Optimizing thin film solar voltaic components

Transcription

1 Optimizing thin film solar voltaic components Peter Monk University of Delaware, USA Research support in part by an NSF-SOLAR grant

2 Outline This talk will be about designing multiplasmonic solar cells and solar concentrators, and beam splitters: Introduction to the project and plasmonics Design Problems Validation for the solar cell Numerical analysis Optimizing the beam splitter The time domain

3 Surface Plasmons Suppose we have an interface 1 between two materials at z = with electromagnetic parameters µ and ɛ i, i = 1, 2 respectively. z= E 2, H 2 E 1, H1 z> z< 1 E i H i = ɛ i c t E i = 1 H i c t Assuming p-polarization, we seek solutions localized at the boundary z =, E i = (E i x,, E i z)e κ i z e i(qx ωt) and H i = (, H i y, )e κ i z e i(qx ωt) To satisfy Maxwell: κ i = q 2 ɛ i ω 2 /c 2. 1 This introduction is taken from J. Pitarke et al, Rep. Prog. Phys., 7 (27) 1-87

4 Plasmon condition Continuity at z = of tangential components requires, ɛ 1 κ 1 + ɛ 2 κ 2 = Suppose ɛ 2 = 1 and the metal is described by a Drude model: ɛ 2 = 1 ω 2 p ω(ω + iη) The plot shows ω versus R(q) when η =. Red line: ω s = ω p / 2, Green line: ω = cq. Note R(ɛ 2 ) < for ω small enough. Angular frequency ω Wave vector q 9

5 SPPs in photovoltaics 2 Motivation: To absorb light well, Silicon solar cells need to be thick. But efficiency (and cost!) motivate thin film cells. Light is poorly absorbed in a thin-film solar cell Solar spectrum absorbed in 2 m thick Si If incident light couples into SPP modes, these modes might trap and guide light allowing a thinner or more efficient cell. 2 From: H. Atwater and A. Polman, Nature Materials, 9 (21)

6 Nature M SPPs in photovoltaics Light trapping in a thin-film solar A problem: Incident light will not couple to SPP waves in a planar structure. A metal grating can be used to generate surface waves.

7 SOLAR project Our project 3 started with the work of M. Faryad and A. Lakhtakia 4 showing that if ɛ 1 is periodic in z multiple SPP modes exist at each frequency. Penn State Delaware T. Mallouk (PI) Chemistry Monk Math T. Mayer Electrical Engineering M. Solano Math A. Lakhtakia Eng. Science and Mech, L. Fan Math G. Barber Chemistry M. Faryad Eng. Science and Mech. L. Liu Eng. Science and Mech. S. Hall Chemistry 3 NSF-DMR See for example, M. Faryad and A. Lakhtakia, J. Opt. Soc. Am. B. 27 (21)

8 Typical Structures Wavelength of light of interest: 4-12nm Period of the grating: 4nm Height of structure: 2nm Incident light -1 z= -2 θ d T ε T Reflected light +1 d 3p d 3i d 3n d 2p ε 3p ε 3i ε 3n ε 2p L d d 2i ε 2i d 2n d 1p d 1i ε 2n ε 1p ε 1i d 1n ε 1n L g L m L 1 d a ε r1 Metal ε a L Transmitted light Thin film solar cell 5. Concentrator. 6 Beam splitter 7 5 M. Solano et al, Applied Optics, 52 (213) M. Solano et al., Appl. Phys. Lett. 13, , L. Fan et al., SPIE San Diego, 214

9 Basic Multi-SPP cell z= L d L g L m Incident light -2-2 L 1 θ d T d 3p d 3i d 3n d 2p d 2i d 2n d 1p d 1i d 1n d a -1-1 ε T ε 3p ε 3i ε 3n ε 2p ε 2i ε 2n ε 1p ε 1i ε 1n ε r1 Metal Reflected light ε a Transmitted light L Incident field: Let k = ω/c u i = exp(ik(x sin θ + z cos θ)). Equations: For s polarization: A = 1, n = ɛ r, u = E 2 For p-polarization: A = 1/ɛ r, n = 1, u = H 2. A u + k 2 nu = in (, L) R e ikl sin θ u(x + L, z) = u(x, z) for all x, z u i + u s = u in (, L) R. and u s satisfies upward and downward propagating wave conditions above and below the cell.

10 Above and below the grating For z < : u s = [ ( un s exp i n Z k (n) x )] x k (n) z z, where k (n) x = k sin θ + 2πn/L, k (n) k 2 (k (n) x ) 2, k 2 > (k (n) z = i x ) 2, (k (n) x ) 2 k 2, k 2 < (k (n) x ) 2 Similarly for the transmitted wave in z > and z large enough. u t = [ ( )] un t exp i k (n) x x + k (n) z z, n Z

11 Computed Quantities For s-polarized incidence, we compute the modal reflectances [ ] = un s 2 Re /(k cos θ) R (n) s and modal transmittances [ = un t 2 Re T (n) s k (n) z k (n) z ] /(k cos θ). The absorptance for s-polarized incident waves is given by A s = 1 N t n= N t [ R (n) s ] + T (n) s. Similar definitions hold for p-polarized waves.

12 Variational Formulation Let Ω = (, L) (, H) where H is the height of the cell H 1 qp(ω) = {u H 1 (Ω) u(l, ) = exp(ikl sin θ)u(, )} Seek u H 1 qp(ω) such that Ω A u v k 2 nuv T H uv + Γ H (T (u u i ) + u i )v = Γ for all v H 1 qp(ω) where T H is the Dirichlet to Neumann map on the upper surface Γ H and similarly T on Γ. These are computed using the Fourier representation of the solution above and below the grating.

13 Numerical Method (I): Finite Element Methods Let S h H 1 qp(ω) Seek u h,n S h such that A u h,n v k 2 nu h,n v da Ω T H P N u h,n v ds + (T P N (u h,n u i ) + u i )v ds = Γ H Γ for all v S h where P N is L 2 projection onto the space spanned by Fourier basis functions of degree n, N n N. Note: this requires the mesh on the left and right edges to be identical.

14 Theoretical papers Uniqueness problems: Uniqueness except at a countable set of exceptional frequencies 8 Grating theory and numerical analysis. 9 Goal oriented adaptivity A.S. Bonnet-Bendhia and F. Starling, Mathematical Methods in the Applied Sciences, 17 (1994), A. Kirsch: Diffraction by periodic structures, Inverse Problems in Mathematical Physics (1993), L. Paivarinta and E. Somersalo, Eds., pp , A. Kirsch and P. Monk, IMA J. Numer. Anal., 1(199) , G.C. Hsiao, N. Nigam and J. Pasciak, J. Comp. Appl. Math. 235(211) , J. Elschner and G.Schmidt, Math. Meth. Appl. Sci., 21(1998), (1998) 1 N.H.Lord and A.J. Mulholland, Proc. Roy. Soc. 469(213),

15 Numerical Method (II): Rigorous Coupled Wave Expansion (RCWA) RCWA 11 is a commonly used scheme for this type of problem. The relative permittivity in the region z H is expanded as a Fourier series with respect to x: ɛ(x, z) = n Z ɛ (n) (z) exp(i2πnx/l), z [, H], where the z-dependent coefficients ɛ (n) (z) are known. The electric field is similarly expanded u(x, z) = n Zu (n) (z) exp (i2πnx/l), z [, H]. 11 L. Li, J. Opt. Soc. Am. A 12, 2581 (1993).

16 Discretization in z The Fourier series are substituted into the Maxwell system and the results truncated to order N The grating is divided into horizontal layers, and ɛ (n) (z) is approximated by its mid-layer value. The resulting coupled system of ordinary differential equations in z is solved exactly and the map between incident, transmitted and reflected fields can be computed. The layers are combined by marching using a splitting into upward and downward going waves to help stabilize the procedure.

17 Basic Multi-SPP cell Incident light -1 z= -2 θ d T ε T Reflected light +1 d 3p d 3i ε 3p ε 3i d 3n ε 3n d 2p ε 2p L d d 2i ε 2i d 2n ε 2n d 1p ε 1p d 1i ε 1i L g L m L 1 d 1n d a ε 1n ε r1 Metal ε a L Transmitted light

18 Frequency dependence of coefficients 24 1n, 2n and 3n layers 22 1n, 2n and 3n layers p, 2p and 3p layers 1i layer 2i layer 3i layer p, 2p and 3p layers 1i layer 2i layer 3i layer Re (ε d ) 16 Im (ε d ) λ (nm) λ (nm) (Left) Real and (right) imaginary parts of the relative permittivities of all semiconductor layers as functions of the free-space wavelength.

19 Relative permittivity of aluminum Re (ε m ), Im (ε m ) Re(ε m ) Im(ε m ) λ (nm) Real and imaginary parts of the relative permittivity of aluminum as functions of the free-space wavelength.

20 Grating profiles I Experimental setup 12 f maximizing the conicity. It is used in metal stalline silicon ( 5 μm.5 μm thick), 1 which h thicker ( 1 μm) terface of a 1D metallic electric material occurs with its magnetic field lines) but not with vector parallel to the wave mode is p-polar-. The optical energy is ickness in the dielectric e SPP-wave mode can ce wavelength λ o over f incidence. 1,11 Hence, (A =1 R T, where transmittance, 12 A. Shoji respec- Hall et al., Figure ACS NANO 1. 1D 7(213) metallic grating coupled to a 1D photonic ARTICLE

21 Grating profiles II x1 x2 L Semiconductor Lg Metal Lm L1 x1 x2 L x1 x3 x4 x2 L Semiconductor Semiconductor L2 Lg Metal Lg Metal Lm Lm L1 Unit cell of a surface-relief grating with a (top) rectangular corrugation, (bottom left) sinusoidal corrugation, or (bottom right) trapezoidal corrugation. L1

22 RCWA validation 1.9 p polarization, 43 deg Nt = 12 Nt = 2 Nt = 3 Nt = s polarization, 43 deg Nt = 12 Nt = Absorptance.5.4 Absorptance λ (nm) λ (nm) Trapezoidal grating. Duty cycle.86. Upper dimension = 251 nm, θ = 43

23 FEM Validation 1.9 p polarization, 43 deg FEM coarse mesh FEM fine mesh Absorptance λ (nm) Trapezoidal grating. Duty cycle.86. Upper base = 251 nm, θ = 43. FEM convergence (left)

24 FEM/RCWA Validation 1.9 p polarization, 43 deg RCWA Nt = 4 FEM Shoji experiment 1.9 s polarization, 43 deg RCWA Nt = 12 FEM Shoji experiment Absorptance.5.4 Absorptance λ (nm) λ (nm) Trapezoidal grating. Duty cycle.86. Upper base = 251 nm, θ = 43, experimental results by Dr. Shoji Hall.

25 θ θ Comparison to experiment: s-polarization 55 S pol, experiment Shoji 1 55 S pol, RCWA Manuel λ nm λ nm Left panel: experiments by S. Hall. Right Panel: RCWA by M. Solano. Two periods of photonic crystal..1

26 θ θ Comparison to experiment: p-polarization 55 P pol, experiment Shoji 1 55 P pol, RCWA Manuel λ nm λ nm Left panel: experiments by S. Hall. Right Panel: RCWA by M. Solano.

27 FE Convergence - planar back reflector Errors and rates of convergence for the FEM when the metallic backreflector is planar and θ =. l N e e Ap r Ap e As r As

28 FE/RCWA convergence for rectangular grating Θ ẽap ẽas Θ Errors (top) ẽ As and (bottom) ẽ Ap versus Θ calculated with the RCWA (+) and FEM ( ) when θ =. Θ = N for RCWA, Θ = N dof for finite elements (uniform mesh).

29 Source of the FE error The solution looses regularity at the metal-dielectric corners. Expect O(h.6 ) in H 1 norm (a) (b) (c) FE solution: refine near the offending point. 13 J. Elschner and G. Schmidt, Math. Meth. Appl. Sci. 21, (1998).

30 Refinement improves reliability Θ Errors ẽ Ep (+) and ẽ Ap ( ) versus Θ calculated with the refined meshes when θ =

31 Splitter geometry n a (air) d g n g d r n r q L n 2 L n t (glass) Optimize over L, d r, d g, q L, n 2, n g (each in a limited range).

32 Optimization of the Splitter We want to send waves at frequencies below a cutoff frequency c in the specular direction and waves above that frequency in non-specular directions with no loss due to reflection. We set λmax F =.1 λ min [.9 [ (T ss(λ) H(λ c)) 2 + (T n ss (λ) (1 H(λ c))) 2] (T pp(λ) H(λ c)) 2 + (T n pp (λ) (1 H(λ c))) 2] dλ where H(x) is the Heaviside step function and c = 65nm.

33 Differential Evolution Algorithm We use a genetic algorithm called the Differential Evolution Algorithm 14. Suppose we want to maximize f (v). DEA requires three parameters C (, 1) the crossover probability, α (, 2) the differential weight and M the number of vector initial guesses (we use C =.7, α =.8 M = 1#parameters). The steps to find v opt S are given on the next slide mutation recombination selection 14 Storn, R. and Price, K. (1997), Differential Evolution - A Simple and Efficient Heuristic for Global Optimization over Continuous Spaces, Journal of Global Optimization, 11, pp

34 DEA 1: Randomly initialize a set of M vectors: {v (m) } M m=1 S 2: while stopping criterion is not satisfied do 3: for each v (ν), ν {1, M}, do 4: Randomly choose three different v (m1), v (m2), v (m 3) 5: Choose a random index j {1,..., n} 6: for all l {1,..., n} do 7: Pick r l U(, 1) to be a uniform random number in (,1). 8: if r l < C or l = j then 9: w l v (m 1) l + α(v (m 2) l v (m 3) l ) 1: else 11: w l v (ν) l 12: end if 13: end for 14: if f (w) > f (v (ν) ) then 15: v (ν) w 16: end if 17: end for 18: v opt is the vector for which f (v opt ) f (v (ν) ) ν {1,..., M} 19: end while

35 Transmittance / Reflectance T 2 T n 2 R 2 R n (nm) Before the optimization: L = 5, q L = 25, d t = 3, d r = 1, n t = 1.9 and the incident angle is 6.

36 Differential Evolution optimization starts with: L [3 nm, 6 nm] d t [5 nm, 3 nm] d r [5 nm, 3 nm] n t [1.1, 1.9] q L /L [.1,.5]

37 Optimized Results Optimized Results, L = 6, q L = 24, d t = 3, d r = 255, n t = 1.9 Transmittance / Reflectance T 2 T n 2 R 2 R n 2 Transmittance / Reflectance T 2 T n 2 R 2 R n (nm) Before (nm) After

38 Objective function value of objective function number of iteration

39 The Time Domain Let S = (, L) R. Taking the inverse Fourier transform of the s-polarized time harmonic problem (setting n real and ignoring frequency dependence!!) we get that the scattered field u s = u s (x, z, t) satisfies n c 2 us tt = u s + n 1 c 2 utt i in S R u s (, ) = in S u s t (, ) = in S u s (L, z, t) = u s (, y, t d 1 L/c) in R R. where d = (sin θ, cos θ). We assume that u i (x, z, t) = f (t x d/c) where f is such that f (ct x d) = for t < and (x, y) [, L] [, H] (e.g. a windowed and translated power of the sine function).

40 Change of variables Common to use the change of variables w(x, z, t) = u s (x, z, t + (x L)d 1 /c) Then recalling S = ((, L) R) we see that n d1 2 c 2 w tt = w 2 d 1 c w xt + n 1 c 2 wtt i in S R w(, ) = in S w t (, ) = in S w(l, z, t) = w(, z, t) in R R. Here w i (x, z, t) = f (t d 2 z/c).

41 Analysis in the time domain Typically implicit methods are used to discretize in time. We can provide an analysis for a small class of methods. To do this we take the Laplace transform ŵ(x, z, s) = w(x, y, t) exp( st) dt, s = σ iω, where σ R is fixed and σ >, while ω R.

42 Laplace domain problem Find ŵ H 1 p(s) such that s 2 n d 2 1 c 2 ŵ = ŵ 2s d 1 c ŵx + s 2 n 1 c 2 ŵ i in (S) R. Here ŵ i (x, y) = ˆf (s) exp( sd 1 y/c).

43 The weak Laplace domain problem For ŵ, ξ H 1 p(s) define a(ŵ, ξ) = S ( F(ξ) = s 2 Then ŵ H 1 p(s) satisfies S ŵ ξ + s 2 n d 2 1 c 2 ŵξ + 2s d 1 c ŵxξ n 1 c 2 ŵ i ξ da a(ŵ, ξ) = F(ξ), for all ξ H 1 p(s). ) da

44 Coercivity and continuity For ŵ, ξ Hp(S) 1 define ( ) a(ŵ, sŵ) = s ŵ 2 + s s 2 n d 1 2 c 2 ŵ s 2 d 1 c ŵxŵ da S Then ( ) Ra(ŵ, sŵ) = σ s ŵ 2 + s 2 n d 1 2 S c 2 ŵ 2 da so provided n d 2 1 > α > for some constant α Ra(ŵ, sŵ) σ min(1, α) ŵ 2 where ŵ 2 1 = S ) ( ŵ 2 + s 2 c 2 ŵ 2 da

45 Laplace domain result Obviously F(sŵ) C s 2 ŵ i L 2 ŵ 1 Lemma Suppose α >. For each s = σ iω, σ >, there exists a unique solution ŵ H 1 p(s) of the Laplace domain problem and ŵ 1 C s 2 σ ŵ i L 2 Taking the inverse Laplace transform establishes existence of the time domain solution in suitable space-time function spaces (note the exponential weight exp( 2σt) in the time direction). A good choice might be σ = 1/T.

46 Time domain result Using Lubich s convolution quadrature theory 15 if Backward Euler (p=1) or BDF2 (p=2) are used to discretize the problem (leaving space continuous) at time steps t n = n t, and if a sufficiently smooth (in time) incident field is used, then (w(, t n ) wn t ) + w(, t n ) wn t = O( t p ), < t < T, where t > is the time step size and wn t Hp(S) 1 is the discrete in time solution at t n. We have yet to analyze the spatial discretization. 15 C. Lubich, Numer. Math. 67, (1994).

47 Current work Optimization of the concentrator with matching layers, and other splitter designs. 3D grating structure Complete analysis of the time domain problem with frequency dependent materials properties.