1. Introduction

The ever-increasing demand for high speed data communications and the imminent breaching of the fundamental physical laws that limit the size (and consequently performance) of electrical circuit elements has motivated extensive research in the field of photonics in recent years [1,2]. Silicon based optical interconnects have emerged a promising solution to the communication bottleneck by virtue of their low intrinsic losses, high bandwidths and cost advantages. Consequently high speed monolithically integrated photodetectors are increasingly sought-after as the receiver end of high performance optical circuits and interconnects [3–6].

In particular, we consider the design of state-of-the-art normally incident (NI) germanium (Ge) photodetectors. A significant part of the challenge of realizing high speed, bandwidth and responsitivity photodetectors arises from the inherent tradeoffs that occur between these performance parameters: for NI devices where the light propagation direction is parallel to that of photocarrier motion, a large bandwidth implies a short carrier transit time and low device capacitance, which necessitates low device thickness; on the other hand, a thin device sacrifices external quantum efficiency (overall light absorption), resulting in low responsitivity values [3,7]. These facts are especially relevant for Ge devices: for example, the relatively low absorption coefficient of ~460 cm⁻¹ for intrinsic Ge at 1550 nm requires film thicknesses on the order of microns for sufficient light absorption, which results in extremely low 3-dB bandwidths × efficiency figure-of-merit (FOM) of several GHz [3].

Efforts to increase detector performance have primarily been focused on increasing the external quantum efficiency (i.e. external light absorption) for a thin film of given thickness. Similar to methods of enhancing thin film photovoltaic cells, typical techniques to increase light absorption include the introduction of plasmonic structures such as metallic gratings or nanoparticles, which generate intense, strongly localized near fields (near field enhancement) [8–13]; the use of periodic metal or dielectric elements that couple NI light into guided modes that propagate parallel to the plane of the active layer [9,10,14,15]; or the incorporation of backreflectors, anti-reflection coatings, optical cavities or some combination thereof to suppress transmitted and reflected power [7,9,16–20].

Here we propose an alternative technique of enhancing the photodetection capabilities of Ge for normally incident light at 1550 nm using dielectric grating structures that, somewhat counter-intuitively, strongly couple the excited, guided modes in the Ge layer to external radiative modes. This can also be understood as the chip-scale generalization and scaling up of the 1D leaky mode resonance phenomenon in semiconductor nanowires that have been reported in the literature [21]. We exploit the fact that these coupled leaky mode resonances generate enhanced near fields which possess a large spatial extent compared to their plasmonic counterparts, resulting in significant interaction and modal overlap with the absorbing Ge film. By optimizing necessary parameters and using full-field electromagnetic simulation techniques, it is found that a 200 nm thick patterned Ge layer can absorb up to 43% of NI light at 1550 nm, over an order of magnitude larger than a pristine film of equal thickness (~4%). This is also broadly comparable to enhancement factors achievable using plasmonic structures [8–15].

A further advantage of the proposed design is the relatively slow rate of scaling of absorption enhancement factor with active (grating) area, due to the nature of the radiative “leaky” near-fields. A significant enhancement factor is still possible even for a grating area substantially smaller than the incident beam spot size. We accentuate this superior absorption-versus-area scaling of the device by contrasting the results with a reference case comprising an optimized, low loss Ag plasmonic grating on a Ge film of equal thickness: it is found that the former exhibits on average 50% greater absorption than Ag gratings under similar conditions. Thus even with CMOS compatibility issues aside the proposed design enables a significant reduction of active device area, suppressing noise and increasing the bandwidth. In particular, we find a greatly improved theoretical 3 dB bandwidth$\times$efficiency FOM of ~58 GHz, an approximately fourfold improvement over typical NI detectors and comparable to waveguide photodetectors [3].

2. Proposed structure and design

The proposed design is shown in Fig. 1. It comprises of a straightforward grating structure formed by partially etching (50 nm depth) square holes into a thin (200 nm) Ge film (see Fig. 1(a)). The square “mesh” geometry was chosen for polarization independence under normal incidence conditions. The dimensions shown in Fig. 1(b) are optimized for maximum light absorption at the target wavelength of 1550 nm; more details are presented in subsequent parts of this paper. For the purposes of this study and without loss of generality, we consider a simple Ge-on-insulator (GeOI) structure comprised of a partially etched Ge thin film overlaid on buried oxide layer on a wafer substrate. The design can be easily adapted to accommodate, for instance, a Ge-on-Si architecture, with accompanying changes to the optimized design parameters due to different sub- and superstrate refractive indices.

 

Fig. 1 (a) Rough 3D schematic of active region of the proposed design. Here for simplicity, and without loss of generality we consider a Ge –on-insulator (GeOI) structure, consisting of partially etched gratings in a 200 nm thick Ge film on buried oxide. Definitions for angled incident light and associated polarizations are also shown in the coordinate system above. (b) Top view of one unit cell. The optimal design parameters under conditions for this particular study are shown: the period of the cell is 700 nm and the etched area is a square of side 410 nm. (c) Cross-sectional view showing relevant parameters in greater detail: a Ge film thickness of 200 nm and etch depth of 50 nm is found to yield maximum absorption at target wavelength of 1550 nm.

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We emphasize that the entirety of the proposed design is fully compatible with standard CMOS processes. High quality GeOI wafers are now commercially available and possess excellent uniformity and exceptionally low threading dislocation densities [22]; similarly there has also been significant progress for achieving mono-crystalline Ge films on Si via epitaxial growth [23–25]. Additionally the 410/700 nm feature sizes of the array (miniumum inter-feature size of 290 nm) are easily achievable using existing KrF 248 nm deep UV optical lithography, and Ge can be etched using standard fluorine-based reactive ion etching (RIE) processes with great selectivity against Si [26]. Finally a zero-bias charge separating P-I-N configuration with a depletion zone spanning the thickness of the intrinsic, light-active region can be straightforwardly achieved by selective doping of the Ge film and/or substrate [27–29].

3. Theory and optimization

The concept of exciting waveguide modes from incident light and subsequently coupling these modes to free space using a single index and phase-matched grating layer is not new: this phenomenon is known as guided mode resonance (GMR) [30,31]. Such structures have seen significant applications in the form of bandpass filters [32], and various forms of sensors [33,34] due to the very narrow linewidths (typically tens of nm) and high Q and Purcell factors that can be obtained with these resonance modes. Here, instead of focusing on manipulating the reflected and/or transmitted intensities, we exploit the locally enhanced near fields, which, despite having a significantly weaker intensities than their plasmonically generated counterparts, possess a considerably larger spatial extent due to the nature of the “leaky” mode resonance [35]. Thus by optimizing relevant structural parameters accordingly, we find that these electric fields can also significantly enhance absorption by over an order of magnitude in thin film absorbers.

Since the resonance modes arise fundamentally from guided modes in the Ge mesh structure which exhibits 2D periodicity, we can adopt a band structure formalism similar to that commonly used in the analysis of photonic crystals [36] to help us estimate initial device parameters (such as overall film thickness and grating periodicity), and also to gain an intuitive understanding of these modes. For simplicity here the corrugated Ge film is approximated as a continuous slab with a homogeneous effective index n_eff, obtained from averaging the index contributions of Ge and the superstrate (air in this case). The relevant E-k dispersion relations are therefore [37]:

The resultant band diagram for the 1st BZ under normal incidence $\left(\theta =\varphi =0\right)$is shown in Fig. 2(a). Here the periodicity is 710 nm, overall Ge film thickness is 200 nm and the grating etch depth is 50 nm, which are optimized parameters obtained from rigorous electromagnetic calculations using finite difference time domain (FDTD) techniques. The size of the grating was taken into account in the calculation of the averaged index of the structure (which was found to be 3.98, compared to 4.3 of unpatterned intrinsic Ge). Reasonable agreement is found (with deviations of ~10-15%) between the predicted resonance positions from the band diagram (the intersections of the dispersion lines with the $\Gamma$axis in Fig. 2(a)) and those resulting from FDTD calculations, demonstrating that the homogeneous slab and average index approximations can be used together with analytical theory (Eqs. (1)-(3)) for initial, ballpark estimations of required device parameters. Here we note that while analytical theories such as scattering matrix methods can be employed to calculate the dispersions of these Fano modes to excellent detail [38], it detracts from our primary purpose here in using the simplified band picture as an initial estimate of device response.

 

Fig. 2 (a) Band diagram in the 1st BZ constructed by approximating the partially etched structure as a homogeneous slab with an averaged index, and using Eqs. (1)-(3) together with Bloch boundary conditions. “Guided” modes lie above the light line and are thus coupled Fano resonant modes. (b) FDTD simulation of spectra, response of the optimized design. The maximum absorption at target wavelength of 1.55 is ~43%, an enhancement of ~11 times compared to pristine Ge of equal (200 nm un-etched) thickness. The absorption of a pristine Ge film of equal thickness is shown as the green dashed line, (c) FDTD derived, source-normalized near field intensities of the y-z plane cross-section at 1550 nm. (d) The surface of a unit cell of the structure at 1550 nm. (e) y-z plane cross section electric field for an optimized, low loss Ag grating structure on Ge. White dotted lines trace the boundaries of the patterned structure.

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Furthermore, from this simplified band diagram we see that all TE and TM “guided” discrete modes in this structure lie above the light line, indicative of coupling with the free space continuum, resulting in the distinctive Fano resonance lineshapes and “leaky mode” field profiles shown in Figs. 2(b) and 2(c) respectively. In particular, it is observed that the absorption spectrum in Fig. 2(b) has a fundamental resonance mode at approximately 1650 nm, corresponding to band 1 in Fig. 2(a), and a stronger, second order peak at the targeted 1550 nm (band 2 in Fig. 2(a)), which provides an enhancement factor of ~11 over the absorption of a pristine 200 nm thick Ge layer (green dashed line in Fig. 2(b)).

To better understand the mechanism of the enhancement associated with these GMR grating structures, we consider the near field profiles of the structure at 1550 nm (Figs. 2(c) and 2(d)). Crucially, the enhanced electric field intensities of the radiative modes extend for up to approximately 100 nm beyond the surface of the grating (Fig. 2(c)), resulting in significant interaction length with the Ge film. In contrast, the near field enhancement from typical plasmon-based enhancement techniques using metal gratings, while generally much more intense, usually extend for only up to ~10 nm [8,12,13] due to the intrinsic, strongly confined nature of the surface plasmons. As a reference, we performed similar optimization and calculation procedures for a low loss silver (Ag) plasmonically active grating structure ($\Lambda$ = 430 nm, W = 110 nm, d = 30 nm) on the same 200 nm thick Ge film at 1550 nm, and find 40% absorption in the Ge layer, compared to 43% for the GMR structure; detailed results are presented in a later section. The same near field cross-section plot is shown in Fig. 2(d); results are in agreement with the afore-referenced literature. Therefore despite the higher values of field enhancement that are obtained with plasmonic metal gratings, it is possible to achieve comparable net absorption values using dielectric grating couplers that also operate based on a near-field enhancement principle, which was in turn a key motivation for this study. Nevertheless we caution that for such dielectric structures to be viable, especially when compared to their metallic counterparts, the active layer needs to be sufficiently thick (on the order of the leaky mode field decay length); for active layers which are very thin, on order of the plasmon near field decay length, metal structures are unarguably superior.

From the basic working principles behind the GMR structure outlined previously, we can understand the dependence of overall absorption enhancement (i.e. the ratio of enhanced versus original absorption for a film of equal thickness) on the various parameters, such as grating period, etch depth and patterned width. Figure 3 illustrates the influence of various parameters on this enhancement. In Fig. 3(a), the peak positions are observed to redshift with increasing periodicity, as is expected since the periodicity directly affects the magnitude of the coupled in-plane wavevector (Eq. (2)). A larger contribution from the grating results in fulfillment of the periodic grating conditions and a resonance feature at smaller incident wavevectors, i.e. a redshift in spectral peak position. A similar, albeit inverse correlation between peak position and grating size (represented as the width of the square array W) is observed in Fig. 3(b): larger pattern sizes result in a blueshift of peak positions, and vice versa. This can be understood by returning to the simplified band diagram in Fig. 2(a): the change in area fill factor translates to a change in the effective index of the structure, which shifts the slopes of the dispersion lines and hence, the intercepts. Specifically, an increase in fill factor lowers the effective index; dispersion lines therefore increase in slope and intersect the Bloch unit cell boundaries at higher energies, resulting in a blueshift of resonant energies. In Figs. 3(c) and 3(d) we plot the enhancement factor at 1550 nm (the ratio of absorption of patterned against unpatterned Ge films of equal thickness) against grating depth, film thickness, and width respectively. This parametrization was broken into two sequential steps instead of a single 3D step for clarity, and also because the thickness parameter can be largely considered to be an independent variable practically (typically limited by device speed, dark current, or fabrication process concerns). Intuitively, the optimal enhancement factor for the design should occur for film thicknesses similar to the resonant near-field decay length (~100 nm as shown in Fig. 2(c)): this is corroborated by Fig. (3), where little enhancement due to the leaky mode resonances occur for thicknesses larger than approximately 250 nm and below a grating depth of approximately 0.6. Somewhat surprisingly, we observe significant enhancements for grating depths above 0.6 for numerous thicknesses, up to approximately 350 nm: this occurs due to the coupling of incident modes into guided modes in the structure, as opposed to the leaky, radiative modes that are present at smaller depths. This phenomenon can be understood by considering the overall device structure as consisting of a fully etched “grating layer” overlaid on a slab waveguide; in order to couple guided modes into free-space radiative modes, the grating layer must be index and phase matched with the propagating modes inside the waveguide. Altering the grating depth destroys the matching process, resulting in true guided modes with varying degrees of confinement; since a significant part of the incident light is coupled into highly confined guided modes propagating along the slab waveguides, this fraction can be considered to be fully attenuated, giving rise to the generally high enhancement values observed for grating depth > 0.6 in Fig. 3(c). This interpretation of a leaky-to-guided mode transition modulated by the grating depth has been verified by studying the mode profiles of the structures which are not shown.

 

Fig. 3 Absorption spectra as a function of (a) grating period and (b) grating width, cyan lines (periodicity of 700 nm and width of 410 nm) represent optimized absorption at 1550 nm. Successive curves are offset by 0.5, and dashed lines are visual guides for the tuning of the resonance peaks. (c) Contour plot of enhancement factor as a function of grating depth and film thickness, and (d) as a function of grating depth and grating width. Data sets for (c) and (d) have a resolution of half a major tick length, and have been smoothed to fit their respective contours.

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The trend for Fig. 3(d) can be understood in exactly the same manner: the increasing size of the patterned area across the bottom axis lowers the effective index of the grating layer, which is compensated for by a reduced grating depth to maintain the necessary conditions for coupling light out of the structure. A leaky mode “branch” where significant enhancement factors occur for various pattern widths is thus visible in Fig. 3(d) as moving diagonally down towards the right of the bottom axis. Similarly, the reverse process occurs for grating depths above 0.6, where the enhancement is due to guided modes: two distinct branches corresponding to different guided mode orders can be identified, pointing diagonally upwards to the right.

From these figures we can identify the optimal device parameters as being a 200 nm thick film of periodicity 700 nm, patterned width 410 nm and an etch depth of 50 nm (for an etched fraction of 0.25), as presented at the beginning of this paper.

4. Device performance metrics

As mentioned in the introduction, the overall absorption of a photodetector device is but one aspect of the performance metric; a better indicator is the bandwidth-efficiency product, where gains in efficiency are balanced against device speed. For our proposed design, we thus seek to characterize device performance as a function of its area (or equivalently, the number of periods for a given source spot size). We find that a significant percentage of the enhancement is retained in very few periods, whose total area is approximately equal to the beam spot size (on the order of a few microns). In particular, we contrast the results against the same reference case of an optimized Ag grating overlaid on Ge, and find the absorption of the latter structure to be significantly (~10% in absolute terms) smaller for the same number of periods. This improved scaling of absorption efficiency versus active device area results in a significant reduction of active device area, suppressing noise and increasing the bandwidth; consequently we report a theoretical bandwidth-efficiency product of ~58 GHz, a figure comparable to (the theoretical results of) waveguide photodetectors. We also explore the variation in absorption with the source angle of incidence and find that the targeted 1550 nm resonance peak remains essentially unperturbed up till large incidence angles for TM polarized light, since the resonance arises from the coupling of TE guided modes to free space (Fig. 2(a)). This implies that the device could be used for polarization-sensitive photodetection applications as well by controlling the nature of the optically active mode at a desired wavelength.

Figure 4 presents the results of the parameterization of absorption against the total number of periods in the device for a normally incident plane-wave light source 5 × 5 μm wide, for both the dielectric and Ag gratings. The polarization direction was arbitrarily chosen as TE (here, electric field parallel to x-axis) since the device structure is polarization insensitive at normal incidence. For the former structure, beginning with the infinitely periodic case and progressively reducing the total number of periods, it is found that for a sufficient number of periods (approximately >7, for a total area of approximately 5 × 5 μm²) within the beam spot the absorbed light intensity is almost the same as the infinitely periodic case (~37% vs 43%). It is only when the number of periods approaches the beam spot size that absorption begins to noticeably decrease. For example, one is still able to achieve 25% absorption for 5 grating periods, or approximately 12-μm² active area. In contrast the absorption of the optimized Ag grating structure for the same area is only about 13%, i.e. half that of the Ge structure.

 

Fig. 4 (a) Parametrization of Ge absorption enhancement (against reference absorption of 4%) against the number of periods at 1.55 μm at normal incidence. Solid line corresponds to the proposed GMR structure; dashed line corresponds to an optimized Ag plasmonic grating overlaid on equal thickness of Ge. The Ag grating is found to exhibit significantly lower absorption (~10%) for the same number of periods. (b) Cross-sectional and top view electric field profile diagrams normalized to a common source intensity at 1.55 μm. Source dimensions and structural features are outlined in white dashed lines. Note that the resonant modes decay (due to material and radiative losses) in just a few periods.

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This somewhat surprising result can be understood by considering that under normal incidence, light can interact with the GMR structure in only 3 possible ways: (1) normal transmission (after reflection) and consequently attenuation through the structure; (2) coupling into guided modes which are then attenuated as they propagate along the ridges of the structure; (3) coupling into guided modes and re-emitted into free space upon scattering off each subsequent grating encountered. Therefore as long as the number of periods exceed the spot size, the overall absorption of the device remains approximately constant since (2) and (3) remain essentially unchanged as significant decay of the field intensity is observed in just a few periods (Fig. 4(b)), and (1) is almost independent of the number of periods due to the very small absorption coefficient of Ge (field enhanced regions are also strongly centered around the gratings in x-y space). In particular, it can be observed from Fig. 4(b) that the average propagation length of the quasi-guided modes in the structure is just 2 or 3 periods as a result of intrinsic material losses and the strong coupling to the radiative modes. Conversely for the Ag grating, the source of enhancement is the presence of confined surface plasmons which span the Ag grating surface (for a beam spot size equal to or larger than active area). Thus any reduction in effective area directly decreases the amount of interaction of the incident light with the surface plasmons.

The key implication of this result (relatively high absorption with small device area) can be understood by considering the bandwidth-efficiency product FOM. We evaluate the industry standard 3 dB bandwidth by considering both carrier transit-time limited and capacitance limited bandwidths [3,7]:

 

Fig. 5 (a) Transit time and capacitance limited bandwidth as a function of depletion zone thickness. Different device areas with their corresponding number of grating periods are shown. (b) Calculated bandwidth-efficiency products for various types of Ge NI detectors. Relevant data is obtained from Ref [3]. The proposed design exhibits significantly improved performance over regular normal incidence detectors and is comparable to waveguide detectors for some size regimes.

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As expected, due to its relatively small size, the proposed device with optimized parameters exhibits a high bandwidth of ~170 GHz and a peak bandwidth-efficiency product of ~58 GHz. This is significantly higher compared to typical CMOS compatible Ge NI detectors reported in the literature [3]. In fact, it boasts an increase in the FOM for all size regimes up till approximately 1000 μm² where the capacitance limited response time dominates. As mentioned previously, this is due to the much higher specific external quantum efficiency (light absorption per unit area or volume) obtained from the strong coupling between guided and radiative modes, especially when compared to usual techniques of enhancement such as resonant cavity structures which require large film thicknesses. From Fig. 5(b), it is seen that the proposed design significantly bridges the gap between normal incidence and waveguide integrated photodetectors, with the latter still possessing markedly higher FOM values due to the fundamental decoupling of the carrier transition path and light propagation direction. Nevertheless we expect our design to help make normal incidence detectors more relevant and applicable, such as for packaged devices where the light source is normally incident by default. We also note that the values used for FOM comparisons in Fig. 5(b) are all theoretical values derived from Eqs. (4) and (5) or otherwise sourced from the literature [3], to ensure a fair and consistent basis for comparison.

Furthermore, due to the smaller absolute area and volume of the proposed design, dark current and relevant noise figures (which are based on absolute magnitudes of the dark current, and not the density) should also be correspondingly smaller.

5. Angled source injection

Finally we characterize device performance at non-zero incidence angles, since in addition to device fabrication or packaging errors such as misalignment of the source fiber, for NI detectors the source fiber is typically inclined at a small angle to avoid interference from reflection off the structure and/or substrate. Gratings in general (in particular, 1D gratings) are known for their extreme sensitivity to source angle due to strict cavity boundary conditions. However for a 2D periodic structure, even by utilizing a single, uniform periodicity, we can control the optical activities of the resonant modes across a wide spectrum and hence achieve significant angular tolerance. For instance, the main resonance peak can be designed to arise from purely TE or TM modes (such as bands 1 and 2 in Fig. 2(a)) for polarization selectivity, or a hybrid mode with both TE and TM branches for polarization insensitivity (such as band 3 in Fig. 2(a)). Then standard grating physics applies: for an optically active mode, incident light of similar polarization will cause significant shifts in spectral response for varying angles. Optically inactive modes naturally remain largely unperturbed.

Figure 6 illustrates the change in spectral response of the proposed device using our optimized absorption parameters under angled source injections for both polarizations. From Fig. 6(a), it is observed that for TM polarized light the resonant peak at targeted 1550 nm wavelength changes negligibly even at large incidence angles of up to 15° (Δλ = 3 nm for $\Delta \theta$ = 10$°$). The fundamental and higher order resonances split into 2 separate, smaller peaks. Under TE illumination, however, the targeted resonance peak tunes strongly with varying angle of incidence, shifting by its FWHM at only about 5° inclination. As mentioned in the preceding paragraph, these seemingly complex behaviors can be fundamentally understood by returning to the simplified band diagram of Fig. 2(a), and taking into account the non-zero wave vectors in the x and y direction that arise from the angled incident light as well as the optical (in) activity of the resonance modes concerned. We see that non-zero incidence angles break the degeneracy between the x and y propagating modes, resulting a splitting of the intercepts at the $\Gamma$axis. Figure 6(b) illustrates this degeneracy-breaking; for clarity both x and y propagating modes are portrayed as having shifted from the (degenerate) normal incidence mode (1 into 1’ and 1”), corresponding to an angle of incidence rotated by both$\theta$ and$\varphi$(Fig. 1(a) for coordinate references). In addition, since the targeted resonance mode at 1550 nm is the second order resonance, it corresponds to a pure TE excited mode in the structure and is therefore optically inactive and largely unperturbed under TM incident light.

 

Fig. 6 (a) Spectral response of the device under TE and TM polarizations with varying angle of incidence, negligible shifts to the 1550 nm resonance peak is observed for TM light up to 10$°$ angle of incidence. For TE light the peak rapidly splits into two smaller peaks. (b) Band diagram reproduced from Fig. 2(a), showing the breaking of degeneracy (manifested as splitting of resonance peaks) for oblique incidences due to the introduction of non-zero x and y direction wavevectors.

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However for an actual corrugated structure the simplified band diagram results cannot be taken too literally: realistically, the oblique TM incident light can result in complicated oblique scattering/reflection events in the grating cavities and result in perturbations to the spectral response. This discrepancy between the actual structure and the simplified homogeneous slab assumption is most likely the reason for the differences between FDTD derived spectra and the band diagram predictions, notably for the fundamental resonance frequency, where we observe splitting of the peak under TM incidence, despite it being a pure TE resonance mode and possessing much lower energy than the TM modes.

6. Conclusion

In summary, we have proposed and theoretically characterized an efficient and ultra-compact Ge photodetector based on the guided mode resonance phenomenon, i.e. by using local near field enhancements arising from coupling of guided modes to free space radiative modes. The nature of the enhancement mechanism results in rapid field decay after only a few grating periods, enabling relatively high external quantum efficiencies in small device areas. This translates to a high bandwidth-efficiency product of ~58 GHz. In addition, the device can be customized for polarization sensitivity with good angular tolerance. The proposed device architecture is fully compatible with existing Si infrastructure.