One-Dimensional Fast Transient Simulator for Modeling CdS/CdTe Solar Cells. Da Guo

Transcription

1 One-Dmensonal Fast Transent Smulator for Modelng CdS/CdTe Solar Cells by Da Guo A Thess Presented n Partal Fulfllment of the Requrements for the Degree Master of Scence Approved Aprl 2013 by the Graduate Supervsory Commttee: Dragca Vasleska, Char Igor Sankn Stephen Goodnck ARIZONA STATE UNIVERSITY May 2013

2 ABSTRACT Solar energy, ncludng solar heatng, solar archtecture, solar thermal electrcty and solar photovoltacs, s one of the prmary alternatve energy sources to fossl fuel. Beng one of the most mportant technques, sgnfcant research has been conducted n solar cell effcency mprovement. Smulaton of varous structures and materals of solar cells provdes a deeper understandng of devce operaton and ways to mprove ther effcency. Over the last two decades, polycrystallne thn-flm Cadmum-Sulfde and Cadmum-Tellurde (CdS/CdTe) solar cells fabrcated on glass substrates have been consdered as one of the most promsng canddate n the photovoltac technologes, for ther smlar effcency and low costs when compared to tradtonal slcon-based solar cells. In ths work a fast one dmensonal tme-dependent/steady-state drft-dffuson smulator, accelerated by adaptve non-unform mesh and automatc tme-step control, for modelng solar cells has been developed and has been used to smulate a CdS/CdTe solar cell. These models are used to reproduce transents of carrer transport n response to step-functon sgnals of dfferent bas and vared lght ntensty. The tme-step control models are also used to help convergence n steady-state smulatons where constraned materal constants, such as carrer lfetmes n the order of nanosecond and carrer moblty n the order of 100 cm 2 /Vs, must be appled.

3 Dedcated to Mom and the memory of Dad

4 ACKNOWLEDGMENTS Ths work s motvated and supported by my advsor, Dr. Dragca Vasleska, wthout whom ths project would never take place. I thank her for recrutng me nto her research group at the tme I lost my bearngs, her treless gudance n ths project, and her contnuous trust placed n me. I am grateful to my graduate supervsory commttee members Dr. Stephen M. Goodnck and Dr. Igor Sankn for ther useful nformaton regardng ths research. I would lke to extend my apprecaton to the School of Electrcal, Computer and Energy Engneerng at Arzona State Unversty and Frst Solar Inc., for provdng me ths opportunty to pursue Master s degree and supportng me n ths research. I would also lke to say thanks to my grlfrend back home, Meng Zhang, for her effort on keepng ths long dstance relatonshp durng my study. I must thank my mom, who sturdly backed me up and encouraged me to acheve hgher goals after the msfortune loss of my dad.

5 TABLE OF CONTENTS Page LIST OF FIGURES...v LIST OF TABLES..x CHAPTER: 1. INTRODUCTION Solar Energy Solar Cell Operatons CdS/CdTe Solar Cells Semconductor Devce Smulatons Importance of Smulaton General Devce Smulaton Framework NUMERICAL METHODS Posson Equaton Normalzaton of the Posson s Equaton Dscretzaton of the Posson s Equaton Dscretzaton of Contnuty Equatons Incorporaton wth Heterojunctons Posson s Equaton for Heterojunctons Contnuty Equatons for Heterojunctons Non-Unform Mesh Automatc Tme Step Control Numercal Soluton Technques v

6 CHAPTER: Page LU Decomposton Method Gummel s Iteraton Method PHYSICAL MODELS Generaton and Recombnaton Mechansms Shockley Read Hall Recombnaton Optcal Generaton Band-to-Band Recombnaton Surface Recombnaton Ohmc and Schotky Contact Modelng of Ohmc Contact Modelng of Schottky Contact Partal Ionzaton of Dopants SIMULATION RESULTS Equlbrum Smulaton Results Steady-State Smulaton Results Under dark Under llumnaton Transent Smulaton Results Step Bas Response Photocurrent Transent CONCLUSIONS AND FUTURE WORK Conclusons v

7 CHAPTER: Page 5.2. Future Work REFERENCES..70 v

8 LIST OF FIGURES Fgure Page 1.1. Average solar energy on Earth An llustraton of Concentrated Solar Power Systems Three steps of the operaton of solar cells Typcal IV characterstcs of solar cells Smulated IV characterstcs of solar cells Detaled balance for AM0 and AM Typcal confguraton of a CdTe solar cell Schematc descrpton of the devce smulaton sequence Equlbrum energy band dagram of an abrupt hereostructure Meshng factor vs. Electrc feld Grd spacng for a typcal SnO/CdS/CdTe/ZnTe solar cell dt1 vs. pulse wdths Automatc tme step Δt vs. current changes Gummel s teraton scheme Classfcaton of the generaton/recombnaton process Graphcal descrptons of the SRH recombnaton Photon absorptons n drect bangdap and ndrect gap semconductors Absorpton coeffcents SnO, CdS and CdTe Accumulaton type Ohmc contact P-doped depleton type Schottky contact Soluton of wth partal onzed dopants and Ferm-Drac statstcs..45 v

9 Fgure Page 4.1. Equlbrum energy band dagram Electrc feld profle for unform and non-unform mesh at equlbrum Carrer dstrbutons at Equlbrum Equlbrum band dagram wth Schottky contact appled Electrc feld profles at equlbrum wth Schottky contact appled Carrer dstrbutons at equlbrum wth Schottky contact appled Equlbrum band dagram wth accumulaton type Ohmc contact Comparsons between dark IV-characterstcs Sem log plot of dark IV-characterstcs Current conservaton along the entre devce Illumnated IV characterstcs of CdTe solar cell The dfference n carrer denstes at strong bas Power voltage characterstcs of CdTe solar cell The effect of CdTe thckness on the solar cells IV characterstcs under dfferent temperature Power voltage characterstcs under dfferent temperature Current transents for small pulse sgnals Current transents for small ΔV and larger pulse wdths Reverse recovery transent observed for turn-off Effects of a step turn-off transent on mnorty carrers n P-type CdTe Effects of a step turn-on transent on mnorty carrers n P-type CdTe Photolumnescence currents decay v

10 Fgure Page Exponental decay of photolumnescence currents Current transent under 10 Sun llumnaton wth 30 ns pulse wdth Majorty carrers transent near juncton n P-type CdTe Majorty carrers transent near contact n P-type CdTe Turn-off transent of majorty carrers near juncton n P-type CdTe Turn-off transent of majorty carrers near contact n P-type CdTe Transent of photolumnescence current for a short lght pulse Transent of excess holes for a short lght pulse The polycrystallne nature of the CdS and CdTe layers are ndcated schematcally and are not to scale Schematc drawng of the band dagram for CdS/CdTe solar cells usng dfferent back contact strateges..68 x

11 LIST OF TABLES Table Page 1.1.Performance of common PV cells Devce parameters for equlbrum smulatons Materals propertes for non-equlbrum smulatons Schottky contact s effect on key performance characterstcs..53 x

12 Chapter 1 INTRODUCTION 1.1. Solar Energy As the ultmate source of energy, the Sun shaped ths blue planet called home. It generates atmospherc currents, drves rver flow and provdes energy n photo-synthess, whch converts solar energy drectly nto the chemcal energy that fuels all lvng thngs on Earth. The annual amount of energy consumed by humans on Earth, roughly joules, can be delvered by the Sun n an hour. The enormous power suppled contnuously by Sun, terawatts, dwarfs every other energy source, renewable or fossl fuel. It dramatcally exceeds the 13 TW power that human cvlzaton produces[1]. Fgure 1.1 Average solar energy on Earth. 1 (Courtesy of Wkpeda) As of today, solar energy technologes nclude solar heatng and coolng, solar

13 thermal electrcty and solar photovoltacs. Solar thermal technologes can be used for water heatng, spacng heatng, space coolng and process heat generaton. Solar thermal electrcty technology, also known as concentrated solar power systems, use mrrors and trackng systems to focus a large area of sunlght nto a small beam, as llustrated below n Fgure 1.2. The concentrated heat s then used as a source of a conventonal power plant, where steam drves generators for electrcty. Fgure 1.2 An llustraton of Concentrated Solar Power Systems. A photovoltac cell, or solar cell, s a devce that converts lght nto electrc current drectly by utlzng photoelectrc effect. Although the hstory of solar cells can be dated back to 1880s, Pearson, Fuller and Chapn started the whole new chapter of photovoltacs by creatng the frst slcon solar cell n 1954[2] Solar Cell Operatons Solar cell works n three steps. Frst, Photons n sunlght are absorbed by sold 2

14 state materals, such as slcon and CdTe, known as photoelectrc effect. Electron hole pars are generated wth the absorpton of photons. Secondly, carrers are separated by the bult-n potental (or the depleton regon) of pn junctons. At last, current flows when the separated carrers are extracted to external crcuts. Fgure 1.3 descrbed these steps graphcally. Fgure 1.3 Three steps of the operaton of solar cells. (Courtesy of Dr. Schroder) The letter A, B and C n the mddle of Fgure 1.3, denote three operatng condtons of solar cells. For condton A, where no bas or load s appled to the devce, known as the deal short crcut condton, carrers are beng separated and extracted by 3

15 the pn juncton tself. In ths case, negatve short crcut current, I sc or J sc, the largest operatng current of a solar cell, can be produced. If we apply a certan forward bas, V oc, to the dode that flat carrer denstes are generated, Case C, the open crcut condton, can be acheved wth zero current. Case C can also be nterpreted as dark current balancng out the short crcut current. Theoretcally, open crcut can be produced by applyng an nfntely large load resultng n nfntesmally small current flowng. In realty, only a fnte value of external load can be appled nto the crcuts. Thus the solar cell wll operate between these two condtons, as shown by letter B n Fgure 1.4, where postve bas appled and negatve current flows, resultng n negatve power consumpton, whch also means power s generated by the solar cell under llumnatons. Fgure 1.4 Typcal IV characterstcs of solar cells. (Courtesy of Dr. Schroder) The operatng regme of solar cells s the range of bas, from 0 to V oc, n whch the cell generates power. The power reaches a maxmum at the maxmum power pont, as marked P MAX n Fgure 1.5. Ths occurs at a voltage V mp wth a correspondng current densty J mp, also shown below. 4

16 Fgure 1.5 Smulated IV characterstcs of solar cells. The effcency,, one of the most mportant propertes of solar cells, s the power densty generated at the maxmum power pont as a fracton of the ncdent solar rradance power densty, P s, PMAX P s J V mp mp P s (1.1) The effcency s related to J sc and V oc, P MAX P s JscVoc FF P s (1.2) where FF s the fll factor, whch descrbes the squareness of the J-V characterstcs, s defned as the rato, FF J V mp mp J V sc oc (1.3) 5

17 These four quanttes: J sc, V oc, FF and are the key performance characterstcs of a solar cell. Typcal numbers of these PV cell characterstcs are shown below from Green[3]. Table 1.1 Performance of common PV cells. Cell Type Voc (V) Jsc (ma/cm2) FF Effcency (%) crystallne S thn flm GaAs CIGS CdTe CdS/CdTe Solar Cells Fgure 1.6 Detaled balance for AM0 and AM1.5. As the most commercally successful thn flm solar cell, cadmum tellurde 6

18 (CdTe) has a market share of around 8% n the PV ndustry; ths exceeds all other nonslcon solar cells. Research n CdTe dates back to the 1950s, after whch the 1.5 ev bandgap of CdTe was found to be almost perfectly matched to the solar spectrum n terms of optmal converson to electrcty[4]. Fgure 1.6 above shows the theoretcal maxmum effcency one can get for dfferent bandgaps. Another advantage of CdTe solar cells s the short optcal absorpton lengths. Two mcrometer thck CdTe s able to absorb 99 percent of photons under AM1.5G solar llumnaton, whle hundreds of mcrons of slcon s requred. Due to the poor qualty of n-typed dopng of CdTe, a smple heterojuncton desgn evolved n early 1960s n whch p-type CdTe was matched wth n-type cadmum sulfde (CdS) as wndow layer. A thn CdS layer (less than a mcron) developed n the 1990s by Chu[5, 6] and Brtt[7] n order to allow more photons passng through, resulted n 15% effcency, a great success n terms of commercal potental. A transparent conductng oxde (TCO) layer was added to CdTe solar cells, to facltate the movement of currents across the top of the cell as the cells were beng scaled up n sze for large area products called modules. In ths smulator, tn oxde, the most popular TCO materal, was employed. And we arrved at the standard confguraton of CdTe solar cells, as depcted n Fgure 1.7. Many mprovements have been developed durng the last two decades for hgher converson effcency of CdTe solar cells, such as better juncton qualty, longer carrer lfetme and new bufferng layers n laboratores; these mprovements have resulted n 18.3% converson effcency achevement by GE Global Research and NREL s 7

19 confrmaton n 2012[8]. In the commercal productons, average module effcency of 11.7% has been clamed by Frst Solar[9]. Fgure 1.7 Typcal confguraton of a CdTe solar cell 1.4. Semconductor Devce Smulatons Semconductor devce smulatons provde an understandng of actual operatons of sold state devces, wth the necessary level of sophstcaton to capture the essental physcs whle at the same tme mnmzng the computatonal burden so that the results can be obtaned wthn a reasonable tme frame Importance of Smulaton Due to ncreasng costs for R&D and producton facltes wth shorter process 8

20 technology lfe cycles, devce smulaton tools have been developed tremendously wthn the semconductor ndustry. Devce modelng offers many advantages such as: provdng problem dagnostcs, provdng full-feld n-depth understandng, provdng nsght nto extremely complex product sets where no drect characterzaton can be conducted, evaluatng what-f scenaros rapdly, decreasng desgn cycle tme and decreasng tme to market. Smulatons requre enormous techncal depth and expertse not only n smulaton technques and tools but also n the felds of physcs and chemstry. Laboratory nfrastructure and expermental expertse are essental for both model verfcaton and nput parameter evaluaton n order to ensure truly effectve and predctve smulatons. The developer of smulaton tools needs to be closely ted to the development actvtes n the research, the laboratores and commercal productons n ndustry General Devce Smulaton Framework Fgure 1.8 Schematc descrpton of the devce smulaton sequence (Courtesy of Dr. Vasleska & Dr. Goodnck) 9

21 Fgure 1.8 shows the man components of semconductor devce smulatons at any level. It begns wth the electronc propertes of sold state materals. The two man kernels, transport equatons that govern charge flow and electromagnetc felds that drve charge flow, must be solved self-consstently and smultaneously wth one and another, due to ther strong couplng. The soluton of transport equatons, carrer dstrbuton, can be used to evaluatng the electromagnetc felds by solvng Posson s equaton n the quas-statc approxmaton. Electrc feld profles are essental to obtan current and carrer densty profles from the transport equatons. Although advanced models such as hydrodynamc equatons, Monte Carlo method and Green s Functon method have been developed, drft dffuson equatons were employed for the transport equatons n ths project, due to ts smplcty for mplementaton, ts relatvely small computatonal burden and ts accuracy for devces larger than 0.5 mcrons. Implementaton of the Posson s equaton, drft dffuson equatons and ther soluton technques wll be dscussed n the next chapter. We wll ntroduce other physcal models, such as generaton/recombnaton mechansms and metal contacts n Chapter 3. In Chapter 4, our smulator wll be compared wth the results from commercal software such as Atlas, Slvaco. Interestng results, especally the transent behavors of CdTe solar cells, wll also be analyzed n Chapter 4. Transent smulator s mplemented because of two reasons: (a) to get more accurate steady-state results wth regard to the current conservaton; and (b) to study true transents n the devce that allow one to extract mnorty carrer lfetmes, etc. Fnally, ths dssertaton work was performed to provde better and more flexble solver from what currently exsts n the academa and ndustry n terms of TCAD. 10

22 Chapter 2 NUMERICAL METHODS In ths chapter, the dscretzed form of Posson s equaton and contnuty equatons wll be derved for heterojunctons Posson Equaton Posson s equaton descrbes the relatonshp between electron charge and the electrostatc potentals[10]: ( ( )) q( p n N N ) (2.1) D A where s the spatally varyng electrostatc potental, s the permttvty, q s the fundamental charge, p s the hole densty, n s electron densty, N + D s the onzed donor - concentraton and N A s the actvated acceptor concentraton. In ths smulator, nstead of Boltzmann s statstcs, Ferm-Drac statstcs are consdered and the followng equatons are employed: n p E E kt F C NCF1/2 E E kt V F NV F1/2 (2.2) where N C and N V are the conducton and valence band effectve densty of states, E F s the Ferm energy level, E C s the conducton band energy level, E V s the valence band energy level, k s the Boltzmann s constant, T s the lattce temperature and kt s the thermal energy n the system. Instead of solvng the Ferm ntegral, a smple analytcal approxmaton was employed to estmate the ntegral of the Ferm-Drac dstrbuton functon[11]. 11

23 Normalzaton of the Posson s Equaton Under equlbrum condtons, wth all parameters as gven above, 1D Posson s equaton can be rewrtten as equaton 2.3: d d ( ) q( p n ND N A) dx dx (2.3) Now consder Boltzmann statstcs here, n and p can be defned by the equatons 2.4, n n pn exp( ) V T exp( ) V T (2.4) where n s the ntrnsc carrer densty. Assumng that = + δ, applyng e ±δ = 1 ± δ when δ s small, substtutng ths n equaton 2.4 and usng ( N D + - N A - ) / n = C, equaton 2.5 reads, d d V ( ) (e T V e T V ) (e T V qn C qn e T ) dx dx (2.5) Substtutng δ = φ new -φ old, we get new d d VT VT new VT VT VT VT ( ) qn (e e ) qn (e e C) qn (e e ) dx dx old (2.6) Rewrtng n terms of n and p gves, new d d new ( ) q( p n) q( p n Cn ) q( p n) dx dx old (2.7) Changng permttvty to the relatve permttvty, normalzng x wth L D, wth V T, p and 12

24 n wth n, we wll have the normalzed form[12] of the Posson s Equaton: new d d new ( r ) ( p n) ( p n C) ( p n) dx dx old (2.8) Although assumpton of Boltzmann statstcs has been made, the normalzed Posson s equaton s sutable for Ferm-Drac statstcs[13] Dscretzaton of the Posson s Equaton Usng Selberherr s central dfference scheme[14], dφ/dx, d 2 φ/dx 2, dε r /dx can wrtten as followng equaton, d 1/2 1/2 dx 1 ( dx 1 dx ) 2 2 d dx 1 1 ( dx dx ) dx ( dx dx ) dx 2 2 d r 12 1 dx dx dx (2.9) Expandng Equaton 2.8 wth Equaton 2.9, we get, a b c f new new new a dx dx 1 1 ( ) / ( 1) dx dx 1 dx 1 2 b p n dx 1dx1 2 c dx dx 1 1 ( ) / ( 1) dx dx 1 dx old f n p C ( n p ) (2.10) where represents the number of the grd pont. 13

25 The fnte dfference dscretzaton of the one-dmensonal Posson s equaton leads to the coeffcent matrx as the follows, wth Drchlet Boundary condtons (Ohmc Contact n ths case) appled, new f1 new a2 b2 c2 2 f2 a3 b3 c 3 a n2 bn2 cn 1 new an 1 bn 1 cn 1 1 f n n new n fn (2.11) The soluton technque of ths coeffcent matrx wll be dscussed n secton Dscretzaton of Contnuty Equatons To dscretze the contnuty equatons, the determnaton of the currents on the md-pont of each neghborng grd ponts s requred. Snce all data are accessble only at the grd nodes, nterpolaton schemes must be employed. In consstency wth Posson s equaton we dscussed, t s a common assumpton that the potental vares lnearly between connectng ponts; ths s based on another assumpton that constant feld s observed between neghborng nodes. In addton, nterpolaton of carrer denstes at the md-ponts s also necessary to calculate the current. One smple way to evaluate the carrer densty s to take the arthmetc average between two nodes under the assumpton of lnear varaton of carrer denstes. However, the exponental relatonshps between carrer densty and electrostatc potentals make the lnear varaton vald only at small potental dfference and near zero electrc feld between nodes, whch s not acceptable 14

26 for non-equlbrum devce smulaton. An optonal approach s provded by Scharfetter and Gummel to solve ths problem[15] wth the acceptable lnear potental varaton between neghborng mesh nodes. Consder the one-dmensonal electron current contnuty equaton: n 1 Jn U t q (2.12) whch, by usng the half-pont dfference gves: J J t q ( dx dx ) / 2 n n n 1 1/2 1/2 1 U (2.13) where Drft Dffuson model s beng employed and we have, J qn E qd n n n 1/2 1/2 1/2 1/2 dn dx (2.14) where s the carrer moblty, E s electrc feld strength and D s the dffuson coeffcents. Equaton 2.14 can be wrtten as: n n 1/2 J 1/2 ne 1/2 n n 1/2 1/2 dn dx D qd (2.15) Knowng that D=V T and E +1/2 =( +1 - )/dx, one arrves at, dn 1 J n dx V dx qd n 1 1/2 n T 1/2 (2.16) nterpreted by the followng equaton, dn dn d 1 dn dx d dx dx d (2.17) Equaton 2.17 s next summarzed as: 15

27 dn n dx J d V qd n 1/2 n T 1 1/2 (2.18) Usng Laplace transformaton, we get: 1 n n 1 g n g ( ) (2.19) where g() s known as the growth functon. Therefore, the Drft-Dffuson model electron current can be wrtten as: n n qd 1/2 1 1 J 1/2 ( n 1B n B ) dx VT VT (2.20) Smlarly, n n qd 1/2 1 1 J 1/2 ( n B n 1B ) dx 1 VT VT (2.21) where B s the Bernoull functon, x, x x1 x, x1 x x x 2 e 1 x 1, x2 x x3 Bx ( ) 2 x xe, x x x x 1 e x xe, x4 x x5 0, x x5 3 4 (2.22) Substtutng Equaton 2.20 and 2.21 nto Equaton 2.14, wth mplct Euler method appled for the tme dscretzaton [16], gves the dscretzed electron contnuty equaton: 16

28 an n bn n cn n fn new new new 1 1 n 2D 1/2 1 an B( ) dx ( dx dx ) V 1 1 T n n 2 D 1/2 1 D 1/2 1 1 bn ( B( ) B( )) dx dx dx V dx V t 1 1 T T n 2D 1/2 1 cn B( ) dx ( dx dx ) V fn U 1 T n t old (2.23) where U s the net generaton rate of carrers, Δt s the tme step nterval, and 2 2 n n n n 1/2 T 1/2 T 1 D V V n n n n 1/2 T 1/2 T 1 D V V (2.24) Smlarly, dscretzed contnuty equaton of hole s derved to gve ap p bp p cp p fp ap new new new 1 1 p 2D 1/2 1 B( ) dx ( dx dx ) V 1 1 T p p 2 D 1/2 1 D 1/2 1 1 bp ( B( ) B( )) dx dx dx V dx V t cp fp 1 1 T T p 2D 1/2 1 B( ) dx ( dx dx ) V U 1 T p t old (2.25) The calculaton of tme step nterval wll be dscussed n secton 2.5, whle the physcal models mplemented for the net generaton rate wll be evaluated n secton 3.1. Equatons 2.23 and 2.25, wth approprate boundary condtons appled, whch wll be 17

29 dscussed n secton 3.2, can be solved teratvely[17] as descrbed n secton Incorporaton wth Heterojunctons We have already dscussed the dscretzaton of Posson s equaton and contnuty equaton for homojunctons n secton 2.1 and 2.2. For heterostructures, modfcatons must be made to the electrostatc potental n order to adjust band offsets between dfferent materals. Otherwse, abrupt electrostatc potental leads to dvergence n the teratve solver. In ths project, Band Parameter Approach[18] s employed to take nto account the dfferent band parameters, ncludng bandgaps and electron affntes, to ensure ϕ vares contnuously along the heterojunctons. Fgure 2.1 Equlbrum energy band dagram of an abrupt heterostructure. 18

30 Wth the ad of Fgure 2.1, whch shows us a band dagram of heterostructure comprsng of two materals, we can have a better understandng of ths band parameter approach. Let s begn wth the conducton band and the valence band, E q q C 0 E q q E V 0 C G C (2.26) As stated before, the carrer concentraton should be wrtten as: n N exp ( E E ) / K T C Fn C B p NV exp ( EV EFp ) / KBT (2.27) Equaton 2.27 can also be expressed as: q C n NC exp n 0 KBT q q C EG p NV exp 0 p KBT q q (2.28) Also we have, N N C V N C n2 exp ln n 2 N V n2 exp ln n 2 (2.29) substtutng Equaton 2.29 nto 2.28, q C KT B N n n2exp n 0 ln K BT q q n q E KT B N p n2exp 0 p ln K BT q q q n C 2 C G V 2 (2.30) Next we choose materal 2, whch most lkely wll be CdTe n CdS/CdTe solar cells, as 19

31 the reference pont, hence we could assume the band parameters to be zero for ths materal: C KT B NC n 0 ln 0 q q n 2 C EG KT B NV p 0 ln 0 q q q n 2 (2.31) ψ 0 can be solved as: 0 0 K T N E K T N ln ln q q n q q q n C 2 B C 2 C 2 G B V K T N E K T N ln ln q q n q q q n C 2 B C 2 C 2 G B V (2.32) where subscrpt 2, refers to materal 2. And the band parameter of materal 1 can now be wrtten as: n KT N ln q q N C C 2 B C C 2 E E KT N p ln q q q N C 2 C g2 g B V V 2 (2.33) Equaton 2.33 gves the band parameters consstent wth the Boltzmann statstcs. The band parameters for Ferm Drac Statstcs had been derved by Lundstrom[19] as the followng: kt kt kt NV x f1/2 V p ( x r ) EG x EGr log V q NVr q e kt NC x f1/2 C n ( x r) log log C q NCr q e (2.34) 20

32 2.3.1 Posson s Equaton for Heterojunctons We already dscussed n secton 2.1 the dscretzed Posson s equaton for homojunctons wth permttvty vared. For complete heteroscturctures, except band parameters for adjustng quas Ferm level and carrer densty, addtonal work s requred to make the electrostatc potental to denote the relatve energy level. Another band parameter, ntrnsc level offset, whch represents the dfference between ntrnsc levels of dfferent materals, has been employed as below: offsets C2 1 E C g2 Eg (2.35) q 2 q By deductng these offsets before evaluatng Equaton 2.12, ϕ can be vared smoothly n an abrupt heterojuncton. Also we need to restore the real electrostatc potentals by addng these offsets back when convergence s acheved for band structures plottng Contnuty Equatons for Heterojunctons In secton 2.2 we have derved the dscretzed contnuty equaton for nonunform meshed homojunctons. Smlar procedures wll be conducted for the heterojuncton equatons, wth appearance of new terms. Recall that the Drft Dffuson current equaton s of the form, dn (2.36) Jn qnne qdn dx Where the electrc feld strength, E, should be based on the Ferm level, as below: defn dec KT B 1 dn dnc dx dx q n dx dx (2.37) 21

33 Equaton 2.37 can be expanded as, defn dc KBT dn KBT dnc qe (2.38) dx dx n dx N dx C Combnng Equaton 2.36 and 2.38, the tradton drft dffuson terms and the effect of spatally vared electron affnty and densty of states can be both reserved n Equaton 2.39: d dn Jn qnn n KBT n dx dx d dp J p qn p p KBT p dx dx (2.39) Smlarly as descrbed n secton 2.2, employng the Scharfetter and Gummel scheme after nsertng Equaton 2.39 nto Contnuty Equatons, we obtan Equaton 2.40 and Note that ϕ n =ϕ+θ n for electrons. an n bn n cn n fn new new new 1 1 n n n 2D 1/2 1 an B( ) dx ( dx dx ) V 1 1 T n n n n n n 2 D 1/2 1 D 1/2 1 1 bn ( B( ) B( )) dx dx dx V dx V t 1 1 T T n n n 2D 1/2 1 cn B( ) dx ( dx dx ) V fn U 1 T old n t (2.40) And ϕ p =ϕ-θ p for holes, 22

34 ap p bp p cp p fp ap new new new 1 1 p p p 2D 1/2 1 B( ) dx ( dx dx ) V 1 1 T p p p p p p 2 D 1/2 1 D 1/2 1 1 bp ( B( ) B( )) dx dx dx V dx V t cp fp 1 1 T T p p p 2D 1/2 1 B( ) dx ( dx dx ) V U 1 T old p t (2.41) 2.4. Non-Unform Mesh Less grd ponts would accelerate the smulatons, smply by lowerng the number of calculatons. It also means less accuracy, especally n heterojunctons, where crucal electrostatc potental and electrc feld exst. Non-unform mesh s employed n ths solver to relax the grd spacng where low electrc feld s observed and to retan dense meshng at junctons for hgh accuracy. In ths secton, the generaton mechansm of nonunform mesh wll be ntroduced. Intally a unform mesh (mesh1) based on the Debye length crteron s generated and the equlbrum potental and electrc feld profles (efeld1) are solved teratvely, as wll be dscussed n secton 2.7. Then, the mesh refnement (generaton of non-unform mesh) was done based on the unform mesh electrc feld profle under equlbrum condton. The meshng factor was calculated usng: mesh2 fhlog ( efeld1 1) fl (2.42) 10 where efeld1 s the unform mesh electrc feld under equlbrum n unt of V/m, fh s the 23

35 meshng coeffcent for hgh feld part (whch also represents depleton regon), and fl s the tunng factor for low feld regon. As shown below (Fgure 2.2), larger fh gves hgher meshng factor for hgh electrc feld whle smaller fl gves lower factor for low feld. In order to avod negatve numbers n takng the log functon, absolute value of the electrc feld plus one s employed. Fgure 2.2 Meshng factor vs. Electrc feld. Once the meshng factor profle s evaluated, the grd spacng s determned by local Debye Length and the meshng factor together, dx L / mesh2 (2.43) D It s clear from the results presented n Fgure 2.2 that the factors are hgher than 1 for hgh electrc feld, whch wll guarantee the accuracy of ths meshng strategy by meetng the general meshng crtera of grd spacng smaller than the Debye Length[20]. 24

36 Snce abrupt grd spacng would ntroduce spkes on the electrostatc potental profles, keepng dx contnuous s mportant for accurate smulatons. The smple approach s to use one sngle Debye length for the entre meshng, whch s convenent but not effcent. Employng the smallest Debye lengths from dfferent materals leads to huge number of grd ponts, whle usng the largest wll cause napproprately coarse mesh and poor accuracy. Fgure 2.3 Grd spacng for a typcal SnO/CdS/CdTe/ZnTe solar cell. A more complcated approach s beng mplemented n ths solver to ensure dx vares smoothly at the junctons. We determned the local Debye length on ts relatve poston and the L D of neghborng layer materals. As n the frst half of CdTe layer n a typcal SnO/CdS/CdTe/ZnTe solar cell, the local Debye length was evaluated by ts own L D, L D of neghborng CdS layer and the relatve poston towards the CdS/CdTe heterojuncton. Whle n the second half, the local Debye length should be calculated by 25

37 the Debye lengths of CdTe, neghborng ZnTe and the relatve poston towards the CdTe/ZnTe juncton. Smlar procedure was appled for the entre devce, resultng n a meshng as depcted n the Fgure 2.3 above, where comparson between electrc feld strength and grd spacng has been made. The smooth varaton of dx has also been observed clearly Automatc Tme Step Control Smlarly to the dea of non-unform mesh, automatc tme step control was developed for ths solver, n order to accelerate the transent smulatons. The general mechansm s to make tme steps larger when the current changes wth respect to tme are smaller and to make tme steps smaller when current changes sgnfcantly. To ensure that the solver converges, we ntroduced dt1 and dt3 as convergence protecton n ths smulator. The tme step used at the begnnng and end of a pulse sgnal, n our case dt1, should be small so that the solver could converge. If the tme step s too small, extra useless computatons wll be conducted. Thus dt1 must be determned by the pulse wdth and lmted by 10ps; the result s a sem-emprcal number whch can assure convergence for most cases: (2.44) 11 dt1 10 pulse _ wdth Fgure below shows the dt1 for varety of pulse wdths from 1us to 1ps. It s clear that f we make dt1 as the constant tme step n ths solver, 100,000 teratons wll be conducted for a 1us pulse, regardless of an even longer tme perod of current or 26

38 Photocurrent decay. Fgure 2.4 dt1 vs. pulse wdths. As mentoned above, dt1 wll be used at the begnnng and end of pulse sgnals where sgnfcant current changes can be predcted. Thus dt3, the tmng of the actvaton of automatc tme step control, s crucal to ths solver. Smlarly to dt1, dt3 was evaluated by one tenth of the pulse wdths and a sem-emprcal 10ns tme nterval wthn whch 90% of current changes can be fnshed, as the followng equaton descrbes: (2.45) 8 dt3 10 pulse _ wdth As tme s passng through dt3, automatc tme step control s actvated to accelerate the smulatons. The same method s used n the evaluaton of the automatc tme step, Δt: 27

39 1 1 dj ( t) t 10 dt (2.46) where 1ns s the tme step used n steady states smulatons for fast convergence, and the second term descrbes the current change wth respect to tme. Fgure 2.5 below shows how the automatc tme step vares for current changes orders of magntude. Fgure 2.5 Automatc tme step Δt vs. current changes Numercal Soluton Technques In ths secton, LU Decomposton method, as the soluton technque of dscretzed dfferental equatons such as the dscretzed Posson s equaton and contnuty equatons, wll be dscussed. The mplementaton of Gummel s scheme wll also be ntroduced. 28

40 2.6.1 LU Decomposton Method As derved above n secton 2.2 and 2.3, both Posson s equaton and contnuty equatons can be wrtten n matrx form as, [ A][ x] [ F] (2.47) where A s the coeffcents matrx, x s the soluton and F s the forcng functon. Gauss Elmnaton Method[21] can be employed to solve ths matrx equaton. However t has the dsadvantage that all rght-hand sdes, the forcng functon n ths case, must be known n advance for the elmnaton steps to proceed. The LU Decomposton Method[22] has the property that the matrx decomposton step can be performed ndependent of the forcng functons. Square matrx equatons as n Equaton 2.12 and 2.48, can be solved by breakng the trdagonal square coeffcent matrx [A] nto lower trangular and upper trangular matrces, usually named as [L] and [U] matrces, [ A] [ L][ U] (2.48) And the orgnal Equaton 2.48 becomes, [ L][ U][ x] [ F] (2.49) Equaton 2.50 can be further broken nto two problems, [ L][ y] [ F] [ U ][ x] [ y] (2.50) [y] can be solved wth a smple forward substtuton step at frst and then [x] can be evaluated by a backward substtuton algorthm easly. Recall the dscretzed Matrx form of Posson s equaton, 29

41 b c a b c a b c 0 new new f f an 2 bn 2 c n1 new an 1 bn 1 cn 1 f n1 n1 new an b n f n n (2.51) Decompose the coeffcent matrx nto a product of lower and upper trangular matrces: [ A] [ ][ ] c c c L U n n (2.52) where, b 1 1 a / k k k 1 b c k k k k 1 k 2,3,..., n (2.53) Now we can have the [L][y]=[F] as followng, L y F 1 y1 f y2 f2 0 n 1 1 n 1 y n f n (2.54) Usng forward substtuton, solutons can be easly obtaned: 30

42 y f 1 1 y f y 1 2,3,..., n (2.55) Then we have the matrx of [U][ϕ]=[y] as, U y 1 c1 1 y1 2 c2 0 y2 0 n 1 cn 1 n1 n n y n (2.56) Smlarly to the soluton of the lower trangular matrx, [U] can be solved by backward substtuton, yn n y n c 1 (2.57) Thus we have the soluton for [ϕ] at each grd node usng ths method. The carrer densty of electrons and holes that appear n the forcng functon [f] shall be updated mmedately. Accurate soluton s acheved wthn several teratons Gummel s Iteraton Method Gummel s method solves the coupled set of carrer contnuty equatons together wth the Posson s equaton va a decoupled procedure. The potental profle obtaned from equlbrum smulatons s substtuted nto the contnuty equatons (Equaton 2.41 and 2.42), for carrers dstrbuton profle calculaton. The result s then sent back nto 31

43 Posson s equaton (Equaton 2.11) to update the forcng functon and the central coeffcents for new electrostatc energy profles. Ths process s repeated untl convergence requrement s acheved, as shown n Fgure Fgure 2.6 Gummel s teraton scheme. Chapter 3 PHYSICAL MODELS In ths chapter, dfferent knds of physcal models that we mplemented n ths 32

44 smulator, such as generaton/recombnaton mechansms, Schottky contact and partally onzed dopants wll be dscussed Generaton and Recombnaton Mechansms Generaton/recombnaton events take place when the devce s under the nfluence of bas or llumnaton. They determne the performance and characterstcs of devces. The smplest classfcaton of generaton and recombnaton mechansms starts from the number of partcles nvolved n the process, as shown n Fgure 3.1. Two partcle Energy-level consderaton One step (Drect) Two-step (ndrect) Photogeneraton Radatve recombnaton Drect thermal generaton Drect thermal recombnaton Shockley-Read-Hall (SRH) generaton-recombnaton Surface generatonrecombnaton Three partcle Impact onzaton Auger Pure generaton process Electron emsson Hole emsson Electron capture Hole capture Fgure 3.1 Classfcaton of the generaton/recombnaton process. In ths secton the numercal expresson for U, the net recombnaton rate n the contnuty equatons, wll be evaluated. Shockley Read - Hall recombnaton, as well as trap-assted recombnaton, radatve (band-to-band) recombnaton, surface recombnaton and optcal generaton wll be ntroduced. Auger recombnaton has not been mplemented snce t does not domnate the bulk recombnaton mechansms n 33

45 CdTe materal Shockley Read Hall Recombnaton The Shockley Read Hall (SRH) model was frst ntroduced n 1952 by Shockley, Read[23] and Hall[24] to descrbe the statstcs of recombnaton and generaton of carrers n semconductors occurrng through traps, whch exst n every semconductors. These trap levels, known as recombnaton-generaton centers, lay wthn the forbdden band. Trap levels are caused by crystal lattce mperfecton such as doped mpurtes and vacances; they facltate the recombnaton of carrers, snce the jump can be splt nto two parts, requrng lower energy, as llustrated below. Fgure 3.2 Graphcal descrptons of the SRH recombnaton. (Courtesy of Dr. Schroder) These mechansms consst of: (a) electron capture (a free electron moves from the 34

46 conducton band to an unoccuped trap level), (b) electron emsson (a trapped electron jumps to the conducton band), (c) hole capture (a hole recombnes wth an electron trapped n the bandgap and (d) hole emsson (a trapped hole jumps to the valence band). Physcal models for these processes nvolve equatons for electron densty n the conducton band, holes n the valence band, ther capture probabltes, trapped carrer densty, and relatve emsson rates. The conventonal SRH net recombnaton rate s gven by: R SRH 2 pn n Et E Et E p( n n exp( )) n( p n exp( )) kt kt (3.1) where, E t s the energy level of traps, E s the ntrnsc Ferm level, n and p are the mnorty carrer lfetmes whch are heavly dependent on the densty of trap centers, σ s the capture cross secton for dfferent type of carrers and V th, s the thermal velocty, 1 n V N n th T 1 p V N p th T (3.2) A smple model could be employed wthout the consderaton of trap energy levels and the occupatons of trap states when E t =E : R SRH 2 pn n ( n n ) ( p n ) p n (3.3) Optcal Generaton 35

47 Carrers can be generated n semconductors by llumnaton wth lght; ths process s called photo-generaton. An ncomng photon wth suffcent energy can excte electrons from valence band nto the conducton band. Optcal absorpton can be a drect or ndrect process, dependng on the band structure of the semconductor. Wth ndrect bangdap materals, as Slcon, addtonal phonons are requred to conserve the momentum n the process of carrer generaton, as shown n the rght panel of Fgure 3.3, whle phonons do not play bg roles n the absorpton of drect bandgap materals, such as GaAs, Germanum and CdTe, as llustrated n the left panel of Fgure 3.3. For ths reason, the band edge absorpton coeffcent for drect bandgap semconductors s sgnfcantly larger than that of ndrect gap materals. E E Vrtual states Phonon emsson E c E g Phonon absorpton E g E V Drect band-gap SCs k Indrect band-gap SCs k p E f f p E E ph p E f f p E p s E s E ph fnal ntal photon fnal ntal phonon Fgure 3.3 Photon absorptons n drect bangdap and ndrect gap semconductors. Usually, the suffcent energy to excte electrons must be the bandgap energy, E g. Due to the exstence of Urbach tal[25], photons wth energy less than bandgap can be 36

48 absorbed, as the large absorpton coeffcents below bandgap depcted n Fgure 3.4. The generaton was determned by the absorpton coeffcents of the materals[26]: (1 R( E)) ( E) G( E, x) Popt ( E)exp( ( E) x) EA (3.4) where E s the ncomng photon energy, α(e) s the absorpton coeffcent of the materal at the ncomng photon energy, R(E) s the reflecton rate of certan photon energy at the front surface, A s the area of the llumnaton, P opt (E) s the lght ntensty for photons at the front surface. By numercal ntegrals over photon energy, we can determne the total generaton rate, G(x). Fgure 3.4 Absorpton coeffcents SnO, CdS and CdTe Band-to-Band Recombnaton 37

49 Contrary to optcal generaton, Band-to-band recombnaton annhlates electronhole pars, wth photons generated at the bandgap energy[27] ths s how LEDs and semconductor lasers operate. It s also the major mechansm for Photolumnescence decay. Snce both electrons and holes are requred n ths process, the recombnaton rate s proportonal to the excess carrer densty, and can be expressed as, U b np n (3.5) 2 bb rad ( ) where b rad s the bmolecular recombnaton constant. Snce these generated photons have energy near the bandgap, t s possble to reabsorb these photons before they ext the solar cell. A well desgned drect bandgap photovoltac solar cell can take advantage of ths photon recyclng and ncrease the carrer lfetmes[28]. Wth the dervaton of radatve recombnaton, the net generaton rates term n contnuty equatons, can be fnally expressed as, U R SRH U bb G opt (3.6) Surface Recombnaton In real devces, defects are much more lkely to stay at the nterface between dfferent crystal lattces, for example, we mght have broken bonds at semconductormetal contacts. In such cases, the trap states are constraned onto a 2D surface rather than 3D bulk. It s also much more meanngful to express these r g centers n terms of densty per unt area than per unt volume. Unlke SRH recombnaton and optcal generaton, the relevant quantty that determnes recombnaton velocty should be flux, nstead of a volume recombnaton rate. 38

50 Assume a surface contans Ns traps per unt area, then wthn an nfnte thn layer δx around the surface, the recombnaton flux should be U x s 2 ns ps n 1 1 ( ps pt ) ( ns n t ) S S n p (3.7) per unt area, where n s, n p are the carrer denstes at the surface, S n, S p are the surface recombnaton velocty n unt of meter per second, defned as, S S B N n n s B N p p s (3.8) In p-type CdTe materal, Equaton 3.6 can be reduced to U x S ( n n ) (3.9) s n n 0 where n 0 s the mnorty carrer concentraton, electron densty n ths case, under equlbrum. Ths leakage of mnorty carrers to the surface results n surface recombnaton current, derved as, J ( x ) qs ( n n ) (3.10) n s n n 0 Substtute Equaton 3.9 nto the electrons contnuty equaton, n n n n n n qd 1/2 1 1 J 1/2 ( n B n 1B ) dx 1 VT VT (3.11) to arrve at n n n n n D 1/2 1 1 Sn( n n 0) ( nb n 1B ) (3.12) dx 1 VT VT 39

51 Whch can be wrtten as, D B n D B dx n dx n n n n n 1 1 1/2 n 1 [ 1/2 Sn 1] Sn 1n0 VT VT (3.13) The coeffcents n Equaton 3.13 should be the boundary condtons of the dscretzed electron contnuty equaton, of the form an D n n n 1 1/2B VT n n n 1 [ 1/2 Sn 1] VT 1 0 bn D B dx cn 0 fn S dx n n (3.14) where denotes the surface node n the meshng. Smlar dervaton can be made for the hole contnuty equaton. It s mportant to menton that ths mechansm must be combned wth a Schottky contact, snce a partcular value of surface recombnaton velocty results n excess mnorty carrers at the contact; ths contradcts wth the Ohmc contact model that assumes no excess mnorty carrers exsts at the boundary Ohmc and Schotky Contact Many of useful propertes of p-n junctons can be acheved by formng dfferent metal-semconductor contacts[29]. The major dfference between ohmc and Schottky contact s the Schottky barrer heght, B, s non-postve or postve. For Ohmc contacts, the barrer heght should be near zero or negatve, formng accumulaton type contacts, thus the majorty carrers are free to flow out the semconductors, as shown below n 40

52 Fgure 3.5. Whle for Schottky contacts, on the contrary, the barrer heght would be postve, formng depleton type contacts, so that the majorty carrers cannot be absorbed freely due to the band bendng caused by postve barrer heght. Hence, the way we mplemented them n our smulator s dfferent. Fgure 3.5 Accumulaton type Ohmc contact Modelng of Ohmc Contact Although the barrer heght could be negatve for an Ohmc contact, we can treat them smply as flat band, wth the carrer concentraton and electrostatc potental under equlbrum. For the dscretzed Posson s Equaton, we appled the Drchlet Boundary Condtons[30], that the coeffcent matrx elements at boundary are fxed as the equlbrum results, durng the teratve calculatons, as the followng Equaton 3.15, where 1 denotes the frst grd nodes and nmax represents the last grd pont. 41

53 a 0 1 b 1 1 nmax nmax 1 nmax a b f 0 1 c 0 c 0 f 1 1 nmax nmax (3.15) The carrer concentratons at the contact have been fxed at the equlbrum value for the dscretzed contnuty equaton, preventng any excess mnorty carrers exstence. The followng s an example of boundary condtons for the electron contnuty equaton of a p-n dode, an 0 an 0 1 nmax bn 1 bn 1 1 nmax cn 0 cn 0 1 nmax fn n fn n 1 p0 nmax n0 (3.16) where n p0 s the equlbrum electron concentraton (mnorty carrer dentsy) at p-type materal and n n0 s the equlbrum majorty concentraton n the n sde. Smlar boundary condtons can be defned for holes. ap 0 ap 0 1 nmax bp 1 bp 1 1 nmax cp 0 cp 0 1 nmax fp p fp p 1 p0 nmax n0 (3.17) Modelng of Schottky Contact Due to the self-compensaton mechansm[31], CdTe s usually lghtly doped n solar cells. Thus a Schottky must be formed due to the dfference n workng functons of metal contact and CdTe. A Schottky contact, smlar to an Ohmc contact, s a Drchlet 42

54 boundary contton n the dscretzed Posson s equaton. The electrostatc potental s fxed at a certan value hence the dervaton of the forcng functon s crucal for the Schottky contact model. Let s begn wth the barrer heght, E E (3.18) B F V It can be wrtten as, ( E E ) ( E E ) (3.19) B F V Fgure 3.6 P-doped depleton type Schottky contact. knowng that =E F -E and E -E V =E g /2, Equaton 3.18 arrves, nmax E g B 2 (3.20) Thus the forcng functon at the Schottky contact for Posson s equaton wll be, E f max g n B (3.21) 2 All the other elements n the matrx form of Posson s equaton should retan the same as Ohmc contact case. Whle for non-equlbrum, surface recombnaton can be appled 43

55 together wth Schottky contact Partal Ionzaton of Dopants As one knd of mpurtes, dopants are governed by dstrbuton functon of mpurtes, whch dffers from the Ferm - Drac dstrbuton. Due to the electron spn state, a modfed dstrbuton functon for dopants can be gven by, f f donor acceptor 1 ( ED) EF ED 1 gd exp( ) V 1 ( EA) EA EF 1 g A exp( ) V T T (3.22) Hence we can have the actvated dopants densty, N N D A ND EF ED 1 gd exp( ) V N A EA EF 1 g A exp( ) V T T (3.23) where g D and g A are the degeneracy factor for donors and acceptors, E D and E A are the actual dopants energy level. The partal onzaton of dopants works wth Ferm Drac statstcs n the charge neutralty equatons for the soluton of electrostatc potentals, whch wll be the ntal guess n the Posson equaton solver. For a pece of n-type semconductor, the charge neutralty equaton s, n N 0 (3.24) D 44

56 takng the Ferm Drac dstrbuton of electrons and partal onzaton donors nto consderaton, we get, ND ( EF EC N ) 0 CF1/2 EF ED 1 g exp( ) VT D V T (3.25) Equaton 3.25 can be solved teratvely or graphcally, as shown below. Fgure 3.7 Soluton of wth partal onzed dopants and Ferm-Drac statstcs. A smlar evaluaton of holes and acceptors can be made. It s necessary to update the dopant profle, C n Posson s equaton, wth any new electrostatc potental profles obtaned, both n the equlbrum solver and n non-equlbrum smulators. In ths project, copper was employed as the acceptors n P-type CdTe[32]. And ts actvaton energy level was set to be 0.1 ev above the valence band. 45

57 Chapter 4 SIMULATION RESULTS It s mportant for smulators to have accurate materal parameters to generate approprate results. Much research has been conducted for cadmum tellurde solar cell materals electronc propertes recently. Thus we combned a varety of sources[33-35], and came up wth a set of reasonable numbers for common CdTe solar cells. Table 4.1 below shows the standard devce confguraton and the materal parameters for equlbrum smulatons. Table 4.1 Devce parameters for equlbrum smulatons. Temperature = 300 K SnO2 CdS CdTe Layer thckness (μm) Bandgap (ev) Electron Affnty (ev) Dopng Densty (cm -3 ) N-type: N-type: P-type: Relatve Permttvty Conducton Band DOS (cm -3 ) Valence Band DOS (cm -3 ) Dopants Actvaton Energy (ev) Schottky Barrer Heght (ev) For non-equlbrum smulatons, typcal materal propertes for CdTe solar cells are shown n Table 4.2. In ths project, all parameters used are gven n these two tables, unless mentoned otherwse. 46

58 Table 4.2 Materals propertes for non-equlbrum smulatons. Temperature = 300 K SnO2 CdS CdTe Electron Moblty (cm 2 /Vs) Hole Moblty (cm 2 /Vs) Electron Lfetme (s) Hole Lfetme (s) Radatve Recombnaton Rate (cm 3 /s) Surface Recombnaton Velocty (cm/s) Equlbrum Smulaton Results Fgure 4.1 Equlbrum energy band dagram. By solvng the Posson equaton solely, equlbrum results can be acheved. Shown n Fgures are the energy band dagram, the electrc feld profle and the carrer denstes. The back contact was assumed to be Ohmc, so that flat band s 47

59 observed. All energy levels above are referenced wth respect to the Ferm level, whch equals to zero along the entre devce. Fgure 4.2 Electrc feld profle for unform and non-unform mesh at equlbrum. Fgure 4.3 Carrer dstrbutons at Equlbrum. The number of grd ponts has been reduced from 688 to 74 wth the non-unform 48

60 strategy we developed. We also modeled the equlbrum wth Schottky contact and the results of these smulatons are as shown n Fgure The number of grd ponts ncreased to 79 due to the ncreasng electrc feld near Schottky contact. Both band bendng and depleted majorty carrer concentraton were observed. Fgure 4.4 Equlbrum band dagram wth Schottky contact appled. Fgure 4.5 Electrc feld profles at equlbrum wth Schottky contact appled. 49

61 Fgure 4.6 Carrer dstrbutons at equlbrum wth Schottky contact appled. We could also acheve the accumulaton type Ohmc contact by adjustng the barrer heght to near zero value, as depcted n Fgure 4.7 below. Fgure 4.7 Equlbrum band dagram wth accumulaton type Ohmc contact 50

62 4.2. Steady-State Smulaton Results In ths secton, the current voltage characterstcs of the standard cadmum tellurde solar cell wll be smulated both under dark and under AM1.5G solar spectrum. The results wll also be compared wth Atlas Under dark Fgure 4.8 Comparsons between dark IV-characterstcs. As shown n Fgure 4.8, the flat band Ohmc contact model was equvalent to the accumulatve type Ohmc contact, whle the Schottky contact reduced the current densty at strong bases sgnfcantly[36, 37]. The Schottky barrer also helped the solver at small bas by avodng negatve currents near zero. As llustrated n Fgure 4.9 below, both Ohmc contact models experenced unstable current below 0.3 V forward bas, whch probably s caused by ther more conductng nature. Also, the exponental relatonshp between bas and current s well observed for Schottky contact below threshold and for 51

63 Ohmc contact above 0.3 V. Fgure 4.10 depcted the achevement of current conservaton along the entre devce. Fgure 4.9 Sem log plot of dark IV-characterstcs. Fgure 4.10 Current conservaton along the entre devce. 52

64 4.2.2 Under llumnaton For the SnO/CdS/CdTe standard confguraton wth thckness of 0.1/0.2/3.6 mcron and dopng concentratons of /10 17 / cm -3, the llumnated IV characterstcs are shown below n Fgure Current degradaton caused by depleted Schottky contact s well observed, whch can be explaned by the carrer dstrbuton fgure below. The major performance characterstcs shown n Table 4.3 are consstent wth those from Table 1.1. Fgure 4.11 Illumnated IV characterstcs of CdTe solar cell. Table 4.3 Schottky contact s effect on key performance characterstcs. J sc (ma/cm 2 ) V oc (V) Effcency (%) Fll Factor Ohmc Schottky

65 Fgure 4.12 The dfference n carrer denstes at strong bas. As llustrated above majorty carrer, holes were depleted near the contact, whle large amount of excess mnorty carrers exsted due to the band bendng caused by the Schottky contact. The shft n the maxmum power ponts s depcted n Fgure Fgure 4.13 Power voltage characterstcs of CdTe solar cell 54

66 Fgure 4.14 The effect of CdTe thckness on the solar cells The CdTe layer thckness was changed from 0.5 m to 5 m; results are shown n Fgure All solar cell characterstcs were kept almost unchanged at the thckness of 2 5 m. However, due to the lack of the absorpton of long wavelength photons, both V oc and J sc decreased drastcally below the thckness of 2 m, whch eventually leads to the reducton n effcency. Also, thnner CdTe layer, representng shorter length n the 55

67 drecton of current flow, led to smaller nternal seres resstance of the solar cell, governed by the relatonshp between resstvty and resstance; ths results n flat currents at weak bas, resultng n hgher fll factor for smaller CdTe thckness. Fgure 4.15 IV characterstcs under dfferent temperature. Fgure 4.16 Power voltage characterstcs under dfferent temperature. 56

68 Due to the mplementaton of Ferm Drac statstcs, we were able to produce IV characterstcs under dfferent temperature. The degradaton of the devce performance caused by hgh temperature s well observed, as shown above n Fgure Transent Smulaton Results Step Bas Response The classc step functon current denstes of a p-n juncton are reproduced by ths smulator n ths secton. Due to small ΔV and pulse wdth appled, current overshoot s barely observed n Fgure Fgure 4.17 Current transents for small pulse sgnals. Wth smlar ΔV and larger pulse wdths appled, the turn on characterstcs of p-n dodes are well observed: t only took several nanoseconds to reach 90 percent of the current ncrements, as llustrated n Fgure

69 Fgure 4.18 Current transents for small ΔV and larger pulse wdths. The current overshoot observed for a strong pulse sgnal, as n Fgure 4.19 below, can be explaned by the storage charges. These excess carrers near the juncton wll be swept nto the other sde of the juncton by the strong electrc feld n the depleton regon. Hence a large reverse current wll flow temporarly. Fgure 4.19 Reverse recovery transent observed for turn-off. 58

70 Fgure 4.20 Effects of a step turn-off transent on mnorty carrers n P-type CdTe. Fgure 4.20 shows the excess mnorty carrers drftng back to the n sde of the juncton. As can be seen, the electron concentraton below 0.5 m actually ncreased n the frst 0.5 ns due to the drft n the depleton regon. Fgure 4.21 Effects of a step turn-on transent on mnorty carrers n P-type CdTe As for the turn-on transent, the dffuson process of the mnorty carrers s 59

71 produced as depcted n Fgure Carrers stored n the n sde space charge regon, dffusng nto p sde, caused hgh carrer concentraton below 1 m n the frst half nanosecond. These carrers can be further dffused nto the entre CdTe layer, as shown for 2 ns, n whch case the lower concentraton below 1 m, ndcates the number of njected carrers beng reduced to a normal level, eventually resultng n the electron dstrbuton of 100 ns Photocurrent Transent Smlarly to the step bas response, a varety step functons of llumnaton have been appled to the standard solar cell under short crcut condtons, so that the current decay and carrers transents can be analyzed. Shown below are the chargng and dschargng processes n solar cells due to on and off llumnaton. The natural decay of current has been reproduced n Fgure Fgure 4.22 Photocurrent transents. 60

72 Fgure 4.23 Exponental decay of photocurrents. Fgure 4.24 Current transent under 10 Sun llumnaton wth 30 ns pulse wdth. A ten Sun concentrated AM1.5G spectrum lastng 30 nanoseconds, was tested for a clearer vew on the majorty carrer transents on P-type CdTe. Fgure 4.25 shows the process of holes beng optcally generated and drftng from the depleton regon. The black dash lne represents the generated holes densty wthn the frst 0.01 ns, whch 61

73 matched perfectly wth the carrers dstrbuton at 0.01 ns, except the carrers drfted due to strong electrc feld below 0.4 m. It s clear that the hole concentraton near juncton ncreased to the magntude of the 30ns dstrbuton, whch can be seen as steady state values here, wthn 0.5 ns. Fgure 4.25 Majorty carrers transent near juncton n P-type CdTe. Fgure 4.26 Majorty carrers transent near contact n P-type CdTe. 62

74 Whle at the contact regon, where optcal generaton rate can be neglected, t took at least 10 ns for the carrer densty to get close to the steady-state value, as llustrated n Fgure 4.26; ths can also be seen as the carrers generated at the deleton regon got drftng to the contact, as current flows. Fgure 4.27 Turn-off transent of majorty carrers near juncton n P-type CdTe. Fgure 4.28 Turn-off transent of majorty carrers near contact n P-type CdTe. As for the turn-off transent after 30ns n Fgure 4.24, the reverse processes of the 63

75 turn-on transent were reproduced: the majorty carrers near juncton drfted away from the depleton regon wthn 5 nanoseconds, whle majorty carrers at the contact regon returned to equlbrum value wth at least 30 nanoseconds after the turn-off, as shown above n Fgure 4.27 and For a more clear vew of the drft of holes under llumnaton, a 10 sun llumnaton was appled to the devce for 0.1 ns, resultng n the followng plot. Fgure 4.29 Transent of photolumnescence current for a short lght pulse. The postve current n the frst 0.1 ns depcted n the left panel of the above graph, was due to the mmedate collecton of photo-generated holes at the N-type front contact and carrer separaton caused by the TO/CdS heterojuncton. The negatve current after the turn-off of the llumnaton s the collecton of excess holes at the back P-type contact, whch s usually called lght current at steady state. Due to the confguraton of the solar cell, t may take some tme for the holes to travel through the entre devce and get 64

76 collected at the back contact, thus the decay process of the lght current took hundreds of nanoseconds, as shown n the left panel of Fgure As shown n the excess carrer dstrbuton pcture below, the lght sgnal had been transferred nto electrc sgnal n terms of carrers movement. Photo generaton had been captured well at the frst 0.1 ns, whle the movement of concentraton peaks ndcated the drft of carrers wthn the frst nanosecond after the shutdown of llumnatons. Also the reducton n carrer velocty was observed, as the dsplacement of concentraton peaks shrnks between ns and ns tme nterval, due to the deeper poston n the P-type layer, where electrc feld was sgnfcantly weaker than the juncton area. Fgure 4.30 Transent of excess holes for a short lght pulse. 65

77 Chapter 5 CONCLUSIONS AND FUTURE WORK Ths chapter summarzes the key features of ths thess project and ts results, followed by the plan for future research nto the role of the defects n CdTe solar cells Conclusons To conclude, a drft dffuson model has been developed from scratch to smulate the steady state and transent operaton of CdS/CdTe solar cells. The selfconsstent solutons of potental and carrer dstrbutons are obtaned by solvng the coupled Possons equaton and the contnuty equatons. The confguraton of the solar cell s a SnO/CdS/CdTe heterostructure, wth an n + -n + -p dopng profle. The effect of Schottky contact was observed n both dark current and lght current smulatons. Ths smulator has been tested for solar cells under dark, and compared to the dark current obtaned from other commercal tools wth acceptable dfferences. The converson effcency of the devce changes wth the absorber s thckness due to ts ablty to capture long wavelength photons but the effcency starts decreasng after a crtcal length, due to the loss of uncollected carrers. The capablty of modelng the devce at low temperature has been certfed for temperatures down to 220K. Hgh temperature degradaton effect on the devce performance was also shown clearly va the smulatons presented. The step functon bas turn-on characterstcs and the effects of storage charges on the turn-off transent, usually called current overshoot, has been reproduced by ths solver. The chargng and dschargng processes caused by llumnaton were also smulated. Natural decay of photocurrent has been generated. The mechansms behnd these characterstcs have been analyzed wth the correspondng carrer transents generated by ths very 66

78 smulator Future Work The smulatons presented here have been done on a standard SnO/CdS/CdTe abrupt heterojuncton solar cell, but the code s capable of modelng graded heterojunctons constructed wth other materals. Photon recyclng has not been mplemented n the current verson of code. For complete smulatons of Photolumnescence, the absorpton of regenerated photons wll be mplemented n the next verson of the code. Fgure 5.1 The polycrystallne nature of the CdS and CdTe layers are ndcated schematcally and are not to scale. Many of the physcal propertes of crystallne solds depend on the presence of natve or foregn pont defects and gran boundares (see Fgure 5.1). In pure compound crystals the natve defects are atoms mssng from lattce stes where, accordng to the crystal structure, atoms should be (vacances); atoms present at stes where atoms should not be (ntersttals); and atoms occupyng stes normally occuped by other atoms (msplaced atoms). In addton, there may be defects n the electronc structure: quas-free 67