**1. Introduction**

The SC gap, which characterizes the energy cost for breaking a Cooper pair, is an important quantity when clarifying the SC mechanism. The gap size and its momentum dependence reflect the strength and anisotropy of the pairing interactions, respectively. Some experiment executed by Li *et al*. [1] in response to a suggestion by Klemm [2] tested the phase of the wave function in Bi2Sr2CaCu2O8 and revived the *s*-wave viewpoint [3, 4]], which, although championed by Dynes's group [4], had been out of favor even for Bi2Sr2CaCu2O8, although not disproven. This experiment once more created uncertainty over whether the superconducting pairs are consistent with *s*-wave or *d*-wave superconductivity (Van Harlingen [5], Ginsberg [6], Tsuei and Kirtley [7]).

The discovery of Fe-based superconductors [8] generated intensive debate on the superconducting (SC) mechanism. Motivated by high-*Tc* values up to 56 K [9], the possibility of unconventional superconductivity has been intensively discussed. A plausible candidate is the SC pairing mediated by antiferromagnetic (AFM) interactions. Two different approaches, based on the itinerant spin fluctuations promoted by Fermi-surface (FS) nesting [10, 11], and the local AFM exchange couplings [12], predict the so-called *s*±-wave pairing state, in which the gap shows a *s*-wave symmetry that changes sign between different FSs. Owing to the multiorbital nature and the characteristic crystal symmetry of Fe-based superconductors, *s*++-wave pairing without sign reversal originating from novel orbital fluctuations has also been proposed [13, 14]. The unconventional nature of the superconductivity is supported by experimental observations such as strongly FS-dependent anomalously large SC gaps [15–17] and the possible sign change in the gap function [18, 19] on moderately doped BaFe2As2, NdFeAsO and FeTe1−*x*Se*x*. However, a resonance like peak structure, observed by neutron scattering measurements [18], is reproduced by considering the strong correlation effect via quasiparticle damping, without the necessity of sign reversal in the SC gap [20]. Although the *s*±-wave state is expected to be very fragile as regards impurities due to the interband scattering [21], the superconducting state is remarkably robust regarding impurities and *α*-particle irradiation [22].

©2012 Belova et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2012 Belova et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

There is growing evidence that the superconducting gap structure is not universal in the iron-based superconductors [23, 24]. In certain materials, such as optimally doped BaKFe2As2 and BaFeCo2As2, strong evidence for a fully gapped superconducting state has been observed from several low-energy quasiparticle excitation probes, including magnetic penetration depth [25, 26], and thermal conductivity measurements [27]. In contrast, significant excitations at low temperatures due to nodes in the energy gap have been detected in several Fe-pnictide superconductors. These include LaFePO (*Tc* = 6 K) [28, 29], BaFe2AsP2 (*Tc* = 31 K) [30–32], and KFe2As2 (*Tc* = 4 K) [33, 34].

of up to 20 GPa [49]. Resistivity and susceptibility as well as *μ*-spin rotation experiments show no evidence of magnetic transition [50, 51]. Only a weak magnetic background [51] and field induced magnetism in the doped compound have been detected [50]. What was identified was a notable absence of the Fermi surface nesting, a strong renormalization of the conduction bands by a factor of 3, a high density of states at the Fermi level caused by a van Hove singularity, and no evidence of either a static or a fluctuating order; although superconductivity with in-plane isotropic energy gaps have been found implying the *s*++ pairing state [52]. However, a gap anisotropy along the Fermi surface up to ∼ 30% was observed in Ref. [53]. Thus, the type of the superconducting gap symmetry in LiFeAs is still

Eilenberger Approach to the Vortex State in Iron Pnictide Superconductors 201

The aim of our paper is to apply quasiclassical Eilenberger approach to the vortex state considering *<sup>s</sup>*±, *<sup>s</sup>*++ and *dx*<sup>2</sup>−*y*<sup>2</sup> -wave pairing symmetries as presumable states for the different levels of impurity scattering rates Γ∗, to calculate the cutoff parameter *ξ<sup>h</sup>* [54, 55] and to compare results with experimental data for iron pnictides. As described in Ref. [56], *ξ<sup>h</sup>* is important for the description of the muon spin rotation (*μ*SR) experiments and can be

The London model used for the analysis of the experimental data does not account for the spatial dependence of the superconducting order parameter and it fails down at distances of the order of coherence length from the vortex core center, *i*.*e*., *B*(*r*) logarithmically diverges as *r* → 0. To correct this, the **G** sum in the expression for the vortex lattice free energy can be truncated by multiplying each term by a cutoff function *F*(*G*). Here, **G** is a reciprocal vortex lattice vector. In this method the sum is cut off at high *Gmax* ≈ 2*π*/*ξh*, where *ξ<sup>h</sup>* is the cutoff parameter. The characteristic length *ξ<sup>h</sup>* accommodates a number of inherent uncertainties of the London approach; the question was discussed originally by de Gennes group [57] and discussed in some detail in Ref. [58]. It is important to stress that the appropriate form of *F*(*G*) depends on the precise spatial dependence of the order parameter in the the vortex core

A smooth Gaussian cutoff factor *<sup>F</sup>*(*G*) = *exp*(−*αG*2*ξ*2) was phenomenologically suggested. Here, *ξ* is the Gizburg-Landau coherence length. If there is no dependence of the superconducting coherence length on temperature and magnetic field, then changes in the spatial dependence of the order parameter around a vortex correspond to changes in *α*. By solving the Ginzburg-Landau (GL) equations, Brandt determined that *α* = 1/2 at fields near *Bc*<sup>2</sup> [59], and arbitarily determined it to be *α* ≈ 2 at fields immediately above *Bc*<sup>1</sup> [60]. For an isolated vortex in an isotropic extreme (the GL parameter *κGL* � 1) *s*-wave superconductor, *α* was obtained by numerical calculation of GL equations. It was found that *α* decreases smoothly from *α* = 1 at *Bc*<sup>1</sup> to *α* ≈ 0.2 at *Bc*<sup>2</sup> [61]. The analytical GL expression was obtained by [62] for isotropic superconductors at low inductions *B* � *Bc*2. Using a Lorentzian trial function for the order parameter of an isolated vortex, Clem found for large *κGL* � 1 that *F*(*G*) is proportional to the modified Bessel function. In Ref. [63], the Clem model [62] was extended to larger magnetic fields up to *Bc*<sup>2</sup> through the linear superposition of the field profiles of individual vortices. In this model, the Clem trial function [62] is multiplied by a second variational parameter *f*∞ to account for the suppression of the order parameter due to the overlapping of vortex cores. This model gave the method for calculating the magnetization of type-II superconductors in the full range *Bc*<sup>1</sup> < *B* < *Bc*2. Their analytical formula is in a good agreement with the well-known Abrikosov high-field result and considerably corrects the results obtained with an exponential cutoff function at

region, and this, in general, depends on the temperature and the magnetic field.

an open question.

directly measured.

At a very early stage, it was realized that electron and hole doping can have qualitatively different effects in the pnictides [35]. Hole doping should increase the propensity to a nodeless (*s*±) SC phase. The qualitative picture applies to both the "122" as the "1111" compounds: As the Fermi level is lowered, the *M h* pocket becomes more relevant and the *M* ↔ *X* scattering adds to the (*π*, 0)/(0, *π*) scattering from Γ to *X*. As such, the anisotropy-driving scattering, such as interelectron pocket scattering, becomes less relevant and yields a nodeless, less anisotropic, and more stable *s*± [36]. This picture is qualitatively confirmed by experiments. While thermoelectric, transport, and specific heat measurements have been performed for <sup>K</sup>*x*Ba1−*<sup>x</sup>*Fe2As2 from *<sup>x</sup>* = 0 to the strongly hole-doped case *<sup>x</sup>* = 1 [37, 38], more detailed studies have previously focused on the optimally doped case *x* = 0.4 with *Tc* = 37 K, where all measurements such as penetration depth and thermal conductivity find indication for a moderately anisotropic nodeless gap [39, 40]. Similarly, angle-resolved photoemission spectroscopy (ARPES) on doped BaFe2As2 reveals a nodeless SC gap [16, 41].

The experimental findings for the SC phase in KFe2As2 were surprising. Thermal conductivity [33], penetration depth [34], and NMR [42] provide a clear indication of nodal SC. The critical temperature for KFe2As2 is ∼ 3 K, an order of magnitude less than the optimally doped samples. ARPES measurements [43] show that the *e* pockets have nearly disappeared, while the *h* pockets at the folded Γ point are large and have a linear dimension close to *π*/*a*. A detailed picture of how the SC phase evolves under hole doping in K*x*Ba1−*<sup>x</sup>*Fe2As2 was found and that the nodal phase observed for *x* = 1 is of the (extended) *d*-wave type [44]. The functional renormalization group was used to investigate how the SC form factor evolves under doping from the nodeless anisotropic *s*± in the moderately hole-doped regime to a *d*-wave in the strongly hole-doped regime, where the *e* pockets are assumed to be gapped out. The *d*-wave SC minimizes the on-pocket hole interaction energy. It was found that the critical divergence scale to be of an order of magnitude lower than for the optimally doped *s*± scenario, which is consistent with experimental evidence [44].

The synthesis of another iron superconductor immediately attracted much attention for several reasons [9, 45]. LiFeAs is one of the few superconductors which does not require additional charge carriers and is characterized by *Tc* approaching the boiling point of hydrogen. Similar to AeFe2As2 ( Ae = Ba, Sr, Ca "122") and LnOFeAs ("1111") parent compounds, LiFeAs (*Tc* = 18 K) consists of nearly identical (Fe2As2) <sup>2</sup><sup>−</sup> structural units and all three are isoelectronic, though the former do not superconduct. The band structure calculations unanimously yield the same shapes for the FS, as well as very similar densities of states, and low energy electronic dispersions [46, 47]. Moreover the calculations even find in LiFeAs an energetically favorable magnetic solution which exactly corresponds to the famous stripelike antiferromagnetic order in "122" and "1111" systems [46, 48]. The experiments, however, show a rather different situation. The structural transition peculiar to "122" and "1111" families is remarkably absent in LiFeAs and is not observed under an applied pressure of up to 20 GPa [49]. Resistivity and susceptibility as well as *μ*-spin rotation experiments show no evidence of magnetic transition [50, 51]. Only a weak magnetic background [51] and field induced magnetism in the doped compound have been detected [50]. What was identified was a notable absence of the Fermi surface nesting, a strong renormalization of the conduction bands by a factor of 3, a high density of states at the Fermi level caused by a van Hove singularity, and no evidence of either a static or a fluctuating order; although superconductivity with in-plane isotropic energy gaps have been found implying the *s*++ pairing state [52]. However, a gap anisotropy along the Fermi surface up to ∼ 30% was observed in Ref. [53]. Thus, the type of the superconducting gap symmetry in LiFeAs is still an open question.

2 Will-be-set-by-IN-TECH

There is growing evidence that the superconducting gap structure is not universal in the iron-based superconductors [23, 24]. In certain materials, such as optimally doped BaKFe2As2 and BaFeCo2As2, strong evidence for a fully gapped superconducting state has been observed from several low-energy quasiparticle excitation probes, including magnetic penetration depth [25, 26], and thermal conductivity measurements [27]. In contrast, significant excitations at low temperatures due to nodes in the energy gap have been detected in several Fe-pnictide superconductors. These include LaFePO (*Tc* = 6 K) [28, 29], BaFe2AsP2 (*Tc* = 31

At a very early stage, it was realized that electron and hole doping can have qualitatively different effects in the pnictides [35]. Hole doping should increase the propensity to a nodeless (*s*±) SC phase. The qualitative picture applies to both the "122" as the "1111" compounds: As the Fermi level is lowered, the *M h* pocket becomes more relevant and the *M* ↔ *X* scattering adds to the (*π*, 0)/(0, *π*) scattering from Γ to *X*. As such, the anisotropy-driving scattering, such as interelectron pocket scattering, becomes less relevant and yields a nodeless, less anisotropic, and more stable *s*± [36]. This picture is qualitatively confirmed by experiments. While thermoelectric, transport, and specific heat measurements have been performed for <sup>K</sup>*x*Ba1−*<sup>x</sup>*Fe2As2 from *<sup>x</sup>* = 0 to the strongly hole-doped case *<sup>x</sup>* = 1 [37, 38], more detailed studies have previously focused on the optimally doped case *x* = 0.4 with *Tc* = 37 K, where all measurements such as penetration depth and thermal conductivity find indication for a moderately anisotropic nodeless gap [39, 40]. Similarly, angle-resolved photoemission

The experimental findings for the SC phase in KFe2As2 were surprising. Thermal conductivity [33], penetration depth [34], and NMR [42] provide a clear indication of nodal SC. The critical temperature for KFe2As2 is ∼ 3 K, an order of magnitude less than the optimally doped samples. ARPES measurements [43] show that the *e* pockets have nearly disappeared, while the *h* pockets at the folded Γ point are large and have a linear dimension close to *π*/*a*. A detailed picture of how the SC phase evolves under hole doping in K*x*Ba1−*<sup>x</sup>*Fe2As2 was found and that the nodal phase observed for *x* = 1 is of the (extended) *d*-wave type [44]. The functional renormalization group was used to investigate how the SC form factor evolves under doping from the nodeless anisotropic *s*± in the moderately hole-doped regime to a *d*-wave in the strongly hole-doped regime, where the *e* pockets are assumed to be gapped out. The *d*-wave SC minimizes the on-pocket hole interaction energy. It was found that the critical divergence scale to be of an order of magnitude lower than for the optimally doped *s*±

The synthesis of another iron superconductor immediately attracted much attention for several reasons [9, 45]. LiFeAs is one of the few superconductors which does not require additional charge carriers and is characterized by *Tc* approaching the boiling point of hydrogen. Similar to AeFe2As2 ( Ae = Ba, Sr, Ca "122") and LnOFeAs ("1111") parent

and all three are isoelectronic, though the former do not superconduct. The band structure calculations unanimously yield the same shapes for the FS, as well as very similar densities of states, and low energy electronic dispersions [46, 47]. Moreover the calculations even find in LiFeAs an energetically favorable magnetic solution which exactly corresponds to the famous stripelike antiferromagnetic order in "122" and "1111" systems [46, 48]. The experiments, however, show a rather different situation. The structural transition peculiar to "122" and "1111" families is remarkably absent in LiFeAs and is not observed under an applied pressure

<sup>2</sup><sup>−</sup> structural units

spectroscopy (ARPES) on doped BaFe2As2 reveals a nodeless SC gap [16, 41].

scenario, which is consistent with experimental evidence [44].

compounds, LiFeAs (*Tc* = 18 K) consists of nearly identical (Fe2As2)

K) [30–32], and KFe2As2 (*Tc* = 4 K) [33, 34].

The aim of our paper is to apply quasiclassical Eilenberger approach to the vortex state considering *<sup>s</sup>*±, *<sup>s</sup>*++ and *dx*<sup>2</sup>−*y*<sup>2</sup> -wave pairing symmetries as presumable states for the different levels of impurity scattering rates Γ∗, to calculate the cutoff parameter *ξ<sup>h</sup>* [54, 55] and to compare results with experimental data for iron pnictides. As described in Ref. [56], *ξ<sup>h</sup>* is important for the description of the muon spin rotation (*μ*SR) experiments and can be directly measured.

The London model used for the analysis of the experimental data does not account for the spatial dependence of the superconducting order parameter and it fails down at distances of the order of coherence length from the vortex core center, *i*.*e*., *B*(*r*) logarithmically diverges as *r* → 0. To correct this, the **G** sum in the expression for the vortex lattice free energy can be truncated by multiplying each term by a cutoff function *F*(*G*). Here, **G** is a reciprocal vortex lattice vector. In this method the sum is cut off at high *Gmax* ≈ 2*π*/*ξh*, where *ξ<sup>h</sup>* is the cutoff parameter. The characteristic length *ξ<sup>h</sup>* accommodates a number of inherent uncertainties of the London approach; the question was discussed originally by de Gennes group [57] and discussed in some detail in Ref. [58]. It is important to stress that the appropriate form of *F*(*G*) depends on the precise spatial dependence of the order parameter in the the vortex core region, and this, in general, depends on the temperature and the magnetic field.

A smooth Gaussian cutoff factor *<sup>F</sup>*(*G*) = *exp*(−*αG*2*ξ*2) was phenomenologically suggested. Here, *ξ* is the Gizburg-Landau coherence length. If there is no dependence of the superconducting coherence length on temperature and magnetic field, then changes in the spatial dependence of the order parameter around a vortex correspond to changes in *α*. By solving the Ginzburg-Landau (GL) equations, Brandt determined that *α* = 1/2 at fields near *Bc*<sup>2</sup> [59], and arbitarily determined it to be *α* ≈ 2 at fields immediately above *Bc*<sup>1</sup> [60]. For an isolated vortex in an isotropic extreme (the GL parameter *κGL* � 1) *s*-wave superconductor, *α* was obtained by numerical calculation of GL equations. It was found that *α* decreases smoothly from *α* = 1 at *Bc*<sup>1</sup> to *α* ≈ 0.2 at *Bc*<sup>2</sup> [61]. The analytical GL expression was obtained by [62] for isotropic superconductors at low inductions *B* � *Bc*2. Using a Lorentzian trial function for the order parameter of an isolated vortex, Clem found for large *κGL* � 1 that *F*(*G*) is proportional to the modified Bessel function. In Ref. [63], the Clem model [62] was extended to larger magnetic fields up to *Bc*<sup>2</sup> through the linear superposition of the field profiles of individual vortices. In this model, the Clem trial function [62] is multiplied by a second variational parameter *f*∞ to account for the suppression of the order parameter due to the overlapping of vortex cores. This model gave the method for calculating the magnetization of type-II superconductors in the full range *Bc*<sup>1</sup> < *B* < *Bc*2. Their analytical formula is in a good agreement with the well-known Abrikosov high-field result and considerably corrects the results obtained with an exponential cutoff function at

#### 4 Will-be-set-by-IN-TECH 202 Superconductors – Materials, Properties and Applications Eilenberger Approach to the Vortex State in Iron Pnictide Superconductors <sup>5</sup>

low fields [64]. This approximation was widely used for the analysis of the experimental data on magnetization of type-II superconductors (see references 27-29 in Ref. [65]). The improved approximate Ginzburg-Landau solution for the regular flux-line lattice using circular cell method was obtained in Ref. [65]. This solution gives better correlation with the numerical solution of GL equations.

The Ginzburg-Landau theory, strictly speaking, is only valid near *Tc* but it is often used in the whole temperature range taking the cutoff parameter *ξ<sup>h</sup>* and penetration depth *λ* as a fitting parameters. Recently, an effective London model with the effective cutoff parameter *ξh*(*B*) as a fitting parameter was obtained for clean [54] and dirty [55] superconductors, using self-consistent solution of quasiclassical nonlinear Eilenberger equations. In this approach, *λ* is not a fitting parameter but calculated from the microscopical theory of the Meissner state. As was shown in Ref. [66], the reduction of the amount of the fitting parameters to one, considerably simplifies the fitting procedure. In this method, the cutoff parameter obtained from the Ginzburg-Landau model was extended over the whole field and temperature ranges. In this case, the effects of the bound states in the vortex cores lead to the Kramer-Pesch effect [67], i.e. delocalization between the vortices [68, 69], nonlocal electrodynamic [58] and nonlinear effects [70] being self-consistently included.

Following the microscopical Eilenberger theory, *ξ<sup>h</sup>* can be found from the fitting of the calculated magnetic field distribution *hE*(**r**) to the Eilenberger - Hao-Clem (EHC) field distribution *hEHC*(**r**) [54, 55]

$$h\_{EH\mathbf{C}}(\mathbf{r}) = \frac{\Phi\_0}{\mathbf{S}} \sum\_{\mathbf{G}} \frac{F(\mathbf{G})e^{i\mathbf{G}\mathbf{r}}}{1 + \lambda^2 G^2} \tag{1}$$

where

$$F(\mathbf{G}) = \mathfrak{u}K\_1(\mathfrak{u}),\tag{2}$$

**Figure 1.** (Color online) The temperature dependence of superfluid density *ρS*(*T*)/*ρS*<sup>0</sup> at (a) interband scattering rate Γ*<sup>π</sup>* = 0 with different values of intraband scattering Γ<sup>0</sup> and (b) intraband scattering rate

Fermi surface, i.e. a two dimensional limit of the five-band model [74]. In Eq. (1) *λ*(*T*) is the

Δ¯ 2 *n*

*<sup>n</sup>* + *ω*<sup>2</sup>

*VSC*Δ¯ *<sup>n</sup>* Δ¯ <sup>2</sup>

<sup>1</sup> <sup>+</sup> *ab* , *<sup>g</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> *ab*

*<sup>n</sup>* + *ω*<sup>2</sup> *n*

*<sup>F</sup>N*0)1/2 is the London penetration depth at *<sup>T</sup>* <sup>=</sup> 0 including

<sup>2</sup> and <sup>Γ</sup>*<sup>π</sup>* <sup>=</sup> *<sup>π</sup>niNF*|*uπ*<sup>|</sup>

Eilenberger Approach to the Vortex State in Iron Pnictide Superconductors 203

*<sup>n</sup>* for the *s*<sup>±</sup> pairing and Δ¯ *<sup>n</sup>* = Δ(*T*) for the

. (4)

*<sup>L</sup>*0/*λ*2(*T*), with different values of

<sup>1</sup> <sup>+</sup> *ab* , (5)

*<sup>n</sup>*)3/2 , (3)

<sup>2</sup> are the intra-

*ηn*(Δ¯ <sup>2</sup>

*ωn*>0

the Fermi velocity *vF* and the density of states *N*<sup>0</sup> at the Fermi surface and *η<sup>n</sup>* = 1 +

and interband impurity scattering rates, respectively (*u*0,*<sup>π</sup>* are impurity scattering amplitudes with correspondingly small, or close to *π* = (*π*,*π*), momentum transfer). In this work, we investigate the field distribution in the vortex lattice by systematically changing the impurity concentration in the Born approximation, and analyzing the field dependence of the cutoff parameter. In particular, we consider two limits: small Γ<sup>∗</sup> � 1 (referred to as the "stoichiometric" case) and relatively high Γ<sup>∗</sup> ≥ 1 ("nonstoichiometric" case). Here, Γ<sup>∗</sup> is measured in the units of 2*πTc*0. We consider Γ∗ as intraband scattering Γ<sup>0</sup> with constant

penetration depth in the Meissner state. In this model *λ*(*T*) is given as

*<sup>λ</sup>*2(*T*) <sup>=</sup> <sup>2</sup>*π<sup>T</sup>* ∑

*<sup>n</sup>*). Here, Γ<sup>0</sup> = *πniNF*|*u*0|

Δ¯ <sup>2</sup>

Δ(*T*) = 2*πT* ∑

<sup>1</sup> <sup>+</sup> *ab* , *<sup>f</sup>* † <sup>=</sup> <sup>2</sup>*<sup>b</sup>*

dependence of the superfluid density *ρS*(*T*)/*ρS*<sup>0</sup> = *λ*<sup>2</sup>

¯ *<sup>f</sup>* <sup>=</sup> <sup>2</sup>*<sup>a</sup>* *<sup>n</sup>* + *ω*<sup>2</sup>

*s*++ pairing symmetry. The order parameter Δ(*T*) in Meissner state is determined by the

0<*ωn*<*ω<sup>c</sup>*

Experimentally, *λ*(*T*) can be obtained by radio-frequency measurements [75] and magnetization measurements of nanoparticles [76]. Fig. 1 shows the calculated temperature

impurity scattering Γ for *s*±-wave pairing symmetry. With the Riccati transformation of the Eilenberger equations, quasiclassical Green functions *f* and *g* can be parameterized via

*λ*2 *L*0

Γ<sup>0</sup> = 3 with different values of interband scattering Γ*π*.

where *λL*<sup>0</sup> = (*c*2/4*πe*2*v*<sup>2</sup>

interband scattering Γ*<sup>π</sup>* = 0.

self-consistent equation

functions *a* and *b* [77]

In Eq. (3), <sup>Δ</sup>¯ *<sup>n</sup>* <sup>=</sup> <sup>Δ</sup>(*T*) <sup>−</sup> <sup>4</sup>*π*Γ*π*Δ¯ *<sup>n</sup>*/

Δ¯ <sup>2</sup>

*<sup>n</sup>* + *ω*<sup>2</sup>

2*π*(Γ<sup>0</sup> + Γ*π*)/(

where *K*1(*u*) is modified Bessel function, *u* = *ξhG* and *S* is the area of the vortex lattice unit cell. It is important to note that *ξ<sup>h</sup>* in Eq. (1) is obtained from solving the Eilenberger equations and does not coincide with the variational parameter *ξv* of the analytical Ginzburg-Landau (AGL) model.

In **chapter 2** and **<sup>3</sup>** we solve the Eilenberger equations for *<sup>s</sup>*±, *<sup>s</sup>*++ and *dx*<sup>2</sup>−*y*<sup>2</sup> -wave pairing symmetries, fit the solution to Eq. (1) and find the cutoff parameter *ξh*. In this approach all nonlinear and nonlocal effects connected with vortex core and extended quasiclassical states are described by one effective cutoff parameter *ξh*. The nonlocal generalized London equation with separated quasiclassical states was also developed as regards the description of the mixed state in high-*Tc* superconductors such as YBa2Cu3O7−*<sup>δ</sup>* compounds (the Amin-Franz-Affleck (AFA) model) [70, 71]. In this case, fourfold anisotropy arises from *d*-wave pairing. This theory was applied to the investigation of the flux line lattice (FLL) structures [72] and effective penetration depth measured by *μ*SR experiments [73]. This approach will be considered in **chapter 4**.
