Dielectric Waveguides and their Use in Solar Cells Zachariah Peterson ABSTRACT In this paper, I discuss a brief history of waveguide technologies, followed by a derivation of the behavior of light in a cylindrical dielectric. The cutoff conditions and an expression relating wavelength modes and wavelength radius will be derived. Next, the effectiveness of nanowire thin films as waveguides for increasing light trapping in solar cells will be discussed. Experimental results will be presented showing decreased light reflection from Si solar cells covered with ZnO nanowire arrays.
1. Introduction Waveguides are structures that have long been known to force wave disturbances to propagate along a specified direction. The practical applications for waveguides have varied through history; as technology has advanced, so have the applications. Today, the use of waveguides has found a home in fiber optics, telecommunications, and information transfer over long distances. There are currently more than 6 transatlantic fiber optic cables in operation. [1] Advanced deposition and material growth techniques have made possible the growth of nanostructures with cross sectional length on the order of the wavelength of visible light, making waveguides useful as nanoscale photonic circuit elements. There has been much research focusing on the use of nanowire films as active circuit elements in solar cells. Metal oxide nanowire films have also been shown to enhance light adsorption in solar cells, essentially functioning as an anti-reflective (AR) coating. [9, 10] This study will highlight the history of waveguide technologies, give an overview of electromagnetic wave behavior in a cylindrical dielectric, and show the analytical solutions. We will consider individual nanowires to be described by the wave equation in cylindrical coordinates, with the electric and magnetic fields being coupled by Maxwell s Equations. Numerous processes and materials for waveguide fabrication as an antireflective coating in solar cells will be discussed. The dispersion relation for a cylindrical dielectric waveguide will be derived and the allowed modes of propagation will be determined. Experimental results showing the effectiveness of nanowire films as AR coatings in solar will be presented and discussed.
2. Historical background discovery and applications of waveguides The ability to force a wave disturbance to travel along a long channel of dielectric was discovered in the 19 th century. John Tyndall showed that light could be confined to travel along a thin stream of water in 1870. [2] Tyndall realized that total internal reflection of light due to refraction at the air-water interface was responsible for guiding the light in this case. He wrote about total internal reflection in his book, Notes on Light: "When the light passes from air into water, the refracted ray is bent towards the perpendicular... When the ray passes from water to air it is bent from the perpendicular... If the angle which the ray in water encloses with the perpendicular to the surface be greater than 48½ [degrees], the ray will not quit the water at all: it will be totally reflected at the surface [3] Tyndall performed an experiment before the Royal Society showing that light could be bent around a corner as it travelled through a stream of running water. [4] The light followed a zig-zag path similar to what is shown in the figure below: Figure 1: Path travelled by light rays in a flowing jet of water. [5]
J. J. Thomson was the first to propose a structure for guiding waves in 1893, and O. J. Lodge tested these ideas experimentally in 1894. In 1897, Lord Rayleigh performed the first mathematical analysis of electromagnetic waves propagating in a conducting cylinder. [6] Later, theoretical investigations on the dielectric waveguide were conducted by Hondros and Debye, followed by experimental investigations by Schriever. [7] Over the years, glass light pipes were developed to further demonstrate the phenomenon. Early in the 19 th century, the trapping of light along a stream of water was used as a decorative effect for water fountains. In the late 19 th and early 20 th century, there were many practical applications, and a number of light-guiding devices were invented. Dentists and doctors used light pipes to illuminate body cavities. [5] About ten years after Tyndall s experiment in front of the Royal Society, a Massachusetts engineer named William Wheeler invented a system for piping light through buildings. Glass rods with reflective linings guided light that was emitted from an arc discharge lamp in the basement of the building. Although the glass pipes probably did not reflect enough light to illuminate many rooms, his idea was the basis for the optical fiber. [4] It wasn t until the 1950 s that serious research into the transfer of information over long distances was being conducted. In 1952, physicist Narinder Singh Kapany conducted experiments that would lead to the invention of the optical fiber. Modern optical fibers, which employ a glass core surrounded by a cladding of lower refractive index, began to appear later that decade. [8] Improvements in the fabrication of high purity glass helped reduce the signal attenuation of transmitted light, and once signal attenuation in optical fibers was reduced below 20 db/km, optical fibers became a practical source of communication over large distances. [5] For some time, the use of waveguides was confined to the area of telecommunications and information transfer. The principle of trapping and transporting light along a one-dimensional structure is now being used to guide light in nanoscale devices. New technology has made possible the fabrication
of waveguides with diameter on the order of the wavelength of visible light. As a result, nanowire arrays and other 1-dimensional nanostructures can act as an anti-reflection (AR) coating, which increases transmission of light into the solar cell. [9] The AR layers can be used to increase light transmission in many optical devices, such as planar displays, prisms, and even sunglasses. As a result, solid-state metal, metal-oxide, and semiconductor materials which form nanowires have gained much attention for their AR properties. [9, 10] 3. EM Waves in a Cylindrical Dielectric All waveguides are structures which have been specifically designed to force a wave disturbance to propagate in one direction. Waveguides all have a similar structure in that one of the dimensions is much greater than both the transverse dimensions. [11] Typical waveguides are long, hollow cylindrical tubes. The materials needed for construction and the dimensions of any waveguide will depend on the frequency of the wave that propagates in the waveguide. Furthermore, when the transverse dimensions are comparable to the wavelength, only certain field distributions will satisfy Maxwell s equations and the associated boundary conditions. [7] Waveguides allow specific frequencies, or modes, to propagate. While theoretically an infinite number of modes can exist in the waveguide, only a finite number have their fields localized in the core. Higher order modes will have lower intensities, and will contribute less to the signal that is received at the other end of the waveguide. While waveguide modes are permitted to exist in larger structures, there are so many modes that describing the field using geometric optics is more useful. [7] There are two types of waveguides for light: metallic and dielectric. The difference between them is the mechanism that is responsible for wave propagation. In metallic waveguides, light is reflected from a conductor at the waveguide boundary. In dielectric waveguides, total internal reflection
is responsible for guiding the wave, and requires that the core dielectric of the waveguide be surrounded by a cladding material with lower refractive index than the core material. [7] We will consider the latter case, which will require that the field be continuous at the boundary of the core dielectric. The electromagnetic field obeys the wave equation, with the fields being coupled by Maxwell s equations. From the property of the homogeneous wave equation that functions of the form f(x ct) are solutions, we know the form of the solution for the electromagnetic wave to be exp[i(k r ωt)], with r oriented along the guide axis. In most wave propagation problems, one finds that solutions to the wave equation with finite boundary conditions require that the wave frequency ω be a real function of the wavenumber k: ω = ω(k) (1) Equation (1) is called the dispersion relation for multidimensional problems. This is because in general, the wavenumber is a vector, and ω = ω(k x, k y, k z ). For a wave phenomenon to be considered nondispersive, the phase velocity ω/k = constant and the group velocity ω/ k = constant. [11] The cylindrical dielectric waveguide we consider consists of a core with high dielectric constant ε 1 = ε 0 (1 + χ 1 ) and radius a, which is surrounded by a cladding of dielectric constant ε 2 = ε 0 (1 + χ 2 ). [11, 13] In this problem, we assume the time component of the electromagnetic field to vary with frequency ω such that E and B are proportional to exp[-iωt]. [11] Placing this condition on the electric field guarantees that the solution to Maxwell s equations will also satisfy the wave equation. Maxwell s equations in matter are:
(2) In (2), we have tentatively identified ε = ε 0 (1 + χ), allowing for polarizable media. [13] Since we would like to model the behavior of a solar cell, we must include ρ and J in our formulation. If we assume that the current is photo-induced, we require that ρ varies sinusoidally with frequency equal to the frequency of the field oscillations (ρ exp[-iωt]). Thus it follows from the continuity equation div(j) = iωρ. Also from the definition of charge density, J = σe, so that the charge density ρ = -(iσ/ω)*div(e). Since we have assumed the electromagnetic field has harmonic dependence with frequency ω, the time derivatives of E and B are easily obtained. Let t denote the partial derivative with respect to time, so that t B = -iωb, and t E = -iωe. This gives Maxwell s equations in the following form: (3) Using the equations in (3), we can derive wave equations for the electric and magnetic fields respectively. First, take the time derivative of the fourth equation, followed by the curl of the second equation. Using a vector product rule on the second equation, and recognizing that div(e) = 0 from (3), we can obtain the following relationships between the electric and magnetic fields:
(4) From Fubini s theorem, and because the derivative with respect to time is a linear operator, we know that it does not matter which order the differential operators are applied in the expressions in (4), therefore t div(b) = div( t B) (where t represents differentiation with respect to time), so we can set the two equations in (4) equal to each other, giving an equation for the electric field. Using a complimentary method on the second and fourth Maxwell equations gives an equation for the magnetic field, and thus the following wave-like equations for the electric and magnetic fields: (5) Since we already assume the solution to be proportional to exp[-iωt], we can derive the following relationship between the derivatives of the fields:. Substitution into (5) yields the wave equation for the electromagnetic field in a dielectric: (6) In (6), the speed of light in the dielectric is immediately identified: (7)
To describe the spatial behavior of light in the waveguide, a cylindrical coordinate system is chosen with the z dimension aligned along the guide axis. The orientation of the system, the lack of spatial boundary conditions on the z-coordinate, and the form of the wave equation requires the z and time dependence to take the form exp[i(kz ωt)]. [11] Because the waveguide is cylindrically symmetric, the radial and angular components of the field can be expressed in terms of the z-component of the electric and magnetic fields, E Z and H Z. [12] These equations are shown below: (8) Since the field must be continuous at the interface between the two dielectrics [11], this requires the following boundary conditions on the components tangential to the radial coordinate [7]: (9) With equations (6) through (9), we can completely describe the behavior of light entirely in terms of the z-components of the electric and magnetic fields. The following solutions to the field z-components in the waveguide core satisfy the wave equation in cylindrical coordinates: (10) where J n is the Bessel function of order n. Outside the core, the J n in (10) needs to be replaced with K n, the nth order modified Hankel function of the first kind. A n and B n are also replaced with C n and D n. φ n and ψ n are phase factors which are determined from the boundary conditions, however, they are
unnecessary for determining the allowed modes. While the Bessel function of the second kind is also a solution to (6), it has been ignored because it goes to infinity as r goes to 0. For wave propagation problems, we associate the angular frequency with the frequency of incident wave. [11] The eigenvalues of (8) and (9) are related by the dispersion relation:, (11) where c 1 and c 2 is the speed of light in the core and cladding respectively, ω is the incident photon frequency, and k is the wavenumber of the transmitted wave. Both λ s must be real numbers for a 2 propagating mode; if λ 1 is too large, k will be a positive complex number. In this case, the solution in the core will decay exponentially in z. [7, 11] Finally, the dispersion relation is determined from matching the solutions at core boundary subject to the boundary conditions in (9). The boundary conditions on E z and E θ give: (12) (13) Also, the boundary conditions on B give: (14) (15) Let us introduce two new symbols related to the Bessel and Hankel functions:
, (16) The primes in (12) through (16) denote differentiation with respect to r. By substituting (12) and (14) into (13) and (15), we can now derive the following relation which includes the quantities in (16): (17) The left hand side of equation (17) is a constant and independent of the angular coordinate and the phases. In order for the right hand side to have these same conditions, the phases must be related by ψ n - φ n = ±π/2. As a result the right hand side of (17) is equal to 1, and we arrive at the dispersion relation: (18) The allowed values of k are found by taking the limit λ 2 0 in equations (11) and (18). The result is k (ω/c 2 ), and when substituted into λ 1 in (11), we arrive at an expression relating the allowed value of λ 1 a in terms of the wavelength: (19) The limit on λ 2 in (18) will give the allowed values of the product λ 1 a. From the property of the Bessel and Hankel functions, it is more convenient to rewrite the dispersion relation in terms of:, (20) Let us define two new quantities ζ 1 and ζ 2 : (21) Using the relations in (21) we can derive the following equation, which is quadratic in ζ 1 :,
(22.a) With the parameters α, β, and γ given by the following: (22.b) The solution to (22.a) is then: (23) Now that the solution to ζ 1 has been obtained, we now ask: what happens to the parameters in (22.b) as we examine the limit λ 2 0? From the asymptotic properties of the Hankel functions, we can evaluate for different values of n:, for n = 1 (24), for n > 1 β 2 is proportional to (1/λ 2 ) 4, while γ is only proportional to (1/λ 2 ) 2. Therefore, β 2 goes to infinity much faster than γ and we may ignore the 4αγ term in (23), and the only solution is ζ 1 infinity. This implies the following condition on the Bessel function: (25)
Therefore, λ 1 a is equal to the mth zero of the Bessel function of order n. Using a table of Bessel function roots, the cutoff wavelength for mode n-m can be determined using (19). 4. Zinc Oxide Nanowires in Silicon Solar Cells Zinc oxide (ZnO) is an attractive dielectric for use in photovoltaic (PV) devices due to its ease of deposition and doping, transparency, high refractive index (n = 2), and its morphology. ZnO nanowires can be grown via electrodeposition in solution on a variety of conducting or semiconducting substrates. Because nanowires can be grown in solution, they can be easily doped to either p or n type by including additives in the growth solution. The nanowires have diameters such that they can accommodate modes corresponding to visible wavelengths, although the allowed modes tend to be closer to the blue end of the spectrum. [14] A schematic of a silicon PV device and SEM images of ZnO nanowires can be seen below: Figure 2: Left) ZnO nanowire-zno-si solar cell schematic. Right) SEM image of ZnO nanowires. Scale bar is 200 nm. A large scale image can be found in the appendix. ZnO nanowires can have diameters with a distribution from 10 nm to as large as 250 nm, with the average diameter depending on growth time and growth conditions. Nanowire lengths are between 1 and 2 microns. Deposition of a ZnO thin film by spray catalysis (prior to electrodeposition) creates an effective seed layer, resulting in a film with higher density of nanowires. Changing the growth
parameters of ZnO nanowires not only affects doping, but also the film morphology. Growth in solution at high temperatures (~80 C) results in deposition of highly aligned wires with flat tops. When grown at ~92.5 C and with the addition of 1,3-diaminopropane to the growth solution, the tips of the ZnO nanowires change from flat topped to a highly tapered morphology with tip diameter of 10 nm. [14] The effectiveness of nanowires as waveguides in solar cells is determined by measuring either reflectance or transmittance of light incident on the cell. The exact type of measurement taken depends on whether or not the substrate is reflective, however in silicon cells, total reflectance and reflectivity at different wavelengths are measured. For the nanowires to function as an antireflective layer, we should see less light reflected from solar cells covered with nanowire films. Figure 3 shows total reflectance measurements from silicon coated with a ZnO nanowire coating compared to measurements from a solar cell without a nanowire layer: Figure 3: Reflectance measurements for light reflected off a silicon substrate. A solar without ZnO is compared with a solar cell with a ZnO nanowire layer. [9]
From the above figure, we can see that silicon coated with ZnO nanowire layers reflect less light than uncoated silicon, especially at smaller wavelengths. A minimum reflectance in the red curve can be seen at ~350 nm; a mode with this wavelength corresponds to a nanowire radius of 77 nm. When incident light has smaller wavelengths, there is much less reflectance off of ZnO than from bare silicon. At wavelengths greater than ~400 nm, the reflectance is less than the reflectance from bare silicon, yet still follows the same trend as bare silicon. Both these results show that nanowire thin films enhance the transmission of light into silicon.
References 1. Hayes, Jeremiah. A History of Transatlantic Cables. IEEE Communications Magazine (September 2008). 2. Hecht, Eugene. Optics: 4 th Edition. Addison-Wesley, 2002. 3. Tyndall, John. Total Reflexion. Notes on Light. 1870. (Notes 123, 124, and 142) 4. Ballou, Glen. Handbook for Sound Engineers. Gulf Professional Publishing, 2005. 5. Hecht, Jeff. City of light: the story of fiber optics. Oxford University Press US, 2004. 6. N. W. McLachlan. Theory and Applications of Mathieu Functions, p. 8 (1947) (reprinted by Dover: New York, 1964). 7. Snitzer, Elias. Cylindrical Dielectric Waveguide Modes. American Optical Company. April, 1961. 8. Bates, Regis J. Optical Switching and Networking Handbook. New York: McGraw-Hill, 2001. p. 10. 9. J.Y. Chen, K.W. Sun. Growth of vertically aligned ZnO nanorod arrays as antireflection layer on silicon solar cells. Solar Energy Materials & Solar Cells 94 (2010), 930 934. 10. Barrelet, C. J.; Greytak, A. B.; Lieber, C. M. Nanowire Photonic Circuit Elements. Nano Letters (2004). Vol. 4, No. 10, 1981-1985. 11. Haberman, Richard. Applied Partial Differential Equations, 4 th ed. Pages 628-38. Pearson Prentice Hall: New Jersey, 2004. Chap. XIV. 12. Stratton, J. A. Electromagnetic Theory. (McGraw-Hill Book Co., Inc., New York, 1941.), Chap. V. 13. Jackson, J. D. Classical Electrodynamics 3 rd ed. (Wiley & Sons, Inc., 1999). 14. Y. Lee, D. Ruby, D. W. Peters, B. B. McKenzie, J. W. P. Hsu. ZnO Nanostructures as Efficient Antireflection Layers in Solar Cells. Nano Letters, Vol. 8, No. 5, 2008.