1.Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 2.School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China 3.Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai 200050, China 4.Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China 5.Songshan Lake Materials Laboratory, Dongguan 523808, China
Abstract:The magnetic penetration depth (λ) of a superconductor is an important parameter which connects the macroscopic electrodynamics with the microscopic mechanism of superconductivity. High-accuracy measurement of λ is of great significance for revealing the pairing mechanism of superconductivity and exploring the applications of superconductors. Among various methods used to measure λ of superconducting films, the two-coil mutual inductance (MI) technique has been widely adopted due to its high precision and simplicity. In this paper, we start with introducing the principle of MI technique and pointing out that its accuracy is mainly limited by the uncertainties in the geometric parameters (e.g. the distance between two coils) and the leakage flux around the film edge. On this basis, we build a homemade transmission-type MI device with a delicate design to achieve high-accuracy. Two coils are fixed by a single-crystal sapphire block machined with high precisions to minimize the uncertainty in geometry. As a result, the reproducibility in induced voltage measured with sample remounted is better than 4%. Besides, the flux leakage around the film edge is accurately determined by measuring a thick Nb film and Nb foils. The voltage induced by leakage flux is only around 1% of that measured in the normal state. Therefore, the absolute value of λ can be accurately extracted after flux leakage subtraction and normalization. It is shown that the error of the measured λ is less than 10% for a typical superconducting film with a thickness of 100 nm and a penetration depth of 150 nm. Furthermore, the performance of our apparatus is tested on epitaxial NbN films with thickness of 6.5 nm. The results show that the low temperature variation of superfluid density is well described by the dirty s-wave BCS theory, and at temperatures close to Tc, the superfluid density decrease drastically, owing to the Berezinski-Kosterlitz-Thouless transition transition. Moreover, the zero-temperature magnetic penetration depth and the superconducting energy gap extracted from the fitting parameters are both consistent with the reported values. Our device provides an ideal platform for carrying out detailed studies of the dependence of λ on temperature, chemical composition and epitaxial strain, etc. It could also be utilized to characterize other parameters of superconductor such as the critical current density, and when combined with the ionic liquid gating technique, our device offers an efficient route for revealing the microscopic mechanism of superconductivity. Keywords:magnetic penetration depth/ two-coil mutual inductance technique/ NbN superconducting film/ Meissner effect
前文中提到, 合理的线圈参数是保证扣漏磁方法有效的重要前提. 上世纪九十年代末, Claassen等[28]给出线圈设计上的两条指导性原则: 第一, 为了减小漏磁并提高信号强度, 线圈间距应当尽可能小; 第二, 尽可能减小线圈的高度. 经过综合考虑, 我们确定线圈间距h为0.9 mm, 线圈内直径为0.5 mm, 外直径为1.3 mm, 高度为1.6 mm, 共300匝. 下面利用文献[30]中的数值方法验证线圈参数设计的合理性. 为方便计算, 考虑一厚度d = 100 nm、半径为R、穿透深度λ = 150 nm 的圆形超导薄膜. 利用矩阵法[30]对积分方程(2)进行求解, 并计算出该系统的互感系数Msample(R), 如图2(a)中的红色曲线所示. 取λ = 0, 就得到了该薄膜的漏磁M1(R), 如图2(a)中的蓝色曲线所示. 若(4)式成立, ${M_{{\rm{sample}}}}\left( R \right) - {M_1}\left( R \right)$等于无限大尺寸薄膜的互感系数, 应是一个不随薄膜半径变化的常数. 我们的计算结果的确与之相符, 如图2(a)中的黑色曲线所示. 此外, 为了模拟实际的数据处理过程, 我们将Msample(R)及M1(R)代入(7)式, 反解出穿透深度计算值λcal(R). 图2(b)中的黑色曲线是利用本文装置的实际测量值h = 0.9 mm计算得到的λcal(R)曲线, 在R = 2.5-5 mm范围内均与实际穿透深度150 nm(图2(b)中的虚线)十分接近. 作为对比, 图2(b)也给出了将线圈间距扩大为原来的5倍(h = 4.5 mm)和10倍(h = 9 mm)后计算得到的λcal(R)曲线, 两种情况均在样品半径较小时明显偏离穿透深度实际值. 这是由于当线圈间距较大时, 样品边缘处的屏蔽电流对互感系数的贡献不可忽略, 此时互感系数Msample不再能简单地分解为M∞与M1之和[29]. 总之, 图2中的数值计算结果均表明本文装置线圈参数的选择是合理的. 图 2 (a) d = 100 nm, λ = 150 nm的超导薄膜的互感系数随薄膜半径R的变化曲线; (b)基于不同的线圈间距(h = 0.9, 4.5, 9.0 mm) 得到的穿透深度计算值随薄膜半径R的变化曲线, 虚线代表实际穿透深度λ = 150 nm Figure2. (a) The mutual inductance as a function of film radii R calculated for the typical superconducting film with d = 100 nm, λ = 150 nm; (b) calculations of penetration depth λcal vs film radii R for different spacings between two coils (h = 0.9, 4.5, 9.0 mm). The real penetration depth (λ = 150 nm) is indicated by the dotted line.
23.3.系统校验及误差分析 -->
3.3.系统校验及误差分析
图3(a)给出两次测量同一铌膜得到的感生电压-温度曲线Vx, 1(T)及Vx, 2(T). 铌膜使用磁控溅射方法生长, 厚度为350 nm, 衬底为5 × 5 × 0.5 mm3 MgO单晶. 结果显示Vx的重复率达到96%以上. 由于铌膜的厚度远大于其穿透深度(约40 nm), 因此可以认为此时感生电压的实部Vx(T ≈ 4.5 K)就是系统的漏磁[32]. 经过多次重复测量, 得到系统漏磁M1 = 7.32 ± 0.05 nH, 其平均值$\overline {{M_1}} = 7.32\;{\rm{nH}}$, 仅占正常态互感的1.01%, 误差ΔM1 = 0.05 nH, 仅占平均值$\overline {{M_1}} $的0.68%, 主要来自于铌膜与线圈相对位置的变化及仪表的噪声. 为了进一步验证所测漏磁的可靠性, 我们还多次测量了尺寸相同而厚度为0.22 mm的铌箔的低温互感实部, 与铌膜给出的漏磁值仅相差1.8%. 图 3 (a)两次测量同一片铌膜得到的感生电压Vx, 1(T)及Vx, 2(T); (b)铌膜的感生电压V(T = 4.5 K)随频率的依赖关系 Figure3. (a) The induced voltage data Vx, 1(T) and Vx, 2(T) taken from the same Nb film with sample remounted; (b) the frequency dependence of induced voltage V(T = 4.5 K) for the Nb film.
4.NbN薄膜的磁场穿透深度测量为了检验装置的精度, 我们测量了研究较多的s波BCS超导体NbN. NbN超薄膜使用磁控溅射方法生长, 厚度为6.5 ± 0.2 nm, 衬底为 5 × 5 × 0.5 mm3 MgO单晶. 图4(a)给出了NbN薄膜的感生电压-温度曲线. 可以看出, 当样品进入超导态时, 感生电压实部Vx迅速下降, 这反映了薄膜的抗磁性; 感生电压虚部Vy则呈现峰状, 其展宽能够反映样品的均匀性. 我们利用前述方法得到了样品的穿透深度-温度曲线λ(T), 如图4(b)中的黑色圆圈所示, 可以发现低温段的λ(T)十分平缓, 这意味着材料的超导能隙没有节点. 图4(b)中的其他数据来自同批生长的另外3片NbN薄膜, 结果非常接近. 图 4 NbN薄膜(NbN#1, NbN#2, NbN#3, NbN#4)的双线圈互感测量结果 (a) NbN#1样品的感生电压曲线Vx(T)及Vy(T); (b)四个样品的穿透深度随温度变化曲线λ(T); (c) NbN#1样品的超流密度-温度曲线${{\rm{\lambda }}^{ - 2}}\left( T \right) \propto {n_{\rm{s}}}\left( T \right)$, 黑色实线是脏极限BCS理论的拟合结果; (d)四块样品的穿透深度零温外延值λ (T → 0)与Tc的关系, 符合文献报道趋势[38], 误差棒的长度小于数据点的标记尺寸 Figure4. Two-coil mutual inductance measurement results of NbN films (NbN#1, NbN#2, NbN#3, NbN#4): (a) Temperature dependence of induced voltage Vx(T) and Vy(T) for NbN#1; (b) temperature-dependent penetration depth λ(T) of four NbN films; (c) temperature variation in superfluid density ${{\rm{\lambda }}^{ - 2}}\left( T \right) \propto {n_{\rm{s}}}\left( T \right)$ for NbN#1. The black line shows the dirty s-wave BCS theory fit to the data; (d) the value of λ (T → 0) for four NbN films, which shows a good agreement with the published value[38]. The length of error bar is shorter than the symbol size.
图 1 (a)双线圈互感装置示意图; (b)等效电路图
图 2 (a) d = 100 nm, λ = 150 nm的超导薄膜的互感系数随薄膜半径R的变化曲线; (b)基于不同的线圈间距(h = 0.9, 4.5, 9.0 mm) 得到的穿透深度计算值随薄膜半径R的变化曲线, 虚线代表实际穿透深度λ = 150 nm
图 3 (a)两次测量同一片铌膜得到的感生电压Vx, 1(T)及Vx, 2(T); (b)铌膜的感生电压V(T = 4.5 K)随频率的依赖关系
图 4 NbN薄膜(NbN#1, NbN#2, NbN#3, NbN#4)的双线圈互感测量结果 (a) NbN#1样品的感生电压曲线Vx(T)及Vy(T); (b)四个样品的穿透深度随温度变化曲线λ(T); (c) NbN#1样品的超流密度-温度曲线
