Real-space probing of the local magnetic response of thin-film superconductors usingsingle spin magnetometry

D. Rohner,1, ∗ L. Thiel,1, ∗ B. Muller,2 M. Kasperczyk,1 R. Kleiner,2 D. Koelle,2 and P. Maletinsky1, †

1Department of Physics, University of Basel, Klingelbergstrasse 82, Basel CH-4056, Switzerland2Physikalisches Institut and Center for Quantum Science (CQ) in LISA+,

Universitat Tubingen, Auf der Morgenstelle 14, D-72076 Tubingen, Germany(Dated: July 23, 2018)

We report on direct, real-space imaging of the stray magnetic field above a micro-scale disc ofa thin film of the high-temperature superconductor YBa2Cu3O7−δ (YBCO) using scanning singlespin magnetometry. Our experiments yield a direct measurement of the sample’s local Londonpenetration depth and allow for a quantitative reconstruction of the supercurrents flowing in thesample as a result of Meissner screening. These results show the potential of scanning single spinmagnetometry for studies of the nanoscale magnetic properties of thin-film superconductors, whichcould be readily extended to elevated temperatures or magnetic fields.

I. INTRODUCTION

Thin-film superconductors are of scientific interest andever-increasing technological importance. For example,such thin films offer possibilities to systematically explorefundamental properties of cuprate superconductors [1] orallow for exquisite tunability of fundamental propertiessuch as the superconductors critical temperature [2]. Inthe emerging field of quantum technologies, thin-film su-perconductors form the basis for superconducting quan-tum circuits, such as quantum bits or low-loss, microwaveresonators [3, 4] and they are also becoming increasinglyimportant in single-photon detection [5]. These devel-opments and especially the increased technological rele-vance of micro- and nano-structured superconductors callfor novel tools to locally probe superconductivity in suchmaterials with high sensitivity and nanoscale resolution.

Magnetic probes are particularly well suited for thistask, as they offer direct access to the local magneticsusceptibility and thereby a defining feature, i.e. per-fect diamagnetism, of the superconducting state. Indeed,various such probes, including magnetic force microscopy(MFM) [6–8], scanning Hall-probes [9, 10] and supercon-ducting quantum interference device (SQUID) [11–13]magnetometry have been developed and successfully em-ployed for this purpose. Despite the remarkable insight,such experiments offered into microscopic properties ofsuperconductors, they still suffer from their high inva-siveness (MFM), reduced spatial resolution (Hall probes)or the limited range of temperature and magnetic field, inwhich they operate (SQUID). These limitations warrantthe exploration of novel approaches to probe the localmagnetic response of superconductors, which would al-low, e.g. for studying them close to or above their criticaltemperature with nanoscale resolution.

In this work, we introduce a possible solution to thisbottleneck by employing scanning single spin magnetom-

∗ These authors contributed equally to this work† patrick.maletinsky@unibas.ch

etry using a Nitrogen-Vacancy (NV) electronic spin indiamond [14] for magnetic probing of a micro-structuredthin-film superconductor. We demonstrate nanoscaleimaging of the Meissner screening fields above a super-conducting disc and employ a quantitative model to ex-tract the local value of the London penetration depth λin our sample. Furthermore, we employ reverse propaga-tion of the experimentally measured magnetic field mapto reconstruct the screening supercurrents circulating inthe sample and thereby address spatial inhomogeneitiesin the superconducting state. Our findings complementrecent advances in using ensemble NV magnetometryto study Meissner screening or vortices in superconduc-tors [15–18] and promote this promising sensing technol-ogy to applications on the nanoscale. While we havepreviously reported on nanoscale studies of vortices intype-II superconductors [19], we here focus on the mag-netic screening properties of a thin-film superconductor– a technique which in contrast to vortex imaging can beapplied to both type-I and type-II superconductors.

II. EXPERIMENT

Our experiments exploit the electronic spin degrees offreedom of a single NV center in diamond – a latticepoint defect formed by a Nitrogen atom adjacent to alattice vacancy – as a nanoscale magnetometer. TheNV center orbital ground state forms an electronic spintriplet consisting of the magnetic sublevels |ms = 0〉 and|ms = ±1〉 (Fig. 1(a)), where ms denotes the magneticquantum number along the Nitrogen-Vacancy bindingaxis. At zero magnetic field, the states |ms = ±1〉 aredegenerate and split by a frequency D0 = 2.87 GHzfrom |ms = 0〉. The application of a magnetic fieldBNV along the NV spin quantisation axis induces a Zee-man splitting 2γNVBNV of the states |ms = ±1〉, whereγNV = 28 GHz/T is the gyromagnetic ratio. Conversely,transverse fields couple to the spin degree of freedom onlyto second order and will be neglected here. The NV spincan be initialised and read out optically, since optical

arX

iv:1

807.

0758

4v1

[co

nd-m

at.s

upr-

con]

19

Jul 2

018

2

a)

D0 =

2.8

7 G

Hz

2γN

VB

NV

νESR

ms = -1

ms = +1

ms = 0

b)

2γNVBNV

Scan plane

zStep

zNV

c)

STO

YBCOB

θNV

d)

eNV B 10 um

FIG. 1. (a) Nitrogen-Vacancy (NV) center ground statespin levels with zero-field frequency splitting D0 and Zeemansplitting 2γNVBNV (with γNV = 28 GHz/T and BNV the mag-netic field along the NV axis). As indicated by the red circles,the |ms = 0〉 spin sub-level exhibits a higher fluorescence ratethan |ms = ±1〉. (b) Optically detected electron spin reso-nance (ESR) of a single scanning NV center. (c) Schematiccross-section of the sample and the scanning probe hostingthe NV center. The NV is stabilised at a distance zNV fromthe superconductor surface using atomic-force distance con-trol. (d) Top view of the micro-structured YBCO sample.Blue (grey) represents regions of YBCO (substrate), respec-tively. The highlighted disk with 6 µm diameter is used hereto study Meissner screening and a nearby four-point bridgeto determine sample resistance.

excitation with 532 nm light results in spin-dependentfluorescence rates (as indicated in Fig. 1(a)) and spin-pumping into |ms = 0〉 [20]. These combined propertiesenable optical detection of electron spin resonance (ESR)of NV centers, where after initialisation into the (bright)|ms = 0〉 state, a microwave driving field resonant witheither of the |ms = 0〉 → |ms = ±1〉 transitions popu-lates the less fluorescent |ms = ±1〉 states, resulting ina significant drop in NV fluorescence (Fig. 1(b)). Suchoptically detected ESR thus yields a direct measure ofBNV, i.e. the magnetic field projection onto the NV axis.

In order to measure stray magnetic fields above thesample surface, we employ a single NV center lo-cated in the tip of a single-crystalline diamond scanningprobe [21]. This approach yields optimised sensing per-formance, maximal robustness and NV centers which areoriented at an angle of 54.7◦ with respect to the scan-ning probe (set by the crystal orientation of the tips),corresponding to an angle θNV ≈ 55◦ with respect tothe sample normal (Fig. 1(c)). Atomic force microscopy(AFM) feedback is used to stabilise the NV center abovethe sample surface at typical distances ∼ 50 nm, whiletip and sample are positioned and scanned using piezoactuators. The NV spin is addressed optically with a con-focal microscope and manipulated with microwave mag-netic fields generated by a close-by bonding wire spanned

across the sample. The experiment is located in a low-vibration Helium bath cryostat with a base temperatureof 4.2 K, equipped with a superconducting vector mag-net reaching fields up to 0.5 T. To investigate the mag-netic response of our sample, we apply an external biasmagnetic field Bz of 1.7 mT perpendicular to the samplesurface after cooling the superconducting film throughits critical temperature Tc in zero field. The continuous-wave microwave driving we employ leads to an elevatedtemperature ∼ 8 K (as determined by a nearby resistivethermometer) – a heating-mechanism which does not af-fect our results obtained on a high-Tc superconductor andwhich could be readily alleviated by employing pulsedmethods for ESR driving [22] in case of more fragile su-perconducting samples.

The sample under study is a thin film of the type-IIsuperconductor YBa2Cu3O7−δ (YBCO) [23], patternedinto lateral structures using optical lithography and Arion milling (Fig. 1(d)). The c-axis-oriented YBCO filmwas grown to a thickness dYBCO of ∼ 119 nm on top of a(001)-oriented single-crystal SrTiO3 (STO) substrate us-ing pulsed laser deposition and was covered with ∼ 16 nmof STO to avoid oxygen diffusion out of the sample [24].The film shows a Tc ≈ 90 K as measured in-situ by a four-probe conductance measurement. The data presented inthis work were taken on the 6 µm diameter disk high-lighted in Fig. 1(d).

III. RESULTS AND DISCUSSION

To study the nanoscale magnetic response of our sam-ple, we first measure a full, two-dimensional stray mag-netic field map of BNV above the 6 µm diameter YBCOdisk (Fig. 2 (a)). Magnetic field maps are created by mea-suring the entire ESR spectrum at each point of the scanand by determining BNV in a subsequent fit. Our datashow a strong reduction of magnetic field above the disk,while an enhanced field is measured close to the edgesof the disk. This observation is in qualitative accordancewith Meissner screening, where circular supercurrents in-side the superconductor lead to a magnetisation oppos-ing the external magnetic field and therefore to a reducedfield in the disk center and a compression of field linestowards the disk edge. The orientation of the NV spinquantisation axis with respect to the sample normal thenresults in the BNV map not being rotationally symmetricaround the disk center.

In order to gain further understanding of these ob-servations, we numerically calculate the current densityjs(r, z, λ) induced in a superconducting disk exposed toan external perpendicular magnetic field. Due to thecylindrical symmetry of our sample, it is sufficient to onlyconsider the z-component of the second London equationin cylindrical coordinates [25]

− Bzµ0λ2

= (∇×~js)z =(~js)φr

+∂(~js)φ∂r

, (1)

3

eNV B

FIG. 2. (a) Quantitative map of the magnetic field BNV,measured with the scanning NV spin above the YBCO diskin an external magnetic field of 1.7 mT applied perpendicularto the sample. Low magnetic fields are observed in the centerof the disk due to Meissner screening in the superconductorand maximal fields at the edges of the disk due to compres-sion of the field lines expelled from the disk. The observedabsence of rotational symmetry of BNV around the disk centeris a result of the NV orientation being away from the sam-ple normal. The data were acquired with a pixel dwell-timeof 12 s resulting in a scan time of 8 h for the entire scan.The green readout laser was set to a power of 350 µW witha microwave power of ∼ 15 dBm sent into the cryostat. Thedashed line indicates the position of the line scan in Fig. 3.(b) Calculation of BNV using the numerical model describedin the text, with λ = 250 nm and zNV = 100 nm as manuallyset input parameters.

where the subscripts indicate the vectorial components incylindrical coordinates and µ0 = 4π × 10−7 H/m is thevacuum permeability. Since only the supercurrent den-sity in azimuthal direction (~js)φ appears in this equation,we denote this quantity as js in the following. We find jsby numerically solving Eq. (1) in a disk with a diameter of6 µm and a thickness of 119 nm using a grid of 300× 12elements, corresponding to a resolution of 10 nm. Asboundary condition, we set js(r = 0) = 0 in the center ofthe disk and determine js(r, z) with the Euler method.Our calculation of js converges by iteratively calculatingjs(r, z), updating Bz(r, z) and then recalculating js(r, z).The magnetic field BNV is then calculated at the positionof the sensor using Biot-Savart’s law, taking into accountthe topography of the disk. Our procedure yields goodqualitative agreement with our data when using a pene-tration depth λ = 250 nm determined earlier [19] and anestimated NV-to-superconductor distance zNV = 100 nm(Fig. 2(b)). This agreement motivates further quantita-tive studies for an independent determination of λ fromour data, as discussed in the following.

In order to quantitatively analyse our data and deter-mine the penetration depth λ, we perform a high res-olution line scan of BNV across the YBCO disk, alongthe trajectory indicated in Fig. 2 (a). The resulting data(Fig. 3) show the same global features already discussedfor the 2D map, all while providing greater detail and

a higher sampling rate for subsequent, quantitative fit-ting using the same method as described above. For thefew data points (marked in blue in Fig. 3) where the ob-served NV ESR splitting falls short of the ESR linewidthof ∼ 8 MHz, our determination of BNV is unreliable dueto poor fitting quality and the effect of local strain onthe NV spin splitting [26]. We therefore excluded thesepoints in the subsequent analysis. From a fit to thesedata, we seek to extract the three key unknown param-eters λ, zNV, and θNV. To obtain the best estimatefor their values and respective errors, we include in ourmodel three additional nuisance parameters, namely thelateral position of the superconductor within the scan,the magnitude of the externally applied field, and a cali-bration factor to scale the x-axis. These nuisance param-eters are required for the model and the error estimationbut carry little information about the final values of λ,zNV, and θNV. The resulting model fit (Fig. 3) shows ex-cellent agreement with data and yields λ = 249 ± 5 nm,zNV = 70±10 nm and θNV = 55.3±0.7◦. Here, the errorbars denote 2σ confidence intervals, which we calculatefrom the diagonal elements of the covariance matrix ofthe free parameters, i.e. the width of the marginal prob-ability density function for each parameter. The errorbars therefore in principle account for correlations be-tween all free parameters. We find, however, significantcorrelations only between λ and zNV. The value of λ wefind with this approach agrees well with a previous andindependent measurement we performed in the same ma-terial [19], which gives further confidence in the validityof our approach. Taking into account the 16 nm thickSTO capping layer, we reach a NV-to-sample distance of∼ 50 nm, which constitutes an important benchmark forthe resolution capability of our setup.

Lastly, we take advantage of the one-to-one mappingbetween a two-dimensional current distribution and amagnetic field map in a plane away from the field sourceand use the 2D data presented in Fig. 2 (a) to determinethe supercurrent distribution in the YBCO disk which isresponsible for Meissner screening at first place. To thatend, we employ a well-established reverse-propagationmethod [27] to determine js in the superconductor. Inshort, the current density is obtained by applying theinverse Biot-Savart law to the measured Oersted mag-netic field in Fourier space. The method assumes noz-dependence of the current distribution inside the disk,which, however, is a valid approximation in our case sincedYBCO � λ. This reverse propagation process has atendency to exponentially amplify high-frequency com-ponents, and therefore noise, in the imaging data. Tocounteract this detrimental tendency, it is common prac-tice [27] to filter out these high-frequency components us-ing a low-pass filter with a typical cutoff frequency cor-responding to the inverse sample-to-sensor distance zNV.In our case, the sampling distance in the image was com-parable to zNV and therefore provided a suitable filter onits own and no further numerical filtering was performed.The resulting reconstructed current density shows the ex-

4

YBCOYBCO

a)

b)

FIG. 3. (a) Measurement of BNV across the YBCO diskalong the trajectory indicated in Fig. 2 (a). The black linein (a) shows the fit with the numerical model which yields apenetration depth of λ = 249 ± 5 nm. Data points markedin blue were excluded from the fit (see text). The data inte-gration time was 24 s per point resulting in 80 min for theentire scan. A laser power of 470 µW and a microwave powerof ∼ 15 dBm were applied to the NV spin. (b) Magnetic fieldlines around the YBCO disk in the Meissner state, along witha sketch of the experimental setup. The field lines togetherwith the NV spin quantisation direction (purple arrows) high-lighted for various positions illustrate the asymmetry in BNV

observed in the data. The dashed line illustrates the topogra-phy of the scan which is taken into account in the calculationof BNV.

pected circular screening currents inside the YBCO disk(Fig. 4). Moreover, the current reconstruction reveals theposition and shape of a region in the sample where su-perconductivity is suppressed and the current thereforedeviates from the circular shape of the disk. Signaturesof this were already visible in Fig. 2 (a), but we now vi-sualise it more directly through the impact it has on thecurrent flow in the superconductor. Note that no hintof a defect was visible in the topography of the sample.While further studies are required to assess the origin ofthis feature in js, we speculate that it may either con-stitute a vortex trapped at the edge of the YBCO discor a defect connected to material impurities introducedduring growth or micro-patterning.

For the circulalry symmetric part of the current dis-tribution, we can compare js as obtained from reverse-propagation to the one calculated by our numerical modeldescribed above. For this, we determine the azimuthalaverage of the reconstructed map js(r), while excludingcurrents close to the irregularity discussed in the pre-vious paragraph (Fig. 4(b)). Close to the center of thedisk, we observe a linear increase of js(r) with r, corre-

sponding to a constant curl of ~js due to a homogeneousfield Bz > 0 (c.f. Eq. (1)). Conversely, the supercur-rent increases super-linearly within a distance Λ fromthe edge of the disk, where Λ = 2λ2/dYBCO ≈ 1.04 µm is

a) b)

FIG. 4. (a) Current density js reconstructed by magneticfield reverse-propagation from the data in Fig. 2 (a). The cir-cular currents generate a magnetic field counteracting the ex-ternal magnetic field. (b) Azimuthal average of the currentdensity js(r) as a function of distance to the disk center, alongwith the current density calculated in the numerical model.

the effective penetration depth in a superconductor withthickness dYBCO [25].

IV. CONCLUSION AND SUMMARY

In conclusion, we have experimentally demonstratedthe use of scanning NV magnetometry to addressnanoscale properties of superconductors through straymagnetic field imaging. Specifically, our experiments per-formed on micron-scale YBCO disks allowed us to quan-titatively determine the local value of the London pene-tration depth in the material and to assess the impact ofdisorder in our thin superconducting film through super-current reconstruction. These results establish NV mag-netometry as an attractive resource to study magneticproperties of superconductors and open the way to ad-dressing open questions in mesoscopic superconductivity.Specifically, our quantitative and high-resolution imagingmethod could be employed to study supercurrents be-yond simple Meissner screening, such as presented in re-cent studies on superconductor-ferromagnet heterostruc-tures [28] or anomalous surface currents arising from An-dreev bound states [29], which may enable determinationof the superconductors order parameter symmetry. Ourdirect access to the nanoscale magnetic response of super-conductors offers attractive prospects to study the pre-cursor phase of superconductivity at elevated tempera-tures above Tc, where island superconductivity could beobserved in real space using our approach [30]. Further-more, novel quantum sensing techniques recently devel-oped for NV magnetometry offer attractive prospects inthe present context. For example, noise spectroscopy [31]and NV relaxometry [32, 33] could be used to assess mag-netic flux noise in superconductors – a general problemthat plagues superconductor-based quantum devices [4]

5

and that NV magnetometry could help alleviate in thefuture.

V. ACKNOWLEDGEMENTS

We thank D. Budker for fruitful discussions. Wegratefully acknowledge financial support through the

NCCR QSIT, a competence center funded by the SwissNSF, through the Swiss Nanoscience Institute (SNI) andthrough SNF Grant Nos. 142697 and 155845. This re-search has been partially funded by the European Com-missions 7. Framework Program (FP7/ 2007-2013) un-der Grant Agreement No. 611143 (DIADEMS). We alsoacknowledge support by the COST action NANOCOHY-BRI (CA16218).

[1] I. Bozovic, X. He, J. Wu, and A. T. Bollinger, “Depen-dence of the critical temperature in overdoped copperoxides on superfluid density,” Nature 536, 309 (2016).

[2] A. D. Caviglia, S. Gariglio, N. Reyren, D. Jac-card, T. Schneider, M. Gabay, S. Thiel, G. Hammerl,J. Mannhart, and J. M. Triscone, “Electric field controlof the LaAlO3/SrTiO3 interface ground state,” Nature456, 624 (2008).

[3] G. Wendin, “Quantum information processing with su-perconducting circuits: a review,” Rep. Prog. Phys. 80,106001 (2017).

[4] C. Muller, J. H. Cole, and J. Lisenfeld, “Towards under-standing two-level-systems in amorphous solids-insightsfrom quantum devices,” arXiv preprint arXiv:1705.01108(2017).

[5] E. A. Dauler, M. E. Grein, A. J. Kerman, F. Marsili,S. Miki, S. W. Nam, M. D. Shaw, H. Terai, V. B. Verma,and T. Yamashita, “Review of superconducting nanowiresingle-photon detector system design options and demon-strated performance,” Optical Engineering 53, 081907(2014).

[6] A. Badıa and H. C. Freyhardt, “Meissner state proper-ties of a superconducting disk in a non-uniform magneticfield,” Journal of Applied Physics 83, 2681–2688 (1998).

[7] J. Kim, L. Civale, E. Nazaretski, N. Haberkorn, F. Ron-ning, A. S. Sefat, T. Tajima, B. H. Moeckly, J. D.Thompson, and R. Movshovich, “Direct measurementof the magnetic penetration depth by magnetic force mi-croscopy,” Superconductor Science and Technology 25,112001 (2012).

[8] L. Luan, O. M. Auslaender, T. M. Lippman, C. W.Hicks, B. Kalisky, J.-H. Chu, J. G. Analytis, I. R. Fisher,J. R. Kirtley, and K. A. Moler, “Local measurementof the penetration depth in the pnictide superconduc-tor Ba(Fe0.95Co0.05)2As2,” Physical Review B 81, 100501(2010).

[9] A. M. Chang, H. D. Hallen, H. F. Hess, H. L. Kao,J. Kwo, A. Sudbø, and T. Y. Chang, “Scanning hall-probe microscopy of a vortex and field fluctuations inLa1.85Sr0.15CuO4 films,” Europhysics Letters (EPL) 20,645–650 (1992).

[10] J. Brisbois, B. Raes, J. Van de Vondel, V. V.Moshchalkov, and A. V. Silhanek, “Determination ofthe magnetic penetration depth in a superconducting Pbfilm,” Journal of Applied Physics 115, 103906 (2014).

[11] H. T. Lin, S. L. Wu, J. W. Wang, T. J. Chen, M. J.Wang, J. C. Chen, M. K. Wu, and C. C. Chi, “Determi-nation of the London penetration depth of FeSe0.3Te0.7thin films by scanning SQUID microscope,” Supercon-ductor Science and Technology 28, 085006 (2015).

[12] L. Embon, Y. Anahory, Zelko L Jelic, E. O. Lachman,Y. Myasoedov, M. E. Huber, G. P. Mikitik, A. V. Sil-hanek, M. V. Milosevic, A. Gurevich, et al., “Imagingof super-fast dynamics and flow instabilities of supercon-ducting vortices,” Nature Communications 8, 85 (2017).

[13] D. Vasyukov, Y. Anahory, L. Embon, D. Halbertal,J. Cuppens, L. Neeman, A. Finkler, Y. Segev, Y. Mya-soedov, M. L. Rappaport, M. E. Huber, and E. Zeldov,“A scanning superconducting quantum interference de-vice with single electron spin sensitivity,” Nature Nan-otechnology 8, 639 (2013).

[14] L. Rondin, J.-P. Tetienne, T. Hingant, J.-F. Roch,P. Maletinsky, and V. Jacques, “Magnetometry withnitrogen-vacancy defects in diamond,” Reports onProgress in Physics 77, 056503 (2014).

[15] A. Waxman, Y. Schlussel, D. Groswasser, V. M. Acosta,L.-S. Bouchard, D. Budker, and R. Folman, “Diamondmagnetometry of superconducting thin films,” PhysicalReview B 89, 054509 (2014).

[16] N. Alfasi, S. Masis, O. Shtempluck, V. Kochetok, andE. Buks, “Diamond magnetometry of Meissner currentsin a superconducting film,” AIP Advances 6, 075311(2016).

[17] N. M. Nusran, K. R. Joshi, K. Cho, M. A. Tanatar, W. R.Meier, S. L. Bud’ko, P. C. Canfield, Y. Liu, T. A. Lo-grasso, and R. Prozorov, “Spatially-resolved study of themeissner effect in superconductors using NV-centers-in-diamond optical magnetometry,” New Journal of Physics20, 043010 (2018).

[18] L.-S. Bouchard, V. M. Acosta, E. Bauch, and D. Budker,“Detection of the Meissner effect with a diamond mag-netometer,” New Journal of Physics 13, 025017 (2011).

[19] L. Thiel, D. Rohner, M. Ganzhorn, P. Appel, E. Neu,B. Muller, R. Kleiner, D. Koelle, and P. Maletinsky,“Quantitative nanoscale vortex imaging using a cryo-genic quantum magnetometer,” Nature Nanotechnology11, 677–681 (2016).

[20] A. Gruber, A. Drabenstedt, C. Tietz, L. Fleury,J. Wrachtrup, and C. von Borczyskowski, “Scanningconfocal optical microscopy and magnetic resonance onsingle defect centers,” Science 276, 2012–2014 (1997).

[21] P. Appel, E. Neu, M. Ganzhorn, A. Barfuss, M. Batzer,M. Gratz, A. Tschope, and P. Maletinsky, “Fabricationof all diamond scanning probes for nanoscale magnetome-try,” Review of Scientific Instruments 87, 063703 (2016).

[22] A. Dreau, M. Lesik, L. Rondin, P. Spinicelli, O. Arcizet,J.-F. Roch, and V. Jacques, “Avoiding power broad-ening in optically detected magnetic resonance of singleNV defects for enhanced dc magnetic field sensitivity,”Physical Review B 84, 195204 (2011).

6

[23] M. K. Wu, J. R. Ashburn, C. J. Torng, P. H. Hor, R. L.Meng, L. Gao, Z. J. Huang, Y. Q. Wang, and C. W. Chu,“Superconductivity at 93 K in a new mixed-phase Y-Ba-Cu-O compound system at ambient pressure,” PhysicalReview Letters 58, 908–910 (1987).

[24] S. Scharinger, M. Turad, A. Stohr, V. Leca, E. Goldobin,R. G. Mints, D. Koelle, and R. Kleiner, “Mag-netic field dependence of the critical current inYBa2Cu3O7−δ/Au/Nb ramp-zigzag Josephson junc-tions,” Physical Review B 86, 144531 (2012).

[25] M. Tinkham, Introduction to Superconductivity (DoverPublications Inc., 2004).

[26] M. W. Doherty, N. B. Manson, P. Delaney, F. Jelezko,J. Wrachtrup, and L. C.L. Hollenberg, “The nitrogen-vacancy colour centre in diamond,” Physics Reports 528,1–45 (2013).

[27] B. J. Roth, N. G. Sepulveda, and J. P. Wikswo, “Using amagnetometer to image a two-dimensional current distri-bution,” Journal of Applied Physics 65, 361–372 (1989).

[28] M. G. Flokstra, R. Stewart, N. Satchell, G. Burnell,H. Luetkens, T. Prokscha, A. Suter, E. Morenzoni,S. Langridge, and S. L. Lee, “Observation of anoma-lous Meissner screening in Cu/Nb and Cu/Nb/Co thinfilms,” Physical Review Letters 120, 247001 (2018).

[29] C. Iniotakis, T. Dahm, and N. Schopohl, “Effect of sur-face Andreev bound states on the Bean-Livingston bar-rier in d-wave superconductors,” Physical Review Letters100, 037002 (2008).

[30] V. Z. Kresin, Y. N. Ovchinnikov, and S. A. Wolf, “Inho-mogeneous superconductivity and the “pseudogap” stateof novel superconductors,” Physics Reports 431, 231(2006).

[31] N. Bar-Gill, L. M. Pham, C. Belthangady, D. Le Sage,P. Cappellaro, J. R. Maze, M. D. Lukin, A. Yacoby, andR. Walsworth, “Suppression of spin-bath dynamics forimproved coherence of multi-spin-qubit systems,” NatureCommunications 3, 858 (2012).

[32] J.-P. Tetienne, T. Hingant, L. Rondin, A. Cavailles,L. Mayer, G. Dantelle, T. Gacoin, J. Wrachtrup, J.-F. Roch, and V. Jacques, “Spin relaxometry of sin-gle nitrogen-vacancy defects in diamond nanocrystals formagnetic noise sensing,” Physical Review B 87, 235436(2013).

[33] D. Schmid-Lorch, T. Haberle, F. Reinhard, A. Zappe,M. Slota, L. Bogani, A. Finkler, and J. Wrachtrup, “Re-laxometry and dephasing imaging of superparamagneticmagnetite nanoparticles using a single qubit,” Nano Let-ters 15, 4942 (2015).