Physica B 334 (2003) 88–97 Variable 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 r 2003 Elsevier Science B.V All rights reserved PACS: 71.55.Jv; 73.50.Àh Keywords: Variable range hopping; Finite 1D systems; Anisotropic 2D systems Introduction The variable-range hopping (VRH) conduction in low-dimensional systems has recently received a significal attention [1–3] The interest stems not only from the fundamental physics aspects of the problem, but also from the potential practical applications associated with the so-called nano*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) structures, i.e quantum wells (two dimensional— 2D), quantum wires (one dimensional—1D) and quantum dots As it is well known, the Mott law for the temperature dependence of VRH resistivity has the general form [4] rTị ẳ r0 expT0 =Tị1=dỵ1ị ; T0 ẳ b=kB g0 xd ; 1ị where r0 is 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 0921-4526/03/$ - see front matter r 2003 Elsevier Science B.V All rights reserved doi:10.1016/S0921-4526(03)00021-8 V.L Nguyen, D.-T Dang / Physica B 334 (2003) 88–97 density of localized states near the Fermi level is constant, gðEÞ g0 ¼ 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 rðTÞ-dependence should behave as rTị ẳ r0 expT1 =Tị 2ị with T1 ¼ a=kB g0 x; and a ¼ 1=4: Later, Raikh and Ruzin [7] by optimizing the hopping rate in the ðr À 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 rðTÞ; 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 rðTÞ seems to have a Mott-like form lnðrðTÞ=r0 ÞpT À1=2 ; but the resistivity then becomes depending on the length L of the measured chain, r rðT; LÞ: Treating the problem by the way of percolation method Lee et al [10,11] showed that to the first approximation 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 /lnr=r0 ịS ẳ T2 =Tị1=2 ẵln2L=xị1=2 ; 3ị where /?S implies an ensemble average, which was assumed to be tantamount to an average over chemical potential [10]; T2 EðkB g0 xÞÀ1 : A similar expression was later reported by Hunt [12], but with a slight difference in the argument of logfunction: the factor 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=r0 ÞSpT À1=2 and, on the other hand, at a given temperature the length dependence of the VRH resistance has the form /lnr=r0 ịS pẵln2L=xị1=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 89 Restricting our attention to the plane systems, within this model, it was shown that due to a finite interchain coupling the low temperature conduction becomes isotropically 2D with the power n in ln rðTÞpT Àn equaling 1=3 (2D Mott-like) in the low temperature limit and weakly increasing with temperature to the value E1=2:7 [13] An isotropicity, 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;>Þ ðTÞ always follow the 2D Mott law Furthermore, there may exist the ‘‘impurities’’ in the spaces between chains, which effectively support interchain transitions, as noticed in various experiments [14–16] The aim of this work is to simulate the VRH resistances in a more realistically anisotropic 2D model, which is similar to Shante’s model consisting of parallel conducting chains, but with additional ‘‘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 The model and calculation The simulated model is schematically drawn in Fig 1, where it is assumed a ðL  LÞ-square sample with sides parallel to the x- and y-axis of the deCartesien coordinates system containing n linear conducting chains, arranged in the way that they are parallel with the x-direction and regularly V.L Nguyen, D.-T Dang / Physica B 334 (2003) 88–97 L, y 90 d L, x so1 (the linear concentration of sites in chains is equal to 1) In the limit of s ¼ the model is reduced to the case considered in Refs [8,13] if L is infinitely large, and, particularly, to a single finite chain of [10] if n ¼ and L is finite In the other limit of s-1 we will practically deal with a quasiisotropic 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 dependences the problem of 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 Fig The model: the parallel conducting chains are coupled weakly to each other via rare ‘‘impurities’’ modelled by the random dots in spaces between chains ri À ~ rj j=x ỵ jEi mj ỵ jEj mj lnRij =R0 ị ẳ 2j~ ỵ jEi Ej jị=2kB T Zij ; separated from each 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 (the length units in the simulation) The interchain-coupling ‘‘impurities’’ (or ‘‘small metallic islands’’) are modelled by sites added at random into the interchain spaces with a concentration 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 ~ ri (both at chains or between them) is given an energy Ei chosen randomly with a uniform distribution in the range ½ÀW =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] Hereafter, each configuration of random site coordinates ½~ ri 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 parameters: the sample size L; distance between chains d ðdb1Þ; and interchain impurity concentration where ~ ri and Ei are 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 À0:5 to 0:5 (in units of W ) Following the percolation approach [17,18], the exponent Zc in the percolative resistance r ¼ r0 expðZc Þ of the network of resistors (4) can be determined as the threshold in the continuum percolation problem with the criterion of Zij pZc ; where Zij is defined in Eq (4) In our model, due to a strong anisotropicity in the topological structure, perhaps, the percolation in the direction, longitudinal to the chain direction, should proceed easier than that in the perpendicular direction In other words, for a given sample the LR rjj should be smaller than the TR r> ; or in terms of percolation threshold, Zjjc ¼ lnrjj =r0 ịoZ> c ẳ > > lnr =r0 ị: The ratio Zc =Zjjc can be therefore ð4Þ V.L Nguyen, D.-T Dang / Physica B 334 (2003) 88–97 thought of as a measure for the anisotropicity in VRH resistance of systems under study Thus, given parameters L; d; s; and dimensionless temperature t kB T=W ; two quantities Zðjj;>Þ c 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, from the right to left edge (for Zjjc ) and from the upper to lower edge (for Z> c ) of the sample (see, Fig 1), using the standard percolation-checking procedure as discussed in detail in Ref [19] In order to obtain ensemble averaging values /Zðjj;>Þ S the percolation thresholds Zðjj;>Þ c c are then averaged over many random ðr; EÞrealizations 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 The number of realizations used for averaging Zðjj;>Þ is k ¼ 5000 for L ¼ 100 c and decreases as L increases in such a way that the number k  L2 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 problem of 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 transverse, are always equivalent, regardless of s: In the limiting case of finite 1D chains, as was pointed out by Lee [10], the hopping system is 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 /Zc S (the same 91 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 À0:4 to 0.4, and then, the obtained m-averaging percolation thresholds are further averaged over a number of random site realizations Note that for a given value of m the percolation threshold strongly fluctuates from one 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 /Zc Sm over site realizations is always very weak Therefore, it is not necessary to examine many realizations to find good double-averaging values /Zc S: 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) Numerical results and discussion In this section we present simulation results for exponents of the VRH resistance, /Zc S for 1D and /Zðjj;>Þ S for 2D dependent on the temperac ture, sample size, and other parameters (i.e d and s in the 2D case) The 1D data, presented in Figs and 3, will be discussed in comparison with analytical expressions Concerning the 2D case, the data presented in Figs 4–6 are mainly concentrated 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 In Fig we show, for example, the quantities /Zc S /lnðr=r0 ÞS; plotted against tÀ1=2 ðt kB T=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 /Zc ðtÞS-dependence is that the simulation points follow very well the relation /Zc ðtÞSptÀ1=2 ; in some temperature range (approximately described by the fitting solid straight lines in Fig 2), then V.L Nguyen, D.-T Dang / Physica B 334 (2003) 88–97 92 25 tc × 103 100 80 20 60 0.5 1.5 q(L, ξ) × 104 40 15 20 2.2 tc 0 20 2.4 2.6 2.8 [ ln( 2L/ ξ ) ]1/2 40 60 80 100 -1/2 t Fig The finite 1D chains: /Zc S /lnðr=r0 ÞS versus tÀ1=2 for chains of different ðL; xÞ; from top: (64,000, 20); (16,000, 20); (4000, 20); (1000, 20); and (1000, 50) The fitting straight solid lines correspond to the tÀ1=2 -law of Eq (3) For each sample the dashed lines freely connect simulation points outside the tÀ1=2 -range Inset: tc is plotted against qðL; xÞ xLÀ2 lnð2L=xÞ for samples of ðL; xÞ; from top: (1000,50); (1000,40); (1000,30); (1000,20); (2000,50); (2000,20); and (4000,50) become relatively downward at lower temperatures (the dashed curves) Thus, our simulation results, on the one hand, well support the T À1=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 observed, 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 tc clearly depends on the chain length L as well as the localization length x: tc decreases as L increases (compare four upper data in Fig Fig The nite 1D chains: /Zc S versus ẵln2L=xị1=2 at t ẳ 0:002 (in the tÀ1=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) for samples with the same x; but with different L) and for a given L it increases with increasing x (compare two lower data in Fig for samples of the same L ẳ 1000; but with different x: 20ị and 50Wị) The fact that there exists such a low temperature limit in observing the law (3) can be easily understood qualitatively from the expression for the typical hopping distance, corresponding to the VRH regime of Eq (3) [10,17]: rh E xðT2 =TÞ1=2 : ð5Þ 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 Tc such 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 V.L Nguyen, D.-T Dang / Physica B 334 (2003) 88–97 93 (a) 2.2 0.8 sc (L) 2.0 2.0 0.6 0.4 1.8 0.2 1.6 ln (arbitrary shift) ln (arbitrary shift) 2.2 s = 0.1 1.4 1.8 1.6 1.8 0 0.004 0.008 0.012 L-1 1.6 1.4 t-1/3 -1/3 1.4 t 1.2 1.2 s = 0.4 -3 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 Fig ( and Â) and ln/Z> c S (3 and W) are plotted versus ln t for samples of the same L ¼ 100; but with different d: 10 ( and 3) and 20 ( and W) and different s: 0.1 (a) (upper frame) and 0.4 (b) (lower frame, where straight solid line represents the 2D Mott law) Note on an arbitrary shift in both axes ln/Zjjc S ‘‘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 ðpT À1 Þ 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 localization length From the fact that the hopping distance rh (5) is limited by the chain length at ToTc ; we can suggest the simple scaling relation Tc p T2 ðL=xÞÀ2 lnð2L=xÞ; which is rewritten in the dimensionless form of simulations as tc pxLÀ2 lnð2L=xÞ; ð6Þ -2 -1.5 -1 -0.5 lnt (arbitrary shift) lnt (arbitrary shift) (b) -2.5 Fig Similar to Fig 4, but for sample of L ¼ 1000; d ¼ 10 and different s (from top): 0.1, 0.2, 0.3, and 0.4 The symbols and are for /Zjjc 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: sc versus L1 for d ẳ 10 ị and 20 ị; repeated partly from Ref [20] where T2 ¼ W =ðkB xÞ was used The scaling relation (6) is ready to be compared with simulation data Such a comparison is given in the inset of Fig 2, where the simulation values tc are plotted versus the quantity qðL; xÞ xLÀ2 lnð2L=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 /Zc ðtÞS-dependence in some large range of temperatures.1 Obviously, despite considerable errors, the simulation points in the inset of Fig lie quite well along a straight line in For this end, we calculated /Zc S 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 Lp4000: V.L Nguyen, D.-T Dang / Physica B 334 (2003) 88–97 94 belonging to the tÀ1=2 -regime The data used for this figure are exactly those used in Fig for chains with lengths between 1000 and 64,000 and two different localization lengths: x ẳ 20 Jị and 50 Wị: In both cases, clearly, the simulation points follow very well the linear relation /lnr=r0 ịSpẵln2L=xị1=2 as given in Eq (3) Hence, on the whole, our simulation data in Figs and support both aspects, the temperature 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 tÀ1=2 -range and the expression (3), as a whole, is valid only in chains with L=x large enough 1.03 η⊥ /η|| c c 1.02 1.01 1.0 3.2 Anisotropic 2D systems 0.002 0.004 0.006 -1 L jj Z> c =Zc Fig The ratio is plotted versus LÀ1 for some values of s (from top): 0.1, 0.2, 0.3, and 0.4 d ẳ 10ị: The dashed straight lines are the best fits of data to the suggested linear relation (see the text) supporting 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 from Fig 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 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 Fig is devoted to examining another aspect of the expression (3), the length dependence of the VRH resistance For this end the quantities /Zc S /lnðr=r0 ịS are presented versus ẵln2L=xị1=2 at the temperature t ẳ 0:002; 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: Znij x=2ịZij ẳ j~ ri ~ rj j ỵ Eij =tn ; where Eij ẳ jEi mj ỵ jEj mj ỵ jEi Ej jị=2 and tn ¼ ðx=2ÞkB T=W ; and then solve the corresponding percolation problem Znij pZnc : 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, contains about  105 sites It should be, however, 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 problem can 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, from here on, for simplicity, the star V.L Nguyen, D.-T Dang / Physica B 334 (2003) 88–97 symbol (*) 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 resistance exponents /Zjj;>ị S ẳ /lnrjj;>ị =r0 ÞS as c well as their relative magnitude for samples with different L; d; and s and then analyze the role of finite size effects In Fig we plot quantities ln/Zjjc S ( and Â) 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 (upper points,  and W), and different concentrations 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/Zjj;>ị S ẳ n ln t: The constants n; c 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 Zjjc the slopes njj slightly decreases with decreasing d and/or increasing s: njj E0:38 and E0:36 for s ¼ 0:1 and d ¼ 20 and 10, respectively (frame (a)); and njj E0:34 for s ¼ 0:4 and d ¼ 10 (frame (b)) There is an argument to believe that njj should tend to the value E1=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 jj in the upper frame (a) for s ¼ 0:1; clearly, Z> c > Zc ; in the lower frame (b) for s ẳ 0:4 two points of Zjj;>ị for the same d (W and  or and ) are very c 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 for L ¼ 100 is also observed in simulation samples of other sizes up to 1000 The behaviours of Zðjj;>Þ as a function c 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 the data for the cases L ¼ 1000; d ¼ 10; but with different s: 95 s from topị ẳ 0:1; 0.2, 0.3, and 0.4 For each s the symbols and 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;>Þ are everywhere correspondingly coinc 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 system could be considered isotropic 2D, when the impurity concentration is about 0.2 and higher jj Examining accurately the ratio Z> c =Zc as the concentration s varies, we recognized that consistent with what reported in Ref [20] for the corresponding r-percolation problem, given d; for each sample of size L; there exists a ‘‘critical’’ jj impurity concentration sc in the sense that Z> c =Zc > and njj > n> E1=3 at all sosc ; while for all sXsc jj two resistance exponents are coincident: Z> c =Zc ¼ and njj ¼ n> E1=3: The values sc obtained for different L are exactly those presented in Fig of Ref [20], which is for convenience repeated in the inset of Fig for two cases of d ẳ 10 ị and 20 ị: 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 Particularly, the VRH longitudinal resistance of finite samples may experience a crossover from the finite 1D-like behaviour, i.e ln Zjjc ptÀnjj with njj being close to the 1D-value 1/2 of Eq (3), to the 2D Mott behaviour of njj E1=3: We assume that this suggested crossover might be useful in understanding 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 jj ratio Z> c =Zc changes with L for a given s: The former is already shown in the inset of Fig The fact that sc monotonically decreases with increasing L can be merely used as an argument in suggesting that in the limit of L-N; any system 96 V.L Nguyen, D.-T Dang / Physica B 334 (2003) 88–97 should always be isotropic and obey the 2D Mott T À1=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 jj À1 present in Fig the ratio Z> c =Zc ; plotted versus L for samples with d ¼ 10 and different s (from top): 0.1, 0.2, 0.3, and 0.4 Obviously, for any s under jj study the ratio Z> c =Zc always tends to a unit as L-N: 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 of Fig and Fig 6, strongly support the idea that, as discussed in detail in Ref [20], the observed crossover is simply a consequence of finite size effects There is no crossover in the relative jj magnitude of the resistance exponents Z> c =Zc as well as the ‘‘dimensional crossover’’ in the temperature-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 L ẳ 100ị and the largest L ẳ 1000ị from simulated samples, have been specially chosen to be presented in Figs and The data for samples of other sizes, 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 of Fig 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 rpT À1=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 system of finite size there exists an impurity-induced crossover in the relative magnitude between the LR and TR from the regime where TR > LT to the regime, where two resistances are coincident; (2) at the crossover the exponent njj in the LR, ln rjj pT Ànjj ; changes from the 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 hope that these results may be useful in describing the impurity-induced dimensional crossover 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 becomes 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, in Refs [22,23] for the 2D case Acknowledgements Interestingly, the data in Fig can be approximately jj À1 with gpsÀ1 (see, dashed fitting expressed as Z> c =Zc E1 ỵ gL straight lines) One of us (V.L.N.) thanks the National Center for Theoretical Sciences (Hsinchu, Taiwan) for V.L Nguyen, D.-T Dang / Physica B 334 (2003) 88–97 kind 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 References [1] I Shlimak, S.I Khondaker, M Pepper, D.A Ritchie, Phys Rev B 61 (2000) 7253 [2] A.I Yakimov, et al., Phys Rev B 61 (2000) 10868 [3] E Abrahams, S.V Kravchenko, M.P Sarachik, Rev Mod Phys 73 (2001) 251 [4] N.F Mott, J Non-cryst Solids (1968) [5] V.L Nguyen, R Rosenbaum, Phys Rev B 56 (1997) 14960; V.L Nguyen, R Rosenbaum, J Phys.: Condens Matter 10 (1998) 6083 [6] J Kurkijarvi, Phys Rev B (1973) 922 [7] M.E Raikh, I.M Ruzin, in: B.A Altshuler, P.A Lee, R.A Webb (Eds.), Mesoscopic Phenomena in Solids, North-Holland, Amsterdam, 1991 [8] I.P Zvyagin, Sov Phys.—JETP 80 (1995) 93 97 [9] W Brenig, G.H Dohler, H Heyszenau, Philos Mag 27 (1973) 1093 [10] P.A Lee, Phys Rev Lett 53 (1984) 2042 [11] R.A Serota, R.K Kalia, P.A Lee, Phys Rev B 33 (1986) 8441 [12] A Hunt, Solid State Commun 86 (1993) 765 [13] V.K.S Shante, Phys Rev B 16 (1973) 2597 [14] A.N Samukhin, V.N Prigodin, L Jastrabik, Phys Rev Lett 78 (1997) 326 [15] J.A Reedijk, H.C.F Martens, H.B Brom, Phys Rev Lett 83 (1999) 3904 [16] A.I Yakimov, et al., J Phys.: Condens Matter (1997) 889 [17] B.I Shklovskii, A.L Efros, Electronic Properties in Doped Semiconductors, Springer-Verlag, Berlin, 1984 [18] V Ambegaokar, B.I Halperin, J.S Langer, Phys Rev B (1971) 2612 [19] V.L Nguyen, B.I Shklovskii, A.L Efros, Sov Phys.Semicond 13 (1979) 1281 [20] V.L Nguyen, D Toi Dang, H Nam Nguyen, Physica A 316 (2002) [21] R.M Ziff, C.D Lorenz, P Kleban, Physica A 266 (1999) 17 [22] V.L Nguyen, Sov Phys.- Semicond 18 (1984) 207; V.L Nguyen, Phys Lett A 207 (1995) 379 [23] D.N Tsigankov, A.L Efros, Phys Rev Lett 88 (2002) 176602 ... other in such a way that the interchain hops are also allowed, though rare 89 Restricting our attention to the plane systems, within this model, it was shown that due to a finite interchain coupling... Concerning the 2D case, the data presented in Figs 4–6 are mainly concentrated in examining an anisotropicity of the VRH resistance and the role of finite size effects in the model under study 3.1 Single... quasi-1D to 2D crossover in the percolation radius of finite systems as the impurity concentration s increases, in the limit of in nite systems two percolations, longitudinal and transverse, are always