Electron glass effects in amorphous NbSi films
J. Delahaye, T. Grenet, C.A Marrache-Kikuchi, V. Humbert, L. Bergé, L. Dumoulin
SSciPost Physics Submission
Electron glass effects in amorphous NbSi films
J. Delahaye , T. Grenet , C.A Marrache-Kikuchi , V. Humbert , L. Berg´e , L. Dumoulin Univ. Grenoble Alpes, CNRS, Institut N´eel, 38000 Grenoble, France CSNSM, Universit´e Paris-Sud, Orsay, F-91405, FranceMarch 18, 2020
Abstract
We report on non equilibrium field effect in insulating amorphous NbSi thin filmshaving different Nb contents and thicknesses. The hallmark of an electron glass,namely the logarithmic growth of a memory dip in conductance versus gate volt-age curves, is observed in all the films after a cooling from room temperatureto 4.2 K. A very rich phenomenology is demonstrated. While the memory dipwidth is found to strongly vary with the film parameters, as was also observedin amorphous indium oxide films, screening lengths and temperature dependenceof the dynamics are closer to what is observed in granular Al films. Our resultsdemonstrate that the differentiation between continuous and discontinuous sys-tems is not relevant to understand the discrepancies reported between varioussystems in the electron glass features. We suggest instead that they are not offundamental nature and stem from differences in the protocols used and in theelectrical inhomogeneity length scales within each material.
Contents a r X i v : . [ c ond - m a t . d i s - nn ] M a r ciPost Physics Submission In the past two decades, intriguing out-of-equilibrium phenomena have been reported in theelectrical conductance G of several disordered insulating systems [1]. After they are cooledfrom room temperature to liquid He, G decreases logarithmically with the time elapsed sincethe cooling [2, 3]. In most cases no saturation to some equilibrium value can be observed, evenafter weeks of measurements. A fruitful way to investigate this effect is to make MOSFETstructures whose (weakly) conducting channels are thins films of the materials under study.After such a device has been allowed to relax for some time under a given gate voltage ( V g )at constant temperature, a change of V g triggers an immediate increase of the conductance,followed by a new slow downward relaxation. Fast V g scans show that the relaxation historyremains printed in the G ( V g ) curves as memory dips (MDs) centred on V g values at which thedevice was allowed to relax for some time [4, 2]. MDs are slowly erased with time when V g ismaintained out of these values. In granular Al films it was demonstrated that the conductancerelaxation triggered by a V g change to a pristine value is not exactly logarithmic with time anddepends on the total time elapsed since the cooling [5]. This age dependence of the dynamicsor ageing effect underlines the strong similarity between these out-of equilibrium phenomenaand what is seen in structural and spin glasses.The origin of the glassy features of disordered insulators has been the subject of manyexperimental and theoretical efforts. It was soon suggested that they may reflect the existenceof an electron glass, a glassy state of the carriers first predicted in the 80’s and induced by thecoexistence of disorder and ill-screened electron-electron interactions [6, 7, 8]. Many numericalworks have studied the out-of-equilibrium physics of the Efros-Shlovskii Coulomb glass andrelated models of disordered insulators [1]. Mean field approaches have suggested a glass phasetransition concomitant with the formation of the Coulomb gap which needs the absence ofscreening present in the glass phase in order to fully develop [9, 10]. The occurrence of sucha transition was recently found in numerical simulations of the 3D Coulomb glass model [11]but with a transition temperature much smaller than the mean field prediction.The electron glass scenario was recently strengthened in amorphous indium oxide (a-InOx) films by the demonstration that the duration of the logarithmic relaxation depends onthe charge carrier density n (as estimated from room temperature Hall effect) and electricalresistance of the samples. The higher time cutoff of the logarithmic relaxation could bereduced down to measurable times in the low doping and low resistance (sheet resistance R s close to quantum resistance R Q ) regime of insulating InOx films [12]. This offers a cleardemonstration of a correlation between the electronic properties and the relaxation timedistribution.But other experimental issues remain controversial or are waiting for a satisfactory ex-planation. For example, the connection between the MD and Coulomb correlations still needto be clarified. In InOx films, the MD width was found to be insensitive to the sample re-sistance but increases systematically with n [13]. It was proposed that this n dependence isuniversal [14] and includes Be [15], Tl O − x [16] and Ge-Te alloys films [17, 18, 19], althoughthe carrier concentration cannot be conveniently changed in all of them. Using a percolationapproach on a classical disordered model, Lebanon and Muller [20] were able to reproduce the n dependence of the MD width but not its temperature dependence experimentally observedat low temperature in granular Al films [21, 2].The controversial results reported on the temperature dependence of the glassy dynamics2 ciPost Physics Submission also raise serious questions. In discontinuous metal films (gold, etc.), a marked slowdownupon cooling was highlighted, giving rise to the formation of large frozen MDs [22, 23, 24].More recently, we demonstrated that at low temperature the MD dynamics of granular Alfilms is thermally activated [25]. In contrast, in the case of InOx, it was argued that thedynamics is not activated and may even accelerate upon cooling in the less doped samples,which was explained by the quantum nature of the glass [26]. However, the protocols used toquantify the dynamics are more indirect [13, 26] and we have questioned their relevance [27].Another debated issue has recently emerged about the length over which the glassy featuresare perturbed upon the application of a gate voltage change, hereinafter called the penetrationlength . In granular Al films, only the 10 nm-thick layer of the film closest to the gate insulatoris electrically disturbed by a V g change [28], while the disturbance extends to more than 70 nmin a-InOx films [29], indicating markedly different length scales in the two systems. The largepenetration length observed in InOx films was taken as an experimental evidence for electronicavalanches found after a charge injection in numerical simulations of Coulomb glass models[30, 31]. However, they seem to contradict the very short screening lengths sometimes invokedin this system [13, 29].In light of these contradictory results, one may wonder whether the glassy physics couldhave a different origin in discontinuous/granular systems and continuous disordered systemslike a-InOx, or if the observed differences are specific to the protocols used or to peculiarcharacteristics of InOx. To answer this question, it is of prime importance to study otheramorphous systems which electrical properties can be tuned within a significant range bychanging the thickness or the chemical composition of the films. This is what is done in thepresent study. In a preliminary work on a-NbSi films, we found a strong slow down of thedynamics and the freezing of prominent broad MDs upon cooling from room temperature[32], evidencing activated dynamics in a continuous amorphous system. Here we report on acomprehensive study of electron glassiness in a series of NbSi films of different Nb contentsand thicknesses which confirms the general character of the activated dynamics and shed somelight on the glassy phenomenology in this system.The paper is organized as follows. The measurement set-up and film parameters arepresented first. Then, we discuss the penetration lengths and how the MD width dependson the film parameters. Last, time and temperature dependencies of the glassy dynamics areexplored in detail. The NbSi results are discussed in light of what has been found in otherdisordered insulating systems. The studied a-NbSi films were synthesized by the co-deposition of Nb and Si on substratesat room temperature and under ultra-high vacuum (typically a few 10 − mbar) as describedelsewhere [33]. Structural investigations have shown that such films are amorphous and contin-uous at least down to a thickness of (cid:39) (the gate insulator). Shadow masks were used to define a filmof typical size 0 . × . ciPost Physics Submission glassy behavior described below [32].The electrical conductance was measured in a two-point contact configuration. An AC ora DC bias voltage is applied and the resulting current is detected using a low noise currentamplifier. The bias voltage was kept sufficiently small to remain in the ohmic regime (in AC,the bias voltage should also be much smaller than the V g width of the MD). V g was allowedto vary between -30 V and 30 V, far enough from the practical breakdown limit of our SiO gate insulating barrier (around 50 V). No significant leaking current can be measured withinthis V g range.Six samples with two different thicknesses T h (2.5 nm and 12.5 nm) and three differentNb contents for each thickness were measured. Their names used hereafter and their generalelectrical parameters (resistance per square R s at 300 K and 4.2 K, resistance ratio RRR = R K /R K ) are given in Table 1.Name Thickness % Nb R s K R s K RRR
A1 2.5 nm 13 16 . . . . . . . (cid:39)
100 nm (3Dsamples), the films are metallic or superconducting when the Nb content is above (cid:39)
9% andinsulating when it is below. When the thickness is smaller than 100 nm, the insulating stateis observed up to larger Nb concentration: up to (cid:39)
14% for a thickness of 12.5 nm and upto (cid:39)
20% for a thickness of 2.5 nm. All the samples measured in this study thus lie onthe insulating side of the metal- or superconductor-to-insulator transition. The insulatingcharacter of the films is confirmed by their resistance versus temperature dependence. InFigure 1, we show Arrhenius plots of the low T parts of the samples’ resistances. In therange 4 K - 50 K, they are well described by an exponential-like divergence R ∝ exp( T /T ) α with an exponent α which varies between 0.5 and 0.8 depending on the samples, a behaviourcommonly observed in disordered insulating materials [34, 37, 25]. Let us first present how the samples conductance evolves after they are cooled down to liquidHe under a fixed V g . The films are first cooled from room temperature to 4.2 K, which takesabout 15 min with our experimental set-up. During the cooling, V g is kept constant and equalto V geq . Once at 4.2 K, G is measured as a function of V g as sweeps are performed around V geq . Between each sweep, V g is maintained at V geq for a waiting time at least ten times aslong as the sweep duration. The conductance variations induced by the small temperaturedrifts of the He bath are corrected [38] and we end up with the set of G ( V g , t ) curves plotted4 ciPost Physics Submission Figure 1: Log of R s versus 1 /T between (cid:39)
50 K and 4.2 K for the six a-NbSi films used inthis study. The lines are guides for the eye. Full symbols: 2.5 nm thick films; empty symbols:12.5 nm thick films.in Figure 2.Beyond the growth of prominent MDs centred on V geq , what is remarkable is the widevariety of G ( V g ) shapes and evolutions observed for the different films. First, an overalldownward drift of the curves called hereinafter background relaxation is observed to variousdegrees, from almost absent in the 2.5 nm-thick films to very pronounced in 12.5 nm-thickfilms. Second, the MD width becomes significantly larger when the R s value decreases. Wenow discuss these two features and their possible significance. The existence of a background relaxation can be best visualized by plotting G ( t ) curves atdifferent V g from the data of Figure 2, as shown in Figure 3. In 2.5 nm thick films, G relaxations are significant at V geq but much smaller far from it ( G is even constant at 30 Vin sample A1, left panel of Figure 3). In 12.5 nm thick films however, the relaxations remainlarge even 30 V away from V geq clearly demonstrating the background relaxation effect.Similar features were also observed in granular Al films [28] and are believed to showup when the static screening length L sc becomes smaller than the film thickness T h . Theevolution of the conductance observed at a given V g can then be seen as the sum of twoseparate contributions: one coming from the part of the film within L sc from the gate insulatorwhich is influenced by the applied V g (the V g -sensitive layer), and another one coming fromthe rest of the film which is insensitive to V g changes (the V g -insensitive layer) and whichpursues it relaxation induced by the initial cooling. Note that according to this interpretation,the penetration length introduced earlier is nothing else than the screening length of thesystem. As long as the conductance is measured at V geq , both layers contribute to the observedrelaxation. But for the conductance observed at a V g value far enough from V geq , the V g -sensitive layer equilibrated at V geq has its excited and time-independent conductance (see the5 ciPost Physics Submission Figure 2: Conductance versus V g curves measured at 4.2 K and at times t after a cooling fromroom temperature. V geq was fixed to 0 V during and after the cooling for all the samples, andthe total sweep time was at least 10 times smaller than the waiting time between two sweeps.Left column: 2.5 nm thick films; right column: 12.5 nm thick films. Refer to Table 1 for thesamples parameters.sample A1 at 30 V in Figures 2 and 3), so that the only contribution to the relaxation thatremains is the one of the V g -insensitive layer. The relaxation of the V g -sensitive layer results6 ciPost Physics Submission Figure 3: Two examples of G ( t ) sets of curves measured at different V g values at 4.2 K aftera cooling from room temperature under V geq = 0 V (same data as Figure 2). Left side panel:sample A1; Right side panel: B2.in the growth of the MD at V geq . Assuming that the relaxation modes are homogeneouslydistributed throughout the thickness of the films (see Figure 4), the G vs. ln( t ) relaxationslopes should be proportional to the thickness of the relaxing layer, i.e. T h when measured at V geq and ( T h − L sc ) far enough from V geq when the slope is (almost) V g -independent [28]. Bycomparing the G vs. ln( t ) relaxation slopes at V geq = 0 V and at 30 V in our 12.5 nm thickfilms, we get L sc estimates of 1.3 - 1.9 nm for sample B3, 4 nm for B2 and 7 nm for B1 (theinteratomic distance d Nb − Si is around 0 .
26 nm [36]). In 2.5 nm thick films, the relaxationslope at 30 V is at least 5 times smaller than at 0 V, which means that L sc is of the order orlarger than 2.5 nm. In granular Al films, a screening length of about 10 nm was extracted from20 and 100 nm thick films background relaxations, which was found to be roughly constantwith R s in the samples investigated, probably due to the granular nature of the system [28].In InOx films, no background relaxations were reported even in films as thick as 75 nm[29], which in our picture would imply significantly larger screening lengths. Several otherexperimental results shed some light on this specificity of InOx films as compared to othersystems. Transmission electron microscope investigations in NbSi and InOx films have shownthat, although these two systems are amorphous from the structural point of view, their chem-ical disorder length scales are markedly different. NbSi thin films are highly homogeneous,with compositional fluctuations of less than 0.1% down to 1 nm [36]. By comparison, thevariations of the O/In ratio in InOx films can be as large as 15-40% for a sampling area of5 nm × R s values[41]. The bias voltage extent of the ohmic regime is found to be systematically smaller inInOx films as compared to the granular Al case [2, 40, 29] and points in the same direction.More recently, the smallness of the volume occupied by the current carrying network hasbeen outlined in InOx films [42], most of the remaining volume being attributed to (much)more insulating zones where the very slow electron dynamics is believed to take place. In a7 ciPost Physics Submission MOSFET device, these highly insulating zones will allow the penetration of the unscreenedelectrical field over large distances. All these findings suggest that a material specific longrange electrical inhomogeneity may explain why the penetration length is larger in InOx thanin granular Al and NbSi films.In contradiction with the preceding, it was suggested that in InOx films, the screeninglength is as small as (cid:39) T . Mean field approaches predictthat screening is suppressed at low T in the glassy phase, which preserves the long-rangepart of Coulomb interactions and allow the opening of the Efros-Shklovskii Coulomb gap[10, 43]. When thermal effect and Coulomb interactions compete, the density-of-states at theFermi level remains finite and the naive application of Thomas-Fermi theory gives a screeninglength that goes like 1 /T [43]. Numerical simulations at T = 0 found that a charge injectionin an electron glass triggers an electronic avalanche which size scales with the size of thesystem [30, 31]. This divergence of the screening length at low T was actually experimentallyinferred from the T study of the background relaxation in granular Al films [38]. It thusseems dubious to us that very short screening lengths exist at low temperature in insulatingInOx films. These controversial results emphasize that the physical origin of the penetrationlength, and especially of the large values observed in InOx films, remains an open and debatedissue. Screening length determination by other means than gate voltage induced conductancerelaxations would be very helpful in order to confirm our interpretation.We end this section with a note on relaxation amplitudes. It is seen in Figure 4 that theamplitudes in % of the G relaxations at V geq after a quench to 4.2 K are essentially determinedby the R s value and not by the Nb content or the thickness of the films. The larger R s , thelarger the amplitude. From the above, the relaxation at V geq is the sum of the ongoing MDgrowth and the background relaxation and it thus reflects the amplitude relaxation of wholevolume of the sample. The fact that all the points fall on a smooth curve although thesamples have different T h /L sc values, shows that the relaxing modes are distributed in thewhole film thickness, ruling out mechanisms like e.g. charge relaxation in the substrate or atthe interface. Note that this R s trend is common to all the disordered systems in which MDsand conductance relaxations have been observed [1], even if the charge carrier density wasalso found to play a role in the MD amplitude of InOx films [12]. Let us now focus on the MD centred on V geq that is visible for all the films in Figure 2. Itactually results from the growth of a narrow 4.2 K contribution (hereafter called the 4.2 KMD) on top of a broad and time independent contribution inherited from the cooling historyof the sample [32]. If we plot the differences between the G ( V g ) curve measured at the longesttime and the ones measured at different times after cooling, they can be superposed for eachsample on a single curve simply by using an ad hoc vertical scaling (see Figure 5). This showsthat the evolution of the total MD at V geq does not result from a slow “thermalization” ofthe broader MD formed during the cooling of the samples but from the growth of the 4.2 KMD on top of it. Note that the shapes of the 4.2 K MDs do not depend on the V geq value:8 ciPost Physics Submission Figure 4: Relative amplitude of the ln( t ) conductance relaxations measured at V geq after aquench to 4.2 K over three decades in time as a function of R s K .if the same protocol is applied under a different V geq , the same shapes are observed after thecooling, the MDs being then centered on the new V geq value.Figure 5: Time independence of the MD shape after a cooling from room temperature to4.2 K ( V geq = 0 V). The plotted curves correspond to the difference between the G ( V g ) curvesmeasured a time t after the cooling, and the last G ( V g ) curve measured (long-time reference).These difference curves are shifted to 0 at large V g values in order to remove the backgrounddrift discussed in Section 4 and the value at 0 V ( V geq ) is normalized to 1 in order to comparethe MD shapes (the vertical axes are reversed).Building on this uniformity in shape of the MDs, we can characterize the influence of theapplication of V geq on a given sample by the width of the MD. For the two most resistivesamples A1 and B1, 4.2 K MDs are fully visible in the 60 V window of the experiment andtheir widths at half maximum ∆ V g can be deduced directly from the top panels of Figure 6.For the other samples, the 4.2 K MDs extend to larger V g values (see the remaining slope d ∆ G/dV g at −
30V and 30V) and a more subtle procedure needs to be applied. As it is shownin the lower panels of Figure 6, the different 4.2 K MDs corresponding to the various Nbcontent for a given thickness can be overlaid by using ad hoc vertical and horizontal ( V g )9 ciPost Physics Submission scaling factors. ∆ V g estimates for A2 and A3 (resp. B2 and B3) can thus be obtained bymultiplying A1- (resp. B1-) ∆ V g by the corresponding V g scaling factors. The ∆ V g valuesof our NbSi films are displayed in Table 2 and found to vary between 0.7 V and 12 V. Ifwe scale the values reported in the literature for other systems so that they correspond toa 100 nm-thick SiO gate insulator , we get: 0.2 V – 2 V in InOx films [13, 12], 1 V – 2 Vin 20 nm thick granular Al films [28], 0.1 V in Be films [15], 1 V in Tl O − x films [16] and0.6 V - 2 V in Ge-Te alloys [17, 18, 19]. The value of 12 V found in films A3 and B3 is bycomparison extremely large, while the MD width in film A1 is similar to values observed in10 nm thick granular Al films having similar R s . We should note that the existence of a frozencontribution to the MD is usually overlooked in the literature, although it might influence thedetermination of the MD width if care is not taken to only measure the contribution of thenarrow growing MD. For example in Be films, only a narrow central part of the MD close to V geq is modified when the sweeping parameters are changed (see Figure 5 of Ref. [15]), whichindicates that the measured total MD may results from the superposition of a wide frozencontribution and a narrower time-dependent one.Name Thickness % Nb R s K ∆ V g L sc A1 2.5 nm 13 97 MΩ 0.7 V ≥ . . ≥ . ≥ . (cid:39) (cid:39) . − . V g and screening length values L sc for the different a-NbSifilms.How can one interpret the MD width variations we observed? As can be seen in Figure 7, in12.5 nm-thick films for which the screening length values L sc can be estimated (see Section 4),a correlation ∆ V g ∝ /L sc is found between L sc and the MD width. Interestingly enough,the extrapolation of NbSi data to thinner MDs using this law includes granular Al results[28] and predicts a screening length larger than (cid:39)
60 nm for the InOx films of Ref. [29], inrough agreement with the absence of background relaxations in films thinner than 75 nm.If we assume that the screening is provided by the electrons populating an Efros-ShklovskiiCoulomb gap [44, 45], the density of thermally excited electrons at a finite T , n ( T ), is givenby: n ( T ) (cid:39) α/L sc ( T ) (1)where α is a constant of order unity. In the percolation approach of the MD proposed byLebanon and M¨uller [20], the instability scale ∆ V g for the conductance is obtained when thedensity of carriers induced by the gate voltage change, n (∆ V g ), becomes larger than n ( T ).For films thicker than L sc : n (∆ V g ) (cid:39) ( C ∆ V g ) /L sc (2)where C is the capacitance per area between the metallic gate and the film. According toEquations 1 and 2, the instability criterion n (∆ V g ) = n ( T ) thus leads to:∆ V g = [ n ( T ) L sc ] /C (cid:39) ( α/C )(1 /L sc ) (3) For a gate insulator with a dielectric constant (cid:15) and a thickness d , ∆ V g is multiplied by ( (cid:15)/ × (100 nm/d ). ciPost Physics Submission Figure 6: Shape of the 4.2 K contributions ∆ G to the MD as a function of V g (top panels)and of V g /N (bottom panels). Top panels: ∆ G was fixed to 1 at 30 V and 0 at 0 V for all thesamples. Bottom panels: vertical and horizontal scalings were applied for A2 and A3 (resp.B2 and B3) curves so that they can be superimposed on that of A1 (resp. B1). The numbersindicated for N are the factors by which the A2 and A3 (resp. B2 and B3) V g scales werecompressed. See the text for details. Left side panels: 2.5 nm thick films. Right side panels:12.5 nm thick films. V geq = 0 V for all the films.This prediction disagrees with the experimental data of Figure 7, which suggests that eitherthe samples are not in the Efros-Shklovskii Coulomb gap regime (the sample B3 is indeedvery close to the metal-insulator transition), or the instability criterion for the MD should bereconsidered. Note that Equation 3 predicts a T dependence for ∆ V g ( L sc ∝ /T ), insteadof the T dependence experimentally observed [2].In InOx films, the MD width was shown to depend systematically on n , the charge carrierdensity deduced from Hall effect measurement at room T , but not on the R s value of the films[13]. In our system, we find a seemingly contradictory result, namely the fact that the MDwidth is determined by the sample’s resistance. More precisely the quantity ∆ V g / min( L sc , T h )correlates very well with R s (4 . n dependence may not be a11 ciPost Physics Submission Figure 7: ∆ V g versus L sc for a-NbSi (samples B), granular Al (Ref. [28]) and InOx films(samples of Figure 8 Ref. [29]). The doted line represents the law ∆ V g ∝ /L sc .proof of a dependence on the carrier density per se, but rather on the disorder strength. Indeedboth are believed to be closely related in samples that are close enough to the metal-insulatortransition (MIT), which is presumably the case for insulating samples having measurableresistances at low temperature. Second, in NbSi films, the interrelationship between R s , thechemical composition, the thickness and the charge carrier density n is still poorly understood.However, it seems reasonable to assume that for a given thickness, the smaller R s (the largerthe Nb content), the larger n . In 100 nm-thick films, a direct estimate of n by Hall effectmeasurements give n = 3 . × e/ cm for a metallic film with 26% of Nb [46], compared toa value 5 × e/ cm obtained from the linear term of specific heat for a weakly insulatingfilm with 9% of Nb [34]. Note that this last value is larger than the highest n reported inInOx films, in qualitative agreement with the large MD widths observed in low R s NbSi films.Figure 8: ∆ V g /T h for samples A, ∆ V g /L sc for samples B, as a function of R s at 4.2 K. Seethe text for details. 12 ciPost Physics Submission In spite of the large differences in their widths, the 4.2 K MDs all grow in a similar (log) waywith the time elapsed since the cooling in all samples studied, with no signs of saturation up tothe longest times achieved ( (cid:39) × s, see Figure 9). Such a logarithmic time dependence iscommonly observed in all other disordered insulating systems where electrical glassy featureshave been found [1]. It can be simply described as resulting from the sum of independentdegrees of freedom having a log-normal distribution of relaxation times [47, 2, 48]. Note thatthis logarithmic relaxation includes sample B3 which has a rather small R s value of 28 k Ωat 4.2 K, putting it very close to the MIT: the amplitude of the MD is very small but thetime scales over which the relaxation is measurable are large. This shows that, as we alreadystressed in another context [27], the amplitudes of the conductance relaxations cannot directlyquantify the dynamics and allow comparisons between systems (like for instance in [29]).Actually one expects that approaching the MIT from the insulating side, the sensitivity of ourexperimental probe (the conductance) to any changes occurring in the sample, does vanish.Only direct measurements of relaxation times are meaningful. These were only recentlyachieved in InOx films [12] and phenomenologically showed that both low R s and low carrierdensities ( n ≤ e/ cm ) are needed to get measurable shorter relaxation times. The carrierdensity value of 5 × e/ cm deduced from the linear term of the specific heat in a weaklyinsulating NbSi film [34] suggests that our B3 sample, although of rather small resistance,has a charge carrier density that is most probably too large for the higher time cutoff of thelogarithmic relaxation to occur within measurable times.Figure 9: Growth of the MD amplitude as a function of the time elapsed after a cooling to4.2 K. The conductance is measured at constant time intervals under V gref = −
20 V, andcompared to the V geq = 0 V value. Left side panel: sample A1; right side panel: sample B3.In sample B3, the large noise level is due to the smallness of the MD (the MD amplitude issmaller than 0.1% after 2 × s).The dynamic response to a V g change was tested using the so-called erasure protocol. Thesample is first prepared in a given initial state by cooling it from room T to 4.2 K under V g = V g and by maintaining it under this V g value for a time t a (step 0). Then, V g ischanged to V g and a new MD centred on V g is formed during a time t w (step 1). Last, V g is changed to V g and the erasure of the V g MD is measured as a function of time (step13 ciPost Physics Submission V g = V g = 0 V and V g = 20 V. At constant time intervals, G is measured at V g = −
20 V (reference value), 0 V ( V g value) and 20 V ( V g value) and[ G ( − − G (20V)] /G ( − G/G of the 20 VMD . In granular Al films, it was shown that if all the protocol was performed at a fixed T (isothermal protocol) and if t a (cid:29) t w , the erasure curve of the MD formed at V g is welldescribed by the law: ∆ G ( t, t w ) = A ( T ) ln(1 + t w /t ) (4)Equation 4 was shown to result from a log-normal distribution of relaxations times switchingback and forth under V g changes [2, 48]. The curve thus scales with t w and the typicalerasure time t eras , given by the intercept of the logarithmic dependence at short times withthe ∆ G = 0 line, is equal to t w . Note that in granular Al films, departures from Equation 4are observed when t a < t w and when t eras becomes larger than t w [5]. This dependence of thedynamics with the time spent since the cooling of the sample, i.e. the age of the system, aproperty called ageing, is the hallmark of glasses. In our NbSi films, we observed significantand unexplained departures from the analytical time dependence of Equation 4 around t (cid:39) t w (see Figure 10) but the t w scaling and ageing effects are still present.Figure 10: Influence of the waiting times t a and t w on the 4.2 K erasure curves for samplesB2 (left panel) and A1 (right panel). See the text for details.In its isothermal form, the erasure protocol described above cannot reveal the T depen-dence of the dynamics [2]. In order to do so, one has to use a different version implying twotemperatures [2, 32, 25]: the MD formation (step 1) is performed at T , and the MD erasure(step 2) at T (cid:54) = T (the V g change from V g to V g is done just before the T change from T to T ). One wants to know whether the time needed to erase the V g MD depends on therespective values of T and T . The results of such a protocol for T = 4 . T = 4 . t eras as the intercept between the timeaxis and the short time linear parts of the curves ( t/t w < .
03 for T = 4 . t/t w < . Note that defined in this way, ∆
G/G doesn’t go to zero when the time goes to infinity since there is alwaysa time independent difference between the conductance at -20 V and 20 V, coming for example from a normalfield effect contribution. This asymmetry was measured during the initial cooling under V geq = 0 V (no MDat V g ) and subtracted from the ∆ G/G values. ciPost Physics Submission Figure 11: 4.2 K erasures of MDs formed during a time t w = 20000 s at different temperatures T . In our set-up the cooling takes (cid:39)
30 s from 9 K and (cid:39)
100 s from 20 K. Intersectionsbetween the straight lines and the ∆
G/G = 0 line give an estimate of the typical erasure times t eras . Note that due to the smallness of the MD amplitudes, the noise level is high for thesample B3 data. The relaxation measurements also start at longer times so that the straightlines of the T = 4 . T = 9 K and almost all the time window for T = 20 K). The dependence of t eras with T reflects the temperature dependence of the conductance relaxation dynamics. If T = 4 . ciPost Physics Submission we get the usual isothermal result t eras = t w while a marked increase of t eras is observedwhen T > . t eras varies between 4 t w and 10 t w for T = 9 Kand between (cid:39) t w and (cid:39) t w for T = 20 K. This clearly shows that the dynamics isthermally activated and slows down upon cooling.The comparison of the different MD amplitudes provide a natural explanation for thevarious aspects of the curves shown in Figure 2. Looking at the curves of Figure 11 atshort times, one sees that depending on the samples, the amplitudes of the MDs formed at T > . R s , the larger they are compared to 4.2 K MDs. This trend is especiallypronounced in 12.5 nm thick films (B films). Consequently, when the high temperature MDsare much smaller than the 4.2 K contributions (film B1), the “frozen” MD accumulated duringthe cooling under V geq = 0 V is almost invisible, thus the flat background. In opposite cases,the “frozen” MD can be the dominant contribution and the growing 4.2 K contribution canbe hardly discernable (see film B3 in Figure 2).We have tried to explain quantitatively the erasure curves of Figure 11, assuming thatthe same modes are relaxing back and forth during the V g changes at T and T . If therelaxation times obey an Arrhenius dependence of the form τ i ( T ) ∝ exp(∆ E/T ) with a singleactivation energy ∆ E , then t eras /t w = exp [∆ E (1 /T − /T )]. But as it is seen in Table 3,the ∆ E values corresponding to T = 20 K data are from 4 to 10 times larger than the onescorresponding to T = 9 K. Such a simple Arrhenius model can thus not describe the dramaticincrease of the erasure times observed in a given sample when T is changed from 9 K to 20 K.We have tested other simulations with a distribution of activation energies, but without beingable to replicate the experimental data of Figure 11. The T dependence dynamics of the NbSifilms is thus non trivial and would require a deeper change in our theoretical assumptions.Arrhenius law hypothesisName ∆ E ( T = 9 K) ∆ E ( T = 20 K)A1 13 K - 15 K 50 K - 70 KA2 14 K - 17 K 70 K - 90 KA3 9 K - 13 K 80 K - 90 KB1 16 K - 20 K 50 K - 70 KB2 15 K - 20 K 70 K - 120 KB3 100 K - 170 KTable 3: Activation energies ∆ E deduced from the erasure times of Figure 11 ( T = 4 . T is lowered [25].However, the T variation of the slow dynamics was found to be essentially of the Arrheniustype, with an activation energy of the order of 30 K which does not depend much on theresistance of the films. In discontinuous gold films, a different protocol has been used: theerasure of a MD formed at T and during the cooling to T is measured at T and comparedwith the formation of a new isothermal MD at T [22, 23]. Signs of a strong temperaturedependence of the dynamics are present since the MDs formed upon cooling between T and T cannot be significantly erased by subsequent V g changes at low temperatures, even after daysof measurements. But the differences in the protocols make the comparison with amorphous16 ciPost Physics Submission NbSi and granular Al films difficult. In InOx, the situation is still unclear. No temperaturedependence or even an acceleration of the dynamics when the temperature is lowered werereported [13, 26] but we have questioned the protocols used during these experiments [25]. Ourresults clearly stress that the activated character of the dynamics is not limited to granularor discontinuous systems and call for reconsidering InOx results. In order to clarify theexperimental situation and to make progress on theoretical models, the erasure protocolsdescribed in the present study should be applied to the other systems in which electricalglassy effects have been found and in a T range as large as possible. Preliminary results showthat the MD dynamics is indeed also thermally activated in amorphous InOx films [49] . In conclusion, our field effect measurements on amorphous NbSi films lead to the followingresults. First, we observed significant background conductance relaxations in 12.5 nm thickfilms, which reflect a spatial extent of the gate voltage disturbance of only few nanometres. Weinterpret this length as the screening length of the films and we suggest that the difference withthe lower bound of 75 nm reported in amorphous indium oxide films stem from the chemicalinhomogeneity of the films giving rise to electrical inhomogeneities. Second, the widths of thememory dips formed at 4.2 K vary strongly with the film parameters. They correlate withthe R s values of the films, the smaller R s , the wider the dip, and with the screening length ofthe films, which may help to better understand the physical mechanisms involved. Third, theconductance relaxations after cooling are logarithmic in time over days with no cutoff of thetime distribution visible, even in low R s films in close vicinity to the metal-insulator transition.When triggered by V g changes, the relaxations display ageing and scale with the waiting timespent under a new V g value. Last, the temperature dependence of the glassy dynamics wastested between 4.2 K and 20 K by applying a non-isothermal erasure protocol. We foundin all the films a strong slowdown of the dynamics upon cooling, in qualitative agreementwith recent results in granular Al films. The thermally activated character of the dynamics isthus not limited to granular or discontinuous films, which calls for a reexamination of the T dependence of the dynamics in indium oxide films. More generally, our results demonstratethat there is no clear line between continuous and discontinuous systems, which is a majorstep towards an universal vision of the electron glass effects. Acknowledgements
We gratefully acknowledge discussions with M. M¨uller.
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