## 2 Junctions and Methods  
### Junctions  
We used the valley type of bicrystal high-\(T_{c}\) JJs with mutually tilted \(c\)-axes [21] schematically shown in Figure 1a. Each high-\(T_{c}\) electrode looks like a stack of superconducting layers along the \(a\)-\(b\) planes. Thin-film electrodes to the left and the right of the bicrystal boundary (plane \(x=0\)) have the c-axes tilted toward the boundary by the same angle \(\alpha\), and the \(a\)-axes are parallel to the boundary. Previously, there were attempts to assume that the \(a\)- or \(b\)-axis passes along the bicrystal boundary and to consider films with a tilt of the \(c\)-axis due to rotation around the [100]- or [010]-axis, respectively. However, our YBa\({}_{2}\)Cu\({}_{3}\)O\({}_{7-x}\) thin films consist of twin domains with a set of twin boundaries, where the \(a\)- and \(b\)-orientations of the film change directions. Therefore, we call such junctions \"bicrystal junctions with mutually tilted \(c\)-axes.\" Because of twinning in the junction electrodes, it can be expected that a high-frequency electromagnetic electric field generated perpendicular to the bicrystal boundary can excite collective modes in the electrodes both in \(a\)- and \(b\)-directions. The same remark is also valid for the mode excitation along the \(c\)-axis. Valley-type JJs were fabricated by epitaxial growth of YBa\({}_{2}\)Cu\({}_{3}\)O\({}_{7-x}\) thin films on bicrystal substrates made from NdGaO\({}_{3}\), SrTiO\({}_{3}\) or NdGaO\({}_{3}\) with a thin (\(\approx\)10 nm) buffer layer of SrTiO\({}_{3}\). The bicrystal parts of the substrates were oriented to get the mutual tilt of the \(c\)-axes of the counter electrodes [20; 21]. For the deposition of thin films, we used dc sputtering from a stoichiometric target at a high oxygen pressure of 3.0-3.4 mbar and heater temperatures of 930\({}^{\circ}\)C - 970\({}^{\circ}\)C. The deposition rate was about 1 nm per minute. After deposition, the films were at an oxygen pressure of 1 bar and a heater temperature of 550\({}^{\circ}\)C for 30 min. Figure 1b shows the AFM amplitude image for a typical YBa\({}_{2}\)Cu\({}_{3}\)O\({}_{7-x}\) thin film  
Figure 1: **(a)** Schematic representation of a bicrystal high-\(T_{c}\) junction of the “valley” type with mutually tilted \(c\)-axes. **(b)** AFM amplitude image (10 \(\times\) 10 μm) of a typical fabricated YBa\({}_{2}\)Cu\({}_{3}\)O\({}_{7-x}\) thin film on a 2 \(\times\) 12\({}^{\circ}\) SrTiO\({}_{3}\) bicrystal substrate consisting of two parts with mutually tilted \(c\)-axes. Sample yd107201. **(c)** AFM topographic image (9 \(\times\) 9 μm) of an YBa\({}_{2}\)Cu\({}_{3}\)O\({}_{7-x}\) thin-film bridge across the bicrystal boundary on a SrTiO\({}_{3}\) bicrystal substrate with mutually tilted \(c\)-axes. Sample yd10711. **(d)** Profile of the film surface during scanning across the bicrystal boundary. The plane z = 0 is the plane of the substrate. fabricated on a bicrystal SrTiO\({}_{3}\) substrate with the \(c\)-axes tilted to the boundary by \(\pm 12^{\circ}\). The film regions remote from the bicrystal boundary (x = 0) consist of grains extended up to 1-2 um in the y-direction along the boundary and compressed to approximately 0.2 um in the x-direction perpendicular to the boundary. In the region of the bicrystal boundary, the deposited thin YBa\({}_{2}\)Cu\({}_{3}\)O\({}_{7-\mathrm{x}}\) film has a regular V-shaped profile, which is the result of the nucleation of YBa\({}_{2}\)Cu\({}_{3}\)O\({}_{7-\mathrm{x}}\) at the boundary of the substrate bicrystal (x = 0) and the predominant growth of YBa\({}_{2}\)Cu\({}_{3}\)O\({}_{7-\mathrm{x}}\) along the \(a\)-\(b\) plane in the left and right parts of the bicrystal substrates. The bicrystal boundary in a thin film here deviates from the substrate boundary by no more than \(\pm 20\) nm, which is approximately an order of magnitude less than the meandering of grain boundaries in conventional bicrystal JJs made from \(c\)-oriented films.