Investigation of defects formed by ion implantation of H2+ into silicon

Transcription

1 Rochstr Istitut of chology RI Scholar Works hss hsis/dissrtatio Collctios Ivstigatio of dfcts formd by io imlatatio of H2+ ito silico Patrick Whitig Follow this ad additioal works at: htt://scholarworks.rit.du/thss Rcommdd Citatio Whitig, Patrick, "Ivstigatio of dfcts formd by io imlatatio of H2+ ito silico" (2009). hsis. Rochstr Istitut of chology. Accssd from his hsis is brought to you for fr ad o accss by th hsis/dissrtatio Collctios at RI Scholar Works. It has b acctd for iclusio i hss by a authorizd admiistrator of RI Scholar Works. For mor iformatio, las cotact ritscholarworks@rit.du.

2 Ivstigatio of Dfcts Formd by Io Imlatatio of H 2 + ito Silico By: Patrick Whitig A hsis Submittd i Partial Fulfillmt of th Rquirmts for th Dgr of Mastr of Scic I Matrials Scic ad Egirig Arovd By: Dr. Karl Hirschma, Advisor Dr. Satosh Kuric, Committ Mmbr Dr. Robrt Parso, Committ Mmbr Dr. Kalathur Sathaam MAERIALS SCIENCE AND ENGINEERING COLLEGE OF SCIENCE ROCHESER INSIUE OF ECHNOLOGY ROCHESER, NY AUGUS 2009 i

3 Forword Uo comltig this thsis ad starig at th titl ag, I ralizd that thr ar so may ol that dsrv my thaks for thir assistac i this thsis. Firstly, I ow a grat dbt to Karl Hirschma, my advisor. Without Karl, I d still b lookig for my first co-o ad I would hav vr had a chac to work at Sulliva Park (Corig Icororatd s rsarch facility i Paitd Post, Nw York), which brigs m to th scod rso I wat to thak. Dick Maschmyr, formrly of Corig, Ic. (ow rtird) has likly b th biggst ifluc o my rofssioal dvlomt to dat. O my first day of work, durig luch if I rmmbr corrctly, I mtiod to Dick that I was itrstd i larig thrmodyamics. Dick lauchd ito a imassiod sch about what a grat subjct it was ad I foud what would bcom my thr favorit txtbooks o my dsk th day aftr which Dick lt to m for th summr. Without Dick s hl, I doubt that I d hav lard th thrmodyamics ad kitics cssary for th work I did i this thsis. So, it s ot a mbllishmt to say that if it wr t for Dick, this thsis would t hav v b udrtak. hird, fourth ad fifth o th list ar Johas Moll, Grg Couillard ad Jayatha Sawirat, all of Corig, Icororatd. Johas actd as my survisor durig th riod whil I was dvloig th small sigal modl for DLS aalysis of MIS Caacitors ad was th guy I wt to wh I had idas or simly watd a brak to talk. Grg actd as scodary advisor to m as my work slowly startd to mov away from thory ad towards xrimtatio. H was my rimary sourc for matrials ad for guidac o my xrimts (as wll as my uofficial mailbox at Corig thaks dud). Ovr th ast yar, I do t thik I v gott mor tha a half-doz mails from him that did t iclud th hras, If thr s aythig you d from m, lt m kow. Jayatha was my sourc for DLS. If it wr t for his takig th tim to ru my samls ad collaborat o basli xrimts oft comig i o a wkd to ru xrimts, my DLS work would hav vr b comltd. Guys, this thsis is as much a tstamt to your ow hard work, ddicatio ad brilliac as it is to my ow hard work. I ow you all a hug dbt. hak you. ii

4 Abstract Io cuttig achivd by th imlatatio of hydrog or th co-imlatatio of hydrog ad hlium is amog th most commo mthods for th formatio of Silico o Isulator (SOI) structurs usd i th smicoductor idustry. I this mthod, hydrog is imlatd ito silico at a high fluc ad is hatd i ordr to iduc a xfoliatio vt. Durig this xfoliatio vt, a silico wafr is clavd alog th dth at which th maximum damag coctratio occurs, ad th clavd matrial bods chmically to ay udrlyig matrial big usd as a hadl substrat. h io imlatatio rocss iducs a varity of dfct scis which volv as thy ar aald at varyig tmraturs ad tims ad th charactristics of ths dfcts ad th ractios which domiat thir formatio ar critical for low tmratur substrats such as LCD glass. his study obsrvs th aalig charactristics of a varity of structural ad lctroic dfcts iducd by io imlatatio, icludig hydrog dcoratd moovacacis ad hydrog dcoratd itrstitials. h stats arisig from ths dcoratd oit dfcts wr aalyzd usig Multil Itral rasmissio Ifrard Sctroscoy (MI-IR) ad D Lvl rasit Sctroscoy (DLS). A mthod for obsrvig imlat-rlatd dfcts o a MOS Caacitor usig a DLS masurmt was dvlod. A w mthod for xtractig th activatio rgy ad th catur cross sctio of stats obsrvd with DLS through th us of th Full Width at N th Maximum was also dvlod. MI-IR sctra rsultig from io imlatatio wr aalyzd usig a ovl mthod to xtract th activatio rgy, ractio vlocity ad ordr of a solid stat ractio, trmd Kitic Diffrtial Aalysis. Aalysis usig th mthods dscribd abov allowd for th idtificatio of fiv tra rgy lvls associatd with hydrog io imlatatio which wr ttativly assigd to VH 2 (.15V) VH 3 (~.54V) ad IH x (.16V ad.19v) dfcts. Kitic Diffrtial Aalysis of MI- IR sctra has idtifid ractio athways associatd with th dcay of dcoratd moovacacy dfcts hs chmical ractios hav kitic ractio ordrs of aroximatly 1.5, idicatig a scodary ractio which cotributs to th dcay as wll as som gral itractio btw ractats durig th dcay rocss. iii

5 abl of Cotts itl Pag.i Forword..ii Abstract...iii abl of Cotts iv List of Figurs.vi List of abls.vii 1. Itroductio Motivatio Shockly Rad Hall (SRH) Rcombiatio D Lvl rasit Sctroscoy (DLS) Multil Itral rasmissio Ifrard (MI-IR) Sctroscoy Hydrog Imlatatio Dfcts i Smicoductors DLS Fittig Fuctios Simlificatios to th Charg rasit DLS Aalysis of MIS Caacitor Structurs h Structur of MIS Caacitors DLS Small Sigal Aalysis A Sctral Fittig Program h Half Width at Nth Maximum Molcular Hydrog Io Imlatatio Exrimts Kitic Diffrtial Aalysis Exrimtal Procdur Rsults ad Discussio Coclusio iv

6 5.1. Aalytical Mthods Io Imlatatio Rlatd Dfcts Futur Work..110 Bibliograhy 111 v

7 List of Figurs Fig. Dscritio Pg. 1.1 h Hydrog Io Cuttig chiqu Shockly Rad Hall Rcombiatio Gratio of a D Lvl rasit Sctroscoy Sigal Majority ad Miority Carrir rad Charg Variatio With im Fr Carrir Bhavior Durig DLS Aalysis Arrhius Plottig of a DLS Sigal MI-IR Saml Dsig MI-IR Bvl Gomtry MI-IR Bam Pathlgth Drivatio Aalig Bhavior of H2+ Imlat Elctro ras DLS Sctra from a P-y Silico Saml Imlatd with H DLS Sctra from a P-y Silico Saml Imlatd With Si Miority Carrir Coditios Rquird for Sigl Catur Rat Domiac Maximum Hol Coctratio Rquird for Emissio Maximum Elctro Coctratio Rquird for Emissio Maximum Rsolvabl Ergy for Hol Emissio Miimum Rsolvabl Ergy for Elctro Emissio Maximum Rsolvabl Hol ad Elctro Ergy Gas for Prfrtial Emissio MIS Caacitor Charg Structur MIS Caacitor DLS Small Sigal Modl DLS Itrfac Stat ad Bulk ra Stat Sigal Variatio with Isulator hickss DLS Itrfac Stat ad Bulk ra Stat Sigal Variatio with Substrat Rsistivity DLS Sctra of a P-y Silico Saml Imlatd with P Program Dsig for Smi Automatd Fittig Variatio of C/C O with Emissio Rat Variatio of N with k O DLS Sctra of a P-y Silico Saml Imlatd with B HWNM Aalysis of DLS Sctra of a P-y Silico Saml Imlatd with B Variatio of N /N O for a Gratio/Dcay Ractio Caacitac-Voltag Data of MIS Caacitors for DLS MI-IR Sctra for Silico Samls Imlatd with H MI-IR Sctral Itsitis with Rsct to mratur Normalizd MI-IR Sctral Itsitis for VH 3 ad VH Normalizd MI-IR Sctral Itsitis for VH 2 ad VH Normalizd MI-IR Sctral Itsitis for IH Normalizd DLS Itsity Data mratur Ddac of Hydrogatd Dfcts 105 vi

8 List of abls abl Dscritio Pg. 4.1 Saml Praratio for DLS Samls IH X ad VH X Rsoat Stats for MI-IR KDA Fittig Rsults for MI-IR Sctra Obsrvd ra Scis Obsrvd ra Scis 109 vii

9 Chatr 1: Itroductio 1.1: Motivatio h cotiuig driv for high rformac Comlmtary Mtal Oxid Silico (CMOS) rasistors has driv matrial costraits for Silico substrats to th rquirmt of Silico o Isulator (SOI) structurs. hs structurs cosist of a thi layr of Silico o to of a much largr isulatig substrat. his substrat may b a isulatig crystal, as is th cas i Silico o Sahir matrials, whr Silico is grow itaxially o to of a Sahir substrat. Altrativly, th substrat may b formd from a Silico wafr that is oxidizd to form a thi isulatig layr [1]. A third otio which is far lss rvalt tha th us of a oxidizd substrat is th us of a amorhous isulatig matrial such as glass. his last otio is articularly usful for lctroics alicatios whr a rlativly trasart substrat is rquird, as is th cas for dislay lctroics. I th cas of structur formatio o a oxidizd crystal or a amorhous substrat, th mthod for joiig th smicoductig layr to th isulatig substrat is a bodig tchiqu. yically, this is accomlishd by joiig a Silico wafr to th isulatig substrat at room tmratur. Va Dr Waal s forcs grat a wak bod which holds th two substrats togthr rior to a thrmal tratmt. his thrmal tratmt iducs much strogr chmical bodig btw th isulator (which is tyically SiO 2 ) ad th silico substrat. his bodig tchiqu rsults i a thick smicoductor substrat bodd to a thick isulatig substrat which, alo, is ot a imrovmt ovr covtioal bulk smicoductors. h smicoductor layr must b thid dow. A thi layr may b formd by laig ad chmical tchig of th thick smicoductor layr util th dsird thickss is achivd. 1

10 Aothr commoly favord tchiqu is th us of a tch-back, whr th smicoductig layr is xosd to a RF lasma ad a d ractiv io tch is rformd util th dsird thickss is achivd. Both laig ad tch-back tchiqus ar ifficit, at bst, bcaus thy ffctivly wast a tir wafr to form a SOI substrat. hy ar also ihrtly slow rocsss bcaus of th tim rquird to tch through or olish dow samls which may b a half-millimtr thick or mor, as is tyically th cas for silico wafrs. Aothr otio which has gaid oularity rctly is a Io-Cuttig tchiqu [1]. Prior to bodig, th smicoductor substrat gos through a Io Imlatatio st, whr Hydrog or a combiatio of Hydrog ad Hlium is imlatd ito th smicoductor. his givs ris to rimary imlat-rlatd dfcts such as slf-itrstitials ad vacacis as wll as som hydrog-dfct comlxs. Followig this io imlatatio st, th ormal bodig rocdur is followd, but th tmratur is raisd aftr th bodig rocss is comlt. Wh xosd to highr tmraturs, hydrog-dfct comlxs will grow ad agglomrat, formig a sris of hydrog-trmiatd dislocatio loos ad stackig faults, calld blistrs. As aalig rogrsss, hydrog ad hlium will coalsc at ths blistr sits ad th risig gas rssur du to highr coctratios withi th blistrs as wll as risig tmratur will caus a fractur vt which will clav th wafr. h dth at which fractur ad clavig occurs is th dth ito th imlatd smicoductor saml whr th umbr of rcoils ad corrsodig umbr of iducd vacacis ar maximizd [1]. 2

11 Figur 1: h Hydrog Io Cuttig chiqu Hydrog or Hydrog-Hlium is ijctd ito th smicoductor via io imlatatio (1). h smicoductor is lacd o a isulatig substrat, imlatd sid dow (2). h smicoductor substrat is aald at lvatd tmraturs, iducig chmical bodig of th smicoductor to th isulator ad xfoliatig th imlatd layr (3). I all SOI tchiqus that utiliz bodig for saml formatio, a critical factor which costrais th ffctiv tmratur cilig for rocssig aftr saml formatio is th thrmal xasio cofficit mismatch btw th various thi film layrs of th saml. I th cas of SOI whr silico is bodd to a oxidizd wafr, this is ot a sigificat cocr. h thi oxid itrmdiary btw th two silico layrs caot grat ough strss to iduc d- 3

12 lamiatio of th smicoductor film. hus, lad, RIE ad Io-Cut SOI cratd o a oxidizd silico substrat ca tyically b aald u to tmraturs tyical for bulk smicoductor rocssig. I th cas of a saml formd whr th substrat is a amorhous matrial, this mismatch of th thrmal xasio cofficit ca lad to d-lamiatio at lvatd tmraturs, so th cilig for rocssig tmratur is much lowr [1]. Bcaus of th rducd tolrac for high tmraturs, samls formd by Io-Cuttig o a amorhous substrat may ot b aald so as to comltly limiat all dfcts withi th saml. May of th oit dfcts ad hydrog-dfct comlxs formd durig th xfoliatio vt may rmai rst withi th saml cratd v aftr a thrmal tratmt at th availabl (albit low) maximum tmratur [2]. hs dfcts lad to dgradd dvic rformac rimarily through rcombiatio ffcts. Scifically, th dfcts formd durig imlatatio which ract ad form scodary dfcts aftr aalig, act as ctrs for Shockly Rad Hall (SRH) Rcombiatio vts [3]. Ddig uo th charactristics of th tra gratd, ths ctrs may act as sudo-doats, which ca drastically affct th rsistivity of th thi film silico layr or may, altrativly, act as sourcs of lakag at mtallurgical juctios. his scod situatio rsults i a icras i th off-stat currt of dvics fabricatd o SOI structurs, forcig circuit dsigrs to work aroud highr owr cosumtio o-substrat as wll as ossibly dsig far mor comlx circuits to accout for th oor switchig quality of thir dvics. If commrcial roducts formd from Io-Cut rocsss with amorhous isulators ar to b ixsiv ad ultimatly succssful, it is imrativ that th dfcts causd by Io-Cut rocsss b idtifid ad cotrolld. o this d, this thsis will sk to idtify lctrical dfcts causd by th Io-Cut rocss, corrlat ths dfcts with associatd abormalitis i 4

13 crystal structur ad roos a mas for udrstadig th ractios that cotribut to th formatio ad dissolutio of ths dfcts durig th aalig rocss. his thsis will mak us of D Lvl rasit Sctroscoy, a tchiqu dvlod by D.V. Lag i 1974 [4] as wll as Multil Itral rasmissio Ifrard (MI-IR) Sctroscoy [5]. If th ffcts of structural dfcts associatd with io cuttig tchiqus ar to b corrlatd with lctroic traig ffcts, svral othr tchical issus must b workd out i advac. Firstly, a ffctiv mthod must b foud which ca rlat th aalig dyamics of structural dfcts to th corrsodig aalig dyamics of lctroic dfcts i ordr to mak ositiv corrlatios btw th two. o this d, fidig a ratabl mthod of formig a saml which ca grat a DLS sigal with miimal ois is of th utmost imortac. his mthod must allow for th saml to b imlatd with hydrog ad aald for oly low tmraturs i th rag of 300 C to 600 C or, rfrably, ot aald at all. h mthod of formatio xlord i this txt will b a Mtal-Isulator-Smicoductor (MOS) Caacitor. Additioally, a mthod for aalysis of DLS sigals ast th tyical mthodology associatd with DLS is rquird i ordr to xtract th most data from all masurmts. his w mthod of aalysis will likly rquir th dfiitio of a thortical fuctio rrstig a DLS sigal ad may of th o-idalitis coutrd wh modlig a DLS ak with a simlistic modl. his w mthod may also rquir w mirical aalysis mthods i ordr to aalyz DLS sctra i a mor fficit, accurat fashio. 5

14 1.2: Shockly Rad Hall (SRH) Rcombiatio Shockly Rad Hall (SRH) Rcombiatio is a four-athway kitic rocss for charg traig du to rcombiatio ctrs xistig i a smicoductor [6]. hs rcombiatio ctrs ar causd by imrfctios withi th smicoductor crystal, such as thos causd by Io Imlatatio. Ulik dirct mthods of rcombiatio, whr lctro-hol airs ar limiatd, roducig a hoto of rgy qual to th combid rgy lost by th two carrirs, SRH Rcombiatio is a idirct rocss. Elctros ad Hols rcombi by mas of momtum loss i a ilastic collisio at a tra ctr. h rgy dissiatd i this rcombiatio vt may b giv off as a lattic vibratio, or hoo, which roagats from th tra through th crystal. I SRH Rcombiatio, th four kitic athways which xist may b drivd from th bhavior of lattic imrfctios i th rsc of chargd articls. Most lattic imrfctios may b modld as sigl-chargd dfcts. hat is, thy may xist ithr as a utral scis or as a ioizd scis. his bhavior is similar to that of doat atoms which ar rcombiatio ctrs with vry shallow ioizatio rgis. Ioizatio or utralizatio may b achivd i two ways. A charg may b gaid or may b lost. A dfct which ioizs to form a ositiv (gativ) charg may ithr los a lctro (hol) or gai a hol (lctro) to bcom ioizd. o utraliz, this dfct must los a hol (lctro) or gai a lctro (hol). h act of gaiig a charg is rfrrd to as a catur vt. h act of losig a charg is rfrrd to as missio. Mor comlx ioizatio cascads do xist whr a dfct may hav mor tha o chargd stat. Howvr, ths cascads ar rar ad aalysis is comlx i comariso to th rlativly simlistic SRH kitics. yically, dfcts which xhibit ths cascads ar tratd as 6

15 clos ough to a sigl-ioizatio dfct to b aalyzd usig tyical SRH Rcombiatio statistics, articularly at cryogic tmraturs. Figur 2: A diagram of Shockly Rad Hall (SRH) Rcombiatio for a hol tra ctr - Elctros may b caturd (c ) from or mittd ( ) to th coductio bad. Hols may b caturd (c ) from or mittd ( ) to th valc bad. h kitics of catur ad missio ar vry similar. All dfcts hav som ara associatd with thm which rrsts thir shr of ifluc, calld a catur cross-sctio. If a carrir imigs withi this ara, it will itract with th dfct. his costituts a collisio btw th carrir ad th dfct. h umbr of collisios btw a carrir ad a dfct may b dfid as th umbr of istacs r uit tim whr th ara swt out by a dfct coms ito cotact with a carrir as th carrir udrgos its radom walk through th crystal lattic. It stads to raso, thrfor, that for a sigl dfct stat, th rat of collisios (f or f h ) is qual to th thrmal vlocity (v or v h ) ad satial dsity ( or ) of th carrirs multilid by th catur cross-sctio (σ or σ h ) of th dfct stat i qustio. It is imortat to 7

16 ot that th satial dsity of carrirs is, of yt, udfid bcaus th bad i qustio is ot yt dfid. I th cas of catur vts, fr carrirs (hols i th valc bad or lctros i th coductio bad) ar of itrst. I th cas of missio vts, boud carrirs (lctros i th valc bad or hols i th coductio bad) ar of itrst. h catur cross-sctio varis basd o th carrir ty. A dfct will hav a diffrtly sizd ara of ifluc for a hol as comard to a lctro. (1-1) (1-2) Of th collisios that ha i ay giv riod of tim, oly a crtai umbr will rsult i a ractio btw th carrir ad th dfct stat. h liklihood of this ractio is dtrmid by th statistical quatum mchaics of th ractio which is to tak lac. I th cas of carrir catur, th robability of a ractio occurrig durig a collisio is ffctivly uity bcaus carrirs fall ito a ottial wll. hs carrirs must b fr to mov about th crystal, so th dsity of carrirs which is of itrst for catur kitics is th dsity of fr carrirs, commoly rfrrd to as for lctros ad for hols. h carrir catur kitics ar as follows. (2-1) 8

17 (2-2) his is ot th cas for missio vts. Durig a missio vt, carrirs ar abl to ovrcom th cofis of a ottial wll ad ar mad fr to mov about th crystal lattic. his ffct imlis a thrmalizatio vt which must b rgulatd by som statistical mchaical distributio which is ddt uo th dth of th ottial wll as wll as th tmratur at which th smicoductor xists. Bcaus wh o carrir is mittd, it lavs bhid a carrir of th oosit ty, th missio of a carrir from a tra stat to a giv bad may b viwd as th catur of a oosit tra stat from th sam bad. So, th missio of a lctro (hol) to th coductio (valc) bad corrsods to th catur of a hol (lctro) from th coductio (valc) bad. h dsity of oosit carrirs withi th bad associatd with missio dtrmis th dsity which is usd dtrmi to th rat at which missio occurs. It is assumd that both carrir tys hav th sam dsity withi a giv bad. hus, th dsity of miority carrirs withi a giv bad, b it th coductio or valc bad is th sam as th dsity of majority carrirs withi that sam bad. h rsultig quatio for missio is as follows, whr P() or P(h) ar th robability fuctios associatd with th missio vt, N C is th dsity of stats i th coductio bad ad N V is th dsity of stats i th valc bad. (3-1) (3-2) 9

18 yically, carrir ad tra dsitis ar low withi a smicoductor ad classical Boltzma Statistics may b alid to th missio robability. h tra lvl is assumd to b th groud ottial i this cas, with th rgy of th barrir qual to th diffrc i rgy btw th bad to which th missio is occurrig ad th tra rgy. h rsult is th fial xrssio for hol ad lctro missio robability, whr E is th tra rgy, E C is th coductio bad dg rgy, E V is th valc bad dg rgy, k is Boltzma s Costat ad is th tmratur of th smicoductor i Klvis. h diffrc btw E ad E C or E V may also b xrssd as qual to a activatio rgy, E. (4-1) (4-2) Giv th costraits o SRH kitics listd abov, th diffrtial quatios dfiig th trasit charactristics of hol-filld ad lctro-filld traig stats may b dscribd as follows whr th variabl t is th tim associatd with th trasit rocss. It is imortat to ot that ths trms dscrib th kitics of formatio assumig a coctratio of tra stats qual to uity. d dt = (5-1) ( + c ) ( + c ) 10

19 d dt = (5-2) ( + c ) + ( + c ) I ordr to dtrmi th tru rats ad th tru trasit coctratio of tra stats, th rats abov ad thir associatd coctratios must b multilid by th tru coctratio of traig stats. For th sak of simlicity, w rdfi th rcdig diffrtial quatios as follows, assumig that ay tra which is ot filld with a hol must b filld with a lctro. d dt d dt = K ) ( K + K ) (6-1) ( = K ) ( K + K ) (6-2) ( I th simlst cas of SRH Rcombiatio, oly two charg stats ar rmittd for a sigl dfct structur, a utral stat ad a ositivly or gativly chargd stat. I som cass, most otably for vacacis i smicoductors, it is ossibl for multil charg stats to xist ast th utral stat. hs cass rrst dyamics which rquir far mor comlx aalysis ad will ot b tratd i this txt. 1.3: D Lvl rasit Sctroscoy (DLS) D Lvl rasit Sctroscoy (DLS) bga as a diffrtial caacitiv tchiqu dvlod by D.V. Lag i 1974 [4] ad has sic xtdd to a varity of othr diffrtial charg ssig mthods, icludig Charg rasit Sctroscoy (QS), Costat Caacitac 11

20 D Lvl rasit Sctroscoy (CCDLS), Currt rasit Sctroscoy (CS) ad Photo-Iducd rasit Sctroscoy (PIS). hs various ssig mthods may b alid to a larg varity of dvics icludig Schottky Diods, PN Juctio Diods, MOS Caacitors, MOS rasistors, Biolar Juctio rasistors ad so o. h mthodology commo to ths tchiqus is th obsrvatio of charg dcay i a smicoductor at cryogic tmraturs. Figur 3: Gratio of a D Lvl rasit Sctroscoy sigal. his is achivd by moitorig th variatio i th chag i th dcayig ormalizd caacitac ( C/C O ) dfid by th discrt diffrtial with rsct to tim samld at two sarat tims, t 1 t D ad t 2 t D (whr t 1 ad t 2 ar costats ad t D is th tim dlay ). his sigal ca b siusoidal, as i th cas of a lock-i tchiqu, or a squar wav of variabl duty cycl, as is th cas i a boxcar diffrtiatio tchiqu. Durig boxcar 12

21 diffrtiatio, th sigal is dsigd to bias th structur i such a way that tras fill durig a short fillig uls (labld hraftr as t f ) ad th mit wh th sigal trasitios to a much logr missio uls (hraftr labld t ). For th uross of modlig, this missio uls may b assumd to b ifiit. Figur 4: h ottial sttigs (yllow), majority carrir tra (rd) ad miority carrir (blu) traig robabilitis for a gric DLS trasit. Masurmt of th charg trasit occurs durig th missio uls, as it is tyically domiatd by thrmalizatio of carrirs from filld tra stats. Boxcar diffrtiatio is achivd by masurig th charg lvls rst withi th smicoductor at two sarat oits, t 1 ad t 2, i ordr to driv a xrssio for th discrt diffrtial charg. I this situatio, th fillig uls acts as a mas of dfiig th iitial coctratio of trad charg, th trasit of which is obsrvd durig th missio uls. With ths costraits i mid, w may driv a xrssio for th obsrvabl dcay i filld tra stats ovr tim. W obsrv i quatios (7) ad (8) that th diffrtial 13

22 corrsodig to this dcay is a liar 1 st Ordr Diffrtial Equatio ad solv it i trms of boudary coditios dfid by th fillig uls, 0 ad 0. K K + K + [ ( K+ K 0 ) t(1,2)] = (7-1) K [ ( K+ K ) t(1,2)] = + 0 (7-2) K + K h quilibrium coditio attaid as th masurmt tim is allowd to rocd to ifiity is th first trm i th sums listd abov. h scodary trm rrsts th trasit dcay of th charg stat from th coditios st u by th fillig uls to th quilibrium coditios ihrt to th coditios dscribd as art of th missio uls. hus, th magituds of th trasit costats, 0 or 0, ar qual to th diffrc btw th iitial trad charg coctratio (dfid by th fillig uls) ad th fial coctratio which th missio trasit aroachs asymtotically as tim icrass to ifiity. hs trasit costats may b dfid as is show blow. It is imortat to ot that th kitic rats (K f ad K ) ar dfid by th carrir coctratio xistt durig fillig ad missio ulss ad, thrfor, hav diffrt valus durig ths diffrt ulss. K 0 = FP (8-1) K + K 14

23 15 N K K K F + = 0 (8-2) h trms F P ad F N ar th trad carrir coctratios achivd at th d of th fillig uls trasit (wh th trasitio is mad from th fillig uls to th missio uls). h coctratio at th d of th fillig uls may b dfid as follows. ] ) ( [ f f f t K K P f f f P A K K K F = (9-1) ] ) ( [ f f f t K K N f f f N A K K K F = (9-2) A N ad A P rrst th diffrc btw th trad charg coctratio rst at th start of th fillig uls (which is qual to th stady stat coditio of th missio uls) ad th stady-stat trad charg coctratio that would rsult from a ifiit fillig uls. + + = f f f P K K K K K K A (10-1) + + = f f f N K K K K K K A (10-2) With th quatios listd abov, th trasit obsrvd durig aalysis with DLS may b fully dfid for ay tim durig th missio uls as wll as for ay fillig uls width

24 rovidd that th missio uls coditios ar xistt for a log ough tim that th stat of th smicoductor may b cosidrd to b i thrmal quilibrium. his charg trasit is diffrtiatd, tyically by ssig th diffrtial charg at two tims durig th dcay ad subtractig th two masurmts. yically, th assumtio of comlt fillig rior to th missio uls is mad ad th quatio most tyically usd to dfi th rsultig diffrtial sigal from this charg dcay is as follows. (11-1) (11-2) Figur 5: Diagrams of th fr carrir bhavior of a sigl-sidd juctio durig th thrmal quilibrium iducd by th Fillig Puls ad durig th trasit stat of o-quilibrium immdiatly followig th alicatio of th Emissio Puls. hs diagrams rrst th bhavior of a hol tra. I a lctro tra, trad carrirs would b mittd to th Coductio Bad. 16

25 A sctrum is cratd by lottig this diffrtial ovr a wid rag of tmraturs (as i Figur 3) with ach ak i th sctrum rrstig a diffrt tra scis. By itratig a sca with variabl valus of t 1 ad t 2, a family of sctra may b cratd. A imortat mtric that dfis ths sctra is th rat widow, τ, which is a fuctio of t 1 ad t 2. his rat widow corrsods to th ivrs missio rat at th ak maximum. It is dfid as follows. (12) h activatio rgis ad catur cross-sctios of ths tra scis ca b dtrmid via a Arrhius Plot of th tmratur corrsodig to th ak maximum, τ m. I this Arrhius Plot, th atural logarithm of m 2 τ m is lottd agaist 1/k m. h rsult of this lot is a xrssio for th diffrc btw th tra rgy ad th bad rgy, E, which is rrstd by th slo of th Arrhius Plot. h lot also yilds th Catur Cross-Sctio of th tra, which corrsods to th oit at which th Arrhius lot itrcts a valu of 1/k m qual to zro. 17

26 Figur 6: Arrhius lottig of a sigal arisig from a sigl d lvl tra obsrvd usig D Lvl rasit Sctroscoy. h slo of th drivd Arrhius Plot yilds th activatio rgy of th d lvl tra whil th oit whr th d lvl tra itrcts th 2 τ axis yilds th catur cross-sctio. (Courtsy J. Sawirat of Corig, Icororatd) A valu for th coctratio of tra stats rst withi th smicoductor may b drivd from th xrssio for th dltio caacitac of a sigl-sidd juctio. It is as follows, whr C o is th caacitac at ifiit tim, C M is th chag i caacitac obsrvd at th maximum of th DLS sigal, N is th tra coctratio ad N A is th doat coctratio i th smicoductor. (13) 18

27 1.4: Multil Itral rasmissio Ifrard (MI-IR) Sctroscoy Multil Itral rasmissio Ifrard (MI-IR) Sctroscoy is a Fourir rasform Ifrard (FIR) Sctroscoy tchiqu which is xctioally wll-suitd to th masurmt ad obsrvatio of bodig ad structural dfcts both at matrial itrfacs as wll as withi bulk matrials [5]. MI-IR couls light gratd by a ifrard sourc ito a bulk matrial through th us of a rism with its dgs bvld at a rcis agl. his rism has a matchig rfractiv idx i comariso to th idx of th matrial big obsrvd, oft bcaus it is formd from th sam matrial. Bcaus of this, light may b asily could ito th saml big valuatd aftr big trasmittd ito th rism. As this light asss through th saml, it vtually will coutr th back itrfac btw th saml ad th ambit ad rflct back i th dirctio of th rism. h rism itslf is tchd i th ctr i ordr to cofi th light to th saml util it rachs th othr sid of th rgio big robd, at which oit th light couls back ito th rism ad th back out ito th ambit. At this oit it is samld by a dtctor. h itsity of th light rflctig through th saml is dissiatd by th rflctiv losss causd by imrfct rflctio at saml itrfacs, to trasmissio loss of icidt light flowig ito ad out of th rism ad, most imortatly, to absortio of th ifrard bam withi th saml du to both th saml matrial itslf as wll as dfcts iducd by io imlatatio withi th saml. 19

28 Figur 7: MI-IR Saml Dsig h guidd ifrard bam striks th rism at a icidt agl, θ. h agl of th rism ormal with rsct to th saml, α, dtrmis th agl at which th bam striks th bottom surfac of th saml, Φ. R dfis th xtt of th imlat damag rofil. rasmissio ito ad out of th rism as wll as rflctios off of ach to ad bottom surfac of th saml cotribut to attuatio. h itsity loss du to ach of ths comots is ddt uo th umbr of boucs th ifrard bam udrgos durig its trasit through th saml. h ffctiv athlgth that th light travls ovr durig its boucig trasit through th saml is dtrmid by th icidt agl of th light i comariso to th bvl, th bvl agl i comariso to th surfacs of th rism ad th dimsios of th saml ad th rism thmslvs. I ordr to rorly aalyz th attuatio xricd by th ifrard bam durig its trasit through th combiatio of th rism ad saml, th ffctiv ath-lgth of th light must b stimatd. I ordr to rform this stimatio, th icidt agl of th ifrard bam i rlatio to th ormal of th bvld dg of th rism, θ, must b kow. h agl of th bvl i rlatio to th bottom surfac of th rism, α, must also b kow alog with th lgth of th bottom surfac of th rism, d, ad th thickss of th saml, t. h ath that th icidt bam taks aftr assig through th rism bvl may b calculatd by makig us of Sll s Law with th imlicit assumtio that th xtictio cofficit is much lowr i magitud tha th rfractiv 20

29 idx. I this drivatio, V is th rfractiv idx of th ambit closig th rism ad saml ad P is th rfractiv idx of th rism ad, by associatio, th rfractiv idx of th saml. By otig that th itrsctio of th bvl with th icidt bam ad with th bottom surfac of th rism forms a triagl, th followig quatio may b statd rlatig th agl of th trasmittd bam with th bottom surfac of th rism, ϕ [7]. (14) Figur 8: h gomtry formd by a ifrard bam as it asss through th combiatio of a rism ad saml with similar rfractiv idics. h ath of th ifrard bam through th rism ad through th itrfac btw th rism ad saml dfis a triagl ad this gomtry may b usd to calculat Φ. 21

30 his agl dfis th agl at which th ifrard bam boucs off of ach horizotal itrfac o th saml. From th agl, ϕ, th ath lgth may b calculatd to a first ordr by calculatig, first, th distac travld by th ifrard bam i th saml durig a sigl bouc ad, scodly, calculatig th umbr of boucs that th ifrard bam udrgos off of ithr surfac of th saml. h ath lgth r bouc of th ifrard bam is simly th thickss of th saml dividd by th cosi of th agl. From th assumtio that th icidt bam trs ad lavs th saml at th dgs of th rism, th umbr of boucs,, may b rlatd to this ath lgth r bouc by cosidrig th horizotal distac travld r bouc, which is simly th ath lgth of a sigl bouc multilid by th si of th icidt agl, ϕ. Dividig th total lgth of th bottom surfac of th rism by this distac travld for a sigl bouc yilds th followig xrssio for by virtu of th obsrvatio that a bouc off of th bottom surfac is a mirror imag of a bouc off of th to surfac. (15) A bouc is a discrt valu ad a v umbr of boucs ar rquird for th bam to trasit through th saml ad b collctd by th rism at th othr d, so may b roudd dow to th arst v itgr. h total ath lgth may b rlatd to th umbr of boucs by rcogizig that th umbr of tims that th bam trasits from th bottom of th saml to th to or from th to to th bottom is roughly qual to. hus, th total ath lgth, x, may b xrssd as follows. 22

31 (16) Figur 9: h drivatio of th athlgth of th ifrard bam through th saml big valuatd usig MI-IR Sctroscoy. h athlgth is qual to th athlgth associatd with a sigl bouc, x, multilid by th umbr of boucs,. th athlgth r bouc is dfid by th horizotal travl r bouc, dtrmid by Φ, as wll as th saml thickss. h attuatio of th ifrard itsity du to th ath tak by th ifrard bam through th saml may b xrssd by cosidrig svral diffrt factors. h ratio of itsity btw th icidt ifrard bam ad th bam trasmittd through th rism is a fudamtal limit to th obsrvd itsity. At ach bouc, oly a giv amout of ifrard rgy will b rflctd back ito th saml. A rasoabl first ordr assumtio is that all rgy ot rtaid by total itral rflctio withi th saml is lost. For a giv matrialrism systm, ddig uo th olarizatio of th ifrard bam as wll as th surfac roughss of th itrfacs, this ratio of th rflctd itsity to th trasmittd itsity will 23

32 vary. By usig samls olishd o both sids, th ffcts of surfac roughss ca b gatd, lavig oly a ddc uo th olarizatio of th ifrard bam. h rsultig variatio i rflctd itsity for th itrfac btw th rism or saml ad th ambit is dfid as follows, whr Rs is th rflctd itsity ratio for a rdicularly olarizd bam ad R is th rflctd itsity ratio for a aralll olarizd bam. As bfor, it is assumd that th xtictio cofficit is much smallr i magitud tha th rfractiv idx ad, as such, dos t hav a ffct o th obsrvd rflctivity. h trm ϕ rrsts th agl of th trasmittd bam which may b xrssd usig Sll s Law [8] whil V is th rfractiv idx of th ambit ad P is th rfractiv idx of th rism. (17) (18) h rasmissio ito ad out of th rism may b xrssd usig ths sam rflctivity trms as th trasmissio is simly qual to 1-R. I th cas of trasmissio ito th rism, th icidt agl of th ifrard bam rlacs th rflctd agl off th bottom ad to surfacs of th rism ad V ad P ar switchd bcaus th trasmittd bam o logr travls from th rism ito th ambit, but from th ambit ito th rism. I th cas of trasmissio from th rism to th dtctor, V ad P rmai as i th quatio ad th agl of itrst is qual to th diffrc btw th bvl agl, α, ad th rflctd agl, ϕ. his is quivalt to th agl (ϕ ) rsultig from th rfractio of th icidt bam as it asss from th ambit 24

33 to th rism. his xrssio may b drivd usig Sll s Law for th agl of th icidt bam with rsct to th rism bvl, ϕ. h attuatio of th ifrard bam du to th bulk of th saml may b dtrmid by alyig th Lambrt-Br Law for itsity loss i a absorbig mdium. I this drivatio, two sarat absortio vts occur. h saml itslf, irrsctiv of ay xistig Io Imlat Damag will absorb radiatio as this radiatio asss through it. h rat at which this absortio occurs ovr a uit ath lgth is qual to th trm γ, which is rlatd to th absortio factor of th rfractiv idx, k, as follows, whr λ is th wavlgth of th light gratd i th ambit. Imlatatio. I additio, radiatio will b absorbd by ay damag rsultig from Io h rat at which radiatio is absorbd by Io Imlatatio Damag r uit athlgth may b xrssd as qual to th damag coctratio, N, multilid by a catur cross sctio, σ. I additio, a scalig costat must b multilid i to accout for th ratio of th athlgth ovr which Io Imlat Damag cotributs to absortio as comard to th total athlgth. his trm may b xrssd as th ratio btw th Projctd Rag of th imlat, R, ad th total saml thickss, t. h sum of ths two trms rrsts a xrssio for th total attuatio of radiatio for a giv uit athlgth [9]. (19) (20) 25

34 h quatity of light trasmittd ovr th full athlgth of th ifrard bam i th saml may b xrssd as follows. I this quatio, I 0 rrsts th bam itsity rior to travlig through th rism whil I rrsts th bam itsity oly du to Lambrt-Br attuatio aftr travlig through th saml. (21) h total attuatio for a o-absorbig saml may b xrssd by multilyig th loss du to trasmissio ito ad out of th rism by th attuatio du to th absortio of th saml as wll as by th total umbr of rflctios off th to ad bottom saml surfacs. I th cas of a saml with Io Imlat Damag, th Br-Lambrt Law rquirs that both th trm rrstig th absortio of th bulk saml, γ, ad th trm rrstig th absortio du to Io Imlat Damag b icludd. his is ot th cas for a basli saml, whr o Io Imlatatio occurs ad oly γ must b icludd. h rsultig trms for th attuatio of th bam itsity ar as follows, whr I BL is th itsity associatd with a basli saml whil I S is th itsity gratd by th io imlatd saml. (22) (23) 26

35 It is worth otig that divisio of th sctrum of a saml with th stats of itrst by a basli stadard without ay stats will rsult i a diffrtial sctrum which may b usd for aalysis. R, t, x (dfid i Equatio 10) ad, idally, σ, may b cosidrd costat. hus, aftr takig th atural logarithm of th ratio btw th ifrard sigal gratd by absortio i th imlatd saml ad th basli ifrard sigal, a xrssio may b drivd that dirctly corrlats th dsity of articular scis of io damag to th itsity ratio of a io imlatd saml to a basli saml. his xrssio is as follows. (24) 1.5: Hydrog Imlat Dfcts i Smicoductors A varity of studis hav b rformd i ordr to obsrv th structural ad lctroic dfcts arisig from Hydrog Imlatatio or Hydrog-Hlium Co-Imlatatio, both for lowfluc studis as wll as for io cuttig rlatd studis. Structural dfct studis hav icludd such aalytical tchiqus as rasmissio Elctro Microscoy (EM), Atomic Forc Microscoy (AFM), Ruthrford Backscattrig Sctroscoy (RBS), ad Multil Itral rasmissio Ifrard (MI-IR) Sctroscoy. h ffcts of Hydrog Imlatatio ad Hydrog-Hlium Co-Imlatatio as studid by MI-IR wr of articular itrst i rlatio to this study MI-IR has ickd u vibratioal mods ad bdig mods of a varity of stats. hs stats iclud trad molcular hydrog ad hydrog-bodd itrfacs corrsodig to crystal facts of dislocatio loos. hs stats 27

36 corrsod to 2084 wavumbrs for th <111> fact ad 2110 for th <100> fact. I additio, AR-IR Sctroscoy has dtctd th rsc of moovacacis dcoratd with aywhr from o to four hydrog atoms as wll as dcoratd multivacacis, articularly hydrogdcoratd divacacis ossssig aywhr from o to six bodd hydrog atoms. VH, VH2, VH3, ad VH4 hav b obsrvd to ossss strtchig mods i th 2025, 2125, 2161/2188 ad th 2219 wavumbr rags, rsctivly. h rsc of hydrog-dcoratd silico itrstitial dfcts has also b ostulatd basd o th rsc of rsoat stats i th 2000 to 1950 wavumbr rag [10]. Elctroic dfct studis hav icludd studis at room tmratur of sudo-doig ffcts usig tchiqus such as Hall Effct Rsistivity ad Hall Effct Mobility as wll as Four Poit or wo Poit Prob Rsistivity. hs studis hav obsrvd svral gral trds i carrir dsity with rsct to aalig tmratur. Hall Effct Rsistivity studis hav dmostratd that -ty samls imlatd with Hydrog or a Hydrog-Hlium combiatio at high flucs maitai thir -ty bhavior u util aroximatly 450 C, at which oit a rdomiac of -ty tras, calld thrmal doors, comsat ad ivrt th smicoductor matrial. his ffct is maitaid util aroximatly 500 C, at which oit th majority of thrmal doors bgi to dissociat, lavig i thir wak th backgroud -ty doig of th smicoductor as wll as a host of thrmal acctors which rmai stabl at lvatd aalig tmraturs u to aroud 700 C. h rsult of this ffct is th rsc of hol carrir coctratios i xcss of th backgroud doig. Cryogic Studis hav mad us of Elctro Paramagtic Rsoac as wll as Lalac DLS ad traditioal DLS mthods to dtct scific rgy lvls ad catur crosssctios associatd with d lvl tras [3]. Aalysis of Hydrog Imlatatio dfcts has 28

37 largly b cofid to study of lctro traig i -ty matrials. Hydrog imlatd - ty matrials maifst thmslvs i a varity of cofiguratios. VH 1, a sigl vacacy dcoratd with a sigl hydrog atom maifsts itslf as a lctro tra with a activatio rgy for missio corrsodig to.43v to.44v. A lctro tra with a missio activatio rgy of.49v has b validatd xrimtally ad thortically as a V 2 H dcoratd divacacy dfct. h VO dfct, corrsodig to a vacacy-oxyg bod is trasformd from a.15v lctro tra ctr to a.32v lctro tra aftr thrmal aalig. VH, ad V2H hav b obsrvd to dcay aftr thrmal aalig i th rag of 500 C to 550 C whil VOH rquirs mor thrmal rgy for th dissociatio ractio to tak lac, ot aalig out util aroud 600C. VH osssss vry similar symmtry to a similar dfct, VP, which is th dfct arisig from icomlt activatio of a hoshorus atom i a vacacy sit. his has ld to som sculatio that VH acts as a Psudo Grou V Elmt [11]. 29

38 Figur 10: Aalig bhavior of lctro tras gratd by H + imlatatio [11]. Som ivstigatio of hydrog-iducd acctor stats has occurrd. Howvr, th acctor stats arisig from hydrog imlatatio hav ot b studid i arly as much dtail. hr is littl udrstadig of th structural imrfctios that lad to ths dfcts. A dfct at.28v abov th valc bad has b obsrvd by multil authors ad th ossibility of this dfct big causd by hydrog bodig has b discussd. Howvr th similaritis i aalig charactristics btw this dfct ad th Ci-Oi (or carbo-oxyg itrstitial bod) dfct has ruld this ossibility out [12]. A additioal dfct at.51v abov th valc bad has also b obsrvd. his dfct is currtly attributd to a boro-hydrog itrstitial comlx [13]. wo additioal tras xistig at.33v ad.66v hav b xtractd usig 30

39 DLS ad th roosal has b mad that ths dfcts corrsod to VH 2 ad VH 3 dfcts, rsctivly [14]. Figur 11: DLS Sctra of a as-formd P-y saml imlatd with H+ ios at 300 KV [13]. h rsc of itrstitial Carbo-Oxyg as wll as uboudd Vacacy dfcts may also b xctd to cotribut to sigals arisig from Hydrog or Hydrog-Hlium Imlatatio xrimts by virtu of th fact that ay hysical bombardmt of a Cz-grow saml will iduc oit dfcts icludig vacacis, silico itrstitials, itrstitial doat atoms ad itrstitial carbo ad oxyg. Vacacy dfcts do ot hav a sigl charg stat. Rathr, thy may xist i a ositiv stat, which dcays to utrality (V + corrsodig to.05v+e V ) or gativ charg stat, which dcays to utrality (V - corrsodig to.32v-e C ) as wll as a doubl ositiv charg stat which dcays to a ositiv stat (V ++ corrsodig to 31

40 .13V+E V ) [3]. As statd arlir, itrstitial carbo has b foud to xist i a stat corrsodig to.28v abov th valc bad as wll as i a stat corrsodig to.1v blow th coductio bad [12]. Itrstitial carbo may also bod with substitutioal carbo to form a C i -C s bod with a rgy corrsodig to.33v abov th valc bad [15]. A additioal stat, corrsodig to a substitutioal carbo bod with a divacacy xists at.09v blow th coductio bad ad has dmostratd bistability [16]. Aftr aalig at a tmratur of 300 C or mor, this stat dcays, yildig a w stat at.17v blow th coductio bad [17]. Carbo may bod with Oxyg i a itrstitial cofiguratio to form a C i -O i bod corrsodig to a stat at.38v abov th valc bad [18]. Figur 11: DLS Sctra of hol tras i P-y Silico silico imlatd with Silico at 1MV [3]. 32

41 Chatr 2: DLS Fittig Fuctios 2.1: Simlificatios to th Charg rasit h dual-boxcar tchiqu usd i DLS to roduc a sigal is a mthod of discrt diffrtiatio with rsct to tim. his is accomlishd by subtractig th sigal big masurd at two sarat tims. With this i mid, it stads to raso that a fittig fuctio for th rsultig discrt diffrtial could b roducd simly by diffrtiatig th tra catur/missio fuctio listd abov with rsct to th missio tim ad that this fittig fuctio may b accurat as a first ordr aroximatio. h rsult of such a diffrtiatio is as follows, barig i mid that th kitic rats dscribd blow oly aly to coditios ihrt to th missio uls, ot th fillig uls. d [ ( K K ) tdt(1,2)] = 0 ( K+ K ) + (1-1) dt d [ ( K K ) tdt(1,2)] = 0 ( K+ K ) + (1-2) dt hs quatios ar adquat to dscrib th rsultig trasit of a I-DLS masurmt sic th currt is litrally th diffrtial of th trad charg coctratio with rsct to tim. I th cas of a caacitiv or voltag-basd masurmt, this diffrtiatio must b covrtd to a chag i trad carrir coctratio alo, which is accomlishd by saratig th chag i tim from th chag i trad carrir coctratio as follows whr 33

42 34 th chag i tim, t, is qual to th diffrc btw t D t 1 ad t D t 2. I this cas, th fuctio, t, is th avrag of t D t 1 ad t D t 2. t K K K K t ) ( 0 ) ( + + = (2-1) t K K K K t ) ( 0 ) ( + + = (2-2) h rsult of this diffrtiatio is a akd fuctio. It is radily obsrvabl that th fuctio dscribd hr has a magitud which is ddat ot oly o th coditios ad duratio of th fillig uls (idicatd by 0 ad 0 ), but also by th rat of catur ad missio of carrirs durig th missio uls. his fuctio is a rasoabl first-ordr aroximatio of th tru DLS sigal rsultig from a voltag-basd or caacitiv masurmt, which is a discrt diffrtial. his quatio is as follows. ( ) ] ) ( [ ] ) ( [ 0 2 t 1 t K K t t K K D D + + = (3-1) ( ) ] ) ( [ ] ) ( [ 0 2 t 1 t K K t t K K D D + + = (3-2) I this discrt form, t 1 ad t 2 rrst th first ad scod tims at which th missio trasit is samld through boxcar itgratio.

43 hs quatios rmai comlx, v wh simlifid to th first ordr aroximatio giv by th diffrtial form listd abov. If furthr rductio of th quatio is rquird, it is rasoabl i som circumstacs to assum that th rats dscribd abov may b simlifid by rducig th four-rat quatios abov dow to two or ossibly o rat. h law of mass actio cssitats that i ay smicoductor whr th Frmi Ergy is sigificatly dislacd from th itrisic rgy oly o carrir ty will b rst i larg umbrs withi th matrial. hus, i th cas of a tra xistig at a oit whr th Frmi Ergy is sigificatly dislacd from th itrisic rgy, oly o catur rat will domiat ad th othr may b glctd. h critria for this trasitio from a two-rat catur kitic to a sigl-rat kitic is a oit whr th catur rat for o carrir is lss tha o tth of th catur rat for th othr carrir. By ivokig th law of mass actio ad saratig th catur kitics ito thir costitut catur cross-sctios ad thrmal vlocitis, th followig quatios may b dfid for th hol coctratio (lctro coctratio) rquird for th domiac of th lctro catur kitic (hol catur kitic). σ 10 v < i σ v 1/ 2 (4-1) σ v < i 10σ v 1/ 2 (4-2) 35

44 Figur 1: h maximum miority carrir coctratios rmissibl i a smicoductor i ordr for o catur rat to domiat i a DLS masurmt (4-1 ad 4-2). his ddc is rlatd to th ratio btw th hol catur cross sctio ad th lctro catur cross sctio ad corrsods to th maximum carrir coctratios rmissibl for carrirs to fill tra stats durig a missio uls. It is ossibl for both catur trms to rmai domiat i comariso to o aothr i cass whr th Frmi Ergy aroachs th itrisic rgy. Howvr, i such a situatio, th missio rats associatd with thrmal missio of carrirs from th tra will domiat th rats associatd with carrir catur, ad both catur rats may b igord. h critria for a cas whr th catur rats for a giv tra ar sufficitly small i comariso to thir corrsodig missio rats is whr th catur rat is lss tha o tth of its corrsodig missio rat. his may b xrssd i trms of carrir coctratios or i trms of th rgy lvl of th tra i qustio. Exrssig this critria i trms of th lctro or hol coctratio yilds th followig quatios, whr th trm γ is th roduct of th thrmal vlocity, th dsity of stats ad th squard ivrs tmratur. 36

45 γ < 10 v γ < 10 v 1 2 ( E EV ) / k 1 2 ( E E ) / k C (5-1) (5-2) Figur 2: h maximum hol coctratio for tra missio to occur with varyig tmratur ad varyig tra rgy barrirs (5-1 ad 5-2). Giv th bhavior of Boltzma Statistics, th obsrvd trds (amly that th maximum hol coctratio icrass with tmratur ad dcrass with rgy barrir) ar to b xctd. Figur 3: h maximum lctro coctratio for tra missio to occur with varyig tmratur ad varyig tra rgy barrir (5-1 ad 5-2). Giv th bhavior of Boltzma Statistics, th obsrvd trds (amly that th maximum lctro coctratio icrass with tmratur ad dcrass with rgy barrir) ar to b xctd. 37

46 It is imortat to ot that th domiac of ay missio trm ovr its corrsodig catur trm is comltly iddt of th catur cross-sctio of th tra scis i qustio. h rcdig critria may b r-xrssd for th tra rgy as follows. 2 [ l(10 ) l( v ] EV γ (6-1) E < k ) + E 2 [ l(10 ) l( v ) ] > E k γ (6-2) C Figur 4: Variatio of th maximum rgy lvl rsolvabl with rsct to th valc bad for thrmal acctor missio with rsct to tmratur ad hol coctratio withi smicoductor (6-1 ad 6-2). As tmratur icrass, th maximum rsolvabl rgy lvl also icrass, as xctd. It is also xctd that th maximum rsolvabl rgy lvl dcrass with rsct to hol coctratio. 38

47 Figur 5: Variatio of th miimum rgy lvl rsolvabl with rsct to th coductio bad for thrmal door missio with rsct to tmratur ad lctro coctratio withi smicoductor (6-1 ad 6-2). As tmratur icrass, th miimum rsolvabl rgy lvl also dcrass, as xctd. It is also xctd that th maximum rsolvabl rgy lvl icrass with rsct to lctro coctratio. It is tyically th cas that o missio trm will domiat ovr th othr trm for ay tra with a rgy lvl sufficitly largr or smallr tha th rgy associatd with mid-ga. For stats xistig ar th mid-ga rgy or for stats with sufficitly larg diffrcs i hol ad lctro catur cross-sctios, it may ot b radily aart as to whthr o missio trm domiats ovr th othr or if both missio trms rmai domiat. It may b assumd that oly o missio rat will rmai domiat if o of th two missio rats is lss tha o tth th magitud of th othr rat. With this critrio i mid, th tra rgy rquird for o missio rat to domiat is as follows. E < 1 Ec 2 + E V γ σ l (7-1) 10γ σ 39

48 E < 1 Ec 2 + E V γ σ l 10γ σ (7-2) Figur 6: h maximum rsolvabl rgy ga for a hol or lctro tra with rsct to th catur cross sctio diffrc associatd with th hol ad lctro missio rats (7-1 ad 7-2). It is to b xctd that as th ratio btw hol ad lctro catur cross sctio icrass, th maximum rsolvabl rgy ga for hols icrass whil th maximum rsolvabl rgy ga for lctros dcrass. I th cas of a mtallurgical juctio whr ioizd charg xists largly o o sid or th othr, two sarat rgios may b rsumd to xist. Whr th dltio aroximatio is accurat, a rgio whr missio domiats ovr catur xists. I this rgio, th miority carrir catur rat may also domiat, scially ar th mtallurgical juctio. Howvr, bcaus th miority carrir catur cross-sctio is tyically quit small, th miority carrir catur rat is tyically domiatd by th majority carrir missio rat. As a cosquc, oly o rat is domiat i this rgio. For tra lvls with rgis away from mid-ga whr th dltio aroximatio is valid, th followig quatios ar alicabl. 40

49 41 t t = 0 (8-1) t t = 0 (8-2) Similarly, th followig quatios rsult for th discrt form of th DLS sigal. ( ) t t t t D D = (9-1) ( ) t t t t D D = (9-2) As th dltio aroximatio bgis to fail ar th dg of th dltio rgio, a scod rgio xists whr catur may o-logr b glctd. I th cas whr th dltio aroximatio is ot valid, th majority carrir catur rat may ot b igord ad th followig quatios aly. It is imortat to ot that whil this is, tchically th form obsrvd for rgios ar th dltio dg, th fillig dyamic of th tra lvls withi this rgio lads to icomlt or o fillig. t c c t ) ( 0 ) ( + + = (10-1) t c c t ) ( 0 ) ( + + = (10-2) h followig quatios rsult for th discrt form of th DLS sigal. ( ) 2 1 ) ( ) ( 0 t c t c + + = (11-1)

50 0 ( + c ) t1 ( + c ) t2 ( ) = (11-2) It is imortat to ot that i th cas of a tra which xists clos to mid-ga, o assumtios ca b mad rgardig th domiac of missio or catur rats for ithr carrir, ad th full form of th quatio must b usd. h trms 0 ad 0 rmai comlx du i larg art to th trasit homa ivolvd i th fillig uls. Svral simlificatios may b mad to ths trms i ordr to rduc th ovrhad ihrt i calculatio. h trms dscribd may b simlifid substatially if flatbad coditios ar assumd durig th fillig uls. h fillig uls is tyically ot rformd such that flat-bad coditios ar achivd throughout th smicoductor. Oft, som dltio ffcts rmai at th mtallurgical juctio. As a rsult, aras xist whr tras ar ot filld durig th fillig uls ad, as a rsult, o missio occurs. hr ar also rgios whr fillig is ot comlt as a cosquc of artial dltio ad, as a rsult, attuatio of th DLS sigal occurs. hs two cass ar structural cosidratios ad may b artially aroximatd with a attuatio factor, as will b discussd i a latr sctio. I th cas of fillig of majority carrirs ad missio of miority carrirs, it may b assumd that all majority carrir tras will b filld ad all miority carrir tras will b mtid durig th catur uls coditios giv a ifiit tim for fillig. A rlatd assumtio is that all miority carrir tras will b filld ad all majority carrir tras will b mtid durig th missio uls if th uls is allowd to ru to comltio. h assumtios ar rvrsd for th fillig of miority carrirs ad missio of majority carrirs. 42