Superconductors have two defining properties: (1) vanishing of electrical resistivity below a critical transition temperature (T_c) and (2) expulsion of magnetic flux below a critical field (H_c). The former was discovered by Kamerlingh–Onnes in 1911, and the latter by Meissner and Ochsenfeld in 1933. YBa₂Cu₃O_7−δ (YBCO) was the first discovered superconductor that has T_c above the liquid N₂ temperature (77 K) [1]. YBCO’s discovery fueled a great deal of research activity in superconductivity. Initially, most research focused on developing superconductors with a higher T_c and critical current density (J_c). Subsequently, other superconductors such as HgBa₂Ca₂Cu₃O_8+δ [2] (HgBCCO), Tl₂Ba₂Ca₂Cu₃O_10+δ [3] (TlBCCO), and Bi₂Sr₂Ca₂Cu₃O_10+δ [4] (BSCCO) have also been discovered that had T_cs above 77 K. Since the common property of all these materials was the layered crystal structure that contained one or more CuO₂ planes, they were called “cuprates”, which are structurally, chemically, and/or electronically inhomogeneous at the nanoscale.

It is believed that high–temperature superconductors (HTSs) are most suitable for fabricating a lot of potential applications, including motors, magnetic and flywheel energy storage systems, wind generators, magnetic resonance image (MRI), fault current limiters, filters, superconducting quantum interference devices (SQUIDs), and so on ^[5][6][7][5,6,7]. The field of applied, technological superconductivity is now moving beyond these preliminary demonstrators to the industrial development of commercially viable machines and devices. Strain is a key parameter for understanding many physical phenomena at the nanoscale. The mechanical and electronic properties of a material are directly related to the strain in the material, and the response of a material to an applied strain is fundamental to the engineering of mechanical or electronic properties. Superconducting performance is strongly influenced by strain, which is an unexpected phenomenon [8]. Researchers have spent a considerable amount of time investigating the strain to gain a deeper understanding of its relationship with the film’s superconducting performance [9]. For La_1.9Sr_0.1CuO₄ superconducting films deposited on LaSrAlO₄ substrates, the compressive epitaxial strain caused by lattice mismatch doubled T_c [10]. Researchers have observed that anomalous strain fields surrounding misfit dislocations can change the functional properties of oxide heterointerface structures ^[11][12][11,12]. A decrease in J_c with increasing axial tension or compression in self-field has also been observed in YBCO ^[13][14][15][13,14,15]. It has been shown in theoretical studies that modulating the heterointerface strain can either improve ^[16][17][18][16,17,18] or demolish the superconductivity [19]. Therefore, the widespread application of HTSs requires an in-depth understanding of the relationship between composition, functionality, and microstructure.

2. Strain Measurement Techniques

At present, the strain measurement techniques for HTS are relatively mature, mainly including X-ray diffraction (XRD)/ Transmission electron microscope (TEM)/Raman/Neutron Diffraction/Electron backscatter diffraction (EBSD), etc. In this section, rwesearchers briefly summarize the relevant applications in strain studies of superconductors.

2.1. XRD

Jingfeng Yu et al. ^[20][23] investigated YBCO films prepared on (LaAlO₃)_0.3-(Sr₂AlTaO₆)_0.7 (LSAT), SrTiO₃, and LaAlO₃ substrates, and the corresponding influence on superconductivity. Internal strain and residual stress were measured by an extended small-angle sin²ψ method and the Williamson-Hall plot method, respectively. It was confirmed that YBCO/LaAlO₃ samples exhibited compressive stress, but YBCO/LSAT and YBCO/SrTiO₃ samples exhibited tensile stress. The results of their study demonstrated that T_c had the same variation trend as strain. They suggested that controlling adequate microscopic internal strain and macroscopic residual stress are critical to tailoring microstructures and superconductivity. Ziliang Li et al. ^[21][24] studied the superconducting properties and microstructure of chemical solution deposited (CSD) YBCO films by XRD measurements and magnetic susceptibility. With film thickness decreasing down to 5 nm, the effect of Y₂Ba₄Cu₈O₁₆ intergrowth in films was revealed. Ultrathin films offer a unique opportunity to investigate the superconducting properties of highly concentrated nanoscale defects due to the elastic energy related to the misfit strain. Their results demonstrated that superconducting volume decreased strongly correlated with intergrowth volume fraction increasing. Furthermore, they proved that these intergrowths were non-superconducting nanoscale regions that suppressed Cooper pair formation, supporting their role as vortex pining in YBCO films and coated conductors.

2.2. Neutron Diffraction

Kozo Osamura et al. ^[22][25] precisely investigated the strain of YBCO coated conductors and their corresponding effect on critical current (I_c). The internal strain of the YBCO layer was characterized at 77 K by the neutron diffraction technique. The force-free strain (A_ff) is defined as the point in the YBCO layer at which the internal uniaxial stress becomes zero, where the initial compressive strain decreases during tensile loading and changes to a tensile component. The A_ff was assessed to be about 0.19 ~ 0.21% at 77 K. A uniaxial tensile load was used to measure I_c at 77 K. I_c maximum was observed at 0.035% for the strain dependence. As a result, YBCO-coated conductors’ strain at the maximum I_c does not correlate with their A_ff.

2.3. Raman Spectroscopy

Currently, Raman spectroscopy has been used to probe phonons and other types of low-energy excitations in HTS, which can be applied to analyze internal strains and evaluate possible oxygen loss. For example, Sofia Favre et al. ^[23][26] deposited YBCO films by Pulsed Laser Deposition (PLD) and studied the connection between superconductivity and strain. They also studied oxygen loss effects and strain by Raman spectroscopy. The films exhibited an in-plane residual compressive strain, whose degree was determined by film thickness and growth conditions, which influenced the superconductivity. An explanation for the nearly linear depression of T_c with c-axis expansion has been proposed using mutual Coulomb screening between consecutive CuO₂ planes in the structure.

2.4. EBSD

Anjela Koblischka-Veneva et al. ^[24][27] performed a meticulous analysis of transmission Kikuchi diffraction (TKD) and electron backscatter diffraction (EBSD) data acquired on various YBCO samples doped with Y₂BaCuO₅ (Y-211) nanoparticles. The crystallographic parameter difference between Y-211 and YBCO will introduce residual strain around such embedded Y-211 nanoparticles within the YBCO matrix down to the nanometer scale. Researchers found that the strain around clusters and Y-211 nanoparticles was so large that subgrains formed in YBCO matrix. In addition, they discussed the effect of strain distribution on vortex pinning because stress or strain can provide another source of vortex pinning.

2.5. TEM

Currently, TEM has been widely applied in the field of microscale strain measurement, and the strain of various materials has been well studied. In previous research, the authors analyzed the heterointerface structure and strain of Bi-based HTS films deposited on SrTiO₃ ^[25][28] and MgO ^[26][29] single crystal substrates and demonstrated that the combined effect of lattice mismatch and thermal expansion mismatch caused HTS films to behave differently from the expected strain state (expected tensile strain, but actually compressive strain). In general, four main TEM techniques have been created to measure strain, including high-resolution (S)TEM ^{[25][26][27][28][29][30]}[28,29,30,31,32,33], nanobeam electron diffraction (NBED) ^[31][32][34,35], convergent-beam electron diffraction (CBED) ^[33][36] and dark-field electron holography (DFEH) ^[34][37]. Strain measurements depend on their spatial resolution for precision and accuracy. As spatial resolution increases, the technique becomes less accurate. The corresponding information is shown in Table 1. For a specific discussion, see previous reviews, e.g., Martin J. Hÿtch et al. ^[35][38], David Cooper et al. ^[36][39], and A. Béché et al. ^[37][40].