Study of the Casimir-Lifshitz force in the plane parallel geometry

Study of the Casimir-Lifshitz force in the plane-parallel geometry Victoria Esteso Carrizo List of abbreviations FC−LCasimir-Lifshitz force FgGravity force FBBuoyancy force Fel Electrostatic force EM field Electromagnetic field E field Electric field TE Transversal Electric mode TM Transversal Magnetic mode rSimple Fresnel coefficient RMultiple Fresnel coefficient jPolarization index j= TM or TE lLayer material index mMatsubara frequency index ˜nComplex refractive index nReal part of refractive index kImaginary part of refractive index 0Static dielectric constant Dielectric function ~Reduced Planck constant ωAngular frequency ξMatsubara frequency TTemperature iii ∆Increment in the contiguous magnitude d0Separation distance between the thin film and the substrate or equivalently thickness of layer l= 0 MS Multilayer Structure iv Contents 1 Introduction 1 1.1 Dispersion forces: from Casimir effect to Lifshitz theory 4 1.2 CasimirEffect....................... 6 1.3 Casimir-Lifshitz force . . . . . . . . . . . . . . . . . . . 8 1.4 Repulsive Casimir-Lifshitz force and levitation . . . . . 11 1.5 Levitation phenomenon in the plane-parallel geometry . 14 1.6 Casimir-Lifshitz force and Materials Science . . . . . . 16 1.7 Thisthesis......................... 18 2 Methods and Materials 20 2.1 Expressions of the Casimir-Lifshitz force in the planeparallelgeometry ..................... 21 2.1.1 Rotation from real frequencies (ω) to Matsubara frequencies (ξm) at T6=0K ........... 23 2.1.2 Systems with arbitrary number of layers . . . . . 28 2.1.3 Dielectric function of materials . . . . . . . . . 30 2.1.4 Semi-analytical calculations . . . . . . . . . . . 33 2.2 Electrostatic force . . . . . . . . . . . . . . . . . . . . . 36 2.3 Roughness and patch potentials effect . . . . . . . . . . 37 2.4 Transfer Matrix Method . . . . . . . . . . . . . . . . . . 39 2.5 Field distribution and power absorbed . . . . . . . . . . 43 v 3 Levitation of thin films due to Casimir-Lifshitz interactions. Analysis of temperature effects 46 3.1 Introduction........................ 47 3.2 Levitation Phenomena in plane-parallel systems due to Casimir-Lifshitz and gravity force balance . . . . . . . . 47 3.3 Temperature dependence of equilibrium distance at room temperature ........................ 47 3.4 Simple rules to predict temperature variation effects on deq 47 3.5 Conclusions........................ 47 4 Optical resonators based on Casimir-Lifshitz forces 48 4.1 Introduction........................ 49 4.2 Fabry-P´ erot optical cavities . . . . . . . . . . . . . . . . 49 4.3 CasimirLifshitz Force Based Optical Resonators . . . . . 49 4.4 High Q-factor optical resonators for accurate measurements of repulsive FC−L................. 49 4.5 Effect of room temperature variations on the force balance and the optical cavity characterization . . . . . . . 49 4.6 Conclusions........................ 49 5 Optical Interference Effects on Casimir-Lifshitz Forces 50 5.1 Introduction........................ 51 5.2 Periodic multilayer structures and their optical response . 51 5.3 Effect of thickness and number of layers on Casimir-Lifshitz force............................ 51 5.4 Tuning the Casimir-Lifshitz force with multilayer nanostructures.......................... 51 5.5 Conclusions........................ 51 Bibliography 52 vi Chapter 1 Introduction It might sound surprising that the gecko’s ability to adhere to surfaces [1], the friction that so often plagues tiny components in nanomachines [2, 3], the density anomaly of water [4], and the early stages of planets formation [5] have a common origin in the zero-point fluctuations of quantum fields [6]. These fluctuations are the basis of the weakest type of the interatomic and intermolecular forces which allow the formation of condensed matter, named dispersion forces [7]. Dispersion forces are inherently present in our daily life. In nature, they are at the heart of important phenomena in chemistry, physics and biology, being the cause of adhesion, surface tension, strength of solids and stability of both colloidal suspensions or biological membranes [8, 9, 10, 11], amongst others. They are also responsible for the organization of bio-systems like cellulose, lignin, and proteins, and they are the reason why geckos can climb in walls and ceilings [12, 13, 14, 15]. In nowadays technology, accelerometers integrated in smartphones or high precision positioning devices are examples of specialized cases in which dispersion forces play a main role, becoming more important as the distance between the interacting bodies is reduced [16, 17, 18, 19, 20]. Continuous miniaturization of devices has been a common fact in 1 the last decades. With this miniaturization, high accuracy in the control of forces acting on, for example, micro- and nanoelectromechanical systems (MEMS and NEMS), has become essential for their optimized design. Another fascinating phenomenon fruit of the confinement of quantum fluctuations is the Casimir effect [21]. Briefly speaking, the Casimir effect occurs when two or more bodies are placed at nano- and micro-scale distances perturbing the available vacuum modes. Since the vacuum energy density varies when one of the objects is displaced with respect to the others, a force results. Such force is the so-called Casimir force. As it will be described in detail along this thesis, this force was originally predicted for perfectly conducting plates, but when realistic materials are considered, such force remains. In what follows, this force occurring amongst macroscopic bodies at nano- and micro-scale distances will generally be referred to as the Casimir-Lifshitz force, FC−L. In this regard, the influence of attractive FC−Lin NEMS and MEMS, which are the basis of computers, smartphones, and similar electronic devices, has been particularly analyzed in literature since they are the cause of main failures due to stiction between the components. One of the first evidences of this undesirable attractive interaction in devices was reported by Srivastava in 1985 [22], pointing that the Casimir-Lifshitz energy in the gate-region capacitor of a field-effect transistor device was 10 %of the electrostatic energy. By then, the scale of the systems was too large to have a critical impact, but nowadays, with the shrinking of devices, the role of the Casimir-Lifshitz interaction becomes decisive in the stability and device performance [16, 23, 24, 25]. Typically, the effect of FC−L on micro and nanodevices is studied in simple systems like parallel plate capacitive switches (Fig. 1.1(a)), and cantilevers (Fig. 1.1(b)). In them, external forces such as FC−Lor electrostatic forces, affect the dynamic of a movable part of the device which in turn is restored to its original position due to an elastic force. The stability is given by the balance between the elastic force and the applied external forces, and when the 2 Figure 1.1: (a) Scheme of a MEMS consisting on a plane-parallel capacitive switch with one of the plates attached to a spring. (b) Scheme of a cantilever deflected an angle θwith respect to a reference position when no external forces are applied. latter overcomes the former, the movable part collapses and irreversible adheres to the opposite surface. This phenomenon is known as jump into contact and results in the uselessness of the device. Repulsive Casimir- Lifshitz interactions or reducing the attractive intensity of FC−Lcould avoid adhesion and permanent stiction between movable parts in devices. Because of that, a deep knowledge of the underlying fundamental concepts of such forces is essential for understanding nature mechanisms, as well as for the future of advanced technologies, such as MEMS and NEMS, optoelectromechanics and microfluidic devices, sensors, or optical communications. Recent experimental advances have allowed acquiring novel insight into the nature of such dispersion forces [26] while proving their potential to attain quantum levitation between macroscopic bodies [27], settling this topic in the focus of the scientific community and making the comprehension of this quantum interaction essential for the development of the forthcoming tinier technology. Interestingly, the nature (attractive or repulsive) and strength of the 3 analytical expression is only provided for the plane-parallel configuration. For curved surfaces, for instance, as that would be the case of colloidal particles or gas bubbles, it is possible to use the so called proximity force approximation (PFA) which was first applied by Derjaguin and collaborators [35] in FC−Lmeasurements between two fused quartz surfaces, one flat and the other with spherical shape. They showed that the force acting between a plate and a body of finite curvature can be expressed in terms of the force between two planar semi-infinite walls. It was also demonstrated that the force is much more intense in the plane-parallel configuration than in geometries incorporating spheres. The first experimental attempt to measure the Casimir-Lifshitz force with real materials was reported by M. Sparnaay in 1958 [36], who observed, through capacitance measurements, the attractive force between pairs of metallic plates made of aluminium, chromium, and steel and chromium. However, results were not conclusive due to, among other difficulties, the inaccuracy in the plates alignment. Such difficulty was overcome by Dejarguin and co-workers [35], by considering curved surfaces of different radii to avoid limitations on the parallelism between the interacting bodies. After some other attempts without conclusive measurements of the Casimir-Lifshitz force[37, 38, 39], some relevant experimental results were finally attained several decades after the theoretical prediction thanks to the improvement in precision and sensitivity of the techniques employed. Some examples are the experiments by S. Lamoreaux et al. [40], who measured the force between a lens and a plate both coated with gold, using a torsion pendulum; by U. Mohideen et al. [41], who employed for the first time an atomic force microscope and measured the force between a sapphire plate and a sphere attached to the cantilever coated with metals such as aluminium, chromium and gold; or the work by by R. Decca et al. [42], who achieved the highest experimental precision of the Casimir-Lifshitz force employing a micromechanical torsional oscillator. Interestingly, the spanning of the Casimir- 10 Figure 1.4: Scheme of a plane-parallel system with a material (that might be different from vacuum) between the interacting bodies. Lifshitz interaction to current technologies was reported in 2001 with the first MEMS actuated by FC−L[43]. In fact, few examples demonstrating repulsive Casimir-Lifshitz forces have been reported, being one of the reasons the lack of knowledge about the value of the dielectric permittivity at ’all’ frequencies (essential for evaluation of Casimir-Lifshitz forces), which is only known for a few solids and liquids. Among them, a limited number of material combinations that are chemically stable when arranged together may display repulsive forces of this kind. Additional outstanding experimental demonstrations of thermal contributions to the Casimir-Lifshitz force [44] and the critical Casimir force [45] were also achieved in the following decade. Further details about the experimental force measurements progress including real materials are provided in Chapter 4. 1.4 Repulsive Casimir-Lifshitz force and levitation One of the most interesting results derived from the Lifshitz theory is the fact that FC−Lmay be repulsive, having important implications in 11 fundamental physics and technology [32, 46, 47], since, for instance, it would prevent adhesion and stiction between movable parts in NEMS and MEMS which are the reason of main failures in this kind of devices [48, 3]. Moreover, the possibility of having repulsive FC−Lrapidly led to the idea of quantum trapping, a phenomenon in which FC−Levolves from repulsion to attraction as the separation distance between bodies increases. Moreover, if the repulsive FC−Lis strong enough to counteract the effect of gravity if the system is in presence of the gravitational field, quantum levitation, might occur. Until the extended theory was developed, the Casimir-Lifshitz interaction was predicted to be always attractive between parallel plates, typically separated by vacuum or air (i.e ε00 = 1). Actually, the interaction between two plates made of the same material will always be attractive. Several approaches might be applied to obtain repulsive Casimir-Lifshitz forces between two bodies in vacuum. Examples cover the consideration of interacting objects with alternative geometries, such as an elongated metallic particle approaching a metal plate with a hole drilled on it [49], the employment of metamaterials, like chiral or magnetodielectric metamaterials carefully designed [50], or the operation with materials whose dielectric functions fulfill certain relations, as it will be extensively explained in the coming sections. This thesis will focus on the latter approach and will consider the plane-parallel geometry based on stratified realistic materials having a liquid medium between the interacting plates. Figure 1.4 displays a scheme of a general plane-parallel system consisting of two bodies whose dielectric functions are ε1and ε3, separated by a liquid with ε2. According to Lifshitz results, repulsive forces might be attained if the dielectric function of the interacting materials fulfill one of the following conditions: ε1< ε2< ε3(1.3) 12 or ε1> ε2> ε3(1.4) Importantly, these conditions must be satisfied in a wide range of the so-called Matsubara frequencies (further details are provided in Chapter 2). In short, an integral over the full spectral frequency range (ω∈[0,∞]) of the dielectric function of materials is computed, making it difficult to find combinations of materials that obey the previous conditions. The first experimental results of repulsive Casimir-Lifshitz interactions were measured six decades after the theoretical prediction with the use of an atomic force microscope (AFM) between the tip of the cantilever, or a sphere attached to it, and a flat substrate, all immersed in a liquid at small separation distances (below 10 nm) [51, 52, 53, 54, 55]. It was also demonstrated that the inclusion of a fluid different than air between the interacting materials decreased the intensity of the force [56]. For larger separations distances, above tens of nanometers, repulsive FC−Lwas first theoretically predicted [57] and afterwards measured with the use of an AFM [58] between a gold sphere and a silica substrate all drowned in bromobenzene. Several considerations were needed to successfully perform this experiment: i) calibrating the AFM measurements in liquids [56], ii) studying the effect of discrepancies among theoretical and real dielectric properties of materials [59], and iii) analyzing the effect of residual electrostatic forces [60]. Most of the experimental measurements of repulsive FC−Lwere performed in the sphere-plane geometry in order to avoid alignment problems. Concerning the plane-parallel geometry, the first experimental evidence was recently attained [27] by using a completely different approach based on spectroscopic measurements in a quantum trapping scenario. In it, an optical resonator was built with a thin metallic film immersed in a fluid over a substrate due to the balance of gravity and repulsive FC−L. Casimir-Lifshitz forces were indirectly attained through reflectance measurements by adjusting the equilibrium distance at which the metallic film 13 levitated. All those advances prove the possibility of manipulating the strength and nature of FC−L, shedding light on fundamental aspects of FC−Land laying the groundwork for technological applications. Particularly, stable quantum levitation can be used as a platform for a variety of applications such as contact-free nanomachines, ultrasensitive force sensors, and nanoscale manipulations. Next, levitation will be deeply analyzed in a plane-parallel geometry. 1.5 Levitation phenomenon in the plane-parallel geometry Let us consider a simple plane-parallel system with two plates (one of them a self-standing thin film) facing each other in a fluid as it is displayed schematically in Fig. 1.5. Typically, the thickness of the thin film (d) and the separation distance (d0) between it and the substrate, range from tens to hundreds of nanometers. In this simple case, it is considered that the forces acting on the thin film are gravity (Fg), buoyancy (FB), and FC−L. It might also be taken into account the effect of other forces acting on the system, such as the double layer force (Fdl), arising near charged surfaces in liquid solution, or the effect of surface roughness and patch potentials, described in detail in Section 2.2 and Section 2.3, respectively. However, for the sake of simplicity, in what follows, those effects will be neglected to analyze fundamental concepts of the levitation phenomenon. Definitions of Fg,FB, and FC−Lare given in Eq. 1.5, 1.6, and Eq. 2.1 (extensively explained in Section 2.1), respectively. Fg=−gρfilmd(1.5) FB=gρfluidd(1.6) 14 Figure 1.5: Scheme of a plane-parallel system consisting of a selfstanding thin film of thickness dimmersed in a fluid standing above a substrate at a separation distance d0. In the above expressions, gis the standard gravity, and ρfilm and ρfluid, the density of the suspended film and the fluid, correspondingly. Note that units of ρare kg/m because of the infinite area considered of both the substrate and the thin film. Following the sign convention of the Casimir effect, positive values of the force correspond to repulsive forces, and negative ones to attractive FC−L. The expression of the force balance at equilibrium is then given by: FT(deq, T) = Fg+FB+FC−L(deq, T) = 0 (1.7) with FTthe total force acting on the suspended film, and deq the equilibrium separation distance at which the thin film stands over the substrate in thermal equilibrium at a fixed temperature, T. In Eq. 1.7, the balance of forces is zero and the system is at mechanical equilibrium. This equilibrium can be either stable or unstable, depending on the sign of the forces acting on the thin film. Figure 1.6 illustrates both scenarios of stability. Panel (a) displays a situation in which the suspended film is at stable equilibrium. In this case, the density of the levitating film is higher than that of the fluid, and so, the intensity of gravity is larger than that of the buoyancy force. In this situation, if the thin film approaches the substrate, the repulsive FC−Lrestores the equilibrium position deq, whereas, if the thin film moves away from the substrate, Fg brings it back to deq. Panel (b) presents the opposite case, in which FC−L 15 Figure 1.6: (a) Stable equilibrium distance (b) Unstable equilibrium distance. between the thin film and the substrate is attractive, but the buoyancy force compensates both gravity and FC−L, as the density of the fluid is larger than that of the thin film. In this case, the system is at an unstable equilibrium position, and if the film deviates from deq it will stick to the substrate due to the attractive FC−Lif d0< deq, or it will float if d0> deq. In this thesis, the balance of FC−L,Fg, and FBis theoretically analyzed in plane-parallel geometries based on stratified materials that can be easily found in nature and whose dielectric functions relations enable repulsive Casimir-Lifshitz forces. 1.6 Casimir-Lifshitz force and Materials Science Materials Science is an interdisciplinary branch of Science based on Physics, Chemistry and Engineering, focused on the development of novel materials. Originally, Material Science was devoted to the study of metallurgy an ceramics, focusing on the relation between the process of fabrication, the microstructure and the resulting properties of materials. However, 16 soon those analyses were extended to all kinds of materials, including semiconductors, polymers, biomaterials, nanomaterials, and optical and magnetic materials, among others. Designs of such materials are commonly inspired by the surrounding world, mimicking or reproducing the same structures or complexity found in nature, but also new materials are created by adding advanced functionalities to the already existing ones. Examples of outstanding materials developed in Material Science are graphene [61], furellenes [62], photonic and acoustic crystals [63, 64], doped semiconductors with desired electronic properties[65], perovskites [66], biocompatible polymers [67], quantum dots [68], spin valves [69], and a large etcetera, which have applications on diverse fields such as photovoltaics [70], tissue regeneration [71], magnetic storage [72], drugs delivery [73], light emitting devices [74] or quantum computation [75]. Of particular interest for the thesis work herein presented are the multifunctional optical materials. Materials with designed and engineered optical properties have allowed the control of the absorption, emission or propagation of light in optical fibers [76], wave guides [77], antireflecting coatings [78], plasmonic structures [79], photonic crystals [80] or negative refractive index materials [81]. The possibility of tailoring electromagnetic boundary conditions enables a variety of optical phenomena and related applications, and in addition, it allows to engineer dispersion forces, such as the Casimir-Lifshitz force. Particularly, the intensity and nature of FC−Lcan be tuned by controllably modifying the optical properties of the interacting materials. In this regard, optical design may allow the development of materials with specific functionalities and characteristics, allowing a further comprehension of the underlying physical and chemical processes in dispersion forces. For instance, based on attractive van der Waals forces, Material Science has allowed the creation of bio-inspired materials that let humans climb vertical walls as geckos do [82], and new functionalities have been added to already existing nanomachines designed to be 17 driven by the Casimir-Lifshitz force [43]. However, as it was mentioned before, it is not straightforward to find combinations of materials presenting repulsive FC−Las intuition cannot be applied when Matsubara frequencies are computed. Moreover, since the dielectric function of all materials in the system need to be known over an ’infinite’ frequency range, restrictions on the possible materials to be employed are maximized. Ultimately, these compact materials must be chemically compatible with the fluid in which they are immersed, and their thickness need to be easily determined in plausible experiment. Different examples of materials combinations have been proposed in literature [83, 84, 85], including the use of chiral materials[86, 87, 88], magnetodielectric metamaterials [50, 85, 89], superconductors [90], and graphene [91]. In the same direction, another important challenge is to reduce the strength of the attractive interaction to maximize the durability of MEMS and NEMS. Many different approaches have been proposed in literature, such as nanostructuring the surface of materials [92], modifying the dielectric function of materials by changing the crystalline phase with temperature [93], through the change of the carrier density shining silicon [94, 95] or by means of external magnetic fields [96]. 1.7 This thesis This thesis tackles the theoretical study of the Casimir-Lifshitz force in the plane-parallel geometry from the Materials Science point of view, paying specific attention to the optical properties of the interacting materials. The main interest will be to control the intensity and sign of FC−L, but also to find novel material designs that will provide insight into the underlying fundamental physics, with potential applications in nanotechnology. This thesis has a profound theoretical basis, but the systems under 18 study have been conceived bearing in mind potential experimental applications, i.e., taking in consideration available materials and optical characterization techniques. The devised self-standing thin films (in both single or multilayer configurations) need to be compact, mechanically stable, of smooth surfaces, of controlled thickness, and chemically compatible with the fluid in which they are immersed. In Chapter 2, the theoretical tools used along the thesis are presented. Specifically, the methods applied for the calculation of FC−Lin the planeparallel geometry and for the simulation of the optical response of the systems studied are described. The Lifshitz expression for the FC−Land the Transfer Matrix method will be discussed in detail. Furthermore, semianalytical calculations of the EM field and the spatial distribution of absorption inside the optical materials under consideration are explained. In Chapter 3, the effect on FC−Lof temperature variation around room temperature has been investigated in systems which present levitation due to repulsive FC−L. At the end of the chapter, the connection between the effect of temperature variations and the intrinsic optical properties of the materials forming the system is analyzed. In Chapter 4, based on the levitating systems analyzed in Chapter 3, a careful design of an optical resonator is carried out in order to propose a levitating system in which measurements of repulsive FC−Lare possible through optical spectroscopy. In addition, the effect of temperature in this levitating system is analyzed. In Chapter 5, the focus is put on how to control the strength and nature (attractive or repulsive) of the FC−Lby nanostructuring onoe of the interacting planar bodies. The geometrical parameters of the nanostructure allows to tune FC−Lin intensity and sign, and its variations are explained in terms of the optical interference occurring inside the material, which gives rise to specific EM field distribution and absorption profiles inside the nanostructure. 19 r(0,±1) TE (iξ, k⊥) = k0(iξ, k⊥)−k±1(iξ, k⊥) k0(iξ, k⊥) + k±1(iξ, k⊥)(2.11) Up to now, FC−Lexpression are valid T= 0 K. In that case, the only contribution to the fluctuations of the EM field are due to the zero-point energy. At a finite temperature T > 0 K, thermal fluctuations of the EM field also contribute to the Casimir-Lifshitz force. In what follows, it will be shown how to calculate FC−Lat any temperature T≥0 K. Note that, although the dielectric function of materials may be affected by temperature changes, for the materials and temperatures considered in this thesis, variations with Twill be negligible. Therefore, dielectric functions will be considered unaltered under temperature changes. According to Bose-Einstein statistics, the population of an EM field mode of frequency ωat finite temperature Tis: p(ω) = 1 2+1 e~ω/kbT−1=1 2coth ~ω 2kBT(2.12) where the 1/2 term is associated to zero-point fluctuations, and 1 e~ω/kbT−1 to thermal fluctuations. Taking all this into account, the expression of the Casimir-Lifshitz force at finite temperature expressed in real frequencies can be rewritten as: FC−L(d0, T) = −~ 2π2Z∞ 0 k⊥dk⊥Z∞ 0 dωcoth ~ω 2kBT Im    k0X j=TE,T M "e2k0d0 r(0,1) j(ω, k⊥)·r(0,−1) j(ω, k⊥)−1#−1   (2.13) This expression is equivalent to that given in Eq. 2.1. Again, FC−L 26 depends exponentially on kl=0, and so, rapid oscillations are included in the integrand. They can be eliminated by performing the integral in the complex plane of frequencies Ω, as it was shown in Eq. 2.6. However, in this case, the function coth ~Ω 2kBTin the integrand introduces an infinite number of poles for Ω = iξm, with ξmdefined as: ξm=2πkBT ~·m(2.14) being m= 0, 1, 2,... ∞. These discrete ξmfrequencies are commonly named Matsubara frequencies. The fact of having those poles in the imaginary axis (showed as points in Fig. 2.2(b)) leads to the calculation of residues to compute the integral along the straight segment L2. Because of that, the integral in frequencies ωin Eq. 2.13 is replaced by a summatory over Matsubara frequencies iξm. Explicitly, the modification in the expression is: ~ 2πZ∞ 0 dξ ←→ kBT ∞ X m=0 0(2.15) where the prime (’) indicates that the term with m=0 is multiplied by 1/2. This substitution gives rise to the expression of FC−Lat T > 0 K: FC−L(d0, T) = −kBT π ∞ X m=0 0Z∞ 0 km 0k⊥dk⊥· ·X j=TE,T M "e2km 0d0 r(0,1) j(iξm, k⊥)·r(0,−1) j(iξm, k⊥)−1#−1 (2.16) 27 with km l=kl(iξm, k⊥) = k2 ⊥+εl(iξm)ξ2 m c21/2 (2.17) In these expressions, the simple Fresnel coefficients r(0,1) j(iξm, k⊥) and r(0,−1) j(iξm, k⊥)evaluated at the specific frequency iξm, given in Eqs. 2.17, 2.10, and 2.11 also apply here. 2.1.2 Systems with arbitrary number of layers Diverse physical situations in which Casimir-Lifshitz forces play an important role involve systems comprising several materials, and sometimes, these materials can be considered to be built in layers. The expressions for FC−Lpresented in previous sections can be generalized [98, 99, 100] to systems with an arbitrary number of layers, each one labeled with l= −L, ..., L (L= 1, 2, ...∞), and with the total number of layers given by 2L+1. Layers are symmetrically piled up above (positive subscripts) and below (negative subscripts) the central material (l=0), which is the one that mediates the interaction between top layers and bottom layers (see scheme in Fig. 2.3). Each layer is described by its dielectric function εl, density ρl, and thickness dl, being dL+1 and d−L−1equal to infinite as they account for semi-infinite materials. In this configuration, the reflection coefficients on the top and bottom interfaces of layer l=0, now named multiple Fresnel coefficients R(+) jand R(−) j, respectively, are expressed as an iterative function of the corresponding simple Fresnel reflection coefficients of the interfaces above and below such material. The new generalized expression of FC−Lat a finite temperature is written as: 28 Figure 2.3: Scheme of a mutilayer structure of an arbitrary number of layers for the FC−Lcalculation. Each layer is characterized by its dielectric function εl, density ρland thickness dl, where layer with l= 0 stands for the material mediating the interaction between top layers and bottom layers. R± jare the multiple Fresnel coefficients on the top (+) and the bottom (-) interfaces of the material mediating the interaction. 29 FC−L(d0, T) = −kBT π ∞ X m=0 0Z∞ 0 km 0k⊥dk⊥·X j=TE,T M "e2km 0d0 R(+) j·R(−) j −1#−1 (2.18) with R(±) jdefined as: R(±) j(m, k⊥) = Γ(±) l=0(m, k⊥)(2.19) and the Γ(±) lfunctions expressed as: Γ(±) l(m, k⊥) = r(l,l±1) j+ Γl±1e−2km l±1dl±1 1 + r(l,l±1) jΓl±1e−2km l±1dl±1 (2.20) 2.1.3 Dielectric function of materials The dielectric function of a material, ε(ω), is a magnitude that describes the response of a material to an external EM field of frequency ω. In particular, it is the factor that relates the electric displacement field (−→ D) with the applied electric field (−→ E): −→ D=ε−→ E=0ε−→ E=0 −→ E+−→ P(2.21) being 0the vacuum permittivity, εthe relative permittivity or dielectric function, and −→ Pthe polarization density. As the materials response to an external EM field is not instantaneous and it obeys a causal process described by the Kramers-Kronig relations, the dielectric function is expressed as a complex function, ε(ω) = ε0(ω)−iε00(ω), in order to specify the magnitude and the phase difference 30 between −→ Dand −→ E. It is important to note that the dielectric function and the refractive index of a material are related through the relation ε= ˜n2with ˜n=n−ik, so that ε0=n2−k2, and ε00 =2nk. The imaginary part of ˜n,kis named extinction coefficient and it accounts for the specific absorption loss of the material. The dielectric function is an inherent material property, and it has been demonstrated its strong dependence on the sample preparation process [101, 102]. The dielectric function can be experimentally measured employing different techniques, and the technique to be applied will depend on the frequency range of interest. Additionally, it can also be modeled from limited tabulated experimental data by using a simple oscillator models [103, 104, 105]. They are based on the equation of motion of classical electrons harmonically bounded to atoms interacting with an electric field, resulting in a sum of absorption peaks with a Gaussian or Lorentzian shape. These models were originally used when dealing with dispersion forces because dielectric data at vacuum ultraviolet ranges were not able to be measured or were not precisely known. Only recently, accurate measurements have been achieved. In Lifshitz theory, which solves the microscopic Maxwell’s equations, the dielectric function enters directly in the formalism, playing a central role as it is the term which contains the information about the interacting materials. As it was pointed out in the previous section, Lifshitz theory is much simpler to compute considering the complex plane of frequencies than the real axis. Because of that, also the dielectric function evaluated at imaginary frequencies using Eq. 2.9 should be employed in order to simplify the calculation of FC−L. Figure 2.4 illustrates examples of how the dielectric function changes when real frequencies ω(panel (a)) rotate to imaginary frequencies iξ (panel (b)), for SiO2(orange line) and PS (light blue line) materials. Their 31 Figure 2.4: (a) Dielectric function of SiO2(orange) and PS (blue) at real frequencies and (b) dielectric function evaluated at Matsubara frequencies for several materials. tabulated dielectric data in real frequencies are found in references [106, 107, 108, 109]. In addition, in panel (b) dielectric functions evaluated directly at Matsubara frequencies are also shown for some other reported materials [105]. In Fig. 2.5 it is shown the extinction coefficient k(panel (a)), and the dielectric function evaluated at Matsubara frequencies (iξ) (panel (b)) of SiO2, as a function of the angular frequencies. High k values, which account for high absorption of the material, correspond to high negative slopes of ε(iξ). In the figure, such behaviour is observed in the UV and the IR frequency ranges, i.e., from 1013 to 1015 rad·s−1, and from 1016 to 1017 rad·s−1. On the contrary, transparent frequency ranges with k≈0, correspond to zero slope regions in the ε(iξ)curve. As it was mentioned in previous sections, the nature and intensity of the Casimir-Lifshitz force strongly depend on the dielectric function of the materials involved and their relation to one another. Some predictions on the attractive or repulsive nature of FC−Lmight be established when comparing the dielectric functions in Matsubara frequencies before performing full calculations. Along this thesis, one of the main goals will be to find materials combinations displaying repulsive Casimir-Lifshitz forces. To this end, the 32 Figure 2.5: (a) Imaginary part of the refractive index, kand (b) dielectric function evaluated at the Matsubara frequencies ε(iξ)for SiO2. comparative relation expressed in Eqs. 1.3 and 1.4 for the dielectric functions of the materials in the system will be pursued. 2.1.4 Semi-analytical calculations Concerning the computation of FC−L, results reported by Elbaum and Schick in 1990 [110] will be first reproduced. In their study, it is shown how the Casimir-Lifshitz force would promote the growth of water as a nanometer thick layer above an ice surface in air at T= 273.15 K. Three materials compose the multilayer scheme, and FC−Lcan be estimated through Eqs. 2.16 for 3 materials, or 2.18 for a generalized multilayer structure. In the latter case, Lwould be equal to 1. Both expressions present an integral in k⊥from 0 to ∞and a summatory over the Matsubara frequencies. Therefore, the convergence of the integrand must be ensured for the upper limits of the integrals over ξmand k⊥, for the range of separation distances considered. In Fig. 2.6(a), materials labeled as l= 1, 0, -1 will correspond to air, water and ice, respectively. The values of the dielectric functions for those materials are the same used in Elbaum and Schick’s paper and they are shown in Fig. 2.6(b). Figure 2.7(a) shows, on the one hand, an example of one integrand as 33 Figure 2.6: (a) Scheme of the three materials system consisting in a d0 thick layer of water in between semi-infinite air and ice slabs. (b) Dielectric function of the three materials evaluated at Matsubara frequencies extracted from Ref. [110] Figure 2.7: (a) Integrand of FC−Lin Eq. 2.16 as a function of k⊥for m= 0 and d0= 2 nm. (b) Contribution of each Matsubara frequency to FC−L. These contributions correspond to each term in the summatory in Eq. 2.16, evaluating m= [1-10,000]. For m= 0 the contribution is 672 N/m². Positive and negative values will produce repulsive and attractive contributions, correspondingly. 34 Figure 2.8: FC−Las a function of d0for the system considered in Fig. 2.6(a). Black solid line represents the resulting FC−Lfrom Ref. [110], whereas red circles display the numerical calculation carried out applying Eq. 2.16. a function of k⊥for a given Matsubara frequency. The frequency selected is the first mvalue defining Matsubara frequencies, i.e., m= 0, which represents the largest contribution to FC−L. Calculations are performed for an arbitrary separation distance of d0= 2 nm. As it can be observed, the integrand converges at large k⊥values, using steps of 104m−1. On the other hand, panel (b) displays, as a function of Matsubara frequencies, the discrete values obtained after the integration of the curves equivalent to the one shown in panel (a), but for all mvalues, i.e., for m= 0, 1, ... ∞. Note that the contribution of m= 0 is 672 N/m2, a value not shown in panel (b) due to the logarithmic scale. For large Matsubara frequencies (as large as that given for m= 104), the curve in panel (b) decays to zero. The final FC−Lvalue will be the summatory of all discrete points in panel (b). In this case, FC−Lis 1347 N/m2at d0= 2 nm. Original results in Ref. [110] are expressed in energy instead of force. For comparison, those results have been here transformed to force values and are shown in Fig. 2.8 for separation distances d0= [1-4] nm. Recall that both energy and force magnitudes are given per unit area. Reproduc- 35 with Pl=eiφl0 0e−iφl(2.31) being φlthe phase at the interface x=dlbetween layer land l+ 1 φl=klx ·dl(2.32) Therefore, each layer in the system introduces two interfaces, being the propagation of light through each layer expressed by DlPlD−1 l· The subsequent product of matrices for the whole structure can be rewritten as an unique matrix with elements Mij, relating amplitudes from both sides of the stratified media as it shows Eq. 2.33 and 2.34. a0 b0=M11 M12 M21 M22 aL+1 bL+1 (2.33) M11 M12 M21 M22 =D−1 0"L Y l=1 DlPlD−1 l#DL+1 (2.34) From the matrix elements Mij, one can extract information about the optical response of the whole stratified system knowing which fraction of the incident radiation is reflected Ror transmitted T. R=|r|2= M21 M11  2 (2.35) T=n0cos(θ0) n0cos(θ0)|t|2=n0cos(θ0) n0cos(θ0) 1 M11  2 (2.36) From conservation of energy, light that is not reflected nor transmitted must be absorbed by the subsequent layers. Thus, the absorption of the 42 system can be expressed as A= 1 −R−T(2.37) 2.5 Field distribution and power absorbed In order to gain physical insight into the light-matter interaction, the spatial distribution of the electric field intensity (|E(ω, r)|2) across materials, and the optical power absorbed (Pabs) by each slab is inspected for stratified structures in this thesis. First, the spatial and spectral distribution of the electric field intensity is obtained using the TMM described in the previous section. After that, the spatial and spectral optical power absorbed by the material is extracted by using the following expression: PAbs(ω) = 1 2ZV 0 ω0ε00 l(ω, r)|E(ω, r)|2dV (2.38) where ε00 l(ω)is the imaginary part of the dielectric function of the material in layer l, and Vis the total volume occupied by the whole stratified structure. Generally, the absorptance (A) of any system can be seen as the fraction of the incident power (P0) that has been absorbed. A=PAbs P0 (2.39) being possible to separate the absorptance in each layer from the total Aby computing the path integral of the power absorbed per unit volume (δA) between the corresponding limits of each layer. At normal incidence and being zthe perpendicular axis to the slab surface, the absorptance is 43 given by: A=ZδA dz (2.40) In Chapter 5 it will be shown that this information sheds light into results that, otherwise, could not be retrieved experimentally. For the moment, as an example, Fig. 2.11 illustrates the kind of analysis to be performed by means of the TMM. In it, as it is depicted in panel (a), a stratified thin film comprising four layers of 10 nm thick, two made of silicon dioxide (SiO2) (in orange) and two made of polystyrene (PS) (in blue), are considered. Results for the spatial and spectral |E|2and δA distributions are shown in panels (b) and (c), respectively, and the spectral power absorbed in regions or layers denoted as I, II, III and IV are displayed in (d). Regions I and III correspond to SiO2layers and II and IV to PS slabs. Notice that in panels (b) and (c) light is considered to propagate from the bottom (negative values of position) to the top (positive values of position) of the panel. To guide the eye, white horizontal lines mark the limits of each layer. As it will be discussed in Chapter 5, materials only absorb light at wavelengths in which ε00 l(ω)6= 0 and |E|2is sufficiently intense, resulting in the film absorptance (black line in pannel (d)) as the summatory of the power absorbed by each layer (colored lines in pannel (d)). 44 Figure 2.11: (a) Scheme of the 40 nm thick thin film consisting in two layers of SiO2(orange) and two layers of PS (blue) interleaved. (b) Spectral and spatial electric field intensity |E|2distribution. White horizontal lines delimit layer interfaces. (c) Spectral and spatial power absorbed per unit volume. Labels I, II, III, IV stand for the four regions of space occupied by material layers. (d) Spectral absorptance of each region of space (colored lines) and spectral total absorptance of the thin film (black line). 45 Chapter 3 Levitation of thin films due to Casimir-Lifshitz interactions. Analysis of temperature effects The results of this chapter are gathered and published in the following reference: V. Esteso, S. Carretero-Palacios, and H. M´ ıguez. Effect of temperature variations on equilibrium distances in levitating parallel dielectric plates interacting through Casimir forces.Journal of Applied Physics, 119(14), 144301, 2016. 46 3.1 Introduction 3.2 Levitation Phenomena in plane-parallel systems due to Casimir-Lifshitz and gravity force balance 3.3 Temperature dependence of equilibrium distance at room temperature 3.4 Simple rules to predict temperature variation effects on deq 3.5 Conclusions 47 Chapter 4 Optical resonators based on Casimir-Lifshitz forces The results of this chapter are gathered and published in the following reference: V. Esteso, S. Carretero-Palacios, and H. M´ ıguez. 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