VN U . JO U R N A L OF SCIENCE, M athem atics - Physics. T . x x , NqI - 2004
O N S E V E R A L C O R R E L A T I O N I N T E G R A L S OF T H E D E E P L E V E L T R A N S I E N T S *
H o a n g N a m N h a t, P h a m Q uoc T rieu Department o f Physics, College o f Sciences, V N U
A b s tra c t.T h is works presents the theoretical study on several correlation integrals of the capacitance transients of the deep levels. The deconvolution of the transient signals was the major subject of the number of methods referred to under the common name as the Deep Level Transient Spectroscopy methods. In general the separation of the overlapping exponential decays C(t) does not provide a unique solution, so the detection of the closely spaced deep levels by these transients should base mainly on the temperature dependence C(T), not only on the C(t). The results show that the average emission factor en is obtainable directly from various correlation integrals of the capacitance transients and the average activation energy E of the deep levels is detectable via the shift operators of the transients according to the temperature. A new scanning technique is suggested.
K e y w o rd s : Deep level. Transient, DLTS. Correlation N o m e n c la tu re
We call a normalized capacitance C n (t) at fixed tem perature T the function Cn{t) — C q 1 X [C(t) - C l ] , where Co is C(t) at t - 0 and Cl is C(t) a t t = 0 0. For 0 < t < 0 0, c„{t) always specifies the relation 0 < C 7l(t) < 1 , i.e. L(t) = Ln[C n (t)\ has definite and negative value within (0.1). Taking L n oil L{t) is not possible b u t M( t ) — Ln[—Ln[Cn (t)]]
has definite values. By replacing t by T we have a normalized capacitance Cn (T) at fixed gate this Cn (T) is not exponential. For section 3, w = E / k where E is the activation energy of the deep level and k is th e Boltzman co n stan t .
1. In trod u ction
The problem of separation of the closely spaced levels in Deep Level Transient Spectroscopy has last for decades but no clear answer is available until now. Several people suspected th at the problem is ill-posed in principle [4]. By its n a tu re the problem finalizes in the search for a unique decomposition of any exponential decay into a finite combination of several other ones. Theoretically all exponential functions is expandable into an infinite series of the other exponential components, so we may expect the existence of many finite approxim ations which satisfy the precision lim it. T h u s th e critica l question follows: if there exist man y decompositions so which o f them, preserves the physical meaning or is that the true physical reality what the D L TS discovers?
*1991 M ath em a tics S u b je c t C lassification P A C S : 85.30 De, 71.55 C n , 72.15 Jf, 72.20 J v
T y p e se t by AjK (<S-TgX
14
The DLTS in its natu re is not a purely measurement technique but involves a lot of analysis, so the obtained results can hardly be attached directly to the physical reality without some theoretical presumptions. One of the basic. presumptions was about the duality of the DLTS peaks and the emission factors of the deep centers. This was so evident th at needed no explicit statem ent, however we must re-state here th at the word
"emission center" must be understood statistically, i.e. the DLTS peak reflects only the statistical behaviour of the underlying physical reality and not the physical reality (of the emission centers) themselves. Thus giving a set of emission centers, the appearance of DLTS peaks follows the statistical rules for composition of the output signals which must not be a simple one-to-one relation in principle.
There are known various interpretations of the level resolution [2]. Providing that the deep levels are s ta tis tic a l term s, th e level closeness m ust also be u nderstood sta tis
tically. i.e. closeness as th e sm allest observable variation in th e sta tistic a l behaviours of the levels. We may call this closeness the local closeness. The physical meaning OÍ this definition is that in some tem p eratu re range the statistical contributions of the levels to the output signals are not comparable, whereas in some other range they are quite unclis- tinguishable. Some levels m ay be seen as th e closely spaced w ithin one T range b u t not out of it. Consequently, if only one level contributes to the output signal while all others remain inactive then this level is far from the rest and is separable by the sense of the underlying DLTS technique. As the evident consequence, the local closeness will require more smoother tem p eratu re scan th an for the normal techniques (e.g. for D .v. Lang’s classical DLTS [6] or for s. Weiss and R. Kassing’s Fourier DLTS [7]).
Now let us tu rn to the problem of the statistical rules for composition of the final output signals. Until now no one hesitates to write:
n
c „ ( 0 = (1.1)
2 — 1
which automatically assumes the composition additiveness. Although this relation has provided satisfied results when the levels are well separated, there is no clear reason for the additiveness in case of the closely spaced levels. As known, for the perfect exponential decays, the level contributions are nọt statistically independent (the known Levy theorem states th a t for any two Gaussian distributions, and thus the exponential ones, the final composition is always Gaussian, see [5]). IÍ one assumes the final emission factor to be efinal = Ì t 'PÁe )em w here Pi(e) is th e s ta tis tic a l weight (em ission facto r’s density
2 = 1
distribution) then final o u tp u t Cn (i) evidently becomes:
n
c n (t) = e~ íe/”‘a' = Ị Ị e - ‘Pi(e)e"i (1.2) 1=1
It is questionable w hether (1.1) or (1.2) takes place. W ith regards to the above stated problems, we present in this article the study on several correlation integrals of the output signals C( t ) and C ( T ) and suggest, a scan technique which bases mainly on scanning the C( T) according to the tem perature. As shown below, the emission factor
16 Hoang N a m N h a t, P h a m Quoc Trieu and activation energy may be retrieved directly from the statistical study of the output.
Evidently these values correspond to the locally closed states if these states exist.
2. C o r re la tio n in tegrals
By definition all correlation functions R{ r ) of a signal F(t) involve the shift operator of th at signal T r [F(£)] = F ( t — r) and represent the average of signal over the preset period T. N aturally, th e co rrelatio n functions filter noise, b u t also rem ove th e fine local stru ctu re of the signals (see e.g. [3]).
2.1. C o r re la tio n in te g ra ls at fixed T.
Taking correlations a t fixed T defacto means averaging signal according to time t. At fixed tem p eratu re the development of signal after a time t wholly depends on the emission constant en and has a simple exponential form e ~Cllt. Let A be a period width and integrate through the whole period.
2.1.1. Cross-correlation otion of Cn (t) and c~^(t)’
A / 2
- A / 2
(for all T)
2.1.2. Autocorrelation of L(t).
A / 2
R ( r ) = J
J
( - e nt ) ( - e n t + en r)d t =— A / 2
2.1.3. Autocorrelation of — .
A / 2
R { t } = \ J ( t )( i-T )dt = e”
- A / 2
O ne m ay also check t h a t a cross-correlation of L ( t ) ~ i a n d — X is always 1.
A / 2
w - i
- A / 2
2.2. C o r re la tio n in te g ra ls a t fixed t.
This class of correlation functions illustrates the averaging process according to tem perature T. i.e. the process of filtering the tem perature noise. Write Cn ( T) = e~tpT e V L ~ l/ l (T) — p T 2e~w/ T and p u t M ( T) = Ln[ —L ( T) ~ ỉ ^t ] = Ln( p) + 2L n ( T ) — lu/ T . The correlation is considered within one segment of T so we integrate from T\ to TV Let A T = T2 - Tỵ.
The cross-correlation between M ( T) and 1 / M ( T ) is:
T2
R ( r ) = PT2e - * p { T - r ) 2e - t t d T = T,
= J _ /’[_Z_]2e-(T - T ^ d T
A T J T — T
T i
S u b stitu te X = ị , X i = - f r , x 2 = T~ a n d A x = x 2 - X ị . In u n it of X an d replace Ơ for r as a new shift:
C(x) = e~ tpx~2e~'1’*, L ( x ) = - t p x ~ 2e~UJX, L ( x ) ~ i = p x ~ 2e ~ UJX a n d th e correlation integral is:
X 2 _ x 2
1 f p x ~ 2e~“x ^ e~wơ /*■ Ơ 2
R( ơ) = —— --- --- ^---7——zd x = —— / (1 - - ) dx
Ax
J
(p(x — ơ) 2e w(x A xJ
XX \ x l
After integration:
R( a ) = e - ^ ị ĩ + 2 ơ L n ^ - — + ơ 2( - ^ — )]
v ' 1 A x X1X2
By p u ttin g A = B = ^ we o b tain : 1 r 1 + 2<J A -b B , a; = - L n ---— T---
Therefore the correlation integral directly determines the activation energy of the deep level.
3. Shift o p e r a to r Cn {T X p) = Tp[Cn(T)]
According to te m p e ra tu re T th e shift o p e ra to r Tp moves C n (T)<f)n to Cn ( T X p) for a real positive multiplicative constant p : C n (T X p) = T p [C n (T)]. In the following sections Tp will be derived (many aspects of the shift operators are explored in [1]).
3.1. Shift o p e r a to r o f CnỰT), L n ( T ) and M n ( T ) .
D i v i d i n g C n ( T x p ) = e - t p ( T x p )2e - T ^ b y C n ự j = e [ - t p T * e ~ * I l e a d s to:
Cn( T x p ) = T p [Cn (T)} = lCn ( T ) / e ~ f{lp~1}
Similarly:
UJ (1 T Vp(Ỉ-D- L ( T x p ) = T p [L (r)] = [L(T)]p2e
M ( T x p ) = Tp[M{T) } = M (T ) + 2Ln(p) _ ^ ( 1 - 1)
(3.1.1)
18 Hoang N a m N h a t, P h a m Quoc Trieu 3.2. C a lc u la tio n o f e n erg y using shift o p era to rs.
With UJ = E / k and by relation (3.1.1):
(3.2.1) So an arbitrary shift from L( T) to L ( T x p ) determines a energy constant significant within this tem perature shift [T —> Txp] . A technique for detection and analysis of closely spaced levels by scanning CJ in various tem perature range is suggested. We call it the
’Selective Tem perature Scan Technique’ (STST).
4. C o n clu sio n
Not only u but also p is detectable via shift operators. For calculation purposes we have simplified (3.2.1) to the following relations. Suppose a shift from (Ti,ti) to (Tk,tk) and use short notes Li = L( Ti ) , Lk = L(Tịc).
In practical case we put ti = tịc and chose T/c as close as possible to Tx , i.e. Tfc displaces from Tj only by one scan step. The displacement Tk — Tt refers to the sensibility limit of the tem perature setting and the scanning range [T1...T2] refers to the statistical weight. For general case where t i f a k , th e tk —ti m eans th e tim e v a ria tio n of th e statistical weight. The functionality of method is adjustable by these three parameters. Note that th e above relatio n s hold only for cases where th e s ta tis tic a l c o n trib u tio n of one deep center (in the scanning tem perature range) is significantly greater th an of all the rest.
A c k n o w l e d g e m e n t s . The authors would like to thank the Center for Materials Science, HUS - VNU HN for supports regarding the work on the DLTS and other equipments R eferen ces
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