**4. The quasiclassical approach to the Amin-Franz-Affleck model and the effective penetration depth in the mixed state in** *dx*<sup>2</sup>−*y*<sup>2</sup> **-wave pairing symmetry**

In this chapter, we construct a model where the nonlinear corrections arising from the Doppler energy shift of the quasiparticle states by the supercurrent [85] and effects of the vortex core states are described by an effective cutoff function. Nonlocal effects of the extended quasiparticle states are included in our model explicitly, i. e. instead of *λ*(*T*) in Eq. (1) we use an analytically obtained anisotropic electromagnetic response tensor [70, 72, 73]. Because the nonlocal effects are assumed to be effective in clean superconductors we limit our consideration to the case Γ = 0.

For a better comparison with the nonlocal generalized London equation (NGLE) and the AGL theory we used another normalization of the cutoff parameter in Eq. (1), *u* = *k*<sup>1</sup> <sup>√</sup>2*ξBCSG*. This form of *F*(*G*) correctly describes the high temperature regime. We compare our results with those obtained from the NGLE theory in a wide field and temperature range considering *k*<sup>1</sup> as the fitting parameter.

The magnetic field distribution in the mixed state in the NGLE approximation is given by [72]

$$h\_{NGLE}(\mathbf{r}) = \frac{\Phi\_0}{S} \sum\_{\mathbf{G}} \frac{F(G)e^{i\mathbf{G}\mathbf{r}}}{1 + L\_{lj}(\mathbf{G})G\_l G\_j} \tag{27}$$

where

$$L\_{ij}(\mathbf{G}) = \frac{Q\_{ij}(\mathbf{G})}{\det \hat{\mathbf{Q}}(\mathbf{G})}. \tag{28}$$

**Figure 9.** (Color online) The temperature dependence of the coefficient *k*<sup>1</sup> in the NGLE model obtained at *κ* = 10 and *B* = 0.1, 1, 2, 3, 5 from a fitting made with the solution of the Eilenberger equations.

Eilenberger Approach to the Vortex State in Iron Pnictide Superconductors 213

where *γ***<sup>G</sup>** = **v***<sup>F</sup>* · **G**/2. In Eq. (29) the term with *γ***<sup>G</sup>** describes the nonlocal correction to the London equation. Putting *γ***<sup>G</sup>** = 0 we obtain the London result *Lij*(**G**) = *λ*(*T*)2*δij*. We use the same shape of the cutoff function as in Eq. (1) but the values of the cutoff parameters are different because of fitting them to the various field distributions. In presentation of *hNGLE*

**Figure 10.** (Color online) Field dependence of *k*<sup>1</sup> at *T* = 0.75 and 0.8 obtained from the fitting to the Eilenberger equations. The inset shows *k*1(*B*) calculated from the Hao-Clem theory at *T* = 0.95.

the Ginzburg-Landau equations for the single vortex [62] gives *k*<sup>1</sup> = *π*/

Fig. 9 shows the *k*1(*T*) dependence in the NGLE model obtained at *κ* = 10 and *B* = 0.1, 1, 2, 3, 5 from the fitting to the solution of the Eilenberger equations. As can be seen from Fig. 9 the coefficient *k*<sup>1</sup> is strongly reduced at low temperatures. This is a reminiscent of the Kramer-Pesch result for *s*-wave superconductors (shrinking of the vortex core with decreasing temperature) [95]. It is also found that *k*<sup>1</sup> is a decreasing function of *B*. This can be explained by reduction of the vortex core size by the field [68]. The topmost curve in Fig. 9 gives the values of *k*<sup>1</sup> calculated for a single vortex [96]. At high temperatures the Ginzburg-Landau theory can be applied. Using the values of the parameters of this theory

reasonable agreement with the high temperature limit of *k*<sup>1</sup> for a single vortex in Fig. 9. Another interesting observation is the nonmonotonic behavior of *k*1(*B*) in low fields at high

<sup>√</sup>3 is obtained. A variational approach of

<sup>√</sup><sup>3</sup> <sup>≈</sup> 1.81 is in

the anisotropy effects of the Eilenberger theory remain.

for *d*-wave superconductors [97] *ξGL* = *ξBCSπ*/

The anisotropic electromagnetic response tensor is defined by

$$Q\_{ij}(\mathbf{G}) = \frac{4\pi T}{\lambda\_{L0}^2} \sum\_{\omega\_n > 0} \int\_0^{2\pi} \frac{d\theta}{2\pi} \frac{\Delta(\theta)^2 \theta\_{Fi} \theta\_{Fj}}{\sqrt{\omega\_n^2 + |\Delta(\theta)^2|} (\omega\_n^2 + |\Delta(\theta)^2| + \gamma\_\mathbf{G}^2)},\tag{29}$$

14 Will-be-set-by-IN-TECH

pairing with *B* = 5. For clean superconductors (Fig. 8 (a)) *ξh*/*ξc*<sup>2</sup> has a minimum in its field dependence similar to usual *s*-wave superconductors [93]. However, this ratio decreases with temperature due to Kramer-Pesch effect. It was demonstrated theoretically and experimentally that the low energy density of states *N*(*E*) is described by the same singular *<sup>V</sup>*-shape form *<sup>N</sup>*(*E*) = *<sup>N</sup>*0(*H*) + *<sup>α</sup>*|*E*<sup>|</sup> <sup>+</sup> *<sup>O</sup>*(*E*2) for all clean superconductors in a vortex state, irrespective of the underlying gap structure [94]. This explains the similarity in the behavior

The difference between pairing symmetries reveals itself in impurity scattering dependence *ξh*/*ξc*2. In *s*++ symmetry *ξh*/*ξc*<sup>2</sup> always decreases with impurity scattering rate Γ (Fig. 3 (a)), in *s*± symmetry its behavior depends on the field range and relative values of intraband and interband impurity scattering rates: it can be a decreasing function of Γ*<sup>π</sup>* (Fig. 4 (b)) or an increasing function of Γ*<sup>π</sup>* (Fig. 3 (b)). In *d*-wave superconductors *ξh*/*ξc*<sup>2</sup> always increases with Γ (Fig. 8 (b)) similar to the case of *s*± symmetry with Γ<sup>0</sup> = Γ*<sup>π</sup>* ( Fig. 5). This can be understood from the comparison of the Ricatti equations of the *s*± and *d*-wave pairing. In both cases the renormalization factor *F* = 0 due to a cancelation of the intraband and interband impurity

**4. The quasiclassical approach to the Amin-Franz-Affleck model and the**

In this chapter, we construct a model where the nonlinear corrections arising from the Doppler energy shift of the quasiparticle states by the supercurrent [85] and effects of the vortex core states are described by an effective cutoff function. Nonlocal effects of the extended quasiparticle states are included in our model explicitly, i. e. instead of *λ*(*T*) in Eq. (1) we use an analytically obtained anisotropic electromagnetic response tensor [70, 72, 73]. Because the nonlocal effects are assumed to be effective in clean superconductors we limit

For a better comparison with the nonlocal generalized London equation (NGLE) and the AGL

This form of *F*(*G*) correctly describes the high temperature regime. We compare our results with those obtained from the NGLE theory in a wide field and temperature range considering

The magnetic field distribution in the mixed state in the NGLE approximation is given by [72]

*F*(*G*)*ei***Gr** 1 + *Lij*(**G**)*GiGj*

Δ(*θ*)2*v*ˆ*Fiv*ˆ*Fj*

*<sup>S</sup>* ∑ **G**

*Lij*(**G**) = *Qij*(**G**)

det**Qˆ** (**G**)

*<sup>n</sup>* <sup>+</sup> <sup>|</sup>Δ(*θ*)2|(*ω*<sup>2</sup>

<sup>√</sup>2*ξBCSG*.

, (27)

. (28)

**G**)

, (29)

*<sup>n</sup>* <sup>+</sup> <sup>|</sup>Δ(*θ*)2<sup>|</sup> <sup>+</sup> *<sup>γ</sup>*<sup>2</sup>

theory we used another normalization of the cutoff parameter in Eq. (1), *u* = *k*<sup>1</sup>

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

 2*π* 0

*dθ* 2*π*

 *ω*2

The anisotropic electromagnetic response tensor is defined by

∑ *ωn*>0

*Qij*(**G**) = <sup>4</sup>*π<sup>T</sup>*

*λ*2 *L*0

**effective penetration depth in the mixed state in** *dx*<sup>2</sup>−*y*<sup>2</sup> **-wave pairing**

scattering rates in *s*<sup>±</sup> pairing or symmetry reason �*f*� = 0 for *d*-wave pairing.

between *s*- and *d*-wave pairing symmetries.

**symmetry**

our consideration to the case Γ = 0.

*k*<sup>1</sup> as the fitting parameter.

where

**Figure 9.** (Color online) The temperature dependence of the coefficient *k*<sup>1</sup> in the NGLE model obtained at *κ* = 10 and *B* = 0.1, 1, 2, 3, 5 from a fitting made with the solution of the Eilenberger equations.

where *γ***<sup>G</sup>** = **v***<sup>F</sup>* · **G**/2. In Eq. (29) the term with *γ***<sup>G</sup>** describes the nonlocal correction to the London equation. Putting *γ***<sup>G</sup>** = 0 we obtain the London result *Lij*(**G**) = *λ*(*T*)2*δij*. We use the same shape of the cutoff function as in Eq. (1) but the values of the cutoff parameters are different because of fitting them to the various field distributions. In presentation of *hNGLE* the anisotropy effects of the Eilenberger theory remain.

**Figure 10.** (Color online) Field dependence of *k*<sup>1</sup> at *T* = 0.75 and 0.8 obtained from the fitting to the Eilenberger equations. The inset shows *k*1(*B*) calculated from the Hao-Clem theory at *T* = 0.95.

Fig. 9 shows the *k*1(*T*) dependence in the NGLE model obtained at *κ* = 10 and *B* = 0.1, 1, 2, 3, 5 from the fitting to the solution of the Eilenberger equations. As can be seen from Fig. 9 the coefficient *k*<sup>1</sup> is strongly reduced at low temperatures. This is a reminiscent of the Kramer-Pesch result for *s*-wave superconductors (shrinking of the vortex core with decreasing temperature) [95]. It is also found that *k*<sup>1</sup> is a decreasing function of *B*. This can be explained by reduction of the vortex core size by the field [68]. The topmost curve in Fig. 9 gives the values of *k*<sup>1</sup> calculated for a single vortex [96]. At high temperatures the Ginzburg-Landau theory can be applied. Using the values of the parameters of this theory for *d*-wave superconductors [97] *ξGL* = *ξBCSπ*/ <sup>√</sup>3 is obtained. A variational approach of the Ginzburg-Landau equations for the single vortex [62] gives *k*<sup>1</sup> = *π*/ <sup>√</sup><sup>3</sup> <sup>≈</sup> 1.81 is in reasonable agreement with the high temperature limit of *k*<sup>1</sup> for a single vortex in Fig. 9. Another interesting observation is the nonmonotonic behavior of *k*1(*B*) in low fields at high

the temperature dependence of the ratio �*δh*<sup>2</sup>

from Fig. 9.

core solving.

Here, <sup>|</sup>*δh*<sup>2</sup>

suggested in the AFA model [70, 73]:

in the field distribution *hNGLE*(**r**).

*NGLE*� with the cutoff parameter obtained from

Eilenberger Approach to the Vortex State in Iron Pnictide Superconductors 215

1/4. (30)

*eff* calculated from the

0/*λ*<sup>2</sup>

the solution of the Eilenberger equations to that with *k*<sup>1</sup> = 1. From the data presented in Fig. 12, it can be sen that this ratio deviates considerably from unity when the temperature is lowered, which points to the importance of the proper determination of the value for the cutoff parameter. For the magnetic field distribution, obtained from solving the NGLE, we also calculate the mean-square deviation of this distribution from the origin (the Eilenberger equations solution). The inset demonstrates this deviation for fixed and fitted parameter *k*1.

**Figure 13.** (Color online) The ratio of *λ*<sup>0</sup> to *λeff* calculated from the NGLE equation with *k*<sup>1</sup> = 1 and *k*<sup>1</sup>

This consideration proves that the nonlocal generalized London model with *hNGLE*(**r**) distribution also needs the properly determined cutoff parameter *k*1, *i*.*e*. introducing only nonlocal extended electronic states does not allow the avoidance of the problem of vortex

In the analysis of the experimental *μ*SR and SANS data the field dependent penetration depth *λeff*(*B*) is often introduced [56]. It has physical sense even if it is not dependent on the core effects, *i*.*e*. it should be an invariant of the cutoff parameter. One such way of doing this was

0|

<sup>0</sup>| is the variance of the magnetic field *h*0(**r**) obtained by applying the ordinary

*<sup>λ</sup>* = ( <sup>|</sup>*δh*<sup>2</sup>

<sup>|</sup>*δh*<sup>2</sup> *NGLE*| )

London model with the same average field *B* and *λ* and with the same cutoff parameter as

*hNGLE* distribution with *k*<sup>1</sup> = 1 and with Fit *k*<sup>1</sup> from the solution of Eilenberger equations for the different field value. The obtained *λeff*(*B*) dependences are quite similar in these cases. The low-field result (*B*/*B*<sup>0</sup> = 0.1) for *λeff* is close to *λ*(*T*) in the Meissner state. This demonstrates that *λeff* is determined by a large scale of the order of FLL period and is not very sensitive to details of the microscopical core structure and the cutoff parameter [98]. The AFA model was originally developed in order to explain the structural transition in FLL in *d*-wave superconductors where anisotropy and nonlocal effects arise from nodes in the gap at the Fermi surface and the appearance there of the long extending electronic states [72].

*λeff*

In Fig. 13 establishes the temperature dependence of the ratio *λ*<sup>2</sup>

**Figure 11.** (Color online) Normalized differences between the fields calculated with the London model (NGLE) and the Eilenberger equation (ELENB) for *B* = 1 and *T* = 0.6. The scales of lengths are those of the flux line lattice unit vectors.

temperatures. Fig. 10 depicts the field dependence of *k*<sup>1</sup> at *T* = 0.75 and 0.8 showing a minimum which moves to lower fields with increasing of the temperature. This result agrees qualitatively with the Hao-Clem theory [63] which also predicts a minimum in the *k*1(*B*) dependence. This is demonstrated in the inset to Fig. 10, where *k*1(*B*) is shown at *T* = 0.95.

**Figure 12.** (Color online) Temperature dependence of the ratio of the second moment of the magnetic field distributions obtained from the NGLE model with the fixed and fitted parameter *k*<sup>1</sup> (see the text below). The inset shows the mean-square deviation of the magnetic field distribution from the origin for parameter *k*<sup>1</sup> set to unity (solid line) and fitted (dotted line).

The quality of the fitting can be seen from Fig. 11 where the normalized difference between the fields calculated in the NGLE model and the Eilenberger equations at *B* = 1, *T* = 0.6 and *κ* = 10 is shown. The accuracy of the fitting is about 1 percent. Thus, there is only a little improvement in the Eilenberger equations fitting to NGLE theory in comparison with local London theory (Eq. (1)). The similarity of the field and temperature dependences of the cutoff parameter in these theories are shown in Fig. 9 and Fig. 10.

To show the influence of the magnetic field and temperature on *k*<sup>1</sup> dependence, we calculate the values of *<sup>δ</sup>h*<sup>2</sup> *NGLE* using the field distribution obtained in the Eq. 27. Fig. 12 shows the temperature dependence of the ratio �*δh*<sup>2</sup> *NGLE*� with the cutoff parameter obtained from the solution of the Eilenberger equations to that with *k*<sup>1</sup> = 1. From the data presented in Fig. 12, it can be sen that this ratio deviates considerably from unity when the temperature is lowered, which points to the importance of the proper determination of the value for the cutoff parameter. For the magnetic field distribution, obtained from solving the NGLE, we also calculate the mean-square deviation of this distribution from the origin (the Eilenberger equations solution). The inset demonstrates this deviation for fixed and fitted parameter *k*1.

16 Will-be-set-by-IN-TECH

**Figure 11.** (Color online) Normalized differences between the fields calculated with the London model (NGLE) and the Eilenberger equation (ELENB) for *B* = 1 and *T* = 0.6. The scales of lengths are those of

temperatures. Fig. 10 depicts the field dependence of *k*<sup>1</sup> at *T* = 0.75 and 0.8 showing a minimum which moves to lower fields with increasing of the temperature. This result agrees qualitatively with the Hao-Clem theory [63] which also predicts a minimum in the *k*1(*B*) dependence. This is demonstrated in the inset to Fig. 10, where *k*1(*B*) is shown at *T* = 0.95.

**Figure 12.** (Color online) Temperature dependence of the ratio of the second moment of the magnetic field distributions obtained from the NGLE model with the fixed and fitted parameter *k*<sup>1</sup> (see the text below). The inset shows the mean-square deviation of the magnetic field distribution from the origin for

The quality of the fitting can be seen from Fig. 11 where the normalized difference between the fields calculated in the NGLE model and the Eilenberger equations at *B* = 1, *T* = 0.6 and *κ* = 10 is shown. The accuracy of the fitting is about 1 percent. Thus, there is only a little improvement in the Eilenberger equations fitting to NGLE theory in comparison with local London theory (Eq. (1)). The similarity of the field and temperature dependences of the cutoff

To show the influence of the magnetic field and temperature on *k*<sup>1</sup> dependence, we calculate

*NGLE* using the field distribution obtained in the Eq. 27. Fig. 12 shows

parameter *k*<sup>1</sup> set to unity (solid line) and fitted (dotted line).

parameter in these theories are shown in Fig. 9 and Fig. 10.

the values of *<sup>δ</sup>h*<sup>2</sup>

the flux line lattice unit vectors.

**Figure 13.** (Color online) The ratio of *λ*<sup>0</sup> to *λeff* calculated from the NGLE equation with *k*<sup>1</sup> = 1 and *k*<sup>1</sup> from Fig. 9.

This consideration proves that the nonlocal generalized London model with *hNGLE*(**r**) distribution also needs the properly determined cutoff parameter *k*1, *i*.*e*. introducing only nonlocal extended electronic states does not allow the avoidance of the problem of vortex core solving.

In the analysis of the experimental *μ*SR and SANS data the field dependent penetration depth *λeff*(*B*) is often introduced [56]. It has physical sense even if it is not dependent on the core effects, *i*.*e*. it should be an invariant of the cutoff parameter. One such way of doing this was suggested in the AFA model [70, 73]:

$$\frac{\lambda\_{eff}}{\lambda} = (\frac{|\delta h\_0^2|}{|\delta h\_{NGLE}^2|})^{1/4}.\tag{30}$$

Here, <sup>|</sup>*δh*<sup>2</sup> <sup>0</sup>| is the variance of the magnetic field *h*0(**r**) obtained by applying the ordinary London model with the same average field *B* and *λ* and with the same cutoff parameter as in the field distribution *hNGLE*(**r**).

In Fig. 13 establishes the temperature dependence of the ratio *λ*<sup>2</sup> 0/*λ*<sup>2</sup> *eff* calculated from the *hNGLE* distribution with *k*<sup>1</sup> = 1 and with Fit *k*<sup>1</sup> from the solution of Eilenberger equations for the different field value. The obtained *λeff*(*B*) dependences are quite similar in these cases. The low-field result (*B*/*B*<sup>0</sup> = 0.1) for *λeff* is close to *λ*(*T*) in the Meissner state. This demonstrates that *λeff* is determined by a large scale of the order of FLL period and is not very sensitive to details of the microscopical core structure and the cutoff parameter [98]. The AFA model was originally developed in order to explain the structural transition in FLL in *d*-wave superconductors where anisotropy and nonlocal effects arise from nodes in the gap at the Fermi surface and the appearance there of the long extending electronic states [72]. The obtained anisotropy of superconducting current around the single vortex in AFA theory agrees reasonably with that found from the Eilenberger equations [96]. Extending electronic states also results in the observed field dependent flattening of *λeff*(*B*) at low temperatures [73]. Thus, our microscopical consideration justifies the phenomenological AFA model and the separation between localized and extended states appears to be quite reasonable.

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