ARTICLE IN PRESS

0378-4371/$ - se

doi:10.1016/j.ph

�CorrespondE-mail add

(R. Brederlow)

Physica A 362 (2006) 277–288

www.elsevier.com/locate/physa

Novel analytical and numerical approach to modelinglow-frequency noise in semiconductor devices

Roberto da Silvaa,�, Gilson I. Wirthb, Ralf Brederlowc

aDepartamento de Informatica Teorica, Instituto de Informatica, Universidade Federal do Rio Grande do Sul, Porto Alegre, Brazil, RSbUniversidade Estadual do Rio grande do Sul, Guaıba, Brazil, RScInfineon Technologies, Corporate Research, Munich, Germany

Received 29 September 2005

Available online 28 November 2005

Abstract

In this work, based on robust experimental results, we derive a novel theoretical probabilistic approach to model the

low-frequency noise power Sð f Þ in semiconductor devices. Using the proposed approach we obtained, analytically, the

average of the integrated noise power W p ¼R f H

f LSð f Þdf as a function of the frequency bandwidth ½ f L; f H �, showing that

hW pi / ln2ð f H=f LÞ. The second moment and the relative error in W p are also calculated. A numerical fit for the relative

error was performed, showing that lower and upper bound estimates to noise power depend only on the ratio f H=f L.

r 2005 Elsevier B.V. All rights reserved.

Keywords: Numerical modeling; 1/f noise; CMOS devices

1. Introduction

Low-frequency (LF) noise in semiconductor devices is a performance limiting factor in analog and radiofrequency (RF) circuits. Therefore its understanding, modeling and control is a key point to the design ofhigh-performance analog and RF products. Recent experimental works have shown that the LF-noiseperformance of modern small-area MOS devices is dominated by random telegraph signal (RTS) noise [1,2].

This noise arises from the capture and subsequent emission of charge carriers at discrete trap levels near theSi–SiO2 interface. Noise performance may strongly vary between different devices onto one chip, andmoreover even between different operation points of a single device. Although LF-noise has deserved greatattention, no detailed theoretical statistical model has been proposed so far.

The strong variations in noise performance of CMOS devices pose great challenges to the design for highyield of low-noise analog and RF circuits in advanced CMOS technologies, since active device area oftransistors used in these circuits will become smaller and smaller with each new technology generation.

e front matter r 2005 Elsevier B.V. All rights reserved.

ysa.2005.11.014

ing author. Tel.: +5551 3316 7772; fax: +5551 3316 7308.

resses: rdasilva@inf.ufrgs.br (R. da Silva), gilson-wirth@uergs.edu.br (G.I. Wirth), ralf.brederlow@infineon.com

.

ARTICLE IN PRESSR. da Silva et al. / Physica A 362 (2006) 277–288278

In this paper our objective is to supply a theoretical statistical modeling of LF noise in a transistor. For thispurpose, it is important to understand its origin using a simple model. However, the model has to reproducethe experimental data and properly model the physical phenomena involved. In this work, the transistor ismodeled by the parallel plate capacitor composed by the polysilicon or metal gate electrode, the gate oxide(insulator) and the channel inversion layer as can be seen at Fig. 1. Under voltage, a laminar current appearson the inversion layer. The gate oxide ðSi–SiO2Þ has many imperfections, represented by traps distributed in allextension of the volume of the insulator. The traps are spatially distributed according to a Poisson distribution(this also is strongly corroborated by experimental results [1–3]). A good analogy for that is to see as driedgrapes are distributed in a bread.

The observed noise in the laminar current originates from the interaction between traps in the insulator andelectrons in the inversion layer. Each time an electron is ‘‘captured’’ by a trap jointly with the subsequentemission, one current fluctuation appears in the laminar current [4,5]. The behavior of the alternating‘‘captures and emissions’’ produce a noise in the current, that can be seen in an experimental plot of thecurrent of the device as a function of the time (see Fig. 2).

In this paper a novel theoretical approach, initially proposed by the same authors in Ref. [6], is extended todescribe and include all theoretical details to modeling the statistical properties of the LF noise power of the

Inversion Layer

Trap

oxide (gate)

Capture + emission Eelectron

Fig. 1. Model to device: Insulating layer (Si–SiO2) and channel inversion layer. The traps in the insulating layer interact with the electrons,

generating the noise in the current. For the sake of simplicity, the upper electrode (polysilicon gate) is not shown.

5.0 5.1 5.2 5.3 5.41.15

1.20

1.25

1.30

1.35

1.40

1.45

I(t)

/10-7

(A)

t (s)

Fig. 2. Plot of current as a function of time. The discrete current fluctuations correspond to capture and emission of a electron of the

inversion layer (electrons sea) by a particular trap.

ARTICLE IN PRESSR. da Silva et al. / Physica A 362 (2006) 277–288 279

laminar current. To achieve this goal, analytical and numerical rigorous calculation are performed. For thispurpose some robust experimental hypothesis was used to make the calculations in the paper. The schedule ofthe paper follows 3 main parts:

1.

Define the approach and to measure its implications in the theory of the MOS devices. 2. Using the approach to explore some known results to show the consistence of the novel ideas employed. 3. Exploring the ideas in new and precise theoretical estimation of noise in transistors.

We have obtained rigorous estimates to the average, upper and lower bound of the integrated noise powerin MOS transistors. The results are expressed as simple functions of the ratio between the upper and lowerlimit of the circuit frequency bandwidth.

In Section 2, we present the novel statistical modeling approach used to describe the noise power. Here, weshow that the 1=f behavior in noise performance follows naturally as a consequence from the modelingapproach. Following in Section 3, we show results and estimates to the average noise power, as well asanalytical calculation of the variance and the relative error in this quantity. In this same section, numericalresults for simple modeling of the relative error in the integrated noise power are derived. Some suitable fits aretested and the error is expressed as a Boltzmann function in a log–log scale, using a single variable, the ratiobetween the upper and lower limit of the circuit frequency bandwidth.

Section 4 with summary and conclusions finishes the paper, compiling the main results with short finalcomments.

2. The two important steps of the novel approach to modeling the noise power in a transistor

2.1. Step 1—Lorentzian spectra and the superposition over many traps

Supported by experimental results (see for example Ref. [7]), we suppose that the noise power due a singletrap, is distributed according to a Cauchy distribution in the frequency f written as

Sið f Þ ¼ A2i

1

f i

1

1þ ð f =f iÞ2,

where f i defines the corner frequency of the Lorentzian spectrum produced by the trap i. Here Ai areamplitudes, with i ¼ 1; . . . ;Ntr, where Ntr is the number of traps.

The superposition over all traps is calculated to give an expression to the total noise power, after theequation

Sð f ; f 1; . . . ; f Ntr;A1; . . . ;ANtr

Þ ¼XNtr

i¼1

Sið f Þ ¼XNtr

i¼1

A2i

1

f i

1

1þ ð f = f iÞ2.

In the next section a probabilistic formulation to noise power will be developed. And its implicationsfollows in the future sections of the paper.

2.2. Step 2—a probabilistic formulation for the problem

Let us suppose that f f igNtr

i¼1 are random variables independent and identically distributed (i.i.d.) andstatistically independent of another set of (i.i.d.) random variables fAig

Ntr

i¼1. Then we can write the jointprobability density function (jpdf) as

Pðf f igNtr

i¼1; fAigNtr

i¼1Þ ¼YNtr

i¼1

P1ð f iÞP2ðAiÞ, (1)

where P1ð f iÞ is the pdf of corner frequency f i and P2ðAiÞ is the pdf of amplitude Ai.

ARTICLE IN PRESSR. da Silva et al. / Physica A 362 (2006) 277–288280

The corner frequencies are distributed according to f i ¼ 10ci [3], where ci is uniformly distributed. If ci isrestricted to the interval ½log10 f min; log10 f max�, the distribution over ci is as follows:

fðciÞ ¼log�1

f max

f min

� �if log10 f minpciplog10 f max;

0 else;

8><>:

obtained by the normalization condition, i.e.,R log10 f max

log10 f minfðciÞdci ¼ 1.

Being the probability of f i 2 ½ f i; f i þ Df i� equal to the probability of ci 2 ½ci; ci þ Dci�, we must haveP1ð f iÞdf i ¼ fðciÞdci, what provides a simple formulae

P1ðf iÞ ¼ Cf �1i ,

where C ¼ ½lnð f max=f minÞ��1.

On the other hand, the distribution for the Ai’s, is more complex and experimental evidences not yet bear inmind a suitable modeling of this parameter. However, it is important to say that the formulation here deriveddoes not require explicit knowledge about the distribution over amplitudes (Ai’s). Nevertheless, investigativeexperiments should be better explored. In Section 3.1, we perform, using the approach here developed, theanalytical evaluation of average of the noise power S, over the distribution given by (1), corroborating theknown expected behavior 1=f to noise power, as a form to verify our approach. In the following section weobtain the standard deviation and consequently the relative error, analytically too. Similar results are obtainedfor the integrated noise power, Section 3.2. Finishing the section, in Section 3.3, numerical explorations andmultiple fits are experimented in analysis of the integrated noise power as a function of ratio ð f L=f H Þ betweenthe extremes of the bandwidth ½ f L; f H �.

3. Results

3.1. The 1=f noise—an analytical revisiting and new estimates to relative error in the noise power

First of all, we analyze the noise power Sð f Þ, using the statistical approach developed in this paper, toobtain some analytical results, which was only explored from an experimental point of view, nowadays. Themain contribution of this section is the analytical derivation of the average and variance of noise power ðSÞ,according to the model of the transistor as depicted in Fig. 1. This analytical approach has motivated theauthors for new investigations via Monte Carlo simulations and experiments for Sð f Þ [8].

We will show that the noise behaves as inverse to frequency, showing the compatibility of the knownexperimental data to our analytical approach.

By this time, let us write the average of S, keeping Ntr fixed, as described by

hðSÞNtri ¼

YNtr

i¼1

Z f max

f min

Z Amax

Amin

df i dAi P1ðf iÞP2ðAiÞ

" #Sð f ; f 1; . . . ; f Ntr

;A1; . . . ;ANtrÞ.

For that, a normalization is required:R f max

f minP1ðf iÞdf i ¼

RAmax

AminP2ðAiÞdAi ¼ 1, following that

hðSÞNtri ¼ C

XNtr

i¼1

hA2i i

Z f max

f min

df i

1

f 2i

1

1þ ðf =f iÞ2

¼NtrChA

2i

f

Z f max

f min

df i

1

1þ ðf i=f Þ2

¼NtrChA

2i

ftan�1

f max

f

� �� tan�1

f min

f

� �� �. ð2Þ

ARTICLE IN PRESSR. da Silva et al. / Physica A 362 (2006) 277–288 281

Evaluation of the average over Ntr which is a Poisson random variable with parameter l ¼ N, leads to

hSi ¼X1

Ntr¼0

hSiNtr

NNtre�N

Ntr!

¼ChA2i

ftan�1

f max

f

� �� tan�1

f min

f

� �� � X1Ntr¼0

NNtre�N

Ntr!Ntr

¼ChA2i

ftan�1

f max

f

� �� tan�1

f min

f

� �� �N.

The average of the number of traps N is proportional to the active device area of the transistor described bythe WL dimensions (see Fig. 1 again) and equal to

N ¼ Ndec lnf max

f min

� �WL.

Here r ¼ ln 10 �Ndec is the trap density per unit area and frequency decade and N is the average of thenumber of traps with corner frequencies between f min and f max.

Looking at hSi at limits f max!1 and f min! 0, tan�1ð f max=f Þ ! p=2 and tan�1ð f min=f Þ ! 0 once fa0,we arrive on

hSi !pNdecWL

2fhA2i,

showing that the noise in this statistic approach supplies the expected behavior 1=f to noise power.Following the computations, other analytical estimates can be explored as for example the variance and

consequently, a estimate to relative error in S. Therefore, we also have to calculate the second moment hS2i,starting from

hðS2ÞNtri ¼

XNtr

i¼1

hA4i i

Z f max

f min

df i

P1ð f iÞ

f 2i ½1þ ð f =f iÞ

2�2

þXNtr

i¼1

XNtr

j¼1

iaj

hA2i ihA

2j i

Z f max

f min

Z f max

f min

df i df j

P1ð f iÞ

f i½1þ ð f =f iÞ2�

P1ð f jÞ

f j½1þ ð f =f jÞ2�

¼ CNtrhA4i

Z f max

f min

dx

x3½1þ ð f =xÞ2�2þ C2NtrðNtr � 1ÞhA2i2

Z f max

f min

dx

x2½1þ ð f =xÞ2�

!2

,

where hA4i i ¼ hA

4i and hA2i A2

j i ¼ hA2i ihA

2j i ¼ hA

2ihA2i ¼ hA2i2, which follows from the statistic independenceof the variables.

The first and the second integral are easily calculated, after the variable substitution u ¼ f =x and subsequentseparation in partial fractions,Z f max

f min

dx

x3½1þ ð f =xÞ2�2¼

ð f 2max � f 2

minÞ

2ð f 2þ f 2

maxÞð f2þ f 2

minÞ,

Z f max

f min

dx

x2½1þ ð f =xÞ2�¼

1

f½tan�1ð f max=f Þ � tan�1ð f min=f Þ�.

From the same procedures described above, our estimates can be calculated taking the average over Ntr thatis a Poisson random variable, obtaining

hS2i ¼ChA4ið f 2

max � f 2minÞN

2ð f 2þ f 2

maxÞð f2þ f 2

minÞþ

C2N2hA2i2

f 2tan�1

f max

f

� �� tan�1

f min

f

� �� �2

¼ChA4ið f 2

max � f 2minÞN

2ð f 2þ f 2

maxÞð f2þ f 2

minÞþ hSi2

ARTICLE IN PRESSR. da Silva et al. / Physica A 362 (2006) 277–288282

once that

X1Ntr¼0

NNtre�N

Ntr!NtrðNtr � 1Þ ¼ N2.

The variance of variable S (varðSÞ) can be written at limits f max!1 and f min! 0. Explicitly we have

varðSÞ ¼ hS2i � hSi2

¼ChA4ið f 2

max � f 2minÞN

2ð f 2þ f 2

maxÞð f2þ f 2

minÞ!

f min!0

f max!1

NdecWLhA4i

2f 2

resulting in a relative error (at limits f max!1 and f min ! 0) defined by eðSÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffivarðSÞ

p=hSi, as a function

of parameters of the transistor (W and L) and also of fourth and second moments of the noise amplitude A

(respectively hA2i and hA4i) and of the density of the traps (Ndec):

eðSÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2hA4i

p2NdecWLhA2i2

s.

It is interesting to note that the relative error is inversely proportional to square root of trap density,eðSÞ / N

�1=2dec , although

ffiffiffiffiffiffiffiffiffiffiffiffiffivarðSÞ

pis directly proportional to Ndec what is reasonable and consistent.

Other measures can be explored besides S and its error estimates. An interesting quantity to analyze is thenoise power integrated in a given frequency window ½ f L; f H �, where f Lpf H . This measure is important fromthe experimental point of view, since it supplies a suitable comparison with practical measures. Its analyticaldetermination remains a lack in the literature. In the next subsection, we calculate this not yet publishedquantity as well as the relative errors associated to it.

3.2. Noise power integrated over a window frequency

The noise power integrated over a bandwidth ½ f L; f H �, is defined as a marginal distribution

W pð f 1; . . . ; f Ntr;A1; . . . ;ANtr

Þ ¼

Z f H

f L

Sð f ; f 1; . . . ; f Ntr;A1; . . . ;ANtr

Þdf .

Such quantity measures the noise power in the circuit frequency bandwidth, which is an importantparameter in circuit design.

The theoretical mean of the integrated noise for Ntr fixed is defined as the expected value of W p over thedistribution (1). It is computed by means of

hðW pÞNtri ¼

Z f H

f L

dfYNtr

i¼1

Z f max

f min

df i

Z Amax

Amin

dAi P1ð f iÞP2ðAiÞ

" #Sð f ; f 1; . . . ; f Ntr

;A1; . . . ;ANtrÞ

¼ CXNtr

i¼1

hA2i i

Z f H

f L

df

Z f max

f min

df i

1

f 2i

1

1þ ð f =f iÞ2

¼ CNtrhA2i

Z f max

f min

df i

f i

tan�1f H

f i

� ��

Z f max

f min

df i

f i

tan�1f L

f i

� �" #

considering thatR f max

f minP1ð f iÞdf i ¼

RAmax

AminP2ðAiÞdAi ¼ 1.

Making the change of variable u ¼ f H=f i in the first integral and u ¼ f L=f i in the second one, we have

hðW pÞNtri ¼ CNtrhA

2i

Z f H=f min

f H=f max

du

utan�1 u�

Z f L=f min

f L=f max

du

utan�1 u

" #.

ARTICLE IN PRESSR. da Silva et al. / Physica A 362 (2006) 277–288 283

Observing the inequalities f L=f maxof H=f max and f L=f minof H=f min, after some cancellations, we have newintegration intervals, such that

hðW pÞNtri ¼ CNtrhA

2i

Z f H=f min

f L=f min

dutan�1 u

u�

Z f H=f max

f L=f max

dutan�1 u

u

" #. (3)

A Taylor expansion can be written for the two different cases of the convergence:

tan�1 u ¼

p2þX1n¼0

ð�1Þnþ1

ð2nþ 1Þu2nþ1if u41;

P1n¼0

ð�1Þnu2nþ1

ð2nþ 1Þif uo1:

8>>>><>>>>:

Observing that in Eq. (3) the first integral corresponds to the situation u41 and the second one to uo1, wededuce that

hðW pÞNtri ¼ CNtrhA

2ip2ln

f H

f L

� �þX1n¼0

ð�1Þn

ð2nþ 1Þ2

"

�f min

f H

� �2nþ1

�f min

f L

� �2nþ1

�f H

f max

� �2nþ1

þf L

f max

� �2nþ1 !#

.

At limits f min! 0 and f max !1 and averaging over Ntr we can conclude

hW pi fmin!0

fmax!1

¼p2

NdecWLhA2i lnf H

f L

� �.

This important result shows that the integrated noise behaves as W p / lnð f H=f LÞ. By the same way asbefore performed in this paper to obtain the relative error of the S, we can study the relative error associatedto the integrated noise power W p.

The second moment, keeping the Ntr fixed is computed as

hðW 2pÞNtri ¼

Z f H

f L

Sð f ; f 1; . . . ; f Ntr;A1; . . . ;ANtr

Þdf

!2* +

¼

Z f H

f L

Z f H

f L

hSð f ÞSð f 0ÞiNtrdf df 0

¼ NtrhA4iC

Z f H

f L

Z f H

f L

Z f max

f min

f i

½ f 2i þ f 2

�½ f 2i þ f 02�

df i

!df df 0

þNtrðNtr � 1ÞC2hA2i2Z f H

f L

Z f H

f L

Z f max

f min

df i

f 2i ½1þ ð f =f iÞ

2�

!

�

Z f max

f min

df j

f 2j ½1þ ð f =f jÞ

2�df j

!df df 0.

Again, by taking the average over the distribution in Ntr we can write

hW 2pi ¼ NhA4iC

Z f H

f L

Z f H

f L

Z f max

f min

f i

½ f 2i þ f 2

�½ f 2i þ f 02�

df i

!df df 0

þN2C2hA2i2Z f H

f L

Z f max

f min

dx

x2½1þ ð f =xÞ2�dxdf

!2

¼ NhA4iC

Z f H

f L

Z f H

f L

Z f max

f min

x

½x2 þ f 2�½x2 þ f 02�

dx

!df df 0 þ hW pi

2.

ARTICLE IN PRESSR. da Silva et al. / Physica A 362 (2006) 277–288284

Making u ¼ x2 and considering the partial fraction separation

1

ðuþ f 2Þ þ ðuþ f 02Þ

¼1

f 02 � f 2

1

ðuþ f 2Þ�

1

ðuþ f 02Þ

� �,

the integration over u leads to

hW 2pi ¼ NhA4iC

Z f H

f L

Z f H

f L

1

2ð f 02 � f 2Þlnð f 2

max þ f 2Þ

ð f 2min þ f 2

Þ

ð f 2min þ f 02Þ

ð f 2max þ f 02Þ

" #df df 0 þ hW pi

2.

On the limits f max!1 and f min! 0, we can conclude that

varðW pÞ ¼ hW2pi � hW pi

2

¼ NdecWLhA4i

Z f H

f L

Z f H

f L

lnðf 0=f Þ

ð f 02 � f 2Þdf df 0.

Leading to the expression

eðW pÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivarðW pÞ

phW pi

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4

p2NdecWL

s ffiffiffiffiffiffiffiffiffihA4i

hAi2

sGð f L; f H Þ, ð4Þ

where we define

Gðf L; f H Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiIð f L; f H Þ

ln2ð f H=f LÞ

s

with

Ið f L; f H Þ ¼

Z f H

f L

Z f H

f L

lnð f =f 0Þ

ð f 02 � f 2Þdf df 0.

At the present time, it is easy to show that the integral depends only of the ratio z ¼ f H=f L. For that, wemake a change of the variables according to u ¼ f =f L and w ¼ f 0=f L. This change of variables provides asimple Jacobian Jðu;wÞ ¼ 1=f 2

L. In this manner

Ið f L; f H Þ ¼

Z z

1

Z z

1

lnðu=wÞ

u2 � w2dudw ¼ IðzÞ,

it makes our analysis a little more simple, i.e., a reduction of the problem to one variable:

Gð f L; f H Þ ¼ GðzÞ ¼1

ln z

ffiffiffiffiffiffiffiffiIðzÞ

p.

We can conclude that eðW pÞ depends solely of the ratio z ¼ f H=f L. However, this must be better explored inour analysis. This will be made in the next subsection. We will analyze the dependency of GðzÞ numerically andlook for more suitable formulas for eðW pÞ as a function of z, by suitable numerical fits of GðzÞ.

3.3. Numerical results

In this section, we analyze numerically the function GðzÞ that exactly describes the relative error eðW pÞ. Firstof all, it is interesting to analyze the double integral IðzÞ. After an integration over w we have obtained a moreappropriate function, from a numerical point of view

IðzÞ ¼ �1

4

Z z

1

1

y2dilog

z

y

� �þ 2dilog

yþ z

y

� �þ 2 ln

z

y

� �ln

yþ z

y

� ��

þ 2dilogðyÞ þ ln2 y� 2dilogyþ f L

y

� �þ ln

1

y

� �ln

yþ 1

y

� ��dy, ð5Þ

where

dilogðxÞ ¼

Z x

1

ln t

1� tdt.

ARTICLE IN PRESSR. da Silva et al. / Physica A 362 (2006) 277–288 285

Using simple programming in maple, we perform numerically GðzÞ, calculating the integral (5), sweeping thevalue of z in the interval I � ½zmin; zmax�. A curve for GðzÞ as a function of z was obtained, drawn as the curvemarked with circles in Fig. 3, where zmin ¼ 2 and zmax ¼ 105. In order to span the largest range possible,satisfying the numerical limitation, since a large number of integrals must be performed, we have partitionedthe interval in five subintervals, I i ¼ ½2� 10i�1; 10i�, indexed by i ¼ 1; 2; 3; 4; 5, i.e., I1 ¼ ½2; 10�, I2 ¼ ½20; 100�,I3 ¼ ½200; 1000�, I4 ¼ ½2000; 10 000� and I5 ¼ ½20 000; 100 000� with subdivisions ðDzÞi ¼ 10i, respectively.Hence, we fit GðzÞ as simple curves observing the error for each curve family fitted.

We start numerically analyzing exponential fits. More precisely, in normal scale, we try a family ofexponential curves of order n, defined by

GneðzÞ ¼ A0 þ A1 expð�z=t1Þ þ A2 expð�z=t2Þ þ � � � þ An expð�z=tnÞ.

For the purpose of describing the function as a simple formulae, we only analyze the first three ordersn ¼ 1; 2; 3. Here the fit error was defined simply as the sum of squared deviations d ¼

PNi¼1ðG

neðzÞ � GðzÞÞ2,

where the number of points used was N ¼P5

i¼1jI ij ¼ 45.Clearly, a simple exponential decay (order 1) is not sufficient to model our curve, as can be seen in

Fig. 3. From this same figure, one can conclude that n ¼ 3 is a suitable fit and certainly this becomes better(less error) when the order grows (see Table 1). However, simpler curve fits may be found and have to beexplored.

In a ln–ln scale, polynomial and sigmoid functions can become good candidates, due to two reasons; first,because, as said before, there is no good fit with a simple exponential, second, such curves have a very simpleform in a ln–ln scale.

After the exponential tests, we restart with a polynomial family of curves. In this case the fit is defined as

lnGnpðzÞ ¼ a0 þ a1 ln zþ a2ðln zÞ2 þ � � � þ anðln zÞn,

1000 10000 1000000.40

0.45

0.50

0.55

0.60

0.65

Results of the G(fH/fL) obtaineddirectly from integration

order 3

order 2

order 1

log(fH/fL)

G(f

H/f

L)

Fig. 3. Exponential fit. The exponentials of order 1, 2, 3 are plotted in comparison with the theoretical curve.

Table 1

Fit errors for different orders of exponencial fits

n A0 A1 A2 A3 t1 t2 t3 Fit error

1 0.462(6) 0.196(9) 0 0 4:8ð8Þ � 102 0 0 2:87� 10�2

2 0.437(3) 0.147(5) 0.106(4) 0 69(6) 6:1ð7Þ � 103 0 2:71� 10�3

3 0.432(2) 0.102(3) 0.072(3) 0.098(4) 24(2) 1:5ð2Þ � 104 5:2ð5Þ � 102 4:84� 10�4

ARTICLE IN PRESSR. da Silva et al. / Physica A 362 (2006) 277–288286

where n ¼ 1; 2; 3 corresponds, respectively, to linear, quadratic and cubic fit. The resulting fit GnpðzÞ would then

be written as

GnpðzÞ ¼ Czða1þa2 ln zþa3 ln

2 zþ���þan lnn�1 zÞ

with the constant written as a function of a0, in accordance to C ¼ expða0Þ. In the linear fit case we have asimple power-law

G1pðzÞ ¼ Cza1 .

The polynomial fit was performed for the cases n ¼ 1; 2; 3; 4 for a error comparison. The fits can be seen inFig. 4. The errors corresponding to respective orders used can be seen in Table 2.

It is important to say that for the first order (linear approximation) the polynomial behavior is better in theln–ln scale than the exponential in normal scale. This corroborates much more a power law behavior than anexponential one for low orders and is desired from a practical and technological point of view.

At this point a little of intuition is necessary. Looking at the curve in ln–ln scale in Fig. 4, we observe twosegments of the curve breaking the power-law behavior, the first segment at the left and the other one at theright part of the plot. This empirical analysis suggests that a good function to fit the points could be a sigmoid.

For this purpose we consider the 4-parameters Boltzmann function

yðxÞ ¼A1 � A2

1þ exp½ðx� x0Þ=d�þ A2,

1080 2 4 6 12-1.0

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

Theoretical

linear fit

quadratic fit

cubic fit

fourth order fit

lnG

(fH

/fL)

ln(fH/fL)

Fig. 4. Polynomial fit. The exponentials of order 1, 2, 3, 4 are plotted for comparison to the theoretical curve.

Table 2

Error fits for different orders of polynomial fits

n Error fit

1 3:6� 10�3

2 1:7� 10�3

3 3:8� 10�4

4 3:1� 10�5

ARTICLE IN PRESS

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

ln G

(z)

ln z

Theoretical

sigmoidal fit (Boltzmann function)

0 2 4 6 8 10 12

Fig. 5. Sigmoidal fit. The Boltzmann function was calculated in comparison to the theoretical curve.

R. da Silva et al. / Physica A 362 (2006) 277–288 287

where A1, A2, x0 and d are fit parameters. Considering y ¼ lnG and x ¼ ln z, and making x0 ¼ ln z0; we arriveat

lnG ¼A1 � A2

1þ exp½ðln z� ln z0Þ=d�þ A2

¼A1 � A2

1þ exp½lnðz=z0Þ1=d�þ A2

¼A1 � A2

1þ ðz=z0Þ1=dþ A2

with

GðzÞ ¼ K expA1 � A2

1þ ðz=z0Þ1=d

" #,

where K ¼ expðA2Þ.After performing the Boltzmann fit, a fit error equal to 3:15� 10�4 is obtained. This is better than all

exponential fits (low orders) performed in this work and better than the polynomial fit up to n ¼ 3, showing agood approximation for a non-trivial function. The values found in the fit are A1 ¼ �0:12ð2Þ, A2 ¼ �1:00ð1Þ,x0 ¼ 5:1ð1Þ and d ¼ 4:1ð2Þ. A plot derived from this fit can be seen in Fig. 5.

4. Summary and conclusions

In this work, a new probabilistic approach is introduced as an alternative to understand the low-frequencynoise and its implications in microelectronic devices. First of all, the approach was explored to find relationsthat show coherence between modeling approach and experimental data. Analytical expressions for noisebehavior were provided. A formulation for the dependence of noise power integrated over frequency window(bandwidth) ½f L; f H � as a function of frequency ratio z ¼ f H=f L was derived. This result shows that for anyfrequency range region the mean noise as well as the relative error are invariant under a scale transformationf H ! cf H and f L ! cf L, where c is a constant.

New estimates to lower and upper bound of relative error were calculated as a function of knownparameters of the device. Numerical calculations were performed to fit a function that properly describes therelative error in the noise. From the explored curve fits, the best fits were chosen by looking at the error of each

ARTICLE IN PRESS

Table 3

Summary of results, for A1 ¼ �0:12ð2Þ, A2 ¼ �1:00ð1Þ, x0 ¼ 5:1ð1Þ, d ¼ 4:1ð2Þ, K ¼ expðA2Þ and z0 ¼ expðx0Þ

hSi eðSÞ ¼ hSi=ffiffiffiffiffiffiffiffiffiffiffiffiffivarðSÞ

phW pi hW pi=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivarðW pÞ

ppNdecWL

2fhA2i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2hA4i

p2NdecWLhA2i2

sp2

NdecWLhA2i lnf H

f L

� �K

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4

p2NdecWL

r ffiffiffiffiffiffiffiffiffihA4i

hAi2

sexp

A1 � A2

1þ ðf H=f L=z0Þ1=d

" #

R. da Silva et al. / Physica A 362 (2006) 277–288288

performed fit. In accordance to the classification criteria, a summary of important formulas established in thispaper are listed in Table 3.

Acknowledgements

Gilson Wirth thanks to CNPq and Fapergs and Roberto da Silva thanks to CAPES and CNPq for partialfinancial support.

References

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