1 Physica C ) 349±356 Quantum detection by current carrying superconducting lm Alex D. Semenov *,1, Gregory N. GolÕtsman, Alexander A. Korneev Department of Physics, State Pedagogical University of Moscow, Moscow, Russian Federation Received 18 July 2000; received in revised form 9 October 2000; accepted 11 October 2000 Abstract We describe a novel quantum detection mechanism in the superconducting lm carrying supercurrent. The mechanism incorporates growing normal domain and breaking of superconductivity by the bias current. A single photon absorbed in the lm creates transient normal spot that causes redistribution of the current and, consequently, increase of the current density in superconducting areas. When the current density exceeds the critical value, the lm switches into resistive state and generates the voltage pulse. Analysis shows that a submicron-wide lm of conventional low temperature superconductor operated in liquid helium may detect single far-infrared photon. The amplitude and duration of the voltage pulse are in the millivolt and picosecond range, respectively. The quantitative model is presented that allows simulation of the detector utilizing this detection mechanism. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: Pb; F Keywords: Quantum detection; Superconducting lm; Quasiparticle di usion; Phase slip centers 1. Introduction The superconducting quantum photodetectors have been developed for X-ray spectroscopy as early as in 70s see e.g. Ref. ). Modern devices typically represent an absorber from superconducting lm to which one or more tunnel junctions are connected. Photon absorbed in the lm creates a high-energy electron that relaxes to equilibrium producing nonequilibrium quasiparticles. At the * Corresponding author. address: A.D. Semenov). 1 Present address: Deutsches Zentrum fur Luft & Raumfahrt, DLRInstitut fur Weltraumsensorik und Planetenerkundung, Rutherfordstrasse 2, Berlin, Germany. operation temperature of few hundreds millikelvin, losses due to electron±phonon interaction and phonon escape from the lm are su ciently small, so that almost all nonequilibrium quasiparticles are in time to di use to tunnel junctions and contribute to the junction current. The integral of the current pulse through the junction measures the total charge of nonequilibrium quasiparticles, which were produced due to absorption of a single photon. This quantity, the collected charge, is proportional to the photon energy that provides energy resolution of the detector. Because of the low temperature and large lm area, the duty cycle of the detector lasts typically few microseconds that limits the count rate. To realize the intrinsic sensitivity of the detector, FET or SQUID readout of the tunneling current should be used that complicates the technology. Alternatively, the /01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S )
2 350 A.D. Semenov et al. / Physica C ) 349±356 transition edge microbolometer in the voltage bias regime has been used  as the photon counter. The bolometer was kept in the resistive state in the middle of the transition and served itself as an absorber. Although this approach allows one to get red of the tunnel junctions, it su ers the same limitations as the tunnel junction detectors. In these two types of quantum detector the nonequilibrium area is spread over the whole detector. Another recently proposed regime  of the quantum detection by the transition edge bolometer relies on the localized hotspot. If the photon creates a hotspot with the elevated resistivity in the lm, whose size is larger than the diameter of the hotspot, there still appears an overall increase of the lm resistance. The magnitude of the e ect scales as squared ratio of the spot diameter to the lm size. To be sensitive to the appearance of the hotspot, the detector must be maintained in the resistive state. At the transition temperature of conventional superconductors such detector should have a duty cycle of few picoseconds corresponding to the energy relaxation time of nonequilibrium electrons. The disadvantage of the approach steams from the small current density that the lm can stand without been driven into the normal state. The response of the detector is proportional to the bias current; thus, the small operating current requires a complicated readout scheme. In this paper we describe quantum detection regime in the superconducting lm well below the transition temperature that carries the bias current slightly smaller than the critical value at the operation temperature. Appearance of the normal spot at the position where the photon has been absorbed forces the supercurrent to ow around the spot through those parts of the lm, which remain superconducting. With the increase of the diameter of the normal spot the current density in the suprconducting portion of the lm increases and reaches the critical value. At this very moment the resistive ``bridge'' is formed across the width of the lm, giving rise to the voltage pulse with the magnitude proportional to the bias current. The physical di erence of the proposed regime is that the resistivity and, thus, the response appear due to collaborative e ect of the bias current and the growing normal domain. As we will show, this regime is essentially nonlinear. The magnitude of the voltage pulse does not up to certain extent depend on the quantum energy, although, the duration does that provides the basis for spectral sensitivity. 2. Response mechanism We rst describe the formation of the normal spot in the superconducting lm at the position where the photon is absorbed. For the sake of simplicity, we consider two-dimensional problem. More detailed three-dimensional analysis including the substrate can be found in the review article by Gross and Koelle . Let the lm have the thickness d and the normal state di usivity D such that the thickness is small compared to the thermalization length L th, i.e. d L th ˆ Ds th 1=2 where s th is the electron thermalization time. Then the concentration of nonequilibrium quasiparticles is uniform through the lm thickness. The lm is kept well below the transition temperature T C at the bath temperature T that determines the magnitude of the critical current density j C, the energy gap D and the equilibrium concentration C 0 of unpaired electrons. The ow of nonequilibrium quasiparticles produced by the absorbed photon is described by the two-dimensional di usion equation oc ot ˆ D 1 r oc or r o2 C C C 0 or 2 s 1 with C r; t the quasiparticle concentration, r the distance from the point where the photon has been absorbed and 1=s the rate of quasiparticle decay via recombination and phonon escape into the substrate. Inside the normal spot the decay time reduces to the electron cooling time s ep c e =c p s es where s ep is the electron±phonon interaction time, s es is the time of phonon escape to the substrate, c e and c p are the electron and phonon speci c heat, respectively. Assuming that the di usion and thermalization are independent processes and that the photon is absorbed at the time t ˆ 0, we arrive to the solution of Eq. 1) in the form
3 A.D. Semenov et al. / Physica C ) 349± C r; t ˆ M t exp r2 4Dt exp t the supercurrent is expelled from the spot and is C 0 ; 2 concentrated in the sidewalks between the spot 4pDd t s and the edges of the lm. The process is shown where M t is the time dependent multiplication schematically in the inset of Fig. 1 together with factor. The maximum value that M t reaches the quasiparticle concentration pro le at di erent during thermalization is commonly called the moments. quantum yield. This is the maximum number of The lm possesses no resistance unless the excess quasiparticles produced per one absorbed current density in sidewalks becomes larger than photon. For model simulations we will use the the critical value. At this point the superconductivity in the sidewalks turns to be destroyed. There exact integral expression that was obtained  assuming that the di usion time of quasiparticles appear constrictions similar to phase slip centers in the energy space is proportional to the reciprocal square of the energy E [6,7] and that the PSCs)  in the narrow superconducting channel biased with supercritical current in the vicinity of nonequilibrium distribution function has rectangular shape. For the quantum energy hm the mul- the transition. Within the supercurrent response time s j 2k B T C h= pd 2 or Maxwell relaxation tiplication factor is given by time s m l 0 rdq, whatever is longer, the bias current redistributes between the central normal portion of the lm and sidewalks according to their 2 M t ˆ q D hm=d 1 2 resistance. Both characteristic times appear to be 1 negligibly small, so that the static description can Z E D hm D E D p be used. Fig. 2 depicts the structure of the arising t E D 2 =s th D 2 1 E D D p 4D t E D 2 =s th D 2 1 E p de: 3 E 2 D 2 Although, rather precise quantitative results can also be obtained with the simple analytical form M t ˆK 1 exp t=s th where K is the experimental value of the quantum yield. In deriving the Eq. 3), the loss of the photon energy due to generation of subgap phonons was not taken into account. Although, subgap phonons are lost for multiplication only if the energy gap does not collapse. In our case the normal state with zero energy gap is achieved. Therefore, low energy phonons, which are idle at the initial stage of thermalization, will contribute at a later stage when the energy gap becomes small enough. These low energy phonons do not disappear from the normal spot since both the di usion time and escape time are much larger than the thermalization time. The radius r n of the normal cylindrical spot is determined by the condition C r n ; t ˆC n where C n ˆ N 0 K B T C is the concentration of equilibrium quasiparticle at the transition temperature and N 0) is the normal metal density of states at the Fermi level. After the normal spot has sprung up, Fig. 1. Concentration of nonequilibrium quasiparticles across the width of the lm at di erent moments after the photon has been absorbed. Time delays are 0.8, 2.0 and 5.0 measured in units of the thermalization time. Distance from the absorption site is shown in units of the thermalization length. Inset illustrates redistribution of supercurrent in the superconducting lm with the normal spot ± the basis of quantum detection. It shows the cross-section of the lm drawn through the point where photon has been absorbed.
4 352 A.D. Semenov et al. / Physica C ) 349±356 Fig. 2. Schematic of the resistive state formed in the lm after the current density in sidewalks has exceeded the critical value. The dark circle represents the normal spot; gray zones correspond to the area of superconductor with penetrating electric eld. Pro les of the electric eld E) and the energy gap D) are shown along lines crossing the normal spot a) and the sidewalk b). resistive state. The resistance of the central part consists of the normal spot resistance plus the resistance, which arises due to penetration of the electric eld into the superconductor at the boundary with the normal spot. The electric eld penetrates the superconductor over the distance L E ˆ Ds Q 1=2 where s Q is the relaxation time of the charge imbalance. Presence of the bias current modi es relaxation of the charge imbalance. For the current close to the critical value, corresponding time is given  by s Q ˆ 4k B T C pd 3s e s j 1=2. We identify here the inelastic electron scattering time s e with the electron±phonon interaction time. The portion of the superconducting lm with the nonzero electric eld contributes the resistance ql E F T =S where q is the resistivity of the normal lm and S is the cross-section area. F T < 1 accounts for the portion of the supercurrent that is directly converted into the normal current by means of Andreev re ection and, thus, generates no electric eld. We admit that the resistance of either sidewalk is just the resistance contributed by the PSCs alone. Although, for a bias current close to the critical current at the ambient temperature, the energy deposited in the PSC due to Joule dissipation may cause  the growing of the hotspot and consequent switching of the lm into the normal state. If the lifetime of the resistive state is long enough, the normal domain may become slightly larger than the coherence length. This would increase the resistance contributed by sidewalks. We approximate the circular normal spot with the radius r n by the circumscribing square. Further assuming that the photon is absorbed at a distance from the lm edge larger than the maximum diameter of the spot, we obtain the time dependent resistance of the lm and the density of the current owing through the sidewalk R ˆ q d j ˆ 2F T L E 1 F T L E rn ; 4 w 1 F T L E rn 2r n RI 4qF T L E ; where I is the bias current and w is the lm width. This approach is valid until the eld penetration length is larger than the thermal healing length L T ˆ Ds 1=2 and heating of the PSC by the current can be neglected. The resistance disappears and the lm switches back into the superconducting state when either the critical current in sidewalks drops below the critical value or the quasiparticle concentration in the center of the normal spot decreases beneath the critical value N 0 k B T C, whatever occurs rst. The resistance transient results in the voltage pulse U t ˆR t I developing between lm edges. From this simple consideration we see that the ability of the lm to detect single photon is the trade-o between the quantum energy, bias current, and the width of the lm. The value of the bias current is limited by thermal uctuations. The di erence between the critical current and the bias current should be at least few times the root mean square uctuation of the critical current dj C ˆ dj C =dt dt where dt ˆ 4k B T 2 = cv 1=2, c ˆ c e exp D= k B T C, and c e is the electron speci c
5 A.D. Semenov et al. / Physica C ) 349± heat at the transition temperature. The relevant volume V ˆ wdn is the portion of the lm with the length equal to the coherence length n. Decrease of the critical current density in this volume below the density of the bias current would lead to the loss of phase coherence between adjacent superconducting parts and, consequently, to the appearance of the PSC. From the exterior, the voltage pulse generated by this event cannot be distinguished from the voltage pulse produced by the absorbed photon. For the xed density of the bias current, the smallest energy of the photon that still can be detected decreases with the decrease of the lm width. There is, although, the physical limit for the single photon detection regime that does not depend on the lm width. Appearance of the normal spot is only possible if the rate of quasiparticle multiplication exceeds the rate of out di usion. Equating C 0; t from Eq. 2) to N 0 k B T C, using analytical expression for the multiplication factor M t ˆK 1 exp t=s th, and neglecting C 0 we get for times t=s th < 1 the minimal value of the quantum yield K that is su cient to get momentarily the quasiparticle concentration at r ˆ 0 equal to their equilibrium concentration at the transition temperature. Usually, experimental value of the quantum yield for the speci c material at speci c frequency is somewhat smaller than the upper theoretical limit hm=d, i.e. K ˆ 1=n hm=d. The factor n accounts, for example, for the energy lost due to generation of subgap phonons, which are unable to create quasiparticles unless the gap disappears. For given n, the smallest photon energy that is su cient for single photon detection regime can be estimated as e ndn 0 k B T C Dds th. It is instructive to see how multi-photon processes may manifest themselves. If for either reason the lm does not react to single photon, there is certain probability that two, three, or even more photons are simultaneously absorbed in the lm and give rise to the voltage pulse. In order to produce cumulative e ect in the lm, these N photons should be con ned in the volume restricted by the lm width, thermalization length, and the optical path corresponding to the thermalization time. Let us consider the weak photon ux in which the mean number of photons u in this volume is less that unity. Fluctuations in such low-density photon gas can be treated classically. The probability of large uctuations is given by Poisson distribution u N exp u = N!. Since u is proportional to the intensity of radiation, the experimental dependence of the count rate on radiation intensity makes it possible to determine the number of photons responsible for one count event. Thus, until u 1 and, consequently, exp u 1, for single photon detection regime N ˆ 1 the count rate should be proportional to the radiation intensity. Two photon events N ˆ 2 would result in the count rate proportional to squared radiation intensity and so on. 3. Simulation results Finally, we provide a quantitative check of the model based on real material. We chose niobium nitride, although that can be any of type II dirty superconductor like niobium, lead, or one of the A 3 B 5 compounds. The reason is that among others this material has known parameters and established thin lm technology. Let the lm be 6 nm thick that is the smallest thickness at which lms have material constants close to those of bulk material. Parameters of NbN, which we used for simulations, are listed in Table 1. Temperature dependent parameters were calculated in the framework of the Bardeen± Cooper±Schrie er BCS) theory for the dirty superconductor. Experimental value of the energy gap  was used instead of the standard BCS value. We also used experimental temperature dependence  of the electron±phonon interaction time. Temperature dependence of relevant parameters is given by n T ˆn 0 1 T =T C 1=2 ; h D T ˆ2:15k B T C 1 T =T C 2i ; s e / T 1:6 ; c e / T ; c p / T 3 ; 1=2 j C ˆ 8p3 c p 7f 3 3 k B T C D en 0 D 0 3 h h 1 T =T C 2i h 1 T =T C 4i 1=2 ; r pdt C 0 ˆ 2N 0 e D=T : 2 5
6 354 A.D. Semenov et al. / Physica C ) 349±356 Table 1 Material constants of NbN q lx m) n 0) nm) T C K) D cm 2 s ±1 ) N 0) m ±3 K ±1 ) s th ps) c e mj cm ±3 K ±1 ) c p mj cm ±3 K ±1 ) a a b 7 c 2.4 b 9.8 d 17 d 78 e a Coherence length at zero temperature and di usion constant for electrons are both extracted from measurements  of the second critical magnetic eld. b Density of states at the Fermi level and electron speci c heat at 10 K are calculated from q and D. c Experimental data . d Experimental data  scaled to 10 K. e Experimental data  scaled to the thickness 6 nm. s e ps) s es ps) We chose operation temperature T ˆ 0:5 T C in order to have reasonably large critical current density. For the strip with a width of 150 nm, which is readily achievable by common technology, the theoretical value of the critical current is 118 la. Temperature uctuations in the relevant volume wdn) see discussion in Section 2) have a magnitude of approximately 0.5 K that should result in uctuations of the critical current with a relative magnitude of 8%. To noticeably decrease the dark count rate one should operate the detector at a current which is more than 8% smaller that the critical current. For low temperature Josephson junctions with approximately same volume, the rule of thumb is that for reliable operation the bias current should be approximately 20% smaller than the critical current. Therefore, we simulate the response to ultra-violet photons of the 150 nm wide NbN lm operated at the temperature 5 K and biased with the current 100 la. The result is shown in Fig. 3 for photon energies 19, 12, and 5.8 ev. The remarkable feature is the dependence of the pulse height and duration on the photon energy that provides the spectral resolution of the detection mechanism. As it is typical for quantum detectors, the best resolution can be achieved by measuring the time integral of the pulse. Fourier transform of the voltage transients in Fig. 3 gives the frequency bandwidth varying from 20 to 10 GHz, which the registration electronics should have in order to record the signal. According to material parameters and the analytical expression for the multiplication factor with n ˆ 1, the smallest photon energy su cient for single quantum detection is as low as 130 mev. This is the energy of a photon that still creates a normal spot in the supercondutcing lm. Using numerical simulations and more accurate expression 3) for the multiplication factor, we estimate the smallest energy 85 mev corresponding wavelength 15 lm). Since the Eq. 3) gives the value of the quantum yield for NbN coinciding with the experimental value , the latter estimate seems to be more reliable. However, since the roll-o energy of the single photon detection regime increases with the increase of the lm width, practical realization of the quantum detector for this photon energy may require unrealistic lm dimensions. Fig. 4 shows the simulated roll-o wavelength of the single photon detection regime in a 150-nm width lm for two bias currents. For a Fig. 3. Simulated voltage transients for photon energies a) 5.8 ev, b) 12 ev, and c) 19 ev. Time is shown in units of the thermalization time. The bias current equals 100 la.
7 A.D. Semenov et al. / Physica C ) 349± large lm the roll-o occurs when the density of supercurrent in the sidewalks fails to reach the critical value. One can see that in a practical device this happens at almost an order of magnitude smaller wavelength that the wavelength allowing for hotspot formation. The direction that could provide detection of less energetic photons is the use of low temperature superconducting materials with low electron di usivity and small energy gap. The detector can be optimized for detection of photons with speci c energy. In particular, the decrease of the absorptivity of the lm for photons with large energy can be partly compensated by the increase of the lm thickness that would also increase the roll-o frequency. However, the lm thickness should not be larger than the thermal healing length. This puts the upper frequency limit for our device that is better suitable for photon counting in the spectral range from UV to infrared. Patterning the detector upon a meander line that covers the area of about squared wavelength can increase optical coupling for infrared photons. Thus far we have ignored self-heating due to Joule power dissipated in the normal spot by the bias current. It is known from the dynamic analysis of the heat di usion in a current biased superconducting strip  that if the large enough thermal energy is released within the short time dt s in the short portion of the strip with the length d L T, the normal electro-thermal domain appears and expands in nitely even without any further external perturbation. Smaller energy deposition also creates a normal domain but it disappears soon after the perturbation is over. For conditions speci ed above, the critical energy Q that demarcates these two regimes does not depend on the details of the deposition process and is given by fc T j T g 3=2 Q ˆ dwl T fqj 2 s 4c T j T g ; 6 1=2 where T j is the temperature at which the critical current of the lm equals the actual bias current. An estimate for our device gives the critical energy 10 ±20 J while the Joule energy, that should be dissipated by the bias current during the voltage transient Fig. 3), is an order of magnitude larger. The observation suggests that the self-heating should play a signi cant role in the detector operation. Although, rigorous analysis is required in order to quantitatively evaluate the contribution of this e ect. In a real device, the Joule energy is dissipated within the length not less than L E that is larger that the thermal healing length. Since spread energy deposition relaxes the requirement to the critical energy, the e ect may be much weaker than Eq. 6) predicts. However, self-heating brings out at least one consequence. To operate the detector without an external reset, one should provide voltage bias regime at least for a.c. current. The self-heating e ect will then smear the falling part of the voltage transient. Preliminary analysis shows that this may increase the response time by approximately 0.2s and hamper the energy resolution of the detector. 4. Summary Fig. 4. Simulated roll-o wavelength of the single photon regime for two bias currents as function of the lm width. The critical current equals 120 la. In conclusion, we have introduced a novel mechanism of quantum detection by a superconducting lm carrying the current close to the critical value. The mechanism suggests an increase of
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