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Light management in nanostructures: nanowire solar cells and optical epitaxial growth
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Light management in nanostructures: nanowire solar cells and optical epitaxial growth
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Content
LIGHT MANAGEMENT IN NANOSTRUCTURES:
NANOWIRE SOLAR CELLS AND OPTICAL EPITAXIAL GROWTH
by
Ningfeng Huang
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(ELECTRICAL ENGINEERING - ELECTROPHYSICS)
December 2015
Copyright 2015 Ningfeng Huang
Dedication
To my parents, Baojian Huang, Ju Zhuang and Chen Zhang.
Thank you for all of your support along the way.
ii
Acknowledgments
First of all, I would like to sincerely thank my research advisor, Professor
Michelle Povinelli, for the privilege of starting my research career; I not only
gained invaluable knowledge in the field of photonics, but also learned how to be a
good scholar. Under her guidance, I was given the freedom to pursue my research
in my own way. I wouldn’t have joined the LATS project and written the second
part of this dissertation without such freedom. It was always a great pleasure to
work and discuss with her. Thanks for supporting and encouraging me through
these years.
I would also like to thank the former and current Nanophotonics Group mem-
bers. Chenxi Lin initially guided me into the solar cell modeling project and
constantly provided comments and help until his graduation. He also served as a
role model of a systematic and diligent researcher to me. Luis Javier Martínez,
Eric Jaquay and Camilo Mejia achieved great progress on the LATS project, which
makes the second half of this dissertation possible. They were also great mentors
to lead me into this project. Luis and Eric spent a lot of time training me on
the machines in the clean room and the optical setup in our lab. Luis also shared
with me his expertise in photonic crystal design. Camilo always came up with
fascinating ideas and insightful questions on the project. I also enjoyed working
with Mehmet Solmaz, who put a great effort in emphasizing the importance of
iii
experimental work; Jing Ma, who always impressed me with her confidence and
efficiency; Roshni Biswas and Duke Anderson, who shared and discussed with me
their very interesting work; and Shao-Hua Wu and Aravind Krishnan, who worked
with me very hard in the last year to pick up the optical assembly project from
me quickly.
I feel tremendously lucky to have had the opportunity to work with many
outstanding professors and scholars at USC. I would like to express my gratitude
towards them. In regards to the study on nanowire solar cell in Center for Energy
Nanoscience (CEN), I thank Professor Dan Dapkus, Professor Chongwu Zhou and
Professor Stephen Cronin as well as their group members for the continuous effort
in growing, testing and improving nanowires. They continuously provided insight
and comment from the experimental side which pushed my modeling work forward
and towards a more realistic and accurate direction. I would also like to thank
Maoqing Yao and Chunyung Chi from Dr Dapkus’ group, Sen Cong and Anuj
Madaria from Dr Zhou’s lab, and Chia-Chi Chang and Shermin Arab from Dr.
Cronin’s group for their help and support in collaborative projects. For the optical
epitaxial growth project, I would like to thank Professor Aiichiro Nakano’s input
on the simulation. I also would like to thank Professor Anthony Levi and Professor
Wei Wu for serving on my qualifying exam and thesis defense committee.
Last, but certainly not least, I would like to thank my dear wife, Chen Zhang,
for her unconditional love, support and encouragement. I also owe a great debt of
gratitude to my parents, Baojian Huang and Ju Zhuang. They are always ready
to offer whatever they have to support me moving forward, and I dedicate this
dissertation to them.
iv
Contents
Dedication ii
Acknowledgments iii
List of Tables viii
List of Figures ix
Abstract xvii
I Nanowire Solar Cells 1
1 Introduction 2
1.1 Solar energy and photovoltaic cells . . . . . . . . . . . . . . . . . . 2
1.2 Nanostructured solar cells . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Outline of PART I . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Broadband absorption of semiconductor nanowire arrays for pho-
tovoltaic applications 7
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Limiting efficiencies of tandem solar cells consisting of III-V
nanowire arrays on silicon 20
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Optical absorption modeling and detailed balance analysis . . . . . 22
3.2.1 III-V nanowire array on silicon structure . . . . . . . . . . . 22
3.2.2 Detailed balance analysis . . . . . . . . . . . . . . . . . . . . 23
3.3 Electrical transport modeling . . . . . . . . . . . . . . . . . . . . . 32
3.3.1 Junction geometry . . . . . . . . . . . . . . . . . . . . . . . 32
v
3.3.2 Electrical transport simulation methods . . . . . . . . . . . 33
3.3.3 Design examples and results . . . . . . . . . . . . . . . . . . 39
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4 DesignofpassivationlayersonaxialjunctionGaAsnanowiresolar
cells 44
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2 Passivation structure design . . . . . . . . . . . . . . . . . . . . . . 46
4.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3.1 Optical simulation method . . . . . . . . . . . . . . . . . . . 48
4.3.2 Device simulation . . . . . . . . . . . . . . . . . . . . . . . . 49
4.4 Absorption properties of the design . . . . . . . . . . . . . . . . . . 51
4.5 Carrier transport in the design . . . . . . . . . . . . . . . . . . . . . 53
4.5.1 Band diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.5.2 J−V response . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5 Conclusion and outlook 61
5.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
II Optical Epitaxial Growth 64
6 Introduction 65
6.1 Building photonic matter from the bottom up . . . . . . . . . . . . 65
6.2 Optical binding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.3 Optics matter and optical epitaxial growth . . . . . . . . . . . . . . 66
6.4 Outline of PART II . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
7 Monte Carlo simulation of optical epitaxial growth process 70
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
7.2 Interparticle interaction between particles in light field . . . . . . . 71
7.3 Induced energy shift in optical epitaxial growth . . . . . . . . . . . 75
7.4 Monte Carlo simulation of optical epitaxial growth . . . . . . . . . 79
7.5 Monte Carlo simulation result . . . . . . . . . . . . . . . . . . . . . 81
7.6 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.6.1 Effect of the growth substrate . . . . . . . . . . . . . . . . . 84
7.6.2 Phase transition . . . . . . . . . . . . . . . . . . . . . . . . . 86
7.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
vi
8 Optical epitaxial growth of a gold nanoparticle array 89
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
8.2 Epitaxial growth template design . . . . . . . . . . . . . . . . . . . 89
8.3 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
8.4 Optical epitaxial growth experiment . . . . . . . . . . . . . . . . . . 93
8.5 Discussion and conclusion . . . . . . . . . . . . . . . . . . . . . . . 97
9 Conclusions and outlook 100
Reference List 103
A Supporting Information of Optical Potential Calculation 116
A.1 Numerical Calculation of the Polarizability . . . . . . . . . . . . . . 116
A.2 Full-vector force simulation versus the dipole approximation . . . . 117
A.3 The effect of holes on the photonic-crystal slab . . . . . . . . . . . . 119
vii
List of Tables
3.1 Device simulation parameters for nanowire top cells . . . . . . . . . 39
4.1 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . 50
viii
List of Figures
1.1 Rendering of a vertically-aligned semiconductor nanowire array. . . 4
2.1 Schematic of a vertically-aligned semiconductor nanowire array. (a)
Perspective view. (b) Cross-sectional view. . . . . . . . . . . . . . 10
2.2 Materials’ optical properties (a) Refractive indices (b) Absorption
lengths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Direct+circumsolarspectralirradianceAM1.5(blackline),reprinted
from Ref. [1], and the perfect-absorption limit of the ultimate effi-
ciency (blue line). . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Optimizationoftheultimateefficiencywithrespecttothestructural
parameters for a silicon nanowire array of 3 μm height. . . . . . . 13
2.5 Optimized ultimate efficiencies for nanowire arrays composed of dif-
ferent materials: (a) shown as a function of array height; (b) shown
as a function of the bandgap energy of the material. . . . . . . . . 14
2.6 Optimalstructuralparametersfordifferentheightsofdifferentmate-
rials: (a) lattice constant; (b)a/d ratio. . . . . . . . . . . . . . . . 15
ix
2.7 Comparison of the ultimate efficiencies of optimized (a) Si, (b) Ge,
(c) GaAs, (d) InP, (e) InGaP, and (f) CdTe nanowire arrays with
the ultimate efficiencies of thin films made of the same material and
of the same height. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.8 Effect of the substrate on the absorption in nanowire arrays. . . . 18
3.1 Schematic of a vertically aligned III-V nanowire array on a semiin-
finite substrate. (a) Perspective view and (b) Top view. . . . . . . 23
3.2 (a) Limiting detailed balance efficiency for double junction solar
cells as a function of band gap energies assuming perfect absorption
in the top cell. (b) Optimal fractional absorption in the top cell
that maximizes the total efficiency. (c) Limiting detailed balance
efficiency with optimal fractional absorption in the top cell given
in (b). (d) Limiting efficiencies (left axis) and optimal fractional
absorption (right axis) as a function of the band gap of the top
cell, for a fixed bottom cell band gap energy of 1.1 eV (silicon),
corresponding to the white, dashed lines in (a)-(c). . . . . . . . . . 27
3.3 (a) Optimized detailed balance efficiency as a function of nanowire
height for different nanowire materials. (b) Optimized detailed
balance efficiency as a function of nanowire band gap energy for
nanowire arrays of different heights. The solid lines give the lim-
iting detailed balance efficiency, identical to the red dashed line in
Figure 3.2(d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4 Optimal structural parameters for different materials as a function
of nanowire height: (a) lattice constant a, (b)d/a ratio, and (c)
diameter d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
x
3.5 Relationship of photon absorption to detailed balance efficiency for
materials with band gaps above and below the optimal value. (a)
and(b)1800nm-tallGaAsNWarrayonsilicon. (c)and(d)1800nm
tall GaInP NW on silicon. (a) and (c) The percentage of photons
with energy above the bandgap of silicon absorbed by the nanowire
array (blue) and silicon layer (red) as functions of lattice constant
and d/a ratio. (b) The detailed balance efficiency for the GaAs NW
tandem cell. The white line indicates the structural parameters for
which the percentage of photons absorbed in GaAs and silicon is
equal. (d)ThedetailedbalanceefficiencyfortheGaInPNWtandem
cell. Overlaid contours show the percentage of photons absorbed in
GaInP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.6 JunctiondesignsforIII-VNWonsilicontandemcellswith(a)radial
junctionand(b)axialjunctioninthenanowire. Thethicknessofthe
silicon substrate is not to scale. (c) Carrier generation rate profile
in GaAs nanowire. . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.7 Simulated structures for (a) radial junction and (b) axial junction
GaAsnanowiretopcell. (c)J−V curvesforaxialandradialjunction
geometries for varying SRV. . . . . . . . . . . . . . . . . . . . . . . 34
3.8 Normalized power of nanowire cells as functions of junction depth
for (a) radial junction and (b) axial junction. . . . . . . . . . . . . 42
4.1 (a) Schematic of the axial junction GaAs nanowire-array solar cell
on substrate. (b) Detailed illustration of the p-n junction and the
direction of the generated carriers. (Insets show the band diagrams
across the p and n regions.) . . . . . . . . . . . . . . . . . . . . . . 46
xi
4.2 Specific passivation layer design and the doping concentration of
each part. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3 Maximum achievable short-circuit current of a square array of 3-μm
tall GaAs nanowires as a function of lattice constanta andd/a ratio. 51
4.4 (a) Carrier generation rate profile in a bare GaAs nanowire. (b)
Carrier generation rate profile in a passivated GaAs nanowire. (c)
Magnified view of carrier generation rate profile near top of passi-
vated GaAs nanowire. . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.5 Effect of passivation layer on absorption in the nanowires. (a)
Absorption spectra of GaAs NW array. (b) Absorbed photon flux
densities (blue and red curves) shown relative to the incident photon
flux density (black curve). . . . . . . . . . . . . . . . . . . . . . . . 53
4.6 Band diagrams of the cross sections of (a) the intrinsic region, (b)
the n-region, (c) below the top contact, and (d) near the junction,
corresponding to dashed lines labeled 1, 2, 3, and 4 in Figure 4.2. . 54
4.7 (a)J−V curves for nanowire arrays without (blue) and with (red)
the passivation layer. (b) Efficiencies as functions of junction depths. 56
4.8 (a) Short-circuit current, (b) open-circuit voltage, and (c) efficiency
as functions of surface state density. . . . . . . . . . . . . . . . . . 57
4.9 Efficienciesof(a)thepassivatednanowiresolarcellsand(b)thebare
nanowire solar cells with different bulk SRH lifetimes as functions
of the junction depth. . . . . . . . . . . . . . . . . . . . . . . . . . 59
xii
6.1 Pattern formation due to optical trapping and optical binding. (a)
A periodic optical potential traps weakly interacting particles at
intensity maxima, due to the optical gradient force. (b) Optical
binding causes strongly interacting particles to arrange themselves
into well-defined patterns, even in a nearly uniform light field. (c)
Competition between a periodic optical potential and optical bind-
ing can give rise to pattern formation, preventing particles from
trapping at every site. Here, particles form 1-D chains. . . . . . . . 67
6.2 Comparisons between “electrical matter” and “optical matter”. (a-
b) A unit cell of gallium arsenide crystal and electron band diagram
of gallium arsenide [2]. (c) Schematic of molecular beam epitaxy.
(d-e) A photonic-crystal slab and photonic band diagram of the
photonic-crystal slab [3]. (f) Optical epitaxial growth of a single
layer of “optical matter”. . . . . . . . . . . . . . . . . . . . . . . . . 69
7.1 Simulation setup for Green’s functions (a) simulation of the first
and second columns in Green’s function tensor. (b) simulation of
the third column in Green’s function tensor. . . . . . . . . . . . . . 73
7.2 Schematic of electric field distribution in a photonic-crystal slab
optical epitaxial growth substrate. . . . . . . . . . . . . . . . . . . 76
7.3 Graphical illustration of the energy shift in a two-particle system.
(a) The xy-plane cross-section (b) The xz-plane cross-section. . . . 77
7.4 Induced energy shift due to interaction. (a) Energy shift as function
of spacing between two particles for the polarization parallel to the
particle pair (black) and perpendicular to the particle pair (red).
(b) log-log plot of the energy shift magnitude. The function 1/r is
plotted for reference. . . . . . . . . . . . . . . . . . . . . . . . . . . 78
xiii
7.5 A typical Monte Carlo simulation. (a) Average energy shift normal-
izedtotheincidentpowerasafunctionofsimulationstep. Theinset
shows the first 100,000 steps (shadowed area in the main figure). (b)
The histogram of the normalized average energy shift over the last
4.5 million steps. (c-d) The configurations at step 1000 and step
5,000,000 step, respectively. The color map shows the normalized
energy shift for individual particles. . . . . . . . . . . . . . . . . . . 81
7.6 Strong interaction-induced pattern formation during optical epitax-
ial growth. (a-b) Square and hexagonal lattices, each with lattice
constant a. (c) Monte-Carlo simulation results of energy shift per
particle as a function of lattice constant. The solid lines are guides
to the eye. Energy is plotted in arbitrary units. . . . . . . . . . . . 82
7.7 (a-c) Particle configurations obtained for square lattices with vary-
inglatticeconstants. (d-f)Particleconfigurationsobtainedforhexag-
onallatticeswithvaryinglatticeconstants. Thecirclesshowa50μm
diameter device area. . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.8 Energy shift in a two-particle system in different environments. (a)
water. (b) 125 nm above the surface of a 250 nm thick SOI slab.
(c) 125 nm above the surface of a 340 nm thick SOI slab. . . . . . 85
7.9 Mean value and standard deviation of the average energy shift per
particle as a function of the incident power. The simulation is on a
square lattice with a = 0.93λ and a coverage of 5%. . . . . . . . . . 87
7.10 Mean values (a,c) and standard deviations (b,d) of the average
energy shift as functions of power and lattice constant. The upper
row shows the result for square lattices and the bottom row shows
the result for hexagonal lattices. . . . . . . . . . . . . . . . . . . . 88
xiv
8.1 Designofthephotoniccrystaldeviceforexperiment. (a)Ascanning
electron micrograph (SEM) image of the fabricated photonic crystal
slab device. The 200 nm diameter gold particles are also shown in
this graph. (b) The measured transmission spectrum through the
device. (c) The simulated electric field profile (|E|
2
/|E
0
|
2
) of the
guided-resonance mode used in the experiments on a plane 125 nm
above the slab. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
8.2 Calculated force and potential of the photonic crystal trap. (a)
The normalized vertical force exerted on a 200 nm gold particle at
different positions along a plane 40 nm above the photonic-crystal
slab. Negative force indicates attraction toward the slab. (b) The
normalized in-plane force. The blue scale bar indicates Fc/P = 2. 92
8.3 Schematicoftheopticalsetupusedinopticalepitaxialgrowthexper-
iments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
8.4 Opticalepitaxialgrowthofagoldparticlearray. Opticalmicroscope
images show 200-nm diameter gold nanoparticles trapped on the
photonic crystal slab, visible in the background. (a-f) Snapshots
taken with the laser power on; elapsed time is shown below image.
(g,h) Snapshots taken with laser power off. The scale bar indicates
5 μm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
8.5 Optical epitaxial growth experiments in a microfluidic channel. (a-
c) The snapshots when the beam is on for 0 s, 150 s and 300 s. (d)
The number of trapped particles (black triangles) and the standard
deviation of the particle positions (blue squares) as functions of
time. The scale bar indicates 5 μm. . . . . . . . . . . . . . . . . . 96
xv
8.6 Energy lowering during the growth process. (a-c) The energy shift
of each particle. (d) Average energy shift per particle as a function
of time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
8.7 Comparison of optical forces on gold and polystyrene nanoparticles.
(a)Calculated, normalizedverticalforceexertedon200nmgoldand
polystyrene particles at different heights. The force is in dimension-
less units of Fc/P, where c is the speed of the light and P is the
incident optical power within one unit cell. Positive force indicates
attraction toward the slab. (b-d) Field intensity profiles for gold
(b),(d), and polysytrene (a),(c), particles at different heights. . . . 99
9.1 Hybridization of the photonic mode with the coupled plasmonic
mode of the gold nanoparticle array. (a) Electric field intensity of
a guided resonance mode in the photonic crystal slab. (b) Electric
field intensity of a coupled plasmonic mode in the gold nanoparti-
cle array. (c) Electric field intensity of a hybrid photonic-plasmonic
mode between the trapped gold nanoparticle array and the pho-
tonic crystal trapping template. (d) The extinction cross-section
spectrum of the structures shown in (a-c). . . . . . . . . . . . . . . 102
A.1 Comparisonbetweenthefull-vectorforcesimulationversusthedipole
approximation energy formula. (a) the optical force between two
particles in water. (b) the optical force between two particles in
water on a plane 125 nm above a 250 nm SOI. . . . . . . . . . . . . 118
A.2 ComparisonbetweentheGreen’sfunctiononaphotonic-crystalslab
(a) and on an uniform slab (b). . . . . . . . . . . . . . . . . . . . . 120
xvi
Abstract
This dissertation work studies the use of nanostructures to control the flow of
light in two application areas: photovoltaics and material self-assembly, which are
discussed in I and Part II, respectively.
The work in photovoltaics focuses on designing high-efficiency nanowire array
solar cells. Nanowire arrays are a promising candidate for the next generation
of low-cost, high efficiency, flexible photovoltaic cells. Nanowire structures relax
the lattice-matching constraints and allow the usage of materials with different
lattice constants for multijunction cells. This opens up a much wider range of
materials choices than for traditional, planar cells. The design of a high-efficiency
cell involves two factors: optical absorption and carrier collection. In this dis-
sertation, I first use the full-wave electromagnetic simulation to investigate the
absorption properties of periodic nanowire arrays and provide the optimal designs
in a single junction and a tandem wire-on-substrate cell configurations. I then
study strategies for optimal carrier collection by finite-element method electronic
device simulations. I optimize the p-n junction geometry, doping parameters, and
surface passivation scheme. This work not only establishes the fundamental limits
of nanowire solar cells’ designs but also provides practical guidelines and solutions
for high performance nanowire solar cell devices.
xvii
The work of material self-assembly is based on the light-assisted, templated
self-assembly (LATS) technique developed in our group. In this method, we shine
light through a photonic crystal (or template) to create an array of optical traps.
The traps drive the self-assembly of nanoparticles into regular patterns. In this
dissertation work, I discover the crucial effect of inter-particle interactions on the
pattern formation of metallic particles in the LATS system. I envision the analogy
between the optical assembly of nanoparticles and atomic level epitaxial growth
and suggest that the system can be viewed as “optical eptaxial growth”. I develop
the modeling and simulation technique to explain and predict experimental results.
Such model leads to the first successful demonstration of the optical assembly of
a 2D periodic gold nanoparticle array.
xviii
Part I
Nanowire Solar Cells
1
Chapter 1
Introduction
1.1 Solar energy and photovoltaic cells
Energy is one of the most important technical issues nowadays. The power
generation nowadays mainly relies on the burning of fossil fuels, which emits large
amount of carbon dioxide and other pollution. More importantly, fossil fuel will
eventually run out in the near future. In order to make sustainable development of
the civilization and cause less harm to the environment, new sources of substitute
clean energy are required.
Solar energy is an important alternative energy source to fossil fuel. Solar
energy is the planet’s most plentiful and widely distributed renewable energy
source. All wind, fossil fuel, hydro and biomass energy have their origins in sun-
light. The earth receives 120 petawatts (PW =10
15
W) of incoming solar radiation
on the surface [4], which means the solar energy captured by the Earth in 80
minutes equals to the annual global energy consumption in the year 2012 [5].
A range of technologies have been developed to harness the abundant solar
energy, such as solar heating, solar thermal energy, artificial photosynthesis and
solar photovoltaics [6]. Solar photovoltaics technology is one of the most promising
of them, which directly converts sunlight to electricity without any moving parts
or environmental emissions. Solar photovoltaic power plants can be deployed com-
parativelyfasterandeasierthanotherrenewableenergypowerplantssuchaswind,
hydro and solar thermal power. Moreover, since arid or semi-arid areas usually
2
are more suitable for building solar power stations, the occupation of large areas
of land by large utility stations will not be a serious problem.
Despite all the technological advantages that a photovoltaics system has and
dramatic market growth, total PV power is now around 138.9 GW (data from 2013
[7]), which is still small relative to the world’s 5550 GW of installed electric gen-
eration capacity (2.55% for data from 2012 [8]). The most critical issue becomes
economics. Over the past 20 years, the photovoltaic power industry has experi-
enced drastic technology advances and price drops, however, the levelized cost of
energy (LCOE) for PV plants is still above the range of conventional generation
options and other renewable generation options such as wind plants, even taking
into account the impact of the U.S. federal 30% investment tax credit [9].
The difficulty has always been converting solar energy in an efficient and cost-
effectiveway. Ononehand, theadvanceoftechnologiesleadstoveryhighefficiency
solar cell devices with efficiencies very close to the Shockley-Queisser detailed bal-
ance limit. More sophisticated multiple junction solar cells have efficiencies far
beyond the SQ limit. However, most high efficiency devices rely on expensive
crystalline materials. On the other hand, emerging thin-film technologies pro-
duce lower cost but lower energy conversion efficiency. The lower efficiency results
in higher installation cost, with the result that there is near cost parity at the
installed-system level. So far, a solar photovoltaics system with high efficiency at
low cost, which can challenge the LCOE of conventional generation options, has
not been achieved.
3
1.2 Nanostructured solar cells
The nanostructured solar cells discussed in this dissertation are nanowire array
solar cells,showninFigure1.1. Theyaremadeofaperiodicarrayofsemiconductor
nanowires. Both the diameter and spacing between wires are below 1 micrometer.
Because of the extended periodic structure, the device is fully compatible with
current solar panels for large area solar energy harvesting.
Figure 1.1: Rendering of a vertically-aligned semiconductor nanowire array.
The nanowire structure has been studied extensively in recent years, and it is
advantageous for photovoltaic applications in several aspects. Firstly, by proper
design, thenanowirestructurecanabsorbthesameamountorevenalargeramount
of light than a planar structure while using much less material, which greatly
reduces the material and process cost. Secondly, the nanowire structure can relax
the atomic lattice-matching constraints, enabling wider and more optimal material
choices for multi-junction solar cells and yielding a higher efficiency. Non-lattice-
matched growth of nanowire arrays also makes possible the growth of high quality
4
optoelectronic material such as gallium arsenide on a relative cheaper substrate
such as silicon. Thirdly, the ultra-thin nanowire array structures are flexible and
stretchable, which can enable novel applications in addition to the conventional
solar power plant electricity generation.
Designing and modeling these miniature sub-micrometer nanowire solar cell
devices are challenging. Optically, the sub-wavelength structure has intriguing and
counter-intuitivepropertieswhichcannotbeexplainedbythetraditionalray-optics
or effective-medium modeling approach. Full-wave electromagnetic simulation is
required to get accurate results. Electrically, the nanowires are inherently three
dimensional structures, different from the traditional planar, one-dimensional cells.
This opens up larger design parameter space. Furthermore, nanowire structures
have a much larger surface-to-volume ratio. Surface effects must be considered
carefully, and a robust surface passivation scheme is required for high performance
devices.
1.3 Outline of PART I
In the first part of the dissertation, I use accurate electromagnetic simulations
coupled with electrical device simulations to provide design guidelines for nanowire
arraysolarcells. Boththetheoreticallimitofperfectcarriercollectionandpractical
designs that consider non-ideal bulk and surface properties are discussed.
Chapter 2 focuses on a systematic study of the optical properties of semi-
conductor nanowire arrays made of various semiconductor materials for photo-
voltaic applications. It is shown that by optimizing the structural parameters, the
nanowire array can absorb more sunlight than an equally-thick thin film made of
5
the same material. The height dependence of the absorption is also discussed,
which is important for minimizing the material usage.
Chapter 3 presents the study of a III-V nanowire array on silicon, double-
junction solar cell. Such a tandem solar cell is an ideal proof-of-concept model and
key intermediate goal towards high efficiency, low cost, multijunction nanowire-
array solar cells on cheap and lattice-mismatched substrates. In this chapter, I
address the importance of current matching in nanowire-array-on-silicon solar cells
and how to tune the parameter of the nanowire array to achieve current matching.
The electrical consideration of the nanowire solar cell design is also discussed. The
junction position and geometry affect the efficiency strongly. The radial junction
outperforms the axial junction geometry in the case with bad surface quality.
To solve the issue of the inferior performance of the axial junction nanowire
solar cells, which are essential building blocks for high efficiency multi-junction
nanowire solar cells, in Chapter 4, I propose a high performance and robust surface
passivation design to mitigate undesired surface effects.
6
Chapter 2
Broadband absorption of
semiconductor nanowire arrays
for photovoltaic applications
A version of the results in this chapter was published as Ref. [10].
2.1 Introduction
Semiconductor nanowire solar cells are promising candidates for next-
generation, thin-filmphotovoltaicdevicesduetotheirattractiveanti-reflectionand
light-trapping properties. Recent experimental work has demonstrated vertically
aligned semiconductor nanowire arrays in silicon [11, 12, 13, 14, 15, 16, 17, 18, 19],
germanium [20], various direct band gap materials [21, 22, 23, 24, 25, 26, 27, 28],
and combined systems [29, 30, 31]. Nanowire arrays can be fabricated by either
top-down [11, 12, 32] or bottom-up [13, 20, 22, 31] methodologies. By using
different patterning techniques [20, 24, 25, 27, 33] regular arrays of nanowires
have been achieved. Junctions have been made between semiconductor nanowires
and substrate [22] and between the core and shell of semiconductor nanowires
[23, 33]. Experiments on hybrid nanowire/polymer systems have also been con-
ducted [34, 35].
7
In this section, we use electromagnetic simulations to map out the limiting
efficiencies of nanowire solar cells. We focus on structures for which the nanowires
themselves function as the broadband absorber. Ideally, a photovoltaic cell will
absorb as large a fraction of incident photons as possible over the entire solar
spectrum. Previousworkhasshownthatthestructuralparametersofthenanowire
array strongly influence the broadband absorption [36, 37]. Given proper design,
light-trapping effects yield high broadband absorption, even for nanowire heights
shorter than the bulk absorption length. Specifically, work from my group has
shown that the ultimate efficiency of an optimized silicon nanowire array exceeds
that of an equal-height thin film, even though it contains less absorptive material
[37]. Similar optimization work has been carried out for silicon nanowires on silicon
thin films [38], as well as for InP/InAs [39], InP [40] and GaAs/AlGaAs nanowire
arrays [41].
However, previous work has been restricted either to a fixed height or to a very
limited height range. The dependence of broadband absorption on height has not
been determined. It is important to determine the extent to which light trapping
can be used to minimize material usage while maintaining acceptably high photo-
voltaic efficiency. Material usage can have important implications for the cost of
a process. For bottom-up growth methods such as MOCVD, for example, it is of
particular interest to know what heights are sufficient to guarantee acceptable effi-
ciencies, given the potentially time-consuming and expensive nature of the growth
process. For photovoltaic space applications, the material volume affects the total
weight, which correlates with the launch cost. From a scientific standpoint, it is of
interest to determine how fast the efficiency degrades as the height of a nanowire
array is reduced, in order to determine whether optimized structures will allow
approach to an ‘ultra-thin’ film limit.
8
In this section, we systematically study the broadband absorption of vertically-
aligned nanowire arrays made of six common photovoltaic materials. For each
material, we study how the ultimate efficiency depends on the height of the array.
At each value of height, we optimize the structural parameters of the array to max-
imize the broadband absorption. Thus, the results we present concisely describe
the trade-offs between material usage and maximum achievable efficiency in semi-
conductor nanowire array solar cells. We further compare the optimized nanowire
arrays to thin films of the same height and show that for all six materials, and over
the entire range of heights tested (100 nm - 100μm), the ultimate efficiencies of the
arrays exceed those of equal-height thin films. Our results suggest that nanowire
array solar cells hold strong potential for the development of next-generation, thin-
film solar cells.
2.2 Methods
Figure 2.1 shows a schematic of a vertically-aligned semiconductor nanowire
array. The array is illuminated by sunlight from the top, as indicated by the red
arrow in Figure 2.1(a). The electric field of the incident light is polarized in either
the x- or the y-direction. As shown in Figure 2.1(b), nanowires with diameter d
are arranged in a hexagonal lattice with lattice constant a.
Weconsidernanowirearrayscomposedofoneofsixcommonphotovoltaicmate-
rials. Among the materials considered, silicon and germanium are indirect band
gap materials, while GaAs, InP, In
0.48
Ga
0.52
P, and CdTe are direct band gap mate-
rials. The optical constants (refractive indices and absorption lengths) are taken
from the literature: Si [42], Ge [42], GaAs [42], InP [43], In
0.48
Ga
0.52
P [44], and
9
d
a
a
a
h
a b
x
y
z
x
y
Figure 2.1: Schematic of a vertically-aligned semiconductor nanowire array. (a)
Perspective view. (b) Cross-sectional view.
CdTe [45] (Figure 2.2). Si has a relatively large absorption length in the 400-
1100 nm wavelength range compared to the other materials.
a b
Figure 2.2: Materials’ optical properties (a) Refractive indices (b) Absorption
lengths.
We use the ISU-TMM simulation package [46, 47], an implementation of the
transfer matrix method [48, 49], to calculate the broadband absorption of semi-
conductor nanowire arrays. The software can determine the wavelength-dependent
10
transmittance T (λ) and reflectance R(λ). The absorptance spectrum A(λ) is
obtained from the relation A(λ) = 1− T (λ)− R(λ). We average the results
for x- and y-polarized incident light. The spectral resolution is chosen to be 5 nm
in the wavelength range of interest.
We use the ultimate efficiency [50] to quantify the broadband absorption, as in
previous work [36, 37]. The ultimate efficiency is given by
η =
R
λg
300 nm
I(λ)A(λ)
λ
λg
dλ
R
4000 nm
300 nm
I(λ)dλ
(2.1)
where λ is the wavelength, and λ
g
is the wavelength corresponding to the band
gap of the semiconductor. I(λ) is the ASTM AM1.5 solar spectral irradiance [1],
which is plotted as a black line in Figure 2.3. A(λ) is the absorption spectrum.
We set the lower limit of integration to 300 nm in Equation 2.1 because the solar
irradiance is negligible below this value. The ultimate efficiency is an upper bound
on the achievable efficiency of a solar cell, assuming that each absorbed photon
with energy greater than the band gap produces exactly one electron-hole pair at
the energy of the gap, E
g
= hc/λ
g
. The ultimate efficiency can be related to the
maximumshortcircuitcurrentbyassumingperfectcarriercollectionefficiency, i.e.,
every photogenerated carrier can reach the electrodes and contribute to the pho-
tocurrent. Within this approximation, we do not explicitly consider the junction
geometry. In this case,
J
sc
=
Z
λg
300 nm
I(λ)A(λ)
eλ
hc
dλ =η
eλ
g
hc
Z
4000 nm
300 nm
I(λ)dλ (2.2)
Inthecaseofperfectabsorption, wemaysetA(λ) = 1inEquation2.1toobtain
the limiting value of ultimate efficiency η
max
[50]. The value of η
max
is plotted as
11
0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Band gap energy (eV)
Spectral Irradiance (W m
-2
nm
-1
)
0
10
20
30
40
50
Max ultimate efficiency (%)
1400 1000 800
W avelength (nm)
600
Figure 2.3: Direct+circumsolar spectral irradiance AM1.5 (black line), reprinted
from Ref. [1], and the perfect-absorption limit of the ultimate efficiency (blue line).
a function of E
g
in Figure 2.3 and obtains a maximum of 49% for a band gap of
1.12 eV.
For nanowire arrays, as for any realistic structure, the ultimate efficiency will
always be less than η
max
due to incomplete absorption. Below, we determine the
extent to which structural optimization of a nanowire array can yield efficiency
values approaching the perfect-absorption limit. Moreover, we determine the opti-
mized ultimate efficiency as a function of nanowire height.
We optimize the ultimate efficiencies of nanowire arrays with respect to the
structural parameters as follows. For fixed nanowire height h, we vary the lattice
constant a and the ratio of the lattice constant to diameter, a/d. Seven height
values were used: 100 nm, 300 nm, 1 μm, 3 μm, 10 μm, 30 μm and 100 μm.
Figure 2.4 shows an example of a parameter sweep for 3 μm long silicon nanowire
arrays. The lattice constant was varied from 200 to 1000 nm in steps of 100 nm
and the a/d parameter was varied from 1 (for which the nanowires touch) to 4
12
in steps of 0.2. The value of the ultimate efficiency is given by the color bar and
depends strongly on the structural parameters. In this specific case, the ultimate
efficiency varies from a minimum of 5% to a maximum of 26% in the parameter
space we consider. Below, we refer to the maximum value of ultimate efficiency,
optimized over a and a/d for a particular material and nanowire height, as the
optimized ultimate efficiency.
a/d
La ce constant a (nm)
1 2 3 4
200
300
400
500
600
700
800
900
1000
10
15
20
25
Ul"mate efficency(%)
Figure 2.4: Optimization of the ultimate efficiency with respect to the structural
parameters for a silicon nanowire array of 3 μm height.
2.3 Results and discussion
In Figure 2.5(a), we plot the optimized ultimate efficiency as a function of
height for all six materials considered. For each material, the optimized ultimate
efficiency increases with increasing height of the nanowire array. For the direct
band gap materials (GaAs, InP, InGaP, CdTe), the ultimate efficiency increases
quickly and is relatively flat for heights above 5 μm. For Si, which is an indirect
band gap material, the ultimate efficiency slowly increases over the entire range
shown (up to 100 μm). The data for Ge represent an intermediate case.
13
In Figure 2.5(b), we plot the optimized ultimate efficiency as a function of
bandgap energy. The black solid line is the perfect-absorption limit of ultimate
efficiency (η
max
), identical to the blue line in Figure 2.3. The colored symbols
represent identical data to Figure 2.5(a). The data for each set of nanowire arrays
are aligned to the bandgap energy of the constituent material. Different colors
represent different heights of the nanowire array. From this plot, the saturation
behavior of the ultimate efficiency may be clearly observed. For direct band gap
materials, theoptimizedultimateefficiencyapproachestheperfect-absorptionlimit
much more quickly than for indirect band gap materials. For the materials studied,
an optimized nanowire array of 3μm height provides an ultimate efficiency that is
>92% of the perfect-absorption limit for InP, GaAs, CdTe, and GaInP, while only
53% for Si and 78% for Ge.
a b
Figure 2.5: Optimized ultimate efficiencies for nanowire arrays composed of differ-
ent materials: (a) shown as a function of array height; (b) shown as a function of
the bandgap energy of the material.
Each data point in Figures 2.5(a) and (b) is obtained by optimizing the struc-
tural parameters of the nanowire array to maximize the ultimate efficiency. The
14
optimal values ofa anda/d are shown in Figures 2.6(a) and (b), respectively. Dif-
ferent color lines represent different materials. From Figure 2.6(a), we observe that
there is no obvious trend in the optimal lattice constant. However, most values
are comparable to the wavelength of visible light (∼ 400− 700 nm). Figure 2.6(b)
shows that the optimal value of a/d increases with increasing nanowire height.
Larger values of a/d correspond to sparser arrays. Intuitively, as the nanowire
height increases, it becomes comparable to or larger than the absorption length in
the material. In this limit that the nanowire height exceeds the absorption length
over the whole solar spectrum, the ultimate efficiency will be maximized by min-
imizing the reflection from the top surface, which may be achieved by increasing
a/d. From Figure 2.6(b), it may also be observed that the optimal value of a/d
increases more quickly with length for direct band gap materials than for Si and
Ge.
a b
Figure 2.6: Optimal structural parameters for different heights of different mate-
rials: (a) lattice constant; (b)a/d ratio.
In Figure 2.7, we compare the performance of optimal nanowire arrays to thin
films of the same height. The plot shows that in the entire height range we con-
sider, optimal nanowire arrays have higher ultimate efficiencies than their thin-film
15
counterparts. This holds true even though the nanowires contain a smaller volume
of absorbing materials than the thin films. Nanowire structures tend to have a
lower reflection from their top surface than a thin film. Moreover, the nanowires
can also couple normally incident light into modes that propagate in the plane of
the array, a form of light trapping.
Above, we have considered free-standing nanowire arrays in air. The methods
we use can be straightforwardly adapted to model particular choices of substrate
material, contact geometries and materials. As one example of a more complex
geometry, weconsidertheeffectofasubstrateontheultimateefficiency. Wechoose
GaAs as a representative direct band gap material and compare three possible
substrate choices (no substrate, GaAs substrate, and glass substrate) using FDTD
calculations. We assume that the solar cell is designed such that photogenerated
carriers are collected only from the nanowire region. Hence, we calculate the
ultimate efficiency using A(λ) for the nanowire region alone. Absorption spectra,
A(λ), are obtained from the FDTD simulation by monitoring the flux difference
between flux planes located above and below the nanowire array. We vary the
height of the nanowire array as in the calculations above. For each height, we use
the optimal structural parameters from Figure 2.6, which were obtained for the
free-floating structure.
The black line in Figure 2.8 shows the ultimate efficiency of GaAs nanowire
arrays without a substrate, corresponding to the green dotted line in Figure 2.5(a).
The red and blue lines represent the ultimate efficiencies achieved for a GaAs and
a glass substrate (n = 1.55) underneath the nanowires, respectively. As one would
expect, the glass substrate has a smaller effect on the ultimate efficiency than
the higher index, GaAs substrate. Moreover, we observe that when the height
of the nanowire array is longer than about 3 μm, the effect of a glass or GaAs
16
a b
c d
e f
Si Ge
GaAs InP
InGaP
CdTe
Figure 2.7: Comparison of the ultimate efficiencies of optimized (a) Si, (b) Ge,
(c) GaAs, (d) InP, (e) InGaP, and (f) CdTe nanowire arrays with the ultimate
efficiencies of thin films made of the same material and of the same height.
17
substrate on the ultimate efficiency is minimal. Intuitively, when the height of
the nanowire array is large enough, most light will be absorbed without reflecting
from the interface between the nanowires and the substrate. We observe that this
height scale is similar to that at which the ultimate efficiency of the nanowire
array approaches the perfect-absorption limit (Figure 2.5(b)). We note that the
ultimateefficienciesshownbytheblueandredlinesinFigure2.8arenotnecessarily
the maximum values achievable in the presence of a substrate; re-optimizing the
structural parameters for a particular substrate of interest is likely to increase the
ultimate efficiency.
Figure 2.8: Effect of the substrate on the absorption in nanowire arrays.
2.4 Summary
In summary, we use electromagnetic simulation tools to quantitatively deter-
mine how the ultimate efficiency of optimized semiconductor nanowire arrays
approaches the perfect-absorption limit as a function of nanowire height. More-
over, we demonstrate that optimized nanowire arrays in six different materials and
for a range of heights from 100 nm to 100μm all outperform unpatterned thin films
18
of equal height. The results will assist in the design of highly-efficient nanowire
solar cells, while minimizing material usage.
In this chapter, we have considered nanowire arrays consisting of a single semi-
conductor material. Of the materials considered, GaAs, In
0.48
Ga
0.52
P and Ge are
commonly used in triple-junction solar cells. In the next chapter, we will extend
ourmodelingtoaIII-Vnanowirearrayonsilicondual-junctionnanowiregeometry.
We will also incorporate both optical and electrical properties in a more detailed
model to guide cell design.
19
Chapter 3
Limiting efficiencies of tandem
solar cells consisting of III-V
nanowire arrays on silicon
A version of the results in this chapter was published as Ref.[51]
3.1 Introduction
In the previous chapter, we have optimized the optical properties of single-
junction nanowire arrays with respect to nanowire size and spacing. We showed
that optimized arrays have a higher broadband absorption than a thin film of the
same height, increasing efficiency while shrinking material usage.
Nanowire tandem cells promise further increases in photovoltaic efficiency
[52, 53, 54]. Tandem solar cells use multiple semiconductor materials with differ-
ent band gaps to selectively absorb different wavelength ranges of the solar spec-
trum, reducing thermal loss and increasing efficiency [55]. However, the choice of
materials is traditionally constrained by lattice-matching requirements. Nanowire
structures [56, 57, 58] have a high tolerance of lattice mismatch, enabling more
material choices. Moreover, large-area fabrication of III-V nanowire arrays with
controlled structural parameters has been successfully demonstrated using scalable
patterning techniques such as nanosphere lithography [59].
20
In this chapter, we focus on double-junction cells in which the top junction is
formed in a III-V nanowire array, and the bottom junction is formed in a silicon
substrate. This system is promising for several reasons. First, III-V direct gap
alloys are highly absorptive and provide a wide range of band gap choices. Recent
experiments [53, 60, 61, 62, 63] have successfully demonstrated the fabrication
of III-V nanowire arrays on relatively inexpensive silicon substrates. Second, this
approachleveragespreviousdevelopmentofhighlyoptimized, singlejunction, crys-
talline silicon cells, used here as the bottom cell. Previous work in the literature
has simulated a nanowire on silicon tandem cell for a III-V material with 1.7 eV
band gap, assuming a simple Beer-Lambert model of absorption. Experiments
have demonstrated a nanowire on silicon tandem cell in InGaAs [64], a material
with a suboptimal bandgap. However, a number of important general questions
remain. How should the nanowire array be designed to achieve current match-
ing between top and bottom cell? How does the achievable efficiency vary with
the band gap of the top material, and what nanowire size and spacing optimizes
it? And lastly, how should the nanowire p-n junction be designed to achieve high
carrier collection efficiency?
In this chapter, we first calculate the limiting efficiency of III-V nanowire on
silicon tandem solar cells, using realistic 3D electromagnetic simulations to model
absorption and a detailed-balance model to estimate current collection. We con-
sider several choices of III-V material with different band gaps. For each, we find
the optimal structural parameters (lattice constant and nanowire diameter) as a
function of nanowire height. We discuss the implications of current matching on
the array design. We then conduct more accurate device simulations, using finite
element modeling of the drift-diffusion equations. We study the effect of the p-n
junction design and surface recombination on carrier collection and compare to
21
the results of the detailed-balance model. Taken together, our results provide a
comprehensive set of design guidelines for the optimization of III-V nanowire on
silicon tandem cells.
3.2 Optical absorption modeling and detailed
balance analysis
Inthissection,weuseopticalabsorptionmodelinganddetailedbalanceanalysis
[50] to estimate an upper bound on the efficiency of III-V nanowire on silicon
tandem solar cells. We first consider the detailed balance limit for an idealized
tandem cell. We then use three-dimensional, full vectorial simulations of Maxwell’s
equationstoobtaintheabsorptanceofvariousactualnanowirearrays. Weoptimize
the nanowire structural parameters so as to provide tandem cell efficiencies as
close as possible to the ideal limit. We describe how current matching conditions
affect the optimal parameters for nanowire arrays composed of different materials,
specifically those with band gaps above and below the ideal value.
3.2.1 III-V nanowire array on silicon structure
For the purpose of optical modeling, we consider the structure shown in Fig-
ure 3.1. The nanowires form a vertically aligned, hexagonal array, characterized
by the distance between adjacent nanowiresa, nanowire diameterd, and nanowire
heighth. We assume a semi-infinite silicon substrate. In practice, this means that
the bottom silicon cell is assumed to be thick enough to absorb any light above its
band gap that is not absorbed in the nanowire array. It is assumed that each sub-
cell (nanowire array, silicon cell) contains a p-n junction, and that the two subcells
are connected in series.
22
Semi-infinite substrate (bottom material)
III-V nanowire array (top material)
h
d
a
x
y
z
a b
Figure 3.1: Schematic of a vertically aligned III-V nanowire array on a semiinfinite
substrate. (a) Perspective view and (b) Top view.
We consider four different III-V materials for the nanowire array, each with a
different band gap energy (InP 1.34 eV, GaAs 1.43 eV, Al
0.2
Ga
0.8
As 1.72 eV, and
Ga
0.5
In
0.5
P 1.9 eV).
3.2.2 Detailed balance analysis
We use a modified detailed balance analysis to estimate the efficiency of the
nanowire tandem solar cell. In each subcell, the short circuit current can be related
to the absorptance by
J
sc,i
=
e
hc
Z
λ
g,i
310 nm
I(λ)A
i
(λ)λdλ (3.1)
where the subscription i=1,2 represents the nanowire array or silicon substrate,
respectively. Here, λ is the wavelength, λ
g
is the wavelength corresponding to
the band gap of the absorbing material, I(λ) is the ASTM AM1.5D solar spectral
irradiance [1], and A
i
(λ) is the absorptance spectrum of the subcell. Equation 3.1
23
assumes one absorbed photon can generate one electron hole pair and perfect
carrier collection, so represents an upper bound on the short circuit current.
We use the J−V characteristic of an ideal diode to describe the electrical
properties of each subcell
V
i
(J) =
k
B
T
e
ln
J
sc,i
−J
J
0,i
+ 1
!
, (3.2)
whereJ
0,i
isthereversesaturationcurrentdensity. Inthedetailed-balanceanalysis,
J
0,i
is calculated by assuming that the only loss process in the dark is radiative
relaxation of electrons through spontaneous emission, which is in detailed balance
with the absorption of ambient blackbody radiation at room temperature [50].
Absorption is assumed to occur at all photon energies above the semiconductor
material’s band gap. The reverse saturation current density can then be written
as an integral of the Planck distribution,
J
0,i
=
2πe
h
3
c
2
Z
∞
E
g,i
E
2
exp(E/k
B
T )− 1
dE (3.3)
The current matching condition dictates that because the two subcells are
connected in series, the current density J is the same for both. The total J−V
curve can be obtained from
V
total
(J) =V
1
(J) +V
2
(J). (3.4)
J cannot exceed the minimumJ
sc
of the two cells. GivenA
i
(λ) for each subcell, we
can find the maximum power point on the J−V curve numerically to determine
the efficiency of the structure. Below, we calculate the efficiencies obtained from
two models for A
i
(λ): an idealized model based only on the band gap of the
24
material, and an accurate model based on full, electromagnetic simulations of the
nanowire-on-silicon structure.
Detailed balance efficiency limit: Idealized model
Ideally, any incident photons with energy above the band gap of either the
nanowire material or the silicon would be absorbed by the tandem cell structure
and contribute to the current. However, in an actual structure, some photons
are always reflected from the device and do not contribute. We first study ideal
conditions to determine an upper bound on the tandem cell efficiency. Following
the analysis in Ref. [65], we consider two cases: “perfect” absorption, where the
nanowire array absorbs all incident photons with energy above its band gap, and
“imperfect” absorption, where the nanowire array absorbs only a portion of such
photons. We calculate the efficiencies for these two cases using the detailed balance
analysis described above.
Figure 3.2(a) shows the case of perfect absorption. The efficiency is plotted as
a function of the band gap energies of the top and bottom subcells. The highest
limiting efficiency (45.3%) occurs when the band gaps of the top and bottom
subcells are 1.57 eV and 0.935 eV, respectively. The dashed line shows the band
gap for silicon (1.1 eV). For certain band gap combinations, shown as the blue,
solid line in graph, the number of photons absorbed in the top layer is the same
as the number of photons absorbed in the bottom layer. These cases provide
current matching and yield higher efficiency. For band gap combinations below
and to the right of the current matching line, the top cell’s band gap is higher
than optimal, and it limits the current due to insufficient absorption. For the
band gap combinations above and to the left of current matching line, the top cell
25
absorbs too much light, limiting the amount of photons available for the bottom
cell. In this case, the bottom cell limits the current.
We next consider the case of imperfect absorption. We choose the value of
fractional absorption in the top cell to maximize the efficiency. The fractional
absorption is defined as the percentage of photons with energy above the band gap
of the top cell that are absorbed therein. Results are shown in Figure 3.2(b). In
the region of the plot below the current matching line, 100% absorption is optimal.
Above the line, imperfect absorption maximizes the efficiency. Figure 3.2(c) shows
the efficiency obtained using the fractional absorption of Figure 3.2 (b). From
the figure, it can be seen that in the region below and to the right of the current
matching line, the limiting efficiencies are the same as those for perfect absorption
case. However, in the region above and to the left of the current matching line, the
efficiency is higher than for the case of perfect absorption. This is due to current
matching constraints. By decreasing the absorption in the top cell, the solar flux
can be evenly divided between the two subcells, yielding higher efficiency.
In Figure 3.2(d), we plot the efficiency values along the dashed lines in Figures
3.2(a)and3.2(c), correspondingtothespecificcasewherethebottomcellissilicon.
We see that the ideal band gap value of the top cell is close to 1.7 eV. For a range
of values below 1.7 eV, the limiting efficiency can be higher than 30% provided
that the absorption in the top cell is optimized. This can be achieved in an actual
nanowire structure by tuning the structural parameters, as investigated below.
Detailed balance efficiency limit: Full electromagnetic simulations
We next determine the efficiency limit of the nanowireon-silicon tandem cell
using the calculated absorptance spectra for realistic nanowire structures. We
optimize the detailed balance efficiency via an exhaustive scan over the structural
26
E
g,top
(eV)
E
g,bot
(eV)
1 1.4 1.8 2.2 2.2
0.4
0.8
1.2
1.6
0
10
20
30
40
E
g,top
(eV)
E
g,bot
(eV)
1 1.4 1.8 2.2 2.2
0.4
0.8
1.2
1.6
0
10
20
30
40
E
g,top
(eV)
E
g,bot
(eV)
1 1.4 1.8 2.2 2.2
0.4
0.8
1.2
1.6
50
60
70
80
90
100
0
10
20
30
40
E!ciency (%)
E
g,top
(eV)
40
60
80
100
Perfect absorption
Imperfect absorption
Fractional absorption (%)
1 1.4 1.8 2.2
a b
c d
Figure 3.2: (a) Limiting detailed balance efficiency for double junction solar cells
as a function of band gap energies assuming perfect absorption in the top cell. (b)
Optimal fractional absorption in the top cell that maximizes the total efficiency.
(c) Limiting detailed balance efficiency with optimal fractional absorption in the
top cell given in (b). (d) Limiting efficiencies (left axis) and optimal fractional
absorption (right axis) as a function of the band gap of the top cell, for a fixed
bottom cell band gap energy of 1.1 eV (silicon), corresponding to the white, dashed
lines in (a)-(c).
parameters (lattice constant a and d/a ratio) for each nanowire height h and
for each nanowire material considered. We use a modified version of the ISU-
TMMsimulationpackage[46], animplementationofthescatteringmatrixmethod,
to calculate the wavelength-dependent absorptance in both the nanowire array
(A
NW
(λ)) and the silicon substrate (A
S
(λ)). The optical constants of each material
(refractive index and absorption length) used in the simulation are taken from
Ref. [45].
27
InFigure3.3(a), weplottheoptimizeddetailedbalanceefficienciesasafunction
of height for all four nanowire materials considered. We see that the efficiencies
of InP and GaAs nanowire arrays peak at a relatively short height of∼ 550 nm,
while the efficiencies of the other two materials continue to increase with nanowire
height, up to heights of 10 μm. Figure 3.3(b) shows the same set of data, where
the efficiency values for each material are aligned with the band gap energy of the
material. Different symbols represent different nanowire heights. The solid lines
show the limiting efficiency.
0.1 1 10
10
20
30
40
InP
GaAs
Al
0.2
Ga
0.8
As
Ga
0.5
In
0.5
P
Efficiency η (%)
Height h (μm)
1.0 1.2 1.4 1.6 1.8 2.0 2.2
0
10
20
30
40
50
Ga
0.5
In
0.5
P
Al
0.2
Ga
0.8
As
GaAs
Limiting efficiency for planar cells
h =100nm h =300nm h =1μm
h =3μm h =10μm
Efficiency η (%)
Top cell band gap energy (eV)
InP
a b
Figure 3.3: (a) Optimized detailed balance efficiency as a function of nanowire
height for different nanowire materials. (b) Optimized detailed balance efficiency
as a function of nanowire band gap energy for nanowire arrays of different heights.
The solid lines give the limiting detailed balance efficiency, identical to the red
dashed line in Figure 3.2(d).
Optical structural parameters
The optimized efficiencies in Figure 3.3 correspond to particular values of the
nanowire structural parameters. These values are shown in Figure 3.4. For all
four materials, the optimal lattice constant increases with height. For InP and
GaAs, the optimal lattice constant increases more rapidly with height than for
28
AlGaAs and GaInP. The ratio of the diameter to the lattice constant (d/a) tends
to decrease with height for all four materials. The optimal diameter has relatively
little variation with nanowire height. The results indicate that as the nanowire
height increases, the optimal array is sparser, corresponding to a smaller filling
fraction.
For InP and GaAs, as the height increases, making the array sparser helps to
reduce the fractional absorption in the top cell to the optimal value (shown in
Figure 3.2(b)). For AlGaAs and GaInP, 100% absorption is desired. However, for
larger heights, high absorption can be achieved with sparser wires, which simulta-
neously reduce reflection from the top surface.
The optimal structural parameters depend on the current matching constraint.
In Figure 3.5, we consider 1.8 μm-tall GaAs and GaInP nanowire arrays as illus-
trative examples. GaAs has a bandgap energy of 1.43 eV, below the optimal band
gap (1.7 eV), while GaInP has a bandgap of 1.9 eV, larger than optimal.
Figure 3.5(a) shows the percentage of photons with energy above the band gap
of Si (1.12 eV) that are absorbed by the GaAs nanowire array (blue) and the silicon
substrate (red) as a function of lattice constant (a) and d/a ratio. Depending on
the structural parameters, the nanowire array absorbs either more or less light
than the silicon substrate. Figure 3.5 shows the detailed balance efficiency map
as a function of the structural parameters. Along the white line, the number of
photons absorbed in the nanowire array and in the substrate is the same, and the
efficiency is high. Note that the white line corresponds to the curve along which
the two surfaces in Figure 3.5(a) cross.
For the GaInP case in Figure 3.5(c), no matter how we change the structural
parameters, the absorption in the GaInP nanowire array is lower than the absorp-
tion in the silicon substrate. Figure 3.5(d) shows the efficiency map as a function
29
0.1 1 10
0
400
800
1200
1600
InP
GaAs
Al
0.2
Ga
0.8
As
Ga
0.5
In
0.5
P
Lattice constant a (nm)
Height h (μm)
0.1 1 10
0.0
0.2
0.4
0.6
0.8
1.0
InP
GaAs
Al
0.2
Ga
0.8
As
Ga
0.5
In
0.5
P
d /a ratio
Height h (μm)
0.1 1 10
0
100
200
300
400
InP GaAs
Al
0.2
Ga
0.8
As Ga
0.5
In
0.5
P
Diameter d (nm)
Height h (μm)
a b
c
Figure 3.4: Optimal structural parameters for different materials as a function of
nanowire height: (a) lattice constant a, (b)d/a ratio, and (c) diameter d.
of the structural parameters (efficiency value indicated by color bar). Overlaid on
this plot are contours indicating the fractional absorption in the GaInP nanowire
array (white lines/numbers). Note that the contour lines correspond to the data
shown by the blue surface in Figure 3.5(c). The highest efficiencies occur when the
absorption in the nanowire array is highest.
30
a
c
d/a ratio
Lattice constant a (nm)
Detailed balance e!ciency (%)
0.2 0.4 0.6 0.8
200
400
600
800
10
15
20
25
30
d/a ratio
Lattice constant a (nm)
Detailed balance e!ciency (%)
10
20
20
30
30
30
30
30
30
33
33
33
33
33
33
33
35
35
0.2 0.4 0.6 0.8
200
400
600
800
10
15
20
25
30
35
Photon absorption (%)
Photon absorption (%)
GaAs/Si
GaInP/Si
III-V
Silicon
GaAs/Si
GaInP/Si
b
d
Figure 3.5: Relationship of photon absorption to detailed balance efficiency for
materials with band gaps above and below the optimal value. (a) and (b) 1800 nm-
tall GaAs NW array on silicon. (c) and (d) 1800 nm tall GaInP NW on silicon.
(a) and (c) The percentage of photons with energy above the bandgap of silicon
absorbed by the nanowire array (blue) and silicon layer (red) as functions of lattice
constant and d/a ratio. (b) The detailed balance efficiency for the GaAs NW
tandem cell. The white line indicates the structural parameters for which the
percentage of photons absorbed in GaAs and silicon is equal. (d) The detailed
balance efficiency for the GaInP NW tandem cell. Overlaid contours show the
percentage of photons absorbed in GaInP.
31
We can relate the insight gained from Figure 3.5 to the difference in satura-
tion behavior with height seen in Figure 3.3(a). For GaAs and InP, the highest
efficiencies are obtained when the absorption in each subcell is the same, and per-
fect absorption in the nanowire array is not required. As a result, large nanowire
heights are not necessary. For AlGaAs and GaInP, the largest efficiencies are
obtained when the absorption in the nanowire array is maximized. This absorp-
tion increases with increasing height.
3.3 Electrical transport modeling
Above, we have used detailed balance analysis to find an upper limit on the
efficiency of nanowire-on-silicon tandem cells, and we have found the nanowire
structural parameters that optimize this efficiency. In this section, we show how
electrical transport modeling can be used to design nanowire p-n junctions with
efficiencies as close as possible to the detailed-balance limit. In particular, we
compare radial and axial junction designs for a sample, GaAs nanowire array and
compare the effect of surface recombination in the two types of junction.
3.3.1 Junction geometry
Figure 3.6 shows two possible junction geometries in the nanowires, radial
(a) and axial (b). Transparent conductive oxide (TCO) and metal contacts are
placed at the top and bottom of the structure. The structural parameters used in
simulation come from the optimization of a GaAs nanowire array on silicon based
on the detailed-balance model (a=560 nm, d=268 nm, and h=1 μm). Single-
junction silicon solar cells have been the subject of extensive development. We
thus focus on the transport properties of the upper GaAs nanowire cell, as shown
32
in Figures 3.7(a) and 3.7(b). In electrical simulations below, for the purpose of
calculating nanowireJ−V curves, we will assume ideal ohmic contacts at the top
and bottom of the nanowire.
TCO contact
n III-V
p III-V
Tunneling
junction
n Si
p Si
Back contact
Oxide
a b c
1 μm
134 nm
log (m s )
10
-3 -1
Figure 3.6: Junction designs for III-V NW on silicon tandem cells with (a) radial
junction and (b) axial junction in the nanowire. The thickness of the silicon sub-
strate is not to scale. (c) Carrier generation rate profile in GaAs nanowire.
3.3.2 Electrical transport simulation methods
In order to accurately simulate the J−V response of nanowire solar cell, we
first need to calculate the positiondependent carrier generation rate. The position-
dependent absorptance normalized to incident power (P
in
) in unit of (m
−3
) is
A(r,λ) =
1
2
ω
00
|E(r,λ)|
2
/P
in
(3.5)
where
00
is the imaginary part of the position-dependent permittivity. The electric
field intensity is obtained by finitedifference time domain (FDTD) simulation using
33
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0
5
10
15
20
Current density (mA/cm
2
)
Voltage (V)
Radial SRV = 0 cm/s
Radial SRV = 3000 cm/s
Radial SRV = 30000 cm/s
Axial SRV = 0 cm/s
Axial SRV = 3000 cm/s
AxialSRV = 30000 cm/s
Detailed balance limit
n++
a
Ohmic contact
n
p
p++
Ohmic contact
b c
Figure 3.7: Simulated structures for (a) radial junction and (b) axial junction
GaAs nanowire top cell. (c) J−V curves for axial and radial junction geometries
for varying SRV.
theLumericalsoftwarepackage. Assumingthatonephotongeneratesoneelectron-
hole pair, the position-dependent carrier generation rate can be calculated as
G(r) =
Z
867 nm
310 nm
A(r,λ)I(λ)Sλ
hc
dλ (3.6)
where S is the unit cell area, I(λ) is the AM1.5D solar irradiance, and hc/λ is
the photon energy. We make the approximation that the field distribution in the
nanowire is rotationally symmetric in order to reduce the problem to 2D. The
2D carrier generation rate is calculated using a circularly polarized source, which
effectively time averages different incident polarizations. Figure 3.6 (c) shows the
carrier generation rate profile. The effect of the contacts is not included in the
optical simulations. We can see clearly that there is a highly concentrated hotspot
at around 100 nm from the top surface.
34
We use the finite element method to calculate a realisticJ−V relation for the
cell, given different p-n junction designs. In this part, we solve two current conti-
nuity equations (3.7) and (3.8) coupled with Poisson’s equation (3.9) in COMSOL.
∇·J
n
=∇· (nμ
n
∇E
F,n
) =−e(G−R), (3.7)
∇·J
p
=∇· (pμ
p
∇E
F,p
) =e(G−R), (3.8)
∇
2
ψ =−e(p−n +N
+
D
−N
−
A
). (3.9)
Here,μ
n
,μ
p
are the electron and hole mobilities, andN
+
D
andN
−
A
are the donor and
acceptor doping concentrations. G the carrier generation rate, from Equation 3.6.
R the recombination rate. Here, only SRH recombination is taken into account in
the simulation, and traps are assumed to be at mid-gap. SRH recombination can
be written as
R
SRH
=
np−n
2
i
τ
p
(n +n
i
) +τ
n
(p +n
i
)
(3.10)
wheren
i
istheintrinsiccarrierdensityandτ
n
andτ
p
aretherecombinationlifetimes
forelectronsandholes,respectively. E
F,n
andE
F,p
arethequasi-Fermienergylevels
for electrons and holes. Under Fermi statistics, the relation between electron and
hole densities (n andp) and electron and hole quasi-Fermi energy levels (E
F,n
and
E
F,p
) are
n =N
c
γ
n
exp
E
F,n
−E
c
k
B
T
, (3.11)
p =N
v
γ
p
exp
E
v
−E
F,p
k
B
T
, (3.12)
35
where N
c
and N
v
are effective densities of states for the conduction band and
valence band, respectively. E
c
andE
v
are the conduction and valence band edges.
γ
n
and γ
p
are called “degeneration factors” and are defined as
γ
n
=F
1/2
E
F,n
−E
c
k
B
T
,
exp
E
F,n
−E
c
k
B
T
, (3.13)
γ
p
=F
1/2
E
v
−E
F,p
k
B
T
,
exp
E
v
−E
F,p
k
B
T
, (3.14)
where F
1/2
is the Fermi integral of order 1/2. Then Equations 3.7 and 3.8 can be
expanded into
∇·J
n
−e
=∇· (−D
n
∇n +D
n
n∇lnγ
n
+μ
n
n∇ψ) =G−R, (3.15)
∇·J
p
e
=∇· (−D
p
∇p +D
p
p∇lnγ
p
+μ
p
p∇ψ) =G−R, (3.16)
whereD
n
andD
p
are diffusion coefficients defined asD
n,p
=μ
n,p
k
B
T/e. Equations
3.9, 3.14, and 3.16 are three equations with three unknowns: electron density (n),
hole density (p), and electrostatic potential (ψ). With proper boundary conditions,
these coupled equations can be solved and the current-voltage response can be
extracted from the solution. Note that γ
n
and γ
p
are defined by Equations 3.11-
3.14.
The boundary conditions for electron and hole densities can be written as
ˆ n·J
n
=−ev
th
(n−n
0
), (3.17)
ˆ n·J
p
=ev
th
(p−p
0
), (3.18)
36
where ˆ n is the outward normal to the nanowire surface. We have assumed that the
carrier extractions are equal to the thermal velocity v
th
and are set to 10
7
cm/s
for both electrons and holes.
For an ideal ohmic contact, the potential ψ fixed by the external bias voltage
(V
a
),
ψ = ψ
0
,at the n contact (3.19)
ψ = ψ
0
+V
a
,at the p contact (3.20)
where ψ
0
, n
0
, and p
0
are the electrostatic potential, electron density, and hole
density in thermal equilibrium, which can be solved by using Equations 3.9, 3.1,
and 3.12, assuming that E
F,n
=E
F,p
=E
F
.
At the nanowire surface, the boundary conditions for n and p are
ˆ n·J
n
=−eR
surf
(3.21)
ˆ n·J
p
=eR
surf
(3.22)
whereR
surf
is the surface recombination rate. We assume SRH recombination with
the traps at midgap. Then
R
surf
=
v
th
σ
n
σ
p
(N
dt
+N
at
)(np−n
2
i
)
σ
p
(p +n
i
) +σ
n
(n +n
i
)
, (3.23)
where v
th
is the thermal velocity of the carriers, σ is the trap cross section, and
N
dt
and N
at
are the donor and acceptor-like trap density.
37
Because of the surface recombination, there is charge accumulation on the
surface. The positive charges at the surface are caused by occupation of the donor-
like states by holes and have the form
p
s
=
N
dt
(σ
n
n
i
+σ
p
p)
σ
p
(p +n
i
) +σ
n
(n +n
i
)
. (3.24)
The negative charges at the surface are caused by occupation of the acceptor-like
states by electrons and have the form
n
s
=
N
at
(σ
p
n
i
+σ
n
n)
σ
p
(p +n
i
) +σ
n
(n +n
i
)
. (3.25)
The net surface charge densityQ
SS
is equal toe(p
s
−n
s
). The boundary condition
for the electrostatic potential is
∇ψ =−
Q
ss
0
r
, (3.26)
from Gauss’ law.
In summary, solving the equations above yields the position-dependent poten-
tial (ψ), which is related to the applied voltage V and carrier concentrations (n
and p), which can be used to calculate the current density J. Thus, the J−V
curve and efficiency can be obtained.
We carry out simulations using the parameters listed in Table 3.1. The electron
and hole mobilities are assumed to have the same values as bulk materials, which
are taken from Ref. [66]. SRH recombination lifetimes are taken from Ref. [67].
This set of parameters gives diffusion lengths of 15 μm for electrons and 3.87 μm
for holes. The donor and acceptor concentrations are fixed to 1× 10
18
cm
−3
.
10 nm-thick minority carrier reflectors with 1× 10
19
cm
−3
doping concentration
38
Table 3.1: Device simulation parameters for nanowire top cells
Parameter Description Nominal values
d
NW
Diameter of nanowire 269 nm
h
NW
Height of nanowire 1000 nm
N
c
Effectivedensityofstatesinconductionband 3.97× 10
17
cm
−3
N
v
Effective density of states in valance band 9.68× 10
18
cm
−3
μ
n
Electron mobility 2500 cm
2
/Vs
μ
p
Hole mobility 150 cm
2
/Vs
τ
n
=τ
p
SRH recombination lifetimes 1 ns
N
+
D
Donor concentration (n-doping) 1× 10
18
cm
−3
N
−
A
Acceptor concentration (p-doping) 1× 10
18
cm
−3
N
dt
Surface donor like trap density 0, 1.5×10
11
cm
−2
,1.5×
10
12
cm
−2
N
at
Surface acceptor like trap density 0, 1.5×10
11
cm
−2
,1.5×
10
12
cm
−2
v
n,p
th
Thermal velocities for carriers 1× 10
7
cm/s
σ
n,p
Trap cross-sections for carriers 1× 10
−15
cm
2
r
Relative permittivity for GaAs 13.2
are put just below/above the top/bottom contacts to reduce recombination (shown
as n++ and p++ regions in Figures 3.7(a)and 3.7(b)). We consider three kinds
of surfaces here. The first case is a perfectly passivated nanowire surface. The
second case is a surface with both donor and acceptor-like traps with densities
of 1.5× 10
11
cm
−2
. For this case, the surface recombination velocity (SRV) is
3000 cm/s, which has been demonstrated experimentally by AlGaAs passivation
[68, 69]. The third case is a surface with poorer surface passivation, with donor
and acceptor-like trap densities of 1.5× 10
12
cm
−2
.
3.3.3 Design examples and results
Figure 3.7(c) shows sample J−V curves for radial and axial junction geome-
tries. For reference, we also plot the J−V curve given by the detailed balance
39
limit. The junction depth is selected to be 35 nm for the radial junction and
100 nm for the axial junction. For no surface recombination (SRV=0 cm/s) and
for low surface recombination (SRV =3000 cm/s), the radial junction has a higher
short circuit current, while the axial junction has a higher open circuit voltage.
The current density at zero voltage is higher for the radial junction because the
distance required for carriers to diffuse to the junction is shorter, improving car-
rier extraction efficiency. However, the junction area is much larger for the radial
junction than the axial junction. Therefore, under forward bias, recombination in
the radial junction is more severe, reducing the open circuit voltage. When the
surface recombination is severe (SRV=30,000 cm/s), the short circuit current of
the radial junction is much larger than that of the axial junction. The open circuit
voltages for both junctions have similar values.
From Figure 3.7(c), we can see that the radial junction is more tolerant to
surface recombination than the axial junction. If the n-type shell is thin, most of
the excess carriers are generated in p-region, which is protected from the surface.
Once these electrons diffuse across the junction to become majority carriers, the
effect of surface recombination is negligible. In contrast, for the axial junction,
both n and p regions are exposed to surface.
Given theJ−V curve for a particular nanowire junction geometry, we calculate
themaximumpowerandnormalizebythemaximumpowerforthedetailedbalance
limit.
Figure 3.8 shows the normalized maximum power as a function of junction
depth for radial and axial junctions from side wall and top, respectively, with
the three cases of surface passivation considered above. In the case of no surface
recombination, the radial junction’s power increases with junction depth over the
range shown. The axial junction’s power reaches a maximum of 0.75 at a junction
40
depth of 260 nm and decreases as the junction depth is further increased. For
low surface recombination (SRV=3000 cm/s), these trends remain without much
power reduction. However, the drop in power in the axial junction is larger than in
the radial junction. In the case with severe surface recombination, the difference is
clearer. The normalized power of the radial junction is around 0.55 for a junction
depth of 35 nm and drops with increasing junction depth. The axial junction can
only achieve a normalized power of less than 0.35 with a junction depth of 100 nm.
The optimal junction depth for the axial junction is close to the position of the
absorption hotspot seen in Figure 3.6(c).
Two things are worth noticing here for the axial junction design. (1) Since
the axial junction requires a long diffusion length of photogenerated carriers, high
material quality and a carefully designed surface passivation technique may be
required to achieve high efficiency. In this case, materials with band gap energy
less than 1.7 eV are preferred because of their short optimal heights. (2) The
performance of axial junction is sensitive to junction position, especially for the
severe surface recombination case. Since the position of hot spot in Figure 3.6 (c)
may vary in different structures, the optimal junction position may also vary.
3.4 Summary
In this chapter, we first use a perfect diode model to calculate the detailed-
balance efficiency of III-V nanowire on silicon tandem solar cells. Our optimiza-
tion results show that for all four III-V materials considered, a larger than 30%
detailed-balance efficiency can be achieved by using 1μm tall nanowire arrays with
optimized lattice constants and diameters. For materials with bandgap smaller
than the optimal value (1.7 eV), it is crucial to tune the structural parameters
41
20 40 60 80 100
0.3
0.5
0.7
0.9
Normalized power
Junction depth (nm)
No surface recombination
SRV = 3000 cm/s
SRV = 30000 cm/s
0 100 200 300 400
0.2
0.3
0.7
0.8
Normalized power
Junction depth (nm)
No surface recombination
SRV = 3000 cm/s
SRV = 30000 cm/s
a b
Figure 3.8: Normalized power of nanowire cells as functions of junction depth for
(a) radial junction and (b) axial junction.
such that the number of absorbed photons in the top cell is no larger than the
number absorbed in the silicon bottom cell. For materials with bandgap equal to
or larger than optimal, the structural parameters should be tuned to maximize
absorption in the nanowire cell.
We then conducted electrical transport simulations to illustrate how the output
power of particular junction geometries, either radial or axial, compares to the
detailed-balance limit. The simulation method allows for evaluation of the effects
of surface recombination on output power. We find that the radial junction is more
robust to the effects of surface recombination than the axial junction. For radial
junction structures and a surface recombination velocity of 3000 cm/s, which has
been achieved in experiments, the normalized power is decreased by less than 1.4%
comparedtothecaseofperfectsurfacepassivation(nosurfacerecombination). Our
results indicate strong promise for high efficiency nanowire III-V tandem cells on
silicon.
However, for the multiple junction nanowire solar cell, an axial junction is still
preferable to a radial one. It is easier to stack different segments of the sub-cells in
42
the axial direction than to squeeze the different layers in the radial direction with
only∼200nmdiameter. TheresultinFigure3.7alsoshowsthattheaxialjunction
can potentially have higher open-circuit voltage if the surface issue is solved. In
the next chapter, I will propose a design for a passivation layer on axial junction
GaAs nanowire solar cells to mitigate the surface effect.
43
Chapter 4
Design of passivation layers on
axial junction GaAs nanowire
solar cells
A version of the results in this chapter was published as Ref.[70].
4.1 Introduction
Chapter2showedthatperiodicnanowirearrayspossessidealopticalabsorption
properties for photovoltaic applications. Chapter 3 demonstrated that with the
capability to provide natural strain relief, III-V nanowire arrays can be grown
on lattice-mismatched silicon to construct double-junction solar cells with greatly
enhanced efficiency. To further extend this approach, epitaxial growth techniques
might be used to grow several segments of different III-V materials along the
nanowire length. The material composition in each segment would be selected
to obtain the ideal combination of bandgap energies for a multijunction cell [55,
71, 72]. Within each segment, the doping would be switched from p-type to n-
type during growth to form an axial p-n junction. Such multijunction nanowire
solar cells could potentially offer extremely high efficiencies, while relieving the
lattice-matching constraints associated with planar junctions.
44
Toachievethisvision,however,potentialsurfaceeffectsinthenanowiresrequire
careful consideration. Many III-V compounds such as gallium arsenide (GaAs)
have a large density of surface states within the electronic bandgap. These mid-
gap surface states have several negative effects on photovoltaic device performance
[73, 74, 68, 69]. To date, experimental work has focused on the achievement of
single-junction nanowire cells. Most reported works have adopted radial junctions
[75, 76, 23, 77, 78, 79] to diminish surface effects. High-performance axial-junction
nanowire solar cells have only been reported by using indium phosphide (InP)
[80, 81, 53], a material known for its surface quality [82]. For GaAs, there has
been previous investigation of surface passivation strategies to reduce detrimental
surface effects [69, 76, 83, 84, 85, 86]. However, most of these efforts have been
devoted to radial junction nanowires or uniformly doped nanowires only; little
work has been done to deliberately investigate the effect of a surface passivation
layer on an optimized axial p-n junction.
In this chapter, we demonstrate that high-efficiency axial junctions are fea-
sible even for a material such as GaAs with poor surface quality. Specifically,
we design a passivation scheme for axial junction GaAs nanowire-array solar cells
using an AlGaAs shell layer. Using coupled optical and electrical simulations, we
show that our passivation design greatly improves both the short-circuit current
and open-circuit voltage relative to an unpassivated nanowire device, increasing
efficiency by a factor of 2.7. Our surface passivation scheme should also enhance
the performance of other axial junction nanowire devices, such as photodetectors,
light-emitting diodes, field effect transistors, and tunneling diodes. Importantly,
an extension of this approach to nanowires containing multiple junctions could
open a path toward high-efficiency multijunction nanowire cells.
45
4.2 Passivation structure design
Figure 4.1(a) shows an axial-junction nanowire-array solar cell. We consider a
periodic array of GaAs nanowires. Within each nanowire, an axial p-n junction
is used to separate the photogenerated electron-hole pairs. Both the p-region and
the n-region have exposed surfaces.
h
a
d
GaAs NW array
a b
Ec
EF
Ev
Ec
EF
Ev
Figure 4.1: (a) Schematic of the axial junction GaAs nanowire-array solar cell on
substrate. (b) Detailed illustration of the p-n junction and the direction of the
generated carriers. (Insets show the band diagrams across the p and n regions.)
Due to the large surface-to-volume ratio of the nanowires, the surface has sev-
eral negative effects on device performance. GaAs and many other III-V com-
pounds have surface energy states within the bulk bandgap. The surface states
deplete the majority carriers, bending the bands so as to shift the Fermi energy
toward midgap, as shown in the insets to Figure 4.1(b). This effect increases the
series resistance [69]. The induced field due to band bending also drives photogen-
erated minority carriers toward the surface, where they recombine, as shown by
46
the arrows in Figure 4.1(b). Due to these various surface effects, the efficiency of
the nanowire solar cell decreases.
The goal of this chapter is to passivate the surface of an axial junction nanowire
solar cell with a single epitaxial layer of uniformly doped material. We choose
aluminum gallium arsenide (AlGaAs) for this purpose. Figure 4.2 shows our
detailed design. For the core region, which is made of GaAs, we choose an
intrinsic top segment and an n-type doped (N
D
= 10
18
cm
−3
) bottom segment.
The core region is covered with a 20-nm thick p-type doped (N
A
= 10
18
cm
−3
)
Al
0.8
Ga
0.2
As shell. Carrier reflectors, made of 20-nm-thick heavily doped p and n
GaAs (N
A
=N
D
=10
19
cm
−3
), are placed below/above the top/bottom contacts to
reduce the recombination at the contacts.
p++ GaAs
Intrinsic GaAs
n GaAs
n++ GaAs
19 -3
(NA = 10 cm )
p Al Ga As
0.8 0.2
18 -3
(NA = 10 cm )
18 -3
(ND = 10 cm )
19 -3
(ND = 10 cm )
1
2
3
4
Figure 4.2: Specific passivation layer design and the doping concentration of each
part.
As we show below, this design has the following features. First, AlGaAs with
a high aluminum mole fraction has a large bandgap, which minimizes the optical
loss in the shell. Second, AlGaAs creates large barriers for both electrons and holes
throughout the structure, preventing the minority carriers from being recombined
on the surface. Finally, the heterojunction between the p-type AlGaAs and the
47
intrinsic GaAs creates a p-type modulation doping in the GaAs region. The use
of an intrinsic material maximizes the minority carrier diffusion length, ensuring a
large short-circuit current.
In the following sections, we use coupled electromagnetic-electronic simulations
to validate this design. We first introduce the modeling methods in Section 4.3.
The optical performance of the passivation layer is discussed in Section 4.4. The
device performance and the effects of the junction depth and surface recombination
are shown in Section 4.5. Section 4.6 discusses the effects of the bulk lifetime on
efficiency.
4.3 Methods
4.3.1 Optical simulation method
We use the finite-difference time-domain method (Lumerical Solutions, Inc.) to
calculate the absorption profile in the nanowire as a function of position and wave-
length, denoted A(r,λ). The optical constants of GaAs and AlGaAs used in the
simulation (refractive index and absorption length) are taken from Ref. [45]. We
assume the nanowires are placed on a semiinfinite GaAs substrate. The absorption
spectrum A(λ) is found by integrating over the nanowire:
A(λ) =
ZZZ
GaAs NW
A(r,λ)dr. (4.1)
To obtain the carrier generation profile G(r), we weight the absorption pro-
file by the solar spectrum and assume that each photon absorbed generates one
electron-hole pair:
G(r) =
Z
λg
300 nm
A(r,λ)λ
hc
I(λ)Sdλ (4.2)
48
where λ
g
is the wavelength that corresponds to the semiconductor bandgap, h is
Planck’s constant, c is the speed of light in vacuum, I(λ) is the AM 1.5D solar
spectrum [1], and S is the area of the unit cell.
For a square array of nanowires, the absorption spectrum is independent of
polarization at normal incidence. To reduce the size of the computation, we use
a circularly polarized incident plane wave and approximate the generation profile
as cylindrically symmetric. The generation profile is then exported to the device
simulator as an excitation term.
4.3.2 Device simulation
We use Synopsys Sentaurus to model electrical transport in the device. The
homojunction (core region) is simulated using a drift-diffusion model. The detailed
simulation method for homojunctions is explained in Chapter 3. The heterojunc-
tion between AlGaAs and GaAs is modeled using the thermionic emission model
[66]. The electron and hole currents (J
n
and J
p
) across the heterostructure have
an exponential relation with the barrier heights (ΔE
c
and ΔE
v
):
J
n
=a
n
q
"
v
n,2
n
2
−
m
n,2
m
n,1
v
n,1
n
1
exp
−
ΔE
c
k
B
T
!#
(4.3)
wherea
n
is an dimensionless coefficient,v
n
is the emission velocity of the electrons
(calculated using the equation v
n
=
q
k
B
T/2πm
n
),n is the electron density, and
m
n
is the effective mass of the electrons. k
B
is the Boltzmann constant, and T is
the temperature set to be room temperature in the simulation. The subscripts 1
49
Table 4.1: Simulation parameters
Symbol Description Nominal values
μ
n
,μ
p
electron and hole mobility doping dependent [66]
ΔE
c
conduction band discontinuity between
Al
0.8
Ga
0.2
As and GaAs
0.315 eV [88]
ΔE
v
valance band discontinuity between
Al
0.8
Ga
0.2
As and GaAs
0.31 eV
E
g,GaAs
band gap of GaAs 1.43 eV
E
g,AlGaAs
band gap of Al
0.8
Ga
0.2
As 2.1 eV
τ
e
,τ
h
SRH recombination lifetimes for electrons
and holes
1 ns except in Fig-
ure 4.9
D
dt
,D
at
surface donor, acceptor like trap densities 1.5×10
12
cm
−2
except
in Figure 4.8
a
n
,a
p
thermionic current coefficients 2
m
n,GaAs
GaAs electron relative effective mass 0.067 m
0
a
m
p,GaAs
GaAs hole relative effective mass 0.485 m
0
m
n,AlGaAs
AlGaAs electron relative effective mass 0.115 m
0
m
p,AlGaAs
AlGaAs hole relative effective mass 0.598 m
0
a
m
0
= 9.1× 10
−31
kg is the electron mass.
and 2 represent the materials with the lower and higher conduction band edges,
respectively. A similar equation for the hole thermionic current is shown as follows:
J
p
=−a
p
q
"
v
p,2
p
2
−
m
p,2
m
p,1
v
p,1
p
1
exp
−
ΔE
v
k
B
T
!#
(4.4)
We assume that the interface between AlGaAs and GaAs is perfect without
any additional recombination centers. This is usually valid for the lattice-matched
epitaxy of AlGaAs on GaAs [87]. Surface recombination on the AlGaAs surface is
modeled by surface traps at mid gap [51]. The donor-like and acceptor-like surface
states are taken to have the same densities.
The parameters used in the device simulation are listed in Table 4.1.
50
4.4 Absorption properties of the design
We optimize the diameter and lattice constant of the nanowire array to max-
imize solar absorption. Figure 4.3 shows a map of the maximum achievable
short-circuit current of a 3-μm-tall square array of GaAs nanowires on a semiinfi-
nite GaAs substrate as a function of lattice constant (a) and diameter-to-lattice-
constant (d/a) ratio. Only the absorption in the nanowire is considered here. The
highest absorption is∼25.93 mA/cm
2
, which is achieved with a 320-nm lattice
constant and a 160-nm diameter. This is close to the perfect absorption value of
GaAs (27.94 mA/cm
2
). We, therefore, adopt these parameters in our design.
Figure 4.3: Maximum achievable short-circuit current of a square array of 3-μm
tall GaAs nanowires as a function of lattice constant a and d/a ratio.
Figure 4.4 shows the carrier generation rate profiles for a nanowire in the array
for a bare nanowire (a) and for a nanowire with a 20-nm passivation layer (b).
We observe that the two structures have similar generation profiles within the core
region. The generation rate in the AlGaAs shell is lower than that in the GaAs
core. Figure 4.4(c) shows a magnified view of the top portion of the passivated
51
nanowire, indicated with the red dashed box in (b). We note that there is signifi-
cant absorption in the thin p++ GaAs layer below the top contact. To investigate
the spectral response, we plot the absorption spectra for the bare and passivated
nanowires in Figure 4.5(a). Only the absorption in the core region is included.
There is a large discrepancy between the two curves below 550 nm. However, the
difference in absorption does not strongly affect carrier generation when the struc-
tures are illuminated by the solar spectrum. Figure 4.5(b) shows the absorption
of the two structures weighted by the AM1.5D photon flux density. The photon
flux under the blue curve (bare nanowire) is equivalent to a maximum current den-
sity of 25.93 mA/cm
2
(as shown in Figure 4.3), while the one under the red curve
(passivated NW) has a current density of 23.13 mA/cm
2
.
a b c
Figure 4.4: (a) Carrier generation rate profile in a bare GaAs nanowire. (b) Carrier
generationrateprofileinapassivatedGaAsnanowire. (c)Magnifiedviewofcarrier
generation rate profile near top of passivated GaAs nanowire.
52
a b
Figure 4.5: Effect of passivation layer on absorption in the nanowires. (a) Absorp-
tion spectra of GaAs NW array. (b) Absorbed photon flux densities (blue and red
curves) shown relative to the incident photon flux density (black curve).
4.5 Carrier transport in the design
4.5.1 Band diagrams
We expect that the AlGaAs shell layer will form a barrier protecting both
electrons and holes from reaching the nanowire surface. For Al
0.8
Ga
0.2
As, the
commonly accepted value of the band offset at the GaAs/AlGaAs interface is
approximately 0.3 eV for both the valence and conduction bands [88] (see Table
4.1). To investigate the effect of the shell layer on carrier transport, we first plot
the electrostatic band diagrams in thermal equilibrium along the dashed lines in
Figure 4.2. Figure 4.5(a)-(d) shows the diagrams for the lines running across the
intrinsic region (dashed line 1), n-region (line 2), near the top contact (line 3), and
near the junction (line 4), respectively.
It can be seen from Figure 4.6(a) that even though the core region (cyan) is
intrinsic, the electrostatic Fermi level, shown as the dashed line, is very close to
the valence band edge, indicating that the core region is effectively p-type doped.
53
a b
c d
Figure 4.6: Band diagrams of the cross sections of (a) the intrinsic region, (b) the
n-region, (c) below the top contact, and (d) near the junction, corresponding to
dashed lines labeled 1, 2, 3, and 4 in Figure 4.2.
The effective doping concentration is 10
16
cm
−3
. This phenomenon is similar to
modulation doping [89]. Because of the limited diameter of the core region (only
∼150 nm), the modulation dopants in the AlGaAs shell layer (yellow) dope the
whole core, instead of only creating a doped layer near the surface. By placing
the dopants in the shell, rather than in the carrier transport region, the design
provides p-type behavior in the core with the long minority carrier diffusion length
characteristic of an intrinsic material.
The band offset of the conduction band at the interface prevents generated
minority electrons from being transported toward the surface, reducing surface
recombination. Figure 4.6(b) shows the band diagram of the n-region (orange),
54
which is also covered by the p-type doped AlGaAs shell (yellow). The GaAs bands
bend upward near the interface. However, due to the heavy doping of the core
region, the space charge region only extends 10 nm below the interface. Most of
the core region is still heavily doped and highly conductive.
Figure 4.6(c) shows the band diagram below the top contact. The thin layer of
AlGaAs (yellow) between the nanowire (cyan) and the top contact (blue) serves as
a carrier reflector to reflect generated minority electrons, reducing recombination
at the top contact. There is a smaller barrier for the majority carriers (holes)
below the top contact.
Figure 4.6(d) shows the band diagram near the interface between the n-type
segment (orange) and the intrinsic segment (cyan). The interface is effectively a
p-n junction, ensuring high open circuit voltage.
4.5.2 J−V response
Figure 4.7(a) shows the J−V curves for an optimized bare GaAs nanowire
array device (blue line) and the device with the passivation layer (red line). For
the passivated NW design, both the short-circuit current and the open-circuit
voltage are much higher than for the bare nanowire(see Figure 4.7(a)). The surface
passivation results in a 2.7 times higher efficiency. The optimized bare nanowire
design had a top-n/bottom-p structure with 10
17
cm
−3
doping concentrations for
both n- and p-regions. The junction depth was optimized with respect to power
conversionefficiency, asshowninFigure4.7(b). Forthebarenanowire, theoptimal
junction depth is very close to the top surface at∼ 200 nm, close to the generation
hot spot shown in Figure 4.4(b). The power conversion efficiency drops quickly
with the junction depth. For the passivated NW, the efficiency is less sensitive to
the junction depth, and the optimal depth is around 1.8 μm.
55
Figure 4.7(a) also shows the J−V curves for the Shockley-Queisser limit [50],
either for perfect absorption (gray, dashed line) or simulated absorption (black,
solid line). The short-circuit current of the passivated NW design is close to the
Shockley-Queisser limit with simulated absorption, indicating efficient carrier col-
lection. However, the open-circuit voltage is substantially lower than the Shockley-
Queisser limit. This is mainly due to the low effective doping (10
16
cm
−3
) in the
intrinsic segment, which reduces the voltage drop across the p-n junction. How-
ever, the mobility increases with decreasing doping. As a result, the use of an
intrinsic region, rather than a p-type doped region, also increases the minority
carrier diffusion length and, thus, improves the minority carrier collection. The
design represents a trade off between these two effects. We note that a high shell
doping concentration is essential for achieving high efficiency. For a reduced shell
doping concentration of 10
17
cm
−3
, the efficiency drops to 18.56% (compared with
21.26% at a shell doping of 10
18
cm
−3
). There is also a large discrepancy between
the short-circuit current of the passivated NW design and the Shockley-Queisser
limit assuming perfect absorption of photons above the bandgap of GaAs. This is
mainly due to the incomplete absorption of the photons [10].
a b
Figure 4.7: (a)J−V curves for nanowire arrays without (blue) and with (red) the
passivation layer. (b) Efficiencies as functions of junction depths.
56
Figure 4.8 shows the short-circuit current, open-circuit voltage, and efficiency
as functions of the surface state density. We vary the surface state density from
1.5× 10
9
to 1.5× 10
15
cm
−2
, corresponding to surface recombination velocities
from 3 to 3× 10
7
cm/s. All three characteristics vary weakly as the surface state
density changes by six orders of magnitude.
a b c
Figure 4.8: (a) Short-circuit current, (b) open-circuit voltage, and (c) efficiency as
functions of surface state density.
4.6 Discussion
So far, we have assumed that the Shockley-Read-Hall recombination life time
within the bulk is 1 ns, which is a reasonable estimate [86]. State-of-the-art growth
techniques, such as MOCVD, can produce GaAs with high material quality, and
the longest life time achieved in bulk structures is much greater than 1 ns [90]. The
bulk diffusion lengths of the minority electrons and holes used in the calculations
above are 4.6μm and 935 nm, respectively, and are comparable with the nanowire
height. However, for the bare nanowire, surface recombination effectively reduces
the diffusion length below the bulk value, preventing carriers from being collected
at the contacts. The key function of the AlGaAs passivation layer is to prevent
surface recombination, increasing collected current.
57
For shorter values of the bulk diffusion lengths, the efficiency of the solar cell
will decrease. To illustrate this, Figure 4.9 shows the power conversion efficiency
as a function of junction depth for different bulk SRH recombination lifetimes.
Figure 4.9(a) shows the efficiency for the passivated nanowire structure. The blue
curves are the same as the curves in Figure 4.7(b). The red and black curves show
the efficiency when the SRH lifetime is reduced to 0.1 and 0.01 ns, respectively.
The corresponding minority electron diffusion lengths are 1.45 μm and 460 nm,
respectively. The highest achievable efficiency drops with decrease in bulk diffusion
length, and the optimal junction depth moves toward the top contact. When the
diffusion length is shorter than the nanowire, placing the junction near the position
where the most carriers are generated results in the largest current. Conversely, for
larger values of the bulk diffusion length, the efficiency of the solar cell increases,
and the optimal junction position moves toward the bottom of the wire, as shown
by the green curve in Figure 4.9(a). For the bare nanowire (shown in Figure
4.9(b)),the efficiencies are lower than for the passivated nanowire, and this holds
true for any value of the SRH lifetime. We conclude that surface passivation is a
robust strategy for increasing efficiency across a range of material qualities.
One possible concern with a bare high-Al content AlGaAs layer is natural
oxidation in the atmosphere. If this is a problem, one possible strategy is to
reduce the Al content near the outer surface. Adding a thin GaAs outer layer is
also an option [86]. The final device may also employ a polymer infill between the
nanowires [91], which may inhibit nanowire oxidation.
58
a b
Figure 4.9: Efficiencies of (a) the passivated nanowire solar cells and (b) the bare
nanowire solar cells with different bulk SRH lifetimes as functions of the junction
depth.
4.7 Summary
In this chapter, we have designed a passivation scheme for axial junction
nanowire-array solar cells. We used coupled optical and electrical transport sim-
ulations to validate our approach. We found that the absorption properties of
the GaAs nanowire array with an AlGaAs passivation layer do not deviate much
from the optimal bare GaAs nanowire design, due to the high bandgap energy of
AlGaAs. Ourdesignexhibitsa21.26%powerconversionefficiencyforthenanowire
array with nonideal surface conditions, which is 2.7 times higher than the optimal
bare nanowire design with the same surface conditions. The efficiency enhance-
ment comes from the better protection of the generated minority carriers as well
as the longer minority carrier diffusion length achieved by the modulation dop-
ing. The design also has great tolerance to the active layer doping concentration,
surface conditions, and junction depth.
59
In this study, we designed a passivation layer specifically on a single-junction
GaAs nanowire solar cell. The same concept can be applied to a nanowire mul-
tijunction tandem solar cell, with reoptimization for desirable band alignment by
adjusting the shell’s composition and doping. Furthermore, since this design can
greatly enhance effective minority carrier lifetime and conductivity, it can also be
used in other nanowire applications such as photodetectors, light-emitting diodes,
field effect transistors, and tunneling diodes.
60
Chapter 5
Conclusion and outlook
5.1 Conclusion
The main contribution of the first part of this dissertation can be listed as
following. Firstly, I used accurate electromagnetic simulations to carry out a sys-
tematic study of the optical absorption properties of vertically-aligned semicon-
ductor nanowire arrays for photovoltaic applications. In general, I demonstrated
that optimized nanowire arrays in six different materials and for a range of heights
from 100 nm to 100 μm all outperform unpatterned thin films of equal height.
More specifically, I found that the absorption properties depend strongly on the
structural parameters of the nanowire arrays (wire diameter d, spacing between
wires a and wire height h). For a certain semiconductor material, the absorption
saturates to unity by a certain height of the nanowire array with the optimized
diameter and spacing. Such saturation height is as short as 3 μm for the four
direct band gap materials considered. This result is important for minimizing the
material usage as well as the cost and time of the process. It is also important for
the electrical solar cell device design when the efficiency is limited by the diffusion
length of the minority carriers.
Secondly, I designed a III-V-nanowire-array-on-silicon double junction solar
cell. I addressed the importance of the current matching in such design and opti-
mize the structure to achieve such matching condition. For all four III-V materials
considered, a larger than 30% detailed-balance efficiency can be achieved by using
61
1 μm-tall nanowire arrays with optimized lattice constants and diameters. This
approachleveragespreviousdevelopmentofhighlyoptimized, single-junction, crys-
talline silicon solar cells, used here as the bottom cell. The efficiency of the silicon
cell can be boosted above 30% by a very thin film (∼ 1μm) lattice-mismatched III-
V nanowire array, which is impossible for a planar structure. The design methods
I use in this work can also be extended to the designing of multijunction nanowire
tandem cells.
Last but not the least, I used coupled optical and electrical transport simula-
tions to design a high performance and robust surface passivation scheme on axial
junction GaAs nanowire solar cells. This design exhibits a 21.26% power con-
version efficiency for the nanowire array with nonideal surface conditions, which
is 2.7 times higher than the optimal bare nanowire design with the same surface
conditions. This design solves the common issue of the large surface-to-volume
ratio nanowire semiconductor devices. Axial junction geometry is essential for the
practical multiple junction nanowire solar cell device. The small junction area can
potentially achieve high open circuit voltage for a photovoltaic cell and high speed
for the nanowire photodetector.
5.2 Outlook
The theoretical work indicates a bright future for nanowire array photovoltaics,
and recent experimental achievements also confirm that. Our collaborators at
USC (Dapkus group and Zhou group) have demonstrated a single junction GaAs
nanowire array solar cell [91] and a GaAs-on-silicon dual junction solar cell [92],
following the designs described in Chapter 2 and Chapter 3. Both works feature
highshortcircuitcurrentswhichareclosetothetheoreticallimit. Thisconfirmsthe
62
accuracy of the electromagnetic modeling and the excellent absorption properties
of nanowire arrays without the need of an anti-reflective layer. Efficiency wise,
the energy conversion efficiencies of 7.68% for single-junction cell and 11.4% for
dual-junction cell are still far lower than the best existing cells in similar material
systems. There is still plenty of room to improve the device performance. The
open circuit voltage obtained in the experiments is far below the state-of-art GaAs
solar cell. The main reason for the low open circuit voltage might be the large
surface recombination velocity. This was observed in the electrical simulation
result shown in Figure 4.7. I expect that the application of the surface passivation
scheme discussed in Chapter 4 will greatly enhance the open circuit voltage.
In the simulation part, although we are confident enough about the accuracy
of the electromagnetic simulation, it is always challenging to accurately decide
the material parameters used in electrical simulations, such as mobilities, minority
carrierlifetimes,dopingconcentrationsandsurfacerecombinationvelocities. While
greatefforthasbeenputintoexperimentallycharacterizethevaluesoftheelectrical
transport parameters, a design which is robust for a wide range of the parameter is
also desirable. In Chapter 4, I gave an example of such design, whose performance
depends only weakly on the surface quality.
63
Part II
Optical Epitaxial Growth
64
Chapter 6
Introduction
6.1 Building photonic matter from the bottom
up
Opticalfieldsinducemechanicalforcesonmatter,capableofpullingobjectsinto
precise positions [93, 94, 95]. Single-particle optical traps based on this principle
have been demonstrated for biological cells [94, 96, 97, 98], dielectric particles [99,
100] and metal particles [101, 102, 103] and have widespread application in physics
andbiology. Forsmallparticles, theopticalforceonaparticleisproportionaltothe
gradient of the light intensity. Arrays of optical traps can be created by varying the
optical field intensity in space, creating a periodic optical potential (Figure 6.2(a)).
Holographic optical traps [104], interference optical lattices [105, 106], micro-optics
arrays [107] and microphotonic near-field approaches [108, 109, 110] have all been
used to trap arrays of objects. These techniques suggest the exciting possibility of
on-demand assembly of photonic matter “from the bottom up.” Nano- and micro-
scale constituents such as dielectric particles [104], particle-molecule complexes
[109], biological cells [107, 111], and microlenses [112] have been assembled into
arraystoachievevariousfunctionalities. Althoughinitialattemptshavebeenmade
to assemble metal particles [113, 114], optical trapping of a 2D periodic array has
not been achieved. Periodic metallic nanoparticle arrays have unique and tunable
optical properties with applications in Raman amplification [115], sensing [116]
65
and lasing [117], and the ability to assemble such structures optically would enable
far-reaching dynamic control.
6.2 Optical binding
Metallic particles interact strongly through coherent light scattering, compli-
cating optical trapping techniques. Scattering produces interparticle forces, a phe-
nomenon known as optical binding [105]. Optical binding (Figure 6.1(b)) drives
the formation of certain well-defined particle arrangements within an optical trap
[105, 118, 119, 120, 121]; for extended, multi-particle patterns, the arrangements
are in general aperiodic. To assemble metallic particles in a periodic optical trap
array, the optical binding effect between particles in different trapping sites must
be considered. In recent experiments, we used a photonic crystal slab to create a
2D periodic trap array [108, 122] (Figure 6.1(c)). While polystrene nanoparticles
trap at every site in the array [108], gold nanoparticles trap in one-dimensional
chains [113]. This difference can be attributed to stronger optical binding in the
metallic nanoparticle system, which competes with trapping forces to prevent the
filling of every trap site. However, because previous work on optical binding has
not studied binding effects within multiple traps, no general theory is available to
explain or predict the observed behavior.
6.3 Optics matter and optical epitaxial growth
Many of the breakthroughs in modern technology have resulted from the suc-
cessful development of semiconductor materials and the control of their electrical
properties. Due to the natural proclivity, the atoms in a semiconductor organize
themselves into periodic structures. Figure 6.2(a) shows a unit cell of gallium
66
a
b c
Figure 6.1: Pattern formation due to optical trapping and optical binding. (a) A
periodic optical potential traps weakly interacting particles at intensity maxima,
due to the optical gradient force. (b) Optical binding causes strongly interacting
particles to arrange themselves into well-defined patterns, even in a nearly uniform
lightfield. (c)Competitionbetweenaperiodicopticalpotentialandopticalbinding
can give rise to pattern formation, preventing particles from trapping at every site.
Here, particles form 1-D chains.
arsenide crystal. This unit cell repeats periodically in space. Periodic arrange-
ment of atoms presents a periodic potential to electrons propagating through it
andcreatesenergybandsshowninFigure6.2(b). Modernadvancedepitaxygrowth
techniques such as molecular beam epitaxy (MBE) enable us to place these atoms
into precise positions layer by layer (Figure 6.2(c)).
Photoniccrystalsareartificialopticalanaloguesofsemiconductormaterialused
to precisely control the optical properties of materials [123, 124]. In a photonic
crystal, shown in Figure 6.2(d), materials with different dielectric constants are
placed into periodic structures on the length scale of the wavelength. Similar to
electron band gaps in semiconductor materials, photonic crystals have photonic
band gaps (Figure 6.2(e)), which prevent light from propagating in certain direc-
tions with specified frequencies (i.e., color of light).
67
Here we suggest that the analogy between natural crystalline materials and
photonic crystals can be extended to the growth mechanism. In traditional epi-
taxial growth of “electronic matter”, the growth substrate provides a 2D periodic
potential landscape and atoms are essentially organized and held together by the
exchange of electrons with both the growth substrate and the atoms on the same
growth layer. Similarly, “optical matter” could be organized and held together
by photons. A periodic array of optical traps serves as the growth substrate, with
individualparticlesabletotrapatanysite. Particlesalsointeractwithoneanother
via optical binding. Both effects determine the final arrangement of particles. To
grow particular structures of interest, one must determine how to design the sub-
strate so that optical trapping and optical binding forces cooperate, rather than
compete.
6.4 Outline of PART II
In this part of the dissertation, I first introduce a Monte Carlo model to study
the energetics of growth in Chapter 7. For square lattices, we find that 1D particle
chains form perpendicular to the incident light polarization, in agreement with
previous experiments [113]. For hexagonal lattices, we predict that a 2D parti-
cle array will be formed. Drawing upon this prediction, in Chapter 8, I design
and fabricate a photonic crystal template with hexagonal symmetry. Using the
template, I demonstrate low power optical trapping of a 2D periodic array of over
fifty gold nanoparticles with spacing comparable to the wavelength. Due to optical
binding interactions, the stability of the 2D array is much greater than for a single,
trapped particle. To our knowledge, this is the first experimental demonstration
of a periodic array of gold nanoparticles via optical trapping methods.
68
a b c
d
e f
Figure 6.2: Comparisons between “electrical matter” and “optical matter”. (a-b) A
unit cell of gallium arsenide crystal and electron band diagram of gallium arsenide
[2]. (c) Schematic of molecular beam epitaxy. (d-e) A photonic-crystal slab and
photonicbanddiagramofthephotonic-crystalslab[3]. (f)Opticalepitaxialgrowth
of a single layer of “optical matter”.
69
Chapter 7
Monte Carlo simulation of optical
epitaxial growth process
A version of the results in this chapter was published as Ref.[125].
7.1 Introduction
The optical manipulations of nano-objects are often within two very distinct
categories. The first one treats each optical trap individually [93, 94]. The opti-
cal force relies on the strong gradient of the optical intensities. One particle can
be stably trapped near the strongest light intensity position. Even though many
works involve multiple optical traps [104, 105, 106, 107], the interaction between
particles trapped at different trapping sites were often neglected. The second cat-
egory investigates the interaction between multiple particles with a single trap
(a.k.a. optical binding), where the gradient force is comparatively weaker than
the interaction force between particles [105, 118, 119, 121, 120]. Due to the mul-
tiple scattering of light, particles can arrange themselves into well-spaced regular
patterns.
As introduced in Chapter 6, in this dissertation work, I am investigating a
system involves an array of optical traps and strongly scattering metallic nanopar-
ticles. The gradient force generated by a single optical trap can hold a metallic
nanoparticle in place. At the same time, the particles trapped on the different
70
traps interact with each other by the scattering of the light. Two effects compete
with each other and prevent the particle from filling of every trap site. So far, no
general theory is available to explain or predict the observed behavior.
Here we suggest that the behavior of strongly interacting particles within a
periodic array of optical traps can be viewed as an optical analogue to epitaxial
growth. The array serves as the growth substrate, with individual particles able to
trap at any site. Particles also interact with one another via optical binding. Both
effects determine the final arrangement of particles. To grow particular structures
ofinterest, onemustdeterminehowtodesignthesubstratesothatopticaltrapping
and optical binding forces cooperate, rather than compete. We introduce a Monte
Carlo model to study the energetics of growth. For square lattices, we find that 1D
particle chains form perpendicular to the incident light polarization, in agreement
with previous experiments. For hexagonal lattices, we predict that a 2D particle
array will be formed.
7.2 Interparticle interaction between particles in
light field
To model the interparticle interaction between particles in light field, we first
study the scattering of the electric field by the single particle. If the particle
dimension is small compare to the wavelength of the light, the particle can be
modeled as oscillating dipoles.
For a dipole source (p) at position (r
A
), the emitted electric field (E) at arbi-
trary position (r
B
) can be written as
E(r) =G(r
B
,r
A
)p(r
A
) (7.1)
71
where G(r
B
,r
A
) is the dyadic Green’s function, a tensor. For a dipole in a com-
plicated environment (e.g. on top of a photonic-crystal slab), G(r
B
,r
A
) can be
solved numerically using the finite-difference time-domain (FDTD) method. A
dipole source is placed at a given positionr
A
with fixed polarization, and the field
produced is monitored throughout the structure. At each position r
B
, the three
columns of the tensor G correspond to the fields due to an x-, y-, z-polarized
source, respectively. The full function G can be mapped out by repeating the
FDTD calculation for each source locationr
A
. The calculation time can generally
be reduced by using any symmetries of the structure (e.g. mirror, translational)
to reduce the total number of source locations required.
Figure 7.1 shows the detailed simulation geometry to obtain the Green’s func-
tions. I consider a silicon slab with 250 nm thickness. If we consider the transla-
tional symmetry in x- and y-directions and only at a certain height, the Green’s
function can be reduced to a two-dimensional tensor which only depends on the in-
plane displacement (G(r
k
) =G(x,y)). Two different simulations are performed to
get all nine components in Green’s function (G
xx
(x,y), G
xy
(x,y), G
xz
(x,y) etc.).
In the first simulation, shown in Figure 7.1(a), the dipole (red arrow) is positioned
along x direction and 125 nm above the slab. We perform FDTD simulation and
monitor the vectorial electrical field E
x
(x,y), E
y
(x,y) and E
z
(x,y) at the same
plane as the dipole, shown as a yellow plane. The first column components in the
Green’s function correspond to the field:
G
xx
(x,y) = E
x
(x,y) (7.2)
G
xy
(x,y) = E
y
(x,y) (7.3)
G
xz
(x,y) = E
z
(x,y) (7.4)
72
Byusingrotationalsymmetry, theresultcanbeconvertedtothesecondcolumn
in the Green’s function. Another simulation with the dipole polarized along z
direction is performed for the third column in the Green’s function (G
zx
,G
zy
and
G
zz
). The simulation mesh can be further reduced by considering the mirror
symmetry of the field and the geometry. As shown in Figure 7.1, in the first
simulation, symmetric and anti-symmetric symmetries have been applied along
y- and x-axis to reduce the simulation mesh to one quarter. Similarly, symmetric
boundaryconditioncanbeusedinbothx-andy-directioninthez-polarizeddipole
case.
x
y
z
symmetric
an -symmetric
symmetric
symmetric
a b
electric field monitor
Figure 7.1: Simulation setup for Green’s functions (a) simulation of the first and
second columns in Green’s function tensor. (b) simulation of the third column in
Green’s function tensor.
Once we obtain the scattered field from a particle, to investigate the inter-
particle interaction, we need to investigate the mechanical effect (optical force)
of the scattered field to other particles. In the dipole approximation, the time-
averaged total force on a sphere in a time-harmonic electromagnetic fieldE(r,t) =
< [E
0
exp(−iωt)] can be expressed as
D
F
i
E
=
1
2
<
h
αE
0j
∂
i
(E
j
0
)
∗
i
, (7.5)
73
where i and j are each one of the Cartesian components. For a complex polariz-
ability α =α
0
+iα
00
, the time-averaged force can be written as
hFi =
1
4
α
0
∇E
2
+
1
2
α
00
E
2
∇φ, (7.6)
whereφisthepositiondependentphaseoftheelectricfield. Thefirsttermdepends
on the spatial variation of the intensity and is known as the gradient force. The
second term depends on the spatial variation in the phase and is known as the
dissipative or absorption force. For the particles considered in this work, the
imaginary part of the polarizability is two orders of magnitude smaller than the
real part, and so we will neglect the second term.
From the first term, we can define a potential energy
U =−
1
4
α
0
E
2
. (7.7)
Suppose one particle (A) is sitting at position r
A
and another particle (B) at
position r
B
, denote the initial electric field at these two positions by E
A
and E
B
,
respectively. The scattered field from particle A at position B is
ΔE = G(r
B
,r
A
)α
B
E
A
(7.8)
= G(x,y)α
A
E
A
, (7.9)
where (x,y) is the displacement from particle B from particle A (r
k
=r
B
−r
A
). If
we only consider the first scattering process, the total field at position B is
E =E
A
+ ΔE (7.10)
74
and the potential energy for the two particle system is
U =−
1
4
α
0
B
E
2
B
+ 2<(E
∗
B
ΔE) + ΔE
2
. (7.11)
7.3 Induced energy shift in optical epitaxial
growth
The equations derived in the last section (Equation 7.11) are universally correct
as long as the local electric field (E
A
and E
B
) at position r
A
and r
B
is carefully
characterized and the Green’s function G(r
B
,r
A
) is accurately simulated. In this
section, I will discuss how to use the equation to calculate the induced energy shift
in the optical epitaxial growth process.
Figure 7.2 shows a schematic of an optical epitaxial growth experiment. We
use a 2D photonic-crystal slab as a growth substrate. The figure shows a side
view of it. When light with the resonance wavelength is shined onto the photonic-
crystal slab, the electric field intensity is greatly enhanced within the holes in the
photonic-crystal slab, shown as red hot spots. The enhanced electric field exerts
force and drag the particles towards the surface. Particle A and B are stably
trapped particles and particle C is flowing around. This figure also shows the
main characteristics of the electric field distribution of the guided resonance mode,
plotted along two dashed lines. In the z-direction through the center of holes, the
electric field peaks at the center of the slab and exponentially decays above and
below the slab. The trapping of the particles rely on the tail of the exponentially
decaying field at around 100 nm above the slab. When the position is very far from
the slab (typically several hundred of nanometers), the electric field is negligible.
In the x-direction across the trapped particle plane (125 nm above the slab in
75
the model), there is still significant field enhancement (typically|E|
2
/|E
0
|
2
> 50).
The intensity oscillates periodically to create a 2D optical potential landscape
(trapping sites). Because of the unique feature of the guided-resonance mode, the
local electric field is in-phase at each trapping site. This fact is very important for
simplifying the following simulation.
E
x
z
E
A
B
C
Figure 7.2: Schematic of electric field distribution in a photonic-crystal slab optical
epitaxial growth substrate.
In this work, we are mainly interested in the particles which are stably trapped
on the photonic-crystal slab (particle A and B). In this case, the local electric
field (E
A
andE
B
) are about the same. The scattered field from dipole A reaching
position r
B
is much smaller than the original field E
B
, by neglecting the second
order term ΔE
2
, the induced energy shift of the two-particle system due to the
scattered field is
ΔU≈−
1
2
α
0
A
< (E
∗
A
G(x,y)α
B
E
B
). (7.12)
For the same reason, we can neglect the secondary scattering process and only
consider the first scattering (a numerical example can be found in Appendix A).
76
Figure 7.3 is the graphical illustration of Equation 7.12. The particle A is posi-
tionedatoriginwhiletheparticleBismovedthroughoutthe2Dplane. Weapprox-
imate the Green’s function by that of a uniform dielectric slab (see Appendix A
for validation of this approximation). The color at different positions shows the
induced energy shift when particle B is positioned at that position. The elec-
tric field is polarized along x-direction with the same intensity and phase at the
positions of particle A and B. The electromagnetic wavelength used to obtain the
Green’s function is 1550 nm in vacuum. The dipole is placed 125 nm above a 250
nm thick silicon slab (n
Si
= 3.49). The silicon slab is on top of a semi-infinite
silicon dioxide substrate (n
SiO
2
= 1.55) and immersed in water (n
water
= 1.318).
x(μ m)
y(μ m)
−2 0 2
−3
−2
−1
0
1
2
3 −1
−0.5
0
0.5
1
k T/(mW/μm )
B
2
Figure 7.3: Graphical illustration of the energy shift in a two-particle system. (a)
The xy-plane cross-section (b) The xz-plane cross-section.
Due to the scattering of the light, the energy shift of two-particle system can
either be a positive value (less stable) or a negative value (more stable), depending
on the relative position of the two particles. Figure 7.4 shows two cross-sections of
Figure 7.3 along the x-axis (black curve) and y-axis (red curve). The interaction
energy shift is non-monotonic and oscillates around zero. The interaction is long-
range with a decaying envelope, as can be seen in Figure 7.4(b).
77
0.1 1 10
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
Parallel
Perpendicular
1/r
|ΔE|
Spacing (μm)
0 10 20 30 40 50
-0.4
-0.2
0.0
0.2
0.4
Parallel
Perpendicular
B
T/(mW/μm
2
))
Spacing (μm)
b
1 2 3 4 5
-0.4
-0.2
0.0
0.2
0.4
S p acing (μm)
a
Figure 7.4: Induced energy shift due to interaction. (a) Energy shift as function
of spacing between two particles for the polarization parallel to the particle pair
(black) and perpendicular to the particle pair (red). (b) log-log plot of the energy
shift magnitude. The function 1/r is plotted for reference.
For multiple particle system, because the scattered field is much smaller than
the unperturbed field, the interference between the scattered field (ΔE
∗
i
ΔE
j
) can
also be neglect. The energy shift of particle i induced by all other particles can be
written as
ΔU =−
1
2
<(α
i
)
X
j6=i
< (E
∗
i
G(R,k)α
j
E
j
), (7.13)
where α
i
and α
j
are the polarizabilities of particle i and particle j, and are the
unperturbed local electric fields at particle i and particle j, R is the vector from
particle i to particle j, k is the wave vector, and G(R,k) is the dyadic Green’s
function.
78
7.4 Monte Carlo simulation of optical epitaxial
growth
The process of optical epitaxial growth can be described as following. The
photonic-crystal slab creates a 2D array of viable trapping sites. When a single
particle is considered, the particle can be trapped equally at each trapping site.
The probability for this trapped particle to release from the trap depends on the
energy created by the photonic-crystal slab (P∝ exp(U/k
B
T )), where U depends
on the local electric field intensity and the polarizability of the particle (Equation
7.7). When multiple particles are trapped on the photonic-crystal slab, because
of the scattering of the light, the energy barrier for a certain particle to release
can shift either higher or lower, depending on the relative position of all the other
particles. Because the less stable particles (with higher energy) are more likely to
release from the surface or hop to the nearest trapping sites, the trapped particle
will evolve to a more stable configuration with lower energy.
We use the Monte Carlo method to simulate this process and identify low-
energy configurations. The Monte Carlo method only considers the local equilib-
rium state: in this case, trapped particles on a periodic lattice, and the transition
between different states: hopping or release of the particles. We begin by randomly
placing the particles at trapping sites. The energy shift for the particle ensemble
(ΔE =
P
i
ΔU
i
) is calculated from Equation 7.13. One particle is then selected
at random. The particle is either moved to one of the nearest neighbor sites or
released from the surface, with each possible option having equal probability. If a
particle is released from the surface or hops outside of the device region, it is relo-
cated at a random empty site. If the total energy of the new configuration ΔE
new
is lower than the old one ΔE
old
, the new configuration will be unconditionally
79
accepted. Otherwise, the new configuration will be accepted with the probability
exp (−(ΔE
new
− ΔE
old
))/k
B
T ) , wherek
B
is the Boltzmann constant andT is the
temperature.
The simulation parameters are chosen to be closest to our typical experiments.
The simulations used a 50 μm diameter area lattice with fixed 5% site coverage.
The unperturbed field values were set assuming horizontal polarization, constant
phase at all trapping sites, and overall amplitude given by a two-dimensional Gaus-
sian envelope (I =I
0
exp(−
x
2
+y
2
w
2
0
)), with the beam waistw
0
set to be 15μm. The
total incident power is assumed to be 15 mW, similar to experiment. The electric
field intensity on each trapped site is assumed to be 50× higher than the incident
planewave, basedonelectromagneticsimulations(seeFigure8.1). Eachsimulation
is run for 5 million steps.
Figure 7.5 shows the result of a typical simulation. It is a square lattice with
the lattice constant of 0.93λ. An initial transient is observed in the energy shift,
corresponding to loss of memory of the random initial configuration (shadowed
area in Figure 7.5(a)). The energy then fluctuates around an equilibrium value.
The statistics of energy shift (mean value and standard deviation) is calculated
over the last 4.5 million steps. Figure 7.5 (b) shows the histogram of the energy
shift over the last 4.5 million steps. Figure 7.5 (c) and (d) shows the configuration
as well as the energy shift for individual particles at step 1000 and at the last step,
respectively. Initially, the particles are at random positions and the energy shift is
close to zero. At the end of the simulation, the particles arrange themselves into
well-organized pattern with much lower energy, indicated as dark red.
80
-20 -10 0 10 20
-20
-10
0
10
20
E/P
total
(k
B
T/mW )
y ( m)
x ( m)
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
-20 -10 0 10 20
-20
-10
0
10
20
y ( m)
x ( m)
S tep 5, 000, 000
S tep 1000
0 1x10
6
2x10
6
3x10
6
4x10
6
5x10
6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
avg ( E)/P
total
(k
B
T/mW)
Steps
d c
a b
-0.44 -0.42 -0.40 -0.38 -0.36
0.0
5.0x10
4
1.0x10
5
1.5x10
5
2.0x10
5
2.5x10
5
3.0x10
5
3.5x10
5
4.0x10
5
Count
avg ( E)/P
total
0.0 2.0x10
4
4.0x10
4
6.0x10
4
8.0x10
4
1.0x10
5
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
avg ( E)/P
total
Steps
Figure 7.5: A typical Monte Carlo simulation. (a) Average energy shift normal-
ized to the incident power as a function of simulation step. The inset shows the
first 100,000 steps (shadowed area in the main figure). (b) The histogram of the
normalized average energy shift over the last 4.5 million steps. (c-d) The config-
urations at step 1000 and step 5,000,000 step, respectively. The color map shows
the normalized energy shift for individual particles.
7.5 Monte Carlo simulation result
Two lattice symmetries are considered in this study. They are the square lattice
and hexagonal lattice, which are shown in Figure 7.6(a) and (b), respectively. The
average energy shift per particle is plotted in Figure 7.6(c) as a function of lattice
constant, for both the square and hexagonal arrays. The energy strongly depends
on the lattice constant. Several local minima (marked with arrows) can be seen.
81
For the hexagonal lattice, there is a pronounced dip when the lattice constant is
similar to the wavelength (a/λ∼ 1).
a
b
c
a
a
x
y
Figure 7.6: Strong interaction-induced pattern formation during optical epitaxial
growth. (a-b) Square and hexagonal lattices, each with lattice constant a. (c)
Monte-Carlo simulation results of energy shift per particle as a function of lattice
constant. The solid lines are guides to the eye. Energy is plotted in arbitrary
units.
Particle configurations corresponding to each of the local minima in Figure
7.6(c) are plotted in Figure 7.7(a-f). The configurations change dramatically with
lattice type and lattice constant. For the square template (Figure 7.7(a-c)), we
observe vertical chains with different spacings. The simulation result agrees well
with our previously reported experiments, in which we observed chain formation
on a lattice with a/λ ∼ 0.94 [113]. In both simulation and experiments, the
chains are oriented perpendicular to the polarization direction. For the hexagonal
template, oblique chains are observed at a/λ = 0.80 and a/λ = 0.90. However, at
the lowest-energy local minimum (a/λ = 1.0), we observe a close-packed array, or
cluster. We have verified that an array is also formed for a wide range of lattice
82
constants nearby, from∼ 0.94 to∼ 1.1. The results suggest that for appropriate
choice of lattice type and lattice constant, interparticle interactions act to stabilize
the array, which is the lowest energy particle configuration of any shown in Figure
7.6(c).
f e d
c b a
a/ =1.00 a/ =0.90 a/ =0.85
a/ =0.99
a/ =0.93 a/ =0.88
Figure 7.7: (a-c) Particle configurations obtained for square lattices with varying
lattice constants. (d-f) Particle configurations obtained for hexagonal lattices with
varying lattice constants. The circles show a 50 μm diameter device area.
83
7.6 Discussions
7.6.1 Effect of the growth substrate
It is worth noticing here that the simulation result shown before is only true
for a certain wavelength of light (1550 nm in vacuum and 1176 nm in water) and a
certain configuration (250 nm silicon-on-insulator wafer immersed in water). This
is the configuration in our experimental work. The result is in general not valid for
another configuration at another wavelength, and new simulation of the Green’s
function (Figure 7.4) is required. I illustrate this point in Figure 7.8. Here I show
the energy shift of a two-particle system in different environments. Figure 7.8 (a)
shows the energy shift in free space in water, while Figure 7.8(b) and (c) show the
energy shifts of two particles which are 125 nm above a 250 nm and 340 nm SOI
wafer, respectively. Figure 7.8(b) is the reproduction of Figure 7.4. The energy
shift in all three cases oscillates around zeros throughout the 2D space. However,
the oscillating behaviors are essentially different for the three cases shown. For
clear illustration, a square dot array with the lattice constant of 1.1 μm are shown
on top of each graphs. The oscillation along y-axis in case b is slightly faster than
the one in case a, while the oscillation is x-axis is much faster (about two times).
The thickness of the silicon layer also affects the energy shift. In Figure 7.8(b),
when the particle is placed at (0, 1.1)μm, the energy shift is negative, indicating
a more stable configuration. However, if a particle is placed at the same position
above a 340 nm SOI (shown in Figure 7.8(c)), the energy shift is positive, which
means a less stable structure.
This phenomenon enables a large design space of the optical epitaxial growth
substrate, where we can not only design a symmetry and lattice constant, but
also engineer the interaction (Green’s function) between the particles. In this and
84
x(μm)
y(μm)
−2 0 2
−3
−2
−1
0
1
2
3 −1
−0.5
0
0.5
1
x(μm)
y(μm)
−2 0 2
−3
−2
−1
0
1
2
3 −1
−0.5
0
0.5
1
x(μm)
y(μm)
−2 0 2
−3
−2
−1
0
1
2
3 −1
−0.5
0
0.5
1
a
b c
water
250 nm SOI in water 340 nm SOI in water
Figure 7.8: Energy shift in a two-particle system in different environments. (a)
water. (b) 125 nm above the surface of a 250 nm thick SOI slab. (c) 125 nm above
the surface of a 340 nm thick SOI slab.
the next chapters, I focus on design and growth on a 250 nm SOI platform. The
result shows that chains can be formed on a square lattice where clusters can be
formed on a hexagonal lattice. By using other thicknesses of the silicon layer, we
might get a completely different result. When the optical trapping lattices are
generated by free space optics such as a holographic optical tweezers array, it is
also possible to engineer the particle-particle interactions. Different materials and
different thicknesses of the substrate can be used.
85
7.6.2 Phase transition
Figure 7.9 shows the mean value and standard deviation of the average energy
shiftperparticleasafunctionoftheincidentpower. Theenergyshiftisnormalized
bytheincidentpowerandhasunitsofk
B
T mW. Asharptransitioncanbeobserved
on the mean value curve (black) when the power is around 10 mW. The average
energy per particle is close to zero when the incident power is below 7 mW, and the
energy shift per particle saturates to around -0.43 k
B
T mW for power larger than
20 mW. These are the two distinct phases for the grown crystalline optical matter.
At the low power end, the particle positions are random, similar to those shown
in Figure 7.5(c). The system stays in this amorphous phase without forming any
particular pattern. When the power increases to a certain value (around 7 mW),
the interaction between particles start to bring particles together to form a chains
of three or four particles. Occasionally, the chains can grow quite long. However,
the interaction is still not strong enough to overcome thermal fluctuations. The
system switches back and forth between the random amorphous phase and the
well-patterned crystalline phase and cause the standard deviation of the energy
shift to peak at this energy. When the power is large enough, the particles form
a pattern with the lowest possible energy with only slight fluctuations around this
state. Such phase transition behavior further supports the analogy between optical
eptaxial growth and crystal growth in general. High incident power here can be
seen as the low “growth temperature”. This analogy suggests various strategies
for improving the quality of the photonic material. Optimization of the “growth
temperature” (laser intensity) and growth rate (particle delivery rate), along with
the use of annealing, may increase the size and quality of the particle array.
In Figure 7.10 we plot two dimensional maps of the mean values and standard
deviationsasfunctionsofpowerandlatticeconstantforboththesquarelatticeand
86
0 10 20 30 40 50 60 70
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
avg( E)/P
total
power (mw)
mean value
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
standard deviation
Figure 7.9: Mean value and standard deviation of the average energy shift per
particle as a function of the incident power. The simulation is on a square lattice
with a = 0.93λ and a coverage of 5%.
hexagonal lattice. Cross-sections indicated by the white-dashed lines are plotted as
the two curves in Figure 7.6(c). In general, the deeper the energy shift, the lower
the power required for the phase transition. For all the local minima discussed in
Figure 7.7, the system has already transited to a well-defined crystalline state with
30 mW power.
7.7 Summary
In summary, we have studied a system in which strong optical binding interac-
tions between particles drive pattern formation within a periodic optical trapping
potential. Monte Carlo simulations predict that patterns such as 1-D chains and
2-D arrays may be formed, depending on the lattice constant and lattice symmetry.
87
power (mW)
a/λ
20 40 60
0.8
0.9
1
1.1
1.2 −0.5
−0.4
−0.3
−0.2
−0.1
0
power (mW)
a/λ
20 40 60
0.8
0.9
1
1.1
1.2 0.001
0.01
0.1
power (mW)
a/λ
20 40 60
0.8
0.9
1
1.1
1.2
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
power (mW)
a/λ
20 40 60
0.8
0.9
1
1.1
1.2 0.001
0.01
0.1
a b
c d
square mean square std
hexagonal mean hexagonal std
Figure 7.10: Mean values (a,c) and standard deviations (b,d) of the average energy
shift as functions of power and lattice constant. The upper row shows the result
for square lattices and the bottom row shows the result for hexagonal lattices.
88
Chapter 8
Optical epitaxial growth of a gold
nanoparticle array
A version of the results in this chapter was published as Ref.[125].
8.1 Introduction
The Monte Carlo simulation results in the last chapter indicate the important
effect of inter-particle interactions (optical binding) on pattern formation in the
optical epitaxial growth process. The simulation nicely explains the previously
observed abnormal chain formation behavior in the trapping experiments of gold
particlesonasquarelatticewithacertainlatticeconstant. Italsosuggeststhatthe
particles trapped by a hexagonal symmetry template with a lattice constant close
to the wavelength will form a closely packet cluster with lower energy (i.e. higher
stability). In this chapter, I will discuss the photonic-crystal template design and
the experiments to achieve the trapping of a 2D array of gold nanoparticles.
8.2 Epitaxial growth template design
Following the prediction of the simulation, we designed a photonic crystal slab
with hexagonal symmetry and a/λ∼ 1. We patterned the 250 nm thick silicon
device layer on a silicon-on-insulator wafer into a hybrid triangular-graphite lattice
[126]. The larger holes form a hexagonal lattice, while the smaller holes form a
89
graphite lattice. Such design ensures the hole sizes on the photonic crystal are
smaller than the size of the gold particles (d = 200 nm) we used in the experi-
ment, which prevents the particles from sinking into the holes completely. This
allows better visualization of trapped particles by optical microscopy. This design
supports several low-dispersion photonic bands with high quality factors. The fab-
ricated device is shown in Figure 8.1(a). The silicon layer is 250 nm. The spacing
between the larger holes (i.e. lattice constant a) is 1.166 μm, and the laser wave-
length in vacuum was 1550 nm; adjusting for the refractive index of water, a/λ
= 0.99. The diameters of the larger and smaller holes are 0.134 a (156 nm) and
0.094 a (110 nm), respectively.
Figure 8.1(b) shows the measured transmission spectrum through the photonic-
crystal slab. The spectrum shows a Fano resonance feature with an asymmetric
dip. We fit this spectrum to the Fano formula as shown by the red curve [127].
The resonant wavelength is 1549 nm and the quality factor is 630.
Figure8.1(c)showsthesimulatedelectricfieldintensityoftheguided-resonance
mode shown in Figure 8.1(b). The field is taken on a plane which is 125 nm above
the slab. The electric field of the mode is enhanced in the proximity of the holes.
I employed the Maxwell stress-tensor [128] to calculate the optical force acting
upon a gold nanoparticle when the particle is in the proximity to the surface
of the photonic-crystal slab. Finite-difference time-domain (FDTD) simulation
is performed with the periodic boundary condition in x- and y-directions, and
perfect matching layer (PML) boundary condition in z-direction in a supercell
area of 3a×2
√
3a , wherea is the spacing between two adjacent large holes. Large
supercell size minimizes the influence from the periodic boundary conditions.
Figure 8.2(a) and 8.2(b) show the calculated optical force exerted on a 200 nm
diametergoldparticleplacedatdifferentpositions,wherethebottomoftheparticle
90
c
1530 1540 1550 1560
0.1
1
10
Transmission (μW)
Wavelength (nm)
b a
|E|
2
/|E
2
0
|
x (μm)
y (μ m)
−2 −1 0 1 2
−1.5
−1
−0.5
0
0.5
1
1.5
0
10
20
30
40
50
Figure 8.1: Design of the photonic crystal device for experiment. (a) A scanning
electron micrograph (SEM) image of the fabricated photonic crystal slab device.
The200nmdiametergoldparticlesarealsoshowninthisgraph. (b)Themeasured
transmission spectrum through the device. (c) The simulated electric field profile
(|E|
2
/|E
0
|
2
) of the guided-resonance mode used in the experiments on a plane 125
nm above the slab.
is 40 nm above the top surface of the photonic-crystal slab. The force is normalized
to P/c, where P is the incident power per unit cell and c is the speed of the light
in vacuum. The vertical force is strongly attractive in the proximity of the holes.
The force at the position of the larger holes is stronger than that near the smaller
holes. Strong repulsive forces can also be observed in the space between holes.
Within the plane parallel to the photonic crystal slab, the trapping is stable in
91
the vicinities of the holes in bothx- andy-directions. The in-plane forces near the
larger holes are stronger than those near the smaller holes.
a b
−500 0 500
x(nm)
F
//
c/P
x(nm)
y(nm)
F
z
c/P
−500 0 500
−800
−600
−400
−200
0
200
400
600
800
−2
−1
0
1
2
Figure 8.2: Calculated force and potential of the photonic crystal trap. (a) The
normalized vertical force exerted on a 200 nm gold particle at different positions
along a plane 40 nm above the photonic-crystal slab. Negative force indicates
attraction toward the slab. (b) The normalized in-plane force. The blue scale bar
indicates Fc/P = 2.
8.3 Experimental setup
Figure8.3showstheopticalsetupusedforphotonic-crystaldevicecharacteriza-
tionandopticalepitaxialgrowthexperiments. Asingle-mode-fibercoupledtunable
laser with the tuning range from 1500 nm to 1620 nm is collimated to free-space by
anasphericallens(f = 11mm). Thenapolarizerisusedtoimprovethelinearityof
the polarization and a half-wave plate is used to control the polarization of the inci-
dent light beam. Then the linearly polarized light with certain polarization angle
is focused by a achromatic doublet lens (f = 30 mm) onto the photonic-crystal
device which is mounted on a translation stage. The beam spot at the device
92
position has a 2D Gaussian profile (I = I
0
exp[−2(x
2
+y
2
)/w
2
0
]) with w
0
mea-
sured to be∼ 14 μm. The transmitted light through the photonic-crystal device
is collected by another lens or a microscope objective. The collected transmission
beam passes another polarization control set (half-wave plate and polarizer) and
then be focused by another apherical lens to a fiber coupled photodetector. The
transmission spectrum is obtained by swiping the laser wavelength and read the
photodetector power accordingly. In the optical epitaxial growth experiment, the
laser wavelength is tuned to the resonant wavelength of the photonic-crystal device
and operates in a continuous wave (CW) mode. When the high power (P>15 mW)
is desired, the laser output is first amplified by an erbium-doped fiber amplifier
(EDFA) with the bandwidth from 1530 nm to 1560 nm. The peak power of the
EDFA is around 300 mW. A polarization controller is used to gradually control
the output power from the EDFA.
Figure 8.3 also shows the illumination and imaging configuration. A
multimode-fiber coupled white light source is coupled to free-space and then is
refocused onto the device by the microscope objective. The images and videos of
the epitaxial growth are captured by a high pixel density CCD camera.
8.4 Optical epitaxial growth experiment
I perform the optical epitaxial growth experiment using the photonic-crystal
template designed in Section 8.2. To clearly illustrate the growth process, a low
density gold nanoparticle colloidal solution is used. The laser is tuned to 1549 nm,
whichistheresonantwavelengthofthephotonic-crystaldevice. Thepoweristuned
to the minimal trapping threshold power (∼ 30 mW), when the particles just start
to be trapped. The low power lets the system freely evolve to the global energy
93
TL
LS PC HWP
MO
TS
MO
HWP PC
BS DM
LS
PD
LP
CMR
IR Excitation
Visible illumination
TL: Tunable light source BS: Beam splitter
LS: Lens DM: Dichroic mirror
PC: Polarizer LP: Lamp
HWP: Half wave plate CMR: Camera
TS: Translation stage PD: Photodetector
MO: Microscope objective
Figure 8.3: Schematic of the optical setup used in optical epitaxial growth experi-
ments.
minimum without getting stuck in local energy minima. A shallow microfluidic
chamber with the thickness around 500 nm is used to ensure the two-dimensional
environment and minimize the thermal convection out of plane.
Figure 8.4 shows a series of optical microscope images from a typical experi-
ment. The diameter of the gold nanoparticles is 200 nm, and a red arrow indicates
incident light polarization. When the laser power was turned on, particles imme-
diately began to trap near the slab (Figures 8.4(a,b)). The trapping sites are not
close-packed, and release and re-trapping of particles were frequently observed. As
time passed, the trapping area became smaller and the cluster became more filled-
in (Figures 8.4(c-f)). Particles also appeared to be trapped more stably. After 40
minutes, a highly stable, close-packed array of nanoparticles was formed. When
the laser is turned off, the particles released immediately and diffused back into
solution (Figures 8.4(g,h)).
94
The final array shows evidence of extremely strong optical trapping. Movement
of particles within each trapping site was negligible when seen by the naked eye.
Analysis of video data for the experiment found that the stiffness of an individual
trap was greater than 82.5 pN/nm/W, two orders of magnitude higher than values
reported in previous experiments on polystyrene particles [108]. The total power
usedintheexperimentwasonly30mW,foramaximumpowerof85μWperoptical
trap. The power per trap is orders of magnitude lower than for previously reported
metallic nanoparticle traps. In particular, it is around 2,000 times lower than for
standard optical tweezers, given similar particle size [101, 102, 103, 119, 120, 129]
a b c d
e f c h
power on 0.5 min. 10 min. 20 min.
30 min. 40 min. power off 15 s a"er power off
Figure 8.4: Optical epitaxial growth of a gold particle array. Optical microscope
images show 200-nm diameter gold nanoparticles trapped on the photonic crystal
slab, visible in the background. (a-f) Snapshots taken with the laser power on;
elapsed time is shown below image. (g,h) Snapshots taken with laser power off.
The scale bar indicates 5 μm.
The formation of the gold nanoparticle array in still solution relies on the
Brownian motion of the particles to bring them close enough to the holes to get
95
trapped, which makes the assembly process slow. Moreover, in a still solution,
the particle density drops as the trapping experiment progresses and eventually
limits the size of the trapped gold cluster. To solve these issues, we fabricated
a microfluidic flow channel to deliver particles. Figures 8.5(a-c) show snapshots
during a LATS experiment in a microfluidic channel with flow. At the beginning,
the diameter of the trapping region is over 25 μm, which is close to the incident
beam size. The flow brings a large amount of gold particles to the center region of
thephotoniccrystaldevicewherethelaserbeamisfocused. Thenumberoftrapped
particles increases rapidly with time while the cluster becomes more close-packed.
In six minutes, more than 120 particles are trapped in a preliminary cluster.
a b c
d
0 s 150 s 300 s
0 75 150 225 300
10
30
50
70
90
110
130
time (second)
number of particles
4000
5000
6000
7000
8000
STD (nm)
Figure 8.5: Optical epitaxial growth experiments in a microfluidic channel. (a-c)
The snapshots when the beam is on for 0 s, 150 s and 300 s. (d) The number
of trapped particles (black triangles) and the standard deviation of the particle
positions (blue squares) as functions of time. The scale bar indicates 5 μm.
96
8.5 Discussion and conclusion
In this chapter, we use the prediction of the Monte Carlo simulation shown in
the last chapter to successfully demonstrate growth of a periodic, hexagonal array
of closely-spaced gold nanoparticles on top of a photonic crystal template. In this
system, the optical force of the template on the particle cooperates with the optical
binding interaction to produce a pattern that is energetically favorable and highly
stable.
We attribute the ultra low threshold trapping and highly stable particle arrays
to two facts. The first one is the cooperative interactions between particles, which
hasbeenshowninFigure7.6(c). Tofurtherunderstandtheenergeticsofthegrowth
process, wecalculatedtheenergyshiftasafunctionoftime. Particlepositionswere
extracted from the video, and the energy shift was calculated from Equation 7.13.
Figures 8.6(a-c) show the energy shift for each particle at three different snapshots
in the growth process, corresponding to the microscope images in Figures 8.4(d-
f). At early times (Figure 8.6(a)), the particles sparsely attached to the substrate
over a large area, and the energy was high (light red). Some oblique chains were
formed, and the particles at the center of the chains had lower energy. As time
passed (Figure 8.6(b)), the trapping area became smaller and the cluster became
more filled-in. Meanwhile, the energy was greatly reduced (dark red), especially
for particles at the center of the cluster (Figure 8.6(c)). This result correlates with
the observation of greater trapping stability as the experiment proceeded. The
average energy shift per particle is plotted in Figure 8.6(d), which confirms that
the formation of the nanoparticle array was accompanied by minimization of the
energy.
The second fact which contributes to the stable particle array is the strong
interaction between the particle and photonic-crystal slab growth substrate. In
97
-1.3
-1.0
-0.6
-0.3
-0.0
ΔU
30 min. 20 min.
0 10 20 30 40
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
ΔU
tot
/N
Time (min.)
d
c b a
10 min.
Figure 8.6: Energy lowering during the growth process. (a-c) The energy shift of
each particle. (d) Average energy shift per particle as a function of time.
the last chapter’s simulation, we only assume that the particles can be stably
trapped by the substrate and polarized by the field. Here we perform detailed full-
wave electromagnetic simulation to characterize the particle-template interaction.
Calculations show that the optical force on a gold particle is much stronger than on
apolystyreneparticleofequalsize. Figure8.7(a)plotsthenormalizedverticalforce
for a particle centered above one of the bigger holes in the template. As the gold
particle approaches the slab, the electromagnetic field is significantly redistributed
(Figures 8.7(b),(d)). This effect is present to a much lesser degree for polystyrene
(Figures 8.7(c),(e)), which experiences an order of magnitude weaker optical force.
The results suggest that the gold particle does not simply experience a “gradient”
force due to the evanescent tail of the resonant mode of the photonic-crystal slab;
the strong attraction is a cooperative effect between particle and slab, mediated by
98
field redistribution. Here, the incident laser wavelength is at 1550 nm, far above
the plasmon resonance of the gold particle.
x (nm)
z (nm)
−200 0 200
−200
0
200
400
0
200
400
600
800
x (nm)
z (nm)
−200 0 200
−200
0
200
400
0
200
400
600
800
x (nm)
z (nm)
−200 0 200
−200
0
200
400
0
1
2
3
x 10
4
x (nm)
z (nm)
−200 0 200
−200
0
200
400
0
500
1000
1500
a b c
d e
gold polystyrene
-40 -20 0 20 40 60
0.1
1
10
100
gold
polystyrene
0.74 e
-z/43.67(nm)
0.67 e
-z/8.81(nm)
+ 7.27 e
-z/35.69(nm)
F
z
c/P
z (nm)
Figure 8.7: Comparison of optical forces on gold and polystyrene nanoparticles.
(a) Calculated, normalized vertical force exerted on 200 nm gold and polystyrene
particles at different heights. The force is in dimensionless units of Fc/P, wherec
is the speed of the light and P is the incident optical power within one unit cell.
Positive force indicates attraction toward the slab. (b-d) Field intensity profiles
for gold (b),(d), and polysytrene (a),(c), particles at different heights.
99
Chapter 9
Conclusions and outlook
The main contribution of the second part of this dissertation is to, for the first
time, investigate an optical trapping system for assembling an array of nanoparti-
cles, where both the particle-trap interaction and particle-particle interaction play
important roles. I also suggest that the system can be seen as “optical epitax-
ial growth”. An array of optical traps serves as a growth substrate to provide a
two-dimensional potential. However, the stable growth structure depends not only
on the particle-substrate interaction but also the particle-particle interactions. I
develop the model and simulation techniques to investigate the pattern formation
and verify the theory by the experiment.
This work is the initial exploration of “optical epitaxial growth”, a rich and
intriguing system which has both scientific and technological interest. The anal-
ogybetweenoursystemandtraditionalepitaxialgrowthsuggestsvariousstrategies
for improving the quality of the photonic material. Optimization of the “growth
temperature” (laser intensity) and growth rate (particle delivery rate), along with
the use of annealing, may increase the size and quality of the particle array.
An expanded range of complex materials could be grown using different types
of nanoparticles (dielectric, metal, semiconductor, magnetic) as building blocks,
offering a wide range of valuable physical properties. Moreover, the ability to
observe the optical epitaxial growth process directly in both space and time sug-
gests that our system could form a versatile model for studying interaction-driven
dynamics in microscopic systems, such as cold atoms in optical lattices [130].
100
In this work, I mainly investigated the in-plane scattering (interaction) between
particles and the formation of a two-dimensional array. However, the electromag-
netic scattering process is a three-dimensional phenomenon. The interference of
the scattered field from first layer of trapped particles may create strong optical
trapping sites on top of them. The strongly trapped second layer particles have
alreadybeen observedin boththe chain formation[113]and clusterformation [125]
experiments. Careful design of the template (substrate) may repeatedly foster the
scattering process and enable three-dimensional optical epitaxial growth.
In the force analysis in Figure 8.7, strong field redistribution leads to strong
optical force. Strong field redistribution effects can also be exploited to design
hybrid modes at shorter wavelengths, close to the plasmon resonance. Figure
9.1(a) shows a resonant mode of the photonic-crystal slab close to 1320 nm, while
Figure 9.1(b) shows the coupled localized surface plasmon resonance of the gold
nanoparticle array. Hybridization [131] of the resonances (Figure 9.1(c)) produces
a mode with enhanced field intensity and deep subwavelength mode volume. Anal-
ysis of the spectral response (Figure 9.1(d)) shows that the lineshape of the hybrid
mode is narrower than either the photonic or plasmonic resonances. We note that
the plasmonic resonance of the gold nanoparticle array is itself narrower than that
of an isolated gold nanoparticle [132]. The entire system, namely the gold nanopar-
ticle array assembled on top of the photonic-crystal template, can thus be viewed
as a form of all-optically tunable, hybridized photonic-plasmonic matter.
101
0
50
100
150
200
0
1000
2000
3000
4000
5000
a b c
0
1
2
3
4
x 10
4
d
1300 1325 1350 1375 1400
0.0
0.5
1.0
1.5
Au array
PhC
Au array+PhC
σ
ext
/A
Wavelength (nm)
a
c
b
Figure 9.1: Hybridization of the photonic mode with the coupled plasmonic mode
of the gold nanoparticle array. (a) Electric field intensity of a guided resonance
mode in the photonic crystal slab. (b) Electric field intensity of a coupled plas-
monic mode in the gold nanoparticle array. (c) Electric field intensity of a hybrid
photonic-plasmonic mode between the trapped gold nanoparticle array and the
photonic crystal trapping template. (d) The extinction cross-section spectrum of
the structures shown in (a-c).
102
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particles. John Wiley & Sons, 2008.
115
Appendix A
Supporting Information of
Optical Potential Calculation
A.1 Numerical Calculation of the Polarizability
Whenasmallparticleispositionedinelectromagneticfield, thefieldwillinduce
dipole moment at the particle’s position. The relation between the induced dipole
moment (p) and the local electric field (E
0
) is defined as
p =αE
0
(A.1)
where α is the polarizability of the particle. In electrostatic approximation, the
polarizability can be written using the permittivities of the particle material (
1
)
and the surrounding medium (
m
).
α = 4π
m
a
3
1
−
m
1
+ 2
m
(A.2)
Equation A.2 is only valid for small particles with diameter much smaller than
the wavelength of the light. For larger particle which is considered in this disser-
tation, this equation need to be corrected by considering the radiation reaction
[133]. Full-vector FDTD simulations can also be used to obtain a more accurate
value of polarizability [134].
116
In the electrostatics approxmiation, the absorption and scattering efficiencies
of a small sphere may be written [135]
Q
abs
= 4x=
1
−
m
1
+ 2
m
(A.3)
Q
sca
=
8
3
x
4
1
−
m
1
+ 2
m
2
(A.4)
wherex =ka = 2πNa/λ is the size parameter. The cross-sections can be obtained
by numerical (FDTD) simulation by monitoring the scattered power around a
particle. Once the absorption and scattering efficiencies are obtained, we can
calculate the polarizability by using Equation A.2, A.3 and A.4. The polarizability
of a 200 nm diameter gold particle in water is calculated to be (2.37− 0.033j)×
10
−31
(Cm
2
/V).
A.2 Full-vector force simulation versus the
dipole approximation
In Chapter 7, we characterize the interaction between particles by introducing
the energy shift written as Equation 7.13. The derivative of the potential energy
is the optical force. In this section, we compare the force generated by taking the
derivative of Equation 7.13 and the force simulated by FDTD and Maxwell stress-
tensor formula. For the full-vector FDTD simulation, we put a 200 nm diameter
gold particle in the simulation region and put a total-field-scattered-field source to
shine the plane-wave onto it. A cubic field monitor is put around the particle to
monitor the electromagnetic field and the force is calculated by the Maxwell stress-
tensor formula. For the dipole approximation, we take the derivative of Equation
7.12.
117
Figure A.1 shows the comparison result. Figure A.1 (a) shows the optical force
between two particles in water, which is the same case shown in Figure 7.8 (a). The
result for the light polarization along both directions is shown. In this case, the
forces calculated by FDTD simulation and by the dipole approximation are nearly
identicalwitheachotherforbothpolarizationcaseswithoutanycorrection. Figure
A.1 (b) show the optical force between particles with the center 125 nm above a
250 nm SOI wafer. For a 200 nm diameter particle, the bottom of the particle
is 25 nm above the slab. For the dipole approximation, the Green’s function is
calculated by putting the dipole 125 nm above the slab. The FDTD simulation
and the dipole approximation are still similar, with all the features recovered.
2000 4000 6000
-2.0x 10
-23
-1.5x 10
-23
-1.0x 10
-23
-5.0x 10
-24
0.0
5.0x 10
-24
1.0x 10
-23
1.5x 10
-23
2.0x 10
-23
FDTD 200 nm Au (perpendicular)
dipole approximation
FDTD (parallel)
dipole approximation
F/(P/A) (NW
-1
m
2
)
Spacing (nm)
2000 4000 6000
-2.0x 10
-23
-1.5x 10
-23
-1.0x 10
-23
-5.0x 10
-24
0.0
5.0x 10
-24
1.0x 10
-23
1.5x 10
-23
2.0x 10
-23
FDTD 200 nm Au (perpendicular)
dipole approximation
FDTD (parallel)
dipole approximation
F/(P/A) (NW
-1
m
2
)
Spacing (nm)
b a
Figure A.1: Comparison between the full-vector force simulation versus the dipole
approximationenergyformula. (a)theopticalforcebetweentwoparticlesinwater.
(b) the optical force between two particles in water on a plane 125 nm above a 250
nm SOI.
This comparison validates the usage of the dipole approximation (Equation
7.12 and 7.13) in this work. It also shows that the secondary multiple scatter-
ing is negligible. In Figure A.1, only the first scattering is considered in dipole
approximation.
118
A.3 The effect of holes on the photonic-crystal
slab
In this work, we approximate the Green’s function of a photonic-crystal slab
to a Green’s function on an uniform slab without holes in it. The simulation
method introduced in Section 7.2 can be used to simulate the Green’s function in
any kind of environment including the case on top of a photonic-crystal slab. The
reason that I made this approximation is that in Chapter 7, I investigate photonic-
crystal templates with a wide range of lattice constant (from 0.8a to 1.1a). For
a certain lattice constant, the photonic-crystal slab needs to be carefully designed
to support a guided-resonance mode around the operating wavelength of the laser
(∼ 1550 nm), and sometimes it becomes very challenging. The approximation to
an uniform slab makes the systematic study shown in Figure 7.6 possible.
To validate this approximation, we design a photonic crystal template with
a lattice constant of 1.1 μm that supports a guided resonance at 1558 nm. The
template has a perturbed square lattice similar to the one used in Ref [113]. The
radii of the big and small holes are 80 nm and 70 nm, respectively. Figure A.2 (a)
showstheGreen’sfunctionforadipoleat125nmabovethesurfaceofthephotonic-
crystal slab. The holes are also drawn on top of the Green’s function. Figure A.2
(b) shows the Green’s function for a dipole 125 nm above an uniform slab at the
same wavelength. The dots only indicate the position of the large holes. The holes
on the photonic-crystal slab perturb the Green’s function slightly, however, the
main features, especially for the position closest to the center dipole, remain. The
amplitude is also similar.
119
x(μm)
y(μm)
−3 −2 −1 0 1 2 3
−3
−2
−1
0
1
2
3 −1
−0.5
0
0.5
1
x(μm)
y(μm)
−3 −2 −1 0 1 2 3
−3
−2
−1
0
1
2
3 −1
−0.5
0
0.5
1
k T/(mW/μm )
B
2
k T/(mW/μm )
B
2
a b
Figure A.2: Comparison between the Green’s function on a photonic-crystal slab
(a) and on an uniform slab (b).
120
Abstract (if available)
Abstract
This dissertation work studies the use of nanostructures to control the flow of light in two application areas: photovoltaics and material self-assembly, which are discussed in I and Part II, respectively. ❧ The work in photovoltaics focuses on designing high-efficiency nanowire array solar cells. Nanowire arrays are a promising candidate for the next generation of low-cost, high efficiency, flexible photovoltaic cells. Nanowire structures relax the lattice-matching constraints and allow the usage of materials with different lattice constants for multijunction cells. This opens up a much wider range of materials choices than for traditional, planar cells. The design of a high-efficiency cell involves two factors: optical absorption and carrier collection. In this dissertation, I first use the full-wave electromagnetic simulation to investigate the absorption properties of periodic nanowire arrays and provide the optimal designs in a single junction and a tandem wire-on-substrate cell configurations. I then study strategies for optimal carrier collection by finite-element method electronic device simulations. I optimize the p-n junction geometry, doping parameters, and surface passivation scheme. This work not only establishes the fundamental limits of nanowire solar cells’ designs but also provides practical guidelines and solutions for high performance nanowire solar cell devices. ❧ The work of material self-assembly is based on the light-assisted, templated self-assembly (LATS) technique developed in our group. In this method, we shine light through a photonic crystal (or template) to create an array of optical traps. The traps drive the self-assembly of nanoparticles into regular patterns. In this dissertation work, I discover the crucial effect of inter-particle interactions on the pattern formation of metallic particles in the LATS system. I envision the analogy between the optical assembly of nanoparticles and atomic level epitaxial growth and suggest that the system can be viewed as “optical eptaxial growth”. I develop the modeling and simulation technique to explain and predict experimental results. Such model leads to the first successful demonstration of the optical assembly of a 2D periodic gold nanoparticle array.
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Creator
Huang, Ningfeng
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Core Title
Light management in nanostructures: nanowire solar cells and optical epitaxial growth
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Electrical Engineering
Publication Date
08/20/2015
Defense Date
08/20/2015
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tandem solar cell