1140 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 64, NO. 3, MARCH 2017

A Linear Equivalent Circuit Modelfor Depletion-Type Silicon

Microring ModulatorsMyungjin Shin, Yoojin Ban, Byung-Min Yu, Min-Hyeong Kim, Jinsoo Rhim, Student Member, IEEE,

Lars Zimmermann, and Woo-Young Choi, Member, IEEE

Abstract— We present a linear equivalent circuit modelfor the depletion-type Si microring modulator (MRM). Ourmodel consists of three blocks: one for parasitic compo-nents due to interconnects and pads, one for the electricalelements of the core p-n junction, and the third for a lossyLC tank representing Si MRM optical modulation character-istics. Model parameter values are extracted from measure-ment of a fabricated Si MRM device. Simulated modulationcharacteristics with our equivalent circuit show very goodagreement with measured results. Using our model, we cananalyze Si MRM modulation frequency response character-istics and perform gain-bandwidth product optimization ofthe entire Si photonic transmitter composed of a Si MRMand electrical driver circuits.

Index Terms— Equivalent circuit model, opticalinterconnect, silicon photonics, Si ring modulator.

I. INTRODUCTION

S i PHOTONICS is attracting a great deal of research anddevelopment efforts, since it can provide cost-effective,

high-bandwidth, and small-footprint optical interconnect solu-tions [1]. Among several photonic devices that can be usedfor Si photonic interconnect systems, the depletion-type Simicroring modulator (MRM) is of special interest as it haslarge modulation bandwidth, a small footprint, and low-powerconsumption [2]–[7]. In designing optimal Si MRMs for targetapplications, a model that can accurately predict Si MRM

Manuscript received August 16, 2016; revised November 25, 2016;accepted January 2, 2017. Date of publication January 27, 2017; date ofcurrent version February 24, 2017. This work was supported in part by theNational Research Foundation of Korea through the Korean Ministry ofScience, ICT, and Future Planning under Grant 2015R1A2A2A01007772and in part by the Materials and Parts Technology Research andDevelopment Program through the Korean Ministry of Trade, Industry &Energy under Project 10065666. The review of this paper was arrangedby Editor E. G. Johnson.

M. Shin, B.-M. Yu, M.-H. Kim, and W.-Y. Choi are with the Departmentof Electrical and Electronic Engineering, Yonsei University, Seoul 03722,South Korea (e-mail: wchoi@yonsei.ac.kr).

Y. Ban was with the Department of Electrical and Electronic Engineer-ing, Yonsei University, Seoul 03722, South Korea. She is now with imec,B-3001 Leuven, Belgium.

J. Rhim was with the Department of Electrical and Electronic Engineer-ing, Yonsei University, Seoul 03722, South Korea. He is now with HewlettPackard Laboratories, Palo Alto, CA 94304 Jinsoo USA.

L. Zimmermann is with Innovations for High Performance Microelec-tronics, 15236 Frankfurt, Germany.

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TED.2017.2648861

modulation characteristics is very important. There have beenseveral reports on Si MRM modulation characteristics analysesusing either the time-dependent dynamics model [8] or themodel based on the coupled-mode theory [9]–[15]. The time-dependent dynamics model provides very accurate results butusing it can be computationally intensive. The model based onthe coupled-mode theory is computationally less intensive, andits numerical solution has been implemented in Verilog-A [15],a popular behavior-level simulator among circuit designers.These numerical approaches, however, are not very convenientto use for design optimization.

In this paper, we introduce a linear equivalent circuit whoseoutput voltage models the normalized output optical powerof Si MRM in small-signal modulation condition. Althoughthere have been several reports on the Si MRM circuitmodel [13]–[16], ours is the first model that can simulate bothelectrical and optical characteristics of the Si MRM entirelybased on circuit parameters. In particular, using the circuitmodel for the Si MRM modulation characteristics, instead ofscattering parameters or transfer functions within the circuitsimulator, allows faster simulation, especially when designingintegrated circuits containing several Si MRMs and electroniccircuits. In addition, values of all our circuit parameters canbe extracted in a straightforward way as will be demonstratedin this paper.

This paper is organized as follows. In Section II, wegive details of our circuit model. In Section III, we explainhow model parameter values are extracted from measurement.We also confirm the accuracy of our model by comparingSi MRM modulation characteristics simulated with the cir-cuit model in Cadence Spectre, a very popular electricalcircuit simulator, with measurement results. In Section IV,we demonstrate the usefulness of our model by analyzing thegain-bandwidth product of the entire Si photonic transmittercomposed of a Si MRM and driver electronics. Section Vconcludes this paper.

II. SI MRM EQUIVALENT CIRCUIT MODEL

Fig. 1(a) and (b) shows the device structure and the crosssection of a depletion-type Si MRM, respectively. The SiMRM device used for this paper has been fabricated through SiPIC MPW provided by IHP. The ring waveguide is divided inton- and p-regions with the nominal peak doping concentrationof 7×1017 cm−3 for p-region and 3×1018 cm−3 for n-region.

0018-9383 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

SHIN et al.: LINEAR EQUIVALENT CIRCUIT MODEL FOR DEPLETION-TYPE SILICON MRMs 1141

Fig. 1. (a) Structure, (b) cross section, and core electrical model of SiMRM. (c) Block diagram of Si MRM equivalent circuit.

The nominal width ratio is 2:3 for n- and p-regions.Similar devices having different doping concentrations havebeen used for parametric characterization of Si MRM self-heating [17]. In a depletion-type Si MRM, its p-n junctionreverse bias voltage is modulated by external voltage signals,which causes the effective index modulation for the ringwaveguide [18]. With this, the Si MRM resonance wavelengthshifts and the amount of light coupled into the optical outputport is modulated. The device can be modeled with threeblocks, as shown in Fig. 1(c): BPara for parasitic componentsfor pads and interconnects, BCore for electrical components ofthe Si MRM core, and BOpt for a lossy LC tank, representingsmall-signal optical modulation characteristics of Si MRM.Detailed explanations for each block are given as follows.

A. BPara: Pads and Interconnects

In previously reported models for Si MRMs, the influence ofpads and interconnects has been considered together with theSi MRM core and simply modeled by one capacitor [14]–[16].However, as the modulation speed increases, the effect of theseparasitic components becomes more significant and providingan accurate model for them becomes necessary. Fig. 2(a)shows the parasitic components that should be considered.Cpad is the capacitance between two signal pads as wellas metallic interconnects lines, and Cox1 and Cox2 are thecapacitance between pad and silicon substrate for n-port andp-port, respectively. Rsub is the resistance through the siliconsubstrate between n-port and p-port pads through Cox1 andCox2. L int1, L int2, Rint1, and Rint2 are inductances and resis-tances of interconnect lines for n-port and p-port terminals.Cc-c represents the capacitance between two metal openings

Fig. 2. (a) Parasitic components for pads and interconnects and (b) theirequivalent circuit (BPara).

Fig. 3. Equivalent circuit for Si MRM Core (BCore).

to n+ and p+ regions. The output voltage of this block isthe voltage across the Si MRM p-n junction (Vp-n). Sincethe same amount of current flows in both n-port and p-portinterconnect lines in this two-terminal device, we can simplifythe circuit by using Cox = Cox1||Cox2,L int = L int1+ L int2, andRint = Rint1 + Rint2. Fig. 2(b) shows the resulting equivalentcircuit for BPara.

B. BCore: Si MRM Core

As shown in Fig. 1(b), the Si MRM core can be modeledwith RC components [14]–[16]. C j represents the junctioncapacitance for p-n junction, Rn and Rp represent the resis-tances in the doped Si layers, Cn and Cp represent the capac-itance between doped silicon layers and the silicon substrate,and Rsi represents the resistance of the silicon substrate belowthe device. Fig. 3 shows the equivalent circuit of the Si MRM

1142 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 64, NO. 3, MARCH 2017

Fig. 4. Equivalent circuit for Si MRM optical modulation characteris-tics (BOpt).

core where Rs = Rn + Rp and Csi = Cn ||Cp . The outputsignal of this block is the voltage across C j , since light in theSi MRM is modulated by the effective index variation due tothe junction voltage (Vj ) variation.

C. BOpt: Optical Modulation Characteristics

When the Si MRM is modulated with a small-signal junc-tion voltage given as Vj (t) = v0cos(ωmt) around the biasvoltage, the change in the effective index with time, η(t), canbe expressed as

η(t) = ∂η

∂Vjv0 cos(ωmt). (1)

Considering only the linear response, the output opticalsignal is also modulated with the same angular frequency ωm .The modulated optical signal is detected by a photodetector,generating photocurrents that are proportional to output opticalpower, Pout, of the Si MRM. This process can be modeled inthe s-domain based on the coupled-mode equation as [11], [12]

�O(s) = Pout/Pin(s)

Vj (s)

= 4

η0· ∂η

∂Vj· ωr D/τe

D2 + 1/τ 2 · s + 2/τl

s2 + (2/τ)s + D2 + 1/τ 2

(2)

where D(= |ωin − ωr|), the detuning parameter, representshow much the input light angular frequency ωin is detunedfrom the ring resonance angular frequency ωr , and and η0represent the effective index of the ring waveguide at thegiven bias voltage. τl and τe are decay time constants ofthe ring resonator due to the round-trip loss and ring-buscoupling, respectively. They satisfy 1/τl + 1/τe = 1/τ , whereτ is the total decay time constant for the ring resonator.Equation (2) represents a two-pole and one-zero linear systemwhose damping factor ζ and natural frequency ωn are givenas ζ = (1/((1 + D2τ 2)1/2)) and ωn = (D2 + (1/τ 2))1/2.Since ζ is smaller than 1 unless D = 0, the Si MRM hasthe underdamped modulation frequency response except at theresonance frequency.

Fig. 4 shows BOpt that models the transfer functiongiven in (2), where Vout represents normalized output power(Pout/Pin). It is composed of a lossy LC tank with a voltage-controlled current source having transconductance g and twoadditional resistors, R1 and R2. The transfer function for thiscircuit is given as

�E (s) = VOut(s)

Vj (s)= g

C

s + R2L

s2 +(

1C R1

+ R2L

)s + 1

LC

(R2R1

+ 1) .

(3)

Fig. 5. Measured (circles) and simulated (colored solid lines) S11 of SiMRM at VBias = −1 V.

By comparing (2) and (3), we can express circuit parametersin terms of τe, τl , and D as

R1C = τe

2(4)

L

R2= τl

2(5)

R1

R2= [(1/τ 2 + D2)τeτl/4 − 1] (6)

and

g = 2

η0· δη

δv· ωr D

D2 + 1/τ 2

1

R1U. (7)

The unit conversion factor U(= 1 V) is added in (7),so that g can have the correct unit of transconductance.Our model preserves the physical meaning of the two timeconstants of the ring resonator by representing τe withR1C and τl with L/R2. The factor of two shows upin (4) and (5), because τe and τl are time constants for electri-cal fields whereas the equivalent circuit models the opticalpower. Damping factor ζ and natural frequency ωn

are given as ζ = (C R1 R2 + L)/(2(LC R1(R1 + R2))1/2)

and

ωn =√

1

LC

(R2

R1+ 1

)= ((1/(LC))((R2/R1) + 1))

12 .

III. PARAMETER VALUE EXTRACTION AND

MODEL VERIFICATION

BPara can be determined from electrical S11 measurementof short and open test patterns having the same pad andinterconnect structures as the target Si MRM fabricated onthe same die [19]. In this paper, open and short test patternshave the same structure from pad to Metal 1. Since neithershort nor open test pattern includes Cc-c, we first determined

SHIN et al.: LINEAR EQUIVALENT CIRCUIT MODEL FOR DEPLETION-TYPE SILICON MRMs 1143

TABLE IEXTRACTED PARAMETER VALUES FOR BPARA AND BCORE

Fig. 6. Measured and simulated transmission characteristics of Si MRMat VBias = −1 V.

the value of Cc-c by EM simulation. Then, by fitting simu-lated S11 values to measured results, we can determine thenumerical value for each component in BPara, as shown inTable I. With BPara known, BCore can be determined byfitting simulated S11 values for BPara and BCore into measuredvalues. Fig. 5 shows the magnitude and phase response ofmeasured (circles) and simulated (colored solid lines) S11values at VBias = −1 V with extracted parameter values listedin Table I.

For BOpt, values for L, C , R1, and R2 can be determinedfrom τe, τl , and D using (4)–(6). With the knowledge of R1value, g can be determined from optical parameters, η0, D,ωr , τ , and δη/δv. For determination of η0, the ring resonancemode number m can be first identified as 85 from waveguidesimulation of the Si MRM. With this, η0 can be determinedas 2.637149 at VBias = −1 V from the resonance conditionmλres = Lη0 with measured λres. The value of τe and τl , canbe determined by fitting the Si MRM steady-state transmissioncharacteristic given as [20], [21]

T = Pout

Pin=

∣∣∣∣jω − jωr + 1/τl − 1/τe

jω − jωr + 1/τl + 1/τe

∣∣∣∣2

. (8)

to the measured transmission characteristics, as shown inFig. 6, at VBias = −1 V with the minimum mean squarederror [22].

TABLE IIBOPT PARAMETER VALUES

Fig. 7. Measured (circles) and simulated (solid lines) relative frequencyresponses at three different detuning values (VBias = −1 V).

Fig. 8. Simulated normalized frequency responses of BPara, BPara, andBCore, and BPara, BCore, and BOpt at three different detuning values.

For this measure, input optical power after the gratingcoupler is minimized to −15 dBm in order to avoid any self-heating. With this fitting, τe = 24.635 ps and τl = 22.882 psare determined. The value for g can be determinedfrom (7) using the measured value of δη/δv of 2.2 × 10−5

at VBias = −1 V. The corresponding BOpt circuit parametervalues at three different detuning values are listed in Table II,where R2 is fixed at 10 k for convenience.

The simulated Si MRM modulation frequency responses atVBias = −1 V for three different detuning values are shown inFig. 7 along with the measured results. Both simulation andmeasured results are normalized to the low-frequency valueof case (A). The simulation is done with Cadence Spectre(Virtuoso Version No. 14.1.0.459), using circuit parameters

1144 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 64, NO. 3, MARCH 2017

Fig. 9. (a) Schematic of Si MRM transmitter. (b) Simulatedgain-bandwidth product of the transmitter with different driver loadimpedance (RL) and D values. Simulated transmitter frequencyresponses at (c) three different detuning values with RL = 100 Ω and(d) four different RL values with D = 90 GHz.

values given in Tables I and II. Clearly, different modulationfrequency responses of Si MRM at different detuning condi-tions are accurately modeled with our equivalent circuit.

IV. APPLICATION OF THE EQUIVALENT CIRCUIT

Using our model, we can identify how much each ofBpara, BCore, and BOpt contributes to the modulation frequencyresponse. Fig. 8 shows the simulated normalized modulationfrequency responses for only parasitic components (BPara),including parasitic and core components (BPara and BCore), andfull equivalent circuit model (BPara, BCore, and BOpt) for threedifferent detuning conditions. For the Si MRM device underinvestigation, the 3-dB modulation bandwidth due to electricalcomponents is approximately 65 GHz, and consequently, themodulation frequency response is mainly limited by BOpt.It is possible to enlarge the bandwidth of BOpt by reducingQ-factor of the ring resonator, or by designing a Si MRMhaving lower value for τ but at the cost of reduced modulationgain as can be seen in (7).

The real advantage of the Si MRM equivalent circuit isthe ease with which it can be used for cosimulation ofelectronic circuits and Si MRMs in a similar manner ashas been demonstrated for Mach–Zehnder modulators andelectroabsorption modulators [23], [24]. Fig. 9(a) shows aschematic for an integrated Si photonic transmitter based ona pseudodifferential cascode common-emitter driver and aSi MRM. Such an integrated circuit can be fabricated withIHP’s EPIC technology, which provides monolithic integrationof 0.25-μm SiGe BiCMOS and Si PIC technologies [25].In this transmitter, the value of load resistance RL shouldbe carefully optimized. Fig. 9(b) shows the simulated gain-bandwidth products as a function of RL and D. Simulatedfrequency responses for selected points are shown in Fig. 9(c)for different D values and in Fig. 9(d) for different RL

values. Clearly, the gain-bandwidth product is a sensitivefunction of electrical characteristics represented by RL andoptical characteristics represented by D, and our Si MRMcircuit model allows its optimization in a very straightforwardmanner.

V. CONCLUSION

We presented an equivalent circuit model for the depletion-type Si MRM. The model has three blocks, each of whichmodels parasitic components, core Si MRM p-n junction, andthe modulation characteristics of the Si MRM. The circuitmodel parameters are extracted from relatively simple devicemeasurement and simulation. The accuracy of our model isconfirmed by measurement. Our circuit model allows efficientcosimulation of Si MRMs with electronic circuits in the stan-dard electronic circuit simulation environment, which shouldbe of great help for efficient design of Si electronic-photonicintegrated circuits containing Si MRMs.

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Myungjin Shin is currently pursuing theM.S. degree with Yonsei University, Seoul, SouthKorea.

His current research interests include design,modeling, and optimization of Si microringmodulators.

Yoojin Ban received the M.S. degree fromYonsei University, Seoul, South Korea, in 2015.

She joined imec, Leuven, Belgium, whereshe is currently involved in high-performanceSi modulators.

Byung-Min Yu is currently pursuing thePh.D. degree with Yonsei University, Seoul,South Korea.

His current research interests include design,modeling, and optimization of silicon photonicdevices.

Min-Hyeong Kim is currently pursuing thePh.D. degree with Yonsei University, Seoul,South Korea.

His current research interests include high-speed interface circuits for optical interconnects.

Jinsoo Rhim (S’17) received the Ph.D. degreefrom Yonsei University, Seoul, South Korea, in2016.

He joined Hewlett Packard Laboratories, PaloAlto, CA, USA. His current research inter-ests include photonic device characterizationand modeling, Si-based electronic-photonic ICdesign, and high-speed interface circuit designin advanced CMOS/BiCMOS technology.

Lars Zimmermann received the Ph.D. degreein electrical engineering from Katholieke Univer-siteit Leuven, Leuven, Belgium, focused on near-infrared sensor arrays, which he conducted atIMEC.

He was with IHP, Frankfurt (Oder), Germany, in2008. He is currently a Team Leader with SiliconPhotonics Technologies, California, USA. He isalso with TU Berlin, Berlin, Germany, coordinat-ing the cooperation with IHP in the field of Siphotonics.

Woo-Young Choi (M’92) received the B.S.,M.S., and Ph.D. degrees in electrical engineer-ing and computer science from the Massa-chusetts Institute of Technology, Cambridge, MA,USA, focused on molecular-beam epitaxy-grownInGaAlAs laser diode.

He is currently a Professor of Electrical andElectronic Engineering, Yonsei University, Seoul,Korea. His current research interests includehigh-speed circuits and systems, that are Siphotonics and high-speed Si integrated circuits

for interface applications.