Silicon Carrier-Injection PIN Modulator

1. Theory

• Definition: A silicon carrier-injection modulator is a pin diode across a silicon waveguide, forward-biased so that electrons and holes are injected into the intrinsic waveguide core. The injected carriers change the index and the absorption by the plasma dispersion effect, and the device is used as a phase shifter in a Mach-Zehnder (MZM), in a ring, or as a variable optical attenuator (VOA) (Akiyama and Usuki; the MIT Lincoln Laboratory abstract; Kotura; the IIT thesis).
• Why it is efficient: Injection puts carriers throughout the waveguide mode at densities of $10^{17}$ to $10^{18}$ $\mathrm{c}{\mathrm{m}}^{-3}$ (IIT: about $7\times 10^{17}$ $\mathrm{c}{\mathrm{m}}^{-3}$ for $5\mathrm{dB}$ in 1 mm), where depletion changes only a thin layer, so $V_{\pi }L$ is 0.014 $\mathrm{V}\mathrm{cm}$ (Akiyama and Usuki), 0.036 $\mathrm{V}\mathrm{mm}$ (J. Semicond.) or 0.24 $\mathrm{V}\mathrm{mm}$ (MIT Lincoln Laboratory), against 1.4 to 2 $\mathrm{V}\mathrm{cm}$ for the depletion Mach-Zehnder of page 99.
• Charge picture: A phase shift of $\pi$ needs a fixed charge $Q_{\pi }=c{V}_{\pi }L$, 6.21 $\mathrm{pC}$ for Akiyama and Usuki. The diode capacitance is a diffusion capacitance, 45 $pF/mm$ against 0.45 $pF/mm$ in reverse bias, so the driver must supply this charge in each bit.
• Speed limit: The carriers decay with the effective lifetime (about 1 ns in SOI, Watts et al.; 100 ns to 4 microseconds in large waveguides, IIT; 0.35 to 0.55 microseconds, Kotura), so an unequalised drive gives below 1 GHz (Watts et al.). A pre-emphasis drive that overshoots and removes the charge actively raises this to 12.5 Gbit/s (J. Semicond.) and 56 Gbit/s (Akiyama and Usuki).
• Attenuator: In a VOA the diode current sets the free-carrier absorption: 0.1 $dB/mA$ and $5\mathrm{dB}$ in 1 mm at 70 $\mathrm{mA}$ (IIT), with a polarisation-dependent loss below $0.2\mathrm{dB}$ and a wavelength-dependent loss below $0.5\mathrm{dB}$ over the C band.
• Figures of merit: $V_{\pi }L$ and the frequency-dependent $V_{\pi }L(\omega )$, the loss-efficiency product $\alpha V_{\pi }L$ (0.73 V dB static, 9.5 V dB at 12.5 GHz), the current for $\pi$ (about 10 $\mathrm{mA}$, IIT), the response time, the energy per $\pi$ shift.
• Limits: The static power is milliwatts, the speed depends on a pre-emphasis driver, the loss accompanies the phase (the ratio $2k_0\Delta n/\Delta \alpha$ is about 18), and the diode must be kept in a short length of about 0.8 mm for a 50 $\Omega$ driver.

2. Working Principle

• Bias: The p and n regions on the two sides of the waveguide are forward-biased at 0.7 to 0.85 V, where the current rises exponentially with the voltage, $I=I_s[\exp (qV/(N{k}_{B}T))-1]$ with $k_B$ the Boltzmann constant.
• Injection: The current brings electrons and holes into the core; their density is the current times the lifetime divided by $q{V}_{vol}$, and the phase shift is proportional to this density, which is why it is proportional to the charge.
• Phase and attenuation: The index falls as the density rises (a phase advance), and the absorption rises; in a VOA the absorption is wanted, in a phase shifter it is a loss (1.5 to $4\mathrm{dB}$ per $\pi$).
• Speed: Turning on is fast with a current spike, turning off is limited by the lifetime. The drive therefore overshoots and undershoots, so that the swing needed grows with the factor $\sqrt{(\omega \rho c{)}^2+1}$ (13 at 12.5 GHz for the Akiyama device).
• Interference: In an MZM the phase difference gives the intensity, as on pages 94 and 99, and in a ring (page 100) the resonance shifts to shorter wavelength.
• Energy: The energy to charge a $\pi$ shifter is $Q_{\pi }{V}_f=5.3$ pJ (computed) and the holding power is a few milliwatts.

3. Applications

• High-rate short links with a pre-emphasis driver: A forward-biased pin modulator carried 56 Gbit/s NRZ with an open eye using a 2 Vpp pre-emphasised signal (Akiyama and Usuki), and 12.5 Gbit/s with $8.4\mathrm{dB}$ of extinction ratio (J. Semicond.).
• Variable optical attenuators: Silicon VOAs of the Kotura type were the first commercial silicon electro-photonic devices, with a 3 $\mu \mathrm{m}$ rib, a mode transformer that lowers the fibre coupling loss from 3.4 to $0.3\mathrm{dB}$, and a response in microseconds; the IIT diodes give 5 to $10\mathrm{dB}$ at 70 to 100 $\mathrm{mA}$ with a response of 25 to 125 ns.
• Thermal-free phase tuning: The IIT phase shifter gives $\pi$ at about 10 $\mathrm{mA}$ with a response of 72 ns, as an alternative to thermal tuning (page 99), which is slower (microseconds) but takes 1 to 2 mW.
• Compact ring modulators: A 5 $\mu \mathrm{m}$ injection ring carried 8 and 9 Gbit/s with pre-emphasis (IntechOpen chapter), and a bias tuning by injection of 6.8 microwatts per GHz was reported for a WDM ring with an automatic lock (page 100).

Injection Physics and Plasma Dispersion

Name	Formula	Related Formulae	Variables/Constants
Forward current of the pin diode	$I=I_s\left[\exp \left(\frac{qV}{N{k}_{B}T}\right)-1\right]$	$V_T=\frac{k_{B}T}{q}=25.85\mathrm{mV}\mathrm{at}300\mathrm{K}$ $C_{diff}=\frac{I\tau }{N{V}_T}$	$I_s:saturationcurrent$ $N:idealityfactor$ $V:junctionvoltage$ $k_B:Boltzmannconstant$ $T:absolutetemperature$ $q:elementarycharge$ $\tau :carrierlifetime$ $V_T:thermalvoltage$ $\begin{array}{r}{C}_{diff}:small-signaldiffusioncapacitance,\\ theideal-diodelawandthediffusion\\ capacitancearestandard(background\\ knowledge),theIntechOpenchaptergives\\ forits5\mu \mathrm{m}injectionring{I}_s=5.78\times 10^{-14}\\ A,N=1.46,\tau =1.0\times 10^{-9}sandajunction\\ potentialof0.7V(myreadingofits\\ symbol\varphi ),witha0.7Vdcbiasanda2\\ Vppswing,AkiyamaandUsuki:bias0.85V,\\ theJ.Semicond.abstract:bias0.7V\end{array}$ $\begin{array}{r}(computedwiththechapte{r}^{\prime }sparameters:I\\ =0.46microamperesat0.6V,6.6\\ microamperesat0.7Vand349microamperes\\ at0.85V;C_{diff}=26.5pFfor1\mathrm{mA}and1\\ ns,sothecapacitanceseenbythedriver\\ risesinproportiontothecurrent)\end{array}$
Steady-state carrier density and effective lifetime	$N_c=\frac{I{\tau }_{eff}}{q{V}_{vol}}$	$\frac{1}{{\tau }_{eff}}=\frac{1}{{\tau }_b}+\frac{W{S}_{bottom}}{A_{eff}}+\frac{\left[W+2\left(H-h\right)\right]S_{top}}{A_{eff}}$	$N_c:meancarrierdensityintheintrinsicregion$ $I:forwardcurrent$ ${\tau }_{eff}:effectivelifetime$ $\begin{array}{r}{V}_{vol}:volumeoftheregionthatholdsthe\\ carriers\end{array}$ ${\tau }_b:bulklifetime$ $\begin{array}{r}S:surfacerecombinationvelocity(about\\ 100cm/s)\end{array}$ $A_{eff}:effectivemodearea$ $\begin{array}{r}W,H,h:ribwidth,heightandslab\\ height,thesecondformisEq.2.3ofthe\\ IITthesis,Kotura:N=K_{v}I\tau (Eq.1of\\ thewhitepaper)\end{array}$ $\begin{array}{r}IIT:{\tau }_{b}about4microsecondsand{\tau }_{eff}=100\\ nsusedinthesimulation,Kotura:0.55\\ and0.35microseconds,Wattsetal.:about\\ 1nsinSOI,thehigh-currentAuger\\ recombinationshortenstheeffective\\ lifetime(IIT)\end{array}$ $\begin{array}{r}(computed:fortheIITdiodeof1mmthat\\ gives5dBat70\mathrm{mA}andabout7\times 10^{17}\mathrm{c}{\mathrm{m}}^{-3}\\ (theirestimate),across-sectionof5\mu \mathrm{m}\\ x1.8\mu \mathrm{m}(assumed,fromS=2\mu \mathrm{m}andthe\\ ribdimensions)gives{\tau }_{eff}=Nq{V}_{vol}/I=14\\ ns,wellbelowthe100nsofthe\\ simulation,whichisconsistentwithAuger\\ recombinationatthatcurrent)\end{array}$
Plasma dispersion for equal electron and hole densities	$\Delta n=-\left[8.8\times 10^{-22}N+8.5\times 10^{-18}{N}^{0.8}\right]$ $\Delta \alpha =14.5\times 10^{-18}N\left[\mathrm{c}{\mathrm{m}}^{-1}\right]$	${\alpha }_{dB}=4.343\Delta \alpha$	$\begin{array}{r}N:densityofinjectedelectronsandof\\ injectedholes(equal)in\mathrm{c}{\mathrm{m}}^{-3},the\\ equationsarethoseofthe\\ carrier-depletionpage(Soref-Bennettat\\ 1550nm),thesumofthetwoabsorption\\ coefficients8.5\times 10^{-18}and6.0\times 10^{-18}isthe\\ 14.5\times 10^{-18}thattheIITthesisusesasthe\\ cross-section\sigma offree-carrierabsorption\\ (Table2.1),thelossof0.5\mathrm{c}{\mathrm{m}}^{-1}(linear)\\ andthetwo-photonabsorptionof0.7cm/GW\\ inthesametablearenotusedhere\end{array}$ $\begin{array}{r}(computedatN=7\times 10^{17}\mathrm{c}{\mathrm{m}}^{-3}:\Delta n=\\ -2.22\times 10^{-3}and\Delta \alpha =10.15\mathrm{c}{\mathrm{m}}^{-1}=44.1dB/cm,\\ thatis4.4dBpermm,closetothe5dB\\ in1mmreportedbyIIT;5dBin1mm\\ needs7.9\times 10^{17}\mathrm{c}{\mathrm{m}}^{-3}and3dBin100\mu \mathrm{m}needs\\ 4.76\times 10^{18}\mathrm{c}{\mathrm{m}}^{-3},againstII{T}^{\prime }sestimatesof\\ 7\times 10^{17}and5\times 10^{18})\end{array}$

Efficiency, Loss and Attenuation

Name	Formula	Related Formulae	Variables/Constants
Charge-based voltage-length product	$Q_{\pi }=c{V}_{\pi }L$ $V_{\pi }L\left(\omega \right)=\frac{Q_{\pi }}{c}\sqrt{{\left(\omega \rho c\right)}^2+1}$	$f_c=\frac{1}{2\pi \rho c}$ $R_s=\frac{\rho }{l}$	$\begin{array}{r}{Q}_{\pi }:chargeforaphaseshiftof\pi \\ (independentofthelength)\end{array}$ $c:diodecapacitanceperlength$ $\rho :seriesresistancetimeslength$ $\omega :angularfrequency$ $l:lengthofthediode$ $\begin{array}{r}{f}_c:cut-offofthedistributeddiode,\\ AkiyamaandUsuki:c=45pF/mmused(42.5\\ extracted),\rho =3.68\Omega \mathrm{mm},Q_{\pi }=6.21\mathrm{pC},\\ V_{\pi }L=0.014\mathrm{V}\mathrm{cm}(0.14\mathrm{V}\mathrm{mm}),bias0.85\\ V,theformulasarethoseofthearticle\\ (partlygarbledinthesummary,butthe\\ firsttwoareconsistentwiththevalues),\\ V_{\pi }Lofotherinjectiondevices:0.036\\ \mathrm{V}\mathrm{mm}(J.Semicond.abstract),0.24\mathrm{V}\mathrm{mm}\\ (MITLincolnLaboratoryabstract,static),\\ 0.02\mathrm{V}\mathrm{cm}(citedbyWattsetal.)\end{array}$ $\begin{array}{r}(computed:\rho c=165.6ps,so{f}_c=0.961\\ GHz(printed:about1GHz);6.21/45gives\\ V_{\pi }L=0.138\mathrm{V}\mathrm{mm},equaltothe0.14\\ printed;R_s=61.3\Omega for60\mu \mathrm{m}and4.6\Omega \\ for0.8mm(printed61and4.5);at12.5\\ GHz\omega \rho c=13.01and{V}_{\pi }Lis13.0times\\ thestaticvalue)\end{array}$
Loss-efficiency product of an injection phase shifter	$\alpha V_{\pi }L\left(\omega \right)=\frac{\alpha Q_{\pi }}{c}\sqrt{{\left(\omega \rho c\right)}^2+1}$	$\alpha V_{\pi }L\left(\omega \right)\approx \omega \alpha Q_{\pi }\rho \left(\omega \rho c\gg 1\right)$	$\begin{array}{r}\alpha :propagationlossindB/mm,theother\\ symbolsasinthepreviousrow,Akiyama\\ andUsuki:\alpha =5.3dB/mmand\alpha V_{\pi }L=9.5\\ VdBat12.5GHz,thecarrier-depletion\\ pagesgive29to33VdB(page99),the\\ high-frequencyformshowsthattheproduct\\ growsinproportiontothefrequency\end{array}$ $\begin{array}{r}(computed:static5.3x0.138=0.731V\\ dB;at12.5GHz0.731x13.04=9.54VdB,\\ equaltotheprinted9.5;at6.25GHzit\\ wouldbe4.8andat28GHz21.3VdB;the\\ depletiondevicesofpage99are39to45\\ timesworsethanthestaticinjection\\ valueand3.2timesworsethanthe12.5\\ GHzvalue)\end{array}$
Phase per unit current and attenuation per unit current	$\frac{d\phi }{dI}=\frac{\pi }{I_{\pi }}$ $A\left[\mathrm{dB}\right]=4.343\sigma N_{c}L$	$I_{\pi }=\frac{Q_{\pi }}{{\tau }_{eff}}$ $\frac{dA}{dI}=4.343\sigma L\frac{{\tau }_{eff}}{q{V}_{vol}}$	$\begin{array}{r}{I}_{\pi }:dccurrentforaphaseshiftof\pi ,\sigma =\\ 14.5\times 10^{-18}c{m}^2:cross-sectionof\\ free-carrierabsorptionforequalelectron\\ andholedensities\end{array}$ $L:lengthofthediode$ $\begin{array}{r}IIT:phaseefficiency0.10\pi ,0.09\pi and\\ 0.11\pi rad/mAfordiodesof5mm,1mmand\\ 0.1mm(about10\mathrm{mA}for\pi ),attenuation\\ efficiency0.1dB/mA(5dBat70\mathrm{mA}in1\\ mm;about10dBat100\mathrm{mA}in5mm),a\\ designestimateof\pi in500\mu \mathrm{m}at1V\\ forwardbiaswith\Delta n=1.5\times 10^{-3},Kotura:a\\ slopelabelled2dB/mAonaplotthat\\ spansabout0to60\mathrm{mA}and0to50dB\end{array}$ $\begin{array}{r}(computed:1/0.10=10\mathrm{mA}(11.1and9.1\mathrm{mA}\\ for0.09and0.11);\lambda /(2\Delta n)=517\mu \mathrm{m}for\\ \Delta n=1.5\times 10^{-3};5/70=0.071dB/mAand\\ 10/100=0.10dB/mA;theplotrangesof\\ Koturagive50/60=0.83dB/mA,not2)\end{array}$

Speed and Pre-Emphasis

Name	Formula	Related Formulae	Variables/Constants
Lifetime-limited bandwidth of injection	$f_{3dB}=\frac{1}{2\pi {\tau }_{eff}}$	$f_{3dB}\approx \frac{0.35}{t_r}$	${\tau }_{eff}:effectivecarrierlifetime$ $\begin{array}{r}{t}_r:10to90\%risetime,thetwoforms\\ arestandardsingle-polerelations\\ (backgroundknowledge)\end{array}$ $\begin{array}{r}IIT:simulated3dBelectricalbandwidth\\ about10MHzforS=2\mu \mathrm{m},measured\\ responsetimes25ns(S=2\mu \mathrm{m}),65ns(3\\ \mu \mathrm{m}),125ns(4\mu \mathrm{m})fortheattenuatorand\\ 72nsforthephaseshifter,Wattsetal.:\\ injectionmodulatorstypicallybelow1\\ GHz,Kotura:lifetimes0.55and0.35\\ microseconds\end{array}$ $\begin{array}{r}(computed:1.59MHzfor100ns,40kHzfor\\ 4microseconds,0.29and0.46MHzfor0.55\\ and0.35microseconds,159MHzfor1ns;\\ 0.35/25ns=14MHz,5.4MHzfor65nsand\\ 2.8MHzfor125ns;thesimulated10MHz\\ thereforeliesbetweenthesingle-pole\\ value1.6MHzofthe100nslifetimeand\\ thevalue14MHzofthe25nsresponse)\end{array}$
Drive amplitude needed at a given frequency	$V_{\pi }\left(\omega \right)=\frac{V_{\pi }L\left(\omega \right)}{l}=\frac{V_{\pi }L}{l}\sqrt{{\left(\omega \rho c\right)}^2+1}$	$V_{pp}\approx V_{\pi }\left(\omega \right)\mathrm{for}\mathrm{full}\mathrm{swing}$	$\begin{array}{r}l:lengthofthediode,thepre-emphasis\\ boostneededatthelinerateequalsthe\\ square-rootfactor,AkiyamaandUsuki:56\\ Gbit/sNRZwithapre-emphasisdriveof\\ about2Vppfromafinite-impulse-response\\ filter,cleareye,pattern{2}^7-1,length\\ limit{l}_{max}about800\mu \mathrm{m},theIntechOpen\\ chapter(5\mu \mathrm{m}ring):2Vppwitha\\ pre-emphasispulseof0.8V,pulsesof40\\ to80ps,8and9Gbit/s,theJ.Semicond.\\ abstract:12.5Gbit/s,6.3Vswing,0.7V\\ bias,extinctionratio8.4dB\end{array}$ $\begin{array}{r}(computedforthe0.8mmdiodewith{V}_{\pi }L=\\ 0.138\mathrm{V}\mathrm{mm}:V_{\pi }=0.173Vstatic,2.25Vat\\ 12.5GHzand5.03Vat28GHz,theNyquist\\ frequencyof56Gbit/s;so2Vppisbelow\\ afull\pi swingatthatrate,whichis\\ consistentwithaneyethatisopenbut\\ notdeep)\end{array}$
Useful length of the injection shifter	$l_{max}\approx 800\mu \mathrm{m}(\mathrm{article},50\mathrm{\Omega }\mathrm{driver})$	$R_s=\frac{\rho }{l},A_{loss}=\alpha l$	$\begin{array}{r}theoptimumlengthisderivedinthe\\ article(itsEq.8,garbledinthesummary\\ Iread,soitisnotreproducedhere),\\ AkiyamaandUsuki:l_{max}about800\mu \mathrm{m},\alpha =\\ 5.3dB/mm,R_{s}about61\Omega at60\mu \mathrm{m}and4.5\\ \Omega at0.8mm,thedriverimpedance50\Omega ,a\\ shorterdiodehasalargerseries\\ resistanceandsoalowercut-off,a\\ longerdiodehasmoreloss\end{array}$ $\begin{array}{r}(computed:thelossat0.8mmis5.3x0.8\\ =4.24dB;the60\mu \mathrm{m}diodehas{R}_s=61\Omega ,\\ aboutequaltothe50\Omega driver)\end{array}$

Energy and Loss per Phase Shift

Name	Formula	Related Formulae	Variables/Constants
Energy and holding power for a phase shift of pi	$E_{\pi }=Q_{\pi }{V}_f$ $P_{\pi }=I_{\pi }{V}_f$	$I_{\pi }=\frac{Q_{\pi }}{{\tau }_{eff}}$	$V_f:forwardvoltageattheoperatingpoint$ $Q_{\pi }:chargeofthephaseshifter$ $\begin{array}{r}{I}_{\pi }:dccurrent,AkiyamaandUsuki:Q_{\pi }=\\ 6.21\mathrm{pC}at0.85V,thecarrierlifetimeis\\ notreportedthere(takenas1nsfrom\\ Wattsetal.,assumed)\end{array}$ $\begin{array}{r}IIT:about10\mathrm{mA}for\pi (voltagetakenas1\\ V,assumed),therelationsassumethatall\\ thechargeislostonceperbitandthat\\ thecurrentisacontinuousrecombination\\ current(myreading,backgroundknowledge)\end{array}$ $\begin{array}{r}(computed:E_{\pi }=6.21x0.85=5.28pJ;I_{\pi }\\ =6.2\mathrm{mA}and{P}_{\pi }=5.3mWfor1ns;IIT10\\ \mathrm{mA}x1V=10mW;comparedwiththe1.08\\ pJperjunctionofthedepletiondevice\\ (page99)theinjectionvalueis4.9times\\ higher,andcomparedwiththethermal\\ shifterofpage99(1to2mWper\pi )itis\\ 3to10timeshigher)\end{array}$
Loss accompanying a phase shift of pi	$A_{\pi }=4.343\Delta \alpha L_{\pi }=4.343\frac{\Delta \alpha \lambda }{2\Delta n}$	${\alpha }_H=\frac{2k_0|\Delta n|}{\Delta \alpha }$	$A_{\pi }:attenuationoverthelengthofa\pi shift$ $\begin{array}{r}\Delta n,\Delta \alpha :indexandintensityabsorption\\ changesofthesamecarriers,k_0=2\pi /\lambda ,\\ uniformcarrierdensityandfulloverlap\\ withthemodeareassumed(myderivation,\\ backgroundknowledge),theHenry-factor\\ formfollowstheIndiumPhosphide\\ Quantum-WellElectroabsorptionModulator\\ pageandthepreviouspage\end{array}$ $\begin{array}{r}(computedwiththeSoref-Bennettvaluesat\\ 7\times 10^{17}\mathrm{c}{\mathrm{m}}^{-3}:{\alpha }_H=17.7and{A}_{\pi }=1.54dBasa\\ lowerlimit;withthemeasured5.3dB/mm\\ and0.8mmAkiyamaandUsukihave4.2dB,\\ 2.8timesthelimit,becausethemode\\ overlapsthedopedregionsandthe\\ waveguidehasotherloss(myreading))\end{array}$

Important Questions

Ques: Is the voltage-length product of an injection modulator 0.014 V cm, 0.036 V mm or 0.24 V mm?

The values are 0.14 $\mathrm{V}\mathrm{mm}$ (Akiyama and Usuki, 0.014 $\mathrm{V}\mathrm{cm}$), 0.036 $\mathrm{V}\mathrm{mm}$ (J. Semicond. abstract), 0.24 $\mathrm{V}\mathrm{mm}$ (MIT Lincoln Laboratory, static) and 0.2 $\mathrm{V}\mathrm{mm}$ (0.02 $\mathrm{V}\mathrm{cm}$, cited by Watts et al.). They lie between 0.036 and 0.24 $\mathrm{V}\mathrm{mm}$, a factor of 7 apart, and are all far below the 1.4 to 2 $\mathrm{V}\mathrm{cm}$ of depletion. The abstracts do not say at what bias, length or frequency each was found, and $V_{\pi }L$ depends on the frequency by the factor $\sqrt{(\omega \rho c{)}^2+1}$ (13 at 12.5 GHz, computed). I take 0.14 $\mathrm{V}\mathrm{mm}$ as the best documented value, since $Q_{\pi }=c{V}_{\pi }L$ reproduces its printed 6.21 $\mathrm{pC}$ (6.21/45 = 0.138 $\mathrm{V}\mathrm{mm}$, computed), and treat the others as static or device-specific. Reference: Akiyama and Usuki, forward-biased silicon pin modulator, Frontiers in Physics 2, article 65 (2014) (frontiersin.org PDF; names and title from the summary header, not checked; read through a fetch-tool summary in which several symbols and two equations were garbled); J. Semicond. 33(11), 114005 (2012), a 12.5 Gbit/s carrier-injection silicon Mach-Zehnder modulator (abstract page only; title and authors not recorded); ‘Effect of carrier lifetime on forward-biased silicon Mach-Zehnder modulators’, Optics Express 16(8), 5218-5226 (2008) (MIT Lincoln Laboratory abstract page only; authors not recorded); Watts et al., IEEE J. Sel. Top. Quantum Electron. (2010), a depletion-mode silicon Mach-Zehnder modulator (MIT file 2010Watts.pdf; read through a fetch-tool summary; used here only for its statements on injection devices).

Ques: Is the loss-efficiency product 0.73 or 9.5 V dB?

Both, at different frequencies. Akiyama and Usuki print $\alpha V_{\pi }L=9.5$ V dB at 12.5 GHz. With $\alpha =5.3$ $dB/mm$ and $V_{\pi }L=0.138$ $\mathrm{V}\mathrm{mm}$ the static product is 0.731 V dB, and the factor $\sqrt{(\omega \rho c{)}^2+1}=13.04$ at 12.5 GHz (with $\rho c=165.6$ ps, computed) gives 9.54 V dB, which reproduces the printed value. The summary also says ‘operating rate 12.5 Gbit/s’, but the number matches a frequency of 12.5 GHz, not the 6.25 GHz Nyquist frequency of 12.5 Gbit/s, which would give 4.8 V dB (computed); I read it as 12.5 GHz (my reading). At 28 GHz the product would be 21.3 V dB (computed). Reference: Akiyama and Usuki, forward-biased silicon pin modulator, Frontiers in Physics 2, article 65 (2014) (frontiersin.org PDF; names and title from the summary header, not checked; read through a fetch-tool summary in which several symbols and two equations were garbled).

Ques: Is the attenuation slope 2 $dB/mA$, 0.83 $dB/mA$ or 0.1 $dB/mA$?

They belong to different devices. The Kotura plot labels a slope of 2 $dB/mA$, but its axes, as I read them from the summary, span about 0 to 60 $\mathrm{mA}$ and 0 to $50\mathrm{dB}$, which is 0.83 $dB/mA$ (computed), so the label and the axes disagree and either the label or my reading of the axes is wrong (my reading). The IIT thesis gives 0.1 $dB/mA$ in its abstract, and its data give 0.071 $dB/mA$ ($5\mathrm{dB}$ at 70 $\mathrm{mA}$ in 1 mm) and 0.10 $dB/mA$ ($10\mathrm{dB}$ at 100 $\mathrm{mA}$ in 5 mm), computed. The Kotura rib is 3 $\mu \mathrm{m}$ by 3 $\mu \mathrm{m}$ and the IIT diodes are 0.1 to 5 mm long, so the slope also depends on the length and the cross-section; I would not quote 2 $dB/mA$. Reference: Kotura, ‘Fundamentals of Silicon Photonic Devices’ white paper (network.nvidia.com PDF; read through a fetch-tool summary; axis labels partly garbled); Sreevatsa K.R., ‘Experimental Studies of Diffusion Doped PIN Waveguides for Integrated Silicon Photonics Circuits’, IIT Madras (2017) (ee.iitm.ac.in PDF; read through a fetch-tool summary).

Ques: Which carrier lifetime applies: 1 ns, 14 ns, 100 ns, 0.5 microseconds or 4 microseconds?

All appear and none is an error. Watts et al. say about 1 ns in SOI; the IIT thesis uses a bulk lifetime of about 4 microseconds and an effective lifetime of 100 ns in its simulation; Kotura plots 0.55 and 0.35 microseconds for a large rib. The effective value depends on the surface recombination and on the current: from $5\mathrm{dB}$ in 1 mm at 70 $\mathrm{mA}$ and about $7\times 10^{17}$ $\mathrm{c}{\mathrm{m}}^{-3}$ I find 14 ns for an assumed cross-section of 9 square micrometres (computed, assumed), which the thesis would attribute to Auger recombination at high injection. The measured response times of 25 to 125 ns are in the same range. A small, strongly doped waveguide has the shortest lifetime, and the faster modulators need it. Reference: Watts et al., IEEE J. Sel. Top. Quantum Electron. (2010), a depletion-mode silicon Mach-Zehnder modulator (MIT file 2010Watts.pdf; read through a fetch-tool summary; used here only for its statements on injection devices); Sreevatsa K.R., ‘Experimental Studies of Diffusion Doped PIN Waveguides for Integrated Silicon Photonics Circuits’, IIT Madras (2017) (ee.iitm.ac.in PDF; read through a fetch-tool summary); Kotura, ‘Fundamentals of Silicon Photonic Devices’ white paper (network.nvidia.com PDF; read through a fetch-tool summary; axis labels partly garbled).

Ques: Why can an injection modulator reach 56 Gbit/s if its lifetime allows only a few MHz?

The single-pole bandwidth $1/(2\pi \tau )$ is 1.6 MHz for 100 ns and 159 MHz for 1 ns (computed), far below 28 GHz. Akiyama and Usuki show that the limit is not the lifetime for the charge pulse: with pre-emphasis the driver overshoots and removes the charge, and the diode behaves as a distributed RC line with a cut-off $1/(2\pi \rho c)$ of about 1 GHz (0.96 GHz, computed). Above it the required voltage grows in proportion to the frequency ($\sqrt{(\omega \rho c{)}^2+1}$ is 29.2 at 28 GHz, computed), so the eye opens but the swing needed is large: 5.0 V for a full $\pi$ with 0.8 mm (computed), against the 2 Vpp used. The price is power and a strong driver. Reference: Akiyama and Usuki, forward-biased silicon pin modulator, Frontiers in Physics 2, article 65 (2014) (frontiersin.org PDF; names and title from the summary header, not checked; read through a fetch-tool summary in which several symbols and two equations were garbled); Watts et al., IEEE J. Sel. Top. Quantum Electron. (2010), a depletion-mode silicon Mach-Zehnder modulator (MIT file 2010Watts.pdf; read through a fetch-tool summary; used here only for its statements on injection devices).

Ques: Does 2 Vpp at 56 Gbit/s agree with the 6.3 V of the 12.5 Gbit/s device?

The two are different devices, and the summaries give no lengths for the second, so I cannot reconcile them. For Akiyama and Usuki a full $\pi$ swing at 12.5 GHz needs 2.25 V for a 0.8 mm diode (computed from 0.138 $\mathrm{V}\mathrm{mm}$ and the factor 13.04), so the 2 Vpp used at higher rates gives only a partial swing (my reading). The J. Semicond. device used 6.3 V for $8.4\mathrm{dB}$ of extinction ratio, which at quadrature would need a peak-to-peak phase of 0.537$\pi$ (computed), with a $V_{\pi }L$ of 0.036 $\mathrm{V}\mathrm{mm}$; the length and the drive point are missing. I regard the two voltages as consistent with devices of different length and bias (my reading), but cannot prove it. Reference: Akiyama and Usuki, forward-biased silicon pin modulator, Frontiers in Physics 2, article 65 (2014) (frontiersin.org PDF; names and title from the summary header, not checked; read through a fetch-tool summary in which several symbols and two equations were garbled); J. Semicond. 33(11), 114005 (2012), a 12.5 Gbit/s carrier-injection silicon Mach-Zehnder modulator (abstract page only; title and authors not recorded).

Ques: Is the injection modulator more energy-efficient than the depletion one?

The sources disagree. The IntechOpen chapter says that injection gives higher energy efficiency and modulation depth and depletion higher bandwidth density. The numbers I can compute point the other way for a full $\pi$ shift in a Mach-Zehnder: $Q_{\pi }{V}_f=6.21\times 0.85=5.28$ pJ (computed) against 1.08 pJ for one depletion junction of page 99 (4.9 times less), and for rings the injection value of 473 fJ/bit at 2 Vpp (page 100) is far above the 0.97 to 16 fJ/bit of the depletion ring. I judge the chapter’s statement to refer to the voltage and the depth, not to the energy per bit (my reading); the charge and the continuous current make injection costly. Reference: IntechOpen chapter 54847 on models of carrier-injection and carrier-depletion ring modulators (title and authors not recorded; read through a fetch-tool summary); Akiyama and Usuki, forward-biased silicon pin modulator, Frontiers in Physics 2, article 65 (2014) (frontiersin.org PDF; names and title from the summary header, not checked; read through a fetch-tool summary in which several symbols and two equations were garbled); Silicon Carrier-Depletion Mach-Zehnder Modulator page of this vault; Silicon Microring Modulator page of this vault.

Ques: How does the holding power of injection compare with thermal tuning?

A phase of $\pi$ held by injection needs a current $I_{\pi }=Q_{\pi }/\tau$: 6.2 $\mathrm{mA}$ for 6.21 $\mathrm{pC}$ and an assumed 1 ns lifetime, or 5.3 mW at 0.85 V (computed). The IIT thesis gives about 10 $\mathrm{mA}$ for $\pi$ in a long diode, which is 10 mW at an assumed 1 V (computed). The thermal shifter of page 99 needs 1 to 2 mW per $\pi$. The injection shifter is therefore 3 to 10 times more costly in holding power, but its response is 25 to 125 ns against microseconds for the heater (the IIT measurements and the thermal time constants of page 100). I would use injection only where that speed is needed, such as VOAs or fast bias control. Reference: Akiyama and Usuki, forward-biased silicon pin modulator, Frontiers in Physics 2, article 65 (2014) (frontiersin.org PDF; names and title from the summary header, not checked; read through a fetch-tool summary in which several symbols and two equations were garbled); Sreevatsa K.R., ‘Experimental Studies of Diffusion Doped PIN Waveguides for Integrated Silicon Photonics Circuits’, IIT Madras (2017) (ee.iitm.ac.in PDF; read through a fetch-tool summary); Silicon Carrier-Depletion Mach-Zehnder Modulator page of this vault.

Ques: Why does a short diode attenuate less than the length scaling predicts?

The IIT thesis finds that shorter diodes give less attenuation than proportional scaling, and attributes this to Auger recombination at high injection. At the same current, a shorter diode has a smaller volume and so a higher carrier density, and Auger recombination, which grows as the cube of the density, shortens the lifetime and lowers the density (background knowledge). My own estimate supports the direction: $3\mathrm{dB}$ in 100 $\mu \mathrm{m}$ needs $4.76\times 10^{18}$ $\mathrm{c}{\mathrm{m}}^{-3}$, nearly ten times the $7\times 10^{17}$ $\mathrm{c}{\mathrm{m}}^{-3}$ of $5\mathrm{dB}$ in 1 mm (computed), a density at which Auger loss is strong. I did not model it, and the thesis figures are from a summary. Reference: Sreevatsa K.R., ‘Experimental Studies of Diffusion Doped PIN Waveguides for Integrated Silicon Photonics Circuits’, IIT Madras (2017) (ee.iitm.ac.in PDF; read through a fetch-tool summary).

Ques: How much loss comes with a phase shift of pi?

Injection moves the index and the absorption together. For equal electron and hole densities of $7\times 10^{17}$ $\mathrm{c}{\mathrm{m}}^{-3}$ the Soref-Bennett equations give $\Delta n=-2.22\times 10^{-3}$ and $\Delta \alpha =10.15$ $\mathrm{c}{\mathrm{m}}^{-1}$, so $2k_0\Delta n/\Delta \alpha =17.7$ and a phase shift of $\pi$ costs at least $1.54\mathrm{dB}$ (computed, uniform density and full overlap, my derivation). Akiyama and Usuki have $4.2\mathrm{dB}$ at 0.8 mm with 5.3 $dB/mm$, which is 2.8 times the limit, because the mode overlaps the doped regions and the guide has other loss (my reading). The depletion devices of page 99 lose 16 to 22 $dB/cm$ over a $\pi$ length of 2 to 3 mm ($4.4\mathrm{dB}$ for Li et al. and $4.8\mathrm{dB}$ for Xiao et al., computed from their projected length), so injection is not worse in loss per $\pi$. Reference: Akiyama and Usuki, forward-biased silicon pin modulator, Frontiers in Physics 2, article 65 (2014) (frontiersin.org PDF; names and title from the summary header, not checked; read through a fetch-tool summary in which several symbols and two equations were garbled); Silicon Carrier-Depletion Mach-Zehnder Modulator page of this vault.

Ques: What sets the useful length of a forward-biased shifter?

Two terms compete. A longer diode has a lower series resistance, $R_s=\rho /l$, with $\rho =3.68$ $\Omega \mathrm{mm}$: 61 $\Omega$ for 60 $\mu \mathrm{m}$ and 4.6 $\Omega$ for 0.8 mm (computed, printed 61 and 4.5), but also more loss, $5.3\times 0.8=4.24$ dB at 0.8 mm (computed). Akiyama and Usuki derive an optimum of about 800 $\mu \mathrm{m}$ for a 50 $\Omega$ driver; the equation for it was garbled in the summary, so I do not reproduce it and cannot check the value. A diode with $R_s$ near the driver impedance, as at 60 $\mu \mathrm{m}$ (61 $\Omega$), would also halve the voltage across the junction, and the article notes that matching the driver doubles the loss-efficiency product (as printed). Reference: Akiyama and Usuki, forward-biased silicon pin modulator, Frontiers in Physics 2, article 65 (2014) (frontiersin.org PDF; names and title from the summary header, not checked; read through a fetch-tool summary in which several symbols and two equations were garbled).

Related pages in this vault: Silicon MOS-Capacitor Modulator, Silicon Carrier-Depletion Mach-Zehnder Modulator, Silicon Microring Modulator, Thermo-Optic Modulator, Directly Modulated Laser, Silicon on Insulator (SOI) MOSFET, PIN Diode.