Solar Cell Loss Mechanisms and Characterization

1. Theory

• Definition: The efficiency of a real cell is the ideal one minus the losses of three kinds: optical (light that creates no carrier), recombination (carriers that are not collected) and resistive (voltage and power lost on the way out); the characterization measures each of them through the I-V curve, the quantum efficiency and the lifetime.
• Optical losses: PVEducation lists the reflection at the front surface (over 30% for bare silicon), the shading by the top contact and the incomplete absorption; the remedies are an anti-reflection coating, a random-pyramid texture and light trapping, where good trapping may give an optical path of 50 thicknesses.
• Recombination: Carriers are lost by Shockley-Read-Hall (defect), radiative and Auger recombination in the bulk and by recombination at the surfaces and in the space-charge region; thermal oxide, silicon nitride, a back surface field and heavy doping under the contacts are the means to passivate the surfaces.
• Collection: A carrier generated within a diffusion length of the junction is collected with a probability near one, so that blue light is sensitive to the front surface, green light to the bulk and the long wavelengths to the rear surface and the thickness.
• Resistive losses: The emitter and the top grid dominate the series resistance; the loss of the grid is the sum of the fingers, the emitter, the contact, the busbars and the shading, and the finger spacing is chosen to balance shading and resistance.
• Quantum efficiency: The external quantum efficiency counts the collected carriers per incident photon, the internal one excludes the reflected and transmitted light, and the spectral response is the current per incident power.
• Characterization: The light I-V curve is taken under standard conditions (1000 $W/{\mathrm{m}}^2$, AM1.5, 25 °C) with a classified simulator; the quantum efficiency is measured with a chopped monochromatic beam and a lock-in amplifier; the lifetime is derived from the photoconductance or from the decay of the voltage.
• Benchmark: A record-class silicon cell in the paper of Kostylyov et al. has 41.45 $mA/c{\mathrm{m}}^2$, 751.4 mV, a fill factor of 86.07% and an efficiency of 26.81%, with a series resistance of 0.125 $\Omega \mathrm{c}{\mathrm{m}}^2$.

2. Working Principle

• Optical channel: Reflection and shading lower the photon flux that enters, which scales the whole quantum-efficiency curve down and lowers $J_{sc}$ in proportion; the coating uses destructive interference with a quarter-wave layer of index $\sqrt{n_0{n}_2}$.
• Recombination channel: A short lifetime or a fast surface lowers the collection probability near the surface or far from the junction, which shows as a drop of the quantum efficiency in the blue or in the red, and raises the saturation current, which lowers $V_{oc}$.
• Resistive channel: The current crosses the emitter to the finger and the finger to the busbar, and the loss $I^{2}R$ grows with the square of the current; it lowers the fill factor and leaves $V_{oc}$ and, until it is very large, $J_{sc}$ unchanged.
• Reading the curves: The area above the quantum-efficiency curve is the current that is lost, the slope of the I-V curve at the two ends gives the resistances, and the implied voltage from the lifetime is an upper limit of $V_{oc}$.

3. Applications

• Cell design: The choice of the coating, the texture, the passivation and the grid is made with the formulas of this page; the trade-off of the grid spacing is a standard exercise (the numbers on this page are assumed examples).
• Loss analysis: The splitting of the difference between the ideal and the measured cell into optical, recombination and resistive parts, as in the analysis of record-class silicon cells.
• Process control: Lifetime and implied voltage of wafers and precursors, light I-V and quantum efficiency of finished cells (background, the pages read describe the methods).
• Rating and certification: The classified simulator, a calibrated reference cell and a spectral-mismatch correction give the rated power; the temperature coefficient converts it to operating conditions.

Optical Losses and Light Trapping

Name	Formula	Related Formulae	Variables/Constants
Reflection at the front surface	$R_T=\frac{R_{\perp }+R_{\parallel }}{2},R_{\perp }=\frac{{\sin }^2\left({\theta }_1-{\theta }_2\right)}{{\sin }^2\left({\theta }_1+{\theta }_2\right)},R_{\parallel }=\frac{{\tan }^2\left({\theta }_1-{\theta }_2\right)}{{\tan }^2\left({\theta }_1+{\theta }_2\right)}$	$n_1\sin {\theta }_1=n_2\sin {\theta }_2$ $R_0={\left(\frac{n_0-n_2}{n_0+n_2}\right)}^2$ ${\theta }_c=\arcsin \frac{n_2}{n_1}$	$R_T:reflectanceforunpolarisedlight$ $R_{\perp },R_{\parallel }:reflectancesforthetwopolarisations$ ${\theta }_1,{\theta }_2:anglesofincidenceandrefraction$ $n_1,n_2:refractiveindicesofthetwomedia$ $\begin{array}{r}{R}_0:reflectanceatnormalincidencefroma\\ medium{n}_{0}ontosilicon{n}_2(the\\ normal-incidenceformisthelimitofthe\\ Fresnelequations,notprintedinthetext\\ thatIread)\end{array}$ $\begin{array}{r}{\theta }_c:criticalanglefortotalinternal\\ reflection(n_2<n_1)\end{array}$ $\begin{array}{r}(thepagetakesaconstant{n}_2=3.5for\\ silicon(asimplification):R_0=30.9\%in\\ airand16.0\%underacoverof{n}_0=1.5,\\ and{\theta }_c=16.6^{\circ }forsilicontoair,\\ computed;PVEducation:over30\%forbare\\ silicon)\end{array}$
Single-layer anti-reflection coating	$R=\frac{r_1^2+r_2^2+2r_1{r}_2\cos 2\theta }{1+r_1^2{r}_2^2+2r_1{r}_2\cos 2\theta }$	$r_1=\frac{n_0-n_1}{n_0+n_1},r_2=\frac{n_1-n_2}{n_1+n_2},\theta =\frac{2\pi n_1{t}_1}{\lambda }$ $n_1=\sqrt{n_0{n}_2},d_1=\frac{{\lambda }_0}{4n_1}$	$R:reflectanceatnormalincidence$ $\begin{array}{r}{r}_1,r_2:amplitudereflectioncoefficients\\ oftheouterandtheinnerinterface\end{array}$ $\begin{array}{r}{n}_0,n_1,n_2:indicesofthesurroundings,\\ ofthecoatingandofsilicon\end{array}$ $t_1:thicknessofthecoating$ $\theta :phaseofthecoating$ ${\lambda }_0:designwavelengthinvacuum$ $\begin{array}{r}{d}_1:quarter-wavethickness,Rvanishesat\\ thedesignwavelengthfortheoptimum\\ index\end{array}$ $\begin{array}{r}({\lambda }_0=600nmand{n}_2=3.5:n_0=1gives{n}_1\\ =1.871and{d}_1=80.2nm;n_0=1.5gives\\ n_1=2.2913and{d}_1=65.5nm;\\ photon-weightedoverAM1.5Gupto1107nm\\ theresidualreflectionis7.1\%and3.3\%\\ against30.9\%and16.0\%bare,i.e.J_{sc}=\\ 40.7and42.4against30.3and36.8mA/c{m}^2\\ fora43.8mA/c{m}^{2}cell(constant-index\\ model,computed))\end{array}$
Light trapping and path length	$EQ{E}_l\left(\lambda \right)=\frac{f}{1+b{\left(4n_r^2\alpha d\right)}^{-1}}$	$Z=\frac{L_{opt}}{W}\le 4n_r^2\text{(Lambertian limit)}$ $Z_{eff}=\frac{4n_r^2}{b}$	$\begin{array}{r}EQ{E}_l:externalquantumefficiencyatlong\\ wavelengths(above800nminthesource)\end{array}$ $\begin{array}{r}f:factorthatmatchesittothe\\ short-wavelengthpartat800nm\end{array}$ $\begin{array}{r}b:light-trappingparameter(1foran\\ ideallydiffusivesurface,1.6to1.8\\ fittedfortworecord-classcells)\end{array}$ $n_r:refractiveindexofsilicon$ $\alpha :absorptioncoefficient$ $d:thickness$ $\begin{array}{r}Z:enhancementoftheopticalpathlength\\ L_{opt}overthethicknessW(thelimit4n_r^2\\ isbackgroundknowledge,notwrittenon\\ thePVEducationpage)\end{array}$ $\begin{array}{r}{Z}_{eff}:effectiveenhancementforweak\\ absorption(myreadingofthemodel)\end{array}$ $\begin{array}{r}(n_r=3.5gives4n_r^2=49;PVEducation\\ saysthatgoodlighttrappingmayreacha\\ pathlengthof50thicknesses,andnone\\ aboutone;b=1.6and1.8give{Z}_{eff}=30.6\\ and27.2,computed)\end{array}$

Recombination and Carrier Collection

Name	Formula	Related Formulae	Variables/Constants
Recombination rates	$U_{SRH}=\frac{np-n_i^2}{{\tau }_p\left(n+n_1\right)+{\tau }_n\left(p+p_1\right)}$	$U_{rad}=B\left(np-n_i^2\right)$ $U_{Auger}=\left(C_{n}n+C_{p}p\right)\left(np-n_i^2\right)$	$U:netrecombinationrateperunitvolume$ $n,p:electronandholeconcentrations$ $n_i:intrinsicconcentration$ $\begin{array}{r}{\tau }_n,{\tau }_p:lifetimesofelectronsandholes\\ forthetraps\end{array}$ $\begin{array}{r}{n}_1,p_1:concentrationsfortheFermilevel\\ atthetraplevel\end{array}$ $B:radiativecoefficient$ $\begin{array}{r}{C}_n,C_p:Augercoefficients(insilicon\\ theydependondopingandinjection,see\\ theparametrizationofRichteretal.);\\ thethreeformsarethestandardtextbook\\ onesandnotcopiedfromapagethatI\\ read\end{array}$ $\begin{array}{r}(atequilibriumnp=n_i^{2}andU=0;the\\ SRHrateislargestfortrapsnearmid-gap\\ (n_1,p_1\approx n_i),andtheAugerrategrows\\ withthethirdpowerofthecarrier\\ concentration)\end{array}$
Lifetimes in low and high injection	${\tau }_{SRH}=\frac{1}{\sigma v_{th}{N}_t}\text{(low injection, mid-gap trap)}$	${\tau }_{rad}=\frac{1}{B\left(n_0+p_0+\Delta n\right)}$ ${\tau }_{Auger}\approx \frac{1}{C_p{p}_0^2}\text{(p-type, low)},{\tau }_{Auger}\approx \frac{1}{C_a\Delta n^2}\text{(high)}$	${\tau }_{SRH}:Shockley-Read-Halllifetime$ $\sigma :capturecross-section$ $v_{th}:thermalvelocity$ $N_t:trapconcentration$ $n_0,p_0:equilibriumconcentrations$ $\Delta n:excesscarrierconcentration$ $\begin{array}{r}{C}_a:ambipolarAugercoefficient,the\\ limitingformsarestandardandarenot\\ takenfromapagethatIread\end{array}$ $\begin{array}{r}(assumed\sigma =10^{-15}c{m}^2,v_{th}=10^{7}cm/sand\\ N_t=10^{11}c{m}^{-3}give{\tau }_{SRH}=1ms;withB=\\ 4.7\times 10^{-15}c{m}^3/s(recalledvalueforsilicon\\ at300K,notfromasourceread)andN=\\ 10^{16}c{m}^{-3}{\tau }_{rad}=21ms;tentimesmoredoping\\ lowers{\tau }_{rad}by10and{\tau }_{Auger}by100)\end{array}$
Effective lifetime of a cell	$\frac{1}{{\tau }_{eff}}=\frac{1}{{\tau }_{intr}}+\frac{1}{{\tau }_{extr}}$	$\frac{1}{{\tau }_{intr}}=\frac{1}{{\tau }_r}+\frac{1}{{\tau }_A},\frac{1}{{\tau }_{extr}}=\frac{1}{{\tau }_{SRH}}+\frac{1}{{\tau }_S}+\frac{1}{{\tau }_{exc}}+\frac{1}{{\tau }_{SCR}}$ ${\tau }_S=\frac{d}{S},{\tau }_{SCR}=\frac{d}{S_{SCR}},{\tau }_{exc}={\tau }_{SRH}\frac{n_x}{n_0+\Delta n},{\tau }_{SRH,eff}=\frac{{\tau }_{SRH}}{1+n_0/n_x}$ ${\tau }_r^{-1}=B\left(1-P_{PR}\right)\left(n_0+\Delta n\right)$	${\tau }_{eff}:effectivelifetime$ ${\tau }_{intr}:intrinsic(radiativeandAuger)$ $\begin{array}{r}{\tau }_{extr}:extrinsic(SRH,surface,\\ exciton-mediatedAugerandspace-charge\\ region)\end{array}$ $d:thickness$ $S:surfacerecombinationvelocity$ $\begin{array}{r}{S}_{SCR}:equivalentvelocityofthe\\ space-chargeregion\end{array}$ $\begin{array}{r}{n}_x:concentrationoftheexcitonterm\\ (8.2\times 10^{15}c{m}^{-3}inthesource)\end{array}$ $\begin{array}{r}{P}_{PR}:probabilityofphotonrecycling,the\\ formsarethoseofKostylyovetal.(the\\ factor2fortwofacesdependsonhowSis\\ defined)\end{array}$ $\begin{array}{r}(therecord-classcellofthesource:\\ measured{\tau }_{eff}=5.15msagainsta\\ theoreticalmaximumof18.7ms;S=0.017\\ cm/sandd=130\mu \mathrm{m}give{\tau }_S=0.76s,and\\ anassumedS=10cm/sgives1.3ms;n_0=\\ 3.23\times 10^{15}c{m}^{-3}gives{\tau }_{SRH,eff}=0.717{\tau }_{SRH},\\ computed)\end{array}$
Diffusion length and collection probability	$CP\left(x\right)=\cosh \frac{x}{L}-\frac{\frac{SL}{D}\cosh \frac{W}{L}+\sinh \frac{W}{L}}{\frac{SL}{D}\sinh \frac{W}{L}+\cosh \frac{W}{L}}\sinh \frac{x}{L}$	$L=\sqrt{D\tau },CP\approx \exp \left(-\frac{x}{L}\right)$ $J_L=q{\int }_0^{W}G\left(x\right)CP\left(x\right)dx$	$\begin{array}{r}CP:collectionprobability(unityinthe\\ depletionregion)\end{array}$ $x:distancefromthejunction$ $L:diffusionlength$ $S:surfacerecombinationvelocity$ $D:diffusivity$ $W:widthofthelayer$ $\tau :lifetime$ $G:generationrate$ $\begin{array}{r}{J}_L:light-generatedcurrentdensity,the\\ exponentialformisforL<<WorSL/D=1,\\ theexpressionL=\sqrt{D\tau }isstandard\end{array}$ $\begin{array}{r}(D=27c{m}^2/s(thebasevalueofthe\\ PVEducationfigure)and\tau =1,5.15and\\ 18.7msgiveL=1.64,3.73and7.11mm,\\ farabovea130\mu \mathrm{m}wafer,sothe\\ exponentialformdoesnotapplytosucha\\ cell;forL<<W,CP\left(L\right)=0.368,computed)\end{array}$

Resistive and Shading Losses of the Grid

Name	Formula	Related Formulae	Variables/Constants
Emitter resistance loss	$p_{emitter}=\frac{P_{loss}}{P_{gen}}=\frac{{\rho }_{\square }{S}^2{J}_{mp}}{12V_{mp}}$	$P_{loss}=\frac{J^{2}b{\rho }_{\square }{S}^3}{24}$ $S_{max}=\sqrt{\frac{12p{V}_{mp}}{{\rho }_{\square }{J}_{mp}}},R_{s,emitter}=\frac{{\rho }_{\square }{S}^2}{12}$	$\begin{array}{r}{p}_{emitter}:fractionalpowerlossinthe\\ emitterbetweenthefingers\end{array}$ ${\rho }_{\square }:sheetresistanceoftheemitter$ $S:fingerspacing$ $\begin{array}{r}{J}_{mp},V_{mp}:currentdensityandvoltageatthe\\ maximumpowerpoint\end{array}$ $b:lengthofthefingerconsidered$ $p:allowedloss$ $\begin{array}{r}{R}_{s,emitter}:seriesresistancecontribution\\ (productofpand{V}_{mp}/J_{mp},mystep)\end{array}$ $\begin{array}{r}(PVEducation:{\rho }_{\square }=40\Omega /sq,J_{mp}=30\\ mA/c{m}^2,V_{mp}=450mVandalossbelow4\%\\ needS<4mm;theformulagives4.24mm,\\ and3.6\%at4mm;assumed100\Omega /sqandS\\ =2mmgive2.3\%and0.333\Omega c{m}^{2}forthe\\ examplecell,computed)\end{array}$
Finger resistance loss	$p_{finger}=\frac{L^2{J}_{mp}{S}_f{\rho }_f}{3V_{mp}{w}_f{d}_f}$	$P_{finger}=\frac{1}{3}{L}^3{J}_{mp}^2{S}_f^2\frac{{\rho }_f}{w_f{d}_f}$ $R_{s,finger}=\frac{L^2{S}_f{\rho }_f}{3w_f{d}_f}$	$p_{finger}:fractionalpowerlossinthefingers$ $\begin{array}{r}L:fingerlength(halfthebusbarspacing\\ ifthecurrentflowstothenearest\\ busbar)\end{array}$ $S_f:fingerspacing$ ${\rho }_f:effectiveresistivityofthemetal$ $\begin{array}{r}{w}_f,d_f:widthandheightofafinger,the\\ pageassumesuniformgeneration,a\\ constantwidthandperpendicularentryof\\ thecurrent\end{array}$ $\begin{array}{r}(assumedL=2.6cm,S_f=2mm,w_f=50\\ \mu \mathrm{m},d_f=15\mu \mathrm{m}and{\rho }_f=3\times 10^{-6}\Omega cmgive\\ 1.25\%withtheexamplecell{J}_{mp}=38.2\\ mA/c{m}^{2}and{V}_{mp}=0.5508V,computed)\end{array}$
Contact resistance loss	$R_{s,contact}=\frac{S_f{W}_T{R}_{sheet}\coth \left(\frac{W_f}{2W_T}\right)}{2}$	$W_T=\sqrt{\frac{{\rho }_c}{R_{sheet}}}$ $R_c\simeq \frac{{\rho }_c}{L_f{W}_T}\left(W_T<\frac{W_f}{2}\right),R_c\simeq \frac{2{\rho }_c}{L_f{W}_f}\left(W_T>2W_f\right)$	$\begin{array}{r}{R}_{s,contact}:seriesresistanceperunitcell\\ areafromthemetal-semiconductorcontact\end{array}$ $W_T:transferwidth$ ${\rho }_c:specificcontactresistivity$ $\begin{array}{r}{R}_{sheet}:sheetresistanceofthelayerunder\\ thecontact\end{array}$ $W_f,L_f:widthandlengthofthefinger$ $S_f:fingerspacing$ $\begin{array}{r}{R}_c:resistanceofonesideofafinger\\ contact(currententeringfromoneedgeinto\\ halfthewidth,hencethelimits{\rho }_c/(L_f{W}_T)\\ and2{\rho }_c/(L_f{W}_f));thetwosidesinparallel\\ givethefactor1/2ofthefirstformula\\ (auditcheck),thesourcegivesnotypical\\ values\end{array}$ $\begin{array}{r}(assumed{\rho }_c=10^{-3}\Omega c{m}^{2}and{R}_{sheet}=100\\ \Omega /sqgive{W}_T=31.6\mu \mathrm{m}(W_f=50\mu \mathrm{m},S_f=2\\ mm),\coth =1.52and{R}_{s,contact}=0.048\Omega c{m}^2,\\ i.e.0.33\%,computed)\end{array}$
Total grid loss and optimum finger spacing	$P_{loss,total}=P_{fingers}+P_{emitter}+P_{contact}+P_{busbars}+P_{shading}$	$p_{shading}\approx \frac{w_f}{S_f}$ $\frac{d}{d{S}_f}\left(p_{shading}+p_{emitter}+p_{finger}+p_{contact}\right)=0$	$\begin{array}{r}{P}_{loss,total}:totallossofthetopcontact\\ (alltermsbuttheshadingareresistive)\end{array}$ $\begin{array}{r}{p}_{shading}:fractionoftheareacoveredbythe\\ fingers(myformforthefingersalone;\\ thebusbarsadd)\end{array}$ $w_f:fingerwidth$ $\begin{array}{r}{S}_f:fingerspacing,theshadingfallswith\\ thespacingandtheresistivetermsrise,\\ sothereisanoptimum\end{array}$ $\begin{array}{r}(withtheassumedvaluesofthethreerows\\ aboveandnobusbars:S_f=2mmgives\\ 2.5\%shading,2.3\%emitter,1.25\%\\ fingersand0.33\%contact,atotalof\\ 6.4\%;theoptimumis{S}_f=1.43mmwith\\ 5.8\%,computed)\end{array}$

Quantum Efficiency and Characterization

Name	Formula	Related Formulae	Variables/Constants
External and internal quantum efficiency and spectral response	$EQE\left(\lambda \right)=\frac{I_{ph}/q}{\Phi \left(\lambda \right)},SR\left(\lambda \right)=\frac{q\lambda }{hc}EQE\left(\lambda \right)$	$SR\left[A/W\right]=\frac{EQE\lambda \left[\mathrm{nm}\right]}{1239.8}$ $IQE=\frac{EQE}{1-R}$	$\begin{array}{r}EQE:externalquantumefficiency\\ (collectedcarriersperincidentphoton)\end{array}$ $I_{ph}:photocurrent$ $\Phi :photonfluxatthewavelength\lambda$ $\begin{array}{r}SR:spectralresponse(currentper\\ incidentpower)\end{array}$ $IQE:internalquantumefficiency$ $\begin{array}{r}R:reflectance,IQE=EQE/(1-R)holdswhen\\ nothingistransmitted(theslidesgiveit\\ withagarbledlayout,thepagecorrects\\ bymeasuringreflectionandtransmission)\end{array}$ $h:Planckconstant$ $c:speedoflight$ $\begin{array}{r}(anidealcell(EQE=1)hasSR=0.484\\ A/Wat600nm,0.807A/Wat1000nmand\\ 0.887A/Wat1100nm,computed;bluelight\\ isaffectedbyfrontrecombination,green\\ lightbythediffusionlengthandtherear\\ surfaceinthickcells,asthepagesays)\end{array}$
Short-circuit current from the quantum efficiency	$J_{sc}=q\int EQE\left(\lambda \right){\Phi }_{AM1.5}\left(\lambda \right)d\lambda$	$\Delta J_{sc,loss}=q\int \left[1-EQE\left(\lambda \right)\right]{\Phi }_{AM1.5}\left(\lambda \right)d\lambda$ $J_{sc,corrected}=\frac{J_{sc,measured}}{M}$	$\begin{array}{r}{\Phi }_{AM1.5}:photonfluxdensityofthereference\\ spectrum,\Delta J_{sc,loss}:thepartofthecurrent\\ thattheareaabovetheEQEcurve\\ represents\end{array}$ $\begin{array}{r}M:spectralmismatchfactorofthe\\ simulator(thelecturesaysthatthe\\ EQE-based{J}_{sc}correctsthemismatch;the\\ expressionofMisnotinthesourcesthat\\ Iread),thesourcegivesnoexplicit\\ equationfor{J}_{sc}\end{array}$ $\begin{array}{r}(each1\%ofEQEoverthewholebandupto\\ 1107nmis0.438mA/c{m}^{2}ofthe43.8;the\\ videoexamplecellgivesabout34mA/c{m}^2,\\ andtherecord-classcellofthesource\\ 41.45mA/c{m}^2,i.e.94.6\%ofthelimit,\\ computed)\end{array}$
Light I-V measurement	$\eta =\frac{V_{oc}{J}_{sc}FF}{G_{STC}},G_{STC}=1000W/{\mathrm{m}}^2$	$R_s\approx -{\frac{dV}{dJ}|}_{V_{oc}},R_{sh}\approx -{\frac{dV}{dJ}|}_{J_{sc}}$ $T_{cell}=25{}^{\circ }\mathrm{C},AM1.5G$	$\begin{array}{r}{G}_{STC}:irradianceatstandardtest\\ conditions(IEC61215:1kW/m^2,AM1.5,25\\ C),theslopesatopenandshortcircuit\\ estimatetheresistances(PVEducation\\ namesthemethod;thesignsandthesign\\ conventionaremine),thesimulator\\ classesofIEC60904-9(spectralmatch,\\ non-uniformity,instability):A+\\ 0.875-1.125,1\%,1\%(300-1200nm);A\\ 0.75-1.25,2\%,2\%;B0.6-1.4,5\%,5\%;\\ C0.4-2.0,10\%,10\%\end{array}$ $\begin{array}{r}(NRELmeasureswithfour-terminalKelvin\\ connectionsandmonitorcells,a3kW\\ xenonsimulatorcovers0.2to10sunswith\\ \pm 3\%uniformity,theefficiency\\ uncertaintyis\pm 2\%to\pm 5\%andthatofa\\ referencecell\pm 1\%(reportof2000))\end{array}$
Quantum-efficiency measurement	$EQ{E}_{dut}\left(\lambda \right)=EQ{E}_{ref}\left(\lambda \right)\frac{I_{dut}\left(\lambda \right)}{I_{ref}\left(\lambda \right)}$	$I\left(\lambda \right)\propto V_{lock-in}\left(\lambda \right)/R_{shunt}$ $f_{chop}\ne k\cdot 50\mathrm{Hz}$	$\begin{array}{r}EQ{E}_{dut},EQ{E}_{ref}:quantumefficienciesofthe\\ cellundertestandofthecalibrated\\ referencedetector\end{array}$ $\begin{array}{r}{I}_{dut},I_{ref}:photocurrentsforthesame\\ monochromaticbeam(theratioformisthe\\ generalpracticeandisnotwritteninthe\\ sources;equalareasandthesamebeamare\\ assumed)\end{array}$ $V_{lock-in}:lock-insignalacrosstheshunt{R}_{shunt}$ $\begin{array}{r}{f}_{chop}:chopperfrequency,chosensothatit\\ isnotaharmonicofthemains\end{array}$ $\begin{array}{r}(TUDelft:xenonlamp,order-sorting\\ filter,chopperatmostcommonly123Hz,\\ monochromator,shuntandlock-in\\ amplifier,whitebiaslightofonesun,\\ calibratedreferencemeasuredfirst;NREL:\\ 61filtersofabout10nmfrom280to1900\\ nm(15to100Hz),oragratingfrom300\\ to3000nm,anda1-sunbiasof10to40\\ mAfor1c{m}^2)\end{array}$
Lifetime and implied voltage	$V_{oc,implied}=\frac{k_{B}T}{q}\ln \left[\frac{\Delta n\left(N_A+\Delta n\right)}{n_i^2}\right]$	${\tau }_{eff}=\frac{\Delta n}{G-d\Delta n/dt}\text{(QSS and transient)}$ ${\tau }_{OCVD}=-\frac{k_{B}T}{q}{\left(\frac{d{V}_{oc}}{dt}\right)}^{-1},{\tau }_{eff}=\frac{qd\Delta n_{oc}}{J_{sc}}$	$\begin{array}{r}{V}_{oc,implied}:voltagethatthecarrier\\ densitieswouldgive(anupperlimit\\ becausemetal-siliconrecombinationisnot\\ included)\end{array}$ $\Delta n:excesscarrierconcentration$ $N_A:doping$ $G:generationrate$ $\begin{array}{r}{\tau }_{OCVD}:lifetimefromthedecayofthe\\ open-circuitvoltage(thestandardform;\\ thepagethatIreadnamesthevariables\\ anddropstheequation)\end{array}$ $\begin{array}{r}\Delta n_{oc}:excessconcentrationatopen\\ circuit,thefirstgeneralisedformis\\ frommemory,becausetheequationsofthe\\ pagewerestripped,andthelastisthe\\ Jsc-VocmethodofKostylyovetal.\end{array}$ $k_B:Boltzmannconstant$ $\begin{array}{r}(theintrinsic-limitcellofthesource\\ (J_{sc}=43.4mA/c{m}^2,d=63.3\mu \mathrm{m},\Delta n_{oc}\approx 2.7\\ to3.0\times 10^{16}c{m}^{-3})gives{\tau }_{eff}=0.67msfor\\ 2.85\times 10^{16},computed)\end{array}$
Temperature rating	$P\left(T\right)=P_{STC}\left[1+\gamma \left(T-25\right)\right]$	$\gamma \approx -0.0045{\mathrm{K}}^{-1}$ $\Delta V_{oc}\approx \frac{d{V}_{oc}}{dT}\Delta T$	$P_{STC}:poweratstandardtestconditions$ $T:celltemperatureinC$ $\begin{array}{r}\gamma :relativetemperaturecoefficientofthe\\ power(Wikipedia:about0.45\%perdegree,\\ withoutsayingwhetheritisrelative;\\ PVEducation:0.4\%to0.5\%perdegreefor\\ silicon;thelinearformismine,since\\ theformulaofthepagewasstripped)\end{array}$ $\begin{array}{r}d{V}_{oc}/dT:voltagecoefficientofthe\\ fundamentalspage\end{array}$ $\begin{array}{r}(acellat65Closes18\%oftherated\\ powerfor\gamma =-0.0045K^{-1}(computed,taken\\ asrelative);theWikipediaexampleof\\ 0.60Vat25Cand0.55Vatabout45Cis\\ 2.5mV/K)\end{array}$

Important Questions

Ques: What is the quarter-wave thickness of the coating: 0.75 µm from the PVEducation calculator, or 65 nm?

The quarter-wave rule gives 65 nm for the calculator’s own outputs. The page gives $d_1={\lambda }_0/(4n_1)$, and its example prints $n_1=2.2913$ and $d_1=0.75\mu \mathrm{m}$ for a design wavelength of 0.6 µm. But $0.6/(4\times 2.2913)=0.0655\mu \mathrm{m}$ (computed), which is 65.5 nm and 11 times smaller. The index $2.2913=\sqrt{1.5\times 3.5}$ shows that the example is an encapsulated cell ($n_0=1.5$), and $0.75$ µm is not a quarter wave of anything near 0.6 µm. The thickness 0.75 µm is probably a unit or input slip of the calculator’s output, which is my reading, because the page does not give the other inputs. Use 65.5 nm for 600 nm in glass and 80.2 nm in air. Reference: PVEducation, pveducation.org/pvcdrom/design-of-silicon-cells/anti-reflection-coatings.

Ques: How much current does an anti-reflection coating recover?

With the constant index $n_2=3.5$ of the page and the AM1.5G photon flux up to 1107 nm, I find that bare silicon in air loses 30.9 % of the photons, so that a 43.8 $mA/c{\mathrm{m}}^2$ cell gives 30.3 $mA/c{\mathrm{m}}^2$; a quarter-wave coating for 600 nm leaves 7.1 % and gives 40.7 $mA/c{\mathrm{m}}^2$. Under a cover of index 1.5 the bare loss is 16.0 % (36.8 $mA/c{\mathrm{m}}^2$) and the coating leaves 3.3 % (42.4 $mA/c{\mathrm{m}}^2$). A coating of index 2.0 and 75 nm (assumed) in air leaves 7.5 % (computed). The residue comes from the wavelengths away from the design value; a real silicon index varies with the wavelength, so these numbers show the size of the effect and are not a design value. Reference: PVEducation, pveducation.org/pvcdrom/design-of-silicon-cells/anti-reflection-coatings; ASTM G173 reference spectra as distributed in pvlib 0.16.1, integrated by me.

Ques: Is the finger spacing of 4 mm correct for the emitter example of PVEducation?

It is correct but conservative. The page says that ${\rho }_{\square }=40\Omega /\mathrm{sq}$, $J_{mp}=30$ $mA/c{\mathrm{m}}^2$ and $V_{mp}=450$ mV keep the emitter loss under 4 % for a spacing below 4 mm. Solving $p={\rho }_{\square }{S}^2{J}_{mp}/(12V_{mp})$ for 4 % gives 4.24 mm, and 4 mm gives 3.6 % (computed). So the page rounds down, and both numbers agree with the formula. The slides of MIT give the same example and the same limit. The page writes the numerator once with a different symbol for the sheet resistance, which looks like a typo of the text (my reading). The result is for the emitter alone; the shading, the fingers and the contact must be added, and they move the optimum spacing. Reference: PVEducation, pveducation.org/pvcdrom/design-of-silicon-cells/emitter-resistance; MIT 2.627 (Fall 2013), Lecture 7, ‘Toward a 1D Device Model Part 1: Device Fundamentals’ (slide text partly garbled).

Ques: Where is the optimum finger spacing for an example grid?

For the assumed values (fingers of 50 µm by 15 µm with ${\rho }_f=3\times 10^{-6}\Omega \mathrm{cm}$, length 2.6 cm, emitter 100 $\Omega$/sq, ${\rho }_c=10^{-3}$ $\Omega \mathrm{c}{\mathrm{m}}^2$, no busbars), a spacing of 2 mm gives 2.5 % shading, 2.3 % emitter, 1.25 % finger and 0.33 % contact loss, in total 6.4 % (computed). The sum has its minimum at 1.43 mm with 5.8 %: 3.5 % shading, 1.2 % emitter, 0.9 % finger and 0.2 % contact. The gain is 0.6 % of the generated power, about 0.13 points for a 21 % cell. The values are assumed, the formula of the shading is mine, and the busbars would add to the shading and shift the optimum, so only the method is from the sources. Reference: PVEducation, pveducation.org/pvcdrom/design-of-silicon-cells/series-resistance, /emitter-resistance, /contact-resistance and pveducation.org/node/200 (finger resistance), pveducation.org/node/195.

Ques: What limits the effective lifetime of a record-class silicon cell?

According to Kostylyov et al., recombination in the space-charge region, not at the surface, is mainly responsible for the reduced effective lifetime. For the record cell the measured ${\tau }_{eff}$ is 5.15 ms against a theoretical maximum of 18.7 ms (3.6 times), and the surface velocity is 0.017 cm/s, which gives $d/S=0.76$ s for 130 µm (computed), far above 5 ms. For another cell, neglecting the space-charge region raises the efficiency by about 1 %, neglecting the surface by 0.12 %. The paper’s fits are of its own model, with six channels, and I did not check them. An assumed S of 10 cm/s would give 1.3 ms, so the surface matters only for ordinary passivation. Reference: Kostylyov, Sachenko, Evstigneev, Sokolovskyi and Shkrebtii, ‘Characterization and optimization of high-efficiency crystalline silicon solar cells’, arXiv:2409.01101.

Ques: Does the record cell obey the arithmetic and the ideal fill factor?

The arithmetic does: 751.4 mV, 41.45 $mA/c{\mathrm{m}}^2$ and 86.07 % give 26.81 %, and 740.3 mV, 42.5 $mA/c{\mathrm{m}}^2$ and 84.64 % give 26.63 % (computed). The fill factor is more interesting. The empirical ideal fill factor for $n=1$ and 751.4 mV is 0.854 (computed), below the measured 0.8607. With $n=2/3$, the Auger value, the same voltage gives 0.893 (computed). A cell that is limited by Auger recombination at high injection can thus have a fill factor above the $n=1$ value; that is my reading, as the paper does not compare. It also means that the formula of the fundamentals page is a limit for $n=1$ only. Reference: Kostylyov, Sachenko, Evstigneev, Sokolovskyi and Shkrebtii, ‘Characterization and optimization of high-efficiency crystalline silicon solar cells’, arXiv:2409.01101; Solar Cell Fundamentals page of this vault.

Ques: Where is the missing current of the record cell: 41.45 against 43.8 $mA/c{\mathrm{m}}^2$?

The cell reaches 94.6 % of the maximum that I computed for a step absorber at 1.12 eV (cutoff 1107 nm, AM1.5G of ASTM G173). The paper cites only 1 % of total optical loss for the best cell, so that most of the 2.4 $mA/c{\mathrm{m}}^2$ must be something else. One part is the model: silicon is an indirect absorber, so that light from 1000 to 1200 nm is weakly absorbed, and the long-wavelength model $EQ{E}_l=f/[1+b(4n_r^2\alpha d{)}^{-1}]$ with $b=1.6$ to 1.8 allows only about 28 to 31 passes against 49 for the ideal one (computed). The other part is the collection of the carriers. The paper does not give a breakdown, so the split is not known to me. Reference: Kostylyov, Sachenko, Evstigneev, Sokolovskyi and Shkrebtii, ‘Characterization and optimization of high-efficiency crystalline silicon solar cells’, arXiv:2409.01101; ASTM G173 reference spectra as distributed in pvlib 0.16.1, integrated by me.

Ques: Is the light-trapping path length 50 or 49 thicknesses?

Both are the same number at the precision of the sources. PVEducation says that a cell with good light trapping may have an optical path length of 50, and a plain cell one thickness. The Lambertian limit $4n_r^2$ gives 49 for $n_r=3.5$ (computed), a limit that the page does not write; I add it from background knowledge. The Kostylyov et al. model contains the same $4n_r^2$ and the parameter $b$, which is 1 for an ideally diffusive surface and was fitted to 1.6 and 1.8 for two cells, so that the effective enhancement is 30.6 and 27.2 (computed, my reading of the model). The critical angle for silicon to air is 16.6° (computed), which is why the light is trapped. Real cells therefore reach about 55 to 62 % of the ideal path. Reference: PVEducation, pveducation.org/pvcdrom/design-of-silicon-cells/light-trapping; Kostylyov, Sachenko, Evstigneev, Sokolovskyi and Shkrebtii, ‘Characterization and optimization of high-efficiency crystalline silicon solar cells’, arXiv:2409.01101.

Ques: Is a simulator that is called Class A by NREL consistent with the limit of 2 %?

Not with the present table. NREL’s report of 2000 describes its X-25 simulator as filtered to AM1.5G and ASTM E927 Class A, with a uniformity of ±3 % at one sun. The present IEC 60904-9 and ASTM E927-19 give for Class A a spectral match of 0.75 to 1.25, a non-uniformity of 2 % and a temporal instability of 2 %, as listed by Wikipedia, which adds Class A+ at 1 % and 0.875 to 1.125 (for 300 to 1200 nm). The older edition of the standard probably had a looser limit, which is my guess; I did not read it. So ±3 % may have been Class A in 2000 and would be Class B (5 %) today. The quoted efficiency uncertainty of 2 % to 5 % includes the reference cell, the spectrum and the mismatch. Reference: NREL, ‘Measurements and Characterization’, BR-530-22215 (March 2000); Wikipedia, ‘Solar simulator’.

Ques: Is the temperature behaviour of Wikipedia’s example consistent with the coefficients?

Roughly. The example, 0.60 V at 25 °C and 0.55 V at about 45 °C, is 2.5 $mV/K$, against -2.26 $mV/K$ that I computed for 0.6 V with the PVEducation formula, a difference of 11 % (the temperature of 45 °C is approximate). The power coefficient of about 0.45 % per degree gives an 18 % loss at 65 °C if it is relative, which the text does not say; PVEducation gives 0.4 % to 0.5 %. The same page says that $V_{oc}$ may drop only 10 % for an 80 % fall of the illumination. For $n=1$ the fall is $(k_{B}T/q)\ln 5=41.6$ mV, 6.6 % of 0.631 V, and for $n=2$ it is 83 mV, 13 % (computed), so the 10 % lies between the two cases. Reference: Wikipedia, ‘Solar cell efficiency’; Solar Cell Fundamentals page of this vault.

Related pages in this vault: Solar Cell Fundamentals, Interdigitated Back-Contact (IBC) Silicon Cell, Aluminium Back-Surface-Field (Al-BSF) Silicon Cell, Concentrator Photovoltaic (CPV) Solar Cell, PERC and PERT Silicon Cell, Lifetime Testers, HgCdTe Photodiode, Organic Photodetector (OPD), EBIC and EBAC.