A multiple field coupled resistive transition model for superconducting Nb3Sn A multiple field coupled resistive transition model for superconducting Nb3Sn Lin Yang, He Ding, Xin Zhang, and Li Qiao Ci[.]
A multiple-field coupled resistive transition model for superconducting Nb3Sn Lin Yang, He Ding, Xin Zhang, and Li Qiao Citation: AIP Advances 6, 125101 (2016); doi: 10.1063/1.4971214 View online: http://dx.doi.org/10.1063/1.4971214 View Table of Contents: http://aip.scitation.org/toc/adv/6/12 Published by the American Institute of Physics AIP ADVANCES 6, 125101 (2016) A multiple-field coupled resistive transition model for superconducting Nb3 Sn Lin Yang,1 He Ding,2 Xin Zhang,2 and Li Qiao2,a College of Materials Science and Engineering, Taiyuan University of Technology, Taiyuan, Shanxi 030024, People’s Republic of China Institute of Applied Mechanics and Biomedical Engineering, College of Mechanics, Shanxi Key Laboratory of Material Strength and Structural Impact, Taiyuan University of Technology, Shanxi 030024, People’s Republic of China (Received 11 September 2016; accepted 16 November 2016; published online December 2016) A study on the superconducting transition width as functions of the applied magnetic field and strain is performed in superconducting Nb3 Sn A quantitative, yet universal phenomenological resistivity model is proposed The numerical simulation by the proposed model shows predicted resistive transition characteristics under variable magnetic fields and strain, which in good agreement with the experimental observations Furthermore, a temperature-modulated magnetoresistance transition behavior in filamentary Nb3 Sn conductors can also be well described by the given model The multiple-field coupled resistive transition model is helpful for making objective determinations of the high-dimensional critical surface of Nb3 Sn in the multi-parameter space, offering some preliminary information about the basic vortexpinning mechanisms, and guiding the design of the quench protection system of Nb3 Sn superconducting magnets © 2016 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/) [http://dx.doi.org/10.1063/1.4971214] I INTRODUCTION As one of the prominent materials for the high-field applications, the A15 phase Nb3 Sn compound has great application prospects in International Thermonuclear Experimental Reactor (ITER),1 nuclear magnetic resonance (NMR) spectroscopy,2 and high energy physics (HEP).3 The electrical transport properties are the most popular and among the most extensively investigated features of the Nb3 Sn superconductor, which is of both scientific and technological interest The anomalous transport properties of the compound Nb3 Sn under the application of magnetic field, temperature and strain are invaluable for understanding the combined multiple-field coupled sensitivity of the material, the importance of which is reflected in a variety of ways: (a) for making objective determinations of the high-dimensional critical surface of Nb3 Sn in the multi-parameter space, multiple field-induced resistive transition characteristics should be studied and modeled; (b) the electrical resistivity behavior of Nb3 Sn above the critical surface has a direct link to the superconducting performance below it Something about the generic nature of the Nb3 Sn can be drawn from the interpretations on the anomalies in the multiple-field-dependent normal state electrical resistivity The normal-state electrical resistivity is strongly dependent on the characteristics controlling the superconducting properties, which is difficult to identify by other means A broadening of the superconducting transition, as shown in resistivity measurements in the presence of magnetic fields, temperature, and strain, implies important information on the basic vortex-pinning mechanisms controlling the critical current density; (c) the resistive transition from the superconducting state to the normal state is usually known as a quench, which can potentially cause damage in Nb3 Sn superconducting magnet systems by overheating and excessive voltages a Author to whom correspondence should be addressed; electronic mail: qiaoli@tyut.edu.cn 2158-3226/2016/6(12)/125101/8 6, 125101-1 © Author(s) 2016 125101-2 Yang et al AIP Advances 6, 125101 (2016) A preliminary study conducted by Mentink et al.4 experimentally determines the low temperature resistivity of bulk Nb3 Sn at various magnetic fields and strain states, and it is shown that the resistive curves exhibit three distinct features: (a) before the onset of the superconductivity, the decrease of the normal-state resistivity is nonlinear, and multi-field coupling; (b) under the combined application of magnetic field and strain, the zero resistive transition (tail part) becomes broadened; and (c) the application of multi-field drastically modifies the complete shape of the resistive transition, which shows high nonlinear characteristics and complicated coupling among the field variables Similar resistive transformations have also been found in pure Nb3 Sn single crystals.5 The same transition characteristics can also be acquired in magneto-resistance measurements of Nb3 Sn The experiments show that, for Nb3 Sn thin films6 and multifilamentary Nb3 Sn wires,7 near the critical magnetic field regions, the resistance initially drops almost linearly with magnetic field A significant deviation of the resistance from this magnetic-field dependence accompanying a sharp drop subsequently appears, and eventually a “zero-resistance” is achieved The experiments show that the transitions tend to broaden in high magnetic fields, and the shape of the magneto-resistive transition is apparently temperature dependent In the later study by Oguro et al.,8 the strain sensitivity of the magnetic resistance of CuNb/Nb3 Sn composite wire at the temperature of 4.2K is revealed A considerable increase of the nonlinearity in the resistive transition zone is observed when the Nb3 Sn material is under severe multi-field environment It is crucial to capture and model the multicoupling properties and performance of superconducting Nb3 Sn Godeke et al.7,9 suggests a resistivity model which can well describe characteristics of representative magneto-resistive transitions for the multifilamentary Nb3 Sn wires Unfortunately, the model only enable us to consider the single-field case, a general model accounting for the multi-field coupling characteristics needs to be defined In order to understand the multi-field coupled effects in Nb3 Sn, identify the unified scaling law for the superconducting properties, and yield insights into the origin of the multi-field coupling effects in the quench process of the Nb3 Sn-based superconducting magnet, having a comprehensive model for the strong nonlinear resistive transition characteristics helps, which provides a basis for many important scientific and technological problems related to high field magnet development Under the multi-field coupled environment, the resistive transition response features highly complex nonlinear characteristics, which arise from two principle sources: (a) as the Nb3 Sn material undergoes external magnetic fields and applied stresses, changes of the contribution from the electronelectron scattering, the electron-phonon scattering, and the electron density of states (DOS) at the Fermi surface to the temperature-dependent resistivity would be expected to occur, this contributes to the observed nonlinearities of the multi-field coupled normal-state resistivity responses;10 and (b) the nonlinear effects of transition broadening can be attributed to the coupling response of vortex state to the multi-field in Nb3 Sn, fluctuations typically dominating the superconducting transition included in the family of second-order phase transition for which the order parameter is continuous at the phase transition point, and so on.11,12 The peculiar coupling of the multiple fields complicates the construction of unified modeling and prediction of the anomalous resistive transition behavior as well as the normal-state transport behavior of Nb3 Sn material Development of a phenomenological modeling approach can balance the demand for a much better founded formalism and practical applicability to experimental observations A multi-field coupled phenomenological model for the resistive transition of Nb3 Sn is established in this paper, which makes good agreement with the observed experimental data The rest of the paper is structured as follows In Section II, the basic characteristics and details of the proposed model are described A verification of the model by comparing its predictions with the experimental literature data acquired on bulk Nb3 Sn and Nb3 Sn multifilamentary wire is shown in Sec III The relevant analysis of the parameterization of available sets of experimental data is also dealt with in this section Finally, Sec IV is devoted to summarize the conclusions II A MULTI-FIELD COUPLED RESISTIVE TRANSITION MODEL FOR SUPERCONDUCTING Nb3 Sn Inspired by the phenomenological success of the simple model proposed by Wiesmann et al.13 for the accurate description of normal-state electrical resistivity over a broad temperature range, it is 125101-3 Yang et al AIP Advances 6, 125101 (2016) assumed in this paper that the transition resistivity ρ can be described by a competition between the normal-state dependence, ρn , and some limiting value given by ρs representing the superconductingstate dependence, which is essentially “shunting” ρn A superconductor in parallel with a normal resistor is served as the basic model system, the resistivity of which thus can be written as ρ , = ρn + Π (1) with Π = ρn ρs A The multi-field coupled expression of the dimensionless resistance function Π Under applied multiple fields, when the Nb3 Sn sample undergoes a superconducting-to-normal transition, as mostly seen in the transport critical current measurements,7–10 accompanying with the complete disappearance of the superconducting phase, the increasing of the voltage typically follows the approximate exponential behavior Moreover, as the normal resistive behavior occurs, the initial exponential characteristics tail-off, allowing for the production of multiple-field dependent nonlinear resistivity responses From the shape of the typical resistivity-temperature transition curve, it clearly appears that the multiple field-induced resistive transition, situated around a midpoint temperature of T1/ (H, ε) (which is always used to define the superconducting transition temperature, and exhibits magnetic field H, and strain ε dependence), and with a coupled-field dependent width ∆TW (H, ε), is between the two limit behaviors: as the dimensionless temperature t ≡ T/T1/ (H, ε) varies from to 0, the transition has reached the superconducting state, indicating that the dimensionless resistance function Π should vary continuously and monotonically from to ∞; and with t increasingly deviating from 1, the transition reaches the normal-state resistance behavior, showing that the Π function should gradually step down from to in the range of half a transition width The scaling Π function which determines the ratio of the two extremes should vary smoothly across the transition interval Hinted by Kim et al.14 and Godeke et al.,9 a rewriting of the Arrhenius law gives 1−t ), (2) Π = exp( κ with the dimensionless parameter κ ≡ ∆TW (H, ε) T1/ (H, ε) describing the width of the transition zone The shifts in the midpoint temperature T1/ and the transition interval width ∆TW arise due to the combined application of magnetic field and strain distortion, which contribute to the complexity of the resistive shape contour and the shape area The magnetic field effect on the spin part of the Cooper pair is modulated by the spin-orbit scattering The sufficiently strong spin-orbit scattering produces the continuous transition to the normal state The spin-orbit scattering counteracts the effects of the spin paramagnetism in limiting the magnetic-field-dependent critical temperature T1/ (H).15 Moreover, the sub-lattice distortion of Nb chains will be induced by external application of strain,16 which modifies the phonon density of states and electronic band structures, leading to the intrinsic strain sensitivity in Nb3 Sn Noticing that the strain dependence of the critical surface in Nb3 Sn superconductors is through strain-dependent critical temperature T1/ and critical magnetic field H1/ ,17 it allows us to take into account the coupled effects of magnetic field and strain on T1/ in an implicit relation of the Werthamer, Helfand, and Hohenberg (WHH) theory,15 an empirically derived explicit form of which is given by T1/ (H, ε) T1/ (0, 0) = s(ε)1/ (1 − h)1/ 1.52 , (3a) with the dimensionless magnetic field h ≡ H H1/ (0, ε), in which H1/ (0, ε) H1/ (0, 0) = s(ε), (3b) here,T1/ (0, 0) and H1/ (0, 0), respectively, refers to the midpoint temperature and the midpoint magnetic field under no applied fields, and the function s(ε) accounts for the strain dependence of the critical properties in superconducting Nb3 Sn, and is evaluated with Ekin’s power-law description:18 s(ε) = − a |ε| u with a and u being the parameters defining the Ekin’s scaling law (4) 125101-4 Yang et al AIP Advances 6, 125101 (2016) Another important parameter defining the resistive transition is the transition width ∆TW , which is usually defined as19 ∆TW = T90% − T10% , here T90% and T10% are the temperatures corresponding to 90% and 10% of the resistivity jump, respectively The presence of multiple field induces a broadening of the transition width, and it may be qualitatively ascribed to a variety of causes, such as flux creep effects, fluctuation effects, the two gap superconductivity, and so on.19–21 Quantitative analysis on the coupled-field dependence of the resistive broadening behavior is imperative in deconvolving the data of the transition response and to reach an accurate description Based on the unified construction of the critical surface for Nb3 Sn, the role of strain is introduced through modification of the critical temperature and the upper critical field, a result of dimensional analysis in terms of key separation variables indicates ∆TW (H, ε) = KT1/ (0, ε)g(H, H1/ (0, ε)), where K is a dimensionless proportionality constant, and g(H, H1/ (0, ε)) is a non-dimensional function of the applied magnetic field H, and the strain-dependent upper critical field H1/ (0, ε) at zero temperature Drawing lessons from the Josephson-coupling model14 and the flux-dynamics-based Tinkham model22 proposed for the study of resistive transition in high-temperature superconductors, the magnetic field dependence of the transition width ∆TW ∝ (H1/ (0, ε) − H)α (where α is a fitting parameter whose physical meaning is related to the activation energy which must be overcome to allow flux motion and hence resistance) is assumed, giving that the expected function g is proportional to (1 − h)α The above analysis leads to the following expression for κ: α ¯ κ = Ks(ε) (1 − h)α−1/ 1.52 , (5) in which the non-dimensional parameter K¯ correlates to T1/ (0, 0) and H1/ (0, 0) B The expression for ρn characterizing the multi-field coupled effects on the normal-state electrical resistivity The temperature dependence of the normal-state resistivity ρn (T ) above the transition region, but typically below the martensitic temperature of about 50K, can be closely described by an expression of the form ρn (T ) = ρ0 + AT ,23–25 in which ρ0 is the residual resistivity and A is the quadratic coefficient The T dependence in the low temperature resistivity of A15 superconducting materials has been shown to be an intrinsic effect The mechanisms such as the electron-electron scattering23 and the electron-phonon interband scattering with non-Debye-like phonon structure25 have been argued to be inadequate to explain the dependence.24 A common consensus has not been arrived at concerning this relationship since which is the form expected from the mutual scattering of fermion quasiparticles within the standard theories of metals The normal-state of superconducting Nb3 Sn is a Fermi liquid with quasiparticle energies agreeing fairly well with eigenvalues obtained from the local-densityfunctional theory, whose transport properties can be described by the Bloch-Boltzmann theory26 resting on the Migdal approximation, in which the electron-phonon as the dominate interaction is presumed Multiple field-induced changes of the effects of the electron-electron scattering, the electron-phonon scattering, and the electron DOS at the Fermi level on the normal-state resistivity behavior happen, a detailed physical explanation for the coupling response is not yet clear but some plausible assumptions can be put forward, based on the experimental phenomenon and understanding gained from previous investigations on strain effects10 and magnetic field effects.11 It is postulated that the following assumptions of the normal-state resistivity model are fulfilled: (a) below the strainand magnetic field-dependent martensitic temperature, the T dependence still holds, the parameters ρ0 and A are not constants but vary with strain (the two parameters are expected to be magnetic field independent); and (b) effect of magnetic field on the temperature dependence is through the modified onset of the superconducting-to-normal transition T1/ , which is also influenced by strain exerting on the material The assumptions give f g ρn (T , H, ε) = ρ0 (ε) + ρn (T1/ (H, ε) − ρ0 (ε) t , (6a) in which ρn (T1/ (H, ε)) = ρ0 (ε) + A(ε)T12/2 (H, ε) (6b) Following and extending the proposal developed in previous researches on the normal-state transport properties of fullerene superconductors,27 the expression ρ0 (ε) = ρ00 + ρ01 exp(−k0 ε) (here, 125101-5 Yang et al AIP Advances 6, 125101 (2016) ρ00 and ρ01 are strain-independent constants, and k0 is a dimensionless proportionality constant) is here proposed to characterize the strain dependence of the residual resistivity in Nb3 Sn The strain dependence of the quadratic coefficient A is simply approximated by a linear function, ie, A(ε) = A(0) + kA ε (in which A(0) is the quadratic coefficient under no application of strain field, and kA is a constant characterizing the linear approximation) III DISCUSSION ON THE PREDICTIVE ABILITY OF THE PROPOSED MODEL To search for the reasons of the strain sensitivity in Nb3 Sn, an experiment conducted to measure the effects of coupled fields on the resistive transition is discussed in Ref The proposed model is utilized to determine the coupled effects of strain states, magnetic field, and temperature on the resistive transition behavior in bulk Nb3 Sn superconductor, a comparison of its predictions with experimental observations is presented in Figures and The optimized curves fitting yields sets of values for the parameters shown in Table I The value of the two parameters appearing in the Ekin strain FIG Low temperature resistivity at different magnetic fields and strain states The continuous lines represent the experimental observations(carried out by Mentink et al.4 ) fit using the developed model FIG The low temperature resistivity (at 19K) is nonlinearly reduced as the bulk Nb3 Sn sample is compressed The continuous line represents the experimental observations(carried out by Mentink et al.4 ) fit using the developed model 125101-6 Yang et al AIP Advances 6, 125101 (2016) TABLE I Fitting parameters used for the application of the proposed model to experimental data reported by Mentink et al.4 T1/ (0, 0) (K) H1/ (0, 0) (T) K¯ α 25.43 0.13 × 10−2 −2.10 17.82 ρ00 (µΩcm) 11.02 ρ01 (µΩcm) k0 A(0) (µΩcmK−2 ) kA (àcmK2 ) 2.60 86.58 0.58 ì 102 0.51 scaling law are set to be a = 1250 (the generally used value for the compressive-strained case) and u = 1.9 (it is a little higher than the commonly used 1.7 for multifilamentary Nb3 Sn wire, this quantitative difference between the bulk Nb3 Sn and the multifilamentary one has been demonstrated by the first principle calculations28 ) The extrapolated critical temperature and upper critical magnetic field are 17.82K and 25.43T, respectively, which is consistent with the reported data for bulk Nb3 Sn.17 The obtained resistivity of undistorted Nb3 Sn superconductors at 19K is 15.7 µΩcm, showing accordance with the literature results.29,30 It can be observed from the figures that the trends of the calculated curves of resistive transition to the normal state agree well with the experimental findings by Mentink et al The multi-field coupling characteristics of resistive transition in Nb3 Sn can be well reproduced by the proposed model For the magneto-resistive transition from normal to superconductive state, the core structure of the model (Eqs (1) and (2)) still holds The resistivity can be modeled by replacing the dimensionless temperature t with the dimensionless magnetic field h (h ≡ H/H1/ (T , ε), in which the coupled temperature and strain dependence of the midpoint magnetic field can be expressed as: H1/ (T , ε) = H1/ (0, 0)(1 − t 1.52 )s(ε)10,17 ) Generally speaking, the non-dimensional transition width for the magneto-resistive transition (κ ≡ ∆HW (T , ε)/H1/ (T , ε)) is temperature- and strain-dependent It is assumed here, for simplicity, that κ is a constant and adequately characterizes the transition at moderate temperatures and in low strain states Following the discussion on the detailed magnetotransport data on dense wires of MgB2 ,31 the magnetic field dependence of the normal-state resistivity ρn (H) is assumed to take the form ∆ρn (H) ρ0 ∝ H β , where β is a fitting parameter By performing an analogous analysis as above in the Section II B, it is found that the expression for the magneto-resistive transition can be written as ρ(H, T , ε) , (7a) = ρ0 (ε) + (ρn (H1/ (T , ε)) − ρ0 (ε))hβ + exp( −κ h ) where β ρn (H1/ (T , ε)) = ρ0 (ε) + B(ε)H1 (T , ε) (7b) / Being similar with A(ε), the format of the function B(ε) is assumed to be a linear function under low strain levels Here, ρn denotes the normal-state resistivity of the Nb3 Sn composites, which includes the contribution from the copper magnetoresistance,17 since in the resistive transition measurements, it is practically impossible to measure the resistive transition without the cooper sheath around the Nb3 Sn superconductor The simplified forms of the derived Eqs (7a) and (7b) are directly utilized to characterize the experimental observations given by Godeke et al.,7 in which the magneto-resistive transition of filamentary Nb3 Sn conductor in different temperatures (under no applied strain) is studied (see Figure 3) The values of the optimized parameters for the proposed model are itemized in Table II The deduced values of the critical temperature and the upper critical field are, respectively, 17.57K and 29.36, which are in accord with the reported experimental results.7 The obtained residual resistivity parameter ρ(0) for the multifilamentary Nb3 Sn composite conductor is far lower than the one determined from the experiment carried out by Mentink et al It is owing to that the residual resistivity (denoted by ρ00 + ρ01 ) is sample-dependent, which has been shown by many experiments.32–35 The shape of the magneto-resistive transition described by Eqs 7(a) and 7(b) fits well to the measured curves It is well to mention that although phenomenological in origin, the numerical simulations performed adopting the described model show reasonably good agreements with the experimental 125101-7 Yang et al AIP Advances 6, 125101 (2016) FIG The magneto-resistivity at various temperatures The continuous lines represent the experimental observations(carried out by Godeke et al.7 ) fit using the developed model TABLE II Fitting parameters used for the application of the proposed model to experimental data reported by Godeke et al.7 T1/ (0, 0) (K) H1/ (0, 0) (T) κ ρ(0) (µΩcm) B(0) (µΩcmT−2 ) β 29.36 0.01 0.06 0.13 0.7 17.57 observations presented by Mentink et al.4 and by Godeke et al.7 The apparent simplicity with which the proposed model puts the identification of effects of both the coupling response of vortex state to the multi-field and the multiple field-induced phonon disorder and structural disorder on the resistive transition in Nb3 Sn on a similar basis makes the authors believe that there is some physical basis in the introduced phenomenological scheme It should be also noted that the intermetallic phases Nb3 Sn are formed by a diffusion reaction, which necessitates the Sn composition gradients across the Nb3 Sn layer This composition effects on the resistive and magnetic transition should be included in the future studies for the further comprehensive understanding of the Nb3 Sn material, it is another essential part of the research The resistivity as function of atomic Sn content can be well described by a fourth power fit proposed by Flăukiger et al,36 the combination of which and the model development in this paper is an important character to reflect the complex coupling in the practical engineering applications of superconducting Nb3 Sn IV CONCLUSIONS The following are the main conclusions of this paper: (1) A resistive transition model for describing the multi-field coupling characteristics of superconducting Nb3 Sn is proposed A good agreement of the theoretical predictions and experimental observations on the resistive transition behavior of bulk Nb3 Sn and multifilamentary Nb3 Sn composite wire is achieved The findings reported here offer a modeling approach for understanding the complex multi-field coupled effects in superconducting Nb3 Sn (2) The model implies some preliminary information about the basic vortex-pinning mechanisms governing the critical current density stored in the coupled experimental data (3) The proposed resistive transition model is useful for making objective determinations of the high-dimensional critical surface of Nb3 Sn in the multi-parameter space, and enabling a better understanding of the design of the quench protection system of Nb3 Sn superconducting magnets 125101-8 Yang et al AIP Advances 6, 125101 (2016) ACKNOWLEDGMENTS This work is supported by the National Natural Science Foundation of China(Grant Nos 11402159 and 51402206) A Devred, I Backbier, D Bessette, G Bevillard, M Gardner, M Jewell, N Mitchell, I Pong, and A Vostner, IEEE Trans Appl Supercond 22, 4804909 (2012) Wada and T Kiyoshi, IEEE Trans Appl Supercond 12, 715 (2002) V Zlobin, N Andreev, G Apollinari, B Auchmann, E Barzi, S Izquierdo Bermudez, R Bossert, M Buehler, G Chlachidze, J DiMarco, M Karppinen, F Nobrega, I Novitski, L Rossi, D Smekens, M Tartaglia, D Turrioni, and G Velev, IEEE Trans Appl Supercond 25, 4002209 (2015) M G T Mentink, M M J Dhalle, D R Dietderich, A Godeke, W Goldacker, F Hellman, and H H J ten Kate, AIP Conf Proc 1435, 225 (2012) M Reibelt, A Schilling, and N Toyota, Phys Rev B 81, 094510 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L Qiao, L Yang, and X J Zheng, J Appl Phys 114, 033905 (2013) 11 R Kadono, K H Satoh, A Koda, T Nagata, H Kawano-Furukawa, J Suzuki, M Matsuda, K Ohishi, W Higemoto, S Kuroiwa, H Takagiwa, and... simplicity, that κ is a constant and adequately characterizes the transition at moderate temperatures and in low strain states Following the discussion on the detailed magnetotransport data on dense