Invited paper

A general model of 1=f c noiseBruno 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 coe�cient between local and output ¯uctuations, the dependence of the defect occupation

factor on s, the frequency exponent c and the coe�cient 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 ex-

periments, 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�t�i, (ii) the relevant power spectral density S of

i�t�=I is given by

S � af d

a

Nf c; �1�

where f � x=2p �fa� is the frequency (the central fre-

quency of the interval in which we perform the mea-

surement) and N is the total number of free electrons in

the device, (iii) the dimensionless coe�cient a is spread

in the very wide range from 10ÿ8 to 1 [2], (iv) the fre-

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), experi-

mental 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/ � 4u�1ÿ u� s1� s2x2

: �2�

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, re-

spectively, a relaxation time

s � ucn� 1ÿ u

e� 1ÿ u

sÿ1o exp �ÿ�EN ÿ E�=vkT � ; �3�

and an average value u � 1=f1� exp ��E ÿ EF�=kT �ggiven by the Fermi±Dirac statistics, EF, k and T being

Microelectronics Reliability 40 (2000) 1775±1780

www.elsevier.com/locate/microrel

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PII: S0 02 6 -2 71 4 (00 )0 0 06 1 -5

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 conduc-

tion 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 � sÿ1o exp �ÿ�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 e�ects 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 electrokine-

matics theorem that, in the case of electrode voltages

kept at a constant value, allows us to write i in the form

[4]

i � ÿqZ

n0�r; t�v0�r; t� � F�r�d3r; �4�

where v0�r; t� is the mean velocity of the n0�r; t�d3r elec-

trons in the volume element d3r 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�rÿ rt� of the den-

sity of the free electrons generated by the capture and

emission of one electron by the defect at rt deter-

mines 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 coe�cient

C is given by

C�rt� � ÿv�rt� � F�rt�Z

n�r�v�r� � F�r�d3r

�� ÿ Fx�rt�

�nZ

Fx�r�d3r �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 ¯uctu-

ation Di=I due to all the defects, of density nt�rt�, in the

form

S � 4

Z Z Zu�1ÿ u� s

1� s2x2C2�rt�nt�rt�

� D�E; s; rt�d3rt 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

S � 4

Zs

1� s2x2Gse�s�ds; �8�

where Gse�s� is a probability-density-like function for sde®ned by

Gse �Z Z

u�1ÿ u�C2�rt�nt�rt�D0�s; p; rt�d3rt dnp; �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 di�-

culties: the ®rst one is physical and derives from the fact

that D or D0, which depend on many microscopic pa-

rameters, are usually unknown. The second di�culty 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 pro-

cedure. 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�o ln�f =fr� � 1� d; �10�

so that in a proper frequency interval around fa, we have

1776 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 � sÿ1r exphx and

s � sr exph in Eq. (8), which becomes

S � 2sr exp�ÿhx�Z

Ghe�h�cosh�h� hx� dh; �12�

where now, Ghe�h� � Gse�s�h��sr exph is a probability-

density-like function for h; moreover, in general, in the

following, we shall set hh � ln�sh=sr� � ÿ ln�2psrfh�.From Eqs. (10) and (12), we get

d �Z

tanh�h� hx�H�h�dh � tanh�hm � hx�

� �o lnGhe�h�=oh�jh�hm; �13�

where the ``weight'' function H�h� of tanh�h� hx� at his given by

H�h� � Ghe�h�cosh�h� hx�

�ZGhe�h�

cosh�h� hx�dh; �14�

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 ntD0 into more parts, in

any way, i.e., ntD0 �P�ntD0�i �

Rnt�g�D0�g�dg in Eqs.

(8) and (9), from Eqs. (12)±(14) we get

d �X

di�Si=S� �Z

d�g��Sg�g�=S�dg; �15�

where di and Si are the values of d and S, respectively,

relevant to �ntD0�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�j6 1 and 06H�h�6 1, we have jdj6 1 and 06 c6 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� hxjP 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 hwith respect to 1=cosh�h� hx�, the ``weight'' function

H(h), due to 1=cosh�h� hx�, tends to have a symmet-

rical 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

times s spread at least up to sn is necessary to have

¯icker noise down to a minimum frequency fL: for in-

stance, 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

sn � so exp�x=k� with k � 0:5 �A for rectangular ones, we

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 dis-

tance 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 ob-

tain by eliminating their dependence on E. Now, in or-

der 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

su�ciently general, of D0�s; p; rt� � Dp�p�Ds�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 � �EÿEF�=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 conse-

quence of the central limit theorem, a good approxi-

mation for Dh � Ds�s�h��sr exph is given by the normal

law,

Dh � D0 exp �ÿ�hÿ 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=������2pp

r.

Actually, we can always decompose Dh into a ®nite or

in®nite number of components Dh / exp�jajhÿ jbjh2�,i.e., of the Gaussian type (16), and then we can add the

B. Pellegrini / Microelectronics Reliability 40 (2000) 1775±1780 1777

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�1ÿ u� � s=ss� �l�s� � sr=ss� �l�s�h�� expfl�s�h��hg; �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 ex-

pression 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�E��1ÿ u�E�� and s(E), we obtain a (complex) equa-

tion 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 func-

tion 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 � ssu1=v�1ÿ u��vÿ1�=vfrom which, and from Eq. (17),

we have l � v > 0 for defects G1 with E > EF � 3kT ,

l � �vÿ 1�=v6 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 lon h in a linear form l � lo3 � l0�hÿ h0o�. (In particular

for THE, i.e., for v � 1, from the above equation, or

directly from Eq. (3), we have s � ssu 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), re-

spectively.

From Eqs. (9), (12), (16) and (17), we get

S � 2

xu�Ed��1ÿ u�Ed��

ZC2�rt�nt�rt�d3rt

� exp�v2r2=2�Z

D0 exp �ÿ�hÿ he�2=2r2�cosh�h� hx� dh

� 4�p=x�u�Ed��1ÿ u�Ed��Z

C2�rt�nt�rt�d3rt

� exp�v2r2=2�D0 exp �ÿ�hx ÿ he�2=2r2�; �18�

where he � hd � vr2 and, according to Eq. (17), we have

set u�Ed��1ÿ u�Ed�� � svdh1=sv

si, 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�1ÿ 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 mul-

tiplied by the factor exp�v2r2=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 su�ce to

generate ¯icker noise.

From the comparison between Eqs. (12) and (18), we

obtain Ghe / exp �ÿ�hÿ he�2=2r2�, and then from Eq.

(13), we obtain [3]

d � hx � he

1� r2� ln�f =fe�

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 d16 d6 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�=2ÿ 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

dj j6 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 pre-

vious models with v � 0.

We obtain more meaningful results for the case of

TUE, for which we have dj j6 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 coe�cient

a � 2u�Ed��1ÿ u�Ed��NZ

C2�rt�nt�rt�d3rt

� D0 expfv2r2=2ÿ � ln�fa=fe��2=2r2g; �20�

which can be drastically simpli®ed in su�ciently general

cases.

Indeed, we can set exp�ÿ�ln�fa=fe��2=2r2� � 1 with

an error jgj6 0:2 in interval in an interval f 0a, f 00a of the

frequency fa around fe of log�f 00a =f 0a� � 2�����2gp

r log e de-

1778 B. Pellegrini / Microelectronics Reliability 40 (2000) 1775±1780

cades, i.e., for instance for jgj � 0:2, of 2.47 (5.5) de-

cades for THE (TUE). Moreover, in the case of cylin-

drical and homogeneous devices, we have C � ÿ1=N ,

and, if the defects have a uniform distribution over a

volume Xt 6X of the device (for instance that of the

oxide in a MOS transistor), the integral of Eq. (20) be-

comes equal to ntXt=N 2. Furthermore, if Ed > EF � 3kT ,

we have u�Ed��1ÿ u�Ed��=n�EF� � 1=n�Ed�, where

nd � n�Ed� is the electron concentration that we should

have if the Fermi level coincides with Ed . Finally, for

D0 � 1=������2pp

r; we have

a � 2

p

� �1=2exp�v2r2=2�

rnt

nd

Xt

X; �21�

where Xt=X � 1 when the defects are uniformly dis-

tributed over all the device; for THE, it is also

1=nd � sdhci.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 � ntXt � 800 defects only in a device with a

volume X � Xs � 1 mm3 are su�cient to generate

¯icker noise.

We obtain a larger density nt � 1:9� 1012 �1:7�1011� 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

su�cient 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 experimen-

tally 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 experi-

mentally measured spread of a can be accounted for.

In order to show the e�ectiveness in generating 1=f c

noise of an ensemble A of defects with dispersed relax-

ation times and concentration nt, let us compare it with

the g±r noise originated by an ensemble B of equal de-

fects 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 � 4ntb

nbNsb

1� s2bx2� 2

pntb

fbnbN; �22�

where 1=nb � u�Eb��1ÿ u�Eb��=n and the second equal-

ity holds true for sbx� 1. From Eq. (1), for d � 0, Eqs.

(21) and (22) we get S�fu� � Sg±r for a frequency fu given

by

fu

fb� p

2

� �1=2 nb

nd

nt

ntb

exp�v2r2=2�r

: �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 e�ective in generating 1=f c noise (especially for

THE) than defects B in producing g±r noise. This dif-

ference is due to the fact that, according to Eqs. (2) and

(17), one half of the power h�D/�2i � u�1ÿ u� � �s=ss�lis 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 s1�l,

i.e., the defects with a larger s at the origin of 1=f c noise

are more e�ective 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-energy-

level 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, di-

rectly leads to the spectrum dependence on the recip-

rocal of the total number N of electrons.

An important new element of the model is the de-

pendence of the defect occupation factor on the relax-

ation time s, which greatly increases the weight of the

largest sÕs or, equivalently, the e�ective 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 hy-

perbolic 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 coe�cient has shown that

it can vary in a very wide range and its typical experi-

mental values are such that the number of defects as low

as a few tens can be su�cient 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

B. Pellegrini / Microelectronics Reliability 40 (2000) 1775±1780 1779

measured is due to the inevitable existence in the con-

ducting media of defects, which have arbitrarily large

and distributed relaxation times and have random lo-

calisation 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 exis-

tence.

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, di�erent from the

current, which are sensitive to the ¯uctuations of the

carrier number, and, moreover, in general, to any sys-

tem, even di�erent from physical ones, in which a few or

many independent traps of elements that are character-

ised 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.

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