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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4. A General Description of Hopping Conduction in
Lightl y Doped Semiconductors
T his chap te r is conce rned with the ba sic experi me nta l facts related to hoppin g
conduction, and the simplest models used in their interpretation. Section 4.1
describ es the range of temp erature s and degrees of compe nsa tion for which
electrical conduction in semiconductors occurs by the hopping mechanism. It
a lso shows t he typic al dep endences of hopp ing cond uctivity on th e
tempe rature and imp urity concentra tion. Secti on 4.2 shows how, following
Miller and Abra ham s, one can reduce t he prob lem of calculating the hopping
condu ct ivity to th at of ca lcu lating the conductivity of a ra ndom network of
resistors connecting donor pairs. N aive approaches to that problem based on
averag ing eit her resistances or conductances are critica lly co nsidered .
4.1 Basic Experimental Facts
At high temperat ures semiconductors possess an intrinsic e lec trica l
conductivity du e to th erma l ac tivation of ca rriers across t he gap separating
the va lence and the conduction bands. The intrinsi c carrier conce ntrations n
of electrons and p of holes arc exponen tial fun ctions of temperature [4. 11 ,
viz.:
(4. 1.1)
Here me and lrlh arc the masses of electrons and holes, respectively, and Eg is
the width of the forb idden gap.
Due to the large activation energy Eg/ 2, the intrinsic carrier concentration
decreases very rapidly with temperatu re. At sufficiently low temperatures it
becomes less t ha n t he concen tra tion contributed by impurit ies. In th is region
the conduction is enti rely determined by the nat ure an d concent ration of
impurities, and is t herefore called extrinsic.
Figure 4.1 shows schema tically OIi a semilog plot the inverse temperat ur e
dependence of t he resistivity of a lightly doped semiconductor. The
temperatu re range A (where the slope is nearl y vertical) corresponds to
intrinsic conduction, while ranges B - D correspond to extrinsic conduction.
If we ar c dealing with sha llow imp uri ties whose io ~i zatio n ener gy is much
lower t han Eg , t hen there exists a temperat ure range B, ca lled th e sa turation
range, in which all the impurities are ionized and hence the carrier
4 , 1 Basic Experimental Facts
Ig p
75
Fig. 4 . 1. Schematic temp erature dependence o r the resistivity of a lightly dop ed semico nducto r (A) Intrinsic
co nduction range . (8) Saturatio n range o r impurity con du ct ion . (C) Freeze-out range. (D) Ho pping co nduction
range
A
B
C
D
conce nt ra tion in th e ba nd is independe nt of tempera tur e. In th is range the
tempera ture dep endence of the resistivity is entire ly det ermi ned by that of th e
mobility. For exa mp le, the decrea se in resistivity with a lowering of t he
temperature is associated with a weaker phonon scattering.
II fur ther decrease in tempera tur e (ran ge C) leads to a gra dua l freezingout of impurity elect rons, i.e., th ey a rc reca ptured by do nors (for concreteness
we shall be speak ing of a n ,, -type semiconductor) . In thi s region the
temperature dependence of the elect rica l cond uctivity is entirely due to a
ra pid dec rease in the free elec tro n conce nt ration . T heoretical ca lcula tion of
th e temperat ur e dependence of the concent rat ion in region C is
strai ghtforward if one neglects dispersi on of the impuri ty levels (the impuri ty
band widt h) and assumes th at all donors have eq ua l ionizati on energy Eo.
The det ai ls of thi s calcula tion can be found in the book by Blakemor e [ 4.2].
The results essent ially depend on the relati on between t he number of empty
donor positions KN 0 du e to compensation a nd th e num ber of emp ty positions
,, ( T ) due to the thermal exci ta tion of electrons into the cond uction band .
For KN o «,, ( T ) the Fermi level is locate d pr act ica lly midway between
the do nor level an d th e bott om of the conduction band, whence
,,(T)
~
[ N o2Nc ] " e-f.',I2kT ,
( 4.l.2 a)
where
(27rm, k T) J/l
Nc -
(4.l.2b)
411' J li J
«
For" (T)
() =
IIT
K N 0 t he Fermi level is close to the isolated-donor level, and
Nc(No -NA) -e.tsr
e'
2N
A
'
(413)
. .
Dependence (4. 1.2) holds in th e region where th e tem peratu re is nei ther
too low nor too high , i.e., KN o « ,, (T) « N o . Th us, its ran ge of valid ity
is generally narrow, and does not exist at all unless the compensation is low
enoug h.
Experime nta lly, th e te mperatu re depend ence of the resistivity in
region C often shows only one activation energy Ell viz.
4. A General Description of H opping Conduction in Lightly Doped Semico nductors
76
p(1-)
where
=
1: 1
PI
.,1kT
e
(4. 1.4)
is close to the ionization energy Eo of an isolated donor, as
determined, for exa mple, from opti cal mca suremcnt s. On ly in very weakly
compensated semiconductors can one discern a narrow range with an
ac tiva t ion energy of order £0/2 in the high -temperature part of region e.
Th e gra dua l freezing -out of cond uction elect rons wit h decr easin g
temperature eventually leads to a situation in which the main contribution to
the elec t rica l conduct ivity comes from electrons hoppin g direct ly between
impurities without any excursion to the conduction band. This is called
hopping conductivity. Elect ron s j ump from occupied donors to empt y ones,
and therefore the presence of empty positions on donors is a necessary
co nd it ion.
At low temperatu res thi s cond it ion ca n be fulfill ed on ly by
compensation.
The hopping mech ani sm of conduct ion corres ponds to a ver y low mobi lit y,
since the electron jumps arc associated with a weak overlap of wave-function
tai ls from neighb oring donors. Neverthe less, it wins in the com petition with
band conduction, because only an exponentially small number of free carriers
ca n participat e in the latter. In Fig . 4.1 hoppin g conduction corres ponds to
range D . In germanium with a concentaration of shallow donors
N D == 1015 cm - 3, th e tr a nsiti on to hopping conduct ion occ urs at T == 7 K.
Ran ge e correspond s to temper atures
7 < T < 50 K, ra nge B to
50 < T < 400K, a nd range A to T > 400K .
The possibi lit y of hopp ing co nduc tion was th eo reti ca lly predi cted by
Gudden and Sc hottky [4 .3], and t he first ex perime nta l dat a on it wer e
obtai ned by Bu sh a nd Labhort [4.4] for silicon ca rbide and by ll ung and
Gliessman [4.5] for ge rma nium and silico n. Hopping conduct ivity was
subsequently studied in numerous investigations with germanium, silicon,
indium antimonide, gallium arsenide, gallium antimonide, indium phosphide,
cadmium sulfide, diamond, and other semiconductors. Below we discuss the
main experi mental facts, using the data on germanium as an example. We
begin with th e temper a tur e depe ndence of hopping co nd uct ion.
Figure 4.2 shows th e results obta ined by Fritzsche and Cuevas [4.61 for
the dependence of resistivity on inverse temperature in p-type neutron-doped
germanium with a degree of compensation K = 0.4. Each curve corresponds
to a difTerent acce pto r (gallium) conce nt ra tio n. With good accura cy all t he
curves can be approximated by the expression
(4. 1.5)
The first term co rres ponds to the band conduct ion. It is practi cally
ind ep endent of th e accept or conce ntra tio n. Ind eed , as is seen from (4. 1.3),
the free-carrier concentration depends only on the degree of compensation,
which was chose n to be the sa me for a ll sa m ples. T he weak dep endence of
th e m ob ility on im purity concentration is not noti ceable on th e sca le of
Fig. 4 .2.
4. 1 Basic Experimental Fact s
T IKI
1010300 10
5 L
3
2
1.5
1.2 5
1
.'ig. 4.2. Resistivity o f co mpensated
p-germanium fo r K = 0.4 (4.6). The
curves co rres pond 10 the following
accepto r concentratio ns (in cm -) :
( I) 7.5 x 1014
(2) 1.4
10'
E
(4)
(5 1
(61
(7)
(8)
10'
~
z-
x lOI S
(J ) I.5x IO"
2.7 x 10' 5
3.6 x 10'5
4.9 x 10' 5
7.2 x lOIS
9 x 10"
(9 ) t.a x 10 16
£:
:~
77
(10) 2.4 X 10"
(II) 3.5 X 1016
10'
(11 ) 7.3xlO 16
;;;
."•
(lJ) 1.0 x l0 17
(14 ) 1.5 x 1017
(15) 5.3 x 1017
(16 ) 1.35 x 1018
0:
10'
13
lL
15 16
10"'
0
0.2
O.L
0.6
0.8
r'IK" '1
The seco nd term correspond s to the hopping conduction (it will become
cle ar be low why t his term , "jumping the que ue ", is give n the number 3) . W e
sec th at t he hopping co nd uc t ion has a noticea ble ac tiva t ion e ne rg y ' 3, th ough
it is small compared to
EI .
It arises due to the dispersion of impurity levels.
In hoppin g ove r donor s, an elect ron e mits a nd a bsor bs phonons. T his result s
in an exponential dependence of the electrical conductivity on temperature.
\ Vh cn the impurit y concentration is increased, it first somewhat enhances
t he ac tiva tion e nergy. This is assoc ia ted with t he inc rea sing ra ndo m Co ulomb
potential of ch arged impurities. However, a further increa se in the
concentra tion enhances the wave -funct ion overlap of neighboring centers and
leads to a sma ller ' 3. A t N A - 10 " c m- 3 t he ac tiva tion ene rgy '3 van ishes.
This concentration corresponds to the transition between the activated and the
me ta llic
types
of conduct ivity
described
in
C ha p. 3.
Th e
ra nge
of
concentrations close to the dielectric-metal transition is difficu lt to interpret
quantitatively. For this reason. when discussing hopping conduction, we shall
always be dealing with much lower concentrations.
Another characteristic feat ure of hopp ing conduction which is evide nt
fro m F ig. 4 .2 is a n extr em ely st ro ng depend en ce of the qua ntity Pl on the
impurity co ncentra t ion. The va lue of P3 ca n be determined by ext rapolat ing
t he low-tem pera ture (linear) portion of th e curves in Fig . 4.2 to
0, i.e.,
by find ing the intersect ion of this line with th e ord ina te axis. Increa sing t he
r:' -
78
4 . A General Description o r Hop ping Co nduct ion in Lightly Doped Semicond uctors
impurity conce nt ra tio n by a fact or of 30 results in a valu e of Pl which is 107
times lowe r. It is na tura l to regard such a shar p depende nce as exponent ia l
a nd wri te Pl in th e form
P3 """ PO]
e
[<N.>
(4.1.6)
,
where P Ol and f (N D) a re powe r-law funct ions of the impurity concentrat ion.
The phy sical reason for the exponentia l depend ence (4.1.6) is clear. Th e
prob ability of a ju mp between two im purities is determined by th e wave -
function overlap. \Ve are interested in the case where separations between
impurities a re much larger than th e Bohr rad ius of th e impuri ty state. At
such dist ances the wave functi ons fall exponentiall y (Chap . I J. Therefore the
overla p integrals also dr op exponenti ally with increasing dist an ce betw een
impurities.
As the concentration decreases, the mean inter-impurity
sepa ra tion increases, with the jump probability and hen ce the electrica l
conduc tivity decre asing ex ponentia lly. The exponentia l depend ence of
conductivity on the impurity concentration is usually considered the main
ex peri menta l evide nce in favor of the hopping mecha nism.
1.5,.,.--- - - - - - .,
a5
Flg, 4.3 . Expe rimen tal dependence or the activatio n
energy c) o n the degree o r compe nsatio n K {4.7J
o!C-;!-;;--f,----,S---+;----;',
az 0.4 0.6 0.8 1.0
K
The q ua nu nes
'l
a nd
Pl
depend
on t he degre e o f compe nsation.
Figure 4.3 shows experimental values of the activation energy
EJ
for various
com pen sati ons
in
p-t ype
sa mples
doped
to
the
sa me
level
N A - 2.66 . 10 l5cm - l [4.71. As th e degree of compensat ion inc reases, ' l is
seen to decrease sha rply at first, reachin g a minimum a round K - 0.4 , a nd
then to g row rapidly as K - 1.
The experimen tal evidenc e is cont radictory on the dep end ence Pl(K) In
the re view by MOil a nd Twos. [4.7] it is asse rted t hat Pl incre ases
monot oni call y with K. On the othe r han d, Dobrego a nd Ermo layev [4.8 J
observed a decr ease in Pl as K va ried from 0 to 0.2 in german ium sam ples of
a pprox imately the sa me dopant conce nt rat ion. Similar behavior a t K < 0.2
has been obser ved ea rlier by Davis a nd Comp ton [4.91. The resistivi ty P l
reac hed a minim um at K == 0.2 and then it grew with increasing K [4.91.
4 .1 Basic Experimental Facts
79
In add ition to the band a nd the hopping mech anisms of conduction,
semicond uctors with low compensa tion (K < 0.2) display a nother activated
mechanism which manifests itself in a limited range of concentra tions near
the Mott transition. This mechanism contributes one more term of the form
pi ' exp (-,,/k T) 10 the temperature dep endence of the conductivity (4.1.5>'
Since PI « P2 « PJ and {) > {2 > {J, this mechanism works in the
intermed iate tem perat ure range between the band a nd the hopping
conductivity regimes.
T he ' 2 conduction has bee n exte nsively stud ied in the literatu re (c.g.,
14. 10, III >. Most a uthors believe that the mech anism of e cond uctivity is
connected with th e motion of electrons over singly fi lled neut ral donors.
Gershenzo n et al. [4.12] showed that neutral donors in germanium possess a
second electr onic state (the D- sta te) with a binding ene rgy of order 0.1 Eo.
T he rad ius of D - states is large. a nd th ey overla p strongly a t int ermediate
imp urity concen tra tions. A wide band ( Fig. 4.4) a ppea rs as a consequence.
c
."jg. 4.4 . Electro n density o r stales in an 'H ype semiconductor (4.1 3].
The impurity band is split into two subbands: £ ) (0) and £2(b) . Range
(c) co rrespo nds to the conductio n band
T his ba nd is a nalogous to the upp er Hu bbar d subba nd we talked about in
Cha p. 2. with one importa nt difference. In the present case the ba nd arises in
a d isordered system. Disorder ca n lead to a complete or part ial locali zati on of
sta tes in the D- ban d. Th ere is no doub t, however, tha t even with localizati on
th e D - ba nd has substa ntially higher mobility than hopping cond uction over
empty sta tes, because of th e la rge radi us of D- states (i.e., lar ge bandwidth) .
Therefor e it is conceivable that despite the exponen tia lly sma ll number of
electrons, E2 conductivity can exceed the EJ • conductivity in a narrow range of
te mpera tu res. O n the other han d. because of th e lar ge width of th e D- ba nd
its lower mobility edge ca n be much closer to th e Ferm i level tha n the bott om
of the con duction band (cf', Fig. 4.4>' As a result th e D - band can also win
in the co mpetition with the conduction ban d. which, of course, has a lar ger
mobility but a substantially lower electron concentra tion.
80
4 . A General Descriptio n o f Hopping Co nducriou in Lightly Doped Semico nductors
The fact th at th e ' 2 co nduct ivity is obse rved only when th e impurit y
co nce nt ra t ion is hig h a nd the compensa tion is low supports th e D- ba nd
conce pt. Indeed, lower ing the dopi ng level should lead to a significant
nar row ing of the D - band, with its energy a pproachi ng th ai of a n isolat ed D ia n. A s a lready menti oned , thi s en ergy is of order -0. 1 E o. Hence th e
ac tiva t ion e ne rgy '2 in th e limit of low co ncent ra tions should a pproa ch 0 .9 Eo,
i.e., become very close to fl . Naturally, in this case f2 conduction would not
be competitive with band conduction.
On th e othe r hand, it is clea r th at th e most favorab le co nditions (largest
neutral donor concentration) for E2 conduction arc in the absence of any
compensation. when f] condu ctivity vanishes. Increasing the compensation
significantly improves the conditions for f] conductivity and worsens those for
E2 conductivity. Consequently, the temperature ra nge in which the former is
less favo rab le th a n th e latte r di sa ppea rs.
Des pite t he fac t th at th e D- band mod el provides a quit e reasona ble
interpretation of the experimental data, there is an objection to it.
In
ge rma nium the observed values of '2 a rc of orde r 1/ 3 Eo. Thi s imp lies that
du e to wave-fu nct ion overla p, th e D- ba nd en er gy is lower ed from -0 .1 E o to
- 2/ 3 Eo. There is no a ppa re nt reason to believe th at th e width of the Dband wou ld t hen sta y na rrow as shown in Fig. 4.4 . On th e cont rar y, it is
more logica l tha t the D- band should becom e wider by a n amou nt of order
Eo, ove rla p th e conduct ion band, a nd thu s lose a ll structu re. But then it
would be hard to explain th e origin of a definite ac t iva tio n energy '2 '
O ne wou ld like to sett le t hese questi ons wit h th e help of a qua nt ita t ive
th eor y. However, deve lop ing such a th eory for ' 2 conduc tion is ver y d iffi cu lt,
mainly because one must work in the region of concentrations where wavefu ncti on overla p pla ys a role com pa ra ble to th at of Co ulomb int er act ion of
electron s with imp urities a nd with each ot he r. The difficulty is co mpounded
by t he disorde r in t he im pu rity distribu tion . We sha ll not re tu rn to ' 2
co ndu ct ion; whe n spea king below of "hopp ing cond uc tio n" we sha ll be
referring only to f] conduction.
It is well known th at in the temperature ranges A , B , a nd C , where
conduction is governed by band electrons, complementary measurements of
elec trica l conductivi ty a nd t he Hall effec t make it possib le to determine
se pa ra te ly the co nce nt ra tion a nd the mob ility of ca rriers a nd hen ce esta blish
the electron scattering mechanism. Let us see what can be learned about
hopping co nd uction from t he Hall effect.
Figure 4.5 shows t he depen de nce on T - 1 of th e Ha ll co efficient R for p type germa nium (the resist ivity da ta for th ese sa m ples were shown in
Fig. 4 .2) . We see th at in ran ge C th e Hall coeffi cient grows exponentia lly
d ue to free zing- out of the free -electron concent ra tion /I (T), while the mob ility
rema ins high .
T he sha rp d rop in th e Ha ll coefficient in ra nge D ca n be explained with
the help of the so-ca lled two-band model, in which all electrons pa rti cipating
in electrical conduction are divided into two groups: conduction-band electrons
4, 1 Basic Exp eriment a l Facts
R.
JIJ(J
10
5
J
~
,
Z
FiK · 4.5. Inverse temperature dependence of the Hall
coeffici ent R (in cml xC - 1) for some of the p-Ge
T[KJ
samples (4 .6) who se characteristics were given in the
caption 10 Fig. 4.2
(\
10'
81
/0" -
wit h co nduc tivity u, - ,, (T ) ell , a nd Hall coefficient R, - (" (T) ee J-' , where
,, (T ) is th e ban d elec tr on concent ration, a nd impurity -ba nd elec t rons with
hopping conduc tivity Uh a nd Hall coefficient R h • A sim ple phe mom en ological
ca lcula t ion gives th e following express ion for th e observed Ha ll coeffi cient:
R -
R, u;
(u,
+ Rhul
+ Uh)
2
(4.1.7 )
•
Th is expression explains the temperatu re dependence o f R, assuming that the
hop pin g Hall mobilit y Ilh - eRhuh is very sma ll co mpa red to th e ba nd
mobility Jlc "'" c R cuc . For not-too-low temperatures, the second tcrm in the
num er at or of (4. 1.7) ca n be negle ct ed , and th e formula gives a sha rp
maximum at
Uc
=:::
(Jh .
where the conductivity mechanism changes from band
to hoppin g. T o th e left of th e ma ximum, when u,
R - R, - (II (T) ee 1-'
<X
R,u;
R - - - -Il;,e, ( T )
ul
cal
<X
Uh ,
(4.1.8)
e' ,lk T ,
a nd to the right of th e ma ximum (u,
»
«
Uh ) ,
ex p [2' kJT"
1
(4. 1.9)
82
4, A General Descript ion of Hopping Conduction in Lightly Doped Semiconductors
If '3 is neglected in compa rison with ' " the va lue of R declines to the rig ht of
the maxi mum as exp(- "lkTl . T his explains why th e curves have simila r
slopes on both sides of the ma ximum .
We sec th at measuring the Ha ll coe fficient in the region of validity of
(4.1.9) does not permit us to sepa ra te out the Hall mobilit y a nd the
concentra tion fact ors in u,. Thi s would be possible if we could move into the
reg ion o f even lower temperatures wher e the second term in the numera tor of
(4.1.7) become s ta ngible. However. this ca nnot usually be done, becau se of
the high resist ivity of the sa mples. Co nseq uently, the only hard fact known
abou t the Hall mob ility is that it is ve ry s mall. Th eorie s of the Hall effect in
th e hopping condu ct ivity regime were proposed in [4. 14 - 171.
T he drift mobility can be estimated if one assume s that practically all
ca rriers locat ed on im purites take part in hopping co nducti on . For insta nce,
for Sample 4 of p-lype germanium a t 2.5 K (of. Fig. 4.2) we find
(4. 1.10)
Such a low value of mobility tota lly exclud es a ny possib ility of inter preting
the hopp ing conduction on the basis of a picture of randomly sca ttered (say,
by ph onons) qu asi-free electrons; i.e., Bolt zmann's kinetic eq ua tion brea ks
down hopelessly in the ca lculatio n of u. Indeed, suppose for a momen t th at
we could usc th at picture a nd write Il in the form Il ~ erim . For relaxat ion
time Tusing (4.1.10) a nd the free- electron mass. we find T - 4 ·\O- l8s. The
energy unce rt ainty lilT corresponding to such a value of T is hundreds of
electron volts. i.e., it much exceed s the cha rac teristic a tomi c energies. This,
of cour se, is con trary to the free-electron picture a nd rende rs it inva lid.
Thus. a th eory of hopp ing conduction should be based on different
concepts, the most important of which should clearly be tha t of local ized
elect ron sta tes . Interacti on with phonons a nd overlap of the wave functions of
locali zed sta tes give rise to infr equent j ump s from one state to another.
In th e next section we sha ll formu la te a n apparatus which uses these
concepts and forms the basis of the hopp ing conduction theo ry.
4.2 The Resistor Network Model Proposed by Miller
and Abrahams
Miller and Abra hams [4. 18J suggested the following approa ch. Starting with
electron wave funct ions localized on individual donor s, calculate the
probability that a n electron tr ansition will occur between two donors i and j
with the em ission or a bsorption of a phonon. The n calcu late the number of
tra nsitions i - j per unit time. In the a bsence of a n electric field, an equa l
number of electr ons undergo the reverse tran sition , i.e., there is a deta iled
ba lance. In a weak electric field the forward a nd reverse tr a nsitions will not
be bal an ced, giving rise to a curre nt propor tional to the field . Evalua ting th is
curren t yields th e resistance R'j of a given tran sition, a nd thu s the whole
4 .2 The Resistor Netwo rk Model Pro posed by Miller and Abrahams
83
problem is redu ced to calculating the elect rical conductivity of a n equiva lent
net wor k of ra ndom resistors. T his will be traced in de tail below.
Co nsider two donors i and j having the coord ina tes r, a nd rj a nd sha ring
one elect ron . Let '!' (r- ri) == '!' i (r) be th e grou nd-st at e wave function of an
isola ted donor , sa tisfying (1.2.5) wit h U - - e'/K lr- ril. We sha ll conside r
the distance r lj - Ir;-rj l to be much larger than the characteristic wavefunction size, so t hat the overla p of func tions '!'i and '!' j will be weak.
Interaction betwe en the elect ron a nd both donors produ ces a splitt ing of t he
degenerate sta te. Within the fram ewor k of the LCA O method (linear
combination of a to mic orbi tals) the split-state wave fun ctions represent a
symmetric and an antisymmetric combination of the atomic functions, viz.
'!'
__ _ _ '!'~i__'±"___'!'
:...L
j
_
2 1/2 (J ±
'!';'!'j dr ) 1/2 .
f
-
I.' -
(4.2. \)
Evalua tion of the energy of S tates 1 a nd 2 wit h the Hamiltonian
H - Ho -
e2
e2
Klr- ri l
Klr- rj l
(4.2.2)
gives
E ' .I
~ -
.
e'
(4.2.3)
Eo - - - ± lij .
Kr ;j
Here fl o is t he elect ron Hamilt onia n in the host cr ystal, - Eo is t he energy
level of a n isolated donor , a nd I'j is t he ene rgy overlap integ ra l given by
lij ~
f
·
'!' i '!' j
K
I
e'
r- rj
I dr -
f '!'i"'!'jdr f
e'l'!' i l'
I
I dr
K
(4.2.4)
r -rj
Co nsid er th e simple case where the donor sta te is connected with a single
extremu m at th e ce nter of th e Brillouin zone, i.e., when '!' , (r) is of th e form
(1.2.2 1). S ubstituting (1.2 .21) int o (4 .2.4) , we split each int egr al int o a sum
of integral s over elementary crystal cells. Using the fact that the envelope
functions a re pract ically constant on the sca le of one latt ice constant 0 0, we
ca n ta ke the m out of the sig n of the integr al over one cell. The sa me can be
done wit h th e factors Ir - rj/-l in the importan t region of inte gr ation where
Ir- rjl »
00. T he cell integra ls of lu• .ol' give unity in accordance with
(J .2.10) . G oing back from t he sums over ce lls to integ ra ls we obta in
lij -
f
()
F, r Fj(r)
K
Ir e'-r · I
dr -
f
'
) ' f e'IF,' (r)I
r-r ·
Fi( r)Fj(r d r
J
K
J
dr .
(4.2.5)
For a hydrogenlike fun ction (1.2.17) eva lution of (4.2.5) yields the following
simple expression:
f; {// )
Ii'
lij -
~
[:; ] [ r: ] exp [
-7].
(4.2.6)
84
4. A General Description o f Hopping Conduction in Lightly Doped Semiconductors
It sho uld be note d tha t there is a more acc urate ca lcula tion of the energy
splitting (4.2.3) which goes beyond the LCAO met hod by taki ng into acco unt
the deform ed shape of the wave functi on mid way bet ween two impuri ty
centers, where their potentials are comparable. This calculation results in
replacing the fact or 2/3 in (4 .2.6) by 2/ e [Ref. 1.8, p. 3631.
It has a lready been mentioned that in a lig htly doped semicond uctor the
above resonant situation is not realized in practice, because of the strong
interaction W (r ) between t he electron and t he ch arged impurities su rrounding
th e donors i a nd j . For almost every pair of donors the following inequa lity
holds:
(4.2.7)
By including t he pote ntia l energy W (r ) in t he Ha miltonia n a nd performi ng a
variational calculation with the function
(4.2.8)
one ca n show [4. 18] tha t when (4.2.7) is satisfied, the wave functions of the
two lowest states are of the form:
'1'; - '1', + -~Jlij.- 'l' i
i
(4.2.9)
'
(4.2. 10)
T hey d ilfer lillie from th e isolated -donor wave functions, which is not
surprising since accordi ng to (4.2.7) we arc in the complete Anderson
locali zat ion limit. Th erefore in cont rast to (4 .2. 1), it is meanin gful to keep
th e indic es i and j on th e '1" functions in (4.2.9) and (4.2.10). A n electro n
transition from state '1'; to '1'; clearly implies a transfer of charge
distance r i j ' Such transitions give rise to a current.
-r
e on the
When (4.2.7) is sa tisfied, th e energ y difference between the states (4.2.9)
and (4. 2. 10) coincides with t,i, to within a small quan tity I,)/D/ ,. Hence.the
energy of a phonon a bsorbed in the tr an siti on i-:-j equa ls tJ. i , . Inasmuch as
tJ.i , is usu ally of the orde r of a few milliclcctronvolts. the tran sition i ~j
requires the part icipat ion of a long-wave acoustic phonon. Following Mi ller
a nd Abrah am s, we ass ume for simpilicity th at elect rons interact only with one
acoustic branch whose spectrum is isotropic.
The tran sition probabili ty i ~j with the absorptio n of one phonon is given
by
(4 .2.11)
wher e V o is the volume of the crysta l, s is th e speed of sou nd, a nd q is the
phonon wave vector. M q' which is given by
4.2 The Resistor Network Model Pr oposed by Miller and Abrahams
li N ] 1/ 2
-.!L.L
[ 2 Vosd
J '1" . eiq-r '1': d r
J
85
(4.2.12)
,
is the matrix element or electron-phonon interaction. where E I is a constant
descri bing th e deform ation potenti al, d is the crysta l den sit y, a nd N q is t he
num ber of phonons with momen tum q. If wc substit ute th e wave functions
(4.2.9) a nd (4.2.10) into (4.2. 12) a nd eliminate the per iodic functions 11".0 by
using (\ .2.21), as we did when going from (4.2.4) to (4 .2.5) , t he resulting
expression is of th e form
li N
M .... i E 1 -.!L.L
[ 2 Vos d
•
]'/2 {~
/" [J F f. eiqrd r 6/ /
J
J F · eiq-rd r ]
2
'
(4.2.13)
For what follows it is importa nt to estimate th e mag nitud e of the
dimensionless parameters
6/ j r jj
qr ·· - '1
lis-
a nd
qa
---
a6/;
Ti s
Co nside ring that t;j , is of the order of e'N}/\- 1 an d rij ::: N ;;'/J , it is
easy to sec th at t he pa ra meter qr'j is la rge (::: 20-30) , while qa is of t he
orde r of unity. Th e cha rac te ristic lengt h of t he vari a tion of Fi Fj across the
line connec ting the two cente rs is .jr/ja . Hence the last integra l in (4.2. 13)
is grea lly reduced due to the oscillatory factor , a nd it ca n be neglect ed .
T akin g into account tha t Fj(r- r'j) - F,(r), we ca n rewrite the first sq ua rebracketed expression in the form
(4.2. 14)
For a hydrogenlike funct ion, ( 1.2. 17), one has
(4.2. 15)
T hus,
S ubstituti ng (4.2. 16) into (4.2 .11) , we ca n neglect the integra l involving
cos q'r/j when integ ra ting over the d irect ions of q beca use of th e large va lue
of t he param eter qr/j . Moreover, we shall conside r the elect ric field to be
sma ll enoug h not to dist ur b the phonon equilib rium. T hus Nq repr esents th e
equ ilibr ium Pla nck d ist ribut ion func tion. From (4.2. II ) we hence obt ain the
M6
4. A General Description o f Hopping Conduction in Lightly Doped Semiconductors
transiti on proba bility in th e form:
Y'i -
ye exp(-2' ij/ a) N(t;i,),
(4 .2. J 7)
whe re
(4 .2.18)
a nd
N(t;i, ) -
[ exp [
~; 1
I
r
(4 .2. 19)
Let 11, - (0, 1) be th e ;-th donor occ upa tion nu mber which fluctuat es in time.
i~j is possible only when 11, -1 and "i -O. Th erefore th e
number of electrons ma king thi s tr an siti on per unit time is given by
The tr an siti on
(4.2.20)
where the averaging is over time. The quantit y 'Yij also fluctuates in time.
This happens becau se of the fluctu ating occupa tion num ber s for donors
neighbori ng; a nd j. Vari at ions with time in the potenti al of these donors
gives rise to fluctuations in 6/; and hence in "'Ilj '
We shall now mak e a very im port ant simplifying assum ption. We sha ll
ass ume thai th e site occupa tion numbers a nd e nerg ies do not fluctuat e in
tim e, but remain eq ual to th eir avera ge values. In othe r words, the system
will be described in a self-consiste nt field a pproxi ma tion a na logous to the
Hartree a pproxi mat ion. T his a pproxima tion consists in t he following: (J )
Eac h donor is c ha racte rize d by th e avera ge occupation number < 11, > =1,.
(2) Correspond ing to eac h donor th ere is a time-aver aged electronic level
ener gy in the field of a ll other impurities and electrons, viz.
acc
e'
do. e' (J -I,)
,, - ~
~ K I " -, , I
,K\" - ,,I ,,,,,
(4.2.2 1)
Her e the first sum exte nds over all acce ptors, a nd the seco nd over all donors
except i; th e qu anti ty (J - j,) e represent s th e ave rage c ha rge of donor k . 0)
The phonon e nergy a bso rbed in the transiti on i ~j is tak en to be t;i , - 'F"!
In thi s approximatio n we find
(4.2.22)
It is a nontrivial mailer to pass from (4.2.20) to (4.2.22), a nd the proced ure
ca nnot be ju st ified in general. In C ha p. 10 we sha ll a nalyze the validity of
thi s a pproxima tion a nd show that man y of the result s obta ined wit h it a re
correc t. For exam ple, t he exponential dependence of Plan concentrati on is
4 .2 The Resisto r Network Model Propo sed by Millcr and Abrahams
87
not at all sensitive to this approximation. This can be seen even now. since
the exponential factor exp(-2rij/a) responsible for the concentra tio n
dependence, being time indepe nden t, is not a ffected by the averaging in
(4.2.20>' As will be see n in C ha p. 10, this approxima tion a lso correctly
accounts for the activation energy 1:) in the limits of both very low and very
high compensation. However, for intermediate compensation or for variablera nge hopping conductivity (cf. C ha p. 9) t he abo ve a pproxima tion is not
justifi able. In our view this leads to an unknown numerical factor in the
expo nent
cha rac terizing
the
temperat ur e dependence of
the elect rical
conductivity.
For th e reverse proce ss, j-r-I , with the emission of a phonon, we have, by
a na logy wit h (4.2.22):
f' jo
- ,,8 exp(- 2rij / a )
[N(' r 'i )
+
1] Ii O - l i ) '
(4 .2.23)
Using rij and rji we can write the current between the donors i and j in the
form:
J ij - - e (f' ij -f'j,)
(4 .2.24)
In th e ab sence of a n electr ic field £ , the functions
Ii
are given by the
equilibri um expression
Ii - 1,0 - [ "2I exp [,P-Il
kT ]
+Ir
(4.2.25)
where EP is the average energy on site i at E """ 0, and the factor Ih is
associated with two possible spin stales for an occupied site. The quantities EP
a nd 1,0 a re mad e self-consistent by (4.2.2)) . At t he low tem peratures we a rc
interested in, th e energies ,p arc only slightly d ifferen t from those
corresponding to zero temperature, whose distribution was studied in detail in
C hap. 3.
It is easy to see that in eq uilibri um there is a detailed ba lance between t he
tran sitions i - j an d j -rt , Indeed, using (4.2.19) and (4.2.23 - 25) we see
tha t l'ij - l'j i a t E - 0 a nd hence also J'j = O. This ba la nce is, of
course, des t royed by a n electric field. T his occurs as follows. Firstly, the field
redist ribu tes electr ons over donors, creatin g correc tions bf, to (4.2.25) . We
il,:
express th ese correcti ons by introd ucing small quan tities O
1'8
1,(£)
=
1,0 + 01, I
[I + \-> exp [,p - k;Oil"- Il ]]
-'
Second ly, the field affects th e donor-level energ ies e a nd 'j , viz.
(4.2 .26)
88
4. A General Description o f Hopping Conduction in Lightly Doped Semiconductors
(4.2.27)
T he first ter m in b" desc ribes the di rect ac tion of th e exte rna l field E, a nd the
second the variation in the Coulomb potential due to a redistribution of
electrons [of. (4.2.2 1)1. T his gives rise to a c ha nge in the a bsorbed phonon
energy 'j-" which enters the a rg ume nt of t he Pla nck function.
If the extern al circ uit is broke n <the sa mple is located between ca paci tor
pla tes), then t he elec t ric field results in a new equilibriu m, in which
OPi = -Oli _ To calculate the electrical conductivity, one should consider the
case of a closed circuit. In this case the equilibrium is broken and
51" ;" -5" . If th e elect ric field is so sma ll tha t the corre ctions bl" a nd 5" a rc
sma ll compa red to k T, then one ca n expa nd the fun ct ions I" /j , a nd
N( Ej-Ej ) in the expressions for l'ij and Ij; as power series in these
corrections. In an approximation wh ich is linear in the external Held. we find ,
usin g simple a lgebra ic manipulations with (4.2.22 - 25) , th e current J /j in th e
form
e r~·
J IJ. - ''
kT
[5u . + b, · r-)
J
(bu
.
r- I
+ b,,) ]
(4.2.28)
•
1'3
wh ere
is the freq uency of t ra nsitions ; - j a nd j »-l in equili bri um . T his
exp ression for the curr ent ca n be put in the form of Ohm' s la w:
J 'j -
tu. -
Rij l
Vj )
,
(4.2.29)
and
(4.2.30)
whe re
kT
e'rO.
"
e 2 don
- eV, - 5" + 51" - e E .
r, + bill + -
~
K
(4.2.31)
Jc~ j
Th e qu antity - eVi ca n be regard ed as a local value of th e elect rochemica l
pot cntial on donor i , co unted from the elec tron che mica l potenti al 1'. Th en
Vi - U, ca n be inter preted as a voltage dr op on the tr an sition i - j , a nd R'j as
the resistance of this transition. Turning now from the case of two donors to
tha t o f a rea l system having many donors, we see that th e curre nt between
a ny two donors ca n be ca lcula ted , provided we ca n somehow calculate the
quan tities Vi. The tota l curr ent thr ough the sa m ple is given by t he sum of a ll
currents piercing any cross-section, We see that it is not necessary to know
all of th e indi vidu al terms in (4 .2.3 I) in orde r to ca lculate th e c urrent.
The voltages V, a re, in prin cipl e, determined as follows. For donors
imm ed iatel y adjace nt to thc metallic elect rodes, V, is tak en to be cqua l to th e
elec t rode poten tial. If th e sa m ple has length L the a pplied voltage bet ween
terminal electrodes is eE L . It is th rough these boundar y conditions that th e
exte rna l field E e nte rs the problem. All other voltages ar c determined by th e
4.2 The Resistor Network Model Proposed by Miller and Abrahams
89
condition of equ a lity of curre nts flowing in a nd out of eac h donor, i.e., from
the stead y-sta te conditions for I, .
T hus th e conductivity of th e sa mple is completely de te rmined by the
resistan ces R'j' T he hopping conduct ivity problem is, in fact, redu ced to t hat
of ca lcula ti ng the cond uct ivity of a rand om net work which has its vertices a t
th e donors a nd in which resistan ce (4.2.30) connects each pair of vertices
(F ig. 4.6).
FiSt.
~. 6.
Random resistance network proposed by Miller and
Abraham s (4 .18)
For what follows it is convenient to separate two factors in the expression
(4.2.30) for Rij' one depending exponentially on r ij a nd the donor energ ies,
a nd the othe r containing all wea ker, power- law , dep endences on these
param eters. By substituting the equilibrium functi ons N , 1,0, a nd lio into th e
expression for
it is easy to confirm that a t suffi ciently low te mpera tures,
when k T « I" - ,71, I,P-Ill. 1'7-1l1, the q ua ntit y r'j ca n be written in the
form
"t
"3~ 'Y3ex p( -2r'j / a ) ex p ( -" j / kT )
,p a nd ,7 with respect
y, [I "~ - 'j i + k - III + i' j - Ili J
for a n a rbit rary posit ion of
"j -
(4.2.32 )
to the Ferm i level. Here
(4.2.33)
Th en , acc o rding to (4.2.30) we have
R ij - R8 exp (~ij )
(4.2.34)
where
(4.2.35)
a nd
RB
kT
=
e2"'V ~
"J
•
(4 .2.36)
For the sa ke of simplicity, in (4.2.33) a nd everywhe re below we omit th e
superscript 0 when dealing with equ ilibriu m energ ies made self-consistent by
(4 .2.21) an d (4.2.25) . We st ress agai n th at since we arc inte rested in
90
4 , A General Descriptio n of Hopping Co nduction in Lightly Doped Semico nductors
te mperatures for whic h the impu rit y-level dispersion is much grea ter than kT ,
th e energies ente ring (4.2.33) have the sa me meaning as the donor-level
ene rg ies in the grou nd sta te of t he whole system, whose dist ributi on was
st ud ied in C hap . 3.
A n import ant feat ure of th e resistance netwo rk (4.2.34) is its ext remely
wide spect rum of resista nces R ij. In typica l samples used for studying
hop pin g cond uctivity in lightl y doge d sem iconduc tors , th e mea n separation
bet ween dono rs ro ~ (41fNo / 3 ) - 1 3 constitutes abo ut 6 to 12 Boh r rad ii a .
The resis ta nce of a pair wit h rij - 2ro differs from th at with rij ~ ro by a
factor of e 12 to e24 . At sufficiently low temperatures the energy term in
(4.2.35) a lso lead s to a strong resista nce dispersion .
Eva luati ng th e elect rica l conductivity of a rand om network with an
exponentially wide spec t rum of resistan ces is the most comp licated part of
ca lcula ting the hopping con duc tivity . Th e genera l meth od of solving t his
problem will be formul at ed in Sec ts. 5.6, 6. 1. Let us fi rst cons ide r certain
incorrect approaches which are nevertheless instructive. Having understood
the weak points of t hese a tt emp ts, we sha ll develop a better feel for th e
esse nce of t he problem.
At first glance it appears th at a reasonabl e approx imation ca n be obtained
by neglect ing the second and t hird terms in (4.2.3 I) . T hen all t he currents
would be known:
Jij
=
Rij l Tij E cos Oij ,
(4.2 .37)
where O'j is th e a ngle betwee n rlj a nd E. T he following trick ca n be used to
calcu late the current density . Let P be th e polarizati on (d ipole moment per
unit volume). T hen the curre nt de nsity j can be written in t he form
J > ddPt _ VOl",
":J
[- eddtf
j
]
r,
(4.2 .38)
) ,
)
wher e Vo is t he volume of th e system , an d the su mma tion exte nds over a ll
donor s. Taking int o account th at - e df , / d t - ~ Jij and Jij - - J j I we find
(4.2. 39)
Pr oj ecting (4.2.39) onto th e field direction and using (4.2.37) we find
p- I -
I
'"
R ij-I rij2 cos 2 0ij'
-2V
":J
o
(4 ..
2 40)
j ~j
Si nce most of t he inverse resistan ces Ri; 1 in (4 .2.40) are exponent ially sma ll,
th e su m will be de te rmine d by a few ter ms with rij '" a and 'ij '" kT .
Co nseq uently, p will be rather sma ll and will not depend expo nentia lly on t he
conce ntra tion and te mpera ture. This resu lt is clea rly wrong . Ind eed, at th e
values of ro we are interested in, ro
»
a , pairs having rij ::::: a occur very
4 ,2 The Resistor Network Model Prop o sed by Miller and A brahams
91
seldo m. They look like sma ll, isolated met al lic island s surrounded by a sea of
rela tively poor conductivity. Obv iously th ese island s ca nnot determine the
electrical conduct ivity of the system. The origin of the mista ke is also
evident. In reality, voltage drops U; -Uj on "metallic pair s" a re very small
compa red to e E· r;j . In ot her words, for the se pairs our initia l assumption
break s down .
The above ca lculation consisted in averaging local va lues of conduc tivity.
M ill er and Abrahams [4.18] took the opposite approa ch, averaging the
resistances Rij . Let us consider a simple interpretation of their calculation for
relatively high temperatures, when the exponent in the temperature factor in
(4.2.32) is of t he order of unit y and
(4.2.4 1)
We sha ll, in fac t, ca lculate th e resisuvity P3 in (4.1. 5) . Miller and
Abrahams ass umed that a rand om network is effect ively equ ivalent to a set of
independent chai ns of resistances, each cha in passing t hroug h the enti re
sa mple from one electr ode to the ot her. The resistances of a single chai n are
in series, and hence the cha in resistan ce is given by the su m of individua l
resistances RU' N ext, they pointed out that due to ra ndom impurity
dist ribu tion in the crysta l, there are regions of size r grea ter th an Nii ' /3
where ch ain s break (Fig. 4.7).
o
o
0
o
o
o
o
o
O, / .... - -- - . . . -,
o
I
/
I
0
\
'.
,
\
o \,
o
o
'\
I
o
o
'
./
' 0
/ 0
0- - -- / 0
o
0 0
0
o
!"I~ . 4.7. Illustratio n o f the
derivation of (4 .2.44). The
ci rcles represent do nors. The
so lid line sho ws a path o f rela rively low resistance bypassing a rando m void. The
arrow shows how an electron
crosses the vo id, acco rding to
Miller and A brahams 14 .181
T he probability of occurrence of a spherica l void of rad ius r is given by
Poisson's d ist ribut ion a nd equa ls ex p(- 4trN Dr 3 / 3). Mi ller and Abrah ams
ass umed th at every chai n connec ting t he terminal elect rode s must cross these
voids. It is very unlikely for a n electron to hop over such a void; eac h void
has a large resistan ce, of orde r
exp(2r [a) , The contri bution of the voids,
given by
RB
f R8 exp (2r [a ) exp (-4trNDr
3/3)
dr
(4.2 .42)
4 . A General Description o f Ho pping Co nduction in Lig h tly Do ped Semiconductors
92
dom inates the cha in resistan ce. Th e produ ct of exponent s in th e integ ra nd of
(4 .2.42) has a sha rp maximum a t r - r«. where
r;
=
-Jf
ro [
r:
' /2
]
» r»
[
=
47T N D
3
]-'/3
(4.2.4 3)
Thus, the main contribution to the chain resistance is due to elements
who se length is close to r no' Subst itut ing (4.2.4 3) into (4.2.4 1) we find the
res ista nce of a typic al c ha in. Assuming th at a ll cha ins have th e sam e
resistance, we find the resistivity P1 in the form
(4.2.44 )
Th e reason ing leading to (4 .2.44) has a n obvious wea kness. Ind eed, we found
that the resistance is dominated by voids of radius r m which occur with an
expo ne ntially sma ll prob ability exp (-4" N Dr ,!,! 3) . Su ch voids occupy a n
infinitesimal portion of th e tota l volum e. Th er efore besides the cha ins which
go th rough th ese voids, th er e a re numerous chai ns which bypass th em
(cf. Fig. 4.7), T hese cha ins have much lower resist a nce. It is as if we were
deal ing with di electrical isla nds imm ersed in a conducting sea . Su ch isla nds
clea rly ca nnot play a domin ant role in th e effective conductivity. T his
approach fails because it assumes that large voids are always connected in
series, so th at th e c ur re nt which flows th rough th em is of the sa me order as
the c urre nt through typical resist an ces of le ngth ro . In rea lity these voids a re
con nec ted pa ra llel to the mai n current-ca rry ing paths, an d the curren t
th roug h th em is va nishing ly sma ll. Ng uyen an d S hk lovski i [4.191 have shown
tha t a n ap proac h similar to th at of Mill er a nd Abrah am s is a pplica ble only
for hopping cond uction in a n ext reme ly stro ng elec tric field at low
concentrations of electrons in the impurity band.
W e have considered two opposite , tha t is to say ext re me, a pproac hes to
ca lcu la ti ng the effec tive conduc tivity of a ra ndom netw ork : ave rag ing
conductances and averaging resistances. Both approaches suffer from a
common fault: they overemphasize the role of anomalously rare resistances.
A possible me rit of the se ca lculations is th at they give a n upper a nd a lowe r
bound on th e conductivity, althoug h th ese constra ints a re wea k. Anoth er
method of ca lcu lating the effectiv e elect rica l conductivity of the Mill erAbrah am s network was sugges ted by Pippard a nd car ried out by T wose [4.71.
Th e Pi ppard-Twose meth od is more balanced in th e sense th at it avoids th e
above extre mes. It repr esent s a variation of the effective-medium meth od
which is widely used for averaging various characteristics of inhomogeneous
medi a [4.201. The prob lem is to ca lculate the resistivity of a ra ndo m netw ork
mad e up of resista nces (4 .2.4
First, the single unit resist anc e R ;j is
re placed by a sphe re of radi us ru a nd conductivity (R uru )- ' . Thi s sphere is
n.
assumed to be embedded in a continuous medium whose macroscopic
conduct ivity pj ' coincides with that we seek to ca lculate . N ext, we find the
4.2 The Resistor Netwo rk Model Proposed by Miller and Abrahams
93
p,'
current de nsity inside t he sphe re as a function of rij a nd
a nd average it
over a d istribution of length s r ij . Exp ressing in th is way t he average curre nt
density th rough the average conductivity
we find a n int egr a l equa tion for
PJ. Its solution is of the for m
p,',
(4 .2.45)
PJ cc exp (O .9INlPa) .
T he resistivity (4 .2.4 5) rep resen ts an inte rme dia te value between th e above
two ext reme results. Accord ing to (4.2.45) t he conductivity is domin ated by
jumps ove r an e ffec tive length of order ro , which a ppea rs q uite reasonable.
None th eless, t he nu merical coeffi cient in t he exponent is not reliab le. Indeed,
t he effect ive-med ium the ory gives accura te results only in t he cas e of a weak
relative inh omogeneit y. T his approac h, by its very essence, ca nnot lay cla im
to good resu lts for a strongly (exponen tially) in homogeneo us medi u m. In
Cha p. 6 we will obta in a bette r estim a te for the num erical coefficient in the
expo nen t of PJ, one which is almost twice t hat in (4.2.45) . A t the low val ues
of co ncentration we are interested in. N
a :::::: 0 .1. this discrepancy is very
significant.
A new met hod of calc ulat ing t he ra ndom network conducti vity, based on
the mat hema tica l theory of percolati on, was proposed independe ntly by
A mbegaak ar et al. [4.2 1], Pollak [4.22), a nd Shk /ovskii and Efros [4.23J.
T his method made it possible to develop a quant ita tive t heo ry of hoppi ng
cond uction. On e of the main aims of t he present book is to give a deta iled
descr ipt ion of th is theory a nd its result s. Si nce percolati on theory has not yet
become a widely fam iliar bra nch of ma t hem a tics, our next cha pter will give a
t horou gh intr oduction to it.
l/J