Magnetic field expulsion in optically driven YBa2Cu3O6.48

Sample preparation and experimental geometries

A YBCO₇ thin film grown on two-side polished Al₂O₃ was patterned into a half-disc shape using a laser lithography process based on an AZ1512 photoresist mask. After exposure and lift-off, wet etching of the sample was done using a 1% H₃PO₄ solution. After etching, the residual photoresist was removed using acetone and isopropanol. Extended Data Fig. 2a shows a micrograph of the YBCO₇ film after patterning. The thin films and GaP (100) detector were then mounted onto an Al₂O₃ plate that could be fixed directly on the cold finger of the cryostat. Al₂O₃ was used to make the sample holder to minimize the effect of eddy currents on the applied field while ensuring good cooling power. A 75-μm thick GaP (100) crystal (SurfaceNet) was used as a detector and put in close contact with the sample (Extended Data Fig. 2b). The detector was polished with a wedge angle of about 1.5° to spatially separate the reflections from the front and back surfaces. This enabled us to detect exclusively the reflection from the back surface, which accumulated the Faraday polarization rotation during its propagation across the detector thickness. Moreover, the GaP back surface and the sample plane were not coplanar, to avoid interference of the reflections from these two surfaces (Extended Data Fig. 2c). The gap between the detector and the patterned YBCO₇ film was about 10 μm. This experimental geometry was used for the equilibrium superconductivity and disruption measurements reported in Fig. 1. Further details and calibrations regarding the use of GaP(100) as a magneto-optic detector are provided in Supplementary Information section 4.

The YBCO_6.48 single crystals were polished after growth to expose an ac-oriented surface that allowed access to the crystal c-axis. The single crystal sample was glued on the edge of a half-disc-shaped Al₂O₃ plate. On the top face of the same Al₂O₃ plate, a GaP (100) detector analogous to the one used for the disruption measurements was placed in contact with the YBCO_6.48 crystal. A 30-μm thick Al₂O₃ crystal was placed on top of the GaP crystal and acted as a shield preventing the 15-μm wavelength pump pulses from reaching the GaP detector. A second 30-μm thick Al₂O₃ crystal was placed on the side to also protect the detector from the side. Although Al₂O₃ is transparent for light at 800 nm wavelength, it is an almost perfect reflector for 15 μm wavelength pulses and features vanishingly small transmission⁴⁶ making it the perfect choice for this application. The two thin Al₂O₃ crystals also have the function of shaping the pump beam into a half-Gaussian with a well-defined edge. A sketch of this experimental geometry is shown in Extended Data Fig. 2d, alongside a top-view micrograph of the detector and sample assembly in Extended Data Fig. 2e. Also, in this case, the GaP detector was wedged and the YBCO_6.48 single crystal as well as the Al₂O₃ thin crystals were angled to prevent spurious reflections from reaching the polarization analyser (see Supplementary Information section 2 for more details on the experimental setup). Further details on the growth and characterization of these samples are given in Supplementary Information section 1.

Magnetostatic calculations

The changes in the magnetic field surrounding the sample were calculated in COMSOL using a finite element method to solve Maxwell’s equations taking into account the geometry of the experiment. The solution domain was defined as a spherical region of 1 mm radius in which a constant uniform magnetic field was applied. A half-disc-shaped region characterized by a constant, field-independent, spatially homogeneous magnetic susceptibility χ_v was placed in the centre of the spherical region and was used to model the magnetic response of either the patterned YBCO₇ thin film or the photo-excited region in YBCO_6.48.

Although the size of the half-disc in the simulation was exactly matched to the one used in the experiments for the patterned YBCO₇, assumptions had to be made regarding the size of the photo-excited region in YBCO_6.48. The latter was modelled as a half-disc of 375 μm diameter, coinciding with the measured 15-μm pump beam spot size, using different thickness values corresponding to different assumptions on the pump penetration depth as discussed below. The weak magnetic response of the substrate or the unperturbed YBCO_6.48 bulk was not included in the modelling as these are expected to be several orders of magnitude smaller because of their much lower magnetic susceptibility. To account for the detector response that, as written in the main text, generates a polarization rotation that is proportional to the average of the magnetic field in the volume probed by the light pulse, the results of the calculation were integrated along the detector thickness. This yielded two-dimensional maps of the spatially resolved magnetic field that were then convoluted with a two-dimensional Gaussian function to account for the spatial resolution given by the finite size of the focus of the probe beam.

Extended Data Fig. 3a,b shows a comparison between a line scan measured across the straight edge of the YBCO₇ half-disc and the results of a magnetostatic calculation performed using geometrical parameters that reflect the experimental conditions. In this simulation, χ_v was varied to achieve the best agreement with the experimental data. We extracted a value for χ_v ≈ −1 that is compatible with the zero-field-cooled magnetic properties of YBCO₇ thin film⁴⁷.

Extended Data Fig. 3c shows the temperature dependence of the magnetic field measured at equilibrium on top of a YBCO_6.48 single crystal, using a magneto-optic detector of thickness 250 μm. The geometry of this experiment is analogous to that shown at constant temperature in Fig. 2. At T_c ≈ 55 K, as the sample turns superconducting, a sudden decrease in the measured magnetic field is observed. Magnetostatic calculations were used to link the measured magnetic field expulsion to the magnetic susceptibility of the YBCO_6.48 sample. Extended Data Fig. 3d shows a comparison of the extracted magnetic susceptibility χ_v with that measured on the same sample with a commercial d.c. SQUID magnetometer. The agreement between these two measurements is very good, validating this approach.

Similar calculations were used to quantify the magnetic susceptibility that the photo-excited region in YBCO_6.48 should acquire after photo-excitation to produce a magnetic field change equal to that measured at the peak of the pump–probe response. This was achieved by running the calculations for a set of χ_v values and thicknesses of the photo-excited region to obtain calibration curves that related the average magnetic field expulsion measured 50 μm away from the edge to the susceptibility χ_v.

The curve shown in Fig. 2c is calculated under the assumption of a thickness d of the photo-excited region equal to 2 μm, corresponding to the electric field penetration depth of the pump, defined as \(d=\frac{c}{\omega \cdot {\rm{Im}}\left[{\widetilde{n}}_{0}\right]}\), where \({\widetilde{n}}_{0}\) is the stationary complex refractive index of YBCO_6.48 along the c-axis⁴⁸ at the pump frequency. This assumption is justified given the sublinear fluence dependence reported in Extended Data Fig. 9.

Extended Data Fig. 4 shows the dependence of the extracted χ_v on the assumption for the thickness d of the photo-excited region used in the magnetostatic calculations. Three different assumptions are considered:

• d = 1 μm, corresponding to the intensity penetration depth of the pump;

• d = 2 μm, corresponding to the electric field penetration depth of the pump; and

• d = 5 μm, corresponding to the region in which about 99% of the pump energy is absorbed.

We stress that independently of the chosen value for the penetration depth d, the retrieved value of χ_v remains in the 10⁻¹ range, several orders of magnitude higher than the strongest diamagnetic response observed in metallic systems such as graphite (Fig. 2c).

Additional spatially resolved equilibrium scans

The data shown in Fig. 1 report spatially dependent scans of the magnetic field surrounding the sample at a single temperature T = 30 K, which is less than T_c. These measurements were performed in a 2-mT applied magnetic field that switched polarity at 450 Hz frequency, scanning the probe beam across the edge of the YBCO thin film half-disc (Extended Data Fig. 5a). Below T_c, an enhancement of the magnetic field close to the edge and a reduction above the sample centre were observed (red and blue symbols in Extended Data Fig. 5b), both associated with a magnetic field exclusion from the sample interior. At T = 100 K > T_c, no changes in the local magnetic field were observed (grey symbols). This confirms that the presence of a physical edge does not introduce spurious effects (for example, multiple reflections) in our measurement.

This type of measurement was also carried out in the same geometry of Fig. 2 on a bulk YBCO_6.48 single crystal at a fixed temperature T = 25 K < T_c (Extended Data Fig. 5c,d). Similar to the data collected for the YBCO₇ thin film, a reduction in the magnetic field above the sample centre and an enhancement near the edge were observed. This confirms that the experimental geometry does not qualitatively affect our observations. In these data, the amount of field enhancement measured near the edge peaks at around 2%, roughly one order of magnitude higher than what was observed in the out-of-equilibrium measurements on the same sample.

This quantitative difference between out-of-equilibrium and equilibrium measurements is well understood based on the following:

1. 1.

   In the equilibrium measurements, the magnetic field polarity is changed periodically while the sample is in the superconducting state making them effectively analogous to a zero-field-cooled (ZFC) measurement. In the out-of-equilibrium measurements, the field is constant during the photo-excitation of the material, and hence these are the equivalent to a field-cooled type of measurement. ZFC measurements generally give rise to larger signals compared with the field-cooled ones. For example, magnetic susceptibility measurements (Supplementary Information section 1) performed on the same YBCO_6.48 single crystal show that, even at the lowest temperature, the ZFC susceptibility is about 50 times larger than that of the field-cooled one.

2. 2.

   The geometry of the two experiments is different. The YBCO_6.48 sample is a bulk single crystal (2 mm × 0.5 mm and 2 mm thick) and when cooled below T_c, the sample becomes superconducting throughout the whole volume. By contrast, in the non-equilibrium case, we expect the magnetic field expulsion to appear only in the photo-excited region, a half-disc (radius of about 150 μm), with a thickness determined by the penetration depth of the pump pulse (around 2 μm). The reduced thickness in the out-of-equilibrium case is expected to give rise to smaller signals (Extended Data Fig. 4) because of different field profiles generated in the vicinity of the sample.

Spatially resolved pump–probe scans in YBCO₇

The data in Fig. 1c,d were acquired in two different positions (above the sample centre and outside of it near the edge) as a function of the time delay between the 800-nm probe pulse and the ultraviolet (400 nm) pump pulse, used to disrupt superconductivity in YBCO₇. In Extended Data Fig. 6, we report spatial-dependent measurements of the pump-induced magnetic field changes at one selected time delay t = 10 ps. The pump–probe signal shows a spatial dependence similar to that of the static magnetic field expulsion. On the edge of the superconductor, where an enhanced magnetic field is observed at equilibrium, the destruction of superconductivity induced a negative magnetic field change, indicating that the applied magnetic field penetrated back into the sample. By contrast, above the sample centre, a reduced magnetic field is observed at equilibrium and disruption of superconductivity induced a positive signal, indicating that the magnetic field shielding ceased after photo-excitation. Owing to the specific pulse sequence used for the measurement (Supplementary Information section 3), spatially inhomogeneous trapped magnetic flux was present and the amplitude of the magnetic field change is slightly altered compared with what we would expect from the equilibrium measurements.

Temperature-dependent delay scans in YBCO_6.48

A time delay scan, similar to the scans shown in Fig. 2b, was acquired for each temperature. The peak value of each of these scans was extracted by a Gaussian fit of the data, and the peak value was plotted as a function of temperature in Fig. 4a. The data reported in Extended Data Fig. 7 show that the dynamics of the magnetic field expulsion is mostly independent of temperature and only the peak value reduces as the temperature is increased.

Additional spatial dependences in YBCO_6.48

Figure 3 shows a probe beam spatial dependence performed by moving the probe at progressively longer distances from the edge of the photo-excited region in the YBCO_6.48 crystal. In Extended Data Fig. 8, we report additional measurements that were performed at a 50-µm constant distance between the probe and the centre of the excitation beam, moving both of them together from a position where the YBCO_6.48 crystal is present underneath the GaP detector layer to one where it is not (Extended Data Fig. 8a). The result of this scan is shown in Extended Data Fig. 8b. As both the pump and the probe were moved on the detector away from the sample, the signal vanished confirming that this signal arose from a magnetic field expulsion in the YBCO_6.48 crystal following photo-excitation and not from spurious interactions in the magneto-optic detection crystal or the Al₂O₃ filter. This proves that the effect does not originate from the pump pulse itself, rather it is given by photo-excitation of the YBCO_6.48 sample in an applied magnetic field.

Fluence-dependent measurements in YBCO_6.48

In Extended Data Fig. 9, we report the dependence of the measured magnetic field expulsion on the excitation fluence. These data are taken at the peak of the response, at a temperature T = 100 K and with an applied magnetic field of 10 mT, after photo-excitation with 15 µm centre wavelength, 1 ps long pulses. The observed scaling seems to be sublinear in fluence. The data in the Figs. 2–4 were acquired at a fluence of around 14 mJ cm⁻², corresponding in this case to a peak electric field of about 2.5 MV cm⁻¹.