1 Introduction

Unique properties of high-temperature superconductors (HTSs) promise new capabilities for superconducting electronics and novel optoelectronic devices. Therefore, the search for reliable methods of fabrication of the HTS-based electronic circuits is a very important and highly studied research area.

Because of its well-developed technology and excellent superconducting properties, the YBa2Cu3O7−x (YBCO) perovskite material, especially is a form of a thin film, has been widely accepted as the superconductor of choice for HTS-based electronics. YBCO thin films are being grown epitaxially and are patterned by means of many various techniques such as wet-chemical etching [1], ion-beam patterning [2], laser ablation [3], laser direct writing [4]. In the laser direct writing technique, the oxygen content is locally decreased/increased by laser-induced heating in the inert gas/oxygen gas ambient atmosphere [5]. The stoichiometric YBCO (x < 0.1) exhibits an orthorhombic structure and shows a metal-like resistivity behavior with the onset of the superconducting state at the critical temperature T onc of 92 K [6]. On the other hand, oxygen-poor films with x ≥ 0.6 are tetragonal and show no evidence of superconductivity even at lowest temperatures.

In type-II superconductors at temperatures lower than T c, a current self-produced magnetic field H I starts to penetrate the volume of a superconducting film when it value exceeds the first critical magnetic field H c1 of the material. First, the magnetic flux penetrates the film’s regions of the weakest superconductivity (i.e., exhibiting the lowest local T onc ) in a form of quantized magnetic flux lines (Abrikosov magnetic vortices), each embracing a single flux quantum Φo = 2.07 × 10−15 Wb [7], and results in a so-called mixed state (i.e., the state in which both the superconducting and nonsuperconducting phases coexist). Due to the Meissner-Ochsenfeld effect (i.e., the effect of absolute diamagnetism) [8], magnetic flux lines penetrate at one edge of the superconducting film in a form of vortices and as antivortices at the opposite one. Next, the Lorentz force F L forces the magnetic flux lines (vortices and antivortices) to move from the film’s edges toward its center [9]. In actual superconducting materials, there are, however, imperfections, such as, for example, point defects [10], grain boundaries [11], or screw dislocations [12] that resist the flux drift, forming a pinning force F p, which tends to stop flux lines (so-called thermally activated flux-flow regime) or bundles of vortices (so-called flux creep regime). If F L > > F p, the vortex–antivortex pairs nucleated at the opposite edges move nonstop toward the film center, and this vortex motion is called the flux-flow regime.

Nucleation, motion, and annihilation of Abrikosov vortex–antivortex pairs express themselves as energy dissipation in a superconductor at temperatures T < T onc and at the zero external magnetic field. The flux-flow regime can be used for generation of high power and jitter-free ultrafast electric transients [13], for example, for applications in electromagnetic acceleration; however, in most cases, especially in low-power superconducting electronic devices, flux drift negatively affects their operation by producing excessive electronic noise and can even lead to an irreversible damage of device [14].

The most interesting type of movement of Abrikosov vortices in superconducting devices is a so-called coherent motion, occurring when vortices are tightly squeezed and strongly interact between each other [15], or, when they interact with a spatially periodic pinning landscape [16]. The appearance of voltage steps (a Josephson-like effect) on a current–voltage (I–U) characteristics of a microbridge-type, thin-film device is a hallmark of a coherent motion of Abrikosov magnetic vortices. The steps appear, when the inverse of the time of flight of vortices, across the half-width of the bridge, matches the frequency of the vortex nucleation at the device edges [17].

Our work reports on results of laser-written (LW), partially oxygen-depleted channels in YBCO microbridges that were defined by means of a continuous-wave (cw) laser illumination. We present electric and magnetic properties of such devices and the analysis of energy dissipation due to a coherent vortex motion in our artificially formed LW channels. The level of energy dissipation and the type of motion of vortices, confined to a given, partly deoxygenated channel area, have been evaluated by means of the analysis of I–U characteristics of microbridges with either one or two channels.

2 Experiment

2.1 Samples preparation

Our fabrication process starts with high quality, 0.3-μm-thick, epitaxial YBCO films grown by means of metalorganic chemical vapor deposition (MOCVD) technique onto crystalline LaAlO3 substrates at a substrate temperature of 830 °C and, subsequently, post-annealed at 350 °C in pure oxygen gas. More details of the MOCVD method and the superconducting and structural properties of resulting YBCO films are given in [18].

The 50-μm-wide and 100-μm-long superconducting YBCO microbridges were laser-patterned (LP) at room temperature in an ambient nitrogen atmosphere (see Fig. 1) using a green light, Ar-ion, cw laser beam focused into a spot diameter of approximately 5 μm. For patterning, YBCO films were attached perpendicularly to the laser beam’s direction on a computer-controlled X–Y translation stage.

Fig. 1
figure 1

Schematics of the experimental setup for LP and LW processing of YBCO thin films

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The laser power of 2.0–2.3 W and the X–Y stage scanning speed of 5 μm/s ensured the full deoxygenation (i.e., x > 0.6) of the laser-illuminated regions of the patterned film, converting them into an insulator [19]. In this way, we produced a set of six microbridges, each containing large, Ag/In contact pad areas for the 4-probe electric measurement. From the set of six identical microbridges, five were subsequently embedded with either one or two, partially oxygen-depleted channels for vortex flow using a LW technique and one was left as a pristine, reference device. In two-channel LW-defined structures, the channel separation distance was 10 μm.

During the LW procedure, the illumination laser power was lowered to 0.1–0.3 W, and a writing velocity was increased to ~ 50 μm/s [5], resulting in 5-μm-wide, only partially deoxygenated (i.e., x ~ 0.2) channels, perpendicular to the bridge main axis. The channels exhibited T c, H c1, and the critical current I c values comparatively lower than those of the laser-untreated reference sample [6].

2.2 The electric and magnetic properties of the laser-patterned YBa2Cu3O7−x devices

The resistivity versus temperature (R–T) and I–U characteristics of our superconducting devices were measured in a 4-probe, dc biasing arrangement at a zero external magnetic field, as is schematically shown in Fig. 2. The devices were thermally anchored to a copper holder inside a temperature-controlled liquid nitrogen cryostat, operating at a temperature range from 70 to 300 K. Temperatures below 78 K were achieved by means of a vacuum pumping of liquid nitrogen.

Fig. 2
figure 2

Schematics of a laser-processed (LP and LW) YBCO microbridge with a single vortex channel. A 4-probe (ampermeter A and voltmeter V for current and voltage measurements, respectively) measurement configuration and corresponding Ag/In contact pads are also shown

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Measurements of the reference (microbridge #6) device showed its excellent superconducting properties with a metallic-like R–T dependence, with a residual resistivity ratio R(300 K)/R(100 K) = 2.74, T onc  = 91.2 K, and the superconducting transition width of 0.4 K. The critical current density J c was 5 × 106 A/cm2 at T = 78 K, measured by means of a nanosecond-pulsed electric current [14]. The current self-induced Joule heating at the device contacts was negligible and did not affect our electric measurements in a whole range of tested temperatures and currents.

The quality of our LP and LW patterns was verified, in turn, by magneto-optical (MO) measurements. The analysis of MO images of the magnetic field penetration into our LW channel (see Fig. 3) demonstrated that both the magnetic field of the first flux entry into the channel’s area and H c1 of the superconductor were considerably smaller than those measured in the laser-untreated areas of the microbridge, or in the reference, that is, channel-free sample. From Fig. 3, we could also conclude that despite producing an increased concentration of oxygen vacancies, the LW procedure, actually, resulted in a decrease of the overall pinning strength, facilitating an easy vortex penetration and, subsequent, motion in LW-formed channels [20].

Fig. 3
figure 3

An MO image of a magnetic field inductance B = 0.028 T penetration into a single LW channel of a YBCO microbridge, observed at T = 0.658 T onc . Brighter areas in the picture represent the higher intensity of the penetrated field, clearly visualizing the channel

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3 Results and discussion

3.1 I–U dependences and electric noise measurements

Figure 4 presents two nonlinear I–U dependences: curve 1 for a bridge with a single LW channel and curve 2 for a reference sample. We note that the LW device exhibits voltage steps, associated with a coherent vortex motion in the bridge [21], measured at T = 0.962 T onc and the bias current I ≥ I c. Voltage steps can be clearly identified up to I ≈ 20 I c. The I c value of the sample was determined from the I–U curve by means of a 10-μV-voltage criterion fulfilled due to nucleation and motion of Abrikosov magnetic vortices. For larger currents, that is, for I > 20 I c, at T = 0.962 T onc , the amplitudes of the voltage steps fell below the resolution limit of our experimental setup. The stepped-like I–U dependence for this channeled device has been observed in a range of reduced temperatures 0.943 < T/T onc  < 0.98. At the same time, the channel-free (reference) bridge (curve 2 in Fig. 4), as well as a two-channel structure not presented here, did not show any evidence of voltage steps in their I–U curves in the whole range of tested temperatures and biasing currents.

Fig. 4
figure 4

A typical I–U characteristics of a 0.3 × 50 × 100 μm3 YBCO device with a single channel for easy vortex motion (curve 1) and a channel-free, reference device (curve 2). Both curves were collected at T = 0.967 T onc . Note that scales of both axes are logarithmic

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A family of I–U characteristics measured for another single LW channel device, collected in a 0.943 < T/T onc  < 0.962 range, is shown in Fig. 5. In this case, the I–U curves were measured by stepping-up the temperature of 0.2 K. The voltage steps on the I–U characteristics, especially for the ones collected in the low end of the temperature range (see, for example, curves 9 through 5), clearly start above a certain, specific biasing current I onst  > I c, which can be associated with an onset of the coherent vortex motion in our artificial LW channel. The parameter I onst decreases with the increase of the operating temperature as it is demonstrated in Fig. 6. Data for the particular set of I onst values (curve 1 in Fig. 6) were taken directly from the experimental I–U characteristics shown in Fig. 5. At the high-temperature end (curves 1, 2, and 3 in Fig. 5), on the other hand, even a comparatively very weak current (I ~ I c) creates favorable conditions for coherent vortex motion. As it was mentioned before, at those temperatures, vortex pinning in the LW channel is extremely low; therefore, any current-produced magnetic field is able to create vortex–antivortex pairs in the channel, leading to the onset of the flux-flow regime.

Fig. 5
figure 5

A family of I–U characteristics of a YBCO microbridge containing a single LW channel for easy vortex motion at zero external magnetic field and at temperatures starting with (1) at T = 0.962 T onc and ending with (9) at T = 0.943 T onc . The temperature interval between any of two nearest I–U curves was 0.2 K

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Fig. 6
figure 6

The I onst parameter of the coherent vortex motion for I-U dependences presented in Fig. 5 (curve 1) as a function of the reduced temperature. For comparison, curves 2 and 3 present independently measured I c(T) dependences for a micro bridge with the LW channel and the reference sample, respectively

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Figure 6 shows that I onst monotonically decreases with the increase of the device reduced temperature T/T onc , exhibiting an initial fast drop in the 0.952 < T/T onc  < 0.957, followed by a saturation region that approaches the standard I c(T) dependence (curve 2), independently measured for the same sample. The fast drop I onst (T/T onc ) dependence can be attributed to the temperature-affected change of the vortex-pinning mechanism in the LW channel volume. The increase of vortex pining with the temperature decrease has been already demonstrated in our earlier work [20]. We have also shown in [20] that microbridges containing LW channels exhibit two-step R–T transitions, with the first step at T = T onc and the second one corresponding to the current-dependent resistivity tail. The resistivity tail is due to the existence of partly deoxygenated YBCO material in the LW channel, as it can be clearly seen by comparing the I c(T) dependences for the bridges with and without the LW channel, that is, curves 2 and 3 in Fig. 6, respectively. Therefore, the saturation tails present in both curves 1 and 2 can be associated with thermally activated flux flow and random pinning/de-pinning of Abrikosov magnetic vortices in the LW channel. The latter corroborate with our MO observations (see Fig. 3) that indicated a strong penetration of the magnetic flux into the LW channel area. Thus, we can conclude that vortex motion in our studied microbridge is solely confined to the LW channel.

Another independent evidence of the coherent motion of Abrikosov vortices along the LW channels is shown in Fig. 7, where we plot time evolution of vortex-motion-induced voltage fluctuations in a microbridge with two LW channels. The observed a two-level, electric noise signal can be attributed to vortex channel intermittency. When a channel is open for an easy vortex flow, Abrikosov vortices move with velocity v, generating a voltage signal U = vnB, where n is the number of vortex–antivortex pairs in the channel and B is the magnetic inductance of the current self-produced magnetic field. When the magnetic vortices get pinned, their motion stops and the voltage amplitude drops down to zero. However, random thermal fluctuations are able to restore the vortex motion and reopen the channel again. Since in Fig. 7, the tested device contains two LW channels, besides the zero voltage level (no vortex flow), we observe two additional, equally spaced voltage levels (lower: 0.3 μV and upper: 0.6 μV), corresponding to either one or both channels opened, respectively. In the case of a microbridge with only one LW channel, we observed similar voltage fluctuations (not shown here), but they were one level only, while, as expected, the reference device did not exhibit any voltage fluctuations.

Fig. 7
figure 7

Real-time record of vortex-motion-induced, two-level voltage fluctuations in a YBCO microbridge with two parallel to each other LW channels for easy vortex motion spaced at distance of 10 μm. The electric signal has been recorded at T = 0.87 T onc and I ≈ I c = 2 μA

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3.2 Analysis of voltage steps

The voltage steps on I–U dependences can be attributed to energy dissipation caused by the coherent motion of Abrikosov vortices in the LW channel. For a given step, at a fixed temperature, its voltage amplitude U st consists of two parts, denoted in Fig. 8, as U vp and U vv, respectively.

Fig. 8
figure 8

A zoomed segment of the stepped I–U dependence, earlier shown as curve 2 in Fig. 5, measured at T = 0.959 T onc . U vp and U vv represent the electric voltage amplitudes associated with appearance of an additional vortex pair moving in the LW channel and the increase of the vortex velocity already moving along a given channel, respectively

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3.2.1 Analysis of voltage Uvp in the I–U curves

Analyzing Fig. 8, we have concluded that the value of U vp measured for any single step does not increase with the increase of I in the tested range of bias currents. For example, at T = 0.959 T c, the averaged value for U vp ~ 7.6 μV stays constant up to the voltage step number N = 100, while the bias current has been increased from 0.1 mA to 0.6 mA (i.e., 6 times). The latter reflects the fact that U vp is associated with the increase of energy dissipation in a superconductor, due to the appearance of an additional vortex–antivortex pair in the LW channel. During their coherent motion in the channel, vortices are arranged into a triangle magnetic lattice [22]; thus, an additional vortex–antivortex pair can be introduced into the LW channel’s area only if vortices can get more tightly squeezed, making some additional space for a new flux line.

Moving coherently, Abrikosov vortices are spaced at a distance \(d = \sqrt {2\Upphi_{0} /\left( {B\sqrt 3 } \right)} ,\) which is inversely proportional to \(\sqrt B .\) Thus, any additional vortex–antivortex pair entering into the LW channel induces a new voltage step with the total amplitude U st. The number of vortex–antivortex pairs n in the channel can be estimated from the ratio of dissipation (dynamic electrical resistance) in the vicinity of the N and N + 1 steps, for a given, fixed device’s temperature and biasing conditions. Following this ratio, one can estimate the n value even in the case of thermally activated flux-flow regime.

Finally, energy dissipation in the LW channel can be also affected by vortex pinning, for example, on screw dislocations or grain boundaries, or due to oxygen vacancies. The above mentioned defects are quite common in YBCO films produced by MOCVD [23]; however, as it has already been demonstrated in [20], their impact on the vortex dynamics in LW channels is very limited (< 5 %). Thus, we believe that the main contribution to the energy dissipation in LW channels actually comes from an increased number of oxygen vacancies, which behave like extended defects along the length of channel. They reduce the efficiency of pinning centers associated with the film growth defects, making pinning force F p close to zero.

3.2.2 Analysis of voltage U vv in the I–U curves

The energy dissipation due to change in the vortex velocity can be determined as a subtraction U vv = U st − U vp. The energy of vortices ηv \(\propto\) F L − F p should increase with increasing the Lorentz force F L ~ IΦo ~ ma, where m is the “effective mass” of the Abrikosov magnetic vortex, a is the vortex acceleration, and η = f(B,T) is the viscosity coefficient of the superconducting medium and is a function of B and T. The dissipative coherent vortex motion develops a voltage U vv ~ vNΦ0 = vB ~ ma. Thus, U vv increases with increasing I within a single voltage step on the I–U characteristics and, in this way, adds some small input to overall value of U st, as is especially clearly visible at low bias currents, for example, for I ≤ 10·I c (see Fig. 8). Further increase in device’s bias current affects appearance of an additional vortex–antivortex pair in the LW channel (i.e., appears as an additional mass m of the vortex matter) and, at that moment, the acceleration a of magnetic vortices sharply drops down. Subsequently, it, of course, increases again with the bias current increase until the current self-produced magnetic field creates one more vortex–antivortex pair in the channel. We believe that this is the reason why U st stays almost constant in the I-U dependences in a wide range of bias currents, for example, I < 10 I c at T = 0.959 T c. At the high bias range, that is, 10 I c < I < 19 I c, the input from U vv decreases below the resolution limit of our measurement setup.

4 Conclusions

In summary, this work reports on results of investigation of electric and magnetic properties of partially oxygen-depleted channels for easy vortex motion in superconducting YBCO thin films kept at temperatures below T onc . The YBCO epitaxial thin film can be either fully (LP mode resulting in x > 0.6) or partially (LW mode, resulting in x < 0.2) deoxygenated under a controlled laser illumination in an inert gas ambient. Our results demonstrate that this technique can be very effectively used for forming artificial channels for easy vortex flow. The LW channels allow for a precise control and study of vortex nucleation and, subsequent, coherent motion in pre-assigned regions of the YBCO film.

Magnetic field penetration into LW channels is in a form of Abrikosov vortices, and their coherent motion along the channel was observed in our samples at a very narrow temperature range (from 0.945 T c to 0.98 T c). The vortex coherent motion manifested itself as a rich, stepped-voltage structure (quasi-Josephson effect) on our experimental I–U dependences. The level of energy dissipation and the coherent character of vortex motion was evaluated by analysis of the I–U characteristics and the electric noise measurements in microbridges with either single or double channels for easy vortex motion. For practical applications, our superconducting microbridges with artificial channels for easy vortex motion can be implemented as sensitive imagers of magnetic field distributions with a measurement precision reaching single magnetic flux, that is, ± Φ0.