From previous investigations on the inuence of various ALD parameters on the lm properties [169, 171], the strong dependence of the NbN lm quality on the hydrogen-plasma condition (hydrogen ow rate, exposure time) has become evident. In this section, the plasma inuence is further investigated by varying the vertical distance between the remote ICP source and the substrate (Fig. 3.1), thereby modifying how the plasma aects the lm growth, while keeping the other deposition parameters the same (see the caveats to this in the discussion below). By bringing the sample closer to the ICP, the interaction of the plasma species (ions, radicals) with the precursor layer on the substrate surface should be amplied. This method was also chosen because in the current ALD process the ICP is already set to the highest possible power (300 W) of the particular source.

**Figure 3.4:** Properties of ALD-NbN stripes dependent on the distance between the inductively coupled plasma (ICP) source and the substrate: residual resistivity ρn, critical temperature Tc (a), and switching current density jsw (b). Each NbN stripe is 10 nm thick, 200 nm wide and 2 µm long [see inset in (b)].

The vertical distance of the ICP from the substrate table amounts to 23 cm. By placing the samples on aluminum pedestals of various heights (between 0.6 cm and 7.9 cm), this distance was reduced. Each pedestal was placed in the center of the ALD sample table concentrically below the ICP, and each sample was deposited in a separate run. As substrates, we used 10 mm × 10 mm silicon chips with 300 nm thermal oxide and a 20 nm thick Al2O3 layer deposited by ALD beforehand and in parallel for all substrates. Each NbN layer had a nominal thickness of 10 nm.

After deposition, nanobridges for four-point measurements were patterned from each lm by EBL and ion milling (same process as described in Section 3.2). The following measurements were conducted on bridges with a size w ×l = 0.2 µm × 2 µm. Similar to Section 3.2, standard R(T ) and I (V ) measurements were conducted to extract the basic superconducting properties of the samples.

The results are depicted in Fig. 3.4. As it is clearly visible, there is a linear dependence of all parameters on the distance of the sample to the ICP source. The "best" properties (lowest ρn, highest Tc and jsw) were obtained on the sample the furthest away from the ICP source. Extrapolating the measured dependence, these values are consistent with previously measured ones from samples placed directly on the same table at distance 23 cm but on sapphire and quartz substrates (compare to Figs. 3.2 and 3.3). The closer the substrate was placed to the plasma, the stronger the degradation of the superconducting properties. At the closest distance of 15.1 cm, the residual resistivity has increased up to 690 µΩ cm (up from 293 µΩ cm), while Tc has decreased to 8.95 K (down from 10.57 K) and jsw to 1.32 MA/cm2 (down from ≈5.5 MA/cm2 ).

One reason that could explain the observed degradation is the higher plasma power which might lead to hydrogenation of the deposited lm (NbHn). These hydrides can oxidize to compounds like Nb2O5, potentially leading to a signicant amount of oxygen impurities and a degradation of the NbN lm properties. This assumption could be substantiated by studying the compositional changes of the lms and, specically, a possible increase in oxygen content in lms on substrates placed closer to the ICP.

However, there are other factors beyond the immediate plasma inuence in this particular parameter-variation experiment which could explain the observed lm degradation. The substrate table is heated to 380 ◦C. The aluminum pedestals which are used to vary the height, however, likely cause the substrate temperature to be signicantly lower. A lower deposition temperature was found to have a strong inuence on the lm properties before [169]. On the other side, there is a stronger heat contribution on the substrate surface by the plasma itself for samples closer to the ICP, which might have an inuence on the lm growth as well. Another factor could be a dierent precursor-gas exposure of the substrates at various heights, which is, however, highly unlikely given that the ALD process is mostly independent of the specic gas distribution.

The inuence of the substrate's proximity to the remote plasma is therefore not yet conclusively evaluated and further investigations are necessary to isolate the various possible contributions to the lm growth. Based on these results and considering that the distance is limited by the physical dimensions of the ALD chamber, lowering the plasma power might be worth to consider for future process optimization.

## **3.4 Spatial uniformity of ALD-NbN**

This section contains parts from previously published content: [219] E. Knehr, M. Ziegler, S. Linzen, et al., "Wafer-level uniformity of atomic-layer-deposited niobium nitride thin lms for quantum devices", Journal of Vacuum Science & Technology A 39, 052401 (2021).

Permission to include adapted text was obtained from the authors. The experimental concept was developed by E. Knehr together with S. Linzen. The lms were deposited by M. Ziegler and V. Ripka, bridges were patterned by M. Ziegler (EBL) and E. Knehr (etching). The measurements were performed by M. Ziegler (AFM), U. Brückner, J. Plentz (SIMS), S. Linzen, M. Diegel (XRD), P. Schanz, and E. Knehr (electrical). The experimental results were analyzed by E. Knehr with contributions from M. Ziegler, S. Linzen, J. Plentz, and K. Ilin.

The lm uniformity is crucial for the fabrication of SNSPDs. Nanowire constrictions limit the photon sensitivity of the detector as they reduce the maximum current the nanowire can carry, thus, only a fraction of the nanowire is optimally biased. Apart from the eect on the minimal detectable photon energy, there's also a detrimental impact on the detector's timing jitter [114, 220, 221]. These constrictions are not only caused by patterning imperfections, i.e., edge roughness, but also small variations in thickness and lm-intrinsic uctuations in composition and parameters like the superconducting gap [75, 222, 223].

Another factor with similar consequences on the detector performance is the large-area lm uniformity. This becomes important when scaling SNSPDs to larger areas and pixel numbers. A degradation in lm properties at a distance from the optimum area (e.g., the center of the wafer table in a deposition chamber) can be compensated for only by few multiplexing architectures and also only to a certain degree.

As discussed before, ALD lends itself well to deposit smooth and uniform thin lms over large areas due to its self-limiting, surface-controlled growth mechanism [207]. Nevertheless, there are limitations, e.g., imposed by the plasma distribution of the PEALD process used for NbN. For this reason, the parameter spread of NbN thin lms regarding their morphology, composition and superconducting properties over the equivalent area of a 6-in. wafer is studied in this section.

The same deposition parameters as specied in Tab. 3.1 are employed. As substrates, we use (100) silicon with 300 nm thermal oxide. A total of 25 silicon chips of 10 mm × 10 mm size were arranged in a cross-like shape with the remote plasma source (75 mm diameter)

**Figure 3.5:** (a) Dimensions of the ICP conguration in the ALD chamber (not to scale). (b) Chip distribution across the ALD wafer table. Each grey square represents a 10 mm × 10 mm chip. (c), (d), (e) Film-layer stacks used to investigate the spread in electrical parameters and surface roughness (c), in lm thickness (d), and in composition and crystal structure (e); lm thicknesses not to scale.

concentrically placed 23 cm above the center. In the following, the x- and y-coordinates correspond to this position during deposition, x,y = (0, 0) being dened as the center of the substrate table. Three depositions were conducted: one with a target thickness of 6 nm (130 ALD-cycles) for atomic force microscopy (AFM) and electrical measurements, another 6 nm deposition with a 20 nm thick etch-stop Al2O3 layer deposited prior to NbN for thickness measurements, and one with a thickness of 20 nm (435 ALD-cycles) for x-ray diraction (XRD) and secondary ion mass spectroscopy (SIMS) measurements. The dierent lm-layer stacks of each deposition together with schematics of the sample arrangements are shown in Fig. 3.5.

Electrical four-point probe measurements were conducted on microbridges patterned from the NbN lms to determine temperature-dependent resistivity and switching current. The NbN bridges are 1 µm wide and 10 µm long (see geometry in the inset of Fig. 3.8). This comparatively wide structure was chosen to separate lm-inherent inhomogeneities from any adverse eect caused by the patterning process. To avoid current crowding and maximize the switching current [82], the voltage leads to the microbridge are rounded o. The patterns were dened by electron-beam lithography using 130 nm thick ma-N 2401 resist exposed to electrons of acceleration voltage 50 kV with a dose density of 495 µC/cm2 . After development, the written structures were transferred into the NbN layer by Ar-ion milling.

The NbN lm thickness was measured on steps etched into the samples with the additional Al2O3 layer by reactive-ion etching (RIE) using CF4 and oxygen. With an etch rate of

**Figure 3.6:** (a) Spread of lm thickness over the position across the wafer table. The error bars correspond to 95 % condence intervals. The horizontal dashed line represents the mean value of 6.1 nm within ±50 mm from (0, 0). (b) Distribution of root mean square (RMS) surface roughness over a 6-in. area. The horizontal dashed line represents the mean value of 0.82 nm within ±50 mm. (c) Exemplary 3 µm × 3 µm AFM image at position y = −37.5 mm with the mean value as zero level.

0.05 nm/min compared to 18 nm/min of NbN for etching conditions at hand, the Al2O3 layer acts as an etch-stop layer and therefore allows for a well-dened step.

#### **3.4.1 Morphological uniformity**

To evaluate geometric lm homogeneity, we measured the lm thickness and surface roughness using an FRT MicroProf tool equipped with an atomic force microscope. The surface was scanned in the non-contact mode with a line speed of 9 µm/s. The lm thickness was dened as the mean value of 21 individually measured proles of the patterned steps on each chip. The surface roughness was taken as the root mean square (RMS) roughness over a scanned area of 5 µm × 5 µm at various positions.

Figure 3.6a shows the distribution of lm thickness along the x- and y-axis of the examined area. Within a diameter of 4 in. ( x,y  ≤ 50 mm), there is a negligible variation in thickness around a mean value of 6.1 nm, equivalent to a lm inhomogeneity of 3.6 %. This corresponds to a growth rate of 0.469Å/cycle, which is comparable to previously reported values [170, 171]. Increasing the area to 6 in. leads to 9.1 % inhomogeneity and a maximum thickness of (7.0 ± 0.3) nm (y = −75 mm).

The NbN surface roughness is shown in Fig. 3.6b, with an exemplary AFM image in Fig. 3.6c. The RMS roughness Rq uctuates around a mean of 0.82 nm, with a maximum of 0.89 nm at x = +75 mm. Rq corresponds to roughly two lattice constants of NbN (0.45 nm [224]). A reference measurement of the bare substrate led to similar Rq = 0.8 nm. This indicates that ALD-NbN roughness is mainly determined by the surface roughness of the substrate below.

**Figure 3.7:** (a) SIMS: distribution of nitrogen and carbon counts relative to the respective niobium count at each position. Both relative curves are normalized, i.e., the absolute values of nitrogen and carbon cannot be compared. The dashed lines are guides to the eye. (b) XRD: 2Θ-scan (ω xed at 0.4°) of 20 nm thick ALD-NbN at three positions along the x-axis of the wafer table. The two peaks around 35.5° and 41.5° correspond to orientations (111) and (200) of cubic δ -NbN, respectively.

#### **3.4.2 Compositional uniformity and crystal structure**

To analyze the chemical composition of the deposited lms, we used a Hiden Analytical MAXIM SIMS equipped with a cesium source and with a spot diameter of 20 µm. The 20 nm thick NbN lms were sputtered with a rate of 5 nm/s and the composition of the sputtered material was analyzed using a time-of-ight analytical detector. We traced the most relevant elements present in the lms: Nb, N, C, and O. Plateaus in the recorded depth proles indicate the deposited thin lm and were used to extract the intensity of each element. The resulting counts of carbon, nitrogen, and oxygen were normalized with respect to the niobium count at each position to compensate for slight variations of the primary ion current during the measurement. Niobium was chosen as a reference because of two reasons. First, we assume a constant niobium content over the whole area because of the self-limiting ALD process and the saturation of the surface with TBTDEN. In previous studies, we found that the main impact on lm composition stems from the plasma step [171, 172]. Second, the sputter rate of niobium should be less dependent on the surrounding chemical bonds because of its higher atomic weight as opposed to carbon, nitrogen, and oxygen. Because the ionization probability of each element is dierent, the curves of N and C were normalized and absolute values cannot be compared between the elements.

In Fig. 3.7a, the normalized intensities of nitrogen and carbon are depicted. It can be seen that the nitrogen concentration slightly varies over an area of 4-in. diameter, whereas there are larger deviations further from the center. The concentration of nitrogen compared to niobium decreases in the range from ±50 mm to ±75 mm in all directions. A similar behavior is observed for carbon. These declining concentrations indicate a niobium-rich composition at the edges of a 6-in. area.

The presence of oxygen was recorded as well but proved to be widely uctuating and not suited to evaluate the oxygen content within the lms. The penetration depth of the impinging Cs+ primary ions is in the range of the NbN lm thickness of 20 nm. When the surface is sputtered, there is therefore an inuence of the SiO2 layer below (see layer stack in Fig. 3.5e), which is assumed to have the biggest eect on the measured oxygen counts. A reliable measurement of the oxygen concentration would be possible by using pure silicon substrates and removing the native oxide surface layer on the substrate immediately before the deposition. Nevertheless, previous analyses have disclosed a considerable portion of oxygen impurities in the ALD-NbN lms [171, 172].

The changes in crystallinity and lattice parameter across the deposition area were evaluated using a PANalytical X'Pert Pro x-ray diractometer equipped with a Cu Kα source. Because of the small lm thickness, the samples were mainly analyzed in grazing incidence (ω = 0.4°) with a parallel beam setup instead of standard Bragg-Brentano conguration. The scan time in the 2Θ-range (33 to 44)° was kept constant at 40 min for each sample.

In Fig. 3.7b, the diraction peaks of three samples along the x-axis are given. In the center (x = 0 mm), two peaks at 35.49° and 41.34° are detected and can be assigned to the orientations (111) and (200) of δ-NbN, respectively. The full width at half maximum (FWHM) of both peaks is rather large (1.02° and 0.84°), indicating a nanocrystalline lm structure and a distinct lattice parameter spread [172]. Using the Scherrer equation, crystallite sizes of (9 to 11) nm can be estimated.

At position x = 40 mm, the peaks shift to slightly larger angles, but the FWHMs (0.98° and 0.88°) remain nearly constant. Further from the center at x = 60 mm, there is a more distinct shift to larger angles, with the peaks at 36.08° (111) and 41.91° (200). Notably, the peak intensity of the (111)-orientation is only slightly dependent on the sample position, while the intensity of (200)-peaks is strongly reduced toward the edge. The disproportionally strong decrease of [100]-crystallites also becomes apparent from the peak-area ratios given in Tab. 3.2.

Within the 2Θ-range (20 to 80)°, ve diraction peaks from cubic NbN were recorded. Only in the center (x = 0 mm), one very small peak at ≈51° from another phase was observed, either from Nb4N5 or from niobium oxynitride. Nevertheless, previous investigations revealed the presence of impurities within the deposited lms [171, 172]. Therefore, the coexistence of amorphous phases of niobium oxides, niobium carbides, and niobium oxynitrides might be possible. The amorphous content, however, should be very small because diractograms recorded in Bragg-Brentano conguration do not show the occurrence of characteristic "amorphous humps".

**Table 3.2:** Peak position, FWHM, and peak-area ratio of recorded XRD peaks. For the area ratios of the (111)- to (200)-peaks, we subtracted the baseline of each curve and integrated the peaks in the ranges (34 to 37.5)° and (40 to 43)°, respectively.

**Figure 3.8:** Distribution of critical temperature (a), superconducting transition width (b), residual resistivity at 25 K (c), and switching current density at 4.2 K (d) over the sample's position on the wafer along the x- and y-axis. The measurements were conducted on microbridges of 1 µm width and 6.1 nm thickness (see the scanning electron micrograph in the inset). The vertical dashed lines illustrate wafer edges of the denoted sizes.

#### **3.4.3 Spread of superconducting properties**

The superconductivity parameters were measured on a total of 68 microbridges (four per chip) distributed along the x- and y-axis of the examined area. For Tc, ∆T , and ρn, the temperaturedependent resistance was recorded, while current-voltage curves at 4.2 K were used to get switching current density jsw. In addition, the temperature dependence of jsw between Tc and 4.2 K was measured on several samples. For ρn and jsw, we took cross-sections obtained by SEM and AFM measurements.

The spread of these superconductivity metrics is displayed in Fig. 3.8. Across a 2-in. diameter, only minimal variation is visible and the electrical parameters approximately amount to Tc = 10.3 K, ∆T = 1.38 K, ρn = 260 µΩ cm, and jsw = 5.7 MA/cm2 . Further distant from the center, the lm quality is degrading by varying degree of each property. At x,y = ±50 mm (corresponding to the edges of a 4-in. wafer), the resistivity is on average increased by 53 % compared to the center. Similarly, the switching current density decreases by 45 %. The critical temperature is shown to be more stable over the deposition area compared to ρn and jsw and decreases by 10 % only. This is expected since Tc is mainly determined by any contiguous, high-quality region along the microbridge. To evaluate lm homogeneity, the transition width is better suited because it encompasses the spread of Tc of various regions in the lm. From the center to the 4-in. wafer edge, ∆T increases by ≈26 %. This points to a degrading homogeneity of dierent grains and regions of the thin lm.

Within the uniform area in the center, the critical temperature is in line with the measurements in Section 3.2 for a thickness of 6 nm but on sapphire and quartz substrates. From the other metrics, however, ALD-NbN on thermal oxide seems to be a slightly better superconducting material, with a 20 % higher jsw and ≈20 % lower values of ρn and ∆T , respectively. Compared to properties of unpatterned, slightly thicker ALD-NbN on SiNx reported by Cheng et al. [225], we obtained very similar values for ρn, while Tc is (2 to 3) K higher and ∆T about 0.3 K lower in our case. Reasons for this discrepancy could be dierent deposition temperatures (380 ◦C vs. 300 ◦C), plasma gas compositions (pure H2 vs. mixed N2/H2), and substrate materials (thermally oxidized Si vs. LPCVD-grown SiNx ).

We analyzed whether the decreasing jsw toward the wafer edges can be entirely explained by the change of Tc and ρn over the same distance. This can be accomplished by comparing the ratios of the measured switching currents density jsw to the theoretical depairing current density jdep (0) = 0.74∆(0) 3/2 /(eρn √ ~D), with Eq. 31 in Ref. [226], the energy gap ∆(0) = 2.05kBTc, and the density of states N (0) = 1/(2e 2 ρnD).

For the micrometer-wide bridges like in our case, the barrier for vortex entry is considerably reduced compared to (20 to 100) nm wide bridges (see also results and discussion in Section 3.2.2). This decreases the measurable jsw for T/Tc . 0.8. Although this eect should be similar for all bridges with the same dimensions, it introduces an uncertainty because any edge defect can reduce the entry barrier further in a hardly predictable way. This is why we did not take the measured jsw values at 4.2 K directly but instead recorded jsw(T ) curves of each sample and t the experimental data with

$$j\_{\rm fit}(T) = j\_{\rm fit}(0) \cdot \left[1 - (T/T\_c)^2\right]^{3/2} \tag{3.8}$$

**Figure 3.9:** (a) Distribution of the ratio of t current density jt to theoretical depairing current density jdep at 0 K over the wafer area. (b), (c) Exemplary jsw(T ) curves at x = −47.8 mm (b) and x = 0.7 mm (c). The dots represent jsw measured on bridges of 1 µm width, the dashed curves denote ts by Eq. 3.8 to the measured data in the range of 0.8 < T /Tc < 1.0 (indicated by the shaded area), and the solid line shows the depairing current density jdep, obtained from jdep (0) and temperature dependence in Eq. 3.8.

in the temperature range T ≥ 0.8Tc (Bardeen approach, see Eq. 5.26 in Ref. [227]; compare to Eq. 3.5 in Section 3.2.2). In this range close to Tc, the coherence length diverges and jsw is mostly determined by the depairing mechanism [228, 229]. The t parameter jt(0) is then compared to the theoretical value of jdep (0), calculated using the respective Tc and ρn at each x-/y-position. The coecient D ≈ 0.3 cm2 /s is taken from previous measurements on NbN of the same thickness, albeit on a crystalline quartz substrate (see Fig. 3.2e), and is assumed to be constant over the wafer area.

As shown in Fig. 3.9a, the ratio jt(0)/jdep (0) amounts to ≈0.6 in the center. This value is close to the ratio shown previously in Fig. 3.3 for w = 100 nm. Over the examined area, the current ratio is almost constant with a maximum deviation of 10 % compared to the center. A constant ratio jt(0)/jdep (0) means that jt ∝ T 3/2 c /ρn and the measured drop of jsw is in line with the changes in Tc and ρn at each position. The fact that the ratio slightly increases toward the edges is most likely because the assumption of constant diusion coecient D is not applicable. Instead, D should decrease toward the edges as it is expected for higher resistivity [156], and therefore, the ratio jt(0)/jdep (0) should be smaller than calculated.

The degradation in lm quality starting at ≈25 mm from the center can not be explained by morphological changes because both the lm thickness and surface roughness are fairly uniform within a 4-in. area ( x,y  ≤ 50 mm). A decreasing nitrogen content in the lm relative to niobium was observed, but this could only explain a change of the superconducting properties further distant from the center (>50 mm).

Instead, XRD measurements suggest changes in the lm structure as a possible reason for this drop. The diminishing peak intensity, especially of the (200)-oriented grains, signies a declining portion of cubic NbN toward the wafer edges. The detected 2Θ-peak shift with increasing distance from the center corresponds to a decrease in the lattice parameter from 4.37Å to 4.31Å. According to the inuence of this lattice parameter reduction on the superconducting properties described by Linzen et al. [172], a decrease in Tc by (2 to 3) K is expected.

The maximum temperature deviation over the substrate table is specied to be ±2 ◦C. Hence, we do not expect inhomogeneities in deposition temperature to be a factor contributing to lm degradation toward the edges.

In previous studies, we found plasma parameters to be crucial for nal lm composition and electrical properties. A decrease in hydrogen pressure resulted in higher Tc and jsw values [171]. In addition, low hydrogen pressure led to an increase in plasma radicals near the substrate surface [169]. This suggests that in the downstream region, the remote plasma could rather act as an active plasma. Moreover, the low pressure additionally prevented interactions of radicals within the downstream due to the increased mean free path, leading to a rather directional plasma exposure. The plasma source in our setup has a diameter of 75 mm and is concentrically aligned above the substrate holder (see sketch in Fig. 3.5a). Therefore, the directional plasma is expected to have its highest energy impact on the substrate within a region of approximately ±40 mm. This is consistent with our results indicating that the lm composition, the lattice parameter, and the superconducting properties are signicantly altered at distances ≥40 mm from the center in all directions.

Based on these results, the large-area homogeneity of ALD-NbN might be improved by using a larger-diameter plasma source and increasing the ICP-to-substrate table distance (see also results from Section 3.3). This would result in an extended area of uniform plasma distribution and reaction with the deposited precursor.

## **3.5 Summary**

In this chapter, the deposition of polycrystalline, superconducting NbN thin lms using ALD was investigated and evaluated in view of its suitability for the fabrication of SNSPDs.

**Table 3.3:** Summarized parameter spread of ALD-NbN.

Basic transport measurements on lms of (3 to 21) nm thickness on sapphire and quartz substrates revealed mostly constant properties down to a thickness of ≈5 nm, with a residual resistivity ρn below 400 µΩ cm and a critical temperature Tc above 10 K for lms on sapphire. At d ≤ 5 nm, the lm quality is rapidly degrading, likely owing to the shrinking superconducting core between the lm surface and the substrate interface. Qualitatively, lms on quartz exhibit a similar thickness dependence, but overall slightly "worse" superconducting properties compared to sapphire (although the dierence diminishes for thicker lms). With an electron density of states N (0) ∝ (ρnD) −1 comparable to sputtered lms, the heat capacity ratio Ce/Cph of ALD-NbN should also be similar. Compared to reference values, the diusion coecient of ALD-NbN is rather small at D = 0.32 cm2 /s for 10 nm on sapphire, which is advantageous for SNSPDs.

Further material properties relevant for SNSPDs were obtained by measurements on microand nanobridges of width (0.1 to 10.0) µm to extract the temperature- and width-dependent switching current density jsw. Although the jsw values of ALD-NbN bridges were smaller compared to sputtered ones in absolute terms, in conjunction with the higher ρn and lower Tc of ALD-NbN, the most crucial switching to depairing current ratio (with Idep obtained from the model by Kupriyanov and Lukichev [218]) is very similar at values close to Isw/Idep = 0.6. With a rather large ρn and smaller Tc and jsw, the investigated ALD-NbN lms are "worse" superconductors compared to NbN lm sputtered at high temperatures [85], but meet the specic requirements of SNSPDs very well (high ratio Isw/Idep, small D, moderate Tc).

Continuing previous investigations of the inuence of various plasma parameters on the lm properties [171], the interaction between the plasma and the substrate surface was modied by varying the distance between the sample and the inductively coupled plasma source. This resulted in gradually worse properties (higher ρn, lower Tc and jsw), suggesting that a reduction of the plasma power might actually be benecial and worth to pursue.

The main goal of employing ALD for the deposition of NbN, enabling highly uniform thin lms over large areas, was investigated by measuring the spread of various lm parameters over the area of a 6-in. wafer. All observed lm properties including the most decisive switching current density are near-constant over a diameter of 50 mm. Further from the center, at a distance of ±50 mm, we observe larger variations of the electrical properties: +50 % (ρn), −10 % (Tc), +30 % (∆T ), and −45 % (jsw), while the lm thickness and surface roughness are mostly unchanged (see also summary in Tab. 3.3). XRD measurements suggest that this change of parameters is most likely caused by a shrinking crystalline portion of cubic NbN together with a decrease in the lattice constant. Correlations with the lateral dimension of the downstream region of the plasma suggest that the lm uniformity could be extended to larger areas by increasing the diameter of the plasma source.

Overall, the presented ALD-NbN thin lm properties indicate a high suitability for SNSPDs, allowing for large-area detector arrays on a 2-in. wafer with atomic-level thickness control and potentially increasing the fabrication yield compared to conventional reactive magnetron sputtering.

# **4 ALD-NbN SNSPDs**

The transport measurements of ALD-NbN in the previous chapter revealed promising properties for the use in nanowire detectors. In this chapter, actual ALD-NbN-based SNSPDs are fabricated and characterized under optical illumination.

In Section 4.1, the detector layout and sample preparation with the focus on the patterning process is described. This is followed by the experimental setup in Section 4.2. The results of the optical detector characterization in terms of bias, photon-rate, and spectral dependence of the detector response are presented together with a discussion of the lm-thickness inuence in Section 4.3.

## **4.1 Detector design and fabrication**

The optical characterizations were carried out on detectors from ALD-NbN lms deposited using the process parameters as specied in Section 3.1. To investigate the inuence of the lm thickness on the detector properties, both 5 nm and 10 nm thick detectors were fabricated. For both depositions, 10 mm × 10 mm R-plane sapphire served as substrate. In the following, the layout and patterning process of the detectors is introduced.

#### **4.1.1 Detector design**

An overview of a single detector chip together with zooms on the nanowire is shown in Fig. 4.1. Nine of these devices were placed on a 10 mm × 10 mm area for parallel fabrication.

Straight nanowires with a width of ≈100 nm and a length of 5 µm dene the active area of the devices. Although this basic nanowire geometry is not suited for ecient photon coupling, it serves the purpose of characterizing the fundamental single-photon response of ALD-NbN-SNSPDs by minimizing patterning defects commonly found in large-area detectors. To reduce the current crowding eect (see Section 2.3.1 and Ref. [82]), the nanowire is connected by small tapers on each end, as seen in the inset of Fig. 4.1b.

**Figure 4.1:** Design of the SNSPD chip. (a) Detector chip with a size 3 mm × 3 mm. The SNSPD and inductor in the center are connected to CPWs on both sides. The gray areas denote the NbN layer. (b) Zoom of chip center with the straight nanowire (inset) and a series inductor. (c) SEM micrograph of the fabricated device.

Since the small kinetic inductance of the short nanowires would lead to detector latching (described in Section 2.2.3 and later investigated in Section 5.2.1), a 1 µm wide and 1 mm long inductor was placed in series with the nanowire. With 1000 squares, this adds an inductance of about 46 nH for d = 10 nm. The structure much wider than the nanowire ensures that, at currents below the switching current of the nanowire, the inductor itself is not sensitive to photons but only the nanowire (at least in terms of the SNSPD detection mechanism).

The nanowire and inductor are connected to coplanar waveguides (CPW) from the same NbN lm for bias and readout. The 250 µm wide, inner conductor was designed wide enough to enable conveniently connecting the CPW by aluminum wire bonds to a printed circuit board (PCB) for readout. To set an impedance of 50 Ω, the inner conductor of the CPW is separated by 140 µm wide gaps to the ground planes on each side. Both ground planes feature integrated, etched rings with an outer diameter of 3 mm. These markers are used to facilitate the manual positioning of the chip in the detector housing. At a minimum distance of 700 µm from the CPW gaps (ve times the gap size), these structures have a negligible impact on the CPW impedance.

#### **4.1.2 Detector patterning**

The detectors were patterned out of NbN lms via electron-beam lithography (EBL) and Ar-ion milling.

The EBL comprised three separate steps. First, the nanowires and alignment marks (not visible in Fig. 4.1) were dened in a negative-tone process. The chip was spin-coated with poly (methyl methacrylate) (PMMA, 950k) and baked at 150 ◦C for 5 min to get a resist thickness of ≈80 nm. The structures were then written by EBL at 15 kV acceleration voltage, a beam current of 14 pA, and a high dose of 10 mC/cm2 , which causes cross-linking of the exposed polymers and PMMA eectively acting as a negative-tone resist [230]. Nanowires fabricated by this process were found to have smoother edges and enhanced superconducting properties due to a more homogeneous cross-section [87]. After exposure, the resist was developed by submersion in acetone (1 min) and rinsing in isopropanol (IPA; 30 s). In the second and third lithography step, the coarse and mid-size structures were patterned using a standard positivetone process to avoid the impractically long exposure times which would be necessary for the larger areas exposed with the same high dose as the nanowires. These subsequent writing steps were aligned to the nanowires using the previously written alignment marks. To write the coarse structures on the scale of ≈100 µm including the CPWs, chip-positioning rings, and chip labels, the same type of PMMA resist was spin-coated with the same parameters. For illumination, however, a larger beam current of 4.5 nA and a much lower dose of 130 µC/cm2 were used, corresponding to the regular positive-tone process for PMMA. The exposed resist was developed in methyl isobutyl ketone (MIBK):IPA 1:3 for 35 s before rinsing in IPA for 10 s. Lastly, without applying a new resist layer, the mid-size patterns on the scale of ≈1 µm, namely the inductors, were dened in a third exposure at the same dose as before but at a lower beam current of 120 pA. The lower current allowed for a higher resolution and a lower edge roughness of the micrometer-wide inductors than what was necessary for the coarse structures. After lithography, the resist was again developed in MIBK:IPA 1:3 but for a slightly shorter time of 25 s. The nal PMMA structure comprised the nanowire, written in the negative process, as well as the CPW and inductor, both written using a positive process.

Following the lithography and development, the patterned resist structures were transferred to the NbN layer by Ar-ion milling. For this process, the samples were placed in a vacuum chamber below an RF-coupled ion source on a rotating sample holder to ensure uniform etching. The argon pressure was set to ≈2.8 × 10−4 mbar before igniting the plasma. The generated ions were accelerated toward the sample by the electrostatic eld of two electrode grids which were biased by DC voltages of +200 V (inner electrode) and −250 V (outer electrode). The RF plasma power was adjusted to achieve a constant current at the positive electrode grid of 90 mA, which is proportional to the ion-beam current. Using these parameters, the

etching rates of PMMA and NbN were previously determined to be around 7 nm/min and 3 nm/min, respectively. The cross-linked PMMA (negative-tone EBL) is more resistant to the ion impact and was found to have an etch rate similar to NbN. For the 5 nm and 10 nm thick samples, etch times of 6 min and 10 min were chosen, respectively. The etch times were much longer than required according to the etch rates in order to ensure that the NbN layers were etched through. Because of the single-layer detectors, an overetching into the substrate had no detrimental eect on the samples. Moreover, there is only minimal etching into the hard sapphire substrate. To prevent overheating of the NbN layer due to the ion-beam impact, the sample holder was cooled to ≈0 ◦C. As an additional procedure to reduce the heat load, the sample was not etched continuously but in intervals of 30 s with 30 s long periods in between during which the shutter between the ion source and the sample was closed.

After etching, the remaining resist was removed by submersion in acetone, and the 10 mm × 10 mm substrates were each diced into nine individual chips. Because the cross-linked PMMA is insoluble in acetone, it was left on the nanowires as a protective layer. An SEM micrograph of the fabricated SNSPD is shown in Fig. 4.1.

## **4.2 Experimental setup**

For the optical characterization, the detectors were mounted in a LHe-cooled dipstick cryostat equipped with an optical ber and both RF and DC lines. All measurements were conducted at 4.2 K. The detectors were biased by low-noise voltage source with low-pass RC lters (≈72 Hz edge frequency), connected to the DC port of a bias tee. The detector signal at the RF output of the bias tee was amplied at room temperature by a two-stage amplier with a total gain of ≈67 dB. Depending on the applied bias current, the resulting pulse height after amplication was in the range (200 to 900) mV. A 32 GHz real-time oscilloscope and a pulse counter for count rates up to 300 MHz were used for readout.

For illumination, the output of various light sources was coupled to the detectors at the bottom of the dipstick using a step-index multimode ber specied for the wavelength range (400 to 2100) nm. The detectors were illuminated from the top, normal to the substrate plane. In order to ensure a uniform illumination, the detectors were placed at a distance of (4 to 5) mm from the ber tip, resulting in a widely illuminated area with a radius of about 1 mm. For measurements in dependence of the bias current and optical power, we used an LED at 385 nm and a laser diode at 658 nm wavelength. The output power of both sources could be swept over a broad range by the applied current. For spectral measurements in the wavelength range (400 to 2000) nm, a broadband halogen lamp in conjunction with a Czerny-Turner

monochromator were used. The monochromator has two diraction gratings with dierent blaze wavelengths: one with 1200 G/mm (grooves per millimeter) for the wavelength range (400 to 1100) nm, and another one with 600 G/mm for (1100 to 2000) nm. In order to block light of higher-order diraction (e.g., at 400 nm if the grating is set to 800 nm), two longpass lters were used: one with a cut-on wavelength of 750 nm for the wavelength range (800 to 1400) nm, and one with a cut-on wavelength of 1050 nm for (1400 to 2000) nm.

All of the light sources were calibrated using a powermeter equipped with Si- and InGaAsbased photodiodes for the wavelength range (350 to 1100) nm and (800 to 1700) nm (Thorlabs S150C and S154C), respectively. The calibration was conducted in the detector plane at the bottom of the dipstick in order to take into account any wavelength-dependent loss contribution from the vacuum ber feedthrough, the ber itself, and the output of the stripped ber.

## **4.3 Optical detector characterization**

This section contains parts from previously published content: [209] E. Knehr, A. Kuzmin, D. Y. Vodolazov, et al., "Nanowire singlephoton detectors made of atomic layer-deposited niobium nitride", Superconductor Science and Technology 32, 125007 (2019). Permission to include adapted text was obtained from the authors. The lms were deposited by M. Ziegler and S. Goerke at Leibniz-IPHT. E. Knehr designed, patterned, and characterized the detectors and analyzed the results with contributions from A. Kuzmin, K. Ilin, and D. Vodolazov.

Before photon illumination, the basic superconducting properties of the fabricated detectors were measured. The results are summarized in Tab. 4.1. While the values for d = 5 nm are in line with the results from Fig. 3.2, the 10 nm sample has slightly smaller values for both ρn and Tc. Ratio Isw/Idep was obtained via Eq. 3.5 (with diusion coecient D from Fig. 3.2e and β0 = 2.05) and is consistent with the result of Fig. 3.3b for the same thickness of 10 nm, whereas a smaller ratio Isw/Idep = 0.4 is found for the thinner sample.

#### **4.3.1 Bias and photon-rate dependence**

The bias-dependent count rates for dierent lm thicknesses, photon wavelengths, and optical power levels are shown in Fig. 4.2. As summarized in Section 2.3.1, SNSPDs exhibit three

**Table 4.1:** Superconducting properties of SNSPDs with 5 nm and 10 nm thickness. Values of Isw, jsw, and Isw/Idep are obtained at 4.2 K.

**Figure 4.2:** Count rate over relative bias current Ib/Isw for illumination with photons at 385 nm (a) and 658 nm (b) wavelength and dierent optical power levels. The curves at 0 nW correspond to dark-count rates. (Insets) Respective plots on the linear scale for the highest power level.

distinct regimes dependent on the applied bias current, which is reected in the experimental results. At low currents, e.g., below 0.4Isw for λ = 385 nm and d = 5 nm, there is an exponential increase in count rate with the bias. The strong bias dependence is associated with a probabilistic or uctuation-assisted photon detection. At higher currents, the curves saturate (in logarithmic scale) which is attributed to the deterministic detection regime.

At λ = 385 nm (Fig. 4.2a), all measured curves exhibit saturating count rates. As indicated by the prolonged plateau in CR(Ib), the thinner sample shows a deterministic regime over a wider bias range. The saturating curves in linear scale (see inset) signify near-unity internal detection eciencies at Ib close to Isw at this photon energy [46]. For the longer wavelength (λ = 658 nm, Fig. 4.2b), the plateaus are less pronounced (5 nm) or not existing anymore (10 nm). This is again better illustrated by the curves on the linear scale (inset). Comparing curves of both thicknesses at the same optical power, the level of saturation (extrapolated in the case of d = 10 nm and λ = 658 nm) is largely independent of the detector thickness, further emphasizing the near-unity internal eciency. Small dierences in the saturation level might come from the fact that both detectors were measured in separate cool-downs

**Figure 4.3:** Count rate over incoming photon rate (λ = 385 nm) for dierent bias currents of 5 nm (a) and 10 nm (b) thick detectors. The solid lines are ts according to CR ∝ PRn to the respective data. The dashed horizontal line in (a) indicates the DCR at Ib = 0.92Isw. (Inset) Basic output pulse of the 10 nm thick detector.

with potential deviations in the optical coupling, i.e., dierent photon rates impinging the detector at the same optical input power.

The dark-count rate (DCR) was measured by blocking the ber feedthrough of the dipstick. Dark counts were found to appear at currents above 0.8Isw and to grow exponentially with increasing current. This, however, was only observed for d = 5 nm, whereas the 10 nm sample did not exhibit an increasing DCR above a current-independent noise level of ≤3 Hz up to Isw.

As pointed out in Section 2.3.1, the operation temperature has a profound inuence on the photon sensitivity of SNSPDs. Although the measurements were conducted at T = 4.2 K < 0.5Tc and the short nanowire is benecial for detecting low-energy photons (minimal current crowding, small probability of nonuniformities), it is obvious that the operation conditions at hand do not allow for an ecient photon detection with saturated CR(Ib) curves at longer wavelengths, i.e., in the infrared range. The temperature inuence on the shape of CR(Ib) is, e.g., shown in the extended data Fig. 1a of Ref. [8], which demonstrates much longer count-rate plateaus when lowering the temperature from 4 K to even 2 K.

Figure 4.3 depicts measured CR(PR) curves at λ = 385 nm. It is seen that the count rates increase with the incoming photon rate PR as CR ∝ PRn with an exponent n = 1.00 ± 0.01. This applies to both thicknesses and all Ib/Isw ratios corresponding to dierent bias regimes. Similar results were also obtained for a photon wavelength of 658 nm. The linear dependence CR ∝ PR over many orders of magnitude is a strong indication of single-photon detection [4, 231]. A saturation in count rate approaching the maximum count rate of the detectors could not be observed because of the weak optical coupling from the ber to the active area and,

**Figure 4.4:** Spectral dependence of the normalized count rate. (a) Normalized count rate over photon wavelength at dierent bias currents for the 5 nm thick detector. The solid lines are ts via Eq. 2.12. Denoted cut-o wavelengths are obtained from the ts. (b) Comparison of wavelength-dependent count rate of the 5 nm and 10 nm thick detectors at the same current ratio Ib/Isw = 0.8.

consequently, heating of the sample which suppressed Isw and therefore limited Ib. As seen in Fig. 4.3a for d = 5 nm, the count rate at Ib = 0.92Isw saturates to DCR at low photon rates. In all cases, a divergence between the experimental data and the linear progression is visible at low photon rates, which is likely due to the calibration precision of the optical powermeter. The specied minimum power of about 100 pW corresponds to PR ≈ 2 × 108 Hz (3 × 108 Hz) at λ = 385 nm (658 nm).

#### **4.3.2 Spectral dependence**

Spectral measurements were conducted by sweeping the monochromator's output wavelength in steps of 10 nm to 100 nm and recording count rates at each wavelength. The output power of the monochromator was strongly dependent on the wavelength. In order to keep the detector in the same detection regime during spectral measurements, the count rate was kept at values below 1 × 104 Hz by manually adjusting the input slit width of the monochromator accordingly. Furthermore, because of dierent gratings and lters employed over the whole wavelength range, several wavelength sweeps with partially overlapping data points were conducted. These were then used to stitch the count rates and the optical power levels starting at the shortest wavelength. All curves were normalized to the count rate at short wavelengths.

In Fig. 4.4a, the normalized count rate is plotted as a function of the incident photon wavelength from 0.6 µm to 2.0 µm. Below ≈1.0 µm, there is a plateau in the recorded count rates. This indicates a deterministic detector response and, thus, matches the results of Fig. 4.2. The fact that the saturation level slightly increases with higher count rates likely implies that an increased portion of the nanowire is active, which ts to the near-saturated CR(Ib) curves in linear scale (and therefore ηint . 1) in the insets of Fig. 4.2.

Above λ ≈ 1 µm, the count rates at a xed ratio Ib/Isw drop rapidly with the photon wavelength. To obtain the cut-o wavelength λc, each curve was t by Eq. 2.12 with λc, slope p, and a prefactor denoting the saturation level as tting factors. At the highest bias current (Ib = 0.90Isw), we obtained λc = 1.25 µm (see the annotations in Fig. 4.4a). Similar measurements on the 10 nm sample resulted in λc = 0.88 µm at Ib = 0.98Isw. It should be noted that there are dierent ways to dene the cut-o wavelength. Besides the t used here, an approximation of the dependence by two straight lines and dening λc at the intersection is sometimes used as well [85]. In general, the latter denition leads to slightly smaller values of λc. As it can be seen, increasing the bias current goes along with a longer cut-o wavelength and a shallower drop in count rate above λc. Dependence λc (Ib) is described by various detection models [40, 46, 48, 232] and was also probed experimentally [79, 80, 85]. By increasing the bias current, the energy barrier for vortex entry is reduced and the superconducting state becomes unstable at lower photon energies [42, 95, 233]. Also, larger portions of the nanowire contribute to photon detection at higher bias currents (related to the position-dependent detection current pointed out in Section 2.3.1).

Similarly, CR(λ) depends on the lm thickness. As illustrated in Fig. 4.4b, we measured an increase in cut-o wavelength by a factor of 1.6 for the same ratio Ib/Isw = 0.8 when reducing the thickness from 10 nm to 5 nm. This is approximately in line with the proportionality λc ∝ d −1 since less demand on ratio Ib/Isw is made for detectors with smaller cross-sections (see again Section 2.3.1 and discussion below). The change in λc by a factor smaller than the ratio of two in lm thickness is at least partly due to the fact that the material parameters also change with the thickness. The smaller diusion coecient D (see Fig. 3.2e), smaller energy gap ∆ ∝ Tc, and larger current ratio Isw/Idep (see Tab. 4.1) of the 10 nm sample all favor photon sensitivity and cause a longer cut-o wavelength. Hence, the dierence in λc of both detectors is smaller than expected from the ratio in lm thickness.

#### **4.3.3 Thickness-dependent dark and photon counts**

According to existing SNSPD detection models, both photon counts in the probabilistic regime and dark counts can be explained by uctuations, e.g., vortex hopping [41, 47], vortexantivortex-pair (VAP) unbinding [39, 45, 48, 234–236], Fano uctuations (in the case of photon counts) [72], and local thermal uctuations [73]. Here, the inuence of the lm thickness on

dark and photon counts is estimated. Because the energy for VAP creation is almost double the energy for a single vortex [41, 233], only single-vortex entry is considered.

According to Eq. 2 in Ref. [233], the energy barrier for vortex entry of superconducting stripes without defects at currents close to the depairing current can be expressed by

$$\mathrm{d}F = \mathrm{d}F\_0 \left( 1 - \frac{I}{I\_{\mathrm{dep}}} \right),\tag{4.1}$$

with current-independent factor

$$\mathrm{d}F\_0 = \frac{\Phi\_0^2}{2\pi\mu\_0\Lambda}.\tag{4.2}$$

Via Pearl length Λ = 2λ 2 l /d, dF and dF0 scale linearly with thickness d. The vortex-hopping rate (or DCR) depends on the energy barrier dF and is estimated by

$$
\Gamma \propto \frac{l}{\xi\_{\rm GL}} \nu \cdot \exp\left(-\frac{\rm{d}F}{k\_{\rm B}T}\right),
\tag{4.3}
$$

with the attempt frequency

$$
\Delta \nu \propto \frac{\Delta}{\hbar} \sqrt{\frac{\text{d}F}{k\_{\text{B}}T}}.\tag{4.4}
$$

This is in analogy to the theory of phase slips in one-dimensional superconducting wires developed in Refs. [237, 238]. Although this theory assumes a defectless stripe and describes thermally activated uctuations only close to Tc (within the Ginzburg-Landau model), it was found in Refs. [233, 239] that it is valid also for lower temperatures well below Tc and holds also in the presence of defects with minor modications. Furthermore, it was expanded to two-dimensional stripes in Ref. [240].

In Eq. 4.3, factorl/ξGL is the number of independent points for vortex entry along the nanowire and amounts to ≈1000 (see the measured ξGL in Fig. 3.2f). However, because vortex entry is considered to originate predominantly from only a few weak spots of size ≈ξGL along the stripe where the energy barrier is lowered, it can be assumed to be on the order of one. Moreover, we take a value of 1 × 1012 Hz for ∆/~ in Eq. 4.4 as a typical order of magnitude for NbN with a critical temperature of 10 K.

Using ρn and Tc from Tab. 4.1 together with λl from Eq. 3.1, the Pearl lengths of the 5 nm and 10 nm detectors amount to Λ ≈ 144 µm and 65 µm, respectively. With Eq. 4.2, this corresponds to dF0/kBT ≈ 65 for 5 nm and dF0/kBT ≈ 144 for 10 nm. From Eq. 4.1, the dierent dF0 should go along with a dierence in switching current Isw relative to Idep. For example, if we assume

**Figure 4.5:** Count rate over relative bias current Isw/Idep for illumination with photons at 385 nm (a) and 658 nm (b) wavelength (equivalent to Fig. 4.2 but with the current scaled to units of Idep instead of Isw). The vertical dashed lines denote the respective switching current.

the vortex-hopping rate at which the superconducting stripe switches to the resistive state to be Γ ≈ 10 000 Hz (dF/kBT ≈ 20), we obtain switching currents of Isw ≈ 0.69Idep (d = 5 nm) and Isw ≈ 0.86Idep (d = 10 nm).

In the presence of edge defects, the current-dependent energy barrier may dier from the simple dependence in Eq. 4.1, as shown in Refs. [233, 239]. Nevertheless, assuming otherwise constant material properties, dF is directly proportional to d and we expect higher relative switching currents Isw/Idep (not just absolute values of Isw) for the thicker detector. As it is seen from the values in Tab. 4.1, we indeed observe an increase in Isw/Idep by 20 % when going from 5 nm to 10 nm. This is also visualized in Fig. 4.5 which shows the data from Fig. 4.2 but with the bias current scaled to units of Idep. Besides the larger ratio Isw/Idep, the thicker detector is also expected to exhibit dark counts in a narrower relative current interval. Experimentally, however, we only observed an exponential dark-count rate for the 5 nm sample. The absence of dark counts in the case of d = 10 nm (see open symbols in Fig. 4.5) is unclear at this point.

The thickness-dependent energy barrier has not only consequences on dark counts and the switching current but also on uctuation-assisted photon counts. From the considerations above, one would expect a narrower probabilistic detection regime in the case of thicker detectors, in line with the expected narrower current interval for dark counts. However, as seen from the sharper curve of the 5 nm detector in Fig. 4.5, we actually observe the thinner detector to have an exponential count-rate increase within a narrower relative current interval. Further measurements of detectors with dierent lm thickness are needed to probe the

theoretically expected dependences and could also prove an optimal detector thickness with minimum dark counts and high-enough photon counts.

## **4.4 Summary**

This chapter evaluated the performance of SNSPDs made from ALD-NbN thin lms. At an operating temperature of 4.2 K, the 5 nm thick detector exhibits photon detection in the deterministic regime above Ib ≈ 0.4Isw at λ = 385 nm and above ≈ 0.6Isw at 658 nm, as observable from the saturating count rates over bias current (in logarithmic scale). Because of the larger cross-section (see Chapter 2), the onset of count-rate saturation of the 10 nm thick sample is shifted to higher relative bias currents. The photon-count rates of both detectors are linearly dependent on the incoming photon rate (CR ∝ PR) which corresponds to a constant detection eciency independent of the optical power and is a strong indication for single-photon detection. Spectral characterizations show a count-rate plateau below λ ≈ 1 µm and a strong decrease in CR when detecting longer-wavelength photons. This is in accordance with the expected bias- and wavelength-dependent minimal photon energy detectable by SNSPDs. Comparing 5 nm and 10 nm thick detectors, the thinner SNSPD exhibits an increase in cut-o wavelength by a factor of 1.6 at the same relative bias current.

Estimations of the thickness-dependent vortex-entry barrier together with the obtained experimental results have shown that, due to the dependence of the energy barrier dF ∝ d, thicker detectors can be operated closer to the depairing current and potentially feature fewer dark counts. For the same reason of suppressed thermally activated vortices, however, detectors from thicker lms are less sensitive to photons of lower energy, which poses a compromise in detector development.

Compared to reported detectors out of sputtered NbN with similar cross-sections and operating temperatures, the ALD-NbN-SNSPDs show slightly longer cut-o wavelengths [85]. Signicantly higher λc values and saturated count rates up to around 3 µm were demonstrated for sputtered, NbN-based detectors before, albeit at a much lower operating temperature of 1.5 K and for a smaller nanowire cross-section of 5.5 nm × 30 nm [10].

Overall, the results depict ALD-based nanowire detectors as a promising alternative to sputtered detectors, especially in conjunction with the large-area uniformity results from Chapter 3. Nevertheless, while the investigated SNSPDs show saturated single-photon detection up to near-infrared range, further thin-lm development is necessary to optimize ALD-NbN for SNSPDs. Besides, operating the detectors below 4.2 K will also be needed to probe the performance limit.

# **5 Frequency-multiplexed SNSPDs**

Exceptional detection properties have been shown by single-pixel SNSPDs. For many practical applications like imaging and spectroscopy, however, arrays of SNSPDs are required. An SNSPD array which can take advantage of the outstanding timing precision and count rates combined with high eciencies from visible light to mid-IR would enable new elds of applications for SNSPDs.

As outlined in Section 2.5, frequency-domain multiplexing (FDM) is one of the various architectures that have been shown for such arrays [186]. These so-called RF-SNSPDs can be biased individually, read out by a single RF line, and oer spatial resolution of multi-photon events. However, there are also several shortcomings of this multiplexing technique which will be addressed in this chapter. The oscillating bias current is expected to have several consequences on the detector operation. On the one hand, it should lead to a stable operation without the risk of the nanowire latching into the resistive state, which will be the subject of Section 5.2. On the other hand, the microwave bias is also expected to modulate the intrinsic detection eciency ηint(Ib) accordingly, leading to a reduced duty cycle of the detector. A detection scheme which takes advantage of the modulated eciency will be introduced and analyzed in Section 5.3. Another disadvantage is the elaborate readout necessary to dierentiate between detection events of dierent pixels. In Section 5.4, an alternative readout method based on direct digitization and matched lters will be investigated.1

## **5.1 Array design and fabrication**

The devices used in this chapter were designed and fabricated by A. Kuzmin and originally intended for experiments with microlenses (see Section 6.1). Here, we use the same layout to analyze the frequency-domain multiplexing scheme instead.

1 A third shortcoming of RF-SNSPDs, the low ll factor owing to the large resonator, will be addressed in Section 6.1 of the next chapter.

**Figure 5.1:** Design of a ve-pixel RF-SNSPD array. (a) Array consisting of ve pixels connected to a common transmission line from port 1 to port 2. (b) Layout of pixel 1 with a coupling capacitor (red), an LC parallel circuit (blue) and a nanowire (green). (c) Equivalent circuit of a single pixel with colors matching the highlighted elements in (b).

Throughout this chapter, an RF-SNSPD array with ve pixels in the frequency range (1.9 to 3.5) GHz is used as depicted in Fig. 5.1. The pixels each have a target resonance width of ∆f ≈ 15 MHz and are all connected to a common 50 Ω transmission line of width 250 µm and with a gap of 75 µm.

A close-up of pixel 1 is shown in Fig. 5.1b. The resonant circuit (highlighted in blue) consists of a meandered inductor and an interdigitated capacitor which dene Lr and Cr , respectively. Both are connected in parallel and are dened by 1 µm-wide structures and gaps. The inductor length is kept constant at 1105 µm for all pixels. With a square kinetic inductance of Lk,sq ≈ 45 pH of the 5 nm thick NbN in use, Lr amounts to about 50 nH. To vary the resonance frequency, parallel capacitance Cr is tuned by modifying the nger number and length across the pixels. This is in contrast to Ref. [188] and [186] where a variable inductance Lr was used to set fr . There, it was supposed, however, that the contribution of the inductor's turns to Lr lead to insuciently controllable values of fr . As a result, the resonance frequencies of the fabricated device were not distributed with equal distance as designed. In an eort to mitigate this problem here, Cr was chosen instead as the parameter to vary fr .

Each pixel is coupled to the transmission line by an interdigitated capacitor (shaded red in Fig. 5.1b) of 1 µm in both width and gap. In order to keep the resonance depth constant over all pixels, Cc was adjusted in conjunction with Cr with the help of simulations in Sonnet em.

The meandered nanowire (shaded green), which constitutes the active area of each pixel and connects each resonant circuit to the ground plane, has a width of ≈100 nm and a length of ≈15 µm. With 150 squares, Lk ≈ 7 nH which corresponds to a ratio Lr/Lk ≈ 7 and ensures that the strong bias dependence of Lk close to the switching current of the nanowire does not signicantly inuence the resonance frequency (see explanation in Section 2.5.2).

The array is fabricated from a single, 5 nm thick NbN layer on an R-plane sapphire substrate. Similar to the process described in Section 4.1.2, the device is patterned by EBL and Arion milling. Dierent from previous chapters, detectors characterized in this chapter were fabricated from sputtered NbN lms. Using ALD-NbN for the detectors would have been possible as well. In this case, the higher resistivity and smaller energy gap of the ALD-NbN lms would have resulted a slightly higher kinetic inductance (Eq. 2.4) and therefore in a shift of fr toward higher frequencies for the same device geometry. This could have been compensated by adjusting the resonator dimensions.

Two nominally identical devices were fabricated, featuring slightly dierent resonance frequencies. Results obtained from one device are shown in Sections 5.2 and 5.3, whereas results of the other device are presented in Section 5.4 (and later in Section 6.1).

## **5.2 Self-resetting behavior**

This section contains parts from previously published content: [241] E. Knehr, A. Kuzmin, S. Doerner, et al., "Synchronous singlephoton detection with self-resetting, GHz-gated superconducting NbN nanowires", Applied Physics Letters 117, 132602 (2020). Permission to include adapted text was obtained from the authors.

When a photon is absorbed by a DC-biased nanowire, a normal-conducting domain (hotspot) can form and its resistance manifests itself as a voltage pulse. If the nanowire's kinetic inductance is large enough (the nanowire is suciently long), the superconductive state can be restored and the detector resets. Latching into the normal state can occur if the nanowire is too short (see also Section 2.2.3). In this case, a stable negative electrothermal feedback by the load impedance keeps the induced hotspot inside the nanowire in a self-heating state [53]. The hotspot's lifetime strongly depends on the applied bias current [54, 242]. For currents below the retrapping current, the hotspot rapidly collapses. The latching behavior of the detector also depends on the readout circuit. The commonly used AC-coupling of the readout amplier has several consequences on the detector behavior and can lead to latching at high

**Figure 5.2:** Self-resetting behavior of an RF-SNSPD pixel. (a) Dark-count rate over relative bias current in semi-log and linear scale (inset). (b) Recorded waveforms of the detector signal at the bias levels highlighted in (a). (c) Microwave transmission of the RF-SNSPD pixel over frequency f − fr at bias levels marked in (a).

photon rates when biased below but close to the switching current. This is described in detail in Ref. [107].

For applications, detector latching poses a problem, for example, in laser ranging of space debris where intense photon backscattering at lower altitudes can cause the SNSPD to permanently switch to the resistive state [243]. In order to overcome this problem, Zhang et al. showed a DC-SNSPD operation with periodical o-states using a programmed bias source to reset the detector [243]. Another example are blinding attacks in SNSPD-based quantum key distribution (QKD) systems, provoked by bright light pulses. As a possible countermeasure, Elezov et al. have tested an auto-reset system for SNSPDs using a voltage-comparator [244].

In the case of an RF-SNSPD, the nanowire is biased by a microwave current Irf at the resonance frequency fr of the nesting lumped-element resonator, which is capacitively coupled to a 50 Ω feed line. After photon absorption, a growing hotspot damps the resonator and therefore leads to an increased transmission coecient |S21 | on the feed line at fr , which is used as the detector response.

With this scheme, self-resetting of the nanowire is enforced by the oscillating current and the decay time of the resonator τd. The hotspot growth time is limited to few periods of the RF current, since the hotspot collapses at low currents on a timescale of ≈10 ps. Additionally, the damping of the resonant circuit upon hotspot generation causes an RF current redirection out of the nanowire to the feed line for at least several periods. This eectively resets the detector after every detection event.

The self-resetting behavior leads to a stable operation of the RF-SNSPD also at bias amplitudes close to the switching current. This is demonstrated by measuring the bias-dependent dark-count rate of a single pixel of the array (Fig. 5.2a). Below a certain current amplitude (associated with switching current Isw and shown as 1.0), there is an exponential dependence of the DCR on the bias and dark counts are random. At Isw, DCR reaches a value of ≈1/τd. For relative bias currents above 1.0, however, the DCR does not drop to zero but linearly rises and the pulses are periodic. We associate this operation with a bolometric regime in which the RF-SNSPD demonstrates a self-oscillatory behavior, seen as strictly periodic "dark counts" with a rate linearly proportional to the ratio of the bias amplitude to the switching current (DCR ∝ Irf/Isw, as seen on the linear scale in the inset of Fig. 5.2a). If any excess optical power heats the nanowire, Isw is reduced and the apparent DCR increases correspondingly. This way, the RF-SNSPD with DCR > 1/τd operates as a bolometric detector with "power-tofrequency" responsivity, similarly to a superconducting noise bolometer [245]. For the bias levels highlighted in Fig. 5.2a, recorded RF pulses are depicted in Fig. 5.2b. One can see darkcount pulses with varying pulse-to-pulse times in (i), whereas at relative bias currents above 1.0 the pulses become periodic with a pulse rate ∝ Ib in (ii) and (iii). In addition, microwave transmission measurements of the RF-SNSPD at the corresponding bias levels are shown in Fig. 5.2c. Curve (i), for which the current amplitude through the nanowire is still below Isw, exhibits a resonance depth close to the value measured at much lower RF power levels. When the power is increased and the current amplitude exceeds Isw, the nite resistance of the nanowire leads to an increased transmission |S21 | at fr which is seen as a rapidly vanishing resonance. However, the cratered resonance does not indicate that the nanowire is locked in the normal state but, in fact, periodically switches between superconducting and normal states.

#### **5.2.1 Simulation of DC- and RF-biased detectors**

From the previous section, it is clear that the oscillating bias current through the nanowire eectively resets the detector even for Irf with an amplitude above Isw. The investigated RF-SNSPD pixel, however, has a comparatively long reset time (ringdown time of the resonator) of ≈20 ns. During normal operation, most DC-biased SNSPDs from common materials do not exhibit latching at this reset time either. In order to substantiate the latching-free behavior of the RF-SNSPD, the operation of an RF-biased nanowire with much shorter ringdown time on the order of 1 ns was simulated using an electrothermal model of the detector, and compared to the behavior of a DC-biased nanowire. To this end, we implemented the hotspot dynamics provided by Kerman/Berggren (see Section 2.2.3 and Refs. [53, 60]) using the Simscape Electrical library of Simulink.

**Figure 5.3:** Simscape schematic to simulate the latching behavior of DC- and RF-biased nanowires. The DC-biased nanowire was simulated using the schematic in (a). For the RF-biased nanowire, the kinetic inductance and bias tee [blue background in (a)] were substituted by the capacitively coupled inductance in (b).

In Fig. 5.3 the schematic of the model is shown. The hotspot dynamics according to Eqs. 2.6 and 2.7 are implemented by the block labelled as "electrothermal model". It provides the hotspot resistance Rhs from nanowire current In and an additionally injected current dI. The following device parameters are used: Isw = 30 µA, Rsh = 350 Ω, w = 110 nm, ν0 = 236 m/s, and ψ = 76. A photon detection event in the nanowire causes a local suppression of the switching current, leading to a detection event if Isw is reduced below the applied current through the nanowire. Following Ref. [60], we simulate the detection event by injecting an excess current dI which shortly increases the current through the nanowire beyond its switching current Isw. For this purpose, the block named "photon pulses" emits current pulses of amplitude 10 µA.

Two dierent congurations, DC and RF bias, were simulated. For the DC-biased nanowire (Fig. 5.3a), the kinetic inductance was set to Lk1 = 40 nH, resulting in a fall time of ≈0.8 ns. It was biased by a constant current Ib = 0.95Isw through the inductive port of a bias tee (LBT = 1 µH) and read out through capacitanceCBT = 4 nF. The RF-biased nanowire (Fig. 5.3b) with the same switching current was modeled by an LC resonator consisting of the nanowire's kinetic inductance Lk2 = 5 nH and capacitance Cc = 1.4 pF. With fr = 1/(2π √ LC), the device exhibits a resonance at ≈1.9 GHz and a simulated resonance width of ≈860 MHz. The latter

corresponds to a quality factor of 2.2 and a ringdown time of 0.4 ns, about half the value of the DC-biased detector. The nanowire was biased by an alternating current with an amplitude close to the switching current of the nanowire. Note that this conguration diers from the RF-SNSPD circuit used in the other sections of this chapter (see schematic in Fig. 5.1c). Here, the nanowire's kinetic inductance is directly used as the inductive part of a series LC resonator and the separate, parallel LC circuit is omitted.

In both congurations (DC and RF bias), dependence Lk (Ib) was neglected. Typically, the maximum variation of the inductance is ≤20 % over the bias range (see Ref. [60], or Fig. 7.4 in Section 7.3) and therefore does not change the time scales of the simulated DC-biased nanowire behavior signicantly. For the RF-bias case, Lk (Ib) can, in general, prohibit the detector operation when fr (Lk) leads to a large enough resonance shift such that the bias power at a specic frequency does not couple into the resonant circuit anymore after a detection event. However, due to the large resonance width ∆f of this particular circuit (larger than the expected shift in fr ), the Lk (Ib) dependence is not expected to hinder the detector operation and neglecting the dependence is therefore adequate.

Depending on the conguration, the pulse period of dI slightly varied: in the case of DC bias, the period was set to τ = 5 ns, while for the RF bias τ = 10/fr . The latter was chosen to avoid problems with the reduced duty cycle of the nanowire with an oscillating bias current. This will be investigated in Section 5.3.3. By dening the photon pulse period by means of the resonance frequency, the phase of the bias current at the time of the photon event is kept constant. If the phase is chosen such that dI is injected at bias currents close to the maximum, it is ensured that every photon event leads to a growing hotspot.

Figure 5.4 shows simulated time traces over 100 ns of the detector signal2 , the hotspot resistance Rhs, and current Isim = |In| + dI for both DC (left column) and RF bias (right column). Photon pulses are injected after a delay of 10 ns to account for the settling time of both circuits. These pulses are seen as short peaks of ∆I = 10 µA in the bias traces.

The DC-biased nanowire exhibits periodically emerging hotspots as expected, with Rhs ≈ 850 Ω and a relaxation time of ≈230 ps. Over the load resistance of 50 Ω, this is seen as voltage pulses of ≈1.2 mV amplitude and ≈1 ns decay time. About 2 ns after every detection, the bias current through the nanowire has restored. Notably, the voltage signal depicted in Fig. 5.4a returns to a value below zero after every detection event, equivalent to a negative load current. This is the result of the AC-coupled readout, namely the discharge of the capacitor used to readout the detector (CBT in Fig. 5.3), and is also known as "back-biasing" [60]. The eect was

2 To approximate a real-world setup, the voltage across the load resistance of 50 Ω was ltered by a second-order Butterworth band-pass lter with edge frequencies of 15 MHz and 8 GHz.

**Figure 5.4:** Simulation of the detector signal, hotspot resistance Rhs, and current Isim = |In | + dI for DC bias (left column) and RF bias (right column) using the models from Fig. 5.3. The plots in the lower half are zooms of the respective time spans denoted in the graphs above. In (f) and (l), dashed lines indicate bias current Ib and bias amplitude ˆIrf , set by the respective bias source.

previously described by Kerman et al. (compare Fig. 1f in Ref. [107]), arises at high count rates, and also causes a higher current through the nanowire In > Ib than what is set by the current source. Indeed, in Fig. 5.4f, a higher current In through the nanowire than Ib = 28.5 µA can be discerned before the event at t = 70 ns. Eventually, the back-biasing of the nanowire can lead to relaxation oscillations and latching into a stable resistive state. For the particular

operation parameters here, only a single oscillation at t ≈ 73.5 ns between two consecutive photon pulses is observed, before the detector latches after the following photon pulse at t = 75 ns. This manifests itself as a steadily growing hotspot resistance, which results in a small constant current through the nanowire (disregarding the periodic dI pulses) and a permanent voltage across the load resistance. The particular time at which the nanowire oscillates and latches depends on the applied bias current Ib, the kinetic inductance Lk, the readout capacitance (here only CBT), as well as the readout resistance Rl . For a larger kinetic inductance, we observed multiple relaxation oscillations with decreasing period before the detector latches, comparable to Fig. E2 in the supplementary of Ref. [60]. Similarly, a bigger readout capacitance also prolongs the time span over which oscillations occur before the detector latches. Conversely, the detector operation is more prone to latching for bias currents closer to Isw or a higher load resistance (see, e.g., Fig. 7 in Ref. [60]).

Time traces over the same span for the RF-biased nanowire are shown in Figs. 5.4g–l. The respective detector pulses are slightly shifted with respect to the DC bias because of the dierent photon-pulse period (see above). From Figs. 5.4k and 5.4l, it is visible that the maximum hotspot resistance at approximately 300 Ω is smaller than the one simulated for the DC-biased nanowire, and drops rapidly as the amplitude of the RF current through the nanowire decreases. This is both due to the oscillation phase and the fast redirection of the RF current to the 50 Ω load resistance. It is also seen that, for the device parameters in use, the hotspot only persists over half a period of the RF current. This behavior depends on the quality factor of the resonator. For a larger quality factor, it was found that the hotspot can persist over few bias oscillations as well. Following a detection event, the bias current through the nanowire is restored after around three full oscillations of Irf (see Fig. 5.4l). Compared to the DC-biased nanowire, the operation is not locked into the resistive state of the nanowire at any time and the detector is eectively reset after every detection event, enabling a stable operation.

It should be noted that the detector signals were not amplied in these simulations. Needless to say, a capacitively coupled amplier would add a capacitor in series to the load also for the RF-biased nanowire, even in the absence of a bias tee. Nevertheless, even if the amplitude of |Irf | would exceed the switching current of the nanowire, the resulting oscillatory behavior would not lead to the instability of the device, i.e., latching (see Fig. 5.2).

## **5.3 Synchronized gated detection**

This section contains parts from previously published content: [241] E. Knehr, A. Kuzmin, S. Doerner, et al., "Synchronous singlephoton detection with self-resetting, GHz-gated superconducting NbN nanowires", Applied Physics Letters 117, 132602 (2020). Permission to include adapted text was obtained from the authors.

The intrinsic detection eciency of an SNSPD depends on the applied bias current. DC-biased SNSPDs are biased by a constant current Ib close to Isw. If the PCR(Ib) curve is saturated in this bias regime, one can assume ηint ≈ 1. This free-running operation of SNSPDs is an advantage over competing types of single-photon detectors like single-photon avalanche diodes (SPADs), which require a gated operation to mitigate afterpulsing [246].3 The only detrimental eect on ηint during the normal SNSPD operation is due to the nite current recovery time to the nanowire after a detection event, and becomes relevant only at high count rates. This dependence was investigated in Ref. [52] (see also SDE(CR) measurements in Fig. 4d of Ref. [106]).

On the contrary, RF-SNSPDs are operated by an oscillating bias current Irf only. Because of dependence ηint[Irf (t)], the RF-SNSPD is not expected to be active all the time, i.e., it is in gated mode. In the case of random photon arrivals, this should lead to a reduced detection eciency compared to a similar DC-biased SNSPD. For known photon-arrival gates, on the other hand, the detection eciency can be optimized by synchronizing the RF-SNSPD to the base clock of the optical signal. As a result, the RF-SNSPD's detection eciency should become comparable to a DC-biased SNSPD. In Ref. [189], the direct comparison of a DCand RF-biased SNSPD has demonstrated that a similar asynchronous ηint can be achieved. However, the comparison requires an accurate determination of the RF current through the nanowire which is dicult. Moreover, ηint of an RF-SNSPD synchronized to a modulated light source at half of the gating frequency has revealed a sensitivity to the relative phase between the RF bias and optical modulation, although the change in ηint was not well understood. In this section, results of an RF-SNSPD fully synchronized to a modulated light source in GHz range are presented. Based on these results, the inuence of the sigmoidal ηint(Ib) dependence on the duty cycle is estimated in the end.

3 For dierent purposes, a gated or quasi-gated approach for SNSPDs has been adopted before. Besides the aim to mitigate the latching problem (see previous section), Akhlaghi and Majedi presented a gated mode at 625 MHz with a DC oset to increase the maximum count rate [247].

**Figure 5.5:** (a) Schematic of the experimental setup for synchronized photon detection. The RF-SNSPD (lumpedelement model in green) is biased by a single tone from a signal generator and its signal is read out both directly as well as down-mixed. The VCSEL is biased by a second, phase-locked signal generator with an RF current oset by a DC source. (b) Sketch of the VCSEL operation with DC (red) and RF bias (blue). The black line is loosely based on the output power vs. forward current graph provided in the data sheet.

For the following investigations, pixel 1 from a ve-pixel array (see Fig. 5.1) is used and operated at its resonance frequency fr = 1.9 GHz. Thus, the gating frequency is fg = 2fr = 3.8 GHz.

#### **5.3.1 Experimental setup**

The measurement setup for the synchronized photon detection is depicted in Fig. 5.5a. As a GHz-modulated light source, we use a vertical-cavity surface-emitting laser diode (VCSEL, Optek Technology OPV314YAT) emitting light at a wavelength of 850 nm and allowing for modulation frequencies fopt . 4.5 GHz.

Two phase-locked RF signal generators provide the RF bias currents for both the laser diode and the RF-SNSPD. The laser diode is biased by a DC current of 3 mA and an RF current with an amplitude large enough to turn it o at the minimum, resulting in an optical output power modulated in GHz range similar to on-o keying (OOK). A visualization of this is shown in the output power vs. forward bias sketch of the VCSEL operation in Fig. 5.5b. A fast photodiode (Thorlabs PDA8GS, DC–9.5 GHz) was used to accurately measure the spectrum of the optical

**Figure 5.6:** Current-dependent photon-count rate and detector pulse. (a) Measured photon-count rate dependent on the relative bias current. The solid line is a t by Eq. 2.10. (b) Single RF (blue) and downmixed (red envelope) pulse upon photon absorption. The trigger levels indicated by dashed lines are used to record the graph in Fig. 5.7.

modulation of the VCSEL. After setting the appropriate bias level, the VCSEL output was optically coupled to the detector using a multi-mode ber (Thorlabs FG105LCA).

The RF-SNSPD is biased by an RF current only. The output power of the signal generator is set to 21 dBm, such that, combined with the loss of −6 dB from the splitter, the mixer is fed with a power within the specied LO (local oscillator) drive level range of (13 to 18) dBm. The exact bias power level of the RF-SNSPD pixel is then set by a combination of a variable attenuator for coarse adjustment and the signal generator for ne-tuning (within ±1 dBm). The pulses from the detector are amplied by 37 dB using a room-temperature amplier and read out by a 32 GHz real-time oscilloscope (port 2). Additionally, the RF pulses are downmixed by an analog mixer using the RF port as input, the IF (intermediate frequency) port as output and the bias signal as LO (homodyne mixing scheme). The resulting envelope is read out with the oscilloscope (port 1) and an ordinary pulse counter to measure count rates.

#### **5.3.2 Phase-sensitive detection eiciency**

The current-dependent photon-count rate (PCR) of the detector was measured at a constant VCSEL output power (Idc = 3 mA) and is plotted in Fig. 5.6a. As it is seen, the curve does not saturate up to a current at which the DCR becomes dominant.

In Fig. 5.6b, a single detector response upon photon absorption and its envelope are shown. The decay time τd = Q/(π fr ) ≈ 20 ns is determined by the loaded quality factor Q of the resonant circuit and is consistent with the target resonance width of 15 MHz (Q ≈ 120).

**Figure 5.7:** Time traces and histogram of pulse envelopes. (a) Rising edge of multiple pulse envelopes measured by triggering on the RF pulses at the trigger levels indicated in Fig. 5.6b. (b) Normalized histogram of the number of counts over time at the rising edge of the pulses [indicated by the semitransparent rectangle in (a)] with measurements in blue and Gaussian ts in red. Time dierence td is taken as the dierence between the mean values of both ts.

Figure 5.7a shows multiple traces of the envelope (channel 1 of the oscilloscope, Fig. 5.5), which are obtained with constant optical power by triggering on the original RF pulse (channel 2) exceeding a dened range (−Vtrig to +Vtrig, see Fig. 5.6b). The polarity and amplitude of the rst oscillation of the RF pulse depends on the phase of the bias current at the time of photon absorption. With this trigger method, one can visualize the distribution of count events relative to the phase of the bias. Two distinct groups of envelopes can be observed. This is also seen from the histogram of rising edges which is depicted in Fig. 5.7b. The time dierence between the two peaks amounts to td ≈ 263 ps, which corresponds to a frequency f ≈ 3.8 GHz or twice the resonance frequency.

The grouping of pulses shown in Fig. 5.7a can be explained by the nonsaturated count rate characteristic CR (Ib) of the NbN detector at λ = 850 nm and T = 4.2 K (Fig. 5.6a). Since, in this case, ηint does not reach 100%, the majority of the detection events will originate from bias points near the extrema (±Irf,max). The measured grouping of pulses around these two bias points of opposite polarity implies that the detector is active twice in a period of the RF bias. The active time span (gate duration) can be estimated by the time during which |Irf | ≥ Idet, with the minimal detection current Idet for a particular photon energy.

In order to show that the RF-SNSPD's detection eciency might be optimized by synchronizing the detector to the optical signal, the phase-sensitive changes of the count rate are investigated. The measurements are conducted with the light modulated either with the resonance frequency of the RF-SNSPD (fopt/fr = 1) or with its gating frequency (fopt/fr = 2). To slowly sweep the relative phase between the RF bias and the optical signal, the frequency

**Figure 5.8:** Measurement and simulation of modulated count rate CR(t) for a synchronized detection and beat frequency δ f = 10 mHz. (a), (b) Experimental count rates CR(t) for fopt/fr = 1 and fopt/fr = 2 compared to simulated curves obtained from Eq. 5.4. (c), (d) Simulation of optical signal Popt(t) and instantaneous detection eciency η(t) for both modulation frequencies fopt. In each case, full alignment and misalignment between Popt(t) and η(t) are depicted [in (d) corresponding to phases 0° and 180° of η(t) relative to Popt(t)].

of the RF bias is detuned with respect to the modulation frequency of the laser diode by δ f = 10 mHz. This small detuning (δ f /fr ≈ 10−12) should lead to a slow periodic modulation of the count rate with the beat frequency.

The phase-sensitive detection eciency was analyzed experimentally (Figs. 5.8a and 5.8b), and oscillations of the count rate with a period of 50 s are indeed visible. The measured count rate modulation index m = (CRmax − CRmin)/(CRmax + CRmin) amounts to m ≈ 0.71 for fopt/fr = 2. For the case fopt/fr = 1, the modulation index is smaller (m ≈ 0.30) and depends on the modulation amplitude of the laser diode. The modulation period of 50 s observed for a detuning δ f = 10 mHz corresponds to a detuning δ fg = 20 mHz in gating frequency, which is in accordance with the detector being active twice each bias period (Fig. 5.7).

To describe the measurement results, we simulate the CR(t) dependence using the instantaneous detection eciency η(t) obtained from the t of CR(Ib). For this, we take a sinusoidal RF bias current

$$I\_{\rm rf}(t) = \sin\left[\left(\omega\_{\rm r} + \delta\omega\right)t\right],\tag{5.1}$$

with ωr = 2π fr = 2π × 1.9 GHz and δω = 2πδ f = 2π × 10 mHz. The instantaneous detection eciency η(t) is then obtained by tting

$$\eta(t) = a \left[ 1 + \text{erf} \left( b \left| I\_{\text{rf}}(t) \right| - c \right) \right] \tag{5.2}$$

to the experimental data [Fig. 5.6a, assuming Ib = Irf (t)] with tting parameters a, b, and c (equivalent to Eq. 2.10). Since η(t) in Eq. (5.2) depends on the modulus of the RF bias |Irf | ∝ p 1 − cos (2ωrt), it oscillates with the frequency fg = 2fr .

The amplitude-modulated optical signal is modeled as

$$P\_{\rm opt}(t) = a\_0 + a\_1 \cos\left(\omega\_{\rm opt} t\right) + a\_2 \cos\left(2\omega\_{\rm opt} t\right),\tag{5.3}$$

with the modulation frequency of the laser diode ωopt = 2π fopt and experimentally measured spectral coecients a0, a1, and a2 for our particular laser diode at the used operation point and both frequencies fopt. Both the instantaneous detection eciency η(t) and the normalized optical power Popt(t) are plotted over time in Figs. 5.8c and 5.8d for operation frequencies fopt/fr = 1 and fopt/fr = 2, respectively.

The time-dependent count rate is estimated using Mandel's formula (Eq. 8 in Ref. [248]) by

$$\text{CR}(t) \propto \int\_{t}^{t+\tau} \eta(t') \cdot P\_{\text{opt}}(t') \text{dt'} \tag{5.4}$$

for the mean number of photon clicks in time interval τ .

A comparison of the simulated and measured count rate CR(t) is shown in Figs. 5.8a and 5.8b. As indicated by the arrows, the maxima and minima of CR(t) correspond to detector gates fully aligned and misaligned to the optical pulses, depicted in Figs. 5.8c and 5.8d.

For fopt/fr = 2, in which case the detector and the laser diode are operated with the same frequency and phase, the measurement data coincide well with the simulated curve in both beat frequency and modulation indexm. In this case, the amplitude a2 of the second harmonic of Popt(t) was found to be very small. The large modulation index signies that the detection eciency can be maximized by synchronizing the detector to the light source. The attenuation of stray light and suppression of dark counts can be quantied by the factor (1 + m)·(1 − m) −1 , with a value of about 6.0 for fopt/fr = 2.

For fopt/fr = 1, the modulation of CR(t) is only due to the second harmonic of Popt(t), which has a signicant amplitude a2 originating from the nonlinear current-voltage characteristic of the laser diode. The latter also explains the observed dependence of modulation index m on the bias amplitude of the diode. If a2 was negligible, the shared area under the curves Popt(t)

**Figure 5.9:** Simulation of dierent CR(Ib) curves (a) and corresponding instantaneous detection eciencies ηint(t) according to Eq. 5.2 over one period of fr (b). The black dots in (a) indicate measurement results. Average detection eciencies ηint in (b) are calculated from the integral of ηint(t).

and η(t) in Fig. 5.8c would be constant over time and, thus, no modulation of CR(t) would be observed for fopt/fr = 1.

All measurements in this section were conducted using a single pixel. A gated operation of a whole array of pixels with dierent resonance frequencies is only possible within limits. Assuming xed frequencies fr,i for pixels i = 1, . . . ,n of the array, the highest operation frequency of the detection system is double the fundamental frequency, i.e., the greatest common divisor (GCD) of all fr,i . However, the operating frequency of each pixel could be slightly adjusted around its fr by few MHz (depending on the resonance width) and the GCD of all gating frequencies thus optimized.

#### **5.3.3 Duty cycle**

According to our simple model, the duty cycle of the gated detector is highly dependent on the CR(Ib) characteristic. With a nonsaturated CR(Ib) curve, the investigated detector exhibits a gated detection with photon detection mainly occurring for a current Irf near the extrema ±Irf,max when |Irf | ≥ Idet, as discussed before (see solid, red curves in Fig. 5.9). In the case of a pronounced count-rate plateau, however, the active time spans can be much longer, which becomes important for detecting photons with random arrival times. This is simulated for two saturated CR(Ib) curves (see dashed and dotted curves in Fig. 5.9). For a broader saturation in Fig. 5.9a, the gate duration increases accordingly, as seen by the instantaneous detection eciencies ηint(t) in Fig. 5.9b. The average detection eciency ηint can be estimated from the integral of ηint(t), which results in values of 0.30, 0.57, and 0.88 compared to a DC-biased SNSPD constantly operated at ηint = 1. For the broadest simulated CR(Ib) curve, the pure RF bias leads to an only slightly decreased eciency for detecting random photon arrivals and,

therefore, becomes comparable to free-running mode. Thus, the duty cycle of an RF-SNSPD with a pronounced count rate saturation can be tuned between gated and quasi-free-running mode by choosing an appropriate RF bias amplitude.

## **5.4 Matched-filter array readout**

In the previous sections, a single pixel of an RF-SNSPD array was operated by a single-tone probe signal from a signal generator. The same microwave signal was also used to downmix the RF pulses coming from the detector, essentially resulting in pulses equivalent to those of conventional DC-biased SNSPDs (see Fig. 5.6b).

To operate a whole array of RF-SNSPD pixels, a frequency comb containing each of the pixel's resonance frequency is necessary. In Ref. [186], an arbitrary waveform generator (AWG) was used to generate the bias signal containing the tone of every pixel. However, the detector signal was only analyzed in frequency-domain by continuously transforming individual time spans of the digitized detector signal using a fast Fourier transform (FFT) and comparing the result to threshold amplitudes for each pixel. The precise timing information of the photon event could be extracted from the RF pulse itself or from the phase spectrum of the complex FFT.

An alternative readout method is explored in this section by using correlation lters instead. In the frequency-domain multiplexing scheme, each pixel operates at a unique frequency. Therefore, templates of each pixel could be used as matched lters to detect the presence of respective detection events in the signal on the common transmission line. Compared to the FFT method used before, the matched lters are applied in time domain and could increase the signal-to-noise ratio (SNR). Matched lters [249] are commonly used to detect known templates in an acquired, noisy signal, for example to detect radar [250] or gravitational-wave signals [251].

#### **5.4.1 Concept**

The matched lter is applied by cross-correlating a time trace with a pre-recorded template of the respective pixel. For a discrete trace s and template tp of pixel p, this is expressed by

$$s(t\_{\rm p} \star s)[n] = \sum\_{k=-\infty}^{\infty} t\_{\rm p}^\*[k] \, s[k+n],\tag{5.5}$$

**Figure 5.10:** Microwave transmission |S21 | over frequency of the RF-SNSPD array. Resonances corresponding to all ve pixels between 2 GHz and 4 GHz are visible. For all further measurements, the highlighted pixels 1, 3, and 4 were used.

where t ∗ p [k] denotes the complex conjugate of tp[k]. For nite-length signals tp[i], 0 ≤ i ≤ K, s[i], 0 ≤ i ≤ M, the boundaries of the summation are dened as k = max(0, −n) to min(K, M − n). From the convolution theorem, this is equivalent to F −1 ( F ∗ (tp) · F (s) ) , where F (x) is the Fourier transform of x. Because the matched lter weights the frequency components of the signal according to the spectrum of the template, the inuence of stochastic noise on the signal is reduced and the SNR maximized (in the case of Gaussian noise). The result of cross-correlating the detector signal with a pulse templates are separate traces of detector pulses for each pixel.

#### **5.4.2 Proof-of-principle experiment**

A proof-of-principle of the matched lter readout is shown using the ve-pixel RF-SNSPD array described in Section 5.1 (see layout in Fig. 5.1). Two pixels of the array are biased simultaneously and the detector signal is directly digitized. The correlation with pulse templates is then carried out by post-processing the recorded signal.

Figure 5.10 depicts the microwave transmission |S21 | over frequency recorded with a vector network analyzer (VNA). All ve pixels exhibit dened resonances in the range (2.0 to 3.6) GHz with a spacing of ≈400 MHz, except for pixel 4 which is shifted to a slightly smaller frequency than designed. For the following measurements, pixels 1 and 3 were biased, while a third pixel (pixel 4) was not biased and only used to check for false-positive detection-event attributions. The pixels exhibit resonance depths of  S21,min,px1 = −38 dB (fr,px1 = 2.01 GHz), S21,min,px3 = −33 dB (fr,px3 = 2.85 GHz), and  S21,min,px4 = −31 dB (fr,px4 = 3.08 GHz),

**Figure 5.11:** Measurement setup to test the matched-lter readout. Two signal generators provide the harmonic signals (tones) with frequencies matching the resonance frequency of the respective pixels. Both bias tones are combined using a signal combiner and coupled into the feedline. The amplied signal on the feedline is directly recorded and digitized by a real-time oscilloscope. Two light sources were utilized: a continuous-wave VCSEL diode at 850 nm wavelength and a pulsed laser at 1550 nm with 100 MHz repetition frequency.

respectively. The resonance depth directly translates to the transmitted power upon photon absorption and therefore the pulse height of a detector click. For each pixel, the loaded bandwidth ∆f at −3 dB was ≈20 MHz. Therefore, the decay time of a single pulse resulted in τd = (π∆f ) −1 ≈ 16 ns.

To bias and readout the detector array, we use the setup shown in Fig. 5.11. The two pixels operating at fr = 2.01 GHz and 2.85 GHz were biased simultaneously through a common feed line by two separate signal generators and a wideband power combiner. The bias signal was attenuated by 40 dB, which helps to reduce noise on the bias line. As light sources, we used a pulsed laser at 1550 nm wavelength with 100 MHz repetition frequency to obtain the individual detection events in Fig. 5.12, and a VCSEL diode at 850 nm wavelength in continuous-wave mode for photon statistics (Fig. 5.13). The signal at the end of the feed line was amplied by 37 dB by a cryogenic amplier and directly recorded using a real-time oscilloscope at 10 GSa/s sampling rate.

Before the simultaneous operation, we obtained pulse templates of every pixel by biasing only one pixel at a time and record single-shot pulses. After template acquisition, pixels 1 and 3 were simultaneously biased and illuminated. Because of the large ber-to-detector distance of 40 mm and the small pitch of 200 µm between both pixels, a homogeneous illumination is assumed, however not strictly necessary for the purpose of the proof-of-principle demonstration.

Waveforms of three recorded events are depicted in the upper row of Fig. 5.12. The increased transmission upon photon absorption is seen as a transient oscillation (RF pulse). These

**Figure 5.12:** Three dierent recorded waveforms with pixels 1 and 3 triggering at the same time (event 1), pixel 3 triggering two laser repetition periods before pixel 1 (event 2), and pixel 1 triggering one laser repetition period before pixel 3 (event 3). The dashed lines in the upper row denote the timing of the laser pulses with a repetition period τrep = 10 ns. The lower row shows upper envelopes of the normalized cross-correlations of the respective event with templates of three dierent RF-SNSPD pixels. Pixel 4 was not biased during acquisition of the waveforms.

waveforms s[i] were then correlated with templates tp[i] of each pixel according to Eq. 5.5. The normalized upper envelopes of the results are shown in the bottom row of Fig. 5.12. Each pulse can be assigned to a specic pixel. The events depict two overlapping pulses from pixels 1 and 3 ("event 1"), pixel 1 triggering two laser periods after pixel 3 ("event 2"), and pixel 1 triggering one laser period before pixel 3 ("event 3"). As it can be seen, the matched-lter readout is suited for the attribution of the RF signal to individual pixels, also in the case of two overlapping pulses as shown in "event 1". Depending on the normalization of the cross-correlation, also the amplitude of the RF pulses are replicated by the matched-lter output.

To check for false positives, we used a pre-recorded template of an additional pixel at fr = 3.06 GHz ("pixel 4"), which was not biased during the time of measurement. Pixel 4 is separated by 230 MHz from pixel 3. Only a slight inuence of the events attributed to pixel 3 is seen on the matched-lter output of pixel 4.

The minimum achievable frequency separation between two adjacent pixels while maintaining distinguishable matched-lter outputs was tested by simulations with articial pulse templates at fr,sim = fr,px3 + (0 to 230) MHz. For this purpose, a sinusoidal signal of the respective frequency was modulated by a simulated DC pulse with τd = 20 ns. At frequencies fr,sim = fr,px3 and fr,sim = fr,px4, the matched-lter output closely resembles the one obtained using

**Figure 5.13:** Photon-count statistics using a coherent laser diode as photon source and the matched-lter readout of two pixels of the RF-SNSPD array at dierent bias levels. (a) An exemplary, 1 ms-long window showing separate time traces of both pixels. (b) Probability density of inter-pulse time δ t obtained from a 1 s-long time trace. The black, dashed lines are exponential ts by Eq. 5.6 to the corresponding curve.

the respective measured pulse templates. The frequency of the simulated pulse was then swept between these two limits, and a matched-lter output peak of half the value of the adjacent pixel (in this case pixel 3) was taken as the maximum value in order to still being able to distinguish between both pulses. With this criterion, the minimum fr separation was simulated to be ≈30 MHz. Assuming an unchanged loaded bandwidth of 20 MHz per pixel and smaller separation bands of 30 MHz between them, a total number of 100 pixels per channel could be realized within a bandwidth of 5 GHz and read out by matched lters.

Beyond attributing single detection events to the correct pixel, we derived photon-count statistics with this method for the two simultaneously biased pixels over a longer acquisition time. To this end, a 1 s-long time trace of detector events on the transmission line was recorded. Both pixels were ood-illuminated by the 850 nm VCSEL diode. In order to compare count statistics at dierent ηint, both pixels were intentionally biased at unequal bias points relative to Isw, with pixel 1 operated at much lower RF bias power than pixel 3. The recorded trace was then correlated with the individual pulse templates of both pixels, which results in two separate time traces (see Fig. 5.13a). The pulse-arrival times of each trace was obtained by setting the trigger level to half the maximum pulse height. From these traces, we derived the probability density of time δt between two sequential pulses, which is shown in Fig. 5.13b.

The coherent photon emission from the laser diode resembles a Poisson process and the probability density function of the photon inter-arrival time is expressed by

$$f(\delta t) = \chi e^{-\chi \cdot \delta t} \tag{5.6}$$

with mean photon rate γ . For accurate measurements, it is important that the photon rate is low enough such that the detector's dead time is signicantly smaller than the photon inter-arrival time. As seen from ts to both curves with tting parameter γ in Fig. 5.13b, the inter-pulse distributions of both pixels follow the expected photon distribution very well, which means that both detectors operate in single-photon regime.4 The mean detection rates amount to γpx1 ≈ 10 ms−1 and γpx3 ≈ 138 ms−1 , respectively. The large dierence of γ between pixel 1 and 3 stems from the dierent bias levels and therefore ηint between these two pixels. Nevertheless, both probability densities of δt in Fig. 5.13b are well described by Eq. 5.6, independent of the specic bias point.

#### **5.4.3 Future readout of RF-SNSPD arrays**

For the proof-of-principle demonstration, the results herein were obtained by two separate analog signal generators as bias sources and post-processing on recorded time traces. In the future, the operation and readout of RF-SNSPD arrays could be implemented in real time on a software-dened radio (SDR) platform, similar to what has been demonstrated for MKID arrays [253].

The probe tones matching the resonance frequencies of an extended RF-SNSPD array would be generated by an FPGA-based AWG. After digital-to-analog conversion, the tone comb can be up-converted into the appropriate frequency band. The amplied output signal of the array would then be down-converted into the IF band and digitized. To extract the spatial information of each pulse from its IF-band frequency, either an FFT-based detection scheme or matched lters running on FPGAs could be used.

It remains to be seen how the extraction of timing information from the matched-lter output aects the timing jitter. In Ref. [254], a matched-lter approach was adopted to improve the readout of an SNSPD array with varying pulse height. With pulse-arrival times taken at the peak of the matched lter outputs (at the highest SNR), the timing jitter could be slightly improved compared to a conventional readout. For RF-SNSPD, however, it could also be necessary to extract timing information directly from the RF signal and to only determine the corresponding spatial information from the matched-lter output.

4 Similar photon statistics recorded with SNSPDs have been presented in Refs. [49, 73, 252].

## **5.5 Summary**

In this chapter, the concept of RF-SNSPD arrays, previously introduced by Doerner et al. [186], was investigated further, mainly with regard to the consequences of the oscillating bias current for the detector operation.

The RF bias leads to a stable operation without the risk of the detector latching into the resistive state. The self-resetting behavior is the consequence of the limited hotspot growth because of the alternating current through the nanowire and the current redirection out of the nanowire due to the damped resonance in the case of a nite hotspot resistance. Measurements of the bias-dependent dark-count rate show that even if the detector is biased by a current with an amplitude above Isw, the device does not latch but exhibits self-oscillations at a pulse rate proportional to the current amplitude. In addition, an RF-biased nanowire with a ringdown time of only 0.4 ns was simulated and compared to DC bias at high photon-count rates using an electrothermal model of Kerman/Berggren [53, 60]. While the DC-biased operation led to relaxation oscillations and eventually latching due to back-biasing caused by the capacitive readout, the RF bias enabled a reliable restoration of the superconducting state after every detection event.

Another consequence of the RF bias is the gated detector operation because the microwave current modulates the instantaneous intrinsic detection eciency ηint[Irf (t)]. This was conrmed by analyzing the pulse envelopes when triggering on both rising edges of the RF pulses, showing a gated detection at the frequency fg = 2fr = 3.8 GHz. Although gating is not necessary for the SNSPD operation, in applications with known photon arrival time spans, a gated operation is benecial since background noise can be suppressed and the signal-to-noise ratio therefore improved. Synchronizing the RF-SNSPD to a fast VCSEL diode modulated with fopt = fg revealed phase-sensitive changes of the photon-count rate with a modulation index of 0.71, which agrees well with a simple model of the operation conditions. In this way, we showed that the detection eciency of RF-SNSPDs can be maximized and the inuence of stray light minimized (here by a factor ≈6.0) by using a phase-locked light modulated with the detector's gating frequency. However, the oscillating intrinsic detection eciency also implies a reduced duty cycle of the detector for photons with random arrival times, depending on the bias-dependent count rate. In our particular case, the average detection eciency amounted to 30 %. For devices with a pronounced count-rate plateau, the duty cycle of the detector can be much larger and tuned over a broad range by setting the bias amplitude.

Lastly, a proof-of-concept readout of an RF-SNSPD array using pre-recorded templates of each pixel's response as matched lters showed the reliable attribution of each RF pulse on the common transmission line to the respective pixel, thereby enabling spatial resolution of the count events. Photon-count statistics over 1 s, extracted from two pixels simultaneously, revealed an exponential distribution of the inter-pulse time in accordance with the Poissonian photon-number distribution of the coherent light source.

# **6 Optical coupling**

By its nature, a nanowire only covers a small area, which makes ecient photon guiding a challenge. Typically, SNSPDs consist of meandered nanowires covering a larger area on the order of several µm side length, which, however, requires long nanowires. This impairs the main benecial properties of SNSPDs in timing and maximum count rate. Long nanowires are also more prone to constrictions, which potentially limit the spectral bandwidth of the detector. Therefore, the nanowire should be as short as possible while still allowing for ecient optical coupling from the system input. This trade-o not only aects free-space but also ber coupling. To account for typical mode-eld diameters of a telecom single-mode ber (≈10 µm) and a potential misalignment between the ber core and the detector, the nanowire is typically >1 mm long [252].

In Section 2.6, existing methods to improve the coupling eciency for SNSPDs have been presented, for free-space as well as for ber and on-chip coupling. In this chapter, the integration of SNSPDs with free-form polymer structures will be investigated in order to achieve high coupling eciencies with comparatively short nanowires.

These polymer structures are fabricated by 3D direct laser writing (DLW), a technique which uses two-photon polymerization of a photoresist in the focal region of a near-infrared, femtosecond-pulsed laser. By moving the focal region of the beam through the resist, the material gets selectively polymerized within a small volume (voxel) of typically <1 µm in diameter. Because of the squared intensity dependence of the photon absorption rate in the resist and therefore also the exposure dose, arbitrary 3D structures can be written with high accuracy even in axial direction (see the explanation in Ref. [255]). An overview of this technology is given in Refs. [256, 257]. Using 3D DLW, beam-shaping elements [193, 258] and optical interconnects [201, 202, 204, 205] have been demonstrated before.

Typically, these structures are used for applications at room temperature, e.g., in photonic circuits. Extending the 3D-DLW technique to optical coupling of SNSPDs (or superconducting photon detectors in general) poses questions about the mechanical stability of the polymer structures during cryo-cycling, potential harm to the detectors during laser lithography, and, ultimately, the achievable coupling eciency.

The investigation of SNSPDs integrated with these optical elements is separated in two parts of this chapter. Section 6.1 deals with microlenses on top of an RF-SNSPD array, mainly targeting free-space coupling. As outlined in Section 2.5.2, RF-SNSPDs have the drawback of a low ll factor of the photon-sensitive area and could therefore particularly benet from enlarged light-collection areas. In Section 6.2, photonic wire bonds will be explored for the coupling of a single-mode ber array to a DC-biased SNSPD array consisting of detectors with comparatively short (≈105 µm long) nanowires.

## **6.1 Microlens-integrated RF-SNSPD**

The research on microlens-integrated SNSPDs was conducted in collaboration with the Institute of Photonics and Quantum Electronics (IPQ) at KIT. An analysis of the lens design and experimental data on a DC-biased SNSPD were published in:

[259] Y. Xu, A. Kuzmin, E. Knehr, et al., "Superconducting nanowire single-photon detector with 3D-printed free-form microlenses", Optics Express 29, 27708 (2021).

In addition to the results presented there, the same lens concept was also applied to RF-SNSPDs and is shown in this section. Y. Xu designed and fabricated the lenses. A. Kuzmin and E. Knehr performed the lens characterization and analyzed the experimental data.

In free-space applications of SNSPDs, photons are usually impinging the detector with an incidence normal to the substrate plane. The focussing optics and alignment method of the detection system therefore directly determine the area the nanowire has to cover for an ecient coupling.

In Ref. [259], we demonstrated the feasibility of integrating SNSPDs with polymer microlenses fabricated by 3D DLW. These lenses focus incident photons onto the active area of the detector in the focal plane of the lens. The eective collection area is increased from the detector area Ad to the collection area of the lens Ac = πr 2 c with radius rc. The optical coupling eciency is not bound to the length of the nanowire anymore and the microlenses therefore enable the combination of a short nanowire with a large light-collecting area. The eective lens gain Gd = Ac/Ad could be arbitrarily increased by scaling the lens dimensions. However, a larger physical size of the lens goes along with a larger lateral shift of the focal point for light with an incidence direction nonparallel to the optical axis of the lens (see Eq. 9 in Ref. [259]). Depending on the detector size and the tolerable angular misalignment, this might set an

**Figure 6.1:** Microlens concept and dimensions. (a) Side-view sketch of a microlens (rotationally symmetric around the optical axis denoted as a dashed line) which focuses incoming light with incidence normal to the detector plane onto the active area Ad. The lens has a total height of h = 70 µm (equivalent to focal length f ), a physical radius ra = 30 µm, and an eective collection radius rc . ra. (b) Top view of the active area Ad of an RF-SNSPD pixel and approximate dimension of the spot size in the detector plane.

upper limit to the feasible lens gain. In a proof-of-pinciple device, we demonstrated a gain of approximately two orders of magnitude for a lens diameter of 60 µm and an SNSPD area of 4.5 µm × 4.5 µm, which is reasonably consistent with the simulated value of Gd = 117. The experiment was conducted using two DC-biased SNSPD in close distance, one with a lens on top and the other as a reference detector without a lens.

In this section, the lens concept is applied to frequency-multiplexed SNSPDs. Beyond the importance of short nanowires for DC-biased SNSPDs, increasing the eective light-collection area for these RF-SNSPD arrays is of particular importance because the active areas are small compared to the size of each pixel, which includes the large resonator. As described in Section 2.5.2, the length of the nanowire is limited due to the requirement Lk Lr in order to have a resonance frequency which is near-independent of the bias-dependent kinetic inductance Lk (Ib).

In the following, the basic lens concept and fabrication are briey described before going into detail of the experimental characterization of a microlens-integrated RF-SNSPD.

#### **6.1.1 Microlens concept and fabrication**

A sketch of the lens concept is shown in Fig. 6.1a. The polymer has a refractive index of nlens ≈ 1.53 at 1550 nm (VanCoreA, Vanguard Automation GmbH). Incoming light within the

**Figure 6.2:** SEM micrograph of the RF-SNSPD sample. (a) SEM micrograph of the RF-SNSPD array with the examined pixels 1, 3, and 5. The rst two serve as reference detectors while the latter has a lens on top. Each active area and part of the resonant circuit is covered by a square AlN cover. Light gray areas show the NbN layer. (b) Close-up of the microlens on pixel 5.

radius rc is refracted onto the detector of size Ad. In general, rc . ra (ra: physical radius of the cylinder) because of Fresnel reections at steeper angles of the incident light with respect to the lens surface [259]. A detailed analysis of lens shapes is given in Section 3 of Ref. [259]. The particular design used here (identical to the one in the experimental section of Ref. [259]), is a polynomial approximation of a spheroid with a minor half axis length of 32 µm, clipped at ra = 30 µm. The lens height h is designed to match its focal length f = h = 70 µm.