Long-term Correlations and 1/f^alpha Noise in the Steady States of Multi-Species Resistor Networks

1.2 MB·English

by  C. Pennetta

Long-term Correlations and 1/fα Noise in the Steady States
of Multi-Species Resistor Networks
C. Pennetta, E. Alfinito, and L. Reggiani
Dipartimento di Ingegneria dell’Innovazione, Universita` del Salento and Consorzio Nazionale
∗
Interun. per le Scienze Fisiche della Materia (CNISM), Via Arnesano, I-73100, Lecce, Italy.
7 (Dated: February 6, 2008)
0
We introduce a multi-species network model which describes the resistance fluctuations of a
0
resistorinanon-equilibriumstationarystate. Moreprecisely,athinresistorcharacterizedbya1/fα
2
resistance noise is described as a two-dimensional network made by different species of elementary

n resistors. Theresistorspeciesaredistinguishedbytheirresistancesandbytheirenergiesassociated
a withthermallyactivatedprocessesofbreakingandrecovery. Dependingontheexternalconditions,
J
stationarystatesofthenetworkcanariseasaresultofthecompetitionbetweentheseprocesses. The

9 properties of the network are studied as a function of thetemperature by Monte Carlo simulations
2 carried out in thetemperaturerange 300÷800 K.Atlow temperatures, theresistance fluctuations
display long-term correlations expressed by a power-law behavior of the auto-correlation function
] andbyavalue≈1oftheα-exponentofthespectraldensity. Onthecontrary,athightemperatures
i
c theresistancefluctuationsexhibitafiniteandprogressivelysmallercorrelationtimeassociatedwith
s a non-exponentialdecay of correlations and with a value of the α-exponent smaller than one. This
-
l temperaturedependenceoftheαcoefficientreproducesqualitativelywelltheexperimentalfindings.
r
t
m PACSnumbers: 05.40.-a,72.70.+m,61.43.-j,64.60.Ak
Keywords: Resistornetworks,fluctuation phenomena, 1/fnoise,disorderedmaterials
.
t
a
m
I. INTRODUCTION resistors(highresistivityresistors)andabiasedrecovery
- process(whichrestoresthe“regular”resistorswith“nor-
d
mal” resistivity) [19, 24]. The SBRN model provides a
n
Sincealongtimetheanalysisoftheresistancefluctua-
o gooddescriptionofmanyfeaturesassociatedwiththere-
tionshasturnedouttobeaverypowerfultoolforprobing
c sistance fluctuations in nonequilibrium stationary states
[ various condensed matter systems [1, 2] and, in partic- of disordered resistors [6, 12, 19, 20, 25, 29], with resis-
ular, disordered materials [1, 2, 3, 4, 5] like conductor- tor trimming processes [30] and with agglomerationand
1
insulator composites [6], granular systems [7], porous [8]
v thermal instability phenomena in metallic films [31, 32].
or amorphous materials [9, 10, 11] and organic conduct-
2 Further successful applications of the model concern the
1 ing blends [12]. Therefore, many experimental and the- study of electrical breakdown phenomena in composite
7 oretical investigations have been devoted to study the and granular materials [6, 19, 20, 25], including electro-
1 properties of the resistance noise as a function of tem-
migration damage of metallic lines [16, 26, 33].
0 perature, bias strength and of the main material param-
However,thismodelonlyappliestosystemscharacter-
7
eters [1, 2, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17].
0 ized by a Lorentzian noise [1], i.e. by a power spectral
/ Only few studies have considered the behavior of dis- density of the resistance fluctuations scaling as 1/f2 at
t ordered materials over the full range of bias values, by
a high frequencies and becoming flat at frequencies below
m analyzing under a unified approach the linear regime the so called corner value f . The existence of a cor-
c
and the nonlinear regime up to the threshold for elec-
- ner frequency in the spectrum corresponds in the time
d trical breakdown [6, 18, 19, 20], even though this kind domain to the existence of a well defined characteristic
n of studies can provide significant insights about basic
time, τ (correlation time), associated with the decay of
o properties of the response of nonequilibrium systems
the correlations in the resistance fluctuations [1]. In the
c [6, 17, 18, 19, 20, 21, 22, 23].
: particularcaseofLorentziannoisethisdecayisexponen-
v
In previous works we have introduced [24] and de- tial, i.e. the auto-correlation function of the resistance
i
X veloped the Stationary and Biased Resistor Network fluctuations is given by [1]:
r (SBRN)model[19,20,25,26]whichisbasedonaresistor C (t)≡hδR(t)δR(t+τ)i=h(∆R)2iexp(−t/τ) (1)
a networkapproach[3, 4, 5] and which allowsthe study of δR
the electrical conduction and noise of disorderedmateri- where h(∆R)2i is the variance of the resistance fluctu-
alsoverthefullrangeofbiasvalues. Thismodeldescribes ations. On the other hand, it is well known [1] that
a disordered conducting material as a resistor network at low frequencies many condensed matter systems dis-
whose structure is determined by a stochastic competi- play 1/f resistance noise, i.e. a spectral density scal-
tion between a biased percolation [27, 28] of “broken” ing over several frequency decades as 1/fα, with α ≈ 1.
The amazing presence of 1/f noise in a large variety of
natural phenomena represents a puzzling problem not
yet fully solved. Therefore, a huge quantity of mod-
∗Electronicaddress: [email protected] els providing signals with 1/f noise have been proposed
2
[1, 2, 7, 9, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43]. In model [1], we take the breaking and recovery activation
particular, the ubiquity of the 1/f noise has given rise energies distributed in a broad range of values. A choice
to many attempts to explain it in terms of a univer- physically justified in disordered systems, like granular
sal law. One simple way to obtain a 1/f spectrum is and amorphous materials, composites, etc. where the
by superimposing a large number of Lorentzian spectra orientational disorder present inside these materials can
withanappropriatedistributionofthecorrelationtimes: give rise to different energy barriersfor the electron flow
D(τ) ∝ 1/τ, distributed over many orders of magnitude along the different conducting paths [1, 3, 4, 34]. As a
of τ values [1]. In some cases, the distribution of the result, the resistance fluctuations of the MSN exhibit a
correlation times of the elementary processes contribut- 1/fα spectrum, where the value of α depends on the ex-
ing to the 1/f spectrum can be derived from the dis- ternal conditions (temperature and electrical bias). In
tributionofsomevariableonwhichthecorrelationtimes particular, in this paper, we discuss the states of a MSN
themselvesdepend,asinthecaseofthepioneeringworks asafunctionofthetemperatureandinthelinearregime
of Mc Whorter [1] and Dutta and Horn [1]. In particu- of the external bias, when Joule heating effects are neg-
lar, these authorsproposed[1] that the originof the 1/f ligible.
noise can be attributed to a thermally activated expres- The paper is organised as follows: in Sect. II we il-
sion of the correlation times, associated with a broad lustrate the MSN model, while in Sect. III we report
distribution of the corresponding activation energies, an the results by focusing in particular on the role played
assumption physically plausible for many systems. On by the temperature on the fluctuations properties of the
the other hand, many other important contributions to multi-species network. Finally, in Sect. IV we draw the
the understanding of the 1/f noise have been advanced conclusions of this study.
inthelasttwentyyears,whichhaveshownthatthepres-
ence of a 1/f spectrum can also arise from other basic
II. MODEL
reasons [2, 7, 9, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43].
The dynamical random network model [34] and other
spin-glasses models [1, 34] provide a good explanation According to the SBRN model [19, 20] a conducting
of the 1/f noise in conducting random magnetic mate- thin film with granular structure is described as a two-
rials. Dissipative self-organised criticality (SOC) mod- dimensional square-lattice resistor network [3, 5]. For
els, started from the famous work of Bak, Tang and simplicity here we study a network N × N (where N
Wiesenfeld[35],clarifytheoriginof1/f spectraincertain is the linear size of the network), even if networks with
dissipative dynamical systems naturally evolving into a a different geometry can be considered [26]. The net-
critical state. Avalanche models [36], clustering models workisbiasedbyanexternalconstantcurrent,I,applied
[37] and percolative models [7, 9, 38, 39, 40] represent through perfectly conducting bars placed at the left and
other relevant classes of theoretical approaches explain- righthandsidesanditliesonaninsulatingsubstrateata
ing the appearence of 1/f noise in a variety of systems. giventemperatureT,whichactsasathermalbath. Each
Thus, the conclusion, now largely accepted in the liter- resistor can be in two different states: i) regular, corre-
ature [1, 2, 40], is that a unique, universal origin of 1/f sponding to a resistance rn =r0[1+αT(Tn−T)], where
does not exist, even though classes of systems can share α is the temperature coefficient of the resistance and
T
a common basic origin of 1/f noise. T is the local temperature and ii) broken, correspond-
n
ing to an effectively “infinite” resistance, r = 109r .
The aimofthis paperisto presentanew modelwhich OP n
Resistors in the broken state will be called defects. The
includes and extends to systems characterized by 1/fα
temperature T is expressed as [28]:
noise the main feature of the SBRN model, i.e. the n
stochastic competition between thermally activated and D
2 2 2
T =T +A[r i + (r i −r i )] (2)
biased processes of breaking and recoveryof the elemen- n n n N l l n n
neig X
taryresistorsofanetwork[19]. Actually,thisfeaturehas l
turned out to be essential for the success of the SBRN where D = 3/4, the sum is performed over the N
neig
modelinthedescriptionofthewidephenomelogyassoci- nearest neighbors of the n-th resistor and i is the cur-
n
ated with the resistance fluctuations of disordered resis- rent flowing in it. The above expression of T takes into
n
tors, in both nonequilibrium stationary states and non- accountthe Joule heating of the n-thresistor[27, 28, 44]
stationaryfailurestates[6,12,26,29,30,31,32]. Tothis and the thermal exchanges with its neighbors [28]. The
purpose,herewestudyanetworkmadebyseveralspecies importance of both these effects is controlled by the pa-
ofresistors,whereeachspeciesischaracterizedbyaresis- rameter A, thermal resistance, which describes the heat
tancevalueandbyapairofactivationenergiesassociated coupling of each resistor with the substrate [19, 27, 44].
with the breaking and the recovery processes. For this WenotethatbyadoptingEq.(2)weareassumingforsim-
reasonwecallthismodel“multi-speciesnetwork”(MSN) plicity an instantaneous thermalization of each resistor
model. The states, either stationaryor nonstationary,of and we are neglecting time dependent effects discussed
the MSN resultfromthe stochasticcompetitionbetween in Ref. [44].
theseprocessesofbreakingandrecoveryinvolvingtheel- In the initial state of the network all the resistors are
ementary resistors. In analogywith the Dutta and Horn taken to be identical: rn ≡ r0. The SBRM model then
3
assumes that the evolution of the network is determined of resistance values: r0,i ∈ [rmin,rmax], while other op-
by the competition between two biased stochastic pro- tionsareaswellreasonable. Welimitthepresentstudyto
cesses: one of breaking and the other of recovery. The thelinearregimeintheexternalbias,whenJouleheating
formerprocessconsistsinthetransitionofanelementary effects are negligible, i.e. T ≡ T, ∀n and consequently
n
resistorfromtheregulartothebrokenstateanditoccurs rn,i ≡r0,i, ∀n and ∀i. Alternatively, we can say that we
with a thermally activated probability [27]: study a system characterized by negligibly small values
of the thermal resistance and of the resistance temper-
WDn =exp(−ED/kBTn) (3) ature coefficients: A = 0 and αT,i = 0, ∀i. We stress
that the linear regime condition corresponds to uniform
thelatterprocessconsistsinthereversetransitionandit probabilities of breaking and recovery for the resistors
occurs with probability [19, 24]: belonging to each given species [28, 45]. Therefore, the
resultsthat wewillreportinSect. III directly generalize
WRn =exp(−ER/kBTn) (4) the results of Ref. [45] to systems characterizedby 1/fα
resistance noise. The limitation to the linear regime of
where E and E are the activation energies of the two
D R currentshasbeenadoptedhereforsimplicityandalsobe-
processesandk theBoltzmannconstant. Thetimeevo-
B causewe aimto focus onthe temperaturedependence of
lution of the network is then obtained by Monte Carlo
the correlation properties of the resistance fluctuations.
simulations which update the network resistance after a
However, we underline that there is no special difficulty
sweep of breaking and recovery processes, according to
in extending this study to the nonlinear regime in the
an iterative procedure detailed in Ref. [19]. The se-
applied bias. In this respect, it must be noted that we
quence of successive network configurations provides a
expect that all the other significant features associated
resistancesignal,R(t),afteranappropriatecalibrationof
with the SBRN model (existence of a threshold current
the time scale. Then, depending onthe stress conditions for breakdown [19, 20], dependence of hRi and h(∆R)2i
(I and T) and on the network parameters (size, activa-
on the bias [19], times to failure distribution [26], non-
tionenergies,r0,αT andA),thenetworkeitherreachesa Gaussianityofthe resistancefluctuations [20,25], roleof
steadystateorundergoesanirreversibleelectricalfailure
the networkgeometry[26], etc.) remainunchangedfor a
[19, 20, 25, 26]. This latter possibility is associated with
MSN network.
the achievement of the percolation threshold, p , for the
c The values of the activation energies E and E
fractionofbrokenresistors[5]. Therefore,foragivennet- D,i R,i
for the different resistor species have been chosen as fol-
work at a given temperature, a threshold current value,
lows. In Ref. [45] it was proved that for a single-species
I , exists above which electrical breakdown occurs [19].
B network, when the average fraction of defects hpi is suf-
For values of the applied current below this threshold,
ficiently far from the percolation threshold (p = 0.5 for
the steady state of the network is characterized by fluc- c
aN×N square-latticenetwork[5]), the correlationtime
tuationsofthefractionofbrokenresistors,δp,andofthe
of the resistance fluctuations is approximated by:
resistance,δR,aroundtheirrespectiveaveragevalueshpi
and hRi. 1 W W
R D
=W + = (5)
The network described above, as provided by the D
τ (1−W ) hpi
R
SBRN model, is made by a single species of resistors,
all characterized by the same values of r0, αT, ED and where WD and WR are the thermal activated proba-
E . Thus, we can speak of a “single-species network”. bilities defined by the Eqs. (3)-(4) and calculated for
R
The disorder inside this network only arises from the T = T. In other terms, Eq. (5) implies that at a given
n
fact that some resistors can be in the broken state. As temperature,bothhpiandτ onlydependonE andE .
D R
a consequence, current and temperature are not homo- Thisequationcanthusbe invertedto determineE and
D
geneously distributed [26, 27], thus implying inhomoge- E as functions of hpi and τ. These considerations can
R
neousprobabilitiesofbreakingandrecoveryand,through be generalized to the case of a multi-species network by
the temperature coefficient α , different resistance val- giving the criterionfor a convenientchoice of the activa-
T
ues of the regular resistors. Furthermore, by analogy tionenergies. Tothispurpose,firstwechooseareference
with generation-recombination noise in a two-levels sys- value of the substrate temperature, T , and a set of
ref
tem, this model describes a system characterized by a values of the averagedefect fraction, hp i, corresponding
i
single time scale and thus by Lorentzian noise. To over- tothedifferentresistorspecies(wherep ≡N /N
i brok,i tot,i
come the above limitations here we consider a network withi=1,...,N ). Second,byconsideringthecorrela-
spec
made by N species of resistors, i.e. a “multi-species tion times, τ , associatedwith the fluctuations [46] of p ,
spec i i
network”(MSN). Eachspecies is characterizedby differ- we choose the values of the activation energies E and
D,i
ent values of the parameters: r0,i, αT,i, ED,i and ER,i ER,i in such a way to obtain a logarithmic distribution
with i = 1,...N . Then, an “identity” (an index i) is of τ inside a sufficiently wide interval: τ ∈[τ ,τ ].
spec i i min max
attributedatrandomtoeachresistor. Intheinitialstate The different τ are then attributed to the different re-
i
of the network the species are taken to be present with sistor species by adopting the criterion that increasing
the samenumberofresistors. As areasonablechoice,we values of τi are paired with increasing values of r0,i. Al-
havetakenr0,i uniformlydistributedinsideagivenrange ternative choices are of course possible.
4
Alltheresultsreportedhereareobtainedbyapplyinga 3.0
2.1
biascurrentI =1.0(A)toanetworkofsizes75×75made
byNspec =15resistorspeciesandbychoosingrmin =0.5 1.9
Ω and r = 1.5 Ω. Furthermore we take: T = 300 2.5
max ref )
K, hpii ≈ 0.25 ∀i, τmin ≈ 2 and τmax ≈ 5×105 (where Ωe ( 1.75.98 5.99 6.00
times are expressed in units of iterative steps). We un- c
n 2.0
derlinethatthechoicemadeaboveforthevaluesofTref, ta
hpii, τmin and τmax, determines once for all (i.e. for all sis
thetemperatures)thesetofactivationenergiesE and e
D,i R 1.5
E . In the present case E and E are in the range
R,i D,i R,i
58÷375 meV and 37÷346 meV, respectively. In fact,
thevaluesofED,i andER,i,togetherwithr0,i andNspec, 1.0
determine the intrinsic properties of the material under 0 5x105 1x106
test. Therefore,whenperformingasimulationattemper- Time (arb. units)
atures different from T , the values of τ recalculated
ref i
fromEq. (5) are the input parametersofthe simulation,
FIG.1: Resistanceevolutionofamulti-speciesnetwork(MSN
while the values of hp i only serve as a check of consis-
i model) calculated at 300 K. The resistance is espressed in
tencywiththoseobtaineddirectlyfromtheoutputofthe
Ohmandthetimeiniterativesteps. Theinsethighlightsthe
simulation. As a general trend, by increasing the tem-
resistance fluctuations on an enlarged time scale. In particu-
perature the range of values of τi becomes more narrow lar, the time unitsare divided by a factor 10−5.
and the values of hp i increase progressively.
i
The time evolution of the network is then obtained 3.0
1.74
by Monte Carlo simulations according to the iterative
procedure described in details in Ref. [19]. The auto-
1.72
correlation functions and the power spectral densities of 2.5
)
the resistance fluctuations are calculated by analyzing Ω 1.70
(
stationary signals R(t) consisting of 2×106 records. e 5.98 5.99 6.00
c
n 2.0
a
t
s
si
III. RESULTS e
R 1.5
Theresistanceevolutionofamulti-speciesnetworkcal-
culated at a temperature of 300 K is reported in Fig. 1. 1.0
Theinsetdisplaysasmallpartofthesameevolutionover 0 5x105 1x106
anenlargedtimescale. Theco-existenceofdifferentchar- Time (arb. units)
acteristic time scales in the R(t) signal is clearly shown
inFig. 1,where the longrelaxationtime associatedwith
FIG. 2: Resistance evolution of a single-species network
the achievement of the steady state, τ , co-exists with
rel (SBRN model) calculated at 300 K. The resistance is
the shorter times characterizing the resistance fluctua- espressed in Ohm and the time in iterative steps. The in-
tions displayed in the inset. For comparison, we report set displays the resistance fluctuations on the same enlarged
in Fig. 2 the time evolution of the resistance of a single- time scale of the inset in Fig. 1. (the vertical scales of the
species network obtained at T = 300 K by the SBRN insets in thetwo figures are different).
model. In this case, the values of the activation energies
(E =350 meV and E =310 meV) are chosen to give
D R
solid and short dashed lines represent the best-fits to
a relaxation time comparable with that of the signal in
the two C functions carried out respectively with a
Fig. 1. Thebehavioroftheresistanceisnowdetermined δR
power-lawand with an exponential law. The fitting pro-
by a single time scale (τ ≈τ ) and thus it is less noisy
rel
cedureconfirmstheexponentialdecayofthecorrelations
and essentially flat over time scales shorter than τ. This
in the resistance fluctuations of the single-species net-
is emphasized by the inset of Fig. 2 where the stochas-
work, described by Eq. (1), and points out the existence
tic signal resembles that of a few level system (it must
oflong-termcorrelationsinthe resistancefluctuations of
be noted that the vertical scale of the inset in Fig. 2 is
the multi-species-network, characterized by a power-law
significantly enhanced with respect to that of Fig. 1).
decay of the auto-correlationfunction:
Figure 3 reports the auto-correlation functions of the
resistance fluctuations on a log-log plot. More precisely, C (t)∼t−γ (6)
δR
theauto-correlationfunctioncorrespondingtothemulti-
species network (MSN model) is shown by black trian- with 0 < γ < 1. In particular, for the case of Fig.
gles while that corresponding to the single-species net- 3 we have found a value γ = 0.22 for the correlation
work (SBRN model) is reported by black squares. The exponent. We stress that the above expression of the
5
SBRN model are displayed in Fig. 4. The grey solid
τ = 5.8x104 line represents the best-fit with a power-lawto the MSN
100 spectrum. The grey dashed curve is the best-fit with a
LorentziantotheSBRNspectrum. Thecornerfrequency
of the Lorentzian, f = 4.0×10−6 (arbitrary units), is
c
0) SBRN consistent with the correlation time reported in Fig. 3
(
CδR and obtained by the best-fit of the corresponding auto-
)/ MSN correlation function. Thus, Fig. 4 shows that at 300 K
t
C(δR γ = 0.22 tehxehibreitsiastpanowceerflsupcetcutartailodnesnosfitythsecamliunlgtia-ssp1e/cfieαsonveetrwsoervk-
eral decades of frequency, with a value α = 0.94. This
10−1 result is a consequence of the envelope of the different
100 101 102 103 104 105 time scales associated with the different resistor species.
Time (arb. units) Now, we will analyze the effect of the temperature on
theresistancefluctuationsofamulti-speciesnetwork. To
this purpose, Fig. 5 displays the resistance evolutions
FIG. 3: Auto-correlation functions of the resistance fluctu-
calculatedatincreasingtemperatures: T =400K(lower
ations calculated for a multi-species network (MSN model,
curve) and T =600 K (upper curve). Already the qual-
black triangles) and for a single-species network (SBRN
itative comparison of the R(t) signals in Figs. 1 and
model, black squares). Both functions are obtained at 300
K. The solid and short dashed grey lines show the best-fit 5, shows that at increasing temperature there is a sig-
respectively with apower-law of exponentγ =0.22 and with nificant growth of both the average resistance and the
an exponential function with correlation time τ =5.8×104. variance of the resistance fluctuations (we will discuss in
The time is expressed in iterative steps. detail the behavior of these quantities in the following).
Moreover, these figures point out a drastic reduction of
therelaxationtimeatincreasingtemperature(morethan
one order of magnitude when T rises from 300 K to 400
10−2
K).
MSN
nits) 10−5 slope=−0.94 6.0
u
b. 5.0
r T=600 K
(f) (aR 10−8 Ωce () 4.0
Sδ n
10−11 a
t
s 3.0
SBRN si
e
R
10−1410−6 10−5 10−4 10−3 10−2 10−1 2.0 T=400 K
Frequency (arb. units)
1.0
0. 0.5x105 1.0x105 1.5x105
FIG. 4: Power spectral density of the resistance fluctua- Time (arb. units)
tionsat300Kcalculatedforthemulti-speciesnetwork(MSN
model, solid line) and for the single-species network (SBRN
model, dotted line). The grey solid line shows the best-fit to FIG. 5: Resistance evolution of a multi-species network at
theMSNspectrumwithapower-lawofslope-0.94. Thegrey 400Kand600K.TheresistanceisespressedinOhmandthe
dashed curverepresents thebest-fit with a Lorentzian to the time in iterative steps.
SBRN spectrum.
Thetemperatureisalsofoundtoaffectthedistribution
of the resistance fluctuations by increasing its skewness,
auto-correlationfunctionimplies adivergenceofthe cor-
as shown in Fig. 6, which reports on a lin-log plot the
relation time, as can be easily seen by considering the
probability density function (PDF, here denoted by φ),
following general expression of the correlation time [47]:
of the distribution of the resistance fluctuations. A nor-
∞ C (t) malizedrepresentationhasbeenadoptedforconvenience
δR
τ = dt (7) (here σ is the rootmean square deviation from the aver-
Z0 CδR(0)
ageresistance). InFig. 6,thedifferentsymbolsrepresent
The power spectral densities of the resistance fluctua- the PDFs calculated in the steady state of the multi-
tions calculated at 300 K by the MSN model and by the species network at 300 K (full diamonds), 400 K (open
6
100 u → ∞ Eq. (8) becomes a power-law while for h → 0 it
Gaussian describes an exponential decay.

10−1 Actually, Fig. 7 makes evident a strong reduction of
thecorrelationtimeofthefluctuationswhenthetemper-
ature increases. This result can be easily understood in
10−2
terms of Eq. (5), by considering the thermally activated
Φ
σ expressionsofthebreakingandrecoveryprobabilities: in
10−3 fact the increase of temperature implies a significant re-
duction of the ratio τ /τ . Therefore the interval
min max
10−4 [τmin,τmax] where the τi are distributed becomes more
andmorenarrow,destroyingthepower-lawdecayofcor-
10−5 relations and giving rise to an exponential tail. It must
−7 −5 −3 −1 1 3 5 be notedthat,forsimilarreasons,whenthe temperature
(<R>−R)/σ decreases below the reference value, T < Tref, the ratio
τ /τ strongly increases and the correlations in the
min max
resistance fluctuations keep their power-law decay over
FIG. 6: Normalized probability density functions of the re-
wider time scales (for example, when T =200 K the ra-
sistance fluctuations of a multi-species network calculated at tio τ /τ becomes 7.5×107). T is thus closely
300 K (full diamonds), 400 K (open circles), 500 K (open min max ref
∗
related to the transition temperature, T , from a long-
up-triangles) and 600 K (full down-triangles); σ is the root-
mean-squaredeviationfromtheaverageresistancevalue. The termcorrelatedbehaviori.e. abehaviorcharacterizedby
∗
solidblackcurveshowsthenormalizedGaussiandistribution. a divergent correlation time and occurring for T < T ,
to a behavior characterized by a finite correlation time
∗
and occurring for T >T .
circles),500K(openup-triangles)and600K(fulldown- The correlationtime of the resistance fluctuations can
triangles). Forcomparison,thenormalizedGaussiandis- be directly estimated at T > T by considering Eq.
ref
tributionhasbeenalsoreportedasasolidline. Thefigure (7) and making use of Eq. (8). We obtain the following
showsasignificantnon-Gaussianityoftheresistancefluc- expressionof τ in terms of the best-fit parameters of the
tuations, which becomes stronger at high temperatures, auto-correlationfunction:
meaning that the system is approaching failure condi-
tions. Thedistributionofthe resistancefluctuations and τ =u1−hΓ(1−h) (9)
theroleplayedonthenon-Gaussianitybythesize,shape
where Γ is the Gamma function. The values of τ calcu-
anddisorderofthenetworkhasbeeninvestigatedinRefs.
lated in such a way are reported in Fig. 8 as a function
[20, 25], wherethe link with the universaldistributionof
of the temperature. The figure shows a strong increase
fluctuationsofBramwell,HoldsworthandPinton[48,49]
ofτ atdecreasingtemperatures,whenT approchesT ,
has been discussed. ref
in fact for this value of temperature τ is expected to
Figure 7 reports the auto-correlation functions of the
diverge consistently with the power-law behavior of the
resistance fluctuations of a multi-species network calcu-
auto-correlationfunction. Tohighlightthedependenceof
latedat400Kand600K.Forcomparison,theC func-
δR
τ onthetemperature,wereportinFig. 9alog-logplotof
tion obtained at 300 K (already shown in Fig. 3) has
∗
thecorrelationtimeasafunctionofthedifferenceT−T ,
been drawn again in Fig. 7 with the solid grey line still
∗
where the value of the transition temperature T = 306
giving the best-fit with a power-law. The dashed grey
K is determined by a best-fit to the data in Fig. 8 with
curvesinsteadrepresentthebest-fittotheC functions
δR the power-law: τ ∼(T −T∗)−θ. The fitting curve is dis-
at 400 K and 600 K with the expression:
played in Fig. 9 by the dashed straight line. We find a
value θ = 2.7 for the exponent. Therefore Fig. 9 shows
CδR(t)=C0t−hexp(−t/u) (8) that the dependence of the correlation time on temper-
∗
ature is well described by a power of T −T , where the
∗
The values of the best-fit parameters are: C0 = 1.13, transition temperature T ≈Tref.
h = 0.30 and u = 1.42 × 104 for T = 400 K and The power spectral densities of the resistance fluctu-
C′ = 0.965, h′ = 0.46 and u′ = 5.57×102 for T = 600 ations calculated at the temperatures of 400 K and 600
0
K. As pointed out by Fig. 7, the fit to C with Eq. (8) K are reported in Fig. 10. Here the grey lines represent
δR
is very satisfying. We thus conclude that at high tem- the best-fitwith power-lawsofslopes-0.87and-0.78,re-
peratures the auto-correlation function of the resistance spectively. Thus we find that the power spectrum keeps
fluctuationsofthemulti-speciesnetworkiswelldescribed the 1/fα form also at high temperatures. However, the
by a power-law with an exponential cut-off. Such kinds value of the exponent α decreases to values well below
∗
of mixed decays ofthe correlations,non-exponentialand unity when T > T . This dependence of α on tempera-
non power-law, are often found in the transition of a ture has been studied since a long time [1, 10]. A quali-
complex system from short-termcorrelatedto long-term tative description of this dependence is provided by the
correlated regimes [50, 51]. It should be noted that for well known Dutta-Horn relation [1]. On the other hand,
7
100 104
)
s
T=300 K t
ni
b. u 103
r
(0) e (a
CδR 10−1 m 102
C(t)/δR T=400 K ation ti 101 τ ~ (T−T*)−θ
el
T=600 K rr θ=2.7 , T*=306 K
o
C
10−2 100
100 101 102 103 104 10 100 1000
Time (arb. units) T− T* (K)
FIG. 7: Auto-correlation functions of the resistance fluctua- FIG. 9: Correlation time of the resistance fluctuations as a
tions of a multi-species network calculated at different tem- function of the difference T −T∗. The time is expressed in
peratures. The solid grey curve shows the best-fit with a iterative steps and the temperature in K. The value of T∗ is
power-lawtotheauto-correlationfunctionat300K(thesame reported in the figure. The dashed line shows the fit with a
of Fig. 3). Thedashedgrey curvesdisplaythebest-fittothe power-law of exponent θ=2.7.
auto-correlation functions at 400 and 600 K with the func-
tion: C(t) = C0t−hexp[−t/u] (see the text for the values of 10−4
thefit parameters). The time is expressed in iterative steps.
T = 600 K
1250
s) s) 10−5 slope=−0.78
nit 1000 nit
u
rb. b. u 10−6
a 750 r
( a
me ) (
f
on ti 500 S (δR 10−7 T = 400 K
ati 250 slope=−0.87
el
rr 10−8
Co 0 10−4 10−3 10−2 10−1
Frequency (arb. units)
200 400 600 800 1000
T (K)
FIG.10: Powerspectraldensityoftheresistancefluctuations
of the multi-species network at T = 400 K and T = 600 K.
FIG. 8: Correlation time of the resistance fluctuations as a
The grey lines show the best-fit with power-laws of slopes
functionofthetemperature. Thetimeisexpressediniterative
-0.87 and -0.78, respectively.
steps and the temperature in K. The dotted line connecting
thediamonds is a guideto theeyes.
alsodependsontemperaturecangiverisetoamorecom-
manyexperimentshavepointedoutastrongdependence plicated behavior of α versus temperature.
of the detailed behavior of α(T) on the particular mate- Now we consider the dependence on temperature of
rial [1, 2, 10]. In this respect, it should be noted that a the average resistance and of the variance of the resis-
decreaseoftheαexponentfromα≈1andα≈0.8−0.5is tance fluctuations. We call R0 the average value of the
frequentlyobservedintheexperiments[1,2,10]atinter- resistance in the limit T = 0. This value represents the
mediate temperatures above 100-200 K, similarly to our resistanceofanetworkfreeofbrokenresistorsbutinany
∗
case where this decrease occurs above T ≈ T ≈ 300 case disordered (due to the presence of different resis-
ref
K. It should be noted, however, that in our model Tref tor species). Thus R0 not only depends on the network
merelyplaystheroleofaninputparameterwhichcanbe size and on the set of values {r0,i} with i = 1,...Nspec,
adapted to fit the experiments. Furthermore, we stress but also on the particular random distribution of the
∗
that the monotonic decrease of α for T >T is obtained r0,i within the network. Figure 11 displays the differ-
hereinthelinearregimeofcurrents,i.e. neglectingJoule ence hRi−R0 as a function of the temperature. A log-
heatingeffects. Actually,theseeffects,whoseimportance log representation is adopted here for convenience. The
8
dashed straight line represents the best-fit of the data 0.010
wAit=h 1th.3e5e×xp1r0e−ss2ioΩn/:KhRγian=dRR00+=A0T.5γ2, Ωw.ithWγe =can0.s8e3e, <( ∆ R)2> ~ <R>S
2
thatthis expressiondescribes ratherwellthe behaviorof R> s=2.6
the average resistance. <
>/
2
)
R
10 ∆(
<
γ
<R> = R + AT
0
0.001
) γ=0.8
Ω
(
0
R
− 1 10
> Average resistance (Ω)
R
<
FIG. 13: Relative variance of the resistance fluctuations as
a function of the average resistance (this last is expressed in
1 Ohm). The dashed line shows a best-fit with a power-law of
200 400 1000 exponents=2.6.
Temperature (K)
Finally, Fig. 13 shows in a log-log plot the relative
FIG. 11: Variation of the average resistance as a function
varianceoftheresistancefluctuationsasafunctionofthe
of the temperature. The resistance is espressed in Ohm and
averageresistance. Inthiscasethedashedlinerepresents
thetemperatureinK.Thedashedstraightlinerepresentsthe
the best-fit with a power-law of exponent 2.6. In other
best-fit with theexpression reported in the figure.
terms, Fig. 13 shows that:
h(∆R)2i/hRi2 ∼hRis (10)
10−1
<(∆R)2>=<(∆R)2> + BTη wheres=2.6. Tobeconsistentwiththebehaviorsofthe
0
average resistance and of the variance of the resistance
2Ω) η=3.5 fluctuations reported in Figs. 11-12, the following scal-
(
0 ingrelationshouldholdwithintheexponentss,η andγ:
>
2 s=η/γ−2. Byusingthepreviouslyreportedvaluesofη
)
∆R10−2 andγ weobtainforsthe value2.4,inagreement(within
(
< the error) with the result s = 2.6 obtained by directly
−
> consideringthedependenceofh(∆R)2i/hRi2 onhRi. We
2
)
R underlinethatboth,thebehavioroftherelativevariance
∆
( and the value of s, are perfectly consistent with the re-
<
sults reportedin Ref. [45]. This agreementconfirms and
10−3
pointsoutthefactthatthemodeldiscussedheredirectly
200 400 1000
Temperature (K) generalizes the results of Ref. [45] to resistors character-
ized by 1/fα resistance noise.
FIG.12: Changeofthevarianceoftheresistancefluctuations IV. CONCLUSIONS
asafunctionofthetemperature. Thevarianceisespressedin We have developed a stochastic model to investigate
Ω2 andthetemperatureinK.Thedashedstraightlineshows the 1/fα resistance noise in disordered materials. More
a best-fit with theexpression reported in the figure.
precisely, we have considered the resistance fluctuations
of a thin resistor with granular structure in a non-
The change with the temperature of the variance of equilibriumstationarystate. Thissystemhasbeenmod-
the resistance fluctuations is reported in Fig. 12. Here eled as a two-dimensional network made by different
h(∆R)2i0 is the limit value of the variance at T = 0. species of elementary resistors. The steady state of this
This value mainly depends on the network size [45] and multi-species network is determined by the competition
itvanishesforN →∞. Again,alog-logrepresentationis among different thermally activated and stochastic pro-
used in this figure. The dashed straightline displays the cesses of breaking and recovery of the elementary resis-
best-fitwiththe expression: h(∆R)2i=h(∆R)2i0+BTη tors. The network properties have then been studied by
which is found to provide a good description of the data Monte Carlo simulations as a function of the tempera-
for η = 3.5, B = 7.32×10−12 Ω2/Kη and h(∆R)2i0 = ture and in the linear regime of the external bias. By
5.0×10−7 Ω2 . a suitable choice of the values of the parameters, the
9
model gives rise to resistance fluctuations with different aslow,power-lawdecayofthe correlationsat300K to a
power spectra, thus providing a unified approach to the fasterbutnon-exponentialdecayathighertemperatures.
study of materials exhibiting either Lorentzian noise or
1/fα noise. In particular, by increasing the tempera-
ture in the range from 300 to 800 K the α exponent
decreases from values near to 1 down to values around
ACKNOWLEDGMENTS
0.5, in qualitative agreement with experimental findings
available from the literature [1, 2, 10]. This behavior,
indicative of a drastic reduction of the correlation time SupportfromMIURcofin-05project”Strumentazione
of the resistance fluctuations, has been also analysed in elettronica integrata per lo studio di variazioni confor-
terms of the auto-correlation functions. These last pro- mazionali di proteine tramite misure elettriche” is ac-
gressively change at increasing temperature, going from knowledged.
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