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Superconducting current–phase relation in high-Tc symmetrical bicrystal junction
Physica C 368 (2002) 328–331 www.elsevier.com/locate/physc
Superconducting current–phase relation in high-Tc symmetrical bicrystal junction I.V. Borisenko a,*, P.B. Mozhaev a, G.A. Ovsyannikov a, K.Y. Constantinian a, E.A. Stepantsov b a
Institute of Radio Engineering and Electronics, RAS, GSP-9 Mokhovaya 11-7, 1011999 Moscow, Russia b Institute of Crystallography, RAS, 117333 Moscow, Russia
Abstract The current–phase relation (CPR) of a YBa2 Cu3 Ox Josephson junction (JJ) on a bicrystal sapphire substrate was determined using the dependences of the amplitudes of the harmonic and subharmonic Shapiro steps and the critical current dependences of the DC-SQUIDs on external magnetic field. The deviations of CPR from sinusoidal one, caused by asymmetric dc biasing, result in an appearance of subharmonic Shapiro steps and in asymmetric shape of the SQUID modulation of the Ic ðH Þ dependence. Precise measurements suppose sinusoidal CPR for symmetrical bicrystal junctions, while for asymmetric current flow in JJ the contribution of second harmonic in CPR increases monotonously. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 85.25 Keywords: Josephson junction; Grain boundary; Current–phase relation; DC-SQUID
1. Introduction Sinusoidal current–phase relation (CPR) IS ðuÞ ¼ Ic sin u (Ic is the critical current) is observed for tunnel Josephson junctions (JJs) between ordinary (s-wave) superconductors (SIS) in a wide range of temperatures [1]. However, JJ with direct (nontunnel) conductivity like point contacts (ScS) usually show sawtooth-like dependence of IS ðuÞ at low temperature. The possible reason is contribution of multiple Andreev reflections in the super*
Corresponding author. Tel.: +7-95-203-0935; fax: +7-095203-8414. E-mail address: [email protected] (I.V. Borisenko).
conducting current. Almost for all types of JJs Andreev’s levels are described by formula [2,3]: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eb ¼ D ð1 D sin2 ðu=2ÞÞ; ð1Þ where D is the superconducting gap, and D is the barrier transparency. The levels are close to D when the transparency of the barrier is small ðD 1Þ. In this case deviations from tunnelling behaviour and, hence, from sinusoidal CPR, are rarely observed in experiments. The situation considerably changes when energy states e < D are present, as well as for JJs of high-Tc superconductors, that are generally assumed to be d-wave superconductors [3–5]. In this paper we present our experimental investigation of electrical
0921-4534/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 ( 0 1 ) 0 1 1 9 1 - 1
I.V. Borisenko et al. / Physica C 368 (2002) 328–331
329
properties of the high-Tc bicrystal JJs that can result from deviations of the junction CPR from the sine form. 2. Experimental The JJs were fabricated as YBa2 Cu3 Ox (YBCO) bridges, crossing the symmetrical bicrystal boundary of the r-cut sapphire or neodymium gallate bicrystal substrates with misorientation angles of 2a ¼ 28°, 36° and 66°. The detail of fabrication technique was presented elsewhere [6–8]. The YBCO bridges were inclined to the boundary normal n by the angle c ¼ 0–70°. Typical current density of 104 –105 A/cm2 and typical V0 ¼ Ic RN of 0.5–2 mV at T ¼ 4:2 K were obtained. 2.1. Shapiro step measurements For CPR evaluation we used a method based on measurement of the junction critical current and Shapiro steps as a function of the amplitude of external monochromatic electromagnetic radiation at frequency fe . Changes in Shapiro steps were first used to estimate CPR in superconducting bridges [9] and were then applied to the high-Tc structure [10]. Within the RSJ model for x ¼ hfe = 2eIc RN P 1 the amplitudes of the harmonic components dn unambiguously determine the maximum values of the subharmonic Shapiro steps. Supposing dn ¼ 0 for n > 2 the amplitudes of the steps can be written as [8,10]: I1 ðaÞ ¼ max Ic f½ð1 d2 ÞJ1 ða=xÞ sin H þ d2 J2 ð2a=xÞ sin 2H g;
ð2Þ
I1=2 ðaÞ ¼ max Ic ½d2 J1 ð2a=xÞ sin 2H ; where the maximum is determined over the phase shift H between the self-induced oscillation and the external signal, Jn are the Bessel functions of the order n and a ¼ ARF =Ic is the normalised amplitude of the external radiation. To estimate the deviation from IS ðuÞ ¼ Ic sin u we have measured I–V curves under applied monochromatic mm wave radiation of fe ¼ 40– 100 GHz. Fig. 1 shows the variation of I1 ðAÞ and subharmonic Shapiro step I1=2 ðAÞ for two junctions with symmetrical and asymmetrical current flow
Fig. 1. Normalised RF current dependence of the first and n ¼ 1=2 Shapiro steps for two junctions with symmetric (c ¼ 0, ), and asymmetric (c ¼ 54°, ) biasing. Dashed and solid lines show the calculated curves for d2 ¼ 0 and d2 ¼ 0:2, correspondingly. The CPRs IS ðuÞ for these two cases are shown in the inset.
(c ¼ 0° and 54°). The calculated form of IS ðuÞ using the RSJ model for fe > 2eIc RN =h in the case of IS ðuÞ ¼ Ic sin u and IS ðuÞ ¼ ð1 d2 ÞIc sin u þ d2 Ic sin 2u at d ¼ 0:2 are presented on Fig. 1. For d2 < 1 the difference between two theoretical dependences of I1 ðAÞ is small and both curves fit the experimental data fairly well. On the other hand, small deviations of IS ðuÞ from sinusoidal CPR result in appearance of subharmonic Shapiro steps. The maximum amplitudes of subharmonic steps Im=n are proportional to sinðnuÞ harmonic amplitudes in IS ðuÞ, where m=n is the step order. Precise measurements of In ðAÞ, as well as Im=n ðAÞ at T ¼ 4:2 K (T =Tc 0:05) allow us to suppose that sinð2uÞ components of CPR are absent for symmetrically biased JJs (c ¼ 0–36°) with an accuracy of at least 5%. For c > 40° I1=2 increases monotonously up to I1=2 =Ic ¼ 0:25 at c ¼ 54° (Fig. 2). 2.2. DC-SQUID measurements The Ic ðBÞ dependencies of DC-SQUID were studied for signs of deviation from the sinusoidal CPR in the bicrystal JJs. Symmetrical (c1 ¼ c2 ) and asymmetrical (c1 ¼ 0°, c2 > 45°) geometry of the DC-SQUID were applied, with the same junction width and the same area of the SQUID
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Fig. 2. Dependencies of the maximum of the Shapiro subharmonic step and the critical current density on the angle of the deviation of the current direction from the normal. The solid curve gives the theoretical dependence of the critical current of JJ as a function of the angle of asymmetry of the bicrystal interface [12].
loop for both cases. The angle between the SQUID bridge directions was set to be 45° or higher; both symmetrical and asymmetrical SQUIDs were prepared on the same substrate for each specified sum of inclination angles c1 þ c2 . Fig. 3 shows typical critical current dependence on applied magnetic field for an asymmetric DCSQUID. Zero value of magnetic field was assigned
Fig. 3. The critical current dependence on applied magnetic field for asymmetric DC interferometer. The geometry of the SQUID is shown in the inset.
to the Ic ðBÞ envelope maximum [11]. For CPR evaluations we used only Ic ðBÞ dependencies with symmetrical envelopes to avoid effects of asymmetrical biasing of the SQUID and freezing of magnetic field in the junctions. For all SQUIDs containing a bridge with high asymmetric biasing inclination angle (c2 > 45°) the SQUID modulation peaks of the Ic ðBÞ dependence had asymmetric form, one of its slopes being more shallow than another. Some SQUIDs showed even an additional modulation maximum in vicinity of the main SQUID modulation peak. Moreover, the SQUID modulation is shifted so that neither maximum, nor minimum corresponds to the zero external magnetic field. These distortions are observed for the same inclination angle range (c > 45°), in which the subharmonic Shapiro step was observed for the irradiated single junctions. We calculated the Ic ðBÞ dependencies in supposition that the CPR of one of the junctions includes small second harmonics part: Iðu; ue Þ ¼ ð1 d2 ÞIc1 sin u d2 Ic1 sinð2uÞ þ Ic2 sinðu þ ue Þ;
ð3Þ
where Ic1 , Ic2 the critical currents of first and second junctions in the DC-SQUID, u the phase of the superconducting order parameter on one of the junctions, ue the external phase, introduced into the second junction by external magnetic field, and d2 is the part of the second harmonics in the first junction’s CPR. The second component in the sum is taken negative after [12], where the deviation from the CPR was reconstructed after resonant RF-SQUID measurements. The simulated Ic ðBÞ dependences are shown in Fig 4. Two main changes can be seen for the ‘2u’-SQUID compared to the ordinary sine CPR-SQUID: (i) the maximum Ic shifts from the zero magnetic field, and (ii) the shape of the dependence becomes asymmetric. For high value of the second harmonic current even additional peak can be observed; note, that the additional peak is not equidistant from the main modulation peaks. Both features, characteristic for the 2u-SQUID, can be observed for all SQUIDs, incorporating a bridge with a high inclination angle (Fig. 3).
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Therefore, for description Andreev’s level in symmetric (a ¼ b) junction in wide range of a ¼ 35– 45° we can use as an approximation the following equation [3–5]: qffiffiffiffiffiffiffiffiffiffi ð4Þ Eb ¼ DðhÞ cos½ðu pÞ=2 DðhÞ:
Fig. 4. Simulated dependencies of critical current on external magnetic field of a DC-SQUID including a JJ with a non-sine CPR at different values d2 of sin 2u component in the junction.
3. Discussion A superconducting order parameter with dwave symmetry changes sign in a–b plane, when rotated 90° around c-axis. Since the quasiparticle changes its momentum when scattered at the bicrystal boundary, and there is a sign difference of the order parameter before and after scattering, a bound state appears. An electron travelling towards the surface of d-wave superconductor, which is not parallel to a crystal axis, is reflected back into d-superconductor and is subsequently Andreev reflected into hole by the positive pair potential. In the next step the hole follows the same path backwards, reflected at the surface, finally Andreev reflected into another electron by negative pair potential [3–5]. This is the essential physical difference between tunnel JJs in s- and d-wave superconductors. For tunnel junction in d-wave superconductors (DID) the bound energy level is very close to Fermi level while for s-wave superconductor the energy is always close to the gap. For a DID junction with order parameters at the left and right banks DRðLÞ ¼ D0 cosð2h þ 2aðbÞÞ the energy of the bound states Eb depends on four angles: quasiparticle incident angle h, phase u and misorientation angles aðbÞ. For a ¼ 35–45° a small amount of quasiparticles in the range h ¼ 0–10° satisfies the condition max jEb j > 0:1D0 [8].
The IS ðuÞ can be determined from the energy of the bound Andreev levels Eb in the junction. Our calculations of IS ðuÞ for a ¼ 45° give d2 ¼ 0:2 in comparison with experimental d2 < 0:05. The actual mechanism leading to distortion of CPR remains unclear. The faceting of the interface boundary and twinning of the high-Tc films could be the reasons for sin u CPR for symmetrical biasing. All facets give different contribution to CPR of the junction which lead to sin u CPR during averaging. For asymmetric biasing the contribution for different oriented facets is nonequal [3–5].
Acknowledgements Authors would like to thank Dr. Kornev for fruitful discussion. The work was supported in part by Russian State Program ‘‘Modern Problems of the Solid State Physics’’, ‘‘Superconductivity’’ division, Russian Foundation for Basic Research, grant no. 00-02-1746, NATO ‘‘Science for Peace’’ program, grant no. 973559, and ISTC grant no. 1199.
References [1] K.K. Likharev, Rev. Mod. Phys. 51 (1979) 102. [2] A. Furusaki, M. Tsukada, Phys. Rev. B 43 (1991) 10164. [3] Y.S. Barash, H. Burkhardt, D. Rainer, Phys. Rev. Lett. 77 (1996) 4070. [4] Y. Tanaka, S. Kashiwaya, Phys. Rev. B 56 (1997) 892. [5] R.A. Riedel, P.F. Bagwell, Phys. Rev. B 57 (1198) 6084. [6] A.D. Mashtakov et al., Tech. Phys. Lett. 25 (1999) 249. [7] G.A. Ovsyannikov et al., J. Low Temp. Phys. 117 (1999) 605. [8] G.A. Ovsyannikov et al., Tech. Phys. Lett. 25 (1999) 913. [9] V.N. Gubankov, V.P. Koshelets, G.A. Ovsyannikov, Sov. Phys. J. Expt. Theor. Phys. 44 (1976) 181. [10] R. Kleiner et al., Phys. Rev. Lett. 76 (1996) 2161. [11] R.R. Schulz et al., Appl. Phys. Lett. 76 (2000) 912. [12] E. Il’ichev et al., Phys. Rev Lett. 81 (1999) 894.
Superconducting current–phase relation in high-Tc symmetrical bicrystal junction I.V. Borisenko a,*, P.B. Mozhaev a, G.A. Ovsyannikov a, K.Y. Constantinian a, E.A. Stepantsov b a
Institute of Radio Engineering and Electronics, RAS, GSP-9 Mokhovaya 11-7, 1011999 Moscow, Russia b Institute of Crystallography, RAS, 117333 Moscow, Russia
Abstract The current–phase relation (CPR) of a YBa2 Cu3 Ox Josephson junction (JJ) on a bicrystal sapphire substrate was determined using the dependences of the amplitudes of the harmonic and subharmonic Shapiro steps and the critical current dependences of the DC-SQUIDs on external magnetic field. The deviations of CPR from sinusoidal one, caused by asymmetric dc biasing, result in an appearance of subharmonic Shapiro steps and in asymmetric shape of the SQUID modulation of the Ic ðH Þ dependence. Precise measurements suppose sinusoidal CPR for symmetrical bicrystal junctions, while for asymmetric current flow in JJ the contribution of second harmonic in CPR increases monotonously. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 85.25 Keywords: Josephson junction; Grain boundary; Current–phase relation; DC-SQUID
1. Introduction Sinusoidal current–phase relation (CPR) IS ðuÞ ¼ Ic sin u (Ic is the critical current) is observed for tunnel Josephson junctions (JJs) between ordinary (s-wave) superconductors (SIS) in a wide range of temperatures [1]. However, JJ with direct (nontunnel) conductivity like point contacts (ScS) usually show sawtooth-like dependence of IS ðuÞ at low temperature. The possible reason is contribution of multiple Andreev reflections in the super*
Corresponding author. Tel.: +7-95-203-0935; fax: +7-095203-8414. E-mail address: [email protected] (I.V. Borisenko).
conducting current. Almost for all types of JJs Andreev’s levels are described by formula [2,3]: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eb ¼ D ð1 D sin2 ðu=2ÞÞ; ð1Þ where D is the superconducting gap, and D is the barrier transparency. The levels are close to D when the transparency of the barrier is small ðD 1Þ. In this case deviations from tunnelling behaviour and, hence, from sinusoidal CPR, are rarely observed in experiments. The situation considerably changes when energy states e < D are present, as well as for JJs of high-Tc superconductors, that are generally assumed to be d-wave superconductors [3–5]. In this paper we present our experimental investigation of electrical
0921-4534/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 ( 0 1 ) 0 1 1 9 1 - 1
I.V. Borisenko et al. / Physica C 368 (2002) 328–331
329
properties of the high-Tc bicrystal JJs that can result from deviations of the junction CPR from the sine form. 2. Experimental The JJs were fabricated as YBa2 Cu3 Ox (YBCO) bridges, crossing the symmetrical bicrystal boundary of the r-cut sapphire or neodymium gallate bicrystal substrates with misorientation angles of 2a ¼ 28°, 36° and 66°. The detail of fabrication technique was presented elsewhere [6–8]. The YBCO bridges were inclined to the boundary normal n by the angle c ¼ 0–70°. Typical current density of 104 –105 A/cm2 and typical V0 ¼ Ic RN of 0.5–2 mV at T ¼ 4:2 K were obtained. 2.1. Shapiro step measurements For CPR evaluation we used a method based on measurement of the junction critical current and Shapiro steps as a function of the amplitude of external monochromatic electromagnetic radiation at frequency fe . Changes in Shapiro steps were first used to estimate CPR in superconducting bridges [9] and were then applied to the high-Tc structure [10]. Within the RSJ model for x ¼ hfe = 2eIc RN P 1 the amplitudes of the harmonic components dn unambiguously determine the maximum values of the subharmonic Shapiro steps. Supposing dn ¼ 0 for n > 2 the amplitudes of the steps can be written as [8,10]: I1 ðaÞ ¼ max Ic f½ð1 d2 ÞJ1 ða=xÞ sin H þ d2 J2 ð2a=xÞ sin 2H g;
ð2Þ
I1=2 ðaÞ ¼ max Ic ½d2 J1 ð2a=xÞ sin 2H ; where the maximum is determined over the phase shift H between the self-induced oscillation and the external signal, Jn are the Bessel functions of the order n and a ¼ ARF =Ic is the normalised amplitude of the external radiation. To estimate the deviation from IS ðuÞ ¼ Ic sin u we have measured I–V curves under applied monochromatic mm wave radiation of fe ¼ 40– 100 GHz. Fig. 1 shows the variation of I1 ðAÞ and subharmonic Shapiro step I1=2 ðAÞ for two junctions with symmetrical and asymmetrical current flow
Fig. 1. Normalised RF current dependence of the first and n ¼ 1=2 Shapiro steps for two junctions with symmetric (c ¼ 0, ), and asymmetric (c ¼ 54°, ) biasing. Dashed and solid lines show the calculated curves for d2 ¼ 0 and d2 ¼ 0:2, correspondingly. The CPRs IS ðuÞ for these two cases are shown in the inset.
(c ¼ 0° and 54°). The calculated form of IS ðuÞ using the RSJ model for fe > 2eIc RN =h in the case of IS ðuÞ ¼ Ic sin u and IS ðuÞ ¼ ð1 d2 ÞIc sin u þ d2 Ic sin 2u at d ¼ 0:2 are presented on Fig. 1. For d2 < 1 the difference between two theoretical dependences of I1 ðAÞ is small and both curves fit the experimental data fairly well. On the other hand, small deviations of IS ðuÞ from sinusoidal CPR result in appearance of subharmonic Shapiro steps. The maximum amplitudes of subharmonic steps Im=n are proportional to sinðnuÞ harmonic amplitudes in IS ðuÞ, where m=n is the step order. Precise measurements of In ðAÞ, as well as Im=n ðAÞ at T ¼ 4:2 K (T =Tc 0:05) allow us to suppose that sinð2uÞ components of CPR are absent for symmetrically biased JJs (c ¼ 0–36°) with an accuracy of at least 5%. For c > 40° I1=2 increases monotonously up to I1=2 =Ic ¼ 0:25 at c ¼ 54° (Fig. 2). 2.2. DC-SQUID measurements The Ic ðBÞ dependencies of DC-SQUID were studied for signs of deviation from the sinusoidal CPR in the bicrystal JJs. Symmetrical (c1 ¼ c2 ) and asymmetrical (c1 ¼ 0°, c2 > 45°) geometry of the DC-SQUID were applied, with the same junction width and the same area of the SQUID
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Fig. 2. Dependencies of the maximum of the Shapiro subharmonic step and the critical current density on the angle of the deviation of the current direction from the normal. The solid curve gives the theoretical dependence of the critical current of JJ as a function of the angle of asymmetry of the bicrystal interface [12].
loop for both cases. The angle between the SQUID bridge directions was set to be 45° or higher; both symmetrical and asymmetrical SQUIDs were prepared on the same substrate for each specified sum of inclination angles c1 þ c2 . Fig. 3 shows typical critical current dependence on applied magnetic field for an asymmetric DCSQUID. Zero value of magnetic field was assigned
Fig. 3. The critical current dependence on applied magnetic field for asymmetric DC interferometer. The geometry of the SQUID is shown in the inset.
to the Ic ðBÞ envelope maximum [11]. For CPR evaluations we used only Ic ðBÞ dependencies with symmetrical envelopes to avoid effects of asymmetrical biasing of the SQUID and freezing of magnetic field in the junctions. For all SQUIDs containing a bridge with high asymmetric biasing inclination angle (c2 > 45°) the SQUID modulation peaks of the Ic ðBÞ dependence had asymmetric form, one of its slopes being more shallow than another. Some SQUIDs showed even an additional modulation maximum in vicinity of the main SQUID modulation peak. Moreover, the SQUID modulation is shifted so that neither maximum, nor minimum corresponds to the zero external magnetic field. These distortions are observed for the same inclination angle range (c > 45°), in which the subharmonic Shapiro step was observed for the irradiated single junctions. We calculated the Ic ðBÞ dependencies in supposition that the CPR of one of the junctions includes small second harmonics part: Iðu; ue Þ ¼ ð1 d2 ÞIc1 sin u d2 Ic1 sinð2uÞ þ Ic2 sinðu þ ue Þ;
ð3Þ
where Ic1 , Ic2 the critical currents of first and second junctions in the DC-SQUID, u the phase of the superconducting order parameter on one of the junctions, ue the external phase, introduced into the second junction by external magnetic field, and d2 is the part of the second harmonics in the first junction’s CPR. The second component in the sum is taken negative after [12], where the deviation from the CPR was reconstructed after resonant RF-SQUID measurements. The simulated Ic ðBÞ dependences are shown in Fig 4. Two main changes can be seen for the ‘2u’-SQUID compared to the ordinary sine CPR-SQUID: (i) the maximum Ic shifts from the zero magnetic field, and (ii) the shape of the dependence becomes asymmetric. For high value of the second harmonic current even additional peak can be observed; note, that the additional peak is not equidistant from the main modulation peaks. Both features, characteristic for the 2u-SQUID, can be observed for all SQUIDs, incorporating a bridge with a high inclination angle (Fig. 3).
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Therefore, for description Andreev’s level in symmetric (a ¼ b) junction in wide range of a ¼ 35– 45° we can use as an approximation the following equation [3–5]: qffiffiffiffiffiffiffiffiffiffi ð4Þ Eb ¼ DðhÞ cos½ðu pÞ=2 DðhÞ:
Fig. 4. Simulated dependencies of critical current on external magnetic field of a DC-SQUID including a JJ with a non-sine CPR at different values d2 of sin 2u component in the junction.
3. Discussion A superconducting order parameter with dwave symmetry changes sign in a–b plane, when rotated 90° around c-axis. Since the quasiparticle changes its momentum when scattered at the bicrystal boundary, and there is a sign difference of the order parameter before and after scattering, a bound state appears. An electron travelling towards the surface of d-wave superconductor, which is not parallel to a crystal axis, is reflected back into d-superconductor and is subsequently Andreev reflected into hole by the positive pair potential. In the next step the hole follows the same path backwards, reflected at the surface, finally Andreev reflected into another electron by negative pair potential [3–5]. This is the essential physical difference between tunnel JJs in s- and d-wave superconductors. For tunnel junction in d-wave superconductors (DID) the bound energy level is very close to Fermi level while for s-wave superconductor the energy is always close to the gap. For a DID junction with order parameters at the left and right banks DRðLÞ ¼ D0 cosð2h þ 2aðbÞÞ the energy of the bound states Eb depends on four angles: quasiparticle incident angle h, phase u and misorientation angles aðbÞ. For a ¼ 35–45° a small amount of quasiparticles in the range h ¼ 0–10° satisfies the condition max jEb j > 0:1D0 [8].
The IS ðuÞ can be determined from the energy of the bound Andreev levels Eb in the junction. Our calculations of IS ðuÞ for a ¼ 45° give d2 ¼ 0:2 in comparison with experimental d2 < 0:05. The actual mechanism leading to distortion of CPR remains unclear. The faceting of the interface boundary and twinning of the high-Tc films could be the reasons for sin u CPR for symmetrical biasing. All facets give different contribution to CPR of the junction which lead to sin u CPR during averaging. For asymmetric biasing the contribution for different oriented facets is nonequal [3–5].
Acknowledgements Authors would like to thank Dr. Kornev for fruitful discussion. The work was supported in part by Russian State Program ‘‘Modern Problems of the Solid State Physics’’, ‘‘Superconductivity’’ division, Russian Foundation for Basic Research, grant no. 00-02-1746, NATO ‘‘Science for Peace’’ program, grant no. 973559, and ISTC grant no. 1199.
References [1] K.K. Likharev, Rev. Mod. Phys. 51 (1979) 102. [2] A. Furusaki, M. Tsukada, Phys. Rev. B 43 (1991) 10164. [3] Y.S. Barash, H. Burkhardt, D. Rainer, Phys. Rev. Lett. 77 (1996) 4070. [4] Y. Tanaka, S. Kashiwaya, Phys. Rev. B 56 (1997) 892. [5] R.A. Riedel, P.F. Bagwell, Phys. Rev. B 57 (1198) 6084. [6] A.D. Mashtakov et al., Tech. Phys. Lett. 25 (1999) 249. [7] G.A. Ovsyannikov et al., J. Low Temp. Phys. 117 (1999) 605. [8] G.A. Ovsyannikov et al., Tech. Phys. Lett. 25 (1999) 913. [9] V.N. Gubankov, V.P. Koshelets, G.A. Ovsyannikov, Sov. Phys. J. Expt. Theor. Phys. 44 (1976) 181. [10] R. Kleiner et al., Phys. Rev. Lett. 76 (1996) 2161. [11] R.R. Schulz et al., Appl. Phys. Lett. 76 (2000) 912. [12] E. Il’ichev et al., Phys. Rev Lett. 81 (1999) 894.