Strain relaxation behaviour of vortices in a multiferroic superconductor

RUS is a well-established method for determining the elastic and anelastic properties of samples with dimensions in the range ~1–5 mm and has been described in detail by Migliori and Sarrao [32]. It was already used for Ba(Fe_1−xCo_x)₂As₂ pnictides by Fernandes et al [33] and Carpenter et al [20].

In the present study, Crystal 1 was placed with its largest faces resting lightly between piezoelectric transducers in an RUS head described by McKnight et al [34], the only difference being that the steel component was replaced by copper. The sample holder was lowered into an Oxford Instruments Teslatron cryostat with a temperature range of 1.5–300 K and a superconducting magnet capable of delivering a magnetic field of up to 14 T [35, 36]. In this configuration, the field was aligned parallel to the c-axis of the crystal (H//c). The sample chamber was first evacuated and then filled with a few millibars of helium as exchange gas. Spectra were generally measured in the frequency range 10–500 kHz, with a maximum applied voltage of 2 V and 35 000 data points per spectrum, using purpose built electronics produced by Migliori in Los Alamos. Individual spectra were collected under conditions of fixed temperature and field, with a period of 60 s allowed for thermal equilibration at each fixed point. Programmed sequences had small changes of temperature under constant field or small changes in field at constant temperature.

Quantitative analysis of the temperature and field dependence of selected peaks in the primary spectra was achieved by fitting with an asymmetric Lorentzian function using the software package Igor (Wavemetrics) to give the peak frequency, f , and width at half maximum height, Δf . Values of f ² scale with the combinations of elastic constants which determine each resonance mode (f ² $\propto$ elastic moduli). The inverse mechanical quality factor, expressed here as Q⁻¹  =  Δf /f , is a measure of acoustic loss.

The heat capacity of Crystal 2 was measured as a function of temperature in a range of externally applied magnetic fields using the heat capacity option of a Quantum Design physical properties measurement system (PPMS). The field was applied perpendicular to the large faces of the crystal (H//c). In-plane electrical resistivity was measured with four electrodes attached to the surface of the same crystal using the Electrical Transport Option of the PPMS. Temperature was varied between 2 and 20 K in magnetic fields ranging between 0 and 12.5 T (H//c). DC magnetic properties, including the temperature dependence of moment and hysteresis loops to  ±6.7 T, were measured in a Quantum Design magnetic properties measurements system (MPMS) XL squid magnetometer, again with H//c. AC magnetic properties were measured using the same crystal with the AC measurement System option of a PPMS instrument at frequencies of between 0.01 and 10 kHz in fields of up to 12.5 T. Complete details are given in the appendix.

The temperature dependences of f ² and Q⁻¹ for selected peaks which, on the basis of large softening through the ferroelastic transition have been determined to be dependent predominantly on C₆₆, are shown in figure 4. Full reversibility between heating and cooling is demonstrated in the close up view of a resonance peak with frequency near 140 kHz, as measured in a field of 12.5 T (figure 4(a)). The temperature interval of steep stiffening with falling temperature is marked also by variations in peak widths. These variations are shown quantitatively for a resonance mode near 330 kHz in different applied field strengths (figure 4(b)). The degree of softening in zero field reduces at 1 and 2.5 T, and becomes stiffening in fields of 5 T and above. Q⁻¹ remains at low values through the temperature interval of the normal–superconducting transition in zero field but develops up to three successive Debye-like peaks when the transition is followed in an applied magnetic field.

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The extent of stiffening varies between resonances but the greatest increase, by up to ~200%, occurs for a resonance near 25 kHz (figure 4(c)). The increased stiffness at 1.5 K scales with applied field as f ² $\propto$ (µ_oH)² (figure 4(d)).

Details of the field dependence at 11 K of a resonance peak with frequency near 90 kHz are illustrated in figure 5(a). This mode is determined predominantly by C₆₆ and, in addition to having changes in frequency and peak widths, reveals the clear hysteresis between increasing and decreasing field.

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Figure 5(b) shows f ² and Q⁻¹ data obtained at 12.3 K for a resonance with frequency near 46 kHz. Specific features of the data are indicated by labels A, B and C. A marks the field at which a minimum in f ² occurs at low values of µ_oH when the field is being ramped up. The loci of equivalent points at other temperatures are referred to below as 'H of low field anomaly, constant T'. The field at which the change from irreversible to reversible evolution occurs is labeled B and the loci of such points are referred to below as 'H_irr, constant T'. C marks the field at which there is a Debye-like peak in Q⁻¹ and a change in the trend of f ². This is referred to below as 'Q⁻¹ peak, constant T'.

Figure 5(c) contains a compilation of f ² and Q⁻¹ data for the resonance with frequency near 90 kHz measured at temperatures between 10 and 14 K. It shows that the peak in Q⁻¹ reduces in magnitude as temperature is increased and disappears altogether by 14 K. The amount of stiffening with reducing field associated with the loss peak diminishes at progressively higher temperatures, also disappearing by 14 K. Finally, the field at which the peak evolution changes from reversible to irreversible also reduces with increasing temperature.

Variations of f ² and Q⁻¹ below the limit of reversibility become more complex at progressively lower temperatures, as illustrated by an additional data set given in figure A1 for the resonance peak with frequency near 330 kHz.

Values of the temperature and field at which the first loss peak with falling temperature under the condition of constant field (figure 4(b)) coincide with those for the single, broader acoustic loss peak observed when varying field at constant temperature (figure 5(c)). Values of the field at which the change from reversible to irreversible behaviour occurs correlate with field and temperature values for the second loss peak observed with falling temperature at constant field.

Some resonances of the sample showed no overt dependence on temperature through T_S, T_N and T_c. The peak with frequency near 175 kHz in figure 2 is an example of this for zero field and has been assigned to a primary dependence on C₁₁–C₁₂. Variations of f ² and Q⁻¹ for this resonance mode as a function of field at different temperatures are featureless up to 8 T, apart from a slight trend of increasing Q⁻¹, as seen in figure 6. Resonances attributed to shearing controlled by C₄₄ at higher temperatures [20] were too weak to allow them to be followed as a function of changing field.

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In the interests of brevity, details of a formal analysis of the loss peaks are given in section A.2 and only fits to the data are shown in figure 7(a). Values of E_a/R extracted from the fitting, where E_a is the activation energy, R is the gas constant and Arrhenius thermal activation has been assumed (equation (A.1)), fall in the range ~250–600 K for the first peak, ~200–350 K for the second peak and ~100–200 K for the third peak (table A1). There are correlations with field strength and frequency but the strongest correlation is with temperature, such that the overall pattern is of a succession of anelastic pinning/freezing events with progressively smaller activation energy barriers as temperature reduces. Separate Arrhenius treatment of the frequency-temperature dependence of peak 1 with an external field of 7.5 T (figures 7(b) and (c)) gives E_a/R  =  353  ±  3 K. f _o values fall in the range 10¹⁵–10²⁷ Hz, which is rather large to be physically realistic. Smaller values would be obtained for Vogel–Fulcher behaviour (equation (A.9)), but the frequency range is too narrow to allow this to be tested.

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High resolution thermal expansion data for a crystal with composition Ba(Fe_0.945Co_0.055)₂As₂ are reproduced from Meingast et al [29] in figure 8(a). They reveal the overall pattern of strain variations which accompany the structural, magnetic and superconducting transitions. Changes of the a and c lattice parameters give strains Δa/a and Δc/c of up to ~0.0001 below T_S  ≈  41 K, implying only very small changes in components e₁ and e₃ of the spontaneous strain tensor defined with respect to a parent tetragonal structure. By way of contrast, the value of e₆, obtained by detwinning the crystal with an externally applied shear stress, increases to a maximum of ~0.0008 (figure 8(a), and see details in section A.3). The evolution of e₆ can be understood in terms of its bilinear coupling with Q_E, λe₆Q_E, though the tail above T_S is due to the applied shear stress. The coefficient, λ, specifies the strength of coupling. There are no obvious deviations near T_N (~29 K for the Co content of 0.055), implying that coupling of strains with the magnetic order parameter is very weak/absent.

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An obvious and discrete reversal of all the trends in figure 8(a) occurs below T_c  ≈  22 K, consistent with an effective coupling of e₁, e₃ and e₆ with Q_SC. This is expected to be of the form λe_i,SC$Q_{{\rm SC}}^{2}$, where e_i,SC (i  =  1, 3, 6) are components of the spontaneous strain defined with respect to the parent (normally conducting) orthorhombic crystal.

A formal analysis of less complete thermal expansion data for crystals from the same batch as used for RUS is given in section A.2 and the key results are shown in figure 8(b). The linear strain e_1,SC has a temperature dependence which is consistent with a second order transition as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{e}_{1,{\rm SC}}}\propto Q_{{\rm SC}}^{2}\propto \left[\coth \left(\frac{{{\Theta}_{{\rm so}}}}{{{T}_{{\rm c}}}} \right)-{\rm coth}\left(\frac{{{\Theta}_{{\rm so}}}}{T} \right) \right].\nonumber \end{align} \tag{ 1 }$$

Following Salje et al [37], Θ_so is the order parameter saturation temperature. Figure 8(b) shows that changes in e_6,SC are an order of magnitude larger than for e_1,SC (up to ~8  ×  10⁻⁵ instead of ~5  ×  10⁻⁶), but follow the same trend, consistent with the expected relationships e_1,SC ($\propto$e_3,SC) $\propto$ e_6,SC $\propto$ $Q_{{\rm SC}}^{2}$.

The observed stepwise softening of C₆₆, C₁₁ and C₃₃ through T_c in zero field (figure 4(b) and [16, 18, 19]) is typical of classical strain/order parameter coupling due to terms of the form λeQ² at a second order transition (e.g. [38–40]). A step, with magnitude expected to scale with λ², occurs between more or less constant values above and below T_c.

The pattern of slight elastic softening shown by C₆₆ at 0, 1, 2.5 T, without overt changes in acoustic loss, becomes a pattern of stiffening at 5, 7.5, 12.5 T accompanied by successive peaks in Q⁻¹ (figures 4(b) and 7(a)). The overall Debye-like form of this is typical for freezing or pinning processes of defects coupled with strain [41]. In this case the relevant strain must again be e₆. Elastic stiffening has also been observed below the superconducting transition in cuprates, but the increases in elastic constants were orders of magnitude smaller than seen here and were accompanied by only a single loss peak [42].

Figure 9 contains a comparison of elastic softening in zero field, as expressed by variations of f ² and Q⁻¹ for selected resonances which depend predominantly on C₆₆, with particular temperatures extracted from the measurements of heat capacity, electrical conductivity and magnetism that are set out in the appendix.

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A step in the heat capacity at ~12.3 K (figure A3) is consistent with a discrete second order transition. This value of T_c is not distinguishable from the value of T_c  =  12.5 K deduced from the Landau description of e₁ (figure 8(b)).

Variations of electrical resistivity through T_c are shown in figure A4 and the temperatures at which it falls to 95% and 1% of the value for the normally conducting state are listed in table A2. In zero field, these are 19.1 and 14.8 K, respectively. There is no elastic anomaly near 19.1 K, but the 1% limit coincides with the onset of softening ahead of T_c (figure 9).

Two particular temperatures have been extracted from the variations of DC moment as a function of temperature shown in figures A5(a) and (b). The onset of a measurable diamagnetic moment in a field of 0.002 T, T_Monset, is 14.0 K, which falls just ahead of the steepest segment of the softening trends in figure 9. Extrapolation of the linear segment of the increasing diamagnetic moment with falling temperature back to zero, T_Mextrap, gives 12.6 K, which is within experimental uncertainty of T_c determined from measurements of strain and heat capacity.

A steep change in the real part of AC magnetic moment, χ', and the accompanying peak in the imaginary part, χ'', are illustrated for selected frequencies and field strengths in figure A6. The peak in χ'' measured in a field of 0.002 T occurs at a temperature which is very slightly dependent on frequency (~13.0 K at 10 Hz, ~13.15 K at 10 kHz) and coincides with the midpoint of the steep elastic softening trend in figure 9.

In summary, measurements of bulk properties, ie spontaneous strain, heat capacity and DC moment, show that the normal—superconducting transition occurs at ~12.5 K. This is just below the temperature interval in which steep elastic softening of C₆₆ occurs. The electrical conductivity measurements show the onset of superconductivity as occurring ahead of transformation in the bulk. AC magnetic anomalies correlate closely with the steepest part of the elastic softening.

Values of temperature and field for the peaks in Q⁻¹ and initial breaks in slope of f ² (ie of C₆₆) illustrated in figures 4, 5, 7 and A1, together with additional data given in table A1, are combined in figure 10(a). Whether obtained in sequences of varying temperature at constant field or varying field at constant temperature, the data produce three clearly defined trends. With falling temperature at high fields, the first trend is for a peak in Q⁻¹ (peak 1 in figure 7(a), C in figure 5(b)) accompanied by elastic stiffening characteristic of a Debye-like anelastic freezing process. The second trend is defined by the second acoustic loss peak seen with falling temperature (peak 2 in figure 7(a)) and the field, H_irr, at which a change from reversible to irreversible evolution of f ² occurs (B in figure 5(b)). The third trend is at low fields and is given by the locus of the minimum in f ² observed with increasing field strength at constant temperature below T_c (A in figure 5(b)). Peak 3 of Q⁻¹ in figure 7(a) was only observed at the highest field strength.

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Figure 10(b) shows how the elastic anomalies correlate with changes in other properties as a function of temperature and field strength. The line for 95% resistivity is systematically ~4 K above the first elastic anomalies, while the line for 1% resistivity coincides with the trend of Debye-like freezing represented by peak 1 of Q⁻¹ at field strengths of ~5 T and above. Below 2.5 T, the 1% line diverges and is ~2 K higher than the elastic anomalies in zero field.

AC magnetic data collected at frequencies of 0.01, 0.1, 1, 10 kHz in DC fields of 1, 2.5, 5, 7.5, 10 T all have a peak in the imaginary part of the susceptibility, χ'', accompanied by a steep change in the real part, χ', with a strong dependence on frequency (figures A6(b) and (c)). A more complete analysis of these is given in section A.7 and is discussed below. Here it is necessary only to note that this pattern is explained in the literature as being due to glassy freezing of vortices [43–45] and that the temperature-field locus of the anomalies overlaps at the high frequency end with the locus of acoustic loss peak 2 and H_irr of f ² from the RUS data. The RUS data are for frequencies in the range ~25–350 kHz.

Values of T_Mextrap and T_Monset from DC measurements of magnetic moment are listed in table A3 and define two curves in figure 10(b). At the lowest measuring field (0.002 T), T_Monset is ~1 K above the temperature at which anomalies in the AC response occur but, at fields of 1 T and above, the curve for T_Monset lies parallel to and just below the curves for AC magnetic loss. On this basis, T_Monset appears to correspond to a static (zero frequency) measurement of the same process being sampled in the AC measurements and the acoustic loss process detected in the RUS measurements. The curve for T_Mextrap occurs systematically ~1–2 K below the curve for T_Monset and does not obviously correlate with any changes in elastic properties, apart from its extrapolation passing close to acoustic loss peak 3 at 12.5 T.

Magnetic hysteresis loops measured to  ±6.7 T show the fishtail pattern characteristic of an unconventional superconductor (figure A5(c)). Three specific points which occur with increasing field in these loops are H_p, the field at which there a maximum in moment occurs during the first increase from zero, H_2p, the field at which there is a rounded maximum in the moment and H_irr, the field at which the change from reversible to irreversible evolution occurs. H_p is understood to be the field at which vortices nucleated at the surface of a crystal reach its centre [46] and H_2p has been explained in terms of a change in pinning properties of the vortices (e.g. [47]). Irregular variations of f ² evident at low fields below the interval of vortex freezing include minima (A in figure 5(b)) which are in the same part of the phase diagram as H_2p and H_p, though the correlation is not exact (figure 10(b)). Values of H_irr for the evolution of moment in the hysteresis loops differed between two different series of measurements (see section A.6 below), but nearly all fall between T_Monset and T_Mextrap. This is just below the temperature range in which the loss process indicated by the AC magnetism and RUS data occurs.

The most fundamental consequence of coupling of order parameters with strain is suppression of fluctuations and promotion of mean field behaviour due to the long ranging nature of strain fields. Data for heat capacity, evolution of the spontaneous strain and softening of C₆₆ are all consistent with a description of the normal-superconducting transition in zero field as being a classical second order phase transition with coupling of the form λe₆$Q_{{\rm SC}}^{2}$.

If there is coupling between strain and order parameters of the form λe₆$Q_{{\rm SC}}^{2}$ and λe₆Q_E, there must also be coupling of the order parameters with the form λ$Q_{{\rm SC}}^{2}$Q_E via the common strain. If there is also coupling of Q_E with Q_M it follows that all three order parameters are coupled, so defining the existence of multiferroic superconductivity in a single phase. In the case of Ba(Fe_0.957Co_0.043)₂As₂, the significant coupling is between Q_SC and Q_M and all three order parameters vary together. By way of contrast, YBCO does not belong to this class, in effect, because the tetragonal–orthorhombic transition occurs hundreds of degrees above T_c [48, 49] and antiferromagnetic ordering also appears to be quite separate (e.g. [50]).

More generally, linear-quadratic coupling occurs when one of the instabilities is proper or pseudoproper ferroelastic and leads to distinctive phase diagram topologies [51, 52], which must be a significant factor in the phenomenological richness of the behaviour of pnictides with respect to changes in composition, temperature, magnetic field, pressure, stress, etc.

The existence of a contrast in the shear strain e₆ between normal and superconducting states of Ba(Fe_0.957Co_0.043)₂As₂ means that there must also be shear strain contrast between vortex cores and the superconducting matrix. It follows inevitably that interactions between vortices and of vortices with their surrounds will be mediated by long ranging strain fields, in much the same manner as is evident in the way that twin walls in ferroelastic materials interact with each other over lengths scales of up to ~0.1 µm (see, for example, chapter 7 of [53]). Relatively stiff vortices in a relatively soft superconducting matrix would lead to an increase in overall stiffness with increasing field simply because of the expected increase in the density of vortices. However, the strain fields around each vortex will give rise to additional interactions between them. Strong repulsive forces arising from unfavourable overlapping of these would give an additional increase in resistance to specific orientations of shear stress and are likely to be a significant factor in the steep increases in stiffness evident in figure 4. Changes in elastic stiffness will depend on the orientation of the vortices which, in turn, depends on the orientation of the magnetic field. In the present study, the vortices will have been aligned parallel to [0 0 1] of the parent tetragonal structure and the observed changes are in elastic constants are for C₆₆. This means that the relevant repulsive forces between vortices will have been within the (0 0 1) plane, ie perpendicular to their lengths.

Whatever microscopic model is developed to account for the added stiffness per vortex, a linear dependence on field strength is not expected because reversing the sign of the field would not change their density and distribution. Instead, the maximum amount of stiffening, i.e. for the fully frozen state at 1.5 K, displays a parabolic dependence on applied field (figure 3(d)), as previously reported for YBCO [14].

The complete analysis of AC magnetic data in section A.7 is consistent with views from the literature that the Debye-like anomalies in χ' and χ'' in an applied magnetic field are due to freezing of vortices. Analysis based on Vogel–Fulcher dynamics (equation (A.9)) and the Havrilian–Negami equation (equation (A.10)) provide a reasonable description of the frequency dependence for a process which involves a small spread of relaxation times. In an RUS experiment the range of accessible frequencies corresponds to only ~1 log unit but data for Q⁻¹ of peak 2 measured at 10 and 12.5 T fall on extrapolations to data for χ'' plotted in Arrhenius form (figure 11). This shows that the same liquid—glass transition is being detected and demonstrates that the vortices couple significantly with strain.

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Values of field for DC measurements of the onset of irreversibility in hysteresis loops, H_irr in figures A8 and 10(b), and of temperature for the first appearance of a diamagnetic moment, T_Monset, fall just below the dynamic magnetic measurements, consistent with their marking the static limit of the freezing process. With respect to elasticity, the limit of reversibility in variations of f ² with changing field at constant temperature, H_irr in figure 10(a), corresponds to the freezing point, as measured at frequencies in the vicinity of ~100 kHz.

Values of ~200–300 K (~17–26 meV) for the activation energy estimated from Arrhenius treatment of acoustic loss peak 2 (table A1) compare with ~10–70 K (~1–6 meV) from Vogel–Fulcher analysis of the AC magnetic loss peaks (section A.7). These are comparable with a previous determination of 120  ±  20 K for vortex freezing in Ba(Fe_0.93Rh_0.07)₂As₂ [54].

The structural landscape through which vortices move in Ba(Fe_1−xCo_x)₂As₂ is believed to be homogeneous in the sense that antiferromagnetism and superconductivity coexist without phase separation [3–8, 55]. However, while it may be single phase, local strain heterogeneities must exist on a unit cell scale due to the size difference of Co substituting for Fe [56]. The existence of local electronic inhomogeneity has already been discussed in the context of other measurements, including, for example, heat capacity and NMR line broadening [57–59], as well as magnetic dependence of critical current density [60]. This local strain heterogeneity is likely to influence the effective viscosity experienced by mobile vortices and the strength of pinning processes within the glass state.

Activation energies from Arrhenius and Vogel–Fulcher treatments of the Q⁻¹ and AC magnetic data (figure 7 and sections A.2 and A.7) all fall in the range ~50–600 K, which overlaps with the range of values estimated for the pinning of ferroelastic twin walls (400–750 K) at higher temperatures [20]. This is also the same energy scale as reported for resistivity and acoustic attenuation in other superconductors (e.g. [61]), including ~200 K for polaron-like conductivity in BaFe₂As₂ [62] and ~850 K for anelastic loss in YBCO [63]. It appears that the pinning/freezing processes are controlled by essentially the same activation barriers as the mobility of polarons, for which coupling with a local strain cloud provides the most likely constraint. The magnitude of the activation energy estimated from Q⁻¹ peaks 1–3 reduces with falling temperature (table A1), showing that there is a sequence of pinning/freezing mechanisms associated with progressively lower pinning potentials.

RUS data collected as a function of temperature at constant field are relatively featureless below the vortex liquid–vortex glass transition and the evolution of f ² for all resonances is fully reversible (figure 4). This implies that a fixed density of vortices is established in the liquid state and that their configuration does not then change in response to increasing or decreasing temperature. Increases in the height of the broad loss peak at progressively higher fields (figure 4(c)) can be understood simply in terms of the increase in the number of vortices present.

Significant variations in elastic properties occur when the field is increased at constant temperature below their freezing point, due to changes in both density and configuration. H_p from hysteresis loops (figure A5) can be understood as marking the field at which vortices nucleated at the surface of a crystal first penetrate to its centre [46]. Anomalies in f ² do not correlate exactly with these (figure 10(b)), but are sufficiently close to be understandable on the same basis. With increasing field, the density of vortices will increase but their distribution will be fixed by pinning points. At sufficiently high fields the glass transition point is exceeded and the vortices become mobile, but hysteresis in the elastic properties of the crystal as a whole below H_irr implies that they acquire a different configuration on refreezing.

H_2p has been interpreted as a change in pinning properties of the vortices [47] but there does not appear to be any equivalent anomaly in elastic or anelastic properties associated with this (figure 10(b)). The low stress experienced within a resonating crystal during an RUS experiment [64, 65] is presumably well below the critical stress that would be needed to unpin the vortices.

There is, perhaps, a slight decrease in the rate of elastic stiffening with falling temperature which correlates with the 95% resistivity limit (figures 7(a) and (b)), but there is no other obvious anomaly at this point. If so, the softening mechanism is likely to be analogous to the effect of polar nanoregions in relaxor ferroelectrics such as PbMg_1/3Nb_2/3O₃ where acoustic phonons interact with a dynamical local domain microstructure (e.g. [66]). Local dynamical regions of superconductivity would generate locally fluctuating strain fields which, in turn, would interact with acoustic phonons to give the slight softening.

Evidence from strain, heat capacity and AC magnetism in zero field is for a discrete phase transition ~2 K below the temperature at which resistivity reduces to 1% of its value in the normally conducting state (figure 8). This value is generally taken as marking the onset of superconductivity but it is not accompanied by any elastic anomaly beyond a slight acceleration of the softening trend. On the other hand, the line for 1% resistivity at high fields correlates closely with the Debye-like combination of elastic softening and acoustic loss referred to in figures 7 and 10 as peak 1. This is ~2 K ahead of the interval of vortex freezing, and data of Ni et al [30] have shown the same discrepancy systematically in other underdoped crystals. The onset of superconductivity and the occurrence of some pinning/freezing process responsible for the acoustic loss are within what is expected to be the stability field of vortex liquid. However, changes in magnetic properties which would indicate superconductivity throughout the bulk of the crystal only appear at lower temperatures and two obvious explanations are that an initial reduction in resistance takes place in surface layers only or that some percolative superconducting pathways develop through the crystal.

Given that the microstructure of Co-doped orthorhombic crystals in the underdoped range is of two sets of orthogonal twin walls, with a tweed like pattern on a scale of  <2 µm for individual domains [67], an interesting possibility is that the twin walls could provide the percolating pathways of superconductivity. This would be consistent with previous observations of enhanced superfluid density on twin walls below T_c [68, 69]. The associated loss peak would be understandable in terms of pinning or freezing of vortices close to the superconducting twin walls.

The lesson from ferroelastic materials with more than one order parameter is that twin walls can have structures and properties which are distinct from both the parent phase and the matrix in which they lie. A perovskite example of this is the development of ferrielectric dipoles at the ferroelastic twin walls in CaTiO₃ [70]. Linear-quadratic coupling generates new possibilities for a variety of exotic domain walls [71]. In the present case, there must be strain-mediated interactions between vortices and ferroelastic twin walls via all of the common strains e₁, e₃ and e₆. The evidence from a crystal with x  =  0.055 (figure 8(a)) is that, like e₆, the sign of linear strains which couple with the ferroelastic order parameter is the reverse of the linear strains with couple with the order parameter for superconductivity. Superconductivity should therefore be favoured in regions of the crystal where these strains are smallest, ie within the ferroelastic twin walls. Vortices will be favoured in regions where the ferroelastic strain is large, ie away from the ferroelastic twin walls. This strain argument is consistent also with the fact that vortices in Ba(Fe_1−xCo_x)₂As₂ are repelled from the twin walls [69] and that the twin walls influence the vortex glass transition itself at low fields [72].

In the context of ferroelastic properties, YBCO again provides a contrast in that vortices have been observed to decorate the twin walls [73–75] rather than being repelled from them. The implication is that the flux density will be reduced along the walls in comparison with the bulk and, according to the arguments presented here, this is likely to be a consequence of favourable strain coupling between the walls and the vortices. FeSe is more like Ba(Fe_1−xCo_x)₂As₂ than YBCO in that ferroelasticity and superconductivity are coupled. However, vortices accumulate on the twin walls and superconductivity along the walls is degraded [76].

This aspect of the pnictides remains to be fully explored but it must be expected that the superconducting properties of the ferroelastic twin walls could be engineered by choices of chemistry, temperature and field. With respect to the possibilities for domain wall engineering, the interesting question is whether it might be possible to produce a wide field of stability for a pnictide with 2D superconductivity on the twin walls.

In combination, the elasticity and magnetism data in figure 10(b) define two expected phase boundaries. Firstly, there is the onset of changes in resistivity. The line marking 90 or 95% of normal resistivity has previously been taken as an estimate for H_c2, (e.g. [60, 77–80]). When measured on the same crystal, values of H_c2 defined in terms of resistivity do not necessarily coincide with values defined in terms of magnetic susceptibility (e.g. [30, 78]). If this determination of H_c2 is correct, condensation of the mixed phase of mobile vortices in a superconducting matrix (vortex liquid) takes place between 5 and 10 K above the temperature at which the vortex glass transition occurs, as also reported by Shen et al [77] for a crystal with x  =  0.06. The 95% line is interpreted here as representing a diffuse boundary between normally conducting behaviour and the development of dynamical regions, with local superconductivity and/or mobile vortices, which couple with acoustic phonons sufficiently to cause an onset of elastic softening. Interpretation of the second boundary due to the vortex liquid–vortex glass transition, as defined by frequency dependent magnetic and acoustic loss peaks, is more secure. It is closely followed by changes in DC magnetic properties that mark the static limit of the freezing process.

A third boundary, seen in both magnetic and elastic properties at low fields within the stability field of the vortex glass, is interpreted as defining the limit with increasing field where vortices nucleated at the surface of the crystal penetrate through to its centre. However, the exact location of this boundary would be expected to vary with the thickness of the crystal and adjustment by a factor of 7.4 in the position of H_p in figure 10(b) has been made to allow for the difference in thickness of the crystals used for the magnetic and RUS measurements.

The physical origin of the fourth boundary, marked by the overlap of acoustic loss peak 1 and the line for 1% resistivity, is less obvious as it appears to be located in what would be expected to be the stability field for a vortex liquid. The divergence of these two lines below ~2 T suggests that the acoustic loss is related to the presence of vortices. As discussed above, the most interesting possibility is that it relates to the development of percolative pathways for superconductivity along ferroelastic twin walls. In order to test this, it will be necessary to investigate the elastic properties of pnictide crystals which remain tetragonal through the normal–superconducting transition and would not, therefore, contain ferroelastic twins. Otherwise, the superconducting pathway could be in the surface layers of the crystal.

Figure A1 shows data for the field dependence of f ² and Q⁻¹ at nine different temperatures from a resonance peak near 330 kHz, which predominantly reflects the variation of C₆₆. The variations become increasingly hysteretic as temperature is reduced.

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Peaks in Q⁻¹ and the accompanying increases in f ² with falling temperature appear to be consistent with Debye-like loss processes associated with freezing of defects. On this basis, the temperature dependence of the loss can be described by [81–84]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{Q}^{-1}}\left(T \right)=Q_{\rm max}^{-1}{{\left[{\rm cosh}\left\{\frac{{{E}_{{\rm a}}}}{{\rm R}{{r}_{2}}\left(\gamma \right)}\left(\frac{1}{T}-\frac{1}{{{T}_{\rm max}}} \right) \right\} \right]}^{-1}},\nonumber \end{align} \tag{ A.1 }$$

where the maximum value of Q⁻¹, $Q_{{\rm max}}^{-{\rm 1}}$, occurs at temperature T_max, E_a is an activation energy conforming to equation (A.8), below, and r₂(γ) is a width parameter. Values of r₂(γ) specify a spread of relaxation times, as set out in tables 4-2 of Nowick and Berry [41]. It is unity for a single relaxation time (γ  =  0), and increases as the spread increases.

Table A1 contains parameters obtained by fitting equation (A.1) to a selection of the acoustic loss peaks, as illustrated in figures 7(a) and (b). Values of E_a/R fall in the range ~250–600 K for the peak associated most closely with the normal-superconducting transition (peak 1), ~200–350 K for the peak associated with the vortex glass transition (peak 2) and ~100–200 K for peaks which fall at lower temperatures than these (peak 3). There are trends with external field and frequency but the dominant correlation is with values of T_max such that loss peaks at the highest temperatures have the highest effective thermal barriers, while those at the lowest temperature have the lowest barriers. Variations of Q⁻¹ for peak 1 at 7.5 T are sufficiently well constrained to give lnf _o  =  43.04  ±  0.24 (f _o  =  4.9  ×  10¹⁸ Hz) and E_a/R  =  353  ±  3 K from a conventional Arrhenius treatment (figures 7(b) and (c)). This compares with 596, 278 and 265 K (average 380 K) from fitting of the individual peaks at the three different frequencies. γ  =  1 or 2, for which r₂(γ)  =  1.26 or 1.74 [41], would give values of E_a/R extracted from the individual peaks that are 26 and 74% higher, respectively, which would take them increasingly out of the range of the conventional Arrhenius result. Thus, if there is a spread of relaxation times associated with the acoustic loss mechanism at the normal-superconducting transition it is only small, which is consistent with the result from the Havrilian–Negami treatment of AC magnetic data for vortex freezing, below. Values of f _o were obtained by inserting values of E_a/R into equation (A.8).

A formal treatment of selected spontaneous strains accompanying the phase transitions in Ba(Fe_1−xCo_x)₂As₂ at compositions near x  =  0.45 has been given elsewhere [20]. There are two non-symmetry breaking strains, e₁ (=e₂) and e₃, and a symmetry breaking shear strain, e₆. These are related to the structural/electronic order parameter, Q_E, as e₆ $\propto$ Q_E and e₁ $\propto$ e₃ $\propto$ $Q_{{\rm E}}^{2}$ due to coupling terms of the form λe₆Q_E, λe₁$Q_{{\rm E}}^{2}$ and λe₃$Q_{{\rm E}}^{2}$. The equivalent coupling terms for a scalar order parameter due to the superconducting transition, Q_SC, are λe_6,SC$Q_{{\rm SC}}^{2}$, λe_1,SC$Q_{{\rm SC}}^{2}$ and λe_3,SC$Q_{{\rm SC}}^{2}$, so the contributions to e₆, e₁ and e₃ below T_c, e_6,SC, e_1,SC, e_3,SC, are expected to scale with $Q_{{\rm SC}}^{2}$.

The conventional way of determining variations of these strains with temperature would be from lattice parameter data obtained by diffraction methods. However, data can also be extracted from changes in the linear dimensions of a single crystal with respect to a reference state at a temperature above T_S, according to

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{e}_{1}}=\frac{\Delta a}{a}-{{\left(\frac{\Delta a}{a} \right)}_{{\rm o}}}\nonumber \end{align} \tag{ A.2 }$$

and

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{e}_{3}}=\frac{\Delta c}{c}-{{\left(\frac{\Delta c}{c} \right)}_{{\rm o}}}.\nonumber \end{align} \tag{ A.3 }$$

Δa/a and Δc/c are the relative length changes within and perpendicular to the (0 0 1) plane, respectively, of a twinned single crystal of the orthorhombic phase. (Δa/a)_o, (Δc/c)_o are relative length changes of the tetragonal parent crystal extrapolated to temperatures below T_S. Changes in volume below T_S can be taken as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{\Delta V}{V}=2\frac{\Delta a}{a}+\frac{\Delta c}{c}.\nonumber \end{align} \tag{ A.4 }$$

The symmetry breaking shear strain, e₆, due to the tetragonal–orthorhombic transition is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{e}_{6}}\approx \frac{2\left(a-b \right)}{\left(a+b \right)},\nonumber \end{align} \tag{ A.5 }$$

where a and b are lattice parameters of the orthorhombic structure. Linear thermal expansion data for both Δa/a and Δb/b can be obtained by applying a detwinning shear stress to the crystal [85]. Values of e₆ are then given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{e}_{6}}=\left[\frac{\Delta a}{a}-{{\left(\frac{\Delta a}{a} \right)}_{o}} \right]-\left[\frac{\Delta b}{b}-{{\left(\frac{\Delta a}{a} \right)}_{o}} \right]=\frac{\Delta a}{a}-\frac{\Delta b}{b}.\nonumber \end{align} \tag{ A.6 }$$

As shown in figure 4 of Böhmer et al [85] for a different 122 phase, this reveals the form of the lattice distortion below T_S. The detwinning stress results in an additional contribution to e₆, which appears as a tail stretching to higher temperatures.

Variations of linear strains and the volume strain just above and through T_c are given for a crystal with x  =  0.055 in figure 8(a), based on primary data from Meingast et al [29]. Equivalent data are not available for crystals with x  =  0.043, apart from the variations of Δa/a and e₆ shown in figures A2(a) and (c), which were obtained using the techniques described in Böhmer et al [85] for a crystal taken from the same batch as used for the present study. In order to determine e₁ due to the superconducting transition alone, e_1,SC, values of (Δa/a)_o, have been obtained by fitting the function

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\left(\frac{\Delta a}{a} \right)}_{{\rm o}}}={{a}_{1}}+{{a}_{2}}{{\Theta}_{{\rm s}}}{\rm coth}\left(\frac{{{\Theta}_{{\rm s}}}}{T} \right)\nonumber \end{align} \tag{ A.7 }$$

to data in the temperature interval immediately above T_c and extrapolating to lower temperatures (figure A2(a)). Resulting changes in e_1,SC, which should scale with $Q_{{\rm S}}^{2}$, are shown in figure A2(b).

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A fit of equation (1) to the data for e_1,SC gives T_c  =  12.5  ±  ~0.1 K (figure 8(b)). The straight line fit to data for the DC magnetic moment measured at 0.002 T (section A.6, below) in the small temperature interval where they vary steeply extrapolates to zero at 12.6  ±  ~0.1 K. Not only do both sets of data give the same value of T_c, but they also have a tail that extends up to between ~13.5 and ~14.0 K (figure A2(b)). Figure A2(c) shows the variation of e₆ in the lowest temperature range, with a curve based on equation (A.1) fit to data in the interval 15–41 K and extrapolated down to 5 K. Values of shear strain due to the superconducting transition, e_6,SC, have been taken as the difference between the extrapolation and measured values. Variations of e_1,SC and e_6,SC with temperature are shown together in figure 8(b).

The magnitudes of strains associated with the superconducting transition are substantially smaller than those associated with the ferroelastic transition. For, example, by coupling with the ferroelastic order parameter e₁ reaches values of ~  −1.6  ×  10⁻⁴ (at x  =  0.047) and e₆ reaches values of ~2.7  ×  10⁻³ (at x  =  0.045) [20]. Below the superconducting transition, the change in e_1,SC for the crystal with x  =  0.043 is up to ~6  ×  10⁻⁶ (figure A2(b)) and the change in e_6,SC for is up to ~  −7  ×  10⁻⁵ (figure A2(c)). It follows that changes in elastic constants due to strain coupling are expected to be substantially smaller for the normal–superconducting transition than for the ferroelastic transition and that changes in C₁₁, C₁₂ and C₃₃ associated with the superconductivity are expected to be substantially smaller than those associated with C₆₆.

The heat capacity of Crystal 2 was measured using the heat capacity option of a Quantum Design PPMS. Data collected in 0.1 K steps through the range 8–18 K from two separate runs are shown in figure A3. An arbitrary polynomial function has been fit to data in the higher temperature range to highlight the small anomaly in the vicinity of 12.5 K. The form of the anomaly is closely similar to that shown elsewhere for Ba(Fe_1−xCo_x)₂As₂ crystals with ranges of composition between x  =  0.038 and 0.14 [18, 28, 30, 86–88]. It is in the form of a small, rounded step, with a slight tail to higher temperatures, consistent with expectations for a second order phase transition. The step, ΔC_p, shown at T_c  ≈  12.3 K is ~0.05 J mol⁻¹ K⁻¹. In the treatment of Ni et al [30], T_c was taken as the temperature at which the tail reached baseline values, as determined by comparing with data collected at 9 T, which would give a slightly higher value than the definition used here.

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Four electrical leads were attached to the surface of Crystal 2 for measurement of in-plane resistivity (ρ) using the Electrical Transport Option of a 14 T PPMS from Quantum Design, with an AC current of 1 mA at 18.1 Hz. The temperature sweep rate was 0.3 K min⁻¹, with steps of 0.15 K during heating and cooling through the temperature interval 2–20 K. These measurements were repeated in successively increasing magnetic field strengths of 0, 0.002, 1, 2.5, 5, 7.5, 10 and 12.5 T applied perpendicular to the large faces of the crystal (H//c). Results are shown in figure A4, and the temperatures listed in table A2 were determined by drawing horizontal lines at values of ρ corresponding to 1, 10, 90 and 95% of the resistivity of the normal phase, ρ_n. The difference in temperature between 10% and 90% of ρ_n in zero field is 2.6 K and remains less than 3 K at all fields, consistent with results reported for a crystal with x  =  0.1 by Yamamoto et al [60]. The zero-field value reported by Sefat et al [89] for a crystal also with x  =  0.1 was 0.6 K.

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Measurements of DC magnetic moment made on Crystal 2 over a wide temperature interval have been reported previously [20]. For the present study, a new set of measurements were made in a Quantum Design MPMS XL squid magnetometer with H//c. Cooling was in zero field followed by heating with the field on. The data shown in figure A5 are from heating sequences between 2 and 20 K in steps of 0.2 K at progressively higher fields (0, 0.002, 1, 2.5, 5, 7 T). These are shown as susceptibility, M/H, in figure A5(a) and were used to determine the two separate temperatures listed in table A3. The first is the temperature at which extrapolation of a linear segment of the variation of diamagnetic moment with temperature extrapolated to zero, T_Mextrap, and the second is the onset temperature for a diamagnetic response with falling temperature, T_Monset. A measure of the sharpness of the transition at low fields is provided by the difference between the temperatures at which the values of M/H are 10 and 90% of that of the superconducting phase at low temperatures. Here this value is 0.8 K, which is comparable with 0.5 K reported by Marsik et al [3] for a crystal with x  =  0.055.

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Variations of DC moment and resistivity in the vicinity of T_c are shown together in figure A5(b). Under the influence of external magnetic fields through the range 0.002–5 T there is a difference of ~2–3 K between the temperatures at which the resistivity tends to zero and the DC moment tends to its high temperature value, which is also close to zero.

Magnetic hysteresis loops (H//c) measured to  ±6.7 T are reproduced from Carpenter et al [20] in figure A5(c). They show the typical fishtail pattern of an unconventional superconductor. M–H loops collected at higher temperatures were virtually indistinguishable from those collected at 15 K. A weakly ferromagnetic component is present and is attributed here to local moments rather than the presence of a ferromagnetic impurity phase. Values of H_2p, the field at which the second peak occurs, and of H_irr, the field at which the top and bottom segments of each loop merge on increasing or decreasing field, are listed in table A4. Crystal 2 was used again for further hysteresis measurements at 7 and 9.5 K, with a focus on the initial magnetization using field steps of 0.0071 T in order to obtain estimates for the position of the first peak, H_p, as shown in figure A5(d). Values of H_p, H_2p and H_irr estimated from all the hysteresis loops are given in table A4. H_irr values for 5 and 7 K fall outside the range of field strengths attainable in the magnetometer and have been estimated by extrapolation.

As previously described [20], AC magnetic properties were measured from Crystal 2 using the AC Measurement System option on a Quantum Design PPMS instrument. The data presented here were collected with H//c in static DC fields of 0.002–7.5 T and with an amplitude of 3 Oe for the driving AC field. The AC frequencies were 0.01, 0.1, 1, 10 kHz, with heating in steps of 0.2 K between 2 and 20 K. Additional measurements were made at 10, 10.5, 11 and 12.5 T in a PPMS instrument with a larger magnet. Variations of the real and imaginary components of the magnetic susceptibility, χ' and χ'', are illustrated in figures A6(a) and (b) for DC fields of 0.002 and 10 T. The same anomalies were not seen at 10.5, 11 or 12.5 T, but the possibilities of experimental issues being responsible for the difference were not explored.

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χ' and χ'' exhibit forms of variation which are typical of what might be expected for a Debye relaxation process, and each peak in χ'' is therefore assumed to occur at a temperature, ${{T}_{{\rm max}{{\chi}^{\prime\prime}}}}$' where the applied frequency, ω, and relaxation time, τ, are related by ωτ  =  1. Values of T_max obtained by fits of an asymmetric Lorentzian function to all the data for χ'' are similar to other reports for pinning or freezing of vortices in Ba(Fe_1−xCo_x)₂As₂ [43, 44] and (Ba_1−xK_x)Fe₂As₂ [45]. They display a frequency dependence which reduces significantly at the lowest applied DC fields (figure A6(c)).

The Debye loss process being sampled by the AC data can be represented in the simplest manner as a thermally activated (Arrhenius) pinning process according to

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle f={{f}_{{\rm o}}}{\rm exp}\left(-\frac{{{E}_{{\rm a}}}}{{\rm R}{{T}_{{\rm max}}}} \right),\nonumber \end{align} \tag{ A.8 }$$

or in terms of a glassy (Vogel–Fulcher) freezing process according to

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle f={{f}_{{\rm o}}}\exp \left(-\frac{{{E}_{{\rm a}}}}{{\rm R}\left({{T}_{{\rm max}}}-{{T}_{{\rm VF}}} \right)} \right),\nonumber \end{align} \tag{ A.9 }$$

T_max is the temperature at which the peak in χ'' occurs, f  is the applied frequency, f _o is an attempt frequency, E_a is an activation energy, R is the gas constant and T_VF is the Vogel–Fulcher (zero frequency) freezing temperature. An Arrhenius plot of lnf  versus 1/${{T}_{{\rm max}{{\chi}^{\prime\prime}}}}$ gives values of E_a/R and lnf _o which have a strong dependence on field (figures A7(a) and (b)). The results are very similar to those obtained for (Ba_1−xK_x)Fe₂As₂ by Nikolo et al [45] but values of f _o  >  10³⁹ do not make much physical sense. On the other hand, a Vogel–Fulcher plot, using values of T_VF taken to be T_Mextrap from the DC magnetic data above, gives E_a/R values of ~10–70 K (E_a ~ 1–6 meV) and lnf _o in a narrower range of 25–35 (f _o ~ 10¹¹–10¹⁵ Hz), which are physically much more reasonable (figures A7(c) and (d)) and in good agreement with f _o ~ 10¹³–10¹⁵ Hz for the glass transition model presented by Prando et al [44] of vortex freezing in Ba(Fe_0.9Co_0.1)₂As₂. Bossoni et al [54] obtained a similar fit to data for vortex freezing in Ba(Fe_0.93Rh_0.07)₂As₂, with E_a/R  =  120  ±  20 K.

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If there is only one characteristic relaxation time, the Debye equations give a semicircle for the relationship between for χ' and χ'' collected as a function of frequency at constant temperature. The Cole–Cole plot for all the data, collected as a function of temperature at constant frequency and at different field strengths in this case, define a single, slightly flattened and asymmetric curve with some slight scatter (figure A6(d)). This pattern is normally represented by the Havrilian–Negami equation

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\chi}^{\prime *}}\left(\omega \right)=\chi _{\infty}^{\prime}+\frac{\chi _{0}^{\prime}-\chi _{\infty}^{\prime}}{{{\left(1+{{\left({\rm i}\omega \tau \right)}^{\alpha}} \right)}^{\beta}}}\quad \quad 0<\alpha <1,\,\beta <1\nonumber \end{align} \tag{ A.10 }$$

where ${{\chi}^{\prime *}}(\omega)$ is the complex susceptibility, $\chi _{0}^{\prime}$ and $\chi _{\infty}^{\prime}$ the real components measured at zero and infinite frequency, α is a broadening parameter and β is a skew parameter. Debye equations correspond to α  =  1, β  =  1. The formal treatment applies strictly only for data collected as a function of frequency, but a graphical treatment of the data in figure A6(d) (following figure 1 of [90]) gives values of α ~ 0.9, β ~ 0.7 which can be taken as some effective average and as being indicative of a small spread of relaxation times. This result is consistent with the report of a narrow distribution of correlation times in Ba(Fe_0.9Co_0.1)₂As₂ by Prando et al [44], based on frequency dependencies.

Results from the magnetic and resistivity data for Crystal 2 have been combined in figure A8. Their general form is similar to what would be expected from other reports in the literature for Ba(Fe_1−xCo_x)₂As₂ (e.g. [30, 47, 60, 77, 91]), except that values of H_irr from the magnetic hysteresis loops collected in three separate experimental runs are somewhat erratic. Centering of the sample in the magnetometer may have been an issue and identifying H_irr as the point of convergence between increasing and decreasing fields in M–H loops is subject to significant uncertainty. It is also possible that the thermal history of the sample is important, particularly in relation to the configuration and density of twin walls, which need not be the same after each cycle through the tetragonal–orthorhombic transition. The observed values of H_irr all fall between or close to the values of T_Mextrap and T_Monset for the DC magnetic moment measured as a function of temperature, as expected if the development of a diamagnetic moment and the vortex freezing process are closely related.

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The location of the second peak, H_2p, is considered to mark a change in pinning properties of the vortices (e.g. [31, 47, 91]) and the first peak, H_p, is considered to mark the field at which the magnetic flux lines penetrate all the way from the surface to the core of the crystal [46]. The locus of H_2p in figure A8 is at lower fields than previously reported for crystals with x between 0.07 and 0.1 [47, 60, 91], as expected given that these have higher values of T_c. H_p is expected to vary with thickness and, as a first approximation, has been taken here to scale with the thickness of the crystal such that it will have values which are a factor of 0.35/0.047  =  7.4 greater for Crystal 1 than Crystal 2.

Temperatures for the loci of 1% ρ_n and 95% ρ_n fall above the temperatures where well defined anomalies in magnetic properties occur. These temperatures, or slight variants on how they are defined, have previously been taken as estimates of H_irr and H_c2, respectively (e.g. [59, 77–80]).