VNU JOURNAL OF SCIENCE Mathematics - Physics T XVIII N0 - 2002 A N E W M E T H O D F O R S E P A R A T IO N O F R A N D O M N O IS E F R O M C A P A C I T A N C E S IG N A L IN D L T S M E A S U R E M E N T Hoang Nam Nhat, Pham Quoc Trieu D epartm ent o f P h y sics y College o f S cien ce - V N U A bstract We introduce a new statistical method f o r separation o f random noise fro m capacitance signal in D L T S m easurement F o r the in te rfe re n ce o f a white random no ise £ with capacitance signals (7(0 of general expo nen tial fo rm % we show that, noise £ and e m issio n fa c t o r are statistically different and can be well separated each fro m other Theoretical fo rm a lis m f o r re co n stru ctio n o j noise-free capacitance signals based on d ete rm in a tio n o f e m is s io n fa c to r is presented The method has been tested f o r v a n o u s sign a l-to -n oise ratio s fro m 1000 down to 10 S im u la tio n and examples are given Abbreviations T tem perature t tim e C n (t) normalized capacitance at certain fixed T L ( t ) L n ( C n ) e.g natural logarithm of normalized capacitance at fixed T p(£) density probability of random variable £ P (0 cumulative probability of random variable £ , emission factor of a deep center I i?, activation energy of a deep center i LJi ratio E i / k between E i and Boltzmann constant k for a deep center i Definition of terms We will work with a so-called n o rm a lized capacitance C n at certain temperature T defined as C n { t ) = Cq X [C(/) — Cl), where C{) is C ( t ) at t = and C \ is C ( t ) at t = oo For < t < 0 , C n (t) always specifies relation < c n (t) < , this means that L n ( Cri) has definite and negative value within this range Taking Ln on Ln { Cn) is not possible bu t L n \ - L n ( C Ti)} has definite values The average value of a variable X defined on the probability distribution p(£) of a random variable £ will be denoted by < X Practically we will consider :.he average values of X = Exp { - E / k T ) and L n ( X ) according to probability distribution of emission factor p (e ) Generally, the small p — s denotes density function where the capital p - s means cum ulative probability A random noise with uniform probability distribution in whole range of frequen cies is called white ran d om noise White random noise in restricted area of frequencies will be called white gaussian n oise if possessing Gaussian distribution 32 A n e w m e th o d fo r se p a tio n o f r a n d o m n o ise fr o m 33 I Introduction Tilt oenu rrnec of noise always disturbs the signals and lowers the* quality of mea surement or rvrii it iiujx>ssiK>lo In a fine-tuned measurement system like DLTS the or.nuTPWP of noisr is rxtmnolv critical for many important cases Doolittle Kf, Rohatgi lirivi* trslrd thí' iuiK t ionnlit V of various techniques when noise iii1 ori’(T n and - < E > r? / k T must be identical To determine the density probability function p(r/) we perform the calculation for all measured t (with respect to that e = p T E x p ( - E / k T ) where p is a constant): { v = L n { - r l T - * L n C n ( t ) } } t = { L n ( p ) - E / k T } t As seen, p(rj) does not reveal < E > v directly but < L n ( p ) - E / k T > v holds fixed we may suppose that: < L n ( p ) - E / k T > f;= L n (p )- < E / k T As consequence p (£ ) = p(rỗ) ( b l ) in ease L n ( p ) L n (p )- < £ > „ /fcT (b.2) However, statistics (b.l) always produces < L n ( p ) - E / k T > ff not < E > n in general c ) R e l a t i o n be tween p (f ) a n d p { E ) Suppose that (b.2) holds e.g p ( E ) = p(f?) Ill term of < E >,), the average < L n X >TJ reads: < L n X > „ = - < E > n / k T = ~ ^ Vl{ E ) E J k T i (c.l) Emission factor becomes < e > , = p T 2Exp(< L n X Whi le in term of < X >€,< e >= — p T Y ì i P Ì { e) X i — p T < X > e Comparing these two relations leads to: L n < X > E = < L n X >r> (c.2) We use this relation to check how much p(e) and p ( E ) differ each from other If they differ too much then the relation (b.2 ) may not hold for the case under investigation The physical meaning of (b.2 ) is that the noise effecting activation energy does not influence level concentration and capture cross-section, th at is to say, E and L n ( p ) are statistically independent d ) S t a t i s t i c s o f e m i s s i o n f a c t o r p(e) i n o c c u r r e n c e o f w h it e r a n d o m n o is e With existence of a random white noise, capacitance signal has the form: C n = Noise-1-Exp[— < e > t\ (d.1 ) A new m e th o d f o r s e p a m tio n o f r a n d o m 35 n o ise fr o m Rr-write C n to: Cn ICxpf— < ( > /]( + Noise/Exp[- < € > t\) and put: Noise1 K.Exp[- < > /ỊExpị—£/], (ii.2) when' K is constant aiK £ is i\ random variable Wo havo f'n a's: c u - Exp[— < f > t](l + /cExp(-£fị) D r n o tr o , = Exp- — < f > c\ r„ = C Q ( -r/\E x p - £ / ; ) and — (C{ - )/k : o rC + n = c < (1 + k Q „ ) ((1.3) This moans that the cnpacit.rUKT transicMit in occurrriier of noisr follows relation (a.2 ) for closo-spacrd deep centfTs r.u, random noise behaves as if it is a drop miter This would not he true if £ does not have; (Irnsity prob ability similar to On Fortunately, for ar bitrary positive noise lcvrl Noise] (‘quation (d.2) always has solution £ Lĩ ỉ ( Noise/*;)' 1/f — < ( > If [Noise is a random noiso with uniform density, than £ has density proba Í [a.u] bility of L//(Noìs('/k) ]/t -~ < < > which is practically the same as C n (Sor Fig.l) Fig Density probability p(£) of £=Ln Clearly, for all measured / th(‘ statistics p { f ) : (Noise/K)'1/l- { Ỉ M C r > r l / t }t = { - < f > + f£}r, (d.4) where = { L n ( \ -f tfKxpi—£/j) f} will re veal average value of { — < ( > -fir} which differs generally from (a I) Fig.2 shows p(e) for different T As seen, while at the middle T the real f peak is high and proportional to the noise peak , at the high T the real e peak is much smaller than tho noise peak The side-effect of is that it widdens the width of a delta-like (a I) peak with the amount proportional to < > One mav expect that if < e > and are absolutely ad ditive than the distribution spectrum of (cl.4) Fig,2 Spectrum |Ln(Cn)'ỉ/f)ị at will contain only one smooth Gaussian peak various temperature T Noise=2% However Fig.3 shows two different areas, one of Cn unit corresponds to < e > and the other to €£ This separation is true with two exceptions, the first occurs at low T when C n ispractically equal 1and the second occurs at high T when Cn is near In both cases, noise so dominating that spectrum { L n ( C n ) ~ i ^t}t contains only values of becom es Hoang N a m N h a t , P h a m Quoc Tineu 36 e) T e m p e r a t u r e d e p e n d e n c e o f s i g n a l n o i s e s e p a r a t i o n i n { L n { C n) ~ ì ỉ t }t s p e c trum There exists a threshold temperature Tcrit where < e > is small enough and can not be distinguished from €£ Let ị and ị be variance of < > and < >, the criteria for threshold temperature T crtt is that at Tcrit the displacement < € > — < € $ > becomes proportional to (ơf — ị ) / This relation is used to filter-off the noise where no signal structure is seen: Fig The exitstence of two different area for and e* at noise level 5%, 10% and % of Cn unit (< e > - < * > ) Trrtt fe.l) III Simulation and measurement a ) P r o c e d u r e f o r the r e c o n s t r u c t i o n o f n o i s e - f r e e c a p a c i t a n c e s i g n a l Data in the capacitance transient measurement are usually c o l l e c t e d at preset tem perature T when the emission factor e can be considered as constant To obtain the statistical characteristics of e we should measure C n { t ) as dense as possible However the number of several hundreds data is adequate and 0 recorded data provide quite satisfied results on simulation At the first step a logic circuit should be available' to transform C n (t) into Ln[Crn(/-)'~1^] and then into L n \ —t ~ l T ~ 2L n C n {t)}- This is easy w ith com puter The sta tistics p(c) is ob tained after recording all L n [ C n ( t ) ~ l / i ] and sim ilarly p(rj) by all L n [ —t ~ l T ~ 2L n C n (t)} Two statistics are then checked against each other using relation (c.2) to see if p { E ) can be set equal to p ( 7 ) If p ( E ) = /;(?/) holds we hâve a simple case of one noise-free center, other wise overlapped centers occur and noise should be filtered A numeric calculation of deriva tion [dp(e)/de] should provide peak value fmax of p(e) As noted before, we have two differ ent cases: i) at the extremely low and high end T there is only one e.£ peak This noise-driven exponentially-distributed peak should be removed Fig (a) Un-filtered signal Cn(t); (b) Cn(t)c reconstructed by Enwu; (c) Cn(t)i obtained using lock-in; (d) Cn(t)ỗ reconstructed bv Noise = 2% of c„ unit A n e w m e th o d fo r se p a tio n o f r a n d o m n o ise fr o m 37 since it does not rorn spuinl to signals and contains no information about omission factor; ii) i\i the middlo nuụ>, ) / inshows the false ones Noise=2% of Cn Stead of real value4 f max Once max is €