1. www.unamur.be Temporal incoherence of solar radiation: First-principle theory & application to solar cell optical simulations Olivier Deparis, Michaël Sarrazin, Aline Herman Project review meeting, 23-24 April 2014

2. www.unamur.be Outline Problem statement Existing computational methods A method derived from first principles A simplified case study The full treatment Application to solar cell optical simulations Apologies to those who do not like (too much) mathematics after lunch

3. www.unamur.be Sunlight has both spatial and temporal incoherence Spatial incoherence coherence length estimated to 60 m not critical in thin films with lateral/vertical dimensions of the order of 1-10 m Temporal incoherence coherence time estimated to 3 fs implicitly taken into account in solar cell efficiency measurements (solar simulators) usually not taken into account in solar-cell optical simulations Impact of temporal incoherence on efficiency still unclear Rarity of theoretical investigations due to large computational demands with existing methods

4. www.unamur.be Existing methods perform statistics on coherent calculations Multiple runs of coherent calculations + statistical averaging Each optical carrier frequency treated independently (assumption!) Incident carrier phase selected randomly at each run Statistical averaging of many independent runs (large computational demand) Each coherent run relies on solutions of Maxwell’s equations in laterally periodic stratified media using standard methods (RCWA, FDTD) Hundreds of runs required to simulate temporal (phase) incoherence W. Lee, S.-Y. Lee, J. Kim, S. C. Kim, B. Lee Optics Express 20 (2012) A941A953 Practical limitations (CPU time) for complex solar cell structures Time required to compute one run increases dramatically with solar cell complexity Because of both of solar cell structure complexity and statistical treatment, accurate modeling of state-of-the-art solar cells under incoherent light is a formidable task!

5. www.unamur.be A question of methodology Quantity of interest (accessible to measurements): photocurrent (100% internal conversion efficiency and perfect carrier collection assumed) Illumination: solar power spectral density spectrum S( ) Intermediate quantity : absorption spectrum A( ) in active layer Coherent illumination: A() represents the “coherent absorption” Incoherent illumination: A() represents the “incoherent absorption” Would it be possible to compute A incoh without multiple runs and therefore to deduce directly the photocurrent under incoherent illumination?

6. www.unamur.be A direct method The method requires only one single run of coherent calculation! Two independent steps: 1.calculation of the coherent absorption A coh at each carrier wavelength  c 2.incoherent absorption spectrum A incoh ( c ) deduced directly from A coh ( c ) convolution product in frequency domain with the power spectral density (PSD) of the random process spectrum PSD is assumed to be Gaussian, the same for each carrier frequency and depends solely on the sunlight coherence time  c Step #1 is time-consuming but performed once for all (no multiple runs) Step #2 is straightforward and fast  : convolution product M. Sarrazin, A. Herman, O. Deparis, Optics Express 20 (2012) A941A953  cc  1/  c

7. www.unamur.be Preliminary remarks Principle of the method can be catch by establishing the response of a linear system with 1 input/1 output channels in the frame of random signal theory Generalization to a linear system in scattering configuration (1 input/2 outputs) is tedious (only the great lines will be highlighted hereafter) Each individual frequency component of the solar spectrum can be regarded as a quasi-monochromatic signal whose spectral width is defined by random process All random processes related to each carrier frequency are independent (each carrier frequency can be treated individually) This is the basic assumption made in multiple run statistical methods cc

8. www.unamur.be Incoherent response of a linear system with 1 input/1 output G(ω): transfer function g(t): impulse response G(ω): transfer function g(t): impulse response Linear system Input signal (excitation) Output signal (response) Calculation of the response in time or frequency domain

9. www.unamur.be Linking the incoherent response of a linear system to the coherent response: theoretical framework Random signal theory basic concepts Real stationary random signal: x(t) (electric field of the electromagnetic radiation) Autocorrelation function of the random signal E[]: expectation value (ensemble average) Mean square value (average power transported by the optical carrier wave) Power spectral density (PSD) = Fourier transform of autocorrelation (Wiener-Khinchine) In order to define the PSD, the signal must be truncated within a span of time T, i.e. the sampling interval: x T (t) is one realization of the random signal Stochastic quantity corresponding to the Fourier transform of the truncated signal For T large enough, it can be shown that Normalized average power (deduced by integrating PSD)

10. www.unamur.be Remark about the sampling interval The sampling interval T is used to define a realization of the random signal In the context of solar cells, T is effectively the photo-detector response time which is very long at the time scale of the random process  assuming T is large is fully satisfied t T

11. www.unamur.be Trivial case of coherent excitation Linear response Coherent input signal Coherent output power G(ω): transfer function g(t): impulse response G(ω): transfer function g(t): impulse response Linear system Input signal (excitation) Output signal (response)  c : carrier frequency

12. www.unamur.be Non-trivial case of incoherent excitation Incoherent input signal: carrier with randomly modulated amplitude Incoherent input power (random process PSD) (normalization condition*) *Both coherent and incoherent input signals must have the same power

13. www.unamur.be Linking incoherent and coherent output powers: the convolution formula Incoherent output power Linking this to coherent output power with (normalized random process PSD)

14. www.unamur.be Generalization to the scattering problem We have considered the transfer function of a linear system with 1 input/1output This formalism obviously does not allow us to calculate reflectance (R), transmittance (T), hence absorption (A =1−R−T) To do this, we must consider the scattering matrix of the linear system Though the derivation is complicated, it also ends up with a convolution formula! |F in > |F sca,R > |F sca,T > S(  ) S: scattering matrix |F  : field super-vector

15. www.unamur.be Scattering matrix formalism (e.g. RCWA method) Applicable to laterally periodic, arbitrarily stratified medium Field expanded in spatial Fourier series according to lateral periodicity of dielectric constant Fourier components of the field expansion gathered in a super-vector Maxwell’s eqs recast in matrix form relating incident and scattered super-vectors Quantity of interest in “photonic” solar cells: photocurrent Intermediate quantities: Poynting vector fluxes, reflectance, transmittance, absorption z0z0 z1z1 zLzL z j-1 zjzj k=1 k=L k=j z x y   I: incidence medium (z<z 0 ) II: laterally periodic stratified medium (z 0 <z<z L ) III: emergence medium (z>z L ) Unit cell pattern periodically repeated in x,y directions

16. www.unamur.be Coherent response (deterministic process) General response (coherent or incoherent) of the linear system in the frequency domain: Incident flux Scattered fluxes (X=R: reflected, X=T: transmitted) Coherent incident field Coherent scattered fields C X : connection matrix between stratified medium and incidence (emergence) medium S: scattering matrix of stratified medium  : angular frequency  : unit cell area  : incidence angle where Carrier

17. www.unamur.be Coherent reflectance/transmittance Power fluxes Reflectance (X=R) & transmittance (X=T) Remark: the time-averaged incident flux is constant in the coherent case

18. www.unamur.be Incoherent response (random process) General response (coherent or incoherent) of the linear system in the time domain: Incoherent incident field (=randomly amplitude modulated carrier) Incoherent scattered fields Power spectral density: Fourier transform: Random process ! Carrier

19. www.unamur.be Temporal averaging The device response under incoherent excitation (finite coherence time  c ) is the time averaged value of its response recorded during a sampling time T c >>  c In a solar cell device: excitation=sunlight, response=photo-generated current T c is fixed by recombination/generation time of charged carriers (0.1 ns to 1 ms in Silicon) since  c 3fs, T c >> c is fully satisfied Temporal averaging of scattered fluxes Since J X has spectral width  1/  c (around  =0) and since T c >>  c, we can take the limit T c  

20. www.unamur.be Incoherent time-averaged scattered fluxes Instantaneous scattered fluxes Time-averaged scattered fluxes Fourier transform: From Fourier transform of U X (t): developing explicitly the convolution product and then setting  =0: and since m(t) and S X (t) are real functions: we find finally PSD of random process

21. www.unamur.be Incoherent time-averaged incident flux Instantaneous incident flux Time-averaged incident flux Coherent incident flux Integration of the PSD of the random process leads to time- averaged incoherent flux Fourier transform:

22. www.unamur.be Putting all together … Ratio of time-averaged incident/scattered leads to X=R or T Link with coherent R or T At each carrier frequency, the incoherent R/T is given by the convolution product (in frequency domain) between the coherent R/T and the normalized PSD of the random process

23. www.unamur.be Photocurrent Incoherent absorption spectrum Photocurrent directly deduced from coherent absorption spectrum and PSD of random process (convolution product in frequency domain)

24. www.unamur.be Case study: flat/1D grating structures FDTD+statistical analysis RCWA+direct method Sarrazin M, Herman A and Deparis O 2013 Optics Express 21 A616 Lee W, Lee S.-Y, Kim J, Kim S. C and Lee B 2012 Optics Express 20 A941A953

25. www.unamur.be Solar cell thin corrugated active layer Sarrazin M, Herman A and Deparis O 2013 Optics Express 21 A616

26. www.unamur.be Further reading