approximation, the absorption coefficient due to the light-induced
intersubband transitions can be calculated: by absorbing a photon, a carrier at a
certain subband jumps to another subband. The absorption coefficient takes into
account all the possible transitions, and the selection rules governing the
transitions are obtained: some of the transitions are allowed while others are not
allowed under the dipole-dipole approximation. The spectrum of the calculated
absorption coefficient demonstrated that the active intersubband absorption occurs
over the entire terahertz range, although the spectrum is uneven in the terahertz
range and contains enhanced peaks. The fact that the absorption spectrum
contains enhanced regions would be useful for realizing light detection at specific
frequency, and would also be unwanted for implementing a wideband detection.
This drawback is considered and addressed in the present work at the stage of
device design. A satisfactory solution based on an array of gradually rescaled
14
nanopore superlattices is obtained both for wideband detection and an accurate
spectroscopy.
However, as discussed in the preceding section, before one is able to
actually design a terahertz detector based on the nanopore superlattices, an
obvious hurdle to be overcome lies in that the information about the light induced
intersubband absorption should be extracted reliably and as simply as possible. In
previous designs based on individual or an ensemble of quantum dots and wells,
such as cascaded quantum-well light detection, the information were usually
extracted from induced currents stemming from individual quantum wells. In
other words, the information relates to intersubband transitions that could only be
extracted globally from the entire thin film. It appears therefore necessary to solve
Boltzmann transport equation for the system.
In the literature [ZIM01, KIT96], linear electric conductivity has been
extensively studied for various nanometric systems by solving the Boltzmann
equation within the relaxation time approximation and with the omission of
higher orders of change as a function of energy or momentum. The omission
results in a linear conductivity that is not a function of the external electric field
and the modification of electron energy and momentum. Therefore, the
modification of electron energy due to light-induced intersubband transitions
cannot be extracted from the current measurement. In order to capture the
modification, one goes beyond the approximations to solve the Boltzmann
transport equation rigorously.
15
The Boltzmann equation can be solved rigorously beyond the linear
conductivity regime. The solution can be carried out with accurate numerical
calculations. Preliminary results have demonstrated that when the bias voltage
continues to increase after the linear conduction range, the conductivity becomes
a function of the applied voltage, i.e. the nonlinear conductivity, and the bias
current shows a significant response to the change of electron distribution on the
subbands. This is a critical finding for realizing light detection with subband
systems. The detection roadmap looks then as follows: A direct current (DC) bias
is applied to the nanopore thin film and the bias is pre-adjusted to an optimized
point in the nonlinear region. Light with frequencies corresponding to the
intersubband absorption spectrum is incident on the film, and electron distribution
on the subbands is modified. The modification of the electron distribution is
eventually read out from the measured current.
In addition, there are a number of issues that need to be addressed. For a
precise calculation of the light induced intersubband transitions, a self-consistent
solution would be useful to replace the solution based on the first Born
approximation. At the end of PART I, a systematic tailoring on the nanopore
superlattice is carried out and discussed extensively in order to provide a guide for
device optimization.
1.6 Proposed model device
In the last stage of the work, distinguishable from previous designs in the
literature [LLS07, VIT12], a design of terahertz detection based on the above
16
findings is proposed, and properties and capabilities of the detection are analyzed
and discussed. It is found that the detection is suitable for high speed analog and
digital terahertz detection and it is expected to extend the operation temperature
towards room temperature. As mentioned in previous sections, the proposed
model device makes use of biased nanopore superlattices. A picture to illuminate
the idea is drawn in Figure 1.3 below:
Figure 1.3: Picture to illuminate the idea for terahertz detection.
In Figure 1.3, one sees that a nanopore superlattice is biased by a DC voltage. The
nanopore superlattice is assumed to be GaAs/InGaAs/GaAs as shown in Figure
1.1. Light in terahertz frequency is incident on the nanopore superlattice. The
external bias voltage is applied across the superlattice, and the bias current is
measured as the readout of the device.
The properties and capabilities of the proposed model device are carefully
analyzed. An important property of a detector is the noise level. There are several
sources of noise that may be significantly related to the proposed device, namely
thermal noise, shot noise and generation-recombination noise, etc. An estimation
17
on the significance of the various noises is useful for optimizing the device design
and also for predicting the possible capabilities of the device. Apart from the
noise, estimations of two important figure-of-merits, namely signal noise rate
(SNR) and bit error rate (BER), are provided. To further evaluate the feasibility of
the device, the possible saturation limit and the likely transition time, which are
important to determine the speed and sizes of the detector, are studied extensively.
At the end of PART I of the dissertation, a design is also proposed to
construct a wideband detector based on an array of gradually rescaled lattices.
This extra design is to address the issue that since the absorption spectrum of
nanopore superlattice is uneven and contains enhanced peaks, a single lattice
detector cannot be calibrated if one does not already know the frequency of the
incoming light. In other words, the multi-lattice design aims to realizing a
wideband detection and an accurate frequency analysis, simultaneously or
separately.
18
Chapter 2
Subbands in nanopore superlattice
2.1 Model for nanopore superlattice
A 3D schematic diagram of the nanopore superlattice is shown in Figure 2.1(a)
where the InGaAs thin film including the GaAs cylinders is sandwiched in the z-
direction by GaAs background. In order to model the nanopore superlattice, 2D
schematic diagrams of the cross-sections with more details are presented in Figure
2.1(b). The nanopore structure is arranged in the periodic triangular cell
configuration with a lattice constant a , as shown in Figure 2.1(a).
This problem can be approximately solved by breaking this problem into a
combination of a 2D problem and a 1D problem. However, this problem is not
rigorously separable and 3D simulations here based on finite difference method
are applied. In the 2D+1D estimation, the 2D Bloch periodic condition and the 1D
finite well quantum confinement can be independently evaluated, so that the
method requires less computation power and is suitable for the cases where the
cylinders are thin and long. On the other hand, in the 3D finite difference method
simulations, the 2D periodic conditions are fully coupled with the confinement in
the z-direction. The method deals with the leaky modes accurately, requires
higher computation power, and is more suitable for cases where the cylinders are
thick and short.
19
Figure 2.1: (a) Schematic diagram of the nanopore superlattice. (b) Top view of
the 2D triangular cell lattice, where R is the nanopore radius, a the lattice
constant. (c) The parallelogram and the rectangular unit cells, and the 2 / a
circularly shifted clone of the rectangular unit cell. (d) The first Brillouin zone,
the irreducible Brillouin zone and the highly symmetric points.
In the present work, the emphasis is on the conduction band. Therefore, the
model is limited to a thin GaAs/InGaAs layer of thickness nm L 8 = , and a 3D
finite difference method is used to solve the band structures. In this case, one
expects that the main contribution to the formation of subband structures stems
both from the interactions between the electrons and the lattice, and from the local
confinement in-between the nanopores.
20
2.2 Modified Schrödinger equation
Since the effective mass for the electrons in the conduction band is different in the
two zones, GaAs and InGaAs, one has to take into account the position-dependent
effective mass and properly modify the Schrödinger equation. In addition, the
offset potential created as a result of the unbalanced energy gaps in the two
materials as well as the boundary strains is also essentially position-dependent
throughout the lattice. In other words, the Schrödinger equation also needs to
incorporate the position-dependency of the potential difference between the two
materials. The properly modified Schrodinger equation takes the form below:
( )
( ) ( ) ( ) ( ) r E r r V r
r m m
e
ψ ψ ψ = +
∇ ⋅ ∇ −
1
2
2
(2.1)
where r
indicates a position,
e
m electron mass, ( ) ( )
e
m r m r m /
*
≡ the position-
dependent ratio between the effective electron mass and the electron mass, and
( ) r V
the position-dependent offset potential. After the gradient operator is
applied to both the wave function and the mass ratio, the Schrödinger equation in
Equation (2.1) becomes,
( ) ( )
( ) ( ) ( ) r E r r V
r m r m m
e
ψ ψ ψ ψ = +
∇ + ∇ ⋅ ∇ −
2
2
1 1
2
(2.2)
which is to be solved numerically with the 3D finite difference method.
21
2.3 The 3D finite difference method
To proceed from here, one has to draw a unit cell, and perform the discretization
in the unit cell for numerical simulations. It is convenient to choose a unit cell to
match exactly with the first Brillouin zone, so as to avoid unnecessary folding.
For a triangular lattice configuration, there are two commonly adapted unit cells,
namely the parallelogram and rectangular unit cells. A parallelogram unit cell is
shown in Figure 2.1(c) where the two arms are of the lattice vectors
1
a
and
2
a
.
For triangular lattice, one has a a a = =
2 1
and the angle between the two vectors
is ( )
o
a a 60 ,
2 1
= ∠
. The parallelogram unit cell is widely used in the literature and
the periodic conditions can be easily enforced at all sides of the cell. However, a
shortcoming of the parallelogram unit cell is the need to modify the differential
operators. In the present work, due to the introduction of the position dependent
effective mass and potential, both the gradient operator and the Laplace operator
must be modified as shown in Equation (2.2) in the standard Cartesian coordinate
system, which would bring about errors in a large-scale computation.
In the present work, the rectangular unit cell, which is shown also in Figure 2.1(c)
with dotted lines, is employed. One may consider that the triangle on the right
side of the parallelogram is replaced by an equal but upside-down triangle on the
left side of the parallelogram. The two arms of the rectangle are a in x-direction
and 2 3 a in y-direction. The area of the rectangular unit cell
2 3 2 3
2
rectangle
a a a S = × = is the same as that of the parallelogram unit cell
2 3 60 sin
2 2
2 1 ram parallelog
a a a a S
o
= = × =
. Therefore, the rectangular unit cell
22
matches exactly with the first Brillouin zone too. In selecting the rectangular unit
cell, one avoids modifying any vector differential operators and is thus able to
proceed with a BenDaniel-Duke discretization scheme.
One denotes a point in the unit cell with three indices ( ) k j i , , for the mass
ratio
k j i
m
, ,
, potential
k j i
V
, ,
, and wave function
k j i , ,
ψ . The differential operations
in Equation (2.2) appear then as
( )
+ −
∆
+
+ −
∆
+
+ −
∆
= ∇
− + − + − + 1 , , , , 1 , , , 1 , , , , 1 , , , 1 , , , , 1
1 2 1
2
ˆ 1 2 1
2
ˆ 1 2 1
2
ˆ 1
k j i k j i k j i z k j i k j i k j i y k j i k j i k j i x
m m m
z
m m m
y
m m m
x
r m
(2.3)
( ) ( ) ( ) ( )
1 , , , , 1 , , , 1 , , , , 1 , , , 1 , , , , 1
2
2
ˆ
2
2
ˆ
2
2
ˆ
− + − + − +
+ −
∆
+ + −
∆
+ + −
∆
= ∇
k j i k j i k j i
z
k j i k j i k j i
y
k j i k j i k j i
x
z y x
r ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ
(2.4)
which leads to
( )
( )
( )
( )
( )
+ −
+ −
∆
+
+ −
+ −
∆
+
+ −
+ −
∆
= ∇ ⋅ ∇
− +
− +
− +
− +
− +
− +
1 , , , , 1 , ,
1 , , , , 1 , ,
, 1 , , , , 1 ,
, 1 , , , , 1 ,
, , 1 , , , , 1
, , 1 , , , , 1
2
1 2 1
2
1
2
1 2 1
2
1
2
1 2 1
2
1
1
k j i k j i k j i
k j i k j i k j i z
k j i k j i k j i
k j i k j i k j i y
k j i k j i k j i
k j i k j i k j i x
m m m
m m m
m m m
r
r m
ψ ψ ψ
ψ ψ ψ
ψ ψ ψ
ψ
(2.5)
and
23
( )
( )
( )
( )
( )
+ −
∆
+
+ −
∆
+
+ −
∆
= ∇
− +
− +
− +
1 , , , , 1 , , 2
, 1 , , , , 1 , 2
, , 1 , , , , 1 2
, ,
2
2
1
2
1
2
1
1 1
k j i k j i k j i
z
k j i k j i k j i
y
k j i k j i k j i
x
k j i
m
r
r m
ψ ψ ψ
ψ ψ ψ
ψ ψ ψ
ψ
(2.6)
where
x
∆
,
y
∆
, and
z
∆
are the step lengths in x, y, and z directions , respectively.
Note that the step length in the z direction has to be varied according to the
position for the sake of computation efficiency. Substituting Equations (2.5) and
(2.6) into Equation (2.2), one obtains
( )
( )
( )
( )
k j i
k j i e
k j i k j i k j i
z k j i k j i k j i k j i z
k j i k j i k j i
y k j i k j i k j i k j i y
k j i k j i k j i
x k j i k j i k j i k j i x
E V m
m m m m
m m m m
m m m m
. . 2
. .
1 , , , , 1 , ,
, , 1 , , , , 1 , ,
, 1 , , , , 1 ,
, , , 1 , , , , 1 ,
, , 1 , , , , 1
, , , , 1 , , , , 1
2
2
2 1 2 1
2
1
2
2 1 2 1
2
1
2
2 1 2 1
2
1
ψ
ψ ψ ψ
ψ ψ ψ
ψ ψ ψ
−
=
+ −
∆
+ + −
∆
+
+ −
∆
+ + −
∆
+
+ −
∆
+ + −
∆
− +
− +
− +
− +
− +
− +
(2.7)
The equation in Equation (2.7) is further modified with the proper boundary
conditions. In what follows,
x
n ,
y
n and
z
n are used to denote the number of steps
in the x, y and z directions respectively. In the z-direction, the range is truncated.
The range in the z direction is extended to 40nm away from the nanopores.
Further extending the range in the z direction does not result in significant
changes in the band structures.
In the x-y plane, Bloch periodic conditions are applied so that the unit cell is
repeated for the whole surface. In applying the periodic condition in the x-
24
direction, one replaces directly the indexes ( ) k j n
x
, , with ( ) k j, , 1 − , and ( ) k j, , 0
with ( ) k j n
x
, , 1 + . In applying the periodic condition in the y-direction, one
replaces the indexes ( ) k n i n
y x
, , 2 / + with ( ) k i , 1 , − , and ( ) k i n
x
, 0 , 2 / + with
( ) k n i
y
, 1 , + . The situation in the y-direction is schematically shown in Figure
2.1(c) where a clone of the original rectangle unit cell is pushed forward at the y
extremes along the x-direction starting from the middle of the cell. Indeed, as
shown in Figure 2.1(c), the rectangular unit cells are just arranged like a brick
wall. In this way, the periodic condition along
2
a
is strictly enforced.
While implementing the numerical differentiations, one multiplies a phase
term, ( )
x x
ik ∆ ± exp or ( )
y y
ik ∆ ± exp to the forward and backward elements in both
x and y directions. The wave vector components in the x and y directions,
x
k , or,
y
k , are thus introduced. For each point in the 2D Brillouin zone ( y k x k k
y x
ˆ ˆ
//
+ =
),
one writes down
z y x
n n n N × × = equations as in Equation (2.7) for each point
( ) k j i , , . Therefore, a N N × matrix is formed for each point in the k-space, and an
eigenequation set is to be solved numerically, which, rewritten in matrix format,
is
( ) 0
,
3
= − ψ E M
y x
k k
D
(2.8)
The solution of the equation gives energy dispersion ( )
y x
k k E , , and eigenvectors,
i.e. the wave function in real space ( ) z y x , , ψ . The path for the dispersion relation
along the high symmetry points ( ) X J X − − Γ − , in the irreducible Brillouin
zone, is schematically shown in Figure 2.1(d).
25
2.4 Subband structure and density of states
Let us now present some numerical results for the subband structure (conduction
subband dispersion) and the density of states. In the present work, the
configuration is of triangular cells with cylindrical barriers. The detailed changes
in energy band structures due to modifications of the cell form, materials and
barrier size, etc, are not considered. In the numerical simulations, the following
fixed parameters are used:
e
m m 0453 . 0
*
InGaAs
= , and
e GaAs
m m 067 . 0
*
= , nm L 8 = ,
nm a 80 = , nm R 20 = , and the offset potential meV V V
GaAs InGaAs
160 = − [ DRE01].
The first numerical results and also the most important ones for the present
work are the subband structure of the nanopore superlattice. The energy
dispersion, together with the corresponding density of states, is presented in
Figures 2.2(a) and 2.2(b), respectively.
35
40
45
50
55
60
k
//
E (meV)
X Γ J X
(a) Band structure
0 1 2 3 4 5
35
40
45
50
55
60
DOS (10
20
cm
-3
meV
-1
)
E (meV)
(b) DOS
Figure 2.2: (a) Energy band structure as a function of the 2D wave number
//
k ,
along the boundary of the irreducible Brillouin zone (see Figure 2.5) between the
high symmetry points for filling factor and cylinder radius from 2267 . 0 = f and
nm R 20 = . (b) The corresponding density of states, normalized to the volume and
the energy interval used in the calculations, and in units of ( ) ( )
1 3 − −
meV cm .
Since the radius of the cylinders is chosen nm R 20 = in the sandwiched
26
GaAs/InGaAs film, there is less than ¼ of the area occupied by GaAs. In this case
the electrons are still able to exist in-between the cylinders. Therefore, both local
confinement and scattering of electrons by the Bloch lattices contribute
substantially to the subband construction. There are subband gaps opened at the
high symmetry points, but the gaps are mostly smaller than 1 meV, whereas at
other points in the Brillouin zone, the gaps vary up to 20 meV. The subband
structure and the density of states curves are consistent with the experimental and
numerical report in Ref. [VEC09]. Note in particular that the fact that the subband
structure varies dramatically along the inside of the first Brillouin zone implies
that the splitting of the conduction band is largely due to the 2D periodic lattice.
Throughout the subband curves, it is noticeable that the lowest band 1 = n
moves steadily up from about meV 35 . The energy is measured from the bottom
edge of the conduction band. For very small filling factors, the z-direction
confinement can be considered as largely due to a 1D quantum well of finite
potential barriers with finite thickness. The potential height is at meV 160 and the
thickness of the quantum well is chosen to be nm 8 . There exist two energy levels
in this well, the ground state is located at about meV 45 and the first excited state
is close to the top of the quantum well at meV 155 . Note that if the width of the
quantum well is reduced, to about nm 7 , there remains only one energy level
inside the well.
The wave functions (conduction subband envelope functions) are also
analyzed. The eigenvectors of the eigenequation in Equation (2.8) is the
eigenfunction ( ) z y x , , ψ , namely the wave function. It is interesting to look
27
closely at some of the probability distributions, so that one understands how the
electrons are distributed at microscopic scales.
Figure 2.3: Electron probability distribution ( )
2
, , z y x ψ normalized over a unit
cell, k at Γ point, (a) for 2267 . 0 = f and 1 = n , (b) for 2267 . 0 = f and 2 = n ,
(c) for 6 . 0 = f and 1 = n , (d) for 6 . 0 = f and 2 = n . The dotted circles are a
guide for the eye indicating the InGaAs cylinders.
In Figure 2.3, probability distributions are plotted respectively for the Γ
point in k-space [see Figure 2.1(d)]. At this point, 0 = =
y x
k k , the differences due
to the polarization of the wave vector are avoided. In Figures 2.3(a) and 2.3(b),
the filling factor is chosen 2267 . 0 = f . In this case, the electron is likely found
around the cylinders and the distributions appear symmetric. When the filling
factor is increased to 6 . 0 = f , as shown in Figure 2.3(c) and 2.3(d), the
distributions contain more oscillations around the cylinders. This is expected as
the electrons are more confined locally in the narrow space in-between the
28
cylinders. This suggests that more open and flat subband structures would be
obtained with higher filling factors.
The above microscopic pictures are useful especially when one considers
some of the macroscopic optical and electric properties, such as the optical
absorption due to transitions between the subbands. For example, for dipole-
dipole transitions, the significance of a specific transition is largely affected by
the dipolar orientations of the two subbands involved.
2.5 Intersubband absorption
One of the most important macroscopic properties of a semiconductor device is
the optical absorption [PIP03, LUN00]. Here one considers absorption due to
harmonic electric field written as
( ) ( ) t r k E k E ω ω − ⋅ =
exp ,
0
(2.9)
where the field is polarized in
0
E
direction with wave vector k
and angular
frequency ω . The optical field interacts with the electrons in the sample. The
model of the electron-photon interaction is limited to the dipole-dipole transitions
only. In this case, the Hamiltonian of electron-photon interaction is simplified as
r E e H
dipole dipole
⋅ − =
−
(2.10)
where e denotes the carrier charge. According to Fermi's golden rule, the
transition rate between a subband that bears a quantum number m and another
29
subband that bears a quantum number n can be written as
( ) ( ) ω δ
π
− ∆ − =
− ∑
E f f n H m W
m n dipole dipole
m n
nm
2
,
2
2 (2.11)
where one factor of 2 is added due to the spin degeneracy, ( )
m n
f f − is the
difference between the Fermi-Dirac distribution function for the two states, and
the delta function ( ) ω δ − ∆E requires the photon energy ω to match exactly
with the energy interval between the two states
n m
E E E − = ∆ . Introducing a
damping factor Γ , one rewrites the delta function in a Lorentzian function as
( )
( ) ( )
2 2
2 /
2 /
;
Γ + − ∆
Γ
= Γ − ∆
ω
π
ω δ
E
E (2.12)
Therefore, the photon absorption coefficient, defined as IV W / ω α = ,
where V is the volume, I the time averaged intensity, becomes eventually,
( )
( ) ( )
2 2
2
0
, 0
2 /
2 / 2
Γ + − ∆
Γ
− ⋅
=
∑
ω
π
ε
πω
α
E
f f n r E m
cV n
m n
m n
(2.13)
where n is the optical index, c light speed in vacuum, and
0
ε vacuum dielectric
function. Note that the absorption depends on the polarization of the input field,
or, in other words, the absorption coefficient is a dyadic tensor. In the present
work, two elements are studied:
||
α , where the field is polarized along the x-y
plane, also known as s-polarization, and
⊥
α , where the field is polarized in the z-
direction, also known as p-polarization.
30
In the following, some numerical results obtained with Equation (2.13) are
presented. In the simulations, the mean optical index is chosen to be 3 . 3 = n , the
damping factor meV 1 = Γ corresponds to a relaxation time of ps 01 . 0 = τ . The
calculations were carried out for
Γ
k point, i.e. 0 = =
y x
k k , In addition, for
simplicity, the dependence of the damping factor on the temperature and
frequency is not considered. In calculating the intersubband absorption, there are
a number of other key parameters to be considered, namely the injected current,
the Fermi level
f
E , and the operation temperature T . Three operation
temperatures: K T 0 = , K T 77 = , and K T 300 = are considered. The injected
carrier density and the Fermi level are dependent on each other. Choosing a Fermi
level determines the injected carrier density. In actual operation, one is able to
adjust the current bias so as to move the Fermi level within the range considered.
In our numerical simulation, it is convenient to choose a Fermi level for the
calculation.
Some numerical results for absorption are presented in Figure 2.4. In Figure
2.4(a) and 2.4(b), the Fermi level was chosen at meV E
f
40 = , which indicates a
rather low carrier density, and the input field was taken to be s-polarized. One can
see that the main transition peaks are concentrated in the range from meV 5 to
meV 130 , and strongly pronounced peaks are found at about meV 40 and
meV 80 . It is also shown that when the temperature is increased from liquid
nitrogen temperature K T 77 = to room temperature K T 300 = , the overall
absorption peaks are reduced, but the transitions of higher energies become
enhanced and the shape of the spectrum is modified. This is understandable
31
because at higher temperature the carrier distribution is more diffused.
In Figure 2.4(c) and 2.4(d), the Fermi level is increased to meV E
f
60 = ,
which roughly corresponds to a carrier density
3 13
/ 10 0 . 1 cm × . One observes that
the absorption peaks are enhanced especially for higher energy transitions. The
maximum absorption peak is almost doubled in comparison with the curves in
Figure 2.4(a) and 2.4(b).
In Figure 2.4(e) and 2.4(f), the polarization of the input field is changed to
be perpendicular to the surface plane. It is interesting to note that while most of
the absorption peaks are dramatically attenuated compared to the peaks in s-
polarization, strong peaks stand out at about meV 80 (other peaks still exist but the
strength is dramatically reduced). A closer look at the strong peak reveals that the
peak is actually formed with many transitions. This can be understood based on
the 1D quantum well model in the z-direction. As already mentioned, for a
meV 160 finite potential barriers and a thickness of nm 8 , there exist two bound
states in the quantum well. Therefore, as far as z-polarization is concerned, the
significant transitions occur between a subband associated with the first energy
level and another subband associated with the second energy level.
It is also worth mentioning that the obviously different spectral patterns
between s- and p-polarization absorption would be useful to develop polarization
tunable devices and switches.
32
Figure 2.4: Optical absorption coefficient for (a) 2267 . 0 = f ,
Γ
k , meV E
f
40 = ,
meV 1 = Γ , s-polarization, at K T 77 = , (b) 2267 . 0 = f ,
Γ
k , meV E
f
40 = ,
meV 1 = Γ , s-polarization, at K T 300 = , (c) 2267 . 0 = f ,
Γ
k , meV E
f
60 = ,
meV 1 = Γ , s-polarization, at K T 77 = , (d) 2267 . 0 = f ,
Γ
k , meV E
f
60 = ,
meV 1 = Γ , s-polarization, at
K T 300 = , (e) 2267 . 0 = f ,
Γ
k , meV E
f
60 = ,
meV 1 = Γ , p-polarization at K T 77 = , (f) 2267 . 0 = f ,
Γ
k , meV E
f
60 = ,
meV 1 = Γ , p-polarization at K T 300 = .
2.6 Selection rules
In relation to the intersubband absorption in Figure 2.4, one is particularly
interested in the individual contribution from each transition. It is understandable
33
that due to the various shapes of the wave functions, the strength of the dipole-
dipole coupling varies and is largely determined by the matrix elements between
two states. The dipole matrix elements for intersubband transitions depend on the
symmetries of the electron wave functions in both the initial and final subbands.
The variations in dipole matrix elements across the Brillouin zone must be
considered because each point in the Brillouin zone contributes differently to the
absorption.
Figure 2.5: Individual contributions to the absorption curve in Figure 2.4(a) for
meV 25 < ω , corresponding to transitions from the ground state
1 = n
E . The two
numbers marked for each peak indicate the two states involved in the specific transition,
with green background for transitions from
1 = n
E and pink background for other
transitions. (a) K T 0 = (b) K T 77 = .
In Figure 2.5(a) and 2.5(b), the individual contributions of each transition in
comparison with the overall absorption curve are presented. The calculations were
carried out for
Γ
k point, i.e. 0 = =
y x
k k . In order to have a clearer picture, the
range of the photon energy is limited within meV 25 < ω , so that the individual
transitions can be better observed.
The transitions for zero temperature K T 0 = are first analyzed and the
results are plotted in Figure 2.5(a). Since the Fermi energy is chosen slightly
34
higher than the ground state
1 = n
E , the transitions between two excited states are
limited by the Fermi-Dirac distribution. In this case, one can only observe the
transitions between the ground state
1 = n
E and an excited state
1 > n
E . In Figure
2.5(b), pronounced transitions exist for 1-3, 1-5, 1-7, 1-9, 1-11, the other
transitions such as 1-2, 1-4, 1-6, 2-8 etc. are forbidden. The observation reveals an
important parity regulation that the transitions between the ground state
1 = n
E and
an even state
... 9 , 7 , 5 , 3 = n
E are allowed, whereas the transitions between the ground
state
1 = n
E and an odd state
... 8 , 6 , 4 , 2 = n
E are forbidden.
The transitions for higher temperature K T 77 = are also tested and the
results are plotted in Figure 2.5(b). Since the Fermi-Dirac distribution became
non-zero for finite temperature, the transitions between two excited states are
allowed. In this case, one can observe the transitions between the ground state
1 = n
E and an excited state
1 > n
E as well as the transitions between two excited
states. In Figure 2.5(b), it is shown that in addition to the transitions between the
ground state and an excited state, i.e.1-3, 1-5, 1-7, 1-9, 1-11 (marked in green),
some of the transitions between two excited states, such as 2-6, 3-7, 4-10, etc.
appear possible. This extends the parity regulation to include all transitions, i.e.
transitions between an even state and another even state or between an odd state
and another odd state are possible, whereas transitions between an even state and
an odd state are not possible.
35
Chapter 3
Electrical transport
3.1 Boltzmann equation
From the first stage of the work, one understands that terahertz light induced
absorption occurs in the nanopore thin films. However, before one is able to
actually design a terahertz detector based on the nanopore superlattices, an
obvious hurdle to be overcome lies in that the information about the light induced
inter-subband absorption should be extracted reliably and as simply as possible. In
previous designs based on individual or an ensemble of quantum dots and wells,
such as cascaded quantum-well light detection, the information was usually
extracted from induced currents stemming from individual quantum wells.
However, in the case of nanopore configuration, the subbands are created by the
interaction of electrons with the whole lattice. It appears therefore necessary to
solve for the system the Boltzmann transport equation [ZIM01, KIT96]. The
Boltzmann equation under relaxation time approximation can be written as
[ZIM01, CL11],
( )
( )
( ) ( ) ( ) ( )
τ
E r f t k r f
k
t k r f
t
k
r
t k r f
k v
t
t k r f , , , , , , , , ,
0
−
− =
∂
∂
⋅
∂
∂
+
∂
∂
⋅ +
∂
∂
(3.1)
where τ is the mean relaxation time, and the group velocity can be obtained from
36
the dispersion relation ( )
( )
k
k E
k v
∂
∂
=
1
, and
( )
1
0
1 exp
−
+
−
=
T k
E E
E f
B
F
is the
Fermi-Dirac distribution in equilibrium, with
B
k Boltzmann constant,
F
E Fermi
energy, and T the temperature. If the electrons are driven by an external field E
,
according to Newton’s law E
e
t
k
=
∂
∂
. Furthermore, let us consider only the
steady state, i.e. the time independent solution, and quasi-homogeneous solid, i.e.
the spatial change of the distribution is assumed insignificant. With the
assumptions, the equation in Equation 3.1 becomes,
( )
( ) ( ) 0
0
= − +
∂
∂
E f k f
k
k f e
E τ
(3.2)
This equation can be solved numerically with finite difference method. In this
case, the equation extends into a large-scale linear equation. Another approach is
to have a Taylor expansion to account for the higher orders of the distribution.
3.2 Intrasubband current
Once the distribution is solved, the overall intersubband current can then be
conveniently integrated in k space,
( ) ( ) k d k f k v
e
j
3
3
4
∫
=
π
(3.3)
where
3
8 π is the volume in k space and spin of 2 is considered.
37
Replacing ( ) k f
in the above equation with the Boltzmann equation in
Equation 3.2, one obtains the formula for the current as,
( ) k d k f v
e
j
k
3
3
2
4
∇ ⋅ − =
∫
π
τE
(3.4)
If the Taylor expansion is truncated to include only the first order differentiation,
the current becomes linear as the conductivity is a constant,
( ) k d f v
e
r j
k
3
0 3
2
1
4
∇ ⋅ − =
∫
π
τE
(3.5)
and
( ) k d f v
e
r
k
3
0
3
2
1
4
∇ − =
∫
π
τ
σ (3.6)
which is the formula extensively adapted in the literature [ZIM01, KIT96, CL11].
The omission of higher order change as a function of energy or momentum
results in a linear conductivity that is not a function of the external electric field.
Therefore, the modification of electron energy due to light induced intersubband
transitions cannot be extracted from the linear current. In order to capture the
modification, one goes beyond the approximations to solve the Boltzmann
transport equation accurately. Numerical results demonstrated that when the bias
voltage continued to increase after the linear conduction range, the conductivity
becomes a function of the applied voltage, i.e. the nonlinear conductivity.
38
3.3 Numerical solution with finite difference
method
In order to go beyond the linear current regime, it is necessary to solve the
Boltzmann equation rigorously or as accurately as possible. The equation is
written in Equation 3.2. This equation can be either solved numerically with the
finite difference method or analytically with a Taylor expansion. In the following,
mathematic processes for both approaches are explained in details.
Let us show first the process for finite difference method. Before writing
down the process for the solution, let us define a constant to simplify our writing:
a e
a
τ
≡
0
(3.7)
The equation appears now
( )
( ) ( ) 0
0 0
= − +
∂
∂
E f k f
k
k f
a
E (3.8)
In finite difference method approach, one digitalizes the k space into a set of
discrete numbers, and writes down for each of the point in the k space the
equation in finite difference method form
0
2
2
2
2
, 0 ,
1 , , 1 , , 1 , , 1
0
= − +
∆
+ −
+
∆
+ −
− + − +
j i j i
y
j i j i j i
y
x
j i j i j i
x
f f
f f f f f f
a E E (3.9)
For each point one has such an equation. For the whole set of discrete points, one
39
obtains the following linear equation set as
( ) ( )
j i j i
y
j i j i j i
y
x
j i j i j i
x
f f
f f f f f f
a
, , 0 ,
1 , , 1 , , 1 , , 1
0
1
2
2
2
2
=
+
∆
+ −
+
∆
+ −
− + − +
E E
(3.10)
This is a linear equation set of the form symbolically
0
F MF = and the solution is
straightforwardly with the inverse of the matrix
0
1
F M F
−
= .
When the distribution ) (k f
is solved numerically by finite difference
method, one obtains the current directly with the integration in k space as shown
in Equation (3.3). Note in passing that since the equation is directly solved, the
resulting current and thus the conductivity becomes a nonlinear function of the
external field, which differs from the linear approximation widely adapted in the
literature. The linear approximation avoids solving the Boltzmann equation, but
leaves the nonlinear dependence unaccounted for. In other words, one expects that
the nonlinear dependence of the current on the field would bring about more
significant changes so that light induced intersubband absorptions can be read out
from measuring the current. On the other hand, since the accuracy of the approach
depends mainly on the number of points used in the finite difference method
approach, in general the more points the higher accuracy.
3.4 Rigorous solution with a Taylor expansion
Alternatively, in the present work a novel approach for solving the Boltzmann
40
equation is introduced. In this approach, the resulting solution for the current
becomes a Taylor expansion based on the derivations of all orders of the
distribution. Since the derivations can be analytically obtained and also since the
convergence of the Taylor expansion is usually rapid, the solution becomes
rigorous, which helps us to calculate the current accurately and to avoid large
scale finite difference method computation.
The approach is based on the integral solution of the standard linear
differential equation, symbolically as:
) ( ) ( x Z y x H
dx
dy
= + (3.11)
which has the solution:
dx x Z e e y
x U x U
∫
−
= ) (
) ( ) (
(3.12)
where
dx x H x U
∫
= ) ( ) (
(3.13)
with the help of integration by parts
∫ ∫
− = BdA AB AdB (3.14)
One identifies from Equation 3.5 the following functions
41
E
τ e
x H = ) ( , (3.15)
( )
E
τ e
k f
x Z
0
) ( = , (3.16)
E E
τ τ e
k
k d
e
dx x H x U = = =
∫ ∫
) ( ) ( , (3.17)
( ) k f A
0
≡ , (3.18)
and
E
τ e
k
e B ≡ . (3.19)
The process of integration by part can be repeatedly applied, and the first few
resulting integrations become as follows
( )
=
∫
−
E E
τ τ e
k
e
k
e d k f e f
0
k d
k d
f d
e e
e
k d
df e
f f
e
k
e
k
2
0
2
0
0
E E
E E
τ τ
τ τ
∫
−
+ − =
−
+ − =
∫
−
2
0
2
2
2
0
2
2
0
0
k d
f d
d e e
e
k d
f d e
k d
df e
f f
e
k
e
k
E E
E E E
τ τ
τ τ τ
+
−
+ − =
∫
−
E E
E E E E
τ τ
τ τ τ τ
e
k
e
k
e d
k d
f d
e
e
k d
f d e
k d
f d e
k d
df e
f f
4
0
4
4
3
0
3
3
2
0
2
2
0
0
k d
k d
f d
e e
e
k d
f d e
k d
f d e
k d
f d e
k d
df e
f f
e
k
e
k
−
+
−
+ − =
∫
−
5
0
5
4
4
0
4
4
3
0
3
3
2
0
2
2
0
0
E E
E E E E E
τ τ
τ τ τ τ τ
42
(3.20)
Therefore it appears
+ ∇
+ ∇
− ∇
+ ∇
− =
0
4
4
0
3
3
0
2
2
0 0
f
e
f
e
f
e
f
e
f f
k k k k
E E E E τ τ τ τ
(3.21)
and finally the Taylor expansion appears as
( )
0
0
1 f
e
f
n
k
n
n
n
∇
− =
∑
∞
=
E τ
(3.22)
Substituting the distribution in Equation 3.22 into Equation 3.2, one has
eventually
( ) ( ) ( ) k d f
e
k v
e
r j
n
k
n
n
n
3
0
0
3
1
4
∇
− =
∑
∫
∞
=
E τ
π
(3.23)
An important message one receives from Equations (3.22) and (3.23) is that the
electric distribution and the current are now explicitly dependent on all orders of
field, i.e. , ,
2
E E j ∝ , provided that the corresponding order of gradient, i.e.
the coefficients of the Taylor series,
, , ,
0
2
0 0
f f f
k k
∇ ∇ , remains significant. In
other words, the developed formalism would be suitable for accounting for the
nonlinear dependence of the current.
3.5 Nonlinear dependence of intrasubband current
It turns out that the distribution depends on all order gradients of
0
f . In most
43
cases, the higher order gradients will vanish after a few derivatives. Consider the
dispersion relation ( ) k E
already obtained in the first stage of the present work
(see Chapter 2), the gradient
n
n
n
k
k d
f d
f
0
0
= ∇ can be broken down into two
derivatives
m
m
m
m
n
n
k d
E d
and
dE
f d
k d
f d
0 0
⇒ . (3.24)
In Equation (3.24), the second derivative has to be done numerically with the
dispersion relation obtained previously in Chapter 2, whereas the first derivative
can be analytically obtained assuming the Fermi-Dirac distribution in equilibrium
( )
1 exp
1
0
+
−
=
T k
E E
E f
B
F
(3.25)
An important remark is that the break down shown in Equation (3.24) is rather
complicated. In the literature, general formulae for high order derivative for
composed function, i.e. the chain rule, exist, which is referred as Faá di Bruno
formula [BRU1955]
( ) ( )
( )
( ) ( )
( ) ( )
( )
( )
n
k
n
k k
k
n
n
n
n
t g t g t g
t g f
k k k
n
t g f
dt
d
=
∑
! ! 2
' '
! 1
'
! ! !
!
2 1
2 1
(3.26)
where the sum is over all nonnegative integer of the Diophantine equation
[MOR69]
44
n nk k k
n
= + +
2 1
2 (3.27)
and
n
k k k k + + =
2 1
: (3.28)
It is interesting to take notes on the convergence of the expansion. The
distribution ( ) k f
is usually a polynomial function with respect to the incoming
electric field, and the coefficients of the polynomial are the gradients
n
n
n
k
k d
f d
f
0
0
= ∇ . Since ( ) k f
depends on the dispersion relation ( ) k E
, the gradient
becomes a combination of
m
m
dE
f d
0
and
m
m
k d
E d
. While the derivatives
m
m
dE
f d
0
exist for
∞ ~ m , the gradient for the energy bands
m
m
k d
E d
vanishes after a few terms.
Therefore the solution is actually rigorous and convergent in most common
situations such as that of the present work.
45
Chapter 4
Photocurrent
4.1 Redistribution of carriers
In the preceding chapter, it is shown that the current could be rigorously
calculated when both the dispersion relation and the electrical distribution at the
subbands are known. The rigorousness enables the nonlinear part of the current to
be calculated, which is usually omitted in the literature. Since the current changes
when the electron distribution on the subbands is modified, the current bears a
significant response to the change of electron energy or momentum. This is a
critical finding for the realization of light detection, i.e. the information on the
intersubband absorption can be extracted from the bias current. The roadmap of
the detection is as follows: a DC bias voltage is applied across the nanopore thin
film and the bias is pre-adjusted to an optimized point in the nonlinear region. The
film is illuminated with radiation of frequencies corresponding to the
intersubband absorption spectrum, and the electron distribution on subbands is
modified. The modification of the electron distribution is eventually read out from
the measured current.
The modification of the electron distribution is calculated for each of the
various subbands. The simulation process runs as follows. The intersubband
absorption is calculated for a given light intensity according to the formulae given
46
in the previous chapter in Equation (2.11) and Equation (2.13). The transition rate
can be either calculated directly based on Equation (2.11), or related to the data
for the absorption coefficients in Equation (2.13). The modified electron
distribution is introduced in Equation (3.23) to calculate the induced current,
which should contain the changes due to the light induced intersubband
absorptions.
Before presenting the numerical results for the bias current responses to the
light induced intersubband absorption, a discussion on the intersubband transition
is presented. The primary purpose of the present chapter is to demonstrate the fact
that the light induced intersubband absorptions bring about significant changes in
the bias current, which is taken as the foundation of our proposal for light
detection in terahertz, and gives guidance in the device designs to be shown in the
succeeding chapter. Therefore, a simple and clear approach is employed to take
into account the light induced intersubband transitions. In such an approach, once
the transition rates are determined, assuming an adequate transition time, one is
able to calculate the modified electron distribution at each subband. On the other
hand, one should be fully aware of the degree of the complexity involved in the
calculation of the transition. More effects and influences, though negligible in
most situations, have to be considered before one is able to have an accurate
picture of the transition. Among others, the effects include electron-electron and
electron-phonon interactions, which could reduce the relaxation time and thus
complicate the transitions. In the succeeding chapter, the relaxation time based on
some experimental data in the literature will be discussed.
47
4.2 Light induced current
In what follows, simulation results for the bias current are presented. The key
questions to be answered are: 1) how the intrasubband current behaves when a
bias field is applied; 2) how the current responds to the change of electron
distribution over the subbands; and 3) how the current responds to the
intersubband absorption. The numerical results are presented in Figure 4.1, Figure
4.2, Figure 4.3, and Figure 4.4, respectively for different dependences of the
current.
In Figure 4.1, the bias current dependences on the bias voltage are drawn for
a given photon energy and for various light intensities, including the case where
there is no light at all. Plotted are the current density and the electric field, i.e. the
current and the voltage are both normalized to the sizes (length and area
respectively), which is to avoid involving the sizes at the present stage, so that the
results are more general. The sizes will be discussed in the succeeding chapter in
a different context.
20 40 60 80 100 120 140 160 180 200
-500
0
500
1000
1500
2000
2500
3000
3500
4000
E (mV/ µm)
j (nA/ µm
2
)
I = 0 nW/ µm
2
I = 0.1 mW/ µm
2
I = 0.3 nW/ µm
2
I = 1 nW/ µm
2
Figure 4.1: Intrasubband current as a function of bias field for various light
intensities. The photon energy is 15meV. The temperature is assumed K T 77 =
.
48
One takes note from curves in Figure 4.1, where the intrasubband current is
plotted as a function of the bias voltage (i.e. the I-V curve). The generated current
becomes nonlinear when the bias voltage increases. The nonlinear terms are kept
because the Boltzmann equation was solved rigorously without taking the usual
linear approximation. As expected, after a linear regime, when the bias voltage
augments further, the current increases nonlinearly and the conductivity becomes
a function of the applied field.
In Figure 4.2, the bias current dependence on the light intensity for various
bias voltages is presented. The curves increase with the intensity of the light,
which means that it is possible to read out the optical intersubband absorption
from the bias current.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
200
400
600
800
1000
1200
1400
Light intensity (nW/ µm
2
)
j (nA/ µm
2
)
E = 99 mV/ µm
E = 115 mV/ µm
E = 132 mV/ µm
E = 148 mV/ µm
Figure 4.2: Intrasubband current as a function of light intensity for various bias
fields. The operation temperature is assumed K T 77 =
.
The fact that the bias current changes significantly with the light intensity is
taken as the foundation of the proposed light detection in terahertz. Under
different biases, the I-V curve varies. Therefore, the bias should be preset to an
optimized value so as to have the detection with the highest sensitivity.
49
Another important parameter is the light frequency, namely the photon
energy. It has already been demonstrated that the nanopore superlattices support
subbands and the gaps between the subbands cover the terahertz range. In general,
the electron distribution over the subbands is a function of many factors, Fermi
energy, temperature, external fields, and carrier injection etc. These factors are
usually related to each other. Since the intrasubband current responds to the
changes in the electron distribution over the subbands, the same current should
also respond to the light induce intersubband absorptions. It was shown
previously in Figure 2.4 that intersubband absorption occurs between the
subbands, and the electron distribution is modified. Obviously, the spectral
response of the intrasubband current to the intersubband absorption would follow
more or less similar spectral curves as those in Figure 2.4. The bias current
dependence on the photon energy is demonstrated in Figure 4.3 below.
In passing, one understands that the intersubband absorption can now be
read out from the current measurement for light of terahertz frequencies, which is
the key concept in the present proposal. On the other hand, the spectral response
varies significantly with the frequency, which reflects the nature of intersubband
transitions. In the succeeding chapter, a proposal to construct a wideband detector
will be introduced.
50
Figure 4.3: The intrasubband current as a function of photon energy for different
bias voltages. The operation temperature is assumed K T 77 =
.
Finally, one should take into account the operation temperature. It has
always been a challenge to have terahertz devices work at room temperature. In
fact, the operation temperature of various terahertz devices especially those based
on optoelectronic concepts, is kept far below the room temperature, even below
the liquid nitride temperature. Here, one should have a test on the operation
temperature, based on the computational process introduced in this chapter.
Obviously, at room temperature, the thermal fluctuation appears of the same
strength as that of the intersubband gaps, which explains the reason for the
quantum cascade terahertz lasers not to perform at high temperature.
51
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
30
40
50
60
70
80
90
100
110
120
130
h ν=15meV, T=300k
Light intensity (nW/ µm
2
)
j (nA/ µ m
2
)
E = 99 mV/ µm
E = 115 mV/ µm
E = 132 mV/ µm
E = 148 mV/ µm
Figure 4.4: Intrasubband current as a function of light intensity for various bias
fields. The operation temperature is assumed K T 300 =
.
It is interesting to note that as shown in Figure 4.4 the device performs even
at room temperature, despite a notable reduction on sensitivity.
4.3 Self-consistent approach
In previous simulations, an approximation was adapted, i.e. to treat the
intersubband transitions only once. This is indeed the first Born approximation.
Below, let us consider a self-consistent solution that goes beyond the
approximation.
What is to be calculated is the carrier distribution under the light-induced
intersubband transitions. The Fermi-Dirac distribution ( ) E f
0
in Equation (3.25)
is used to account for the distribution when there is no light received. Once the
light is incident, intersubband absorption occurs and the distribution is modified
to ( ) E f
I
, where the subscript I indicates the distribution is a function of the
incoming light intensity. Another input parameter is the transition time τ which
52
is determined by the device itself, i.e. an intrinsic parameter.
The degree of the transition strength depends on the transition rate between
the two states involved:
( ) ( ) ω δ
ε
π
− ∆ −
=
∑
E f f P
cn
I
w
m n mn
m n
2
, 0
2 2
2
(4.1)
One can see from Equation (4.1) that the transition rate depends on the light
intensity. From a state
i
to another state
j
, the total transition is
τ
ij
w
, where
τ
is the transition time. For the sake of simplicity, let us define τ
ij ij
w W ≡ , and
call
ij
W the coupling. That means, the state
i
loses
ij
W while the state
j
gains
ij
W .
ij ji
W W − = can be assumed, therefore it is only necessary to calculate the
loss. Also 0 =
ii
W is assumed. Let us write down the possible intersubband
transitions for all the subbands (and all the states),
nn
I
n n
I
n
I
n
I
n
I
n
n
I
n
I I I I
n
I
n
I I I I
n
I
n
I I I I
W f W f W f W f f f
W f W f W f W f f f
W f W f W f W f f f
W f W f W f W f f f
+ + + + =
+ + + + =
+ + + + =
+ + + + =
3 3 2 2 1 1
0
3 33 3 32 2 31 1
0
3 3
2 23 3 22 2 21 1
0
2 2
1 13 3 12 2 11 1
0
1 1
(4.2)
Strictly speaking, the linear equation set in Equation (4.2) can hardly be
solved rigorously as it may involve too many subbands. However, it is reasonable
to assert that the contribution from subbands far beyond the Fermi level are
53
insignificant and therefore can be neglected. That means one can truncate the
equation set so that the size of the coupling matrix
ij
W becomes accessible for
numerical calculations. In the present work, n = 100 subbands is used. The
neglected transitions bear couplings that are too weak in the frequency range of
interest for the present work. The above is indeed a linear equation set, in matrix
format it looks,
+
=
I
n
I
I
I
nn n n n
n
n
n
n
I
n
I
I
I
f
f
f
f
W W W W
W W W W
W W W W
W W W W
f
f
f
f
f
f
f
f
3
2
1
3 2 1
3 33 32 31
2 23 22 21
1 13 12 11
0
0
3
0
2
0
1
3
2
1
(4.3)
or, writing in matrix and vector format,
I I
WF F F + =
0
(4.4)
which appears already self-consistent. To solve the linear equation set in Equation
(4.4), one has usually three options:
a ) First Born approximation: In this approximation one replaces
0
F F
I
⇒ on
the right side of the equation in Equation (4.4), and the equation becomes
0 0
WF F F
I
+ = (4.5)
This is indeed what was done so far in the previous part of the work. The
approach works well particularly when the intersubband couplings are not
54
strong enough to cause resonant interactions. However, since the overall
coupling is calculated only once, the nature of multiple transitions as well
as the proportion of the importance of each coupling cannot be described
accurately. Therefore, although the numerical simulations based on the
approximation would be sufficient to propose the terahertz detection and
to design for a workable detector, approaches to go beyond the first Born
approximation would be necessary to describe the light induced
intersubband transitions rigorously, not limited to the present nanopore
system.
b ) Iterative solution: A widely adapted approach to go beyond the first Born
approximation is the iterative solution as follows
1 0
2 0 3
1 0 2
0 0 1
−
+ =
+ =
+ =
+ =
n n
WF F F
WF F F
WF F F
WF F F
(4.6)
Obviously, the accuracy of the solution depends on the number of
iterations. It is worth noting that the increase of accuracy in the iterative
approach does not remove the restriction that the solutions have to be
convergent. In other words, it cannot describe any resonant interactions. A
solution that removes the restriction and deals with resonances has to be
self-consistent.
c ) Self-consistent solution: A rigorous self-consistent solution is to solve the
linear equation set directly by taking the inverse of the matrix, as
55
( )
0
1
F W U F
I
−
− =
, (4.6)
where U is a unit matrix. This is the standard solution of a linear equation
set in the textbooks. Numerically, it involves a matrix inversion. Since it is
rigorous, the self-consistent solution contains naturally all the information,
including the possible resonances.
In order to have a comparison between the first Born approximation and the self-
consistent solution, as far as the nanopore system in the present work is
concerned, numerical simulations based on the self-consistent solution in
Equation (4.6) have been carried out using the same parameters as those in
Figures 4.1 and 4.2. The numerical results are presented below in Figures 4.5 and
4.6, respectively.
Figure 4.5: Intrasubband current as a function of bias field for various light
intensities. The photon energy is 15meV. The operation temperature is assumed
K T 77 =
. The redistribution of carriers is calculated with the self-consistent
solution.
56
Figure 4.6: Intrasubband current as a function of light intensity for various bias
fields. The operation temperature is assumed K T 77 =
. The redistribution of
carriers is calculated with the self-consistent solution.
In Figure 4.5, intrasubband current is plotted as a function of the bias
voltage (i.e. the I-V curve), while in Figure 4.6, the bias current dependence on
the light intensity for various bias voltages is presented. One compares the curves
in Figures 4.5 and 4.6 with the curves in Figures 4.1 and 4.2, and concludes that
the self-consistent calculation gives in general similar curves as those obtained by
the first Born approximation. Therefore, the observations given previously for
Figures 4.1 and 4.2 remain valid. On the other hand, some deviations are however
noticeable, such as the modified current scale and the shape of the curves. A
remarkable difference is the dip at the beginning of the curves in Figure 4.6, i.e.
the resonances. As already discussed, an advantage to make use of the self-
consistent solution is the inclusion of the cases where resonant interaction takes
place. The curves in Figure 4.6 indicate that when the incoming light is weak, the
bias current decreases with light intensity, but after a short range, the bias current
increases with light intensity in a way similar to the curves in Figure 4.2.
57
Chapter 5
Device design
5.1 A model device for terahertz detection
So far in the preceding chapters, it is conceptually and numerically demonstrated
that: 1) the nanopore superlattices support subbands and intersubband absorption
occurs when the thin film is illuminated with light in the terahertz range; 2) with
an external bias electric field along the nanopore thin film, the information about
the terahertz light induced inter-subband absorption can be extracted by
measuring the induced bias current. The above findings suggest that terahertz
detection is feasible with the nanopore structure. In the present section, in terms
of instrumentation, the properties and capabilities of the device will be
considered. The proposed model device is schematically drawn in Figure 5.1.
In Figure 5.1, a DC external voltage is applied along a GaAs/InGaAs
nanopore superlattice of which the detailed structures were shown in Chapter 2 in
Figure 2.1. The current and light intensity dependence is shown in Figures 4.1 and
4.2 respectively.
Before explaining in detail the operation of the proposed model device, it is
necessary to clarify certain points. At this stage of design, the actual sizes can
hardly be fixed. Therefore, instead of size-dependent quantities, current and
voltage, normalized quantities are used for the curves in Figures 4.1 and 4.2. In
58
some specific calculations, specific sizes for the thin film were chosen, which
provided the possibility of comparison with data in the literature. On the other
hand, these numbers do not necessarily represent the optimized situations for the
device, and are subject to change for an experimentally oriented projection.
Figure 5.1: Schematics showing a model device for terahertz detection. A bias
voltage is applied to the nanopore thin film and the current is measured. Light to
be detected comes in vertically of s-polarization with photon energy ω
and light
intensity I.
The proposed model device in Figure 5.1 operates as follows. Before
detecting the light, the device needs to be calibrated by adjusting the bias voltage
V. In Figure 4.1, the I-V curves show that after some value the I-V dependence
becomes nonlinear. In this region, the current changes significantly with respect
to the incoming light intensity. Once the device is adjusted to an optimized bias, it
is ready to detect the incoming light. In general, the bias current is proportional to
the light intensity. Note also that, as already shown in Figure 4.3, the current and
light intensity curve depends on the photon energy, i.e. for light with different
frequency the current has a different response curve, which is a challenge for
wideband detection and also for precise frequency analysis. At the end of the
59
present chapter, a design to overcome this hurdle will be proposed.
5.2 Noise analysis
For device development, one is concerned mainly with the intrinsic noise, which
is due to physical properties of the detector, and causes fluctuation in the detector
outputs. There are five primary noise sources in detector systems: thermal noise,
photon noise, shot noise, modulation noise (1/f), generation-recombination noise,
and temperature noise. As far as integrated compact semiconductor devices are
concerned, three noise sources are usually studied: generation-recombination
noise, thermal noise, and shot noise. In general, generation-recombination (G-R)
and shot noise are intrinsic and usually significant for semiconductor components,
whereas thermal noise is often considered the main noise for long wavelength
optical detectors, especially at room temperature. It is interesting to test the noise
current densities
2 2 2
, ,
thermal shot GR
i i i against the frequency f ∆ , which is the
half width of the measured pulses [LUN00].
At the present stage, for the purpose of analyzing the feasibility of the
device it is possible to estimate three primary noise sources, namely generation-
recombination noise, shot noise, and thermal noise (Johnson noise). In
instrumentation, the noise is usually described with the spectral density function
or the mean square current. The spectral density function can be obtained from the
noise current in the frequency domain averaged by the period, and the mean
square current is integrated over the spectral density function [LUN00],
60
( )
2 2
) ( f i
T
f S = (5.1)
And
( )df f S i
f
T
∫
∆
=
0
2
(5.2)
In the following, let us calculate the mean square current for the three noise
sources.
a ) Generation-recombination noise
Generation-recombination noise is due to fluctuations in the rates at which
charges are generated and recombined [LUN00]. It depends upon the type of
detector (extrinsic, intrinsic, photovoltaic, photoconductive). The noise
current can be evaluated as below,
( )
( )
2
2
2 1
4
τ πf
f egI
f i
i
i
R G
+
∆
=
∑ −
(5.3)
where
transit
life
g
τ
τ
= is the photoconductive gain. Generation-recombination
noise is larger at low frequencies, and with faster time constants. For our
model device, the summation is over 100 energy levels. At each level, the
intrasubband current is calculated individually.
b) Shot noise
The short noise does not arise due to imperfections in the detector or the
electronics but from the detection process. Shot noise is due to the
61
randomness of the emission of the electrons in the detector [LUN00,
YAR85]. It follows Poisson statistics (variance = mean). The variance can
be shown to be:
f qI i
DC
shot ∆ = 2
2
(5.4)
where q is the charge of an electron, i
shot
the photocurrent, and I
DC
the
average photocurrent. Here, the key element is that the noise (standard
deviation) is proportional to the square root of the variance and thus the
square root of the inputs (mean signal). The output or the signal is
proportional to the input. Therefore, the noise does not increase as rapidly as
the signal. This is important later when discussing the ratio of the signal to
the noise. The spectral density and the mean square current for shot noise
can be shown respectively as
( )
p shot
qI f S 2 = (5.5)
and
f qI i
p
shot ∆ = 2
2
(5.6)
where q is the charge of an electron, and I
p
the photocurrent. Note also that
there is no dependency on frequency (i.e. white noise).
c ) Thermal noise
Thermal noise, also called Nyquist or Johnson noise, is due to the random
62
motion of charge carriers in the conductor. The power spectral density and
noise current for thermal noise can be expressed respectively as,
( )
1
4
/
−
=
T k hf
B
e
hf
R
f S (5.7)
and
( )df f S i
f
T
∫
∆
=
0
2
(5.8)
where R is the resistance of the conductor. At low frequencies, the power
spectral density, mean square noise voltage and current can be carried out
respectively as,
( ) TR k f S
B johnson
4 = (5.9)
f TR k V
B
johnson ∆ = 4
2
(5.10)
and
R
f T k
i
B
johnson
∆
=
4
2
(5.11)
Note also that there is no dependency on frequency (i.e. white noise). The
thermal current varies with the temperature of the system and resistance.
63
d ) The total noise
One difficulty for dealing with several noise sources is how to determine the
overall noise in the system. One method is to simply determine the largest
noise source and ignore all others but this is not recommended. Another
method is to assume that all noise sources in the system are independent.
Taking the square root of the sum of the squares,
2 2 2
mod
2 2
_ e temperatur R G ulation shot johnson total noise
i i i i i i + + + + =
−
(5.12)
In the following, the noise spectra is presented for the three types of noises
(generation-recombination, thermal and shot noise) discussed above.
Figure 5.2. Spectra of generation-recombination noise, shot noise and thermal
noise at 77K and 300K.
One notices from Figure 5.2 that for the parameters used in the present example,
the highest noise level is the generation-recombination noise, which is followed
by the shot noise. It is interesting to also note that as expected the thermal noise
64
level is higher for T=300K than for T=77K, and both thermal noise levels are
noticeably lower than the level of the other two noises.
5.3 Figures of merit
In this section, two important figures of merit are discussed, namely the signal-to-
noise ratio (SNR) and the bit error rate (BER). Both quantities are critical for the
device to perform and especially useful for asserting the capability of the device
to operate in the high frequency digital detection regime.
a) Signal-to-noise ratio
The most straightforward figure of merit is the signal-to-noise ratio (SNR) which
is the ratio of the signal to the noise [LUN00, YAR85]. The typical method for
SNR is to take the average output as the signal and the standard deviation of the
measurements as the noise,
2 2 2
2
shot R G thermal
p
power
i i i
i
N
S
+ +
=
−
(5.13)
As an example, for the parameters used in the present design, the SNR spectrums
are calculated in dB for various light intensities. It appears that the SNR can still
be larger than 10dB for frequency near 20GHz, and the higher light intensity the
larger SNR. Therefore, as far as high frequency digital detection is concerned, the
device maintains operation at least for signals with up to 20GHz modulation
frequency.
65
2 4 6 8 10 12 14 16 18 20
10
15
20
25
30
35
40
SNR dB spectrum
Frequency ∆f (GHz)
SNR in dB
I=0.29nW/ µm
2
I=0.39nW/ µm
2
I=0.69nW/ µm
2
Figure 5.3. Spectrums of SNR in dB for various light intensities. The other
parameters are assumed the same as those for Figure 4.1.
b) Bit error rate
The performance of digital receivers is governed by the bit-error-rate (BER),
which is the probability of incorrect identification of a bit by the decision circuit
of the receiver [YAR85]. BER is related to SNR as follows [BMS02],
=
N
S
i
i
erfc BER
2 2 2
1
(5.14)
which assumes that the noise current i
N
is a random Gaussian variable.
In Figure 5.4. some numerical examples for BER are presented. The
calculations used the same parameters previously employed for Figure 4.1. It
shows that the BER maintains lower than 10
-10
for modulation frequency up to
20GHz, and the higher light intensity the lower BER. Therefore, as far as high
frequency digital detection is concerned, the device maintains operation at least
for signals with up to 20GHz modulation frequency.
66
2 4 6 8 10 12 14 16 18 20
-150
-100
-50
0
BER dB spectrum
Frequency ∆f (GHz)
BER in dB
I=0.29nW/ µm
2
I=0.39nW/ µm
2
I=0.69nW/ µm
2
Figure 5.4. Spectrums of BER in dB for various light intensities. The other
parameters are assumed the same as those for Figure 4.1.
From an application point of view, in addition to the noise and figure of
merit, an important issue is to estimate the saturation limit on the carrier speed, or
equivalently the saturation bias voltage, which in turn provides guidance on
optimizing the device size, and predicts the possible high speed and high
temperature performance of the device. In a two-dimensional electron gas, the
mobility μ of electrons is limited at high temperature by phonon scattering and at
low temperature by impurities. For InGaAs, the mobility at room temperature was
reported within the range of Vs cm / 10
2 4
[ BMM07]. On the other hand, the
saturation field can be inferred from the reported experimental results on InP
photonic crystal thin film in Ref. [HAP88], where the saturation voltage about
10V was found over a m µ 40 long photonic crystal of triangular hole lattice with a
filling factor 2 . 0 ~ . This gives a saturation field m mV d V E µ / 250 / = = , and
eventually leads to a saturation velocity s cm E v
s
/ 10 2
7
× = = µ for the
InGaAs/GaAs nanopore superlattice considered in the present work, which is also
67
within the range reported earlier [ZI91]. For a transition time ns 1 < , the size limit
falls in a range m µ 200 < . If a much larger light window needs to be considered
due to diffraction limit, interdigitated contacts would be a convenient and
practical solution [MMS06, GKP91]. Note also that similar saturation was
observed in magneto-transport experiments on 2D lateral superlattices [ASW99,
AGR01].
5.5 Wideband detection
In preceding sections, a model device for terahertz detection based on nanopore
superlattices is proposed and the performance and capabilities are evaluated.
However, due to the complicated intersubband absorption spectrum, for light of
unknown frequency, it is difficult to read out the light intensity with a universal
calibration. Therefore, the detector is not yet suitable for wideband light power
measurements, nor for precise frequency analyses. To mitigate this, a design is
proposed to construct a wideband detector based on an array of gradually rescaled
lattices as shown schematically in Figure 5.5, i.e. all lattices bear the same
configuration (as shown in Figure 5.1), whereas the sizes are gradually rescaled.
Accordingly, the energy is scaled to
2 −
a , and the photocurrent is proportional to
E a
. For example, if the lattice constant a is changed from 80nm to 113nm as
shown in Figure 5.5, the photon energy is reduced by 50% and the bias voltage is
lowered by 1/3. The bias voltage was calibrated separately at the 6 lattices so that
the responses become the same for all parts, which would also be useful for
reducing artifacts due to fabrication errors.
68
Figure 5.5: Schematic of a probe integrated with 6 gradually rescaled chips. The
bias voltage at each chip can be calibrated individually so that the spectral
response to a scanning Gaussian light beam of half width 10meV appears like the
simulated curves in the upper right inset. The photocurrent from each chip is
compared to select the highest, so that the probe spectrum becomes wideband as
indicated in the bottom-right inset.
The response to an incident terahertz beam corresponding to a Gaussian-
distributed (in frequency domain) light beam of various half widths was
simulated. A curve is presented in Figure 5.5, where responses from the 6 lattices
appear the same, but the energy scale is gradually reduced. If the strongest signal
is always selected, one is to obtain the spectrum as shown in Figure 5.5, i.e. a
uniform wideband response. Note that in principle there is no limit to the number
of integrated lattices. Considering the usual e-beam fabrication limit, the idea is
illustrated with 6 lattice periods.
5.6 Tailoring the nanopore superlattice
In the present work, the main goal is to propose a scheme for terahertz detection
based on the subband electronic structures created in GaAs/InGaAs nanopore
69
superlattices in the triangular cell configuration. Throughout the work in the
numerical simulations, the same geometric parameters for the calculations have
been used. Specifically, the lattice constant, i.e. the distance between the centers
of two neighboring cylinders nm a 80 =
,
and the radius of the cylinders nm R 20 = .
A filling factor f
is also defined to connect the two lengths. The filling factor is
the proportionality between the areas of two different materials. In our case, the
filling factor can be calculated 2267 . 0 = f .
Now, the design is finished and the most important conclusion of the work
is reached: detecting in terahertz is feasible with biased nanopore superlattices. In
this section, some numerical results that were obtained with modified geometric
parameters, i.e. tailoring of the nanopore superlattice, are presented. One expects
reasonably that when the size and shape are modified, the subband structures
should be modified too, and consequently the intersubband absorption as well as
the intrasubband current is changed accordingly. A systematic tailoring on the
geometric parameters will be useful for finding the optimized system and guiding
the designs.
There are a number of geometric parameters for the tailoring, and each time
one can change only one parameter and keep the rest unchanged. A convenient
parameter is the filling factor f . If the lattice constant nm a 80 =
is maintained, a
change in the filling factor modifies the radius of the cylinders. For examples,
when 2267 . 0 = f , nm R 20 = , and when 4 . 0 = f , nm R 565 . 26 = . In the
following, two kinds of changes in the filling factor are presented, namely small
70
changes and large changes, so as to see how the outputs are changed and what
conclusions one can reach.
a) Small changes in the filling factor:
Firstly, the filling factor is modified by a small percentage, i.e. % 10 − .
Accordingly, the radius of the cylinders becomes nm R 19 = , i.e. a reduction of
nm 1
in radius. In Figure 5.6(a) and 5.6(b), the subband structures and density of
states spectrum are respectively presented.
Figure 5.6: (a) Energy band structure as a function of the 2D wave number
//
k ,
along the boundary of the irreducible Brillouin zone (see Figure 2.5) between the
high symmetry points for filling factor and cylinder radius from
% 90 2267 . 0 × = f and nm R 19 = . (b) The corresponding density of states,
normalized to the volume and the energy interval used in the calculations, and in
units of ( ) ( )
1 3 − −
meV cm .
One realizes clearly, with a close comparison with the figures in Chapter 2, that
although there are some deviations, the differences are not significant, both for the
band structure and the density of states spectrum in Chapter 2. Therefore, for
small changes in the parameters, the results are not to be modified dramatically,
which is actually a positive finding as it may bring about larger fabrication
tolerances.
71
b) Large changes in the filling factor:
Secondly, the filling factor was changed significantly. Three filling factors were
explored 4 . 0 = f ( nm R 6 . 26 = ), 6 . 0 = f ( nm R 5 . 32 = ), and 8 . 0 = f
( nm R 6 . 37 = ). Note that when 5 . 0 > f the cylinders occupied more area than the
background, and the strength of the confinement to the carriers increases
dramatically. One realizes from Figure 2.4 that the carriers are limited mostly in-
between the cylinders. If the cylinders become bigger, the carriers are further
confined into a quantum well. In other words, the contribution of the lattice
modes becomes less important. Therefore, one expects that the subbands are
separated with bigger gaps, and fine details of the subbands turn into flat lines just
like in a simple quantum well.
72
Figure 5.7: Energy band structure as a function of the 2D wave number
//
k , along
the boundary of the irreducible Brillouin zone (see Figure 2.5) between the high
symmetry points for filling factor and cylinder radius and the corresponding
density of states, normalized to the volume and the energy interval used in the
calculations, and in units of ( ) ( )
1 3 − −
meV cm , for (a)(b) 4 . 0 = f , nm R 6 . 26 = ,
(c)(d) 6 . 0 = f , nm R 5 . 32 = , and (e)(f) 8 . 0 = f , nm R 6 . 37 = .
In Figure 5.7, numerical results for the subband structures and density of
states are presented respectively for 4 . 0 = f ( nm R 6 . 26 = ), 6 . 0 = f
( nm R 5 . 32 = ), and 8 . 0 = f ( nm R 6 . 37 = ). In comparison with the figures in
Figure 2.2, one sees from Figure 5.7(a) that the gaps between subbands are
73
dramatically enlarged, which can also be observed in the density of states
spectrum in Figure 5.7(b). The density of states spectrum becomes simpler and
the gaps are more obvious. When the filling factor is further increased to 0.6, one
sees in Figure 5.7(c) that not only the gaps between the subbands are enlarged, but
also the levels become flatter. The density of states spectrum in Figure 5.7(d)
shows clearly the flattening of the curves as the peaks in density of states
spectrum appear much narrower. Finally, when the filling factor increases to 0.8,
as shown in Figures 5.7(e) and 5.7(f), the local confinement to the carriers
increases dramatically. The band structures become quantum-well like, i.e.
several otherwise separated subbands merge together to form a degenerate flat
line and the density of states spectrum looks composed of fine peaks. This is
expected because the subband structure supported by the lattice network is now
less important.
With the dramatic changes in the band structure and in density of states
spectrum, one expects to have also dramatical changes in the transport properties,
i.e. the bias current. Actually, the intrasubband current should be much smaller,
because the energy level is now flattened. Since the current depends on the group
velocity and the effective mass of the carrier, in general, the flatter the energy
level, the weaker the carrier velocity and the heavier the effective mass. Therefore,
one is to expect the bias current to decrease with the increase of the filling factor.
The calculated I-V curves for the case of 6 . 0 = f ( nm R 5 . 32 = ), and the results
are presented in Figure 5.8 and Figure 5.9 respectively for the bias dependence
and light intensity dependence. In comparison with the curves in Figure 4.1 and
74
4.2, one understands immediately that the current response is much weaker for
large filling factors.
Figure 5.8: Intrasubband current as a function of bias field for various light
intensities. The photon energy is 15meV. The operation temperature is assumed
K T 77 =
.
Figure 5.9: Intrasubband current as a function of light intensity for various bias
fields. The operation temperature is assumed K T 77 =
.
Actually, this work demonstrated an understanding as follows. The reason
for our model device to be able to detect light in terahertz is because the subbands
are supported by the lattice network. If the contribution of the lattice mode is
replaced by quantum-well like band structures, the idea of using biased nanopore
75
superlattice will not work, as the bias current would be too weak to be measured.
5.7 Conclusions of PART I
In conclusion, in PART I of the dissertation, a proposal to realize optical terahertz
detection based on externally biased GaAs/InGaAs nanopore superlattice has been
presented.
In the first stage, GaAs/InGaAs nanopore superlattices in the triangular cell
configuration are analyzed with numerical simulations based on finite difference
method solutions of 3D Schrödinger equation within the position-dependent
effective mass and offset potential approximations. It has been shown that
subbands exist in the conduction band and the interaction between mobile
electrons and lattice modes are mainly responsible to the formation of the
subbands. Once the subband structures are created, the optical intersubband
absorption can be calculated within the dipole-dipole approximation. The
absorption spectrum appears to cover the terahertz frequency range, which lays
down the basic foundation of the present proposal.
In the second stage, the electrical transport along the GaAs/InGaAs
nanopore superlattice has been studied in the case where the superlattice is
externally biased with a voltage. Emphases have been on the nonlinear regime,
that is, where the measured bias current changes with the bias voltage. This
requires a rigorous solution of the Boltzmann equation under the relaxation time
approximation. For this purpose, a rigorous solution based on a Taylor expansion
of the gradients is developed, which enables the transport equation can be solved
76
with accurate numerical simulations. It has been shown that the bias current
responds actively to the optical intersubband absorption, which provides the
second important foundation of the present work. It is worth noting that, as far as
the light induced intersubband transition is concerned, a self-consistent approach
to solve the light induced redistribution of carriers is developed, which allows for
more accurate calculation that reflects the nature of multiple transitions.
In the final stage, a scheme for optical terahertz detection has been proposed
based on the above foundations, namely the subbands supported by the nanopore
superlattice and the response of the intrasubband current to the intersubband
absorption. A conceptual design of the terahertz detector has been presented, and
the device properties including noise levels and figure of merits are estimated. A
scheme for wideband detection is proposed based on an array of gradually
rescaled lattices. The detection becomes uniformly wideband and the accurate
frequency analysis can also be carried out. Finally, a systematic tailoring of the
nanopore superlattice is presented and discussed, which provides a guidance on
the optimization of the device.
77
PART II
PROPAGATION IN SLOT W A VEGUIDE
78
Chapter 6
Introduction
6.1 Abstract of PART II
In the second part of the dissertation, studies on light propagation in slot
waveguides are presented. Firstly, the background, methods and main results of
the studies are introduced in the following sections in the present introduction
chapter.
In Chapter 7, a design of a slot waveguide in which the core layer is
orthogonally slotted to form a rectangular sub-core is presented. The device is
referred as dual slot waveguide as it outputs a two-dimensionally confined light
spot. In this configuration, while the overall guiding and coupling efficiency
remains the same as a conventional slot waveguide, the field confinement is
enhanced and appears two-dimensional. The performance as well as optical
properties of the proposed dual slot waveguide is controllable by selecting the
intermediate index as well as various geometrical parameters.
In Chapter 8, extensive studies on the dual slot waveguide are presented
with the focus on engineering the frequency response of the waveguide for
various applications. By changing different variables, the linear/nonlinear
dispersion and birefringence can be tailored with extended ranges. Zero-
dispersion points, where the dispersion vanishes, are achieved, and constant-
dispersion points, where the dispersion is insensitive to size changes, are also
79
demonstrated.
Finally, also in Chapter 8, two studies on dispersion tailoring of slot
waveguides are introduced, including a design to construct slot waveguides with
high dispersion and another design to realize slot waveguides with near-zero
dispersion.
In the first study, a slot waveguide with high dispersion is proposed based
on cascading a slot waveguide with a strip waveguide. The strong coupling
between the slot waveguide and the strip waveguide enables extended dispersion
range. A negative dispersion as large as -181520 ps/nm/km is feasible thanks to a
strong interaction between the slot and strip modes, and near constant dispersion
at lower degree can be achieved within a relatively wide frequency band.
Specifically, -31300 ps/nm/km dispersion over 147-nm bandwidth with <1%
variance is shown. This design is to be useful for on-chip dispersion
compensation and other signal processing applications.
In the other study, designs of highly nonlinear slot waveguides with constant
and low dispersion over a wide wavelength range are investigated. Silicon nano-
crystal and chalcogenide glass are considered respectively as the slab materials,
which are both of relatively high nonlinear coefficients. The new design enables
the device to behave as a wideband low dispersion waveguide. Specifically, over
a 244-nm bandwidth, dispersion of 0±0.16 ps/(nm•m) is achieved in the silicon
nano-crystal slot waveguide, with a nonlinear coefficient of 2874 /(W•m),
whereas the chalcogenide glass As
2
S
3
slot waveguide exhibits a dispersion of
0±0.17 ps/(nm•m) over a bandwidth of 249 nm, with a nonlinear coefficient 16
80
times larger than that of As
2
S
3
rib waveguides and a nonlinear figure of merit
three times larger than that of silicon strip waveguides.
6.2 Slot waveguide
As already mentioned in the preceding section, work reported in PART II of the
present dissertation is all about slot waveguides. Therefore, in the present section,
the slot waveguide is introduced, and the physics and operation principle of the
waveguide are explained.
In photonics, there has been an increasing demand for highly efficient
compact waveguides that can be integrated on a chip. Among others, slot
waveguides become recently promising for various applications in integrated
photonics. [KLR98, GGT00] A slot waveguide transmits light with a strong field
confinement across the slot. The confinement is based on the large discontinuity
of the electric field at a high-index-contrast interface. On the low-index side of the
interface and within a small fraction of the wavelength, the electric field, mostly
of evanescent components, is enhanced dramatically. In the slot waveguide
configuration in which a low-index slot layer is sandwiched by high-index layers
and the slot layer is sufficiently thin, the enhanced fields at both interfaces meet,
interact, and form a highly enhanced beam within the slot core. This guiding
mechanism is fundamentally different from the widely adapted high-index-core
waveguides based on total internal reflection (TIR), and enables light guiding at
the nanometer scale [AXB04,XAP04]. Because of the tight spacing, slot
waveguides are finding applications in various compact photonic designs [FSK07,
81
ZYX09]. In general, the slot waveguide might be useful for various applications
including on-chip communications, microscopy, spectroscopy, sensing, and data
storage [DP07].
In addition, since slot waveguides transport high field amplitude and optical
power at levels that cannot be achieved with conventional strip waveguides, a slot
waveguide with this unique property allows highly efficient interaction between
fields and active materials, which may lead to all-optical switching, optical
amplification and optical detection on integrated photonics. In general, strong
field confinement enables light to be localized in a nanometric low-index region,
which is useful to greatly increase the sensitivity of compact optical sensing
devices. At terahertz frequencies, a slot waveguide based splitter can be designed
which allows for low loss propagation of terahertz waves. Therefore, a potential
application of slot waveguide is to be used as the entrance of on-chip compact
light detectors, such as the terahertz detector that is proposed in PART I of the
present dissertation. A more attractive design would be a combination of a slot
waveguide with the proposed nanopore superlattice terahertz detector. In more
ambitious designs, the slot waveguide can even be incorporated with photon
detectors.
A schematic of a typical slot waveguide is shown in Figure 6.1. For
comparison, a strip waveguide is also presented in Figure 6.1. Both waveguides
are already assumed to be fabricated on a silica substrate, merely as an example.
One understands from Figure 6.1 that the slot waveguide could be directly
converted from a strip waveguide by cutting it in two parts ( To avoid
82
unnecessary confusion, in what follows let us refer these two parts as slabs ). In
Figure 6.1, the slot region is assumed to be of air, which can be replaced with any
material that has an optical index lower than that of the two slabs. Note that to
explain the principle of operation for the slot waveguide, it is not necessary to
include the substrate. However, in a realistic case, the slot waveguide has to be
fabricated on a substrate.
Figure 6.1: Schematic showing the difference between a strip waveguide and a
slot waveguide, and how slot waveguide can be converted from a strip waveguide.
The substrate material is assumed of silica SiO
2
, and the strip material be of
silicon Si.
Before going further, it is worth noting that a slot waveguide does not have to be
vertically placed as shown in the above figure in Figure 6.1. The waveguide can
actually be placed horizontally. In the present work, slot waveguides with both
vertical and horizontal positions are presented. In this introduction, let us keep the
slot waveguide in vertical position as shown in Figure 6.1. Note also that the
height of the slot waveguide ( or the width of the slot waveguide if the waveguide
is horizontally placed ) is limited as shown in the figures in Figure 6.1. Therefore,
83
one expects that the true slot modes would exist only in the center part of the slot
region, and at the two ends of the slot, there exist more and more leaking and
radiating modes.
In an ideal case, one can however consider the height of the slot waveguide
being sufficiently long so that one can have an analytic formalism for the field
distribution. In the ideal case, one expects that light is to be localized in the slot
between the two slabs, due to the large discontinuity of the electric field at high-
refractive-index-contrast interfaces. In the slot region, the large discontinuity of
the electric field at both sides may result in an enhancement of the field in the
narrow area. In general, evanescent field decays exponentially at the interface and
vanishes within a fraction of the wavelength of the light. However, in the case of
slot waveguide, since the slot region is in nanometers, the evanescent fields from
both sides meet in the center of the slot region before the fields start to decay
exponentially.
Maxwell’s equations state that, to satisfy the continuity of the normal
component of the electric displacement field D
at an interface, the corresponding
electric field must undergo a discontinuity with higher amplitude in the low-
refractive-index side. It is possible to analytically solve the field distribution if
one assumes an infinite height for the waveguide. Let us assume the coordinators
as follows: the cross section is in x direction, the vertical height is in y direction,
and the light is propagating in z direction, as shown in Figure 6.2. The origin of
the coordinates is chosen in the center of the slot region.
84
Figure 6.2: Schematic showing a cross section of a slot waveguide, where the
refractive indexes are respectively marked as in the slot region, in the high-index
slabs region, and in the cladding region. The cross section is assumed in x
direction and the origin of the coordinate is located at the center. The width of the
slot is a 2 , and the width of the waveguide including the two slabs and the slot is
assumed b 2 . The refractive indexes in the three regions are respectively, in the
slot region
S
n , in the strip regions
H
n , and in the cladding region
C
n .
Based on the arrangement in Figure 6.2, one writes immediately the x component
of the electric field in the three regions, namely, the slot region, the high-index
slabs region and the cladding region, as [AXB04]
( )
( )
( ) ( ) [ ] ( ) [ ]
( ) ( ) [ ]
( ) [ ]
( ) [ ]
> − −
− +
−
< < − + −
<
×
−
=
b x b x
a b
n
n
a b a
n
b x a a x
n
a x a
n
a x x
n
k
n k
E x E
C
H
H S
S H
H S
C
H
H S
S
H S
H
S
S
H H
x
γ
κ
κ
γ
κ γ
κ
κ
γ
κ γ
γ
κ
exp
sinh
cos cosh
1
sinh cos cosh
1
cosh
1
2
2
2
2 2
2
0
2 2 2
0
0
(6.2)
85
where
0 0
/ 2 λ π = k is the wave number in vacuum, and
H
κ ,
S
γ and
C
γ are
respectively the wave number in the high-index slabs and the decay coefficient in
the slot and cladding regions. Note that the use of decay coefficients in the slot
and cladding regions indicates actually imaginary wave numbers and the field
becomes evanescent in these regions.
The boundary conditions should satisfy the continuity of the normal
component of the electric displacement field D
at all interfaces, which leads to
the eigenmode propagation constant of the waveguide β . This is actually the
common wave number component in the propagating direction, i.e. the z
direction in Figure 6.2, as
+
+
−
=
2 2 2
0
2 2 2
0
2 2 2
0
2
S S
C C
H H
n k
n k
n k
κ
γ
κ
β
(6.3)
The values of the various wave numbers are to be solved implicitly from the
transcendental characteristic equation:
( ) [ ] ( ) a
n
n
a b
S
S H
H S
H
γ
κ
γ
ϕ κ tanh tan
2
2
= − − (6.4)
with the phase factor defined as
=
−
2
2
1
tan
C H
H C
n
n
κ
γ
ϕ (6.5)
86
One realizes that the highest field amplitude occurs at the internal sides of
the interfaces between the slabs and the slot region a x = , which is
2 2
/
S H
n n times
enhanced with respect to the field amplitude in the external side of the interface.
For the specific materials indicated in Figure 6.1, i.e. silicon slabs and air slot, one
estimates the enhancement is 12 times ( 48 . 3 =
Si
n and 1 =
air
n ). If the slot is filled
with silica ( 44 . 1
2
=
SiO
n ) the enhancement becomes 6 times. Since the slot region
is narrow, the enhanced field at both sides is added in the center of the slot region
before it decays exponentially.
A typical field scan along the cross section of a slot waveguide is shown in
Figure 6.3 below. The calculation uses the analytic formalism presented in
Equations (6.1) to (6.5). The field distribution can be solved analytically because
the slot waveguide is assumed to have an infinite height (The vertical direction
along the slot as indicated in Figure 6.2).
Figure 6.3: Normalized field profile along a cross section of a slot waveguide.
The width of the slot is a 2 , and the width of the waveguide including the two
stabs and the slot is assumed b 2 .
In a realistic case where the length of the slabs along the slot is finite, one has to
87
employ a numerical method to solve the field distribution. In the present work,
COMSOL, a 3D solver for differential equations has been employed for the
computation task. In Figure 6.4 below, an example of the field distribution is
presented. To simplify the system, the substrate is removed and the material of
low refractive index in the slot is chosen the same as the cladding material and the
corresponding refractive index is marked as
L
n .
One can see from Figure 6.4 that along the vertical direction, the highest
field is at the center of the slot, and the field is reduced at the edges. Since the slot
height is finite, there is considerable field leakage at the two sides of the slot. In
the next Chapter, more field distributions will be presented in 3D format based on
numerical simulations with COMSOL.
Figure 6.4: Normalized field distribution along a cross section of a slot
waveguide. The height of the slot waveguide is finite ( The vertical direction as
indicated in the figure ).
It is worth noting that since the wave propagation is due to total internal
reflection in the separated two slabs, there is no interference effect involved and
the slot-structure exhibits very low wavelength sensitivity [AXB04].
88
So far, I have introduced the slot waveguide, and explained in details how a
slot waveguide works and what the physics behind the operation of the waveguide
is. In the next sections, work on the slot waveguide is described. The main
contribution in this document is a proposal for a dual slot waveguide that, instead
of one-dimensional field confinement, confines light in two dimensions. The other
studies involve mostly designs of slot waveguides for specific dispersion profiles.
In the first study, the dispersion of the proposed dual slot waveguide is tailored
and discussed. In the second study, a slot waveguide with strong dispersion is
proposed based on cascading the slot waveguide with a strip waveguide. The third
study is a design with geometric parameters and different slot materials to achieve
highly nonlinear slot waveguides with wideband low dispersion waveguiding
capability.
6.3 Dual slot waveguide
In this section, a novel design of slot waveguide is introduced [XO12,XO14],
namely the dual slot waveguide. The design is based on numerical simulations
with COMSOL. The purpose of the study is to enable a slot waveguide to
transport two-dimensionally confined light beams, instead of one-dimensionally
confined beams that are realizable with conventional slot waveguides.
So far, as shown in the previous section, slot waveguides transmit only one-
dimensional sheet-like beams and the transmission is usually polarization
dependent. On the other hand, in many applications two-dimensionally confined
spot-like light beams are desired. In the literature, in order to deliver
89
subwavelength light spots, both ridged apertures and tapered slots were proposed
[TMS06,TIS08]. Note also that enhanced subwavelength beaming was achieved
with metallic slits and apertures, and the transmission was mediated by the
propagation of surface plasmons under cavity enhancement [YCB08, FK06].
In the present work, a design in which a slot waveguide can create and guide
two-dimensionally confined beams is described. In this new design, the main
configuration remains a conventional slot waveguide. Therefore, one expects that
the overall guiding and coupling efficiency (both for the input and output) would
be retained. The modification introduced in the new design is such that an
additional confining core region is incorporated in the slot. The sub-core region
can be filled with materials (or simply air) that have a refractive index lower than
the core layer of the slot waveguide. In other words, the core of the slot
waveguide is again slotted in the orthogonal direction. Since the waveguide has a
two-dimensional geometry, there exist quasi-TM modes. Due to the fact that the
sub-core is orthogonally slotted twice, there are discontinuities of the field at all
boundaries. Considering the geometry, one would expect the transmitted light
beam to be highly confined in the direction across the slot, as well as in the
direction along the slot with a reduced degree of confinement. Based on
numerical simulations, it has been found that the waveguide can guide and deliver
two-dimensionally and nanometrically confined light beams. The property of the
waveguide depends on the intermediate slot index as well as various geometrical
parameters.
90
6.4 Dispersion tailoring of slot waveguides
Apart from the capability to guide light two-dimensionally, the proposed dual slot
waveguide has some other distinguishable photonic properties. As already
mentioned, since the field in the slot region is highly enhanced, some otherwise
intrinsic properties of the material, such as chromatic dispersion, birefringence,
and nonlinearity, would be significant. Therefore, comparing to the conventional
slot waveguide, the proposed dual slot waveguide should be more capable to
achieve desired dispersion, birefringence and nonlinearity profiles. In other
words, one has larger room for tailoring. It is shown in the present work that by
changing different variables, optical properties, such as linear/nonlinear
dispersion and birefringence, of the proposed dual slot waveguide can be tailored
with increased ranges compared to conventional slot waveguides. This is due to
the increased confinement in the dual slot waveguide. Specifically, constant
dispersion points where the dispersion is insensitive to size changes are
demonstrated. At the constant dispersion points, one expects that as far as the
dispersion is concerned the fabrication tolerance on the geometric parameters
would be dramatically increased.
In addition to tailoring the proposed dual slot waveguide, progress was also
made on designs of slot waveguides with specific properties, such as dispersion,
birefringence and nonlinearity etc. The studies were motivated by some on-going
research activities on slot waveguides. Recently, slot waveguides have been
proposed [AXB04] for on-chip applications, exhibiting enhanced birefringence
[YCB08], nonlinearity [FK06] as well as improved modulation efficiency
91
[BHW05]. A slotted structure could provide some design freedom to tailor the
waveguide dispersion [ZIL08]. A potential application that makes use of the slot
waveguides would be to form on-chip highly dispersive elements with a large
dispersion-bandwidth product.
One of the studies is to design dispersive slot waveguides that are able to
produce extremely high negative dispersion [ZYX09]. The design is based on a
special configuration where a slot waveguide is tightly combined with a strip
waveguide. It is shown that the dispersion is dramatically increased due to the
strong coupling between the slot waveguide and the strip waveguide. Highly
dispersive waveguides are useful in photonics, especially for dispersion
compensation in optical communication. There is a potential advantage to achieve
various on-chip optical functions. One of the basic issues in optical fiber
communications is combating chromatic dispersion. A solution to reduce the
chromatic dispersion is to insert a dispersion compensating device. A dispersion
compensating device bears a dispersion that is opposite to the chromatic
dispersion that one intends to correct. A common method for dispersion
compensation is to make use of dispersion compensating fibers. However, since
the dispersion compensating fibers bear a small dispersion, in order to compensate
a large dispersion, it is often needed to use a very long dispersion compensating
fiber. On the other hand, since the fiber is long, it produces a large bending radius
which also distorts the signals. An alternative way for dispersion compensation is
to use fiber Bragg gratings. Although fiber Bragg gratings are much smaller, the
total dispersion is somewhat traded off with bandwidth [KLR1998, GGT00].
92
Therefore, there is a need to reduce the dispersion by connecting the devices to an
on-chip waveguide that bears large opposite dispersion so that the dispersion is
canceled within a short distance. On the other hand, for high-speed signals, the
deployed dispersion compensating fibers may not provide accurate compensation,
which would also raise the need for integrated receiver-based compensation for
the residual dispersion that may be performed on a per-channel basis. A laudable
goal would be to design on-chip dispersion compensators that are capable of
producing a large and tailorable dispersion [TTS08]. Moreover, the dispersive
elements could also be essential in various signal processing functions, such as
analog-to-digital conversion [BCB1998], tunable delays [WYY05, SOH05] and
optical correlation [GSF08]. Reports of integrated dispersion elements include: (i)
ring resonators that have bandwidths on the order of tens of GHz [MLB1999,
ZSY08], (ii) wideband silicon waveguides that have dispersions of ~4000
ps/nm/km [DXS06,TMS06], and (iii) sidewall-modulated Bragg gratings that
exhibit a large dispersion and good bandwidth in reflection [TIS08].
In the present work, a dispersive slot waveguide is proposed based on a
strongly wavelength dependent coupling between a slot waveguide and a strip
waveguide. On a silicon-on-insulator platform, the two waveguides are vertically
coupled, producing a strong negative dispersion due to an anticrossing of the two
modes [PPL1995, SLJ07]. The waveguide structure as well as the numerical
model for the proposed dispersive slot waveguide are to be presented in Chapter 8
and the operation of the waveguide is to be discussed in detail there.
93
The proposed dispersive slot waveguide has been studied with numerical
simulations with COMSOL. The simulations show that a negative dispersion of
up to -181520 ps/nm/km is obtained. Therefore, the slot waveguide is useful for
on-chip dispersion compensation. It is demonstrated that dispersion compensation
for a 320 Gb/s return-to-zero signal has an eye opening penalty of ~4 dB after
10.87-km single-mode fiber transmission. On the other hand, the dispersion
bandwidth is greatly increased to 147 nm for a dispersion of -31300 ps/nm/km,
with a <1% variance, which corresponds to a 6.3-ns tunable delay given by a 1-m-
long slot waveguide.
Another study is to investigate the possibility of achieving low (near-zero)
dispersion slot waveguide with nonlinear slot waveguides. By using silicon nano-
crystal and chalcogenide glass for the slots, low dispersion in slot waveguides was
achieved, and the low dispersion persisted over a wide frequency range.
Indeed, wideband flat and low dispersion is desired in various applications
in photonics. Integrated highly nonlinear waveguides and photonic nano-wires
can form the backbone of on-chip optical signal processing [KJP07, LPA07],
especially for high bit-rate signals [GXM09]. Such waveguides can be composed
of silicon [KJP07, LPA07], silicon nitride [ISA08, LGT09], Si nano-crystals (Si-
nc or Si-rich Si dioxide) [SDC09, YAD09], III-V compound semiconductors
[WYN1969, LC1991], and chalcogenide glasses [GXM09, TBF07]. Flat and low
chromatic dispersion over a wide wavelength range is crucial for enhancing
nonlinear interactions of optical waves. Recently, there has been much interest in
tailoring dispersion in the waveguides [DXS06, YLA06]. A tight confinement of
94
light with high index contrast may introduce strong and fast-changing waveguide
dispersion, making it difficult to obtain a wideband and flat dispersion profile. A
slotted waveguide structure [LSE07] could provide extra design freedom to tailor
the waveguide dispersion [AXB04, JTH07], while keeping a large fraction of the
guided mode in a thin slot layer. However, there has been little reported in the
literature on the wideband low dispersion in these waveguides.
Table 6.1. Dispersion and nonlinearity comparison between the published and
proposed waveguides.
In the present work, highly nonlinear slot waveguides for achieving a flat
and near-zero dispersion profile is described. Two types of nonlinear materials are
considered:
(i) Chalcogenide glass (As
2
S
3
), which reduces the overall two-photon-
absorption (TPA) coefficient and
(ii) Silicon nano-crystal (Si-nc), which greatly increases the nonlinear
Kerr coefficient compared to silicon strip waveguides.
The waveguide structure and the numerical model of the proposed highly
nonlinear slot waveguide are to be presented in Chapter 10 and the principle of
operation of the waveguide is discussed in detail there. Basically, the slot
waveguide is horizontally placed on a silica substrate. The slots are assumed to
95
be made of either silicon nano-crystal or chalcogenide glass. The slabs are silicon
layers.
Based on numerical simulations with COMSOL, the chalcogenide glass
As
2
S
3
slot waveguide exhibits a flat dispersion of 0±0.17 ps/(nm•m) over a
bandwidth of 249 nm, with a nonlinear coefficient of 160 /(W•m) and a
significantly improved nonlinear figure of merit (FOM). The silicon nano-crystal
slot waveguide features a flat dispersion of 0±0.16 ps/(nm•m) over a bandwidth of
244 nm, with a high nonlinear coefficient of 2874 /(W•m) at 1550 nm. The
proposed waveguides are compared with previously published results in Table
6.1, in terms of dispersion and nonlinearity properties. Accumulated dispersion in
the waveguides is low, since, for most on-chip applications, the waveguides are
typically a few centimeters in length.
96
Chapter 7
Dual slot waveguide
7.1 Model and simulation
In this section, the model for the proposed dual slot waveguide is presented. The
new design in a finite element solver (COMSOL) program is simulated. In the
simulations, all the field components, i.e. E
x
, E
y
, and E
z
are taken into account.
Since E
z
is insignificant and E
y
is considerably larger than E
x
, the waveguiding
modes involved in the simulations are of quasi-TM modes. For comparison,
numerical simulations for the original slot waveguide before the sub-core slot was
incorporated were also carried out. The basic system is schematically shown in
Figure 7.1(a) where a silicon-silica-silicon slot waveguide is embedded on a silica
substrate. The light wavelength is assumed to be 1550nm and the geometrical
parameters were chosen, i.e. the width of the waveguide 375nm, the thickness of
the slot layer 21nm, the thickness of both upper and lower silicon layers 135nm.
Constant refractive indices were selected as n
silicon
= 3.476 and n
silica
= 1.444. In
parallel, the model incorporated the sub-core in the original slot waveguide. The
modified system is schematically shown in Figure 7.1(b) where a sub-core slot of
air (n
air
= 1) is incorporated in the center of the basic system with the same
geometric parameters and refractive indices as those in Figure 7.1(a), except that
the slot material is replaced by silicon nitride and the corresponding refractive
97
index is selected as n
silicon nitride
= 2.1. The reason for choosing silicon nitride for
the slot layer is that it has been known that the refractive indices of silicon nitride
films are controllable within a wide range (1.9~2.8) by varying deposition
parameters [NAY02]. Note that the above sizes are chosen at the nanometer scale
and other sizes were also used in the present work. The influence of the size
changes on the waveguiding is to be presented in the next sections. The sub-core
was assumed to be air for simplicity in simulations, and the air void could be
formed by chemical etching. It will be shown that by tuning the slot index, one
can optimize the field confinement. In the modified system, the width of the sub-
core slot is assumed to be 50nm. Therefore, the sub-core forms a rectangular slot
along the waveguide. Since the refractive indices are chosen such that n
cladding
>
n
core
> n
sub-core
, one has actually two orthogonally placed slot waveguides.
Mediated by the tight coupling between the two slot states [RHH07],
nanometrically and two-dimensionally confined light beams are guided within the
sub-core slot. In the following, the numerical results both for the original and the
modified slot waveguides are presented. In the rest of the chapter, these two
waveguides are referred to as the slot waveguide and the dual slot waveguide,
respectively.
98
Figure 7.1. Schematics of the slot waveguide (a) and the dual slot waveguide (b).
In the text, the geometric terms, width and thickness, are respectively referred to
the lateral and vertical lengths of the layers described in these schematics.
The field intensity distribution is shown in different formats in Figure 7.2,
both for the slot waveguide [Figure 7.2(a) and 7.2(c) in the left column] and the
dual slot waveguide [Figures 7.2(b) and 7.2(d) in the right column]. In Figure 7.3,
cross-sectional curves from the intensity distributions in Figure 7.2 are plotted
along two directions. For the slot waveguide, Figures 7.2 and 7.3 show that the
guided light is mostly confined in the slot layer, and the confinement is only in the
direction across the slot. Since the waveguide has a finite width, the beam is,
however, curved in the direction along the slot. From the field intensity
distribution for the dual slot waveguide in Figures 7.2 and 7.3, it is apparent that a
two-dimensionally confined light beam is transmitted in the sub-core slot. In other
words, in comparison to the slot waveguide, the dual slot waveguide supports an
additional field confinement in the orthogonal direction. Note that the point-like
beam in the sub-core region is on top of the sheet-like beam in the original slot.
Hence, the overall guiding and coupling efficiency of the waveguide remains the
same as that of the original slot waveguide. What happens is that the field in the
original slot is redistributed with a concentration in the sub-core region. The
99
degree of the concentration would depend on the selection of the refractive
indices of the materials and the width of the sub-core slot. Below, the influences
of changing the refractive index of the slot and varying the sub-core width is
analyzed.
Figure 7.2. Intensity distribution of the slot waveguide (a) and the dual slot
waveguide (b); Intensity distribution in surf format of the slot waveguide (c) and
the dual slot waveguide (d). The sizes are chosen waveguide width 375nm, slot
thickness 21nm, and sub-core width 50nm.
100
Figure 7.3. Cross-sectional curve of the intensity distribution in Fig. 7.2 along the
center axis of the slot of the slot waveguide (a) and the dual slot waveguide (b);
Cross-sectional curve of the intensity distribution in Figure 7.2 across the slot of
the slot waveguide (c) and the dual slot waveguide (d).
7.2 Index and size dependence
Starting from this section, the numerical results from tailoring the dual slot
waveguide are presented. Recall that the degree of the field confinement in the
slot waveguides depends on the index contrast. For silicon based devices such as
the system studied in the present work, the index contrast between the silicon
layers and the sub-core is fixed. What can be selected is the refractive index of the
original slot layer. Note that, by changing the slot index, one can actually adjust
the competing contributions between the original and the orthogonal
confinements. The dependence of the confined beam on the refractive index in the
slot has been investigated. As a function of the slot index, the average peak
intensity, the normalized peak height, and the full width half maximum (FWHM)
101
of the peak are plotted in Figures 7.4(a) and 7.4(b). Here, the average peak
intensity is defined as the intensity average within the sub-core. The peak height
is referred as the extra-peak above the peak formed without the sub-core and is
normalized to that peak as well. From the curve in Figure 7.4(a), it can be seen
that the peak intensity reaches its maximum when the slot index is about two. The
strongest peak occurs when contributions from the original and orthogonal
confinements are balanced, which is realized by selecting an appropriate
intermediate material for the slot layer. For the specific example in the present
work, the optimized slot index is found to be n
slot
~2.1, which is in the range of the
possible index of silicon nitride films. Further increasing the slot index causes a
significant reduction of peak intensity. On the other hand, as shown in Figure
7.4(a), peak height increases with the slot index. This implies that, with an
increase in the slot index, the contribution from the original slot decreases. Figure
7.4(b) shows the dependence of FWHM on the slot index. In comparison with the
slot waveguide (dot curve), the FWHM decreases with the slot index (square
curve) for the dual slot waveguide since light is confined in both directions.
102
Figure 7.4. (a) Average peak intensity and peak height normalized to the peak of
the slot waveguide as a function of the slot refractive index; (b) FWHM of the slot
waveguide (dot curve) and FWHM of the dual slot waveguide (square curve) as a
function of the slot refractive index. The sizes chosen are waveguide width
375nm, slot thickness 21nm, and sub-core width 50nm.
Next, the dependence of the confinement on the width of the sub-core is
analyzed and the numerical results are presented in Figure 7.5. In Figure 7.5(a),
the average intensity in the sub-core normalized to the average intensity in the slot
region is plotted as a function of the sub-core width, whereas, in Figure 7.5(b), the
power ratio between light in the sub-core and light in the whole slot region is
presented. One can see that for the specific waveguide the intensity peaks at about
50~70nm, which shows that selection of geometrical parameters would enable the
transmission and confinement to be optimized. In general, when the width of the
sub-core slot decreases from the optimized peak value, the field intensity
increases but there is less light to pass through as the cross-section area becomes
smaller. On the other hand, when the width of the sub-core slot increases,
although more light passes through, the confinement actually decreases as the
peak intensity is reduced. Note that the power ratio curve in Figure 7.5(b)
provides an important measurement of the additional field confinement stemming
103
from the secondary slot. The curve in Figure 7.5(b) reveals that the modified slot
waveguide provides an additional confinement of a similar or higher degree. One
finds in Figure 7.5(b) that a power ratio of approximately 15% is achievable for a
sub-core that is 50nm in width. This is about the position for the peak intensity to
occur. [See the curve in Figure 7.5(a)]. Therefore, further increasing the width of
the sub-core will bring more light in the sub-core region, but the achievable peak
intensity is reduced.
Figure 7.5. (a) Average intensity in the sub-core normalized to average intensity
in the slot as a function of sub-core width for fixed sub-core thickness 21nm; (b)
Power ratio between light in the sub-core and light in the slot.
104
Chapter 8
Engineering a slot waveguide
8.1 Properties of dual slot waveguides
Since the confinement can be realized two-dimensionally, the proposed dual slot
waveguide would find applications in compact all-optical photonic devices. For
example, since the field confinement is enhanced, various effects, such as
linear/nonlinear dispersion and birefringence, are expected to increase
significantly. In order to demonstrate the usefulness of the proposed dual slot
waveguide, various effects have been investigated, namely, (1) the group velocity
dispersion (GVD) stemming from the combined influence of the material
dispersion and the waveguide modal property, (2) the nonlinear coefficient γ(λ)
dispersion, and (3) the birefringence dispersion. In the calculations, standard
definitions for the three dispersive functions are used [YLA06, TMS06]. In the
simulations, the material dispersions n(λ) are introduced by Sellmeier empirical
equations with properly selected fitting parameters for silicon, silica, and silicon
nitride [PA1998].
In Figure 8.1, for different sub-core sizes, numerical results are presented
for these aforementioned dispersions, namely, the group velocity dispersion, the
nonlinear coefficient dispersion, and the birefringence dispersion, respectively.
For comparison, the results for the slot waveguide are also included.
105
Figure 8.1: (a) Group velocity dispersion; (b) Nonlinear coefficient dispersion; (c)
Birefringence dispersion.
In Figure 8.1(a), the dispersion of the group velocity appears significantly
enhanced when the sub-core width increases. Without the sub-core, the dispersion
curve crosses the zero dispersion line into the positive dispersion zone and
contains two points of zero-group velocity-dispersion (ZGVD). A close look at
the dispersion curves in Figure 8.1(a) reveals two special points, namely, the zero-
dispersion point and the constant-dispersion point. At the zero-dispersion point,
the dispersion vanishes and the waveguide does not cause any additional
dispersion to the guided beams. It is well-known [YLA06, TMS06] that the zero-
dispersion point shifts according to the effective area of the waveguide. Once the
spectrum position is fixed, according to the curves shown in Figure 8.1(a), one
only needs to adjust the sub-core to make sure that there would be merely one
zero-dispersion point. In the case of Figure 8.1, the single zero-dispersion curve is
106
achieved when the sub-core width is 20nm. The usefulness of the single zero-
dispersion curve was discussed in Ref [TMS06].
Further increasing the sub-core size pushes the curve down dramatically.
For the 150nm dual slot waveguide, the minimum dispersion point appears at
about 3000ps/nm/km lower than the zero-dispersion line, which is considerably
more significant than what was achieved with silicon-in-insulator waveguide
[TMS06]. In the case of dual slot waveguide, in order to achieve this range of
dispersion tailoring, it is sufficient to modify only the sub-core size. On the other
hand, as shown in Figure 8.1(b), the nonlinear coefficient dispersion reduces and
becomes flatter when the sub-core width increases. For the 150nm dual slot
waveguide, the nonlinear coefficient dispersion reduces from 200 W
−1
m
−1
to 80
W
−1
m
−1
at the communication wavelength λ = 1550nm. The reduction and
flattening should be helpful for some nonlinear optical processes. Meanwhile, as
shown in Figure 8.1(c), the birefringence increases especially for the shorter
wavelengths. In particular, the enhanced birefringence reaches maximum (up to
0.4 for the 50nm curve) at about 1550nm in wavelength, which shows a great
possibility to make use of the birefringence in photonic applications.
In Figure 8.2, more extensive dispersion curves are presented. Curves in
Figure 8.2(a) are partially extracted from Figure 8.1(a), but the spectral curves are
extended up to λ = 1700nm. Dispersion curves in Figure 8.2(b) are obtained with
an increased slot thickness 50nm, whereas curves in Figure 8.2(c) are calculated
with a fixed sub-core width 60nm while varying slot thickness. By changing the
geometric parameters, significantly shifted spectra were obtained, with respect to
107
the curves in Figure 8.2(a), both toward shorter wavelengths (blue shift) and
larger wavelengths (red shift). The data in Figure 8.2(b) show a blue shift of the
minimum dispersion point at about λ = 1350nm, whereas a red shift of the
minimum dispersion point appears in Figure 8.2(c) at λ = 1500nm. The curves in
Figure 7.7 demonstrate that, in general, it is sufficient to tailor only one geometric
parameter for achieving relatively large spectral shifts.
Figure 8.2: Dispersions of dual slot waveguides with (a) fixed slot thickness
21nm, (b) fixed slot thickness 50nm, and (c) fixed sub-core width 60nm.
An important observation one has from the curves in Figure 8.2 is the
constant-dispersion point (marked with circles) where different dispersion curves
for various sizes meet each other at a common point. This implies that in these
zones the dispersion is insensitive to the change of slot thickness or sub-core
width, and consequently the tolerance on the fabrication variations and defects is
increased. These special points have been investigated and it was found that the
constant dispersion point can be established and shifted according to the
108
geometric parameters. In Figure 8.2, three cases are presented. In Figure 8.2(a)
the constant-dispersion point occurs at about the communication frequency λ =
1650nm, whereas in Figure 8.2(b) and Figure 8.2(c) the constant dispersion point
occurs respectively at λ = 1520nm and λ = 1750nm. In all the three cases, the
dispersion at the constant-dispersion point becomes insensitive to the change of
either sub-core thickness or sub-core width.
8.2 Properties of slot waveguides
Apart from the proposed dual slot waveguide, various slot waveguides have been
investigated. In the following, these studies are presented.
a ) Highly dispersive slot waveguide
The proposed structure consists of a strip waveguide and a slot waveguide that are
vertically coupled to each other. The effective index of a quasi-TM mode
(vertically polarized) in the strip waveguide decreases with wavelength faster than
that in the slot waveguide, and a strong mode coupling occurs around a certain
wavelength where the effective indices are close to each other, as conceptually
shown in Figure 8.3. At the crossing-point, composite modes, including
symmetric and anti-symmetric modes, experience a sharp transition of mode
shape from short to long wavelength in Figure 8.3, which induces a high
dispersion [PPL1995, SLJ07]. The strip waveguide with a thickness of 255 nm
and width of 500 nm is placed on the top of the slot waveguide, separated by a
500-nm-thick silica base layer. The low-index slot is a 40-nm silica layer,
109
surrounded by two 160-nm silicon layers. Such a vertical placement allows for
more accurate control of the structural parameters during fabrication and reduced
propagation loss to several dB/cm [SDF07]. Since the symmetric mode is used, a
mode converter design was implemented to excite it [PPL1995]. At the crossing
wavelength 1.489-μm, the index difference of the symmetric and anti-symmetric
modes equals 0.019, which produces the coupling length from a slot mode to a
strip mode for 100% coupling of 39.2 μm. To excite the symmetric mode, ~50%
coupling is needed, and mode converter length is ~20 microns. A mode solver
based on finite element algorithm was used to obtain effective index with an
element size equal to 3, 5, and 15 nm for slot, silicon and other parts. Material
dispersion is considered using Sellmeier equations for both silicon and silica, and
dispersion profiles are calculated from the effective index [ZIL08].
Figure 8.3: Slot and strip modes strongly interact with each other due to index-
matching at the crossing point, producing a sharp index change of symmetric and
anti-symmetric modes. Modal power distributions of the symmetric mode at
different wavelengths (Media 1).
110
Figure 8.4: (a) Dispersion profiles of the symmetric and anti-symmetric modes,
and a negative dispersion of -181520 ps/nm/km can be obtained from the
symmetric mode. (b) The dispersion profile red-shifts with a small peak value as
the slot thickness increases.
At the crossing wavelength 1.489-μm shown in Figure 8.4(a), the symmetric
mode has -181520 ps/nm/km dispersion. It is obtained over a relatively small
wavelength range of 3.5 nm for a 1% dispersion variation. The anti-symmetric
mode has almost the same amount of positive dispersion. Keeping all other
parameters the same, it is shown in Figure 8.4(b) that an increment in slot
thickness (ST) shifts the dispersion profile towards a longer wavelength, and the
symmetric mode becomes less dispersive. Figure 8.5(a) shows that dispersion
peak wavelength almost linearly shifts by 77 nm, as the slot increases from 40 to
60 nm. Accordingly, the dispersion value decreases by 38%. This can be
attributed to the fact that the increased slot thickness lowers the effective index of
the slot mode and red-shifts the crossing point, where the indices of the two
guided modes have closer slopes over wavelength. This causes a less dispersive
symmetric mode. The silicon layers surrounding the slot also modify the
dispersion properties of the proposed structure. In Figure 8.5(b), an increment
(from 150 to 170 nm, with a 56-nm slot) in the thickness of the silicon layers
111
blue-shifts the dispersion by 103 nm and makes dispersion more negative, from –
79683 to –182800 ps/nm/km. It is important to mention that the dispersion profile
becomes wider, while the symmetric mode is less dispersive.
Although changing the slot thickness or silicon-layer thickness can slightly
modify the peak value of dispersion, it might be desirable to dramatically change
dispersion for various applications. The thickness of the silica base between the
strip and slot waveguides plays a critical role in tuning the dispersion while
almost keeping the dispersion peak wavelength. In Figure 8.6(a), the dispersion
decreases greatly from -181520 to -28473 ps/nm/km, as the base thickness varies
from 500 to 200 nm. To keep the same peak wavelength, one has to change the
thickness of the strip waveguide to 255, 255.5, 257 and 259 nm for the silica base
of 500, 400, 300 and 200 nm, respectively. As the base decreases, effective
indices of the slot and strip waveguides increase, causing a small shift of the
crossing wavelength. The strip thickness is thus changed to balance this effect. A
trade-off occurs [PPL1995] between peak dispersion and dispersion’s full width at
half maximum (FWHM), from Figure 8.6(b). The dispersion’s FWHM drops
from 177 to 26 nm as the waveguide is more dispersive. A thicker base helps
separate two modes well, and thus the strong interaction between them occurs at a
smaller wavelength band where their effective indices are very close to each
other. The symmetric mode thus experiences a sharper transition from the strip
mode to slot mode and higher dispersion.
112
Figure 8.5: Dispersion value and peak wavelength are examined as functions of
the slot thickness (a) and the silicon-layer thickness (b), respectively.
Figure 8.6: (a) The dispersion profile is fixed at the same wavelength as the
dispersion’s peak value is changed from -181520 to -28473 ps/nm/km by varying
the silica base thickness. (b) A trade-off is found between the dispersion peak
value and dispersion bandwidth.
It may be desirable to design a dispersion-shifted profile without changing
its peak value and bandwidth. One can achieve this by thickening both the slot
and the silica base. A thickened slot makes the dispersion profile red-shifted but
reduces its peak value as well, while a larger silica base makes the dispersion
deeper and mostly keeps its peak wavelength. In Figure 8.7(a), the dispersion
profile is shifted relative to the original 40-nm slot by 37 and 77 nm, with slot
thickness changed to 48 and 60 nm, respectively. The base thicknesses are 500,
113
555 and 606 nm, and the maximum dispersion values are -181520, -182585, and -
182028 ps/nm/km, respectively. Figure 8.7(b) shows that shrinking the waveguide
width from 500 to 200 nm can also vary the dispersion, but it is less sensitive to
the width change as compared to changing the vertical dimensions, since
horizontal light confinement for the y-polarization state is not as tight as in the
vertical direction for the 500-nm width. The dispersion compensation of high-
speed return-to-zero on-off-keying signals is simulated after 10.87-km single-
mode fiber transmission, using a 1-m-long waveguide. Data rate ranges from 160
to 400 Gb/s. The waveguide width is 500 nm, and the strip waveguide thickness is
255 nm. The slot is 48-nm thick, surrounded by two 160-nm silicon layers. The
silica base layer is 555 nm. Signal carrier is aligned to the peak dispersion
wavelength, where the single mode fiber has a dispersion of 14.4 ps/nm/km. In
Figure 8.8, the dispersion compensator works for the 160 Gb/s signal with almost
no penalty. However, with the increased bit rate, the signals suffer from the third-
order dispersion in the fibers, which would not be compensated by the waveguide
in this case. An eye-opening penalty of 4 dB is induced to a 320 Gb/s return-to-
zero signal, relative to the case at 160 Gb/s.
114
Figure 8.7: (a) Dispersion shifts with different slot thicknesses, exhibiting almost
unchanged dispersion value and bandwidth. (b) Dispersion properties change with
the waveguide width.
Figure 8.8: Dispersion compensation for very high-speed signals transmitted over
11.4-km single mode fiber. Eye-opening penalty increases with bit rate. Eye-
diagrams are in the same scale.
b ) Dispersion-flattening slot waveguide
Broadband and strong dispersion could be useful in both telecom systems and
optical signal processing, e.g., for achieving multi-channel dispersion
compensation or a tunable optical delay. Dispersion can be flattened by a cascade
of the waveguide sections with the modified structural parameter, each with a
shifted dispersion profile. As an example, Figure 8.7(a) shows that the strip
115
waveguide has a slightly tailored waveguide thickness (WT) by depositing
silicon, and the dispersion curve shifts over wavelength, as shown in Figure
8.9(b). A similar trend can be seen in Figure 8.4(b) and Figure 8.7(b) as other
structural parameters are changed. The length of each modified waveguide section
is calculated by solving the following linear equations:
(8.1)
where D
i
(λ) (i = 1, 2, …, n) is the dispersion profile of the ith waveguide section;
D
0
is the desired dispersion value after flattening. Length coefficients c
1
, c
2
, … ,
and c
n
are solved to determine the length ratio of each modified section to the
total waveguide. This forms a dispersion profile with n dispersion values clamped
to D
0
at wavelengths λ
1
, λ
2
, … , and λ
n
.
The waveguide width is 500 nm. The slot is 40-nm thick and surrounded by
two 150-nmthick silicon layers. The silica base is 500 nm. There are six
waveguide sections. Figure 8.9(b) shows the shifted dispersion profiles when
waveguide thickness are 246, 249, 252, 255, 259 and 265 nm, respectively. It is
important to choose the waveguide thickness values according to the clamping
wavelengths to make the overall dispersion as flat as possible. The length ratios
are 15%, 11%, 12%, 12%, 17% and 33% accordingly. In Figure 8.9(c), a
dispersion of -31300 ps/nm/km is obtained over 147 nm bandwidth, with a
variance of 305.5 ps/nm/km, <1% of the mean dispersion. Such dispersive media
116
could be used to introduce a tunable optical delay of 6.3 ns/m for on-chip signal
processing by converting wavelength from 1450 to 1680 nm, as shown in Figure
8.9(c). Flat dispersion can also be obtained by varying the waveguide width. With
the same vertical dimensions of the slot-waveguide and silica base, we keep
waveguide thickness = 255 nm and vary the width from 565 to 500, 445, 390 and
340 nm, and flat dispersion of -46100 ps/nm/km is from 1473 to 1564 nm. The
length ratios are 26%, 17%, 17%, 7% and 33%, respectively, with a variance is
623 ps/nm/km, <1.4% of the mean dispersion. Varying the waveguide width is
more fabrication friendly and can be easily realized by lithography [YS07].
The dispersion bandwidth is extendable if more sections are added. The
overall dispersion can be tailored in other ways, e.g., changing with wavelength
linearly when the structural parameters are chosen appropriately. Nevertheless, it
is still challenging in current fabrication technology to make signal loss in a long
waveguide (a few meters) acceptably low. It might be inevitable to adopt optical
amplification [RKX06] in the waveguides to mitigate this problem.
Figure 8.9. (a) Waveguides with variable waveguide thickness or width are
cascaded. (b). The dispersion shifts over wavelength by changing waveguide
thickness. (c). Flat dispersion of -31300 ps/nm/km over 147 nm. 6.3 ns/m tunable
delay can be obtained by 230-nm wavelength conversion.
117
c ) Nonlinear slot waveguide
In this section, an analysis of nonlinear slot waveguides is presented. A schematic
of the waveguide structure is shown in Figure 8.10(a), where a horizontal slot is
surrounded by two silicon layers with air cladding, and the waveguide substrate is
2μm of buried oxide. For the quasi-TM mode (vertically polarized), due to the
discontinuity of its electric field at the interfaces of the slot and the silicon layers,
a large fraction of the guided mode is confined in the slot layer [JTH07]. The
effective index of quasi-TM mode as a function of wavelength is obtained by
using a finite-element mode solver (COMSOL Multiphysics 3.4), with an element
size of 5, 40, and 100 nm for slot, silicon and other regions, respectively. Material
dispersion is taken into account for the slot (As
2
S
3
[PAL1998] and Si nano-crystal
[ZIL08]), silicon [BRD07] and silica substrate. Group velocity dispersion,
D = -(c/λ)•(d
2
n/dλ
2
), is calculated, where n is the effective index of refraction, and
c and λ are the speed of light and wavelength in vacuum, respectively.
Figure 8.10: (a) Slot waveguide with silicon layers surrounding a highly nonlinear
slot layer. (b) Dispersion profiles in 10 cm long chalcogenide slot waveguides
with different slot heights.
118
The Kerr nonlinear index of refraction n
2
and two-photon-absorption coefficient
β
TPA
, corresponding to the real and imaginary parts of nonlinear coefficient γ
respectively, vary with wavelength [LZP07, SQL98]. For silicon, the
measurement results given in [LZP07] and [SQL98] are fitted using six-order
polynomials and averaged to take the dispersion of nonlinearity into account,
when γ is calculated as a function of wavelength. For As
2
S
3
, different n
2
and β
TPA
values have been reported [AKK1992, BBK01] at individual wavelengths, and the
measured n
2
is 2.5×10
−18
[AKK1992], (averaged) 3×10
−18
[BBK01, RLL05],
2.93×10
−18
[RLJ04], and 3.05×10
−18
m
2
/W [AM09] at 1.064, 1.31, 1.54 and 1.57
μm, respectively, which is roughly unchanged. Here constant n
2
=3×10
−18
m
2
/W
and β
TPA
= 6.2×10
−15
m/W for these simulations at all wavelengths. For Si nano-
crystal with 8% silicon excess, annealed at 800 °C, n
2
= 4.8×10
−17
m
2
/W and β
TPA
= 7×10
−11
m/W at 1550 nm [SDC09]. For silica, n
2
= 2.6×10
−20
m
2
/W is used, and
β
TPA
is neglected. The nonlinear coefficient γ is computed with a space step of 1
nm using a full vector model [AM09], in which the contributions of different
materials to nonlinearity are weighted by the optical mode distribution, and the
longitudinal electric field of the mode is also considered. We use a figure of merit
defined as γ's real part divided by γ's imaginary part times 4π, i.e., γ
re
/4πγ
im
. In a
scalar model with a single nonlinear material, γ
re
= 2πn
2
/λA
eff
and γ
im
=β
TPA
/2A
eff
,
where A
eff
is effective mode area. The figure of merit that was used here is
equivalent to the widely used figure of merit n
2
/λβ
TPA
[MDS89].
119
Figure 8.11: Dispersion profiles in 10-cm-long chalcogenide slot waveguides with
different (a) waveguide widths and (b) upper silicon heights.
d ) Chacogenide/Si nano-crystal slot waveguides
For As
2
S
3
slot waveguides, W = 280 nm, H
u
= H
l
= 180 nm, and H
s
= 115 nm.
Figure 8,10(b), shows a flat dispersion profile within 0±0.017 ps/nm obtained
from 1460 to 1709 nm wavelength (bandwidth of 249 nm) for a 10-cm-long
waveguide on chip. There are two zero dispersion wavelengths, located at 1500
and 1677 nm, respectively. The maximum dispersion of 0.1695 ps/(nm•m) occurs
at 1595-nm wavelength. The flat and low dispersion is obtained by employing slot
structures. The effective index of a slot waveguide can change rapidly with
wavelength, and this makes the effective index of the slot mode closer to that of a
substrate mode at long wavelengths (e.g., around 2100 nm in this case), causing
mode coupling and negative dispersion [ZYX06]. Thus, the total dispersion curve
is bent and becomes small and flat at the wavelength of interest near 1550 nm.
Varying slot height can significantly increase the dispersion, and Figure
8.10(b) shows an increase in the value of the dispersion peak from 0.0971 to
0.3301 ps/(nm•m) as the slot height decreases from 120 to 105 nm. Accordingly,
the dispersion peak blue-shifts from 1600 to 1580 nm. Dispersion can also be
120
tailored by changing the waveguide width, as shown in Figure 8.11(a). Widening
the waveguide produces a dispersion curve that is shifted to longer wavelength.
The right zero dispersion wavelength red-shifts by 111 nm, from 1677 to 1788 nm
as the waveguide width is changed from 280 to 310 nm, while the left zero
dispersion wavelength blue-shifts by only 32 nm. Changing the slot height and
waveguide width together enables tailoring the center wavelength and zero
dispersion wavelengths while keeping flat and low dispersion. For example, one
can start with the structure parameters given above and then increase both the
waveguide width and the slot height. These modifications produce a red-shift of
dispersion curves while maintaining the low peak value of dispersion. Figure
8.11(b) shows the tailored dispersion as the upper Si height H
u
is reduced from
190 to 160 nm. It is noted that the right zero dispersion wavelength has a shift
towards long wavelength by 65 nm, larger than the left zero dispersion
wavelength. Moreover, the peak dispersion value is relatively tolerant to the
change of H
u
, and the dispersion flatness is improved by increasing H
u
.
The nonlinear coefficient γ and figure of merit is considered as a function of
wavelength with varied slot height in Figure 8.12(a). For a slot height H
s
= 115
nm, γ decreases from 178.3 to 135.6 /(W•m) as wavelength increases from 1400
to 1800 nm. A similar trend is found for other slot heights. In contrast, the figure
of merit increases with wavelength from 0.87 to 1.52 for H
s
= 115 nm. This is
explained as follows. First, the dispersion of nonlinearity in silicon is considered.
From 1400 to 1800 nm, silicon's material figure of merit defined as n
2
/λβ
TPA
increases from 0.252 to 0.739. Second, we assume that n
2
and β
TPA
in As
2
S
3
do
121
not change with wavelength, so As
2
S
3
material figure of merit decreases with
wavelength. Third, the material index and mode distribution change with
wavelength, and contributions of different materials to figure of merit depend on
wavelength. With Hs of around 120 nm and a smaller index contrast between Si
and As
2
S
3
(compared to Si and Si-nc), the field enhancement in the As
2
S
3
slot is
less than that shown in Figure 8.14 insets for an 120-nm Si-nc slot, and thus
contribution of silicon layers to figure of merit is more than that of the As
2
S
3
slot.
This is why the overall figure of merit increases with wavelength. The third factor
has limited effect. For a larger slot height, more power is confined in the As
2
S
3
slot, and the figure of merit value becomes higher. There are similar changes in γ
and figure of merit versus wavelength in Figure 8.12(b), with the waveguide
width changed, but the figure of merit is insensitive to the width change.
Figure 8.12: For chalcogenide slot waveguides, nonlinear coefficient γ and figure
of merit are examined over wavelength with different (a) slot heights and (b)
waveguide widths, respectively.
122
Figure 8.13: For 10-cm-long Si nano-crystal slot waveguides, (a) dispersion
profiles change with slot height. (b). Dispersion profile red-shifts as lower silicon
height increases.
For silicon nano-crystal slot waveguides, analyzed here W = 500 nm, H
u
=
H
l
= 180 nm, and H
s
= 47 nm. Figure 8.13(a) shows a flat dispersion profile
within 0±0.16 ps/(nm•m) obtained over a 244-nm wavelength range, from 1539 to
1783 nm. There are two zero dispersion wavelengths at 1580 and 1751 nm,
respectively. The peak dispersion of 0.156 ps/(nm•m) is found at 1670 nm. Figure
8.13(a) shows that the dispersion peak value is decreased from 0.2101 to 0.0508
ps/(nm•m) as slot height H
s
varies from 46 to 49 nm, at a rate of 0.053 ps/(nm•m)
per nm. As compared to As
2
S
3
slot waveguides, the Si nano-crystal slot
waveguides have a larger index contrast between the slot and silicon layers and a
smaller slot height, and this causes stronger field enhancement in the slot. The
overall dispersion is dominated by waveguide dispersion, which results in the
high sensitivity of dispersion to the slot height. The dispersion change caused by
increasing the lower silicon height H
l
is shown in Figure 8.13(b). The thicker the
lower silicon layer, the flatter the dispersion profile near its peak value. The
dispersion curve is red-shifted for a larger H
l
. The right zero dispersion
123
wavelength shifts by 106 nm, from 1696 to 1802 nm as H
l
is changed from 170 to
190 nm. Generally, similar trends of dispersion tailoring are found for the As
2
S
3
and Si nano-crystal slot waveguides as a structural parameter is changed.
However, for a smaller slot height, dispersion is more sensitive to a change in the
slot height. As an example, Fig. 5 shows dispersion profiles modified by a 10-nm
change of Hs in a 10-cm-long Si nano-crystal slot waveguide when H
s
= 40, 80
and 120 nm, (W = 500 nm, H
u
= H
l
= 180 nm), and accordingly dispersion value
is changed by 0.698, 0.339, and 0.195 ps/(nm•m) at 1650-nm wavelength. A
small H
s
induces strong field enhancement shown in Figure 8.14 insets, and light
is tightly confined in a small area, which causes an increased dispersion
sensitivity to the slot height.
Figure 8.14: For 10-cm Si nano-crystal slot waveguides, dispersion sensitivity
changes with Hs.
Calculated nonlinear coefficient γ and figure of merit with the slot height H
s
of 47
nm are 2874 /(W•m) and 0.447, respectively, at 1550-nm wavelength. A small
change in the slot height H
s
, from 46 to 49 nm, does not change γ and figure of
merit much. Due to the strong field enhancement in the slot, figure of merit is
124
dominated by the material properties of the Si nano-crystal slot. This is confirmed
by noting that silicon's material figure of merit is 0.352 at 1550-nm wavelength,
but the Si nano-crystal’s material figure of merit n
2
/λβ
TPA
is 0.4424, which is
close to the computed figure of merit. Due to the lack of measurements on the
dispersion of nonlinearity in Si nano-crystal, γ is not calculated as a function of
wavelength.
8.3 Conclusions of PART II
In PART II of the dissertation, analysis and design of a slot waveguide in which
the core layer is orthogonally slotted to form a rectangular sub-core are presented.
Based on numerical simulations that the modified slot waveguide is able to guide
light with confinement in two-dimension. The properties of the waveguide can be
controlled by varying the intermediate index as well as the sizes. The
linear/nonlinear dispersion and birefringence can be tailored by changing various
geometric parameters with extended ranges. Finally, constant-dispersion points,
where the dispersion is insensitive to size changes, are demonstrated and
discussed.
Highly dispersive slot waveguides are proposed for telecom and on-chip
signal processing applications. Negative dispersion of -181520 ps/nm/km is
introduced by the strong interaction between the slot and strip modes in the
waveguide coupler. It has been shown that a dispersion-flattening slot waveguide
can be designed by cascading a series of the waveguide sections with a slightly
125
modified strip waveguide thickness or waveguide width. -31300 ps/nm/km is
obtained over 147-nm wavelength range, with a <1% variance.
Chalcogenide glasses are good photonic platforms for on-chip nonlinear
signal processing [TBF07, PTF08]. Typically, the nonlinear coefficient γ of As
2
S
3
rib waveguides is found to be ~10/(W•m) [PTF08, LLC08], which is much lower
than that of silicon waveguides [KJP07, OPD09]. This is partially because the
As
2
S
3
waveguides have a smaller index contrast and larger mode area, compared
to silicon waveguides. However, the relatively small refractive index in As
2
S
3
allows the formation of a chalcogenide slot waveguide, greatly enhancing light
confinement and increasing γ from ~10 to 160 /(W•m). Moreover, a benefit of
introducing the As
2
S
3
slot is that it increases the waveguide figure of merit from
~0.35 (for Si) to 1.15 at 1.55 μm, which approaches saturation of nonlinear
performance [TBF07]. By properly combining different materials, one can have a
slot waveguide with a designable (instead of geometry independent) figure of
merit. Though the Si nano-crystal slot waveguides have relatively small figures of
merit, the high nonlinear coefficient γ allows a great reduction of pump power for
nonlinear optical signal processing on a chip. Nonlinear length L
NL
= 1/γP may
still be comparable to dispersion length for high-speed signals [GXM09], which
requires the optimization of the dispersion properties of the waveguides.
Highly nonlinear slot waveguides with flat and low dispersion for on-chip
nonlinear signal processing have been proposed. Flat dispersion within 0±0.16
ps/(nm•m) is obtained over a 244-nm wavelength range for Si nano-crystal slot
waveguides, and nonlinear coefficient is 2874 /(m•W) at 1550 nm. As
2
S
3
slot
126
waveguides can produce a dispersion profile within 0±0.17 ps/(nm•m) over a 249-
nm bandwidth, with greatly increased nonlinear coefficient as compared with
previously reported chalcogenide rib waveguides.
127
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Abstract (if available)
Abstract
This PhD dissertation is divided in two parts. In the first part of the dissertation, a proposal is presented for optical detection in terahertz based on subbands supported by nanopore superlattices. In the second part of the dissertation, studies on light propagation in slot waveguides are presented. The two parts are presented with separate introductions and conclusions. The numbering of the chapters is however unified, so are the figure list and the reference list. Some contents of the dissertation have been published. The relevant publications are listed in the reference list and cited respectively. ❧ In PART I of the present PhD dissertation, a design for terahertz detection is proposed based on externally biased nanopore superlattices. In Chapter 1, important aspects related to terahertz detection are introduced, including the current status of available techniques and the challenges for terahertz detection, the challenges with solid state integrated photonic devices for terahertz detection, an introduction of nanopore superlattices and their possible usefulness for terahertz detection, the methods to study the electronic subband structure as well as the electrical transport in the nanopore superlattices, and the proposed model device for terahertz detection. ❧ After the introduction in Chapter 1, theoretical foundations, simulation results, and device designs are presented in detail in Chapters 2 ~ 5. Firstly, electronic subband structures of GaAs/InGaAs nanopore superlattices in the triangular cell configuration are analyzed by solving the 3D Schrödinger equation with the finite difference method. The results demonstrated that subband gaps could be observed ranging from less than 1meV up to 20meV. The corresponding density of states spectrum has been calculated, which also shows promise for terahertz detection. Optical absorption due to intersubband transitions is also studied within the dipole-dipole approximation. Strong and narrow absorption peaks covering the entire terahertz and far-infrared ranges are observed at liquid nitrogen temperature (77K) as well as at room temperature (300K). Selection rules governing the intersubband transitions are discovered and analyzed. ❧ Secondly, external electric field induced intrasubband currents are analyzed by solving the Boltzmann transport equation rigorously, and it is found that as the bias voltage increases, after a linear conductivity region, the current increases with higher orders of the external bias field, i.e. the nonlinear conductivity region. It is further revealed that the light-induced intersubband absorption brings about changes in the external-electric-field-induced nonlinear current. Therefore, detection of terahertz light becomes apparently feasible. In addition, a self-consistent approach is implemented in treating the light-induced intersubband absorption while keeping the total carriers conserved, so that the intersubband absorption process is described more precisely. ❧ Finally, based on the above findings, a model device for terahertz detection is proposed. A tunable bias voltage is applied to the nanopore structure and the induced current is measured. The device is found suitable for high speed analog and digital terahertz detection. The main properties and capabilities of the proposed detector are analyzed. Noises, such as generation-recombination, thermal, and shot noises, as well as figures of merit, such as signal to noise rate and bit error rate, are estimated. A design is also proposed to construct a wideband detector based on an array of gradually rescaled lattices. The wideband design is also capable of analyzing the frequency of the incoming light. At the end of Chapter 5, a number of numerical simulations that tailor the nanopore superlattice with the key parameter, namely the filing factor, are presented, and the results are discussed with the emphasis on the influences of the parameter tailoring to the subband structures, the light-induced intersubband transitions, and the intrasubband current. ❧ In the second part of the dissertation, studies on light propagation in slot waveguides are presented. Firstly, the background, methods and main results of the studies are introduced in the following sections in the present introduction chapter. ❧ In Chapter 7, a design of a slot waveguide in which the core layer is orthogonally slotted to form a rectangular sub-core is presented. The device is referred as dual slot waveguide as it outputs a two-dimensionally confined light spot. In this configuration, while the overall guiding and coupling efficiency remains the same as a conventional slot waveguide, the field confinement is enhanced and appears two-dimensional. The performance as well as optical properties of the proposed dual slot waveguide is controllable by selecting the intermediate index as well as various geometrical parameters. ❧ In Chapter 8, extensive studies on the dual slot waveguide are presented with the focus on engineering the frequency response of the waveguide for various applications. By changing different variables, the linear/nonlinear dispersion and birefringence can be tailored with extended ranges. Zero-dispersion points, where the dispersion vanishes, are achieved, and constant-dispersion points, where the dispersion is insensitive to size changes, are also demonstrated. ❧ Finally, also in Chapter 8, two studies on dispersion tailoring of slot waveguides are introduced, including a design to construct slot waveguides with high dispersion and another design to realize slot waveguides with near-zero dispersion. ❧ In the first study, a slot waveguide with high dispersion is proposed based on cascading a slot waveguide with a strip waveguide. The strong coupling between the slot waveguide and the strip waveguide enables extended dispersion range. A negative dispersion as large as -181520 ps/nm/km is feasible thanks to a strong interaction between the slot and strip modes, and near constant dispersion at lower degree can be achieved within a relatively wide frequency band. Specifically, -31300 ps/nm/km dispersion over 147-nm bandwidth with <1% variance is shown. This design is to be useful for on-chip dispersion compensation and other signal processing applications. ❧ In the other study, designs of highly nonlinear slot waveguides with constant and low dispersion over a wide wavelength range are investigated. Silicon nano-crystal and chalcogenide glass are considered respectively as the slab materials, which are both of relatively high nonlinear coefficients. The new design enables the device to behave as a wideband low dispersion waveguide. Specifically, over a 244-nm bandwidth, dispersion of 0±0.16 ps/(nm•m) is achieved in the silicon nano-crystal slot waveguide, with a nonlinear coefficient of 2874 /(W•m), whereas the chalcogenide glass As₂S₃ slot waveguide exhibits a dispersion of 0±0.17 ps/(nm•m) over a bandwidth of 249 nm, with a nonlinear coefficient 16 times larger than that of As₂S₃ rib waveguides and a nonlinear figure of merit three times larger than that of silicon strip waveguides.
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University of Southern California Dissertations and Theses
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Asset Metadata
Creator
Xiao Li, Yinying
(author)
Core Title
Optical studies in photonics: terahertz detection and propagation in slot waveguide
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Electrical Engineering
Defense Date
07/28/2014
Publisher
University of Southern California
(original),
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OAI-PMH Harvest,photonics,slot waveguide,terahertz detection
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English
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O'Brien, John D. (
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), Feinberg, Jack (
committee member
), Wei, Wu (
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yxiaoli@usc.edu
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https://doi.org/10.25549/usctheses-c3-472136
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Xiao Li, Yinying
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Tags
photonics
slot waveguide
terahertz detection