Thermodynamic Limits of Solar Cells with Non-ideal Optical Response

Transcription

1 Absorptance A(E Thermodynamic imits of Solar Cells with Non-ideal Optical Response M Ryyan Khan Peter Bermel and Muhammad A Alam Electrical and Computer Enineerin epartment Purdue University West afayette IN United States Abstract The Shockley-Queser ( theory defines the thermodynamic upper limits for Jscoc FF and iciency of a solar cell The classical calculation assumes an abrupt onset of absorption at the band-ede perfect absorption for all eneries above the bandap and absence of non-radiative recombination These assumptions are never satfied for any practical solar cell In th paper we explain how the limits are redefined in the presence of the non-ideal ical ects and we provide closedform analytical expressions for the new limits for Jscoc and FF Remarkably these new limits can be achieved to a very hih deree even with sinificantly imperfect materials Index Terms photovoltaic cell thermodynamic limit incomplete absorption I INTROUCTION Recent developments in the photovoltaic (P technoloy has resulted in hihly icient cells operatin close to the Shockley-Queser ( limit [] The thermodynamic limits to the various performance matrices such as short circuit current J open circuit voltae fill-factor FF and iciency have been ensively analyzed in the literature [ 5] However these analyses do not consider the intrinsic non-ideal ical responses of the absorber material associated with finite film thickness or weak absorption at the band ede There have been considerable recent ort to develop advanced ical desins to improve cell iciency[6 8] It therefore important to quantify the ects of the non-ideal ical response on the thermodynamic limits of P performance matrices and establh new limits if any due to intrinsic constraints In th paper we consider the non-ideal ical ects for derivin the thermodynamics limits for solar cells We account for two aspects of th non-ideal response in practical cells: incomplete absorption near the band ede and non-radiative recombination or less than unity ernal fluorescence iciency (EFE The EFE defined as the fraction of the total recombination that contributes to ernal radiation from the solar cell We analytically derive the new limits for P performance matrices ie J and FF includin the ical non-idealities dcussed above The analytical expressions are in excellent areement with the correspondin numerical model Our results imply that it possible to operate close to th new limit despite imperfect absorption; further ical desin aimin towards complete absorptance (e ical black hole [9] would yield neliible improvement in cell iciency Also low can derade P iciency by reducin The conclusions apply and are accurate for broad-rane of bandaps ie E 5 e These predictions from our analytical calculations are also supported by the study based on the wellknown P material allium arsenide (GaAs Finally we dcuss the practical implications of low to illustrate the scope for improvement in P iciency II AYTICA EXPRESSIONS FOR P PARAMETERS We assume that sunliht incident onto the solar cell with a 5 very small solid anle ( S ~ 68 0 see Fi (a An anti-reflection coatin (ARC the blue layer in Fi (a suppresses reflection from the top surface Only a fraction of the liht AE ( absorbed due to imperfect liht trappin The unabsorbed liht ( AE ( bounces out of the solar cell If non-radiative recombination absent (ie EFE the sum of carriers racted and photons emitted from the solar cell must equal the number of absorbed incident photons for all voltae These emitted photons are dtributed over a much larer solid anle ( ~ yellow hemphere in Fi (a compared to that of the incident rays -A(E n in n out ( J/q A(E (a Perfect abs - =6e (0μm c-si - =30e (00μm c-si Photon Enery E Fi (a Schematic outline of the P operation (see t for details (b Absorptance as a function of enery (at E e for 6 / e (blue line and 30 / e (red line The absorptance for 30 / e (red line matches very well with the absorptance of 00μm c-si (red dotted line Correspondinly 6 / e fits 0μm c-si absorptance (blue solid and dotted lines The black dashed line shows perfect absorption Also see Fi 5 for the absorptance of GaAs The empirically fitted absorptance (for eneries above the bandap: E E of the described solar cell approximated as follows [0] (also see part chapter 5 in []: ( A( E e E E ( We assume no absorption for eneries below the bandap E E Here denotes the combined ect of the ( (b

2 material parameter ( related to the absorption coicient and the ective absorption path (which includes the ect of liht trappin For example absorptance of ~ 00μm in c-si ( E e ASi ( E exp( ( E where ( E the absorption coicient of c-si [] Th absorptance spectrum can be approximately fitted by ( usin ~ 30 / e and E e ASi ( E and the fitted spectrum are shown in Fi (b as red dotted and solid lines respectively irect bandap materials (e GaAs E ~ 4e have much hiher absorption coicients near the band ede For instance absorptance for ~μm GaAs can be approximated with 3 ~ 0 / e and E ~ 4e Fi (b illustrates the absorptance profile AE ( for two different Note that the form for AE ( in ( based on the theory of absorption [0] in indirect band-ap materials (e c-si therefore it shows ood fits for such materials However th formula can also be used (approximately for direct band-ap materials so lon the sufficiently lare to allow moderately ood absorption in these films For example Eq ( fits reasonably well for absorptance spectra for GaAs films thicker than - μm a dimension typical of practical GaAs solar cells Th lenth scale also appropriate for the semi-classical calculations used in th paper In the followin we derive expressions for short circuit current J open circuit voltae fill factor FF and iciency considerin imperfect absorption near the band ede and deraded ernal fluorescence iciency ( Note that the ernal fluorescence iciency iven by the fraction of recombination which emitted as radiation from the solar cell [3] A The J- relationship The photon flux per enery radiated from a blackbody with chemical potential at temperature T iven by [4] n ( E T E rad 3 ( E / kt c h e Here the solid anle covered by the concernin radiation c the speed of liht in free space h the Planck s constant and k the Boltzmann constant The photon flux per enery incident on the solar cell from the sun taken to be nin ( E nrad ( E TS 0 S Th idealized spectrum resembles the standard raterrestrial (AM0 solar spectrum Now the emsion from the solar cell operatin at voltae would be characterized by nout ( E q nrad ( E T q In our calculations we assume TS 6000 K and T 300 K [4] The principle of etailed balance ensures that the number of carriers racted from the solar cell equals the absorbed and emitted photons Thus the net current iven by: nout ( E q J( q nin ( E A( E de ( E nout ( E q q nin ( E de E n ( out E q ( EE q nin ( E e de E J ( J ( Here the ernal fluorescence iciency Equation ( was used in the second line in the above expressions Note that J the current correspondin to perfect absorption above E The ect of imperfect absorption reflected in the second term J We find that the J relationship can be obtained analytically by usin Boltzmann approximation as follows: E / kt q / kt E / kt q ( S T e e J( qs ( TS e (3 The last term in Eq (3 hihlihts the importance of improvin for hihly icient solar cell Here ( T the contribution from the 3 photonic density of states and ( T ( T ( T ( T kt 3 E kte k T ch kt ( T ( 3 E c h Here we define an absorption non-ideality term as (/ kt e ( ( / kt Remarkably the ect of incomplete absorption accounted for by a simple multiplicative factor ( For the small values of associated with poor absorbers the parameter ( For very stron absorption (hih the parameter ( 0 Note that by allowin ( 0 and we return to the limits exactly as presented in Ref 4 B Short circuit current and open circuit voltae Short circuit current can be obtained from (3 by settin 0 : / / ( E kt S E kt J qs TS e q ( T e (4 The open circuit voltae obtained by settin J ( 0 in (3: T ( T q E kt ln kt ln ( TS S TS kt ln kt ln (5 Th important eneralization of the expressions for iven in Refs 4 5 and 8 The loarithmic suppression of due to poor clearly indicated Physically low reflects reduction in carrier buildup (and hence lowered

3 J (ma/cm ( analytical/num analytical/num due to non-radiative recombination Also note that the last term reflects chane in due to incomplete absorption ( 0 Interestinly increases from its value due to imperfect absorption at the band ede (which causes ective widenin of the ical bandap However th increase in counterbalanced by a reduction in J so that the overall iciency remain below the classical limit See Sec III for additional dcussion on th topic C and Fill-Factor The imum iciency of a solar cell written as the ratio of the output power ( J at imum operatin condition to the incident solar power P in : J P in 4 3 The input solar power Pin S ( kts / ( c h The imum operatin condition can be derived by imizin iciency ( J / P in with respect to By settin d / d 0 we solve for to find : T q E kt ln TS S kt ln kt ln (6 Note that as seen from (5 and (6 both and are shifted by the same amount due to the non-ideal ical ects considered here (ect of both and ( 0 The imum current found from J( in (3: E / kt / q / kt S ( J ( E kt qs TS e q T e e E / kt q S Se ( TS ( T (7 The second line utilizes (6 for the final expression of J As explained earlier ( for smaller values of Hence / ( very hih for low saturatin rapidly to a low value with increasin Th means that while increases for low absorption J decreases rapidly Indeed since the decrease in J (decrease ~ / ( faster compared to increase in (increase ~ ln[/ ( ] with lowered absorption toether they derade ( J below the limit The fill-factor J FF J (8 sc can be obtained from previously derived expressions for J J sc and We find that all our analytically derived values from (4-(8 are accurate within 7% to the numerical thermodynamic calculation results for practical solar cell bandap rane of e E e The results of th comparon are summarized in Fi which can be used to translate the analytical results to accurate numerical results with appropriate scalin 05 5 E (a : / ev 5 E (b : 000 to Fi Accuracy of the analytical model in terms of ratio of iciencies calculated analytically to those found numerically: (a for various values and (b for various The errors for various and are within 7% of the numerically calculated values for e E e III IUSSION A Insihts from the analytical relationships Fi 3 shows that poor absorption with low (blue curves derades P performance in terms of J and iciency but counter-intuitively it improves slihtly compared to hiher absorption (red curves Recall that determined by the emsion spectrum ( nout AE ( of the solar cell; the weihtin of the emsion by AE ( shifts the emsion peak away from E towards hiher eneries Th ective widenin of the ical bandap increases Note that th ective widenin of the ical bandap essentially will shift the iciency vs E curve to the left yieldin slihtly improved P performance for the smaller bandap solar cells (where E 35e Th concept could be useful for P materials with bandap lower than the imum ( E 35e =50e =5e E =5e - =50e 05 5 E =50e - =5e E Fi 3 Analytically derived solar cell parameter ( J and limits at 5 / e (blue line and 50 / e (red line The limit shown as the black dashed lines (corresponds to For these results we assume What would be the implication of dramatically improvin absorption (parameterized by for a cell that oriinally had very poor absorption? Consider a GaAs solar cell with E 4e and The blue solid curves in Fi 4 show that the ect of saturates very quickly after ~ 000 / e Th translates to ~μm for GaAs After th critical point J increases very slowly decreases very slowly and saturates close to the

4 J (ma/cm ( J (ma/cm ( (/m (/m A(E limit Th indicates that the iciency would essentially saturate for a GaAs solar cell with finite absorptance Althouh J has not reached its imum possible value at th point the increased compensates for the lowered J to yield an approachin value from below due to imperfect absorption Fiure 4 also demonstrates the ect of reduced ernal fluorescence iciency Only a small fraction of the recombination in indirect bandap materials occurs radiatively As seen in Fi 4 for 0 [4] the iciency limit lowered even for perfect absorption As expected does not affect J Th because only modifies the emsion process and thus alters the The non-radiative recombination ( reduces the which in turn derades =0 = (/e 5 5 =0 = (/e = = (/e Fi 4 Effect of varyin on solar cell parameter ( J and limits at (blue line and 0 (red dotted line The black dashed lines ive the limit B Practical implications: incomplete absorption and Urbach Tails From a practical perspective let us investiate incomplete absorption in GaAs The absorption coicient of GaAs shown in Fi 5(a has been obtained from [] Althouh E 4 e for GaAs the absorption in the Urbach tail [5] indicated by the shaded reion in the inset plot of Fi 5(a -- makes non-zero even for E E The correspondin absorptance AE ( exp( (E of a GaAs film of thickness shown in Fi 5(b For smaller values (μm and 0μm represented by blue and red lines respectively the exponentially small in the Urbach tail ensures that the contribution from these states to overall absorption neliible However for a very thick film ( 0 3 μm represented by the black line the relative contribution by the Urbach tail increases sinificantly especially for the photons with 4 E 3e Th can be thouht of as ective lowerin of the ical bandap below 4 e We do expect some deviations in the P performance parameters for real materials (e GaAs from the estimates obtained usin our simplified analytical absorption model However the predicted trends are surprinly robust which we will show for practical case of GaAs solar cells Fi 6 shows the P parameters ( J and iciency as a function of GaAs film thickness The J- relationships and the correspondin P parameters in th case have been calculated numerically based on ( with AM5 illumination For (blue line Fi 6 initially J res very quickly with however the rate of increase decreases sharply once 3μm see Fi 6(a We also observe continually decreasin as a function of as dcussed earlier Finally we observe that the iciency quickly res for very thin solar cells but then saturates for 3μm Beyond th point radual increase in J counterbalanced by a radual decrease in resultin in a saturated versus relationship Note that the J ain shown in the shaded reion of Fi 6(a contributed by the Urbach tail however the absorption in the Urbach tail reduces the ective ical bandap yieldin in a deraded The increased J compromed by a correspondin deraded keepin the iciency approximately constant in these values e e E E (a =0 3 µm =µm =0µm Photon enery Fi 5 (a Absorption coicient spectrum of GaAs The shaded reion in the inset plot shows the Urbach tail (b The absorptance spectrum of a GaAs film of thickness 0μm (red μm (blue and 3 0 μm (black The ect of non-radiative recombination essentially the same as dcussed earlier in Sec IIIA Reduced ( 0 decreases without havin any ect on J thus deradin the iciency (see red curves in Fi 6 In short our analys of GaAs predicts that we do not require very hih liht harvestin to reach the ultimate ( solar cell iciency A GaAs solar cell of thickness ~3 μm can very closely approach the performance limit associated with th material bandap 30 0 =0 = (m = = (m (b = = (m (a (b (c Fi 6 Effect of varyin on GaAs solar cell parameter ( J and limits at (blue line and 0 (red dotted line The black dashed lines in (a (c the iciency limit

5 C Practical implications: non-radiative recombination Equation (5 suests that reduced by kt ln(/ in presence of non-radiative recombination ie for Th provides us with the opportunity for possible improvement in by increasin Table estimates th value for some well-known solar cells [4] To stress the importance of improved it obvious that the increase in by ~70m in GaAs (Alta cells compared to GaAs (ISE cells can be explained exclusively by the enhancement in Th improved yielded the hihest iciency GaAs solar cell by Alta evices [6] Room for improvement by enhancin in Si-UNSW and Si-SPWR devices remarkably close (see Table However these Si solar cells have different values which can be attributed to other losses in the devices CIGS (alon with most other thin-film materials still has the reatest potential for improvement Table : The and values for various available solar cell devices [4] are shown here The room for improvement associated with estimated in the riht-most column evice (m (% I CONCUSIONS We have derived the thermodynamics performance limit of solar cells in the presence of imperfect absorption and nonideal ernal fluorescence iciency The expressions illustrate in a compact analytical form the ects of imperfect ical absorption and non-radiative recombination We find that approachin limit does not require perfect absorption and therefore the need for perfect ical desin can be relaxed considerably Numerical analys based GaAs further reinforced th conclusion Finally solar cells with low have room for improvement in and thus in iciency Opto-electronic desin aimin towards devices with predominantly radiative recombination would be of prime interest for th purpose ACKNOWEGEMENTS ( EFE kt ln( / (m Si UNSW Si SPWR GaAs Alta GaAs ISE CIGS(NRE Th material based upon work supported as part of the Center for Re-efinin Photovoltaic Throuh Molecule Scale Control an Enery Frontier Research Center funded by the US epartment of Enery Office of Science Office of Basic Enery Sciences under Award Number E The computational resources for th work were provided by the Network of Computational Nanotechnoloy under NSF Award EEC Th was also supported by the Bay Area P Consortium a epartment of Enery project with Prime Award number E-EE REFERENCES [] W Shockley and H J Queser etailed Balance imit of of p-n Junction Solar Cells J Appl Phys vol 3 no 3 p [] A uque and S Heedus Eds Handbook of Photovoltaic Science and Enineerin Second Edition 0 [3] J Nelson The physics of solar cells ondon: Imperial Collee Press 004 [4] C Hirst and N J Ekins-aukes Fundamental losses in solar cells Pro Photovolt: Res Appl vol 9 no 3 pp [5] M A Alam and M R Khan Fundamentals of P Interpreted by a Two-evel Model arxiv:05665 May 0 [6] J N Munday M Callahan and H A Atwater iht trappin beyond the 4n limit in thin waveuides Applied Physics etters vol 00 no p 4 Mar 0 [7] M Callahan J N Munday and H A Atwater Solar Cell iht Trappin beyond the Ray Optic imit Nano ett vol no pp [8] A Polman and H A Atwater Photonic desin principles for ultrahih-iciency photovoltaics Nat Mater vol no 3 pp Mar 0 [9] E E Narimanov and A Kildhev Optical black hole: Broadband omnidirectional liht absorber Applied Physics etters vol 95 no 4 p [0] S atta Quantum Phenomena Addon-Wesley 989 [] M S resselhaus 673 SOI STATE PHYSICS [Online] Available: [Accessed: 08-Jun-03] [] Refractive Index atabase Table of Refractive Index alues for Thin Film Thickness Measurement [Online] Available: [Accessed: 09-Jul-0] [3] O Miller E Yablonovitch and S R Kurtz Intense Internal and External Fluorescence as Solar Cells Approach the Shockley-Queser imit arxiv:06603v3 Jun 0 [4] M A Green Radiative iciency of state-of-the-art photovoltaic cells Proress in Photovoltaics: Research and Applications 0 [5] F Urbach The on-wavelenth Ede of Photoraphic Sensitivity and of the Electronic Absorption of Solids Phys Rev vol 9 no 5 p ec 953 [6] Alta evices: Findin a Solar Solution - Technoloy Review Technoloy Review [Online] Available: [Accessed: 7-Mar-0]