Transport mechanisms of electrons and holes in dielectric films

Transcription

1 Pysics ± Uspki 56 (0) 999 ± 0 (03) #03 Uspki Fizicskik Nauk, Russian Acadmy of Scincs FROM THE CURRENT LITERATURE PACS numbrs: 7.0.Ht, 7.0.Jv, 7.80.Sk, Sx Transport mcanisms of lctrons and ols in dilctric films K A Nasyrov, V A Gritsnko DOI: /UFN Contnts. Introduction: t importanc of undrstanding lctron and ol transport mcanisms in silicon dvic dilctrics 999. Two-band conduction in dilctrics Contact-limitd mcanism of lctron and ol injction Fowlr±Nordim tunnling injction at a contact; 3. Trmally assistd contact tunnling; 3.3 Trmoionic contact mission: t Scottky ffct; 3.4 Trap mdiatd injction in silicon nitrid 4. Trap ionization modls Trap ionization by t Frnkl mcanism; 4. Multiponon ionization of traps; 4.3 Carg transport by trap-to-trap tunnling 5. Spac carg in a dilctric and t Sockly±Rd±Hall quations Two band conduction and trap paramtrs in Si 3 N Conduction in silicon-nricd silicon nitrid Monopolar lctron conduction in Al O Conclusion 0 Rfrncs 0 Abstract. Elctron and ol transport mcanisms in amorpous silicon oxid, silicon nitrid, and aluminum oxid, dilctric matrials of ig rlvanc to silicon dvic tcnology, ar rviwd. It is stablisd tat t widly accptd Frnkl modl provids a formal dscription of transport in trap-containing insulators, but nonpysical modl paramtrs must b introducd in ordr to obtain quantitativ agrmnt. It is sown tat t multiponon modl of trap ionization consistntly dscribs carg transport in insulators wit traps.. Introduction: t importanc of undrstanding lctron and ol transport mcanisms in silicon dvic dilctrics Dilctric films in silicon dvics av a ticknss in t rang ±300 nm. T lctric fild in a dilctric is in t rang Vcm for a voltag of V. Altoug t most frquntly usd dilctrics ar amorpous films, polycrystallin films av also found applications in rcnt yars. T K A Nasyrov Institut of Automation and Elctromtry, Sibrian Branc of Russian Acadmy of Scincs, prosp. Akadmika Koptyuga, Novosibirsk, Russian Fdration Fax +7 (383) nasyrov@ia.nsk.su V A Gritsnko Rzanov Institut of Smiconductor Pysics, Sibrian Branc of Russian Acadmy of Scincs, prosp. Lavrnt'va 3, Novosibirsk, Russian Fdration Tl. +7 (383) Fax +7 (383) grits@isp.nsc.ru Rcivd 4 Marc 03 Uspki Fizicskik Nauk 83 (0) 099 ± 4 (03) DOI: /UFNr Translatd by E G Strl'cnko; ditd by A M Smikatov most important silicon dvic dilctrics ar t amorpous silicon compounds SiO (dioxid), Si 3 N 4 (nitrid), and SiO x N y (oxynitrid) []. Bcaus lctrons scattr strongly in amorpous matrials lacking a long-rang ordr, amorpous films av ig brakdown filds, as is xmplifid by t valu : 0 7 Vcm for SiO. Tis is too muc for a crystallin smiconductor to witstand bcaus, du to t larg man fr pat, t avalanc multiplication of carrirs rsults in brakdown filds 0 5 Vcm. Dilctrics placd in a strong lctric fild profusly xibit nonlinar pnomna tat ar difficult to obsrv in crystallin smiconductors, suc as tunnl injction of carrirs in contacts, Frnkl trap ionization, and multiponon trap ionization. T following applications of dilctrics crucially rquir undrstanding transport mcanisms for t dvlopmnt of rliabl silicon dvics. Currntly (as of 03), silicon fild ffct transistors basd on mtal±dilctric±smiconductor (MDS) structurs routinly us 63 m as a caractristic transistor siz (t lngt of t transistor cannl is about 70% of tis numbr), t subgat silicon oxid bing. nm tick. At t gat voltag V, t fild in t dilctric is clos to 0 7 Vcm, and t tunnl currnt racs A cm. A ig lakag currnt lads to t injction of lctrons from t invrsion layr to t gat, tus dcrasing t slop of t transistor I±V curv and lading to additional powr disprsion (and nc to t ovrating of t microcircuit). A furtr dcras in t transistor dsign rul to 45, 3,, and 6 nm rquirs a corrsponding incras in t capacity of t subgat dilctric of t MDS transistor. For four dcads, MDS transistors av usd trmal silicon oxid SiO on silicon as a subgat dilctric. T capacity of subgat dilctrics was incrasd by dcrasing t SiO ticknss. As notd abov, incrasing t subgat

2 000 K A Nasyrov, V A Gritsnko Pysics ± Uspki 56 (0) capacity furtr by dcrasing t SiO ticknss is impossibl bcaus of t proibitivly larg tunnling currnt. T most gnral approac to solving tis problm is to rplac SiO by so-calld altrnativ dilctrics, or dilctrics wit ig dilctric constants (ig-k dilctrics). W not tat t dilctric constant of silicon dioxid is ˆ 3:9. T currnt ig-k dilctric candidats ar Al O 3 ( 0), Hf x Si y O ( 5), Hf x Al y O z ( 5), HfO x N y ( 5), HfO ( 5), ZrO ( 40), and TiO ( 80) [±5]. Using ig-k dilctrics can incras t pysical ticknss wil kping t capacity, and can trfor supprss t tunnling currnt in an MDS transistor. Similarly to silicon nitrid Si 3 N 4, ig-k dilctrics gnrally av a ig dnsity (0 9 cm 3 ) of lctron and/or ol traps. T injction of lctrons and ols and tir subsqunt localization on (captur by) traps lads to t accumulation of carg in ig-k dilctrics, sifting t transistor trsold voltag, and dgrading t dvic prformanc. W not t xtrmly tigt stability rquirmnts for t MDS transistor trsold voltag. For xampl, vn aftr 0 yars of t continuous opration of a microprocssor, t accumulation of lctrons (ols) on traps sould not xcd t trsold by mor tan 0 mv. Trfor, prdicting t stability and rliability of a ig-k MDS transistor rquirs a dtaild undrstanding of t mcanisms of carg carrir injction at t contacts and t captur and trap ionization procss. Anotr important application of dilctrics is as insulators in RAM storag capacitors. For carg accumulation on t capacitor plats not to caus any loss of information during t ms rprogramming cycl, t dilctric lakag currnt at a voltag of V sould not xcd 0 7 A cm [3, 4]. T first matrial usd as an insulator in a RAM storag capacitor was SiO, followd by silicon nitrid Si 3 N 4 ( ˆ 7) as an lmnt in oxid±nitrid±oxid (ONO) structurs (oxid r maning SiO ). Currntly, Al O 3, Ta O 5, HfO, ZrO, TiO, and SrTiO 3 ( 000) ar bing studid for t rol of storag capacitor insulator. Using insulators in a storag capacitor rquirs t dvlopmnt of tcnology for syntsizing low-conductivity dilctrics, wic implis t nd for undrstanding t conduction mcanism and t dpndnc of t conductivity on t concntration, nrgy, and trap captur cross sction (undrstanding tis allows dirctionally optimizing, i.., dcrasing, t conductivity of t RAM capacitor dilctric). In addition to t RAM storag capacitor, passiv capacitors ar involvd in t dsign of silicon intgratd circuits, wic typically us SiO and Si 3 N 4 as an insulator. Incrasing t information capacity of a silicon dvic dcrass t capacitor ara. Currntly, ig-k dilctrics (Al O 3, Ta O 5, HfO, ZrO, TiO, and SrTiO 3 ) ar considrd as candidats for suc capacitors. Flas mmory dsign is yt anotr major application of dilctrics [5±0]. A flas mmory cll is an MDS transistor wos trsold voltag is variabl du to t accumulation of lctrons and/or ols in t storag mdium. T trsold voltag rtntion tim is 0 yars at 85 C. T storag mdium is insulatd from t smiconductor cannl by t tunnling dilctric troug wic lctrons and ols ar injctd (rad±ras systm). T gat and t storag mdium ar also insulatd from ac otr by a dilctric. Flas mmory storag mdia includ polysilicon `floating' gats, silicon nitrid wit a ig concntration of lctron and ol traps, and smiconductor (Si, G, SiG) or mtallic (Ni, Au, Pt) nanoclustrs. T matrials usd as flas mmory tunnl dilctrics ar trmal silicon dioxid (SiO ) and silicon oxynitrid (SiO x N y ). Figur prsnts t nrgy diagram of TaN ± Al O 3 ± Si 3 N 4 ± SiO ± Si (TANOS) flas mmory for diffrnt stats [9, 0]. In t data rcording mod, t conducting TaN-basd gat (TaN as mtallic proprtis) is subjctd to a positiv potntial wit rspct to t silicon substrat (Figs a, b). Silicon's lctrons prform Fowlr±Nordim tunnling troug t tunnling oxid into t silicon nitrid, to b capturd tr by dp lctron traps (W t :5 V). T ssntial rquirmnt r is to block t parasitic injction of ols from t gat to t silicon nitrid, for wic tr ar two rasons: t ig barrir to ol injction and t fact tat du to t larg dilctric constant in t blocking dilctric (Al O 3 ), t lctric fild in it is low. In t storag mod, t ngativ lctron carg in sort-circuitd silicon nitrid inducs an nricmnt layr in t p-silicon substrat (Fig. c), and trfor tr is no conducting cannl in t transistor (logic 0). T logic 0 old tim is limitd by t ionization of lctron traps in silicon nitrid, a procss wic (similarly to ol traps) occurs by itr t Frnkl mcanism or t multiponon mcanism. In t ras mod, t tunnling mcanism rsults, for a ngativ potntial on t silicon gat, in ols bing injctd troug t silicon oxid into t silicon nitrid, wr ty ar capturd by ol traps (Fig. d). (Importantly, t parasitic injction of lctrons into t storag mdium sould b blockd.) In t data stor mod, an invrsion layr (conducting cannl) is inducd in t silicon substrat (logic, Fig. ). T data storag tim (for logic ) is dtrmind by t ionization of ol traps in silicon nitrid. Parasitic lctron injction from t silicon and from t gat is also possibl (Fig. ). In t writ±ras mod, knowing t carrir injction mcanisms from t smiconductor and t conducting gat (r, t conducting tantalum nitrid TaN) to t dilctric (r, SiO and Al O 3 ) is ssntial. W not tat t flas mmory data (logic 0 and logic ) rtntion tim sould b 0 yars at 85 C. To prdict t rtntion tim, a study is mad of t acclratd sprad of lctrons and ols in a TANOS flas at incrasd tmpraturs (00±300 C). Clarly, a rliabl dtrmination of t rtntion tim rquirs a dtaild undrstanding of t ionization mcanisms of lctron and ol traps in a storag mdium (r, silicon nitrid). In t stor mod, t lctron and ol cargs localizd on traps in t silicon nitrid can b compnsatd by carrirs parasitically injctd from t silicon and t gat. Prdicting tis pnomnon rquirs knowing t lctron and ol injction mcanisms in dilctrics in wak lctric filds. Blocking layrs usd in TANOS flas mmoris ar tos of amorpous aluminum oxid Al O 3, a matrial wic, as sown by xprinc, contains traps. In t logic 0 and logic storag mods, a carg flow can occur via Al O 3 traps from traps in t silicon nitrid to t gat, rsulting in limiting t data rtntion tim. In tis connction, t conduction mcanism in aluminum oxid is important to know. In a yt anotr flas mmory typ tat is currntly dominating t markt, t mmory mdium is a so-calld `floating' polysilicon gat, sparatd from t cannl by a trmal SiO layr and from t conducting silicon gat by an ONO structur. Cycling writ±ras procsss involv t

3 Octobr 03 Transport mcanisms of lctrons and ols in dilctric élms V.6 V p-si SiO 4.5 V Si3N 4 AlO 3 TaN 3.8 V 8.0 V.5 V Figur. Enrgy diagram of a TaN ± Al O 3 ± Si 3 N 4 ± SiO ± Si (TANOS) flas mmory wit a silicon nitrid storag mdium: (a) witout applid voltag, (b) positiv potntial on mtal, lctron injction into silicon nitrid, (c) closd-circuit lctron-accumulating stat, information storag, logic 0, (d) ngativ potntial on mtal, ol injction into silicon nitrid, () closd-circuit ol-accumulating stat, information storag, logic. V V a b c d formation of lctron traps in t tunnling oxid of suc a flas mmory. Wn in t stor mod, t lctrons rsiding in t floating gat can flow to t silicon via traps in t tunnling oxid by t trmally assistd tunnling mcanism [8±], dcrasing t carg accumulatd in t floating gat. T trmally assistd tunnling limits t numbr of writ±ras cycls in t floating gat flas mmory structur. T numbr of flas mmory rprogramming cycls lis in t rang 0 4 ± 0 6. T limitd numbr of writ±ras cycls is a ky disadvantag of flas mmory vrsus magntic (ard disc) mmory. T mcanism of trmally assistd tunnling is currntly t subjct of intns rsarc [, ]. A silicon intgratd circuit as its spd dtrmind by t signal dlay in t circuit wir layout. Signal transfr btwn instrumnts occurs by mans of a multilvl mtal layout, t numbr of mtallization lvls racing igt in currnt microprocssors. T dlay t of a signal passing troug a mtal lin is dtrmind by t lin rsistanc R tims t parasitic capacitanc C btwn nigboring lins, t ˆ RC. T spd is incrasd by dcrasing bot t rsistanc of t conducting lins and t parasitic capacitanc btwn nigboring lins. Wras arly intgratd circuits usd aluminum lins, lowr-rsistivity coppr is currntly mployd. T microcircuit insulator usd arly on was pyrolitic aluminum dioxid SiO ( 3:9). Incrasing t intrlin parasitic capacitanc rquirs an incras in t dilctric constant of t insulator (`low-k' insulators). T low-k insulator currntly in us is SiO x C y ± SiO x F y [3]. T lakag currnt of a low-k dilctric sould not xcd a spcifid valu. In rcnt yars, muc work as bn don toward dvloping t nxt gnration of ig-spd (rsistiv) mmory tcnology [4±3]. Rsistiv mmory (also known as a mmristor) consists of a mtal±dilctric±mtal-basd capacitor wos dilctric as its rsistanc varid and mmorizd as t applid voltag is varid. A ky problm in t pysics of t dvic is t carg transport mcanism in ig- and low-rsistiv stats. Rsistiv mmory dilctrics usd includ stoiciomtric SiO x, GO x, TiO x, HfO x, TaO x, ZrO x, and AlO x. T conductivity of a dilctric sould, by dfinition, b zro. Howvr, in 97, Pool obsrvd an incras in t conductivity of mica wit an lctric fild [4]. In 938, Frnkl proposd a tortical xplanation of t xponntial lctric fild dpndnc of t currnt in a dilctric wit traps [5, 6]. Currntly, Frnkl's modl is ubiquitously usd for xplaining carg transport in a wid class of dilctrics, including nonstoiciomtric oxid SiO x (x < ), silicon nitrid SiN x (x 4 4=3) [7], ig-dilctric-constant dilctrics [8±3], and frrolctrics [3]. Howvr, as an xplanation of trap ionization in smiconductors, t Frnkl ffct can only b usd for wak lctric filds (< 0 3 V cm ) [33]. For strong lctric filds, t multiponon mcanism sould b usd [34, 35]. In two xtrm cass, t conductivity of a dilctric can b limitd itr by contact carrir injction (Fig. a) or by trapmdiatd transport in a bulk dilctric. Tis classification is oftn vagu, dpnding on ow strongly t lctric fild on t contact is scrnd by t spac carg accumulatd in t traps (Fig. b). Early work assumd tat t conductivity of a dilctric is du to lctrons, wtr monopolar or singl band in natur. Figur 3a prsnts t nrgy diagram of an MDS structur in t absnc of an applid voltag, and Figur 3b sows an MDS structur for a monopolar (lctron) conduction modl. Mor dtaild studis av sown tat similarly to crystallin dilctrics, ols can contribut to t conductivity of a dilctric. In gnral, dilctrics xibit bipolar, two-band

4 00 K A Nasyrov, V A Gritsnko Pysics ± Uspki 56 (0) Figur. Elctron injction at a contact and lctron transport in a bulk dilctric for a dilctric (a) wit and (b) witout t lctron spac carg. conduction (Fig. 3c) wit lctrons and ols rspctivly injctd from t ngativly biasd contact (catod) and from t positivly biasd contact (anod), and possibly wit lctron±ol rcombination in trap cntrs nar contacts.. Two-band conduction in dilctrics J Figur 3. Enrgy diagram of an MDS structur in t absnc of an applid voltag: (a) monopolar (lctron), (b) bipolar, and (c) dilctric conduction modl. As is known, conduction in crystallin smiconductors usually occurs via fr lctrons and/or ols tat aris from t ionization of sallow donors and accptors or rsults from band-to-band xcitation du to t small widt of t gap. At low tmpraturs (T40 K), wn carrirs ar frozn out to donors and accptors, t opping mcanism sts in. Wit tir wid band gap (3±8 V), dilctrics av a ngligibly low concntration of intrinsic carrirs. Wit t xcption of trmal SiO oxid on silicon, all t rmaining dilctrics (t oxids SiO x, Al O 3, Ta O 5, HfO, ZrO, TiO ; t nitrids Si 3 N 4, G 3 N 4, BN) av a ig dnsity of traps, cm 3. Amorpous silicon nitrid Si 3 N 4, a typical ig-trap-dnsity matrial, is currntly a modl J J J J a a b b c dilctric to study lctron and ol transport mcanisms and t natur of lctron and ol localizd stats (traps). T liftim of localizd (trappd) lctrons and ols in silicon nitrid is 0 yars at 400 K (t so-calld mmory ffct, wic in Si 3 N 4 is usd in flas mmory dvics). T nrgy of lctron and ol traps in dilctrics lis in t rang 0.5±.0 V. Also, t xprimntal vidnc is tat wn in t original stat, traps in a dilctric ar mpty of lctrons and ols. For ts rasons, and unlik donors and accptors in smiconductors at rlativly ig tmpraturs (sufficint to ioniz donors and accptors), traps do not supply fr lctrons and ols to t conduction band. As sown by xprimnt, conduction in dilctrics is du to t nonquilibrium lctrons and ols injctd from a contact. In smiconductors, t sign of carg carrirs (lctrons and ols) is dtrmind from t Hall ffct (or trmo-mf) masurmnts. For dilctrics, wit tir ngligibly low concntration of fr carrirs (s abov), tis is not possibl. Silicon nitrid is t first matrial for wic mtods for sparating t lctron and ol currnt componnts wr dvlopd. Tr suc mtods ar currntly known. T first mtod is basd on t accumulation in traps of lctrons or ols injctd from silicon or mtal [36±40], wil t otr two rly on dtrmining t sign of carg carrirs at t silicon±dilctric intrfac, rspctivly, by masuring t lg J V and C V curvs in t rgim of nonquilibrium dpltion of silicon [4±44] or by sparating t lctron and ol currnt componnts in an MDS transistor [45]. As alrady notd, silicon nitrid xibits bipolar twoband conductivity, wit lctrons and ols rspctivly injctd from t ngativly and positivly biasd contacts. In gnral, for itr potntial polarity, bipolar injction of lctrons from t catod and of ols from t anod occurs (Fig. 3c). T lctron and ol contributions to t injction currnt can vary strongly dpnding on t following factors: t igt of t contact±dilctric barrir, t ticknss of t tunnling oxid, t puls duration, t polarity of t potntial, t ticknss of t silicon nitrid. 3. Contact-limitd mcanism of lctron and ol injction As notd in t Introduction, two mcanisms can b convntionally idntifid for t conductivity of tin dilctric films: carrir injction from a contact into t dilctric and carg transfr in t bulk dilctric. In t formr cas, tr injction mcanisms ar usually considrd: t Fowlr±Nordim mcanism, trmally assistd tunnling, and Scottky trmolctron mission (Fig. 4) 3. Fowlr±Nordim tunnling injction at a contact T Fowlr±Nordim mcanism (autolctron mission) (Fig. 4, arrow ) dominats in ig lctric filds and at low tmpraturs, at wic carg carrirs can tunnl from t conduction band of a (mtal or smiconductor) contact into t conduction band of t dilctric. Altoug tunnling is of a quantum mcanical (and nc tmpratur-indpndnt) natur, t numbr of fr carrirs at t Frmi lvl dpnds on t tmpratur, lading to a wak tmpratur dpndnc in t lctric currnt [46±50].

5 Octobr 03 Transport mcanisms of lctrons and ols in dilctric élms F 0 3 4p 0 x 0 6 Mg Al Si Fx J, A cm Figur 4. Elctron injction mcanisms at a mtal±smiconductor contact:, Fowlr±Nordim tunnling;, trmally assistd tunnling; 3, ovr-barrir mission; 4, Scottky ffct. T atcd rgion is a mtal. For t lctron injction from a mtal, t currnt is givn by [48] J ˆ AF xp B p m 3= F 0 ; F A ˆ 3 ; B ˆ 8p p 8pF 0 3 : wr J is t currnt dnsity, F is t lctric fild strngt, F 0 is t igt of t triangl potntial barrir at t lctrod±dilctric intrfac, and m is t ffctiv tunnling mass. Bcaus t Fowlr±Nordim mcanism lads to a wak tmpratur dpndnc of t currnt, it manifsts itslf at ig lctric filds suc tat t barrir for tunnling bcoms sufficintly narrow. T Fowlr±Nordim ffct is obsrvd at sufficintly larg contact±dilctric barrir igts. It as bn stablisd tat t Fowlr±Nordim ffct limits t lctron injction in Si ± SiO ± M structurs [48]. Rprsntd in Fig. 5 ar t V±I curvs of suc structurs wit diffrnt mtal contacts. T igt of t Si ± SiO ± M barrir for lctrons dpnds on t work function of t mtal, and t lakag currnt dpnds xponntially on t work function of t mtal. T V±I curvs ar straigtnd in t coordinats tat corrspond to xprssion () (Fig. 6) [48]. In accordanc wit tortical prdictions, t tunnling currnt in silicon oxid dpnds wakly on t tmpratur. T ffctiv tunnling lctron mass in silicon oxid stimatd from t slop of t V±I curv in Fig. 6 is m 0:4m 0, wr m 0 is t fr lctron mass. In [49], lctron and ol injction from silicon to silicon oxid wr studid in t cas wr t uppr contact matrial was takn to b corona discarg plasma ratr tan a mtal. It was found tat for a positiv potntial on t silicon, ol injction from t silicon to t silicon oxid occurs. T Fowlr±Nordim ffct was obsrvd at liquid nitrogn and room tmpraturs for lctron injction from aluminum to silicon oxid (Fig. 7) [39]. T ffctiv tunnling lctron mass in silicon nitrid dtrmind from t slop of t V±I curv is m 0:4m 0. Obsrvations at liquid nitrogn tmpraturs wr mad for ol tunnling injction from gold to silicon nitrid (s Fig. 7) [39]. T ffctiv tunnling ol mass in silicon nitrid as dtrmind from t slop of t V±I curv is m 0:35m F, 0 6 V cm Figur 5. V±I caractristics of Si ± SiO ± M structurs for lctron injction from mtals wit diffrnt work functions. J=F, A V Trmally assistd contact tunnling T procss of trmally assistd tunnling in a contact (Fig. 4, arrow ) can b dividd somwat loosly into two stags. T first is t ponon-absorption-assistd xcitation of carg carrirs to a crtain nrgy lvl, and at t scond stag, t carrirs tunnl troug a triangl barrir. For t carrir injction from a mtal, t xprssion for t currnt as t form [47, 5] J ˆ CF xp Si m m ˆ 0.4 m m ˆ 0.96 wr C ˆ pm kt = =. Al m m ˆ 0.39 p F 0 =6 F = 4pkT m kt Au Mg m m ˆ =F, 0 7 cm V Figur 6. V±I caractristics of Si ± SiO ± M structurs in Fowlr± Nordim coordinats. ;

6 004 K A Nasyrov, V A Gritsnko Pysics ± Uspki 56 (0) a b J=F, A V E Al F E Au F Al Au Si 3N 4 Si 3N 4 J=F, A cm V F 0, V cm F = 0 3, V = cm = Figur 8. V±I caractristics for ol injction from gold to silicon nitrid in coordinats corrsponding to (a) trmally assistd tunnling coordinats and (b) t Scottky ffct. Curv 5: tortical dpndnc of t Scottky ffct at 53 K. Curvs:, T ˆ 593K;, 4, T ˆ 53 K; 3, T ˆ 77 K =F, 0 7 cm V Figur 7. Fowlr±Nordim ffct for () lctron injction from aluminum into silicon nitrid and () ol injction from gold into silicon nitrid. Trmally assistd tunnling dominats at mdium filds and mdium tmpraturs (wn t trmal nrgy is insufficint to ovrcom t contact barrir by t Scottky mcanism) and as t following faturs: t currnt dnsity dpnds xponntially on t squar of t fild, t activation nrgy incrass wit t tmpratur, and t currnt strongly (xponntially) dpnds on t mtal work function. T ffct was prdictd for t vacuum±mtal systm [47, 5] and analyzd tortically (for lctrons) for a mtal± dilctric intrfac [5]. Exprimntally, trmally assistd injction of ols was obsrvd at t gold±silicon-nitrid intrfac in Rf. [39]. Figur 8 prsnts V±I curvs for ol injction from gold into silicon nitrid in coordinats corrsponding to t mcanism of trmally assistd tunnling (Fig. 8a) and t Scottky ffct (Fig. 8b). In cas (a), t V±I curvs ar straigtnd at tmpraturs of 53 K and 593 K, and in cas (b) ar not straigtnd at all. 3.3 Trmoionic contact mission: t Scottky ffct Unlik t Fowlr±Nordim mission, t trmolctron Scottky mission (Fig. 4, arrow 3) dominats at incrasd tmpraturs at wic trmal nrgy is sufficint for ovrcoming t junction barrir, and carg carrirs can b xcitd from t contact Frmi lvl to t dilctric conduction band. T strong fild dpndnc in t Scottky ffct occurs bcaus of t lowring of t barrir du to imaging forcs. T xprssion for t currnt for t mtal-to-dilctric injction as t form [47, 5] J ˆ AT xp F 0 b S F = ; 3 kt wr Aˆ 4pm k = m ˆ0m =m [A cm dg ] is t Ricardson±Dusman constant, b S ˆ 3 = 4p 0 Š = is t Scottky constant, and is t ig-frquncy dilctric constant. A caractristic fatur of t Scottky mission is tat t currnt dpnds xponntially on bot t tmpratur and t lctric p fild. T V±I curvs plottd in t coordinats lg J ˆ f F ar straigt lins. T Scottky ffct is caractrizd by an xponntial variation of t currnt wit t work function of t contact (i.., t contact±dilctric barrir igt) and dominats at ig tmpraturs and low valus of t potntial barrirs at t MDS contact. 3.4 Trap mdiatd injction in silicon nitrid If t silicon oxid layr at t silicon intrfac is tick (> 5 nm), tn carg carrirs can b injctd troug traps containd in t silicon oxid (a carg first tunnls from t silicon to a trap in t bulk silicon oxid and tn from t trap to t fr band of t silicon nitrid). Du to t lowr subbarrir factor, tis procss, occurring in stags, can b mor advantagous, for xampl, tan t Fowlr± Nordim injction troug a triangl barrir. Tortical and xprimntal studis of trmally assistd tunnling for lctron injction ar citd in Rf. [53]. W not, owvr, tat tis injction mcanism also applis to ols. T xprssion [54] J TAT ˆ FP t =3 m 6 xp m = F 3= 4 3 F can b usd to dscrib t trmally assistd injction of lctrons and ols. T notation r is as follows: F is t barrir igt (for a ol or an lctron) and P t is t dnsity of t nrgy and volum distributions of traps (assumd to b uniform in t drivation). It is to b notd tat t undrbarrir factor in t xponnt turns is two tims lss tan t on for t Fowlr±Nordim law. For tis

7 Octobr 03 Transport mcanisms of lctrons and ols in dilctric élms 005 mcanism to b fficint, t traps must av t dnsity gratr tan 0 8 cm 3 for t givn valu (8 V) of t silicon oxid gap widt. 4. Trap ionization modls If traps av a ig concntration in a dilctric, t volum carg on tm limits injction from t contact, and t conductivity is dtrmind by t trap ionization rat. Tr ar currntly two trap ionization mcanisms dominating t litratur on trap mods: t Frnkl ffct and t multiponon ionization. In smiconductors containing sallow traps, opping conductivity is obsrvd at low tmpraturs, wn t carrirs ar localizd on traps [55]. 4. Trap ionization by t Frnkl mcanism T Frnkl ffct is t rduction in t Coulomb potntial of an isolatd trap du to an applid lctrical fild (Fig. 9). It is assumd tat tr is a trap (localizd stat) wit a capturd lctron in t band gap of t dilctric and tat tis lctron intracts via t Coulomb law wit a positivly cargd dfct [5, 6]. T lctric fild lowrs t potntial barrir, incrasing t probability of an lctron to undrgo a transition from a localizd stat to t conduction band. Tr trap ionization mcanisms ar possibl:. Ovr-barrir ionization dominating at sufficintly ig tmpraturs and rlativly low filds, for wic t barrir widt is important and t barrir tunnling probability is low (Fig. 9, cas ). It is tis cas tat was tratd in t original work of Frnkl [5, 6].. Trmally assistd tunnling (Fig. 9, cas ), t intrmdiat cas btwn tunnling and ovr-barrir ionization. 3. Tunnling troug a potntial barrir (Fig. 9, cas 3) dominats in ig filds (for wic t potntial barrir is narrow noug to allow tunnling) and at low tmpraturs. T trmal ionization of traps, wic is xactly wat is calld t Frnkl ffct, yilds t following xprssion for t trap ionization rat: P ˆ n xp W b p Fr F ; 5 kt qfr E wr n is t frquncy factor and b Fr ˆ 3 = p 0 = is t Frnkl constant. For trmally assistd tunnling, t trap ionization probability pr unit tim taks t form [56] p W b F P ˆ n xp n F p W b 0 kt xp kt x x V x ˆ W Fx ; 4p 0 x kt p m V x dx d ; wr is t nrgy of t xcitd lvl, and x and x ar t classical turning points! = x ; ˆ W F F p 0 W 6 : 7 Tunnling troug t Coulomb potntial barrir is obsrvd at low tmpraturs in strong lctric filds. A ig concntration of traps rsults in lowring t barrir du to an ovrlap of t Coulomb potntials of nigboring traps (Fig. 0) [57]. At ig tmpraturs, t xprssion for t V±I caractristic in t on-dimnsional modl as t form J ˆ nn =3 xp W kt sin F N =3 kt : 8 In strong filds, t logaritm of t currnt is proportional to t lctric fild strngt (Pool's law). 4. Multiponon ionization of traps Tis modl assums tat a fr carg can b capturd by (gnrally nutral) lattic dfcts. As notd abov, traps in dilctrics ar dp, and tir capturd lctrons av small localization rgions. Suc a concntratd carg dforms t narst vicinity of t trap, wic in turn affcts t nrgy lvl of t capturd carg. It is tus assumd tat t nrgy of t trap and t dilctric lattic ar strongly rlatd (t so-calld ponon-coupld trap). Bcaus t binding nrgy of lctrons in dp cntrs gratly xcds t man ponon nrgy, only multiponon = Fa r F t Ovr-barrir mission Trmally assistd tunnling 3 Tunnling troug potntial barrir W Figur 9. Frnkl ffct: lctric-fild-assistd lowring of t trap Coulomb potntial. Arrows indicat possibl ionization mcanisms:, trmal ovr-barrir mission;, trmally assistd tunnling; 3, tunnling ionization. q is t lctron carg and r is t spatial coordinat. a= a= Figur 0. Potntial barrir rduction du to an ovrlap of t Coulomb potntials of nigboring traps (Hill modl).

8 006 K A Nasyrov, V A Gritsnko Pysics ± Uspki 56 (0) procsss can provid trmal ionization. T tory of multiponon ionization uss configuration diagrams to dscrib lctron transitions from a localizd to a dlocalizd stat. Vibrational motions of t dfct and tos of t narst surroundings of t trap ar dscribd by a cang in t coordinat Q. Clarly, tis on-dimnsional dscription is approximat if t trap as many narst nigbors. Figur a sows t configuration diagram of a dfct (trap) for an mpty trap and for t trap wit a capturd lctron. T potntial curv U f Q corrsponds to t trap wit a capturd lctron, and U 0 Q corrsponds to t mpty trap. T tiltd lin is t dpndnc of t nrgy of t capturd lctron Q on t configuration coordinat. Wn Q < E c (E c is t dilctric conduction band dg), t lctron is considrd to b capturd; o is t dfct vibration frquncy and W p ˆ o is t local ponon nrgy. As can b sn from Fig., tr ar two ways in wic a trap can b ionizd. In t first cas, t transition from t ground to an xcitd stat occurs du to a vrtical (optical) transition; suc a transition occurs at a fixd valu of Q and rquirs t optical trap ionization nrgy W opt. In an optical transition, t lctron bcoms fr, and t narst vicinity of t dfct rmains dformd at t first instant. Tis dformation is tn carrid by ponons trougout t dilctric. In t scond cas, an lctron is xcitd by absorbing ponons wit a cang in t configuration coordinat (to nsur tat Q > E c ). T rquird nrgy U 0 Q E c Q b W opt U f Q Figur. (a) Configuration diagram of a trap in t tory of multiponon ionization. Potntials U f and U 0 rspctivly corrspond to a trap wit a capturd lctron and an ionizd trap. (b) Oscillator modl illustrating t multiponon trap ionization, quilibrium stat. (b) Prturbd stat, transition to t continuous spctrum. W T a Q c is t trmal ionization nrgy W T ; W opt always xcds W T. A trap wit a capturd lctron as t minimum nrgy at Q 0 ˆ W opt W T Š =. In t simplst cas, a dp trap as on bound stat. T position of t local lvl is dtrmind by t dfct-inducd potntial and dpnds ssntially on t distanc btwn t dfct and t nigboring atoms. Tus, t vibrations of t dfct modulat t position of t lvl of t localizd lctron (Fig. b). In t cas of strong trmal vibrations, t lvl can mov to t continuous spctrum, lading to t ionization of t trap [58]. A trap can also b ionizd by a DC lctric fild, rsulting in a strongly tmpratur-dpndnt ionization rat. Trating dfct vibrations in t smiclassical approximation, t ionization probability pr unit tim is givn by [35] F P ˆ p m W opt 4 W opt W T W opt xp 4 3 m o p m F W 3= opt Wopt F cot W p : 9 T T corrsponding quantum mcanical rsult is [4] P ˆ X nwp xp kt W opt W T cot W p W p kt n W opt W T I n P i W T nw p ; W p sin W p =kt 0 F P i W ˆ p xp 4 p m m W 3 F W 3= : Hr, I n is t modifid Bssl function and P i W is t tunnling probability troug a triangl barrir of igt W. 4.3 Carg transport by trap-to-trap tunnling T Sockly±Rd±Hall transport tory assums tat a carg (wic w spcify to b an lctron) is injctd from a contact into t dilctric conduction band and can b capturd by a trap in t bulk dilctric. Tis capturd lctron can tn scap into t conduction band and b again capturd by anotr mpty trap lowr down in t fild. A transitional trap filling procss occurs and continus until t filling rat of mpty traps bcoms qual to t ionization rat of t filld ons. Following tis, a stady-stat carg transport procss sts in. But wn t trap concntration is ig and t distancs btwn tm ar small, tr is a possibility for t lctron to tunnl btwn traps witout bing xcitd to t conduction band. Figur sows a diagram of lctron tunnling from on ponon-coupld trap to anotr troug a distanc a in t prsnc of an lctric fild F. Solid and dasd lins rspctivly rprsnt t original (prior to tunnling) and final (post-tunnling) stats of t systm. In an xtrnal fild, t lctron lvls in nigboring traps ar not coincidnt, and it trfor follows tat inlastic procsss Ð t mission and absorption of lattic ponons Ð sould b involvd in t transition to compnsat t nrgy diffrnc. T ponon-coupld trap modl asily taks tis into account. Witin tis modl, t tunnling transition rat btwn traps

9 Octobr 03 Transport mcanisms of lctrons and ols in dilctric élms 007 U f Q E c is givn by [59] p p WT P ˆ p m a Q 0 kt xp a p m W T xp W opt W T Fa kt : T numrical transport modl uss t following quation to calculat t trap-filling rat in t bulk of t dilctric: q qt n i ˆ P i ; i n i n i N t P i ; i n i n i N t P i; i n i n i N t P i; i n i n i N t : 3 Hr, N t is t total trap dnsity and n i is t dnsity of lctron-filld traps on sit i of t modl lattic. T distanc btwn sits is qual to t man distanc btwn traps,, and P i ; i is t tunnling rat from sit i to sit i; for t invrs procss, t tunnling rat is P i; i. T filling of t vry first trap occurs from t contact and is dscribd by a ˆ N =3 t 0 Q 0 Q U f Q Fa q qt n ˆ v inj N t n v ion n P ; n P ; n n N t n N t ; 4 wr v inj is t rat of trap filling by lctrons from t contact and v ion is t invrs ionization of t trap into t contact: v inj ˆ xp F W T v ion ; kt V out v ion ˆ p z pkt xp Q Q 0 4 m = 3= Fz 3= kt 3 F xp F Fz dq; 5 kt ˆ Q 0 Q Q 0 W opt : W opt Figur. Diagram of lctron tunnling from on ponon-coupld trap to anotr ovr a distanc a in t lctric fild F. Solid lins: initial stat of t systm bfor tunnling. Dasd lins: final stat of t systm aftr tunnling. Horizontal dasd lin: t tunnling transition of an lctron from on trap to anotr. W T Q Hr, F is t nrgy diffrnc btwn t conduction band bottom of t dilctric and t Frmi lvl of t contact, z is t trap±contact distanc, and V out is t vlocity of t scapd lctron in t contact. T boundary condition assumd for t opposit nd of t dilctric is tat t trap is mpty nar t contact. 5. Spac carg in a dilctric and t Sockly±Rd±Hall quations T xprimntal vidnc is tat in t unprturbd stat, lctron and ol traps in a dilctric ar nutral and unfilld. As lctrons and ols ar accumulatd on traps, a spac carg forms in t dilctric. Carg transport, t captur of lctrons and ols injctd from t contact to t traps, and t ionization of traps ar dscribd by t Sockly± Rd±Hall quations qn x; t qt qn t x; t qt qp x; t qt ˆ qj x; t svn x; t Nt qx n t x; t n t x; t P x; t s r vn x; t p t x; t ; 6 ˆ svn x; t N t n t x; t n t x; t P x; t s r vp x; t n t x; t ; ˆ qj p x; t svp x; t Nt qx p t x; t 7 p t x; t P x; t s r vp x; t n t x; t ; 8 qp t qt ˆ svp x; t Nt p t x; t p t x; t P x; t s r vp x; t n t x; t ; 9 wr n (n t ) is t concntration of fr (trappd) lctrons, p (p t ) is t concntration of fr (trappd) ols, Nt (Nt ) is t concntration of lctron (ol) traps, F x; t is t local lctric fild, is t lctron carg, s is t trap captur cross sction, s r is t rcombination cross sction btwn t fr and trappd carrirs of t opposit sign, v is t drift vlocity, is t low-frquncy dilctric constant, and P is t probability of t ionization of filld traps pr unit tim. T lctron drift vlocity is rlatd to t currnt dnsity as J ˆ nv. T corrsponding laws of injction (Fowlr±Nordim injction, t Scottky ffct, trmally assistd tunnling, tc.) ar usd as boundary conditions, and t Poisson law qf qx ˆ n t x; t p t x; t 0 0 is usd to dscrib t spac carg of trappd lctrons and ols. 6. Two-band conduction and trap paramtrs in Si 3 N 4 Amorpous smiconductors and amorpous dilctrics sar t ability to localiz lctrons and ols. Amorpous silicon nitrid Si 3 N 4 can localiz lctrons and ols for long priods of tim (a olding tim of 0 yars at 400 K [6]). T mmory

10 008 K A Nasyrov, V A Gritsnko Pysics ± Uspki 56 (0) ffct in Si 3 N 4 is widly usd in nrgy-autonomous mmory microcircuits. T mcanism govrning carg transport in silicon±oxid±nitrid±oxid±silicon (SONOS) mmory dvics is t ionization of dp cntrs in Si 3 N 4. T dominant currnt viw is tat t ionization mcanism in Si 3 N 4 is du to t Frnkl ffct. In tis sction, xprimntal rsults on carg transport in Si 3 N 4 ar prsntd for a wid rang of lctric filds and tmpraturs and ar quantitativly compard wit t prdictions of t Frnkl modl and wit tos of t quantum modl of t multiponon trap ionization mcanism. In Rf. [56], a quantitativ comparison is mad btwn t xprimntal rsults on t conductivity of mtal±nitrid± oxid±smiconductor (MNOS) structurs and t tory of multiponon ionization for a monopolar conduction modl, wr only t lctron injction from silicon is considrd and t ol injction from a mtal is nglctd. Hr, xprimntal rsults ar compard wit a mor gnral bipolar modl, wic allows t lctron injction from silicon and t ol injction from t mtal and in wic fr lctrons rcombin wit localizd ols and fr ols rcombin wit localizd lctrons (Fig. 3). For a MNOS structur, t tmpratur dpndnc of t currnt (Fig. 4) was masurd for diffrnt voltags, and V±I caractristics wr masurd for diffrnt tmpraturs (Fig. 5). In all masurmnts, t voltag on t mtal was positiv, a polarity at wic bot lctron injction from t silicon substrat and ol injction from aluminum occur (Fig. 3b). For comparison wit xprimnt, a on-dimnsional bipolar modl of conduction in Si 3 N 4 [56] was usd. Carg transport was dscribd using Sockly±Rd±Hall quations (6)±(9) and Poisson quation (0), wic taks t nonuniform distribution of t lctric fild in silicon nitrid into account. T captur and rcombination cross sctions in silicon nitrid wr takn to av t valus st ˆ st ˆ s r ˆ cm [6, 36, 54, 56]. T trap ionization probability P in Si 3 N 4 is stimatd using itr t Frnkl modl in (5)±(7) or t multiponon ionization modl in (0). For modling purposs, lctrons and ols wr takn to av t sam ffctiv mass 0:5m 0. T lctron and ol injction from t silicon substrat and t aluminum lctrod wr rspctivly calculatd for t Fowlr±Nordim mcanism. T tmpratur dpndnc of t currnt was masurd in t tmpratur rang 77±40 K for voltags of 44, 37, and 30 V. Rfrring to Fig. 4, wic plots t rsults in t Arrnius coordinats lg J T, t currnt dpnds wakly on tmpratur for T < 00 K, suggsting tunnling as t trap ionization mcanism. Using first t Frnkl modl wit trmally assistd tunnling to dscrib t xprimntal data, t bst fit is obtaind for lctron and ol traps aving t sam nrgis (W ˆ W ˆ W ˆ : V), t sam ffctiv masss (m ˆ m ˆ m ), and t sam concntrations (N t ˆ N t ˆ cm 3 ) (dasd lins in Fig. 4). Complt agrmnt was obtaind only by using an unpysically small frquncy factor, n ˆ s (compard to n ˆ W= s originally stimatd by Frnkl [5]). Earlir work [60±63] also usd a lowrd frquncy factor, n ˆ s, to obtain agrmnt wit xprimnt. T low-tmpratur rang is dominatd by t tunnling ffct, and of ky importanc is t tunnling ffctiv mass. Maximizing t agrmnt btwn xprimnt and calculations rquird using diffrnt ffctiv masss for diffrnt voltags: m ˆ 4:8m 0 for Si SiO Si 3N J p J n Figur 3. Enrgy diagram of an MNOS structur (a) in t flat band rgim and (b) for a positiv potntial on t mtal. J, A cm m ˆ.0 m () V ˆ 44 V () V ˆ 37 V (3) V ˆ 30 V m ˆ 4.8 m m ˆ 3.8 m Al =T, K Figur 4. Tmpratur dpndnc of t currnt in an MNOS structur for diffrnt positiv potntials on t mtal. Dots: xprimnt, solid lins: multiponon modl calculations. Modl paramtrs: m ˆ m ˆ 0:5m 0, W opt ˆ :8 V, W T ˆ :4 V, W p ˆ 60 mv, st ˆ st ˆ s r ˆ cm. Dasd lins: Frnkl modl for qual trap nrgis, W ˆ W ˆ W ˆ : V, qual ffctiv masss m ˆ m ˆ m and qual concntrations Nt ˆN t ˆ709 cm 3 for lctron and ol traps. + a b

11 Octobr 03 Transport mcanisms of lctrons and ols in dilctric élms 009 J, A cm V ˆ 44 V, m ˆ3:8m 0 for Vˆ37 V, and m ˆ :0m 0 for V ˆ 30 V. T ffctiv mass valus usd in t calculations ar an ordr of magnitud gratr tan t lctron and ol tunnling ffctiv masss 0.3±0.6m 0 found xprimntally for Si 3 N 4 [39]. W trfor conclud tat t Frnkl modl dos dscrib carg transport in Si 3 N 4 formally, but rquirs an unpysically small frquncy factor and an anomalously larg tunnling ffctiv mass for t quantitativ agrmnt. At t nxt stag, t xprimntal and calculatd rsults wr compard using t multiponon ionization tory in Rf. [34] (solid lins in Fig. 4). It is stablisd tat t agrmnt is closst wn t paramtrs of t lctron and ols traps ar takn to b t sam: WT ˆ W T ˆ :4 V, W opt ˆ W opt ˆ :8 V, N t ˆ Nt ˆ cm 3, W p ˆ 0:06 V. Calculations using t multiponon tory wr prformd to obtain V±I caractristics for T ˆ 77, 300, and 40 K (Fig. 5, solid lins) and tmpratur dpndncs of t currnt for V ˆ 44, 37, and 30 V (Fig. 4, solid lins). Ovrall, ratr good agrmnt is obsrvd btwn t modld and masurd rsults. T small discrpancy obsrvd for voltags blow 35 V may b du to t slow rlaxation of t currnt in silicon nitrid, a pnomnon wos natur is not yt undrstood [64]. Trfor, t tory of multiponon trap ionization quantitativly dscribs xprimntal carg transport in silicon nitrid in a wid rang of lctric filds and tmpraturs. T rsults suggst tat t tory of multiponon ionization fits t data bst if t concntration, t captur cross sction, and t optical and trmal nrgy ar takn to b t sam for lctron and ol traps. T ida of qual paramtrs was first introducd in Rf. [38] to xplain xprimntal rsults on t draining of carg at diffrnt `contracting' potntials and was also ypotsizd in Rfs [65, 66] (s also Rf. [67]) basd on luminscnc xprimnts and quantum mcanical modling. T calculations prdict t optical nrgy to b twic t trmal nrgy, W opt ˆ W T, suggsting tat t activation nrgy for t captur of an lctron or a ol is nar zro. 40 () T ˆ 40 K () T ˆ 300 K (3) T ˆ 77 K 50 V, Figur 5. V±I caractristics of Si 3 N 4 masurd at diffrnt tmpraturs (dots) for a positiv potntial on Al and calculatd (solid lins) from t multiponon ionization tory. T paramtrs ar t sam as in Fig Tis fact is consistnt wit t obsrvd wak tmpratur dpndnc of t captur cross sction for lctron and ols in Si 3 N 4 [36]. T calculatd ponon nrgy W p ˆ 60 V is qual to t ponon nrgy in amorpous silicon obtaind from Raman scattring spctra, idntifying amorpous silicon nanoclustrs in silicon nitrid as likly candidats for lctron and ol traps. Quantum siz ffcts in amorpous silicon dots in silicon nitrid wr obsrvd in optical absorption and potoluminscnc xprimnts [68, 69]. Amorpous Si nanoclustrs can captur lctrons and ols in silicon nitrid [70]. T larg diffrnc btwn t trmal and optical ionization nrgis prdictd by t multiponon tory is possibly du to t strong dformation of t Si 3 N 4 lattic. T captur of lctrons and ols in Si 3 N 4 probably occurs on t minimal silicon clustr, t Si±Si bond. Tis modl assums tat t Si±Si bond or, altrnativly, a clustr of a fw silicon atoms, is a dp cntr for lctrons and ols and tat it also maks an ionization cntr. T quantum mcanical modling of t Si±Si bond in Si 3 N 4 supports tis ypotsis qualitativly [66]. 7. Conduction in silicon-nricd silicon nitrid T tory of transport by tunnling btwn traps was tstd by comparing its prdictions wit xprimntal rsults on t conductivity of silicon-nricd amorpous films of SiN x grown at various SiH 4 =NH 3 ratios [7]. T xprimnt masurd t dpndnc of t currnt on t voltag applid to a dilctric wit a ticknss of 00 nm [6]. It is known tat as t SiH 4 =NH 3 ratio incrass during t syntsis of Si 3 N 4, and t paramtr x dcrass, signifying an incras in t concntration of stoiciomtrically xcssiv silicon. Incrasing t SiH 4 =NH 3 ratio incrass t rfractiv indx n (s t caption to Fig. 7 blow for its valu). In Fig. 6, t masurd room-tmpratur V±I caractristics of stoiciomtric () and silicon-nricd () silicon nitrid ar prsntd in coordinats corrsponding to t Frnkl law and in coordinats corrsponding to Pool's law, lg J F. Worty of not, t currnt in siliconnricd nitrid is muc largr tan in its stoiciomtric countrpart. Furtrmor, t formr compound conducts a noticabl currnt vn at low voltags, wras in t lattr tis is only possibl aftr a crtain voltag trsold is racd Ð a striking contrast, wic w bliv is du to t ig concntration of traps in t nricd compound. Tis is consistnt wit t arlir ypotsis [65] tat t natur of traps in silicon nitrid rlats to silicon clustrs and, in particular, t Si±Si bond can srv as a trap in t compound, and a local ponon is associatd wit tis bond. To obtain t voltag dpndnc of t currnt, w prformd numrical calculations tat allowd t ionization of nutral traps, but not tunnling btwn tm. T calculations sowd good agrmnt wit xprimntal data for nar stoiciomtric nitrid (curvs, in Fig. 7), but undr no conditions could ty rproduc t rsults obtaind for silicon-nricd nitrid. Tis could only b satisfactorily don by introducing trap-to-trap tunnling (curvs 3, 4 in Fig. 7). To aciv good agrmnt btwn tory and xprimnt, a ig concntration (> 0 cm 3 ) of traps is ndd.

12 00 K A Nasyrov, V A Gritsnko Pysics ± Uspki 56 (0) 0 a b 0 4 J, A cm J, A cm p F 0 3, (V cm ) = F 0 6, V cm Figur 6. (a) V±I caractristics of silicon-nricd silicon nitrid in coordinats corrsponding to t Frnkl law: SiH 4 =NH 3 ˆ 0:0 (); 0. (); 0. (3); 0.33 (4); 0.5 (5). (b) T sam in coordinats corrsponding to t Pool law: SiH 4 =NH 3 ˆ 0:0 ( 0 ); 0. ( 0 ); 0.33 (3 0 ). J, A cm V, V 8. Monopolar lctron conduction in Al O 3 Figur 7. Exprimntal (dots) and calculatd (solid lins) V±I caractristics of Si 3 N 4 for diffrnt trap concntrations. T trap paramtrs usd in t calculations ar () SiH 4 =NH 3 ˆ 0:0, nˆ:96; otr paramtrs: N t ˆ0 9 cm 3, W T ˆ:7 V, W opt ˆ3:5 V, Fˆ:3 V, m ˆ 0:5m 0 ; () SiH 4 =NH 3 ˆ 0:, n ˆ :05, N t ˆ 0 9 cm 3, W T ˆ :5 V, W opt ˆ 3 V, F ˆ :8 V, m ˆ 0:5m 0 ; (3) SiH 4 =NH 3 ˆ 0:, n ˆ :, N t ˆ 3 0 cm 3, W T ˆ :55 V, W opt ˆ 3 V, F ˆ : V, m ˆ 0:5m 0 ; (4) SiH 4 =NH 3 ˆ 0:5, n ˆ :35, N t ˆ30 cm 3, W T ˆ :7 V, W opt ˆ 3 V, F ˆ : V, m ˆ 0:5m 0. Hr, F is t barrir igt for lctron injction from t silicon substrat into nitrid (t nrgy diffrnc btwn conduction bands in silicon nitrid and silicon). Calculations ar don by varying N t, W T, W opt, and F wil kping t ffctiv lctron mass fixd. In amorpous aluminum oxid, t band gap is E g 6: V and t barrir igt for lctrons at t Si=Al O 3 intrfac is f :0 V [5]. Al O 3 films av a rlativly low concntration of traps, and nc tir lakag currnt is lowr tan in HfO. Using Al O 3 as a blocking layr in silicon nitrid quantum dot flas mmoris was proposd in [6, 9, 0]. Conduction in Al O 3 as bn intrprtd by many in trms of t Pool±Frnkl modl [7,73]. T potntial barrir for ols at t Si=Al O 3 intrfac quals 3. V, wic is muc largr tan t barrir for lctrons at t Si=Al O 3 and Al=Al O 3 intrfacs. Trfor, if t Al lctrod is at a ngativ potntial, t dominant mcanism is t lctron injction from t uppr lctrod (i.., from aluminum) [74]. For tis rason, monopolar conduction in Al O 3 du to lctrons is considrd. T injction currnt at t Al=Al O 3 intrfac was calculatd using t Fowlr±Nordim mcanism, wit t lctron ffctiv mass takn to b 0:45m 0 [75±77]. T xprimntal and prdictd rsults (t Frnkl modl wit trmally assistd tunnling) on t tmpratur dpndnc of t currnt in and V±I caractristics of Al O 3 ar compard in Figs 8 and 9. T calculations wr prformd for t captur cross sction s ˆ cm and t lctron trap concntration N t ˆ cm 3. At ig tmpraturs in wak lctric filds, t conductivity dpnds on t tmpratur xponntially. T trap nrgy is stimatd as W ˆ :4 V from t slop of t lg J T curv. T bst fitting valu of t frquncy factor was n ˆ 0 9 s. Tis valu is also anomalously small. At low tmpraturs, t currnt dpnds wakly on tmpratur and is dtrmind by t tunnling mcanism. T bst fitting tunnling lctron ffctiv mass is m ˆ 3:5m 0. W not t ncssity of using diffrnt lctron ffctiv masss at t boundary and in t bulk of Al O 3 (m ˆ 3:5m 0 in t bulk and m ˆ 0:45m 0 at t Si=Al O 3 intrfac). Using t bulk valu m ˆ 0:45m 0 at t boundary rsults in a contactlimitd conductivity. In Fig. 0, t xprimntal tmpratur dpndncs of t V±I caractristics (dots) ar compard wit tos calculatd for t multiponon ionization in Al O 3 (solid lins) [34] using a cross sction of cm and a concntration of 0 0 cm 3 for nutral traps. T lctron ffctiv mass and t magnitud of W opt wr stimatd undr low-tmpratur, ig-fild conditions to giv t lctron ffctiv mass 0:4m, W opt ˆ 3:0 V, W T ˆ :5 V, and W p ˆ 0:05 V as t bst fitting valus for liquid nitrogn tmpratur. A dtaild study of optical transition on a ydrogn vacancy in amorpous aluminum oxid yildd t lumins-

13 Octobr 03 Transport mcanisms of lctrons and ols in dilctric élms Frnkl ffct 0 4 Multiponon ionization J, A cm V ˆ. V 0 V =T, K Figur 8. Tmpratur dpndncs of t currnt in Al O 3 masurd for diffrnt ngativ potntials (dots) on Al and calculatd using t Frnkl modl including trmally assistd tunnling (solid lins). Trap paramtrs: Wˆ:4 V, m ˆ 3:5m, nˆ0 9 s, s ˆ cm, N t ˆ cm 3. J, A cm T ˆ 400 K V, V Figur 0. V±I caractristics of Al O 3 masurd at diffrnt tmpraturs for ngativ potntial on Al (dots) and calculatd using t tory of multiponon ionization (solid lins). Trap paramtrs: W T ˆ :5 V, W opt ˆ 3:0 V, W p ˆ 0:05 V, m ˆ 0:4m, N t ˆ 0 0 cm 3, s ˆ cm. 8 J, A cm Frnkl ffct T ˆ 400 K cnc nrgy 3.0 V and t xcitation nrgy 6.0 V [78±80]. T alf of t luminscnc Stoks sift is.5 V, wic quals t trmal nrgy of t trap. Tis idntifis oxygn vacancis as traps for lctrons in amorpous aluminum oxid. In amorpous aluminum oxid, t ratio W opt =W T ˆ corrsponds to t zro activation nrgy for trapping. Prviously, t sam valu of tis ratio was obtaind for Si 3 N 4 [6]. Tus, just as for silicon nitrid, t Frnkl modl quantitativly (formally) dscribs carg transport in a wid rang of lctric filds and tmpraturs, but rquirs an anomalously larg ffctiv mass and an anomalously small frquncy factor to fit t data. At t sam tim, t multiponon conduction modl is capabl of quantitativly (consistntly) dscribing t transport of lctrons in amorpous aluminum oxid V, V Figur 9. V±I caractristics of Al O 3 masurd (dots) at diffrnt tmpraturs and calculatd (solid lins) using t Frnkl modl including trmally assistd tunnling. Trap paramtrs: W ˆ :4 V, m ˆ 3:5m, n ˆ 0 9 s, s ˆ cm, N t ˆ ˆ cm Conclusion Wit t xcption of trmal oxid on silicon, t majority of dilctrics av tir conduction limitd by t ionization of traps. T widly accptd Frnkl modl basd on t ionization of cargd Coulomb traps dscribs xprimnt quantitativly (formally) ovr a wid rang of filds and tmpraturs. Howvr, quantitativ agrmnt wit xprimnt rquirs an unpysically larg frquncy factor and an anomalously larg ffctiv mass, an ordr of magnitud largr tan in xprimnt. Exprimntal V±I dpndncs ar dscribd quantitativly by t tory of multiponon ionization of nutral traps in a wid tmpratur rang. Dpnding on t barrir igt for lctrons and ols, a dilctric can b a monopolar conductor, as xmplifid by amorpous Al O 3, for wic ol barrirs ar muc igr tan tos for lctrons. Al O 3 xibits monopolar lctron conduction, and in amorpous silicon nitrid, wit its narly qual-igt barrirs for lctrons and ols, two-band bipolar conduction occurs, wit lctrons (ols) injctd from t ngativly (positivly) biasd lctrod. In t bulk of t dilctric, two rcombination procsss occur: btwn fr lctrons and localizd ols, and btwn fr ols and localizd lctrons. Exprimnts on transport ar quantitativly xplaind by assuming tat traps for lctrons and ols ar nutral. As sown by xprimnt, concntrations, nrgis, and cross sctions ar t sam for lctron and ol traps in silicon nitrid. T trmal activation nrgy is twic t optical ionization nrgy, suggsting tat tr ar no barrirs for t captur of lctrons and ols onto traps. Tis dos not contradict t wak tmpratur dpndnc of t cross sction for t captur of lctrons and ols in silicon nitrid. Rfrncs. Gritsnko V A Pys. Usp (008) [Usp. Fiz. Nauk (008)]. Kingon A I, Maria J-P, Striffr S K Natur (000) 3. Wilk G D, Wallac R M, Antony J M J. Appl. Pys (00) 4. Robrtson J Eur. Pys. J. Appl. Pys (004)