Overactivated transport in the localized phase of the superconductor-insulator transition
V. Humbert, M. Ortũno, A. M. Somoza, L. Bergé, L. Dumoulin, C. A Marrache-Kikuchi
AARTICLE
Overactivated transport in the localized phase ofthe superconductor-insulator transition
V. Humbert , , M. Ortu˜no * , A. M. Somoza , L. Berg´e , L. Dumoulin , C. A Marrache-Kikuchi † Universit´e Paris-Saclay, CNRS/IN2P3, CSNSM, 91405 Orsay, France Unit´eMixte de Physique, CNRS, Thales, Universit´e Paris-Saclay, 91767, Palaiseau,France Departamento de F´ısica - CIOyN, Universidad de Murcia, Murcia30071, Spain Universit´e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay,France
Beyond a critical disorder, two-dimensional (2D) superconductorsbecome insulating. In this Superconductor-Insulator Transition(SIT), the nature of the insulator is still controversial. Here, wepresent an extensive experimental study on insulating Nb x Si − x close to the SIT, as well as corresponding numerical simulationsof the electrical conductivity. At low temperatures, we show thatelectronic transport is activated and dominated by charging ener-gies. The sample thickness variation results in a large spread ofactivation temperatures, fine-tuned via disorder. We show numer-ically and experimentally that the localization length varies expo-nentially with thickness. At the lowest temperatures, overactivatedbehavior is observed in the vicinity of the SIT and the increase inthe activation energy can be attributed to the superconducting gap.We derive a relation between the increase in activation energy andthe temperature below which overactivated behavior is observed.This relation is verified by many different quasi-2D systems. The recent interest in low dimensional systems primarily stemsfrom the many exotic electronic phases they exhibit, due to theirextreme sensitivity to any emerging order . The Superconductor-Insulator Transition (SIT) in two-dimensional disordered materials isone such famous example. There, superconductivity competes with lo-calization and Coulomb interactions to give rise to unusual electronicphases .Despite several decades of investigation of the SIT, the nature ofthe insulator is still a subject of debate. Some argue that Coulombinteractions and disorder prevail so that the insulator is fermionic ,whereas others claim that localized Cooper pairs exist even if global su-perconducting coherence is suppressed . Moreover, there is a con-troversy on whether the system is electronically homogeneous, granu-lar or fractal . Especially intriguing are the reports of very stronginsulating behaviors in the immediate vicinity of the SIT, with acti-vated or even overactivated temperature dependence of the resistivityat the lowest experimentally accessible temperatures . These find-ings have prompted fierce debate as to their possible interpretation.Arrhenius, or activated, behavior is found in the electronic transportof many insulators. In this case, the temperature dependence of theresistance is of the form: R = R exp (cid:18) T T (cid:19) , (1)where T is the activation temperature. It is usually associated eitherwith a band gap or with nearest neighbor hopping. In disordered ma-terials, the insulating behavior originates from charge carriers beingspatially localized and activated behavior usually takes place at rel- atively high temperatures . At low temperatures, electrons have tocompromise between tunneling to close neighbors at the price of anenergy mismatch, or traveling further where the hopping energy differ-ence may be smaller. This process is known as variable-range hopping(VRH) and results in a temperature dependence of the resistance ofthe form: R = R exp (cid:18) T T (cid:19) α . (2)For non-interacting systems, Mott predicted an exponent α = D/ ( D + 1) , with D the system dimensionality. Efros and Shklovskii extended the argument to Coulomb interacting systems and obtainedan exponent of / , independent of dimensionality. In both cases, thetemperature dependence is subactivated, i.e. α < . Experimentallyhowever, in systems close to the SIT, activated and, in some cases,overactivated dependencies are found at very low temperatures .This unfamiliar behavior calls for investigation and is the subject ofthe present work.Here, we investigate, both experimentally and numerically, trans-port properties of quasi-2D systems on the insulating side of the SIT.We have measured over 80 insulating Nb x Si − x films down to 7 mKto span several orders of magnitude in activation temperatures. Weperform Monte Carlo simulations of VRH conductivity on quasi-2Dsystems, so as to reproduce the experimental situation. We analyze atwhich conditions activated behavior is present at very low temperaturesand investigate the different scenarii leading to overactivated behavior.We compare our experimental results with the outcome of our simu-lations to extract the main physics of conduction mechanisms in theinsulating state of the SIT. Our main conclusion is that, close to theSIT, appearance of Cooper pairing enhances the activation temperatureat low temperature. Thin metal alloy films consti-tute model systems to study the zero-magnetic field superconductor-insulator transition. In these compounds, the SIT can be driven eitherby a reduction of the sample thickness or by a variation in stoichiometrywhich directly affects the amount of disorder in the system. In amor-phous Nb x Si − x (a-NbSi) which we consider, thermal treatment is yetanother parameter that enables us to very finely and progressively tunea single sample disorder . The combination of these experimentaltuning parameters allows us to study the effects of thickness and disor-der independently.We have considered over 80 insulating a-NbSi thin films with com-positions ( x = 9 % and . %), thicknesses ( d ∈ [20 , ˚A )and heat treatments ( θ ht ∈ [70 , ◦ C) that allowed us to investi-gate localized regimes with activation temperatures spanning over fourdecades.The as-deposited sample conductivity has been studied from roomtemperature down to 7 mK (see Materials and Methods). The corre-sponding temperature dependence of the samples sheet resistance is1 a r X i v : . [ c ond - m a t . d i s - nn ] F e b hown in figure 1 (panel (a) x = 13 . % batch, panel (c) x = 9 %batch). The films considered in this work are all insulators and, asexpected, a thickness reduction drives the system deeper into the in-sulating regime. All x = 13 . % samples have been progressivelyannealed from 70 to 160 ◦ C, to slightly change the amount of disorderin the films. Panels (d) to (f) of figure 1 show the effect of successiveheat treatments on a single sample: in our case, the higher the heattreatment temperature θ ht , the more insulating the film becomes .For all films and all heat treatments, the low temperature sheet re-sistance follows an Arrhenius-type law as given by equation (1). Thisis particularly visible in figure 1 (panels (b) and (c)). The x = 9 %samples show a single activated behavior, while the x = 13 . % sam-ples deviate from this law at the lowest temperatures. Previous workshave reported such a behavior which have been referred to as overac-tivation and associated with the presence of superconductivity in thesystem . However, this intriguing feature has not been system-atically addressed experimentally. As can be seen from figure 1.g, theresistance actually undergoes a crossover from an activated regime toan overactivated regime at very low temperatures, with a slightly largercharacteristic energy T (cid:48) . A quantitative understanding of both regimesis one of the main goals of this work. Let us first study the activated regimeand derive the expression for its characteristic temperature. Close tothe SIT, one expects thin films to have very large localization lengths ξ loc and therefore very high dielectric constants κ . As a consequence,the electric field lines will tend to stay within the film, resulting in anapproximately logarithmic effective interaction : V ( r ) ≈ e π(cid:15) κd ln r max r , (3)for distances r between r min and r max . r min corresponds to a typicalgrain radius in granular materials and to ξ loc otherwise. r max usuallycorresponds to the electrostatic screening length κd . Such long-rangeCoulomb interactions give rise to an exponential gap in the density-of-states and to a charging energy: E ≈ e π(cid:15) κd ln r max r min , (4)where the dielectric constant is given by : κ = κ + 4 πβ e a N ( E F ) ξ , (5)where β ≈ . , a is the typical interatomic distance and N ( E F ) isthe 2D density of states at the Fermi level. Extending Mott’s VRHargument to logarithmic interactions (Eq. (3)), one predicts an activatedconductivity with logarithmic corrections . As a consequence, alow temperature activated behavior can be the result of either transportdominated by charging energies or VRH in quasi-2D materials withhigh dielectric constants.To study these two scenarii, we model the electrical transport inthese disordered films by a 2D random capacitor network schematizedin figure 2.a (Materials and Methods and Ref. [36]). Grains are inter-connected by random capacitors with average capacitance C and con-nected to the gate by a capacitance C . Conduction is by hops of quan-tized charges between grains. By a proper choice of parameters, themodel can generate regimes either controlled by charging energies, ordominated by logarithmic Coulomb interactions between charges. Agate capacitance C (cid:28) C results in a regime dominated by long-rangelogarithmic Coulomb interactions with a screening length, κd , givenby (cid:112) C/C . If C (cid:46) C the only relevant energies involved arethe grain charging energies. The hopping conductivity in the system iscalculated using a kinetic Monte Carlo method . The results of the simulations are shown in Fig. 2.b, where we rep-resent the resistance for three values of the gate capacitance. Solid linescorrespond to clean samples and dashed lines to disordered sampleswhere 5 % of the nodes have fixed charges ± , at random, not con-tributing to the current and creating a random on-site potential. We ob-serve a roughly activated behavior at low T in all cases. The activationenergy increases as C decreases and the screening length increases.Nevertheless, the introduction of disorder screens out the long-range contribution to the activation energy, so that r max in equations(3) and (4) then corresponds to the inter-impurity distance. All sam-ples then end up with very similar activation temperatures, of the orderof kT = E , where k is Boltzmann constant.Logarithmic interactions in disordered systems imply the existenceof charging energies, and so effective electronic granularity, irrespec-tive of the film morphology . We can think of our samples as formedby effective grains with a charging energy given by Eq. (4). Let us come back to the experimental situation.In figure 2.c, the activation temperature T is plotted as a function ofsample thickness for x = 13 . % and all heat treatment temperatures.The first striking feature of this plot is that T varies over almost fourorders of magnitude. Except for the thinnest sample ( d = 20 ˚A), T presents an exponential dependence on sample thickness ( T ∝ e − ζd ),with ζ that depends on the heat treatment temperature, i.e. the degreeof disorder.According to Eqs. (4) and (5), when the localization length is large, T ∼ ( κd ) − ∼ ξ − d − . To explain the exponential dependence of T on thickness observed experimentally, ξ loc must increase exponentiallywith thickness. To investigate this dependence, we have calculated ξ loc for square samples of finite thickness in an Anderson model, through aone-parameter scaling analysis of their conductance. The main resultsof these simulations are presented in figure 2.d where we plot ξ loc /a asa function of W/t , ( W is the disorder level and t the kinetic energy ofthe electrons) for thicknesses ranging from d = 1 to 7 layers.For small disorder levels, one can appreciate the strong dependenceof ξ loc on both W and d . More quantitatively, our numerical simula-tions establish that in this regime the localization length is of the form(Fig. 2.e): ξ loc ≈ A exp (cid:26) ηdW (cid:27) (6)with A and η approximately constants, but non-universal. The smallshift between data for different thicknesses can be taken into accountwith a thickness-dependent prefactor.Although the exponential dependence of the localization lengthwith thickness has been observed in many systems, it is often attributedto a thickness-induced change in the amount of the disorder W . Weare here able to distinguish the quite distinct effects of both parame-ters. Equation (6) is in line with the self-consistent theory of Andersonlocalization that predicts this exponential dependence of the localiza-tion length ξ on disorder W for 2D systems. It is also in agreementwith analytical self-consistent results in weakly localized 2D systems and with numerical simulations performed on quasi-1D systems . Anexponential dependence of the localization length with thickness wasobserved in Mo-C films in VRH regimes.The activation temperature therefore exponentially decreases withthickness as shown by the straight lines in Fig. 2c. Moreover, we expecteach θ ht to correspond to a single value of W , so that, the disorder-induced spread in T should linearly depend on the thickness, whichis experimentally the case in our films. For small thicknesses, the con-tribution of the host dielectric constant κ becomes non-negligible and T is smaller than the exponential behavior, as shown by the solid line.2 igure 1 | Low temperature resistance measurements. (a) Temperature dependence of the sheet resistance for the x = 13 . as-deposited batch. (b) and (c) Sheetresistance as a function of /T on a semilogarithmic scale to highlight the activated behavior for the x = 13 . (b) and x = 9% (c) as-deposited samples. (d) to (f)Effect of successive heat treatments on the sheet resistance of the 13.5% 40 ˚A (d), 35 ˚A (e) and 30 ˚A (f) samples. The insets show the R (cid:3) ( T ) , while the main panelsdisplay the same data as a function of /T . (g) At low temperature, at about 0.9 K for the 13.5% 30 ˚A as-deposited film, there is a crossover between two activatedregimes. Figure 2 | (a) A sketch of the model used for numerical simulations. The electrode connected to the left edge is grounded, the electrode at the right edge has a potential V . The bottom panel shows a lateral view of the model where the capacitors C connecting the system to the gate can be seen. (b) Numerical results of the resistance asa function of inverse temperature. Each color corresponds to a different value of C . The solid lines correspond to the clean case whereas the dashed lines correspondto the case where disorder is added (5% of the nodes have fixed charges ± at random). (c) Characteristic temperature T as a function of the sample thickness (in ˚A)for different heat treatments ( x = 13 . % sample). The solid line corresponds to equations (4)-(6) with κ (cid:39) . For each thickness, the disorder level was averagedover the different heat treatments. Each dashed line corresponds to the exponential thickness dependence for a given heat treatment (as-deposited and θ ht = 160 ◦ C). (d) Localization length on a logarithmic scale as a function of the disorder level W (relative to the kinetic energy t ) for thicknesses ranging from 1 to 7 layers. (e)Exponential dependence of ξ loc with d/W . Overactivation.
We will now turn to the overactivated regime,i.e. the increase in the activation energy at the lowest temperatures, inthe immediate vicinity of the SIT.The thinnest ( d = 20 ˚A ) samples do not exhibit any overactivatedregime. However, the rest of the samples do. For d = 45 ˚A the wholeactivated regime is actually overactivated. To appreciate the system-aticity of the overactivated behavior, we scaled the high temperatureactivated regime for the 13.5% 35 ˚A -thick sample (figure 3.a) for thedifferent heat treatment temperatures (i.e. disorder). By doing so, theoveractivated regimes cannot be made to scale. Moreover, the ratioof the overactivated characteristic temperature over the activated onedecreases with the disorder level.We have extracted T (cid:48) , whenever an overactivated was observed,by fitting the low temperature data to equation 1. The transition tem-peratures T cross between the two activated regimes were also extractedfrom the crossing point between the two fits. The corresponding resultsare plotted in figure 3.d as a function of the disorder level, quantified bythe high temperature (200 K) resistance R K . We note that these tem-peratures vary over almost three orders of magnitude. Moreover, T cross gets closer to T as we approach the SIT, at about R K (cid:39) Ω .Our aim is now to understand the relation between these characteristictemperatures.since they shine light on the conduction mechanisms.Since the overactivated regime is only found close to the SIT, it isreasonable to assume it is related to superconductivity. Let us exam-ine what will happen if, due to the proximity to the SIT, superconduc-tivity sets in locally in some (effective) grains. In this case, Cooperpairs will be phase coherent on a short range, but there will not beany global phase coherence. There are then two different possibilities.The first one is that conduction is still dominated by one-electron pro-cesses. Then, when a single electron enters a superconducting grain, ithas to pay an extra energy penalty associated with the superconductinggap. The second possibility is that conduction between superconduct-ing grains is ensured by tunneling of Cooper pairs. In this case theenergy penalty for a pair entering a superconducting grain is four timesthe charging energy, since the grain becomes doubly charged. In bothcases, the crossover temperature to the overactivated regime must beof the order of the pairing energy. In our experimental case, T cross ismuch smaller than the activation energy and since T ∼ E , we con-clude that the pairing energy is much smaller than the charging energy,so that conduction by single electrons is more likely to be the mainconduction mechanism.We have performed Monte Carlo simulations of the conductivity onour random capacitor model to quantify the increase in activation en-ergy when the grains become superconducting. We consider the modelwith only charging energies and no long-range interactions, since, aswe have seen, both cases produce similar results, and the higher com-putational efficiency of the former is convenient in this highly demand-ing calculation.We assumed that grains become superconducting at the criticaltemperature T cross . Below this temperature, the new energy penaltyto charge a superconducting grain is: E (cid:48) = E + ∆( T ) , (7)where ∆ is the superconducting gap. Assuming a BCS ratio of 2, wehave : E (cid:48) = E + 2 kT c (cid:114) T cross − TT cross . (8)The results of the simulations are shown in figure 3.b, where weplot R versus /T for several values of T cross (in units of T ). Theblack curve corresponds to the non-superconducting case (pure acti-vated behavior) and the rest to different values of T cross /T , chosen toreproduce the experimental situation. At temperatures below T cross , weobtain a new roughly activated regime, as experimentally observed. As expected, the transition temperature between both regimes coincideswith T cross .In figure 3.c we compare our numerical simulations for T cross =0 . T with the experimental data for the 35 ˚A sample ( θ ht = 110 ◦ )by scaling the theoretical curve so that both coincide in the activatedregime. The overactivated behavior is fairly well reproduced by oursimulations.Extracting characteristic temperatures from our numerical resultsin the same way as from the experimental data, we have established therelation: T (cid:48) − T = γT cross (9)and found γ = 1 . . For each experimental point, we derived the valueof γ given by Eq. (9) and represent it as an orange triangle (bottompanel of figure 3.d). On average, we find γ = 1 . (black line). Theagreement with theoretical predictions is fairly good, clearly indicatingthat the overactivated regime is a consequence of electrons having topay the superconducting gap penalty, proportional to T cross . Close tothe SIT, uncertainties on γ are large, since the activated regime expandson a very narrow temperature range.We have also reanalyzed the overactivated behavior found in theliterature and compared the experimental data to our prediction. Thiscan be seen in figure 3.e where we represent γ obtained from equation(9). The value of γ is fairly constant for all systems, and of the order ofunity. This is remarkable given the variety of systems considered, andthe three orders of magnitude over which T extends.Let us note that the theoretical prediction depends both on the BCSratio linking T c and ∆ , which we took to be 2 following STM data fora-NbSi , and on the proportion of superconducting grains. If, insteadof having all grains superconducting, only a fraction p of them are, T (cid:48) − T should be proportional to p . From the agreement between thesimulations and the experiment, we can conclude that a large fractionof the grains become superconducting.We can explain the dependance of T cross on disorder (Figure 3.d)by assuming that the superconducting gap is of the form ∆ = ∆ D ξ SC d (cid:18) ξ ξ (cid:19) ∼ T cross (10)Indeed, close to the SIT, equation (10) is in agreement with the exper-imental dependence of T cross with d . As the system moves awayfrom the SIT, equation (10) reflects how the superconducting gap in-creases in small grains as the inverse of the volume over which Cooperpairs are forced to be confined, along similar lines to ref. [53]. Athigh disorder, the dependence of T cross and T on the fastest changingparameter, ξ loc , is similar ( T ∝ ξ − through equations (4) and (5)).Strictly speaking, ξ loc in Eq. (10) should be the Cooper pair localiza-tion length, that, due to interactions, may be larger than the one particlelocalization length, but this difference is small in the regime consideredand we have assumed that both were equal.According to (10), T cross should be of the form a/d + bT . As canbe seen figure 3.d, there is a good agreement between the experimen-tal T cross and . /d [ ˚ A ] + 0 . × T (green open squares). However,close to the SIT, the temperature range over which T is observed van-ishes (it is the case for our 45 ˚A samples). Since T and T (cid:48) have similardependencies, T cross should then be of the form a/d + b (cid:48) T (cid:48) . This is ex-perimentally the case (green full squares). In this regime, T (cid:48) is almostconstant because the charging energy is smaller than the superconduct-ing gap which is constant and equal to ( ξ SC /d )∆ . We have shown that the activated transport behav-ior observed in thin films close to the SIT is due to transport dominatedby charging energies. In homogeneous systems, the electronic gran-ularity is a consequence of a diverging localization length. We have4 igure 3 | (a) Scaled resistance as a function of T /T so that the activated regime overlaps for all disorder levels (tuned by the successive heat treatments) for the 35˚A -thick 13.5% sample. (b) Simulation results for the resistance as a function of /T , on a semilogarithmic scale. The black line corresponds to activated behavior,with no superconductivity, and the other curves to overactivation due to superconductivity for several values of the ratio T cross /T . (c) Resistance as a function ofthe inverse temperature, /T , on a semilogarithmic scale, for the 13.5% 35 ˚A -thick sample (heat treatment of 110 ◦ C, black dots). The continuous blue curve is oursimulation without superconductivity effects, while the red curve is with grain superconductivity for the case T cross /T = 0 . . (d) Characteristic temperatures T , T (cid:48) and T cross for all measured samples. The bottom panel shows the value of γ = ( T (cid:48) − T ) /T cross as given by equation (9). The black line then corresponds tothe average γ = 1 . . The green squares correspond to T cross given by equation (10). Far from the SIT, T cross (cid:39) . /d [ ˚ A ] + 0 . × T (green open squares),and closer to the SIT, T cross (cid:39) . /d [ ˚ A ] + 0 . × T (cid:48) (green full squares). (e) Characteristic temperatures T , T (cid:48) and T cross of the activated regimes in severaldifferent experiments. The bottom panel gives the value of γ , determined by equation (9). The black line corresponds to γ = 1 . . The experimental data are fromRef. [15] for Sn and Pb, [16] for Bi, [32] for Al, [17] for InOx/In, [46] for InOx, and [22] for β (cid:48) -(BEDT-TFF) ICl . ξ loc depends exponen-tially on the film thickness. The overactivated regime observed closeto the SIT is a cross-over to a regime governed both by the chargingenergy and the superconducting gap. Our results indicate that the latterdepends critically on the grains size. The overall conclusion is that, inthe insulating regime close to the SIT, localized Cooper pairs exist butelectronic transport is still dominated by single electrons. The authors thank Zvi Ovadyahu forvaluable discussions and guidance, and Dan Shahar’s group for shar-ing their experimental data on InO x . This work has been par-tially supported by the ANR (grant No. ANR 2010 BLAN 040301). M.O. and A.S. acknowledge support by Fundaci´on S´eneca grant19907/GERM/15 and AEI (Spain) grant PID2019-104272RB-C52.V.H., L.B., L.D. and C.A.M.K acknowledge the support of the ANR(grant ANR 2010 BLAN 0403 01). Experimental details.
The a-NbSi films have been grown at roomtemperature by e-beam co-deposition of Niobium and Silicon underultra-high vacuum (the chamber pressure during the deposition wastypically of a few 10 − mbar). The film composition was fixed bythe respective evaporation rates of Nb and Si (both of the order of 1˚A.s − ) and monitored in situ by a set of dedicated piezoelectric quartzcrystals. The sample thickness was determined by the duration of thedeposition. Both parameters have been checked ex situ by RutherfordBackscattering Spectroscopy (RBS).The samples have been deposited onto sapphire substrates coatedwith a 25-nm-thick SiO underlayer designed to smooth the substratesurface. They were also protected from oxidation by a 25 nm-thickSiO overlayer. a-NbSi films of similar compositions and thicknesseshave been measured to be continuous, amorphous and homogeneous atleast down to a thickness 2.5 nm .The transport characteristics of a-NbSi thin films are mainly deter-mined by their composition x and their thickness d . However, an ad-ditional thermal treatment can also microscopically modify the systemdisorder while keeping x and d constant. a-NbSi becomes more insu-lating as the heat treatment temperature increases without any changein the sample morphology . Thermal treatments and the film compo-sition have an analogous effect on the disorder level: films of similarsheet resistance R (cid:3) have the same transport characteristics. On theother hand, we have shown that the effect of the thickness is distinct .In the present case, we have considered as-desposited films which pa-rameters are listed in table 1. The composition x has been chosen sothat the samples are close to the SIT ( x = 13 . %) or further within theinsulating regime ( x = 9 %). The sample thickness has been variedbetween d = 20 and 50 ˚A for x = 13 . %. For this stoichiometry,the critical thickness at which the system undergoes a thickness-tunedSIT is about 140 ˚A . a-Nb Si is not superconducting, even for bulksamples , and in this work, we considered thicknesses of 125 and250 ˚A . Each batch having a constant composition is considered tohave the same disorder level ( W constant). We can therefore directlyevaluate the effect of the thickness within each batch.The samples resistances have been measured in a dilution refriger-ator with a base temperature of 7 mK. In the case of the x = 13 . batch, two regions of each sample, labeled left and right , could beprobed independently. We used standard low noise transport measure-ment techniques to ensure the samples were measured without electri-cal heating of the electronic bath. In other words, we made sure viaappropriate filtering that the sample electron temperature was the sameas its phononic temperature. Moreover, we have checked that the ap-plied bias was sufficiently low for the resistance measurement to be inthe ohmic regime. Name x (%) d ( ˚A) R K CKSAS43 α left 13.5% 20 111 G Ω CKSAS43 α right 13.5% 20 n.dCKSAS61 α left 13.5% 30 162 k Ω CKSAS61 α right 13.5% 30 151 k Ω CKSAS61 β left 13.5% 35 18.8 k Ω CKSAS61 β right 13.5% 35 17.8 k Ω CKSAS61 γ left 13.5% 40 10.8 k Ω CKSAS61 γ right 13.5% 40 10.5 k Ω CKSAS61 δ left 13.5% 45 7.73 k Ω CKSAS61 δ right 13.5% 45 7.50 k Ω CK8 γ
9% 125 53.3 k Ω CK9 γ
9% 125 52.6 k Ω CK9 β
9% 250 14.1 k Ω Table 1 | Characteristics of the different a-NbSi samples: composition x , thick-ness d , low temperature sheet resistance R K evaluated at 4.2 K except for the20 ˚A-thick sample for which they have been measured at 5 K. R T
20 40 60 80 100 120
Figure 4 | Numerical results.
Resistance as a function of inverse temperature forthe model with (thick curves) and without (thin curves) long-range interactions.
Capacitor model
To calculate the hopping conductance of quasi-2Dhigh dielectric constant disordered systems, we consider a 2D randomcapacitor model in order to get a consistent set of energies. Sites(nodes) are randomly distributed and the corresponding junction net-work without crossings can be constructed using a Delaunay triangu-larization algorithm . Capacitors with randomly chosen capacitanceswere placed at the links between adjacent nodes, as shown in Fig. 3.a.The capacitances C i,j assume random values drawn from the distribu-tion C i,j = Ce ϕ , where ϕ ∈ [ − B/ , B/ . We have chosen C = 1 (in units of (cid:15) a , where a is the lattice constant) and B = 2 . The leftbank is connected to the ground, while the right bank is at the potential V , and there are periodic boundary conditions in the lateral direction.In order to take into account the leakage of field lines to outside the 2Dsystem due to the finite value of κ , we introduce capacitances to theground, C , as shown in the bottom panel of Fig. 3.a.The plates of each capacitor carry opposite charges and the totalcharge on each site Q i is the sum of the charges on the plates of thecapacitors connected to it, Q i = (cid:80) j q i,j . Here, q i,j is the charge onthe plates of the capacitor connecting nodes i and j and satisfies q i,j = (cid:88) j C i,j ( V i − V j ) . (11)We construct a vector Q , whose components are the charges of thenodes in the sample, and a vector V , whose components are the node6otentials. Both vectors are related by Q = C V where the capacitancematrix C has components equal to [ C ] i.j = (cid:88) k (cid:54) = i C i,k δ i,j − C i,j . (12)The total electrostatic energy of this system can be obtained in acompact form through the inverse of the capacitance matrix H = 12 Q C − Q T . (13)The matrix C − plays the role of an interaction matrix. In the contin-uous limit, the average interaction between two charges separated bya distance r is proportional to the modified Bessel function K ( r/ Λ) ,where the screening length is Λ = r (cid:112) C/C . In the limit of small r ( r (cid:28) r (cid:28) Λ ) we have K ( x ) ≈ − ln( x/ , and the effectiveinteraction is given by Eq. (3) provided that we identify the screeninglength with Λ = κd , and choose C = (cid:15) κd and C = (cid:15) r / ( κd ) . Inthe simulation of hopping transport, r is our unit of distance, whichfor random nodes is defined by r = L/N / , N being the number ofnodes, L the size of the system, and e / ( r (cid:15) ) is our unit of energy.The charging energy, i.e., the energy cost for putting a unit charge at agiven site i , is equal C − i,i / .Carriers hop from site to site transferring a quantized unity charge.They can hop over many sites with a probability that exponentiallydecays with distance, thus modeling charge transfer in experimentalsystems mediated by the cotunneling mechanism. The transition ratebetween sites i and j can be expressed as: Γ i,j = τ − e − r i,j /ξ loc e − ∆ i,j /kT (14)where τ − is the phonon frequency, r i,j the hopping distance, ξ loc thelocalization length and ∆ i,j the transition energy, given by our capac-itor model. To simulate hopping conductivity in the system of inter-acting electrons, we employ a kinetic Monte Carlo method . Theallowed node charges are 0 and ± . At each Monte Carlo step, the al-gorithm chooses a pair of sites ( i, j ) with the probability proportionalto exp( − r ij /ξ ) , Ref. [38]. Doing so, the time step associated with ahop attempt per site is τ / (cid:80) ij exp( − r ij /ξ ) , where τ is the inversephonon frequency. The algorithm first checks whether the transfer of aunit charge from site i to j is compatible with the allowed node charges.Then it calculates the transition energy ∆ i,j = V j − V i − C − i,j + 12 (cid:2) C − i,i + C − j,j (cid:3) , (15)where V i = (cid:80) j C − i,j Q j is the potential at i , and the hop is performedwhen ∆ i,j is negative or with probability exp( − ∆ i,j /T ) otherwise.All site potentials V i are recalculated after every successful transition.The last term in (15) is the charging energy of the nodes involved. Theelectric current is generated by the potential difference V lead betweenthe leads at the opposite sides of the sample, which is reflected in thesite potentials by extending the Q vector to include nodes j in the leadconnected to nodes i in the sample and associating a charge − C i,j V lead with each of them .The algorithm starts from an initial random charge configurationand follows the dynamics at a given temperature. Once the system isin a stationary situation, the conductivity of each sample is calculatedfrom the number of electrons crossing to one of the leads. The numberof Monte Carlo steps performed in this calculation drastically increaseswith decreasing T , and it is determined by the condition that the netcharge crossing one of the leads is on the order of 1000. The numberof samples considered is 100. Finally, we averaged ln σ over the set ofsamples (an ensemble averaging). The main results of the simulations were shown in Fig. 2.b. Fromthem, we concluded that, in the presence of disorder, the activationenergy is fairly independent of the screening length or, equivalently,of the ratio C /C . To simulate the overactivated regime, we have touse the most efficient numerical procedure in order to reach very lowtemperatures. To this end, in Fig. 4 we compare the resistance for theexact interacting potential in the C = 1 case (thick curves) with theresults including only the charging energy (thin curves). Again, thicklines correspond to the case without site disorder, while thin lines re-flect samples with 5 % of the nodes having fixed random charges. Wesee that the activation energies are similar and conclude that a modelwith charging energies only, without the logarithmic contribution, isadequate to simulate conductivity in disordered samples. Localization length.
For the calculation of the localization length, weconsider the standard Anderson Hamiltonian for spinless particles H = (cid:88) i (cid:15) i n i + (cid:88) (cid:104) i,j (cid:105) t ( c † j c i + c † i c j ) (16)where c † i is the creation operator on site i and n i = c † i c i is the numberoperator. We consider a transfer energy t = − , which sets our unit ofenergy, and a disordered site energy (cid:15) i ∈ [ − W/ , W/ . The doublesum runs over nearest neighbors. We have considered square samplesof finite thickness with dimensions L × L × d . All calculations are doneat an energy equal to . , to avoid possible specific effects associatedwith the center of the band.We have studied numerically the zero temperature conductance g proportional to the transmission coefficient T between two semi–infinite leads attached to opposite sides of the sample, g = 2 e h T, (17)where the factor of 2 comes from spin. We measure the conductancein units of e /h . We calculate the transmission coefficient from theGreen function, which is obtained propagating layer by layer with therecursive Green function method . We can solve samples with lateralsection up to L × d = 400 . The number of different realizations em-ployed is for most values of the parameters. We have consideredwide leads with the same section as the samples, represented by thesame hamiltonian as the system, but without diagonal disorder. We usecyclic periodic boundary conditions in the long direction perpendicularto the leads, and hard wall conditions in the narrow traversal direction.According to single-parameter scaling, the conductance is a func-tion of the ratio of the two relevant lengths of our problem, g = g f ( L/ξ loc ( W, d )) , where L is the lateral system size and ξ the correlation length, whichcarries the dependence on W and d . Data for different W and d canbe made to overlap by an adequate choice of ξ loc ( W, d ) . We use thisidea in both the diffusive and the localized regimes. In the former theconductance depends logarithmically on system size g = g − π log Lξ loc , (18)where the factor /π has been obtained by diagrammatic perturbationtheory . In the strongly localized regime, the conductance dependsexponentially on the system size g = c exp − Lξ loc . (19)The results for ξ loc ( W, d ) are shown in Fig. 2d.To analyze quantitatively the overall behavior of ξ loc , we have per-formed a new one-parameter scaling analysis of the data plotted7 igure 5 | Evolution of the 3D localization length with disorder. Each color cor-responds to a given disorder. There is a localized-delocalized transition arounda critical disorder W c ≈ . . Inset: ξ loc /d as a function of the localizationlength that the corresponding bulk (3D) system would have, on a double logarith-mic scale. The localized-delocalized transition occurs for log ( ξ loc / d) (cid:39) . . in figure 2.d. The idea is to plot log( ξ loc /d ) as a function of log( d ) and shift the data horizontally by the disorder-dependent amount thatbest overlaps the data. This shifting quantity corresponds to the three-dimensional correlation length ξ D cor ( W ) which is either the metalliccorrelation length if the bulk system with the same amount of disor-der W is delocalized or corresponds to the three-dimensional local-ization length ξ D loc ( W ) if it is localized (Fig. 5). The two branchestherefore correspond to the well-known three-dimensional localization-delocalization transition at a critical disorder of about W c ≈ . . Thecorresponding scaling is shown in the inset of Fig. 5. For small (resp.large) disorder levels, the system is in the upper (resp. lower) branchand the corresponding 3D system is delocalized (resp. localized): ξ loc increases (resp. decreases) faster than thickness. The transition be-tween those two regimes, for our model, occurs when: log (cid:18) ξ loc d (cid:19) ≈ . . (20)This implies that if for a given thickness and amount of disorder ξ loc > d , the system will tend to metallic when its thickness increases and toan insulator when ξ loc < d .Usually in experimental situations, one does not know with enoughprecision the relation between d and ξ loc to decide if a sample for agiven disorder will become extended or localized when its thicknessincreases according to criteria (20). We can establish a new criteria topredict the 3D character of a system. If the 3D system corresponding toa given disorder value is extended, as its thickness decreases, the ratio ξ loc /d decreases, meaning that the localization length decreases fasterthan the thickness: ξ loc , ξ loc , > d d , (21)if d > d . Conversely, if the 3D system corresponding to a givendisorder value is localized: ξ loc , ξ loc , < d d , (22)for d > d . Let us apply criterion (21) to our experimental samples.For the most disordered x = 13 . % samples ( θ ht = 150 o C), onecan estimate the ratio of the localization lengths through the relation ξ loc ∝ T − / (equations (4) and (5) ): ξ loc , ξ loc , ≈ (cid:114) T T = 6 > d d = 43 . (23)for d = 40 ˚A and d = 30 ˚A for instance. The correspond-ing 3D system is therefore on the metallic side of the Metal-InsulatorTransition. All considered x = 13 . % samples exhibit ξ loc (cid:29) d .The x = 9 % samples are also extended, but closer to the transition: ξ loc , /ξ loc , = 2 . for d /d = 2 .
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