Wide-field quantitative magnetic imaging of superconducting vortices using perfectly aligned quantum sensors

Superconducting vortex, as a manifestation of macroscopic quantum effects, is one of the central subjects in the physics of superconductivity. Diverse vortex phases, such as vortex lattice, vortex liquid, and Bragg glass, appear in type-II superconductors' mixed state.^1–3 Those phases and vortex dynamics lead to bulk electromagnetic responses of superconductors and, thus, have been under vigorous investigation. In addition, since the flux quantization in superconducting vortices originates from the gap symmetry, anomalous quantization, such as a half-quantum vortex in p–wave superconductors,^4,5 is proposed to emerge as a signature of unconventional pairing symmetry. Therefore, techniques that can quantitatively image quantum vortices under various temperatures, pressures, and magnetic fields would help to probe a wide variety of superconductivity with open questions.

Several techniques are available to visualize local magnetic fields.^6–9 In particular, scanning techniques using sensor chips are widely used for quantitative measurements of magnetic flux density.^8,10–12 In such scanning techniques, superconducting quantum interference devices (SQUIDs)^10,11 and nitrogen-vacancy (NV) centers in diamonds^13,14 are prominent as sensors. While SQUIDs have excellent sensitivity, NV centers operate under severe environments, such as high temperatures and high magnetic fields.^15,16 Scanning microscopy provides nanoscale spatial resolution and high accuracy.¹² As for the NV-center technique, alternatively, imaging with a wide field of view exceeding (100 × 100 μm²) is possible with a camera and NV ensemble sensors.^9,17 This technique is beneficial in terms of high throughput.⁹ Furthermore, it can be introduced into extreme environments such as ultrahigh pressure,^18–20 which are not accessible by the scanning technique. Thus, it aids in researching unconventional superconductors at high temperatures and pressures.^21,22 Using this technique, efforts have been made particularly to image the stray magnetic fields of superconducting quantum vortices,^23,24 but achieving magnetic accuracy close to the scanning technique¹³ has been challenging. The issue arises primarily due to the fact that the measurement of superconductors is conducted in a low magnetic field, where the inhomogeneity of the sensor's strain parameter²⁵ and the signal overlap resulting from a diamond sensor ensemble with four NV axes render quantitative analysis to extract field component perpendicular to the superconductors' surface practically impossible.

Here, we address these issues by utilizing a perfectly aligned NV ensemble sensor^26–30 and implementing an analysis that eliminates sensor inhomogeneities resulting from strain distribution, complemented by reference measurements in a zero magnetic field. Consequently, we report a quantitative wide-field magnetic imaging of superconducting vortices in a thin film of a typical high-T_c superconductor YBa₂Cu₃O_7−δ (YBCO). The combination of the inherent high throughput of wide-field NV microscopy and achieved quantitativeness enables statistical analysis. The obtained statistics is consistent with the single quantization of vortices. Moreover, the stray magnetic field distribution aligns well with theoretical models, offering an alternative method for estimating the magnetic field penetration depth. Our technique, which combines high throughput and accuracy, is helpful for comprehensive characterization, including exploring unconventional superconductors.^31–33 

We use NV ensemble sensors at the diamond surface to visualize vortex stray magnetic fields. Figure 1(a) is the measurement schematic. The sensors are located in a thin film grown on a (111) Ib diamond substrate (⁠ $1 × 1 × 0.5$ mm³) using a chemical vapor deposition (CVD) technique.^26–30 The symmetry axis of the NV center (NV axis) is perfectly aligned perpendicular to the diamond surface. The CVD-grown NV layer thickness is 2.3 μm [Fig. 1(a)], measured by secondary ion mass spectroscopy. The areal density of NV centers and the density of nitrogen atoms are estimated to be $2.1 × 10 5$ μm^–2 and $3 × 10 19 cm − 3$⁠, respectively. We adhere the diamond chip to a YBCO thin film by varnish [Fig. 1(b)]. The stray field from the vortices is detected at the NV centers in the CVD layer at a distance d away [Fig. 1(a)]. The YBCO sample is a (100) thin film (S-type) on a MgO substrate purchased from Ceraco Ceramic Coating GmbH. The nominal YBCO thickness is $t sc = 250 nm$⁠. The critical temperature is estimated to be $T c = 88.7 K$ from the temperature-dependent sheet resistance shown in Fig. 1(c).

Our microscope system is shown in Fig. 1(d). The sample is fixed with vacuum grease to a stage in an optical cryostat (Montana Instruments Cryostation s50). The sample temperature is controlled by a heater and monitored by a thermometer of the stage. Hereafter, we use the stage thermometer value as the temperature. We expand a green laser (532 nm, 120 mW) onto the diamond to image the photoluminescence (PL) of the NV centers. We image the wavelength range of the NV center (⁠ $λ NV = 650$–750 nm) with a CMOS camera and optical filters. The optical diffraction limit is estimated to be $0.61 λ NV NA ∼ 750$ nm. Since we acquire the images through the diamond, the optical resolution becomes 0.9 μm due to optical aberration.³⁴ We use a loop microwave antenna³⁵ fixed on the optical window of the cryostat to manipulate the NV centers. A coil applies a spatially uniform static magnetic field in the direction perpendicular to the YBCO surface, parallel to the NV axis. We perform field-cooling (FC) to generate the vortices by cooling down the stage temperature from $90 K ( > T c )$ to the desired temperature. At the same time, we modulate their density by tuning the field generated by the coil.

Figures 2(a) and 2(b) show the distributions of $Δ f$ obtained under FC conditions of $B z ( T > T c ) = − 1.1 μ T$ and $B z ( T > T c ) = 3.8 μ T$⁠, respectively. We obtain these images at 40 K. There are multiple point-shaped magnetic field distributions at the larger field [Fig. 2(b)], while no such distributions at the smaller field [Fig. 2(a)]. Each of these points is a superconducting vortex. Later we prove that they are genuinely single vortices. The absence of such a feature in Fig. 2(a) indicates no vortices in this view, implying minuscule magnetic fields are realized in the cooldown process. We define this condition as zero-field cooling.

We examine the relation between the number of vortices and the magnetic flux density during FC. We count the number of the vortices in the field of view to obtain vortex areal density, as shown in Fig. 2(f). The vortex density increases linearly with the absolute value of the magnetic field. A superconducting vortex has a single flux quantum $Φ 0 = h / 2 e = 2068 μ T μ m 2$ (where h is Planck's constant and e is the elementary charge). The vortex density corresponds to the magnetic flux density. Thus, in Fig. 2(f), the proportionality coefficient should be $β = 4.84 × 10 − 4 μ m − 2 / μ T$⁠. The solid black line is the theoretical fitting based on the calibration in Fig. 1(f), consistent with the experimental result within the error bars. As shown by the vertical dashed line in the inset of Fig. 2(f), the zero field calibration is carried out within $− 0.7 μ T$⁠, corresponding to the exact residual field of $B resid = 37.5 μ T$⁠, including geomagnetism. These results prove that the observed vortices have a single flux quantum.

The present method, which observes many vortices in a wide field of view quantitatively and simultaneously, enables us to make a statistical analysis. The inset of Fig. 3(a) depicts the distribution of $Δ f$ for a typical vortex. Thus, the magnetic field is isotropically distributed concerning the distance r from the vortex center. We rely on Eq. (3) to extract the field, where we define $Δ f 0$ as an average of $Δ f$ far away from the vortex center (specifically, $4.8 μ m < r < 5.0 μ m$⁠) to avoid the effect of drift during FC cycles. There are 290 vortices in the results obtained under FC of several $B z ( T > T c )$ between −13.9 and 5.3 μT. We estimate the center-of-mass positions of these vortices by Gaussian fitting. Among them, we extract 190 vortices, located away from large inhomogeneity and separated by more than 8 μm to avoid the effect of drift and the influence of stray fields from neighboring vortices.

Figure 3(a) shows the obtained distribution of the magnetic field of a vortex as a function of r. The error bar reflects the standard deviation concerning the 190 vortices used in the analysis. The magnetic field just above the vortex center is 51.1 μT, while the error bars are kept as small as $± 5.47$ μT.

Figure 3(b) shows the magnetic flux projection obtained by integrating each vortex field over the region of $r < 2.5$ μm, as indicated by the arrow in the top left of Fig. 3(a). The histogram forms a Gaussian distribution, meaning that all the single vortices are accurately captured as having the same flux. The magnetic flux's average and standard deviation is $0.295 Φ 0$ and $0.029 Φ 0$⁠, respectively, showing that the present technique has a precision of 10%. The statistical uniformity also guarantees that our analysis has removed the observed inhomogeneities. $0.295 Φ 0$ is smaller than $Φ 0$ because the integration range is limited to $r < 2.5$ μm and only the field component parallel to the NV axis is detected, as schematically shown in Fig. 1(a).

We investigate the temperature dependence of λ. Figure 3(c) shows the $λ ( T )$ from fitting the experimental result $B z ( T < T c ) ( r ; d , λ )$ at each temperature obtained by raising temperature after FC of $B T > T c = − 20.8 μ T$ (see supplementary material for full data). The resulting $λ ( T )$ remains at $∼ 100$ nm from $10$ to $30 K$ but dramatically increases above $40 K$⁠, reaching $∼ 500 nm$ at $55 K$⁠. The vortex disappears at $T c ′$ between $55$ and $60 K$ [a gray area in Fig. 3(c)], lower than the original $T c = 88.7$ K, due to the local heating by laser irradiation.

To conclude, we have quantitatively established the wide-field imaging of superconducting vortices using a perfectly aligned diamond quantum sensor. By eliminating the effect of inhomogeneity, the magnetic flux of a single vortex in a YBCO thin film was visualized with an accuracy of ±10%. In addition, we demonstrate the quantitative method to examine the penetration depth. We can further improve sensitivity and accuracy by combining techniques such as multi-frequency magnetic resonance⁵² and thinner CVD layers.²⁸ The demonstrated precise high throughput method, applicable over a wide temperature range, helps to explore various superconducting properties and statistical evaluation, including their MHz–GHz dynamics.⁵³ For example, it could apply to investigating an anomalous quantum vortex, such as a half-integer one, and to the high-pressure superconductivity in diamond anvil cells.^18–20