**2. The cutoff parameter for the field distribution in the mixed states of** *s*± **and** *s*++**-wave pairing symmetries**

In this chapter, we consider the model of the iron pnictides, where the Fermi surface is approximated by two cylindrical pockets centered at Γ (hole) and M (electron) points of the

4 Will-be-set-by-IN-TECH

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

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

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

> *<sup>S</sup>* ∑ **G**

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

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

**2. The cutoff parameter for the field distribution in the mixed states of** *s*±**-**

In this chapter, we consider the model of the iron pnictides, where the Fermi surface is approximated by two cylindrical pockets centered at Γ (hole) and M (electron) points of the

*F*(*G*)*ei***Gr**

<sup>1</sup> <sup>+</sup> *<sup>λ</sup>*2*G*<sup>2</sup> , (1)

*F*(**G**) = *uK*1(*u*), (2)

*hEHC*(**r**) = <sup>Φ</sup><sup>0</sup>

solution of GL equations.

distribution *hEHC*(**r**) [54, 55]

approach will be considered in **chapter 4**.

**and** *s*++**-wave pairing symmetries**

where

(AGL) model.

nonlinear effects [70] being self-consistently included.

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

Fermi surface, i.e. a two dimensional limit of the five-band model [74]. In Eq. (1) *λ*(*T*) is the penetration depth in the Meissner state. In this model *λ*(*T*) is given as

$$\frac{\lambda\_{L0}^2}{\lambda^2(T)} = 2\pi T \sum\_{\omega\_n > 0} \frac{\bar{\Delta}\_n^2}{\eta\_n (\bar{\Delta}\_n^2 + \omega\_n^2)^{3/2}},\tag{3}$$

where *λL*<sup>0</sup> = (*c*2/4*πe*2*v*<sup>2</sup> *<sup>F</sup>N*0)1/2 is the London penetration depth at *<sup>T</sup>* <sup>=</sup> 0 including the Fermi velocity *vF* and the density of states *N*<sup>0</sup> at the Fermi surface and *η<sup>n</sup>* = 1 + 2*π*(Γ<sup>0</sup> + Γ*π*)/( Δ¯ <sup>2</sup> *<sup>n</sup>* + *ω*<sup>2</sup> *<sup>n</sup>*). Here, Γ<sup>0</sup> = *πniNF*|*u*0| <sup>2</sup> and <sup>Γ</sup>*<sup>π</sup>* <sup>=</sup> *<sup>π</sup>niNF*|*uπ*<sup>|</sup> <sup>2</sup> are the intraand 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 interband scattering Γ*<sup>π</sup>* = 0.

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> *<sup>n</sup>* for the *s*<sup>±</sup> pairing and Δ¯ *<sup>n</sup>* = Δ(*T*) for the *s*++ pairing symmetry. The order parameter Δ(*T*) in Meissner state is determined by the self-consistent equation

$$\Delta(T) = 2\pi T \sum\_{0 < \omega\_n < \omega\_\varepsilon} \frac{V^{SC} \bar{\Delta}\_n}{\sqrt{\bar{\Delta}\_n^2 + \omega\_n^2}}. \tag{4}$$

Experimentally, *λ*(*T*) can be obtained by radio-frequency measurements [75] and magnetization measurements of nanoparticles [76]. Fig. 1 shows the calculated temperature dependence of the superfluid density *ρS*(*T*)/*ρS*<sup>0</sup> = *λ*<sup>2</sup> *<sup>L</sup>*0/*λ*2(*T*), with different values of 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 functions *a* and *b* [77]

$$f = \frac{2a}{1 + ab'} \quad f^\dagger = \frac{2b}{1 + ab'} \quad g = \frac{1 - ab}{1 + ab'} \tag{5}$$

**Figure 2.** (Color online) (a) The temperature dependence of the upper critical field *Bc*<sup>2</sup> at interband scattering Γ*<sup>π</sup>* = 0 with different values of intraband scattering values Γ0. (b) The calculated temperature dependence of *Bc*<sup>2</sup> at intraband scattering rate Γ<sup>0</sup> = 3 with different values of interband scattering Γ*π*.

satisfying the nonlinear Riccati equations. In Born approximation for impurity scattering we have

$$\mathbf{u} \cdot \nabla a = -a \left[ 2(\omega\_{\mathrm{fl}} + G) + i \mathbf{u} \cdot \mathbf{A}\_{\mathrm{s}} \right] + (\Delta + F) - a^2 (\Delta^\* + F^\*), \tag{6}$$

$$\mathbf{u} \cdot \nabla b = b \left[ 2(\omega\_{\mathrm{il}} + \mathbb{G}) + i \mathbf{u} \cdot \mathbf{A}\_{\mathrm{s}} \right] - (\Delta^\* + F^\*) + b^2 (\Delta + F), \tag{7}$$

where *ω<sup>n</sup>* = *πT*(2*n* + 1), *G* = 2*π* � *g* �(Γ<sup>0</sup> + Γ*π*) ≡ 2*π* � *g* �Γ∗, *F* = 2*π* � *f* �(Γ<sup>0</sup> − Γ*π*) for *s*<sup>±</sup> pairing symmetry and *F* = 2*π* � *f* �Γ<sup>∗</sup> for the *s*++ pairing symmetry. Here, **u** is a unit vector of the Fermi velocity. In the new gauge vector-potential **A***<sup>s</sup>* = **A** − ∇*φ* is proportional to the superfluid velocity. It diverges as 1/*r* at the vortex center (index *s* is put to denote its singular nature). The FLL creates the anisotropy of the electron spectrum. Therefore, the impurity renormalization correction in Eqs. (6) and (7), averaged over the Fermi surface, can be reduced to averages over the polar angle *θ*, i.e. �...� = (1/2*π*) ... *dθ*.

To take into account the influence of screening the vector potential **A**(**r**) in Eqs. (6) and (7) is obtained from the equation

$$
\nabla \times \nabla \times \mathbf{A}\_{\mathbf{E}} = \frac{4}{\kappa^2} \mathbf{J}\_{\prime} \tag{8}
$$

The self-consistent condition for the pairing potential Δ(**r**) in the vortex state is given by

*ω<sup>c</sup>* ∑ *ωn*>0

where *VSC* is the coupling constant and *ω<sup>c</sup>* is the ultraviolet cutoff determining *Tc*<sup>0</sup> [55]. Consistently throughout our paper energy, temperature, and length are measured in units of *Tc*<sup>0</sup> and the coherence length *ξ*<sup>0</sup> = *vF*/*Tc*0, where *vF* is the Fermi velocity. The magnetic

To obtain the quasiclassical Green function, the Riccati equations [Eq. (6, 7)] are solved by the Fast Fourier Transform (FFT) method for triangular FLL [55]. This method is reasonable for the dense FLL, discussed in this paper. In the high field the pinning effects are weak and they are not considered in our paper. To study the high field regime we needed to calculate the upper critical field *Bc*2(*T*). This was found from using the similarity of the considered model

[*ω*−<sup>1</sup>

*<sup>D</sup>*1(*ωn*, *Bc*2) = *<sup>J</sup>*(*ωn*, *Bc*2) <sup>×</sup> [<sup>1</sup> <sup>−</sup> <sup>2</sup>(Γ<sup>0</sup> <sup>−</sup> <sup>Γ</sup>*π*)*J*(*ωn*, *Bc*2)]<sup>−</sup>1, (13)

*dy* exp (−*y*) arctan [

)(<sup>1</sup> <sup>+</sup> *<sup>b</sup>*4)1/2[<sup>1</sup> <sup>−</sup> <sup>2</sup>*b*(<sup>1</sup> <sup>−</sup> *<sup>b</sup>*)2]

calculations the ratio *κ* = *λL*0/*ξ*<sup>0</sup> = 10 is used. It corresponds to *κGL* = 43.3 [77].

 2*π* 0

*dθ*

<sup>2</sup>*<sup>π</sup> <sup>f</sup>*(*ωn*, *<sup>θ</sup>*,**r**), (11)

<sup>0</sup>. The impurity scattering rates are in units of 2*πTc*0. In

Eilenberger Approach to the Vortex State in Iron Pnictide Superconductors 205

*<sup>n</sup>* − 2*D*1(*ωn*, *Bc*2)], (12)

(*Bc*2*y*)1/2

*<sup>α</sup>* ], (14)

1/2. (15)

Δ(**r**) = *VSC*2*πT*

to the model of spin-flip superconductors from the equations [78]

*πBc*<sup>2</sup> ) 1/2 <sup>×</sup>

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

*n*≥0

 ∞ 0

Fig. 2 shows *Bc*2(*T*) dependences at (a) Γ*<sup>π</sup>* = 0, Γ<sup>0</sup> = 0, 1, 2, 3, 4, 5, 6 and (b) Γ<sup>0</sup> = 3, Γ*<sup>π</sup>* = 0.01, 0.02, 0.03, 0.04, 0.05, 0.06 calculated from Eqs. (12-14). In Fig. 2 the different influence of the intraband and interband scattering on *Bc*2(*T*) dependence can be seen. The *Bc*2(*T*) curve increases with Γ<sup>0</sup> (*ξc*<sup>2</sup> decreases with Γ0), but Γ*<sup>π</sup>* results in decreasing *Bc*2(*T*) (increasing of

Fig. 3 (a) shows magnetic field dependence *ξh*(*B*) in reduced units at *T*/*Tc*<sup>0</sup> = 0.5 for the *s*<sup>±</sup> pairing with Γ<sup>0</sup> = 3, Γ*<sup>π</sup>* = 0.02 and Γ<sup>0</sup> = 0.5, Γ*<sup>π</sup>* = 0.03 and "clean" case (solid lines) and for the *s*++ pairing with Γ∗= 0.5 and Γ∗ = 3 (dotted lines). The dashed line shows the analytical

This dependence with *ξc*<sup>2</sup> as a fitting parameter is often used for the description of the experimental *μ*SR results [56, 79]. As can be seen from Fig. 3 (a), the magnetic field dependence of *ξh*/*ξc*<sup>2</sup> is nonuniversal because it depends not only on *B*/*Bc*<sup>2</sup> (as in the AGL theory, dashed line in Fig. 3 (a)), but also on interband and intraband impurity scattering parameters. In the cases where Γ<sup>0</sup> = Γ*<sup>π</sup>* = 0, the results are the same for *s*± and *s*++ pairing symmetries. We indicated that this curve is "clean" one. In this figure, the case Γ<sup>0</sup> � Γ*<sup>π</sup>* is considered

ln( *Tc*<sup>0</sup>

*<sup>J</sup>*(*ωn*, *Bc*2)=( <sup>4</sup>

*ξ<sup>v</sup>* = *ξc*2(

√

<sup>2</sup> <sup>−</sup> 0.75 *κGL*

where *α* = 2(*ω<sup>n</sup>* + Γ<sup>0</sup> + Γ*π*).

solution of the AGL theory [63]

field **h** is given in units of Φ0/2*πξ*<sup>2</sup>

where

*ξc*2).

where the supercurrent **J**(**r**) is given in terms of *g*(*ωn*, *θ*,**r**) by

$$\mathbf{J}(\mathbf{r}) = 2\pi T \sum\_{\omega\_{\rm n}>0} \int\_0^{2\pi} \frac{d\theta}{2\pi} \frac{\hat{\mathbf{k}}}{\mathbf{i}} \mathbf{g}(\omega\_{\rm n}, \theta, \mathbf{r}). \tag{9}$$

Here **A** and **J** are measured in units of Φ0/2*πξ*<sup>0</sup> and 2*evFN*0*Tc*, respectively. The spatial variation of the internal field *h*(**r**) is determined through

$$\nabla \times \mathbf{A} = \mathbf{h}(\mathbf{r}),\tag{10}$$

where **h** is measured in units of Φ0/2*πξ*<sup>2</sup> 0. The self-consistent condition for the pairing potential Δ(**r**) in the vortex state is given by

$$\Delta(\mathbf{r}) = V^{\mathbf{SC}} 2\pi T \sum\_{\omega\_n > 0}^{\omega\_\ell} \int\_0^{2\pi} \frac{d\theta}{2\pi} f(\omega\_{\mathbf{n}\prime} \theta, \mathbf{r}),\tag{11}$$

where *VSC* is the coupling constant and *ω<sup>c</sup>* is the ultraviolet cutoff determining *Tc*<sup>0</sup> [55]. Consistently throughout our paper energy, temperature, and length are measured in units of *Tc*<sup>0</sup> and the coherence length *ξ*<sup>0</sup> = *vF*/*Tc*0, where *vF* is the Fermi velocity. The magnetic field **h** is given in units of Φ0/2*πξ*<sup>2</sup> <sup>0</sup>. The impurity scattering rates are in units of 2*πTc*0. In calculations the ratio *κ* = *λL*0/*ξ*<sup>0</sup> = 10 is used. It corresponds to *κGL* = 43.3 [77].

To obtain the quasiclassical Green function, the Riccati equations [Eq. (6, 7)] are solved by the Fast Fourier Transform (FFT) method for triangular FLL [55]. This method is reasonable for the dense FLL, discussed in this paper. In the high field the pinning effects are weak and they are not considered in our paper. To study the high field regime we needed to calculate the upper critical field *Bc*2(*T*). This was found from using the similarity of the considered model to the model of spin-flip superconductors from the equations [78]

$$\ln(\frac{T\_{c0}}{T}) = 2\pi T \sum\_{n\geq 0} [\omega\_n^{-1} - 2D\_1(\omega\_{\text{fl}}, \mathcal{B}\_{c2})] \,\tag{12}$$

where

6 Will-be-set-by-IN-TECH

**Figure 2.** (Color online) (a) The temperature dependence of the upper critical field *Bc*<sup>2</sup> at interband scattering Γ*<sup>π</sup>* = 0 with different values of intraband scattering values Γ0. (b) The calculated temperature dependence of *Bc*<sup>2</sup> at intraband scattering rate Γ<sup>0</sup> = 3 with different values of interband scattering Γ*π*.

have

obtained from the equation

satisfying the nonlinear Riccati equations. In Born approximation for impurity scattering we

where *ω<sup>n</sup>* = *πT*(2*n* + 1), *G* = 2*π* � *g* �(Γ<sup>0</sup> + Γ*π*) ≡ 2*π* � *g* �Γ∗, *F* = 2*π* � *f* �(Γ<sup>0</sup> − Γ*π*) for *s*<sup>±</sup> pairing symmetry and *F* = 2*π* � *f* �Γ<sup>∗</sup> for the *s*++ pairing symmetry. Here, **u** is a unit vector of the Fermi velocity. In the new gauge vector-potential **A***<sup>s</sup>* = **A** − ∇*φ* is proportional to the superfluid velocity. It diverges as 1/*r* at the vortex center (index *s* is put to denote its singular nature). The FLL creates the anisotropy of the electron spectrum. Therefore, the impurity renormalization correction in Eqs. (6) and (7), averaged over the Fermi surface, can

To take into account the influence of screening the vector potential **A**(**r**) in Eqs. (6) and (7) is

 2*π* 0

Here **A** and **J** are measured in units of Φ0/2*πξ*<sup>0</sup> and 2*evFN*0*Tc*, respectively. The spatial

*dθ* 2*π* **kˆ** *i*

∇×∇× **AE** <sup>=</sup> <sup>4</sup>

*ωn*>0

0.

be reduced to averages over the polar angle *θ*, i.e. �...� = (1/2*π*)

where the supercurrent **J**(**r**) is given in terms of *g*(*ωn*, *θ*,**r**) by

variation of the internal field *h*(**r**) is determined through

where **h** is measured in units of Φ0/2*πξ*<sup>2</sup>

**J**(**r**) = 2*πT* ∑

**<sup>u</sup>** · ∇*<sup>a</sup>* <sup>=</sup> <sup>−</sup>*<sup>a</sup>* [2(*ω<sup>n</sup>* <sup>+</sup> *<sup>G</sup>*) + *<sup>i</sup>***<sup>u</sup>** · **<sup>A</sup>***s*] + (<sup>Δ</sup> <sup>+</sup> *<sup>F</sup>*) <sup>−</sup> *<sup>a</sup>*2(Δ<sup>∗</sup> <sup>+</sup> *<sup>F</sup>*∗), (6)

**<sup>u</sup>** · ∇*<sup>b</sup>* <sup>=</sup> *<sup>b</sup>* [2(*ω<sup>n</sup>* <sup>+</sup> *<sup>G</sup>*) + *<sup>i</sup>***<sup>u</sup>** · **<sup>A</sup>***s*] <sup>−</sup> (Δ<sup>∗</sup> <sup>+</sup> *<sup>F</sup>*∗) + *<sup>b</sup>*2(<sup>Δ</sup> <sup>+</sup> *<sup>F</sup>*), (7)

... *dθ*.

*<sup>κ</sup>*<sup>2</sup> **<sup>J</sup>**, (8)

*g*(*ωn*, *θ*,**r**). (9)

∇ × **A** = **h**(**r**), (10)

$$D\_1(\omega\_{\nu\nu}B\_{c2}) = f(\omega\_{\nu\nu}B\_{c2}) \times [1 - 2(\Gamma\_0 - \Gamma\_\pi)f(\omega\_{\nu\nu}B\_{c2})]^{-1},\tag{13}$$

$$\mathcal{J}(\omega\_{\rm n}, \mathcal{B}\_{\rm c2}) = (\frac{4}{\pi \mathcal{B}\_{\rm c2}})^{1/2} \times \int\_0^\infty dy \, \exp\left(-y\right) \arctan\left[\frac{(\mathcal{B}\_{\rm c2}y)^{1/2}}{a}\right] \tag{14}$$

where *α* = 2(*ω<sup>n</sup>* + Γ<sup>0</sup> + Γ*π*).

Fig. 2 shows *Bc*2(*T*) dependences at (a) Γ*<sup>π</sup>* = 0, Γ<sup>0</sup> = 0, 1, 2, 3, 4, 5, 6 and (b) Γ<sup>0</sup> = 3, Γ*<sup>π</sup>* = 0.01, 0.02, 0.03, 0.04, 0.05, 0.06 calculated from Eqs. (12-14). In Fig. 2 the different influence of the intraband and interband scattering on *Bc*2(*T*) dependence can be seen. The *Bc*2(*T*) curve increases with Γ<sup>0</sup> (*ξc*<sup>2</sup> decreases with Γ0), but Γ*<sup>π</sup>* results in decreasing *Bc*2(*T*) (increasing of *ξc*2).

Fig. 3 (a) shows magnetic field dependence *ξh*(*B*) in reduced units at *T*/*Tc*<sup>0</sup> = 0.5 for the *s*<sup>±</sup> pairing with Γ<sup>0</sup> = 3, Γ*<sup>π</sup>* = 0.02 and Γ<sup>0</sup> = 0.5, Γ*<sup>π</sup>* = 0.03 and "clean" case (solid lines) and for the *s*++ pairing with Γ∗= 0.5 and Γ∗ = 3 (dotted lines). The dashed line shows the analytical solution of the AGL theory [63]

$$\xi\_{\mathcal{V}} = \xi\_{c2} (\sqrt{2} - \underbrace{0.75}\_{\mathbb{K}\_{GL}}) (1 + b^4)^{1/2} [1 - 2b(1 - b)^2]^{1/2}. \tag{15}$$

This dependence with *ξc*<sup>2</sup> as a fitting parameter is often used for the description of the experimental *μ*SR results [56, 79]. As can be seen from Fig. 3 (a), the magnetic field dependence of *ξh*/*ξc*<sup>2</sup> is nonuniversal because it depends not only on *B*/*Bc*<sup>2</sup> (as in the AGL theory, dashed line in Fig. 3 (a)), but also on interband and intraband impurity scattering parameters. In the cases where Γ<sup>0</sup> = Γ*<sup>π</sup>* = 0, the results are the same for *s*± and *s*++ pairing symmetries. We indicated that this curve is "clean" one. In this figure, the case Γ<sup>0</sup> � Γ*<sup>π</sup>* is considered

**Figure 3.** (Color online) (a) The magnetic field dependence of *ξh*/*ξc*<sup>2</sup> for superconductors with impurity scattering. The solid lines represent our solution of Eilenberger equations at *T*/*Tc*<sup>0</sup> = 0.5 for "clean" case (Γ<sup>0</sup> = Γ*<sup>π</sup>* = 0) and *s*<sup>±</sup> model (Γ<sup>0</sup> = 0.5, Γ*<sup>π</sup>* = 0.03 and Γ<sup>0</sup> = 3, Γ*<sup>π</sup>* = 0.02). The dotted lines show result for *s*++ model (Γ<sup>∗</sup> = 0.5 and Γ<sup>∗</sup> = 3). Dashed line demonstrates the result of the AGL theory for *ξ<sup>v</sup>* from Eq. 15. The inset shows the magnetic field dependence of mean square deviation of the *hEHC* distribution from the Eilenberger distribution normalized by the variance of the Eilenberger distribution, *ε*, for *T*/*Tc*<sup>0</sup> = 0.5 at Γ<sup>0</sup> = Γ*<sup>π</sup>* = 0 ("clean"); Γ<sup>0</sup> = 3, Γ*<sup>π</sup>* = 0.02 and Γ<sup>0</sup> = 0.5, Γ*<sup>π</sup>* = 0.03. (b) The interband scattering Γ*<sup>π</sup>* dependence of *ξh*/*ξc*<sup>2</sup> at different temperatures *T*/*Tc*<sup>0</sup> (intraband scattering Γ<sup>0</sup> = 0.5 and *B* = 5) for the *s*± pairing.

**Figure 4.** (Color online) (a) The magnetic field dependence of cutoff parameter *ξh*/*ξc*<sup>2</sup> at different temperatures (*T*/*Tc*<sup>0</sup> = 0.2, 0.3, 0.4, 0.5) for *s*<sup>±</sup> pairing with Γ<sup>0</sup> = 3, Γ*<sup>π</sup>* = 0.04. (b) The magnetic field dependence of *ξh*/*ξc*<sup>2</sup> for *s*<sup>±</sup> model (Γ<sup>0</sup> = 3, Γ*<sup>π</sup>* = 0.04, solid line) and *s*++ model (Γ<sup>∗</sup> = 3, dotted line) at

The superfluid density in iron pnictides often shows a power law dependence with theexponent, which is approximately equal to two at low temperatures [39, 74]. This law was explained by *s*<sup>±</sup> model with parameters Γ<sup>0</sup> = 3 and Γ*<sup>π</sup>* = 0.04 − 0.06. Fig. 4 (a) shows *ξh*/*ξc*2(*B*/*Bc*2) dependence with Γ<sup>0</sup> = 3 and Γ*<sup>π</sup>* = 0.04 at different temperatures. All curves demonstrate rising behavior with values much less than one in the whole field range, i.e. they are under the AGL curve of *ξv*. The small value of the cutoff parameter was observed in iron pnictide BaFe1.82Co0.18As, where *ξh*/*ξc*2(∼ 0.4) < 1 [80]. Fig. 4 (b) shows *ξh*/*ξc*2(*B*/*Bc*2) for Γ<sup>0</sup> = 3, Γ*<sup>π</sup>* = 0.04 (*s*± pairing) and Γ∗ = 3 (*s*++ pairing). It can be seen from the graph that *ξh*/*ξc*<sup>2</sup> is strongly suppressed in *s*<sup>±</sup> pairing with comparison to the *s*++ pairing. This can be explained by the fact that in superconductors, without interband pair breaking, the increase in high field is connected with the field-dependent pair breaking, as the upper critical field is approached. The physics of unconventional superconductors depends on impurity pair breaking and introducing characteristic field *B*∗ in the field dependence by the substitution *B*/*Bc*<sup>2</sup> → (*B* + *B*∗(Γ*π*))/*Bc*2(Γ*π*). The crossing point between *s*<sup>±</sup> and *s*++ curves depends on Γ*<sup>π</sup>* and it shifts to the lower field in comparison with case Γ*<sup>π</sup>* = 0.02 shown in Fig. 3 (a).

Eilenberger Approach to the Vortex State in Iron Pnictide Superconductors 207

**Figure 5.** (Color online) The magnetic field dependence of the cutoff parameter at *T*/*Tc*<sup>0</sup> = 0.15 with the

same values of intraband Γ<sup>0</sup> and interband Γ*<sup>π</sup>* scattering rate Γ (Γ = 0 for "clean" case and Γ = 0.05, 0.06, 0.065 for the *s*<sup>±</sup> pairing). Dotted line shows result for *s*++ model (Γ<sup>∗</sup> = 0.25).

*T*/*Tc*<sup>0</sup> = 0.5.

and the value of *ξ<sup>h</sup>* is reduced considerably in comparison with the clean case. One can compare the observed behavior with that in *s*++ pairing model. In *s*++ pairing symmetry the intraband and interband scattering rates act in a similar way and *ξh*/*ξc*<sup>2</sup> decreases always with impurity scattering. In contrast, in *s*<sup>±</sup> model *ξh*/*ξc*2(*B*/*Bc*2) dependences show different forms of behavior with Γ*π*. Here, *ξh*/*ξc*<sup>2</sup> increases with Γ*<sup>π</sup>* at *B*/*Bc*<sup>2</sup> < 0.8 and decreases at higher fields, i.e. the curves become more flattened. A crossing point appears in the dependences *ξh*/*ξc*2(*B*/*Bc*2) for *s*<sup>±</sup> and *s*++ pairing. We also calculated the magnetic field dependence of mean square deviation of *hEHC* distribution of the magnetic field from the Eilenberger distribution normalized by the variance of the Eilenberger distribution *ε* =

 (*hE* − *hEHC*)2/(*hE* − *<sup>B</sup>*)2, where ··· is the average over a unit vortex cell. The inset to Fig. 3 (a) demonstrates *ε*(*B*) dependence for *T*/*Tc*<sup>0</sup> = 0.5 at Γ<sup>0</sup> = 0, Γ*<sup>π</sup>* = 0; Γ<sup>0</sup> = 3, Γ*<sup>π</sup>* = 0.02 and Γ<sup>0</sup> = 0.5, Γ*<sup>π</sup>* = 0.03. From this figure, it can be seen that the accuracy of effective London model is deteriorating as the magnetic field increases; however, in superconductors with impurity scattering the accuracy is below 6% even when it is close to the second critical field (the inset to Fig. 3 (a)).

In Fig. 3 (b), the interband scattering Γ*<sup>π</sup>* dependences of *ξ<sup>h</sup>* are presented in low fields for the *s*<sup>±</sup> pairing at different temperatures *T*. As can be seen *ξh*/*ξc*<sup>2</sup> increases with the interband scattering rate Γ*π*. Strong decreasing of *ξh*/*ξc*<sup>2</sup> with a decrease in the temperature can be explained by the Kramer-Pesch effect [67]. It should be noted that the normalization constant *ξc*<sup>2</sup> increases with Γ*<sup>π</sup>* because Γ*<sup>π</sup>* suppress *Tc* similar to superconductors with spin-flip scattering (violation of the Anderson theorem). Thus, the rising *ξh*/*ξc*<sup>2</sup> implies more strong growth of *ξ<sup>h</sup>* than *ξc*<sup>2</sup> (from GL theory one can expect *ξh*/*ξc*<sup>2</sup> = *Const*). Qualitatively, it can be explained by the strong temperature dependence of *ξh*(*B*, *T*/*Tc* ), which is connected to the Kramer-Pesch effect [67]. Increasing Γ*<sup>π</sup>* results in suppression of *Tc*, i.e. effective increasing of *T* and *ξh*(*T*/*Tc* ). *ξc*2(*T*/*Tc* ) has not such a strong *Tc* dependence, thus leading to the increasing of the ratio *ξh*/*ξc*<sup>2</sup> with Γ*π*.

8 Will-be-set-by-IN-TECH

**Figure 3.** (Color online) (a) The magnetic field dependence of *ξh*/*ξc*<sup>2</sup> for superconductors with impurity scattering. The solid lines represent our solution of Eilenberger equations at *T*/*Tc*<sup>0</sup> = 0.5 for "clean" case (Γ<sup>0</sup> = Γ*<sup>π</sup>* = 0) and *s*<sup>±</sup> model (Γ<sup>0</sup> = 0.5, Γ*<sup>π</sup>* = 0.03 and Γ<sup>0</sup> = 3, Γ*<sup>π</sup>* = 0.02). The dotted lines show result for *s*++ model (Γ<sup>∗</sup> = 0.5 and Γ<sup>∗</sup> = 3). Dashed line demonstrates the result of the AGL theory for *ξ<sup>v</sup>* from Eq. 15. The inset shows the magnetic field dependence of mean square deviation of the *hEHC* distribution from the Eilenberger distribution normalized by the variance of the Eilenberger distribution, *ε*, for *T*/*Tc*<sup>0</sup> = 0.5 at Γ<sup>0</sup> = Γ*<sup>π</sup>* = 0 ("clean"); Γ<sup>0</sup> = 3, Γ*<sup>π</sup>* = 0.02 and Γ<sup>0</sup> = 0.5, Γ*<sup>π</sup>* = 0.03. (b) The interband scattering Γ*<sup>π</sup>* dependence of *ξh*/*ξc*<sup>2</sup> at different temperatures *T*/*Tc*<sup>0</sup> (intraband scattering Γ<sup>0</sup> = 0.5 and

and the value of *ξ<sup>h</sup>* is reduced considerably in comparison with the clean case. One can compare the observed behavior with that in *s*++ pairing model. In *s*++ pairing symmetry the intraband and interband scattering rates act in a similar way and *ξh*/*ξc*<sup>2</sup> decreases always with impurity scattering. In contrast, in *s*<sup>±</sup> model *ξh*/*ξc*2(*B*/*Bc*2) dependences show different forms of behavior with Γ*π*. Here, *ξh*/*ξc*<sup>2</sup> increases with Γ*<sup>π</sup>* at *B*/*Bc*<sup>2</sup> < 0.8 and decreases at higher fields, i.e. the curves become more flattened. A crossing point appears in the dependences *ξh*/*ξc*2(*B*/*Bc*2) for *s*<sup>±</sup> and *s*++ pairing. We also calculated the magnetic field dependence of mean square deviation of *hEHC* distribution of the magnetic field from the Eilenberger distribution normalized by the variance of the Eilenberger distribution

(*hE* − *hEHC*)2/(*hE* − *<sup>B</sup>*)2, where ··· is the average over a unit vortex cell. The inset to Fig. 3 (a) demonstrates *ε*(*B*) dependence for *T*/*Tc*<sup>0</sup> = 0.5 at Γ<sup>0</sup> = 0, Γ*<sup>π</sup>* = 0; Γ<sup>0</sup> = 3, Γ*<sup>π</sup>* = 0.02 and Γ<sup>0</sup> = 0.5, Γ*<sup>π</sup>* = 0.03. From this figure, it can be seen that the accuracy of effective London model is deteriorating as the magnetic field increases; however, in superconductors with impurity scattering the accuracy is below 6% even when it is close to the second critical

In Fig. 3 (b), the interband scattering Γ*<sup>π</sup>* dependences of *ξ<sup>h</sup>* are presented in low fields for the *s*<sup>±</sup> pairing at different temperatures *T*. As can be seen *ξh*/*ξc*<sup>2</sup> increases with the interband scattering rate Γ*π*. Strong decreasing of *ξh*/*ξc*<sup>2</sup> with a decrease in the temperature can be explained by the Kramer-Pesch effect [67]. It should be noted that the normalization constant *ξc*<sup>2</sup> increases with Γ*<sup>π</sup>* because Γ*<sup>π</sup>* suppress *Tc* similar to superconductors with spin-flip scattering (violation of the Anderson theorem). Thus, the rising *ξh*/*ξc*<sup>2</sup> implies more strong growth of *ξ<sup>h</sup>* than *ξc*<sup>2</sup> (from GL theory one can expect *ξh*/*ξc*<sup>2</sup> = *Const*). Qualitatively, it can be explained by the strong temperature dependence of *ξh*(*B*, *T*/*Tc* ), which is connected to the Kramer-Pesch effect [67]. Increasing Γ*<sup>π</sup>* results in suppression of *Tc*, i.e. effective increasing of *T* and *ξh*(*T*/*Tc* ). *ξc*2(*T*/*Tc* ) has not such a strong *Tc* dependence, thus leading to the increasing

*ε* =

*B* = 5) for the *s*± pairing.

field (the inset to Fig. 3 (a)).

of the ratio *ξh*/*ξc*<sup>2</sup> with Γ*π*.

**Figure 4.** (Color online) (a) The magnetic field dependence of cutoff parameter *ξh*/*ξc*<sup>2</sup> at different temperatures (*T*/*Tc*<sup>0</sup> = 0.2, 0.3, 0.4, 0.5) for *s*<sup>±</sup> pairing with Γ<sup>0</sup> = 3, Γ*<sup>π</sup>* = 0.04. (b) The magnetic field dependence of *ξh*/*ξc*<sup>2</sup> for *s*<sup>±</sup> model (Γ<sup>0</sup> = 3, Γ*<sup>π</sup>* = 0.04, solid line) and *s*++ model (Γ<sup>∗</sup> = 3, dotted line) at *T*/*Tc*<sup>0</sup> = 0.5.

The superfluid density in iron pnictides often shows a power law dependence with theexponent, which is approximately equal to two at low temperatures [39, 74]. This law was explained by *s*<sup>±</sup> model with parameters Γ<sup>0</sup> = 3 and Γ*<sup>π</sup>* = 0.04 − 0.06. Fig. 4 (a) shows *ξh*/*ξc*2(*B*/*Bc*2) dependence with Γ<sup>0</sup> = 3 and Γ*<sup>π</sup>* = 0.04 at different temperatures. All curves demonstrate rising behavior with values much less than one in the whole field range, i.e. they are under the AGL curve of *ξv*. The small value of the cutoff parameter was observed in iron pnictide BaFe1.82Co0.18As, where *ξh*/*ξc*2(∼ 0.4) < 1 [80]. Fig. 4 (b) shows *ξh*/*ξc*2(*B*/*Bc*2) for Γ<sup>0</sup> = 3, Γ*<sup>π</sup>* = 0.04 (*s*± pairing) and Γ∗ = 3 (*s*++ pairing). It can be seen from the graph that *ξh*/*ξc*<sup>2</sup> is strongly suppressed in *s*<sup>±</sup> pairing with comparison to the *s*++ pairing. This can be explained by the fact that in superconductors, without interband pair breaking, the increase in high field is connected with the field-dependent pair breaking, as the upper critical field is approached. The physics of unconventional superconductors depends on impurity pair breaking and introducing characteristic field *B*∗ in the field dependence by the substitution *B*/*Bc*<sup>2</sup> → (*B* + *B*∗(Γ*π*))/*Bc*2(Γ*π*). The crossing point between *s*<sup>±</sup> and *s*++ curves depends on Γ*<sup>π</sup>* and it shifts to the lower field in comparison with case Γ*<sup>π</sup>* = 0.02 shown in Fig. 3 (a).

**Figure 5.** (Color online) The magnetic field dependence of the cutoff parameter at *T*/*Tc*<sup>0</sup> = 0.15 with the same values of intraband Γ<sup>0</sup> and interband Γ*<sup>π</sup>* scattering rate Γ (Γ = 0 for "clean" case and Γ = 0.05, 0.06, 0.065 for the *s*<sup>±</sup> pairing). Dotted line shows result for *s*++ model (Γ<sup>∗</sup> = 0.25).

The case of weak intraband scattering was also studied. This case can be realized in stoichiometrical pnictides such as LiFeAs. Fig. 5 presents the magnetic field dependence of *ξh*/*ξc*<sup>2</sup> with scattering parameters Γ<sup>0</sup> = Γ*<sup>π</sup>* = Γ equal to 0, 0.05, 0.06 and 0.065 at *T*/*Tc*<sup>0</sup> = 0.15. The dotted line shows the result for *s*++ model (Γ<sup>∗</sup> = 0.25). The *ξh*(*B*) dependence shifts upward from the "clean" curve and has a higher values in *s*<sup>±</sup> model. In contrast, the *ξh*/*ξc*<sup>2</sup> curve shifts downward with impurity scattering in *s*++ model. The high values of *ξ<sup>h</sup>* observed in *μ*SR measurements in LiFeAs [81] supports the *s*± pairing.
