**3. The cutoff parameter in the mixed state of** *dx*<sup>2</sup>−*y*<sup>2</sup> **-wave pairing**

### **symmetry**

A nontrivial orbital structure of the order parameter, in particular the presence of the gap nodes, leads to an effect in which the disorder is much richer in *dx*<sup>2</sup>−*y*<sup>2</sup> -wave superconductors than in conventional materials. For instance, in contrast to the *s*-wave case, the Anderson theorem does not work, and nonmagnetic impurities exhibit a strong pair-breaking effect. In addition, a finite concentration of disorder produces a nonzero density of quasiparticle states at zero energy, which results in a considerable modification of the thermodynamic and transport properties at low temperatures. For a pure superconductor in a *d*-wave-like state at temperatures *T* well below the critical temperature *Tc*, the deviation Δ*λ* of the penetration depth from its zero-temperature value *λ*(0) is proportional to *T*. When the concentration *ni* of strongly scattering impurities is nonzero, <sup>Δ</sup>*<sup>λ</sup>* <sup>∝</sup> *<sup>T</sup>n*, where *<sup>n</sup>* <sup>=</sup> 2 for *<sup>T</sup>* <sup>&</sup>lt; *<sup>T</sup>*<sup>∗</sup> � *Tc* and *<sup>n</sup>* <sup>=</sup> <sup>1</sup> for *T*<sup>∗</sup> < *T* � *Tc* [24]. Unlike *s*-wave superconductor, impurity scattering suppresses both the transition temperature *Tc* and the upper critical field *Hc*2(*T*) [82].

The presence of the nodes in the superconducting gap can also result in unusual properties of the vortex state in *dx*<sup>2</sup>−*y*<sup>2</sup> -wave superconductors. At intermediate fields *Hc*<sup>1</sup> <sup>&</sup>lt; *<sup>H</sup>* � *Hc*2, properties of the flux lattice are determined primarily by the superfluid response of the condensate, i.e., by the relation between the supercurrent�*j* and the superfluid velocity �*vs*. In conventional isotropic strong type-II superconductors, this relation is to a good approximation that of simple proportionality,

$$
\vec{j} = -e\rho\_{\text{s}}\vec{v}\_{\text{s}} \tag{16}
$$

Nonlocal corrections to Eq. (16) become important for the response of electrons with momenta on the Fermi surface close to the gap nodes, even in strongly type-II materials. This can be understood by realizing that the coherence length, being inversely proportional to the gap [85], becomes very large close to the node and formally diverges at the nodal point. Thus, quite generally, there exists a locus of points on the Fermi surface where *ξ* � *λ*<sup>0</sup> and the response becomes highly nonlocal. This effect was first discussed in Refs. [72, 86] in the mixed state. Similarly, the nonlinear corrections become important in a *d*-wave superconductors. Eq. (17) indicates that finite areas of gapless excitations appear near the node for arbitrarily small *vs*. Low temperature physics of the vortex state in *s*-wave superconductors is connected with the nature of the current-carrying quantum states of the quasiparticles in the vortex core (formed due to particle-hole coherence and Andreev reflection [87]). The current distribution can be decomposed in terms of bound states and extended states contributions [88]. Close to the vortex core, the current density arises mainly from the occupation of the bound states. The effect of extended states becomes important only at distances larger than the coherence length. The bound states and the extended states contributions to the current density have opposite signs. The current density originating from the bound states is paramagnetic, whereas extended states contribute a diamagnetic term. At distances larger than the penetration depth, the paramagnetic and diamagnetic parts essentially cancel out each other, resulting in exponential decay of the total current density. The vortex core structure in the *d*-wave superconductors can be more complicated because there are important contributions coming from core states, which extend far from the vortex core into the nodal directions and significantly effect the density of states at low energy [89]. The possibility of the bound states forming in the vortex core of *d*-wave superconductors was widely discussed in terms of the Bogoliubov-de Gennes equation. For example, Franz and Tesanovi ˆ c claimed ˆ that there should be no bound states [90]. However, a considerable number of bound states were found in Ref.[91] which were localized around the vortex core. Extended states, which are rather uniform, for |*E*| < Δ where *E* is the quasiparticle energy and Δ is the asymptotic value of the order parameter, were also found far away from the vortex. In the problem of the bound states, the conservation of the angular momentum around the vortex is important. In spite of the strict conservation of the angular momentum it is broken due to the fourfold symmetry of Δ(*k*), however, the angular momentum is still conserved by modulo 4, and this

Eilenberger Approach to the Vortex State in Iron Pnictide Superconductors 209

Taking into account all these effects, the applicability of EHC theory regarding the description of the vortex state in *dx*<sup>2</sup>−*y*<sup>2</sup> -wave superconductors is not evident *apriori*. In this chapter, we numerically solve the quasiclassical Eilenberger equations for the mixed state of a *dx*<sup>2</sup>−*y*<sup>2</sup> -wave superconductor for the pairing potential Δ(*θ*,**r**) = Δ(**r**) cos (2*θ*), where *θ* is the angle between the **k** vector and the *a* axis (or *x* axis). We check the applicability of Eq. (1) and find the cutoff parameter *ξh*. The anisotropic extension of Eq. (1) to Amin-Franz-Affleck will be discussed in

To consider the mixed state of a *d*-wave superconductor we take the center of the vortex as the origin and assume that the Fermi surface is isotropic and cylindrical. The Riccatti equations

**<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>+</sup> <sup>Δ</sup> <sup>−</sup> *<sup>a</sup>*2Δ∗, (18)

**<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> <sup>Δ</sup><sup>∗</sup> <sup>+</sup> *<sup>b</sup>*2Δ, (19)

is adequate to guarantee the presence of bound states.

for *dx*<sup>2</sup>−*y*<sup>2</sup> -wave superconductivity are [92]

chapter 4.

where *ρs* is a superfluid density. More generally, however, this relation can be both nonlocal and nonlinear. The concept of nonlocal response dates is a return to the ideas of Pippard [83] and is related to the fact that the current response must be averaged over the finite size of the Cooper pair given by the coherence length *ξ*0. In strongly type-II materials the magnetic field varies on a length scale given by the London penetration depth *λ*0, which is much larger than *ξ*<sup>0</sup> and, therefore, nonlocality is typically unimportant unless there exist strong anisotropies in the electronic band structure [84]. Nonlinear corrections arise from the change of quasiparticle population due to superflow which, to the leading order, modifies the excitation spectrum by a quasiclassical Doppler shift [85]

$$
\varepsilon\_k = E\_k + \vec{v}\_f \vec{v}\_{s\prime} \tag{17}
$$

where *Ek* = *�*2 *<sup>k</sup>* <sup>+</sup> <sup>Δ</sup><sup>2</sup> *<sup>k</sup>* is the BCS energy. Once again, in clean, fully gapped conventional superconductors, this effect is typically negligible except when the current approaches the pair breaking value. In the mixed state, this happens only in the close vicinity of the vortex cores that occupy a small fraction of the total sample volume at fields well below *Hc*2. The situation changes dramatically when the order parameter has nodes, such as in *dx*<sup>2</sup>−*y*<sup>2</sup> superconductors. Nonlocal corrections to Eq. (16) become important for the response of electrons with momenta on the Fermi surface close to the gap nodes, even in strongly type-II materials. This can be understood by realizing that the coherence length, being inversely proportional to the gap [85], becomes very large close to the node and formally diverges at the nodal point. Thus, quite generally, there exists a locus of points on the Fermi surface where *ξ* � *λ*<sup>0</sup> and the response becomes highly nonlocal. This effect was first discussed in Refs. [72, 86] in the mixed state. Similarly, the nonlinear corrections become important in a *d*-wave superconductors. Eq. (17) indicates that finite areas of gapless excitations appear near the node for arbitrarily small *vs*.

10 Will-be-set-by-IN-TECH

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

A nontrivial orbital structure of the order parameter, in particular the presence of the gap nodes, leads to an effect in which the disorder is much richer in *dx*<sup>2</sup>−*y*<sup>2</sup> -wave superconductors than in conventional materials. For instance, in contrast to the *s*-wave case, the Anderson theorem does not work, and nonmagnetic impurities exhibit a strong pair-breaking effect. In addition, a finite concentration of disorder produces a nonzero density of quasiparticle states at zero energy, which results in a considerable modification of the thermodynamic and transport properties at low temperatures. For a pure superconductor in a *d*-wave-like state at temperatures *T* well below the critical temperature *Tc*, the deviation Δ*λ* of the penetration depth from its zero-temperature value *λ*(0) is proportional to *T*. When the concentration *ni* of strongly scattering impurities is nonzero, <sup>Δ</sup>*<sup>λ</sup>* <sup>∝</sup> *<sup>T</sup>n*, where *<sup>n</sup>* <sup>=</sup> 2 for *<sup>T</sup>* <sup>&</sup>lt; *<sup>T</sup>*<sup>∗</sup> � *Tc* and *<sup>n</sup>* <sup>=</sup> <sup>1</sup> for *T*<sup>∗</sup> < *T* � *Tc* [24]. Unlike *s*-wave superconductor, impurity scattering suppresses both the

The presence of the nodes in the superconducting gap can also result in unusual properties of the vortex state in *dx*<sup>2</sup>−*y*<sup>2</sup> -wave superconductors. At intermediate fields *Hc*<sup>1</sup> <sup>&</sup>lt; *<sup>H</sup>* � *Hc*2, properties of the flux lattice are determined primarily by the superfluid response of the condensate, i.e., by the relation between the supercurrent�*j* and the superfluid velocity �*vs*. In conventional isotropic strong type-II superconductors, this relation is to a good approximation

where *ρs* is a superfluid density. More generally, however, this relation can be both nonlocal and nonlinear. The concept of nonlocal response dates is a return to the ideas of Pippard [83] and is related to the fact that the current response must be averaged over the finite size of the Cooper pair given by the coherence length *ξ*0. In strongly type-II materials the magnetic field varies on a length scale given by the London penetration depth *λ*0, which is much larger than *ξ*<sup>0</sup> and, therefore, nonlocality is typically unimportant unless there exist strong anisotropies in the electronic band structure [84]. Nonlinear corrections arise from the change of quasiparticle population due to superflow which, to the leading order, modifies the excitation spectrum by

superconductors, this effect is typically negligible except when the current approaches the pair breaking value. In the mixed state, this happens only in the close vicinity of the vortex cores that occupy a small fraction of the total sample volume at fields well below *Hc*2. The situation changes dramatically when the order parameter has nodes, such as in *dx*<sup>2</sup>−*y*<sup>2</sup> superconductors.

�*<sup>j</sup>* <sup>=</sup> <sup>−</sup>*eρs*�*vs*, (16)

*ε<sup>k</sup>* = *Ek* +�*v <sup>f</sup>*�*vs*, (17)

*<sup>k</sup>* is the BCS energy. Once again, in clean, fully gapped conventional

**3. The cutoff parameter in the mixed state of** *dx*<sup>2</sup>−*y*<sup>2</sup> **-wave pairing**

in *μ*SR measurements in LiFeAs [81] supports the *s*± pairing.

transition temperature *Tc* and the upper critical field *Hc*2(*T*) [82].

**symmetry**

that of simple proportionality,

a quasiclassical Doppler shift [85]

 *�*2 *<sup>k</sup>* <sup>+</sup> <sup>Δ</sup><sup>2</sup>

where *Ek* =

Low temperature physics of the vortex state in *s*-wave superconductors is connected with the nature of the current-carrying quantum states of the quasiparticles in the vortex core (formed due to particle-hole coherence and Andreev reflection [87]). The current distribution can be decomposed in terms of bound states and extended states contributions [88]. Close to the vortex core, the current density arises mainly from the occupation of the bound states. The effect of extended states becomes important only at distances larger than the coherence length. The bound states and the extended states contributions to the current density have opposite signs. The current density originating from the bound states is paramagnetic, whereas extended states contribute a diamagnetic term. At distances larger than the penetration depth, the paramagnetic and diamagnetic parts essentially cancel out each other, resulting in exponential decay of the total current density. The vortex core structure in the *d*-wave superconductors can be more complicated because there are important contributions coming from core states, which extend far from the vortex core into the nodal directions and significantly effect the density of states at low energy [89]. The possibility of the bound states forming in the vortex core of *d*-wave superconductors was widely discussed in terms of the Bogoliubov-de Gennes equation. For example, Franz and Tesanovi ˆ c claimed ˆ that there should be no bound states [90]. However, a considerable number of bound states were found in Ref.[91] which were localized around the vortex core. Extended states, which are rather uniform, for |*E*| < Δ where *E* is the quasiparticle energy and Δ is the asymptotic value of the order parameter, were also found far away from the vortex. In the problem of the bound states, the conservation of the angular momentum around the vortex is important. In spite of the strict conservation of the angular momentum it is broken due to the fourfold symmetry of Δ(*k*), however, the angular momentum is still conserved by modulo 4, and this is adequate to guarantee the presence of bound states.

Taking into account all these effects, the applicability of EHC theory regarding the description of the vortex state in *dx*<sup>2</sup>−*y*<sup>2</sup> -wave superconductors is not evident *apriori*. In this chapter, we numerically solve the quasiclassical Eilenberger equations for the mixed state of a *dx*<sup>2</sup>−*y*<sup>2</sup> -wave superconductor for the pairing potential Δ(*θ*,**r**) = Δ(**r**) cos (2*θ*), where *θ* is the angle between the **k** vector and the *a* axis (or *x* axis). We check the applicability of Eq. (1) and find the cutoff parameter *ξh*. The anisotropic extension of Eq. (1) to Amin-Franz-Affleck will be discussed in chapter 4.

To consider the mixed state of a *d*-wave superconductor we take the center of the vortex as the origin and assume that the Fermi surface is isotropic and cylindrical. The Riccatti equations for *dx*<sup>2</sup>−*y*<sup>2</sup> -wave superconductivity are [92]

$$\mathbf{u} \cdot \nabla a = -a \left[ 2(\omega\_{\mathrm{il}} + \mathbf{G}) + i \mathbf{u} \cdot \mathbf{A}\_{\mathrm{s}} \right] + \Delta - a^2 \Delta^\* \,\mathrm{\,} \tag{18}$$

$$\mathbf{u} \cdot \nabla b = b \left[ 2(\omega\_{\rm nl} + G) + i \mathbf{u} \cdot \mathbf{A}\_{\rm s} \right] - \Delta^\* + b^2 \Delta,\tag{19}$$

#### 12 Will-be-set-by-IN-TECH 210 Superconductors – Materials, Properties and Applications Eilenberger Approach to the Vortex State in Iron Pnictide Superconductors <sup>13</sup>

where *G* = 2*π* � *g*�Γ with *d*-wave pairing potential Δ(*r*)

$$\Delta(\theta, \mathbf{r}) = V\_{d\_{\mathbf{z}^2 - \mathbf{y}^2}}^{\text{SC}} 2\pi T \cos \left(2\theta \right) \sum\_{\omega\_n > 0}^{\omega\_\varepsilon} \int\_0^{2\pi} \frac{d\bar{\theta}}{2\pi} f(\omega\_{\text{n}}, \bar{\theta}, \mathbf{r}) \cos(2\bar{\theta}), \tag{20}$$

where *VSC dx*<sup>2</sup>−*y*<sup>2</sup> is a coupling constant in the *dx*<sup>2</sup>−*y*<sup>2</sup> pairing channel. The obtained solution is fitted to Eq. (1) giving the value of cutoff parameter *<sup>ξ</sup><sup>h</sup>* for *dx*<sup>2</sup>−*y*<sup>2</sup> -wave pairing symmetry.

**Figure 6.** (Color online) (a) The temperature dependence of superfluid density *ρS*(*T*)/*ρS*<sup>0</sup> with different values of impurity scattering Γ. (b) The temperature dependence of the upper critical field *Bc*<sup>2</sup> with different values of impurity scattering Γ.

In *dx*<sup>2</sup>−*y*<sup>2</sup> -wave superconductor *<sup>λ</sup>*(*T*) in Eq. (1) is given as [85]

$$\frac{\lambda\_{L0}^2}{\lambda^2(T)} = 2\pi T \oint \frac{d\theta}{2\pi} \sum\_{\omega\_n > 0} \frac{|\tilde{\Delta}(\theta)|^2}{(\tilde{\omega}\_n^2 + |\tilde{\Delta}(\theta)|^2)^{3/2}},\tag{21}$$

**Figure 7.** (Color online) Normalized differences between the fields calculated with the London model and the Eilenberger equation for *dx*<sup>2</sup>−*y*<sup>2</sup> -wave pairing with <sup>Γ</sup> <sup>=</sup> 0.03, *<sup>B</sup>*/*Bc*<sup>2</sup> <sup>=</sup> 0.1 and *<sup>T</sup>*/*Tc*<sup>0</sup> <sup>=</sup> 0.3.

where *<sup>v</sup>* <sup>=</sup> <sup>2</sup>Γ, *<sup>t</sup>* <sup>=</sup> *<sup>T</sup>*/*Tc*0, *tc* <sup>=</sup> *Tc*/*Tc*<sup>0</sup> and *<sup>x</sup>* <sup>=</sup> *<sup>ρ</sup>u*2(<sup>1</sup> <sup>−</sup> *<sup>z</sup>*2), *<sup>ρ</sup>* <sup>=</sup> *<sup>B</sup>*/(4*πt*)2. Fig. 6 (b) depicts the temperature dependence of the upper critical field *Bc*<sup>2</sup> with different values of impurity scattering Γ. Figs. 6 (a) and (b) are similar to those in *s*±-wave superconductors. *Tc* is suppressed by impurity scattering resulting in the same expressions for *s*± and *d*-wave

**Figure 8.** (Color online) (a) The magnetic field dependence of the cutoff parameter *ξh*/*ξc*<sup>2</sup> with different temperatures (*T*/*Tc*<sup>0</sup> <sup>=</sup> 0.2, 0.3, 0.4, 0.5, 0.7, 0.8) for *dx*<sup>2</sup>−*y*<sup>2</sup> pairing with <sup>Γ</sup> <sup>=</sup> 0. (b) The impurity scattering

Fig. 7 shows the normalized differences between the fields calculated with the London model and the Eilenberger equations for *dx*<sup>2</sup>−*y*<sup>2</sup> -wave pairing symmetry for the values of <sup>Γ</sup> <sup>=</sup> 0.03,

Fig. 8 (a) demonstrates the magnetic field dependence of cutoff parameter *ξh*/*ξc*<sup>2</sup> at different temperatures (*T*/*Tc*<sup>0</sup> <sup>=</sup> 0.2, 0.3, 0.4, 0.5, 0.7, 0.8) for *dx*<sup>2</sup>−*y*<sup>2</sup> pairing with <sup>Γ</sup> <sup>=</sup> 0. Fig. 8 (b) shows the impurity scattering <sup>Γ</sup> dependence of *<sup>ξ</sup>h*/*ξc*<sup>2</sup> at different temperatures for *dx*<sup>2</sup>−*y*<sup>2</sup>

<sup>Γ</sup> dependence of *<sup>ξ</sup>h*/*ξc*<sup>2</sup> at different temperatures for *dx*<sup>2</sup>−*y*<sup>2</sup> pairing with *<sup>B</sup>* <sup>=</sup> 5.

*B*/*Bc*<sup>2</sup> = 0.1 and *T*/*Tc*<sup>0</sup> = 0.3. The accuracy of the fitting is better than 2%.

) + Ψ( 1 2 + *v* 2*t* )] =

*dz*|(<sup>1</sup> <sup>−</sup> *<sup>z</sup>*2)[*e*−*x*(−*<sup>x</sup>* <sup>+</sup> *<sup>c</sup>*(<sup>1</sup> <sup>−</sup> <sup>4</sup>*<sup>x</sup>* <sup>+</sup> <sup>2</sup>*x*2)) <sup>−</sup> *<sup>c</sup>*]*e*<sup>−</sup> *<sup>v</sup>*

Eilenberger Approach to the Vortex State in Iron Pnictide Superconductors 211

*<sup>t</sup> <sup>u</sup>*, (26)

*<sup>c</sup>*[ln( *<sup>T</sup> Tc*

> 1 0

*du shu <sup>x</sup>*

superconductors with replacing Γ*<sup>π</sup>* → Γ/2.

<sup>=</sup> <sup>3</sup> 2 ∞ 0

) − Ψ( 1 2 + *v* 2*tc*

where

$$
\tilde{\omega}\_n = \omega\_n + \Gamma \langle \frac{\tilde{\omega}\_n}{\sqrt{\tilde{\omega}\_n^2 + |\tilde{\Delta}(\tilde{p}\_f'; \omega\_n)|^2}} \rangle\_{\tilde{p}\_f'} \tag{22}
$$

$$
\tilde{\Delta}(\vec{p}\_{f'};\omega\_{\mathfrak{n}}) = \Delta(\vec{p}\_{f}) + \Gamma \langle \frac{\tilde{\Delta}(\vec{p}'\_{f'};\omega\_{\mathfrak{n}})}{\sqrt{\tilde{\omega}\_{\mathfrak{n}}^2 + |\tilde{\Delta}(\vec{p}'\_{f'};\omega\_{\mathfrak{n}})|^2}} \rangle\_{\vec{p}'\_{f}} \tag{23}
$$

$$\Delta(\vec{p}\_f) = \int d\vec{p}'\_f V(\vec{p}\_f, \vec{p}'\_f) \pi T \sum\_{\omega\_n}^{|\omega\_n| < \omega\_\varepsilon} \frac{\tilde{\Delta}(\vec{p}'\_f)}{\sqrt{\tilde{\omega}\_n^2 + |\tilde{\Delta}(\vec{p}'\_f)|^2}}. \tag{24}$$

Because of the symmetry of *dx*<sup>2</sup>−*y*<sup>2</sup> -wave pairing the impurity induced corrections for the pairing potential in Eq. (23) are zero and Δ˜ = Δ. This is different from the *s*±- and *s*++ cases, where the corrections are not zero. Fig. 6 (a) 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 <sup>Γ</sup> for *dx*<sup>2</sup>−*y*<sup>2</sup> -wave pairing symmetry.

To study high the field regime we need to calculate the upper critical field *Bc*2(*T*). For *dx*<sup>2</sup>−*y*<sup>2</sup> -wave *Bc*2(*T*) is given as [82]

$$\ln(\frac{T}{T\_c}) - \Psi(\frac{1}{2} + \frac{v}{2t\_c}) + \Psi(\frac{1}{2} + \frac{v}{2t}) = \frac{3}{2} \int\_0^\infty \frac{du}{\text{shu}} \int\_0^1 dz (1 - z^2) [e^{-x} (1 - 2xc)^{-1}] e^{-\frac{v}{t}u},\tag{25}$$

12 Will-be-set-by-IN-TECH

*ω<sup>c</sup>* ∑ *ωn*>0

fitted to Eq. (1) giving the value of cutoff parameter *<sup>ξ</sup><sup>h</sup>* for *dx*<sup>2</sup>−*y*<sup>2</sup> -wave pairing symmetry.

**Figure 6.** (Color online) (a) The temperature dependence of superfluid density *ρS*(*T*)/*ρS*<sup>0</sup> with different values of impurity scattering Γ. (b) The temperature dependence of the upper critical field *Bc*<sup>2</sup> with

> *dθ* <sup>2</sup>*<sup>π</sup>* ∑ *ωn*>0

*<sup>ω</sup>*˜ *<sup>n</sup>* <sup>=</sup> *<sup>ω</sup><sup>n</sup>* <sup>+</sup> <sup>Γ</sup>� *<sup>ω</sup>*˜ *<sup>n</sup>*

<sup>Δ</sup>˜(�*<sup>p</sup> <sup>f</sup>* ; *<sup>ω</sup>n*) = <sup>Δ</sup>(�*<sup>p</sup> <sup>f</sup>*) + <sup>Γ</sup>�

*ω*˜ 2 *<sup>n</sup>* <sup>+</sup> <sup>|</sup>Δ˜ (�*<sup>p</sup>* � *f* ; *<sup>ω</sup>n*)|<sup>2</sup> � �*p* � *f*

> *ω*˜ 2 *<sup>n</sup>* <sup>+</sup> <sup>|</sup>Δ˜ (�*<sup>p</sup>* � *f* ; *<sup>ω</sup>n*)|<sup>2</sup> � �*p* � *f*

Because of the symmetry of *dx*<sup>2</sup>−*y*<sup>2</sup> -wave pairing the impurity induced corrections for the pairing potential in Eq. (23) are zero and Δ˜ = Δ. This is different from the *s*±- and *s*++ cases, where the corrections are not zero. Fig. 6 (a) shows the calculated temperature dependence of

To study high the field regime we need to calculate the upper critical field *Bc*2(*T*). For

*du shu*  1 0

 2*π* 0

*dx*<sup>2</sup>−*y*<sup>2</sup> is a coupling constant in the *dx*<sup>2</sup>−*y*<sup>2</sup> pairing channel. The obtained solution is

*d* ¯ *θ* <sup>2</sup>*<sup>π</sup> <sup>f</sup>*(*ωn*, ¯

<sup>|</sup>Δ˜(*θ*)<sup>|</sup> 2

> Δ˜ (�*p* � *f*)

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

*dz*(<sup>1</sup> <sup>−</sup> *<sup>z</sup>*2)[*e*−*x*(<sup>1</sup> <sup>−</sup> <sup>2</sup>*xc*)−1]*e*<sup>−</sup> *<sup>v</sup>*

*<sup>n</sup>* <sup>+</sup> <sup>|</sup>Δ˜ (*θ*)|2)3/2 , (21)

, (22)

, (23)

. (24)

*<sup>t</sup> <sup>u</sup>*, (25)

(*ω*˜ <sup>2</sup>

Δ˜ (�*p* � *<sup>f</sup>* ; *ωn*)

> *ω*˜ 2 *<sup>n</sup>* <sup>+</sup> <sup>|</sup>Δ˜ (�*<sup>p</sup>* � *f* )|2


*θ*,**r**)*cos*(2 ¯

*θ*), (20)

where *G* = 2*π* � *g*�Γ with *d*-wave pairing potential Δ(*r*)

*dx*<sup>2</sup>−*y*<sup>2</sup>

In *dx*<sup>2</sup>−*y*<sup>2</sup> -wave superconductor *<sup>λ</sup>*(*T*) in Eq. (1) is given as [85]

*λ*2 *L*0 *<sup>λ</sup>*2(*T*) <sup>=</sup> <sup>2</sup>*π<sup>T</sup>*

Δ(�*p <sup>f</sup>*) =

) + Ψ( 1 2 + *v* 2*t* ) = <sup>3</sup> 2 ∞ 0

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

for *dx*<sup>2</sup>−*y*<sup>2</sup> -wave pairing symmetry.

*dx*<sup>2</sup>−*y*<sup>2</sup> -wave *Bc*2(*T*) is given as [82]

 *d*�*p* � *<sup>f</sup> V*(�*p <sup>f</sup>* ,�*p* � *<sup>f</sup>*)*πT*

2*πT* cos (2*θ*)

Δ(*θ*,**r**) = *VSC*

different values of impurity scattering Γ.

where *VSC*

where

ln( *<sup>T</sup> Tc* ) − Ψ( 1 2 + *v* 2*tc*

**Figure 7.** (Color online) Normalized differences between the fields calculated with the London model and the Eilenberger equation for *dx*<sup>2</sup>−*y*<sup>2</sup> -wave pairing with <sup>Γ</sup> <sup>=</sup> 0.03, *<sup>B</sup>*/*Bc*<sup>2</sup> <sup>=</sup> 0.1 and *<sup>T</sup>*/*Tc*<sup>0</sup> <sup>=</sup> 0.3.

$$c[\ln(\frac{T}{T\_c}) - \Psi(\frac{1}{2} + \frac{v}{2t\_c}) + \Psi(\frac{1}{2} + \frac{v}{2t})] = $$

$$c = \frac{3}{2} \int\_0^\infty \frac{du}{du} \ge \int\_0^1 dz \left| (1 - z^2) [e^{-x}(-x + c(1 - 4x + 2x^2)) - c] e^{-\frac{v}{7}u} \right| \tag{26}$$

where *<sup>v</sup>* <sup>=</sup> <sup>2</sup>Γ, *<sup>t</sup>* <sup>=</sup> *<sup>T</sup>*/*Tc*0, *tc* <sup>=</sup> *Tc*/*Tc*<sup>0</sup> and *<sup>x</sup>* <sup>=</sup> *<sup>ρ</sup>u*2(<sup>1</sup> <sup>−</sup> *<sup>z</sup>*2), *<sup>ρ</sup>* <sup>=</sup> *<sup>B</sup>*/(4*πt*)2. Fig. 6 (b) depicts the temperature dependence of the upper critical field *Bc*<sup>2</sup> with different values of impurity scattering Γ. Figs. 6 (a) and (b) are similar to those in *s*±-wave superconductors. *Tc* is suppressed by impurity scattering resulting in the same expressions for *s*± and *d*-wave superconductors with replacing Γ*<sup>π</sup>* → Γ/2.

**Figure 8.** (Color online) (a) The magnetic field dependence of the cutoff parameter *ξh*/*ξc*<sup>2</sup> with different temperatures (*T*/*Tc*<sup>0</sup> <sup>=</sup> 0.2, 0.3, 0.4, 0.5, 0.7, 0.8) for *dx*<sup>2</sup>−*y*<sup>2</sup> pairing with <sup>Γ</sup> <sup>=</sup> 0. (b) The impurity scattering <sup>Γ</sup> dependence of *<sup>ξ</sup>h*/*ξc*<sup>2</sup> at different temperatures for *dx*<sup>2</sup>−*y*<sup>2</sup> pairing with *<sup>B</sup>* <sup>=</sup> 5.

Fig. 7 shows the normalized differences between the fields calculated with the London model and the Eilenberger equations for *dx*<sup>2</sup>−*y*<sup>2</sup> -wave pairing symmetry for the values of <sup>Γ</sup> <sup>=</sup> 0.03, *B*/*Bc*<sup>2</sup> = 0.1 and *T*/*Tc*<sup>0</sup> = 0.3. The accuracy of the fitting is better than 2%.

Fig. 8 (a) demonstrates the magnetic field dependence of cutoff parameter *ξh*/*ξc*<sup>2</sup> at different temperatures (*T*/*Tc*<sup>0</sup> <sup>=</sup> 0.2, 0.3, 0.4, 0.5, 0.7, 0.8) for *dx*<sup>2</sup>−*y*<sup>2</sup> pairing with <sup>Γ</sup> <sup>=</sup> 0. Fig. 8 (b) shows the impurity scattering <sup>Γ</sup> dependence of *<sup>ξ</sup>h*/*ξc*<sup>2</sup> at different temperatures for *dx*<sup>2</sup>−*y*<sup>2</sup>

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

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