Waveguide Photodiode

1. Theory

• Definition: A waveguide photodiode (WGPD) is a photodiode in which the light travels along a waveguide and is absorbed over the length $L$ of an adjacent or embedded absorbing layer, so that the absorbed fraction $\eta =1-e^{-\Gamma \alpha L}$ is set by the length, while the carriers move over the short distance across the thin layer.
• Why it is used: In a top-illuminated diode the thickness must be $1/\alpha$ or more for a high efficiency, which lengthens the transit time; waveguide structures overcome the bandwidth limits imposed by the responsivity maximization and the finite drift times of vertical structures, and are the natural match for planar integration, as the book states for bit rates of 10 Gb/s and beyond.
• Structures: The book shows (a) normal incidence, (b) edge-coupled waveguide, (c) tapered evanescently coupled waveguide and (d) multimode input waveguide evanescently coupled photodiodes with optical matching layers; the evanescently coupled diodes allow about four times higher saturation currents than the edge-coupled ones.
• III-V diodes: An InGaAs/InP evanescent diode uses a short multimode waveguide of 20 μm with a diluted waveguide (InGaAsP layers of 20 nm to 140 nm separated by 400 nm InP layers) and optical matching layers, a partly p-doped InGaAs absorber, and a semi-insulating InP substrate; a 400 nm intrinsic InGaAs absorber bonded on a silicon-on-insulator guide gave 67 GHz.
• Ge-on-Si diodes: A Ge absorber of about 600 nm to 700 nm on a silicon rib or slab waveguide forms a vertical or lateral PIN diode, with an absorption coefficient of the Ge of $920\mathrm{c}{\mathrm{m}}^{-1}$ at 1550 nm and $7956\mathrm{c}{\mathrm{m}}^{-1}$ at 1310 nm; the length is a few micrometres to 40 μm.
• Receiver design: The length also fixes the capacitance (about 1 fF/μm), so the efficiency, the capacitance, the load resistor and the sensitivity of the receiver have to be traded off together.
• Traveling-wave devices: The bandwidth of long detectors is extended with traveling-wave electrodes that match the velocity of the electrical wave to that of the optical wave, or with discrete short detectors connected by delay lines.
• High power: The absorption is highest at the front end, which limits the saturation current, and a directional-coupler diode with a lateral input guide reached more than 60 mA at $-4$ V and 0.80 A/W.

2. Working Principle

• Guided absorption: The power in the absorber decreases exponentially along the guide, $P(z)=P_0{e}^{-\Gamma \alpha z}$, and each absorbed photon creates an electron-hole pair that is separated by the reverse-bias field of the PIN junction.
• Coupling: The light is brought to the absorber by butt coupling at the facet, or evanescently from a lower guide of larger mode size; in the InP diode the coupling length is 20 μm, against 700 μm for tapered evanescent diodes.
• Collection: The carriers cross the thickness $d$ of the thin absorber (400 nm to 600 nm) or the lateral gap of a lateral diode, so that the transit-time limit is $f_{tr}\approx 0.56v/d$ for a uniform generation.
• Electrical limit: The junction capacitance grows with the length, $C_D=c^{\prime }L$, and the RC limit $f_{RC}$ combines with $f_{tr}$ to give the bandwidth.
• Saturation: The photocurrent density is highest near the front end, where the space charge lowers the field first, so that the saturation current falls as the peak-to-average absorption ratio ${\rho }_{pk}$ rises.
• Mode control: In a high-power diode the excitation of a strongly absorbing mode of large overlap places the absorption peak close to the front end, and a lateral directional coupler suppresses this mode (peak-to-average ratio 1.5 in the simulated design).
• Traveling-wave operation: The photocurrent excited along a long absorber adds in phase on the electrode only when the electrical and optical velocities match, $v_{\mu }=v_{opt}$; otherwise each detector must be shorter than a quarter of the microwave wavelength.
• Peaking: Bonding wires of about 1 nH/mm act as an inductor in series with the detector capacitance and increased the bandwidth of a Ge detector from a typical 30 GHz to about 60 GHz (calculated).

3. Applications

• Receivers in photonic integrated circuits: The photodiode of a preamplified receiver and of receiver arrays, such as the 40-channel receiver with an arrayed-waveguide-grating demultiplexer and 40 Gb/s diodes of the book, where most photodiodes used today are reverse-biased PIN structures.
• Silicon-photonic receivers: Ge-on-Si waveguide diodes with a responsivity of 0.5 A/W to 1 A/W and bandwidths of 20 GHz to 60 GHz are the detector of the silicon photonics process.
• Photocurrent-driven transceivers: The high-speed photodiode of a traveling-wave electroabsorption transmitter shares its electrode with the modulator, as in the transceiver and wavelength converter of the book that operated up to 40 Gb/s.
• High-power analog links: Diodes with controlled absorption density deliver more than 60 mA for analog optical links, where the saturation current limits the linearity.

Name	Formula	Related Formulae	Variables/Constants
Absorption along the guide, efficiency and length	${\eta }_{int}=1-e^{-\Gamma \alpha L},L_D=\frac{1}{\Gamma \alpha }\ln \frac{1}{1-\eta }$	$I_p=\mathcal{R}{P}_{in},\eta =\frac{I_p/q}{P_{in}/h\nu }=\mathcal{R}\frac{h\nu }{q}=\frac{\mathcal{R}1.24}{\lambda }$ ${\eta }_{ext}={\eta }_c{\eta }_{int}$	$\begin{array}{r}\Gamma :overlap(confinementfactor)ofthe\\ guidedmodewiththeabsorber\end{array}$ $\alpha :absorptioncoefficientoftheabsorber$ $L:absorberlengthalongtheguide$ $L_D:lengthfortheefficiency\eta$ $\begin{array}{r}{\eta }_c:couplingefficiencyofthelightinto\\ theguide\end{array}$ $\begin{array}{r}(bookexample:\Gamma =5\%,\alpha =5000c{m}^{-1},\eta =\\ 90\%gives{L}_D=92\mu \mathrm{m}and{I}_p=1.125\mu \mathrm{A}\\ for1\mu \mathrm{W};aGeguidewith\Gamma =0.86and\alpha =\\ 920c{m}^{-1}gives29\mu \mathrm{m}for90\%,and\mathcal{R}=1.20\\ A/Wfor40\mu \mathrm{m}against1.2A/Wmeasured,\\ and0.68A/Wfor10\mu \mathrm{m}against0.5A/W\\ measuredand0.63A/Wcalculated,\\ computed)\end{array}$
Capacitance, load resistor and signal voltage of the receiver	$C_D=c^{\prime }L,R_L=\frac{1}{2\pi B{C}_D}$	$V_G=I_p{R}_L$ $f_{RC}=\frac{1}{2\pi \left(R_L+R_s\right)C}$	$\begin{array}{r}{c}^{\prime }:capacitanceperunitlengthofthe\\ guide(about1fF/\mu \mathrm{m}inthebook)\end{array}$ $C_D:detectorcapacitance$ $\begin{array}{r}B:bitrate,takenequaltothebandwidth\\ ofanRC-limitedreceiver\end{array}$ $R_L:loadresistoroftheFETstage$ $V_G:signalvoltageattheFETgate$ $\begin{array}{r}(bookexample:L=92\mu \mathrm{m}gives{C}_D=92fF,\\ R_L=1.73k\Omega at1Gb/sand{V}_G=1.95mV;\\ thequantumefficiencyandthecapacitance\\ bothgrowwiththelength,sothe\\ sensitivityfallsasthelengthis\\ increased)\end{array}$
Transit-limited and overall bandwidth	$f_{tr}\approx \frac{3.5}{2\pi \tau }=\frac{0.56v}{d},\frac{1}{f_{3dB}^2}\approx \frac{1}{f_{tr}^2}+\frac{1}{f_{RC}^2}$	$f_{tr}^{sheet}\approx \frac{0.44v}{d}$ $\tau =\frac{d}{v}$	$\begin{array}{r}d:distancefromthegenerationsiteto\\ thecollectingelectrode(theabsorber\\ thicknessinaverticaldiode,thegapin\\ alateraldiode)\end{array}$ $v:averagecarriervelocity$ $\begin{array}{r}{f}_{tr}:transit-timelimitforageneration\\ thatisuniformoverd\end{array}$ $\begin{array}{r}{f}_{tr}^{sheet}:limitforcarriersgeneratedatone\\ planeandcrossingthewholedistance(the\\ factor0.44ofthePINPhotodiodepage);a\\ numericalFouriertransformgives0.443\\ forthesheetand0.553fortheuniform\\ generation,closeto3.5/2\pi =0.557\\ (computed)\end{array}$ $\begin{array}{r}(InPevanescentdiode:d=400nmandv=\\ 5.3\times 10^{6}cm/sgive74GHz,with{f}_{RC}=130\\ GHzthecombinationis64GHzagainst67\\ GHzmeasured,computed)\end{array}$
Evanescent coupling and absorption of two guided modes	$L_c=\frac{\lambda }{2\left(n_s-n_a\right)}$	$P_{abs}\left(z\right)\approx {\chi }_0^2{\Gamma }_0{e}^{-{\alpha }_{0}z}+{\chi }_1^2{\Gamma }_1{e}^{-{\alpha }_{1}z}-2{\chi }_0{\chi }_1\cos \left(\frac{2\pi }{\lambda }\mathrm{Re}\left(n_{eff,0}-n_{eff,1}\right)z\right)\left|\iint {\Phi }_0{\Phi }_1^{\ast }dxdy\right|e^{-\left({\alpha }_0+{\alpha }_1\right)z/2}$	$\begin{array}{r}{L}_c:lengthforthefulltransferofpower\\ betweentwoparallelguides\end{array}$ $\begin{array}{r}{n}_s,n_a:effectiveindicesofthesymmetric\\ andantisymmetricsupermodes\end{array}$ $\begin{array}{r}{\chi }_0,{\chi }_1:excitationcoefficientsofthetwo\\ guidedmodesoftheabsorbersection(set\\ bythezeropoweratthefrontend)\end{array}$ ${\Gamma }_i:overlapwiththeabsorber$ ${\alpha }_i:modalabsorptioncoefficient$ ${\Phi }_i:modefields$ $n_{eff,i}:effectiveindices$ $\begin{array}{r}(thesecondformisthemodaldescription\\ ofawaveguidephotodiodewithastrongly\\ absorbinghigh-overlapmodeandaweak\\ mode,andthedifferenceofthetwosigns\\ ofthefieldsintheabsorbergivesthe\\ minussignoftheinterferenceterm)\end{array}$
Absorption density, peak-to-average ratio and saturation current	${\rho }_{pk}=\frac{\Gamma \alpha L}{1-e^{-\Gamma \alpha L}}=\frac{\ln \left[1/\left(1-\eta \right)\right]}{\eta }$	$I_{sat}\approx \frac{I_{sat,0}}{{\rho }_{pk}}$ $P_{abs}\left(z\right)\propto e^{-\Gamma \alpha z}$	$\begin{array}{r}{\rho }_{pk}:ratioofthepeakabsorbedpowerper\\ unitlength(atthefrontend)tothe\\ averageoverthelength,forasingle\\ exponentiallydecayingmode\end{array}$ $\begin{array}{r}{I}_{sat,0}:saturationcurrentforauniform\\ absorption(space-chargelimited,ason\\ theUTC-PDpage)\end{array}$ $\begin{array}{r}({\rho }_{pk}=2.56for\eta =90\%and4.65for99\%;\\ adirectional-couplerdiodewitha\\ simulatedpeak-to-averageratioof1.5\\ gavemorethan60mA,about50\%morethan\\ aconventionalguide,againstaratio\\ 2.56/1.5=1.7expectedfromtheformula,\\ computed)\end{array}$
Velocity matching and segment length of the traveling-wave detector	$v_{\mu }=\frac{c}{\sqrt{{\epsilon }_{eff}}}=v_{opt}=\frac{c}{n_g}$	${\ell }_{max}=\frac{{\lambda }_{\mu }}{4}=\frac{c}{4f_{max}\sqrt{{\epsilon }_{eff}}}$ $f_B=\frac{v_{\mu }}{\pi d_p}$	$\begin{array}{r}{v}_{\mu }:phasevelocityofthemicrowavewave\\ ontheelectrode(coplanarline)\end{array}$ ${\epsilon }_{eff}:effectivepermittivityoftheline$ $v_{opt}:groupvelocityofthelightintheguide$ $n_g:groupindex$ $\begin{array}{r}{\ell }_{max}:lengthofadiscretedetector\\ (quarterofthemicrowavewavelengthat\\ thehighestfrequency{f}_{max})\end{array}$ $\begin{array}{r}{f}_B:cutofffrequencyofalinethatis\\ loadedperiodicallywiththeperiod{d}_p\end{array}$ $\begin{array}{r}(for{\epsilon }_{eff}=6.7(assumed){\ell }_{max}=1.45mmat\\ 20GHzand0.29mmat100GHz,againstthe\\ under1mmofthepatent;aperiodof150\\ \mu \mathrm{m}gives{f}_B=246GHzagainstabout300\\ GHzinthequoteddevice,computed)\end{array}$
Ge absorber with a metal contact in the optical path	$\mathrm{IQE}=\frac{{\alpha }_{Ge}{L}_{Ge}}{{\alpha }_{Ge}{L}_{Ge}+{\alpha }_m{L}_m}\left[1-e^{-\left({\alpha }_{Ge}{L}_{Ge}+{\alpha }_m{L}_m\right)}\right]$	$f_t\approx \frac{3.5v}{2\pi d_{abs}}$ $f_{RC}=\frac{1}{2\pi R_{eff}{C}_{pd}}$	$\begin{array}{r}{\alpha }_{Ge},{\alpha }_m:absorptioncoefficientsoftheGe\\ andofthemetalinthemode\end{array}$ $\begin{array}{r}{L}_{Ge},L_m:lengthsoverwhichthemodesees\\ theGeandthemetal\end{array}$ $\begin{array}{r}{d}_{abs}:distancefromthegenerationsiteto\\ thecollectingelectrode\end{array}$ $\begin{array}{r}(valuesofthesource:{\alpha }_{Ge}=7956and920\\ c{m}^{-1}and{\alpha }_{Au}=182and130c{m}^{-1}at1310nm\\ and1550nm,sothatGeabsorbs7to43\\ timesmorethanAu;{\alpha }_{Ge}/{\alpha }_{Au}=7.1at1550\\ nm,computed)\end{array}$

Important Questions

Ques: Why does the waveguide geometry remove the trade-off between efficiency and speed of a vertical photodiode?

In a vertical diode the light crosses the absorber along the carrier path, so a high efficiency needs a thickness of several $1/\alpha$ and the transit time grows with it. For $\alpha =5000\mathrm{c}{\mathrm{m}}^{-1}$ a 90% efficiency needs $\ln 10/\alpha =4.6$ μm, and with $v=5\times 10^{6}cm/s$ (assumed) the limit $0.56v/d$ is only 6 GHz (computed), whereas a 400 nm absorber gave 67 GHz. In the waveguide diode the light travels along the absorber and the efficiency depends on the length, $\eta =1-e^{-\Gamma \alpha L}$, so the thickness can be chosen for the speed. The book example with an overlap of 5% needs a length of 92 μm for 90%, which is the price: the length adds capacitance, about 1 fF/μm. Reference: L. A. Coldren, S. W. Corzine and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed., Wiley, 2012, Sec. 8.3.1, Eqs. 8.20-8.22, Fig. 8.23 and Example 8.8; Heterogeneously Integrated InP-Based Evanescently Coupled p-i-n Photodiode on SOI (siliconphotonics.ece.ucsb.edu, C975.pdf).

Ques: Why do the length and the load resistor of a waveguide receiver have to be chosen together?

Both the quantum efficiency and the capacitance grow with the length, while the sensitivity of the receiver, which depends on the load resistance, is inversely proportional to the capacitance, since the load must pass the highest frequency of the data. For an RC-limited receiver with a bandwidth equal to the bit rate $B$, $R_L=1/(2\pi B{C}_D)$. In the book example the length of 92 μm gives 92 fF, and for 1 Gb/s the load is 1.73 kΩ, with a photocurrent of 1.125 μA for 1 μW and a gate voltage of 1.95 mV (reproduced). A shorter diode lowers the capacitance and allows a larger load and a larger voltage, but gives a smaller efficiency. The 400 nm InP diode on silicon-on-insulator had a junction capacitance of 17.5 fF, with an RC limit of about 130 GHz. Reference: L. A. Coldren, S. W. Corzine and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed., Wiley, 2012, Sec. 8.3.1, Eqs. 8.20-8.22, Fig. 8.23 and Example 8.8; Heterogeneously Integrated InP-Based Evanescently Coupled p-i-n Photodiode on SOI (siliconphotonics.ece.ucsb.edu, C975.pdf).

Ques: Why can an evanescently coupled photodiode be short and still reach a responsivity of 1 A/W?

In an edge-coupled diode all the light enters through one small facet and the absorption peaks at the front. In the evanescent design a dilute multimode guide with optical matching layers carries the light, and the mode is transferred gradually into the absorber below it, so that the absorption is spread along the guide and the large mode size eases the fibre coupling. The InP diode with a multimode guide of 20 μm reached 1.01 A/W for an absorption length above 20 μm, against a coupling length of 700 μm for the tapered devices, and a 1-dB compression current above 23 mA. The book gives about four times the saturation current of the edge-coupled structure. The coupling loss itself was not quantified in this source. Reference: Wu et al., High-Performance Evanescently Edge Coupled Photodiodes With Partially p-Doped Photoabsorption Layer at 1.55-μm Wavelength, IEEE Photon. Technol. Lett. 17 (2005) (staff.ee.ncu.edu.tw/jwshi); L. A. Coldren, S. W. Corzine and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed., Wiley, 2012, Sec. 8.3.1, Eqs. 8.20-8.22, Fig. 8.23 and Example 8.8.

Ques: Is the transit-time limit $0.44v/d$ or $3.5v/(2\pi d)$?

The PIN Photodiode page uses $0.44v/W$, whereas the Ge plasmonic-detector paper uses $f_t\approx 3.5v/(2\pi d_{abs})=0.56v/d_{abs}$, and the two differ by 27%. A numerical Fourier transform gives 0.443 for carriers released at one plane that cross the whole distance, and 0.553 for a generation that is uniform over the distance, which is the case of a thin absorber illuminated along the guide (computed). With the 400 nm InP absorber and $v=5.3\times 10^{6}cm/s$, $0.56v/d=74$ GHz, the value that the source gave, whereas 0.44 would give 58 GHz. The factor 0.56 is therefore the more likely for waveguide absorbers, and 0.44 for a charge sheet such as the collector of a UTC-PD. Reference: High bandwidth waveguide-integrated plasmonic germanium photodetector, arXiv:1906.04676; Heterogeneously Integrated InP-Based Evanescently Coupled p-i-n Photodiode on SOI (siliconphotonics.ece.ucsb.edu, C975.pdf).

Ques: Why does the saturation current of a waveguide photodiode depend on how the absorption is distributed?

For one exponentially decaying mode the absorbed power per length is highest at the front end, where the space charge first lowers the field, so the saturation current is that of a uniform absorption divided by ${\rho }_{pk}=\ln [1/(1-\eta )]/\eta$, which is 2.56 for 90% and 4.65 for 99% (computed). A strongly confined mode places the peak close to the front end: the conventional design had its absorption peak about 10 μm from the front, and the failed device was overheated over 12 μm. A lateral directional coupler suppressed the mode, gave a simulated peak-to-average ratio of 1.5 and more than 60 mA at $-4$ V, about 50% more than the conventional guide. The formula predicts 1.7 times (computed), so it gives the right order. Reference: Absorption density control in waveguide photodiode - analysis, design, and demonstration, Frontiers of Optoelectronics (journal.hep.com.cn).

Ques: Why must the electrical and optical velocities of a traveling-wave detector be matched, and how long may a lumped detector be?

In a long absorber the current is generated along the guide at the speed of the light, and the contributions add in phase on the electrode only if the electrical wave travels at the same speed; otherwise they cancel at high frequency. In the distributed detector the coplanar line alone was 31.8% faster than the light in the guide, and the periodic loading with the photodiodes slowed it down to the match. A discrete detector avoids the problem if it is shorter than a quarter of the microwave wavelength, ${\ell }_{max}=c/(4f_{max}\sqrt{{\epsilon }_{eff}})$, which is 1.45 mm at 20 GHz for ${\epsilon }_{eff}=6.7$ (assumed), consistent with the under 1 mm stated in the patent. The loaded line has its own cutoff, 246 GHz for a period of 150 μm (computed). Reference: US Patent 6,418,248, Traveling-wave photodetector (Google Patents); velocity-matched distributed balanced photodetector paper, IEEE Trans. Microwave Theory Tech. (file MTT-99, ece.ucdavis.edu/~saif); L. A. Coldren, S. W. Corzine and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed., Wiley, 2012, Sec. 8.3.1, Eqs. 8.20-8.22, Fig. 8.23 and Example 8.8.

Ques: Which gain of the bandwidth of the vertical Ge-on-Si diode is right: about 20 GHz or about 26 GHz?

The simulation of vertical and lateral Ge-on-Si diodes gives 42.98 GHz for the p-type Ge absorber (V-PIN 1) and 69.57 GHz for the n-type Ge absorber (V-PIN 2), and the abstract calls the gain approximately 20 GHz, whereas the results section says about 26 GHz. The difference of the two reported values is 26.6 GHz (computed), so the figure of 26 GHz is the more likely, and the abstract is probably in error. The source does not explain the cause, and the full-width values quoted with them (19.44 ps and 39.20 ps) are not consistent with the order of the bandwidths, a value 39.20 ps being also repeated for a lateral device. The values should therefore be taken as a simulation result only. Reference: Frontiers in Physics (2023), doi 10.3389/fphy.2023.1150684 (vertical and lateral Ge-on-Si PIN simulation).

Ques: How can bond wires double the bandwidth of a vertical Ge-on-Si detector?

The article attributes the rise from about 30 GHz to about 60 GHz (calculated, with a response measured past 60 GHz) to two bonded gold wires of about 450 μm in total that act as an inductor, about 1 nH/mm, between the discrete ground electrodes. An inductor in series with the junction capacitance forms a resonant circuit, $f_{res}=1/(2\pi \sqrt{LC})$, which peaks the response above the RC corner; for 0.45 nH and an assumed 20 fF it is 53 GHz (computed), the capacitance not being given. The 30 GHz is the article’s typical value for vertical Ge detectors, not a measurement of the same device. The responsivity fell from 1 A/W to 0.85 A/W and the dark current was 64 nA and 61 nA at $-0.5$ V, the cause of the drop not being stated. Reference: Laser Focus World, Silicon Photonics: Standard wire-bonding technique doubles waveguide Ge detector bandwidth; Wafer-scale High-bandwidth Germanium on Silicon Photodetectors for Communications Applications, University of Washington thesis (abstract).

Related pages in this vault: PIN Photodiode, Uni-Traveling-Carrier Photodiode (UTC-PD), PN Photodiode, Metal-Semiconductor-Metal Photodetector (MSM), Resonant-Cavity-Enhanced Photodetector (RCE), Photodetectors (PIN and Avalanche Photodiodes), Lithium Niobate Mach-Zehnder Modulator, Germanium and Silicon-Germanium Electroabsorption Modulator, Indium Phosphide Quantum-Well Electroabsorption Modulator.