Higher Order Effects in the Dielectric Constant of Percolative Metal-Insulator Systems above the Critical Point

5 Pages·0.13 MB·English

by  W. D. Heiss

Higher Order Effects in the Dielectric Constant of
Percolative Metal-Insulator Systems above the Critical Point
W.D. Heiss, D.S. McLachlan and C. Chiteme
Department of Physics
University of the Witwatersrand, PO Wits 2050, Johannesburg, South Africa
PACS: 64.60.Ak, 71.30.+h, 77.90.+k symmetric media equation [7]. Equation (1) yields the
two limits
φs
The dielectric constant of a conductor–insulator mixture |σ |→∞: σ =σ c φ<φ (2)
C M I(φ −φ)s c
0 showsapronouncedmaximumabovethecriticalvolumecon- c
0 centration. Furtherexperimentalevidenceispresentedaswell
0 as a theoretical consideration based on a phenomenological
(φ−φ )t
2 equation. Explicit expressions are given for the position of |σ |→0: σ =σ c φ>φ (3)
n the maximum in terms of scaling parameters and the (com- I M C(1−φc)t c
a plex)conductancesoftheconductorandinsulator. Inorderto
whichcharacterizetheexponentssandt. NotethatEqs.
J fit some of the data, a volume fraction dependent expression
9 fortheconductivityofthemorehighlyconductivecomponent (2)and(3)arethe normalizedpercolationequations. At
1 is introduced. φ=φc,
The ac and dc conductivities of resistor and resistor-
1 capacitor (RC) networks and continuum conductor- σMC =σC/A(st)/(s+t)(σI/σC)t/(s+t) (4)
v insulator composites have been extensively studied for
1 up to higher order terms in σ /σ .
many years. In systems where there is a very sharp I C
6 Thus,inthecrossoverregionwhereφliesbetweenφ −
change (metal-insulator transition or MIT) in the dc c
12 conductivity at a critical volume fraction or percolation (σI/σC)1/(s+t) and φc+(σI/σC)1/(s+t) we have [2]- [4]
0 threshold denoted by φc, the most successful models, for
0 both the dc and ac properties, have proved to be ap- σM ∝σIt/(s+t)σCs/(s+t). (5)
0 proaches using percolation theory. Early work concen-
t/ tratedonthedcproperties,butsinceitwasrealizedthat When using the above equation to analyze ac systems
a the percolationthresholdisa criticalpoint, andthatthe the complex conductivities (σx =σxr−iωǫoǫrx, where ω
m
percolation equations could be arrived at from a scaling istheangularfrequency,ǫo the permittivityoffreespace
- relation, several papers ( [1] and the references therein), and ǫrx the real relative dielectric constant of the com-
d
n reporting experimental results on the ac conductivity ponent)mustbeinsertedinEq.(1). Belowφc theleading
o have appeared. Review articles, containing the theory term yields the imaginary conductivity or real dielectric
c and some experimental results, on the complex ac con- constant, while the next order term gives the real con-
v: ductivity and other properties of binary metal-insulator ductivity. In turn, above φc the real conductivity is the
i systems include [2]- [4]. leading term and the real dielectric constant appears in
X In[1],[5]and[6]thefollowingequationwasintroduced the next order term, upon which the present paper fo-
r cuses. In many instances the approximation σC = σcr
a
(1−φ)(σ1/s−σ1/s) φ(σ1/t−σ1/t) and σI = −iωǫoǫr is made [2]- [4]. In this case, the
I M + C M =0, (1) crossover region ranges over φ between φ −(ωǫ ǫ /σ )
(σ1/s+Aσ1/s) (σ1/t+Aσ1/t) c o r C
I M C M and φc+(ωǫoǫr/σC). This means that, as a function of
frequency, a sample with φ close to φ will enter an ex-
which gives a phenomenological relationship between c
perimentally observable crossover region for sufficiently
σ ,σ , and σ . They are, respectively, the conductivi-
C I M large ω and eventually obey Eq. (5) with a frequency
ties of the conducting andinsulating componentandthe
dependence of ωt/(s+t) for both the real and imaginary
mixtureofthetwocomponents. Notethatallthreequan-
parts.
tities σ ,σ , and σ can be realor complex numbers in
C I M In a series of papers reporting on the dc [5] and ac
Eq.(1). The conducting volume fraction φ ranges be-
[1,6] conductivities of three Graphite-Boron Nitride sys-
tween0 and1 with φ=0 characterizingthe pure insula-
tem systems it was shown theoretically that
tor substance (σ ≡σ ) and φ = 1 the pure conductor
M I
substance (σM ≡σC). The critical volume fraction, or 1. the scaling relations derived from the first order
percolation threshold is denoted by φc, where a transi- terms of Eq.(1), above and below φc, had the fre-
tion from an essentially insulating to an essentially con- quency dependencies requiredby the classicalscal-
ducting medium takes place, and A = (1−φc)/φc. For ing ansatz [2]- [4] (dc scaling is dealt with in [8]);
s = t = 1 the equation is equivalent to the Bruggeman
1
2
2. the second order scaling relations, derived from s+t(1−φc)(1−2φc) − 2s |σI| s+t
Eq.(1), in the crossover region, had the frequency φmax =φc+ A s+t (7)
2t φ σ
and(φ−φ )dependenciesrequiredbytheclassical c (cid:18) C (cid:19)
c 3
scaling ansatz [2]- [4]; σI s+t
+O( ).
σ
3. the second order scaling relations, in the region (cid:18) C(cid:19)
where the first order terms obey Eqs.(2) and (3), We emphasize that the deviation of φ from φ
max c
were not in agreement with the classical scaling is obtained only when the expansion in powers of
ansatz [6]; (σ /σ )1/(s+t) is taken beyond the first power; in fact,
I C
the result given can be obtained only if the expressions
4. the loss or conductivity term in the dielectric state
are expanded up to the third power.
hadafrequencyexponentof(1+t)/tandnot2as
required by the classical ansatz [2]- [4]. ε ε
11
Experimentally it was shown that 60
9
40
1. the first order experimental results for the G-BN 7
systemscouldbescaledontoseparatecurvesabove 20
5
φ φ
andbelowφ andthatthesescaledcurvescouldbe
c 0 0.05 0.1 0 0.05 0.1
fittedbythe scalingfunctions generatedbyEq.(1),
FIG.1. ComparisonoftheanalyticexpressionσM(φ)with
forallφandω values[5]. Thiswasalsofoundtobe
thenumericalsolutionfor10Hz(left)and1Mhz(right). The
the case by Chiteme and McLachlan [9] for ”cellu-
curve for the analytic σM(φ) has been drawn only for values
lar”percolationsystems, some of the dc properties
of φ where theanalytic expression for δ(φ) is supposed to be
of which are given in [10];
reliable. The parameters used are given in thetext.
2. after taking the loss (conductivity) term of the di-
We have checked the reliability of this analytic ex-
electric component into account the exponent for
pression by comparing with numerical solutions over a
the loss was much closer to (1+t)/t than to 2 [6];
widerangeoffrequenciesandaresatisfiedwithitsperfor-
3. therealdielectricconstant,whenplottedasafunc- mance (see Figs.1). The parameters chosen in the illus-
tion of φ for the fixed ω, does not peak at φ but tration adhere to the Niobium Carbide system [6] which
c
continues to increase reaching a peak at a value exhibits pronounced maxima beyond φc = 0.065; they
of φ greater than φc. The resultant curve has a aret=5.46(4.71),s=0.75(0.40),σC =1.07·104(Ωm)−1
hump like appearanceandthe positionof the peak and σI = −iωǫ0ǫr with ǫr = 5.4(6.7) for the left (right)
depends on ω, ǫ and σ . It was then shown [6] case. The difference between the two curves, even at
ir C
that this behavior could be qualitatively modeled φ = φc, reflects the error made by considering linear
by Eq.(1). terms in δ only. Nevertheless,the analytic expressional-
lows important conclusions to be drawn for the position
In the present paper progress is reported on this last ofthemaximum. Notethatφ doesnotdependonin-
max
point. We begin with a further theoretical investigation dividualvaluesofσ orσ ,inotherwordsthemaximum
I C
ofthe behaviorofthe dielectricconstantǫjust aboveφc. positiondependsonlyontheratioσI/σC (orequivalently
Since σM is in general the result of finding numerically on ωǫ0ǫr/σC). Note in particular that
the root of a transcendental equation, this can only be
achieved by expanding σM around φc. With the ansatz 1. thedeviationofφmaxfromφcstartswiththesecond
σM(φ)1/t =σM1/Ct +δ we obtain from Eq.(1) order term in (σI/σC)1/(s+t); recall that this first
order term determines the width of the cross over
1/t 1/s 1/t 1/s region [2]- [4];
σ σ (A−1)+∆φσ σ
δ = MC I C MC
A(t/s+1)σM1/Cs −t/s∆φσC1/tσM1/Cs−1/t−(A−1)σI1/s 2. the larger the ratio σI/σC the further is the maxi-
mum pushedawayfromthe transitionpoint φ . In
c
(6)
turn, the position of the maximum tends towards
φ for σ /σ →0 as it should;
with∆φ=(φ−φ )/φ . Notethatδdoesnotvanishwhen c I C
c c
∆φ=0 aswe haveto take intoaccountadditionalterms 3. for finite values of the ratio σ /σ the distance
I C
of lower order which do not vanish for φ = φc. These φmax−φc increasesthe morerapidlythe morepro-
terms are omitted in Eq.(4) but they are of the same nounced the inequality s+t>2;
orderasterms linearinφ−φ andmustbe incorporated
c
forreasonsofconsistency. Insertingtherighthandsideof 4. to lowestorder,thefrequencydependence ofofthe
Eq.(4) for σMC it is now straightforward,albeit tedious, distance φmax−φc is given by ωs+2t.
to obtain the position of the maximum of the imaginary
part. The calculation yields for the maximum
2
The derivative of ℑσ at φ is easier to obtain, but the with r>0. The solutionofEq.(1)shouldyielda contin-
M c
result is slightly more involved. We here report the es- uous σ across φ , which means that the conductivity
M c
sential result σ should nowhere vanish. As a consequence, σ must
C C
reach a non–zero value denoted by σ at φ which is
dℑσM| ∝ℑ σIst s+1t σs1σ1t . expected to be considerably lower tha00n σc0.cTo avoid
dφ φc  σst ! I C toomany parametersthe simplestassumptionwasmade
C for φ < φ , that is σ ≡ σ This should be reasonably
c C 00
  valid just below φ where our interest is focussed. We
We stress that the derivatives of ℜσ and ℑσ at φ c
M M c
stressthatthe effectofthe modificationbyEq.(8)is vir-
arealwayscontinuousandtendto zerofor (σ /σ )→0.
I C
tually indiscernible with regard to first order effects for
ε ε the solution σ of Eq.(1) or the corresponding percola-
M
tion power laws (Eqs. (2) and (3)) in the region where
70
40 the power laws are obeyed. Computer simulations show
50
30
that using Eq.(8) or a constant σ in Eq.(1) causes a
30 20 C
differenceinσ (φ−φ )(φ>φ )orǫ (φ −φ)(φ >φ)
10 M c c M c c
10
φ φ whichis toominute to be resolvedfromavailableexperi-
0 0.05 0.1 0 0.05 0.1
mentaldata. Thesameholdsfordispersionplotsagainst
ε ε
ω of σ above φ and ǫ below φ as given in [1]. For
M c M c
25 thisreasonithasneverbeennecessarypreviouslytocon-
20 7 sider the modification given by Eq.(8). However, in our
15 6 context the higher order effects are discernible in the di-
10
5 electric constant just beyond φ and can be fitted much
c
φ φ
more satisfactorily than in [6] using Eqs.(1) and (8).
0 0.05 0.1 0 0.05 0.1
FIG.2. Fits of experimental data at 1 Hz, 10 Hz, 1 KHz ε ε
and 1 MHz (top left to bottom right) for conducting Fe3O4 400
grains in a wax coated talcum powder matrix. The parame-
60 300
ters used are given in thetext.
40 200
20 100
As it has now been established and confirmed exper-
φ φ
imentally in the present paper and in [6], Eq.(1) gives 0 0.05 0.1 0 0.05 0.1
rise to the hump inthe dielectric constantjust aboveφ . ε ε
c
However, as can be seen in [6], Eq.(1), as it stands, only
40 40
qualitatively models the experimental data. In order to
30 30
better fit the dielectric data we propose an effective de- 20 20
pendence of σC on φ that is based in spirit on a model 10 10
due to Balberg [11] for non–universal values of t. The φ φ
0 0.05 0.1 0 0.05 0.1
model accounts for values of t higher than those allowed
by the Random Void (RV) and Inverse Random Void FIG. 3. Fits of experimental data at 1 Hz for T = 250
(topleft)andT =1200 (topright). Thebottomrowdisplays
model(IRV)[12,13]. InthismodelBalbergassumesthat
corresponding results for 1 KHz. The system is the same as
the resistance distribution function h(ǫ), where ǫ is the
proximity parameter, has the form ǫ−w as ǫ → 0 and in Fig.2. The parameter valuesare given in thetext.
does not tend towards a constant as in the RV and IRV
InFig.2wedisplaydataandcorrespondingfitsforcon-
models. Using the approach of [13,14] and keeping the
ductingFe O grainsinawaxcoatedtalcumpowderma-
underlying node links and blobs model [15], Balberg de- 3 4
trix[9,10]. The fits arefar superiorthanthose presented
rives an expression for a non-universal t which is equal
in [6]. We have chosen (somewhat arbitrarily) σ =
to the one obtained from the RV model when w = 0 00
σ /10 with the values σ = σ = 2.63·10−1(Ωm)−1
but can give a larger t for w > 0. For w > 0 the model C C c0
obtained from extrapolating dc experimental results to
alsoshowsthattheaverageresistanceinthenetworkcan
φ = 1, and σ = −iωǫ ǫ where ǫ is measured sep-
diverge when φ → φ . The increase in t beyond its uni- I 0 r r
c
arately at each frequency. Good fits are obtained for
versalvalue t as found by Balberg can be lumped into
un
(t−r,r,s) = (4.7,0.5,0.97),(4.3,0.7,0.98),(4.4,0.6,1.0)
t = t +t +r where r is the extra contribution due
un nun
and (5.6,0.4,1.6) when moving from the top left to the
to the characteristic resistance of the network diverging
bottom right display in Fig.2. The value φ = 0.025,
at φ → φ . This increase in characteristic resistance is c
c
obtained from dc measurements, is used in Figs.2 and 3.
incorporatedinto Eq.(1)andhence alsointoEqs.(2)and
Note that t=t +t +r is somewhatlarger than the
(3), by substituting σ with un nun
C
separately measured non-universal dc value of 4.2. Also
φ−φ r note that in all cases it turned out that r < 1. This
σCeff =σ00+σc0(cid:18)1−φcc(cid:19) , φ>φc (8) implies a steep increase of σCeff just above φc. The fits
3
are not necessarily optimal as the multi-parameter land- quotedpapers. The fitting curvesuses=1.15(1.35)and
scapeofthe leastsquareexpression|σexp−σnum|2 inthe t = 1.85(2.1). Note that the values of t are close to the
M M
parameterst,s andr (for fixedσ )has many localmin- universal value. Although better fits should be obtained
00
ima. However,thequalityofthefitsatthedifferentlocal withamorepreciseknowledgeofthesystemparameters,
minima does not vary greatly for acceptable fits. these resultsshowthatusing Eq.(8)asinput inEq.(1)is
While this second order behavior of ǫ is in principle a necessarytofitthedielectricdataonlywhenabnormally
complicated function of the various parameters the po- high values of t are observed in the dc data. Note that
sition of the maximum at φ is essentially dependent these data have been previously analyzed using Eq.(1)
max
onlyonthequotientσ /σ . Thisimpliesthatadecrease with a single exponent (s=t) in [19].
I C
of frequency (which is a decrease of σ ) has an effect To summarize: the pronounced hump observed exper-
I
similar to a corresponding increase of σ , which can be imentally for the dielectric constant of percolative metal
C
obtainedby anincreaseoftemperature. This is convinc- insulator systems just above the critical concentration
ingly demonstrated in Fig.3 where fits are obtained for can be modeled using the higher order terms of Eq.(1)
thesamesystemasinFig.2atdifferenttemperatures,i.e. near φ . In the cases where t is well above the universal
c
at different values of σ and ǫ . The experimental data value Eq.(8) must be used to quantitatively model the
C r
and the theoretical curves show that an increase in σ dielectric data. Note also that all previous experiments
C
is equivalent to a decrease in ω (or σ ). The experimen- on the dielectric constant ”below” φ , where φ has not
I c c
tal data appear somewhat erratic as they usually do for been independently measured by dc conductivity mea-
the low frequency chosen (1 Hz). The data in the first surements, have probably incorrectly identified values of
column of Fig.3 are slightly different from those in the φ .
c
first column of Fig.2 as the former are taken in a tem-
perature controlled oven. Much better fits are obtained
for higherfrequencies asillustratedin the bottom rowof
Fig.3 (1 KHz) as the dielectric data are more reliable at
higher frequencies. Similar to Fig.2, experimental values
areusedfor ǫ (ω,T)with σ =σ =3.22·10−1(Ωm)−1
r c0 C
at 250C and σ = σ = 1.42(Ωm)−1 at 1200C. The [1] Junjie Wu and D S McLachlan, Phys. Rev. B58, 14880
c0 C
fits were obtained with the parameters (t − r,r,s) = (1998).
(4.65,0.5,1.0),(3.4,0.6,1.0),(4.4,0.6,1.0),(3.5,0.9,1.0) [2] J P Clerc, G Girand, J M Langier and J M Luck, Adv.
again choosing the fixed value σ =σ /10. Phys. 39, 191 (1990).
00 C [3] D J Bergman and D Stroud, Solid State Physics 46,
ε ε editedbyHEhrenreichandDTurnbull(AcademicPress,
40 San Diego, 1992), p147.
80 [4] Ce-Wen Nan,Prog. Mater. Sci. 37, 1 (1993).
60 30
[5] JWuandDSMcLachlan,Phys.Rev.B56,1236(1997).
40 20
[6] D S McLachlan, W D Heiss, C Chiteme and Junjie Wu,
20 10
φ φ Phys. Rev.B58 ,13558 (1998).
0 0.1 0.2 0.3 0 0.05 0.1 [7] DSMcLachlan,MBlaskiewics andRNewnham,J.Am.
FIG. 4. Fits of experimental data at 100 KHz of a wa- Ceram. Soc. 73 , 2187 (1990).
ter–oil emulsion using a hydrophil agent for uniform droplet [8] D S McLachlan, Physica B254, 249 (1998).
size. [9] C Chiteme and D S McLachlan, Physica A. to be pub-
lished.
In continuum systems, all models for a non univer- [10] C Chiteme and D S McLachlan, in Electrically Based
sal t are based on the premise [11]- [13] that a distri- Microstructure Characterization, MRS Proceedings 500,
bution of conductances exists in the conducting compo- editedbyRAGerhardt,MAAlimandSRTaylor(Ma-
terials Research Society, Pittsburgh, 1998), p357.
nent which has a power law singularity. In a system
[11] I Balberg, Phys. Rev.B57, 13351 (1998).
where no such singularity occurs it should be possible
[12] B I Halperin, S Feng and P N Sen, Phys. Rev. Lett.54,
to fit the data using Eq.(1), with t values closer to 2
2391 (1985).
and without the use of Eq.(8). A system par excellence
[13] SFeng,B.I.HalperinandPNSen,Phys.Rev.B35,1197
which satisfies this requirement is a water-oil emulsion,
(1987).
using a surfactant to ensure uniform drop size. Data [14] A-MSTremblay,SFengandPBreton,Phys.Rev.B33,
for such systems has been published in [16] -[18] and we
R2077 (1986).
presentinFig.4datafromFig.1of[16]andfromFig.4of
[15] D Stauffer and A Aharony Introduction to Percolation
[17], which have been fitted using Eq.(1) with the same
Theory, Taylor and Francis, London 1994.
value of σ above and below φ . The values used, i.e.
C c [16] M A van Dijk,Phys. Rev.Letts. 55, 1003 (1985).
σC =1.9(1.5)(Ωm)−1, σI =−i2·10−11ω, φc =0.2(0.06) [17] M A van Dijk, G Castleliejn, J G H Joosten and Y K
for the left (right) display and ν = 105Hz are based Levine, J Chem. Phys. 85, 626 (1986).
on the dielectric and conductivity curves given in the [18] H F Eicke, S Geiger, F A Sauer and H Thomas, Ber.
4
Besenges. Phys. Chem. 90, 872 (1986).
[19] D S McLachlan, Solid State Comm. 69, 925 (1989).
5