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Stability of a (2G) Coated, Thin-Film YBaCuO 123 Superconductor: Intermediate Summary

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Abstract

This paper summarises results obtained with stability calculations of thin films and multi-filamentary superconductors. In a series of papers, all the contributions have been published in this journal. We now extend our previous investigations to the temporal aspect of the internal heat transfer and to the material homogeneity problem. Within multi-component heat transfer (solid conduction, radiation), the standard theory of radiative transfer in a coated, thin-film, YBaCuO3 123 superconductor correctly treats the energetic aspects of radiation propagation; this is the actual core of stability models. But a rigorous solution of the temporal aspect still is missing. It is the study of this aspect that would provide a new access to the physics of superconductor stability, in particular if after a disturbance the system is already close to a phase transition. A matrix formulation, using a combination of Monte Carlo and radiative transfer calculations, is suggested to circumvent the temporal solid conduction/radiative transfer problem in multi-component heat flow. As an important result, quench is not an event that proceeds instantaneously. Instead, it is a process the speed of which decreases the more, the closer the superconductor temperature approaches critical temperature until the residual number of electron pairs becomes too small to support critical current. The stability of superconductors and thermal fluctuations might reflect a common background: the relaxation time of the density of electron pairs after disturbances.

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Correspondence to Harald Reiss.

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Personal note: A minus sign is missing in Eq. (2a,b) in [12], as well as the albedo. Experts among the esteemed readers certainly have discovered this lapsus and might think “Since decades he has been working with radiative transfer and meanwhile really should know how the ERT looks like”. Kindly please accept my apologies.

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Appendices

Appendix 1. Monte Carlo approximation

The Monte Carlo approach considers a large number M of beams that after emission experience a total N of interactions on their transit path of total length, L, through an object. Each beam shall be emitted from any position within the target, at z = 0 (Fig. 6, part b) and under arbitrary angles, θ, against the surface normal. Its transit time can be calculated for arbitrary events of absorption/remission and scattering interactions. In the Monte Carlo language, beams are called “bundles”. For more descriptions of Monte Carlo simulations applied to radiative heat transfer, see Chap. 11 in [21] or, in short, Chap. 18 in [29].

Monte Carlo and finite element simulations have to be performed in appropriately chosen time steps.

The part Ω (the albedo) and its impact on transit time is responsible for the energy remaining from the bundle after absorption/remission processes; this part again is scattered and the transit time accordingly has to be simulated by the Monte Carlo approach. Also propagation of purely scattered radiation (Ω = 1) can be described as a diffusion-like process because of a multiple of scattering interactions, but a radiative conductivity cannot exist in this case (we do not have absorption/remission and thus no radiative equilibrium within the object).

For the Monte Carlo method, the scattering angle, θ, under elastic scattering can be defined as a random variable by

$$ \theta =\operatorname{arccos}\ \left[1-R\left(\theta \right)/{m}_{\mathrm{S}}\right] $$
(16)

in which R(θ) denotes the probability to find a bundle emitted or scattered at particular angle θ from the total set 0 ≤ θ ≤ π. Large mS indicate strong forward scattering; compare the spider diagram in Figure A1 in [29]. In the same reference (Fig. 7), it has already been shown that the angular distributions of scattered radiation in ceramic samples approach the better the theoretical cos(φ)-distribution (when bundles leave the rear sample surface) the larger the extinction coefficient and the larger the number of bundles.

The effect of variations of the extinction coefficient on the distribution of the scattered bundles is shown in Fig. 18. Convergence of the solid symbols to the theoretical cos(φ) curve locally becomes weak due to interferences of the results obtained in the three optically different layers (Fig. 9a). But the overall result confirms applicability of the diffusion model.

The angular distribution of scattered bundles leaving the thin-film sample at the rear position, D = 2 μm, is presented in Figs. 16, 17 and 18. The total thickness of the film is divided into three sections. Results in Fig. 16 are obtained assuming mS-factors and albedo identical in each of the layers. In all cases, the solid symbols approach the theoretical cos(φ)-distribution (open circles).

As a more realistic case, values mS different in each layer are applied in Fig. 17, and Fig. 18 shows results obtained with different extinction coefficients in the thin surface layers and the core of the thin film.

If not just single, individual bundles but the total set of M = 105 bundles is applied, and if mS is large, all M bundles are sharply focussed to small forward angles, Fig. 8a.

The scattering phase function, Ψ(μ), is obtained from counting the number of bundles, n(θ), that as a result of the Monte Carlo simulation are scattered into intervals in steps of Δθ = 10 deg (with the angle θ vs. surface normal between 0 and 90 deg) and for different values of the (scattering) anisotropy parameter mS. We have Ψ(μ) = n(θ)/M, with μ = cos(θ) given by the midpoints in the intervals.

Figure 8b shows the mean value, μm = cos(θ)m, to be used in the radiative conductivity, Eqs. (12a) and (12b). It is calculated from

$$ {\mu}_{\mathrm{m}}=1/2\int \varPsi \left(\mu \right)\ \mu\ d\mu $$
(17)

as the weighted mean of the individual cos(θ), with the weights provided by the scattering phase function in Fig. 8a.

Appendix 2. Calculation of the Stability Function

If the critical current density, JCrit, in a superconductor depends only on temperature (neglecting its dependence on magnetic field), and when assuming uniform temperature in the total superconductor cross section, its critical current density is uniform, too.

If at a time t > t0 losses (like Ohmic) come up in the superconductor, its temperature increases from T(t0) to T(t). We then write the stability function, Φ(t), as

$$ \varPhi (t)=1-{J}_{\mathrm{Crit}}\ \left[T\left(t>{t}_0\right)\right]\ A/{J}_{\mathrm{Crit}}\ \left[T\left({t}_0\right)\right]\ A $$
(18)

with A the total conductor cross section. This can be rewritten as

$$ \varPhi (t)=\left\{{J}_{\mathrm{Crit}}\ \left[T\left({t}_0\right)\right]\ A-{J}_{\mathrm{Crit}}\ \left[T\left(t>{t}_0\right)\right]\ A\ \right\}/{J}_{\mathrm{Crit}}\ \left[T\left({t}_0\right)\right]\ A $$

Because of the dependency of JCrit on T, we have JCrit [T(t0)] > JCrit [T(t > t0)].

Recalling that any current in a superconductor flows with critical current density, the stability function in this simple case accordingly reads

$$ \varPhi (t)=\left\{{I}_{\mathrm{Crit}}\ \left[T\left({t}_0\right)\right]\kern0.37em -{I}_{\mathrm{Crit}}\ \left[T\left(t>{t}_0\right)\right]\right\}/{I}_{\mathrm{Crit}}\ \left[T\left({t}_0\right)\right] $$

In other words, Φ(t) under losses (strictly speaking: under any kind of source functions) is the ratio of zero-loss transport current at time t > t0 to the critical current at the start temperature, t0. Without losses, T(t) remains constant, ICrit [T(t0)] = ICrit [T(t > t0)] and Φ(t) = 0.

In this simple case, we have assumed that transport current, I, would occupy the total superconductor cross section. This is not necessarily the case: Since I flows with critical current density, part of the conductor cross section might be sufficient to carry the (full) transport current. In the remaining part of A, ICrit might be very small, which means there is a non-zero current distribution in all parts of the cross section (the transport current circumvents those parts of the cross section that are resistive). Regions with zero resistances, flux flow and Ohmic resistances thus may exist in parallel. Current transport (its distribution in the cross section) is an optimisation problem.

But in real situations, superconductor temperature is not uniform, and critical current density and critical temperature depend on time (since losses depend on time like in AC applications). If we again assume that critical current density depends on only temperature (again neglecting the dependence on the magnetic field), we need the temperature distribution, T(x, y, t), in the conductor cross section to obtain the distribution of critical current density, JCrit(x, y, t). The temperature distribution results from local (Ohmic or other) losses or from boundary conditions like cooling. The obtained temperature field, T(x, y, t), accordingly is mapped onto the field of critical current density, JCrit(x, y, t). This mapping is not bijective.

There are well-known relations between JCrit and T that apply approximations to experimental data. If at any local position, temperature exceeds critical temperature, TCrit, the critical current density at this position is zero, with corresponding variations of the stability function.

If the dependency of JCrit also on the magnetic field (flux density, B) is considered, the stability function reads, in analogy to Eq. (16),

$$ \varPhi (t)=1-\left\{\int {J}_{\mathrm{Crit}}\ \left[T\left(x,y,t\right),B\left(x,y,t\right)\right]\ dA\right\}/\left\{\int {J}_{\mathrm{Crit}}\ \left[T\left(x,y,{t}_0\right),B\left(x,y,{t}_0\right)\right]\ dA\right\} $$

Solutions of this equation cannot be found with analytical methods but have to be obtained numerically.

The first step is calculation of the temperature field T(x, y, t) by e.g. finite differences or finite element (FE) methods to solve Fourier’s differential equation. For this purpose, the total conductor cross section, A, is divided into a large number (a set) of geometrical, i.e. area elements, ΔAi. The ΔAi not necessarily would be of equal size.

The division of A into the set ΔAi has to be made fine enough to approximate the continuous variation of the field T(x, y, t) by a discrete distribution TiAi, t). The finite element solution delivers local, discrete nodal values (temperature at the corners of e.g. a plane area element) from which the element temperature, TiAi), is obtained as the arithmetic mean, or solutions are provided at the centroid of the elements.

The set TiAi, t) replaces the continuous distribution, T(x, y, t). Roughly speaking, the finer the division of A into the set ΔAi, the more exact is the finally obtained solution (but the longer is the computation time). The TiAi, t) are obtained under convergence criteria in standard FE computer programs. In our previous papers, you will find information on the finite element mesh, the solution (integration) procedures, convergence criteria and, occasionally, enormous computation times with standard, 4-core PC under Windows 7.

The next step is calculation of the field JCrit,iAi). Relations like JCrit(T) = JCrit(T0) (1 − T/TCrit)n are frequently reported in the literature, with JCrit(T0) the value at t = t0 and with the exponent n as a material specific property. In applications of high-temperature superconductors, we for example find n = 2. But JCrit(T0) is not the same in each of the area elements; it fluctuates against a mean value (like in Fig. 19) because of experimental uncertainties, or from tolerances in conductor manufacturing and handling.

We accordingly have JCrit,i(T0) given as random values in the corresponding ΔAi, usually with i ≤ N a very large number. The same applies to critical temperature, TCrit,i and the lower and upper critical magnetic field, BCrit,1,2,i, and to the anisotropy ratio, r. Random variations ΔTCrit, ΔBCrit,2 and ΔJCrit of the electrical/magnetic critical parameters against the conventional values of YBaCuO 123 in the present paper (92 K, 240 Tesla at T = 0 and 3 1010 A/m2 at T = 77 K) are within ± 1 K, ± 5 Tesla and ± 1%, respectively.

The anisotropy ratio, r, of thermal diffusivity and of JCrit is taken into account in directions parallel to the crystallographic ab-plane against the c-axis, respectively. In the present paper, we have applied r = 10 constant, from experimental tests, with fluctuations Δr of ± 0.5.

Once JCrit,i(T0) and JCrit,i(T0) are available, the critical current density has to be transformed to its magnetic field dependence, JCrit,i [x, y, t, B(t)]. We consider each of the area elements, ΔAi, as tiny conductors. The field dependence is taken into account using standard relations (found in tables in volumes on electrotechnical problems) for calculation of the magnetic field of conductors of simple geometry (self-fields and fields from neighbouring conductors that are added to obtain the local field for each ΔAi). Insertion of B(x, y, t) yields JCrit [x, y, t, B(x, y, t)] of the superconductor, as indicated in Eqs. (11a) and (11b).

The stability function (area element index i omitted) then reads

$$ \varPhi (t)=1-\left\{\Sigma {J}_{\mathrm{Crit}}\ \left[T\left(x,y,t\right),B\left(x,y,t\right)\right]\Delta A\right\}/\left\{\Sigma {J}_{\mathrm{Crit}}\ \left[T\left(x,y,{t}_0\right),B\left(x,y,{t}_0\right)\right]\Delta A\right\}, $$

with the summations taken over all ΔAi (the x,y-co-ordinates are the centroids of the area elements, ΔAi).

In most stability calculations and application of high-temperature superconductors, it is only current flow parallel to the ab-plane that is relevant for superconductor technical applications and their stability. An example for the stability function also in c-axis direction is reported in the present paper.

Calculation of the TiAi), in each of the area elements ΔAi, takes into account magnitude and distribution of losses that may depend on time (like in an AC application), experimental values of the thermal diffusivity of the superconductor material, and boundary conditions.

The thermal diffusivity of YBaCuO 123 in the ab-plane is shown in Figure 5 of [7].

Calculation of the JCrit,iAi) applies penetration depth of the magnetic field, with B estimated using standard relations for single conductors of given (ΔAi) cross sections; with these results, the Meissner effect is checked in each of the ΔAi. For this purpose, critical temperature, TCrit,i in each of the ΔAi is taken not as constant but in dependence of local magnetic induction, Bi.

For the estimate of B, the transport current distribution in the conductor cross section has to be known. It is obtained from the electric resistances that result separately for each of the area elements ΔAi. This takes into account Ohmic and flux flow resistances, the former in dependence of temperature and with experimental values of the specific resistance, the latter from comparison of JCrit vs. JTransport as described in the standard literature or using a new flux flow resistance model described in [9, 11]. Once the resistances are obtained, the elements ΔAi as mentioned are considered as tiny current-conducting transport channels all directed in parallel along the length of the conductor. Current distribution then follows from the distribution of the individual resistances of the ΔAi in an iterative procedure and from application of Kirchhoff’s law.

An interesting practical problem is to calculate the stability function in case there is current-sharing, but this is beyond the scope of the present paper.

All electric/magnetic and thermal material parameters are considered as dependent on temperature. Since temperature under losses or from boundary conditions is transient, all obtained results and the calculated stability functions depend on time.

Note that calculation of the stability function does not imply an actual transport current and its actual distribution. The stability function only delivers a criterion to decide whether, and to which extent, zero-loss current transport remains possible under disturbances (here Ohmic and flux flow losses). It is then the task of the designer to decide by this tool which percentage of critical current can be tolerated as transport current in a technical application of superconductivity.

Fig. 16
figure 16

Angular distribution, n(φ), calculated at z = 2 μm of in total M = 5 104 bundles emerging from x = y = 0, z = 0 (compare Fig. 6, part a and b) that leave the rear surface (thin-film sample of the YBaCuO 123 superconductor). Results are shown vs. angle φ against surface normal of the volume elements (concentric rings generated by rotation of the area elements of Fig. 6, part b) and in dependence of the (scattering) anisotropy factor mS. The curves (solid diamonds) are indexed as mS1, mS2, mS3 (running from 2 to 18; these are the same mS that were applied in the spider diagram shown in Fig. A1 in [28]). The mS factors in this figure are assumed as identical in the inner (index 1) and outer, about 0.1 μm thin boundary layers (3) and in the 1.8-μm-thick central core (2) of the thin film. The same applies to values of the albedo (Ω = 0.912 in the three layers; the value is taken from [11]). Extinction coefficients in the three layers are E1 = E3 = 3.417 106 and E2 = 1.409 107 1/m, respectively, again from [11]. The solid symbols approach the theoretical cos(φ) distribution (open circles) of the residual beams leaving the sample on the rear surface

Fig. 17
figure 17

Angular distribution, n(φ), again of M = 5 104 bundles. The curves are calculated as before (albedo, Ω, and extinction coefficients, E, are the same as in Fig. 16), but the mS values are different in the three layers: mS1 = 6, mS2 = 2 and mS3 = 6 (red diamonds) and mS1, mS2, mS3 = 2, 6, 2 (lilac-brown diamonds, indexed as before). Local deviations of both curves from the theoretical cos(φ) distribution and in particular the dip of the solid red symbols at scattering angles between − 25 and + 25 degrees result from interferences caused by the strongly different optical parameters of the three layers. The overall tendency to approach the theoretical distribution is confirmed also in these cases

Fig. 18
figure 18

Angular distribution, n(φ), of in total M = 5 104 bundles. The curves are calculated as before (Figs. 16 and 17) using identical mS values (mS1 = mS2 = mS3 = 6) and the same albedo (Ω = 0.972), but here with a variation of the extinction coefficients: E1 = E3 = 3.417 106 and E2 = 1.409 107 (solid lilac diamonds) and, reversely, E1 = E3 = 1.409 107 and E2 = 3.417 106 1/m (green diamonds). Indices are as before

Fig. 19
figure 19

Existence diagram of type II superconductivity (schematic, not to scale; the lower critical magnetic field is not shown). The dashed blue line and the open blue circles in this figure denote the conventional region of existence of superconductivity (the open blue circles accordingly are located on the corresponding axes of the diagram). Random variations of TCrit(B), JCrit(B) and JCrit(T) against this (conventional) region are indicated by small black dots; this applies (schematically) to the existence diagrams of all elements in the finite element scheme. These random variations (taken as standard variations in each of the area elements) ΔTCrit, ΔBCrit,2 and ΔJCrit of the electrical/magnetic critical parameters against the conventional values of YBaCuO 123 in the present paper are within ± 1 K, ± 5 Tesla and ± 1%, respectively. For a particular element number, jj, as an example, its region of superconductivity existence is indicated by the coloured quadrants that in this single, special case are (exaggerated) located all within the conventional region (the dashed blue curves). Thick black solid circles indicate for this element the critical values TCrit, BCrit, JCrit that, again exaggerated, are shifted against the conventional values. The figure is copied from [15], here with slight modifications. Reprinted from [15]. With kind permission of Old City Publishing Inc., Philadelphia PA 19123, USA

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Reiss, H. Stability of a (2G) Coated, Thin-Film YBaCuO 123 Superconductor: Intermediate Summary. J Supercond Nov Magn 33, 3279–3311 (2020). https://doi.org/10.1007/s10948-020-05590-3

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