In the limiting case of single finite chains the results are in good agreement with existing analytical expressions for both the length and temperature dependences, but with a low tempera
Trang 1Variable range hopping in finite one-dimensional and
anisotropic two-dimensional systems Van Lien Nguyena,b,*, Dinh-Toi Dangc a
Theoretical Department, Institute of Physics, P.O Box 429 Bo Ho, Hanoi 10000, Viet Nam b
Physics Division, National Center for Theoretical Sciences, Hsinchu, Taiwan c
Physics Faculty, Hanoi State University, 90 Nguyen-Trai, Hanoi, Viet Nam
Received 15 August 2002
Abstract
The variable range hopping conduction is simulated in a strongly anisotropic two-dimensional (2D) percolation model, which consists of parallel conducting chains coupled to each other weakly via rare ‘‘impurities’’ The exponential temperature dependence of resistance has been calculated for samples of different size, interchain distance, and impurity concentration in two directions, longitudinal and perpendicular to the chain direction In the limiting case of single finite chains the results are in good agreement with existing analytical expressions for both the length and temperature dependences, but with a low temperature limit, depending on the chain length and localization length In the 2D case it was shown that there exists a crossover in relative behaviour between the longitudinal and transverse resistance of a finite system, which however disappears from the limit of infinite systems, where the hopping conduction should be always isotropic and obeys the 2D Mott law
r2003 Elsevier Science B.V All rights reserved
PACS: 71.55.Jv; 73.50.h
Keywords: Variable range hopping; Finite 1D systems; Anisotropic 2D systems
1 Introduction
The variable-range hopping (VRH) conduction
in low-dimensional systems has recently received a
significal attention [1–3] The interest stems not
only fromthe fundamental physics aspects of the
problem, but also from the potential practical
applications associated with the so-called
nano-structures, i.e quantumwells (two dimensional— 2D), quantumwires (one dimensional—1D) and quantumdots As it is well known, the Mott law for the temperature dependence of VRH resistivity has the general form[4]
rðT Þ ¼ r0expðT0=T Þ1=ðdþ1Þ; T0¼ b=kBg0xd; ð1Þ where r0is a prefactor, x is the localization length,
d is dimensionality, and b is a constant coefficient:
b ¼ 18:1 and 27=p for 3D and 2D systems, respectively (see, for example,Ref [5]) The Mott law was obtained on the assumption that the
*Corresponding author Theoretical Department, Institute of
Physics, P.O Box 429 Bo Ho, Hanoi 10000, Viet Nam Tel.:
+84-4-843-5917; fax: +84-4-8349050.
E-mail address: nvlien@iop.ncst.ac.vn (V.L Nguyen).
0921-4526/03/$ - see front matter r 2003 Elsevier Science B.V All rights reserved.
doi:10.1016/S0921-4526(03)00021-8
Trang 2density of localized states near the Fermi level is
constant, gðEÞ g0¼ const:
The general law of Eq (1) however cannot be
directly applied to the 1D case Kurkijarvi[6]was
the first to show that due to a divergence of the
spatial factor for a single infinite chain, when the
temperature T approaches zero, the asymptotic
VRH rðT Þ-dependence should behave as
rðTÞ ¼ r0expðT1=TÞ ð2Þ
with T1¼ a=kBg0x; and a ¼ 1=4: Later, Raikh and
Ruzin [7] by optimizing the hopping rate in the
ðr EÞ-space have also arrived at the expression
(2), but with a ¼ 1=2: The same value of a ¼ 1=2
can be found in Ref [8]
We recall that the activation behaviour of rðT Þ;
as given in Eq (2), was suggested for a single
infinite chain in the low temperature limit For
finite chains Brenig et al.[9]have first pointed out
that the temperature dependence rðT Þ seems to
have a Mott-like formlnðrðT Þ=r0ÞpT1=2; but the
resistivity then becomes depending on the length L
of the measured chain, r rðT ; LÞ: Treating the
problemby the way of percolation method Lee
et al [10,11] showed that to the first
approxima-tion in the low temperature limit the temperature
and length dependences of the VRH resistivity in a
single finite chain can be expressed in the form
/lnðr=r0ÞS ¼ ðT2=T Þ1=2½lnð2L=xÞ1=2; ð3Þ
where /?S implies an ensemble average, which
was assumed to be tantamount to an average over
chemical potential [10]; T2EðkBg0xÞ1: A similar
expression was later reported by Hunt [12], but
with a slight difference in the argument of
log-function: the factor 2 was replaced by e ¼ 2:718y
Thus, following Eq (3), for chains of a given L=x
the VRH resistance depends on the temperature as
/lnðr=r0ÞSpT1=2 and, on the other hand, at
a given temperature the length dependence of
the VRH resistance has the form /lnðr=r0ÞS
p½lnð2L=xÞ1=2: The expression (3) tends to the
Kurkijarvi’s activation behaviour of Eq (2) as L is
large and the temperature is low enough [12]
Shante [13] considered a more complicated
model of a large number of parallel chains coupled
weakly to each other in such a way that the
interchain hops are also allowed, though rare
Restricting our attention to the plane systems, within this model, it was shown that due to a finite interchain coupling the low temperature conduc-tion becomes isotropically 2D with the power n in
ln rðT ÞpTn equaling 1=3 (2D Mott-like) in the low temperature limit and weakly increasing with temperature to the value E1=2:7 [13] An iso-tropicity, i.e two equal exponents of resistances, measured in directions, longitudinal (longitudinal resistance (LR), rjj) and perpendicular (transverse resistance (TR), r>) to the chain direction, may be however violated in certain interval of relatively high temperatures as noted later by Zvyagin[8] It should be emphasized here that all that is discussed in Refs [8,13] is exclusively concerned with infinite samples, where the topologically anisotropic details may be averaged out, leading
to an isotropicity in hopping conductions at low temperatures, when the typical hopping distance is about or exceeds the distance between adjacent chains Some questions may then be raised, for example, if such an isotropicity is still held in real systems with finite sizes or if both resistances
rðjj;>ÞðTÞ always follow the 2D Mott law Further-more, there may exist the ‘‘impurities’’ in the spaces between chains, which effectively support interchain transitions, as noticed in various experiments[14–16]
The aimof this work is to simulate the VRH resistances in a more realistically anisotropic 2D model, which is similar to Shante’s model consist-ing of parallel conductconsist-ing chains, but with addi-tional ‘‘impurities’’ located in the interchain spaces and with a special emphasis on finite size effects A single finite 1D chain then simply appears as the limiting case of the model, when the simulation results could be compared with existing theoretical expressions
2 The model and calculation The simulated model is schematically drawn in
Fig 1
sample with sides parallel to the x- and y-axis of the deCartesien coordinates systemcontaining n linear conducting chains, arranged in the way that they are parallel with the x-direction and regularly
Trang 3separated fromeach other Along each conducting
chain the sites, modelling the centres of localized
states, are randomly arranged by the Poisson
distribution with an average separation being
equal to 1 (the length units in the simulation)
The interchain-coupling ‘‘impurities’’ (or ‘‘small
metallic islands’’) are modelled by sites added at
randominto the interchain spaces with a
concen-tration so1: The distance between adjacent
chains, d; is assumed to be always much larger
than the average separation between sites in a
chain, db1: Further, each site i at location ~rri(both
at chains or between them) is given an energy Ei
chosen randomly with a uniform distribution in
the range ½W =2; W =2: As a result, we have a
typical model for studying the 2D Mott VRH[17]
with a particular emphasis on an anisotropicity in
the topological structure of systems in accordance
with some experimental descriptions[15,16]
Here-after, each configuration of randomsite
coordi-nates ½~rri and associated energies ½Ei will be as
usual referred to as a site realization
Within the suggested model each simulation
sample is characterized by three physical
para-meters: the sample size L; distance between chains
d ðdb1Þ; and interchain impurity concentration
so1 (the linear concentration of sites in chains is equal to 1) In the limit of s ¼ 0 the model is reduced to the case considered inRefs [8,13]if L is infinitely large, and, particularly, to a single finite chain of [10]if n ¼ 1 and L is finite In the other limit of s-1 we will practically deal with a quasi-isotropic 2D system In general, for 0os51; the simulated sample is strongly anisotropic in the sense that it is merely easier to ‘‘hop’’ along chains than in the perpendicular direction
As is well known, to the exponential depen-dences the problemof calculating the VRH resistivity in the present model can be reduced to calculating the equivalent resistance of the Miller– Abrahams random resistor network [17,18], in which the hopping between two sites i and j is equivalent to having a resistor Rij such that
lnðRij=R0Þ ¼ 2j~rri ~rrjj=x þ ðjEi mj þ jEj mj
þ jEi EjjÞ=2kBT Zij; ð4Þ
where ~rri and Eiare the position vector and energy
of the site i; m is the chemical potential; R0 is the pre-exponential factor, whose temperature and position dependences are relatively weak and often assumed to be neglected
In practical simulations the expression (4) of course should be rewritten in a dimensionless form In this work the average distance between sites in a chain and the band width W are chosen
to be the length and energy units, respectively The chemical potential m can then take any value in the range from 0:5 to 0:5 (in units of W )
Following the percolation approach[17,18], the exponent Zc in the percolative resistance r ¼
r0expðZcÞ of the network of resistors (4) can be determined as the threshold in the continuum percolation problemwith the criterion of ZijpZc; where Zijis defined in Eq (4) In our model, due to
a strong anisotropicity in the topological structure, perhaps, the percolation in the direction, long-itudinal to the chain direction, should proceed easier than that in the perpendicular direction In other words, for a given sample the LR rjjshould
be smaller than the TR r>; or in terms of percolation threshold, Zjjc¼ lnðrjj=r0ÞoZ>
lnðr>=rÞ: The ratio Z>=Zjj can be therefore
d
L, x
Fig 1 The model: the parallel conducting chains are coupled
weakly to each other via rare ‘‘impurities’’ modelled by the
randomdots in spaces between chains.
Trang 4thought of as a measure for the anisotropicity in
VRH resistance of systems under study
Thus, given parameters L; d; s; and
dimension-less temperature t kBT=W ; two quantities Zðjj;>Þc
have to be calculated for each site realization and
each value of chemical potential m: For this end,
the percolations are always examined along two
directions, fromthe right to left edge (for Zjjc) and
fromthe upper to lower edge (for Z>c ) of the
sample (see, Fig 1), using the standard
percola-tion-checking procedure as discussed in detail in
values /Zðjj;>Þ
c S the percolation thresholds Zðjj;>Þ
c
are then averaged over many random ðr;
EÞ-realizations The calculations have been performed
for samples of L ¼ 100; 200, 400, 600, 800, and
1000 with d ¼ 10 and 20, and different values of s
between zero and 1 The number of realizations
used for averaging Zðjj;>Þc is k ¼ 5000 for L ¼ 100
and decreases as L increases in such a way that the
2 are almost constant for all the
cases under study
We like to mention that the present 2D
calculation is in fact a generalized extension of
our recent work [20], where the study was
restricted to the so-called ‘‘r-percolation’’, i.e to
solving the percolation problemof Eq (4) for the
same model, but only in the limit of infinite
temperature T-N: There it was shown that while
there is a quasi-1D to 2D crossover in the
percolation radius of finite systems as the impurity
concentration s increases, in the limit of infinite
systems two percolations, longitudinal and
trans-verse, are always equivalent, regardless of s:
In the limiting case of finite 1D chains, as was
pointed out by Lee[10], the hopping systemis not
self-averaging, so that the realization averaging
alone is not acceptable for the experimental
comparisons To obtain ensemble quantities,
which could be compared with analytical and
experimental results, Lee suggested an argument,
which consists in combining two averages of Zc;
over chemical potential positions m and over site
realizations The former originates from the fact
that in experiments the measurement results are
averaged over gate voltages[10,11], while the latter
is the standard ensemble averaging In the present
study, all double-averaging values /ZS (the same
notation /?S will be used for short) for the 1D case have been obtained in the following way: first, for each site realization we average Zc over 200 values of m; ranging regularly from 0:4 to 0.4, and then, the obtained m-averaging percolation thresholds are further averaged over a number of randomsite realizations Note that for a given value of m the percolation threshold strongly fluctuates fromone realization to another (much stronger than fluctuations over m for a given site realization) However, after averaging over 200 values of m; the fluctuation of m-averaged /ZcSm over site realizations is always very weak There-fore, it is not necessary to examine many realiza-tions to find good double-averaging values /ZcS:
In this work the 1D calculations have been performed for samples of length L ¼ 1000; 2000,
4000, 8000, 16,000, 32,000, and 64,000 The number of random site realizations used for obtaining double-averaging quantities is about 30–1000 (inversely proportional to L)
3 Numerical results and discussion
In this section we present simulation results for exponents of the VRH resistance, /ZcS for 1D and /Zðjj;>Þ
c S for 2D dependent on the tempera-ture, sample size, and other parameters (i.e d and
s in the 2D case) The 1D data, presented inFigs 2
analytical expressions Concerning the 2D case, the data presented in Figs 4–6 are mainly con-centrated in examining an anisotropicity of the VRH resistance and the role of finite size effects in the model under study
3.1 Single finite 1D chains
/ZcS /lnðr=r0ÞS; plotted against t1=2 ðt
kBT=W Þ for some chains of different ðL; xÞ in a large range of temperatures t: For all samples under study, the typical feature of the obtained /ZcðtÞS-dependence is that the simulation points follow very well the relation /ZcðtÞSpt1=2; in some temperature range (approximately described
by the fitting solid straight lines in Fig 2), then
Trang 5become relatively downward at lower
tempera-tures (the dashed curves) Thus, our simulation
results, on the one hand, well support the T1=2
-behaviour of the temperature dependence of VRH
resistivity in finite chains as given in Eq (3), and
on the other hand, show that the range of
temperatures, where this behaviour can be
ob-served, does not extend to the limit of zero
temperature as generally suggested [10,11], but
has some low limit tc(shown, for example, by the
arrow to the lowest curve in Fig 2) This
temperature tcclearly depends on the chain length
L as well as the localization length x: tc decreases
as L increases (compare four upper data inFig 2
for samples with the same x; but with different L) and for a given L it increases with increasing x (compare two lower data inFig 2for samples of 50ðWÞ) The fact that there exists such a low temperature limit in observing the law (3) can be easily understood qualitatively fromthe expression for the typical hopping distance, corresponding to the VRH regime of Eq (3)[10,17]:
rhE1
2xðT2=TÞ
This relation means a continuous increase of rh with decreasing T : However, since rh cannot,
of course, exceed the sample chain L; there should exist some temperature Tcsuch that for all ToTc
the hopping distance rh; limited by L; ceases to increase with lowering T: The relation (5) is then violated, the conduction is no more
15 20 25
ηc
Fig 3 The finite 1D chains: /ZcS versus ½lnð2L=xÞ 1=2 at t ¼ 0:002 (in the t 1=2 -regime) The simulation points are data for samples of L ¼ 1000; 2000, 4000, 8000, 16,000, 32,000, and 64,000 and with two localization lengths, x ¼ 20 ð3Þ and 50ðWÞ: The dashed lines—fitting straight lines of data to the length dependence of Eq (3).
0
20
40
60
80
100
t -1/2
η c
t c
1
2
t c
Fig 2 The finite 1D chains: /ZcS /lnðr=r0ÞS versus t 1=2
for chains of different ðL; xÞ; fromtop: (64,000, 20); (16,000,
20); (4000, 20); (1000, 20); and (1000, 50) The fitting straight
solid lines correspond to the t 1=2 -law of Eq (3) For each
sample the dashed lines freely connect simulation points
out-side the t 1=2 -range Inset: t c is plotted against qðL; xÞ
xL 2 lnð2L=xÞ for samples of ðL; xÞ; fromtop: (1000,50);
(1000,40); (1000,30); (1000,20); (2000,50); (2000,20); and
(4000,50).
Trang 6‘‘variable-range’’ hopping, and therefore, the
expression of Eq (3) is no more a matter of
interest Here it is necessary to note that, though at
ToTc the hopping distance is already limited by
L; due to the energy term ðpT1Þ in the
expression (4) the sample resistance still increases
with lowering T (see, dashed lines in Fig 2)
Besides, the factor x in the relation (5) explains
why Tc becomes higher with increasing
localiza-tion length
Fromthe fact that the hopping distance rh (5)
is limited by the chain length at ToTc; we
can suggest the simple scaling relation Tcp
T2ðL=xÞ2lnð2L=xÞ; which is rewritten in the
dimensionless form of simulations as
tcpxL2lnð2L=xÞ; ð6Þ
where T2¼ W =ðkBxÞ was used The scaling rela-tion (6) is ready to be compared with simularela-tion data Such a comparison is given in the inset of
versus the quantity qðL; xÞ xL2lnð2L=xÞ for samples with different L and x: Here we like to mention that in order to get an acceptably accurate value of tc for each sample we have to carefully examine the /ZcðtÞS-dependence in some large range of temperatures.1 Obviously, despite con-siderable errors, the simulation points in the inset
1.4
1.6
1.8
2.0
2.2
ηc
s = 0.1
1.2
1.4
1.6
1.8
lnt (arbitrary shift)
s = 0.4
(a)
(b)
Fig 4 ln/Zjj >cS (3 and W) are plotted
versus ln t for samples of the same L ¼ 100; but with different
frame) and 0.4 (b) (lower frame, where straight solid line
represents the 2D Mott law) Note on an arbitrary shift in both
axes.
1.2 1.4 1.6 1.8 2.0 2.2
lnt (arbitrary shift)
ηc
0 0.2 0.4 0.6 0.8 1
s c
Fig 5 Similar to Fig 4 , but for sample of L ¼ 1000; d ¼ 10 and different s (fromtop): 0.1, 0.2, 0.3, and 0.4 The symbols and 3 are for /Z jj S and /Z >
c S; respectively The dashed lines are the best fitting straight lines of simulation points The solid straight line represents the 2D Mott law Inset: s c versus L 1 for
Ref [20]
1 For this end, we calculated /ZcS at about ten temperature points around the expected one, then used the MATLAB functions to extend the data and to fix tc: Such a procedure costs much computer time, and therefore the calculations have been restricted to chains of L p4000:
Trang 7supporting the scaling relation (6) This relation is
assumed to be useful whenever one likes to observe
the law (3)
One more consequence can be found fromFig 2
in relating to the question about how long the
simulation chain should be for observing the
activation behaviour of Eq (2) [10] Basically,
the activation behaviour is expected to be observed
only in the limit of T-0 [6] However, as
discussed above, Fig 2 demonstrates that the
VRH mechanism cannot be realized in a finite
chain at any low temperature So, in reality, our
simulation results suggest that it is practically
impossible to observe the activation behaviour in
the T -dependence of VRH resistivity in any finite
simulation chain
of the expression (3), the length dependence
of the VRH resistance For this end the
quantities /ZcS /lnðr=r0ÞS are presented
ver-sus ½lnð2L=xÞ1=2 at the temperature t ¼ 0:002;
belonging to the t1=2-regime The data used for this figure are exactly those used in Fig 2 for chains with lengths between 1000 and 64,000 and two different localization lengths: x ¼ 20 ðJÞ and
50 ðWÞ: In both cases, clearly, the simulation points follow very well the linear relation /lnðr=r0ÞSp½lnð2L=xÞ1=2 as given in Eq (3) Hence, on the whole, our simulation data in
tempera-ture and length dependences, of the expression (3), but with a low temperature limit Tc; depending on
L and x: Certainly, the length dependence, as observed in Fig 3, can only be realized at temperatures in the t1=2-range and the expression (3), as a whole, is valid only in chains with L=x large enough
3.2 Anisotropic 2D systems Before carrying out simulations in the 2D case,
we realized that since we are only interested on the relatively exponential temperature dependence of VRH resistances, it is reasonable to rewrite the relation of Eq (4) in the form: Zn
ij ðx=2ÞZij¼ j~rri ~rrjj þ Eij=tn
; where Eij¼ ðjEi mj þ jEj mj þ
jEi EjjÞ=2 and tn
¼ ðx=2ÞkBT=W ; and then solve the corresponding percolation problem Zn
ijpZn
c: The localization length is now simply an implicit parameter, and therefore, we can avoid the need to work with too large systems, since, otherwise, the sample size must be large ðLbxÞ; while a sample of
L ¼ 1000; d ¼ 10; and s ¼ 0:2; for example,
5 sites It should be, how-ever, emphasized that we are now not interested in the value of resistances, but only their relatively temperature-dependent behaviours, which should not be affected by such a linear ‘‘re-scaling’’ of the percolation relation, as originally stated by Sinai (Sinai’s theorem, see Ref [17], Chapter 5) Certainly, the localization length is here taken to
be the same for both the longitudinal and transverse conductions
We like to emphasize also that all that is discussed above relating to the Z-percolation problemcan be equally applied to the re-scaling
Zn
-problem Moreover, because the difference between ðZ; tÞ and ðZn
; tn
Þ is nothing but a change
of scale, fromhere on, for simplicity, the star
1.0
1.01
1.02
1.03
L -1
η c
Fig 6 The ratio Z>c=Z jj is plotted versus L 1 for some values of
s (fromtop): 0.1, 0.2, 0.3, and 0.4 ðd ¼ 10Þ: The dashed straight
lines are the best fits of data to the suggested linear relation (see
the text).
Trang 8symbol (*) will be removed from everywhere,
assuming that both Zc and t are counted with an
arbitrary scale Thus, we at first examine the
temperature-dependent behaviours of two
resis-tance exponents /Zðjj;>Þc S ¼ /lnðrðjj;>Þ=r0ÞS as
well as their relative magnitude for samples with
different L; d; and s and then analyze the role of
finite size effects
and ln/Z>c S (3 and W) versus ln t for samples of
the same L ¼ 100; but with different interchain
distances d; d ¼ 10 (lower points, and 3) and 20
tions s: s ¼ 0:1 (upper frame, (a)) and 0.4 (lower
frame, (b)) The most important feature observed
in this figure is that for all the studied cases the low
temperature simulation points always follow an
universal behaviour, which fits well to the linear
relation ln/Zðjj;>Þc S ¼ n ln t: The constants n;
measured by the slopes of fitting straight (solid
for Zjjc and dashed for Z>c ) lines are however
different While the slopes of all dashed fitting
lines of TR data, regardless of d and s; fall well
into a narrow value range n>¼ 0:32570:005; for
the solid lines of LR Zjj
c the slopes njj slightly decreases with decreasing d and/or increasing s:
njjE0:38 and E0:36 for s ¼ 0:1 and d ¼ 20 and 10,
respectively (frame (a)); and njjE0:34 for s ¼ 0:4
and d ¼ 10 (frame (b)) There is an argument to
believe that njjshould tend to the valueE1=2 for a
single finite chain in the limit of large d and s-0:
On the other hand, an increase of s results in a
rapid reduction of not only njj; but also the relative
difference between two resistance exponents: while
in the upper frame (a) for s ¼ 0:1; clearly, Z>c > Zjjc;
in the lower frame (b) for s ¼ 0:4 two points of
Zðjj;>Þc
close to each other and both data follow well the
Mott law (straight solid line), especially at low
temperatures
All that is recognized in Fig 4 for L ¼ 100 is
also observed in simulation samples of other sizes
up to 1000 The behaviours of Zðjj;>Þ
c as a function
of t are qualitatively the same, but the larger L; the
role of d and s as well as the discrepancy between
two resistance exponents become weaker As
an example, we show in Fig 5 the data for
the cases L ¼ 1000; d ¼ 10; but with different s:
s ðfromtopÞ ¼ 0:1; 0.2, 0.3, and 0.4 For each s the symbols and 3 represent simulation points for Zjjc and Z>c ; respectively Obviously, except the case of smallest s; s ¼ 0:1; practically, the two data-points for Zðjj;>Þc are everywhere correspondingly coin-cident in obeying the Mott law (straight solid line) One can then roughly assume that for such large samples of L ¼ 1000 and d ¼ 10 the LR and TR could be seen to be equal, i.e the systemcould be considered isotropic 2D, when the impurity con-centration is about 0.2 and higher
Examining accurately the ratio Z>c =Zjj
c as the concentration s varies, we recognized that consis-tent with what reported in Ref [20] for the corresponding r-percolation problem, given d; for each sample of size L; there exists a ‘‘critical’’ impurity concentration scin the sense that Z>c =Zjjc>
1 and njj> n>E1=3 at all sosc; while for all sXsc
two resistance exponents are coincident: Z>c =Zjjc¼
1 and njj¼ n>E1=3: The values sc obtained for different L are exactly those presented inFig 6of
inset of Fig 5
20 ð Þ: Thus, in general, for any sample of finite size our simulations suggest an existence of the well-defined critical impurity concentration, where the VRH conduction experiences a crossover from the anisotropic regime to the isotropic one with the 2D Mott temperature dependent behaviour Parti-cularly, the VRH longitudinal resistance of finite samples may experience a crossover from the finite 1D-like behaviour, i.e ln Zjjcptnjj with njj being close to the 1D-value 1/2 of Eq (3), to the 2D Mott behaviour of njjE1=3: We assume that this suggested crossover might be useful in under-standing the dimensional crossover induced by impurities observed in the VRH conduction in various compounds[14,15]
However, to our understanding, the recognized crossover should be the property of only finite systems To verify this, we have examined, on the one hand, how the ‘‘critical’’ concentration sc depends on L; and, on the other hand, how the ratio Z>c =Zjj
c changes with L for a given s: The former is already shown in the inset ofFig 5 The fact that sc monotonically decreases with increas-ing L can be merely used as an argument in suggesting that in the limit of L-N; any system
Trang 9should always be isotropic and obey the 2D Mott
T1=3-law whatever small s be In finite systems an
anisotropicity in material structure, in sample size,
or in boundary conditions may unequally affect the
conductions, measured along different directions
[21] In infinite systems such an anisotropicity
should be however averaged out, giving rise to an
isotropicity of the percolation conduction with
respect to measuring directions For the latter, we
present inFig 6the ratio Z>c =Zjj
c; plotted versus L1
for samples with d ¼ 10 and different s (fromtop):
0.1, 0.2, 0.3, and 0.4 Obviously, for any s under
study the ratio Z>c =Zjj
c always tends to a unit as
L-N:2
This unambiguously implies that in infinite
systems two VRH resistances, TR and LR, are
always coincident, i.e two conducting directions
are always equivalent, regardless of s: Thus, both
data, the inset ofFig 5andFig 6, strongly support
the idea that, as discussed in detail inRef [20], the
observed crossover is simply a consequence of finite
size effects There is no crossover in the relative
magnitude of the resistance exponents Z>c =Zjjc as
well as the ‘‘dimensional crossover’’ in the
tem-perature-dependent behaviour of LR Zjjc in infinite
systems, where the VRH conduction is always
isotropic and obeys the 2D Mott law
Lastly, it should be mentioned that two sets of
data, corresponding to the smallest ðL ¼ 100Þ and
the largest ðL ¼ 1000Þ from simulated samples,
have been specially chosen to be presented in
or s and d exhibit similar behaviours Besides,
since all presented averaging values are so accurate
that the error bars nowhere exceed the symbol
sizes, they are, therefore, not shown in all the
figures, except the tc-data in the inset ofFig 2
4 Conclusion
We have simulated the Mott VRH conduction
in a strongly anisotropic 2D model, which consists
of parallel conducting chains coupled to each other
weakly via rare impurities The exponents of the
longitudinal and transverse percolation resistances
ln rðjj;>Þ have been calculated for samples of different sizes L; interchain distances d; and impurity concentrations s and in a large range of temperatures T ; corresponding to the VRH regime
In the limiting case of single finite 1D chains the results, on the one hand, support well existing analytical expressions for both the temperature and length dependences of VRH resistivity and, on the other hand, show that the temperature range in observing the law ln rpT1=2 does not extend to the limit of T-0; but has a low temperature limit, depending on the chain length and the localization length
In the general anisotropic 2D case it was shown that (1) for each systemof finite size there exists an impurity-induced crossover in the relative magnitude between the LR and TR fromthe regime where
TR > LT to the regime, where two resistances are coincident; (2) at the crossover the exponent njjin the
LR, ln rjjpTnjj; changes fromthe finite 1D-like value to the 2D Mott value (‘‘dimensional’’-like crossover); and ð3Þ as a consequence of finite size effects the observed crossover disappears in the limit
of infinite systems, where the VRH conduction should be always isotropic and obeys the 2D Mott law We do hope that these results may be useful in describing the impurity-induced dimensional cross-over in the temperature dependence of VRH observed in some compounds [15] Concerning the temperature-induced crossover [16] the picture is similar: at relatively high temperature, when the hopping distance is relatively small, the electrons can hop only along chains and the VRH behaves as mostly 1D, while at lower temperatures, when the hopping distance is large enough, the transverse direction becomes conducting and the VRH be-comes 2D Finally, we assume that though this work deals with the Mott VRH only, qualitatively, its results may also be applied for the Efros–Shklovskii Coulomb gap VRH, which is discussed in detail, for instance, inRefs [22,23]for the 2D case
Acknowledgements One of us (V.L.N.) thanks the National Center for Theoretical Sciences (Hsinchu, Taiwan) for
2 Interestingly, the data in Fig 6 can be approximately
expressed as Z>c=Z jj E1 þ gL 1 with gps 1 (see, dashed fitting
straight lines).
Trang 10kind financial support to his visit during which this
work was completed The Computer Center for
Nuclear Science (INST, Hanoi) is acknowledged
for generous computer facility This work was in
part supported by the collaboration fund from
Solid State Group of Lund University (Sweden)
and by Natural Science Council of Vietnam
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