**How to Distinguish the Mixture of Two D-wave States from Pure D-wave State of HTSC**

Peter Brusov and Tatiana Filatova

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/59180

#### **1. Introduction**

The chapter is devoted to the very important problem of the actual symmetry of the order parameter in cuprates, and, more generally, in unconventional superconductors. We adopt the concept of the mixed-order parameter. We calculated collective modes in the *dx2–y2 +idxy* case and found them to differ from those appropriate to the pure situation. The results are useful for both theoretical as well as experimental aspects of the discussed problem.

Several groups have worked in this direction and we can indicate the contributions of Tony Leggett, James Annett, David Pines, Doug Scalapino, and Sasha Balatsky et al. in the theory of the problem and of John Ketterson et al. in its experimental study.

In this chapter we look at collective excitations in unconventional superconductors (USC). This unusual topic was chosen for two reasons: 1) there is no superconductor for which unconven‐ tional pairing has been exactly established, while there is some evidence of nontrivial pairing in HFSC and HTSC; 2)the existence of collective excitations in superconductors is questionable.

Within the last couple of decades the situation has drastically changed, collective excitations becoming increasingly important in studies and experiments on USC.

An amplitude mode has been observed in ordinary superconductor films with frequency of order 2∆. Furthermore, the type of pairing has been established for most superconductors. There is an s-pairing in electron-type HTSC and in ordinary superconductors, a p-pairing in pure 3 He; 3 He in aerogel, Sr2RuO4 (HTSC), UPt3 (HFSC), and d-pairing in hole-type HTSC, organic superconductors, and some HFSC (UPd2Al3, CePd2Si2, CeIn3, CeNi2Ge2, etc.).

In 2002 a microwave surface-impedance study of the HFSC UBe13 demonstrated an absorption peak [16]. The frequency-dependence as well as temperature-dependence of this peak scales

with the BCS gap function ∆(T). This means that this was the first direct observation of the resonant absorption into a collective mode (CM). The CM energy turned out to be proportional to the superconducting gap. This was therefore a new page in the study of collective excitations in USC.

**Figure 1.** Temperature dependence of surface impedance in heavy-fermion superconductor UBe13 [Ref. 16].

**Figure 2.** Normalized frequency of collective mode in heavy-fermion superconductor UBe13 [Ref. 16]

#### **2. Two d-wave states mixture**

Ashasbeenshownbothexperimentally[1] andtheoretically,[2, 3]inHTSCamixingofdifferent d-wave states takes place. The collective-mode spectrum in a mixed *dx2–y2 +idxy* state of HTSC was calculatedbyPaulBrusovandPeterBrusov[4]forthe firsttime.Withinthed-pairingmodel for superconductiveandsuperfluidFermi-systems (HTSC,HFSC,etc.)earliercreatedbyBrusov and Brusov within the path integration technique, [5-7] they showed that the spectrum in mixture *dx2–y2 +idxy* state turns out to be quite different from spectra in both states *dx2–y2* and *dxy* in spite of the fact that both spectra are identical. [8, 10] Thus the ultrasound and/or microwave absorption experiments could be used as the probe of the CM spectrum, allowing the pure dwave states to be distinguished from the mixture of the two d-wave states [18].

While most scientists believe that there is a *d*-wave pairing in HTSC, there is still an active debate over different ideas concerning mixture of *s*- and *d*-states, extended s-wave pairing, and mixture of different *d*-states [18]. The main cause of this is the absence of answers to the question of whether there is an exact-zero gap along some chosen lines in momentum space (as in the case of *dx2–y2*) or an anisotropic gap that remains nonzero everywhere (except maybe at some points). There is no certain answer to this question that has been yielded by existing experiments (tunnelling, etc.) but the answer is quite principled. However, there are some experiments [1] which, suggesting realization in HTSC in a mixed state like *dx2–y2+idxy*, could provide an explanation [3]. The possibility of a mixture of different d-wave states in HTSC has been considered by Annett et al., [2] who came to the conclusion that the most likely state is *dx2–y2+idxy*. [18]

**Figure 3.** Gap symmetry in pure *d*-wave state (*dxy* -state).

with the BCS gap function ∆(T). This means that this was the first direct observation of the resonant absorption into a collective mode (CM). The CM energy turned out to be proportional to the superconducting gap. This was therefore a new page in the study of collective excitations

**Figure 1.** Temperature dependence of surface impedance in heavy-fermion superconductor UBe13 [Ref. 16].

**Figure 2.** Normalized frequency of collective mode in heavy-fermion superconductor UBe13 [Ref. 16]

Ashasbeenshownbothexperimentally[1] andtheoretically,[2, 3]inHTSCamixingofdifferent d-wave states takes place. The collective-mode spectrum in a mixed *dx2–y2 +idxy* state of HTSC was calculatedbyPaulBrusovandPeterBrusov[4]forthe firsttime.Withinthed-pairingmodel for superconductiveandsuperfluidFermi-systems (HTSC,HFSC,etc.)earliercreatedbyBrusov and Brusov within the path integration technique, [5-7] they showed that the spectrum in

**2. Two d-wave states mixture**

in USC.

184 Superconductors – New Developments

One possible way to distinguish pure *d*-states from mixture was suggested by Peter Brusov and Pavel Brusov[4], who considered the mixed *dx2–y2+idxy* state and calculated the spectrum of collective modes in this state. The comparison of the spectrum of a pure d-wave state of HTSC with the spectrum of the mixed *dx2–y2+idxy* state shows that they are significantly different. This means that they could be used as the probe of the symmetry of the order parameter in HTSC. [18]

**Figure 4.** Gap symmetry in the mixed s-wave and d-wave states (*s+idxy*).

#### **3. Collective-Modes Spectrum Equations in a Mixed d-Wave State**

#### **3.1. Mixed d-wave-state model**

Brusov et al.'s [4] study is generalized here for the case of arbitrary *dxy*-state admixture. The mixed (1*–γ*)*dx 2 –y2 + iγdxy* state in high-temperature superconductors is considered here and a full set of equations for the collective-modes spectrum in mixed d-wave state with arbitrary admixture of dxy-state is derived.

**Figure 5.** Gap symmetry in the mixed (1*–γ*)*dx 2 –y2 + iγdxy* state.

We have used the model of *d*-pairing in high-temperature superconductors and HFSC, created by Brusov et al.[5, 11]

It is described by the effective functional of action:

$$\mathcal{S}\_{\text{eff}} = g^{-1} \sum\_{p,i,\boldsymbol{\mu}} c\_{\text{ia}}^{+}(p) c\_{\text{ia}}(p) + \frac{1}{2} \ln \det \frac{\hat{M}\left(c\_{\text{ia}}, c\_{\text{ia}}^{+}\right)}{\hat{M}\left(c\_{\text{ia}}^{(0)}, c\_{\text{ia}}^{(0)\*}\right)}\,\text{}\tag{1}$$

where *cia* (0) is the condensate value of Bose-fields *cia* (symmetric traceless matrix) and *<sup>M</sup>* ^ (*cia*, *cia* + ) is the 4×4 depending on Bose-fields and parameters of quasi-fermions matrix.

In the case of the *d*-pairing the number of degrees of freedom is equal to 10. This means that we have five complex canonical variables:

$$\begin{aligned} \mathcal{c}\_1 &= \mathcal{c}\_{11} + \mathcal{c}\_{22}, \mathcal{c}\_2 = \mathcal{c}\_{11} - \mathcal{c}\_{22}, \mathcal{c}\_3 = \mathcal{c}\_{12} + \mathcal{c}\_{21}, \\ \mathcal{c}\_4 &= \mathcal{c}\_{13} + \mathcal{c}\_{31'}, \mathcal{c}\_5 = \mathcal{c}\_{23} + \mathcal{c}\_{32}. \end{aligned}$$

The effective action becomes equal to:

$$S\_{\rm eff} = \left(2\,\mathrm{g}\right)^{-1} \sum\_{p,j} c\_{j}^{+}\left(p\right) c\_{j}\left(p\right)\left(1 + 2\,\delta\_{j1}\right) + \frac{1}{2} \ln \det \frac{\hat{M}\left(c\_{j}^{+}, c\_{j}^{+}\right)}{\hat{M}\left(c\_{j}^{+\left(0\right)}, c\_{j}^{\left(0\right)}\right)}\,\mathrm{}\tag{2}$$

where

**3. Collective-Modes Spectrum Equations in a Mixed d-Wave State**

*2 –y2*

*ia ia*

is the 4×4 depending on Bose-fields and parameters of quasi-fermions matrix.

4 13 31 5 23 32

*c c cc c c*

=+ =+

*S g c pc p*


*+ iγdxy* state.

We have used the model of *d*-pairing in high-temperature superconductors and HFSC, created

() () ( ) () () ( ) <sup>1</sup> eff 0 0 , ,

*pia ia ia*

(0) is the condensate value of Bose-fields *cia* (symmetric traceless matrix) and *<sup>M</sup>*

In the case of the *d*-pairing the number of degrees of freedom is equal to 10. This means that

1 11 22 2 11 22 3 12 21

=+ =- =+

, . *c c cc c c c c c*

( ) () ()( ) ( ) () () ( ) <sup>1</sup>

d

*p j j j*

eff <sup>1</sup> 0 0 , <sup>ˆ</sup> , <sup>1</sup> <sup>2</sup> 1 2 ln det , <sup>2</sup> <sup>ˆ</sup> ,

*jj j*

,, ,

<sup>+</sup> <sup>=</sup> å + + (2)

<sup>ˆ</sup> , <sup>1</sup> ln det , <sup>2</sup> <sup>ˆ</sup> ,

*ia ia*

+

*j j*

*Mc c*

+

*Mc c*

^ (*cia*, *cia* + )

*Mc c*

<sup>+</sup> = + å (1)

*Mc c*

Brusov et al.'s [4] study is generalized here for the case of arbitrary *dxy*-state admixture. The

full set of equations for the collective-modes spectrum in mixed d-wave state with arbitrary

*–y<sup>2</sup> + iγdxy* state in high-temperature superconductors is considered here and a

**3.1. Mixed d-wave-state model**

186 Superconductors – New Developments

*2*

admixture of dxy-state is derived.

**Figure 5.** Gap symmetry in the mixed (1*–γ*)*dx*

It is described by the effective functional of action:

we have five complex canonical variables:

The effective action becomes equal to:

*S g c pc p*


by Brusov et al.[5, 11]

where *cia*

mixed (1*–γ*)*dx*

$$\begin{aligned} M\_{11} &= \boldsymbol{Z}^{-1} \Big[ i\boldsymbol{o}\boldsymbol{o} + \boldsymbol{\xi} - \mu \left( \mathbf{H} \boldsymbol{\sigma} \right) \Big] \delta\_{p\_1 p\_2} \\ M\_{22} &= \boldsymbol{Z}^{-1} \Big[ -i\boldsymbol{o}\boldsymbol{o} + \boldsymbol{\xi} + \mu \left( \mathbf{H} \boldsymbol{\sigma} \right) \Big] \delta\_{p\_3 p\_2} \\ M\_{12} &= M\_{21}^\* = \left( \boldsymbol{\beta} \boldsymbol{V} \right)^{-1/2} \left( \frac{15}{32\pi} \right)^{1/2} \left[ c\_1 \left( 1 - 3 \cos^2 \theta \right) + \\ &+ c\_2 \sin^2 \theta \cos 2\phi + c\_3 \sin^2 \theta \sin 2\phi + \\ &+ c\_4 \sin 2\theta \cos \phi + c\_5 \sin 2\theta \sin \phi \right] \end{aligned} \tag{3}$$

here *H*-magnetic field, σ=(σ1, σ2, σ3) – Pauli-matrices, *p* =(**k**, *ω*) ; *ω* =(2*n* + 1)*πT* are Fermifrequencies and *x* =(**x**, *τ*), ξ is the kinetic energy with respect to Fermi-level, μ-magnetic moment of quasi-fermion.

All the properties of the model superconducting Fermi-system with *d*-pairing are determined by functional (2).

Let us consider the mixed (1*–γ*)*dx 2 –y2 + iγdxy* state of high-temperature superconductors. The order parameter in this state takes the following form:

$$
\Delta\_0 \begin{pmatrix} T \\ \end{pmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} + i\gamma \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} \tag{4}$$

or in canonical variables:

$$
\Delta\_o(T) \Big( 0; (1 - \gamma) \sin^2 \theta \cos 2\varphi; i\gamma \sin^2 \theta \sin 2\varphi; 0; 0\Big) \tag{5}
$$

The gap equation has the following form:

$$g^{-1} + \frac{a^2 Z^2}{2\beta V} \sum\_{\boldsymbol{\rho}} \frac{\sin^4 \Theta \left[\boldsymbol{\gamma}^2 + \cos^2 2\phi \left(1 - 2\boldsymbol{\gamma}\right)\right]}{a^2 + \boldsymbol{\xi}^2 + \boldsymbol{\Lambda}\_0^2 \sin^4 \Theta \left[\boldsymbol{\gamma}^2 + \cos^2 2\phi \left(1 - 2\boldsymbol{\gamma}\right)\right]} = 0,\tag{6}$$

where

*<sup>Δ</sup>*<sup>0</sup> <sup>=</sup>2*cZα*, *<sup>α</sup>* <sup>=</sup>(15 / 32*π*)1/2 and gap

$$
\Delta^2 \left( T \right) = \Delta\_0^2 \sin^4 \Theta \left[ \gamma^2 + \cos^2 2\phi (1 - 2\gamma) \right]. \tag{7}
$$

For a limited case of γ=0, we obtain *dx2–y2* state with order parameter (0; sin<sup>2</sup> *θ*cos2*ϕ*;0;0;0). The gap equation in this case has the following form:

$$\log^{-1} + \frac{a^2 Z^2}{2\beta V} \sum\_{\rho} \frac{\sin^4 \Theta \cos^2 2\phi}{\alpha^2 + \xi^2 + \Lambda\_0^2 \sin^4 \Theta \cos^2 2\phi} = 0 \tag{8}$$

and gap

$$
\Delta^2(T) = \Delta\_0^2 \sin^4 \Theta \cos^2 2\phi. \tag{9}
$$

For limited case γ=1 one gets *dxy-*state with order parameter ∆0(*T*) (0;0;*i*sin<sup>2</sup> *θ*sin2*ϕ*;0;0). The gap equation has the following form:

$$\log^{-1} + \frac{\alpha^2 Z^2}{2\beta V} \sum\_{p} \frac{\sin^4 \Theta \sin^2 2\phi}{\alpha^2 + \xi^2 + \Delta\_0^2 \sin^4 \Theta \sin^2 2\phi} = 0 \tag{10}$$

and gap

$$
\Delta^2(T) = \Delta\_0^2 \sin^4 \Theta \sin^2 2\phi \tag{11}
$$

Brusov et al.[8] case of equal admixtures of *dx2–y2 -* an*d dxy*-states in our consideration corre‐ sponds to the case γ=1/2. The order parameter takes the following form:

$$
\Lambda\_0 \binom{T}{0} \left( 0; \sin^2 \theta \cos 2\varphi; i \sin^2 \theta \sin 2\varphi; 0; 0 \right) \tag{12}
$$

The gap equation has the following form:

$$\log^{-1} + \frac{\alpha^2 Z^2}{2\beta V} \sum\_{p} \frac{\sin^4 \Theta}{\alpha^2 + \xi^2 + \Lambda\_0^2 \sin^4 \Theta} = 0 \tag{13}$$

and gap ∆(*T*)=∆0(*T*) sin<sup>2</sup> θ.

#### **3.2. Equations for collective-modes spectrum in a mixed d-wave state at arbitrary admixture of** *dxy***-state**

In the first approximations the spectrum of collective excitations is determined by the quadratic part of *Seff*, obtained after shift *cj* → *cj +cj* o . Here *cj o* are the condensate values of *cj* of the following form:[5, 11]

$$c\_{\nearrow}^{0}(p) = \left(\beta V\right)^{\mathsf{yl}2} c \delta\_{p0} b\_{\nearrow}^{0} \text{ and } b\_{2}^{0} = 2\left(1 - \gamma\right), \ b\_{3}^{0} = 2i\gamma^{0}$$

with all remaining components of *bj* 0 equal to zero.

Excluding terms involving *g* <sup>−</sup><sup>1</sup> by gap equation, one obtains the following form for the quadratic part of *Sh*:

$$\begin{split} S\_{H} &= \frac{\alpha^{2} Z^{2}}{8 \beta V} \sum\_{p} \frac{\left[c^{0} Y^{\*}\right] \left[c^{-0} Y\right]}{\alpha^{2} + \xi^{2} + \left[c^{0} Y^{\*}\right] \left[c^{-0} Y\right]} \sum\_{f} \left(1 + 2 \delta f \mathbf{1}\right) c^{+}\_{f} \left(p\right) c\_{f} \left(p\right) \\ &+ 2^{2} \Big/ 4 \beta V \sum\_{p + p = 2-p} \frac{1}{M\_{1} M\_{2}} \Big/ \left[i a\_{0} + \xi\_{1}\right] \left(i a\_{2} + \xi\_{2}\right) \left[\left(c^{+} \left(p\right) Y \left(p\_{2}\right)\right)\right] \\ & \left[c \left(p\right) Y \left(p\_{1}\right)\right] + \left[c^{+} \left(p\right) Y \left(p\_{1}\right)\right] \left[c \left(p\right) Y \left(p\_{2}\right)\right] - \Delta^{2} \left[c^{+} \left(p\right) Y \left(-p\_{1}\right)\right] \\ & \left[c^{+} \left(-p\right) Y \left(-p\_{2}\right)\right] - \Delta^{2} \left[c \left(p\right) Y \left(-p\_{1}\right) \left[\left[c \left(-p\right) Y \left(-p\_{2}\right)\right]\right]\right] \end{split} \tag{14}$$

Here,

For a limited case of γ=0, we obtain *dx2–y2* state with order parameter (0; sin<sup>2</sup>

w x

For limited case γ=1 one gets *dxy-*state with order parameter ∆0(*T*) (0;0;*i*sin<sup>2</sup>

w x

sponds to the case γ=1/2. The order parameter takes the following form:


2 2 4 2

2 *<sup>p</sup>* sin sin 2

2 24 2 <sup>0</sup> D =D Q ( ) sin sin 2 *T*

+ =


2 2 4 2

2 *<sup>p</sup>* sin cos 2

2 24 2 <sup>0</sup> D =D Q ( ) sin cos 2 . *T*

+ =

2 2 24 2 0

2 2 24 2 0

Brusov et al.[8] case of equal admixtures of *dx2–y2 -* an*d dxy*-states in our consideration corre‐

 qj

2 2 24 0 sin <sup>0</sup>

**3.2. Equations for collective-modes spectrum in a mixed d-wave state at arbitrary admixture**

In the first approximations the spectrum of collective excitations is determined by the quadratic

( )( ) <sup>2</sup>

<sup>2</sup> D 0;sin cos2 ; sin sin 2 ;0;0 qj

2 2 4

w x


*+cj* o . Here *cj o*

2 *<sup>p</sup>* sin

+ =

sin cos 2 <sup>0</sup>

f

f

sin sin 2 <sup>0</sup>

f

f

f

*iT* (12)

+ +D Q å (13)

are the condensate values of *cj*

of the following

+ +D Q å (10)

f

+ +D Q å (8)

(9)

(11)

The gap equation in this case has the following form:

*<sup>Z</sup> <sup>g</sup> <sup>V</sup>* a

b

1

The gap equation has the following form:

1

0

1

θ.

*<sup>Z</sup> <sup>g</sup> <sup>V</sup>* a

b

The gap equation has the following form:

and gap ∆(*T*)=∆0(*T*) sin<sup>2</sup>

part of *Seff*, obtained after shift *cj* → *cj*

**of** *dxy***-state**

form:[5, 11]

*<sup>Z</sup> <sup>g</sup> <sup>V</sup>* a

b

and gap

188 Superconductors – New Developments

and gap

*θ*cos2*ϕ*;0;0;0).

*θ*sin2*ϕ*;0;0).

$$\begin{aligned} \left[c\boldsymbol{Y}^\*\right] &= c\_1 \left(1 - 3\cos^2\theta\right) + c\_2 \sin^2\theta\cos 2\phi + c\_3 \sin^2\theta\sin 2\phi + \\ &+ c\_4 \sin 2\theta\cos\phi + c\_5 \sin 2\theta\sin \phi. \end{aligned}$$

The quadratic form coefficients are proportional to the sums of the Green's functions of quasifermions products. One can go from summation to integration at low temperatures (*Tc* −*T* ~*Tc*) by the following rule:

$$\frac{1}{\beta V} \sum\_{p} \rightarrow \frac{1}{\left(2\pi\right)^{4}} \frac{k\_{\text{F}}^{2}}{c\_{\text{F}}} \int dod\xi d\Omega. \tag{15}$$

To evaluate these integrals it is useful to use the Feynman equality:

$$\begin{aligned} \left[ \left( \alpha\_1^2 + \underline{\xi}\_1^2 + \Delta^2 \right) \left( \alpha\_2^2 + \underline{\xi}\_2^2 + \Delta^2 \right) \right]^{-1} &= \\ \left[ d\lambda \left[ \alpha \left( \alpha\_1^2 + \underline{\xi}\_1^2 + \Delta^2 \right) + \left( 1 - \lambda \right) \left( \alpha\_2^2 + \underline{\xi}\_2^2 + \Delta^2 \right) \right]^2 \right. \end{aligned} \tag{16}$$

It is easy to evaluate the integrals with respect to variables ω andξ can also be easy evaluated. Then, they can be evaluated with respect to parameter α and the angular variables.

We obtain the following set of ten equations for the whole spectrum of the collective modes for (1*–γ*)*dx 2 –y2 + iγdxy* state at arbitrary γ after calculating all integrals, except over the angular variables and equating the determinant of the resulting quadratic form to zero:

{ ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( )} ( ) <sup>2</sup> <sup>2</sup> 22 2 <sup>1</sup> 0 <sup>2</sup> <sup>2</sup> 22 2 2 2 2 2 <sup>2</sup> <sup>2</sup> 22 2 2 2 <sup>2</sup> 2 22 22 2 <sup>2</sup> <sup>2</sup> 22 2 4 1 cos 2 1 2 4 1 cos 2 (1 2 ) ln 1 3 [1 3 4 1 cos 2 (1 2 ) <sup>3</sup> 1 3 3 1 cos 2 ]ln 1 cos 2 1 2 0 8 { 4 1 cos 2 (1 2 ) *x dx d x x x x xx x dx d x* w g fg j w w g f gw w g f gw f g fg w j w g fg +- + - é ù ë û ´ +- + - + é ù ë û ´ - +- - +- + - - é ù ë û æ ö - - +- - + -= é ù ç ÷ ë û è ø +- + - é ò ò ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) <sup>1</sup> <sup>2</sup> <sup>2</sup> 0 <sup>2</sup> <sup>2</sup> 22 2 <sup>2</sup> <sup>2</sup> <sup>2</sup> <sup>2</sup> 22 2 2 2 <sup>2</sup> 2 22 22 2 1 3 4 1 cos 2 1 2 ln [1 3 4 1 cos 2 (1 2 ) <sup>3</sup> 1 3 3 1 cos 2 ]ln 1 cos 2 1 2 } 0 <sup>8</sup> *x x x x xx x* w g f gw w g f gw f g fg ×- ´ <sup>ù</sup> ë û +- + - + é ù ë û ´ +- - +- + - - é ù ë û æ ö - - +- - + -= é ù ç ÷ ë û è ø ò ò

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) <sup>2</sup> <sup>2</sup> 22 2 <sup>1</sup> <sup>2</sup> 2 2 0 <sup>2</sup> <sup>2</sup> 22 2 <sup>2</sup> 2 2 2 22 2 2 2 2 222 2 2 2 <sup>2</sup> <sup>2</sup> 22 2 4 1 cos 2 1 2 { 1 cos 2 4 1 cos 2 1 2 ln [ 1 cos 2 4(1 ) [ cos 2 (1 2 )] 2 1 cos ]ln 1 cos 2 1 2 } 0 { 1 4 1 cos 2 1 2 *x dx d x x x x xx x dx d x* w g fg j f w w g f gw f w g f gw f g fg w j w g fg +- + - é ù ë û ×- ´ +- + - + é ù ë û ´ +- - +- + - - - - - + -= é ù ë û × +- + - é ù ë û ò ò ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) <sup>1</sup> <sup>2</sup> 2 2 0 <sup>2</sup> <sup>2</sup> 22 2 <sup>2</sup> 2 2 <sup>2</sup> <sup>2</sup> 22 2 2 2 <sup>222</sup> 22 2 <sup>2</sup> <sup>2</sup> 22 2 <sup>1</sup> <sup>2</sup> <sup>222</sup> 0 <sup>2</sup> <sup>2</sup> 2 2 cos 2 4 1 cos 2 1 2 ln [ 1 cos 2 4 1 cos 2 (1 2 ) 2 1 cos )]ln 1 cos 2 (1 2 ) } 0 4 1 cos 2 1 2 { 4 1 cos 4 1 ln *x x x x x x x x dx d x x x* f w g f gw f w g f gw f g fg w g fg j f w w g - ´ +- + - + é ù ë û ´ +- - +- + - - é ù ë û - - - + -= é ù ë û +- + - é ù ë û ×- ´ + - ´ ò ò ò ò ( ) ( ) ( ) ( ) ( ) 2 <sup>2</sup> <sup>2</sup> <sup>2</sup> <sup>2</sup> 22 2 2 2 22 222 2 2 2 cos 2 1 2 [4 1 4 1 cos 2 (1 2 ) cos 2 1 cos ]ln 1 cos 2 (1 2 ) } 0 *x x x xx x* f gw w g f gw f f g fg é ù + -+ ë û +-´ +- + - - é ù ë û ´ -- - + -= é ù ë û

How to Distinguish the Mixture of Two D-wave States from Pure D-wave State of HTSC http://dx.doi.org/10.5772/59180 191

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) <sup>1</sup> <sup>2</sup> <sup>222</sup> <sup>2</sup> <sup>2</sup> 22 2 <sup>0</sup> <sup>2</sup> <sup>2</sup> 22 2 22 2 2 <sup>2</sup> <sup>2</sup> 22 2 <sup>2</sup> 22 2 2 22 2 <sup>2</sup> <sup>2</sup> 22 2 { 4 1 cos 4 1 cos 2 1 2 4 1 cos 2 1 2 ln [4(1 ) cos 4 1 cos 2 1 2 2(1 ) cos )]ln 1 cos 2 1 2 } 0 4 1 cos 2 (1 2 { *dx d x x x x x x x x x x x dx d* w j f w g fg w g f gw f w g f gw f g fg w g f j ×- ´ +- + - é ù ë û +- + - + é ù ë û ´ +- - +- + - - é ù ë û - - - + -= é ù ë û +- + ò ò ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) <sup>1</sup> <sup>2</sup> <sup>222</sup> 0 <sup>2</sup> <sup>2</sup> 22 2 <sup>2</sup> <sup>222</sup> <sup>2</sup> <sup>2</sup> 22 2 <sup>2</sup> 22 2 2 22 2 <sup>1</sup> <sup>2</sup> <sup>222</sup> <sup>2</sup> <sup>2</sup> 22 2 <sup>0</sup> ) 4 1 sin 4 1 cos 2 1 2 ln [4 1 sin 4 1 cos 2 (1 2 ) 2(1 ) cos ]ln 1 cos 2 1 2 } 0 { 4 1 sin 4 1 cos 2 1 2 l *x x x x x x xx x dx d x x x* g f w w g f gw f w g f gw f g fg w j f w g fg é ù ë û ×- ´ +- + - + é ù ë û ´ +- - +- + - - é ù ë û - - - + -= é ù ë û ×- ´ +- + - é ù ë û ´ ò ò ò ò ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) <sup>2</sup> <sup>2</sup> 22 2 <sup>2</sup> <sup>222</sup> <sup>2</sup> <sup>2</sup> 22 2 2 2 <sup>222</sup> 22 2 4 1 cos 2 1 2 n [4 1 sin 4 1 cos 2 1 2 2 1 cos )]ln 1 cos 2 1 2 } 0 *x x x x x x x* w g f gw f w g f gw f g fg +- + - + é ù ë û +- - +- + - - é ù ë û - - - + -= é ù ë û

{ ( ) ( )

g

w

4 1 cos 2 (1 2 )

+- + - + é ù

*x*

g

g

4 1 cos 2 (1 2 )

+- + - - é ù ë û

*xx x*

w

<sup>1</sup> <sup>2</sup> <sup>2</sup>

g

( ) ( )

+- + - - é ù ë û

+- + - + é ù ë û ´ +- -

*xx x*

4 1 cos 2 1 2 ln [1 3 4 1 cos 2 (1 2 )

+- + - é

4 1 cos 2 1 2

+- + - é ù

( ) ( )( )

 g

 g

ë û ×- ´

( )

*x*

×- ´

1 3

 gw

 gw

 fg

ë û ´

2 2 2 2

*x x*

 fg

<sup>2</sup> <sup>2</sup>

 fg

*x*

( )

( )

( )

*x*

*x*

( )

[4 1

<sup>2</sup> <sup>2</sup>

*x*

 fg

ë û

*x*

<sup>2</sup> 2 2

<sup>2</sup> 2 2

cos 2


 f

f

f

f

 f

( )( ) ( ) ( )}

æ ö - - +- - + -= é ù ç ÷ ë û è ø

 fg

<sup>ù</sup> ë û

 gw

 gw

( ) ( )

2 2 <sup>2</sup> 2 22 22 2

( ) ( )

ln [ 1 cos 2

ë û ´ +- -

 g

 f

 f

<sup>2</sup> <sup>2</sup> 22 2 <sup>1</sup> <sup>2</sup> 2 2

4 1 cos 2 1 2

+- + - é ù

*dx d x*

w

g

( ) ( )

+- + - + é ù

4(1 ) [ cos 2 (1 2 )]

+- + - - - - - + -= é ù

4 1 cos 2 1 2

( ) ( ) ( )

w

g

( ) ( )

+- + - + é ù

4 1 cos 2 1 2

2 2 <sup>222</sup> 22 2

4 1 cos 2 (1 2 )

+- + - - é ù ë û

2 1 cos ]ln 1 cos 2 1 2 } 0

{ 1 4 1 cos 2 1 2

2 2 222 2 2 2

<sup>2</sup> <sup>2</sup> 22 2

*x*

g

g

( ) ( )

<sup>1</sup> <sup>2</sup> 2 2

( ) ( )

*dx d x x*

2 1 cos )]ln 1 cos 2 (1 2 ) } 0


4 1 cos 2 1 2

+- + - é ù

2

f

g

w

( ) ( )

2 2 22 222 2 2 2

4 1 cos 2 (1 2 )

+- + - - é ù ë û ´ -- - + -= é ù

<sup>2</sup> <sup>2</sup> 22 2 <sup>1</sup> <sup>2</sup> <sup>222</sup>

cos 2 1 2

 f

f

 g

{ 4 1 cos

 fg

( ) ( ) ( )

 gw

é ù + -+ ë û +-´

 gw

cos 2 1 cos ]ln 1 cos 2 (1 2 ) } 0

+- + - é ù

f

( )( ) ( ) ( )

æ ö - - +- - + -= é ù ç ÷ ë û è ø

<sup>3</sup> 1 3 3 1 cos 2 ]ln 1 cos 2 1 2 } 0 <sup>8</sup>

{ 1 cos 2

 fg

> gw

 gw

 fg

> gw

 gw

ë û

( ) ( )

ln [ 1 cos 2

ë û ´ +- -

 f

 f  fg

 fg

ë û ×- ´

ë û

 g ×

ë û

<sup>3</sup> 1 3 3 1 cos 2 ]ln 1 cos 2 1 2 0

2 2 <sup>2</sup> 2 22 22 2

f

4 1 cos 2 (1 2 )

 f

> f

ln 1 3 [1 3

ë û ´ - +- -

 f

 f

( )

w

*x*

*x*

0

190 Superconductors – New Developments

*dx d*

ò ò

j

w

w

8

ò ò

0

*dx d*

{

w

j

w

> w

0

ò ò

*dx d*

ò ò

0

0

´

ln

ò ò

<sup>2</sup> <sup>2</sup> 22 2 <sup>1</sup>

<sup>2</sup> <sup>2</sup> 22 2

<sup>2</sup> <sup>2</sup> 22 2

( )

<sup>2</sup> <sup>2</sup> 22 2

*x*

*x*

w

j

w

> w

j

w

> w

> > j

w

> w

> > f

w

<sup>2</sup> <sup>2</sup> 22 2

*x*

g

g

<sup>2</sup> <sup>2</sup> 22 2

*x*

*x xx x*

( ) ( )

*x x x*

f

*x*

g

g

*x*

( )

*x*

*x*

<sup>2</sup> <sup>2</sup> 22 2

*x xx x*

4 1

+ -

w

<sup>2</sup> <sup>2</sup> 2 2

*x*

g

g

2 22 2 2

f

<sup>2</sup> <sup>2</sup> 22 2

*x*

<sup>2</sup> <sup>2</sup> 22 2

<sup>2</sup> <sup>2</sup> 22 2

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) <sup>2</sup> <sup>2</sup> 22 2 <sup>1</sup> <sup>2</sup> 2 2 0 <sup>2</sup> <sup>2</sup> 22 2 <sup>2</sup> 2 2 <sup>2</sup> <sup>2</sup> 22 2 2 2 222 2 2 2 1 <sup>2</sup> <sup>2</sup> 22 2 <sup>0</sup> 4 1 cos 2 1 2 { 1 sin 4 1 cos 2 1 2 ln [ 1 sin 4 1 cos 2 1 2 2 1 cos ]ln 1 cos 2 (1 2 ) } 0 { 4 1 cos 2 1 2 *x dx d x x x x xx x dx d x* w g fg j f w w g f gw f w g f gw f g fg w j w g fg +- + - é ù ë û ×- ´ +- + - + é ù ë û ´ +- - +- + - - é ù ë û - - - + -= é ù ë û × +- + - é ù ë û ò ò ò ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) <sup>2</sup> 2 2 <sup>2</sup> <sup>2</sup> 22 2 <sup>2</sup> 2 2 <sup>2</sup> <sup>2</sup> 22 2 2 2 <sup>222</sup> 22 2 1 sin 4 1 cos 2 1 2 ln [ 1 sin 4 1 cos 2 (1 2 ) 2 1 cos )]ln 1 cos 2 1 2 } 0. *x x x x x x x* f w g f gw f w g f gw f g fg - ´ +- + - + é ù ë û ´ +- - +- + - - é ù ë û - - - + -= é ù ë û ò (17)

We have used the following substitutions: cosθ=*x*, ω=ω/∆0.

The whole spectrum of collective modes in mixed (1*–γ*)*dx 2 –y<sup>2</sup> + iγdxy* state of HTSC and arbitrary admixture of *dxy*-state is determined by equations (17). For interpretation of the sound attenuation and microwave absorption data as well as for identification of the type of pairing and order parameter in unconventional superconductors, knowledge of the collective-mode spectrum could be used. They allow, in particular, evaluation of the extent of admixture of a *dxy*-state in a mixed state.

Suppose that the dominant state is *dx 2 –y2* -state and admixture of *dxy*-state is small, say 5–10%; thus, the most interesting case turns out to be the case of small γ. One could in this case expand all expressions in powers of small γ and obtain the corrections to the spectrum of pure *dx 2 –y*2 state, as found previously. [4, 5, 11]

#### **3.3. Equations for collective-modes spectrum in a mixed-d-wave state with equal admixtures of** *dx 2 –y***2 - and** *dxy***-states**

Brusov et al. [4] supposed equal admixtures of *dx 2 –y2 -* and *dxy-*states – in our consideration this corresponds to the case γ =1/2 – and derived the following equations:

$$\begin{aligned} \text{i } i &= 1\\ \int\_0^1 d\mathbf{x} \int d\boldsymbol{\rho} \mathbf{\dot{\ell}} \frac{\sqrt{\boldsymbol{\alpha}^2 + 4\boldsymbol{f}}}{\boldsymbol{\alpha}} \ln \frac{\sqrt{\boldsymbol{\alpha}^2 + 4\boldsymbol{f}} + \boldsymbol{\alpha}}{\sqrt{\boldsymbol{\alpha}^2 + 4\boldsymbol{f} - \boldsymbol{\alpha}}} \mathbf{g}\_1 + \left(\mathbf{g}\_1 - \frac{3}{2}\boldsymbol{f}\_1\right) \mathbf{\ln}\, f\mathbf{\dot{\ell}} = \mathbf{0} \\ \int\_0 d\mathbf{x} \int d\boldsymbol{\rho} \mathbf{\dot{\ell}} \frac{\boldsymbol{\alpha}\mathbf{\dot{\ell}}}{\sqrt{\boldsymbol{\alpha}^2 + 4\boldsymbol{f}} \boldsymbol{\ell}} \ln \frac{\sqrt{\boldsymbol{\alpha}^2 + 4\boldsymbol{f}} + \boldsymbol{\alpha}}{\sqrt{\boldsymbol{\alpha}^2 + 4\boldsymbol{f}} - \boldsymbol{\alpha}} \mathbf{g}\_1 + \left(\mathbf{g}\_1 - \frac{3}{2}\boldsymbol{f}\_1\right) \mathbf{\ln}\, f\mathbf{\dot{\ell}} = \mathbf{0} \end{aligned} \tag{18}$$

$$\begin{aligned} &i = 2, 3, 4, 5\\ &\int\_0^1 d\phi \sqrt{\frac{\alpha^2 + 4f}{\alpha}} \ln \frac{\sqrt{\alpha^2 + 4f} + \alpha}{\sqrt{\alpha^2 + 4f} - \alpha} \mathcal{G}\_i + \left(\mathcal{g}\_i - \frac{1}{2}\mathcal{g}\right) \ln f\rangle = 0\\ &\int\_0^1 d\phi \{d\phi \{\frac{\alpha \alpha}{\sqrt{\alpha^2 + 4f}} \ln \frac{\sqrt{\alpha^2 + 4f} + \alpha}{\sqrt{\alpha^2 + 4f} - \alpha} \mathcal{G}\_i + \left(\mathcal{g}\_i - \frac{1}{2}\mathcal{g}\right) \ln f\} = 0. \end{aligned} \tag{19}$$

Here,

$$\begin{aligned} g\_1 &= \left(1 - 3\mathbf{x}^2\right)^2; \\ g\_2 &= \left(1 - \mathbf{x}^2\right)^2 \cos^2 2\rho; g\_3 = g = 4\left(1 - \mathbf{x}^2\right)\mathbf{x}^2 \cos^2 \rho; \\ g\_4 &= 4\left(1 - \mathbf{x}^2\right)\mathbf{x}^2 \sin^2 \rho; g\_5 = \left(1 - \mathbf{x}^2\right)^2 \sin^2 \rho; \\ f\_1 &= \left(1/4\right)\left[\left(1 - 3\mathbf{x}^2\right)^2 + 3\left(1 - \mathbf{x}^2\right)^2 \cos^2 2\rho\right]; f = \left(1 - \mathbf{x}^2\right)^2. \end{aligned} \tag{20}$$

These equations were solved numerically[4, 11] and the first equations appear to give either Goldstone modes or modes with vanishing energies (of order 0.03 ∆0(*T*) – 0.08 ∆0(*T*)), while five high-frequency modes in each state were obtained from the second equations.

We have used the following substitutions: cosθ=*x*, ω=ω/∆0.

*2*

admixture of *dxy*-state is determined by equations (17). For interpretation of the sound attenuation and microwave absorption data as well as for identification of the type of pairing and order parameter in unconventional superconductors, knowledge of the collective-mode spectrum could be used. They allow, in particular, evaluation of the extent of admixture of a

thus, the most interesting case turns out to be the case of small γ. One could in this case expand all expressions in powers of small γ and obtain the corrections to the spectrum of pure *dx*

**3.3. Equations for collective-modes spectrum in a mixed-d-wave state with equal admixtures**

4 4 <sup>3</sup> { ln ln } 0 4 2

+ ++ æ ö

w

+- = ç ÷

+- = ç ÷

+- = ç ÷

+- = ç ÷

 j

w

+ - è ø + + æ ö

*i i*

+ - è ø + + æ ö

*i i*

<sup>4</sup> <sup>3</sup> { ln ln } 0 4 4 2

+ +- è ø

w

w

4 4 <sup>1</sup> { ln ln } 0 4 2

+ ++ æ ö

w

w

<sup>4</sup> <sup>1</sup> { ln ln } 0. 4 4 2

w

w

+ +- è ø

( ) ( )( ) ( )

1 4 1 3 3 1 cos 2 ; 1 .

2 2 <sup>2</sup> 2 22 <sup>2</sup>

 j

j

<sup>2</sup> 2 2 22 2

1 cos 2 ; 4 1 cos ;

<sup>2</sup> 22 2 2 2

*f x x fx*

é ù = - +- = - ê ú ë û

4 1 sin ; 1 sin ,

*2 –y2*

*2 –y2*

corresponds to the case γ =1/2 – and derived the following equations:

11 1 <sup>2</sup> <sup>0</sup>

w

w

 w

*f f dx d gg f <sup>f</sup> f*

*<sup>f</sup> dx d gg f <sup>f</sup>*

*f f dx d gg g <sup>f</sup> f*

*<sup>f</sup> dx d gg g <sup>f</sup>*

*f f*

( ) ( )

( ) ( )

j

*g x g g xx*

j

= - = = -

 w

11 1 2 2 <sup>0</sup>

*f f*

 w

 w

w

w *–y2 + iγdxy* state of HTSC and arbitrary

*2 –y*2 -

(18)

(19)

(20)


*-* and *dxy-*states – in our consideration this

The whole spectrum of collective modes in mixed (1*–γ*)*dx*

*dxy*-state in a mixed state.

192 Superconductors – New Developments

**of** *dx 2 –y***2**

Here,

Suppose that the dominant state is *dx*

state, as found previously. [4, 5, 11]

1

j

j

2,3,4,5

j

j

Brusov et al. [4] supposed equal admixtures of *dx*

1 2 2

w

1 2

1 2 2 <sup>2</sup> <sup>0</sup> 1 2 2 2 <sup>0</sup>

w

w

w

w

( )

*g x*

= -

13 ;

<sup>2</sup> <sup>2</sup>

2 3

4 5

*g xx g x*

=- = -

w

w

w

 **- and** *dxy***-states**

*i*

=

ò ò

ò ò

*i*

=

ò ò

ò ò

1

1

We shall now present results for high-frequency modes (*Ei* is the energy (frequency) of the *i* th branch).

$$\begin{aligned} E\_{1,2} &= \Lambda\_0 \left( T \right) \left( 1.93 - i \; 0.41 \right); \\ E\_3 &= \Lambda\_0 \left( T \right) \left( 1.62 - i \; 0.75 \right); \\ E\_{4,5} &= \Lambda\_0 \left( T \right) \left( 1.59 - i \; 0.83 \right) \end{aligned} \tag{21}$$

Comparison of these results with spectrum of pure *dx 2 –y2 -* and *dxy-*states, obtained by Brusov et al. as follows: [4, 11]

$$\begin{aligned} E\_1 &= \Lambda\_0 \begin{pmatrix} T \\ \end{pmatrix} \begin{pmatrix} 1.88 - i \ 0.79 \end{pmatrix}; \\ E\_2 &= \Lambda\_0 \begin{pmatrix} T \end{pmatrix} \begin{pmatrix} 1.66 - i \ 0.50 \end{pmatrix}; \\ E\_3 &= \Lambda\_0 \begin{pmatrix} T \end{pmatrix} \begin{pmatrix} 1.40 - i \ 0.68 \end{pmatrix}; \\ E\_4 &= \Lambda\_0 \begin{pmatrix} T \end{pmatrix} \begin{pmatrix} 1.13 - i \ 0.71 \end{pmatrix}; \\ E\_5 &= \Lambda\_0 \begin{pmatrix} T \end{pmatrix} \begin{pmatrix} 1.10 - i \ 0.65 \end{pmatrix} \end{aligned} \tag{22}$$

These results led them to conclusion that in spite of the fact that spectra in both pure states, *dx 2 –y2* and *dxy*, turn out to be identical, the spectrum in mixed *dx 2 –y2 -* and *dxy-*-state is quite different from that in pure states. All modes are non-degenerated in pure states, but two high-frequency modes are twice degenerated in mixed state. The collective modes have higher frequencies in mixed state – from 1.59 *Δ*<sup>0</sup> (*T* ) up to 1.93 *Δ*<sup>0</sup> (*T* ), compared to 1.1 *Δ*<sup>0</sup> (*T* ) up to 1.88 *Δ*<sup>0</sup> (*T* ) in pure state. In pure d-wave states, damping of collective modes is also bigger than in mixed state (in pure states Im *Ei* is from 30% to 65% and in mixed state from 20% to 50%). This is because the gap vanishes in pure states along chosen lines while it vanishes just at two points (poles) in mixed state. [4]

Such difference in the spectra of collective modes in pure *d* -wave states and in mixed *d* -wave state enables us to probe the symmetry of states using ultrasound and/or microwave absorp‐ tion experiments. These experiments require high frequencies (in the order of tens of GHz as in the experiment of Feller et al. [16]), but there are no principle restrictions for ultrasound (microwave) frequencies: frequencies of collective modes are proportional to gap *Δ*<sup>0</sup> (*T* ), and the gap vanishes at *Tc*, so one could in principle use any frequency approaching *Tc*.

Note that the case of the *dx 2 –y2 + idxy-*state was also considered by Balatsky et al., [17] who studied one of the possible collective modes in this state. They considered a superconducting state with mixed*-*symmetry-order parameter components, e.g., *d*+*is* and *dx 2 –y2* +*idxy* and argued for the existence of a new orbital magnetization mode which corresponds to oscillations of relative phase φ between two components around an equilibrium value of φ = π / 2. The analogue of this mode is the so-called clapping mode of the superfluid A-phase in <sup>3</sup> He. The frequency of this mode ω0(*B*, *T*) depending on the field and temperature for the specific case of magneticfield-induced *dxy*-state has been estimated. This mode is tunable with a magnetic field with ω0(*B*, *T*) *∞ B*Δ0, where Δ<sup>0</sup> is the amplitude of gap of the order parameter in *d*-wave state. The velocity *s*(*B*, *T*) of this mode has been estimated as well.

#### **3.4.** *dx 2 –y2* **-state of high-temperature superconductors with a small admixture of** *dxy***-state**

The whole spectrum of collective modes in mixed *dx 2 –y<sup>2</sup> + iεdxy* state of HTSC with arbitrary admixture of *dxy*-state is determined by the above equations (17). For interpretation of the sound attenuation and microwave absorption data as well as for identification of the type of pairing and order parameter in unconventional superconductors knowledge of the collective-mode spectrum could be used. In particular, these experiments allow the estimation in a possible mixed state of the extent of admixture of a *dxy*-state.

The case of small γ is, however, the most interesting case: one supposes that *dx 2 –y2* -state is the dominant state and the admixture of *dxy*-state is small, say 3–10%. For this case one can expand all expressions in powers of small *γ* and obtain the corrections to the equations for the spectrum of pure *dx 2 –y*2 -state, which has been found before [11, 13].

Let us consider the case of small admixture of *dxy*-state (small ε): we suppose that dominant state is *dx 2 –y2* -state and admixture of *dxy*-state is small, about 3–10%. After expanding of all expressions in powers of small *ε* we obtain the below corrections.

We use the following notations:

$$\begin{aligned} a &= \left(1 - \chi^2\right)^2 \cos^2 2\phi; \\ b &= \left(1 - \chi^2\right)^2 \sin^2 2\phi; \\ a\_1 &= a + o^2 \end{aligned} \tag{23}$$

Here, we obtain the following expressions:

$$
\sqrt{\rho^2 + 4(1 - \chi^2)^2 [\cos^2 2\phi + \gamma^2 \sin^2 2\phi]} \approx \sqrt{a\_1} \left( 1 + \gamma^2 \frac{b}{2a\_1} \right) \tag{24}
$$

$$
\ln f \approx \ln a + \gamma^2 \frac{b}{a} \tag{25}
$$

$$\frac{\sqrt{\alpha^2 + 4\left(1 - \mathbf{x}^2\right)^2 \left[\cos^2 2\rho + \gamma^2 \sin^2 2\rho\right]} + o}{\sqrt{\alpha^2 + 4\left(1 - \mathbf{x}^2\right)^2 \left[\cos^2 2\rho + \gamma^2 \sin^2 2\rho\right]}} \approx A + \gamma^2 B,\tag{26}$$

where

phase φ between two components around an equilibrium value of φ = π / 2. The analogue of

this mode ω0(*B*, *T*) depending on the field and temperature for the specific case of magneticfield-induced *dxy*-state has been estimated. This mode is tunable with a magnetic field with ω0(*B*, *T*) *∞ B*Δ0, where Δ<sup>0</sup> is the amplitude of gap of the order parameter in *d*-wave state. The

**-state of high-temperature superconductors with a small admixture of** *dxy***-state**

admixture of *dxy*-state is determined by the above equations (17). For interpretation of the sound attenuation and microwave absorption data as well as for identification of the type of pairing and order parameter in unconventional superconductors knowledge of the collective-mode spectrum could be used. In particular, these experiments allow the estimation in a possible

dominant state and the admixture of *dxy*-state is small, say 3–10%. For this case one can expand all expressions in powers of small *γ* and obtain the corrections to the equations for the spectrum

Let us consider the case of small admixture of *dxy*-state (small ε): we suppose that dominant

<sup>2</sup> 2 2

1 cos 2 ;

<sup>2</sup> 2 2

1 sin 2 ; ,


f

f

 f

*a*

 j w

ë û » +

 j w

1

1

(25)

*b*

 gæ ö

è ø

*a*

2

*A B*

g

,

The case of small γ is, however, the most interesting case: one supposes that *dx*

( )

*a x*

= -

*b x a a*

= - = +

( )

2

w

2 22 2 2 2 2

fg

*x a*

<sup>2</sup> ln ln *<sup>b</sup> f a*

» + g

jg

jg

ë û

4(1 ) [cos 2 sin 2 ] 1 <sup>2</sup>

+- + » + ç ÷


expressions in powers of small *ε* we obtain the below corrections.

1

( )

*x*

<sup>2</sup> <sup>2</sup> 2 2 22

4 1 cos 2 sin 2

+- + + é ù

+- + - é ù

4 1 cos 2 sin 2

<sup>2</sup> <sup>2</sup> 2 2 22

( )

*x*

*2*

He. The frequency of

*–y2 + iεdxy* state of HTSC with arbitrary

*2 –y2*


(23)

(24)

(26)

this mode is the so-called clapping mode of the superfluid A-phase in <sup>3</sup>

velocity *s*(*B*, *T*) of this mode has been estimated as well.

The whole spectrum of collective modes in mixed *dx*

mixed state of the extent of admixture of a *dxy*-state.

**3.4.** *dx 2 –y2*

of pure *dx*

state is *dx*

*2 –y*2

194 Superconductors – New Developments

*2 –y2*

We use the following notations:

Here, we obtain the following expressions:

w

> w

w

$$\begin{aligned} A &= \frac{\left(\sqrt{a\_1} + \alpha\right)^2}{a}; \\ B &= \frac{b}{a^2} \frac{\sqrt{a\_1} + \alpha}{\sqrt{a\_1}} \left[ a + \sqrt{a\_1} \left( \alpha + \sqrt{a\_1} \right) \right] \end{aligned} \tag{27}$$

$$\begin{split} &\ln \frac{\sqrt{\alpha^2 + 4\left(1 - \mathbf{x}^2\right)^2 \left[\cos^2 2\phi + \gamma^2 \sin^2 2\phi\right] + o}}{\sqrt{\alpha^2 + 4\left(1 - \mathbf{x}^2\right)^2 \left[\cos^2 2\phi + \gamma^2 \sin^2 2\phi\right]} + o} \approx\\ &\ln \frac{\sqrt{\alpha^2 + 4\left(1 - \mathbf{x}^2\right)^2 \cos^2 2\phi} + o}{\sqrt{\alpha^2 + 4\left(1 - \mathbf{x}^2\right)^2 \cos^2 2\phi} + \gamma^2 C\_1} + \gamma^2 C\_2 \end{split} \tag{28}$$

where

$$C = \frac{b\left(\sqrt{a\_1} + a\right)}{\sqrt{a}} + \frac{b}{a}.\tag{29}$$

For other expressions, we obtain, with the accuracy of order *γ* <sup>2</sup> :

$$\begin{split} &\frac{\sqrt{o^{2}+4\left(1-\mathbf{x}^{2}\right)^{2}\left[\cos^{2}2\rho+\gamma^{2}\sin^{2}2\rho\right]}}{\alpha} \times \\ &\times \ln\frac{\sqrt{o^{2}+4\left(1-\mathbf{x}^{2}\right)^{2}\left[\cos^{2}2\rho+\gamma^{2}\sin^{2}2\rho\right]}+\alpha}{\sqrt{o^{2}+4\left(1-\mathbf{x}^{2}\right)^{2}\left[\cos^{2}2\rho+\gamma^{2}\sin^{2}2\rho\right]-\alpha}} \approx\\ &\frac{\sqrt{o^{2}+4\left(1-\mathbf{x}^{2}\right)^{2}\cos^{2}2\rho}}{\alpha}\ln\frac{\sqrt{o^{2}+4\left(1-\mathbf{x}^{2}\right)^{2}\cos^{2}2\rho}+\alpha}{\sqrt{o^{2}+4\left(1-\mathbf{x}^{2}\right)^{2}\cos^{2}2\rho-\alpha}} +\\ &+\frac{\gamma^{2}}{\alpha}\sqrt{a\_{1}\left(C+\frac{b}{2a\_{1}}\ln\frac{\sqrt{o^{2}+4\left(1-\mathbf{x}^{2}\right)^{2}\cos^{2}2\rho}+\alpha}{\sqrt{o^{2}+4\left(1-\mathbf{x}^{2}\right)^{2}\cos^{2}2\rho-\alpha}}}\Bigg|;\end{split} \tag{30}$$

$$\begin{aligned} &\frac{\alpha}{\sqrt{\alpha^2 + 4\left(1 - \mathbf{x}^2\right)^2 \left[\cos^2 2\phi + \rho^2 \sin^2 2\phi\right]}} \times \\ &\times \ln \frac{\sqrt{\alpha^2 + 4\left(1 - \mathbf{x}^2\right)^2 \left[\cos^2 2\phi + \rho^2 \sin^2 2\phi\right]} + \alpha}{\sqrt{\alpha^2 + 4\left(1 - \mathbf{x}^2\right)^2 \left[\cos^2 2\phi + \rho^2 \sin^2 2\phi\right]} - \alpha} \approx \\ &\frac{\alpha}{\sqrt{\alpha^2 + 4\left(1 - \mathbf{x}^2\right)^2 \cos^2 2\phi} + \sqrt{\frac{\alpha^2 + 4\left(1 - \mathbf{x}^2\right)^2 \cos^2 2\phi}{\sqrt{\alpha^2 + 4\left(1 - \mathbf{x}^2\right)^2 \cos^2 2\phi} - \alpha}} + \\ &+ \gamma^2 \frac{\alpha}{\sqrt{a\_1}} \left(C - \frac{b}{2a\_1} \ln \sqrt{\frac{\alpha^2 + 4\left(1 - \mathbf{x}^2\right)^2 \cos^2 2\phi}{\sqrt{\alpha^2 + 4\left(1 - \mathbf{x}^2\right)^2 \cos^2 2\phi} - \alpha}}.\end{aligned} \tag{31}$$

Putting all expressions (23)–(31) into (17) one obtains the whole set of equations for the collective-mode spectrum of mixed *dx 2 –y2 + iεdxy* state with small admixture of *dxy*-state to *dx 2 –y2 -*state.

#### **4. Conclusions**

In order to solve one of the problems of unconventional superconductivity – the exact form of the order parameter – we consider the mixed (1*–γ*)*dx 2 –y<sup>2</sup> + iγdxy* state in HTSC. A full set of equations for the collective-modes spectrum in a mixed d-wave state with an arbitrary admixture of dxy-state has been derived. These equations have been solved for the case of equal admixtures *dx 2 –y2* - and *dxy*-states by Brusov et al. [4, 11, 14] It has been shown that the difference of collective excitations spectrum in mixed *d* -wave states and in pure state provide the possibility of probing the symmetry of the state by ultrasound and/or microwave absorption experiments.

The case of small γ is the most interesting case: one supposes that *d*<sup>x</sup> 2 –y2 -state is the dominant state and admixture of *d*xy-state is small, say 3–10%. In this case all expressions can be expanded in powers of small γ and the corrections to the spectrum of pure dx 2 –y2 -state, which has been found before, can be obtained. [18]

For identification of the type of pairing and determination of the exact form of the order parameter in unconventional superconductors the results obtained could be quite useful. They allow evaluation of the extent of an admixture of a dxy-state in relation to the spectrum of pure dx 2 –y2 -state in a possible mixed state. Obtained equations allow the calculation of the whole collective mode. This spectrum could be used for interpretation of the sound attenuation and microwave absorption data. [18]

The results obtained could also allow three quite important questions to be answered: [18]


#### **Author details**

( )

*x*

( )

w

*a a*

the order parameter – we consider the mixed (1*–γ*)*dx*

The case of small γ is the most interesting case: one supposes that *d*<sup>x</sup>

in powers of small γ and the corrections to the spectrum of pure dx

2

g

w

collective-mode spectrum of mixed *dx*

*dx 2 –y2 -*state.

**4. Conclusions**

admixtures *dx*

experiments.

dx 2 –y2 *2 –y2*

found before, can be obtained. [18]

microwave absorption data. [18]

+ - ç

æ

w

ln

w

w

w

196 Superconductors – New Developments

( )

*x*

<sup>2</sup> <sup>2</sup> 2 2 22

w

<sup>2</sup> <sup>2</sup> 2 2 22

4 1 cos 2 sin 2

fg

fg

fg

ë û +- + + é ù ë û ´ » +- + - é ù

4 1 cos 2 sin 2

4 1 cos 2 sin 2

+- + é ù

<sup>2</sup> <sup>2</sup> 2 2 22

( )

*x*

( )

 f w

´

 f w

<sup>2</sup> <sup>2</sup> 2 2

*x*

4 1 cos 2

+- +

f w

f w

*2*


f w

+

(31)

f w

*–y<sup>2</sup> + iεdxy* state with small admixture of *dxy*-state to

*–y2 + iγdxy* state in HTSC. A full set of



2 –y2

2 –y2

. ö ÷

( )

( )

 w

+- + +- -

<sup>2</sup> <sup>2</sup> 2 2

w

ë û

4 1 cos 2

 f

2 2 2 2 2 2 2 2

ln 4 1 cos 2 4 1 cos 2

*x x*

ç +- +

<sup>2</sup> 4 1 cos 2

<sup>÷</sup> ç ÷ø

+- - <sup>è</sup>

*2*

<sup>2</sup> <sup>2</sup> 2 2 <sup>1</sup> <sup>1</sup>

f

*<sup>x</sup> <sup>b</sup> <sup>C</sup>*

w

w

ln

( )

*x*

Putting all expressions (23)–(31) into (17) one obtains the whole set of equations for the

In order to solve one of the problems of unconventional superconductivity – the exact form of

equations for the collective-modes spectrum in a mixed d-wave state with an arbitrary admixture of dxy-state has been derived. These equations have been solved for the case of equal

of collective excitations spectrum in mixed *d* -wave states and in pure state provide the possibility of probing the symmetry of the state by ultrasound and/or microwave absorption

state and admixture of *d*xy-state is small, say 3–10%. In this case all expressions can be expanded

For identification of the type of pairing and determination of the exact form of the order parameter in unconventional superconductors the results obtained could be quite useful. They allow evaluation of the extent of an admixture of a dxy-state in relation to the spectrum of pure

The results obtained could also allow three quite important questions to be answered: [18]


\*Address all correspondence to: pnb1983@yahoo.com

Financial University under the Government of Russian Federation, Russia

#### **References**


#### **Measurements of Stationary Josephson Current between High-***Tc* **Oxides as a Tool to Detect Charge Density Waves Measurements of Stationary Josephson Current between High-***Tc* **Oxides as a Tool to Detect Charge Density Waves**

Alexander M. Gabovich, Mai Suan Li, Henryk Szymczak and Alexander I. Voitenko Alexander M. Gabovich\*1, Suan Li Mai2, Henryk Szymczak2 and Alexander I. Voitenko1

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/59590 10.5772/59590

#### **1. Introduction**

[12] Brusov, P. N., Brusov, P.P. (2001), "Collective properties of superconductors with nontrivial pairing", Sov. Phys. JETP, 119, 913–930 [Zh. Eksp. Theor. Phys. v.92, p.

[13] Brusov, P. N., Brusova, N. P. and Brusov, P. P. (1997), The collective excitations of the order parameter in HTSC and heavy fermion superconductors (HFSC) under d-pair‐

[14] Brusov P., Brusov, Majumdar, P. and Orehova, N. (2003), "Ultrasound attenuation and collective modes in mixed *d* x2–y2*+d* xy state of unconventional superconductors",

[15] Brusov, P., Brusov, P. and Lee, C. (2004), Collective properties of unconventional su‐

[18] Brusov, P., Brusov, P. (2009) Collective Excitations in Unconventional Superconduc‐

[16] Feller J. R., Tsai, C. C., Ketterson, J. B. et al. (2002), Phys. Rev. Lett. 88, 247005. [17] Balatsky, A. V., Kumar, P. and Schrieffer, J. R. (2000), Phys. Rev. Lett. 84, 4445.

tors and Superfluids, pp. 860, World Scientific Publishing, Singapore

795].

198 Superconductors – New Developments

ing. J. Low Temp. Phys. 108, 143.

Brazilian Journal of Physics, 33, 729–732.

perconductors, Int. J. Mod. Phys. B, v.18, 867–882.

Since an unexpected and brilliant discovery of high-*Tc* superconductivity in cuprates in 1986 [8], experts have been trying to find the origin of superconductivity in them, but in vain. There are several problems that are interconnected and probably cannot be solved independently. But they are so complex that researchers are forced to consider them separately in order to find the key concepts and express key ideas explaining the huge totality of experimental data. General discussion and the analysis of high-*Tc*-oxide superconductivity can be found in comprehensive reviews [21, 25, 42, 50, 53, 65, 73, 75, 78, 80, 86, 89, 104]. In particular, the main questions to be solved are as follows: (i) Is superconductivity in cuprates a conventional one based on the Cooper pairing concept? (ii) If the answer to the first question is positive, what is the mechanism of superconductivity, i.e., what are the virtual bosons that glue electrons in pairs? (iii) Which is the symmetry of the superconducting order parameter? This question remains unanswered, although the majority of the researchers in the field think believe that the problem is already resolved (namely, *dx*2−*y*<sup>2</sup> -one, see, e.g., Refs. [54, 90])? (iv) What is the role of the intrinsic disorder and non-stoichiometry in the superconducting properties [2, 28, 43, 69, 70, 75, 103, 105]? (v) What is the origin of the symmetry loss and, specifically, the emerging nematicity [28, 58, 70, 75, 88, 103]? (vi) What is the origin of the so-called pseudogap [42, 68, 75, 94, 98, 104]? (vii) What is the role of spinand charge- density waves (SDWs and CDWs) both in the normal and superconducting states of cuprates? The role of various electron spectrum instabilities competing with the Cooper pairing below the critical temperature *Tc* is a part of the more general problem: How can certain anomalous high-*Tc* oxide properties above *Tc* be explained, e.g., the linear behavior of the resistivity [66, 91]? In this connection, a quite reasonable viewpoint was expressed that if one understands the normal state of cuprates, the superconducting state properties will be perceived [42, 46]. Here, it is also worth to mention a possible failure [91, 92] of the

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. © 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

©2012 Author(s), licensee InTech. This is an open access chapter distributed under the terms of the Creative

Fermi liquid concept belonging to Landau [1] and the role of strong electron correlations [18, 48, 61, 62].

During last decades we have been developing a phenomenological theory to elucidate the influence of CDWs on superconductivity of high-*Tc* oxides, since the CDWs were observed in a number of those materials [23, 33, 35, 37–39]. We identified the CDW energy gap with the pseudogap mentioned above. Such an identification is based, in particular, on the appearance of CDWs only below the approximate border of the pseudogapped region in La2−*x*Sr*x*CuO4 [14] and YBa2Cu3O7−*<sup>δ</sup>* [5, 45]. Moreover, the symmetry of the pseudogap order parameter (isotropic) differs from that for the superconducting one (*dx*2−*y*<sup>2</sup> ) in Bi2Sr2CaCuO8+*<sup>δ</sup>* [79], superconductivity in Bi2Sr2−*x*La*x*CuO6+*<sup>δ</sup>* emerges with doping when the (nodal) pseudogap disappears [72], the pseudogap competes with the superconducting gap at antinodes in (Bi,Pb)2(Sr,La)2CuO6+*<sup>δ</sup>* [43], and the interplay of pseudogapping and superconductivity among different members of the oxide family (Ca*x*La1−*x*)(Ba1.75−*x*La0.25+*x*)Cu3O*<sup>y</sup>* is not the same for varying dopings *x* [15]. It is worthy of note that both angle-resolved photoemission spectroscopy (ARPES) and scanning tunnel microscopy (STM) experiments allow one to measure only overall energy gaps whatever their microscopic origin. That is why it is usually difficult to distinguish for sure between superconducting, SDW, and CDW gaps even in the case when they manifest themselves separately in certain momentum ranges each [11, 42].

As for direct experiments confirming the existence of CDWs competing with superconductivity in cuprates, CDWs have been shown to be a more important factor in this sense than SDWs, the remnants of which survive far from the antiferromagnetic state appropriate to zero-doped samples of superconducting families [20]. It is useful to shortly summarize the main new findings in this area.

X-ray scattering experiments in YBa2Cu3O6+*<sup>x</sup>* revealed the CDW ordering at temperatures lower than those of the pseudogap formation, giant phonon anomalies, and elastic central peak induced by nanodomain CDWs [9, 10, 45, 59]. The CDW correlation length increases with the temperature, *T*, lowering. However, the competing superconducting order parameter, which emerges below *Tc*, so depresses CDWs that the true CDW long-range order does not develop, as was shown by Raman scattering [5]. Suppression of CDWs by Cooper pairing was also found in x-ray measurements of La2−*x*Sr*x*CuO4 [14].

The well-known CDW manifestations in Bi2Sr2−*x*La*x*CuO6+*<sup>δ</sup>* were recently confirmed by complex X-ray, ARPES, and STM studies [13]. Those authors associate CDWs with pseudogapping, but argue that the CDW wave vector connects the Fermi arc tips rather than the antinodal Fermi surface (FS) sections, as stems from the Peierls-insulator scenario [27, 41]. This conclusion, if being true, makes the whole picture even more enigmatic than in the conventional density-wave approach to pseudogaps either in the mean-field approximation or taking into account fluctuations.

The electron-hole asymmetric CDW ordering was demonstrated by STM and resonant elastic x-ray scattering measurements [17] for Bi2Sr2CaCuO8+*<sup>δ</sup>* samples, with the pseudogapping in the antinodal momentum region. As was shown in those experiments, CDWs and concomitant periodic crystal lattice distortions, PLDs can be observed directly, whereas their interplay with superconductivity manifestations can be seen only indirectly, e.g., as anticorrelations between *Tc* and the structural, *Ts*, or CDW, *T*CDW, transition temperature.(There is a viewpoint [19] that the strong interrelation between electronic CDW modulations and PLDs [27], inherent, e.g., to the Peierls model of the structural phase transition [41], does not exist, and PLDs can emerge without electronic contributions, which seems strange in the context of indispensable Coulomb forces.). This fact is well known, say, for superconducting transition metal dichalcogenides [49] or pseudoternary systems (Lu1−*x*Sc*x*)5Ir4Si10 [102]. Therefore, it seems interesting to propose such studies of superconducting properties, which would demonstrate manifestations of CDW existence, although the CDW gapping is an insulating rather than a superconducting one. In a number of publications, we suggested that certain measurements of the stationary Josephson critical current, *Ic*, between quasi-two-dimensional CDW superconductors with the *dx*2−*y*<sup>2</sup> order parameter symmetry (inherent to cuprates) can conspicuously reveal such dependences that would reflect CDW gapping as well or at least demonstrate that the actual gapping symmetry differs from the pure *dx*2−*y*<sup>2</sup> one [29–31, 34, 35]. Below, we present further theoretical studies in this direction, which put forward even more effective experiments.

#### **2. Formulation**

2 Superconductors

[18, 48, 61, 62].

summarize the main new findings in this area.

or taking into account fluctuations.

Fermi liquid concept belonging to Landau [1] and the role of strong electron correlations

During last decades we have been developing a phenomenological theory to elucidate the influence of CDWs on superconductivity of high-*Tc* oxides, since the CDWs were observed in a number of those materials [23, 33, 35, 37–39]. We identified the CDW energy gap with the pseudogap mentioned above. Such an identification is based, in particular, on the appearance of CDWs only below the approximate border of the pseudogapped region in La2−*x*Sr*x*CuO4 [14] and YBa2Cu3O7−*<sup>δ</sup>* [5, 45]. Moreover, the symmetry of the pseudogap order parameter (isotropic) differs from that for the superconducting one (*dx*2−*y*<sup>2</sup> ) in Bi2Sr2CaCuO8+*<sup>δ</sup>* [79], superconductivity in Bi2Sr2−*x*La*x*CuO6+*<sup>δ</sup>* emerges with doping when the (nodal) pseudogap disappears [72], the pseudogap competes with the superconducting gap at antinodes in (Bi,Pb)2(Sr,La)2CuO6+*<sup>δ</sup>* [43], and the interplay of pseudogapping and superconductivity among different members of the oxide family (Ca*x*La1−*x*)(Ba1.75−*x*La0.25+*x*)Cu3O*<sup>y</sup>* is not the same for varying dopings *x* [15]. It is worthy of note that both angle-resolved photoemission spectroscopy (ARPES) and scanning tunnel microscopy (STM) experiments allow one to measure only overall energy gaps whatever their microscopic origin. That is why it is usually difficult to distinguish for sure between superconducting, SDW, and CDW gaps even in the case when they manifest themselves separately in certain momentum ranges each [11, 42]. As for direct experiments confirming the existence of CDWs competing with superconductivity in cuprates, CDWs have been shown to be a more important factor in this sense than SDWs, the remnants of which survive far from the antiferromagnetic state appropriate to zero-doped samples of superconducting families [20]. It is useful to shortly

X-ray scattering experiments in YBa2Cu3O6+*<sup>x</sup>* revealed the CDW ordering at temperatures lower than those of the pseudogap formation, giant phonon anomalies, and elastic central peak induced by nanodomain CDWs [9, 10, 45, 59]. The CDW correlation length increases with the temperature, *T*, lowering. However, the competing superconducting order parameter, which emerges below *Tc*, so depresses CDWs that the true CDW long-range order does not develop, as was shown by Raman scattering [5]. Suppression of CDWs by

The well-known CDW manifestations in Bi2Sr2−*x*La*x*CuO6+*<sup>δ</sup>* were recently confirmed by complex X-ray, ARPES, and STM studies [13]. Those authors associate CDWs with pseudogapping, but argue that the CDW wave vector connects the Fermi arc tips rather than the antinodal Fermi surface (FS) sections, as stems from the Peierls-insulator scenario [27, 41]. This conclusion, if being true, makes the whole picture even more enigmatic than in the conventional density-wave approach to pseudogaps either in the mean-field approximation

The electron-hole asymmetric CDW ordering was demonstrated by STM and resonant elastic x-ray scattering measurements [17] for Bi2Sr2CaCuO8+*<sup>δ</sup>* samples, with the pseudogapping in the antinodal momentum region. As was shown in those experiments, CDWs and concomitant periodic crystal lattice distortions, PLDs can be observed directly, whereas their interplay with superconductivity manifestations can be seen only indirectly, e.g., as anticorrelations between *Tc* and the structural, *Ts*, or CDW, *T*CDW, transition temperature.(There is a viewpoint [19] that the strong interrelation between electronic CDW modulations and PLDs [27], inherent, e.g., to the Peierls model of the structural phase transition [41], does not exist, and PLDs can emerge without electronic contributions,

Cooper pairing was also found in x-ray measurements of La2−*x*Sr*x*CuO4 [14].

Following the dominating idea (see our previous publications [29–31, 34, 35, 95, 96] and references therein) concerning the electron spectrum of high-*Tc* oxides identified as partially gapped CDW superconductors, CDWSs, we restrict our consideration to the two-dimensional case with the corresponding FS shown in Fig. 1a. The superconducting *d*-wave order parameter ∆ is assumed to span the whole FS, whereas the *s*-wave mean-field dielectric (CDW) order parameter Σ develops only on the nested (dielectrized, d) FS sections. There are *N* = 4 or 2 of the latter (the checkerboard and unidirectional configurations, respectively), and they are connected in pairs by the CDW-vectors **Q**'s in the momentum space. The non-nested sections remain non-dielectrized (nd). The orientations of **Q**'s are assumed to be fixed with respect to the crystal lattice. In particular, they are considered to be directed along the **k***x*- and **k***y*-axes in the momentum space (anti-nodal nesting) [39, 67, 74]. The same orientation along **k***x*- and **k***y*-axes is also appropriate to ∆-lobes, so that we confine ourselves to the *dx*2−*y*<sup>2</sup> -wave symmetry of the superconducting order parameter as the only one found in the experiments for cuprates. Hence, the profile of the *d*-wave superconducting order parameter over the FS is written down in the form

$$
\bar{\Delta}(T,\theta) = \Delta(T) f\_{\Delta}(\theta). \tag{1}
$$

The function ∆(*T*) is the *T*-dependent magnitude of the superconducting gap, and the angular factor *f*∆(*θ*) looks like

$$f\_{\Delta}(\theta) = \cos 2\theta. \tag{2}$$

In the case *N* = 4, the experimentally measured magnitudes of the CDW order parameter Σ in high-*Tc* oxides are identical in all four CDW sectors, and the corresponding sector-connecting **Q** vectors are oriented normally to each other. Therefore, we assume the CDWs to possess the four- (the checkerboard configuration) or the two-fold (the unidirectional configuration) symmetry [3, 23, 24, 30, 39, 44, 93]. The latter is frequently associated with the electronic nematic, smectic or more complex ordering [16, 22, 26, 28, 52, 70, 83, 84, 88, 97, 99, 100]). The opening angle of each CDW sector, where Σ �= 0, equals 2*α*. Such a profile of Σ over the FS can also be described in the factorized form

**Figure 1.** (a) Superconducting, ∆¯(*θ*), and dielectric, Σ¯(*θ*), order parameter profiles of the partially gapped *d*-wave charge-density-wave (CDW) superconductor. *N* is the number of CDW sectors with the width 2*α* each. (b) The corresponding energy-gap contours (gap roses).

as

Σ¯(*T*, *θ*) = Σ(*T*)*f*Σ(*θ*), (3)

where Σ(*T*) is the *T*-dependent CDW order parameter, and the angular factor

$$f\_{\Sigma}(\theta) = \begin{cases} 1 \text{ for } |\theta - k\Omega| < \mathfrak{a} \text{ (d section)},\\ 0 \text{ otherwise} & \text{(nd section)}. \end{cases} \tag{4}$$

Here, *k* is an integer number, and the parameter Ω = *π*/2 for *N* = 4 and *π* for *N* = 2.

The both gapping mechanisms (superconducting and CDW-driven) suppress each other, because they compete for the same quasiparticle states near the FS. As a result, a combined gap (the gap rose in the momentum space, see Fig. 1b)

$$
\bar{D}(T,\theta) = \sqrt{\bar{\Sigma}^2(T,\theta) + \bar{\Delta}^2(T,\theta)},\tag{5}
$$

arises on the FS. The actual ∆(*T*)- and Σ(*T*)-values are determined from a system of self-consistent equations. The relevant initial parameters, besides *N* and *α*, include the constants of superconducting and electron-hole couplings recalculated into the pure BCS (no CDWs) and CDW (no superconductivity) limiting cases as the corresponding ∆<sup>0</sup> and Σ<sup>0</sup> order parameters at *T* = 0. It should be emphasized that our model is a simplified, generic one, because real CDWs are complex objects, which behave differently on the crystal surfaces and in the bulk [77]. Thus, it is quite natural that they are not identical for various high-*Tc* oxides [15]. Nevertheless, the presented model allows the main features of the materials concerned to be taken into account. For brevity, we mark the CDW *d*-wave superconductor with *N* CDW sectors as *S<sup>d</sup>* CDW*N*.

The *s*-wave BCS superconductor is described in the framework of the standard BCS theory. Its characteristic parameter is the value of the corresponding superconducting order parameter ∆BCS at *T* = 0. Also for the sake of brevity, it will be marked below as *S<sup>s</sup>* BCS.

4 Superconductors

as

*N* = 2.

with *N* CDW sectors as *S<sup>d</sup>*

(a) (b)

**Figure 1.** (a) Superconducting, ∆¯(*θ*), and dielectric, Σ¯(*θ*), order parameter profiles of the partially gapped *d*-wave charge-density-wave (CDW) superconductor. *N* is the number of CDW sectors with the width 2*α* each.

1 for |*θ* − *k*Ω| < *α* (d section),

Here, *k* is an integer number, and the parameter Ω = *π*/2 for *N* = 4 and *π* for

The both gapping mechanisms (superconducting and CDW-driven) suppress each other, because they compete for the same quasiparticle states near the FS. As a result, a combined

arises on the FS. The actual ∆(*T*)- and Σ(*T*)-values are determined from a system of self-consistent equations. The relevant initial parameters, besides *N* and *α*, include the constants of superconducting and electron-hole couplings recalculated into the pure BCS (no CDWs) and CDW (no superconductivity) limiting cases as the corresponding ∆<sup>0</sup> and Σ<sup>0</sup> order parameters at *T* = 0. It should be emphasized that our model is a simplified, generic one, because real CDWs are complex objects, which behave differently on the crystal surfaces and in the bulk [77]. Thus, it is quite natural that they are not identical for various high-*Tc* oxides [15]. Nevertheless, the presented model allows the main features of the materials concerned to be taken into account. For brevity, we mark the CDW *d*-wave superconductor

where Σ(*T*) is the *T*-dependent CDW order parameter, and the angular factor

Σ¯(*T*, *θ*) = Σ(*T*)*f*Σ(*θ*), (3)

0 otherwise (nd section). (4)

Σ¯ <sup>2</sup>(*T*, *θ*) + ∆¯ <sup>2</sup>(*T*, *θ*), (5)

(b) The corresponding energy-gap contours (gap roses).

*f*Σ(*θ*) =

gap (the gap rose in the momentum space, see Fig. 1b)

CDW*N*.

*D*¯ (*T*, *θ*) =

In the tunnel Hamiltonian approximation, the stationary Josephson critical current is given by the formula [7, 57, 85]

$$I\_{\mathfrak{C}}(T) = 4\varepsilon T \sum\_{\mathbf{p}\mathbf{q}} \left| \tilde{T}\_{\mathbf{p}\mathbf{q}} \right|^{2} \sum\_{\omega\_{\mathfrak{I}}} \mathsf{F}^{+}(\mathbf{p};\omega\_{\mathfrak{n}}) \mathsf{F}'(\mathbf{q};-\omega\_{\mathfrak{n}}).\tag{6}$$

Here, *<sup>T</sup>***pq** are the tunnel Hamiltonian matrix elements, **<sup>p</sup>** and **<sup>q</sup>** are the transferred momenta; *<sup>e</sup>* <sup>&</sup>gt; 0 is the elementary electrical charge, and <sup>F</sup>(**p**;*ωn*) and <sup>F</sup>′ (**q**; −*ωn*) are Gor'kov Green's functions for superconductors to the left and to the right, respectively, from the tunnel barrier (hereafter, all primed quantities are associated with the right hand side electrode). The internal summation is carried out over the discrete fermionic "frequencies" *ω<sup>n</sup>* = (2*n* + 1) *πT*, *n* = 0, ±1, ±2, . . .. Below, we consider tunnel junctions of two types: symmetric *Sd* CDW*<sup>N</sup>* <sup>−</sup> *<sup>I</sup>* <sup>−</sup> *<sup>S</sup><sup>d</sup>* CDW*<sup>N</sup>* between two identical CDWSs, and nonsymmetric *<sup>S</sup><sup>d</sup>* CDW*<sup>N</sup>* <sup>−</sup> *<sup>I</sup>* <sup>−</sup> *<sup>S</sup><sup>s</sup>* BCS between a CDWS as the left electrode and an *s*-wave BCS superconductor as the right one (here, *I* stands for the insulator). Expressions for the corresponding Green's functions can be found elsewhere [30, 35]. Since CDWS electrodes are anisotropic, their orientations with respect to the junction plane will be characterized by the angles *<sup>γ</sup>* and *<sup>γ</sup>*′ (the latter appears only in the symmetric case), i.e. the deflections of the "positive" <sup>∆</sup>- and <sup>∆</sup>′ -lobes from the normal **n** to the junction (Fig. 2). Accordingly, the angular dependences *f*∆(*θ*) and *f*Σ(*θ*) of the corresponding order parameters (see formulas (2) and (4, respectively) should be modified by changing *<sup>θ</sup>* to *<sup>θ</sup>* − *<sup>γ</sup>* or *<sup>θ</sup>* − *<sup>γ</sup>*′ .

**Figure 2.** Configuration of symmetric Josephson junction between identical S*<sup>d</sup>* CDW4's. See further explanations in the text.

An important issue while calculating the Josephson current is tunnel directionality [101], which should be taken into consideration in the tunnel Hamiltonian *<sup>T</sup>***pq**. Indeed, if we calculate *Ic* between, e.g., pure BCS *d*-wave superconductors, *S<sup>d</sup>* BCS, making no allowance for this factor, formula (6) would produce an exact zero. It is so because, owing to the alternating signs of superconducting lobes, the current contributions from the FS points described by the angles *θ* and *θ* + *π* <sup>2</sup> would exactly compensate each other in this case. The same situation also takes place in the case of a junction with *S<sup>d</sup>* CDW4. For a junction with *<sup>S</sup><sup>d</sup>* CDW2, it is not so, but, in the framework of the general approach, we have to introduce tunnel directionality in this case as well.

Here, we briefly consider three factors responsible for tunnel directionality (see a more thorough discussion in Ref. [31]). First, the velocity component normal to the junction should be taken into account. This circumstance is reflected by the cos *θ*-factor in the integrand and an angle-independent factor that can be incorporated into the junction normal-state resistance *RN* [51, 64]. Second, superconducting pairs that cross the barrier at different angles penetrate through barriers with different effective widths [12] (the height of the junction barrier is assumed to be much larger than the relevant quasiparticle energies, so that this height may be considered constant). Since the actual *<sup>θ</sup>*-dependences of *<sup>T</sup>***pq** for realistic junctions are not known, we simulate the barrier-associated directionality by the phenomenological function

$$w(\theta) = \exp\left[-\left(\frac{\tan\theta}{\tan\theta\_0}\right)^2 \ln 2\right],\tag{7}$$

This means that the effective opening of relevant tunnel angles equals 2*θ*0. The barrier transparency is normalized by the maximum value obtained for the normal tunneling with respect to the junction plane and included into the junction resistance *RN*. Hence, *w*(*θ* = 0) = 1. The multiplier ln 2 in (7) was selected to provide *w*(*θ* = *θ*0) = <sup>1</sup> <sup>2</sup> . Third, we use the model of coherent tunneling [12, 56, 60], when the superconducting pairs are allowed to tunnel between the points on the FSs of different electrodes characterized by the same angle *θ*.

As a result of the standard calculation procedure [7, 57] applied to formula (6) and in the framework of the approximations made above, we obtain the following formula for the stationary Josephson critical current across the tunnel junction:

$$\begin{split} I\_{\mathbb{C}}(T,\gamma\_{\prime}\gamma^{\prime}) &= \frac{1}{2\varepsilon\mathbb{R}\_{N}} \\ &\times \frac{1}{\pi} \int\_{-\pi/2}^{\pi/2} \cos\theta \ w(\theta) \, P(T,\theta,\gamma\_{\prime}\gamma^{\prime}) \mathrm{d}\theta\_{\prime} \end{split} \tag{8}$$

where [32, 40]

$$P(T, \theta, \gamma, \gamma') = \bar{\Lambda} \bar{\Lambda}' \int\_{\min\{D, \mathcal{D}'\}}^{\max\{\bar{D}, \bar{D}'\}} \frac{\tanh \frac{\chi}{2T} \mathrm{d}x}{\sqrt{\left(\mathfrak{x}^2 - \bar{D}^2\right) \left(\bar{D}'^2 - \mathfrak{x}^2\right)}}.\tag{9}$$

Here, for brevity, we omitted the arguments in the dependences <sup>∆</sup>¯(*T*, *<sup>θ</sup>* − *<sup>γ</sup>*), <sup>∆</sup>¯ ′ (*T*, *<sup>θ</sup>* − *<sup>γ</sup>*′ ), *<sup>D</sup>*¯ (*T*, *<sup>θ</sup>* − *<sup>γ</sup>*), and *<sup>D</sup>*¯ ′ (*T*, *<sup>θ</sup>* − *<sup>γ</sup>*′ ). Integration over *θ* in Eq. (8) is carried out within the interval −*<sup>π</sup>* <sup>2</sup> <sup>≤</sup> *<sup>θ</sup>* <sup>≤</sup> *<sup>π</sup>* <sup>2</sup> , i.e. over the "FS semicircle" turned towards the junction plane. If any directionality and CDW gapping are excluded (so that the integration over *θ* is reduced to a factor of *<sup>π</sup>*) and the angular factors *<sup>f</sup>*<sup>∆</sup> and *<sup>f</sup>* ′ ∆ remain preserved, we arrive at the Sigrist–Rice model [81].

#### **3. Results and their discussion**

6 Superconductors

phenomenological function

same angle *θ*.

where [32, 40]

*<sup>D</sup>*¯ (*T*, *<sup>θ</sup>* − *<sup>γ</sup>*), and *<sup>D</sup>*¯ ′

<sup>2</sup> <sup>≤</sup> *<sup>θ</sup>* <sup>≤</sup> *<sup>π</sup>*

model [81].

−*<sup>π</sup>*

Here, we briefly consider three factors responsible for tunnel directionality (see a more thorough discussion in Ref. [31]). First, the velocity component normal to the junction should be taken into account. This circumstance is reflected by the cos *θ*-factor in the integrand and an angle-independent factor that can be incorporated into the junction normal-state resistance *RN* [51, 64]. Second, superconducting pairs that cross the barrier at different angles penetrate through barriers with different effective widths [12] (the height of the junction barrier is assumed to be much larger than the relevant quasiparticle energies, so that this height may be considered constant). Since the actual *<sup>θ</sup>*-dependences of *<sup>T</sup>***pq** for realistic junctions are not known, we simulate the barrier-associated directionality by the

*w*(*θ*) = exp

stationary Josephson critical current across the tunnel junction:

) = <sup>1</sup> 2*eRN*

> *<sup>π</sup>*/2 −*π*/2

> > max{*D*¯ ,*D*¯ ′ }

min{*D*¯ ,*D*¯ ′ }

directionality and CDW gapping are excluded (so that the integration over *θ* is reduced to a

Here, for brevity, we omitted the arguments in the dependences <sup>∆</sup>¯(*T*, *<sup>θ</sup>* − *<sup>γ</sup>*), <sup>∆</sup>¯ ′

× 1 *π*

) = ∆¯ ∆¯ ′

*Ic*(*T*, *<sup>γ</sup>*, *<sup>γ</sup>*′

*<sup>P</sup>*(*T*, *<sup>θ</sup>*, *<sup>γ</sup>*, *<sup>γ</sup>*′

(*T*, *<sup>θ</sup>* − *<sup>γ</sup>*′

factor of *<sup>π</sup>*) and the angular factors *<sup>f</sup>*<sup>∆</sup> and *<sup>f</sup>* ′

 −

*w*(*θ* = 0) = 1. The multiplier ln 2 in (7) was selected to provide *w*(*θ* = *θ*0) = <sup>1</sup>

 tan *θ* tan *θ*<sup>0</sup>

This means that the effective opening of relevant tunnel angles equals 2*θ*0. The barrier transparency is normalized by the maximum value obtained for the normal tunneling with respect to the junction plane and included into the junction resistance *RN*. Hence,

we use the model of coherent tunneling [12, 56, 60], when the superconducting pairs are allowed to tunnel between the points on the FSs of different electrodes characterized by the

As a result of the standard calculation procedure [7, 57] applied to formula (6) and in the framework of the approximations made above, we obtain the following formula for the

2 ln 2 

cos *<sup>θ</sup> <sup>w</sup>*(*θ*) *<sup>P</sup>*(*T*, *<sup>θ</sup>*, *<sup>γ</sup>*, *<sup>γ</sup>*′

tanh *<sup>x</sup>* <sup>2</sup>*<sup>T</sup>* d*x* (*x*<sup>2</sup> − *D*¯ <sup>2</sup>) (*D*¯ ′<sup>2</sup> − *x*2)

). Integration over *θ* in Eq. (8) is carried out within the interval

∆ remain preserved, we arrive at the Sigrist–Rice

<sup>2</sup> , i.e. over the "FS semicircle" turned towards the junction plane. If any

, (7)

)d*θ*, (8)

. (9)

(*T*, *<sup>θ</sup>* − *<sup>γ</sup>*′

),

<sup>2</sup> . Third,

The influence of various problem parameters on the critical stationary Josephson curent in the symmetric, *S<sup>d</sup>* CDW*<sup>N</sup>* <sup>−</sup> *<sup>I</sup>* <sup>−</sup> *<sup>S</sup><sup>d</sup>* CDW*N*, and nonsymmetric, *<sup>S</sup><sup>d</sup>* CDW*<sup>N</sup>* <sup>−</sup> *<sup>I</sup>* <sup>−</sup> *<sup>S</sup><sup>s</sup>* BCS, junctions was analyzed in detail in works [30, 31]. Here, we attract attention to the problem of CDW detection in high-*Tc* oxides.

The number of problem parameters can be diminished by normalizing the "order parameter" quantities by one of them. For such a normalization, we selected the parameter ∆<sup>0</sup> and introduced the dimensionless order parameters *<sup>σ</sup>*<sup>0</sup> = <sup>Σ</sup>0/∆<sup>0</sup> and *<sup>δ</sup>*BCS = <sup>∆</sup>BCS(*<sup>T</sup>* → <sup>0</sup>)/∆<sup>0</sup> (for the superconducting order parameter of CDWS, *δ*<sup>0</sup> = ∆0/∆<sup>0</sup> = 1). With regard to experimental needs, we also introduced the reduced temperature *τ* = *T*/*Tc*. Here *Tc* is the actual critical temperature of the CDWS. In the framework of our theory, it has to be found from the system of equations for the CDWS indicated above. For the Josephson current amplitude *Ic*, we introduced the dimensionless combination *ic* = *IceRN*/∆0.

One more preliminary remark concerns the parameter of effective tunnel directionality *θ*<sup>0</sup> (see formula (7)). Our calculations [30, 31] showed that its choice is very important. On the one hand, large values of this parameter correspond to thin junctions and large values of the tunnel current, which is beneficial for the experiment. However, in this case, the predicted phenomena become effectively smoothed out up to their disappearance. On the other hand, narrow tunnel cones (small *θ*0-values) provide well pronounced effects, but correspond to thick interelectrode layers and, as a result, small tunnel currents. Hence, in the real experiment, a reasonable compromise should be found between those two extremes.

#### **3.1. Electrode rotation**

While examining Fig. 2, it becomes clear that the clearest way to prove that electrons in high-*Tc* oxides undergo an additional pairing of some origin besides the *d*-wave BCS one is to demonstrate that the gap rose differs from that in the *S<sup>d</sup>* BCS superconductor. The case in question concerns pairing symmetries, which may be different from the *d*-wave one or/and extend over only certain FS regions. In the framework of the tunnel technique, the most direct way to perform the search is to fix one electrode and rotate the other one (e.g., *<sup>γ</sup>*′ = const and *γ* = var). In the case of *S<sup>d</sup>* BCS <sup>−</sup> *<sup>I</sup>* <sup>−</sup> *<sup>S</sup><sup>d</sup>* BCS junction, the corresponding *ic*(*γ*) dependences are known to have a cosine profile stemming from dependence (2) for the superconducting order parameter ∆ and, since any other gapping is absent, for the corresponding gap rose (*D*¯ (*T*, *θ*) = |∆(*T*, *θ*)|). Any deviations of the gap rose from this behavior will testify in favor of the existence of additional order parameter(s). Certainly, averaging the current over the FS will smooth the relevant peculiarities and making allowance for tunnel directionality will distort them. Nevertheless, the proposed method will be sufficient to detect the competing pairing without its ultimate identification.

In Fig. 3, the corresponding normalized *ic*(*γ*) dependences calculated for the symmetric S*d* CDW*<sup>N</sup>* <sup>−</sup> *<sup>I</sup>*−S*<sup>d</sup>* CDW*<sup>N</sup>* junction and the CDW geometries *<sup>N</sup>* <sup>=</sup> 2 and 4, as well as the reference *<sup>d</sup>*-wave BCS curve, are shown. The tunnel directionality parameter *<sup>θ</sup>*<sup>0</sup> <sup>=</sup> <sup>10</sup>◦ was assumed. A more detailed analysis of *ic*(*γ*) dependences and their relations with other problem parameters can be found in work [30]. The results obtained testify that the formulated task is feasible. An attractive feature of this technique is that, instead of the fixed *S<sup>d</sup>* CDW*<sup>N</sup>* electrode, we may use the *S<sup>s</sup>* BCS one as well, which might be more convenient from the experimental point of view.

**Figure 3.** Orientation dependences of the reduced critical Josephson current for the symmetric junction.

#### **3.2. Anomalous temperature dependence of** *Ic*

The measurement of the temperature dependences of the critical Josephson tunnel current *Ic*(*T*) seems to be the most easily realizable method of those proposed in this work. The dependence *Ic*(*T*) in the symmetric *S<sup>s</sup>* BCS <sup>−</sup> *<sup>I</sup>* <sup>−</sup> *<sup>S</sup><sup>s</sup>* BCS junctions has a monotonic convex shape. Among other things, this fact is associated with the constant sign of order parameter over the whole FS. However, in the case of symmetric *S<sup>d</sup>* BCS <sup>−</sup> *<sup>I</sup>* <sup>−</sup> *<sup>S</sup><sup>d</sup>* BCS junctions, the situation may change. Indeed, for junctions involving YBa2Cu3O7−*δ*, nonmonotonic *Ic*(*T*)-dependences and even the change of *Ic* sign, i.e. the transformation of the 0-junction into the *π*-one or vice versa were observed [47, 87]. Such a phenomenon was not found for other cuprates. However, it is extremely difficult to produce Josephson junctions made of other materials than YBa2Cu3O7−*δ*. Therefore, further technological breakthrough is needed to make sure that the non-monotonic behavior is a general phenomenon inherent to all high-*Tc* oxides with *d*-wave superconducting order parameter.

It should be noted that, in the measurements concerned, the electrodes remained fixed, so that the peculiar behavior of *Ic*(*T*) could not result from the change of overlapping between the superconducting lobes with different signs. There is an explanation based on the existence of the bound states in the junction due to the Andreev–Saint-James effect [51, 64]. This theory predicts that the current *Ic*(*T*) between *d*-wave superconductors must exhibit a singularity at *T* → 0. Nevertheless, the latter has not been observed experimentally until now. Probably, this effect is wiped out by the roughness of the interfaces in the oxide junctions [6, 76] and therefore may be of academic interest.

Earlier we suggested a different scenario [36]. Namely, we showed that, at some relative orientations of *S<sup>d</sup>* BCS <sup>−</sup> *<sup>I</sup>* <sup>−</sup> *<sup>S</sup><sup>d</sup>* BCS junction electrodes, one of them can play a role of differential detector, which enables tiny effects connected with the thermally induced repopulation of quasiparticle levels near the FS to be observed. In our approach, no zero-*T* singularity of the current could arise.

A similar situation takes place for CDWSs. Although we cannot assign a definite sign to the combined gap *D*¯ (see Eq. (5), the corresponding unambiguously signed ∆ enters the expression for the calculation of *Ic* (formulas (8) and 9). In this sense, the FS of the CDWS "remembers" the specific ∆-sign at every of its points and, thus, can also serve as a differential detector of the current at definite electrode orientations. As a result, the dependences *Ic*(*T*)

both for symmetric *S<sup>d</sup>* CDW*<sup>N</sup>* <sup>−</sup> *<sup>I</sup>* <sup>−</sup> *<sup>S</sup><sup>d</sup>* CDW*<sup>N</sup>* and nonsymmetric *<sup>S</sup><sup>d</sup>* CDW*<sup>N</sup>* <sup>−</sup> *<sup>I</sup>* <sup>−</sup> *<sup>S</sup><sup>d</sup>* BCS junctions can also by nonmonotonic and even sign-changing functions. Unlike the *S<sup>d</sup>* BCS <sup>−</sup> *<sup>I</sup>* <sup>−</sup> *<sup>S</sup><sup>d</sup>* BCS junctions, for which the *Ic*(*T*)-behavior could depend only on the orientation angles of both electrodes (*<sup>γ</sup>* and *<sup>γ</sup>*′ ), now the other parameters responsible for the superconducting and combined gaps—these are *σ*<sup>0</sup> and *α*—become relevant. In Figs. 4 and 5, the *ic*(*τ*) dependences are shown for various fixed *α* and *σ*0, respectively, both for the "checkerboard" and "unidirectional" CDW geometry. We would like to attract attention to the fact that those dependences are rather sensitive to the electrode orientations (see the relevant illustration in Fig. 6), so that it might be laborious to find a suitable experimental configuration.

The key issue is that the parameters *σ*<sup>0</sup> and/or *α* can be (simultaneously) varied by doping. Hence, doping CDWS electrodes and keeping their orientations fixed, we could change even the character of the *Ic*(*T*) dependence: monotonic, nonmonotonic, and sign-changing. Provided the corresponding set of parameters, we could transform the same junction, say, from the 0-state into the *π*-one by varying the temperature only.

#### **3.3. Anomalous doping dependence of** *Ic*

8 Superconductors

**Figure 3.** Orientation dependences of the reduced critical Josephson current for the symmetric junction.

BCS <sup>−</sup> *<sup>I</sup>* <sup>−</sup> *<sup>S</sup><sup>s</sup>*

The measurement of the temperature dependences of the critical Josephson tunnel current *Ic*(*T*) seems to be the most easily realizable method of those proposed in this work. The

Among other things, this fact is associated with the constant sign of order parameter over

change. Indeed, for junctions involving YBa2Cu3O7−*δ*, nonmonotonic *Ic*(*T*)-dependences and even the change of *Ic* sign, i.e. the transformation of the 0-junction into the *π*-one or vice versa were observed [47, 87]. Such a phenomenon was not found for other cuprates. However, it is extremely difficult to produce Josephson junctions made of other materials than YBa2Cu3O7−*δ*. Therefore, further technological breakthrough is needed to make sure that the non-monotonic behavior is a general phenomenon inherent to all high-*Tc* oxides with

It should be noted that, in the measurements concerned, the electrodes remained fixed, so that the peculiar behavior of *Ic*(*T*) could not result from the change of overlapping between the superconducting lobes with different signs. There is an explanation based on the existence of the bound states in the junction due to the Andreev–Saint-James effect [51, 64]. This theory predicts that the current *Ic*(*T*) between *d*-wave superconductors must exhibit a singularity at *T* → 0. Nevertheless, the latter has not been observed experimentally until now. Probably, this effect is wiped out by the roughness of the interfaces in the oxide junctions

Earlier we suggested a different scenario [36]. Namely, we showed that, at some relative

detector, which enables tiny effects connected with the thermally induced repopulation of quasiparticle levels near the FS to be observed. In our approach, no zero-*T* singularity of the

A similar situation takes place for CDWSs. Although we cannot assign a definite sign to the combined gap *D*¯ (see Eq. (5), the corresponding unambiguously signed ∆ enters the expression for the calculation of *Ic* (formulas (8) and 9). In this sense, the FS of the CDWS "remembers" the specific ∆-sign at every of its points and, thus, can also serve as a differential detector of the current at definite electrode orientations. As a result, the dependences *Ic*(*T*)

BCS <sup>−</sup> *<sup>I</sup>* <sup>−</sup> *<sup>S</sup><sup>d</sup>*

BCS junction electrodes, one of them can play a role of differential

BCS junctions has a monotonic convex shape.

BCS junctions, the situation may

**3.2. Anomalous temperature dependence of** *Ic*

the whole FS. However, in the case of symmetric *S<sup>d</sup>*

dependence *Ic*(*T*) in the symmetric *S<sup>s</sup>*

*d*-wave superconducting order parameter.

[6, 76] and therefore may be of academic interest.

BCS <sup>−</sup> *<sup>I</sup>* <sup>−</sup> *<sup>S</sup><sup>d</sup>*

orientations of *S<sup>d</sup>*

current could arise.

Now, let the electrode orientations be fixed by the experimentalist [55, 82] and the temperature be zero (for simplicity), but the both parameters *α* and *σ*<sup>0</sup> can be varied (by doping). In Figs. 7 and 8, the dependences of the dimensionless order parameters *δ*(0) = ∆(*T* = 0)/∆<sup>0</sup> and *σ*(0) = Σ(*T* = 0)/∆<sup>0</sup> on *α* and *σ*<sup>0</sup> are exhibited for both analyzed CDW structures. One can see that, in every cross-section *α* = const or *σ*<sup>0</sup> = const, both *δ*(0) and *σ*(0) profiles are monotonic. At first glance, the Josephson tunnel current should also demonstrate such a behavior. However, our previous calculations [30, 31, 35] showed that it is so when the orientations of *S<sup>d</sup>* CDW*<sup>N</sup>* electrodes in the *<sup>S</sup><sup>d</sup>* CDW*<sup>N</sup>* <sup>−</sup> *<sup>I</sup>* <sup>−</sup> *<sup>S</sup><sup>d</sup>* CDW*<sup>N</sup>* junction are close or rotated by about 90◦ with respect to each other, i.e. when the superconducting lobes strongly overlap in the momentum space and make contributions of the same sign to the current. But if they are oriented in such a way that mutually form a kind of differential detector for monitoring the states at the gapped and non-gapped FS sections, contributions with different signs cancel each other and more tiny effects become observable. Such a conclusion can already be made from Figs. 4 and 5.

Really, as is illustrated by Figs. 7 and 8, in the limiting cases—*σ*<sup>0</sup> → <sup>∞</sup> for both kinds of CDWs, and, if *<sup>σ</sup>*<sup>0</sup> <sup>≥</sup> <sup>√</sup>e/2 <sup>≈</sup> 0.824 (here, e is the Euler constant), *<sup>α</sup>* <sup>→</sup> *<sup>π</sup>*/4 at *<sup>N</sup>* <sup>=</sup> 4 or *π*/2 at *N* = 2 [24]—we have *δ*(0) → 0. Then, according to formulas (8) and 9), *Ic* also vanishes. Therefore, if the current crosses the point *ic* = 0 at some values of parameters *α* or *σ*<sup>0</sup> different from their limiting ones, (i) the current behavior becomes nontrivial, because larger values of *α* and *σ*0, which are accompanied by smaller values of the superconducting order parameter *δ*, lead to the current enhancement. Nevertheless, as *α* or *σ*<sup>0</sup> grows further towards its corresponding limiting values, the current must sooner or later begin to decrease by the absolute value.

This conclusion is confirmed by Figs. 9 and 10, where the dependences *ic*(*σ*0, *α* = const) and *ic*(*α*, *σ*<sup>0</sup> = const) at *T* = 0 are shown. While analyzing those figures, the following consideration should be taken into account. Namely, we suppose that gradual doping monotonically affects the parameters *α* and *σ*<sup>0</sup> of *S<sup>d</sup>* CDW*<sup>N</sup>* superconductors. Specific calculations (Figs. 9 and 10) were made assuming that only one of the control parameters, *α*

**Figure 4.** Temperature dependences of the Josephson current for various numbers of CDW sectors *N* = 4 (a) and 2 (b), and their widths *α* = 0 (solid), 5 (dashed), 10 (dotted), 15 (dash-dotted), 20 (dash-dot-dotted), 25 (short-dashed), and <sup>30</sup>◦ (dash-dash-dotted). *<sup>σ</sup>*<sup>0</sup> <sup>=</sup> 1.3, *<sup>γ</sup>* <sup>=</sup> <sup>15</sup>◦, *<sup>γ</sup>*′ <sup>=</sup> <sup>45</sup>◦, *<sup>θ</sup>*<sup>0</sup> <sup>=</sup> <sup>10</sup>◦. See further explanations in the text.

**Figure 5.** The same as in Fig. 4 but for *<sup>α</sup>* <sup>=</sup> <sup>15</sup>◦ and various *<sup>σ</sup>*<sup>0</sup> <sup>=</sup> 0.9 (solid), 1 (dashed), 1.1 (dotted), 1.3 (dash-dotted), 1.5 (dash-dot-dotted), and 3 (short-dashed).

or *σ*0, changes, which is most likely not true in the real experiment. However, the presented results testify that each of those parameters differently affects the current. Moreover, underdoping is usually accompanied by the increase of both *α* and Σ (proportional to the structural phase transition temperature, i.e. the pseudogap appearance temperature, *<sup>T</sup>*∗) [39, 42, 63, 94]. Therefore, the situation when the doping-induced simultaneous changes in the values of *α* and Σ<sup>0</sup> would lead to their mutual compensation seems improbable. Accordingly, we believe that the proposed experiments may be useful in one more, this time indirect, technique to probe CDWs in high-*Tc* oxides. In particular, the oscillating dependences *ic*(*α*) depicted in Fig. 10b, if reproduced in the experiment, will be certain to prove the interplay between the superconducting order parameter and another, competing, one; here, the latter is considered theoretically to be associated with CDWs.

<sup>208</sup> Superconductors – New Developments Measurements of Stationary Josephson Current between High-*Tc* Oxides as a Tool to Detect Charge Density Waves 11 10.5772/59590 Measurements of Stationary Josephson Current between High-*Tc* Oxides as a Tool to Detect Charge Density Waves http://dx.doi.org/10.5772/59590 209

**Figure 6.** The same as in Fig. 4a but for *<sup>α</sup>* <sup>=</sup> <sup>15</sup>◦ and *<sup>σ</sup>*<sup>0</sup> <sup>=</sup> 1.1 and various <sup>10</sup>◦ <sup>≤</sup> *<sup>γ</sup>* <sup>≤</sup> <sup>20</sup>◦.

10 Superconductors

in the text.

(a) (b)

**Figure 4.** Temperature dependences of the Josephson current for various numbers of CDW sectors *N* = 4 (a) and 2 (b), and their widths *α* = 0 (solid), 5 (dashed), 10 (dotted), 15 (dash-dotted), 20 (dash-dot-dotted), 25 (short-dashed), and <sup>30</sup>◦ (dash-dash-dotted). *<sup>σ</sup>*<sup>0</sup> <sup>=</sup> 1.3, *<sup>γ</sup>* <sup>=</sup> <sup>15</sup>◦, *<sup>γ</sup>*′ <sup>=</sup> <sup>45</sup>◦, *<sup>θ</sup>*<sup>0</sup> <sup>=</sup> <sup>10</sup>◦. See further explanations

(a) (b)

**Figure 5.** The same as in Fig. 4 but for *<sup>α</sup>* <sup>=</sup> <sup>15</sup>◦ and various *<sup>σ</sup>*<sup>0</sup> <sup>=</sup> 0.9 (solid), 1 (dashed), 1.1 (dotted), 1.3

or *σ*0, changes, which is most likely not true in the real experiment. However, the presented results testify that each of those parameters differently affects the current. Moreover, underdoping is usually accompanied by the increase of both *α* and Σ (proportional to the structural phase transition temperature, i.e. the pseudogap appearance temperature, *<sup>T</sup>*∗) [39, 42, 63, 94]. Therefore, the situation when the doping-induced simultaneous changes in the values of *α* and Σ<sup>0</sup> would lead to their mutual compensation seems improbable. Accordingly, we believe that the proposed experiments may be useful in one more, this time indirect, technique to probe CDWs in high-*Tc* oxides. In particular, the oscillating dependences *ic*(*α*) depicted in Fig. 10b, if reproduced in the experiment, will be certain to prove the interplay between the superconducting order parameter and another, competing,

one; here, the latter is considered theoretically to be associated with CDWs.

(dash-dotted), 1.5 (dash-dot-dotted), and 3 (short-dashed).

**Figure 7.** Dependences of the normalized zero-temperature order parameters *δ*(0) (a) and *σ*(0) (b) for the S*d* CDW4 superconductor on *<sup>α</sup>* and *<sup>σ</sup>*0.

**Figure 8.** The same as in Fig. 7 but for the S*<sup>d</sup>* CDW2 superconductor.

**Figure 9.** Dependences of the normalized zero-temperature Josephson current on *σ*<sup>0</sup> for *N* = 4 (a) and *N* = 2 (b) CDW configurations and various *α*'s: (a) *α* = 5 (solid), 10 (dashed), 15 (dotted), 20 (dash-dotted), and 25◦ (dash-dot-dotted); (b) *α* = 5 (solid), 15 (dashed), 25 (dotted), 35 (dash-dotted), 45 (dash-dot-dotted), and 55◦ (short-dashed).

**Figure 10.** Dependences of the normalized zero-temperature Josephson current on *α* for *N* = 4 (a) and *N* = 2 (b) CDW configurations and various *σ*<sup>0</sup> = 0.9 (solid), 1,1 (dashed), 1.3 (dotted), 1.5 (dash-dotted). *γ* = 15◦ and *γ* = 45◦.

#### **4. Conclusions**

In the two-dimensional model appropriate for cuprates, we calculated the dependences of the stationary critical Josephson tunnel current *Ic* in junctions involving *d*-wave superconductors with CDWs on the temperature, the CDW parameters, and the electrode orientation angles with respect to the junction plane. It was shown that the intertwining of the CDW and superconducting order parameters leads to peculiar dependences of *Ic*, which reflect the existence of CDW gapping. The peculiarities become especially salient when the crystal configurations on the both sides of the sandwich make the overall current extremely sensitive to the overlap between the superconducting lobes and the CDW sectors. In this case, the whole structure can be considered as a differential tool suitable to detect CDWs. Doping serves here as a control process to reveal the CDW manifestations. Such configurations have already been created for YBa2Cu3O7−*<sup>δ</sup>* [55, 82] and may be used to check the predictions of our theory.

#### **Acknowledgements**

12 Superconductors

(short-dashed).

*γ* = 45◦.

**4. Conclusions**

0

(a) (b)

**Figure 9.** Dependences of the normalized zero-temperature Josephson current on *σ*<sup>0</sup> for *N* = 4 (a) and *N* = 2 (b) CDW configurations and various *α*'s: (a) *α* = 5 (solid), 10 (dashed), 15 (dotted), 20 (dash-dotted), and 25◦ (dash-dot-dotted); (b) *α* = 5 (solid), 15 (dashed), 25 (dotted), 35 (dash-dotted), 45 (dash-dot-dotted), and 55◦

0

(a) (b)

**Figure 10.** Dependences of the normalized zero-temperature Josephson current on *α* for *N* = 4 (a) and *N* = 2 (b) CDW configurations and various *σ*<sup>0</sup> = 0.9 (solid), 1,1 (dashed), 1.3 (dotted), 1.5 (dash-dotted). *γ* = 15◦ and

In the two-dimensional model appropriate for cuprates, we calculated the dependences of the stationary critical Josephson tunnel current *Ic* in junctions involving *d*-wave superconductors with CDWs on the temperature, the CDW parameters, and the electrode orientation angles with respect to the junction plane. It was shown that the intertwining of the CDW and superconducting order parameters leads to peculiar dependences of *Ic*, which reflect the existence of CDW gapping. The peculiarities become especially salient when the crystal configurations on the both sides of the sandwich make the overall current extremely sensitive to the overlap between the superconducting lobes and the CDW sectors. In this case, the The work was partially supported by the Project N 8 of the 2012–2014 Scientific Cooperation Agreement between Poland and Ukraine. MSL was also supported by the Narodowe Centrum Nauki in Poland (grant No. 2011/01/B/NZ1/01622). The authors are grateful to Alexander Kasatkin (Institute of Metal Physics, Kyiv), and Fedor Kusmartsev and Boris Chesca (Loughborough University, Loughborough) for useful discussion.

### **Author details**

Alexander M. Gabovich\*1, Mai Suan Li2, Henryk Szymczak2 and Alexander I. Voitenko<sup>1</sup>

\*Address all correspondence to: gabovich@iop.kiev.ua

1 Institute of Physics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

2 Institute of Physics, Polish Academy of Sciences, Warsaw, Poland

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## **Impurity Effects in Iron Pnictide Superconductors**

Yuriy G. Pogorelov, Mario C. Santos and Vadim M. Loktev

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/59629

#### **1. Introduction**

The recent discovery of superconductivity (SC) with rather high critical temperature in the family of doped iron pnictide compounds [1, 2], has motivated a great interest to these materials (see the reviews [3, 4]). Unlike the extensively studied cuprate family [5], that present insulating properties in their initial undoped state, the undoped LaOFeAs compound is a semimetal. As was established by the previous physical and chemical studies (see, e.g., [6, 7]), this material has a layered structure, where the SC state is supported by the FeAs layer with a 2D square lattice of Fe atoms and with As atoms located out of plane, above or below the centers of square cells (Fig. 1). Its electronic structure, relevant for constructing microscopic SC models, have been explored with high-resolution angle-resolved photoemission spectro‐ scopy (ARPES) techniques [8, 9]. Their results indicate the multiple connected structure of Fermi surface, consisting of electron and hole pockets and absence of nodes in both electron and whole spectrum gaps [8], suggesting these systems to display the so-called extended *s*wave (also called *s*±-wave) SC order, changing the order parameter sign between electron and whole segments [10].

To study the band structure, the first principles numeric calculations are commonly used, outlining the importance of Fe atomic *d*-orbitals. The calculations show that SC in these materials is associated with Fe atoms in the layer plane, represented in Fig. 1 by their orbitals and the related hopping amplitudes. The dominance of Fe atomic 3*d* orbitals in the density of states of LaOFeAs compound near its Fermi surface was demonstrated by the local density approximation (LDA) calculations [10-15]. It was then concluded that the multi-orbital effects are important for electronic excitation spectrum in the SC state, causing formation of two spectrum gaps: by electron and hole pockets at the Fermi surface. To explain the observed SC properties, an unconventional pairing mechanism, beyond the common electron-phonon

© 2015 The Author(s). Licensee InTech. This chapter is 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.

*dyz* (white) Fe orbitals and the Fe-Fe hopping parameters in the minimal coupling model. Note that the hoppings between next near neighbors (*t*3,4) are mediated by the As orbitals (out of Fe plane) **Figure 1.** Schematics of a FeAs layer in the LaoFeAs compound with *dxz* (dark) and *dyz* (white) Fe orbitals and the Fe-Fe hopping parameters in the minimal coupling model. Note that the hoppings between next near neighbors (*t*3, 4) are mediated by the As orbitals (out of Fe plane)

atoms and the out-of-plane As atoms (above and below the centers of squares, Fig. 1) that mainly

Fig. 1. Schematics of a FeAs layer in the LaoFeAs compound with *dxz* (dark) and

scheme, was suggested for these materials [16-19]. In general, the total of 5 atomic orbitals for each iron in the LaOFeAs compound can be involved, however the ways to reduce this basis are sought, in order to simplify analytical and computational work. Some authors [20, 21] have suggested that it is sufficient to consider only the *dxz* and *dyz* orbitals. Building such minimal coupling model based on two orbitals, one is able to adjust the model parameters (energy hopping and chemical potential) to obtain the Fermi surface with the same topology that in the first principles calculations of band structure. Even though it fails to reproduce some finer features of the electronic spectrum [22, 23], this minimal coupling scheme is favored by its technical simplicity to be chosen as a basis for study of impurity effects in LaOFeAs which could be hardly tractable in more involved frameworks. define a proper topology of electronic spectrum for the SC transition. The related microscopic SC models were chosen based on the results of high-resolution angle-resolved photoemission spectroscopy (ARPES) 8,9 that display a multiply connected Fermi surface with electron and hole pockets and the SC transition produces nodeless band gaps on both of them 8. This suggests the so-called extended *s*-wave (also called *s*-wave) SC order in such systems, with opposite signs of the order parameter on electron and hole segments 10. The most common first principles approach to the band structure of these materials uses numeric calculations with most importance for Fe atomic *d*-orbitals. Thus, the local density approximation (LDA) calculations 10-15 confirm predominance of Fe atomic 3*d* orbitals (Fig. 1) in the LaOFeAs density of states near the Fermi level and so in the SC state. The next general conclusion is that the most relevant for the SC state feature of two spectrum gaps, by electron and hole pockets at the Fermi surface, is due to the multi-orbital effects. The observed SC properties are considered to follow from an unconventional pairing mechanism 16-19, beyond

Having established the SC state parameters, important effects of disorder, in particular by impurities, on the system electronic properties, have been studied for doped iron pnictides. Alike the situation in doped perovskite cuprates, impurity centers can result here either from dopants, necessary to form the very SC state, or from foreign atoms and other local defects. Within the minimal coupling model, an interesting possibility for localized impurity levels within SC gaps in doped LaOFeAs was indicated, even for the simplest, so-called isotopic (or non-magnetic) type of impurity perturbation [24, 25]. This finding marks an essential differ‐ ence from the traditional SC systems with \emph{s}-wave gap on a single-connected Fermi surface, were such perturbations are known not to produce localized impurity states and thus to have no sizeable effect on SC order, accordingly to the Anderson theorem [26]. the common electron-phonon scheme. From the total of 5 atomic orbitals that can be involved for each iron in the LaOFeAs compound, a certain reduced basis was sought, in order to simplify analytical and computational work, and it was suggested 20,21 that the choice of only *dxz* and *dyz* orbitals is sufficient. With such minimal coupling model based on two orbitals, the reduced

In presence of localized quasiparticle states by isolated impurity centers, the next impor‐ tant issue is the possibility for collective behavior of such states at finite (but low enough) impurity concentration. They are expected to give rise to some resonance effects like those well studied in semiconductors at low doping concentrations [27]. This possibility was studied long ago for electronic quasiparticles in doped semiconducting systems [28] and also for other types of quasiparticles in pnononic, magnonic, excitonic, etc. spectra under impurities [29], establishing conditions for collective (including coherent) behavior of impurity excitations. Thus, indirect interactions between impurity centers of certain type (the so-called deep levels at high enough concentrations) in doped semiconductors can lead to formation of collective band-like states [28, 30]. This corresponds to the Anderson transition in a general disordered system [31], and the emerging new band of quasiparti‐ cles in the spectrum can essentially change thermodynamics and transport in the doped material [32]. In this course, the fundamental distinction between two possible types of states is done on the basis of general Ioffe-Regel-Mott (IRM) criterion that a given excita‐ tion has a long enough lifetime compared to its oscillation period [32, 33].

Analogous effects in superconductors were theoretically predicted and experimentally discovered for magnetic impurities, both in BCS systems [34-36] and in the two-band MgB2 system [37, 38]. In all those cases, the breakdown of the Anderson's theorem is only due to the breakdown of the spin-singlet symmetry of an *s*-wave Cooper pair by a spin-polarized impurity. This limitation does not apply to the high-*Tc* doped cuprates, however their *d*-wave symmetry of SC order only permits impurity resonances in the spectrum of quasiparticles [39, 40], not their true localization, and hinders notable collective effects on their observable properties.

Therefore the main physical interest in SC iron pnictides from the point of view of disorder in general is the possibility for pair-breaking even on non-magnetic impurity [41-43] and for related localized in-gap states [21, 44-46]. This theoretical prediction was confirmed by the observations of various effects from localized impurity states, for instance, in the superfluid density (observed through the London penetration length) [47, 48], transition critical temper‐ ature [49, 50] and electronic specific heat [51], all mainly due to an emerging spike of electronic density of states against its zero value in the initial band gap. An intriguing possibility for banding of impurity levels within the SC gap [38, 52], similar to that in the above mentioned normal systems, was recently discussed for doped iron pnictides [53]. Here a more detailed analysis of the band-like impurity states is also focused on their observable effects that cannot be produced by localized impurity states.

We apply the Green function (GF) techniques, similar to those for doped cuprate SC systems [54], using the minimal coupling model by two orbitals for host electronic structure and the simplest isotopic type for impurity perturbation. The energy spectrum near in-gap impurity levels at finite impurity concentrations, emergence of specific branches of collective excitations in this range, and expected observable effects of such spectrum restructuring are discussed. Then specific GFs for SC quasiparticles are used in the general Kubo-Greenwood formalism [55, 56] to obtain the temperature and frequency dependences of optical conductivity. These results are compared with available experimental data and some suggestions are done on possible practical applications.

#### **2. Model Hamiltonian and Green functions**

scheme, was suggested for these materials [16-19]. In general, the total of 5 atomic orbitals for each iron in the LaOFeAs compound can be involved, however the ways to reduce this basis are sought, in order to simplify analytical and computational work. Some authors [20, 21] have suggested that it is sufficient to consider only the *dxz* and *dyz* orbitals. Building such minimal coupling model based on two orbitals, one is able to adjust the model parameters (energy hopping and chemical potential) to obtain the Fermi surface with the same topology that in the first principles calculations of band structure. Even though it fails to reproduce some finer features of the electronic spectrum [22, 23], this minimal coupling scheme is favored by its technical simplicity to be chosen as a basis for study of impurity effects in LaOFeAs which

**Figure 1.** Schematics of a FeAs layer in the LaoFeAs compound with *dxz* (dark) and *dyz* (white) Fe orbitals and the Fe-Fe hopping parameters in the minimal coupling model. Note that the hoppings between next near neighbors (*t*3, 4) are

*y*

Fe

As

*t*1 *t*2

*t*3 *t*4

opposite signs of the order parameter on electron and hole segments 10.

the As orbitals (out of Fe plane)

mediated by the As orbitals (out of Fe plane)

222 Superconductors – New Developments

atoms and the out-of-plane As atoms (above and below the centers of squares, Fig. 1) that mainly define a proper topology of electronic spectrum for the SC transition. The related microscopic SC models were chosen based on the results of high-resolution angle-resolved photoemission spectroscopy (ARPES) 8,9 that display a multiply connected Fermi surface with electron and hole pockets and the SC transition produces nodeless band gaps on both of them 8. This suggests the so-called extended *s*-wave (also called *s*-wave) SC order in such systems, with

Fig. 1. Schematics of a FeAs layer in the LaoFeAs compound with *dxz* (dark) and *dyz* (white) Fe orbitals and the Fe-Fe hopping parameters in the minimal coupling model. Note that the hoppings between next near neighbors (*t*3,4) are mediated by

*x*

The most common first principles approach to the band structure of these materials uses numeric calculations with most importance for Fe atomic *d*-orbitals. Thus, the local density approximation (LDA) calculations 10-15 confirm predominance of Fe atomic 3*d* orbitals (Fig. 1) in the LaOFeAs density of states near the Fermi level and so in the SC state. The next general conclusion is that the most relevant for the SC state feature of two spectrum gaps, by electron and hole pockets at the Fermi surface, is due to the multi-orbital effects. The observed SC properties are considered to follow from an unconventional pairing mechanism 16-19, beyond the common electron-phonon scheme. From the total of 5 atomic orbitals that can be involved for each iron in the LaOFeAs compound, a certain reduced basis was sought, in order to simplify analytical and computational work, and it was suggested 20,21 that the choice of only *dxz* and *dyz* orbitals is sufficient. With such minimal coupling model based on two orbitals, the reduced

Having established the SC state parameters, important effects of disorder, in particular by impurities, on the system electronic properties, have been studied for doped iron pnictides. Alike the situation in doped perovskite cuprates, impurity centers can result here either from dopants, necessary to form the very SC state, or from foreign atoms and other local defects. Within the minimal coupling model, an interesting possibility for localized impurity levels within SC gaps in doped LaOFeAs was indicated, even for the simplest, so-called isotopic (or non-magnetic) type of impurity perturbation [24, 25]. This finding marks an essential differ‐ ence from the traditional SC systems with \emph{s}-wave gap on a single-connected Fermi surface, were such perturbations are known not to produce localized impurity states and thus

In presence of localized quasiparticle states by isolated impurity centers, the next impor‐ tant issue is the possibility for collective behavior of such states at finite (but low enough) impurity concentration. They are expected to give rise to some resonance effects like those well studied in semiconductors at low doping concentrations [27]. This possibility was studied long ago for electronic quasiparticles in doped semiconducting systems [28] and also for other types of quasiparticles in pnononic, magnonic, excitonic, etc. spectra under impurities [29], establishing conditions for collective (including coherent) behavior of impurity excitations. Thus, indirect interactions between impurity centers of certain type (the so-called deep levels at high enough concentrations) in doped semiconductors can lead

to have no sizeable effect on SC order, accordingly to the Anderson theorem [26].

could be hardly tractable in more involved frameworks.

For the minimal coupling model of Fig. 1, the hopping Hamiltonian *Ht* is written in the local orbital basis as:

s ds s ds s ds s ds s s d ds s d ds s d ds s d ds s d ds <sup>é</sup> æ ö æ ö <sup>ê</sup> ç ÷ ç ÷ = - <sup>å</sup> <sup>+</sup> + + <sup>+</sup> <sup>+</sup> <sup>ê</sup> ç ÷ ++ ++ ç ÷ êë è ø è ø <sup>æ</sup> <sup>ö</sup> <sup>ç</sup> <sup>÷</sup> + ++++ <sup>ç</sup> + + + - + + + - <sup>÷</sup> <sup>ç</sup> <sup>÷</sup> <sup>è</sup> <sup>ø</sup> + + + + †† †† . . . . 1, , , , 2, , , , , †††† . . 3, , , , , , , , † 4, , *H t x x y y hc t x x y y hc <sup>t</sup> xy yx tx x x x y y y y h c x y x y x y x y tx y y x y nn nn nn nn <sup>n</sup> n n n n n n n n n n n* s d ds s d ds s d ds <sup>æ</sup> öù <sup>ç</sup> ÷ú --+ <sup>ç</sup> + + + - + - ÷ú <sup>ç</sup> <sup>÷</sup> <sup>è</sup> øúû ††† .. . , ,, ,, , *<sup>x</sup> x y y x h c x y x y x y <sup>n</sup> n n n n* (1)

where *xn*, *<sup>σ</sup>* and *yn*, *<sup>σ</sup>* are the Fermi operators for *dxz* and *dyz* Fe orbitals with spin *σ* on *n* lattice site and the vectors *δx, y* point to its nearest neighbors in the square lattice. Passing to the operators of orbital plane waves *x<sup>k</sup>* ,*<sup>σ</sup>* <sup>=</sup> *<sup>N</sup>* <sup>−</sup>1/2 ∑*<sup>n</sup>* <sup>e</sup>*i<sup>k</sup>* <sup>⋅</sup>*nxn*,*<sup>σ</sup>* (*N* is the number of lattice cells) and analogous *yk*, *<sup>σ</sup>* and defining an "orbital" 2-spinor *ψk*, *<sup>σ</sup>* = (*xk*, *<sup>σ</sup>*, *yk*, *<sup>σ</sup>*), one expands the spinor Hamiltonian in quasimomentum:

$$H\_t = \sum\_{\mathbf{k}, \sigma} \boldsymbol{\nu}\_{\mathbf{k}, \sigma} \prescript{\star}{}{\hat{h}}\_t(\mathbf{k}) \boldsymbol{\nu}\_{\mathbf{k}, \sigma}. \tag{2}$$

Here the 2×2 matrix

$$
\hat{h}\_t\left(\mathbf{k}\right) = \varepsilon\_{\*,k}\hat{\sigma}\_0 + \varepsilon\_{-,k}\hat{\sigma}\_3 + \varepsilon\_{xy,k}\hat{\sigma}\_1\tag{3}
$$

includes the Pauli matrices *σ* ^ *i* acting on the orbital indices and the energy functions

$$
\varepsilon\_{\pm,k} = \frac{\varepsilon\_{x,k} \pm \varepsilon\_{y,k}}{2},
\tag{4}
$$

with

$$\begin{aligned} \varepsilon\_{x,k} &= -2t\_1 \cos k\_x a - 2t\_2 \cos k\_y a - 4t\_3 \cos k\_x a \cos k\_y a\_\prime \\ \varepsilon\_{y,k} &= -2t\_1 \cos k\_y a - 2t\_2 \cos k\_x a - 4t\_3 \cos k\_x a \cos k\_y a\_\prime \\ \varepsilon\_{xy,k} &= -4t\_4 \sin k\_x a \sin k\_y a\_\prime \end{aligned}$$

(*a* is the distance between nearest neighbor Fe). An optimum fit for the calculated band structure in the minimum coupling model is with the hopping parameters (in |*t*1| units): *t*1 = –1.0, *t*2 = 1.3, *t*3 = *t*<sup>4</sup> = –0.85, and with the Fermi energy *ε*F = 1.45 [15]. A unitary transformation brings *h* ^ *t* (*k*) from orbital to diagonal subband basis:

$$
\hat{\boldsymbol{h}}\_{\boldsymbol{b}}\left(\boldsymbol{k}\right) = \hat{\boldsymbol{L}}^{\dagger}\left(\boldsymbol{k}\right)\hat{\boldsymbol{h}}\_{\boldsymbol{t}}\left(\boldsymbol{k}\right)\hat{\boldsymbol{L}}\left(\boldsymbol{k}\right) = \boldsymbol{\varepsilon}\_{\boldsymbol{\varepsilon},\boldsymbol{k}}\hat{\boldsymbol{\sigma}}\_{\boldsymbol{\ast}} + \boldsymbol{\varepsilon}\_{\boldsymbol{\varepsilon},\boldsymbol{k}}\hat{\boldsymbol{\sigma}}\_{-\boldsymbol{\prime}} \qquad \qquad \hat{\boldsymbol{L}}\_{\boldsymbol{k}} = \mathbf{e}^{-i\boldsymbol{\theta}\_{2}\boldsymbol{\theta}\_{k}/2}.\tag{5}
$$

Here *θk* = arctan (*εxy*, *k*/ *ε*–, *k*), *σ*± = (*σ*0 ±*σ*3)/2, and the energy eigenvalues:

Impurity Effects in Iron Pnictide Superconductors http://dx.doi.org/10.5772/59629 225

(3)

), one expands the spinor Hamiltonian

*k* (2)

on *n* lattice site

$$
\mathcal{E}\_{h,\varepsilon} \left( \mathbf{k} \right) = \mathcal{E}\_{\circ,\mathrm{k}} \pm \sqrt{\mathcal{E}\_{\mathrm{xy},\mathrm{k}}^{\circ 2} + \mathcal{E}\_{-,\mathrm{k}}^{\circ 2}} \, \mathrm{} \, \tag{6}
$$

correspond to the two subbands in the normal state spectrum that respectively define electron and hole pockets of the Fermi surface. There are two segments of each type, defined by the equations *εe, h*(*k*) = *μ*, as shown in Fig. 2. We note that both functions cos*θk* and sin*θk* change their sign around each of these segments, corresponding to their "azimuthal dependencies" around the characteristic points of the 2D Brillouin zone (Fig. 2), so that integrals of these functions with some azimuthal-independent factors over the relevant vicinity of Fermi surface practically vanish and are neglected beside the integrals of fully azimuthal-independent functions in the analysis below. , , <sup>2</sup> *k k <sup>k</sup>* (4) with ,1 2 3 ,1 2 3 , 4 2 cos 2 cos 4 cos cos , 2 cos 2 cos 4 cos cos , 4 sin sin , *x x y xy y y x xy xy x y t ka t ka t ka ka t ka t ka t ka ka t ka ka k k k* (*a* is the distance between nearest neighbor Fe). An optimum fit for the calculated band structure in the minimum coupling model is with the hopping parameters (in |*t*1| units): *t*1 = 1.0, *t*2 = 1.3,

are the Fermi operators for *dxz* and *dyz* Fe orbitals with spin

, *y<sup>k</sup>*,

 = (*xk*,

 † , ,

,0 ,3 ,1 <sup>ˆ</sup> ˆˆ ˆ *<sup>t</sup> xy <sup>h</sup> <sup>k</sup>*

 *k k*

*kk k*

, ,

*x y* 

<sup>ˆ</sup> . *H h t t* 

> 

*x,y* point to its nearest neighbors in the square lattice. Passing to the operators of

*k n <sup>n</sup>* (*N* is the number of lattice cells) and analogous *y<sup>k</sup>*,

Fig. 2: Electron (pink) and hole (blue) segments of the Fermi surface in the normal state of model system with electronic spectrum by Eq. 5. The dashed line around the point marks a circular approximation (see after Eq. 11). **Figure 2.** Electron (pink) and hole (blue) segments of the Fermi surface in the normal state of model system with elec‐ tronic spectrum by Eq. 5. The dashed line around the Γ point marks a circular approximation (see after Eq. 11).

The adequate basis for constructing the SC state is generated by the operators of electron and hole subbands:

$$\begin{aligned} \alpha\_{\mathbf{k},\sigma} &= \mathbf{x}\_{\mathbf{k},\sigma} \cos \theta\_{\mathbf{k}} / 2 - y\_{\mathbf{k},\sigma} \sin \theta\_{\mathbf{k}} / 2, \\ \beta\_{\mathbf{k},\sigma} &= y\_{\mathbf{k},\sigma} \cos \theta\_{\mathbf{k}} / 2 + \mathbf{x}\_{\mathbf{k},\sigma} \sin \theta\_{\mathbf{k}} / 2, \end{aligned} \tag{7}$$

giving rise to the "multiband-Nambu" 4-spinors *Ψ<sup>k</sup>* √ = (*αk*, ↑ √ , *α*–*k*, ↓, *βk*, ↑ √ , *β*–*k*, ↓) and to a 4×4 extension of the Hamiltonian Eq. (2) in the form:

$$H\_s = \sum\_{k,\sigma} \Psi\_k^\dagger \hat{h}\_s(\mathbf{k}) \Psi\_{k'} \tag{8}$$

where the 4×4 matrix

where *xn*, and *yn*,

and the vectors

in quasimomentum:

Here the 22 matrix

orbital plane waves 1/2

and defining an "orbital" 2-spinor

, , e*<sup>i</sup> x N x*

*k n*

*k*,

,

 

*k*

s

 d ds

224 Superconductors – New Developments

 d ds

*n n n*

of orbital plane waves *x<sup>k</sup>* ,*<sup>σ</sup>* <sup>=</sup> *<sup>N</sup>* <sup>−</sup>1/2

*tx y y x y*

+ + + +

s

s

s

† 4, ,

quasimomentum:

Here the 2×2 matrix

with

brings *h* ^ *t*

includes the Pauli matrices *σ*

 ds

 s

> d ds

 d ds

 s

s

 ds

*n n n n n n n n*

s

††† .. . , ,, ,, , *<sup>x</sup> x y y x h c x y x y x y <sup>n</sup> n n n n*

 d ds

 d ds

 s( )

> e s

 y † , ,

*xy yx*

 s

 s

where *xn*, *<sup>σ</sup>* and *yn*, *<sup>σ</sup>* are the Fermi operators for *dxz* and *dyz* Fe orbitals with spin *σ* on *n* lattice site and the vectors *δx, y* point to its nearest neighbors in the square lattice. Passing to the operators

*yk*, *<sup>σ</sup>* and defining an "orbital" 2-spinor *ψk*, *<sup>σ</sup>* = (*xk*, *<sup>σ</sup>*, *yk*, *<sup>σ</sup>*), one expands the spinor Hamiltonian in

s

<sup>ˆ</sup> . *H h t t k k*

 es

e

=- - - =- - -

4 sin sin ,

*t ka ka*

es

*xy x y*

Here *θk* = arctan (*εxy*, *k*/ *ε*–, *k*), *σ*± = (*σ*0 ±*σ*3)/2, and the energy eigenvalues:

,1 2 3 ,1 2 3

<sup>±</sup> <sup>=</sup> , , , , <sup>2</sup> *x y k k*

+ - =++ ,0 ,3 ,1

 e

*x x y xy y y x xy*

(*a* is the distance between nearest neighbor Fe). An optimum fit for the calculated band structure in the minimum coupling model is with the hopping parameters (in |*t*1| units): *t*1 = –1.0, *t*2 = 1.3, *t*3 = *t*<sup>4</sup> = –0.85, and with the Fermi energy *ε*F = 1.45 [15]. A unitary transformation

> es

+ - = =+ = <sup>2</sup> † <sup>ˆ</sup> / 2 , , ˆ ˆ ˆ ˆ ˆ ˆ ˆ , e . *<sup>i</sup> b t eh h U hU U <sup>k</sup>*

2 cos 2 cos 4 cos cos , 2 cos 2 cos 4 cos cos ,

*t ka t ka t ka ka t ka t ka t ka ka*

s

,

es

e±

<sup>=</sup> åy

*k*

( )

^ *i*

e

*k k k*

e

e

= -

, 4

(*k*) from orbital to diagonal subband basis:

( ) ( ) ( ) ( )

<sup>æ</sup> öù <sup>ç</sup> ÷ú --+ <sup>ç</sup> + + + - + - ÷ú <sup>ç</sup> <sup>÷</sup> <sup>è</sup> øúû

<sup>æ</sup> <sup>ö</sup> <sup>ç</sup> <sup>÷</sup> + ++++ <sup>ç</sup> + + + - + + + - <sup>÷</sup> <sup>ç</sup> <sup>÷</sup> <sup>è</sup> <sup>ø</sup>

*tx x x x y y y y h c x y x y x y x y*

*H t x x y y hc t x x y y hc <sup>t</sup>*

*nn nn nn nn <sup>n</sup>*

†††† . . 3, , , , , , , ,

<sup>é</sup> æ ö æ ö <sup>ê</sup> ç ÷ ç ÷ = - <sup>å</sup> <sup>+</sup> + + <sup>+</sup> <sup>+</sup> <sup>ê</sup> ç ÷ ++ ++ ç ÷ êë è ø è ø

†† †† . . . . 1, , , , 2, , , , ,

 ds

 s

> d ds

> d ds

**k** (2)

 s

 s

∑*<sup>n</sup>* <sup>e</sup>*i<sup>k</sup>* <sup>⋅</sup>*nxn*,*<sup>σ</sup>* (*N* is the number of lattice cells) and analogous

<sup>ˆ</sup> ˆˆ ˆ *<sup>t</sup> xy <sup>h</sup> kk k* **<sup>k</sup>** (3)

*<sup>k</sup>* (4)

s q


*kk k k kkk* (5)

acting on the orbital indices and the energy functions

 ds

(1)

$$
\hat{h}\_s\left(\mathbf{k}\right) = \hat{h}\_b\left(\mathbf{k}\right) \otimes \hat{\boldsymbol{\tau}}\_3 + \Delta\_k \hat{\sigma}\_0 \otimes \hat{\boldsymbol{\tau}}\_1 \boldsymbol{\tau}
$$

includes the Pauli matrices *τ* ^ *<sup>i</sup>* acting on the Nambu (particle-antiparticle) indices in *Ψ*-spinors and *h* ^ *b* (*k*) is defined by Eq. (5). The simplified form for the extended *s*-wave SC order is realized with the definition of the gap function by constant values, ∆*k* = ∆ on the electron segments and ∆*k* = –∆ on the hole segments.

The electronic dynamics of this system is determined by the (Fourier transformed) GF 4×4 matrices [58, 29, 54]:

$$
\hat{G}\_{\mathbf{k},\mathbf{k}^\circ} = \left\langle \left\langle \Psi\_{\mathbf{k}} \right| \Psi\_{\mathbf{k}^\circ} \right\rangle \Big| \right\rangle = \mathbf{i} \int\_{-\infty}^{0} dt \mathbf{e}^{i\mathbf{x}t/\hbar} \left\langle \Psi\_{\mathbf{k}} \left(t\right), \Psi\_{\mathbf{k}^\circ} \left(0\right)^\dagger \right\rangle,\tag{9}
$$

whose energy argument *ε* is understood as *ε* – *i*0 and <*A*(*t*), *B*(0> is the quantum statistical average with Hamiltonian *H* of the anticommutator of Heisenberg operators. From the equation of motion:

$$\boldsymbol{\delta}\mathcal{L}\hat{\mathbf{C}}\_{\mathbf{k},\mathbf{k}^\*} = \hbar\boldsymbol{\delta}\_{\mathbf{k},\mathbf{k}^\*}\hat{\sigma}\_0 \otimes \hat{\boldsymbol{\tau}}\_0 + \left\langle \left[ \left[ \boldsymbol{\Psi}\_{\mathbf{k}^\*} \boldsymbol{H} \right] \mathbb{I} \,\Psi\_{\mathbf{k}^\*} \right] \right\rangle\_{\mathbf{k}^\*} \tag{10}$$

the explicit GF for the unperturbed SC system with the Hamiltonian *Hs*, Eq. (8), is diagonal in quasimomentum, *G* ^ *k* ,*k* ' =*δ<sup>k</sup>* ,*<sup>k</sup>* ' *G* ^ *k* <sup>0</sup> , with the diagonal term:

$$\hat{\mathbf{G}}\_{k}^{0} = \frac{\varepsilon \hat{\boldsymbol{\tau}}\_{0} + \boldsymbol{\varepsilon}\_{\varepsilon,k} \hat{\boldsymbol{\tau}}\_{3} + \boldsymbol{\Delta} \hat{\boldsymbol{\tau}}\_{1}}{2D\_{\varepsilon,k}} \otimes \hat{\boldsymbol{\sigma}}\_{\*} + \frac{\varepsilon \hat{\boldsymbol{\tau}}\_{0} + \boldsymbol{\varepsilon}\_{h,k} \hat{\boldsymbol{\tau}}\_{3} - \boldsymbol{\Delta} \hat{\boldsymbol{\tau}}\_{1}}{2D\_{h,k}} \otimes \hat{\boldsymbol{\sigma}}\_{-} \,\tag{11}$$

where the denominators *Di*, *k* = *ε*<sup>2</sup> – *ε<sup>i</sup>* 2 (*k*) – ∆<sup>2</sup> for *i* = *e*, *h*. Below we refer energy to the Fermi level *ε*F, approximate the segments of Fermi surface by circles of radius *ki* around the charac‐ teristic points *Ki* in the Brillouin zone, and linearize the dispersion laws near the Fermi level as *ε<sup>j</sup>* (*k*) = *ε*F + *ξ<sup>j</sup>*, *k* with *ξ<sup>j</sup>*, *k* ≈ ℏ*vj* (|*k* – *K<sup>j</sup>* | – *kj* ). Though the Fermi wavenumbers *kj* and related Fermi velocities *vj* for *j* = *e*, *h* can somewhat differ at given hopping parameters and chemical potential, we shall neglect this difference and consider single values *kj* = *k*F and *vj* = *v*F.

#### **3. Impurity perturbation and self-energy**

We pass to the impurity problem where local perturbation terms due to non-magnetic impurities [24] on random sites *p* in Fe square lattice with an on-site energy shift *V*:

$$H\_{\rm imp} = V \sum\_{p,\sigma} (\mathbf{x}\_{p,\sigma}{}^{\dagger} \mathbf{x}\_{p,\sigma} + y\_{p,\sigma}{}^{\dagger} y\_{p,\sigma}) \, \tag{12}$$

are added to the Hamiltonian *Hs*. Without loss of generality, the parameter *V* can be taken positive, and this perturbation is suitably expressed in the multiband-Nambu basis:

Impurity Effects in Iron Pnictide Superconductors http://dx.doi.org/10.5772/59629 227

$$H\_{\rm imp} = \frac{1}{N} \sum\_{p,k,k'} \mathbf{e}^{i(k'-k)\cdot p} \Psi\_k \, ^\dagger \hat{V}\_{k,k'} \Psi\_{k'}.\tag{13}$$

through the 4×4 scattering matrix *V* ^ *k* ,*k* ' =*V U* ^ *<sup>k</sup>* †*U* ^ *<sup>k</sup>* ' ⊗ *τ* ^ <sup>3</sup>. From Eq. (5) for *U* ^ *<sup>k</sup>* , this matrix involves either "intraband" and "interband" elements [41]. The latter scattering could lead to a transition from *s*± to a competing *s*++ SC order (with the same sign of order parameter on both Fermi pockets) under impurity effect [43]. However, as shown below, such a possibility is effectively eliminated for the chosen local perturbation type, due to the specific quasimomentum *k*dependence of the scattering elements, unlike their constancy postulated in Ref. [43].

Following Refs. [29, 53], the solution for Eq. (10) with the perturbed Hamiltonian *Hs + Hi* can be obtained in different forms, suitable for different types of states, band-like (extended) or localized. All these forms result from the basic equation of motion:

$$
\hat{G}\_{\mathbf{k},\mathbf{k}^\*} = \delta\_{\mathbf{k},\mathbf{k}} \hat{G}\_{\mathbf{k}}^0 + \frac{1}{N} \sum\_{p,\mathbf{k}^\*} \mathbf{e}^{i(\mathbf{k}^\* - \mathbf{k}) \cdot p} \hat{G}\_{\mathbf{k}}^0 \hat{V}\_{\mathbf{k},\mathbf{k}^\*} \hat{G}\_{\mathbf{k}^\*,\mathbf{k}^\* \prime} \tag{14}
$$

by specific routines of its iterating for the "scattered" GF's *G* ^ *k* '',*k* ' . Thus, the algorithm, where the next iteration step *never* applies to the scattered GF's already present after previous steps, e.g., to that with *k*'' = *k* in Eq. (14), leads to the so-called fully renormalized form (RF), suitable for band-like states. Its result for the most relevant diagonal GF *G* ^ *<sup>k</sup>* ≡*G* ^ *<sup>k</sup>* ,*k* reads:

$$
\hat{\mathbf{G}}\_{\mathbf{k}} = \left[ \left( \hat{\mathbf{G}}\_{\mathbf{k}}^{0} \right)^{-1} - \hat{\boldsymbol{\Sigma}}\_{\mathbf{k}} \right]^{-1} \tag{15}
$$

where the self-energy matrix *Σ* ^ *<sup>k</sup>* is expressed by the related group expansion (GE):

$$
\hat{\Sigma}\_{\mathbf{k}} = c \hat{T}\_{\mathbf{k}} \Big( \mathbf{1} + c \hat{B}\_{\mathbf{k}} + \dots \Big). \tag{16}
$$

Here *c* = ∑*<sup>p</sup> N*–1 is the impurity concentration (per Fe site) and the T-matrix results from all the multiple scatterings by a single impurity:

$$
\hat{T}\_{\mathbf{k}} = \hat{V}\_{\mathbf{k},\mathbf{k}} + \frac{1}{N} \sum\_{\mathbf{k}^{\prime} \prec \mathbf{k}} \hat{V}\_{\mathbf{k},\mathbf{k}^{\prime}} \hat{G}\_{\mathbf{k}^{\prime},\mathbf{k}}^{0} \hat{V}\_{\mathbf{k}^{\prime},\mathbf{k}} + \frac{1}{N^{2}} \sum\_{\mathbf{k}^{\prime} \prec \mathbf{k}} \sum\_{\mathbf{k}^{\prime},\mathbf{k}^{\prime}} \hat{V}\_{\mathbf{k},\mathbf{k}^{\prime}} \hat{G}\_{\mathbf{k}^{\prime},\mathbf{k}^{\prime}}^{0} \hat{V}\_{\mathbf{k}^{\prime},\mathbf{k}} \hat{V}\_{\mathbf{k}^{\prime},\mathbf{k}} + \dots \tag{17}
$$

The next term to the unity in the brackets in Eq. (16):

( ) = Ä +D Ä ( ) t

includes the Pauli matrices *τ*

226 Superconductors – New Developments

∆*k* = –∆ on the hole segments.

matrices [58, 29, 54]:

equation of motion:

quasimomentum, *G*

Fermi velocities *vj*

as *ε<sup>j</sup>*

and *h* ^ *b* ^

e

et e t

**3. Impurity perturbation and self-energy**

^ *k* ,*k* ' =*δ<sup>k</sup>* ,*<sup>k</sup>* ' *G* ^ *k*

*G*

where the denominators *Di*, *k* = *ε*<sup>2</sup>

(*k*) = *ε*F + *ξ<sup>j</sup>*, *k* with *ξ<sup>j</sup>*, *k* ≈ ℏ*vj*

*k*

ˆ

 ds

ˆ ˆ ˆ ˆˆ , *s b h h <sup>k</sup> k k*

with the definition of the gap function by constant values, ∆*k* = ∆ on the electron segments and

The electronic dynamics of this system is determined by the (Fourier transformed) GF 4×4

e

whose energy argument *ε* is understood as *ε* – *i*0 and <*A*(*t*), *B*(0> is the quantum statistical average with Hamiltonian *H* of the anticommutator of Heisenberg operators. From the

the explicit GF for the unperturbed SC system with the Hamiltonian *Hs*, Eq. (8), is diagonal in

 et e t

teristic points *Ki* in the Brillouin zone, and linearize the dispersion laws near the Fermi level

We pass to the impurity problem where local perturbation terms due to non-magnetic

are added to the Hamiltonian *Hs*. Without loss of generality, the parameter *V* can be taken

 ss)

,, ,,

impurities [24] on random sites *p* in Fe square lattice with an on-site energy shift *V*:

ss

<sup>=</sup> å <sup>+</sup> † †

positive, and this perturbation is suitably expressed in the multiband-Nambu basis:

, *H V xx yy imp pp pp*

(

s

,

<sup>0</sup> † † /


, ' ' '

 t = Ä+ Y Y é ù ë û <sup>h</sup> † ,' ,' 0 0 '

> t

 – *ε<sup>i</sup>* 2 (*k*) – ∆<sup>2</sup>

(|*k* – *K<sup>j</sup>*

potential, we shall neglect this difference and consider single values *kj*

level *ε*F, approximate the segments of Fermi surface by circles of radius *ki*

<sup>0</sup> , with the diagonal term:

s

*D D k k*


*k k*

+ +D + -D <sup>=</sup> Ä + <sup>Ä</sup> <sup>0</sup> 0 ,3 1 0 ,3 1 , , ˆ ˆˆ ˆ ˆˆ <sup>ˆ</sup> ˆ ˆ , 2 2 *e h e h*

(*k*) is defined by Eq. (5). The simplified form for the extended *s*-wave SC order is realized

 st3 01

*<sup>i</sup>* acting on the Nambu (particle-antiparticle) indices in *Ψ*-spinors

( ) ( )

*G H* ˆ ˆ ,| , *kk kk <sup>k</sup> <sup>k</sup>* (10)

 t

+ -

for *j* = *e*, *h* can somewhat differ at given hopping parameters and chemical

 s

). Though the Fermi wavenumbers *kj*

*<sup>p</sup>* (12)

for *i* = *e*, *h*. Below we refer energy to the Fermi

= *k*F and *vj*

(11)

around the charac‐

= *v*F.

and related

<sup>ˆ</sup> e , 0, *i t <sup>G</sup> i dt t kk k k k k* (9)

$$
\hat{B}\_{\mathbf{k}} = \sum\_{\mathbf{n}} \left( \hat{A}\_{\mathbf{n}} \mathbf{e}^{-i\mathbf{k}\cdot\mathbf{n}} + \hat{A}\_{\mathbf{n}} \hat{A}\_{-\mathbf{n}} \right) \left( \mathbf{1} + \hat{A}\_{\mathbf{n}} \hat{A}\_{-\mathbf{n}} \right)^{-1}\,,\tag{18}
$$

describes the effects of indirect interactions in pairs of impurities, separated by vector *n*, in terms of interaction matrices *A* ^ *<sup>n</sup>* =*T* ^ *<sup>k</sup>*∑*<sup>k</sup>* '≠*<sup>k</sup>* <sup>e</sup>*i<sup>k</sup>* '⋅*n<sup>G</sup>* ^ *k* ' . Besides this restriction on summation, multiple sums in the products like *A* ^ *nA* ^ <sup>−</sup>*n* never contain coincident quasimomenta. Eq. (18) presents the first non-trivial GE term and its other terms omitted in Eq. (16) relate to the groups of three and more impurities [29].

An alternative iteration routine applies Eq. (14) to *all* the scattered GF's, leading to the so-called non-renormalized form (NRF), suitable for localized states:

$$
\hat{\mathbf{G}}\_{\mathbf{k}} = \hat{\mathbf{G}}\_{\mathbf{k}}^{0} + \hat{\mathbf{G}}\_{\mathbf{k}}^{0} \hat{\Sigma}\_{\mathbf{k}}^{0} \hat{\mathbf{G}}\_{\mathbf{k}\prime}^{0} \tag{19}
$$

The NRF self-energy admits a GE: *Σ* ^ *k* <sup>0</sup> =*cT* ^ *k* <sup>0</sup> (1 + *cB* ^ *k* <sup>0</sup> + …), that differs from the RF one by no restrictions in *k*-sums for T-matrix, interaction matrices *A* ^ *n* <sup>0</sup> =*T* ^ *k* <sup>0</sup>∑*<sup>k</sup>* ' e*i<sup>k</sup>* '⋅*nG* ^ *k* ' 0 and their products.

At the first step, we restrict GE to the common T-matrix level to find the possibilities for localized quasiparticle states and related in-gap energy levels by single impurities [21]. Next, at finite impurity concentrations, formation of (narrow) energy bands of specific collective states near these levels is studied. Finally, the criteria for such collective states to really exist in the disordered SC system follow from the analysis of non-trivial GE terms. We notice that RF for GF's *G* ^ *k* ' in the above interaction matrices is just necessary for adequate treatment of interaction effects within the in-gap bands. To simplify the T-matrix, Eq. (17), note that *V* ^ *<sup>k</sup>* ,*<sup>k</sup>* =*V σ* ^ <sup>0</sup> ⊗ *τ* ^ <sup>3</sup> and use the integrated GF matrix:

$$
\hat{G}\_0 = \frac{1}{N} \sum\_k \hat{\mathcal{U}}\_k \hat{G}\_k^0 \hat{\mathcal{U}}\_k^\dagger = \varepsilon \left[ \mathcal{G}\_\varepsilon \left( \boldsymbol{\varepsilon} \right) \hat{\sigma}\_\ast + \mathcal{g}\_h \left( \boldsymbol{\varepsilon} \right) \hat{\sigma}\_- \right] \otimes \hat{\tau}\_0.
$$

This diagonal form (restricted only to the "intraband" matrix elements) follows from the aforementioned cancellation of integrals with cos*θk* and sin*θk* in the "interband" matrix elements of *U* ^ *kG* ^ *k* <sup>0</sup>*U* ^ *k* † . This permits to consider below the SC order unchanged under the impurity effects.

The functions *gj* (*ε*) = *N*–1∑*<sup>k</sup> Dj, <sup>k</sup>* –1 for *j* = *e*, *h* are approximated near *ε*F, |*ε* – *ε*F| δ ∆, as:

$$\log\_{\rangle}(\varepsilon) \approx -\frac{\pi \rho\_{\rangle}}{\sqrt{\Delta^2 - \varepsilon^2}}.\tag{20}$$

Here *ρ<sup>j</sup>* = *mj a*2 /(2*π*ℏ<sup>2</sup> ) are the Fermi densities of states for respective subbands (in parabolic approximation for their dispersion laws), and by the assumed identity of the Fermi segments they can be also considered identical *ρ<sup>j</sup>* = *ρ*F, so that *gj* (*ε*) = *g*(*ε*) = –*πρ*F/ *Δ* <sup>2</sup> −*ε* <sup>2</sup> . Omitted terms in Eq. (20) are of higher orders in the small parameter *ε*/*ε*<sup>F</sup> ≪ 1. Then the momentum inde‐ pendent T-matrix is explicitly written as:

th<\$%&?>characteristic<\$%&?>point<\$%&?>in<\$%&?>the<\$%&?>Brillouin<\$%&?>zone<\$%&?>(see<\$%&?>in<\$%&?>the<\$%&?>text),<\$

impurity<\$%&?>localized<\$%&?>levels<\$%&?>are<\$%&?>shown<\$%&?>with<\$%&?>dashed<\$%&?>lines.<\$%&?>The<\$%&?>narrow<

whose<\$%&?>main<\$%&?>difference<\$%&?>from<\$%&?>the<\$%&?>unperturbed<\$%&?>SC<\$%&?>bands<\$%&?>with

energy,<\$%&?>Eq.<\$%&?>(16).<\$%&?>It<\$%&?>should<\$%&?>be<\$%&?>noted<\$%&?>that<\$%&?>these<\$%&?>subban ds<\$%&?>for<\$%&?>opposite<\$%&?>signs<\$%&?>of<\$%&?>their<\$%&?>argument<\$%&?>*ξ*<\$%&?>in<\$%&?>fact<\$%& ?>refer<\$%&?>to<\$%&?>excitations<\$%&?>around<\$%&?>different<\$%&?>Fermi<\$%&?>segments<\$%&?>(by<\$%&?>el ectron<\$%&?>and<\$%&?>holes),<\$%&?>but<\$%&?>for<\$%&?>clarity<\$%&?>all<\$%&?>four<\$%&?>*εb*<\$%&?>bands<\$%

reference.<\$%&?>Then<\$%&?>from<\$%&?>the<\$%&?>IRM<\$%&?>criterion<\$%&?>of<\$%&?>band-like<\$%&?>states:

the<\$%&?>position<\$%&?>of<\$%&?>mobility<\$%&?>edge<\$%&?>*εc*<\$%&?>for<\$%&?>these<\$%&?>bands<\$%&?>is<\$%

<\$%&?>Δ<\$%&?>~<\$%&?>(*c*/*c*1)2Δ,<\$%&?>with<\$%&?>*c*1<\$%&?>=<\$%&?>*πρ*FΔ<\$%&?>'<\$%&?>1.<\$%&?>Besides<\$%&?>

gap<\$%&?>*εi*<\$%&?>bands,<\$%&?>generated<\$%&?>close<\$%&?>to<\$%&?>*ε*0<\$%&?>at<\$%&?>finite<\$%&?>concentrat

the<\$%&?>*εb*<\$%&?>bands,<\$%&?>there<\$%&?>appear<\$%&?>also<\$%&?>four<\$%&?>(narrow)<\$%&?>in-

<\$%&?>is<\$%&?>a<\$%&?>finite<\$%&?>linewidth<\$%&?>Γ(*ξ*)<\$%&?>∼<\$%&

,<\$%&?>defined<\$%&?>by<\$%&?>the<\$%&

0 0

like<\$%&?>Fermi<\$%&?>segments<\$%&?>and<\$%&?>red<\$%&?>lines<\$%&?>do<\$%&?>for<\$%&?>those<\$%&?>near<\$%&?>hole-

band<\$%&?>(shown<\$%&?>by<\$%&?>the<\$%&?>arrow)<\$%&?>delimits<\$%&?>the<\$%&?>region<\$%&?>in<\$%&?>Fig.<\$%&?>5.

%&?>blue<\$%&?>lines<\$%&?>are<\$%&?>for<\$%&?>impurity<\$%&?>bands<\$%&?>near<\$%&?>electron-

First<\$%&?>of<\$%&?>all,<\$%&?>there<\$%&?>are<\$%&?>modified<\$%&?>initial<\$%&?>bands

&?>are<\$%&?>set<\$%&?>in<\$%&?>Fig.<\$%&?>3<\$%&?>in<\$%&?>the<\$%&?>same<\$%&?>*ξ*-

 

?>*cVvξ*Δ/[(1<\$%&?>+<\$%&?>*v*2)(*ξ*2<\$%&?>+<\$%&?>ξ02)],<\$%&?> 2 22

$$\hat{T}\left(\boldsymbol{\varepsilon}\right) = \frac{V}{1+\upsilon^2} \frac{\boldsymbol{\varepsilon}^2 - \Delta^2 + \upsilon\varepsilon\sqrt{\Delta^2 - \varepsilon^2 \hat{\boldsymbol{\tau}}\_3}}{\boldsymbol{\varepsilon}^2 - \varepsilon\_0^2},\tag{21}$$

it presents two symmetric poles within the gap, at *<sup>ε</sup>* <sup>=</sup> <sup>±</sup> *<sup>ε</sup>*<sup>0</sup> <sup>=</sup> <sup>±</sup> *<sup>Δ</sup>* / <sup>1</sup> <sup>+</sup> *<sup>v</sup>* <sup>2</sup> [21], with the dimen‐ sionless impurity perturbation parameter *v* = *πρ*F*V*. Near these poles, Eq. (21) can be approxi‐ mated as:

describes the effects of indirect interactions in pairs of impurities, separated by vector *n*, in

presents the first non-trivial GE term and its other terms omitted in Eq. (16) relate to the groups

An alternative iteration routine applies Eq. (14) to *all* the scattered GF's, leading to the so-called

At the first step, we restrict GE to the common T-matrix level to find the possibilities for localized quasiparticle states and related in-gap energy levels by single impurities [21]. Next, at finite impurity concentrations, formation of (narrow) energy bands of specific collective states near these levels is studied. Finally, the criteria for such collective states to really exist in the disordered SC system follow from the analysis of non-trivial GE terms. We notice that

interaction effects within the in-gap bands. To simplify the T-matrix, Eq. (17), note that

= = +Ä e

This diagonal form (restricted only to the "intraband" matrix elements) follows from the aforementioned cancellation of integrals with cos*θk* and sin*θk* in the "interband" matrix

pr

approximation for their dispersion laws), and by the assumed identity of the Fermi segments

= *ρ*F, so that *gj*

in Eq. (20) are of higher orders in the small parameter *ε*/*ε*<sup>F</sup> ≪ 1. Then the momentum inde‐

e

D -2 2 . *<sup>j</sup>*

 es

 é ù ( ) + - ( ) å ë û 0 † 0 0 <sup>1</sup> ˆ ˆ ˆ ˆ ˆ ˆˆ . *G UGU g g e h <sup>N</sup> kk k*

^ *k* '

. Besides this restriction on summation,

<sup>0</sup> + …), that differs from the RF one by no

^ *k* '

0 and their products.

. Omitted terms

<sup>−</sup>*n* never contain coincident quasimomenta. Eq. (18)

= +S ˆˆˆˆ 0 00 0 <sup>ˆ</sup> *GGGG* , *k k kkk* (19)

<sup>0</sup>∑*<sup>k</sup>* ' e*i<sup>k</sup>* '⋅*nG*

 t

^ *n* <sup>0</sup> =*T* ^ *k*

in the above interaction matrices is just necessary for adequate treatment of

 es

† . This permits to consider below the SC order unchanged under the

–1 for *j* = *e*, *h* are approximated near *ε*F, |*ε* – *ε*F| δ ∆, as:

) are the Fermi densities of states for respective subbands (in parabolic

*<sup>j</sup> g* (20)

(*ε*) = *g*(*ε*) = –*πρ*F/ *Δ* <sup>2</sup> −*ε* <sup>2</sup>

*<sup>k</sup>*∑*<sup>k</sup>* '≠*<sup>k</sup>* <sup>e</sup>*i<sup>k</sup>* '⋅*n<sup>G</sup>*

^ *<sup>n</sup>* =*T* ^

non-renormalized form (NRF), suitable for localized states:

restrictions in *k*-sums for T-matrix, interaction matrices *A*

3 and use the integrated GF matrix:

*k*

( )

e

» -

^ *nA* ^

^ *k* <sup>0</sup> =*cT* ^ *k* <sup>0</sup> (1 + *cB* ^ *k*

terms of interaction matrices *A*

228 Superconductors – New Developments

of three and more impurities [29].

multiple sums in the products like *A*

The NRF self-energy admits a GE: *Σ*

RF for GF's *G*

elements of *U*

impurity effects.

The functions *gj*

 = *mj a*2 /(2*π*ℏ<sup>2</sup>

Here *ρ<sup>j</sup>*

^ *kG* ^ *k* <sup>0</sup>*U* ^ *k*

(*ε*) = *N*–1∑*<sup>k</sup> Dj, <sup>k</sup>*

they can be also considered identical *ρ<sup>j</sup>*

pendent T-matrix is explicitly written as:

*<sup>k</sup>* ,*<sup>k</sup>* =*V σ* ^ <sup>0</sup> ⊗ *τ* ^

*V* ^ ^ *k* '

$$\hat{T}\left(\boldsymbol{\varepsilon}\right) \approx \mathcal{V} \frac{\boldsymbol{\varepsilon} - \boldsymbol{\varepsilon}\_0 \hat{\boldsymbol{\tau}}\_3}{\boldsymbol{\varepsilon}^2 - \boldsymbol{\varepsilon}\_0^{-2}},\tag{22}$$

where *γ*<sup>2</sup> = *V*∆*v*<sup>2</sup> /(1 + *v*<sup>2</sup> ) 3/2 is the effective constant of coupling between localized and band quasiparticles. At finite *c*, this T-matrix can be used in Eqs. (16) and (15) and then in the formal dispersion equation: Redet*G* ^ *<sup>k</sup>* <sup>−</sup><sup>1</sup> =0 [57], to obtain the dispersion laws of perturbed SC system in terms of the normal quasiparticles dispersion *ξk* = *εk* – *ε*F ≡ *ξ*. They follow from the general expression: Re(*ε*˜ <sup>2</sup> <sup>−</sup>*<sup>Δ</sup>* <sup>2</sup> <sup>−</sup>*ξ*˜ 2) =0, with *ε*˜ <sup>=</sup>*ε*(1−*cVv* <sup>1</sup>−*<sup>ε</sup>* <sup>2</sup> / *<sup>Δ</sup>* <sup>2</sup> / (*<sup>ε</sup>* <sup>2</sup> / *<sup>ε</sup>*<sup>0</sup> <sup>2</sup> −1)) and *<sup>ξ</sup>*˜ <sup>=</sup>*<sup>ξ</sup>* <sup>+</sup> *cV* (*<sup>ε</sup>* <sup>2</sup> / *<sup>Δ</sup>* <sup>2</sup> <sup>−</sup>1)/(*<sup>ε</sup>* <sup>2</sup> / *<sup>ε</sup>*<sup>0</sup> <sup>2</sup> −1), and display a peculiar multiband structure shown in Fig. 3.

Figure 3. Dispersion<\$%&?>laws<\$%&?>for<\$%&?>band-like<\$%&?>quasiparticles<\$%&?>in<\$%&?>the<\$%&?>Tmatrix<\$%&?>approximation,<\$%&?>neglecting<\$%&?>their<\$%&?>finite<\$%&?>lifetime,<\$%&?>at<\$%&?>a<\$%&?>specific<\$%&?>ch oice<\$%&?>of<\$%&?>impurity<\$%&?>parameters<\$%&?>*v*<\$%&?>=<\$%&?>1,<\$%&?>*c*<\$%&?>=<\$%&?>0.1Δ2/*γ*<\$%&?>2.<\$%&?>The<\$ %&?>argument<\$%&?>*ξ*<\$%&?>composes<\$%&?>all<\$%&?>specific<\$%&?>*ξj*<\$%&?>=<\$%&?>η*v*F<\$%&?>(|*k<\$%&?>*– <\$%&?>*Kj*|<\$%&?>-<\$%&?>*k*F)<\$%&?>for<\$%&?>*k*<\$%&?>near<\$%&?>each<\$%&?>*j-***Figure 3.** Dispersion laws for band-like quasiparticles in the T-matrix approximation, neglecting their finite lifetime, at a specific choice of impurity parameters *v* = 1, *c* = 0.1∆<sup>2</sup> /*γ* <sup>2</sup> . The argument *ξ* composes all specific *ξ<sup>j</sup>* = ℏ*v*F(|*k* – *K<sup>j</sup>* | - *k*F) for *k* near each *j-*th characteristic point in the Brillouin zone (see in the text), blue lines are for impurity bands near electron-like Fermi segments and red lines do for those near hole-like segments. The single-impurity localized levels are shown with dashed lines. The narrow rectangle around the top of *ε<sup>i</sup>* -band (shown by the arrow) delimits the region in Fig. 5.

\$%&?>rectangle<\$%&?>around<\$%&?>the<\$%&?>top<\$%&?>of<\$%&?>*εi*-

like<\$%&?>segments.<\$%&?>The<\$%&?>single-

 , *<sup>b</sup> i* <\$%&?>(23)

<\$%&?>dispersion<\$%&?> 2 2

 

?>T-matrix<\$%&?>term<\$%&?>of<\$%&?>the<\$%&?>self-

 

> 

 ,

 <sup>2</sup> <sup>0</sup> 0 2 2

*<sup>b</sup>* <\$%&?>(24)

&?>estimated<\$%&?>as:<\$%&?>*εc*<\$%&?>–

0 ,

 *<sup>i</sup> c* <\$%&?>(25)

 

ion<\$%&?>of<\$%&?>impurities,<\$%&?>accordingly<\$%&?>to:

First of all, there are modified initial bands

$$
\varepsilon\_b \left( \xi \right) \approx \varepsilon \left( \xi \right) + i \Gamma \left( \xi \right), \tag{23}
$$

whose main difference from the unperturbed SC bands with dispersion *ε*(*ξ*)= *Δ* <sup>2</sup> + *ξ* <sup>2</sup> is a finite linewidth Γ(*ξ*) ∼ *cVvξ*∆/[(1 + *v*<sup>2</sup> )(*ξ*<sup>2</sup> + *ξ*<sup>0</sup> 2 )], *ξ*<sup>0</sup> <sup>2</sup> <sup>=</sup>*<sup>Δ</sup>* <sup>2</sup> <sup>−</sup>*ε*<sup>0</sup> 2, defined by the T-matrix term of the selfenergy, Eq. (16). It should be noted that these subbands for opposite signs of their argument *ξ* in fact refer to excitations around different Fermi segments (by electron and holes), but for clarity all four *ε<sup>b</sup>* bands are set in Fig. 3 in the same *ξ*-reference. Then from the IRM criterion of band-like states:

$$
\xi \frac{\partial}{\partial \xi} \varepsilon\_b \left( \xi \right)\_\leftrightarrow \Gamma \left( \xi \right)\_\prime \tag{24}
$$

the position of mobility edge *εc* for these bands is estimated as: *εc* – ∆ ~ (*c*/*c*1) 2 ∆, with *c*1 = *πρ*F∆ ≪ 1. Besides the *ε<sup>b</sup>* bands, there appear also four (narrow) in-gap *ε<sup>i</sup>* bands, generated close to *ε*0 at finite concentration of impurities, accordingly to:

$$
\varepsilon\_i \left( \underline{\xi} \right) \approx \varepsilon\_0 + c\gamma^2 \frac{\underline{\xi} - \varepsilon\_0}{\underline{\xi}^2 + \underline{\xi}\_0^{\prime 2}} \, \, \, \tag{25}
$$

As follows from Eq. (25), the *ε<sup>i</sup>* band has its extrema *εmax* = *ε*0 + *cγ*<sup>2</sup> /(∆ + *ε*0) at *ξ*+ = ∆ + *ε*0 and *εmin* = *ε*0 – *cγ*<sup>2</sup> /(∆ – *ε*0) at *ξ*– = ∆ – *ε*0. The energy and momentum shifts of these extremal points seen in Fig. 3 specify the impurity effect on a multiband initial spectrum, compared to a simpler situation for an impurity level near the edge of a single quasiparticle band [29].

All these spectrum bands would contribute to the overall density of states (DOS) by the related quasiparticles: *ρ*(*ε*) = (4*πN*) –1 Im Tr ∑*<sup>k</sup> G* ^ *<sup>k</sup>* . More common contributions there come from the *εb* bands and they can be expressed through the Bardeen-Cooper-Schrieffer (BCS) DOS in pure crystal [57]: *ρ*BCS(*ε*, ∆) = *ρ*F*ε*/ *ε* <sup>2</sup> −*Δ* <sup>2</sup> , as follows:

$$\rho\_{\rm b}(\boldsymbol{\varepsilon}) \approx \rho\_{\rm RCS}(\boldsymbol{\varepsilon}, \boldsymbol{\Delta}) - \frac{2c\boldsymbol{\upsilon}^{2}\boldsymbol{\varepsilon}\sqrt{\boldsymbol{\varepsilon}^{2} - \boldsymbol{\Delta}^{2}}}{\pi \boldsymbol{\varepsilon}\_{\rm F} \left(1 + \boldsymbol{\upsilon}^{2}\right) \left(\boldsymbol{\varepsilon}^{2} - \boldsymbol{\varepsilon}\_{0}\right)^{2}}\,\tag{26}$$

at *ε*<sup>2</sup> <sup>↔</sup> *ε<sup>c</sup>* 2 . The second term in the r.h.s. of Eq. (26) describes a certain reduction of the BCS DOS at energies farther from the gap limits.

More peculiar contributions to DOS come from the *ε<sup>i</sup>* bands, written as:

#### more involved analysis of non-trivial GE terms for self-energy. Impurity Effects in Iron Pnictide Superconductors http://dx.doi.org/10.5772/59629 231

 < 

*max* is within

*n*

), Eq. (25). Then

(29)

$$\rho\_i(\boldsymbol{\varepsilon}) \approx \frac{\rho\_{\rm F}}{\upsilon} \frac{\boldsymbol{\varepsilon}^2 - \boldsymbol{\varepsilon}\_0^2 - c\gamma^2}{\sqrt{\left(\boldsymbol{\varepsilon}\_{\max}\,^2 - \boldsymbol{\varepsilon}^2\right) \left(\boldsymbol{\varepsilon}^2 - \boldsymbol{\varepsilon}\_{\min}\right)^2}},\tag{27}$$

) to the dispersion law

*i* band. If the actual energy

 = *i*(

*i*

criteria should be satisfied, especially near the limits of corresponding bands and this requires a

at *εmin*<sup>2</sup> ↑ *ε*<sup>2</sup> ↑ *εmax*<sup>2</sup> , and presented in Fig. 4. the band-like range, the RF self-energy matrix, Eq. (16), can be used up to its pair term, <sup>2</sup> ˆ ˆ *c TB<sup>k</sup>* ,

definiteness a vicinity of the upper limit

that will add a certain finite imaginary part *i*(

As follows from Eq. (25), the

0

line) for the case by Fig. 3.

2

*min* = <sup>0</sup> *c*

matrices at

 = 

from the 

at 

 *c* 2

First of all, there are modified initial bands

linewidth Γ(*ξ*) ∼ *cVvξ*∆/[(1 + *v*<sup>2</sup>

230 Superconductors – New Developments

of band-like states:

= *ε*0 – *cγ*<sup>2</sup>

at *ε*<sup>2</sup> <sup>↔</sup> *ε<sup>c</sup>* 2

quasiparticles: *ρ*(*ε*) = (4*πN*)

crystal [57]: *ρ*BCS(*ε*, ∆) = *ρ*F*ε*/ *ε* <sup>2</sup> −*Δ* <sup>2</sup>

at energies farther from the gap limits.

e x ex

)(*ξ*<sup>2</sup> + *ξ*<sup>0</sup> 2 )], *ξ*<sup>0</sup>

> x ex

≪ 1. Besides the *ε<sup>b</sup>* bands, there appear also four (narrow) in-gap *ε<sup>i</sup>*

( )

ex

As follows from Eq. (25), the *ε<sup>i</sup>* band has its extrema *εmax* = *ε*0 + *cγ*<sup>2</sup>

–1 Im Tr ∑*<sup>k</sup> G*

( ) ( )

 r e

re

More peculiar contributions to DOS come from the *ε<sup>i</sup>*

*ε*0 at finite concentration of impurities, accordingly to:

x « ¶ <sup>G</sup>

the position of mobility edge *εc* for these bands is estimated as: *εc* – ∆ ~ (*c*/*c*1)

 e g x x

situation for an impurity level near the edge of a single quasiparticle band [29].

^

, as follows:

pe

<sup>2</sup> , , <sup>1</sup> *<sup>b</sup> cv v*

BCS 22 2


 x

whose main difference from the unperturbed SC bands with dispersion *ε*(*ξ*)= *Δ* <sup>2</sup> + *ξ* <sup>2</sup> is a finite

energy, Eq. (16). It should be noted that these subbands for opposite signs of their argument *ξ* in fact refer to excitations around different Fermi segments (by electron and holes), but for clarity all four *ε<sup>b</sup>* bands are set in Fig. 3 in the same *ξ*-reference. Then from the IRM criterion

<sup>2</sup> <sup>=</sup>*<sup>Δ</sup>* <sup>2</sup> <sup>−</sup>*ε*<sup>0</sup>

 x( ) ( )

> x e

in Fig. 3 specify the impurity effect on a multiband initial spectrum, compared to a simpler

All these spectrum bands would contribute to the overall density of states (DOS) by the related

*εb* bands and they can be expressed through the Bardeen-Cooper-Schrieffer (BCS) DOS in pure

+ 2 0 0 2 2

0

/(∆ – *ε*0) at *ξ*– = ∆ – *ε*0. The energy and momentum shifts of these extremal points seen

( )( )

+ - 2 22

F 0

. The second term in the r.h.s. of Eq. (26) describes a certain reduction of the BCS DOS

 e e

bands, written as:

e e


( ) » +G ( ) ( ), *<sup>b</sup> i* (23)

¶ , *<sup>b</sup>* (24)

, *<sup>i</sup> c* (25)

*<sup>k</sup>* . More common contributions there come from the

2, defined by the T-matrix term of the self-

2

bands, generated close to

/(∆ + *ε*0) at *ξ*+ = ∆ + *ε*0 and *εmin*

∆, with *c*1 = *πρ*F∆

(26)

*max* of the

Fig. 4: Density of states in the narrow in-gap band near the impurity level 0 (dashed **Figure 4.** Density of states in the narrow in-gap band near the impurity level *ε*0 (dashed line) for the case by Fig. 3. order Bessel function: <sup>2</sup> cos 0 e 2 *ix d Jx* . Since *x* = *n*(*k*F + *v*F) is typically big, *x* 1, the

*i* band has its extrema

*max* = <sup>0</sup> + *c*

<sup>2</sup> /( + 0) at + = 0 and

at energies farther from the gap limits. Fig. 5: The dispersion law (solid line) near the top of impurity band, within the region indicated by a thin rectangle in Fig. 3, and its parabolic approximation (dashed line). **Figure 5.** Parabolic approximation (dashed line) for the dispersion law near the top of impurity band (solid line), with‐ in the region indicated by a small rectangle in Fig. 3.

More peculiar contributions to DOS come from the

. The second term in the r.h.s. of Eq. (26) describes a certain reduction of the BCS DOS

properties of a disordered SC system. But for the related quasiparticles to really exist, some

*<sup>i</sup>* bands, written as:

 22 2 F 0 222 2 , *<sup>i</sup> max min c v* (27) at *min*<sup>2</sup> 2 *max*<sup>2</sup> \$, and presented in Fig. 4. As it will be shown below, formation of the *<sup>i</sup>* bands can strongly influence the physical Formation of the *ε<sup>i</sup>* bands can have important repercussions in the physical behavior of a disordered SC system and they will be considered below. But before this, we need to analyze the criteria for the considered quasiparticles to really exist, especially in closeness to the limits of corresponding bands and this requires a more involved analysis of non-trivial GE terms for self-energy.

#### **4. Group expansion and Ioffe-Regel-Mott criteria**

Let us now study the crossover from band-like to localized states near the limits of *ε<sup>i</sup>* bands, say for definiteness, near its upper limit *εmax*. Supposing the actual energy *ε* < *εmax* to be within the range of band-like states, we use the RF self-energy matrix, Eq. (16), up to the GE pair term, *c* 2 *T* ^ *B* ^ *<sup>k</sup>* , that will add a certain finite imaginary part Γ*<sup>i</sup>* (*ε*) to the dispersion law *ε* = *ε<sup>i</sup>* (*ξ*), Eq. (25). Then the IRM criterion for a state at this energy be really band-like (also called extended) is written as:

$$
\varepsilon\_{\text{max}} - \varepsilon \gg \Gamma\_{\text{/}} (\varepsilon). \tag{28}
$$

To simplify calculation of the scalar function Γ*<sup>i</sup>* (*ε*), we fix the energy argument in the numer‐ ators of T-matrix and interaction matrices at *ε* = *ε*0, obtaining their forms:

$$
\hat{T}\left(\boldsymbol{\varepsilon}\right) \approx \frac{\boldsymbol{\gamma}^{2}\boldsymbol{\varepsilon}\_{0}}{\boldsymbol{\varepsilon}^{2} - \boldsymbol{\varepsilon}\_{0}^{2}} \hat{m}\_{\*} \quad \hat{A}\_{\text{u}}\left(\boldsymbol{\varepsilon}\right) \approx \hat{T}\left(\boldsymbol{\varepsilon}\right) \frac{\boldsymbol{\varepsilon}}{N} \sum\_{k} \frac{\mathbf{e}^{\boldsymbol{k}\cdot\boldsymbol{u}}}{D\_{k}\left(\boldsymbol{\varepsilon}\right)}\tag{29}
$$

both proportional to the matrix *m* ^ <sup>+</sup> =*σ* ^ <sup>0</sup> ⊗ (*τ* ^ <sup>0</sup> + *τ* ^ 3) with important multiplicative property: *m* ^ + 2 =2*m* ^ +. The *k*-summation (integration) in Eq. (29) is suitably done in polar coordinates over the circular segments of Fermi surface. Here the azimuthal integration only refers to the phase

of numerator, resulting in zeroth order Bessel function: *∫* 0 2*π* <sup>e</sup>*ix*cos*θd<sup>θ</sup>* =2*πJ*<sup>0</sup> (*x*). Since *x* = *n*(*k*F + *ξ*/

ℏ*v*F) is typically big, *x* ≫ 1, the asymptotical formula applies: *J*0(*x*) ≈ 2 / *πx* cos(*x* – *π*/4). Then, for radial integration in *ξ* around the extremum point *ξ*+, it is convenient to decompose this function into fast and slow oscillating factors: *J*0(*x*) ≈ 2 /(*πk*+*n*) cos(*k*+*n* – *π*/4)cos[(*ξ* – *ξ*+)*n*/ℏ*v*F] with the fast wavenumber *k*+ = *k*F + *ξ*+ /ℏ*v*F ≈ *k*F, and to write the denominator in the parabolic approximation: *Dξ*(*ε*) ≈ (*ξ* – *ξ*+) 2 – *δ*<sup>2</sup> (*ε*), with *δ*<sup>2</sup> (*ε*) = 4∆(∆ + *ε*0) 2 (*εmax* – *ε*)/(2*cγ*<sup>2</sup> ) (see Fig. 5). Thus, the interaction matrix *A* ^ *<sup>n</sup>*(*ε*)= *An*(*ε*)*m* ^ <sup>+</sup> only depends on the distance *n* between impurities, and, for *ε* close to *εmax*, this dependence can be expressed as:

$$A\_r(x) \approx \sqrt{\frac{r\_c}{r}} \sin k\_x r \cos k\_\mathrm{F} r \tag{30}$$

where the length scales both for the monotonous decay:

$$r\_{\varepsilon} = \frac{2\pi}{k\_{\text{F}}} \left[ \frac{\varepsilon\_{0}\rho\_{\text{F}}\left(\Delta + \varepsilon\_{0}\right)}{c\delta\left(\varepsilon\right)} \right]^{2} \text{ } \epsilon$$

and for the sine factor: *k<sup>ε</sup>* –1 = ℏ*v*F/*δ*(*ε*), are much longer than *k*<sup>F</sup> –1 for the fast cosine. The latter fast oscillation is specific for the interactions mediated by Fermi quasiparticles (like the known RKKY mechanism), unlike the monotonous or slowly oscillating interactions between impurities in semiconductors or in bosonic systems [29].

Now calculation of Γ*<sup>i</sup>* (*ε*) = *c*<sup>2</sup> *T*(*ε*) Im *B*(*ε*) mainly concerns the dominant scalar part of the GE pair term:

$$B\left(\varepsilon\right) \approx \frac{2\pi}{a^2} \int\_{\frac{r}{a}}^{r\_\varepsilon} \frac{r dr}{1 - 4A\_r r^2 \left(\varepsilon\right)},\tag{31}$$

(since the *k*-dependent term in Eq. (18) turns to be negligible beside this). The upper integration imit in Eq. (31) refers to the fact that its integrand only has poles for *r* < *rε*. With respect to the slow and fast modes in the function, Eq. (30), the integration is naturally divided in two stages. At the first stage, integration over each *m-*th period of fast cosine, around *rm* = 2*πm*/*k*F, is done setting the slow factors, \$*r* ≈ *rm* and sin*kεr* ≈ sin*kεrm* constant, and using the explicit formula:

$$\operatorname{Im} \int\_{-\pi}^{\pi} \frac{d\mathbf{x}}{1 - 4A^2 \cos^2 \mathbf{x}} = \operatorname{Im} \frac{\pi}{\sqrt{1 - A^2}} \cdot \frac{1}{2}$$

At next stage, summation of these results in *m* is approximated by integration in slow variable:

$$\frac{\pi}{k\_{\text{p}}} \text{Im} \sum\_{m} \frac{r\_{m}^{\text{3/2}}}{\sqrt{r\_{m} - r\_{c} \sin^{2} k\_{c} r\_{m}}} \approx \text{Im} \int\_{a}^{r\_{c}} \frac{r^{\text{3/2}}}{\sqrt{r - r\_{c} \sin^{2} k\_{c} r}} \cdot \tag{32}$$

Numerical calculation of the latter integral results in:

**4. Group expansion and Ioffe-Regel-Mott criteria**

*<sup>k</sup>* , that will add a certain finite imaginary part Γ*<sup>i</sup>*

g e

0 2 2 , 0

*T mA T*

^ <sup>+</sup> =*σ* ^ <sup>0</sup> ⊗ (*τ* ^ <sup>0</sup> + *τ* ^

e e

of numerator, resulting in zeroth order Bessel function: *∫*

2 – *δ*<sup>2</sup>

^

( )

e

e

*<sup>n</sup>*(*ε*)= *An*(*ε*)*m*

for *ε* close to *εmax*, this dependence can be expressed as:

where the length scales both for the monotonous decay:

^

To simplify calculation of the scalar function Γ*<sup>i</sup>*

e

both proportional to the matrix *m*

approximation: *Dξ*(*ε*) ≈ (*ξ* – *ξ*+)

the interaction matrix *A*

*c* 2 *T* ^ *B* ^

*m* ^ + 2 =2*m* ^

is written as:

232 Superconductors – New Developments

Let us now study the crossover from band-like to localized states near the limits of *ε<sup>i</sup>*

ee

ators of T-matrix and interaction matrices at *ε* = *ε*0, obtaining their forms:

( ) () () ( )

*n*

<sup>e</sup> <sup>ˆ</sup> ˆ ˆ ˆ ,

<sup>+</sup> » » - <sup>å</sup> <sup>2</sup>

e

say for definiteness, near its upper limit *εmax*. Supposing the actual energy *ε* < *εmax* to be within the range of band-like states, we use the RF self-energy matrix, Eq. (16), up to the GE pair term,

(25). Then the IRM criterion for a state at this energy be really band-like (also called extended)

 e

e

*N D*

0

2

(*ε*) = 4∆(∆ + *ε*0)

( ) ( )

d e

0F 0

é ù D + <sup>=</sup> ê ú ê ú ë û

*<sup>r</sup>* , *<sup>k</sup> <sup>c</sup>*

 e 2

2*π*

*k k*

 e

<sup>+</sup>. The *k*-summation (integration) in Eq. (29) is suitably done in polar coordinates over

the circular segments of Fermi surface. Here the azimuthal integration only refers to the phase

ℏ*v*F) is typically big, *x* ≫ 1, the asymptotical formula applies: *J*0(*x*) ≈ 2 / *πx* cos(*x* – *π*/4). Then, for radial integration in *ξ* around the extremum point *ξ*+, it is convenient to decompose this function into fast and slow oscillating factors: *J*0(*x*) ≈ 2 /(*πk*+*n*) cos(*k*+*n* – *π*/4)cos[(*ξ* – *ξ*+)*n*/ℏ*v*F] with the fast wavenumber *k*+ = *k*F + *ξ*+ /ℏ*v*F ≈ *k*F, and to write the denominator in the parabolic

(*ε*), with *δ*<sup>2</sup>

e

p er

F 2

 » <sup>F</sup> sin cos , *<sup>r</sup> <sup>r</sup> <sup>A</sup> kr kr r*

e bands,

(*ξ*), Eq.

(29)

(*x*). Since *x* = *n*(*k*F + *ξ*/

) (see Fig. 5). Thus,

(30)

(*ε*) to the dispersion law *ε* = *ε<sup>i</sup>*

(*ε*), we fix the energy argument in the numer‐

3) with important multiplicative property:

*max* - G ? ( ). *<sup>i</sup>* (28)

e

<sup>e</sup>*ix*cos*θd<sup>θ</sup>* =2*πJ*<sup>0</sup>

(*εmax* – *ε*)/(2*cγ*<sup>2</sup>

<sup>+</sup> only depends on the distance *n* between impurities, and,

×

*k n*

*i*

$$\operatorname{Im} B = \frac{r\_{\varepsilon}^{\cdot^2}}{a^2} f\left(k\_{\varepsilon} r\_{\varepsilon}\right),\tag{33}$$

where the function *f*(*z*) is zero for *z* < *z*0 ≈ 1.3585, and monotonously grows for *z* > *z*0, rapidly reaching the asymptotic constant value: *fas* ≈ 1.1478, for *z* ≫ *z*0. Then the IRM criterion, Eq. (28), at *ε* so close to *εmax* that *k<sup>ε</sup> rε* ≫ *z*0, is expressed as:

$$
\varepsilon\_{\text{max}} - \varepsilon \approx \frac{c\gamma^2}{\varepsilon\_{\text{max}} - \varepsilon\_0} \frac{r\_\varepsilon^2}{a^2} \,\, ^\prime \tag{34}
$$

giving an (*c*-independent) estimate for the range of extended states within the impurity band:

$$
\varepsilon\_{\max} - \varepsilon \gg \Gamma\_0 = \frac{\left(\upsilon \varepsilon\_0\right)^{3/2}}{ak\_{\rm F}} \sqrt{\frac{2\pi \rho\_{\rm F}}{1 + \upsilon^2}}.\tag{35}
$$

Its comparison with the full extension of this band, *εmax* – *εmin* = *cγ*<sup>2</sup> (1 + *v*<sup>2</sup> )/(*v*<sup>2</sup> ∆), would suggest a possibility for such extended states to really exist if the impurity concentration surpasses the characteristic (small) value:

$$
\omega \approx \varepsilon\_0 = \frac{\left(\pi \rho\_{\rm F} \varepsilon\_0\right)^{3/2}}{a k\_{\rm F}} \sqrt{\frac{2v}{1 + v^2}}.\tag{36}
$$

For typical values of *ρ*<sup>F</sup> –1 ~ 2 eV, *ak*F ~ 1, and ∆ ~ 10 meV in LaOFeAs system [8, 13, 58], and supposing a plausible impurity perturbation *v* ~ 1, we estimate *c*0 ≈ 8 10–4, manifesting impor‐ tant impurity effects already at their very low content.

However, the r. h. s. of Eq. (33) vanishes at *kεrε* < *z*0, which occurs beyond the vicinity of the band top:

$$
\varepsilon\_{\max} - \varepsilon > \left(\frac{c\_0}{c}\right)^3 \Gamma\_0. \tag{37}
$$

Under the condition of Eq. (36), this vicinity is yet narrower than Γ0 by Eq. (35), defining the true, even wider, range of extended states within the impurity band.

Otherwise, for *c* ≪ *c*0, the impurity band does not exist at all, then we analyze the energy range near the impurity level with the NRF GE and write an approximate criterion for its convergence as *c*|*B*<sup>0</sup> | ≪ 1. This calculation is done in a similar way to as before but replacing the interaction function, Eq. (29), by its NRF version:

$$A\_r^0(\varepsilon) \approx \sqrt{\frac{\mathcal{R}\_\varepsilon}{r}} \mathbf{e}^{-r/r\_0} \cos k\_\mathbb{F} r\_\prime \tag{38}$$

with *k*F*Rε* = 2*π*(*ε*0 /|*ε* – *ε*0|)2 and *k*F*r*0 = 2*ε*F/*ξ*0. Then the above GE convergence criterion is assured beyond the following vicinity of impurity level:

$$\left|\boldsymbol{\varepsilon} - \boldsymbol{\varepsilon}\_{0}\right| \gg \Gamma\_{0} \exp\left(-\frac{c\_{0}^{4/3}}{c}\right),\tag{39}$$

defining its broadening due to inter-impurity interactions. Within this range, the DOS function for localized states can be only estimated by the order of magnitude, but outside it is given by:

$$\begin{split} \left| \rho\_{\rm loc} \left( \boldsymbol{\varepsilon} \right) \approx \frac{c^2}{c\_0^{4/3}} \middle| \boldsymbol{\varepsilon} - \boldsymbol{\varepsilon}\_0 \right|, \quad \text{ for} \quad \boldsymbol{\Gamma}\_c \ll \left| \boldsymbol{\varepsilon} - \boldsymbol{\varepsilon}\_0 \right| \ll \boldsymbol{\Gamma}\_{0'} \\ \left| \rho\_{\rm loc} \left( \boldsymbol{\varepsilon} \right) \approx \frac{c^2 \boldsymbol{\varepsilon}\_0^4}{\left| \boldsymbol{\varepsilon} - \boldsymbol{\varepsilon}\_0 \right|^5}, \quad \text{ for} \quad \boldsymbol{\Gamma}\_0 \ll \left| \boldsymbol{\varepsilon} - \boldsymbol{\varepsilon}\_0 \right|. \end{split} \tag{40}$$

Notably, the total number of states near the impurity level is 3 *c* (37)

of the band top:

We note the *f*-function in Eq. (33) to vanish at *k*

( )

*v*

e

a possibility for such extended states to really exist if the impurity concentration surpasses the

3/2 F 0 0 2 F

supposing a plausible impurity perturbation *v* ~ 1, we estimate *c*0 ≈ 8 10–4, manifesting impor‐

However, the r. h. s. of Eq. (33) vanishes at *kεrε* < *z*0, which occurs beyond the vicinity of the

Under the condition of Eq. (36), this vicinity is yet narrower than Γ0 by Eq. (35), defining the

Otherwise, for *c* ≪ *c*0, the impurity band does not exist at all, then we analyze the energy range near the impurity level with the NRF GE and write an approximate criterion for its convergence

> e

*<sup>R</sup> <sup>A</sup> k r r*


0 0 exp , *c c*

defining its broadening due to inter-impurity interactions. Within this range, the DOS function for localized states can be only estimated by the order of magnitude, but outside it is given by:

» - G-G

5 0 0

, for ,

4/3 0 0 0

» G-


?


<sup>F</sup> e cos , *r r*

æ ö

4/3 0

> e e

, for ,

= =

*<sup>c</sup>* (40)

e e

=

è ø

and *k*F*r*0 = 2*ε*F/*ξ*0. Then the above GE convergence criterion is assured

3 0 <sup>0</sup> . *<sup>c</sup> c*

æ ö -> G ç ÷ è ø

*ak v*

( ) pr e<sup>=</sup> <sup>+</sup>

3/2 0 F 0 2 F

*ak v*

pr

<sup>2</sup> . <sup>1</sup>

(1 + *v*<sup>2</sup>

)/(*v*<sup>2</sup>

(35)

(36)

(37)

(38)

(39)

∆), would suggest

+

<sup>2</sup> . <sup>1</sup> *v*

–1 ~ 2 eV, *ak*F ~ 1, and ∆ ~ 10 meV in LaOFeAs system [8, 13, 58], and

*max*

 e- G=

Its comparison with the full extension of this band, *εmax* – *εmin* = *cγ*<sup>2</sup>

?

*max* e

true, even wider, range of extended states within the impurity band.

( )

*r*

e e

> ee

*loc c*

0

e


2

*c c*

> e e

e

 e

tant impurity effects already at their very low content.

*c c*

?

e

characteristic (small) value:

234 Superconductors – New Developments

For typical values of *ρ*<sup>F</sup>

band top:

as *c*|*B*<sup>0</sup>

function, Eq. (29), by its NRF version:

beyond the following vicinity of impurity level:

( )

r e

r e

*loc*

( )

with *k*F*Rε* = 2*π*(*ε*0 /|*ε* – *ε*0|)2

$$\int \rho\_{\rm loc}(\boldsymbol{\varepsilon})d\boldsymbol{\varepsilon} \cdot \boldsymbol{c},$$

0 0. *max*

*c*

limit for extended states within the impurity band results even closer to this band top.

alike that of extended states in the impurity band by Eq. (26). The summary of evolution of this area of quasiparticle spectrum in function of impurity concentration is shown in Fig. 7. Otherwise, at *c c*0, the very concept of impurity band does not make sense at all, and the energy range near the impurity level should be described using the NRF GE, until the

Fig. 6: Interaction function *A*<sup>2</sup> *r*() by Eq. 29 for the parameters *max* = 0.1 and F = 5·10<sup>2</sup> combines slower sine oscillations (solid line) with the monotonic decay function (dashed line). The contributions to Im *B* only come from the shadowed intervals where (*r*/*r*)sin2 *kr* > 1. The inset presents the expansion of a narrow rectangle in the main panel to **Figure 6.** Interaction function *A*<sup>2</sup> *<sup>r</sup>*(*ε*) by Eq. 29 at the choice of parameters *εmax* – *ε* = 0.1 and ∆/*ε*<sup>F</sup> = 5 10–2 displays slower sine oscillations (solid line) and the monotonous envelope function (dashed line). The shadowed intervals are those contributing to Im *B*, accordingly to the condition (*rε*/*r*)sin2 *kεr* > 1. Inset: the expansion of the rectangle in the main pan‐ el shows also faster oscillations by the cosine.

for localized states can be only estimated by the order of magnitude, but outside it is given by:

giving an (*c*-independent) estimate for the range of extended states within the impurity band: Fig. 7. Composition of the energy spectrum near the impurity level in function of impurity concentration. **Figure 7.** Composition of the energy spectrum near the impurity level *ε*0 in function of impurity concentration.

#### Its comparison with the full extension of this band, *max min* = *c* 2 **5. Impurity effects on superconducting characteristics**

indicating a possible extreme sensitivity to impurity effects.

F

at 

*k*F*R* = 2(<sup>0</sup> /|

so close to

For typical values of

show also the faster cosine oscillations.

characteristic (small) value: 3/2 F 0 0 2 <sup>2</sup> . <sup>1</sup> *<sup>v</sup> c c ak v* (36) The above results on the quasiparticle spectrum in the disordered SC system can be immedi‐ ately used for calculation of impurity effects on its observable characteristics. Thus the fundamental SC order parameter ∆ is estimated from the modified gap equation:

 3/2 0 F 0 2 F <sup>2</sup> . <sup>1</sup> *max v*

a possibility for such extended states to really exist if the impurity concentration surpasses the

F

reasonable choice of impurity perturbation *v* ~ 1, this value is estimated as low as *c*<sup>0</sup> 8·10,

*ak v*

(35)

(1 + *v* 2 )/(*v* 2

), would suggest

~ 2 eV, *ak*F ~ 1, and ~ 10 meV in LaOFeAs system 8,13,59, with a

$$\mathcal{A}^{-1} = \bigcap\_{\varepsilon \atop 0}^{\varepsilon\_{\mathbb{D}}} \operatorname{Im} \operatorname{Tr} \hat{G} \left( \varepsilon \right) \hat{\tau}\_1 d\varepsilon,\tag{41}$$

where *G* ^ (*ε*)= *<sup>N</sup>* <sup>−</sup><sup>1</sup> ∑*<sup>k</sup> G* ^ *<sup>k</sup>* (*ε*), *λ* = *ρ*F*V*SC is the (small) dimensionless SC pairing constant, and the Debye energy *ε*<sup>D</sup> restricts the energy range of its action. In absence of impurities, *c* = 0, using of Eq. (11) leads straightforwardly to the known result for the non-perturbed ∆0 value: *λ*–1 = arccosh (*ε*D/∆0) and thus ∆0 ≈ 2*ε*D e–1/*<sup>λ</sup>*.

For finite *c*, contributions to Eq. (41) come both from the main band, Eq. (25), and from the impurity band (or level), Eqs. (26) (or (40)). The latter contribution is ~ *c*, accordingly to the previous discussion, defining a small correction beside *λ*–1 ≫ 1. But a much stronger *c*dependent correction comes from the modified main band (limited to its range of extended states):

$$\int\_{\varepsilon\_c}^{\varepsilon\_D} \frac{d\varepsilon}{\sqrt{\varepsilon^2 - \Delta^2}} = \arccos \frac{\varepsilon\_D}{\Delta} - \arccos \frac{\varepsilon\_c}{\Delta}.$$

For *εc* – ∆ ∼ (*c*/*c*1) 2 ∆ (see after Eq. (24)) and *c* ` *c*1, the last term is well approximated by:

$$
\operatorname{arccosh}\frac{\varepsilon\_c}{\Delta} \approx \sqrt{2}\frac{c}{c\_1},
$$

and leads to the modified gap equation as:

$$\arccos \frac{\varepsilon\_{\rm D}}{\Lambda} - \arccos \frac{\varepsilon\_{\rm D}}{\Lambda\_0} \approx \sqrt{2} \frac{c}{c\_1} \tag{42}$$

Its approximate solution for *c* ≪ *c*1 describes the initial decay of SC order parameter with impurity concentration as:

$$
\Delta \approx \Delta\_0 \left( 1 - \sqrt{2} \frac{c}{c\_1} \right) \tag{43}
$$

and for the values of *ρ*F and ∆0 used above, the estimate of characteristic concentration *c*1 is quite low: *c*1 ~ 10–2 (though higher than *c*0 by Eq. (36)), suggesting a very strong impurity effect. With further growing *c* up to *c* ≺ *c*1, the value of ∆ from Eq. (43) would formally tend to zero as ≈ ∆0(*c*1/*c*) 2 /2. However, concentrations *c* ~ *c*1 would already correspond to the impurity band as wide as the gap itself; this goes beyond the validity of the present approach and needs a special treatment.

To study another important dependence, that of the SC transition temperature *Tc* on concen‐ tration *c*, one has, strictly speaking, to extend the above GF techniques for finite temperatures, but a very simple estimate can be done, supposing that the BCS relation ∆/*Tc* ≈ 1.76 still holds in the presence of impurities. Then the r.h.s. of Eq. (44) would also describe the decay of *Tc*/*Tc*0.

It is of interest to compare the present results with the known Abrikosov-Gor'kov solution for BCS SC with paramagnetic impurities in the Born approximation [59, 60]. In that approxima‐ tion, the only perturbation parameter is the (constant) quasiparticle lifetime *τ*. In our frame‐ work, *τ*–1 can be related to Im Σ(*ε*) at a proper choice of energy, *ε* ~ |∆ – *ε*| ~ ∆. Then, in the self-consistent T-matrix approximation \cite{psl}, we estimate *τ*–1 ~ *c*∆/*c*1 which leads to the relation *τTc* ~ *c*1/(1.76*c*), reaching at *c* ~ *c*1 a good agreement with the Abrikosov-Gor'kov universal criterion for complete SC suppression *τTc* < 0.567. <sup>B</sup> B 0 B 2 0 B / / 2/ 0 F e . e 1 <sup>1</sup> e 1 *k T s s k T k T <sup>d</sup> cv k T <sup>c</sup> n T n T <sup>v</sup>* (45) When compared to its unperturbed value in the pure SC system

Also, a notable impurity effect is expected on the London penetration depth *λ*L ~ *ns* 1/2, as follows from the temperature dependence of superfluid density: <sup>B</sup> B 0 B / F 1 / 2 2 e , e 1 2 *k T <sup>s</sup> k T <sup>d</sup> k T n T <sup>c</sup>* 

the effect by the last term in Eq. (45) produces a considerable slowing down of the low-

$$m\_s\left(T\right) = \int\_0^{\upsilon} \frac{\rho\left(\boldsymbol{\varepsilon}\right)d\boldsymbol{\varepsilon}}{\mathbf{e}^{\boldsymbol{\varepsilon}/k\_B T} + 1} \approx n\_s^0\left(T\right) - \frac{c\upsilon^2}{1 + \upsilon^2} \frac{\Delta}{\sigma\_\mathrm{F}} \sqrt{\frac{k\_\mathrm{B}T}{\pi\Delta}} \mathbf{e}^{-\Lambda/k\_B T} + \frac{c}{\mathbf{e}^{\boldsymbol{\varepsilon}\_0/k\_B T} + 1}.\tag{44}$$

Finally, the impurity effect on the electronic specific heat in the SC state can be obtained

*d*

0

0 , 2cosh / 2

= 1/(*k*B*T*):

> ln(*c*1/*c* 1)/(

*i* and *<sup>b</sup>* states:

0) where it is

When compared to its unperturbed value in the pure SC system in a similar way, considering its dependence on inverse temperature 

( )

Debye energy *ε*<sup>D</sup> restricts the energy range of its action. In absence of impurities, *c* = 0, using of Eq. (11) leads straightforwardly to the known result for the non-perturbed ∆0 value: *λ*–1 =

For finite *c*, contributions to Eq. (41) come both from the main band, Eq. (25), and from the impurity band (or level), Eqs. (26) (or (40)). The latter contribution is ~ *c*, accordingly to the previous discussion, defining a small correction beside *λ*–1 ≫ 1. But a much stronger *c*dependent correction comes from the modified main band (limited to its range of extended

> e

*<sup>c</sup> d*

D 2 2 arccosh arccosh .

∆ (see after Eq. (24)) and *c* ` *c*1, the last term is well approximated by:

*c*

0 1

 » <sup>D</sup> <sup>1</sup> arccosh 2 , *<sup>c</sup> <sup>c</sup>*

> e

Its approximate solution for *c* ≪ *c*1 describes the initial decay of SC order parameter with

1 12 , *<sup>c</sup> c*

/2. However, concentrations *c* ~ *c*1 would already correspond to the impurity band

and for the values of *ρ*F and ∆0 used above, the estimate of characteristic concentration *c*1 is quite low: *c*1 ~ 10–2 (though higher than *c*0 by Eq. (36)), suggesting a very strong impurity effect. With further growing *c* up to *c* ≺ *c*1, the value of ∆ from Eq. (43) would formally tend to zero

as wide as the gap itself; this goes beyond the validity of the present approach and needs a

 - » D D D D

arccosh arccosh 2 , *<sup>c</sup>*

æ ö D»D - ç ÷ è ø

0

= - - D D D ò

e


et e

1

*<sup>k</sup>* (*ε*), *λ* = *ρ*F*V*SC is the (small) dimensionless SC pairing constant, and the

e

*<sup>c</sup>* (42)

(43)

ˆ Im Tr , *G d*ˆ (41)

e

D 1

0

l

e

D

e

e

e

e

*c*

where *G*

states):

For *εc* – ∆ ∼ (*c*/*c*1)

^ (*ε*)= *<sup>N</sup>* <sup>−</sup><sup>1</sup>

236 Superconductors – New Developments

∑*<sup>k</sup> G* ^

arccosh (*ε*D/∆0) and thus ∆0 ≈ 2*ε*D e–1/*<sup>λ</sup>*.

2

impurity concentration as:

as ≈ ∆0(*c*1/*c*)

2

special treatment.

and leads to the modified gap equation as:

$$m\_s^0 \left( T \right) = \rho\_{\rm F} \int\_{\Delta}^{\alpha} \frac{\varepsilon d\varepsilon}{\left( \mathbf{e}^{\varepsilon/k\_B T} + 1 \right) \sqrt{\varepsilon^2 - \Delta^2}} \approx c\_1 \sqrt{\frac{k\_B T}{2\pi \Delta}} \mathbf{e}^{-\Lambda/k\_B T},$$

*C*

*C ck <sup>i</sup>*

perturbed SC system shows an important slowing down at

dominated by the exponent ~ exp(

the effect by the last term in Eq. (45) produces a considerable slowing down of the lowtemperature decay of the difference *λ*(*T*)/*λ*(0) – 1 (Fig. 8), in a reasonable agreement with recent experimental observations for SC iron pnictides under doping [47]. 3/2 B 1 exp . *C k c cv <sup>b</sup>* Also its comparison with the known low temperature behavior *C*0() ~ exp() for non-

B

Fig. 8. The London penetration depth for a SC with impurities (solid line) has a slower low-temperature decay of its difference than that in absence of impurities (dashed line). **Figure 8.** Low-temperature decay of the London penetration depth difference for a SC with impurities (solid line) is slower than that in absence of impurities (dashed line).

0) as seen in Fig. 9.

this approach as:

for ' = 

function:

occupation function *f*(

 

Finally, a similar analysis can be applied for the impurity effect on the electronic specific heat in the SC state, whose dependence on inverse temperature *β* = 1/(*k*B*T*) is represented as: **6. Kubo-Greenwood formalism for multiband superconductor** The relevant kinetic coefficients for electronic processes in the considered disordered

superconductor follow from the general Kubo-Greenwood formulation 56,57, adapted here to

effect, such as, e.g., heat conductivity, differential conductivity of scanning tunneling

The further extension of this approach is for kinetic properties of SC under impurity

$$\mathbf{C}\left(\boldsymbol{\beta}\right) = \frac{\partial}{\partial \boldsymbol{\beta}} \Big|\_{0}^{\pi} \frac{\varepsilon \rho\left(\boldsymbol{\varepsilon}\right) d\boldsymbol{\varepsilon}}{\mathbf{e}^{\beta \varepsilon} + 1},\tag{45}$$

and naturally divided in two characteristic contributions, *C* = *Ci* + *Cb*, from *ρ<sup>i</sup>* and *ρ <sup>b</sup>* states: <sup>2</sup> ' ˆ ˆ , , , ' Tr Im Im ' , *x x <sup>e</sup> f f T d dv v G G k k kk k* (47)

$$\begin{aligned} \mathbb{C}\_i(\boldsymbol{\beta}) &\approx c k\_{\mathbb{B}} \left[ \frac{\beta \varepsilon\_0}{2 \cosh \left( \beta \varepsilon\_0 / \, 2 \right)} \right]^2, \\\\ \mathbb{C}\_b(\boldsymbol{\beta}) &\approx k\_{\mathbb{B}} \left( c\_1 - c \right) v \left( \beta \Delta \right)^{3/2} \exp \left( -\beta \Delta \right). \end{aligned}$$

The resulting function *C*(*β*) deviates from the known low temperature behavior *C*0(*β*) ~ exp(– *β*∆) for non-perturbed SC system at *β* > ln(*c*1/*c* – 1)/(∆ – *ε*0), where the characteristic exponent is changed to a slower ~ exp(–*βε*0) as seen in Fig. 9. This function is defined in the whole , plane in a way to coincide with the physical quasiparticle velocities for each particular band, Eqs. (23,24), along the corresponding dispersion

Fig. 9. The crossover in the characteristic exponent of specific heat for a SC with impurities from high-temperature (dashed line) to low-temperature <sup>0</sup> after the **Figure 9.** Temperature behavior of specific heat for a SC with impurities presents a crossover from *β*∆ exponent (dash‐ ed line) to *βε*0 at low enough temperature (high enough *β* = 1/(*k*B*T*)).

impurity dependent threshold > ln(*c*1/*c* 1)/( 0). The same approach can be then used for other observable characteristics for SC under impurity effect, such as, e.g., heat conductivity, differential conductivity for scanning tunneling spectroscopy or absorption coefficient for far infrared radiation that is done in the next section.

#### **6. Kubo-Greenwood formalism for multiband superconductor**

The relevant kinetic coefficients for electronic processes in the considered disordered super‐ conductor follow from the general Kubo-Greenwood formulation [55, 56], adapted here to the specific multiband structure of Green function matrices. Thus, one of the basic transport characteristics, the (frequency and temperature dependent) electrical conductivity is expressed in this approach as:

Finally, a similar analysis can be applied for the impurity effect on the electronic specific heat in the SC state, whose dependence on inverse temperature *β* = 1/(*k*B*T*) is represented as:

effect, such as, e.g., heat conductivity, differential conductivity of scanning tunneling spectroscopy or absorption coefficient for far infrared radiation is done in the next section.

The further extension of this approach is for kinetic properties of SC under impurity

be

 <sup>2</sup> ' ˆ ˆ , , , ' Tr Im Im ' , *x x <sup>e</sup> f f T d dv v G G*

, e 1 *d*

The relevant kinetic coefficients for electronic processes in the considered disordered

be

*k k kk k* (47)

é ù » ê ú ê ú ë û

be

0 , 2cosh / 2

bb

*D*

*k k v k* (48)

0

and the electric field applied along the *x*-axis. Besides the common Fermi

*C* (45)

2

, the above formula involves the generalized velocity

plane in a way to coincide with the physical

<sup>0</sup> after the

+ *Cb*, from *ρ<sup>i</sup>*

and *ρ <sup>b</sup>* states:

er e e

superconductor follow from the general Kubo-Greenwood formulation 56,57, adapted here to the specific multiband structure of Green function matrices. Thus, one of the basic transport characteristics, the (frequency and temperature dependent) electrical conductivity is expressed in

( ) ( )

( ) ( )( ) ( )

,

Fig. 9. The crossover in the characteristic exponent of specific heat for a SC with

**Figure 9.** Temperature behavior of specific heat for a SC with impurities presents a crossover from *β*∆ exponent (dash‐

The same approach can be then used for other observable characteristics for SC under impurity effect, such as, e.g., heat conductivity, differential conductivity for scanning tunneling spectroscopy or absorption coefficient for far infrared radiation that is done in the next section.

The relevant kinetic coefficients for electronic processes in the considered disordered super‐ conductor follow from the general Kubo-Greenwood formulation [55, 56], adapted here to the

> ln(*c*1/*c* 1)/(

(dashed line) to low-temperature

0).

**6. Kubo-Greenwood formalism for multiband superconductor**

 *k*

The resulting function *C*(*β*) deviates from the known low temperature behavior *C*0(*β*) ~ exp(– *β*∆) for non-perturbed SC system at *β* > ln(*c*1/*c* – 1)/(∆ – *ε*0), where the characteristic exponent

quasiparticle velocities for each particular band, Eqs. (23,24), along the corresponding dispersion

 » - D -D 3/2 B 1 exp . *C k c cv <sup>b</sup>*

1 Re , Re . *D*

¥ ¶ <sup>=</sup> ¶ <sup>+</sup> <sup>ò</sup> 0

( ) ( )

b

B

b

**6. Kubo-Greenwood formalism for multiband superconductor**

b

*C ck <sup>i</sup>*

and naturally divided in two characteristic contributions, *C* = *Ci*

 

) = (e+ 1)

this approach as:

238 Superconductors – New Developments

 

occupation function *f*(

for ' = 

function:

b

is changed to a slower ~ exp(–*βε*0) as seen in Fig. 9.

impurities from high-temperature

ed line) to *βε*0 at low enough temperature (high enough *β* = 1/(*k*B*T*)).

impurity dependent threshold

This function is defined in the whole

$$\sigma\left(o\nu,T\right) = \frac{e^2}{\pi} \left[d\varepsilon \frac{f\left(\varepsilon\right) - f\left(\varepsilon^\circ\right)}{o\nu}\right] d\mathbf{k} \upsilon\_x\left(\mathbf{k},\varepsilon\right) \upsilon\_x\left(\mathbf{k},\varepsilon^\circ\right) \text{Tr}\left[\text{Im }\hat{G}\_k\left(\varepsilon\right)\text{Im }\hat{G}\_k\left(\varepsilon^\circ\right)\right],\tag{46}$$

for *ε*' = *ε* – ℏ*ω* and the electric field applied along the *x*-axis. Besides the common Fermi occupation function *f*(*ε*) = (e*βε* + 1)–1, the above formula involves the generalized velocity function:

$$\text{tr}\left(\mathbf{k},\boldsymbol{\varepsilon}\right) = \left(\hbar \frac{\partial \text{Re}\,D\_{k}\left(\boldsymbol{\varepsilon}\right)}{\partial \boldsymbol{\varepsilon}}\right)^{-1} \nabla\_{k} \text{Re}\,D\_{k}\left(\boldsymbol{\varepsilon}\right). \tag{47}$$

This function is defined in the whole *ξ*, *ε* plane in a way to coincide with the physical quasi‐ particle velocities for each particular band, Eqs. (23, 24), along the corresponding dispersion laws: *v*(*k*, *ε<sup>j</sup>* (*k*)) = ℏ–1∇*kε<sup>j</sup>* (*k*) = *vj*, *<sup>k</sup>*, *j* = *b*, *i*. The conductivity resulting from Eq. (48) can be then used for calculation of optical reflectivity.

Other relevant quantities are the static (but temperature dependent) transport coefficients, as the heat conductivity:

$$\kappa\left(T\right) = \frac{\hbar}{\pi} \int d\varepsilon \frac{\partial f\left(\varepsilon\right)}{\partial \varepsilon} \int d\mathbf{k} \left[v\_x\left(\mathbf{k}, \varepsilon\right)\right]^2 \text{Tr}\left[\text{Im }\hat{G}\_k\left(\varepsilon\right)\right]^2\right.\tag{48}$$

and the thermoelectric coefficients associated with the static electrical conductivity *σ*(*T*) ≡ *σ*(0, *T*) [62], the Peltier coefficient:

$$\Pi\left(T\right) = \frac{\hbar e}{\pi \sigma\left(T\right)} \int d\varepsilon \frac{\partial f\left(\varepsilon\right)}{\partial \varepsilon} \varepsilon \left[d\mathbf{k}\right] \overline{v\_{\chi}\left(\mathbf{k},\varepsilon\right)}\right]^2 \text{Tr}\left[\text{Im }\hat{G}\_k\left(\varepsilon\right)\right]^2\tag{49}$$

and the Seebeck coefficient *S*(*T*) = Π(*T*)/*T*. All these transport characteristics, though being relatively more complicated from the theoretical point of view than the purely thermodynam‐ ical quantities from the previous section, permit an easier and more reliable experimental verification and so could be of higher interest for practical applications of the considered impurity effects in the multiband superconductors.

It is worth to recall that the above formulae are only contributed by the band-like states, that is the energy arguments *ε*, *ε*' in Eqs. (47, 49, 50) are delimited by the relevant mobility edges. This is the main distinction of our approach from the existing treatments of impurity effects on transport in iron pnictide superconductors using the T-matrix approximation to a solution like Eq. (15) for the whole energy spectrum [62], even for its ranges where the very concept of velocity, as Eq. (48), ceases to be valid.

Next, we consider the particular calculation algorithms for the expressions, Eqs. (47, 49, 50), for the more involved case of dynamical conductivity, Eq. (47), that can be then reduced to simpler static quantities, Eqs. (49, 50).

#### **7. Optical conductivity**

spells as *d*

where *v*(

Here *v* ª *v*(,), *v*' ª *v*(',

( ,) = |*vk*(

but within the *i*-band, at

*dk* = 2(*hv*F)

velocities) and the most important radial

related to band-like states with positive

). For energies within the *b*-band,

1,2

 

and *h*-segments. The azimuthal

integrand in particular pole terms:

 

*dd*

)| and ˆ ˆ *G G* ,*<sup>k</sup>*

1,2 

2

*c*

*c*, < || < 

The integral in Eq. (47) is dominated by the contributions from *δ*-like peaks of the Im *G* ^ *<sup>k</sup>* (*ε*) and Im *G* ^ *<sup>k</sup>* (*ε* ') matrix elements. These peaks arise from the above dispersion laws, Eqs. (23), (24), thus restricting the energy integration to the band-like ranges: |*ε*| > *εc* for the *b*-bands and *εc*, – < |*ε*| < *ε<sup>c</sup>*, + for the *i*-bands. Regarding the occupation numbers *f*(*ε*) and *f*(*ε*') at reasonably low temperatures *k*B*T* ≪ ∆, *ε*0, the most effective contributions correspond to positive *ε* values, either from *b*- or *i*-bands, and to negative *ε*' values from their negative counterparts, *b*' or *i*'. There are three general kinds of such contributions: i) *b*-*b*', due to transitions between the main bands, similar to those in optical conductivity by the pure crystal (but with a slightly shifted frequency threshold: ℏ*ω* <sup>→</sup> 2*εc*), ii) *b*-*i*' (or *i*-*b*'), due to combined transitions between the principal and impurity bands within the frequency range ℏ*ω* <sup>→</sup> *εc* + *εc*, –, and iii) *i*-*i*', due to transitions between the impurity bands within a narrow frequency range of 2*ε<sup>c</sup>*, – < ℏ*ω* < 2*εc*, +. The frequency-momentum relations for these processes and corresponding peaks are dis‐ played in Fig. 10. The resulting optical conductivity reads *σ*(*ω*, *T*) = ∑*<sup>α</sup> σα*(*ω*, *T*) with *α* = *b*-*b*', *ii*', and *i*-*b*'.

Fig. 10. Configuration of the poles *<sup>j</sup>* of GF's contributing to different types of optical **Figure 10.** Configuration of the poles *ξ<sup>j</sup>* of GF's contributing to different types of optical conductivity processes (over electronic pocket of the quasiparticle spectrum in Fig. 3).

conductivity processes (over electronic pocket of the quasiparticle spectrum in Fig. 3).

the quasimomentum integration in Eq. (47) under the above chosen symmetry of Fermi segments

ˆ ˆ *vv G G A* , , ' Tr Im , Im , ' , '

 

> 

, 

 

 

 

 

), and the indices

 > 

 

0

 

 

0 <sup>2</sup> . <sup>2</sup>

<sup>2</sup>

*A vv* ,' ' .

 

functions. As seen from Eqs. (23, 25) and Fig. 10, there can be two such poles of ˆ*G*

 

 

*-*integration contributes by the factor of

where the factor 2 accounts for identical contributions from *e*-

(52)

 

*<sup>c</sup>*, they are symmetrical:

*<sup>c</sup>*,, their positions are asymmetrical:

2 '

define the respective residues:


(53)

and respective quasi-momentum values denoted as

<sup>2</sup> , (54)

(55)

run over all the poles of the two Green

). Now,

> ,

(from *x*-projections of

, 

is a smooth enough function while the above referred peaks result from zeros of Re *Dk*(

For practical calculation of each contribution, the relevant matrix Im *G* ^ *<sup>k</sup>* (*ε*) (within the bandlike energy ranges) can be presented as Im *G* ^ *<sup>k</sup>* (*ε*)= *N* ^ (*ε*, *<sup>ξ</sup>*) Im *<sup>D</sup><sup>k</sup>* (*ε*)−<sup>1</sup> where the numerator matrix:

$$\hat{N}\left(\boldsymbol{\varepsilon},\boldsymbol{\xi}\right) = \text{Re}\left(\boldsymbol{\tilde{\varepsilon}} + \boldsymbol{\tilde{\xi}}\boldsymbol{\tau}\_{3} + \Delta\boldsymbol{\tau}\_{1}\right),\tag{50}$$

is a smooth enough function while the above referred peaks result from zeros of Re *Dk*(*ε*). Now, the quasimomentum integration in Eq. (47) under the above chosen symmetry of Fermi segments spells as ∫*dφ*∫*dk* = 2(*hv*F) –1∫*dφ*∫*dξ* where the factor 2 accounts for identical contributions from *e*- and *h*-segments. The azimuthal *φ-*integration contributes by the factor of *π* (from *x*projections of velocities) and the most important radial *ξ*-integration is readily done after expanding its integrand in particular pole terms:

$$
\sigma\left(\boldsymbol{\xi},\boldsymbol{\varepsilon}\right)\upsilon\left(\boldsymbol{\xi},\boldsymbol{\varepsilon}'\right)\operatorname{Tr}\left[\operatorname{Im}\,\hat{\operatorname{G}}\left(\boldsymbol{\xi},\boldsymbol{\varepsilon}\right)\operatorname{Im}\,\hat{\operatorname{G}}\left(\boldsymbol{\xi},\boldsymbol{\varepsilon}'\right)\right] = \sum\_{a} A\_{a}\left(\boldsymbol{\varepsilon},\boldsymbol{\varepsilon}'\right)\delta\left(\boldsymbol{\xi} - \boldsymbol{\xi}\_{a}\right),\tag{51}
$$

where *v*(*ξ*, *ε*) = |*vk*(*ε*)| and *G* ^ (*ξ*, *<sup>ε</sup>*)≡*<sup>G</sup>* ^ *<sup>k</sup>* (*ε*) define the respective residues:

This is the main distinction of our approach from the existing treatments of impurity effects on transport in iron pnictide superconductors using the T-matrix approximation to a solution like Eq. (15) for the whole energy spectrum [62], even for its ranges where the very concept of

Next, we consider the particular calculation algorithms for the expressions, Eqs. (47, 49, 50), for the more involved case of dynamical conductivity, Eq. (47), that can be then reduced to

The integral in Eq. (47) is dominated by the contributions from *δ*-like peaks of the Im *G*

is a smooth enough function while the above referred peaks result from zeros of Re *Dk*(

'

 *b*

 

*<sup>i</sup>*

the quasimomentum integration in Eq. (47) under the above chosen symmetry of Fermi segments

conductivity processes (over electronic pocket of the quasiparticle spectrum in Fig. 3).

ˆ ˆ *vv G G A* , , ' Tr Im , Im , ' , '

 

> 

, 

 

 

 

 

), and the indices

 > 

 

0

 

 

0 <sup>2</sup> . <sup>2</sup>

<sup>2</sup>

*A vv* ,' ' .

 

functions. As seen from Eqs. (23, 25) and Fig. 10, there can be two such poles of ˆ*G*

 

 

*-*integration contributes by the factor of

where the factor 2 accounts for identical contributions from *e*-

(52)

 

*<sup>c</sup>*, they are symmetrical:

*<sup>c</sup>*,, their positions are asymmetrical:

2 '

define the respective residues:

*<sup>j</sup>* of GF's contributing to different types of optical

of GF's contributing to different types of optical conductivity processes (over


(53)

and respective quasi-momentum values denoted as

<sup>2</sup> , (54)

(55)

run over all the poles of the two Green

(24), thus restricting the energy integration to the band-like ranges: |*ε*| > *εc* for the *b*-bands and *εc*, – < |*ε*| < *ε<sup>c</sup>*, + for the *i*-bands. Regarding the occupation numbers *f*(*ε*) and *f*(*ε*') at reasonably low temperatures *k*B*T* ≪ ∆, *ε*0, the most effective contributions correspond to positive *ε* values, either from *b*- or *i*-bands, and to negative *ε*' values from their negative counterparts, *b*' or *i*'. There are three general kinds of such contributions: i) *b*-*b*', due to transitions between the main bands, similar to those in optical conductivity by the pure crystal (but with a slightly shifted frequency threshold: ℏ*ω* <sup>→</sup> 2*εc*), ii) *b*-*i*' (or *i*-*b*'), due to combined transitions between the principal and impurity bands within the frequency range ℏ*ω* <sup>→</sup> *εc* + *εc*, –, and iii) *i*-*i*', due to transitions between the impurity bands within a narrow frequency range of 2*ε<sup>c</sup>*, – < ℏ*ω* < 2*εc*, +. The frequency-momentum relations for these processes and corresponding peaks are dis‐ played in Fig. 10. The resulting optical conductivity reads *σ*(*ω*, *T*) = ∑*<sup>α</sup> σα*(*ω*, *T*) with *α* = *b*-*b*', *i*-

*<sup>k</sup>* (*ε* ') matrix elements. These peaks arise from the above dispersion laws, Eqs. (23),

^ *<sup>k</sup>* (*ε*)

). Now,

> ,

(from *x*-projections of

, 

velocity, as Eq. (48), ceases to be valid.

simpler static quantities, Eqs. (49, 50).

**7. Optical conductivity**

240 Superconductors – New Developments

and Im *G* ^

*i*', and *i*-*b*'.

spells as *d*

where *v*(

Here *v* ª *v*(,), *v*' ª *v*(',

( ,) = |*vk*(

but within the *i*-band, at

*dk* = 2(*hv*F)

**Figure 10.** Configuration of the poles *ξ<sup>j</sup>*

Fig. 10. Configuration of the poles

electronic pocket of the quasiparticle spectrum in Fig. 3).

velocities) and the most important radial

related to band-like states with positive

). For energies within the *b*-band,

1,2

 

and *h*-segments. The azimuthal

integrand in particular pole terms:

 

*dd*

)| and ˆ ˆ *G G* ,*<sup>k</sup>*

1,2 

2

*c*

*c*, < || < 

$$A\_{\alpha} \left( \varepsilon, \varepsilon' \right) = \pi \upsilon\_{\alpha} \upsilon'\_{\alpha} \frac{\tilde{\varepsilon} \tilde{\varepsilon} + \tilde{\xi} \tilde{\xi}' + \Delta^{2}}{\prod\_{\beta \neq a} \left( \tilde{\xi}\_{a} - \tilde{\xi}\_{\beta} \right)}. \tag{52}$$

Here *v<sup>α</sup>* ≡ *v*(*ε*, *ξα*), *v*'*<sup>α</sup>* ≡ *v*(*ε*', *ξα*), and the indices *α*, *β* run over all the poles of the two Green functions. As seen from Eqs. (23, 24) and Fig. 10, there can be two such poles of *G* ^ (*ξ*, *<sup>ε</sup>*) related to band-like states with positive *ε* and respective quasi-momentum values denoted as *ξ*1,2(*ε*). For energies within the *b*-band, *ε* > *εc*, they are symmetrical:

$$
\zeta\_{1,2}^{\varepsilon} \left( \varepsilon \right) \approx \pm \sqrt{\varepsilon^2 - \Lambda^2} \,, \tag{53}
$$

but within the *i*-band, at *εc*, – < |*ε*| < *εc*, +, their positions are asymmetrical:

$$\xi\_{1,2}\left(\varepsilon\right) \approx \frac{c\gamma^2 \mp 2\varepsilon\_0 \sqrt{\left(\varepsilon\_+ - \varepsilon\right)\left(\varepsilon - \varepsilon\_-\right)}}{2\left(\varepsilon - \varepsilon\_0\right)}.\tag{54}$$

Within the *i*-band, there is a narrow vicinity of *ε*<sup>0</sup> of ~ *c*<sup>0</sup> 1/3(*c*0/*c*) 3 *ε*<sup>0</sup> width where only the *ξ*1 pole by Eq. (54) is meaningful and the other contradicts the IRM criterion (so that there is no bandlike states with those formal *ξ*2 values in this energy range). Analogous poles of *G* ^ (*ξ*, *<sup>ε</sup>* ') at negative *ε*' are referred to as *ξ*3,4(*ε*') in what follows. Taking into account a non-zero Im *Dk*(*ε*) (for the *i*-band, it is due to the non-trivial terms in the group expansion, Eq. (16)), each *α*-th pole becomes a *δ*-like peak with an effective linewidth Γ*<sup>α</sup>* but this value turns to be essential only at calculation of static coefficients like Eqs. (49, 50).

Since four peaks in Eq. (47) are well separated, the *ξ*-integration is done considering them true *δ*-functions, then the particular terms in *σ*(*ω*, *T*) are given by the energy integrals:

$$\sigma\_{\nu}\left(o\nu,T\right) = 2e^{2} \int\_{c\_{\nu}}^{c\_{\nu}} d\varepsilon \frac{f\left(\varepsilon\right) - f\left(\varepsilon^{\prime}\right)}{o\nu} \sum\_{a=1}^{4} A\_{a\prime} \tag{55}$$

for *ν* = *b*-*b*', *i*-*b*', or *i*-*i*' and the limits *εν*, ± should assure both *ε* and *ε*' to be within the band-like ranges. Thus, in the *b*-*b*' term, the symmetry of the poles *ξ*1, 2(*ε*) and *ξ*1, 2(*ε*') by Eq. (55) and the symmetry of *b*- and *b*'-bands themselves defines their equal contributions, then using simplic‐ ity of the function *v*(*ε*, *ξ*) = *ξ*/*ε* and non-renormalized energy *ε*˜ → *ε* and momentum *ξ*˜ → *ξ*, integration between *εb-b'*, – = *εc* and *εb-b'*, + = ℏ*ω* – *εc* gives an analytic form *σ<sup>b</sup>*-*b'*(*ω*, *T*) = *σ<sup>b</sup>*-*b'*(*ω*, 0) – *σ<sup>b</sup>*-*b'*, *<sup>T</sup>*(*ω*). Here the zero-temperature limit value is:

$$\begin{split} \sigma\_{\mathbb{H}^{\circ}}(o,0) \approx \sigma\_{\mathbb{O}} \frac{2\alpha\_{c}}{\alpha^{2}} \Bigg[ \sqrt{4\alpha^{2} - \alpha\_{c}^{2}} \ln \left[ 2 \frac{o(2\alpha - \alpha\_{c}) + \sqrt{o(\left(o - \alpha\_{c}\right)\left(4\alpha^{2} - \alpha\_{c}^{2}\right)}}{\alpha\_{c}^{2}} - 1 \right] \\ + 2o \ln \left[ 2 \frac{o - \sqrt{o(\left(o - \alpha\_{c}\right)}}{\alpha\_{c}} - 1 \right] - 2\sqrt{o(\left(o - \alpha\_{c}\right)}} \right], \end{split} \tag{56}$$

with characteristic scale *σ*0 = *e*<sup>2</sup> /∆<sup>2</sup> and simple asymptotics:

$$\sigma\_{b\cdot b^{\cdot}}(o\nu,0) \approx \sigma\_{\psi} \begin{cases} \frac{2}{3} \left(\frac{o\nu}{o\nu\_{\c}} - 1\right)^{3/2}, & o - o\_{\circledast} \ll o\_{\epsilon\prime\prime}, \\\frac{32o\nu\_{\circledast}}{o\nu} \ln \frac{2o\nu}{o\nu\_{\circledast}}, & o \gg o\_{\epsilon\prime\prime}, \end{cases}$$

vs the threshold frequency *ωc* = 2*εc*/ℏ, reaching the maximum value ≈ 1.19*σ*0 at *ω* ≈ 2.12 *ω<sup>c</sup>* (Fig. 11). The (small) finite-temperature correction to Eq. (57) is:

$$\begin{split} \sigma\_{b \cdot \mathbf{\hat{n}}', \Gamma} (\boldsymbol{\alpha}) \approx & \sigma\_0 \frac{2 \boldsymbol{\alpha}\_c^2 \mathbf{e}^{-\beta \mathbf{h}}}{\beta \boldsymbol{\alpha} (\boldsymbol{\alpha} - \boldsymbol{\alpha}\_c) \sqrt{\Delta}} \left[ \frac{\sqrt{\hbar \boldsymbol{\alpha}}}{\Delta} \left[ 1 - \frac{\mathrm{F} \left( \sqrt{\beta \hbar (\boldsymbol{\alpha} - \boldsymbol{\alpha}\_c)} \right)}{\sqrt{\beta \hbar (\boldsymbol{\alpha} - \boldsymbol{\alpha}\_c)}} \right] \right] \\ & + \frac{\sqrt{2 \Delta}}{\hbar \boldsymbol{\alpha} - \Delta} \left[ \frac{\sqrt{\pi}}{2} \frac{\mathrm{erf} \left( \sqrt{\beta \hbar (\boldsymbol{\alpha} - \boldsymbol{\alpha}\_c)} \right)}{\sqrt{\beta \hbar (\boldsymbol{\alpha} - \boldsymbol{\alpha}\_c)}} - \mathrm{e}^{-\beta \hbar (\boldsymbol{\alpha} - \boldsymbol{\alpha}\_c)} \right], \end{split} \tag{57}$$

with the Dawson function *F*(*z*) = *π*e−*<sup>z</sup>* <sup>2</sup> erf(*iz*)/ (2*i*) and the error function erf(*z*) [64].

Since four peaks in Eq. (47) are well separated, the *ξ*-integration is done considering them true

e

for *ν* = *b*-*b*', *i*-*b*', or *i*-*i*' and the limits *εν*, ± should assure both *ε* and *ε*' to be within the band-like ranges. Thus, in the *b*-*b*' term, the symmetry of the poles *ξ*1, 2(*ε*) and *ξ*1, 2(*ε*') by Eq. (55) and the symmetry of *b*- and *b*'-bands themselves defines their equal contributions, then using simplic‐ ity of the function *v*(*ε*, *ξ*) = *ξ*/*ε* and non-renormalized energy *ε*˜ → *ε* and momentum *ξ*˜ → *ξ*, integration between *εb-b'*, – = *εc* and *εb-b'*, + = ℏ*ω* – *εc* gives an analytic form *σ<sup>b</sup>*-*b'*(*ω*, *T*) = *σ<sup>b</sup>*-*b'*(*ω*, 0)

<sup>4</sup> <sup>2</sup>

w


 a

, 2 , *f f T ed <sup>A</sup>* (55)

( ) ( )

 w

*c*

( )

*c*

b ww


h

=

*c c*

?

ww w

*c*

(56)

(57)

2 2

a

1

 e '

> ww w w w

w

*c*

î ë û

 w

*c*

é ù <sup>ü</sup> - - <sup>ï</sup> <sup>+</sup> ê ú -- - <sup>ý</sup>

*c*

<sup>ï</sup> ë û <sup>þ</sup>

ww

( ( ))

bww

*c*

*c*

h

bww

( ( )) ( )

bww

é ùü - <sup>D</sup> ê úï + - <sup>ý</sup> - D - <sup>ï</sup> ë ûþ

h

bww

( )

w w

*δ*-functions, then the particular terms in *σ*(*ω*, *T*) are given by the energy integrals:

( ) ( ) ( ) n

+

 e

( ) ( ) ( )( )

w ww w

and simple asymptotics:

w

*c*

<sup>ì</sup> æ ö <sup>ï</sup> ç ÷ - - <sup>ï</sup> » è ø <sup>í</sup>

3/2

<sup>2</sup> 1 , , <sup>3</sup> ,0

<sup>32</sup> <sup>2</sup> ln , ,

 w

vs the threshold frequency *ωc* = 2*εc*/ℏ, reaching the maximum value ≈ 1.19*σ*0 at *ω* ≈ 2.12 *ω<sup>c</sup>* (Fig.

w

*<sup>c</sup> <sup>c</sup>*

*c c*

<sup>h</sup> <sup>h</sup>

*F*

<sup>ì</sup> é ù - <sup>ï</sup> ê ú » - <sup>í</sup> - D <sup>ï</sup> <sup>D</sup> - <sup>î</sup> ë û

p

erf 2 e , <sup>2</sup>

h h

<sup>ì</sup> é ù -+ - - <sup>ï</sup> ê ú » - <sup>í</sup> -

*c cc <sup>c</sup>*

2 4 <sup>2</sup> ,0 4 ln 2 <sup>1</sup>

w w w

w

2 ln 2 1 2 ,

w

w

w

*c*

w

n

,

e

e

n

– *σ<sup>b</sup>*-*b'*, *<sup>T</sup>*(*ω*). Here the zero-temperature limit value is:

 ww

2 2 - ' 0 2 2

/∆<sup>2</sup>

( )

11). The (small) finite-temperature correction to Eq. (57) is:

w 2

( ) ( )

bw w w


 s

b

2 e <sup>1</sup>


w

*<sup>c</sup> b b*

ï ï î

sw

w

ï

w

*b b c*

 s

242 Superconductors – New Developments

with characteristic scale *σ*0 = *e*<sup>2</sup>

s

*bb T*

 ws


s w sw

Calculation of the *i*-*b*'-term is more complicated since asymmetry of the *i*-band poles *ξ*1, 2(*ε*) by Eq. (55) and their non-equivalence to the symmetric poles *ξ*3, 4(*ε*') of the *b*'-band analogous to Eq. (54). Also the generalized velocity function within the *i*-band range:

$$
\hbar \upsilon \left( \xi', \varepsilon \right) = \frac{c\gamma^2 - \xi \left( \varepsilon - \varepsilon\_0 \right)}{\varepsilon \left( \varepsilon - \varepsilon\_0 - c\gamma^2 \mid \varepsilon\_0 \right)} \,, \tag{58}
$$

and the energy integration limits: *ε*'*i*-*<sup>b</sup>*', – = *εc*, – and *ε*'*i*-*<sup>b</sup>*', + = min *εc*,+, ℏ*ω* −*ε<sup>c</sup>* are more complicated. Then *σ<sup>i</sup>*-*b*' (*ω*, *T*) follows from numerical integration in Eq. (56); as seen in Fig. 11, it has a lower threshold frequency *ω<sup>c</sup>*' = *εc* + *εc*,– than the *b*-*b*'-term. Above this threshold, it grows linearly as ~ (*ω*/*ω<sup>c</sup>*' – 1)*c*5/2*c*<sup>0</sup> –5/3*σ*0 and, for "safe" impurity concentrations *c* ≪ *c*1 ~ *c*<sup>0</sup> 2/3, becomes fully dominated by the *b*-*b*' term, Eq. (57) above its threshold *ωc*. Finally, the *i*-*i*'-term from a similar numerical routine on Eq. (56) within integration limits *ε<sup>i</sup>*-*<sup>i</sup>*',– = *εc*,– and *ε<sup>i</sup>*-*<sup>i</sup>*',+ = min[*ε<sup>c</sup>*,+, ℏ*ω* – *εc*, –], using Eq. (55) for the poles *ξ*1, 2(*ε*) and *ξ*3, 4(*ε*') and Eq. (59) for generalized velocities. The resulting *σ<sup>i</sup>*-*i*' (*ω*, *T*) occupies a narrow frequency band from *ω<sup>i</sup>*-*i*' = 2*ε<sup>c</sup>*,–/ℏ to *ω<sup>i</sup>*-*<sup>i</sup>*', + = 2*ε<sup>c</sup>*,+/ℏ (Fig. 11) with asymptotics near these thresholds in the zero-temperature limit:

$$
\sigma\_{i,j}\left(\alpha,0\right) \approx \sigma\_0 \frac{16c^{7/2}\mathcal{V}}{3\sqrt{2}\mathcal{E}\_{-}^{\mathcal{F}}} \left(\frac{\alpha - \alpha\_{-}}{\alpha\_{-}}\right)^{3/2},\tag{59}
$$

at 0 < *ω* – *ω*– ≪ *ω*–, and alike for 0 < *ω*+ – *ω* ≪ *ω*+ with only changes: : *ξ*– → *ξ*+, *ω*– → *ω*+.

Extrapolation of these asymptotics to the center of impurity band gives an estimate for the maximum of *i*-*i*' term: *σi*-*i*', *max* ~ *c*<sup>5</sup> *c*0 –10/3(*ξ*+/*ξ*–) 7/2*σ*0. It shows that the narrow *i*-*i*' peak of optical conductivity around *ω* ≈ 2*ε*0/ℏ, unlike the "combined" *i*-*b*' term, can become as intense as the "main" *b*-*b*' intensity, Eq. (58), if the small factor ~ (*c*/*c*1) 5 be overweighted by the next factor (*ξ* +/*ξ*–) 7/2. It is only possible for *weak* enough impurity perturbation: *v* ≪ 1. Then the ratio *ξ*+/*ξ*– ≈ (2/*v*) 2 ≺ 1 can really overweight the concentration factor if *c* reaches ~ *c*1(*v*/2)7/5 ≪ *c*1, that is quite realistic within the "safety" range *c* ≪ 1. The overall picture of optical conductivity for an example of weakly coupled, *v* = 0.25, impurities at high enough *c* = 4*c*0 is shown in Fig. 11. The effect of "giant" optical conductivity by the in-gap impurity excitations could be compared with the known Rashba enhancement of optical luminescence by impurity levels near the edge of excitonic band [64] or with huge impurity spin resonances in magnetic crystals [29], but here it appears in a two-particle process instead of the above mentioned single-particle ones.

To emphasize, the considered impurity features in optical conductivity cannot be simply treated as optical transitions between localized impurity states (or between these and main complicated. Then

linearly as ~ (

 

Fig. 11. General picture of the optical conductivity showing three types of **Figure 11.** General picture of the optical conductivity showing three types of contributions.

with the Dawson function *F*(*z*) = <sup>2</sup>

*v*

 

> *c*' = *c* +

fully dominated by the *b*-*b*' term, Eq. (57) above its threshold

similar numerical routine on Eq. (56) within integration limits

 


 

and the energy integration limits:

has a lower threshold frequency

/*<sup>c</sup>*' 1)*c* 5/2 *c*0 5/3

*i*-*b*' (

*c*,], using Eq. (55) for the poles

*bb T*

1,2( 2

 ) by Eq. (55) and their non-equivalence to the symmetric poles

2 e <sup>1</sup>

erf 2 e , <sup>2</sup>

e erf / 2 *<sup>z</sup>*

analogous to Eq. (54). Also the generalized velocity function within the *i*-band range:

 

*c*

 

> 1,2() and 3,4(

'*i*-*b*', =

2

*c*, and

*c*

 

, , /

*F*

*<sup>c</sup> <sup>c</sup>*

*c c*

 

Calculation of the *i*-*b*'-term is more complicated since asymmetry of the *i*-band poles

0 2 0 0

> '

<sup>0</sup> and, for "safe" impurity concentrations *c c*1 ~ *c*<sup>0</sup>

 

*c*

*c*

 

 

*iz i* and the error function erf(*z*) 64.

(59)

,*T*) follows from numerical integration in Eq. (56); as seen in Fig. 11, it

 

3,4(

*<sup>i</sup>*-*b*', = min , , *c c* 

*<sup>c</sup>* than the *b*-*b*'-term. Above this threshold, it grows

*<sup>i</sup>*-*i*' = *c* and

 are more

*<sup>c</sup>*. Finally, the *i*-*i*'-term from a

') and Eq. (59) for generalized velocities.

*c*

(58)

') of the *b*'-band

2/3, becomes

*<sup>c</sup>*,

*<sup>i</sup>*-*i*' = min[

bands) since localized states can not contribute to currents. Such effects only appear at high enough impurity concentrations, *c* <sup>↔</sup> *c*0, when the impurity banding takes place. contributions.

#### **8. Concluding remarks**

Resuming, the GF analysis of quasiparticle spectra in SC iron pnictides with impurities of simplest (local and non-magnetic) perturbation type permits to describe formation of impurity localized levels within SC gap and, with growing impurity concentration, their evolution to specific bands of extended quasiparticle states, approximately described by quasimomentum but mainly supported by the impurity centers. Explicit dispersion laws and densities of states are obtained for the modified main bands and impurity bands. Further specification of the nature of all the states in different energy ranges within the SC gap is obtained through analysis of different types of GEs for self-energy matrix, revealing a complex oscillatory structure of indirect interactions between impurity centers and, after their proper summation, resulting in criteria for crossovers between localized and extended states. The found spectral characteris‐ tics are applied for prediction of several observable impurity effects.

Besides the thermodynamical effects, expected to appear at all impurity concentrations, that is either due to localized or band-like impurity states, a special interest is seen in the impurity effects on electronic transport in such systems, only affected by the impurity band-like states. It is shown that the latter effects can be very strongly pronounced, either for high-frequency transport and for static transport processes. In the first case, the strongest impurity effect is expected in a narrow peak of optical conductance near the edge of conductance band in nonperturbed crystal, resembling the known resonance enhancement of impurity absorption (or emission) near the edge of quasiparticle band in normal systems. The static transport coeffi‐ cients at overcritical impurity concentrations are also expected to be strongly enhanced compared to those in a non-perturbed system, including the thermoelectric Peltier and Seebeck coefficients.

The proposed treatment can be adapted for more involved impurity perturbations in SC iron pnictides, including magnetic and non-local perturbations, and for more realistic multiorbital structures of the initial iron pnictide system. Despite some quantitative modifications of the results, their main qualitative features as possibility for new narrow in-gap quasiparticle bands and related sharp resonant peaks in transport coefficients should be still present. The experi‐ mental verifications of such predictions would be of evident interest, also for important practical applications, e.g., in narrow-band microwave devices or advanced low-temperature sensors, though this would impose rather hard requirements on quality and composition of the samples, to be extremely pure aside the extremely low (by common standards) and well controlled contents of specially chosen and uniformly distributed impurity centers. This can be compared to the requirements on doped semiconductor devices and hopefully should not be a real problem for modern lab technologies.

#### **Acknowledgements**

bands) since localized states can not contribute to currents. Such effects only appear at high

Fig. 11. General picture of the optical conductivity showing three types of

Resuming, the GF analysis of quasiparticle spectra in SC iron pnictides with impurities of simplest (local and non-magnetic) perturbation type permits to describe formation of impurity localized levels within SC gap and, with growing impurity concentration, their evolution to specific bands of extended quasiparticle states, approximately described by quasimomentum but mainly supported by the impurity centers. Explicit dispersion laws and densities of states are obtained for the modified main bands and impurity bands. Further specification of the nature of all the states in different energy ranges within the SC gap is obtained through analysis of different types of GEs for self-energy matrix, revealing a complex oscillatory structure of indirect interactions between impurity centers and, after their proper summation, resulting in criteria for crossovers between localized and extended states. The found spectral characteris‐

Besides the thermodynamical effects, expected to appear at all impurity concentrations, that is either due to localized or band-like impurity states, a special interest is seen in the impurity effects on electronic transport in such systems, only affected by the impurity band-like states. It is shown that the latter effects can be very strongly pronounced, either for high-frequency transport and for static transport processes. In the first case, the strongest impurity effect is expected in a narrow peak of optical conductance near the edge of conductance band in nonperturbed crystal, resembling the known resonance enhancement of impurity absorption (or emission) near the edge of quasiparticle band in normal systems. The static transport coeffi‐ cients at overcritical impurity concentrations are also expected to be strongly enhanced compared to those in a non-perturbed system, including the thermoelectric Peltier and Seebeck

enough impurity concentrations, *c* <sup>↔</sup> *c*0, when the impurity banding takes place.

**Figure 11.** General picture of the optical conductivity showing three types of contributions.

with the Dawson function *F*(*z*) = <sup>2</sup>

*v*

 

> *c*' = *c* +

fully dominated by the *b*-*b*' term, Eq. (57) above its threshold

similar numerical routine on Eq. (56) within integration limits

 


 

and the energy integration limits:

has a lower threshold frequency

/*<sup>c</sup>*' 1)*c* 5/2 *c*0 5/3

244 Superconductors – New Developments

*i*-*b*' (

*c*,], using Eq. (55) for the poles

complicated. Then

linearly as ~ (

  *bb T*

1,2( 2

 ) by Eq. (55) and their non-equivalence to the symmetric poles

2 e <sup>1</sup>

erf 2 e , <sup>2</sup>

e erf / 2 *<sup>z</sup>*

analogous to Eq. (54). Also the generalized velocity function within the *i*-band range:

 

*c*

 

> 1,2() and 3,4(

'*i*-*b*', =

2

*c*, and

*c*

 

, , /

*F*

*<sup>c</sup> <sup>c</sup>*

*c c*

 

Calculation of the *i*-*b*'-term is more complicated since asymmetry of the *i*-band poles

0 2 0 0

> '

<sup>0</sup> and, for "safe" impurity concentrations *c c*1 ~ *c*<sup>0</sup>

 

*c*

*c*

 

 

*iz i* and the error function erf(*z*) 64.

(59)

,*T*) follows from numerical integration in Eq. (56); as seen in Fig. 11, it

 

3,4(

*<sup>i</sup>*-*b*', = min , , *c c* 

*<sup>c</sup>* than the *b*-*b*'-term. Above this threshold, it grows

*<sup>i</sup>*-*i*' = *c* and

 are more

*<sup>c</sup>*. Finally, the *i*-*i*'-term from a

') and Eq. (59) for generalized velocities.

*c*

(58)

') of the *b*'-band

2/3, becomes

*<sup>c</sup>*,

*<sup>i</sup>*-*i*' = min[

tics are applied for prediction of several observable impurity effects.

**8. Concluding remarks**

contributions.

coefficients.

Y.G.P. and M.C.S. acknowledge the support of this work through the Portuguese FCT project PTDC/FIS/101126/2008. V.M.L. is grateful to the Special Program of Fundumental Research of NAS of Ukraine.

Parts of the chapter are reproduced from the authors' previous publication [53]

#### **Author details**

Yuriy G. Pogorelov1\*, Mario C. Santos2 and Vadim M. Loktev3

\*Address all correspondence to: ypogorel@fc.up.pt

1 IFIMUP-IN, Departamento de Física, Universidade do Porto, Porto, Portugal

2 Departamento de Física, Universidade de Coimbra, R. Larga, Coimbra, Portugal

3 Bogolyubov Institute for Theoretical Physics, NAN of Ukraine, Kiev, Ukraine

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#### **Theory of Flux Cutting for Type-II Superconducting Plates at Critical State Theory of Flux Cutting for Type-II Superconducting Plates at Critical State**

Carolina Romero-Salazar and Omar Augusto Hernández-Flores Carolina Romero-Salazar\* and Omar Augusto Hernández-Flores

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/59512

## **1. Introduction**

10.5772/59512

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[58] Mazin II, Schmalian J. Pairing symmetry and pairing state in ferropnictides: Theoreti‐ cal overview. Physica C 2009;469:614-627. DOI:10.1016/j.physc.2009.03.019

[59] DeWeert MJ. Proximity-effect bilayers with magnetic impurities: The Abrikosov-Gor'kov limit. Phys. Rev. B 1988;38:732-734. DOI: http://dx.doi.org/10.1103/Phys‐

[60] Srivastava RVA, Teizer W. Analytical density of states in the Abrikosov-Gorkov theory. Solid State Commun. 2008;145:512-513. DOI: 10.1016/j.ssc.2007.11.030

[61] Note that this static limit of Eq. (47) only defines the conductivity by normal quasi‐ particles, seen e.g. in normal resistivity by the magnetic flux flow in the mixed state, but otherwise short circuited by the infinite static conductivity due to supercurrents.

[62] Dolgov OV, Efremov DV, Korshunov MM, Charnukha A, Boris AV, Golubov AA. Multiband Description of Optical Conductivity in Ferropnictide Superconductors. J.

[63] Abramowitz MVL, Stegun IA, editors. Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables. Natl. Bureau of Standards;1964. 1046 p.

[64] Broude, Prikhot'ko AF, Rashba EI. Some problems of crystal luminescence. Sov. Phys. Uspekhi 1959;2:38-49. DOI: http://dx.doi.org/10.1070/

Supercond. Nov. Magn. 2013;26:2637-2640. DOI: 10.1007/s10948-013-2150-3

ISBN 0-486-61272-4 DOI: 10.2307/2008636

PU1959v002n01ABEH003107

World Scientific;2007. p. 443-494. ISBN 978-981-270-572-3. cap. 17

1957 ;12 :570-586. DOI: http://dx.doi.org/10.1143/JPSJ.12.570

10.1063/1.1401186

250 Superconductors – New Developments

RevB.84.144510

DOI: 10.1007/3-540-28841-4

RevB.38.732

With the discovery in 1986 of high critical temperature superconductors *Tc* ≥ 77*K* –which belong to the type-II classification– efforts have been made to recognize which mechanism rules its current carrying capacity in order to expand knowledge of the vortex state and, moreover, devise new and better technological applications. Critical-state phenomenological models for such materials have been a feasible alternative for the theoretical study of the magnetic properties of high- or low-*Tc* type-II superconductors. Here we present a brief revision of macroscopic critical-state models; following a chronological order, we will begin with the Bean model, moving on with the generalized double-critical state model, the two-velocity hydrodynamic model, and finalizing with the Elliptic Flux-Line Cutting Critical-State Model (ECSM). It will be described further the main features of type-II superconductors, the physical meaning of the critical state and the flux-line cutting phenomenon.

#### **1.1. Type-II superconductor critical state**

In 1911 Kammerlingh Onnes discovered the superconductivity of mercury at very low temperature. Nowadays, the characteristics of superconductors are well established: their electric resistance abruptly drops to zero as temperature decreases through a critical temperature value designated as *Tc*. They show the *Meissner-Ochsenfeld effect*, that is, they completely expel a weak magnetic field as temperature decreases through the transition point. Depending on how this diamagnetic phenomenon is destroyed, superconductors can be classified as type I or II. Type-I superconductors are perfect diamagnets below a critical field *Hc*. Because their coherence length *ξ* exceeds the penetration length *λ*, it is energetically unfavorable for borders to be formed between the normal and superconductive phases. However, when a type-II superconductor is subjected to a magnetic field **H***a*, free energy can diminish, thus generating normal matter domains that contain trapped flux, with low-energy

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. © 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

©2012 Author(s), licensee InTech. This is an open access chapter distributed under the terms of the Creative

borders created between the normal core and the superconductive surroundings. When the applied magnetic field exceeds the lower critical field *Hc*1, the magnetic flux penetrates in quantized units Φ0, forming cylindrical domains called **vortices**. As **H***<sup>a</sup>* increases, vortices will overlap increasing the interior field until the material gently enters the normal state, once **H***<sup>a</sup>* has reached the upper critical field *Hc*2. Between the fields *Hc*<sup>1</sup> and *Hc*<sup>2</sup> the superconductor state coexists with the magnetic state in a **mixed state** or **vortex state**.

Another characteristic of superconductors is the presence of a gap, just below the Fermi energy, the energy of conduction electrons. BCS superconductivity theory demonstrated that electrons in the vicinity of the Fermi level are grouped in the so-called Cooper pairs. In addition, the junction of two superconductors –separated by a thin insulating layer– shows the *DC Josephson effect*, in which the superconductor current tunneling is caused by the tunneling of Cooper pairs. This effect demonstrated that the superconductor state is a coherent state, which is associated to a macroscopic uniform-phase wave function; this function corresponds to the order parameter *κ* = *λ*/*ξ* in the Ginzburg-Landau theory. Finally, they also show the *AC Josephson effect* that describes the relation between the time variation of the macroscopic wave function with the voltage produced across the junction. This voltage arises from the quantized magnetic flux movement, and is identical to the macroscopic voltage observed in type-II superconductors in flux flow state.

Indeed, it is well established that type-II superconductors posses a stationary vortex spatial arrangement only if the total force over each vortex is null. If an electric current is applied with **J** density, vortices move at a velocity **v** with a direction determined by the Hall angle. If both the Magnus force and the Hall effect on the material are neglected, equilibrium between the Lorentz force **<sup>F</sup>***<sup>L</sup>* = **<sup>J</sup>** × **<sup>B</sup>** and the pinning force **<sup>F</sup>***<sup>p</sup>* will exist:

$$(\mathbf{J} \times \mathbf{B}) - \mathbf{F}\_p = \mathbf{0}.\tag{1}$$

In addition to being able to describe the vortex dynamics under the transport current influence, the equation (1) can be used for the time-variable external magnetic case, in absence of transport currents.

Indeed, **F***p* opposes the magnetic flux velocity **v** due to the local depression of the Gibbs free energy of each vortex. This potential well may be due to inhomogeneities, defects, or material grains. Therefore, magnetic flux movement will occur if the Lorentz force exceed the pinning force. Any electromotive force, even small, causes the vortices to move further into the material, inducing a local current. Initially, this superconductive current flows in regions close to the superconductor surface because pinning centers near the sample surface can catch the vortices in such a way that, in the interior, the Meissner state is preserved, thus the sample is partially penetrated. For higher external magnetic field values, vortices will completely penetrate the material.

#### **1.2. Bean model for a type-II superconductor in critical state**

Half a century ago, Charles Bean approached — with great physics intuition and from a macroscopic point of view — the study of the magnetic properties of superconductors made with impure metals or alloys. Bean modeled the spatial distribution of the magnetic flux, for partial and totally penetrated states, a couple of years before having experimental evidence of the mixed or vortex state predicted by A. A. Abrikosov.

2 Superconductors

borders created between the normal core and the superconductive surroundings. When the applied magnetic field exceeds the lower critical field *Hc*1, the magnetic flux penetrates in quantized units Φ0, forming cylindrical domains called **vortices**. As **H***<sup>a</sup>* increases, vortices will overlap increasing the interior field until the material gently enters the normal state, once **H***<sup>a</sup>* has reached the upper critical field *Hc*2. Between the fields *Hc*<sup>1</sup> and *Hc*<sup>2</sup> the superconductor state coexists with the magnetic state in a **mixed state** or **vortex state**.

Another characteristic of superconductors is the presence of a gap, just below the Fermi energy, the energy of conduction electrons. BCS superconductivity theory demonstrated that electrons in the vicinity of the Fermi level are grouped in the so-called Cooper pairs. In addition, the junction of two superconductors –separated by a thin insulating layer– shows the *DC Josephson effect*, in which the superconductor current tunneling is caused by the tunneling of Cooper pairs. This effect demonstrated that the superconductor state is a coherent state, which is associated to a macroscopic uniform-phase wave function; this function corresponds to the order parameter *κ* = *λ*/*ξ* in the Ginzburg-Landau theory. Finally, they also show the *AC Josephson effect* that describes the relation between the time variation of the macroscopic wave function with the voltage produced across the junction. This voltage arises from the quantized magnetic flux movement, and is identical to the macroscopic

Indeed, it is well established that type-II superconductors posses a stationary vortex spatial arrangement only if the total force over each vortex is null. If an electric current is applied with **J** density, vortices move at a velocity **v** with a direction determined by the Hall angle. If both the Magnus force and the Hall effect on the material are neglected, equilibrium between

In addition to being able to describe the vortex dynamics under the transport current influence, the equation (1) can be used for the time-variable external magnetic case, in

Indeed, **F***p* opposes the magnetic flux velocity **v** due to the local depression of the Gibbs free energy of each vortex. This potential well may be due to inhomogeneities, defects, or material grains. Therefore, magnetic flux movement will occur if the Lorentz force exceed the pinning force. Any electromotive force, even small, causes the vortices to move further into the material, inducing a local current. Initially, this superconductive current flows in regions close to the superconductor surface because pinning centers near the sample surface can catch the vortices in such a way that, in the interior, the Meissner state is preserved, thus the sample is partially penetrated. For higher external magnetic field values, vortices will

Half a century ago, Charles Bean approached — with great physics intuition and from a macroscopic point of view — the study of the magnetic properties of superconductors made with impure metals or alloys. Bean modeled the spatial distribution of the magnetic flux, for

(**J** × **B**) − **F***<sup>p</sup>* = 0. (1)

voltage observed in type-II superconductors in flux flow state.

the Lorentz force **<sup>F</sup>***<sup>L</sup>* = **<sup>J</sup>** × **<sup>B</sup>** and the pinning force **<sup>F</sup>***<sup>p</sup>* will exist:

**1.2. Bean model for a type-II superconductor in critical state**

absence of transport currents.

completely penetrate the material.

He relied on *Mendelssohn sponge model* to describe the magnetic behaviour of such superconductors, supposing they possessed a filamental structure capable of maintaining a maximum macroscopic current *Jc*, without energy dissipation in form of Joule heating, that he called it *critical current density*. Due to the *Jc* dependence on the magnetic field, he considered that such currents were extended into the material, preserving the magnitude *Jc*.

Bean argued that the macroscopic current is a consequence of the magnetic induction gradient penetrating the material, governed by Ampère's law ∇ × **<sup>B</sup>** = *<sup>µ</sup>*0**J**. He also argued that the current originates as the Lorentz force drives the magnetic flux into the interior of the material. Therefore, he considered that any local region where an electric field (related to heat dissipation) is perceived during the process of magnetization, it would originated a critical current density *Jc* which flows in the electric field direction and it keep flowing even if it the electric field was null [6–9]. He synthesized these ideas in the material law:

$$J = J\_{\mathbb{C}}(\mathcal{B}) \operatorname{sign}(E),\tag{2}$$

which is valid for slabs or infinite cylinders subjected to a magnetic field parallel to the superconductor's surface (this is the so-called *parallel geometry*).

To exemplify the Bean model, Figure (1) shows the profile evolution of magnetic induction for a PbBi plate at mixed-state, as it increased (blue curves) or decreased (red curves) the external magnetic field magnitude **H***<sup>a</sup>* parallel to its surface. The hysteresis cycle of the average static magnetization is shown when the **H***<sup>a</sup>* varies a full cycle, from −0.3T to 0.3T. The superconducting plate has a 8mm thickness and a penetration field *µ*0*Hp* = 0.1015T. We have considered the reliance on *B* of the critical current density *Jc*(*B*).

Subsequently, Bean studied energy dissipation in materials subjected to a magnetic field that rotates in the specimen's plane. He extended his arguments assuming the current density **J** and the electric field **E** vectors would be parallel to each other [9]. For this case, the material equation can be written as follows:

$$\mathbf{J} = f\_{\mathbf{c}}(\mathcal{B}) \frac{\mathbf{E}}{E(I)},\tag{3}$$

where it is necessary to model the *E*(*J*) function form. The Bean model, corresponding to the material equation (2) or (3), together with the Ampère's law, have been used to calculate magnetic induction profiles, hysteresis of magnetization cycles, and the energy dissipation of several type-II superconductor materials. In the search for new superconducting alloys that would produce more intense magnetic fields, or a greater current conduction capacity, these materials were simultaneously subjected to a magnetic field and a transport current parallel to each other. It was desirable that the electric current density **J** and the magnetic induction **B** established a force-free configuration, that is, a zero Lorentz force **<sup>F</sup>***<sup>L</sup>* = **<sup>J</sup>** × **<sup>B</sup>** = 0. However, experimental evidence showed that even if **F***<sup>L</sup>* could be considered null, a significant voltage

**Figure 1.** Theoretical curves of a superconductor plate, with thickness *d* = 8mm and penetration field *µ*0*Hp* = *Bp* = 0.1015T, obtained with the Bean critical-state model and considering *Jc* = *Jc* (*B*). (Left) Evolution of magnetic induction *B* as it increases (continuous blue lines) and decreases (red discontinuous lines) the magnitude of the external magnetic field **H***<sup>a</sup>* . When *µ*0*Ha* = *µ*0*Hp*, *B* at the center of the plate is null. (Right) Average static magnetization cycle *µ*0�*M*� against the applied magnetic field *µ*0*Ha* . Since this material is an irreversible type-II superconductor, *µ*0�*M*� describes a hysteresis.

would arise from the material. Therefore, if the vortex velocity is equal to zero in a force-free configuration, what is going on in this type of configuration? This question could not be answered using the Bean model, so it was considered that another phenomenon might be occurring. The answer to this query is the so-called *flux-line cutting* or *flux crossing*, which will be discussed in the next section.

#### **1.3. Flux-Line cutting**

D.G. Walmsley [10] measured the magnetization and the axial resistance of a type-II superconductor –in mixed state and with cylindrical geometry– by subjecting it simultaneously to a magnetic axial field and a transport current, parallel to each other. The objective was to prove under what the Lorentz force density could be null. He found that, at low currents, the potential difference between the extremes of the sample was negligible (∼ 10*µ*V). However, when the superconducting sample conduced a sufficiently high current, it measured a voltage (or a longitudinal electric field), as well as a paramagnetic moment, that is, a positive average magnetization. He then suggested that the force-free structure could not be established on the material's surface, which originated the flux-flow and, consequently, a voltage in the rest of the material.

He intuited that the measurement of a paramagnetic moment suggested a helicoidal vortex distribution. Nonetheless, the voltage produced by the flux flow would imply a permanent increase in the longitudinal magnetic field. For this reason, he supposed the existence of a non-stationary process in which the magnetic flux lines would continually divide each other, only to reconnect afterward.

As a solution for the flux-flow movement contradiction in a force-free configuration, Clem was the first to suggest that the helicoidal instabilities were precisely the precursors of vortex cutting or crossing; that is, considering the elastic properties of flux lines, he proposed that they could stay fixed but they would be able to bend, calling this phenomenon *flux-line cutting*.

Even though Josephson had already established that cutting or crossing of vortices could not occur because they were energetically too expensive, theoretical calculations done by Brandt, Clem, and Walmsley — using the London and Ginzburg-Landau theories — proved that the threshold of cutting of a pair of rigid flux lines was possible since the characteristics energies of a type-II superconductor.

4 Superconductors

0 0.2 0.4 0.6 0.8 1

PbBi


PbBi

70Ha(T)


70

**Figure 1.** Theoretical curves of a superconductor plate, with thickness *d* = 8mm and penetration field *µ*0*Hp* = *Bp* = 0.1015T, obtained with the Bean critical-state model and considering *Jc* = *Jc* (*B*). (Left) Evolution of magnetic induction *B* as it increases (continuous blue lines) and decreases (red discontinuous lines) the magnitude of the external magnetic field **H***<sup>a</sup>* . When *µ*0*Ha* = *µ*0*Hp*, *B* at the center of the plate is null. (Right) Average static magnetization cycle *µ*0�*M*� against the applied magnetic field

would arise from the material. Therefore, if the vortex velocity is equal to zero in a force-free configuration, what is going on in this type of configuration? This question could not be answered using the Bean model, so it was considered that another phenomenon might be occurring. The answer to this query is the so-called *flux-line cutting* or *flux crossing*, which

D.G. Walmsley [10] measured the magnetization and the axial resistance of a type-II superconductor –in mixed state and with cylindrical geometry– by subjecting it simultaneously to a magnetic axial field and a transport current, parallel to each other. The objective was to prove under what the Lorentz force density could be null. He found that, at low currents, the potential difference between the extremes of the sample was negligible (∼ 10*µ*V). However, when the superconducting sample conduced a sufficiently high current, it measured a voltage (or a longitudinal electric field), as well as a paramagnetic moment, that is, a positive average magnetization. He then suggested that the force-free structure could not be established on the material's surface, which originated the flux-flow and, consequently, a

He intuited that the measurement of a paramagnetic moment suggested a helicoidal vortex distribution. Nonetheless, the voltage produced by the flux flow would imply a permanent increase in the longitudinal magnetic field. For this reason, he supposed the existence of a non-stationary process in which the magnetic flux lines would continually divide each other,

As a solution for the flux-flow movement contradiction in a force-free configuration, Clem was the first to suggest that the helicoidal instabilities were precisely the precursors of vortex cutting or crossing; that is, considering the elastic properties of flux lines, he proposed that they could stay fixed but they would be able to bend, calling this phenomenon *flux-line*

<M> (T)

x/d

*µ*0*Ha* . Since this material is an irreversible type-II superconductor, *µ*0�*M*� describes a hysteresis.

0

will be discussed in the next section.

voltage in the rest of the material.

only to reconnect afterward.

*cutting*.

**1.3. Flux-Line cutting**

0.05

7

H0

a(T)

0.1

0.15

0.2

Subsequently, Brandt and Sudbo extended these results for the case of a pair of twisted flux lines, they considered the tension and interaction between each flux-line or vortex. Given that cutting is energetically plausible, they found that flux-line cutting is an effective disentanglement mechanism of flux lines; and that the cutting energy barrier, in the case of twisted flux lines, is lower for the rigid two-flux-lines case [11, 12]. Several groups, for example, M.A.R. LeBlanc *et. al* [13–15], have done experiments in the last decades that have shown flux-cutting presence in materials with low or high *κ*, and low or high *Tc*. In the Clem *et. al* [16–18] and Brandt [19] papers, we can see flux-line cutting diagrams for the case of rigid-vortices arrangement. Furthermore, it includes diagrams for the first theoretical formulations for this often-studied and not completely understood phenomenon.

More recent theoretical studies have studied the scattering dynamics of vortices, and the resulting topology after a collision between two flux lines generated by an applied current. Using the time-dependent Ginzburg-Landau equations, numerical results yielded two generic collision types dependent on the initial angle: one local collision that induces changes in topology through recombination, and a double collision that can occur due to geometrical restrictions. The second case leads to a vortex-crossing type configuration, that is, it seems as if two vortices, while interacting, would cut themselves and join again. This can be seen in the simulations shown in paper [20]. Experiments have been proposed using a magnetic force microscope to monitor vortex-line dynamics and prove if these cut through each other when they are in a liquid-vortex phase [21].

In 2008, A. Palau *et al.* reported results with superconductive heterostructures subjected to an external magnetic field at a *θ* angle respect to the sample's normal. For this, they designed a device made out of a thin film of low-pinning amorphous material (*Mo*82*Si*18), sandwiched between two *Nb* films– a material characterized for strongly pinning the vortices.

They measured the critical current density *Jc* obtained as a function of *θ*, the applied field *µ*0*Ha*, and temperature *T*. Once obtained, they calculated the balance force between the Lorentz force, the pinning force, and a so-called breaking force. They found that the latter was necessary in order to consider vortex deformation and destabilization. Results showed that the breaking force is independent of *B*, and that the following cross joining neither limit vortex movement nor increase *Jc*. Even if a flux-line segment is strongly pinned to the area where *Nb* material is found, the cut induces other vortex segments to be liberated, thus reducing *Jc* [22].

Furthermore, Campbell's revision paper can be consulted to know the state of the art about experiments and critical state theories for flux-cutting in superconductors [23]. He included the last proposal of Clem to determine the electric field direction, for the flux transport regime in a type-II superconductor.

Here it is presented three critical state models created for the phenomenological study of type-II superconductors subject to magnetic fields that vary not only in magnitude, but also in direction. All models consider that flux pinning and flux-line cutting govern their answer. It is undeniable that both phenomena can occur in cases when a sample oscillates in presence of a static magnetic field, or when it is subjected to a DC magnetic field and a transversal sweeping magnetic field is superimposed.

#### **2. Other critical state models**

#### **2.1. Generalized Double Critical-State Model**

LeBlanc *et. al* proposed a model containing two critical-state equations based on their experimental observations on the magnetic response of a disc oscillating at low frequency, in presence of a magnetic field [24]:

$$
\frac{d\mathbf{B}}{d\mathbf{x}} = \pm \frac{F\_p(\mathbf{B})}{B}, \qquad \frac{d\mathbf{a}}{d\mathbf{x}} = \pm k(\mathbf{B}) \frac{d\mathbf{B}}{d\mathbf{x}}.\tag{4}
$$

This pair of equations is known as the Double Critical-State Model (DCSM). In their construction, the fact that the magnetic induction *B* and the orientation *α* of flux-lines varied spatially was considered. They assumed as well that gradients existed in critical states. In his model, *Fp*(*B*) is a parameter that characterizes pinning intensity; *k* is associated to the shearing coefficient of the flux lattice in the superconductive sample.

Clem and Perez-Gonzalez extended the DCSM based on the assumption that intersection and cross-joining of adjacent non-parallel vortices generate a electric field different to the electric field **E** = **B** × **v**. The latter field is associated to the flux flow with velocity **v**, for the case in which **J** is perpendicular to **B**. For this, they proposed a pair of constitutive laws of the form:

$$J\_{\perp} = J\_{\mathfrak{c},\perp} \operatorname{sign} E\_{\perp}$$

$$J\_{\parallel} = J\_{\mathfrak{c},\parallel} \operatorname{sign} E\_{\parallel \prime} \tag{5}$$

where parameters *Jc*,<sup>⊥</sup> and *Jc*,� correspond to the depinning and the flux-line cutting thresholds, respectively. They considered that the electric field **E** components obey independently the vertical laws:

$$E\_{\perp} = \begin{cases} \rho\_{\perp} [|f\_{\perp}| - f\_{\text{c,\perp}}] \text{sign} \,(f\_{\perp})\_{\prime} & |f\_{\perp}| > f\_{\text{c,\perp}} \\ 0, & 0 \le |f\_{\perp}| \le f\_{\text{c,\perp}} \end{cases} \tag{6}$$

$$E\_{\parallel} = \begin{cases} \rho\_{\parallel} [||f\_{\parallel}| - f\_{c,\parallel}] \text{sign} \,(f\_{\parallel}), & |f\_{\parallel}| > f\_{c,\parallel} \\ 0, & 0 \le |f\_{\parallel}| \le f\_{c,\parallel} \text{'} \end{cases} \tag{7}$$

here, *ρ*<sup>⊥</sup> and *ρ*� are the resistivities caused by flux transport and superconductor flux-line cutting, respectively. The group of equations (5)-(7) constitutes the Generalized Double Critical-State Model (GDCSM.) Clem and Perez-Gonzalez did numerical calculations considering as possible values for magnitude *Jc* all those defined within a rectangular region of *Jc*,� and *Jc*,<sup>⊥</sup> sides. The model reproduced successfully the experimental distributions of magnetic induction and magnetization, when the magnetic field oscillates at great amplitudes [25].

The GDCSM was also used to try to reproduce *Magnetization Collapse* and *Paramagnetism*, phenomena encountered when a type-II superconductor is subjected first to a DC magnetic field on which an oscillating low-frequency magnetic field is superposed, perpendicular to the former.

#### **2.2. Two-velocities Hydrodynamic Model (TVHM)**

This macroscopic model considers that electrodynamics of a type-II superconductor depends on the translation of vortex planes and the interaction between them. It establishes two vortex subsystems, assuming they posses no elastic properties and that the flux cutting consists of the disappearance of interacting vortices, creating new vortices on a plane with an orientation different to the previous one. Gibbs energy varies through small disturbances on the vortices' coordinates considering the following: 1) magnetic energy; 2) work done by pinning forces given the translation of the vortex network; and 3) the work done by the pinning forces to straighten a vortex after its crossing [26–28]. Thus, the model is conformed by a continuity equation for total vortex density *n*(*x*, *t*), and the average angular distribution *α*(*x*, *t*) of the vortex planes:

$$\frac{\partial n}{\partial t} = -\frac{\partial}{\partial \mathbf{x}} \left[ n \frac{V\_A + V\_B}{2} \right],\tag{8}$$

$$\frac{\partial(na)}{\partial t} = -\frac{1}{2} \frac{\partial[na(V\_A + V\_B)]}{\partial \mathbf{x}} - \frac{1}{4} \frac{\partial[na(V\_A - V\_B)]}{\partial \mathbf{x}},\tag{9}$$

where

6 Superconductors

the form:

**2. Other critical state models**

presence of a magnetic field [24]:

independently the vertical laws:

*E*<sup>⊥</sup> =

*E*� =

 

> 

**2.1. Generalized Double Critical-State Model**

*dB dx* <sup>=</sup> <sup>±</sup>

shearing coefficient of the flux lattice in the superconductive sample.

LeBlanc *et. al* proposed a model containing two critical-state equations based on their experimental observations on the magnetic response of a disc oscillating at low frequency, in

*<sup>B</sup>* , *<sup>d</sup><sup>α</sup>*

This pair of equations is known as the Double Critical-State Model (DCSM). In their construction, the fact that the magnetic induction *B* and the orientation *α* of flux-lines varied spatially was considered. They assumed as well that gradients existed in critical states. In his model, *Fp*(*B*) is a parameter that characterizes pinning intensity; *k* is associated to the

Clem and Perez-Gonzalez extended the DCSM based on the assumption that intersection and cross-joining of adjacent non-parallel vortices generate a electric field different to the electric field **E** = **B** × **v**. The latter field is associated to the flux flow with velocity **v**, for the case in which **J** is perpendicular to **B**. For this, they proposed a pair of constitutive laws of

*J*<sup>⊥</sup> = *Jc*,<sup>⊥</sup> sign *E*<sup>⊥</sup>

where parameters *Jc*,<sup>⊥</sup> and *Jc*,� correspond to the depinning and the flux-line cutting thresholds, respectively. They considered that the electric field **E** components obey

*<sup>ρ</sup>*⊥[|*J*⊥| − *Jc*,⊥] sign (*J*⊥), |*J*⊥| > *Jc*,<sup>⊥</sup>

0, <sup>0</sup> |*J*⊥| *Jc*,<sup>⊥</sup>

*<sup>ρ</sup>*�[|*J*�| − *Jc*,�] sign (*J*�), <sup>|</sup>*J*�<sup>|</sup> <sup>&</sup>gt; *Jc*,�

0, <sup>0</sup> <sup>|</sup>*J*�<sup>|</sup> *Jc*,�;

here, *ρ*<sup>⊥</sup> and *ρ*� are the resistivities caused by flux transport and superconductor flux-line cutting, respectively. The group of equations (5)-(7) constitutes the Generalized Double Critical-State Model (GDCSM.) Clem and Perez-Gonzalez did numerical calculations considering as possible values for magnitude *Jc* all those defined within a rectangular region of *Jc*,� and *Jc*,<sup>⊥</sup> sides. The model reproduced successfully the experimental distributions

*dx* <sup>=</sup> <sup>±</sup>*k*(*B*)

*dB*

*J*� = *Jc*,� sign *E*�, (5)

*dx* . (4)

(6)

(7)

*Fp*(*B*)

$$V\_A = V + \frac{U}{2}' \qquad V\_B = V - \frac{U}{2}' \tag{10}$$

correspond to the velocities of subsystems *A* and *B*, *V*(*x*, *t*) is the mean hydrodynamic velocity, and *U*(*x*, *t*) is the relative velocity. The TVHM requires additionally two equations obtained from force balance conditions in a superconductor, defined for the magnetic induction gradient and for angular distribution:

$$\begin{split} \frac{\partial B}{\partial \mathbf{x}} &= -\frac{\mu\_0 I\_{\mathcal{E}, \perp}}{2} [F(V\_A) + F(V\_B)]\_{\prime} \\ \Delta a B \frac{\partial \mathbf{a}}{\partial \mathbf{x}} &+ p \sqrt{\frac{n}{8}} \left[ B - \mu\_0 H\_{\mathcal{E}} \cos(\mathfrak{a} - \mathfrak{a}\_0) \right] \Delta a^2 \text{sign}(V\_A - V\_B)\_{\prime} \\ &= -\mu\_0 I\_{\mathcal{E}, \perp} [F(V\_A) - F(V\_B) + p \text{sign}(V\_A - V\_B)]\_{\prime} \end{split} \tag{12}$$

here *p* corresponds to the probability that flux-line cutting occurs. Finally, to resolve the equation system (8)-(12) for variables *VA*, *VB*, *B* = *n*Φ0, *α* and ∆*α*, it is required to introduce a phenomenological equation that relates ∆*α* to the mean orientation's spatial derivative of the form:

$$
\Delta \mathfrak{a} = -l \operatorname{sign}(V\_A - V\_B) \frac{\partial \mathfrak{a}}{\partial \mathfrak{x}'} \tag{13}
$$

where *l* is the vortex mean free path between two successive cuttings or crossings.

#### **3. Theoretical Description of the ECSM**

Now we introduce the characteristics of the Elliptic Critical-State Model (ECSM) used in this chapter.

#### **3.1. Geometrical aspects of a superconductor plate**

**Figure 2.** Diagram of a portion of infinite superconducting plate. It is shown, in an instant of time *t*, the orientation of the external magnetic field **H***<sup>a</sup>* , which is always parallel to plane *yz*.

The study system is a superconducting plate possessing an infinite surface parallel to a plane *yz* and a finite thickness 0 ≤ *x* ≤ *D*, as it is shown in Figure (2). The plate is subjected to a magnetic field **H***a* parallel to plane *yz* given by the expression:

$$\mathbf{H}\_{a} = H\_{ay}\mathbf{\hat{y}} + H\_{a2}\mathbf{\hat{z}} = H\_{a}(\sin\alpha\_{a}\mathbf{\hat{y}} + \cos\alpha\_{a}\mathbf{\hat{z}}),\tag{14}$$

where *α<sup>a</sup>* is an angle measured relative to *z* axis . This problem pertains to the *parallel geometry*. Demagnetization effects are not present, the current density **J**, electric field **E**, and magnetic induction **B** vectors are all coplanar with their components *y* and *z* depending only on variable *x* and time *t*. Given the applied magnetic field **H***a*, local magnetic induction in a superconducting sample is:

$$\mathbf{B} = B(\mathbf{x}, t)\mathbf{\hat{e}}\_{\parallel} = B(\sin a\mathbf{\hat{y}} + \cos a\mathbf{\hat{z}}).\tag{15}$$

The superconducting current circulates around planes *x* = constant. In region *x* ∈ [0, *D*/2) it circulates in the opposite direction from the one in region *x* ∈ (*D*/2, *D*].

Since we are studying flux-line cutting, ECSM postulates that the current density **J** and electric field **E** can possess components both parallel and perpendicular to **B**. The perpendicular component is associated to the flux-pinning effect, while parallel components are associated to flux-line cutting. Therefore, we wish to calculate how large is the force-free component parallel to B of the current density, and if the latter reaches its critical value *Jc*�.

It is convenient to work with a reference system that rotates (follows) with the (the) magnetic induction **B**. The current density and electric field components –parallel and perpendicular to **B**– are monitored:

$$\mathbf{J} = J\_{\parallel}\mathbf{\dot{e}}\_{\parallel} + J\_{\perp}\mathbf{\dot{e}}\_{\perp \prime} \qquad \mathbf{E} = E\_{\parallel}\mathbf{\dot{e}}\_{\parallel} + E\_{\perp}\mathbf{\dot{e}}\_{\perp \prime} \tag{16}$$

where the unit vector **ˆe**<sup>⊥</sup> is built as **ˆe**<sup>⊥</sup> = **ˆe**� × **ˆx** = cos *α* **ˆy** − sin *α***ˆz**.

8 Superconductors

the form:

chapter.

here *p* corresponds to the probability that flux-line cutting occurs. Finally, to resolve the equation system (8)-(12) for variables *VA*, *VB*, *B* = *n*Φ0, *α* and ∆*α*, it is required to introduce a phenomenological equation that relates ∆*α* to the mean orientation's spatial derivative of

<sup>∆</sup>*<sup>α</sup>* <sup>=</sup> <sup>−</sup>*<sup>l</sup>* sign(*VA* <sup>−</sup> *VB*) *∂α*

Now we introduce the characteristics of the Elliptic Critical-State Model (ECSM) used in this

**j**

**Figure 2.** Diagram of a portion of infinite superconducting plate. It is shown, in an instant of time *t*, the orientation of the

The study system is a superconducting plate possessing an infinite surface parallel to a plane *yz* and a finite thickness 0 ≤ *x* ≤ *D*, as it is shown in Figure (2). The plate is subjected to a

where *α<sup>a</sup>* is an angle measured relative to *z* axis . This problem pertains to the *parallel geometry*. Demagnetization effects are not present, the current density **J**, electric field **E**, and magnetic induction **B** vectors are all coplanar with their components *y* and *z* depending only on variable *x* and time *t*. Given the applied magnetic field **H***a*, local magnetic induction in a

where *l* is the vortex mean free path between two successive cuttings or crossings.

z

*D*

**3. Theoretical Description of the ECSM**

**3.1. Geometrical aspects of a superconductor plate**

x

magnetic field **H***a* parallel to plane *yz* given by the expression:

external magnetic field **H***<sup>a</sup>* , which is always parallel to plane *yz*.

superconducting sample is:

**H**

*∂x*

y

*x=D*

**H***<sup>a</sup>* = *Hay* **ˆy** + *Haz* **ˆz** = *Ha*(sin *α<sup>a</sup>* **ˆy** + cos *α<sup>a</sup>* **ˆz**), (14)

**j**

*x*=0

, (13)

#### **3.2. Physical considerations for a Type-II superconductor**

Type-II superconductor materials are found in vortex state and contain a dense distribution and random pinning centers. When applying a magnetic field **H***a*, Eq. (14), it is considered that the material is in a critic state (metastable state) when there is a balance between Lorentz force density **<sup>F</sup>***<sup>L</sup>* = **<sup>J</sup>***c*<sup>⊥</sup> × **<sup>B</sup>** and the average pinning force. If the perpendicular component of the current overcomes its critical value *Jc*⊥, flux transport is begun until another material metastable state is reached. Also, vortex avalanches can occur; however, we will not cover here this phenomenon [29]. In addition, given that a **J** component is parallel to **B**, magnetic flux distribution will depend on flux line cutting, only if *J*� exceeds its critical value *Jc*�. Finally, the magnitude of an applied magnetic field is considered to be much larger than the first critical field *Hc*1, that is, the Meissner currents on the material's surface are neglected.

We are interested in the macroscopic electrodynamics of a type-II superconductor; thus, we work with Maxwell equations, given that low-frequency magnetic fields are considered, displacement current is neglected. In order to establish the critical state, this model considers quasi-stationary electromagnetic fields. This is why first the Faraday law is used, and subsequently stationary solutions are sought. Having chosen a reference system that rotates with magnetic induction **B**, Ampère and Faraday laws are written as follows:

$$-\frac{\partial B}{\partial \mathbf{x}} = \mu\_0 \mathbf{J}\_{\perp} \qquad \mathcal{B} \frac{\partial \mathbf{a}}{\partial \mathbf{x}} = \mu\_0 \mathbf{J}\_{\parallel \prime} \tag{17}$$

$$\frac{\partial E\_{\perp}}{\partial \mathbf{x}} + E\_{\parallel} \frac{\partial \alpha}{\partial \mathbf{x}} = -\frac{\partial B}{\partial t}, \qquad E\_{\perp} \frac{\partial \alpha}{\partial \mathbf{x}} - \frac{\partial E\_{\parallel}}{\partial \mathbf{x}} = -B \frac{\partial \alpha}{\partial t}. \tag{18}$$

In absence of a demagnetization factor, and considering that the applied magnetic field **H***a* is much larger than *Hc*1, the constitutive relation between **B** and **H** is modeled with the linear approximation **B** = *µ*0**H**, where the Bean-Livingston superficial barrier is neglected. Therefore, the boundary condition is written as follows:

$$
\mu\_0 H\_{ay}(t) = B\_y(0; t) = B\_y(D; t), \tag{19}
$$

$$
\mu\_0 H\_{a\mathbb{Z}}(t) = B\_{\mathbb{Z}}(0; t) = B\_{\mathbb{Z}}(D; t). \tag{20}
$$

#### **3.3. Elliptic Flux-Cutting Critical-State Model for a type-II superconductor in critical state**

The Elliptic Flux-Cutting Critical-State Model (ECSM) [1] considered the presence of an electric field in non-critical states. It models the material as a highly non-linear conductor, using a constitutive relation **E** = **E**(**J**, **B**), which completes the system of equations (17)-(18). Furthermore, it monitors the electric field's decrease through a vertical law. The model supposes that the material reaches a critical state when the magnetic induction distribution enters a stationary state where the electric field has decreased to zero.

The material law of the elliptic model considers, in contrast to multicomponent Bean and double critical-state models, that the critical current density is a rank-2 tensor determined by the equation:

$$\mathbf{J}\_{\mathbf{i}} = (\mathcal{J}\_{\mathbf{c}})\_{\mathbf{i}\mathbf{k}} \frac{E\_{\mathbf{k}}}{E}, \qquad (\mathcal{J}\_{\mathbf{c}})\_{\mathbf{i}\mathbf{k}} = \mathbf{J}\_{\mathbf{c},\mathbf{i}}(\mathcal{B}) \delta\_{\mathbf{i}\mathbf{k}\prime} \qquad \mathbf{i}, \mathbf{k} = \boldsymbol{\perp}, \parallel. \tag{21}$$

or, by the vector relation:

$$\mathbf{J} = J\_{c\parallel}(\mathcal{B})\frac{\mathbf{E}\_{\parallel}}{E} + J\_{c\perp}(\mathcal{B})\frac{\mathbf{E}\_{\perp}}{E}. \tag{22}$$

In critical state, the magnitude of **J**, that is, the critical current density *Jc*(*B*, *ϕ*) traces an ellipse on the plane *J*� − *J*⊥, defined by the equation:

$$J\_c(B, \varphi) = \left[ \left( \frac{\cos \varphi}{J\_{c\parallel}(B)} \right)^2 + \left( \frac{\sin \varphi}{J\_{c\perp}(B)} \right)^2 \right]^{-1/2} \text{.} \tag{23}$$

where the angle *ϕ* is measured respect to **B**. Notice that the behaviour of the vortex distribution is a function of the magnitude of **J** and its relative orientation respect to **B**. The model postulates that *Jc*(*B*, *<sup>ϕ</sup>*) is defined in terms of critical values *Jc*<sup>⊥</sup> and *Jc*�, that is, *Jc*(*B*, *ϕ*) is restricted to the critical ellipse (23). The fact that *J*� and *J*<sup>⊥</sup> are different produces an anisotropy on the plane *J*� − *J*<sup>⊥</sup> because of the external magnetic field **H***<sup>a</sup>* and since such components are associated with different physical phenomena which are still not fully understood. Lastly, a vertical law is used to relate the current density and the electric field magnitudes as:

10 Superconductors

**critical state**

the equation:

or, by the vector relation:

*∂E*<sup>⊥</sup> *∂x*

+ *E*� *∂α <sup>∂</sup><sup>x</sup>* <sup>=</sup> <sup>−</sup>*∂<sup>B</sup> ∂t*

Therefore, the boundary condition is written as follows:

, *E*⊥

In absence of a demagnetization factor, and considering that the applied magnetic field **H***a* is much larger than *Hc*1, the constitutive relation between **B** and **H** is modeled with the linear approximation **B** = *µ*0**H**, where the Bean-Livingston superficial barrier is neglected.

**3.3. Elliptic Flux-Cutting Critical-State Model for a type-II superconductor in**

enters a stationary state where the electric field has decreased to zero.

*Ek*

**J** = *Jc*�(*B*)

 � cos *ϕ Jc*�(*B*)

**E**�

In critical state, the magnitude of **J**, that is, the critical current density *Jc*(*B*, *ϕ*) traces an

�2 +

*<sup>E</sup>* <sup>+</sup> *Jc*⊥(*B*)

*Ji* = (J*c*)*ik*

ellipse on the plane *J*� − *J*⊥, defined by the equation:

*Jc*(*B*, *ϕ*) =

The Elliptic Flux-Cutting Critical-State Model (ECSM) [1] considered the presence of an electric field in non-critical states. It models the material as a highly non-linear conductor, using a constitutive relation **E** = **E**(**J**, **B**), which completes the system of equations (17)-(18). Furthermore, it monitors the electric field's decrease through a vertical law. The model supposes that the material reaches a critical state when the magnetic induction distribution

The material law of the elliptic model considers, in contrast to multicomponent Bean and double critical-state models, that the critical current density is a rank-2 tensor determined by

*∂α <sup>∂</sup><sup>x</sup>* <sup>−</sup> *<sup>∂</sup>E*�

*<sup>∂</sup><sup>x</sup>* <sup>=</sup> <sup>−</sup>*<sup>B</sup> ∂α*

*µ*0*Hay*(*t*) = *By*(0; *t*) = *By*(*D*; *t*), (19) *µ*0*Haz*(*t*) = *Bz*(0; *t*) = *Bz*(*D*; *t*). (20)

*<sup>E</sup>* , (J*c*)*ik* <sup>=</sup> *Jc*,*i*(*B*)*δik*, *<sup>i</sup>*, *<sup>k</sup>* <sup>=</sup>⊥, �. (21)

*<sup>E</sup>* . (22)

, (23)

**E**⊥

� sin *ϕ Jc*⊥(*B*) �2 

−1/2

*∂t*

. (18)

$$E(f) = \begin{cases} 0, & \text{for} \\ \rho(f - f\_c(B, \varphi)), & \text{for } f > f\_c(B, \varphi). \end{cases} \tag{24}$$

It can be considered that an electric field can exist in the superconductor only if the current density exceeds its critical value *Jc*(*B*, *ϕ*). Conventionally the parameter *ρ* acts as a resistivity, however, here it serves as an auxiliary parameter that counteracts the difference *J* − *Jc*(*B*, *ϕ*). The empirical relation between the critical current density **J***c* and magnetic induction **B** used is known as generalized relation, it is an adaptation to such deduced by Y.B. Kim *et al.* [30], and was proposed by M. Xu *et al.* [31], its explicit form is as follows:

*Jc*(*B*) = *Jc*(0)/ 1 + *B B*∗ *n* , (25)

where *<sup>n</sup>* and *<sup>B</sup>*<sup>∗</sup> are fitting parameters, and the maximum critical current density value *Jc*(*B* = 0) is defined by the following relation:

$$J\_c(0) = \left[ \left( 1 + \frac{B\_p}{B^\*} \right)^{n+1} \right] \frac{2B^\*}{(n+1)\mu\_0 d}. \tag{26}$$

In our case, temperature dependency remains implicit in the parameters election.

All solutions that satisfy the system of equations formed by the Ampère's law (17), the Faraday's law (18), the critical current density *Jc*(*B*, *ϕ*) (23), and the material law (24) establish the superconductor's critical state. Such system is designated the *Elliptic Flux-Cutting Critical-State Model* (ECSM) [1].

#### **4. Results using ECSM for a type-II superconductor in vortex state**

Here we present theoretical curves obtained with the ECSM, corresponding to experimental results which depend on vortex dynamics. Specifically, numerical results are shown for magnetic-moment curves and magnetic-induction evolution, using type-II superconductor samples in mixed state, subject to transverse fields, or to a low-frequency rotating field.

#### **4.1. Type-II superconductor plates subjected to transverse DC and AC fields**

In this section we present the effect of an AC magnetic field on the static magnetization of a type-II superconductor plate. The purpose was to compare and explain experimental results - obtained by Fisher *et al.*- from YBCO samples subjected to two fields perpendicular to each other, in parallel geometry [28, 33].

**Figure 3.** Theoretical average magnetization curves �*µ*0*Mz* � vs *Hy* of an YBCO plate. The red (blue) curves were calculated for an initial paramagnetic (diamagnetic) state. In each figure, the magnitude of magnetic fields to which the sample was subjected to is shown: a static one *Hz* , and one with oscillation amplitude *Hy*,*max*. These fields are perpendicular to each other and coplanar to the specimen. We observe the dependency of �*µ*0*Mz* � to the first cycles of *Hy*. Results correspond to a texturized YBCO plate, cooled down on field, a *d* = 3mm thickness and a *µ*0*Hp* = 0.0856T penetration field. The specimen's larger sides are parallel to the crystallographic plane **ab**.

In the experiments of Fisher *et al.*, plates were cut out from the homogenous part of melt-textured YCBO ingots. This was done in such a way that the larger-sized sides were parallel to the crystallographic **ab** plane, which possesses isotropic properties respect to current conduction capacity. First they applied on the sample an *Hz* field generated by direct current, with a direction such that it was parallel to the plate's main surface. Afterward they applied a second field *Hy*, oscillating at low frequencies -of the order 1kOe- coplanar to the sample's surface and perpendicular to the first field. Under this configuration, and depending on the order of magnitude for both fields, they observed phenomena denominated *Magnetization Collapse* and *Paramagnetism*.

Results presented in Figure (3) correspond to a plate with a *d* = 3mm thickness that possesses a penetration field *<sup>µ</sup>*0*Hp* = 0.0856T. This last value is very close to the experimental *<sup>µ</sup>*0*Hp* ∼ 0.1T value for the YBCO samples [33]. The parameters used to model the perpendicular component of current critical density *Jc*⊥(*B*) (see Eq. 25) are: *<sup>n</sup>*<sup>⊥</sup> <sup>=</sup> 0.5 y *<sup>B</sup>*<sup>∗</sup> <sup>⊥</sup> = 0.2T, and a maximum value *Jc*⊥(0) = <sup>5</sup> <sup>×</sup> 108A/m2 (see Eq. 26). The parallel component *Jc*�(*B*) was modeled just as *Jc*⊥(*B*). Therefore, we use *<sup>B</sup>*<sup>∗</sup> � <sup>=</sup> *<sup>B</sup>*<sup>∗</sup> <sup>⊥</sup>, *n*� = *n*⊥. The best fitting to the experimental curves was achieved assuming *Jc*�(0) = 2*Jc*⊥(0).

12 Superconductors

other, in parallel geometry [28, 33].



0

My/Hp

0.5

My/Hp

0

H<sup>z</sup> = 5H<sup>p</sup> |Hy,max| = 0.06(T)

H<sup>z</sup> = 10H<sup>p</sup> |Hy,max| = 0.07(T)

larger sides are parallel to the crystallographic plane **ab**.

0.5


Hy/Hy,max


Hy/Hy,max

phenomena denominated *Magnetization Collapse* and *Paramagnetism*.

component of current critical density *Jc*⊥(*B*) (see Eq. 25) are: *<sup>n</sup>*<sup>⊥</sup> <sup>=</sup> 0.5 y *<sup>B</sup>*<sup>∗</sup>

**4.1. Type-II superconductor plates subjected to transverse DC and AC fields**

(A) (B)

(C) (D)

In this section we present the effect of an AC magnetic field on the static magnetization of a type-II superconductor plate. The purpose was to compare and explain experimental results - obtained by Fisher *et al.*- from YBCO samples subjected to two fields perpendicular to each



H<sup>z</sup> = H<sup>p</sup> |Hy,max| = H<sup>p</sup>

0

H<sup>z</sup> = 10H<sup>p</sup> |Hy,max| = 0.06(T)

My/Hp

My/Hp

**Figure 3.** Theoretical average magnetization curves �*µ*0*Mz* � vs *Hy* of an YBCO plate. The red (blue) curves were calculated for an initial paramagnetic (diamagnetic) state. In each figure, the magnitude of magnetic fields to which the sample was subjected to is shown: a static one *Hz* , and one with oscillation amplitude *Hy*,*max*. These fields are perpendicular to each other and coplanar to the specimen. We observe the dependency of �*µ*0*Mz* � to the first cycles of *Hy*. Results correspond to a texturized YBCO plate, cooled down on field, a *d* = 3mm thickness and a *µ*0*Hp* = 0.0856T penetration field. The specimen's

In the experiments of Fisher *et al.*, plates were cut out from the homogenous part of melt-textured YCBO ingots. This was done in such a way that the larger-sized sides were parallel to the crystallographic **ab** plane, which possesses isotropic properties respect to current conduction capacity. First they applied on the sample an *Hz* field generated by direct current, with a direction such that it was parallel to the plate's main surface. Afterward they applied a second field *Hy*, oscillating at low frequencies -of the order 1kOe- coplanar to the sample's surface and perpendicular to the first field. Under this configuration, and depending on the order of magnitude for both fields, they observed

Results presented in Figure (3) correspond to a plate with a *d* = 3mm thickness that possesses a penetration field *<sup>µ</sup>*0*Hp* = 0.0856T. This last value is very close to the experimental *<sup>µ</sup>*0*Hp* ∼ 0.1T value for the YBCO samples [33]. The parameters used to model the perpendicular

a maximum value *Jc*⊥(0) = <sup>5</sup> <sup>×</sup> 108A/m2 (see Eq. 26). The parallel component *Jc*�(*B*) was

0.5


Hy/Hy,max


Hy/Hy,max

<sup>⊥</sup> = 0.2T, and

Results for �*Mz*(*Hy*)� during the first cycles of the oscillating magnetic field *Hy* are shown in Figure (3). One can observe in (A)-(C) graphics the symmetric reduction of the average magnetization for both the diamagnetic initial state (blue curves) and paramagnetic initial state (red curves). These three cases are characterized by the fact that static magnetic field *Hz* is greater than *Hp* (*Hz* = 5, 10*Hp*), whereas the AC magnetic field oscillates at small amplitudes (*Hy*,*max* ≪ *Hz*). In graphic (D) we appreciate a different behavior of the diamagnetic branch of �*Mz*(*Hy*)� (blue curve). This is due to the order of magnitude of fields *Hz* and *Hy*, which have the same order of magnitude of the sample penetration field *Hp*. One can observe in graphics (A)-(C) that for the same number of cycles of the oscillating field *Hy*, the magnetization reduction prevails, however, it is asymmetric. Specifically, the diamagnetic branch of the average magnetization changes sign, this behavior is the so-called paramagnetism. All theoretical results successfully agree with experiments for large (|*Hy*| ∼ *Hz*) and small (|*Hy*| ≪ *Hz*) values of the transverse field amplitude.

The asymmetrical suppression of average magnetization �*Mz*(*Hy*)� — caused by an AC magnetic field for diamagnetic and paramagnetic initial states as well as the change of sign in its diamagnetic branch — is only possible to reproduce if the effects of flux-line cutting and anisotropy between the perpendicular and parallel components of **J***c*(*B*) are incorporated. Also, it is necessary to consider that the effect of flux-line cutting has greater influence on the sample's magnetic induction behavior than the pinning effect.

**Figure 4.** Theoretical magnetic induction curves *Bz*/*µ*0*Hp* vs *x*/*d* for a YBCO plate (in Fig. (3) specimen characteristics are specified). The red curves (blue) were calculated for an initial paramagnetic (diamagnetic) state. The magnitude of the static magnetic field *Hz* , and the amplitude of the oscillating magnetic field |*Hy* | are shown. In all panels (A)-(D) it is initiated with a magnetic field *Hy* = *Hy*,*max*, each profile is generated every half cycle of *Hy*.

We can observe in Figure (4) the evolution of the magnetic induction profiles *Bz*(*x*) as *Hy* ocillates. They correspond to the paramagnetic and diamagnetic branches of the average magnetization �*Mz*� (see Fig.(3). In graphics (A)-(C) we can see the reduction of magnetic induction component *Bz*(*x*) as the transverse field *Hy* describes small-amplitude cycles (|*Hy*| < *Hp* ≪ *Hz*). The ECSM predicts that *Bz*(*x*) has a tendency towards a fixed distribution (see (A), (B)) or quasi-uniform (see (C)), regardless of the number of cycles. Close to the sample's borders, the distribution *Bz*(*x*) collapses to an almost homogeneous value (*Bz* ∼ *B*) as *Hy* oscillates. However, the graphic (D) shows that when the magnitude of *Hz* and *Hy* are comparable to *Hp* (|*Hy*| ∼ *Hz* = *Hp*), the slope of *Bz*(*x*) changes (conserves) its sign close to the plate borders if the initial state is diamagnetic (paramagnetic). Moreover, *Bz* collapse areas are not present in the region close to the plate border. Therefore, the average magnetization value �*Mz*�, after several *Hy* cycles becomes positive for the initial diamagnetic state.

In more recent experiments using the magneto-optic technique and Hall sensors, the effect of crossed magnetic fields in the perpendicular geometry was studied [34]. In this case, the sample is pre-magnetized in z-direction. Once the magnetic field along the z-axis is removed, an AC magnetic field, parallel to the sample's *ab* plane, is applied. Experimental results show that in this perpendicular geometry, given the magnitude of external magnetic fields *Hy* and *Hz*, only the symmetrical collapse effect of the average magnetization �*Mz*� is present. This occurs regardless of the transverse field *Hy* magnitude respect to the sample's penetration field *µ*0*Hp*. It was employed a finite-element numerical model and a power law function *E*(*J*). They reproduced the experimental curves of �*Mz*� considering that the flux-line cutting effect is not present in the superconductor dynamics. Indeed, their results do not contradict those obtained with the ESCM, because we have reproduced symmetrical collapse of �*Mz*� in a type-II superconductor with the Bean multicomponent model, which does not consider flux-line cutting.

It has also studied results of magnetization collapse for dissipative states using both ECSM [1] and the **extended ECSM**, the latter proposed by Clem [32]. In the experiments, it was measured the remanent magnetization of a PbBi superconductor disk. First, the disk is magnetized with a single pulse of a *Hz* field, and subsequently it is subjected to an oscillating magnetic field *Hy*. Even though the theoretical curves �*Mz*� vs *Hy* obtained with each model are very similar, the profile evolution of *Bz* obtained by using the extended ECSM contradicts the cases when the magnitude of fields *Hy* and *Hz* are comparable to *Hp* [5].

#### **4.2. Rotating type-II superconductors in presence of a magnetic field**

Another instance in which flux-line cutting occurs is when a superconductor sample rotates or oscillates in presence of an external magnetic field. We will focus on experiments conducted by Cave and LeBlanc [24] where a low-*Tc* type-II superconductor disk oscillates slowly in presence of a static magnetic field in parallel geometry. Cave and LeBlanc realized extensive research on the magnetic behavior of such sample, for a series of rotation angles, different initial states i.e. magnetic, non-magnetic, diamagnetic, paramagnetic and hybrid, and investigated the energy dissipation during oscillations.

Here we present theoretical calculations both for magnitude and direction of the magnetic induction **B**, as the sample rotates under different angles, also present the average component �*Bz*� and the hysteresis cycles. Results were obtained considering that the parallel and perpendicular components of current density possess a dependency on the magnetic induction of the form:

$$J\_{c\perp(\parallel)}(B) = J\_{c\perp(\parallel)}(0)\left(1 - \frac{B}{B\_{c2}}\right). \tag{27}$$

We used the second critical field *Bc*<sup>2</sup> = 0.35T reported in Ref.[24], and the maximum value of critical current density *Jc*⊥(0) = 1.6 × 109A/m2. We found that, with the anisotropy parameter *Jc*�(0)/*Jc*⊥(0) = 4, we obtain the best agreement with the experimental curves.

14 Superconductors

state.

does not consider flux-line cutting.

induction of the form:

induction component *Bz*(*x*) as the transverse field *Hy* describes small-amplitude cycles (|*Hy*| < *Hp* ≪ *Hz*). The ECSM predicts that *Bz*(*x*) has a tendency towards a fixed distribution (see (A), (B)) or quasi-uniform (see (C)), regardless of the number of cycles. Close to the sample's borders, the distribution *Bz*(*x*) collapses to an almost homogeneous value (*Bz* ∼ *B*) as *Hy* oscillates. However, the graphic (D) shows that when the magnitude of *Hz* and *Hy* are comparable to *Hp* (|*Hy*| ∼ *Hz* = *Hp*), the slope of *Bz*(*x*) changes (conserves) its sign close to the plate borders if the initial state is diamagnetic (paramagnetic). Moreover, *Bz* collapse areas are not present in the region close to the plate border. Therefore, the average magnetization value �*Mz*�, after several *Hy* cycles becomes positive for the initial diamagnetic

In more recent experiments using the magneto-optic technique and Hall sensors, the effect of crossed magnetic fields in the perpendicular geometry was studied [34]. In this case, the sample is pre-magnetized in z-direction. Once the magnetic field along the z-axis is removed, an AC magnetic field, parallel to the sample's *ab* plane, is applied. Experimental results show that in this perpendicular geometry, given the magnitude of external magnetic fields *Hy* and *Hz*, only the symmetrical collapse effect of the average magnetization �*Mz*� is present. This occurs regardless of the transverse field *Hy* magnitude respect to the sample's penetration field *µ*0*Hp*. It was employed a finite-element numerical model and a power law function *E*(*J*). They reproduced the experimental curves of �*Mz*� considering that the flux-line cutting effect is not present in the superconductor dynamics. Indeed, their results do not contradict those obtained with the ESCM, because we have reproduced symmetrical collapse of �*Mz*� in a type-II superconductor with the Bean multicomponent model, which

It has also studied results of magnetization collapse for dissipative states using both ECSM [1] and the **extended ECSM**, the latter proposed by Clem [32]. In the experiments, it was measured the remanent magnetization of a PbBi superconductor disk. First, the disk is magnetized with a single pulse of a *Hz* field, and subsequently it is subjected to an oscillating magnetic field *Hy*. Even though the theoretical curves �*Mz*� vs *Hy* obtained with each model are very similar, the profile evolution of *Bz* obtained by using the extended ECSM contradicts

Another instance in which flux-line cutting occurs is when a superconductor sample rotates or oscillates in presence of an external magnetic field. We will focus on experiments conducted by Cave and LeBlanc [24] where a low-*Tc* type-II superconductor disk oscillates slowly in presence of a static magnetic field in parallel geometry. Cave and LeBlanc realized extensive research on the magnetic behavior of such sample, for a series of rotation angles, different initial states i.e. magnetic, non-magnetic, diamagnetic, paramagnetic and hybrid,

Here we present theoretical calculations both for magnitude and direction of the magnetic induction **B**, as the sample rotates under different angles, also present the average component �*Bz*� and the hysteresis cycles. Results were obtained considering that the parallel and perpendicular components of current density possess a dependency on the magnetic

the cases when the magnitude of fields *Hy* and *Hz* are comparable to *Hp* [5].

**4.2. Rotating type-II superconductors in presence of a magnetic field**

and investigated the energy dissipation during oscillations.

Figures (5) and (6) show the theoretical curves of �*By*(*θ*)� and �*Mz*� = *<sup>µ</sup>*0*Ha* − �*Bz*� for a non magnetic initial state and four rotation angles *θmax* = 45*o*, 120*o*, 270*o*, 360*o*. All theoretical curves show the main characteristics of the corresponding experimental measurements.

**Figure 5.** Theoretical curves of �*Bz* � vs *θ* for a non magnetic initial state for an *Nb* plate with thickness *d* = 0.25mm, *Bc*<sup>2</sup> = 0.25T , and *Jc*⊥(0) = 1.6 × <sup>10</sup>9A/m2. The relation *Jc*�(0)/*Jc*⊥(0) = <sup>4</sup> was used. Curves are shown for different static external magnetic fields and rotation angles: *µ*0*Ha* = 261mT and *θmax* = 45*<sup>o</sup>* ; *µ*0*Ha* = 149mT and *θmax* = 120*<sup>o</sup>* ; *µ*0*Ha* = 172mT and *θmax* = 270*<sup>o</sup>* ; *µ*0*Ha* = 149mT and *θmax* = 360*<sup>o</sup>* .

The most outstanding phenomenon in this experiment is the *magnetic flux consumption*. This phenomenon is analyzed with the help of the spatial evolution distribution of *B*(*x*) and *α*(*x*) predicted with the ESCM. For instance, Fig. 7 presents the results for an amplitude oscillation *θmax* = 45*<sup>o</sup>* of the sample in presence of a static field *µ*0*Ha* = 261mT. We can observe in the left panel that, as the sample rotates, the two U-shaped minima of the profile of *B*(*x*) move away from the surface of the sample, as well as the local magnetic flux reduction. The right panel shows how *α*(*x*)(B) decreases from a maximum value at the sample's surface to a zero value at the region 0.4 *x*/*d* 0.6. Therefore, the ECSM can predict the existence of three areas within the material: (1) an exterior area where both flux cutting and flux transport occur, (*α* �= 0 y *dB*/*dx* �= 0); (2) an internal area where only flux transport occurs (*α* = 0 y *dB*/*dx* �= 0) and (3) a central area were neither transport flux nor flux cutting occur (*α* = *dB*/*dx* = 0).

**Figure 6.** Theoretical curves of −*µ*0�*Mz* � vs *θ* corresponding to the graphs presented in Fig.(5).

**Figure 7.** Theoretical curves for the evolution of *B*(*x*) vs *x*/*d* and *θ*(*x*) vs *x*/*d* corresponding to *µ*0*Ha* = 261mT y *θmax* = 45*<sup>o</sup>* .

## **5. Conclusions and expectations**

16 Superconductors

(*α* = *dB*/*dx* = 0).

0.01 0.02 0.03 0.04

0

0.05

0.252 0.253 0.254 0.255 0.256 0.257 0.258 0.259 0.26 0.261 0.262

B(T)

!

70Mz(T)

0.1

0.15

!

70Mz(T)

(*α* = 0 y *dB*/*dx* �= 0) and (3) a central area were neither transport flux nor flux cutting occur

0.05

0.05


**Figure 7.** Theoretical curves for the evolution of *B*(*x*) vs *x*/*d* and *θ*(*x*) vs *x*/*d* corresponding to *µ*0*Ha* = 261mT y *θmax* = 45*<sup>o</sup>* .


R1

R0

R3


0

3(degrees)

0.2

0.4

0.6

!

70Mz(T)

0.1

0.15

!

70Mz(T)

0.1

0.15


3max = 120

3max = 360

3(degrees)


3(degrees)

0 0.5 1

R2

R4

x/d


3max = 45

3(degrees)

3max = 270


3(degrees)

0 0.5 1

R5

R3

x/d

**Figure 6.** Theoretical curves of −*µ*0�*Mz* � vs *θ* corresponding to the graphs presented in Fig.(5).

R1

As J.R. Clem [32] emphasized, in contrast with other critical-state models, the Elliptic Flux-line Cutting Critical-state Model (ECSM) contains important new physical characteristics: it considers that the depinning threshold decreases as magnitude *J*� increases and, in the same way, the flux cutting threshold decreases as magnitude *J*⊥ increases. The model also reproduces the experimentally observed soft angular dependency of *Jc*(*B*). In this chapter, we have shown that the ECSM satisfactorily reproduces the response of type-II superconductors subjected to external magnetic field varying in magnitude and direction when the flux-line cutting effect is considered. Following this line of research, and based on recently results reported [32, 35, 36], the effects of the flux transport will be studied, as well as energy dissipation. This can be achieved incorporating an electric field with a well-defined direction. Our next study will consider the anisotropy in the current carrying capacity, since this characteristic is present in high-*Tc* superconductors, and flux line cutting as well. It is also desirable to study more complex phenomena in high-*Tc* superconductors, such as Meissner holes or the turbulent structures that occur in the perpendicular geometry when the external magnetic field is rotated. For this, it is essential to implement the ECSM for the case of finite geometries.

#### **Acknowledgments**

This work was supported by SEP-CONACYT under Grant No.CB-2008-01-106433

## **Author details**

Carolina Romero-Salazar\* and Omar Augusto Hernández-Flores

School of Sciences, Autonumous University "Benito Juárez" of Oaxaca, Oaxaca, Mexico

<sup>∗</sup>Address all correspondence to cromeros@ifuap.buap.mx

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[36] Cortés-Maldonado R., Espinosa-Reyes J.E., Carballo-Sánchez A.F., and Pérez-Rodríguez F. Flux-cutting and Flux-transport Effects in Type-II Superconductor Slabs in a Parallel Rotating Magnetic Field. Low Temperature Physics 2011; 37(11), 947.

20 Superconductors

270 Superconductors – New Developments

[36] Cortés-Maldonado R., Espinosa-Reyes J.E., Carballo-Sánchez A.F., and Pérez-Rodríguez F. Flux-cutting and Flux-transport Effects in Type-II Superconductor Slabs in a Parallel

Rotating Magnetic Field. Low Temperature Physics 2011; 37(11), 947.

## *Edited by Alexander Gabovich*

The chapters included in the book describe recent developments in the field of superconductivity. The book deals with both the experiment and the theory. Superconducting and normal-state properties are studied by various methods. The authors presented investigations of traditional and new materials. In particular, studies of oxides, pnictides, chalcogenides and intermetallic compounds are included. The superconducting order parameter symmetry is discussed and consequences of its actual non-conventional symmetry are studied. Impurity and tunneling effects (both quasiparticle and Josephson ones) are among topics covered in the chapters. Special attention is paid to the competition between superconductivity and other instabilities, which lead to the Fermi surface gapping.

Superconductors - New Developments

Superconductors

New Developments

*Edited by Alexander Gabovich*

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