A Unified Mathematical Framework for Intermediate Band Solar Cells

Transcription

1 A Uified Mathematical Framework for termediate Bad Solar Cells Albert S. Li 1, Sze-Mig Fu 1, Ya-Kai Zhog 1 stitute of lectroic gieerig, Natioal Chiao-Tug Uiversity, Hsichu, Taiwa Departmet of lectro-physics, Natioal Chiao-Tug Uiversity, Hsichu, Taiwa Abstract The modelig of itermediate bad solar cell has bee developed sice 90 s ad cotiued effort is made to facilitate the realizatio of this ovel device. Two formulatio has bee used to model the geeratio recombiatio rate of BSC icludig covetioally available modified-shockley-reed-hall formulatio or later proposed BSC formulatio (Luque ad Mart PRL This paper proves that these two formulatios are actually mathematically equivalet ad actually oe ca be derived from the other. A uified mathematical framework ca thus be established ad the covetioal driftdiffusio model ca thus be employed for modelig ovel BSC with the iclusio of ew model for itermediate bad carrier trasport. The debate whether the additio of impurity atoms would decrease the efficiecy by shorter recombiatio lifetime or icrease the efficiecy by more absorptio is studied, ad results cofirm that the efficiet removal of photo-geerated carriers from valece ad coductio bads ad solar cocetratio is the key to the success of subbadgap photovoltaics. dex Terms itermediate bad solar cell, impurity photovoltaic effect, Shockley-Reed-Hall, drift-diffusio model.. NTRODUCTON mpurity photovoltaics (PV has bee proposed sice 1960[1], ad its formulatio is primarily based o so-called modified Shockley-Read-Hall (SRH formulatio [, 3]. Some studies show icreased J sc [3] while others show miimal improvemet or eve degradatio i V oc ad i charge collectio due to excessive trap states [4-6]. 1997, Luque et. al. proposed the cocepts of itermediate bad solar cells (BSC[7] based o detailed balace calculatio where a ultra-high efficiecy of 63.% iitiates eormous amout of research effort due to the potetial of itermediate bad devices to replace covetioal multi-uctio cells[8-6]. The derivatio of BSC formulatio is i geeral log ad mathematically very complex. Basically it is derived alog the lie of Fermi golde rule ad photo rate equatios. The result shows the efficiecy uder full cocetratio sulight ca be comparable to the efficiecy of triple-uctio tadem cells ad the differece is that the improvemet will be observed i J sc rather tha V oc. Sice the proposal of the cocept of BSC, the formulatio based o modified SRH[] ad later proposed BSC formulatio by Luque et al. [7] have all bee employed for modelig purpose. this paper, it is show that the optical geeratio-recombiatio rates based o BSC formulatio ad traditioal modified SRH formulatio are actually mathematically equivalet ad thus it is proposed that a uified mathematical formulatio ca be applied for all subbadgap absorptio solar cells, regardless of subbadgap badwidth, state desity, or whether itermediate state carrier trasport exists or ot. Sice a uified framework is proposed, the practicability of subbadgap absorptio solar cells ca be evaluated. Subbadgap photovoltaics (BSC or PV is mostly criticized by the fact that addig itermediate states i the fudametal badgap would shorte the carrier lifetime i coductio or valece bad. This work cofirms that through efficiet photo-carrier removal from coductio ad valece bad ad through solar cocetrators, the beefit of subbadgap states will certaily outperforms its disadvatages ad the predicted >60% coversio efficiecy ca be potetially achieved with solar cocetrator techology.. A UNFD MATHMATCAL FORMULATON The derivatio ca begi with the review of modified SRH formulatio[, 3] which is a well-established semicoductor physics: G R ( e + e,opt N f CN (1 f ( e + e N (1 f C pn f p p,opt p ad p is electro ad hole cocetratio. e ad e p is electro ad hole emissio coefficiet, C ad C p is electro ad hole capture coefficiet, ad N is itermediate state cocetratio, f is itermediate state fillig factor, ad most importatly e,opt ad e p,opt is the electro ad hole optical emissio coefficiet, which characterize the optical geeratio via the trap levels. The celebrated thermodyamic cosistecy relatio betwee carrier emissio ad capture, where the emissio ad capture rate is esured to be equal, ca be writte as: e C (1 B C V B exp(, ep p V exp( CN CN ( N C ad N V is coductio bad effective desity of state, C ad V is coductio ad valece bad edge eergy, B is itermediate bad eergy level. k is Boltzma costat, ad T is absolute temperature. The origial formulatio based o A. Luque et.al. s paper starts from photo rate equatio[7]:

2 dν [ hcv( ν + 1 fci (1 fv hv,ciν(1 fci fv] dx c + [ hc( ν + 1 fci (1 f h,ciν(1 fci f] c + [ hv( ν + 1 fi (1 fv hv,iν(1 fi fv] c ν is photo umber, h XY (X,YC,,V is the matrix elemet betwee coductio bad (CB,valece bad (VB ad itermediate bad (B. f is state occupatio, c is the speed of light i vacuum ad is refractive idex. Defie absorptio coefficiet betwee XC,,V ad YC,,V α h ( f f XY X Y Y Xi Note that the sig of f Y ad f Xi is reversed from the origial paper due to the defiitio here that absorptio coefficiet is egative for f Y < f Xi ad positive for f Y < f Xi. This poit does ot matter i most paper i BSC sice a positive, costat value is assumed for the absorptio coefficiet i those works. (4 X ad Y ca be C,, or V, represets coductio, valece, or itermediate bad. By (4 ad with some arragemet, (3 ca be rewritte as dν dx α ( ν ν + α ( ν ν + α ( ν ν CV CV C C V V ν is photo umber i (3 ad α XY is the absorptio coefficiet defied i (4, ν XY is defied as ν XY 1 XY μxy 1 Y μy μ μ fxi (1 fy f f Xi X Y Y Y Xi where f is state occupatio as i (3. μ XY is the separatio betwee electro, hole, or itermediate states quasi-fermi levels, ad XY is trasitio eergy. Notice that here ooverlapped scheme is assumed. The mathematical equivalecy betwee BSC (Luque s ad PV (modified SRH ca also be established for the case of spectrally overlapped absorptio, though the mathematical derivatio is very complicated ad is still uder ivestigatio. additio, the spectrally NOV coefficiet is the key requiremet for the success of subbadgap photovoltaics, i order to miimize thermalizatio. (3 (4 (5 (6 The recombiatio betwee CB ad VB ca be expressed as[9, 6] R 1 1 μcv 1 1 αcvd μcv i i μcv Br i i B p CV 3 G 3 αcvd/ i G (7 r ( i i is itrisic carrier cocetratio, ad B r is the radiative recombiatio coefficiet i stadard semicoductor device physics. The expressio ca be arraged ito the familiar bad to bad radiative recombiatio formulatio ad this poit has bee clear i [9, 6]. The itermediate bad recombiatio rate ca be writte as 1 1 RC μ 3 C 1 1 (8a α d Substitute the expressio for α XY i (4, R 1 ν C 1 h f f C 3 C C, i ( Ci d (8b Arragig the above equatio ad usig Boltzma approximatio for the term 1/(exp(/-1, as i the origial formulatio [9, 7] [ ( ν ] RC h 3 C i, f fci C d h c 3 [ ( exp ( ] hc i, f fci d (8c Substitute the expressio for ν C from (6, the first term (F.T. i (10a, ca be arraged as

3 FTq.. ( [ 3 ( ] f (1 f [ ( ] 3 [ (1 ] hc, i f fci ν C d Ci hc f fci d f fci hc fci f 3 d 3 fc, i hc f d [ (1 ] (8d the secod equality above it is assumed that f C,i for trasitio of a specific eergy is the same for all state k i, ad this is true for most semicoductor sice the occupatio depeds oly o eergy. Therefore, f C,i ca actually be take out of the summatio sice it does ot deped o subscript i. The subscript i is still maitaied for f C,i to distiguish it from coductio bad edge C. Usig Boltzma approximatio for f C,i FC C,i 3 [ h (1 f ] d C (8e order to be i-lie with the assumptio employed i modified SRH formulatio, B desity of state is assumed to be a delta fuctio located at. t is ot difficult to geeralize the derivatio here to a arbitrary desity of state fuctio as i the case of SRH derivatio. Notice the term exp( FC - C,i / is separated ito two terms i order to factor out exp( FC - C /, which times N C equals to electro cocetratio,. FC C C C, i 3 (1 f C d FC C NC N(1 f NN α N ( 1 f C where the defiitio C C, i 3 C0 C h C, i α C0 h d (8f (8g is cosistet with (4 ad is the absorptio coefficiet whe the CB is totally empty ad B is totally full. The secod term (S.T. i (8c ca be arraged as STq.. ( exp 3 [ hc, i( f fci ( ] d (8h Here agai similar to SRH formulatio i (1, the electro emissio from itermediate subbadgap state to CB is assumed ot depedet o electro cocetratio. This is due to the fact that CB state-fillig factor uder o-degeerate case is low ad abudat state is available for such trasitios to happe [ h 3 C( exp ( ] d f (8i f [ 3 hc] d h c Similarly α C0 is defied similar to (8g N f N f e 3 N α C0 d (8 t should be addressed that various geeratio ad recombiatio rates formulated above ca be icorporated ito the carrier cotiuity equatios i drift diffusio model, ad the itermediate bad carrier trasport should be icluded i the drift-diffusio model for completeess. Commercial Drift-Diffusio model simulatio software that ormally equipped with modified SRH formulatio capability ca thus be used to study the ovel third geeratio BSC. The modelig capability of hoppig coductio ad tuelig betwee subbadgap states i commercial device simulatio software is suitable for modelig the B carrier trasport i Quatum Dot-BSC, ad ew model ca be added to the existig software package for modelig carrier trasport i high B desity metallic itermediate bad solar cell where the coductio i B is likely to be drift-domiat.. TH FASBLTY OF SUBANDGAP PHOTOVOLTACS The full solutio of itermediate bad devices requires drift-diffusio model, which has bee the active research subect recetly. The developmet of complete drift-diffusio model has still ecouter several problems, maily due to the gap betwee the formulatio of ovel BSC ad covetioal drift-diffusio model formula. Here the goal is ot to calculate the bad diagram of itermediate bad devices, ad alteratively, illustrative study is doe by first calculatig the efficiecy depedece o carrier cocetratio i CB ad VB, i order to clarify the debatig questio whether the additio

4 of impurity bad would icrease or decrease the carrier geeratio ad efficiecy. Defie optical pumpig factor e, opt ep, opt γ (9 e e p γ is determied by may factors icludig subbadgap state optical matrix elemet, B desity of state, light trappig capability of the device, ad solar cocetratio. The parameters used i this study is N C N V 10 0 cm -3 for the effective desity of state of may semicoductors are aroud this order of magitude, G 1.1eV as i silico, ad C C p cm 3 s -1. The capture coefficiet varies sigificatly i differet materials ad i geeral it depeds o wave fuctio overlap, B eergy positio, ad asymmetry i electro ad hole capture processes. Here stadard values are assumed to facilitate the coduct of geeric study. this study, the carrier geeratio recombiatio pheomeo is assumed to take place i high carrier mobility material. High mobility material is a essece for the success of itermediate bad photovoltaics where efficiet photo-excited carrier removal from CB ad VB esures the et geeratio rate is positive ad this poit will be clear later. calculatio below, firstly the electro cocetratio is assumed ad correspodig et geeratio rate G C -R C from B to CB is calculated. The hole cocetratio, that is low eough to sustai such et G-R ca thus the be derived. tur meas high carrier mobility is advatageous where photo-geerated carriers are ot accumulated i the base regio of solar cells. Strog optical pumpig relaxes the costrait of high mobility, ad thus solar cocetratio is the key to the success of subbadgap photovoltaics. V. CONCLUSON A uified mathematical framework for subbadgap photovoltaics is provided, which shows that the PV based o modified SRH formulatio ad BSC based o referece [7]is mathematically cosistet. The device simulatio for ovel BSC is thus made possible sice the poit relatio for geeratio-recombiatio based o modified SRH formulatio is geerally available for most device simulators, ad itermediate bad carrier trasport ca be modeled as successive hoppig ad tuelig betwee adacet states or as ewly added user physical model iterface for high desity metallic itermediate mii-bad. This work cofirms that high carrier mobility ad solar cocetratio is the key to the success of subbadgap photovoltaics. ACKNOWLDGMNT We ackowledge the support of high speed computig facilities from Natioal Ceter of High-Performace Computig (NCHC, Hsichu, Taiwa. The work is fuded by Natioal Sciece Coucil (NSC, Taiwa. Net Optical Pumpig, G C -R C (cm -3 s γ 0.1 γ 100 γ γ γ geeratio domiat recombiatio domiat lectro Cocetratio, (cm -3 Fig. 1. Net optical pumpig rate versus electro cocetratio for various optical pumpig factors with f 0.5. B occupatio ca be chaged ad clamped by compesatio dopig [6]. From Fig. 1, Low electro ad hole cocetratio beefit the device operatio by movig the G-R to geeratio-domiat regime. Therefore, empty coductio bad ad filled valece bad is desirable, which i RFRNCS [1] M. Wolf, Proc. R, vol. 48, p. 146, [] M. J. Keevers ad M. A. Gree, "fficiecy improvemets of silico solar cells by the mpurity photovoltaic effect," J. Appl. Phys., vol. 75, pp , [3] M. Schmeits ad A. A. Ma "mpurity photovoltaic effect i c -Si solar cells. A umerical study," J. Appl. Phys., vol. 85, pp. 07-1, [4] Schokley ad Queisser, "Detailed balace limit of efficiecy of p- uctio solar cell," J. Appl. Phys., vol. 3, pp , [5] G. Giittler ad H. J. Queisser, ergy Covers., vol. 10, p. 51, [6] P. Wiirfel, Sol. ergy Mater. Sol. Cells, vol. 9, p. 403, [7] A. Luque ad A. Martı, "creasig the fficiecy of deal Solar Cells by Photo duced Trasitios at termediate Levels," Phys. Rev. Lett., vol. 78, p. 5014, [8] A. Luque, et al., "Thermodyamic Cosistecy of Sub- Badgap Absorbig Solar Cell Proposals," Tra. lectro Dev., vol. 49, pp , 001. [9] A. Martí, et al., "Quasi-Drift Diffusio Model for the Quatum Dot termediate Bad Solar Cell," Tra. lectro Dev., vol. 49, p. 163, 00. [10] J. Wu, et al., "Bad aticrossig i highly mismatched V semicoductor alloys," Semicod. Sci. Techol., vol. 17, pp , 00. [11] A. Alguo, et al., "haced quatum efficiecy of solar cells with self-assembled Ge dots stacked i multilayer structure," Appl. Phys. Lett., vol. 83, pp , 003.

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