generic impulse response function for mos systems and its application to linear response analysis

1178 IEEE Transactions on Nuclear Science, Vol. 35, No. 6, December 1988

GENERIC IMPULSE RESPONSE FUNCTION FOR MOS SYSTEMS AND ITS APPLICATION TO LINEAR RESPONSE ANALYSIS"

F. Barry McLean Harry Diamond Labora to r i e s

2800 Powder Mill Road Adelphi, MD 20783-1197

ABSTRACT

A gene ra l i zed response func t ion is presented which can desc r ibe s e v e r a l of t h e p r a c t i c a l l y impor tan t t r an - s i e n t response f e a t u r e s o f MOS systems. I t a l lows for dev ia t ions from strict loga r i thmic time dependencies y e t is mathematically t r a c t a b l e i n performing l i n e a r response ana lyses . F i t s o f the gene r i c response func- t i o n to experimental d a t a are d iscussed , inc luding t h e short-term recovery due to hole t r a n s p o r t , t h e long- term recovery due t o t rapped ho le annea l , and t h e long- term, time-dependent buildup of i n t e r f a c e t r a p s . Analy t ic r e s u l t s f o r t h e convolution i n t e g r a l o f l i n e a r response theory are der ived for a square i r r a d i a t i o n p u l s e , and some simple a p p l i c a t i o n s are d iscussed .

I. INTRODUCTION

Measurements of metal-oxide-semiconductor (MOS) device response a t t h e end of i r r a d i a t i o n t o t h e same t o t a l dose but de l ive red at d i f f e r e n t dose rates u s u a l l y w i l l show d i f f e r e n t resu l t s - -a consequence of t h e complex time h i s t o r y of MOS response i n which d i f - f e r e n t amounts of annea l ing occur dur ing i r r a d i a t i o n f o r d i f f e r e n t exposure times. To d e a l wi th t h i s problem a n a l y t i c a l l y , one can resort t o t h e techniques of l i n e a r response theory , which are v a l i d as long as the response is l i n e a r i n dose. I f one knows t h e impulse response func t ion , A V ( t ) , say t h e th re sho ld vo l t age response t o an i n f in iEes imal ly s h o r t i r r a d i a t i o n pulse of u n i t dose, then t h e gene ra l response (e.g. , th resh- o l d vo l t age s h i f t ) t o an a r b i t r a r y i r r a d i a t i o n descr ibed by t h e dose rate func t ion + ( t ) may be obta ined through t h e convolu t ion i n t e g r a l

where it is assumed t h a t +( t ) = 0 for t < 0. Previous au tho r s Cl-31 have app l i ed Eq. ( 1 ) t o t h e a n a l y s i s of t h e long-term pos t - i r r ad ia t ion response of MOS dev ices sub jec t ed t o cons t an t dose rate i r r a d i a t i o n s o f vary ing exposure times i n which t h e impulse response func t ion was taken to have a loga r i thmic time dependence. I n t h i s case , Eq. ( 1 ) can be i n t e g r a t e d i n c losed form, and t h e a n a l y s i s is reasonably s t r a igh t fo rward . Obviously, such ana lyses are app l i cab le to s i t u a t i o n s i n which t h e time dependence of t h e impulse response is approximately logar i thmic over t h e time regime of i n t e r e s t .

Indeed. t h e va r ious phys ica l p rocesses under ly ing t h e t o t a l dose i o n i z a t i o n response of MOS systems gene ra l ly r e s u l t [ 4 ] i n approximately loga r i thmic time dependencies of phys ica l q u a n t i t i e s of i n t e r e s t over l i m i t e d time regimes. P e r t i n e n t phenomena here inc lude t h e shor t - te rm th re sho ld vo l t age recovery due t o t r ans - p o r t of the rad ia t ion-genera ted holes through t h e oxide films, t h e long-term recovery due t o annea l ing of deeply t rapped ho le s near t h e S i02/Si i n t e r f a c e , and t h e delayed, two-stage time-dependent bu i ldup of i n t e r - face t r a p s (ANIT) which induces s t r e t c h o u t i n device c h a r a c t e r i s t i c s . However, a l though t h e r e s u l t i n g time dependencies are l o g time i n qlzero order , " s t r i c t log- time behavior does no t gene ra l ly hold over more than s e v e r a l decades i n time. Therefore , i n o rde r t o apply

* Supported i n p a r t by t h e U.S. Army S t r a t e g i c Defense Command under t h e SAT 8.1 Program.

Eq. ( 1 ) w i t h improved q u a n t i t a t i v e accuracy and over longer time regimes, one must cons ider dev ia t ions of t h e impulse response from s t r i c t log-time behavior.

Th i s po in t was emphasized r e c e n t l y i n a s tudy 151 of t h e long-term, pos t - i r r ad ia t ion annea l ing of s e v e r a l MOSFETs (MOS f i e l d - e f f e c t t r a n s i s t o r s ) of vary ing r ad i - a t i o n s e n s i t i v i t y , i n which it was shown t h a t t h e more rad ia t ion-hard devices exh ib i t ed r e l a t i v e l y l a r g e r d e v i a t i o n s from t h e s t r i c t log-time behavior and t h a t s i g n i f i c a n t e r r o r s could r e s u l t i n p r e d i c t i o n s of t h e long-term response of t h e s e devices i f t h e dev ia t ions were not accounted f o r . It was argued t h a t t h e devia- t i o n s from log-time behavior r e s u l t from nonuniform t rapped hole d i s t r i b u t i o n s i n t h e g a t e oxides near t h e S i02/Si i n t e r f a c e , and t h a t t h e g r e a t e r dev ia t ions from l o g time i n t h e harder devices r e f l e c t p r imar i ly nar- rower t rapped hole d i s t r i b u t i o n s . Assuming t h a t t h e t rapped ho le s are exponen t i a l ly d i s t r i b u t e d from t h e i n t e r f a c e and t h a t t h e annea l ing t akes p l ace v i a a tunnel/recombination process , we der ived a s u i t a b l e response func t ion for t h e long-term th re sho ld vo l t age s h i f t t h a t accounted very well f o r t h e dev ia t ions from log-time behavior. as well as f o r t h e q u a l i t a t i v e d i f - f e r ences i n t h e response between t h e va r ious devices . However, t h e s tudy [5] d i d not p re sen t t h e l i n e a r response a n a l y s i s us ing t h e der ived response func t ion i n the convolu t ion i n t e g r a l .

I n t h i s paper we present t h e l i n e a r response a n a l y s i s employing an impulse response func t ion which accounts i n a genera l way for f i r s t - o r d e r dev ia t ions of t h e response from s t r ic t log-time behavior and which is mathematically t r a c t a b l e i n performing t h e convolu t ion i n t e g r a l embodied i n Eq. ( 1 ) . We show t h a t t h e gene ra l form of t he response func t ion is broadly app l i cab le t o a range of MOS response problems, i nc lud ing not on ly t h e long-term recovery a s soc ia t ed with t h e tunnel annea l of deeply t rapped holes , bu t also t h e short-term recovery due t o hole t r a n s p o r t and t h e long-term e f f e c t s o f t h e delayed time-dependent bu i ldup of i n t e r - face t r a p s .

We set t h e s t a g e i n t h e next s e c t i o n with a b r i e f review of t h e p e r t i n e n t f e a t u r e s of the time-dependent MOS response. I n Sec t ion I11 we in t roduce our simple gene r i c approximate impulse response func t ion , d i scuss its p r o p e r t i e s , and show t h a t it desc r ibes t h e exper i - menta l ly observed time-dependent MOS responses . Then, i n Sec t ion I V we present the a n a l y t i c r e s u l t s Of t h e convolu t ion i n t e g r a l us ing t h e gene r i c impulse response func t ion f o r t h e case of a square i r r a d i a t i o n pulse , and we d i s c u s s a few s imple a p p l i c a t i o n s t o l i n e a r response ana lys i s . We also b r i e f l y d i scuss some l i m i - t a t i o n s o f l i n e a r response a n a l y s i s of which t o be aware. F i n a l l y , we summarize our r e s u l t s i n the l a s t sect ion.

11. TIME DEPENDENT RESPONSE OF noS SYSTEHS

The c e n t r a l problem addressed he re concerns t h e time-dependent response of a MOS system sub jec t ed t o a s h o r t pu l se o f i on iz ing r a d i a t i o n . For a r ecen t gen- eral review of t h e time-dependent MOS response , t h e reader is r e f e r r e d t o Ref. 4 and t o the r e fe rences l i s t e d t h e r e i n . Here we simply po in t o u t t h e primary f e a t u r e s o f t h e response. F igure 1 i l l u s t r a t e s sche- ma t i ca l ly [4] t he rad ia t ion- induced s h i f t i n t h r shold 8 vo l t age as a func t ion o f log-time from I O + t o I O s

0018-9499/88/1200-1178$01.00 0 1988 IEEE

1179

f o r a rad ia t ion-hardened n-channel MOSFET under posi- t i v e g a t e b i a s a t room tempera ture a f t e r exposure t o an ion iz ing r a d i a t i o n pulse of -1 ps i n dura t ion . This f i g u r e is schemat ic i n t h a t it does not show real d a t a (over t h e enormous t i m e regime dep ic t ed ) . y e t it is b a s i c a l l y r e p r e s e n t a t i v e of t h e composite response of an a c t u a l hardened n-channel device. As such it is a schematic r e p r e s e n t a t i o n of t h e gene ra l impulse response func t ion o f an MOS system.

There are f o u r major b a s i c processes c o n t r i b u t i n g t o t h e r a d i a t i o n response of t h e MOS system, which are ind ica t ed on Fig. 1 i n t h e appropr i a t e time regimes where they are important. ( = AVT(10-6s) i n Fig. 1 ) . which is also t h e maximum s h i f t , is e s s e n t i a l l y determined by t h e e l ec t ron /ho le p a i r c r e a t i o n i n t h e S i02 bulk and by t h e i n i t i a l recombinat i o n occur r ing between t h e e l e c t r o n s and ho le s i n t h e few picoseconds before t h e r e l a t i v e l y h igh ly mobile e l e c t r o n s are c o l l e c t e d a t t h e g a t e [6-lo]. The h o l e s which escape t h e i n i t i a l recombination are rela- t i v e l y immobile and remain behind near their po in t s of gene ra t ion [11,12], caus ing t h e commonly observed nega- t i v e vo l t age s h i f t s i n t h e e lectr ical characteristics of MOS devices (e.g., i n t h e th re sho ld vo l t age of a MOSFET as ind ica t ed i n Fig. 1 ) . The shor t - te rm annealing t h a t is shown occur r ing o u t t o about s i n Fig. 1 is then due t o t h e $low, d i spe r s ive hopping t r a n s p o r t of ho les through t h e oxide l a y e r toward t h e S i02/Si i n t e r f a c e ( f o r p o s i t i v e g a t e b i a s ) . This ho le t r ans - p o r t phase has been success fu l ly descr ibed by a con- t i nuous time-random-walk (CTRW) model C12-173 of ho le s hopping between randomly d i s t r i b u t e d l o c a l i z e d gap states i n t h e Si02. When t h e ho le s reach t h e Si02 i n t e r f a c e , some f r a c t i o n of them are captured i n long- term t r app ing sites, and cause a remnant nega t ive vo l t age s h i f t which can p e r s i s t i n t i m e for hours t o years . I n F ig . 1 , t h e s h i f t i n VT remaining a t s is pr imar i ly t h a t due t o t h e deep ho le t r app ing nea r the i n t e r f a c e . which then anneals o u t slowly i n t i m e ( a t room tempera ture) v i a a process involv ing e l e c t r o n s tunne l ing i n t o t h e oxide from t h e S i s u b s t r a t e and e i t h e r recombining wi th or compensating the t rapped ho le s [5,16,18-241. The s o l i d curve i n Fig. 1 cor re- sponds t o t r a n s p o r t , t r app ing , and annea l ing of ho le s alone.

The i n i t i a l s h i f t AVO

I n add i t ion t o t h e long-term annea l ing of t rapped h o l e s , however, a long-term bui ldup o f r ad ia t ion - induced i n t e r f a c e t r a p s may occur [25-293, t y p i c a l l y i n t h e time regime between 10' and lo5 s, which is indi - ca t ed by t h e dashed curve i n Fig. 1. ( I n t e r f a c e t r a p s are l o c a l i z e d s ta tes r i g h t a t t h e S i02/Si i n t e r f a c e having energy l e v e l s wi th in t h e S i bandgap and whose occupancy depends upon t h e s i l i c o n s u r f a c e p o t e n t i a l and hence upon the app l i ed bias. For an n-channel de- v i c e , t h e i n t e r f a c e - s t a t e con t r ibu t ion t o VT is posi- t i v e , corresponding t o n e t nega t ive i n t e r f a c e t r a p charge a t t h e th re sho ld vol tage . ) The long-term buildup of i n t e r f a c e t r a p s appa ren t ly r e s u l t s C30-331 from p o s i t i v e i o n s (probably H+) being r e l eased i n t h e S i02 bulk dur ing t h e ho le t r a n s p o r t phase, which then d r i f t slowly t o t h e S i02/Si i n t e r f a c e and undergo a subse- quent i n t e r a c t i o n t h e r e c r e a t i n g the AN Note t h a t i f t h e i n t e r f a c e t r a p con t r ibu t ion is re3S i ive ly l a r g e , the threshold vo l t age may a c t u a l l y recover p a s t its p r e - i r r a d i a t i o n va lue (AV going p o s i t i v e ) as t h e t rapped holes annea l , g iv ixg rise to what is c a l l e d super recovery or rebound [34,35].

I n a t tempt ing to d e a l a n a l y t i c a l l y wi th a complex composite response such as ind ica t ed i n Fig. 1 , we can make l i f e easier by breaking t h e response down i n t o its components. Then one can combine t h e components as necessary i n dea l ing wi th a p a r t i c u l a r s i t u a t i o n . ( I n

t

io-' io-' io-' id id io' id id TIME AFlER RADIATION PUw

Fig. 1. Schematic time-dependent th reshold-vol tage recovery of n-channel MOSFET fo l lowing bulsed i r r a d i a - t i o n , r e l a t i n g major f e a t u r e s o f response t o under ly ing phys ica l p rocesses .

p r a c t i c e , f o r l i m i t e d time regimes one usua l ly has t o cons ider on ly one or two of t h e component responses . ) Based on Fig. 1 and t h e d i scuss ion above, we break t h e r ad ia t ion - induced thresh0 I d v o l t age s h i f t i n t o t h r e e time-dependent components:

.. A V T ( t ) = A V S T ( t J + A V O T ( t ) + A V I T ( t ) , ( 2 )

where AV ( t ) is t h e (short-term) con t r ibu t ion from t h e rad ia t ion-genera ted mobile ho le s t r a n s p o r t i n g i n t h e oxide bulk, A V O T ( t ) is due t o t h e bui ldup of deeply t rapped ho le s near t h e i n t e r f a c e and t h e i r long-term annea l ing , and AV ( t ) is t h e con t r ibu t ion from t h e buildup of ChargedITnterface t r a p s . Note t h a t t h e i n i - t i a l processes of pair genera t ion and recombination do not con t r ibu te a time-dependent component i n a prac- t i c a l sense s i n c e they occur i n a picosecond time frame; r a t h e r , t h e i n i t i a l ho le y i e l d sets the vo l t age scale f o r t h e response , i . e . , determines t h e i n i t i a l cond i t ion on A V T , which is given by [8-101

ST

-AVO = 1.9 x dgx f ( E ) D . Y ox ( 3 )

Here, t h e u n i t s o f AV are v o l t s i f t h e oxide th i ckness do, is expressed ino nanometers and t h e dose D i n r ad (S i0 1. The func t ion f (Eox) is t h e f r a c t i o n a l ho le y i e l d ? f r a c t i o n of orig?nal ho le s escaping i n i t i a l recombination) and is a func t ion o f t h e oxide e lectr ic f i e l d , , and of t h e inc iden t p a r t i c l e type and en&?gy. Eo#or h igh energy e l e c t r o n s ( > 100 keV) or f o r Cd gamma i r r ad ia t ion , f ( E ) can be expressed a n a l y t i c a l l y as [ f (Eox)] = 1 + 0.2707/(EOx + 0.0837), where Eo, is i n megavolts pe r cent imeter [lo].

We now simply assert t h a t t h e time-dependent response func t ions f o r t h e t h r e e phys ica l p rocesses inc luded i n Eq. (2 ) -- ho le t r anspor t , t rapped ho le annea l , and i n t e r f a c e t r a p bui ldup -- a l l e x h i b i t e s s e n t i a l l y t h e same gene r i c form wi th q u a l i t a t i v e l y Qimilar c h a r a c t e r i s t i c s as a func t ion o f log-time. This gene r i c form for t h e component impulse response is depic ted schemat i ca l ly i n Fig. 2. We w i l l suppor t t h e above a s s e r t i o n i n t h e next s e c t i o n and i l l u s t r a t e some of t h e p e r t i n e n t d e t a i l s o f t h e component responses when we d i scuss t h e fits of ou r approximate response func t ion t o exper imenta l ly observed da ta . For t h e moment we simply note t h e p r i n c i p a l q u a l i t a t i v e f e a t u r e s of t h e gene r i c response depic ted i n Fig. 2: t h e response starts o u t f l a t i n log-time a t some i n i - t i a l va lue (denoted by AV i n t h e f i g u r e ) , and begins t o recover (bend upwards? around some c h a r a c t e r i s t i c

Y ox

1180

We note t h e fo l lowing p r o p e r t i e s o f t h e response func- t ion:

Fig. 2. Generic form f o r MOS component impulse response func t ion .

time to. For times beyond to, t h e response becomes approximately loga r i thmic over some time regime, bu t then d e v i a t e s from log-time a t longer times, and f in - a l l y approaches a li-miting s a t u r a t i o n va lue (AV-1 i n a power law fash ion ( t ') as t + i n f i n i t y . The power law behavior a t l a te time is e x p l i c i t l y ind ica t ed in t h e cases of ho le t r a n s p o r t [13,17] and t rapped hole annea l v i a tunnel ing [5. 181. For t h e case of ANIT buildup, i t seems a t least t o be ind ica t ed by t h e experimental d a t a [26]. The simple form f o r t h e gene r i c impulse response dep ic t ed i n Fig. 2 provides t h e mot iva t ion for our choice of approximate impulse response func t ion t h a t we s h a l l use.

The q u a l i t a t i v e s i m i l a r i t y of t h e response i n a l l t h r e e cases is a r e s u l t of t h e d i s p e r s i v e na tu re of t h e underlying phys ica l p rocesses . That is, t h e phys ica l phenomena r e spons ib l e f o r t h e t h r e e sepa ra t e response components a l l have wide ind iv idua l microscopic event time d i s t r i b u t i o n s , which g ive r ise t o observed responses occurr ing over many decades i n t i m e and which may appear nea r ly loga r i thmic over l imi t ed time regimes. For ho le t r a n s p o r t t h e d i spe r s ion arises from a wide d i s t r i b u t i o n of hopping times between t h e randomly loca ted hopping sites i n t h e S i02 112,171; t h e d i s p e r s i o n t h e AVOT recovery r e s u l t s from t h e wide d i s t r i b u t i o n of t unne l ing times t o a s p a t i a l d i s t r i b u - t i o n of t rapped ho le s near t h e S i02 / s i i n t e r f a c e [5,181; and i n t h e case of i n t e r f a c e t r a p buildup t h e d i spe r s ion appa ren t ly o r i g i n a t e s from t h e d r i f t / d i f f u - s i o n of an i o n i c s p e c i e s (probably H + ) t o t h e i n t e r f a c e a f te r being r e l eased i n t h e bulk by t h e ho le t r a n s p o r t Phase C30-333. Therefore , even though t h e t h r e e wmpo- nent time-dependent MOS responses o r i g i n a t e from d i s - t i n c t phys ica l phenomena, t h e fact that each has a wide ( d i s p e r s i v e ) microscopic event time d i s t r i b u t i o n l e a d s i n each case t o an observed response q u a l i t a t i v e l y descr ibed by t h e gene r i c impulse response i n Fig. 2.

111. APPROXIHATE IMPULSE RESPONSE FUNCTION

With t h e d i scuss ion of t h e prev ious s e c t i o n as background, we can proceed quick ly t o our c e n t r a l purpose. We want t o choose t h e s imples t poss ib l e a n a l y t i c func t ion having t h e q u a l i t a t i v e charac te r - ist ics exh ib i t ed i n Fig. 2 and which is a l s o mathe- ma t i ca l ly t r a c t a b l e i n eva lua t ing t h e convolu t ion i n t e g r a l , Eq. ( 1 ) . Our choice f o r such a func t ion is of t h e fo l lowing form:

A V R ( t ) = -C + Af,(y), (4)

where C , A , v , and to are cons tan ts , y = 1 + t/to, and f v ( y ) is de f ined by

( a ) By our choice o f t h e argument y, to sets t h e time scale f o r the response. Note t h a t f v ( l ) 0 and the re fo re AVR(0) = -C. Also note for t >> to, y = t/to.

( b ) For v > 0, f v ( y -* - ) = l / v , so t h a t t h e long-term l i m i t i n g va lue of t h e response is -C + A/v. (For v < 0, f v ( y ) does n o t have a l i m i t i n g va lue as y + -. However, f o r a l l ou r app l i ca t ions we are assuming the gene r i c form i l l u s t r a t e d schemat ica l ly by F ig . 2 , which r e q u i r e s t h a t v be pos i t i ve . )

( c ) f v ( y ) has , the series r e p r e s e n t a t i o n

1 1 21 3! (6) f v ( y ) = l ny C I - - v lny + - (u lny )2 - ---I.

i .e . , b a s i c a l l y , a power series i n vlny.

( d ) I n p a r t i c u l a r , l i m f v ( y ) = lny , i n which case t h e response func t ionvr&er t s f o r t >> to t o t h e form used i n t h e earlier l i n e a r response ana lyses [ l - 31. That is, t h e pu re ly log-time response is a s p e c i a l case of Eq. ( 4 ) . We note from p r o p e r t i e s (a ) and ( b ) t h a t i f we le t C = AVO and -C + A/v = AV-, Eqs. ( 4 ) and ( 5 ) can be combined i n t o t h e simple power law form

- A V , ( t ) = AVm + ( A V - AVm)/(l + t/to)' . ( 7 )

The reason we have chosen t o s ta r t wi th t h e d e f i n i t i o n s Eqs. (4) and ( 5 ) is simply because o f Eq. ( 6 ) . which e x p l i c i t l y shows t h e r e l a t i o n s h i p o f t h e response t o log-time and, t he re fo re , y i e l d s a one-to-one correspon- dence wi th prev ious work u t i l i z i n g a s t r ic t logar i thmic annea l ing func t ion .

We note t h a t Eq. ( 4 ) (or Eq. ( 7 ) ) has f o u r con- s t a n t s t o be determined by appropr i a t e f i t t i n g o f experimental dd ta . As such, it a l lows f o r t h e follow- i n g gene ra l q u a l i t a t i v e f e a t u r e s of an observed response which are a l s o c o n s i s t e n t with Fig. 2: it is cons t an t at t h e value -C a t e a r l y times t << to, bend- ing upwards a t t - to t o an approximate l n ( t ) form wi th s lope A ( e s p e c i a l l y f o r small va lues of v ) , then devi- a t i n g from t h e l n ( t ) response a t longer times ( f o r t such t h a t v ln(1 + t/to) > l ) , and f i n a l l y , f o r v > 0, approaching a l i m i t i n g , s a t u r a t i o n va lue -C + A/v a t long times. Loosely speaking, t h e fou r cons t an t s are used t o determine t h e time scale f o r t h e response , t h e i n i t i a l va lue of t h e response , t h e s lope i n t h e logar - ithmic regime, and t h e dev ia t ion from t h e log-time response l ead ing ( f o r v > 0) t o t h e f i n a l s a t u r a t i o n va lue of t h e response. We note a l s o t h a t Eq. (4) is e s s e n t i a l l y t h e form der ived i n Ref. 5 for AVT anneal- i n g v i a tunne l ing of e l e c t r o n s t o 'an exponent ia l d i s - t r i b u t i o n of t rapped holes . ( I n Ref. 5 t h e equat ions apply only for t>>to, and t h e s i g n o f t h e exponent v t h e r e is def ined oppos i te to t h a t adopted here . ) We now go on t o desc r ibe t h e r e s u l t s of f i t t i n g t h e response func t ion Eq. ( 4 ) to t r a n s i e n t response da t a f o r t h e t h r e e p r a c t i c a l l y impor tan t components of MOS response, desc r ib ing i n each case t h e f i t t i n g procedure used and its r e l a t i o n s h i p t o t h e under ly ing phys ica l model.

Hole Transpor t . Fig. 3 shows t h e f i t of Eq. ( 4 ) t o short-term recovery da ta due to ho le t r a n s p o r t , where t h e t r anspor t da t a have been obta ined from a "un ive r sa l curve," i n which t h e recovery da ta taken a t a series of tempera tures , fo l lowing pulsed 4 - p ~ LINAC i r r a d i a t i o n . are p l o t t e d versus time sca l ed t o t h e time, tlI2, a t which h a l f recovery occurs C171. I n such a manner the experimental impulse response curve is obta ined over

1181

0 I I I I I I I I

10-2 100 102 104 1oC l o C 8ul.d Tbna (7 = t/h)

1 .o

Fig. 3. Short-term recovery da ta due t o hole t r ans - p o r t . So l id curve is approximation to impulse response us ing Eq. ( 4 ) . Time is s c a l e d t o to i n lower scale and t o ha l f - recovery time t i n upper scale. Voltage s h i f t is normalized t o ilL&al s h i f t , Eq. ( 3 ) , immedi- a t e l y fo l lowing pulsed LINAC i r r a d i a t i o n . (Transpor t d a t a from Ref. 17.)

many decades i n time from e a r l y t o l a t e sca l ed time values. The s o l i d curve i n Fig. 3 is t h e f i t of t h e approximate impulse response, Eq. (4). The vo l t age s h i f t f o r both t h e da t a and t h e f i t t e d curve is sca l ed t o t h e i n i t i a l s h i f t A V given by Eq. (31, correspond- i n g t o choosing C = 1 ir? Eq. ( 4 ) . The power exponent v is taken t o be a, t he CTRW d i so rde r parameter, because t h e CTRW model of t r a n s p o r t p r e d i c t s a power law dependence of th re sho ld vo l t sge recovery a t late time wi th a as t h e exponent [361. This late-time behavior has been v e r i f i e d exper imenta l ly [13,171, wi th an a-value of 0.25 being determined f o r t h e t r a n s p o r t d a t a used i n Fig. 3. (a-values i n t h e range 0.15 t o 0.35 have been exper imenta l ly determined f o r d i f f e r e n t oxides. How- e v e r , f o r rad ia t ion-hardened oxides less than 100 nm i n th i ckness , t h e observed va lves have c l u s t e r e d around 0.25.) The time scale parameter, t , i n Eq. ( 4 ) is chosen t o f o r c e a f i t of t h e approxfmate func t ion t o t h e da t a a t t h e time of half-recovery, tlI2. This requirement y i e l d s

t = t / ( 2 l l a - 1 ) , 0 112

which f o r a = 0.25 g i v e s t We no te t h a t the lower time scale i n Ffg. 3 is in u n i t s of to, and t h e upper scale i n u n i t s of tlI2. These sca l ed times can be transformed i n t o real time by us ing t h e func- t i o n a l dependence of t on tempera ture ( T ) , oxide f i e l d ( E o x ) , and oxide '&ickness (dox) t h a t has been found from t h e CTFJW a n a l y s i s of t h e t r a n s p o r t da t a [17], which, f o r T >-140 K , is

= 0.067 tlI2.

( 9 )

Here til2 is a cons tan t , kg is t h e Boltzmann cons t an t , a is t e average hole hopping d i s t ance , and t h e f i e l d - dependent a c t i v a t i o n energy is A ( E ) = A - bEox, where b is a cons t an t , and A is t h e lE$ f i e l a l i m i t of A(Eox). For c lean , ha rdene8S i02 , a = 0.25, a = 1 nm,

A. = 0.65 eV, b = 0.05 eV/MV/cm, and to 3 x 10-23S- With these parameter va lues , t h e time/%o reach ha l f - recovery f o r a 100-nm oxide a t room temperature and f o r

a 1-MV/cm f i e l d is -1 x s. Fina l ly , i n Fig. 3 a long-term ho le t r app ing f r a c t i o n , fT,. of 0.02 was used based on t h e t r a n s p o r t d a t a [17], which corresponds t o t h e choice AV- = 0.04 A V (or A = v ( l - 2 fT) = 0.24). i n d i c a t i n g very l i t t l e fong-term t r app ing o f h o l e s i n t h i s p a r t i c u l a r sample. The use of a cons t an t t r app ing f r a c t i o n t o produce t h e f i n a l , s a t u r a t e d va lue o f t h e

th re sho ld vol tage s h i f t accounts f o r the trapped hole bui ldup a t t h e i n t e r f a c e , bu t n e g l e c t s the effect o f the annea l ing o f t h e t rapped h o l e s , an approximation which is s e n s i b l e on ly f o r a hard oxide i n t h e sho r t - term, hole t r anspor t recovery regime. (The combined response of ho le t r anspor t / t r apped ho le annea l w i l l be t r e a t e d next . ) We note t h a t with t h e above f i t t i n g procedure, Eq. ( 4 ) provides a good d e s c r i p t i o n of t h e shor t - te rm response due t o hole t r a n s p o r t , e s s e n t i a l l y as good as provided by t h e more complicated CTRW model f i t L14.17.361. The only s i g n i f i c a n t discrepancy between t h e approximate response func t ion and t h e da t a i n Fig. 3. occu r s a t very e a r l y s c a l e d time, and prob- ab ly r e s u l t s from a tunne l ing con t r ibu t ion which is s i g n i f i c a n t on ly a t low tempera ture f o r which t h e hole t r a n s p o r t is a c t u a l l y very slow i n real time.

Trapped Hole Anneal. A s po in ted o u t i n t h e In t ro - duc t ion , t h e long-term annea l ing of ho le s trapped near t h e S i 0 / S i i n t e r f a c e was t h e main focus o f our r e c e n t s tudy &], where we showed t h a t dev ia t ions of t h e annea l ing curves from s t r i c t log-time behavior r e s u l t from nonuniform s p a t i a l d i s t r i b u t i o n s of t h e trapped holes . I n t h a t s tudy it was assumed t h a t t h e holes t rapped near t h e S i s u b s t r a t e annea l by a tunnel ing/ re - combination process involv ing e l e c t r o n s from t h e S i va lence band l e v e l s . We used t h e not ion of a time- dependent " tunnel ing f r o n t " C191. A t a given t i m e t , t h e tunne l ing t r a n s i t i o n rate is sharp ly peaked spa- t i a l l y (wi th a width of -0.2 nm) f o r t r a n s i t i o n s t o t r a p s a t a depth X,(t) from t h e S i /S i02 i n t e r f a c e , which inc reases loga r i thmica l ly wi th time, as X , ( t ) = ( 1 / 2 B ) i n ( t / t ). Here, t h e tunnel ing parameter B is r e l a t e d t o %he b a r r i e r he igh t f ac ing t h e tunne l ing e l e c t r o n s , and to, which sets t h e time scale f o r t h e process , is r e l a t e d t o t h e t r a n s i t i o n rate t o t h e c l o s e s t t r a p s . P r a c t i c a l l y , then , t h e tunnel ing pro- ceeds v i a a sha rp s p a t i a l f r o n t , moving i n t o t h e insu- l a t o r reg ion as i n ( t ) , with t r a p s c lose r than X , ( t ) annealed o u t a t time t and those beyond , X , ( t ) still occupied. The qtlogar--thmic ve loc i tx ; o f t h e tunne l ing f r o n t is AX = ( 2 8 ) i n 10 = 1.158 per decade i n time, which fir thermal S i02- i s abouJ 0.2 nm per decade a t room tempera ture E201 ( 8 = 5.8 nm 1. 1

To model t h e experimental da t a i n Ref. 5, we assumed a s imple exponent ia l form f o r t h e trapped ho le d i s t r i b u t i o n : N(x) = Noexp[-A ( x - xo)] f o r x > xo (and N(x) = 0 f o r x < xo) , where Ffo is t h e t r a p dens i ty at the d i s t ance xo cor responding t o t h e p o s i t i o n o f t h e tunne l ing f r o n t a t time to, which we took t o be t h e beginning o f t h e p o s t - i r r a d i a t i o n measurements. I n t h i s way t h e annea l ing which occurred before to is accounted f o r . Then we der ived an annea l ing func t ion e s s e n t i a l l y i d e n t i c a l i n form t o Eq. (41 , except t h a t t /to r e p l a c e s 1 + t/to i n Eq. ( 4 ) and time is re- s t r i c t e d t o be g r e a t e r than t/to. I n t h i s case, the power law exponent v = A/2$, C is t h e rad ia t ion- induced vo l t age s h i f t a t time t r e l a t i v e t o t h e pre- i r rad ia- t i o n va lue , and A = qNo?2BCox is t h e i n i t i a l annea l ing s lope r i g h t a f t e r to, where C is t h e oxide capaci- t ance . (We note aga in t h a t Ref. 5 v is def ined oppos i te i n s i g n t o the d e f i n i t i o n adopted here . ) F igure 4 is reproduced from Ref. 5 showing t h e r e s u l t s of t h e model f i ts t o t h e experimental long-term response data f o r three oxides of vary ing r a d i a t i o n s e n s i t i v i t y . For t h e s e d a t a t h e parameters C and A were obta ined d i r e c t l y from t h e e a r l y time measurements ( t 5 t ) , and t h e power law exponent v (or A), which is a mea&e of t h e nonlogar i thmic behavior of t h e re - sponse, was chosen t o produce agreement wi th t h e measurements a t l a te time. For t h e r a d i a t i o n hard sample, A = 1.1 nm-' (or v = 0.1). -7nd f o r the i n t e r - mediate hardness sample, A = 0.9 nm ( v = 0.08). For t h e s o f t samples, A = v = 0 , i n d i c a t i n g a pu re ly logar- ithmic annea l ing over t h e time regime s tud ied and cor-

1182

- TUNNU MODEL RESULTS -0.4 -

E

a e >

-0.6 -

-0.8 -

I I I I I 10s 104 10' 106 10'

-1.01 102

TIME AFTER l R R " ($1

Fig. 4. Long-term annea l ing da ta f o r t h r e e MOSFETs of vary ing r a d i a t i o n hardness. S o l i d curves are f i t s o f Eq. ( 4 ) t o da ta . (From Ref. 5.)

responding t o a f l a t t rapped hole d i s t r i b u t i o n sampled by t h e tunne l ing process dur ing t h e measurement time.

Obviously, t h e response d a t a shown i n Fig. 4 do no t appear t o be of the gene r i c form i l l u s t r a t e d i n Fig. 2. I n f ac t , t h e da t a i n Fig. 4 are only a po r t ion of t h e impulse response curve f o r t rapped ho le annealing. One d i f f i c u l t y i n e s t a b l i s h i n g exper imenta l ly t h e e n t i r e impulse response for t h e t rapQed ho le annea l is the enormous time regime over which it occurs ; gene ra l ly , on ly a l i m i t e d time regime is a c c e s s i b l e exper imenta l ly over which t h e annea l ing is approximately loga r i thmic bu t which, neve r the l e s s , is only a por t ion of t h e e n t i r e annea l ing curve. To be s p e c i f i c , a t the earl- iest measurement times shown @ Fig. 4, a l l t h e ho le s t rapped i n the r eg ion x < xo = 3 nm have a l r eady been annealed by tunnel ing . Another compl ica t ion , which occurs a t e a r l y times i n t h e recovery , is t h a t t h e t rapped ho le annea l process cannot be e a s i l y sepa ra t ed from the hole t r a n s p o r t phase. Even f o r an impulse i r r a d i a t i o n it t a k e s a f i n i t e time f o r the holes t o t r a n s p o r t ac ross t h e oxide f i l m (see Eq. ( 9 ) ) , and while the number of t rapped holes bu i ld up as t h e t r a n s p o r t i n g holes reach t h e i n t e r f a c e and some f rac- t i o n are captured i n deep t r app ing s i tes , t h e r e is a l r eady some annea l ing which is occurr ing .

To d e a l wi th t h e s e compl ica t ions and t o i l l u s t r a t e t h e e n t i r e t rapped ho le response curve, we must d e a l e x p l i c i t l y wi th t h e combined ho le t r anspor t / t r apped ho le response. This is important i n a p r a c t i c a l s ense whenever one is i n t e r e s t e d i n the e a r l y time response (e.g. , t < s) and i f t h e hole t r app ing f r a c t i o n is g r e a t e r than a few pe rcen t . We proceed i n t h e follow- i n g f a sh ion , r e f e r r i n g t o Fig. 5 f o r i l l u s t r a t i o n . We write t h e th re sho ld vo l t age s h i f t as

(10)

where & V H T ( t ) is t h e response due t o hole t r a n s p o r t a l o n e , assuming immediate c o l l e c t i o n of a l l the ho le s by t h e S i s u b s t r a t e as they r each t h e i n t e r f a c e (i.e., t h e response i n t h e absence of long-term ho le t rap- p i n g ) , and AV ( t ) is t h e t rapped ho le response , i nc lud ing both'ihe buildup dur ing the hole t r a n s p o r t phase and the annea l ing of t h e t rapped holes . S ince t h e buildup of the t rapped ho le s is d i r e c t l y c o r r e l a t e d t o t h e ho le t r a n s p o r t , we scale time ' e x p l i c i t l y t o t l I2, t h e ha l f - recovery time for hole t r anspor t a lone (Eq. ( 9 ) ) , and we scale t h e vo l t age s h i f t t o t h e i n i -

t i a l s h i f t , AV as given by Eq. ( 3 ) . Based on our earlier d iscusgion of ho le t r a n s p o r t , A V ( t ) is given HT by

-bVHT(t) = 1 - a f (1 + et / t , , , ) , (11)

where 0 A V O T ( t ) as

= 2'" - 1 (see Eq. (8 ) ) , and we write

- A V O T ( t ) = 2fTafa(l + e t / t l I 2 ) - Afv(l + t/to). ( 1 2 )

The first term i n Eq. (12) desc r ibes t h e buildup o f t h e t rapped h o l e s , and t h e second term t h e i r annea l ing w i t h time; f T is t h e deep hole t r app ing f r a c t i o n , and A , v , and to are parameters desc r ib ing t h e t rapped hole annea l ing . (The factor o f two i n t h e f i r s t term is s imply t h e "moment arm effect" o f t h e holes being t rapped a t t h e i n t e r f a c e r e l a t i v e t o an i n i t i a l l y uni- form d i s t r i b u t i o n . ) I f we assume t h e r e is a permanent t rapped ho le component which does not anneal a t any t i m e , s ay given by AVm = 6(2fTAVo), then from Eq. ( 5 ) we have

A = 2fTv(1+6) ( 1 3 )

( A normalized i n u n i t s of AV 1. From Fig. 5 we see t h a t AV ach ieves a maximum i% abso lu te va lue near t h e end o f %le hole t r a n s p o r t phase. We choose to as t h e time t h a t t h i s maximum i n IAV I occurs , i.e., by t h e condi t ion -[-AV ( t ) ] = 0. T@s y i e l d s d

d t OT

tlI2 21+v l / a t = - o e [dl (14)

where we assume t h a t to >> t / e (where t l I 2 / 8 is t h e time scale parameter i n Eq. Ed? for ho le t r a n s p o r t ) . a cond i t ion which is e s s e n t i a l l y always s a t i s f i e d . The corresponding maximum value of l A V I a t t = to can be r e l a t e d to t h e i n i t i a l s h i f t by an e f f e c t i v e ho le t rapping f r a c t i o n 2f: A V ~ . where

OT

(15 )

F i n a l l y , t h e power law exponent v and t h e permanent ( i n f i n i t e time) t r app ing parameter 6 can be determined from t h e long-term response da ta , as i n Fig. 4.

a = 0.25 Y = 0.10

Gff = 0.15 fT = 0.20

- Id 10'0

Fig. 5. Combined impulse response for hole t r a n s p o r t and deep hole t rapping . T h e dot-dash curve is impulse response for AVoT a lone , i nc lud ing both trapped hole bui ldup and annea .

1183

For t h e example c a l c u l a t i o n i l l u s t r a t e d i n Fig. 5 , we take a = 0.25 ( t h e value found f o r t h e CTRW disper- s i o n parameter i n hard ox ides ) ; f T = 0.2, corresponding t o 20 percent of t h e h o l e s being t rapped as they reach the i n t e r f a c e region; 6 = 0, which assumes a l l t h e t r apped ho le s even tua l ly anneal ou t ; and v - 0.1, corresponding approximately t o t h e long-term power law exponent found i n Fig. 4 C51 f o r t h e hard oxides . With t h e s e parameter va lues we have A = 0.04, to =

and f;ff 0.75fT = 0.15. Recal l ing t h e t y p i c a l time s c a l e s f o r hole t r a n s p o r t a t room temperature and f o r a l-MV/cm oxide f i e l d ( see Eq. (9) f o r t l I 2 ) , we f i n d to = 1.6 x s f o r a 25-nm oxide, We no te t h a t i f we are only i n t e r - e s t e d i n times t > tb , then t h e t rapped hole response can indeed be taken as the gene r i c form sho%qfin Fig. 2, and as expressed by Eq. ( 4 ) . where C = 2 f AV and to is given by Eq. ( 1 4 ) (and A and v are det8rmingd by t h e fits t o t h e response a t long-times t >> to). How eve r , t h e complete impulse response f u n c t i o n f o r AVOT is t h a t depicted by t h e dot-dash curve in Fig. 5, and, of course, t h e s o l i d l i n e is t h e t o t a l composite recov- e r y curve, i nc lud ing ho le t r a n s p o r t recovery, AV as wel l as AV . F i n a l l y , we add t h a t a t t h e us a1"e;mes

vo l t age s h i f t f o r t h e example i n Fig. 5 has recovered t o about 15 percent of t h e i n i t i a l s h i f t f o r a 100-nm oxide and t o l e s s t han 10 pe rcen t o f AV f o r a 25-nm oxide. These va lues of AVT correspond 'to e f f e c t i v e t r app ing f r a c t i o n s of -0.08 and 0.05 f o r t h e 100-nm and 25-nm oxides , r e s p e c t i v e l y , so t h a t t h i s example case would indeed correspond t o a hard oxide a t t h e usua l measurment times.

5 5 t ~ / 2 s

s f o r a 100-nm oxide and to = 6.6 x

f o r e a r l y o l e s t measurements, e.g.. l o 2 - 10 Y s, t h e

I n t e r f a c e Trap Buildup. The impulse response f o r t h e long-term bui ldup o f i n t e r f a c e t r a p s , ANIT, is much more a c c e s s i b l e experimental ly than t h e t rapped h o l e anneal response, a s most of t h e buildup t akes p l ace t y p i c a l l y over on ly 3 t o 5 decades i n t ime. The d a t a p o i n t s in Fig. 6 are such time-dependent ANIT buildup d a t a taken from Ref. 26 f o r Al-gate wet-oxide capaci- t o r s f o r a s ries of oxide f i e l d s between 2 and 6 MV/cm t o almost 10 s fol lowing 0.8-Mrad pulsed LINAC i r r a d i - a t i o n . There is c l e a r l y a s t r o n g f i e l d dependence on both t h e rate of bui ldup and on t h e f i n a l , s a t u r a t e d value of NIT. (We no te t h e d a t a i n Fig. 6 are not normalized.) For a l l f i e l d s , t h e buildup begins on t h e order of seconds, con t inu ing f o r s e v e r a l hundred seconds a t t h e h ighes t f i e l d ( 6 MV/cm) before l e v e l i n g o f f . For t h e lower f i e l d va lue ( 2 MV/cm) t h e genera-

f

Tk* * P& (s)

Fig. 6 . Long-term bui ldup of i n t e r f a c e t r a p s fol lowing pulsed electron-beam i r r a d i a t i o n f o r t h r e e va lues o f oxide f i e l d . S o l i d curves a r e fits of Eq. ( 4 ) t o data . Point i nd ica t ed a t lo-' s is p r e - i r r a d i a t i o n value o f Nit . (Data from Ref. 26.)

t i o n r a t e is much lower, but NIT seems t o be l e v e l i n g o f f a t t h e l a t e r measurement t imes ( - lo5 s) . We no te t h a t t h e NIT bui ldup d a t a shown i n Fig. 6 a r e almost s o l e l y due t o t h e slow, time-dependent bui ldup process ( t h e po in t a t 0.1 s is a l s o the p r e - i r r a d i a t i o n NIT va lue ) . A prompt radiat ion-induced component, p re sen t a t t h e e a r l i e s t measurement times, has a l s o been r epor t ed in some cases [29,33,37]. However, t h e prompt c o n t r i b u t i o n is g e n e r a l l y on ly a small f r a c t i o n ( < 10 pe rcen t ) of the f i n a l number of ANIT. I f p re sen t , t h e r e would simply be a r i g i d upward t r a n s l a t i o n of t h e response curves i n Fig. 6 , and i n t h e composite response i l l u s t r a t e d in Fig. 1 ; i n e f f e c t , t h e i n i t i a l s h i f t a s given by Eq. ( 3 ) would have t o be modified t o account f o r t h e prompt ANIT.

The s o l i d curves i n Fig. 6 a r e the f i t s of Eq. ( 4 ) t o t h e da t a . Since a t p re sen t we do not have as q u a n t i t a t i v e a phys ica l model t o guide u9 f o r NIT bui ldup as we d id f o r t h e cases o f ho le t r a n s p o r t and t rapped hole annea l , t h e f i t s here a r e more empi r i ca l i n na tu re . I n each case we take C t o be t h e prg- i r ra- d i a t i o n value, 0.2 x 10" where we estimate the f i n a l , s a t u r a t e d bui ldup va lues from t h e da t a ; and we choose to i n t h e same fa sh ion we d i d f o r ho le t r a n s p o r t , namely, by r e q u i r i n g t h e calcu- l a t e d curves t o equal t h e observed NIT va lues a t t h e half-bui ldup po in t s . Th i s procedure then l eaves j u s t t h e power law exponent v t o be chosen fo r t h e bes t o v e r a l l f i t . The fol lowing t a b l e lists t h e parameter va lues used f o r t h e f i t s shown i n Fig. 6:

Eox (MV/ cm) A N ; ~ ( I O ~ ~ cm-2) to(s) V

we take A = vANIT,

- -- 2 4 6

1.3 30 0.30 3.5 10 0.38 5.2 3 0.47

The f i t s t o t h e da t a a r e c l e a r l y very good, i n d i c a t i n g t h a t t h e ion d r i f t / d i f f u s i o n process r e spons ib l e f o r t h e long-term NIT bui ldup [30-333 is very poss ib ly descr ibed by a CTRW-like, s t o c h a s t i c model s i m i l a r t o t h a t f o r hole t r a n s p o r t .

F i n a l l y , we simply note t h a t t h e r e is appa ren t ly no d i r e c t c o r r e l a t i o n between t h e long-term NIT bui ldup and t h e t rapped hole anneal ing. Hence, t h e combined response func t ion i n t h e long-time regime would be t h e simple a d d i t i o n of t h e two component responses AVOT and "IT'

I V . APPLICATIOU TO L I H U RESPONSE ANALYSIS

A primary motivat ion f o r i n t roduc ing t h e s imple a n a l y t i c form Eq. ( 4 ) t o desc r ibe t h e gene r i c MOS response (Fig. 2 ) is t o enable us t o perform simple l i n e a r response types o f ana lyses using t h e convolut ion i n t e g r a l , Eq. ( l ) , when t h e impulse response func t ion d e v i a t e s from strict log-time behavior over t i m e regimes o f i n t e r e s t . We have descr ibed i n d e t a i l i n t h e previous s e c t i o n t h a t , wi th t h e appropr i a t e choice o f parameters, Eq. ( 4 ) provides a good d e s c r i p t i o n Of t h e response data f o r each of t h e t h r e e p r a c t i c a l l y important components of MOS response, and the re fo re can be used with accep tab le accuracy i n l i n e a r response ana lyses . Furthermore, it is s u f f i c i e n t l y s imple t o be t r a c t a b l e i n c a r r y i n g o u t t h e i n t e g r a t i o n i n Eq. ( 1 ) .

We restrict o u r a t t e n t i o n he re to the s p e c i a l but p r a c t i c a l l y important cape of a square i r r a d i a t i o n func t ion , namely, +( t ) = Y = constant f o r o < t < tE and ?(t) = 0 otherwise. &en t h e convolut ion i n t e g r a l Eq. ( 1 ) with t h e gene r i c response func t ion Eq. ( 4 ) can be eva lua ted i n c losed form. The f i n a l r e s u l t s a r e

1184

f o r T 7 T~ and

A A V T ( T ) = D I C - F v x [ g v ( T ) - ~ ( T - T E ) - T E l ) (17)

f o r T > T ~ . T = t/to, and T~ = tE/to. We no te simply wi thout proof he re t h a t f o r t >> to t h e s e r e s u l t s agree with prev ious work [l-31 i n t h e l i m i t v + 0 f o r t h e s t r i c t l y log-time impulse response.

Here, gv(T) = (1 + T)f,(l + T ) , D = Y o t E , I - > a I

- - (t/trn) lo-' l@ l# lo'

0.0

02 -

0.4 - Y = 025

We now b r i e f l y d i s c u s s t h e a p p l i c a t i o n of t h e s e

lo' ld r e s u l t s t o a few s imple problems. We w i l l concen t r a t e 0.8 on examples f o r shor t - te rm recovery due t o hole t rans- ld Id p o r t and f o r t h e long-term AN bu i ldup , as t h e pre- suw T*u(r = 14)

vious l i n e a r response work [l-33 focused on t h e case of t rapped hole anneal. Also, we w i l l defer d e t a i l e d Fig. 7. Normalized convolu t ion response for shor t - te rm comparisons wi th exper imenta l da t a t o o t h e r publica- recovery due t o h o l e t r a n s p o r t , u s ing approximate t i o n s , and simply present here a few i l l u s t r a t i v e impulse response curve from Fig. 3. Resu l t s for two c a l c u l a t i o n s us ing t h e appropr i a t e response func t ion exposure times are shown ( t E = t,/2 and 100 t lI2). parameters a sce r t a ined i n Sec t ion 111.

Figure 7 shows examples of t h e convolved response f o r t h e case o f shor t - te rm recovery due t o hole t r ans - p o r t f o r which t h e impulse response shown i n Fig. 3 was used. The r e s u l t s f o r two exposure times ( t E = t ,/2 and 100 t l I2, where t is t h e half-recovery time for

ized t o t h e i n i t i a l s h i f t s g iven by Eq. (3) . As i n Fig. 3 a hole t r app ing f r a c t i o n o f 2 percent is assumed. wi th no account t aken o f t h e annea l ing of t h e deeply trapped ho le s a t t h e i n t e r f a c e (an effect which is not r e a l l y s i g n i f i c a n t for such a low t r app ing frac- t i o n at e a r l y times). Again, t h e upper time scale is i n u n i t s o f t lI2, and t h e lower scale i n u n i t s of to.

We emphasize t ha t i n Fig. 7 un i ty is t h e no- annea l -va lue of t h e response (ho le s f rozen i n p lace a t po in t of genera t ion) w i th the d i f f e r e n c e between un i ty and t h e curves a t t r i b u t e d t o ho le annea l ing v i a t r ans - p o r t and c o l l e c t i o n for times up to t. The curves i n

t h e impulse response)'$e shown wi th t h e s h i f t s normal- OJ- &.,,-6WV/m

% 01-

t h i s f i g u r e can be immediately- t r a n s c r i b e d to vo l t age and time scales f o r s p e c i f i e d va lues of t h e p e r t i n e n t parameters. The Voltage scale is given by Eq. ( 3 ) once t h e dose, ox ide th i ckness , and type and energy o f i nc iden t r a d i a t i o n are s p e c i f i e d . Likewise,

Fig. 8. Convolution response f o r long-term i n t e r f a c e t r a p buildup for exposure time of 30 s us ing impulse response f o r ox ide f i e l d of 6 MV/m from Fig. 6. is normalized t o f i n a l , s a t u r a t e d value.

ANIT

the time scale is known from Eqs. (8) and ( 9 ) f o r given oxide t r a n s p o r t parameters (a and f T ) once t h e oxide th i ckness , f i e l d , and tempera ture are spec i f i ed . We also reiterate t h a t t h e nonlogar i thmic f e a t u r e s of t h e hole transport recovery (Fig. 3) is explicitly p r a c t i c e for which non l inea r e f f e c t s are important,

and, f o r t h e sake of awareness, we po in t ou t a few. F i r s t , f o r ho le t r a n s p o r t one must be concerned t h a t accounted f o r i n these ca l cu la t ions .

As a f i n a l example, t h e convolved response f o r i n t e r f a c e t r a p bui ldup ve r sus time is shown i n Fig. 8 f o r an exposure time of 30 s us ing t h e impulse response curve f o r 6 MV/cm from Fig. 6. Here, t h e i n t e r f a c e t r a p dens i ty is normalized to its late-time s a t u r a t e d va lue , bu t t h e time scale is shown i n real time. The dashed curve is t h e impulse response , so t h a t t h e f ig - ure simply i l l u s t r a t e s the time l a g between t h e a c t u a l NIT bu i ldup fo r a f i n i t e exposure and t h a t which would occur f o r an impulse i r r a d i a t i o n (which i n t h i s case is any pulsewidth less than about 0.1 9) . Note t h a t t h e l a g is g r e a t e s t between -1 and 50 s and t h a t t h e r e is very l i t t l e d i f f e r e n c e i n t h e cu rves f o r t 5 100 s.

As a f i n a l no te we cau t ion t h a t t h e use of a l i n e a r response type of a n a l y s i s impl ies j u s t t h a t : namely, t h e response is assumed t o be l i n e a r -- with dose i n ou r case. Therefore , its a p p l i c a t i o n is obvious ly l i m i t e d to cases where nonl inear e f f e c t s can be neglec ted t o whatever degree of accuracy t h a t is des i r ed . There are c e r t a i n l y s i t u a t i o n s which ar ise i n

t h e space charge f i e l d s due to t h e t r a n s p o r t i n g ho le d i s t r i b u t i o n s do not s i g n i f i c a n t l y pe r tu rb t h e app l i ed oxide f i e l d and cause a d i s t o r t i o n i n t h e impulse response curve. A t s u f f i c i e n t l y h igh dose, f i e l d col- l a p s e o r even f i e l d r e v e r s a l may occur. Second, for t rapped hole annea l one must avoid dose regimes where s a t u r a t i o n of t h e deep ho le t r app ing occurs. Such sat- u r a t i o n may r e s u l t from a c t u a l t r a p f i l l i n g , from space charge effects , or from recombination wi th r ad ia t ion - induced e l e c t r o n s moving through t h e t rapped hole d is - t r i b u t i o n C381. Thi rd , i n t h e case o f r ad ia t ion - induced i n t e r f a c e t r a p s . t h e r e have been s e v e r a l obser- va t ions [26,28,39,401 of a s u b l i n e a r bu i ldup wi th dose even at moderate dose l e v e l s (> 100 k rads ) . Although t h i s remains an open ques t ion , t h e r e is present ev i - dence t h a t t h e s u b l i n e a r dose dependence may be l i nked t o e x i s t i n g (p re rad ) de fec t r eg ions i n t h e oxide , and t h a t f o r a "defect-freeTt dev ice t h e i n t e r f a c e t r a p buildup is l i n e a r wi th dose up t o about 3 Mrads(Si02) C411.

v. SUHHARY

In summary, we presented a gene ra l i zed form (Eq. ( 4 ) ) f o r t h e impulse response func t ion which can be used t o desc r ibe s e v e r a l of t he p r a c t i c a l l y important t r a n s i e n t r a d i a t i o n response f e a t u r e s o f MOS systems, i nc lud ing short- term recovery due t o hole t r a n s p o r t , long-term recovery due t o annea l ing of deeply t rapped h o l e s , and t h e long-term bui ldup of i n t e r f a c e t r a p s . The gene r i c response func t ion inco rpora t e s t h e major f e a t u r e s of t h e experimental ly observed responses , i nc lud ing s p e c i f i c a l l y t h e dev ia t ions from log-time behavior ; and with t h e appropr i a t e choice of parameters, t h e gene r i c response func t ion provides a quant i - t a t i v e l y accu ra t e d e s c r i p t i o n of t h e response in each case. Yet i t is s u f f i c i e n t l y s imple t h a t t he convolu- t i o n i n t e g r a l o f l i n e a r response theo ry can be evalu- a t e d a n a l y t i c a l l y -- a t l e a s t f o r t h e p r a c t i c a l l y important case of a square i r r a d i a t i o n p u l s e -- and we presented and discussed a few simple examples. The r e s u l t s presented i n t h i s paper are, use fu l whenever a l i n e a r response a n a l y s i s is necessary over time regimes where t h e impulse response func t ion d e v i a t e s s i g n i f i - c a n t l y from log-time behavior. The use of a complete impulse response func t ion with t h e d e t a i l contained i n Eq. ( 4 ) is e s p e c i a l l y important i n doing l a r g e extrapo- l a t i o n s of test d a t a (usua l ly taken over l i m i t e d time regimes) t o times o f o p e r a t i o n a l i n t e r e s t .

ACKNOWLEDGEMENTS

The author wishes t o thank h i s co l l eagues Ed Boesch and T i m Oldham f o r many h e l p f u l d i scuss ions , JOaMe Hartman f o r a s s i s t a n c e with some of t h e computer c a l c u l a t i o n s , and Anita Soencksen and Sandy Herrmann f o r typing the manuscript;

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generic impulse response function for mos systems and its application to linear response analysis

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