Djeber Guendouz
email: djeber.guendouz@ims-bordeaux.fr

Mukherjee Chhandak
email: chhandak.mukherjee@ims-bordeaux.fr

Marina Deng
email: marina.deng@ims-bordeaux.fr

Magali De Matos

Christophe Caillaud
email: christophe.caillaud@3-5lab.fr

Hervé Bertin
email: herve.bertin@3-5lab.fr

Antoine Bobin
email: antoine.bobin@3-5lab.fr

Nicolas Vaissière
email: nicolas.vaissiere@3-5lab.fr

Karim Mekhazni
email: karim.mekhazni@3-5lab.fr

Franck Mallecot
email: franck.mallecot@3-5lab.fr

Chhandak Mukherjee

Magali De Matos
email: magali.dematos@ims-bordeaux.fr

Akshay M Arabhavi
email: akshaym@mwe.ee.ethz.ch

Rimjhim Chaudhary
email: rchaudhary@mwe.ee.ethz.ch

Olivier Ostinelli
email: oolivier@mwe.ee.ethz.ch

Colombo Bolognesi
email: bolognesi@mwe.ee.ethz.ch

Patrick Mounaix
email: patrick.mounaix@ims-bordeaux.fr

Cristell Maneux
email: cristell.maneux@ims-bordeaux.fr

Multiscale compact modelling of UTC-photodiodes enabling monolithic terahertz communication systems design

Keywords: uni-traveling carrier photodiode, compact model, SPICE, terahertz, deembedding, on-wafer characterization, bandwidth

come

Introduction

Data traffic is surging with the emergence of new applications such as virtual and augmented reality, autonomous driving and Internet of Things (IoT) [START_REF] Akyildiz | 6G and Beyond: The Future of Wireless Communications Systems[END_REF]. To meet these new requirements in terms of bandwidth and power level, new high-speed communication technologies must be developed. Indeed, the minimum required bandwidth for a high-speed communication system must reach several tens of GHz, which requires an even higher carrier frequency in the THz range (0.1-10 THz). To manage this data traffic, monolithic integration of Opto-Electronic Integrated Circuits (OEICs) (see Figure 1) is a viable solution and very promising among the new emerging THz technologies [START_REF] Nagatsuma | Advances in terahertz communications accelerated by photonics[END_REF]. The monolithic integration of optical and high-speed electronic devices on the same InP substrate minimizes the signal propagation losses between the two device types. The use of a common substrate could also offer a more compact footprint, simplified testing and reduced costs [START_REF] Mukherjee | Towards Monolithic Indium Phosphide (InP)-Based Electronic Photonic Technologies for beyond 5G Communication Systems[END_REF][START_REF] Koch | A Monolithic Bipolar CMOS Electronic-Plasmonic High-Speed Transmitter[END_REF]. The architecture of OEI circuits and systems varies across application types and can rapidly become complex. The need for a co-design and verification infrastructure for such systems capable of supporting billions of devices has therefore become imperative. In addition, co-design is highly dependent on the accuracy of compact models of each block. The compact models are in fact complete physical models converted into simpler and easy-to-solve analytical equations through reasonable assumptions and approximations. These models are commonly transcribed into programming languages such as Verilog-A, compatible with SPICE circuit design tools and software (Cadence, Mentor Graphics, etc.). In electronics, compact models have already reached a significant degree of maturity and computational efficiency over the last few decades. To enable design of high-speed integrated communication technologies, the photonics community has also started to develop compact models for optoelectronic devices, in order to unify electrical, photonic and electromagnetic modeling using the same existing software design tools [START_REF] Sorace-Agaskar | Electro-Optical Co-Simulation for Integrated CMOS Photonic Circuits with VerilogA[END_REF][START_REF] Mukherjee | Reliability-Aware Circuit Design Methodology for beyond-5g Communication Systems[END_REF][START_REF] Shawon | Rapid Simulation of Photonic Integrated Circuits Using Verilog-A Compact Models[END_REF]. It is important to note that a scalable compact model offers two-fold benefit. At the device level, a scalable compact model is a powerful tool for predicting performances for varying device geometries, it also provides a quick feedback on the impact of each technological and physical parameter. At the circuit/system level, compact models enable optimization of bias points and interconnections between photonic and electronic components, providing guidelines for predicting the performances of new OEIC designs.

One of the key components capable of bridging these two worlds of optics and electronics is a photodiode. It allows to recover the information carried by an optical signal in the form of electrical information. There are several photodiode technologies such as conventional PIN photodiodes and avalanche photodiodes. The uni-traveling carrier photodiodes (UTC-PD) represent the most promising one. In particular, UTC-PDs have been widely studied for ultrafast optoelectronic applications, especially for the generation of THz signals [START_REF] Song | Handbook of terahertz technologies: Devices and applications[END_REF]. High performances have been reported demonstrating bandwidths of more than 600 GHz reached with good power levels [START_REF] Song | Handbook of terahertz technologies: Devices and applications[END_REF]. By relying mainly on electron transport, this architecture significantly improves the bandwidth and the power of the UTC-PDs compared to the conventional PIN photodiodes. As a result, UTC-PDs have become a key element in the convergence of electronic and photonic technologies in the millimeter wave band [START_REF] Ishibashi | Uni-Travelling-Carrier Photodiodes[END_REF][START_REF] Ito | High-Power Photonic Millimetre Wave Generation at 100GHz Using Matching-Circuit-Integrated Uni-Travelling-Carrier Photodiodes[END_REF].

With the intention of developing a SPICE compatible model for UTC-PDs, a first compact modeling effort was presented in [START_REF] Mukherjee | Efficient compact modelling of UTC-photodiode towards terahertz communication system design[END_REF]. This compact model has been validated on type-І heterostructure UTC-PDs. Moreover, it is compatible with the Verilog-A design tools. In a previous study, a validation of the HiCuM/L2 compact model, developed mainly for SiGe HBT, on III-V type І (InGaAs/InP-based) and type ΙΙ (GaAsSb/InP-based) double heterojunction bipolar transistors (DHBTs) [START_REF] Schroter | SiGeC and InP HBT Compact Modeling for Mm-Wave and THz Applications[END_REF][START_REF] Mukherjee | Scalable Compact Modeling of III-V DHBTs: Prospective Figures of Merit Toward Terahertz Operation[END_REF] has also been presented in a first attempt to combine the two technological paradigms. Since UTC-PDs and DHBTs could share the same InP substrate and possibly base-collector epitaxial heterostructure, this combined modeling approach can pave the way for the design of monolithic integrated OEICs.

In continuation of previous works, this study presents improvements of the first UTC-PD compact model. We have introduced more physical equations describing, in particular, the dark current, the intrinsic series resistance and the junction capacitance as well as their dependence on the applied bias. As a consequence, the new model now takes into account more material parameters such as doping, mobility and permittivity. Scaling laws have also been implemented by analyzing the geometry of the UTC-PDs. In addition, new accurate de-embedding methods are proposed to overcome the limitations of conventional methods. A scalable parameter extraction methodology is presented for type I and type II heterostructure UTC-PDs, i.e. InGaAs/InGaAsP and GaInAsSb/InP, respectively, providing very good results over several geometries, bias points and frequencies. It is important to note that the model validation does not focus primarily on the type of heterostructure but on the design of the UTC-PDs, test structures as well as the frequency used in the study.

This paper is organized as follows: section 2 provides the formulation of the improvements that have been implemented in the new model and presents the parameter extraction methodology. Section 3 reports the compact model validation and the results, including the extraction of the junction capacitances, intrinsic series resistances, current parameters, and the bandwidths. Finally, Section 4 concludes the study and discusses future perspectives.

Compact model development

Model description

The UTC-PD compact model can be represented by an equivalent circuit with three terminals. Among these terminals, two are for the electrical part and one for the optical input signal. The model translates the analytical equations based on the physics of UTC-PDs into an equivalent electrical circuit representation. Figure 2a shows a SEM micrograph o taken after the UTC-PD mesa formation. Figure 2b illustrates the schematic of an intrinsic UTC-PD mesa structure, showing different epitaxial layers. On this schematic, the small signal electrical model is superimposed depicting different electrical elements of the equivalent circuit representation. The active region consists principally of the epitaxial p-doped absorption layer and a slightly n-doped collection layer. The electrical representation of this region is composed of a junction capacitance, 𝐶𝑗, which describes the absorber-collector junction. In parallel to this capacitance, three current sources model the forward current, the reverse current and the photocurrent [START_REF] Ishibashi | Uni-Traveling-Carrier Photodiodes[END_REF], respectively. Note that here the photocurrent implementation in Verilog-A is more complex than simple current source representations since it includes transfer functions (Figure 2c). In series with the active region, there are resistances representing the resistive contribution of the non-depleted part of the collector, the sheet resistances and the resistances of the metal contacts. The device can be biased via the doped P+ and N+ contacts at the top and the bottom, respectively. On both sides of the mesa, there are the access lines, which are connected to the Ground-Signal-Ground (GSG) pads (Figure 2d).

Model formulation

The first compact model developed in [START_REF] Mukherjee | Efficient compact modelling of UTC-photodiode towards terahertz communication system design[END_REF] serves as the starting point of the modeling framework presented in this work. In order to make the model versatile across different technologies and geometries, improvements in model equations and implementations have been made. In the following sections, the details of the principal improvements are presented.

Firstly, the direct current contribution, 𝐼 𝐹 , has been slightly modified to take into account the two operating regimes: low and high injections [START_REF] Banáš | Accurate Diode Behavioral Model with Reverse Recovery[END_REF]. Thus, the direct current equation becomes the following:

𝐼 𝐹 = 𝐼 𝐷 1 + √ 𝐼 𝐷 𝐼 𝐾 (1)
Here 𝐼 𝐷 is the classic diode equation [START_REF] Brinson | Qucs: A GPL Software Package for Circuit Simulation, Compact Device Modelling and Circuit Macromodelling from DC to RF and Beyond[END_REF] and 𝐼 𝐾 is the knee current parameter that signifies the transition between the medium and high injection regimes.

Among other improvements, we incorporate a more detailed description of the of the series resistances through decorrelation of the resistive contributions as functions of the applied bias and device geometry. In the first version of the model, only one series resistance, independent of the bias voltage, was considered. It reflects the resistance of the contacts and the spreading resistance. However, the thickness and the doping level of the collector varies from one technology to another (usually 𝑁 -doped). When the collector is not fully depleted, an additional bias-dependent resistance appears which, when the collector is fully depleted, becomes negligible. In our new model, therefore, these two contributions are separated. The dependence of the resistance of the active region, 𝑅 𝑁 -, on the applied voltage, 𝑉 𝑑 , can be expressed as follows:

𝑅 𝑁 -(𝑉 𝑑 ) = 𝑅 𝑁 -(0) 𝑤 𝑇 -𝑤 𝑠𝑐𝑟 (𝑉 𝑑 ) 𝑤 𝑇 -𝑤 𝑆𝐶𝑅0 (2)
Where 𝑅 𝑁 -(0) is the resistance of the undepleted collector at 𝑧𝑒𝑟𝑜 bias, 𝑤 𝑇 is the width of the collector and 𝑤 𝑆𝐶𝑅0 is the width of the space charge region at 𝑧𝑒𝑟𝑜 bias. 𝑤 𝑠𝑐𝑟 (𝑉 𝑑 ) and 𝑅 𝑁 -(0) can be expressed with the following set of equations [START_REF] Scholten | The Physical Background of JUNGAP2[END_REF]:

𝑤 𝑠𝑐𝑟 (𝑉 𝑑 ) = 𝑤 𝑠𝑐𝑟0 (1 - 𝑉 𝑑 𝑉 𝑗 ) 𝑀𝑚𝑟 (3)
𝑅 𝑁 -(0) = 𝑤 𝑇 -𝑤 𝑆𝐶𝑅0 𝑞. µ. 𝑁 -. 𝐴 (4)
Where 𝑉 𝑗 is the junction potential and 𝑀 𝑚𝑟 is the grading coefficient at medium reverse bias of the 𝑃-𝑁 -junction, 𝑞 is the electronic charge, µ is the electron carrier mobility in the collector, 𝑁 -is the doping concentration in the collector and 𝐴 is the UTC-PD active area.

The classical model of the bias-dependence of the junction capacitance has evolved into a more mature form in the HiCuM/L2 model. We refer here mainly to the formulation for the base-collector junction. Considering different thicknesses, the collector may not be completely depleted at equilibrium. In this case, the junction capacitance follows the conventional model [START_REF] Scholten | The Physical Background of JUNGAP2[END_REF] until the collector is completely depleted, the voltage drop becomes small due to the high doping level of the waveguide (or n-contact). In reverse bias, the model of the junction capacitance thus consists of two contributions:

𝐶 𝑗𝑇 (𝑉 𝑑 ) = 𝐶 𝑗𝑚 (𝑉 𝑑 ) + 𝐶 𝑗𝑙 (𝑉 𝑑 ) (5)
Where the first contribution, 𝐶 𝑗𝑚 (𝑉 𝑑 ), represents the contribution at medium reverse bias, described by the classical equation:

𝐶 𝑗𝑚 (𝑉 𝑑 ) = 𝐶 𝑗0𝑚 (1 - 𝑉 𝑑 𝑉 𝑗 ) -𝑀 𝑚𝑟 (6)
The second part, 𝐶 𝑗𝑙 (𝑉 𝑑 ), represents the contribution around and beyond the condition of complete depletion of the collector. It can be expressed as follows:

𝐶 𝑗𝑙 (𝑉 𝑑 ) = 𝐶 𝑗0𝑙 (1 - 𝑉 𝑑 𝑉 𝑗 ) -𝑀 𝑙𝑟 (7)
Where 𝐶 𝑗0𝑙 and 𝑀 𝑙𝑟 can be calculated internally from the punch-through bias, 𝑉 𝑃𝑇 = 𝑞𝑁 - 2𝜀 𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑜𝑟 ⋅ 𝑊 𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑜𝑟 2 and 𝑀 𝑚𝑟 respectively [START_REF] Schroter | HICUM/L2 A Geometry Scalable Physics-Based Compact Bipolar Transistor Model[END_REF].

In standard compact modeling practices, the time derivatives of the charges are used for the capacitance model. Thus, in our case, the two contributing charges of the capacitance model are multiplied by smoothing functions (or weights) that allow a continuously differentiable transition across the two adjacent bias regions.

Moreover, in our model we also calculate the electric field, 𝐸 𝑚𝑎𝑥 (𝑣 𝑑 ), at the input of the collector which is included in the expressions of the tunneling currents. We can then use these expressions to calculate the transit time of electrons in the collector, 𝜏 𝑐 , from their drift velocity, which can be expressed by the following empirical expression [START_REF] Eastman | Design Criteria for GaAs MESFETs Related to Stationary High Field Domains[END_REF]:

𝑣(𝐸) = 𝑣 𝑠𝑎𝑡 (1 + 𝐸 𝐸 𝑠𝑐𝑎𝑙𝑒 -1 1 + 𝐴( 𝐸 𝐸 𝑠𝑐𝑎𝑙𝑒 ) 𝑡 ) (8)
Where 𝑣 𝑠𝑎𝑡 is the saturation velocity, 𝐸 𝑠𝑐𝑎𝑙𝑒 = 𝑣 𝑠𝑎𝑡 µ , 𝐴 𝑎𝑛𝑑 𝑡 are parameters to be determined from the material used. The parameters of eq. ( 8) can be determined from a fit on the data of each material used.

Parameter extraction methodology

The extraction method has been developed in accordance with the standard extraction flow used for bipolar transistor technologies. To begin the extraction process, an initialization of the physical and technological parameters is necessary. Next, a sequence of measurements and parameter extractions are followed that leads to the extractions of the intrinsic and extrinsic parameters as shown in the flowchart in Figure 3.

The measurement of one-port 𝑆-parameters of the test structures are first measured to extract the electrical equivalent circuit elements of the RF pads and access lines. It is important to note that the design of the test structures can vary from one technology to another. Therefore, the de-embedding method must be adapted to the available test structures. Subsequently, one-port 𝑆-parameter measurements at various biases followed by a de-embedding process allow accessing the intrinsic 𝑆-parameters of the UTC-PDs. By converting these Sparameters to 𝑌 or 𝑍-parameters, the elements of the UTC-PD equivalent electrical circuit can be extracted. The parameters of the junction, 𝐶 𝑗0 ,𝑉 𝑗 and 𝑀 𝑚𝑟 can be extracted by taking the logarithm on both sides of Equation ( 6) in order to linearize it and then by performing an iteration on 𝑉𝑗, at low reverse voltage to obtain the best linear regression. Once 𝐶 𝑗0 is determined, the set of parameters: 𝑊 𝑆𝐶𝑅0 , 𝐸 𝑚𝑎𝑥 (𝑣 𝑑 = 0) and 𝑅 𝑁 -(0) can be calculated. An optimization on the 𝑉 𝑃𝑇 value, at high voltage, allows to determine 𝐶 𝑗0𝑙 . After determining the junction parameters, forward and reverse current measurements without illumination are used to extract the DC characteristics of the UTC-PD junction as well as the reverse current parameters (band-to-band tunnel current and trap assisted tunnel current parameters) [START_REF] Scholten | The Physical Background of JUNGAP2[END_REF][START_REF] Verdun | Dark Current Investigation in Thin P-i-N InGaAs Photodiodes for Nano-Resonators[END_REF].

Finally, static photocurrent and bandwidth measurements are exploited to determine the optical parameters such as responsivity, 𝑅, and the transit times. The expression of the photo current includes two transit times [START_REF] Ishibashi | Uni-Traveling-Carrier Photodiodes[END_REF]: the transit time in the absorber, 𝜏 𝑎 , and the transit time in the collector,𝜏 𝑐 . The calculation of 𝜏 𝑐 via the electric field, 𝐸 𝑚𝑎𝑥 (𝑣 𝑑 ), at the collector input, leaves only one transit time, 𝜏 𝑎 , to be extracted, which can be determined using the bandwidth measurements. It is important to note that, unlike conventional one-port Sparameter measurements that can be de-embedded, the bandwidth measurements still include the contributions from the pads and access lines. The extracted values of these extrinsic parasitic contributions are therefore reinjected into the final model to obtain realistic results.

Compact model validation

For the validation of the model scalability and affirming model versatility, both type I and II [START_REF] Milnes | Heterojunctions and Metal Semiconductor Junctions[END_REF] UTC-PDs were measured. More specifically, two varieties of type I UTC-PDs have been characterized: (i) an InGaAs/InGaAsP UTC-PD with rectangular waveguides, through which the light is guided evanescently into the active region [START_REF] Anagnosti | Optimized High Speed UTC Photodiode for 100 Gbit/s Applications[END_REF]; (ii) an InGaAs/InP UTC-PD with a simple topilluminated structure. The type II device consist of a GaInAsSb/InP UTC-PD also with a simple top-illuminated structure [START_REF] Arabhavi | Type-II GaInAsSb/InP Uniform Absorber High Speed Uni-Traveling Carrier Photodiodes[END_REF]. Several geometries of the UTC-PDs were investigated in the study, with their active areas listed in Table 1. Dedicated test structures associated with the UTC-PDs were available for extraction of their intrinsic parameters. At high frequencies (high data rates), the impact of the parasitic interconnects becomes even more important and must be accurately modelled. In our compact modelling work, this step is thus very important. Accessing the intrinsic physical parameters of the UTC-PD requires careful modeling of the RF pads and access lines. To achieve this goal, test structures reproducing the pads and access lines are required, such as the Open and the Short test structures presented in Figure 4. Indeed, there are several de-embedding techniques that can be applied to on-wafer devices, as reported in the literature [START_REF] Xu | A De-Embedding Procedure for One-Port Active Mm-Wave Devices[END_REF]. A comprehensive investigation of the validity of these methods has been reported in [START_REF] Konstantinou | Investigation of De-Embedding Techniques Applied on Uni-Traveling Carrier Photodiodes[END_REF]. The de-embedding methods are generally based on on-wafer 𝑆parameter measurements of different dedicated test structures, which are then subtracted from the global S-parameter measurements of the devices under test. The number of test structures and the complexity of the de-embedding method mainly depend on the frequency at which the device operates. In this article, we show the limitations of the basic Open-Short de-embedding method at high frequencies (110 GHz). We then introduce a new de-embedding method that we have developed, which can be adapted to different designs of test structures and the frequency range of use. In order to apply the de-embedding method on the InGaAs/InGaAsP UTC-PDs, one-port S-parameter measurements were performed up to 110 GHz on dedicated test structures, which consist of an Open, a Short and some other complementary test structures that are presented later. The S-parameter measurements were preceded by an off-wafer SOLT calibration on a CS-5 substrate, in order to bring the measurement reference planes at the probe tips (Picoprobe tips). The One-port 𝑆-parameter measurements were performed using an Agilent E8361 PNA, with a frequency extender for the 67-110 GHz frequency range.

The Open and Short test structures are usually modeled by a single capacitance, an inductance and a resistance. Due to the specific design of the InGaAs/InGaAsP UTC-PDs, this representation at high frequency is no longer valid as shown by the frequency-dependent behaviors shown in Figure 5a. Consequently, complementary structures been designed (Figure 5b), that reproduce the pads and the access lines. Along with these, an on-wafer characterization and a step-by-step reconstruction of the distributed network allows to extract the intrinsic characteristics of the UTC-PDs. With the equivalent circuits of the pads and access lines determined, the 𝑆-parameter matrix can be simulated and the reflection coefficient of the UTC-PD can therefore be calculated by the following expression:

𝛤 𝑈𝑇𝐶-𝑃𝐷 = 𝑆 11 -𝛤 𝑀 𝑆 11 𝑆 22 -𝑆 22 𝛤 𝑀 -𝑆 12 𝑆 21 (9)
Where, 𝑆 ij are the S-matrix elements of the pads and access lines and 𝛤 𝑀 is the measured total reflection coefficient. The overall equivalent circuit of the pads and access lines has three sections, as shown in Figure 5c,d. The first section represents the equivalent electrical circuit of the pads. It includes an inductance, 𝐿 𝑃𝑎𝑑 , and a capacitance, 𝐶 𝑃𝑎𝑑 , of the pads. The lumped parasitic elements of the pads can be determined from measurements of the reflection coefficients of the Pads-Short, Г 𝑃𝑎𝑑_𝑆ℎ𝑜𝑟𝑡 , and Pads-Open, Г 𝑃𝑎𝑑_𝑂𝑝𝑒𝑛 , test structures. Extracting the inductive element, 𝐿 𝑃𝑎𝑑 , from Г 𝑃𝑎𝑑_𝑆ℎ𝑜𝑟𝑡 , yields an overestimated value (Figure 6a), as it actually includes a ground return inductance. A in the value of 𝐿 𝑃𝑎𝑑 can be made by optimizing the simulation on the imaginary part of 𝑌 𝑃𝑎𝑑_𝑆ℎ𝑜𝑟𝑡 /𝜔 extracted from measurement at high frequency. The capacitance of the pads, 𝐶 𝑃𝑎𝑑 , can be extracted from 𝑌 𝑃𝑎𝑑_𝑂𝑝𝑒𝑛 /𝜔 at low frequency (Figure 6a).

The same process was performed to reconstruct the other two sections using the mesa-free Open and Short test structures and the earlier test structures (Open and Short) that include the mesa accesses. The equivalent capacitances of the mesa-free access, 𝐶 𝑎𝑐𝑐𝑒𝑠𝑠 , and mesa access, 𝐶 𝑚𝑒𝑠𝑎 , can be determined at low frequency from the imaginary part of 𝑌 𝑚𝑒𝑠𝑎-𝑓𝑟𝑒𝑒_𝑂𝑝𝑒𝑛 and 𝑌 𝑂𝑝𝑒𝑛 , respectively (Figure 6b). The equivalent inductance of the mesa-free access, 𝐿 𝑎𝑐𝑐𝑒𝑠𝑠 , and the mesa inductances, 𝐿 𝑚𝑒𝑠𝑎 , were determined by optimization on the high frequency region of the measured 𝑌-parameters. The access resistance, 𝑅 𝑎𝑐𝑐𝑒𝑠𝑠 , can be determined by averaging over the real part of 𝑍 𝑚𝑒𝑠𝑎-𝑓𝑟𝑒𝑒_𝑆ℎ𝑜𝑟𝑡 . The complete set of parasitic lumped elements are summarized in Table 2.

Table 2. Extracted values of the lumped electrical elements, equivalent to the pads and access lines for the InGaAs/InGaAsP UTC-PDs. It is worth noting that this contribution was accepted at the EuMW 2022 conference [START_REF] Guendouz | Access Modelling-Based De-Embedding Method for High-Frequency Characterization of Uni-Traveling Carrier Photodiodes[END_REF].

GaInAsSb/InP UTC-PD de-embedding

In order to apply de-embedding on the GaInAsSb/InP UTC-PDs, one-port S-parameter measurements were performed up to 67 GHz on Open and Short test structures. The 𝑆-parameter measurements were preceded by an off-wafer LRRM (line/reflection/match) calibration [START_REF] Davidson | LRM and LRRM Calibrations with Automatic Determination of Load Inductance[END_REF] in order to bring the measurement reference plane up to the RF probe tips. For the GaInAsSb/InP UTC-PDs, the electrical elements equivalent to the Open and Short test structures (Figure 7a) are found to be almost frequency independent in the frequency range of study (up to 67 GHz) (Figure 7b). The use of the classical Open-Short/Short-Open deembedding methods allows extraction of the junction capacitance, but they do not provide reliable results for the compact model validation over the entire frequency range. This is because of the fact that the use of the Short test structure leads to an under/over de-embedding at high frequency, which in turn results in an intrinsic junction capacitance that depends on the frequency as shown in Figure 7c. This is an unphysical behavior and cannot be captured by the UTC-PD compact model. Modeling the pads and access lines using only single 𝑅𝐿𝐶 network (Open-Short case) is not adequate enough to capture the actual frequency dependence of the UTC-PDs. Therefore, the pads and access lines were modeled by a cascaded 𝑅 𝑛 𝐿 𝑛 𝐶 𝑛 network consisting of 𝑁 sections as shown in Figure 8a. Such implementation has been demonstrated up to 40 GHz in [START_REF] Ehsan | Development of an Equivalent Circuit Model of a Finite Ground Coplanar Waveguide Interconnect in MIS System for Ultra-Broadband Monolithic ICs[END_REF]. In the present study, three cascaded 𝑅 𝑛 𝐿 𝑛 𝐶 𝑛 elements were identified and this was achieved by ensuring that the resulting intrinsic junction capacitance of the UTC-PDs does not exhibit a frequency dependence. The 𝑅 value was extracted from the real part of 𝑍 𝑆ℎ𝑜𝑟𝑡 . 𝐿 and 𝐶 values were determined by optimization on the imaginary part of 𝑌 𝑂𝑝𝑒𝑛 as shown in the Figure 8b, leading to three cascaded 𝑅 𝑛 𝐿 𝑛 𝐶 𝑛 elements. As presented in Figure 8b, the sum of the capacitances, ∑ 𝐶/𝑁 𝑁 𝑖=1

, tends to 𝑌 𝑂𝑝𝑒𝑛 /𝜔 at low frequencies and the sum of the inductances, ∑ 𝐿/𝑁 𝑁 𝑖=1

, was determined through an optimization at high frequency. The extracted parameter values are listed in Table 3. Once the equivalent network is accurately modeled, two-port 𝑆-parameter simulations of the cascaded network are used to determine the intrinsic parameters of the UTC-PD using eq. ( 9). Figure 8c shows The method of de-embedding therefore can be generalized and extended to a wide range of frequency and various Open test structure designs. Taking advantage of this methodology, a dedicated InGaAs/InP UTC-PD Open test structure was characterized using the same setup and measurement procedure as the one used for the test structures of the InGaAs/InGaAsP UTC-PDs. The Open test structure (Figure 9a) was modelled by a 𝑅 𝑛 𝐿 𝑛 𝐶 𝑛 cascade of nine sections. This was achieved by optimizing the ∑ 𝐶/𝑁 𝑁=9 𝑖=1

and ∑ 𝐿/𝑁 𝑁=9 𝑖=1

values on the imaginary part of 𝑌 𝑂𝑝𝑒𝑛 /𝜔, as shown in Figure 9c. Note here that since we did not have a Short test structure, we used a Thru test structure (Figure 9b) to determine the series resistance, ∑ 𝑅/𝑁 𝑁=9 𝑖=1

. This was achieved by converting the measured tow-port S-parameters into the transfer matrix 𝐴𝐵𝐶𝐷, then by determining the complex propagation constant, 𝛾 = 𝛼 + 𝑗𝛽, and the characteristic impedance of the Thru test structure, 𝑍 𝑐 , from the 𝐴𝐵𝐶𝐷 matrix (Figure 9d). Finally, the linear series resistance, 𝑅 𝑇ℎ𝑟𝑢 , (Figure 9e) can be determined from the real part of the product 𝛾 • 𝑍 𝑐 [START_REF] Pozar | Microwave Engineering[END_REF]. By multiplying the 𝑅 𝑇ℎ𝑟𝑢 by the length (278 µm) of the pads and the access lines of the InGaAs/InGaAsP UTC-PD, we obtain ∑ 𝑅/𝑁 𝑁=9 𝑖=1

. The number of the 𝑅 𝑛 𝐿 𝑛 𝐶 𝑛 cascade is a bit high in this case since, on the one hand, we work at higher frequencies (up to 110 GHz), and on the other hand, the access lines of the test structures are relatively longer than the previous case. The values of these elements are reported in Table 4. Once the electrical equivalent network is accurately modeled, two-port 𝑆parameter simulations of the distributed 𝑅𝐿𝐶 network are used to determine the intrinsic parameters of the UTC-PDs by applying eq. ( 9).

Intrinsic junction Capacitance extraction

In order to validate the dynamic behavior of the UTC-PD compact model, one-port 𝑆-parameter measurements were performed. Measurements up to 110 GHz under a reverse bias ranging from 0 to -2𝑉 were carried on the InGaAs/InGaAsP and InGaAs/InP UTC-PDs. Measurements up to 67 GHz under reverse bias ranging from 0 to -2.5 𝑉 were performed on the GaInAsSb/InP UTC-PDs. After applying the suitable de-embedding method for each UTC-PD, following the procedure explained in Section 2, we can validate the compact model on the intrinsic 𝑆-parameters.

Figures 10 and11 show scalable compact model validation against Sparameter measurements of the amplitude and the phase of the different intrinsic UTD-PDs under study. Excellent agreement is observed for all geometries and all UTC-PD technologies affirming the scalability and versatility of the compact model. The new implementation of the junction capacitance model (eqs. ( 5)-( 7)) captures the physics of UTC-PDs much better than the conventional model and thereby offers excellent scalability and agreement with measurement results as depicted in Fig. 11.

Intrinsic series resistance extraction

The overall series resistance can be extracted from the real part of the intrinsic 𝑍 11 -parameter. In the case of the rectangular InGaAs/InGaAsP UTC-PDs, the contact covers the entire mesa, which makes it easy to determine the scaling law by analyzing the effective area of the device geometry. Figure 12 shows the overall series resistance extracted at 80 GHz for the three geometries at various biases. Since the collector is 425 nm thick, it is not completely depleted. As the reverse voltage increases, the collector resistance decreases until it disappears at full collector depletion. The resistance values are extracted for the GaInAsSb/InP UTCPDs from the real part of 𝑍 11 -parameter at a frequency of 60 GHz. The extracted values are: 11.6 Ω, 10.7 Ω and 12.4 Ω for the UTC-PD geometries of 64 µm 2 , 79 µm 2 and 113 µm 2 active area respectively. Contrary to the rectangular waveguide InGaAs/ InGaAsP UTC-PDs, the metallic contacts of the GaInAsSb/InP UTCPDs are located at the edge of the mesa. In addition, the contact width varies for each geometry. It is therefore not surprising that the resistance does not scale with the mesa area. It is important to note that no bias dependence was observed as the collector is already fully depleted at zero bias, due to the small thickness of the collector.

The InGaAs/InP UTC-PDs have a structure similar to the one of the GaInAsSb/InP UTC-PDs where the metal contact is located at the edge of the mesa. A series resistance of 51.3 Ω was extracted at 80 GHz from the real part of 𝑍 11 -parameter.

Validation of DC Characteristics

The validation of the compact model against the dark current characteristics was performed following the extracting of relevant model parameters from the forward and reverse current measurements. These measurements were performed in the bias range of (-3, 1) V, and (-5, 1) V for InGaAs/InGaAsP and GaInAsSb/InP UTC-PDs, respectively. All measurements were performed at room temperature. Figure 13a,b compare the measured DC characteristics and the compact model simulation, demonstrating good accuracy of the model over the entire forward and reverse bias ranges. All UTC-PDs under test show very low dark current levels (below 25 nA at -2V). The low values of the dark current implies a higher quality of the passivation of the junction in the UTC-PDs. We note here that a study of the dark currents at different temperatures along with compact model validation has been presented in a previous study where the model also showed excellent results [START_REF] Mukherjee | Towards Monolithic Indium Phosphide (InP)-Based Electronic Photonic Technologies for beyond 5G Communication Systems[END_REF].

Bandwidth validation

Once the validation of the compact model of the UTC-PD and electrical model parameter extraction have been performed on the DC and RF characteristics under non-illuminated operating conditions, the dynamic photoresponse of the compact model has then been verified for both types of UTC PD technologies, for all available geometries under different applied bias conditions. For the InGaAs/InGaAsP UTC-PDs, the optical frequency response of the photodiodes was measured with a heterodyne measurement bench. A high responsivity of 0.79 A/W, at λ= 1550 nm, has been reported for these devices [START_REF] Mukherjee | Towards Monolithic Indium Phosphide (InP)-Based Electronic Photonic Technologies for beyond 5G Communication Systems[END_REF]. The optical frequency response of the UTC-PDs are measured at -2V bias under an optical power of 3 dBm. -3dB extrinsic bandwidths ranging from 40 GHz to 47 GHz for all geometries studied can be observed in Figure 14a. The intrinsic bandwidth can be determined using the calculated/extracted transit times using the compact model. In addition, intrinsic bandwidths ranging from 45 GHz to 56 GHz are determined, indicating the detrimental impact of the pads and access lines on the bandwidth.

For the optical characterization of the GaInAsSb/InP UTC-PDs, an optical signal (λ = 1550 nm), of 3 dBm power, modulated by a Mach Zehnder modulator (MZM) nm was used. Measurements were performed with a bias ranging from 0 to -2.5 V. Figure 14b shows an excellent agreement between the measurements and the model predictions over the entire frequency and voltage ranges. It should also be noted that the measured bandwidths of the UTC-PDs do not change significantly compared to their intrinsic bandwidth of the component since the values of the junction capacitances are much higher than those of the pads and the access lines. In other words, the pad parasitic elements are less significant for these type of UTC-PDs.

Conclusion

In this work, we presented improvements on the first compact model developed for the UTC-PDs. The model now incorporates more physical parameters and scaling laws. This is particular important for predictive design and optimization of emerging optoelectronic integrated circuits. An extraction method has been developed based on the parameter extraction strategy for bipolar electronic technologies. A new and improved de-embedding methodology that provides access to the intrinsic parameters of the UTC-PD has been demonstrated and incorporated within the parameter extraction flow. The static and dynamic behavior of the model have been validated against on-wafer measurements up to 67 GHz and 110 GHz on three UTC-PD technologies developed on the InP substrates. The model has demonstrated excellent versatility and scalability with respect to two types of UTC-PDs and several geometries under study under a wide range of bias conditions. These results appear very promising, laying an extensive groundwork for the design of OEIC systems. Model compatibility with existing EDA framework also marks an important step for realizing a single simulator based optoelectronic system design. Further advancements through multi-physics co-simulation and design optimization of the two technological modules will be the subject of future work.

Figure 1 .

 1 Figure 1. Schematic view showing future monolithic integration of the Uni-traveling-Carrier photodiodes (UTC-PD) with Double Hetero junction Bipolar Transistors (DHBT) to be used, for instance, in integrated THz transceivers.

Figure 2 .

 2 Figure 2. (a) SEM micrograph of a GaInAsSb/InP UTC-PD (b) Schematic of the intrinsic mesa UTC-PD showing the location of the equivalent circuit elements (c) the Verilog-A topology of the UTC-PD photocurrent transfer function (d) Top-view SEM micrograph of a GaInAsSb/InP UTC-PD connected to the GSG pads.

Figure 2 .

 2 Figure 2. Methodology flow of the measurements and the parameter extraction of the UTC-PD.

Figure 4 .

 4 Figure 4. Schematic of: (a) Open and Short test structures (b) schematic of the UTC-PD connected to the pads and access lines.

Figure 5 .

 5 Figure 5. (a) Extraction of capacitive, inductive and resistive parasitic equivalents to the pads and access lines (b) schematic of the complimentary test structures (c) schematic of the UTC-PD depicting the three parasitic sections (d) global electrical circuit equivalent to the pads and access lines.

Figure 6 .

 6 Figure 6. Extraction of parasitics equivalent for: (a) the Pad Open and the Pad Short (b) the mesa-free access lines.

Figure 7 .

 7 Figure 7. (a) Open and Short test structures (b) Extraction of parasitic capacitance, inductance and resistance equivalents to the Open and Short test structures (b) Extraction of the UTC-PD junction capacitance.

Figure 8 .

 8 Figure 8. (a) Equivalent circuit of the UTC-PD connected to cascaded pads and access lines (b) extracted parasitic elements associated with the Pads and access lines (c) extracted UTC-PD junction capacitances at various reverse bias.

Figure 9 .

 9 Figure 9. (a) Microscope image of the Open test structure (b) microscope image of the Thru test structure (c) extracted 𝐿𝐶 parasitics of the equivalent circuit for the InGaAs/InP UTC-PD Open test structure (d) extracted attenuation constant 𝛼, and phase constant 𝛽 of the Thru test structure (e) extracted characteristic impedance, 𝑍 𝑐 , and calculated linear series resistance 𝑅 𝑇ℎ𝑟𝑢 , of the Thru test structure.

Figure 10 .

 10 Figure 10. Measured (symbol) and simulated (lines) intrinsic S11-parameter for all the UTC-PDs technologies: (a-c-e) magnitude in dB of InGaAs/InGaAsP UTC-PD with actives areas of 100, 125 and 150 µm 2 (b-d-f) phase in degree of InGaAs/InGaAsP UTC-PD with active area of 100, 125 and 150 µm 2 (g-i-k) magnitude in dB of GaInAsSb/InP UTC-PD with active area of 64, 79 and 100 µm 2 (h-j-l) phase in degree of GaInAsSb/InP UTC-PD with active area of 64, 79 and 100 µm 2 (m) magnitude in dB of InGaAs/InP UTC-PD with a 13 µm 2 active area (n) phase in degree of InGaAs/InP UTC-PD of 13 µm 2 of active area.

Figure 11 .

 11 Figure 11. UTC-PD junction capacitance as functions of the active device area and the applied reverse bias: (a) InGaAs/InGaAsP UTC-PD (b) GaInAsSb/InP UTC-PD.

Figure 12 .

 12 Figure 12. The extracted total series resistance at various reverse biases and for several lengths of the InGaAs/ InGaAsP UTC-PDs.

Figure 13 .

 13 Figure 13. Measured (symbol) and simulated (lines) dark current-voltage characteristics of UTC-PDs with several active areas for (a) InGaAs/InGaAsP UTC-PD (b) GaInAsSb/InP UTC-PD.

Figure 14 .

 14 Figure 14. Bandwidths of the UTC-PDs: (a) InGaAs/InGaAsP UTC-PDs (b) GaInAsSb/InP UTC-PD.

Table 1 .

 1 Geometries of the UTC-PDs under study.

	UTC-PD technology	Drawn active area (µm 2 )
	UTC-PD type I	InGaAs/InGaAsP InGaAs/InP	100, 125 and 150 13
	UTC-PD type II	GaInAsSb/InP	64, 79 and 113
	3.1. De-embedding procedure
	InGaAs/InGaAsP UTC-PD de-embedding

Table 3 .

 3 Extracted RLC lumped elements values from GaInAsSb/InP UTC-PD Open and Thru test structures.

	𝑹	𝐋	𝐂
	0.19 Ω	47.55 pH	11.49 fF
	InGaAs/InP UTC-PD de-embedding

Table 4 .

 4 Extracted RLC lumped element values for InGaAs/InP UTC-PD Open test structure.

	𝑹	𝐋	𝐂
	1.5 Ω	106.74 pH	48.06 fF
	3.2. Validation of RF Characteristics