Equivalent Circuit Model of the Carrier-Depletion-Based Push-Pull Silicon Optical Modulators With T-Rail Slow Wave Electrodes

The t-rail electrode is an effective method to enhance the silicon optoelectronic modulator's performance. To design and optimize T-rail electrodes, engineers often rely on finite-element numerical simulations that require complex device modeling and enormous computing resources. In this paper, we present an equivalent circuit model for carrier-depletion-based push-pull silicon modulators with T-rail electrodes. The analytical solution for the bandwidth of the modulator can be derived from the equivalent circuit. The utilization of the analytical solution offers advantages in terms of memory conservation and flexibility. The values calculated by the equivalent circuit model are in excellent agreement with the numerical full-wave HFSS simulations. Hence, the proposed model can accurately and efficiently develop silicon optical modulators.

tempting for short-reach optical interconnects due to its complementary metal oxide semiconductor (CMOS) compatibility.The silicon modulator is a critical component of SiP communication connections [1].Data transmission and reception rely heavily on high-bandwidth electro-optic (EO) modulators.An ideal modulator should possess a wide bandwidth, a low half-wave voltage (Vπ), and be compatible with standard CMOS technology.
The maximum bandwidth of modulators based on silicon is currently 110 GHz [2].The slow light effect in a coupledresonator optical waveguide (CROW) architecture is utilized.The device efficiency factor is 0.013 π/V, which indicates an extremely large Vπ.Although it has a record-breaking bandwidth, this modulator design requires special production methods and possesses considerable Vπ.A substrate-removed 2 mm-long Mach-Zehnder modulator (MZM) with a modulation bandwidth exceeding 50 GHz was reported, the modulator made by changing the substrate material has a suitable Vπ of 7V [3].However, this type of structure hinders the fabrication process and increases the sensitivity of the device to mechanical vibrations.Based on compatible and common CMOS technology, improving the performance of modulators is usually achieved by changing the electrode structure and introducing a slight impedance mismatch.Two traditional traveling wave electrodes (TWE) are coplanar waveguides (CPW) and coplanar strip lines (CPS) [4], [5], [6], [7], [8], [9].The radio frequency (RF) index, characteristic impedance, and conductor loss are all influenced by variations in the width, gap, and thickness of CPS and CPW.It is unable to meet the requirements of high refractive index, impedance matching, and low loss simultaneously [4], [9].Hence, the performances of the silicon photonic modulators based on CPS and CPW are limited.The reported bandwidths of silicon optoelectronic modulators based on these two electrode configurations are both generally less than 40 GHz [10], [11], [12], [13], [14], [15].For example, at a bias voltage of 0.5 V, a 2 mm-long modulator based on CPS has a relative low bandwidth of only 10 GHz and a relative low Vπ of 2 V [16].Another 3.3 mm-long modulator based on CPS possesses a smaller bandwidth of 15 GHz and a smaller Vπ of 3.35 V [11].A 2 mm-long modulator using CPW electrodes can achieve a bandwidth of 20 GHz and a Vπ of 6.4 V [17].The 4.2 mm-long modulator using CPS and segmented PN junction has a high bandwidth of 35 GHz and a Vπ up to 7.5 V [14].Despite substantial attempts to improve the performance of silicon modulators, such as shortening the device length to 1 mm [7] and implementing an impedance mismatch [6], the electro-optic (EO) bandwidth of modulators based on CPS and CPW has been limited to a maximum value of 46 GHz [6].The 2.5 mm-long modulator is based on CPS and a slightly impedance mismatch and has a Vπ of 7.6 V. T-rail slow wave electrodes (TSWE) have been empirically demonstrated to enhance the refractive index of microwaves, thereby leading to an expansion in the bandwidth of optical modulators [8], [18], [19], [20], [21], [22].In the same device length, the T-rail electrode modulator achieved a bandwidth of 41 GHz, which is substantially higher than that of the modulators based on CPS and CPW [8].The 4.25 mm-long modulator using T-rail electrodes has a 41 GHz load and a Vπ of 8 V under 8 V bias voltage.Based on this T-rail device design scheme, a substantial bandwidth greater than 67 GHz was successfully achieved by shortening the device length and adopting a segmented structure.T-rail electrode modulators produced under general technology can possess a bandwidth beyond 67 GHz and Vπ of 5 V [23].The T-track electrode arrangement is an increasingly common choice for high-performance modulators.
To develop a high-performance modulator, a design method for a suitable T-track traveling wave electrode is crucial.The application of finite element method (FEM) simulations is a common method for reaching a high degree of accuracy in complex 2D or 3D device modeling.Nevertheless, this approach exhibits a weakness in terms of efficacy due to its protracted duration for numerical computation and substantial demand for computational resources.Furthermore, the comprehension of simulation outcomes might be difficult in the absence of an established physical structure.Another method entails the creation of an equivalent electrical circuit that aligns with the physical architecture of the modulator.The utilization of an equivalent model offers significant benefits owing to its lower computational demands in comparison to FEM [9], [24], [25], [26].Furthermore, it provides valuable insight into the physical characteristics of the transmission line.Although the viability and efficiency of this method of similar circuit analysis have been demonstrated on CPS and CPW [24], [25], [27], [28], there is a lack of corresponding research on T-rail electrode structures.Thus, the equivalent circuit model for the T-rail electrode is essential for designing silicon modulators.
In this paper, straightforward closed-form equations for the propagation parameters of TSWE are provided.Various variables are considered in the design process, including electrode thickness, asymmetric schemes, and two-layer substrates, among others.The theoretical capacitance of the diode is incorporated into the capacitance of the TSWE to assess the overall impact.The modulator parameters calculated by using the proposed equivalent circuit model are highly consistent to the HFSS simulation.Due to the accuracy and usability of this circuit model, it is believed that these formulas possess the ability to serve as valuable references for designs.The modulator was fabricated in a high-resistivity handle wafer (>750 ohm-cm).In this case, the space charge area exhibits a minor dielectric loss tangent at the operating frequency because of its light doping.As a result, silicon in the region of PN doping functions as a dielectric.The silicon is permeable to both transverse magnetic and transverse electric fields."Dielectric quasi-TEM mode" is the propagation mode.In the slow-wave mode guided by a micron-sized electrode, quasi-TEM analysis is valid since the very small cross-section of the transmission line ensures that the transverse fields are essentially quasi-stationary [23], [26], [27], [28].As the circuit parameters generated during microwave propagation can be calculated by conformal mapping, partial capacitance method, and other methods, TSWE can be described by equivalent circuits.

A. Transmission Line
Periodic T-rail loading mainly increases the capacitance per unit length [8], [22], [26].The equivalent circuit model is shown in Fig. 2  circuit and the physical device.The above constituent circuit components can be transformed into a two-port transmission line model with effective components RLGC model [9], [24], [25], [27] as shown in Fig. 2(c According to the transmission line model, the capacitance is inferred from the electric field created by the potential difference between the transmission line's two conductors.The inductance represents the generation of a magnetic field due to the current flow in the line.The resistance serves as a representation of the conductor loss.The conductance is used to model the loss in the dielectric between the conductors.The characteristic impedance Z is expressed as: The complex propagation constant γ is: The radio frequency (RF) index n RF is: where c 0 is the speed of light.Three major concerns must be considered to maximize the electro-optical bandwidth of the traveling wave modulator.First of all, microwave attenuation should be as small as possible.Second, the group velocity of the optical and microwave waves must be matched.Research on silicon electro-optic modulators was founded on the assumption that microwave signals propagate in TEM mode without dispersion, that is, that the effective refractive index and the microwave group velocity are equal [29].Hence, the group velocity of light and the microwave index n RF are required to be matched.Third, the transmission line characteristic impedance should be equal to or greater than the source impedance Z s and terminal impedance Z t .Generally, in the conventional microwave communication system, source impedance Z s and terminal impedance Z t are 50 Ω.In order to reduce reflections that may induce inter-symbol interference and achieve the largest feasible voltage drop at both ends of the modulator.The electrode structure can be split into two portions to calculate capacitance.A conformal mapping method can be used to perform part of the capacitance calculation for asymmetric coplanar strips (ACPS) with a gap of S 1 and widths of W s1 and W g1 .The other component is the T-shaped structure's capacitance C trail , which can be computed easily using the plate capacitance formula.We first calculate the capacitance of the transmission line with W s1 , W g1 , and S 1 .
Considering the metallization caused by electrode thickness, some authors suggest including the effect of electrode thickness in a similar way to that used for microstrip [30].The effect of conductor thickness can be modeled by an effective increment of conductor width and a reduction of the gap.We can write the gap as: Therefore, the effective widths of the source and the ground electrode are: where Δ accounts for the effect of metallization thickness.The expression for Δ can be expressed by [30], [31]: where t is the thickness of the electrode, ε r is the relative permittivity of the metal.We can determine the capacitance associated with each layer using the partial-capacitance method and conformal mapping.Fig. 3 illustrates the conformal mapping for the CPS.The geometry of t-plane can be mapped to the w-plane by using the mapping function as follows: Fig. 3. Coplanar plate capacitor in t-plane and transformed parallel capacitor using conformal mapping in w-plane.
By conformal mapping, the calculation of CPS capacitance can be transformed into the calculation of parallel plate capacitance, and then the capacitance expression of CPS can be obtained through inverse transformation [32].The electrode plate is placed in a multi-layer medium.The electric field lines of CPS are distributed in each layer.The capacitance of each dielectric layer is obtained by adding or subtracting the capacitance of different dielectric layers separately [33], [34].The following definition applies to the geometrical factor for a conformally mapped air-filled CPS: where K(k i ) is the complete elliptic integral of the first kind.When i = 0, k 0 is given by: When i>0, k i is given by: where h i is the thickness label in Fig. 1(c).Based on the partial-capacitance technique and the conformal mapping, we can derive the capacitance of the asymmetric coplanar strip line (ACPS) associated with each layer.We use the derivation in [25], [35] to calculate the capacitances.The capacitance C air for the air space above the device can be written as: where ε 0 is the permittivity in vacuum.Analogically, the capacitance of the buried oxide (BOX) layer C box is given by: where ε ox is the silicon dioxide relative permittivity.Contrary to the thick BOX layer, the silicon substrate is a semiconducting layer with nonzero conductivity.This dielectric layer should be described by C sub , C subs , C subl , and G sub as shown in Fig. 2(b).The transverse conductive and displacement currents in the silicon substrate are represented by the capacitor C sub and the conductor G sub , respectively.These two parameters can be written as: where σ sub is the conductivity of the silicon substrate, and ε si is the relative permittivity of the silicon substrate.The capacitance between the signal metal strip and the silicon substrate is represented by the capacitor C subs , which is given by: In the high-frequency range, an extra capacitor C sub1 is present to cause the overall capacitance to converge to C sub .As a result, we have the expression as follows: Fringe capacitance is considered when calculating the capacitance of the CPS transmission line.However, the capacitance caused by the T-shaped load is relatively small, so the fringe capacitance can be ignored.C trail can be directly calculated using the flat plate capacitance calculation formula, and the expression is as follows: After measuring the capacitance, it is required to figure out the transmission line's resistance and inductance.In the lowfrequency range, the line resistance can be calculated directly by Ohm's law, while the line conductance can be derived from common magneto-static theory.As the frequency increases, they become frequency-dependent due to the skin effect of the imperfect metal.When there is an alternating current or electromagnetic field in a conductor, the current distribution inside the conductor is uneven.The current density is the largest near the surface of the conductor.We simulated the electrode using HFSS and demonstrated the skin effect in the electrode in Fig. 4(a).We can see the strong current volume distribution is caused by the skin effect.As such, the resistance of the conductor increases.Fig. 4(b) shows the PN junction unloaded cross-section view of the electric field distribution corresponding to Fig. 1(c).The T-rail electrode's electric field distribution is a non-uniform electric field pointing from the signal to the ground.
The skin depth δ is defined as the depth where current density drops to 1/e of the value at the conductor's surface and is given by: With ω as the angular frequency in radians, μ 0 as the permeability of the vacuum, μ r as the relative permeability of the conductor, and σ as the conductivity of the electrode.For the CPS electrode with W s1 , W g1 and S 1 .The resistance R cps is the sum of R c and R g .R c is the series resistance in ohms per Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.unit length of the source strip conductor and can be measured by [31]: The term R S is the skin effect surface resistance in ohms and is given by: R g is the distributed series resistance in ohms per unit length of the ground planes and is given by: As the capacitance calculation process mentioned above, the T-track can be considered as the independent CPS transmission line when calculating the resistance.By incorporating w ts , w tg , and S 2 into formulas 20-22, the resistance R trail introduced by the T-track can be calculated.The inductance per unit length of the T-rail slow-wave transmission line L cps is equal to the inductance of the electrode with W s1 , W g1 , and S 1 [8], [18], [26].In the skin-effect region, L cps can be divided into the internal inductance L int and the external inductance L ext .The external inductance is frequency independent, which can be obtained from the capacitance of an air-filled C air and is given by: The internal inductance can be calculated by Wheeler's incremental inductance rule [26].Whether the transmission line is symmetrical or not, the internal inductance can be calculated by: where term η can be calculated by the formulas given in [27].All circuit components in Fig. 2(a) except for the PN junction have been determined now.Fig. 5 shows the variation of RLGC in the T-rail slow-wave electrodes as a function of W s1 , W g1 , and S 1 .Resistance and inductance decrease while conductance and capacitance rise when Ws1 and Wg1 increase.Resistance, conductance, and capacitance all decrease, and inductance rises when S1 increases.Certain discrete sections exhibit uniform responses due to the nonlinear variations in the conductor's charge generated by the skin effect.If no additional circuit elements are added, the properties during microwave propagation can be calculated from the electrode parameters obtained now.As the PN junction can cause changes in capacitance and resistance values, the overall performance parameters must be combined with the part of the PN junction.Silicon-based EO modulators require capacitance and resistance Fig. 6.Cross-sections of the n-type carrier concentration of the PN junction under a bias voltage of (a) 0 V, (b) 2 V, and (c) 4 V. P-type doping is found on the left side, while n-type doping is seen on the right.from the PN junction to obtain microwave propagation data [8], [22].

B. PN Junction
We use a CHARGE solver to simulate the carrier concentration as a function of the voltage at the PN junction.Fig. 6. shows the change in n-type carrier concentration at various reverse voltages of the PN junction.Increasing the reverse bias voltage widens the depletion region and affects the resistance and capacitance of the PN junction, thereby impacting the modulator performance.
The resistance of the PN junction mainly depends on the doping concentration.The approximate value of resistance and capacitance of the PN junction can be obtained by assuming uniform doping of the PN junction.The resistance introduced by the PN junction can be calculated from the square resistance and volume of each region.The resistance of the PN junction R p and R n are in series, the resistance can be calculated by [36]: where w rib is rib width, t rib is rib height, and R srn , R srp , R ssn , and R ssp are the sheet resistances of the n-doped rib, p-doped rib, n-doped slab, and p-doped slab, respectively.With W d is determined by the impurity densities (N A for P doped area and N D for N doped area), as well as the applied voltage V, and is given by: The capacitance of the p-n junction is given by: where V bi is the built-in or diffusion potential of the junction given by: where n i is intrinsic charge carriers in m −3 , k B is Boltzmann constant in J/K, T t is the temperature in K.The resistance and capacitance of PN junctions of various sizes at two doping levels are depicted in Figs.7(a) and (b) respectively.An increase in the bias voltage causes the capacitance to increase and the resistance to slightly decrease.The transmission line's overall performance parameters can be computed by adding the electrode's RLGC parameters to the capacitance and resistance produced by the PN junction.The loss will increase due to the resistance produced by the PN junction.The intermediate doping layer can be added to reduce resistance and thereby reduce microwave losses.If the PN junction is too narrow, there may be numerous electrode design restrictions that could compromise the device's overall performance [15].With a larger PN junction, it is possible to gain high design flexibility for the electrodes, but the larger doping zone also results in more optical loss.Tradeoffs must be made during the design process.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

III. SILICON OPTICAL MODULATORS DESCRIPTION AND ANALYSIS
The electrode design parameters from Ref [8] were used to verify the accuracy of our equivalent circuit.We employed our equivalent circuit model and HFSS software to compute the primary parameters of the t-shaped electrode.The outcomes of our calculations are shown in Fig. 8.The data suggests that the results obtained from our circuit model are consistent with the outcomes obtained from HFSS.Fig. 8(a) depicts the microwave loss feature of the transmission line.In this instance, the primary source of loss is usually determined to be conductor loss rather than dielectric loss.The microwave index of the modulator can be seen in Fig. 7(b), which is close to the optical group index in the silicon waveguide (3.8-4) [22].Characteristic impedance Z 0 results for the reference modulator against frequency for the equivalent circuit model approach and the HFSS simulator are shown in Fig. 8(c).As expected, Z 0 decreases as the frequency is increased due to the skin effect [28].The characteristic impedances are extremely close to 50Ω, which is the optimal state for microwave systems.The circuit model maintains an accuracy of 2% when compared to the HFSS simulation.Additionally, as pointed out before, the quasi-static approximation can maintain a capacitance deviation of less than 6% when the frequency is less than 100 GHz [33].
The modulator's small signal electro-optic response as a function of velocity mismatch and total microwave loss can be calculated by [14]: where α 0 is the field attenuation coefficient, n RF is the effective refractive index of the microwave, n opt is the group effective refractive index of the optical wave, and L ef f is the effective length, respectively, with  The optical group index in silicon-based modulators is typically 3.8-4 and is influenced by factors like doping concentration in the PN junction [22].We used the CHARGE solver and the MODE solver to simulate the optical group index at various reverse bias voltages and presented the results in Fig. 9.The group refractive index is around 3.9.The response of the EO modulator, as calculated by the use of HFSS and the equivalent circuit model, is depicted in Fig.The EO response estimated using the equivalent circuit model, exhibits similarities to its HFSS counterpart.The simulated phase shift as a function of the reverse bias voltage is illustrated in Fig. 11.These results translate to a DC Vπ of 5.76 V and 15.36 V for the 4 mm-long and 1.5 mmlong modulator respectively at a bias of −1 V.The efficiency decreased as the increase in reverse bias [16].When biased at -2 V, -3 V, and -4 V, the DC Vπ of the 4 mm effective long device are 6.67 V, 7.57 V, and 8.48 V respectively.Meanwhile, the DC Vπ of the 1.5 mm effective long device is 17.79 V, 20.19 V, and 22.61 V respectively.

IV. CONCLUSION
The equivalent circuit model of the carrier-depletion-based silicon modulators with slow-wave transmission lines is presented.The circuit model of the T-shaped electrode is commonly used in silicon photonics modulators, III-V compound semiconductor electro-optic modulators, and other traveling wave devices.Our equivalent circuit model can accurately estimate key parameters related to TSWE.The acquired results are wellmatched to the HFSS simulation results.We concluded that these formulas serve as an efficient design and optimization tool in various applications due to their correctness, simplicity, and ability to handle a wide range of T-rail electrode structures.

Fig. 1 .
Fig. 1.(a) Overview of the depletion-based silicon optical modulator with segmented electrode.(b) Top-down detail view of the electrode.(c) Cross section of a typical carrier-depletion-based optical modulator in SOI with the TSWE (drawing not to scale).

Fig. 1 (
Fig. 1(a) shows the schematic of the modulator that consists of a traveling-wave electrode and a carrier-depletion-based p-n junction.The electrode is a ground-signal (GS) metal periodically loaded by T-shape capacitors.Figs.1(b) and 1(c) show the top view and the cross-sectional view of the modulator.A lateral p-n junction is embedded in the center of a rib waveguide in this arrangement.Highly doped P++ and N++ regions are employed for ohmic connections.The fixed device parameters used in the simulation are h sub = 750 μm, h box = 2 μm, h clad = 1.5 μm, h slab = 90 nm, and t = 2 μm.The modulator was fabricated in a high-resistivity handle wafer (>750 ohm-cm).In this case, the space charge area exhibits a minor dielectric loss tangent at the operating frequency because of its light doping.As a result, silicon in the region of PN doping functions as a dielectric.The silicon is permeable to both transverse magnetic and transverse electric fields."Dielectric quasi-TEM mode" is the propagation mode.In the slow-wave mode guided by a micron-sized electrode, quasi-TEM analysis is valid since the very small cross-section of the transmission line ensures that the transverse fields are essentially quasi-stationary[23],[26],[27],[28].As the circuit parameters generated during microwave propagation can be calculated by conformal mapping, partial capacitance method, and other methods, TSWE can be described by equivalent circuits.Periodic T-rail loading mainly increases the capacitance per unit length[8],[22],[26].The equivalent circuit model is shown in Fig.2(a).R trail and C trail are the resistance and capacitance per unit length due to the T-rail.C j and R j represent the depletion capacitance and series resistance per unit length of the p-n junction.The remaining components align with the CPS transmission line.Fig. 2(b) shows the relationship between the Fig. 1(a) shows the schematic of the modulator that consists of a traveling-wave electrode and a carrier-depletion-based p-n junction.The electrode is a ground-signal (GS) metal periodically loaded by T-shape capacitors.Figs.1(b) and 1(c) show the top view and the cross-sectional view of the modulator.A lateral p-n junction is embedded in the center of a rib waveguide in this arrangement.Highly doped P++ and N++ regions are employed for ohmic connections.The fixed device parameters used in the simulation are h sub = 750 μm, h box = 2 μm, h clad = 1.5 μm, h slab = 90 nm, and t = 2 μm.The modulator was fabricated in a high-resistivity handle wafer (>750 ohm-cm).In this case, the space charge area exhibits a minor dielectric loss tangent at the operating frequency because of its light doping.As a result, silicon in the region of PN doping functions as a dielectric.The silicon is permeable to both transverse magnetic and transverse electric fields."Dielectric quasi-TEM mode" is the propagation mode.In the slow-wave mode guided by a micron-sized electrode, quasi-TEM analysis is valid since the very small cross-section of the transmission line ensures that the transverse fields are essentially quasi-stationary[23],[26],[27],[28].As the circuit parameters generated during microwave propagation can be calculated by conformal mapping, partial capacitance method, and other methods, TSWE can be described by equivalent circuits.Periodic T-rail loading mainly increases the capacitance per unit length[8],[22],[26].The equivalent circuit model is shown in Fig.2(a).R trail and C trail are the resistance and capacitance per unit length due to the T-rail.C j and R j represent the depletion capacitance and series resistance per unit length of the p-n junction.The remaining components align with the CPS transmission line.Fig. 2(b) shows the relationship between the

Fig. 2 .
Fig. 2. (a) Equivalent circuit model of the TSWE based EO modulator.(b) Relationship between the circuit and the physical device.(c) Equivalent RLGC model for unit transmission lines.

Fig. 4 .
Fig. 4. (a) Top view of current volume distribution in the electrodes at 40 GHz.(b) Cross-section view of the electric field distribution corresponding to Fig. 1(c) without PN junction.

Fig. 7 .
Fig. 7. (a) Resistance of the lateral p-n junction vs. voltage.(b) The capacitance of the lateral p-n junction vs. voltage.

Fig. 9 .
Fig. 9.The simulated optical group index as a function of the reverse bias voltage.

Fig. 10 .
Fig. 10.(a) EO frequency responses using equivalent circuit model and HFSS under different bias voltages when the effective length is 4 mm.(b) EO frequency responses using equivalent circuit model and HFSS under different bias voltages when the effective length is 1.5 mm.

10
. The material of the electrode is aluminum.The electrode design parameters are: W s1 = W g1 = 120 μm, S 1 = 51 μm, W ts = W tg = 10 μm, g = 9.2 μm, W r = 2 μm, L T1 = 47 μm, L T2 = 3 μm.The parameters of PN junction are N A = N D = 4 × 10 17 /cm 3 , x n++ = x p++ = 1.2 μm.The increase in bias voltage given to the PN junction leads to an observed increase in the bandwidth of the electro-optic modulator, as demonstrated by both calculation approaches.

Fig. 11 .
Fig. 11.The simulated phase shift as a function of the reverse bias voltage.

Fig. 10 (
a) shows that the 4 mm effective long modulator provides 43-GHz bandwidth.We further model a modulator with 100-GHz bandwidth by setting the effective length to 1.5 mm in our proposed method.The simulation results of the proposed model and the HFSS are shown in Fig. 10(b).The simulation results of the proposed model are highly consistent to the result of the HFSS.