Validity of simplified Shockley-Read-Hall statistics for modeling carrier lifetimes in crystalline silicon

Transcription

1 Phys Rev B, (23) Validity of simlified Shockley-Read-Hall statistics for modelig carrier lifetimes i crystallie silico Daiel Macdoald * ad Adrés Cuevas Deartmet of Egieerig, FEIT, The Australia Natioal Uiversity, Acto ACT, 2, Australia Abstract - The Shockley-Read-Hall model, i its simlest ad most commo form, is ofte used to describe both ijectio- ad temerature-deedet carrier lifetime measuremets Such lifetime modelig has rovided the basis for ovel, ultra-sesitive sectroscoic techiques for the study of recombiatio cetres i crystallie silico However, this aroximate model is oly valid whe the desity of recombiatio cetres is small eough to avoid traig effects, which cause distortios i the excess mobile carrier cocetratios I this work, the simlified Shockley-Read-Hall model is comared with a more geeral solutio of the cotiuity equatios that takes accout of carrier traig This comariso leads to a exressio for the uer limit o the recombiatio cetre desity for which the simlified Shockley-Read- Hall model remais accurate The limit deeds ot oly o the doat desity, but also o the eergy level ad electro ad hole cature cross-sectios for a give tye of recombiatio cetre The results allow exerimetal coditios that do ot ivalidate the use of the simlified Shockley-Read-Hall model to be determied INTRODUCTION New techiques based o carrier lifetime measuremets have bee develoed recetly with the commo goal of characterisig imurities i crystallie silico Oe aroach, which has bee referred to as Ijectio Deedet Lifetime Sectroscoy (IDLS), ivolves fittig Shockley- Read-Hall theory to ijectio-level deedet lifetime measuremets Withi certai costraits, such a rocedure allows the eergy level ad carrier cature cross sectios of a imurity to be determied, ofte with suerior sesitivity ad accuracy to covetioal techiques such as Dee-Level Trasiet Sectroscoy (DLTS) The IDLS techique has recetly bee alied to boro-oxyge comlexes i Czochralski silico 2,3 ad also to iro-boro airs i silico 4-6, as well as other metallic secies such as chromium, molybdeum ad titaium 7,8 Previously, it had bee emloyed i aalysis of recombiatio cetres at silico dioxide/silico iterfaces 9 ad silico itride/silico iterfaces 9,, which are of critical imortace i solar cell alicatios It has also bee used i characterisig recombiatio cetres i electro-irradiated silico,2 I fact, simlified variatios of the techique effectively formed the basis of well kow, ioeerig studies of iro-boro 3 ad chromium-boro 4 airs i silico A secod grou of lifetime-based techiques, recetly dubbed Temerature Deedet Lifetime Sectroscoy (TDLS), has also bee develoed By aalysig the temerature deedece of the low-ijectio lifetime, the eergy level of the domiat recombiatio cetres ca be iferred So far it has foud alicatio i aalysis of irorelated cetres 5-7 ad various other metallic cetres 8, as well as electro-irradiated silico,2 It aears certai that lifetime-based techiques such as IDLS ad TDLS will become more widesread as icreasigly sesitive techiques are eeded to track ever-dimiishig quatities of imurities, ad as the requiremets of electroic devices become ever more striget These two techiques rely heavily o the use of Shockley-Read-Hall (SRH) 9,2 statistics to model the behaviour of the recombiatio lifetime as a fuctio of either ijectio-level or temerature However, they geerally make use of a simlified versio of SRH theory that ivolves a umber of assumtios 2,22 that must be satisfied whe alyig the model to exerimetal data The most restrictive of these requires almost equal excess utraed electro ad hole cocetratios As the recombiatio cetre desity N aroaches a critical value N crit, this simlified SRH model becomes icreasigly iaccurate, due to the fact that the recombiatio cetres i geeral also act as tras Whe this occurs, carriers sed some fiite time traed at the cetre before either recombiig or beig ejected back ito the bad from which they came Such traig alters the overall carrier dyamics, ofte very sigificatly, ad eve if oly oe tye of carrier is traed This restrictio o the SRH model is ofte exressed as simly requirig that N be small The urose of this aer is to determie exactly how small N must be to esure the simlified SRH model remais accurate, therefore

2 Phys Rev B, (23) eablig lifetime-based sectroscoic techiques to be safely emloyed This is achieved by comarig it with a more geeral model that exlicitly accouts for the traig roerties of the defects ad hece is valid for ay value of N, rovided that steady-state coditios revail This geeral model is used to fid a exressio for N crit i terms of the doat ad excess carrier desities, ad also the eergy level ad cature cross-sectios of a give recombiatio cetre These latter arameters tur out to have a sigificat imact o the rage of validity of the simlified SRH theory Firstly, however, we begi with the cotiuity equatios, which rovide a geeral descritio of carrier dyamics withi a semicoductor, ad which are the basis for all the recombiatio models cosidered 2 RECOMBINATION MODELS I a o-degeerate semicoductor uder the ifluece of exteral illumiatio, the excess cocetratios of free, utraed electros ad holes are described by the followig cotiuity equatios 2 : g E d dt ( d ( + g E dt + + )( ) + N + + )( ) + (2) N + Here, g E is the rate of electro-hole air geeratio by the illumiatio, ad are the electro ad hole cocetratios i thermal equilibrium, ad are the electro ad hole lifetimes ad N is the recombiatio cetre desity The cature time-costats for electros ad holes, ad, are related to the cature cross-sectios σ ad σ of the recombiatio cetre, ad also to the recombiatio cetre desity ad the thermal velocities of electros ad holes v th, via the followig exressios: Nσ v Nσ v th The assumtio of o-degeeracy is required to rovide a uambiguous meaig to the cature cross-sectios, sice they rereset average cross-sectios for carrier cature from all ossible bad-states 2 The statistical factors ad are the equilibrium cocetratios of electros ad holes resectively whe the Fermi th (3) () level coicides with the eergy of the recombiatio cetre E T : EC EG ET NV ex, kt ET EC NC ex kt where E C is the coductio bad edge eergy ad E G the eergy ga The desities of states at the coductio ad valece bad edges N C ad N V are N C ad N V 3 9 cm -3 for silico at room temerature 23 The cotiuity equatios () ad (2) allow for all the ossible iteractios betwee the bads ad the recombiatio cetres, amely recombiatio via cature of a electro (or hole) ad subsequet cature of a hole (or electro), or traig via cature ad release of a electro (or hole) The tedecy of a secific recombiatio cetre to act rimarily as a tra or a recombiatio site, or as some combiatio of both, will deed o its eergy ad cature cross-sectios, ad also o the Fermi level As discussed by Blakemore 2, the timedeedet cotiuity equatios () ad (2) ca oly be solved aalytically uder very secial coditios which are ot of great ractical iterest I the geeral case, comlicatios arise due to the couled ature of these exressios, which is a direct cosequece of the fact that the cetres may act as both recombiatio sites ad as tras Uder steady-state coditios, however, the time-deedet terms are zero, the equatios are less strogly couled, ad geeral solutios do exist This was exloited by Shockley ad Read 9, whose Equatio 44 is recisely such a solutio, the oly uderlyig assumtios beig those of o-degeeracy ad steady-state coditios Cast i this form, SRH theory is ideed quite geeral However, i subsequet sectios of their aer, Shockley ad Read wet o to develo oly cases with restricted alicatio The first of these, i their sectios 5 ad 6, was for arbitrary modulatio (values of ad ) but oly small values of N A later case, i Aedix A, was for arbitrary N but oly small modulatio It is the first of these restricted maifestatios of SRH theory, ie for small N, which is most widely used today We shall refer to it as the simlified SRH model, followig the headig of Shockley ad Read s ow Sectio 5 Oe route to the simlified SRH exressio from the cotiuity equatios ivolves (4)

3 Phys Rev B, (23) elimiatig N by substitutio from Equatio () ito Equatio (2) uder steady-state coditios After some re-arragig, ad uo use of the idetities /( + ) /( + ) ad, the result is: g E + + ( + + ) + ( + + ) Further maiulatio of this exressio, alog with the assumtio of vaishig excess carrier desities, leads to Equatios (A7) of Shockley ad Read, which is the case for arbitrary N but restricted to small modulatio If, o the other had, arbitrary excess carrier desities are allowed, but we isist that N is small eough to ot oticeably erturb them, so that, the we are led directly to Equatio 53 from Shockley ad Read, the simlified SRH model: SRH ( + + ) + ( (6) Note that the simlifyig assumtio merely requires that, but it is ot clear exactly how small N must be to esure this is so This geerates some ucertaity i alyig the simlified SRH model to exerimetal data As metioed, however, this exressio is widely emloyed i modelig carrier recombiatio for alicatios such as IDLS ad TDLS I order to quatify the limitatios of the SRH model, it is useful to comare it with a more geeral solutio to the cotiuity equatios Blakemore has show that geeral solutios of the cotiuity equatios ca be relatively easily foud for the case of arbitrary excess carrier desities ad values of N, makig the assumtios i the simlified SRH model largely uecessary The oly disadvatage of these geeral solutios over the simlified SRH model is that they are log Whe steady-state coditios revail, the cotiuity equatios () ad (2) may be writte as: ) (5) alyig the idetities /( + ) /( + ) ad, the solutio may be writte as 2 : 4 k + 2 [ A + ( k + ) ] N A + ( k ) where we have defied k / ad: N A + k( + ) [ ], 2 By choosig a rage of values for, we may calculate the corresodig values firstly for, the g E, ad fially the carrier lifetimes via /g E ad /g E The ijectio-level deedet lifetime curves obtaied i this way may the be comared with the simlified SRH model Of course it is also ossible to exress directly i terms of ad N, similarly for i terms of ad N As exected, these exressios reduce to the familiar simlified SRH model whe N is small However, they are legthy both to derive ad exress, as illustrated by Blakemore 2 i his Sectio 84, while the more comact aroach show here is more easily comuted Note that if N is small the Equatio (8) simlifies to, as exected Uder these coditios the results will be idetical to the simlified SRH model However, i the geeral case ad therefore This oit has recetly bee highlighted by Karazhaov 24,25, who used Equatios (5) i cojuctio with the electroeutrality coditio to calculate ad ad therefore ad I cotrast, the aroach used here ivolves direct solutio of Equatios () ad (2), which imlicitly cotai the electroeutrality coditio It may seem logical at this oit to suggest relacig the use of the simlified SRH model i lifetime sectroscoy alicatios with this more geeral solutio of the cotiuity equatio Ideed, this would remove etirely the eed to be watchful for traig effects However, the geeral solutio requires exlicit kowledge of the value of N, while the simlified SRH model (8) (9) ( + + )( ) g E + N + This exressio reresets two equatios cotaiig three ukows:, ad g E Maiulatig the two exressios o the right had side of Equatio (7) gives a quadratic equatio ivolvig oly ad Uo ( + + )( ) + N + does ot (although eve i the simlified case N is required to determie the cature cross-sectios, but ot the cature time-costats) I may ractical cases the value of N is ot kow, ad so the simlified model may be the most useful, desite the restricted rage over which it is valid (7)

4 Phys Rev B, (23) A fial oit to ote regardig these models is that they both eglect the ossibility of tuelig trasitios betwee recombiatio cetres If the wave-fuctios of these localised states overla to a areciable extet, the such tuelig trasitios ca affect roerties such as lifetime ad mobility A well kow examle is amorhous silico, which cotais bad-tail states of sufficietly high cocetratio to allow hoig betwee them Hattori et al 26 recetly develoed a recombiatio model for amorhous silico that cotaied three tyes of electroic state: the bads themselves, the bad-tails, ad dee states Their model allowed trasitios betwee the bads ad both the bad-tail ad dee states, as well as betwee the bad-tail states ad the dee states They also icluded tuelig trasitios betwee bad-tail states, ad foud that these were largely resosible for the form of measured drift mobility data They did ot however iclude tuelig trasitios betwee dee states, sice their desity was ket below 7 cm -3, ad hece their wave-fuctios are very ulikely to overla Similarly, i this work, we eglect tuelig trasitios betwee the dee states, ad ay imact this might have o the lifetime, sice i almost all ractical cases the desity of recombiatio cetres will be well below 7 cm -3 Bad-tails do ot ormally exist i crystallie silico, uless it is extremely heavily doed, ad so we do ot iclude their effects here either 3 COMPARISON OF THE TWO MODELS Figure rovides a direct comariso betwee the geeral steady-state solutio derived above ad the SRH model for the secific case of iterstitial iro i crystallie silico This articular recombiatio cetre, which has a eergy level 27 of E V +38eV ad cature crosssectios of σ 5-4 ad σ 7-7 cm -2, was chose because it rovides a good examle of the behaviour of a dee cetre Three differet values of the iro cocetratio are show o the lot For each of these, four lifetime curves are lotted: the simlified SRH lifetime, the electro ad hole lifetimes from the geeral steady-state solutio, ad a mobility-weighted combiatio of these electro ad hole lifetimes The latter weighted curve will most accurately reflect lifetime measuremets from hotocoductacebased methods, which are uable to discrimiate betwee carrier tye sice they merely record chages i coductivity The weighted lifetime is calculated via wtd wtd /g E where the mobilityweighted excess carrier desity wtd is give by: µ + µ wtd µ + µ () The four curves that rereset the lowest cocetratio of iro are all i good agreemet I this case, the small cocetratio of cetres causes egligible traig, or, i other words, a miimal iequality betwee excess mobile electro ad hole cocetratios The result is electro ad hole lifetimes that are almost equal at all ijectio-levels, ad hece oe may safely aly the simlified SRH model i this case As the iro cocetratio icreases, however, the electro ad hole lifetimes start to become sigificatly differet, esecially at the lower ijectio-levels This reflects a icreasig breakdow of the simlified SRH model The mobility-weighted lifetime curves reveal that the large hole lifetime causes the simlified SRH model to uderestimate data obtaied by hotocoductace-based techiques Note however that the curves are still i good agreemet at higher ijectio levels The treds show i Figure are characteristic of dee recombiatio cetres, which always geerate lifetimes that icrease (or remai costat) with icreasig ijectio-level I cotrast, shallow cetres ca result i either icreasig or decreasig lifetimes, deedig o the doat desity ad the exact value of the eergy level 2 FeB airs rovide a examle of shallow cetres i silico Figure 2 gives a comariso of the SRH model ad the geeral steady-state solutio for FeB airs, with a eergy level 27 of E C -23eV ad cature cross-sectios 6 of σ 3-4 ad σ 2-5 cm -2 Oce agai, the results show that the SRH theory becomes icreasigly iaccurate as the recombiatio cetre desity icreases For moder sectroscoy techiques that rely o the use of SRH theory, the imortat questio the becomes what is the critical recombiatio cetre desity above which the simlified SRH model will be iaccurate? This questio is addressed i the followig sectio 4 THE CRITICAL RECOMBINATION CENTRE DENSITY The uer limit o the recombiatio cetre desity N for which the simlified model is accurate will deed o the articular roerties

5 Phys Rev B, (23) of the recombiatio cetre (ie the eergy level ad cature cross-sectios), ad also o the doat desity I geeral, it will also deed o the cocetratio of excess carriers, sice, if sufficietly umerous, these may overwhelm ay traig effects A coveiet way to exress this limit is by defiig a arameter N crit, such that the simlified SRH model is oly valid if N«N crit A direct aroach to calculatig N crit is to cosider the excess carrier ratio / as determied by the geeral steady-state model above For coditios uder which the simle SRH model is accurate, traig must be egligible, ad this ratio will be close to uity A exressio for / ca be arrived at from Equatio (7): k( + + ) + kn ( + ) () k( + + ) + N ( + ) Firstly, it is clear from this exressio that the right had side will always equal uity whe If this coditio holds, traig will always be egligible, irresective of the recombiatio cetre desity N I ractice however, this coditio is oly satisfied by rare coicidece I hysical terms, it reresets the uique combiatio of eergy level ad cature crosssectios that cause a erfect balace betwee majority ad miority carrier traig 2 I the more geeral case, the right-had side of Equatio () will also be close to uity whe the two terms ivolvig N either do ot cotribute much to the umerator or deomiator, or cotribute very similarly A coveiet way to exress this i a sigle iequality is to state that the magitude of their differece must be much less tha the other terms After some rearragig, we the arrive at the followig exressio for N crit : N crit ( k + ) [ k( + + ) ] k If we require the simlified SRH model to be accurate to withi %, the N must be at least a order of magitude smaller tha N crit Uder high-ijectio coditios, a greater cocetratio of recombiatio cetres ca be tolerated, sice N crit icreases due to the ad terms (2) (3) However, it is ofte the value of N crit uder lowijectio coditios which is of iterest For a - tye samle with N A such that 2 i / is egligible, the relevat exressio for N crit i lowijectio is: ( N A + )( + kn A + k) N crit N k (4) This exressio ca be simlified further if the recombiatio cetres are dee, meaig that both ad <<N A I such cases, rovided k is ot excessively large, the: N crit N A o A This fial exressio i fact reresets the lowest ossible, ad hece most restrictive, value for N crit for ay recombiatio cetre eergy, dee or shallow I this sese, dee levels are the most likely to cause the simlified model to become ivalid for a give value of N This does ot ecessarily mea, however, that they will always be the most difficult to study usig the simlified SRH model I geeral, fewer dee recombiatio cetres are required to roduce a give lifetime, couterbalacig the fact that N crit is smaller for such cetres Aother imortat feature of this fial exressio is that N crit deeds oly o the ratio of ad, ad ot o their magitudes This reflects the fact that the cetres must ot cause sigificat traig of carriers, as has bee discussed above The extet to which this traig occurs is characterised by the ratio /, sice this is essetially a measure of the roortio of time that the carriers remai utraed It is ot surrisig the that this arameter has a heavy bearig o the boudaries of accuracy of the simle SRH model It is worth otig at this oit that Shockley ad Read also cosidered uder which coditios their simlified theory would be valid (as discussed i their Aedix A) They cocluded that the simlificatio would hold rovided N is small comared with ay oe of,, or However, the coditios uder which they arrived at this coclusio were oly valid for ifiitesimal modulatio I cotrast, the exressio for N crit give i this work also accouts for arbitrary modulatio, hece the aearace of Furthermore, the exressio derived here also accouts for the otetial imact of highly asymmetric cross-sectios, as (5)

6 Phys Rev B, (23) maifested i the value of k, which was ot cosidered i Shockley ad Read s aroximate aroach Figure 3 deicts the deedece of the low-ijectio limit of N crit, ormalised with resect to N A, as a fuctio of the recombiatio cetre eergy, which aears idirectly i Equatio (4) through ad Five curves are show, corresodig to differet values of the cature cross-sectio ratio k The sikes o the curves corresod to the coditios of Equatio (2) a very secial ad arrow rage of eergies for which the effect of traig will be egligible Of more geeral iterest however, the curves show that there is a broad rage of eergy levels ear the middle of the bad-ga that has the most detrimetal imact o N crit These are referred to as dee cetres, ad the magitude of their imact o N crit is determied directly by the value of k Figure 4 shows that the breadth of the flat regio due to dee cetres is determied by the doat desity For lightly-doed samles, the rage of recombiatio eergies which geerates the most severe traig is reduced Furthermore, shallow cetres of a give eergy level imose a less striget limit o N crit i more lightly-doed material Figures 3 ad 4 reveal that whether a cetre behaves as a dee cetre or ot deeds ot oly o its eergy, but also o the doat desity ad the value of k Therefore, care must be take to avoid alyig Equatio (5) to cetres which, desite havig eergy levels far from the bad edges, may ot behave as dee states It should be oted that i both Figures 3 ad 4, the value of N crit exceeds 7 cm -3 at some oits It is ossible that tuelig trasitios betwee the dee states would sigificatly alter the carrier dyamics if the actual value of recombiatio cetres, N, also exceeded this value I such cases a more comlex model such as that develoed by Hattori et al would be required If a recombiatio cetre is kow to be dee, the N crit is give by Equatio (5) for lowijectio coditios As discussed above, iterstitial iro is a dee cetre i silico, ad its low value of k results i a limit of Ν crit 4 2 cm -3 for the doat desity i Figure (N A 5 cm -3 ) The lots i Figure show that whe N«N crit, (ie the N cm -3 case), the SRH model is accurate across all ijectio-levels For higher recombiatio cetre desities, mid- to high-ijectio lifetime measuremets may still be accurately modeled with the SRH theory, rovided the excess carrier cocetratio is high eough to esure that N«N crit, where N crit is give by Equatio (3) For shallow cetres, the value of N crit deeds o the factors ad also A examle of such a shallow cetre is give by FeB airs, which give a value of Ν crit 36 5 cm -3 for the coditios i Figure 2 (N A 5 cm -3 ) Oce agai, the curves reveal that the value of N crit rovides a effective demarcatio betwee the regios over which SRH theory is valid for all ijectio-levels ad those over which it is ot It should be oted that, while referred to as shallow, FeB airs are still some cosiderable distace from the coductio bad edge As a result, the terms cotaiig ad i Equatio (3) moderate the value of N crit, but ot to a massive extet However, for very shallow cetres, N crit is likely to be much larger, due to the overwhelmig magitude of either or A Modelig more tha oe tye of cetre simultaeously I may ractical situatios, more tha oe tye of recombiatio cetre may be reset i sigificat quatities As metioed, oe such examle is rovided by iterstitial iro ad FeB airs Both cetres may occur with similar cocetratios, ad both may imact sigificatly o the overall, or effective, recombiatio lifetime A questio aturally arises as to how to correctly deal with such cases A commo aroach is to treat the differet tyes of recombiatio cetres as ideedet, ad the roceed to sum the iverse lifetimes due to each tye of cetre to obtai the effective lifetime I usig this aroach, it is assumed that ay additioal terms arisig from iteractios betwee the various tyes of cetres are egligible 2 If beig used i cojuctio with the simlified SRH model, it is oly valid whe all tyes of recombiatio cetre are reset i small eough quatities to avoid traig, that is they must all satisfy the requiremet N«N crit Eve if oly oe such cetre tras carriers to ay sigificat extet, the resultig chage i the free carrier oulatios ca cause very large chages to the recombiatio dyamics of all the cetres A extreme case is described by the Horbeck- Hayes model 28, which was develoed for the situatio i which oe tye of cetre acts oly as a recombiatio cetre, ad aother oly as a tra This model shows that the traig cetres do ideed have a dramatic imact o the behaviour of

7 Phys Rev B, (23) the recombiatio cetres, ad has bee used to exlai uusual hotocoductace measuremets i sigle-crystal silico ad germaium 28-3 ad multicrystallie silico 3 B A Exerimetal Examle Figure 5 shows hotocoductace-based lifetime measuremets, take from Ref 6, of a silico samle with a total Fe cocetratio of 25 3 cm -3 The iro was itroduced by ioimlatatio, ad the samle aealed to distribute the iro uiformly throughout the bulk, takig care to avoid reciitatio 6 The lifetime measuremet was erformed usig the quasisteady-state hotocoductace (QSSPC) techique 32 Ufortuately, it was ot ossible to measure the iro-related lifetime at lower carrier desities tha show, because of the existece of traig cetres associated with the surface assivatig film which overwhelm the hotocoductace 6 I this samle, oly 2% of the iro was reset i iterstitial form, the rest was aired with boro doat ios Also show are the simlified SRH lifetimes for the relevat cocetratios of Fe i ad FeB airs, calculated usig the cature crosssectios ad eergy levels give above, as well as the combied lifetime achieved by addig the iverses of these two comoets The geeral, mobility-weighted steady-state solutios are also show The geeral solutio ad the simlified model are i good agreemet for all ijectiolevels for the FeB airs, but a discreacy emerges at lower carrier cocetratio for Fe i As a result, a similar disagreemet aears for the combied recombiatio lifetimes The values of N crit for this doat desity (9 3 cm -3 ) are 6 ad 36 5 cm -3 for iterstitial iro ad FeB airs resectively Therefore we exect the simlified theory to be valid for all ijectio-levels for the FeB air comoet, sice N FeB «N crit-feb This is clearly ot true for the iterstitial iro comoet however As a result, the simle SRH model is ot valid for this samle across all ijectio-levels However, at higher ijectio-levels the simlified model remais valid due to the swamig of the traig effects by excess carriers For traig effects to imact o the measuremet by o more tha %, we require that N crit-fei be a order of magitude larger tha N Fei 5 2 cm -3 From Equatio (3), this is satisfied whe 45 3 cm -3 Figure 5 shows that this is ideed the regio i which the geeral model ad the simlified SRH model coverge withi % for the Fe i curves Sice iterstitial iro accouts for about oly half of the total recombiatio lifetime at this carrier desity, the effective value for is aroud 2 3 cm -3 4 CONCLUSIONS The simlified Shockley-Read-Hall model is a owerful ad widely-used tool i lifetime-based techiques for characterisig imurities ad defects i semicoductors However, to be cofidet that its use is valid, it is ecessary to verify that the recombiatio cetre desity N is small eough to avoid excessive traig effects A exressio for the critical value of N, above which the simlified SRH model becomes ivalid, has bee reseted It reveals that the most severe restrictios o the regio of validity of the simlified SRH model occur for dee cetres with highly asymmetric cature cross-sectios These restrictios should be cosidered routiely whe usig sectroscoic techiques based o either ijectio-level or temerature deedet lifetime modelig ACKNOWLEDGEMENTS This work has bee suorted by the Australia Research Coucil The authors wish to ackowledge the assistace of J Wog-Leug, H Ta ad C Jagadish of the Research School of Physical Scieces ad Egieerig, ANU, for assistace with samle rearatio REFERENCES S Rei, T Rehrl, W Warta ad S W Gluz, J Al Phys 9, 259 (22) 2 J Schmidt, C Berge ad A G Aberle, Al Phys Lett 73, 267 (998) 3 J Schmidt ad A Cuevas, J Al Phys 86, 375 (999) 4 D Walz, J-P Poly ad G Kamarios, Al Phys A 62, 345 (996) 5 R Falster ad G Barioetti, i Recombiatio Lifetime Measuremets i Silico, ASTM STP 34, edited by D C Guta, F R Bacher ad W M Hughes (America Society for Testig ad Materials, West Coshohocke, 998), D Macdoald, A Cuevas ad J Wog-Leug, J Al Phys 89, 7932 (2) 7 M L Poligao, E Balladi, D Lodi, F Piia, A Sabbadii, F Zaderigo, G Queirolo ad F Priolo, Mat Sci ad Egieerig B55, 2 (998)

8 Phys Rev B, (23) 8 P Eichiger, i Recombiatio Lifetime Measuremets i Silico, ASTM STP 34, edited by D C Guta, F R Bacher ad W M Hughes (America Society fo testig ad Materials, West Coshohocke, 998), 9 A G Aberle, Crystallie silico solar cells: advaced surface assivatio ad aalysis (Cetre for Photovoltaic Egieerig, Uiversity of New South Wales, Sydey, Australia, 999) J R Elmiger, R Schieck ad M Kust, J Vac Sci Techol A 5, 248 (997) H Bleicher, P Josso, N Keskitalo ad E Nordlader, J Al Phys 79, 942 (996) 2 N Keskitalo, P Josso, K Nordgre, H Bleicher ad E Nordlader, J Al Phys 83, 426 (998) 3 G Zoth ad W Bergholz, J Al Phys 67, 6764 (99) 4 K Mishra, Al Phys Lett 68, 328 (996) 5 Y Hayamizu, T Hamaguchi, S Ushio ad T Abe, J Al Phys 69, 377 (99) 6 A Kaiava, E Gaubas, J Vaitkus, J Vahellemot ad A L P Rotodaro, Mat Sci ad Tech, 67 (995) 7 A Kaiava, A L P Rotodaro, J Vahellemot, U Meczigar ad E Gaubas, Al Phys Lett 67, 393 (995) 8 S Rei, T Rehrl, J Iseberg, W Warta ad S W Gluz, i Proc 6th Euro Photovoltaic Solar Eergy Cof, Glasgow, UK, 2, edited by 2), W Shockley ad W T Read, Phys Rev 87, 835 (952) 2 R N Hall, Phys Rev 87, 387 (952) 2 J S Blakemore, Semicoductor Statistics (Pergamo Press, Oxford, 962) 22 R A Smith, Semicoductors (Cambridge Uiversity Press, Cambridge, 959) 23 M A Gree, J Al Phys 67, 2944 (99) 24 S Z Karazhaov, Semicod Sci Techol 6, 276 (2) 25 S Z Karazhaov, J Al Phys 88, 394 (2) 26 K Hattori, T Hirao, Y Musa ad H Okamoto, Phys Rev B 64, 2528 (2) 27 A A Istratov, H Hieslmair ad E R Weber, Al Phys A 69, 3 (999) 28 J A Horbeck ad J R Hayes, Phys Rev 97, 3 (955) 29 J R Hayes ad J A Horbeck, Phys Rev 9, 52 (953) 3 J R Hayes ad J A Horbeck, Phys Rev, 66 (955) 3 D Macdoald ad A Cuevas, Al Phys Lett 74, 7 (999) 32 R A Sito ad A Cuevas, Al Phys Lett 69, 25 (996) Lifetime (s) Fe i, N A 5, N crit 4x 2 cm -3 N Fei SRH vs vs vs wtd vs wtd , wtd (cm -3 ) N Fei 2 N Fei 3 FIG Comariso betwee the ijectio-level deedece of the simlified SRH lifetime ad the electro ad hole lifetimes as calculated from the geeral steady-state solutio, usig the recombiatio arameters of iterstitial iro Curves are show for three values of the iro cocetratio The electro ad hole lifetimes are lotted agaist the excess hole cocetratio Also show for each iro cocetratio is a fourth curve, wtd, which reresets the mobilityweighted combiatio of ad lotted agaist a similarly weighted excess carrier cocetratio wtd Lifetime (s) FeB airs, N A 5, N crit 36x 5 cm -3 SRH vs vs vs wtd vs wtd , wtd (cm -3 ) N FeB 4 N FeB 5 N FeB 6 FIG 2 Comariso betwee the ijectio-level deedece of the simlified SRH lifetime ad the electro ad hole lifetimes as calculated from the geeral steady-state solutio, usig the recombiatio arameters of FeB airs

9 5 4 -tye Si, N A 5 cm -3 Phys Rev B, (23) N A 9x 3 cm -3 [Fe i ]5x 2 [FeB]2x 3 N crit /N A k E C -E T (ev) FIG 3 Values of the low-ijectio limit of N crit, ormalised with resect to N A 5 cm -3, as a fuctio of the recombiatio cetre eergy level E C -E T Curves are show for differet values of the cature cross-sectio ratio k Lifetime (s) -4-5 FeB , wtd (cm -3 ) FIG 5 Comariso betwee exerimetal data (from Ref 6) ad modeled lifetime curves for a silico samle cotaiig iterstitial Fe ad FeB airs as the domiat recombiatio sources Fe i Fe i ad FeB combied SRH vs wtd vs wtd QSSPC data N crit /N A tye Si, k N 3 cm -3 A E -E (ev) C T FIG 4 Values of the low-ijectio limit of N crit, ormalised with resect to N A, as a fuctio of the recombiatio cetre eergy level E C -E T Curves are show for differet values of N A The cature crosssectio ratio k is uity