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fγ noise
Microelectronics Reliability 40 (2000) 1775±1780
Invited paper
www.elsevier.com/locate/microrel
A general model of 1=f c noise Bruno Pellegrini Dipartimento di Ingegneria dell'Informazione: Elettronica, Informatica e Telecomunicazioni Universit a degli Studi di Pisa, CSMDR of CNR, Via Diotisalvi 2, I±56126 Pisa, Italy Received 29 November 1999
Abstract A model of 1=f c noise is proposed that accounts for all its experimental properties. Its origin is found to be in the charge ¯uctuations in few defects, even less than 100, with relaxation times s arbitrarily distributed in a wide interval, up to large values. The coupling coecient between local and output ¯uctuations, the dependence of the defect occupation factor on s, the frequency exponent c and the coecient a of the spectrum are computed in a general and simple way. Published by Elsevier Science Ltd.
1. Premise As is well known, since its experimental discovery in vacuum tubes by Johnson in 1925 [1], the so-called ¯icker or 1=f c noise has been the subject of many experiments, models and discussions, and, unlike what happens for all the other noise types, i.e., thermal, shot, generation±recombination (g±r), and burst noises, a universally accepted theory about it does not seem to exist, so that it is meaningful to ask, ``Is 1=f c noise an unsolved problem''? We try to give a negative answer to this question by presenting a general model of 1=f c noise that is based on the charge ¯uctuations of defects that have arbitrarily distributed relaxation times s in a wide interval, up to very large values, and are randomly scattered, even in very low concentrations, in the electron device. The model can be generalised to a system of any nature. According to the experimental ®ndings, (i) 1=f c noise exists in any device traversed by a steady average current I hi ti, (ii) the relevant power spectral density S of i t=I is given by Sa
fad ; Nf c
1
where f x=2p fa is the frequency (the central frequency of the interval in which we perform the measurement) and N is the total number of free electrons in the device, (iii) the dimensionless coecient a is spread in the very wide range from 10ÿ8 to 1 [2], (iv) the fre0026-2714/00/$ - see front matter Published by Elsevier Science Ltd. PII: S 0 0 2 6 - 2 7 1 4 ( 0 0 ) 0 0 0 6 1 - 5
quency exponent c 1 d ranges between 0.8 and 1.2 over many frequency decades and (v) down to any low frequency at which the spectrum is measurable. The theory that we propose, by developing previous results [3,4], accounts for all the ®ve, (i)±(v), experimental properties of 1=f c noise. 2. Fluctuations in the defects and at the output of the device As is well known, a random telegraph signal /(t) that stays at level 1 (0) an average time s sÿ and, as a consequence, has a mean value u s = s sÿ and a correlation time s s sÿ = s sÿ , is characterised by the Lorentzian spectrum [5], s S/ 4u 1 ÿ u : 2 1 s2 x2 On the other hand, a defect at rt with a single energy level E in a unipolar device characterised by an average density n r of free electrons has an electron number /(t) with times s 1=e and sÿ 1=cn, e and c being the emission and capture probability of an electron, respectively, a relaxation time s
u 1ÿu 1ÿu ; ÿ1 cn e so exp ÿ EN ÿ E=vkT
3
and an average value u 1=f1 exp E ÿ EF =kT g given by the Fermi±Dirac statistics, EF , k and T being
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B. Pellegrini / Microelectronics Reliability 40 (2000) 1775±1780
the quasi-Fermi level, the Boltzmann constant and the temperature, respectively. The fourth term of Eq. (3) takes into account that for both a thermal emission (THE) of the electron from the defect into the conduction band, with a bottom energy EC , and a tunnel emission (TUE) across a parabolic energy barrier with a peak value VM , which cover most of the practical cases, we have e sÿ1 o expÿ EN ÿ E=vkT , where v 1 and EN EC for THE, 1=5 < v < 1=2 and EN VM for TUE, and so 1=2pfo 10ÿ14 s, fo being the electron oscillation frequency in the defect [3]. We now have to determine the eects of the local charge ¯uctuations in the defect on the output current i(t) ¯owing through the device electrode that we are considering. To this end, we exploit the electrokinematics theorem that, in the case of electrode voltages kept at a constant value, allows us to write i in the form [4] Z i ÿq
n0 r; tv0 r; t F r d3 r;
4
where v0 r; t is the mean velocity of the n0 r; td3 r electrons in the volume element d3 r at t, while F ÿrW r is an irrotational vector de®ned by r eF 0 over the volume X of the device, W 1 on the electrode surface and W 0 on the remaining part of device surface, e r being the electric permittivity. The integral of Eq. (4) in the space r (and rt in the following) is computed over X. Apart from the ¯uctuation of i(t) due to those of v0 r; t and n0 r; t around their time average values v and n, respectively, and produced by the electron motion and scattering, the variation Dn ÿD/d r ÿ rt of the density of the free electrons generated by the capture and emission of one electron by the defect at rt determines a ¯uctuation of i(t), which, according to Eq. (4), becomes Di qF rt v rt :
5
If we replace v0 with v and n0 with n in Eq. (4), we obtain I and Di=I CD/; where the coupling coecient C is given by Z
C rt ÿv rt F rt n rv r F r d3 r Z ÿ Fx rt n Fx r d3 r
6
in which the third term holds true in the particular case of a cylindrical and homogeneous device with a velocity that is parallel to the cylinder axis x, and, as well as n, is independent of r; we have C ÿ1=N , if, as possible [4], we choose a vector F independent of r and parallel to the cylinder axis.
3. Frequency exponent of the noise spectrum From Eqs. (2) and (6), in the case of uncorrelated defects, we obtain the spectrum S of the current ¯uctuation Di=I due to all the defects, of density nt rt , in the form Z Z Z s S4 u 1 ÿ u C 2 rt nt rt 1 s2 x2 D E; s; rt d3 rt dE ds;
7
where D E; s; rt is the joint probability density function of E and s at rt . Since s E; p depends on E and on other parameters p1 ; . . . ; pn p independent of E (e.g., in the case dealt with in the preceding section, p1 EN , p2 so , p3 v) we can express E E s; p, u E and D E; s; rt through the independent variables s, p and rt , and we can write Eq. (7) in the form Z s S4 Gse s ds; 8 1 s2 x2 where Gse s is a probability-density-like function for s de®ned by Z Z Gse u 1 ÿ uC 2 rt nt rt D0 s; p; rt d3 rt dn p; 9 where D0 s; p; rt is the joint probability density function of s and p at rt . From Eq. (8), we directly obtain S / 1=f c in the case of Gse / 1=s2ÿc and 0 < c < 2. From a physical point of view, we get this result, for c 1, in the very particular case, dealt with by Mc Worter [6], of electron TUE from defects across rectangular energy barriers of constant height and uniformly distributed width. Moreover, in general, in computing S by means of Eqs. (7) or (8) and (9), we encounter two serious diculties: the ®rst one is physical and derives from the fact that D or D0 , which depend on many microscopic parameters, are usually unknown. The second diculty is mathematical and consists of the fact that the integrals of Eqs. (7)±(9) rarely lead to analytical results in closed form. However, several important properties of S can be deduced from Eq. (8) according to the following procedure. Let us perform the Taylor series expansion ln S f fr ln S fa fr ÿ c fa ln f =fr ÿ ln fa =fr about ln fa =fr of the function ln S f fr of the variable ln f =fr where fr 1=2psr is an arbitrary reference frequency and c f ÿ
o ln S f fr 1 d; o ln f =fr
10
so that in a proper frequency interval around fa , we have
B. Pellegrini / Microelectronics Reliability 40 (2000) 1775±1780
S f S fa
fa f
c fa
:
11
In order to compute the frequency exponent c, let us perform the variable changes x sÿ1 and r exphx s sr exph in Eq. (8), which becomes Z Ghe h S 2sr exp ÿhx dh; 12 cosh h hx where now, Ghe h Gse s hsr exp h is a probabilitydensity-like function for h; moreover, in general, in the following, we shall set hh ln sh =sr ÿ ln 2psr fh . From Eqs. (10) and (12), we get Z d tanh h hx H h dh tanh hm hx o ln Ghe h=ohjhhm ;
13
where the ``weight'' function H h of tanh h hx at h is given by Z Ghe h Ghe h H h dh; 14 cosh h hx cosh h hx and hm , de®ned by the third equality of Eq. (13), is the value of h at which H h reaches a maximum [3]. It is worth noting that d has an additive property, in the sense that if we decompose nRt D0 into more parts, in P any way, i.e., nt D0 nt D0 i nt gD0 g dg in Eqs. (8) and (9), from Eqs. (12)±(14) we get Z X d di Si =S d gSg g=S dg; 15 where di and Si are the values of d and S, respectively, relevant to nt D0 i and g is a parameter. Therefore, from Eq. (13), d is the ``mean'' value of tanh h hx and, since j tanh h hx j 6 1 and 0 6 H h 6 1, we have jdj 6 1 and 0 6 c 6 2, as it has to be in a superposition of Lorentzian spectra. If s is spread in an interval between sl sr exp hl 1=2pfH and sn sr exp hn 1=2pfL , since j tanh h hx j 1 for jh hx j P 1:5, from Eqs. (13) and (14) we have d ÿ1 and c 0 for f < fL =4:5 and d 1 and c 2 for f > 4:5fH . In the frequency band fL , fH , i.e., for hl < hx < hn , on the contrary, if Gse h is a slowly varying function of h with respect to 1= cosh h hx , the ``weight'' function H(h), due to 1= cosh h hx , tends to have a symmetrical maximum at hm hx , around which tanh h hx is an odd function, so that, according to the second and third term of Eq. (13), jdj ! 0. (Such properties of H(h) and of tanh h hx and Eq. (14), in particular, allow us to get d in the form given by the third and fourth term of Eq. (13).) Therefore, in practice for any s dispersion, except for particular cases with sharp peaks, we have c ! 1 in the band fL , fH and the existence of defects with relaxation
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times s spread at least up to sn is necessary to have ¯icker noise down to a minimum frequency fL : for instance, up to sn 160 s for fL 10ÿ3 Hz and up to sn 3:2 105 s 3:7 days for the extreme experimental case of fL 5 10ÿ7 Hz obtained in MOS transistors [7] (if the method used to combine ¯uctuation records obtained from several equal devices is reliable). In thick ®lm resistors, a THE activation energy of EC ÿ E=kT 38 was found at T 300 K [3], so that, being sn so exp EC ÿ E=kT we obtained sn 320 s. For TUE, being sn so exp VM ÿ E=vkT with vkT 75 150 K for parabolic energy barriers and for rectangular ones, we sn so exp x=k with k 0:5 A have sn 3:2 105 s for a barrier height VM ÿ E 0:29 eV in the ®rst case (for vkT 75 K) and for x 2:25 nm in the second case in which, for instance, x is the distance of an oxide defect from the interface of a MOS transistor. All these values are experimentally likely. Relaxation times of 1000 s have been measured in p±n junctions [8].
4. Gaussian dispersion of the parameters, analytical and numerical results Up to this moment, we have considered the general case of a generic and unknown joint probability density function D0 s; p; rt of s and p at rt and of an unspeci®ed relationship between u(E) and s(E) that we should obtain by eliminating their dependence on E. Now, in order to obtain a few analytical and numerical results that indicate the principal trends of the model, let us consider such a relationship and the case, that, however, is still suciently general, of D0 s; p; rt Dp pDs s, i.e., in which s and p are uncorrelated and their probability density functions are independent of rt . Moreover, from Eq. (3), we have h h0 ln so =sr EN ÿ E=vkT for E > EF and h h0 E ÿ EF =kT for E < EF , so that the probability density function Dh h of h is the result of those of the defect energy E and of the quantities EN and ln so =sr , which are random variables depending on several physical properties and quantities. Therefore, also as a consequence of the central limit theorem, a good approximation for Dh Ds s hsr exp h is given by the normal law, Dh D0 exp ÿ h ÿ hd 2 =2r2 ;
16
where hd hhi and r2 are the mean value and the variance of h, respectively, and D0 is the normalisation factor, which, for hn ÿhl ! 1, becomes D0 1= p 2pr. Actually, we can always decompose Dh into a ®nite or in®nite number of components Dh / exp jajh ÿ jbjh2 , i.e., of the Gaussian type (16), and then we can add the
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relevant results for d and S according to Eq. (15) and the following Eq. (18), respectively. On the other hand, about the relationship between u E and s(E), we can set u 1 ÿ u s=ss l s sr =ss ls h expfls hhg;
17
where ss so exp EN ÿ EF =vkT . In fact, by computing exp E ÿ EF =kT as a function of s=ss l from the expression of u E and from Eq. (17) and by using it in the fourth term of Eq. (3), i.e., by eliminating E between u E1 ÿ u E and s(E), we obtain a (complex) equation that allows us to compute l as a function of s and of other parameters. If, instead, we compute exp E ÿ EF =kT as a function of u from its expression and we use it again in Eq. (3), i.e., by eliminating E between u(E) and s(E), we get s ss u1=v 1 ÿ u vÿ1=v from which, and from Eq. (17), we have l v > 0 for defects G1 with E > EF 3kT , l v ÿ 1=v 6 0 for defects G3 with E < EF ÿ 3kT , whereas for the remaining defects G2 with EF ÿ 3kT < E < EF 3kT we can approximate the dependence of l on h in a linear form l lo3 l0 h ÿ h0o . (In particular for THE, i.e., for v 1, from the above equation, or directly from Eq. (3), we have s ss u for any E). We shall develop the calculations for the defects G1 with l v > 0; the analytical results hold true also for G2 and, with little modi®cations, even for G3 . Then, also in this case, we can add the relevant results for d and S according to Eq. (15) and the following Eq. (18), respectively. From Eqs. (9), (12), (16) and (17), we get Z 2 S u Ed 1 ÿ u Ed C 2 rt nt rt d3 rt x Z D0 expÿ h ÿ he 2 =2r2 exp v2 r2 =2 dh cosh h hx Z 4 p=xu Ed 1 ÿ u Ed C 2 rt nt rt d3 rt exp v2 r2 =2D0 exp ÿ hx ÿ he 2 =2r2 ;
18
where he hd vr2 and, according to Eq. (17), we have set u Ed 1 ÿ u Ed svd h1=svs i, to de®ne the ``average'' energy Ed of the defects; the second equality holds true for r > 3. For the following numerical examples, the likely values could be r 4:5 and r 10 for THE and TUE [3], respectively, and fd 1=2pso exp ÿ EN ÿ Ed = vkT , so that, for instance, by setting EN ÿ Ed =vkT 9, we can consider fd 2:5 109 Hz. From Eq. (18), we obtain two important results that can account for some properties of ¯icker noise and that derive from the increase in the weight of the large sÕs due to the dependence of u 1 ÿ u on s itself, according to Eq. (17).
The ®rst result is the shift toward a larger value he hd vr2 for h, i.e., se sd exp vr2 for s, to which the peak of the equivalent probability density function of h that appears in Eq. (18) corresponds. Indeed, this result provides an explanation of the existence of ¯icker noise down to very low frequencies. The other relevant result consists of the fact that the defect density nt rt generating the ¯icker noise is multiplied by the factor exp v2 r2 =2 that, as an example, is equal to 2:5 104 for THE, and to 22.8 for v 1=4 [258 for v 1=3] in the case of TUE. This fact indicates that even a small density of defects with large s can suce to generate ¯icker noise. From the comparison between Eqs. (12) and (18), we obtain Ghe / expÿ h ÿ he 2 =2r2 , and then from Eq. (13), we obtain [3] d
hx he ln f =fe 1 r2 1 r2
19
(whereas from Eq. (10) and the spectrum approximation given by the third member of Eq. (18), we directly could obtain d hx he =r2 ). Therefore, from Eq. (19), we have d1 6 d 6 d2 in a frequency interval f1 , f2 that has a width of log f2 = f1 1 r2 d2 ÿ d1 log e decades and is centred around the frequency fc fd exp 1 r2 d1 d1 = 2 ÿ vr2 ; for d2 ÿd1 d they become log f2 =f1 1 r2 d log e and fc fe fd exp ÿvr2 (e being the base of Neperian logarithms). From a numerical point of view, for THE, we have jdj 6 0:13 over a range of 2.4 decades centred around fc fe 4 Hz which is shifted 8.8 decades below the value fd 2:5 109 Hz that we would have with previous models with v 0. We obtain more meaningful results for the case of TUE, for which we have jdj 6 0:10 (0.05) over 8.8 (4.4) decades centred around fc fe 3:5 10ÿ2 Hz which is shifted 10.8 decades below fd . Therefore, it is the dependence of u on s that allows us to explain the fundamental property of 1=f c noise, i.e., its existence down to the lowest frequencies at which it has been found experimentally [7], provided that, as shown in the previous section, defects with the relevant relaxation times s with large value exist. Finally, from Eqs. (1) and (11) and the third term of Eq. (18), we get the coecient Z a 2u Ed 1 ÿ u Ed N C 2 rt nt rt d3 rt D0 expfv2 r2 =2 ÿ ln fa =fe 2 =2r2 g;
20
which can be drastically simpli®ed in suciently general cases. Indeed, we can set expÿ ln fa =fe 2 =2r2 1 with an error jgj 6 0:2 in interval in an interval fa0 , fa00 of the p frequency fa around fe of log fa00 =fa0 2 2gr log e de-
B. Pellegrini / Microelectronics Reliability 40 (2000) 1775±1780
cades, i.e., for instance for jgj 0:2, of 2.47 (5.5) decades for THE (TUE). Moreover, in the case of cylindrical and homogeneous devices, we have C ÿ1=N, and, if the defects have a uniform distribution over a volume Xt 6 X of the device (for instance that of the oxide in a MOS transistor), the integral of Eq. (20) becomes equal to nt Xt =N 2 . Furthermore, if Ed > EF 3kT , we have u Ed 1 ÿ u Ed =n EF 1=n Ed , where nd n Ed is the electron concentration that we should have if p the Fermi level coincides with Ed . Finally, for D0 1= 2pr; we have a
1=2 2 exp v2 r2 =2 nt Xt ; p r nd X
21
where Xt =X 1 when the defects are uniformly distributed over all the device; for THE, it is also 1=nd sd hci. For a (typical) value 10ÿ3 of a, Xt X and for nd 3:5 1015 cmÿ3 which, for instance, we have in silicon for EC ÿ Ed =kT 9 (while for EC ÿ EF =kT 12 the electron density would be n 1:7 1014 cmÿ3 ), from Eq. (21), we obtain nt 8 108 cmÿ3 for the THE case, i.e., Nt nt Xt 800 defects only in a device with a volume X Xs 1 mm3 are sucient to generate ¯icker noise. We obtain a larger density nt 1:9 1012 1:7 11 10 cmÿ3 in the case of TUE for v 1=4 v 1=3 which for a microelectronic device of volume X 200 lmÿ3 , would lead to Nt 380 [34] defects, that are sucient to generate measurable 1=f c noise and that can be located both in the device bulk or on its surface, for instance, in the oxide (or at the interface) of a MOS transistor. In submicron MOS transistors Nt may reach even values of just a few units, so that random telegraph ¯uctuations due to the single defects can be experimentally detected. On the other hand, since we have several quantities that, according to Eq. (21), determine a and which can vary independently in a range of decades, even among macroscopically identical devices, the widely experimentally measured spread of a can be accounted for. In order to show the eectiveness in generating 1=f c noise of an ensemble A of defects with dispersed relaxation times and concentration nt , let us compare it with the g±r noise originated by an ensemble B of equal defects with concentration ntb that are characterised by an energy Eb and a relaxation time sb 1=2pfb . From Eq. (7), in the case of C ÿ1=N, we obtain a g±r noise with spectrum Sg±r 4
ntb sb 2 ntb ; nb N 1 s2b x2 p fb nb N
22
where 1=nb u Eb 1 ÿ u Eb =n and the second equality holds true for sb x 1. From Eq. (1), for d 0, Eqs.
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(21) and (22) we get S fu Sg±r for a frequency fu given by fu p 1=2 nb nt exp v2 r2 =2 : 2 r fb nd ntb
23
By choosing Eb Ed , i.e., nb nd , from Eq. (23), we obtain ntb =nt 4:4 103 fb =fu for THE and ntb =nt 1:8 fb =fu 22:4 fb =fu for TUE. Since (fb =fu ) can reach values of several decades, we see how defects A are much more eective in generating 1=f c noise (especially for THE) than defects B in producing g±r noise. This difference is due to the fact that, according to Eqs. (2) and (17), one half of the power h D/2 i u 1 ÿ u s=ss l is distributed in the frequency interval 0, 1/2ps, so that in such a band the power spectral density, as can be seen directly from Eqs. (2) and (17), is proportional to s1l , i.e., the defects with a larger s at the origin of 1=f c noise are more eective in contributing to the total noise spectrum. The signi®cance of its contribution is further increased by the fact that in the present new model, at least for the defect group G1 , it is l v > 0.
5. Conclusions We have shown a complete model of the ¯icker noise that is based on charge ¯uctuations of single-energylevel defects. The problem of coupling between them and the output current is solved in a general and simple way by means of the electrokinematics theorem that, in the particular case of cylindrical homogeneous devices, directly leads to the spectrum dependence on the reciprocal of the total number N of electrons. An important new element of the model is the dependence of the defect occupation factor on the relaxation time s, which greatly increases the weight of the largest sÕs or, equivalently, the eective number of the corresponding defects that are usually less numerous and, at the same time, account for 1=f c noise down to the lowest frequencies. We have shown that in the frequency exponent c 1 d, the contribution d is the average of an hyperbolic tangent and, as a consequence, d 0 and c 1 at any frequency for any reasonably smooth probability density function of s. The computation of the a coecient has shown that it can vary in a very wide range and its typical experimental values are such that the number of defects as low as a few tens can be sucient for generating 1=f c noise, whereas, orders of magnitude more defects of the same type are needed to generate g±r noise. On the basis of the presented model and data, we can claim that the ubiquity of the 1=f c noise with c 1 down to the lowest frequency at which it can be reliably
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measured is due to the inevitable existence in the conducting media of defects, which have arbitrarily large and distributed relaxation times and have random localisation in the device, both on its surface and in the bulk. Their number can become so low that the 1=f c noise is the only phenomenon able to reveal their existence. The model can be extended to bipolar devices as well as to nanoelectronic devices, by computing the current with the extension of the electrokinematics theorem to quantum mechanics [9], to quantities, dierent from the current, which are sensitive to the ¯uctuations of the carrier number, and, moreover, in general, to any system, even dierent from physical ones, in which a few or many independent traps of elements that are characterised by highly spread hold times and Lorentzian spectra interact with an observable quantity through the ¯ux or the number of such stored elements. In the light of all the above presented arguments, 1=f c noise would seem a solved problem.
Acknowledgements The present work has been supported by the Ministry for the University and the Scienti®c and Technological Research of Italy, through the National Project ``Silicon based nanoelectronic technologies and devices'', and by
the National Research Council (CNR) of Italy, through the CSMDR Centre and the Project ``Material and Devices for Solid-State Electronics''. The author also wishes to thank Prof. M. Macucci for useful discussions.
References [1] Johnson B. The Schottky eect in low frequency circuits. Phys Rev 1925;26:71±85. [2] Bishop J. Ph.D. Thesis, University of Eindhoven, 1983. p. 82. [3] Pellegrini B. 1=f 0 noise from single-energy-level defects. Phys Rev B 1987;35:571±80. [4] Pellegrini B. Electric charge motion, induced current, energy balance and noise. Phys Rev B 1986;34:5921±4. [5] Machlup S. Noise in semiconductors spectrum of two parameter random signal. J Appl Phys 1954;25:341±3. [6] Mc Worter AL. Lincoln Laboratory Technical Report No. 80. MIT, 1955. [7] Caloyannides MN. Microcycle spectral estimation of 1=f noise in semiconductors. J Appl Phys Rev 1974;45: 307±12. [8] Basso G, Pellegrini B. Photocurrent transients in almost ideal silicon p±n junctions. J Appl Phys 1995;78:2504±8. [9] Pellegrini B. Extension of the electro-kinematics theorem to the electromagnetic ®eld and quantum mechanics. Il Nuovo Cimento D 1993;15:855±79.
Invited paper
www.elsevier.com/locate/microrel
A general model of 1=f c noise Bruno Pellegrini Dipartimento di Ingegneria dell'Informazione: Elettronica, Informatica e Telecomunicazioni Universit a degli Studi di Pisa, CSMDR of CNR, Via Diotisalvi 2, I±56126 Pisa, Italy Received 29 November 1999
Abstract A model of 1=f c noise is proposed that accounts for all its experimental properties. Its origin is found to be in the charge ¯uctuations in few defects, even less than 100, with relaxation times s arbitrarily distributed in a wide interval, up to large values. The coupling coecient between local and output ¯uctuations, the dependence of the defect occupation factor on s, the frequency exponent c and the coecient a of the spectrum are computed in a general and simple way. Published by Elsevier Science Ltd.
1. Premise As is well known, since its experimental discovery in vacuum tubes by Johnson in 1925 [1], the so-called ¯icker or 1=f c noise has been the subject of many experiments, models and discussions, and, unlike what happens for all the other noise types, i.e., thermal, shot, generation±recombination (g±r), and burst noises, a universally accepted theory about it does not seem to exist, so that it is meaningful to ask, ``Is 1=f c noise an unsolved problem''? We try to give a negative answer to this question by presenting a general model of 1=f c noise that is based on the charge ¯uctuations of defects that have arbitrarily distributed relaxation times s in a wide interval, up to very large values, and are randomly scattered, even in very low concentrations, in the electron device. The model can be generalised to a system of any nature. According to the experimental ®ndings, (i) 1=f c noise exists in any device traversed by a steady average current I hi ti, (ii) the relevant power spectral density S of i t=I is given by Sa
fad ; Nf c
1
where f x=2p fa is the frequency (the central frequency of the interval in which we perform the measurement) and N is the total number of free electrons in the device, (iii) the dimensionless coecient a is spread in the very wide range from 10ÿ8 to 1 [2], (iv) the fre0026-2714/00/$ - see front matter Published by Elsevier Science Ltd. PII: S 0 0 2 6 - 2 7 1 4 ( 0 0 ) 0 0 0 6 1 - 5
quency exponent c 1 d ranges between 0.8 and 1.2 over many frequency decades and (v) down to any low frequency at which the spectrum is measurable. The theory that we propose, by developing previous results [3,4], accounts for all the ®ve, (i)±(v), experimental properties of 1=f c noise. 2. Fluctuations in the defects and at the output of the device As is well known, a random telegraph signal /(t) that stays at level 1 (0) an average time s sÿ and, as a consequence, has a mean value u s = s sÿ and a correlation time s s sÿ = s sÿ , is characterised by the Lorentzian spectrum [5], s S/ 4u 1 ÿ u : 2 1 s2 x2 On the other hand, a defect at rt with a single energy level E in a unipolar device characterised by an average density n r of free electrons has an electron number /(t) with times s 1=e and sÿ 1=cn, e and c being the emission and capture probability of an electron, respectively, a relaxation time s
u 1ÿu 1ÿu ; ÿ1 cn e so exp ÿ EN ÿ E=vkT
3
and an average value u 1=f1 exp E ÿ EF =kT g given by the Fermi±Dirac statistics, EF , k and T being
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B. Pellegrini / Microelectronics Reliability 40 (2000) 1775±1780
the quasi-Fermi level, the Boltzmann constant and the temperature, respectively. The fourth term of Eq. (3) takes into account that for both a thermal emission (THE) of the electron from the defect into the conduction band, with a bottom energy EC , and a tunnel emission (TUE) across a parabolic energy barrier with a peak value VM , which cover most of the practical cases, we have e sÿ1 o expÿ EN ÿ E=vkT , where v 1 and EN EC for THE, 1=5 < v < 1=2 and EN VM for TUE, and so 1=2pfo 10ÿ14 s, fo being the electron oscillation frequency in the defect [3]. We now have to determine the eects of the local charge ¯uctuations in the defect on the output current i(t) ¯owing through the device electrode that we are considering. To this end, we exploit the electrokinematics theorem that, in the case of electrode voltages kept at a constant value, allows us to write i in the form [4] Z i ÿq
n0 r; tv0 r; t F r d3 r;
4
where v0 r; t is the mean velocity of the n0 r; td3 r electrons in the volume element d3 r at t, while F ÿrW r is an irrotational vector de®ned by r eF 0 over the volume X of the device, W 1 on the electrode surface and W 0 on the remaining part of device surface, e r being the electric permittivity. The integral of Eq. (4) in the space r (and rt in the following) is computed over X. Apart from the ¯uctuation of i(t) due to those of v0 r; t and n0 r; t around their time average values v and n, respectively, and produced by the electron motion and scattering, the variation Dn ÿD/d r ÿ rt of the density of the free electrons generated by the capture and emission of one electron by the defect at rt determines a ¯uctuation of i(t), which, according to Eq. (4), becomes Di qF rt v rt :
5
If we replace v0 with v and n0 with n in Eq. (4), we obtain I and Di=I CD/; where the coupling coecient C is given by Z
C rt ÿv rt F rt n rv r F r d3 r Z ÿ Fx rt n Fx r d3 r
6
in which the third term holds true in the particular case of a cylindrical and homogeneous device with a velocity that is parallel to the cylinder axis x, and, as well as n, is independent of r; we have C ÿ1=N , if, as possible [4], we choose a vector F independent of r and parallel to the cylinder axis.
3. Frequency exponent of the noise spectrum From Eqs. (2) and (6), in the case of uncorrelated defects, we obtain the spectrum S of the current ¯uctuation Di=I due to all the defects, of density nt rt , in the form Z Z Z s S4 u 1 ÿ u C 2 rt nt rt 1 s2 x2 D E; s; rt d3 rt dE ds;
7
where D E; s; rt is the joint probability density function of E and s at rt . Since s E; p depends on E and on other parameters p1 ; . . . ; pn p independent of E (e.g., in the case dealt with in the preceding section, p1 EN , p2 so , p3 v) we can express E E s; p, u E and D E; s; rt through the independent variables s, p and rt , and we can write Eq. (7) in the form Z s S4 Gse s ds; 8 1 s2 x2 where Gse s is a probability-density-like function for s de®ned by Z Z Gse u 1 ÿ uC 2 rt nt rt D0 s; p; rt d3 rt dn p; 9 where D0 s; p; rt is the joint probability density function of s and p at rt . From Eq. (8), we directly obtain S / 1=f c in the case of Gse / 1=s2ÿc and 0 < c < 2. From a physical point of view, we get this result, for c 1, in the very particular case, dealt with by Mc Worter [6], of electron TUE from defects across rectangular energy barriers of constant height and uniformly distributed width. Moreover, in general, in computing S by means of Eqs. (7) or (8) and (9), we encounter two serious diculties: the ®rst one is physical and derives from the fact that D or D0 , which depend on many microscopic parameters, are usually unknown. The second diculty is mathematical and consists of the fact that the integrals of Eqs. (7)±(9) rarely lead to analytical results in closed form. However, several important properties of S can be deduced from Eq. (8) according to the following procedure. Let us perform the Taylor series expansion ln S f fr ln S fa fr ÿ c fa ln f =fr ÿ ln fa =fr about ln fa =fr of the function ln S f fr of the variable ln f =fr where fr 1=2psr is an arbitrary reference frequency and c f ÿ
o ln S f fr 1 d; o ln f =fr
10
so that in a proper frequency interval around fa , we have
B. Pellegrini / Microelectronics Reliability 40 (2000) 1775±1780
S f S fa
fa f
c fa
:
11
In order to compute the frequency exponent c, let us perform the variable changes x sÿ1 and r exphx s sr exph in Eq. (8), which becomes Z Ghe h S 2sr exp ÿhx dh; 12 cosh h hx where now, Ghe h Gse s hsr exp h is a probabilitydensity-like function for h; moreover, in general, in the following, we shall set hh ln sh =sr ÿ ln 2psr fh . From Eqs. (10) and (12), we get Z d tanh h hx H h dh tanh hm hx o ln Ghe h=ohjhhm ;
13
where the ``weight'' function H h of tanh h hx at h is given by Z Ghe h Ghe h H h dh; 14 cosh h hx cosh h hx and hm , de®ned by the third equality of Eq. (13), is the value of h at which H h reaches a maximum [3]. It is worth noting that d has an additive property, in the sense that if we decompose nRt D0 into more parts, in P any way, i.e., nt D0 nt D0 i nt gD0 g dg in Eqs. (8) and (9), from Eqs. (12)±(14) we get Z X d di Si =S d gSg g=S dg; 15 where di and Si are the values of d and S, respectively, relevant to nt D0 i and g is a parameter. Therefore, from Eq. (13), d is the ``mean'' value of tanh h hx and, since j tanh h hx j 6 1 and 0 6 H h 6 1, we have jdj 6 1 and 0 6 c 6 2, as it has to be in a superposition of Lorentzian spectra. If s is spread in an interval between sl sr exp hl 1=2pfH and sn sr exp hn 1=2pfL , since j tanh h hx j 1 for jh hx j P 1:5, from Eqs. (13) and (14) we have d ÿ1 and c 0 for f < fL =4:5 and d 1 and c 2 for f > 4:5fH . In the frequency band fL , fH , i.e., for hl < hx < hn , on the contrary, if Gse h is a slowly varying function of h with respect to 1= cosh h hx , the ``weight'' function H(h), due to 1= cosh h hx , tends to have a symmetrical maximum at hm hx , around which tanh h hx is an odd function, so that, according to the second and third term of Eq. (13), jdj ! 0. (Such properties of H(h) and of tanh h hx and Eq. (14), in particular, allow us to get d in the form given by the third and fourth term of Eq. (13).) Therefore, in practice for any s dispersion, except for particular cases with sharp peaks, we have c ! 1 in the band fL , fH and the existence of defects with relaxation
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times s spread at least up to sn is necessary to have ¯icker noise down to a minimum frequency fL : for instance, up to sn 160 s for fL 10ÿ3 Hz and up to sn 3:2 105 s 3:7 days for the extreme experimental case of fL 5 10ÿ7 Hz obtained in MOS transistors [7] (if the method used to combine ¯uctuation records obtained from several equal devices is reliable). In thick ®lm resistors, a THE activation energy of EC ÿ E=kT 38 was found at T 300 K [3], so that, being sn so exp EC ÿ E=kT we obtained sn 320 s. For TUE, being sn so exp VM ÿ E=vkT with vkT 75 150 K for parabolic energy barriers and for rectangular ones, we sn so exp x=k with k 0:5 A have sn 3:2 105 s for a barrier height VM ÿ E 0:29 eV in the ®rst case (for vkT 75 K) and for x 2:25 nm in the second case in which, for instance, x is the distance of an oxide defect from the interface of a MOS transistor. All these values are experimentally likely. Relaxation times of 1000 s have been measured in p±n junctions [8].
4. Gaussian dispersion of the parameters, analytical and numerical results Up to this moment, we have considered the general case of a generic and unknown joint probability density function D0 s; p; rt of s and p at rt and of an unspeci®ed relationship between u(E) and s(E) that we should obtain by eliminating their dependence on E. Now, in order to obtain a few analytical and numerical results that indicate the principal trends of the model, let us consider such a relationship and the case, that, however, is still suciently general, of D0 s; p; rt Dp pDs s, i.e., in which s and p are uncorrelated and their probability density functions are independent of rt . Moreover, from Eq. (3), we have h h0 ln so =sr EN ÿ E=vkT for E > EF and h h0 E ÿ EF =kT for E < EF , so that the probability density function Dh h of h is the result of those of the defect energy E and of the quantities EN and ln so =sr , which are random variables depending on several physical properties and quantities. Therefore, also as a consequence of the central limit theorem, a good approximation for Dh Ds s hsr exp h is given by the normal law, Dh D0 exp ÿ h ÿ hd 2 =2r2 ;
16
where hd hhi and r2 are the mean value and the variance of h, respectively, and D0 is the normalisation factor, which, for hn ÿhl ! 1, becomes D0 1= p 2pr. Actually, we can always decompose Dh into a ®nite or in®nite number of components Dh / exp jajh ÿ jbjh2 , i.e., of the Gaussian type (16), and then we can add the
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B. Pellegrini / Microelectronics Reliability 40 (2000) 1775±1780
relevant results for d and S according to Eq. (15) and the following Eq. (18), respectively. On the other hand, about the relationship between u E and s(E), we can set u 1 ÿ u s=ss l s sr =ss ls h expfls hhg;
17
where ss so exp EN ÿ EF =vkT . In fact, by computing exp E ÿ EF =kT as a function of s=ss l from the expression of u E and from Eq. (17) and by using it in the fourth term of Eq. (3), i.e., by eliminating E between u E1 ÿ u E and s(E), we obtain a (complex) equation that allows us to compute l as a function of s and of other parameters. If, instead, we compute exp E ÿ EF =kT as a function of u from its expression and we use it again in Eq. (3), i.e., by eliminating E between u(E) and s(E), we get s ss u1=v 1 ÿ u vÿ1=v from which, and from Eq. (17), we have l v > 0 for defects G1 with E > EF 3kT , l v ÿ 1=v 6 0 for defects G3 with E < EF ÿ 3kT , whereas for the remaining defects G2 with EF ÿ 3kT < E < EF 3kT we can approximate the dependence of l on h in a linear form l lo3 l0 h ÿ h0o . (In particular for THE, i.e., for v 1, from the above equation, or directly from Eq. (3), we have s ss u for any E). We shall develop the calculations for the defects G1 with l v > 0; the analytical results hold true also for G2 and, with little modi®cations, even for G3 . Then, also in this case, we can add the relevant results for d and S according to Eq. (15) and the following Eq. (18), respectively. From Eqs. (9), (12), (16) and (17), we get Z 2 S u Ed 1 ÿ u Ed C 2 rt nt rt d3 rt x Z D0 expÿ h ÿ he 2 =2r2 exp v2 r2 =2 dh cosh h hx Z 4 p=xu Ed 1 ÿ u Ed C 2 rt nt rt d3 rt exp v2 r2 =2D0 exp ÿ hx ÿ he 2 =2r2 ;
18
where he hd vr2 and, according to Eq. (17), we have set u Ed 1 ÿ u Ed svd h1=svs i, to de®ne the ``average'' energy Ed of the defects; the second equality holds true for r > 3. For the following numerical examples, the likely values could be r 4:5 and r 10 for THE and TUE [3], respectively, and fd 1=2pso exp ÿ EN ÿ Ed = vkT , so that, for instance, by setting EN ÿ Ed =vkT 9, we can consider fd 2:5 109 Hz. From Eq. (18), we obtain two important results that can account for some properties of ¯icker noise and that derive from the increase in the weight of the large sÕs due to the dependence of u 1 ÿ u on s itself, according to Eq. (17).
The ®rst result is the shift toward a larger value he hd vr2 for h, i.e., se sd exp vr2 for s, to which the peak of the equivalent probability density function of h that appears in Eq. (18) corresponds. Indeed, this result provides an explanation of the existence of ¯icker noise down to very low frequencies. The other relevant result consists of the fact that the defect density nt rt generating the ¯icker noise is multiplied by the factor exp v2 r2 =2 that, as an example, is equal to 2:5 104 for THE, and to 22.8 for v 1=4 [258 for v 1=3] in the case of TUE. This fact indicates that even a small density of defects with large s can suce to generate ¯icker noise. From the comparison between Eqs. (12) and (18), we obtain Ghe / expÿ h ÿ he 2 =2r2 , and then from Eq. (13), we obtain [3] d
hx he ln f =fe 1 r2 1 r2
19
(whereas from Eq. (10) and the spectrum approximation given by the third member of Eq. (18), we directly could obtain d hx he =r2 ). Therefore, from Eq. (19), we have d1 6 d 6 d2 in a frequency interval f1 , f2 that has a width of log f2 = f1 1 r2 d2 ÿ d1 log e decades and is centred around the frequency fc fd exp 1 r2 d1 d1 = 2 ÿ vr2 ; for d2 ÿd1 d they become log f2 =f1 1 r2 d log e and fc fe fd exp ÿvr2 (e being the base of Neperian logarithms). From a numerical point of view, for THE, we have jdj 6 0:13 over a range of 2.4 decades centred around fc fe 4 Hz which is shifted 8.8 decades below the value fd 2:5 109 Hz that we would have with previous models with v 0. We obtain more meaningful results for the case of TUE, for which we have jdj 6 0:10 (0.05) over 8.8 (4.4) decades centred around fc fe 3:5 10ÿ2 Hz which is shifted 10.8 decades below fd . Therefore, it is the dependence of u on s that allows us to explain the fundamental property of 1=f c noise, i.e., its existence down to the lowest frequencies at which it has been found experimentally [7], provided that, as shown in the previous section, defects with the relevant relaxation times s with large value exist. Finally, from Eqs. (1) and (11) and the third term of Eq. (18), we get the coecient Z a 2u Ed 1 ÿ u Ed N C 2 rt nt rt d3 rt D0 expfv2 r2 =2 ÿ ln fa =fe 2 =2r2 g;
20
which can be drastically simpli®ed in suciently general cases. Indeed, we can set expÿ ln fa =fe 2 =2r2 1 with an error jgj 6 0:2 in interval in an interval fa0 , fa00 of the p frequency fa around fe of log fa00 =fa0 2 2gr log e de-
B. Pellegrini / Microelectronics Reliability 40 (2000) 1775±1780
cades, i.e., for instance for jgj 0:2, of 2.47 (5.5) decades for THE (TUE). Moreover, in the case of cylindrical and homogeneous devices, we have C ÿ1=N, and, if the defects have a uniform distribution over a volume Xt 6 X of the device (for instance that of the oxide in a MOS transistor), the integral of Eq. (20) becomes equal to nt Xt =N 2 . Furthermore, if Ed > EF 3kT , we have u Ed 1 ÿ u Ed =n EF 1=n Ed , where nd n Ed is the electron concentration that we should have if p the Fermi level coincides with Ed . Finally, for D0 1= 2pr; we have a
1=2 2 exp v2 r2 =2 nt Xt ; p r nd X
21
where Xt =X 1 when the defects are uniformly distributed over all the device; for THE, it is also 1=nd sd hci. For a (typical) value 10ÿ3 of a, Xt X and for nd 3:5 1015 cmÿ3 which, for instance, we have in silicon for EC ÿ Ed =kT 9 (while for EC ÿ EF =kT 12 the electron density would be n 1:7 1014 cmÿ3 ), from Eq. (21), we obtain nt 8 108 cmÿ3 for the THE case, i.e., Nt nt Xt 800 defects only in a device with a volume X Xs 1 mm3 are sucient to generate ¯icker noise. We obtain a larger density nt 1:9 1012 1:7 11 10 cmÿ3 in the case of TUE for v 1=4 v 1=3 which for a microelectronic device of volume X 200 lmÿ3 , would lead to Nt 380 [34] defects, that are sucient to generate measurable 1=f c noise and that can be located both in the device bulk or on its surface, for instance, in the oxide (or at the interface) of a MOS transistor. In submicron MOS transistors Nt may reach even values of just a few units, so that random telegraph ¯uctuations due to the single defects can be experimentally detected. On the other hand, since we have several quantities that, according to Eq. (21), determine a and which can vary independently in a range of decades, even among macroscopically identical devices, the widely experimentally measured spread of a can be accounted for. In order to show the eectiveness in generating 1=f c noise of an ensemble A of defects with dispersed relaxation times and concentration nt , let us compare it with the g±r noise originated by an ensemble B of equal defects with concentration ntb that are characterised by an energy Eb and a relaxation time sb 1=2pfb . From Eq. (7), in the case of C ÿ1=N, we obtain a g±r noise with spectrum Sg±r 4
ntb sb 2 ntb ; nb N 1 s2b x2 p fb nb N
22
where 1=nb u Eb 1 ÿ u Eb =n and the second equality holds true for sb x 1. From Eq. (1), for d 0, Eqs.
1779
(21) and (22) we get S fu Sg±r for a frequency fu given by fu p 1=2 nb nt exp v2 r2 =2 : 2 r fb nd ntb
23
By choosing Eb Ed , i.e., nb nd , from Eq. (23), we obtain ntb =nt 4:4 103 fb =fu for THE and ntb =nt 1:8 fb =fu 22:4 fb =fu for TUE. Since (fb =fu ) can reach values of several decades, we see how defects A are much more eective in generating 1=f c noise (especially for THE) than defects B in producing g±r noise. This difference is due to the fact that, according to Eqs. (2) and (17), one half of the power h D/2 i u 1 ÿ u s=ss l is distributed in the frequency interval 0, 1/2ps, so that in such a band the power spectral density, as can be seen directly from Eqs. (2) and (17), is proportional to s1l , i.e., the defects with a larger s at the origin of 1=f c noise are more eective in contributing to the total noise spectrum. The signi®cance of its contribution is further increased by the fact that in the present new model, at least for the defect group G1 , it is l v > 0.
5. Conclusions We have shown a complete model of the ¯icker noise that is based on charge ¯uctuations of single-energylevel defects. The problem of coupling between them and the output current is solved in a general and simple way by means of the electrokinematics theorem that, in the particular case of cylindrical homogeneous devices, directly leads to the spectrum dependence on the reciprocal of the total number N of electrons. An important new element of the model is the dependence of the defect occupation factor on the relaxation time s, which greatly increases the weight of the largest sÕs or, equivalently, the eective number of the corresponding defects that are usually less numerous and, at the same time, account for 1=f c noise down to the lowest frequencies. We have shown that in the frequency exponent c 1 d, the contribution d is the average of an hyperbolic tangent and, as a consequence, d 0 and c 1 at any frequency for any reasonably smooth probability density function of s. The computation of the a coecient has shown that it can vary in a very wide range and its typical experimental values are such that the number of defects as low as a few tens can be sucient for generating 1=f c noise, whereas, orders of magnitude more defects of the same type are needed to generate g±r noise. On the basis of the presented model and data, we can claim that the ubiquity of the 1=f c noise with c 1 down to the lowest frequency at which it can be reliably
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B. Pellegrini / Microelectronics Reliability 40 (2000) 1775±1780
measured is due to the inevitable existence in the conducting media of defects, which have arbitrarily large and distributed relaxation times and have random localisation in the device, both on its surface and in the bulk. Their number can become so low that the 1=f c noise is the only phenomenon able to reveal their existence. The model can be extended to bipolar devices as well as to nanoelectronic devices, by computing the current with the extension of the electrokinematics theorem to quantum mechanics [9], to quantities, dierent from the current, which are sensitive to the ¯uctuations of the carrier number, and, moreover, in general, to any system, even dierent from physical ones, in which a few or many independent traps of elements that are characterised by highly spread hold times and Lorentzian spectra interact with an observable quantity through the ¯ux or the number of such stored elements. In the light of all the above presented arguments, 1=f c noise would seem a solved problem.
Acknowledgements The present work has been supported by the Ministry for the University and the Scienti®c and Technological Research of Italy, through the National Project ``Silicon based nanoelectronic technologies and devices'', and by
the National Research Council (CNR) of Italy, through the CSMDR Centre and the Project ``Material and Devices for Solid-State Electronics''. The author also wishes to thank Prof. M. Macucci for useful discussions.
References [1] Johnson B. The Schottky eect in low frequency circuits. Phys Rev 1925;26:71±85. [2] Bishop J. Ph.D. Thesis, University of Eindhoven, 1983. p. 82. [3] Pellegrini B. 1=f 0 noise from single-energy-level defects. Phys Rev B 1987;35:571±80. [4] Pellegrini B. Electric charge motion, induced current, energy balance and noise. Phys Rev B 1986;34:5921±4. [5] Machlup S. Noise in semiconductors spectrum of two parameter random signal. J Appl Phys 1954;25:341±3. [6] Mc Worter AL. Lincoln Laboratory Technical Report No. 80. MIT, 1955. [7] Caloyannides MN. Microcycle spectral estimation of 1=f noise in semiconductors. J Appl Phys Rev 1974;45: 307±12. [8] Basso G, Pellegrini B. Photocurrent transients in almost ideal silicon p±n junctions. J Appl Phys 1995;78:2504±8. [9] Pellegrini B. Extension of the electro-kinematics theorem to the electromagnetic ®eld and quantum mechanics. Il Nuovo Cimento D 1993;15:855±79.