10.1 Photodetectors (p-i-n, avalanche)

Photodetector fundamentals

Photodetectors convert light into electrical signals through photon absorption and electron-hole pair generation. They're central to optical communication, imaging, and sensing technologies. The core challenge in photodetector design is balancing sensitivity, speed, and noise for a given application.

This section covers two major types: p-i-n and avalanche photodetectors (APDs). Both rely on the same absorption physics, but they differ in how they handle the resulting carriers and what trade-offs they offer.

Photon absorption and electron-hole pair generation

When a photon with energy greater than the semiconductor's bandgap is absorbed, it excites an electron from the valence band to the conduction band, creating an electron-hole pair. An applied electric field then separates and collects these carriers before they recombine.

• The photon energy must satisfy $E_{photon} = \frac{hc}{\lambda} \geq E_g$, where $E_g$ is the bandgap energy
• The absorption coefficient ($\alpha$) of the material determines how deeply photons penetrate before being absorbed. Materials like germanium have higher $\alpha$ at telecom wavelengths than silicon does, which is why material choice depends on the target wavelength.
• Light intensity decays exponentially with depth according to Beer-Lambert: $I(x) = I_0 e^{-\alpha x}$

Quantum efficiency and responsivity

Quantum efficiency ($\eta$) is the fraction of incident photons that successfully produce collected electron-hole pairs. It depends on how much light is absorbed (set by $\alpha$ and the active layer thickness) and how efficiently the generated carriers are collected before recombining.

Responsivity ($R$) tells you how much photocurrent you get per watt of incident optical power, measured in A/W. It connects directly to quantum efficiency through:

$R = \frac{\eta q \lambda}{hc}$

where $q$ is the electron charge, $\lambda$ is the wavelength, $h$ is Planck's constant, and $c$ is the speed of light. Notice that responsivity increases with wavelength for a fixed $\eta$, since longer-wavelength photons carry less energy, so each watt of light contains more photons.

Response time and bandwidth

The response time is how quickly the photocurrent reaches its final value (typically defined as the 10%-to-90% rise time) after the light switches on or off. Three factors limit it:

1. Carrier transit time: the time for photogenerated carriers to drift across the depletion region. A wider depletion region means longer transit time.
2. RC time constant: the product of the junction capacitance and the load resistance. A wider depletion region reduces capacitance, so there's a direct trade-off with transit time.
3. Carrier diffusion: carriers generated outside the depletion region diffuse slowly toward it, adding a slow tail to the response.

Bandwidth is inversely related to response time. A detector with a 10 ps response time can handle modulation frequencies up to roughly tens of GHz.

p-i-n photodetectors

The p-i-n photodetector is the workhorse of fiber-optic communications and high-speed detection. Its structure is straightforward: a p-type region, a wide intrinsic (undoped or lightly doped) region, and an n-type region.

p-i-n structure and operation

The intrinsic (i) region is where most photon absorption and carrier generation happen. Here's why the structure works well:

1. Under reverse bias, the electric field extends across the entire intrinsic region because it's undoped (no free carriers to screen the field).
2. Photons absorbed in the i-region generate electron-hole pairs that are immediately swept apart by this field: electrons drift toward the n-side, holes toward the p-side.
3. The i-region can be made thick enough to absorb most incoming photons, boosting responsivity.

The key design trade-off is the thickness of the i-region. A thicker i-region absorbs more light (higher responsivity) but increases carrier transit time (lower bandwidth). A thinner i-region is faster but absorbs less light and has higher capacitance.

Depletion region and electric field

In a p-i-n structure, the depletion region essentially spans the entire intrinsic layer under reverse bias. This is a major advantage over a simple p-n junction, where the depletion width is narrow and depends heavily on doping.

• The electric field across the i-region is nearly uniform, which ensures consistent carrier drift velocity.
• Increasing the reverse bias strengthens the field, speeding up carrier collection. At high enough fields, carriers reach their saturation velocity, and further increases in bias don't improve transit time.

Photocurrent generation and collection

The photocurrent is directly proportional to the incident optical power (for a given responsivity):

$I_p = R \cdot P_{opt}$

Collection efficiency depends on:

• Absorption coefficient at the operating wavelength: determines what fraction of photons are absorbed in the i-region
• Carrier lifetime and mobility: carriers must be collected before they recombine
• Surface recombination: carriers generated near the device surface can recombine before reaching the contacts

Reverse bias and dark current

p-i-n detectors operate under reverse bias for two reasons: it creates the strong drift field needed for fast carrier collection, and it reduces junction capacitance (improving bandwidth).

The downside is dark current, the current that flows even with no light present. Dark current comes from:

• Thermally generated carriers within the depletion region
• Surface leakage currents
• Tunneling currents (significant at very high bias)

Dark current sets a floor on the minimum detectable signal because it adds shot noise. It increases with temperature and with reverse bias voltage.

p-i-n photodetector characteristics and performance

Typical performance numbers for p-i-n photodetectors:

Avalanche photodetectors (APDs)

APDs add internal gain to the photodetection process through avalanche multiplication. This makes them significantly more sensitive than p-i-n detectors, which is why they're preferred for long-distance optical communications, low-light imaging, and single-photon detection. The trade-off is increased noise and more complex biasing requirements.

Avalanche multiplication process

The multiplication process works in stages:

1. A photon is absorbed in the absorption region, generating a primary electron-hole pair (just like in a p-i-n detector).
2. The carrier (usually the electron) drifts into a high-field multiplication region.
3. In this region, the carrier accelerates until it gains enough kinetic energy to knock a bound electron out of the lattice through impact ionization, creating a secondary electron-hole pair.
4. The secondary carriers also accelerate and can trigger further impact ionization events.
5. This chain reaction produces an "avalanche" of carriers, amplifying the original signal.

The multiplication factor (or gain, $M$) is the ratio of total collected carriers to primary photogenerated carriers. Typical values range from $M = 10$ to $M = 100$ or more, depending on the bias voltage.

Impact ionization and gain

Impact ionization rates differ for electrons and holes. The ionization coefficients $\alpha$ (for electrons) and $\beta$ (for holes) represent the number of ionizing collisions per unit distance.

• If only one carrier type ionizes efficiently (say $\alpha \gg \beta$), the avalanche is more deterministic and produces less noise.
• The gain depends on the ionization coefficients and the width of the multiplication region ($w$). A simplified expression for the case where electrons initiate multiplication:

$M = \frac{1}{1 - \int_0^w \alpha \, dx}$

As the integral approaches 1, the gain diverges toward infinity, which corresponds to avalanche breakdown. APDs are biased just below this breakdown voltage to achieve high but stable gain.

Excess noise factor

Avalanche multiplication is inherently random: each primary carrier may produce a different number of secondary carriers. This randomness introduces excess noise beyond the shot noise you'd expect from the multiplied photocurrent.

The excess noise factor $F$ quantifies this additional noise:

$F = k_{eff} M + (1 - k_{eff})\left(2 - \frac{1}{M}\right)$

where $k_{eff}$ is the effective ratio of ionization coefficients ($k = \beta / \alpha$ if electrons initiate).

• When $k_{eff} \rightarrow 0$ (only one carrier type ionizes), $F \approx 2$, which is the best case.
• When $k_{eff} \rightarrow 1$ (both carriers ionize equally), $F \approx M$, which is very noisy.

This is why silicon APDs perform well at shorter wavelengths: silicon has a large asymmetry between $\alpha$ and $\beta$ ($k \approx 0.02$), giving low excess noise. III-V materials like InGaAs have $k$ closer to 0.5, producing more noise at the same gain.

Gain-bandwidth product

There's a fundamental trade-off between gain and speed in APDs. Higher gain requires more avalanche buildup time, which slows the detector. The gain-bandwidth product (GBP) captures this:

$\text{GBP} = M \times \text{Bandwidth} \approx \text{constant}$

If you double the gain, you roughly halve the bandwidth. Typical GBP values:

• Si APDs: 100-300 GHz
• InGaAs/InP APDs: 100-200 GHz (some advanced designs exceed 300 GHz)
• III-V APDs with optimized structures: can exceed 1 THz

APD structure and operation

An APD has two functionally distinct regions:

• Absorption region: where photons are absorbed and primary carriers are generated (similar to a p-i-n structure)
• Multiplication region: a thin, high-field zone where impact ionization produces gain

APDs require a high reverse bias voltage (often 20-200 V depending on the material and structure) to sustain the strong electric field needed for impact ionization. The bias must be carefully controlled because gain is very sensitive to voltage near breakdown, and temperature changes shift the breakdown voltage.

Reach-through and separate absorption and multiplication (SAM) APDs

Reach-through APDs have a single continuous depletion region spanning both the absorption and multiplication zones. Photogenerated carriers drift through the absorption region and directly enter the multiplication region. This design is simpler but offers less control over the electric field profile.

SAM (Separate Absorption and Multiplication) APDs decouple the two regions, allowing each to be independently optimized:

• The absorption region uses a material with high absorption at the target wavelength (e.g., InGaAs for 1550 nm)
• The multiplication region uses a material with favorable ionization coefficient ratio (e.g., InP, where $k$ is lower)

SAM structures achieve higher gain-bandwidth products and lower excess noise because the field in each region can be tuned separately. Most modern telecom APDs use SAM or SACM (Separate Absorption, Charge, and Multiplication) designs, where an additional charge layer controls the field transition between regions.

Photodetector materials and technologies

The choice of semiconductor material determines the wavelength range, speed, and noise characteristics of a photodetector. Each material has a cutoff wavelength set by its bandgap: $\lambda_{cutoff} = \frac{hc}{E_g}$. Photons with wavelengths longer than this cutoff won't be absorbed.

Silicon (Si) photodetectors

Silicon covers visible and near-infrared wavelengths from about 400 nm to 1100 nm (bandgap of 1.12 eV at room temperature).

• Mature CMOS fabrication technology makes Si detectors inexpensive and easy to integrate with electronics
• Excellent ionization coefficient ratio ($k \approx 0.02$) makes Si APDs among the lowest-noise avalanche detectors available
• Used in short-reach fiber optics (850 nm multimode fiber), digital cameras, and LiDAR
• Si is an indirect bandgap semiconductor, so its absorption coefficient is relatively low compared to III-V materials, requiring thicker active regions

Germanium (Ge) photodetectors

Germanium extends detection into the near-infrared and short-wave infrared, covering roughly 800-1600 nm (bandgap of 0.67 eV).

• Covers the critical telecom windows at 1310 nm and 1550 nm
• Can be grown on silicon substrates using heterogeneous integration, enabling Ge-on-Si photodetectors for silicon photonics platforms
• Higher dark current than III-V detectors at the same wavelengths due to the smaller bandgap
• Ge APDs have less favorable ionization coefficient ratios than Si APDs

III-V compound semiconductor photodetectors

III-V semiconductors (GaAs, InGaAs, InP, and their alloys) offer the most flexibility for photodetector design.

• InGaAs ($In_{0.53}Ga_{0.47}As$ lattice-matched to InP) is the standard material for 1310 nm and 1550 nm telecom detection, with high absorption coefficients and responsivities near 1.0 A/W
• GaAs detectors operate at shorter wavelengths (~870 nm) and are used with VCSELs in data center links
• Bandgap engineering through compositional tuning allows the cutoff wavelength to be adjusted continuously
• III-V materials are direct bandgap semiconductors, giving them much higher absorption coefficients than Si or Ge at their respective operating wavelengths

Heterojunction and quantum well photodetectors

These advanced structures use layered combinations of different semiconductors to enhance performance beyond what a single material can achieve.

Heterojunction photodetectors place the absorption and collection functions in different material layers. For example, an InGaAs absorption layer paired with an InP multiplication layer in a SAM APD exploits the strengths of each material.

Quantum well infrared photodetectors (QWIPs) use thin semiconductor layers (typically a few nanometers) sandwiched between wider-bandgap barriers. Carriers are confined in discrete energy levels within the wells, and intersubband transitions allow detection of mid-infrared and far-infrared wavelengths that would otherwise require exotic narrow-gap materials. QWIPs offer low dark current due to carrier confinement but typically have lower quantum efficiency than bulk detectors.

Photodetector applications

Optical fiber communications

Photodetectors sit at the receiver end of every fiber-optic link, converting optical signals back to electrical form.

• Short-reach links (data centers, local networks at 850 nm): Si or GaAs p-i-n photodetectors, chosen for speed and low cost
• Long-haul links (telecom at 1310/1550 nm): InGaAs p-i-n detectors for moderate distances; InGaAs/InP APDs for longer spans where the extra sensitivity from gain is needed to detect weak signals
• Modern coherent optical systems use balanced photodetectors (pairs of p-i-n detectors) to reject common-mode noise

Imaging and sensing

• Digital cameras: Si CMOS image sensors and CCDs detect visible light
• Night vision and thermal imaging: InGaAs detectors (SWIR), HgCdTe detectors (MWIR/LWIR), and microbolometers cover different infrared bands
• Spectroscopic sensing: photodetectors tuned to specific absorption wavelengths identify chemical and biological substances in medical diagnostics, environmental monitoring, and industrial process control

High-speed and high-sensitivity applications

• High-speed data links: Traveling-wave photodetectors and uni-traveling-carrier (UTC) photodiodes achieve bandwidths exceeding 100 GHz for next-generation optical interconnects
• Single-photon detection: Single-photon avalanche diodes (SPADs) operate above breakdown voltage in Geiger mode, producing a large output pulse from a single absorbed photon. Used in quantum key distribution, fluorescence lifetime imaging, and LiDAR.
• LiDAR: Both APDs (for analog detection) and SPADs (for photon counting) measure time-of-flight of laser pulses for 3D mapping

Photodetector noise and signal-to-noise ratio (SNR)

Noise sets the ultimate limit on how weak a signal a photodetector can detect. Every photodetector system has multiple noise sources that add together, and understanding them is essential for system design.

Shot noise and thermal noise

Shot noise arises because photocurrent consists of discrete charge carriers arriving at random times. The mean-square shot noise current is:

$i_{shot}^2 = 2q(I_p + I_d)B$

where $I_p$ is the signal photocurrent, $I_d$ is the dark current, and $B$ is the electrical bandwidth. Shot noise is fundamental and cannot be eliminated.

Thermal noise (Johnson-Nyquist noise) comes from random thermal motion of electrons in the load resistor and circuit elements:

$i_{thermal}^2 = \frac{4k_BT}{R_L}B$

where $k_B$ is Boltzmann's constant, $T$ is the absolute temperature, and $R_L$ is the load resistance. Thermal noise can be reduced by increasing $R_L$ (at the cost of bandwidth) or cooling the detector.

Amplifier noise and total noise

The transimpedance amplifier (TIA) that follows the photodetector adds its own noise, characterized by input-referred noise current and voltage spectral densities.

The total mean-square noise current in the system is:

$i_n^2 = 2q(I_p + I_d)B + \frac{4k_BT}{R_L}B + i_{n,amp}^2 B$

For p-i-n detectors, thermal and amplifier noise often dominate because there's no internal gain. For APDs, the multiplied shot noise (scaled by $M^2 F$) can become the dominant term, which is why there's an optimal gain that maximizes SNR: too little gain and thermal/amplifier noise dominates; too much gain and excess noise from multiplication takes over.

SNR and detectivity

The signal-to-noise ratio is:

$SNR = \frac{(M \cdot I_p)^2}{i_n^2}$

For a p-i-n detector ($M = 1$), this simplifies to $SNR = I_p^2 / i_n^2$.

Detectivity ($D^*$) normalizes sensitivity to detector area and bandwidth, allowing fair comparison between detectors of different sizes:

$D^* = \frac{\sqrt{A \cdot B}}{NEP}$

where $A$ is the detector area, $B$ is the bandwidth, and $NEP$ is the noise equivalent power. Units are cm·$\sqrt{Hz}$/W (sometimes called "Jones"). Higher $D^*$ means better sensitivity.

Noise equivalent power (NEP)

NEP is the optical power that produces an SNR of exactly 1 in a 1 Hz bandwidth. It represents the minimum detectable signal:

$NEP = \frac{i_n}{R}$

where $i_n$ is the total noise current spectral density (in A/$\sqrt{Hz}$) and $R$ is the responsivity. Lower NEP means the detector can sense weaker signals. Typical NEP values range from $10^{-12}$ to $10^{-15}$ W/$\sqrt{Hz}$ depending on the detector type and operating conditions.