J Le Cochec

F Ladieu

P Pari

Magnetic field effect on the dielectric constant of glasses : an evidence of disorder within tunneling barriers ?

Keywords: numbers: 61.43.Fs, 77.22.Gm, 72.20.Ht, 72.20.My

The magnetic field dependence of the low frequency dielectric constant ǫr(H) of a structural glass a -SiO2+xCyHz was studied from 400 mK to 50 mK and for H up to 3 T. Measurement of both the real and the imaginary parts of ǫr is used to eliminate the difficult question of keeping constant the temperature of the sample while increasing H: a non-zero ǫr(H) dependence is reported in the same range as that one very recently reported on multicomponent glasses. In addition to the recently proposed explanation based on interactions, the reported ǫr(H) is interpreted quantitatively as a consequence of the disorder lying within the nanometric barriers of the elementary tunneling systems of the glass.

I. INTRODUCTION

Since the seventies [1], it is well established that at low temperature T , the properties of amorphous solids differ strongly from their crystaline counterparts. This is currently explained [2] within the tunneling two-level system (TLS) model, stating that the only relevant degrees of freedom at low T are particles going back and forth between two neighboring sites of equilibrium. Indeed, the TLS model accounts for various properties of glasses below ∼ 1 K. A closer look, however, reveals that the TLS model often does not quantitatively account for the measured quantities. Since the late eighties, considerable efforts have been made to include interactions between TLS's in order to explain such discrepancies [3] and justify the TLS model from first principles [4], [5] or, at least, to give a more consistent picture of TLS's [6], [7]. However, the lack of TLS microscopic information allowed to consider interactions either to be weak [5] or strong [6], [7], [8] : these opposite starting points echo Leggett and Yu's question : "Is there anything really tunneling in glasses ?" [9].

The purpose of this work is to gain some microscopic information on TLS's by studying the magnetic field effect on a glassy property, namely the 1 kHz dielectric constant ǫ r . Indeed, for a charge Q, the tunneling transparency of a barrier of size a is strongly modified by magnetic fields H of the order of H c = h Qa 2 : this is due to the (H dependent) difference of quantum phase picked up by the various paths under the barrier [7]. However, setting the usual TLS parameters, a ≃ 0.2 nm and Q = e, with e the electronic charge, leads to H c ≃ 10 4 T, a value so large that it seemed to explain Frossati et al 's results [10] reporting no detectable H effects on ǫ r . Recently, however, detectable ǫ r (H) effects were reported below 80 * ladieu@drecam.cea.fr mK (and down to 1 mK) on a multicomponent glass [8], [11]. Very strikingly, ǫ r (H) is peaked around H c ≃ 0.03 T, a very low value accounted for by using interactions : at low T , it was argued that interactions correlate a large number N ≫ 1 of TLS's and that the resulting spectrum resembles that of a single TLS with a charge Q = N e (and a renormalized tunneling energy ∆ 0 ). In this work, a detectable ǫ r (H) effect is reported on a a-SiO 2+x C y H z glass for 50 mK ≤ T ≤ 400 mK and H up to 3 T. Even if our glass is not really a model glass (such as the various forms of vitreous silica, e.g. Suprasil), the ǫ r (H) dependence reported here shows that these effects are not restricted to the multicomponent glass studied previously [8], [11], favoring the idea that such effects are general in structural glasses. Furthermore our work shows, as stated previously [12], that such measurements demand special experimental care since ǫ r (H) effects are quite small: δǫ ′ r (H)/ǫ ′ r lies typically in the 10 -6 -10 -5 range (ǫ ′ r is the real part of ǫ r ). This is why, after a brief description of the experimental setup, it is explained how to overcome the difficult question of keeping T constant with great accuracy, while increasing H. The resulting ǫ ′ r (H) effect is then accounted for by using numerical calculations of the H effect on the tunneling transparency of a disordered nanometric barrier. As explained below, this additional mechanism should not contradict the one based upon interactions.

II. EXPERIMENTAL SETUP

The sample was deposited on a a -SiO 2 substrate as follows: i) a 0.1 µm thick Cu layer was first evaporated, ii) a L = 0.8 µm thick a -SiO 2+x C y H z layer was then deposited by a 13 MHz plasma vapor technique using TetraEthylOrthoSilane mixed with He [13] iii) a 0.08 µm thick Cu layer, followed by a 0.04 µm Au layer was evaporated. Both electrodes are 3 mm large 10 mm long ribbons recovering each other along ≃ 3 mm (i.e., the capacitance surface is S ≃ 9 mm 2 ) : the opposite ends of the electrodes are free of glass deposit allowing an ohmic contact to be made with a Cu wire glued with Ag paste. These two Cu wires are soldered to the inner wires of cryogenic coaxial cables going from room temperature to the cold copper box embedding the sample : this box is related to the mixing chamber of the dilution fridge by a thermal impedance and contains a RuO 2 resistive thermometer whose resistance will be called θ RuO2 throughout this work (to avoid any confusion with the parallel resistance R of the glass sample). The sample substrate was glued with varnish on a copper sample holder for thermalization. To decrease possible H-dependent heating in the electrodes, H was set parallel to the electrodes (i.e., perpendicular to the 1 kHz measuring field E). Semi-rigid home-made coaxial cables were installed at the top of the fridge and directly plugged in the Andeen-2500A capacitance bridge. The capacitance bridge was used in its parallel option, yielding C ∝ ǫ ′ r and G ∝ ǫ ′′ r , but throughout this work it was chosen to report on C and R = 1/G due to the great similarity of the C(T ) and R(T ) curves (see Fig. 1) : this strongly confirms that the reported R at low T is not some parasitic edge effect but has the same origin that the reported C, namely the TLS's.

From 300 K to 4.2 K, the capacitance C typically halves, while the parallel resistance R increases by a factor ≃ 100. Below 4.2 K, C decreases, reaches its minimum for T rev. and then increases below T rev. before reaching a ultra low T saturation value C sat . According to the standard TLS model the C decrease above T rev. is due to progressive freezing of the diagonal (or relaxational) part of the response, while the C increase below T rev. comes from the induced off-diagonal (or resonant) part of the susceptibility : this effect enlarges as T decreases as do all quantum effects.

III. OBTAINING δǫ ′ r (H)

A. Proof that a non-zero δǫ ′ r (H) exists

Figure 1 shows C(T ) and R(T ) curves when a 1 kHz field E = 3V /0.8 µm = 3.75 MV/m is used. The high value of T rev. ≃ 0.3 K ≃ 8T rev. (E → 0) reveals that such fields correspond to the strongly nonlinear regime [14]. Such large fields were used with a large integration time (100 s) so as to reduce the bridge uncertainties c noise (resp. r noise ) over C (resp. R) measurements down to c noise /C = ±10 -7 (resp. r noise /R = ±10 -5 ). Large fields were also used in [11].

The second key point to investigate small ǫ r (H) effects is to be able to increase H while keeping T accurately constant. It demands either to set up a H-independent thermometry, which is extremely difficult over an extended T range (it may have been the way used in [8], [11]); or, at least, to correct T of the magnetoresistance of the thermometer. However, such a correction cannot be done with great accuracy since the standard technique for measuring the θ RuO2 (H) dependence is as follows [17] : i) For H = 0 the relationship between the power P injected in the fridge and the resulting T is first recorded ; ii) For a given P, H is slowly raised to a given value, and the measured θ RuO2 (H) -θ RuO2 (0) at thermal equilibrium is attributed to the magnetoresistance of the thermometer : the corresponding results are shown in the lower inset of Fig. 1 and lead to corrections of the order of 2 -5% on T when H is raised up to a few T. However, these θ RuO2 (H) measurements lie on the key assumption that the P(T ) relationship does not depend at all on H : this cannot hold accurately, e.g., due to the dissipation of energy resulting from the vibrations of the fridge within the magnetic field. These small and complicated effects cannot be ignored when seeking small ǫ r (H) effects which, translated into a thermal equivalent, amount to an effective variation of T lying in the 0.1 -5% range (see B)2) below).

Going back to the C, R measurements, the fact that both P(T ) and θ RuO2 (T ) relationships sligthly depend on H implies that T cannot be made accurately constant when H is increased. For practical reasons, the C(H), R(H) measurements were made letting P automatically adjust so as to keep the θ RuO2 value fixed at any H value : raising H while keeping θ RuO2 constant leads to a small shift δT (H) of the temperature of the experiment as depicted in the upper inset of Fig. 1.

The measured variations δC(H) = [C(H) -C(0)] constant θRuO 2 are displayed in Fig. 2 for various T values measured at H = 0. The key point is that the

-1.2E-03 -6.0E-04 0.0E+00 6.0E-04 0 1 2 3 H [ T ] 50mK 70mK 100mK 200mK 400mK H [ T ] [C(H) -C(0)] constant θ RuO2 , [pF] 0 3 2 1 -0.005 0.000 0.005 R(H)-R(0), [GΩ] 2 H [T] 0 FIG. 2: δC(H)
measured by keeping θRuO 2 constant while increasing H (the various T labeling the curves are at H = 0). Inset: Corresponding data for δR(H) (same symbols). The fact that δR(H) and δC(H) do not vanish at the same H cannot be explained by using only δT (H) effects but implies that a δǫr(H) dependence exists in the sample. The error bars cnoise = ±10 -5 pF and rnoise = ±10 -5 GΩ are much smaller than the data symbols.

δT (H) dependence cannot explain all the qualitative features of Fig. 2. For example, whatever δT (H) may be, if δǫ r (H) were zero, δC(H) and δR(H) would vanish at the same H : this is clearly contradicted by the 50 mK, 100 mK and 400 mK data reported on Fig. 2. This proves that a ǫ r (H) dependence exists in our sample but that an accurate correction of the δT (H) effects must be done on the data of Fig. 2. This correction cannot be done with the data of the inset of Fig. 1 since, as above evoked, the sought ǫ r (H) is smaller than the accuracy of the δT (H) estimate.

B. Obtaining the order of magnitude of δǫ ′ r (H) At this step, the existence of a detectable δǫ ′ r (H) dependence is established but an accurate suppression of δT (H) effects is still missing. It is shown here that using both δR(H) and δC(H) allows to draw from Fig. 2 the order of magnitude of δǫ ′ r (H), i.e., to know its value within a scale factor 0.1 < η < 10 (see Eq. (2) below) : this interval might seem to be very large, but we will see in the next section that the physical interpretation only involves ln(η).

Overview of the method

The key idea is that since the relative variations of all the involved quantities are small, first order expan-sions are accurate. Defining δX(H) = X(H) -X(0) for any quantity X measured as a function of H keeping the θ RuO2 value constant, one gets :

δC(H) = ∂C ∂T δT (H) + δC int. (H), (1a)
where the measured δC(H) is shown on Fig. 2, δC int. (H) is proportionnal to the sought intrinsic variation of δǫ ′ r (H) and throughout this work, all the partial derivatives with respect to T are taken at H = 0. Introducing the corresponding quantity δR int. (H), an equation similar to Eq. (1a) is obtained for R measurements.

δR(H) = ∂R ∂T δT (H) + δR int. (H). ( 1b
)
Combining Eqs. (1a) and (1b) allows to accurately eliminate δT (H), and to get :

δC int. (H) = η ∂C ∂T ∂R ∂T δR(H) -δC(H) , (2a)
with η = 1 (α-1) and α is defined by :

α = δR int. (H) δC int. (H) ∂C ∂T ∂R ∂T . (2b)
α is introduced here as an attempt to eliminate one of the three unknown quantities among δT (H), δC int. (H) and δR int. (H) since only two quantities, δC(H) and δR(H), are measured. α cannot be predicted since, for strong E, a comprehensive treatment of the TLS dynamics, including quantum coherence effects, is still missing : at present, even the C(T ) curve is not well accounted for [14]. However, within the framework of the standard TLS model, α and η cannot depend on H since, as we show below, ∆ 0,max is the only parameter of the TLS model which might slightly depend on H : neglecting, as above, any second order terms, both the real and the imaginary parts of δǫ r (H) (i.e., both δC int. (H) and δR int. (H)) are expected to be proportionnal to the same small δ∆ 0,max (H)/∆ 0,max (see Eq. ( 4)).

There is no similar argument stating that, generally, α and η = 1/(α -1) cannot depend on T . It is shown in the section A of the Appendix, by using the shape of the curves of Fig. 2 and quite general arguments, that η lies, at any reported T , in [0.1, 10], an interval which will turn out to be "small" enough to neglect any η(T ) variations (see section IV). Basically, it is shown in the Appendix that the δT (H) influence on the measured δC(H) and δR(H) enlarges as H is increased : the quasilinear decrease of δC(H), δR(H) seen in Fig. 2 at large H thus comes from δT (H) effects (at 400 mK it turns to a quasilinear increase due to the fact that ∂C/∂T > 0 contrarily to the case of lower T data). 3)) is reported only for one H at each T : it is much stronger at 400 mK than below 100 mK where error bars hardly exceed the size of the data symbols. Inset: δC(H) measured just at Trev. (with a larger field E = 8.75 MV/m to further increase the signal to noise ratio) : due to the ∂C/∂T = 0 at Trev. the use of Eq. (2a) is not necessary at low H (at higher H, δT (H) is no longer negligible and produces the δC(H) increase). Comparison of the data at Trev. to those at 400 mK supports the idea that η > 0 at 400 mK.

The most salient features of δCint.(H)

Since 0.1 < η < 10, the order of magnitude of δC int. (H) is obtained by setting η = 1 and disregarding any possible η(T ) variations within this interval : the resulting transformation of the data of Fig. 2 through Eq. (2a) is shown on Fig. 3. δC int. (H) is positive for the lowest temperatures (except for three low fields at 50 mK, see below) and negative at higher T , i.e., in the vicinity of T rev. . The 200 mK data were not reported on Fig. 3 since they lie very close to zero. Indeed, it seemed more important to report, in the inset of Fig. 3, on δC(H) measured just at T rev. (measured with a larger field E = 8.75 MV/m to further increase the signal to noise ratio). For the inset of Fig. 3, at low H, the δT (H) contribution (see Eq. (1a)) to δC(H) is negligible since ∂C/∂T = 0 just at T rev. (at large H this is no longer true due to the increase of δT (H), yielding the quasilinear increase of δC(H)). The important point here is the decrease of δC(H, T rev. ) at low H : it further confirms that δC int. (H) is negative in the vicinity of T rev. and that η > 0 at 400 mK. Note that the results of Fig. 3 were reproduced on a second similar sample.

Let us translate the data of Fig. 3 to a thermal equivalent by using δT int.C = δC int. (H)/( ∂C ∂T ) (remember that δC int. (H) are calculated by setting η = 1) : it is found that δT int.C (H) reaches at most 7% of T at 50 mK, 2% at 70 mK, 1% at 100 mK and 4% at 400 mK. Beyond chemical differencies between glasses, this might explain why, with an uncertainty on T of ±5%, previous studies reported undetectable ǫ r (H) effects [10]. Adding quadratically the uncertainties on both terms of the right handside of Eq. (2a), the final uncertainty c err about the δC int. (H) reported on Fig. 3 reads :

c err = η (c noise ) 2 + (r noise ∂C ∂T ) 2 ( ∂R ∂T ) -2 . (3)
On Fig. 3, c err was reported for each T : due to the strong decrease of ∂R/∂T , it is much stronger at 0.4 K than at low T where it hardly exceeds the symbol size. However, even at 0.4 K, c err amounts to an uncertainty over T of ±0.5%, ten times smaller than in [10].

Let us note that δC int. (H)/c err does not depend on the chosen η = 1 : the trends of δC int. (H) reported on Fig. 3 are thus reliable at each T , even if the possible η(T ) dependence forbids any accurate comparison of δC int. (H) at different T . Comparing Fig. 3 with Strehlow's results [11], our δǫ ′ r (H) is somewhat smaller but lies in the same range. As in [11], our δǫ ′ r (H) might be peaked around H ≃ 0.02 T for our lowest T , but this effect is too close to our c err to be systematically studied.

IV. PHYSICAL INTERPRETATION

Let us move to the interpretation of Fig. 3. First, note that, due to the T -dependence of the data of Fig. 2 and Fig. 3, the reported δǫ ′ r (H) can be attributed neither to the coaxial cables nor to the "matrix" surrounding TLS's. Furthermore, the change of sign of δC int. (H) in the vicinity of T rev. excludes a role of the electrode-glass interfaces and further confirms that TLS's are responsible for the reported δǫ ′ r (H). Last, it is shown in the section B of the Appendix that spin effects, if any, can be ruled out at low enough T , i.e., well below T rev. : this is why we will focus on the positive δǫ ′ r (H) observed at T ≤ 200 mK.

A. Going from δCint.(H) to δ∆0,max(H)

For the part of the capacitance C T LS coming from TLS's, the usual result [14] below T rev. can thus be used :

C T LS = C ln (E max /k B T ) where k B is Boltzmann con- stant, E max = ∆ 2
max + ∆ 2 0,max is the maximum interlevel spacing of TLS's, and the constant C ≃ 0.03 pF is determined by fitting the increase of C(T ) of Fig. 1 below T rev. (details are given below).

Since both ∆ (see section B of the Appendix) and C [16] are H-independent, the reported δǫ ′ r (H) is interpreted as a ∆ 0,max (H) dependence, yielding :

2δC int. (H) C = δ∆ 0,max (H) ∆ 0,max (0) = ln ( ∆ 0,max (H) ∆ 0,max (0) ), (4)
where the standard assumption ∆ max = ∆ 0,max is responsible for the factor 2 in the left hand-side of Eq. ( 4). Let us note that the expression of C T LS used to derive Eq. ( 4) only holds accurately between 90 mK and 200 mK since, at lower T , the C(T ) curve tends to saturate. This failure of the fit at low T is due to the fact that the above expression of C T LS is well established only in the linear regime, and that a comprehensive treatment for the strong E used here is still missing. However, this formula should capture the essential part of the physics since it expresses that, well below T rev. , the TLS susceptibility is due to the quantum transitions coherently induced by E. Since, physically, it is likely that tunneling plays a key role in the nonlinear C T LS at low T , Eq. ( 4) should finally hold, up to a multiplicative prefactor which can be disregarded just like η (see below).

B. A possible microscopic explanation of δ∆0,max(H)

A ∆ 0,max (H) dependence means that the tunnel transparency is affected by H. This is well known in another kind of disordered insulators, namely Anderson insulators (e.g. lightly doped semiconductors) where electronic states are localized due to potential disorder. At very low frequency, transport in Anderson insulators occurs by hopping on a fractal percolating path strongly affected by H. H effects are smaller at high frequencies where the conducting path is broken since only the fastest hops contribute : note that, since they are disconnected from each other, each of these fast back-and-forth hops is a TLS. In both limits, H effects were accounted for [17], [18] by studying the tunnel transparency ∆ 0 between two sites separated by a disordered barrier depicted in the inset of Fig. 4 : tunneling between the two sites of the (quasisymmetric) TLS separated by the distance a is strongly affected by the potential disorder coming from the structural disorder within the tunneling barrier itself. This is modeled by a three dimensional network of "impurities" (whose unit length is the elastic mean free path λ el ) and ∆ 0 results from the coherent sum of all quantum paths along the impurity network : at H = 0, a path picks up a phase -1 each time it diffuses on an impurity whose energy e i is smaller than that of the TLS. Since the e i are drawn at random, ∆ 0 must be first averaged over disorder (noted hereafter with < >).

Using this model to account for the measured δǫ ′ r (H) is obviously an oversimplification. Indeed, TLS's might be more complicated than a unique charge going back and forth between two sites separated by a: a TLS could be a group of charges, a 'molecule', and more generally any charged species with two spatially separated potential minima. However, we show now that this model ac- Main figure: ∆0 results from the coherent sum of all the paths linking the TLS sites : due to the additional quantum phase introduced by H for each path, < ln ∆0 > increases with H, the H 2 dependence at very low H (unreported) turning continuously to a H 0.5 behavior at higher fields (ensemble averaging over the possible "impurities" configurations is indicated by < > and Hc = h eλ 2 el ). The three curves for a/λ el ≥ 100 are calculations drawn from Medina et al 's work [19] and the 100 mK data of Fig. 3 are reported by using Eq. ( 4). Both the sign and the sublinear behavior of our data are well accounted for by the calculations. Extrapolation from the logarithmic behavior at large a/λ el gives a/λ el ≃ 5 -10 for our sample, yielding an elementary dipole p0 ∼ 10 D not that far from the one drawn from echo measurements.

1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E-06 1.E-05 1.E-04 1.E-03 H / H c < LN[ ∆ 0 (H) / ∆ 0 (0) ] > 100mK data of
counts for most of the salient features of the low-T δǫ ′ r (H) reported in Fig. 3.

C. Accounting for the data by using the disordered tunneling barrier model ), due to the additional quantum phase introduced by H. Using Eq. ( 4) with C = 0.03 pF and λ el = 0.2 nm, the positive δǫ ′ r (H) reported at 100 mK on Fig. 3 can be displayed on Fig. 4. Comparison with the calculations, made for large a/λ el. , reveals that both the sign and the sublinear dependence δǫ ′ r (H) ∼ H 0.6±0.2 observed on the data are compatible with disorder effects. Moreover, using the roughly logarithmic decrease of < δ ln(∆ 0 (H)) > with a/λ el , the experimental data are quantitatively compatible with a ≃ 5λ el ≃ 1 nm, yielding an elementary dipole p 0 ≃ 10 D [START_REF] Since Δcint | ∝ η, the more general result is a/λ el ≃ 5 + ln(η), yielding negligible corrections whatever η in its[END_REF]: this is the good order of magnitude with respect to the 4 D value drawn from pulse echo experiments [START_REF] Baier | [END_REF]. Last, let us note that, once reported on Fig. 4, the 50 mK and 70 mK data lie very close to the 100 mK data, yielding the same δǫ ′ r (H) ∼ H 0.6±0.2 as well as the same order of magnitude for a/λ el. : these low T data were not reported on Fig. 4 since the expression of C T LS used to derive eq. ( 4) does not hold precisely below 90 mK.

To summarize, the microscopic model presented here allows to fit quantitatively the data with very reasonnable parameters. This is due to the extreme Hsensitivity of tunneling through a disordered barrier which yields non negligible effects even for H ≪ H c . Indeed, suppressing disorder and taking a = λ el = 0.2 nm, would lead, by standard tunneling calculations, to < δ ln(∆ 0 (H)) >≃ -(H/H c ) 2 ≃ -10 -8 , i.e., to an undetectable quadratic negative effect, at odds with experiments.

Despite its success in accounting for our data, a limit of our mechanism is that it yields a monotonous H-behavior of δǫ ′ r (H), i.e., it cannot account for the peaked structure around H = 0.03 T reported in [11]. However, even if our scenario is based on disorder effects, it might not contradict the interaction picture used in [11]. Indeed, the disorder explanation lies upon a ∆ 0,max (H) effect with a very large H c ≃ 10 4 T while the interaction mechanism involves a ∆ 0,min (H) effect with a very small H c ≃ 10 -2 T (due to the very large number N of correlated TLS's). In the Fig. 3 of [12], it seems that collective tunneling can be viewed simply as a simultaneous tunneling of N elementary dipoles. If it is so, both mechanisms should simply add up due to their very different H c . It is well admitted [11] that averaging problems in the interaction picture demand further progress. It should explain, e.g. why, in [11], δǫ ′ r (H) is peaked around a H c field decreasing when T increases , which naively amounts to a counter-intuitive increase of N when T increases.

V. CONCLUSION

To summarize, a ǫ r (H) dependence was shown in a a -SiO 2+x C y H z glass for H ∼ 1 T and T ≤ 0.4 K. This demanded to decrease the T uncertainty to less than 0.5% : beyond chemical differences between glasses, this might explain why previous studies reported undetectable ǫ r (H) effects. At low T , the reported ǫ r (H) effects were interpreted assuming that tunneling is affected by some disorder within the elementary tunneling barriers of size a ≃ 1 nm. This scenario might simply add up to the interacting one previously proposed. It is argued here that, at any reported T , η lies in the interval [0.1, 10]. Defining the exponents s and t by δT (1 T ≤ H ≤ 3 T) ∼ H t and δC int. (1 T ≤ H ≤ 3 T) ∼ H s , we will first assume that ts ≥ 0.3 and show, from the data of Fig. 2, that this implies 0.1 ≤ η ≤ 10 at any reported T . Reciprocally, assuming 0.1 ≤ η ≤ 10, we will show, by using the trends of Fig. 4 and Fig. 1 that ts > 0.3. Note that s and t are not necessarily integers, due to subtle disorder averaging effects, such as those studied in Medina et al 's work [19] (see section IV).

Since we must precisely compare the relative importances in eqs. (1a)-(1b) of the first term containing δT (H) and the second one containing δǫ r (H), it is more convenient to translate the measured δC(H) and the sought δC int. (H) into thermal equivalents by introducing δT C (H) = δC(H)/( ∂C ∂T ) and δT int.C (H) = δC int. (H)/( ∂C ∂T ). Eq. (1a) thus amounts to:

δT C (H) = δT (H) + δT int.C (H), (A1)
while defining the corresponding quantities δT R (H) and δT int.R (H) for R measurements yields from Eq. (1b):

δT R (H) = δT (H) + δT int.R (H). ( A2
)
With these definitions note that the definition of α given in Eq. (2b) becomes α = δT int.R /δT int.C with still η = 1/(α -1).

1. Assuming ts > 0.3 implies 0.1 ≤ η ≤ 10 at any reported T i) η < 10 (i.e., α > 1.1) . Consider the magnitudes of the δC(H) and δR(H) bumps reported on Fig. 2 : for example at 50 mK, these positive bumps around H = 0.5 T amount to δT C ≃ -6 mK and to δT R ≃ -7.6 mK. These two values are ten times larger than that derived from the small negative magnetoresistance of the RuO 2 thermometer (see inset of Fig. 1) and, even if the inset data cannot be accurately trusted, these δT C , δT R values are anyway too large to be entirely attributed to the RuO 2 thermometer [15]. This means that both δT int.C (H) and δT int.R (H) are negative at 50 mK, i.e., α > 0 at 50 mK and both δC int. (H) and δR int. (H) positive in the Tesla range : with the above definition of s, δC int. (H) and δR int. (H) are expected to be proportionnal to the same H s with positive coefficients. Thus δC int. (H) and δR int. (H) increase with H : the (quasilinear) decrease seen on Fig. 2 at large H thus comes from RuO 2 effects, i.e., δT (H) ∝ H t and t > s. Setting these H expansions in Eqs. (A1),(A2) leads to α = (H R /H C ) t-s with H R the magnetic field where δR(H) vanishes and H C the corresponding one for δC(H) : at 50 mK, H R ≃ 4.2 T and H C ≃ 2.9 T, which with the above assumption t-s > 0.3 leads to α > 1.1, i.e., η < 10. Finally, the same argument can be made at T > 50 mK where H R /H C ratios are larger, reaching 2.5 at 400 mK.

ii) η > 0.1 (i.e., α < 10). The assumption t > s+0.3 implies that the low field region H < ∼ 1 T is that where the relative importance of δT int.C (resp. δT int.R ) in the measured δT C (resp. δT R ) is the largest. In this range where δC(H) and δR(H) have the same sign, the data of Fig. 2 correspond, for any given T , to values of δT C and δT R which differ by less than a factor 2. This means that δT int.C and δT int.R are of the same order of magnitude (this might be seen also in [11]), hence α < 10 i.e, η > 0.1: this result holds even at large H since α and η do not depend on H. This assumption on η allows Fig. 3 and Fig. 4 to be drawn from Fig. 2, yielding s = 0.6 ± 0.2 for H > ∼ 1 T. Besides, the data of the lower inset of Fig. 1 about θ RuO2 (H) effects amount to t > ∼ 1.2 for 1 T < ∼ H ≤ 3 T (note that we do not rely here on the precise values of θ RuO2 (H) effects but only on their general trend). Hence, the data ensure ts > 0.3.

B. Ruling out spin effects at low T

Due to the lack of microscopic information on TLS's, spin effects cannot be ruled out at all T . Indeed, if each TLS contains a single electronic spin, the non negligible Zeeman energy δE Z (H) ≃ 0.7 K/T comes into play : each TLS becomes a four-level system. This complicates greatly the ǫ ′ r calculation in the range T > ∼ T rev. where the relaxational (i.e., "thermodynamic") contribution plays an important role. This is why the negative δǫ ′ r (H) close to 400 mK has not been interpreted and the efforts were focused on the positive δǫ ′ r (H) at low T . Indeed, well below T rev. , the TLS susceptibility is driven by the transitions coherently induced by E, and such quantum transitions are forbidden between states of different spins : TLS's can thus be separated into two independent subclasses of a given spin state whose susceptibilities can be added. Thus, for a given spin state, the asymmetry energy ∆ between the two potential wells of given TLS is effectively independent of δE Z (H).

FIG. 3 :

 3 FIG. 3: δCint.(H) ∝ δǫ ′r drawn from Fig.2data by using Eq. (2a) with η = 1. For clarity the error bar (see Eq. (3)) is reported only for one H at each T : it is much stronger at 400 mK than below 100 mK where error bars hardly exceed the size of the data symbols. Inset: δC(H) measured just at Trev. (with a larger field E = 8.75 MV/m to further increase the signal to noise ratio) : due to the ∂C/∂T = 0 at Trev. the use of Eq. (2a) is not necessary at low H (at higher H, δT (H) is no longer negligible and produces the δC(H) increase). Comparison of the data at Trev. to those at 400 mK supports the idea that η > 0 at 400 mK.

FIG. 4 :

 4 FIG.4: Inset: Schematic view of the disorder lying within the tunnel barrier of size ∼ a between the two sites (gray circles) of a TLS : the potential fluctuations are modeled by a set of "impurities" of interspacing λ el whose energies are drawn at random either well above or well below that of the TLS. Main figure: ∆0 results from the coherent sum of all the paths linking the TLS sites : due to the additional quantum phase introduced by H for each path, < ln ∆0 > increases with H, the H 2 dependence at very low H (unreported) turning continuously to a H 0.5 behavior at higher fields (ensemble averaging over the possible "impurities" configurations is indicated by < > and Hc = h

Figure 4

 4 Figure 4 is drawn from Medina et al 's calculations [19] and shows that < ln ∆ 0 > increases with H (in Fig. 4 H c = h eλ 2 el

  that η lies in the interval [0.1; 10]

2 .

 2 Assuming 0.1 ≤ η ≤ 10 implies ts > 0.3

Acknowledgments

Many thanks to P. Ailloud (CNRS/LPS) and P. Trouslard (CEA/INSTN/LVdG) for samples realisation, and to P. Forget for cryogenic help. Useful discussions with D. Boutard, M. Ocio, M. Rotter, J. Joffrin and J.-Y. Prieur are greatly acknowledged.


Kilian Hett

Vinh-Thong Ta

Rémi Giraud

Mary Mondino

José V Manjón

Pierrick Coupé

Patch-based DTI grading: Application to Alzheimer's disease classication

Keywords: Patch-based grading, Alzheimer's disease classication, DTI, DWI, Mild Cognitive Impairment

Early diagnosis is one of the most important challenges related to Alzheimer's disease (AD). To address this issue, numerous studies proposed biomarkers based on anatomical MRI. Among them, patchbased grading demonstrated state-of-the-art results when applied to T1weighted MRI. In this work, we propose to use a similar framework on dierent diusion parameters extracted from DTI. We also propose to use a fast patch-based search strategy to provide novel biomarkers for the early detection of AD. We intensively compare our new grading-based DTI features with basic MRI/DTI biomarkers and evaluate our method within a cross validation classication framework. Finally, we demonstrate that the proposed biomarkers obtain competitive results for the identication of the dierent stages of AD.

Introduction

Alzheimer's disease (AD) and its prodromal phase are the most prevalent neurodegenerative disorders for the elderly people. The neurodegeneration caused by AD leads to an irreversible decline of memory and cognition abilities. One of the most important challenges is to nd relevant biomarkers that could help for early diagnosis and prognosis of AD. Such imaging biomarkers can make easier the design of clinical trials that would allow faster development of new therapies.

Over the last years, numerous works proposed new imaging biomarkers to perform early diagnosis of AD. Among them, MRI-based methods showed that Data used in preparation of this article were obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu). As such, the investigators within the ADNI contributed to the design and implementation of ADNI and/or provided data but did not participate in analysis or writing of this report. A complete listing of ADNI investigators can be found at: http://adni.loni.usc.edu/wp-content/uploads/how_to_apply/ADNI_Acknowledgement_List.pdf.

the atrophy of medial temporal lobes is one of the most predictive biomarker candidates [START_REF] Frisoni | The clinical use of structural MRI in Alzheimer disease[END_REF]. The volume of hippocampus (HC) is now considered as an important biomarker for mild cognitive impairment (MCI), the early stage of AD.

HC-based biomarkers demonstrated state-of-the-art performances in AD classication as shown in several extensive comparisons (e.g., [START_REF] Cuingnet | Automatic classication of patients with Alzheimer's disease from structural MRI: a comparison of ten methods using the ADNI database[END_REF][START_REF] Wolz | Multi-method analysis of MRI images in early diagnostics of Alzheimer's disease[END_REF]). Recently, advanced patch-based grading within HC or over the whole brain were explored and demonstrated competitive performance [47]. These studies showed that patchbased grading methods are able to detect subtle hippocampal alterations several years before diagnosis [START_REF] Coupé | Detection of Alzheimer's disease signature in MR images seven years before conversion to dementia: Toward an early individual prognosis[END_REF] and can be useful to perform dierential diagnosis [START_REF] Koikkalainen | Dierential diagnosis of neurodegenerative diseases using structural MRI data[END_REF].

Besides anatomical MRI, the use of DTI has been proposed to detect the rst signs of microstructure alterations. Microstructural modications are considered to occur before the atrophy measured by anatomical MRI. Therefore, we assume that DTI could be used as an earlier biomarker. DTI-based studies showed modications of diusion parameters in AD patients for structures such as corpus callosum, fornix, cingulum, hippocampus [1013]. Moreover, it has been shown that HC mean diusivity (MD) increase is related to pathology evolution [START_REF] Fellgiebel | Diusion tensor imaging of the hippocampus in MCI and early Alzheimer's disease[END_REF] and thus could be used as an ecient biomarker of AD. Recently, more advanced methods using brain connectivity have been proposed to better capture white matter alterations [START_REF] Liu | Diusion tensor imaging and tract-based spatial statistics in Alzheimer's disease and mild cognitive impairment[END_REF][START_REF] Prasad | Brain connectivity and novel network measures for Alzheimer's disease classication[END_REF]. These studies showed that brain connectivity is modied in the earliest stages of the pathologies.

Contributions: In this paper, we propose to apply the grading-based framework [START_REF] Coupé | Scoring by nonlocal image patch estimator for early detection of Alzheimer's disease[END_REF] on the dierent diusion parameters extracted from DTI for AD classication. To reduce the computational burden, a fast patch-based search strategy is involved in the proposed method. In this work, we propose to compare the performances of basic DTI-based hippocampal features (e.g., mean MD) with hippocampal volume to demonstrate the eciency of DTI biomarkers. Next, we study the performance of the proposed patch-based DTI grading compared to MRI grading. Finally, we demonstrate the eciency of these new biomarkers to identify the dierent stages of AD. MRI processing: T1-weighted (T1w) images were processed using the vol-

Brain system [START_REF] Manjón | volbrain: An online MRI brain volumetry system[END_REF] 3 . This system is based on an advanced pipeline providing automatic segmentation of dierent brain structures from T1w MRI. The preprocessing is based on: (a) a denoising step with an adaptive non-local mean lter [START_REF] Manjón | Adaptive non-local means denoising of MR images with spatially varying noise levels[END_REF], (b) an ane registration in the MNI space [START_REF] Avants | A reproducible evaluation of ANTs similarity metric performance in brain image registration[END_REF], (c) a correction of the image inhomogeneities [START_REF] Tustison | N4itk: improved n3 bias correction[END_REF], and (d) an intensity normalization [START_REF] Manjón | NICE: Non-local Intracranial Cavity Extraction[END_REF]. Afterwards, MRI are segmented in the MNI space using non-local patch-based multi-atlas methods [START_REF] Coupé | Patch-based segmentation using expert priors: Application to hippocampus and ventricle segmentation[END_REF]. The obtained hippocampus are segmented with the EADC protocol [START_REF] Boccardi | Training labels for hippocampal segmentation based on the EADC-ADNI harmonized hippocampal protocol[END_REF] designed for AD studies.

DTI processing: The preprocessing of the DWI images is based on: (a) a denoising step with a LPCA lter [START_REF] Manjón | Diusion weighted image denoising using overcomplete local PCA[END_REF], (b) a correction of the head motion using an ane registration, and (c) an ane and a non-rigid registration to the T1w MRI in the MNI space [START_REF] Avants | A reproducible evaluation of ANTs similarity metric performance in brain image registration[END_REF]. Afterwards, a single tensor model [START_REF] Basser | MR diusion tensor spectroscopy and imaging[END_REF] is estimated at each voxel using the Dipy software [START_REF] Garyfallidis | Dipy, a library for the analysis of diusion MRI data[END_REF] 4 . Next, we estimate the fractional anisotropy (FA), the axial diusivity (AxD), the radial diusivity (RD) and the mean diusivity (MD) within the hippocampus structure with a mask provided by the volBrain system. These measures correspond to: the degree of diusion anisotropy for the FA, the longitudinal diusivity along the axonal bers for the AxD, the diusivity orthogonal to the axonal bers for the RD and the mean diusivity level in the three spatial directions for the MD.

Proposed Method

It is known that AD impacts specic brain areas especially the hippocampus.

It has been shown that the hippocampal atrophy estimated on anatomical T1w

MRI can help to classify the dierent stages of AD. However, a deterioration of the microstructure should precede this atrophy measured with anatomical MRI. To capture these early alterations, we propose to apply the Scoring by

Nonlocal Image Patch Estimator (SNIPE) [START_REF] Coupé | Scoring by nonlocal image patch estimator for early detection of Alzheimer's disease[END_REF] on DTI images. This patch-based method estimates a map of scores representing a CN/AD likeness that reects the amount of anatomical alterations caused by AD. Previous works showed that SNIPE provides competitive results when applied on HC [START_REF] Tong | Multiple instance learning for classication of dementia in brain MRI[END_REF][START_REF] Coupé | Detection of Alzheimer's disease signature in MR images seven years before conversion to dementia: Toward an early individual prognosis[END_REF]. In this paper, we propose for each subject to apply SNIPE on MD, RD, AxD and FA maps as well as T1w images. To reduce the resulting computational burden, we use the Optimized PAtchMatch Label fusion (OPAL) method, which is a fast approximate of K-nearest neigbhors search strategy designed for large 3D template library.

This approach enables to perform structure grading in few seconds. Therefore, we can quickly estimate grading maps for FA, MD, AxD and RD and T1w MRI.

Finally, we use the mean scores within HC mask for each grading maps to identify the dierent stages of AD.

Fast patch-based grading: Contrary to SNIPE that computes the distance of all the patches in a neighborhood surrounding the position of the considered voxel, the used of the OPAL strategy enables to directly obtain the good approximated candidates. The patch surrounding each voxel is used to estimate the anatomical pattern similarity between the considered subject i and a training library of templates denoted L composed of CN and AD patients. Once the set of the closest patches K i is obtained with OPAL, the estimation of the grading value g can be computed for each voxel x i as follows:

g(x i ) = xj,t∈Ki w(x i , x j,t )p t xj,t∈Ki w(x i , x j,t )

, ( 1
)
where w(x i , x j,t ) = e -d(P (x i ),P (x j,t )) 2 h 2

is the weight assigned to p t in the grading value estimation. p t corresponds to the pathological status of the template t, the value -1 is aected to an AD patient and 1 to a CN subject. The weight function w depends on the similarity between the patch P (x i ) and P (x j,t ) centered on the voxels x i and x j,t , respectively. This similarity is estimated with a distance measure d. We use h = min xj,t d(x i , x j,t ) + , with → 0. For the T1w images, the distance between two patches is provided by a zero mean normalized sum of squared dierences (ZNSSD) dened as:

ZN SSD (P (x i ) , P (x j,t )) = P (x i ) -µ P (xi) σ P (xi) - P (x j,t ) -µ P (xj,t) σ P (xj,t) 2 2 , (2)
where µ and σ are the mean and the unbiased standard deviation of the considered patch, respectively. The advantage of the ZNSSD is to compute a robust inter-patches distance while addressing the local inhomogeneities problem. Contrary to anatomical MRI, DTI is a quantitative imaging. To preserve the quantitative information provided by each diusion map we use the sum of squared dierences (SSD) dened as:

SSD (P (x i ) , P (x j,t )) = ||P (x i ) -P (x j,t )|| 2 2 , (3)
Features extraction: Mean value of each DTI map is measured within the hippocampus masks provided by the volBrain system in the MNI space. The hippocampus masks are used to compute the average grading values. Since the features are correlated with age, we performed a correction as done in [START_REF] Dukart | Age correction in dementiamatching to a healthy brain[END_REF].

Validation framework: The classication process is performed in a leave-one- The patch size is set to 5×5×5 voxels. 3 Results

Figure 1 shows the hippocampus segmentation, the T1 and MD grading maps for the considered groups. For the grading images, the blue and the red colors correspond to the AD and the CN similarity, respectively. As shown in Figure 1, grading values decrease in accordance to the dierent stages of AD and quantify impacts of the disease in each voxel. The T1w and MD grading maps show that for the CN subject, almost all the voxels are estimated as healthy while for the AD patient the majority of the voxels are detected to be impacted by AD. The results of the comparisons between basic biomarkers are summarized at the top of

  out cross validation procedure. A linear discriminant analysis (LDA) is used to classify each test subjects. The results of each experiment are compared in terms of area under curve (AUC) and accuracy (ACC). The AUC is estimated with the a posteriori probabilities provided by the LDA. We carried out several experiments: CN vs. AD, CN vs. MCI, AD vs. MCI and eMCI vs. lMCI. Parameter settings: The grading method is computed with a training library of 80 subjects composed of 40 AD and 40 CN templates. For each voxel that belongs to the HC area 100 patches are extracted from the training templates.

Fig. 1 .

 1 Fig. 1. From top to bottom: HC mask on the T1w images, T1w MRI grading and MD grading. For grading images blue and red colors correspond to the AD and the CN similarity, respectively.

Table 1 .

 1 Dataset description

	2 Materials and Methods
	2.1	Dataset

Data used in this work were obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset 1 . ADNI is a North American campaign launched in 2003 with aims to provide and test MRI, PET, clinical and neurological measures and other biomarkers. This dataset includes AD patients, MCI subjects and elderly controls (CN). Table 1 shows the distribution of the data for each group. The data include 60 CN, 110 MCI composed of 74 early mild cognitive impairment (eMCI), 36 late mild cognitive impairment (lMCI) and 48 AD.

chosen to optimize the signal-to-noise ratio in a xed scan time

[START_REF] Jahanshad | Diusion tensor imaging in seven minutes: determining trade-os between spatial and directional resolution[END_REF]

.

Table 2 .

 2 Comparison of the considered biomarkers. Underlined results correspond to the individual basic or grading based features. Bold-faced values correspond to the best ACC over all the features. The results of each column correspond to the AUC / ACC in percentage.

	Features	CN vs. AD	CN vs. MCI	AD vs. MCI	eMCI vs. lMCI
	Volume	88.4 / 83.1	69.5 / 63.9	71.1 / 67.2	67.2 / 63.7
	Mean FA	64.2 / 59.2	57.7 / 56.1	54.0 / 52.7	38.2 / 43.1
	Mean MD	85.7 / 80.3	66.0 / 62.6	75.0 / 72.5	67.6 / 62.8
	Mean AxD	83.5 / 81.4	63.5 / 58.0	74.3 / 70.2	68.9 / 66.8
	Mean RD	86.2 / 79.2	66.5 / 62.3	74.8 / 70.5	66.0 / 61.5
	T1 Grading	93.4 / 87.8	71.3 / 64.1	82.0 / 73.4	68.7 / 66.2
	FA Grading	85.0 / 80.1	63.5 / 60.1	74.9 / 70.3	63.0 / 60.7
	MD Grading	90.6 / 86.5	68.8 / 60.7	80.4 / 76.3	70.4 / 65.8
	AxD Grading	91.1 / 85.8	68.7 / 59.6	80.2 / 73.1	71.8 / 67.6
	RD Grading	90.3 / 85.1	68.9 / 61.0	80.0 / 76.5	69.3 / 65.4

Table 2 .

 2 HC volume provides the best accuracy for comparisons of CN vs.AD and CN vs. MCI with an ACC of 83.1% and 63.9%, an AUC of 88.4% and 69.5%, respectively. The DTI-based features provide best results for AD vs. MCI and eMCI vs. lMCI. For AD vs. MCI the mean MD provides 72.5% of ACC and 75% of AUC. For eMCI vs. lMCI the mean AxD provides 66.8% of ACC and 68.9% of AUC.The lower part of Table2shows the results of the biomarkers based on grading. The grading-based biomarkers provide better results than the basic ones for all comparisons. The best result for the CN vs. AD is obtained by the T1w grading with 87.8% of ACC and 93.4% of AUC. The grading of this modality also provides the best results for the CN vs. MCI with an ACC of 64.1% and an AUC of 71.3%. The DTI-based grading features provide best ACC for AD vs. MCI and eMCI vs. lMCI. For AD vs. MCI the RD grading provides 76.5% of ACC and 80% of AUC. For eMCI vs. lMCI the MD grading provides 67.6% of ACC and 71.8% of AUC. It is interesting to note that results obtained with basic and grading-based features are in line. In fact, T1w-based features provide better results for AD vs. CN and CN vs. MCI while DTI-based features obtain better results for AD vs. MCI and eMCI vs. lMCI. These results demonstrate the interest of DTI-based features to analyze AD evolution along eMCI, lMCI and

http://adni.loni.ucla.edu

https://adni.loni.usc.edu/wp-content/uploads/2010/05/ADNI2_GE_3T_22.0_ T2.pdf

http://volbrain.upv.es

http://nipy.org/dipy/

AD stages. Recently, an advanced method based on brain connectivity has been evaluated on a similar dataset [START_REF] Prasad | Brain connectivity and novel network measures for Alzheimer's disease classication[END_REF]. This approach obtained 78.5% of ACC for the CN vs. AD while MD grading obtains 86.5%. In addition, for the eMCI vs. lMCI, [START_REF] Prasad | Brain connectivity and novel network measures for Alzheimer's disease classication[END_REF] obtained 63.4% while AxD grading obtains 67.6%. This demonstrates the competitive performance of the proposed DTI-based grading biomarkers.

Conclusion

In this work, we proposed a novel patch-based grading framework on dierent diusion parameters extracted from DTI. The proposed method enables a fast feature extraction by using an optimized patch-based search strategy. We compare our new biomarkers with state-of-the-art MRI biomarkers and demonstrated that DTI grading provides ecient features for AD detection. Finally, we obtained competitive results to identify the dierent stages of AD. In a further work, we will study the complementarity of the grading based on T1w and DTI.


F Ladieu
email: ladieu@drecam.cea.fr

D L'hôte

R Tourbot

Non-Ohmic hopping transport in a-YSi: from isotropic to directed percolation

Keywords: numbers: 05, 60+w -64, 60, Ak -72, 15, Rn -72, 20, Ht

Electrical transport has been investigated in amorphous Y 0.19 Si 0.81 , from 30 mK to room temperature. Below 2 K, the conductance G exhibits Efros-Shklovskii behavior G ∼ exp[-(T 0 /T ) 1/2 ] at zero electric field, where conduction is expected to occur along a very sinuous path (isotropic percolation). The non-linear I-V characteristics are systematically studied up to very high fields, for which the conductance no longer depends on T and for which the current paths are expected to be almost straight (directed percolation). We show that the contributions of electronic and sample heating to those non-linearities are negligible. Then, we show that the conductance dependence as a function of low electric fields (F/T < 5000 V m -1 K -1 ) is given by G(F, T ) = G(0, T ) × exp(-eF L/k B T ). The order of magnitude (5-10 nm) and the T dependence (∼ T -1/2 ) of L agrees with theoretical predictions. From the T 0 value and the length characterizing the intermediate field regime, we extract an estimate of the dielectric constant of our system. The very high electric field data do not agree with the prediction I(F ) ∼ exp[-(F 0 /F ) γ′ ] with γ′ = 1/2 : we find a F dependence of γ′ which could be partly due to tunneling across the mobility edge. In the intermediate electric field domain, we claim that our data evidence both the enhancement of the hopping probability with the field and the influence of the straightening of the paths. The latter effect is due to the gradual transition from isotropic to directed percolation and depends essentially on the statistical properties of the "returns" i.e. of the segments of the paths where the current flows against the electrical force. The critical exponent of this returns contribution which, up to now was unknown both theoretically and experimentally, is found to be β = 1.15 ± 0.10. An estimation of the returns length is also given.

I. INTRODUCTION

The electrical transport in disordered insulators such as doped semiconductors or amorphous systems, has been widely investigated for several decades as a convenient way to probe the properties of Anderson insulators [1]. At finite temperature T , the conduction results a priori from all the electronic hops between any localized states i and j separated by a distance r ij and a difference in energy E ij . Starting from the probability of such a hop p ij ∝ exp -(r ij /ξ + E ij /k B T ) (with ξ the localization length and k B the Boltzmann constant), Mott [2] introduced the key concept of variable-range hopping (VRH) stating that transport is dominated by the most probable hops whose characteristic length r m ∝ T -1/(d+1) and energy E m ∝ T d/(d+1) depend on T . As a result, the conductance G is predicted to follow

G = G 0 exp - T 0 T γ (1)
where G 0 and T 0 are material dependent constants, and γ = 1/(d + 1) depends on the dimensionality d of the sample. Later, the percolation theory has been used to derive more rigorously Eq. ( 1) [3] [4]. In addition Efros and Shklovskii [START_REF] Shklovskii | Electronic properties of doped Semiconductors[END_REF] found that electron interactions result in a "soft" Coulomb gap at the Fermi level, leading to γ = 1/2. It is currently admitted now [6] that Mott and Efros-Shklovskii laws are limiting cases of VRH. Both of them were observed experimentally as their relevance depends on the relative magnitudes of the hopping energy and Coulomb gap.

A. Electric field effects in hopping transport

The case of very high fields F , eF ξ/(k B T ) ≫ 1 (e = electron charge) was studied through different theoretical approaches [7]- [12] which all predict that the current I should no longer depend on the temperature (activationless hopping) and behaves according to

I = I 0 exp - F 0 F γ′ (2)
where I 0 and F 0 are constants and γ′ = γ (see Eq. ( 1)). In spite of their similarity, Eqs. ( 1) and ( 2) correspond to completely different current path topologies. Eq. ( 2) is obtained by considering the shortest hops along which the electrostatic energy gain overcomes the energy fluctuations due to disorder [13], [7]. Thus in the very high field case, the hops occur in the direction of the field (directed percolation). On the contrary, for F → 0, the current flows through a network of random impedances along very sinuous paths whose returns and meanders are such that any hop more resistive than the VRH prediction is forbidden.

A quantitative description of these paths can be obtained by using the isotropic percolation theory [14] [15]. The intermediate field case eF ξ/(k B T ) < 1, where neither T nor F can be neglected is the most difficult one to handle theoretically, and the various models lead to different predictions that we summarize in the general equation giving the current

I(F, T ) = I 1 exp - T 0 T γ 1 -A (eF ξ) α (k B T ) α′ + B eF ξ k B T β (3)
where A, B, α, α′ and β are parameters, A and B being positive. The first term A(eF ξ) α /(k B T ) α ′ expresses the enhancement of the hopping probability when F grows. Depending on the authors, α and α′ range from α = α ′ = 1/(ν + 1) [10] (where ν ≃ 0.88 for d = 3 is the critical index of the correlation radius) to α = α ′ = 1 [9], [13], and even α = 2 and α ′ = 9/4 for d = 3 [8], [12]. The discrepancies between these results come from the fact that the net current I ij between two sites may increase exponentially with F or be insensitive to F , depending on the relative positions of the site energies. The second term B(eF ξ/(k B T )) β was predicted only by Böttger, Bryksin et al. [16]- [18] who addressed the additional problem of the "returns" , i.e. of the segments of the paths where the current flows against the electrical force. Their model states that the length of these returns decreases gradually when F increases. We show in this paper that our data evidence the existence of both terms of Eq. ( 3) with α = α ′ = 1 and β = 1.15 ± 0.10.

B. Theories of hopping transport at intermediate fields

We recall here some theoretical points which are important for the interpretation of our data. The very sinuous current paths at F → 0 can be pictured by using the "nodes-links-blobs" concept [14] [15], derived from the isotropic percolation theory : at small scales the current path consists of a fractal-like network of connected conductances up to the "blob" scale L p , and the blobs are the links of a homogeneous network. In the linear regime, the conductances inside a blob are much larger than the key conductances G key ∼ exp -(T 0 /T ) γ lying at the end of each blob. G key sets the overall sample conductivity since at scales larger than L p the system is homogeneous. For the intermediate field case, where neither T nor F can be neglected, three main effects have to be considered :

(i) for a site i belonging to the percolation path, the density of sites j yielding a non negligible conductance G ij is modified by F ;

(ii) a field-induced rearrangement of the charge along the current paths can take place : the local chemical potential µ i can fluctuate widely around the mean value given by eF r i ;

(iii) the lengths of the "returns", i.e. the segments where the current has to go against the electrical force, decrease when F increases.

Most authors disregarded (iii) and focused upon (i) and (ii), yielding predictions summarized by Eq. ( 3) with B ≡ 0. However, they gave diverging predictions depending on the relative influence given to (i) and (ii). For example, Pollak and Riess [9] find α = α ′ = 1 and A = 3/32 in a model where (i) plays the central role, and the correlations between nearest neighbors conductances are taken into account. They find that (ii) is quite negligible (δµ i ≤ k B T ). This contradicts Shklovskii model [10] (A ≃ 1 and α = α ′ = 1/(1 + ν) ≃ 0.53 for d = 3) which emphasizes (ii) (δµ i ≫ k B T ) by concentrating almost all the electrical potential drops existing at the blob scale L p upon the key conductance. The disagreement between these predictions was never completely explained [19] even if numerical simulations by Levin and Shklovskii [20] suggested that the Pollak and Riess result holds in usual experimental conditions (T 0 /T ) 1/(d+1) ≤ 70 while Shklovskii result would hold in the opposite case, i.e. at exceedingly low temperatures. Two other groups found different results : Apsley et al. [8] and Van der Meer [12] obtained α = 2, α′ = 9/4, but their methods were criticized since the former did not take into account the directed percolation requirement, while the latter uses dimensional "invariants" which were shown later to be F dependent [21].

Taking (iii) into account [16][21] [22] prevents the current from increasing too fast : the corresponding second term B(eF ξ/k B T ) β in Eq. (3) reduces the exponential increase of I due to -A(eF ξ) α /(k B T ) α′ . The physical reason is that the returns act as "bottlenecks" for the current. The problem being exceedingly complicated, the precise values of α, β, A, B were found to depend on the various methods used by Böttger, Bryksin et al. The last and more accurate results were obtained numerically [22][17] [18] : α = α′ = 1, A = 1/6 , B ≃ 0.02, 1/β ≃ 1.1 (this latter parameter is the critical exponent of the returns length Λ, see below Eq. ( 12)). The theoretical value of β is given within an error of 50% due to numerical uncertainties. Yet, the precise value of β is of major importance since for β < α, the differential conductivity σ = ∂I/∂V decreases at low fields to reach an absolute minimum at a finite F . On the contrary, in the case β > α, σ increases at low fields and has no absolute minimum at finite field.

The above mentioned theories disregard the possible F -dependence of the carrier density due to trapping in dead ends of paths, as well as the possible F -dependence of the localization length ξ. Fortunately, both effects should be negligible in our case. Indeed, statistically as many holes as electrons become trapped when F increases, as in our case the density of states is symmetric around the Fermi level (parabolic Coulomb gap) [START_REF] Talamantes | [END_REF]. The ξ(F ) dependence should be negligible [24] as long as E F ≫ eF ξ where E F is the kinetic energy at the Fermi level: this is the case in our sample [START_REF] We Have E F = E Mob -K B T | 1 (see Sec. III) with the mobility edge E Mob given by the Ioffe-Regel criterion (2mE Mob ) 1/2 λ el = where λ el is the elastic mean free path. Since our sample is amorphous, λ el ≃ 0.3 nm) yielding E Mob ≃ 0.35 eV ≃ E F . With ξ = 2 -6 nm (see Sec. V)[END_REF].

C. The experimental situation

On the experimental side, I-V non linearities were investigated both in amorphous materials [START_REF] Morgan | [END_REF]- [34] and in doped crystalline semiconductors [35]- [47]. In most of these works, the authors focussed either on very high fields or on intermediate fields. For intermediate fields, the data [28] [29] [31] [35]- [42] [45] were analyzed using the electric field dependence predicted by Hill [13], Pollak and Riess [9], and Shklovskii [10] ln

G(F, T ) = ln G(0, T ) + eF L k B T = ln (G 0 ) - T 0 T γ + eF L k B T (4)
where L is a length related to the hopping distance r m = (ξ/2)(T 0 /T ) γ . Hill [13] and Pollak and Riess [9] predict L = C.(ξ/2)(T 0 /T ) γ with C = 0.75 or 0.18, while Shklovskii predicts L ∝ T -(1+ν)γ [10] with ν = 0.88. Eq. ( 4) results from Eq. ( 3) by neglecting the returns term (B = 0) and by assuming that the F and T dependences of G and I are close to each other. The experimental values of L and their T dependence often disagree with the theoretical predictions [31] [36]- [38][40]. The claim by Wang et al. [44] that the non-linearities in Ge:Ga were due to carriers heating [64] rather than to the preferential hopping along the electric field which leads to Eqs. (3) or (4) triggered hot-electron analyses of the data [47]- [53]. They show that the relative contribution of carriers heating and electric field effects has to be considered at very low temperature. In very high electric fields, the expected activationless conduction (see Eq. ( 2)) has been observed by several authors [START_REF] Nair | [END_REF]- [32] [42] [43] [46]. The equality of the VRH and activationless exponents (γ and γ′ in Eqs. ( 1) and ( 2)) has been checked for γ = 1/2 in several systems [32] [43], but γ′ = 1/4 and γ = 1/2 has been found in amorphous GeCu [31].

In this work we systematically investigate the whole set of non linearities ranging from the linear VRH regime at F → 0, to the activationless conduction at very high fields : in our low temperature domain (0.03 K ≤ T ≤ 1.3 K), F/T varies from 1.5 × 10 2 V m -1 K -1 to 1.4 × 10 6 V m -1 K -1 . Our samples are made of amorphous Y 0.19 Si 0.81 . In comparison to most of the experimental works performed in this low temperature range ( < 1 K) [37] [43]- [47], we go to higher values of F and F/T , thus allowing an investigation of both the intermediate and the very high field regimes. At very low temperatures, this has been done by Rosenbaum et al. on Si:P [37], however the authors found an abnormally large value of L. At higher temperatures, these two regimes were investigated for a-GeCu at 1.6 K < T < 110 K [31] and ZnSe at 1.6 K < T < 4.2 K [42]. The first paper raised the question of unexpected exponents values (in the L vs. T and ln(G) vs. F dependences) while the second one found a puzzling decrease of L when F increases. Clearly, new data are needed. In this work, we test the validity of Eqs. ( 1)-( 4) and show that they are in good agreement with our data for intermediate fields. In particular we find L ≃ C.r m [13] [9]. We also check that the contribution of heating effects to the non-linearities is negligible. Our main result is that Eq. ( 3) with α = α′ is more suitable than Eq. ( 4) to fit the data : this allows us to estimate the size of the returns and the value of the related critical exponent β[16]- [18]. Finally, we show that in the activationless region, the I vs F dependence is more intricate than the law given by Eq. ( 2).

The paper is organized as follows. In section II we describe the experimental setup and method. The conductance as a function of the temperature in the region F → 0 is presented in section III. In section IV, we consider the very high field case. Having extracted the physics of the two extreme F regimes, we finally turn in section V to the intermediate field case. Finally, we summarize our main results in section VI. In appendix A we show that heating mechanisms cannot explain the I -V non-linearities, while appendix B is devoted to the extraction of the localization length from the data.

II. EXPERIMENT

As shown in Fig. 1, each amorphous Y x Si 1-x sample studied (x ≃ 0.19) is deposited on a sapphire substrate on which two interdigited gold electrodes were evaporated and etched. The spacing between the electrodes is l = 128 µm. The Y x Si 1-x layers were obtained by Argon plasma sputtering on the sapphire substrate (cooled at 77 K to prevent Y or Si aggregates formation), using a Y x Si 1-x source. The thickness of the layers is 9 µm, much larger than any hopping length, hence electrical transport is three dimensional. Previous studies of electrical transport in such Y x Si 1-x samples [54]- [59], have shown that a small variation of x allows to cross the metal-insulator transition : above x ≃ 0.22 the samples are metallic; while for lower values of x, the samples exhibit an insulating behavior, i.e. a divergence of the resistance for T → 0, in agreement with the VRH predictions [55]- [58], except for the very weakly insulating samples at the lowest temperatures [59]. In the present work x ≃ 0.19 and the samples are strongly insulating, exhibiting VRH below T ≃ 2 K. Since the two samples we used behaved similarly, we report only the data of one of them.

As shown in Fig. 1, a great attention was paid to the thermalization of the sample because of a possible heating interpretation of the I-V curves non-linearities. As explained in Appendix A, we measured the thermal conductances involved in our sample and concluded that heating effects were irrelevant in our sample: neither heating of the whole sample nor carrier heating can explain the I-V non linearities reported here.

Electrical contacts were made by ultrasonic soldering of two gold wires on the evaporated electrodes. The data were obtained by setting a given voltage V = F l on the sample, and by waiting for a time τ before measuring the current I. We paid the greatest attention to choosing τ large enough to let the current settle, e.g. we took τ (T < 800 mK) ≥ 100 s. The temperature was registered for each I(V ) point, and its stability proved to be better than 1%. The coaxial cables used were previously tested alone (in an open circuit, at all temperatures) to check that their leakage current was negligible. Let us note that it is unlikely that the gold-YSi electrical contact resistance plays a role since the contact area is very large and in previous similar works [58], it was shown to play no role. The fact that the temperature dependence G(0, T ) we observe follows a VRH law (see next section) is an additional indication that the contact resistance is negligible. Finally, the symmetry of the I-V curves with respect to current and voltage reversing was checked.

III. BEHAVIOR AT F → 0

The I-V characteristics are shown in Fig. 2 where, for clarity, we report only the curves for 18 different temperatures out of the 36 ones we measured. As shown in the inset, the resistance (R) -temperature law is simply activated from room temperature to ≃ 10 K : we find R = R 1 exp(-T 1 /T ) with T 1 = 40 ± 4 K and R 1 ≃ 88 ± 2 Ω. For T < 8 K the divergence of the resistance when T → 0 becomes weaker (see inset of Fig. 2), which is a standard indication that VRH takes place at low temperature [START_REF] Shklovskii | Electronic properties of doped Semiconductors[END_REF]. Let us note that simple activation for T ≥ 10 K is usually interpreted as due to thermal excitation from the Fermi level E F to the mobility edge E Mob where electronic states are delocalized; thus

k B T 1 = E Mob -E F .
To study the temperature dependence of the conductance at F → 0, we plotted our data in the (F/T , ln (V /I)) plane (see Fig. 3). The G(F → 0) values were obtained from an extrapolation of the curves towards low fields. To allow for a precise extrapolation, this procedure was restricted to the 0.4 K < T < 1.3 K cases. The extrapolated values were obtained from a linear fit of the first points in each curve (lower F/T values). We checked that those values did not depend significantly on the number of points selected in the fit. The expected VRH temperature-conductance law is given by Eq. ( 1) where γ = 1/2 or γ = 1/4. Discriminating between these exponents is hardly achieved by looking for a straight line in the ln(G(F → 0)) versus T -γ plot, as the lines always seem straight. The comparison between the different exponents γ is improved by plotting a normalized value of

T -γ , i.e. (T -γ -T -γ Min )/(T -γ Max -T -γ
Min ) where T Min and T Max are the two extreme values of T selected for the plot. In Fig. 4, we see that γ = 1/2 is favored in comparison with γ = 1/4. A fit of ln(R(F → 0)) vs. T using Eq. ( 1) in which G 0 and T 0 are free parameters yields a residue (measured minus fitted value) shown in the inset of Fig. 4 : clearly, there is a correlation between the residue and T only for γ = 1/4. The normalized χ 2 per point is 1.1 for γ = 1/2, and 9.0 for γ = 1/4. If γ is a parameter of the fit, we find γ = 0.47 ± .02. We thus conclude that γ = 1/2, and using this value in the fit, we get T 0 = 257 ± 1.5 K and R 0 = 1/G 0 = 92 ± 5 Ω. Following Efros and Shklovskii,

k B T 0 = 2.9e 2 4πǫ 0 ǫ r ξ (5)
where ǫ 0 ǫ r is the dielectric constant of the system. We shall see in Appendix B that the non-linearities analysis together with Eq. ( 5) allow to extract an information on ξ and ǫ r separately.

IV. RESULTS AT VERY HIGH FIELDS

In the very high fields case, eF ξ/(k B T ) ≫ 1, the theory [7] states that the current results from hops between sites i and j, such that E ij = eF r ij . For these activationless hops, I ij ∝ exp (-2r ij /ξ), which, using a maximization procedure leads to Eq. ( 2) with [7]

F 0 = N 2 k B T 0 eξ (6)
where N 2 is a numerical constant which can be calculated within the directed percolation theory. For Mott's VRH (γ = 1/4) Pollak and Riess obtain N 2 = 4.8. For the Efros-Shlovskii VRH (γ = 1/2), N 2 has not been calculated to our knowledge.

Our lowest temperature data should belong to the very high field domain, as the two I-V curves corresponding to T = 29 mK and T = 49 mK are identical. Indeed, we shall see that the low and intermediate field data analysis lead to ξ ≈ 7 nm (see Appendix B), hence eF ξ/(k B T ) ≥ 50 for T = 29 mK. The 29 mK data can be rather well fitted by using Eq. ( 2) with γ′ = 1/2, I 0 and F 0 being free parameters. It yields F 0 = (3.8 ± .05) × 10 7 V/m, hence 4 ≤ N 2 ≤ 11 by using the upper and lower values of ξ. The normalized χ 2 per point is however 290. Thus, in spite of the reasonable extracted F 0 and N 2 values, the relevance of the fit must be questioned. Indeed, if we fit the whole I(F, 29 mK) data using Eq. ( 2), γ′ being a free parameter, we get γ′ = 0.65 ± 0.01 and a χ 2 per point of 94.

To investigate the problem, we give in Fig. 5 the F dependence of the effective exponent γ ef f defined by

γ ef f = - ∂ ln ∂ ln I ∂ ln F ∂ ln F . ( 7
)
If I(F ) is given by Eq. ( 2), then γ ef f = γ′ . For T > 200 mK, γ ef f is always negative, while for T < 200 mK, γ ef f it is negative for the lower F values. This can be related to the fact that in the intermediate field region, the I(F ) dependence should obey Eq. ( 3) for which the expected γ ef f is F -dependent. For the lowest temperatures, γ ef f becomes positive when F increases, but there is no saturation at γ ef f = 0.5. Instead, a maximum is reached at a value between 0.5 and 1, even if a convergence of γ ef f towards 1/2 might exist at fields larger than 4 × 10 4 V/m. We suggest that these values of γ ef f between 0.5 and 1 are due to the fact that the very high field conduction results from an interplay between activationless hopping (for which γ ef f = 1/2) and tunneling across the mobility edge. Indeed, the latter process leads to the simple activated law [33], [START_REF] Sze | Physics of Semiconductor Devices[END_REF]

I ∼ exp -( 4 3eF ) 2m(k B T 1 ) 3 2 = exp - F 1 F (8)
where k B T 1 is the energy difference between the Fermi level and the mobility edge. Since T 1 ≃ 40 K (see Sec. III), F 1 = 1.4 × 10 6 V/m. The fit of the I(F, 29 mK) data for the interval F > 2.8 × 10 4 V/m where γ ef f > 0.5, using I = I 1 exp (-F 1 /F ), yields a normalized χ 2 per point of 40 and F 1 = (5.5±0.05) 10 5 V/m, which is only a factor 2.5 below the calculated value. We cannot interpret the γ ef f (F ) curves more precisely since, to our knowledge, there exists no theory taking into account both the disorder in the localized band states and the tunneling through the Mobility Edge.

It is interesting to note that if activationless hopping was the only transport mechanism in the very high field regime, the critical value (F/T ) c at which the transition from Eq. (3) to Eq. (2) occurs would be temperature independent : this can be readily seen by equating the rhs of Eqs. ( 2) and (3) at F/T = (F/T ) c and disregarding pre-exponential factors. If we define, for any given T the experimental critical field value F c by I(F c , T ) = k I(F c , 29 mK), with k = 1.1, we find that in our sample, (F/T ) c is clearly T-dependent: it increases by a factor 3.5 when the temperature decreases from 413 mK to 74 mK.

V. RESULTS AT INTERMEDIATE FIELDS

We can see on Fig. 3 that our experimental points line up in almost parallel curves in the (ln (R), F/T ) plane. Clearly, they are not really straight lines, hence Eq. ( 4) does not hold with precision. However, in the spirit of the majority of previous works, we first use Eq.( 4) and extract the length L(T ) defined by ln [G(F )/G(F → 0)] = eF L(T )/(k B T ) for F just above the linear regime. We show in a second step that Eq. ( 3) is much more relevant to account for our data.

A. Extraction of L(T ) using Eq. ( 4)

The slopes of the ln (R) vs. F/T curves for F → 0 were extracted for the different temperatures where it was possible, i.e. 0.4 K < T < 1.3 K (see Fig. 3). A linear fit yielded a normalized χ 2 per point close to 1 only for F/T < 5000 V m -1 K -1 . We restricted the L extraction to this interval where the curves are linear. The result is shown in Fig. 6. A fit of these points using the law L = L 0 (T 0 /T ) ζ where L 0 and ζ are free parameters gives ζ ≃ 0.65 with a normalized χ 2 per point of 0.7. Constraining the fit with ζ = 0.5 leads to L 0 = (0.52 ± 0.01) nm and a normalized χ 2 per point of 1.1 : the resulting curve is shown in Fig. 6. As a consequence, we can state that the Shklovskii prediction [10] ζ = 2γ = 1 does not hold for our system, while the predictions of Hill [13] and Pollak and Riess [9], L ∝ r m leading to ζ = γ = 1/2, are in agreement with our results. It is interesting to note that the values of L(T ) remain roughly the same if they are extracted from linear fits of the whole ln(R) vs. F/T curves, although these fits yield a normalized χ 2 per point varying between 3 and 250. The localization length ξ extracted from L 0 = C.(ξ/2) with the Pollak and Riess value C = 0.17 [9] yields ξ ≃ 6 nm. We shall come to this point more in detail later.

B. Extraction of the parameters of Eq. (3)

We turn to the main result of our study, namely the relevance of the Böttger et al. prediction [17] concerning the "returns" and summarized in Eq. (3).

B.1. F/T is the unique relevant parameter

We define, so as to test Eq. ( 3),

Ψ(F, T ) = T T 0 ln I(F, T ) I E.S. (F, T ) , with I E.S. (F, T ) = V R 0 exp [-( T 0 T ) 1/2 ] ( 9
)
where V = F l, R 0 = 92 Ω and T 0 = 257 K (see Sec. III). It is easy to show that if Eq. ( 3) holds in what concerns the G(T, F ) dependence (instead of

I(T, F )), Ψ is just equal to A(eF ξ) α /(k B T ) α′ -B(eF ξ/k B T ) β .
We have chosen to perform our analysis on G(T, F ) = I(T, F )/(F l) rather than on I(T, F ) because all the above depicted theories do not allow to recover a linear regime when F → 0 : in this case, Eq. ( 3) gives I(F → 0) = I 1 exp [-(T 0 /T ) γ ] which is not consistent with Eq. ( 1). Note that this difficulty has been ignored in almost all experimental works dealing with the intermediate fields case, as they focus upon G(F ) rather than on I(F ). Numerically, our choice is justified by the fact that the exponential I(F ) dependence in Eq. ( 3) is much faster than the contribution of the denominator of G(F ) ∼ I(F )/F (see also [START_REF]The choice of Ψ was also motivated by the fact that we first tried to analyze our I(F, T ) dependence by using Φ(F, T ) = (T /T 0 )[END_REF]).

To perform our analysis of the intermediate field case, we suppressed the data points corresponding to the very high field case (see Sec. IV). This was done by selecting 7164 I(F, T ) points among the whole set of 8189 points, on the criterion I(F, T )/I V HF (F ) -1 ≥ 10%, where I V HF (F ) is the very high field current defined by I V HF (F ) = I(F, T = 29 mK). The 10% criterion is rather arbitrary, but we checked that our results do not depend on its precise value, provided it remains above 5%. Then, Ψ is calculated for each of these 7164 points. As shown on Fig. 7,Ψ(F,T ) is a universal function of F/T over more than 3 orders of magnitude, while the 7164 experimental points correspond to 36 different temperatures ranging from 49 mK to 1.29 K.

The fact that Ψ(F, T ) depends only on F/T implies α = α ′ , which definitely shows that both Apsley [8] and Van der Meer [12] models are not relevant for our sample. Moreover, the log-log slope of Ψ(F/T ) at the lowest F/T values is very close to 1 and clearly larger than 1/(ν + 1) ≃ 0.53 which means that Shklovskii prediction [10] does not hold. This is not surprising since in our experiment (T 0 /T ) 1/2 ≤ 70 which is a domain where the numerical work of Levin and Shklovskii [20] leads to α = 1. However our data are poorly fitted by the Hill [13] and the Pollak and Riess [9] prediction Ψ ∼ F/T since the log-log slope of Ψ(F/T ) decreases as F/T increases. More generally, a fit of Ψ(F/T ) with a unique power law (F/T ) α ′′ gives a poor agreement whatever α ′′ , especially because of the points in the region F/T ≥ 3 × 10 4 V K -1 m -1 . The only remaining prediction is thus the one of Böttger et al. where α = 1 and β is unknown theoretically. In the next section this prediction will be assumed and the best value of β will be sought. Quite interestingly, the fit of Ψ we obtain in the next section is, by far, much better than all the "reasonably simple" fits we tried: for example attempting a polynomial fit with Σ m i=1 a i (F/T ) i gives a poor agreement even if m is as large as 5. B.2. Extraction of the critical exponent β assuming α = 1. We thus have to fit the Ψ(F/T ) points using the function

Γ F T -∆ F T β ( 10
)
with Γ, ∆, β as free parameters. Since the Ψ(F/T ) values extends over several orders of magnitude, we perform the fit on ln (Ψ), i.e. we request that a relative error E per point is minimal, with E defined by

E 2 = 1 7164s 2 j=1,7164 (1 -x) ln Ψ (F/T ) j Γ(F/T ) j -∆(F/T ) β j 2 +x ln Γ(F/T ) j -Ψ (F/T ) j ∆(F/T ) β j 2 , ( 11
)
where x is a weight to be chosen in the (0, 1) interval and s 2 = (ln (Ψ)-< ln (Ψ) >) 2 ≃ (0.08) 2 is the variance of ln(Ψ) drawn from the data. Note that the first term in the rhs of Eq. ( 11) is the relative error per point on Ψ while the second one is the relative error per point on the "Böttger second term" of Eq.

(3), ∆(F/T ) β . The minimization of those two terms has opposite effects on the optimal value of β. Indeed keeping only the first one (i.e. setting x = 0) leads to β = 1.38 ± 0.03 and yields a good agreement between the Ψ((F/T ) j ) points and the fitted curve, but leads to a discrepancy between Γ(F/T ) j -Ψ (F/T ) j and ∆(F/T ) β j which becomes clearly too large when F/T < 5 × 10 4 V K -1 m -1 . Conversely, setting x = 1 leads to β = 1.03 ± 0.03, with Γ(F/T ) j -Ψ (F/T ) j ≃ ∆(F/T ) β j , but a discrepancy appears between the Ψ (F/T ) j points and the fitted curve, especially at high F/T values. Keeping x in the (0.05, 0.95) interval leads to optimal values β(x) in the (1.05, 1.15) range, i.e. to an uncertainty on β not very much larger than the one obtained for a given x. We finally keep the larger error bar on β and find :

β = 1.15 ± 0.1, Γ = (1.25 ± 0.5) × 10 -5 K m/V and ∆ = 1.4 × 10 -6 (m K/V) β (∆ 1/β /Γ = 0.70 ± 0.07).
The solid line on Figs. 7 and8 is the fit resulting from β = 1.15; Γ = 1.25 × 10 -5 K m V -1 ; ∆ = 1.4 × 10 -6 (K m V -1 ) β : the agreement with the data is good (E(x = 0.5) = 0.72) and the only systematic deviation occurs at the very few highest F/T values where the transition to the very high fields regime occurs.

The comparison of Eqs. ( 3) and ( 10) gives Γ = Aeξ/k B . From Γ = 1.25 × 10 -5 m K/V we deduce Aξ = 1.1 nm. Unfortunately, the precise value of A is not known in our case of Efros Shklovskii's hopping (γ = 1/2). As shown in Appendix B, using T 0 = 257 K, Aξ = 1.1 nm as well as considerations about the dielectric constant ǫ r we conclude that ξ ≈ 7 nm. We cannot be more precise due to the various unknown numerical factors involved in the predictions we used, but we note that these values of ξ compare favorably to previous results obtained on Y x Si 1-x samples much closer to the metal-insulator transition [58], [59] where larger values of ξ were found (a few tens of nm).

We compare now the analysis performed just above with the one carried out using Eq. ( 4) which yielded L = L 0 (T 0 /T ) 1/2 with L 0 = 0.52 nm, and ξ ≃ 6 nm, using L 0 = C (ξ/2) and C = 0.18 (see Sec. B.). Neglecting the weak difference between the G(T, F ) and the I(T, F ) dependences, the comparison of Eqs. ( 3) and (4) leads to L 0 = Aξ if the additional term B(eF ξ/(k B T )) β in Eq. ( 3) is assumed not to change L 0 . Clearly, we have a discrepancy by a factor 2 between L 0 and Aξ = 1.1 nm : this was checked to come from the B(eF ξ/(k B T )) β term in Eq. (3). Surprisingly both analysis yield roughly the same ξ ≈ 6 -7 nm for A = 1/6 and C = 0.18.

B.3. Extraction of the return length Λ.

We now turn to the analysis of the "returns" size. As explained above, β and ∆ give an information about the typical length Λ of the returns which decreases gradually as F increases. Indeed, according to Böttger et al. [22], if ρ = ln [I(F, T )/I 1 ] (ρ is the argument of the exponential in Eq. ( 3)), Λ is given by

Λ(ρ) ≃ 2δr m ρ c ρ -ρ c 1 β ρ m -ρ ρ c µ (12)
where µ = 1.0 ± 0.3 and δ ≃ 0.25 are numerical constants [22], ρ m is the directed percolation threshold, and ρ c = (T 0 /T ) γ . In the linear regime it is found that ρ(F → 0) ≃ ρ c + 1 [14], yielding a finite value of Λ(F → 0). Increasing F up to the intermediate field region leads to a ρ increase, thus to a Λ decrease. Increasing F further, ρ reaches ρ m and Λ vanishes indicating that the current paths are directed. The maximum length Λ max of the returns is thus obtained just at the onset of the non linear regime, where we can put in Eq. ( 12) ρ = (T 0 /T ) γ + 1 and (ρ m -ρ)/ρ c ≃ (ρ c -ρ m )/ρ c = N 4 with N 4 ≃ 0.07 as predicted in [22]. Hence we get

Λ max = ξδN µ 4 T 0 T γ(1+ 1 β ) = ∆ 1 β k B e T 0 T γ(1+ 1 β ) (13)
where the second equality is obtained by using the Böttger et al. prediction ∆ = (N µ 4 δeξ/k B ) β [22]. Using Eq. ( 15), we can compare Λ max to the blob length L p = ξ(T 0 /T ) γ(1+ν) [14] :

Λ max L p = ∆ 1 β k B eξ T 0 T γ( 1 β -ν) ≃ ∆ 1 β k B eξ (14)
where the second equality comes from the fact that in our case, due to the experimental value of β, Λ max and L p have critical exponents very close to each other since 1/βν is in the (-0.08, 0.07) interval. We thus find that Λ max (T )/L p (T ) does not depend on the temperature. By inserting k B /(eξ) = A/Γ in Eq. ( 15), we find Λ max (T )/L p (T ) = A∆ 1 β /Γ = (0.70 ± 0.07)A , using our fit result ∆ 1 β /Γ = 0.70 ± 0.07. Hence, Λ max (T )/L p (T ) is not much smaller than 1 which is the highest possible theoretical value. We thus conclude that in our sample the importance of the returns is strong.

This importance of the returns must also play a role on the onset of the non linear behavior which occurs at a field F lim (T ). Indeed, according to Böttger et al. F lim /T = k B /eΛ max [22]. According to Pollak et al., F lim /T = k B /er m with r m = (ξ/2)(T 0 /T ) γ the hopping length. In our case, this latter prediction amounts to F lim /T ≥ 1000 V K m -1 for T ≥ 500 mK (for lower temperatures we cannot go to F → 0 because of too low currents). We clearly see on Figs. 7 and 8 that the non linear fit obtained above extends down to F/T ≃ 150 V K -1 m -1 . This is one order of magnitude smaller than the Pollak et al. prediction and in quite good agreement with the Böttger et al. result. We thus conclude that the F lim behavior we find is consistent with the above derived results on the importance of returns.

VI. CONCLUSION

Our study of the electric field effects in variable-range hopping transport for amorphous Y 0.19 Si 0.81 below 2 K exhibits several important features. First, we find that the length L characterizing the intermediate field regime has the order of magnitude and T dependence (≃ T -γ , with γ = 1/2) which is expected in the VRH models of Hill [13] or Pollak and Riess [9] stating that L ≃ r m . Even analyzing our data in the framework of the Böttger, Bryksin et al. predictions (Eq. ( 3)) this result remains true, as the addition of the B(eF ξ/(k B T )) β term in Eq. (3) does not change the order of magnitude of this characteristic length. Second, our most important result is that Eq. ( 3) is much more relevant than Eq. ( 4) to analyze our data, and this shows the importance of the "returns" in the percolation paths of VRH. Furthermore, we were able to extract an information on the length of these returns from our experimental results. Indeed, our data indicate that the critical exponent of the returns contribution 1/β is very close to the one of the blob length : the returns represent an appreciable T -independent fraction of the whole percolation paths lengths at intermediate fields. Third, our very high field data do not follow the expected activationless law given by Eq. ( 2). This could be due to the onset of tunneling across the mobility edge whose interplay with activationless hopping has not been theoretically studied yet.

where g and η are parameters characterizing the thermal conductance between the electron bath and the cryostat, and P p is the parasitic power injected in the sample due to the limited rf shielding etc. As we have excluded a possible heating effect of the sapphire substrate (see previous section), the thermal resistance to consider is either the Kapitza one at the boundary between the YSi sample and the sapphire, or the thermal resistance due to electron-phonon coupling in the YSi itself. In the first case η = 4 [63], while in the second one, η = 5 to 6 [44][47]- [53] [64]. We investigated the experimental values of dP/dT e , which would be equal to gηT η-1 e , hence would not depend on T and P p . Within the heating model, T e can be extracted for each (I,V) point using Eq. (A1). Fig. 10 shows dP/dT e as a function of T e . The fact that most of the curves depend on T is a strong argument against the heating model. However, we can see a trend towards a T independence for low T values. As indicated by the straight lines corresponding to power laws, these low T curves are not compatible with the heating model because they correspond to η values which depend on T e and may be much larger than 4 to 6. Finally, we note that in materials close to a-YSi, the electron-phonon coupling parameter g was found experimentally to be of the order of 10 3 W K -6 cm -3 (in NbSi, with η = 6) [52] or 200 W K -5 cm -3 (in AuGe, with η = 5) [64]. As a consequence, for our 7.2 × 10 -5 cm 3 sample, we expect dP/dT e ≃ 0.4 T 5 e W/K or dP/dT e ≃ 0.07 T 4 e W/K which is several orders of magnitude larger than what we find (see Fig. 10). If we assume that the possible heating effect is due to the YSi-sapphire Kapitza resistance, we can use the experimental g ≃ (1 -10) × 10 -3 W K -3 cm -2 values for sapphire-metal interfaces [START_REF] Yvon | [END_REF], which lead to dP/dT e values of the order of 10 -3 T 3 e W/K, again several orders of magnitude larger than our experimental values. Yet, those very large discrepancies guarantee that the heating effects are negligible. In comparison to data from authors who see heating effects, this can be explained by : i) our very resistive sample, ii) the low electron-phonon and Kapitza resistances due to the geometry of the sample.

FIG. 1 .

 1 FIG. 1. Experimental setup: the 9 µm thick Y 0.19 Si 0.81 layer is deposited onto a sapphire substrate with two interdigited gold electrodes. Thermalization of the sample was ensured by 43 gold wires (4 represented) bonded on the left gold pad. Upper right corner : small 1 MΩ chip heater allowing thermal resistance measurements at low temperature (see Appendix A). FIG. 2. Current as a function of electric field at various temperatures below 1.3 K : for clarity only 18 out of the 36 different curves were represented. Inset: R(F → 0) vs. 1/T (dots), for temperatures above 4 K. The dashed line corresponds to an activated law with a characteristic temperature of 40 K. FIG. 3. Logarithm of the resistance as a function the electric field to temperature ratio, for 14 temperatures among the 28 measured ones. FIG. 4. Normalized value of T -γ for 3 values of γ as a function of the logarithm of the resistance at zero electric field. The normalized T -γ is (T -γ -T -γ Min )/(T -γ Max -T -γ Min ) where T Min and T Max are the two extreme values of the temperature T selected for the plot. Inset : residue (i.e. measured minus fitted value) of the linear fit of ln[R(F → 0)] as a function of T -γ , for γ = 1/4 (open circles) and γ = 1/4 (closed circles). FIG. 5. Effective exponent of the current vs electric field law, as a function of the electric field, for 6 temperatures. FIG. 6. The length L extracted from the slopes of the ln(V /I) vs F/T lines at F → 0 using Eq. (4) (dots), and the fitted L = L 0 (T 0 /T ) 1/2 law (continuous line). FIG. 7. The Ψ(F, T ) function (given by (T /T 0 ) 1/2 ln(I/I E.S. ), with I E.S. = (V /R 0 ) exp[-(T 0 /T ) 1/2 ]) as a function of F/T for all the data except the very high field points. Ψ increases more slowly than F/T (the dashed line corresponds to Ψ ∝ F/T ) and its evolution can be fitted (solid line) using the Böttger et al. prediction Ψ = Γ(F/T ) -∆(F/T ) β with β = 1.15 ± 0.1, Γ = (1.25 ± 0.5) × 10 -5 K m V -1 and ∆ 1/β /Γ = 0.70 ± 0.07. Inset : same Ψ vs. F/T dependence with linear scales showing that the ∆(F/T ) β term plays a role at low F/T values; the dashed line corresponding to Ψ ∝ F/T . FIG. 8. F/T dependence of Γ(F/T ) -Ψ(F, T ) , with Γ = 1.25 × 10 -5 K m V -1 , for all the data points except the very high field ones. The dashed line corresponds to a ∆(F/T ) β dependence with β = 1, and shows that β > 1. Solid line : fit with β = 1.15 and ∆ = 1.4 × 10 -6 (Km/V) β . Inset : same data with linear scales, the dashed line corresponding to a ∼ F/T dependence. FIG. 9. Thermal resistance R th (T ) measured by dissipating a controlled power in the small 1 MΩ chip (see Fig. 1). Dashed line : fit of the data with a T -3 law. FIG. 10. Derivative of the power dissipated in the YSi sample with respect to the effective electron temperature T e , as a function of T e . For each (I, V ) point, T e is calculated using Eq. (A1). The points line up in curves corresponding to different cryostat temperatures among which 5 are indicated in the figure. For comparison, the three straight lines correspond to 3 power laws dP/dT e ∝ T (γ-1) e with γ -1 = 3, 5 and 9.

  Figure 3

  Figure 4

Figure 6 1Figure 10

 610 Figure 6

ACKNOWLEDGEMENTS

We want to thank J.P. Bouchaud, J.L. Pichard and M. Sanquer for fruitful discussions. Many thanks also to Y. Imry for explaining us Ref. [62].

APPENDIX A: IRRELEVANCE OF THE SAMPLE OR CARRIERS HEATING MECHANISM

In this Appendix, we first describe the experimental setup allowing thermal conductances measurements and then show that heating is irrelevant to account for our I-V non linearities.

The sapphire substrate was hold between four sapphire balls which represent a negligible thermal conductance. The thermalization was realised by ultrasonically bonding 43 gold wires (50 µm in diameter, a few millimeters long) between the cryostat ("Heat sink" on Fig. 1) and a 7 mm 2 gold pad schematized on the left of the substrate on Fig. 1. The resulting calculated thermal resistance between the gold pad and the cryostat is smaller than 10 4 K/W at T = 50 mK and 10 3 K/W at T = 500 mK, leading to a temperature difference lower than 0.5 mK between those two points in our experiments. To measure the thermal resistance between the cryostat and the sapphire substrate or the sample, we used the 1 MΩ heating resistor chip shown on the right up corner of Fig. 1. This 0.01 mm 3 device is held only by two Al wires (25 µm in diameter, 5 mm long) which are superconducting below ≃ 1 K. At low temperature, their thermal conductance is thus negligible and the electrical power dissipated in the resistor flows to the substrate through the unique gold wire (25 µm in diameter, 3 mm long) ultrasonically bonded between the resistor and the gold pad evaporated on the sapphire substrate (on the right in Fig. 1). By measuring the temperature of the sample when a given power is injected in the resistor, this system gives an overestimation of the thermal resistance between the sapphire substrate and the cryostat. It has the advantage of being reversible in comparison with a resistor deposited on the sapphire substrate, as the gold wire can be easily removed.

We show now that heating phenomena are irrelevant to account for the I-V non linearities. We first focus on the possible heating of the whole sample together with its sapphire substrate, and then turn to "electronic heating".

A. Heating of the whole sample and substrate

The measurement of the thermal resistance R th between the cryostat on one hand and the sample with the sapphire substrate on the other hand was carried out as follows. A constant voltage V = F l was applied to the Y x Si 1-x sample, leading to a current I(F, T ) depending on the cryostat temperature T . A given electrical power 1 nW ≤ P ≤ 100 nW was then dissipated in the small 1 MΩ resistor. This power flowed to the heat sink through the substrate whose temperature was then increased by δT , leading to an increase δI of the current. We checked that δI was proportional to P. Assuming for a while that the I-V non linearities do not result from sample heating, we extracted, from the data of Fig. 2, δT from δI and deduced R th = δT /P. Fig. 9 gives the resulting R th as a function of T . From it, we get R th (74 mK) ≃ 10 mK/nW, while we can see on Fig. 2 that the I(F, 74 mK) and I(F, 124 mK) curves begin to merge for IV ≃ 0.04 nW which corresponds to δT ≃ 0.4 mK much lower than the 50 mK separating these two I(F ) characteristics. This confirms the above assumption of irrelevance of sample heating; and it can be easily shown to be true at any temperature. Let us note that R th exhibits a R th ∼ T -3 behavior which characterizes the Kapitza thermal resistance at the boundary between two materials. In our case, they are the sapphire substrate and the gold pad thermally connected to the cryostat.

B. Electronic heating

In this section, we show that the non-linearities of our I-V curves cannot be ascribed to a heating effect as found by Wang et al. for NTD sensors [44]. Assuming the validity of such a model for our sample, the power P = IV = IF l injected in the YSi "electron bath" would increase its temperature to a value T e larger than the cryostat temperature T , leading to a YSi electrical resistance

with T e given by

APPENDIX B: ESTIMATION OF ξ AND ǫ r FROM T 0 AND Γ.

The detailed calculations of A, B, α, α′ and β were performed by Böttger et al. assuming Mott VRH (γ = 1/(d + 1)) while we have γ = 1/2. Thus our fit result Aξ = 1.1 nm cannot be used straightforwardly to get ξ. In this appendix, we first try to estimate in which interval A must lie in the case of γ = 1/2 and then we use the value of T 0 as an additional experimental constraint on ξ.

It appears that the exponents α, α ′ and β should remain unchanged when going from γ = 1/(d + 1) to γ = 1/2. This can be readily seen for α = α ′ = 1 in the framework of Pollak's calculations [9]. Moreover, the fact that α = 1 for γ = 1/2 is confirmed by many analyses of experimental results using Eq. ( 4) [START_REF] Elliott | American Institute of Physics Conf. Proc[END_REF] [31] [35] [38] [40] [41] [42] [45]; while, in what concerns α′ the experimental situation is unclear (see Sec. I). However, the prefactors A and B are likely to be changed by the presence of a Coulomb gap. In this case, we can state at least 1). Such a substitution obviously overestimates I in the presence of an electric field since it neglects both the influence of returns and the insensitivity of a large number of pair currents with respect to F . From A ≤ 1/2 we deduce ξ ≥ 2.2 nm. Using the Böttger, Bryksin et al. value [22] [17] [18] A = 1/6, we find ξ = 6.6 nm which, using Eq. ( 5) gives ǫ r = 29, while ξ ≥ 2.2 nm leads to ǫ r ≤ 86.

According to Imry et al. [START_REF] Imry | [END_REF], one expects ǫ r = N 1 (ξ/λ T.F. ) 2 where λ T.F. is the Thomas Fermi screening length and N 1 = 1 according to Abrahams and Lee [65]. Thus, using Eq. ( 5), λ T.F. ≃ 0.08A -3/2 nm which leads to λ T.F. = 1.2 nm for A = 1/6 and λ T.F. ≥ 0.23 nm for A ≤ 1/2. In crystalline metals with a concentration n of one electron per atom we have typically λ T.F. ≃ 0.06 nm. Here we expect n to be of the order of 0.1 and since the standard screening theory yields λ T.F. ∝ n -1/6 , the value of 0.23 nm for our sample is plausible. However, we cannot exclude that the upper value of 1.2 nm is plausible too. Indeed, it was theoretically found [66] that close to the transition, electronic diffusion is considerably lowered, which should reflect in a lowering of screening, i.e. in an (unknown) increase of λ T.F. . Moreover, if as recently suggested [67], many body effects come into play in Efros-Shklovskii VRH, one expect both a reduction of the 2.9 factor in Eq. ( 5) and a change of the A value. Finally, considering all these unknown effects, Aξ is found to have the correct order of magnitude and we estimate that ξ ≈ 7 nm.

The determination of the dielectric constant ǫ r of the system is a very interesting working direction for the future due to the fundamental interest of ǫ r in localization and MIT studies. This is also a strong argument in favor of a precise theoretical determination of the numerical parameters in Eq. (3). Even if those parameters are not completely known, the relative evolutions of ǫ r as a function of the dopant concentration, magnetic field, etc. should be reachable with the ǫ r extraction method we used.