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Tatur, Kevin.
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Theoretical studies of long range interactions in quasione dimensional cylindrical structures
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by Kevin Tatur.
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Dissertation (Ph.D.)University of South Florida, 2009.
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Text (Electronic dissertation) in PDF format.
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ABSTRACT: Casimir forces originating from vacuum fluctuations of the electromagnetic fields are of increasing importance in many scientific and technological areas. The manifestations of these longrange forces at the nanoscale have led to the need of better understanding of their contribution in relation to the stability of different physical systems as well as the operation of various technological components and devices. This dissertation presents mathematical and theoretical methods to calculate the Casimir interaction in various infinitely long cylindrical nanostructures. A dielectricdiamagnetic cylindrical layer immersed in a medium is first considered. The layer has a finite thickness characterized with specific dielectric and magnetic properties. Another system considered is that of perfectly conducting concentric cylindrical shells immersed in a medium. The electromagnetic energy between two infinitely long straight parallel dielectricdiamagnetic cylinders immersed in a medium is also considered. The mode summation method is used to calculate the Casimir energy of all these systems. The energy dependence on the cylindrical radial curvature and dielectric response of the cylinders is investigated. The fundamental effects of these long range interactions are studied in the form of excitonplasmon interactions in carbon nanotubes and this is achieved by looking at the dielectric response of carbon nanotubes.
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Mode of access: World Wide Web.
System requirements: World Wide Web browser and PDF reader.
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Advisor: Lilia M. Woods, Ph.D.
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Electromagnetic interactions
Casimir force
Nanotechnology
Mathematical methods
Carbon nanotubes
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Dissertations, Academic
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x Physics
Doctoral.
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t USF Electronic Theses and Dissertations.
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u http://digital.lib.usf.edu/?e14.3135
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Theoretical Studies of LongRange Interactio ns in QuasiOne Dimensional Cylindrical Structures by Kevin Tatur A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Physics College of Arts and Sciences University of South Florida Major Professor: Li lia M. Woods, Ph.D. Inna Ponomareva, Ph.D. Sagar A. Pandit, Ph.D. Norma A. Alcantar, Ph.D. Date of Approval: October 7, 2009 Keywords: Electromagnetic interactions, Casimir force, nanotechnology, mathematical methods, carbon nanotubes Copyright 2009, Kevin Tatur
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Dedications To my mother, father, Meenakshi and Tanushri.
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ACKNOWLEDGMENTS I would like to thank my family for all their support, love and encouragement throughout my Ph.D. studies. I had the pleasure of work ing and learning at the side of my Ph.D. advisor, Dr. Lilia Woods, who was an insp iration and a mentor during my studies. I would like to express my gratitude towards he r for her patience and tu telage. I would also like to thank Dr. Sagar Pandit, Dr. Inna P onomareva, Dr. Norma Alcantar and Dr. Razvan Theodorescu for agreeing to be in Ph.D. com mittee. Lastly, I would like to thank God for helping me achieve what I had set out to accomplish.
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i TABLE OF CONTENTS LIST OF FIGURES iv ABSTRACT viii CHAPTER 1 INTRODUCTION AND BACKGROUND 1 1.1 Long Range Dispersion Forces 1 1.2 Dissertation Outline 5 CHAPTER 2 THEORETICAL IN VESTIGATIONS OF CASIMIR ENERGY 7 2.1 Important Thoretical results 7 2.2 Experimental Observations 11 CHAPTER 3 QUASI ONEDIMENSIONAL ST RUCTURES 23 3.1 Cylindrical Nanostructures 23 3.2 Carbon Nanotubes (CNTs) and Devices 25 CHAPTER 4 ZERO POINT ENERGY AND MODE SUMMATION METHOD 29 4.1 Theoretical Investigations of Zero Point (Casimir) Energy 29 4.2 Overview of Theoretical Methods 30 4.3 Mode Summation Method Applied to Cylindrical Geometries 33 4.3.1 Zeta Function Regularization 38
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ii CHAPTER 5 CYLINDRICAL DIELEC TRICDIAMAGNETIC LAYER OF FINITE THICKNESS 40 5.1 Cylindrical Model 40 5.2 Electromagnetic Modes 41 5.3 Casimir Energy of DielectricDia magnetic Cylindrical Layer 45 5.3.1 The case of 1 m m 46 5.3.1.1 Limiting Cases 53 5.3.1.2 Numerical Results 54 5.3.2 The case of 1 m m 58 5.3.2.1 Numerical Results 61 CHAPTER 6 N PERFECTLY CONDUCTING CYLINDRICAL SHELLS 65 6.1 Cylindrical Model 65 6.2 Electromagnetic Modes 66 6.3 Casimir Energy of N Perfectly Metallic Cylindrical Shells 69 6.3.1 Limiting Cases 72 6.3.2 Numerical Results 73 CHAPTER 7 TWO PARAL LEL DIELECTRICDIAMAGNETIC CYLINDERS 80 7.1 Cylindrical Model 80 7.2 Electromagnetic Modes 81
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iii 7.3 Casimir Energy of Two Straight Parallel Cylinders 85 7.3.1 Limiting Cases 88 7.3.2 Numerical Results 90 CHAPTER 8 EXCITONPLASMON COUPLING IN CARBON NANOTUBES 97 8.1 Fundamental Effects of Long Range Interactions 97 8.2 Electronic Structure of Ca rbon Nanotubes (CNTs) 99 8.3 Dielectric response of Carbon Nanotubes (CNTs) 105 8.4 ExcitonPlasmon Interaction in Semiconducting Single wall CNTs 110 CHAPTER 9 SUMMARY 116 9.1 Overview and Conclusion 116 9.2 DielectricDiamagnetic Cylindrical Layer of Finite Thickness 117 9.3 N Perfectly Conducting Cylindrical Shells 119 9.4 Two Straight and Parallel Diel ectricDiamagnetic Cylinders 119 9.5 ExcitonPlasmon Coupling Effect 120 REFERENCES 121 ABOUT THE AUTHOR End Page
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iv LIST OF FIGURES Figure 2.1 Schematic of LamoreauxÂ’s experiment [29]. 11 Figure 2.2 Measured force as a func tion of absolute separation [29]. 14 Figure 2.3 Schematic of Mohideen and RoyÂ’ s experiment [30]. 15 Figure 2.4 Measured Casimir force as a function of platesphere separation [30]. 16 Figure 2.5 Schematic of ChanÂ’s experiment [37]. 19 Figure 2.6 Measured angles of rotation of the plate and the Casimir force as a function of platesphere separa tion [37]. The red line represents the Casimir force after taking the roughness of metal lic surfaces into account. The green line repr esents the electrostatic force to demonstrate the operation of the device. 20 Figure 2.7 (a) Schematic of MundayÂ’s experiment. (b) Quantum levitation when 2 3 1 is satisfied. Repulsion is achieved when the dielectric response of the medium3 is between the dielectric responses of the materials, 1 and2 [38]. 21 Figure 2.8 Measured Casimir force as a function of sphereplate distance. The blue (orange) circles repres ent the force between the gold sphere and a silica (gold) plate in bromobenzene. Repulsion only
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v occurs between the gold sphere and silica plate [38]. 22 Figure 3.1 CNT structure is fully sp ecified by a twodimensional chiral vector 2 1ma na Ch T is the translati onal vector along the axial direction [44]. 24 Figure 3.2 Schematic of a nanooscillat or. Figure E shows the inner tube being brought back in by van der Waal s interaction [20]. 27 Figure 3.3 Bundles of carbon nanotubes used for hydrogen storage [41]. 28 Figure 4.1 Contour in the complex fr equency plane with poles along the real axis. 37 Figure 5.1 Cylindrical laye r of finite thickness with its axial direction perpendicular to the page is immersed in an infinite medium. The layer has permittivity and permeability ,, respectively, and the medium m m ,. The interfaces are denoted as I and II. 40 Figure 5.2 The Casimir energy per unit length for a cylindrical dielectricdiamagnetic la yer as a function of R2/R1. 56 Figure 5.3 The Casimir energy per unit length for the same layer as a function of inner radius R1. 57 Figure 5.4 The Casimir energy per un it length for the same layer as a function of 1 2/ R R 62 Figure 6.1 Infinitely long perfectly conducting and c oncentric cylindrical shells immersed in an infin ite medium. The axial direction
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vi is perpendicular to the page. The radii of the shells are iR where i=1,2,Â…N. 65 Figure 6.2 The Casimir energy for the case of N=3 shells as a function of the inner radius R1. 74 Figure 6.3 The Casimir energy for the case of N=3 shells as a function of the radius of the second shell R2. 76 Figure 6.4 The Casimir energy for the case of N=3 shells as a function of the separation between the outer tw o shells. 77 Figure 6.5 The Casimir energy as a func tion of the number of concentric cylindrical shells. 79 Figure 7.1 Two infinitely long parallel cylinders of radii1Rand2R with centertocenter separation R immersed in an infinite medium. The cylindrical axis is perpendicular to the page. 80 Figure 7.2 Dimensionless interaction energy as a function of surfaceto surface separation defined as2 1R R R d 0CE is defined as 2 1/ R cl For all cases, 22 1 and153 93 Figure 7.3 Dimensionless interaction ener gy as a function of the dielectric function of the medium. 0CE is defined as 2 1/ R cl The cylinders have equal radii, nm 12 1 R Rand centertocenter separation, nm 2 2 R. 94 Figure 7.4 Dimensionless interaction ener gy as a function of the dielectric
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vii function of one cylinder. 0CE is defined as 2 1/ R cl The cylinders have equal radii, nm 12 1 R Rand centertocenter separation, nm 2 2 R. 95 Figure 8.1 Band Structure of the (8,0) CNT. 102 Figure 8.2 Band Structure of the (10,0) CNT. 103 Figure 8.3 Band Structure of the (11,0) CNT. 104 Figure 8.4 Dielectric Response of the (8 ,0) CNT. Frequency is measured in eV and the dielectric function is dimensionless. 107 Figure 8.5 Dielectric Response of the (8,0) CNT in the low energy regime. Frequency is measured in eV a nd the dielectric function is dimensionless. 108 Figure 8.6 Dielectric Response of the ( 10,0) CNT. Frequency is measured in eV and the dielectric function is dimensionless. 109 Figure 8.7 Dimensionless conductivity of the (11,0) CNT as a function of dimensionless energy. 111 Figure 8.8 Dimensionless conductivity of the (10,0) CNT as a function of dimensionless energy. 112 Figure 8.9 Dispersion curves of th e exciton and plasmon for the (11,0) CNT. 114 Figure 8.10 Dispersion curves of the exciton and plasmon for the (10,0) CNT. 115
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viii THEORETICAL STUDIES OF LONGRA NGE INTERACTIONS IN QUASIONE DIMENSIONAL CYLINDRICAL STRUCTURES Kevin Tatur ABSTRACT Casimir forces originating from vacuum fl uctuations of the electromagnetic fields are of increasing importance in many scientific and technological areas. The manifestations of these longrange forces at the nanoscale have led to the need of better understanding of their contribu tion in relation to the stab ility of different physical systems as well as the operation of various technological components and devices. This dissertation presents mathematical and theo retical methods to calculate the Casimir interaction in various infinitely long cylindr ical nanostructures. A dielectricdiamagnetic cylindrical layer immersed in a medium is first considered. The layer has a finite thickness characterized with specific dielectr ic and magnetic proper ties. Another system considered is that of perfectly conducting co ncentric cylindrical shells immersed in a medium. The electromagnetic energy between two infinitely long straight parallel dielectricdiamagnetic cylinders immersed in a medium is also considered. The mode summation method is used to ca lculate the Casimir energy of all these systems. The energy dependence on the cylindrical radial curvature and dielectric response of the
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ix cylinders is investigated. The fundamental effects of these long range inte ractions are studied in the form of excitonplasmon in teractions in carbon nanotubes and this is achieved by looking at the dielectri c response of carbon nanotubes.
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1 CHAPTER 1 INTRODUCTION AND BACKGROUND 1.1 Long Range Dispersion Forces Long range dispersion forces are forces that originate from the electromagnetic interaction between electrica lly neutral objects with no permanent electric and/or magnetic moments and are quantum mechanical in nature [16]. H.B.G. Casimir was the first to explain the consequences of electromagnetic zeropoint energy in real macroscopic objects [7]. He predicted the attr action between a pair of neutral, parallel conducting plates in vacuum. This attraction was a consequence of electromagnetic field fluctuations in vacuum and referred as th e now wellknown Casimir effect. The advent of dispersion forces is regarded as one of the most important achievements in quantum electrodynamics (QED), where th e origin of the forces is explained by the groundstate (zeropoint) fluctuations. Thes e long range dispersion forces manifest themselves as van der Waals, CasimirPolder, CasimirLifschitz, and Casimir forces. The concept of long range dispersion for ces was first explaine d by the LifschitzÂ’s theory of the van der Waals forces [1] by cons idering objects as a di stribution of neutral, polarizable atoms. By surface integration th e contribution of all atoms on the object was obtained. However, dispersion forces are not additive and this prevents objects to be
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2 treated as composed of elementary const ituents. Although the Casimir effect can be explained by LifschitzÂ’s theory of the van de r Waals forces, modern quantum field theory offers a more basic explanati on from first principles of th e long range forces, due to a change of the vacuum energy. In other words, a deviation of the zer opoint energy caused by the presence of boundaries gives rise to Casimir for ces. The zeropoint energy investigations have led to the physical va cuum energy of a quantized field to be calculated as the difference be tween the zeropoint energy corresponding to the vacuum configuration with constraints and the free vacu um configuration with no constraints [1]. Dispersion forces exist on different scales and levels. For example, properties of weakly bound molecules are affected by disper sion forces on a microscopic level [8,9]. On a macroscopic level, prope rties of solids and liquids ar e influenced by dispersion forces [1012]. Capillarity and flocculation are examples of such effects where atomsurface dispersion interactions influence atomssurface propert ies, such as the wetting properties of liquids on surfaces [1315]. In cosmology, the formation of planets around stars is a consequence of dust aggregation th at is initiated by dispersion forces [16]. Dispersion forces have also become of si gnificant importance to science and technology, predominantly nanotechnology. As devices are fabricated on the micron and submicron scale, the long range dispersion forces b ecome more pronounced in this process of miniaturization. There are two sides to the im portance of long range dispersion forces in nanotechnological devices. On one hand, the stability of many nanostructured materials relies on these long range forces. Cylindr ically wrapped graphene sheets in carbon
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3 nanotubes and boron nitride layers in boron nitride nanotubes are examples of such materials [1719]. In additi on, oscillating carbon nanotub es or buckyballs inside a stationary carbon nanotube have been demonstrat ed and related to those long range forces [20,21]. Also, with the advent of nanostructured devices and the increase in the number of experiments performed to demonstrate the Casimir effect, there is now the means to improve our fundamental knowledge and studie s of longrange dispersion forces in nontrivial geometries as well as studying how thes e forces manifest them selves in practical devices. On the other hand, dispersion forces can become a hindrance and cause drastic effects causing undesired and permanent stic king of parts in nanostructures [2224]. Microand nano electromechanical system s (MEMS and NEMS) are examples of such devices where the Casimir force can cause components to collapse and stick to one another and thereby resulting in the machin es behaving erratically, a phenomenon called stiction. Dispersion forces play a very important role in biology. These forces are found to be responsible for the intera ction of molecules with cell membranes and for cell adhesion driven by mutual cellmembrane interactions [25]. The stability of hydrocarbon cylinders, viruses, muscle protein, and helical assemblie s in aqueous solutions has been linked to long range forces. Another key role of long range dispersion forces is the interaction between colloids and solvents The van der Waals forces ha ve become recognized as the dominant interaction regarding the stability of colloids [26]. As colloids consist of many atoms, there are dispersion forces acting be tween them. Usually colloids are placed in
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4 solvents and this results in colloidsolvent and solventsolvent interactions due to dispersion forces. Therefore it can be seen that long range forces play a key role in various fields and as a result it becomes im portant to understand th e behavior of long range interactions in nanostructures. In this work, the goal is to provide a framework where one can obtain a better understanding of how long range interactions behave in cylindrical nanostructures by taking into account the finite speed of light and dielectric constant of the considered structures. We consider the th eoretical studies of long range interactions or iginating from the electromagnetic field fluctuations in vari ous cylindrical nanostructures, such as, a dielectricdiamagnetic cylindrical layer of finite thickness, multiple concentric cylindrical metallic shells, and between two parallel dielectricdiamagnetic cylinders, all immersed in an infinite medium. Such systems are of pa rticular interest because they can serve as models to study the Casimir interaction in cy lindrical tubular structures, such as carbon nanotubes, boron nitride nanotubes, nanowires DNA, etcÂ…It can also provide a test ground of how curvature effects coupled with dielectric and/or magnetic properties influence the mutual interaction in cyli nders. We adapt an intuitive and elegant theoretical model to in vestigate the long range interactions in cylindr ical nanostructures.
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5 1.2 Dissertation Outline Chapter 1 gives an overview of the motivation of the dissertation. More precisely, a background of how long range dispersion forces influence the behavior of nanostructured devices is given. The origin of long range dispersion forces is discussed and their manifestations in the form of Casimir effects and van der Waals forces are described in Chapter 2. A brief history of the theoretical and experimental milestones of long range interactions is given. The manifest ation and use of long ra nge interactions in quasi onedimensional structures are described in Chapter 3. In Chapter 4, a brief overview of the theo retical methods that have been used to calculate the Casimir energy of various sy stems is given and their limitation are explained. A more detailed explanation of th e mode summation method is given since it is the method of choice in the Casimir en ergy calculations in this dissertation. In Chapter 5, the Casimir energy of a system of a dielectricdiamagnetic cylindrical layer of finite thickness immersed in an infinite medium is calculated. Again, the methodology of applying the mode summation method is described and the techniques of how to remove the divergen ces are explained. Various limiting cases are investigated and numeri cal results presented. Chapter 6 deals with the Casimir energy of N perfectly metallic cylindrical shells immersed in an infinite medium. Again the mode summation approach is explained with
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6 regards to this particular m odel and limiting cases and numeri cal results are considered in terms of radial dimensions and material composition. In Chapter 7, the Casimir energy of tw o straight and para llel dielectricdiamagnetic cylinders is calcu lated and the numerical results explained with respect to radial dimensions and material composition. Chapter 8 describes the fundamental eff ects of longrange interactions in carbon nanotubes. These interactions ex ist in the form of excitonp hoton coupling that result in excitonplasmon coupling in smalldiam eter semiconducting carbon nanotubes. The plasmonic nature of nanotubes is explained in this chapter by investigating the dielectric response of carbon nanotubes using the random phase approximation. Finally, Chapter 9 gives the conclusion to this dissertation.
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7 CHAPTER 2 THEORETICAL RESULTS AND EXPERIMENTAL OBSERVATIONS 2.1 Important Theoretical Results The history of dispersion fo rces can be traced back to the work of D. van der Waals [27] where the weak attractive forces between neutral molecules were introduced. Later, F. London [28] gave a precise formula tion of the nature and strength of van der Waals forces as due to the interactions of the fluctuating electric dipole moments of the neutral molecules resulting in the neutral bodi es attracting each other. He expressed the potential energy of two isotropi c groundstate atoms as follows: 6) ( r C r U (2.1) with k k k kE E E E k k C ) ( d Âˆ 0 d Âˆ 0 24 10 0 2 2 2 0 2 (2.2) where d Âˆandd Âˆ are the electric dipoles, k and k are the eigenstates, kE and kE are the energies of the unperturbed atoms [1].
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8 The works of Casimir and Polder addresse d a more technical point, the fact that the polarization of the neighboring molecules or atoms induced by a given molecule or atom is delayed as a consequence of the veloc ity of light being finite. These forces are the long range retarded dispersion van der Waals forces. Casimir and Polder formulated the potential energy corres ponding to two atoms sepa rated by a distance r and with static polarizabilities 1 and 2 as: 7 2 14 23 ) ( r c r U (2.3) This was then extended to macroscopic sc ale by performing surface integration to obtain the force per unit area be tween two neutral, parallel, perfectly conducting metallic plates in vacuum. This extension was done under the hypothesis that the macroscopic plates comprise of a distribution of neutra l, polarizable atoms. Another formulation obtained by Casimir and Polder was that of the potential energy of a particle of polarizability inside a cavity of a perfectly conducting material where the particle was separated from the flat wall by a distance r : 48 3 ) ( r c r U (2.4) Two major difficulties arise from the works of London and Casimir and Polder. LondonÂ’s theory did not take into account the finite velocity of propagation of the electromagnetic interaction. Although this had been taken into account in the works of Casimir and Polder, there was still the fact that they viewed macr oscopic objects as a
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9 collection of neutral, polarizable atoms. Th is hypothesis is flawed since van der Waals forces are not additive thereby preventing us to treat macroscopic objects as a collection of neutral atoms. Lifschitz approached this problem by treating matter as a continuum with a welldefined, frequencydependent dielectric susceptibility. LifschitzÂ’s theory was a closed theory th at could deal with any kind of material bodies and it also explained in a precise wa y the transition from one power law to the other due to retardation effect s. It also contained the fo rmulas of London and that of Casimir and Polder. LifschitzÂ’s expressed th e formula for the force per unit area between two parallel infinitely conducti ng plates separated by a distance r as: 4 2240 r c F (2.5) For the case of two identical di electrics of dielectric constant0 the force per unit area was: ) ( 1 1 2400 2 0 0 4 2 r c F (2.6) where ) (0 is a function which has the following behavior for 0 : 6 7 ln 11 1 1 ) (0 0 0 (2.7) For the case of a metal and a dielectric of constant 0 the force per unit area is:
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10 ) ( 1 1 2400 0 0 4 2 r c F (2.8) The important point about LifschitzÂ’s theo ry is that the only information required to calculating the dispersion force is that of the dielectric properties of the body, in particular the dielectric susceptibil ity as a function of frequency.
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11 2.2 Experimental Observations Figure 2.1 Schematic of Lamorea uxÂ’s experiment [29].
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12 Although the Casimir effect was first pr oposed in 1948, its first experimental observation was shown by S. K. Lamoreaux in 1997 [29]. Very fe w experiments had been performed before to verify the Casimir effect mostly due to difficulty in keeping two plates parallel with each other. Also there was a lack of sensitive equipments available at the time to measure the Casimir force. Since those early days, sophisticated equipment has made it much easier to study the Casimir effect and a new generation of experiments and measurements began in 1997 w ith Lamoreaux. The experimental setup is shown in Figure 2.1. A gold coated spherica l lens of 4cm diameter is brought close to a flat plate by means of a piezo stack. The flat plat is mounted on one arm of the torsion balance. The other arm of the torsion balance formed the central electrode of a pair of parallel plate capacitors. By applying voltages to the capacitors the restoring force required to compensate for a ny torque in the torsion bala nce could be measured, as a measure of the Casimir force. The Casimir fo rce was measured by applying the voltage to the piezo stack through discrete and constant steps and at each step, the restoring force was measured. The relative displacement betw een the plates was measured as a function of the discrete steps by using a laser in terferometer. It was found that the average displacement per 5.75 volt step was about 0.75 m and it was ensured that the system was in equilibrium between each measurement. Th e piezo stacks gave very accurate results for the relative plate separation and the ab solute plate separation was determined by measuring the residual electrical attraction betw een the plates as a function of separation. The experimental results agreed with the theo ry to a level of 5%. The closest approach
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13 between the plates was found to be 0.6m and the experimental Casimir force measurements are shown in Figure 2.2. Inspired by LamoreauxÂ’s breakthrough, Mohideen and Roy performed an important experiment to estimate the Casimir force using atomic force microscope [30]. The experimental setup is shown in Figure 2.3 where the force between an aluminium coated polystyrene sphere 200m in diameter and an optically polished flat sapphire disk, also aluminium coated, was measured by the de flection of a laser beam. An atomic force microscope (AFM) was used to measure the fo rce between the sphere and plate and in the AFM, the force was measured by the deflectio n of its tip. A force on the sphere would cause the laser beam to be reflected from the cantilever on which the sphere is mounted due to deflection, and the posi tion of the reflected beam dete rmined by the output of a pair of photodiodes. A piezo stack is used to br ing the flat disk close to the sphere in 3.6 nm steps. The Casimir force measurements obt ained are shown in Fi gure 2.4. Here, the experimental results agreed w ith the theory to 1% when th e sphere was brought to within 0.1m to the plate. Other experimental dem onstrations of the Casimir energy have been performed using atomic force microscope a nd a good review is available in references [3136].
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14 Figure 2.2 Measured force as a function of absolute separation [29].
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15 Figure 2.3 Schematic of Mohideen and RoyÂ’s experiment [30].
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16 Figure 2.4 Measured Casimir force as a functio n of platesphere separation [30].
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17 The first manifestation of the Casimir e ffect in microelectromechanical sytems was demonstrated experimentally by Chan et al. at Bell Labs Luci ent Technologies where they demonstrated the interaction of the Ca simir force with a ME MS device in 2001 [37]. By bringing a 200 m diameter gold coated sphere close to a polysilicon plate suspended 2 m above a substrate, on thin torsional rods, they were able to rotate the plate when the sphere came to within a few 100 nm. The sc hematic of this experiment is shown in Figure 2.5. The attraction between the polysilicon plate and the sphere results in a torque that rotates the plate about the torsional rods. The capacitance between the plate and electrodes placed underneath the plate allows the rotation of the plate to be detected. In the presence of an external torque, the pl ate tilts causing an increase on one of the capacitances and a decrease on the other. Th is difference in capacitance gives the angle of rotation of the plate in response to the at tractive force. The micromachined device is placed on a piezoelectric translation stage with the sphere close to one side of the plate and the plate is brought close to the sphere by extending the piezo. The measured angles of rotation and subsequent Casimir forces are shown in Figure 2.6. The experiments performed for the Casimir force measurements in various systems have so far reported attractive intera ctions. However, recently J. N. Munday, F. Capasso and A. Parsegian have been able to demonstrate a repulsive Casimir force [38]. Repulsive Casimir energy has been predicte d theoretically [39] but no experimental verifications had been performed until lately. The experimental setup is shown in Figure 2.7. The measurements are performed between a 39.8 m gold coated sphere attached to
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18 a cantilever and mounted on an atomic force mi croscope and a large silica plate separated by a fluid, bromobenzene. To detect the be nding of the cantilever, light form a super luminescent diode is reflected off the back of the cantilever. An As ylum Research linear variable differential transformer is used to control a piezo column and this enables the sphere to be moved towards the plate. Any interaction between the sphere and the plate will result in the bending of the cantilever which in turn will cause a change in the detector signal that controls the difference in light intensity between the top half and the bottom half of the detector. The difference in signal is found to be proportional the force and repulsion is achieved when the dielectric response of the medium3 is between the dielectric responses of the materials, 1 and2 as shown in Figure 2.8.
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19 Figure 2.5 Schematic of ChanÂ’s experiment [37].
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20 Figure 2.6 Measured angles of rotation of th e plate and the Casimir force as a function of platesphere se paration [37]. The red line represents the Casimir force after taking the roughness of metallic surfaces in to account. The green line represents the electrostatic force to demonstrate the opera tion of the device.
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21 (a) (b) Figure 2.7 (a) Schematic of MundayÂ’s experiment. (b) Quantum levitation when 2 3 1 is satisfied. Repulsion is achieved when the dielectric response of the medium3 is between the dielectric responses of the materials, 1 and2 [38].
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22 Figure 2.8 Measured Casimir force as a function of sphereplate distance. The blue (orange) circles represent the force betw een the gold sphere and a silica (gold) plate in bromobenzene. Repulsion only occu rs between the gold sphere and silica plate [38].
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23 CHAPTER 3 QUASI ONEDIMENSIONAL STRUCTURES 3.1 Cylindrical Nanostructures The field of nanotechnology is rapidly growing and a multitude of cylindrical nanostructures have become key components in a wide variety of nanotechnological devices. The stability of many nanostructured ma terials is also found to be influenced by the long range dispersion for ces. Cylindrically wrapped graphene sheets in carbon nanotubes, boron nitride layers in boron n itride nanotubes, nanotube bundles, and ropes are examples of such materials [1719,20, 40,41]. Carbon nanotubes have been shown to be applicable in the form of nanooscillators [20] and nano rotors [40]. They have also been proposed to be of importance in ar eas such as energy storage [41] and in optoelectronic device applica tions areas such as nanophot onics and cavity quantum electrodynamics [42,43].
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24 Figure 3.1 CNT structure is fully specified by a twodimensional chiral vector 2 1ma na Ch T is the translational vector along the axial direction [44].
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25 3.2 Carbon Nanotubes (CNTs) and Devices A single wall (n, m) carbon nanotube is a rolledup gr aphite sheet into a seamless cylinder, the structure of which is fully specified by a twodimensional chiral vector2 1ma na Ch where 1a and 2a are primitive lattice vectors of a graphite sheet and n, m are integers defining the structure of the nanotube as shown in Figure 3.1. A nanotube can be characterized as a metal or semiconductor depending on radius and chirality [45]. They have typica l diameters of the order of ~1 nm to ~10 nm and are up to 1 cm in length [42]. As mentioned earlier, carbon nanotubes have been shown to be useful in nanotechnological devices such as nanooscillators [20] Nanooscillators are designed using multiwalled carbon nanotubes, in which one nanotube is found to oscillate in an out of an outer nanotube, a process called telescoping. A nanomanipulator is brought into contact with the i nner tube as shown in Figure 3. 2 (C) and is spotwelded to the inner tube by means of a short, controlled el ectric pulse. In this way, the inner tube is brought in and out of the outer tube and th ereby the atomicscale nanotube surface wear and fatigue is studied. Finally, when the na nomanipulator is disengaged from the inner tube, the latter is brought back in by the inne rtube van der Waals force as seen in Figure 3.2 (E). A nanorotor is another device that has been demonstrated to make use of nanotubes [40]. The rotor consists basically of a gold rotor on a nanotube shaft that spins between electrodes. Such motors could be used in optical circuits to redirect light, a process called optical switching. The rotor could also be used to create a microwave
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26 oscillator, and the spinning rotor could be us ed to mix liquids in microfluidic devices. Since these rotors make use of mutiwall nanotubes, friction between the nested nanotubes can become a major problem to th e smooth running of the device and therefore a good understanding of how long range interact ions influence those devices in needed. Carbon nanotubes have also been proposed to be very useful in storing hydrogen atoms on the surface of nanotubes by adsorp tion [41]. Singlewall and multiwall nanotubes have been found to be able to store significant amounts of hydrogen at room temperature. The hydrogen molecules are ad sorbed on the walls of the nanotubes as shown in Figure 3.3 and interactions between the molecules and the surface rely on long range dispersion forces. It is clear that long ra nge interactions play an important role in various nanotechnological devices which ar e cylindrical in geometry. Therefore one needs to study and understand the behavior of longrange interactio ns in cylindrical nanostructures so that there can be improvements leading to more efficient operation of nanotechnological devices. This is the goal of this dissertation where the Casimir energy in various cylindrical nanostructures, such as a dielectricdiamagnetic cylindrical layer, multiple concentric cylindrical metallic shells, and two parallel dielectricdiamagnetic cylinders, all immersed in an infinite medium will be calculated.
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27 Figure 3.2 Schematic of a nanooscillator. Figure E shows the inner tube being brought back in by van der Waals interaction [20].
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28 Figure 3.3 Bundles of carbon nanotubes used for hydrogen storage [41].
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29 CHAPTER 4 ZERO POINT ENERGY AND MODE SUMMATION METHOD 4.1 Theoretical Investigati ons of Zero Point (Casimir) Energy The importance of the Casimir effect in nanostructured devices and the increase in the number of experiments performed to demons trate the Casimir effect have led to many theoretical investigatio ns using a wide array of methods. These theoretical investigations have been performed on a variety of geomet ries and materials that closely resemble nanotechnological devices and components. The Casimir force is found to depend strongly on the shape, geometry/topology a nd material composition of the boundary or system under consideration. The magnitude and sign of the energy is found to change according to the geometry, topology and type of material constituting the system. Depending on the system, the Casimir energy can be positive (repulsive), negative (attractive), or even zero. For systems in Ref. [7,27,46,47], the energy was found to be attractive. Similarly, for a metallic conducti ng cylindrical shell and eccentric metallic cylinders, the energy is also found to be attractive [3,5,6 ]. However, for a dielectric ball, the energy was calculated to be repulsive [ 48] and for a dielectric cylinder the energy came out to be zero [3,5]. There is an in teresting case where the sign of the energy changes as a function of the dimensions of a box [49,50]. For a medi a of excited atoms,
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30 the Casimir energy can be repulsive or attr active depending on the resonance frequency [51] and in the case of anisotropic and permeab le rectangular plates, the force is repulsive [52]. By altering the material composition of the nanostructures, the sign of the Casimir force can be switched [38, 53]. It ha s been predicted that the interaction between planar materials immersed in a medium can be repulsive if the value of the dielectric constant of the medium is between the values of the dielectric constants of the materials [39]. Recent measurements of the Casimir force between a large sphere and a plate covered with a layer of silica, for which this condition for the dielec tric properties is satisfied, demonstrate that th e interaction can be repulsive [38]. Metamaterials can be used to modify the Casimir energy from attr active to repulsive by placing them between two conducting plates [38]. This can lead to a phenomenon called quantum levitation, where the repulsive Casimir force causes an objec t to Â“floatÂ” or levitate on top of the lefthanded material. This can significantly reduce friction in nanostructured devices (MEMS and NEMS) and resulting in a smoother and more efficient operation of devices. 4.2 Overview of Theoretical Methods There has been a variety of theoretical methods used to calculate long range forces in nonplanar geometries. One of the methods used involved pairwise integration. The potential energies of interaction be tween two parallel, in finitely long carbon nanotubes of the same diameter in various arrangements were computed by making use
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31 of pairwise integration [54]. A continuous di stribution of atoms on the tube surface is assumed and the LennardJones (LJ) carbon carbon potential was used. Although this method was used to calculate the van der Waal s interactions in graphitic structures for various configurations of C60 molecules interacting with ca rbon single walled nanotubes, it was flawed due to the fact that va n der Walls energy is not additive. Long range forces, in particular Casimir forces in certain systems can also be calculated using the semiclassical method. In this method, the limit0 is taken and the energy is evaluated as a su m over the periodic orbits th at reflect the surfaces of the system [4,55]. By considering periodic orb its that make contact with the boundary surface, the infinities never appear. This me thod has been used to calculate the Casimir energy in nonsymmetric configurations, be tween a plane and a sphere, and between concentric cylinders and spheres. Howeve r the semiclassical method is not purely quantum mechanical and as a result it does not give a very good approximation for the Casimir forces. Optical approximation is another theoretical method used to calculate Casimir forces [56]. The optical approxima tion takes into account the optical paths between surfaces, surfaces which are curved, an d optical paths through points that do not lie on straight lines normal to one surface or the other. This method is mostly suitable when calculating the Casimir energy between planar structures. Proximity force approximation (PFA) is yet another technique used for Casimir energy calculations [57]. This method is valid for surfaces where the separation is much
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32 smaller than typical local curvatures. It ha s been successfully used to calculate the Casimir energy of a sphere and cylinder in front of a plane. Roughly speaking, the proximity force approximation maps the Casi mir effect of an arbitrary geometry onto CasimirÂ’s parallelplate configuration. The major problem with this approach is that curvature effects are being neglected and th ereby limiting the Casimir force calculations. One of the most advanced and a recent approach is based on macroscopic quantum electrodynamics of dispersing and ab sorbing media [see Ref. [1] for latest review. There, the vacuum electric and magnetic fields are considered as primary physical observables whose quantum mechani cal operators are give n by the convolution of the Green tensor of the Fourierdomain Maxwell equations with appropriately chosen bosonic field operators for creation and annihilation of the mediumassisted electromagnetic field quanta. In this appr oach, the Green tensor and the (complex) mediumassisted dielectric and magnetic perm eabilities for a particular geometry are responsible for the dissipation processes. Although this appr oach allows one to use to take into account realistic pr operties of materials such as frequencydependent dielectric permittivities, the problem becomes far too complicated for cylindrical geometries where the Green tensor used becomes too big to keep track of. This method had been successfully utilized in dealing with spherical geometries where the Green tensor is not as huge.
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33 4.3 Mode Summation Method Applied to Cylindrical Geometries The theoretical approach that gives us an exact Casimir energy for geometries with curvature effects is the mode summa tion method. The mode summation approach involves representing the Casimir energy in a very intuitive manner as a sum of the zeropoint energies of the electromagnetic exci tations supported by the system. One obtains these photon energies from the dispersion re lation that corresponds to the system under consideration. The photon energi es or electromagnetic modes are obtained by solving the MaxwellÂ’s equations with appropriate boundary conditions. The Casimir energy can then be expressed by the sum over all modes, as follows [36]: } {2p p CE (4.1) The term p represents the eigenfrequencies satisfying the dispersion relation supported by the system. Here { p } are the complete set of quantum numbers determined by the geometry of system under consid eration. For a cylindrical structure, zk m n p, } { where n is the order of the a ppropriate Bessel functions, m denotes the number of roots of the dispersion relation, and zk is continuous co rresponding to wave vector along the infinite axial direction of the cylinder. When solving MaxwellÂ’s equations with respect to cylindrical geom etries, the dispersion relations are found to contain Bessel functions and the order of th e appropriate Bessel functions is denoted by n
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34 The sum in Equation (4.1) is an infinite one and in order to remove the occurring divergence, the zeropoint energy of the in finite homogeneous space is subtracted off from the energy of the system [3]: } {~ 2p p p CE (4.2) The term p ~ represents the eigenfrequencies corres ponding to the reference vacuum with no boundaries present. Since zk is continuous, we can express the Casimir energy per unit length as: m n z m n z m n z Ck k dk E, ,) ( ~ ) ( 2 2 (4.3) Both sums in Equation (4.3) are divergent and each sum will be treated individually and means to remove any occurring divergences will be explained. For the sum over m, we will make use of ResidueÂ’s theorem to treat the divergence, and depending on the system, the sum over n will be evalua ted correspondingl y. For instance in the case of a cylindrical dielectricdiamagnetic layer, the Riemann Zeta function regularization procedure is used whereas in the case of multiple perfectly conducting cylindrical shells, the energy of isolates metall ic shells is subtracted from the energy of the whole system. Before we do any of th ese regularizations, we first introduce the parameter s and express the Casimir energy as follows: m n z s m n z s m n z Ck k dk s E, ,) ( ~ ) ( 2 2 ) ( (4.4) ) ( lim1s E EC s C (4.5)
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35 where s in general is a complex number, allo wing one to perform the regularization procedure rigorously. By using2 2 2 zk where and are the dielectric and magnetic function of the system respectively, Casimir energy ) ( s ECcan be expressed as: nm s z nm s z nm z s Ck k dk c s E1 2 / 2 2 2 / 2 2) ( ) ( 2 2 ) ( (4.6) The term denotes the radius of the cylindrical nanostructures. For a cylinder, we have1R whereas for N multiple cylinders,NR R R,....., ,2 1 Using the ResidueÂ’s theorem, the infinite sum over m can be converted in terms of a contour integral in the comple x plane for any given value of n andzk [35, 58, 59]: n TM n TE n TM n TE n s z z s Cf f f f d d k d dk i c s E) ( ) ( ) ( ) ( ln ) ( 2 4 ) (2 / 2 2 C (4.7) where the contour C is along the imaginary axis ) ( i i and an infinite semicircle closed in the right ha lf of the complex plane with poles on the real axis as shown in Figure 4.1. ) (,TM TE nf are the dispersion relations of the system whereas ) (,TM TE nf are the dispersion relations when there is no bounda ries in the system. The only contribution from the contour integration comes from th e imaginary yaxis. Therefore, making the substitution Im y, ) ( s ECis reduced to: n TM n TE n TM n TE n k s z z s Ci f i f y i f y i f dy d k y dy dk s c s Ez0 2 / 2 2 2 ) ( ) ( ) ( ) ( ln ) ( 2 sin 2 ) ( (4.8)
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36 The order of integration in Equation (4.8) is changed and the integration over the zkvariable is performed, giving: n TM n TE n TM n TE n s s Ci f i f y i f y i f dy d dyy s s c s E0 1) ( ) ( ) ( ) ( ln 2 3 2 4 ) ( (4.9) where ) ( s is the gamma function. The exact Ca simir energy is obtained by taking the limit) ( lim1s E EC s C Equation (4.9) represents the Casimir energy expression in terms of the electromagnetic modes of the system. The el ectromagnetic modes are obtained by solving the MaxwellÂ’s equation with appropriate boundary conditions. We will make use of Equation (4.9) to calculate the Casimir energy for the system of a cylindrical dielectricdiamagnetic layer, multiple concentric metallic shells, and two parallel dielectric cylinders.
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37 Figure 4.1 Contour in the complex frequency plane with poles along the real axis.
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38 4.3.1 Zeta Function Regularization As mentioned in the previous section, the zeropoint energy is divergent and a means of regularization is required to change the infinite quantity into a finite one. Some of the most important regularization schemes are the frequency cutoff, point splitting, zeta function, heat kernel, proper time, and Fu jikawa method [60, 61]. In regards to the cylindrical geometries used in this dissertat ion, the zeta function regularization procedure is found to be convenient. In the zeta function regularization, the Casimir energy in Equation (4.1) becomes temporarily of the form: } {2 ) (p s p Cs E (4.10) where the exact Casimir energy is obtained on removing the regularization in the limit 1 s. This regularization is called the Riem ann zeta function regularization because the zeropoint energy ) ( s ECis given by: ) ( 2 ) ( s s EC (4.11) Equation (4.11) is expressed in terms of the Riemann zeta function as follows: } {) (p s ps (4.12) The parameter s is considered to belong in the region of the complex s plane where Re1s. An analytic continuation of Equation (4.12) to the point 1 s should therefore be constructed.
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39 The calculations involving the zeta functions have all been performed at zero temperature. However, standard cosmology predicts matter at extremely high temperatures in the early universe [60]. Dolan and jackiw [62] have been the first to study the symmetry behavior of Quantum Field Theory at finite temperature. They showed that in massless electrodynamics in two dimens ions, the gauge boson acquired a mass which was independent of temperature. Reuter an d Dittrich [63] rederived these results by utilizing the zeta function procedure. A good review of finite temperature formalism in Quantum Field Theory is found in reference [64].
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40 CHAPTER 5 CYLINDRICAL DIELECTRICDIAMA GNETIC LAYER OF FINITE THICKNESS 5.1 Cylindrical Model ) (, m m ) ( 1R ) (, m m I 2R I I Figure 5.1 Cylindrical layer of finite thickness w ith its axial direction perpendicular to the page is immersed in an infinite medium. The layer has permittivity and permeability ,, respectively, and the medium m m ,. The interfaces are denoted as I and II.
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41 The first model we consider is that of a dielectricdiamagnetic cylindrical layer immersed in an infinite medium. Here we apply the mode summation method to calculate the Casimir energy of such a layer. The syst em under consideration is a cylindrical layer with an inner radius1Rand outer radius2R with an infinite axial direction perpendicular to the page. The dielectr ic layer has dielectric and magnetic functions and respectively, and it is placed in an infinite me dium of dielectric and magnetic functionsm andm respectively, as illustrated in Figure 5.1. 5.2 Electromagnetic Modes The electromagnetic modes supported by the dielectricdiamagnetic layer imbedded in an infinite medium are obtained by solving the MaxwellÂ’s equations [65] with appropriate boundary c onditions across interfaces I and II Once the electromagnetic modes are obtained, one can then make us e of the mode summation method for the Casimir energy calculations. The Ma xwellÂ’s equations are defined as: 0 B (5.1) 0 E (5.2) t B E (5.3) t E B (5.4)
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42 Solving the MaxwellÂ’s equations will give us the electric and magnetic fields supported by the system. The components of the electric ( E ) and magnetic fields ( B ) are expressed in cylindrical coordinates z, as follows [65]: n t z k i in n j n n j n n j n n j n jze e H B J A ik H D J C n z E) ( ) 1 ( ) 1 ( 2, (5.5) n t z k i in n j n n j n n j n n j n jze e H D J C i H B J A nk z E) ( ) 1 ( ) 1 ( 2, (5.6) n t z k i in n j n n j n j zze e H B J A z E) ( ) 1 (, (5.7) n t z k i in n j n n j n n j n n j n jze e H D J C ik H B J A n z B) ( ) 1 ( ) 1 ( 2, (5.8) n t z k i in n j n n j n n j n n j n jze e H B J A i H D J C nk z B) ( ) 1 ( ) 1 ( 2, (5.9) n t z k i in n j n n j n j zze e H D J C z B) ( ) 1 (, (5.10) where3 2 1j stands for the three regions separated by the interfaces I and II in Figure 5.1, zk is the wavevector along the z direction, is the frequency of the electromagnetic excitations and2 2 2zk Also, ) ( n nJ J and ) () 1 ( ) 1 (n nH H are the Bessel function of first kind of order n and the Hankel functi ons of the first kind of order n respectively. In addition, dx x dJ x J Jn n n) ( ) ( and dx x dH x H Hn n n) ( ) () 1 ( ) 1 ( ) 1 (
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43 The unknown coefficients j n j n j n j nD C B A , are related by im posing the boundary conditions for the continuity of j z j j z j j jB E E E 1 , across each interface of the cylindrical layer giving the dispersion rela tion for the electromagnetic modes supported by this system. It is important to note that the permittivity and permeability for the layer, and the permittivity m and permeability m for the medium are taken to be constants. To account for the dielectric proper ties in the Casimir energy turns out to be difficult even for arbitrary dispersionless permittivity and permeability. This situation also occurs when calculating the Casimir eff ect for a ball [5, 48] or a cylinder [3, 5], where the condition of constant light velocity across the interface is imposed. Here we also impose the same condition2 cm m where c is the speed of light. The 2 cm m condition is referred to as the dielec tricdiamagnetic case as opposed to the purely dielectric one in which12 1 The physical implicati on of this condition is that the electric and magnetic fields are tr eated in a symmetric way, thus there is no preference for the electric field as it o ccurs for the dielectri c cylinder [6668]. Now that we have applied the boundary conditions across each interface of the cylindrical layer to the electric and magnetic fields, we obtain after imposing the constant speed of light condition th e following expressions: )] ( ) ( [ ) (1 ) 1 ( 2 1 2 1 1R H B R J A R J An n n n n n m (5.11) )] ( ) ( [ ) (2 ) 1 ( 2 2 2 2 ) 1 ( 3R H B R J A R H Bn n n n n n m (5.12)
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44 ) ( ) ( ) (1 ) 1 ( 2 1 2 1 1R H B R J A R J An n n n n n (5.13) ) ( ) ( ) (2 ) 1 ( 2 2 2 2 ) 1 ( 3R H B R J A R H Bn n n n n n (5.14) )] ( ) ( [ ) (1 ) 1 ( 2 1 2 1 1R H D R J C R J Cn n n n m n n (5.15) )] ( ) ( [ ) (2 ) 1 ( 2 2 2 2 ) 1 ( 3R H D R J C R H Dn n n n m n n (5.16) ) ( ) ( ) (1 ) 1 ( 2 1 2 1 1R H D R J C R J Cn n n n n n (5.17) ) ( ) ( ) (2 ) 1 ( 2 2 2 2 ) 1 ( 3R H D R J C R H Dn n n n n n (5.18) In general, the dispersion e quation for cylindrical stru ctures can be defined in terms of a determinant of a 4 x 4 matrix where separation between pure magnetic (TE) and pure electric (TM) modes is not possible except for the n =0 case [69,70]. Equations (5.115.14) are the expressions for the transverse magnetic (TM) modes whereas Equations (5.155.18) are those for the tran sverse electric (TE) modes. After some algebra, Equations (5.115.18) can be reduces to two expressions: 0 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , (2 2 1 1 2 2 2 1 1 2 1 R H R H R J R J R R R R fn n n n m TE n (5.19) 0 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , (2 2 1 1 2 2 2 1 1 2 1 R H R H R J R J R R R R fn n n n m TM n (5.20) for each ,... 2 1 0 n
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45 We have used the following notations: ) ( ) ( ) ( ) ( ) (r n r n m r n r n r rR H R J R H R J R (5.21) ) ( ) ( ) ( ) ( ) (r n r n m r n r n r rR H R J R H R J R (5.22) where r= 1, 2. Here 0 ) , (2 1 R R fTE n and 0 ) , (2 1 R R fTM n are the dispersion relations for the transverse electric (TE) and transverse magnetic (TM) modes respectively. Now that we have the dispersion relations, we can proceed in applying the mode summation method. Using Equation (4.9), the Casimir ener gy for a dielectricdiamagnetic cylindrical layer of finite thickness is expressed as: n TM n TE n TM n TE n s s Ci f i f y iR y iR f y iR y iR f dy d dyy s s c s E0 2 1 2 1 1) ( ) ( ) ( ) ( ln 2 3 2 4 ) ( (5.23) with the dispersion relationsTM TE nf, given by Equations (5.19) and (5.20). 5.3 Casimir Energy of Diel ectricDiamagnetic Cylindrical Layer In order to perform the calculations of the Casimir energy for the dielectricdiamagnetic cylindrical layer, it is co nvenient to first introduce the parameter m m This parameter allows the Casimir energy of the cylindrical layer to be represented as an infinite series in terms of powers of 2in analogy to the calculations of the Casimir
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46 energy for a solid ball or cylinder [3,5]. The2expansion allows one to consider explicitly each term in the series, a nd thus establish various limiting cases. By introducing the parameter m m we can follow the works done for compact dielectricdiamagnetic cylinders and sphe rical balls [3,5,71,72]. The parameter can be categorized into two very important cases for the cylindrical layer: the 1 case and the 1 case. The 1 case corresponds to an optically dilute dielectricdiamagnetic cylindrical layer with properties not very different from the dielectric and magnetic properties of the medium. The 1 case corresponds to tw o perfectly conducting concentric cylindrical shells. Introducing the parameter also allows us to apply the Riemann Zeta function regularization procedur e to remove the occurring divergences in the considered2terms thus extending its application to cylindrical layer systems. 5.3.1 The case of 1 m m This case corresponds to th e case when the dielectric (magnetic) properties of the surrounding medium and material of the layer do not differ much. After using standard properties of the Bessel functions [ 73], the dispersion relati ons are expressed as follows:
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47 2 4 4 1 3 2 2 2 2 2 1 4 4 2 2 2 1 2 1) 1 ( 16 ) 1 ( ) 1 ( 4 1 1 1 16 ) ( ) ( R R y y iR y iR f y iR y iR fm TM n TE n (5.24) The terms 1 and 2 are defined as follows: 2 1 2 2 2 2 2 2 1 1 1 1 2 1 2 2 2 1 1 1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 yR yR K yR K yR I yR I yR yR K yR K yR I yR I R R y yR K yR K yR I yR In n n n n n n n n n n n (5.25) ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (2 1 1 2 2 1 1 2 2 2 1 1 2yR K yR I yR K yR I yR K yR I yR K yR I yR K yR K yR I yR In n n n n n n n n n n n (5.26) Here the functions) ( iRy Jnand) ( iRy Hnhave been replaced by the modified Bessel functions using ) ( ) ( iy J i y In n n and ) ( 2 ) (1iy H i y Kn n n Now we can perform a series expansion of E quation (5.23) with respect to Taking into consideration that n nI Iand n nK K and retaining only the leading term in the expansion2~, the exact energyCE is expressed in the following way:
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48 )] ( ) ( [ lim 42 1 1 2s s c Es C (5.27) The Casimir energy has been separated into two parts, ) (1s and ) (2s Since the sum over n in Equation (5.23) is still present, we find it convenient to break it into the 0 n term and the 0 n term. ) (1s collects the 0 n terms from the summation: ) ( 2 0 0 0 0 2 1 2 1 1 0 2 0 2 0 1 0 1 0 2 1 2 12 11 ) ( ) ( ) ( ) ( 1 1 2 ) ( ) ( ) ( ) ( 4 ) (R R y s s se y K y K y I y I y R R yR K yR K yR I yR I R R y y dy s (5.28) ) (2s collects all 0 n terms from the summation: ) ( 2 2 1 2 1 1 1 0 2 2 1 1 2 1 2 22 12 ) ( ) ( ) ( ) ( 1 1 4 ) ( ) ( ) ( ) ( 8 ) (R R y n n n n s s n n n n n se y K y K y I y I y R R yR K yR K yR I yR I R R y y dy s (5.29) We evaluate the terms in ) (1s and ) (2s by making use of uniform asymptotic expansion of Bessel functions and applyi ng Riemann Zeta f unction. We start by evaluating the terms in) (2s The first term in ) (2s is evaluated by making use of the change of variables ,... 3 2 1 n nz y The uniform asymptotic expansion of the Bessel
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49 functions [73] is performed and the dominant term is retained. We obtain for the first term of) (2s : 1 1 0 ) ( 2 1 0 2 2 1 1 2 1 22 12 ) ( ) ( ) ( ) ( 8n n n s s n n n n sdy e y n dy yR K yR K yR I yR I R R y (5.30) where 2 2) ( 1 1 ln ) ( 1 y R y R y Ri i i ifor 2 1 i. Interchanging the integration and the summation in Equation (5.30) and taking the limit1 s, one obtains: 00 3   2   2   2 1 ) ( 2 2 2 1 2) 1 ( ) 1 ( ) ( 4 12 1 2 1 2 1 2 1dy e e e y dy e yn n (5.31) The second term in) (2s is also evaluated by making use of the change of variables ,... 3 2 1 n nz y Again, we perform the uniform asymptotic expansion valid for the Bessel functions of large order [73] and this time we retain the first three terms (4/ 1 ~ n) of the expansion. This le ads to the ex pression: 2 1 2 5 4 1 2 1 2 3 2 1 2 1 2 1 1 2 1 1 8 1 ) 1 ( 1 ) 1 ( 2 1 1 1 2 1 1 8 1 ) ( ) ( ) ( ) ( 1 2 1 1 21 1 1 2 1 1 0 3 2 2 2 2 1 1 1 2 1 1 0 2 1 1 1 2 1 1s s s s s s n R R dz z z z z n R R dy y K y K y I y I y n R Rn s s s s n s s s n n n n s n s s s (5.32)
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50 The term 1 1 n sn in Equation (5.32) is the Riemann Zeta function: ) 1 (1 1s nn s (5.33) Also, we have added 1 to 1 12n sn in Equation (5.32). This is compensated by subtracting the appr opriate terms from ) (1s later. Now it can be seen how the divergent sum over n can be regularized using the Riemann Zeta function as mentioned earlier. It follows that: ). 2 ln( 2 ) 1 ( 1 2 1 ) ) 1 (( ) 1 )( 0 ( 2 ) 0 ( 2 lim 2 1 ) 1 ) 1 ( 2 ( lim 2 1 1 22 1 1 1 1 s s s s s s s ns s n s (5.34) where is the Euler constant and ) 1 ( s represents the higher or der terms. In Equation (5.34), ) 1 ( s is expanded in Taylor se ries and we make use of the properties of the Riemann Zeta function, ) 2 ln( 2 1 ) 0 ( 2 1 ) 0 ( This technique allows us to remove the divergences from the poles of the Gamma function that come up in the calculations of the Casimir energy for a di electric cylinder. The remaining terms in
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51 Equation (5.32) are finite. The last term in ) (2s is solved exactly in the 1 slimit resulting in: 2 2 1 1 0 ) ( 2) ( 2 ) 0 ( 22 1R R dyyen yR yR (5.35) where 2 1 ) 0 ( Therefore, we were able to remove the occurring divergences in) (2s with all the terms in Equation (5.29) be ing finite and convergent: 2 2 1 2 2 2 1 0 3   2   2   2 2) ( 4 1 1 1 4 ) 2 ln( ) 1 ( ) 1 ( ) 1 (2 1 2 1 2 1R R R R dy e e e y s (5.36) Now that we have evaluated all the terms in) (2s we can evaluate the terms in) (1s The first and second terms have in) (1s cannot be evaluate d analytically and numerical methods need to be used. The third term however can be solved analytically in the limit 1 s giving: 2 2 1 0 ) ( 2) ( 4 12 1R R dyyeyR yR (5.37) We can therefore write ) 1 (1 sas follows:
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52 2 2 1 0 3 2 2 2 2 0 0 0 0 3 2 2 2 1 0 2 0 2 0 1 0 1 0 2 1 3 1) ( 4 1 2 1 1 1 2 1 1 8 2 1 1 ) ( ) ( ) ( ) ( 4 ) 1 ( R R y y y y y (y) K (y) (y)K I (y) I y dy R R yR K yR K yR I yR I R R y dy s (5.38) All the terms in Equation (5 .38) are finite and converg ent. Combining Equations (5.36) and (5.38) into Equation (5.27), we obt ain an expression for the Casimir energy for an optically dilute cylindrical layer: 2 2 2 1 3 2 2 2 2 0 0 0 0 3 2 2 2 1 2 0 2 0 1 0 1 0 2 1 3 0 3   2   2   2 21 1 8 ) 2 ln( 4 1 1 1 2 1 1 16 1 1 ) ( ) ( ) ( ) ( 2 ) 1 ( 2 ) 1 ( 22 1 2 1 2 1R R y y y y y (y) K (y) (y)K I (y) I y R R yR K yR K yR I yR I R R y e e ye dy c EC (5.39) It is very important to note that all terms in Equation (5.39) are finite and convergent. This shows that we were able to apply the mode summation technique to the
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53 system of cylindrical layer and our endea vor of removing occurring divergences was a successful one. The sum over m was treated using ResidueÂ’s theorem described in section 4.3 while the sum over n was evaluated using the Riemann Zeta function by introducing two parameters, and s as described in section 5.3. 5.3.1.1 Limiting Cases In this section, the Casimir energy for various limits of the cylindrical layer is considered. We first start by investigating the limit for a layer of very small thickness,1 ~ where1 2/ R R In this limit, we find that the dominant contribution in Casimir energy comes from the term in Equati on (5.31) and in this limit the Casimir energy per unit length behaves as: 3 2 1 2) 1 ( 076 0 R c EC (5.40) On the other hand, in the limit of large radii 2 1, R R, while keeping the separation between the two cylinders fixed,const R R d 1 2 we obtain a system that corresponds to an infinite dielectricdiamagne tic plate. In this limit, the Casimir energy per unit area behaves as: 3 2 28 d c EC (5.41) This result is exactly what Klich et al. reported in Ref. [74].
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54 In the limit 2R, we have a system that corre sponds to a compact dielectric cylinder of radius 1R and dielectric and magnetic properties ) ( imbedded in an infinite medium with dielec tric and magnetic properties) (m m with the speed of light being constant across the interface at1R. In this case, only the terms 2 1/ 1 ~ Rin the expressions for 1and 2contribute toCE. However, we find it essential to evaluate the next term (6/ 1 ~ n) in the Bessel functions asympt otic uniform expansion in the calculation of Equation (5.32) Due to the occurring cancellations the value of the Casimir energy the latter becomes zero in the limit of 2R thus recovering the results reported in Refs. [3,5]. It can be seen that we are able to recover some very important results by considering various limiting cas es in our cylindrical layer. All these recovered results were performed theoretically and now we can move onto the numerical implementation of the Casimir energy and confirm those limits we just obtained. 5.3.1.2 Numerical Results After implementing Equation (5.39) numeri cally and varying the thickness of the layer, we find that the Casimir energy behaves as shown in Figure 5.2. It can be seen that in the limit1 the energy behaves as: 3) 1 ( 1 ~  CE (5.42)
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55 This clearly resembles the th eoretical result obtained in the same limit in Equation (5.40). Another case observed is that for7 the layer is already in the limit of large and0CE, thus the system behaves as a compact cylinder with) ( characteristics imbedded in an infinite medium with) (m m characteristics. Again this confirms the theoretical result obtained in Section 5.3.1. 1. Another important obs ervation is that the Casimir energy is negative implying an attractive energy. Next, the interaction energy as a function of the inner radius is investigated. By changing the thickness of the layer, we can see how the Casimir energy behaves and the result is depicted in Figure 5.3. The most significant changes are found when the thickness of the layer is small due to the fact that for small thicknesses the dominant contribution comes from Equation (5.31). As the thickness is increased, the system behaves more like a compact cylinder of radius 1R and dielectric and magnetic properties ) ( imbedded in an infinite medium with dielectric and magnetic properties) (m m with the speed of light being constant across the interface at1R. Here it is clear that 0 CE again confirming our theore tical calculations.
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56 Figure 5.2 The Casimir energy per unit length for a cylindrical dielectricdiamagnetic layer as a function of R2/R1. 13579 0.3 0.2 0.1 0.0 EC/hc2 = R2/R1
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57 159131721 0.20 0.15 0.10 0.05 0.00 EC/hc2 =2 =2.5 =3 =4 =5 =6 =8 =11R1 (nm) Figure 5.3 The Casimir energy per unit length for the same layer as a function of inner radius R1.
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58 5.3.2 The case of 1 m m This case corresponds to two perfectly conducting cylindric al shells if one takes c =1. In this model, the expression for the Casimir energy is given by: n TM n TE n TM n TE n Ci f i f y iR y iR f y iR y iR f dy d dyy c s E0 2 1 2 1 2) ( ) ( ) ( ) ( ln 8 ) 1 ( (5.43) As seen in Equation (5.43) we still have the infinite sum over n which is divergent. In order to remove this diverg ence, we find it convenient to consider the difference between the energy of the concentric cylindrical shells and the energy of a single isolated shell. In our case, since we ha ve two isolated cylindrical shells, we use the following: ) 1 ( ) 1 ( ) 1 ( ) 1 ( ~) 2 ( ) 1 ( s E s E s E s EC C C C (5.44) where ) 1 ( s EC is given by Equation (5.43) and ) 1 () 2 1 ( s ECare the energies for the single isolated cylindrical shells with radius 2 1, R R, respectively. The energies for the isolated shells are alread y known from Ref. [3,5], 2 2 1 ) 2 1 (01356 0 ) 1 ( R c s EC (5.45) Therefore, all we need to figure out is ) 1 ( ~ s ECusing the dispersion relations of the system of cylindrical shel ls. Using the dispersion relations ) 1 ( ~ s EC becomes:
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59 n TM n TE n TM n TE n TM n TE n TM n TE n TM n TE n Ci f i f y iR f y iR f y iR f y iR f i f i f y iR y iR f y iR y iR f dy d dyy c s E0 2 1 2 1 1 1 1 1 2 1 1 2 1 2 1 2) ( ) ( ) ( ) ( ) ( ) ( )) ( ) ( )( ( ) ( ln 8 ) 1 ( ~ (5.46) Here ) (2 1y iR y iR fTE n) (2 1y iR y iR fTM nare the dispersion relations defined with1 In this case only the term containing the 2 expression in Equation (5.24) remain. 2 is the dispersion relation for the el ectromagnetic modes for two infinitely thin concentric perfectly conduc ting cylindrical shells and this result agrees with Ref.[4, 65]. Therefore we have the dispersion relation as: ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (2 1 1 2 2 1 1 2 2 2 1 1 2 1 2 1yR K yR I yR K yR I yR K yR I yR K yR I yR K yR K yR I yR I y iR y iR f y iR y iR fn n n n n n n n n n n n TM n TE n (5.47) ) ( ), (2 1 1 1 ,y iR f y iR fTM TE n TM TE nare the dispersion relations for a single isolated cylinder with radii2 1,R R, respectively. When 1 and1 c, ) (2 1 1 ,y iR fTM TE nbecome equivalent to the dispersion relations of a si ngle infinitely thin perfectly conducting shell as shown in Ref. [3,5]. ) (, i fTM TE nand ) (, 1 ,i fTM TE n are the dispersion relations with no boundaries present. Inserting all the dispersion relations into Equation (5.46), we end up with:
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60 1 0 1 2 2 1 1 2 2 1 0 1 0 2 0 2 0 1 0 1 0 2 0 2 0 1 0) ( ) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( 1 ln 2 ) ( ) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( 1 ln 4 ) 1 ( ~n n n n n n n n n Cdy yR K yR I yR K yR I yR K yR I yR K yR I y dy yR K yR I yR K yR I yR K yR I yR K yR I y c s E (5.48) Now we have to evaluate the terms in Equation (5.48). The first term is evaluated numerically while the second term can be eval uated analytically. For the second term, we first make use of the uniform asymptotic expansion of the Bessel functions for large orders followed by a Taylor series expansion of the logarithmic func tion. This results in the following: 1 0   2 1 0) 1 ( 4 ~ 1 1 ln 22 1m m n n n n ndy e m y dy K I K I y (5.49) Equation (5.48) now becomes: 1 0   2 0 1 0 2 0 2 0 1 0 1 0 2 0 2 0 1 0) 1 ( 4 ) ( ) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( 1 ln 4 ) 1 ( ~2 1m m Cdy e m y dy yR K yR I yR K yR I yR K yR I yR K yR I y c s E (5.50) Now that) 1 ( ~ s EC has been evaluated, using Equa tions (5.50) and (5.45) into Equation (5.44), we obtain an expression for the Casimir energy of perfectly conducting cylindrical shells:
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61 2 2 2 1 0 1 0   2 0 0 0 01 1 01356 0 ) 1 ( 4 1 1 ln 4 ) 1 (2 1R R c dy e m y dy K I K I y c s Em m C (5.51) All the terms in Equation (5.51) are physic ally convergent and finite. The integrals present can be solved numerically. 5.3.2.1 Numerical Results Equation (5.51) is implemented numerically and the behavior of the energy as a function of the separation of th e cylindrical shells is inves tigated. The behavior is shown in Figure 5.4. When the separation between the shells is small,1 where 1 2/R R the energy can be expressed analytically in the following form: 3 2 1) 1 ( 0862 0 R c EC (5.52)
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62 12345 15 10 5 0 EC/hc = R2/ R1 Figure 5.4 The Casimir energy per unit length fo r the same layer as a function of 1 2/R R
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63 In the case where the radii of the shells are very large, 2 1,R Rand the separation between the shells is kept fixed, const R R d 1 2, the system now corresponds to two parallel perfectly conducti ng plates, similar the system studied by Casimir [7]. In this case, we are able to recover the wellknown formula for the Casimir energy per unit area for two perfectly conducting plates: 3 2720 d c EC (5.53) As the radius of the second shell is increased, the system now behaves as a conducting cylindrical shell immersed in a me dium and we see from Figure 5.4 that the energy converges to a constant. For large the only nonzero term in Equation (5.51) is 2 1/ 1 ~ R thus giving the energy for one perf ectly conducting cylindrical shell. Therefore, we have been able to calc ulate the Casimir energy for a dielectricdiamagnetic cylindrical layer immersed in an infinite medium by ma king use of the mode summation method. Two separate cases have been investigated, one where the system behaves like an optically dilute dielectricdiamagnetic cylindrical layer with properties not very different from the dielectric and magnetic properties of the medium and one where the system corresponds to that of two perfectly conducting cylindrical shells. In both cases, the divergences were removed and the numerical values obtained were physically finite and convergent.
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64 The case of two perfectly conducting concentr ic cylinders might be of particular interest as a qualitative model of the Casi mir interactions in a doublewall metallic carbon nanotube system. Previous theoretical studies, howe ver, have shown that the perfectly conducting metallic cylinder mo del does not describe correctly the electrodynamical processes closely related to the Casimir interacti on, such as atomic spontaneous decay [75,76], atomnanotube va n der Waal coupling [77], atomic light absorption [78], and atomic entanglement [79]) near carbon nanotubes. This means that one needs to take into account realistic electromagnetic properties and the strong modification of the photonic density of states due to the increasing role of the interface photonic modes near the nanotube surface. Fo r example, in Carbon nanotubes, the interface photonic modes in the longitudinal direction are plasmons. These plasmons couple strongly with the excitons in the na notube by long range in teractions [43]. The perfectconductor case might serve as a qua litative model of a doublewall metallic nanotube system in the limit of largely different radii, as the role of the interface photonic modes decreases with increasing intertube separation. This model serves as a formulation for future more advanced theore tical techniques which should be able to reproduce the results obtained in this work. The results described here have been published in the paper [80].
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65 CHAPTER 6 N PERFECTLY CONDUCTING CYLINDRICAL SHELLS 6.1 Cylindrical Model Figure 6.1 Infinitely long perfectly conducting and concentric cylindrical shells immersed in an infinite medium. The axial direction is perpendicular to the page. The radii of the shells are iR where i=1,2,Â…N.
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66 In this work, the Casimir energy for a syst em of infinitely long, infinitely thin, and perfectly conducting concentr ic cylindrical shells immers ed an infinite medium is calculated using the mode summation method. The system consists of N multiple shells with radii iR where i=1,2,Â…N as shown in Figure 6.1. This kind of system is of interest to us because it resembles cylindrical structures made of metallic shel ls, such as metallic multiwall nanotubes [81,82]. 6.2 Electromagnetic Modes To calculate the Casimir energy of this system of multiple concentric metallic shells, we again make use of the mode summat ion method. This requires us to find the electromagnetic modes of the system by solv ing MaxwellÂ’s equations with appropriate boundary conditions across each in terface of the shel ls [65]. It follows that the zcomponent of the electric and magnetic fields in cylindrical coordina tes are respectively given as: n t z k i in n n n n zze e H B J A E) ( ) 1 () ( ) ( (6.1) n t z k i in n n n n zze e H D J C B) ( ) 1 () ( ) ( (6.2) where zk is the wavevector along the z direction, is the frequency of the electromagnetic excitations and2 2 2zk Also, ) ( nJ and ) () 1 (nH are the
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67 Bessel function of first kind of order n and the Hankel functions of the first kind of order n respectively. The remaining components of the electri c and magnetic fields can be derived from MaxwellÂ’s equations [65]. The Diri chlet and Neumann boundary conditions are used here to find the unknown coefficientsn n n nD C B A , ,. These boundary conditions require that the zcomponent of the elect ric field and the normal component of the magnetic field to vanish at each surface, respectively [65,83]. After applying the boundary conditions, we obtain the following: 0 ) ( ) () 1 ( i n n i n nR H B R J A (6.3) 0 ) ( ) () 1 ( i n n i n nR H D R J C (6.4) where i=1,2,Â…N and dx x dJ x Jn n) ( ) ( dx x dH x Hn n) ( ) () 1 ( ) 1 ( The dispersion relations for this system are obtained by solving Equations (6.3) and (6.4). The electromagnetic modes can then be classified as transverse electric (TE) and transverse magnetic (TM) modes [ 69,70]. For the TE modes, the frequency ) (zk is found by solving: 1 1 0 ) ( R r R Jn 1 ) 1 ( 1 1 ) 1 ( 0 ) ( ) ( ) ( ) ( j j j n j n j n j nR r R R H R J R H R J (6.5) N N nR r R H 0 ) () 1 (
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68 where j = 1,2,Â…,N1 For the TM modes, the frequency) (zk is found by solving: 1 1 0 ) ( R r R Jn 1 ) 1 ( 1 1 ) 1 ( 0 ) ( ) ( ) ( ) ( j j j n j n j n j nR r R R H R J R H R J (6.6) N N nR r R H 0 ) () 1 ( Thus the dispersion relations for the electromagnetic modes for the N perfectly conducting metallic shells are obtained as follows: ) ( ) ( ) ( ) ( ) ( ) ( ) ,... , () 1 ( 1 1 ) 1 ( 1 1 ) 1 ( 1 2 1j n j n j n j n N j N n n N TE nR H R J R H R J R H R J R R R f (6.7) ) ( ) ( ) ( ) ( ) ( ) ( ) ,... , () 1 ( 1 1 ) 1 ( 1 1 ) 1 ( 1 2 1j n j n j n j n N j N n n N TM nR H R J R H R J R H R J R R R f (6.8) Now that the dispersion rela tions are obtained, we can now make use of Equation (4.9) to express the Casimir energy for the system of multiple concentric metallic shells: n TM n TE n N TM n N TE n Ci f i f y iR y iR y iR f y iR y iR y iR f dy d dyy c E0 2 1 2 1 2) ( ) ( ) ,..., ( ) ,..., ( ln 8 (6.9)
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69 Here, the functions ) ( iyR Jn and ) () 1 (iyR Hn are replaced by the modified Bessel functions using ) ( ) ( iy J i y In n n and) ( ) 2 / ( ) () 1 ( 1iy H i y Kn n n. 6.3 Casimir Energy of N Perfectly Metallic Cylindrical Shells In order to proceed with the calculation of the Casimir energy, we still need to deal with the infinite sum over n in Equation (6.9). To treat this divergence, we find it convenient to take the difference between the energy of the system of concentric shells and the energy of the individual isolated cylindr ical shells. In this way the Casimir energy is expressed in a more transparent way and the remaining divergences are cancelled out [4,80]. Thus for a system of N infinitely long metallic cyli ndrical shells we consider N i i C C CE E E1 ) (~ (6.10) where CE is given by Equation (6.9) and ) (i CE are the energies of the single isolated cylindrical shells with radii NR R R ,..., ,2 1. The energies ) ( i CE have already been derived [3,5] and here we use their final result 2 ) (/ 01356 0i i CR c E From Equations. (6.9) and (6.10) one obtains: n TM n TE n i TM i n i TE i n N i N TM i n TE i n N TM n N TE n Ci f i f y iR f y iR f i f i f y iR y iR y iR f y iR y iR y iR f dy d dyy c E0 , 1 , 2 1 2 1 2) ( ) ( ) ( ) ( ) ( ) ( ) ,..., ( ) ,..., ( ln 8 ~ (6.11)
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70 Here ) ,..., (2 1 ,y iR y iR y iR fN TM TE nare the dispersion relations defined by Equations (6.7) and (6.8) with Imy. ) (, ,y iR fi TM TE i nare the dispersion re lations for a single cylindrical shell with radius iR and ) (, i fTM TE i n are the dispersion relations with no boundaries present. Making the appropriate substitutions in to Equation (6.11) for the dispersion relations from Equations (6.7), (6.8) and for the dispersion relations of single isolated cylindrical shells, we obtain: 1 0 1 1 1 1 1 1 0 1 1 0 1 0 1 0 0 0 1 0 1 0 0) ( ) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( 1 ln 2 ) ( ) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( 1 ln 4 ~n N j j n j n j n j n j n j n j n j n N j j j j j j j j j CyR K yR I yR K yR I yR K yR I yR K yR I dyy yR K yR I yR K yR I yR K yR I yR K yR I dyy c E (6.12) where the0 nand0 nterms in the modified Bessel functions have been separated. For a given number of cylindrical shells N, the first term in the expression of CE ~ can be evaluated numerically. The evaluation of the second term is facilitated by the use of the uniform asymptotic expansion of the modified Bessel functions for large orders [73] and also by using the Taylor series expansion of the logarithmic function ) 1 ln(jx for1 jx, where   21 j jn je x [4,80]. This leads to the second term being expressed as:
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71 1 0 1 1 2 1 0 1 1 1 1 1 1) 1 ( 4 ~ ) ( ) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( 1 ln 21m N j m n N j j n j n j n j n j n j n j n j nj je m y dy yR K yR I yR K yR I yR K yR I yR K yR I dyy (6.13) Combining Equations (6.13), (6.12) and 2 ) (/ 01356 0i i CR c E into Equation (6.10), the Casimir energy of th e considered system of con centric conducting shells is found to be: N i i m N j m N j j j j j j j j j CR c e m y dy yR K yR I yR K yR I yR K yR I yR K yR I dyy c Ej j1 2 1 0 1 1 2 0 1 1 0 1 0 1 0 0 0 1 0 1 0 01 01356 0 ) 1 ( 4 ) ( ) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( 1 ln 41 (6.14) for 1 ,..., 2 1 N j where 2 2) ( 1 1 ln ) ( 1 y R y R y Rj j j j. All the terms in Equation (6.14) can be evaluated numerically. They are convergent and provide finite, physically mean ingful values for the Casimir energy for the perfectly conducting system considered here. This demonstrates that we were successful in applying the mode summation method to calculat e the Casimir energy of infinitely long, perfec tly conducting concentric cylindrical shells.
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72 6.3.1 Limiting Cases In this section we investig ate various limiting cases fo r the Casimir energy with respect to the separation of the shells. First when2 N, we recover the energy obtained in Refs. [4,6,80] for the case of two perfect ly conducing concentric shells described by Equation (5.51). Furthermore, in the limit of infinitely clos e cylinders, the dominant term comes from Equation (6. 13), and we find that: 1 1 3 2) 1 ( 1 0862 0N j j j CR c E (6.15) wherej j jR R/1 Thus for j=1, we obtain the result for two infinitely close shells [4,80]. The limit of large radii and constant shell separation, jRand const R R dj j j 1 for all j, we have a system that corresponds to N infinite parallel conducting plates. For this limit, we obt ain the Casimir energy per unit area as: 1 1 3 2720 1N j j Cd c E (6.16) thereby recovering the well know n formula for the Casimir en ergy per unit area of two perfectly conducting plates (j=1) same as Equation (5.53)) 720 /(3 2d c EC [7].
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73 6.3.2 Numerical Results The considered system implies that as N increases, greater possibilities for radial size and curvature variations of one or more cylindrical shells become available. We illustrate this point by showing numerical results for N=3 shells. One particular case is when the radii of the shells are varied in such a way as to keep the distance between each shell constant. In Figure 6.2 we show th e behavior of the Casimir energy when the separation of the shells are fixed, 2 1 Here, 1 2 1/ R R and2 3 2/R R In the limit of 3 2 1, ,R R Rand const d d 2 1where 1 2 1R R d and2 3 2R R d we obtain the Casimir energy per unit area as 3 2 3 1 2720 / 1 720 / 1 d d c EC confirming a result obtained in Ref. [84].
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74 Figure 6.2 The Casimir energy for the case of N=3 shells as a function of the inner radius R1.
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75 Increasing the radius of the outer two she lls, their separation be ing constant, while keeping the inner radius fixed provides us with a very interesting case for the Casimir energy. This is illustrated in Figure 6.3 where 11 Rnm and 2R increasing withconst 2 We see that  CEis larger for smaller 2 indicating the dominant term from Equation (6.13). As 2R increases, the Casimir en ergy becomes practically a constant with our system be having more like a si ngle cylindrical shell and two perfectly conducting plates. The electromagnetic interac tions between the inne r shell and the outer shells become smaller as the separation increases a nd therefore this le ads to a constant value for the Casimir energy. Another interesting case i nvolves keeping the two inne r radii constant and the outer one 3R is varied. Figure 6.4 shows the behavi or of the Casimir energy in such a case. For the 3Rlimit, the energy approaches that of two perfectly conducting cylindrical shells infinitely sep arated from a conducting parallel plate. Again, we see that  CEis large for small 1 due to the main contribution from Equation (6.13). As 2 increases, the Casimir energy becomes practic ally a constant approaching the limit for two cylindrical shells and a parallel plate.
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76 Figure 6.3 The Casimir energy for the case of N=3 shells as a function of the radius of the second shell R2.
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77 Figure 6.4 The Casimir energy for the case of N=3 shells as a function of the separation between the outer two shells.
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78 Finally, we investigate the relations hip between the number of shells N and the corresponding Casimir energy. In Figure 6.5, we show results CE for various i as a function of N. We find that  CE increases nonlinearly and it is larger for smallers indicating the dominant contribution from Equatio n (6.13). The higher the number of shells implies that there are mo re electromagnetic interactions between the surfaces resulting in a higher Casimir energy. The cl oser the shells will result in a higher Casimir energy due to high electromagnetic in teraction and the dominant contribution of Equation (6.13). As the separation be tween the shells becomes larger,  CEdecreases and it approaches the limit of the sum of energi es of single cylindrical shells with the appropriate radii [3,5].
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79 Figure 6.5 The Casimir energy as a f unction of the number of concentric cylindrical shells. 2468101214161820 6 4 2 0 ddd ddd ddd ddd 4EC/hcN
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80 CHAPTER 7 TWO PARALLEL DIELECTRICD IAMAGNETIC CYLINDERS 7.1 Cylindrical Model R 1R2R) (1 1 ) (2 2 ) (3 3 Figure 7.1 Two infinitely long parallel cylinders of radii1Rand2R with centertocenter separation R immersed in an infinite me dium. The cylindr ical axis is perpendicular to the page.
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81 The system under considerati on consists of two infinite ly long straight parallel circular cylinders of radii1Rand2R with centertocenter separation R The permittivity and permeability of the first cylinder is 1 and 1 respectively and for the second cylinder the permittivity and permeability is 2 and 2 respectively. The cylinders are placed in an infinite medium of dielectric and magnetic functions 3 and3 All dielectric and magnetic characterist ics are taken to be constants. In this work, we consider the interact ion originating from the electromagnetic field fluctuations between infi nitely long parallel circular dielectricdiamagnetic cylinders immersed in a medium. Such a system is of particular interest because it can serve as a model to study the Casimir inte raction between cylindrical tubular structures, such as carbon nanotubes, boron nitride nanotubes, na nowires, DNA, etcÂ…It can also provide a test ground of how curvature e ffects coupled with dielectric and/or magnetic properties influence the mutual interac tion between the cylinders. 7.2 Electromagnetic Modes In order to utilize the mode summa tion method, one needs to find the electromagnetic modes by solving the Maxwel lÂ’s equations [65] w ith the appropriate boundary conditions across the interfaces of each cylinder. In this problem, the electric and magnetic fields are expressed with resp ect to two separate sets of cylindrical coordinates, one for each cylinder. The solution to the wave equation is satisfied in three
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82 regions and consequently, the zcomponent of the electric and ma gnetic fields (axial direction perpendicular to the page ) is defined in these three regions: 2 2 1 1 R R (7.1) 2 2 1 1 R R (7.2) 2 2 1 1 R R (7.3) Therefore, we can write the zcomponent of the electric and magnetic fields as follows: ) ( 2 3 ) 1 ( ) 3 ( 1 1 ) 1 ( ) 1 (2 1) ( ) (t z k i n in n n in n n zze e H B e J A E (7.4) ) ( 2 3 ) 1 ( ) 3 ( 1 1 ) 1 ( ) 1 (2 1) ( ) (t z k i n in n n in n n zze e H D e J C B (7.5) ) ( 1 3 ) 1 ( ) 4 ( 2 2 ) 2 ( ) 2 (1 2) ( ) (t z k i n in n n in n n zze e H B e J A E (7.6) ) ( 1 3 ) 1 ( ) 4 ( 2 2 ) 2 ( ) 2 (1 2) ( ) (t z k i n in n n in n n zze e H D e J C B (7.7) ) ( 2 3 ) 1 ( ) 3 ( 1 3 ) 1 ( ) 4 ( ) 3 (2 1) ( ) (t z k i n in n n in n n zze e H B e H B E (7.8) ) ( 2 3 ) 1 ( ) 3 ( 1 3 ) 1 ( ) 4 ( ) 3 (2 1) ( ) (t z k i n in n n in n n zze e H D e H D B (7.9) where zk is the wavevector along the z direction, the frequency of the electromagnetic excitations, and2 2 2 z j j jk with j=1,2,3. ) ( j nJ and) () 1 ( j nH are the Bessel functions of first kind of order n and the Hankel functions
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83 of the first kind of order n, respectively. Also, regions (1 ), (2) and (3) are defined by Equations (7.1), (7.2) and (7.3) respectively. It can be seen that there are two cylindr ical coordinate systems associated with each cylinder zi i, with i=1, 2 in Equations (7.47.9). In order to be able to solve for the unknown coefficients, we need to be ab le to express the el ectric and magnetic fields with respect to a single cylindrical coordinate system. This is achieved by making use of the additi on theorem [73]: j ij j j n in ne J R H e H1 2) ( ) ( ) (1 3 3 ) 1 ( 2 3 ) 1 ( (7.10) Equation (7.10) enables us to express the second coordinate system in terms of the first coordinate system and a similar relation is used with2 1 2 1 to express the first coordinate system in terms of th e second coordinate system. We can now express the electric and magnetic fields in terms of only one coordinate system as follows with i=1, 2: ) ( 3 3 ) 1 ( ) 2 ( ) ( ) () ( ) ( ) (t z k i nj ij i j j n i n in i i n i n i zz i ie e J R H B e J A E (7.11) ) ( 3 3 ) 1 ( ) 2 ( ) ( ) () ( ) ( ) (t z k i nj ij i j j n i n in i i n i n i zz i ie e J R H D e J C B (7.12) ) ( 2 3 ) 1 ( ) 3 ( 1 3 ) 1 ( ) 4 ( ) 3 (2 1) ( ) (t z k i n in n n in n n zze e H B e H B E (7.13)
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84 ) ( 2 3 ) 1 ( ) 3 ( 1 3 ) 1 ( ) 4 ( ) 3 (2 1) ( ) (t z k i n in n n in n n zze e H D e H D B (7.14) The electric ) 3 (zEand magnetic ) 3 (zBfields for the medium contain one term expressed in the coordinate set of one cylin der and another term Â– in the coordinate system of the other cylinder. Again using th e addition theorem, we can express the fields in terms of a single coordinate system. The and components of the fields can be easily obtained using MaxwellÂ’s equa tions [65]. The unknown coefficients ) 2 ( ) 2 ( ) ( ) ( ) ( ) (, , , i n i n i n i n i n i nD B D C B A are related by imposing the boundary conditions for the continuity of j j z j z j j jB E E Ei i / , ,) ( ) ( ) ( ) ( across the interface of each cylinder giving the dispersion relation for the electromagnetic mod es supported by this system. In the general case, the dispersion relation is complicated and the calculations of th e interaction energy are not feasible, however significant simplif ications occur when the speed of light c is constant everywhere. The dispersion relations can now be obtained by applying the boundary conditions and keeping2 cj j This leads to the disper sion relations defined by: 0 ) ~ 1 ( ) , (2 1 1 Det R R R i f (7.15) 0 ) ~ 1 ( ) , (2 1 2 Det R R R i f (7.16) where the substitution Im has been made. ~ and ~are matrices with elements:
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85 j j j j j n n n n l j j n l n nlR I R K R I R K R I R K R I R K R K R K i ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) )( (1 1 3 1 1 1 2 2 3 2 2 2 2 3 1 3 (7.17) j j j j j n n n n l j j n l n nlR I R K R I R K R I R K R I R K R K R K i ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) )( (1 1 1 1 1 3 2 2 2 2 2 3 2 3 1 3 (7.18) wheredx x dI x Ij n j n/ ) ( ) (, anddx x dK x Kj n j n/ ) ( ) (, The electromagnetic modes supported by the system of two straight, parallel dielectricdiamagnetic cylinders are obtaine d from Equations (7.15) and (7.16). Using Equation (4.9), the Casimir energy of two straight, para llel dielectricdiamagnetic cylinders is given as: 0 2 1 2 1 2 2 1 1 2) ( ) ( ) , ( ) , ( ln 8 i f i f R R R i f R R R i f d d d c EC (7.19) where ) (1 i fand) (2 i fare the dispersion relations with no boundaries present. 7.3 Casimir Energy of Two Straight Parallel Cylinders In order to evaluate Equation (7.19), we find it convenient to use the logdet approximation [85]: n nnTr Det R R R i f } ~ { )} ~ 1 ( ln{ )} , ( ln{2 1 1 (7.20) n nnTr Det R R R i f } ~ { )} ~ 1 ( ln{ )} , ( ln{2 1 2 (7.21)
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86 ) ( )} ( ~ { ))} ( ~ 1 ( ln{ )} ( ln{1 n nnTr Det i f (7.22) ) ( )} ( ~ { ))} ( ~ 1 ( ln{ )} ( ln{2 n nnTr Det i f (7.23) We use only the leading term of the logdet approximation since it gives the contribution to the leading order. This expansion gives the leading term whenR R R 2 1,, which is the case here. The Casimir energy can now be expressed in the following form: 0 2) ( ) ( 8n nn nn nn nn Cd d d c E (7.24) Here, since we are dealing with the traces of the matrices, n=l For all terms occurring in Equation (7.24), we find it convenient to arrange them into four parts,) 0 0 ( j n,) 0 0 ( j n,) 0 0 ( j n, and) 0 0 ( j ngroups. The terms with ) 0 0 ( j n are evaluated by making use of the change of variables, ,..., 3 2 1 j jz and performing the uniform asymptotic expansion of the Bessel functions [73]. The terms with ) 0 0 ( j n are evaluated by making use of the change of variables, ,..., 3 2 1 n nz and again performing the uniform asymptotic expansion of the Bessel functions. The terms with0 0 j nare evaluated using large order expansion for the Bessel functions. This is a technique used in other works for the interaction energy in cylindrical structur es [2,3,80,86]. After doing the appropriate changes, we obtain the following convergent e xpression for the zero point energy per unit length l, for the case of two parallel cylinders:
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87 1 1 ) ( 2 ) ( 2 2 ) ( 2 ) ( 2 2 ) ( 2 ) ( 2 0 2 2 2 3 1 3 2 3 1 3 1 0 1 0 1 1 0 1 0 3 2 0 2 0 2 2 0 2 0 3 2 0 2 3 1 3 0 1 0 1 0 3 1 0 1 0 1 2 0 2 0 3 2 0 2 0 2 2 0 2 31 3) ( 2 ) 1 ( ) 1 ( 1 ) )( ( ) )( ( 2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) )( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) )( ( 41 2 1 2 1 2 2 1 2 1j n j n R R R R Cj n e e e e e e R d R I R K R I R K R I R K R I R K R K R I R K R I R K R I R K R I R K R K d c E (7.25) where 2 2) ( 1 1 / ln ) ( 1 R R R and 2 2 1 2 1 2 2 1 2 1) ( 1 1 / ln ) ( 1 R R R
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88 7.3.1 Limiting Cases In this section we evaluate the Casimir energy for two straight parallel cylinders in the limit for largely separated cylinders,2 1, R R R This particular limit can be used as a qualitative model to inve stigate the interacti on of two tubular systems separated by large distances. To simplify the evaluations, we take the cylinders to have equal radii 2 1R R and the same dielectric 2 1 and magnetic 2 1 properties. When2 1R R R the first two terms in Equation (7.25) are approxi mated by taking a small argument expansion with respect to 2 1R in the denominators. After making the appropriate modifications, we obtain the first term of the Casmir energy from Equation (7.25) to be: 0 1 2 0 7 0 2 0 7 2 1 1 3 0 2 0 5 4 1 2 1 2 1 3 0 2 1 0 1 0 3 1 0 1 0 1 2 0 2 1 3ln ) ( ln ) ( 2 ) ( 4 ) ( ) ( ) ( ) ( ) ( ) ( ) ( R R K d R K d R R K d R R I R K R I R K R K d (7.26) The integrals in Equation (7.26) can be evaluated using the following identities [82]: s s B s R R K ds s s2 1 2 1 2 1 2 ) (2 3 0 2 1 (7.27)
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89 0 2 1 0 2 1) ( ) ( ) ln( R K d s R K ds s (7.28) where B is a beta function. After doing the modifications, the first term ends up as: 420 451 2 ln 35 288 15 16 4 ) ( ) ( ) ( ) ( ) ( ) (1 8 2 1 1 3 6 4 1 2 1 2 1 3 0 2 1 0 1 0 3 1 0 1 0 1 2 0 2 1 3 R R R R R R R I R K R I R K R K d (7.29) where is EulerÂ’s constant. In similar fashion, the second term of the Casimir energy in Equation (7.25) is found to be: 420 451 2 ln 35 288 15 16 4 ) ( ) ( ) ( ) ( ) ( ) (1 8 2 1 3 1 6 4 1 2 3 2 1 3 0 2 1 0 1 0 1 1 0 1 0 3 2 0 2 1 3 R R R R R R R I R K R I R K R K d (7.30) The third and the fourth term of Equation (7.25) become the same when2 1R R and in the limit of2 1, R R R we have:
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90 4 2 1 0 2 ) ( 2 2 2 ) ( 24 1 ) 1 ( 1 42 1 2 1R R dz e R eR R (7.31) The last term of the Casimir en ergy in Equation (7.25) becomes: ...... 1 23328 5 1 526 3 ) ( 1 48 6 1 6 4 1 11 0 2 2 ) )( ( 22 R R R R d j n R enj j n n (7.32) where we have retained only the n=1, j=1,2 terms. Combining Equations (7.29) (7.32) in to Equation (7.25), we have for the limit2 1, R R R where2 1R R the energy per unit length of the system defined as: 6 4 1 2 3 1 2 3 2 1 4 2 1 2 3 1 2 1 31 512 3 1 1 15 8 2 1 8 R R R R c EC (7.33) The energy is expressed as a sum of terms proportional to powers of theR R /1 ratio and we have considered the first two dominant terms. The other terms in CE have higher powers of 2 1/R R indicating that their contribution is small. 7.3.2 Numerical Results In this section we evaluate the Casimir energy for two straight parallel cylinders numerically and we present numerical and anal ytical results for various interesting cases by modifying the radial dimensi on, curvature, and material composition of our system. In order to have a better unders tanding of how the Casimir en ergy behaves for a system of two straight, parallel dielectr icdiamagnetic cylinders, it is essential to investigate the
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91 dependence of the interaction en ergy on the radial dimensions of the cylinders and their separation. This is achieved by ev aluating Equation (7.25) numerically. First we investigate the dependence of the Casimir energy over the surfacetosurface separation. Figure 7.2 shows the behavi or of the interacti on energy as a function of surfacetosurface separation d withnm 11 R. One sees that as 2R is increased the energy diverges slower as a function of d when compared to the one for2 1~ R R. At larger d separations,  CE decreases slower when 1 2R R as displayed in the inset of Figure 7.2. The crossover occurs at d~0.4 nm. The calculations reveal that the dominant contribution for the cases of 2 1~ R Rcomes from the third and fourth terms of Equation (7.25). As 2R becomes larger, the fourth term in Equation (7.25) remains dominant, while the third one becomes less important. Th e reason for the crossover (inset in Figure 5.2) for small and large 2R can be traced to the exponent ial distance dependences in these terms. There is a faster decrease in  CEfor small d when1 2R R as compared to the cases of small d and2 1~ R Rdue to the faster decrease in the exponential factors. But for large d, the situation is reversed as indica ted by the inset of Figure 7.2. When 1 2~ R R d there are two large distance scales. Because of their dominance through the and2 1 factors (which are subtracted), the expone ntial decrease in the fourth term is actually slower than the case for 1 2~ R R d The results shown in Figure 7.2 are independent of the choices for the dielectric responses of the cy linders and medium.
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92 Another very important case study concer ns the dependence of the interaction energy with respect to the dielectric respons es of the cylinders and medium. This study can provide us with very important info rmation about how ma terial composition influences the Casimir energy of a system. It has been predicted that the interaction between planar materials 1 and 2 immersed in medium 3 can be repulsive if the value of the dielectric constant of the me dium is between the values of the dielectric constants of the materials [39]. Recent meas urements of the Casimir for ce between a large sphere and a plate covered with a layer of silica, for which this c ondition for the dielectric properties is satisfied, demonstrate that the interaction can be repulsive [38]. We first investigated the case when the di electric constant of the environment is changed while the dielectric constants of the cylinders 2 1 are constant and secondly when the dielectric constant of one of the cylinders is varied and 3 2 are constant. The results are shown in Figures 7.3 and 7.4.
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93 0.100.150.200.250.30 0.30 0.24 0.18 0.12 0.06 0.30.40.50.60.70.8 0.05 0.04 0.03 0.02 0.01 R2=1 nm R2=3 nm R2=50 nmEC/EC0d(nm) d(nm) EC/hc Figure 7.2 Dimensionless interaction energy as a function of surfacetosurface separation defined as2 1R R R d 0 CE is defined as 2 1/R cl. For all cases, 22 1 and153
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94 1=2=2 1=2=15 1=15;2=20481216 0.12 0.08 0.04 0.00 0.04 EC/EC0 Figure 7.3 Dimensionless interaction energy as a function of th e dielectric function of the medium. 0 CEis defined as 2 1/R cl. The cylinders have equal radii, nm 12 1 R Rand centertocenter separation,nm 2 2 R.
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95 0481216 0.04 0.00 0.04 2=2;3=5 2=7;3=3 EC/EC0 Figure 7.4 Dimensionless interactio n energy as a function of the dielectric function of one cylinder. 0 CEis defined as 2 1/R cl. The cylinders have equal radii, nm 12 1 R Rand centertocenter separation,nm 2 2 R.
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96 The results in Figures 7.3 and 7.4 are easily understood by re alizing that the dominant contribution in CE comes from the third and fourth terms in Equation (7.25) since2 1R R The rest of the terms ar e found to be at least one or more orders smaller. Thus the interaction energy will be dominated by the prefactor ) )( /( ) )( (3 2 3 1 2 3 1 3 For2 1 is always positive meaning that 0 CEand the corresponding force is attractive. This is what is shown in Figure 7.3 for15 22 1 On the other hand, when) ( ) (1 2 3 2 1 the prefactor becomes negative, thus 0 CEwhich corresponds to a repulsi ve force. The maximum in CE as a function of 3 is found when 2 1 max 3 This is seen on Figure 7.3, which shows that5 5max 3 for2 152 1 Another very important scenario that yi elds positive Casimir energy is when we change the composition of one (or both) cylinde rs while keeping the medium fixed. This case is shown in Figure 7.4 where for 22 and53 0 CEas long as2 3 1 is satisfied. Also, for72 and33 0 CEwhen1 3 2 is satisfied. Furthermore, since the prefactor controls the magnitude of th e interaction energy, when the dielectric constants of the cylinders and the environment are such that 3 1 and 3 2 are small,  CEis also small in magnitude. However, when the dielectric properties are very differe nt in value, then  CEbecomes large.
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97 CHAPTER 8 EXCITONPLASMON INTERACTI ON IN CARBON NANOTUBES 8.1 Fundamental Eff ects of Long Range Interactions An emerging area of research in carbon na notubes is that of optoelectronics. The interest in carbon nanotube optoelectronics arises because nanotubes have several properties that make them exce llent optoelectronic materials. For instance, an important characteristic of optoelectronic materials is the presence of a direct bandgap which allows electronic transitions between th e valence and conduction bands to proceed without the intervention of phonons. Carbon nanotubes are ma terials possessing direct band gaps, thereby allowing multiple bands to participate in optoelectronic events spanning a wide range of ener gies. The purpose of this chapter is to investigate the viability of carbon nanotubes as candidates for novel optoelectronic devices. The work is characterized in three aspects: (1) the problem is formulated by bringing an excited atom in the vicinity of a carbon nanotube; (2) th e electronic structure of carbon nanotubes, and (3) the dielectric response of carbon nanotubes. The first part is conceived by defining a Hamiltonian and this was done by Dr. Igor Bondarev [43]. The sec ond and third parts were my work and helped us gain an unde rstanding to carbon nanotubes used in optoelectronic devices.
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98 Quasione dimensional structures such as nanowires and carbon nanotubes exhibit manybody effects that dominate their optical properties. The most important aspect of manybody effects in optical properties are ex citons, which are el ectronhole pairs bound by the Coulomb interaction. The excitation of an electro n across the bandgap by the absorption of a photon leaves behind a positivel y charged hole and they both experience a Coulomb interaction that lead s to a bound state where the ex citon and hole are separated by the exciton radius. The evidence of the importance of excitons in carbon nanotubes has come from both experiment and theory. On the theoretical front, ab initio calculations of the optical spectra of carbon nanotubes incl uding electronhole inte ractions within the Bethe_Salpeter approach [88] has indicated a large exciton binding energy in semiconducting carbon nanotubes and even in meta llic nanotubes. The optical absorption spectrum of semiconducting carbon nanotubes show a series of sharp and pronounced excitonic lines corresponding to bound excitons. Th e excitation excit on energies for different carbon nanotubes have already been reported [88], a nd we will be making use of the lowest exciton energies in this projec t. With the presence of excitons in carbon nanotubes already established, we proceed in investigating the contribution of long range interactions in carbon nanotubes. In this section, we aim to demonstr ate the fundamental e ffects of longrange interactions in carbon nanotubes, more spec ifically in the form of excitonplasmon interactions in carbon nanotubes. The proposed interactions of excitonic states with surface electromagnetic modes of smalldi ameter semiconducting singlewalled carbon
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99 nanotubes can produce strong excitonsurfa ceplasmon coupling. Understanding of electromagnetic interactions between excitons and plasmons in molecular systems and spatially confined quantum systems, such as carbon nanotubes, is both of fundamental interest and of importance in the developm ent of photonic and optoelectronic devices [89]. The interaction strength determines the absorption and emission properties of molecules coupled to nanostructures. The ma nufacture of nanostructures and control of their interaction in devices are some of the challenges researchers are facing today. Organic semiconductors, hybrid semiconductorm etal nanoparticle molecules and carbon nanotubes support excitonic st ates, created by the absorption of photons. Carbon nanotubes also support surface electromagnetic modes, both transversely polarized and longitudinally polarized. The l ongitudinally polarized surfac e electromagnetic modes are generated by the electronic C oulomb potential and result in the plasmonic excitations [90]. Due to the quasione dimensionality of the nanotube, the exciton quasimomentum vector is directed predomin antly along the nanotube axis and this corresponds to the longitudinal exciton. Since the surface plasm ons are also directed along the propagation direction, they couple with the longitudina l excitons on the nano tube surface. It is therefore essential to consid er the correct electronic st ructure of the carbon nanotube since the plasmonic nature of the nanotubes is related to their dielectric function. 8.2 Electronic Struct ure of Carbon Nanotubes (CNTs) The electronic structure of any na notube can be obtained within the band tightbinding (TB) model of graphene by folding the band along a certain direction [91].
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100 However, experimental and theoretical calcul ations [92] suggest that hybridization of graphitic and states should occur because of the curvature of the tubes. The curvature effects induce the mixing of the and bands and these effects are found to alter significantly the electronic st ructure of nanotubes compared to the TB model [93]. The electronic structure of carbon nanotubes is studied within the sp3 tightbinding model [91,94,95] and the randompha se approximation (RPA) [96]. More specifically, we use the sp3 tightbinding model to calculate the electronic structure, and the randomphase approximation to eval uate the dielectric function. The 2pz and (2s, 2px, 2py) orbitals in a CNT, respectively form the and bands. The curvature effects result in the misorientation of the orbitals and ther efore the mixing of the orbitals need to be taken into account in the calcu lations of the band structure. The electron wavefuncti on of a carbon nanotube ) (,rJ k can be represented as linear combinations of basis functions) (,rJ k In the tightbinding method, the basis functions are expressed by atomic orbitals which are centered on atoms with different position vectors. The represen tation of the electron wave function is expressed as: i J ki J ki J kr c r) ( ) (, , (8.1) Using the above representation in the oneelectron equation of Schrodinger, one obtains the matrix equation for the coefficientsJ kic,: 0 ) ( r kl r klr r J kl r klrc S E H (8.2)
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101 In the orthogonal tightbinding models ( TB), the overlap of the atomic orbitals centered on different atoms is i gnored and the overlap matrix SklrÂ’ is approximated by a unit matrix [91]. However, it is clear that the overlap of orbitals on different atoms is not always negligible due to the curvature eff ects and the mixing of the orbitals. Here, the overlap is included in the nonorthogonal TB model and only nearest neighbors are taken into account [92]. Only zigzag nanotubes are considered in our calculations and diagonalizing the matrix defined by Equatio n (8.2) yields the ba nd structure of the nanotubes under consideration. In this work, we consider the following semiconducting zigzag nanotubes: (8,0), (10,0) and (11,0). After diagonalizing the matrix equations for these nanotubes, we obtained their respective band structures shown in Figures (8.1) Â– (8.3).
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102 Figure 8.1 Band Structure of the (8,0) CNT.
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103 Figure 8.2 Band Structure of the (10,0) CNT.
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104 Figure 8.3 Band Structure of the (11,0) CNT.
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105 8.3 Dielectric Respon se of Carbon Nanotubes (CNTs) Now we have obtained the ba nd structure of the nanotubes, we can make use of the eigenstates and energies to calculate th e dielectric response using the random phase approximation (RPA). RPA is one of the most often used methods for describing the dynamic electronic response of sy stems. It is responsible for effects such as screening of external potentials (thos e responsible for scattering in the system) as well as for collective excitations of the free elec tron gas (plasmons in many body theory). The dielectric function within RPA is given as [96]: J k J k J q k F J k F J q k i J J oi E E n n J k e J q k qr K qr I e J w q, , 2 .r q 2, ) ( ) ( 4 ) , ( (8.3) where o is the background dielectric constant;) ( ), (qr K qr IJ Jare the modified Bessel functions of first and second kind of order J respectively; q is the momentum with dimensions 1/r; r is the nanotube radius; is the frequency; is the level broadening parameter; J k is the electronic state described by energiesJ kE; and F J kn, is the FermiDirac distribution. The electronic bands of si ngle wall carbon nanotubes ha ve been assigned by line group symmetry and their corresponding selecti on rules have already been derived [97]. The angular momentum J is found to be a good quantum number and serves as a band index as well [94]. According to the line group theory, the se lection rules for the angular momentum J depend on the direction of the polarizati on. In the direction parallel to the
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106 tube, the selection rule imposes0 J and in the direction perp endicular to the tube we have1   J. Since only the interactions between longitudinally polarized surface electromagnetic modes and longitudinal excito ns are considered in our calculations, due to the quasione dimensionality of the nanotubes, only the condition 0 J is used in the evaluation of the dielectric f unction. The dielectric function, evaluated in both the real and imaginary realms, is e xpected to strongly depend on q and J. Electrons are excited from the occupied valence bands to the unoccupied conduction bands of the same JÂ’s. The results for both the real and imaginary part s of the dielectric f unction for the zigzag nanotubes are shown in Figures (8.4) Â– (8.6).
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107 Figure 8.4 Dielectric Response of the (8,0 ) CNT. Frequency is measured in eV and the dielectric f unction is dimensionless.
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108 Figure 8.5 Dielectric Response of the (8,0) CNT in the low energy regime. Frequency is measured in eV and the dielectric function is dimensionless.
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109 Figure 8.6 Dielectric Response of the (10, 0) CNT. Frequency is measured in eV and the dielectric f unction is dimensionless.
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110 8.4 ExcitonPlasmon Interact ion in Semiconducting Single Wall CNTs After obtaining the dielectric response of the semiconducting nanotubes, we can now proceed in demonstrating the excitonplas mon coupling. In orde r to do so, we first need to find the conductivity of each nanotube and this is easily obtained from the dielectric function using the Drude relation. The plasmons are then located using the loss function ) / 1 Im( or ) / 1 Re( measured in the electron energy loss spectroscopy (EELS) experiments to determine the propertie s of collective electr onic excitations in solids [98]. The plasmons are defined by pronounced peaks occurring in the Plasmon density of states. We find that these peaks take place when the two conditions, 0 ) Im( (or0 ) Re( ) and0 ) Re( (or0 ) Im( ) simultaneously and this is clearly shown in Figures (8.7) and (8.8) for th e (11,0) and (10,0) na notube respectively [42]. Figures (8.7) and (8.8) show the dimensionless conductivity as a function of dimensionless energy and it is observed that the plasmon peaks broaden as the nanotube radius decreases as a result of stronger hybr idization effects in small diameter nanotubes [99]. The peaks in Figures (8.7) a nd (8.8) defined by the loss function) / 1 Re( clearly demonstrate the plasmonic nature of the CNT surface excitations.
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111 Figure 8.7 Dimensionless conductiv ity of the (11,0) CNT as a function of dimensionless energy.
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112 Figure 8.8 Dimensionless conductiv ity of the (10,0) CNT as a function of dimensionless energy.
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113 Now that the plasmonic nature of the nanot ubes is clear, we can proceed to show the excitonplasmon coupling in small diamet er semiconducting nanotubes. In order to examine the excitonplasmon coupling, we need to examine the dispersion curves of the CNTs. In strong coupling regime, the mixing of exciton and plasmon states modifies the dynamics of the system and this appears as an anticrossing of the two coupled modes [42,43]. The dispersion curves are obtai ned by solving the dispersion equation as described in Ref. [42,43] and the dispersion curves are shown in Figures (8.9) and (8.10). The corresponding plasmon peaks are also show n in the figures. The upper branches in Figures (8.9) and (8.10) correspond to th e plamon dispersion curve while the lower branches correspond to the exciton dispersion curve. Fi gures (8.9) and (8.10) clearly demonstrate an anticrossing behavior with (R abi) energy splitting ~0.1 Â– 0.3 eV, thereby indicating a strong surface plasmonexciton co upling. This coupling energy is almost as large as the typical exciton binding energies in similar CN Ts (~0.3 0.8 eV) [100103], and of the same order of magnitude as th e excitonplasmon Rabi splitting in organic semiconductors (~180 meV) [104]. This plas monexciton energy found here for the zigzag nanotubes are however much larger than the excitonpolariton Rabi splitting in semiconductor microcavities (~140400 eV) [105107], or the excitonplasmon Rabi splitting in hybrid semiconductormetal nanopar ticle molecules [108]. The formation of the strongly coupled excitonpl asmon states is only possible if the exciton total energy is in resonance with the energy of the surface plasmon mode.
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114 Figure 8.9 Dispersion curves of the exci ton and plasmon for the (11,0) CNT.
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115 Figure 8.10 Dispersion curves of the exciton and plasmon for the (10,0) CNT.
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116 CHAPTER 9 SUMMARY 9.1 Overview and Conclusion In this dissertation, we we re able to apply the concept of vacuum energy to cylindrical nanostructures using the mode summation method. The mode summation is one of the most convenient and elegant theore tical approaches used when it comes to cylindrical geometries. Although quantum electrodynamics offer a more advanced theoretical technique by taki ng into account realistic materi al properties, it becomes far too complicated when dealing with cylindrical geometries because of the large size of the GreenÂ’s tensor that contains the informa tion of dielectric and magnetic permeabilities. The theoretical challenge of applying the mode summation method to a new set of quasione dimensional cylindrical structures prove d to be a successful one. In doing so, the Casimir energies of a cylindrical dielectricdiamagnetic layer, multiple cylindrical conducting shells and two straight, parallel dielectricdiamagnetic cylinders were calculated and finite and phys ically convergent numerical values were obtained. The models developed in this dissertation can pr ovide a very important and novel insight to long range interactions in cylindrical nanostructures su ch as carbon nanotubes and nanowires. Although qualitative in nature, the models developed here could be of significant interest to ex perimental studies of cy lindrical nanostructures.
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117 In this work, analytical expressions and numerical calculations for various limiting cases were presented in terms of the radial dimension, curvature, and material composition of the cylinder or material. Various limits were investigated and the Casimir energy of certain structures were reproduced such as the Casimir energy of two parallel metallic plates derived by H. G. Casimir was retrieved here. Also, the fundamental effects of long range in teractions were studied in the fo rm of excitonplasmon coupling in smalldiameter semiconducting carbon nanotubes. In this section, we briefly mention the results achieved and the conclusions for the different cylindrical models considered. 9.2 DielectricDiamagnetic Cy lindrical Layer of Finite Thickness In this model, the Casimir interaction en ergy for a cylindrical layer with a finite thickness was calculated using the mode su mmation technique and with the Riemann function regularization procedure. The dielectr ic and magnetic proper ties of the layer and those of the surrounding medi um were assumed to be described by real constant dielectric permittivities and magnetic pe rmeabilities satisfying the relationship of constant speed of light across the interfaces This method proves to be convenient and intuitively easy to follow. It is form ulated in terms of only one parameter ) /( ) (m m which allows us to make analytic al and numerical evaluations for practically important limiting cases such as a dielectricdiamagnetic cylindrical layer as well as two concentric perfectly conducting thin shells.
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118 The case of a dielectricdiamagnetic layer with 1 was considered and evaluated the first nonzero term2in the infinite sum representing the Casimir energy per unit length. The Casimir energy is negative, and it is3 1 2) 1 / /( 1 R Rin the limit1 /1 2 R R. When 2R, the problem becomes equivalent to that of a dielectricdiamagnetic solid cylinder. The Casimir energy2for such a cylinder is zero [3,5], and here we recover this result for 2R. Also recovered was the Casimir energy per unit area of a dielectricdiamagnetic plate [74] in the limit of 2 1, R Rwhen .1 2const R R d Another important case i nvestigated was that of1 describing two th in perfectly conducting concentric cylindrical shells. We find that the Casimir energy in this case is also negative and it is3 1 2) 1 / /( 1 R Rfor1 /1 2 R R, whereas for 2Rit approaches the limit of a single perfectly co nducting cylindrical sh ell [3,5]. We also recover the wellknown Casimir formula for the energy per unit area of two parallel perfectly conducting plates [7] separated by a distance const R R d 1 2in the limit of 2 1, R R.
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119 9.3 N perfectly Conducting Cylindrical Shells The Casimir energy of a system of N perfectly conducting, infinitely long cylindrical shells is also calculated by ma king use of the mode summation method. We present an expression fo r the Casimir energy of N shells and analyzed various limits for which the analytical expressi ons are known. The Casimir formula for the energy per unit area of two parallel perfectly conductin g plates [7] separated by a distance const R R d 1 2in the limit of 2 1, R R was recovered. Also analyzed was various limits in the case of three shells, and it was f ound that our result agrees with Ref. [84] in the limit of 3 2 1, ,R R Rand const d d 2 1where 1 2 1R R d and2 3 2R R d in which case our system corresponds to three parallel plates. The case of N perfectly conducting concentric cylinders mi ght be of particular intere st as a qualitative model of the Casimir interactions in a multiwall meta llic carbon nanotube system. More thorough and realistic analysis is necessary to describe the Casimir interaction in multiwall carbon nanotubes, by taking into account real istic electromagnetic properties. 9.4 Two Straight and Para llel DielectricDiamagnetic Cylinders In this model, the Casimir energy for a system of two infinitely long parallel cylinders immersed in medium is calcula ted using the mode summation method. The analytical expressions deri ved are of particular impor tance for the Casimir energy dependence on the dielectric properties of th e involved objects. The interaction energy is found to be positive (repulsive force) when th e value of the dielectric constant of the medium is between the values of the dielectric constants of the two cylinders. This
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120 particular result may be of significance to re search efforts in quantum levitation or reducing friction between nanosized components in devices when cylindrical objects are involved. It is in teresting to note th at the relation 2 3 1 is the same for a repulsive force between planar systems as described in [38]. These calculations may be viewed as further evidence that the sign of the Casimi r force can be manipulated by changing the dielectric response properties of the involved objects rega rdless of their geometry. 9.5 ExcitonPlasmon Coupling Effect The fundamental effects of longrange in teractions in carbon nanotubes (CNTs) have been studied in the form of excitonpl asmon interactions in CNTs. The interactions of excitonic states with surface electromagne tic modes of smalldiameter semiconducting singlewalled carbon nanotubes have been shown to produce strong excitonsurfaceplasmon coupling. In order to obtain th e excitonplasmon coupling, the dielectric response of the nanotubes was first calcu lated using the random phase approximation (RPA) and both the real and imaginary part s of the dielectric function of zigzag nanotubes were obtained. Th e band structure of the nanotubes was obtained by considering the neares tneighbor nonorthogonal tightbinding model. The plasmonic nature of the semiconducting nanotubes was described by pronounced peaks in the plasmon density of state using the loss functio n. The clear anticrossing behavior in the dispersion curves of the excitons and plas mons provided the evidence of the strong coupling effect.
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ABOUT THE AUTHOR Kevin Tatur was born in the island of Mauritius in the Indian Ocean. He received his BS with honors in Physics at the Univers ity of Mauritius and his MS in Computer Science from the Griffith University in Australia. Kevin then joined the University of South Florida in 2006 where he received his MS in Physic s and worked towards his Ph.D. degree in Physics. Hi s work in the field of long range interactions has been published in high impact journals such as Physical Review A (Rapid Communications), Physical Review B, Physics Letters A, Op tics Communications. Kevin also has various conference proceedings and has given presenta tions at important conferences. Kevin did an internship at the MOFFITT Cancer Center and Research Institute in a medical physics. He was trained in various aspects of medical physics and assisted board certified medical physicists in va rious duties.
