Zurich Open Repository and Archive. Magnetic vortices in superconducting photon detectors

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1 University of Zurich Zurich Open Repository and Archive Winterthurerstr. 190 CH-8057 Zurich Year: 2009 Magnetic vortices in superconducting photon detectors Engel, A; Bartolf, H; Schilling, A; Semenov, A; Hübers, H-W; Il in, K; Siegel, M Engel, A; Bartolf, H; Schilling, A; Semenov, A; Hübers, H-W; Il in, K; Siegel, M (2009). Magnetic vortices in superconducting photon detectors. Journal of Modern Optics, 56(2-3): Postprint available at: Posted at the Zurich Open Repository and Archive, University of Zurich. Originally published at: Journal of Modern Optics 2009, 56(2-3):

2 RESEARCH ARTICLE Magnetic Vortices in Superconducting Photon Detectors A. Engelat, H. BartolF, A. Schillinga, A. Semenovb, P. Haasb, H.-W. Hiibersb, K. Il'inc and M. Siegel C a Physics-Institut of the University of Zurich, Zurich, Switzerland; b Institute of Planetary Research, German Aerospace Center (DLR), Berlin, Germany; C Institute for Micro- and Nanoelectronic Systems, University of Karlsruhe, Karlsruhe, Germany (January 91, 2008) We have measured critical-current densities in micro- and nanometer wide thin-film NbN bridges. These bridges show significant changes in the temperature dependence of the criticalcurrent density depending on the strip width. Taking into account the boundary conditions at the strip edges we can qualitatively describe our data applying a geometric edge-barrier model. We conclude that sub-1jdl wide bridges remain free of single vortices in ambient magnetic fields and at currents up to the depairing critical-current density. This also means that NbN meanders of superconducting single photon detectors should be free of single vortices under normal operating conditions. Keywords: critical-current density, geometric edge-barrier, magnetic vortices, superconducting photon detector 1. Introduction Various schemes of superconducting photon detectors have been proposed and their excellent performances have been demonstrated in a wide range of applications (1, 2). In recent years a detector concept based on a current-carrying superconducting nanowire (3) has attracted a lot of attention. These superconducting single-photon detectors (SSPD) are characterized by very fast response, high sensitivity and good quantum efficiency. This makes them very interesting candidates as single-photon detectors for a broad range of applications, e.g. astronomy, spectroscopy and optical communication, as illustrated by many other papers in this special volume. The superconducting material most commonly used for SSPD is NbN; a classical, strongly type-ii superconductor with a bulk critical temperature Tc of about 17 K. Strongly type-ii superconductors are defined by a Ginzburg-Landau (GL) parameter /'i, == i» 1, where ox is the magnetic penetration depth and e the superconducting coherence length. Their H-T phase diagram in the superconducting state is dominated by the mixed or Shubnikov-phase in which magnetic fields penetrate the material in form of flux lines (vortices) each carrying one quantum of magnetic flux CPo = ~, h being Planck's quantum and e the elementary charge (4). The interaction of vortices with each other, the underlying material and external force fields give rise to many different vortex phases with different technologically important superconducting properties. tcorresponding author. andreas.engel@physik.uzh.ch

3 2 A. Engel et ai. Figure 1. Definition of the coordinate system and layout of the bridges. The dark-shaded region symbolises the geometry of the bridges as they were fabricated from extended NbN films. The drawing is not to scale. In this paper we discuss some aspects of vortex physics and the possible consequences on the performance of SSPD. In order to study the behaviour of vortices we performed 4-point resistivity and critical-current measurements on specifically prepared micro- and nanometer wide thin-film NbN bridges. Analysing these data allowed us to draw conclusions about the presence and the properties of vortices, which we then relate to measurements of SSPD device characteristics. 2. Single Vortices in Micro- and Nanobridges In order to get a better understanding of the behaviour of vortices in micro- and nanobridges we quickly review some important results for thin films (thickness d ;S ~ «>.) with lateral extensions in the x- and y-direction much larger than all other length scales (see also Figure 1 for the definition of the coordinate system used). It follows from pure electrodynamics (5) that in such a situation the magnitude of supercurrents encircling a vortex decay with ~ for ~ «r «A, A = 2~2 being the perpendicular screening length in films with d < >., and <X ;2 for r» A, the latter being in contrast to the exponential decay <X exp( - i) in bulk superconductors (6). If such a vortex is within a distance x 2: A near the edge of the film, boundary conditions apply to the encircling supercurrents, namely the normal component of these currents has to vanish at the border. This can be formally achieved by the addition of an antivortex of opposite polarity (image antivortex) at a distance -x outside the supercurrent. The interaction of the magnetic flux of the vortex with the supercurrents of the virtual anti-vortex leads to an effective attractive force towards the film edge. This leads to the Bean-Livingston barrier for vortex entry in an increasing external magnetic field and for vortex exit in decreasing fields, respectively (1). As a consequence, the first vortex in increasing magnetic fields does not enter the superconductor at the lower critical field Hel as expected when surface and edge effects are neglected. For an ideally straight edge one expects the first vortex to enter the film at a significantly higher field Hs that can be almost as large as the thermodynamic critical field He. Likewise, the energy barrier for vortex exit in decreasing fields vanishes only for zero applied field Geometric Edge-Barrier in Superconducting Bridges If we look at vortices in superconducting thin-film bridges with widths W on the order of A or narrower it is obvious from above discussion that edge effects play an important role and cannot be neglected. First of all it is important to find the condition for which vortices can in principle exist inside such a narrow bridge. It can be shown (8) that for bridges with W > 4.4~ vortices can exist. Consid-

4 ering that e «A this leaves a large range of possible bridge widths for which the reduced geometry should have significant consequences. In the following we will derive the geometric edge-barrier in superconducting bridges following ideas proposed by Clem (9, 10). We assume a long bridge of width W < A «L that contains a single vortex at position e ::; x ::; W - e. In this way we can avoid complications when the vortex core with radius e starts to interact with the bridge edges. Contrary to the situation discussed above when a vortex comes close to the edge of an extended film, we now have to consider two edges, one at x = 0 and the other one at x = W. The boundary conditions of vanishing normal components of the supercurrents have to be fulfilled at both edges. This can be achieved by assuming a series of image antivortices at positions -x + 2nW (n = 0, ±1, ±2,... ) and image vortices at positions x + 2mW (m = ±1, ±2,... ). Each image vortex (antivortex) contributes a force Fi = j(ri)4>od, where j(ri) is the current density of the ith image vortex (antivortex) at a distance ri from its centre. For symmetry reasons all forces are in the positive or negative x-direction. Summation over all image vortices (antivortices) gives the total force on the vortex due to the boundary conditions 4>2 1rX o 2J.LoA W cot W' (1) with J.Lo the permeability of free space. From Fs = - ~ we can then calculate the self-energy of a single vortex inside a bridge of width W: 3 This potential has been normalised to zero at the bridge edges at x = e and W - e and it reaches its maximum at the centre of the strip. The application of an external magnetic field Ba in the positive z-direction induces shielding currents flowing in the bridge. The supercurrents flowing in the y-direction (taking into account W < A) can be approximated as jay(x) = - J.L~~2 (x - :). (3) From Eq. (3) we can again calculate the force on the vortex due to the interaction between the magnetic flux and this induced current, Fax = jay (x) 4>od, and the resulting interaction energy is (2) (4) The total Gibbs free energy of a vortex at position x is then simply the sum of the two contributions Eqs. (2) and (4) (5) for e ::; x ::; W - e. In Figure 2(a) we plotted G(Ba, x) that a single vortex experiences in a 4.9 ~ wide bridge for a number of external magnetic fields (see caption of Figure 2 and below for further parameters of the bridge). With increasing

5 4 A. Engel et al. 500~~~~~~~~~~~~ I :::.'" 0 1--~ <!> mt G.1 mt 0. 2 mt...;:::._o...;.. 3_mT_"--_--I-'---I ", / ',,- / W=4.9J11ll~/ TIT, = Position (~m) (a) Or--~~----~~-~ -200 W=4.9J11ll TIT, = 0.7 B. = 50 ~T Position (~m) (b) Figure 2. Gibbs free energy calculated for a single vortex inside a 4.9 tjlil wide bridge. The film thickness was set to d = 8 nm, the penetration depth A(O) = 250 nm and the coherence length e(o) = 4.5 nm. x = 0 was set to the centre of the bridge. (a): Variation of the Gibbs free energy in units of the thermal energy kbt with increasing applied field. Ba ~ 0.3 mt is the penetration field, Eq. (7). (b): Reduction of the edge barrier with increasing bias current h and a very small magnetic field Ba :;: 50 I-l-T (maximum earth magnetic field). magnetic fields a local minimum develops in the centre of the bridge for Ba larger than At that point the energy barrier for vortex entry is still very large, but one might expect that in field-cooling conditions vortices can already be trapped in the centre ofthe strip. It could be shown in magneto-optical experiments (11) that micrometer wide bridges remain free of vortices in fields significantly exceeding Bm. It turned out that vortices remain trapped inside the bridge only when there is a global minimum in the Gibbs free energy at the centre of the strip. We will call the corresponding critical magnetic field the vortex-penetration field 2q,0 (W) Bpen = 7l'W2 In 7l'e. (7) For experimental zero-field cooling conditions, the bridges may remain free of vortices up to even higher magnetic fields due to the high energy barriers at the edges (see Fig. 2(a)). (6) 2.2. Reduction of Edge-Barrier effects due to Applied Currents With respect to SSPD we have to keep in mind that they are usually operated in the earth ambient magnetic field (;S 50 ~T perpendicular to the NbN film plane depending on detector orientation and local earth magnetic field) which is much smaller than Bm for typical nanowire widths W ~ 100 nm. However, SSPD meanders are biased with large currents. According to the standard detection models (3, 12) the biasing current density has to be a significant fraction of the critical depairing current density ie, typically 90% to 95% (12-14). This means that the magnetic self-field is of the order of the thermodynamic critical field at the edges of the bridge (Silsbee's rule, (15)) and may exceed Bm or even Bpen depending on the bridge width and aspect ratio. The influence of an applied bias current h can be easily included in the edgebarrier model developed in Section 2.1. With our assumption A > W» d we

6 can assume a homogeneous current density ib = ~ in the y-direction. Neglecting pinning forces this current causes a constant force in x-direction driving a vortex across the bridge (and in the opposite direction for an antivortex). The corresponding interaction energy is 5 and the total Gibbs free energy is now the sum of three terms G(Ib, Ba, x) = Us + Ua + Uj = ci>g In (W sin 7rx) _ ci>oba x (W _ x) _ IbcI>ox. (9) 27r/loA 7re W /loa W In Figure 2(b) we plot the Gibbs free energy according to Eq. (9) for a certain range of biasing currents in Ba = 50 ~T, approximately the maximum earth magnetic field. The bias current breaks the geometric symmetry around x = 0, but there remains a relatively large barrier prohibiting the entry of vortices into the bridge. Only for large currents on the order of milliampere this barrier is reduced to a few kbt (kb the Boltzmann constant), so that thermal excitations are sufficient for vortices to overcome the barrier. (8) 3. jc(t, H) in NbN Micro- and Nanobridges Measurements of the critical-current density in narrow thin-film bridges are one possibility to check for the presence of single vortices under conditions which are similar to the operating conditions of SSPD. If a bridge is free of vortices one can expect to measure the maximum possible critical-current density, the depairing critical-current density ie. If vortices are able to penetrate the bridge, the current density at which dissipation starts to become significant is given by the depinning critical-current density ipin, the current density at which vortices start to break free from pinning sites and move across the bridge. Even in strong-pinning films ipin is always significantly less than ie (16, 17). Measuring the critical-current density as a function of temperature and magnetic field can thus give indications on the presence or the absence of vortices. The bridges in this study were prepared from d = 8 nm thick NbN films, which were made by DC magnetron sputtering of pure Nb targets in an Ar+ N 2 atmosphere onto R-plane sapphire substrates. The residual gas pressure was less than 10-6 mbar. The resulting film was patterned into bridges with widths ranging from 100 nm to 10 ~ using e-beam lithography and ion-beam etching (IBE). This fabrication process is analogous to that used for the fabrication of SSPD (12, 18). The bridges were incorporated into a 4-point resistivity setup, with a distance L = low between voltage contacts. The layout and design of the bridges was checked using scanning electron microscopy (SEM) and atomic force microscopy (AFM), see inset of Figure 3(a). Resistivity and critical-current measurements were performed in a Quantum Design Physical Properties Measurement System equipped with a 9 T superconducting solenoid. All bridges were carefully characterized with respect to their normal-state and superconducting properties. Their normal-state resistivity just above the superconducting transition was Pn ~ 150 ~ cm with a zero-field transition temperature Te(O) = 14.0 ± 0.2 K for all bridges. From linear extrapolations of Te(B) in fields

7 6 A. Engel et al. 3.5 N~ 10 E u :;x: '" ," TlTc(O) ~ " * 0.1 mt 1.8 mt 2.5 mt iii 5.3 mt GL T-dep T(K) (a) (b) Figure 3. (a) Experimental critical-current densities of bridges with different widths as a function of the reduced temperature T/Tc(O). At high T all bridges show approximately the same ic(t). Below approximately 0.7Tc(0) the ic(t) values of bridges wider than 1 ~ are lower and show a different T-dependence than narrower bridges. For example ic(t) of the 300 nm-wide bridge can be well described by the GL-depairing critical-current density, Eq. (10). The inset shows an AFM image of the 4.9 ~-wide bridge. (b) Ic(T) of the 4.9 ~-wide bridge in increasing magnetic field. For T ~ 11 K Ic(T) shows no magnetic field dependence within our experimental resolution. At lower temperatures Ic(T) shows a field dependence that can be qualitatively understood as a reduction of the minimum current density needed to sufficiently reduce the edge barrier to allow the entry of vortices. up to 9 T we could determine zero-temperature coherence lengths ~(O) ~ 4.5 nm. Using the relation A(O) = 1.05 x 1O-3VPn/Tc(0) (19) we were able to estimate the zero-temperature penetration depth A(O) ~ 250 nm, which is in line with published data for NbN films (20, 21). With these parameters our bridges fulfill the assumptions made for the edge-barrier model, namely W > 4.4~(T) (except extremely close to Tc(O)) and W < A(O) ~ 15 J.lIll. Critical currents Ic were defined by a voltage criterion Vc = 10 my, typically, corresponding to an ohmic resistance R ;S 10-3 Rn. Special care was taken to reduce the residual magnetic field to less 100 IlT, and we estimate the overall accuracy of our measured critical-current values to be better than ±1O% for T ::; 0.9Tc(0). In Figure 3(a) we show the thus measured critical-current densities as a function of the reduced temperature t = T/Tc(O) for four bridges with widths between 300 nm and 8.9 J.lIll. At temperatures t ~ 0.7 all the ic(t)-curves coincide within our measurement accuracy. At lower temperatures, ic of the 300 nm-wide bridge is significantly larger than for the several micrometer wide bridges, and it can be well described by the mean-field GL depairing critical current (22) (10) over the entire accessible temperature range. The extrapolated zero-temperature critical-current density ic(o) ~ 1.5 X 107 A/cm2 is ~ 40% of the theoretical value using A(O) = 250 nm and ~(O) = 4.5 nm. Taking into account that we used the geometrical cross-section of our bridges, which is certainly larger than the effective superconducting cross-sectional area (23), to calculate the critical-current density from the measured data, the agreement is remarkable. Therefore we conclude that the measured critical-current density is caused by the depairing of Cooper-pairs excluding any influence by the presence of single vortices. For temperatures t ~ 0.7 we can conclude even for the 1.9 Ilm-wide and wider bridges that there are no single vortices present, because all the measured critical-current densities are identical. Only at lower temperatures the experimen-

8 tal critical-current densities are significantly smaller than the expected depairing critical-current density. Here the critical currents of these bridges are already well in the milliampere range, where the edge barrier is reduced to several ket. In our simple edge-barrier model we have assumed perfect edges. Real bridges always show certain variations in W, structural damages or oxidation at the edges, which all together may lead to a reduction of the edge barrier. Therefore, vortices-and anti vortices at the opposite edge-can enter the strip and move across the bridge to the opposite edge, or annihilate each other. This vortex motion leads to a resistive state limiting the critical-current density. We have also measured critical currents as functions of the applied magnetic field in the millitesla range, see Figure 3(b) for the 4.9 j..ull-wide bridge, where we have plotted the measured critical current Ie as a function of T. At temperatures close to the critical temperature, Ie-values are independent of Ba. At lower T and increasing field Ie is progressively reduced below the GL expectations. This is in line with the picture of a reduction of the edge barrier with increasing magnetic field. Thus, at smaller applied currents the edge barrier is sufficiently reduced to allow the entry of vortices Discussion Our simple geometric edge-barrier model seems to capture the main features of the experimental ie(t, H) curves. Bridges with widths narrower than about 1 j..ull remain free of single vortices in zero or in the ambient earth magnetic field even at applied bias currents as high as the depairing critical currents, the maximum dissipation free currents in a superconductor. For wider bridges and below a certain cross-over temperature the vortex penetration current becomes smaller than the depairing critical current. The penetration current is defined as that current value for which the bias current sufficiently reduces the geometric edge-barrier (rv kbt) so that vortices can penetrate the bridge. Our model gives the right order of magnitude (a few ma) for the penetration current in micrometer wide bridges. A more quantitative description cannot be expected from this simple model. It assumes perfect edges, while unavoidable variations in bridge width, structural damages to the edges during IBE, and uncontrolled oxidation of the NbN film at the edges all have an influence on the edge barrier in real bridges. In general, all these effects will lead to a reduction of the edge barrier, similar to the way the Bean Livingston surface barrier is reduced in bulk superconductors (11). Furthermore, we have neglected vortex-vortex interactions in the derivation of the Gibbs free energy for a vortex inside a narrow bridge. If more than one vortex is present in a bridge, such interactions are significant due to the large value of A in thin films. We have also neglected here the actual process of penetration at the edges, i. e. vortex positions closer than e to the edge. The most important result of this study with respect to the operation of SSPD is that SSPDs are free of single vortices for usual operating conditions. We have found strong indications that 300 nm wide thin-film NbN bridges are free of single vortices due to a geometric edge-barrier prohibiting the entry of vortices even for relatively large magnetic self-fields as they occur just below the depairing critical-current density. In SSPD the NbN meander line has a width of typically about 100 nm. In this case the geometric edge-barrier is even more pronounced and the entry of vortices is not possible, even when the magnetic fields caused by neighbouring strips due to the meander geometry are taken into account. Nevertheless, our analysis does not exclude the presence of vortex-antivortex pairs (VAP), excitations commonly found in two-dimensional superconducting

9 8 REFERENCES films. YAPs are topological excitations in two-dimensional systems (24, 25) predicted to occur in superfluids and superconductors. There have indeed been numerous experimental studies that have claimed to have seen clear signs of the presence of YAPs (see Ref. (26) for an early review). The model of geometric edge-barriers does not apply to YAPs, since they are excited inside the superconducting film or bridge and do not have to overcome the energy barrier at the edge. At the temperatures T < TKT, where TKT ;S Tc(O) is the Kosterlitz-Thouless transition temperature, vortices and antivortices constituting YAPs are bound. An applied bias current excerpts equal but oppositely directed Lorentz-forces on them, and the resulting net force is therefore zero. The pair does not move and also does not cause any resistance. In the study of SSPD the existence of YAPs has been proposed to explain an increase in dark-count rates in meanders with a high sheet resistance (27) and a higher than expected quantum efficiency beyond the cut-off wavelength as well as an increased energy resolution (28-30). However, the presence of YAPs in SSPD and in thin-film superconductors in general is not yet conclusive (31, and references therein) and needs further work. In conclusion, we have analysed the temperature and the field dependence of the critical current in NbN micro- and nanobridges. A geometric edge-barrier model predicts that for certain conditions such bridges remain free of single vortices. In particular, NbN meanders in SSPD are most likely free of single vortices in ambient magnetic fields for all temperatures below Tc(O) and biasing current densities less than ic. Acknowledgement This work has been supported by the NCCR Manep of the Swiss National Science Foundation. References (1) Zmuidzinas, J.; Richards, P.L. Superconducting Detectors and Mixers for Millimeter and Submillimeter Astrophysics. IEEE 7hlns. Appl. Supercon. 2004, 92, (2) Irwin, K.D. Seeing with Superconductors. Sci. Am. 2006, (November), 62. (3) Semenov, A.D.; Gol'tsman, G.N.; Korneev, A.A. Quantum detection by current carrying superconducting film. Phys. C 2001, 351, 349. (4) Abrikosov, A.A. On the magnetic properties of superconductors of the second group. Zh. Eksperim. Tear. Fiz. 1957, 92, [SOy. Phys. JETP 5, 1174 (1957)]. (5) Pearl, J. Current Distribution in Superconducting Films Carrying Quantized Fluxoids. Appl. Phys. Lett. 1964, 5, 65. (6) Tinkham, M. Introduction to Superconductivity. 2nd edition; McGraw-Hill: New York, 1996; p lo6. (7) Bean, C.P.; Livingston, J.D. Surface Barrier in Type-II Superconductors. Phys. Rev. Lett. 1964, 12, 14. (8) Likharev, K.K. Superconducting weak links. Rev. Mod. Phys. 1979, 51, lo1. (9) Clem, J.R. In: Bull. Am. Phys. Soc., Vol. 43;, p (lo) Clem, J.R. Vortex Exclusion from Superconductin9 Strips and SQUIDs in Weak Perpendicular Ambient Magnetic Fields;, private communication. (11) Stan, G.; Field, S.B.; Martinis, J.M. Critical field for complete vortex expulsion from narrow superconducting strips. Phys. Rev. Lett. 2004, 92, (12) Semenov, A.; Engel, A.; Hiibers, H.W.; Il'in, K.; et ai. Spectral cut-off in the efficiency of the resistive state formation caused by absorption of a single-photon in current-carrying superconducting nanostrips. Eur. Phys. J. B 2005,47,495. (13) Zhang, J.; Slysz, W.; Pearlman, A.; Verevkin, A.; Sobolewski, R.; Okunev, 0.; Chulkova, G.; et al. Time delay of resistive-state formation in superconducting stripes excited by single optical photons. Phys. Rev. B 2003, 67, (14) Korneev, A.; Kouminov, P.; Matvienko, V.; Chuikova, G.; Smirnov, K.; Voronov, B.; Goltsman, G.N.; Currie, M.; Lo, W.; Wilsher, K.; Zhang, J.; Slysz, W.; Pearlman, A.; Verevkin, A.; et ai. Sensitivity and gigahertz counting performance of NbN superconducting single-photon detectors. Appl. Phys. Lett. 2004, 84, (15) Waldram, J.R. Superconductivity of Metals and Cuprates. lop Publishing: Bristol, 1996; p 3.

10 (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) REFERENOES 9 Blatter, G.; Feigel'mann, M.V.; Geshkenbein, V.B.; Larkin, A.I.; et al. Vortices in high temperature superconductors. Ret!. Mod. Phys. 1994, 66, Matsushita, T. Flux Pinning in Superconductors. Springer: Berlin, 2007; pp 109. Semenov, A.D.; Haas, P.; Gunther, B.; Hubers, H.W.; II'in, K; Siegel, M.; Kirste, A.; Beyer, J.; Drung, D.; Schurig, T.; et a!. An energy-resolving superconducting nanowire photon counter. Supercon. Sci. Technol. 2007, 20,919. Kes, P.H.; Tsuei, C.C. Two-dimensional collective flux pinning, defects, and structural relaxation in amorphous superconducting films. Phys. Ret!. B 1983, 28 (9) (Nov), 512~5139. Kubo, S.; Asahi, M.; Hikita, M.; et al. Magnetic penetration depths in superconducting NbN films prepared by reactive dc magnetron sputtering. Appl. Phys. Lett. 1984,44,258. Oates, D.E.; Anderson, A.C.; Chin, C.C.; Derov, J.S.; Dresselhaus, G.; et al' Surface-impedance measurements of superconducting NbN films. Phys. Ret!. B 1991, 49, Skocpol, W.J.; Beasley, M.R.; Tinkham, M. Self-heating hotspots in superconducting thin-film microbridges. J. Appl. Phys. 1974, 45, II'in, KS.; Siegel, M.j Semenov, A.D.j Engel, A.; et a1. Suppression of superconductivity in Nb and NbN thin-film nano-structures. In: Applied Superconductivity 2009, Pmc. 01 the 6th Eur. Oonl. on Appl. Supercond. Sorrento, Italy, Andreone, A., Pepe, G.P., Cristiano, R., Masullo, G. Eds.; No. 181 in Institute of Physics Conference Series; p Berezinskii, Z.L. Zh. Eksp. Teor. Fiz. [Sot!. Phys. JETP} 1971, 61, Kosterlitz, J.M.; Thouless, D.J. Ordering, metastability and phase transitions in two-dimensional systems. J. Phys , 6, Minnhagen, P. The two-dimensional Coulomb gas, vortex unbinding, and superfluid-superconducting films. Ret!. Mod. Phys. 1987, 59, Engel, A.; Semenov, A.D.; Hubers, H.W.; II'in, K; et a1. Fluctuation effects in superconducting nanostrips. Phys , 444, 12. Haas, P.; Semenov, A.; Hubers, H.W.; Beyer, J.; Kirste, A.j Schurig, T.; II'in, K; Siegel, M.j Engel, A.j et a!. Spectral Sensitivity and Spectral Resolution of Superconducting Single-Photon Detectors. IEEE Trans. Appl. Supercon. 2007, 17,298. Semenov, A.j Haas, P.; Ilin, K; Hubers, H.W.j Siegel, M.; Engel, A.; et al. Energy resolution and sensitivity of a superconducting quantum detector. Phys , , Semenov, A.; Haas, P.; Hiibers, H.W.; Ilin, K; Siegel, M.; Smirnov, A.; Engel, A.; Intrinsic quantum efficiency and electro-thermal response model of a superconducting single-photon detector. J. Mod. Opt. 2008, this volume. Kogan, V.G. Interaction of vortices in thin superconducting films and Berezinskii-Kosterlitz-Thouless transition. Phys. Ret!. B 2007, 75,