Ultra-compact silicon waveguide-integrated Schottky photodetectors using perfect absorption from tapered metal nanobrick arrays

: We propose and numerically analyze an integrated metal-semiconductor Schottky photodetector consisting of a tapered metal nanoblock chain on a silicon ridge waveguide. The metal-semiconductor junctions allow broadband sub-bandgap photodetection through the internal photoemission effects. The tapered array structures with different block widths can gradually tailor the cut-off frequencies and group velocities of the tightly confined plasmonic modes for enhanced light absorption and suppressed reflection of the photonic mode in the silicon waveguide. As a result, according to our simulations, six metal nanobricks with a total device length of 830 nm can almost perfectly absorb the incident sub-bandgap light and subsequently generate photocurrents with a peak responsivity value of 0.125 A/W at 1550 nm. We believe that the proposed design can provide a simple and viable solution for broadband and compact photodetection in the integrated silicon photonics platform.

Silicon waveguide-based Schottky PDs offer a strong confinement of SPPs near the MS interface with a controllable interaction length, thus enabling greater electromagnetic energy to be absorbed in the vicinity of the Schottky contact [13][14][15][16][17]. As a result, 1 μm-long compact waveguide-type Schottky PDs allow a reasonably high responsivity of 12.5 mA/W and a low dark current level of 30 nA at an operating wavelength of 1550 nm [13]. Casalino et al. exploited an asymmetric metal-semiconductor-metal structure and a vertical contact layer to effectively suppress the dark currents [18]. A few-hundreds-micrometer-long Schottky PD employing long-range SPPs has been demonstrated to further increase the responsivity up to ~0.1 A/W by enhancing the photonic-plasmonic mode coupling efficiency with a dark current level on the order of 1 mA [14]. Recently, a metal chain structure integrated with a submicrometer-silicon waveguide was proposed to realize an efficient and compact photonicplasmonic mode converter with a total device length on the order of one micrometer [20][21][22].
In this paper, we propose a design strategy to realize a compact and broadband, Schottky PD consisting of a chain of sub-micrometer-scale metal nanobricks integrated on a typical silicon ridge waveguide. A spatial gradient of polarizabilities from the tapered array of metal nanobricks provides a highly efficient photonic-plasmonic coupling mechanism as well as slow light effects for nearly perfect absorption with little reflection. Strong light confinement and absorption near the MS interface can efficiently generate photocurrents over the wide range of input frequencies. According to our simulation, the maximum absorptivity and the responsivity are 95% and 0.125 A/W, respectively, with a total device length of only 830 nm at an input wavelength of ~1550 nm under the applied reverse bias voltage of 3 V. Furthermore, the extremely small device footprint allows a high operation bandwidth of >40 GHz with a small dark current level of <10 nA. Our design strategy suggests that it is possible to achieve high responsivities and low dark current levels by taking advantages of the perfect absorption principle.   Figure 1(a) shows the schematic view of the proposed broadband photo-detection/absorption structure, which consists of a one-dimensional array of metal nanobricks placed on a silicon ridge waveguide. As an optical input port, we selected a p-type silicon ridge waveguide with a ridge width of Wr = 500 nm, a total height of H = 220 nm, and a slab thickness of Hsl = 90 nm as shown in Fig. 1(b). The access waveguide dimensions were selected to support only one fundamental TE mode for a wide wavelength range of 1450-1700 nm. We also assumed that the metal nanobrick array consisted of six gold (Au) rectangular nanobricks with an equal spacing of P = 150 nm, as shown in Fig. 1(c) (top view). The thickness (T) and short-axis length (S) of the metal nanobricks were fixed at 30 nm and 80 nm, respectively, and the dimension of i th nanobrick is Li × S × T. The width (Li for i = 0, 1, 2, 3, 4, 5) of each nanobrick increased linearly from 128 to 158 nm (Li + 1 -Li = 6 nm). The chain of metal nanobricks was assumed to be aligned along the center of the waveguide (x = 0 line in Fig. 1(c)). As will be explained later, the aspect ratio of the metal nanobricks and their spacing were determined in such a way that the phase-matched plasmonic mode is efficiently excited in the tapered metal chain region through evanescent coupling from the silicon waveguide's optical input.  In our configuration, the metal chain provides strong optical mode coupling between the plasmonic mode and the silicon waveguide's fundamental transverse-electric (TE) mode. The incoming optical energy from the silicon waveguide can be transferred to the metal nanobrick region and then gets absorbed near the metal-silicon interface [20]. Figure 1(d) shows a longitudinal cross-sectional view of the calculated electric-field intensity profile (|E/Einc| 2 where Einc is the incident electric field) when the fundamental TE mode at an input wavelength of 1550 nm (193.5 THz) is excited from the right side and propagates toward the + y direction. Unlike the previous metal chain designs with identical nanobrick widths [20][21][22], our tapered array exhibits nearly perfect absorption near the MS contacts and does not allow recoupling of the plasmonic mode back to the silicon waveguide's photonic mode. The electric field intensity is strongly localized near the MS interface of specific nanobrick positions depending on the input frequency. Moreover, the tapered array configuration helps in achieving broadband antireflection from the silicon waveguide's TE mode due to the gradual change of the effective refractive index, and can also gradually tailor the cut-off frequency for the propagating plasmonic mode to achieve resonance-enhanced light absorption [23,24]. As a result, we realized an ultra-compact perfect absorber design for sub-bandgap Schottky photo-detection that can absorb the incoming optical energy from the access waveguide and convert the absorbed photons to the photo-generated carriers within a very small form factor.

Operation principles
To understand how the guided photonic mode (fundamental TE mode) is localized to a specific metal nanobrick position depending on the input frequency without recoupling or reflection back to the silicon access waveguide, we first analyzed the dispersion relation of a non-tapered infinite array of identical metal nanobricks with a common width of L as described in Fig. 2(a). The other dimensions, such as the thickness (30 nm), short-axis length (80 nm), and block-to-block spacing (150 nm), remain same. Figure 2(b) shows an example of the dispersion relation for an infinite metal nanobrick array with a width of L = 140 nm, which corresponds to the width of the third block (L2) in Fig. 1. The dispersion relationship for an infinite metal chain was calculated by the three-dimensional finite-difference time-domain (FDTD) method (FDTD solutions, Lumerical Inc.) with a periodic boundary condition along the wave propagation (y) direction. The complex refractive index data for Si, SiO2 and Au materials were taken from the literature [25,26]. Three white-dashed lines in Fig. 2(b) represent the light lines for air (left dashed line, nair = 1), silicon dioxide (middle dashed line, nSiO2 = 1.5), and silicon (right dashed line, nSi = 3.48). There exist two mode branches between the light lines of silicon dioxide and silicon. The upper solid curve corresponds to the silicon waveguide's TE mode, while the lower curve represents the plasmonic mode. The splitting of the dispersion curves results from the anti-crossing phenomenon caused by strong coupling between the two propagating modes [20,27]. Our highly-confined waveguide supports only a single plasmonic mode at a given operating wavelength range. Figures 2(c) and (d) show the cross-sectional electric field intensity profile for the silicon waveguide's fundamental TE mode and the plasmonic mode from the metal chain with a constant width of L = 140 nm (L2) at λ = 1530 nm, respectively. The mode mismatch between the photonic and plasmonic modes basically gives rise from the group velocity mismatch. To relieve the mode mismatch, we need to tailor the group velocity of each metal chain.
Figures 2(e)-(g) show the dispersion relations for the non-tapered metal chain with various nanobrick widths as well as the fundamental TE mode for the non-disturbed silicon ridge waveguide. The dispersion curves for the plasmonic modes (circular symbol curves in Figs. 2(e)-(g)) become flatter near the cut-off frequencies, which are strongly dependent on the nanobricks' width, L. Therefore, as the metal nanobrick width increases, the overall dispersion curves shift downward, and the amount of the momentum mismatch between the silicon waveguide's TE mode (triangular symbol curves in Figs. 2(e)-(g)) and the non-tapered metal chain's plasmonic modes increases at a given frequency. This discrepancy results in reduction of the coupling efficiency between the TE and the plasmonic modes. For example, at an input frequency of 193.5 THz, indicated as a red horizontal line in Figs. 2(e)-(g), the guided TE mode from the silicon waveguide can be more efficiently coupled to the metal nanobrick arrays with narrower widths because of smaller momentum mismatch. Moreover, the plasmonic modes for the non-tapered metal chain with a constant nanobrick width of L = L4 is cut-off and cannot propagate as shown in Fig. 2(g). We also would like to note that the group velocity for the plasmonic mode, which can be obtained from the slope of the dispersion curve, decreases with the nanobrick width at a given input frequency as the dispersion curves become more flat toward the cut-off frequencies.
The insights gained from the dispersion analysis of the infinite metal nanobrick array case can be applied to qualitatively explain the coupling behavior of the tapered array. For example, when the silicon waveguide's fundamental TE mode is excited at 193.5 THz for the tapered array structure shown in Fig. 1, it is initially coupled to the metal nanobricks at the front region of the array. Because of relatively small momentum mismatch between two modes, the amount of reflection to the silicon waveguide is low. As the coupled electromagnetic energy propagates toward the wider metal nanobrick region, it gets slowed down because the group velocity decreases with the metal nanobrick width. The electromagnetic field would be eventually reflected when it encounters the metal nanobrick with a width of L4 because the plasmonic mode in the non-tapered metal chain with a width of L = L4 is cut-off at 193.5 THz and cannot propagate further. The reflected field is then tightly confined and dissipated around a cut-off position since the group velocity is significantly reduced. This tapered configuration therefore supports position-dependent resonances for different input frequencies, and results in resonance-enhanced light absorption for a broad frequency range which depends on the nanobricks' size distribution. In addition to the analogy with the non-tapered array case, we also directly analyzed the propagation properties of the tapered metal nanobrick chain by calculating the group delays and group velocities from the scattering parameters. We first calculated the phase difference, φPD, between the input and output plane of the tapered array structure with a total length of 830 nm, and obtained the group delays for the transmitted light signal (τg = dφPD/dω). The group delay from the tapered array and the corresponding group velocity, vg, experiences significant variation over the frequency range of our interest, and even negative values were observed as shown in Figs. 3(a) and (b).
We would like to note that such negative group velocities are mainly resulted from the reflection at the mode cut-off positions, and this phenomenon is a typical feature of light propagation in slow light waveguides and anisotropic metamaterials [23,24,28,29]. For example, slow light waveguides consisting of tapered anisotropic metamaterials have been exploited as broadband perfect absorbers [23,24]. The tapered anisotropic metamaterials supports position-dependent resonances for broadband input frequencies, and thus allows broadband perfect absorption. In our proposed structure, the tapered metal nanobrick array acts as a broadband perfect absorber similar to the anisotropic metamaterials scenario. Once coupled to the tapered metal nanobrick array, the incident guided electromagnetic wave is tightly confined near a specific resonance position and then efficiently absorbed under the similar principle of anisotropic metamaterial perfect absorption [23,24]. As we will later, such broadband high absorption within a very small footprint results in a compact waveguideintegrated photodetection mechanism with a relatively good responsivity over a wide range of input wavelengths.

Results and discussion
To confirm that the silicon waveguide's fundamental TE mode is indeed trapped within the tapered metal nanobrick array structure shown in Fig. 1, we calculated the magnetic field magnitude distribution (|Hz/Hinc| where Hinc is the incident magnetic field) at several input frequencies. Figures 4(a)-(c) show the top views (xy plane) of the magnetic field profile measured at the middle height of the tapered metal chain (z = 235 nm) for the input wavelengths of 1475 nm, 1550 nm, and 1600 nm, respectively. It is seen that the silicon waveguide TE mode is efficiently coupled to the tapered metal nanobrick array and that the electromagnetic field is concentrated around a cut-off position on the tapered array depending on the input frequency/wavelength. For instance, Fig. 4(a) shows that the guided input at a higher frequency (203.4 THz, λ = 1475 nm) is concentrated near the front region of the tapered array with smaller metal nanobricks, and the field distribution becomes weaker toward the end of the array due to the effective cut-off of the propagating plasmonic mode for the wider metal nanobricks. On the other hand, at a lower frequency (187.5 THz, λ = 1600 nm), the guided mode is mainly concentrated at the rear region of the array with wider metal nanobricks as shown in Fig. 4(c).  The trapped electromagnetic field is strongly absorbed within the metal nanobrick array, especially near the MS interface as shown in the inset of Fig. 4(d). To estimate the amount of optical absorption, we calculate the absorptivity of the tapered metal chain when the fundamental TE mode is excited from the silicon ridge waveguide. Figure 4(d) shows the absorptivity spectrum. We obtained high absorptivity values for a wide frequency range for the tapered array design illustrated in Fig. 1. The maximum absorptivity was estimated to be approximately 95% at ~193.5 THz (~1550 nm). Although not explicitly shown here, it is also possible to modify the array design parameters, such as the number of nanobricks and width distributions, to tailor the absorptivity spectrum.
In our previous discussion, we demonstrated that high absorptivity can be achieved over a short distance (<1 μm) due to the slow light effects in the tapered metal chain. The responsivity of a Schottky PD can be estimated from the Schottky barrier photo-detection models. Generally, the PD responsivity can be obtained from its internal quantum efficiency (IQE) and the absorptivity (A), and is given by where q is the elementary charge, h is Planck's constant, and v is the frequency of the incident photon. The IQE of Schottky photo-detection (mainly IPE process) indicates how many of the hot carriers can be transferred over the potential barrier (ФB) into the silicon ridge waveguide. In our design, a p-type silicon ridge waveguide was chosen because the p-type Si/Au Schottky contact has a lower barrier height of ФB = 0.34 eV when compared to the n-type Si/Au contact [30]. To estimate the IQE, we also considered the variation of the Schottky barrier height resulted from the image charge at the MS interface when the reverse bias is applied. In reality, the barrier height lowering (ΔΦB) of our complex 3D case might be different from that of the 1D model, and the actual value can be slightly different. For example, even if ΔΦB is reduced from 0.019 eV to 0 eV (this corresponds to the case when the Schottky barrier lowering is not considered), the IQE would be reduced from 0.101 to 0.100 at the wavelength of 1550 nm. The barrier height lowering for the 1D case is given by [30] where εsi is the silicon permittivity, Wdep is the depletion width, and Vb is the applied bias voltage. Assuming that the doping concentration of p-type silicon is relatively low at 2 × 10 13 cm 3 and the distance between the anode and the cathode is 1 μm, Wdep becomes larger than the anode-to-cathode electrode distance. Thus, Wdep can be approximated to the distance between the anode and cathode metal contacts. The value of ΔΦB is calculated as 0.019 eV at the reverse bias voltage of 3 V, and the corresponding Schottky barrier height is therefore 0.321 eV, which is not significantly different from the original barrier height. We obtained the IQE by following the thin-film Schottky barrier PD model proposed by Scales [31]. This model is developed from the photodetection model by Elabd and Kosonocky [32] for thin metal film Schottky PDs, where the generated hot carriers with isotropic momentum distribution were assumed to get reflected at the internal metal surfaces, and the inelastic scattering mechanisms were considered via a hot carrier attenuation length. Scales et al. extended this model by considering the double Schottky barriers, and verified the extended model with the measurement results [31]. This model has been adopted in a number of previous works on the waveguide-integrated Schottky PDs [16,33]. It should be noted that this model is strictly valid when the temperature approaches 0 K. However, Casalino proved that the proposed IPE model at 0 K could be considered a good approximation at room temperature within 1% when hνΦB  0.46 eV [34]. Since the Schottky barrier of Au/p-Si contact is ΦB = 0.321 eV at a reverse bias of 3 V for our situation and the photon energy is larger than 0.8 eV for the near-infrared light whose wavelength is near 1550 nm, the energy difference becomes sufficiently large (hνΦB~0.48 eV), and the estimated IPE quantum efficiency remains still valid at room temperature. Our calculation results considering only single Schottky barrier case are shown in Fig. 5(f). For comparison with other simpler Schottky PD designs, we calculated and compared the absorptivity values for the proposed tapered metal chain structure, the metal strip, and the metal

Responsivity (A/W)
band with the same device length and thickness on the silicon waveguide. Figure 5(a) shows the schematic views of the three different Schottky PD designs with the same silicon ridge waveguide geometry. Unlike the tapered chain design, significant portion of the input optical energy gets reflected or scattered for the metal strip and band designs due to the abrupt change of the propagating mode, as shown in Fig. 5 (b) and (c), and thus the input optical energy does not contribute to the photocurrents, resulting in lower responsivity values. For the tapered chain design, as shown in Fig. 5 (d) and (e), the absorptivity is much higher than the other structures and the absorbed power density shows field confinement in the vicinity of the Au-Si interface at the wavelength of 1550 nm (193.5 THz). As a result, the responsivity of the tapered chain structure is much higher than that of the other structures as shown in Fig. 5 (g). With the tapered metal chain design, the maximum responsivity of 0.125 A/W was achieved at 193.5 THz. To collect the photo-currents generated from separated metal nanobricks, the metal blocks need to be connected with each other to form a common cathode as schematically described in Fig. 6(a). We assume that the connection metal line is located at the center of the array and its width (W) is much narrower than the nanobrick widths (W<<Li). When the fundamental TE mode is evanescently coupled from the silicon waveguide, the light polarization within the metal nanobricks is also oriented along the x direction [20][21][22]35]. Since the resonance is mainly dependent on the metal nanobrick width, the overall optical properties of the connected metal chain should be similar to those of the discrete tapered metal chain. As shown in Fig.  6(b), the electromagnetic field is still localized at the MS interface between the silicon and the metal nanobricks even in the connected chain case, and the overall electric field intensity distribution profile is nearly identical to the discrete case previously shown in Fig. 1(d). The absorptivity spectra for the connected array case also show the similar trend when compared with the discrete case (Fig. 6(c) and Fig. 4(d)). For the connected metal chain, we expect that the absorption peak is blue-shifted due to the finite width of the connection line. As indicated in Fig. 6(c), the amount of blue shift increases with the connection line width, W. Another important parameter for PDs is the amount of dark currents, which ultimately determine the photo-detection sensitivity as well as the minimum detectable optical power. The dark current from a Schottky PD can be obtained by [30] ( where Carea is the metal-semiconductor Schottky contact area, A** is the effective Richardson constant (32 Acm 2 K 2 for holes in Si), kB is the Boltzmann constant, and T is the temperature (T = 300 K at room temperature). The effective Richardson constant depends on the metal thickness and metal deposition method, and therefore can affect the overall device performance [36]. Because of its extremely small footprint, the dark current was estimated to be as low as 4.46 nA at a reverse bias of 3 V. The total area for the proposed MS Schottky contact is only 0.083 μm 2 , which is a very small value when compared to the previously reported Schottky PDs [37]. Based on the estimated responsivity and dark current values, we can obtain the normalized photocurrent to dark current ratio (NPDR = Responsivity/Idark), which is another important figure of merit to compare the PD's sensitivity performance [38]. The estimated NPDR of the proposed PD is 2.8 × 10 4 mW 1 , and it is larger than typical waveguide-based Schottky PDs. To evaluate our work in broader contexts, we compare our PD's performances to the previously reported experimental results as summarized in Table 1. The 3-dB bandwidth (f3dB) of a PD is typically limited by its RC time delay and the transit time of hot carriers between two electric contacts. The RC-time-limited 3-dB bandwidth is given by fRC = 1/(2πRC), where C is the capacitance (C = εsi Carea /Wdep) and R is the typical load resistance of 50 Ω. Since the Schottky contact area is very small in our proposed structure, the estimated fRC is very large on the order of THz. At the applied reverse bias voltage of 3 V, the transit time bandwidth is given by ftransit = 0.44vsat/Wdep, where vsat is the effective carrier saturation velocity (~10 7 cm/s in Si) and Wdep is the depletion width (equivalent to the distance between the contacts, ~1 μm) [30]. The estimated ftransit is ~44 GHz, and this limits the overall operation bandwidth of the proposed PDs.
Although the silicon waveguide-based Schottky PDs have been demonstrated before [13][14][15][16][17][18], our configuration suggests that it is possible to achieve high responsivities over a broad wavelength range within a very short device length (<1 μm) by taking advantages of the perfect absorption principle. Our design also provides low dark current levels due to its small MS contact area. Because the tapered metal chain consists of multiple metal nanobricks, it is also possible to manipulate the operating frequency ranges by simply varying the number and/or the dimension of the metal nanobrick elements.

Conclusions
In summary, we proposed the perfect-absorber-based waveguide-integrated Schottky PD designs with high responsivities over a broad input frequency range. Realizing near-unity absorptivity within a tapered metal nanobrick chain allows efficient IPE and achieves high responsivity values of up to 0.125 A/W at ~193.5 THz. According to our simulations, the slow light plasmonic modes in the tapered metal chain structure also enable a small device length (<1 μm) and low dark current levels (<10 nA). We believe that our ultra-compact silicon waveguide-based Schottky PD design can provide a solution to develop sub-micrometer-scale high-speed broadband photo-detectors in the integrated silicon photonics platform.