Light-trapping by diffraction gratings in silicon solar cells

Transcription

1 Light-trapping by diffraction gratings in silicon solar cells Silicon: a gift of nature Rudolf Morf, Condensed Matter Theory, Paul Scherrer Institute Benefits from improved absorption Diffraction gratings at RCA labs Maxwell s equations: how to solve? efficient gratings for amorphous Si cells difficulties and pitfalls Outlook Maxwell Symposium 11 October 2014

2 Its abundance Agiftofnature: Silicon

3 Its band gap Its stability: Even used for power electronics - long lasting, reliable even in very high-speed processing chips Very high quality interfaces between Si and SiO 2 permit very high CMOS transistor density

4 But: weak light absorption near bandgap

5 Benefits from reduced cell thickness significantly reduced amount of high-quality silicon material for amorphous Si cells: reduction in deposition time improved collection effciency of photoexcited electron-hole pairs reduced degradation effects in amorphous silicon cells - requires thickness below.300nm The carriers have to diffuse a shorter distance to reach the respective contacts. Increased built-in field However: light absorption is reduced

6 Absorption of a-si layer with ideal planar reflector 0.9 Absorption Thickness of a-si [nm] Need for light trapping

7 diffraction gratings at RCA labs ZOD invention by Knop and Gale: a wavelength selective beam splitter

8 DID by Karl Knop, Mike Gale and RM: Blue Morpho served as objective

9 diffraction for optimized light absorption incident beam outgoing (lost) design gratings to suppress specular reflection angle of first order increases with increasing wavelength: increase of path length as silicon becomes less absorbing very small increase of surface area - structures shallow for crystalline and amorphous Si

10 Maxwell s equations Original text from reference [1]. Note that the symbol curl was not yet used in this original work. I also wish to refer the reader to the wonderful review by C. N. Yang in the November 2014 issue of Physics Today [2].

11 Apply Maxwell s equations to one-dimensional gratings like y λ Θ z c ε inc x ε 2 ε 1 h d ε out In every domain of constant dielectric permittivity, Maxwell s equation can be reduced to the wave equation for electric E(x, ~ y, z) and magnetic field H(x, ~ y, z) separately. Assuming monochromatic light with frequency! and requiring a steady state, where all fields have common time dependence like e i!t, the time derivative just gives rise to a prefactor i!, we can write for both fields the same wave equation (@ x 2 y 2 z) 2 F ~ (x, y, z)+k0 2 i F ~ (x, y, x) =0.

12 At interfaces between different materials, the fields satisfy different boundary conditions. The unit vector normal to the interface is denoted by ~n. Then the boundary conditions demand continuity of E n,,e t H, nh t, where the subscripts n, t refer to the direction normal and tangential to the corresponding interface, respectively. In other words, the normal component of the electric field E as well as the normal derivative of the tangential component of the magnetic field H are discontinuous at an interface, where the electric pemittivity has a discontinuity, + 6=. Let (x, y) = (x +,y) be the position dependent dielectric constant of a one-dimensional grating with grating period. Inside a grating layer (x, y) does not depend on the y-coordinate. For two polarization directions, it suffices to introduce one field component, either E z (x, y) (E-pol) or H z (x, y (H-pol). Within a grating layer, a separation of variables is possible since does not depend on y. Writing ~ H =(0, 0,H z ),with H z = H(x, y) =X(x) Y (y) the wave equation (inside domain in y (x) 0) takes the Y 2 yy (y)+ (x)k 2 0X(x) Y (y) =0

13 Dividing by X(x) Y (y) and separating variables, we 2 xx(x) X(x) + (x)k 2 0 yy (y) Y (y) µ 2 Left hand side is function of x, right hand side is function of y only, must therefore be equal to a constant µ 2. Thus, Y (y) =a exp(±iµy) and µ 2 is an eigenvalue of the Helmholtz equation in the x 2 xx(x)) + (x)k 2 0X(x) =µ 2 X(x). (1) in each domain x"d i of a grating layer separately, and subject to the boundary conditions and periodicity of the grating. Inside a domain D i of constant permeability (x) Di for x"d i, the electromagnetic field is infinitely many times differentiable. This is easily seen by differentiating eq. (1) iteratively. This property guarantees that expansion in terms of basis functions, which are analytic and form a complete set inside the domain D i, converges exponentially to the exact eigenfunction in terms of the number of basis functions used in the expansion. Inserting the expansion into the Helmholtz eigenvalue equation, supplemeted by

14 the appropriate boundary conditions at the interfaces as well as the requirement of periodicity with period leads to an algebraic eigenvalue problem discussed in detail in reference [5]. By contrast, if the eigenfunctions are expanded in term of trigonometric functions which are continuouslky differentiable at the position of the interfaces, a Gibbs phenomenon results which limits the rate of convergence to O(1/N 2 ) or O(1/N 3 ) depending on the polarization of the incident field (cf. reference [5]). The field is then written in terms of a superposition of eigenfunctions as F (x, y) = NX (a + n e iµ ny + a n e iµ ny )X n (x) n=1 Between successive grating layers, again boundary conditions apply, which allow to relate the mode amplitudes a +,a of each layer by a set of couple linear equations (cf. reference [5]).

15 how to solve? algorithm must be efficient: study a large parameter space for optimization stable: to establish convergence as number N of orders N!1(difficulty: evanescent waves with µ p imaginary and large) efficiency use stair-case approximation of grating structure: allows separation of variables expand E(H)-field in eigenmodes of Helmholtz equation same eigenfunctions for structures with different height efficient solution of eigenvalue problem

16 1 1 Results Absorption of a-si layer with ideal planar reflector Absorption H-pol E-pol Absorption height [nm] of 3 step grating with period =400nm Thickness of a-si [nm] absorption spectrum for 3 layer grating mimicking sinusoidal profile: absorption peaks just below =0.7µm. 1 Note the two close Absorption H-pol E-pol planar reflector wavelength [micrometer] which modes contribute to absorption in H-polarization?

17 35 30 modes a + in homogeneous a-si layer modes layer 1 modes layer 2 modes layer 3 modes layer mode amplitude mode 1 mode 2 mode 3 mode 4 mode amplitude wavelength [micrometer] mode number We observe that the enhancement of absorption in the spectral range in which the absorptivity of a-si is small is due to numerous modes which are excited by the grating structure. In the left panel, we show the amplitudes a + of the eigenfunctions with 0,1,2 and 3 nodes as a function of wavelength. In the right panel, we show the amplitudes of all modes for the different layers (1,2,3 are grating layers, 4: homogeneous a-si layer) close to the first high peak around =685nm. Discussions which are limited to the excitation of a single dominant mode cannot be used to reliably determine the light trapping effect of highly efficient light-trapping grating structures.

18 Example: blazed grating for crystalline silicon Absorption [percent] blazed grating rectangular grating planar reflector limit set by front surface reflection Cell thickness [micrometer]

19 grating for crystalline silicon: experiment Reflection [percent] Si-thickness 60 m 177 m 40 m 760 m reflector planar planar blazed grating planar wavelength [nm]

20 Summary the optical absorption of a-si and c-si solar cells can be improved significantly Improvement in H-polarization much better: may two-dimensional gratings further improve unpolarized performance? Difficulties and problem Reflector quality is essential. Roughness of metals leads to parasitic absorption, not light trapping. Outlook interesting further opportunitiies: 2-d gratings what happens if we consider curved interfaces? how can we make structures that are not much affected by metallic absorption

21 References [1] James Clerk Maxwell, "A Dynamical Theory of the Electromagnetic Field", Philosophical Transactions of the Royal Society of London 155, (1865). (This article accompanied a December 8, 1864 presentation by Maxwell to the Royal Society.) [2] C.N. Yang, The conceptual origins of Maxwell s equations and gauge theory, Physics Today, 67(11), 45 (2014) [3] C. Heine and R.H. Morf (1995) Submicron gratings for solar energy applications, Appl. Opt. 34, [4] R.H. Morf, H. Kiess and C. Heine (1997), Diffractive Optics for solar cells, pp , in Diffractive Optics for industrial and commercial applications, J. Turunen and F. Wyrowski, eds., Akademie Verlag Berlin 1997, WILEY-VCH. [5] R.H. Morf, (1995), Exponentially convergent numerically efficient solution of Maxwell s equations for lamellar gratings, J. Opt. Soc. Am. A, 12,