Phase diagram of a current-carrying superconducting film in absence of the magnetic field

It is present the phase diagram for the current states of superconducting films, based on the experimental investigation of the resistive transitions induced by the transport current. It is found that a comparatively narrow film with the width w<5λ⊥(T)(λ⊥ is the penetration depth of the magnetic...

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Date:	2010
Main Authors:	Bezuglyi, E.V., Zolochevskii, I.V.
Format:	Article
Language:	English
Published:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2010
Series:	Физика низких температур
Subjects:	Quantum coherent effects in superconductors and normal metals
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/117527
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Cite this:	Phase diagram of a current-carrying superconducting film in absence of the magnetic field / E.V. Bezuglyi, I.V. Zolochevskii // Физика низких температур. — 2010. — Т. 36, № 10-11. — С. 1248–1252. — Бібліогр.: 26 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-1175272025-02-09T09:36:36Z Phase diagram of a current-carrying superconducting film in absence of the magnetic field Bezuglyi, E.V. Zolochevskii, I.V. Quantum coherent effects in superconductors and normal metals It is present the phase diagram for the current states of superconducting films, based on the experimental investigation of the resistive transitions induced by the transport current. It is found that a comparatively narrow film with the width w<5λ⊥(T)(λ⊥ is the penetration depth of the magnetic field) never enters the vortex state, but experiences direct transition from the purely superconducting state to the resistive state with phase-slip centers as soon as the current exceeds the critical Ginzburg–Landau current IGLc. The Meissner current state of the films of intermediate width, 5λ⊥(T)<w<10λ⊥(T), transforms at I>0.8IGLc to the vortex resistive state which exists within the current interval 0.8IGLc<I<Im, where the value of the upper critical current is in a good agreement with the heory. The vortex state of wide films, w > 10λ⊥(T), is realized within the current region IALc<I<Im, where IALc is the transition point to the vortex state calculated for the limiting case w>>λ⊥.AtI>Im, the films with the width w>5λ⊥(T) enter a vortex-free resistive state with phase-slip lines. 2010 Article Phase diagram of a current-carrying superconducting film in absence of the magnetic field / E.V. Bezuglyi, I.V. Zolochevskii // Физика низких температур. — 2010. — Т. 36, № 10-11. — С. 1248–1252. — Бібліогр.: 26 назв. — англ. 0132-6414 PACS: 74.40.+k, 74.25.Nf https://nasplib.isofts.kiev.ua/handle/123456789/117527 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Quantum coherent effects in superconductors and normal metals Quantum coherent effects in superconductors and normal metals
spellingShingle	Quantum coherent effects in superconductors and normal metals Quantum coherent effects in superconductors and normal metals Bezuglyi, E.V. Zolochevskii, I.V. Phase diagram of a current-carrying superconducting film in absence of the magnetic field Физика низких температур
description	It is present the phase diagram for the current states of superconducting films, based on the experimental investigation of the resistive transitions induced by the transport current. It is found that a comparatively narrow film with the width w<5λ⊥(T)(λ⊥ is the penetration depth of the magnetic field) never enters the vortex state, but experiences direct transition from the purely superconducting state to the resistive state with phase-slip centers as soon as the current exceeds the critical Ginzburg–Landau current IGLc. The Meissner current state of the films of intermediate width, 5λ⊥(T)<w<10λ⊥(T), transforms at I>0.8IGLc to the vortex resistive state which exists within the current interval 0.8IGLc<I<Im, where the value of the upper critical current is in a good agreement with the heory. The vortex state of wide films, w > 10λ⊥(T), is realized within the current region IALc<I<Im, where IALc is the transition point to the vortex state calculated for the limiting case w>>λ⊥.AtI>Im, the films with the width w>5λ⊥(T) enter a vortex-free resistive state with phase-slip lines.
format	Article
author	Bezuglyi, E.V. Zolochevskii, I.V.
author_facet	Bezuglyi, E.V. Zolochevskii, I.V.
author_sort	Bezuglyi, E.V.
title	Phase diagram of a current-carrying superconducting film in absence of the magnetic field
title_short	Phase diagram of a current-carrying superconducting film in absence of the magnetic field
title_full	Phase diagram of a current-carrying superconducting film in absence of the magnetic field
title_fullStr	Phase diagram of a current-carrying superconducting film in absence of the magnetic field
title_full_unstemmed	Phase diagram of a current-carrying superconducting film in absence of the magnetic field
title_sort	phase diagram of a current-carrying superconducting film in absence of the magnetic field
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2010
topic_facet	Quantum coherent effects in superconductors and normal metals
url	https://nasplib.isofts.kiev.ua/handle/123456789/117527
citation_txt	Phase diagram of a current-carrying superconducting film in absence of the magnetic field / E.V. Bezuglyi, I.V. Zolochevskii // Физика низких температур. — 2010. — Т. 36, № 10-11. — С. 1248–1252. — Бібліогр.: 26 назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT bezuglyiev phasediagramofacurrentcarryingsuperconductingfilminabsenceofthemagneticfield AT zolochevskiiiv phasediagramofacurrentcarryingsuperconductingfilminabsenceofthemagneticfield
first_indexed	2025-11-25T10:03:45Z
last_indexed	2025-11-25T10:03:45Z
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fulltext	© E.V. Bezuglyi and I.V. Zolochevskii, 2010 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11, p. 1248–1252 Phase diagram of a current-carrying superconducting film in absence of the magnetic field E.V. Bezuglyi and I.V. Zolochevskii B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine 47 Lenin Ave., Kharkov 61103, Ukraine E-mail: zolochevskii@ilt.kharkov.ua Received April 30, 2010 It is present the phase diagram for the current states of superconducting films, based on the experimental in- vestigation of the resistive transitions induced by the transport current. It is found that a comparatively narrow film with the width w < 5λ⊥ (T) (λ⊥ is the penetration depth of the magnetic field) never enters the vortex state, but experiences direct transition from the purely superconducting state to the resistive state with phase-slip cen- ters as soon as the current exceeds the critical Ginzburg–Landau current Ic GL. The Meissner current state of the films of intermediate width, 5λ⊥(T) < w < 10λ⊥(T), transforms at I > 0.8Ic GL to the vortex resistive state which exists within the current interval 0.8Ic GL < I < Im, where the value of the upper critical current is in a good agreement with the heory. The vortex state of wide films, w > 10λ⊥(T), is realized within the current region Ic AL < I < Im, where Ic AL is the transition point to the vortex state calculated for the limiting case w >> λ⊥. At I > Im, the films with the width w > 5λ⊥(T) enter a vortex-free resistive state with phase-slip lines. PACS 74.40.+k Fluctuations (noise, chaos, nonequilibrium superconductivity, localization, etc.); 74.25.Nf Response to electromagnetic fields (nuclear magnetic resonance, surface impedance, etc.) Keywords: wide superconducting film, vortex resistive state, phase-slip processes, Aslamazov–Lempitskiy theory. According to the Ginzburg–Landau (GL) theory [1], if the transport current through a narrow superconducting film (superconducting channel) exceeds the depairing cur- rent, 0 3/2 2 ( ) (1 / ) 6 3 (0) (0) GL c c c w I T T T ⊥ Φ = − π ξ λ , (1) the superconducting state of the channel is destroyed and transforms to the normal state, as shown in Fig. 1. In (1), 0Φ is the magnetic flux quantum, w is the film width, 2(0) 2 (0)/d⊥λ = λ is the penetration depth of the magnetic field into the superconducting film, (0)ξ and (0)λ are the coherence length and London length, respectively, at zero temperature, and d is the film thickness. Later it was found that the real scenario of the resistive phase transition of the superconducting channel is more complex. Namely, as the transport current exceeds ( )GL cI T , an inhomogene- ous resistive state appears in the channel, consisting of alternating superconducting and quasi-normal regions [2]; the latter are the specific dynamic formations known as phase-slip centers (PSCs). The number of PSCs increases with the transport current, and at GL cn cI I I= the resis- tive state turns to the completely normal state. The basic feature of the current-voltage characteristics (IVCs) of su- perconducting channels in the resistive state are regular voltage steps (Fig. 2), which were first observed in tin whiskers [3,4] and in narrow tin films [5]. We note the following important peculiarities of the step-like IVCs: the Fig. 1. Current-voltage characteristic (IVC) of a narrow super- conducting channel according to the GL theory. nR is the chan- nel resistance in the normal state, and nV is the voltage jump at the point of the resistive transition. 00 11 11 IIcc V /V n V /V n RRnn I/II/Icc VVnn Phase diagram of a current-carrying superconducting film in absence of the magnetic field Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 1249 multiplicity of differential resistances of the sloping IVC parts, the intersection of the continued sloping parts (dashed lines) at a single point on the current axis, and the absence of hysteresis which indicates the non-heating na- ture of the voltage steps. To the present time, the resistive state of narrow super- conducting channels has been rather well studied experi- mentally, and the theory of this state has been universally recognized. In our opinion, this is not the case for the resis- tive state of a wide film, in spite of the fact that the study of current-carrying states in wide films started much earlier than the investigations of superconducting channels. In 1963, Tinkham [6] first involved the vortex conception for calculation of the critical fields of thin films. It was found in [7] that the motion of vortices, induced by the magnetic field of the Earth or the transport current, plays the crucial role in formation of the initial part of the IVCs. In next studies, the resistive state of wide films was associated only with the vortex motion. This looked quite natural be- cause at that time, typical observed IVCs were similar to that shown in Fig. 3, with abrupt transition from the vortex resistive state to the normal state. We believe that in most cases, the origin of such a break-off form of the IVC is imperfection of both the experimental conditions and the samples. We note that in the experimental studies of the resistive state, the heat compliance between the film and the substrate plays an important role [8]. If the choice of the pair «film–substrate» is not optimal, or the adhesion of the film to the substrate is imperfect, the Joule overheating of the film in the resistive state leads to the break-off IVCs shown in Figs. 1 and 3. In 1972, in study of wide tin films sputtered on the quartz substrate which provides optimal heat compliance, the authors of [9] observed not only a typical vortex region of the IVC at small transport current, but also a step-like structure at large current, which obtained no physical ex- planation at that time. Relying on the recommendations given in [8], the authors of [10] fabricated the films having the IVCs similar to that shown in Fig. 4. Then, using their knowledge about the phase-slip processes in narrow chan- nels, they associated the voltage steps in the IVCs of wide films with the creation of phase-slip lines (PSLs). At the present time, it seemed that such a form of the IVC for a wide film, consisting of the vortex region and the step- like part due to the PSLs, is widely recognized; see, e.g., [9–13]. However, the break-off-type IVCs are still fre- quently observed [14-18]. Let us first discuss the theoretical models of the vortex resistive state. We would like to draw one’s attention to the theoretical papers [20,21] which do not explain the whole form of the IVC shown in Fig. 4, but give a rather good description of the vortex part of the IVC. In wide thin su- perconducting films, the magnetic field of the transport current gives rise to the creation of Pearl vortices at the film edges. The motion of the vortices across the film leads to the occurrence of a voltage along the film. On the basis of such a picture of the resistive vortex state, the equation for the critical current was found by Aslamazov and Lem- pitskiy (AL) [20], 1 2( ) 1.5 (0)( (0) ) (1 )AL GL c c cI T I w T T⊥= πλ − , (2) Fig. 2. Typical IVC of a superconducting film channel Sn4 at the temperature / 0.98cT T = . 00 100100 200200 300300 200200 400400 600600 VV ,, VV �� I,I, AA�� IIcc Fig. 3. Break-off-type IVC of a wide superconducting film. 00 11 11 V /V V /V nn RRnn VVnn IIcc I/II/Icc VVmm IImm Fig. 4. Experimental IVC of a wide film SnW13 at T = 0.99 cT . 0,50,5 1,01,0 00 1010 2020 3030 4040 VV ,, VV �� I, mAI, mA TT = 3.798 K= 3.798 K IIcc IImm E.V. Bezuglyi and I.V. Zolochevskii 1250 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 using the condition of stability of the Meissner state with respect to an infinitely small perturbation of the supercon- ducting order parameter. Physically, at AL cI I= , the edge current density approaches the GL critical value, and the edge barrier for the vortex entry disappears. In order to describe the vortex part of the IVC, AL studied the viscous motion of the vortices in the film, using the hydrodynamic approximation which assumes introduction of averaged macroscopic quantities: the density of vortices and the av- eraged current density satisfying the macroscopic equa- tions which connect these quantities with the averaged electric field. According to this theory, the evolution of the resistive vortex state looks as follows. As the current grows, the vortex density increases, and the current distri- bution across the film becomes more homogeneous. At a certain current value, ( 1 2)( ) ( ) ln (2 ( ))GL AL m c cI T C I T w T Iλ− ⊥= , (3) the current density approaches its critical value not only at the film edges, where the vortices are born, but also in the middle of the film cross-section, where the vortices and anti-vortices annihilate. At this moment the vortex state becomes unstable, although the distance between vortices is still larger than the size of the vortex core, and the au- thors of [20] assert that the film undergoes jump-like tran- sition to the normal state, as shown in Fig. 3. The experi- mental study of wide films [12] confirmed much of the statements of the AL theory, including correctness of equa- tions (2) and (3). Besides, these investigations also resulted in considerable refinements of several points of the theory. It turned out [12] that the vortex resistivity occurs only at large enough film width, 5 ( )w T⊥> λ , and at mI I> the film undergoes transition not to the normal state, but to a vortex-free state with PSLs (Fig. 4). Such a picture of the resistive state of a wide film was later recognized in [22] by S.V. Lempitskiy, one of the authors of the resistive vor- tex state theory. He predicted that if the distances between PSLs are larger than the penetration depth for the electric field into the superconductor, then the IVC of a wide film will be described by known equations for a vortex-free narrow superconducting channel [2]. This result was also confirmed experimentally [23,24]. Another approach to the analysis of the vortex mechan- ism of resistivity was used by Vodolazov and Peeters [21] which performed numerical solution of extended time- dependent Ginzburg–Landau equations [25,26] for the vor- tex motion in the superconducting plate, formally consider- ing an infinitely thick superconducting slab which reduces the problem to a two-dimensional one. In this aspect, the problem becomes rather close to the problem of the vortex state in a thin film, although the structure of vortices in these cases is quite different: exponential decay of the magnetic field around the Abrikosov vortices in a massive slab and slow power-like decay around the Pearl vortices in a thin film. The results of [21] generally confirm the macroscopic analysis of the vortex state by AL [20]: when the current exceeds the critical value, at which the surface barrier for the vortex entry is suppressed, the vortices and anti-vortices enter the slab and then, being affected by the transport current, move to the middle of the sample, where they annihilate. This process leads to a maximum of the current density in the middle of the sample, in accordance with macroscopic calculations in [20]. At moderate values of the transport current, the vortex structure of the sample is close to the triangle lattice. However, when the current density in the middle of the sample approaches the depair- ing value, i.e., when the transport current approaches mI , the triangle vortex lattice turns into row-like vortex struc- ture. The authors of [21] interpret this phenomenon as the creation of quasi-PSLs due to acceleration of the vortex– anti-vortex annihilation and anomalously rapid motion of the vortices. According to this theory, the PSLs represent the rows of rapidly moving vortices which occur simulta- neously along the whole length of the sample. This contra- dicts the experimental data shown in Fig. 4 which indicate consequent appearance of the PSLs while the transport current increases. For this reason, in order to confirm their conclusions, the authors of [21] refer only to the experi- ments [14–18], in which the break-off-like IVCs were ob- served. In our opinion, these experiments have mutual drawback: the substrates do not provide a good heat re- moval from the films. This is indicated by hysteresis ob- served, e.g., in [16], or by low quality of the films with numerous pinning centers and bad adhesion with the sub- strate, as in [15]. We believe that for these reasons, the PSLs occur in such films in an avalanche, which is accom- panied by a break-off in the IVC. Our point of view is sup- ported by the results of investigation of the resistive state of wide films by the laser scanning microscope [19]. Due to specifics of this method, the film is overheated by the laser irradiation, which gives rise to the break-off behavior of the IVC (Fig. 3). However, the visualization of the resis- tive state by this microscope shows that at mI I> the number of PSLs increases smoothly with the transport cur- rent, starting from the single PSL at mI I= . Now we proceed to the results of our experimental stu- dies. The samples whose characteristics are given in the Table were fabricated by using an original technique which provides minimization of bulk defects and results in per- fect, almost specular, film edges. All such samples show full GL critical current in a near vicinity of cT , where the condition of narrow channel regime ( )w T⊥< λ is satis- fied; actually, this criterion was used for the selection of samples for next experiments in the resistive state. Special attention was devoted to the quality of the substrate — optically polished crystalline quartz, which seems to be the better material for the heat removal from tin films. Phase diagram of a current-carrying superconducting film in absence of the magnetic field Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 1251 As follows from the AL theory [20] (see (2) and (3)), the reduced critical currents ( )AL GL c cI I T and ( )GL m cI I T for wide films, 1w λ⊥ , must be universal functions of the basic parameter of the theory, ( )w T⊥λ . In other words, the dependencies of the reduced critical currents on this parameter should be not affected either by the geome- try of the films, or by their material properties. Our expe- rimental results completely confirm such universality and, moreover, extend it over the small values of w ⊥λ . Ex- amples of the experimental dependencies of the reduced critical currents are presented in Fig. 5. To avoid overload- ing of the figure, we plot the results obtained on a single pair of essentially different films; other samples demon- strate quite similar behavior. The triangles (direct and in- verted) correspond to the upper boundary of the purely superconducting state. In the regime of narrow film (super- conducting channel), 5w ⊥λ < , the sample is completely superconducting until the transport current reaches the GL depairing current; at GL cI I≥ the film undergoes transition to the resistive state with one-dimensional PSCs. Corres- pondingly, this part of the dependence of GL c cI I can be approximated by horizontal straight line. When the ratio w ⊥λ exceeds 5, an unusual phenomenon is observed: the value of the critical current sharply falls to 0.8 GL cI (verti- cal approximating line) and then holds this value until w ⊥λ reaches 10 (second horizontal line). As soon as the transport current exceeds 0.8 GL cI , the film enters the vor- tex resistive state, in accordance with general conclusions of the theory. However, such behavior of cI is inconsis- tent with the AL theory, even with its generalized version valid for the case of arbitrary value of w ⊥λ and devel- oped by us. Formal consideration of this problem leads to the conclusion that at 5 10w ⊥< λ < the vortices can overcome the edge barrier when the edge current density approaches the value 2(1 )c cj T T−∼ much smaller than the GL critical density 3 2(1 )GL c cj T T−∼ , but the physi- cal mechanism of such anomalous penetration of vortices into the film is unclear. For very wide films with the trans- versal size 10w ⊥> λ , the experimental data excellently agree with the AL theory (curve 1 in Fig. 5). The experi- mental values of the upper boundary of the stability of the vortex state, ( )GL m cI I T , are plotted in Fig. 5 for two dif- ferent films by circles and squares. As it is obvious from Fig. 5, these data well correlate with the result (3) of the AL theory with the fitting parameter 1.6C = (curve 2). The set of lines in Fig. 5 can be considered as critical lines at the phase diagram for thin superconducting films. These lines divide the phase plane «reduced current GL cI I – reduced film width w ⊥λ » into three regions. The lower region, cI I< , corresponds to the completely superconducting state: the homogeneous current state in narrow films ( 5w ⊥< λ ) or the Meissner state in wide films ( 5w ⊥> λ ). For the latter case, there exists the intermediate vortex resistive region, c mI I I< < , where 0.8 GL c cI I= at 5 10w⊥ ⊥λ < < λ , and AL c cI I= for 10w ⊥> λ . Then, at mI I> the wide film ( 5w ⊥> λ ), enters the resistive state with PSLs, while the narrow film ( 5w ⊥< λ ) exhibits di- rect transition at GL cI I> to the resistive state with PSCs. In conclusion, we develop the phase diagram for the current states of superconducting films, based on the expe- rimental investigation of the resistive transitions induced by the transport current in the films. We found that a nar- row film with the width 5w ⊥< λ never enters the vortex state, but experiences direct transition from the purely su- perconducting state to the resistive state with PSCs as soon as the current exceeds the critical GL current. The Meiss- ner current state of the films of intermediate width, Table. Parameters of the film samples Sample L, μm w, μm d, nm R4.2, Ω R , Ω Tc, K li, nm R300, Ω Sn4 30 1 199 1.45 0.048 3.783 131 21.50 SnW9 95 17 159 0.319 0.057 3.825 138 4.900 SnW10 88 7 181 0.487 0.040 3.809 169 9.156 SnW13 90 18 332 0.038 0.008 3.836 466 1.880 C o m m e n t s: Here L, w and d are the length, the width and the thickness of a sample; li is the electron mean free path. Fig. 5. Diagram of the current-carrying states of wide supercon- ducting films in the dependence on the reduced film width and the reduced magnitude of the transport current. Different states are separated by the dependencies of the reduced critical current / GL c cI I for the samples SnW9 ( ), SnW10 ( ) and of the reduced current of the vortex instability / GL m cI I for the sam- ples SnW9 ( ), SnW10 ( ) on the quantity /w ⊥λ . The smooth part of the curve 1 represents the AL theoretical dependence of /AL GL c cI I calculated by (1) and (2); the curve 2 is the AL theo- retical dependence of / GL m cI I calculated by (1) and (3) with 1.6C = . 00 55 1010 1515 2020 0,40,4 0,40,4 0,60,6 0,60,6 0,80,8 0,80,8 1,01,0 1,01,0 22 11 vortex state Ivortex state I < I < I< I < Icc mm superconducting state I < Isuperconducting state I < Icc state with phase-slip processstate with phase-slip process I > II > Imm II /I/I cc ccGG LL II /I/I mm ccGG LL w/w/�� E.V. Bezuglyi and I.V. Zolochevskii 1252 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 5 ( ) 10 ( )T w T⊥ ⊥λ < < λ , transforms at 0.8 GL cI I> to the vortex resistive state which exists within the current inter- val 0.8 GL c mI I I< < , where the value of the upper critical current mI well agrees with the theory [20]. The vortex state of wide films, 10 ( )w T⊥> λ , is realized within the current region AL c mI I I< < , where AL cI is the transition point to the vortex state calculated in [20] for the limiting case w ⊥λ . At mI I> , the films with the width 5 ( )w T⊥> λ enter a vortex-free resistive state with phase- slip lines. 1. V.L. Ginzburg and L.D. Landau, Sov. Phys. JETP 20, 1064 (1950) 2. B.I. Ivlev and N.B. Kopnin, Sov. Phys. Usp. 27, 206 (1984) 3. J.D. Meyer and G.V. Minnigerode, Phys. Lett. A38, 529 (1972). 4. J.D. Meyer, Appl. Phys. 2, 303 (1973) 5. W.J. Skocpol, M.R. Beasley, and M. Tinkham, J. Low Temp. Phys. 16, 145 (1974). 6. M.H. Tinkham, Phys. Rev. 129, 2413 (1963). 7. I.M. Dmitrenko, A.A. Shablo, and L.E. Kolin’ko, Proc. 10th Moscow Int. Conf. Low Temp. Phys. 2, 355 (1966). 8. S.B. Kaplan, J. Low Temp. Phys. 37, 343, (1979). 9. T. Ogushi and Y. Shibuya, J. Phys. Soc. Jpn. 32, 400 (1972). 10. V.G. Volotskaya, I.М. Dmitrenko, L.Е. Musienko, and A.G. Sivakov, Fiz. Nizk. Temp. 7, 383 (1981) [Sov. J. Low Temp. Phys. 7, 188 (1981)]. 11. Е.V. Il’ichev, V.I. Kuznetsov, and V.A. Tulin, JETP Lett. 56, 295 (1992). 12. V.M. Dmitriev and I.V. Zolochevskii, Supercond. Sci. Technol. 19, 342 (2006). 13. A. Kulikovsky, Kh. Erganokov, and H. Bielska-Lewandow- ska, J. Low Temp. Phys. 106, 213 (1997). 14. W. Klein, R.P. Huebener, S. Gauss, and J. Parisi, J. Low Temp. Phys. 61, 413 (1985). 15. L.E. Musienko, I.M. Dmitrenko, and V.G. Volotskaya, JETP Lett. 31, 567 (1980). 16. A.V. Samoilov, M. Konczykowski, N.C. Yeh, S. Berry, and C.C. Tsuei, Phys. Rev. Lett. 75, 4118 (1995). 17. F. Lefloch, C. Hoffmann, and O. Demolliens, Physica C319, 258 (1999). 18. D. Babic, J. Bentner, C. Surgers, and C. Strunk, Phys. Rev. B69, 092510 (2004). 19. A.G. Sivakov, A.P. Zhuravel, O.G. Turutanov, and I.M. Dmitrenko, Appl. Surf. Sci. 106, 390 (1996). 20. L.G. Aslamazov and S.V. Lempitsky, Sov. Phys. JETP 57, 1291 (1983). 21. D.Y. Vodolazov and F.M. Peeters, Phys. Rev. B76, 014521 (2007). 22. S.V. Lempitsky, Sov. Phys. JETP 90, 793 (1986). 23. V.M. Dmitriev, I.V. Zolochevskii, E.V. Bezuglyi, and D.S. Kondrashev, Supercond. Sci. Technol. 20, 891 (2007). 24. V.M. Dmitriev and I.V. Zolochevskii, Fiz. Nizk. Temp. 35, 1187 (2009) [Low Temp. Phys. 35, 922 (2009)]. 25. L. Kramer and R.J. Watts-Tobin, Phys. Rev. Lett. 40, 1041 (1978). 26. R.J. Watts-Tobin, Y. Krähenbühl, and L. Kramer, J. Low Temp. Phys. 42, 459 (1981).

Phase diagram of a current-carrying superconducting film in absence of the magnetic field

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