**2. Theoretical approach**

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

resistivity found already by Kamerlingh-Onnes (sometimes the existence of persistent currents discovered by him in 1914 is considered more prominent and mysterious [27]), (ii) expulsion of a weak magnetic field (the Meissner effect [28]), and (iii) the Josephson effects [29–37], i.e. the possibility of dc or ac super-currents in circuits, containing thin insulating or normal-metal interlayers between macroscopic superconducting segments. Of course, the indicated properties are interrelated. For instance, a macroscopic superconducting loop with three Josephson junctions can exhibit a superposition of two states with persistent currents of

We note that those findings, reflecting a cooperative behavior of conducting electrons (later interpreted in terms of a quantum-mechanical wave function [12, 39–43]), had to be augmented by the observed isotope dependence of *Tc* [44, 45] in order that the first successful semi-microscopic (it is so, because the declared electron-phonon interaction was, in essence, reduced to the phenomenological four-fermion contact one) BCS theory of superconductivity [12] would come into being. Sometimes various ingenious versions of the BCS theory, explicitly taking into account the momentum and energy dependences of interaction matrix elements, as well as the renormalization of relevant normal-state properties by the superconducting reconstruction of the electron spectrum [46–50], are called "the BCS theory". Nevertheless, such extensions of the initial concept, explicitly related to Ref. [12] and results obtained therein, are inappropriate. This circumstance testifies that one should be extremely accurate with scientific terms, since otherwise it may lead to reprehensible

Whatever be a theory referred to as "the BCS one" or as "the theory of superconductivity" [52], we still lack a true consistent microscopic picture scenario (scenarios?) of superconducting pairing in different various classes of superconductors. As a consequence, all existing superconducting criteria [53–72] are empirical rather than microscopic, although based on various relatively well-developed theoretical considerations. Hence, materials scientists must rely on their intuition to find new promising superconductors [73–78], although bearing also

It is no wonder that unusual transport properties of superconductors together with their magnetic-field sensibility led to a number of practically important applications. Namely, features (i) and (ii) indicated above made it possible to manufacture large-scale power cables, fly-wheel energy storage devices, bearings, high field magnets, fault current limiters, superconductor-based transformers, levitated trains, motors and power generators [84–93]. At the same time, the Josephson (weak-coupling) feature (iii) became the basis of small-scale superconducting electronics [88, 94–98], which also uses the emergence of half-integer magnetic flux quantization in circuits with superconducting currents [99, 100]. Smartly designed SQUID devices with several Josephson junctions and a quantized flux serve as sensible detectors of magnetic field and electromagnetic waves, which, in their turn, are utilized in industry, research, and medicine [95–98, 101]. Recently oscillatory effects inherent to superfluid 3He [102–104] and 4He [103–105], which are similar to the Josephson one, were

used to construct superfluid helium quantum interference devices (SHeQUIDs) [106].

High-*Tc* oxide superconductors found in 1986 [107] and including large families of materials with *Tc* ≤ 138 K [108–112] extended the application domain of superconductivity, because,

equal magnitudes and opposite polarity [38].

in mind a deep qualitative theoretical reasoning [43, 79–83].

misunderstandings [51].

## **2.1.** *d***-wave versus** *s***-wave order parameter symmetry**

Coherent properties of Fermi liquids in the paired state are revealed by measurements of dc or ac Josephson tunnel currents between two electrodes possessing such properties. The currents depend on the phase difference between superconducting order parameters of the electrodes involved [30, 31, 119]. Manifestations of the coherent pair tunneling are more complex for superconductors with anisotropic order parameters than for those with an isotropic energy gap. In particular, it is true for *d*-wave superconductors, where the order parameter changes its sign on the Fermi surface (FS) [119, 138–143]. As was indicated above, high-*Tc* oxides are usually considered as such materials, where the *dx*<sup>2</sup>−*y*<sup>2</sup> pairing is usually assumed at least as a dominating one [117, 144–152]. However, conventional *s*-wave contributions were also detected in electron tunneling experiments [153–160] and, probably, in nuclear magnetic resonance (NMR) and nuclear quadrupole resonance measurements [161]. Therefore, only a minority of researchers prefer to accept the isotropic *s*-wave (or extended *s*-wave) nature of superconductivity in cuprates [162–175]. Notwithstanding the existing fundamental controversies, the *d*-wave specificity of high-*Tc* oxide superconductivity has already been used in technical devices [95, 116, 118–120, 122].

## **2.2. Pseudogaps as a manifestation of non-superconducting gapping**

In addition to the complex character of superconducting order parameter, cuprates reveal another intricacy of their electron spectrum. Namely, the pseudogap is observed both below and above *Tc* [176–180]. Here, various phenomena manifesting themselves in resistive, magnetic, optical, photoemission (ARPES), and tunnel (STM and break-junction) measurements are considered as a consequence of the "pseudogap"-induced depletion in the electron density of states, in analogy to what is observed in quasi-one-dimensional compounds above the mean-field phase-transition temperature [181, 182].

order parameter symmetry might be doping-dependent [214]. To obtain some insight into such more cumbersome situations, we treat here the pure isotropic *s*-wave case as well. Other possibilities for predominantly *d*-wave superconductivity coexisting with CDWs lie

The dc Josephson critical current through a tunnel junction between two superconductors,

Here, *T***pq** are matrix elements of the tunnel Hamiltonian corresponding to various combinations of FS sections for superconductors taken on different sides of tunnel junction, **p** and **q** are the transferred momenta, *e* > 0 is the elementary electrical charge, FHTSC(**p**;*ωn*) and FOS(**q**; −*ωn*) are Gor'kov Green's functions for *d*-wave (CDW gapped!) and ordinary *s*-wave superconductor, respectively, and the internal summation is carried out over the discrete fermionic "frequencies" *ω<sup>n</sup>* = (2*n* + 1) *πT*, *n* = 0, ±1, ±2, . . .. The external summation should take into account both the anisotropy of electron spectrum *ξ*(**p**) in a superconductor in the manner suggested long time ago for all kinds of anisotropic superconductors [215], the directionality of tunneling [216–220], and the concomitant dielectric (CDW) gapping of the

Hereafter, we shall assume that the ordinary superconductor has the isotropic order parameter Δ∗(*T*). At the same time, the superconducting order parameter of the high-*Tc* CDW superconductor has the properly rotated (see Figure 1) pure *d*-wave form Δ(*T*) cos [2 (*θ* − *γ*)], the angle *θ* being reckoned from the normal **n** to the junction plane and *γ* is a tilt angle between **n** and the bisectrix of the nearest positive lobe. Note that, for the *s*ext-symmetry, the gap profile is the same as in the *d*-case, but the signs of all lobes are identical rather than alternating (for

The dielectric order parameter Σ(*T*) corresponds to the checkerboard system of mutually perpendicular CDWs (observed in various high-*Tc* oxides [221–223]). In the adopted model, it is nonzero inside four sectors, each of the width 2*α*, with their bisectrices rotated by the angle *β* with respect to the bisectrices of superconducting order parameter lobes [126–128, 210, 211]. Actually, we shall assume *β* to be either 0 or *π*/4. Since the nesting vectors are directed along the **k***x*- and **k***y*-axes in the momentum space [126, 224], the adopted choice corresponds to the choice between *dx*<sup>2</sup>−*y*<sup>2</sup> - and *dxy*-symmetry. Another possible, unidirectional CDW geometry is often observed in cuprates as well [225–227]. It can be treated in a similar way, but we shall

Note also that, in agreement with previous studies [216–220, 228], the tunnel matrix elements *T***pq** in Eq. (1) should make allowance for the tunnel directionality (the angle-dependent probability of penetration through the barrier) [140, 229, 230]. We factorize the corresponding directionality coefficient *w* (*θ*). The weight factor *w* (*θ*) effectively disables the FS outside a certain given sector around **n**, thus governing the magnitude and the sign of the Josephson

HTSC(**p**;*ωn*)FOS(**q**; −*ωn*), (1)

dc Josephson Current Between an Isotropic and a d-Wave

or Extended s-Wave Partially Gapped Charge Density Wave Superconductor

293

somewhere between those pure *s*- and *d*- extremes.

whatever their origin, is given by the general equation [30, 35]

**pq** *T***pq** 2 ∑*ωn* F+

*Ic*(*T*) = 4*eT*∑

**2.4. Formulation of the problem**

nested FS sections [129].

definiteness, let this sign be positive).

not consider it in this work.

Notwithstanding large theoretical and experimental efforts, the pseudogap nature still remains unknown [126–128, 133, 178, 183–201]. Namely, some researchers associate them with precursor order parameter fluctuations, which might be either of a superconducting or some other competing (CDWs, SDWs, etc.) origin. Another viewpoint consists in relating pseudogaps to those competing orderings, but treating them, on the equal footing with superconductivity, as well-developed states that can be made allowance for in the mean field approximation, fluctuation effects being non-crucial. We believe that the available observations support the latter viewpoint (see, e.g., recent experimental evidences of CDW formation in various cuprates [202–205]). Moreover, although undoped cuprates are antiferromagnetic insulators [206], the CDW seems to be a more suitable candidate responsible for the pseudogap phenomena, which competes with Cooper pairing in doped high-*Tc* oxide samples [123–127], contrary to what is the most probable for iron-based pnictides and chalcogenides [78, 207]. Nevertheless, the type of order parameter competing with Cooper pairing in cuprates is not known with certainty. For instance, neutron diffraction studies of a number of various high-*Tc* oxides revealed a nonhomogeneous magnetic ordering (usually associated with SDWs) in the pseudogap state [208, 209].

## **2.3. Superconducting order parameter symmetry scenarios**

Bearing in mind all the aforesaid, we present here the following scenarios of dc Josephson tunneling between a non-conventional partially gapped CDW superconductor and an ordinary *s*-wave one. The Fermi surface (FS) of the former is considered two-dimensional with <sup>a</sup> *dx*<sup>2</sup>−*y*<sup>2</sup> -, *dxy*- or extended *<sup>s</sup>*-wave (with a constant order parameter sign) four-lobe symmetry of superconducting order parameter and a CDW-related doping-dependent dielectric order parameter. The CDWs constitute a system with a four-fold symmetry emerging inside the superconducting lobes in their antinodal directions for cuprates (the *dx*<sup>2</sup>−*y*<sup>2</sup> -geometry of the superconducting order parameter, see Figure 1) or in the nodal directions for another possible configuration allowed by symmetry (the *dxy*-geometry of the superconducting order parameter). (Below, for the sake of brevity, when considering the extended *s*-wave geometries for the superconducting order parameter, we use the corresponding mnemonic notations *s*ext *<sup>x</sup>*<sup>2</sup>−*y*<sup>2</sup> and *<sup>s</sup>*ext *xy* .) Thus, the CDW order parameter Σ competes with its superconducting counterpart Δ over the whole area of their coexistence, which gives rise to an interesting phenomena of temperature- (*T*-) reentrant Σ [126–128, 210, 211]. In this paper, the main objective of studies are the angular dependences, which might be observed in the framework of the adopted model. Of course, any admixture of Cooper pairing with a symmetry different from *dx*<sup>2</sup>−*y*<sup>2</sup> -one [148, 154, 160, 212, 213] may alter the results. Moreover, the superconducting order parameter symmetry might be doping-dependent [214]. To obtain some insight into such more cumbersome situations, we treat here the pure isotropic *s*-wave case as well. Other possibilities for predominantly *d*-wave superconductivity coexisting with CDWs lie somewhere between those pure *s*- and *d*- extremes.

## **2.4. Formulation of the problem**

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

In addition to the complex character of superconducting order parameter, cuprates reveal another intricacy of their electron spectrum. Namely, the pseudogap is observed both below and above *Tc* [176–180]. Here, various phenomena manifesting themselves in resistive, magnetic, optical, photoemission (ARPES), and tunnel (STM and break-junction) measurements are considered as a consequence of the "pseudogap"-induced depletion in the electron density of states, in analogy to what is observed in quasi-one-dimensional

Notwithstanding large theoretical and experimental efforts, the pseudogap nature still remains unknown [126–128, 133, 178, 183–201]. Namely, some researchers associate them with precursor order parameter fluctuations, which might be either of a superconducting or some other competing (CDWs, SDWs, etc.) origin. Another viewpoint consists in relating pseudogaps to those competing orderings, but treating them, on the equal footing with superconductivity, as well-developed states that can be made allowance for in the mean field approximation, fluctuation effects being non-crucial. We believe that the available observations support the latter viewpoint (see, e.g., recent experimental evidences of CDW formation in various cuprates [202–205]). Moreover, although undoped cuprates are antiferromagnetic insulators [206], the CDW seems to be a more suitable candidate responsible for the pseudogap phenomena, which competes with Cooper pairing in doped high-*Tc* oxide samples [123–127], contrary to what is the most probable for iron-based pnictides and chalcogenides [78, 207]. Nevertheless, the type of order parameter competing with Cooper pairing in cuprates is not known with certainty. For instance, neutron diffraction studies of a number of various high-*Tc* oxides revealed a nonhomogeneous magnetic ordering (usually

Bearing in mind all the aforesaid, we present here the following scenarios of dc Josephson tunneling between a non-conventional partially gapped CDW superconductor and an ordinary *s*-wave one. The Fermi surface (FS) of the former is considered two-dimensional with <sup>a</sup> *dx*<sup>2</sup>−*y*<sup>2</sup> -, *dxy*- or extended *<sup>s</sup>*-wave (with a constant order parameter sign) four-lobe symmetry of superconducting order parameter and a CDW-related doping-dependent dielectric order parameter. The CDWs constitute a system with a four-fold symmetry emerging inside the superconducting lobes in their antinodal directions for cuprates (the *dx*<sup>2</sup>−*y*<sup>2</sup> -geometry of the superconducting order parameter, see Figure 1) or in the nodal directions for another possible configuration allowed by symmetry (the *dxy*-geometry of the superconducting order parameter). (Below, for the sake of brevity, when considering the extended *s*-wave geometries for the superconducting order parameter, we use the corresponding mnemonic notations

*xy* .) Thus, the CDW order parameter Σ competes with its superconducting

counterpart Δ over the whole area of their coexistence, which gives rise to an interesting phenomena of temperature- (*T*-) reentrant Σ [126–128, 210, 211]. In this paper, the main objective of studies are the angular dependences, which might be observed in the framework of the adopted model. Of course, any admixture of Cooper pairing with a symmetry different from *dx*<sup>2</sup>−*y*<sup>2</sup> -one [148, 154, 160, 212, 213] may alter the results. Moreover, the superconducting

**2.2. Pseudogaps as a manifestation of non-superconducting gapping**

compounds above the mean-field phase-transition temperature [181, 182].

associated with SDWs) in the pseudogap state [208, 209].

*s*ext

*<sup>x</sup>*<sup>2</sup>−*y*<sup>2</sup> and *<sup>s</sup>*ext

**2.3. Superconducting order parameter symmetry scenarios**

The dc Josephson critical current through a tunnel junction between two superconductors, whatever their origin, is given by the general equation [30, 35]

$$I\_{\mathbb{C}}(T) = 4eT \sum\_{\mathbf{p}\mathbf{q}} \left| \tilde{T}\_{\mathbf{p}\mathbf{q}} \right|^{2} \sum\_{\omega\_{\mathbf{n}}} \mathsf{F}\_{\mathrm{HTSC}}^{+}(\mathbf{p};\omega\_{\mathbf{n}}) \mathsf{F}\_{\mathrm{OS}}(\mathbf{q};-\omega\_{\mathbf{n}}),\tag{1}$$

Here, *T***pq** are matrix elements of the tunnel Hamiltonian corresponding to various combinations of FS sections for superconductors taken on different sides of tunnel junction, **p** and **q** are the transferred momenta, *e* > 0 is the elementary electrical charge, FHTSC(**p**;*ωn*) and FOS(**q**; −*ωn*) are Gor'kov Green's functions for *d*-wave (CDW gapped!) and ordinary *s*-wave superconductor, respectively, and the internal summation is carried out over the discrete fermionic "frequencies" *ω<sup>n</sup>* = (2*n* + 1) *πT*, *n* = 0, ±1, ±2, . . .. The external summation should take into account both the anisotropy of electron spectrum *ξ*(**p**) in a superconductor in the manner suggested long time ago for all kinds of anisotropic superconductors [215], the directionality of tunneling [216–220], and the concomitant dielectric (CDW) gapping of the nested FS sections [129].

Hereafter, we shall assume that the ordinary superconductor has the isotropic order parameter Δ∗(*T*). At the same time, the superconducting order parameter of the high-*Tc* CDW superconductor has the properly rotated (see Figure 1) pure *d*-wave form Δ(*T*) cos [2 (*θ* − *γ*)], the angle *θ* being reckoned from the normal **n** to the junction plane and *γ* is a tilt angle between **n** and the bisectrix of the nearest positive lobe. Note that, for the *s*ext-symmetry, the gap profile is the same as in the *d*-case, but the signs of all lobes are identical rather than alternating (for definiteness, let this sign be positive).

The dielectric order parameter Σ(*T*) corresponds to the checkerboard system of mutually perpendicular CDWs (observed in various high-*Tc* oxides [221–223]). In the adopted model, it is nonzero inside four sectors, each of the width 2*α*, with their bisectrices rotated by the angle *β* with respect to the bisectrices of superconducting order parameter lobes [126–128, 210, 211]. Actually, we shall assume *β* to be either 0 or *π*/4. Since the nesting vectors are directed along the **k***x*- and **k***y*-axes in the momentum space [126, 224], the adopted choice corresponds to the choice between *dx*<sup>2</sup>−*y*<sup>2</sup> - and *dxy*-symmetry. Another possible, unidirectional CDW geometry is often observed in cuprates as well [225–227]. It can be treated in a similar way, but we shall not consider it in this work.

Note also that, in agreement with previous studies [216–220, 228], the tunnel matrix elements *T***pq** in Eq. (1) should make allowance for the tunnel directionality (the angle-dependent probability of penetration through the barrier) [140, 229, 230]. We factorize the corresponding directionality coefficient *w* (*θ*). The weight factor *w* (*θ*) effectively disables the FS outside a certain given sector around **n**, thus governing the magnitude and the sign of the Josephson

**Figure 1.** Geometry of the junction between a conventional *s*-wave superconductor (*s*-BCS) and a *d*-, *s*-extended (*s*ext) or *s*-superconductor partially gapped by charge density waves (CDWs, induced by dielectric, i.e. electron-hole, pairing). The angle *α* denotes the half-width of each of four angular sectors at the Fermi surface, where the CDW gap appears. The gap profiles for the parent CDW insulator (Σ), *s*- (Δ*s*), *d*- (Δ*d*), and *s*-extended (Δ*s*ext) superconductors, and conventional superconductor (Δ∗) are shown. *β* is a misorientation angle between the nearest superconducting lobe and CDW-gapped sector, *γ* is a tilt angle of superconducting lobe with respect to the junction plane determined by the normal **n**, *θ*<sup>0</sup> is a measure of tunneling directionality (see explanations in the text).

tunnel current. Specifically, we used the following model for *w* (*θ*):

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

Here, we explicitly took into account a possible angle deviation *γ* of the Δ-lobe direction, which is governed by the crystal lattice geometry, from the normal **n** to the junction plane; the latter is created artificially and, generally speaking, can be not coinciding with a crystal facet. The concomitant rotation of the CDW sectors is made allowance for implicitly. The quasiparticle spectra *ξd*(**p**) and *ξnd*(**p**) correspond to "hot" and "cold" spots of the cuprate

Substituting Eqs. (2), (3), and (4) into Eq. (1) and carrying out standard transformations [30,

 Δ∗(*T*),

factorized multiplier *w* (*θ*), the integration is carried out over the CDW-gapped and CDW-free FS sections (the FS-arcs *θ<sup>d</sup>* and *θnd*, respectively, in the two-dimensional problem geometry), Δ∗(*T*) is the order parameter of the ordinary isotropic superconductor, whereas the function

Modified Eqs. (3)-(6) turn out valid for the calculation of dc Josephson current through a tunnel junction between an ordinary *s*-wave superconductor and a partially gapped CDW superconductor with an extended *s*-symmetry of superconducting order parameter [142, 241]. For this purpose, it is enough to substitute the cosine functions in Eqs. (3)-(6) by their absolute

At *w* (*θ*) ≡ 1 (the absence of tunnel directionality), Σ ≡ 0 (the absence of CDW-gapping), and putting cos 2 (*θ* − *γ*) ≡ 1 (actually, it is a substitution of an isotropic *s*-superconductor for the *d*-wave one), Eq. (6) expectedly reproduces the famous Ambegaokar–Baratoff result for

Note that, in Eq. (6), the directionality is made allowance for only by introducing the angular function *w* (*θ*) reflecting the angle-dependent tunnel-barrier transparency. On the other hand, the tunneling process, in principle, should also take into account the factors

∇*ξnd* and **v***g*,*<sup>d</sup>* = ∇*ξ<sup>d</sup>* are the quasiparticle group velocities for proper FS sections. Those factors can be considered as proportional to a number of electron attempts to penetrate the barrier [139]. They were introduced decades ago in the framework of general problem dealing with tunneling in heterostructures [243–245]. Nevertheless, we omitted here the

 *<sup>x</sup>*<sup>2</sup> <sup>−</sup> <sup>Δ</sup><sup>2</sup> 1 Δ<sup>2</sup> <sup>2</sup> − *<sup>x</sup>*<sup>2</sup>

*ic* (*T*), (5)

or Extended s-Wave Partially Gapped Charge Density Wave Superconductor

*w*(*θ*) cos [2 (*θ* − *γ*)] *P* [Δ<sup>∗</sup> (*T*), |Δ(*T*) cos 2 (*θ* − *γ*)|] *dθ*. (6)

*dx* tanh *<sup>x</sup>*

, responsible for extra directionality [140, 219, 230], where **<sup>v</sup>***g*,*nd* <sup>=</sup>

2*T*

<sup>Σ</sup><sup>2</sup> <sup>+</sup> <sup>Δ</sup>2(*T*) cos2 [<sup>2</sup> (*<sup>θ</sup>* <sup>−</sup> *<sup>γ</sup>*)]

dc Josephson Current Between an Isotropic and a d-Wave

 *T***pq** 2

. (7)

*dθ*

without the

295

FS, respectively (see, e.g., Refs. [176, 238–240]).

2*eRN*

*P* (Δ1, Δ2) is given by the expression [129, 215]

*P*(Δ1, Δ2) =

tunneling between *s*-wave superconductors [30, 31, 35, 242].

*w*(*θ*) cos [2 (*θ* − *γ*)] *P*

Here, *RN* is the normal-state resistance of the tunnel junction, determined by

max{Δ1,Δ2} 

min{Δ1,Δ2}

*Ic*(*T*) = <sup>Δ</sup> (0) <sup>Δ</sup><sup>∗</sup> (0)

2*π θd*

*ic*(*T*) = <sup>1</sup>

+ 1 2*π θnd*

35], we obtain

values.

 **v***g*,*nd* · **<sup>n</sup>**

 and **v***g*,*<sup>d</sup>* · **<sup>n</sup>** 

where *θ*<sup>0</sup> is an angle describing the effective width of the directionality sector. We emphasize that, for tunneling between two anisotropic superconductors, two different coefficients *w* (*θ*) associated with **p**- and **q**-distributions in the corresponding electrodes come into effect [216].

In accordance with the previous treatment of partially gapped *s*-wave CDW superconductors [123–125, 129, 130, 132, 231–234] and its generalization to their *d*-wave counterparts [126– 128, 210, 211, 235] and in line with the basic theoretical framework for unconventional superconductors [236, 237], the anomalous Gor'kov Green's functions for high-*Tc* oxides are assumed to be different for angular sectors with coexisting CDWs and superconductivity (d sections of the FS) and the "purely superconducting" rest of the FS (nd sections)

$$\mathsf{F}\_{\text{HTSC,nd}}(\mathbf{p}\omega\_{\text{ll}}) = \frac{\Delta(T)\cos\left[2\left(\theta-\gamma\right)\right]}{\omega\_{\text{ll}}^2 + \Delta^2(T)\cos^2\left[2\left(\theta-\gamma\right)\right] + \mathfrak{F}\_{\text{nd}}^2(\mathbf{p})},\tag{3}$$

$$\mathsf{F}\_{\text{HTSC},\text{d}}(\mathbf{p}\omega\_{\text{n}}) = \frac{\Delta(T)\cos\left[2\left(\theta-\gamma\right)\right]}{\omega\_{\text{n}}^{2} + \Delta^{2}(T)\cos^{2}\left[2\left(\theta-\gamma\right)\right] + \Sigma^{2}\left(T\right) + \tilde{\xi}\_{d}^{2}(\mathbf{p})}.\tag{4}$$

Here, we explicitly took into account a possible angle deviation *γ* of the Δ-lobe direction, which is governed by the crystal lattice geometry, from the normal **n** to the junction plane; the latter is created artificially and, generally speaking, can be not coinciding with a crystal facet. The concomitant rotation of the CDW sectors is made allowance for implicitly. The quasiparticle spectra *ξd*(**p**) and *ξnd*(**p**) correspond to "hot" and "cold" spots of the cuprate FS, respectively (see, e.g., Refs. [176, 238–240]).

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

**Figure 1.** Geometry of the junction between a conventional *s*-wave superconductor (*s*-BCS) and a *d*-, *s*-extended (*s*ext) or *s*-superconductor partially gapped by charge density waves (CDWs, induced by dielectric, i.e. electron-hole, pairing). The angle *α* denotes the half-width of each of four angular sectors at the Fermi surface, where the CDW gap appears. The gap profiles for the parent CDW insulator (Σ), *s*- (Δ*s*), *d*- (Δ*d*), and *s*-extended (Δ*s*ext) superconductors, and conventional superconductor (Δ∗) are shown. *β* is a misorientation angle between the nearest superconducting lobe and CDW-gapped sector, *γ* is a tilt angle of superconducting lobe with respect to the junction plane determined by the normal **n**, *θ*<sup>0</sup> is a

> −

where *θ*<sup>0</sup> is an angle describing the effective width of the directionality sector. We emphasize that, for tunneling between two anisotropic superconductors, two different coefficients *w* (*θ*) associated with **p**- and **q**-distributions in the corresponding electrodes come into effect [216]. In accordance with the previous treatment of partially gapped *s*-wave CDW superconductors [123–125, 129, 130, 132, 231–234] and its generalization to their *d*-wave counterparts [126– 128, 210, 211, 235] and in line with the basic theoretical framework for unconventional superconductors [236, 237], the anomalous Gor'kov Green's functions for high-*Tc* oxides are assumed to be different for angular sectors with coexisting CDWs and superconductivity (d

 tan *θ* tan *θ*<sup>0</sup>

*<sup>n</sup>* <sup>+</sup> <sup>Δ</sup>2(*T*) cos2 [<sup>2</sup> (*<sup>θ</sup>* <sup>−</sup> *<sup>γ</sup>*)] <sup>+</sup> *<sup>ξ</sup>*<sup>2</sup>

*<sup>n</sup>* <sup>+</sup> <sup>Δ</sup>2(*T*) cos<sup>2</sup> [<sup>2</sup> (*<sup>θ</sup>* <sup>−</sup> *<sup>γ</sup>*)] <sup>+</sup> <sup>Σ</sup><sup>2</sup> (*T*) <sup>+</sup> *<sup>ξ</sup>*<sup>2</sup>

<sup>2</sup> 

, (2)

*nd*(**p**)

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

, (3)

. (4)

measure of tunneling directionality (see explanations in the text).

tunnel current. Specifically, we used the following model for *w* (*θ*):

*w* (*θ*) = exp

sections of the FS) and the "purely superconducting" rest of the FS (nd sections)

<sup>F</sup>HTSC,d(**p**;*ωn*) = <sup>Δ</sup>(*T*) cos [<sup>2</sup> (*<sup>θ</sup>* <sup>−</sup> *<sup>γ</sup>*)]

*ω*2

<sup>F</sup>HTSC,nd(**p**;*ωn*) = <sup>Δ</sup>(*T*) cos [<sup>2</sup> (*<sup>θ</sup>* <sup>−</sup> *<sup>γ</sup>*)] *ω*2

Substituting Eqs. (2), (3), and (4) into Eq. (1) and carrying out standard transformations [30, 35], we obtain

$$\begin{split} I\_{\mathbf{c}}(T) &= \frac{\Delta\left(0\right)\Delta^{\*}\left(0\right)}{2\epsilon\mathbb{R}\_{N}} i\_{\mathbf{c}}(T), \\\ i\_{\mathbf{c}}(T) &= \frac{1}{2\pi} \int\_{\theta\_{\cdot}} w(\theta) \cos\left[2\left(\theta-\gamma\right)\right] P\left[\Delta^{\*}\left(T\right), \sqrt{\Sigma^{2} + \Delta^{2}\left(T\right)\cos^{2}\left[2\left(\theta-\gamma\right)\right]}\right] d\theta \end{split} \tag{5}$$

$$\begin{aligned} \left[\alpha + 2\pi \int\_{\theta\_{\rm nd}} w(\theta) \cos \left[2\pi(\theta - \gamma)\right] \right] \left[\alpha w(\theta) \sin \left[2\pi(\theta - \gamma)\right] \right] &= \\ + \frac{1}{2\pi} \int\_{\theta\_{\rm nd}} w(\theta) \cos \left[2\left(\theta - \gamma\right)\right] P\left[\Delta^{\ast}\left(T\right), \left|\Delta(T) \cos 2\left(\theta - \gamma\right)\right|\right] d\theta. \end{aligned} \tag{6}$$

Here, *RN* is the normal-state resistance of the tunnel junction, determined by *T***pq** 2 without the factorized multiplier *w* (*θ*), the integration is carried out over the CDW-gapped and CDW-free FS sections (the FS-arcs *θ<sup>d</sup>* and *θnd*, respectively, in the two-dimensional problem geometry), Δ∗(*T*) is the order parameter of the ordinary isotropic superconductor, whereas the function *P* (Δ1, Δ2) is given by the expression [129, 215]

$$P(\Delta\_1, \Delta\_2) = \int\_{\min\{\Delta\_1, \Delta\_2\}}^{\max\{\Delta\_1, \Delta\_2\}} \frac{d\mathfrak{x} \tanh\frac{\mathfrak{x}}{2T}}{\sqrt{\left(\mathfrak{x}^2 - \Delta\_1^2\right)\left(\Delta\_2^2 - \mathfrak{x}^2\right)}}.\tag{7}$$

Modified Eqs. (3)-(6) turn out valid for the calculation of dc Josephson current through a tunnel junction between an ordinary *s*-wave superconductor and a partially gapped CDW superconductor with an extended *s*-symmetry of superconducting order parameter [142, 241]. For this purpose, it is enough to substitute the cosine functions in Eqs. (3)-(6) by their absolute values.

At *w* (*θ*) ≡ 1 (the absence of tunnel directionality), Σ ≡ 0 (the absence of CDW-gapping), and putting cos 2 (*θ* − *γ*) ≡ 1 (actually, it is a substitution of an isotropic *s*-superconductor for the *d*-wave one), Eq. (6) expectedly reproduces the famous Ambegaokar–Baratoff result for tunneling between *s*-wave superconductors [30, 31, 35, 242].

Note that, in Eq. (6), the directionality is made allowance for only by introducing the angular function *w* (*θ*) reflecting the angle-dependent tunnel-barrier transparency. On the other hand, the tunneling process, in principle, should also take into account the factors **v***g*,*nd* · **<sup>n</sup>** and **v***g*,*<sup>d</sup>* · **<sup>n</sup>** , responsible for extra directionality [140, 219, 230], where **<sup>v</sup>***g*,*nd* <sup>=</sup> ∇*ξnd* and **v***g*,*<sup>d</sup>* = ∇*ξ<sup>d</sup>* are the quasiparticle group velocities for proper FS sections. Those factors can be considered as proportional to a number of electron attempts to penetrate the barrier [139]. They were introduced decades ago in the framework of general problem dealing with tunneling in heterostructures [243–245]. Nevertheless, we omitted here the

group-velocity-dependent multiplier, since it requires that the FS shape should be specified, thus going beyond the applied semi-phenomenological scheme, as well as beyond similar semi-phenomenological approaches of other groups [138, 139, 141, 236, 246]. We shall take the additional directionality factor into account in subsequent publications, still being fully aware of the phenomenological nature of both **v***<sup>g</sup>* · **n** and *w* (*θ*) functions.

It is well known [143] that, in the absence of directionality, the Josephson tunneling between *d*- and *s*-wave superconductors is weighted-averaged over the FS, with the cosine multiplier in Eq. (6) playing the role of weight function. In this case, the Josephson current has to be strictly equal to zero. However, it was found experimentally that the dc Josephson current between Bi2Sr2CaCu2O8<sup>+</sup>*<sup>δ</sup>* and Pb [155], Bi2Sr2CaCu2O8<sup>+</sup>*<sup>δ</sup>* and Nb [247], YBa2Cu3O7−*<sup>δ</sup>* and PbIn [248], Y1−*x*Pr*x*Ba2Cu3O7−*<sup>δ</sup>* and Pb [153] differ from zero. Hence, either a subdominant *s*-wave component of the superconducting order parameter does exist in cuprate materials, as was discussed above, or the introduction of directionality is inevitable to reconcile any theory dealing with tunneling of quasiparticles from (to) high-*Tc* oxides and the experiment.

(a) (b)

dc Josephson Current Between an Isotropic and a d-Wave

or Extended s-Wave Partially Gapped Charge Density Wave Superconductor

**Figure 2.** (a) Zero-temperature (*T* = 0) dependences on *γ* of the dimensionless dc Josephson current *ic* for the tunnel junction between an *<sup>s</sup>*-wave superconductor and a CDW *dx*<sup>2</sup>−*y*<sup>2</sup> -wave one (*<sup>β</sup>* <sup>=</sup> <sup>0</sup>◦) for various *θ*0's. The specific gap values for electrodes correspond to the experimental data for Nb (Δ∗(*T* = 0) = 1.4 meV) and Bi2Sr2CaCu2O8<sup>+</sup>*<sup>δ</sup>* (Σ(*T* = 0) = 36.3 meV and Δ(*T* = 0) = 28.3 meV). The calculation parameter *α* = 15◦. See further explanations in the text. (b) The same as in panel (a), but for a

It is evident that, if the sector *θ*<sup>0</sup> of effective tunneling equals zero, the Josephson current vanishes. It is also natural that, in the case of *d*-wave pairing and the absence of tunneling directionality (*θ*<sup>0</sup> = 90◦), the Josephson current disappears due to the exactly mutually compensating contributions from superconducting order parameter lobes with different signs [119, 138, 143]. Intermediate *θ*0's correspond to non-zero Josephson tunnel current of either sign (conventional 0- and *π*-junctions [120, 122, 253]) except at the tilt angle *γ* = 45◦, when *ic* = 0. In this connection, one should recognize that the energy minimum *for non-conventional anisotropic superconductors* can occur, in principle, at any value of the order parameter phase [254]. As is seen from Figure 2(a), the existence of CDWs in cuprates (*α* �= 0, Σ �= 0) influences the *γ*-dependences of *ic*, which become non-monotonic for *θ*<sup>0</sup> close to *α* demonstrating a peculiar resonance between two junction characteristics. The effect appears owing to the actual *dx*<sup>2</sup>−*y*<sup>2</sup> - pattern with the coinciding bisectrices of CDW sectors and superconducting lobes (*β* = 0◦). This circumstance may ensure the finding of CDWs (pseudogaps) by a set of

At the same time, for the hypothetical *dxy* order parameter symmetry (*β* = 45◦, Figure 2(b)), when hot spots lie in the nodal regions, the dependences *ic*(*γ*) become asymmetrical relative

The role of superconducting-lobe and CDW (governed by the crystalline structure) orientation with respect to the junction plane (the angle *γ*) is most clearly seen for varying *α*, which is shown in Figure 3. The indicated above "resonance" between *θ*<sup>0</sup> and *α* is readily seen in Figure 3(a). One also sees that the Josephson current amplitude is expectedly reduced with the increasing *α*, since CDWs suppress superconductivity [123–127, 255]. For *β* = 45◦ (Figure 3(b)), the curves *ic*(*γ*) are non-symmetrical, and their form is distorted by CDWs

The dependence of *ic* on the CDW-sector width, i.e. the degree of dielectric FS gapping, is a rapidly dropping one, which is demonstrated in Figures 4(a) (for *<sup>β</sup>* <sup>=</sup> <sup>0</sup>◦, i.e. for *dx*<sup>2</sup>−*y*<sup>2</sup>

*xy* symmetries) calculated for

297

to *γ* = 90◦ and remain monotonic as for CDW-free *d*-wave superconductors.

relative to the case of "pure" superconducting *d*-wave electrode.

*<sup>x</sup>*<sup>2</sup>−*y*<sup>2</sup> symmetries) and 4(b) (for *<sup>β</sup>* <sup>=</sup> <sup>45</sup>◦, i.e. for *dxy* or *<sup>s</sup>*ext

CDW *dxy*-superconductor (*β* = 45◦).

relatively simple transport measurements.

or *s*ext

We restrict ourselves mostly to the case *T* = 0, when formula (7) is reduced to elliptic functions [30, 249], although some calculations will be performed for *T* �= 0 as well. The reason consists in the smallness of *Tc* for conventional *s*-wave superconductors (in our case, it is Nb, see below) as compared to *Tc* of anisotropic *d*-wave oxides. Hence, all effects concerning *T*-dependent interplay between Δ and Σ including possible reentrance of Σ(*T*) [126–128, 210, 211, 235] become insignificant in the relevant *T*-range cut off by the *s*-wave-electrode order parameter. On the contrary, in the symmetrical case, when one studies tunneling between different high-*Tc*-oxide grains, *T*-dependences of the Josephson current are expected to be very interesting. This more involved situation will be investigated elsewhere.
