-
The
$E\text{-}J$ relationship in conventional conductor generally satisfies the linear Ohm's law. However, the$E\text{-}J$ model in superconductors presents strong nonlinear characteristics, which is significantly different from that of the conventional conductor. According to the nonlinear$E\text{-}J$ power law of superconducting materials, we quantitatively investigate the relationship between the magnetic-thermal stability and the nonlinear constitutive characteristic of superconducting films at different temperatures, magnetic field ramp rates, and critical current densities by using the fast Fourier transform method (FFT). We find that the strong nonlinear electromagnetic constitutive model plays a crucial role responsible for the onset and morphology (tree-like and finger-like) of the magneto-thermal instability of superconducting thin films. In addtion, the reason why similar magneto-thermal instabilities cannot be observed in conventional conductors is also explained. It can be found that the magnetic field on the border of the superconducting film increases rapidly for a larger creep exponent due to the enhancement of diamagnetism, which results in a large magnetic pressure and easily triggering off flux avalanches. Therefore, the threshold field of flux avalanches in the superconducting film decreases with flux creep exponent increasing. Finally, we present the curves that can clearly divide the$n_0\text{-}j_{c0}$ plane and$n_0\text{-}\dot {H}_a$ plane into magneto-thermal stability region and magneto-thermal instability region for superconducting thin film with different levels of nonlinearity.-
Keywords:
- superconducting thin films /
- magneto-thermal instability /
- magnetic flux penetration /
- nonlinear E-J model
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图 1 (a)超导薄膜-基底系统示意图, 外加磁场随着时间线性增加并始终垂直于超导薄膜表面, 其中超导薄膜大小为
$ w\times w $ , 数值模拟区域大小为$ 2 L_x\times 2 L_y $ ; (b)超导材料$E\text{-}J$ 幂次本构关系, n为磁通蠕动指数, 图中黄色填充区域对应常见超导材料的$E\text{-}J$ 本构范围, 常规导体对应n = 1, 即欧姆定律Figure 1. (a) Schematic diagram of the superconducting film-substrate system, where the applied magnetic field increases linearly with time and is always perpendicular to the surface of the superconducting film, where the superconducting film size is w × w and the numerical simulation region size is 2Lx × 2Ly; (b) E-J power instanton relationship for superconducting material, n is the flux creep index. The yellow filled area in the figure corresponds to the E-J instantonal range of common superconducting materials. The conventional conductor corresponds to n = 1, denoting the Ohm’s law
图 2 通过数值模拟得到不同参数
$ n_0 $ 下的超导薄膜分别在外加磁场为$\mu_0 H_{\rm{a}}$ = 1.8 mT ((a), (c), (e))和$\mu_0 H_{\rm{a}}$ = 4.0 mT((b), (d), (f))时的磁场分布. 背景温度$ T_0 $ = 2.5 K, 磁场变化率为5 T/sFigure 2. Flux distributions of superconducting thin films with different parameter
$ n_0 $ at the applied magnetic fields of$ \mu_0 H_{\rm{a}} $ = 1.8 mT ((a), (c), (e)) and$ \mu_0 H_{\rm{a}} $ = 4.0 mT ((b), (d), (f)). The substrate temperature is$ T_0 $ = 2.5 K and the ramp rate is$ \mu_0\dot{H}_{\rm{a}} $ = 5 T/s
图 3 (a)磁通崩塌阈值随着参数
$ n_0 $ 的变化规律, 曲线下方表示薄膜保持磁热稳定状态, 曲线上方表示薄膜出现磁热不稳定; (b)不同$ n_0 $ 下超导薄膜的最大温度随外加磁场的变化; (c)不同$ n_0 $ 下的磁化曲线图. 数值模拟的背景温度为$ T_0 = 2.5 $ K, 磁场变化率为5 T/sFigure 3. (a) The threshold field for the onset of flux avalanches in superconducting films with different
$ n_0 $ . The lower region indicates the film is in magneto-thermal stable state, while the upper region indicates the thermomagnetic instability. (b) Maximum temperature and (c) magnetic moment in superconducting films as a function of increasing applied field for three different$ n_0 $ . The substrate temperature is$ T_0 $ = 2.5 K and the ramp rate is$\mu_0\dot{H}_{\rm{a}}$ = 5 T/s
图 4 数值模拟了不同温度场下(
$ T_0 $ = 1.5, 2.5, 3.0 K), 参数$ n_0 $ 为3, 18, 29时的超导薄膜在相同磁场$\mu_0 H_{\rm{a}}$ = 3.1 mT下的磁场分布Figure 4. Distribution of magnetic field
$ B_z $ in superconducting fillms at the same applied field$\mu_0 H_{\rm{a}}$ = 3.1 mT with$ n_0 $ = 3, 18, 29 and$ T_0 $ = 1.5, 2.5, 3.0 K
图 5 通过数值模拟得到了外加垂直磁场分别为
$\mu_0 H_{\rm{a}}$ = 1.6, 6.2 mT时, 不同临界电流密度下的超导薄膜内磁场分布Figure 5. Magnetic flux distribution in supercondeucting films with different critical current densities at
$\mu_0 H_{\rm{a}}$ = 1.6, 6.2 mT
图 6 超导薄膜内磁通崩塌阈值随临界电流密度
$j_{{\rm{c}}0}$ 的变化. 曲线上方表示超导薄膜内磁热不稳定区域, 曲线下方表示超导薄膜内保持磁热稳定状态Figure 6. The threshold field
$\mu_0 H_{\rm{a}}$ for the onset of flux avalanches as a function of critical current density$j_{{\rm{c}}0}$ The lower region indicates the film is in magneto-thermal stable state, while the upper region indicates the thermomagnetic instability
图 7
$ n_0 $ $\text- j_{{\rm{c}}0}$ 平面内超导薄膜磁热稳定性/不稳定性的范围及分界线, 图中黄色区域表示薄膜磁热不稳定, 青色区域表示薄膜保持磁热稳定状态, 误差棒表示分界线的精度Figure 7. Thermomagnetic stability/instability diagram in the
$ n_0\text{-} j_{{\rm{c}}0} $ planes. Yellow and green denote the regions of flux avalanches and smooth penetration. The error bars show the accuracy of the dividing lines
图 8 磁场变化率为
$ {\mu_0{\dot{{\rm{H}}}}_{\rm{a}} } $ = 2 T/s,$ { \mu_0\dot{{\rm{H}}}_{\rm{a}} } $ = 9 T/s,$ { \mu_0\dot{{\rm{H}}}_{\rm{a}} } $ = 15 T/s情形下的薄膜内磁场分布 (a), (c), (e) 表示外加磁场加载到1.8 mT时的薄膜内部磁场分布; (b), (d), (f)表示外加磁场加载到4.0 mT时的薄膜内部磁场分布. 背景温度场为$ T_0 = 2.5 $ KFigure 8. Magnetic field distribution in thin film at
$ \mu_0 H_a $ = 1.8 mT ((a), (c), (e)) and 4.0 mT ((b), (d), (f)) for$\mu_0\dot{H}_{\rm{a}}$ = 2, 9 and 15 T/s. The substrate temperature is$ T_0 $ = 2.5 K
图 9
$n_0\text{-}{ \mu_0\dot{H}_{\rm{a}}}$ 平面内超导薄膜磁热稳定性/不稳定性的范围及分界线, 黄色区域表示薄膜内部磁热不稳定, 青色区域表示薄膜保持磁热稳定状态, 误差棒表示分界线的精度Figure 9. Thermomagnetic stability/instability diagram in the
$n_0\text{-} { \mu_0\dot{H}_{\rm{a}}}$ planes. Yellow and green denote the regions of flux avalanches and smooth penetration, respectively. The error bars show the accuracy of the dividing lines -
[1] Husse E, Chauvel D, Thiver C, Guilloux-Viry M, Carru J C, Perrin A 1994 Physica C 235 3379
[2] Brinkman A, Veldhuis D, Mijatovic D, Rijnders G, Blank D 2001 Appl. Phys. Lett. 79 2420
Google Scholar
[3] Gallagher W J, Harris E P, Ketchen M B 2012 IEEE Trans. Appl. Supercond. 21 1
[4] Saleh A E, Abu-Samreh M M, Al-Awaysa G M, Kitaneh M L 2008 J. Supercond. Novel Magn. 21 229
Google Scholar
[5] 周又和, 王省哲 2013 中国科学: 物理学 力学 天文学 43 1558
Google Scholar
Zhou Y H, Wang S Z 2013 Sci. China Phys. Mech. Astron. 43 1558
Google Scholar
[6] Altshuler E, Johansen T H 2004 Rev. Mod. Phys. 76 471
Google Scholar
[7] Vestgården J I, Johansen T H, Galperin Y M 2018 J. Low Temp. Phys. 44 460
Google Scholar
[8] Bean C P 1964 Rev. Mod. Phys. 36 886
[9] Kim Y B, Hempstead C F, Strnad A R 1962 Phys. Rev. Lett. 9 306
Google Scholar
[10] McDonald J, Clem J R 1996 Phys. Rev. B 53 8643
Google Scholar
[11] Johansen T H, Bratsberg H 1995 J. Appl. Phys. 77 3945
Google Scholar
[12] Shantsev D V, Galperin Y M, Johansen T H 2000 Phys. Rev. B 61 9699
Google Scholar
[13] Sandvold E, Rossel C 1992 Physica C 190 309
Google Scholar
[14] Konczykowski M, Chikumoto N, Vinokur V M, Feigelman M V 1995 Phys. Rev. B 51 3957
[15] Konczykowski M 1993 Physica C 209 247
Google Scholar
[16] Konczykowski M, Vinokur V M, Rullier-Albenque F, Yeshurun Y, Holtzberg F 1993 Phys. Rev. B 47 5531
Google Scholar
[17] Kupfer A, Kresse R, Meier-Hirmer R, Salama K, Lee D, Selvamanickam V 1993 Physica C 209 243
Google Scholar
[18] Anderson P W 1962 Phys. Rev. Lett. 9 309
Google Scholar
[19] Kim Y B, Hempstead C F, Strnad A R 1963 Phys. Rev. 131 2486
Google Scholar
[20] Zeldov E, Amer N M, Koren G, Gupta A, Mcelfresh M W 1989 Phys. Rev. Lett. 62 3093
Google Scholar
[21] Nideröst M, Suter A, Visani P, Mota A C, Blatter G 1996 Phys. Rev. B 53 9286
Google Scholar
[22] Ban M, Ichiguchi T, Onogi T 1989 Phys. Rev. B 40 4419
Google Scholar
[23] Zeldov E, Amer N M, Koren G, Gupta A, McElfresh M W, Gambino R J 1990 Appl. Phys. Lett. 56 680
Google Scholar
[24] Gurevich A, Brandt E H 1994 Phys. Rev. Lett. 73 178
Google Scholar
[25] Schuster T, Kuhn H, Brandt E H 1995 Phys. Rev. B 51 697
Google Scholar
[26] Schuster T, Kuhn H, Brandt E H, Indenbom M, Koblischka M R, Konczykowski M 1994 Phys. Rev. B 50 16684
Google Scholar
[27] Schuster T, Indenbom M V, Kuhn H, Brandt E H, Konczykowski M 1994 Phys. Rev. Lett. 73 1424
Google Scholar
[28] Schuster T, Kuhn H, Brandt E H, Indenbom M V, Kläser M, Müller-Vogt G, Habermeier H U, Kronmüller H, Forkl A 1995 Phys. Rev. B 52 10375
Google Scholar
[29] Vestgården J, Shantsev D, Galperin Y, Johansen T 2007 Phys. Rev. B 76 174509
Google Scholar
[30] Vestgården J, Shantsev D, Galperin Y, Johansen T 2008 Phys. Rev. B 77 014521
Google Scholar
[31] Brandt E H 1995 Rep. Prog. Phys. 58 1465
Google Scholar
[32] Rhyner J 1993 Physica C 212 292
Google Scholar
[33] Prigozhin L, Sokolovsky V 2011 Supercond. Sci. Technol. 24 075012
Google Scholar
[34] Grilli F, Pardo E, Stenvall A, Nguyen D N, Yuan W, Gömöry F 2014 IEEE Trans. Appl. Supercond. 24 78
Google Scholar
[35] Hong Z, Coombs T A 2010 J. Supercond. Novel Magn. 23 1551
Google Scholar
[36] Vestgården J I, Shantsev D V, Galperin Y M, Johansen T H 2011 Phys. Rev. B 84 054537
Google Scholar
[37] Vestgården J I, Mikheenko P, Galperin Y M, Johansen T H 2013 New J. Phys. 15 093001
Google Scholar
[38] Motta M, Colauto F, Vestgården J I, Fritzsche J, Silhanek A V 2014 Phys. Rev. B 89 134508
Google Scholar
[39] Jing Z, Yong H D, Zhou Y H 2015 Supercond. Sci. Technol. 28 075012
Google Scholar
[40] Kim Y B, Hempstead C F, Strnad A R 1963 Phys. Rev. 129 528
Google Scholar
[41] Mints R G, Rakhmanov A L 2008 Rev. Mod. Phys. 53 551
[42] Nabialek A, Niewczas M, Dabkowska H, Dabkowski A, Castellan J P, Gaulin B D 2003 Phys. Rev. B 67 024518
Google Scholar
[43] Mints R G 1996 Phys. Rev. B 53 12311
Google Scholar
[44] Zhou Y H, Yang X B 2006 Phys. Rev. B 74 054507
Google Scholar
[45] Huang Y, Zhang X, Zhou Y H 2016 Supercond. Sci. Technol. 29 075009
Google Scholar
[46] Zhang W, Xia J, Yong H D, Zhou Y H 2020 AIP Adv. 10 025021
Google Scholar
[47] Gou X F, Zheng X J, Zhou Y H 2007 IEEE Trans. Appl. Supercond. 17 3795
Google Scholar
[48] Wang Q Y, Xue C, Chen Y Q, Ou X J, Wu W, Liu W, Ma P, Sun L T, Zhao H W, Zhou Y H 2022 Physica C 593 1354002
Google Scholar
[49] Denisov D V, Shantsev D V, Galperin Y M, Choi E M, Lee H S, Lee S I, Bobyl A V, Goa P E, Olsen A A F, Johansen T H 2006 Phys. Rev. Lett. 97 077002
Google Scholar
[50] Albrecht J, Matveev A T, Djupmyr M, Schuetz G, Stuhlhofer B, Habermeier H U 2005 Appl. Phys. Lett. 87 182501
Google Scholar
[51] Durán C A, Gammel P L, Miller R E, Bishop D J 1995 Phys. Rev. B 52 75
Google Scholar
[52] Welling M S, Westerwaal R J, Lohstroh W, Wijngaarden R J 2004 Physica C 411 11
Google Scholar
[53] Alvarez S B, Brisbois J, Melinte S, Kramer R B G, Silhanek A V 2019 Sci. Rep. 9 3659
Google Scholar
[54] Leiderer P, Boneberg J, Brüll P, Bujok V, Herminghaus S 1993 Phys. Rev. Lett. 71 2646
Google Scholar
[55] Baruch-El E, Baziljevich M, Shapiro B Y, Johansen T H, Shaulov A, Yeshurun Y 2016 Phys. Rev. B 94 054509
Google Scholar
[56] Biehler B, Runge B U, Wimbush S C, Holzapfel B, Leiderer P 2005 Supercond. Sci. Technol. 18 385
Google Scholar
[57] Wang C H, Liu C, Zhang X Y 2021 AIP Adv. 11 045101
Google Scholar
[58] Wang C H, Liu C, Zhang X Y, Zhou Y H 2021 Exp. Mech. 61 1227
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[65] Lu Y R, Jing Z, Yong H D, Zhou Y H 2016 Proc. R. Soc. A 472 20160469
Google Scholar
[66] Jing Z, Yong H D, Zhou Y H 2016 Supercond. Sci. Technol. 29 105001
Google Scholar
[67] Jing Z, Yong H D, Zhou Y H 2017 J. Appl. Phys. 121 023902
Google Scholar
[68] Jiang L, Xue C, Burger L, Vanderheyden B, Zhou Y H 2020 Phys. Rev. B 101 224505
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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