Critical-field slope reduction and upward curvature of the phase-transition lines of thin disordered superconducting YBa2Cu3O7−x films in strong magnetic fields

Critical-field slope reduction and upward curvature of the phase-transition lines of thin disordered superconducting YBa2 Cu3 O7−x films in strong magnetic fields

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Critical-field slope reduction and upward curvature of the phase-transition lines of thin disordered superconducting YBa2 Cu3 O7−x films in strong magnetic fields A.V. Antonov, A.V. Ikonnikov, D.V. Masterov, A.N. Mikhaylov, S.V. Morozov, Yu.N. Nozdrin, S.A. Pavlov, A.E. Parafin, D.I. Tetel’baum, S.S. Ustavschikov, V.K. Vasiliev, P.A. Yunin, D.A. Savinov PII: DOI: Reference:

S0921-4534(19)30351-X https://doi.org/10.1016/j.physc.2019.1353581 PHYSC 1353581

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Physica C: Superconductivity and its applications

Received date: Accepted date:

7 October 2019 30 November 2019

Please cite this article as: A.V. Antonov, A.V. Ikonnikov, D.V. Masterov, A.N. Mikhaylov, S.V. Morozov, Yu.N. Nozdrin, S.A. Pavlov, A.E. Parafin, D.I. Tetel’baum, S.S. Ustavschikov, V.K. Vasiliev, P.A. Yunin, D.A. Savinov, Critical-field slope reduction and upward curvature of the phase-transition lines of thin disordered superconducting YBa2 Cu3 O7−x films in strong magnetic fields, Physica C: Superconductivity and its applications (2019), doi: https://doi.org/10.1016/j.physc.2019.1353581

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Short Title of the Article

Highlights

• An unconventional critical-field slope reduction of the phase-transition line Hc2(T) was experimentally observed for thin YBCO films by gradual increasing of ion implantation dose. • An upward curvature of temperature dependence of the upper critical field was experimentally found out for thin YBCO films by gradual increasing of ion implantation dose. • Theoretical interpretation of the experimental data was developed in the framework of linearized Ginzburg-Landau theory with an inhomogeneous superconducting coherence length.

First Author et al.: Preprint submitted to Elsevier

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Critical-field slope reduction and upward curvature of the phase-transition lines

Critical-field slope reduction and upward curvature of the phase-transition lines of thin disordered superconducting YBa2 Cu3 O7−x films in strong magnetic fields A.V. Antonov1 , A.V. Ikonnikov2 , D.V. Masterov1 , A.N. Mikhaylov3 , S.V. Morozov1 , Yu.N. Nozdrin1 , S.A. Pavlov1 , A.E. Parafin1,3 , D.I. Tetel’baum1 , S.S. Ustavschikov1,3 , V.K. Vasiliev3 , P.A. Yunin1,3 , D.A. Savinov1,3 1 1 Institute for Physics of Microstructures, Russian Academy of Sciences, 603950 Nizhny Novgorod, GSP-105, Russia 2 Physics Department, Lomonosov Moscow State University, Leninskie Gory, 1 Moscow, 119991, Russia 3 Lobachevsky State University of Nizhny Novgorod, Prospekt Gagarina, 23, Nizhny Novgorod, 603950, Russia Abstract We report the effect of disorder on superconducting phase transition of YBa2 Cu3 O7−x epitaxial thin films in external magnetic fields. The disorder was produced by several successive acts of oxygen ion implantation. Controlling a total accumulated dose of implanted ions 𝑛𝐷 , we carried out transport measurements in 𝑎𝑏-plane for temperatures 𝑇 below 91 K and external magnetic fields 𝐻 up to 11 T. Temperature-field dependencies of in-plane resistivity allow us to analyze 𝐻 −𝑇 phase diagrams for primary compound as well as for disordered structure. Considering the upper critical field 𝐻𝑐2 as the magnetic field which corresponds to a local resistivity drop at the onset of superconducting transition, we found out the following results. By gradual increasing of 𝑛𝐷 , the phase-transition line 𝐻𝑐2 (𝑇 ) suffers an unconventional critical-field slope reduction, while larger defect concentrations usually enhance the upper critical field in the vicinity of 𝑇𝑐0 . Besides, for rather large 𝑛𝐷 , the curvature of 𝐻𝑐2 (𝑇 ) becomes upward for temperatures close to 𝑇𝑐0 . Theoretical interpretation of the experimental data is developed in the framework of linearized Ginzburg-Landau theory with an inhomogeneous superconducting coherence length. A simple expression for the critical temperature 𝑇𝑐 is obtained: 𝑇𝑐 = 𝑇𝑐0 (1 − ℎ + 𝛼ℎ3∕2 ), where ℎ is the dimensionless magnetic field and 𝛼 is a constant which describes the defects in a specimen. The formula nicely fits our experimental results. Keywords: superconducting films, ion implantation, disorder, upper critical field, phase-transition line

1. Introduction

Temperature dependence of the upper critical field, 𝐻𝑐2 (𝑇 ), is one of the most important characteristics of superconductors [1]. Indeed: (i) it provides valuable insight into the peculiarities of superconducting pairing mechanism as well as (ii) the knowledge of the upper critical field and its anisotropy allows one to determine the fundamental superconducting properties such as coherence length scales. Temperature behavior of 𝐻𝑐2 strongly depends on defects in a superconductor. √ The simplest consequence of disorder is associated with renormalization of the superconducting coherence length 𝜉0 : 𝜉0 → 𝜉𝑜 𝓁, where 𝓁 is the electron mean-free path in the normal state. This appears for 𝓁 ≪ 𝜉0 (dirty limit). Therefore, defects should lead to a critical-field slope enhancement of the phase-transition line 𝐻𝑐2 (𝑇 ) in the vicinity of zero-field critical temperature 𝑇𝑐0 : |(𝑑𝐻𝑐2 ∕𝑑𝑇 )|𝑇𝑐0 | ≃ Φ0 ∕(2𝜋𝜉𝑜 𝓁𝑇𝑐0 ), where Φ0 = 𝜋ℏ𝑐∕𝑒 is the magnetic flux quantum. Corresponding dependence 𝐻𝑐2 (𝑇 ) in dirty superconductors is given by the well-known Werthamer-Helfand-Hohenberg theory [2]. The phase-transition line is characterized by negative curvature and saturation of the upper critical field at low temperatures. It has been nicely observed in many superconducting compounds [1]. Though, the experimental data for some other materials, such as high-temperature superconductors (HTSC), MgB2 , iron-based superconductors predict several unconventional features of the phase-transition line in dirty samples (see, e.g., Refs. [3, 4, 5, 6, 7]). In particular, they are an upward curvature of 𝐻𝑐2 (𝑇 ) and unconventional changes in the critical-field slope, disappearance of saturation of the upper critical field at 𝑇 → 0, anomalous metallic state above the upper critical field. Different mechanisms have been considered for the interpretation of these 𝐻𝑐2 (𝑇 ) anomalies (see Refs. [8, 9, 10, 11, 12, 13] and references therein). Moreover, even more striking phenomena have been predicted in recent theoretical works for strongly disordered (𝑘𝐹 𝓁 ≃ 1, where 𝑘𝐹 is the Fermi momentum) d-wave superconductors: (i) application of a small magnetic field enhances superconductivity [14], (ii) critical temperature 𝑇𝑐0 can be increased by disorder [15, 16], (iii) there appears a change in the type of superconducting pairing (from d-wave to s-wave) [17]. However, direct experimental observation of these anomalies remains an important outstanding question. In our work, we investigate magnetic properties of weakly disordered (𝑘𝐹 𝓁 ≫ 1) superconducting films based on HTSC and observe several unconventional peculiarities of the phase-transition line 𝐻𝑐2 (𝑇 ). So, our report appears to be a natural first step for experimental study of disorder-induced anomalies in d-wave superconductors. Considering the case of a weak disorder (𝑘𝐹 𝓁 ≫ 1), the curvature of 𝐻𝑐2 (𝑇 ) can become upward in the vicinity of 𝑇𝑐0 whereas it is negative for lower temperatures closed to zero (see Refs. [10, 11]). Such behavior of the phase-transition line can be derived if one takes into account a spatial dependence of the superconducting coherence length which can result from modulation of disorder characteristics – the diffusion coefficient or local mean-free path. Otherwise, it can also follow from fluctuations in the crystal axes orientation in thin films of anisotropic superconductors. Experimentally both scenarios can be realized in a specimen by the technique of ion implantation. Depending on dose, the ion implantation appears to be a wellknown method which allows (i) to form a certain geometrical structure of the sample, (ii) to fabricate Josephson junctions, and (iii) to modify the superconducting and electron properties of a specimen (see Refs. [7, 18, 19, 20, 21, 22]). Hereafter, the dose corresponds to a number of the implanted ions per square centimeter. The third application of ion implantation is relevant for our study. Indeed, the irradiation results in a certain distribution of point defects which reduces to a random spatial A.V. Antonov et al.: Preprint submitted to Elsevier

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Critical-field slope reduction and upward curvature of the phase-transition lines

dependence of the superconducting coherence length and, therefore, should cause the appearance of 𝐻𝑐2 (𝑇 ) anomalies such as an upward curvature of the phase-transition line. In this report, we investigate and analyze 𝐻 − 𝑇 phase diagrams either for clean compound or for disordered sample. The disorder was produced by a random distribution of defects which were intentionally introduced in a controlled manner by several successive acts of oxygen ion implantation. As a starting material we chose a narrow bridge made of a thin superconducting film of YBCO. The superconducting transition occurred at 91 K. We carried out transport measurements for different temperatures below 91 K, using an external perpendicular magnetic field 𝐻. Temperature-field dependencies of 𝑎𝑏-plane resistivity 𝜌 allow us to analyze 𝐻 − 𝑇 phase diagrams. To clarify the effect of disorder on 𝐻𝑐2 (𝑇 ) line we performed four acts of irradiation. Each act was characterized by a total accumulated dose of implanted ions 𝑛𝐷 . Considering the Drude model for different 𝑛𝐷 , we made typical theoretical estimates for the average mean-free path 𝓁 which allowed us to control the degree of disorder in the film. After each stage of ion implantation we carried out transport measurements in the presence of magnetic field 𝐻. So, we report and interpret measurements of magneto-resistance, superconducting transitions, and temperature dependence of the upper critical field of thin YBCO films which reveal the systematics of how superconductivity is affected by disorder. For each 𝑛𝐷 , we derive the phase-transition lines 𝐻𝑐2 (𝑇 ) obtained for several resistive levels inside the superconducting broadened transition. Therefore, we get a set of various 𝐻𝑐2 (𝑇 ) curves. Note, that these curves have been previously analyzed in YBCO epitaxial thin films without artificially embedded defects in Ref. [23]. In the present work, we are interested in investigation of the disorder effect on 𝐻𝑐2 (𝑇 ). We pay special attention to temperature dependence of the upper critical field 𝐻𝑐2 defined as the magnetic field which corresponds to a local resistivity drop at the onset of superconducting transition. Certainly, considering such high resistive levels, we determine the phase-transition line between the normal state and the superconducting droplet state (as discussed in Ref. [11]), whereas the dependence 𝐻𝑐2 (𝑇 ) commonly describes the second-order phase transition between the normal state and the Abrikosov-vortex state. Indeed, a standard Abrikosov vortex lattice should arise for rather low resistances close to zero, when there appears the formation of a bulk superconducting cluster in a specimen. Therefore, determination of the critical filed such as the magnetic field which corresponds to onset of superconducting transition is sometimes applied for the criterion of the third critical field 𝐻𝑐3 corresponding to a surface superconductivity (see Ref. [24]). Similar criterion is also used for description of the critical fields for localized states in hybrid superconductor/ferromagnetic structures. In particular, domain-wall superconductivity in a Pb thin-film bridge on a ferromagnetic BaFe12 O19 single crystal appears close to onset of superconducting transition which has been visualized from low-temperature scanning laser microscopy and resistive measurements in Ref. [25]. In our report, we also analyze peculiarities of localized superconductivity nucleation when the order parameter arises in the vicinity of region with a locally suppressed coherence length. Following to Ref. [11], we further denote the critical magnetic field of these localized states by 𝐻𝑐2 and the corresponding phase-transition line by 𝐻𝑐2 (𝑇 ). By gradual increasing of a total accumulated dose 𝑛𝐷 from 0.3⋅1013 cm−2 to 4.3⋅1013 cm−2 , we observe a nonmonotonous behavior of the upper critical field in the vicinity of 𝑇𝑐0 . Its enhancement is seen for rather low doses 𝑛𝐷 = 0.3 ⋅ 1013 cm−2 and 0.9 ⋅ 1013 cm−2 . This corresponds to the well-known prediction, when larger defect concentrations usually enhance the upper critical field [2]. However, further increase of 𝑛𝐷 up to the values of 3 ⋅ 1013 cm−2 and 4.3 ⋅ 1013 cm−2 results in unconventional critical-field slope reduction of the phase-transition line 𝐻𝑐2 (𝑇 ). Moreover, the shape of 𝐻𝑐2 (𝑇 ) reveals an upward curvature which may be accompanied by the implantation-induced modulation of superconducting coherence length. For the interpretation of these experimental results, we develop a theoretical model based on Ginzburg-Landau theory. We consider a random spatial modulation of the superconducting coherence length. We also take into account the 𝑇𝑐0 suppression after each of the implantation acts. As a result, we derive a simple expression for the phase-transition line: 1 − 𝑇𝑐 ∕𝑇𝑐0 = ℎ − 𝛼ℎ3∕2 ,

where ℎ is the dimensionless magnetic field. Here, the constant 𝛼 is directly connected with the pinning properties of a superconductor and can be extracted from experiments. We stress that our study refers to the mean-field approximation, and that the role of superconducting fluctuations is a significant unresolved question. However, such approach appears to be in a good accordance with our experimental data.

2. Sample fabrication and ion implantation

The high-quality 𝑐-axis-oriented YBCO epitaxial film with a thickness of 𝑑𝑠 = 200 nm was deposited onto the substrate of lanthanum aluminate LaAlO3 by a magnetron sputtering method [26]. The misorientation of mosaic blocks in the film relative to the 𝑐 axis was characterized by the X-ray rocking curve. Its full width at half maximum (FWHM) Δ𝜔 for the YBCO (006) reflection was 0.14 − 0.18𝑜 . Within the present work, we carried out transport measurements using the bridge made of the grown film (see Fig. 1a.). Its width and length were 𝑤 = 50 𝜇m and 𝐿 = 250 𝜇m, respectively. Other parameters of the bridge were: (i) zero-field critical temperature 𝑇𝑐0 = 91 K, (ii) temperature width of superconducting transition was less than 1 K, and (iii) critical current density 𝑗𝑐 was 4 ⋅ 106 A∕cm2 at 𝑇 = 77 K. In-plane resistivity of YBCO at 𝑇 = 100 K was 100 𝜇Ohm ⋅ cm. This value of the resistivity corresponds to optimal oxygen doping, i.e. we have YBa2 Cu3 O7−x sample with 𝑥 ≃ 0.1 (see the details in Ref. [27]). A.V. Antonov et al.: Preprint submitted to Elsevier

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Critical-field slope reduction and upward curvature of the phase-transition lines

Figure 1: (a) Photo of the bridge under study. (b) Depth distribution of defect concentration derived from the SRIM calculations.

To modify the superconducting coherence length in 𝑎𝑏-plane, we carried out the ion implantation. The implanted ions induce point defects in a superconductor, because the damage is represented principally by lattice vacancies and interstitials caused by atom displacements. The damage is cumulative, and the degree of damage relates to the radiation fluence, i.e., it involves the parameters of ion implantation such as ion energy, ion dose, type of ions, etc. The implantation was performed at room temperature with oxygen ions at energy of 100 keV. Four acts of ion implantation were carried out within this study. The actual implantation doses were: 3 ⋅ 1012 , 6 ⋅ 1012 , 2.1 ⋅ 1013 , and 1.3 ⋅ 1013 cm−2 . So, we achieved the total accumulated dose 𝑛𝐷 = 4.3 ⋅ 1013 cm−2 . Further, we discuss the displacement damage in the bridge caused by the irradiation and present typical estimates for defect concentration as well as for doping level of the compound after its modification. The main components of ion-radiationinduced defects are displacement of oxygen atoms in a crystal structure. This can be obtained from Raman-spectra technique which allows to study a qualitative evaluation of the oxygen content in Cu-O chains as well as in CuO2 planes in YBCO [28, 29]. As it has been studied in Ref. [30], significantly more oxygen defects occur in Cu-O chains rather than in CuO2 planes. This is because the displacement energy of oxygen atoms in the Cu-O chains is much lower than that of oxygen atoms in the CuO2 planes [31]. Considering each of the implantation doses, we carried out the SRIM calculations [32] and obtained the defect concentrations in our specimen numerically. The typical profiles of Frenkel pairs (vacancies and interstitials) are shown in Fig. 1b, respectively for each of 𝑛𝐷 . As it can be seen from this figure, the defect distribution appears to be symmetric with respect to the central plane of the film as well as quasi-homogeneous in 𝑐-axis. Let us now analyze, if the doping level changes due to ion implantation. Our calculations show that the total concentration profiles of implanted oxygen ions 𝑁𝐷 are similarly to those presented in Fig. 1b (therefore, they are omitted here) except for the maximal value. In particular, considering the fourth implantation dose (𝑛𝐷 = 4.3 ⋅ 1013 cm−2 ) we get that the maximal value for 𝑁𝐷 achieves 2 ⋅ 1018 cm−3 instead of 2 ⋅ 1021 cm−3 for the Frenkel pair concentration. This is because one oxygen ion produces almost 1000 defects (see the details in Ref. [32]). Further, we note that typical concentration of oxygen atoms 𝑁𝑂 forming a crystal lattice in a specimen is usually equal to 1022 − 1023 cm−3 . So, we obtain that 𝑁𝐷 is less by 4 − 5 orders of magnitude than 𝑁𝑂 or even much lower. Therefore, we suppose that the ion implantation does not violate the optimal doping condition, and 𝑥 ≃ 0.1 for all implantation acts carried out in our report. Thus, the 𝑇𝑐 suppression introduced by the irradiation can be caused by the appearance of nonmagnetic defects similarly to the case considered in Ref. [33]. Note, that the same approach has been applied for the description of superconducting parameters of thin YBa2 Cu3 O6+x films affected by irradiation [31].

3. Experiment details

The magneto-resistance 𝑅 of the bridge was measured in a variable temperature insert by a standard four-probe technique using a 𝛿-mode regime. The transport current was 10 𝜇A. Transport measurements in the presence of an external magnetic field 𝐻 were carried out in a closed-cycle cryogenic system with two cryostats. One of them contained a superconducting solenoid (Oxford Cryofree SC magnet). The second cryostat (Oxford Optistat PT) with a controllable temperature (from 1.6 K and above) was inserted into the Oxford Cryofree SC magnet. The sample under study was positioned into the Oxford Optistat PT. The temperature was measured by a calibrated thermometer with a resolution of 30 mK. A magnetic field up to 11 T was applied with a resolution of 1.1 mT. First, we performed transport measurements of primary structure, when the bridge had 𝑇𝑐0 = 91 K and 𝑗𝑐 = 4⋅106 A∕cm2 at 𝑇 = 77 K. The measurements were carried out in 𝑎𝑏-plane for temperatures 𝑇 below 91 K in the presence of an external magnetic field 𝐻 applied perpendicular to the film. By changing temperature 𝑇 gradually, we found the experimental deA.V. Antonov et al.: Preprint submitted to Elsevier

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Critical-field slope reduction and upward curvature of the phase-transition lines

700

90K 85K 82K 78K 74K 69K 65K

600 ρ, µΩ · cm

500

400 300

60K

200

56K

100 0

0

2

4

6

H, T

8

10

52K 44K 12

Figure 2: Experimental resistive curves 𝜌(𝐻) obtained for several temperatures 𝑇 and 𝑛𝐷 = 4.3 ⋅ 1013 cm−2 (temperature values are indicated).

700 600

ρ, µΩ · cm

500 400 300

H=0T H=2T H=4T H=6T H=8T H = 10 T

200 100 0 40

50

60

T, K

70

80

90

Figure 3: Experimental resistive curves 𝜌(𝑇 ) presented for given magnetic fields 𝐻 and 𝑛𝐷 = 4.3 ⋅ 1013 cm−2 (magnetic-field values are shown). The symbols present the experimental data, whereas the lines correspond to a spline interpolation.

pendencies of in-plane resistivity 𝜌 = 𝑅 ⋅ 𝑤𝑑𝑠 ∕𝐿 on 𝐻. Then, we carried out similar measurements for the bridge in which defects were intentionally introduced by four acts of ion implantation. As a result, we got 𝜌(𝐻, 𝑇 ) curves after each of the implantation acts. These resistive dependencies allowed us to study the effect of disorder on 𝐻𝑐2 (𝑇 ).

4. Experimental results and data analysis

We start this section from presentation of 𝜌(𝐻) curves corresponding to given temperatures 𝑇 . We restrict ourselves by illustration of these curves only for the case 𝑛𝐷 = 4.3 ⋅ 1013 cm−2 . These data are shown in Fig. 2 by solid lines for several temperatures 𝑇 from 44 K to 90 K. Resistive curves 𝜌(𝐻) for other 𝑛𝐷 appear to be similar to those presented in Fig. 2 and, therefore, they are omitted here. We transform the series of 𝜌(𝐻) curves presented in Fig. 2 to series of 𝜌(𝑇 ) curves corresponding to different values of 𝐻. The result of such transformation is presented in Fig. 3, where the symbols present the experimental data, whereas the lines correspond to a spline interpolation. Obviously, by increasing 𝐻, the superconducting drop shifts towards lower temperatures, because the magnetic field suppresses superconductivity. Let us further determine the onset resistivity 𝜌on as the value of resistivity, when the resistivity change Δ𝜌 = 𝜌(𝐻 = 11 T) − 𝜌(H = 0) becomes less than 5%𝜌(𝐻 = 0). Actually, there is a rather negligible magneto-resistance in YBa2 Cu3 O7−x in the normal state, as we found out for temperatures above 91 K (see also Ref. [34] – Δ𝜌 ≃ 5%𝜌(𝐻 = 0) for the same film with increasing 𝐻 from 0 to 12 T). Certainly, 𝜌on grows with increasing 𝑛𝐷 . In particular, 𝜌on = 105 𝜇Ohm ⋅ cm for the primary structure in contrast to 𝜌on = 120, 160, 376, and 676 𝜇Ohm ⋅ cm after each of four ion-implantation stages, respectively. In Fig. 4, we demonstrate the dependence of the onset resistivity versus 𝑛𝐷 . So, by gradually increasing the implantation dose, we can control the degree of disorder. Indeed, one can consider 𝜌on as a possible estimate for residual A.V. Antonov et al.: Preprint submitted to Elsevier

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Critical-field slope reduction and upward curvature of the phase-transition lines

0.30

Dw, deg

on

0.25

0.20

1

2

13

-2

3

4

5

n D 10 cm

Figure 4: The onset resistivity versus 𝑛𝐷 (the symbols present experimental data while the line corresponds to a spline approximation). Discrete dependence of Δ𝜔 versus 𝑛𝐷 (the symbols show experimental data while the line illustrates a straight connection between neighboring points). Both the lines are guides to the eye.

Figure 5: Typical plots of experimental dependencies 𝐻𝑐2 (𝑇 ), where the upper critical field was defined as the magnetic field that suppresses a certain fraction of resistance drop (resistive levels are presented in fractions relative to 𝜌on ), (a) and (b) correspond to 𝑛𝐷 = 3 ⋅ 1013 cm−2 , 𝜌on = 376 𝜇Ohm ⋅ cm and 4.3 ⋅ 1013 cm−2 , 𝜌on = 676 𝜇Ohm ⋅ cm, respectively. The symbols illustrate experimental data whereas the lines are given from a spline interpolation.

resistivity. Hence, we can derive the electron mean-free time 𝜏 at each stage of ion implantation from the Drude model: 𝜏≃

𝑚 , 𝑒2 𝑛0 𝜌on

(1)

where 𝑛0 is a free charge carrier concentration, and 𝑚 is an effective electron mass in the 𝑎𝑏 plane. The value of 𝑛0 in YBa2 Cu3 O7−x epitaxial thin films was experimentally studied from Hall coefficient measurements in Ref. [35]. Using these data and considering the simplest model of isotropic Fermi surface in the momentum space, we estimate 𝑘𝐹 ≃ 4 nm−1 and 𝑣𝐹 ≃ 4 ⋅ 105 m∕s. Choosing the first dose 𝑛𝐷 , when 𝜌on = 120 𝜇Ohm ⋅ cm, we get 𝑘𝐹 𝓁 ≃ 16. Considering next three implantation doses, we derive 𝑘𝐹 𝓁 ≃ 13, 6, and 3, respectively. Note, that for the fourth dose the product 𝑘𝐹 𝓁 is close to unity and just slightly exceeds it. Though, the above estimate for 𝓁 represents the lower boundary for the average electron mean-free path, because 𝜌on appears to be larger than a genuine residual resistivity. Therefore, the condition of a weak disorder (𝑘𝐹 𝓁 ≫ 1) is satisfied, and we can use a special theoretical technique for the description of our experimental data. The theoretical model is presented in the next section. Let us now analyze the experimental data presented in Figs. 2 and 3. In Figs. 5a and 5b, we present typical phase-transition lines 𝐻𝑐2 (𝑇 ) obtained after the third and fourth acts of ion implantation, respectively. Each plot contains several 𝐻𝑐2 (𝑇 ) curves corresponding to resistive levels close to onset of superconducting transition. While gradual increase in the resistivity, it is seen that the phase-transition lines start to possess an upward curvature. We suggest that such unconventional behavior of A.V. Antonov et al.: Preprint submitted to Elsevier

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Critical-field slope reduction and upward curvature of the phase-transition lines

Figure 6: Typical plots of experimental dependencies 𝐻𝑐2 (𝑇 ) presented for 𝑛𝐷 = 3 ⋅ 1013 cm−2 (curves indexed by 𝑎) and 4.3 ⋅ 1013 cm−2 (curves indexed by 𝑏) in pairs for the same fractions of resistance drop. Here, pairs (1𝑎, 1𝑏), (2𝑎, 2𝑏), (3𝑎, 3𝑏), and (4𝑎, 4𝑏) correspond to resistive levels 0.92𝜌on , 0.93𝜌on , 0.95𝜌on , and 0.96𝜌on , respectively. The lines are obtained from our direct experimental data (the corresponding symbols are omitted here) using spline interpolation, 𝜌on = 376 and 676 𝜇Ohm ⋅ cm for 𝑛𝐷 = 3 ⋅ 1013 and 4.3 ⋅ 1013 , respectively.

𝐻𝑐2 (𝑇 ) can be caused by the inhomogeneous superconducting nucleation in the specimen which is produced by modulation of disorder characteristics after the implantation. Otherwise, it can also follow from the misorientation enhancement of mosaic blocks in the film relative to the 𝑐 axis. Indeed, while 𝑛𝐷 grows, FWHM of X-ray rocking curve increases monotonously as presented in Fig. 4, because the defects produce breaking of mosaic block ordering along the 𝑐 axis. We also make a comparison of the phase-transition lines corresponding to different 𝑛𝐷 and the same fractions of onset resistivity. These results are demonstrated in Figs. 6 for two doses 𝑛𝐷 : 3 ⋅ 1013 and 4.3 ⋅ 1013 cm−2 . Obviously, 𝑇𝑐0 reduces with the rise in 𝑛𝐷 , because there appears an enhancement of the concentration of nonmagnetic defects which usually leads to isotropization of the superconducting energy gap in 𝑑-wave superconductors and, therefore, suppresses the order parameter even in zero magnetic field [33]. Though, we present the plots for 𝑛𝐷 = 3 ⋅ 1013 and 4.3 ⋅ 1013 cm−2 to be shifted to the same point on the abscissa axis for each of the resistive levels. So, with increasing 𝑛𝐷 , we observe an unconventional critical-field slope reduction of the phase-transition lines 𝐻𝑐2 (𝑇 ), while the rise in defect concentration usually enhances the upper critical field in the vicinity of 𝑇𝑐0 (see Ref. [2]).

5. Theoretical model

We proceed with a theoretical model developed within a mean-field approximation. Certainly, for more rigorous description of the phase-transition line one should take into account the effect of fluctuating superconductivity on 𝐻𝑐2 (𝑇 ) (see Ref. [36]. Though, the fluctuating region Δ𝑇 appears to be almost independent of an external magnetic field 𝐻 [37]. Hence, the effect of fluctuations appears to be slightly suppressed for large magnetic fields 𝐻. Indeed, the superconducting drop shifts towards lower temperatures and becomes broader (as demonstrated in Fig. 3). So, by considering a certain constant resistive level for the dependencies presented in Fig. 3, one gets the discrete set of superconducting critical temperatures 𝑇𝑐 which can be beyond of the fluctuating region Δ𝑇 by gradual increase in 𝐻. Therefore, to describe our experimental data, we consider Ginzburg-Landau-type theory. As it was predicted in Ref. [33], the nonmagnetic defects result in the suppression of 𝑑-wave superconductivity in zero magnetic field. Indeed, the decrease of 𝑇𝑐0 should be defined from the equation: ( ) ( ) ( ) 𝑇𝑐0 1 1 ℏ ln =𝜓 , (2) −𝜓 + 𝑡𝑐0 2 2 4𝜋𝑘𝐵 𝜏𝑇𝑐0

where 𝜓 is the digamma function, 𝑡𝑐0 and 𝑇𝑐0 are zero-field critical temperatures for an extremely clean compound and for a specimen with a certain defect concentration defined by the electron mean-free time 𝜏, respectively. The expression (2) also describes the critical-temperature reduction in conventional superconductors with magnetic defects [38]. The critical-temperature dependence on 𝐻 and concentration of magnetic defects was studied in Ref. [39]. Linearization of the resulting phase-transition line 𝐻𝑐2 (𝑇 ) in the vicinity coher√ of 𝑇𝑐0 gives expression for the superconducting −1∕2 ence length which differs from the well-known dirty-limit result 𝜋ℏ𝐷∕8𝑘𝐵 𝑇𝑐0 by factor (2 − 𝑡𝑐0 ∕𝑇𝑐0 ) , where 𝐷 is the A.V. Antonov et al.: Preprint submitted to Elsevier

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diffusion coefficient. Considering 𝑑-wave superconductor with nonmagnetic defects in an external magnetic field 𝐻, let us also take into account such additional renormalization of the superconducting coherence length in 𝑎𝑏-plane: √ √ 𝜋ℏ𝐷 1 𝜋ℏ𝐷 → 𝜉 𝑎𝑏 = √ ⋅ . (3) 𝜉𝑎𝑏 = 8𝑘𝐵 𝑇𝑐0 8𝑘 𝐵 𝑇𝑐0 2 − 𝑡𝑐0 ∕𝑇𝑐0

In the present article, we use expression (3) for description of the coherence-length modification after the irradiation. Indeed, there have been arguments and some evidence in support of the nonmagnetic nature of defects induced by ion irradiation [31]. We also consider the case of randomly distributed 𝐷 = 𝐷(𝐫) to study the standard deviation of the upper critical field induced by modulation of disorder characteristics. As it follows from formula (3), the inhomogeneous distribution of 𝐷 leads to spatially dependence of superconducting coherence length 𝜉 𝑎𝑏 . So, the highest critical temperature 𝑇𝑐 (onset of superconducting transition) appears to be the minimal eigenvalue of the following problem: ) [ ] ( 𝑇𝑐 ̂ 𝜉 2 (𝐫)𝚷Δ(𝐫) ̂ ℏ𝚷 Δ(𝐫) , (4) = 1 − 𝑎𝑏 𝑇𝑐0 ̂ = (𝑖∇ − 2𝜋𝐀∕Φ0 ), 𝐀 is the vector potential, and Δ(𝐫) is the order parameter. The formula (4) presents the linwhere 𝚷 earized Ginzburg-Landau equation in a dirty limit except for three differences: (i) 𝑇𝑐0 is not invariant, but depends on 𝑛𝐷 and can be described by (2); (ii) the superconducting coherence length is renormalized in accordance with (3), and (iii) the superconducting coherence length is spatially dependent. To get an approximate solution of Eq. (4), we will suppose a Gaussian random distribution of the diffusion coefficient: 𝐷(𝐫) = ⟨𝐷⟩ + 𝛿𝐷(𝐫), where ⟨...⟩ stands for the ensemble averaging, ⟨𝐷⟩ = 𝓁𝑣𝐹 ∕3 is the averaged diffusion coefficient, and 𝛿𝐷(𝐫) has the zero mean and the autocorrelation function with a certain correlation length 𝓁𝑐 : ′

2

⟨𝛿𝐷(𝐫)𝛿𝐷(𝐫 )⟩ = ⟨𝐷⟩

(

𝑑 𝓁𝑐

)2

exp(−|𝐫 − 𝐫 ′ |2 ∕𝓁𝑐2 ) ,

where 𝑑 is a phenomenological constant which is associated with the pinning properties of a superconductor and directly determined by a certain dose of ion implantation (see below). Note here, that we restrict ourselves by 2𝐷 dimension of 𝐫 space to get an analytical expression for 𝐻𝑐2 (𝑇 ). Following to the perturbation theory developed in Refs. [10, 11] for the approximate solution of Eq. (4), we get: 1−

𝑇𝑐 𝑑 =ℎ− ℎ3∕2 , 𝑇𝑐0 23∕2 𝜉0

(5)

2

where ℎ = 2𝜋𝜉 0 𝐻∕Φ0 is the dimensionless magnetic field and 𝜉 0 is determined by (3) with 𝐷 = ⟨𝐷⟩: √ 𝜋ℏ𝓁𝑣𝐹 𝜉0 = . 24𝑘𝐵 (2𝑇𝑐0 − 𝑡𝑐0 )

(6)

It should be noted, √ that the formula (5) corresponds to 𝓁𝑐 = 0 which is valid in a small-field regime, when the ratio 𝓁𝑐 ∕𝐿𝐻 → 0, where 𝐿𝐻 = Φ0 ∕2𝜋𝐻 is the magnetic length. We emphasize that formula (5) corresponds to the superconducting nuclei with maximal critical temperature. Therefore, we use this expression for description of our experimental 𝐻𝑐2 (𝑇 ) lines obtained for high resistive levels close to 𝜌on . The fitting is presented in Figs. 7a and 7b for 𝑛𝐷 = 3 ⋅ 1013 and 4.3 ⋅ 1013 cm−2 , respectively. Further, we restrict ourselves by the case of 𝑛𝐷 = 4.3 ⋅ 1013 cm−2 (see Fig. 7b). Considering the resistive levels 0.92𝜌on , 0.93𝜌on , 0.95𝜌on , and 0.96𝜌on , we get different zero-field critical temperature 𝑇𝑐0 = 81.64, 84.23, 85.55, and 86.76 K, respectively, whereas the zero-field critical temperature 𝑡𝑐0 is certainly the same for all phase-transition lines (we take it as 91 K). So, we can find the estimates for 𝜉0 and 𝓁 in accordance with the formula (6). The fitting gives the following ranges for 𝜉0 and 𝓁: 𝜉0 = (1.4 − 1.9) nm, 𝓁 = (0.5 − 1) nm. We also derive the phenomenological constant 𝑑 which determines the upward curvature of 𝐻𝑐2 (𝑇 ) lines. It fluctuates within the interval of (7 − 10) nm.

6. Conclusion

To summarize, we have studied the disorder effect in epitaxial thin films of YBa2 Cu3 O7−x using ion-beam irradiation and external perpendicular magnetic field 𝐻. Considering different doses of ion implantation 𝑛𝐷 , phase diagrams 𝐻 − 𝑇 A.V. Antonov et al.: Preprint submitted to Elsevier

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Critical-field slope reduction and upward curvature of the phase-transition lines

0.92

0.92

0.94

0.94

0.95

0.95

0.96

0.96

Figure 7: Typical plots of experimental dependencies 𝐻𝑐2 (𝑇 ) (symbols) as well as theoretical approximation based on formula (5) depicted by lines: (a) 𝑛𝐷 = 3 ⋅ 1013 cm−2 , 𝜌on = 376𝜇Ohm ⋅ cm and (b) 𝑛𝐷 = 4.3 ⋅ 1013 cm−2 , 𝜌on = 676𝜇Ohm ⋅ cm. Experimental phase-transition lines correspond to resistive levels 0.92𝜌on , 0.93𝜌on , 0.95𝜌on , and 0.96𝜌on . The fitting function and values of fitting parameters 𝑎 and 𝑏 are presented.

have been obtained experimentally and analyzed theoretically. By gradual increasing of 𝑛𝐷 , the phase-transition lines 𝐻𝑐2 (𝑇 ) exhibit an unconventional critical-field slope reduction and positive curvature for temperatures close to 𝑇𝑐0 . We have derived expression (5) which is in a good agreement with our experimental 𝐻𝑐2 (𝑇 ) dependencies. We believe that the disorder plays a significant role in the modification of 𝐻𝑐2 (𝑇 ) line in different HTSCs.

7. Aknowledgments

The authors thank A. S. Mel’nikov for reading the paper and stimulating discussions as well as A. Yu. Aladyshkin for his valuable recommendations. The authors also grateful to V. V. Kurin, V. I. Gavrilenko, D. Yu. Vodolazov, and S. V. Sharov for supporting and valuable comments. This work was supported by the Russian Foundation for Basic Research under Grant No. 18-42-520051 and by the Russian Science Foundation under Grants No. 17-12-01383 and 18-72-10027.

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