**3.1. Total currents**

In what follows, we shall consider in parallel the dc Josephson currents between a more or less conventional (weak-coupling BCS *s*-wave) Nb with a zero-*T* energy gap Δ∗(0) = 1.4 meV and *Tc* <sup>=</sup> 9.2 K [247] and either a *dx*<sup>2</sup>−*y*<sup>2</sup> - or a *dxy*- superconductor (*<sup>β</sup>* <sup>=</sup> 0 and *<sup>π</sup>*/4, respectively). The latter is also possible from the symmetry viewpoint, but have not yet been found among existing classes of CDW superconductors.

The dependences of the dimensionless current *ic* (*T* = 0) on the tilt angle *γ* are shown in Figure 2(a) for *α* = 15◦ and various values of the parameter *θ*<sup>0</sup> describing the degree of directionality. Since *T* = 0, there is no need to solve the equation set for Σ(*T*) and Δ(*T*) for partially CDW-gapped *s*-wave [233] or *d*-wave [128] superconductors self-consistently. Instead, for definiteness, we chose the experimental values Σ(0) = 36.3 meV and Δ(0) = 28.3 meV appropriate to slightly overdoped Bi2Sr2CaCu2O8+*<sup>δ</sup>* samples [250] as input parameters. The half-width *α* of each of the four CDW sectors was *rather arbitrarily* chosen as 15◦. In fact, it is heavily dependent on the doping extent and cannot be unambiguously extracted even from the most precise angle-resolved photoemission spectra (ARPES) [200, 251, 252]. Thus, hereafter we consider the parameter of dielectric FS gapping *α* as a *phenomenological* one on the same footing as the tunneling directionality parameter *θ*0.

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

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

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

and *w* (*θ*) functions.

**v***<sup>g</sup>* · **n** 

dealing with tunneling of quasiparticles from (to) high-*Tc* oxides and the experiment.

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.

In what follows, we shall consider in parallel the dc Josephson currents between a more or less conventional (weak-coupling BCS *s*-wave) Nb with a zero-*T* energy gap Δ∗(0) = 1.4 meV and *Tc* <sup>=</sup> 9.2 K [247] and either a *dx*<sup>2</sup>−*y*<sup>2</sup> - or a *dxy*- superconductor (*<sup>β</sup>* <sup>=</sup> 0 and *<sup>π</sup>*/4, respectively). The latter is also possible from the symmetry viewpoint, but have not yet been found among

The dependences of the dimensionless current *ic* (*T* = 0) on the tilt angle *γ* are shown in Figure 2(a) for *α* = 15◦ and various values of the parameter *θ*<sup>0</sup> describing the degree of directionality. Since *T* = 0, there is no need to solve the equation set for Σ(*T*) and Δ(*T*) for partially CDW-gapped *s*-wave [233] or *d*-wave [128] superconductors self-consistently. Instead, for definiteness, we chose the experimental values Σ(0) = 36.3 meV and Δ(0) = 28.3 meV appropriate to slightly overdoped Bi2Sr2CaCu2O8+*<sup>δ</sup>* samples [250] as input parameters. The half-width *α* of each of the four CDW sectors was *rather arbitrarily* chosen as 15◦. In fact, it is heavily dependent on the doping extent and cannot be unambiguously extracted even from the most precise angle-resolved photoemission spectra (ARPES) [200, 251, 252]. Thus, hereafter we consider the parameter of dielectric FS gapping *α* as a *phenomenological* one on the same footing as the tunneling directionality parameter *θ*0.

of the phenomenological nature of both

**3. Results and discussion**

existing classes of CDW superconductors.

**3.1. Total currents**

**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 CDW *dxy*-superconductor (*β* = 45◦).

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 relatively simple transport measurements.

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

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

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> or *s*ext *<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 *xy* symmetries) calculated for

make sure that this assertion is valid, we calculated the dependences *ic*(*θ*0) for *γ* = 0◦, *α* = 15◦, and *β* = 0◦ and 45◦. The results are presented in Figure 5. Indeed, for *θ*<sup>0</sup> ≥ 30◦, the curves corresponding to *d*-wave and extended *s*-wave superconductors come apart, as it has to be. Thus, Josephson currents between isotropic and CDW *d*-wave superconductors, similarly to the CDW-free case, are non-zero only because the tunneling is non-isotropic.

dc Josephson Current Between an Isotropic and a d-Wave

or Extended s-Wave Partially Gapped Charge Density Wave Superconductor

299

**Figure 5.** Dependences *ic*(*θ*0) for *γ* = 0◦, *β* = 0◦, and *α*<sup>0</sup> = 15◦ for various symmetries of

parameter symmetry for cuprates or other like materials.

It is instructive to compare the tilt-angle-*γ* dependences of the Josephson currents *ic* for possible superconducting order parameter symmetries, which are considered, in particular, for cuprates. The results of calculations are displayed in Figure 7 for *α* = *θ*<sup>0</sup> = 15◦. For an *s*-wave CDW-free superconductor, *ic* (*γ*) = const. The reference curve *ic*(*γ*) for a CDW-free *dx*<sup>2</sup>−*y*<sup>2</sup> -wave superconductor (Figure 7(a)) is periodic and alternating. CDWs distort both curves. Namely, the CDW *dx*<sup>2</sup>−*y*<sup>2</sup> -wave superconductor demonstrates a non-monotonic behavior of *ic* (*γ*), as was indicated above, whereas *ic*(*γ*) for the *s*-wave CDW superconductor becomes a periodic dependence of a constant sign. The curve *ic*(*γ*) for the extended *s*-wave CDW superconductor has a different form than in the *s*-wave case, although being qualitatively similar. The presented data demonstrate that CDWs can significantly alter angle dependences often considered as a smoking gun,when determining the actual order

(a) (b)

**Figure 6.** The same as in Figure 2, but for *θ*<sup>0</sup> = 15◦ and various symmetries of superconducting order

superconducting order parameter.

parameter.

**Figure 3.** The same as in Figure 2, but for *θ*<sup>0</sup> = 15◦ and various *α*'s.

*γ* = 0◦ and *θ*<sup>0</sup> = 15◦. Indeed, for cuprates, where the directions of superconducting lobes and CDW sectors coincide, an extending CDW-induced gap reduces the electron density of states available to superconducting pairing until *α* becomes equal to *θ*<sup>0</sup> (see Figure 4(a)). A further increase of the pseudogapped FS arc has no influence on *ic*, since it falls outside the effective tunneling sector. We note that the *α*-dependence of *ic* for cuprates can be, in principle, non-linearly mapped onto the doping dependence of the pseudogap [200, 251, 252]. It is remarkable that, qualitatively, the results are the same for the extended *s*-symmetry (denoted as *s*ext) of the superconducting order parameter and are very similar to those for the assumed *s*-wave order parameter (curves marked by *s*).

**Figure 4.** Dependences *ic*(*α*) for *γ* = 0◦ and *θ*<sup>0</sup> = 15◦ for *d*-, *s*-extended, and *s*-symmetries of superconducting order parameter.

At the same time, if the CDW sectors are rotated in the momentum space by 45◦ with respect to the superconducting lobes and/or the directional-tunneling *θ*0-cone (see Figure 4(b)), the dependences *ic*(*α*) are very weak at small *α* and become steep for *α* > *θ*0. This result is true for the *dxy*-, rotated extended *s*-, and isotropic *s*-symmetries of the superconducting order parameter coexisting with its dielectric counterpart.

One sees from Figure 4 that, for small *θ*<sup>0</sup> = 15◦, the *d*- and extended *s*-order parameters result in the same *ic*(*α*). Of course, it is no longer true for larger *θ*0, when contributions from different lobes into the total Josephson current start to compensate each other for *d*-wave superconductivity, whereas no compensation occurs for the extended *s*-wave scenario. To make sure that this assertion is valid, we calculated the dependences *ic*(*θ*0) for *γ* = 0◦, *α* = 15◦, and *β* = 0◦ and 45◦. The results are presented in Figure 5. Indeed, for *θ*<sup>0</sup> ≥ 30◦, the curves corresponding to *d*-wave and extended *s*-wave superconductors come apart, as it has to be. Thus, Josephson currents between isotropic and CDW *d*-wave superconductors, similarly to the CDW-free case, are non-zero only because the tunneling is non-isotropic.

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

*γ* = 0◦ and *θ*<sup>0</sup> = 15◦. Indeed, for cuprates, where the directions of superconducting lobes and CDW sectors coincide, an extending CDW-induced gap reduces the electron density of states available to superconducting pairing until *α* becomes equal to *θ*<sup>0</sup> (see Figure 4(a)). A further increase of the pseudogapped FS arc has no influence on *ic*, since it falls outside the effective tunneling sector. We note that the *α*-dependence of *ic* for cuprates can be, in principle, non-linearly mapped onto the doping dependence of the pseudogap [200, 251, 252]. It is remarkable that, qualitatively, the results are the same for the extended *s*-symmetry (denoted as *s*ext) of the superconducting order parameter and are very similar to those for the assumed

**Figure 3.** The same as in Figure 2, but for *θ*<sup>0</sup> = 15◦ and various *α*'s.

*s*-wave order parameter (curves marked by *s*).

superconducting order parameter.

parameter coexisting with its dielectric counterpart.

(a) (b)

(a) (b)

**Figure 4.** Dependences *ic*(*α*) for *γ* = 0◦ and *θ*<sup>0</sup> = 15◦ for *d*-, *s*-extended, and *s*-symmetries of

At the same time, if the CDW sectors are rotated in the momentum space by 45◦ with respect to the superconducting lobes and/or the directional-tunneling *θ*0-cone (see Figure 4(b)), the dependences *ic*(*α*) are very weak at small *α* and become steep for *α* > *θ*0. This result is true for the *dxy*-, rotated extended *s*-, and isotropic *s*-symmetries of the superconducting order

One sees from Figure 4 that, for small *θ*<sup>0</sup> = 15◦, the *d*- and extended *s*-order parameters result in the same *ic*(*α*). Of course, it is no longer true for larger *θ*0, when contributions from different lobes into the total Josephson current start to compensate each other for *d*-wave superconductivity, whereas no compensation occurs for the extended *s*-wave scenario. To

**Figure 5.** Dependences *ic*(*θ*0) for *γ* = 0◦, *β* = 0◦, and *α*<sup>0</sup> = 15◦ for various symmetries of superconducting order parameter.

It is instructive to compare the tilt-angle-*γ* dependences of the Josephson currents *ic* for possible superconducting order parameter symmetries, which are considered, in particular, for cuprates. The results of calculations are displayed in Figure 7 for *α* = *θ*<sup>0</sup> = 15◦. For an *s*-wave CDW-free superconductor, *ic* (*γ*) = const. The reference curve *ic*(*γ*) for a CDW-free *dx*<sup>2</sup>−*y*<sup>2</sup> -wave superconductor (Figure 7(a)) is periodic and alternating. CDWs distort both curves. Namely, the CDW *dx*<sup>2</sup>−*y*<sup>2</sup> -wave superconductor demonstrates a non-monotonic behavior of *ic* (*γ*), as was indicated above, whereas *ic*(*γ*) for the *s*-wave CDW superconductor becomes a periodic dependence of a constant sign. The curve *ic*(*γ*) for the extended *s*-wave CDW superconductor has a different form than in the *s*-wave case, although being qualitatively similar. The presented data demonstrate that CDWs can significantly alter angle dependences often considered as a smoking gun,when determining the actual order parameter symmetry for cuprates or other like materials.

**Figure 6.** The same as in Figure 2, but for *θ*<sup>0</sup> = 15◦ and various symmetries of superconducting order parameter.

(a) (b)

dc Josephson Current Between an Isotropic and a d-Wave

or Extended s-Wave Partially Gapped Charge Density Wave Superconductor

301

(c)

(a) (b)

(c)

**Figure 10.** The same as in Figure 9, but for *dxy* order parameter symmetry.

**Figure 9.** Dependences of *ic* and its d and nd components on *<sup>γ</sup>* for *dx*<sup>2</sup>−*y*<sup>2</sup> order parameter symmetry,

*α*<sup>0</sup> = 15◦, and various *θ*0's (panel a to c).

**Figure 7.** The same as in Figure 2, but for *θ*<sup>0</sup> = 15◦ and various symmetries of superconducting order parameter.

The results for *β* = 45◦ (Figure 7(b)) differ quantitatively from their counterparts found for *β* = 0◦, but qualitative conclusions remain the same.

As was indicated above, the temperature behavior of *ic* between ordinary superconductors and cuprates is determined by the order parameter dependence Δ∗(*T*) for the material with much lower *Tc*, Nb in our case. This is demonstrated in Figure 8 for *d*-, extended *s*- and *s*-wave CDW high-*Tc* superconductors. One sees that all curves *ic* (*T*) are similar, differing only in magnitudes.

**Figure 8.** Dependences *ic*(*T*) for *γ* = 0◦, *θ*<sup>0</sup> = 30◦, *α*<sup>0</sup> = 15◦ and various symmetries of superconducting order parameter.

### **3.2. Analysis of current components**

In Figure 9, the dependences *ic* (*γ*) resolved into d and nd components are shown for CDW *d*-wave superconductors with *β* = 0◦, *α* = 15◦, and various *θ*0's. Note that the order parameter *amplitudes* at *T* = 0 are the same throughout the paper! It comes about that, for a narrow directionality cone *θ*0, the contribution of the nested (d) FS sections has quite a different tilt (*γ*) angle behavior as compared to their nd counterparts. All that gives rise to a non-monotonic pattern seen, e.g., in Figure 2(a).

300 Superconductors – Materials, Properties and Applications dc Josephson Current Between an Isotropic and a d-Wave or Extended s-Wave Partially Gapped Charge Density Wave Superconductor <sup>13</sup> 301 dc Josephson Current Between an Isotropic and a d-Wave or Extended s-Wave Partially Gapped Charge Density Wave Superconductor

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

(a) (b)

**Figure 7.** The same as in Figure 2, but for *θ*<sup>0</sup> = 15◦ and various symmetries of superconducting order

*β* = 0◦, but qualitative conclusions remain the same.

The results for *β* = 45◦ (Figure 7(b)) differ quantitatively from their counterparts found for

As was indicated above, the temperature behavior of *ic* between ordinary superconductors and cuprates is determined by the order parameter dependence Δ∗(*T*) for the material with much lower *Tc*, Nb in our case. This is demonstrated in Figure 8 for *d*-, extended *s*- and *s*-wave CDW high-*Tc* superconductors. One sees that all curves *ic* (*T*) are similar, differing

**Figure 8.** Dependences *ic*(*T*) for *γ* = 0◦, *θ*<sup>0</sup> = 30◦, *α*<sup>0</sup> = 15◦ and various symmetries of superconducting

In Figure 9, the dependences *ic* (*γ*) resolved into d and nd components are shown for CDW *d*-wave superconductors with *β* = 0◦, *α* = 15◦, and various *θ*0's. Note that the order parameter *amplitudes* at *T* = 0 are the same throughout the paper! It comes about that, for a narrow directionality cone *θ*0, the contribution of the nested (d) FS sections has quite a different tilt (*γ*) angle behavior as compared to their nd counterparts. All that gives rise to

parameter.

only in magnitudes.

order parameter.

**3.2. Analysis of current components**

a non-monotonic pattern seen, e.g., in Figure 2(a).

**Figure 9.** Dependences of *ic* and its d and nd components on *<sup>γ</sup>* for *dx*<sup>2</sup>−*y*<sup>2</sup> order parameter symmetry, *α*<sup>0</sup> = 15◦, and various *θ*0's (panel a to c).

**Figure 10.** The same as in Figure 9, but for *dxy* order parameter symmetry.

In Figure 10, the same dependences as in Figure 9 are shown, but for *β* = 45◦. One sees that, whatever complex is the *γ*-angle behavior of d contribution to the overall tunnel currents between a *dxy*-superconductor and Nb, the CDW influence is much weaker in governing the dependences *ic* (*γ*).

It is illustrative to carry out the same analysis in the scenario, when the high-*Tc* CDW superconductor is assumed to be an extended *s*-wave one, i.e. when the sign of superconducting order parameter is the same for all lobes. In the case *β* = 0◦, the corresponding results can be seen in Figure 11, where the *γ*-dependences of d and nd components of *ic*, as well as the total *ic*(*γ*) dependences, are depicted for the same parameter set as in Figure 9. We see that the d and nd contributions oscillate with the varying *γ* almost in antiphase, remaining, nevertheless, positive. For large *θ*<sup>0</sup> = 30◦ (Figure 11(c)), oscillations largely compensate each other making the curve *ic* (*γ*) almost flat, which mimics the behavior appropriate to CDW-free isotropic *s*-wave superconductors. However, we emphasize that this, at the first glance, dull result obtained for a relatively wide CDW sector is actually a consequence of a peculiar superposition involving the periodic dependences of d and nd components on *γ* with rapidly varying amplitudes.

(a) (b)

dc Josephson Current Between an Isotropic and a d-Wave

or Extended s-Wave Partially Gapped Charge Density Wave Superconductor

303

(c)

*xy* order parameter symmetry.

The results obtained confirm that the dc Josephson current, probing coherent superconducting properties [30, 31, 33, 37, 119, 256–258], is always suppressed by the electron-hole CDW pairing, which, in agreement with the totality of experimental data, is assumed here to compete with its superconducting electron-electron (Cooper) counterpart [129, 130, 132, 259–262]. We emphasize that, as concerns the quasiparticle current, the results are more ambiguous. In particular, the states on the FS around the nodes of the *d*-wave superconducting order parameter are engaged into CDW gapping [126–128, 210, 211, 235, 263], so that the ARPES or tunnel spectroscopy feels the overall energy gaps being larger than

Our examination demonstrates that the emerging CDWs should distort the dependences *ic* (*γ*), whatever is the symmetry of superconducting order parameter. It is easily seen that, for equal (or almost equal) *θ*<sup>0</sup> and *α*, CDWs make the *ic* (*γ*) curves non-monotonic and quantitatively different from their CDW-free counterparts. In particular, *ic* values are conspicuously smaller for Σ = 0. The required resonance between *θ*<sup>0</sup> and *α* can be ensured by the proper doping, i.e. a series of samples and respective tunnel junctions should be prepared with attested tilt angles *γ*, and the Josephson current should be measured for them. Of course, such measurements could be very cumbersome, although they may turn out quite realistic to

**Figure 12.** The same as in Figure 11, but for *s*ext

their superconducting constituent.

**4. Conclusions**

be performed.

**Figure 11.** The same as in Figure 9, but for *s*ext *<sup>x</sup>*2−*y*<sup>2</sup> order parameter symmetry.

The same plots as in Figure 11 were calculated for *β* = 45◦ and depicted, in Figure 12. Here, the directionality angle *θ*<sup>0</sup> is the main factor determining the amplitude of *ic*, the role of CDWs being much weaker than in the case *β* = 0◦. It is natural, because now CDW-gapping is concentrated in the nodal regions.

302 Superconductors – Materials, Properties and Applications dc Josephson Current Between an Isotropic and a d-Wave or Extended s-Wave Partially Gapped Charge Density Wave Superconductor <sup>15</sup> 303 dc Josephson Current Between an Isotropic and a d-Wave or Extended s-Wave Partially Gapped Charge Density Wave Superconductor

**Figure 12.** The same as in Figure 11, but for *s*ext *xy* order parameter symmetry.

### **4. Conclusions**

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

In Figure 10, the same dependences as in Figure 9 are shown, but for *β* = 45◦. One sees that, whatever complex is the *γ*-angle behavior of d contribution to the overall tunnel currents between a *dxy*-superconductor and Nb, the CDW influence is much weaker in governing the

It is illustrative to carry out the same analysis in the scenario, when the high-*Tc* CDW superconductor is assumed to be an extended *s*-wave one, i.e. when the sign of superconducting order parameter is the same for all lobes. In the case *β* = 0◦, the corresponding results can be seen in Figure 11, where the *γ*-dependences of d and nd components of *ic*, as well as the total *ic*(*γ*) dependences, are depicted for the same parameter set as in Figure 9. We see that the d and nd contributions oscillate with the varying *γ* almost in antiphase, remaining, nevertheless, positive. For large *θ*<sup>0</sup> = 30◦ (Figure 11(c)), oscillations largely compensate each other making the curve *ic* (*γ*) almost flat, which mimics the behavior appropriate to CDW-free isotropic *s*-wave superconductors. However, we emphasize that this, at the first glance, dull result obtained for a relatively wide CDW sector is actually a consequence of a peculiar superposition involving the periodic dependences of d and nd

(a) (b)

(c)

The same plots as in Figure 11 were calculated for *β* = 45◦ and depicted, in Figure 12. Here, the directionality angle *θ*<sup>0</sup> is the main factor determining the amplitude of *ic*, the role of CDWs being much weaker than in the case *β* = 0◦. It is natural, because now CDW-gapping is

*<sup>x</sup>*2−*y*<sup>2</sup> order parameter symmetry.

dependences *ic* (*γ*).

components on *γ* with rapidly varying amplitudes.

**Figure 11.** The same as in Figure 9, but for *s*ext

concentrated in the nodal regions.

The results obtained confirm that the dc Josephson current, probing coherent superconducting properties [30, 31, 33, 37, 119, 256–258], is always suppressed by the electron-hole CDW pairing, which, in agreement with the totality of experimental data, is assumed here to compete with its superconducting electron-electron (Cooper) counterpart [129, 130, 132, 259–262]. We emphasize that, as concerns the quasiparticle current, the results are more ambiguous. In particular, the states on the FS around the nodes of the *d*-wave superconducting order parameter are engaged into CDW gapping [126–128, 210, 211, 235, 263], so that the ARPES or tunnel spectroscopy feels the overall energy gaps being larger than their superconducting constituent.

Our examination demonstrates that the emerging CDWs should distort the dependences *ic* (*γ*), whatever is the symmetry of superconducting order parameter. It is easily seen that, for equal (or almost equal) *θ*<sup>0</sup> and *α*, CDWs make the *ic* (*γ*) curves non-monotonic and quantitatively different from their CDW-free counterparts. In particular, *ic* values are conspicuously smaller for Σ = 0. The required resonance between *θ*<sup>0</sup> and *α* can be ensured by the proper doping, i.e. a series of samples and respective tunnel junctions should be prepared with attested tilt angles *γ*, and the Josephson current should be measured for them. Of course, such measurements could be very cumbersome, although they may turn out quite realistic to be performed.

At the same time, when an *s*-wave contribution to the actual order parameter in a cuprate sample is dominant up to the complete disappearance of the *d*-wave component, the *ic* (*γ*) dependences for junctions involving CDW superconductors are no longer constant as in the CDW-free case. This prediction can be verified for CDW superconductors with *a fortiori s*-wave order parameters (such materials are quite numerous [123–128]).

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In this paper, our approach was purely theoretical. We did not discuss unavoidable experimental difficulties to face with in fabricating Josephson junctions necessary to check the results obtained here. We are fully aware that the emerging problems can be solved on the basis of already accumulated knowledge concerning the nature of grain boundaries in high-*Tc* oxides [37, 115–119, 122, 264–268]. Note that required junctions can be created at random in an uncontrollable fashion using the break-junction technique [250]. This method allows to comparatively easily detect CDW (pseudogap) influence on the tilt-angle dependences.

To summarize, measurements of the Josephson current between an ordinary superconductor and a *d*-wave or extended *s*-wave one (e.g., a high-*Tc* oxide) would be useful to detect a possible CDW influence on the electron spectrum of the latter. Similar studies of iron-based superconductors with doping-dependent spin density waves (SDWs) would also be of benefit (see, e.g., recent Reviews [78, 269–275]), since CDW and SDW superconductors have similar, although not identical, properties [123–125].
