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Abstract
With the advent of 6G communications and the constant quest for more bandwidth in wireless technologies, the use of frequency bands lying in the terahertz spectrum becomes inevitable. Efficient high-speed modulation and demodulation techniques are necessary for the development of future terahertz communication systems. However, the speed of state-of-the-art terahertz modulators is limited to MHz-GHz; therefore, far away from the requirements of practical high-speed communication systems. In this work, we discuss that lateral Schottky diodes in wide bandgap semiconductors can enable simultaneous high-speed modulation (intrinsic cut-off frequency > 100 GHz), large modulation depth (>10dB), and low-loss (∼1.5dB) in a metamaterial configuration. These devices are lateral and thus do not require complex semiconductor or electromagnetic design or fabrication. Therefore, the proposed modulator design approach can unlock the potentials of the terahertz band for future 6G wireless communications.
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1. Introduction
Effective modulation and demodulation technologies are essential for developing future terahertz communications systems. Similar to what occurs in the optical domain, the development of high-speed electrically driven modulators attaining large modulation depths and exhibiting low loss is necessary to advance communications in the terahertz band. Furthermore, devices capable of efficiently altering the transmission and/or reflection properties of a terahertz beam, although at relatively low speed, could find applications in other important areas such as terahertz imaging. Although tremendous achievements have been reported in terahertz modulators in recent years [1,2], the performance of these in terms of speed is still distant from the state of the art of optical modulators, where speeds above 50 Gbps have been reported [3]. In this context, although recent works have experimentally demonstrated switching speeds on the order of 1 GHz in quasi-optical metamaterial configurations [4], and 14 GHz in on-chip waveguide coupled devices [5], these speeds are still relatively low.
To address this issue, metal-semiconductor-metal cavity arrays, where operation is based on reverse biasing the Schottky junction formed between top metal strips and a semiconductor buried beneath, have been proposed as efficient modulators offering in theory 100% modulation depth, ∼10% insertion loss, and picosecond intrinsic switching times [6]. However, the manufacturing of these devices can be challenging due to their vertical nature. Note that in this device proposal, a relatively thin semiconductor layer, ∼2 µm, is sandwiched vertically between two metal layers, which can be very difficult to fabricate from a practical perspective. By taking advantage of the same phenomenon, that is the intrinsic very high speed of Schottky diodes, we propose in this work a lateral modulator concept capable of attaining similar high-speed performance. We explore the different regimes of operation in these devices and discuss the effect of semiconductor choice on its performance. Our results show that wide bandgap materials, such as GaN, could provide advantages compared to high mobility semiconductors such as GaAs due to a larger breakdown field and relatively good mobility.
2. Analyzed structures and influence of materials
The analyzed structure consists of an array of lateral Schottky diodes with alternating Schottky and Ohmic contacts, as illustrated in Fig. 1(a). As a reverse voltage (V) is applied, the depletion region width, Wdepl, is altered between Wmax = L and Wmin, as depicted in Fig. 1(b), which, under full-depletion approximation, are given by:
Fig. 1. (a) Schematic of the proposed device, which consists of an array of lateral Schottky diodes with periodic Ohmic and Schottky contacts. A terahertz beam is normally incident to the structure. (b) When applying a voltage (electric field) across the adjacent interdigitated electrodes, the width of the depletion region is controlled. (c) Which in turn controls the transmission of a normally incident THz beam, thus modulates the transmitted intensity.
When the device is biased so that Wdepl = Wmin, a substantial charge in the semiconductor leads to a low transmission of a normally incident terahertz beam, whereas when the bias is such that Wdepl = Wmax = L, then the incident terahertz radiation can funnel through the aperture between adjacent metal contacts and lead to high transmission, as illustrated in Fig. 1(c). The modulation of the depletion width, R, is given by R = Wmax/Wmin = (V/Vbi+1)1/2, and the maximum electric field, which takes place at the semiconductor/Schottky metal interface, is given by:
where Ebr represents the breakdown electric field.Therefore, from Eqns. (1) and (2), it results that for a given reverse bias Nd is bounded by:
thus, materials with large Ebr can provide for larger Nd and thus larger effective conductivity swings, which could translate into a significant modulation of the transmitted THz intensity. The intrinsic cut off frequency (f0) can be computed from the inverse of the RC time constant of the device when at Wmin (f0 = 1/(2πRWminCWmin) with RWmin = (Wmax - Wmin)/(qµNd) and CWmin = ɛ/Wmin) and is given by: which therefore also benefits from a large Nd, as allowed by a large Ebr. Here, µ is the carrier mobility. From Eq.ns. (1–4), it is possible to generate contour plots of L, Emax, V, and f0 in the (R, σ) plane, where the conductivity (σ) is given by $\sigma = q\mu {N_d}$. It is to note that mobility µ is dependent on doping and reduces as Nd is increased; to account for this effect, in this work we utilize mobility vs. carrier concentration characteristics reported from the literature. These plots were generated for Si (low-doping µ ∼ 1,500 cm2/V.s, µ vs. carrier concentration extracted from Ref. [7], ɛr = 11.7, Ebr = 0.3 MV/cm), GaAs (low-doping µ ∼ 8,500 cm2/V.s, µ vs. carrier concentration extracted from Ref. [8], ɛr = 12.9, Ebr = 0.4 MV/cm), and GaN (low-doping µ ∼ 1,500 cm2/V.s, µ vs. carrier concentration extracted from Ref. [9], ɛr = 9.0, Ebr = 3.3 MV/cm) and are depicted in Figs. 2(a)-(c). For illustrative and comparative purposes, a uniform value of Vbi, Vbi = 0.7 V, was assumed in all three cases. We chose to represent these curves in the (R, σ) plane since, in principle, from a modulator perspective, in order to enable large modulation depth, large R and large σ are desirable; therefore, superior designs will be at the upper right corner of this plot. Color-shaded in Figs. 2(a)-(c) is the space satisfying L > 100 nm, which is chosen as a lithography fabrication constrain to enable ease of fabrication, Emax < Ebr, and f0 > 100 GHz. This space is identified as a suitable device design space. It is noticed that GaAs can provide device designs with larger R and σ values than Si because of its larger mobility. In contrast, GaN can provide for designs operating at similar conductivity levels as GaAs but at larger R levels because of its larger Ebr (see Fig. 2(d)). In this comparison, shown in Fig. 2(d), we have also included potential design spaces for β-Ga2O3 (low-doping µ ∼ 200 cm2/V.s, µ vs. carrier concentration extracted from Ref. [10], ɛr = 11, Ebr = 8 MV/cm), an emergent ultra-wide bandgap material, as well as for GaN assuming mobility levels consistent with polarization doping in high electron mobility transistor (HEMT) structures (labeled as GaN (p.d.)). A unique property of III-Nitrides is the possibility of enhancing mobility / reducing scattering through polarization doping in graded layers and 2DEGs [11]. In this regard, utilizing mobility values for 2DEGs in AlGaN/GaN QWs on the order of 2,000 cm2/V.s [12] we benchmark in Fig. 2(d) how a lateral device consisting of multiple stacked QWs forming a superlattice, with lateral Ohmic and Schottky contacts to 2DEGs, as proposed in Ref. [13] for microwave varactors, will compare with the bulk devices here discussed (this corresponds to the design space labeled as GaN (p.d.)). We see in this case that a ∼10X larger conductivity is possible vs. GaAs when the voltage swing is limited to 10 V.Fig. 2. Design space exploration for (a) Si, (b) GaAs, and (c) GaN. Green, blue, black, and red colored curves correspond to constant Emax (MV/cm), L (µm), voltage swing (V), and f0 (GHz) designs, respectively. The shaded regions correspond to the design spaces satisfying Emax < Ebr, L > 0.1 µm, and f0 > 100 GHz. (d) Comparison of design spaces in Si, GaAs, GaN, and β-Ga2O3.
It is to note that the analysis herein does not consider effect of parasitics. These could alter the response of the structure to the RF signal. In order to quantify the effect of some of these, we have estimated the parasitic fringing capacitance arising from the electrodes. In a device with W =100 µm and L = 0.3 µm, this will be (roughly) on the order of 1/3 of the junction capacitance. There will be in addition another parasitic effect arising from contact resistances, which will also play a role. However, since the intrinsic cut off frequencies for the devices here discussed are well in excess of 500 GHz (as shown in Fig. 2(c)), we expect that when considering these parasitic effects, the actual cut-off frequency remains above 100 GHz. The devices here discussed, are in essence very similar to the lateral Schottky diodes reported by Shinohara et al so therefore capable of attaining high speed in practice [14]. As such these could be fabricated through a similar process starting from n-GaN epitaxial growth and then two steps of lithography/metal deposition to define the Ohmic and Schottky contacts. Schottky contacts could be made employing Ni/Au whereas Ohmic contacts with Ti/Al.
3. Non-resonant operation
Typical voltage swings for high-speed modulation signals are usually < 10 V. From this perspective, in Fig. 3, we compare device designs in GaAs and GaN under this voltage operation constrain. In this case, the design point leading to superior performance in GaN will be the one corresponding to the intersection of the L = 0.1 µm curve with the V = 10 V curve (R = 3.91), because of the larger conductivity levels attainable in this design. For illustrative purposes, we performed full-wave electromagnetic simulations in COMSOL Multiphysics for GaN modulator designs with L = 0.1, 0.2, 0.3, 0.4, and 0.6 µm. In these designs, in the low transmission state, a quasi-static conductivity of 9.55 × 103, 3.73 × 103, 2.04 × 103, 1.29 × 103, and 0.65 × 103 S/m over a space region from the Ohmic metal to a distance given by L/R from the Schottky metal was assumed. A Drude model is employed to model the frequency dependance of the permittivity and conductivity of this region following our previous discussion in Ref. [15], from the employed mobility vs. carrier concentration characteristics for bulk GaN [9], τ is estimated as 0.064, 0.100, 0.123, 0.140, and 0.159 ps for the designs with L = 0.1, 0.2, 0.3, 0.4, and 0.6 µm, respectively. For illustrative purposes, semi-infinite substrates were considered in these simulations. The thickness of the Schottky region as well as for metal electrodes was taken as 200 nm. In the high transmission state, no charge is assumed in the semiconductor (charge is fully depleted). We simulated several designs by varying the width of the metal contacts. For this purpose, D = W + L was varied in the interval ∼1 to ∼100 µm. In these simulations, the frequency of a normally incident terahertz beam (polarized perpendicular to the metallic stripes) was varied in the 0.1 to 1 THz interval. In this frequency interval, the metal electrodes operate in a broadband extraordinary optical transmission regime [16]. The field enhancement in this regime has been proposed to enhance the modulation depth in devices based on graphene [17] as well as to result in very strong field enhancements [18]. The different points depicted in the modulation depth (MD) vs. insertion loss (IL) plots in Figs. 3(b-f) correspond to different values of W (curves with different colors represent different values of D = W + L for a given L thus variations of W) and different frequencies. Because of the larger conductivity level attainable in the low transmission state, the L = 0.1 µm designs can produce the larger MD levels (maximum MD approaching 85%). However, these large MD occur for designs with the largest W’s. It is to note that because of the small gap and large metal width, the loss resulting from the finite conductivity of the metals, in this case, leads to a relatively large IL. Looking for designs with IL < 50% (3 dB), the possible modulation depth, in this case, is ∼80%. A similar analysis was performed for GaAs, and the results are depicted in Figs. 3(g)-(l), in this case, the breakdown field in GaAs can limit the possible designs. There is a tradeoff between R and σ, and we observe that the best design is when L = 0.6 µm. It is to note that the maximum attainable modulation depth is smaller than that possible in GaN devices (∼75% in GaAs vs. ∼85% in GaN); when comparing the performance of designs attaining < 50% IL, the modulation depths are slightly smaller in GaAs (∼75% in GaAs vs. ∼80% in GaN).
Fig. 3. Modulation performance (MD vs IL) for broadband modulator designs with voltage swing limited to < 10 V. (a-f) GaN-based modulators when varying L in accordance to (a), and (g-l) GaAs-based modulators, when varying L in accordance to (g). The different curves in panels (b-f) and (h-l) represented in different colors correspond to simulations for various values of W (D = W + L is varied at constant L), different points along each curve are associated with different frequencies (in each curve, leftmost points correspond to lower frequencies, rightmost points correspond to highest frequencies). Designs are chosen based on < 10 V operation to enable high speed modulation.
Lifting this restriction on voltage swing, we analyzed as well the response of designs in GaN, assuming these to be breakdown limited. For this purpose, in Fig. 4 we depict designs corresponding to L= 0.1, 0.2, 0.3, 0.4, and 0.6 µm, but operating at values of R limited by Ebr. As a result of this, larger conductivity and R levels are possible than in our analysis in Fig. 3. We observe in this case a tradeoff between IL and MD; whereas, in designs with the smallest L, large modulation levels are possible only at high loss levels; in designs with largest L, large modulation and low loss are simultaneously possible. This illustrates that designs operating at larger voltages but with lower Nd could lead to better performance because of a lower possible loss. These devices can be suitable as broadband modulators for beam-shaping applications, like as spatial light modulators in terahertz imaging [19–21], wherein speed is not a significant consideration.
Fig. 4. Modulation performance for broadband modulator designs in GaN with voltage swing limited to < 100 V. (a-f) MD vs. IL performance of modulators when varying L in accordance to (a). The different curves in panels (b-f) represented in different colors correspond to simulations for various values of W (D = W + L is varied at constant L), different points along each curve are associated with different frequencies (in each curve, leftmost points correspond to lower frequencies, rightmost points correspond to highest frequencies).
4. Resonant operation
We now explore these Schottky diode arrays on a narrowband (resonant) operation. For this purpose, we assume the GaN structure depicted in the insets of Fig. 5(b-c); the substrate thickness is assumed to be 200 µm, and D = W + L is set to 200 µm. The thickness of the Schottky region, as well as metal contacts, is assumed to be 200 nm, as in simulations in previous figures. We analyzed three devices (see Fig. 5(a)), two engineered for 10 V operation with L = 0.1 and 0.3 µm, respectively; another device designed for 100V operation, with L = 0.6 µm. Simulated transmission spectra when Wdepl = Wmax is depicted in Fig. 5(b). Here the finite substrate thickness has an important role. We observe two regimes, one at low frequency, where the response of the structure is the superposition of the low-pass response of the grating and the Fabry-Perot resonances arising from the substrate, and another high-frequency response at which guided modes in the substrate can be excited by diffraction orders. This leads to narrow linewidth-resonances [22]. Field profiles showing the differences in these modes are depicted in the insets below Fig. 5(b). The left inset corresponds to fields at a frequency corresponding to a Fabry Perot resonance, whereas the right inset depicts the fields at the frequency corresponding to the narrow-linewidth resonance. Simulations for Wdepl = Wmin are depicted in Fig. 5(c). Here the transmission spectra are computed for all three cases showing low transmission and, in particular, a much-reduced transmission at the narrow linewidth resonances (than what is observed in Fig. 5(b) at Wdepl = Wmax). From here, we independently depict the modulation attainable in each structure, which is shown in Fig. 5(d). All three analyzed structures can provide for a large modulation depth at resonance (MD > 90%). However, the narrower the gap, the larger the insertion loss. Insertion loss vs. modulation depth at resonance is finally plotted in Fig. 5(e). As mentioned, we observe in all three cases > 90% MD, however, when considering the tradeoff between MD, IL, and speed (voltage operation), the design with L = 0.3 µm provides for the best tradeoff, this design, wherein Nd = 1.17 × 1017 cm-3, can operate at a voltage swing of 10V to switch between Wmin and Wmax and provide for 94% MD (> 10dB) and ∼30% IL (∼1.5dB). It is to notice that this loss arises from Ohmic losses in the metallization; the narrower the gap, i.e., the smallest L, the larger the loss.
Fig. 5. Modulation performance for three GaN based modulators under resonant operation. (a) chosen designs in the R, σ plane. (b-c) simulated transmission spectra at Wmax and Wmin for the three analyzed designs. (d) comparison of transmission modulation. Field profiles are depicted for one of the Fabry-Perot resonances and for the narrow-linewidth resonance at ∼0.52THz. (e) MD vs. IL performance for the three designs.
5. Conclusion and outlook
In synthesis, we have analyzed lateral Schottky diode arrays in bulk GaN and GaN heterostructures as terahertz modulators. Our simulations predict that these structures can enable simultaneous high-speed modulation (intrinsic cut-off frequency above 100 GHz), large modulation depth (> 10 dB), and low-loss (∼1.5 dB) in a metamaterial configuration, which makes these devices attractive for future terahertz communication systems. These devices could be fabricated employing bulk GaN or epitaxial GaN heterostructures with multiple stacked 2DEGs, through the latter approach further superior performance can be in principle attained.
Funding
Air Force Office of Scientific Research (FA9550-18-1-0332).
Disclosures
The authors declare no conflicts of interest.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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