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Abstract
In this work, we study multiple epitaxial layer structures incorporating a resonant tunneling diode photodetector utilizing the In0.53Ga0.47As/InP material system for operation at the near-infrared region of 1.55 and 1.31 micrometers. We study the photodetection speed of response for these devices and the physical limitations affecting their bandwidth. We show that resonant tunneling diode-based photodetectors have bandwidth limitations due to the charge accumulation near the barriers and report on an operating bandwidth reaching up to 1.75 GHz in particular structures, which is the highest number reported for such detectors to the authors’ best knowledge.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Resonant tunneling diodes (RTD) have shown great potential in high-frequency operations where RTDs operating up to 2 THz have been reported [1]. Given that RTDs have an N-shaped current-voltage (IV) curve that exhibits a negative differential resistance (NDR) region, they are capable of providing gain and amplification to electronic signals [2,3]. In this work, we are interested in highlighting the physical limitations on the optical speed of response of RTD-based photodetectors (RTD-PD).
Besides the electronic applications, there has been a considerable effort to take advantage of the high-speed operation, and the NDR properties of the RTD in optoelectronics; such as light emission from unipolar RTD structures [4], and photodetection [5–8]. An RTD-PD contains photo-sensitive material at one or both regions surrounding the double barrier quantum well structure (DBQW) (the emitter and the collector). Utilizing the amplification in the NDR region [3], the need for other circuitry (amplifiers) in photodetector circuits is eliminated, thereby reducing cost, complexity, and power consumption. Additionally, RTDs have shown low noise levels that are characterized by a reduced Fano Factor when biased in the PDR and NDR regions, which further supports their use as detectors [9–11]. In terms of sensitivity, the highest number reported is 3 × 104 A/W for devices with a distributed Bragg reflector [12]. The RTD-PD has also been demonstrated to operate as a single photon detector when integrated with a quantum dot [13–15]. This shows that the RTD-PD has many potential applications, and an investigation into its optical speed of response is important to shed some light on the best applications for such a detector.
Many have shown RTD-PDs operating at various wavelengths in the near and far infrared bands [12,16] but only a handful have reported on the optical speed of response for their devices. The largest bandwidth for such detectors was reported was by Moise et. al. of about 1.5 GHz [17,18].
The reported work on RTD-PDs mainly focuses on a single device topology; when absorption happens at the collector side and generally employs an n-i-n scheme. In this work we characterize RTD-PDs with absorption regions at both the emitter and the collector. Moreover, we report and discuss the results of a p-i-n structure design that we have theoretically analyzed and simulated in [19]. Finally, previous works did not account for the effect of charge generation in the highly doped contact regions. Here, we highlight the importance of such phenomena and how it decreases the bandwidth.
The paper commences by introducing the principle of operation of RTDs and the concept behind the RTD-PD. We then discuss the epitaxial layer structures used, and report on the RTD-PD devices’ microfabrication details. We then characterize the speed of response (bandwidth) of these devices by measuring their impulse response and 3 dB bandwidth. The effect the DBQW has on the overall speed performance of the device is presented by characterizing a double heterojunction diode (DHD) with a similar epitaxial structure as one of the RTD-PD samples but lacking the DBQW. The results are analyzed and contrasted against various physical phenomena to pinpoint those mainly affecting the bandwidth. Finally, we report on the speed limitations of the p-i-n RTD-PD design and detail the modified microfabrication process for this sample.
2. Operation of resonant tunneling diodes
An RTD is a device consisting of a double barrier quantum well (DBQW) between two spacer layers: the emitter and the collector, with ohmic contacts at both ends. Figure 1 depicts a typical RTD structure based on an n-i-DBQW-i-n scheme.
Due to quantum mechanical confinement, energy in the quantum well gets quantized. Figure 2(a) shows a typical RTD conduction band diagram at zero bias. The quantum well has two states E0 and E1, and the Fermi level resides below E0. Due to the presence of the DBQW, the probability of transmission of electrons from the structure is highest at energies corresponding to E0 and E1 as shown in the transmission probability plot in Fig. 2(a). As we apply voltage, the potential drops across the DBQW and bends the energy band diagram, lowering the energy Eigenstates inside the quantum well thereby, reducing the separation between the quasi-Fermi level at the emitter and the first Eigenstate (Fig. 2(b)). The current at this stage resembles the first positive differential resistance region (PDR) of the current-voltage (IV) curve. When the quasi-Fermi level at the emitter aligns with one of the quantized well-states, resonance conditions occur, and the current reaches a peak value (Fig. 2(c)). Increasing the potential further breaks this alignment and reduces the current (Fig. 2(d)). This current resembles the negative differential resistance region (NDR) of the IV curve. Further increment of the voltage will start the second PDR region. Figure 2 depicts the phases of the RTD-IV curve.
Fig. 2. RTD-IV phases: (a) IV curve (red: first PDR, green: peak, and yellow: NDR) and conduction band diagram with transmission probability at zero bias. (b) Conduction band diagram in the first PDR. (c) Conduction band diagram at peak voltage. (d) Conduction band diagram in the NDR. (e) Typical RTD while biased near the peak.
As electrons are injected from the emitter’s contact, and because of the emitter’s barrier, most of the electrons accumulate making an electron accumulation region that eventually turns into a two-dimensional electron gas (2DEG), as in Fig. 2(d). This electron accumulation causes holes to accumulate at the collector’s barrier, making a hole accumulation region and a depletion region [20,21].
The concept of an RTD-PD is the generation of electron-hole pairs by absorbing light at the emitter and the collector. The photo-generated carriers are transported from the point of generation to the collection point through the DBQW, thereby making a photodetector with built-in gain, for high sensitivity and high-speed applications. This is portrayed in Fig. 2(e).
3. Epitaxial layer designs
We begin by introducing the epitaxial layers, which we refer to as samples 1, 2, 3, 4, 5, and 6 hereafters. Tables 1, 2, and 3 show the epitaxial designs of samples 1, 2, and 3/4 respectively. The epi-layers under study have highly doped In0.53Ga0.47As contact regions with an average of 500 nm thickness for the bottom contact. The main difference is the length of the emitter, which varies between 500, 250, and 100 nm. The lightly-doped collector is around 20 nm for samples 2, 3 and 4, and 500 nm for sample 1. Other differences to highlight are the narrower quantum well for sample 2, which will shift the peak voltage to higher values, and the small amount of Al present in sample 1 emitter / collector regions. Samples 5 and 6 are shown later in sections 5 and 6.
Table 1. Epitaxial layer design of sample 1
Table 2. Epitaxial layer design of sample 2
Table 3. Epitaxial layer design of samples 3/4
4. Device fabrication
The RTDs were fabricated using photolithography. Fabrication started with metal evaporation of Ti/Pd/Au to form the top RTD contact. Chemical wet etching (H3PO4:H2O2:H2O = 1:1:38) was used to define the RTD mesa. The bottom contact metal (Ti/Pd/Au) was then evaporated. 2 µm thick Polyimide PI-2545 was used for device passivation. Finally, bond-pad metallization was done using Ti/Au. Figure 3 shows the fabricated RTD-PD devices. The IV curves for fabricated devices are in Fig. 4, noting that the applied bias is such that the top layer is negative (co-planar waveguide (CPW) signal pad). All devices have mesas of 10 × 10 µm2.
5. Optical impulse response
The setup used to measure the optical impulse response of the devices is shown in Fig. 5. Measuring the optical impulse response allows the determination of the speed of the device and provides a means to measure the carrier lifetime under diffusion transport. This is due to the exponential decay of the detected signal with a time constant equal to the minority carrier lifetime (τ) [22].
The laser used is a continuous wave and temperature-controlled source from Thor labs (S3FC1550) operating at 1.55 µm. The arbitrary waveform generator (ARB rider 5000) generates a pulse signal that is 500 ps-wide. This signal modulates the laser through a 10 Gbps Mach-Zehnder modulator. Afterward, the modulated light wave goes through an amplifier (Erbium dopped fiber amplifier JDSU OA400 series) and an optical filter before splitting into two paths. The first goes into a commercial PIN detector (Agilent 83440D high-speed lightwave converter), and the second into the RTD-PD. The light is coupled to the device directly through a bare fiber that we align over the optical window. Then both signals go into a high-frequency oscilloscope (Rhode und Schwarz RTP-16) through a bias network (Keysight 11614A). The impulse response for devices made of these epi-layers is shown in Fig. 6. The bias voltage was kept to a minimum (on average 100 mV), to ensure low electric field values and limit any transport due to drift.
Fig. 6. Optical impulse response of all RTD-PD samples. Inset shows the pulse detected by the commercial PIN-detector (reverse biased at -15 V).
The results clearly show a slow response time characterized by a long-duration tail with a time constant in the tens of nanoseconds depending on the epi-layer used. If the fall-time (τfalltime) of the signal is defined as the duration for the amplitude to decay from 90% to 10% of its maximum value, the 3-dB cut-off frequencies for each of the samples can be given by (1).
For example, using (1) the 3 dB bandwidth for sample 3 would be around 25.5 MHz, and the rest of the samples are also in the 10s of MHz range.
To see the effect of bias voltage on the speed of response, we measured it as a function of the applied voltage for the previous epi-layers as shown in Fig. 7.
Noting that Fig. 7 shows values in the PDR region only. When going into the NDR region, RTDs start to oscillate, making any measurement of the decay time inaccurate. However, the devices are meant to operate in the PDR region, so the oscillations in the NDR are not of a concern for our purposes. The response times obtained are also in the 10s of nanoseconds range, which translates into a bandwidth in the tens of MHz.
Various physical phenomena might be responsible for such low bandwidth. The first is the RC time constant of the circuit. The load resistance of the circuit is 50 Ω, and the capacitance is mainly due to the DBQW structure. To estimate the capacitance of the DBQW, we used the Silvaco ATLAS software that utilizing a non-equilibrium green’s function (NEGF) approach to calculate the charge of the device at two adjacent voltage points. The capacitance is then the derivative of the charge-voltage curve as in Fig. 8 [23–26].
The results suggest that the quantum-well capacitance is in the hundreds of femtofarad range which agrees well with measured values for a similar quantum well design of 3.75 fF/µ2 [27]. This should give a cut-off frequency of about 2.8 GHz. Hence, the RC time constant is not the limiting factor in these devices. Another limitation is the dielectric relaxation time of In0.53Ga0.47As given by (2),
where ɛ is the electric permittivity of the material, e the elementary electron charge, N the charge concentration, and µ the mobility. Using typical numbers for In0.53Ga0.47As [28], the dielectric relaxation time will be around 13.1 femtoseconds for a doping level of 5 × 1016 cm-3 which is extremely fast and cannot be the reason behind such a slow response. The transit time of photo-generated carriers also affects the speed of response of a device. To clarify this point, we examine Fig. 2(e) again, that shows the band diagram of a typical RTD biased near the peak voltage. As we have shown earlier [29], a typical n-i-n RTD device is characterized by an emitter that is mainly screened due to charge accumulation. The screening length is given by the Thomas-Fermi screening length of a 2DEG and the accompanying depletion length in the collector is proportional to the Debye screening length due to the bulk charges in the collector. This means that the lengths of the absorption regions must be tailored to fit these characteristic lengths to avoid electric field dead zones.Considering the doping levels in the epi-structures above, we see that some of them have this problem where there are considerable lengths with low to zero electric fields. Structures with a 20 nm collector will be fully depleted under proper bias, and the emitter will be mostly screened by the 2DEG. With emitter lengths of 100, 250, and 500 nm, the photo-generated carriers within the emitter will need to diffuse and tunnel through the double barrier until they reach the collector where they drift under the influence of the electric field. Taking the longest emitter as an example (500 nm), we estimate the time taken (transit time) for the slowest carrier (hole) to diffuse according to (3), where L is the length to be diffused, and D is the diffusion coefficient of the carrier in question
The diffusion constant for holes in In0.53Ga0.47As is 8 cm2.s-1 [28], which gives a transit time of 312.5 ps and a frequency around 1.6 GHz. Accordingly, shorter emitters of 100 and 250 nm will have transit times of 12.5 and 78.125 ps and bandwidths of 40 GHz and 6.4 GHz respectively. Hence, we cannot explain the slow response of the devices by diffusion from the screened emitter.
The low-bandwidth tallies more with the lifetime of photo-generated carriers in In0.53Ga0.47As. Assuming so implies that photo-generated carriers will not even diffuse due to the lack of a concentration gradient. The highly doped bottom contacts are one such place where this might occur. Even though light absorption is lower in areas of high doping densities, due to the band-filling effect, the value of the absorption coefficient is still considerable for In0.53Ga0.47As operating at 1.55 µm of roughly 2000 cm-1 [30].
The epilayer structures studied have a thick and photo-sensitive bottom contact layer. The n++-n junction at the boundaries of the collector-contact and the collector is usually reverse biased and does not allow the electrons which are photo-generated at the collector-contact to pass through. Only holes that are generated near the junction will drift through, and the holes generated away from it will not. Therefore, most of the carriers generated at the bottom contact will not be able to diffuse nor drift under an electric field, as the electric field in highly doped regions is nearly zero. Hence, these carriers are left there to recombine.
Furthermore, the beam will be absorbed fully in all areas of the device since it goes through multiple passes through reflection at the bottom metal chuck that the samples are tested on. This is shown in Fig. 9.
There have been numerous studies on the carrier lifetime of In0.53Ga0.47As/InP. The results are diverse and span orders of magnitude from picosecond to microsecond range [31–33]. However, our measurements agree well with the results reported by [34].
To test this hypothesis, we need to consider structures that eliminate absorption at the contacts by using high bandgap materials such as InP. Before we move into that, we discuss a test structure that might give more insights into the effect of the DBQW. The structure, in table (4), is a double heterojunction diode (DHD) with a similar epi-structure as sample 1 of the RTD-PDs, without the DBQW (sample 5).
Table 4. Epitaxial layer design of sample 5
We have successfully fabricated devices using the same process and masks as those for the RTD-PDs. The result of the impulse response for a 10 × 10 µm2 device (DHD) is in Fig. 10 (the figure also shows the impulse response of sample 6 (PIN-RTD) which we discuss later). We see that the bandwidth is close to the RTD-PDs supporting our argument.
Fig. 10. Impulse response of sample 5 (DHD) and sample 6 (PIN-RTD). Inset shows the pulse detected by the commercial PIN photodetector.
We now move forward to test our previous hypothesis. We proposed multiple designs having contacts of a higher bandgap material (InP), and appropriate doping levels with bandwidths up to 10 GHz in [19]. One such design is based on a p-i-n scheme where the DBQW is grown inside a PIN structure which we discuss next (sample 6).
6. Sample 6 fabrication and characterization
The epitaxial layer design is shown in table (5). In this design, the emitter is p-doped for two reasons. First, it offers an appreciable built-in potential across the DBQW structure of nearly 800 mV. Second, it delays the formation of the 2DEG near the emitter barrier to higher voltages, thereby, delaying the screening effect of the 2DEG. This causes enough potential drop across the whole structure and ensures a sufficient electric field to drive the photo-generated carriers with near-saturation velocities. Figure 11 is a simulation of the band diagram near the peak voltage (1 V – according to simulations) that shows such depletion.
Table 5. Epitaxial layer design of sample 6
The diagram in Fig. 11 shows the formation of inversion layers (n-region at the p side, and p-region at the n side). These layers are formed due to charge accumulation, effectively making a p-i-n structure at the emitter and another one at the collector.
A final note related to the discontinuity between the InP and In0.53Ga0.47As, particularly that in the valence band. It is known in the literature [35,36] that such high discontinuity might trap charge with high lifetimes, which limits the speed. However, this is mitigated by introducing the grading layers of the quaternary compound InGaAsP with compositions and widths that guarantees band smoothing and eliminates charge trapping by utilizing the theory first presented by Oldham and Milnes [37,38]. More details on the calculations and principle of operation of this device can be obtained from [19].
Sample 6 was fabricated using photolithography. Fabrication started with metal evaporation of Ni/Pt/Au 50/115/100 nm followed by annealing at 380°C for 1 minute to form the top contact to the p-doped InP layer. The mesa was formed starting with chemical wet etching of the top InP layers using 1:4 HCl:H3PO4 followed by 1:1:38 H3PO4:H2O2:H2O to etch the subsequent In0.53Ga0.47As layers. Then Au/Ge/Ni/Au 40/50/25/100 nm was evaporated and annealed at 400°C for 15 seconds to form the contact to the bottom n-doped InP layer. 2 µm thick Polyimide PI-2545 was used for device passivation. Finally, bond-pad metallization was done using Ti/Au. Figure 12 shows the IV curves for multiple devices on the same chip at different locations. The curves show an NDR, though at a higher voltage than anticipated. The reason for such a shift is still unclear, but the contact resistance of the new devices might have a role in that. The IVs differ in terms of peak current and voltage values. It seems likely that, due to variations in the fabrication process, some locations on the sample had higher overall resistance than others. That explains the reduction in current and the shift of the peak to higher values. Interestingly, the peak-to-valley current ratio (PVCR) is drastically affected, with some devices showing milder NDR features. Figure 10 shows the impulse response using the same setup while the device is biased at 1 V.
Fig. 12. IV results for various devices from sample 6 at different locations. Inset shows the NDR of Dev. 4.
The response is faster than the previous samples with a pulse width of circa 500 ps, which supports our hypothesis on the effect of charge generation at the bottom contact. The measured pulse also does not have any significant diffusion tail, which signifies the depletion of the structure (the emitter and the collector). This pulse width is the same as the AWG signal, and the device might perform faster. Since our AWG does not go below 500 ps, we measured the 3 dB bandwidth of the device using a sinusoidal signal. The AWG gives a constant output sinusoidal wave of 4 Vpp up to 2 GHz. Figure 13 depicts the Bode plots for two devices (Dev. 1 -lower plot, and Dev. 4 -upper plot). Starting with Dev. 1 (lower plot of Fig. 13), the electrical 3 dB bandwidth is 370 MHz, and the optical one is 1.23 GHz. The electrical bandwidth corresponds to the 3-dB point of the detected signal’s power, while the optical bandwidth is the 6-dB point. These two bandwidths are typically used by commercial manufacturers to describe their photodetectors. One example being the commercial PIN detector we are using here [39].
Table (6) shows the measured bandwidths for the other samples, noting that these bandwidths were obtained for bias voltages near the peak (around 0.7 V for samples 1, 3 and 4 and 1 V for samples 2 and 5).
Table 6. Measured 3 dB bandwidth for all samples
Despite this being much higher than all the previous wafers, it is still quite low compared with the expected performance. This leads us to believe that there are other reasons related to the DBQW causing the speed limitation. To investigate further, we studied the relationship between the applied voltage and the decay time of the pulse. The voltage was ramped from 0 V to 12 V in steps of 100 mV while recording the decay time. Figure 14 shows the results superimposed on the IV curve. The data in Fig. 13 were obtained at bias points where the response was the fastest as portrayed in Fig. 14.
Fig. 14. IV curve of sample 6 (PIN-RTD) and response time (signal fall time) as a function of voltage. Highlighted section shows when the device is oscillating.
We notice that the response time follows the original electrical pulse at low voltages (up to about 1.5 V) and builds up as voltage increases up to a certain point near the shoulder of the IV curve (near 4 V). This shoulder is a signature of the formation of the 2DEG [40,41]. Afterward, the decay time decreases again until it goes back to the low-bias values at the beginning of the second PDR. Given the above, we believe that the lifetime of the photo-generated carriers, which accumulate at the barriers is the cause for the speed limit. We can interpret the results in Fig. 14 to show a correlation between the lifetime and the accumulation of charge. This is evident by the increase in the fall time until the 2DEG forms (i.e., accumulation is high, and degeneracy occurs). We attribute the reduction in the fall time following the formation of the 2DEG to enhanced tunneling and, eventually, charge leakage from the quantum well leading to recombination.
To investigate further, we measured the 3 dB cut-off frequency for Dev. 4 (upper plot of Fig. 13) from the same sample and found that it operated at a higher bandwidth reaching 1.75 GHz optical and 1.26 GHz electrical. Examining the IVs in Fig. 12 we see that, devices with milder NDRs have lower peak currents and higher peak voltages signifying higher overall resistance. A device with higher resistance will have less current injection and less accumulated charge, which translates into a speedier response.
It might seem counter intuitive at first, since increasing the resistance should increase the RC time, and the device should respond slower. However, this is not the case here. The equivalent circuit of the RTD is shown in Fig. 15, which has a transfer function given by (4)
Notice that for a given L, C, and RNDR, the response time decreases with increasing Rs, which is the case when increasing the contact resistance.
Before we conclude, there are a couple of points we would like to discuss. The first is related to the response time measurement in the NDR region of Fig. 14. RTDs biased in the NDR can oscillate if the conditions for oscillations are satisfied. This usually is the case, and the RTD oscillates with a frequency depending on the RLC elements of the RTD itself and the bias circuitry. The highlighted section in Fig. 14 shows when this particular RTD oscillates, and indeed the response time measurements there are not accurate. However, for the rest of the NDR region the device is stable and detects the light signal as expected. A stable point in the NDR region can indeed exist and forms the basis behind the operation of RTD amplifiers [3].
The second point is related to the electric field switching between the peak and valley points. We have tried to measure the speed of response at the valley points of the above samples. For sample 6 the detected pulse width was around 500 ps (Fig. 14) so a 3 dB bandwidth measure was necessary. The measurement resulted in an optical speed of response of about 770 MHz (lower plot of Fig. 16) which is substantially lower than the measured 1.26 GHz for the same device (Dev. 1). Dev. 4 of the same sample showed a reduction as well from 1.75 GHz to around 1.4 GHz at the valley point (upper plot of Fig. 16). These results are in favor of the explanation provided above, as when biased at the valley point, the electric field at the depleted part of the collector will increase owing to the increase in charge accumulation at the collector barrier.
As we have showed earlier [29], the collector depletion region is controlled by the Debye screening length and any increase in the number of accumulated charges rises the hole concentration in the depleted region of the collector (however the depletion length will remine roughly the same). We tried to do the same for samples 1 - 4, however that was not possible as the devices did not show any measurable light detection at the valley points or at the second PDR. The reason might be that, depending on the level of doping of the collector, and how long it is, the optical response of the device will be affected. In our case (for samples 2, 3 and 4), the collector is roughly 20 nm long and should be fully depleted given the moderate doping levels. At the valley, the increase in hole concentration above degeneracy levels produces a state of self-induced transparency which explains why these devices do not absorb at the valley point or have very low responsivity. For samples 1 and 6 the collector is much longer (500 nm), and hence only a portion of it becomes transparent. If this is true, then these devices should have lower current responsivity at the second PDR. To support this, we have measured the light response under DC illumination of 1 mW for samples 1 and 2 in Fig. 17. The results seem to support the analysis we presented.
The final issue is related to sample 6 (PIN-RTD). Notice that sample 6 changes the emitter to be p-type and removes any charge photo-generation at the contacts by using a higher bandgap material. This raises the question of which change was directly responsible for the speed enhancement? We can answer this question by considering that in an n-i-n RTD device, a PIN-like structure will always be present at the collector side since holes accumulate near the collector barrier and a p-region forms. At the same time, there is an n-region at the boundaries of the collector and contact region, effectively making a PIN structure. This is particularly apparent with the structures having shorter collectors that are fully depleted. Had the presence of a p-region been the main cause of speed enhancement, we should measure lower response times from the n-i-n devices. Additionally, if we assume diffusive transport (which is slower than drift) then, as per the transit time calculations, the response time should be much faster than what we have measured (in the range of picoseconds instead of nanoseconds) for diffusion lengths up to 500 nm. Due to these reasons, we are more inclined towards attributing the speed enhancement to the lack of charge generation at the contacts.
7. Conclusion
In this paper, we studied the physical limitations on the optical speed of response for resonant tunneling diode-based photodetectors. Our approach relied on investigating the optical impulse response of multiple epitaxial layer designs and building physical interpretations around them. The outcomes show that light absorption (charge carrier generation) at highly doped contact regions is detrimental to the speed. Furthermore, the lifetime of the photo-generated charges (particularly the holes), which accumulate at the barriers is key to the speed of response. This time is inversely proportional to the density of the accumulated charges. We show that devices with lower accumulated charges operate at higher bandwidths reaching 1.75 GHz. These devices have higher overall resistance which shows that the performance can be enhanced by proper adjustment of the devices’ parasitics and/or external circuitry. However, this would be at the expense of a reduced peak-to-valley current ratio and reduced gain.
Funding
European Union’s Horizon 2020 research and innovation programme, Marie Skłodowska-Curie (765426 TeraApps); European Union, HORIZON EUROPE Framework Programme (828841 – ChipAI); Fundação para a Ciência e a Tecnologia (UIDB/00099/2020).
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
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