We systematically investigate the resonant behavior of arrays of Ag nano-structures ranging from isolated simple rods, to U-shapes, to single split ring structures. We show that the lowest order plasmonic resonance associated with a rod red shifts as we create a U and SRR into the position normally associated with a simple LC mode. A second mode red shifts and grows in intensity as we extend the arms of the U-shape, and a third mode appears in the spectra as we close the arms and form a split ring structure. We performed simulations of the structures and examine the E-field and current density. The simulations show that the current path is different for these modes. We examine the behavior of the lowest order mode in detail, discuss the effects of skin depth, and present an improved LC model to describe this resonance.
©2008 Optical Society of America
The use of arrays of subwavelength structures for novel optical properties has grown rapidly following John Pendry’s demonstration that for certain combinations of idealized optical “circuit elements” it is possible to achieve simultaneous negative electric permittivity and negative magnetic permeability . There has been intense interest in using arrays of such structures for several applications, including super lenses , optical nanocircuits , and optical absorbing layers . Such devices have been studied intensively in the microwave regime, and a negative index of refraction (NIR) has been demonstrated using combinations of rods and double split ring resonators (SRR) , and other geometries . Achieving negative index of refraction has proven to be a much more daunting task in the visible and near IR regions of the electromagnetic spectrum, both because of difficulties in fabricating the structures of interest precisely at the required very small dimensions and the increased losses in materials in these optical ranges . A number of investigations have centered on the optical response of arrays of SRRs , single split rings [9, 10], “U-shapes” (a special case of the single SRR), cut-wires [12, 13], and “fish-net” structures [14–16], whose dimensions are tuned to achieve resonances in the visible/IR ranges; and indeed a negative index of refraction has been reported in the near IR . Progress in this field has been hampered by the lack of simple predictive models for the visible/IR optical properties of arrays of such structures, necessitating an almost entirely empirical approach. It would be useful to be able to predict how the resonances shift in frequency as the shapes and spacings of the elements are varied. A step toward this was reported by Rockstuhl et al.  who examined the variation of resonances of arrays of shapes ranging from single rods to U-shapes for normal incidence illumination. The plasmonic nature of these resonances was demonstrated in this work. Meyrath et al.  present a numerical RLC circuit calculation in which , but they do not consider the effect of the height (growth thickness) of the metals. In this paper, we extend these approaches by systematically measuring the variation in resonant frequencies for fabricated arrays of Ag shapes ranging from single rods, to U-shapes, to single split rings. We present simulations along with a simple analytical model for the LC resonance that includes the effect of skin depth and non-uniform distribution of charge.
We begin with a brief description of the experimental procedures. For this study, we used an indium tin oxide (ITO)-coated glass substrate. The ITO had a nominal thickness of 45 nm. In order to measure the transmission selectively from the regions of our fabricated structures, we used photolithography to deposit an optically opaque Ti/Au layer everywhere on the substrate except on a five by five array of 40 µm×40 µm windows spaced on 3 mm centers. We next spun on polymethyl methacrylate (PMMA), and used electron beam lithography followed by deposition and lift-off to create arrays of Ag structures, 75 nm in height, within the predefined windows. The individual structures have a horizontal length of 320 nm with approximately 115 nm line width and varying vertical lengths and gaps for the U’s and split rings. The center-to-center spacing between structures (lattice constant) is 600 nm. The transmission measurements were performed on a DA3.02 Bomem FTIR system, with a quartz halogen source. Both Si and InSb detectors were used to extend the range of frequencies over which measurements could be made, although the measurements using the Si detector are not shown here. The transmission measurements are normalized by dividing the measurements for a blank window by the measurements of a patterned window.
Figure 1(a) shows measured normal incidence, horizontal polarization transmission spectra from arrays of single split ring structures as a function of the width of the gap in the broken segment; a scanning electron microscope image of such a structure is shown at the bottom of Fig. 1, second panel from the right. The low frequency peak near 150 THz (labeled as mode 1) has typically been attributed to a LC resonant behavior of the circuit, assuming an external field can induce a circulating current in the coil of the SRR by coupling to a capacitance between the arms [10,11]. As pointed out previously by Enkrich, et al., for normal incidence and polarization perpendicular to the edges of the gap, the electric field can couple to the capacitance, exciting a mode that produces a magnetic-dipole moment normal to the SRR plane, but in the vertical polarization the coupling to this capacitance is absent . Our measured transmission spectra for vertical polarization, shown in Fig. 1(b), are consistent with this. This assignment is also qualitatively consistent with the shift of the resonance to lower frequency as the gap narrows, and its disappearance when a continuous ring is formed, i.e. as the capacitance is shorted out.
The peak labeled mode 2 in Fig. 1(a) near 260 THz has previously been attributed to a Mie resonance within the lower horizontal arm or “cross bar” of the U-structure or SRR . This peak does not shift as the gap closes, but shows a large increase in strength for the continuous ring as both legs now contribute to the resonance. For the continuous square ring the peaks for horizontal and vertical polarization converge as expected from symmetry, as seen by comparing the black curve in Fig. 1(a) with that in Fig. 1(b). The insensitivity of the resonance in Fig. 1(b) to gap width for vertical polarization suggests that it corresponds mainly to a mode localized in the vertical arms and corresponds to the resonance in the simple rod for horizontal polarization and the mode 2 in the horizontal polarization. Finally, there is a third peak near 325 THz that initially red shifts and strengthens as we begin to close the gap, but nearly vanishes when the ring is completely closed. This third mode does not appear in the range of the measured transmission spectra from arrays of U-shape structures (Fig. 1(c)).
To further examine the experimental variation of these modes, we measured the transmission spectra for U-shaped structures with vertical arms of decreasing length; transmission spectra at normal incidence and horizontal polarization are presented in Fig 1(c). The red curve is for the extreme case of a simple rod where we see a single resonance peak near 240 THz. As the vertical arms of the U-structure increase in length this peak red shifts into the position of the peak normally associated with the LC resonance. The higher frequency peak (mode 2), previously associated  with a Mie resonance of the lower horizontal arm does not appear for the simple rod; however a small peak does appear in the spectra as the vertical arms sprout; it then red shifts and grows in intensity as we increase their length. Fig. 1(d) shows measured transmission spectra for vertical polarization. The monotonic red shift with increasing arm length is qualitatively consistent with the expected shift in longitudinal mode plasmon frequency for an anisotropic particle .
3. Simulation results
To interpret the nature of the resonances shown in Fig. 1, we performed simulations of the transmission spectra, local E-fields and currents, using the commercial finite element method software package HFSS. Model structures in the simulations were drawn to match SEM images of the actual structures. The model structures have rounded corners, since proximity effects during e-beam lithography cause rounding instead of sharp edges. In the simulations, the dispersion of Ag was obtained using a Drude model , the dispersion of the SiO2 was obtained from ref. 21, and the ITO was modeled using a Drude term [22, 23] with a high frequency Lorentzian to simultaneously fit the measured transmission curve of a representative bare ITO sample and position-match the experimental position of the resonant peaks of the U and SRR structures. A comparison of the transmission obtained by the simulation and by experiment is shown in Fig. 2 for the SRR with a gap of 55 nm.
Figure 3(a) on the left shows a snapshot of the calculated E-field map for the simple rod at normal incidence and horizontal polarization. The driving frequency in the calculation corresponds to the single resonance observed in the transmission spectrum shown in red in Fig. 1(c). The magnitude of the field is rendered using a color scale, while the direction is indicated by the arrows. The snapshot shown is at a position in which the maximum field is concentrated at the ends of the rod. The rod acts as a simple dipole as the charge oscillates back and forth across the rod, consistent with a plasmonic resonance or the AC current in a dipole antenna. Figure 3 (a) on the right side shows the current density (J) in the rod 90 degrees out of phase with the E-field map on the left. It can be seen that the current is concentrated on the outside edge of the rod in these structures in which the skin depth is smaller than the wire widths.
Figure 3 (b) and (c) shows simulated E-field and J maps for the two lowest modes of a U-shaped structure, again for normal incidence and horizontal polarization. The long wavelength resonance, labeled as mode 1 in Fig. 1(a) and 1(c), is illustrated in Fig. 3(b). The E-field plot on the left shows a single maximum which oscillates from one vertical arm tip to the other, and is analogous to that for the simple rod. This behavior is consistent with similar calculations reported by Rochstuhl  and demonstrates the plasmonic character of this resonance. The field is more intense along the inner edges of the vertical arms of the U-structure, as would be the case for a parallel-plate capacitor, however it is concentrated at the ends of the U arms. The current plot interestingly shows that the current flows mostly on the inner surface of the U. Fig. 3(c) shows the calculated E-field for the mode labeled 2 in Fig. 1(a) and 1(c). In this case, the charge oscillates horizontally across the bottom of the arm, but this is not a simple dipole mode. Instead the opposite corners are in phase, reminiscent of a quadrupole mode, and consistent with simulations by Rochstuhl et al. and interpreted as a higher order plasmonic mode . For this mode, the current flows mostly through the outside of the U, as can be seen in Fig. 3 (c).
Figure 4 shows the results of simulations of the E-field for the single split ring structure, again at normal incidence and horizontal polarization. For mode 1, the most intense regions of the field again oscillate back and forth across the entire SRR structure as was seen for the dipole mode of the simple rod, but the field is more concentrated between the edges of the gap, as would be expected for an LC mode. The current flow is also through the inner part of the SRR, similar to mode 1 in the U structure. Mode 2 for the SRR shows the same general behavior as the U-shape, i.e. opposing corners oscillate in-phase with each other and the current flow is on the outside of the structure. Finally, as shown above in Fig. 1(a), for the SRR structure a third mode appears. The field simulation shows behavior similar to mode 1, but with higher field at the outer edges of the SRR than for the longer wavelength resonance.
4. Analytical models
The position of the resonance for mode 1 is dependent on the total length of the rod or loop (although it goes to zero for the closed loop). For the following discussion, we focus on mode 1 of the SRR structure, and for short hand we use G55, G75, G95, and G125, where the number after the G represents the gap distance in nm. Based on a simple LC model of a magnetic coil and a plate capacitor, an expression for a SRR structure has been derived for the associated resonance frequency . We start with the equation of the inductance of a coil as L=nµoA/h, where n is the number of turns, A is the area enclosed by the loop, and h is the height (see Fig. 8(a)). In the case of the single split ring, n=1 and A=l2, where l is the length of one side, and so L=µol2/h.
The capacitance between the gap (g) of the split ring is given by C=εoA/g. In this case the area for the capacitance is the width of the ring times the height, as seen in Fig. 8(a), and so the capacitance is given by C=εowh/g. From this, we see that LC=εoµol2w/g, and so:
where c is the speed of light in a vacuum and the factor 2 pi is the conversion from angular resonant frequency. We rewrite this equation to include a term for the dispersive ITO substrate:
The term εeff(f) for the dispersive ITO substrate has the form:
In general, we expect α to be a function of the height of the SRR, and it could also depend on frequency since the ITO is dispersive, finite in extent, and coated onto SiO2. By comparing the position of mode 1 of SRR structures in simulations with and without substrates, we estimate α to be about 1/3 in our geometry. Fig. 5 shows the resulting calculations using this simple LC model along with the experimental data. The addition of the dispersive ITO term actually improves the result, but it can be seen that the dependence on the gap does not match the experimental data. Given the non-uniformity of the current distribution, it is not surprising such a simple model fails to describe the mode 1 resonance of the U-structures and split ring resonators.
To improve on this LC model, we first seek to develop a more realistic model of capacitance without having to rely on numerical methods. We do this by modeling the capacitors formed by the ends of the U and SRR shapes as spheres. By adding successive image charges , the capacitance of two spheres can be shown to be approximated by:
where a is the radius of the spheres and d is the center-to-center separation for nonintersecting spheres such that d > 2a. To calculate the capacitance, the ends of the simulation models which are essentially cylinders with height h and radius r are replaced with spheres of equal volume so that:
We expect this two-sphere approximation to work best when the cylinders are 3D, i.e. h ~2r so that the cylinders are neither discs nor wires.
The variable d is chosen such that the edge-to-edge separation for the spheres and cylinders are the same. Using the measured resonance for the G55 structure, the effective dielectric constant is estimated by equation 3 with a frequency of 140 THz, and the capacitance is calculated to be 9.9 aF. Assuming that all the structures G55, G75, G95, and G125 have the same inductance, the LC model predicts that the resonant frequency simply scales as C-1/2. This is also shown in Fig. 5 as the blue curve. The good agreement of this curve with experiment supports the validity of the capacitance approximation. It should be noted that in principle the capacitance between neighboring SRRs should be taken into account, but we have estimated this to be a very small contribution for the current lattice constant (600 nm).
To develop a more realistic model of inductance, we performed simulations in which we varied the skin depth of the SRR material by changing its plasma frequency. The variation of the capacitance with skin depth is much weaker than that of the inductance. This can be understood in terms of the energy stored in the fields. For inductance H|| is continuous across the interface and falls off on the scale of the skin depth, δ, so that there is magnetic field energy in a layer of thickness δ. For capacitance however, Eperp is discontinuous (on the scale of the Thomas-Fermi screening length which is <0.1 nm for Ag) because D·n=∑=εo·E0=εi·Ei. Therefore, the energy associated with the electric field inside the metal is very small and essentially independent of the skin depth except near the plasma frequencies where εo=εi.
In this frequency regime the skin depth is inversely proportional to plasma frequency:
where c is the speed of light and ωP is the plasma frequency . A skin depth of zero was effectively achieved by performing the simulation with a perfectly conducting material. Figure 6 shows the resonant frequency for each mode using HFSS simulations as a function of the inverse plasma frequency normalized to the plasma frequency of silver. All three modes show up in the simulations, as seen in the plot, but there is a noticeable blue shift with increasing plasma frequency. To test the reliability of the simulations, we fabricated additional arrays of structures replacing the silver conductors with aluminum. Al has a higher plasma frequency than Ag, and the simulations predict that its resonances should blue shift. Measured transmission spectra for Al rods are shown by the blue curves in Fig. 7(a), 7(b) and 7(c) for Al rods, U-shapes, and single split rings, respectively; the corresponding spectra for Ag are shown in red In each case the Al resonances have indeed blue shifted from those for Ag. Any realistic model for inductance must be able to explain these shifts.
Simple equations for inductance generally account for the effects of skin depth by assuming that the current flows uniformly in the circuit wires at DC but only on the surface as skin depth goes to zero in the high frequency limit. This is typically a small effect. For instance, the inductance of a wire decreases from 2ℓ(ln(2ℓ/r)-3/4) to 2ℓ(ln(2ℓ/r)-1) when going from DC to the high frequency limit . However, as Fig. 4 shows and as mentioned previously, the current is not uniformly distributed, but is concentrated on the inner surface of the SRR.
To estimate the corresponding inductance of the G55 structure, we first consider a conducting ring of square cross section consisting of four connected rectangular sheets (i.e. the inner ring formed by the split ring structure forms four rectangular sheets). The inductance of such a structure can be easily calculated following the technique of Bueno and Assis [28-29] as:
where Lsheet is the inductance of a rectangular sheet and M pairs is the mutual inductance of opposite and parallel sheets using the Neumann method. Once again we have left out the interaction between neighboring SRRs, since we estimate its contribution to the total inductance to be small. In this case, we take the sheet width to be h, the height of the SRR structures, and the sheet length to be l, the inner length of the SRR cross section as seen in Fig. 8. The width of the wires comprising the SRRs does not enter the equation. For G55, l is about 125 nm. For h/ℓ≈1, equations (2.13) and (3.2) of Bueno and Assis  the inductance for the perfectly conducting SRR is:
Equation 8 predicts an inductance of about 0.14 pH for the perfectly conducting G55 structure. This is much lower than the DC value (which assumes current distributed evenly through the SRR) of 0.36 pH estimated from equation (3.20) of Bueno and Assis , i.e.:
where the length is now the outer length of the SRR ring, and the square cross-section width is replaced with the average of the height and width for our ring with a rectangular cross-section. Although the value obtained from equation 8 takes into account the current flow on the inner surface and is more accurate than equation 9, this value is still about 2 times too large to successfully account for the resonance frequency of ~170 THz in the experimental results and simulations. We attribute the discrepancy to the assumption that all of the current flows on the inner surface of the SRR. Although a majority of the current flows on the inner ring, some current will flow on the outer and side surfaces effectively in parallel with the inner surface current and will reduce the overall inductance. If we correct for this effect by increasing the effective height of the SRR structure as illustrated in Fig. 8 (b), we find that we need to add 125 nm which corresponds roughly to the width of the SRR wires. We arrive at values of 0.070 pH for the inductance and 12.6 aF for the capacitance.
To approximate the correction for the effect of field penetration we add twice the skin depth to the effective length of the ring and also reduce the effective height by twice the skin depth. This is illustrated in Fig. 8 (c). The resonant frequencies calculated in this way for the G55 structure, using equation 4 for the capacitance and taking the inductance as discussed, are shown in Fig. 6 as the green curve as a function of skin depth (plasma frequency), and agree reasonably with the simulations and the experimental results. We expect that as the skin depth approaches the dimensions of the wire and the currents become more uniform, the agreement will begin to worsen significantly.
To investigate the effects of sample height on the LC resonance and to test our model, we performed four series of simulations in which we vary the growth thickness (height) of the G55 structure from 25 nm to 300 nm and compare it to the resonances to that of the 75 nm structure. In the first series, we isolate the effects of the sample height on the inductance and capacitance from substrate and skin depth effects by using a perfectly conducting SRR with no substrate. In these simulations, the shift in the LC peak is less than 0.3% from 25 nm to 150 nm and less than 2% over the full range of height as shown in Fig. 9 (a). In the context of the LC model, this indicates that as the capacitance increases with sample height, the inductance decreases approximately by a proportional amount. To investigate the effects of substrate, we ran a second series of simulations in which we added the ITO and SiO2 substrates while keeping the perfect conductor. These simulations show a significant blue shift with increasing sample height as shown in Fig. 9 (a). This indicates that the effective dielectric constant of the capacitor is decreasing as the sample height increases since a higher percentage of the electric field energy resides in the vacuum between the capacitor electrodes rather than in the substrate. Applying equation 3, we find that α varies from about 0.39 to 0.12 as the height is varied from 25 nm to 300 nm. To investigate the effects of skin depth, we ran a third series of simulations in which we reverted back to our original dispersion for Ag but with no substrate. Compared to the shift for the perfect conductor, there is a significant blue shift with increasing sample height indicating that the increase in capacitance is “outpaced” by the decrease in inductance due to the role of skin depth. As the SRR height increases to several times the skin depth, that role becomes significantly reduced. For instance, the blue shift for the 300 nm structure from that of the 150 nm thick one is 2.7%. For the perfect conductor, this shift corresponds to 1.8%. A fourth series of simulations were run using the substrates and our original dispersion for Ag and the results are shown in Fig. 9 (a). As can be seen, the percentage shifts for sample heights less than 75 nm are less than those for the Ag structures with no substrate. This is due to the dispersive dielectric constant we used for ITO which actually dips below one at the frequencies in question and has the effect of suppressing the shift at small sample heights. By comparing the simulations with and without a substrate, we extract an effective dielectric constant for each height, calculate the capacitance using equations 4 and 5, calculate the inductance using equation 8 with the skin depth correction illustrated in Fig. 8 (c), and calculate the resonance for each height using . The agreement of these calculations to the results of the simulations is reasonable for sample heights appreciably larger than the skin depth as shown in Fig. 9 (b). In the regime of h~< δ, our model for inductance based on surface currents breaks down, and as expected we underestimate the inductance. Additionally, we expect our two-sphere capacitance approximation to work best for cylinder electrodes such that h~2r (where we set r=68 nm) which is not the case for the smallest sample heights.
In summary, we have examined the resonant behavior of arrays of Ag nano-structures ranging from isolated simple rods, to U-shapes, to single split ring structures. The lowest order plasmonic resonance associated with a rod red shifts as we create U and SRR shapes into the position normally associated with a simple LC mode. A second mode red shifts and grows in intensity as we extend the arms of the U-shape, and a third mode appears in the spectra as we close the arms and form a split ring structure. We presented simulations that showed the field distribution for these modes, and show that the current path differs significantly for these modes.
Most significantly, we present an improved LC model to describe the lowest order mode resonance. Further LC modeling of the higher order modes may be possible by taking into account the current path flow, which lowers the inductance and hence raises the resonance frequency. The strength of such a model is that quantitative estimations of the resonant frequencies are easy to perform; we expect it to be useful in predicting the resonant behavior for optical nanocircuits and other possible applications.
This work was supported by the Laboratory for Physical Sciences. The authors want to thank David Schurig, Martin Wegener, and Andre Assis for helpful conversations.
References and links
1. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999). [CrossRef]
4. H. Tao, N. I. Landy, C. M. Bingham, X. Zhang, R. D. Averitt, and W. J. Padilla, “A metamaterial absorber for the terahertz regime: Design, fabrication and characterization,” Opt. Express 10, 7181–7188 (2008). [CrossRef]
5. D. R. Smith, Willie J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Science 84, 4184–4187 (2000).
6. M. Kafesaki, Th. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou, and C. M. Soukoulis, “Left-handed metamaterials: detailed numerical studies of the transmission properties,” J. Opt. A:Pure Appl. Opt. 7, S12–S22 (2005). [CrossRef]
7. M. I. Stockman, “Does nature allow negative refraction with low losses in optical region?” J. Cond. Mat. 14, 0611350 (2006).
8. T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz Magnetic Response from Artificial Materials,” Science 303, 1494–1496 (2004). [CrossRef] [PubMed]
9. N. Katsarakis, G. Konstantininidis, A. Kostopoulos, R. S. Penciu, T. F. Gundogdu, M. Kafesaki, E. N. Economou, Th. Koschny, and C. M. Soukoulis, “Magnetic response of split-ring resonators in the farinfrared frequency regime,” Opt. Lett. 30, 1348–1350 (2005). [CrossRef] [PubMed]
11. C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, Th. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005). [CrossRef] [PubMed]
12. V. Shalaev, W. Cai, U. K. Chettiar, H-K Yuan, A. K. Sarychev, V. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356–3358 (2005). [CrossRef]
13. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Cut-wire pairs and plate pairs as magnetic atoms for optical metamatrerials,” Opt. Lett. 3253–55 (2007). [CrossRef]
15. C. M. Soukoulis, J. Zhou, T. Koschny, M. Kafesaki, and E. N. Economou, “The science of negative index materials,” J. Phys.: Condens. Matter 20, 304217 (2008). [CrossRef]
16. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature Lett. (2008), http://www.nature.com/nature/journal/vaop/ncurrent/abs/nature07247.html. [CrossRef]
17. C. Rockstuhl, F. Lederer, C. Etrich, T. Zentgraf, J. Kuhl, and H. Giessen, “On the reintretation of resonances in split-ring-resonators at normal incidence,” Opt. Express 14, 8827–8836 (2006). [CrossRef] [PubMed]
18. T. P. Meyrath, T. Zentgraf, and H. Giessen, “Lorentz model for metamaterials: Optical frequency resonance circuits,” Phys. Rev. B 75, 205102 (2007).
19. U. Kreibig and M. Vollmer, Optical properties of Metal clusters (Springer-Verlag, 1995).
20. M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, R. W. Alexander Jr., and C. A. Ward, “Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared,” Appl. Opt. 22, 1099–1119 (1983). [CrossRef] [PubMed]
21. E. D. Palik, Handbook of optical constants of solids (Academic, 1985).
22. K. Füchsel, U. Schulz, N. Kaiser, and A. Tünnermann, “Low temperature deposition of indium tin oxide films by plasma ion-assisted evaporation,” Appl. Opt. 47, C297–C302 (2008).
23. J. Ederth, G. A. Niklasson, A. Hultaker, P. Heszler, C. G. Granqvist, A. R. van Doom, M. J. Jongerius, and D. Burgard, “Characterization of porous indium tin oxide films using effective medium theory,” J. Appl. Phys. 93, 984–988 (2003). [CrossRef]
24. S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic Response of Metamaterials at 100 Terahertz,” Science 306, 1351–1353 (2004). Online supplement: http://www.sciencemag.org/cgi/content/full/sci;306/5700/1351/DC1. [CrossRef] [PubMed]
25. C. Wasshuber and W. R. Smythe,http://www.iue.tuwien.ac.at/phd/wasshuber/node77.html, Static and Dynamic Electricity2nd ed. (McGraw-Hill, 1950), in § 5.08 p. 118 the capacitance between two spheres is studied.
26. J. D. Jackson, Classical Electrodynamics2nd ed. (John Wiley and Sons, 1975). Equation 6 is taken from the general expression δ=c/(2πωσ)1/2 and ε-1=ω2p/ω2=4πiσ/ω.
27. F. W. Grover, Inductance Calculations (Dover, 1973), p. 261.
28. M. A. Bueno and A. K. T. Assis, “A new method for inductance calculations,” J. Phys. D Appl. Phys. 281802–1806 (1995). [CrossRef]
29. M. A. Bueno and A. K. T. Assis, Inductance and force calculations in electrical circuits (Nova Science, 2001).
30. M. A. Bueno and A. K. T. Assis, Inductance and force calculations in electrical circuits (Nova Science, 2001), p. 24, 30. See http://www.ifi.unicamp.br/~assis/wbooks.htm for correction to eq. 3.2.
31. M. A Bueno and A. K. T. Assis, Inductance and force calculations in electrical circuits (Nova Science, 2001), p. 40.