Avalanche Photodiodes and Photomultipliers

The most visible modern home of avalanche gain is the LIDAR that lets a car see in three dimensions. A pulsed laser fires into the scene and only a whisper returns from a dark, distant target; a fast APD or Geiger-mode SPAD array recovers it and time-stamps the return to picoseconds, converting flight time into range. Automotive and consumer LIDAR increasingly integrate whole arrays of SPADs in CMOS alongside their timing electronics — the direct-time-of-flight sensors behind smartphone face-unlock and room-scanning apps are exactly this, silicon SPADs firing on single reflected photons. At the other end sit big physics and quantum optics. Photomultipliers still tile neutrino observatories like Super-Kamiokande, tens of thousands of large-area tubes watching for the faint Cherenkov flash of a rare interaction, because nothing beats a PMT for low-noise gain over a square-metre photocathode. In quantum-information labs, cooled gated InGaAs SPADs clock the single photons of fibre quantum-key-distribution at $1550$ nm, and silicon SPADs count photons in fluorescence-lifetime microscopy. In each case the figures of merit shift from analog gain and $F$ to photon-detection efficiency, dark-count rate, jitter and afterpulsing — but the founding idea is unchanged: multiply the carrier before the amplifier can lose it.

Picture the PMT dynode chain as a staircase where every step does two jobs: it accelerates an electron downhill and shatters it into a small cloud when it lands. One photoelectron knocks $\delta$ secondaries out of the first dynode; each is accelerated to the next step and knocks out $\delta$ more, so after $N$ steps the population is $\delta^{N}$. The supply voltage pays for the kinetic energy; the material supplies the multiplication. Nothing is amplified by an external circuit — the macroscopic charge pulse is assembled in vacuum before it reaches the anode, which is exactly why the downstream electronics can be noisy and it will not matter.

An APD trades vacuum for silicon and dynodes for a chain-reaction of collisions. A photon absorbed in the junction makes one electron–hole pair; the reverse bias sweeps the carriers into a thin region where the field is fierce — a hundred kilovolts per centimetre or more. There a carrier accelerates between scattering events until it carries more than the bandgap, then slams into the lattice and promotes a bound electron across the gap: impact ionization, one pair becomes two. The offspring ionize in turn, and multiplication grows geometrically. Below breakdown the avalanche is self-limiting and $M$ is finite; above breakdown a single carrier can ignite a self-sustaining discharge that must be actively quenched — the Geiger mode of a single-photon avalanche diode. The catch is that neither secondary emission nor impact ionization is a clockwork multiplier. The secondaries per dynode, and the ionizations per carrier, are themselves random, so two photons giving the same average gain deliver very different charge packets. This multiplicative jitter is the excess-noise factor $F$, and it grows with $M$: at low gain thermal noise dominates and raising $M$ helps, but at high gain the amplified shot noise $\times F(M)$ takes over and raising $M$ hurts. The sensitivity curve therefore has a minimum — why every APD receiver is designed around a specific operating gain. The second figure is what a receiver engineer reaches for: how fast $F$ climbs, and how a small ionization ratio $k$ tames it.

Start with the PMT, where gain is pure multiplicative bookkeeping. Each of $N$ identical dynodes multiplies the electron population by the secondary-emission ratio $\delta$, so the stages compound into a single power law — a per-stage gain of a few becomes a factory of a million by the anode.

where: $G$ — overall current gain of the tube; $\delta$ — secondary electrons per incident electron at one dynode; $N$ — number of dynode stages. Reading it physically: gain is exponential in stage count, so $\delta=4$, $N=10$ gives $G=4^{10}\approx1.05\times10^{6}$. The per-stage ratio itself scales as $\delta\approx aV_d^{\gamma}$ with $\gamma\approx0.75$, so the total gain runs as $G\propto V_{HV}^{\gamma N}$ with $\gamma N\approx7$–$8$ — a 1 % supply drift shifts the gain by 7–8 %, which is why PMT high-voltage rails must be regulated to a few parts in $10^{4}$. Before the APD can multiply anything it must convert photons to primary carriers, and that conversion is the responsivity that governs every photodiode. It fixes the unmultiplied photocurrent the avalanche then amplifies.

where: $i_{ph}$ — primary photocurrent before gain; $\mathcal{R}$ — responsivity in A/W; $\eta$ — quantum efficiency; $P_{opt}$ — incident optical power; $e,\lambda,h,c$ — electron charge, wavelength, Planck constant, speed of light. Reading it physically: the quantum-limited responsivity is $\lambda(\mu\text{m})/1.24$ A/W, so at $1550$ nm with $\eta=0.8$, $\mathcal{R}\approx1.0$ A/W — one amp per watt, before any gain. The avalanche multiplies that current by a mean factor $M$, set by how close the integrated ionization comes to unity across the high-field width. When the ionizing carriers just replace themselves, $M$ diverges — that is breakdown.

where: $I$ — multiplied output current; $M$ — mean multiplication; $\alpha(x)$ — ionizations per unit length; $w$ — width of the multiplication region. Reading it physically: as $\int\alpha\,\mathrm{d}x\to1$ the denominator collapses and $M\to\infty$ — the mathematical definition of avalanche breakdown. For engineering use that divergence is captured by the empirical Miller relation, which folds the field profile into one breakdown voltage and an exponent — the curve a data sheet plots as gain versus reverse bias.

where: $V$ — reverse bias; $V_{BR}$ — breakdown voltage; $n\approx2$–$6$ — a material/structure exponent. Reading it physically: $V\ll V_{BR}$ gives $M\to1$ and $V\to V_{BR}$ drives $M\to\infty$, so linear-mode operation lives in a narrow, temperature-sensitive window just below breakdown. Because multiplication is random, shot noise is amplified not by $M^{2}$ but by $M^{2}F(M)$, and McIntyre's local-field theory gives the excess factor in closed form for electron injection, with a single parameter, the ionization ratio $k$.

where: $F(M)$ — excess-noise factor ($F=1$ would be noiseless); $k=\alpha_h/\alpha_e\le1$ — ratio of hole to electron ionization coefficients. Reading it physically: with $k\to0$ (only one carrier ionizes) $F\to2-1/M$ and saturates near 2, the quiet regime of silicon; with $k=1$ (both ionize equally) $F=M$, the noisy regime of germanium and, largely, InGaAs. Assembling signal and noise gives the receiver equation: the multiplied shot noise sits on an unmultiplied thermal floor, the two terms scaling differently with $M$ — and the optimal gain is extracted from it.

where: $i_d$ — primary (unmultiplied) dark current; $B$ — electrical bandwidth; $k_B$ — Boltzmann constant; $T$ — temperature; $R_L$ — load resistance. Reading it physically: signal power grows as $M^{2}$ and the shot term as $M^{2}F(M)$, so once $F$ climbs the noise outruns the signal — a best gain exists, beyond which sensitivity falls. Finally, speed and gain are not independent: a larger avalanche needs more ionization generations to build, so bandwidth falls as gain rises and the product is fixed by the device.

where: $f_{3\text{dB}}$ — 3-dB bandwidth; GBP — gain-bandwidth product, set by carrier transit and avalanche buildup time. Reading it physically: a telecom InGaAs APD with GBP $\approx200$ GHz run at $M=15$ has $f_{3\text{dB}}\approx13$ GHz, ample for 10 Gb/s but the reason you cannot simply crank the gain on a fast link.

1 · An APD in a fibre receiver (the concrete one)

A long-haul $1550$ nm link delivers $P_{opt}=100$ nW to an InGaAs APD with quantum efficiency $\eta=0.8$, operated at mean gain $M=10$. Find the responsivity, the primary photocurrent, and the multiplied output current.

Step 1. The quantum-limited responsivity at $1550$ nm is $\lambda(\mu\text{m})/1.24=1.55/1.24=1.25$ A/W; multiply by $\eta$: $\mathcal{R}=\eta\,e\lambda/hc=0.8\times1.25=\mathbf{1.0\ A/W}$ — the charge out per watt in, before any gain.

Step 2. Primary photocurrent is responsivity times power: $i_{ph}=\mathcal{R}P_{opt}=(1.0)(100\times10^{-9})=\mathbf{100\ nA}$ — the current a plain PIN diode would deliver, easily lost in preamplifier noise.

Step 3. The avalanche multiplies it by the mean gain: $I=M\,i_{ph}=10\times100\ \text{nA}=\mathbf{1.0\ \mu A}$. The signal has grown tenfold inside the detector, clearing the amplifier's noise floor — the whole point of an APD over a PIN diode at the end of a lossy link. R = 1.0 A/W, i_ph = 100 nA, I = 1.0 µA.

2 · Why silicon out-whispers InGaAs

Compare the excess-noise factor at $M=10$ for a silicon APD ($k=0.02$) and an InGaAs APD ($k=0.45$). By how much noisier is the InGaAs device?

Step 1. Silicon: $F=kM+(1-k)(2-1/M)=0.02\times10+0.98\times1.9=0.2+1.862=\mathbf{2.06}$. Even at gain 10 the excess factor barely clears its saturated value of 2 — silicon multiplies almost as cleanly as the single-carrier limit allows.

Step 2. InGaAs: $F=0.45\times10+0.55\times1.9=4.5+1.045=\mathbf{5.5}$. Because holes ionize nearly as readily as electrons, two-way feedback inflates the variance to more than five times the ideal.

Step 3. Noise power scales with $F$, so noise amplitude scales with $\sqrt{F}$: the InGaAs device is noisier by $\sqrt{5.5/2.06}=\sqrt{2.67}=\mathbf{1.64\times}$. That is why telecom APDs use SAM structures — absorb the $1550$ nm light in InGaAs but confine the avalanche to a low-$k$ InP layer. F_Si = 2.06, F_InGaAs = 5.5, a 1.64× noise penalty.

3 · The gain a receiver should actually use

For the receiver of Example 1 — $i_{ph}=100$ nA, negligible dark current, load $R_L=1$ kΩ at $T=300$ K, and an InGaAs noise exponent $x=0.7$ — find the optimal gain.

Step 1. The optimum balances multiplied shot noise against thermal noise: $M_{\text{opt}}=\left[4k_BT/(e\,x\,R_L\,i_{ph})\right]^{1/(2+x)}$. Assemble the thermal numerator: $4k_BT=4\times1.381\times10^{-23}\times300=1.657\times10^{-20}$ J.

Step 2. Assemble the denominator: $e\,x\,R_L\,i_{ph}=(1.602\times10^{-19})(0.7)(10^{3})(10^{-7})=1.121\times10^{-23}$. The ratio is $1.657\times10^{-20}/1.121\times10^{-23}=1478$.

Step 3. Raise to the power $1/(2+x)=1/2.7=0.370$: $M_{\text{opt}}=1478^{0.370}\approx\mathbf{15}$. Running near $M=15$ minimizes the receiver's noise; with GBP $\approx200$ GHz the bandwidth is $f_{3\text{dB}}\approx200/15\approx13$ GHz, comfortably supporting a 10 Gb/s link. Pushing to $M=50$ would raise the noise and shrink the bandwidth to a few GHz — worse on both counts. M_opt ≈ 15, giving f_3dB ≈ 13 GHz.

Internal-gain detection is where vacuum-tube ingenuity and semiconductor physics converge on one problem — how to hear a single photon.

Texts. Saleh & Teich, Fundamentals of Photonics, 3rd ed., Ch. 18–19 (Semiconductor Photon Detectors) and Ch. 24 (photoreceivers) · Hecht, Optics, 5th ed., §13 (Detectors) · Loudon, The Quantum Theory of Light, 3rd ed. (photodetection statistics) · Demtröder, Laser Spectroscopy, Vol. 1, Ch. 4 (Photodetectors) · Foot, Atomic Physics (detection of fluorescence).

Provenance. Equations and numbers are cross-checked against Saleh & Teich (Ch. 18–19, 24), Demtröder Laser Spectroscopy Vol. 1 Ch. 4, and McIntyre's 1966 local-field excess-noise theory. A verifier should recompute: $G=\delta^{N}$ for the PMT example ($4^{10}\approx1.05\times10^{6}$) and $F=\delta/(\delta-1)=4/3$; the McIntyre limits $k=0$ ($F\to2-1/M$) and $k=1$ ($F=M$), and the worked values $F(0.02,10)=2.06$, $F(0.45,10)=5.5$; the responsivity chain $\mathcal{R}=1.0$ A/W, $i_{ph}=100$ nA, $I=1.0$ µA at $1550$ nm; and the optimal-gain estimate $M_{\text{opt}}\approx15$. Ionization ratios, gain ranges and gain-bandwidth-product orders of magnitude are standard textbook figures; no device-specific numbers are asserted beyond those.

Builds on. Fibre Attenuation and Loss Mechanisms · Photoelectric Effect and Work Function · Semiconductor $p$–$n$ Junctions and Reverse Bias · Shot Noise and Poisson Statistics. Leads to. Chromatic and Modal Dispersion in Fibres · Integrated Photonic Waveguides and Circuits · Microring Resonators and Optical Cavities on Chip.

Ask the tutor: Want to see how $M_{\text{opt}}$ and the excess-noise penalty shift if you swap a silicon APD for InGaAs on your own receiver's numbers — load resistance, wavelength and target bit rate?