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超导薄膜磁-热不稳定与强非线性电磁本构的关联性

王垚 姜璐 周又和 薛存

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超导薄膜磁-热不稳定与强非线性电磁本构的关联性

王垚, 姜璐, 周又和, 薛存

Dependence of thermomagnetic instability on strong nonlinear E-J models in superconducting films

Wang Yao, Jiang Lu, Zhou You-He, Xue Cun
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  • 常规导体的电磁本构关系一般满足线性欧姆定律, 然而超导体的电磁本构关系呈现很强的非线性特征, 所以与常规导体相比, 超导有截然不同的电磁特性. 本文基于超导材料E-J幂次律本构关系, 采用快速傅里叶变换方法(FFT), 定量研究了不同环境温度、磁场加载速率以及临界电流密度条件下的超导薄膜磁-热不稳定性与非线性本构的关联性, 揭示了强非线性电磁本构是导致超导薄膜磁-热不稳定性(呈现树状、指状磁通崩塌形貌)的重要因素, 同时阐明了常规导体观测不到类似的磁-热不稳定现象的原因. 另外发现由于超导薄膜抗磁性的增强导致超导薄膜边界磁场迅速增大, 较大的磁压极易诱发磁通崩塌, 所以超导薄膜内磁通崩塌阈值随幂指数的增加而降低. 最后给出了$n_0\text{-}j_{{\rm{c}}0}$$n_0\text{-}\mu_0\dot {H}_{\rm{a}}$平面内不同非线性程度下超导薄膜内磁热不稳定状态的分界线.
    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.
      通信作者: 薛存, xuecun@nwpu.edu.cn
    • 基金项目: 国家自然科学基金(批准号: 11972298, 12011530143)资助的课题
      Corresponding author: Xue Cun, xuecun@nwpu.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11972298, 12011530143)
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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

    Fig. 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/s

    Fig. 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/s

    Fig. 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

    Fig. 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

    Fig. 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

    Fig. 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

    Fig. 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 $ K

    Fig. 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

    Fig. 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

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    [2]

    Brinkman A, Veldhuis D, Mijatovic D, Rijnders G, Blank D 2001 Appl. Phys. Lett. 79 2420Google 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 229Google Scholar

    [5]

    周又和, 王省哲 2013 中国科学: 物理学 力学 天文学 43 1558Google Scholar

    Zhou Y H, Wang S Z 2013 Sci. China Phys. Mech. Astron. 43 1558Google Scholar

    [6]

    Altshuler E, Johansen T H 2004 Rev. Mod. Phys. 76 471Google Scholar

    [7]

    Vestgården J I, Johansen T H, Galperin Y M 2018 J. Low Temp. Phys. 44 460Google 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 306Google Scholar

    [10]

    McDonald J, Clem J R 1996 Phys. Rev. B 53 8643Google Scholar

    [11]

    Johansen T H, Bratsberg H 1995 J. Appl. Phys. 77 3945Google Scholar

    [12]

    Shantsev D V, Galperin Y M, Johansen T H 2000 Phys. Rev. B 61 9699Google Scholar

    [13]

    Sandvold E, Rossel C 1992 Physica C 190 309Google 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 247Google Scholar

    [16]

    Konczykowski M, Vinokur V M, Rullier-Albenque F, Yeshurun Y, Holtzberg F 1993 Phys. Rev. B 47 5531Google Scholar

    [17]

    Kupfer A, Kresse R, Meier-Hirmer R, Salama K, Lee D, Selvamanickam V 1993 Physica C 209 243Google Scholar

    [18]

    Anderson P W 1962 Phys. Rev. Lett. 9 309Google Scholar

    [19]

    Kim Y B, Hempstead C F, Strnad A R 1963 Phys. Rev. 131 2486Google Scholar

    [20]

    Zeldov E, Amer N M, Koren G, Gupta A, Mcelfresh M W 1989 Phys. Rev. Lett. 62 3093Google Scholar

    [21]

    Nideröst M, Suter A, Visani P, Mota A C, Blatter G 1996 Phys. Rev. B 53 9286Google Scholar

    [22]

    Ban M, Ichiguchi T, Onogi T 1989 Phys. Rev. B 40 4419Google Scholar

    [23]

    Zeldov E, Amer N M, Koren G, Gupta A, McElfresh M W, Gambino R J 1990 Appl. Phys. Lett. 56 680Google Scholar

    [24]

    Gurevich A, Brandt E H 1994 Phys. Rev. Lett. 73 178Google Scholar

    [25]

    Schuster T, Kuhn H, Brandt E H 1995 Phys. Rev. B 51 697Google Scholar

    [26]

    Schuster T, Kuhn H, Brandt E H, Indenbom M, Koblischka M R, Konczykowski M 1994 Phys. Rev. B 50 16684Google Scholar

    [27]

    Schuster T, Indenbom M V, Kuhn H, Brandt E H, Konczykowski M 1994 Phys. Rev. Lett. 73 1424Google Scholar

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    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 10375Google Scholar

    [29]

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    Vestgården J, Shantsev D, Galperin Y, Johansen T 2008 Phys. Rev. B 77 014521Google Scholar

    [31]

    Brandt E H 1995 Rep. Prog. Phys. 58 1465Google Scholar

    [32]

    Rhyner J 1993 Physica C 212 292Google Scholar

    [33]

    Prigozhin L, Sokolovsky V 2011 Supercond. Sci. Technol. 24 075012Google Scholar

    [34]

    Grilli F, Pardo E, Stenvall A, Nguyen D N, Yuan W, Gömöry F 2014 IEEE Trans. Appl. Supercond. 24 78Google Scholar

    [35]

    Hong Z, Coombs T A 2010 J. Supercond. Novel Magn. 23 1551Google Scholar

    [36]

    Vestgården J I, Shantsev D V, Galperin Y M, Johansen T H 2011 Phys. Rev. B 84 054537Google Scholar

    [37]

    Vestgården J I, Mikheenko P, Galperin Y M, Johansen T H 2013 New J. Phys. 15 093001Google Scholar

    [38]

    Motta M, Colauto F, Vestgården J I, Fritzsche J, Silhanek A V 2014 Phys. Rev. B 89 134508Google Scholar

    [39]

    Jing Z, Yong H D, Zhou Y H 2015 Supercond. Sci. Technol. 28 075012Google Scholar

    [40]

    Kim Y B, Hempstead C F, Strnad A R 1963 Phys. Rev. 129 528Google 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 024518Google Scholar

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    Mints R G 1996 Phys. Rev. B 53 12311Google Scholar

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    Zhou Y H, Yang X B 2006 Phys. Rev. B 74 054507Google Scholar

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出版历程
  • 收稿日期:  2022-02-17
  • 修回日期:  2022-05-01
  • 上网日期:  2022-10-07
  • 刊出日期:  2022-10-20

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