Hole doping dependences of the magnetic penetration depth and vortex core size in YBa{}_{2}Cu{}_{3}O{}_{y}: Evidence for stripe correlations near 1/8 hole doping

Hole doping dependences of the magnetic penetration depth and vortex core size in YBa2Cu3Oy: Evidence for stripe correlations near 1/8 hole doping

Abstract

We report on muon spin rotation (SR) measurements of the internal magnetic field distribution in the vortex solid phase of YBaCuO (YBCO) single crystals, from which we have simultaneously determined the hole doping dependences of the in-plane Ginzburg-Landau (GL) length scales in the underdoped regime. We find that has a sublinear dependence on , where is the in-plane magnetic penetration depth in the extrapolated limits and . The power coefficient of the sublinear dependence is close to that determined in severely underdoped YBCO thin films, indicating that the same relationship between and the superfluid density is maintained throughout the underdoped regime. The GL coherence length (vortex core size) is found to increase with decreasing hole doping concentration, and exhibit a field dependence that is explained by proximity-induced superconductivity on the CuO chains. Both and are enhanced near 1/8 hole doping, supporting the belief by some that stripe correlations are a universal property of high- cuprates.

pacs:
74.25.Qt,74.72.Bk,76.75.+i

I Introduction

Abrikosov vortices in a superconductor are governed by two characteristic length scales. The core of a vortex has a size dependent on the Ginzburg-Landau (GL) coherence length , while the supercurrents circulating around the core decay on the scale of the GL penetration depth . In the early days of high- superconductivity it was common practice to infer the behavior of the in-plane magnetic penetration depth from measurements of the muon depolarization rate in the vortex state of polycrystalline samples. The temperature dependence of was found to be consistent with -wave pairing symmetry (1); (2); (3); (4) and a universal linear scaling of with was observed in the underdoped regime (the so-called ‘Uemura plot’), indicating that , where is the superfluid density.(5); (6) Later, microwave (7) and SR (8) measurements on YBaCuO (YBCO) single crystals in the Meissner and vortex phases established a limiting low-temperature linear dependence of that is consistent with -wave pairing. More recently, non-SR studies of YBCO in the Meissner phase have revealed that has a sublinear dependence on .(9); (10); (11) On the other hand, the relation inferred from the Uemura plot is supported by a recent study of electric-field induced superfluid density modulations in a single underdoped ultra-thin film of LaSrCuO.(12)

The problem with assuming is that there are additional inseparable contributions to from electronic magnetic moments and flux-line lattice (FLL) disorder, which may vary with doping. To circumvent this difficulty we have studied YBCO single crystals. In a single crystal the FLL contribution to the SR line shape is asymmetric and distinct from the other sources of field inhomogeneity.(13) Not only can the behavior of be isolated, but because the finite size of the vortex cores is apparent in a single-crystal measurement of , can be simultaneously determined. While may be accurately determined in conventional superconductors from measurements of the upper critical field , in high- cuprates is generally a very high magnetic field marking the transition from a vortex liquid to the normal phase. Here we present SR measurements that probe and in the bulk of YBCO single crystals deep in the superconducting state. The accuracy of our method was demonstrated in previous studies of conventional superconductors,(14); (15); (16) but is reinforced here through comparisons with the results from other techniques.

Ii Experimental Details

YBCO single crystals with purity of 99.995 % were grown by a self-flux method in fabricated BaZrO crucibles at the University of British Columbia.(17) An exception are single crystals that were grown in yttria-stabilized-zirconia crucibles and characterized by a purity greater than 99.5 %.(18) Single crystals of Ca-doped YBCO were also prepared in BaZrO crucibles. Typical sample sizes consisted of 3 to 5 single crystals from the same growth batch arranged in a mosaic to form a total - surface area of 20-30 mm. The thickness of the crystals are on the order of  mm. The superconducting transition temperatures of the single crystals were measured using a SQUID magnetometer. Twin boundaries were removed from some of the higher-doped single crystals by applying pressure along the or directions at elevated temperature. These basic sample characteristics are summarized in Table 1.

(K) Detwinned
6.60 0.103 62.5
6.57 0.110 59.0
6.67 0.120 66.0
6.75 0.132 74.6
6.80 0.140 84.5
6.95 0.172 93.2
(Y,Ca)6.98 0.192 86.0
Table 1: Characteristics of the YBaCuO and (Y, Ca)BaCuO single crystals. The hole concentration per CuO layer is determined from the dependence of on presented in Ref. (19) for similar YBCO single crystals.

The SR experiments were performed over a 3 year period on the M15 and M20B surface muon beam lines at TRIUMF, Vancouver, Canada. The SR spectra were recorded in a transverse-field (TF) geometry with the applied magnetic field H perpendicular to the initial muon-spin polarization direction, and perpendicular to the -axis of the single crystals. A TF-SR spectrum comprised of 20 to 30 million muon decay events was taken at each temperature and magnetic field.

In a transverse-field, the muon spin precesses in a plane perpendicular to the field direction (which we define here as the -direction). The time evolution of the muon spin polarization is determined from the SR “asymmetry” spectrum formed from the muon decay events detected in opposing positron counters

(1)

where is the time histogram of the temporal dependence of decay positron count rate in the i detector, and is the asymmetry maximum. In our experiments four positron counters were used to completely cover the 360 solid angle in the - plane (see Fig. 1). The muon spin polarization function for the “Left”-“Right” pair of detectors is defined as

(2)

and for the “Up”-“Down” pair as

(3)

where is the muon gyromagnetic ratio, is a phase constant, and

(4)

is the probabilty of finding a local magnetic field in the -direction at an arbitrary position r in the - plane.

Figure 1: (Color online) Schematic of the positron counter arrangement used in the present study. The muon beam axis and the applied magnetic field are perpendicular to the page (i.e. parallel to the -axis). The muon spin precesses in the - plane about the local -component of the magnetic field before undergoing the decay . In our study, the muon spin polarization is formed from approximately 20 to 30 million muon decay events. The decay events detected in the “Left” and “Right” positron counters form the -component of , while decay events detected in the “Up” and “Down” positron counters form the -component.

Iii Data Analysis Method

The TF-SR time spectra for each sample were fit assuming the following analytical solution of the GL equations (20) for the spatial field profile of the ideal FLL

(5)

where G are the reciprocal lattice vectors, is the average internal magnetic field, is a cutoff function for the G sum, is a modified Bessel function, and . The cutoff function depends on the spatial profile of the superconducting order parameter at the center of the vortex core. Consequently, is a measure of the vortex core size. The FLL in all samples was assumed to be hexagonal. Neutron scattering experiments on fully-doped YBCO (21) indicate that the FLL below kOe is only slightly distorted from hexagonal symmetry due to anisotropy . We find that accounting for this small distortion changes the values of and by less than 5 %. Consequently, the FLL was assumed to be hexagonal for all samples studied.

Figure 2: (Color online) SR line shapes in single-crystal YBCO. Fourier transforms of the TF-SR signal from and samples at  K and  kOe (green circles). The righthand peak for is a background signal coming from muons stopping outside the sample (Note, the background and sample peaks nearly coincide in the SR line shape for ). The red curves through the data are the Fourier transforms of the fits in the time domain. The contour plots show the corresponding spatial dependence of the supercurrent density , providing a visual illustration of the change in core size with hole doping.

To avoid the difficulty of modelling the contribution of electronic magnetic moments to the SR line shape, we restricted our study to YBCO crystals free of static or quasistatic spins. For the applied fields considered in this study, this has been determined to be the case for oxygen content .(22)

To properly account for disorder of the FLL, the dimensionality of the vortices must be considered. Josephson plasma resonance measurements on YBaCuO ortho-II single crystals grown by Liang, Bonn and Hardy indicate that the vortices are 3D-like at low temperaturesi,(23) while mutual inductance measurements on thin films by Zuev et al show that even severely underdoped YBCO is quasi-2D only near .(10) Since the focus in the present study is on the variation of and in higher doped samples at low temperatures, the vortices are assumed to be rigid 3D lines of flux. This assumption is also consistent with the observation of highly asymmetric SR line shapes for all of our samples at low (see Fig. 2). For rigid flux lines, random displacements of the vortices from their positions in the ideal hexagonal FLL are accounted for by convoluting the theoretical line shape by a Gaussian distribution of fields.(24) A Gaussian function also describes the local distribution of dipolar fields originating from static nuclear moments.(25) Taking into account both sources of line broadening, the corresponding theoretical polarization functions are

(6)
(7)

where,

(8)

is an effective depolarization rate due to nuclear dipole moments () and FLL disorder (). Values for were obtained by fitting the TF-SR signal above to the theoretical polarization function

(9)

To account for the background signal from muons that did not stop in the sample, an additional term of the form was added to Eq. (7) and to Eq. (9), where is the fraction of muons that stopped inside the sample. Values of for the different samples ranged from 0.8 to 0.9.

In the present work there are two marked improvements over the analysis done in our previous studies of YBCO single crystals (26); (27); (28) that assumed the spatial field profile of Eq. (5): (i) The earlier works used the asymptotic limit () for the Bessel function that appears in the cutoff , whereas here was evaluated numerically. (ii) The second improvement is that due to increased computer speed, was calculated at 15,132 equally-spaced locations in the rhombic unit cell of the hexagonal FLL, compared to 5,628 locations in previous works. Further increasing the number of real-space points sampled in the FLL unit cell did not result in appreciable changes in the fitted parameters. We note that both improvements in our data analysis method influence the absolute values of and , but the temperature and magnetic field dependences of these parameters remain qualitatively similar to that determined in our previous studies.

Iv Results for

iv.1 Temperature dependence

Figure 3 shows the temperature dependence of at low determined at two different values of the applied magnetic field. The solid curves are fits to , where and are field-dependent coefficients. The dependence of on is shown in Fig. 4 for selected values of the applied field. An inflection point at  K is visible in some of the lower field data. This feature was also apparent in our previous measurements of YBaCuO.(28) Harshman et al have argued that the inflection point is caused by thermal depinning of vortices,(29) although an invalid treatment of the data was used to support this assertion.(30) Recently, Khasanov et al have ruled out depinning as the source of a similar inflection point in the temperature dependence of measured in LaSrCuO by TF-SR.(31) Instead they attribute this feature to the occurrence of both a large -wave and a small -wave superconducting gap. As we will show later, the anomalous magnetic field dependence of the vortex core size in YBCO can be explained by an induced superconducting energy gap in the CuO chains that run along the direction. Theoretical calculations by Atkinson and Carbotte (36) for a -wave superconductor with proximity-induced superconductivity in the CuO chains, show that exhibits an inflection point caused by an upturn of at low (where is the penetration depth in the direction).

Figure 3: (Color online) Temperature dependence of at (a)  kOe and (b)  kOe. The solid curves are fits to , where and are field-dependent coefficients.

Figure 4: (Color online) Temperature dependence of at  kOe. The insets show additional low temperature measurements at  kOe or  kOe. The solid curves through the data points are merely guides for the eye.

Figure 5: (Color online) Temperature dependence of at  kOe, normalized to and plotted as a function of reduced temperature . The solid line through the data is a fit given by .

In Fig. 5 it is shown that exhibits a near universal linear temperature dependence at low . We attribute deviations from universal behavior near to softening of the FLL, which narrows the SR line shape and enhances the fitted value of . The universal scaling implies that is a constant,(32) where is the Fermi velocity, is a velocity corresponding to the slope of the gap at the nodes, and is a charge renormalization parameterizing the coupling of the quasiparticles to phase fluctuations. Using values of from thermal conductivity measurements,(33) we find that is bascially doping independent.

iv.2 Magnetic field dependence

Figure 6 shows the magnetic field dependences of . Here we stress that the observed behaviors do not imply that the magnetic penetration depth or superfluid density depend on field in this way. The sublinear dependence of on is primarily due to the failure of Eq. (5) to account for all field-dependent contributions to the internal magnetic field distribution. In Refs. (28); (34) the strong field dependence of in YBaCuO determined by SR was explained by the high anisotropy of the -wave superconducting energy gap not accounted for in Eq. (5). A nonlocal supercurrent response to the applied field in the vicinity of the vortex cores stemming from the divergence of the coherence length at the gap nodes modifies the spatial distribution of field. With increasing , the increased overlap of the regions around the vortex cores reduces the width of the SR line shape. The gap anisotropy also results in a nonlinear supercurrent response to the applied field, resulting from a quasiclassical ‘Doppler shift’ of the quasiparticle energy spectrum by the flow of superfluid around a vortex.(35) When the Doppler shift exceeds the energy gap, Cooper pairs are broken, and increases.

These effects are not restricted to -wave superconductors. Sizeable nonlinear and/or nonlocal effects can also occur in -wave superconductors with a smaller energy gap on one of the Fermi sheets and/or a highly anisotropic Fermi surface. Moreover, these anisotropies result in a rapid delocalization of quasiparticle core states with increasing that modify . Indeed, strong field dependences of from Eq. (5) have been observed in the multi-band superconductor NbSe,(15) and the marginal type-II superconductor V.(16) It has been experimentally established for a variety of materials,(15) including YBCO,(9) that the extrapolated value of agrees with the magnetic penetration depth measured by other techniques in the Meissner phase. Consequently, we stress that only can be considered a “true” measure of the magnetic penetration depth.

Figure 6: (Color online) Magnetic field dependence of the extrapolated zero-temperature value of . The solid curves are fits to , where the coefficient decreases with increasing hole doping.

iv.3 Hole doping dependence

Figure 7: (Color online) Dependence of on . The two data points on the far right are for the overdoped sample (Y,Ca)BaCuO. The solid curves are fits to the data for YBaCuO, yielding  Km and  Km at  kOe and  kOe, respectively.

In Fig. 7 we show as a function of at two different fields. The more inclusive data set at  kOe is described by , which deviates substantially from the linear scaling in the Uemura plot. The power 0.38 is surprisingly close to 0.43 determined by Zuev et al in a Meissner phase study of severely underdoped YBCO thin films.(10) It is surprising because these thin films have a superfluid density that is significantly lower than in single crystals.(11) It is known from microwave studies of YBCO that the doping dependences of and are not the same,(9) due to the conductivity of the CuO chains.(36) Thus there is no reason to expect the power to be universal for the cuprates. While a sublinear dependence of on has also been inferred from more recent SR measurements of the muon depolarization rate in other high- superconductors,(37) the contributions of magnetism and FLL disorder to the SR line shape were not factored out.

Figure 8: (Color online) Dependence of on hole doping concentration at kOe. The dashed curve is the function , where is the critical hole doping concentration for the onset of superconductivity.

Figure 8 shows as a function of hole doping, where the values of are determined from the dependence of on presented in Ref. (19) for similar YBCO single crystals. The behavior is consistent with other studies on cuprates indicating that the maximum value of is not reached before .(38); (39) Our data are in the range and are described by , except near .

Figure 9: (Color online) (a) Hole doping dependences of the Gaussian depolarization rates and at K and kOe. (b) Hole doping dependence of the root-mean-square displacement of the vortices from their positions in the perfect hexagonal FLL at K and kOe plotted as a percentage of the intervortex spacing .

The hole doping dependences of and at kOe are shown in Fig. 9(a). While is independent of , basically tracks . Using the fitted values of and we have calculated the hole doping dependence of the root-mean-square displacement of the vortices from there positions in the perfect hexagonal FLL. As shown in Fig. 9(b), the degree of FLL disorder is small and as expected highest in the Ca-doped sample.

V Results for

v.1 Magnetic field dependence

The field dependences of are shown in Fig. 10. The increase in at low field, which corresponds to an expansion of the vortex cores, was previously reported for YBaCuO (Ref. (26)) and YBaCuO (Refs. (27); (28)). In all samples we find that scales as , which is proportional to the intervortex spacing. In -wave superconductors the field dependence of has been shown to originate from the delocalization of bound quasiparticle core states.(15); (14) This is because a change in the spatial dependence of the pair potential accompanies the change in electronic structure of the vortex cores. In YBCO the low-energy quasiparticle core states should be extended along the nodal directions of the -wave gap function.(40) This allows for a large transfer of low-energy quasiparticles between vortices at low field, which is further enhanced by an increase in vortex density. Hence the vortex core size is predicted to shrink with increasing field.(41); (42) However, the field dependence of in YBCO is considerably stronger than predicted for a pure -wave superconductor. Consequently, we consider an alternative explanation.

Figure 10: Magnetic field dependence of the extrapolated zero-temperature value of . The solid curves are fits to , where the coefficients and depend on the hole doping concentration.

Figure 11: Calculated field-dependent core size relative to the BCS coherence length for single-layer and bilayer models (see Appendix). Also shown is the field dependence of in the sample normalized to the extrapolated value  Å  and plotted as a function of , where .

In Fig. 11 we show that the field-dependence of , in particular the upturn at low field, can be explained by the presence of the CuO chains. The calculations of the core size are based on a semiclassical Doppler-shift approximation (see Appendix) for either a single-layer model representing a superconducting CuO plane or a proximity-coupled model representing a CuO-CuO bilayer. In the bilayer model there are two distinct energy scales for pair breaking: the energy gap associated with Cooper pairs in the CuO planes, and a smaller proximity-induced gap associated with the chains. It is the latter scale which is responsible for the expansion of the vortex cores at low field.

Figure 12: Hole doping dependence of at kOe and kOe. Also shown is data for from Ref. (43) plotted as a function of , rather than .

Figure 13: Plot of as a function of hole doping (red circles), using the values of at kOe. Also shown are the values of (black circles). The black curve is the relation between and from Ref. (19).

v.2 Hole doping dependence

Figure 12 shows the hole doping dependences of at  kOe and  kOe. Qualitatively, the doping dependence of is similar to that reported by Ando et al from magnetoconductance measurements on detwinned YBCO single crystals.(43) This result is also shown in Fig. 12, but plotted as versus . Note that our data must be plotted as versus , because the hole doping concentration of our single crystals (grown in a different kind of crucible than the other samples), is smaller than that of (see Table 1). The general trend of all data sets is an increase of with decreasing . Such behavior has also been observed in the underdoped regime of LaSrCuO,(44); (45); (46) and BiSrCuO.(47) With increasing magnetic field, our values for approach those determined by Ando et al. Note that based on our proximity-induced model for the field dependence of (see Appendix), it is the high-field values of that reflect the intrinsic superconductivity of the CuO planes.

Since the doping dependences of at kOe and kOe in Fig. 12 are similar, we expect the hole doping dependence of to qualitatively resemble that of the upper critical field . Figure 13 shows the hole doping dependence of calculated from the values of at kOe. Consistent with the data of Ando et al, decreases with decreasing in the underdoped regime of YBCO, and displays a dip near 1/8 hole doping.

Vi Summary and Conclusions

We have simultaneously determined the hole doping dependences of the magnetic penetration depth and the GL coherence length in the underdoped regime of YBaCuO. This was achieved by fitting SR measurements in the vortex state to an analytical solution of the GL equations for the internal magnetic field distribution. In this type of analysis the magnetic penetration depth is strictly defined as the extrapolated value of the fitted parameter . The accuracy of this definition was established in previous studies of conventional superconductors.(15); (16) Here we have presented measurements showing a refinement of the Uemura plot for YBaCuO, where is plotted as a function of the isolated quantity , rather than the the muon depolarization rate . We find that exhibits a strong sublinear dependence on , suggesting that is not directly proportional to the superfluid density . This result supports the same conclusion reached in several recent Meissner phase studies of YBCO.

We have reported here a reduction of near 1/8 hole doping concentration. Suppression of or the muon depolarization rate near 1/8 hole doping has only previously been observed in cuprates where is controlled by cation substitution,(38); (39) and was believed to indicate a tendency toward static stripe formation. Static stripes over the doping range investigated here were recently ruled out by inelastic neutron scattering experiments on YBCO.(48) We ourselves find no evidence for static spins in zero-field SR or TF-SR measurements on our samples.(22) However, the suppression of superconductivity near could be caused by fluctuating stripes, recently argued to be relevant in YBCO and other cuprates.(49) Experimental evidence for dynamic stripes in YBCO includes the detection of low-energy one-dimensional incommensurate modulations in YBaCuO by inelastic neutron scattering.(50)

Further evidence for suppression of superconductivity near is found in the hole doping dependence of . In our measurements is a parameter that characterizes the size of the vortex cores. While it mimics the behavior of the GL coherence length, is large at low field due to the contribution of the CuO chains to the spatial dependence of the superconducting order parameter. Enhancement of the GL coherence length or vortex core size near 1/8 hole doping, has also been observed in LaSrCuO.(44) Calculations by Mierzejewski and Máska show that static or quasistatic stripes actually intensify by reducing diamagnetic pair breaking,(51) and hence cannot explain the growth of near . On the other hand, Kadono et al have shown that an expansion of the vortex cores with decreasing hole doping can result from a strengthening of antiferromagnetic correlations competing with superconductivity.(45) Thus, dynamic stripes are a viable explanation for the increased size of the vortex cores near 1/8 hole doping.

Acknowledgements

We gratefully acknowledge D. Broun, I. Vekhter and A. J. Millis for helpful and informative discussions. We also thank Y. Ando for allowing us to reproduce his data here. This work was supported by the Canadian Institute for Advanced Research and the Natural Sciences and Engineering Research Council (NSERC) of Canada.

Appendix: Semiclassical Calculation of the Vortex Core Size

Calculations of the vortex core size are based on a generalisation of the so-called “doppler-shift” approximation for the vortex structure (52); (53) to the case of YBCO, which is a multiband superconductor. In the case of YBCO, there is strong evidence that both the two-dimensional CuO planes and one-dimensional CuO chains superconduct. Furthermore, it is likely that the chains are intrinsically normal, but are driven superconducting by the proximity effect. Proximity models for YBCO have been extensively described elsewhere (36); (54). The essential idea is that the superconductivity originates from a pairing interaction which is confined to the two-dimensional CuO planes, and that the mixing of chain and plane wavefunctions induces superconductivity in the one-dimensional chain layers.

We adopt a simplified bilayer model consisting of a single plane and a single chain, with one Wannier orbital retained per unit cell for each layer. For comparison purposes, calculations are also performed for a single-layer model of an isolated superconducting plane. The Bogoliubov-deGennes Hamiltonian for the bilayer is

(10)

where and () are creation operators for quasiparticles (quasiholes) at lattice site in layer . Here, we take for the plane layer and for the chain layer. The parameters and are the single-electron hopping matrix elments between sites and within and between layers respectively, while is the superconducting order parameter along bonds connecting nearest neighbour sites and . From the form of Eq. (10), it is apparent that only couples quasiparticles belonging to the plane layer. The single-layer Hamiltonian is obtained by setting .

A magnetic field applied perpendicular to the layers induces circulating currents in the superfluid. The superfluid velocity is given by where is the effective mass tensor, is the superfluid momentum, is the magnetic vector potential and is the local phase of the order parameter. In the limit that and are slowly varying functions, one can treat the superflow as uniform in the neighbourhood of . Then, one can make a local gauge transformation (52); (53) such that the phase is removed from the order parameter and appears instead in the hopping matrix elements :

(11)
(12)

where are the hopping matrix elements in zero-field and are the matrix elements of the zero-field quasiparticle velocity. Equation (12) follows from Eq. (11) in the limit that is small. Then, the order parameter takes on the simple -wave form which, in reciprocal space, corresponds to , where is the lattice constant. The local gauge transformation leads to a doppler-shifted spectrum and is exact in the limit of slowly varying superfluid velocity. This procedure has been shown, in many circumstances, to provide a reasonable description of the vortex lattice (52); (53).

We take band structures which are appropriate for YBCO and adopt

(13)

and

(14)

For this work, we measure energies in units of and take . In reciprocal space, the dispersions of the isolated plane and chain layers are then and , respectively. The chain-plane hopping matrix element is not well known in YBCO and is taken to be .

For a slowly varying we can locally Fourier transform the Hamiltonian in the neighbourhood of to give

(15)

where and where is the number of -points in the sum in Eq. (15).

We need to make an ansatz for . For a single vortex in an isotropic medium, one has , where is the azimuthal unit vector and the radius is measured relative to the centre of the vortex (52). For the bilayer model, however, is not isotropic: the chains provide a conduction channel along the direction which is in parallel with the isotropic plane conduction channel. We mimic this anisotropy by assuming that the superfluid momentum will be similar to that of a single-layer superconductor with an anisotropic (diagonal) effective mass tensor with . (For the single-layer model, we take .) We then have two requirements which must be satisfied: and . The first requirement introduces vortex cores at the vortex lattice sites , while the latter incompressibility requirement is strictly true in regions where is uniform. This pair of equations is solved by

(16)

where indicates that is excluded from the sum, are reciprocal lattice vectors of the magnetic unit cell (we assume a square lattice here) with area and magnetic length . The results do not depend strongly on the ratio , which we take to be 0.6 for the parameters chosen above. This choice minimizes , where is the total (plane and chain) current in the bilayer,

(17)

and indicates the expectation value with respect to , Eq. (15). In principle, one could improve on the approximation of Eq. (16) by determining self-consistently from ; however this will not change the qualitative physics of the vortex core expansion.

We then solve self-consistently for the order parameter

(18)

with . Self-consistent solutions find that vanishes near the vortex core centre and obtains an asymptotic value far from the vortex core. In order to measure the vortex core size, we define a quantity . The vortex core size is then defined by the first moment of the radial coordinate with respect to :

(19)

where corresponds to the vortex core centre. For presentation purposes, is shown relative to the BCS coherence length , where is the average of the Fermi velocity on the Fermi surface. The magnetic field is related to the magnetic length by where is the superconducting flux quantum. For presentation purposes, is scaled by the upper critical field, , so that .

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