1. Main text

Photodetectors deliver an integral function of optical-to-electrical (OE) conversion. Unlike solar cells performing this conversion spectrally broadband and temporally at steady-state, photodetectors, and especially photoreceivers are typically designed for high responsivity (gain) at a target wavelength given by the absorbing materials’ bandgap, and by a fast-temporal dynamic response (3 dB bandwidth, BW). Prominent material-spectrum mappings for detectors are GaN for UV (<400 nm) [1], Si for visible to NIR (400-1100 nm) [2], Ge/InGaAs for NIR to MIR (1-5 µm) [3] and HgCdTe for MIR to FIR (>5 µm) [4].

It is the aim of this letter to clarify i) fundamental photodetector tradeoffs, and ii) relate them to design choices, which thence govern detector performance. We show that this correlation is actually not arbitrary and that there exist interdependencies between design-material combinations that yield higher performance than others. The arguments laid out herein, are exemplary made on photonic integrated circuit (PIC)-based detectors, but their generality also holds for free-space coupled devices and applications.

The obtainable signal output from a detector (i.e., photocurrent and available photogain) and its operation response (speed, i.e. bandwidth, BW) are not interdependently optimizable, which can straightforwardly be understood by using the picture of the integration time of a photo generated signal; if the gain is high to increase the detector's sensitivity, then the response rate is slow, and vice versa. An example from daily life is the sluggish smart-phone camera response for taking photos at dim lighting conditions, where the digital signal processing circuit (DSP) increases the gain to enhance the signal-to-noise ratio (SNR), thus reducing the shutter speed.

Most of the photodetectors can be categorized broadly into two classes (Fig. 1(a)); those with high responsivity (R) (but low temporal bandwidth, speed) and vice versa, which relates to the fundamental trade-off gain-bandwidth product (GBP) being a figure-of-merit (FOM) for detectors. Since photodetector gain linearly scales with the responsivity in units of (A/W), here, for discussion purposes, we use gain and responsivity interchangeably. Indeed, the iso-GBP lines show devices cluster in either the upper left (low-BW/high-R) or lower right (high-BW/low-R) quadrant, both corresponding to a GBP ∼ 10⁶ - 10⁹ Hz-A/W (Fig. 1(a)). Naturally, detector performance and next-generation devices scale orthogonally to these two quadrants aiming for simultaneous high-speed and high R. It is the target to design and demonstrate such high-performance photodetectors that guide our motivation for the work presented herein (Fig. 1(a)).

 

Fig. 1. Design strategies for Engineering Gain-Bandwidth-Product (GBP) Photodetectors. a) Photodetector performance cluster into two quadrants; high responsivity yet low speed (bandwidth, BW), and vice versa. Target detectors with a GBP of ∼10¹² - 10¹³ scale orthogonally (upper-right quadrant) to the iso-GBP lines. The values are taken from [5–26]. b) Plot of bandwidth vs. photocarrier-collecting electrical channel length showing two regimes where the transit time-limited bandwidth (BW_transit) is dominant for long-channel detectors (gray shade), whereas for sub 100 nm channel lengths, the bandwidth is limited by RC time (blue shade). BW_transit is estimated for four different mobility values (1 - 1000 cm²/Vs) for a V_sd= 1 V taken from Ref. [19, 27–29]. The RC time-limited bandwidth (BW_RC) is estimated for four different resistance values starting from 0.1−100 kΩ [30–32], where, the electrical capacitance is determined by a parallel-plate model (fringe fields ignored) for a lateral junction. A finding of this parametric study is that for a (arbitrarily selected) target speed of 100 GHz; i) micrometer long channel-based photodetectors required a rather high mobility (<10⁴ cm²/Vs) but with a somewhat relaxed contact resistance RC requirements, yet high mobilities challenges achievable resistance due to scattering events induced by higher doping, while ii) 10’s nm short-channel detectors are only resistance limited, since (near) ballistic transport is given even for poor mobilities (<10 cm²/Vs), and relatively high grain can be achieved by reducing the transit time to ∼ps as supposed to increasing the carrier lifetime. Such short-channel performance-detectors, however, demand optical mode squeezing to adhere to device scale-length theory (SLT) rules which can be achieved with slot-waveguide designs.

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Let us next analyze this fundamental GBP tradeoff in more detail to explore device paradigms that would allow to optimize the GBP beyond what is available with current technology; the photoconducting gain is given by the ratio of the photocarrier lifetime (τ_c) to the photocarrier transit time (τ_t), i.e. the drift time for photocarriers to reach the photocurrent-collecting electrodes. Meaning, if the carrier lifetime exceeds the time for their collection, charge accumulation (gain $ \approx $ τ_c /τ_t) can occur. However, this well-known ‘gain theory’ is highly based on Ohmic contact and on the assumption that the concentration of photogenerated excess carriers in the photoconductor is uniformly distributed which is not true for any photodiode (p-n junction or Schottky junction), where the distribution of photogenerated excess carriers is always non-uniform and therefore the photogenerated excess charge carriers readily skewed by the electric field, resulting in the voltage-dependent excess carrier concentration. This can be seen clearly from the continuity equation, and the total photocurrent will not only be due to the drift current, but also have contributions from the diffusion current. Given that the carrier lifetime in typically used detector semiconductors is about O(∼ns), gains of about O(∼10² - 10³) can be obtained for transit times of 10’s - 100’s of ps corresponding to medium-high mobility materials and micrometer source-drain distances. The transit time, on the other hand, is a function of the electrical transport properties (mobility, μ) of the photo absorbing material and the drift field (V_bias) via ${\tau _t} = {L_{e/h}}v_{drift}^{ - 1}$, where L_e/h where is the distance of the electrodes collecting the electron-hole photocarriers (i.e. channel length, L_e/h).

Parametric BW-scaling of this mobility-limited transit time and RC-delay as a function of photodetector source-drain channel length, L_e/h, reveals several insights (Fig. 1(b)); a) If the channel length is several micrometer long, mobilities of about 1,000 cm²/Vs are required to even achieve 1 GHz fast detection, while RC-delay is not a factor even for low resistivities. b) In the limit for scaled (1 - 10 nm) short channel lengths even a poor mobility of 1 cm²/Vs is not BW limiting up to about 100 GHz, provided at least a resistance of 1 kΩ can be guaranteed. c) A sweet spot for detectors that are aimed for speeds around 10’s of GHz is found for channel lengths near 0.1 µm offering a high design window with relatively relaxed requirements (µ < 10 cm²/Vs and R < 100 kΩ). As an interim conclusion, we can summarize that if high-speed devices (>100 GHz) are desired, then short transit channel lengths (∼10’s of nm) are a possible design option for materials with medium to low mobilities. Interestingly, even high-mobility (∼10,000 cm²/Vs) materials cannot reach such high BW for the case where micrometer wide contacts are used, which is the case for many low-index material waveguides such those based on SiO₂ and also often Lithium-niobate (LN). Furthermore, we see (Fig. 1(b)) that for III-V and silicon-based waveguides a channel length of 500 nm is already challenging, if plasmonic losses from too-closely placed electrical contacts are to be avoided. Thus, for our exemplary 100 GHz goal and micrometer-far apart contacts, mobilities of ∼1,000 cm²/Vs are required. The latter, however, limits the design window for achievable low-resistance (i.e. highly-doped contacts) due to increases scattering events at high doping levels, which increases resistance. Thus, an interesting design option seems to emerge, which is to accept possible optical losses from metal-optics and design 10’s nm narrow slot-waveguides, which can achieve 100 GHz devices with both modest mobilities ∼5 cm²/Vs and resistances ∼5 kΩ. Such material is, for instance, the class of transition metal dichalcogenides (TMDC). In fact, one can show that borrowing device scaling laws from electronics, that ‘flatness’ of the detector material is a key requirement, as discussed further below.

However, since the gain is proportional to both the mobility and the carrier lifetime, the former may be limiting the bandwidth of the detector such as for poor mobilities known form TMDCs [19]. Yet, there are several reports on ultrahigh gain in TMDCs based photodetector where the underlying mechanism is mainly photogating effect instead of multicarrier generation or avalanche amplification [8,9]. In the photogating effect, the charge carriers are trapped in surface impurity states, thus resulting in a DC-slow response time (1 - 10 s), which is too slow for many data communication and signal processing/computing applications, but possibly sufficient for environmental sensing applications [5,6,8,9,24]. Therefore, for long-channel detectors, in order to achieve high GBP, focus should be on improving carrier mobility rather than prolonging the carrier-lifetime to transit-time ratio, while the inverse is required (i.e. focus on improving contact resistance) for short-channel detectors, as further discussed next (Fig. 1(b)).

Borrowing device scaling concepts from electronics, the scale-length theory (SLT) shows that electric field-based device dimensions scale with a nominal factor (σ) such as in FETs both the channel thickness and length [33]. Applying this dimensional scaling concept to the GBP of photodetectors, we find that it scales as GBP ∼ σ^–2 ∼ L_e/h⁻², which suggests that reducing the collection distance rather sensitively (squared) improves detector performance. Hence the question arises, as of why prevailing integrated detector designs do not make use of this scaling performance gains?

The answer can be found in the inability of monolithic material integration to adhere to the SLT scaling; starting our GBP optimization analysis discussion with optical constrains, utilizing this σ⁻² scaling law urges device designers to utilize nanometer (or 10’s of nanometer) short-channel (contact distances) to optimized GBP. This, however, requires a simultaneous ‘squeezing’ of the optical mode to the same order (10’s nm compact), which can be realized with nano-optical light-matter-interaction enhancements such as offered by plasmonics [34] or dielectric discontinuities [35], typically at the cost of a non-zero metal-loss component, however. SLT applied to detectors simultaneously demands nanometer-thin absorbing materials, which is further enforced to ensure a high optical mode overlap (Γ) [36], similar to quantum-well lasers [37]. Realizing such a nanometer-thin absorbing material at high crystalline quality, however, is challenging to be provided by III-VI and IV (e.g. Ge) materials when these materials are heterogeneously integrated in Si or SiN photonic platforms due to the parasitic high defect density during patterning (e.g., reactive ion etching) [38,39]. Epitaxial direct growth of heterojunctions would be the ideal scenario; however, lattice mismatches severely limit this option, along with thermal budgets and possibly low manufacturing throughput. However, a SLT-scaled device detector is synergistic with two-dimensional (2D) materials, as briefly mentioned above and further discussed next, where an interesting design concept emerges that is based on reducing transit time as supposed to increasing carrier lifetime (Fig. 1).

As a possible high GBP-design example, a rather short plasmonic slot atop a silicon on insulator (SOI) waveguide, where L_e/h = W_slot (slot-width ∼10 nm), supports a gap plasmon with an electric field polarization being in-plane with such a nanometer thin material suspending this gap (see Ref. [22,40–41] for design examples). In general, the detector’s external quantum efficiency is given by: EQE = IQE x (1-Reflection-Transmission), where the internal quantum efficiency (IQE) is proportional to the overlap matrix element in the absorption process (typically ∼80’s - 90’s% for band-to-band absorption [42]). The second term relates to the amount of light absorbed by the device, which can be optimized by anti-reflection coatings for free-space coupled devices. In integrated devices, however, this relates to the mode overlap factor (Γ), i.e. the amount of light coupled to the photocarrier-generating material(s). Since a detector’s photo responsivity is proportional to the EQE, and hence to GBP, naturally ensuring a high optical absorption is critical. With 2D materials featuring a rather high intrinsic absorption coefficient (∼10⁵ cm⁻¹), they are an interesting SLT-synergistic material option; furthermore, their atomistically-thin flatness reduces Coulomb scattering effects, thus allowing for a high oscillator strength [43] signified by a high optical index Re{n} > 5 in the visible and NIR bands for TMDCs [44]. Furthermore, such a ∼10 nm short electrical channel, also comes with (near) ballistic electric transport; hence, a high drift mobility would actually not improve the transit time which is ∼1 ps, for V_bias = 1 V and μ_TMDC = 1-10 cm²/Vs, and the low mobility of 2D materials (aside Graphene) is not a limit of an SLT-scaled high-GBP 2D material-based detector designs.

However, improving a material’s mobility does relax length-scaling requirements (const. = GBP → µ ∼ (L_e/h)² (Fig. 1(b)). For instance, some recent undoped TMDC’s have shown mobilities of 10³ cm²/Vs [27,28], which allows increasing L_e/h from 10’s to 100’s nm for high-speed detectors. Channel-length scaling, however, also impacts the detector’s dark current (I_dark); generally speaking, lowering I_dark increases the detectors dynamic range, and improves the noise-equivalent-power (NEP), which is the required optical power to produce an SNR of unity at 1 Hz signal rate. Indeed, we [19] and others [20] have shown, that a TMDCs-based detector is able to operate at 1-10 nA dark currents, while a scaled slot detector has an about 10x higher I_dark. Compared to graphene, TMDCs feature a 10 - 1000x lower dark current due to their 0.5 - 2 eV wide bandgap, whilst even the smallest energy (optical or thermal) will initiate intra-band transitions in a gap-less material such as graphene. Another possibility for reducing the dark current, is to operate the detector in photovoltaic mode, namely at zero bias voltage, instead of the often used 1 - 2 V, which, however, also gives up gain enhancement design options that are based on carrier lifetime-to-transit time ratios. Nonetheless, such a detector can be realized with built-in P/N junction for charge separation polarity; for instance, surveying the literature, points to a potential of 10 - 100x lower I_dark [23,24].

However, 2D materials also have challenges in designing detector (e.g. high resistance), however as discussed before, their low-mobility is not one of them, provided SLT short L_e/h approaches are used. Yet, total resistance of the device comprises of contact resistance (R_c) and channel resistance (R_ch), which is key to achieving high-performance; since R_ch quickly decreases with the channel length scaling, R_c becomes a dominant factor in the total resistance for sub-100 nm channel lengths. For instance, the high contact resistance in kΩ to even MΩ challenges the device response time limiting RC-delay [22] (Fig. 1(b)). A possible solution for this is to design hetero-junctions, where function-material separation can be part of the design process; for instance, a high bandgap TMDC placed atop an SOI waveguide performs the OE conversion, whilst the photocarriers are transferred into graphene layers acting as an electrical conducting channel or as a ∼50 Ω contact resistance to the metal contact pads [30]. Several contact engineering techniques including edge contact geometry, 1T phase contacts, hBN substrate and graphene have been implemented in individual devices, but a scalable route for uniformly realizing these device concepts at the wafer-scale has not yet to be established [45,18].

Taking the above discussed high-GBP design options into account on high gain and responsivity via short transit-times, heterojunction devices for low RC, we can estimate a speed of up to ∼ 100 GHz (limited by RC, not by transit time) and a responsivity of 10 - 100 A/W with a typical TMDC carrier lifetime ∼O(100’s ps) [46], the transit time is only ∼1 ps given a slot width of, exemplary, L_e/h = 33 nm and a mobility of 10 cm²/Vs, (assuming elsewise ideal conditions), thus enabling a GBP approaching 10¹² - 10¹³ Hz-A/W (blue region, Fig. 1(a)). This would be about 4 - 5 orders improvement compared to the average detector performance and 1 - 2 orders enhancement over record demonstrations (Fig. 1(a)) [21–24]. The resulting I_dark would be about O(100’s pA) for a NEP of ∼1 - 10 pW/Hz^0.5 in line with state-of-the-art detectors [11].

Interestingly, 2D materials are being actively investigated to reaching foundry maturity such as pursuit by IMEC [47] or TSMC [48], for example, yet no customer-ready process is available as of this point in time of writing this article. This momentum bears hope for further material improvements with R&D economy-of-scale testing in the near future, and monolithic growth on SOI-oxide surfaces is possible, yet quality levels such as defect density still required attention. While CVD such processes are actively being developed to date, transfer techniques for 2D materials have successfully addressed issues such as deterministic placement [49] and single-flake transfer showing vanishing cross-contamination past transfer [50]. For instance, micro-stamping techniques allow for rapid-prototyping devices benign to PIC heterogeneous 2D material integration enabled by the high spatial selectivity of the micro-stamper [51].

Furthermore, 2D materials offer a higher degree of strain-material adaptation given added range of design flexibility, which we termed ‘strainoptronics’; for instance, straining (tensile or compressive) dramatically (100’s of meV) shifts the band edge of 2D materials leading to enhanced absorption so as responsivity for 2D materials based photodetector [19]. The understanding of the modulation of optical and electronic properties of these materials is key to design improved integrated photonic devices [52–54]. Techniques such as Kelvin probe force microscopy (KPFM) allow for local strain mapping which allows for nanometer-resolution strain and hence bandgap maps, thus allowing to align the photoabsorbing 2D material with photonic structures such as waveguides, or optical cavities precisely [19]. Indeed, the large amount of strain that 2D materials can sustain before rupture occurs along with their large elastic modulus [55], which allows reducing the bandgap of, e.g. MoTe₂, from 1.04 eV down to 0.80 eV, together enable efficient photo conversion at the telecommunication relevant wavelengths of 1550 nm [19].

In conclusion, similar to other opto-electronic fundamental building blocks such as lasers and modulators, the device physics of photodetectors dictate fundamental trade-offs spanning an orthogonal performance space; for a detector, this means that the sensitivity such as the available gain, the responsivity and hence the noise-equivalent power cannot be independently optimized without trade-offs in bandwidth, speed, as both are intercorrelated. Here we discussed photodetector gain-bandwidth-product (GBP) performance in the context of device channel scaling, electrical and opto-electronic performance constrains. We show that state-of-the-art detectors are either optimized for gain or for speed showing an averaged GBP of 10⁶ to 10⁹ Hz-A/W (using responsivity for gain due to linear proportionality amongst them). Further, we discuss strategies for achieving next-generation GBPs of approaching 10¹² to 10¹³ Hz-A/W following scaling length theory, which would constitute an enhancement of one to two orders of magnitude over current best demonstrations. For instance, we show that for achieving 100 GHz-fast detectors, designs using micrometer-wide separated photocarrier-collecting source/drain contacts require exceedingly-high mobilities (∼10⁴ cm²/Vs), which challenges simultaneously achieving low contact resistances in order to ensure short RC-delay. A seemingly more elegant and PIC-density benign approach of achieving high GBP detectors, may be to scale the channel length into the 10’s nanometer-short regime, where transport is near ballistic even for low-mobility (<10 cm²/Vs) materials. Interestingly, we find that the scaling length theory and plasmonic slot-waveguide approaches offer a high-degree of detector design synergies especially for ‘flat’ photo-conversion materials. Such 2D materials may be a solution for such next generation high-GBP devices based on channel length scaling because their low mobility does, in fact, not limit performance, since carrier velocity saturation effects play a minor role and the device response is typically not transit time-limited but determined by RC-delay. However, for such scaled photodetectors, several issues are a limiting factor, foremost of which is the low contact resistance to the channel materials, since capacity increases with scaling, which has to be offset by resistance. The latter is a particular challenge for 2D materials, and future research should place a focus on this. Other challenges, in particular in a context of chip integration, are general scalability and associated yield issues, however with possibilities for tailored doping levels and lack of standardization protocols. Therefore, in order to realize the fullest potential of such device physics-benign emerging materials, constrains from foundry-based mass productions and chip integration should be co-developed to ensure satisfactory and reliable performance.