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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 3 — Feb. 11, 2013
  • pp: 3043–3054

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Plasmonic superconducting nanowire single photon detector

Amin Eftekharian, Haig Atikian, and A. Hamed Majedi  »View Author Affiliations


Optics Express, Vol. 21, Issue 3, pp. 3043-3054 (2013)
http://dx.doi.org/10.1364/OE.21.003043


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Abstract

A theoretical analysis to enhance the quantum efficiency of a meander-line superconducting single photon detector without increasing the length or thickness of the active element is proposed. The general idea is to utilize the plasmonic nature of a superconducting layer to increase the surface absorption of the input optical signal. To satisfy both optical guiding and photon detection considerations of the design, a coefficient is introduced as a measure to maintain the device sensitivity while crossing over from the current crowding to vortex-based detection mechanisms.

© 2013 OSA

1. Introduction

Single-photon detectors are key components in quantum photonic science and technology. Particularly, superconducting nanowire single photon detectors, SNSPDs, are one of the most promising devices with applications ranging from quantum optical computation [1

J. L. O’Brien, A. Furusawa, and J. V. kovic, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009). [CrossRef]

] to single photon source characterization [2

M. J. Stevens, R. H. Hadfield, R. E. Schwall, S. W. Nam, R. P. Mirin, and J. A. Gupta, “Fast lifetime measurements of infrared emitters using a low-jitter superconducting single-photon detector,” Appl. Phys. Lett. 89, 031109 (2006). [CrossRef]

]. They offer wideband optical detection, low dark count rate, low timing jitter, fast response time and present a platform for potential quantum system integration. [3

M. Thompson, A. Politi, J. Matthews, and J. O’Brien, “Integrated waveguide circuits for optical quantum computing,” IET Circuits Devices Syst. 5, 94–102 (2011). [CrossRef]

–5

J. P. Sprengers, A. Gaggero, D. Sahin, S. Jahanmirinejad, G. Frucci, F. Mattioli, R. Leoni, J. Beetz, M. Lermer, M. Kamp, S. Höfling, R. Sanjines, and A. Fiore, “Waveguide superconducting single-photon detectors for integrated quantum photonic circuits,” Appl. Phys. Lett. 99, 181110 (2011).

].

Despite all the promising properties of SNSPDs, low system quantum efficiency remains one of the major obstacles in this class of detectors [6

R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photonics 3, 696–705 (2009). [CrossRef]

–8

R. Sobolewski, A. Verevkin, G. Gol’tsman, A. Lipatov, and K. Wilsher, “Ultrafast superconducting single-photon optical detectors and their applications,” IEEE Trans. App. Supercond. 13, 1151–1157 (2009). [CrossRef]

]. This is mainly due to the input optical wave readily passing through a normal-incidence meander-line structure as a result of the great difference between the thickness of the film and skin depth of the optical wave [9

L. Zhang, L. Kang, J. Chen, Y. Zhong, Q. Zhao, T. Jia, C. Cao, B. Jin, W. Xu, G. Sun, and P. Wu, “Ultra-low dark count rate and high system efficiency single-photon detectors with 50 nm-wide superconducting wires,” Appl. Phys. B 102, 867–871 (2011). [CrossRef]

]. In this paper, we present a novel single photon detection structure to address this issue without increasing the length or the thickness of the active element such that two seemingly contradictory properties, high speed operation (requiring small active area) and high photon absorption probability (requiring large active area), can be maintained simultaneously. The design is based on the plasmonic nature of a superconducting layer [10

A. Hamed Majedi, “Theoretical investigations on THz and optical superconductive surface plasmon interface,” IEEE Trans. App. Supercond. 19, 907–910 (2009). [CrossRef]

] to improve the photon absorption process by exciting the surface plasmon polaritons, SPPs, at the interface of the superconducting layer and a positive index dielectric layer [11

P. Berini, R. Charbonneau, and N. Lahoud, “Long-range surface plasmons on ultrathin membranes,” Nano Lett. 7, 1376–1380 (2007). [CrossRef] [PubMed]

]. The general idea is to deposit and pattern thin film superconducting wires on top of a dielectric slab waveguide. Similar structures based on dielectric slab waveguides have been previously reported, where the optical energy is mostly confined at the center of the high index dielectric layer [5

J. P. Sprengers, A. Gaggero, D. Sahin, S. Jahanmirinejad, G. Frucci, F. Mattioli, R. Leoni, J. Beetz, M. Lermer, M. Kamp, S. Höfling, R. Sanjines, and A. Fiore, “Waveguide superconducting single-photon detectors for integrated quantum photonic circuits,” Appl. Phys. Lett. 99, 181110 (2011).

, 12

C. M. Natarajan, A. Peruzzo, S. Miki, M. Sasaki, Z. Wang, B. Baek, S. Nam, R. H. Hadfield, and J. L. O’Brien, “Operating quantum waveguide circuits with superconducting single-photon detectors,” Appl. Phys. Lett. 96, 211101 (2010). [CrossRef]

]. Consequently, there is a small overlap of the optical field with the active element, and only a small portion of the available power is absorbed. In our plasmonic design however, the presence of a negative index superconducting layer leads to an additional confinement resulting in sub-wavelength mode formations.

Excitation of SSPs in a thin superconducting layer (normally less than 12nm) is the main challenge of this work. We will show that to observe well-defined plasmonic confinement, there is a minimum feature size for the width of both the dielectric waveguide and the superconducting layer. This imposes an undesired limitation on the nanowire from a detection perspective, particularly on single-photon sensitivity, since there is an upper bound for the cross section area of the active element. In general, we will show that there is a trade-off between the probability of photon absorption and the internal quantum efficiency of the device. We present an alternative to decouple these two properties based on the optical design and photon detection mechanisms in SNSPDs (current concentration beyond the depairing current density at the sidewalks upon photon absorption [13

A. D. Semenov, G. N. Gol’tsman, and A. A. Korneev, “Quantum detection by current carrying superconducting film,” Phys. C Supercond. 351, 349–356 (2001). [CrossRef]

, 14

A. J. Annunziata, O. Quaranta, D. F. Santavicca, A. Casaburi, L. Frunzio, M. Ejrnaes, M. J. Rooks, R. Cristiano, S. Pagano, A. Frydman, and D. E. Prober, “Picosecond superconducting single-photon optical detector,” Appl. Phys. Lett. 79, 705–707 (2001). [CrossRef]

] and vortex hopping from one edge across the strip or vortex-antivortex unpinning [15

A. M. Kadin, M. Leung, A. D. Smith, and J. M. Murduck, “Photofluxonic detection: A new mechanism for infrared detection in superconducting thin films,” Appl. Phys. Lett. 57, 2847–2849 (1990). [CrossRef]

, 16

H. Bartolf, A. Engel, A. Schilling, K. Il’in, M. Siegel, H.-W. Hubers, and A. Semenov, “Current-assisted thermally activated flux liberation in ultrathin nanopatterned NbN superconducting meander structures,” Phys. Rev. B 81, 024502 (2010). [CrossRef]

]).

This paper is organized as follows. Section 2 describes the possibility of plasmonic excitation of a thin-film superconductor. In spite of the extremely thin thickness of the superconducting layer in comparison to the other layers, still excitation of the fundamental plasmonic mode is possible with proper design of the thickness and width of all layers. In Section 3, the nonequilibrium to equilibrium relaxation dynamics of the order parameter in a superconducting film after absorption of a single photon is studied. The objective is to find the optimum size of each layer such that the structure can benefit from both photon detection mechanisms mentioned above. Concluding remarks are given in section 4.

2. Plasmonic guiding

This section presents the theoretical investigation on the possibility of SPP formation at the interface of an ultrathin superconducting film with a positive index dielectric medium. The idea of optical confinement through plasmonic excitation arises from the fact that a superconducting film behaves like a negative index medium in the optical bandwidth [10

A. Hamed Majedi, “Theoretical investigations on THz and optical superconductive surface plasmon interface,” IEEE Trans. App. Supercond. 19, 907–910 (2009). [CrossRef]

]. Thus, by having proper dimensions in both transverse and lateral directions in all layers, it is possible to trap optical waves along the interface of these two layers.

Figure 1 illustrates the layered structure of the device. All materials are selected such that the device is compatible with silicon photonics processing. For the superconducting layer, NbTiN is selected for the possibility of thin-film deposition as well as its low sensitivity to latching in the single photon detection regime [17

J. K. W. Yang, A. J. Kerman, E. A. Dauler, V. Anant, K. M. Rosfjord, and K. K. Berggren, “Modeling the electrical and thermal response of superconducting nanowire single-photon detectors,” IEEE Trans. Appl. Supercond. 17, 581–585 (2007). [CrossRef]

,18

A. J. Annunziata, O. Quaranta, D. F. Santavicca, A. Casaburi, L. Frunzio, M. Ejrnaes, M. J. Rooks, R. Cristiano, S. Pagano, A. Frydman, and D. E. Prober, “Reset dynamics and latching in niobium superconducting nanowire single-photon detectors,” J. Appl. Phys. 108, 084507 (2010). [CrossRef]

]. From an optical perspective, the high index dielectric layer placed underneath the superconducting layer will serve as a guiding (core) layer. In the absence of the superconducting layer, the significant difference in the indices of this layer with the surrounding dielectric (cladding) layers permits strong lateral confinement of light on the order of the core dimension. Presence of the superconducting layer alters the lateral power distribution from center of the waveguide towards the superconducting interface and hence can improve the probability of optical absorption. To investigate the possibility of mode formation, a numerical tool is required to accurately determine the complex propagation constant of the optical modes as a function of the dielectric core thickness. There are several approaches available to accomplish this task. In this work, a derivative of the well-known transfer-matrix method, namely the reflection pole method or RPM has been chosen [19

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: Reflection pole method and wavevector density method,” J. Lightwave Technol. 17, 929–941 (1999). [CrossRef]

]. This method is particularly useful for situations where the transfer matrix method results in a complex equation. In the RPM extension, instead of a direct root calculation, changes in the phase of the reflection coefficient as a function of the real part of the effective index is monitored. An extremum in this phase corresponds to the real part of the effective index and the full width at half maximum gives the imaginary part.

Fig. 1 Layered structure of the proposed detector. Superconducting layer (nanowires) with thickness td is placed on top of a Si layer of thickness t and width w in an environment of HSQ/SiO2 cladding layers. The HSQ layer is used during patterning of the wires and can be kept as a protective layer.

Fundamentally, a plasmonic excitation is attainable in configurations where there is an abnormal TM branch with a higher effective index value compared to that of the first normal TE branch in the effective index versus core thickness diagram [20

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70, 1–87 (2007). [CrossRef]

]. This branch is known as the fundamental TM mode curve and when it appears in the structure, there is an optical mode with exponentially decaying tails in all layers [21

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B 61, 10484–10503 (2000). [CrossRef]

, 22

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of asymmetric structures,” Phys. Rev. B 63, 125417 (2001). [CrossRef]

]. Fig. 2 depicts the real part of the effective index versus the core thickness for two different superconducting film thicknesses. As shown in this figure, by decreasing the thickness of the superconducting layer from 100nm to 8nm, the fundamental TM mode shifts to the right which implies lower confinement for the 8nm thick negative index layer. In contrast to the 100nm thick case, in the 8nm layer, the cut-off thickness is also reduced to zero and there is an inflection point in the curve. In a typical structure, where the thickness of dispersive layer is more than the skin depth of the electromagnetic field, the structure is like an asymmetric slab waveguide and inherently has a minimum thickness for the core layer to support an optical mode. However, when the thickness of the superconducting layer is reduced to a value smaller than the skin depth, the structure starts to behave like a symmetric slab waveguide and the cut-off thickness moves towards zero. For the 8nm thick sample, two different regimes of operation are observed for different core thicknesses. It is important to stress that to have plasmonic guiding for very thin superconducting layers, the top and bottom cladding layers must have a similar refractive index such that the effective index seen by the plasmonic mode at either side is almost equal [20

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70, 1–87 (2007). [CrossRef]

]. This implies that the top cladding layer should have an index greater than or equal to than that of the bottom cladding (buried oxide) layer but smaller than the core layer.

Fig. 2 Real part of effective index versus normalized dielectric core thickness (normalized to the free-space wavelength) for 100nm and 8nm thick NbTiN superconducting films. For the 100nm case, the fundamental plasmonic mode has a cut-off thickness which is a direct consequence of the single interface plasmonic excitation at the interface of core and superconducting layers. This behavior can be attributed to the penetration depth of the optical mode being larger than the thickness of the superconducting layer. This condition is not satisfied for the 8nm structure. Therefore, no plasmonic excitation is expected unless the structure looks symmetric to the optical mode which happens only for small core thicknesses. Points P1, P2, and P3 will be used later in Fig. 3 with the same color scheme.

We now consider the 8nm thick superconducting structure. To qualitatively describe the importance of core thickness, without loss of generality, the original four layer structure is considered as a three-layer structure with a semi-infinite boundary at the superconducting layer. With this assumption, surface waves at both interfaces are non-interacting. This is only valid for structures where surface modes are tightly bound to the superconducting interface, or equivalently, the penetration depth of the electromagnetic field in the superconducting layer is smaller than its thickness. Although this condition is clearly violated in this structure, it can still elucidate the intuitive behavior of the system. At the end of this section precise numerical calculations are also provided.

For a very thick dielectric core layer, where neither a TE nor TM normal optical branch could exist, the structure can support a single plasmonic mode similar to one that is supported by a two layer structure made from semi-infinite core/semi-infinite superconducting layers. Decreasing the core thickness to the range of λ0/2ncore where the effect of core thickness is more pronounced, creates a smaller propagation length as the finite thickness of the high index medium forces the optical mode to be more squeezed inside this layer. This is a direct consequence of the fact that the optical wave always tends to distribute itself such that the highest effective index is obtained. The amount of this extra confinement is proportional to the ratio of the optical mode size to the core thickness. Therefore, a smaller propagation length occurs for smaller core thicknesses until some critical point where the optical mode cannot be squeezed any further, reaching its minimum mode field size [23

J. Guo and R. Adato, “Extended long range plasmon waves in finite thickness metal film and layered dielectric materials,” Opt. Express 14, 12409–12418 (2006). [CrossRef] [PubMed]

]. Further decrease of the core thickness would result in an increase of the mode field size approaching to the single plasmonic operation of a semi-infinite top-cladding/semi-infinite superconducting structure. Following this idea, the optimum core thickness which yields the maximum mode squeezing and hence the maximum amount of absorption, can be calculated by searching for the maxima in the imaginary part of effective index versus core thickness diagram.

For the case of an ultrathin superconducting layer, the functionality of the core remains the same as far as plasmonic confinement is possible. This can be done by decreasing the core thickness such that the effective index of the TE0 in the presence of a superconducting layer is larger than that of the TM0 mode in the absence of a superconducting layer. Using the numerical tool introduced at the beginning of this section and the complex refractive index for the NbTiN film [24

V. Anant, A. J. Kerman, E. A. Dauler, J. K. W. Yang, K. M. Rosfjord, and K. K. Berggren, “Optical properties of superconducting nanowire single-photon detectors,” Opt. Express 16, 10750–10761 (2008). [CrossRef] [PubMed]

], the optimum core thickness for the vacuum wavelength of 1310nm is obtained and the results are summarized in Fig. 3. For a fixed core thickness, optical power decays exponentially along the wire due to the absorption in the superconducting film. The amount of absorption is related to the imaginary part of the effective index by e−2Im{ne}β0l where ne is the effective index of the confined mode and l is the length of the nanowires. Due to the exponentially decaying nature of the absorption, a saturation point can be defined for the nanowire length as 2Im{ne}β0lmin > 10. Here, lmin can be considered as the optimum length of the nanowire since further increase of this length just increases the kinetic inductance of the nanowires without significantly improving the absorption. For the optimum core thickness (0.14λ0) where the highest confinement is achieved, the minimum wire length to reach saturation is as short as 7.6μm. This minimum length increases by having less lateral confinement. For instance, if core thickness of 0.2λ0 is chosen, the minimum length would increase to 15.3μm which is still considerably shorter than the normal length of wires used in modern-day SNSPDs. These short wire lengths would not be attainable if a higher quality negative index material (e.g., Gold) had been used in place of the NbTiN layer or the plasmonic nature of the thin-film had not been utilized.

Fig. 3 Imaginary part of effective index versus core thickness (t/λ0) for the fundamental plasmonic mode (blue curve) as well as the transverse mode field diameters versus core width (w/λ0) for three different core thicknesses (green, black and orange curves). The larger the imaginary part of effective index, the stronger the absorption in the dispersive layer. From this figure, the maximum absorption happens at a core thickness of 0.14λ0. This high absorption is achieved at the expense of not having a confined TM0 mode when the dispersive layer is absent. By accepting lower confinement for optical mode, higher transverse confinement is obtainable. For instance, for a 0.2λ0 core thickness, the core width resulting in the minimum transverse mode field size is just 0.09λ0. Note that by getting far from the maximum point of the blue curve, the lateral mode starts to decouple from the transverse mode and hence a smaller variation in the transverse mode field diameter is predicted. This behavior is observed by comparing mode field sizes of 0.17λ0 and 0.2λ0 core thicknesses. Points P1, P2, and P3 refer to three points in Fig. 2.

3. Nonequilibrium dynamics

Thus far, the optimum thickness of the nanowire supported by plasmonic confinement has been calculated. In this calculation, variation of the electromagnetic field in the transverse plane has been neglected for the sake of simplicity. This approximation is only applicable when the lateral mode distribution of the actual hybrid branch is close to the profile of the equivalent two-dimensional TM mode and the optical mode in the transverse plane can be formed on top of the superconducting layer without feeling its limited width.

To investigate the possibility of mode formation, instead of using the actual indices of refraction, the effective indices of all vertical layers are employed to construct a pseudo two-dimensional waveguide. This waveguide is symmetric with respect to the optical axis which implies a zero cutoff frequency for both fundamental TE and TM modes. Therefore, for any arbitrary width of the effective core layer, at least one optical mode is expected for both polarizations. This statement is in fact not entirely true for all thickness to width ratios. The decoupling of the actual hybrid mode into a set of perpendicular TE and TM modes is only legitimate when either the width or the thickness of the core layer can be treated as infinite in the length scale of the mode size.

As a consequence, in the slab configuration, to observe a well-defined plasmonic confinement, there is a lower bound on the feature size for the width of the waveguide dictated by the optical design. This imposes an unwanted limitation on the nanowire width from a detection perspective to achieve acceptable single-photon sensitivity, since there is an upper bound for the cross section area of the active element. Consequently, there is a trade-off between the probability of absorption and the internal quantum efficiency of the device. Although it might seem obvious that the poor optical absorption can be corrected by increasing the length of the nanowire in the direction of the wave propagation, it would come at the expense of having higher kinetic inductance or equivalently a slower operating device.

In this section, we will try to develop an intuitive model for the photon detection process, and from that estimate the optimum width of the nanowire to have reasonable detection probability as well as confined plasmonic guiding. To investigate the transient electrodynamics of the initial stage of hotspot formation, the Rothwarf-Taylor rate equations are utilized [25

N. E. Glass and D. Rogovin, “Transient electrodynamic response of thin-film superconductors to laser radiation,” Phys. Rev. B 39, 11327–11344 (1989). [CrossRef]

]. These rate equations are essentially coupled diffusion-recombination equations based on current continuity equations of participating particles:
[ ∂ ∂t− D QP ∇2] n QP= 2 n ph τB− n QP τR [ ∂ ∂t− D ph ∇2] n ph= n QP/2 τR− [ n ph τB+ n ph− n ph0 τ es]
(1)
here, n and n0 denote the time and position dependent density of the subscript particle in the transient and steady-state regimes, respectively. τB and τes are the time required by a phonon to break a pair or to escape from the hot area if there is an imbalance in the population. τR is regarded as the quasiparticle recombination time. DQP and Dph are phenomenological diffusion coefficients for the quasiparticles and phonons. In this model, there is no constraint on the density of quasiparticles, meaning, the number of superconducting pairs must greatly exceed that of quasiparticles such that there is no limitation on the quasiparticle generation. This condition is only valid when the bath temperature is small compared to the critical temperature and there is no significant local accumulation of quasiparticles. To fulfill these conditions we look beyond the first few picoseconds until the initial photon energy has generated quasiparticles with an average energy of 3Δ. At this point, phonons cannot contribute to quasiparticle generation and hence phonon density can be approximated by the thermodynamic equilibrium value, n ph=τB/τR× n QP0/2. Having this assumption, the population of quasiparticles as a function of time and position can be studied by injecting (h̄ω/Δ)(1 − e−t/τth) δ(r) excess quasiparticles during the initial thermalization time, τth. The resulting quasiparticle distribution can then be used to determine the amount of departure from the equilibrium state by relating the order parameter to the quasiparticle density through the BCS theory considering the strong-coupling regime,
n QP=4×2.08N (0) kB TcI (β)
(2)
where N(0) is the single spin electron density, β is equal to Δ(T)/kBT and
I (β)= ∫0∞ dy 1 1+exp [ β2 ( 1+exp ( −2F (β)) y2)] F (β)=− β2 ∫0∞ dx x sinh −1x sech2 ( [ 1+ x2] 1/2β/2) 1+ x2
(3)
To express the strong coupling behavior of NbTiN in these films the experimentally determined relation Δ(0) = 2.08KBTc(0) for the zero temperature energy gap has been used [26

R. Romestain, B. Delaet, P. Renaud-Goud, I. Wang, C. Jorel, J.-C. Villegier, and J.-P. Poizat, “Fabrication of a superconducting niobium nitride hot electron bolometer for single-photon counting,” New J. Phys. 6, 129–144 (2004). [CrossRef]

]. Combining the normalized quasiparticle density given by the rate equations with Eq. 2, the local increase of temperature and suppression of the order parameter can be monitored over time through I (β)=I ( β0) n QP/ n QP0 and Δ = Δ(0)exp(F(β)).

The nonequilibrium dynamics for absorption of a 1310nm wavelength photon in an 8nm thick NbTiN film is calculated and the results are shown in Fig. 4. According to this figure, the incoming photon creates a finite-sized region with a partially suppressed order parameter within the superconducting film. This region expands over time and destroys the local pair-coherency until the temperature at the center of the spot collapses down to a value less than the superconducting critical temperature. At this point, further suppression of the local order parameter is negligible, defining the end of the first stage of photon detection. Now the question is whether this local perturbation is strong enough to drive the whole system out of its equilibrium state or if the superconducting coherency could dissipate this extra energy without resulting in a phase transition. To answer this question, two main sources of instability must be studied. The first and the most well-known source is current density beyond the depairing current density outside the hot area in the sidewalks. This source is prominent when the effective width of the strip carrying current can be affected by the formation of the initial spot. To observe a voltage pulse between the two ends of the nanowire, one needs to bias the system with a constant current source with a current density lower than the depairing current in the normal condition, but higher than the depairing current density when the formation of the initial hotspot reaches its maximum effective size. The effectiveness of this source wanes as the film gets wider. For the case where changes in the effective width before and after hotspot formation is insignificant, the second source which is either vortex hopping or vortex-antivortex pair unbinding becomes the dominant source of instability [27

A. M. Kadin, M. Leung, and A. D. Smith, “Photon-assisted vortex depairing in two-dimensional superconductors,” Phys. Rev. Lett. 65, 3193–3196 (1990). [CrossRef] [PubMed]

, 28

A. M. Kadin and M. W. Johnson, “Nonequilibrium photon-induced hotspot: A new mechanism for photodetection in ultrathin metallic films,” Appl. Phys. Lett. 69, 3938–3940 (1996). [CrossRef]

]. To make the vortex-based detection an effective mechanism, the Lorentz force between the vortex (antivortex) and the driving current needs to be comparable to the repulsive force due to the barrier shape when no current is flowing.

Fig. 4 (a) Transient electrodynamics of the energy gap and quasiparticle density as a function of time and distance from the hotspot location. (b) Minimum weight coefficient versus wire width for an IR photon with 1310nm wavelength (black line). The weight coefficient can be considered as a measure of how effective the current crowding model is for a specific film width and also provides an approximate normalized current required for unpinning the vortex and antivortex. Red line shows the minimum width requirement for photons with higher energy (1.5 times higher energy compared to the black curve). For these higher energy photons, at a fixed nanowire width, lower current is required to have the same sensitivity.

In conventional SNSPDs, there is no restriction on the minimum width of wires, therefore, to take advantage of the both detection mechanisms, the minimum possible width that can be reliably fabricated (to have vortex-based detection wires need to be wider than 4.4ξ[29

K. K. Likharev, “Superconducting weak links,” Rev. Mod. Phys. 51, 101–159 (1979). [CrossRef]

]) is usually employed. However, in our plasmonic model, the wider the wires are the higher the optical performance. Therefore, based on the non-equilibrium model introduced in this section, we will try to find a measure for the effectiveness of the current crowding model for a given bias condition as a function of film width. The maximum width where current crowding still has an acceptable impact on the detection process is then selected as the optimum width for the structure. Certainly the broadening of the nanowires results in a lower internal quantum efficiency resulting from an ebbed contribution of the current crowding mechanism. However, with recent investigations on the adverse effect of sharp turns in the maximum current achievable in practical devices [30

H. L. Hortensius, E. F. C. Driessen, T. M. Klapwijk, K. K. Berggren, and J. R. Clem, “Critical-current reduction in thin superconducting wires due to current crowding,” Appl. Phys. Lett. 100, 182602 (2012). [CrossRef]

, 31

D. Henrich, P. Reichensperger, M. Hofherr, K. Ilin, M. Siegel, A. Semenov, A. Zotova, and D. Y. Vodolazov, “Geometry-induced reduction of the critical current in superconducting nanowires,” Phys. Rev. B 86, 144504 (2012). [CrossRef]

], this reduction can be compensated by utilizing an optimum bending procedure introduced in reference [32

J. R. Clem and K. K. Berggren, “Geometry-dependent critical currents in superconducting nanocircuits,” Phys. Rev. B 84, 174510 (2011). [CrossRef]

], and also driving the detector with a bias current high enough to have an effective vortex-based detection mechanism.

As a starting point, the current distribution in the nanowire as a function of time and position must be calculated. This can be achieved by writing down the conservation of current density for each time step. To solve this problem, we bound the region with a suppressed superconducting order parameter to a circle with radius R, with dimensions taken from the half width at half maximum of the actual order parameter spatial distribution. Moreover, the magnitude of order parameter is considered to be a step like function with a value equal to the average of actual order parameter inside the circle and the equilibrium value for the rest of the film. Following the same approach as the one introduced in reference [33

A. N. Zotova and D. Y. Vodolazov, “Photon detection by current-carrying superconducting film: A time-dependent Ginzburg-Landau approach,” Phys. Rev. B 85, 024509 (2012). [CrossRef]

], the effect of current crowding in the sidewalks can be expressed in terms of a single coefficient (weight coefficient),
ηw=1− ( αRw)2 ∫ in ds n QP− n QP0 ∫ in ds n QP+ n QP0
(4)
where α varies from 2 to square root of 2 based on the location of the hot region. 2 is for absorption at the center of the film and square root of 2 is for absorption at the film edges. According to this equation, aside from the clear width dependency, the size of the initial spot becomes more pronounced when there is a significant difference in the quasiparticle density inside and outside of the hot region.

In the inset of Fig. 4(b), the variation of the weight coefficient for a 150nm width wire is plotted. According to this figure, the weight coefficient starts to decay by diffusion of excess quasiparticles from the location of the initial hotspot until the average quasiparticle density in the hot region degrades faster than the hotspot growth rate. At this point, the weight coefficient reaches its minimum value and hence the maximum effect. This time step can be regarded as the end of the initial hotspot formation. The value of the weight coefficient at this time step can be considered as a measure of how effectively the initial spot can create a perturbation in the system. In general, two results can be obtained from this coefficient. First, it gives a measure for the effectiveness of the current crowding model for a given film width at a given driving current. Second, it provides an upper bound for the normalized current required for the unpinning of vortex-antivortex pairs from the normal region.

As a conclusion, when the film width increases, the effect of the current crowding model starts to diminish and consequently to keep the film sensitivity, one needs to increase the driving current above the value required by vortex-based detection mechanisms. Based on our simulation, if a photon with a wavelength of 1310nm is absorbed at the center of the wire and the source current is set to 0.7 of the depairing current, only wires with widths smaller than roughly 150nm are well sensitive to photons. This is comparable to the normal maximum achievable current in conventional SNSPDs with sharp turns and 50% filling factor [32

J. R. Clem and K. K. Berggren, “Geometry-dependent critical currents in superconducting nanocircuits,” Phys. Rev. B 84, 174510 (2011). [CrossRef]

]. This minimum sensitivity would jump to 250nm if the driving current can be increased by roughly 20 percent. This can be achieved by replacing all the sharp turns with optimum bends. It is important to stress that from experimental evidence, lower detection efficiency is expected for the vortex-antivortex model [34

M. Hofherr, D. Rall, K. S. Ilin, A. Semenov, N. Gippius, H.-W. Hübers, and M. Siegel, “Superconducting nanowire single-photon detectors: Quantum efficiency vs. film thickness,” J. Phys. 234, 012017 (2010).

]. This can be attributed to the fact that not all nucleations can result in sample heating. Thus, by crossing over from the current crowding to vortex-based detection mechanisms, a gradual reduction in system detection efficiency is inevitable. To take advantage from both detection mechanisms, considering no sharp turns in the structure, wires with width less than 200nm are recommended for infrared photon detection.

According to the optical calculations of section 2, for a waveguide with confined plasmonic operation (required for getting high optical absorption), a very poor transverse confinement is expected. In this case, the size of the optical mode in the transverse plane is actually the minimum width for both the dielectric core layer and the superconducting layer required to keep the mode’s surface confinement. Comparing the results of the nonequilibrium model (maximum acceptable width must be less than 200nm) with the mode field diameter of the first fundamental plasmonic mode, reveals that there is no optimum width for a single nanowire that is wide enough to cover the entire optical mode and still give high photon sensitivity. Although a single wire cannot fulfill both of these requirements simultaneously, a set of wires parallel to the direction of propagation with a spacing much smaller than the transverse mode field size can be used as a uniform dispersive layer from an optical perceptive. The minimum spacing between wires can be as small as a few nanometers with current fabrication technology and is dictated by the resolution of the electron beam tool used for patterning the wires [9

L. Zhang, L. Kang, J. Chen, Y. Zhong, Q. Zhao, T. Jia, C. Cao, B. Jin, W. Xu, G. Sun, and P. Wu, “Ultra-low dark count rate and high system efficiency single-photon detectors with 50 nm-wide superconducting wires,” Appl. Phys. B 102, 867–871 (2011). [CrossRef]

].

Connection of these parallel wires with a high filling factor can generally cause two practical problems as all dimensions are smaller than the Pearl penetration depth (more than 50μm for our device). First, there is no optimum design for bends at this high filling factor resulting in a lower critical current than the one predicted by Kupriyanov and Lukichev [36

M. Kupriyanov and V. Lukichov, “Temperature dependence of the pair-breaking current density in superconductors,” Fiz. Nizk. Temp. 6, 445–453 (1980).

]. This low driving current leads to a quantum efficiency smaller than what can be potentially expected from the device. More importantly, for a fixed bias current, the device exhibits a larger dark count rate in comparison to straight wires with no bends. The high dark count rate can be explained due to the presence of local areas (bend areas) with current density much higher than that of straight sections. These high bias regions are very small in dimension and cannot enhance device sensitivity, but are susceptible enough to produce huge dark counts with either vortex-antivortex unbinding [16

H. Bartolf, A. Engel, A. Schilling, K. Il’in, M. Siegel, H.-W. Hubers, and A. Semenov, “Current-assisted thermally activated flux liberation in ultrathin nanopatterned NbN superconducting meander structures,” Phys. Rev. B 81, 024502 (2010). [CrossRef]

] or vortex hoping origins [37

T. Yamashita, S. Miki, K. Makise, W. Qiu, H. Terai, M. Fujiwara, M. Sasaki, and Z. Wang, “Origin of intrinsic dark count in superconducting nanowire single-photon detectors,” Appl. Phys. Lett. 99, 161105 (2011). [CrossRef]

–39

L. N. Bulaevskii, M. J. Graf, C. D. Batista, and V. G. Kogan, “Vortex-induced dissipation in narrow current-biased thin-film superconducting strips,” Phys. Rev. B 83, 144526 (2011). [CrossRef]

]. To address these issues, we propose connection links with a non-uniform filling factor and optimally rounded bends at all inner corners (Fig. 5). In this design the size of connection links are small enough that they can be neglected in the optical design but can effectively enhance the maximum achievable current of the device.

Fig. 5 Top view of a sample connection link at the input terminal. This link can address the problem of current crowding in a high filling factor device. Optical input terminal can be either an inverse taper coupler or a surface grating coupler [35

M. Antelius, K. B. Gylfason, and H. Sohlström, “An apodized SOI waveguide-to-fiber surface grating coupler for single lithography silicon photonics,” Opt. Express 19, 3592–3598 (2011). [CrossRef] [PubMed]

].

4. Concluding remarks

In this paper, the idea of enhancing quantum efficiency of SNSPDs through plasmonic excitation of wires was proposed. In the first section, we discussed the possibility of plasmonic excitation on a superconducting film only a few nanometers thick. We showed that to get the maximum optical confinement, and hence maximum absorption, one needs to use a very thin high index dielectric material in proximity to a thin film superconductor. The thickness of this core dielectric layer must be smaller than what is required for normal dielectric guiding at the same wavelength. Also, the structure needs to be placed in an environment of a symmetric low index medium. The following section described how the detection mechanism crosses over from the current crowding model to the vortex-antivortex model by increasing the width of the nanowire. A weight coefficient based on nonequilibrium superconductivity was introduced as a measure on the bias condition to keep the device sensitivity when the width is increased. We also demonstrated that there is no optimum width for wires that are wide enough to cover the entire optical mode and still provide high photon sensitivity. To address this issue a set of wires parallel to the direction of propagation with a spacing much smaller than the transverse mode field size with non-uniform filling factors was proposed.

Acknowledgments

The authors would like to acknowledge financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Institute for Quantum Computing (IQC). Amin Eftekharian is supported by the Mike and Ophelia Lazaridis Fellowship, and Haig Atikian is supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).

References and links

1.

J. L. O’Brien, A. Furusawa, and J. V. kovic, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009). [CrossRef]

2.

M. J. Stevens, R. H. Hadfield, R. E. Schwall, S. W. Nam, R. P. Mirin, and J. A. Gupta, “Fast lifetime measurements of infrared emitters using a low-jitter superconducting single-photon detector,” Appl. Phys. Lett. 89, 031109 (2006). [CrossRef]

3.

M. Thompson, A. Politi, J. Matthews, and J. O’Brien, “Integrated waveguide circuits for optical quantum computing,” IET Circuits Devices Syst. 5, 94–102 (2011). [CrossRef]

4.

R. Yan, D. Gargas, and P. Yang, “Nanowire photonics,” Nat. Photonics 3, 569–576 (2009). [CrossRef]

5.

J. P. Sprengers, A. Gaggero, D. Sahin, S. Jahanmirinejad, G. Frucci, F. Mattioli, R. Leoni, J. Beetz, M. Lermer, M. Kamp, S. Höfling, R. Sanjines, and A. Fiore, “Waveguide superconducting single-photon detectors for integrated quantum photonic circuits,” Appl. Phys. Lett. 99, 181110 (2011).

6.

R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photonics 3, 696–705 (2009). [CrossRef]

7.

S. Miki, M. Fujiwara, M. Sasaki, B. Baek, A. J. Miller, R. H. Hadfield, S. W. Nam, and Z. Wang, “Large sensitive-area NbN nanowire superconducting single-photon detectors fabricated on single-crystal MgO substrates,” Appl. Phys. Lett. 92, 061116 (2008). [CrossRef]

8.

R. Sobolewski, A. Verevkin, G. Gol’tsman, A. Lipatov, and K. Wilsher, “Ultrafast superconducting single-photon optical detectors and their applications,” IEEE Trans. App. Supercond. 13, 1151–1157 (2009). [CrossRef]

9.

L. Zhang, L. Kang, J. Chen, Y. Zhong, Q. Zhao, T. Jia, C. Cao, B. Jin, W. Xu, G. Sun, and P. Wu, “Ultra-low dark count rate and high system efficiency single-photon detectors with 50 nm-wide superconducting wires,” Appl. Phys. B 102, 867–871 (2011). [CrossRef]

10.

A. Hamed Majedi, “Theoretical investigations on THz and optical superconductive surface plasmon interface,” IEEE Trans. App. Supercond. 19, 907–910 (2009). [CrossRef]

11.

P. Berini, R. Charbonneau, and N. Lahoud, “Long-range surface plasmons on ultrathin membranes,” Nano Lett. 7, 1376–1380 (2007). [CrossRef] [PubMed]

12.

C. M. Natarajan, A. Peruzzo, S. Miki, M. Sasaki, Z. Wang, B. Baek, S. Nam, R. H. Hadfield, and J. L. O’Brien, “Operating quantum waveguide circuits with superconducting single-photon detectors,” Appl. Phys. Lett. 96, 211101 (2010). [CrossRef]

13.

A. D. Semenov, G. N. Gol’tsman, and A. A. Korneev, “Quantum detection by current carrying superconducting film,” Phys. C Supercond. 351, 349–356 (2001). [CrossRef]

14.

A. J. Annunziata, O. Quaranta, D. F. Santavicca, A. Casaburi, L. Frunzio, M. Ejrnaes, M. J. Rooks, R. Cristiano, S. Pagano, A. Frydman, and D. E. Prober, “Picosecond superconducting single-photon optical detector,” Appl. Phys. Lett. 79, 705–707 (2001). [CrossRef]

15.

A. M. Kadin, M. Leung, A. D. Smith, and J. M. Murduck, “Photofluxonic detection: A new mechanism for infrared detection in superconducting thin films,” Appl. Phys. Lett. 57, 2847–2849 (1990). [CrossRef]

16.

H. Bartolf, A. Engel, A. Schilling, K. Il’in, M. Siegel, H.-W. Hubers, and A. Semenov, “Current-assisted thermally activated flux liberation in ultrathin nanopatterned NbN superconducting meander structures,” Phys. Rev. B 81, 024502 (2010). [CrossRef]

17.

J. K. W. Yang, A. J. Kerman, E. A. Dauler, V. Anant, K. M. Rosfjord, and K. K. Berggren, “Modeling the electrical and thermal response of superconducting nanowire single-photon detectors,” IEEE Trans. Appl. Supercond. 17, 581–585 (2007). [CrossRef]

18.

A. J. Annunziata, O. Quaranta, D. F. Santavicca, A. Casaburi, L. Frunzio, M. Ejrnaes, M. J. Rooks, R. Cristiano, S. Pagano, A. Frydman, and D. E. Prober, “Reset dynamics and latching in niobium superconducting nanowire single-photon detectors,” J. Appl. Phys. 108, 084507 (2010). [CrossRef]

19.

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: Reflection pole method and wavevector density method,” J. Lightwave Technol. 17, 929–941 (1999). [CrossRef]

20.

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70, 1–87 (2007). [CrossRef]

21.

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B 61, 10484–10503 (2000). [CrossRef]

22.

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of asymmetric structures,” Phys. Rev. B 63, 125417 (2001). [CrossRef]

23.

J. Guo and R. Adato, “Extended long range plasmon waves in finite thickness metal film and layered dielectric materials,” Opt. Express 14, 12409–12418 (2006). [CrossRef] [PubMed]

24.

V. Anant, A. J. Kerman, E. A. Dauler, J. K. W. Yang, K. M. Rosfjord, and K. K. Berggren, “Optical properties of superconducting nanowire single-photon detectors,” Opt. Express 16, 10750–10761 (2008). [CrossRef] [PubMed]

25.

N. E. Glass and D. Rogovin, “Transient electrodynamic response of thin-film superconductors to laser radiation,” Phys. Rev. B 39, 11327–11344 (1989). [CrossRef]

26.

R. Romestain, B. Delaet, P. Renaud-Goud, I. Wang, C. Jorel, J.-C. Villegier, and J.-P. Poizat, “Fabrication of a superconducting niobium nitride hot electron bolometer for single-photon counting,” New J. Phys. 6, 129–144 (2004). [CrossRef]

27.

A. M. Kadin, M. Leung, and A. D. Smith, “Photon-assisted vortex depairing in two-dimensional superconductors,” Phys. Rev. Lett. 65, 3193–3196 (1990). [CrossRef] [PubMed]

28.

A. M. Kadin and M. W. Johnson, “Nonequilibrium photon-induced hotspot: A new mechanism for photodetection in ultrathin metallic films,” Appl. Phys. Lett. 69, 3938–3940 (1996). [CrossRef]

29.

K. K. Likharev, “Superconducting weak links,” Rev. Mod. Phys. 51, 101–159 (1979). [CrossRef]

30.

H. L. Hortensius, E. F. C. Driessen, T. M. Klapwijk, K. K. Berggren, and J. R. Clem, “Critical-current reduction in thin superconducting wires due to current crowding,” Appl. Phys. Lett. 100, 182602 (2012). [CrossRef]

31.

D. Henrich, P. Reichensperger, M. Hofherr, K. Ilin, M. Siegel, A. Semenov, A. Zotova, and D. Y. Vodolazov, “Geometry-induced reduction of the critical current in superconducting nanowires,” Phys. Rev. B 86, 144504 (2012). [CrossRef]

32.

J. R. Clem and K. K. Berggren, “Geometry-dependent critical currents in superconducting nanocircuits,” Phys. Rev. B 84, 174510 (2011). [CrossRef]

33.

A. N. Zotova and D. Y. Vodolazov, “Photon detection by current-carrying superconducting film: A time-dependent Ginzburg-Landau approach,” Phys. Rev. B 85, 024509 (2012). [CrossRef]

34.

M. Hofherr, D. Rall, K. S. Ilin, A. Semenov, N. Gippius, H.-W. Hübers, and M. Siegel, “Superconducting nanowire single-photon detectors: Quantum efficiency vs. film thickness,” J. Phys. 234, 012017 (2010).

35.

M. Antelius, K. B. Gylfason, and H. Sohlström, “An apodized SOI waveguide-to-fiber surface grating coupler for single lithography silicon photonics,” Opt. Express 19, 3592–3598 (2011). [CrossRef] [PubMed]

36.

M. Kupriyanov and V. Lukichov, “Temperature dependence of the pair-breaking current density in superconductors,” Fiz. Nizk. Temp. 6, 445–453 (1980).

37.

T. Yamashita, S. Miki, K. Makise, W. Qiu, H. Terai, M. Fujiwara, M. Sasaki, and Z. Wang, “Origin of intrinsic dark count in superconducting nanowire single-photon detectors,” Appl. Phys. Lett. 99, 161105 (2011). [CrossRef]

38.

L. N. Bulaevskii, M. J. Graf, and V. G. Kogan, “Vortex-assisted photon counts and their magnetic field dependence in single-photon superconducting detectors,” Phys. Rev. B 85, 014505 (2012). [CrossRef]

39.

L. N. Bulaevskii, M. J. Graf, C. D. Batista, and V. G. Kogan, “Vortex-induced dissipation in narrow current-biased thin-film superconducting strips,” Phys. Rev. B 83, 144526 (2011). [CrossRef]

OCIS Codes
(040.0040) Detectors : Detectors
(040.5570) Detectors : Quantum detectors
(270.5570) Quantum optics : Quantum detectors

ToC Category:
Detectors

History
Original Manuscript: December 6, 2012
Revised Manuscript: January 22, 2013
Manuscript Accepted: January 23, 2013
Published: January 31, 2013

Citation
Amin Eftekharian, Haig Atikian, and A. Hamed Majedi, "Plasmonic superconducting nanowire single photon detector," Opt. Express 21, 3043-3054 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-3-3043


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References

  1. J. L. O’Brien, A. Furusawa, and J. V. kovic, “Photonic quantum technologies,” Nat. Photonics3, 687–695 (2009). [CrossRef]
  2. M. J. Stevens, R. H. Hadfield, R. E. Schwall, S. W. Nam, R. P. Mirin, and J. A. Gupta, “Fast lifetime measurements of infrared emitters using a low-jitter superconducting single-photon detector,” Appl. Phys. Lett.89, 031109 (2006). [CrossRef]
  3. M. Thompson, A. Politi, J. Matthews, and J. O’Brien, “Integrated waveguide circuits for optical quantum computing,” IET Circuits Devices Syst.5, 94–102 (2011). [CrossRef]
  4. R. Yan, D. Gargas, and P. Yang, “Nanowire photonics,” Nat. Photonics3, 569–576 (2009). [CrossRef]
  5. J. P. Sprengers, A. Gaggero, D. Sahin, S. Jahanmirinejad, G. Frucci, F. Mattioli, R. Leoni, J. Beetz, M. Lermer, M. Kamp, S. Höfling, R. Sanjines, and A. Fiore, “Waveguide superconducting single-photon detectors for integrated quantum photonic circuits,” Appl. Phys. Lett.99, 181110 (2011).
  6. R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photonics3, 696–705 (2009). [CrossRef]
  7. S. Miki, M. Fujiwara, M. Sasaki, B. Baek, A. J. Miller, R. H. Hadfield, S. W. Nam, and Z. Wang, “Large sensitive-area NbN nanowire superconducting single-photon detectors fabricated on single-crystal MgO substrates,” Appl. Phys. Lett.92, 061116 (2008). [CrossRef]
  8. R. Sobolewski, A. Verevkin, G. Gol’tsman, A. Lipatov, and K. Wilsher, “Ultrafast superconducting single-photon optical detectors and their applications,” IEEE Trans. App. Supercond.13, 1151–1157 (2009). [CrossRef]
  9. L. Zhang, L. Kang, J. Chen, Y. Zhong, Q. Zhao, T. Jia, C. Cao, B. Jin, W. Xu, G. Sun, and P. Wu, “Ultra-low dark count rate and high system efficiency single-photon detectors with 50 nm-wide superconducting wires,” Appl. Phys. B102, 867–871 (2011). [CrossRef]
  10. A. Hamed Majedi, “Theoretical investigations on THz and optical superconductive surface plasmon interface,” IEEE Trans. App. Supercond.19, 907–910 (2009). [CrossRef]
  11. P. Berini, R. Charbonneau, and N. Lahoud, “Long-range surface plasmons on ultrathin membranes,” Nano Lett.7, 1376–1380 (2007). [CrossRef] [PubMed]
  12. C. M. Natarajan, A. Peruzzo, S. Miki, M. Sasaki, Z. Wang, B. Baek, S. Nam, R. H. Hadfield, and J. L. O’Brien, “Operating quantum waveguide circuits with superconducting single-photon detectors,” Appl. Phys. Lett.96, 211101 (2010). [CrossRef]
  13. A. D. Semenov, G. N. Gol’tsman, and A. A. Korneev, “Quantum detection by current carrying superconducting film,” Phys. C Supercond.351, 349–356 (2001). [CrossRef]
  14. A. J. Annunziata, O. Quaranta, D. F. Santavicca, A. Casaburi, L. Frunzio, M. Ejrnaes, M. J. Rooks, R. Cristiano, S. Pagano, A. Frydman, and D. E. Prober, “Picosecond superconducting single-photon optical detector,” Appl. Phys. Lett.79, 705–707 (2001). [CrossRef]
  15. A. M. Kadin, M. Leung, A. D. Smith, and J. M. Murduck, “Photofluxonic detection: A new mechanism for infrared detection in superconducting thin films,” Appl. Phys. Lett.57, 2847–2849 (1990). [CrossRef]
  16. H. Bartolf, A. Engel, A. Schilling, K. Il’in, M. Siegel, H.-W. Hubers, and A. Semenov, “Current-assisted thermally activated flux liberation in ultrathin nanopatterned NbN superconducting meander structures,” Phys. Rev. B81, 024502 (2010). [CrossRef]
  17. J. K. W. Yang, A. J. Kerman, E. A. Dauler, V. Anant, K. M. Rosfjord, and K. K. Berggren, “Modeling the electrical and thermal response of superconducting nanowire single-photon detectors,” IEEE Trans. Appl. Supercond.17, 581–585 (2007). [CrossRef]
  18. A. J. Annunziata, O. Quaranta, D. F. Santavicca, A. Casaburi, L. Frunzio, M. Ejrnaes, M. J. Rooks, R. Cristiano, S. Pagano, A. Frydman, and D. E. Prober, “Reset dynamics and latching in niobium superconducting nanowire single-photon detectors,” J. Appl. Phys.108, 084507 (2010). [CrossRef]
  19. E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: Reflection pole method and wavevector density method,” J. Lightwave Technol.17, 929–941 (1999). [CrossRef]
  20. J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys.70, 1–87 (2007). [CrossRef]
  21. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B61, 10484–10503 (2000). [CrossRef]
  22. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of asymmetric structures,” Phys. Rev. B63, 125417 (2001). [CrossRef]
  23. J. Guo and R. Adato, “Extended long range plasmon waves in finite thickness metal film and layered dielectric materials,” Opt. Express14, 12409–12418 (2006). [CrossRef] [PubMed]
  24. V. Anant, A. J. Kerman, E. A. Dauler, J. K. W. Yang, K. M. Rosfjord, and K. K. Berggren, “Optical properties of superconducting nanowire single-photon detectors,” Opt. Express16, 10750–10761 (2008). [CrossRef] [PubMed]
  25. N. E. Glass and D. Rogovin, “Transient electrodynamic response of thin-film superconductors to laser radiation,” Phys. Rev. B39, 11327–11344 (1989). [CrossRef]
  26. R. Romestain, B. Delaet, P. Renaud-Goud, I. Wang, C. Jorel, J.-C. Villegier, and J.-P. Poizat, “Fabrication of a superconducting niobium nitride hot electron bolometer for single-photon counting,” New J. Phys.6, 129–144 (2004). [CrossRef]
  27. A. M. Kadin, M. Leung, and A. D. Smith, “Photon-assisted vortex depairing in two-dimensional superconductors,” Phys. Rev. Lett.65, 3193–3196 (1990). [CrossRef] [PubMed]
  28. A. M. Kadin and M. W. Johnson, “Nonequilibrium photon-induced hotspot: A new mechanism for photodetection in ultrathin metallic films,” Appl. Phys. Lett.69, 3938–3940 (1996). [CrossRef]
  29. K. K. Likharev, “Superconducting weak links,” Rev. Mod. Phys.51, 101–159 (1979). [CrossRef]
  30. H. L. Hortensius, E. F. C. Driessen, T. M. Klapwijk, K. K. Berggren, and J. R. Clem, “Critical-current reduction in thin superconducting wires due to current crowding,” Appl. Phys. Lett.100, 182602 (2012). [CrossRef]
  31. D. Henrich, P. Reichensperger, M. Hofherr, K. Ilin, M. Siegel, A. Semenov, A. Zotova, and D. Y. Vodolazov, “Geometry-induced reduction of the critical current in superconducting nanowires,” Phys. Rev. B86, 144504 (2012). [CrossRef]
  32. J. R. Clem and K. K. Berggren, “Geometry-dependent critical currents in superconducting nanocircuits,” Phys. Rev. B84, 174510 (2011). [CrossRef]
  33. A. N. Zotova and D. Y. Vodolazov, “Photon detection by current-carrying superconducting film: A time-dependent Ginzburg-Landau approach,” Phys. Rev. B85, 024509 (2012). [CrossRef]
  34. M. Hofherr, D. Rall, K. S. Ilin, A. Semenov, N. Gippius, H.-W. Hübers, and M. Siegel, “Superconducting nanowire single-photon detectors: Quantum efficiency vs. film thickness,” J. Phys.234, 012017 (2010).
  35. M. Antelius, K. B. Gylfason, and H. Sohlström, “An apodized SOI waveguide-to-fiber surface grating coupler for single lithography silicon photonics,” Opt. Express19, 3592–3598 (2011). [CrossRef] [PubMed]
  36. M. Kupriyanov and V. Lukichov, “Temperature dependence of the pair-breaking current density in superconductors,” Fiz. Nizk. Temp.6, 445–453 (1980).
  37. T. Yamashita, S. Miki, K. Makise, W. Qiu, H. Terai, M. Fujiwara, M. Sasaki, and Z. Wang, “Origin of intrinsic dark count in superconducting nanowire single-photon detectors,” Appl. Phys. Lett.99, 161105 (2011). [CrossRef]
  38. L. N. Bulaevskii, M. J. Graf, and V. G. Kogan, “Vortex-assisted photon counts and their magnetic field dependence in single-photon superconducting detectors,” Phys. Rev. B85, 014505 (2012). [CrossRef]
  39. L. N. Bulaevskii, M. J. Graf, C. D. Batista, and V. G. Kogan, “Vortex-induced dissipation in narrow current-biased thin-film superconducting strips,” Phys. Rev. B83, 144526 (2011). [CrossRef]

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