Various techniques have been applied to visualize superconducting vortices, providing clues to their electromagnetic response. Here, we present a wide-field, quantitative imaging of the stray field of the vortices in a superconducting thin film using perfectly aligned diamond quantum sensors. Our analysis, which mitigates the influence of the sensor inhomogeneities, visualizes the magnetic flux of single vortices in YBa_{2}Cu_{3}O_{7−δ} with an accuracy of ±10%. The obtained vortex shape is consistent with the theoretical model, and penetration depth and its temperature dependence agree with previous studies, proving our technique's accuracy and broad applicability. This wide-field imaging, which in principle works even under extreme conditions, allows the characterization of various superconductors.

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 *μ*m^{2}) 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 technique^{13} 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 parameter^{25} 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 YBa

_{2}Cu

_{3}O

_{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 \xd7 1 \xd7 0.5$ mm^{3}) 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 *μ*m [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 \xd7 10 5$ *μ*m^{–2} and $ 3 \xd7 10 19 \u2009 cm \u2212 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 \u2009 nm$. The critical temperature is estimated to be $ T c = 88.7 \u2009 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 ( $ \lambda NV = 650$–750 nm) with a CMOS camera and optical filters. The optical diffraction limit is estimated to be $ 0.61 \lambda NV NA \u223c 750$ nm. Since we acquire the images through the diamond, the optical resolution becomes 0.9 *μ*m due to optical aberration.^{34} We use a loop microwave antenna^{35} 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 \u2009 K \u2009 ( > T c )$ to the desired temperature. At the same time, we modulate their density by tuning the field generated by the coil.

^{25}

^{,}Figure 1(e) shows typical ODMR spectra, where the vertical axis is the relative PL intensity with and without microwave irradiation, and the horizontal axis is microwave frequency. There are two dips in each spectrum, which correspond to electron spin resonances of the NV centers. The splitting between the resonance frequencies $\Delta f$ is larger at 1 mT (red squares) than at 0 mT (black circles), reflecting the Zeeman effect. The splitting is given by

^{25}

^{,}

*B*is the magnetic flux density in the direction of the NV axis, $\gamma e=28$ MHz/mT is the gyromagnetic ratio of an electron spin, and

_{z}*E*is a strain parameter, which is position-dependent in the crystals. We fit the ODMR spectrum by two Lorentzian to determine $\Delta f$ and

*E*at each position and convert $\Delta f$ to

*B*using Eq. (1) (described later). Note that such a simple analysis is possible thanks to the perfectly aligned NV centers; ordinary ensemble centers have up to eight resonance signals, complicating the investigation. In addition, the absence of the sensors oriented to other symmetric axes is beneficial for the high sensitivity because it prevents contrast reduction.

_{z}^{28,36}We analyze the data of whole CMOS pixels (1536 × 2048 pixel

^{2}) to obtain the magnetic field distribution. To mitigate the failure of the Lorentzian fitting, we reduce shot noise by smoothing the PL image with a Gaussian filter smaller than the optical resolution (whose $1/e$ decay length is $350\u2009nm=5$ pixels of the camera) (see supplementary material for details.).

*T*to avoid the diamagnetism of superconductivity. $ \Delta f$ increases with increasing the absolute value of the coil voltage, as expected from Ampère's law. The total magnetic field is obtained as

_{c}*B*at the temperature $ T > T c , \u2009 \alpha coil$ is the linear coefficient between coil voltage and magnetic field, and $ B resid$ is a residual magnetic field, including geomagnetism. The solid black line in Fig. 1(f) is the fitting using Eqs. (1) and (2). It agrees well with our experimental result. We obtain the fitting parameters $ \alpha coil = 1.64 \u2009 \mu T / mV$, and $ B resid = 38.2 \u2009 \mu T$. The calibration accuracy of $ B z ( T > T c )$ is $ \xb1 1.2 \u2009 \mu T$ (95% confidence interval).

_{z}Figures 2(a) and 2(b) show the distributions of $ \Delta f$ obtained under FC conditions of $ B z ( T > T c ) = \u2212 1.1 \u2009 \mu T$ and $ B z ( T > T c ) = 3.8 \u2009 \mu 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.

*E*in the diamond crystal. We also observe that $\Delta f$ depends on the excitation light intensity, which can lead to such a distribution

^{37,38}(see supplementary material for details). We find that the latter effect, which is smaller than that of the strain, can be efficiently removed by phenomenologically including it in strain

*E*in the following analysis. We calculate the magnetic field density at each pixel using

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 $ \Phi 0 = h / 2 e = 2068 \u2009 \mu T \u2009 \mu 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 $ \beta = 4.84 \xd7 10 \u2212 4 \u2009 \mu m \u2212 2 / \mu 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 $ \u2212 0.7 \u2009 \mu T$, corresponding to the exact residual field of $ B resid = 37.5 \u2009 \mu 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 $ \Delta 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 $ \Delta f 0$ as an average of $ \Delta f$ far away from the vortex center (specifically, $ 4.8 \u2009 \mu m < r < 5.0 \u2009 \mu 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 *μ*T. 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 *μ*m 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 *μ*T, while the error bars are kept as small as $ \xb1 5.47$ *μ*T.

Figure 3(b) shows the magnetic flux projection obtained by integrating each vortex field over the region of $ r < 2.5$ *μ*m, 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 \u2009 \Phi 0$ and $ 0.029 \u2009 \Phi 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 \u2009 \Phi 0$ is smaller than $ \Phi 0$ because the integration range is limited to $ r < 2.5$ *μ*m and only the field component parallel to the NV axis is detected, as schematically shown in Fig. 1(a).

^{39–41}and thin-film

^{40–42}cases, dictated by the London penetration depth

*λ*and the Pearl length

^{43}

^{,}$\Lambda =2\lambda 2/tsc$, respectively. Given that the thickness is $tsc\u223c250$ nm in our case, comparable to the penetration length

*λ*(a few hundred nm

^{44–46}), we analyze our results using the model derived from Carneiro and Brandt,

^{40}which is applicable to both bulk and thin-film cases (see supplementary material for details):

*J*

_{0}is 0-th order Bessel function of the first kind, $\tau =k2+\lambda \u22122$, and

*λ*is the London penetration depth, which depends on temperature. Our method is subject to the influence of the thickness of the CVD layer and the optical resolution. The solid red line in Fig. 3(a) results from the fitting using a spatially integrated form of Eq. (4) to include these effects, reproducing the experimental result well within the error bars. The calculated flux is also consistent with the statistical results of the magnetic flux shown by the red vertical line in Fig. 3(b). We obtain $\lambda =154\u2009nm$ when we fix $d=1.35$

*μ*m. Since we repeat thermal cycles several times and confirm that two-parameter estimation from fitting both

*d*and

*λ*always yields a value of

*d*around 1.35

*μ*m, we fix

*d*= 1.35

*μ*m hereafter. The vortex size in a superconducting thin film, i.e., the Pearl length, is estimated to be Λ = 190 nm, smaller than the optical resolution. The stray field distribution from the vortex appears larger than Λ because the sensor ensemble is located away by

*d*from the YBCO film, which disperses the magnetic flux, as shown in Fig. 1(a).

We investigate the temperature dependence of *λ*. Figure 3(c) shows the $\lambda (T)$ from fitting the experimental result $Bz(T<Tc)(r;d,\lambda )$ at each temperature obtained by raising temperature after FC of $BT>Tc=\u221220.8\u2009\mu T$ (see supplementary material for full data). The resulting $\lambda (T)$ remains at $\u223c100$ nm from $10$ to $30\u2009K$ but dramatically increases above $40\u2009K$, reaching $\u223c500\u2009nm$ at $55\u2009K$. The vortex disappears at $Tc\u2032$ between $55$ and $60\u2009K$ [a gray area in Fig. 3(c)], lower than the original $Tc=88.7$ K, due to the local heating by laser irradiation.

*λ*varies from a minimum of 130 nm to a maximum of 810 nm.

^{44–46}The observed behavior of $ \lambda ( T )$ is consistent with them. We fit the temperature dependence of

*λ*using the following empirical model for a

*d*-wave superconductor,

^{47–49}

^{,}

^{43,50,51}the covariance of

*d*and

*λ*is large, meaning that

*λ*might vary depending on

*d*(and vice versa), and they might not be well determined by two-parameter fitting. Nevertheless, estimating the scaling behavior of one parameter from the fitting with the other parameter fixed is still meaningful in such a situation. The penetration depth

*λ*is an essential phenomenological parameter in describing superconductivity, and various methods have studied its behavior. Although the present method is not immune from the effect of laser heating, it provides an important alternative to systematically address this parameter under a wide range of experimental conditions.

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 resonance^{52} 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}

## SUPPLEMENTARY MATERIAL

See the supplementary material for all the magnetic imaging data in the present experiment, details of the numerics employed for the analysis, descriptions of the fitting methods, and information regarding the sensitivity.

We appreciate K. M. Itoh (Keio University) for providing the cryostat. The authors acknowledge the support of Grant-in-Aid for Scientific Research (Nos. JP22K03524, JP23H01103, JP19H05826, and JP22H04962) and of the MEXT Quantum Leap Flagship Program (Grant No. JPMXS0118067395). Some parts of this work were conducted at (Takeda Clean Room, Univ. Tokyo and Nanofab, Tokyo Tech), supported by Advanced Research Infrastructure for Materials and Nanotechnology in Japan (ARIM), Grant Nos. JPMXP1222UT1131 and JPMXP1222IT0058. S.N. is supported by the Forefront Physics and Mathematics Program to Drive Transformation (FoPM), WINGS Program, and JSR Fellowship, the University of Tokyo.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Shunsuke Nishimura:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Taku Kobayashi:** Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal). **Daichi Sasaki:** Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Visualization (equal). **Takeyuki Tsuji:** Data curation (equal); Investigation (equal); Methodology (equal); Resources (equal). **Takayuki Iwasaki:** Funding acquisition (supporting); Investigation (equal); Resources (supporting); Supervision (equal); Writing – review & editing (equal). **Mutsuko Hatano:** Funding acquisition (equal); Project administration (equal); Resources (equal); Supervision (equal). **Kento Sasaki:** Conceptualization (equal); Investigation (equal); Methodology (equal); Software (equal); Supervision (lead); Validation (lead); Visualization (lead); Writing – original draft (equal); Writing – review & editing (equal). **Kensuke Kobayashi:** Conceptualization (equal); Project administration (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

## REFERENCES

*Characterization of Materials*