The error function fit which we observe for low photon energies are indicative of Fano fluctuations. However, deviations from this shape at high energies suggests that this not the only mechanism present here. A possible explanation could come in the form of an additional model which predicts position-dependent effects in the nanowire. In this model, different parts of the cross section of the superconducting nanowire become photodetecting at different bias currents, due to an intrinsic position dependence in the fundamental detection mechanism [35, 45]. In such a model, different points in the cross-section of the wire have different energy-current relations. Consequently, this gives rise to additional broadening of the transition (in addition to the Fano fluctuations), where the width of the transition is given by∆Ib =Imin(E)−Imax(E), where Imin and Imax are the threshold currents at the most efficient point (edge) and the least efficient point (middle) along the cross-section of the wire, respectively. For such a model, one expects the width of the transition to increase with higher photon energies [32, 35], which could explain why the error function fit is not as good at higher photon energies. Moreover, due to the sharpening of the error-function transition in Fig. 3.8 at higher photon energies, one would expect any additional effects to be more visible, even if the position dependence effect is weakly dependent on photon energy.
It has to be noted that due to the transition width energy dependence shown in Fig. 3.8, the choice for η affects the shape of energy-current relation. It is therefore interesting to consider different values. The η= 50% value seems to be the optimal choice in the context of the Fano fluctuations model because it corresponds to the inflexion point of the error functions used to fit the PCR curves. The η= 1% relation is also interesting to make a comparison with measurements based on QDT, since the energy-current relation is probed in the rising part of the PCR curves, far below the saturation of the detection efficiency. The choice of η = 1% makes the energy-current relation appears closer to linear, but it remains clearly nonlinear nonetheless. The appropriate choice for ηto study the physical meaning of the energy-current relation is crucial.
Regarding other experiments, for 220 nm-wide NbN SNSPDs made from nanobridges, and also with nanodetectors and meanders, the energy-current relation was found to be linear in the range of 0.75 to 8.26 eV using quantum detector tomography [31]. A result consistent with this was found for TaN detectors [46], and for a series of NbN meanders of varying widths [36]. Nevertheless, a nonlinear behaviour for NbN meanders probed with a filtered black body light source was later observed in the 0.5 to 2.75 eV range [47]
by using the two probability thresholds η= 50%and 90%of the normalised PCR. For the amorphous materials, a previous study with WSi meanders found a linear relation at low energies and a single point deviating from this trend at 1.8 eV [42]. Recently, measurements on 220 nm-wide WSi SNSPDs nanobridges [34] using quantum detector tomography technique have shown a linear behaviour from 0.85 to 2.5 eV, but with a slight deviation from the linear tendency between 0.75 and 0.85 eV. Reviewing this
3.4. Discussion seemingly contradictory evidence, no obvious distinction between the two groups of results presents itself: neither wire width, nor device geometry, nor measurement method, nor the crystallinity of the material. While the results presented in this chapter add additional data for the theorists community, the complete description of the detection mechanism remains an open problem.
4 System detection efficiency
The system detection efficiency (SDE) is defined as the ratio of the number of detected photons over the number of photons sent onto the detector. It is a crucial characteristic of single photon detectors for many different applications [3]. To enhance the absorption of photon in the superconducting nanowire, the detector can be stacked in an optical cavity consisting of dielectric layers and metallic mirrors. By tuning the thickness of the dielectric layers, constructive interferences appear and increase the absorption in the superconducting material. A SDE of 93% has been demonstrated by Marsili et al. [4] with amorphous WSi in 2013. Very recently, >95% has been performed by NIST (National Institute of Standard and Technologies) [6]. To reach near-unity efficiencies, a perfectly optimized optical stack is mandatory.
The SDE can be decomposed in the following way:
η(Ib) =ηcoupl×ηabsorb×ηQE(Ib) (4.1)
whereηcoupl,ηabsorb, andηQE are the coupling efficiency, absorption, and internal quantum efficiency, respectively. Ib is the bias current running through the nanowire. The three components are sketched in Fig. 4.1.
The coupling efficiency is the easiest component to optimise. By using self-aligning package [26], the center of the single mode fibre (SMF) can be aligned with the center of the meander with a precision below±3 µm. As described previously, the absorption can be improved by fabricating detector stacked in a cavity. This part is detailed in the next Section 4.1. The internal detection efficiency depends on many parameters, namely: (i) the nanowire width, (ii) its thickness, (iii) the superconducting material properties, (iv) temperature, (v) photon wavelength, and (vi) bias current. A saturated internal quantum efficiency is mandatory to reach near-unity SDE. When cooled down under 2 K, MoSi nanowires exhibit large plateau for 1550 nm photon wavelength, making it a material of choice for high efficiency detectors at telecom wavelength. While the points (i) and (ii) can be optimized by fabricating different devices, the point (iii) is way more complex and
(a) (b) (c)
Figure 4.1: Schematic representation of the different system detection efficiency compo-nents. (a) Coupling efficiency. (b) Absorption. (c) Internal quantum efficiency.
requires high quality superconducting films. This point is described in Section 2.2.
4.1 Finite-element modelling of detector design
In order to compute and optimize the absorption in the nanowire, the cavity structure was modelled using Comsol MultiPhysics 5.4 software with the wave optics module. The nanowire is represented by an infinitely repeating 2D unit cell as illustrated in Fig. 4.2.
The nanowire is assumed to be infinitely long in the y-direction. In order to optimize the dielectric stack, a specific nanowire design has to be decided beforehand. With a given fill-factor (fraction of active area, ff = gap+widthwidth ) and nanowire width, the simulation is run to vary the dielectric layers thickness aiming for the absorption optimal. For simplicity, the examples shown in this section are done with a fill-factor of 0.5, a width of 150 nm and a photon wavelength of 1550 nm.
The simulated example shown in Fig. 4.3 corresponds to a single top SiO2 layer cavity design. Parameters are varied around the optimal values that gives an absorption of 92% for linear polarization parallel to the nanowire. The design is constructed as follows:
nanowire width = 150 nm, MoSi thickness = 6.5 nm, fill-factor = 0.5,n= 5.5,k= 5.0, SiO2
spacer and top layer thicknesses = 200 and 23 nm, respectively. The Fig. 4.3a represents the photon absorption as a function of the SiO2 spacer and top layer thicknesses. On the nano-fabrication side, the typical precision of a SiO2 thickness deposition is realistically around ±2 nm. We see that even if both layers are slightly shifted form what we expect, the absorption remains high. The Fig. 4.3b illustrates the absorption as a function of the refractive indexnand the extinction ratiokof MoSi. This dependence is more problematic as nandk are determined by an ellipsometry measurement, which is a complex setup, especially when dealing with custom and thin-layers of metals. Uncertainties can easily emerge from the ellipsometry model and caution must be taken. In the worse case
4.1. Finite-element modelling of detector design
0 200
Electric field norm (V/m)
0 100
-100
-200
-300 200 300 400 600
x (nm) z (nm)
Figure 4.2: Unit cell for COMSOL simulation. The nanowire is assumed to be infinitely long in the y-direction, the unit cell is copied and repeated in the x-direction with border conditions that impose continuity at the edges.
scenario, the absorption reduces down to 87% if n, k varies by a factor of 10% in the wrong direction. The Fig. 4.3c plots the absorption as a function of the nanowire thickness.
We estimate the deposition uncertainty to be under ±1 nm. Importantly, it has be noted that increasing the nanowire thickness increases the absorption. While this phenomenon is exploited by other groups to increase the efficiency for single top layer cavity design [48], increasing the thickness affects the thin-film superconducting properties and would result in less sensitive nanowire for 1550 nm photon. The Fig. 4.3d shows the absorption as a function of the fill-factor. The important conclusion of this graphics is that by increasing the fill-factor, the absorption can be enhanced quite easily even with the single top layer cavity design. Benefiting from the great sensitivity of the MoSi material, the fill-factor could be increased up to 0.7 while keeping the saturation regime needed for high efficiencies. This point is detailed in Section 4.3.
To conclude on the simulation part, as we can see from the data set, the absorption is relatively robust to nano-fabrication imperfections (SiO2 thickness, n, k values and MoSi thickness), and we should in theory expect highly efficient detectors. These results define the design parameters for the micro-cavity-enhanced SNSPD fabrication used during this thesis. It must be noted that this simulation assumed a perfect device without any intra-cavity loss. In the comparison with the experimental results in Section 4.3, we will see that the measured detection efficiency is slightly lower, pointing towards imperfections either on the fabrication and/or simulation side.
SiO2 spacer thickness (nm)
SiO2 top layer thickness (nm)
Absorption
(a)
k Absorption
n (b)
MoSi thickness (nm)
Absorption
(c)
Fill-factor
Absorption
(d)
Figure 4.3: Simulation data set for single top layer cavity design. Parameters are varied around the following optimal values: nanowire width = 150 nm, MoSi thickness = 6.5 nm, fill-factor = 0.5, n = 5.5, k = 5.0, SiO2 spacer and top layer thicknesses = 200 and 23 nm, respectively. The polarization is parallel to the nanowire (TE). The corresponding absorption is 92%. (a) Absorption as a function of the SiO2 spacer and top layer thicknesses. (b) Absorption as a function of the refractive indexn and extinction ratiok. (c) Absorption as a function of the MoSi thickness. (d) Absorption as a function of the
fill-factor.