B.I. Shklovskii A.L. Efros
Electronic Properties
of Doped
Semiconductors
With 106 Figures
Springer-Verlag
Berlin Heidelberg NewYork Tokyo 1984
Professor Dr. Boris I. Sh klovskii
Professor Dr. Alex L. Efros
A.F. JO F F E Physico-Techn ical Institute,
Acade my of Sciences o f th e USSR , Politekhn icheskaja ,
Leningrad 194021 , USSR
Translater
Dr, Serge Luryi
Bell La boratories, 600 Mountain Avenu e, Murray Hill, NJ 07974, USA
Series Editors:
Professor Dr. Manu el Cardona
Professor Dr. Peter Fulde
Professor Dr. Hans-Joachim Queisser
Max-Pl an ck-Institut flir Fes tkdrperfurschung , ll eisen bergstrasse I
D-7000 Stuttga rt 80, Fe d. Rep . of Ge rma ny
Title of the original Russian edition :
Etekironntye 5\'0;5Ivo leglrcvanny kh poluprovodnlkov
© by "Nau ka" Pu blish ing House, Mo scow 1979
ISBN 3-540-12995-2 Springer-Verlag Berlin Heidelberg New York Tokyo
ISBN 0-387-12995-2 Springer-Verlag New York Heidelberg Berlin Tokyo
Library of Congress Cataloging in Pub licatio n Data. Shklovskii, B.l. (Boris Ja novich), 1944-. Electro nic
properties of dop ed semiconductors. (Spri ngerseries in solid-state sciences; 45). Translation of: Elektronnye
svcistva legirovannykh poluprovodni kov, Includes bibliograph ical references and ind ex. 1. Doped semiconductors. 2. Electron -electro n interactions. 3. Hopping conduction. 4. Mater ials at low temperatures.
I. Efros, A. L. (Alex L.), 1938-. II. T itle. III. Ser ies. QC611.8.D66S5513 1984 537.6'22 84-5420
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s.
Th e
Percolation Theory
ter m
"perco lati on"
was
introd uce d
in
195 7
by
Broadbent
and
Hamm ersley [5.1 J in connection with th eir new class of math emati cal
probl em s. T hese problems conce rned the flow of a liquid thr ough a random
maze, a nd thu s th e nam e "percolati on the ory."
Co nce ptually, the simplest problems in percolat ion theory are latti ce
problem s, and th ey were th e first to be studied. T hey were th e subjec t of the
maj or part of origina l resea rc h, and wer e essent ial to the developm ent of the
meth ods and ideas of percolat ion theor y. We th erefore begin thi s cha pte r
with latt ice problems. Two other ca tegories of problems exist in percolati on
th eory: continuum problems and rand om site problems. It is these th at are
important in th e theory of impurity conductio n; th ey ar e discussed in
Sects. 5.2 and 5.3. Scalin g the ory, which expla ins the interde pende nce of the
critica l exponents, is exa mined in Sect. 5.4. Section 5.5 is conce rned with
electric conductivity of a resistor netw ork and th e atte nda nt problem of
infinite cluster topology. Fina lly, in Sect. 5.6 we discuss a pract ical
a pplication of percolati on theory: the ca lcula tion of electrical conductivity in
strongly inhomogen eous media . Th e last sect ion of thi s ch apt er plays a very
import ant role in our book.
5:1 Lattice Prob lems
Let us form ulat e the simplest problems in percolat ion th eor y. Imagine an
infinite two-dimensional or three-d imen sional latti ce.
Let the re be bonds
bet ween a ll adjace nt sites th at per mit the flow of liquid in both directions, so
that a ny wet site will instantl y wet all adjacent sites. In suc h a lattice, the
wetting of any site will result in th e wett ing of a ll sites. The vari ous problems
in percolat ion theor y follow from the introduct ion of random ele ments into
such a lattice.
Let us begin with the so-called bond problem. Assume th at eac h bond in
th e lattice ca n be either blocked (i.c., no flow of liquid ca n occu r in either
direction) or unbl ocked. Let the probability of a ny given bond being
unblocked be x , Th en we have a n "idea l d istribution" of blocked and
unb locked bonds: th e relat ive concentra tion of unbl ocked bond s is x , and of
th e blocked bonds I-x . Th is distribution of blocked a nd unblo cked bonds is
fixed a nd rem ains consta nt in tim e.
5.1 Lattice Problems
95
N ow if a ra ndom site is wet, there are two possibili ties. T he initial site
will wet either a finite or a n infinite number of ot her sites. The outcome
dep ends on t he fraction of unbl ocked bonds in the lat tice, Also, the rand om
posit ioning of blocked a nd unbl ocked bonds make s the choice of the initi al site
impo rta nt. T hus in Fig. 5. 1, site A wets a large - a nd possibly infinite number of ot her sites, while site B wets no ot her site at a ll.
Wh en ana lyzing th e system as a whole it is convenient to consider the
proba bility of a rand om initial site wetting an infi nite nu mber of ot her sites,
rather tha n conce ntrating on a give n initia l site. Here it is ve ry important to
note tha t in a n infinite latt ice this probabi lity is independent of th e ac t ua l
positioning of blocked a nd unblocked bonds, but varie s on ly with x . Let us
denote th is probabil ity by p b (x ) , where the supersc ript b indica tes th at we
are dealing with a bond proble m.
I
1.00
p (6J
(X
06
A
05
04
B
0.2
I
Fi2. 5.1. Bond problem on a
square lattice. Broken bonds are
shown by thin lines, conducting
bonds by heavy lines
0
0.04
020
035
0.52
058
s:
U84
Fig. 5.2. Probability p b(X) (infinite du ster density) for the
bond problem on various lattices (5,2]. The lattices are: (1)
face centered cubic, (2) simple cubic. (3) triangular, (4)
tetrahedral, (.5) square, (6) honeycomb
Th e funct ion p b (x) is plotted for d ifferen t lattices in Fig. 5.2. C urves I,
2 a nd 4 corre spo nd to thr ee-d imensional la tt ices, the rest to two-dimensiona l
ones. W hen x is small, pb (x) == 0, as blocked bonds prevent the liquid from
spreading far from t he initial site . On the ot her hand, as x a pproac hes 1, so
does p b (x) . Of greatest impo rt a nce is t he notion of a per colation threshold
(or critica l probability) intro d uced by Broadbenl a nd Ham mersley [5. 11. T he
percola tion threshold x; is defined as t he upper limit of the values of x for
which p b (x) - 0 (see Fig. 5.2>' It is genera lly accepted (in acco rda nce with
compu ter experi ments) that above th e percolation threshold th e pr obability
p b(x ) incr eases continuously until p b( x ) - I. Wh en x -xc « I, one has
(5. 1.1)
where {3 is a critica l exponent, the numerical values of which will be discussed
below.
96
5. Percolation Theory
Thus, the behavior of p b (x) with incre asin g x is simila r to th at of a n
order parameter for a second-order phase transition as the temperature is
lower ed . Su ch , for exa m ple, is the behavior of sponta neous magneti zati on in
ferromagnetism or polarization in ferroelectricity. In connection with the
percolati on thresh old it is often sa id that t here is no per colati on as long as
x < xc. while when x = x; percolation occurs immediately. The existence of
a definite per colati on th resh old x; as well as of th e well-de fined fun cti on
p b (x) , a re du e, of co urs e, to the fac t that we a re conside ring a n infin ite
lattice, wher e a ll rand om reali zati ons of blocked and un blocked bonds for a
given x are equivalent from the point of view of percolation. This also
explai ns the nonan al yt ic nature of p b (x) when x ~ x, . An alogously, it is th e
infi nite size of a thermodynamic system which renders its partition function
and free energy nonanalytic at the point of a second-order phase transition.
In add ition to pb (x ) , wh ich is th e probabilit y of wetting an infinit e
numbe r of sites, we ca n also define th e prob abilit y p,t (x), th at a given site
will wet at least N oth er sites . Th e proba bility p,t (x) differs from zero for
a ll 0 < x < I , alt houg h if N » I a nd x < x; it is exceedi ngly sma ll.
p b (x) ca n be deri ved from p~ (x) by letting N a pp roac h = :
pb(X ) ~ lim p~ (x) .
N-oo
(5.1.2)
Let us consider an exa mple of the bond problem [5.1,31. An orc ha rd is
plann ed in t he form of a squa re lattice, with a tr ee plant ed at each site . It is
known that a diseased tree will contaminate another tree a distance r away
with th e probability f ir) , wher e f ir) is a ra pid ly decrea sin g function of r .
It is necessary to find th e minim um peri od of the latti ce such that a diseased
tree will contaminate only a finit e number of other trees, thus preventing an
epidemic. Fro m th e constrai nts of th e pr oblem it is obvious th at nonadj acent
tr ees pose no threat a nd th at th e probabil ity of conta mina tion of a n adj acent
t ree is [ th); where h is th e peri od of the latti ce. If conta mina tion occ urs , the
bond ca n be defined as unbl ocked, all oth er bond s ca n be defined as blocked ,
a nd we have a bond problem wher e x - j (h ) . Co nsequently, th e required
peri od h min ca n be fou nd from the condition
(5. 1.3)
Let us now consider th e second ba sic problem of perco lation theory - th e
"sit e pr oblem". Here all bond s a re unblo cked, a nd it is the sites th at ca n be
block ed or unbl ocked . Blocked sites permit no now of liquid in eit her
dir ecti on - they ca nnot be wet, nor ca n th ey wet ot her sites. Let x be th e
fracti on of sites t ha t are unbl ocked a nd P' (x) be th e probability of a ra ndo m
site wetting a n infinite number of sites (s indi cat es a site problem) .
Th e fun cti ons P' (x) for three cubic latt ices a re plotted in Fig. 5.3. As in
the bond problem , there is a percolation threshold x, (s ) - the upp er bound
of x va lues for which P' (x) - o.
5. 1 Lanlce Pr oblems
97
fO ~·-------~
0.8
:E: 0.6
a 0.4
0.2
J /I "
o
o.':z:
1,0
Fig. 5.3. Infinite clus ter de nsity
p (J)(x ) for three cubic la ui ces 15.4)=
(I ) simple cubic. (2) body centered
cubic. (3 ) face centered cubic
Fi ~ .
5.... Site pro blem
011
a sq uare la ul ce
--.--,"'-
.'ig. 5.5 . Schematic representation
of an infinite cluster and several
large (critical) finite clusters for
o < x - Xt;<C 1. The lines arc actually
chains of black sites. bUI the resolution chosen is so low that details o f
the order of the orig inalla ttice period
arc not not icea ble. L(x) is the correlation length. The dashed squares
have sides 1< L (x) and I > L(x)
Both the bond and th e site problems are usua lly discussed in terms of
cluster statistics, rat her tha n liqu id flow from site to site. Let us a pply th is
ap proac h to our site problem. Imagin e th at a fract ion x of a ll la tt ice sites are
pa inted black, while the rest are painted whit e. An y two adjacent sites arc
conside red connected if bot h arc black. A cha in of such black sites become s a
cluster. For example, in Fig. 5.4 t here arc three clusters : one consists of five
black sites and two consis t of th ree black sites.
In the lan guage of clusters the appea ra nce of percolati on, as x increases,
can be define d as follows. Wh en x is sma ll, so a rc a ll the cl usters. As we
approac h the percola tion t hreshold, however, severa l clusters may come
toge ther, and the mean cluste r size increases. A t x = x, an infinite black
cl uster is born. This infinite cluster resembles a random networ k which
permeate s the entire space , while smaller, finite clusters ex ist in its "pores"
(Fig. 5.5).
The concept of an infinite cluster allows for a somewhat different
definition of P' (x ) . Sin ce only sites th at belong to th e infinite cluster wet an
infinite nu mber of sites, the probabili ty P' (x) is eq ua l to the ra tio of th e
number of sites in the infinite cluster to the num ber of a ll sites in th e latt ice.
In ot her words, P' (x) is th e densit y of th e infinite cluster . As x inc reases
98
5. Percolatio n Theory
over a nd above th e percolat ion thr eshold, P' (x ) a lso increases, a nd the
infinit e cluster incorp orat es other, finite cluste rs - becomin g denser in the
process. Its "pores" decrease in size and the mean cluster size of finite clusters
decreases accordingly.
We have been discussing the site problem, but the same argu ment applies
to the bond problem : aga in one ca n discu ss finite and infinite cluste rs of bonds
which co nnect th e sites . p b (x) ca n also be defined as the ratio of th e number
of infinite cluster sites to the number of a ll latt ice sites . It should be noted,
however , th at p b (x) ca n be defined in a nother way, with the density of the
infinite cluster p b (x) taken as the ratio of the number of bonds belonging to
the infinite cluster to the number of bonds in th e lattice. Shante and
Kirkpatrick [5.51 provide graphs of p b (x ) according to thi s last definition .
It is genera lly accepted th at th ere ca n be no more th an one infinite cluster
in a lattice. We also subscribe to th is view, although th e proposition has
never been rigorously proven. Our view is based on an extension of Kikuchi's
argume nt [5.61. Let us ass ume the contrary, th at is, th at in the ran ge
x, < x < X I there exist two infinit e cluste rs th at pierc e all spac e. Let
x - x, + 6.12, wher e C> < X I - x,. Th en the two infinit e clust ers have finite
densiti es a nd are at all point s sepa ra ted by a finite distan ce. N ow, increase x
to x, + C> by painting more sites black . Th e prob ability th at these new black
sites will fill any path linkin g th e two infinite clusters is sma ll but fi nite, At
th e same tim e, the number of suc h links for infinite clusters is infinite, Thus
it is certain that when x = x, + D. the two infinite clusters will be joined into
one, and since x; + D. < X 1> we have established a contradiction in our
original assumption.
Th is brings us to an im port ant physical applica tion of the site
problem [5,71. Imagine a crysta lline solution of a fer romagnetic substa nce in
a nonmagnetic host. Let the fraction of the ferro ma gnetic atoms be x.
Assum e th at th e exchan ge int eraction bet ween ferromagnetic atoms, which
tend s to align th eir magneti c moment s, falls off so ra pidly wit h distance th at
it ca n be considered negligible for nonadjacent atoms in the lattice. At zero
tem peratu re a ny pair of nearest neighb ors will exhibit identi cal spin
orienta tions, When x is sma ll thi s obviously lead s to th e formati on of small,
isolated magneti c cluste rs, suc h that all spins within a given cluster a re
parallel. The orienta tion of spins in ditferent clusters, however, will be
different. Th e cond itions necessary for th ese sma ll magneti c cluster s to merge
are preci sely th ose th at govern the form ati on of an infinite black cluste r in th e
site problem , Co nsequently, if the concent ra tion of magnetic atoms is less
than x , (s ) , there will be no ' macro scopic magnetic moment even at T = 0 ,
On the other hand, if x > x, (s) a finite frac tion of th e magnetic atoms will
form an infinite cluster and produce a macroscopic magnetic moment.
Tw o conclusions ma y be dr awn, Fir st, th e magneti c phase tr an sition in a
dilute ferromagnet can occ ur only if x > x, (s). Second , th e sa tura tion
magnetizati on M (T -O) when x > x, es ) ca n be ex pressed in ter ms of the
den sity of the infinit e cluste r:
5. 1 Lattice P roblems
99
where I' is the a tomic magnetic momen t. T he fact thai the temperatu re of
the ferromagnetic phase transition depends on the concentration of magnetic
at oms T , (x) has at tracted considerable a ttention [5.8- IOl. The law acc ording
to which T ,(x) ap proac hes zero as x ~ x , + 0 was der ived in [5. 11,121.
In Cha p. 2 we encounte red another im port a nt applica tion of the site
problem. It concerned our estimation of the Anderson transition point.
Resonant atoms correspond to black sites, nonresonant atoms to white sites.
The appearance of nonlocalized states is equivalent to the formation of an
infi nite black cluster.
Let us now form a lly define P (x ) a nd other qu ant ities of percolation
t heory in t he lan guage of cluster sta tistics. Let II, be t he number of clusters
containing s sites, per site of an infinite lattice. Then the sum
~ S11$
over all
finite 5 is t he fraction of the lattice sites that belongs to finite cluste rs. S ince
th e frac tio n of all black sites is x , the fra ction of a ll latt ice sites belonging to
t he infinite cluster P (x ) is given by
p (x ) - x - ~ 511, .
(5. 1.5)
s
Th e second import ant qua ntity of t he perc olation t heory is t he mean
cluster size:
~ 5 ' 11,
S (x) -
-=':=-~ 511,
(5.1.6)
As in (5. 1.5) , the su mmation extends over a ll finite 5 . Accord ing to
num er ica l ca lculati ons, S (x) becom es infinite as x ~ x, + 0 or x ~ x, - 0:
(5.1.7)
whcre v [li ke (3 in (5.1.1) ] is a critica l exponent of percolation theory.
It is easily seen t hat (5. 1.6) ca lculates a weig hted mean . T he num ber of
sites in a cluster to which a given site belongs is averaged over all sites.
'T herefor e the fact tha t S (x ) lends to infi nit y as Ix - x, I ~ 0 mea ns neither
th at the re a re more large clust ers tha n sma ll ones , nor t ha t th e vas t maj orit y
of t he black sites belong to lar ge clusters. Indeed, from (5.1.5) one has
lim ~ S11$ = X ,
x - x. s
a nd t herefore t he simple mea n ~ 511, / ~ II, rema ins finite when x - x,.
T hus the maj orit y o f black sites rem ai n in c1 usle rs of 5 - I even when
x "'" x c .
Accord ing to numerical calculations t he va lue II, as a funct ion of 5
behaves as follows when Ix - x; I « I. T here exists a critica l number of
100
5. Percolation T heory
sites in the cluster Sc which grows as
x
-
Xc
+ 0 or
x
-
Xc -
0:
(5. 1.8)
If s « s; and s is increasing, 11s decreases according to a power law, while
if 5 » 5" t hen II, decreases exponentia lly. T he num erat or of (5. 1.6) is
determined by clusters with s =::: S CI which we will refer to as critical clusters.
Th e fact that S (x ) d iverges as x ~ x, reflects th e increasing nu mber of sites
in critical clusters as we move away from the exponential drop of lis _
Hereafter. "mean cluster size" will always refer to (5.1.6).
T he thi rd imp ort a nt qu an tity of pe rcola tion th eory is th e corre lation
fun cti on . To int roduce it, let us define a func tion g (r., rj ) as equa l to I if
sites i a nd j are black a nd belong to the sa me finite clus ter, a nd as equa l to
zero in all other cases. No w we can introduce a pair correlation function by
averag ing g (r;, rj ) ove r a ll latt ice sites:
(5.1. 9)
As r ~ 00 the function G (r . x ) tends to zero, beca use the nu mber of
clusters lis of size s decreases as s gets larger. At distances r , which arc
smaller than the average size of critical clusters - which we will denote by
L (x)
t he corre latio n is governe d by clu st ers with 5« 5, (x ) .
Co nseq uently, in t his ra nge of r t he fu nct ion G (r , x ) dec reases as a power
la w with incr easin g r . If r » L (x ), th e expo nentia lly ra re clusters wit h
5 »
5, (x ) a re dom inant , a nd therefore G (r , x ) dec reases ex ponentially as r
increases.
It is natu ral to regard t he average size of thc critica l clu st er, L (x), as the
only character istic length of G (r ,x ) , a nd ca ll th is length t he corre lation
radius. Since the number of sites in a critical cluster tends to infi nity as
x - X c ± 0, the correlation radius L must grow as we approach the
percolat ion threshold:
<I x - x,1«
I)
(5.1.10)
Her e v is the critica l expone nt of th e correlation radi us . It is usually thought
that the critical exponents 'Y, d , and j) arc the same whether x > X c or
x < x ,. Wh e n x > x, the low pr obabi lity of the formation of finite clusters
la rger th an L (x) a lso imp lies a n equa lly low probab ility of pores of that size
in th e infin ite cluster (cf. Fig. 5.5), T hus, when x > x" th e correlation
ra di us can be int erpreted as th e typical size of th e pores in the infinite clust er.
It ca n be sai d that th e correlation radius gives a n indica tion of the average
d ist an ce betwee n the nodes of th e networ k, t hat is, its "peri od."
Let us now form ulate th e site (and bond) pro blem in a t hird , equivalent,
manner, nam ely via per colat ion t hr ough a finite latt ice. Co nsider a finite
latt ice, say a squa re of e sites per side (cf', Fig. 5.4) . Let a ll th e sites initiall y
5. 1 Lattice Pro blems
101
be white. We will ra ndomly pain t some sites black , slowly increa sing the
fraction x of black sites . At some va lue x - x" a chai n of blac k sites will
the
connect th e left side of th e squa re to the rig ht side. We ca n conside r
percola tion t hr eshold of the finite latt ice. If we repe al the proced ure, the
black sites will be positioned dilTerently and x" will genera lly be dilTeren t as
well. Therefore X c( is a ran dom qua ntity. After repeat ing the procedure
x"
many times we can calculate the average value <x, s> , The percolation
th reshold of t he correspond ing infi nite syste m is
Xc
=
lim
t-~
(5.1. 11)
< Xet > .
In three dimensions we can similarly discuss percolation from an edge of a
cube to t he oppos ite edge. Anoth er va ria tion on the sa me theme is to define x
as the probability tha t a given site will be black, rat her th an as th e frac tion
of a ll sites t ha t a rc blac k. For a n infinite lattice these two definitions of x a rc
eq uivalent. In a finite system, however , the two definitions of x lead to
dilTerent va lues of Xc( and <x,t > because of fluctuations. O ne ca n sa y t hat
the first defin ition of < Xct > corresponds to a microcanonical ensemble, and
the second to a ca nonica l ensemble. Of course, the limiting va lue (5. 1.11) is
the sa me in bot h ensembles, notwit hstand ing the dilTerent va lues of x".
Figure 5.6 [5.131 shows the d ist ribution fun ctions I t of x" in a canonical
ensemble corre spo nd ing to the site problem on a sq ua re lattice. Th ese
functions were obta ined by modelling percolati on th rough a finite la tt ice on a
computer using the Monte Ca rlo method. An alg ori thm for th is type of
modelling will be described in Sect. 14.4.
It is a ppa rent that t he d istr ibution functions a rc nea rly Gau ssia n in sha pe.
A deta iled a na lysis of the cu rves [5.131 showed th at within the ma rgin of
"
I[
I[
II
II
II
II
II
I I
JI
I
, - I
• -2
o -J
I
II 'I
I
1
_ _ ",-
x_ _ ",--x
- x-
\,
','--\
0.6
0.7
x el
lJ.8
Fig. 5.6 . The dis tribution fun ction s of values of the percolatio n threshol d for the site problem o n
I xl square latt ices (5.131. The I values in units o f the lattice consta nt are: (I ) 8, (2) 32, (3 ) 128
102
5. Percolation Th eory
acc u racy, no deviatio n of [,(x,,) from a Gau ssian func tion could be observed.
As expected, t he la rger the side of the square latt ice, the lower t he disper sion
w } - < (x" - <x,,> ) 2> . Fa r less t rivial is the fact th at t he mean-square
fl uct uati on W , noticeabl y exceeds the shift in the ma xim um value of [, (x,,)
(or in th e ave rage va lue of <x,,». S imilar behavior is a lso observed in
three d imen sions. In [5.131 it was shown th at the dimi nution of W , can be
descri bed by a power la w:
(5.1.12)
In th e sa me stud y a power extra pola tion of <x,,> was used to find the
percola tion thr eshold x, viz.
(5.1.13)
In three dim ensions it was found th at v == 0.9 a nd A == I.
Let us now show that the exponen t v defined by (5. 1.12) equ als t he critica l
index for th e corr elation radi us [5. 141. To this end , we d efine a func tion
x
p ,(x) -
J [ ,(x') dx ' ,
(5.1.14)
o
which rep resents the probab ility t hat percolation exists a t a given x , From
th e geome tric inter pre ta tion of t he corr ela tion radi us a t x > x, it follows that
for » L (x), percolat ion is prac tica lly a cer ta inty, i.e.,
e
(5 .1.15)
e
T his is easy to und ersta nd by imagining a sq uare of side
cut from t he
« L (x)
infinite latt ice (F ig. 5.5>' Similarly, it can be seen t hat for
per colat ion is no longer a ce rta inty, in othe r words, t he inequa lity (5. 1.15) no
longer holds. Now conside r the case of x < x . , In this case , if » L(x ),
almos t all finite cluste rs arc sma ller than
a nd it is practically certai n th at
no percola tion exists, i.e.,
e,
e
e
(5. 1.16)
T he inequalit y (5.1.16) docs not hold in the opposite limit of t « L (x >.
From t hese proper ties of the corre lation rad ius L , a nd using the numer ical
For exa mple, fixing the value
dat a on [, (x) , we ca n ca lculate L.
x - x ; > 0 a nd gradua lly dim inish ing t ; we obtain t he correlatio n radius L
as th at va lue of
for which the ineq ua lity (5.1.15) bre ak s dow n. T his will
hap pen when t he hal fwid th W, of the funct ion [ ,(x) a pproa ches x - x , .
Using (5.1. 12) , we find
e
(5.1.17)
T hus th e exponent v in (5. 1.12) indeed corre spond s to t he co rrela tion radius.
5.1 Lattice Problems
103
As a n example of percolation be tween opposite fac es for t he threedim ension a l bond problem, let us consider the electrica l cond uctivity of a
large c ube in which unblocked bonds arc represented by sta nda rd resist ances
connec ting neighborin g sites, whi le blocked bond s have infinit e resistance,
Figure 5.7 illustrates this formulation in two dimensions.
Fig. 5.7 . A random resistor network between two elec-
trodes
If the cube is su fficiently lar ge, its effecti ve conductivity «( x) differs from
Numerica l ca lculations show th at
if
ze ro only if x > x, (b) .
o < X -Xc « I, one has
(5. 1.18)
where 1 is yet ano the r critica l exponen t. We shall sec below that the behav ior
of q(x) when x > x c(b) is of interest in connec tion with the conductivity of
disord ered syste ms. Obvi ously q( x ) cannot ge nera lly be ca lculated using only
p b( x); to find q( x ) we also need to know the topologica l st ruct ure of the
infinite cluster.
We have con sidered three formu lations of the site a nd bond proble ms:
"liquid now" th a t wets a n infinit e number of sites, cluster sta tistics, an d
per cola tion th rough a finite latti ce. T he c ritical phenomena arc the same in
. a ll three ca ses - numerical calculations of the perco lation threshold support
t he equivalence of a ll t hree formu la tions for an infinite system.
Th e typ es of que st ions that percolatio n theory must answe r thus become
clear. He re a rc some of them: W ha t is the perc olation threshold xc? W hat
a rc the cr itical exponents 13, 1', v, a nd I ? How do dimensiona lity a nd latt ice
geo met ry affect the per colation thresholds a nd t he critica l ex ponents? What
is t he topology of finite a nd infinite clus te rs?
.
The val ues of the per colat ion threshold for commonly stud ied latt ices a re
summa rized in T able 5. 1. In T able 5.2 we provide the va lues of the most
im port ant critical expo nents : 13, 1', v, a nd I. Exa ct values of per colati on
thresholds Xc for selected two-dime nsiona l la tt ices were found by Sykes a nd
Essam [5. 15, 16] <these values a re unde rlined in T able 5.1). There a re other
rigorous results in percolation theory, notably various inequalities connecting
t he values of X c for d ifferent latt ices a nd problems [5.44]. A number of exact
resu lts were ob tained for !let he latti ces, bot h for the critical expone nts a nd
t he perco la tion th resholds [5.45-48]. A deta iled exposition of exact resu lts in
104
5. Percolation Th eory
Tab le S. J. Percola tion threshold x(" fo r various la ui ces. Exact values are underlined. All other
values were obtai ned by numerical methods
Latt ice
x, (b )
x, {s }
±
±
±
±
Honeycomb
0.6527
0.640 ± 0.0 18
[5.15,161
[5.17- 191
0.700
0.688
0.697
0.698
Square
0.5000
0.493 ± 0.013
0.498 ± 0.017
[5.15,161
[5.17-191
15.21,22,241
0.590 ± 0.0 Ia
0.58 1 ± 0.015
0.591 ± 0.00 1
0.593 ± 0.02
0.594 ± am
0.5898 ± 0.0008
[5.201
[5.17-191
[5.21,221
[5.231
[5. 131
[5.251
0.3473
0.34 1 ± 0.011
0.349 ± 0.010
[5.15,161
15.17-191
[5.21,221
0.5000
0.493 ± 0.0 18
0.500 ± 0.00 I
[5.15,161
[5.17-191
[5.21,221
Diamond
0.388 ± 0.005
0.390 ± 0.0 II
[5.201
[5.17-191
0.425 ± 0.QI5
0.436 ± 0.012
[5.201
[5.21,22J
S imple
cubic
0.247 ± 0.005
0.254 ± 0.013
[5.201
[5. 17· 191
0.307
0.325
0.320
0.312
0.3 12
0.312
±
±
±
±
±
±
0.010
0.023
0.001
0.002
0.003
0.001
[5.201
[5.17· 191
[5.21,221
[5.41
[5.261
[5.271
0. 178 ± 0.005
[5.201
0.243
0.254
0.248
0.248
±
±
±
±
0.010
0.001
0.003
0.003
[5.201
[5.21,221
15.41
[5.26J
0.119 ± 0.002
0. 125 ± 0.005
0.1190 ± 0.0005
[5.201
[5.17·1 9]
[5.28]
0.195
0.199
0.208
0.200
0.198
±
±
±
±
±
0.005
0.008
0.001
0.002
0.003
[5.201
[5.17·1 91
[5.21,221
[5.41
15.291
T riangular
Body
ce nte red
cubic
Face
ce ntered
cubic
.0 10
0.017
0.00 I
0.003
[5.201
[5.17-191
[5.21,221
[5.231
perco lat ion th eor y ca n be found in th e su pe r b revi ews by Essam [5.49 ,501 a nd
Shante a nd Kirkpatrick [5.51.
Th e va st maj orit y of the valu es liste d in Ta bles 5. 1 a nd 5.2 were found
throu gh the usc of num eri ca l m ethods. In rare cases, experimental models
5.1 La ttice Problems
105
Table 5.2. Numerically calculated critical exponents for the infinite cluster density p, mean number
of sites per cluster )I . correlation radius v, and electrical conductivity I (d is the space d imensionality)
Exponent
(3
d - 2
0.14 ± 0.03
0. 138 ± 0.007
0.148 ± 0.004
[5.30]
[5.31]
[5.32]
0.20
0.03
0.02
0.03
0.30
[5.32]
[5.20]
[5.28]
[5.23J
[5.34J
1.34 ± 0.02
1.33 ± 0.04
1.36 ± 0.04
S 1.5
[5.281
[5.131
[5.261
[5.251
1.38 ± 0. 12
1.1 ± 0. 1
1.15 ± 0. 15
1.1 - 1.25
1.31 ± 0.04
1.26 ± 0.03
[5.361
[5.271
[5.371
[5,381
[5.391
[5.401
1.85
2.38
2.38
2.43
2.34
v
±
±
±
±
±
d - 3
0.05
0.02
0.06
0.02
[5.4]
[5.27]
[5.29]
[5.33]
1.69 ± 0.05
1.70 ± 0. 11
1.80 ± 0.05
[5.20]
[5.28]
[5.27]
0.94 ± 0.05
0.82 ± 0.05
0.83 ± 0. 13
[5.13,261
[5.281
[5.351
2
1.725 ± 0.005
1.6 ± 0.1
1.75 ± 0. 10
1.6 ± 0. 1
[5.41!.
[5.421
[5.271
[5.381
[5.431
0.35
0.39
0.42
0.47
±
±
±
±
were used. Th e latter will be covered in Sect. 5.5; for now let us br iefly
examine the numerical methods.
In 1959, Domb [5.51] proposed the use of series to fi nd X C' T he first
calculations were ca rried out by Elliott et al. [5.71 a nd by Dom b a nd
S ykes [5.521. Th e method essentially consists in writ ing the mea n cluster size
S (x) for sma ll x in the form of a series:
(5. 1.19)
n
S ince S (x ) tend s to infinity as x - X c> the percolati on thr eshold x, is found
by locat ing the neare st singula rity point of the series on the real positive axis.
In the theory of fer romagnetism a similar method is used to find the
tempera tur e of the phase tr ansition [5.531 by expan ding the magnetic
106
5. Perco lation Theo ry
susce ptibility in inverse powe rs of t he
tem pe ra ture a nd study ing the
convergence of this series.
S ykes a nd Essam [5.20l were the firs t to ca lcula te the cri tical ex ponent 'Y
for bond a nd site prob lems by using t he series method a nd exa mini ng th e
behavior of S (x) nca r x - x, . T hey found that 'Y values for a ll pla na r
lattices - square, triangular and honeycomb - were similar. The same was
found to be t rue in three dimensions for the various cubic lattices: simple,
bod y ce ntered , fac e ce nte red , a nd d ia mond . On t he basis of these dat a Sykes
a nd Essam concl uded that t he exponent 'Y is uni versal ; that is, give n t he
dim en sion al ity, it is independent of bot h the type of problem 'a nd the kind of
lattice. Fu rthe rmor e, c urre nt dat a support the hypoth esis that all percola tion
theory exponents are universal. From here on we shall use a subscript to
indi cat e th e d imensional ity of th e la tti ce (c.g ., 'Yl , {3,) . In recent yea rs, t he
series meth od has also been used for x > x, . For exam ple, the qu ant ity
p (x) is expan ded in powers of () - x , ) . Th e critica l index {3 is th en
foun d [5 .29 -331 by studying th is series wit h th e Pad e a pprox ima tion [5.53 1.
T he resu lts of these ca lculations a re prese nted in T a ble 5.2.
An ot her produ ctive nume rica l method in percola tion th eory is t he ra ndom
trial me thod, known as the Monte Ca rlo meth od . It includ es practi cally all
computer calculations in which a random number generator is used to any
exte nt. Th e Monte Carlo met hod was first a pplie d to percola tion theory by
Ham mersley [5.2] a nd a gro up of America n resea rchers [5. 17-1 9 1. La te r
Dean [5.2 1] developed a new va ria nt o f t he Monte Ca rlo met hod which
a llowed Dean a nd Bird to obtain very acc ura te results for x, [5.221. Ot her
researchers used yet another improved version, in which an actual simulation
o f per colation throu gh a finite lattice was prog ra m med [5.13, 26, 54-56 1.
Some o f their resu lts were discu ssed ea rlier in th is c hap ter.
Let us now d iscu ss th e num erical values of x, . From T a ble 5.1 it is
evide nt that x; vari es wide ly with t he t ype of problem a nd latti ce. An
a pprox ima te empirica l rul e exists for th e bond problem [5.52, 5.57 ]' where
x, (b ) is expressed in ter ms of th e simplest ch a racte ristic s of a latt ice - its
d imen sionali ty d a nd th e numbe r of adjace nt sites Z :
d
B, - Zx,(b) - -d - '
- 1
(5. 1.20)
where B - Zx is the averag e number of sites con nec ted to a given latt ice site .
T a ble 5.1 shows tha t (5. 1.20) remai ns acc ura te within a few perce nt when Z
an d x, (b ) a re cha nged by a fact or of 3.
On t he ot he r ha nd, for th e site proble m, Zx; ca nnot be conside red even a n
approximate invariant. For this case, however. another approximate invariant
was found by S cher a nd Z allen [5 .58l. Co nst ruct a circle (sphere) a rou nd
eac h site, with the radi us equ al to one-ha lf the distan ce to a ny adjacent site
(F ig. 5.8) . Let f be the frac tion of t he a rea (volume) conta ined by the se
circles (sph ere s) , in ot her word s - th e filling rati o. It ca n be easil y
calcu la ted for a ny la tt ice. S cher a nd Za llen found th e fract ion of th e a rea
5.2 Continuum Problems
107
1'12. 5.8. Scher and Zallen's
construction for a honeycomb
lattice 15.58). Circles co r responding to black SilCS are
shaded
(volu me) conta ined by circl es (spheres) const ruc ted a round black sites a t
x - x . : T o this end th ey ca lculat ed 8, - fx; (s ) :
8, -
l
o. 15 ± o.OI
if d - 3
0.45±O.02
if d - 2 ,
(5.1.2 ])
whic h turn ed out to be consta nt for a ll latti ces of th e sa me d imensionality.
5.2 Continuum Problems
Continuu m prob lem s a rc formulated in th e following man ner. Suppose a
random continuous fun ctio n V (r) is defined in the entire space by its
correlations. W ithout loss of genera lity, we ass ume t hat < V(r) > (w here < . . . > de note s averaging). . Now ta ke a rea l num ber V a nd "paint "
all regi ons whe re V (r) < V black, while all ot her regions remain white .
As V cha nges from - 0 0 to +00 the volum e of black regions cha nges from
zero to a ll space (Fi g. 5.9). Initi ally isola ted black "lakes" me rge, a nd form
an "ocea n" if V is sufficiently la rge. We need to find the lower bound V, of
t hose va lues of V whic h permit one to sta rt from some black point a nd tr avel
infinit ely far via black regions on ly.
Such a prob lem arises, for example, when we must find the minimal
°
energy an electron must have in order to traverse an infinite distance in space
with a pote ntial e nergy profile V (r) , wit hout escaping the classical
bounds [5.57, 59-631. In t his ca se, V, is called the percolati on level. In
add ition to th e a bove formulati on of t he problem , which is a na logous to the
"liquid flow" formulat ion of latt ice problems, it ca n a lso be discussed in term s
a na logous to clust e rs: here percolation is equival ent to the form ation of a
continuou s blac k region of infinite volume. Pe rcolat ion th rough a finite
volum e ca n also be int roduced; here one conside rs "black" percolat ion pat hs
between t he opposite sides of a squa re (or edges of a cu be).
108
5. Percolation Theory
Fig, 5.9. Percolation in a two-dimensional potential
(5.591. The outlines co rrespond to surface
V( r) = consr. "Black" regions IV( r) < VI are shaded.
The three maps correspond 10 three different values o f
V( V1 < V2 < V)
It is apparent that continuum problems are quite similar to the previously
d iscussed lattice problem s. Imagin e a lattice with a period so small that V (r)
docs not va ry a ppreciably over it. T hen fix V an d consider all latt ice sites in
black regions to be black, an d those in white regions to be white. If V
increases, the fraction of black sites increases as well:
x
~ O(V) ==
v
I
F( V') d V' ,
(5.2. 1)
where O( V) is t he fraction of space in which V (r ) < V , and F(V) is th e
dist ribution funct ion of th e potential V. T he percola tion level V, ca n then be
found from t he condi tion tha t the fraction of space occu pied by black regions
a t th e percola tion th reshold ,
v.
0, -
I
F(V)dV ,
(5.2.2)
-~
should equa l the percolation t hreshold x, for t he site problem , which differs
5.2 Con tinu um Problems
109
from th e previously formulat ed condition in th a t the distribution of black and
white sites is correlated ac cording to V (, ) ; i.e., black a nd whit e sites a rc
grouped into clusters. T his proposition is essential to the ca lcula tion of 0,
(see below) .
I ~ add ition to perc ola tion "throug h black " one ca n conside r percolation
"th rou gh whit e." Wh en V is sma ll, white fills a lmost all space, forming a
"contine nt" which enco mpasses black "lak es". Wh en V is la rge, th e
"contine nt" splits into sma ll "islan ds" a nd there is no percola tion t hroug h
white . Let the percolat ion level "through white" be that va lue V - V; a t
which percolation through white disappears. In the two-dim ensional case the
percola tion levels V, a nd V; a re simply related , provided the ra ndom
potent ia l V (r) possesses a finite correlation rad ius.
Indeed , let us exami ne per colation bet ween opposite sides of a sq ua re
whose side s a re m uch large r tha n the cor relat ion rad ius of t he potential.
O bviously, the a bse nce of percola tio n t hrou gh blac k from left to right imp lies
percolation t hrough wh ite fro m top to bottom . Con versely, percolation
t hrou gh wh ite from left to right ens ures th at there is no percolatio n t hrough
black from top to bott om .
Since in ve ry la rge sq ua res percolati on through a give n color occurs at the
sa me va lue of V in both d irecti ons, we ca n drop the ex pressions "left to righ t"
a nd "top to bottom" from t he previous pa rag rap h. We th us a rrive a t the
following conclusion: pe rcola tion through black disappea rs at the sa me value
of V which ind icat es the onse t of per colati on through whi te, i.e.,
V, - V; .
(5.2.3)
From (5 .2.3) we ca n det er mine V, for a pote nti al th at is sym metrical a bout
v -a. For such a potential it is obvious that
V, - - V; ,
(5.2.4)
a nd , tak ing (5.2.3) into acco unt, we have
V, = V; -
a
a nd
0, - 0 .5 .
(5.2.5)
In other words, the a rea is eq ua lly distri buted between black a nd white at the
pe rcolat ion level. T his result was obta ined in [5.59, 64 , 65 1.
For a t hre e-d ime nsiona l potential, as in the two-dimensional case , one
ca nnot have a situa tion in whic h per colati on t hroug h both colors is a bsent. In
thr ee dim ensions, howeve r, per cola tion th rou gh black a nd th rough whit e ca n
coexis t. si nce their respecti ve channels need not intersec t. Consequently, in
three dimensions we only have an inequality:
(5.2 .6)
i.c., th er e a rc three possible ra nges of V from the stan dpoint o f percolation:
I)
2)
3)
V < V, an d t her e is percolation thr ough white , but not throu gh black;
V, "" V < V; an d t here is per colation through bot h blac k a nd white ;
V .. V; a nd t here is percola tion through black , but not th rough white.
110
S. Percolation Theory
In the case of a sta tist ica lly symmetrica l potenti al, (5.2.6) combined with
(5.2.4) produces:
(5.2.7)
Za llen a nd Sc her [5.591 postula ted that the quan tity 0, is muc h less sensitive
than V, to the form of pote ntia l. To evalua te 0, they proposed using the
invaria nt Ix,(s) . According to (5.1. 21), in two d imensions Ix,(s) - 0.45 ,
which is close to th e value 0, - 0.5 for a symmet rica l poten tial. Exact
ag reement between the values could not be expec ted, since in Zallen and
Sc her's constructio n there was no symmetry between points internal and
exte rna l to their circl es or spheres.
Skal et al. [5.54, 551 stud ied contin uum problems numerically , using the
Monte Carl o method. They considered a Gaussian potential defined by the
expression:
JK (r-r)j (r) dr ,
V( r) -
(5.2.8)
where I( r) is a random Ga ussia n function with th e correlator:
</ (r)/ (r » - o(r - r ') ,
(5.2.9)
and th e kernel K (r) drops off rapidly with r beyond the correl atio n distance
roo From (5.2.8) and (5.2.9) it follows th at the d istribution function of the
potential V is of th e form :
F(V) -
;.
exp [- V:],
," , 'Y
'Y
'r' -
2 JK 2 (r)d r .
(5.2. 10)
In orde r to model th e continu um prob lem on a lattice with a period
sma ller th an ro, the values I i were taken ra ndom ly, a nd the values Vi were
found by summation:
Vi = LK(ri- rj ) / j '
j
Then a value of the poten tial V was introd uced, which led to the form at ion of
a n array (Ail. consisting of zeros a nd ones: A i - I if Vi < V, and Ai - 0 if
Vi ~ V. Finally, a finite latt ice with sides muc h grea ter tha n ro was tested
for percolatio n th rough ones.
In two dimensions th e fun ction exp (- r/ro) served as K (rl. If the period
of th e latt ice is tak en to be unit y, then as we move from ro < I to ro » I
the g rad ua lly increas ing corre la tion of ones shou ld ca use the threshold
concentra tion of ones to cha nge from the site-problem va lue x, - 0.59 to the
continuum -problem threshold 0, - 0.50. Such a tra nsition was indeed
observed [5.54, 66, 671.
In an other stu dy [5.551 the quan tity 0, was foun d for three-dimensional
Gaussian poten tia ls using five different funct ions K (r) characteri zed by the
5.3 Random Site Problems
111
correla tion radi us roo Th e results were th en ext ra polated to large va lues of ro,
which yielded the va lue of 0, for th e continuum problem. T his value proved
the sa me, within limits of error, for all five functions, viz.
(5.2. 11)
0, - 0.17 ± 0.0 1 .
This result su pports the idea of Z allen an d Scher , as it is close to the
inva riant valu e (5. 1.2 J) .
Miiller-Krum bhaar [5.681 applied the Monte Ca rlo me thod to a site
problem on the simple eubic lattice where the sites were correlated
therm odynamicall y. li e examined a nondilute Ising ferrom agnet a t T < T,
for percolation th rough nipped spins, i.c, spins pointing in t he opposite
direction from the spontaneous magnetization. Such spins are correlated in
their position due to mutual attraction. As it turned out, even a modest
corre lation - like the one th at exists when the temperature and the
interacti on energy a rc of the sa me orde r of magnitude - will lower x, from
0.3 1 to 0 ,19. Th e gra dua l tr an siti on from a site to a cont inuum problem d ue
to increasing positive correlation between adjacent sites was investigated
num ericall y by Webman et a l. [5.4 31. Th ey concluded t hat as correla tion
increases , x, tend s to the val ue 0, - 0.145 ± 0 ,00 5.
5.3 Random Site Problems
As we sha ll sec lat er , ran dom site problems a rc of the gre at est im porta nce in
hoppin g co nduction theory . Random sites arc chao tically dist ribut ed points in
space. The average number N of sites per unit volume is assumed to be
known. Let tj he some function of the vector
r ij
which connects sites
i and j . Now the sites i and j will be considered bonded if, for some number
~,
(5.3 . 1)
Henceforth the inequ alit y (5 .3. 1) will be known as th e bondi ng crite rion. If
two sites arc bonded , eit her directl y or th rough other bonded sites, t hey
belong to th e same cluster. It is required to find the percolat ion th reshold ~"
th at is, t he lower bound of the pa rameter ~ which still permits an infinite
cluster.
Th e simplest of a ll ran dom site problems is the one with the bonding
criterion
(5.3.2)
T his cond ition is fulfilled if site j is within a sphere of ra di us r a bout site i .
Consequently, thi s proble m has the following geomet ric int erp ret ati on: spheres
of radius r ar c const ruc ted about all sites; we must find the lowest value
S. Percolat ion Theory
112
Fig. 5.10. (a) Inclusive figure
a nd (b ) ove rlapping figure co nstructio ns. Co nnected sites are
joine d by straight line segments
b)
a)
r
o
o
•
E:
r c which allows for an endless chain of sites in which each site lies inside
th e sphere constructed about the preceding site (Fig . 5.IOa),
We will term the percolation threshold re the percolat ion radiu s.
Occasionally it is more convenient to construc t spheres with radius r / 2 and
ca lculate the value of r that a llows for a n infinite cl uster of overla pping
spheres (Fig. 5. ioi». Pike a nd S eager [5.691 na med the fi rst situation the
inclusive figure (I F) construction, and the second the overlapping figure
(O LF) constr ucti on. As we sha ll see lat er, in hopp ing conduction theory the
first co nstruction is of more use; for th is reason it will be discussed in greater
det ail.
T he percolati on rad ius re depends solely on th e site concentration N, a nd
beca use of dimensionality, it is proportional to N - 1!J. Instead of re it is often
instruct ive to conside r th e threshold value
Be - 4/ 3
1r
Nrl
(5.3.3)
of the dimensionless para meter B - 4/ 3 « Nr ", which corr esponds to the mean
number of bonds per site. T he proble m of determinin g re and Be will be
term ed the sphere problem, a nd the sa me problem in two dimensions, will be
referred to as th e circle problem . In th e latt er case th e value to be
calc ulated is
(5.3.4)
it is th e num ber of rand om sites per unit area a nd i e is the percolati on
radius in two dimensions.
Th e sphere problem a rises in studies of the ferromag netism of dil ute
crysta lline systems of magne tic a nd nonmagnetic a toms, where the exchange
forces between magnetic atoms a re modelled by an interac tion having a finite
radius ri nt larger than the period of the lattice. If r int is very large. so much
where
5.3 Random Site Problems
11 3
so th at a large number of sites fall int o the sphere of int eracti on and the
discrete positioning of sites becomes irrelevant, then we have a sphere
probl em, th e only differ ence being th at ri ", [th e righ t hand sid e of (5.3 .2) 1 is
now given a nd we need to find th e concent ra tion of ran do m sites. If the
number of sites that fall into the sphere of int eraction is de noted by Z then
Z x is the mean number of black sites connected to a given site a nd for a ll
lattices of the sa me di me nsionality we obviously have
(5 .3.5)
lim (Z x e) - Be ,
z--
wher e Be is defined by either (5.3.3) or (5.3.4) depend ing on the
dimensionality of the prob lem .
It has been sho wn [5.70-721 t hat one ca n a lso red uce to the sphere
problem the problem of calculating t he C urie temp eratur e and the lowtemperatu re t herm odynamical quan tities for a ferr om agnetic in which the
exchange inte ract ion between magnetic imp urities falls exponentia lly wit h the
sepa ra tion, viz. V = Voex p (- ria) .
T he cr itica l indices of the latt ice site prob lem ca n be a lso int roduced in
the ra ndom site problem with nearl y identi cal results. Let fI, be th e num ber
of clusters of size s in a given volume. Then,
P(B ) = I - ~ Sfl"
S (B) = ~ s 2f1, /~ sfI, .
s
(5.3 .6)
s
Nca r the percolat ion threshold B, one has
P (B)
cc
(B- BY,
S( B )
cc
IB -Bel - ' .
(5.3.7)
In a na logy to the latt ice problem, we ca n introduce the corr elation function
a nd the correlation rad ius L . Nc a r the thr eshold
(5.3.8)
where N - 1/'. th e average distan ce bet ween sites , is ta ken as th e natural length
sca le. The correlation radius still represent s t he characte ristic size of critical
cl uste rs, i.c., the largest clu sters for which th e quantity fI, is not yet
exponentia lly sma ll. Express ions (5.3.7) a nd (5.3.8) ca n be rewritten
substituting th e d ime nsionless vari able (r-re)!re or , more genera lly, (~-~cl/~e
for the difference B-Be. For exa mple,
(5.3 .9)
Sin ce cri tica l exponents do not depend on lattice geometry, we can logically
assume th at the tr an sit ion from lattice to ra ndom site problem s will not affect
them either. T his hypothe sis is in ag reeme nt with existing dat a .
114
5. Percol ation Theory
T he sphere problem is especially signifi ca nt for the theory of hopping
cond uction. For th is reason, we will briefly review the litera ture devoted to
the ca lcula tion of B, a nd the cri tical exponents for the sphere problem.
Domb et a l. [5.73 , 741 used the series method to ca lculate th e percolat ion
threshold x, (s) for the site problem on various lattices, taking into account
the interacti on of nonadjace nt sites. Extra polating the produ ct Z x, (s) to
Z - "", they obta ined the values of B, listed in the first row of Table 5.3.
Man y researchers [5.35, 69, 76-831 ca lculated B, using the Monte Ca rlo
method; their results a rc a lso listed in Table 5.3.
Tab le 5 .3 . Thresho ld values o f the mean number Bc o f bonds per site fo r circle and the sphere problcms
Meth od
B, <Circles)
Se ries
4. 5
[5.7 3,741
Mont e Ca rlo
3.2
[5.751
)) Constructio n of a
cluster includin g th e
central site
3.82
4.1
4.4
2) Percolati on th rough
a finite a rray
4.5
B, (spheres)
2.7
[5.73,741
[5.761
[5.77 1
[5.78J
2.4
2.95
2.4
2.6
2.7
[5.791
[5.801
[5.8 1]
[5.82l
[5.78]
[5.69 l
3.0
2.7
2.77
[5.831
[5 .69l
[5.35l
We should ment ion the stud ies mad e of percolatio n through a finite array
of random sites. Skal a nd S hk lovsk ii [5.831 a nd Pike an d S eager [5.69l
ap proached the sphere proble m by exa mi ning percolati on be tween the opposite
edges of a cu be conta ining a lar ge num ber n of random sites (up to 3000
in [5.831 an d up to 8000 in [5 .69 ]). Pike an d Seager [5.691 a lso studied the
circle prob lem on a squa re contain ing up to 4000 sites. In the se studies two
layers borderi ng on the opposite edge s of a cube (or sides of a squa re) and
having a th ickness slightly less th an the mean distance betwee n sites were
used to ide ntify percolation. Percolation was registered if two sites belonging
to opposite control layers were found to be part of the sa me cluste r - in
othe r word s, if there was a cha nnel from a n edge to th e opposite edge. As in
the lattice problems, extra polati ng th e dat a to an infinite num ber of sites was
crucia l in UJe determinati on of B, . Pike a nd Se ager [5 .691 obta ined two
values of B d , ) av~aged over th eir realizations of percolat ion: B dI OOO) 2.92 ± 0.03 and B d 8000) - 2.80 ± 0.0 2. Th ey extr ap olat ed to n - co
5.3 Random Site Problems
11 5
according to th e law
jjc (n ) ~ Bc + const·,, - 1/1A ,
(5.3.10)
where A ~ I, and found that B, - 2.7. S ka l a nd Shklovskii [5.83], for th eir
pa rt , ob ta ined the values B,0 5OO) - 3.0 ± 0.1 a nd B, oooo) - 2.9 ± 0.1,
wher eup on they a ba ndoned extrap olati on becau se of insufficient accura cy a nd
used th e va lue B, - 3.0. It is evident th at despite the systema tic differ en ce of
orde r 0.1, th e results of [5.831 a nd [5.69] a re quit e close.
Kur kijiir vi [ ) . ~ )] consi dered percolation in a finit e volume with more
com plica ted boundary co ndit ions. In his simula tio n th e coordinates of "
rand om sites were first ge ne ra ted a nd then the cube was peri odicall y ext ended
in all directions. Percolation was said to exist when at least one site within
the cube turn ed out to be connected with its replica in some fixed direction.
It a ppears likely th at thi s a pproac h should yield th e correct value of the
threshold B, when" - 00. In a ny case, Kurk ijar vi's results were clos e to
th ose for percolation between opposite edges . An aly zing th e data for" = 32,
64, 108, 256 , 1000, a nd 2000 random sites on th e basis of (5.3.10), he fou nd
B, ~ 2.77 a nd A - 0.6 ± 0.25. In addit ion, he exa mined t he va riance of th e
,; of th e quantit y B,(,). He obta ined
dispersion W
arc
-
where
a~c
r,
::::::'n
- 1/1
(5.3.11)
"
and orc are the mean -squ are flu ctuations of
~c
and rei and
It ca n
v ~ 0.83 ± 0.13 (we estimat ed th e error from the gra ph in [5.35]) .
easi ly be dem onst rated tha t as in th e la tti ce problem s, the exponent v found
us ing th e expression (5.3.11) is none ot he r th an th e crit ica l ind ex for th e
correlation radius. Thus we see that the correlation radius exponent
v - 0.83 ± 0. 13 for the sphe re probl em coincides, wit hi n th e margin of
acc uracy, with th e exponen t Vl for lattice probl ems (sec Ta ble 5.2>' This was
the first evidence that the universality of exponents encompasses random site
probl em s as well.
Le t us now discuss problems in whi ch th e bondin g crite rion is fulfill ed not
withi n a sphere, but within some co mplex closed surface Qt, whose lin ear
dimensions increase with some par arneter j . These problems are of major
importan ce in th e th eory of hoppi ng conduc tion. Th e bonding cri terio n, in
general, can be written in the form:
(5.3.I 2)
where ~ > 0, 4> is a homogeneous, increasing, positive function of X ij' Yij , Z;j '
an d th e equati on of the surface Qt is given by ~;j == q, (ru ) - ~. We mu st
find ~, - t he lower bou nd of ~ values wh ich a llows for th e existence of a n
infin ite clust er of co nnected sites . In geo met ric terms, an identical surface Q(
11 6
5. Percolation Theory
is constructed about every site, and by increasing ~ , we must find the value
( - (, which corresponds to the form ation of the first infinite chai n, where
each s ite lies inside th e surface constructed about the precedin g site (a n IF
construction). The OLF construction for a n a nisotropic Q, is not usually as
obviou s as in the case of spheres; in some cases (co ncave surfaces) it is
impossible.
Le t V, be the volum e contai ned by the surface QE' Then it is convenien t
to introdu ce a dimen sionless qu anti ty B - N V, ind icat ing the mean number
of bonds per site. The critica l value ( - (, corr esponds t? th e critical value
(5.3. 13)
In par tic ular, when Q, is a sphere, B, is defi ned by (5.3.3), It is of interest to
find out how the shape of the surfa ce affects the value B, . We ca n prove the
following theorem, du e to Ya, G. Si nai: If a s urf ace Qi call be obtained from
a surf ace
Q~
via a linear transformation oj the coordinates
Xk - l; A kt Xt
(5.3.14)
t
that involves b011l rotat ion and dila tation (A kt are CO il S lam s), then the
valu es B, f or the t wo surf aces are identical. Thi s th eorem implies, for
instance, that the B, of a ny ellipsoid is th e same as that of a sphere. T o
prove the theorem, we note that if not only the surface a bout each site, but
also the coordinates of the sites themselves, a rc tran sform ed, th en the
d ist ribution of the sites in space remai ns comp letely random (only the
conce ntra tion of the sites ca n cha nge), Indeed, the density corre iator per)
which describes the ran dom site system is of the form:
< p (r ,) p(r, ) > ~ No(r ,-r,)
(5.3.15)
And from (5.3.14) we get:
< p (rj ) p(ri ) >
~
N
- D b(rj - ri) .
et A
(5.3.16)
The refo re the natu re of the d istributi on remains uncha nged .
A simultaneous transformat ion of surfaces and site coordinates does not
a ffect the topology of the system. If th ere was no percolatio n prior to the
tran sfor mat ion there will be none a fte r, and, simila rly, exist ing percolation
will not disappear as a result of the tr an sformat ion. Th e th reshold situation
before the tr an sform ati on corr esponds to the situation a fter it. Also, th e
mean number of bonds per site, B, is not a ffected. Th us, if B, is the
thres hold for the surface Q" a nd B ~ is the th reshold for the surface Qi, then
Be .- B ~ .
In connection wit h the question of how the surface sha pe (e.g., concavity)
affects percolation, Skal a nd Shk lovskii [5.83) investi gated the relationship
5.3 Random Site Problems
117
between the sha pe of the surfaee Q! and the corresponding value B, . The
val ues of B, were ca lcula ted for various sur faces with 1500 sites. Recall that
for spheres B« l5oo) - 3 ± 0.1. For cubes B « l5oo) a lso turn ed out to be
3 ± 0. \ , and for tetrahedr a B « 1500) - 2.9 ± 0.1. T hese dat a sugges t that the
va lues of B, a re identical within 0. 1 for all convex figures.
Two co ncave sha pes were also studied . O ne of them is releva nt to hoppi ng
cond uction in IH ype germ anium. It represents a n envelope surface of four
identica l ellipsoids of revolution (see Fig. 6.5>' T he cen te r of each ellipsoid is
at the origi n, a nd th e a xes of revolution a re d irec ted a long the body diago nals
of a cube . The scmimi nor axis of eac h ellipsoid coincides with its axis of
revolution and is sma ller th an the semimajor ax is by a fact or of 4.45. For
this "tetracllipsoid" surface B « 1500) - 2.8 ± 0. 1. T hus it can be seen th at
conca vity lowers B, slightly. The ot her ca lculation involved a "threedimen sional cross", formed by three parallelepipeds positioned a long the
coordina te axe s a nd intersect ing a t the origin. T he equa tions of their edges
are: (I ) Ixl - I, Iyl - I , [z ] - 5; (2) Ix l = I, Iyl = 5, Izi = I ; (3) Ixl = 5,
IyI = I, IzI - I. For this "extremely concave" surface 8 «1 500) - 2.4 ± 0. 1,
which is only 20% less than for a sphere . It is likely th at th e value of B, lies
within the se limits for all concave surfaces.
Pike a nd S eager [5 .69 1 stud ied the depend ence of B, on the surface sha pe
in two di me nsions. T hey found 8, to be 4.4 for a squa re, only 2% less than
for a circle . Th us, we see th at given the dimensi ona lity, cha nges in th e shape
of the bond ing criterion surface (5.3.12) have littl e effect on B, .
It is instructive to compare these approximate invariants with the values of
Be for lattice problem s (5.1.20>' As it turn s out, random site problems
req uire a significantly la rger number of bonds per site than latt ice problems.
In our view this can be a tt ributed to the fact that in the form er case ma ny
bonds are of littl e use. For exa mple, isolated pa irs or trip les of close sites
contr ibute to the numb er of bonds but do littl e to promote percolat ion.
We ca n att empt to find a cha ra cte ristic of the ra ndom site problem simila r
to Za llen a nd Scher's lattice inva riant 8, . In the overla pping figure
constru ct ion (O LF) one ca n ca lculat e the fraction of volume th at is enclosed
by th e sphe res at the percolat ion threshold. Co nsider a ra ndom point A a nd
ca lculate th e probab ility that A does not lie inside any sphere of rad ius r, /2
const ructed ab out a ll ra ndom sites. Th is proba bility is identical to the
probabili ty exp [ - (4,,/3) N(r, /2 )' j that no rand om site lies within a sphere
of radiu s rJ 2 centered a t point A. The probabi lity of A being inside a ny
sphere yields the req uired fract ion of volume, which is
8, = I - exp [ - 4" N(r' /2) ' ] - I - e - 8,/8
3
=
0.29 .
(5.3. 17)
Thi s value is q uite d ifferent from the cri tica l volume fra ct ion 8, = 0.15 for
latt ice pro blems. We think this is du e to the more effective use of "allowed
volume" in a lattice, where all bonds are imm ediately esta blished by tan gent
11 8
5. Perco lation Theo ry
spheres (see Fig. 5.8), In the OLF construction, on the othe r ha nd, such
"thin " cha nne ls are very rar e. Thi s hypoth esis was confirmed in experiments
cond ucted by Fitzpa trick et al. [5.84], who studied percolation in ran dom
close -packed mixtures of har d, conduc ting a nd noncond ucting spheres. T hey
determi ned the critic al fraction of conducting sphe res, observing a sha rp
increase in th e cond uctivity of the system, which corre sponded to the
form ati on of a n infinit e cluster. The filling rati o of their syste m was known to
be I - 0.65. T he product lx, in th eir system t urned out to be close to 0. 15.
We conclude th at the la rge increase in the volume fraction (5.3.17) for
random s ite systems is probably due to the frequent proximity of random
sites.
5.4 Theory of Critical Exponent s
In Sect. 5.2 we discussed the crit ica l quantities of percolati on th eory and
introdu ced th e correspo nding critica l exponents (3, 'Y, and v. Th e numeri cal
values of these expo nents, found through various computer expe riments arc
listed in Table 5.2. In the presen t section we discuss a th eory tha t essays to
connect the vario us cri tical exponents. It is well known th at a similar
ap proach to second-order pha se transit ions in th erm odynami cs was very
successful. Thus, we will naturall y seek an a nalogy be twee n percolati on
problem s and second-orde r phase tran sit ions, notably the ferromagn etic
transition.
Let us recall th at P (x ) is equiva lent to an order par ameter such as
magnetizati on. T he qua ntity x is a nalogous to temperature: th e threshold x,
corres ponds to the Curie temperatu re T~ , the region x < x, corresponds to
the param agn etic region T > T" and the region x > x, correspon ds to the
ferr om agnetic regio n T < T, .
In order to broad en the ana logy, we shall introdu ce into percolat ion theory
concepts equ iva lent to the magnet ic field and the free ene rgy th at depends on
that field a nd on the par ameter x . T his was first done by Kasteleyn and
Fortuin [5.85!. We shall illustrat e the techni que in th e instan ce of the site
problem.
Let us first introduce a n additi ona l black site tha t is not part of the latt ice
(Fig. 5.1 1) . Sin ce th is additional site has some unusual propert ies, we shall
design at e it as Kastcleyn a nd Fort uin's "ghost. " By defin ition, th e probability
of th e ghost being connected with a ny given black site in the lattice is
Fig. S. t I , Formation o f an infi nite cluster through ghost
5.4 Th eory of Critical Exponents
119
I - exp(- h ) , where h is a par ameter equiv ale nt to the dim ensionless
mag netic fi eld " H / k T (" is the mag netic spin moment, H the magnet ic
field>' Indeed , if h '" 0, P (x) will differ fro m zero no matter how small x is,
because a n infinite cluster of black sites is formed thr ough th e ghost
( Fig. 5. 11) . For instan ce, when x « I a nd h « I - whe n th e black sites
are isola ted - it is easil y seen th at p (x) ~ hx .
In the presence of the ghost, the mean numbe r of finite clusters per latt ice
site can be written in the form
(5.4. 1)
where li s is the number of finite clusters of size s per lattice site where " = o.
The fac tor exp(- sh) == [exp (- h ) l' indica tes the num ber of cluster s of s
sites none of which is connected with the ghost.
The qua ntity F (x,h ) is a nalogous to the free ene rgy of a ferromagnet.
Indeed, p(x) and S(x) at h - 0 ca n be expre ssed , respectively, as the first
a nd seco nd de rivat ives of F (x ,h) with respec t to h . T he orde r param eter
p (x ) is de termi ned by the deri vati ve 8F /8h at h -0:
P {x} - x - ~ S
II , -
s
X
+ -8F
8h
I
.
(5.4.2)
h- O
. .
fJ 2F
T he mean cluster size S (x) is det ermined bY t he second dcn vauve - -2 a t
fJh
h = 0:
1
x r- Pix)
S(x) -
(5.4 .3)
Accord ing to (5.4.3), the quant ity S (x) is ana logous to magneti c
susceptibility. T his ca n be dem onstra ted directly by calcu lating an increase in
the magnitude of the infinite cluster dP d ue to a sma ll cha nge in the
magnet ic field dh . Th e probabi lity of a new cl uste r of s sites joining the
gh ost is s dh . Th e infinite cluste r will gain s add itional sites. Summing up
all the clusters an d divid ing by the number of lattice sites we get:
- fJ P
fJ h
I
-
h -O
~ S 211, - [x- p (x )] S (x ) .
(5.4.4 )
s
In orde r to further underscore the a nalogy with thermodyna mics, we used f3 ,
'Y, a nd v to denote the critica l exponents of P (x) . S (x) . a nd L (x) - these
sa me symbols a re generall y used for the the rmodynam ic analogs. By
differenti at ing F(x , h) with respect to either va riable, we ca n produce a '
number of functions a nd critica l exponents that will correspond to a ll
thermodynamic functions and exponents. For example, we can produce an
a na log of heat cap acity
120
5. Percolation Theory
a'p-,
C (r, h-O) - -
aT
I
(5.4.5)
11 - 0
where, - x - x, . If we write Flr.h) for, - 0 a nd h
P(r - O,h )
0:
«
I in the form
h Ili,
(5.4.6)
we arrive at yet another critical exponent
o.
Other exponents can also be
deri ved .
A natu ra l way of con necting the exponents is to formulate a sca ling
hypothesis for small values of 1,1 a nd h (I, \.h « I) . The sca ling
hypothesis posit s that for eac h, and h the singula r part of the free energy is
determined by clusters of some characteristic size and number of sites. These
critical clusters shrink as T and h increase. It is assumed that there exist
exponents y a nd z , such th at when, a nd h become T' - rt)' a nd h' - h f '
the characte risti c size of these cri tica l cluste rs shrinks by a factor of f ; i.e.,
the free energy , which by de fi nition equa ls the number of clusters per uni t
volume, is sca led by a fac tor of f d, where d is the dime nsiona lity of the
problem. In other word s, the free energy sa tisfies the expression :
(5.4.7)
An ana logous expression is also assumed for the correlation functio n:
(5.4.8)
where ~ is a new cr itical exponent. Assumptions (5.4.7) and (5.4.8) should
be complemented by a simple rela tion which follows from (5. 1.5), (5. 1.6), a nd
(5. 1.9) :
I
S(x )[ x-P(x) ] - ~ G ( r,x ).
(5.4 .9)
T his expression is completely a nalogous to the relat ionship between the
mag ne tic susceptibility and the spin corr elati on functi on in the the ory of
ferro mag netism . T hus th e formu lation of th e sca ling hypothesis becomes
equi val en t to th at adop ted in phase tran sition th eory (sec Stanley's
book [5.86]) . We ca n th us recover the usual relati ons betwee n the exponents
by expressing them in terms of y and z using (5.4.7-5.4.9) :
a - 2 - dv - 2 - 2/3 - -y ,
(5.4. 10)
0_.1..+ 1
/3
'
(5.4.1 I)
~_2 _.1..,
(5.4 .12)
dv - 2/3 +-y .
(5.4.13)
v
The first partial formul ati on of the sca ling hypoth esis for percolati on
prob lem s - equiva lent to (5.4.7) but not involving the corre lation functio n -
5.4 Th eory of C ritical Exponents
121
is du e to Essam an d Gwilym [ 5.871. A complete formulat ion was developed
in [5.13, 88 1.
Exp re ssion (5 .4. 13) , which ties together t he most thoroughl y studied
cr it ical exp onents, is of grea test im par lance for per colation theory. Wc sha ll
now demonstr at e how it can be obta ined by ass umpt ions equ ivalen t to (5.4 .7)
a nd (5.4 .8), but witho ut invoking the conce pts of free e ne rgy a nd magn eti c
field . Let us proceed fro m t he assum ption th at (5.1.8 ) a nd (5 .1.10) det er mine
the mean number of sites and the size of the critical cluster (the correla tion
radius) in the abse nce of a ma gnet ic field:
(5.4 .14)
We sha ll assume a lso th at th e number of c ritic a l clu st ers per lattice site,
defined , e.g ., as
",
L
II"
is of th e order of [ L (T) J- d .
In ot her word s, we
assume that the distance between two critical clusters is of the same order of
ma gnitude as the clu st er s th emselves, and ther efore the spa ce is den se ly
packed wit h cr itica l clu ste rs. Eva lua ting S (T) a nd using (5.4.3) a nd (5.4 . 14),
we get:
SeT)
==
L -d s } « Td. - ' 6,
th at is,
(5.4 .15)
"I -2c.-dv.
(5.4.16)
H
d. - 6. Logicall y
The density of crit ica l cl usters is of t he order of s.L- d ==
then, t he densit y of t he infini te clu st er, which "a bsorbs" a lmos t a ll clust er s
exceedi ng the cri t ica l size, is of the sa me order of mag nitude:
P (T,O) _ Td. - 6 .
(5.4 . 17)
Th us
(5.4 . 18)
{3- d v - c. .
If we elim ina te th e exponent c. from (5.4.18) a nd (5.4 .16) , we a rrive a t
(5.4 .13) .
Let us sec if t he expo ne nts' numeri ca l values sa tisfy th e sca ling relat ions.
We begin wit h (5.4 .13) , which co nnec ts exponents f3 , "I, a nd v, the values of
which arc listed in T ab le 5.2. In two dime nsions f3, - 0 .14 and "I, - 2.40,
and from (5.4 .13) we find th at
1.34, which ag ree s with the values of
from Table 5.2; in three dimensions (33 - 0.45 a nd "13 = 1.70, a nd we obtai n
V3 = 0 .87, which a lso agrees with the exist ing numerical va lues.
For th e critical exp onent ex , Kirkpa trick [5.271 cited ex, - 0.65 ± 0.0 1,
and if we subst itute
1.34 into (5.4 .10) we obtain ex = 0.62 , which fits
adequa te ly. Gaunt a nd S y kes [5.891 ca lcula ted the ex pone nt 0, for bond a nd
site problems on various two-dimension al lattices; it turned out to be 18.0 ±
0.75. If we su bstit ute va lues fro m Ta ble 5.2 ("I, - 2.40 and f3, - 0.14) int o
(5.4 .11l we obta in a ver y close va lue : 0, - 18.2.
v,-
v,-
v,
122
S. Percolation Theory
An im portant investigati on of th e scaling hypoth esis was ca rr ied out by
et al. [5.281. Th ey used the series meth od to study th e critica l behavior
of mom ent s of the correlation functio n G (r, x):
DUIlIl
I' n (x ) - ~ r nG ( r. x ) .
(5.4.19)
According to (5.4.19), I'o(x ) is ident ical to S (x ) to within a nonessenti a l
fac tor x - P (x) . and hence its critical beh avior is of the form :
(5.4 .20)
Moment s I' n (x) with
11 '"
0 also tend to infi nity as
ITI -
0:
(5.4.2 1)
According to the sca ling hypothesis. the corre lat ion fu nction (5.4.8) at h-O
ca n be written in the form:
(5.4.22)
Co nseq uently. if
'Y(n ) - 'Y
11
+ II V
is increased by one. 'Y(n) should incre ase by v:
(11 -
1.2....) .
(5.4.23)
et a l. [5.281 ca lculated exponents 'Y, 'Yl2l • a nd 'Y (4) for tri angular and
face -ce ntere d-c ubic lattices. T he calc ulated values a rc equidistant and.
therefo re support the sca ling hypoth esis. Th e values of 'Y a nd v obta ined
in [5.281 a rc listed in Ta ble 5.2.
We sha ll confine ourselves to the above exa mples of testin g the sca ling
hypot hesis. T hese questions a rc discussed in grea ter det a il in a n excellent
review by Stauffer [5 .901.
We ca n conclude th at the connections between criti ca l expon ents deri ved
from the sca ling hypoth esis a rc consistent with the exist ing nume rical values
of these exponents.
N ow lei us proceed to theoretical constructs that not only connect critical
expone nts. but also ca n be used to find their numerical values. First. it is
instru ctive to calc ulate the critical exponents by expa nding the free energy
F(T,h) in powers of the orde r pa ram eter P, as it is done in Land au 's th eory
of second-order phase transitions. Unlike the theor y of ferrom agne tism.
percolati on theory is not subject to symmetry considera tions that rule out the
cubic term in this expansion. Co nseq uently. th e values of th e critica l
expone nts in the self-consistent field app roxima tion arc as follows:
DWlII
{3 - I.
'Y - I .
v-I /2 .
(5.4 .24)
(In La nda u's th eory {3 - 1/2,} Th ese exponents differ rad icall y from the
5.5 Electric Co nd uc tivity o f Ran d om Networks o f Co nd ucting Bond s
123
known num erical values for two a nd three dim ensions (see T ab le 5.2) . In the
th eory of ferromagnet ism th e mean-field exponents a lso d iffer from the correct
valu es for two or three d imensions. Land au 's theory docs hold, however, from
a critica l di mensionalit y de = 4 onward; th is cri tical dimensiona lity
corresponds to th at value of d at which exponent s of Landau's theory sa tisfy
th e sca ling rela tion (5.4. 13),
Toulouse [5.9 11 proposed the existe nce of a simila r critica l d imensionality
in perco lation th eory. a nd suggested that it could be found in the sa me
mann er. By substituting exponents (5.4.24) into (5.4.13) he found d, - 6.
His res ult was lat er corrobora ted by Kirkpatrick [5.27], who calcula ted
ex ponents (3 a nd y for the site problem using th e Mont e Ca rlo method on
simple "cubic" latt ices in th ree, fou r, five a nd six dim ensions.
A la rge number of stud ies have been devoted to ca lculating the critica l
exponents using renorma lization group methods (see. for example. [5.32. 9294]) . In the ea rly day s. most of th e authors based the ir a rguments on the
Ham iltoni an formulation of percolat ion problems proved by Kasteleyn a nd
Fortuin [5.851. More rece ntly. however. it has become more popula r to use
the real-space renor maliza tion group method. A de tailed descript ion of th is
method a nd the results obtained with it ca n be foun d in [5.951. T he valu es
found for the exponents using renormali za tion group methods genera lly agree
with the existing calc ulated values listed in T able 5.2.
5.5 Electric Conductivity of Random Networks of Co nducting
Bond s. Infinite Cluster T opology
In Sect. 5.2 we considered the electric conductivity c {x ) of a latt ice with
conduct ing a nd noncondu ctin g bonds (sec Fig. 5.7) . but we hardly discussed
th e values of th e cr itica l exponent t in the formu la
(5.5.1)
\Ve never menti oned a ny connection bet ween / and th e critica l exponents. In
particular. the exponent I had no a nalog among thermodynamic exponents.
Recall. however. tha t our a im is to utilize percolati on theory to ca lculate the
conductivity of stro ngly inhomogeneous systems. Calcul atin g u (x) is the
simplest problem incorporating both conduct ivity a nd percolation. a nd for this
reaso n a lone it deserves specia l att ention. It is convenient to study the
conductivity u(x ) by modelling th e problem di rectly; thi s method is especially
effective in two dim ensions.
Last a nd Th ouless [5.961 were the first to conduct such a study. T hey
pun ched holes in a squa re piece of conducting gra phite pap er in such a way
th at the cente rs of these holes fell on random sites of a squa re lattice. Then
they me a sur ed the resista nce betwee n the opposite sides of the square as a
fun ction of th e numbcr of holes (holes corresponding to adjacent sites
overlappe d),
124
5. Percolation Th eory
Lat er , Walson and Leath [5.36] measured th e conductivity of a sq ua re
met alli c mesh with rand om sites removed. Their result for th e site problem
was / 2 - 1.38 ± 0.12. Levinsht ein [5.371 investiga ted th e cond uctivity of a
conducting latt ice painted on noncondu cting pap er with aquadag (a gra phite
suspensionl. In orde r to model th e bond problem , ra ndom bonds were cut
out; in the case of th e site problem , holes were punched thr ough lattice sites
in suc h a way that a ny hole would cut a ll gra phite lines passing th rough the
site. Wor king wit h squa res of up to 120 x 120, Levinshtein found th at / 2 1.15 ± 0.15 for both the bond a nd the site problem. In three dimensions th e
site problem was mod elled by means of a 16 x 16 x 16 c ube of standard
resisto rs [5.411. All resistor contac ts were removed a t random sites. The
study of cond uctivity as a functi on of blocked sites led to the value / J - 2.
An oth er meth od of looking for t involves computerized ca lculations of
resist an ce using Kirchh off's equati ons. Kir kp atri ck [5.971 used cubes with
25 x 25 x 25 bonds a nd fou nd I ) = 1.6 ± 0. 1 for the bond prob lem a nd I ) 1.5 ± 0.2 for the site problem . In two dimensions, accord ing to Kirkpatrick
[5.27], 12 - 1.1 ± 0.1. Oniz uka [5.42] studied th e site problem in a
50 x 50 x 50 cube a nd found I ) - 1.725 ± 0.005. S traley [5.38] used a
100 x 100 sq ua re to find 1 2 - 1.10 ± 0.05 for the bond problem and ( 2 =
1.25 ± 0.05 for the site problem. In 25 x 25 x 25 a nd 30 x 30 x 30 cubes
he obtained I ) - 1.70 ± 0.05 and t ) - 1.75 ± 0.05 for the bond and site
problem s, respecti vely.
T he above values indica te that the exponent s ( for th e bond an d site
problem s arc close in va lue, i.e ., thei r numerical va lues do not contradic t the
univer salit y hypothe sis. It seems logical th at these values should also hold for
th e continuum version of this problem (Sect. 5.3l.
Let the re be a two-phase system in which one phase has a loca l
cond uc tivity of unit y, while the other docs not cond uct at all. T hen we ca n
express the depend ence of the system's effective conductivity on the fraction
of volume occupied by th e cond ucting phase as follows:
(5.5.2)
There arc man y physical syste ms whose conductivity is interpreted by
envisaging eith er a latt ice of conducti ng a nd nonconducting clemen ts (5.5. 1)
or a continuous two-ph ase system (5.5.2l. T he former incl ude the tun gsten
bronze systems [5.98], and th e latt er includ e expa nded liquid Hg [5.991,
a lka li-am monia mixtures
(5.43, 99 1, gra nula r meta l films [5.1001, a nd
agg regated films [5.10 11. In a ll the se syste ms there is a metal -dielectric
tran siti on as de nsity and composition cha nge. The exponent / was obtained
from exper imental dat a : 1 - 1.8 ± 0.2 and 1.9 ± 0.2 for thr ee-dimensional
systems [5.98, IDOl, a nd 1 - 1.l 5± 0.20 in two d imensions [5.101 1. T hese
results ag ree with the numerical values discu ssed earlier.
In addition to d irect a pplica tions, th ere a rc several othe r physical problems
th at involve the ca lculation of o (x }. O ne is th e problem of long spin wave
5.5 Electric Conductivity of Random Networks of Conducting Bonds
125
spect ra in dilu te Heisenberg ferrom agnets [5.971. Another is a related
probl em of th e beh avior of cri tica l temperature T, (x) of dilut e Heisenberg
ferromag ne ts nea r th e per colati on threshold [5.10-1 21. A third involves th e
elasti cit y of ge ls <elast ic polym er solutio ns) [5.10 2, 1031.
T he u (x ) problem ad mits a very int er esting ge nera liza t ion. Inst ead of th e
co nduc ting a nd insula ting bonds, one ca n int rodu ce two types of bonds whose
resista nces a re very d iffere nt, th ough both a re finite . Alte rnat ively, one ca n
study th e con duct ivity of a random two-p hase mi xtu re, whe re one phase
co nd uct s much better th an th e othe r, i.e., u , » u, . In th ese ca ses u(x) is
finit e at
x
<
X c>
and the growth of
0" (x
) for x -
Xc -0
is described by a
new c r it ica l index. An other index descri bes th e vari ati on of u (x, ) with
va ry ing rati o u ,!u, . These indices can be related to t wit h th e help of the
sca ling the ory [5. 38, 104, 105). At a finite freq uency the cond uctivit ies
a nd
a re complex va riables. The sca ling the ory mak es it possible to find the
crit ica l index which describes the d ivergence of th e effective d ielectric
permittivity near the percolat ion threshold [5.105, 1061. For ani sotropic
co nd uc t ivities add it iona l indices arise which describe weak en ing of the
a nisot ro py as x - x; [5.107- 1\ 01. In a magn eti c field one has indi ces to
describe th e c rit ica l behav ior of th e Hall coe fficient near x, [5.56, \08, \09,
u,
u,
\1 \ I.
T he list of ph ysica l a pplica tions of c (x ) could be exten de d. Despite t he
obvious signifi cance of these applications. it is not our aim here to give a more
deta iled a na lysis. In th is cha pte r th e qu a nti ty u (x) is of interest to us ma inly
as a sou rce of inform at ion a bout the topology of th e infinite cl uste r near th e
per colation th reshold . Thi s topology is of conside ra ble inte rest beca use, as we
sha ll see lat er, ca lculat ing th e conduct ivity of a st rongly inh om ogeneous
medium is equivalent to calculating the conductivity of a certain infin ite
clust er.
Some features of thc in finite c1ustcr wer e disc ussed earlier . Wc know its
density is P (x) a nd the size of its pores - its "period" - equa ls the
co rrela t ion len gth L (x) . Th is infor ma tion is in itself insufficient to ca lculate
u(x) , however . Thus we mu st mak e some assumpt ions as to t he topology of
th e infinite clu ster, or co nside r a n a ppropria te model, a nd the n ca lculate u (x )
expressing exponent t in terms of other infinite cluster exponents. in particular
t he expo nent v. Then by com pa ri ng our results wit h th e known va lue of t , we
ca n selec t t he best model for the infinite clu ster; t his a pproac h is discussed
be low.
5.5. 1 Dead Ends
Let us co nside r th e co nd uc tivity of a large cube of
latti ce units per side.
Suppose th e infinite clu ster resembles a simple cubic la tt ice (or, rather , a
su per-la tt ice bui lt on the init ial la tt ice), T hen \ / 3 of th e bond s belonging to
the infinite clu st e r form ch ain s t ha t connect th e opp osit e ed ges of th e cube;
i.e. , the number of bond s in the se ch ain s is a pproxima te ly 1/3 p(x) e3 Each
cha in h a s e sites, a nd therefore t he number of par allel cha ins is 1/ 3 p(x)e' .
e
126
5. Perco lation Theory
e.
Eac h chain's resistan ce is proporti onal to its length
As a result , the
resistance of the whole cube, R, is prop ortional to [ 1/ 3 P (x )
For the
conductivity we thu s obta in
e
e' ]-'.
(5.5.3)
that is, I - {3. We know, however, that I is much grea ter th an {3 (sec
T abl e 5.2) , and th erefore the conductivity c (x ) near th e percolation threshold
is mu ch lower than should be ex pected from (5.5.3) . Thi s mean s that an
overwhelming portion of the infinite cluster docs not conduc t. l .as t a nd
Thouless [5 .96] concluded that most of the infinite cluster 's mass is in dea dend cha ins (Fig.5 .12a), T hese chai ns contribute to Pl.x) but not to u (x).
Thi s conclusion should not be considered definitive: the inefficiency of the
infinite cluster from th e standpoint of conductivity may be a lso d ue to man y
long chains in par allel with short segments of other chains (Fig. 5.1 2b),
b)
aj
f'i2. 5.12. (a) Dead ends and (b ) redundant
loops on a conducting chain
5.5.2 T he Nodes and li nks Model
T he model of an infinite cluster ncar th e percolati on threshold was fur ther
refined
by Skal
a nd
S hk lovsk ii [5.56]
and
independentl y
by
De Gennes [5.10 21, and it is now often referred to as the SSDG model. Th is
model assumes that the infinite cl uster conta ins a backbone network with a
characte risti c size (mean distance bet ween nodes) of the same orde r of
magnitude as the correlation radiu s L - (x - x, ) -, (Fig. 5. 13) , a nd that this
backbone ca rr ies the dead ends. The part of th e backbone between the
near est nodes was termed a link. It was assumed tha t a backbone could be
isolated in suc h a ma nner th at at least half of the bonds of a link would not
be duplica ted. The researchers ad mitte d the possibility of the link being very
twisted and resembling a polyme r tangle (Fig. 5. 13), In other words, it was
supposed th at the len gth of a link i. could be considera bly gre at er th an the
distance betw een its ends L measured along a stra ight line. In other words,
for x - x,
i ==
«
I, in units of the lattice constant, one has
(x - x,) - t ,
where
I ;;' v .
(5.5.4)
5.5 Electric Conductivity of Random Networks of Conducting Bonds
l
127
I: i ~ . 5. 13. Schematic representation of
the nodes and links (SSDG ) model (without dead ends). The link length along the
twisting curve is l.
Th e a bove nodes a nd links (or SSDG) mode l could neith er be dis proved
nor con vincingly support ed by d irectl y examining t he infinite cluster [5.561.
We would like to emphasize, therefore that the SSD G model ca nnot yet be
conside red firmly estab lished : t he infinite clu ster conta ins st ructures of a ll
shapes a nd it is not evident a priori that suc h a simple structure as the
backb on e ca n be isola ted in it.
T his reser vati on notwith stand ing, let us a tte mpt to compute the quan tity
u(x ) of th e backb one wit hout recour se to add itiona l assumptions. T o t his end
we first find the exponent S thr ough the following rigorous a rgument [5.561.
Let the re be a backb one that corres ponds to some value x > x, . We cut
eac h lat tice bond wit h th e probability (x - x) x -I . As a result , the frac tion
of unbr oken bonds will become x [1- (x - x, )lx J - x" i.e., we will arr ive at
th e percolati on t hreshold of th e origina l latt ice, Since t he links a re not
duplica ted, to cut a link we need only cut one of its constituent lattice bond s.
Th erefor e t he probability of each link bein g broken is proportional to its
length an d equa l to i (x - x,) Ix. If we wish th e backb one to approac h the
percolat ion threshold, we must brea k a certa in fraction of the links y, . T his
qua ntity, like all percolation t hresholds, is of t he order of severa l tenths. If
we write t he condition for th e backbone to break in th e form
x - xc ~
- - - L -Yc
I
X
(5 .5.5)
we find th at
i
<r
(x - X, ) - I ,
(5 .5.6)
a nd therefore S - I . In two d imensions v > J a nd therefore v > S, i.e., the
SSDG mode l docs not hold. In th at case , the d uplicat ion of links is of prime
importa nce. If d ~ 3, t hen v < I an d t here is no contra diction. For d - 3,
in part icular, VJ == 0.9, and from (5.5.6) we find th at i == L 1.1. T he
twisted ne ss of th e links turn s out to be quite minor when compa red to tha t of
128
5. Percolation Theory
ra ndom pat hs (L == L 2) and even when eom pa red to tha t of self-avoiding
pa t hs (L == L I/J).
N ow we ea n ea lcula te uCtl. If th e resist an ee of one bond is R o, th en th e
resist a nce of a cube of side L is of the order of
(5.5.7)
R - R oL .
O n a sca le larger th a n L the infinite clu ster is hom ogeneou s, and the refore
u (x ) can be ca lcula ted as the effective conductivity of a cube of side L eq ua l
to the correlation radius. Throughout this section we measure the correlation
rad ius L and length i in units of the lattice constant to . In dimensional units
t he co rre la tion radius is Leo a nd for uCtl we ge t
u(x)~
I
()d 2 '
R Leo
(5 .5.8)
If we subst itute (5.5.7) an d (5.5.6) int o (5.5 .8) we get
or
/ - I
+ " (d -
2) .
(5.5.9)
T hu s, in th ree dime nsions th e lin k model yields [5.561:
t
s - I + ") .
(5.5.10)
If we subs ti tute t he va lue ") - 0 .9 we find / ) = 1.9, whic h is somew ha t lar ger
th an th e nu merica l value / ) = 1.7. Co nseq ue ntly th e S SDG model is not
ac cu ra te. It is, however, of grea l heurist ic im port a nce because it is simple
a nd g ra phica lly clear. It will be used to illustrat e cer tai n resu lts of th e
hopping conduction th eor y.
T he SSDG model has been used in th e the or y of dilute fe rroma gnet s
[5.1121 a nd granul ar supercon ductors [5.1 13, 1141, as well as in th e t heory of
th e Hall elTect a nd co nd uct ivity a nisot ropy in di sord er ed systems [5.56, 108-
I I 11.
R ecentl y, th e S S DG mod el was further developed by Pik e a nd S tanley
[ 5.115 J. They proposed ca lling a bond of t he backbone red if br eakin g th is
bond break s t he link itself, a nd blu e ot he rwise. T hey fur lher sugges ted th at
th e bl ue bonds form blobs link ed by singly connected c ha ins. Pike a nd
S tanley developed a prog ram which (in th e two-dimensional case) a llowed
them to calcu late the number of red a nd blue bonds in the backbone . It
t urne d ou t th at alt hough most of th e bond s a re blu e, th e num ber of red bonds
in the link between two nodes L I diverges as (x - xc) -l when x - xc. This
appears na tu ral not only for th e two -dimensiona l case but a lso for any
num be r of d imen sions. Indeed, th e blobs do not break if the fraction x of
I
conducting bonds is decreased little. Therefore to evaluate L , one can use an
a rgu me nt sim ilar to th at leadin g to (5.5 .6) , repl acin g L by L' in (5.5.5) a nd
5.5 Electric Conductivity of Random Network s of Conducting Bonds
129
(S.S.6) . A more rigorous de riva tion of the relation L' cc Ix - X C I- I based on
the sa me ideas, was given by Conglio [S.1 161 a nd S traley [S.1171.
T he modification of the SSDG model propo sed by Pik e a nd Stanley mak es
it more flexible. In pa rti cular , the t rue length of a link between two nodes can
now be g rea ter t ha n L', d ue to the la rge nu mbe r a nd size of the blobs. For
example, in the two-dimensional case it can diverge as fast as L or even
faste r. Thus, the objec tion aga inst usin g th is mode l in t wo dimensions, wher e
v > I, is rem oved . An oth er im port a nt ad vant age of Pike a nd St anl ey's mode l
(which is now often called the nodes, link s, a nd blobs model) is th at th e blobs
ca n conta in th e backb one eleme nts (nodes a nd links) of smaller size. T his
provides a self-simila r geometric structure of the backbone as x is varied
[s . 1161 (see a lso below). A self-similar d roplet model of the infinite cluste r
which resem bles Pike an d St a nley' s model was independently proposed by
Sarychev a nd Vinogradov [S.1 18 1. A deta iled discussion of othe r
im proveme nts in the SSDG model ca n be found in the work of S traley
[S.1171.
5 .5.3 The Sca ling Hypot hes is a nd Ca lcula tion of th e Co nd uctivity a(x)
Levinshtein et al. [S.11 91 suggested a n a pproach ba sed on a n extension of t he
sca ling hypot hesis. T hey assu med th at as we a pproach the percolation
t hresho ld the large-scale structure of the network remain s simila r to itself in
th e sen se that its topology rem ains the same a nd its linea r di mensions c hange
proportionally with the correlating distance L = Ii 1-', where 7 - x - xc'
S imply put , they proposed tha t t wo photogr a phs of the infinite cluste r a t
T - T , a nd T - T , will match if thc sca le of one is cha nged by a fac to r of
(r ,/T,) ' . O f course, this does not apply to the small-scale st ruc ture , whic h
conta ins a minima l length correspond ing to t he latt ice consta nt.
The seco nd ass umption mad c in [S.11 91 is th at th e resis ta nce of the
infini te clus te r is de te rm ined by its la rge-scale st ruct ure. T he two
assumptions yield a simple connection between the exponents t and v, Let us
examine our system at
7 "'" 7 1
and
7 "'"
rz. such that
T2
<
il '
As above, we
rep lace it with a syste m of randoml y ta ngled a nd int er conn ected wires. If T
become s sma ller the d imensions of t he syste m (but not t he th ickness of t he
wiresl) increase by a fac tor of L (T, )IL (r ,) - (T,h,) ' . In st udying t he
conductivity we must examine a finite system between two flat metal contacts.
We assu me that t he d ist a nce between th e contacts is a lso scaled by a factor of
(" I, ,)', a nd so is th e pot enti al d iffere nce between th e contac ts . S ince we a re
interested in specific conductivity, a change in the volume of our system is of
no importance.
The curre nt dist ributi on will not be a ffected by cha nges in t he dimen sions
of the syste m: since the electric field rema ins consta nt, so does the curre nt
flowing th rou gh a ny given wire. The macroscopic density of the c urr ent will
fall as, fa lls, however .
Consider an area element that is perpendicular to the average direction of
the c urr ent. T he num ber of channels pie rcing t his ele ment will decr ease by a
.. .
. ... , ... vuU I V Il
I ll t"U l y
fac to r of [ L hI ) / L (" ) J' - (,,/, ,)" beca use of the inc rease in sca le.
Co nseq uently, the cu rren t flowing through thi s clcm ent an d also thc
cond uc tivity dec rea se by a factor of (,,!T I ) " . Th us, in three di mensions one
has:
a (x)
0:.
T 2)' ,
i.e.,
/ - 2 11 .
(5.5 . 11)
S imilar a rgu ments hold in t he two-dim ensiona l cas e, but her e one should
conside r a n eleme nt of leng th ra ther than a rea . T hus, when d - 2:
a(x)o:. T)' ,
i.c.,
/ """V.
(5.5.12)
If we su bstitute the va lues", - 1.33 a nd "J - 0.90 into (5.5. 12) a nd (5.5. 11)
we get / , - 1.33 a nd I J = 1.80, which is close to the ex pe rime ntal values if
slightly highe r (see, however , the most rec ent Monte Ca rlo calcul at ions with
ve ry la rge a rrays, which gave I , - 1.3 1 ± 0.04 [5.39] a nd / , - 1.26 ± 0 .03
[5.40 ]) .
Recen t years have seen furthe r developments in the the ory of infinite clu st er topology, based on sca ling a rgume nts [5. 120, 121 J. Especia lly
interesting is t he a pplication of t he idea of fracta l d imensionalit ies to this
prob lem [5. 1211.
In conclusion, we should ment ion t ha t the ex ponent I ca n a lso be
calcula ted by reno rma liza tio n grou p met hods. T his was first done by Walson
a nd S tinchcomb e [5. 1221. Referen ces to subseque nt work in th is a rea ca n be
found in [5.11 71.
5.6 Percolation Theory and the Electric Co nductivity of
Strongly Inhomogene ou s Media
In S ect. 5. 1 we sa w th at th e ca lculation of hoppin g cond uct ion ca n be redu ced
to the pr ob lem of the cond uctivity of a rand om network (Fi g. 4.6) with a n
exponent ia lly wide ra nge of resist a nces.
An a na logo us problem ca n be formula ted for a continuum. For example,
suppose we mu st find the effective conduc tivity of a med ium whose local
conductivity ,,(r) fluct uat es widel y, i.e.,
(5.6. I)
( [~ (r) - ( ~ ) l ') ~ I .
(5 .6.2)
where ( . .. ) mea ns space averaging. Such, for instance, is the na ture of a semico nd uc tor's local co nduct ivity in the pr esence of la rge-scal e pot enti al flu ctuation s which bend th e co nd uc tio n ba nd (Fig. 3.5) . In th at case ~ ( r) = e(r)/k T,
wher e e (r) is the distan ce fro m the Fe rmi level to th e bo ttom of the co nduction
band .
T he p roblem of a n exponentially wide ra nge of resistivit ies ca n also be
formulated on a latt ice [5.1231. Imagine a simple cubic latti ce with random
5.6 Percolation Theory and the Electric Conductivity
131
resistances betwee n adjace nt sites. We write these resistances in the form
(5.6.3)
where R o is a constant a nd the rand om variable E' is un iformly distr ibu ted
the int er va l - ~o .;; E' .;; ~o, wit h ~o » J. We must find th e cond uctivity
t he latt ice a nd d etermine its de pendence on the pa rameter ~o .
All t he above systems ca n be termed st rongly inhomoge neous med ia
ind ica te t hei r main fea tu re: a n exponentia lly wide range of loca l valu es
in
of
to
of
conductiv ity . T he present section is concerned with developing a method for
ca lcula ting th e cond uctivity of such med ia based on percolat ion th eory. We
shall deal wit h a simple cubic latt ice, for such a latt ice provides a gra phic
example of th e met hod. Also, we believe lattice problem s are simpler to gras p
th an ra ndom site and contin uu m problem s. Fina lly, the problem we have
chosen was subjec ted to deta iled machin e computa tions, which will permi t us
to di rectly tes t ou r met hod of ca lculat ing th e conductivity.
T his section is orga nized as follows. First we will give a derivat ion that
will a llow us to det ermine the exponentia l fac tor in effective cond uctivity.
Then we will compa re our result with numerical ca lcu la tions. Anot her
derivation will then be introduced to find not only the exponent but also the
pree xponen tia l factor acc ura te to a nu merical coefficient. Finally, we will
illust rate ou r results in terms of the SS DG model.
We will use th e following a pproach to find th e condu cti vity expo nent for a
simple cubic latt ice with resista nces (5.6.3) . Select a valu e ~ in t he interval
-~o < ~ < ~o a nd rep lace a ll resistan ces with E' > ~ by infinite resista nces,
thu s breaking th e circuits. Let t he cond uctivity of a lattice th at corres ponds
to a cert ai n ~ be IT (~). Ob viously, th e network cond uctivity we want to fi nd is
IT(~o). T he chose n va lue of ~ determ ines the probabilit y of a random
resistance not being broken:
(
x (~) -
f
(5.6.4)
F(f)d E' ,
- (,
where F (~') is th e dist ribu tion function of ~, whic h is de fined by the natu re of
th e prob lem as follows:
(5.6.5)
I ~I > ~o
Fro m (5.6 .4) a nd (5.6.5) we find
x (~) _ ~o + ~ .
2 ~o
(5.6.6)
If ~ is near to - ~o the quantity x(O is sma ll, and u nbroken resistan ces form
132
5. Percolation Theory
isolated clusters; in thi s case a (~) - O. Let us slowly incre ase the value of ~ .
Wh en ~ reac hes the thr eshold value ~" which cau ses x (~) to equ al the
percol ation thresh old x , (b) for the bond problem , an infinite cluster of
unbroken resistan ce is formed . According to (5.6.6) the critical valu e ~, is
det ermined by th e cond ition
~o
+ ~,
2~0
~ x, (b) .
(5.6.7)
If we furth er increase ~ from ~, to ~, + I , th e presence of an infinite cluste r
will ca use the conduc tivity a (~) to increase ra pidly (Fi g. 5. 14). Th is is due tu
th e rapidly diminishing corr elation radius:
(5.6.8)
eo
is the period of the cubic latt ice. Th erefore th e numbe r of par allel
wher e
conduc ting cha ins in t he infinite clu ster network a lso incre ases rapidl y. At
the sa me time, if ~ cha nges by less than I, the cha nge in the individual
resistances th at make up the cluster is negligible. Thus, when ~ -~, « I
the quantit y a (~) incre ases acco rding to a power law :
aW
cc (~ -
c»:
where b
>
0.
ri~.
o
r
5.14. Schematic dependence of the conductivity
the largest exponent ~ of turned-on resistances
a( ~) a ll
I
Let the infinite cluste r th at is form ed when ~ a nd ~, differ by about I be
ca lled th e critica l subnetwork. The resistan ce of the critical subnetwor k is
determined by its highest resistances (by definition these resistances cannot be
shunted by lower ones, si nce th at would allow for percolation at ~
Co nseq uently, for th e critica l subnetwork conductivity we have
<
~,).
(5 .6.9)
Here we have written the exponential factor to within terms of order unity.
Further incre asing ~ cannot significantly a lte r
aW, even
th ough the den sity of
the infinite cluster network will continue to rise. This is so because if we
increase <- <c several times over, we will introduce resistances exponentially
lar ger th an ex p (~, ). Despit e th eir large number, the new cha ins will not alter
a(V significantly becau se th ey are shunted by the critical subnetwork (see
Fig. 5.14). In thi s manner, the critica l su bnetwork determines th e ord er of
5.6 Percolation Theory and the Electric Conductivity
133
magn it ude of the ra ndom net work's conductivity, i.e.,
(5.6. 10)
We have a rr ived at a conclusion that is both import a nt and, as we shall see
lat er , applicab le to a ll inhomogeneo us systems: if elements of a medium a re
turned on in the order of increasing resistance , then the e ffective
conductivity's exponentia l factor is determined by those clements tha t fi rst
crea te percolat ion. T his conclusion, first formu lat ed in works by Ambegaokar
et a I., Pollak. a nd S hklovskii and Efros [4.21-23], is the found ati on of
hopp ing conduct ion the or y. T he preexponen tia l factor ero is evalu at ed below
[see (5.6 .12) I.
Let us ca rr y our model problem to the end. T abl e 5. \ lists x, (b ) - 0.25
for a simple cubic latt ice. From (5.6.7) we find ~, - -~oI2, a nd conseq uently
the exponential dep endence of condu ctiv ity on ~o ca n be writt en in th e form'
(5.6.11)
This form ula (5.6 . 11) holds only if ~o »
I, beca use only then does the wide
range of resistances permit us to isolate the critical subnetwork and express its
cond uctivity in the form (5.6.9), Wh at is the natur e of the factor ero? We
ca n immedi a tely observe, using the d imensionali ty argument, that
(5.6.12)
where eo is the latti ce period an d c<~o) is a dim ensionless coeffi cient th at may
contain some power of ~o. We will find this power later on; wha t is important
now is th at when ~o is lar ge, ero depends on ~o accor ding to a power law and
therefor e ca nnot com pete with th e exponentia l fac tor in (5.6. 11),
In or der to check the ideas und er lying th e de rivatio n of (5.6. 1I) ,
Kirkp at rick ca lculated the effective conductivity of a \ 5 x 15 x 15 cube as a
function of ~o . T he va riable f was selected by a ra ndom nu mber genera tor ;
the voltage was set a t 1 a nd the opposite edges of the cube were tested for
current. Thi s was done by solving Kirch hoff's eq uations, which dicta te
conserva tion of current in eac h site, on the compu ter. T o check (5.6.11) we
can plot In[ er (~o) /er (Q) 1 versus ~o [here er(Q) == ( R oeOi - ' is the cube 's
conductivi ty at ~o = 0 1. Accordi ng to (5.6. 11), when ~o » I the relatio n
should have a slope of 112. In his first study [5. 1231 Kirkpatrick tested
va lues of up to ~o == 7, a nd he found that the hig hest points fit a slope of 1/3
bett er tha n 1/ 2. T his led to doubts a bout the percola tion met hod.
Subsequent investiga tions by Seager and Pike [5. 1241 and Kirkpatrick
himself [5. 1251, however, dem onst ra ted that a t sufficiently high values of
I.
The negative value of E(' is due to the fact that f varies in the interval - Eo ~ f ~ Eo. In
hopping conduction theory the values Elj in (4.2.34) are positive. and consequently g, > O.
134
5. Perco lation Theory
FiR· 5.15. Dependence o f the conductivity a fo r a mod el la ttice problem
o n the parameter {a. which characterizes the dispersio n o f resistance
expo nents. Circles represent the
results of calculatio ns by Seager and
Pike [5. 124]. The line through the
last point s is almost parallel to the
coo rdinate angle bisecto r ( - . - . - ).
i.e., its slo pe is nearly 112. The
dashed line co rrespo nds to (5.6. 18)
for A = 1
/
/
/
/
/
/
/
/
2
/
/
,.. /'
/
/
/
/
/
/
/
/
0/
/'
o
5
/0
/5
s,
~o(~o ~ 9)
the percolation met hod yields adeq ua te results. T his ca n be seen
in Fig. 5. 15, which is based on the data from [5.1241. Th e coordin a te angle
bisect or has a slope of 112. We see that the da ta points grad ually approac h a
line of a ver y similar slope, although if we plot points when ~o is sma ll a nd
d raw a line, the slope ca n be significa ntly less th a n 112.
Th us we see that the percolation method is quit e acc ura te in predict ing
the exponen tial fac tor. Neve rthe less, qua ntita tively it can only be used in the
presence of a wide range of resistances.
We will now proffer a nother de rivation of th e effective conductivity a(~o)
for th e sa me problem, which will help us determine not only the exponentia l
factor but a lso the power depend ence of the preexponen tial factor ao on ~ o.
Only th e forme r is of interest in th e overwhelming majori ty of experimental
data . Consequently, if the reader finds our first der ivati on convincing and is
not part icula rly interested in the hopping conduct ion's preexponent ial factor,
he may proceed to the next section.
T he alterna tive deri vat ion of a (~o) was fi rst proposed in [5.35, 126],
Imagi ne a finite cube of side e cut from a n infinite la tt ice. Let us break all
resistances within the c ube and then connect them in ascending order until
there is percola tion bet ween opposite faces. Let ~" be the grea test resistance
that has to be connected. If the cube is sufficiently sma ll, th e connected
resistan ces will differ significa ntly. In par ticular , the grea test resistance
~" will be much grea ter th an the next connected resistance a nd will thus
alone de termine the resist ance of the cube. Therefore
r
r-
(5.6.13)
If we calculate th e resistivity of a cube of side
e we get from (5.6. 13);
(5.6.14)
S ince the cube is finite, the value ~" will fluctu at e from one realizatio n to
5.6 Percolation Theory and the Electric Conductivity
135
a nother, depending on the cube' s locati on in th e infinite latt ice. The qu anti ty
1;" is con nected through (5.6.6) to the percolation threshold x" of a finite
was discussed in Sect. 5. 1 and is
cube . The mea n-squa re fluctu ation of
given by ( 5. 1.12) . We find from (5. 1.12) that the mea n-squa re fluctu ati on of
I;,t equals
x"
,
~I;" - B 1;0 [
e
1/,
to
(5.6.15)
]
Here v is the co rrelation radius exponent in three dimen sions (v::::: 0.9), B' is
a numerical coefficient of order unity, a nd unlike (5.1.12), where t was in
units of lattice consta nt to , the length t is in dim ensional unit s.
Let us gra d ually incr ease t . From some point th e maximu m resistance
will not be a lone in det ermining th e cu be's resistance; th e num ber of
significant resistances will grow in proportion with the cube 's volume. When
this happens the qua nt ity Ut will cease to depend on t a nd will equ al the
cond uctivity of the mac roscopic system. Another crit erion for the tran sition
from (5.6 . 14) to macroscopic conductivity, which occurs as t incre ases, is the
disa ppeara nce of relative ly la rge fluct uati ons of Ut . Th is will happen when
th e mea n -squ ar e fluctu at ion of 1;" approaches unity, i.e., acco rdi ng to
(5.6.15), wh en
t '" t o I;ci == L o ·
(5.6. 16)
The resistivity of a cube of side L o can be found from (5.6. 14) if we note th at
in such a cube the dilTerence between 1;" and 1;, is negligible:
-
UL. -
(R 0 L 0 ) - 1c - I. .
(5.6.17)
But as ab ove, the elTective conduc tivity
(JL.,
U
is of the sa me order of magn itude as
that is
u(l;ol
==
(R oL o) - Ie- ("
or
(5.6.18)
where A is a numerica l coefficient of order unit y. Co mparing (5.6.18) a nd
(5.6.10) we see that both derivat ions lead to the sa me exponentia l factor. In
addition we have found the power of 1;0 in the preexponen tia l fac tor of uo.
Note that according to (5.6 .8) the import a nt lengt h Lo is none other th an the
infinite cl uster' s correlation radiu s. Thi s cluster first appears when the
exponential index of connected resista nces I; exceeds 1;, by I. Previously we
termed th is cluster the cri tica l subnetwork. T hus we see once aga in that the
effective c onductiv ity of a lattice is determined by its cr itical subnetwork.
136
5. Percolation T heory
The unknown value of the coefficient A mak es it difficult to compare
(5.6. 18) with numer ical da ta . Noneth eless, we plotte d a (~o) aga inst Eo
accord ing to (5.6. 18) for A = I (Fig . 5. 15, das hed line). It ca n be seen th at
the formula yields good results for bot h the magn itude of a(~ol a nd the cu rve
in ge nera l. In a wide interval of ~o (3 <;; Eo <;; 16) , numerica l values of a(~ol
deviat e from (5.6. 18) wit h A = I by at most 50%. Thus, the agree ment
between (5.6.18) and num erical dat a can be conside red satisfactor y.
In conclusion let us illust ra te the result (5.6. 18) wit h a concrete model of
th e infi nite cl uster - the SSDG model. First we define the critical
subnetwork in terms of this model and then calculate its resistance. The
cri tica l subnetwor k is that part of th e origina l resistor network th at ca rr ies
almost all current and determines the lattice's conductivity. As above, to
create a critical subnetwork we must turn on all resistances with
<;; ~ = ~c + I . We have a lready found t hat the corre latio n radius of t he
critica l subnetwork is L o == eo~o . Eac h link of th e critica l subnetwork is a
cha in of resistan ces in series, and th e length of thi s cha in Lo can be found if
we substitute ~ - ~c - I into the for mula following from (5.5.6) and (5.6.6) :
r
L-
eo(~ - ~c) -I~O.
(5.6.19)
W e get
Lo ==
eo~o .
(5.6.20)
From (5.6.20) we find t hat th ere a re a bout ~o consec utively connected
of
resistan ces in the link cha in of the critical subnetwork. T he exponents
th ese resistances are dist ribut ed in th e inte rval from - ~o to ~c + I - 1/2 ~o + I.
r
It ca n be shown t ha t the values of ~' a re uni formly dist ributed in t his
inter val , and th erefore if we lab el the values of a link in descend ing order:
r
(5.6.2 1)
r
the mean d ifference between consecutive values of
will be of orde r unity.
Th is mea ns t hat th e fi rst resistan ce is rou ghl y e times th e second, the second e
tim es th e third, a nd so fort h. Th e resistan ce of suc h a cha in is det er mined by
t he first <the highest ) resistanc e, which we sha ll often refer to as the critica l
resistance:
(5.6.22)
T h us we have fou nd t he link's resista nce. Thi s resistan ce a lso represent s th at
of a cube of side L o. On a scale larger th an L o t he critica l su bnet work is
homogeneous, a nd hence t he macroscopic conductivi ty is of the sa me order of
magn itude as the conductivit y of the sa id cube. Th erefore
a
==
(RLo) - 1 == (R oLo) -le - /. ,
i.e., we aga in a rrive a t (5.6. 18l.