Marcin Konczykowski

Cornelis Jacominus Van Der Beek

Makariy A Tanatar

Huiqian Luo

Zhaosheng Wang

Bing Shen

Haihu Wen

Ruslan Prozorov

M Kończykowski

C J Van Der Beek

Hai Hu

Vortex creep and critical current densities in

Keywords: numbers: 74.25.Ha, 74.25.Op, 74.25.Wx

I. INTRODUCTION

Recently, the measurement of the critical current density j c of superconducting iron-based compounds has been recognized as a useful tool for the characterization of microscopic 1,2 and nanoscale 1,3 disorder in these materials. However, the study of vortex pinning and the critical current density is compromised by surprisingly large thermally activated flux creep [4][5][6] in the Bean critical state. 7,8 In some materials, such as single crystalline (Ba,K)Fe 2 As 2 , the logarithmic creep rate S ≡ -d ln j/d ln t of the sustainable current density j approaches that previously measured in some of the high temperature cuprate superconductors. [9][10][11] Sizeable creep rates influence the magnitude, and, potentially, the temperature and flux density-dependence j(T, B), thereby compromising the analysis of fundamental vortex pinning mechanisms in the iron-based superconductors 1,2 and the understanding of possible phase transitions of the vortex ensemble. 2,12,13 A detailed analysis of flux creep in the iron-based superconductors is therefore justified. In the cuprates, such analysis has unveiled the non-logarithmic nature of vortex creep, a direct consequence of the non-linearity of the relevant potential barrier U (j) opposing thermally activation as function of the driving force Bj. In turn, this nonlinear behavior arises from the elasticity of the vortex ensemble, i.e., the fact that this can be deformed continuously on very different length scales. In contrast to singleparticle creep, 14 the relevant activation barrier does not depend algebraically on the driving force, but, rather, increases steeply at low driving forces because of the nonlinear increase of the size of the critical nucleus. [15][16][17] In general, the size of the critical nucleus increases as an inverse power-law in j, leading to the well-known relation [15][16][17]

U (j) = U c j c j µ . (1)
The value of the creep exponent µ depends on the dimensionality of the critical nucleus as well as that of the elastic manifold as a whole. 15 In the case of single crystalline iron-based superconductors to be considered here, the latter can be either a single one-dimensional (1D) vortex line, or the 3D vortex ensemble. The law (1) does not extrapolate to a zero activation barrier at large driving force; one therefore frequently resorts to the so-called interpolation formula, 16,17 U (j) = U c j c j µ -1 .

(2)

Among the various methods to experimentally establish the U (j)-dependence, 10,[18][19][20][21][22] the analysis of the logarithmic time dependence of 1/S is the most reliable: 21,22 for Eq. ( 2), one has 1/S ∝ µ ln[(t 0 + t)/τ ], with t 0 a constant determined by transients at the onset of relaxation, and τ = (Λj c /E 0 ) (k B T /U c ) a normalization time determined by the barrier magnitude, a factor E 0 (with dimension of electric field) related to the details of the creep mechanism, and a sample "inductance" Λ = µ 0 a 2 (a is a relevant sample dimension). 23,24 An alternative "dynamic" method based on the dependence of the sustainable current density on the sweep-rate Ḣa of the applied magnetic field in magnetic hysteresis loop measurements was used in Refs. 5,6,20, and 25-27. In principle, the product (T /Q)d ln j/dT , with Q ≡ d ln j/d ln Ḣa , directly yields µ. 5,6,[25][26][27][28] The potential of local magnetic measurements of the magnetic induction B was exploited by Abulafia et al., 29 who reconstructed the current-voltage characteristics of YBa 2 Cu 3 O 7-δ single crystals from Ampère's law j ∼ µ -1 0 dB/dx for the current density and Faraday's law

E(x) = - x 0 (dB/dt)dx (3)
for the electric field (integrating from the sample centre to its perimeter).

As opposed to the cuprates, the dynamic range over which creep data can be collected in the ironbased superconductors is insufficient to reliably determine S(ln t), Q(ln Ḣa ), or the curvature of E(j) = E 0 exp(-U (j)/k B T ). Moreover, the factor d ln j/dT ∝ [1-(k B T /U c )(dU c /dT )] used in the dynamic method admixes the temperature dependence U c (T ) with the creep exponent µ. As a result, it is difficult to distinguish between a logarithmic U ∼ U c ln(j c /j) with a temperaturedependent U c , and a so-called "negative µ"-type barrier, 27,28 One should therefore resort to other constructions, such as that proposed by Maley et al., 5,13,19,23 which combine the results of relaxation measurements at different T .

U = U c [1 -(j/j c ) |µ| ].
Below we show that even though this method is fraught with shortcomings, it can be suitably adapted to yield reliable results. In particular, a plot of the average -k B T d ln(|dj/dt|)/dj vs. the average j unambiguously yields the curvature of the U (j)-relation and therefore the flux creep mechanism. Applying this to single crystalline (Ba,K)Fe 2 As 2 , one finds, first of all, that the creep rate is insufficient to qualitatively modify the field dependence j(B) as obtained from magnetic hysteresis. Second, the low-temperature sustainable current density j is, typically, less than a factor of 2 lower than the value 0 0.5 the critical current density j c would have in the absence of creep. Finally, the obtained µ-values indicate that the same creep mechanism is relevant for all temperatures up to the transformation of the vortex ensemble at the so-called "second magnetization peak". 12 We discuss the µ-value and its increase as function of magnetic field in terms of the interplay of strong pinning by nm-scale heterogeneities and weak collective pinning by atomic scale point defects in the material.

II. EXPERIMENTAL DETAILS

The Ba 0.6 K 0.4 Fe 2 As 2 single crystals with T c ∼ 38.1 K (Fig. 1) were grown by the self-flux method, using FeAs as the self-flux. Details of the growth can be found in Refs. 32,33. For the experiment, crystals were cut to regular rectangles of dimensions 350 × 160 × 70 µm 3 using a wire saw. The spatial distribution of the local induction perpendicular to the surface of the crystals in the critical state was measured using an array of microscopic Hall sensors, fashioned in a pseudomor- for Ha c at T = 10 K. The sweep-rate of the applied magnetic field was 0.8 mT/s for the virgin magnetization loop (up to µ0Ha = 0.5 T), while further measurements were carried out with a sweep-rate of 4 mT/s. The four measurement sequences ( 2, 3, , •) illustrate the decay of the flux gradient at different applied fields, over a period spanning 5 to 5000 s after field-cooling, and reduction of the external field Ha to the measurement field Ha -∆Ha (with ∆Ha = 0.2 T). (b). Time dependence of the local induction gradient for an applied field of 0.6 T and T = 8 K.

phic GaAlAs/GaAs heterostructure using ion implantation. The 10 Hall sensors of the array, spaced by 20 µm, had an active area of 3 × 3 µm 2 , while an 11 th sensor was used for the measurement of the applied field. For measurements with field parallel to ab, an array of sensors spaced by 10 µm was used. The Hall sensor array was placed on the center of the crystal surface, perpendicular to the long crystal edge, and spanning the crystal boundary. In this manner, hysteretic loops of the spatially resolved local induction were measured as function of the applied magnetic field H a , and as function of temperature. The sweep-rate of the applied magnetic field was 40 G/s (µ 0 Ḣa = 4 mT/s). In all experiments, the applied magnetic field was not only much smaller than the upper critical field H c2 , but also smaller than the field at which the second magnetization peak occurs. 12 Under all circumstances, the profiles of the flux density B were well described by the critical state model, 7,8 see Fig. 3, allowing for the straightforward extraction of the local screening current density j ≈ (2/µ 0 )dB/dx. The factor 2 takes the finite dimensions of the crystal into account; namely, the field gradient on the end surface of a semi-infinite bar in the critical state is half the field gradient in the interior. 30 Measurements were done with the Hall array parallel to the ab-plane and H a c, a configuration that measures the usual screening current density in the ab-plane, j = j c ab , and with the array c for H a ab, which yields the ab-plane screening current density j ab ab corresponding to vortices moving parallel to the c-axis (perpendicular to the FeAs layers).

Measurements of the ac screening were performed with the same set-up. A sinusoidally time-varying field of magnitude 1 Oe and frequency f is applied colinearly with the dc field. The ac component B ac (f, T ) of the local induction is then measured using the Hall probe array. Results, presented as the in-phase fundamental ac transmittivity 31 are shown in Figs. 1a and1b respectively. The non-zero value of the latter signals the existence of a non-zero critical current density j c ; the vanishing of B ac (3f ) at high-temperature is used to trace the dc irreversibility field H irr (T ), for dc field aligned along the c-axis and the ab-plane respectively (Fig. 1c).

T H = [B ac (f, T )-B ac (f, T T c )]/[B ac (f, T T c ) -B ac (f, T T c )] and the third harmonic |T 3H | = B ac (3f, T )/[B ac (f, T T c ) -B ac (f, T T c )],
As for the magnetic relaxation experiments, these were carried out on the decreasing field branch magnetization branch only (corresponding to flux exit relaxations), in order to prevent possible influence of surface barrier relaxation. The external magnetic field was applied at a temperature T > T c , at which the Hall probe array was calibrated (with respect to the applied field). The crystal was subsequently field-cooled to the measurement temperature T e < T c , and the field reduced by an amount ∆H a to the measurement field. Care was taken that ∆H a exceeded the the field of full flux penetration, so that a full critical state is established. After waiting 5 s to allow for the settling of the magnet, the flux density values at the different Hall sensor positions were measured as function of time, for a period of 5000 s. Furthermore, flux creep annealing experiments 19 were performed by heating the sample to T a > T e , returning to the experimental temperature T e , and re-measuring S over a period of 600 s. This procedure is equivalent to performing measurements at the effective time t = τ exp[U (j(T a ))/k B T e ], of the order 10 6 s.

III. RESULTS

Fig. 2(a) shows hysteresis loops of the flux density gradient dB/dx as function of the local induction B for H a c, at T = 10 K. The width of the loop is proportional to the screening current density j c ab in the ab-plane, for field c. It has the characteristic shape found in all charge-doped iron-based superconductors: 1 a "central peak" of j around B = 0, followed by a drop

j ∝ B -1/2 characteristic
-T ln ( E/E 0 ) ~ j -0.8 -k B T e ln E ( meV ) dB/dx ( G/µm ) j ( Am -2 )
-k B T e ln E / E 0 (T) ( meV )

FIG. 4: (Color online) Treatment of creep data for µ0Ha = 0.2 T c, using the method of Ref. 29. The data at the bottom of the frame show the j-dependence of the product of Te and the logarithm of the electric field, such as extracted using Eq. ( 3), for different experimental temperatures Te (lefthand axis). The master curve at the top (righthand axis) was compiled from these data using a temperature-and current density-independent E0; however, as in Ref. 29, E0 was assumed to be proportional to B; this yields a µ-value of 0.8.

nm-scale point defects 2,34,35 or heterogeneities. As B increases, the strong pinning-contribution to the critical current becomes irrelevant, and j(B) saturates to a field-independent value determined by weak collective pinning of individual vortices, 36 presumably by the K dopant atoms in the material. 1 The magnitude of j c ab is very similar to that found in other measurements on the same material. 1,38,39 At all magnetic fields, the sustainable current relaxes as function of time, with S ∼ -0.06 [Fig. 2(b)]. Creep is logarithmic in time, with curvature indicative of a nonlinear U (j) relation. However, contrary to the cuprate superconductors, and as illustrated by the near-linear evolution of the experimentally determined flux-creep activation barrier in the inset to Fig. 3, the accessed dynamical range of S(j(t)) is too small in the iron-based superconductors to reliably extract a value of µ directly. The effect of flux creep-annealing is also illustrated in Fig. 3, which shows the relaxation of the trapped fluxprofile at T = 8 K together with profiles obtained after annealing at various temperatures.

The time-dependent current density is analyzed using the method of Maley et al.. 19 This yields the experimental current density-dependent activation barrier as

U e (j) = -k B T e ln dj dt + cT e , (4)
with c a temperature-independent constant, to be chosen so that segments corresponding to the U e (j) relation measured at different temperatures line up to form the U (j) relation representative of the vortex creep mechanism governing the magnetic relaxation. For the collective creep mechanism of Ref. 15,

j = j c {(k B T /U c ) ln [(t 0 + t) /τ ]} 1/µ , (5)
so that the method gives 23 -k B T e ln dj dt = U (j) -k B T e ln k B T e U (j)

j µτ ; (6)
hence, c ≡ k B ln [k B T e j/U (j)µτ ] actually depends logarithmically on temperature. For a putative logarithmic dependence U = U c ln(j c /j), 40,41

yielding j = j c [(t 0 + t) /τ ] -k B T /Uc , one has 23 -k B T e ln dj dt = U (j) -k B T e ln k B T e U c j c τ . (7)
The following shortcomings of the method of Ref. 19 are identified. While, at low temperature, only a single c-value will satisfy the requirement of lining up the measured segments, at higher T the temperature dependence of c becomes important. Second, the prefactor U c (T ) ≡ U c (0)f (T ), as introduced in Eqs. ( 1) and ( 2), itself introduces a more important temperature dependence. This can be recovered by dividing the result of 9), for all temperatures, and the two orientations ( c and ab) of the applied magnetic field.

Eq. ( 4) at each T e by phenomenological factors f (T e ), with, again, a degree of arbitrariness. Correcting for the temperature dependences of c and U c in different ways results in different final results for the compiled U (j)-curve.

Finally, the method supposes that the same U (j) mechanism governs flux creep at all the temperatures used to reconstitute the experimental U (j)-curve. In the present set of experiments, we find that applying different procedures to cope with these different temperature dependences yield an error bar on µ that amounts to 50 to 100 % of its value.

Because it involves the integral (3) rather than the local value dj/dt, the method of Ref. 29 has the merit of yielding more accurate U (j)-values. In our experiments, we determine the electric field by integrating to the sample boundary; in accordance, the relevant value of the current density is that which corresponds to the slope dB/dx at the boundary. Fig. 4 shows -k B T ln E thus determined, for various temperatures, and µ 0 H a = 0.2 T c. However, if one wishes to extract flux-creep activation energies, one is faced with the same arbitrariness concerning the factor c in Ref. 19, now contained by the factor E 0 . Compiling a full U (j)-curve using a temperature-and current-density independent E 0 yields the illustrated master curve, suggesting that U (j) ∝ j -µ with µ = 0.8. However, if one takes E 0 to be proportional to Bj, as in Ref. 29, one obtains a curve that fits U (j) ∝ j -0.25 .

To avoid ambiguities, we evaluate the averaged current density-derivative -k B T e d ln |E|/dj at each T e . This procedure has the advantage of eliminating the prefactor c (or, equivalently, E 0 ) from the analysis. For Eq. ( 1 one has

k B T d ln |E| dj = - (µ + 1)k B T e j - dU (j) dj (8) ≈ µ U c j c j c j 1+µ , (U c k B T )
while for power-law creep 40,41

k B T d ln |E| dj = U c j . (U c k B T ) (9)
A double-logarithmic plot (Fig. 5) of the average -k B T e d ln |E|/dj vs. j for all different temperatures T e therefore directly reveals the creep mechanism, as well as the value of µ, from the deviation of the linear slope from 1. Whether the condition U c k B T is verified can be easily checked from the Bean (straight-line)-like nature of the flux profiles in Fig. 3. 37 From the slopes in Fig. 5, we find that Eqs. ( 1) and (2) describe the data satisfactorily. The corresponding µ-values are rendered in Fig. 6. For H a c, µ = µ c ≈ 0.65 at low fields, slowly increasing towards µ c ≈ 0.8 at higher fields, while for H ab ab, µ ab ≈ 1.5 at low fields, decreasing towards µ ab ≈ 1 at higher fields. The results for H a c are remarkably similar to those found in Ba(Fe 0.92 Co 0.08 ) 2 As 2 by Shen et al.. 6 The obtained µ-values are checked by plotting -k B T e ln |E| vs. j -µ for all temperatures and fields. Fig. 7 shows that this yields straight lines for both field orientations, as required. Deviations from linearity are only apparent below 8 K, and above 24 K, which means that µ is T -independent for the greater part of the investigated temperature range. That is, the same mechanism governs vortex creep for all T < 24 K. The slopes in Fig. 7 yield U c j µ c , while the intercept with the abscissa corresponds to j -µ e , where

j e = j c {U c /k B T e ln [k B T e j/U (j)µτ ]} 1/µ .
With the uncertainty regarding µ removed, the curves obtained for different T e can now be compiled into a "zero-temperature" curve by adding factors k B T e ln E 0 , and dividing by f (T e ). For the lowest two temperatures, we set f (T e ) = 1 to obtain an unequivocal E 0 = 0.019. For higher T e , the value of f (T e ) is adapted in order to line up the relevant data segments to obtain a continuous compiled curve, with continuous derivative. The result is given in Fig. 9, which shows the extracted activation barrier U (j) as function of j -µ (top panel) and as function of j (bottom panel), for various magnetic fields c. The extracted temperature dependence U c j µ c for H a c is depicted in Fig. 8.

H a = 0 0.2 T 0.4T 0.6 T j ( Am -2 ) U c ( T )/k B [ j c ( T )/j c (0) ] µ ( K ) T ( K )

IV. DISCUSSION

Inspection of the creep barrier in Fig. 9 shows deviations from power-law behavior at low temperature < 8 K, and above 24 K. In all cases, the dependence U (j) at the lowest temperature appears to be linear in the current density j. This behavior is interpreted as being due to the proximity of the measured screening current density to the (pinning) critical current density. A linear extrapolation of the U (j) curves to zero (dotted lines in Fig. 9) thus provides an estimate of the critical current density j c (0) in the limit T → 0. The obtained values range between j c (0; 0.6 T) = 6.7×10 9 Am -2 to j c (0; H a = 0) = 9.7×10 9 Am -2 for field c, and j c (0; 0.5 T) ≈ 2.3×10 10 Am -2 for H a ab (Fig. 8). These values can be factored out from the slopes of the curves in Fig. 9 creasing values of field c, see Fig. 8, and U c (0)/k B = 24 K for H a ab. The low-temperature values of the measured screening current density j closely approach j c (0). These values therefore yield a good order-of-magnitude estimate of the bulk pinning force.

The Inset, Fig. 9(c), compares the high-and low current limiting behavior of the current density-dependent creep barrier for µ 0 H a = 0.2 T c to the often-used interpolation formula, Eq. ( 2). It is clear that the current density range over which the experimental barrier crosses over from the high-current, linear-in-j, to the low current-behavior given by Eq. ( 1) is much narrower than what can be described using Eq. ( 2). In fact, the dashed-dotted curve shows that applying the interpolation formula (2) yields a gross overestimate of the pinning critical current density, as well as a possible overestimate of µ. It therefore seems imperative to use Eq. ( 1), which was derived on physical grounds, 15 rather than the phenomenological formula (2).

The extracted parameter values can be used to crosscheck the analysis. The drawn lines in Fig. 8 render a numerical evaluation of Eq. 5 using the experimentally extracted U c j µ c products, and ln(t 0 +t/τ ) = 21. In the intermediate temperature range, at which creep is described by the barrier (1), the agreement with the temperaturedependent screening current density values j(T ) at the onset of the relaxation (such as these might be measured during a field-sweep measurement) is more than satisfactory. Given that all times the current density is, to good approximation, given by the equation U (j) = k B T ln(t 0 +t/τ ), 23,24,41 one has ln(t 0 +t/τ ) = U (j)/k B T , where the numerator and denominator can be simply read from Figs. 4 and9. At the onset of relaxation (t ∼ 5 s), one finds values ranging from 20.5 (for 4.2 to 12 K) to 52 (at T = 26 K). Thus, the value ln(t 0 + t/τ ) = 21 is reasonable in the intermediate temperature range. The expression τ = (Λj c /E 0 )(k B T /U c ) ∼ 10 -8 s allows one to estimate ln(t 0 + t/τ ) independently; for t ∼ 5 s one again has a value of 20. For measurements performed with a commercial superconducting quantum interference device-based magnetometer, t ∼ 100 s, so that ln(t 0 + t/τ ) is slightly larger.

Fig. 8 shows that the field dependence of the screening current density, as expressed by Eq.( 5), is contained by the parameter U c j µ c . Hence, the j(B)-dependence is not the consequence of a field-dependence of the creep process, which would be reflected by a strongly fielddependent µ; rather, it reflects the intrinsic field dependence of the pinning force. The B -1/2 -dependence of the screening current density in iron-based superconductors was recently interpreted in terms of strong pinning 34,35 by nm-sized heterogeneities. [1][2][3] In this respect, the creep exponent µ ∼ 0.6 -0.8 at low fields parallel to the c-axis, comparable to µ = 0.5 found for single vortex creep in the Bragg-glass phase in single crystals of the cuprate high temperature superconductor Bi 2 Sr 2 CaCu 2 O 8+δ , 42,43 comes as somewhat of a surprise. Namely, vortex pinning at low fields in Bi 2 Sr 2 CaCu 2 O 8+δ is thought to be not in the strong pinning limit, but in the opposite, weak pinning limit. 44 In the field-values under scrutiny, the field-dependence observed in Fig. 2 suggests that one is dealing with strong pinning (see also Ref. 13). A possible explanation for the similarity of the creep exponents in the two cases is that the velocity of the flux lines at low and intermediate currents is limited by the progression of roughened vortex segments spanning the region between strong pins through the background of weak pins. This would seem natural given that weak pinning in the iron-pnictide superconductors is thought to be due to the local fluctuations of the dopant atom density, while strong pinning would originate from inhomogeneity of the dopant atom density on a much larger length scale. Thus, vortex segments would progress more or less continuously through areas of homogeneous doping, before becoming stuck on a "strong pin" , that is, a region in which the dopant atom density averaged over several dozen nm significantly deviates from the overall mean. The critical current density, and the screening current density at low temperature, would then be determined by the strong pins, while creep at intermediate and high temperatures would be determined by the weak pinning background. An alternative hypothesis is that the creep exponent would be determined by the shape of the energy distribution of the vortices pinned by largescale heterogeneity, much in the same way as this was proposed for creep through columnar defects in the socalled variable-range hopping regime. 45 Finally, we remark that the consistently larger µvalues found for H a ab are not unexpected, because, in the investigated orientation, j ab but ⊥ H a . Vortex lines are therefore oriented in the ab-plane, but are forced to move parallel to the crystalline c-axis. This is the hard direction for vortex motion, requiring the nucleation of vortex loops in the c-direction, a process limited by the value of the out-of-plane vortex lattice tilt modulus c ⊥ 44 ∼ ε -3 λ c44 . This exceeds the non-local tilt modulus c44 for vortices c by the inverse cube of the anisotropy factor ε λ ∼ 0.4.

V. SUMMARY AND CONCLUSIONS

Iron based-superconductors show a surprisingly large value of the relaxation rate S of the irreversible magnetization. Since the relaxation rate is too small to directly extract the current-density dependence of the flux-creep activation barrier from the time dependence of 1/S, an alternative, straightforward analysis method is proposed, that directly yields the relevant creep mechanism, and the temperature dependence of the pinning parameters in the absence of creep. Applying this to single crystalline Ba 0.6 K 0.4 Fe 2 As 2 , we find evidence for nucleation-type (collective) flux creep, with a weakly field-dependent creep exponent, µ = 0.6-0.8 for magnetic fields oriented along the c-axis, and a slightly larger µ = 1.2 -1.5 for field along ab. Several hypotheses leading to such µvalues, among which, the combined action of strong and weak pinning centers, and a non-trivial pinning energy distribution function are proposed. At low temperature, the screening current approaches the pinning critical current; therefore, meaningful information on flux pinning in the iron-pnictide superconductors can be directly extracted from low-temperature (T < ∼ 5 K) magnetic hysteresis experiments. The field dependence of the screening current density is found to arise from the underlying mechanism of pinning, and not from varying creep rates due to the flux creep process. Finally, it is found that the crossover between low-current and high-current behavior of the flux creep activation barrier is poorly described by the so-called "interpolation formula".

FIG. 1 :

 1 FIG. 1: (Color online) (a,b) In-phase fundamental (T H ) and amplitude of the third harmonic ac transmittivity (|TH3|) as function of the temperature, at an applied field µ0Ha = 1.5 T, for different indicated frequencies of the ac field. (c) Irreversibility field Hirr, as determined from the vanishing of |T3H |, indicated by the arrow in (b), for Ha c and Ha ab respectively.

FIG. 3 :

 3 FIG. 3: (Color online) (a) Relaxation of the critical state flux profile in the Ba0.6K0.4Fe2As2-crystal, for Ha c, at T = 8 K. Open symbols show the flux density profile across the Ba0.6K0.4Fe2As2 single crystal for four successive times, while closed symbols illustrate the flux density profiles obtained after the annealing of the critical state to the indicated temperatures. (b). Flux-creep activation barrier as function of the screening current density, as extracted using Eq. (4).

2 FIG. 5 :

 25 FIG. 5: (Color online) Compiled double-logarithmic plot of d[-kBTed ln(|dj/dt|)]/dj vs. j , as suggested from Eq. (9), for all temperatures, and the two orientations ( c and ab) of the applied magnetic field.

  FIG. 7: (Color online) Current-density dependence of the barrier U (j) for thermally activated vortex creep in the Ba0.6K0.4Fe2As2 crystal, for (a,b) µ0Ha = 0.6 T c, µ = 0.75; and (c,d) µ0Ha = 0.5 T ab, µ = 1.13. (b) and (d) show the (logarithm of) the raw electric field data, multiplied by the experimental temperature, -kBTed ln |E|, versus (dB/dx) -µ ∝ j -µ . (a,c) show the full, compiled curves of the creep barrier, obtained by adding the relevant factor kBTe ln E0 for each temperature.

FIG. 8 :

 8 FIG. 8: (Color online) Comparison of the temperature dependence of the activation barrier Uc(T )[jc(T )/jc(0)] µ (open symbols) extracted from the creep experiments, with that of the measured screening current density j(T ) at the onset of relaxation (closed symbols). The j(T ) values (closed symbols) are tantamount to those one would measure in magnetic hysteresis experiments. Large closed symbols on the lefthand abscissa denote the jc(T = 0)-values, determined from the extrapolation of the activation barrier to zero in Fig. 9. Drawn lines denote Eq. (5), evaluated using the experimental values of the product Ucj µ c , and ln(t0 + t/τ ) = 21. Dashed lines are guides to the eye.

  FIG. 9: (Color online) (a) Plot of -kBTed ln |E| vs.(dB/dx) -µ ∝ j -µ , for different magnetic fields strengths c. (b) Flux-creep activation barrier compiled from relaxation experiments at different temperatures, for the same fields as (a). The dotted lines show the linear extrapolation of the lowtemperature barrier to jc(0). The drawn lines show fits to the collective creep theory, Eq. (1). (c) Data for applied field µ0Ha = 0.2 T, showing the inadequacy of the interpolation formula (2) (dashed-dotted line).

  of strong vortex pinning by

		0	2 10 9	4 10 9		6 10 9
	0.12							0.12
	0.1					4.2K		0.1
						6K
	0.08					8K 10K		0.08
						12K
	0.06					14K 16K		0.06
						18K
						20K
	0.04					22K		0.04
						24K
						26K
	0.02							0.02
	0	0	10	20	30	40	50	0

Acknowledgments

We thank V. Mosser for providing the Hall sensor arrays. This work was supported by the French National Research agency, under grant ANR-07-Blan-0368 "Micromag". The work at Ames Laboratory was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under contract No. DE-AC02-07CH11358. Work at SNU was supported by National Creative Research Initiative (2010-0018300). The work of R. Prozorov in Palaiseau was funded by the St. Gobain chair of the Ecole Polytechnique. C.J. van der Beek and M. Konczykowski acknowledge the hospitality of Ames Lab and Iowa State University during the preparation of the manuscript.


Sylvain Kubler

William Derigent

Eric Rondeau

André Thomas

Éric Rondeau

Storage of information on manufactured products using "communicating materials"

come

Introduction

New challenges and opportunities arise with concepts such as Internet of Things (IoT), Ubiquitous/Pervasive Computing [START_REF] Weiser | The computer for the 21st century[END_REF] or still Artificial Intelligence [START_REF] Russell | Artificial intelligence: A modern approach[END_REF]. Through these concepts, objects of the real world are linked with the virtual world. Thus, connections are not just people to people or people to computers, but people to things and most strikingly, things to things [START_REF] Sundmaeker | Vision and challenges for realising the Internet of Things[END_REF][START_REF] Schneider | The smart pizza packing: An application of object memories[END_REF]. Ley [8] quotes the example of clothes able to carry their own information, and thus enabling the washing machine to automatically adapt its washing program. Such applications rely on ever more complex information systems combined with ever increasing data volumes, which are stored in a large number of information vectors. These vectors may be fixed (computers) or mobile (wireless devices, RFID).

Any product, during its life, passes through numerous companies and undergoes various operations (manufacturing, transportation, recycling. . . ). Technical, semantic and organizational interoperability between these companies is not always ensured, thus, conducing to information loss. If one considers the product as an information vector (on which information could be stored), it would contribute to improve interoperability all along its life cycle. Meyer et al. [START_REF] Meyer | Intelligent products: A survey[END_REF] provides a complete survey on intelligent products, i.e. products carrying their own information and intelligence. Främling et al. [START_REF] Främling | Agent-based model for managing composite product information[END_REF] argue that it is a formidable challenge to link the product related information to the products themselves, making the information of all the product components easily achievable. However, most of the time, products are only given an identifier (e.g. via a RFID tag) which provides a network pointer to a linked database and decision making software agent [START_REF] Morel | Editorial of special issue[END_REF].

Moreover, this kind of product is still limited on some points: risk of tag damage, small memory capacity, problem of data transfer (e.g. when the product is cut), etc.

As a result, we propose a new concept referred to as communicating material, which considers the material as intrinsically communicating. First, recent works carried out on this concept are discussed in section 2. This corpus of works mainly focus on the development of a data dissemination process to identify what the relevant information to users is and where it should be stored: on databases or on the product themselves? One important step of this framework deals with the storage/retrieval of data on/from the communicating materials, which is the subject of the paper. An appropriate architecture of communication is developed in section 3 which aims at splitting data over the communicating material and, at determining where this information is (or will be) located. An applicative scenario is finally presented in section 4.

Data dissemination process

As said previously, the product passes through numerous companies during its life cycle. Each actor/operation requires product related information (e.g., for decision-making, production orders) which is not always available (inaccessible information, e.g. owned by another supplier) and is not always up-to-date [START_REF] Kärkkäinen | Intelligent products-a step towards a more effective project delivery chain[END_REF] (unavailable information, e.g not shared by the supplier). Accordingly, solutions and platforms have emerged to link the product related information to the products themselves such as EPCglobal, ID@URI or WWAI [START_REF] Meyer | Intelligent products: A survey[END_REF]. However, information is generally deported on the network through these solutions because products are memory-constrained and the question of what information is relevant to users and where information needs to be stored is not answered. For this reason, we have proposed in recent publications [START_REF] Kubler | Information dissemination process for context-aware products[END_REF][START_REF] Kubler | Key factors for information dissemination on communicating products and fixed databases[END_REF] a data dissemination process consisting of 3 steps as shown in Figure 1(a): ➫ Process step 1 consists in implementing the database system architecture, which can be either centralized or distributed. Many works in the literature help the designer to choose the more suitable system according to the application constraints [START_REF] Mt Ozsu | Principles of Distributed Database Systems[END_REF].

➫ Process step 2 aims at selecting context-sensitive information. When users want to write information on the product at a given time (e.g. before the product leaves the company), it is necessary to identify information that is relevant (e.g. for subsequent users). This identification is achieved thanks to the process step 2. Our approach, detailed in [START_REF] Kubler | Information dissemination process for context-aware products[END_REF], consists in assessing what is the relevance of storing a given data item on the product according to the user concerns, the environment details, etc. One data item corresponds to a cell of a relational table (i.e. the intersection between a column, named "attribute", and a row, named "tuple") as emphasized in Figure 1(b). For instance, the data item located at row 3-column 1 in table Material, noted T Mat{3,1} , has the value MD06. Only data items which have an interest of being stored on the product are assessed (such a selection is also proposed in [START_REF] Kubler | Information dissemination process for context-aware products[END_REF]). For instance, only the tuples 3 in Material is assessed (see dashed background in Figure 1(b)) and the relevance of each data item from this tuple is computed (e.g. the relevance of T Mat{3,1} = 0.6). To compute the relevance value, the approach uses the notion of priorities which are numerical values either supplied or generated through observation and experimentation and are assigned through a multi criteria evaluation [START_REF] Kubler | Information dissemination process for context-aware products[END_REF]. At the end, all data items from all relational tables of the database are assessed and then, classified in order of relevancy. The higher the relevance value, the higher the probability that these data items will be stored on the product. The storage/retrieval of data items on/from the communicating material is done via the process step 3, as depicted in Figure 1(a).

➫ Two things are needed in process step 3: (i) first, the product must be instrumented in order to carry such information. As introduced previously, [START_REF] Kubler | Prototyping of a communicating textile[END_REF] propose a new concept referred to as communicating material, which considers the material as intrinsically and wholly communicating thanks to a huge amount of RFID µtags scattered in the material. Different textile prototypes were designed 2 through an industrial process with different types of RFID tags (µtags from Hitachi that can only store an identifier or Omron tags that can store up to 64 bytes). An example of communicating textile is shown in Figure 2. (ii) Secondly, it is necessary to design an architecture of communication to be able to store data fragments (i.e. data items) on a communicating material.

3 Data fragment storage/retrieval using communicating materials

Process step 3 of the data dissemination process deals with the storage of data items (identified across the process step 2) on the product. In this section, an appropriate architecture combined with a protocol of communication is developed in order to split data items over the material and then, to rebuild

2 Designed in collaboration with the CETELOR laboratory. them. Subsequently, we develop a method to determine where data are located on the communicating material.

Data storage on communicating materials

First, it is necessary to have a communicating material, as the communicating textile designed in our previous work [START_REF] Kubler | Prototyping of a communicating textile[END_REF].

Let us remind ourselves that a huge quantity of RFID tags are spread over/in the material. Since the RFID tags are memory-constrained, the idea is to split the set of data items over several tags. To do so, a specific architecture must be implemented and a specific application protocol is developed (named protocol of splitting in our paper). Figure 3(a) depicts the global architecture with a RFID reader, a communicating material and a database (containing data which are assessed and which may be store on the material). The protocol of splitting respects the RFID standard ISO/IEC 18000-1 [START_REF]Information technology aidc techniques -rfid for item man-agement -air interface, part 1 -generic parameters for air interface communication[END_REF], which relies on the 1 st , 2 nd and 7 th OSI layer as shown in Figure 3(b). Layer 1 corresponds to the physical part of the RFID (i.e antenna, analog part). Layer 2 deals with the communication protocol and especially the collision mechanisms. Layer 7 deals with the application data (this is the memory portion in which data can be added or modified by users).

A RFID tag may store more or less information according to the technology and, therefore, one data item may require more memory space than that available in a unique tag. That is why we propose a protocol of splitting. This application protocol is obviously defined at layer 7 of the OSI model (cf. 1. Protocol (8 bits): Integer from 0 to 255 which enables to know which fields compose the packet. The value 255 is defined in our application which refers to the frame structure defined in Figure 3(b) (specific to our application), 2. Size (8 bits): Integer from 0 to 255 which indicates the size of data included into the field Data (7 th field), 3. Seq Num (8 bits): Integer from 0 to 255. It provides the sequence number of the current frame (1 frame/tag). The sequence number is used to determine the order of the different frames that have been written over the RFID tags (needed to rebuild the set of data items), 4. Last Num (8 bits): Integer from 0 to 255. It provides the sequence number of the last frame which contains data related to the same writing phase. The notion "same writing phase" is important because one data item may be written at two different times in its life cycle and then, data inconsistency/conflict may occur, 5. ID Phase (64 bits): Integer from 0 to 2 64 which is the identifier of the writing phase (the date of writing is currently used). Several frames may have the same ID Phase but a couple ID Phase/Seq Num is unique. 6. Checksum (32 bits): Integer from 0 to 2 32 . Used for data error-checking (it does not detect errors in the header), 7. Data (n -128 bits): The content of the data item is added in this field, which is a string. This string may or may not be stored integrally in a unique RFID tag according to the technology (where n is the number of writable data bits in one RFID tag). Let us note that an index is added for locating each data item in the database (i.e. the table name, the attribute name and the instance concerned). In our method, the index is coded as follows: Tablename.Attributename.PrimaryKeyValue. For instance, when the data item T Mat{3,3} is written (see Regarding the application layer, 128 bits (32 × 4) out of n useful are used in each tag for the application header. The application logic rebuilds each data item thanks to the Seq Num and the Last Num, which indicate in what order data items have been split. In section 4, the tag memory is itemized after having been written. Let us note that according to the set of data items carried by the product, queries may be answered or unanswered. Nevertheless, some methods can be deployed to know in advance (i.e. before data items will be written on the product) if queries may be answered or not (e.g. by transforming queries to corresponding bitmaps [START_REF] Chan | Context-sensitive mobile database summarisation[END_REF]).

Data location on communicating materials

In this section, we develop an approach to know where information will be written on the material (in case of writing) or where information is located (in case of reading). In certain production processes like textile manufacturing, materials are transformed with highly automatised machines, at high manufacturing speed. However, some defects may occur on specific material zones (e.g., holes, stains). Because machines are not aware of these defaults, end products could not be sold. In our vision, materials are consider hard disks, able to store information about themselves. If a defect is detected, this information is then stored on the material and can be reused ad libitum when needed. The type of data manipulated is arearelated : consider a grease spot on the textile. Information related to the grease spot must be stored as close as possible to the real grease spot. As a result, information must be located. Our method is designed to help locating information on the material, thus enabling machines not to be blindfolded and to adapt their behavior. To do so, a more complex architecture is required than previously. Indeed, in the previous section, the data location over the material was not taking into account and therefore, data was written/read anywhere ('on the fly') as illustrated in Figure 3(a). However, if the user desires to know where a specific information is located over the material, he needs to use a specific method which is detailed hereafter. First, let C be the set of RFID tags present in the material. Let R be the set of RFID readers which are aligned as depicted in Figure 4 (e.g. on a ramp), with R = {r1, r2, r3}). Each RFID tag and RFID reader has a reference number respectively noted IDc and IDr with c ∈ C and r ∈ R. A reader r generates an event er,c if the presence of the tag c is detected. This event is made up of {IDr, IDc, tr,c}, with tr,c the acquisition time of the event. All the events form the set E.

Let us focus now on the algorithm which calculates the theoretical positions of the tags in the material (i.e. ∈ C). First, it is necessary to model the reading zone (i.e. zone in which a RFID reader and a RFID tag can communicate) which depends on the RFID technology. This reading zone must be modeled as a cylinder3 as depicted in Figure 4 (the three RFID readers have the same cylindrical shape). Once the reading zone is modeled, the algorithm for computing the location 2D of tags takes as input the set of events E. Since the reading zone is modeled as a cylinder, it is possible to calculate the tag position based on the chord of the circle. The RFID reader transmits events as long as one tag is in its reading zone and as many events as acquisition cycles (i.e. if a tag c stays during n acquisition cycles, n events are generated). By this way, it is possible to calculate the chord based on the difference between the date of the last and the first event. Thus, we obtain two possible positions on y-axis via equation 1 for each RFID reader : r1 and r2 detect together c2 and compute respectively the right chord (noted chord Rr 1 and chord Rr 2 in Figure 4) and the left chords (noted chord Lr 1 and chord Lr 2 ) so as to locate c2 ; r3 only detects c1 and proceeds to the same computation. Let us note that in computing the average between the two chords, the measured error is reduced by two. The tag coordinate in x axis is calculated via equation 2. The error made on the chord computation depends on both the acquisition time cycle and the conveyor speed. In fact, the higher the acquisition time cycle, the lesser the error. Obviously, the error depends also on the modeled reading zone (modeled as a perfect cylindrical shape). Figure 4 illustrates that the computed chords do not exactly overlap with the RFID tag c1 and c2. By using the chord approach, two possible chords are determined, which means that two possibilities are returned. If one tag is detected by only one RFID reader, it implies two possible coordinates yc t (impossible to know if the real tag is located on the left or right chord). This is illustrated in Figure 4 with the tag c1 (only detected by r3). This problem does not occur when a tag is detected by several readers, as illustrated with c2 (detected by r1 and r2). Indeed, the tag might most likely be located between chr1 r and chr2 l , as shown in Figure 4. The higher the number of readers detect one tag, the higher the precision.

yc t = yr ± rad 2 - tr,c L -tr,c F 2 × v 2 (1)
xc t = tr,c F + tr,c L 2 -ts × v (2)

Applicative scenario

In this scenario, we use the communicating textile reel designed in [START_REF] Kubler | Prototyping of a communicating textile[END_REF] (cf. Figure 2). This textile reel is preparing to leave the company X to a company Y as depicted in Figure 5 (it will be used for designing vehicle headrests). Accordingly, the supplier X decides to implement the data dissemination process for storing on the textile reel, the product related information which are useful for the supplier Y. This is done at Activity X as shown in Figure 5: implementation of step 1, 2 and 3 for identifying and storing the relevant data from the database, on the communicating textile reel. Let us note that the supplier X does not mind where information will be stored on the textile (i.e. it is unnecessary to use the architecture from Figure 4). Then, the textile reel will continue its transformation and will arrive at Activity Y as depicted in Figure 5. In this activity, the machine retrieve information carried by that one and wants to locate information on the textile (i.e. it is necessary to use the architecture from Figure 4). Both activities must implement the protocol of splitting detailed in section 3.1. Moreover, Activity Y must implement the architecture and algorithm detailed in section 3.2 for locating data on the textile. The communicating textile arrives at Activity X, where the machine starts the relevant data identification (i.e. step 2 of the data dissemination process). This step returns the set of data items that need to be stored on the textile. Let us assume that among these data items, the ones composing the tuple 3 of table Material (see Figure 1(b)) are included. The machine then starts to write T Mat{3,1} , T Mat{3,2} and T Mat{3,3} over the communicating textile thanks to the protocol of splitting described in section 3.1, as depicted in Figure 5.

As said before, the supplier in Activity X does not mind where information is written over the textile. As a result, we use the architecture given in Figure 3(a) (i.e. data is written 'on the fly'). Figure 6(a) shows the log events generated when writing the three data items over the textile. It can be observed that 4 tags are required for splitting the three data items. However, 5 tags have been written because one Then, the textile reel arrives at Activity Y in which the machine requires data carried by that one and must identify where it is located over the textile.

Data retrieval and data location:

Activity Y Since information must be located over the textile, the architecture described in Figure 4 is implemented. Then, the communicating textile passes under the ramp of RFID readers and is read. The three data items are retrieved and their content is displayed via an application software (JAVA programming) as shown in Figure 7. Then, services can be programmed: for instance, queries may be directly performed via the JAVA software based on the set of data items retrieved from the communicating textile (unanswered queries could happen). Many services may therefore be imagined as the example of Ley [START_REF] Ley | Ubiquitous computing[END_REF] with the washing machine (cf. section 1). Until this point, data items are retrieved from the communicating textile. To locate them over the textile, we implement the algorithm described in section 3.2. To assess the algorithm precision, the theoretical results will be confronted to the real tag positions. The virtual map given in Figure 8(a) details the real tag position as well as those returned by the algorithm (two possible solutions for one tag). These results have been obtained when the textile is read with a distance of 60mm from the ramp of readers6 . If we look at the detailed case in Figure 8(a), the euclidean distance between the real tag position and those computed by the algorithm is equal to 7.01mm for the nearest position (chord 2) and 33.44mm for the furthest (chord 1). The focus in Figure 8(a) emphasized where the tags e007000002199de6, e007000002199de7, e007000002199ddd and e007000002199dde (which have been used for storing the 3 data items) are located on the textile.

For each tag, the minimal error is measured between the real tag position and the nearest position among the two computed by the algorithm (euclidean distance). The set of all the computed errors obtained is synthesized in a box-and-whisker diagram in Figure 8 (the reader is positioned 60mm above the textile). We can see that the minimal error (i.e. the minimal euclidean distance) is ≃ 7mm. 25% of the errors are inferior to 8mm (cf. the 1 st quartile) and 25% are superior to 31mm (cf. the 3 rd quartile). In average, the error is about 15mm with a reader positioned 60mm above the textile, with the conveyor speed equals to 4m/s and with an acquisition time

Conclusion

New challenges and opportunities arise with concepts such as Internet of Things, Ubiquitous Computing and Artificial Intelligence. Nowadays, products are more and more fitted with electronic devices (e.g. sensors, RFID tags) which give them abilities such as data storage, decision making, monitoring. Some authors argued the usage of intelligent products in the framework of the supply chain management. Indeed, these products are able to control their own life, evolution and could serve as an interoperability hub between the supplier members. However, most of the time, products are only given an identifier (e.g. via a RFID tag) which provides a network pointer to a linked database and decision making software agent. As a result, a new kind of material is discussed in this paper referred to as "communicating material" which allows to embed significant proportion of data directly on the manufactured product. In order to answer the question of what information is relevant to store on the product during its lifecycle, a data dissemination process (consisting of three steps) is presented and makes reference to previous works. One important step deals with the storage and retrieval of data on/from the communicating material. In this paper, an appropriate architecture of communication is developed which aims at splitting information over the "communicating material" and at determining where this information is located.

  high developed pol... 3mm...

Figure 1 .

 1 Figure 1. Overview of the data dissemination process and the data assessment (data from a relational table)

Figure 2 .

 2 Figure2. Prototype communicating textile designed in[START_REF] Kubler | Prototyping of a communicating textile[END_REF]

Figure 3 (Figure 3 .

 33 Figure 3. Protocol of communication developed to read/write data fragments from/on communicating materials data item) and the last one contains the data item value:

Fig- ure 1

 1 (b)), the index Material.Value.MD06 is added. Then, the data item content is added (i.e. 15 • C...).

Figure 4 .

 4 Figure 4. Architecture needed to locate information

  xc t , yc t theoretical coordinates x-y of the tag c (c ∈ C) xr, yr coordinates in x and y-axis of the reader (r ∈ R) tr,c F first date to which the reader r detected the tag c tr,c L last date to which the reader r detected the tag c v conveyor speed (or material speed) ts textile detection date (thanks to a presence sensor) rad radius of the reading zone

3Figure 5 .

 5 Figure 5. Description of the scenario applicative activities

  Writing error: tag e007000002199de8 tag is ignored (Invalid response after a write command) (a) Log events of the writing process DB 0: FF F0 02 03 ÿû.. DB 1: 00 00 01 34 ...4 DB 2: D7 20 8F 6D * m DB 3: 00 00 46 E8 ..Fè DB 4: 31 2E 61 38 Mate DB 5: 2E 31 34 3D rial DB 6: 3F 3B 54 31 .Des DB 7: 2E 61 38 2E crip DB 8: 31 35 3D 53 tion DB 9: 61 77 79 65 .MD0 DB 10: 72 20 3B 54 6=Te DB 11: 31 2E 61 38 xtil DB 12: 2E 31 36 3D e wh DB 13: 44 69 63 6B ich DB 14: 69 6E 73 6F is p DB 15: FF F0 02 03 rovi DB 16: 6E 20 3B 54 ded DB 17: 31 2E 61 38 with DB 18: 72 21 3B 54 two DB 19: 31 2E 61 36 lay DB 20: 72 20 4B 54 ers . . .

Figure 6 .

 6 Figure 6. Results of the data item relevancewrite operation failed (tag e007000002199de8). This highlights that all data items are stored on the material even if writing errors occur 4 . Now, let us focus on the datagram content of a given RFID tag. Figure6(b) details the datagram contained in the tag e007000002199ddd after having written the three data items. It is important to note that a RFID tag memory (whatever the RFID technology) consists of several Data Blocks (denoted DB in Figure6(b)). The RFID tags disseminated in our textile are the Omron's V720-D52P03 and their memory is divided in 64 blocks of size 4 bytes 5 . Accordingly, the datagram header occupies the four first bytes of each tag memory as highlighted in Figure6(b) (DB0 to DB3). Then, the remaining DB are exclusively reserved to the content of data items. However, let us remind that one index is added at beginning of each data item in order to locate it in the database (cf. section 3.1). Figure6(b) shows the index related to T Mat{3,2} , which is Material.Description.MD06. Then, the content of this data item (string of value: Textile which is provided with two layers of protective coatings) is added (cf. Figure6(b)).Then, the textile reel arrives at Activity Y in which the machine requires data carried by that one and must identify where it is located over the textile.

Figure 6 (

 6 b) shows the index related to T Mat{3,2} , which is Material.Description.MD06. Then, the content of this data item (string of value: Textile which is provided with two layers of protective coatings) is added (cf. Figure 6(b)).

Figure 7 .

 7 Figure 7. Display of tuples related to Material

  Errors synthesized in form of whisker-box plots

Figure 8 .

 8 Figure 8. Assessment of the precision of the location algorithm cycle fixed to 5ms (errors depending on these 3 parameters).

Centre de Recherche en Automatique de Nancy, Université de Lorraine, CNRS UMR 7039 Campus Sciences, BP 70239, F-54506 Vandoeuvre-lès-Nancy Cedex, France, email: Sylvain.Kubler@cran.uhp-nancy.fr

A methodology to model the reading zone of a given RFID technology is presented in[START_REF] Kubler | Prototyping of a communicating textile[END_REF].

This depends on the technology but most RFID technologies implement error correction mechanisms.

Let us remind that one ASCII character occupies 1 byte.

The maximum distance depends on the modeled reading zone.