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Optimizing nanoemitters using quasi-aperiodicity and photoelectrochemical etching

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Optimizing nanoemitters using quasi-aperiodicity and photoelectrochemical etching

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Optimizing Nanoemitters using Quasi-
Aperiodicity and Photoelectrochemical Etching

A Dissertation submitted to the faculty of the USC Graduate School
In partial fulfillment of the requirements for the degree of
Doctor of Philosophy
(ELECTRICAL ENGINEERING – ELECTROPHYSICS)

P. Duke Anderson
May 2017

Copyright P. Duke Anderson 2017
1

Acknowledgments

I would like to begin by sincerely thanking my advisor, Dr. Michelle Povinelli, for
her guidance and support the past six years. The dedication she has shown to my
development as a scientist will always be appreciated. I would also like to express
my gratitude to members of the Povinelli Nanophotonics Laboratory, both past and
present. Specifically, I would like to thank Dr. Chenxi Lin for offering his
mentorship my first years within the group, Dr. Roshni Biswas for counseling me
on optical measurements, Dr. Ningfeng Huang for helping maintain the software
integral to so much of my research, Dr. Eric Jaquay, for, I’m not sure what, Dr.
Camillo Mejia for his enthusiasm in discussing research-related problems, and our
previous post-docs, Dr. Luis Martinez Rodriguez and Dr. Mehmet Solmaz, for
cultivating our laboratory’s experimental capabilities. I would also like to thank
Dr. Jing (Maggie) Ma, Shao-hua (Nick) Wu and Aravind Krishnan for always
being supportive of my research. Next, I would like to express sincere gratitude to
my co-adviser at Sandia National Lab, Dr. Ganesh Subramania. Dr. Subramania
offered me a tremendous scientific opportunity and invested enumerable hours into
helping develop my nanofabrication skills. I am also grateful to Dr. Olga Spahn for
formally approving my internship at Sandia and for always providing positive
mentorship during my two-year tenure with her. I would be remiss not to thank Dr.
Art Fischer for the time he took to discuss photonic characterization and III-nitride
devices, Mike Smith for always being available to assist in the lab at a moment’s
notice, Tony Coley for helping keep our photonic characterization lab running,
Diane Gaylord for helping keep my professional life at Sandia in order and all the
amazing members at the Center for Integrated Nanotechnology (CINT). These
members of CINT include Tony James, John Nogan, Bill Ross, Denise Webb,
2

Doug Pete and Joe Lucero. These individuals keep the nanofabrication facilities at
CINT firing on all cylinders, and I owe them a tremendous amount of gratitude. I
would also like to thank Dr. George Wang and Dr. Dan Koleske for their scientific
guidance and contributions. Next, I would like to express gratitude to the following
members of my thesis and qualification exam committees: Dr. Stephen Cronin, Dr.
Wei Wu, Dr. Daniel Dapkus and Dr. Aiichiro Nakano. Next, and of most
importance, I would like to thank my family; my mother Roseanne Anderson,
father Pat Anderson, and sister Kacie Ellis. I have not met better people in my life
(nor will I), and I would have never been able to achieve any of this without their
unconditional love and support. I also owe a great deal of thanks to my brother-in-
law Nate Ellis, close friends Erick Moen, Steph Moen, Andrew Smith, Michael
Smith and former adviser Dr. Marc Christensen and mentor Dr. Nathan Huntoon.
Dr. Christensen and Dr. Huntoon are not only gifted scientists but are also
tremendous people who provided me with tremendous direction. Lastly, I would
like to thank my Lord and Savior, Jesus Christ, for teaching me humility these past
years and helping me find true appreciation for all the ways in which He has
blessed my life.

3

Table of Contents
Chapter 1 – Introduction
1.1 Incandescence and Electroluminescence ............................................................................. 6
1.2 III-Nitride Solid-State Lighting ........................................................................................... 9
1.3 Piezoelectric and Polarization Fields ................................................................................. 10
1.4 Efficiency Droop ................................................................................................................ 12
1.5 Emergent III-Nitride Nanoemitter Platforms ..................................................................... 13
1.6 Thesis Outline .................................................................................................................... 15
Chapter 2 – Effect of Guided Resonance Modes on Emission
from GaN Core-Shell Nanorod Arrays
2.1 III-Nitride Core-shell Nanorod Arrays .............................................................................. 17
2.2 Simulation Methodology ................................................................................................... 18
2.3 Emission Calculations ........................................................................................................ 20
2.4 Internal and External Quantum Efficiency ........................................................................ 23
Chapter 3 – Optimizing Emission in Nanorod Arrays Through
Quasi-Aperiodic Inverse Design
3.1 Simple Periodic and Quasi-Aperiodic Arrays.................................................................... 25
3.2 Forward and Inverse Simulations ...................................................................................... 27
3.3 Random-Walk Optimization .............................................................................................. 28
3.4 Emission Calculatios .......................................................................................................... 29
3.5 Internal and External Quantum Efficiency ........................................................................ 30
4

Chapter 4 – Tailoring Emission in III-Nitride Nanorod Arrays
through Quasi-Aperiodic Design
4.1 Coupled and Uncoupled Modes in Nanoemitters .............................................................. 33
4.2 Epitaxial Growth ................................................................................................................ 35
4.3 Top-Down Nanorod Array Fabrication ............................................................................. 35
4.4 Device Characterization ..................................................................................................... 36
4.5 Photonic Bandstructure and In-Plane Distributed Feedback ............................................. 40
Chapter 5 – Selective Quantum Dot Placement and Preferential
Membrane Formation through Photoelectrochemical Etching
5.1 Introduction ........................................................................................................................ 42
5.2 Photoelectrochemical Etching of III-Nitrides .................................................................... 44
5.3 Fabrication of Quantum Dots with Deterministic Placement ............................................ 45
5.4 Low-Temperature Photoluminescence .............................................................................. 46
5.5 Internal Quantum Efficiency .............................................................................................. 48
5.6 Second-Order Cross-Correlation Measurement ................................................................. 49
5.7 Photonic Crystal Defect Cavity Design ............................................................................. 50
5.8 Preferential Membrane Formation ..................................................................................... 54
5.9 Future Work: Futher Cavity Optimizaion .......................................................................... 56

Chapter 6 – Conclusion…………………………………….………………………..58
Appendix
A.1 Guided Modes, Radiation Modes and Guided-Resonance Modes .................................... 61
A.2 Coupled and Uncoupled Modes in Photonic Crystal Slabs ............................................... 63
5

A.3 Lorentz Reciprocity: Derivation and Relation to Emission Optimization ......................... 65
A.4 Inverse Design: Random-Walk Optimization of Quasi-Aperiodic Arrays ........................ 67

References………………………………………………………………………...………..69

6

Chapter 1 – Introduction

1.1 Incandescence and Electroluminescence

Early light sources, including oil lambs, candle wicks and incandescent bulbs, operate
according to a physical principle referred to as incandescence. The term is derived from a Latin
verb meaning to glow white [1]. Incandescence is a form of thermal radiation which results from
the heating of a material [2]. Incandescent sources have temperature dependent emission spectra,
unrelated to material composition. Incandescent sources and their spectral density distribution
are described by Planck’s Law [3, 4]

𝐵 (𝜈 , 𝑇 ) =
2ℎ 𝜈 3
𝑐 2
1
𝑒 ℎ𝑐 𝑘 𝐵 𝑇 − 1

where 𝜈 represents the frequency of emitted light, T represents the equilibrium temperature of the
material, ℎ is Planck’s constant (unreduced), 𝑐 is the speed of light in a vacuum and 𝑘 𝐵 is
Boltzmann’s constant. Max Planck proposed the law in 1900, resolving the ultraviolet
catastrophe, or Rayleigh-Jeans Catastrophe [5]. The ultraviolet catastrophe resulted from the
Rayleigh-Jeans Law, which accurately predicted emission for longer wavelengths but incorrectly
predicted diverging emission at shorter wavelengths [6]. Planck’s law resolved the issue, having
an energy distribution that tends to Wien’s law in the limit of high frequencies, and an energy
distribution with tends to the Rayleigh-Jeans law in the limit of low frequencies [5, 6]. In fact,
the Rayleigh-Jeans law is the classical limit of Planck’s law and can be derived by taking the
limit as Planck’s constant tends to zero. Materials whose emission spectra follow Planck’s law
are referred to as blackbodies and are considered to be perfect emitters and absorbers of light.
Figure 1.1 shows the spectral irradiance of a series of blackbodies as a function of wavelength
for equilibrium temperatures of 2700 K, 4000 K and 5000 K. The visible portion of the
electromagnetic spectrum is indicated in the figure, falling roughly between λ = 400 nm – 750
nm. For each blackbody, peak emission is indicated by a dot of the corresponding color. As the
temperature of a blackbody increases, the resulting luminescent energy shifts to higher
7

frequencies (shorter wavelengths). However, the radiation is spectrally broad, producing
considerable emission outside the visible window of the electromagnetic spectrum. For a
temperature close to the operating temperature of an incandescent filament (T = 2700 K), the
majority of irradiance falls outside the visible spectrum. Consequently, incandescent sources are
not very efficient lighting sources. A typical incandescent light bulb only converts around 5% of
the total electrical power it consumes into visible light [7].

Fig. 1.1 The radiant flux distribution as a function of wavelength for black bodies at temperatures of 2700 K (green),
4000 K (blue) and 5000 K (purple). Dots of corresponding color indicate the peak irradiance for each blackbody.
The visible portion of the electromagnetic spectrum is also indicated between λ = 400 – 750 nm.

Early in the twentieth century, however, it was discovered light could be emitted from a
solid-state material, under a different principle of operation [8]. The differing principle of
operation is illustrated schematically in Figure 1.2. By passing current through a semiconducting
material, it is possible to excite an atomic state by pushing an electron into the conduction band.
As the electron enters the conduction band it leaves behind a vacancy, or hole, in the valence
band. This bound state is referred to as an exciton, and represents an excited atomic state.
Eventually, the state relaxes, with the electron and hole radiatively recombining in the valence
band. Due to energy conservation, a quantum of light (photon) is emitted, with an energy equal
to the bandgap (E
g
) of the material.
8

Fig. 1.2 Simplified schematic of electron-hole excitation and radiative recombination in a semiconductor.

Around this time, the growing interest in silicon carbide (SiC) unwittingly led to the
discovery of this new light-emitting mechanism. Due to its high crystalline hardness and low
cost of chemical synthesis, SiC had emerged as a choice material for the abrasives industry [8].
The development of SiC manufacturing also permitted researchers to pursue rectifying, solid-
state detectors composed of crystalline materials. The vacuum-tube diodes used for rectification
at the time were expensive and heavy consumers of power. Using metal probes and forming a
Schottky diode, a radio engineer by the name of H. J. Round first demonstrated rectifying
current-voltage behavior on a piece of SiC. The tested samples were subjected to very high
operating voltages (excess of 100 V) and demonstrated photoemission [9]. Figure 1.3 shows the
short publication reporting on the phenomenon Round discovered [9].
Later in 1928, O. V. Lossev presented a more detailed study of SiC metal-semiconductor
rectifiers [10]. Puzzled by the origin of light emission, the researcher placed drops of liquid
benzene on SiC devices and monitored how quickly the drops evaporated in the presence of an
applied voltage. The evaporation process was slow and led Lossev to (correctly) assume that the
photoemission was not related to incandescence [10]. The photoemission process was eventually
shown to be dependent on electrical current injection, and appropriately termed
electroluminescence [8]. Further investigation of SiC light-emitting diodes continued well into
the 1960s. However, being an indirect bandgap material, the efficiencies of SiC devices were
quite low, and attention slowly gravitated towards III-V semiconductors. At this time, III-V
semiconductor research was burgeoning due to the development of chemical processes that
enabled the growth of such materials. Tracking these developments, the Radio Corporation of
9

America (RCA) grew interested in creating a flat-panel, full color image display. To generate the
color display, methods needed to exist for creating blue, green and red pixels. Methods were
already in place for creating both green and red photoemission using semiconductor technologies
[11, 12], but challenges remained for the generation of blue light [8]. J. J. Tietjen, a director in
the Materials Research Division at RCA, started investing considerable resources in the pursuit
of growing highly-crystalline GaN. H. P. Maruska was ultimately assigned to this task and began
a thorough investigation of potential GaN growth methods. Using a vapor-phase epitaxial
technique, RCA ultimately succeeded in growing a film of GaN in 1969 [13]. Soon thereafter,
the research lab demonstrated the first example of electroluminescence from a film of GaN. By
changing the dopant material from Zn to Mg, RCA researchers were able to tune the emission
wavelength from approximately 430 nm to 475 nm [14]; the era of III-nitride light-emitting
diodes had begun.

Fig. 1.3 H. J. Round’s publication reporting on the first observation of electroluminescence
from a SiC LED (Ref. [9]).

1.2 III-Nitride Solid State Lighting

In
x
Ga
1-x
N/GaN materials have emerged as a promising material for high-efficiency lighting
[15, 16]. One reason for the popularity of GaN-based lighting is the material’s direct electronic
bandgap, which makes it well suited for light-emitting applications [17]. Moreover, III-nitride
10

heterostructures have demonstrated wide-tuning of the electronic bandgap energy [18]. By
increasing the incorporation of In within In
x
Ga
1-x
N QWs it is possible to tailor emission
throughout the entire visible spectrum [18]. Finally, incorporating III-nitride light-emitting
diodes (LEDs) with partial, phosphor down-converters produces a whitish glow [19].
Consequently, III-nitride LEDs are currently one of the lead candidates for the future of solid-
state lighting [15, 16]. Since their inception, solid-state lighting based on III-nitride platforms has
grown extremely competitive with traditional light sources [15]. Recently, I. Akasaki, Hirosh
Amano and Shuji Nakamura shared the 2014 Nobel prize in physics for their contributions to the
development of such highly-efficient, III-nitride LEDs [20]. The wide-bandgap of III-nitride
materials also make them well suited for single photon emission [21, 22]. Owing to their large
exciton binding energy, III-nitride QDs can avoid thermionic emission and behave as single
photon sources even at room temperature [22]. Moreover, III-nitride QDs emit in a spectral
region where Si-based photodiodes are highly absorptive. Despite all mentioned advantages, III-
nitride emitters still face several serious challenges related to their quantum efficiency [23 – 26].

1.3 Piezoelectric and Polarization Fields

Due to the high cost associated with specialized substrates, GaN materials are customarily
grown along the Ga polar c-plane atop a substrate of Al
2
O
3
[27, 28]. The relatively large lattice
mismatch (~ 16%) between GaN and Al
2
O
3
leads to a substantial density of growth dislocations
(~ 10
9
cm
-2
) [27]. The growth defects result in material strain, with the material strain producing
large piezoelectric fields [23]. Consider such a piezoelectric field (𝑷 𝑝𝑧
) that is given by

𝑷 𝑝𝑧
= 𝑒 𝑖𝑗
𝜀 𝑗 = 𝑑 𝑖𝑗
𝑐 𝑗𝑘
𝜀 𝑘

where 𝑒 𝑖𝑗
represents the piezoelectric coefficients, 𝜀 𝑗 is the strain tensor, 𝑐 𝑗𝑘
is the elastic tensor
and 𝑑 𝑖𝑗
represents the piezoelectric coefficients relating to the stress tensor 𝜎 𝑗 = 𝑐 𝑗𝑘
𝜀 𝑘 . In
addition to a strain induced piezoelectric field, the lack of inversion symmetry along the c-plane
of wurtzite GaN gives rise to a spontaneous polarization field (𝑷 𝑠𝑝
). The total polarization field
11

is then the summation of both fields (𝑷
= 𝑷 𝑝𝑧
+ 𝑷 𝑠𝑝
). In the absence of an externally applied
electric field it is seen that

𝛻 ⋅ (𝑷 𝑝𝑧
+ 𝑷 𝑠𝑝
) = −𝜌 𝑝𝑜𝑙

where 𝜌 𝑝𝑜𝑙 is the polarization charge density. At an abrupt interface, such as a heterojunction, a
polarization sheet charge density arises from the differences in 𝑷 𝑠𝑝
within each layer and the
abrupt change in strain, which affects 𝑷 𝑝𝑧
. The combined effect of these polarization fields tend
to separate the electron and hole wavefunctions in the QW, reducing the radiative recombination
rate and internal quantum efficiency (IQE). This effect is shown schematically in Figure 1.4(a).
In the present work, strategies for reducing (or altogether eliminating) polarization fields in
QW structures are presented. First, theoretical work examines III-nitride nanorod arrays with
core-shell MQW geometries. Core-shell nanorod arrays increase the quantum efficiency of the
emitting layers in two key ways: First, the smaller nanowire footprint helps reduce the strain in
the QWs. Second, the core-shell MQW geometry is assumed to be grown along the non-polar,
m-plane, and thus avoid spontaneous polarization fields. The electronic bandstructure for a QW
oriented along the m-plane is illustrated in Figure 1.4(b). In the figure, it is readily apparent that
the lack of polarization fields tends to increase the electron-hole wavefunction overlap.
Building upon these nanoemitter architectures, theoretical electromagnetic calculations
reveal a guided-resonance mode ideally suited for core-shell nanorod arrays. Using this optical
mode it is possible to simultaneously enhance the radiative recombination rate and light
extraction efficiency, relative to an infilled reference structure. Next, attention is turned to quasi-
aperiodic arrays. Quasi-aperiodic arrays are arrays which exhibit small scale aperiodicity, but are
themselves repeated on a larger scale. In conjunction with an inverse design optimization
procedure, it is theoretically shown how quasi-aperiodicity can be utilized to further increase
external quantum efficiency (EQE) in nanoemitter architectures. Finally, attention is turned to
quantum-size controlled photoelectrochemical (QSC-PEC) etching of III-nitride In
x
Ga
1-x
N QWs.
Using a top-down QSC-PEC etching technique, quantum dots (QDs) are fabricated and shown to
significantly increase the IQE relative to an unetched QW structure. The increase in IQE is
attributed to both the increased carrier confinement (i.e. increased electron-hole wavefunction
overlap) and strain reduction in the emitting material. Further, this PEC etching technique is
12

shown to be a useful tool for the top-down, deterministic placement of quantum-size controlled
QDs.

Figure 1.4 Electronic bandstructure for a QW oriented along (a) the c-plane and (b) m-plane
of wurtzite GaN.

1.4 Efficiency Droop

A second major challenge facing the development of III-nitride LEDs and laser diodes
(LDs) is the efficiency droop associated with high operating current densities [24]. The internal
quantum efficiency (IQE), or radiative efficiency, is defined as the fraction of electron-hole pairs
that radiatively combine to emit a photon. For low to moderate input current densities we can
define the IQE using the standard ABC model [16]:

𝜂 𝐼 =
𝐵 𝑛 2
𝐴𝑛 + 𝐵 𝑛 2
+ 𝐶 𝑛 3

The first term in the denominator is indicative of Shockley-Read-Hall (SRH) recombination,
where A is defined to be the Shockley-Read-Hall coefficient. Shockley-Read-Hall recombination
is a non-radiative process which varies linearly with carrier density (n). During this
recombination process, a defect introduces an extra energy level within the electronic bandgap,
which a conduction band electron may fill. Eventually the electron recombines with a hole in the
valence band, releasing energy in the form of a quantized lattice vibration (phonon). The second
13

denominator term is band-to-band radiative recombination. Radiative recombination has a
radiative recombination coefficient (B) and is proportional to the square of the carrier density
(n
2
). During this process, an electron in the conduction band recombines radiatively with a hole
in the valence band, producing a quantum of light (photon). The frequency of the emitted photon
is 𝜐 = 𝐸 𝑔 /ℎ, where 𝐸 𝑔 is the energy associated with the electronic bandgap of the material and ℎ
is Planck’s constant. The final recombination process is Auger recombination, which is a non-
radiative process involving three carriers, and therefore proportional to the cube of the carrier
density (n
3
). The Auger recombination coefficient is C. During Auger recombination, an electron
and hole recombine, and the energy is imparted to another conduction band electron.
As noted previously, when the carrier density in a III-nitride heterojunction increases
beyond a threshold value, the IQE begins to decrease exponentially [24, 25]. The mechanism(s)
for the efficiency droop, though debated, are customarily attributed to carrier leakage in thin
active regions (~ 3 nm thick) and the associated Auger recombination at high current densities
[24, 25]. In addition, the onset of efficiency droop tends to occur at lower current injection
densities as the emission wavelength red-shifts from blue wavelengths towards green
wavelengths. This trend has led to the well-known green-gap in III-nitride LEDs [26].
In the present work, potential architectures for reducing efficiency droop are explored.
Specifically, by constructing nanorods with a core-shell MQW geometry, it is possible to
increase the volume of emitting material (provided a fixed chip-size). Increasing active material
volume in this manner is one potential method for emitting more light at lower current densities.
Additionally, designing core-shell nanorod arrays to support the correct guided-resonance mode
offers opportunity in bolsterering both the spontaneous emission rate and light extraction
efficiency. Finally, both QDs and core-shell QWs offer unique geometries for reducing material
strain, and thus are potential architectures for pushing higher IQEs towards the green-gap and
beyond.

1.5 Emergent III-Nitride Nanoemitter Platforms

Recent years have experienced an explosion in the development of bottom-up (growth-
based) and top-down (etchant-based) III-nitride nanoemitter fabrication techniques [27 – 34].
The hike in fabrication capability has opened up new opportunities for increasing the quantum
14

efficiency of III-nitride nanoemitters. For instance, non-polar In
x
Ga
1-x
N/GaN core-shell nanorods
have been fabricated using electron beam lithography coupled with metal organic chemical vapor
deposition (MOCVD) [27]. Another fabrication approach, consisting of a top-down etch process
followed by bottom-up regrowth, was also demonstrated [35]. Figure 1.5 shows non-polar
In
x
Ga
1-x
N/GaN core-shell nanorods fabricated using each approach [27, 35]. Regardless of the
fabrication methodology, the formation of non-polar, core-shell geometries offer many
interesting possibilities. The non-polar QW structures are largely unhampered by polarization
fields which otherwise reduce a device’s radiative efficiency (as discussed in Section 1.3). Core-
shell MQW architectures also provide strain-relaxation, which can be exploited to increase In
incorporation and further red-shift emission. Finally, as previously noted, MQW core-shell
geometries make it possible to drastically increase emitter volume compared with axial junction
devices. Such an increase in emitter volume may produce brighter nanoemitters at lower
operating current densities.

Figure 1.5 Overhead views of MQW core-shell III-nitride nanorods fabricated from a (a) bottom-up and a (b)
top-down, regrowth fabrication methodology (Ref. [27] and Ref. [35], respectively).

Quantum dots (QDs) have also shown great promise in increasing the IQE of III-nitride
nanoemitters [36 – 38]. The increased efficiency can be attributed to the reduction in surface area
and increased carried confinement. QDs help redistribute material strain and reduce piezoelectric
fields that may otherwise be present in QW architectures [36, 37]. It has also been suggested that
QDs can function as better gain mediums than QWs [38]. Owing to their delta-like electronic
density of states (EDOS), carriers are fed into nearly identical energy states, dramatically
increasing material gain [38]. One recent QD fabrication approach has demonstrated uniform
emission from QDs fabricated on epitaxial structures [34]. The QD emission uniformity was
15

achieved using a quantum-size controlled photoelectrochemical (QSC-PEC) etching technique,
similar to techniques used in earlier works on colloidal, solution-based QDs [32, 39]. QSC-PEC
etching utilizes a light-assisted wet etching technique that self-terminates after quantum size
effects suppress light absorption. The previous work demonstrated this technique on an
unpatterned epitaxial GaN film containing a single, uncapped InGaN QW. Figure 1.6 shows a
TEM of fabricated III-nitride QDs following QSC-PEC etching [34].

Figure 1.6 TEMs of III-nitride QDs fabricated through a top-down, QSC-PEC etching technique using
different laser excitation wavelengths (Ref. [34]).

1.6 Thesis Outline

A theme common to all the work presented in this thesis is the optimization of quantum
efficiency and tailoring of emission in III-nitride nanoemitters. Chapter 2 is concerned with
modelling the process of incoherent emission from In
x
Ga
1-x
N QWs in core–shell nanorod arrays
using the finite-difference time-domain (FDTD) method. It is found that high-intensity features
in the emitted far-field correspond to guided-resonance modes near the Γ-point of the photonic
band structure. One Γ-point mode is identified whose electric field intensity profile is ideal for
core–shell nanorod array geometries. Using this mode, it is possible to simultaneously enhance
the radiative recombination rate and extraction efficiency relative to an in-filled slab. The
conditions on radiative and nonradiative recombination rates for which the nanorod array has a
higher IQE and EQE than a reference slab are determined. One nanorod array geometry is
presented that enhances the EQE up to a factor of 25, relative to an unpatterned slab.
Chapter 3 presents a new class of quasi-aperiodic nanorod structures for the enhancement of
quantum efficiency. One optimized structure is identified using an inverse design algorithm and
the FDTD method. Emission calculations are performed on both the optimized structure as well
16

as the simple periodic array from Chapter 2. The optimized structure achieves nearly perfect
light extraction while maintaining a high spontaneous emission rate. Overall, the optimized
structure can produce up to a 42% increase in EQE relative to the simple periodic design
presented in Chapter 2.
In Chapter 4, quasi-aperiodic nanoemitters and their ability to tailor light emission are
experimentally investigated. Highly anisotropic arrays of nanorods with underlying simple
periodic and quasi-aperiodic geometries are fabricated using a top-down fabrication procedure.
Room temperature micro-photoluminescence (μ-PL) measurements reveal resonances in each
structure. While the simple periodic resonance produces a donut beam in the emitted far-field,
the quasi-aperiodic resonance tailors the emitted beam into a uniform pattern. Further, the input-
output power relationship of the simple periodic array is shown to be strongly nonlinear. For the
quasi-aperiodic array, the onset of nonlinear input-output behavior is observed. Photonic
bandstructure calculations reveal that quasi-aperiodicity can also be leveraged to decrease
distributed in-plane feedback, a quantity critical to the design of large-area, photonic-crystal
surface-emitting lasers (PCSELs).
Chapter 5 presents a top-down fabrication procedure capable of deterministically positioning
quantum-size controlled QDs within nanostructures. The QDs are positioned using a quantum-
size controlled photoelectrochemical (QSC-PEC) etch on pre-fabricated nanorods. Following
QSC-PEC etching, low-temperature optical spectroscopy reveals strong QD emission signatures.
IQE measurements are performed, with the QSC-PEC etched QDs exhibiting nearly a 10 X
increase in efficiency, relative to a QW. Further, second-order cross-correlation measurements
reveal strong anti-bunching, single-photon emission in individual nanorods. Next, a method for
incorporating QSC-PEC etched QDs into III-nitride photonic crystal (PhC) membranes is
proposed. An L1 PhC cavity is theoretically optimized and experimentally realized on a III-
nitride platform. Preferential QSC-PEC etching of a MQW structure is demonstrated and shown
not to affect emission from a separate single quantum well (SQW) of differing In composition.
Following preferential membrane formation, room-temperature μ-PL measurements reveal the
presence of a strong defect-mode resonance. The ability to deterministically position QDs and
preferentially form membranes in complex III-nitride structures opens up many interesting
opportunities in the fields of nanophotonics and quantum science.

17

Chapter 2 – Effect of Guided Resonance Modes on Emission
from GaN Core-Shell Nanorod Arrays

2.1 III-Nitride Core-shell Nanorod Arrays

Light-emitting diodes (LEDs) based on GaN materials have emerged as promising
candidates for solid-state lighting (SSL). One attractive feature of these material systems is their
wavelength tunability, which can be achieved by adjusting the indium content within In
x
Ga
1-x
N
quantum wells (QWs) [18]. Recently, there has been much effort invested in growing QWs along
planes that mitigate the effects of strain-induced piezoelectricity [27, 28, 35]. In particular, core–
shell nanorod arrays allow growth along non-polar planes parallel to the nanorod axis [27, 35]. In
these examples, the periodic patterning of the emitting structure leads to the formation of a
photonic crystal (PhC) slab. From previous work on photonic crystal LEDs, it is expected that
the design of the nanorod array (lattice type, rod diameter, height, and spacing) can strongly
impact spontaneous emission and/or light extraction [40 – 49]. It is therefore important to
identify potential mechanisms for optimizing optical performance.
Previous work has examined the role of guided-resonance modes [50 – 51] in nanorod
arrays, demonstrating a correspondence between far-field emission patterns and modes of the
photonic bandstructure [41, 46, 47]. In this study, we focus our attention on modes near the Γ-
point of the band structure (zero parallel wave-vector) which favors emission in the vertical
direction. A version of the work in this chapter was published in Ref. [52]. In the work, we
identify one Γ-point mode whose field intensity profile is ideally suited for In
x
Ga
1-x
N/GaN core–
shell nanorod arrays. Using this mode, we are able to enhance both the radiative recombination
rate and extraction efficiency relative to an in-filled slab. We then provide quantitative
conditions for determining whether or not infilling the nanorod structure with GaN will improve
or degrade the overall quantum efficiency. This framework weighs enhancements in the radiative
recombination rate and extraction efficiency due to nanostructuring against the potential
detriment of increased surface recombination. We present one nanorod array geometry where the
external quantum efficiency (EQE) is enhanced by a factor of up to 25. Our results suggest that
18

engineering of nanostructure can play an important role in improving both the brightness and
efficiency of GaN nanorod LEDs.

2.2 Simulation Methodology

We model the system shown in Figure 2.1(a), consisting of a square array of hexagonal
core–shell nanorods. The lattice spacing is a, the rod height is h and the cross-sectional size of
the nanorod is proportional to the parameter r. Each nanorod contains a hexagonal In
x
Ga
1-x
N
QW (red) with its outline defined by a center-to-vertex length of 0.75r. The nanorods are
surrounded by air.
We model the process of spontaneous emission within the QW using incoherent electric
dipole sources [44, 53, 54]. GaN is modeled as a lossless material with a refractive index n of
2.526, corresponding to a wavelength of 420 nm [55]. Instead of incorporating QWs directly into
our simulations, we model their emission characteristics by placing electric dipole emitters in
regions consistent with their location. Simulations were performed within a computational cell
containing a 13 × 13 array of nanorods with perfectly matched layer (PML) boundary conditions.
Each simulation includes a single dipole source located within the QW. For an infinitely large
array, the power emitted by a dipole source will not depend on which rod it is placed in, though
it may vary with location within the rod. We may therefore, to good approximation, model the
emission properties of the nanorod array using dipoles placed only within the center rod. Due to
symmetry, the positions were restricted to the top half of the nanorod within one quarter of its
cross section. In this manner, we effectively model 72 different QW locations. At each location,
two different polarizations tangential to the QW plane were considered: parallel and
perpendicular to the axis [44, 54]. After all simulations were completed, we incoherently
summed the emission results.

19

Fig. 2.1 (a) Schematic of the simulated core-shell GaN nanorod array where a refers to pitch of the square lattice, h
is the height of the nanorod and r is proportional to the cross-sectional area. (b) Schematic of the nanorod and (c) in-
filled reference slab structures. Dashed black lines indicate flux planes and boxes.

The total dipole power (TDP) is defined as the power emitted by the dipole, indicated by the
flux through a box surrounding it (black, dashed box in Figure 2.1(b)). TDP is proportional to the
radiative recombination rate. The extracted power (EP) is defined as the total power collected
from the top of the array (black, dashed line in Figure 2.1(b)). The flux-plane monitors used to
calculate this quantity extend through the entire computational domain and are situated directly
above and below the nanorod array. In general, EP is lower than TDP due to the fact that light
emitted by the dipole can couple into guided modes of the photonic crystal slab and propagate
laterally through the cell. We compare the emission properties of the nanorod array to that of an
in-filled reference slab, shown in Figure 2.1(c). The locations of the QWs and dipole sources are
preserved, while the regions between the rods are infilled with GaN. The extraction efficiency
(EE), or efficiency of light extracted, is equal to EP/TDP.
The photonic band structure and far-field emission are calculated using the Lumerical FDTD
Solutions software package [56]. The far-field intensity is mapped onto the band structure in the
following manner. For a given frequency ω and parallel wavevector 𝑘 //
= 𝑘 𝑥 𝑥 ̂ + 𝑘 𝑦 𝑦 ̂ in the x–
y plane, we identify an emission angle (𝜃 , 𝜙 ) via the relations 𝜃 = sin
−1
(𝑘 //
𝑘 ⁄ ) , 𝜙 =
tan
−1
(𝑘 𝑦 𝑘 𝑥 ⁄ ) and 𝑘 = 𝜔 𝑐 = √𝑘 //
2
+ 𝑘 𝑧 2
⁄ , and overlay the far-field intensity at this angle on the
band diagram. Wave vectors are restricted to the first irreducible Brillouin zone.

20

2.3 Emission Calculations

Figure 2.2 shows the photonic bandstructure superimposed upon the far-field emission
intensity for a nanorod lattice with r/a = 0.268 and h/a = 0.5. Green and blue symbols indicate
TE- and TM-polarized modes, respectively. Here, we define TE/TM modes as having even/odd
vector symmetry with respect to the lateral midplane of the array.

Fig. 2.2 Photonic bandstructure superimposed upon far-field emission
intensity for a square nanorod array with r/a = 0.268 and h/a = 0.5

The dashed, white line in the figure indicates the light line [57]. Modes beneath the light line
are guided and do not couple to external radiation. Modes above the light line are so-called
guided-resonance modes [50, 51, 57]. These modes couple to external radiation unless forbidden
by symmetry [58]. For a thorough discussion of guided-resonance modes please reference
Section A.1. Modes at the Γ-point have a zero in-plane lattice vector, and (if coupled) radiate in
the vertical direction. Here, the lowest Γ-point has a normalized frequency (𝜔𝑎 2𝜋𝑐 ⁄ = 𝑎 𝜆 ⁄ )
equal to 0.707. A physical wavelength of 420 nm coincides with this lowest order Γ-point mode
for a lattice constant near to 300 nm.
Figure 2.3(a) shows the total dipole power (TDP) for the nanorod array and reference slab.
The results are normalized to the maximum TDP produced by the nanorod array. The dashed-
green line indicates the corresponding normalized frequency of the lowest order Γ-point mode.
The peak TDP amplitude of the nanorod array is nearly 2.8 X larger than that of the reference
21

slab. To understand the reason for this enhancement, consider the spontaneous emission rate 𝛤 𝑆𝐸

(which is proportional to TDP) given by Fermi’s Golden Rule

𝛤 𝑆𝐸
=
ℎ
2
2𝜋 〈|𝜇 ⃗ ∙ 𝐸 ⃗ ⃗
(𝑟⃗)|〉
2
𝜌 (𝜔 )

where h is Planck’s constant, 𝜇 ⃗ is the atomic dipole moment of the electron-hole recombination
event, 𝐸 ⃗ ⃗
(𝑟⃗) is the electric field at the location of the emitter and 𝜌 (𝜔 ) is the photonic density of
states (PDOS). First, designing around a Γ-point (or band edge mode) is useful, as it increases
the PDOS within a narrow frequency range. Second, the Γ-point selected has an ideal transverse
electric field profile for a core-shell geometry (being distributed similar to a TE
01
cylindrical
waveguide mode). Recalling that the dipole moment is tangential to the QW, and seeing how the
in-plane electric field of the Γ-point mode circulates nearly tangential to the QW (Figure 2.4(a)),
it becomes readily apparent that the interaction Hamiltonian in Fermi’s Golden Rule is strongly
enhanced. Looking at the electric field intensity in Figure 2.4(b), it is also apparent that the
electric field intensity of the mode is largest in regions containing the QW. Each of these effects
combine to make this Γ-point mode an ideal candidate for enhancing the spontaneous emission
rate in core-shell nanorod array geometries.

Fig. 2.3 (a) TDP and (b) EE for the square nanorod array (r/a = 0.268, h/a = 0.5) and the in-filled reference slab
(solid and dashed line, respectively).

Figure 2.3(b) shows the extraction efficiency (EE) for both the nanorod array and reference
slab. The EE of the slab is low due the relatively large index contrast between it and the
22

surrounding air. Consequently, all power emitted at an angle 𝜃 𝑐 ≤= sin
−1
(𝑛 𝑎𝑖𝑟 𝑛 𝐺𝑎𝑁
⁄ ) is guided
by total internal reflection (TIR) and unable to escape from within the slab. The EE of the
nanorod array is considerably larger, achieving near-perfect light extraction above 𝜔𝑎 2𝜋𝑐 ⁄ =
0.697. The high light extraction is attributed to the fact that nearly all guided resonance modes
above this onset frequency are above the light-line. However, there is a pronounced dip in the EE
coinciding with the Γ-point mode (green, dashed line). Despite this dip, the total amount of
power collected, or, extracted power (EP = TDP x EE) is nearly 25 X larger than the infilled
reference slab.

Fig. 2.4 (a) In-plane electric field vector (b) electric field intensity and (c) H
z
field profile for the
lowest order Γ-point mode of a square GaN nanorod lattice with r/a = 0.268 and h/a = 0.5

Careful inspection of Figure 2.4(a) (or Figure 2.4(c)) sheds light onto why there is a
pronounced dip in light extraction for this resonant mode. This particular Γ-point mode is
theoretically uncoupled: forbidden by symmetry to couple to normal radiation [58]. Such
uncoupled modes have been shown to produce donut shaped emission beams in the far-field zone
[59]. Please refer to Section A.2 for a more thorough description of uncoupled and coupled
modes.

23

2.4 Internal and External Quantum Efficiency

The TDPs and EEs are directly related to the resulting internal and external quantum
efficiencies. The internal quantum efficiency (IQE) is defined as 𝜂 𝐼 = 𝛤 𝑟 𝛤 𝑟 + 𝛤 𝑛𝑟
⁄ , where 𝛤 𝑟 and
𝛤 𝑛𝑟
refer to the radiative and non-radiative recombination rates. The radiative recombination rate
is proportional to the TDP calculation in the FDTD simulations. A radiative recombination rate
enhancement factor is defined as

𝜒 𝑟 =
𝛤 𝑟 𝑛𝑎𝑛𝑜 𝛤 𝑟 𝑠𝑙𝑎𝑏 =
𝑇𝐷𝑃 𝑛𝑎𝑛𝑜 𝑇𝐷𝑃 𝑠𝑙𝑎𝑏

where the superscripts refer to the nanostructured array and reference slab, respectively. An
internal quantum efficiency enhancement (IQEE) can further be defined as

𝐼𝑄𝐸𝐸 =
𝜂 𝐼 𝑛𝑎𝑛𝑜 𝜂 𝐼 𝑠𝑙𝑎𝑏 = 𝜂 𝐼 𝑛𝑎𝑛𝑜 +
𝜒 𝑟 𝜒 𝑛𝑟
(1 − 𝜂 𝐼 𝑛𝑎𝑛𝑜 )

where 𝜒 𝑛𝑟
= 𝛤 𝑛𝑟
𝑛𝑎𝑛𝑜 𝛤 𝑛𝑟
𝑠𝑙𝑎𝑏 ⁄ . The value of 𝜒 𝑛𝑟
depends upon the bulk and surface defect states in
the two structures. This quantity may be larger in the nanorod system, for instance, due to the
increase in surface area from nanostructuring. For a high-quality material (𝛤 𝑛𝑟
𝑛𝑎𝑛𝑜 ≪ 𝛤 𝑟 𝑛𝑎𝑛𝑜 )
where non-radiative recombination is negligible, 𝜂 𝐼 𝑛𝑎𝑛𝑜 → 1 and the IQEE also approaches one.
This is a consequence of the nanorod array and slab having the same internal quantum
efficiencies. Alternatively, for a low quality material (𝛤 𝑛𝑟
𝑛𝑎𝑛𝑜 ≫ 𝛤 𝑟 𝑛𝑎𝑛𝑜 ) , 𝜂 𝐼 𝑛𝑎𝑛 𝑜 → 0 and the
IQEE is equal to 𝜒 𝑟 𝜒 𝑛𝑟
⁄ . In this case, the nanorod array enhances the IQE, provided that the
radiative enhancement is larger than any non-radiative enhancement.
The external quantum efficiency (EQE) is defined as the product of the IQE and light
extraction efficiency (𝜂 𝐸 = 𝜂 𝐼 × 𝐸𝐸 ). An external quantum efficiency enhancement EQEE is
then defined to be

𝐸𝑄𝐸𝐸 =
𝜂 𝐸 𝑛𝑎𝑛𝑜 𝜂 𝐸 𝑠𝑙𝑎𝑏 = 𝐼𝑄𝐸𝐸 × 𝜒 𝐸𝐸

24

where 𝜒 𝐸𝐸
= 𝐸𝐸
𝑛𝑎𝑛𝑜 𝐸𝐸
𝑠𝑙𝑎𝑏 ⁄ is defined to be the extraction efficiency enhancement. In the
limit of a high quality material, the IQEE approaches one and thus the EQEE is determined
solely by 𝜒 𝐸𝐸
. In this limit, the nanorod array produces an EQEE near to 7 X due to enhanced
light collection. However, for a poor quality material, the EQEE approaches a value equal to
(𝜒 𝑟 𝜒 𝑛𝑟
⁄ )𝜒 𝐸𝐸
. In this case, the nanorod array increases the EQE by increasing both the radiative
recombination rate within the structure in addition to the extraction efficiency. The EQE of the
nanorod structure will be greater than that of the slab provided that any increases in the
nonradiative recombination rate in the nanorod array does not outweigh these combined effects.
For the geometries considered, the EQE of the nanorod structure is enhanced by a factor of 25 X
provided the nonradiative recombination rate is comparable to that of the reference slab (𝜒 𝑛𝑟
=
1) . In this manner, the nanorod array is capable of significantly increasing the EQE by
simultaneously enhancing the radiative recombination rate and light extraction efficiency.
However, it should be noted that the reference slab structure was chosen so that the active area
(or, number of emitters) were equivalent. Compared to axial c-plane geometries, even higher
brightness may be possible due to the potential increase in emitter volume.

25

Chapter 3 – Optimizing Emission in Nanorod Arrays through
Quasi-Aperiodic Inverse Design

3.1 Simple Periodic and Quasi-Aperiodic Arrays

In previous work (please refer to Chapter 2), it was shown that guided-resonance modes can
be leveraged in III-nitride photonic crystal nanorod arrays to enhance the external quantum
efficiency (EQE). This efficiency enhancement was accomplished in two ways. First, high
electric field intensity was concentrated near the active region, thereby enhancing the
spontaneous emission rate and improving the internal quantum efficiency (IQE). Second, the
guided-resonance mode supported by the array was leaky, enabling light to be extracted from the
slab, thereby improving the light extraction efficiency (EE) and EQE. However, the resulting
emission rate enhancement (on resonance) was also met by a sharp dip in extraction efficiency.
This behavior can be attributed to the uncoupled nature of the mode: its field symmetry strictly
prohibits it from coupling to emission along the normal direction. Please refer to Section A.2 for
a more detailed discussion on coupled and uncoupled modes.
Breaking symmetry has recently been shown to couple modes that were otherwise
uncoupled [60]. Common approaches incorporate asymmetry within each unit cell. For instance,
work in photonic crystal surface-emitting lasers (PCSELs) showed that the use of asymmetric
holes increased normal emission [59]. Another approach for breaking symmetry is to use quasi-
aperiodic structures, where the unit cell is aperiodic, but the overall structure remains periodic on
a larger scale. Such structures have been shown to significantly enhance absorption for normally
incident light [61, 62]. The quasi-aperiodic approach is particularly useful for systems where the
basic repeating unit (such as a nanorod) is constrained to be symmetric, for example by material
growth considerations [27]. However, no work has explored the impact of quasi-aperiodic
designs on LED emission.
In this work, we present a quasi-aperiodic nanorod design optimized for emission. A version
of this work was published in Ref. [63]. In the work, we adopt an inverse design procedure
motivated by Lorentz reciprocity and a random walk optimization algorithm. By changing the
26

position of the emitting structures, we are able to achieve nearly perfect light extraction while
maintaining a high spontaneous emission rate. Moreover, the quasi-aperiodic design is shown to
redistribute the original resonance without sacrificing broadband emission. Direct emission
calculations are used to verify that the optimized design produces a 20% – 42% increase in the
EQE relative to a simple periodic design, depending on material quality.

Fig. 3.1(a) Schematic of a core-shell nanorod array. Gray regions are GaN and red regions are InGaN QWs. (b)
Electric field intensity for a mode at ωa ∕ 2πc = 0.697. Solid and dashed hexagonal outlines mark the boundary
of the nanowire and position of the QW, respectively.

We begin by considering the GaN system shown in Figure 3.1(a). The system consists of an
array of hexagonal core-shell GaN nanorods. Each GaN nanorod (gray) contains an In
x
Ga
1-x
N
QW (red). The lattice spacing is a, the rod height is h, and the center-to-vertex length of the
nanorod cross-section is r. For the structure considered here, a = 280 nm, r∕a = 0.268, and h∕a =
0.5. The selection of these lattice parameters aligns a guided resonance mode with the emission
wavelength of the structure (please refer to Chapter 2). Figure 3.1(b) show the electric field
intensity profile of this resonant mode. In the figure, solid black lines mark the outline of four
hexagonal nanorods while dashed black lines indicate the position of four core-shell QWs.
27

Fig. 3.2. (a) Cross-sectional positions of nanorods for a periodic (black hexagons) and complex (red hexagons) unit
cell. (b) FOM for each iteration of the random walk optimization algorithm. (c) Schematic of the optimized, quasi-
aperiodic array. (d) Electric field intensity profile at a frequency of ωa ∕ 2πc = 0.706.

In order to preserve high electric-field intensity within the QW while increasing extraction
efficiency, we consider quasi-aperiodic structures. Starting with a 2 × 2 periodic unit cell, we
shift the rod positions. An example is shown in Figure 3.2(a). In the figure, red lines show the
outline of four nanorods in a quasi-aperiodic, complex unit-cell. While the unit cell appears
aperiodic, the overall structure is now periodic with lattice constant 2a.

3.2 Forward and Inverse Simulations

To model emission from the nanorod structures, we can use a conventional forward
simulation approach, illustrated in Figure 3.3(a). In the forward simulation approach, an electric
dipole source is placed within a QW and the emitted field is observed in the far-field zone [44,
54]. Since LED emission is incoherent, each simulation includes a single dipole. Multiple
simulations are run with the dipole in different locations. Fluxes and field intensities are then
incoherently summed over all simulations. For quasi-aperiodic arrays, the number of forward
simulations scales by a factor of N, the number of rods in the complex unit cell.
28

In order to optimize the quasi-aperiodic arrays, we instead use a quicker inverse simulation
method motivated by Lorentz reciprocity [64 – 66]. A detailed derivation of this electromagnetic
principle is included in Section A.3. Using this reciprocal relationship, the dipole source is now
placed in the far-field zone and the electric field is monitored within the structure, as shown in
Figure 3.3(b). The fields can now be recorded at multiple QW locations in a single simulation,
greatly reducing computational cost. Moreover, we can locally approximate the incident field as
a plane wave and use periodic boundary conditions, appreciably reducing simulation size. This
approach allows us to quickly identify best-guess, optimized structures.

Fig. 3.3 (a) Forward simulation: a dipole models emission from a QW and the electric field is observed in the far-
field zone. (b) Inverse simulation: a dipole emits from the far-field zone and its electric field is observed in the QW.

3.3 Random-Walk Optimization

We use a random walk algorithm to maximize the integrated field intensity within the QWs
in the inverse simulations. A detailed description of the random-walk optimization procedure is
presented in Section A.4. At each step in the random walk, one nanorod is randomly selected and
shifted to a new, random position within the cell. We then run an inverse simulation. As a figure
of merit (FOM), we integrate the electric field intensity over the QW region and normalize by
the same quantity calculated for the original, simple periodic structure. If moving the rod
increases the FOM, the rod is left in the new position. If the FOM decreases, the rod is returned
to its former position. The procedure is iterated until the FOM converges. Simulations are
performed using the FDTD Lumerical Solutions package [56]. The material dispersion of GaN is
29

included in the simulations. For the inverse simulations, we used a plane wave with a normalized
frequency ωa ∕ 2πc = 0.697 and a FWHM Δω ∕ ω
per
= 0.075, where ω
per
refers to the resonant
frequency of the periodic array. We consider three azimuthal polarizations, each with its electric
field tangential to the QW.
Figure 3.2(b) shows the evolution of our FOM as a function of iteration in the random walk.
A maximum value of nearly 1.60 is obtained after approximately 60 iterations. Figure 3.2(c)
shows a schematic of the optimized quasi-aperiodic array. The emergence of selective area
growth, partnered with electron beam lithography, permits the growth of such quasi-aperiodic
arrays [27, 28]. Figure 3.2(d) shows the electric-field intensity distribution for the Γ-point mode
of the optimized geometry. High-intensity features are still present within the QWs, while the
mode profile has lost the mirror symmetry present in the original geometry.

3.4 Emission Calculations

Next, we perform forward simulations to compare the overall emission produced by the
optimized and simple periodic arrays. Please refer to Section 2.2 for a thorough description of
the definitions used in this forward simulation process. For both geometries, we impose perfectly
matched layers on all boundaries. Each finite lattice contains 9 × 9 unit cells. The center
frequency of the dipole source is selected to align with the Γ-point mode of each structure (field
intensity profiles are shown in Figures 3.1(b) and 3.2(d)). The FWHM of each dipole emitter is
Δω ∕ ω
per
= 0.075. Simulations are performed with dipoles at six different cross-sectional
positions and three heights (where symmetry is exploited to reduce the number of computations).
The dipole polarization is chosen to be tangential to the QW.
The radiative recombination rate is proportional to the local density of states (LDOS), which
is given by 4ε ∕π times the total dipole power (TDP). Figure 3.4(a) shows the TDP produced by
both the optimized (red line) and simple periodic (black line) arrays, normalized to the maximum
value of bulk GaN emission at each frequency. Dashed lines indicate the center frequency of
guided-resonance modes in each structure. The periodic array produces a tall sharp peak,
indicative of the uncoupled Γ-point mode in Figure 3.1(b). The guided-resonance mode in the
optimized structure is slightly detuned from the mode of the original structure, with its amplitude
also reduced. However, integrating the TDP over the frequency range shown in the figure yields
30

less than a 1% difference between the two structures; this result suggests the optimized structure
has redistributed the original resonance.

Fig. 3.4 (a) TDP (b) EE and (c) EP for the optimized (red line) and periodic (black line) structures. Black and red
dashed lines indicate the frequency of the Γ-point mode of the periodic and optimized structures.

In addition to maximizing the spontaneous emission rate, it is also essential to extract the
TDP from the structure for use. The extraction efficiency (EE) is a measure of how much TDP
within the QW is collected above the array. We calculate the EE by using flux monitor planes
positioned slightly above and below the structure and dividing their captured flux by the TDP.
Figure 3.4(b) shows the EE for both the optimized and periodic array. A dip in EE is visible at
the resonant frequency of the periodic array. This dip is attributed to the guided-resonance mode
being uncoupled to the normal direction. The EE in the optimized structure is considerably larger
than that of the periodic array. More importantly, due to the symmetry present in our structure,
the optimized array achieves an EE of nearly 100% at its guided-resonance mode frequency.
Figure 3.4(c) shows the extracted power (EP) for both the optimized and periodic array.
EP is the product of TDP and EE and is calculated using the flux monitor planes described
previously. The EP of the optimized structure is significantly larger than that of the periodic
array. If we integrate the total EP over the frequency range shown in the figure, we find that the
optimized structure produces nearly 1.42 X more extracted power than the periodic array.

3.5 Internal and External Quantum Efficiency

Our numerical results allow us to compare the internal and external quantum efficiencies
of the two structures. The internal quantum efficiency (IQE) is again defined as 𝜂 𝐼 =
31

𝛤 𝑟 𝛤 𝑟 + 𝛤 𝑛𝑟
⁄ , where 𝛤 𝑟 and 𝛤 𝑛𝑟
refer to the radiative and non-radiative recombination rates. The
radiative recombination rates are proportional to the TDP produced in the FDTD simulations. An
internal quantum efficiency enhancement (IQEE) is defined as

𝐼𝑄𝐸𝐸 = ∫ 𝜂 𝐼 𝑜𝑝𝑡 𝑑𝜔 ∫ 𝜂 𝐼 𝑝𝑒𝑟 𝑑𝜔 ⁄

This definition is somewhat dissimilar to that presented in Chapter 2, as we are now examining
the contributions of many frequency components (not exclusively the resonant frequency). Since
both the periodic and optimized arrays have the same size and material composition, it is
reasonable to assume a similar 𝛤 𝑛𝑟
in each structure. Figure 3.5(a) shows the IQEE plotted as a
function of 𝛤 𝑛𝑟
𝛤 𝑟 𝑏𝑢 𝑙 𝑘 ⁄ , where 𝛤 𝑟 𝑏𝑢𝑙𝑘 is the radiative recombination rate in bulk GaN. For a high
quality material (𝛤 𝑛𝑟
≪ 𝛤 𝑟 𝑏𝑢𝑙𝑘 ) the IQEE is equal to one. As the material quality decreases, the
IQEE asymptotically approaches a value of nearly 0.99. Overall, the periodic and optimized
structures have very similar IQEs, regardless of material quality. This is expected, however,
since the integrated TDPs were nearly identical and equivalent non-radiative recombination rates
were assumed.

Fig. 3.5 (a) IQEE and (b) EQEE for the competing geometries as a function of the non-radiative rate.

The external quantum efficiency (EQE) is defined similar to before as 𝜂 𝐸 = 𝜂 𝐼 × 𝐸𝐸 . A
quantity referred to as the external quantum efficiency enhancement (EQEE) is further defined as

𝐸𝑄𝐸𝐸 = ∫ 𝜂 𝐼 𝑜𝑝𝑡 × 𝐸𝐸
𝑜𝑝𝑡 𝑑𝜔 ∫ 𝜂 𝐼 𝑝𝑒𝑟 × 𝐸𝐸
𝑝𝑒𝑟 𝑑𝜔 ⁄
32

Figure 3.5(b) shows the plotted EQEE as a function of 𝛤 𝑛𝑟
𝛤 𝑟 𝑏𝑢𝑙𝑘 ⁄ . For a high quality material,
the EQEE is almost 1.20, indicating the optimized structures produces an EQE that is nearly 20%
larger than that of the periodic structure. As the material quality worsens, the EQEE steadily
increases, asymptotically approaching a value of 1.42. This result indicates that the optimized
structure has an EQE that is 20% - 42% better than the periodic structure, depending on material
quality. Consequently, the optimized structure further improves upon the results presented in
Chapter 2, making it an interesting candidate to further enhance the quantum efficiency of
nanoemitter systems.

33

Chapter 4 – Tailoring Emission in III-Nitride Nanorod Arrays
through Quasi-Aperiodic Design

4.1 Coupled and Uncoupled Modes in Nanoemitters

Interestingly, optical modes can exist above the light cone that are forbidden by symmetry to
couple into the radiation continuum [58]. Such, uncoupled modes have found extensive use in
photonic crystal surface emitting lasers (PCSELs), owing to their high quality factors and low
lasing thresholds [59, 67]. For a thorough discussion of coupled and uncoupled modes, please
refer to Section A.2. However, uncoupled modes also yield cylindrical far-field patterns referred
to as donut beams, which are often undesirable. Previous work has demonstrated improving the
far-field uniformity of uncoupled modes by breaking the symmetry of the emitting structure [59,
67]. Specifically, a square array of circular holes in a photonic crystal slab was tailored into a
square array of triangular holes [59, 67]. Recently, much nanoemitter work has turned its
attention to III-nitride material systems [27 – 31].

Both bottom-up and top-down fabrication
strategies have recently been utilized to produce highly anisotropic nanorod arrays in these
material systems [27, 30, 31].

However, using these fabrication methods make it difficult to
arbitrarily change the shapes of nanorods due to growth considerations and etch-plane
competition [27, 30, 31]. Consequently, new methods for improving the far-field uniformity of
uncoupled modes in III-nitride nanorod systems are of interest to a variety of lighting and lasing
applications.
In previous work (please see Chapter 3), we theoretically investigated quasi-aperiodic
nanorod arrays in a III-nitride emitter system.

In quasi-aperiodic arrays, symmetry is broken by
introducing aperiodicity on a small scale and repeating the small-scale aperiodicity on a larger
scale. In our theoretical work, we predicted that quasi-aperiodic arrays could be leveraged to
improve light extraction without sacrificing spontanous emission [63].

In the present work, we
experimentally demonstrate the tailoring of emission using quasi-aperiodic nanorod arrays.
Beginning with a III-nitride epitaxial structure, both simple periodic and quasi-aperiodic nanorod
arrays are fabricated using a top-down fabrication procedure. The competing topologies are then
characterized using room-temperature micro-photoluminescence (µ-PL) measurements. Optical
34

resonances are found in both the simple periodic and quasi-aperiodic arrays, with quality factors
matching theoretical predictions. Next, far-field measurements confirm that quasi-aperiodic
arrays can be leveraged to drastically improve the emitted far-field uniformity. Input vs. output
power measurements are carried out, with the simple periodic resonance demonstrating a strong
non-linear input vs. output power relationship while the quasi-aperiodic resonance reaches the
onset of non-linear behavior. Finally, we theoretically demonstrate how quasi-aperiodic arrays
offer potential for decreasing distributed in-plane feedback, a quantity critical to large area
PCSEL design. Based on our results, we believe that quasi-aperiodic arrays offer a promising
route for increasing the far-field uniformity and decreasing the distributed in-plane feedback
strength in nanoemitter systems.
We begin by considering the III-nitride system shown in Figure 4.1(a). The system consists
of a square array of cylindrical nanorods; we will refer to this structure as a simple periodic
array. Each nanorod is composed of GaN (green) and contains five axial In
x
Ga
1-x
N quantum
wells (black). The lattice spacing is a, the rod height is h, and the nanorod radius is r. For the
structures studied here, we choose a = 240 nm – 280 nm, r/a = 0.167 - 0.25 and h = 600 nm. The
selection of these lattice parameters aligns a series of uncoupled, guided-resonance modes within
the emission bandwidth of our structure.

Fig. 4.1 (a) Simple periodic and (b) quasi-aperiodic array devices. Green indicates GaN, black indicates the
position of InGaN QWs and dashed white lines indicate the boundaries of complex unit cells.

Figure 4.1(b) shows an alternative nanorod system, which we refer to as a quasi-aperiodic
array. In the figure, dashed white lines indicate the boundaries of four complex unit cells, each
containing four nanorods. Each complex unit cell in the quasi-aperiodic array is asymmetric, but
the overall structure remains periodic on a larger scale. The nanorod height and radius are the
35

same as for the simple periodic array, and the lattice constant of the complex unit cell is double
that of the irreducible, simple periodic array.

4.2 Epitaxial Growth

The epitaxial structures used in this work were grown on 2” c-plane sapphire using a Veeco
D-125 metal-organic vapor phase epitaxy (MOVPE) reactor. The epitaxial layers consist of a 4
µm thick GaN buffer layer, a 185 nm InGaN underlayer, a five layer axial In
x
Ga
1-x
N quantum
well (QW) structure emitting near λ = 450 nm and, finally, a 200 nm GaN cap. Device
fabrication is illustrated in Figure 4.2, and follows a top-down recipe developed at Sandia
National Labs [30, 31].

4.3 Top-Down Nanorod Array Fabrication

Fig. 4.2. Device fabrication procedure: (a) EBL patterning of PMMA followed by MIBK:IPA development (b)
Ni deposition followed by liftoff (c) Cl-based dry etch (d) KOH-based wet etch followed by H
2
SO
4
–based Ni
removal. Green indicates GaN, blue indicates PMMA, silver indicates Ni and black lines indicate an InGaN
QW structure.

36

We begin device fabrication by spinning polymethylmethacrylate (PMMA) resist onto our
sample and patterning an array of holes using electron beam lithography (EBL). Following
development, the patterned holes are filled with Ni in an electron beam evaporator. Next, Ni
islands are formed via lift-off in an acetone bath. Lift-off is followed with an inductively coupled
plasma (ICP) etch, utilizing a Cl
2
/BCl
3
-based etch chemistry. The Ni islands function as a hard
mask during the dry etch, leading to the formation of tapered nanorods. To improve nanorod
anisotropy, the devices are immersed in a KOH-based solution at an elevated temperature.
Finally, the remaining Ni is removed using diluted H
2
SO
4
.
Figures 4.3(a) and 4.3(b) show scanning electron micrographs (SEMs) for fabricated simple
periodic and quasi-aperiodic nanorod arrays. Insets show overhead views of the complex unit
cells in each respective structure. Black scale bars represent a physical lengths of 500 nm. SEMs
reveal cylindrical nanorods with smooth vertical sidewalls. Typical device size was 20 μm x 20
μm.

Fig. 4.3 SEMs for a single (a) simple periodic and (b) quasi-aperiodic device. Insets show complex unit cells in each
structure. Black markers indicate a physical length of 500 nm.

4.4 Device Characterization

Fabricated devices were optically characterized at room-temperature using an ultraviolet
micro-photoluminescence (µ-PL) setup. A schematic of the µ-PL setup is included in Figure 4.4.
Our excitation source was a quadrupled Nd:YAG laser with a peak emission wavelength of 266
nm, a 10 kHz repetition rate and pulse lengths of 500 ps. The laser’s peak power density was
37

adjusted using a tunable neutral density filter. A 50X Mitutoyo deep-UV objective focused the
laser to a spot size of nearly 5 µm onto the sample. The device was imaged using the same
objective and directed into a 300 mm spectrometer and liquid N
2
cooled CCD camera.

Fig. 4.4 Experimental setup for the μ-PL illumination path (left) and pump path (right).

Figure 4.5 shows resonances in both a simple periodic (black) and quasi-aperiodic (purple)
structure. The simple periodic resonance is situated near the InGaN underlayer’s gain window
whereas the quasi-aperiodic resonance is red-shifted towards peak QW emission. The quality
factors (Qs) of the resonances differ significantly, with measured values of nearly 390 and 40 for
the simple periodic and quasi-aperiodic resonances, respectively. The significant difference in
quality factors arises from the symmetry of the optical modes that are supported in each
structure. The simple periodic structure supports a resonance which is forbidden by symmetry to
couple out of the structure as normal radiation, and thus lends itself to smaller losses and larger
quality factors; such modes are commonly referred to as uncoupled modes, and manifest
themselves as donut beams in the emitted far-field. The short range aperiodicity in the quasi-
aperiodic structure, however, breaks the mirror symmetry present in the original geometry.
Consequently, the field symmetry of the optical mode is also broken, enabling light to escape
which was otherwise forbidden by symmetry to do so. The increase in light extraction manifests
38

itself as an additional loss, decreasing the overall Q. Theoretical Qs were also calculated using
the finite-difference time-domain method (Lumerical FDTD Solutions [56]). The calculated Qs
are similar to those measured, with values of 442 and 129, respectively. It should be noted that
the quasi-aperiodic array displays a larger discrepancy between its measured and calculated Qs.
This discrepancy likely arises due to close nanowire proximity, which results in additional etch-
plane competition during the KOH-based wet etch. The etch-plane competition effectively
reduces the length of the nanorods, as evidenced from careful inspection of the nanowire bases in
Figure 4.3(b).
The right inset of Figure 4.5 shows the emitted far-fields of each resonance, confirming their
uncoupled and coupled natures directly. The simple periodic resonance (top right), forbidden by
field symmetry to couple light at the Γ-point in reciprocal space, distributes itself as a donut
beam in the emitted far-field. While donut beams have found use in applications which benefit
from vector vortex beams, they are not ideal beam profiles for applications that demand higher
beam quality factors (M
2
s) [67]. Using a quasi-aperiodic array, however, it is possible to
significantly improve light extraction near the Γ-point. Here, the small-scale aperiodicity helps
eliminate the field symmetry present in the original geometry. As a consequence, the resulting
far-field pattern is tailored into a much more uniform profile (bottom right).

Fig. 4.5 (Left) Room temperature µ-PL for a simple periodic (black) and quasi-aperiodic (purple) device. Dashed
lines indicate the center of a resonance in each device. (Right) Emitted far-field patterns for the simple periodic (top)
and quasi-aperiodic (bottom) devices.
The output power of each optical resonance was also studied as a function of input pump
power. Figure 4.6(a) shows an input vs. output power relationship for a simple periodic
resonance. Beginning with a low pump power, the output power of the resonance begins to
39

linearly increase as the pump power increases. Near an input power density of 120 kW/cm
2
,
however, the input-output power relation turns strongly non-linear. The input-output power
relation for a quasi-aperiodic array is shown in Figure 4.6(b). In this case, the input-output power
relationship is linear up to an input power of nearly 1200 kW/cm
2
, and then turns non-linear. The
onset of non-linear input-output power behavior thus requires nearly 10 X more input power than
the simple periodic device. The increased threshold for nonlinear behavior in the quasi-aperiodic
device can be attributed to the additional out-of-plane losses, relative to the periodic structure.
Further increase of the input power resulted in a loss of the quasi-aperiodic resonance, likely due
to optical damage. Recent work, however, has shown that organic sulfide passivation can
improve the PL of III-nitride nanorod arrays by more than an order of magnitude; we believe
such a technique could be used in the future to improve the optical performance of our devices.

Fig. 4.6 Input-output power relations for a resonance in a (a) simple periodic and (b) quasi-aperiodic array. Projected
scale bars mark the peak amplitude of each device resonance.
40

4.5 Photonic Bandstructure and In-Plane Distributed Feedback

Figures 4.7(a) and 4.7(b) show calculated photonic bandstructures for both the simple
periodic and quasi-aperiodic arrays. Here, we assume emission is predominately tangential to the
quantum well plane, and thus calculate the transverse electric (TE), or even-symmetric modes
[68]. The bandstructures are calculated using the finite-difference time-domain method. For the
structures considered, r = 65 nm, h = 200 nm, a = 240 and a
*
= 480 nm. Here the nanorod
heights have been reduced to more clearly illustrate band dispersion. The bands of interest are
shown in red, and extend along the Γ-X direction in reciprocal space. The simple periodic array,
shown in Figure 4.6(a), supports a band which is highly quadratic. Near the Γ-point, the group
velocity (υ
g
= dω/dk) of the highlighted band approaches zero. Such slow light, band-edge modes
have proven to be of great use in nanoemitter design [31, 52, 67]. Aligning the emission
frequency with these modes can significantly improve the spontaneous emission rate [31, 62,
67]. However, slow light modes also pose a unique problem to large area PCSEL design. Since
distributed in-plane feedback is inversely proportional to the group velocity of light, slow group
velocities lend themselves to large in-plane feedback. In effect, this feedback can create separate
areas of coherence, or multiple lasers, within a single device. Solutions have recently been
proposed to decrease distributed in-plane feedback, and enable larger, single areas of coherence
[69, 70].

However, the proposals rely on specialized lattices that introduce an accidental Dirac
point. A possible alternative to this approach, with many degrees of design freedom is, instead,
to use quasi-aperiodic geometries.

Fig. 4.7 Calculated TE modes for the photonic bandstructure of a (a) simple periodic array and (b) quasi-aperiodic
array. Here, r = 65 nm, h = 200 nm, a = 240 and a* = 480 nm, where a refers to the lattice constant of the simple
41

periodic array and a* refers to the lattice constant of the quasi-aperiodic array. Small graphics within each figure
depict the cross-sectional geometries of each respective structure.
Figure 4.6(b) shows the band dispersion for the transverse electric modes in a quasi-
aperiodic array. Here, the bands of interest are outlined in red. As compared to the simple
periodic structure, the dispersion of the quasi-aperiodic array’s modes are much more linear.
This reduction of distributed in-plane feedback likely arises from the doubling of the lattice
constant, relative to the simple periodic array. We note that tailoring photonic bandstructure
dispersion in this manner is not limited to a 2 × 2 complex unit cell, nor to a square lattice. As a
consequence, we believe quasi-aperiodicity offers a unique method for decreasing distributed in-
plane feedback with many degrees of design freedom.

42

Chapter 5 – Selective Quantum Dot Placement and Preferential
Membrane Formation through Photoelectrochemical Etching

5.1 Introduction

Semiconductor quantum dots (QDs) have become an integral light source in the fields of
nanophotonics and quantum science [71 – 75]. Unlike their QW counterparts, III-nitride QDs
possess several advantages unique to their artificial atom-like construction. First, owing to their
small size, which is on the order of a semiconductor’s exciton Bohr radius, quantum-size effects
grow prominent in QD nanostructures. Consequently, QDs contain discrete energy levels that
determine their absorption/emission spectra. Second, QDs can possess a significantly larger
internal quantum efficiency (IQE) resulting from increased carrier confinement and material
strain reduction [36, 37]. As recently demonstrated, these significant IQE enhancements are
possible even for QDs grown along the c-plane [36, 37]. Third, recent work has suggested that
QDs are superior gain mediums, owing to their delta-like electronic density of states [38].
Finally, QDs enable functionality which cannot be tapped into by conventional, classical light
sources. In particular, QDs can exhibit anti-bunched, single photon emission, which enables a
variety of applications in quantum encryption, computing and meteorology.
Currently, several nanofabrication approaches exist for producing semiconductor QDs,
many of which rely upon solution-based, colloidal synthesis [32, 33]. Nevertheless,
nanofabrication techniques that are either bottom-up self-assembly (e.g. Stranksi-Krastanov (S-
K) growth) or top-down are likely to be more suitable for semiconductor-based photonic
applications [33]. However, two principal challenges remain when using such processes: size
control and deterministic placement [33]. For instance, bottom-up growth techniques are
typically accompanied with large fluctuations in QD size and uncertainty in their resulting
locations [33]. As a result, integrating bottom-up growth techniques with nanophotonic cavity
designs face serious challenges and low device yields. Recently, however, one approach has
demonstrated uniform emission from QDs fabricated using a top-down procedure on an epitaxial
structure [34, 76]. QD emission uniformity was achieved using a quantum-size controlled
photoelectrochemical (QSC-PEC) etching technique, similar to those used in earlier solution-
43

based works [32, 33]. QSC-PEC etching utilizes a light-assisted wet etching technique whose
initial step is surface oxidation by photoexcited holes. The photoexcitation depends on
absorption, with absorption depending on the bandgap of the material, and the bandgap
depending on nanostructure size. Thus, QSC-PEC etching is a technique that self-terminates after
quantum-size effects begin to suppress absorption and the photogeneration of carriers. Previous
work demonstrated the feasibility of QSC-PEC etching on an unpatterned III-nitride platform,
containing an uncapped InGaN quantum well (QW) [34]. As a consequence, the resulting QDs
exhibited no preferential placement.
For the first time, we demonstrate deterministic positioning of QDs by first fabricating a
series of III-nitride nanorods. We integrate QDs of controlled size (and hence electronic bandgap
energy) within the pre-patterned nanorods using QSC-PEC etching. An illustration of one
fabricated nanorod is depicted in Figure 5.1. In the figure, the QSC-PEC etching technique has
re-moved the bulk of an axial QW structure, leaving behind a single In
x
Ga
1-x
N QD near its
center. Low-temperature micro-photoluminescence (μ-PL) measurements of fabricated nanorods
reveal sharp spectral signatures (Δλ
FWHM
< 0.3 nm), indicative of QD formation. Further, internal
quantum efficiency (IQE) measurements confirm a near order of magnitude improvement in
emitter efficiency following QSC-PEC etching. Finally, second-order cross-correlation (g
2
(0))
measurements of isolated QD photoluminescence (PL) reveal photon anti-bunching (g
2
(0) ~ 0.5).
Our results illustrate an exciting method for the top-down integration of non-classical light
sources within pre-patterned nanophotonic platforms. The precise spatial and spectral matching
of QDs within high-Q microcavities promises to be of great benefit to the fields of quantum
science and nanophotonics.

Fig. 5.1 Schematic of a fabricated III-nitride nanorod with a QD emitting near its center.

44

5.2 Photoelectrochemical Etching of III-Nitrides

Due to its high chemical stability, GaN is resistive to many wet etches.
Photoelectrochemical etching (PEC) of GaN, however, produces photoexcited electron-hole
pairs which have been shown to assist in electrochemical reactions [78, 79]. When the
semiconductor is immersed in an electrolyte solution there is an exchange of electrons due to the
difference in Fermi levels. When a photon with an energy larger than the bandgap of the
semiconductor is present, the light is absorbed and electron-hole pairs are generated. For n-type
GaN, photogenerated holes accumulate at the surface. The accumulation of holes leads to broken
surface bonds, which contribute to surface oxidation and etching of the GaN. Furthermore, the
intensity of the impinging light can control the speed of the etch process. The following reaction
equation describes this PEC process for GaN [34, 79]:

2GaN + 6h
+
→ 2Ga
3+
+ N
2

A schematic of the QSC-PEC etching setup used in this work is shown in Figure 5.2. During
the etch process, a III-nitride sample is partially submerged in a solution of H
2
SO
4.
An In contact
is bonded to the III-nitride sample and a Pt electrode is suspended in the electrolyte solution. A
tunable, frequency doubled Ti:Sapphire laser with a relatively narrow-bandwidth (~ 1 nm) is
used as the photoexcitation source. A voltage is applied across the In contact and Pt electrode to
attract photoexcited carriers towards the surface of the sample. Since the etching depends on the
photoexcitation of the carriers (and the photoexcitation depends on absorption), QSC-PEC
etching is a bandgap selective etching mechanism.

Fig. 5.2 Schematic of the quantum-size controlled photoeletrochemical (QSC-PEC) etching setup.
45

5.3 Fabrication of Quantum Dots with Deterministic Placement

The epitaxial structures used in the first part of this work were grown using metal-organic
chemical vapor deposition (MOCVD). The wafer consists of a nominally 4.7 μm thick n-GaN
layer grown on a Al
2
O
3
template. This growth is followed with a 185 nm thick In
0.04
Ga
0.96
N
underlayer, a single 2.7nm thick In
0.15
Ga
0.85
N QW layer and capped with 30 nm of n-GaN. For
an unpatterned substrate, the etchant solution can enter through micro/nanoscale growth defects
present in relatively thin GaN capping layers. The QSC-PEC etching of the QW then proceeds
predominately laterally, ultimately creating a non-uniform distribution of QDs with non-
preferential positioning. In the present work, however, we focus our attention on the fabrication
of QDs with predefined locations within a nanostructure. The top-down fabrication process is
illustrated schematically in Figure 5.3. We begin by spinning on polymethylmethacrylate
(PMMA) resist and patterning a series of holes using electron beam lithography (EBL). The
pattern consists of a square array of holes, approximately 100 – 200 nm in diameter and equally
spaced with a pitch equal to 1 μm. Following development, the openings are filled with
approximately 20 nm of Ni via electron beam evaporation. Next, circular Ni islands are formed
by ultrasonicating the sample in an acetone bath. Lift-off is followed by an inductively coupled
plasma (ICP) etch utilizing a Cl
2
/BCl
3
-based etch chemistry, with the Ni islands functioning as a
hard mask during the dry etch. Following the Cl
2
/BCl
3
-based dry etch, the remaining Ni is
removed in a solution of H
2
SO
4
. Next, the nanorod sample is partially submerged into the QSC-
PEC etching setup shown in Figure 5.2. The PEC etch consists of a 0.2 M H
2
SO
4
solution and a
fiber coupled laser source. The excitation source is a frequency doubled Ti:Sapphire laser with a
2 ps pulse width, 82 MHz repetition rate and average output power of 30 mW at a wavelength of
420 nm. The PEC etch was performed for 90 minutes. Scanning electron micrographs (SEMs) of
fabricated samples are included in Figure 5.3(e) and 5.3(f).

46

Fig. 5.3 Device fabrication procedure: (a) EBL patterning of PMMA followed by MIBK:IPA development (b) Ni
deposition followed by liftoff (c) Cl-based dry etch (d) PEC etching. Green indicates GaN, blue indicates PMMA,
silver indicates Ni and a black line indicate an axial InGaN QW. SEMs of (e) an array of nanorods and (f) a
magnified image of a single nanorod. Black scale bars represent a physical length of 500 nm.

5.4 Low-Temperature Photoluminescence

We obtained spectroscopic information of etched nanorods by performing low-temperature
mirco-photoluminescence (μ-PL) measurements. The device sample was housed in a cryostation
that was nominally cooled to a temperature, T = 10 K. The sample was pumped (using the same
tunable laser) at a wavelength of 375 nm and average power of 5 μW. A 50X near-UV Mitutoyo
objective focused the laser to a spot size of nearly 1 µm
2
on the sample. The device was imaged
using the same objective and directed into a 300 mm spectrometer and liquid N
2
cooled charge-
coupled device (CCD) camera.
Emission from a QSC-PEC etched nanorod with a diameter of 200 nm is shown in Figure
5.4(a). In the figure, three peaks are observed at wavelengths of 411.2 nm, 413.4 nm and 417.9
47

nm, with the third peak being most prominent, producing nearly five times the intensity of the
other two when background intensity is subtracted. The narrow linewidths associated with the
peaks (Δλ
FWHM
< 0.3 nm) indicate QD-like emission signatures. Multiple emission peaks,
however, suggest the presence of multiple QDs within a single nanowire. One possible reason
behind this could be due to the relatively large nanorod diameter (~ 200 nm), compared with the
GaN capping layer (~ 30 nm). Consequently, the lateral PEC etch, which is the desired
mechanism, competes with the vertical etching of nitride material that takes place through
unavoidable threading dislocation defects in the GaN capping layer. These competing PEC etch
directions likely results in the formation of multiple QDs in a single nanorod. Furthermore, the
spacing between neighboring nanowires (1 μm) is near the spot-size diameter of the focused
beam. Consequently, misalignment or slight defocusing may contribute to pumping more than
one nanowire simultaneously.
In order to address such challenges, we turn our attention to nanorod arrays featuring smaller
diameters (~ 100 nm) and greater neighbor-to-neighbor spacing (2 μm). Figure 5.4(b) shows the
low-temperature μ-PL associated with one such nanorod. In the figure, a single sharp peak with
narrow peak linewidth (Δλ
FWHM
~ 0.3 nm) is observed at a wavelength near 414.0 nm. This result
highly suggests the formation of a single QD. Further, it should be noted that the QD location
has been fully controlled by the following considerations: 1. The thickness and relative location
of the QW in the original epitaxial structure and 2. The diameter of the nanowire. Careful control
of such parameters offers much promise in spatially and spectrally coupling QSC-PEC etched
QDs within pre-patterned PhC microcavities. In particular, we further note that the diameters of
the nanowires considered here are comparable to the surface areas of L1 PhC defect cavities.
This makes this structure highly suitable for subsequent incorporation into a high-Q PhC cavity.

Fig. 5.4 (a) Low-temperature μ-PL collected from a QSC-PEC etched nanorod with (a) multiple QDs and (b) a
single QD. Dashed blue lines indicate peak emission associated with individual QDs.
48

5.5 Internal Quantum Efficiency

Internal quantum efficiency (IQE) measurements were taken using the low-temperature
μ-PL setup described formerly. To make a comparison between the IQE of QDs and QWs,
Figure 5.5(a) shows the PL corresponding to an unpatterned QSC-PEC etched area (~ 1 μm
2
).
Here, ensembles of QDs are optically pumped so that PL is still observable at room temperature
(T = 300 K). The ensemble QD PL was measured at temperatures of 10 K and 300 K, where it is
assumed that the PL taken at T = 10 K corresponds to perfect IQE. The PL at each temperature is
plotted as a function of pump power (log scale). As the pump power increases the PL intensity
steadily increases, with PL at each temperature saturating near an average input power of 45
mW. Taking the ratio of PL intensities at this input power yields an IQE of nearly 40.2 %. Figure
5.5(b) shows the QD IQE as a function of pump power and compares it against the IQE of an
unpatterned, unetched region of the sample (QW). As the pump power rises, the IQE of each
structure again saturates near an average input power of 45 mW. Upon saturation, the IQE of the
QDs is shown to be nearly an order of magnitude higher than that of the QW. The significant
increase in IQE shows that the etch has assisted in strain relaxation, fundamentally changing the
structure of the emitting material.

Fig. 5.5 (a) Low-temperature and room-temperature PL intensity collected from QD ensembles as a function of
input pump power. (b) IQE comparison for a QD ensemble and QW as a function of input pump power.

49

5.6 Second-Order Cross-Correlation Measurement

The motivation for creating site-controllable quantum dots is for subsequent utilization in a
quantum light source. In order to study that, we investigate the photon statistics of isolated QDs
using a Hanbury-Brown and Twiss (HBT) setup, illustrated in Figure 5.6(a). The setup consists
of low-temperature QD PL that is split into two paths using a 50/50 beamsplitter (BS). Each light
path is optically focused onto two, single-photon avalanche detectors (SPADs). The signal from
each detector is then fed into a time-correlated single photon counting (TCSPC) module. Figure
5.6(b) shows the second-order cross-correlation (g
2
(0)) of low-temperature QD PL collected
under pulsed operation. At a time of zero, there is a large reduction in photon counts, yielding a
g
2
(0) ~ 0.5. The g
2
result clearly indicates photon anti-bunching, a non-classical behavior
associated with single photon emission. However, low light-collection and potential background
emission from defect GaN sites leads to large noise in the measurement. We believe
incorporating QSC-PEC etched QDs into properly designed PhC defect cavities will enable
simultaneous improvement in light-collection and emission rate enhancement, through the
Purcell effect. Improving the emission signatures of QSC-PEC etched QDs using such structures
could greatly improve signal-to-noise ratios and enable the development of room-temperature,
single photon sources.

Fig. 5.6 (a) Hanbury-Brown and Twiss (HBT) setup consisting of QD PL (blue) being split by a 50/50 beamsplitter
(BS) and focused onto two, single-photon avalanche detectors (SPADs). Each signal is fed into a time-correlated
single photon counting (TCSPS) module. (b) Second-order cross-correlation (g2) of low-temperature PL collected
from a single QD. The QD was pumped using a pulsed, tunable laser.

50

5.7 Photonic Crystal Defect Cavity Design

The deterministic positioning of QDs opens the possibility of coupling quantum emitters
with high-Q photonic crystal (PhC) cavities. Interestingly, III-nitride L1 defect cavities possess
diameters comparable to those of the nanorods fabricated in this work. Consequently, L1
photonic crystals offer a potentially exciting route for deterministic placement of QDs within
PhC cavities. Coupling QDs with high-Q cavities promises to greatly enhance the emission rate
of single QDs, enabling strong-coupling interactions and anti-bunched sources with larger
emission signatures. To this end, we begin with a standard hexagonal L1 PhC cavity, whose
parameters are r/a = 0.3 and h/a = 0.93, where r refers to the radius of the holes, h is the height
of the slab and a is the pitch of the lattice. For all theoretical calculations, we make use of the
finite-difference time-domain (FDTD) method for modelling the electromagnetic fields
(Lumerical FDTD Solutions [56]). The PhC lattice is finite in size and terminated with perfectly
matched layers (PMLs). Figure 5.7(a) shows the z-component of the magnetic field for a cavity
defect mode, which is a doubly-degenerate dipole mode. The normalized frequency of the cavity
resonance is 𝑎 𝜆 ⁄ = 0.36. Figure 5.7(b) shows the electric field intensity of the dipole mode. The
mode exhibits an anti-node with high electric field intensity near its center. The Purcell factor of
the mode is given by

𝐹 𝑃 =
3
4𝜋 2
𝑄 𝑉 (
𝜆 𝑛 )
3

where V is the mode volume, Q is the quality factor of the resonance and 𝜆 𝑛 ⁄ is the reduced
wavelength of the cavity resonance. The mode in Figure 5.7 has a Q around 490, producing a
Purcell factor (emission rate enhancement) of nearly 20. However, as shown in Figure 5.7(c), the
emitted far-field distribution of the mode is highly diffractive, which poses serious experimental
challenges on vertical light collection.

51

Fig. 5.7 (a) z-component of the magnetic field (b) electric field intensity and (c) far-field pattern of the original L1
cavity resonance.

In order to improve upon the cavity design, we aim to adjust the position of the six holes
immediately surrounding the cavity, as illustrated in Figure 5.8(a). Such cavity modifications
have been shown to significantly increase the Q-factor of defect PhC modes [80 – 86 ]. The two
holes above and below the cavity are shifted away from the defect in increments along the y-
axis; these shifts are referred to as y-shifts, and effectively increase the size of the cavity.
Similarly, the holes to the left and right of the defect are shifted outward in increments along the
x-axis; these shifts are referred to as x-shifts. Figure 5.8(b) shows the resulting Q-factor of the
mode for various x- and y-shifts. As indicated in the figure, many of the modified cavity designs
produce Q-factors in excess of 3,000 (dark red boxes). However, the Q-factor itself is not the
only metric of interest. Although the Purcell factor (and subsequent spontaneous emission rate)
are proportional to the Q-factor, it alone does not determine how many photons are collected. In
order to complete the picture, we define a second quantity referred to as the collection efficiency
(CE)
𝐶𝐸 =
∬ |𝐸 |
2
𝑠𝑖𝑛 𝜃 𝑑 𝜃 𝑑 𝜑 𝜃 ′
0
∬ |𝐸 |
2
𝑠𝑖𝑛 𝜃 𝑑 𝜃 𝑑 𝜑 𝜋 0

where θ’ is the largest azimuthal angle in the far-field zone from which light can be collected. In
an experiment this upper bound is determined by the numerical aperture of the setup (the sine of
the acceptance angle), which here is assumed to be 0.35.
52

Fig. 5.8 (a) Schematic illustrating the cavity optimization procedure. Holes highlighted in red are shifted outward
tangential to the y-axis while holes highlighted in blue are shifted outward tangential to the x-axis. (b) The Q-factors
(c) collection efficiencies (CEs) and (d) FOMs produced by various L1 cavity modifications. The black dot indicates
the best cavity design.

Figure 5.8(c) shows the CE for the various modified cavities. Interestingly, the CE is found to be
reasonable even in geometries producing high Q-factors. Seeking to weigh the combined effect
of enhanced emission and light collection, we define a figure of merit (FOM) that is equal to the
product of the Q-factor (which approximates the emission rate enhancement) and the CE. In
maximizing this FOM we maximize the total number of photons collected in the far-field. The
resulting FOM is plotted in Figure 5.8(d). The design that maximized the FOM produces a Q-
factor of nearly 3,150.
Figure 5.9(a) shows the emitted far-field pattern produced by the best cavity design. The
emitted far-field is much more uniform than that produced by the original cavity. To directly
calculate how much more light is being collected, we define a quantity called the Far-Field
Enhancement (FFE):
53

𝐹𝐹𝐸 = ∬ |𝐸 |
𝑚𝑜𝑑 2
𝑠𝑖𝑛 𝜃 𝑑𝜃𝑑𝜑 ∬ |𝐸 |
𝑜𝑟𝑖𝑔 2
𝑠𝑖𝑛 𝜃 𝑑𝜃𝑑𝜑 ⁄
The FFE integrates the far-field collected within the modified cavity at a given polar angle and
normalizes it to that collected by the original cavity. Figure 5.9(b) shows the resulting FFE for
various angles. The half angle corresponding to a NA = 0.35 is indicated by a solid black line. As
evident from the figure, the total amount of collected light has significantly improved.

Fig. 5.9 (a) The far-field distribution and (b) Far-Field Enhancement (FFE) for the best cavity design.

Figure 5.10 shows the spatial Fourier transformations of the resonant electric fields in both
the original and modified cavities. In each figure, solid black lines represent the boundary of the
hexagonal Brillouin zone in reciprocal space and dashed black lines indicate the boundary of the
light cone. Larger spatial Fourier components are present within the light cone of the modified
cavity. The larger spatial Fourier components lead to larger light extraction. Here we note that
the distributions within each light cone strongly resemble the corresponding emitted far-field
patterns.

Fig. 5.10 Spatially Fourier transformed modes of the (a) original cavity and (b) best cavity design.
54

5.8 Preferential Membrane Formation

Fig. 5.11 Device fabrication procedure: (a) PECVD of SiO
2
(b) EBL patterning of PMMA followed by MIBK:IPA
development (c) F-based dry etch followed by a BOE bath (d) Cl-based dry etch (e) PEC etching. Light green
indicates GaN, dark green indicates SiO
2
, blue indicates PMMA, a single black line represents a SQW structure and
a series of black lines represents a MQW structure. (f) SEM of a fabricated device. The black scale bar represents a
physical length of 500 nm.

In order to experimentally create a membrane structure, we begin with the epitaxial
structure shown in Figure 5.11(a). The epitaxial structures was grown using metal-organic
chemical vapor deposition (MOCVD). The wafer consists of a nominally 4.7 μm thick n-GaN
layer grown on a Al
2
O
3
template. The buffer layer growth is followed with a 150 nm thick In
0.03

Ga
0.97
N underlayer and a fifteen MQW structure, featuring 2.7 nm thick In
0.16
Ga
0.84
N QW
layers and 8 nm thick n-GaN barriers. Peak emission associated with the MQW is centered near
a wavelength of 460 nm. The MQW growth is followed with a 70 nm thick n-GaN layer. Next, a
single 2.5 nm thick In
0.10
Ga
0.90
QW layer is grown and capped with 70 nm of n-GaN. Peak
emission associated with the SQW is centered near a wavelength of 410 nm.
Device fabrication begins with a plasma-enhanced deposition of SiO
2
. Next, we spin
polymethylmethacrylate (PMMA) resist onto our sample and pattern a series of holes using
55

electron beam lithography (EBL). The patterns considered here are of the modified L1 cavity
design that produced the largest FOM. Following development, a F
3
-based reactive ion etch
(RIE) transfers the pattern from the PMMA to the underlying SiO
2
. Next, an inductively coupled
plasma (ICP) Cl
2
/BCl
3
-based etch transfers the photonic crystal pattern to the III-nitride sample.
Finally, a PEC etch is performed for 150 minutes at a wavelength of 450 nm to selectively etch
the MQW structure. Here, we use the same setup as described in Section 5.2.
Figure 5.12 shows room-temperature μ-PL measurements taken from unpatterned (a)
unetched and (b) PEC etched regions of the sample. The PL was collected at various PEC etch
durations. As shown in Figure 5.12(a), the unetched area produces broad emission peaks
associated with the SQW and MQW structures (wavelengths near 410 nm and 460 nm,
respectively). Even as the PEC etch duration lengthens, no significant differences in emitted PL
are observed. However, Figure 5.12(b) illustrates a different result for the unpatterned PEC
etched region. Before any etching takes place (t = 0 min.), two broad peaks are again observed
near wavelengths of 410 nm and 460 nm. However, as the etch duration lengthens, the peak
associated with the broad MQW emission reduces; the reduction in PL intensity results from the
MQW structure being selectively etched away. Following 150 minutes of etching, emission from
the MQW structure is unobservable, indicating the material has been successfully etched.
Despite significant etching of the MQW structure, emission associated with the SQW structure is
uninhibited by the PEC etches. Consequently, a membrane-like structure has been preferentially
formed through PEC etching. These results suggest an interesting method for preferentially
fabricating membrane structures in III-nitrides while also preserving active layers of lower In
composition(s).

Fig. 5.12 Room-temperature PL measurements of an (a) unetched reference region and (b) etched reference region
after various PEC etch durations. The dashed line in each figure coincides with the wavelength of the laser used
during the PEC etch.
56

Figure 5.13 shows room-temperature μ-PL measurements taken after the complete 150
minute PEC etch duration. PL is collected from an unetched reference region (red), etched
reference region (purple) and etched device region (black). The upper inset in the figure shows a
bright-field optical micrograph (OM) of the sample. Each of the measured regions are indicated
with dots of the associated color. For the unetched reference region, we again observe broad PL
near wavelengths of 410 nm and 460 nm, associated with peak SQW and MQW emission,
respectively. However, the MQW peak is unobservable for the etched reference region, verifying
selective etching of the MQW structure. Finally, PL collected from a PhC device region reveals
the presence of a strong resonance near a wavelength of 385 nm. This resonant peak is a cavity
mode of the L1 PhC and was unobservable preceding membrane formation. Additionally, broad
SQW emission is still present in the PhC device’s PL. Maintaining the SQW widens the gain
window beyond what would be possible with only GaN. Our fabrication strategy allows wide
tuning of cavity resonances, spanning from the near-UV into the far-blue regions of the visible
spectrum.

Fig. 5.13 Room-temperature μ-PL measurements for various regions on a PEC etched sample. Red represents an
unetched reference region, purple represents an etched reference region and black represents an etched device
region. The inset shows an OM of the sample. Red, purple and black markers represent the unetched reference
region, etched reference region and etched device, respectively.

5.9 Future Work: Further Cavity Optimization

To further improve upon the theoretical design, we use a random-walk optimization
algorithm. The optimization process is illustrated in Figure 5.14. Starting with the modified
cavity design (Figure 5.11(f)), certain holes are grouped into families according to symmetry
57

considerations. The three families are indicated by red, blue and purple in the figure. During the
random walk algorithm, a family of holes is randomly selected and their radii are randomly
adjusted. Next, an FDTD simulation calculates the mode’s Q-factor and collection efficiency
(CE). If the random adjustment improves the FOM, the move is retained. If the adjustment
reduces the FOM, the lattice returns to its previous configuration. The optimization algorithm
iterates in this fashion until the FOM converges.

Fig. 5.14 Random walk optimization procedure

Figure 5.15 shows a scanning electron micrograph (SEM) of the optimized cavity (left) as
well as the theoretical far-field pattern its dipole-mode produces (right). The adjustment to the
lattice are slight, involving only small changes in hole size. Although the theoretical collection
efficiency decreases slightly, the Q-factor increases to a value of nearly 8,050. The combined
effect produces a FOM which is nearly 1.7 X better than the previous design.

Fig. 5.15 Fabricated optimized cavity design along with the predicted far-field distribution pattern of the cavity
resonance. Dashed, hexagonal boxes enclose the holes affected by the random-walk optimization.
58

Chapter 6 – Conclusion

In this thesis we have investigated methods for improving the quantum efficiency of III-
nitride nanoemitters. We began by theoretically investigating core-shell, multiple quantum well
(MQW) nanorod arrays. Such structures have generated much interest in the experimental
community due to their potential for avoiding polarization fields that otherwise reduce internal
quantum efficiency (IQE). However, when the arrays maintain pitches on the order of the
wavelength of light, photonic crystal effects dominate. While theoretically investigating one such
square array of nanorods, we identified a guided-resonance mode ideally suited for a core-shell
QW geometry. Using this mode, we were able to simultaneously enhance the spontaneous
emission rate and dramatically increasing the extraction efficiency. Our nanorod array
demonstrated an external quantum efficiency (EQE) enhancement up to 25 X when compared to
an infilled reference slab.
Building upon these theoretical findings, we began to explore potential perturbative methods
for further increasing the EQE of core-shell nanorod arrays. Using an inverse design method and
a random-walk optimization procedure, we were able to identify a quasi-aperiodic structure that
improved upon the quantum efficiency of the first design. Quasi-aperiodic arrays are arrays that
exhibit small-scale aperiodicity but remain periodic on a larger scale. Using an optimized quasi-
aperiodic arrays we were able to demonstrate up to a 42 % further increase in EQE when
compared to the original, simple periodic structure. Our theoretical results marked the first
instance where quasi-aperiodicity was used to improve the EQE of a nanoemitter.
Motivated by our theoretical findings, we then experimentally investigated the potential use
of quasi-aperiodic arrays for tailoring light emission. Using a novel, top-down fabrication
procedure developed at Sandia National Labs, we were able to fabricate highly anisotropic arrays
of simple periodic and quasi-aperiodic nanorod arrays. Starting with a simple periodic array we
used μ-PL measurements to verify the presence of a high-Q uncoupled mode (a mode forbidden
by symmetry to couple to the normal direction). The uncoupled nature of the mode was directly
confirmed by room-temperature micro-photoluminescence (μ-PL) measurements. Being unable
to couple emission along the normal direction, the resonance distributed itself as a donut beam in
the emitted far-field. By introducing small scale aperiodicity, the quasi-aperiodic array broke the
mirror symmetry present in the original array. As a consequence, more light was coupled out of
59

the quasi-aperiodic device, as revealed by a uniform far-field pattern and reduced resonant Q-
factor. Theoretical calculations matched the trends of our measured Qs, further confirming our
results. Next, we performed room-temperature μ-PL input-output power measurements, with the
simple periodic array demonstrating a strong nonlinear relationship, indicative of lasing. The
quasi-aperiodic array reached the onset of a nonlinear input-output power relationship. Finally,
theoretical bandstructure calculations revealed the potential for using quasi-aperiodicity as a tool
for decreasing in-plane distributed feedback, a quantity critical to the design of large-area
photonic crystal surface-emitting lasers (PCSELs). We believe quasi-aperiodic nanoemitters
offer a unique method to tailor the shape of emitted beams and decrease the in-plane feedback of
future PCSELs.
We then turned our attention to quantum-size controlled photoelectrochemical (QSC-PEC)
etching of III-nitride materials. By first pre-patterning nanorods on a III-nitride platform, we
were able to use our wavelength selective QSC-PEC etching method to etch away an In
1
Ga
1-x
N
QW structure. Since QSC-PEC etching depends on absorption, and absorption depends on
bandgap (for small nanostructures), we were able to create size-controlled quantum dots (QDs)
in the center of our nanorods. Low-temperature μ-PL measurements revealed the presence of
sharp spectral features within the QSC-PEC etched nanorods, indicative of QD emission. Next,
measurements revealed that the IQE of the QDs were nearly 10 X larger than that of the
unetched QW; such a result shows the potential in using QDs to reduce strain and the associated
polarization fields in QD nanoemitters. Reducing strain in such nanostructures permits the
increased incorporation of In, effectively shifting higher quantum efficiency to longer
wavelengths. Furher, second-order cross-correlation measurements of single QDs revealed clear
anti-bunching behavior. Our results illustrate an exciting route for the top-down integration of
non-classical light sources within high-Q nanophotonic platforms.
To this end, we next began to theoretically explore the design of an L1 photonic crystal
(PhC) cavity. By introducing an optimization procedure which takes into account the rate of
photon enhancement and efficiency of light collection, we found a modified design which greatly
improved upon our original design. Following our calculations, we started to pursue the
experimental realization of such a cavity. We began with a III-nitride sample consisting of a
large multiple quantum well (MQW) and single quantum well (SQW) structure. Each
heterostructure had different In fractions and associated peak emission wavelengths of nearly
60

460 nm and 410 nm, respectively. Seeking to remove the MQW structure, we PEC etched
samples at a wavelength of 450 nm. Room-temperature μ-PL measurements revealed the
complete removal of emission signatures associated with the MQW structure and preservation of
the SQW emission signatures. Further, we observed strong resonances associated with the
photonic crystal cavity which were not present before the PEC etch. We believe that our methods
involving the deterministic placement of QDs and preferential etching of III-nitride
heterostructures opens up many exciting possibilities for integrating QDs into high-Q
nanophotonic cavities.

61

Appendix

A.1 Guided Modes, Radiation Modes and Guided-Resonance Modes

Consider the propagation of light between an interface of differing dielectric materials. The
refraction of light across the interface can be described using Snell’s law [87]

𝑛 1
sin 𝜃 1
= 𝑛 2
sin 𝜃 2

where 𝑛 𝛼 describes the refractive index of the dielectric and 𝜃 𝛼 describes the angle the light ray
makes with the normal to the interface (𝛼 = 1, 2). Snell’s law follows directly from conservation
of energy and the continuity of the electric field tangential to the interface. In particular, from
phase-matching it can be shown that |𝑘 //
⃗ ⃗ ⃗ ⃗ ⃗⃗
|
1
= |𝑘 //
⃗ ⃗ ⃗ ⃗ ⃗⃗
|
2
or (𝜔 𝑐 ⁄ ) 𝑛 1
sin 𝜃 1
= (𝜔 𝑐 ⁄ ) 𝑛 2
sin 𝜃 2
with
the constant terms of angular frequency (𝜔 ) and the speed of light (𝑐 ) canceling on either side of
the equation. For 𝑛 1
> 𝑛 2
there exists a critical angle of incidence, 𝜃 𝑐 = sin
−1
(𝑛 2
𝑛 1
⁄ ). For
𝜃 1
> 𝜃 𝑐 the solutions for 𝜃 2
become complex. The physical interpretation of such a scenario is
that the light has undergone total internal reflection (TIR) within the higher index medium. Light
which is fully guided within a higher index material (such as a fiber optic cable surrounded by
air) is said to be a guided-mode. Guided modes support regions of harmonic variation within the
higher-index, guided region and exponential decay in the lower-index region. There also exist
radiation modes within such interfaces. Radiation modes do not possess a large enough in-plane
wavenumber (𝑘 //
) along the interface to support index guiding within the higher-index region.
Instead, these modes couple into the radiation continuum. A ray-schematic illustration of both a
guided mode and radiation mode is illustrated in Figure A.1
In a two-dimensional system, the boundary for real and imaginary solutions in the lower
index region (𝑛 2
) is given by 𝑐 = 𝜔 𝑘 //
= 𝜔 √𝑘 𝑥 2
+ 𝑘 𝑦 2
; this boundary is referred to as the light-
line. For regions below the light-line, the only solutions in the lower index medium are
imaginary with 𝑘 𝑧 = ±𝑖 √𝑘 //
2
− 𝑘 2
. These imaginary solutions correspond to the evanescent
fields associated with guided modes. Evanescent fields are exponentially decaying fields that do
62

not produce any net energy flow. For regions above the light-line, the solutions in the lower
index medium are real, with 𝑘 𝑧 = √𝑘 2
− 𝑘 //
2
. These real solutions correspond to propagating
fields in the lower index region.

Fig. A.1 Ray diagram for a guided-mode (TIR satisfied) and unguided or radiation-mode (TIR not satisfied)
along the interface between two dielectrics. The material with the larger index of refraction (n
1
) is shown in blue
while the material with the smaller index of refraction (n
2
) is shown in in grey. In the simplified geometry,
translational invariance is assumed along the x- and y-directions.

In two-dimensional photonic crystal (PhC) slabs, light can be controlled in-plane through the
photonic crystal effect and confined in the transverse diretion through index guiding.
Interestingly, PhC slabs support a third type of mode referred to as a guided-resonance mode
(GRMs). GRMs, similar to guided modes, have much of their optical power confined within the
slab, or patterned region. However, the photonic crystal also behaves as a phase-matching
element, coupling light to external radiation and effectively leaking power out of the slab.
Guided-resonance modes were exploited greatly in this thesis due to key advantages they offer
nanoemitter design. First, slow-light (𝑑𝜔 𝑑𝑘 → 0) ⁄ GRMs greatly increase material interaction
and have been shown to enhance the spontaneous emission rate [31, 52, 68]. Second, band-edge
modes exist near the Γ-point mode in reciprocal space (𝑘 𝑥 ~ 𝑘 𝑦 ~0) and thus contribute to
radiation near the normal of the slab-air interface; collecting radiation along this direction is
beneficial to the design of high beam quality (M
2
) photonic crystal surface-emitting lasers
(PCSELs) and light-emitting diodes [63, 67]. Finally, the grating-like structure offers a useful
mechanism for extracting light and shaping the emitted far-field beam profile [63, 67].

63

A.2 Coupled and Uncoupled Modes in Photonic Crystal Slabs

Let us consider a photonic crystal slab composed of nanorods of finite length. The nanorods
are arranged into a square array that extends infinitely into the x-y plane. Figure A.2 shows the
unit cell of one such array. The lattice contains four mirror symmetries, indicated by black
dashed lines in the figure. In addition to the mirror symmetries, three rotational symmetries are
also present in the geometry.

Fig. A.2 Cross-sectional (x-y plane) schematic of a unit-cell in a square photonic crystal lattice. Blue indicates
a cylindrical nanorod, surrounding grey indicates air and dashed lines indicate the mirror symmetries present in the
geometry.

As shown in previous work [58, 60], the lateral symmetries present in the square array
support optical modes that can be classified according to six mirror symmetries. Figure A.3
shows the symmetries of the six modes supported in the structure. The first two modes are
doubly-degenerate modes (𝐸 (1)
, 𝐸 (2)
), and are 90° rotations of each other. The next four modes
are non-degenerate modes, labeled 𝐴 1
, 𝐴 2
, 𝐵 1
and 𝐵 2
. The symmetries of an x-polarized (𝑃𝑊 𝑥 )
and y-polarized (𝑃𝑊 𝑦 ) plane wave are also included in the figure. The mirror symmetries of
each mode are illustrated with blue + signs and red – signs. Here, the signs relate to the mirror
symmetry of the mode as follows: If the signs on either side of a mirror plane match, the mode is
symmetric along the plane. If the signs on either side of a mirror plane do not match, the mode is
antisymmetric along the plane. For the work presented in Chapter 2 and Chapter 3, we are
interested in a TE mode, and thus interested in the electric field symmetries in the plane parallel
to the slab (x-y plane).
64

Fig. A.3 Left of separation bar: Six possible modes in a square PhC lattice classified according to mirror
symmetries. Right of separation bar: Possible symmetries of a polarized plane wave at normal incidence.

Careful inspection of our TE
01
-like mode field distribution (Figure 2.4) shows that it possesses
the parity of a non-degenerate mode. In PhC slabs composed of square arrays, all non-generate
modes are uncoupled [60]. Uncoupled modes can neither be excited by a plane wave at normal
incidence nor excite a normally-incident plane wave in the far-field zone (the Γ-point, in
reciprocal space). To understand why such modes are uncoupled, consider the symmetry overlap
of 𝐴 2
and both plane-wave polarizations in Figure A.3:

〈𝑃𝑊 𝑥 |𝐴 2
〉 = 〈𝑃𝑊 𝑥 |𝜎 𝑥 †
𝜎 𝑥 |𝐴 2
〉 = (⟨−𝑃𝑊 𝑥 |)(|𝐴 2
⟩) = −〈𝑃𝑊 𝑥 |𝐴 2
〉 = 0
〈𝑃𝑊 𝑦 |𝐴 2
〉 = 〈𝑃𝑊 𝑦 |𝜎 𝑦 †
𝜎 𝑦 |𝐴 2
〉 = (⟨−𝑃𝑊 𝑦 |)(|𝐴 2
⟩) = −〈𝑃𝑊 𝑦 |𝐴 2
〉 = 0

From the above overlap integrals we see that a plane-wave is forbidden by symmetry to couple to
this mode. Similar calculations reveal that only the double-degenerate modes in the figure may
couple to an external plane wave. Please refer to Ref. [58] and Ref. [60] for a more exhaustive
group theory treatment of coupled and uncoupled modes in photonic crystal slabs.

65

A.3 Lorentz Reciprocity: Derivation and Relation to Emission Optimization

First, consider the material volume (V) shown in Figure A.4, bounded by a surface (S). The
material contains two electric current densities, 𝐽 1
⃗⃗⃗
and 𝐽 2
⃗ ⃗ ⃗⃗
. Each current source produces electric
and magnetic fields, observed at points 𝐸 1
⃗ ⃗ ⃗ ⃗⃗
, 𝐻 1
⃗ ⃗ ⃗ ⃗⃗
and 𝐸 2
⃗ ⃗ ⃗ ⃗⃗
, 𝐻 2
⃗⃗ ⃗⃗⃗
, respectively.

Fig. A.4 Schematic of a bound volume containing two electric current sources 𝐽 1
⃗ ⃗⃗
(blue) and 𝐽 2
⃗ ⃗ ⃗⃗
(green). The
current sources produce electric and magnetic fields that are observed in locations marked by their respective colors.

Next, let’s begin to relate the field quantities by considering the following vector operation
∇ ⋅ (𝐸 1
⃗ ⃗ ⃗ ⃗⃗
× 𝐻 2
⃗⃗ ⃗⃗⃗
− 𝐸 2
⃗ ⃗ ⃗ ⃗⃗
× 𝐻 1
⃗ ⃗ ⃗ ⃗⃗
) (1)
Using a vector identity, the operation above can be expanded to
𝐸 2
⃗ ⃗ ⃗ ⃗⃗
⋅ (∇ × 𝐻 1
⃗ ⃗ ⃗ ⃗⃗
) + 𝐻 2
⃗⃗ ⃗⃗⃗
⋅ (∇ × 𝐸 1
⃗ ⃗ ⃗ ⃗⃗
) − 𝐸 1
⃗ ⃗ ⃗ ⃗⃗
⋅ (∇ × 𝐻 2
⃗⃗ ⃗⃗⃗
) − 𝐻 1
⃗ ⃗ ⃗ ⃗⃗
⋅ (∇ × 𝐸 2
⃗ ⃗ ⃗ ⃗⃗
) (2)
Assuming a linear, isotropic and homogeneous material with no additional sources and a
harmonic time dependence of the fields (𝑒 −𝑖𝜔𝑡 ), Maxwell’s curl equations become
∇ × 𝐸 𝛼 ⃗ ⃗ ⃗ ⃗⃗
= −𝑖𝜔𝜇 𝐻 𝛼 ⃗ ⃗ ⃗ ⃗ ⃗⃗
(3)
∇ × 𝐻 𝛼 ⃗ ⃗ ⃗ ⃗ ⃗⃗
= 𝑖𝜔𝜀 𝐸 𝛼 ⃗ ⃗ ⃗ ⃗⃗
+ 𝐽 𝛼 ⃗ ⃗ ⃗⃗
(4)
where α = 1, 2 in the present scenario. Therefore
∇ ⋅ (𝐸 1
⃗ ⃗ ⃗ ⃗⃗
× 𝐻 2
⃗⃗ ⃗⃗⃗
− 𝐸 2
⃗ ⃗ ⃗ ⃗⃗
× 𝐻 1
⃗ ⃗ ⃗ ⃗⃗
) = 𝑖𝜔𝜀 𝐸 1
⃗ ⃗ ⃗ ⃗ ⃗⃗
⋅ 𝐸 2
⃗ ⃗ ⃗ ⃗⃗
+ 𝐽 1
⃗⃗⃗
⋅ 𝐸 2
⃗ ⃗ ⃗ ⃗⃗
− 𝑖𝜔𝜇 𝐻 1
⃗⃗⃗⃗ ⃗⃗
⋅ 𝐻 2
⃗⃗ ⃗⃗⃗
(5)
66

− 𝑖𝜔𝜀 𝐸 1
⃗ ⃗ ⃗ ⃗ ⃗⃗
⋅ 𝐸 2
⃗ ⃗ ⃗ ⃗⃗
− 𝐽 2
⃗ ⃗ ⃗⃗
⋅ 𝐸 1
⃗ ⃗ ⃗ ⃗⃗
+ 𝑖𝜔𝜇 𝐻 1
⃗⃗⃗⃗ ⃗⃗
⋅ 𝐻 2
⃗⃗ ⃗⃗⃗

with the terms in blue and green canceling, leaving
∇ ⋅ (𝐸 1
⃗ ⃗ ⃗ ⃗⃗
× 𝐻 2
⃗⃗ ⃗⃗⃗
− 𝐸 2
⃗ ⃗ ⃗ ⃗⃗
× 𝐻 1
⃗ ⃗ ⃗ ⃗⃗
) = 𝐽 1
⃗⃗⃗
⋅ 𝐸 2
⃗ ⃗ ⃗ ⃗⃗
− 𝐽 2
⃗ ⃗ ⃗⃗
⋅ 𝐸 1
⃗ ⃗ ⃗ ⃗⃗
(6)
Next, let us integrate Equation (6) over the volume of interest
∭{∇ ⋅ (𝐸 1
⃗ ⃗ ⃗ ⃗⃗
× 𝐻 2
⃗⃗ ⃗⃗⃗
− 𝐸 2
⃗ ⃗ ⃗ ⃗⃗
× 𝐻 1
⃗ ⃗ ⃗ ⃗⃗
)} 𝑑 𝑉 ′
= ∭( 𝐽 1
⃗⃗⃗
⋅ 𝐸 2
⃗ ⃗ ⃗ ⃗⃗
− 𝐽 2
⃗ ⃗ ⃗⃗
⋅ 𝐸 1
⃗ ⃗ ⃗ ⃗⃗
)𝑑 𝑉 ′
(7)
Applying the Divergence Theorem, it is seen that
∯(𝐸 1
⃗ ⃗ ⃗ ⃗⃗
× 𝐻 2
⃗⃗ ⃗⃗⃗
− 𝐸 2
⃗ ⃗ ⃗ ⃗⃗
× 𝐻 1
⃗ ⃗ ⃗ ⃗⃗
) ⋅ 𝑑 𝑆 ′
= ∭( 𝐽 1
⃗⃗⃗
⋅ 𝐸 2
⃗ ⃗ ⃗ ⃗⃗
− 𝐽 2
⃗ ⃗ ⃗⃗
⋅ 𝐸 1
⃗ ⃗ ⃗ ⃗⃗
)𝑑 𝑉 ′
(8)
Assuming the fields are observed at very long distances from the sources (far-field zone), we can
think of the fields taking the form of a plane wave. Under this plane wave approximation, the
magnetic field is related to the electric field by
𝐻 𝛼 ⃗ ⃗ ⃗ ⃗ ⃗⃗
= (𝑟⃗ × 𝐸 𝛼 ⃗ ⃗ ⃗ ⃗⃗
) 𝑍 ⁄ (9)
where 𝑟⃗ is the unit vector corresponding to the radial direction of propagation, 𝑍 = √𝜇 𝜀 ⁄ is the
impedance of the surrounding medium, 𝜇 is the magnetic permeability of the medium, 𝜀 is the
electric permittivity of the medium and α = 1, 2. Following this approximation, Equation 8
becomes
∯ (𝐸 1
⃗ ⃗ ⃗ ⃗⃗
× 𝑟⃗ × 𝐸 2
⃗ ⃗ ⃗ ⃗⃗
− 𝐸 2
⃗ ⃗ ⃗ ⃗⃗
× 𝑟⃗ × 𝐸 1
⃗ ⃗ ⃗ ⃗⃗
) 𝑍 ⁄ ⋅ 𝑑 𝑆 ′
= ∭( 𝐽 1
⃗⃗⃗
⋅ 𝐸 2
⃗ ⃗ ⃗ ⃗⃗
− 𝐽 2
⃗ ⃗ ⃗⃗
⋅ 𝐸 1
⃗ ⃗ ⃗ ⃗⃗
)𝑑 𝑉 ′
(10)
Using the vector triple product identity and noting for a plane wave 𝑟⃗ ⋅ 𝐸 𝛼 ⃗ ⃗ ⃗ ⃗⃗
= 0 (α = 1, 2) we find
that surface integrand on the left vanishes, leaving:
∭( 𝐽 1
⃗⃗⃗
⋅ 𝐸 2
⃗ ⃗ ⃗ ⃗⃗
)𝑑 𝑉 ′
= ∭( 𝐽 2
⃗ ⃗ ⃗⃗
⋅ 𝐸 1
⃗ ⃗ ⃗ ⃗⃗
)𝑑 𝑉 ′
(12)
For a delta-function current source, the relationship finally reduces to:
𝐽 1
⃗⃗⃗
⋅ 𝐸 2
⃗ ⃗ ⃗ ⃗⃗
= 𝐽 2
⃗ ⃗ ⃗⃗
⋅ 𝐸 1
⃗ ⃗ ⃗ ⃗⃗
(13)
Equation 13 has powerful implications. Generally, it shows that the relationship between a time-
harmonic electric current and its resulting electric field remains unaffected if the positions of
each are interchanged. For the calculations presented in Chapter 3, is tells us that the relationship
67

between emission in a quantum well (QW) and observation of the electric field in the far-field
zone remains unchanged if we instead place the emitter in the far-field zone and observe the
electric field in the QW. In other words, this concept tells us there is an intimate relationship
between emission and absorption. As shown in Section A.4, we can use this reciprocal
relationship to speed up our quasi-aperiodic, emission optimization procedure. Please refer to
Ref. [88] for a more thorough discussion of Lorentz reciprocity.

A.4 Inverse Design: Random-Walk Optimization of Quasi-Aperiodic Arrays

In the optimization procedure, it is important to first identify an appropriate target or merit
function to maximize. Of particular interest in the present case is maximizing the number of
photons collected near the Γ-point in the emitted far-field. This quantity is affected by two
central aspects: First, the rate of photon generation and second, the light extraction efficiency of
the nanophotonic system. Since the Purcell factor, or enhancement in the photon generation rate,
is directly proportional to the quality factor (Q-factor) of a guided-resonance mode, our two
quantities of interest compete with each other. For instance, higher-Q resonances (such as those
associated with uncoupled Γ-point modes) can enhance the spontaneous emission rate, but at the
cost of reducing light extraction [52, 63]. Of course, you can design for very high light extraction
(low-Q modes), but it may reduce the spontaneous emission rate. In short, it is difficult to create
a very high-Q resonances that simultaneously produce large light extraction.
However, as shown in Chapter 3, it is possible to improve light extraction without
sacrificing broadband emission. In working to this end, we introduce an inverse optimization
procedure. For our optimization calculations, we take advantage of the reciprocal relationship
outlined in Section A.3 (Lorentz reciprocity). As opposed to placing our electric dipole-
excitation source in the QW layer and calculating the emitted far-field at the Γ-point, we place
our dipole in the far-field zone and calculate the electric field inside the QW; from the work
derived in Section A.3, we know the two simulations are equivalent, provided the assumptions
are valid. Moreover, the inverse simulation procedure offers many advantages. First, we can
approximate an in-plane electric dipole emitted at the Γ-point in the far-field zone as a plane
wave normally incident on our PhC slab. Thus, we can now simulate single complex unit-cell
structures, impose bloch boundary conditions and calculate the electric field in all the QW
68

regions of interest. All of these simplifications combine in helping reduce the length and number
of required simulations. However, these advantages do come at the expense of losing
information at other emission angles. If you are only interested in emission near the Γ-point
though, the approach is sufficient.
Next, we aim to maximize the electric field intensity within all the QW field monitors.
Maximizing this quantity is of particular interest as it is related to both the coupling strength of
the plane wave to the guided-resonance mode and resulting light absorption; or reciprocally, the
quantity relates to light extraction and emission. To optimize our figure of merit, we begin with a
2 × 2 complex unit-cell and use a random walk or downhill optimization method [89]. In
particular, we first calculate the integrated electric field intensity for the simple periodic
(unaltered) complex unit-cell. Next, we randomly select one of the four nanorods within the
complex unit-cell, and randomly adjust its position. Here, we only demand that no nanorods
overlap, as this can change the volume of emitting material and make the comparison unfair. For
the new quasi-aperiodic geometry, we run the inverse simulation and again calculate the
intergrated electric field intensity collected by the QW monitors. We normalize this quantity to
the original, unaltered geometry. If this quantity improves the current figure of merit (which is
unity by default), then we keep the move. Conversely, if the move has reduced our figure of
merit, we reduce to the previous geometry. We iterate in this fashion until our figure of merit
converges and remains converged for 100 iterations. The convergence of our figure of merit
indicates that we have reached a local minimum in our optimization procedure.
The random walk optimization is easily implementable. However, for landscapes with few
degrees of design freedom, the procedure can quickly converge onto a local minimum [89].
Nevertheless, integrating this technique with other techniques (such as an annealing technique)
can help improve the figure of merit of our designs by more effectively escaping local minima.
Also, extending the optimization domain to larger complex unit-cells with more degrees of
design freedom (nanorod shape, size, etc.) may offer interesting possibilities for further
optimizing quantum efficiency in quasi-aperiodic nanoemitter structures.

69

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Abstract (if available)

Abstract Photonic structure plays a significant role in determining the brightness and efficiency of nanoemitter systems. Using photonic crystal slabs it is possible to affect these quantities in various ways. First, positioning a leaky mode near the emission frequency allows more light to be extracted from within the slab. Second, concentrating high electric field intensity near emitter locations significantly enhances the spontaneous emission rate. Improving the spontaneous emission rate is essential for systems inhibited by non-radiative recombination mechanisms, such as surface recombination. However, a large body of work has suggested these two contributing factors are in competition, making it difficult to simultaneously achieve high spontaneous emission and light extraction. Here, we begin by investigating the effects of guided-resonance modes on core-shell InGaN nanowire arrays. Using a guided-resonance mode ideal for a core-shell nanowire array, we are able to simultaneously enhance the spontaneous emission rate and light extraction efficiency. Our theoretical results predict nearly a 25× increase in external quantum efficiency, relative to an in-filled reference geometry. We build upon these theoretical findings by introducing quasi-aperiodic nanowire arrays: arrays that exhibit short-range aperiodicity, but themselves are periodic on a larger scale. Using an inverse design optimization procedure, we show that quasi-aperiodic arrays can be leveraged to bolster light extraction efficiency without sacrificing broadband spontaneous emission. Relative to the original, simple periodic geometry, our optimized quasi-aperiodic array improves the external quantum efficiency of the nanowire array up to 42%. Next, we experimentally investigate quasi-aperiodic geometries, fabricating highly-aniostropic nanowire arrays using a novel, top-down fabrication procedure. Room-temperature photoluminescence measurements support our theoretical findings and demonstrate the ability of quasi-aperiodic arrays to greatly improve upon the emitted far-field uniformity of uncoupled modes. Further, we illustrate how quasi-aperiodic arrays can be exploited to decrease in-plane distributed feedback, a quantity critical to the design of large-area photonic crystal surface-emitting lasers (PCSELs). We then turn our attention to the seamless, top-down synthesis of InGaN quantum dots (QDs). Using a quantum-size-controlled photoelectrochemical (QSC-PEC) etching technique we fabricate QDs within pre-fabricated nanowires. Low-temperature photoluminescence measurements reveal sharp spectral signatures within isolated nanowires and a 10× improvement in emitter efficiency following QSC-PEC etching. Second-order cross-correlation measurements of individual nanowires further confirm QD formation in addition to non-classical, anti-bunching emission behavior. Lastly, we show how photoelectrochemical etching can be used to selectively form membranes in complex heterostructures of pre-fabricated photonic crystal defect cavities. Our results offer exciting opportunity in improving the quantum efficiency of nanoemitter systems as well as spectrally and spatially coupling QDs within pre-fabricated nanophotonic structures.

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Asset Metadata

Creator Anderson, Patrick Duke (author)

Core Title Optimizing nanoemitters using quasi-aperiodicity and photoelectrochemical etching

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 05/01/2017

Defense Date 03/16/2017

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag InGaN,nanoemitters,nanophotonics,OAI-PMH Harvest,photonic crystal surface-emitting lasers,photonic crystals,quantum dots,single photon sources

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Povinelli, Michelle L. (committee chair), Cronin, Stephen B. (committee member), Subramania, Ganapathi (committee member), Wu, Wei (committee member)

Creator Email duke.anderson35@gmail.com,pdanders@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-368644

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