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INVENSTIGATION OF SOLAR ELECTRIC SYSTEMS BASED ON NANO RECTENNA
WANG JIAJIE, IVAN
Ph.D
The Hong Kong Polytechnic University
2014
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The Hong Kong Polytechnic University Department of Building Services Engineering
Investigation of Solar Electric Systems Based on Nano Rectenna
WANG Jiajie, Ivan
A thesis submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy
September 2013
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CERTIFICATE OF ORIGINALITY
I hereby declare that this thesis is my own work and that, to the best of my knowledge and belief, it produces no material previously published or written nor material which has been accepted for the award of any other degree or diploma, except where due acknowledgement has been made in the text.
I also declare that the intellectual content of this thesis is the product of my own work, even though I may have received assistance from others on style, presentation and language expression.
______(Signed)
____WANG Jiajie, Ivan______(Name of Student)
Department of Building Services Engineering The Hong Kong Polytechnic University
September 2013
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Table of Content
Investigation of Solar Electric Systems Based on Nano Rectenna ...... III Table of Content ...... VII Publications ...... XI Figures...... XIII Tables ...... XVII Abstract ...... XIX Acknowledgements ...... XXIII 1 Introduction ...... 1 1.1 Research Background ...... 1 1.2 Research Aim and Objectives ...... 5 1.3 Research Scope ...... 7 1.4 Research Methodology ...... 9 1.5 Significance and Value ...... 11 2 Literature Review...... 13 2.1 Development of Energy Transmission Systems ...... 13 2.2 Receiving Nano Antenna ...... 15 2.3 Energy Rectification ...... 18 2.4 Conclusion ...... 20 3 Nano Antenna Design ...... 22 3.1 Basic Parameters for Optical Antenna ...... 22 3.1.1 Radiation Patterns ...... 24 3.1.2 Radiation Intensity ...... 26 3.1.3 Directivity and Gain ...... 26 3.1.4 Antenna Field Zones ...... 27 3.1.5 Input Impedance...... 28 3.1.6 Receiving Antenna Efficiency ...... 29 3.2 Numerical Method for Nano Antenna Design ...... 30 3.2.1 Brief FDTD Method ...... 30 3.2.2 Frequency-Dependent Material in FDTD Method ...... 33 3.3 Choice of Metal Materials for Nano Antenna ...... 36 3.4 Conclusion ...... 39 4 Disk Nano Antenna for Field Enhancement and Broadband ...... 41 4.1 Directional Field Enhancement of Dielectric Nano Optical Disc Antenna Array ...... 41 4.1.1 Analysis of Electrical Field Distributions of Nano Antenna ...... 41 4.1.2 FDTD Simulation of Dielectric Nano Optical Disc Antenna Array for Field Enhancement ...... 45 4.1.3 Conclusion ...... 49 VII
4.2 Broadband Optical Antenna Design and Equivalent Circuit ...... 50 4.2.1 Broadband Optical Antenna Design...... 50 4.2.2 Equivalent Circuit for Broadband Optical Antenna ...... 52 4.2.3 FDTD Simulation for Broadband Optical Antenna ...... 53 4.2.4 Conclusion ...... 56 4.3 Conclusion ...... 56 5 Optical Input Impedance of Nanostrip Antennas ...... 57 5.1 Optical Nanostrip Antenna ...... 57 5.2 Optical Input Impedance and Optical Radiation Efficiency ...... 58 5.3 Antenna Parameters against Nanoantenna Size ...... 63 5.3.1 Gap Distance g ...... 63 5.3.2 Strip Width a ...... 65 5.3.3 Strip Length b ...... 67 5.4 Conclusion ...... 71 6 Rectification Device...... 73 6.1 Metal Insulator Metal Overview ...... 73 6.1.1 Device Structure ...... 74 6.1.2 Functions ...... 75 6.1.3 Difference from Traditional PV in Working Function ...... 77 6.2 Tunneling Effect of the MIM Diode ...... 77 6.3 MIM Diode Without/With External Time Harmonic E-field ...... 83 6.3.1 Unilluminated MIM Diode ...... 87 6.3.2 Illuminated MIM Diode ...... 91 6.4 Conversion Efficiency of Solar Energy ...... 96 6.4.1 Conversion Efficiency of MIM Diode ...... 96 6.4.2 Conversion Efficiency of PV ...... 97 6.5 Selection of Material for the MIM Diodes ...... 98 6.6 Equivalent Circuit of the MIM Diode ...... 103 6.7 Conclusion ...... 104 7 Characteristics of Rectenna System with Rectification Excited by Plane Wave ...... 105 7.1 Rectenna System for Optical Energy Receiving ...... 105 7.2 Rectenna System by Gap Excitation ...... 109 7.2.1 Reference Optical Antenna ...... 109 7.2.2 Optical Antenna with a MIM Diode ...... 111 7.2.3 Optical Antenna with the MIM Diode and the DC Line ...... 114 7.3 Rectenna System Excited by Plane Wave ...... 119 7.3.1 Nano Antenna with the MIM Diode ...... 123 7.3.2 Nano Antenna with the MIM Diode and the DC Line ...... 127 7.4 Energy Coupling from Antenna to the MIM diode ...... 131 7.5 Impedance of Insulator in the MIM Diode ...... 138 7.6 Conclusion ...... 139 8 Effect of an External Load ...... 141 8.1 Equivalent Circuit of a Rectenna System ...... 141
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8.2 Effect of the External Load ...... 145 8.2.1 Transmission of Power ...... 145 8.2.2 Effect of Load Voltage to Antenna Impedance ...... 152 8.3 System Efficiency Calculation for Optimal State ...... 154 8.4 Conclusion ...... 157 9 Rectenna Array for Solar Energy Harvesting ...... 159 9.1 The Construction of Nano Antenna Array ...... 159 9.2 Antenna Array Structure for Field Enhancement ...... 161 9.3 Simulation and Analysis ...... 162 9.3.1 Rectenna Array of 1Row×2Columns ...... 162 9.3.2 Rectenna Array of 1Row×4Columns ...... 168 9.3.3 Rectenna Array of 2Rows×2Columns ...... 174 9.3.4 Calculated Output Power for Rectenna Array ...... 180 9.4 Conclusion ...... 182 10 Conclusion and Suggestions for Future Research ...... 184 10.1 Summary ...... 184 10.1.1 Nano Antenna for Harvesting Solar Energy ...... 184 10.1.2 Optical Energy Rectification ...... 185 10.1.3 Nano Device Integration ...... 186 10.1.4 Nano Rectenna Array ...... 187 10.2 Future Research ...... 188 10.2.1 Nano Antenna for Broadband Harvesting Solar Energy ...... 188 10.2.2 Nano Device Integration for Broadband Rectification ...... 188 10.2.3 Nano Broadband Antenna Array Design for Mass Production Application ...... 188 10.3 Conclusion ...... 189 Reference ...... 190
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Publications
Ivan WANG, Ya-ping Du; Directional field enhancement of dielectric nano optical disc antenna arrays, Optical Materials, Volume 34, Issue 1, November 2011, Pages 126–130 http://dx.doi.org/10.1016/j.optmat.2011.07.027
Ivan WANG, Ya-ping Du; Broadband Optical Antenna with a Disk Structure, Asia Communications and Photonics Conference and Exhibition, Shanghai, China, November 13, 2011 http://www.opticsinfobase.org/abstract.cfm?URI=ACP-2011-830724
Ivan WANG, Ya-ping Du; Optical input impedance of nanostrip antennas, Optical Engineering 51(5), 054002 (May 04, 2012) http://dx.doi.org/10.1117/1.OE.51.5.054002
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Figures
Figure 1 - 1.2.1 Schematic diagram of the whole solar energy harvesting system 5 Figure 2 - 1.2.2 Impedance matching ...... 6 Figure 3 – 1.3.1 Solar radiation spectrum [1] ...... 8 Figure 4 - 2.1.1 Schematic diagram of a simple rectennna system...... 14 Figure 5 - 3.1.1.1 Field pattern of a directional antenna for E or H field [46] .... 25 Figure 6 - 3.1.4.1 Antenna regions for near and far field [46] ...... 28 Figure 7 – 3.2.1.1 Yee cell [55] used by a FDTD algorithm ...... 32 Figure 8 - 4.1.1.1 Cylindrical rod and disc resonator/antennas ...... 43 Figure 9 - 4.1.1.2 EM wave distributions in cross section of the dielectric cylindrical waveguide and resonator, ...... 44 Figure 10 - 4.1.2.1 Configuration of the dielectric nano optical disc antenna arrays ...... 46 Figure 11 - 4.1.2.2 Directivity in the E-plane and H plane of a single disc ...... 47 Figure 12 - 4.1.2.3 Comparison of maximum directivity of different antenna arrays ...... 48 Figure 13 - 4.2.1.1 Three-layer broadband optical antenna (inner-metal, middle-silicon, outer-metal) ...... 51 Figure 14 - 4.2.2.1 Equivalent circuit for a single plasmonic disc ...... 52 Figure 15 - 4.2.2.2 Equivalent circuit for a single plasmonic disc with a dielectric ring ...... 52 Figure 16 - 4.2.2.3 Equivalent circuit for a broadband optical antenna...... 53 Figure 17 - 4.2.3.1 Radiation pattern of a broadband optical antenna by using the Drude model for material; E- and H Planes ...... 55 Figure 18 - 5.1.1 Geometry of a nanostrip antenna ...... 57 Figure 19 – 5.2.1 Optical input impedance and radiation efficiency of a nanostrip antenna in reference size ...... 60 Figure 20 - 5.2.2 Equivalent circuit of a nanostrip antenna for optical resonances ...... 61 Figure 21 - 5.3.1.1 Optical input impedance of a nanostrip antenna with a variable gap distance from 15nm to 25nm (a=15nm, t=15nm, b=50nm) .... 65 Figure 22 - 5.3.2.1 Optical input impedance of a nanostrip antenna with a variable width from 15nm to 25nm (g=15nm, t=15nm, b=50nm) ...... 67 Figure 23 - 5.3.3.1Optical input impedance of a nanostrip antenna with a variable strip length from 45nm to 55nm (a=15nm, t=15nm, g=15nm) ...... 69 Figure 24 – 6.1.1.1 Structure of a MIM diode ...... 75 Figure 25 - 6.1.2.1 States of a MIM diode: ...... 75 Figure 26 – 6.2.1 Schematic diagram of an electron tunneling in a MIM diode . 78 Figure 27 – 6.2.2 Diagram of energy distribution ...... 79 Figure 28 – 6.2.3 Analysis of one meshed part in the process of electron XIII
tunneling ...... 80 Figure 29 – 6.2.3 Energy distribution of a Cr-Cr2O3-Pd diode ...... 82 Figure 30 - 6.2.4 Tunneling probability of an electron in the MIM diode (Cr-Cr2O3-Pd) ...... 83 Figure 31 – 6.3.1 Diagram of energy bands...... 84 Figure 32 – 6.3.2 Electron tunneling through a voltage biased insulator ...... 85 Figure 33 – 6.3.3 Diagram of free state ...... 85 Figure 34 – 6.3.4 Diagram of Vdc=0V state ...... 86 Figure 35 - 6.3.5 Diagram of Vdc≠0 state ...... 87 Figure 36 – 6.3.1.1 Energy band diagram of a MIM diode ...... 88 Figure 37 – 6.3.1.2 Fermni function distribution ...... 89 Figure 38 – 6.4.1.1 Current density v.s. biased voltage ...... 97 Figure 39 – 6.5.1 Comparision of current density of a MIM diode under 660THz ...... 100 Figure 40 – 6.5.2 Current density comparison for the insulator with different thickness ...... 101 Figure 41 - 6.5.3 Quantum energy conversion efficiency of the MIM diode - Cr-Cr2O3-Pd ...... 102 Figure 42 – 6.6.1 Equivalent circuit model of a MIM diode ...... 103 Figure 43 – 7.1.1 Rectenna system ...... 108 Figure 44 – 7.2.1.1 Optical input impedance (gap excitation)...... 110 Figure 45 – 7.2.2 Optical input impedance of a optical antenna with MIM diode, and a variable strip lengthfrom 61nm to 50nm (a=25nm, t=30nm, gap=20nm) ...... 114 Figure 46 - 7.2.3.1 Optical input impedance of a optical antenna with the MIM diode, DC line, and a variable strip length from 55nm to 75nm (a=25nm, t=30nm, gap=20nm) ...... 116 Figure 47 – 7.2.3.2 Optical input impedance of a optical antenna with the MIM diode, variable DC linde from 50nm to 200nm (a=25nm, t=30nm, gap=20nm,b=70nm) ...... 118 Figure 48 - 7.3.1 Electrical field intensity of an optical antenna under a plane wave excitation (a=25nm, b=61nm, t=30nm, gap=20nm)...... 121 Figure 49 – 7.3.2 (a) Equivalent circuit of Rectenna system; (b) Method for V and I calculation ...... 122 Figure 50 – 7.3.1.1 Electrical field intensity of an optical antenna with the MIM diode under plane wave excitation ...... 125 Figure 51 – 7.3.1.2 Optical input impedance of an optical antenna with the MIM diode under plane wave excitation ...... 126 Figure 52 – 7.3.2.1 Electrical field intensity of an optical antenna with the MIM diode, DC line under plane wave excitation ...... 128 Figure 53 – 7.3.2.2 Optical input impedance of a optical antenna with the MIM diode, DC line under plane wave excitation ...... 129 Figure 54 – 7.3.2.3 Electrical field intensity of an optical antenna with the MIM diode, variable DC linde from 50nm to 200nm ...... 130
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Figure 55 – 7.3.2.4 Optical input impedance of a optical antenna with the MIM diode, variable DC line from 50nm to 200nm ...... 131 Figure 56 – 7.4.1 AC power on the MIM diode against frequency ...... 132 Figure 57 – 7.4.2 AC power on the MIM diode against Rmim ...... 134 Figure 58- 7.4.3 Optical input impedance of optical antenna with the MIM diode, DC line under plane wave excitation ...... 136 Figure 59 – 7.3.4 AC power on the MIM diode against frequency and Rmim . 137 Figure 60 – 7.5.1 Impedance of insulator in the MIM diode with antenna (a) even DC line (b) under plane wave excitation ...... 139 Figure 61 – 8.1.1 Rectenna system ...... 142 Figure 62 – 8.1.2 Structure of a MIM diode ...... 143 Figure 63 – 8.1.3 Equivalent circuit of a rectenna system ...... 144 Figure 64 – 8.1.4 Equivalent circuit can be separated into DC and AC parts ... 145 Figure 65 – 8.2.1.1 Rmim vs. the voltage biased (660THz) ...... 149 Figure 66 - 8.2.1.2 Rectified current vs. the voltage applied (660THz) ...... 150 Figure 67 – 8.2.1.3 Rectified current against incident voltage (660THz) ...... 151 Figure 68 – 8.2.1.4 AC power of the MIM diode before rectification (660THz) ...... 151 Figure 69 – 8.2.1.5 Pac to Pdc conversion efficiency (660THz) ...... 152 Figure 70 – 8.2.2.1 Effect of an external load to the antenna impedance ...... 153 Figure 71 – 8.3.1 Input and output power of a MIM diode against incident E-field ...... 155 Figure 72 – 9.1.1 Distribution of nano antennas in an antenna array ...... 160 Figure 73 –9.1.2 Skematic of the nano antenna array for solar enrgy havesting ...... 161 Figure 74 – 9.3.1.1 Sturcture of a rectenna array 1row×2columns...... 163 Figure 75 – 9.3.1.2 Impedance of a nano antenna in the array with the spacing of 50nm ...... 164 Figure 76 - 9.3.1.3 Impedance of the nano antennas in the array with the spacing of 100nm ...... 165 Figure 77 - 9.3.1.4 Impedance of the nano antennas in the array with the spacing of 150nm ...... 166 Figure 78 – 9.3.1.5 Average Vac and Pac for one single Rmim in the array .... 168 Figure 79 - 9.3.2.1 Sturcture of a rectenna array 1row×4columns ...... 169 Figure 80 - 9.3.2.2 Impedance of the nano antennas in the array with the spacing of 50nm ...... 170 Figure 81 - 9.3.2.3 Impedance of the nano antennas in the array with the spacing of 100nm ...... 171 Figure 82 - 9.3.2.4 Impedance of the nano antennas in the array with the spacing of 150nm ...... 172 Figure 83 – 9.3.2.5 Average Vac and Pac for one single Rmim in the array .... 174 Figure 84 - 9.3.3.1 Sturcture of a rectenna array of 2rows×2columns ...... 175 Figure 85 - 9.3.3.2 Impedance of the nano antennas in the array with the spacing of 50nm ...... 176
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Figure 86 - 9.3.3.3 Impedance of nano antenna in the array with the spacing of 100nm ...... 177 Figure 87 - 9.3.3.4 Impedance of nano antennas in the array with the spacing of 150nm ...... 178 Figure 88 – 9.3.3.5 Average Vac and Pac for one single Rmim in the array .... 180 Figure 89 – 9.3.4.1 Comparison of Pac for single ref unit and average value of a single unit in an array ...... 181 Figure 93 – 9.3.4.2 Comparison of Pdc for single ref unit and average value of a single unit in an array ...... 181 Figure 91 – 9.3.4.3 Comparison of total system efficiency for a single unit and the average value of a single unit in an array ...... 182
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Tables
Table 1 – 3.3.1 Comparison of material properties for different metals...... 39 Table 2 - 4.1.2.1 Comparison of directivity in antenna arrays with different numbers of discs and different spacings between the edges, unit is dBi ..... 48 Table 3 - 4.2.3.1 Comparison of gain of a broadband optical antenna by different methods ...... 55 Table 4 – 5.2.1 Antenna parameters of a nanostrip antenna ...... 62 Table 5 - 5.3.1.1 Optical resonant frequency, input resistance and radiation efficiency against gap g ...... 65 Table 6- 5.3.2.1 Optical resonant frequency, input resistance and radiation efficiency against strip width a ...... 67 Table 7 - 5.3.3.1Optical resonant frequency, input resistance and radiation efficiency against strip length b ...... 69 Table 8 - 5.3.3.2 Optical resonant frequency and input resistance variation against strip sizes ...... 70 Table 9 – 6.5.1 Materials’ workfunction [113] ...... 99 Table 10 – 6.5.4 Comparison of conversion efficiency for different materials . 102 Table 11 – 7.2.1.1 Antenna parameters of the reference optical antenna ...... 111 Table 12 – 7.2.2 Optical resonant frequency, input resistance and radiation efficiency against strip length b ...... 114 Table 13 - 7.2.3.1 Optical resonant frequency, input resistance and radiation efficiency against strip length b ...... 116 Table 14 – 7.2.3.2 S11 parameter of DC line for transmitting 600THz optical energy ...... 118 Table 15 -7.2.3.3 Summary of the Experiments results ...... 119 Table 16 – 7.3.1.2 Optical resonant frequency, input resistance and radiation efficiency against strip length b ...... 126 Table 17 – 7.3.2.1 Optical resonant frequency, input resistance and radiation efficiency against strip length b ...... 129 Table 18 – 7.3.2.2 S11 parameter of a DC line for transmitting optical energy 131 Table 19 – 7.5.1 Impedance of Insulator in MIM diode at resonant frequency 139 Table 20 – 8.3.1 Maximum output power tracking point against E-Field ...... 156 Table 21 – Comparison of the system with same structure but different materials for antenna ...... 157 Table 22 –9.3.1.1 Impedance and voltage of the nano antenna at 660THz with the spacing of 50nm ...... 164 Table 23 - 9.3.1.2 Impedance and voltage of the nano antennas at 660THz with the spacing of 100nm ...... 165 Table 24- 9.3.1.1 Impedance and voltage of nano antennas at 660THz with the spacing of 150nm ...... 167 XVII
Table 25 - 9.3.2.1 Impedance and voltage of the nano antennas at 660THz with the spacing of 50nm ...... 170 Table 26 - 9.3.2.3 Impedance and voltage of the nano antennas at 660THz with the spacing of 100nm ...... 171 Table 27 - 9.3.2.3 Impedance and voltage of nano antennas at 660THz with the spacing of 150nm ...... 173 Table 28 - 9.3.3.1 Impedance and voltage of the nano antennas at 660THz with the spacing of 50nm ...... 176 Table 29 - 9.3.3.2 Impedance and voltage of nano antennas at 660THz with the spacing of 100nm ...... 177 Table 30 - 9.3.3.3 Impedance and voltage of nano antennas at 660THz with the spacing of 150nm ...... 178
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Abstract
Abstract of thesis entitled: Investigation of Solar Electric Systems Based on Nano
Rectenna
Submitted by : WANG Jiajie, Ivan
For the degree of : Doctor of Philosophy at The Hong Kong Polytechnic University in September, 2013.
Key words: Optics, Antenna, Quantum Mechanics
Renewable energy, as a complement to traditional fuel energy, has received increasing attention. Solar radiation is one particular source of renewable energy that has been widely applied in vehicles, machines, and buildings, among others. There are generally two different ways in which solar energy is used – heat and electricity.
Electric power can be obtained from converting solar energy through the use of photo voltaic cells, a conversion process that involves stimulating the electrons on PV cells by solar energy to generate electrical current.
Another way of converting solar energy to electrical current is by tapping its electromagnetic wave properties – nano antennas. A nano antenna receives solar/optical energy in a highly efficient way. The energy is then transmitted to the
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MIM (Metal Insulator Metal) diode for rectification. Finally, the DC current is generated by the diode based on the tunnelling effect.
The idea of harvesting solar energy by nano antennas dates back to the end of the
1970’s as a type of theoretical research because of fabrication technology. Even to date, most of the research remains within the realm of theoretical study using computer numerical analysis. Only a fewer researchers have managed to fabricate these nano meter scale devices, but how fabrication technology works is worth studying.
The aim of this thesis is to study a novel technology for harvesting solar energy by using nano antennas. This technology is developed by using plasmatic properties of the metal for collecting solar energy and tunnelling effect in the MIM diode for optical rectification. A solar-energy-harvesting rectenna system is developed. The rectenna is an antenna with a diode. The whole nano rectenna system consists of a nano antenna, a MIM diode and DC output lines, which are mounted on a silicon substrate. The substrate is chosen as the base board, because of its relative low cost and ease of production, while the nano antennas, made of aluminium, are used to collect solar energy for their relatively high efficiency. The MIM diodes are embedded in the gap of nano antennas in order to reduce the optical loss and to directly receive the optical energy for rectification to DC current. The output DC lines are embedded in the substrate to eliminate interference and to transmit the DC current.
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Thus the antenna and the MIM diode are integrated into a complete rectenna system.
An antenna array is composed of multiple single rectenna units to provide more current output. In this thesis, the method of FDTD (finite-difference time-domain) is employed to study the characteristics of nano antennas, and the tunnelling effect of quantum mechanics is employed to study the characteristics of DC rectification.
In summary, this thesis theoretically presents the design and analysis of a rectenna system for harvesting solar energy for electrical application, which is quite different from the traditional photovoltaic technology. It is found that the total conversion efficiency of the rectenna system in an array is 15%. The following contributions have been made in the field of optical nano devices: (1) nano antenna proposed for harvesting solar energy; (2) optical energy rectification; (3) nano device integration;
(4) nano antenna array; (5) a complete design procedure for solar-energy-harvesting using nano antennas.
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Acknowledgements
I an indebted to my supervisor, Professor Du Ya-ping, for his academic guidance, and patience during the whole project. Professor Du’s patient support plays a key role in my research life.
Thanks are also due to Dr Wai-Sang Li for helping me to understand the tunneling effect in quantum mechanics.
I am also grateful to my parents for their encouragement and faith in me.
My gratitude also goes to all those who have helped me in the past four years!
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1 Introduction
1.1 Research Background
Energy is widely recognized as a key driver of civilization and development, and energy consumption is an indicator of advances in social and economic development.
The 1970s saw the beginning of the information era when easy and instant access to information was brought about by technological improvements. This, in turn, has enabled the economy to develop at a faster pace, and fuelled demands for energy and electricity to meet the ever-growing capacities of computers and the Internet, among others. Without electricity, the Internet cannot transmit any information. Similarly, lighting, cooking, heating, and cooling, are dependent on the availability of electricity in contemporary society.
According to the data from International Energy Agency (IEA) [1], global total primary energy supply by fuel accounted for only fewer than 6 000 Mtoe in 1971, but
40 years later, in 2008, the amount increased to more than 12 000 Mtoe. This fuel consists of coal, oil, gas, nuclear, hydro and combustible renewables. While global consumption totalled 4 000 Mtoe in 1971, it jumped to more than 8 000 Mtoe in 2008, doubling the amount of that recorded in 1971. Coal, oil, and gas are the first three kinds of source that are widely used for conversion to other forms such as heat and
1 electricity. It is predicted that - more energy will be consumed in the future.
Coal, oil, and gas, as the traditional source of energy, supply most of the energy for daily use, because they have been researched and used for thousands of years.
However, the byproducts are carbon material, which will lead to greenhouse effects.
Reducing unwanted energy consumption may not be the best way to address the problem of green house effect. Researchers have been trying to identify new sources through different means. Nuclear energy is clean, but it needs radiative sources and huge construction costs to maintain its safety. Hydro energy comes from ionizing-water, but it is not readily accessible in places such as mountainous areas or deserts where water supply is low or limited. Hence, scientists have embarked on research on wind and solar energy, which are believed to be clean and renewable.
Solar energy is clean energy, which was initially used as a heat source. Unlike coal, oil or gas, it has an additional advantage of being easily available for consumers. The main problem, however, lies in making devices that can transform solar energy to heat or electricity. In 1839, a French physicist, A. E. Becquerel, discovered photovoltaic
(PV). In 1883, Charles Fritts investigated the semiconductor material Selenium, and p-n juncitons based on gold to form a very thin layer, which resulted in the invention of the first solar cell, with the conversion efficiency being only 1% [2]. By drawing on quantum theory, traditional solar cells use photovoltaic material to convert solar energy into electricity. Electrons are hit by photons from the sun, and are then turned
2 into a higher state of energy to generate electricity. Solar cells need to be integrated into a module to generate DC, normally 12V or 24V for normal application [3].
Limited by Carnot efficiency, photovoltaic material cannot achieve high efficiency.
Theoretically, this kind of solar cells with single junction has an efficiency of no more than 30%. By adding more layers for conversion, PV solar cell can achieve even more than 60% for efficiency, which leads to a thicker structure and costs more money.
Using nano antennas is an alternative to photovoltaic cells for solar energy harvesting.
Combined with a rectifier, which is used to convert photons to electrons, nano antennas can harvest solar energy more efficiently than traditional PV. Not restricted by Carnot efficiency, reseachers found that nantennas can achieve a theoretical optical energy harvesting efficiency of greater than 85% or even 90% [4], which means that
90% of incident power can be harvested by nano antennas. The basic parameters of traditional solar cells are materials characteristics and corresponding electrical properties. But the resonant frequency and bandwidth of antennas are determined by the antenna structure. The incoming light induces a time varying current on the antenna surface. With proper design of the nano antenna profile and impedance, efficient collection of incident radiation can be achieved. Rectification for this THz current on nano-antennas, however, is necessary. The common rectifier diode of performing high frequency current rectification is the metal- insulator-metal (MIM) diodes. The potential difference of energy levels in different metal junctions leads to non-linear effects, which makes high-speed rectification available. The thinner the
3 insulated layers, the more effevitive they are in realizing higher the non-linear effects.
Theoretical and experiment research has effectively proved the application of the energy conversion. The device has been made in the microwave band with a high efficiency rate [5].
The application of optical antenna has generated much interest in the field of nano-tech research. Research in the interaction between light and metal is well documented. When a nano metallic structure interacts with light, this will lead to the excitation of surface plasmons. As a result, the local field near this metal nano-structure can be enhanced. Normally, the structure can be a wire, nanosphere, cylinder, or even bowtie [6,7]. In a dipole optical antenna, the spacing between two monopoles can determine the strength of local field enhancement [8]. In nanometer scale, carbon nanotube is used in the design of a dipole antenna [9]. A yagi-uda optical antenna can also realize the high gain with an endfire radiation pattern [10].
When dielectric material is applied to make nano antenna arrays, it shows a higher emission quantum efficiency by metallic array structures than a single one, and stronger emission enhancement emission is achieved for poor emitter position and orientation in metallic array structures than a single one [11]. Fractal antenna has been widely used in the design of microwave antenna, which has also been scaled down to optical antenna [12]. In [13], the characteristics of emission in microwave and optical frequency have been extensively studied and compared.
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1.2 Research Aim and Objectives
This thesis aims to develop a new way of solar energy collection, with a high energy conversion efficiency based on the electromagnetic (EM) theory and semiconductor nonlinearity. Most of the solar energy distributes in the visible range [14,15], so this technique covers the electromagnetic frequency from 384THz to 789THz, in wavelength 780nm to 380nm respectively. The harvesting system receives energy within this range. It is also possible to develop a system operating in infrared range, which contains heat energy. In the following chapter, the field enhancement and bandwidth of the antenna will be first discussed to facilitate an understanding of application in harvesting solar energy. Then the recticfication compoment is examined in relation to its effect derived from the connected load.
The whole system consists of three parts, optical antenna, impedance matching, and rectifier. The optical antenna is used for solar energy harvesting; impedance matching is for the maximum energy transferring between the optical antenna and rectifier; the rectifier is to convert solar energy from optical frequency to DC current.
Optical Impedance DC Current Rectification Antenna Matching Output
Figure 1 - 1.2.1 Schematic diagram of the whole solar energy harvesting system
The optical antenna is the receiver of the solar energy at the size of nanometer scale,
5 so an antenna array will be formed with thousands of single units to collect the harvested energy. Similar to microwave broadband antenna, the optical antenna will also cover the visible frequency range. Besides, the radiation pattern will have a wide angle so that the antenna can receive light from different directions.
In EM theory, the impedance value of a device varies with the frequency. Two different devices at fixed frequency - do not necessarily have the same impedance.
This will cause a problem shere the energy from device-1 cannot be fully transmitted to device-2. Therefore, it is important to consider impedance matching between device-1 and device-2 that enables the maximum energy transmission.
Impedance device-1 device-2 Matching
Figure 2 - 1.2.2 Impedance matching
The frequency of energy induced on the optical antenna is very high, so it should be rectified to DC current by using the metal-insulator -metal (MIM) diodes. Different from Schottky diodes, the tunneling function of MIM diodes is not influenced by parasitic capacitors, in that the rectification is based on the electron tunneling. The
MIM diodes, therefore, can show an outstanding operation at frequencies above 150
THz. Other method and material will be studied to meet the need of rectification.
Finally, these three parts will be integrated on an insulated substrate for electricity 6 generation. The MIM diode is put in the gap of the nano antenna with consideration of impedance matching DC rectification. The substrate is the bottom layer. The whole system can be processed by roll-to-roll technology for massive manufacturing.
1.3 Research Scope
The research scope includes: (a) solar radiation as the energy source; (b) EM theory and size scaling for optical antenna structure design; (c) plasma, plasmonic, and dielectric for optical antenna, and (d) impedance matching; (e) nonlinearity of semiconductor material for DC rectification.
Solar energy radiates from the sun’s nuclear fusion in the core. Sun-shine represents the huge loss of this nuclear fusion. Normally, this light is deemed as unpolarized electromagnetic wave [16]. The radiated spectrum is composed of visible light (47%), infrared (45%), and short wave (8%). Thus, the visible light and the infrared contain most of the solar energy, whose wavelength is in the size of nanometer. It has been calculated that about 1353W/m2 solar energy reaches the outside of earth’s atmosphere, which is defined as air mass zero (AM0 is the extraterrestrial intensity)
[17]. Bouger’s law is defined to compute the sum of power above the water level, called aim mass one (AM1, the terrestrial intensity when the sun is overhead) [18]
km Ib I0 e (1.3.1)
In this equation, I0 is the terrestrial intensities of the radiation; Ib is the extraterrestrial
7 intensities of the radiation; k is the absorption constant for the atmosphere; while m is the dimensionless path length of the sunlight through the atmosphere, air mass ratio.
Figure 3 – 1.3.1 Solar radiation spectrum [1]
Metal and dielectric are two main materials for antennas operating in microwave band, whose conductivity or permittivity is constant, and does not change during the whole operating band. However, when it comes up to infrared and visible frequency, the permittivity of dielectric is not a constant; it is a complex number varying with frequency; the real and imaginary of the permitivitty parts are positive [19]. As for the metal at these frequency ranges, it cannot be considered as PEC, but shows the characteristics of plasma, similar to gas plasma. Its permittivity is also a complex number varying with frequency. The difference from dielectric is that the real part is negative [20] which means that the EM wave cannot permeate through the material,
8 only exist along a small depth from the skin, like the skin depth in microwave band.
Metal is called plasmonic material in this band.
As in infrared and optical frequency, dielectric has a positive real part, and metal has a negative real part; the positive imaginary part means the loss of energy. So the dielectric can be viewed as a capacitor, while the plasmonic can be considered as an inductor. Figure 1 [21] provides an explanation of the equivalent circuit: plasmonic with ε>0 is capacitive while ε<0 is inductive. For antenna design, this equivalence contributes to LC resonance. For impedance matching, it is easy to design the network for energy transferring.
The MIM diode is used for rectification of high frequency power to DC power by the nonlinearity of semiconductor. The MIM diode is a tunneling-diode. Unlike the
Schottky diode, it is not influenced by parasitic capacitance in electron tunneling.
Now the structure can operate effectively at frequencies around 660 THz. Other method and material will be studied to meet the need of rectification.
1.4 Research Methodology
The whole system is mainly composed of the nano antenna for solar energy harvesting, and the metal-insulator-metal diodes for current rectification. The characteristics of these components are very important in the system application and
9 evaluation. The research methodology is presented in the ensuing paragraphs.
The nano antenna has the characteristic parameters such as field patterns, field efficiency, field directivity and gains which depend on the distribution of current density on the antenna. Metals, in the THz spectrum, are different from those in microwave band, not as perfect electrical conductor; but as plasmonic material which is very similar to plasma gas. The Drude model is applied to represent the plasmonic material, in which the permittivity is a complex number, the real part means the storage of energy while the imaginary parts means the loss of energy. This makes plasmonic material as lossy dielectric material, and the traditionally conductive current turns into displacement current. The effective permittivity of plasmonic material is complex, even negative for real part in some frequencies. By derivation from motion for an electron of the plasma sea subjected to an external electric field E, the permittivity can be expressed as [20]
2 p p 0 1 (1.4.1) () i
γ is damping effect, ωp plasma frequency of plasmonic material. Based on the characteristic of material, the FDTD (finite-difference time-domain) method is applied in the computer simulation to study the characteristics of nano antenna as well as antenna arrays.
The MIM (Metal Insulator Metal) diode, with the theory of tunnelling effect of quantum mechanics, is designed for the rectification of the optical energy from 10 antenna to DC current. Contrary to traditional theory, the tunnelling effect means an electron with energy tunnels through a barrier with potential, even the energy of the electron is lower than energy of barrier. The application of this theory in a MIM diode means that a stimulated electron emits from one metal into the insulator, then travels into the other metal. A biased voltage is applied to the two metals. Considering the effect of tunneling probability, Fermi distribution functions, and density of state, the relation of current density and applied biased voltage can be expressed as [111]
()/EkT mqkTe 1 z JVTEdE()()lndczz (1.4.2) 230 ()/EqVkTzdc 21 e
T(Ez) is the tunneling probability of electrons transmitting from one metal through the insulator barrier to the other metal; μ is the energy potential of the first metal; Vdc is the applied biased voltage; m is the effective electron mass; Ez is the energy of the tunneling electron, and q is the electron charge. Here a semiclassical model is applied to deal with electric current.
1.5 Significance and Value
This research aims to enhance the efficiency of solar energy harvesting with the theories of electromagnetics, antenna theory and photons. A solar energy harvesting system based on nano-antenna is researched; the energy harvesting and conversion effect are studied as well. Academically, nano antenna for receiving optical energy is examined and analyzed, and then a new design is developed for converting optical energy to DC energy. The results are used to identify an efficient way to collect clean
11 energy from solar radiation mainly in the optical and infrared spectrum, with high rectificication efficiency.
12
2 Literature Review
The notion of harvesting solar energy with nano-antennas can date back to wireless power transmission. In 1899, Nikolas Tesla was the first to conceive the idea of using microwave to transmit electrical power and to conduct the experiment at Colorado by building a gigantic coil. This coil was fed with 300kW electrical power resonating at
150kHz, and it successfully lit 200 incandescent lamps of 50W about 42km away from this coil via wireless transmission [22]. Tesla’s experiment has unshered in new area – that of wireless power transmission, from very low frequency up to very high microwave even optical band. The review in the following aims to provide a better understanding which helps to guide and inform the project in the thesis.
2.1 Development of Energy Transmission Systems
Basically, a wirelss power transmission system has two parts – energy transmission and energy receiving. In the solar energy system, the light source works as a conduit for the energy transmission, which suggests that it is crucial to examine the receiving part – energy harvesting system. The transmission of solar energy or optical energy is similar to that of microwave energy, but it is different from microwave
13 telecommunication, in which signal transmission errors should be avoided.
The structure of a rectenna system for wireless power transmission is shown in Figure
2.1.1; the antenna receives the wireless energy and transmits it to the rectifying device such as a diode for generating DC electricity. This is similar to the structure of a solar enegy harvesting system.
dipole antenna cap
diode Rload
Figure 4 - 2.1.1 Schematic diagram of a simple rectennna system
In 1964, W.C. Brown designed the first airborne microwave power generated helicopter based on the rectenna system. In the 1980s, RFID (Radio Frequency
Identification ) system appeared, using inductive antenna coupling for both power and data transmission at the frequency of 2.45GHz, with the help of integrated circuits and low power technology.
Another pratical application is the solar power satellite erected by Dr Peter Glaser in
1968 which used PV arrays on the satellite to collect solar energy, and then tranmit it 14 in microwave spectrum back to the earth for rectification by rectenna. The whole project was supported by DOE and NASA, which showed a very high efficiency rectenna whose lost energy was caused by the diode and the second harmonic power.
But several issues remained to be addressed, including ways to reduce the dissipated power back-radiated to space, minimize the cost, improve high gain antenna, and prolong work life. In the antenna array, aluminum sheets were used to form three-side waveguide for power transmission, snd the transimission frequency was from
2.45GHz to 35GHz in this freuqnecy range; the GaAs semiconductor material was developed for rectification.
Today fabrication technology has enabled research in the size of nanometer, leading to the development of nanometer-size systems. One of these is the nano atenna with nano rectification device for solar energy harvesting. The research aim of nano antenna is to enhance the near field for harvesting more energy in case of the scattering back to space. On the other hand, the research of rectification is focused on the semiconductor material for even higher optical spectrum rectification than those
GaAs for 35GHz energy rectification.
2.2 Receiving Nano Antenna
Nano antenna can be studied by drawing on the traditional antenna design theory.
However, the operating frequency causes dramatic changes of materials
15 characteristics. The metals are basically turned into a plasma state at optical frequency which is very similar to the gas plasma antenna at microwave band. The review of current nano antenna development is essential for developing a high-efficiency energy harvesting system.
The FDTD method is applied to analyze the nano antenna system with the metal deemed as plasmonic materials [16]. Based on the material characteristics, an equivalent circuit for nano antenna is proposed, with the results that when the imaginary part of material’s permittivity is greater than zero, the material can be deemed as either inductive or capacitive [17].
The characteristic of plasmonic material excited by optical energy allows the effective wavelength of a nano dipole antenna to be shortened [18]. Different from the traditional antenna surface current, the current of nano antenna is mainly the displacement current not conductive current, as shown in [19]. Besides, the definitions of optical input impedance and optical radiation resistance are commonly given based on the concept of traditional antenna theory. One important application of nano dipole antenna is the near field enhancement which can be realized by controlling the spacing of the gap [20]. Numerical simulation shows that the metallo-dielectric nano antnana can also achieve a higher gain and the electromagnetic radiation over a wide range of frequency and compact structure, though it needs high fabrication technology
[23]. Tiwari succeeded in coupling a dipole antenna with a MIM diode (Al-AlOx-Pt)
16 to form a sensor for infrared detection [24]. Different structures of nano antennas were also studied. For example the EI-Shenawee’s torus nano antenna was found to be capable of achieving a better near field enchanement at the selected resonant frequency [25]. Detailed explanation based on the analysis of permittivity is presented in [26] to help to understand the principle of enhanced directivity of nano antenna.
The geometry of nano antenna is studied in [27] with the result indicating that the largest emission enhancement is achieved when the ratio of length to radius of antenna is four-to-one. Strong field enhancement of nano antenna can provide a highly sensitive and compact uncooled infrared dectector [28]. Kim in his research also showed that by changing the nano gap of the dipole, the resonant property of nano antenna was altered [29].
The application of a nano antenna array in collecting optical energy proves a higher efficiency than traditional PV cells; the antenna can harvest 90% of the incident energy [4]. Some researchers focused on the periodically ordered metal nano antenna and found that in some selected frequencies, these arrays could generate enhanced electromagnetic field effects over a broad spectral range [21,22,23]. Field enhancement could be generated at a broad spectral range or in a selected frequency.
Besides, theoretically, it is proved that a plasmonic array can change the radiation field by being placed at different posisitons or modified geometric parameters
[59].The periodic array structure is important in tuning the optical response, plasmon-enhanced effects such as surface-enhanced raman scattering [24,25,26].
17
Similar research in dielectric nano antanena arrays shows that better performance than an isolated nanoantenna in high emission efficiencies by Pellegrini [30]. The nano array can also work as a frequency selective device, and its equivalent circuit has been proposed in [27]. The field enhancement effect can help to collect solar energy better by using an array than by a single nano antenna.
Carbon nanotube is also available in the design of nano antenna, and the characteristic of current distribution and radition pattern have been investigated in [31,32]. A nano antenna array made of carbon nanotube has been simulated and it is found that the antenna’s RCS (radar cross section) is negligible [33].
2.3 Energy Rectification
The nano antenna receives the optical power, and this optical power will be rectified to DC power. The metal-insulator-metal diode works as a rectifier to realize the conversion to DC power. The two metals on the outside determine the Fermi function of the current, while the middle insulator between these two metals determines the tunnelling probability of the electron to realize the rectification. This thesis is mainly focused on the investigation of the work functions of the materials and the thickness and the contact area of the insulator. Sanchez theoretically and experimentally proved the tunnelling effect as the rectification mechanism [34].
18
Compared with a Schottky diode which often works in microwave band, a MIM diode can rectify energy in even higher frequency such as terahertz optical energy [28]. As the current generated by the MIM diode depends on the analysis of the electron tunnelling through the three materials, some researchers focus on the algorithm of the tunneling matrix for probability [29,30,31]. Based on this, the rectification system was examined to find the equivalent circuit model [32,33]. Relevant techniques for the MIM diode fabrication have been shown in [35], whose operating frequency is in infrared spectrum; an array of MIM junctions in 94GHz has also been studied and fabricated [36]. The MIM diode coupled to a tunable antenna has also been investigated for the advantage of uncooled infrared detection especially for biological systems operation [37]. A thinner insulator can bring out more current density, which has been shown in [38]. The research results show that a 3nm thick insulator is much better than insulators of other sizes. Similar results also have been obtained and shown in [39] for the three-layer metal-insulator-semiconductor structure. A thinner insulator helps electrons to tunnel through the insulator. The MIM diode with strongly nonlinear current-voltage characteristics has been researched in [40] and it was shown that negative differential resistance can be appliled as infrared and optical frequency oscillators and amplifiers. Compared with the GaAs Schottky-barrier diode at optical and infrared spectrum, the MIM diode shows a more sensitive detection which means that the conversion of optical energy to low frequency energy is stronger [41].
Some researchers think that a double barrier may help increase the effect of tunnelling,
19 so by controlling the material and the thickness of the insulators; some results are given in [34,35]. Practical applications of the MIM diode have been successful in
60GHz with the high speed performance capability for signal trasnmission, vehicle collision avoidance and navigation, and national defense [42].
Based on quantum mechanics, different algorithms for calculating tunneling effects are studied, multistep potential approximation can be applicable for tunneling probability and current across arbitrary potential barriers [43]. In [44], the current responsivity is presented with the characteristics of the rectified current. Based on the transfer matrix method, the electrons transmission coefficient in arbitrary potentials can be calculated recursively. By discretizing the Schrodinger equation and tight-binding Hamiltonians, Mendez found that this algorithm can achieve a high accuracy rate even with not very small integration steps [45].
2.4 Conclusion
In this chapter, the development of wireless power transmission is first reviewed following the basic system structure. The system is composed of antennas and rectification diodes to generate DC current. As the wavelength of optical energy are in nano meter scale, the nano antenna is studied for this application, where another concept - field enhancement by antenna is introduced for improve the efficiency of harvesting energy. The Metal-Insulator-Metal diode and its basic principle tunneling
20 effect are reviewed for energy rectification after optical energy is harvested for conversion to DC current. The review of the development of WPT systems, nano antennas and rectification diodes serves to illuminate the scientists’ contributions, from which this investigation drew inspiration. In the field of directional field enhancement by antenna, broadband nano atnena and equivalent circuit require further improvement. All of these are scrutinized in this thesis.
21
3 Nano Antenna Design
Contrary to the traditional Photovoltaic cell, which directly seizes the photon and coverts it to the DC current, the nano antenna receives solar energy as electromagnetic energy then transmits it to the MIM diode for rectification. Hence, the role of nano antenna is very important in harvesting optical energy. In this chapter, the design of nano antenna is discussed.
3.1 Basic Parameters for Optical Antenna
In antenna theory, the high frequency AC current induced on the surface of conducting metal material determines the radiation pattern [46]. In microwave band, mostly the metal is considered as a perfect electric conductor (PEC), the current density Jz (in the unit of amperes/m) is induced on the metal surface and is given by
Jqv zsz (3.1.1)
2 where qs (in the unit of coulombs/m ) is the surface charge density with a uniform velocity vz (in the unit of meters/sec) in the z direction. If the metal is thin enough, ideally zero in thickness, the current can be expressed as
Iq v zlz (3. 1.2) where ql (in the unit of coulombs/m) is the charge per unit length.
22
The high-frequency AC current is time-varying, so the derivative of the current of equation (3.1.2) can be written as
dIzz dv ql q l a z dt dt (3.1.3)
2 where az is in the unit of meters/sec . As the current is on a wire of the length l, then equation (3.1.3) can be written as
dIdvzz llqlqallz dtdt (3.1.4)
Equation (3.1.4) is the basic equation about electric charge and current. This equation states the criterion for radiation, that is, a time-varying current or a velocity varying charge. To accelerate electron charges, the wire should be curved, bent, discontinuous or terminated [46]. Periodic charge acceleration or time changing current can be generated by oscillating the charges in a time-harmonic motion of a λ/2.
It is worth noting that moving charges induce current. If the velocity is constant, this is a DC current, and no radiation is generated along a straight wire. However, on a curved bent, discontinuous, terminated, or truncated wire, electromagneitc radiation occurs.
If a charge is in a continuous time-harmonic oscillation, it will form a high-frequency
AC current, so broadband radiation occurs. The external source can provide energy to
23 the charge to sustain this oscillation, so that a radiated field is formed. It is clear that the material and structure of an antenna determine the current distribution on the antenna.
3.1.1 Radiation Patterns
The radiation pattern is usually used to describe the EM energy radiated from or received by the antenna. Theoretically, the radiation and reception of the energy for an antenna is reciprocal [47,48]. And this reciprocal theorem may work in optical band [49]. Antenna radiation patterns are measured at every single frequency, polarization, and at each plane. The patterns are usually plotted in the form of a polar with a unit of dB as strength scale. In radiation patterns, data are normalized with its maximum value.
In Fig 3.1.1.1, a field pattern is presented. The θ component of the electric field as the function of the angles θ and φ, Eθ(θ, φ) (unit V/m); another φ component –of E-field as a function of the angles θ and φ, Eφ(θ, φ) (unit V/m) ; the phases of E-fields as a function of the angels θ and φ, δθ(θ, φ) and δφ(θ, φ) in the unit of rad or degree. It can describe the radiation pattern called the principal plane patterns, which is obtained by plotting the data over the xoz and yoz planes.
24
z θ Main lobe Eθ
Back lobe φ y x Eφ
Figure 5 - 3.1.1.1 Field pattern of a directional antenna for E or H field [46]
The angular beam width at the half-power beam width (HPBW) or -3dB beam width and the beam width between first nulls (FNBW) are two important parameters [48].
They describe the radiation pattern by using the normalizing field, as follows:
E (,) E (,)n (3.1.1.1) E (,)max
The half-power level appears at the angles θ and φ when Eθ(θ, φ)n is set as 0.707; and the normalized power pattern is given by
S(,) Pnn(,) (3.1.1.2) S(,)max
EE22( , )( , ) S(,) (3.1.1.3) Z0
2 S(θ, φ) is Poynting vector, in the unit of W/m . S(θ, φ)max is the maximum value of
S(θ, φ). Z0 is defined as intrinsic impedance of space, and the value of Z0 is 377Ω.
The power pattern is defined as the total power radiated by the source integrated over the surface of the sphere, as expressed in equation (3.1.1.4)
25
PSds(,)(,) (3.1.1.4)
The average power density over a sphere is then given by (3.1.1.5):
1 2 P( , ) P ( , )sin d d av 4 00 (3.1.1.5) 1 Pd( , ) (W/sr) 4 4
3.1.2 Radiation Intensity
Radiation intensity U is the parameter to measure the radiated or received power of an antenna in per unit solid angle, whose unit is watts per steradian or per square degree.
In normalized expression, power pattern Pn(θ, φ) can be described as the ratio of radiation intensity U(θ, φ) to its maximum value. Normalization of power pattern is given by:
US(,)(,) Pn (,) (3.1.2.1) US(,)(,) maxmax
3.1.3 Directivity and Gain
In antenna theory, directivity D and gain G are two main parameters describing antenna characteristics. The antenna directivity is the ratio of the maximum power density P(θ, φ)max to its average value in a sphere in the far field;
P(,) D(,) max (3.1.3.1) P(,)av
The directivity is a ratio whose value is less or equal to 1. It is usually described in the unit of dB.
26
So the directivity can be expressed as
P(,) 1 D(,) max (3.1.3.2) 11(,) P Pdd(,) 44 44(,) P max
Finally it is described by
4 D(,) (3.1.3.3) Pd(,) 4 n
In the transmitting mode of an antenna, if the energy transferred to the antenna is not radiated, it will make the material of antenna heated. In addition, impedance mismatch of the antenna with its power may reduce the gain. Gain is expressed as the ratio of the antenna maximum power density to a reference antenna multiplied by the gain of a reference antenna, in which the reference antenna usually is a dipole antenna:
P (AUT) GainGmax (ref. ant.) (3.1.3.4) Pmax (ref. ant.)
3.1.4 Antenna Field Zones
The EM fields radiated from or receiving to an antenna can be separated as two regions; one is near field or Fresnel zone and the other in a large distance away - the far field or Fraunhofer zone, as seen in Fig.3.1.4.1. The boundary between these two regions is determined by this radius R, which is given by
2L2 R (3.1.4.1) 27 where L is the maximum dimension of the antenna, λ is the wavelength of the frequency.
Near Field To Infinity Fresnel Region
2 R 2L Far Field L dipole Fraunhofer Region
Antenna’s boundary
Fresnel-Fraunhofer boundary sphere
Figure 6 - 3.1.4.1 Antenna regions for near and far field [46]
In the near field zone of an antenna, the longitudinal component of electrical field is dominant beause power flow is not fully radiated. It is also called inductance region, and the shape of the field pattern relies on the distance from the antenna.
In the far region, the dominant components are transverse to the direction in which the antenna radiating with all power flow. The shape of the far field pattern is independent of the distance.
3.1.5 Input Impedance
Input impedance is defined at antenna terminals, by the ratio of the voltage to current,
28 or the ratio of the appropriate components of the electric fields to magnetic fields.
Z RAAA j X (3.1.5.1)
RArL R R (3.1.5.2)
Rr is the antenna radiation resistanc, and RL is the loss resistance of the antenna.
The antenna impedance varies greatly according to its frequency, although the inherent impedance at an antenna feeder remains unchanged. When the antenna is well matched to its feeder within a frequency range, that frequency range is called the antenna impedance bandwidth.
3.1.6 Receiving Antenna Efficiency
The efficiency of an antenna is usually referred to as radiation efficiency, denoting the radiated power to the power received at the feeder from terminals. For receiving antenna, the efficiency means the received power of the antenna to the external power on the antenna. Antenna with a high efficiency can radiate/receive more power while an antenna with a low efficiency lessens the power because of impedance mismatching. The losses can be conduction losses and dielectric losses. The former are due to finite conductivity of a conductor and the latter are due to the conduction within a dielectric.
29
3.2 Numerical Method for Nano Antenna Design
Computer simulation is widely used in antenna design. Numerical algorithms derived from EM theory are very important in improving the efficiency of computation.
Normally, FDTD, MoM, FEM, GTD are commonly used in the antenna design analysis. FDTD is finite differential time domain method, which discretes the
Maxwell’s equations in time domain, replacing the partial differential equation with the differential equation to make numerical computation available [50]. MoM is the method of moments, which discretes the integral equation for numerical computation.
As the integral equation meets the requirements of the radial boundary condition,
MoM is commonly used to solve the open-boundary problem [51]. FEM is the finite element method, which discretes the functional variation equation to solve the eigen function in EM environment [52]. GTD is geometrical theory of diffraction, which applies the geometrical optical theory with the help of the scattering ray to solve the scattering problem usually in the higher frequency of microwave spectrum [53].
3.2.1 Brief FDTD Method
In FDTD method, time domain Maxwell’s equations in free space are expressed as
[54]
E 1 H t 0 (3.2.1.1) H 1 E t 0
30
E and H are spatial vectors. For simplification, one dimension equations of Ex and Hy
are discussed below,
E 1 H x y tz 0 (3.2.1.2) H 1 E y x tz0
Equation (3.3.1.2) depicts a plane wave with an electric component oriented in x direction, and its magnetic component oriented in y direction, travelling in the z direction. The equations 3.2.1.2 can be discreted in the following form,
EkEknn1/21/2()() 1 HkHknn(1/ 2)(1/ 2) xx yy tx 0 (3.2.1.3) HkHknn1(1/ 2)(1/ 2) 1 EkEknn1/21/2(1)() yy xx tx0
The equation 3.2.1.3 can be rearranged as,
t EkEkHkHknnnn1/21/2( )( )[(1/ 2)(1/ 2)] xxyy x 0 (3.2.1.4) nnnn11/21/2 t HkHkEkEkyyxx(1/ 2)(1/ 2)[(1)( )] 0 x
The meshed cell size ∆x is determined by the time step given in the following equation,
x t (3.2.1.5) 2 c0 where c0 is the speed of light in free space. In three dimensions, Maxwell’s equation is expressed by
D 1 H t 00 DE (3.2.1.6) H 1 E t 00 31
Considering a Yee cell (Fig 3.2.1.1) in FDTD algorithm, six scalar equations are generated as follows:
z
Hx
Ez Hy (i,j,k) (i,j+1,k) y Hz Ex Ey (i+1,j,k) x
Figure 7 – 3.2.1.1 Yee cell [55] used by a FDTD algorithm
Dx 1 H H y H x 1 Ey E z z tyz 00 tzy 00
D H y 1 H x H z y 1 Ez Ex tzx 00 txz 00
D 1 H y H x H 1 Ex Ey z z txy 00 tyx 00
(3.2.1.7)
By differentiating the following equation,
(3.2.1.7a)
32
H 1 Ex Ey z (3.2.1.7b) tyx 00 the finite difference approximations are generated as follows,
nn1/21/2 Dijzz kDij( ,,1/ k 2)( ,,1/ 2)
t nn [(1/Hijyy 2,,1/ kHij 2)(1/ k 2,,1/ 2) (3.2.1.8.a) x 00 nn HijkHijkxx( ,1/ 2,1/ 2)( ,1/ 2,1/ 2)] nn1/2 HijkHijkzz(1/ 2,1/ 2,)(1/ 2,1/ 2,)
t nn1/21/2 [(1/EijkEijkyy 2,1/ 2,)( ,1/ 2,) (3.3.1.8.b) x 00 nn1/21/2 EijkEijxx(1/ k 2,1,)(1/ 2,,)]
3.2.2 Frequency-Dependent Material in FDTD Method
In optical spectrum, charateristics of both metal and dielectric are quite different from what they behave in microwave or even lower frequency. The dielectric materials whose relative permittivity varies with frequency can be expressed as [56]:
Ne2 1 ()1 i rrr mi()22 12 000 (3.2.2.1)
Ne2 22 ( ) 1 0 r1 m ()() 2 2 2 2 0 0 0 (3.2.2.2)
Ne2 () r 2 m ()()22 22 0 00 (3.2.2.3)
χ is electric susceptibility, N is the number of atoms, per unit volume, m0- mass of electron, ε0 is the electric constant, γ - damping rate, ω0 – resonant frequency. Clearly, even up to the microwave band, the relative permittivity only has an effective real part whose imaginary part is zero. As the frequency goes up to an infrared, even optical
33 and ultraviolet frequency band, the imaginary part is not zero. The whole relative permittivity is a frequency-varying complex; both real and imaginary parts are positive, according to the measured data [57].
However, a metal in optical spectrum is different from a dielectric; the Drude model can be applied to describe the metal in optical frequency - plasmonic material with a negative real part of its permitivitty - which is different from that of a dielectric in optical frequency. The real part εr1 means the energy that can be stored in the material, while the imaginary part εr2 stands for the loss of energy in the material.
Dielectric material in optical band may dissipate power because of the positive imaginary part εr2 of permittivity. While the real part εr1 is larger than that in microwave band, which means it can make the effective wavelength smaller than that in lower band. As the real part εr1 is still positive, this makes the dielectric material as a resonator in optical band, which is different from plasmonic material whose real part is negative. As a result, no EM wave can penetrate into a plasmonic.
Metal is treated as a plasmonic material in optical spectrum. The plasma frequency, collision frequency, permittivity of a plasmonic material can be expressed as [58]
Ne2 p m 0 (3.2.2.4)
Similar to gas plasma, the Drude model is used to describe the plasmonic material.
The effective permittivity of a plasmonic material is complex, and can be expressed as 34
2 p 1 p 0 () i (3.2.2.5) where γ represents the damping effect [59]. And the conductivity is given by
2 0 p p i i (3.2.2.6)
To modify the FDTD method for a plasmonic material, the frequency-dependent permittivity data for the materials should be made available. Based on the Drude model, the 1st order approximation in one dimension is given by [60]:
D () () s z 1() jE z (3.2.2.7) where εs means ε(0), ε∞ means ε(∞), and τ is the Debye relaxation time constant.
The FDTD equation can be written as
tt 22 2 t EkDkDnnnn11( )( kE)( )( k ) s zzzz 222 ttt sss (3.2.2.8)
The 2nd order approximation in one dimension is expressed as,
2 () D () () 0 s z 22jE2 ( ) 0 z (3.2.2.9)
ω0 is the resonant frequency, and δ is the damping coefficient, the FDTD equation is then expressed by:
22 nn nn110 tt 22 44DEzz Ezz() k2 2 D 2 2 00t ss 2 t 2 t 2 t 2 2t 2 2 t 2 2 t 2 2 t 2 00DEnn11s 2t 2 2 t 2 zz 2 t 2 2 t 2 00ss (3.2.2.10)
35
With the computed electrical field, the voltage is obtained by the line integral as follows
V E d l (3.2.2.11)
Similarly, the current is calculated by integrating the H field along the closed path
I H dl (3.2.2.12)
3.3 Choice of Metal Materials for Nano Antenna
Metal is always as the material for traditional antenna, because of its conductivity and solidity. For a larger antenna even a reflector antenna, metal, such as copper and aluminum, can easily be manufactured into a large structure. Copper is used for micro-strip antennas operating at microwave spectrum. Materials such as gold, silver, copper and aluminum, are the commonly chosen for nano antennas. The characteristics of the four metals are discussed in the ensuing sections.
Gold
As a chemical element, gold is very stable and hard to be oxidized or acidized. But it can dissolve in aqua regia [61]. Besides, gold is very ductile and thin enough as a wire for semiconductor application. Its density is about 19.3g/cm3, relatively high comparing with other metals.
Because gold has a high concentration of free electrons, it is a good electrical
36 conductor. Besides its resistance to oxidation, gold is a versatile material in the application of electronics, such as wires. For example, an audio and video signal transmission cable can effectively restore the original sound of music. Gold is also applied in the switch contacts as it resists corrosion. Clearly, gold could be a choice of material for nano antenna.
Silver
Silver is also a good electrical conductor as well as thermal conductor, besides it is a soft material. Its density is about 10.49 g/cm3, and free form in natural. But it is also a byproduct in the manufacture of other metals like gold and copper. Silver is also ductile but harder than gold [62]. Silver can be diluted in nitrate solutions and form compounds.
Of all the metals, silver is the best one in electrical conductivity, and much better than copper. Although the cost of using silver as wires is much higher than that of other metals, it is still widely used in RF circuits for the improvement of conductivity.
Silver displays the lowest resistance among metals, and is considered the best material for thermal conductivity.
Copper
Copper is also a ductile material with high electrical conductivity and conducts thermal very quickly. But copper is easily oxidized to generate red CuO2, because it
37 dissolves in water and is influenced by the water particles in the air. The density of copper is about 8.96 g/cm3 [63].
Because of low cost and high electrical conductivity, copper is widely used in the electrical circuit. Copper is the material for the cable of electrical power transmission, though the best preferable is aluminum. Copper cables are widely applied in power generation, transmission, distribution; telecommunications, electronics circuit, and numerous types of electrical application [64].
Aluminum
Aluminum is the most abundant metal on the earth. It is soft and ductile. One of its characteristics is that it is lighter than other commonly used metals, about 2.70 g/cm3
[65]. Aluminum has the ability to resist corrosion. As it is nonmagnetic and not easy to ignite, it can serve as a reflector.
The lightness of aluminum makes itself widely used in electrical and thermal industry.
Because the superconductivity of aluminum makes it very suitable for electrical circuits, such as electrical transmission lines for power distribution, it is widely used in semiconductiors.
38
Electrical Thermal Density resistivity (20 °C) conductivity
Gold 19.30 g/cm3 22.14 nΩ·m 318 W/m·K
Silver 10.49 g/cm3 15.87 nΩ·m 429 W/m·K
Copper 8.96 g/cm3 16.78 nΩ·m 401 W/m·K
Aluminum 2.70 g/cm3 28.2 nΩ·m 237 W/m·K
Table 1 – 3.3.1 Comparison of material properties for different metals
Considering all the above factors, aluminium is selected as the material for making nano antennas, because of its lightness, good conductivity, low cost, and resistivity.
As for efficiency, nano antennas made of aluminium can achieve as high as 90% compared with other materials from simulation results, which is very important for optical energy coupling.
3.4 Conclusion
In this chapter, the antenna theory is reviewed for the design of nano antennas. First, relevant parameters, such as radiation patterns, intensity, directivity, field zones, are discussed, becaused they can measure the energy for harvesting. Then the impedance matching is outlined in relation to antenna input impedance to improve the efficiency of energy transmission. The Modified FDTD algorithm is discussed, giving due consideration to plasmonic materials at the optical spectrum which are different from
39 the metral as PEC at microwave band. Finally, the material for antennas is chosen following the three main selection criteria: lightness, availability and reasonable resisivity for application.
40
4 Disk Nano Antenna for Field Enhancement and Broadband
The aim of solar energy harvesting is to collect the energy by antennas in an efficient way. In this chapter the disk nano antenna is selected to study the directional field enhancenment and broadband energy collection. Both radiation pattern (reciprocal principle, radiation equals collection) and impedance are two important parameters discussed in this chapter.
4.1 Directional Field Enhancement of Dielectric Nano Optical Disc Antenna Array
4.1.1 Analysis of Electrical Field Distributions of Nano Antenna
A dielectric disc antenna operates as a radiative dielectric cylindrical resonator, and a resonator is a section of a waveguide. The cylindrical dielectric waveguide is first analyzed. The Dielectric material in optical band can be analyzed with electromagnetic parameter permittivity. The dielectric waveguide theory is a good method to analyze the rod, disk nano antenna made of dielectric. To solve the problem of a dielectric waveguide or resonator [66], the inner electric field distribution should be carefully studied by taking the boundary condition into account. In microwave
41 band, an antenna made of a dielectric material is good at radiating EM waves [67] with the help of dielectric waveguide theory. However due to its negative real part of permittivity, a plasmonic material cannot be considered as a waveguide because no
EM wave in optical band can penetrate into it below plasma frequency.
For the waveguide in microwave band, the EM wave cannot penetrate through metal.
Because of the skin depth, only surface current is found on the metal. On the other hand, for a dielectric, EM wave can penetrate into it. Usually inner E-field will be excited. With this inner E-field, displacement current will be generated according to
Maxwell’s equations. It is the displacement current that determines the radiated field
[68,69].
For the dielectric waveguide, E and H are continuous across the boundaries, that is
EE HH (4.1.1.1) binnerboutbinnerbout which is quite different from the metal waveguide in microwave band.
According to [71]
2 2 2 kkc (4.1.1.2) where propagation constant
γ=α+jβ (4.1.1.3)
α is wave attenuation constant and β is wave phase constant. The cutoff wavenumber is given by
42
kc=(λ/λc)k (4.1.1.4) wavenumber by
k2=(2πf)2με (4.1.1.5) and permittivity by
ε=ε0εr (4.1.1.6)
The wave phase constant β can be derived as
2 (2 f ) 01 rc 1 ( / ) (λc=2π/kc) (4.1.1.7) and wave attenuation constant is given by
(/) / (1(/)) 2 (4.1.1.8) rrc21
For a dielectric material in optical band, r1 >0 and r2 >0. So the value of β depends on the wavelength of the EM wave, as shown in the equation.
εr Waveguide L εr
εout εout (a) Cylindrical Rod (b) Disc Resonator/Antenna
Figure 8 - 4.1.1.1 Cylindrical rod and disc resonator/antennas
In a dielectric circular waveguide [71], if the attenuation constant
22 qf(2) out (4.1.1.9)
(εout is the permittivity of the medium cladding around the dielectric material,
Fig4.1.1.1) is an imaginary number, the EM wave can be effectively radiated from the 43 dielectric material to the out medium. Otherwise, if q is positive number, the EM wave is confined within the waveguide.
Besides, the cutoff wavelength λc (for β) has a relationship with frequency, permittivity, β, and the transmission mode in the circular waveguide [71]. The fundamental transmission mode is HE11, unlike TM01 in the traditional metal waveguide in microwave band. The first three transmission modes in the cross-section dielectric of the cylindrical waveguide and resonator are plotted in the Fig 4.1.1.2, which also exists in a dielectric circular rod.
(a)HE11 (b)TM01 (c)TE01 E-field H-field
Figure 9 - 4.1.1.2 EM wave distributions in cross section of the dielectric cylindrical waveguide and resonator, (a) Fundamental transmission mode HE11; (b) 2nd mode TM01 ; (c) 3rd mode TE01
Normally, a cylindrical resonator is a segment with the length-L of λ/2 from cylindrical waveguide, in Fig 4.1.1.2. When the cylindrical resonator works as a disc antenna for radiating an EM wave, L is much smaller than λ/2, and it is often defined as δ, which does not influence the application of the theory. The E-field distributions in the transverse cross section determine the equivalent magnetic current [70] on the 44 top surface of disc antenna, which is treated as the radiating source. The disc antenna made of dielectric material mainly operates in the mode of HE11δ [71] while the other modes (TE01δ and TM01δ) may also be generated. So in the transverse, the E field distributes differently in different modes, quasi-straight lines with maximum value in the center for HE11δ, radial lines for TM01δ, and circular lines for TE01δ (reaches its maximum value in the center for TE01δ and TM01δ), Fig 4.1.1.1. For a single disc antenna, the mode HE11δ can effectively radiate directionally while the radiation patterns of other modes has wide angles. When the disc antennas form an array, they can enhance the effect of directivity with the mode of HE11δ. Besides, the disc antenna can achieve a wide-band frequency directional field enhancement when the Q-factor of it is lower.
4.1.2 FDTD Simulation of Dielectric Nano Optical Disc Antenna
Array for Field Enhancement
When the dielectric nano optical antenna array is applied for directional field enhancement, it is better to excite the energy as only one mode within the dielectric resonator, because different modes have different E-field distributions which are not easy to control directionally. Disc antenna arrays are modeled to prove the directional field enhancement via FDTD simulation. The material is silicon in the frequency of visible light, whose permittivity is quoted from [72]. The exciter is coupled to the dielectric antenna array by locating it in a place where the electric field is maximum.
45
The excited frequency is set at 500THz to monitor the directional effect of broad band.
The radius of the disc is set according to the EM theory - half wavelength of center frequency 500THz in the dielectric material, λ/2,
/ 0 r (4.1.2.1)
λ0 is the vacuum wavelength, for complex permittivity, real part is only considered.
1-disc 2-disc 4-disc 9-disc
16-disc 25-disc
Figure 10 - 4.1.2.1 Configuration of the dielectric nano optical disc antenna arrays
The single disc is first simulated based on the FDTD method; three basic different modes are excited to observe the effect of directional field enhancement. The radius for the single disc is chosen as 76nm, which is half wavelength of 500THz in Si, and
50nm for the thickness-L. The simulated results for directivity are 2.82dBi for mode
HE11 in cross section, 1.41dBi for TE01 and 1.28dBi for TM01, Fig 4.1.2.2. As the maximum value of directivity is in the center of cross section for the mode HE11δ, this makes it more amenable to directional field enhancement.
46
Figure 11 - 4.1.2.2 Directivity in the E-plane and H plane of a single disc
Therefore based on the FDTD method, all the discs in the antenna arrays are stimulated in the mode of HE11δ simultaneously, which renders the directivity of the array easily controllable. The square arrays, 2-disc, 4-disc, 9-disc, 16-disc and 25-disc,
47 in Fig4.1.2.3, are modelled with varying spacings between edges of discs, λ0/10, λ0/4,
λ0/3, λ0/2, 2λ0/3, 3λ0/4, λ0, λ0 as 600nm is the vacuum wavelength for 500THz, shown in Table 4.1.2.1.
spacing λ0/10 λ0/4 λ0/3 λ0/2 2λ0/3 3λ0/4 λ0
2-disc 4.3 5.3 5.9 6.3 6.1 5.8 5.3
4-disc 5.2 4.1 4.7 5.7 6.1 5.7 5.4
9-disc 9.1 14.1 15.3 12.4 11.7 12.7 14.9
16-disc 14.5 19.4 27.4 28.9 27.9 22.8 22.1
25-disc 22.1 24.3 37.9 59.5 59.9 41.7 45.9
Table 2 - 4.1.2.1 Comparison of directivity in antenna arrays with different numbers of discs and different spacings between the edges, unit is dBi
Figure 12 - 4.1.2.3 Comparison of maximum directivity of different antenna arrays
48
Table 4.1.2.1 shows the results of directional field enhancement- directivity for these models. Approximately, in the same spacing, the directivity increases when the number of discs in array increases. For an antenna array, the directivity shows a periodic trend; if the discs are close to each other, the directivity goes down, which is same when the discs separate to λ0. The maximum directivity appears to be around
λ0/2; this can be explained with mode HE11δ, where the minimum value of it is at the edge of cross section, but it is half of λ0/2 is λ0/4. According to the EM theory, a wave changes the phase to the opposite direction when it passes quarter-wavelength; the field value shifts from minimum to maximum, or vice versa. Around this spacing, in the center of two discs, the field is directionally enhanced. The greater the number of the discs, the higher the directivity is. From the table 4.1.2.1, it can be concluded that the directivity exponentially increases as the number of discs increases in the array.
For 2-disc and 4-disc arrays, the directivity is almost the same. In 4-disc array, the parallel discs enhance the field, but diagonal discs may cancel part of the enhanced field because of their spacing.
4.1.3 Conclusion
In this section, the property of dielectric material is first analyzed, especially in relation to the complex number of the permittivity which determines the transmission of the EM wave. Then the traditional dielectric waveguide and resonator theory is
49 used to analyze the E-field in the transverse plane. By choosing the intrinsic phase constant β and the transmission modes, the EM wave can effectively radiate into the free space. Finally, the idea of directional field enhancement by a dielectric nano optical antenna array is effectively proved by FDTD simulation.
4.2 Broadband Optical Antenna Design and Equivalent Circuit
4.2.1 Broadband Optical Antenna Design
For the present fabrication technology, the planar structure is the most common type for nanoantenna. Though in microwave band, volume antennas are designed [73] to achieve UWB antenna performance, planar circular and elliptical antenna can also make to achieve broad-band radiation [74] [75]. This is because the structure, metal disk-dielectric-metal ground, forms a resonator to meet the requirement of frequency, and size of the metal disk is determined by the center frequency. Additionally, the circular disk enlarges the area of the surface current for radiation, which meets the need of a broadband antenna. Given this, this idea can be applied to antenna designed at optical frequency with modification of scaling down the size [76].
Here, a structure for broadband optical antennas is proposed, Fig 4.2.1.1. In the inner circular disk is metal, whose radius is half-wavelength, or written as λ0/2 of the center frequency, (fhigh+flow)/2. The middle ring is made of a dielectric material, whose inner
50 radius is λ0/2 and outer radius is 3λ0/2. The outer ring is still metal, whose inner radius is 3λ0/2 and outer radius is 2λ0. The whole structure is very similar to those in microwave band. However, the permittivity of materials works differently, which hightlights a need to examine it by using LC resonance. In optical band, as the real part of permittivity for metal is negative, metal can be treated as inductance, while the real part of permittivity for a dielectric material is positive, which still works like a capacitor [77]. The circular disk is better structured for radiation at different frequencies more effectively than rod or particle structures, because it can realize broadband resonance.
t
Figure 13 - 4.2.1.1 Three-layer broadband optical antenna (inner-metal, middle-silicon, outer-metal)
For microwave broadband antennas, the permittivity of the material is a real constant.
In optical band, the permittivity changes with the frequency. Consequently in different frequencies, the wavelength varies greatly in a dielectric material. In plasmonic material, the EM wave is evanescent with a very small depth from the surface. This frequency-variation permittivity will lead to a varying radiation pattern.
51
4.2.2 Equivalent Circuit for Broadband Optical Antenna
A plasmonic material works as an inductor and a dielectric material as a capacitor. An equivalent unit of a single plasmonic disc is composed of 4 elements – inductance and resistance loss for the plasmonic material, capacitance for air, and radiation resistance,
Fig 4.2.2.1. The imaginary part of permittivity stands for the loss of EM energy, which can be treated as a resistance.
L t Pi Rpi RRad Cair
Figure 14 - 4.2.2.1 Equivalent circuit for a single plasmonic disc
When a dielectric ring is added around the plasmonic disc, the capacitance and resistance loss of the dielectric ring should be taken into consideration.
Rd L t Pi Rpi RRad Cd Cair
Figure 15 - 4.2.2.2 Equivalent circuit for a single plasmonic disc with a dielectric ring
52
Rd LPo LPi t Rpi RRad
Cd Rpo Cair Exciter
Figure 16 - 4.2.2.3 Equivalent circuit for a broadband optical antenna
The equivalent induntance is determined by the real part Re[εp] of permittivity of the plasmonic material in its operating frequency fi. The induntance is the function of f (fi,
Re[εp], Vp), where Vp is the volume of the plasmonic material. The equivalent capacitance is set by the real part Re[εd] of permittivity of the dielectric material in its operating frequency fi. The capacitance is the function of f (fi, Re[εd], Vd), where Vd is the volume of the dielectric material. The equivalent resistance is determined by the imaginary part Re[ε] of permittivity of materials in its operating frequency fi. The resistance is the function of f (fi, Im[εp], Im[εd], Vp, Vd), Vp. The equivalent circuit shows the resonance of the broadband antenna, which has a relation with the radiation pattern.
4.2.3 FDTD Simulation for Broadband Optical Antenna
Based on the theory mentioned in the previous sections, a broadband optical antenna is designed to operate in the frequency band from 400THz to 600THz. The basic configuration is shown in Fig 4.2.3.1. The center frequency of the operating band is
53
500THz, so the vacuum wavelength is 600nm. Thus, the radius of the inner circular disk is 300nm with the material of gold-Ag. The inner radius of the Si circular ring is
300nm, while the outer radius is 450nm. As for the outermost gold circular ring, the inner radius is 450nm and the out radius is 600nm. The thickness of the disc is 50nm.
The model is excited to observe the effect of broadband radiation by using the FDTD method. The permittivity of silicon changes with frequency. Drude model is then used with the center frequency of 500THz [78]. The simulated result is shown in figure
4.2.3.1.
54
Figure 17 - 4.2.3.1 Radiation pattern of a broadband optical antenna by using the Drude model for material; E- and H Planes
400THz 450 THz 500 THz 550 THz 600 THz
Drude Model 4.51dB 5.34dB 5.84dB 5.05dB 4.94dB
Table 3 - 4.2.3.1 Comparison of gain of a broadband optical antenna by different methods
The simulations in Drude model show the same varying trend with the frequency. As the frequency increases, the radiation field pattern becomes narrow in E-plane, while in H-plane the radiation pattern becomes wide. This is the same as the change of antenna gain. This phenomenon can be explained with frequency-related permittivity, which changes the wavelength of the EM wave in the material to have an effect on the phase of the EM wave in travelling. Thus, the radiation pattern changes, just like a pad rotating 90 degrees around the axis.
55
4.2.4 Conclusion
In this section, the theory of broadband optical antennas is proposed. In optical frequency, a metal turns into a plasmonic material which is quite different from traditional PEC. Then the microwave UWB antenna is scaled down to realize the optical UWB antenna. This broadband optical antenna is explained with an equivalent circuit. Finally, an example is provided with FDTD simulation to illustrate the theory.
4.3 Conclusion
In this chapter, the disk nano antennas are studied to reveal its relative broadband characteristics. The results show that the disk antenna array can realize the directional field enhancement, and a disk nano antenna is capable of realizing broadband optical energy collection.
56
5 Optical Input Impedance of Nanostrip Antennas
This chapter investigates antanna input impedance for realization of impedance matching. Here, a nano dipole strip atanna is chosen for the analysis of the input impedance and the relationship with the antenna geometry.
5.1 Optical Nanostrip Antenna
The nano antenna under investigation is a dipole made by two nanostrip arms.
Fig5.1.1 shows the geometry of this optical nanostrip antenna. Two strip arms separated by an air gap of distance g have a length of b and a rectangular cross-section area of a × t. Practically, strip arms with rectangular cross section may be favorable to a nanoantenna as they are easy to fabricate, analyze. This nanostrip antenna on a substrate will be discussed in the following section. The nanostrip antenna is fed at its center gap. It may be connected to a source - a receiver or a waveguide at optical frequencies.
gap gap strip arm strip arm a t b
(a) side view (b) top view
Figure 18 - 5.1.1 Geometry of a nanostrip antenna
57
A nanoantenna is usually fabricated with the metal in THz spectrum. Gold is a common material as it has a very stable chemical state. Aluminum can also be the possible material, and is used for the nanostrip antenna discussed in this thesis. In microwave band aluminum is deemed as PEC (perfect electrical conductor). In optical band, the characteristic of metal is described by the Drude model. The complex dielectric function of aluminum is given by
2 ε(f) = ε0(εr1+iεr2) = ε0[ε∞-fp /f(f+iγ)] (5.1.1) where ε∞ is 1 and fp is 3.551PHz and γ is 19.9THz [7,79]. Note that aluminum has interband transitions in the optical regime, notably a strong one at around 800 nm [80].
To avoid such interband transitions the nanostrip antenna discussed in this thesis operates at the wavelength of 600nm or even smaller. Aluminum has a relatively larger (negative) real part εr1 of the complex dielectric function. This real part is associated with the energy stored in the material, while the imaginary part εr2 is with the energy (ohmic) losses in the material.
5.2 Optical Input Impedance and Optical Radiation Efficiency
Input impedance and radiation efficiency are two fundamental parameters used in RF antenna design. They are also fundamentally important for a nano antenna if it serves as a solar energy harvesting device or a device for transmitting a signal to remote
58 devices. Optical input impedance is expressed as the ratio of the exciting optical voltage applied to the antenna gap over the total current at the feeding point. This current includes both displacement current and ohmic current flowing into the antenna
[81]. Note that the nanoantenna may be connected to an external load with impedance other than 50Ω. Radiation efficiency instead of total antenna efficiency is discussed.
Optical radiation efficiency is expressed as a ratio of the power radiated over the power delivered to a nanoantenna.
The FDTD codes were employed to compute both optical input impedance and efficiency of radiation of a nanostrip antenna. The nanostrip antenna is illuminated with a beam of light which is linear polarized. The nanodipole has two strip arms with the dimensions: width is 15nm. Thickness is 15nm, and length is 50nm, which are separated by a 15nm-long gap. These values are considered as reference values of the nanoantenna for the discussion in the following sections. Fig5.2.1 shows the simulation results of optical input impedance and radiated efficiency. As seen in the figure 5.2.1, the nanoantenna can be viewed as a capacitor at low frequencies, which is physically formed by two strip arms. When the frequency increases, the inductive reactance contributed by the strip arms becomes significant. Eventually a series resonance (short circuit oscillation, LC oscillates in series connection) is observed at
564THz, under which the reactance is identical to zero. This is quite similar to that observed in a RF dipole antenna. When the frequency increases further, the capacitance determined by the centre gap couldn’t be ignored. As it is connected in
59 parallel with the series LC circuit (inductor and capacitor resonant circuit) a parallel resonance (open circuit oscillation, LC oscillates in parallel connection) yields at
787THz at the feeding point. These resonances are clearly illustrated by the dash line in Fig5.2.1 (a).
(a) Optical input impedance
(b) Radiation efficiency
Figure 19 – 5.2.1 Optical input impedance and radiation efficiency of a nanostrip antenna in reference size
60
An equivalent circuit of the nanoantenna for this pair of resonances is given in
Fig5.2.2. In the plasmonic strips the total current density is determined by -jωr1E+
ωr2E. These two items are the displacement and ohmic (conductive) current densities, and can be expressed by Eaajωl+ Eaar where l and r the per-unit inductance and resistance. Clearly, both lumped-circuit parameters L and Rloss of the plasmonic strips are connected in parallel, as indicated in Fig5.2.2. Note in Fig5.2.1 that there is a ripple of impedance at the frequency of 950THz. This corresponds to a localized parasitic resonance caused by gap within the nanostrip antenna. As the radiation efficiency drops significantly at this frequency, it seems a series resonance with a serial strip resistance. The parasitic resonance may be due to the parasitic inductance arising from the ohmic (conductive) current path in the strips.
Cair L R Zin loss Cgap
Rrad
Figure 20 - 5.2.2 Equivalent circuit of a nanostrip antenna for optical resonances
Table5.2.1 shows the optical resonant frequencies, input resistance and radiation efficiency of this nanostrip antenna. The input resistance is equal to 31 under the series resonance, and is lower than those of a RF dipole characteritics. The input resistance under the parallel resonance is relatively large and reaches 1260. Note 61 that the nanostrip antenna retains an efficiency of 0.74 (74%) under the series-resonance frequency and 0.8 (80%) under the parallel-resonance frequency.
This may allow the nanostrip antenna to be operated under the series resonance for a low-impedance external device, and under the parallel resonance for a high-impedance external device.
Parameters 1st series resonance 1st parallel resonance
Frequency f (THz) 564 787
Resistance R (Ω) 31 1260
Efficiency ecd 0.74 0.80
Table 4 – 5.2.1 Antenna parameters of a nanostrip antenna
The power delivered to a nanoantenna includes both radiated power and ohmic losses within the nanoantenna. The ohmic losses are primarily due to the ohmic or conductive current within the plasmonic material. Note that the ohmic current in the nanostrip antenna is weak, as the imaginary part r2 of the dielectric function is much smaller around the resonant frequencies. As a result, the ohmic losses within the nanostrip antenna are small and radiation efficiency retains high. Note that the imaginary part r2 of the plasmonic material decreases with increasing frequency. The radiation efficiency at the parallel-resonance frequency is relatively higher. Under these resonances the displacement current in the antenna strips is dominant.
62
It is noted [82] that the effective wavelength on a plasmonic dipole is shorter than its wavelength in free-space. For this nanostrip antenna, the ratio of effective wavelength to free-space wavelength is 0.21. It is significant short due to the negatively large εr1 of the plasmonic material at the optical frequency.
5.3 Antenna Parameters against Nanoantenna Size
Antenna parameters of a nanoantenna are affected by its physical size. For a dipole antenna with strip arms it is possible to vary both strip dimensions and gap distance to achieve optimal performance of the nanostrip antenna.
5.3.1 Gap Distance g
Computer simulation of a nanostrip antenna with the gap distance g varying from
10nm to 20nm has been conducted. The optical input impedance is presented in
Fig5.3.1.1, and a summary of resonant frequencies, input resistance and radiation efficiency are given in Table5.3.1.1. Note that increasing the gap distance leads to an increase of the overall antenna length. This causes both resonance frequencies to decrease, as shown in Table5.3.1.1. The change of these frequencies, however, is not significant (e.g., less than 1.4%). This indicates that the resonant frequency is basically determined by the strip length of the nanoantenna. When the distance g is increased from 10mm- to 20nm the input resistance increases approximately by 18%
63 under the series resonance, and remains almost unchanged under the parallel resonance. Table5.3.1.1 also indicates that varying the gap distance does not affect the radiation efficiency of a nanostrip antenna. The efficiency remains a value around
0.7-0.8 at these resonant frequencies.
The simulation results disclose that it is possible to modify the dimension of a nanostrip antenna for impedance matching with an external feeding or loading network at nano scales. The nanostrip antenna has the dimension: width of antenna is
15nm, length of one pole of antenna is 50nm and gap of antenna is 10nm, which is set as reference antenna, for example, it has an optical input resistance of 28 at the frequency of 567THz under the series resonance. By increasing gap distance to 20nm, the input resistance of this nanoantenna is increased to 33. However, there is also a change of resonant frequency, which is 1.4% approximately.
(a) Resistance
64
(b) Reactance
Figure 21 - 5.3.1.1 Optical input impedance of a nanostrip antenna with a variable gap distance from 15nm to 25nm (a=15nm, t=15nm, b=50nm)
Distance 1st series resonance 1st parallel resonance
g (nm) f (THz) Rin() ecd f (THz) Rin() ecd
10 567 28 0.70 787 1212 0.79
15 564 31 0.74 787 1260 0.80
20 559 33 0.76 778 1235 0.80
Table 5 - 5.3.1.1 Optical resonant frequency, input resistance and radiation efficiency against gap g
5.3.2 Strip Width a
Cross-section area of an antenna strip can be adjusted by varying strip width a. In the simulation the strip thickness is fixed to be 15nm, and its length remains 50nm. The 65 gap distance is equal to 15nm in all cases. Fig5.3.2.1 shows the input impedance of the nanostrip antenna with the strip width varying from 10nm to 20nm.
Table5.3.2.1shows the summary of input resistance and radiation efficiency under the resonant frequencies. Note that increasing the strip width leads to the increase of both resonant frequencies. The input resistance increases as well under the series resonance, but decreases under the parallel resonance. The change of input resistance can reach
263% when strip width is changed from 10nm to 20nm, while the change of resonant frequency is just around 14%. As shown in Table 3, the radiation efficiency increases with increasing strip width, and can reach a value of 0.84 under the parallel resonance.
(a) Resistance
66
(b) Reactance
Figure 22 - 5.3.2.1 Optical input impedance of a nanostrip antenna with a variable width from 15nm to 25nm (g=15nm, t=15nm, b=50nm)
Strip width 1st series resonance 1st parallel resonance
a (nm) f (THz) Rin() ecd f (THz) Rin() ecd
10 514 30 0.59 722 2050 0.68
15 564 31 0.72 787 1260 0.80
20 594 32 0.79 821 780 0.84
Table 6 - 5.3.2.1 Optical resonant frequency, input resistance and radiation efficiency against strip width a
5.3.3 Strip Length b
As indicated in the previous sections, optical input resitance at resoance freqienceis can be adjusted by changing the dimentsions of antena strips. One side effect of such
67 technique is that the resoannt frequenceies will be changed as well, although the change may not be significant. Fortunately, the nanoatenna can be tuned to resonate by varying its strip length, just like a traditional RF metal antenna.
Fig5.3.3.1 shows the optical input impedance of a nanostrip antenna when the strip length is changed from 50nm to 60nm. Other geomtrical parameters of the nanoantenna are fixed to be the corresponding reference values. Similarly to the gap distance, increasing the strip length leads to a decrease of resonant frequency and an increase of input resistance under the series resonance. Note that the resonant frequency is more sensitive to the strip length than the gap distance if the change of input reisistnace remains the same. Under the parallel resonance the resonant frequency decreases but the input resistance increases. This is different from that observed by varying the strip width and the gap distance. The radiation efficiency remains the value of around 0.7 under the series resonance, and around 0.8 under the parallel resonance, table5.3.3.1.
(a) Resistance 68
(b) Reactance
Figure 23 - 5.3.3.1Optical input impedance of a nanostrip antenna with a variable strip length from 45nm to 55nm (a=15nm, t=15nm, g=15nm)
Strip length 1st series resonance 1st parallel resonance
b (nm) f (THz) Rin() ecd f (THz) Rin() ecd
50 564 31 0.72 787 1260 0.80
55 527 32 0.70 740 1295 0.78
60 495 33 0.69 699 1330 0.79
Table 7 - 5.3.3.1Optical resonant frequency, input resistance and radiation efficiency against strip length b
Also, there are a certain frequency, below this frequency at a fixed frequency, the longer the strip length the larger the impedance. But above this frequency, the smaller the impedance of a longer strip length.
69
Table2.5.3.3.3.2 shows the pattern of resonant frequency variation and input resistance variation when the antenna sizes change. The symbol of or indicates that resoant frequency f or input resistance R increases or decreases. The symbol of
or represents a more signficant change than or . The symbol of reprsentes there is almost no change of a parameter. It is noted that under the series resonance increasing strip area in cross section will cause both the resonant frequency and input resistance to incrase. However, the increase of the resonance frequency can be compensated by increasing strip length, which leads to a further increase of input resistance. This allows a fine adjustment of optical input resistance without changing the resonant frequency.
Serial resonance Parallel resonance Change of Parameters f1 Rin,1 f2 Rin,2
Increase of Gap Distance g
Increase of Strip Aera a×t
Increase of Strip Length b
Table 8 - 5.3.3.2 Optical resonant frequency and input resistance variation against strip sizes
Under the parallel resonance the pattern of the parameter change by varying strip area cross section is similar to that by varying strip length (b). Note that the input change made by varying strip area cross section is much significant than by varying strip
70 length (b). Impedance adjustment without changing the resonant frequency is still possible. For example, the nanostrip antenna has the dimension: width of antenna is
20nm, length of one pole of antenna is 50nm and gap of antenna is 15nm, it has a parallel-resonance frequency of 722THz and an input resistance of 2050. Increasing strip width to 15nm leads to the input impedance of 1260, but the resonant frequecy is increased to 787THz. To retain the resonant freqeucny of 722THz, the strip length is incerased to 58nm with the updated input impedance of 1335. This new input impedance is just 65% of the original value.
5.4 Conclusion
This chapter discussed the optical antenna parameters of nanostrip dipole antennas made with a plasmonic material – aluminum at optical band. Using the FDTD simulation, the optical input impedance and radiation efficiency of nanostrip antennas were investigated. An equivalent circuit was presented for the nanostrip antennas at optical resonances. It is found that the optical input resistance is in the order of 30 at the series-resonance frequency, and can reach 2000 at the parallel-resonance frequency. The input impedance can be adjusted by changing the geometric parameters of antenna strips. With simultaneous changes of strip area and strip length or gap distance, optical input resistance can be adjusted to match impedance with an external feeding or loading circuit without changing the optical resonant frequency. It is found again that the optical radiation efficiency does not vary greatly when the
71 dimensions of the nanostrip antenna vary. It retains approximately the value of 0.7 at the series-resonance frequency, and 0.8 at parallel-resonance frequency.
72
6 Rectification Device
In this chapter, rectification made by metal-insulator-metal diodes is discussed in detailed. The following issues are addressed, device structure and materials, energy conversion function, tunneling effect and equivalent circuit when the MIM diode is connected to external devices. The conversion efficiency is discussed by comparing with the traditional PV device.
6.1 Metal Insulator Metal Overview
The research of the MIM (Metal Insulator Metal) diodes stemmed from metal-insulator transitions, as metal has good electrical conductivity, while an insulator does not has such a characteristic for electrical charges. This was first proposed in 1920s by Bethe, Sommerfeld and Bloch [83]. This transition of an electron from a metal to an insulator or vice versa is achieved by external excitation such as electric field, pressure and so on. Later it was found that insulators placed between the lowest empty band and highest filled band would be the material for semiconductors because of thermal excitation of the electrons [84].
After 1960s, the research of MIM focused on the electron current with the theory of tunneling effect [85,86,87]. Scientists used the Sommerfeld and Bethe model with the
73 help of numerical calculation to find the relationship of current density and parameters such as electric field and temperature etc. Considering the effect of quantum mechanics, Tien and Gordon expanded the research to interaction of MIM with an electromagnetic field and proved the rectification of MIM for EM waves [88].
Recently, especially in the past decade, most scientists focused on the material and fabrication technology for manufacturing MIM devices [89,90,91]. In [92] the author made the MIM diodes to rectify microwave energy to DC power, in which the insulator of the MIM diodes should be very thin for electrons to tunnel successfully
[93]. And some scientists tried to make the MIM diodes to work at higher frequency, such as 30THz [94]. The thinner the insulator, the higher frequency energy it can rectify.
6.1.1 Device Structure
Usually, a MIM diode is composed of three parts: a top metal layer as cathode; an oxide middle layer as an insulator barrier and another metal layer on the substrate as anode. The electron can tunnel from the cathode through the insulator to the anode, so the insulator layer must be thin enough, e.g., – 1nm to 5nm.
74
base metal layer top metal layer as anode as cathode insulator barrier
Figure 24 – 6.1.1.1 Structure of a MIM diode
6.1.2 Functions
For two metals with different work functions, electrons can flow through the very thin insulator layer until the two Fermi levels reach the same value – a state of equilibrium, and then no other electron will flow.
current electron
insulator insulator insulator cathode anode cathode anode cathode anode a b c
Figure 25 - 6.1.2.1 States of a MIM diode:
(a) Equilibrium; (b) Forward bias; (c) Reverse bias
There are two kinds of bias – forward, if the positive voltage is applied to anode, the
75 electric potential of anode will rise relatively to cathode; and reverse negative voltage is applied to cathode in order to raise the potential of cathode. The potential difference between the insulator and anode is fixed, while the height of potential of the insulator to the cathode depends on the external applied voltage. In the forward and reverse bias, the electron flows from cathode to anode when the applied voltage varies, because this depends on the potential energy expressed in figure 6.1.2.1. The energy level of an electron in the metal (cathode) is distributed exponentially according to
Boltzmann’s law [95]. So the current in a MIM diode is nonlinear to the applied voltage for rectification.
Unlike traditional theory, the insulator allows the electrons to transmit through the tunneling effect. The electron with potential above the Fermi level of cathode, is able to travel through the cathode without losing any energy to the anode. Aluminum oxide is widely used as insulating material, for manufacturing the MIM diode. For example,
Al-Al2O3-Au means three materials for the anode-insulator-cathode, respectively.
The characteristic of an insulator barrier plays a key role in the tunneling performance of a MIM diode [96]. Besides, the two metals with uniformity will show the better electron transport property than those amorphous metals. And the insulator must be slowly oxidized to be very thin for fluctuation-free emission [97].
76
6.1.3 Difference from Traditional PV in Working Function
A traditional photovoltaic solar cell has suffered from several limitations such as thermodynamic efficiency, quantum efficiency, and fill factors. So considering the
Carnot efficiency, the theoretical limit of a standard solar cell is 30%.
[98,99,100,101,102,103.104] Optical energy excites the electron of a traditional PV solar cell to pass through the band gap of a semiconductor, where some photons with low energy has not enough energy to excite electrons to pass through the band gap.
While other photons with equal to or higher energy than the band gap of the semiconductor can make the electron to pass the band gap to the conduction band. So part of optical energy loses because of the photon with low energy.
A MIM diode harvests the optical energy via rectification not by generating a free carrier. This makes the theoretical conversion rate to be approximately 100%. [105]
The detail of conversion efficiency calculation is given. The MIM diode can make such a high efficiency mainly because the device is much smaller than the photon wavelength and can respond very fast to keep the oscillation of optical energy. As each diode rectifies individually which results in a different voltage and current, the interconnections of them should be considered with these factors.
6.2 Tunneling Effect of the MIM Diode
77
The theory of tunneling effect is usually applied to calculate the tunneling current in the MIM diodes. In this theory, the optical-frequency current induced on the antenna feeds to the metal of the MIM diode. The alternating current is rectified by this device, and the DC current flows out from metals to the load, as shown in figure. [106] In tunneling theory, the tunneling probability of an electron is an important parameter to be determined first.
electron
Insulator Metal 1 Metal 2
Figure 26 – 6.2.1 Schematic diagram of an electron tunneling in a MIM diode
The uncertainty postion of an electron can be expressed in the wave function of a plane wave:
E jkxjt x e (6.2.1) where k is the wave number of the plane wave, Ex is the electron energy (one dimension is only considered), and is normalized planck constant. As it has been mentioned in the previous section, when the wave comes at the intereface of two materials, boundary conditions should be met; the electric field shoud be continuous at zero order and first order derivation of the wave function. While at the same time, the monmentum of an electron is conserved during the tunneling procedure. 78
The probability of the electron’s position is uncertain, but it can be decided by the wave function, which means the position of the plane wave. The magnitude of the plane wave at the two sides of the interface is studied, though the energy potential of the material can be arbitray. In figure 6.2.2, the energy potential is meshed to calculate the tunneling probability. The calculation of transfer matrix is applied here, when the electron travels through an interface of two sections, a transfer matrix is calculated.
L R
)
eV (
- - - - Energy
x1 x2 Length(nm) Figure 27 – 6.2.2 Diagram of energy distribution
For example, a single interface between two adjacent meshed slice is selected. The plane wave on the two side of the interface can be expressed as
AeBejkll xjk x l (6.2.2) jkrr xjk x r CeDe among which -k stand for the forward wave and k stand for the reverse wave. The momentum k is defined as
k2 me ( E V ) / 2 (6.2.3) 79
E is electron energy, while V is the material potential, m is the mass of electron, and e is the charge of an electron.
A matrix is introduced to analyze the wave function at the interface with the consideration of boundary conditions mentioned in the previous section.
j()() kkxjlrlr kkx ()()kkekkelrlr 22kkmm ACCll1112 (6.2.4) j()() kkxjlrlr kkx BDD ()()kkekke mm2122 lrlr 22kkll
In this transmission process, it is assumed that no reflection from right of the interface to left, which means D equals 0, it can be found that the tunneling probability is
2 C 1 PT 2 (6.2.5) Am()11
In detail, in an arbitrary shape of a barrier, the divided parts can be viewed as many rectangular barriers approximately, expressed in figure 6.2.3:
Δx
Aj
Bj
xj-1 xj
Figure 28 – 6.2.3 Analysis of one meshed part in the process of electron tunneling
At the interface xj-1, considering the continuity at boundary,
80
AeBe jkxjkxjrjjrj1,11,1 jrjrjr1,1,1, (6.2.6) jkxjkxj,1,1 ljj lj j,,, lj lj lCeDe
jrjl 1,, (6.2.7) jrjl 1,, '' It follows AeBeCeDejkxjkxjkxjkxjr1,11,1,1,1 jjr jj l jj l j jrjrj1,1,,, lj l (6.2.8) jkxjkxjkxjkxjr1,11,1,1,1 jjr jj l jj l j kAekBekCejkDejrjrjrjrj1,1,1,1,,,,, lj lj lj l
jkjrj1, x 1 jk jrj 1, x 1 jk jlj , x 1 jk jlj , x 1 kAejrjr1, 1, kBe jrjr 1, 1, kCe jrjl 1, , kDe jrjl 1, , (6.2.8) jkjrj1, x 1 jk jrj 1, x 1 jk jlj , x 1 jk jlj , x 1 kAejrjr1, 1, kBe jrjr 1, 1, kCe jljl , , kDe jljl , ,
jkj1, r xjk jjl 1, xjkjjl 1, jjl xjk 1, jjl xjk j 1, x 1 2()()kAekekj1,1,1,,,1,,, r j rjeCkek rj lj lj eD rj lj l
1 j()() kk xjk kx j1, r j , l jj 1,1,1 l j r j (6.2.8) Akkj1,1,,,1,,, rj eCkk rj lj lj rj[()()] lj eD l 2k jr1,
1 j()() kkj1, r xjk j , l kx jj 1,1,1 l j r j Bj1,rj rj lj lj rj[()()] lj lkk1,,,1,,, eCkk eD 2k jr1,
j()() kkj1,,1,1,1 r xjk j l kx jj l j r j Akkj1,1,,1,,, rjekk rj lj1 rj lj eC l ()() (6.2.9) BD2k j()() kkj1,,1,1,1 r xjk j l kx jj l j r j j1,, rj l jr1, ()()kkj1,,1,, rjekk lj rj l e
Similarly, at the interface of xj j()() kj, r k j 1, l x j jk j 1, l k j , r x j Ajr, 1 ()() k jrjl , k 1, e k jrjl , k 1, e C jl 1, (6.2.10) BD2k j()() kj, r k j 1, l x j jk j 1, l k j , r x j j, r jr, ()()kj, r k j 1, l e k j , r k j 1, l e j 1, l Note that in the same section, it is
kkjrjl,. (6.2.11)
xj=xj-1×Δx (6.2.12)
The total tunneling probability of the system is determined by the multiplication of all the single tunneling probability
PPPPPP (6.2.13) TTTTTT1 2 3nn 1
81
A simple case is illustrated in figure 6.2.3. In this case, metal 1 is made of Cr, metal 2 is Pd, and the insulator in the middle is Cr2O3. The reference energy potential is set at
10eV. Based on the characteristic of the material, the potential energy at x1 is 10.74eV; and the potential energy at x2 is 12.36eV. An external biased voltage is set on the left.
)
eV 1.36 (
Energy 0.74 -1 10 x1 x2 Length(nm)
Figure 29 – 6.2.3 Energy distribution of a Cr-Cr2O3-Pd diode
The figure 6.2.4 shows the tunneling probability of an electron in the MIM diode of
Cr-Cr2O3-Pd based on the previous mentioned theory. When the potential energy of the electron is higher than 12eV, the probability of tunneling through the barrier is 1.
82
Figure 30 - 6.2.4 Tunneling probability of an electron in the MIM diode
(Cr-Cr2O3-Pd)
6.3 MIM Diode Without/With External Time Harmonic E-field
The external applied electric field on the MIM diode can be time-independent (DC) and time-harmonic (AC). Physically, the electron acquires energy form an external field for tunneling. The characteritisc of a MIM diode under DC and AC applied field are discussed separately.
The definition of some basic physical parameters should be mentioned before discussion. In the microscope of a MIM diode, several characteristic parameters of the materials have to be introduced with the help of the energy band diagram shown in
83 figure 6.3.1.
Evacuum
χ ΦM EBG
ΦB
M1 I M2
Figure 31 – 6.3.1 Diagram of energy bands
The barrier height, ΦB, is defined as the potential difference between the Fermi energy of the metal and the band edge where the majority carriers reside, in the figure 6.3.1 can be expressed as
BM (6.3.1)
ΦM is the potential barrier height from metal to insulator or the value equals work function, χ is the electron affinity at the metal and insulator interface. EBG (Band Gap) is the difference from the valence band’s top to the bottom of conduction in the insulator.
When an external voltage is applied on the MIM diode, the potential of the material in the device will be changed microscopically. The whole process can be divided into three states: free state; Vdc=0V state; and Vdc≠0 state.
84
J(Current Density) J(Current Density)
17.5
)
V e
( 15
y
g r
e Metal2 n 12.5 T(E) E T(E) n(E) n(E) Metal1 Insulator Metal2 Metal1 Insulator Vbias 10 3nm 3nm no external with external voltage biased voltage biased
Figure 32 – 6.3.2 Electron tunneling through a voltage biased insulator
Free State
In this state, there is no external biased voltage being applied on the two metals. The metal, insulator, and metal are considered as separated items independently. Electrons can escape from the metal with the least energy needed, and the Fermi energy (Ef) of the metals is highest, or Ef is closest to the Evacuum. So the potentials of the two metals are not on the same level.
Evacuum
I2 I’20
M1 F
M1 I1 I M20 M 2
Figure 33 – 6.3.3 Diagram of free state
Vdc=0V State
A MIM diode with zero DC voltage applied means that the whole circuit is closed, or 85 the two metals are short circuited. This is very important because the closed circuit makes the potential of the two metals same. In the figure 6.3.4, the energy rise of
Metal2 from M20 to M21 can be expressed in equation 6.3.3, Ef1 is the potential energy for M21, and Ef2 is the potential energy for M20, shown in the figure 6.3.4. The amount of rising is the same as that of insulator, mathematically, in equation 6.3.2. The energy potential of Metal1, and Insulator I are fixed, the potential energy of Metal2 changes, as described in the following figure,
I’20-M21=I2-I1 (6.3.2)
I’21-I’20=M21-M20=ΔEf=Ef1-Ef2 (6.3.3)
I 21 I2 I 20 M1 F M21 M1 I1 I M20 M2 Vdc=0
Figure 34 – 6.3.4 Diagram of Vdc=0V state
Vdc≠0 State
When a DC biased voltage is applied to the MIM diode, voltage of metal 1 is still considered being fixed, the energy change take places on metal 2. The potential energy of M21 changes to M’21, as shown in the figure 6.3.5. The potential energy of insulator on the right changes, but the relative difference is fixed. The biased voltage only changes the potential energy of metal2.
M’21-M21=qVdc (6.3.4) 86
I’’21-I’’22=ΔEf’ (6.3.5)
I’’20-M’21=I2-I1 (6.3.6)
I 21
I 20
I2 M 21 I qVdc M1 20
M1 I1 I M21 M2
Vdc 0
Figure 35 - 6.3.5 Diagram of Vdc≠0 state
6.3.1 Unilluminated MIM Diode
The MIM diode under the DC voltage biased is first studied. It is also called unillumination, because no time harmonic electric field is applied on this diode. For simplicity, the one dimension tunneling effect is discussed to find the relationship of current density and DC biased voltage.
Here, the MIM diode of Cr-Cr2O3-Pd is chosen as the reference for analysis, in which the insulator part (Cr2O3) is 3nm thick.
87
J(Current Density) J(Current Density)
17.5
)
V e
( 15
y
g r
e Metal2 n 12.5 T(E) E T(E) n(E) n(E) Metal1 Insulator Metal2 Metal1 Insulator Vbias 10 3nm 3nm no external with external voltage biased voltage biased
Figure 36 – 6.3.1.1 Energy band diagram of a MIM diode
The external voltage can be applied on either metal 1 or metal 2. As shown in the figure 6.3.1.1, the tunneling current is same.[106] So the DC voltage is assumed to be applied on the metal 2, in which the electron flows from metal 1 to metal 2 while the current flows in the opposite direction. According to the definition, in a second, the amont of positive charge cariers flowing through a point affects the current density.
The tunneling probability and the concentration of charge carrier N(E) also determine the current density.The Fermi function is used, which means the electron occupies an energy level at energy E in a thermal equiliburium system, in which EF is Fermi energy, k is Boltzmann constant, and T is the absolute temperature in equation
(6.3.1.1). As shown in the figure 6.3.1.1, the external DC voltage is applied on the metal 2, so the Fermi function can be written as
1 1 fE() , fE() (6.3.1.1) 1 EE f 2 E() Ef qV dc e kT 1 e kT 1
88
Figure 37 – 6.3.1.2 Fermni function distribution
The output current density after the electrons have tunneled through the insulator, can be defined as
JqNEv EPE()()() T (6.3.1.2) in which the velocity of electron is v(E) and the charge is |q|; PT(E) is tunneling probability, the concentration of charge carrier N(E) is determined by density of state
(DoS) and Fermi function of electrons tunneling from metal2:
N( EDoS )( )( EfE ) 2 (6.3.1.3)
The density of state can be expressd by electron’s effective mass and energy level with Planck’s constant h.
82 mE3 DoS() E dE (6.3.1.4) h3
Note that the electron’s kinetic energy can be written as 89
1 EmvdEmvdv 2 (6.3.1.5) 2 So the density of state can be further written as,
8m3 DoSEvdv() 2 (6.3.1.6) h3
DoS(E) can be expressed as [106], velocity is composed of vx, vy, vz,
22m3 DoSEdv(), dvdvdvdvdE (6.3.1.7) hm3 xyzyz the expression of concentration of charge carrier N(E), considering the integration of all energy levels, becomes
4m2 NvfEdEdv()() (6.3.1.8) xxh3 2
Here, the current density expression of electrons tunneling from metal 2 to metal 1 is written as
4mq2 JPEdEfE2 12 dE3 Txx()() (6.3.1.9) h 0 Ex The tunneling current density from metal 1 to metal 2 has a similar expression, so the total expression of current density in the system is expressed as,
4mq2 JJJPDCTxxdc E2 11 dEf 221 E qVf3 E dE ()[ ()( )] (6.3.1.10) h 0 Ex The final expression becomes
4 m qkT J 2 DC h3 E E qVE qV x f dcf dc (6.3.1.11) eeekTkTkT PT( EdE xx )[lnln] E qVE qV 0 Ex qVqVdcdc f dcf dc eeeekTkTkTkT
90
6.3.2 Illuminated MIM Diode
When a MIM diode works in the optical spectrum, the quantum effect of the electromagnetic field must be taken into consideration. Because the magnitude of photon energy is comparable to that of biased DC voltage, the influence of optical energy is non-negligible.
Semi-classical theory is applied in the analysis of a MIM diode when it is exposed to optical energy [111]. In this theory, the electric field of incident optical energy should be quantumized with time-dependent perturbation theory. The nano antenna receives the optical energy, then transmits the energy to the MIM diode. During the whole process, in the first phase, the efficiency of power coupled to the antenna is η1; while in the second phase, the power coupled to the MIM diode is η2. In the whole process, this theory only considers the absorption and stimulating emission caused by optical energy. The spontaneous emission is not included.
The following is the Schrodinger wave equation with the semi-classical theory [106]: (,)rt iHr t e ˆ (,) (6.3.2.1) t e ˆ In the equation, H is the Hamiltonian operator and e is the electron wave function including both spatial and time dependent components. The complete system
Hamiltonian can be separated into two parts: time independent un-perturbed tern H0 and a time dependent perturbed term H’(t)
ˆ H( H0 H '( t )) (6.3.2.2) 91
The Schrodinger wave can be solved by using a numerical method. Recast it into vector notation then introduce the matrix formalism of quantum mechanics. iHHt ('()) (6.3.2.3) t 0 The orthonormal eigenfunctions |n> of the unperturbed system can be used to express the eigenfunction of the perturbed sytem |Ψ> as follows,
cn () t n (6.3.2.4) n1
Replace the Schrodinger wave equation with the eigen function of the perturbed system, the inner product with eigenvector m.
m ictnmHHtctnnn( )('( ))( ) 0 (6.3.2.5) t nn11
For simplification of the equation 6.3.2.5, the orthonormality is applied between eigenvector m and n. So the equation is reduced to
ictHctnmnn( )( ) (6.3.2.6) t n1 where cn is the expansion coefficients and Hmn is the Hamiltonian matrix.
The Hamiltonian matrix can be separated into unperturbed and perturbed parts.
1 0 00 HHHH1112131 n 000 HHHH 2 2122232 n
HHHHHmnn0 00 3 3132333 (6.3.2.7) 0 0 0 Hmmmmn123 HHH
As the perturbation part can be zero, only the unperturbed system is left for eigen energy. In the perturbation matrix, the elements along the diagonal can effectively shift the unperturbed energy levels to some fixed potential. The off-diagonal elements produce transitions between energy levels or quantum jumps. The first task in solving 92 the reduced equation is to calculate the explicit form of the system Hamiltonian for a
MIM diode coupled to a sinusoidal perturbation.
The Hamiltonian describing the interaction between a classical electromagnetic field and an electron of mass me and charge q is written as
1 2 H pˆ qA V() r (6.3.2.8) 2me where A( r , t ) is the vector magnetic potential, pˆ is the momentum operator , and
Vr() is the static electric potential. [107] A common electromagnetic gauge is introduced here - A 0 to simplify the analysis. Then the Hamiltonian is separated into the time dependent and perturbation terms:
2 2 2 ˆ pˆ q q A H H0 H()() t V r A pˆ (6.3.2.9) 22me m e m e
Usually, the magnitude of the perturbation is very small, so the quadratic term of the perturbed Hamiltonian may be neglected.
pˆ 2 H0 V() r (6.3.2.10) 2me
q HtAp()ˆ (6.3.2.11) me
The perturbed term can be casted into matric notation
q Hmn A p mn (6.3.2.12) me where pmn is the component of momentum. The vector potential for a plane wave is sinusoidal:
A( t ) | A0 | cos( k r t ) (6.3.2.13) 93
Dipole approximation is supposed before substituting the expression for the vector potential into the system Hamiltonian. This dipole approximation assumes that the spatial dependence of the field can be neglected. So the equation 6.3.2.13 can be written as the following because the space term and the phrase of the electromagnetic wave is set as zero:
AtAt()||cos() 0 (6.3.2.14)
The electric field E can be derived by partial difference of vector magnetic potential,
A EV (6.3.2.15) t
This equation can further be simplified by reducing the potential energy term,
A E (6.3.2.16) t
The direction of the vector potential is in parallel with electric field, so the vector potential can be expressed with electric potential
EAtEt|| sin()||00 sin() (6.3.2.17)
So the vector potential can be written with the magnitude of electric field magnitude 1 A()|| tEt cos() (6.3.2.18) 0 Substituting the expression of vector potential A into the perturbation Hamiltonian matrix element of the equation yields
q HEtpmnmn || cos()0 || (6.3.2.19) me
For the dot product of the equation, its result is not zero only when the vector momentum is parallel with the electric field. Next, convert the momentum matrix element form momentum space to real space. 94
ime mpnmHxnimxxemnmn[,]||||0 ˆ (6.3.2.20)
Inserting the momentum matrix element into the perturbation Hamiltonian matrix, then the complete perturbation Hamiltonian matrix element yields. || HqxEt mn |||| cos() (6.3.2.21) mnmn 0 It can be simplified as
HqVtmnmn cos() (6.3.2.22)
The non-zero perturbation elements only appear on the diagonal terms. So the EM field can be equivalent to the situation that applying a sinusoidal voltage to the structure that modulates the unperturbed single electron energy levels in relation to the optical radiation field.
According to the reference [108], the expansion coefficients have a Bessel function depending on the electric field magnitude
qVph cn J n J n () (6.3.2.23) where Jn stands for the nth order Bessel function. The perturbed electron wave function may be casted into the current voltage expression for the semiclassical illuminated J(V) curve [108] and the positive portion of the summation represents absorption and the negative portion represents the stimulated emission.
qVph J()() VVJJdcphnDC Vn dc (6.3.2.24) n q
95
6.4 Conversion Efficiency of Solar Energy
6.4.1 Conversion Efficiency of MIM Diode
In calculating the conversion rate of a MIM diode, the analysis begins with the responsivity (not resistance), which is derived from the second order of Taylor series and is expressed by [109]
1I '' RV() (6.4.1.1) 2'I VV op
Under illumination, it is expressed as [110]
J() VJ 2 VJ ( V )() J q qq RV()DC ILLUM (6.4.1.2) 1 VIJ VJ V ()() 2 qq
In which ΔJDC-ILLUM means the change of DC current when illuminated. Considering the nonlinearity of MIM diode: J(V + qEph) is much great than - J(V - qEph), in which
Eph is photon energy, and the expression of JV() has been shown in equation q
6.3.2.24. The quantum conversion rate is the product of R(V) multiplied with photon energy, approaches 90% for an unbiased MIM diode when J(qEph) is 20 times - J( - qEph). [111]
96
J(0)
Eph
Figure 38 – 6.4.1.1 Current density v.s. biased voltage
The current density J (V) (the expression has been shown in equation 6.3.2.24) can be a function of responsivity R(V), fig 6.4.1.1. The output power is
PEJArea () (6.4.1.3) outph q while the inicident power from single photon per time is
qEph Pin (6.4.1.4) Tph
2 Tph is photon’s time period, usually described as Tph [106]. So based on the qEph equation (6.3.2.24), the conversion rate can be expressed as
P JqArea(/ )(/ TJqArea ) E out phph (6.4.1.5) Pqin 2
6.4.2 Conversion Efficiency of PV
The traditional PV cell conversion efficiency is defined as maximum power point (Pm)
2 2 per incident optical energy (E, in W/m ) and surface area (Ac in m ) of solar cell [112]
97
P m (6.4.2.1) EA c
Considering quantum efficiency, the maximum conversion effieciency is less than
29%. This is because in traditional PV theory, the number of excited electrons in semiconductor depends on the incident photons with sufficient energy. Those photons with lower energy cannot excite electrons across the bandgap, which is part of the energy that cannot be converted. Besides the photon with higher energy at the same time heat the condution band edge which is another part of energy loss that fails to be converted.
6.5 Selection of Material for the MIM Diodes
In the selection of material for the MIM diode, according to the DC voltage biased current density equation (6.3.1.11), the Fermi function and the tunneling probability should be taken into consideration in calculation for maximum DC power output.
Obviously, the characteristic of MIM diode materials determines the output tunneling current, especially the materials of two metals which determine the tunneling probability.
The metals at the two sides of a MIM diode are always different, which means they have different work functions. The insulator is always made of the oxide of one metal at either side. [106]
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Material Workfunction (eV)
Ag 4.26
Al 4.28
Au 5.1
Cr 4.5
Nb 3.99
Ni 5.15
Pd 5.12
Pt 5.65
Ti 4.33
Table 9 – 6.5.1 Materials’ workfunction [113]
Using the workfucntion of materials provided in the table 6.5.1, the results of tunneling current for several cases are given in figure 6.5.1. These cases have different combinations of metal and insulator, as shown in figures 6.5.1.
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Figure 39 – 6.5.1 Comparision of current density of a MIM diode under 660THz
In the figure 6.5.1, the current densisy is presented in a logrithemic scale. The MIM diode with different structures are biased under the DC voltage from -3V to +3V and
660THz optical illumination. The first picture in the figure 6.5.1 shows the curves of current up to 1013A/m2; while the second picure shows the curves of current density up to 102A/m2. In this comparision, it is clear that the difference of two metals’ work functions determines the output tunneling current density. The lower of the difference of two workfunctions, the more output current will be generated. For example, the 100 combination of Nb (3.99eV) and Ag (4.26eV) lead to the maximum output current.
On the other hand, the combination of Cr (4.5eV) and Pt (5.56eV) leads to the minimum output current. Besides, the thinner the insulator will also lead to a reatively higher current density. Of course, it cannot be zero which means that no tunneling effect for rectification in figure 6.5.2, shows the variation of current density generated with different thickness of the insulator.
Figure 40 – 6.5.2 Current density comparison for the insulator with different
thickness
Based on the equation 6.4.1.5 and considering the model Cr-Cr2O3-Pd, Eph is the 101 muplitcaiton of planck constant with frequency (600THz ~ 700THz); the cross-section area of the MIM diode is 7.5×10-16m2 (width is 25nm and height is
30nm); the conversion rate for the MIM diode is 93% in average in the frequency range of 600THz ~ 700THz, shown as in the figure 6.5.3. The curve is not stable and swings around 0.93 because the current equation is non linear. And a comparison of conversion efficiency for different materials is made in table 6.5.4, clearly the material determines the conversion efficiency.
Figure 41 - 6.5.3 Quantum energy conversion efficiency of the MIM diode -
Cr-Cr2O3-Pd
660THz Cr-Cr2O3-Pd Cr-Cr2O3-Pt Cr-Cr2O3-Cr Nb-NbOx-Ag efficiency 0.93 0.79 0.42 1
Table 10 – 6.5.4 Comparison of conversion efficiency for different materials
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6.6 Equivalent Circuit of the MIM Diode
Usually, the equivalent circuit for a MIM diode is composed of resistance and capacitance [5], as shown in figure 6.6.1.
RMIM
CMIM
Figure 42 – 6.6.1 Equivalent circuit model of a MIM diode
The impedance of the MIM diode results from the effect of electrons tunneling through the potential barrier; and the capacitance is formed by the two metals of the insulator layer, which can be approximated by using the equation for a parallel plate capacitor A C (6.6.1) MIMr 0 d
εr is the relative dielectric constant of the insulator, A is the contact area and d is the thickness of the insulator layer. Usually the AC resistance of the MIM diode is expressed by the following equation [114]
E 2 ph q R (6.6.2) MIM EE JVJV()() phph DC DCDCqq DC
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6.7 Conclusion
In this chaper, the metal-insulator-metal diode is first reviewed at the aspects of structure and characteristics. The tunneling probability is studied in detail. Then the
MIM diode excited by DC voltage and optical energy is examined to study the output current density. The conversion efficiency of the MIM diode is presented, and compared with with the traditional PV cell. Materials and thickness are selected. By calculation, the quantum conversion efficiency of the MIM diode is found to be around 93%. Finally the equivalent circuit for the MIM diode is provided, which will be used for the discussion in later chapters.
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7 Characteristics of Rectenna System with Rectification Excited by Plane Wave
In this chapter, the characteristics of a complete rectenna system, including the optical antenna and the MIM diode for rectification (these two devices comprise rectenna), is mainly discussed using FDTD simulation. Particularly the changes caused by the interaction of different parts are investigated. The analysis is made by exciting the rectenna with a plane wave. This is because the rectenna system operates under receiving mode. In this case, the antenna works as a voltage source, and the MIM diode is employed to provide the DC current.
7.1 Rectenna System for Optical Energy Receiving
Traditionally, especially in microwave band, the metal as PEC is used for antenna design. In a transmitting antenna, the distribution of the current, fed to the antenna from a source, determines the far-field pattern. While in a receiving antenna, the metal interacts with the electromagnetic field and high frequency current is induced on the surface of the metal. Parameters, such as impedance, gain, efficiency etc, show the characteristics of the receiving antenna.
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Impedance of an antenna depends on the voltage and current at the feed of the antenna.
The real part of the impedance is associated with the power to be radiated or absorbed, while the imaginary part is the power stored within the near field. Efficiency is normally used for antennas as the percentage of power radiated to the antenna to the power absorbed by the antenna [115]. The loss of the energy received is mainly caused by reflection of antenna, material loss, and impedance of antenna unmatching to other device such as the MIM diode.
When the frequency goes up to optical band, several hundreds of THz, even beyond visible spectrum, the characteristic of metal is totally different from that at microwave band. At this range, the metal turns into a frequency-dependent plasmonic material, which can be described by the Drude model. For optical energy, the metal works like the dielectric [116]. Both the induced current and displacement current exist, the displacement current in the metal is dominant, and plays a role like those conductive current in microwave band [117]. The traditional parameters can be adopted in the design of nano antenna with metal as the antenna material. As the antenna receives optical energy, and transmits the energy to the feed, the antenna is usually deemed as a voltage source.
A MIM diode is used for the rectification of optical energy received by the nano antenna. The MIM diode has a three-layer structure; the two metals are made by different materials and have different energy levels. The middle is made by insulator
106 which is usually an anodic oxidation of one metal. According to quantum theory, an electron can tunnel through the insulator from one metal to the second when an external field is applied. [118] If the MIM diode is excited by optical energy such as illuminated by an optical wave, the metal parts should also be considered with the drude model.
Obviously the MIM diode can be viewed as a DC current source controlled by the AC voltage supplied by the antenna. After rectification, the DC current flows from the
MIM diode to the external load.
The geometry of the complete rectenna system is shown in the figure 7.1.1, it is comprised of a nano antenna and a MIM diode – receiving the optical energy to generate direct current power. The nano dipole antenna is selected. It is the simplest antenna and the result can be extended to the antenna with other structures. Two strip arms separated by an air gap of distance g. The rectifier (a MIM diode) occupies the whole gap. The strip arms have a length of b and a rectangular cross-section area of the muplication of width a and thickness t. The MIM diode has the same cross-section size as the strip arm. The optical antenna and MIM diode are mounted on a substrate made of silicon. In order to avoid the effect of the THz wave to DC line, the DC line is fabricated within the substrate, and then the DC current will flow along the DC line after being rectified by the MIM diode. The MIM diode is placed in the gap of antenna and tightly connects with antenna arms in order to decrease the possible loss
107 of optical energy. The DC line is embedded in the substrate to avoid the influence of optical field and to transmit the DC current only.
b b a MIM dp t t DC Line depth substrate
a DC Line length
DC Line length
Figure 43 – 7.1.1 Rectenna system
Aluminum is chosen as the material of the nanostrip antenna. In microwave spectrum, aluminum is deemed as PEC; but when the frequency comes to optical band, aluminum turns into a frequency-dependent plasmonic material. It is described by the
Drude model, with ε∞ is 1 and fp is 3.551PHz and γ is 19.9THz. Note that aluminum has interband transitions in the optical regime. To avoid such interband transitions the nanostrip antenna discussed operates at the wavelength of 600nm or below.
The rectenna operates in receiving mode, and is exposed to the solar radiated wave.
According to solar energy distribution of spectrum [119,120,121,122],a plane wave, setting the electric field as 1000V/m and polarized parallel to the nanoantenna, is assumed to illuminiate the antenna. Electric field distribution is investigated to analyze the resonance and impedance of the nano-antenna. The geometry of receiving
108 antenna is same as the transmitting antenna under gap excitation. The receiving antenna resonant at 600THz is excited by a plane wave; energy excites the arms of the antenna.
7.2 Rectenna System by Gap Excitation
7.2.1 Reference Optical Antenna
First, a reference model of the nano optical antenna with the structure given in Fig.
7.1.1 is chosen. The antenna has the following dimensions: the gap in the antenna is
20nm, the thickness of antenna is 30nm, the width of the antenna is 25nm, and the length of one pole in the anntenan is 61nm (Fig. 7.1.1). Without the substrate it oscillates at 600THz under gap excitation - the antenna is embedded in a 200mm thich substract. Fig7.2.1.1 shows the FDTD simulation results of optical input impedance and radiation efficiency under the gap excitation. Optical input impedance is expressed as the ratio of the exciting optical voltage applied to the antenna gap over the total current at the feeding point. This current includes both displacement current and ohmic current flowing along the antenna strip [117]. Optical radiation efficiency is defined by a ratio of the radiated power over the power received by a nanoantenna.
As seen in Fig.7.1.1, the nano antenna can be viewed as a capacitor formed by the two strip arms at low frequencies. While at higher optical frequency, the inductive reactance contributed by the strip arms becomes significant large. Around 500THz, a
109 series resonance (short circuit oscillation, LC oscillates in series connection) is observed, where reactance is identical to zero. This is quite similar to that observed in a RF dipole antenna. When the frequency increases further, the capacitance determined by the center gap couldn’t be ignored. As it is connected in parallel with the series LC circuit a parallel resonance (open circuit oscillation, LC oscillates in parallel connection) yields exactly at 600THz at the feeding point. These resonances are clearly illustrated by the dash line in Fig. 7.2.1.1
Figure 44 – 7.2.1.1 Optical input impedance (gap excitation)
Table 7.2.1.1 shows the optical resonant frequencies, input resistance of this nanostrip antenna with substrate. The optical antenna is set to operate in the parallel resonance mode, where the input resistance under the parallel resonance reaches 146. Here, parallel resonance is chosen for analysis. At the frequency for parallel resonance, the real part of impedance is relatively larger while the imaginary part is zero. [123,124]
Note that the nanostrip antenna retains an efficiency of 0.9281 under the
110 parallel-resonance frequency. It means that 92% energy from the gap can be converted to the radiation power. This may allow the nanostrip antenna to be operated under the parallel resonance with a high-impedance external device. And the gain for this optical antenna is 2.762 dB at the resonant frequency.
Parameters parallel resonance Frequency f (THz) 600 Resistance R (Ω) 146
Efficiency ecd 0.9281 Gain (dB) 2.762
Table 11 – 7.2.1.1 Antenna parameters of the reference optical antenna
7.2.2 Optical Antenna with a MIM Diode
The optical antenna collects the solar energy. The diode of Cr-Cr2O3-Pd,
(8.5nm-3nm-8.5nm, and the same cross-section structure as optical antenna poles) placed in the gap of the antenna rectifies the high frequency optical energy. But this will affect antenna’s characteristic, espeically the resonance frequency of the optical antenna by reflecting the optical energy. As the metal part of the MIM diode is illuminated under the sunlight as well, it may be viewed as the extended part of anatena strips. The strip length of this nanoantena is increased. The Drude model should be used for this extended antenna strip. According to [125], the real length of optical antenna under resonance is shorter than that calculated using the traditional antnena theory. In such high frequency, the conductive current is very small, the 111 transmission of energy depends on the displacement current, which means the optical energy is transmitted in the form of electromagnetic field. Therefore, whether two metals are tightly connected for conductive current transmission is not the point.
Here, the cross section and gap of the optical antenna are fixed. The antenna is excited on the inner side of the gap. The length of the antenna strips is adjusted to investigate the influence of a MIM diode to the electrical length of the antenna. The frequency is fixed at 600THz. The optical antenna with different arm lengths is simulated for investigation of arm length and impedance. Using the previous structure, the characteristics of the optical antenna with strip length varying from 61nm to 50nm is studied. A comparison of input impedance, radiation efficiency and gain are given in
Table 7.2.2. Obviously, the reference antenna with the strip length of 61nm resonants at 600THz. But when the MIM diode is presented, the resonant frequency drops to
564THz. The strip length must be modified to retain the same resonant fruquency.
This is because the metal part of the MIM diode lengthens the strip length. It is clear from the figure that the resonant frequency comes down and is reduced to 600THz again, if the strip length is reduced to 55nm. The efficiency remains 0.93 at frequency of 600THz, while the antenna gain is about 2.73dB. It is obvious that the metal part of the MIM diode affects the resonance frequency of the antenna. The effective length
(Leff) is the antenna length (Lantenna) plus metal length (LMIM-metal) and insulator
(Linsulator) (epsilon also play a role): Leff is the sum of Lantenna, LMIM-metal, and Linsulator.
The aim of introducing this antenna effective length can help to design the antenna
112 with the MIM diode.
The simulation results disclose that it is possible to modify the dimension of a nanostrip antenna for target resonant frequency with the presence of an external loading MIM diode. The nanostrip antenna has the following dimensions: the gap in the antenna is 20nm, the thickness of antenna is 30nm, the width of the antenna is
25nm, and the length of one pole in the anntena is 55nm. Without a MIM diode in the gap, for example, the antenna has an optical input resistance of 87.52Ω at the frequency of 600THz. By decreasing strip length to 50nm, the input resistance of this nano antenna is 103.23 Ω. While the radiation efficiency and gain nearly keeps the same. So the length of arms can change resonant frequency and aslo change the impedance of the antenna.
113
Figure 45 – 7.2.2 Optical input impedance of a optical antenna with MIM diode, and a variable strip lengthfrom 61nm to 50nm (a=25nm, t=30nm, gap=20nm)
Strip length b (nm) Zin() ecd Gain (dB)
50 103.23@656THz 0.93 2.71
55 87.52@600THz 0.92 2.75
60 103.66@573THz 0.92 2.74
61 102.44@564THz 0.93 2.75
Table 12 – 7.2.2 Optical resonant frequency, input resistance and radiation efficiency against strip length b
7.2.3 Optical Antenna with the MIM Diode and the DC Line
The MIM diode is a rectification device.It is connncected to the external load via a conductive line for DC current transmission; the material of ths line is aluminum. It has the cross-section dimension of 6nm×10nm (width×thickness), and the length of the line is 100nm. It is embedded in the substrate with the depth of 5nm from its top 114 surface. This DC line may affect the resonant frequency of the antenna and other antenna characteristics. Fortunately, the nanoatenna can be tuned to resonate by varying its strip length, just like a traditional RF metal antenna.
Fig7.2.3.1 shows the optical input impedance of a nanostrip antenna when the strip length varies from 55nm to 75nm. Other geomtrical parameters of the optical antenna are fixed using their reference values (the gap in the antenna is 20nm, the thickness of antenna is 30nm, the width of the antenna is 25nm, and the strip length is 61nm).
Unlike the previous one, the strip length increases from 55nm to 75nm in order to resonate at 600THz. Increasing length of antenna strips leads to an increase of input resistance under the parallel resonance. This is different from that phenomenon observed in rectenna using the MIM diode, which uses shorter antenna strips at resonant frequency. On the other hand, the radiation efficiency and gain decrease from 0.89 to 0.83, and 2.65dB to 2.55dB respectively, as seen in Table 7.2.3.1. The length of a DC line seems to have a negligible effect on the antenna.
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Figure 46 - 7.2.3.1 Optical input impedance of a optical antenna with the MIM diode, DC line, and a variable strip length from 55nm to 75nm (a=25nm, t=30nm, gap=20nm)
Strip length b (nm) Zin() ecd Gain (dB) 55 53@564THz 0.89 2.65 65 196@613THz 0.86 2.59 70 223@600THz 0.84 2.55 75 240@578THz 0.83 2.56
Table 13 - 7.2.3.1 Optical resonant frequency, input resistance and radiation efficiency against strip length b
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The optical energy transmission in the DC line is also analyzed. Using the same structure, (the gap in the antenna is 20nm, the thickness of antenna is 30nm, the width of the antenna is 25nm, and strip length is 70nm). Fig 7.2.3.2 shows that the resonant frequency and impedance have no change for the optical antnena with a 100nm-length
DC line. Table 7.2.3.2 presents the S11 parameter (input port voltage reflection coefficient – the ratio of reflection voltage to input voltage ) of the DC line when transmitting 600THz optical energy,lower than -10dB, which means the optical energy attenuates significantly along the DC line. The DC line is deemed as a dielectrical waveguide showing a low pass charateristic, making most of the optical energy attenuated. This is certainly an advantage for DC current trasmission and eleminates the high frequency energy influence to the antenna. The DC line length does not have a significant effect, but the joint with metal in the MIM diode have the impact.
117
Figure 47 – 7.2.3.2 Optical input impedance of a optical antenna with the MIM diode, variable DC linde from 50nm to 200nm (a=25nm, t=30nm, gap=20nm,b=70nm)
Length of DC line 0nm 50nm 100nm 150nm 200nm S11(dB)ant -9.8 -12.1 -12.1 -12.1 -12.1
Table 14 – 7.2.3.2 S11 parameter of DC line for transmitting 600THz optical energy
Table 7.2.3.2 presents the summary of all the experiments results when the antenna strip length varies. The symbol of or indicates that the value of the parameters increases or decreases. The symbol of represents there is almost no change of a parameter. It is clear that when an external load is added to the system, the characteristics of the optical antenna will be changed, either in impedance or resonant frequency even efficiency. But changing the length of the antenna strip can effectively make the antenna resonate at desired frequency.
118
Change of Strip Resonant Radiation Impedance Gain Parameters Length Frequency Efficiency Antenna + MIM Antenna + MIM +DC Line
Table 15 -7.2.3.3 Summary of the Experiments results
The reference antenna has a strip length of 61nm. When the MIM diode is mounted, the strip length decreases to 55nm to retain the resonant frequency. With a DC line being added to the system, the strip length increases back to 70nm. This is because the
DC line reduces the effect of the MIM diode as part of the antenna and attenuates the optical energy. On the other hand, the length of a DC line has no effect on the resonant frequency of the system. It effectively attenuates the optical energy up to
-55dB, and plays a role as a DC filter for DC current transmission. Other parameters like antenna efficiency, and gain slightly go down because of the energy loss. The mutual interference makes the resonance goes up.
7.3 Rectenna System Excited by Plane Wave
In this section, the rectenna system is investigated, which includes the nano antenna, the MIM diode, and DC line. When the system is illuminated in a plane wave as the excitation. The retenna models are the same as before. The nano antenna has the same geometry as the model presented in previous section, that is, the gap is equal to 20nm, the thickness is equal to 30nm, the width of strips is equal to 25nm, and the strip
119 length is equal to 61nm. The substrate has the dimensions of 1000nm in width,
1000nm in length and 200nm for thickness. Fig 7.3.1 shows the FDTD simulation results of the far field and near field intensity from the antenna under the plane wave excitation. The figure of the far field electric intensity shows that the peak intensity is found at the frequency of 600THz which corresponds to the resonance of gap excitation. While the near field electric intensity has a peak at 750THz. This shows that a lot of energy is stored in the form of magnetic field in the near field zone. The efficiency of a receiving antenna is defined as the ratio of power collected by the antenna (not including reflected power) to the incident power on the surface of the antenna. In the simulations, the similar value is measured as that of a transmitting antenna for same geometry, which is in accordance with reciprocal thereom of antennas.
120
Figure 48 - 7.3.1 Electrical field intensity of an optical antenna under a plane wave excitation (a=25nm, b=61nm, t=30nm, gap=20nm)
The energy collected by an antenna can be represented as a voltage source. This induced voltage is calculated based on gap voltage of the rectenna and current on the
MIM diode in open circuit; and the circuit current when the gap is shorted. In figure
7.3.2 (a) the open circuit means the circuit is open at the terminal; the short cuicuit means the the antenna gap is shorted directly, current flows not through the MIM diode, only through the voltage source and the antenna. The gap voltage can be calculated by the length of the gap multiplying the induced electric field in the gap of antenna. [115] Similarly, the current of the antenna can be calculated by integrating the incident H field along the integration path of the cross section of the antenna, shown in figure 7.3.2(b).
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Antenna MIM RDC Line Vmim antenna Incident field H V Im E Vs integration path CMIM Incident RMIM Z MIM field cross- Z ANT section of antenna (a) (b)
Figure 49 – 7.3.2 (a) Equivalent circuit of Rectenna system; (b) Method for V and I calculation
Fig 7.3.2 depicts an equivalent circuit for the whole system with the MIM diode and
DC line. In order to calculate the impedance, a group of equations are used
Vac VsMantac I Z V , Vsscant I Z , VacMMIM I Z , Zant (7.3.1) IIscM where Vs is the voltage source determined by electric, IM is the current flows through the MIM diode when the load is open (integration of magnetic field strength H around the MIM diode). Vac is the voltage on the MIM diode when the load is open
(integration of electric field strength E along MIM). ISC is the short circuit current of the antenna when the DC line is short circuited and no current flows across the MIM diode (integration of magnetic field strength H around the antenna); Zant is the impedance of antenna and ZMIM is the impedance of the MIM diode. Because there is no external load connected to the antenna, thus plane wave excitation makes a serial equivalent circuit.
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7.3.1 Nano Antenna with the MIM Diode
The rectenna system is composed of a nano antenna with the mono pole length of
61nm; and a MIM diode. The material for the MIM diode is Cr-Cr2O3-Pd. This three layer structure has a thickness of 8.5nm-3nm-8.5nm, and their cross-section is the same size as optical antenna, as illustrated in fig 7.1.1. A pair of conductive wires connect with the MIM diode for transmitting DC power, the material of this DC line is aluminium, in the cross-section: the width is 6nm and the heighth is 10nm; the length of this DC line is 100nm, about 5nm depth embedded in the substrate. In Fig
7.3.1.1, the symbol – b=61nm measns the antenna monopole length is 61nm. It is the same structure in Fig 7.1.1; the symbol – b=69.5nm means it share a same geometry as Fig7.1.1 but the gap is 3nm, and the strip length is 69.5nm. The expression of
61nm+M+air+M means the material structure is Al-Cr-air-Pd-Al, the corresponding lengths are 61nm, 8.5nm, 3nm, 8.5nm, 61nm. The symbol - b=61nm+MIM means the
MIM diode is embedded in the gap of antenna. In this nano antenna, the impedance is calculated at the two cross section of the two metals of the MIM diode. It has the same structure as M+air+M, but a 3nm insulator replaces the air in the gap.
In Fig 7.3.1.1, the maximum intensity of the far field is mainly in the region of
600THz-700THz which corresponds to the resonant frequency range of the antenna around 600THz. As the strip length of the antenna increases, the far field intensity goes higher; but as the gap narrows, the near field intensity is also enhanced at resonant frequency because the energy is collected by the antenna. On the other hand, 123 the less electrical energy means the relative more magnetic energy in the near field.
The changes of antenna structures lead to a small change in the resonant frequency, especially when the strip length is 61nm with the MIM diode. The insulator also has an effect on the electrical field distribution as well as to the antenna impedance. The simulation results for the antenna with monopole length being 69.5nm and antenna monopole length being 61nm with metal-air-metal structure is almost same and do not make a great change. In table 7.3.1.2, the simulation results are listed. It is found that the resonant frequency is around 600THz, and impedances are on the order of
100ohm. The source voltage is calculated based on equation 7.3.1, and listed in table
7.3.1.2. Compared with the investigation of the antenna excited by gap voltage, the impedance shows a slight difference. This can be explained by that the plane wave excites the antenna only in several angles, which leads to an uneven energy distribution to the antenna.
The second resonant point, or named parallel resonant point is chosen as the operating frequency. This is because, in the first resonant point, the inductive characteristic dominates, and the antenna works as a low pass filter. While in the second resonant point, the capactive characteristic dominates. The low pass filtering function diminishes, and the antenna can collect more power.
The efficiency of a receiving antenna can be calculated by receiving power at the gap to the incident power on the antenna. The power received is calculated by voltage and
124 current at the gap of antenna; while the incident power is determined by electric field on the surface of the antenna. Here, the efficiency for these antennas is around 0.913 at the resonant point. In table 7.3.1.2, it shows that the generated optical voltage source |Vs| by antenna harvesting. When the gap size reduces, the voltage decreases because of the ohmic loss. Besides, the voltage of the model - strip length being 61nm with the MIM diode, is measured at the two sides of the insulator of the MIM diode.
Figure 50 – 7.3.1.1 Electrical field intensity of an optical antenna with the MIM diode under plane wave excitation
125
Figure 51 – 7.3.1.2 Optical input impedance of an optical antenna with the MIM diode under plane wave excitation
resonant Z () |Vs|(mV) frequency in b=61nm 600THz 141.13 7.41 b=69.5nm 578THz 173.61 7.43 b=61nm+M+air+M 586THz 177.62 7.36 b=61nm+MIM 605THz 171.86 7.02
Table 16 – 7.3.1.2 Optical resonant frequency, input resistance and radiation efficiency against strip length b
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7.3.2 Nano Antenna with the MIM Diode and the DC Line
Fig 7.3.2.1,2 shows the electric field intensity and impedance distribution of the nano antenna with the MIM diode and DC line. The length of the DC line is 100nm. Both the MIM diode and the DC line have an effect on the nano antenna. In the far field intensity, antenna strip length being 61nm with the MIM diode and DC line has a little change in the value compared with that of the same antenna without the DCline; this also happens in the near-field intensity, but the simulation valuse are quite different.
In figure 7.3.2.1, it shows the intensity value varying with frequency for the antenna strip length of 69.5nm with and without the MIM diode. Clearly, the MIM diode has changed the frequency, at which the nano antenna can harvesting the power maximumly. The maximum intensity of the far field distribution appears at resonant frequency, around 600THz.
The DC line lowers resonant frequency of an antenna and increases the antenna impedance in both real and imaginary parts. When the DC line is mounted, the efficiency of the antenna comes down. Other parameters from simulation results are listed in the table 7.3.2.1
127
Figure 52 – 7.3.2.1 Electrical field intensity of an optical antenna with the MIM diode, DC line under plane wave excitation
128
Figure 53 – 7.3.2.2 Optical input impedance of a optical antenna with the MIM diode, DC line under plane wave excitation
resonant Z () |Vs|(mV) frequency in b=69.5nm 578THz 173.61 7.43 b=61nm+M+air+M 586THz 177.62 7.36 b=61nm+MIM 605THz 171.86 7.02 b=61nm+MIM+DC 597THz 202.17 7.01 line
Table 17 – 7.3.2.1 Optical resonant frequency, input resistance and radiation efficiency against strip length b
The optical energy transmission in the DC line is also studied using the same antenna structure, that is, the width is 25nm, the height is 30nm, the gap between two pole is
20nm in length, the strip length is 61nm. Fig 7.3.2.1,2 shows the electric field intensity, the resonant frequency and impedance. They almost keep the same when the length of the DC line varies. That line length is 0nm in the figure means the horizontal part of DC line is removed, but the vertical part retains the same as shown in Figure
7.3.2.2. Table 7.3.2.2 presents the S11 parameter (input port voltage reflection
129 coefficient – the ratio of reflection voltage to input voltage ) of the DC line when transmitting 600THz optical energy. It is lower than -10dB. The DC line is deemed as a dielectric waveguide, which shows low frequency pass charateristic, making most of the optical energy attenuated. This is much better for DC current conduction and eleminates the high frequency energy reflecting back to the antenna, as the metal works as PEC for DC current. The efficiency for this antenna with different lengths of the DC line is around 0.87.
Figure 54 – 7.3.2.3 Electrical field intensity of an optical antenna with the MIM diode, variable DC linde from 50nm to 200nm
130
Figure 55 – 7.3.2.4 Optical input impedance of a optical antenna with the MIM diode, variable DC line from 50nm to 200nm
Length of DC line 0nm 50nm 100nm 150nm 200nm S11(dB)ant -9.8 -12.1 -12.1 -12.1 -12.1
Table 18 – 7.3.2.2 S11 parameter of a DC line for transmitting optical energy
7.4 Energy Coupling from Antenna to the MIM diode
Theoretically, the MIM diode can be represented as a resistance in parallel with a
131 capacitance.[114] The capacitance is determined by the insulator of the MIM diode.
The resistance is determined by unilluminated current - I, the recived photon energy -
Eph and the biased voltage - Vdc. Based on the previous research, the full expression of the resistance is shown, as follows. [114]
2(/)Eeph RMIM (7.4.1) IVEeIVEe((/))((/))dcphdcph
Obviously, in the equation 7.4.1, the resistance of a MIM diode is totally controlled by the external circuit, which leads to a variable resistance for the MIM diode and this will affect the output current of rectified DC power.
2 ||Vac Pac (7.4.2) RMIM
Figure 56 – 7.4.1 AC power on the MIM diode against frequency
Based on equation 7.4.2, Vac means the voltage on the MIM diode, Rmim is calculated from equation 7.4.1. Figure 7.4.1 shows the distribution of AC power on 132 the MIM diode, when RMIM varies between 40Ω to 70Ω based on the simulation. The model has the following dimensions: in cross section, the width of antenna is 25nm, height is 30nm, the gap is 20nm, the strip length is 61nm, the length of DCLine is
100nm. The external electric field is set as 1000V/m with the help of solar energy distribution. Under the state of such an external input, the received highest power on the MIM diode shifts from the previous center frequency – 600THz to 660 THz whose bandwith is 20GHz, quite different from the state of the unmounting MIM diode. In the state of only antenna without the MIM diode, the highest output power of the antenna is observed around 600THz. But the maximum output power for rectification is shifted to different frequency when the MIM diode is mounted. So the frequency, at which the maximum power received for the antenna or antenna resonance, may not be the frequency at which the maximum power on the MIM diode.
This is because of the effect on the system from the MIM diode, the impedance of which is composed of resistance and capacitance; this will change the impedance of the whole system which receives the optical energy. The antenna impedance decreases with frequencies increasing.
In the spectrum of solar power distribution, the highest energy radiatated at the sea level is in around the green and blue light region, in the range of 500THz to 700THz.
The energy level can be deemed to be the same within this region. Considering the
AC voltage distribution on the MIM diode, the highest energy can be received by the
MIM diode is that at the frequency of around 660THz. Obviously, the best operating
133 state to receive solar energy is to set the system operating at 660THz to maximumly receive the optical energy on the MIM diode for rectification, the antenna also has an efficiency of 0.87 at 660THz.
Figure 57 – 7.4.2 AC power on the MIM diode against Rmim
In the figures 7.4.2, both the AC power and the AC voltage on Rmim are presented
134 against different external incident electric field. As the value of electric field increses, the voltage on the Rmim goes up, which leads to the power on Rmim (not including
Cmim) goes up at the same Rmim value. But in the comparison of the calculated AC power at different Rmim, it is found that the maximum power on Rmim happends at
Rmim being 40ohm under diffrernt external incident electric field at 660THz. The power on Rmim is only 17% of the total power received by antenna as an AC source by calculation based on equation 7.4.2. This low transmission efficiency is caused by impedance mismatch with the antenna.
Other materials – gold and copper are chosen as the antenna material for comparisons.
Figures 7.4.3 shows the impedance of antennas under the plane wave excitation.
Though the antennas have the same geometry (the gap in the antenna is 20nm, the thickness of the antenna is 30nm, the width of the antenna is 25nm, the length of one pole in the anntenan is 61nm), the impedances are quite different, because these materials have different plasmonic charateristics especially plasmonic frequency. As the research is for aluminium, these parameters are not the optimal ones for gold and copper, so in the range of 580THz to 700THz, antennas made by these two materials do not operate at resonant frequency. The receiving efficiency of an antenna made of gold is 0.53 and the other made of copper is 0.46.
135
Figure 58- 7.4.3 Optical input impedance of optical antenna with the MIM diode, DC line under plane wave excitation
Both the AC voltage and power are plotted in the 7.3.4 for comparison. Compared with the power acquired by the MIM diode, the antenna made of gold shows a much better performance than the other two. For example, the maximum power that the
MIM diode can acquire from the gold antenna is 3.92×10-14W when Rmim is 30ohm, and is about 20% more than aluminium and copper antenna. The efficiency of power coupling to the MIM diode for gold and copper antenna systems is 0.24 and 0.23 respectively. 136
Figure 59 – 7.3.4 AC power on the MIM diode against frequency and Rmim
137
7.5 Impedance of Insulator in the MIM Diode
The rectification made by a MIM diode depends on the theory of tunneling effect, which means electrons tunnel through an insulator from one metal to another one. The characteristic of the insulator therefore determines the rectification, not the length of metal of the MIM diode. Here the impedance and rectified voltage are investigated. In
[120], a similar structure for the MIM diode was presented, but there was no discusson about the antenna characteristic under the plane wave excitation. The simulation was done for the impedance of the insulator in the MIM diode. Clearly the insulator has a capactive characteristic, which varies with frequency. When a DC line is connected, the impedance becomes smaller compared with that of no DC line as seen in Fig 7.5.1. This can be explained by the parallel connenction in the equivalent circuit. Table 7.5.1 shows the impedance of the insulator at nano antenna resonant frequency.
(a)
138
(b)
Figure 60 – 7.5.1 Impedance of insulator in the MIM diode with antenna (a) even DC line (b) under plane wave excitation
Antenna resonant |Vs|(mV) Zinsulator(ohm) frequency b=61nm+MIM 605THz 7.02 -j63.81 b=61nm+MIM+DC 597THz 7.01 -j67.62 line b=61nm+MIM+DC 660THz 7.11 -j53.36 line
Table 19 – 7.5.1 Impedance of Insulator in MIM diode at resonant frequency
7.6 Conclusion
In this chapter, the performance of the rectenna system is investigated, using the gap excitation and plane wave excitation with the FDTD simulation. It is found that under different excitation methods, the characteristics of the same rectena change. The plasmonic characteristic of the metal make it hard to have a perfect impedance matching with the impedance of the MIM diode. Besides, considering the effect of the
139
MIM diode in receiving optical energy, the frequency for maximum harvesting is not the same as the resonant frequency of the antenna standalone. This frequency is shifted to a higher value (10% above the resonant frequency of a standalone unit). By calculation, 17% of the received power is coupled to the MIM diode. Comparison is also made among aluminium, gold, and copper, for the antenna with the same structure. The aluminium shows a better energy coupling effect. Finally the charactereitic of the MIM diode is studied. The reactance of the insulator is found to be aournd -j60ohm.
140
8 Effect of an External Load
The external load which is connected to the optical rectenna system has a great influence on the operation of an optical antenna as well as the current generation of the metal insulator metal diode. In this chapter, with the help of semiclassical quantum mechanical theory, FDTD simulations are conducted to study the effect caused by an external load. As the principle of a rectenna used for receiving and converting optical energy is totally different from a photovoltaic cell, it is necessary to study the whole rectenna system for DC current generation.
8.1 Equivalent Circuit of a Rectenna System
The figure 8.1.1 shows the nano rectenna system. The nano antenna directly interacts with the MIM diode for energy conversion. Silicon is chosen as the material for the substrate holding the nano antenna and the MIM diode. Field enhancement in the gap of the antenna for improving energy harvesting is achieved by controlling antenna geometry, such as the strip length, the gap length. The MIM diode is placed in the gap of the antenna arms, so the electric field received by the antenna can be directly applied on the MIM diode. The DC line transmits the rectified DC current to the Rdc, and at the same time works as a low pass filter to filter the AC power.
141
b b a MIM dp t t DC Line depth substrate
dipole a antenna external
DC Line load M
DC Line length I M
dipole antenna Low Pass cap Filter + - diode Rdc
Figure 61 – 8.1.1 Rectenna system
After the optical energy is collected by the nano antenna, rectification of such high frequency power to DC power is made by a metal-insulator-metal diode. The MIM diode has a three-layer structure; the two metal layers are made of different materials and have different energy levels. The middle layer is made by an anodic oxidation of one metal material. According to quantum theory, an electron can tunnel through the insulator from one metal to the second when an external field is applied.
142
base metal layer top metal layer as anode as cathode insulator barrier
Figure 62 – 8.1.2 Structure of a MIM diode
When biased voltage Vdc is applied to the metals, DC current density from the diode can be expressed by using the applied biased voltage and other parameters as follows
[126]
()/EkT mqkTe 1 z J()()ln VTdczz EdE (8.1.1) 230 ()/Ezdc qVkT 21 e
T(Ez) is the tunneling probability of electrons transmitting from one metal through the insulator barrier to the other metal [127]; μ is the energy potential of the first metal; m is the effective mass of electron; Ez is the range of energy of the tunneling electron, q is the electrical charge of electron. Here the semiclassical model is applied to determine the tunneling current.
Based on the physical characteristics of a MIM diode, an equivalent circuit of the rectenna system with the load is presented in Fig 8.1.3. The equivalent circuit of the
MIM diode/antenna was investigated in [128]. The nano antenna collects solar energy, and works as a voltage source to provide electromagnetic energy to the MIM diode.
The MIM diode rectifies the electromagnetic power to DC power which is transmitted
143 to an external load through DC line. The AC energy from the antenna may leak through the DC line because of the parallel structure. However, as the electromagnetic field on the diode has a very high freqency, it will attenuate significantly when it propogates along the DC line. There is alomost no leakage of AC power via the DC line. On the other hand, because of this parallel structure, when the DC current is applied on the external load, there will be a DC voltage generated on the load. This voltage will also have an effect on the MIM diode through the DC line. The impedance and capacitance of the MIM diode can be calculated by voltage and current flowing through the diode, and the voltage and current are determined by electric and magnetic field received by the antenna. Rmim is determined by both the external optical energy and DC biased voltage.
Antenna MIM RDC Line
V
CMIM RLoad RMIM
ZANT
Figure 63 – 8.1.3 Equivalent circuit of a rectenna system
This equivalent circuit can be divided into DC and AC equivalent circuits for analysis.
Fig 8.1.4 shows the two circuits separately. In the DC equivalent circuit, it only consists a controlled current source and a load. The load can be substituted by a DC voltage source, which is controlled externally. It is easy to understand that in the DC circuit, a capacitor is considered as an open circuit while an inductor is deemed as a
144 short circuit. In the AC equivalent circuit, Zeq is defined as equivalent impedance, which includes all other impedance; the Zmim includes the resistance and capacitance of the MIM diode.
Antenna ZMIM MIM
Z V MIM + I Vdc - Load
Zeq
AC equivalent circuit DC equivalent circuit
Figure 64 – 8.1.4 Equivalent circuit can be separated into DC and AC parts
8.2 Effect of the External Load
The present researches only focus on the generated current of a MIM diode without considering the effect of an external load. This is because the parallel structure of the circuit has the effect from load voltage on the MIM diode. If there is no load, the whole system looks like an open circuit.
8.2.1 Transmission of Power
As a receiving antenna, the nano antenna collects the solar energy on its gap. The enhanced electric field in the gap is directly applied on the MIM diode for tunneling rectification to generate a DC current to the external load, shown in Fig 8.1.1.
145
According to the spectrum of solar energy [129], an electric field of 1000V/m and polarized parallel to the nanoantenna is assumed to apply in far field as excitation. Far field is simulated to study the resonance and impedance of the nano-antenna [130].
The resonance frequency is that under which the far field or near field intensity reaches a local maximum in the spectrum of the antenna. Plane wave excitation is used to study the receiving antenna, the incident electric field to drive the antenna to generate an AC voltage in the gap of the antenna.
A reference model of the nano optical antenna is given in Fig 8.1.1. The gemometry of the nano antenna is as following: the gap length is 20nm, the antenna thickness is
30nm, the arm width is 25nm, the strip length is 61nm, and the substrate length is
1000nm; the substrate width is 1000nm; the substrate thickness is 200nm. The MIM diode in the gap work as a rectification device, and has the structure of Cr-Cr2O3-Pd
(the parameter for the material can be referred in [131]). The thickness of these three layers is 8.5nm-3nm-8.5nm, respectively. The area of the cross-section of the MIM diode is the same as that of the optical antenna - 7.5×10-16m2. A conductive line made of aluminium is connected to the MIM diode for DC current transmission. It has the cross-section dimension: width of 6nm, thickness of 10nm, and length of 100nm. It is embedded in the substrate with the depth of 5nm in figure 8.1.1.
The power received by the antenna is transmitted to the external load as shown in
Fig8.1.3. The nano antenna works as an AC voltage source in the equivalent circuit of
146 the rectenna system – defined as Pant, while the incident energy on the antenna is
Pinc. Then this AC power Pant is transmitted to the MIM diode for recitification.
Before recitification, the power that the MIM diode gets is defined as Pac, and after rectificaiotn, the output power is defined as Pdc.
Tthe first transmission effiency η1 in a rectenna system is the rate of the AC power that the MIM diode gets from the antenna Pant, the efficiency η0 means the ratio of
Pant to the incident power on antenna Pinc. The conversion efficiency of the MIM diode is defined as η2=Pdc/Pac. In the DC equivalent circuit, the MIM diode is deemed as a DC current source. The power relation is list in equation 8.1.2.4,5:
PPantinc0 , PPacant1 , PPdcac2 , (8.2.1.4)
PPdc0 1 2 inc (8.2.1.5)
In the application of a MIM diode for rectification, traditionally, no effect of the external load is taken into consideration. The nano antenna receives optical energy; the energy is coupled to the metals and then displacement current generates within the metals (mainly electrical field not conductive current); and an AC voltage is generated in the gap of the antenna. The MIM diode is in the center of the gap. As the antenna works as a voltage source to provide electric field to the MIM diode, electrons acquire energy from electric field, then tunnel through the insulator of a MIM diode to realize rectification. The generated DC current flows along the DC line to the external load.
In the equivalent circuit, the AC power can leak to the DC line, but as the frequency is
147 high enough to THz so that it will attenuate and nearly no such high frequency energy will flow to external load.
But the effect of an external load to the system especially the current generation cannot be omitted. The equivalent circuit of the whole system clearly shows a parallel structure, which means the voltage on the external load will also have an effect on the
MIM diode and nano antenna. As a DC biased voltage is on the MIM diode for electron tunneling, this DC voltage has the same effect when the system is in a stable state. On the other hand, the antenna will not be affected by this DC voltage because energy with different frequency will not affect each other. So considering the effect of an external load, the current density can be expressed as:
()/E E kT fz 2 qVac mqkTe 1 J(/)()( Eqphdcnzz VJT )ln EdE (8.2.1.6) 230 ((/))/Ef E zphdc q nE q V kT n Eph 2 1 e where Eph is , the photon energy of incident wave; Vdc is the external load induced voltage in the steady state; Vac is the amplitude of the AC voltage applied
2 qVac across the diode at a frequency. Jn () is Bessel function; q is the charge of an Eph electron, 1.6 × 10-19 coulombs; planck constant h is 4.1× 10-15eV∙s; and reduced h Planck constant is . When the 660THz optical frequency is applied, the value 2 of is 2.73eV.
Using the following equation, the relationship of resistance in the MIM diode and the rectified DC voltage is found, as shown in the fig 8.1.1. Idc is the current density multiplied by the cross section area of MIM diode. 148
E 2 ph q R (8.2.1.7) MIM EE IVIV()() phph dcdcdcdc qq
E 2 ph q R (8.2.1.8) MIM ()/EEqVEkT mqkTe 1 fzdcph AreaTEdEMIMzz*()ln 230 ()/EEqVEkTfzdcph 2 1 e
Figure 65 – 8.2.1.1 Rmim vs. the voltage biased (660THz)
As can be seen in Fig 8.2.1.1, when the voltage applied (optical ac source) increases, the Rmim decreases in the equivalent circuit. The AC voltage on the MIM diode is chosen as 5mV, 10mV, 15mV; while the rectified voltage changes from -1.2V ~
+1.2V. Current can be calculated by equation 8.2.1.6 with the area of the MIM diode being 30nm×25nm. The figure shows a decrease from 150ohm to 40ohm for Vdc at the range of -1.2V to -0.5V. Then at the range of - 0.5V to 1.2V, the value of Rmim is fixed around 40ohm, which can be explained in the equation 8.2.1.8 that as Vdc goes up in the positive axis the contribution to the value of Rmim comes down. This figure
149 reflects the effect of the DC biased voltage to the value of Rmim.
Figure 66 - 8.2.1.2 Rectified current vs. the voltage applied (660THz)
The effect of an external load to rectified current is also very clear. When the external incident electric field increases, which leads to an increase of the electrons’ kenetic energy and an increase of Vac, the current increases at the same Vdc. In fig 8.2.1.2, the rectified current deceases from 5×10-12A ~ 6×10-12A to less than 1×10-12A as Vdc increases to 1.2V. Figure 8.2.1.3 shows the rectified DC power distribution when Vdc increases. When Vac is 5μV, the maximum rectified power is 2.27×10-13W with Vdc being 0.2V. But when Vac is 10μV or 15μV, the maximum rectified power is
5.37×10-13W and 1.27×10-12W respectly at Vdc being 0.9V. This is because that the increament of external power leads to the increament of electrons’ kenetic energy, which generates more DC current compared with that of low external power.
150
Figure 67 – 8.2.1.3 Rectified current against incident voltage (660THz)
Figure 68 – 8.2.1.4 AC power of the MIM diode before rectification (660THz)
The figure 8.2.1.4 shows the AC power that applied on the MIM. Vdc determines the value of Rmim thus affects the AC power distribution. Based on these data, the power conversion efficiency is calculated and shown in figure 8.2.1.5. Overally the output power and conversion efficiency keeps stable likes a line with small swing. The conversion efficiency is about 0.93.
151
Figure 69 – 8.2.1.5 Pac to Pdc conversion efficiency (660THz)
Through the output DC power and the conversion efficiency – η2 in equation 8.2.1.4 are stable on the figure 8.2.1.5. The conversion efficiency curve is not sable and swings around 0.93 because the current equation is a non linear formula and particles’ nonlinear movement.
8.2.2 Effect of Load Voltage to Antenna Impedance
The load has an effect to the antenna impedance, so in the following, this effect of the load to the antenna is studied.
152
Figure 70 – 8.2.2.1 Effect of an external load to the antenna impedance
The figure 8.2.2.1 shows that the antenna impedance slightly changes because of the effect of load voltage. On the whole, the resonant frequency is fixed at 660THz, both distributuion of real and imaginary part of the impedance has been moved to lower frequency as the load DC voltage from 0.1V to 0.3V，the antenna impedance slightly increases as the external load increases. The change of the antenna impedance caused by the load has been discussed in detail in ref [132].
153
8.3 System Efficiency Calculation for Optimal State
The rectenna system contains an optical/nano antenna and other rectification components. The interaction of physical parameters between two parts is studied. The nano antenna works as a voltage source, the MIM diode provides the DC current. The
MIM diode is embedded in the gap of the dipole nano antenna to reduce the optical energy loss, under which the DC line is buried in the substrate to transmit the rectified
DC current at the same time to reduce the interference from optical energy. Otherwise, the DC line on the surface of substrate will also receive optical energy. The length of the DC line nearly has no effect on the parameters of the MIM diode and antenna, so
100nm is selected for the length of DC line.
An optimal state of maximum DC output power should be chosen to calculate the system efficiency. In the system, the energy harvesting efficiency of the antenna is
87% when the operating frequency is chosen at 660THz (data from chapter 7), optical energy coupling efficiency of the antenna to the MIM diode is 17% (because of mismatch with the antenna) when the Rmim is chosen as 40ohm. The AC power to
DC power conversion efficiency of the MIM diode is around is 93% with the structure of Cr-C2O3-Pd. Based on equation 8.2.1.5, the total efficiency of rectenna system is
13.6%, when Vdc is set to be 0.47V (Rmim is 40ohm).
The rectenna system under different incident electric fields is simulated to find the input and output power of the MIM diode. Clearly, as the incident power increases,
154 the output DC power also increases, as shown in the figure 8.3.1. Pac is the input power on the MIM diode and Pdc is the output power. The system efficiency keeps around 13% - 14%.
Figure 71 – 8.3.1 Input and output power of a MIM diode against incident E-field
Based on the data under different incident electrical field, it is needed to output the maximum power, as the output curve swings with the Vdc, because of the quantum mechanics effect. The maximum output power in these data occurs when the Vdc is
155 equal to 0.5V and Rmim is 40ohm. So in order to keep the rectenna system in an optimal state for maximum DC power output, the Vdc should be kept at 0.5V and
Rmim being 40ohm.
E-Field Vdc(V) Rmim(ohm)
800V/m 0.47 40.1
900V/m 0.52 38.7
1000V/m 0.47 40.1
1100V/m 0.48 39.8
1200V/m 0.5 39.3
Table 20 – 8.3.1 Maximum output power tracking point against E-Field
This system efficiency is about the same as the present photovoltaic cell, but it can be higher when the impedance of the antenna is matched with Rmim. Thus more power can be transmitted to the MIM diode for rectification. Theorectically, the maximum system efficiency can be 40% or more, when a perfect impedance matching between the antenna and the MIM diode is achieved.
The rectenna system efficiency is defined as the multipilication of the following three parameters: the energy harvesting efficiency of the antenna - η0, the optical energy coupling efficiency of the antenna to the MIM diode - η1, the AC power to DC power conversion efficiency of the MIM diode - η2.
156
The following table shows a comparison of system efficiency using different materials for nano antenna. The antenna structure is same (the width is 25nm, the thickness is 30nm, the gap is 20nm, the strip length is 61nm). The MIM diode strucure is same with the Vdc of 0.47V and Rmim of 40ohm. The antenna made of
Aluminium has a higher system efficiency because of the higher antenna efficiency.
But gold and copper has a higher coupling efficiency to the MIM diode. By changing size of the antenna, they could have a higher system efficiency.
Coupling Impedance Antenna System Material |Vs| efficiency of to @660THz efficiency Efficiency the MIM diode Aluminium 121-j81 0.87 7.1mV 0.17 0.14
Gold 55-j47 0.53 4.1mV 0.24 0.11
Copper 71-j39 0.46 4.0mV 0.23 0.10
Table 21 – Comparison of the system with same structure but different materials for antenna
8.4 Conclusion
In this chapter, the effect of an external load on the optical rectenna system is studied.
The external load is directly connected to the system. When the applied voltage increases, the impedance of the MIM diode decreases first and then to a stable value.
Increasing both input optical voltage and the external load voltage can help to
157 generate more current. The system made of aluminium as the antenna material has a system efficiency of 13%, and relatively higher than those antennas made of gold and copper.
158
9 Rectenna Array for Solar Energy Harvesting
The rectenna array is studied in this chapter, with the knowledge of a single rectenna unit developed in the previous chapters. The array with a large number of rectenna units can provide more DC power for normal application such as mobile phone recharging. The antenna array has a better resonant effect and mutual coupling among units within the array, which enhances the optical energy receiving efficiency and increases the generated optical voltage at the gap of a dipole nano antenna feeding the
MIM diode. The structure of the rectenna antenna array is presented in this chapter.
9.1 The Construction of Nano Antenna Array
The antenna material is aluminum because of its lightness. Both Cr and Pd are selected to be the material for two metals of a MIM diode. Cr2O3 is selected as the material for the insulator. The DC lines are constructed for transmitting the DC power to the load, which is connected parallelly in the array. A single unit of rectenna system is composed of a nano antenna, a MIM diode and a DC line.
The renctenna array is composed of many energy collection units discussed in previous sections. The structure of a single collection unit remains unchanged, and
159 each uint is connected by a DC line embedded in the substrate. All the DC lines are connected to a common busbar. As shown before, the length of the DC line will not affect the characteristics of the nano antenna and the MIM diode. Figure 9.1.1 shows the stuctures of an array in detail. The vertical and horizontal spacing between adjacent units is same. The vertical spacing is also equal to the length of two DC lines.
Controlling the spacing between units makes it possible to achieve field enhancement for improving energy collection.
antenna antenna
substrate DC Line spacing antenna MIM
spacing
DC Line length spacing
Figure 72 – 9.1.1 Distribution of nano antennas in an antenna array
The rectification MIM diode outputs the DC current to the external load. All the units are connected in parallel, which is shown in the following figure. The parallel connection can effectively accumulate more generated DC current. The collection unit is connected in a straight column; then at the end the output current is accumulated to the output port for normal application. 160
Output
. . .
...... Unit ......
......
Figure 73 –9.1.2 Skematic of the nano antenna array for solar enrgy havesting
9.2 Antenna Array Structure for Field Enhancement
The rectanna array is constructed to generate more DC output current using parallel connection of individual units. It can effectively enhance the field strength as well, for improving collection efficiency of individual units in the array.
In the antenna theory, there is a reciprocal principle which means the antenna is equivalent as a transmitter or a receiver. The radiation pattern is identical to receiveing pattern. Based on this principle, the result from transmiting antenna can also be applied to receiving antenna. Some researches focus on the investigation of gain enhancement by using antenna arrays with help of metamaterial [133,134], gain of microstrip antenna array enhancement by electromagnetic band gap structures
161
[135]. This can also be applied to directional receiving, especially for energy collection.
9.3 Simulation and Analysis
The reference nano antenna presented in the previous chapters is selected for building an antenna array. The antenna has the following dimensions – the gap is 20nm, the thickness is 30nm, the width of the antenna is 25nm, and the strtip length is 61nm.
The antenna can oscillate at 600THz with the MIM diode set in the gap of the antenna.
First, the array of two units is studied by varying antenna spacing. Then array of 4 units is investigated. In the discussion, three values of antenna spacing are to be selected, that is, 50nm, 100nm, or 150nm. The incident electric field is set to be
1000V/m.
9.3.1 Rectenna Array of 1Row×2Columns
Figure 9.3.1.2 shows the calculated input impedance and gap voltage of the investigated nano antenna array with 1row×2columns when the spacing is equal to
50nm, 100nm, or 150nm.
162
spacing=50nm,100nm,150nm
antenna MIM
spacing
Figure 74 – 9.3.1.1 Sturcture of a rectenna array 1row×2columns
163
Figure 75 – 9.3.1.2 Impedance of a nano antenna in the array with the spacing of
50nm
Antenna Zin() |Vs|(mV) Upper 142.2-j101.1 7.32 Lower 148.3-j92.6 7.32 ref 120.7-j81.3 7.11
Table 22 –9.3.1.1 Impedance and voltage of the nano antenna at 660THz with the
spacing of 50nm
Figure 9.3.1.2 and table 9.3.1.1 show the impedance and gap voltage of the antenna units in the array. Clearly, the impedances of an antenna unit change slightly from one to another, and a single nano antenna has the impedance of 145-j95 ohm at 660THz.
In the array, two antennas with same geometry oscillate at same frequency, which helps to enhance the electric field and the magnitude of Vs is 7.32mV, which is a little higher than that of the reference antenna (7.11mV). 164
Figure 76 - 9.3.1.3 Impedance of the nano antennas in the array with the spacing of 100nm
Antenna Zin() |Vs|(mV) Upper 139.7-j93.1 7.37 Lower 139.9-j94.6 7.37 ref 120.7-j81.3 7.11
Table 23 - 9.3.1.2 Impedance and voltage of the nano antennas at 660THz with the
spacing of 100nm
165
Similar results are observed when the spacing is equal to 100nm. The antenna impedace becomes small, compared with that of the 50nm spacing. But the received voltage increases about 7%. Note the spacing of 100nm is 1/5 wavelength of the resonance frequency. The close spacing between antenna units show a better mutual coupling performance in field enhancement [136] .
Figure 77 - 9.3.1.4 Impedance of the nano antennas in the array with the spacing of 150nm
166
Antenna Zin() |Vs|(mV) Upper 145.8-j100.1 7.32 Lower 145.9-100.1 7.32 ref 120.7-j81.3 7.11
Table 24- 9.3.1.1 Impedance and voltage of nano antennas at 660THz with the
spacing of 150nm
Figure 9.3.1.4 and table 9.3.1.1 show the results of the array with the spacing of
150nm. The input impedance changes slightly (3%), while the efficiency goes up to
0.9 at 660THz. The voltage of an antenna unit at the gap is around 7.3mV. In these three cases, the spacing of 100nm in the array leads to a relatively higher magnitude of Vs. The reference [136] shows that the close spacing of 0.1~0.3 wavelength can help the mutual resonance. Impedance is approximately 20ohm smaller than others, and at the same time, the voltage is 10% higher than others. The AC voltage on the
MIM diode is approximately 1mV according to section 7.4. The comparision among these three cases shows that better performance is achieved for the array with spacing of 100nm. Figure 9.3.1.5 shows the output average Vac and average Pac on a Rmim
(40ohm). The maximum output AC power is 3.4×10-14W.
167
Figure 78 – 9.3.1.5 Average Vac and Pac for one single Rmim in the array
9.3.2 Rectenna Array of 1Row×4Columns
The following figures show the calculated input impedance and gap voltage of the investigated nano antenna array of 1row×4columns when the spacing is set to be
50nm, 100nm, or 150nm.
168
spacing=50nm,100nm,150nm
antenna MIM
spacing
spacing
spacing
Figure 79 - 9.3.2.1 Sturcture of a rectenna array 1row×4columns
169
Figure 80 - 9.3.2.2 Impedance of the nano antennas in the array with the spacing of 50nm
Antenna Zin() |Vs|(mV) Upper1 147.6-j93.5 7.43 Upper2 142.3-j96.7 7.45 Upper3 142.3-j96.7 7.45 Upper4 147.6-j93.5 7.43 ref 120.7-j81.3 7.11
Table 25 - 9.3.2.1 Impedance and voltage of the nano antennas at 660THz with the
spacing of 50nm
Figure 9.3.2.2 and table 9.3.2.1 show the results for the array with the spacing of
50nm.The impedance of the antenna array is about the same as that of array 1row×
2columns, but the magnitude of Vs is higher than that of the array with 1row×
2columns with the 50nm spacing. This indicates that increasing the number of elements can help the field enhancement. So increasing the number of antenna units leads to an increment of the magnitude of Vs.
170
Figure 81 - 9.3.2.3 Impedance of the nano antennas in the array with the spacing of 100nm
Antenna Zin() |Vs|(mV) Upper1 143.6-j96.4 7.56 Upper2 143.6-j97.1 7.58 Upper3 143.6-j97.1 7.58 Upper4 143.6-j96.4 7.56 ref 120.7-j81.3 7.11
Table 26 - 9.3.2.3 Impedance and voltage of the nano antennas at 660THz with the
spacing of 100nm
171
The impedance is similar for different spacing values, and the gap voltage is slightly higher in the array of 1row×4columns than that in 1row×2columns. The magnitude of Vs inceases to the maximum value of 7.58 mV.
Figure 82 - 9.3.2.4 Impedance of the nano antennas in the array with the spacing of 150nm
172
Antenna Zin() |Vs|(mV) Upper1 145.2-j99.6 7.52 Upper2 135.1-j92.7 7.57 Upper3 135.1-j92.7 7.57 Upper4 145.2-j99.6 7.52 ref 120.7-j81.3 7.11
Table 27 - 9.3.2.3 Impedance and voltage of nano antennas at 660THz with the
spacing of 150nm
Figure 9.3.2.4 and table 9.3.2.3 show the results of the antenna array with the spacing of 100nm. Compared with the other antenna spacing, the impedance is about 210ohm at the resonant frequency of 600THz. And the |Vs| varies from 7.52mV up to the highest 7.57mV. On the other hand, compared with the first group, the array of 1row
×2columns, the average impedance keeps around 260ohm. The magnitude of Vs increases from 7.32mV up to 7.37mV. Clearly, the number of units in the array makes this change. Close spacing helps field enhancement at resonance. The spacing of
100nm leads to a relatively higher magnitude of Vs. The antenna efficiency goes up to
0.91 in an array from 0.87 for a single ref unit. This is also observed in the array of
1rows×2columns. Figure 9.3.2.5 shows the output average Vac and average Pac on a
Rmim (40ohm). The maximum output AC power is 3.6×10-14W.
173
Figure 83 – 9.3.2.5 Average Vac and Pac for one single Rmim in the array
9.3.3 Rectenna Array of 2Rows×2Columns
The following figures show the calculated input impedance and gap voltage of the investigated nano antenna array of 2rows×2columns when the spacing is set to be
50nm, 100nm, or 150nm.
174
spacing=50nm,100nm,150nm antenna MIM
spacing
DC Line length spacing
Figure 84 - 9.3.3.1 Sturcture of a rectenna array of 2rows×2columns
175
Figure 85 - 9.3.3.2 Impedance of the nano antennas in the array with the spacing of 50nm
Antenna Zin() |Vs|(mV) Left Upper 147.1-j93.2 7.60 Left Lower 147.6-j93.5 7.51 Right Upper 147.1-j93.2 7.60 Right Lower 147.6-j93.5 7.51 ref 120.7-j81.3 7.11
Table 28 - 9.3.3.1 Impedance and voltage of the nano antennas at 660THz with the
spacing of 50nm
Figures 9.3.3.2 and table 9.3.3.1 shows the results for the array of 2rows×2columns with the spacing of 50nm.The impedance of the antenna is about the same as that of the previous ones, but the magnitude of Vs is a littele higher. The mutual coupling in these antennas helps to raise the magnitude of Vs.
176
Figure 86 - 9.3.3.3 Impedance of nano antenna in the array with the spacing of 100nm
Antenna Zin() |Vs|(mV) Left Upper 142.8-j96.6 7.53 Left Lower 142.9-j96.7 7.62 Right Upper 142.8-j96.6 7.53 Right Lower 142.9-j96.7 7.62 ref 120.7-j81.3 7.11
Table 29 - 9.3.3.2 Impedance and voltage of nano antennas at 660THz with the
spacing of 100nm
The impedance is similar, and the gap voltage is slightly higher in the array of 100nm
177 unit spacing. The results show that when the spacing is fixed to be 0.2 of the wavelength, the array generate higher Vs.
Figure 87 - 9.3.3.4 Impedance of nano antennas in the array with the spacing of 150nm
Antenna Zin() |Vs|(mV) Left Upper 134.4-j92.2 7.55 Left Lower 134.5-j92.3 7.52 Right Upper 134.4-j92.2 7.55 Right Lower 134.5-j92.3 7.52 ref 120.7-j81.3 7.11 Table 30 - 9.3.3.3 Impedance and voltage of nano antennas at 660THz with the spacing of 150nm
178
Figure 9.3.3.4 and table 9.3.3.3 show the result of the array with 150nm spacing. The impedance is about 260ohm at the resonant frequency of 600THz. This is same as that with 1row×4columns. The magnitude of Vs is about 7.56mV.
It is concluded that the effect of the mutual coupling at resonance depends on the number of antenna uints in the array, which can lead to higher Vs. The array of 2rows
×2columns (plane distribution) can generate higher Vs than that of 1row×4columns though the numbers of them are same. On the other hand, the array with 100nm spacing has a relatively higher Vs. This is because the length is about 0.2 of the wave length, which effectively helps mutual coupling at resonances. AC voltage on the
MIM diode can be up to 1.2mV in the array of 2rows×2columns. Figure 9.3.3.5 shows the average Vac and average Pac on Rmim (40ohm). The maximum output AC power is 3.4×10-14W.
179
Figure 88 – 9.3.3.5 Average Vac and Pac for one single Rmim in the array
9.3.4 Calculated Output Power for Rectenna Array
Based on the results shown in previous sections, the 1rows×4column array with the spacing of 100nm is selected. The array structure shows a field enhancement and a relatively strong mutual coupling under the light illumination compared with a single ref nano antenna.
In the figures 9.3.4.1, a comparison of a single ref unit and one unit in the array is made. The average AC power is 3.5×10-14 W for a single unit and 3.7×10-14 W in the array, before the rectification is made under the incident electric field of 1000V/m at the frequency of 660THz. After rectification, the generated DC power is 3.26×10-14W in a single unit and 3.43×10-14W in the array. Obviously, because of the electric field enhancement phenonenom in the antenna array, the harvested power is higher than that in a single unit. So the generated DC power is two times higher than that of a
180 single unit though the conversion rate is about the same – 93%. The total system efficiency (from optical power to DC power) for a single unit is 13.6% (η0 is 0.87, η1 is 0.17, and η2 is 0.93). While the average value of a single unit in an array is 14.5%
(η0 is 0.92, η1 is 0.17, η2 is 0.93). This means 1% more power is generated by the one unit in an array. The definition of η is defined in equation 8.2.1.4.
Figure 89 – 9.3.4.1 Comparison of Pac for single ref unit and average value of a single unit in an array
Figure 90 – 9.3.4.2 Comparison of Pdc for single ref unit and average value of a single unit in an array
181
Figure 91 – 9.3.4.3 Comparison of total system efficiency for a single unit and the average value of a single unit in an array
9.4 Conclusion
In this chapter, the nano antenna array is studied by varying the spacing between the nano antenna unit (50nm, 100 nm, 150 nm). It is found that the spacing of 1/5 wavelength of the resonance frequency can maximize the near field enhancement of antenna units and the optical energy collection. Narrower spacing causes a disturbance to the resonant frequency of antennas, while larger spacing decreases the number of units in a fixed area and then reduces the output power. Compared with a single antenna unit, the array resonance enhances the energy harvesting for rectification. Simulation results show that field enhancement is observed in an array under the light illumination. The antenna array enhances the near electric field to provide more optical energy for rectification. Based on this result, a comparsion is made that between the average value of one unit in an array and a single unit. More
182
DC power output is provided in an array.
183
10 Conclusion and Suggestions for Future Research
This thesis theoretically studies harvesting solar energy by a nano antenna, which is quite different from traditional photovoltaic technology. The work makes some contributions to the knowledge base in the field of optical nano devices: (1) nano antenna for harvesting solar energy including material selection; (2) optical energy rectification; (3) nano device integration; (4) nano rectenna array.
10.1 Summary
The specific contributions of the work are summarised:
10.1.1 Nano Antenna for Harvesting Solar Energy
In the study of nano antennas for harvesting solar energy at optical spectrum, metal as the material for antennas is considered as a plasmonic material which is very similar to plasma gas. The Drude model is applied to describe the plasmonic material, in which the permittivity is complex. The real part means the storage of energy the while the imaginary parts means the loss of the energy. This makes the plasmonic material to be the lossy dielectric material, and the traditionally conductive current turns into the displacement current. The dipole nano antenna is selected for investigation. By 184 controlling the spacing of two monopoles, maximum near field enhancement is achieved at the gap of the dipole to provide more optical energy for rectification.
Aluminium is selected as the material for a single nano dipole antenna for lightness, good conductiveness, and low cost. As for efficiency, a nano antenna made of aluminium has a higher optical energy harvesting efficiency compared with other materials, which is very important for optical energy coupling. The length of the nano dipole is about a free-space wavelength divided by the square of real part of the relative permittivity. Adjusting the other geometry parameters (width and height) can optimize impedance of the antenna.
10.1.2 Optical Energy Rectification
The MIM (Metal Insulator Metal) diode, based on the theory of tunnelling effect, is adopted for rectifying the optical energy to DC current. The tunnelling effect means that an electron tunnels through a barrier with potential, even the energy of the electron is lower than the potential of a barrier. In the selection of materials for the two metals, the difference of these two band gaps should be significant to generate more current. The thickness of the insulator should be thin enough for electrons tunnelling through. But decreasing the thickness will decrease the capacitance of the
MIM diode, which affects the impedance thus the transmission of optical energy from the optical antenna. The cross section will not affect the tunnelling effect, but a larger area will lead to more current. Like thickness, the impedance is also affected by the
185 area of cross section; the cross section of the MIM diode is the same as the nano antenna. Based on these requirements, Cr2O3 is selected as the insulator. The efficiency of a MIM diode can be nearly 100% based on the result of computer simulation. It can be concluded that the thickness is selected by considering the tunnelling effect and the capacitance of the MIM diode. In addition, the difference of the metal bands gap should be as large as possible.
10.1.3 Nano Device Integration
The rectenna system is made of nano antennas and rectification components. The parameter interaction between two parts is studied. A receiving nano antenna works as a voltage source, and the MIM diode provides the DC current to the external load.
The MIM diode is embedded in the gap of the dipole nano antenna to reduce the optical energy loss. A DC line is buried in the substrate to transmit the rectified DC current and reduces the interference from optical energy. Otherwise, the DC line on the surface of the substrate will also receive optical energy. The length of the DC line nearly has no effect on the operation of the MIM diode and antenna. In the system, the energy harvesting efficiency of the antenna is 87%, the optical energy coupling efficiency of the antenna to the MIM diode is 17%, the AC power to DC power conversion efficiency of the MIM diodeis 93%. So the total efficiency of rectenna system is 14%, which is calculated by the multiplication of the three numbers. In the rectenna system, the generated DC current is provided for an external load. The load
186 voltage has an effect to the rectenna system. The higher AC voltage increases the energy of electrons for tunnelling through the insulator. In order to realize the maximum output power, the DC biased voltage (V) is to be chosen after considering the output DC current (I). It can be concluded that the biased voltage should be selected first to maximum output power is provided. Here, the total power conversion efficiency of a rectenna system is 13.6%, when Vdc is set to be 0.47V, Rmim is
40ohm.
10.1.4 Nano Rectenna Array
The rectenna system is composed of a nano antenna, a MIM diode and a DC line for output, which is in the size of nanometer. The generated DC current is too small. So an array of the rectenna units is formed to provide more power, in which the spacing of units is considered to reduce the unnecessary mutual interference of antenna resonance. Besides, in order to increase the current, parallel connection of units is selected to provide more current for the external loads. The spacing in the nano antenna array means the distance between the antenna units. By comparison, the spacing of 1/5 wavelength of the resonance frequency is set in order to maximize the near field enhancement for optical energy collection. Narrow spacing distance interferes the resonant frequency of antenna, while large spacing decreases the number of units in a fixed area. Compared with a single antenna unit, the array enhances the energy harvesting for rectification. The simulation results shows the
187 field enhancement of single units in an array is better than the only one unit exposed to illumination. The total efficiency of one unit in an array is 1% higher than that of a single unit, and is up to 15%. It can be concluded that the antenna array can enhance the electric near field to provide more optical energy for rectification.
10.2 Future Research
10.2.1 Nano Antenna for Broadband Harvesting Solar Energy
Broadband electromagnetic energy harvesting can bring more energy for rectification by the MIM diode. So a high-efficiency broadband nano antenna is highly expected.
10.2.2 Nano Device Integration for Broadband Rectification
A unit for broadband energy rectification is to be studied, in which parameters such as the broadband impedance matching, and geometry of the MIM diode will be considered.
10.2.3 Nano Broadband Antenna Array Design for Mass
Production Application
Based on the broadband unit, the array can effectively acuiqre more energy than one unit. So the spacing among units is to be investigated for higher field enhancement to
188 increase the energy to be harvested for rectification.
10.3 Conclusion
This project covers many areas such as nano antenna design, semiconductor design, nano device integration, and material engineering. A trade-off plan is needed in manufacturing the whole system, such as cost minimization, quality control for precision of nano devices which are left to be improved in the future. The need for energy brings more and more research attention to energy harvesting. It is hoped that this thesis can bring more and more attention to the energy harvesting by nano antennas.
189
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