Magnetical Response and Mechanical Properties of High Temperature Superconductors , YBaCu 3 O 7X Materials

J.J. Roa1,*, F.T. Dias2 and M. Segarra3 1Institut P' (UPR 3346), Département de Physique et Mécanique des Matériaux, CNRS Université de Poitiers – ENSMA, SP2MI Téléport 2 BP 30179, F-86962 Futuroscope, Chasseneuil Cedex 2Federal University of Pelotas, Physics and Mathematics Institute, Department of Physics, 3DIOPMA Centre. Universitat de Barcelona, Facultat de Quimica, Departament de Ciència dels Materials i Enginyeria Metal·lúrgica, Barcelona 1France 2Brazil 3Spain


Introduction
Since the discovery of the second generation High Superconductor Materials (2G-HTSC) in 1986, their magnetic properties have been widely studied by different research groups around the world.During the last years, the mechanical properties at micro-/ and nanometric scale are starting to be studied in order to know and improve the durability of conventional devices.
In this book chapter, we would like to focus our attention on the magnetical response and also the mechanical properties of 2G-HTSC.
In relation to the magnetic response, the Meissner effect is one important signature of the superconductivity.In this case, a diamagnetic response is observed due to exclusion of the magnetic flux of the interior of the superconducting material when the temperature is below the critical temperature.This important property allows to distinguish a superconducting material from a conducting one.But in several cases the superconducting materials exhibit a paramagnetic response instead of the conventional diamagnetism.This effect is frequently called paramagnetic Meissner effect (PME).In this case, the magnetic flux is not expelled, and a paramagnetic state can be originated.This effect is observed in several magnetic field regimes, and in some cases the paramagnetic response increases with the applied magnetic field, but in others the paramagnetic response decreases when the magnetic field increases.
The paramagnetic Meissner effect (PME) was first observed in polycrystalline samples of high temperature superconductors.The discovery of this effect has originated an enormous effort of investigation by several groups around the world and several theories and models have been proposed to explain this interesting and controversial behavior, but without a

The Meissner effect
The Meissner effect is one important signature of the superconductivity.In this case, a diamagnetic response is observed due to exclusion of the magnetic flux from inside the superconducting material when the temperature is below its critical temperature.This important property allows us to distinguish a superconducting material from a perfectly conducting material.
Type-II superconductors present a perfect diamagnetism (Meissner effect) just when the magnetic applied field is lower than H C1 (T), known as the lower critical field, as showed by Fig. 1.Consequently this region is known as Meissner state.For magnetic applied fields higher than H C1 (T) but lower than H C2 (T), or upper critical field, the magnetic field can penetrate inside the superconductor in the form of vortices.This state is known as mixed state, and the H C1 (T) value determines the penetration of the first magnetic flux-line inside the superconductor.The Meissner effect can be observed by the field-cooled warming (FCW) or field-cooled cooling (FCC) prescriptions.The FCW prescription consists of a first cooling down of the superconducting material to temperatures well below the critical temperature (T C ) in magnetic applied field; then the magnetization can be measured under constant magnetic field while the sample warms up to temperatures above the critical temperature.The FCC prescription is similar, but in this case the magnetization is measured while cooling the sample back to temperatures well below the critical temperature in magnetic applied field.In these conditions the magnetic flux uniformly distributes into the sample, and when the sample is cooled below T C a diamagnetic moment appears as a consequence of the expulsion of magnetic flux from within it.Due to vortex pinning effects, in general the field-cooled (FC) response is lower than the zero-field cooling (ZFC -diamagnetic shielding) response, as showed by Fig. 2. Fig. 2. ZFC and FC curves for a melt-textured YBa 2 Cu 3 O 7-x sample.

Paramagnetic Meissner effect
The Meissner effect is characterized by a diamagnetic response of the superconducting material when a magnetic field is applied, but in several cases this magnetic response in FC experiments can be paramagnetic, challenging the conventional Meissner effect.This effect is known as paramagnetic Meissner effect (PME), but sometimes is called Wohlleben effect in honor of some of its discoverers (Bräuchle et al., 1994;Khomskii, 1994).
The first observations of the PME were reported in a pioneer work by P. Svedlindh et al. (Svedlindh et al., 1989) in the BiSrCaCuO superconductor.The discovery of the PME has prompted an enormous effort of investigation due to the striking contrast of this effect with the expected diamagnetic response.Initially, the effect was credited as an artifact of the measurement procedure (Blunt et al., 1991), but several further results showed that the PME is a genuine response of the superconducting material.
The PME is qualitatively distinct in relation to the applied magnetic field, and can be observed from few Oe up to several kOe.In some cases the effect decreases when the magnetic field increases, as can be seen in Fig. 3 for an YBa 2 Cu 3 O 7-x single-crystal.However, in other cases the effect increases when the magnetic field increases, as showed by Fig. 4 for another YBa 2 Cu 3 O 7-x single-crystal.Fig. 3. Magnetic susceptibility in an YBa 2 Cu 3 O 7-x single crystal.The magnetic susceptibility decreases when the magnetic field increases.From Bräuchle et al., 1994. Fig. 4. FC magnetizations in an overdoped YBa 2 Cu 3 O 7-x single crystal.In this case the magnetization increases when the magnetic field increases.From Kusmartsev et al., 1992.Sometimes the PME presents a curious time dependence of the FC magnetization, as showed by Fig. 5 for a melt-textured YBa 2 Cu 3 O 7-x sample.In Fig. 5 a paramagnetic signal was observed when the FC moment relaxes at constant magnetic field and temperature.The inset of this figure shows the sign reversal obtained in a relaxation experiment in the same sample, but cooled under another rate and magnetic field configuration.
It is possible to observe that an originally negative FC moment relaxes towards positive values and reverses sign after a wait time of about 3000 s.The time dependence of the PME has been related by other authors in different superconducting materials (Pan et al., 2006;Terentiev et al., 1999;Dias et al., 2004).

Models and experimental results
The paramagnetic Meissner effect is a controversial and challenging effect that can be observed in a variety of superconducting systems.However, in several cases there is a strong dependence on the applied magnetic field (magnitude and configuration), cooling rate during measurements, time effects, pinning effects, and others circumstances, and hence there is no a definitive model that can explain the paramagnetic moments observed during the FC experiments.
After the first observations of the PME, an enormous effort of investigation has emerged in order to explain this anomalous effect.The initial motivation was given by its interpretation based on the occurrence of  junctions between the superconducting grains in the material (Kusmartsev et al., 1992).
According to this, the sample showing PME may be modeled as a Josephson medium where the  junctions are randomly distributed.Assuming that closed superconducting loops containing an odd number of these anomalous junctions are likely to occur in these networks, spontaneous and polarizable orbital currents may be generated, giving rise to the PME when very low magnetic fields are applied.The required -phase shifts in the Josephson links were proposed to arise from d-wave pairing symmetry in the high-Tc superconducting materials (Sigrist & Rice, 1995).
The Fig. 6 shows PME results obtained in a polycrystalline Bi 2 Sr 2 CaCu 2 O 8 sample by Braunisch et al. (Braunisch et al., 1992) and explained on the basis of this model.From Braunisch et al., 1992.Several results can be explained based on this model, specially when low magnetic fields are applied.The concept of polarized orbital currents to explain the low field PME was further developed by studies of the characteristic microstructure of samples showing this effect (Freitag et al., 1999), and by investigations on the time dependence of the paramagnetic moment (Magnusson et al., 1995), ac susceptibility (Nordblad et al., 1998) and magnetic aging (Svedlindh et al., 1999).
A controversial point to this model arises from similar results obtained in Nb samples, because some authors (Sigrist & Rice, 1995) attribute that  junctions are a consequence of the d-wave pairing symmetry, which is nonexistent in this system.The PME observed in Nb (Thompson et al., 1996;Minhaj et al., 1994;Kostic et al., 1996) and Nb thin films (Ortiz et al., 2001;Terentiev et al., 1999) was demonstrated to be strongly influenced by the sample geometry and surface.In some cases, the polishing of the surfaces affects the PME significantly, leading to its disappearance.Fig. 7 shows the magnetization results of a Nb disk before and after polishing the sample surface, where it is possible to see that the PME disappears after the sample is polished.A surface dependent PME in YBa 2 Cu 3 O 7-x single crystals was also found (Lucht et al., 1995).
These results motivated the proposal of mechanisms to explain the PME without taking into account effects from intrinsically nonconventional superconductivity, and models based on flux trapping and flux compression effects were proposed.According to these models, a nonequilibrium compressed flux state may be generated and stabilized by inhomogeneous cooling of the sample, so that its surface becomes superconducting prior to the bulk (Koshelev & Larkin, 1995).Then, the magnetic flux may be pushed into the sample, creating a flux compressed state upon further cooling.Fig. 7. FC (a) and ZFC (b) experiments in a Nb disk.The circles were taken before polishing the sample surface, and the triangles were taken after polishing the sample surface.In this case, the polishing of the surface affects the PME, leading to its disappearance.From Kostic´ et al., 1996.Similarly, the formation of a giant vortex state on the sample surface proposed by Moshchalkov et al. (Moshchalkov et al., 1997) may produce a compression of the trapped flux.According to the authors, the PME observed in high-T C and Nb superconducting samples can be explained by the persistence of a giant vortex state with a fixed orbital quantum number L, formed during the FC regime at the third critical field.However, the amplitude of the PME is suppressed by applying a magnetic field.The giant vortex state was also shown to be responsible for PME in small disks of Al samples (Geim et al., 1998), where transitions between metastable giant vortex states having different angular momenta (L) have been observed.The transitions compress the giant vortex into smaller volumes, allowing extra flux to penetrate in the sample surface and producing the paramagnetic moments.
The flux capture by samples with a special shape proposed by Obukhov (Obukhov et al., 1998), may also produce compression of the trapped flux.According to the author, an unusual behavior of the long-time relaxation of the magnetization after a FC procedure may show not only an increase in the magnetization over the time but a change of sign from negative to positive.This interesting result was observed in melt-textured YBa 2 Cu 3 O 7-x samples (Pureur et al., 2001), as shown by the inset in the Fig. 5.
A model to explain the PME observed in Josephson-junctions arrays, such as Nb-Al 2 O 3 -Nb (Nielsen et al., 2000), was proposed to account for the paramagnetism without invoking the role of  junctions or d-wave superconductivity (Araujo-Moreira, 1997; Barbara, 1999;Nielsen et al., 2000).According to this model, the screening of the array of these multiplyconnected superconductors is provided by the loops at its boundaries, and the diamagnetic currents running at the external perimeter induce currents in the opposite sense inside the array.These inner currents generate a paramagnetic scenario under certain conditions.This mechanism is shown in Fig. 8. Fig. 8. Mechanism for generation of a paramagnetic current inside a multiply-connected superconducting sample.From Nielsen et al., 2000.According to this model, only the exterior plaquettes create a diamagnetic screening current on the outside of the sample, and as a consequence a paramagnetic current of the same magnitude is created inside the sample.Simulations taking into account the mutualinductance interactions between loops in the array confirm that PME might be the dominant response, depending on the field strength and parameters as the self-inductance and capacitance of the Josephson junctions (Rotoli et al., 2001).
The granularity in superconductors emulates a disordered network of Josephson junctions, and the paramagnetic moments observed in granular thin films of Nb and YBa 2 Cu 3 O 7-x has also been described as an intrinsic effect of multiply-connected superconductors.
Important results were obtained by Kusmartsev et al. (Kusmartsev et al., 1997) after cooling large YBa 2 Cu 3 O 7-x single crystals in strong applied fields.The authors called this phenomenon the high-field paramagnetic effect (HFPME) so that it may be distinguished 191 from the low-field PME.The HFPME was found to depend on the cooling rate and the sample size, which suggests that the paramagnetic moment in this case might be induced by a flux compression mechanism.The Fig. 4 shows representative results obtained by Kusmartsev et al.The PME at high magnetic fields was also observed by Dias et al. (Dias et al., 2004) in different melt-textured YBa 2 Cu 3 O 7-x samples containing large amounts of the Y 2 BaCuO 5 (Y-211) phase.Magnetic fields up to 50 kOe were applied either parallel or perpendicular to the ab planes and the magnitude of the high-field paramagnetic moment increases when the field is augmented.The Fig. 9 shows FCC and FCW magnetizations as a function of the temperature in a melt-textured YBa 2 Cu 3 O 7-x sample measured in the indicated fields applied parallel and perpendicular to the ab planes.In the Fig. 9a the data are obtained when the field is applied parallel to the ab plane.These results present the general trends of the PME shown by the melt-textured YBaCuO samples.In the Fig. 9b are depicted the peculiar results obtained when the field is applied along the c-axis.In this case the positive contribution to the moment rises abruptly at an almost field-independent temperature around 55 K, then shows a tendency to saturate at low temperatures at values varying roughly proportional to the applied field.When the magnetic field increases the magnetization also increases and for magnetic fields below 5 kOe the magnetization is negative, as shown in the Fig 9b.
The effect shows a strong and anomalous relaxation, such that the paramagnetic moment increases as a function of the time, as shown in Fig. 10.
According to the authors, the pinning by the Y-211 particles is relevant and the formation of a flux-compressed scenario modulated by pinning and a strong flux-creep is a possible origin of the observed PME.Recently, the high-field PME was also observed in an Y 0.98 Ca 0.02 Ba 2 Cu 3 O 7-x single crystal (Dias et al., 2010), and the results were attributed to strong flux compression state into the sample modulated by pinning.An important review of the paramagnetic Meissner effect and related dynamical phenomena was presented by M.S. Li (Li, 2003), with emphasis on the theoretical description of the  junctions.

Theoretical fundamentals
Indentation testing is a simple method that consists essentially of touching the material of interest, whose mechanical properties such as elastic modulus and hardness are unknown, with another material whose properties are known (Fischer-Cripps et al., 2004).The technique has its origins in Mohs' hardness scale of 1822, in which materials that are able to leave a permanent scratch in another were ranked as harder materials, with diamond assigned the maximum value of 10 on the scale.The establishment of the Brinell, Knoop, Vickers and Rockwell tests all follow from a refinement of the method of indentation test, in which the length scale of the penetration is measured in nanometers rather than microns or millimeters, the latter being common in conventional hardness tests.Apart from the displacement scale involved, the distinguishing feature of most nanoindentation testing is the indirect measurement of the contact area.
In conventional hardness tests, the area of contact is calculated from direct measurements of the dimensions of the residual impression in the specimen surface upon the removal of load.In nanoindentation tests, the size of the residual impression is of the order of microns and too small to be conveniently directly measured.Thus, it is usual to determine the area of contact by measuring the depth of indentation of the indenter into the specimen of interest.This, together with the known geometry of the indenter, provides an indirect measurement of the contact area at full load.For this reason, nanoindentation can be considered a special case of depth-sensing indentation or instrumented indentation testing.
In this section, we would like to do a brief explanation of the different hardness tests employed to characterize the YBa 2 Cu 3 O 7-x samples textured by Bridgman and TSMG technique.Concretely, we will explain in detail the micro-, nano-and even picoindentation, according to the characteristic indentation sizes that fall into a particular size range and the applied load employed.

Microindentation
The purpose of microindentation hardness testing is to study fine scale changes in hardness.This technique is also commonly called microhardness testing, although this term is misleading because it implies that the hardness is extremely low.The Vickers and the Knoop tests, which are the two most common microindentation tests, will be described in more detail in the present section.
The term microhardness test usually refers to static indentations made with loads not exceeding 1kgf.The indenter is either the Vickers diamond pyramid or the Knoop elongated diamond pyramid.The surface being tested generally requires a metallographic finishing; the smaller the load used, the higher the surface finish required.
The Knoop indenter is a diamond ground to pyramidal form that produces a diamond shaped indentation, having an approximate ratio between long and short diagonals of 7:1.The depth of indentation is about 1/30 of its length.The Knoop hardness number (KHN) is the ratio of the load applied to the indenter to the unrecovered projected area as: Where F or P is the applied load in kgf, A is the unrecovered projected area of the indentation in mm 2 , L is the measured length of the long diagonal of indentation in mm, and C is a constant of the indenter relating the projected area of the indentation to the square of the length of the long diagonal, and equal to 0.07028.
The Vickers diamond pyramidal indenter is ground in the form of a square pyramid with an angle of 136° between faces.The depth of indentation is about 1/7 of the diagonal length.
The Vickers diamond pyramid hardness number (HV) can be obtained by using the following equation:

Nanoindentation
During the last years, Instrumented Indentation Testing has been widely employed.Also known as depth-sensing indentation, continuous-recording indentation, ultra-low load indentation or nanoindentation, ITT is a relatively new form of mechanical testing, which significantly expands the capabilities of traditional hardness tests at macro-and microscale.It employs high-resolution instrumentation to continuously control and monitor the loads and displacements of an indenter (Golovin et al., 2008;Pharr et al., 1992).This technique presents several advantages compared to the microindentation technique: -It is a local probe that can evaluate the properties of a material in different areas.-It is depth-sensing, so it can characterize a material at different depths, unlike most scanning probe techniques, which usually only work at a given indentation depth.-It can also measure the Elastic modulus (E) of the material.
Magnetical Response and Mechanical Properties of High Temperature Superconductors, YBaCu 3 O 7-X Materials 195 Finally, this technique eliminates the need to visualize the imprints produced during the test for homogeneous materials, which makes the extraction of mechanical properties easier.Due to all these advantages, nanoindentation could be a good and versatile technique for characterizing the mechanical properties of a given component.
The nanoindenter is an equipment which measures load and maximum indentation depth as a function of time during loading and unloading, and displays load-displacement data.
Depending on the details of the specific testing systems, loads as small as 1 nN can be applied, and displacements of 0.1 nm can be measured.Stress-strain curves () and mechanical properties such as hardness (H), elastic modulus (E), fracture toughness (K IC ), yield strength ( ys ), or shear stress (), can be obtained from the P-h data and with the corresponding tip indenter.The different indenters can be classified in four different groups: i. Pyramidal: the most used sharp indenter is the Berkovich.It is a three-sided pyramid with the same depth-to-area relation as the four sided Vickers pyramid commonly used in microhardness.This indenter allows obtaining the H and E .The K IC of brittle materials can also be determined with this indenter without high accuracy.ii.Spherical: for this kind of indenters, the contact stress is initially small and produces only elastic deformation.As the spherical indenter is driven into the surface, a transition from elastic-to-plastic deformation occurs, which can theoretically be used to examine the  ys and obtain the  curve (Field et al., 1993).
iii.Cube-corner: a three-sided pyramid with mutually perpendicular faces arranged in geometry like the corner of a cube.The center-line-to-face angle for this indenter is 34.3°.The sharper cube-corner indenter produces much higher stress and strain in the vicinity of the contact, which is useful, for example, in producing very small and well defined cracks at the corners of the residual imprints in brittle materials.iv.Conical indenters: this type of indenter is interesting due to the existence of stress concentrations at the sharp edges of the indenter.It is difficult to manufacture conical diamonds with sharp tips, making them of little use in nanoindentation technique (Tsui et al., 1997).
Moreover, in order to obtain reproducible results, several factors should be controlled during the indentation tests:  Choosing an appropriate indenter: this is one of the most important steps and requires the consideration of a number of factors.One of them is the strain the tip imposes on the test materials.Although the indentation process produces a complex strain field beneath the indenter, it has been proven to be useful to quantify this field with a single quantity, often termed as the characteristic strain.There are problems, however, in obtaining accurate measurements of H and E with cube-corner indenters (Hay et al., 1999).Although not entirely understood, the problems appear to have two separate origins: first, as the angle of the indenter decreases, friction in the specimen-indenter interphase and its influence on the contact mechanics becomes increasingly important; on the other hand, to obtain the relation between the contact stiffness (S), contact area (A(h c )) and effective elastic modulus (E eff ), corrections are required, and the magnitude of the correction factor depends on the angle of the indenter.


Environmental control: to take full advantage of the fine displacement resolution available in most ITT systems, several precautions must be taken when choosing and preparing the testing environment.Uncertainties and mistakes in measured displacement arise from two separate environmental sources: vibration and temperature.To reduce vibration, the nanoindenter should be located on quite, solid foundation and mounted on vibration-isolation system.Thermal stability can be provided by closing the apparatus in a thermally buffered cabin and controlling the room temperature to within ± 0.5°C.


Surface preparation: one of the most important factors is the surface roughness of the samples, which is extremely important during the nanoindentation test because the mechanical properties are obtained from contact areas, which are calculated from the contact depth and area function.The main problems are found when the characteristic wavelength of the roughness is comparable to the contact diameter.


Testing procedure: to avoid interference from previous assays, and ensure the independence of the measurement, successive indentations should be separated up to 20-30 times the maximum indentation depth (h max ) when Berkovich or Vickers indenters are employed.However, for other geometries, the rule is to perform indentations separated from 7 to 10 times the maximum contact radius.


Detecting the surface: the most important step of any good nanoindentation test procedure is to accurate identificate the location of the specimen's surface.For hard and stiff materials, such as hardened metals and ceramics, the load and/or contact stiffness, both of which increase upon contact, are often employed.However, for soft compliant materials (like polymers and biological tissues), the rate of increase in load and contact stiffness is often too small to allow for accurate surface identification.In these situations, a better method is sometimes offered by dynamic stiffness measurement (Pethica et al., 1989;Lucas et al., 1998).

a. Hardness and Elastic modulus determination
In this section we will explain the method developed by Oliver and Pharr (Pharr et al., 1992;Oliver et al., 1992) to determine the hardness (H) and the elastic modulus (E) from the unloading part of recorded indentation load/unload (or P-h curves, such as Fig. 11).The H is calculated by dividing the load by the projected contact area (A(h c )) at maximum applied load, P max :

S
where the contact depth (h c ) is determined as: being h max the maximum indentation depth,  a parameter approximately equal to 0.75 for a Berkovich tip indenter, and S the unloading contact stiffness at maximum indentation depth (dP/dh, see Fig. 11).In the case of a perfect Berkovich tip, the A(h c ) is determined as: However, in practice the projected contact area differs from this value when the tip is worn out, because the different angles of the faces may be somehow different from the nominal.Hence, the projected contact area must be regularly calibrated.When catastrophic fracture mechanisms occur, the A(h c ) obtained by this theoretical method is overestimated, and results in an H underestimation.To avoid this problem, the real contact area should be determined using a technique of microscopy (as Scanning Electron Microscopy or Atomic Force Microscopy) and the contact area must be recalculated in order to avoid the sink-in or pile-up effects for brittle and/or ductile materials, respectively (Roa et al., 2009d(Roa et al., , 2010c(Roa et al., , 2011b)).
The effective elastic modulus (E eff ) has been expressed in terms of S and A(h c ) as: where  is a geometrical factor (1.034).This parameter takes into account the lack of symmetry of the Berkovich indenter.Then, according to the Oliver and Pharr approach (Oliver et al., 1992), the Elastic modulus can be obtained as: where E and  are the Elastic modulus and the Poisson's ratio (ν YBa2Cu3O7-x = 0.3 (Roa et al, 2007(Roa et al, , 2009d(Roa et al, , 2010a(Roa et al, , 2010c(Roa et al, , 2011b(Roa et al, , 2011d)), respectively.The subindex i denotes the indenter properties (E i = 1141GPa and   ) (Oliver et al., 1992;Pharr et al., 1992).

b. Fracture toughness
Moreover, nanoindentation provides information about contact mechanisms taking place in the material.In brittle materials indented with a Berkovich tip, surface examination of residual imprints has revealed the appearance of several typical features: surface deformation effects such as sink-in (Johnson et al., 1985;Alcala et al., 2000), microcracks or damage inside the imprints (Burghard et al., 2004); and fracture mechanisms such as radial cracks emanating from the imprint corners (Roa et al., 2008a(Roa et al., , 2008b)), and chipping (Lawn et al., 1975).
The resistance of the material against crack propagation, characterised by the fracture toughness K IC (Anderson et al., 2005), is a critical mechanical parameter in brittle materials.
Fracture toughness estimation by microfracture Vickers indentation is a well-known and broadly employed technique in ceramic materials.Usually, a Vickers diamond pyramid indenter at loads greater than 1 N is used to nucleate radial cracks at the imprint corners.The cracks emanating from the corners of a Vickers or Berkovich indenter are arrested when the residual stress driving force at the crack tip is in equilibrium with the K IC .Several expressions are widely employed to extract the fracture toughness, depending on the indenter geometry and crack morphology.In this experimental work, the method we are pursuing is based on the radial cracking which occurs when brittle materials are indented by sharp indenter such as Vickers or Berkovich diamond (Casellas et al., 2007;Jang et al., 2008).K IC is then evaluated using the equation developed by Lawn, Evans, and Marshall (Lawn et al., 1980): where is an empirical constant depending on the indenter geometry (for pyramidal tips), P is the peak indentation load, c is the length of the radial cracks, E is the elastic modulus, and H is the hardness value.The fracture toughness of YBa 2 Cu 3 O 7-x melttextured samples has been determined by Vickers indentation (Leblond-Harnois et al., 2000;Li et al., 1997;Yoshino et al., 2001;Leenders et al., 1999;Cook et al., 1987;Fujimoto et al., 1992), and also by the single-edge notch beam (Joo et al., 1998) and bending (Giese et al., 1990) techniques.Values in the range of K IC =0.4-2.8MPa•m 1/2 have been found, with a scattering attributed to differences between the techniques employed to obtain this parameter.In addition, K IC is influenced by exact matrix composition.For instance, Joo et al. (Joo et al., 1998), reported that the fracture toughness of YBa 2 Cu 3 O 7-x increased from K IC =1.60 MPa•m 1/2 to 2.80 MPa•m 1/2 by adding a 15% in volume of silver to the matrix.

c. Yield strength and stress-strain curve
The stress and deflection arising from the contact between two elastic solids are of particular interest to understand the first steps of indentation testing.The contact depth between a rigid sphere and a flat surface for small indentation depths was described by Hertz (Hertz et al., 1881).This phenomenon is shown in Fig. 12.The radius of the circle of contact is a, the total indentation depth is h max , the depth of the circle of contact from the specimen free surface is h a , and h p is the distance from the bottom of the contact to the contact circle.
When using spherical tips, the contact point is more difficult to be determined, due to the moderate increase in stiffness during the initial contact, and the interaction between the tip and samples before contacting.The indentation load-displacement data were analyzed based on the Hertz equation in the elastic region as follows (Oliver et al., 2004;Field et al., 1993;Field et al., 1995): where h e is the elastic indentation depth.Recently, Barsoum et al. (Moseson et al., 2008) proposed the fitting of the stiffness vs. indentation depth data, and extrapolating to zero, as predicated by a relation between E eff and a, as follows (Roa et al., 2010b(Roa et al., , 2010c(Roa et al., , 2011b)): Hertz found that the radius of the circle of contact, a, is related to the indentation load, P, the indenter radius, R, and the elastic properties for small indentation depths of the material by: The mean contact pressure, p m , between the indenter and the material is the applied load divided by the contact area.For small indentation depths, it can be obtained from the Hertzian equations as: The mean contact pressure is referred to as the "indentation stress" and the quantity a/R as the "indentation strain".This equation allows us to plot the stress-strain curves when we use a spherical tip.The left side of this equation represents the indentation stress or mean contact pressure, also referred to as the Meyer hardness (Basu et al., 2006).The expression in parentheses on the right side of this equation represents the indentation strain (Rayon et al., 2011).
The maximum tensile stress occurs at the edge of the contact circle in the surface and can be expressed as (Fischer-Cripps, 1999): The yield stress of the material,  ys , can be obtained using the following expression: 1.1σ This parameter will give information about the transition from elastic to elastic-to-plastic response (Jiménez-Piqué et al., 2007).The maximum shear stress ( max ) is produced beneath the indentation axis at a depth close to 0.5a, and equal to: The maximum shear stress is also known as the Tresca criterion.This assumes that yield occurs when the shear stress  exceeds the shear yield strength,  ys (Tsui et al., 1997; Roa et   al., 2008a): Where  1 and  3 are the principal stresses (also, can be re-written as  x and  z , respectively).

Picoindentation
The applied load in this case is much lower than in the previous one.Normally, this technique produces some indentations of few nanometers (generally fully elastic) obtaining the Elastic modulus.Picoindentation by means of atomic force microscopy-force spectroscopy (AFM-FS) presents several advantages over standard methods with regard to the quantification of E values: -First, the measurements are extremely local (small contact areas between the sample and the probe, of few nm 2 ) and can be performed on different areas so as to average the mechanical properties of the samples.-Second, it is a form of depth-sensing, so it allows characterization of a material at different indentation depths without surpassing the yield strength (fully elastic regimen).-Finally, it is not necessary to visualize and characterize the imprint produced during the indentation test, which simplifies the calculation in order to extract the E values.
These advantages mean that picoindentation is a suitable and reliable technique for measuring E values while applying loads ranging from a few nN down to the pN level.The study of force versus displacement (F-z) curves can shed new light on the elastic properties of hard materials, as these curves are similar to those obtained with classic microscopic or nanoscopic indentation tests.
The different steps of the method to extract the elastic modulus value using the AFM-FS and Hertz equations are described below (Roa et al., 2011e, 2011f):   F is calculated as (see Fig. 14b): where z is the cantilever deflection, defined as: where V is the increment in photodetector vertical signal, as the tip comes into contact with the sample, and S is the sensitivity, which is the slope of the contact region of a force curve performed on a rigid sample.
The indentation depth () due to F is calculated as (see Fig. 14c): where z represents the piezo-scanner displacement along the axis perpendicular to the sample plane.After puncturing the sample, a topographic image should be captured in www.intechopen.com

Magnetical Response and Mechanical Properties of
High Temperature Superconductors, YBaCu 3 O 7-X Materials 203 order to ensure that the sample had not undergone any plastic deformation.Only F-z curves that presented negligible lateral deflection should be considered to extract the mechanical response.F-z curves obtained at a certain F value were analyzed using the Hertz model in the elastic region (Oliver et al., 2004;Field et al., 1993, Field et al., 1995) by means of equation 20 (see Fig. 14d): where E eff can be obtained using the equation 7.In this case the subindex i corresponds to the mechanical properties of the AFM probe (SiO 2 with E = 76 GPa, and  i = 0.17 (Oliver et al., 1992)).The quality of S value relies on the absence of elastic deformation of the reference sample.

State of the art
The mechanical properties of YBa 2 Cu 3 O 7-x (Y-123 or YBCO) samples have been studied during the last years.The most important properties studied have been the elastic modulus, E, the hardness, H (at micro-/ nanometric scale), the toughness fracture, K IC , and the yield stress value,  ys .Some authors studied the mechanical properties at room or at cryogenic temperatures, also known as work temperature (T w ).Lots of different techniques could be used to determine these parameters.The reported values of E, H, K IC and  ys of YBa 2 Cu 3 O 7-x samples (bulk materials) obtained using indentation techniques (macro-, micro-, nano-and/or picoindentation techniques) are summarized in

Mechanical properties
The elastic modulus values reported by Roa et al. (Roa et al., 2007(Roa et al., , 2009d(Roa et al., , 2011b) ) presented in Table 1 for Bridgman and TSMG texture processes present a dependency with the applied load.This phenomenon is far from what are expected, as elastic modulus is an intrinsic property of each material and only depends on its composition.Then, this difference of elastic modulus is able to be attributed to one of the following factors:  Firstly, the contact area value, which is strongly affected by the presence of macro-/microcracks, porosity, roughness and a wide variety of defects.In order to avoid this effect, the polishing process before the indentation should be carefully done.


Secondly, the elastic modulus is obtained using the unloading slope of the loadingunloading curve, also denoted as S. If the contact point (indenter-sample) is not well estimated, the S value will be underestimated, yielding to a lower E eff . Finally, the  factor could not modify the elastic modulus because it remains equal to 1.034 for each test performed (Oliver et al., 1992).
After considering all the different parameters that can modify the elastic modulus, we believe that the main contribution to change it is the contact area, and then the scattered values presented in the Table 1 and reported by Roa et al. (Roa et al., 2007(Roa et al., , 2008b) ) are attributed to this effect.Moreover, the Bridgman samples present elastic modulus values higher than for TSMG, as this last presents a higher density of macro-and microcracks, which can produce a relaxation of the elastic deformation range, see Fig. 15.66,7 m 66,7 m 66,7 m 66,7 m 66.7 m Moreover, the Table 1 presents several measurements of the elastic modulus for Y-123, which resulted in values scattered within the range between 40 -200 GPa, caused by residual porosity and bad contact between the grains, as reported Johansen et al. (Johansen et al., 2001).If we compare all the elastic modulus reported in Table 1, we can conclude that the broad distribution observed can be attributed to the different measuring techniques (nanoindentation, AFM-FS) and to the different quality of the studied YBa 2 Cu 3 O 7-x samples (grain structure, texture, and others).
Table 2 exhibits the hardness values for YBa 2 Cu 3 O 7-x samples using different techniques.It is well known that the YBa 2 Cu 3 O 7-x samples textured by Bridgman and TSMG techniques present a heterogeneous microstructure with Y-123 matrix and a minor phase (Y-211 or inclusions).Using the conventional micro hardness technique it is not possible to isolate the hardness value for each phase due to that the size of the residual imprint is higher than the size of the different inclusions.However, in 2007, Roa et al. (Roa et al., 2007) using the Nanoindentation technique obtained the hardness value for each phase when the applied load was lower or equal to 10mN.With this applied load, the residual contact area is lower than the size of the inclusions.Then, the mechanical property can be obtained, see Table 2.We can observe that the hardness value for Y-211 is around twice times that for Y-123.This fact could be due to different reasons (Roa et al., 2007): high anisotropy of dislocations confined onto a (001) plane (Sandiumenge et al., 2000), and/or  the melt-processed ceramic composites contain a dense population of fine peritectic particles, which drastically affects the microstructure acting as nucleation sites for dislocations (Sandiumenge et al., 2000).
If the different residual imprints have been performed near the grain boundary, the plastic deformation mechanisms under the imprint could be affected by the Y-123 matrix, and the hardness value would be modified, thus giving us an average of the studied property.
At very small applied loads (lower than 10 mN), the hardness value can be strongly affected by the presence of defects and impurities that can cause almost no change in dislocation movement.In other words, hardness does not remain constant at different applied loads as corroborate Roa et al. in 2007and 2008(Roa et al., 2007, 2008b).Using Nanoindentation technique this effect can be related to one physical phenomenon known as Indentation Size Effect.Normally, hardness presents a maximum value for low loads.However, when the applied load increases this property is widely modified by the superficial defects, such as roughness, macro-cracks, and others.When this phenomenon takes place, the contact area could be over-or underestimated yielding to a false hardness value.
When the applied load using Nanoindentation technique is high (100 mN), overall nanohardness is very similar for both texturing techniques (8.9 GPa and 7.9 GPa for Bridgman and TSMG techniques, respectively).However, these values are lower than the nanohardness value of the Y-123 phase.The different hardness values reported in 2008).Moreover, several studies performed at cryogenic temperature (Güçlü et al., 2005) exhibit that the mechanical properties tend to be higher due to the nitrogen going inside the pre-existing cracks, yielding a homogeneous material.As we can observe in Table 2, the different YBa 2 Cu 3 O 7-x samples studied at room temperature are much brittle than at cryogenic temperature due to the effect explained above.Finally, another mechanical parameter to take into account is the yield stress, see Table 4.
As can be observed in this table, the intrinsic properties obtained using the stress-strain plots present similar values for both samples.The TSMG samples, present a higher density of macro-/microcracks and pore density (see Fig. 15) along the c-axis than Bridgman samples.Then, during the first contact between the indenter and the samples, the energy applied by the tip onto the surface is employed to close the different cracks produced during the oxygenation process.For this reason, TSMG samples present a higher contact radius value than Bridgman samples.

Fracture mechanisms
In this section we have analysed the different brittle effects appearing in as-grown samples textured by Bridgman and TSMG techniques.Fig. 16 and Fig. 17, show Field Emission Scanning Electron Microscopy images of some residual imprints performed at 10 mN of applied load for Bridgman and TSMG samples, respectively (from Roa et al., 2011b).
All the different imprints in Fig. 16 present the sink-in effect, typically found in elastic, brittle materials (Oliver et al., 1992).Sink-in is an elastic displacement of the surface at the contact perimeter.This effect can lead to an overestimation of the contact area, and thus an underestimation of the hardness value obtained by nanoindentation.As we can observe in this Figure, we noted a high density of micrometric pores around the residual imprint.Presumably, this micrometric porosity is created during the recrystalization process.The nanoindentation process would render the pores exposed to the surface, an effect that has been previously described (Gaillard et al., 2008).
In addition, all these images show the presence of microcracks inside the imprints, generated during the indentation process.This kind of damage is found in brittle materials containing inclusions harder than the matrix, in this case Y-211 (see table 2).Upon indentation, microcracks tend to be created at inclusions, and are recognised as microflaws within the imprint when observed from the surface.Another common feature in all imprints is the presence of radial cracks emanating from the corners of the imprints.This fracture mechanism, typical for ceramic materials, is nucleated due to the tensile strength originated during the indentation process.
Another common fracture mechanism is the chipping, i.e. a partial removal of surface material around the imprint.This phenomenon occurs as a result of Palqmvist cracks nucleating at the lateral sides of the imprint, beneath the surface.
In order to understand the fracture mechanisms that take place under the residual imprint, one nanoindentation performed at 10mN has been visualized by FIB, see Fig. 18 (from Roa et al., 2011bRoa et al., , 2011d)).

µm
Fig. 18.Cross-sectioning and imaging of damage under an indentation imprint performed at 10mN of applied load in Bridgman samples using FIB-SEM.Cross-sectioning in the middle of the imprint (from Roa et al., 2011d).
Fig. 18 shows a heterogeneous distribution of Y-211 particles around the matrix, with different sizes from 1 to 5 m.Moreover, no cracks or failure events can be appreciated under the nanoindentation imprint, thus implying that the deformation can be attributed to the closing of the oxygenation micro-cracks.The TSMG samples present the same effect -not shown in this work.
Due to all of these fracture mechanisms, the different values reported in the literature and summarized in the table 3, are not reproducible.

Conclusions
The paramagnetic Meissner effect has been observed in several superconducting materials grown by different techniques since their discovery, and different models and explanations have been proposed in order to explain their origin.Apparently, there is no a definitive model neither a tendency to saturation and the paramagnetic moments persist up to very high magnetic fields.
The mechanical properties (hardness, elastic modulus, toughness fracture and yield stress) for YBCO samples can be determined using the nanoindentation technique.This technique allows us to isolate the mechanical behavior of each phase, independently of the growing technique used.The mechanical properties for Bridgman samples are higher than that for TSMG samples.This phenomenon can be attributed to the texturing process, as TSMG process creates a high density of macro-and microcracks during the texture and oxygenation steps, respectively.
The values for yield stress for both texturing methods (Bridgman and TSMG), cannot be compared to previous works, as scare information in the literature is available.
With a cross-section, and using FIB technique, we can observe that the samples present a heterogenous distribution of Y-211 particles imbedded in the Y-123 matrix.No generation of porosity or microcracking have been observed, so the deformation mechanisms can be attributed to dislocations movement and twin generation in the ab-plane and c-axis, respectively.

Fig. 5 .
Fig. 5. Time dependence of normalized FC magnetization in a melt-textured YBa 2 Cu 3 O 7-x sample exhibiting the PME.The inset shows the sign reversal obtained in a relaxation experiment in the same sample.From Pureur et al., 2001.

Fig. 10 .
Fig. 10.Strong time dependence of the FCC magnetization in a melt-textured YBa 2 Cu 3 O 7-x sample exhibiting the PME.
Cleaning the tip: it is important to remove contaminants covering the tip.First, rinse the tip with miliQ water, acetone and ethanol and dry it under a nitrogen flow.After that, put the tip in an ozone cleaning chamber during 30-35 minutes, and ultraviolet light.2. Measuring the tip radius by means of a NiO Pattern: A NiO sample that features extremely sharp crystallographic ridges is used for measuring the tip radius.Then, as the sample ridges are nominally sharper than the tip apex, it is possible to obtain a "tip image" using blond reconstruction software.AFM images of the pattern should be collected in tapping mode with scan of 0.5µm by 0.5µm, scan rates of 1 Hz and 512 by 512 pixels.The blind reconstruction software employed is Scanning Probe Image Processor, see Fig. 13(Roa et al., 2011e).

Fig. 13 .
Fig. 13.Software reconstruction images of the AFM tip, (a) before the picoindentation test, and (b) after the picoindentation test.From Roa et al., 2011e.

Fig. 14 .
Fig. 14.Images of the different steps taking place during the picoindentation process.(a) AFM image with 40 different spots.(b) F vs. z cantilever deflection.(c) F vs , and (d) F vs. 3/2 from the slope of Elastic modulus, which is calculated using Hertzian equations.FromRoa et al., 2011e.

Fig. 15 .
Fig. 15.FE-SEM image of the ab-plane of TSMG samples, which present a high microcracks density performed during the oxygenation process due to the tetragonal to orthorhombic transition (from Roa et al., 2011b).

Table 1 .
Literature values of E for YBa 2 Cu 3 O 7-x obtained by indentation techniques.

Table 1 .
Literature values of E for YBa 2 Cu 3 O 7-x obtained by indentation techniques (continuation).

Table 2 .
Literature values of H for YBa 2 Cu 3 O 7-x obtained by indentation techniques.

Table 2 .
Literature values of H for YBCO obtained by indentation techniques.(continuation)

Table 3 .
Literature values of toughness fracture, K IC , for YBa 2 Cu 3 O 7-x obtained by indentation techniques.

Table 4 .
Literature values of yield stress,  ys , for YBa 2 Cu 3 O 7-x obtained by indentation techniques.

Table 2
by Micro or Nanohardness tests at room temperature present the same value at high applied loads.However, when the YBa 2 Cu 3 O 7-x samples are doped with a ductile material as silver oxide, the mechanical response tends to decrease, as reported by Foester et al.(Foester et al.,

Table 3
(Foester et al., 2008)ughness fracture values reported in the literature and obtained using Vickers and Nanoindentation tests.The different results reported in this table present a high scattered due to the YBa 2 Cu 3 O 7-x samples are strongly heterogeneous due to the presence of internal cracks, secondary phases and grain boundaries.Due to all this defects the different toughness values do not present any tendency.Lenblond-Harnois et al.(Lenblond-Harnois et al., 2000)and Foester et al.(Foester et al., 2008)obtained a toughness fracture of 1.88 and 1.6 MPam 1/2 for YBa 2 Cu 3 O 7-x with 5 and 10% of Ag and Ag 2 O, respectively.In that case, this value is higher than the others reported in this table due to the silver introduced inside of the YBa 2 Cu 3 O 7-x samples reduce the fragility of these materials.