The following chapter treats and compares aspects of the so called “peak effect” in poly-cristalline NbTi and in single crystalline 2H-NbSe2. The topic of scaling is discussed in both the low temperature and the peak region. Evidence is presented concerning the nature of the transition between the low field and the peak region.
8.1 Introduction to the peak effect
For several decades, the abrupt and strong increase of the critical current density in many type II superconductors, at fields slightly smaller than the upper critical field Hc2, has remained a topic under vivid discussion both experimentally and theoretically.
Multiple ideas have been put forward about the origin of this phenomenon: One of the first tentative explanations, by Pippard [55], assumed that the rigidity of the flux line lattice (FLL), near Hc2, would drop more rapidly than the pinning strength due to inhomogeneities, leading to a better matching of FLL and defects.
In contrast to this, Larkin and Ovchinnikov [56, 5] put forward the idea of weak collective pinning, where large amounts of pinning centers lead to a division of the FLL into small regions of volume Vc with short-range order. At fields just below Bc2, the reduction of Vc leads to a maximization of the pinning force per unit volume.
For the case of superconductors of large spin susceptibility, Tachiki et al. [57] proposed the occurence of a generalized Fulde-Ferrell-Larkin-Ovchinnikov (GFFLO) state as the origin of the peak effect in these systems, based on work by Fulde and Ferrell [58] and by Larkin and Ovchinnikov [59, 60]. In Tachiki et al.’s ansatz, a spatially modulated order parameter leads to flexible vortices readily pinned through the collective pinning mechanism. Modler et al. claim [61] to have found strong evidence for the occurence of such a state in UPd2Al3 and CeRu2.
For the case of NbTi, quite recently, U. Wyder et al. have carried out an investigation using magnetostriction measurements. In their work [11], they mainly focus on the applicability of scaling laws, first introduced by Fietz and Webb [62], to the temperature dependence of the peak region.
A key topic of current interest is the nature of the transition between the low field and the peak effect region. Neutron diffraction studies of Nb single crystals were interpreted in favor of a first order phase transition between a stable ordered Bragg-glass phase and a fully disordered either solid [63] or liquid [6] vortex state.
In the following, we will be considering the dependence of the amplitude, position and width of the peak on temperature, thereby discussing the concept of scaling and
65
66 8.2. GENERAL OBSERVATIONS applying it to the low field region as well. In addition, evidence is reported against a first order transition. Another topic, which has been connected intimately to the peak region up to now, is the phenomenon of thermomagnetic history dependence. It will be discussed separately in the next chapter.
8.2 General observations
Figure8.1shows a detail from figure4.8(a) for temperatures1.5 K< T <4.2 K.
The peak is defined by a local minimum at its onset Bon for increasing field, the irreversibility fieldBirr and an offset field Boff (indicated for T = 1.5 K), corresponding to a local minimum in the descending branch of the curve.
Figure 8.1: Plot of the peak region in magnetostriction on a cylindrical sample for dif-ferent temperatures. The arrow through the curves points in the direction of increasing temperature.
The peak maxima for increasing and decreasing field, Bp,up and Bp,down, are also marked. Since those positions do not differ too much, however, a mean peak fieldBpwas defined as the position of the maximum in the total irreversible magnetostriction defined as
One observes that the positions of the peak maximum for increasing and decreasing field, respectively, do definitely not coincide. Moreover, and presenting a feature more suited for comparison, the on- and offset of the peak on its low-field side occur at different values of the external magnetic field, i.e. the width of the peak region is significantly
CHAPTER 8. THE PEAK EFFECT AND SCALING 67 larger for decreasing than for increasing field. This asymmetry is well illustrated by figure 8.2, where the part of the curve measured for decreasing field was multiplied by (−1)to compare the increasing and decreasing branch in the peak region. The difference in peak positions is not commonly observed in magnetization measurements, whereas a similar asymmetry in the on- and offset has been found in magnetization and transport measurements on high quality single crystals of superconducting UPd2Al3, CeRu2 [61], 2H-NbSe2 [46] as well as in amorphous Nb3Ge and Mo3Si films [64].
Figure 8.2: Comparison of up- (B >˙ 0) and downsweep (B <˙ 0) in the peak effect region in the magnetostriction on a small cylinder. The branch of the curve with B <˙ 0 was multiplied by (−1).
8.3 Temperature dependence of the irreversibility ampli-tudes
To illustrate the qualitatively different behaviour of low field and peak region, figures8.3 and 8.4 present the temperature dependence of the magnetostrictive irreversible signal, for different values of the magnetic field (defined as a percentage of the peak field), for the case of a cubic sample and a thin cylinder, respectively.
The striking differences in the temperature dependence make it clear that we are dealing with two entirely different regimes, independently of the question of the nature
— first order or not — of the transition between both field regions.
The peak amplitude, if considered as a function of temperature, apparently drops to zero at a temperature far belowTc (see e.g. inset of fig.8.3). At least, it decreases more rapidly than the irreversible contribution determined at lower magnetic field and seems to have a qualitatively different temperature dependence.
68 8.3. T DEPENDENCE OF THE IRREVERSIBILITY AMPLITUDES
∆∆ ∆∆
Figure 8.3: Temperature dependence of the magnetostrictive irreversibility for different magnetic fields, chosen as a percentage of the peak field, measured on a cubic sample.
Note that the main graph’sy scale is logarithmic, while the inset shows a linear plot of the peak’s amplitude. In both graphs, the right border corresponds toTc≈9.5 K.
Figure 8.4: Same as above, measured on a thin cylindrical sample
CHAPTER 8. THE PEAK EFFECT AND SCALING 69 This observation, raising the question if the the peak’s amplitude really drops to zero at a temperatureT < Tcor only disappears exponentially or with a high power in(Tc−T) or in some property depending onT, motivates a comparison of the dependences of the amplitudes on a normalized temperaturet=T /Tc for both NbTi and 2H-NbSe2, where all amplitudes are normalized to1at the same (arbitrary) reduced temperature, to com-pare theirT dependence. This is done in figure8.5. Data is taken from magnetostriction measurements on the cylindrical and the cubic sample as well as on 2H-NbSe2. In addi-tion, for comparison, the peak amplitude from magnetization measurements in CeRu2, measured by Kadowaki et al. [65], is also included. In this plot and in the following,Bc2
is used synonymously for Birr, which is justified — in the context of scaling —, since in chapter6, the latter was shown to be an almost fixed fraction of the former.
Figure 8.5: Comparison of the dependences of low field and peak amplitudes on a reduced temperaturet=T /Tc, for both 2H-NbSe2and NbTi. At one (arbitrary) valuet, all curves are normalized to1 for comparison.
As was mentioned before, for both measurements on NbTi samples, the low field (B = 0.2Bp) irreversibility drops to zero less rapidly than the peak amplitudes. In NbSe2, for low temperatures, the low field irreversibility seems to decrease as strongly withT as in the peak region, before theT dependence gets weaker at higher T.
One could be tempted to conclude that the peak disappears at T ≈ (0.7±0.1)Tc, and this more or less universally for all samples examined. However, one might as well assume that the amplitude decreases with a high power in some property depending on T (e.g. Bc2(T)), and that the peak becomes invisible nearTconly due to the limitations in measurement resolution. To tackle this question, in the next section, the possibility of scaling will be discussed.
70 8.4. SCALING
8.4 Scaling
8.4.1 Introduction to the concept of scaling
In order to connect phenomenological models of the critical state to a more microscopic understanding of the pinning mechanisms, Fietz and Webb [62] investigated the field and temperature dependence of dislocation pinning by comparing measurements in a large range inκandT to existing microscopical models of pinning. They extracted the pinning forceFpfrom magnetization measurements, and found a power law dependence ofFp(T) onHc2(T). Their main result is a scaling law
Fp=Kk(κ)g(B/Hc2)[Hc2(T)]p, (8.2) in which the pinning force density depends on magnetic field through some universal function g(B/Hc2) and on temperature only through Hc2(T). k depends only on the Ginzburg-Landau parameterκ, and K is a constant. p was found to be5/2.
Kramer compared measurements on different compounds [66] and thereby also paid attention to the peak effect region, where he distinguished between two types of “peaks”:
A large one at intermediateB and the narrow one just below Bc2 which is known as the
“peak effect”. For the latter one, only few data points were presented (taken from [67]), leading to a scaling exponent p≈2.9 >2.5. It remains unclear, however, if this higher value of the exponent is only caused by uncertainties in its determination.
More recently, U. Wyder et al. have performed a scaling analysis on measurements of magnetostriction on NbTi in the peak region [11], on samples that have been extra cold worked, resulting in a large peak amplitude compared to the low field irreversibility. In this investigation, a much higher value p ≈4.5 was found. This was attributed, using a macroscopical thermodynamical argument, to an additional dependence on B of the elastical constants of the sample:
c(T, B) =c0(b)Bc2−2, (8.3) withb=B/Bc2(T), and wherec0(b) is a function depending only onb.
8.4.2 Scaling in the peak region
A scaling analysis has been performed in the peak region for our magnetostriction mea-surements on both NbTi and NbSe2. In figure8.6, the peak amplitude has been plotted in a double logarithmic plot versus the upper critical field at the same temperature, in order to determine the exponent p, for the cubic NbTi sample (]2). This procedure is justified under the assumption that the peak field Bp is a fixed fraction of Bc2, which is only roughly correct, as we shall see. Nevertheless, we get an estimate of p, which is p≈7.22.5
For further corroboration of this value, the same analysis has been performed for the cylindrical sample (]6, fig. 8.7) and for the anisotropic single crystalline 2H-NbSe2 sample (see chapter 7, fig. 8.8). It yields p ≈ 5.8 and p ≈ 6.2, respectively. These results suggest that scaling in the way defined above does not work in the peak region.
A similar result is obtained by Kadowaki et al. for weak pinning CeRu2 [65], though it is argued there that the non-compliance to the scaling rule is a proof for the hypothesis that the peak in CeRu2 has different origins than in “conventional” superconductors. Our argument is simply that the scaling concept is not applicable in the peak region, be it in “conventional” (NbTi) or weak pinning (NbSe2) superconductors. This point of view
CHAPTER 8. THE PEAK EFFECT AND SCALING 71
∆
∆ ⋅ ⋅
∆
Figure 8.6: Scaling analysis for magnetostriction on a cubic NbTi sample in the peak region. Squares represent values extracted from magnetostriction measurements, the solid line presents a fit of the form(∆L/L) =a·Bc2p .
is further corroborated by the tentative scaling plot (figure 8.9) for the peak region in the irreversible magnetostriction on the thin cylindrical NbTi sample, showing that the curves definitely do not coincide. This remains true even when for normalization of the magnetic field the peak’s maximum fieldBp is chosen instead ofBc2. A similar plot for NbSe2 is shown in figure 8.10, suggesting weakly satisfactory compliance to scaling rules only forT ≤2.5 K.
Figure 8.7: Scaling analysis for magnetostriction on a cylindrical NbTi sample in the peak region. Squares represent values extracted from magnetostriction measurements, the solid line presents a fit of the form(∆L/L) =a·Bc2p .
72 8.4. SCALING
∆
∆ ⋅ ⋅
∆
Figure 8.8: Scaling analysis for magnetostriction on a NbSe2 sample in the peak region.
Squares represent values extracted from magnetostriction measurements, the solid line presents a fit of the form(∆L/L) =a·Bpc2.
Nevertheless, as we have seen above, it seems that, roughly, a power law dependence holds between the peak irreversibility (∆L/L)peak and Bc2. This suggests that, at tem-peratures near Tc, the peak amplitude decreases below the resolution threshold in our measurements instead of disappearing entirely, answering the question raised in section 8.3.
8.4.3 Scaling in the low field region
The same procedure that we applied to the peak region as described above was also used at lower magnetic fields.
For the cubic NbTi sample, the low field irreversibility maximum at B ≈ 0.05Bc2
was used to normalize the data. Figure 8.11 shows the normalized data vs. a reduced magnetic fieldB/Bc2. The curves coincide only for very low field (B <0.1Bc2), but the resulting exponentp≈2.0(fig. 8.12) is much closer to what would be expected.
Since one could assume that the discrepancies are geometry related, and to move towards a more ideal geometry, the same analysis was performed for the cylindrical NbTi sample, using the intermediate field maximum for normalization. Figure 8.13 shows remarkable scaling for temperatures1.5 K< T <4.2 K below the peak region. Even for T = 6.7 K (curve not shown for clarity) the general curve form remains the same, being only slighlty tilted, probably due to the more important background signal from the cell at these small amplitudes.
A scaling analysis (fig. 8.14) reveals p ≈ 2.4, which is also remarkably close to the original expectations.
We can therefore conclude that scaling rules are obeyed in the NbTi alloy under investigation, though only for the magnetic field region below the peak.
Concerning NbSe2, the situation is somewhat less clear. Coincidence of the curves normalized to their value atB = 0.1Bc2 is not convincing (fig. 8.15), the scaling fit done using very few data points (fig. 8.16) gives only a rough value of p ≈ 3.5. This may
CHAPTER 8. THE PEAK EFFECT AND SCALING 73
∆∆
Figure 8.9: Plot of the peak region in irreversible magnetostriction for a cylindrical NbTi sample at different temperatures, renormalized to the respective peak’s amplitude, vs. a reduced magnetic fieldB/Bc2(T). The arrow through the curves points in the direction of increasing temperature.
∆∆
Figure 8.10: Plot of irreversible magnetostriction on a NbSe2 sample at different tem-peratures, renormalized to the respective peak’s amplitude, vs. a reduced magnetic field B/Bc2(T), for a measurement along the a plane. The arrow through the curves points in the direction of increasing temperature.
74 8.4. SCALING
∆ ∆
Figure 8.11: Plot of irreversible magnetostriction on a cubic NbTi sample at different temperatures, renormalized to the low field irreversibility maximum, vs. a reduced mag-netic fieldB/Bc2(T). The arrow through the curves points in the direction of increasing temperature.
∆
∆ ⋅ ⋅
∆
Figure 8.12: Scaling analysis for magnetostriction on a cubic NbTi sample for low B.
Squares represent values extracted from magnetostriction measurements, the solid line presents a fit of the form(∆L/L) =a·Bpc2.
CHAPTER 8. THE PEAK EFFECT AND SCALING 75
Figure 8.13: Plot of magnetostriction on a cylindrical NbTi sample at different temper-atures, renormalized to the irreversibility maximum at intermediate fields, vs. a reduced magnetic field B/Bc2(T). The arrow through the curves points in the direction of in-creasing temperature.
Figure 8.14: Scaling analysis for magnetostriction on a cylindrical NbTi sample for in-termediateB. Squares represent values extracted from magnetostriction measurements, the solid line presents a fit of the form(∆L/L) =a·Bc2p .
76 8.4. SCALING have several reasons. The geometry of the sample was certainly not ideal, though. In addition, possible effects of the system’s strong anisotropy have not been considered.
∆∆
Figure 8.15: Plot of irreversible magnetostriction on a NbSe2 sample at different tem-peratures, renormalized to the value at B = 0.1Bc2(T), vs. a reduced magnetic field B/Bc2(T). The arrow through the curves points in the direction of increasing tempera-ture.
∆
∆ ⋅ ⋅
∆
Figure 8.16: Scaling analysis for magnetostriction on a NbSe2 sample for intermediateB
CHAPTER 8. THE PEAK EFFECT AND SCALING 77
8.5 Dependence of B
on− B
offon the sample diameter
As mentioned in the introduction to this chapter, one of the points most vividly discussed nowadays in the physics of vortex matter is the nature of the transition into the peak regime. In this context, the difference ∆B = Bon−Boff between the onset and offset of the peak region was interpreted in terms of a first order transition, in the sense that a disordered “supercooled” vortex phase coexists with an ordered phase [61, 65].
The very existence of the “supercooling” might mean that the peak effect phenomenon marks a true thermodynamic phase transition and that it is of first order [46]. To test this hypothesis, we have compared the measurements on three cylindrical samples with different diameters: 1.5 mm (]6), 2.8 mm (]3), and 10 mm(]4) (see section4.2.1). The field difference ∆B = Bon−Boff is plotted in Fig. 8.17 as a function of the sample’s diameter. Since only differences of magnetic fields was considered, the slight fluctuations in sample parameters (see sections 4.2.1 and 6.2.2) should not be able to significantly influence these results. ∆Bfollows linearly the transverse dimension of the sample within the accuracy of the measurement. But this geometrical scaling is a strong evidence against the interpretation of ∆B in terms of a first-order thermodynamic transition.
On the other hand, it supports a much simpler interpretation of ∆B, due to a spatial separation of two phases in the framework of the critical state model. A cone-like field profile is generated within the bulk of the material when the field is swept down and out of the peak region. The highest value of the local magnetic induction, as well as its gradient and screening current density are in the center of the sample. The disordered high-field phase is preserved in the interior of the sample as long as a local induction is higher than Bon. Boff indicates the field on the surface of the sample, therefore ∆B scales with the diameter. This means that two phases do not coexist in the same point of theH-T phase diagram; their coexistence at the same applied field is possible due to a spatial variation of the local magnetic induction, simply following the basic idea of the critical state.
∆
Figure 8.17: Scaling of the difference between on- and offset of the peak with the sample diameter
78 8.6. CONCLUSION
8.6 Conclusion
The peak’s amplitude as a function of temperature as well as the low field irreversibil-ity have been compared for different samples. An analysis in terms of scaling shows compliance to classical scaling laws only for the low magnetic field region, invalidating arguments against the universality of the peak effect in different types of superconduc-tors. For higher fields, it suggests the reduction of the peak with a high power (p≈6) inBc2 rather than a complete disappearance atT < Tc. In addition, evidence has been presented that the asymmetry between onset and offset of the peak cannot be interpreted in terms of a first order phase transition.