Ž .Physics Letters A 261 1999 108–113 www.elsevier.nlrlocaterphysleta Coulomb correlation effects in variable-range hopping thermopower a Theoretical Department, Institute of Physics, P.
Trang 1Ž
Physics Letters A 261 1999 108–113
www.elsevier.nlrlocaterphysleta
Coulomb correlation effects in variable-range hopping
thermopower
a
Theoretical Department, Institute of Physics, P.O.Box 429 Bo Ho, Hanoi 10000, Viet Nam
b
Physics Faculty, Hanoi National UniÕersity, 90 Nguyen Trai Str., Thanh-Xuan, Hanoi, Viet Nam
Received 4 May 1999; accepted 24 August 1999 Communicated by J Flouquet
Abstract
Expressions are presented for describing the variable-range hopping thermopower cross-overs from the Mott
T Ž dy 1.r Ž dq 1.
-behaviour to the temperature-independent behaviour as the temperature decreases for both two-dimensional
Žd s 2 and three-dimensional d s 3 cases The cross-overs show a profound manifestation of the Coulomb correlation Ž
along with that observed in resistance cross-overs q 1999 Published by Elsevier Science B.V All rights reserved.
1 Introduction
Ž The variable-range hopping VRH conception
w x was first introduced by Mott 1,2 with his famous
Ty1 rŽ dq1.
-laws for the temperature dependence of
resistivity:
1r dq1
Ž d
R T s R exp TŽ 0 Ž M rT. ,
TMŽ d sbMŽ d rŽk G jB 0 d., Ž 1
where d s 2,3 is the dimensionality, j is the
local-ization length, and bŽ d
are numerical coefficients
M
The Mott optimizing argument in obtaining these
laws consists of minimizing the exponent of the
hopping probability, while the electron–electron
in-teraction is assumed to be neglected, and
conse-quently, the density of localized states is constant
Ž
near the Fermi level, G E ' G s constant.0
1
Corresponding author Fax: q 84-4-8349050; e-mail:
nvlien@bohr.ac.vn
w x Later, it was shown that 3 the Coulomb correla-tion between localized electron states leads to an appearance of a depressed gap in the density of
Ž states DOS at the Fermi level, which, following
Ž w x Efros and Shklovskii ES 4,5 , has the form:
d
< <
G dŽE s a E. d , a s drp d Ž Žkre . , Ž 2
where k is the dielectric constant and e the elemen-tary charge The one-particle energy E is measured
from the Fermi level
The most observable manifestation of the
Ž Coulomb gap of Eq 2 is that the temperature dependence of VRH resistivity should behave as
w4,5 :x
1r2
Ž d
R T s R exp TŽ 0 Ž ESrT. ;
TESŽ d sbESŽ d Že2rk kj ,B Ž 3
Ž instead of the Mott laws of Eq 1
Ž Experimentally, both the Mott laws of Eq 1 and
Ž the ES-laws of Eq 3 have been observed in a great
w x number of measurements for various materials 5–8
0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V All rights reserved.
PII: S 0 3 7 5 - 9 6 0 1 9 9 0 0 6 0 5 - 2
Trang 2Moreover, for some materials the low-temperature
measurements show a smooth cross-over from the
Mott Ty1 rŽ dq1.
-behaviour to the ES Ty 1 r2
-be-Ž
haviour of VRH R T as temperature decreases
Such a cross-over is considered as an evidence of the
role of the Coulomb correlation at low temperatures
To describe observed Mott–ES cross-overs,
theo-w x retically, there exist different arguments 9–12 Our
w x
theory 11,12 is based on using an ‘effective’ DOS
of the form:
< <dy 1
E
G d ŽE s a E. d d E d dy1q<E<dy1, Ž 4
where E is a parameter, a E d d d dy 1 ' G The DOS0
Ž
of Eq 4 tend to the Mott constant DOS in the limit
of E 4 E and to the Coulomb gap DOS of Eq 2 d
in the opposite limit
Ž
On the basis of the DOS Eq 4 , using the
standard Mott optimizing procedure, we obtained the
following expressions for describing Mott–ES
resis-w x tance cross-overs 11,12 :
ydq1 rd
FdŽx ŽdFFdŽx rdx .
1rd
Ž d Ž d
sŽQ pb d M E rk T d B M ŽE rk T , d B Ž 5
1rd
Ž d Ž d
2 rrj s 2 dr pbŽ M E rk T d B M
=ŽFdŽx .y1 rd, Ž 6
h s 2 rrj q E rk T x , d Ž d B Ž 7
where x sgrE with g being the optimum hop- d
2Ž Ž dy 1 d
d s 3 and 1 for d s 2, and where
x t dy 1
FdŽx s H dy1 dt. Ž 8
1 q t
0
For a given value of d 2 or 3 , by solving Eq.
Ž Ž 5 – 8 it is easy to obtain the exponent h of thed
Ž
resistivity as a function of the temperature T : R T
Ž
sR exph T 0 d
Ž Ž The simple expressions of Eq 5 – 8 , as was
shown in Refs 11–14 , describe quite well the
experimental Mott–ES resistance cross-overs
ob-served in a y In O and a y Ni Si x y x 1yx films These
cross-over expressions also predict the cross-over
temperature
Tc sd ŽpbM d r2 ŽE rk T d B M . TM ,
9
Ž
which is in good agreement with measured values
In the limit of the Mott constant DOS, the
sions of Eq 5 – 8 give the Mott laws of Eq 1 with widely acceptable values of the coefficients:
bMŽ2.s27rp and bMŽ3.f18.1 In the opposite limit
of the ES Coulomb gap DOS they give ES laws of
Eq 3 with bESf8 and bESf7.27
A favourite of the cross-over expressions of Eq
Ž Ž 5 – 8 is that in fitting these expressions to experi-mental data with a defined characteristic temperature
T Ž d
deduced from the data, the temperature T)
s
E rk d B is the only adjustable fitting parameter used
w x Note also that the cross-over theory of Refs 11,12 has recently been generalized to describe VRH
resis-tance cross-overs from the Mott Ty1 rŽ dq1.
-be-haviours to the soft gap Tyn
-behaviours with any n
Ž
from 1r d q 1 to 1, including the ES value of
w x
n s 1r2 as a special case 13,14 Thus, the role of the Coulomb correlation effects
in temperature dependence of VRH resistivity is well understood both experimentally and theoretically, while much less is known about the role of this correlation in other transport properties The
ther-w x mopower, as was originally noted by Mott 2 , is sensitive to the material parameters and is expected
to provide a good test of the principle ideas of the transport theory of disordered systems On the other hand, the thermopower is easier to measure than other thermal transport coefficients most commonly studied
In this work we present expressions for describing Mott–ES cross-overs in temperature dependences of
Ž VRH thermopower for both two-dimensional 2D
Ž and three dimensional 3D cases The obtained ex-pressions show a profound manifestation of the Coulomb correlation along with that observed in VRH resistances Besides, they are simple and easy
to be used in comparison with experiments
2 Cross-over expressions
Based on the percolation method the VRH
moelectric power thermopower could be found as
w15–18 :x
Trang 3where the transport energy W is given by
EG E p E dEŽ Ž
H
G E p E dEŽ Ž
H
and where
p E s dr dEŽ H H XG EŽ X.
=
< < < X< < X<
2 r E q E q E y E
u h yž j y 2 k TB /
12
Ž
Here u is the Heaviside step function, h is the
percolation threshold which defines the exponent of
the VRH resistivity, which is hd d s 2,3
formu-lated in the previous section
Ž Ž The expressions Eqs 10 and 11 show that, as
for the VRH resistivity described above, the
ther-mopower SS is entirely determined by the form of
Ž
the density G E of localized states close to the
w x Fermi level Qualitatively, Burn and Chaikin 19
suggested that for the Mott constant DOS the
ther-Ž mopowers depend on temperature as SS Mott A
T Ž dy 1.r Ž dq 1.
, while for the ES electron–electron
cor-Ž relation DOS of Eq 2 the VRH thermopowers are
temperature-independent Hence, the temperature
de-pendence of VRH thermopower seems to be much
more sensitive to the electron–electron correlation
than that of VRH resistivity
To derive the expressions of VRH thermopower
in a large range of temperature, covering both the
high-limit of Mott constant DOS regime and the
low-limit of ES electron–electron correlation
Coulomb gap regime, we start from the ‘effective’
asymmetric DOS of the form:
G E s GŽeff
Ž eff Ž
where the symmetric part G d E is just the DOS
Ž
of Eq 4 This symmetric part of the DOS does not
w x give any contribution to the VRH thermopower 18
The asymmetric correction that responds to the VRH
thermopower is assumed to be small, i.e it is
as-Ž
sumed in Eq 13 that g g< 1, where g is thed
optimum hopping energy
Ž Ž
Using the suggested DOS G E of Eq 13 the d
Ž Ž expressions of Eqs 11 and 12 could be evaluated
simply, though longishly To the terms linear in the small parameter g g the VRH thermopowers ob-d
Ž Ž tained from Eq 11 – 13 are the following:
For the 2D case SS d s 2 ' SS :2
P
P T2Ž
SS s g2 2Žk reB ŽE rk T2 B . , Ž14.
Q
Q T2Ž
where
2 2
P
P s 3600 C q 1 ln2 Ž ŽC q 1.
yŽ120C5q3600C3q12300C2q13200C
q4620 ln C q 1 q 75C Ž 6y76C5
q995C4q3640C3q7290C2q4620C,
Q
Q s 1800 C q 1 ln2 Ž ŽC q 1 y 2400C Ž
q9000C2q10800C q 4200 ln C q 1 Ž
q450C4q3200C3q6900C2q4200C
ln x q 1Ž
1 C 1yrŽ
2
For the 3D case SS d s 3 ' SS :3
PP T3Ž
SS s g3 3Žk reB ŽE rk T3 B . , Ž15.
QQ T3Ž
where P
P s 210C3 Ž 5y7000C3y6930C ln C Ž 2q1.
q yŽ 140C6q7350C4y6930 arctgC.
qŽ75C7y1799C5q4620C3q6930C.
q6300C3ŽI r2 y 2 I y 2 I y I1 2 3 4., Q
Q s 4200C 1 q C3 Ž 2.ln 1 q CŽ 2.
q4200 1 y C4 arctgC q 630C5
y2800C3y4200C q 12600C3ŽI q I2 3., and where
1
I s1 H dr ln ŽD q 1 ;.
0
I s2 H drarctg D;
0
arctg D y xŽ
I s3 H drH 2 dx ;
x q 1
D y x ln x2q1
I s4 H drH 2 dx.
0 0 ŽD y x. q1
Ž
with D ' C 1 y r
Trang 4In all the expressions of PP , Q2 Q , P2 P3 and QQ3
presented above the temperature is just being in the
quantity C defined as C ' h k TrE Since the d B d
exponent hd also depends on the temperature, the
Ž Ž resistance expressions 5 – 8 should be included as
the first part in the full expressions for VRH
ther-mopowers To calculate VRH thermopowers at a
Ž Ž given temperature one has first to solve Eq 5 – 8
in getting h , and afterward, to put the obtainedd
Ž Ž Ž
value of h into Eq 14d or Eq 15 for further
calculating SS The factor g , measuring an asym-d d
metricalness of DOS, should be considered as a
material parameter, which might even be negative
w18 x
We would also note that, consistently, the
resis-Ž tance exponent h in thermopower expressions 14 ,d
Ž15 should be calculated using the same DOS of Eq
Ž13 , including the asymmetric part ; g g Such d
an inclusion, however, will lead to thermopower
Ž 2 corrections, which are ; g gd and which are
therefore assumed to be negligible small
3 Discussion
Žeff Ž
In the high-energy limit, when the DOS G d E
Ž
of Eq 4 tends to the Mott constant one, and
Ž Ž therefore when the resistance expressions 5 – 8
Ž Ž d 1rŽ dq1.
Ž 1 , the expressions 14 , 15 give for 2D and 3DŽ Ž
Mott VRH thermopowers the well-known
sions 15–17 , respectively, as:
1 Ž2.2r3 1r3 2
S
S Mott s g T2Ž 6 2 M T k re ,B Ž16.
5 Ž3.1r2 1r2 2
S
S Mott s3Ž 42g T3 M T k re B Ž17.
Ž
In the opposite limit, when the DOS of Eq 4 takes
Ž the forms of the ES Coulomb gap of Eq 2 , and
therefore when the resistance expressions of Eq
Ž Ž 5 – 8 give h s T d Ž ESŽ d rT.1r2 of the ES law of Eq
Ž 3 , we receive from Eqs 14 and 15 the 2D andŽ Ž
3D Coulomb gap VRH thermopowers, respectively,
as follows:
43 Ž2. 2
S
S ES s2Ž 98g T2 ESk re ,B Ž18.
87 Ž3 2
S
S ES s3Ž 159g T3 ESk re B Ž19.
We would like here to note that while the number
Ž
5r42 in Eq 17 exactly coincided with that
ob-w x tained by Zvyagin 18 and is very close to those
w x obtained by Pollak and Friedman 16 , and by
Over-w x hof and Thomas 17 , the numerical coefficients in
Ž Ž Ž other expressions of Eqs 16 and 18 , and 19 are,
to our knowledge, new Certainly, the values of the
Ž Ž coefficients in all the expressions of Eq 16 – 19 for the limit cases should be independent of the chosen model of DOS
Ž Ž Thus, the obtained expressions of Eq 5 – 8
Ž Ž and Eqs 14 and 15 really describe the smooth VRH thermopower cross-overs from the Mott
Ž the temperature-independent behaviours of Eq 18
Ž
or Eq 19 , respectively, as the temperature
creases The cross-over temperature Tc of Eq 9 should keep having the same sense for the
ther-Ž Ž mopower cross-overs It seems from Eq 16 – 19 that the VRH thermopower cross-overs are more sensitive to the temperature than the VRH resistance cross-overs above mentioned
As an illustration, a solution of the cross-over
Ž Ž Ž Ž Ž expressions 5 – 8 and Eq 14 for 2D or Eq 15
Žfor 3D is presented in Fig 1 together with the limit
expressions of Eqs 16 and 18 or Eqs 17 and
Ž19 , respectively The thermopowers are here mea-
Ž sured in units of SS ' k re g E , and the temper-0 B d d
ature in units of E The values of the parameter d
ŽE rk T d B MŽ d .are arbitrarily chosen for this figure as
E rk TŽ2.s2.10y 2
and E rk TŽ3.s10y 3
Note
Ž
Fig 1 The numerical solutions of cross-over expressions solid
lines are presented in together with the high limits of Eqs 16
and 17 dots and the low limits of Eqs 18 and 19 dashed
lines ; SS ' k r e g E ; the parameters used: E r k T0 B d d 2 B M s
2.10 for 2D , E r k T s 10 for 3D
Trang 5that in practical comparisons of these cross-over
expressions with experiments since the characteristic
temperatures T Ž d
could always be deducted from
M
data; the only adjustable parameter used in fitting is
E d
Fig 1 shows a non-monotonous behaviour of the
VRH thermopower with a slight minimum at some
temperature between two limits for both 2D and 3D
cases We assume that such a minimum may be
resulted from a concurrence of two effects: the first
is related to the hopping energy, which increases as
the temperature increases; the second is related to the
relative role of the asymmetrical part in the DOS,
which is more essential at low temperatures
Experimentally, one might expect that the
ther-Ž Ž
observed along with the Mott conduction laws of Eq
Ž 1 However, there are very few data on VRH
thermopowers could be found in the literature An
early observation of the law SS f T3 1r2 in the
fluo-rine-substituted magnetite Fe O3 4yxF had been re-x
w x ported by Graener et al 20 Recently, measuring
various transport characters of sintered
ing compositions Fe Nb1y xW O in a large temper-x 4
w x ature range, Schmidbauer 21 parallelly observed the
T -law of Eq 17 with a negative sign of the
factor g3 at not very low temperatures up to ; 300
K A positive VRH thermopower with a similar
behavior was recognized in the transmutation-doped
w x
Ge:Ga at T F 2 K by Andreev et al 22
The only 2D data we know are those reported by
w x
Buhannic et al 23 for the parallel thermopower in
the layered intercalation compounds Fe ZrSe withx 2
x s 0.09–0.2 In the temperature range of the Mott
ther-mopower roughly follows the 2D VRH therther-mopower
T -law of Eq 16
There are a number of reports on
temperature-in-dependent-like behaviours of VRH thermopowers at
w x
a low temperature 24 , but we do not find any data
available to compare with the expressions of Eqs
Ž18 and 19 quantitatively The difficulties in ob- Ž
serving low temperature VRH thermopowers might
Ž
be due to: i the magnitude of thermopowers is
Ž < < y1
often so small SS F 20 VK that could even not
Ž
ther-mopower is very sensitive to the conditions in
Ž preparing measurement samples vacuum level, im-purity content, deposition rate, substrate tempera-ture , they might uncontrollably affect the form of
Ž the DOS and the position of the Fermi level ; iii a possible compensation between the thermopower re-lated to the asymmetry of the DOS studied here and the Hubbard correlation contribution associated with the features of the electron distribution function, when two of these parts of thermopowers are
oppo-w x site on sign 22 Regarding all these difficulties, we assume that the cross-over curves in the figure quali-tatively describe the data for Fe ZrSe presented inx 2
Fig 4 for 2D and the data for a–Ge films
sented in Fig 5 for 3D in Ref 18 , including an existence of a shallow minimum
4 Conclusion
We have presented the expressions for describing the VRH thermopower cross-overs from the
well-known Mott T Ž dy 1.r Ž dq 1.
-behaviours to the tempera-ture-independent behaviours as the temperature de-creases The expressions are obtained by using the
Ž
‘effective’ DOS of Eq 4 , which tends to the Mott constant DOS in the high energy limit and tends to the ES Coulomb gap DOS in the opposite limit This form of DOS, on the one hand, is no other than the solution to the first approximation of Efros’s
self-w x Ž consistent equations 25 while the zero
approxima- tion gives the Coulomb gap and, on the other hand, was previously suggested for describing the Mott–ES
w x VRH resistance cross-overs 11,12 The obtained cross-over expressions are simple and show a pro-found manifestation of the Coulomb correlation They could also be extended for the whole class of the Mott to any soft gap regime cross-overs by a way
w x similar to that for VRH resistance cross-overs 13,14 Note again that while the limit expressions of Eq
Ž16 – 19 are well defined, independent of the cho- Ž
Ž
sen model of DOS, the energy E d measure of the
gap width should be used as an adjustable parameter
in fitting theoretical cross-over curves to experimen-tal data
Thus, the Mott–ES VRH thermopower cross-overs could be described by the same way of percolation
Ž methods using the same ‘effective’ DOS of Eq 4
Trang 6and with the same fitting parameter E as for corre- d
sponding VRH resistance cross-overs We do hope
that the present work will stimulate further
investiga-tions of electron–electron correlation effects on the
thermopower as well as other VRH transport
charac-ters which even might promise important
technologi-w x cal applications 26 The interesting behaviours of
w x the Coulomb gap, analyzed in recent works 27,28
should be manifested in the VRH transport
proper-ties To find possible relations between different
w x VRH transport characters as was done in Ref 29
may also be interesting since, as is stated by Polyakov
and Shklovskii 30 , the resistivity of the T
-be-haviour observed at the resistivity minimum in the
quantum Hall effect should be considered as an
effect of the 2D Coulomb gap
Acknowledgements
This work is partly supported by the collaboration
fund from the Solid State Group of Lund University,
Sweden
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