1. Introduction

2. Experimental data of PME

2.1. Comparison of the PME in HTSc and metallic superconductors

We start with a comparison of the PME found in bulk Nb disks (see Figure 1a, [26]) with that of HTSc Bi-2212 polycrystalline ceramics (see Figure 1b, [13]) and an artificially patterned YBCO thin film sample (see Figure 1c, [101,102,103]). For all 3 samples, the magnetization is plotted versus temperature, $m\left(T\right)$. The PME is observed when field-cooling (FC-C) or field-warming (FC-W) the sample in a small applied magnetic field. The third cooling mode is zero-field cooling (ZFC), where the sample is cooled in zero field to the lowest temperature, and then a small magnetic field is applied. Also in this mode, signatures of PME may be observed (lower panel of Figure 1b). The features found in Figure 1a, b and c are, on a first glance, qualitatively the same: In all three cases, there results a positive $m\left(T\right)$ signal, i.e. a paramagnetic signal, when cooling the sample in quite small fields, and the positive $m\left(T\right)$ signal reduces gradually when applying larger magnetic fields. However, it is also quite obvious that the Nb disk exhibits very clear minima ($T_1$)/maxima ($T_p$) on field cooling as well as on warming, whereas these features are less clear in Bi-2212 (Figure 1b) and practically non-existant in the case of the patterned YBCO film (Figure 1c).

In Ref. [104], the authors list some more distinct differences of the PME in Nb and Bi-2212 samples: "Before proceeding to a discussion of our experiment we would like to remark that these two forms of paramagnetism in ceramic Bi-2212 and in a bulk Nb sample can be clearly distinguished in several other ways. For example, the cooling rate affects the magnetic response differently in the two cases. Recent experimental data show significant differences between Nb and granular Bi-2212 HTSc samples. While slow cooling enhances the paramagnetic signal for the granular sample, it is diminished in the Nb sample. This clearly indicates that the equilibrium state of both samples in a small magnetic field is quite different [105]. For the Nb discs Koshelev and Larkin gave an explanation based on the idea that during the cooling process the surface region nucleates superconductivity before the bulk, so that magnetic flux in the sample is compressed and creates an enhanced magnetization [85]. This compressed flux mechanism leads to a metastable state which depends on the cooling procedure whereas the polarization of the spontaneous orbital moments is an equilibrium process. Further, noise measurements of the magnetization of Bi-2212 give signals which are compatible with the presence of spontaneous orbital moments [106]."

The more detailed measurements on the Nb disks performed in the Detroit group in the later years also revealed several important differences between the polycrystalline HTSc samples and the bulk Nb samples like the presence of two characteristic temperatures, $T_1$ and $T_p$ (see the lower panel of Figure 1a), at all applied magnetic fields, and the different shape of the magnetization loops in the temperature range between $T_p$ and $T_c$. This will be discussed in detail in Section 3 below.

Figure 1. Comparison of the experimentally observed PME effect in 3 different types of superconductors. (a). PME in a bulk Nb disk. In the upper graph, the recorded field-cool warming (FC-W) curves are given for small magnetic fields (5 $\mu$T – 966 $\mu$T). The lower graph gives both FC-C and FC-W curves with the applied fields given alongside the respective curves. The arrows indicate the measurement direction. Note here the characteristic minima ($T_1$) and maxima ($T_p$), which are visible up to the highest applied fields. Reprinted with permission from Ref. [26]. (b). PME in a HTSc Bi-2212 polycrystalline, bulk sample. The upper plot shows the field-cool cooling (FC-C) data together with the zero-field cooling (ZFC) curves for several applied magnetic fields. The lower graph of (b) presents the transition region in more detail. Reprinted with permission from Ref. [13,14]. (c). PME observed in an artificially granular HTSc YBCO thin film [101,102,103], see the image of the sample in the upper graph. The lower graph gives the FC-W and ZFC curves obtained on this type of sample. Reprinted with permission from Ref. [101].

Figure 1. Comparison of the experimentally observed PME effect in 3 different types of superconductors. (a). PME in a bulk Nb disk. In the upper graph, the recorded field-cool warming (FC-W) curves are given for small magnetic fields (5 $\mu$T – 966 $\mu$T). The lower graph gives both FC-C and FC-W curves with the applied fields given alongside the respective curves. The arrows indicate the measurement direction. Note here the characteristic minima ($T_1$) and maxima ($T_p$), which are visible up to the highest applied fields. Reprinted with permission from Ref. [26]. (b). PME in a HTSc Bi-2212 polycrystalline, bulk sample. The upper plot shows the field-cool cooling (FC-C) data together with the zero-field cooling (ZFC) curves for several applied magnetic fields. The lower graph of (b) presents the transition region in more detail. Reprinted with permission from Ref. [13,14]. (c). PME observed in an artificially granular HTSc YBCO thin film [101,102,103], see the image of the sample in the upper graph. The lower graph gives the FC-W and ZFC curves obtained on this type of sample. Reprinted with permission from Ref. [101].

The situation of sample (Figure 1c) is completely different: This sample is patterned from an homogeneous YBa${}_2$Cu${}_3$O${}_{7-\delta }$ (YBCO) thin film [101,102], which in its original form did not show any PME. The structure is pictured in the top graph of Figure 1c. The sample does not contain any weak links, which one expects from a granular sample, but the current paths between the individual disks are constricted to a narrow area, where the disks are touching each other. This guarantees the flow of a strong inter-disk current, but with a changed field dependence. Thus, the PME in this sample is only observed at very low applied fields, and entirely caused by trapped flux between the disks. Specific features of the PME like the characteristic temperatures $T_1$ and $T_p$ could not be observed here. Interestingly enough, practically the same situation was encountered recently when investigating the magnetic properties of superconducting YBCO nanofiber mats [107,108], where the supercurrents can flow in local rings as well. Thus, the resulting PME signals resemble the ones of this artificially granular thin film, but with a much smaller signal strength.

In the literature, there is a fourth type of PME reported, the so-called high-field PME or HFPME [109,110,111], which is only observed on HTSc bulk samples of the (RE)BCO-type (where RE= Y, Nd, Eu, Sm, Gd) when measuring superconducting transitions in high applied magnetic fields. The $m\left(T\right)$-signals recorded are first going to negative values at $T_c$, but then start to turn towards positive values at lower temperatures. However, the recorded transitions are fully different from the ones of polycrystalline Bi-2212 [13,14], which suggest a different origin. In Refs. [112,113], we could show that the superconducting transition may be only a small negative contribution to an otherwise positive $m\left(T\right)$-signal when measuring various (RE)BCO bulk superconductors, including GdBCO, NdBCO or a mixture like (Nd,Eu,Gd)BCO (abbreviated: NEG). Especially, GdBCO may show a Néel temperature below 4 K to a ferromagnetic state. So, this high-field PME is caused by such strong paramagnetic magnetic moments inherent to the sample itself.

2.2. Metallic superconducting samples with PME

Figure 2 and Figure 3 present observations of the PME on a variety of metallic superconductors, where FC curves result in positive values of $m\left(T\right)$. In nearly all these measurements, the external magnetic field is applied perpendicular to the sample surface unless noted otherwise (Figure 2a,d and Figure 3a). This also applies to the various measurements on the Nb disks with the exception of Figure 10. However, only some of the authors provide also detailed graphs of the superconducting transition, where some of the characteristic features like in the case of the Nb disks can be seen. Some of the samples investigated are even multilayers where one component may be magnetic, or samples with different amounts of additional doping. Such samples may exhibit positive (paramagnetic) $m\left(T\right)$ signals in the entire temperature range also above the superconducting transition, so the magnetic signals must be worked out properly and be subtracted from the respective plot. This makes the comparison of the curves found in the literature quite difficult. On the other hand, whereever the measurements provide the details close to $T_c$, the characteristic minima and maxima can be observed, which clearly points to the importance of the two characteristic temperatures, $T_1$ and $T_p$.

We further must note here that some of the experiments reported in the literature did only show the presence of PME in their specific system, but some of the experiments done were indeed dedicated and planned experiments to test the theoretical predictions for the occurance of the PME, which nicely reflects that the metallic superconductors are the workhorses for experimentalists.

The PME was observed up to now on a large variety of metallic superconductors including Nb (bulk disks, bars, thin films, as well as nanostructured samples), Pb (on bulk disks, but also in nanowire arrays and Pb-porous glass nanocomposites), mesoscopic structures of Al and Nb [63], Ta, 2H-NbSe${}_2$ single crystals, Nb-AlO${}_x$-Nb tunnel junctions, ZrB${}_{12}$ single crystals, indium-tin oxide films with electrochemical doping (e.g., Mg), Ca${}_3$Rh${}_4$Sn${}_{13}$, Ti-V alloys, Nb-Gd composite thin films, Bi${}_2$Te${}_3$-FeTe heterostructure, Mo${}_{100-x}$Re${}_x$ alloy, Dy${}_{1-x}$Y${}_x$Rh${}_4$B${}_4$, Co/Nb multilayers, Ag/Nb proximity structures, and also the MgB${}_2$ system [70,71,72,73,74,75], which represents the metallic superconductor with the highest transition temperature. Here, we must note that the PME in MgB${}_2$ was always observed in polycrystalline samples (bulks or tapes), where the grain connectivity may be different depending on the chosen preparation route.

Figure 2a–f and Figure 3a–f give several experimental results on various metallic superconductors. Table 1 lists all PME experiments on metallic superconductors, together with the data for the superconducting transition temperature (onset), the recorded transition width $\Delta T_c$, $\kappa$, the references and some remarks to the experiment itself. Especially the PME observations on mesoscopic metallic superconductors (Al, Nb) [63,65] gave very important input to the understanding of the PME. The same applies to the measurements on ZrB${}_{12}$ crystals with an unique vortex-vortex interaction [37], making the material an intermediate 1.5-type superconductor. This we will discuss in Section 5 in detail.

Figure 2. PME observed on various metallic superconducting samples I. (a) Rings of In-Sn, measured in FC and ZFC modes (applied field $H=$ 0.2 mT (2 Oe) with the field applied parallel and perpendicular to the sample surface (ZFC (⊥) •, ZFC (‖)▪ ▪ ▪ ▪ ▪, FC (⊥) ∘). The inset shows details of the superconducting transition. Reprinted with permission from Ref. [49]. (b) Pb-films on PEEK, rolled up into cylinders. Data shown are FC curves in various applied fields (0.01 T – 0.3 T). Graph redrawn from the data presented in Ref. [50]. (c) Pb porous. FC and ZFC data are shown for various applied fields. The inset gives the standard measurement at low applied field. Reprinted with permission from Ref. [33]. (d) Ta foil. Shown are the superconducting transitions at $H=$ 30 mT (300 Oe) and for 3 different angles (0${}^{\circ }$, 45${}^{\circ }$ and 90${}^{\circ }$ of the field to the sample surface). Reprinted with permission from Ref. [35]. (e) ZrB${}_{12}$ crystals, measured at 0.1 mT (1 Oe), 0.5 mT (5 Oe), and 1.5 mT (15 Oe) applied field. The insets give details of the transition at 5 $\mu$T (0.05 Oe) (FFC-W), 0.5 mT (5 Oe) (FFC-W and ZFC) and 5 mT (50 Oe) (SFC-C), where the F stands for `fast’ and S for `slow’. Reprinted with permission from Ref. [37]. (f) Susceptibility of an Al mesoscopic disk (2.5 $\mu$m dia., thickness 0.1 $\mu$m) at applied fields of 0.9, 3.5, 4.0, 6.5 and 12 mT (9, 35, 40, 65 and 120 G). The last 4 curves were multiplied by a factor of 10. The authors point out that there is no hysteresis when cooling/warming. Reprinted with permission from Ref. [63].

Figure 2. PME observed on various metallic superconducting samples I. (a) Rings of In-Sn, measured in FC and ZFC modes (applied field $H=$ 0.2 mT (2 Oe) with the field applied parallel and perpendicular to the sample surface (ZFC (⊥) •, ZFC (‖)▪ ▪ ▪ ▪ ▪, FC (⊥) ∘). The inset shows details of the superconducting transition. Reprinted with permission from Ref. [49]. (b) Pb-films on PEEK, rolled up into cylinders. Data shown are FC curves in various applied fields (0.01 T – 0.3 T). Graph redrawn from the data presented in Ref. [50]. (c) Pb porous. FC and ZFC data are shown for various applied fields. The inset gives the standard measurement at low applied field. Reprinted with permission from Ref. [33]. (d) Ta foil. Shown are the superconducting transitions at $H=$ 30 mT (300 Oe) and for 3 different angles (0${}^{\circ }$, 45${}^{\circ }$ and 90${}^{\circ }$ of the field to the sample surface). Reprinted with permission from Ref. [35]. (e) ZrB${}_{12}$ crystals, measured at 0.1 mT (1 Oe), 0.5 mT (5 Oe), and 1.5 mT (15 Oe) applied field. The insets give details of the transition at 5 $\mu$T (0.05 Oe) (FFC-W), 0.5 mT (5 Oe) (FFC-W and ZFC) and 5 mT (50 Oe) (SFC-C), where the F stands for `fast’ and S for `slow’. Reprinted with permission from Ref. [37]. (f) Susceptibility of an Al mesoscopic disk (2.5 $\mu$m dia., thickness 0.1 $\mu$m) at applied fields of 0.9, 3.5, 4.0, 6.5 and 12 mT (9, 35, 40, 65 and 120 G). The last 4 curves were multiplied by a factor of 10. The authors point out that there is no hysteresis when cooling/warming. Reprinted with permission from Ref. [63].

Table 1. Metallic superconductors exhibiting the PME. Given are the varoius materials, the sample type, the superconducting transition temperature, $T_c$, in Kelvin, the transition width, $\Delta T_c$, the Ginzburg-Landau parameter $\kappa$, the respective citations and some remarks to the experiment. The abbreviation nps denotes nanoparticles, ⌀ stands for diameter and ITO = Indium Tin Oxide.

Table 1. Metallic superconductors exhibiting the PME. Given are the varoius materials, the sample type, the superconducting transition temperature, $T_c$, in Kelvin, the transition width, $\Delta T_c$, the Ginzburg-Landau parameter $\kappa$, the respective citations and some remarks to the experiment. The abbreviation nps denotes nanoparticles, ⌀ stands for diameter and ITO = Indium Tin Oxide.

name	type	$T_c$	$\Delta T_c$	$\kappa$	citation	remarks
Nb bulk	commercial	9.26	∼0.1	–	[25,30,31,32,114,115,116,117]	stationary sample, SQUID
Nb bulk	from ingot	9.25	0.05–0.1	∼1	[27]	several types of samples,
						disk-shaped, 3–6 mm ⌀
Nb	calculated	–	–	–	[39,40]	disks, cylinders
Nb	crystal, poly	9.38	–	–	[35]	Nb crystal bar, polycryst. disks
Ag-Nb	wires	9.2	–	–	[64]	Nb-wires with Al sheath
Nb	films	9.2	0.03	–	[91]	strain-free thin films
Nb	films	8.8/8.3	–	–	[52]	thin films, relaxation
Nb-Gd	films	8.85/4	–	–	[54]	Gd-doped Nb films, various doping
nano-Nb	Nb powder+corund	–	–	–	[44]	granular Nb with various pore sizes
Nb/Cu	multilayers	9.25	0.3	–	[51]	PME, AC frequency dep.
Nb/Co	multilayers	9.2	–	–	[55]	Co-layer top/bottom of Nb (240 nm)
Au-Ho-Nb	trilayer	8.52	0.3	–	[56]	$\mu$SR-study
Nb-AlOx-Nb	multiply connected	–	–	–	[58,59]	Josephson junction arrays
Pb	films on PEEK	7.2	0.1	–	[50]	rolled up as cylinders
Pb-glass	porous glass	7.2	∼0.5	–	[33]	85 % filling of pores with Pb
Pb-nw	NWs 40nm dia	7.2/4	–	–	[53]	filled alumina template
Pb-Co	nanocomposite	6.2	–	–	[62]	Pb thin film with 1 vol-% Co
Al	thin film/disks	1.1	0.7	0.3	[63]	Al and Nb mesoscopic structures
Al	disk 1.5 $\mu$m ⌀	–	–	–	[65]	Al mesoscopic disk, 0.03 K
Ta	foil	4.38	–	1.39	[35]	Ta foil
Bi/Ni	Ni layer on top	3.9	0.1	–	[60]	PME in positive/negative fields
NbSe${}_2$	single crystals	7.15	broad	–	[41]	very clean crystals
Ca${}_3$Rh${}_4$Sn${}_{13}$	single crystals	8.4	∼ 2.5	–	[46]	SQUID-VSM with various amplitudes
Dy${}_{(1-x)}$Y${}_x$Rh${}_4$B${}_4$	crystals	∼ 6	0.5	–	[36]	various contents x tested
LiRhB	polycrystalline	2,4-2,6	–	1	[47]	different composition, partly 2 $T_c$’s
Bi${}_2$Te${}_3$-FeTe	bilayer	∼ 6	–	–	[61]	Bi${}_2$Te${}_3$ (9 nm)/FeTe (140 nm)
In-Sn	cylinders, 3 phases	6.2/4.7/3.7	0.2	–	[34]	$\beta$-InSn, $\gamma$-InSn, $\beta$-Sn → extrinsic PME
In-Sn-O	films, Mg-dop.	4.81	0.09	–	[49]	doped ITO with Mg, 90/10
Mo${}_{100-x}$Re${}_x$	bulk	4.47	–	–	[45]	high-field PME
Ti${}_{0.8}$V${}_{0.2}$	bulk	4.15	0.2	–	[38]	high-field PME
V/Fe	bilayers	3.3–3.5	–	11–20	[57]	40.1 nm V / 1.1 nm Fe
ZrB${}_{12}$	crystals	5.95	0.08	0.8	[37]	type II-1 sc., vortex interaction
B-doped diamond	thin film	5.8–2.1	–	–	[76]	various doping
MgB${}_2$	granular	38.2	∼ 2	–	[70]	bulk/powder
MgB${}_2$	granular/sintered	38	∼ 2	–	[71]	bulk, $\gamma$-irradiation
MgB${}_2$	TiO${}_2$ np	–	–	29.1	[72,73]	2 % TiO${}_2$
MgB${}_2$	tapes	35–29.9	∼5	–	[74]	Fe-sheated tapes with Co${}_3$O${}_4$ nps
MgB${}_2$	MgO	37.1/38.8	15/0.5	–	[75]	MgO ∼ 40%/ ∼ 7.3 %

And, of course, samples of the metallic superconductors are much better suited than HTSc materials (deposition of clean thin films, patterning) to attempt the various types of magnetic imaging of the vortex structures as we will see later in Section below.

After the first observations of the PME in the Bi-2212 HTSc, the work in this direction also continued. Here, we list only some of the contributions as this could be the topic of an own, dedicated review. Even more details on the superconducting transitions of Bi-2212 were worked out [118,119], and several more types of HTScs were found to exhibit the PME like YBCO as thin films, polycrystalline bulks, single crystals or nanofiber mats [17,107,108,120,121,122,123], La${}_{1-x}$Sr${}_x$CuO${}_4$[124], the electron-doped superconductor Nd${}_{2-x}$Ce${}_x$CuO${}_y$[125], the bismuthate Ba${}_{1-x}$K${}_x$BiO${}_3$[126] and RSr${}_2$Cu${}_2$NbO${}_{7-\delta }$ compounds (R= Y, Pr, and Gd), which have a crystallographic structure like YBCO [127]. Furthermore, the PME was observed in measurements of the iron-based superconductors (IBS) like K${}_{0.8}$Fe${}_{1.7}$Se${}_2$[128]. An important issue we must note here is that all these HTSc materials mentioned here are typically granular, and most of the samples studied are such with $T_c<$ 77 K.

Figure 3. PME observed on various metallic superconducting samples II. (a) Pb nanowire arrays. ZFC curves of a sample with $d=$ 40 nm for 5, 8 and 11 mT (50, 80 and 110 Oe). H is applied along the long axis of the wire. Reprinted with permission from Ref. [53]. (b) 2H-NbSe${}_2$ single crystals. H (‖a,b)= 5 mT (50 Oe). The $M\left(T\right)$ plots in the vicinity of $T_c$. Paramagnetic signals are evident in all three modes (ZFC, FC-C, FC-W), and $T_p<T_c$. Reprinted with permission from Ref. [41]. (c) Bi/Ni-bilayers. The FC results are measured in magnetic fields in the range ±5 mT (50 Oe).The signals from the Ni layer and the substrate are subtracted in all data shown. Reprinted with permission from Ref. [60]. (d) MgB${}_2$ polycrystalline bulk sample. FC magnetization curves at different applied fields. The PME is observable up to 25 $\mu$T (250 mOe). Reprinted with permission from Ref. [71]. (e) Superconductor/ferromagnet heterostructure V(40 nm)/Fe(1.1 nm). Temperature dependence of the magnetic moment around $T_c$ measured in different magnetic fields and cooling regimes (FC, ZFC). The inset shows the field dependence of the induced magnetic moment in the FC regime. Reprinted with permission from Ref. [57]. (f) Sn${}_{90}$In${}_{10}$ alloy in form of a cylinder with 3 different phases ($\beta$-InSn, $\gamma$-InSn, $\beta$-Sn). The graph gives the FC curves for two samples annealed at 340 K and 380 K. Reprinted with permission from Ref. [34].

Figure 3. PME observed on various metallic superconducting samples II. (a) Pb nanowire arrays. ZFC curves of a sample with $d=$ 40 nm for 5, 8 and 11 mT (50, 80 and 110 Oe). H is applied along the long axis of the wire. Reprinted with permission from Ref. [53]. (b) 2H-NbSe${}_2$ single crystals. H (‖a,b)= 5 mT (50 Oe). The $M\left(T\right)$ plots in the vicinity of $T_c$. Paramagnetic signals are evident in all three modes (ZFC, FC-C, FC-W), and $T_p<T_c$. Reprinted with permission from Ref. [41]. (c) Bi/Ni-bilayers. The FC results are measured in magnetic fields in the range ±5 mT (50 Oe).The signals from the Ni layer and the substrate are subtracted in all data shown. Reprinted with permission from Ref. [60]. (d) MgB${}_2$ polycrystalline bulk sample. FC magnetization curves at different applied fields. The PME is observable up to 25 $\mu$T (250 mOe). Reprinted with permission from Ref. [71]. (e) Superconductor/ferromagnet heterostructure V(40 nm)/Fe(1.1 nm). Temperature dependence of the magnetic moment around $T_c$ measured in different magnetic fields and cooling regimes (FC, ZFC). The inset shows the field dependence of the induced magnetic moment in the FC regime. Reprinted with permission from Ref. [57]. (f) Sn${}_{90}$In${}_{10}$ alloy in form of a cylinder with 3 different phases ($\beta$-InSn, $\gamma$-InSn, $\beta$-Sn). The graph gives the FC curves for two samples annealed at 340 K and 380 K. Reprinted with permission from Ref. [34].

Thus, the metallic superconductors represent a large playground for the observation of the PME in clearly non d-wave superconductors. This also applies for the measurements in MgB${}_2$, representing a two-band superconductor with all tunable inter-grain properties [69,129,130]. Interestingly enough, there are no observations of the PME reported in granular systems like metallic superconductor wires of NbTi or Nb${}_3$Sn, which may be a consequence of the strong coupling between the grains in these materials [7,131].

3. Specific measurements, details of the superconducting transitions of Nb disks and discussion

3.1. Investigated Nb samples

The samples originally measured in Detroit were Nb disks of 6.4 mm in diameter, which were punched from commercially available, 0.127-mm thick sheets (99.98 % purity, Johnson-Matthey) [25,26]. In addition, the optical and electron microscopy investigations on these Nb disks punched from cold-rolled 0.127-mm thick Nb sheets also indicate the presence of surface defects and voids.

Table 2. Summary of the Nb disks samples measured in Detroit. Given are the sample names, their origin, some comments, the sample dimensions and the measured data for $T_{c,\mathrm{onset}}$, $T_p$ and $T_1$.

Table 2. Summary of the Nb disks samples measured in Detroit. Given are the sample names, their origin, some comments, the sample dimensions and the measured data for $T_{c,\mathrm{onset}}$, $T_p$ and $T_1$.

sample	Nb1	Nb2	Nb3	Nb4
sample origin	D4S2	D2S2	D10S2-1	DI08-1
Comment	“basic”	abraded	edge sand	implanted
PME	yes	no	yes	yes
radiusr (mm)	3.2	3.2	3.2	3.2
thicknesst ($\mu$m)	127	∼110	127	250
$T_{c,\mathrm{onset}}$ (K)	9.20	9.26	9.24	9.28
$T_1$ (K)	9.15	n/a	9.24	9.17(5)
$T_p$ (K)	9.05(5)	n/a	9.06(5)	9.08

The results presented in the following Figure 5, Figure 6, and Figure 7 are for one particular Nb disk (sample Nb1, stemming from Nb sheet D4S2), see also Ref. [25]. One of the original Nb sheets (D4S2) and some punched disks (samples named Nb001–Nb005) were carried to Tokyo (SRL/ISTEC, Div. 3) for further measurements in different magnetometers, and finally to IJL Nancy, France, where the same material was studied again nearly 25 years later using SQUID magnetometry and AC susceptibility measurements.

3.2. Observation of PME – superconducting transitions

In this Section, we focus on the detailed measurements performed on various Nb disk samples, which were found to exhibit the most clear PME behavior. Niobium (Nb) has an bcc lattice ($a=$ 0.33 nm and $\alpha =\beta =\gamma$), is isotropic, and represents a classical s-wave superconductor with a Ginzburg-Landau parameter $\kappa$ close to the border between type-I and type-II superconductors ($\kappa \sim$0.8–1.0).

The Nb sample studied in Detroit (similarly to some other bulk Nb samples seen in the literature) shows a very specific behavior around the superconducting transition region. In previous papers [25,27], this behavior was in detail studied mainly using the field-cooled cooling (FC-C) and field-cooled warming (FC-W) curves, i.e., the temperature dependence of magnetic moment $m\left(T\right)$ was measured at constant magnetic field when temperature is slowly continuously swept from a temperature where the sample is in normal state down to a temperature well below the superconducting transition (FC-C) and then the temperature sweep direction is reversed (FC-W). These scans of the magnetic moment, $m\left(T\right)$, reveal several characteristic temperatures, namely $T_1(<T_{c,\mathrm{onset}})$ describing the minimum of magnetization, and $T_{\mathrm{p}}(<T_1)$, which indicates the maximum of the positive moment [114]. Note here that in some publications, $T_1$ was also called $T_u$. We must note here that these characteristic temperatures are very important for the understanding what is going on in the sample when cooling or warming it in applied magnetic fields.

Figure 5a presents details of $m\left(T\right)$ transitions measured on a punched Nb disk. The superconducting transitions were measured in small applied magnetic fields (10, 50, 100, 200 $\mu$T) perpendicular to the sample surface generated by the Cu coil (using the low-field option with stationary sample). The temperature was continuously swept with a rate of $|\mathrm{d}T/\mathrm{d}t|=$ 38 mK/min. The arrows indicate the direction of the measurement (FC-C and FC-W). The temperature $T_p$ is defined here as the temperature of the maximum $m\left(T\right)$. The inset to (a) shows the definition of the temperature $T_1$ as the lowest $m\left(T\right)$ recorded on the diamagnetic side.

Figure 5b presents a similar situation, but here the measurement is performed at a fixed field ($B_{\mathrm{ext}}=$ 0.5 mT) and the temperature was swept with a rate of $|\mathrm{d}T/\mathrm{d}t|=$ mK/min to reach various minimum values ($T_{\min }$) between $T_1$ and $T_p$ and also slightly below $T_p$, which are indicated in the graph. The inset to (b) gives more details close to $T_1$. Here, one can see that the $m\left(T\right)$ curves are nearly reversible in the temperature range between $T_1$ and $T_c$. Whatever the origin of the paramagnetic moment appearing in the sample between $T_1$ and $T_p$ is, it appears and disappears nearly reversibly with increasing/decreasing the temperature. Furthermore, the onset of the nearly reversible diamagnetic moment can be determined to be at about $T=$ 9.27 K. This moment is fully reversible as long as temperature did not go close to 9.13 K and below. If the sample was cooled below 9.13 K, one can observe a very slight hysteresis of the $m\left(T\right)$ curves even in the temperature range above $T_1$. The $m\left(T\right)$ curves below $T_1$ are fully reversible as long as the temperature $T_{\min }$ is not below about 9.07 K during the cooling and warming cycle. For the $m\left(T\right)$ curves measured at $T_{\min }<$ 9.07 K, one can see during warming a sharper decay of the magnetic moment towards its equilibrium value at $T=T_1$.

Thus, the field-cooled magnetization results for Nb disks show a paramagnetic moment that has a reversible nature as it appears and disappears at the same temperature $T_1$ during cooling and warming, respectively. This moment is superimposed on a typical diamagnetic superconducting behavior below $T_1$ and continues to increase in magnitude until the temperature, $T_{\mathrm{p}}$. Thus, the appearance of a net positive signal at lower temperatures depends on the relative magnitude of the paramagnetic moment of the sample to that of the diamagnetic moment below the temperature $T_{\mathrm{p}}$.

In Figure 6a–c, FC-C and FC-W $m\left(T\right)$-curves are shown in various applied magnetic fields up to 140 mT. A paramagnetic (positive) $m\left(T\right)$ is observed on FC-C for small applied magnetic fields. This positive moment is gradually reducing on increasing field, and the low-T part of each curve is obtained at negative $m\left(T\right)$ for all applied fields above 3 mT. However, the FC-W curves all show the characteristic step in the temperature range 8.8–9.2 K.

Figure 7 presents high-field measurements of the PME on a Nb disk (Nb002) which was carried from Detroit to Tokyo. Using the constant temperature sweep rate of a 7T-XL SQUID [134], the measurements were performed with the superconducting coil in the persistent mode. The measurements were performed after the entire magnetometer had been warmed up to room temperature for servicing. In this way, any spurious trapped fields in the coil system can be safely excluded. From this graph we can see that the $m\left(T\right)$-curves obtained are always on the negative (i.e., diamagnetic) side, but the fingerprints of the PME can be observed as characteristic steps in the FC-W curves up to 200 mT applied magnetic field.

Figure 8 gives the results of an important experiment which we called `shifting the PME’. To estimate the magnitude of the paramagnetic step on the FC-C and FC-W curves measured at $B_{\mathrm{ext}}=$ 0.05 mT (see Figure 8a), both the cooling and warming curves below $T_p$ were shifted to such a position that the magnetization curves above $T_1$ and below $T_p$ point toward each other. The best results at this value of $B_{\mathrm{ext}}$ were obtained for the shift by $m_{\mathrm{shift}}=$ 1.33 × 1$0^{-9}$ Am${}^2$. In the detailed view of the transition given in Figure 8b, one can see that the shifted curves combined with the curves above $T_1$ form a nice, archetypal FC-C and FC-W curve. This is a clear indication that the relative shape of the recorded FC-C and FC-W curves below $T_p$ follows an archetypal shape. The crucial change in the superconducting state takes place between temperatures $T_1$ and $T_p$, where an additional positive moment (superconducting currents) appears in the sample, and this moment stays constant in the sample as long as the temperature stays below $T_p$. It again starts to disappear from the sample when the temperature is raised above $T_p$ and it finally disappears completely above $T_1$.

Below $T_{c,\mathrm{onset}}$, $m\left(T\right)$ becomes more negative upon cooling until $T_1$ is reached (≈ 9.16 K at the lowest fields), where $m\left(T\right)$ abruptly turns towards positive values. This increase in $m\left(T\right)$ continues until another characteristic temperature, $T_{\mathrm{p}}$ (≈ 9.16 K at the lowest fields), is reached, where $m\left(T\right)$ exhibits a cusplike behavior. Below $T_{\mathrm{p}}$, the relative temperature dependence of $m\left(T\right)$ appears to correspond to the archetypal FC-C behavior of a superconducting sample not exhibiting the PME. As discussed in Ref. [25], another characteristic feature of the PME for the Nb disks is the large hysteresis between the FC-C and FC-W curves with the FCW curves being lower than the FC-C curves. Upon warming, the $m\left(T\right)$ curves remain practically constant until the moment turns abruptly towards negative values at a temperature essentially the same as $T_{\mathrm{p}}$. Then, at a temperature identical to $T_1$, $m\left(T\right)$ jumps to a less diamagnetic and more probable equilibrium value as the PME appears to vanish suddenly. The similarity in characteristic temperatures where the positive moment first appears during cooling and then disappears during warming as well as where the positive contribution stops increasing in magnitude during cooling and subsequently begins to decrease upon warming suggests that the PME has a reversible nature.

Figure 9 presents the phase diagram of the PME in the bulk Nb disks, constructed from the temperature variation of the characteristic temperatures $T_1$, $T_p$, and $T_{c,\mathrm{onset}}$. The inset to Figure 9 presents a schematic phase diagram including surface superconductivity as described by $H_{c3}\left(T\right)$ and the flux flow region described by the irreversibility field $H_{\mathrm{irr}}\left(T\right)$. The green arrow indicates where FC-C and FC-W measurements are crossing the phase diagram. $H_{\mathrm{irr}}\left(T\right)$ is for most metallic superconductors very close to $H_{c2}\left(T\right)$ [137]. Note also that in the field-cooling experiments with a magnetic field applied, there is always magnetic flux in the sample, and a pure Meissner state will not be reached even if flux is expelled from the sample. The temperature $T_{c,\mathrm{onset}}$, which is slightly larger as the bulk $T_c$, defines the first onset of superconductivity, i.e., the first recording of a diamagnetic moment which occurs at ≈ 9.23 K for the sample discussed in here. All Nb samples from the same Nb sheet we studied in Detroit, Tokyo and Nancy have $T_c$ between 9.23 and 9.27 K. Also given are $T_c$ data (magnetically measured to determine $H_{c2}$) for pure Nb and Nb treated in N${}_2$ from Ref. [136]. Any differences determined between the FC-C and FC-W runs for these temperatures are within the size of the data points and are the result of a temperature lag between the sample and the thermometer during the cooling and warming processes. One further can note here that the field dependence for all three temperatures is nearly linear: $\mathrm{d}{T}_p/\mathrm{d}{B}_e=$ 13.5 K/T (< 60 mT) and 11 K/T (>60 mT), $\mathrm{d}{T}_1/\mathrm{d}{B}_e=$ 10 K/T, and $\mathrm{d}{T}_{c,\mathrm{onset}}/\mathrm{d}{B}_e=$ 8.7 K/T. Our $T_{c,\mathrm{onset}}$-data are higher and $\mathrm{d}{T}_c=\mathrm{d}{B}_e$ smaller as compared to Weber and Schachinger [136], which may be possible due to different accuracy in measurement/determination of the onset of the diamagnetic signal. However, another explanation is given in Ref. [37]: Our $T_{c,\mathrm{onset}}$ could directly correspond to $H_{c3}\left(T\right)$, i.e., the onset of the surface superconductivity. Then, $T_1$ could correspond to the curve of $H_{c2}\left(T\right)$ and $T_p$ to $H_{\mathrm{irr}}\left(T\right)$, which requires some further analysis using also AC susceptibility measurements (see Section 3.6 below).

Figure 9. Phase diagram for the PME in bulk Nb disks, constructed using the characteristic temperatures $T_1$, $T_p$ and $T_{c,\mathrm{onset}}$. $T_1$ and $T_p$ are taken from both the warming (`w’) and cooling (`c’) data. Also given are the data of $T_c$ from Ref. [136] for pure and N${}_2$-treated Nb samples. The inset shows a schematic phase diagram including surface superconductivity as described by $H_{c3}\left(T\right)$ and the flux flow region described by the irreversibility field $H_{\mathrm{irr}}\left(T\right)$. The green arrow indicates where FC-C and FC-W measurements are crossing the phase diagram.

Figure 9. Phase diagram for the PME in bulk Nb disks, constructed using the characteristic temperatures $T_1$, $T_p$ and $T_{c,\mathrm{onset}}$. $T_1$ and $T_p$ are taken from both the warming (`w’) and cooling (`c’) data. Also given are the data of $T_c$ from Ref. [136] for pure and N${}_2$-treated Nb samples. The inset shows a schematic phase diagram including surface superconductivity as described by $H_{c3}\left(T\right)$ and the flux flow region described by the irreversibility field $H_{\mathrm{irr}}\left(T\right)$. The green arrow indicates where FC-C and FC-W measurements are crossing the phase diagram.

In Figure 10a,b the results of applying a magnetic field parallel to the sample surface are shown. Figure 10a, presenting $M/H_{\perp }\left(T\right)$ and $M/H_{\Vert }\left(T\right)$, indicates the temperatures $T_u$ (which is defined as the first appearance of a paramagnetic moment, and $T_u\le T_1$ as discussed previously) and $T_p$, where the positive contribution stops increasing in magnitude during cooling [116]. The values of the applied field were, however, not given in [116]. It is also evident from Figure 10a that these temperatures can be identified with certain characteristic features in the ZFC data. $T_p$ can be associated with the temperature where the diamagnetic response is at its maximum value and full flux exclusion occurs, while $T_u$ represents the temperature where global diamagnetic screening disappears during warming. On the other hand, the FC $M/H$-data with in-plane field (□) exhibit a more complex but completely diamagnetic behavior, including a hysteresis depending upon whether the sample is being cooled or subsequently warmed through the superconducting transition.

Figure 10b presents QD PPMS-VSM [134] measurements performed in Nancy on a rectangular Nb disk stemming from sheet DS4D with the magnetic field applied parallel to the sample surface. Fields in the range between 1 mT and 100 mT were applied. For fields up to 10 mT applied parallel to the sample surface, we observe a positive (paramagnetic) response, which changes into completely diamagnetic signals on further increase of the applied field. Note here also the shift of $T_c$ towards lower values with increasing the applied field. This observation clearly points out that PME is possible also in the configuration $H\Vert$ sample surface.

Figure 10. PME on Nb disks with perpendicular or parallel magnetic field. (a). $M/H$ recorded in ZFC and FC modes with the magnetic field applied parallel (□) and perpendicular (•) to the Nb disk. The dashed vertical lines indicate the temperatures $T_u\le T_1$ and $T_p$, see text. Reprinted with permission from Ref. [116]. (b). PME on a Nb disk with the applied magnetic field parallel to the sample surface, measured in the year 2023 in Nancy using a QD PPMS-VSM system [134]. The lines serve as guides for the eye.

Figure 10. PME on Nb disks with perpendicular or parallel magnetic field. (a). $M/H$ recorded in ZFC and FC modes with the magnetic field applied parallel (□) and perpendicular (•) to the Nb disk. The dashed vertical lines indicate the temperatures $T_u\le T_1$ and $T_p$, see text. Reprinted with permission from Ref. [116]. (b). PME on a Nb disk with the applied magnetic field parallel to the sample surface, measured in the year 2023 in Nancy using a QD PPMS-VSM system [134]. The lines serve as guides for the eye.

3.3. Magnetization loops close to the superconducting transition

In order to gain more information about the physical origin of this additional paramagnetic moment, we performed detailed magnetization hysteresis loop (MHL) measurements in three distinct temperature ranges. MHLs of samples exhibiting the PME were rarely investigated in literature on both Nb and HTSc materials. Kötzler et al. [22] showed some virgin magnetization curves of their Bi-2212 samples, and Thompson et al. [31] have presented a MHL on Nb. The MHL presented in Ref. [31] showed a quite specific nearly parallelogram-like shape but there was no detailed analysis nor discussion of this feature. Thus, we take here the opportunity to present a detailed experimental analysis of the MHLs measured on Nb samples exhibiting the PME and a correlation of these results to the widely used PME measurements of magnetic moment at constant magnetic field while the temperature changes.

Figure 11a shows MHLs measured in Detroit on one of the Nb disk samples at temperatures between 8.96 and 9.08 K. These are "archetypal" MHLs of a type II superconductor with the shape given by the field-dependent superconducting currents induced in the sample and vortex pinning in this material. Note here the well-developed central peak located at slightly negative values, but close to zero field as it is expected from a bulk, superconducting sample [138,139]. Also shown are the corresponding virgin curves (i.e., the magnetization curves measured when the magnetic field was first applied to the sample after cooling in zero field). The joining of the virgin curve with the main MHL is marked as "A”. All the MHLs presented in Figure 11a were measured in the standard operation mode of the QD MPMS SQUID magnetometer with scan length below 1 cm. In contrast, the MHLs in the transition region between 9.00 K and 9.20 K were measured with temperature steps of 0.01 K using the stationary sample technique. At at field of ∼1 mT, there is still a finite induced magnetic moment and when the direction of the field ramping is reversed, one obtains the so-called "reverse leg" of the MHL given by the process of reversing the induced currents in the sample. This process is qualitatively similar to the process when the virgin curve merges the full MHL.

Figure 11b shows MHLs taken at elevated temperatures in the range 9.05 to 9.10 K, i. e., around the characteristic temperature, $T_{\mathrm{p}}$. While the MHLs measured at 9.00 K and 9.04 K look again more like "archetypal" MHLs, the MHL measured at 9.08 K has apparently a different shape with no clear maximum near to zero field. Even for the MHLs measured at 9.00 K and 9.04 K, one should note that the virgin curves are straight with a very sharp turn when they merge the full MHL (see the area marked as "B" in Figure 11b). All these curves develop a more parallelogram-like shape without any low-field maximum. Note also that the slope of the virgin curve changes between 9.05 K and 9.06 K. There is furthermore a significant change in the shape of the reverse legs of the MHLs. Along with this change also the slope of the reverse leg changes to being steeper at temperatures below 9.05 K (see also the analysis shown in Figure 13 below).

Below $T_p$, the MHL increases significantly and acquires an archetypal shape of MHL loop of a bulk superconductor with maximum close to $B_{\mathrm{ext}}\approx$ 0.

Figure 12a shows MHLs in the very interesting temperature range between 9.07 and 9.14 K, i.e., between $T_1\approx$ 9.16 K and $T_{\mathrm{p}}\approx$ 9.05 K. MHLs at these temperatures are found to exhibit several distinct features that probably have the same physical origin as the appearance of the paramagnetic moment during field-cooling in small applied fields. The MHLs have the shape of a nearly regular parallelogram with all turning points being sharp. Note that the reverse legs are entirely linear, all the turning points are quite sharp, and the size of the parallelograms increases with decreasing temperature. Also, the shape of the initial magnetization curve (virgin curve, abbreviated VC) is very linear and then abruptly merges with the nearly field-independent MHL (see Figure 12b). These features cannot be explained by magnetic flux gradually penetrating the superconducting sample, governed by vortex pinning when the field is changed.

Here it is important to mention again that all MHLs at temperatures close to the transition temperature were determined from the SQUID voltage change generated by stationary sample. This method is very "safe" as it eliminates spurious signals that may be generated when sample is moving during the measurement. However, this method cannot be used at higher fields and consequently at lower temperatures where the induced moments are large and we cannot measure anything but reversible screening currents. Fortunately, at lower temperatures we can use the conventional measurement of the magnetic moments in the point by point method with the sample moving (the scan length should be as short as possible) over a distance in a magnetic field generated by the build-in superconducting magnet. At lower temperatures the induced magnetic moment is quite large and so, the possible spurious signals due to moment motion are relatively small.

Figure 12b shows more details of the virgin curves of the MHLs measured after cooling down in zero field. The slope $S=\mathrm{d}m/\mathrm{d}{H}_{\mathrm{ext}}$ of virgin curves recorded at $T=$ 9.06 K and above is remarkably lower (by about 20 %) as compared to the slope of all virgin curves recorded at $T=$ 9.05 K and below. The major change of the virgin curve slope takes place just below $T_{\mathrm{p}}$ where also a remarkable change in the shape of the MHLs takes place. Additionally, the slope S of the initial curves changes below $T_{\mathrm{p}}$ with the magnitude and temperature dependence of this change corresponding to the step observed in the zero field-cooled susceptibility curves [25]. At the temperature $T_{\mathrm{p}}$ also the VC shape changes. Below $T_p$, the VC bends when approaching the MHL indicating gradual penetration of magnetic flux into the sample.

Finally, Figure 12c illustrates the field reversal on the MHLs at temperatures from 9.05 to 9.12 K. In this temperature range, one can see very straight reversal legs with very sharp and regular turning points at the MHL at field reversal. This behavior is again a very important characteristic feature of the bulk Nb disks, closely related to the presence of the PME.

Figure 13 shows the temperature dependence of the magnitude of MHL (i.e., the maximum induced magnetic moment, $m_{\max }\left(T\right)$). $m_{\max }\left(T\right)$ is nearly linear below the temperature $\approx T_p$ going sharply towards zero at about 9.065 K, as indicated by dashed line (▪ ▪ ▪ ▪ ▪). The inset to Figure 13 presents the same analysis for the temperature range 8.7 $\le T\le$ 9.0 K, i.e., the range where the "normal" MHL shape is found. The slope obtained here and the slope of the magenta dashed line in the main panel are the same. However, above 9.06 K, $m_{\max }\left(T\right)$ abruptly changes the slope and decreases, again nearly linearly, but with a slope being about one order of magnitude smaller, as indicated by the dashed green line (▪ ▪ ▪ ▪ ▪). Thus, in the temperature range $T_p\le T_c$, the PME-MHLs are visible, giving an indication of the giant vortex state.

Figure 13. Evaluated details of the hysteresis measurements. The graph gives the MHL magnitude (maximum induced magnetic moment) as a function of temperature. The blue datapoints (•) give the data obtained from the MHLs, the black crosses (+) represent the data from a second run. The slope from the archtypal MHLs as shown in the inset (━) and the slope of the component C in the main panel (▪ ▪ ▪ ▪ ▪) are the same. It is obvious from this graph that at $T_p$ an abrupt change in the MHL shape takes place, and the normal flux pinning vanishes. At $T>$ 9.06 K, the component P is measured follwing a completely different slope (▪ ▪ ▪ ▪ ▪).

Figure 13. Evaluated details of the hysteresis measurements. The graph gives the MHL magnitude (maximum induced magnetic moment) as a function of temperature. The blue datapoints (•) give the data obtained from the MHLs, the black crosses (+) represent the data from a second run. The slope from the archtypal MHLs as shown in the inset (━) and the slope of the component C in the main panel (▪ ▪ ▪ ▪ ▪) are the same. It is obvious from this graph that at $T_p$ an abrupt change in the MHL shape takes place, and the normal flux pinning vanishes. At $T>$ 9.06 K, the component P is measured follwing a completely different slope (▪ ▪ ▪ ▪ ▪).

It is obvious from this graph that the $m_{\max }\left(T\right)$ dependence can be formally separated into two components, one (we will call this component "C") that goes sharply to zero at $T=$ 9.06 K $\approx T_p$, and another component (we will call this component "P") that has a much weaker temperature dependence. The component P is not "visible" at temperatures below $T_p$, where the P component is overwhelmed by the larger C component and the magnetic moment is given primarily by the dominant component C, which describes the classic situation of flux pinning of Abrikosov vortices. The situation changes below the characteristic temperature, $T_p$, where the magnetic moment from the component C is (practically) zero and the $m_{\max }\left(T\right)$-dependence is entierly given by the component P that diminishes only at temperatures close to $T_1$ or higher.

Considering both the shape and character of MHLs at various temperatures, and also the FC and ZFC curves studied in detail in Refs. [25,114,115], we can attribute the component C to the conventional behavior of a type-II superconductor, where induced currents are governed primarily by vortex pinning. The origin of the component P is still not fully understood. This component is obvious only above $T_p$, where the C component diminishes to zero, but the character of the $m_{\max }\left(T\right)$-dependence indicates that the P component is present in the sample also at temperatures below $T_p$. We thus believe that the component P of the induced magnetic moment is responsible for specific superconducting behavior of Niobium below the transition temperature including (but not limited to) the appearance of that positive magnetic moment when the sample is cooled in small applied magnetic fields (i.e., the Paramagnetic Meissner effect).

4. Magnetic Imaging

Another interesting experiment was carried out by Nusran et al. [150] (Figure 20) employing the non-invasive magnetic field sensing using optically-detected magnetic resonance of nitrogen-vacancy centers in diamond, short NV magnetometry. The experimental apparatus incorporates a confocal microscope optimized for NV fluorescence detection. The fluorescence is stimulated by the green off-resonant 532 nm laser excitation and low-energy levels are populated by the microwave radiation applied using a single silver wire loop antenna coupled to a MW frequency generator. A thin diamond plate with an ensemble of NV centers embedded near the surface (∼20 nm depth) is used as the magneto-optical sensor. The spatial resolution of the sensor is determined by the effective size of the probe, which is essentially a convolution of the focal volume with the NV distribution in the diamond plate. This leads to a disk-like probing volume of thickness ≈20 nm and diameter of ≈500 nm. A detailed review of the NV-centers and NV magnetometry can be found in Refs. [146,147].

Figure 19. MO images obtained at University of Liège of an Nb disk from Detroit. (a). The MO image at $H=$ 5 mT applied field, and the temperature is swept from 10 K to about 8 K. The top L-shape indicates the MO indicator film, the inner L-shape represents the sample edges. The two yellow boxes give the regions ($I_{\mathrm{out}}$, $I_{\mathrm{in}}$), where the profiles are taken for (b). (b). Profiles taken in the two yellow boxes marked in (a). The black curves are for $I_{\mathrm{out}}$, and the red curves for $I_{\mathrm{in}}$.

Figure 19. MO images obtained at University of Liège of an Nb disk from Detroit. (a). The MO image at $H=$ 5 mT applied field, and the temperature is swept from 10 K to about 8 K. The top L-shape indicates the MO indicator film, the inner L-shape represents the sample edges. The two yellow boxes give the regions ($I_{\mathrm{out}}$, $I_{\mathrm{in}}$), where the profiles are taken for (b). (b). Profiles taken in the two yellow boxes marked in (a). The black curves are for $I_{\mathrm{out}}$, and the red curves for $I_{\mathrm{in}}$.

Figure 20. Diamond center imaging by AMES on an Nb sample with PME. (a). Temperature-dependent total magnetic moment measured using Quantum Design MPMS. Shown are ZFC-W and FC-C curves measured in Nb. (b). FC profiles of the magnetic induction for applied magnetic fields of 1.0, 1.2, and 5 mT (10, 12, and 50 Oe) recorded for an Nb disk-shaped crystal. Measurement performed at $T=$ 4.2 K. The signatures of the paramagnetic Meissner effect (PME) are observed in various random regions for low magnetic fields. Cooling in 1 mT (10 Oe, red datapoints) and 1.2 mT (12 Oe, green datapoints) applied magnetic fields results in very similar profiles. The inset to (b) presents a temperature-resolved measurement in a 1 mT (10 Oe) magnetic field upon FC-C. The peak (‘P’) and valley positions (‘V’) of the inset are indicated by arrows in the main panel. Image reproduced with permission from Ref. [150].

Figure 20. Diamond center imaging by AMES on an Nb sample with PME. (a). Temperature-dependent total magnetic moment measured using Quantum Design MPMS. Shown are ZFC-W and FC-C curves measured in Nb. (b). FC profiles of the magnetic induction for applied magnetic fields of 1.0, 1.2, and 5 mT (10, 12, and 50 Oe) recorded for an Nb disk-shaped crystal. Measurement performed at $T=$ 4.2 K. The signatures of the paramagnetic Meissner effect (PME) are observed in various random regions for low magnetic fields. Cooling in 1 mT (10 Oe, red datapoints) and 1.2 mT (12 Oe, green datapoints) applied magnetic fields results in very similar profiles. The inset to (b) presents a temperature-resolved measurement in a 1 mT (10 Oe) magnetic field upon FC-C. The peak (‘P’) and valley positions (‘V’) of the inset are indicated by arrows in the main panel. Image reproduced with permission from Ref. [150].

5. Discussion

The origin of the PME became evident [63] when comparing the field dependence of the Meissner effect discussed above with the magnetization response measured by sweeping the magnetic field at a constant temperature (abbreviated: C-T regime). Instead of a single magnetization curve characteristic of macroscopic superconductors, the spatial confinement of a mesoscopic sample gives rise to an entire family of magnetization curves, which are corresponding to different vortex states. Several superconducting states can be realized at the same applied magnetic field (up to five such states as can be seen in Figure 22b), but only the state with the most negative C-T magnetization is the thermodynamically stable one [154,155]. All other states are metastable and become observable due to the presence of the surface barrier of the Bean-Livingston-type [156]. The recent theory is in good agreement with similar C-T curves as reported previously.

Figure 22. Magnetic susceptibility, $\chi$, of mesoscopic Al disks, see also Figure 2f.

(a). Field dependence of the Meissner response for a Al disk with 1.0 $\mu$m diameter (□) and for one with 2.5 $\mu$m diameter (•). The thickness is for both disks 0.1 $\mu$m. The strongly oscillating behaviour clearly seen (the dashed line is a guide to the eye) for the larger sample is due to size quantization. Each jump corresponds to a change in the number of vortices inside the disk, which can either form an array of single quantum vortices or assemble into a single giant vortex. The inset to (a) compares the field-cooling (FC) and zero-field-cooling (ZFC) magnetization for the 2.5-mm disk at the field where the paramagnetic response is close to its maximum value. The ZFC response is always diamagnetic, and the jumps in the ZFC curve correspond to entry of individual vortices into the disk interior.

(b). Comparison of the magnetization states reached by cooling in a field and by sweeping the field at a constant temperature (arrows) at $T=$ 0.4 K. The field-cooling (FC-C) data shown by □ are for the 2.5-$\mu$m disk shown in (a). The filled boxes (■) indicate the low-temperature states as shown in the inset to (a), ZFC and FC. The inset to (b) illustrates the compression of a giant vortex (T close but below $T_c$) into a smaller volume (T further away from $T_c$,) which enables extra flux to enter the sample at the surface. Image reproduced with permission from Ref. [63].

Figure 22. Magnetic susceptibility, $\chi$, of mesoscopic Al disks, see also Figure 2f.

(a). Field dependence of the Meissner response for a Al disk with 1.0 $\mu$m diameter (□) and for one with 2.5 $\mu$m diameter (•). The thickness is for both disks 0.1 $\mu$m. The strongly oscillating behaviour clearly seen (the dashed line is a guide to the eye) for the larger sample is due to size quantization. Each jump corresponds to a change in the number of vortices inside the disk, which can either form an array of single quantum vortices or assemble into a single giant vortex. The inset to (a) compares the field-cooling (FC) and zero-field-cooling (ZFC) magnetization for the 2.5-mm disk at the field where the paramagnetic response is close to its maximum value. The ZFC response is always diamagnetic, and the jumps in the ZFC curve correspond to entry of individual vortices into the disk interior.

(b). Comparison of the magnetization states reached by cooling in a field and by sweeping the field at a constant temperature (arrows) at $T=$ 0.4 K. The field-cooling (FC-C) data shown by □ are for the 2.5-$\mu$m disk shown in (a). The filled boxes (■) indicate the low-temperature states as shown in the inset to (a), ZFC and FC. The inset to (b) illustrates the compression of a giant vortex (T close but below $T_c$) into a smaller volume (T further away from $T_c$,) which enables extra flux to enter the sample at the surface. Image reproduced with permission from Ref. [63].

Figure 22b demonstrates that the paramagnetic states reached via field-cooling are all metastable. Indeed, the FC data predictably fall on the C-T curves because only these distributions of the order parameter are allowed by quantization. However, among all possible states at a given magnetic field, the system unexpectedly `chooses’ the metastable state with the most positive possible magnetization. Only if we remove the proper screening in the experimental setup, a metastable high-magnetization state can eventually relax to the corresponding stable state on the lowest curve. The same result was obtained when the experiment was carried out in a more controllable manner by applying an oscillating magnetic field at a constant field, H. One can verify that, according to Figure 22b, an oscillating (fluctuating) field moves the system down the ladder of curves towards the equilibrium state. The following consideration may answer the question how, when cooling down, the system can end up in the most thermodynamically unfavourable state. Superconducting states in a confined geometry can be characterized by a quantum number L, which corresponds to the number of nodes in the distribution of the complex order parameter $\Psi$ along the sample circumference. For the case of a giant vortex and an array of single-quantum vortices, L has a simpler meaning: L then represents the angular momentum and the number of vortex cores, respectively. Transitions between the various states with different L are of first order and lead to (little) jumps in the magnetization data.

Figure 23. (a). Temperature dependence of the magnetization in the FFCW mode at 0.5 mT (5 Oe). (b). Phase diagramfor the PME in FFCW mode. The third critical field $H_{c3}$ is defined from the onset of DPE in the in-phase ac susceptibility measurements; $H_{c2}$ is defined from the intersection of two linear fits of the $M\left(T\right)$ curves above and below the onset; $T_{irr}$ is derived as the onset of a diamagnetic signal on the in-phase ac susceptibility curve.The solid and dashed lines are plots of the empirical formula $H\left(T\right)=H_0{[1-{(T/T_c)}^2]}^n$ The crosses show the field locations where the magnetic relaxation curves are measured. Image reproduced with permission from Ref. [37].

Figure 23. (a). Temperature dependence of the magnetization in the FFCW mode at 0.5 mT (5 Oe). (b). Phase diagramfor the PME in FFCW mode. The third critical field $H_{c3}$ is defined from the onset of DPE in the in-phase ac susceptibility measurements; $H_{c2}$ is defined from the intersection of two linear fits of the $M\left(T\right)$ curves above and below the onset; $T_{irr}$ is derived as the onset of a diamagnetic signal on the in-phase ac susceptibility curve.The solid and dashed lines are plots of the empirical formula $H\left(T\right)=H_0{[1-{(T/T_c)}^2]}^n$ The crosses show the field locations where the magnetic relaxation curves are measured. Image reproduced with permission from Ref. [37].

6. Conclusions and outlook

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