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- 1. NCN Summer School: July 2011 Modeling an Simulation ofPhotovoltaic Devices and Systems Prof. Jeffery L. Gray grayj@purdue.edu Electrical and Computer Engineering Purdue University West Lafayette, Indiana USA
- 2. copyright 2011This material is copyrighted by Jeffery L. Gray underthe following Creative Commons license.Conditions for using these materials is described athttp://creativecommons.org/licenses/by-nc-sa/2.5/ Lundstrom 2011 2
- 3. Outline1. Objectives of PV Modeling & Simulation2. PV Device Modeling3. Fundamental Limits4. PV System Modeling (multijunction)5. Detailed Numerical Simulation: “Under the Hood” 3
- 4. Objectives of PV Modeling &Simulation1. Understanding of measured device operation • dependence of terminal characteristics (Voc, Jsc, FF, η) on ◦ Device structure (dimensions, choice of materials, doping, etc.) ◦ Material parameters (mobility, lifetimes, etc.)2. Predictions of performance • Different operation conditions ◦ Temperature, illumination conditions, etc. Leads to improved designs 4
- 5. Compact Models• based on measured terminal characteristics, lumped element equivalent circuit models, and semi-analytical models Bulk and Surface Recombination Dominated ln J Space Charge Recombination Dominated q/2kT lnJ02 q/kT ln J01 Voltage V 5
- 6. Compact Models • useful for representing overall device operation (in SPICE, for example) • provides some physical insight into device performanceI = I SC − I o1e q (V + IRS ) kT − I o 2 e q (V + IRS ) 2 kT − (V + IRS ) Rsh 6
- 7. Analytic Models• based on relevant device physics (minority carrier diffusion equation)• provides deeper insight into device operation and design dependencies• device and material characterization methods typically based on analytic models• limited by simplifying assumptions 7
- 8. Minority Carrier DiffusionEquation: D ∂ m m−m 2 − =−G ( x) o M ∂x 2 τmBoundary Conditions: BSFLaw of the Junction ni2 qV p N (− x N ) = e kT + P ND ni2 qV nP ( xP ) = e kT . NAContacts d∆p S F,eff d∆n S = ∆p (−W N ) ∆n(WP ) = 0 or = ∆n(WP ) − BSF dx Dp dx Dn 8
- 9. It is worth noting that the effective front surface recombination velocity is not independent of the operating condition… W D cosh N WN Lp (1 − s) S F G N τ p cosh − 1 + po (e − 1) s + SF qV Ao kT p Lp Lp W sinh N Lp S F,eff = WN (1 − s ) po (e qV Ao kT − 1) + G N τ p cosh − 1 Lp 9
- 10. Special cases: • No grid (s=0): S F,eff = S F • Full metal (s=1) S F,eff → ∞ S F + s D p WN • Dark S F,eff = 1− s • Short-Circuit S F,eff = S F S F + s D p WN • V large (~Open-Circuit) S F,eff = 1− s 10
- 11. But, I digress… MCDE D ∂ ∆m ∆m 2 − =( x) −G M ∂x 2 τmBoundary Conditions: BSFLaw of the Junction ni2 qV p N (− x N ) = e kT + P ND ni2 qV nP ( xP ) = e kT . NAContacts d∆p S F,eff d∆n S = ∆p (−W N ) ∆n(WP ) = 0 or = ∆n(WP ) − BSF dx Dp dx Dn 11
- 12. We can learn a lot from solvingthe MCDE… ∂ 2 ∆m ∆m DM − =( x) −G ∂x 2 τm∆mM ( x) = ∆mM ogeneous ( x) + ∆mM hom particular ( x) = AM sinh[( x − xM ) Lm ] + BM cosh[( x − xM ) Lm ] + ∆mM particular ( x) 12
- 13. Effects of Base Lifetime onSolar Cell Figures of Merit … 13
- 14. Effects of BSF on Solar CellFigures of Merit … 14
- 15. Spectral Response 15
- 16. What makes a good solar cell?The key is the open-circuit voltage…Consider a solar cell with a perfect BSF and very thinemitter, then • All recombination occurs in the base (minority carrier lifetime is τm) • At open-circuit, minority carrier concentration in the base (width W) is constant wrt position and total recombination must equal total generation ∆m W W q ∫ R( x)dx = q ∫ G ( x)dx → q W = JL 0 0 τm 16
- 17. What makes a good solar cell?Combining the “law of the junction” at open-circuit ni2 qVOC =∆m NB e ( kT −1) J Lτ m with the ∆m = from the previous slide, yields qW 17
- 18. What makes a good solar cell? N Bτ m J L VOC = kT ln qni2W kT VOC − ln[q VOC kT + 0.72]FF = q J SC = J L VOC + kT q VOC FFJ SC η= PinFF expression from: M. A. Green, Solar Cells: Operating Principles, Technology, and System 18Applications, Prentice Hall, 1982.
- 19. What makes a good solar cell? High VOC yields high FF and JSC, hence efficiency N Bτ m J L VOC = kT ln qni2W • Optically thick (light trapping) • Mechanically thin • High doping (trade-off with lifetime and ni {bandgap narrowing}) • Wide bandgap [low ni] (trade-off with JL) • Plus, assumptions of perfect BSF and thin emitter • Slight modifications for high-injection conditions and for other dominant recombination mechanisms (Auger, radiative) 19
- 20. What makes a good solar cell? 20
- 21. What makes a good solar cell? 21
- 22. Fundamental Limits “Ultimate” Efficiency1 But a single junction solar cell does not use all the photons efficiently. JSC=JL FF=1 qVOC=EG1W. Shockley, W. and H. J. Queisser, “Detailed Balance Limit of Efficiency of p-n Junction Solar Cells,” J. 22 ofAppl. Phys., 32(3), 1961, pp. 510-519.
- 23. Carnot Limit (thermodynamic) Tsolar cell η= 1− = 94.8% TSun (~ 5800 K )• More detailed calculations put the limit at ~87% as the number of junctions approaches infinity (~300K)• Efficiency actually peaks for a finite number of junctions and approaches zero as the number of junctions approaches infinity 23
- 24. Fundamental LimitsGray, J.L.;et. al., "Peak efficiency of multijunction photovoltaic systems," Photovoltaic Specialists Conference 24(PVSC), 2010 35th IEEE , pp.002919-002923, 20-25 June 2010
- 25. System Modeling LIGHT Modeling and analysis of multijunction PV systems can benefit from a different view of the efficiency. 1η= Pin ∑ junctons VOC , j FFj J S Cj , 25
- 26. System Efficiencyηsys = ηultimate η photon ηic ∑ β i FFi ηV ,i ηC,i 1ηphoton: efficiency of photon absorption EG,i Igen,i βi = qηic: electrical interconnect efficiency ∑ 1 q EG,i Igen,iηV,i: voltage efficiency (qVOC/EG)ηC,i: collection efficiencyAchievement of a PV system efficiency of greater than 50%requires that the geometric average of these six terms(excluding β) must exceed ( 0.5 ) = 0.891 1 6Gray, J. L.; et.al. , "Efficiency of multijunction photovoltaic systems," Photovoltaic Specialists Conference, 262008. PVSC 08. 33rd IEEE , pp.1-6, 11-16 May 2008.
- 27. Detailed Numerical Simulation• based on more rigorous device physics• numerical solution circumvents need for simplifying assumptions, i.e. allows spatially variable parameters• provides predictive capability o Terminal Characteristics (I-V, SR, C-V, etc.)• provides diagnostic capability o Can examine internal parameters (energy band, recombination, etc.)• Ability to test simplifying assumption in analytic modeling 27
- 28. Historical Overview of Solar CellSimulation at Purdue (not comprehensive) SCAP1D (Lundstrom/Schwartz ~1979) x-Si solar cells (1D) SCAP2D (Gray/Schwartz ~ 1981) x-Si solar cells (2D) PUPHS (Lundstrom, et. al. mid-1980s) III-V heterostructure solar cells (1D) TFSSP (Gray/Schwartz mid-1980s) Amorphous Si solar cells (1D) ADEPT (Gray, et. al. late 1980s to present) A Device Emulation Program and Tool(box) Arbitrary heterostructure solar cells (CIS, CdTe, a-Si, Si, GaAs, AlGaAs, HgCdTe, InGaP, InGaN, …) Fortran version (1D, on nanoHUB ) C versions (1D, 2D -- 3D capable, but not extensively used) MatLab ™ toolbox (under development – 1D, 2D, 3D) 28
- 29. Simulation Inputs solar cell structure: composition, contacts, doping, dimensions material properties: dielectric constant, band gap, electron affinity, other band parameters, absorption coefficients, carrier mobilities, recombination parameters, etc. operating conditions: operating temperature, applied bias, illumination spectrum, small-signal frequency, transient parameters 29
- 30. Simulation InputsThe ADEPT input file consists of a series of diktats: *title simple example mesh nx=500 layer tm=2 nd=1.e17 eg=1.12 ks=11.9 ndx=3.42 + nv=1.83e19 nc=3.22e19 up=400. un=800. layer tm=200 na=1.e16 eg=1.12 ks=11.9 ndx=3.42 + nv=1.83e19 nc=3.22e19 up=400. un=800. genrec gen=dark i-v vstart=0 vstop=.1 dv=.1 solve itmax=100 delmax=1.e-6 30
- 31. Simulation Outputs the numerical solution provides the value of the potential, V, and the carrier concentrations, p and n at every point within the device, from which one can compute and display: • the terminal characteristics, i.e. I-V, cell efficiency, spectral response, etc. [predictive] • a microscopic view of any internal parameter – for example, recombination rate (i.e. losses) [diagnostic] 31
- 32. Sample output: terminalcharacteristics 32
- 33. Sample output:recombination rate 33
- 34. Detailed Numerical Simulation‘Under the Hood’Semiconductor Equations ∇ ⋅ ε∇V = −q ( p − n + N ) ∂ p ∂n∇ ⋅= q G − R p − Jp ∇ ⋅ J n = −q G − Rn − ∂t ∂t J p = µ p ∇ (V − V p ) − kT µ p ∇p −q J n = µn∇ (V + Vn ) + kT µn∇n −qOperating conditions, material properties, and otherphysics are in the B.C. and T, ε, N, G, Rp, Rn, µp, µn, Vp,and Vn. 34
- 35. Numerical Solution Transform differential equations into difference equations on a spatial grid – yields a large set of non- linear difference equations. Use a a generalized Newton method to solve – results in a iterative sequence of matrix equations J (v k )∆v k +1 = (v k ) −F • v = [p n V]; F(vk) is the set of difference equations • J(∆vk) is a sparse block tri-diagonal matrix of order 3n , where n is the number of mesh points (1D) • In 2D (n x m grid), J(∆vk) is a sparse block tri-diagonal matrix of order 3nm 35
- 36. Sparseness of 1D Jacobi matrix 36
- 37. Sparseness of 2D Jacobi matrix 37
- 38. Questions 38

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