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Triangular Quantum Photonic Devices with Integrated Detectors in Silicon Carbide

Triangular cross-section SiC photonic devices have been studied as an efficient and scalable route for integration of color centers into quantum hardware. In this work, we explore efficient collection and detection of color center emission in a triangular cross-section SiC waveguide by introducing a photonic crystal mirror on its one side and a superconducting nanowire single photon detector (SNSPD) on the other. Our modeled triangular cross-section devices with a randomly positioned emitter have a maximum coupling efficiency of 89 % into the desired optical mode and a high coupling efficiency (>75 %) in more than half of the configurations. For the first time, NbTiN thin films were sputtered on 4H-SiC and the electrical and optical properties of the thin films were measured. We found that the transport properties are similar to the case of NbTiN on SiO2 substrates, while the extinction coefficient is up to 50 % higher for 1680 nm wavelength. Finally, we performed Finite-Difference Time-Domain simulations of triangular cross-section waveguide integrated with an SNSPD to identify optimal nanowire geometries for efficient detection of light from TE and TM polarized modes.


Introduction
Color centers in silicon carbide (SiC) have been a prominently studied quantum platform for applications in quantum information processing (QIP). Their unique combination of spectral homogeneity, emission at near infra-red and telecommunication wavelengths, long spin coherence, and spin-photon entangling processes offer key applications in quantum communication, simulation, computing, and sensing [1][2][3][4][5][6][7].
The optical response of a color center has a sharp peak called the zero phonon line (ZPL) and a broad phonon side band (PSB). Emission through ZPL, when there is no dephasing, results in indistinguishable photon creation necessary for QIP applications. Integration of color centers with photonic devices is an important step toward their performance improvement and scalability, as shown in explorations with SiC pillars [8,9], waveguides [10,11], and micro-and nano-resonators [11][12][13][14][15]. Moreover, recent results on growth and patterning of superconducting NbN on 3C-SiC [16,17] for use as the superconducting nanowire single photon detectors (SNSPDs) indicates that quantum circuitry can be further miniaturized and optimized.
The challenges in maintaining the pristine quality of color centers have led photonic integration away from the established nanofabrication processes and toward alternative approaches that require non-standard sample preparation [11,13,18] or angled etching [10,19,20]. The latter, in particular, has been explored for its potential in wafer-scale processing [21]. Angle etching produces triangularly shaped devices whose photonic modes have been recently studied in terms of waveguide propagation [10,22], photonic crystal band gap formation [23], photonic crystal cavity resonances [22], and grating coupler performance [24].
In this paper, we expand on the studies of triangular SiC photonics by focusing on methods to optimize collection and detection of color center light by integration with a photonic crystal mirror (PCM) and an SNSPD, as illustrated in Figure 1. In particular, we study the case of the nitrogen vacancy (NV) in 4H-SiC with emission wavelengths in the range 1176 nm to 1243 nm [25][26][27]. When placed in a symmetric waveguide, at least half of the light is lost in an undesired direction, but this can be circumvented by the addition of a PCM. PCM has a periodic variation of refractive index, which results in the formation of photonic band gaps. Understanding the formation of these band gaps is necessary for applications in selective reflection of the ZPL and PSB emissions with the goal of enhancing light collection efficiency [23]. The other side of the waveguide deals with the efficient detection through hybrid integration with NbTiN SNSPDs. The SNSPDs are the leading detectors in terms of quantum efficiency, low dark counts and low timing jitter. Moreover, they offer an on-chip detection that overcomes the losses involved in the off-chip detection, thus boosting the success rate of QIP protocols [28]. However, depending on the type of excitation, the pump laser used for an off-resonant excitation of the color center can scatter into the detector region and saturate the SNSPD. There have been successful demonstrations to suppress the stray pump laser by coating the backside of the chip with an absorbing material [29] and masking the inactive regions of the chip [30]. To further suppress the pump laser coupled into the waveguide, spectral filters like ring resonators [31,32] and inverse-designed structures [33,34] can be integrated in the waveguide region before the detectors, with implementation in the illustrated cloud segment in Figure 1.
We first find the 1D photonic crystal (PC) conditions for band gap formation in triangular geometry on the desired wavelength domain (NV in 4H-SiC in this work) using the Plane Wave Expansion (PWE) and the Finite-Difference Time-Domain (FDTD) methods. We then utilize the optimal PC design as a PCM in a triangular waveguide and study the role of color center positioning in obtaining high emission coupling to the propagating optical mode.
Finally, in a combination of experimental and modeling approaches, we study the triangular waveguide mode absorption by a NbTiN SNSPD.

Photonic crystals for efficient propagation of color center emission in a triangular waveguide
Color centers have an optical dipole-like emission covering a solid angle of 4π [10]. This causes 50 % higher than in spin-entangled photon loss in the triangular cross-section waveguide resulting in low collection efficiency. An in situ mirror, capable of reflecting color center emission, should theoretically double the collection efficiency. It is possible to design such a structure artificially by periodic variation of the dielectric contrast, known as the photonic crystal [35]. For enhanced collection, we deliberately open up a gap possessing the NV center emission wavelength in the photonic bands, which is achievable by making a particular choice of PC parameters [36].
Waveguides support light propagation in a particular direction via total internal reflection.
We implement 1D PC in our triangular cross-section waveguide to introduce asymmetry for redirecting the color center emission. However, 1D PCMs have mostly been realized in conventional rectangular geometry [37][38][39], and photonics with triangular geometry has been focused on making efficient active photonic devices [7,22,40]. Photonic band structures in triangular geometry have not been studied in detail. In this light, our recent work provides an insightful explanation of the dispersion relations in SiC triangular geometry [23]. From the band structures obtained from Plane Wave Expansion (PWE) method, we use Finite-Difference Time Domain (FDTD) method to analyze the performance of the triangular-cross section 1D PCM and the dependence of the coupling efficiency to the waveguide mode on the emitter position with respect to the PCM.

Considerations for integrating color centers into triangular cross-section devices
Color centers in SiC have multiple orientations due to the presence of inequivalent lattice sites (h, k). The dipole orientations of the color centers can be parallel (silicon vacancy in SiC) or at an angle (NV center in SiC) to the crystal axis (c-axis). In a photonic device fabricated from commonly available 4H-SiC, grown along the c-plane, this would result in emissions strongly coupling to the fundamental TM (f-TM) mode [8]. On the other hand, it has been shown that using 4H-SiC grown along the a-plane, the color center emission can be predominantly coupled to the fundamental TE (f-TE) mode [10].
Triangular cross-section waveguides support f-TE/TM modes and other higher order modes [22]. The fraction of a color center emission coupling to a supported waveguide mode depends on the position of the emitter within the triangular cross-section. Highest coupling to a mode can be achieved by positioning the emitter at the maximum intensity point of that mode. The maximum intensity point of the f-TE mode is located approximately at the centroid of the triangle, which is an optimal color center position for achieving highest coupling efficiency to the f-TE mode [10,22]. The depth of the centroid from the surface depends on the width and etch angle of the triangular cross-section waveguide. These dimensions can be decided on the basis of the implantation capabilities, to ensure that the implanted color centers achieve the highest coupling efficiency to the f-TE mode. For a given emission wavelength, a triangular crosssection waveguide has an optimal width at which it has single mode propagation (f-TE), with high (> 80 %) coupling to the f-TE mode [10]. Single mode propagation in the waveguide is key for applications in QIP [41,42]. The optimal width of the triangular cross-section devices increases with the etch angle and emission wavelength. Hence, triangular cross-section devices with larger etch angles have more fabrication friendly dimensions. Moreover, they require shallow implantation depths, which causes less damage to the crystal and having an emitter closer to the surface is useful for SNSPD absorption. Advantages like single mode propagation with high coupling efficiency, less damage to the crystal and better detection in SNSPD outweighs the overall effect of the color center emission lost into the f-TM or other modes.

Waveguide design with TE and TM band gap
NV center emission in SiC has both TE-and TM-like components [43]. Therefore, a PCM with polarization-independence is necessary to increase collection efficiency. For broadband reflectivity, such a mirror must have as large a complete photonic band gap as possible [44,45].
Single mode propagation [41,42] and positioning of emitter at the centroid of the triangular cross-section [10,22] ensures that most of the color center emission is coupled into the waveguide. It was shown that 45°etch angle has predicted the best results for the complete band gap [23]. But in this study, an etch angle of 60°was chosen because the optimal emitter position is closer to the top surface, offering realistic implantation depths (70 nm) and efficient absorption in the SNSPDs. Moreover, a 60°etch angle supports single mode propagation for larger dimensions than in the case of 45° [10], resulting in a more fabrication friendly waveguide design. The width of the waveguide was set to 800 nm, as it supports the f-TE and f-TM modes only.
The 1D PC is made of cylindrical air holes in a triangular cross-section SiC waveguide as shown in Figure 2a. The parameters are as follows: periodicity along the y axis a, waveguide width w, hole radius r, and etch angle α, which corresponds to half-angle at the apex. The refractive index of SiC is considered 2.6 for all the simulations. In this study, we focused on the dispersion relations for α = 60°w.r.t w and r. We vary w from 1.2a to 2.25a and r from 0.25a to 0.45a. The band structure in Figure 2b exhibits three properties: i) a wide TE band gap, ii) a narrow TM band gap, and iii) a partial overlap between TE and TM band gaps, which is the complete photonic band gap. Figure 2c delineates the changes in the TE/TM gap (c/a) and the corresponding gap size (gap-midgap ratio) with r/a for w values of 1.2a, 1.35a and 1.5a. Midgap frequency increases with r/a for both TE and TM band gaps as volumetric fill factor (VFF) and effective refractive index (n eff ) have an inverse relationship with r/a [23]. Gap size initially increases with r/a as fewer propagating modes are supported, but declines after the maximum value is reached. Though this trend is observed for both TE and TM bands, the TM band gap completely vanishes for larger widths (w > 1.5a).
Moreover, the highest TE gap size (∼22 %) is an order magnitude higher compared to the highest TM gap size (∼2 %), which is attributed to the connectivity between high-dielectric regions [35]. In general, complete band gap mostly occurs for smaller widths with larger hole radii as depicted in Figure 2c. Nevertheless, the choice of parameters for designing the reflector is made in such a way that there is a balance between optimal band gap size and reproducibility of the nanofabrication.

SiC waveguide design with a PCM
In this section, we use the PC parameters from the PWE method and simulate a triangular cross-section waveguide with a reflector, using the Finite-Difference Time-Domain (FDTD) package in Lumerical [10,22]. This allows for a comparison of the band gap position estimated from the two different methods, PWE and FDTD. The FDTD method uses the Fourier transform of the time-domain signal to record the field data in the frequency domain. To increase the collection efficiency of the photons from the ZPL and PSB of the NV center in SiC, the PC parameters were chosen to have a band gap spanning these emission wavelengths.
As mentioned in Section 2.2, the size of the TM band gap (and thus the complete band gap) is significantly smaller than the size of the TE band gap, limiting the reflected wavelengths of both TE-and TM-like components of the NV center emission.
In this study, we focus on two sets of parameters for the PC 1) complete band gap and 2) NV optimized band gap, for a waveguide with w = 800 nm and α = 60°. In the first scenario,  As the number of air holes increases, the maximum reflectivity of the PCM increases and saturates for 20 air holes, as shown in Figure 3a-b. It is observed that the FDTD calculated wavelength for maximum reflectivity deviates from the center of the PWE calculated bandgap (grey region) [46]. The reflectivity curve has a full width half maximum of ∼200 nm, large enough to reflect the ZPL and the entire PSB of the color center emission. At 1230 nm, the NV optimized design has ∼85 % reflectivity for the f-TE mode. The reflectivity at a given wavelength depends on two factors: 1) the effective index of the f-TE mode (n eff ) and 2) the fraction of color center emission coupled into the waveguide. While n eff decreases monotonically with an increased wavelength, the coupling efficiency rises to the optimal point (designed at 1230 nm) and subsequently decreases. The combination of the two effects cause the reflectivity to gradually increase for the short wavelengths and sharply drop for the long wavelengths. In the case of the f-TM modes, the reflectivity values are relatively lower due to the low coupling efficiency to the f-TM mode (∼20 %) and evanescent loss to the surroundings (n eff = 1.09).

Integrated SNSPDs for efficient detection of light in triangular waveguides
For single-photon detection, SNSPDs [47] provide unrivaled performance metrics such as high detection efficiency, including at 1550 nm wavelength [48], high count rate, low dark-count rate [49], and high temporal resolution [50]. As such, they have found many applications in quantum communication and information processing as well as in high energy physics for charged particle detection [51][52][53]. Compared to the top-illumination of meander SNSPDs and the enhancement of their detection efficiency by use of an optical cavity with a mirror underneath [54,55], short waveguide integrated SNSPDs avoid coupling losses when being interfaced with on-chip devices and provide smaller recovery times as a result of their lower kinetic inductance [56,57]. Due to their superior performance metrics, in this section we study the potential of integrating SNSPDs onto triangular cross-section SiC waveguides to detect single-photon emission from color centers embedded in the waveguide. We start by investigating the fundamental properties of NbTiN films on SiC substrates and subsequently provide performance estimates for SNSPDs on SiC. Next, we use FDTD simulations to determine the coupling efficiency from triangular SiC waveguides to SNSPDs integrated on top of the waveguide and its dependence on the detector geometry.

Optical and transport properties of NbTiN thin films on SiC
In this section, we investigate the optical and the transport properties of NbTiN thin films on SiC substrates. For this, we used DC reactive magnetron sputtering at room temperature with an Ar/N 2 atmosphere and a sputtering target with the stoichiometry Nb 0.7 Ti 0.3 to fabricate the NbTiN films studied in the following.
For simulation of absorption in NbTiN SNSPDs on triangular cross-section SiC waveguides, the optical constants of the deposited superconductor are essential, especially because the optical and the transport properties vary with deposition parameters and the underlying substrates [58]. We used a variable angle spectroscopic ellipsometer, Model M-2000 of J.A. Woollam Co., to measure the optical constants over a wavelength range of 210 nm to 1687 nm.
As NbTiN is absorbing at all measurement wavelengths and one cannot resort to a transparent wavelength window, it is difficult to decouple and determine the optical constants and film thickness simultaneously due to correlations between both. However, this correlation can be overcome by measuring with the interference enhancement method, where the film is measured on an additional thick transparent dielectric layer and at multiple angles [59][60][61]. As we measured the NbTiN films on two-sided polished SiC substrates, two additional challenges emerge: The occurrence of reflections on the back side of the substrate, as well as the fact that SiC is a birefringent material. However, similar to the interference enhancement method, the anisotropy helps in decoupling thickness and optical constants for the NbTiN film [62].
Including backside reflections of the uniaxial 4H-SiC substrate in the model, we obtain a fitting of the measured data with a mean square error less than 5, indicating good agreement between the model and the measured data. The resulting optical constants, shown in Fig. 5, reveal a smaller extinction coefficient k and a higher refractive index n for the thicker film and provide the basis for the subsequent FDTD simulations. Compared to the optical constants of a 5.5 nm thick NbTiN film grown by sputtering on silicon substrates in the publication of Banerjee [59], we obtain similar results, although our data shows no features in the range from 500 nm to 1000 nm and our refractive index is~20 % and the extinction coefficient~50 % higher for a wavelength of 1680 nm. This agrees with the fact that the growth and properties of polycrystalline materials such as NbTiN depend on the underlying substrate. it is essential to know the device's critical temperature T c at which it switches from the superconducting to the normal conducting state. As this temperature is reduced for nanowires compared to thin films or bulk material [63,64], the critical temperature of the film provides an upper limit for reachable SNSPD critical temperatures. Furthermore, by measuring fundamental film transport properties such as sheet resistance R sheet , quasiparticle diffusivity D [65,66], and Hall coefficient R H , we can estimate key detector metrics such as depairing  [71,72], the elementary charge e, the resistivity in the normal conducting state ρ nc , and the quasiparticle diffusivity D by A higher switching current of the detector also means a higher amplitude of the detection pulse, which is beneficial for the signal-to-noise ratio.
Moreover, we can calculate the kinetic inductance from the sheet resistance R sheet , the energy gap ∆, and the nanowire width w and length l [65,73].
that can be used to estimate the fall time of a detection pulse [74,75] by with R load denoting the load resistor, which can be either a shunt resistor or the readout electronics (typically 50 Ω). From the fall time, we can also estimate the maximum count rate by 1/τ fall . In Fig. 6 [67,76,77]. However, it seems worthwhile to note that our data for the electron density is lower than the values found in the literature. Nevertheless, when comparing the data in [67,76], the electron density shows a dependence on the stoichiometry of Nb  (measured, e.g., in [74] as well as in our labs). We speculate that the fabrication process of detectors modifies the transport properties of the film such that the kinetic inductivity and thus the fall time are increased compared to the film-derived ones.

SiC waveguide
In  single photon detection efficiency decreases [78], which is critical for applications in QIP.

Discussion
The results presented in this paper demonstrate scalable approaches to efficiently collect and detect color center emission in triangular cross-section SiC devices for applications in QIP. Triangular cross-section devices are promising for chip scale integration of color center photonics. Moreover, these devices have a rich parameter space (etch angle, width) for optimal color center integration. At a given emission wavelength, there exists an optimal width for each triangular cross-section (with an etch angle alpha) such that the waveguide supports single mode propagation [10], which is important for applications in QIP. The polarization of the color center emission depends on the dipole orientation of the color center with respect to the c-axis. Using 4H-SiC grown along a-plane, most of the color center emission (silicon vacancy, divacancy, nitrogen vacancy) can be coupled into the f-TE mode.
Furthermore, the coupling of the color center emission can be improved by positioning the color center at the maximum intensity point of the f-TE mode, which is around the centroid of the triangular cross-section. Color centers are generated in SiC through implantation of high energy projectiles like electrons, protons, neutrons, ions. The implantation energy and the mass of the implanted particle determine the implantation damage done to the crystal, which can only be partially recovered through annealing [79]. Triangular cross-section devices can be designed depending on the implantation energies available, while still maintaining the single mode propagation and emitter's position at the centroid of the triangular cross-section.
The waveguide simulated in this work (w = 800 nm, α = 60°) has single mode propagation, with ∼44 % of the 50 % dipole emission (1230 nm) along one direction of the waveguide coupled to the f-TE mode.
The collection efficiency (∼44 %) on one end of the designed waveguide can be further improved using a photonic crystal mirror. The periodic variation of the dielectric constant (air holes) in a PC results in a photonic band gap for both TE-and TM-like modes in the waveguide. In a parallel work, a thorough photonic bandgap analysis using PWE method shows that the PCs can be used for a variety of applications and that their dimensions can be scaled depending on the color center emission wavelength [23]. In triangular cross-section PCMs, the TE bandgap is an order of magnitude larger than the TM bandgap, consequently a relatively much smaller complete bandgap. Therefore, a more effective approach for using the PCs as reflectors of c-axis oriented color center emission is to fabricate them in a-plane SiC substrates [10]. This would ensure that a large portion of any color center emission is coupled to the f-TE mode, reflecting frequencies spanning the ZPL and PSB of the color center emission. FDTD analysis shows that the position of the emitter relative to the PCM determines whether there is constructive/destructive interference in the waveguide region.
We highlight that using the PCM proposed in this paper, a randomly positioned emitter has a higher collection efficiency compared to a symmetric waveguide (∼44 %) for 75 % of cases.
Efficient detection is another important factor for boosting the success rates of QIP protocols. SNSPDs offer a scalable approach for integrating detectors onto photonic devices.
On-chip detection using SNSPDs can reduce optical losses, latency, and wiring complexity involved in off-chip detection [28]. As a basis for SiC waveguide integrated SNSPDs, we investigated, for the first time, the optical and electrical properties of NbTiN thin films for SNSPD applications on 4H-SiC substrates. While the absorption of the NbTiN films is significantly higher on SiC than on SiO 2 substrates (up to 50 % for photons of 1680 nm), the magneto-electrical properties of the films, and thus presumably also of the detectors, are similar. Provided that the fabrication of SNSPDs on SiC waveguides is of similar quality as on SiO 2 substrates, we expect that they provide comparable electrical performance and simultaneously enhanced system detection efficiency due to higher absorption in the film,

Conflict of Interest
The authors have no conflicts to disclose.

Data Availability Statement
The data that support the findings of this study are available upon reasonable request from the authors.