Photoelectric sensors: photodiodes, phototransistors, LDRs

Why it matters

Light sensors count bottles on a conveyor, read optical encoders, detect smoke, measure blood oxygen, receive fibre-optic data and switch street lights at dusk. Choosing between a photodiode, a phototransistor and an LDR is a trade-off between speed, sensitivity, linearity and cost, and the photodiode's responsivity calculation is a standard numerical.

Key ideas

Photoelectric effects — light delivers energy in photons of E = h·f = h·c/λ. A photon can do useful work in a detector only if its energy exceeds a threshold: the work function (photoemission) or the semiconductor band gap Eg. This sets a cut-off wavelength λc = h·c/Eg (≈ 1.24/Eg µm with Eg in eV): about 1.1 µm for silicon, 1.8 µm for germanium.

Three families:

• Photoemissive: photons eject electrons from a cathode into vacuum (phototube, photomultiplier tube). PMTs have enormous gain and are used for very weak light.
• Photoconductive: photons create carriers that raise the conductivity of a semiconductor — the LDR (light-dependent resistor, usually CdS or CdSe). Its resistance falls from megaohms in darkness to hundreds of ohms in bright light, approximately as R = R1·(L/L1)^(−γ) with γ typically 0.7–0.9 (from the datasheet). CdS has a spectral response close to the human eye (peak around 550 nm). Response is slow (tens to hundreds of ms) because of long carrier lifetimes and trapping, with a "memory" of previous light levels.
• Photovoltaic: photons create electron–hole pairs in a p–n junction's depletion region, and the built-in field separates them — the photodiode and solar cell.

Photodiode.

• Photoconductive mode (reverse biased): the photocurrent is linear in optical power over many decades, the depletion layer is wide and its capacitance small, so it is fast (PIN diodes reach GHz). Dark current (leakage) adds noise and an offset that doubles roughly every 10 °C.
• Photovoltaic mode (zero bias): no dark current, lowest noise, used for precision light meters; the open-circuit voltage is logarithmic in light, so it is read with a short-circuit current (transimpedance) amplifier.
• Responsivity R = I_p/P (A/W) = η·q·λ/(h·c), where η is quantum efficiency. Silicon peaks at about 0.5–0.6 A/W in the near IR.
• Transimpedance amplifier: the photodiode drives an op-amp inverting input with feedback resistor Rf, so Vo = −I_p·Rf, holding the diode at zero (or fixed) bias.
• Avalanche photodiodes add internal multiplication at high reverse bias.

Phototransistor. A bipolar transistor whose collector–base junction acts as a photodiode. The photocurrent becomes base current and is multiplied by the current gain: I_C ≈ (β + 1)·I_p. Hence 100–1000 times the photodiode current, but the large collector–base capacitance is multiplied too (Miller effect), so response is slower (µs) and less linear, and gain varies with temperature and from device to device. Photodarlingtons give even more gain and are slower still.

Optical switching sensors pair an LED (often IR, modulated to reject ambient light) with a detector in through-beam, retro-reflective or diffuse configurations; optocouplers use the same pair for galvanic isolation.

Formulas

E = h·f = h·c/λ E = photon energy (J); h = 6.626 × 10⁻³⁴ J·s; c = 2.998 × 10⁸ m/s; λ = wavelength (m).

λc = h·c/Eg ≈ 1.24/Eg (λc in µm, Eg in eV)

I_p = η·q·P/(h·f) = R·P, R = η·q·λ/(h·c) I_p = photocurrent (A); η = quantum efficiency; q = 1.602 × 10⁻¹⁹ C; P = optical power (W); R = responsivity (A/W).

Vo = −I_p·Rf (transimpedance amplifier) Rf = feedback resistance (Ω).

I_C ≈ (β + 1)·I_p (phototransistor) β = current gain.

R = R1·(L/L1)^(−γ) (LDR) L = illuminance (lux); R1 = resistance at L1; γ = datasheet exponent.

Worked examples

Example 1 — photodiode responsivity and TIA. A silicon photodiode with η = 0.8 receives 100 µW at 850 nm. (a) Find responsivity and photocurrent. (b) It feeds a transimpedance amplifier with Rf = 10 kΩ. Find Vo. (c) Can it detect 1.3 µm light?

1. R = η·q·λ/(h·c) = 0.8 × 1.602 × 10⁻¹⁹ × 850 × 10⁻⁹ / (6.626 × 10⁻³⁴ × 2.998 × 10⁸) = 0.548 A/W.
2. I_p = R·P = 0.548 × 100 × 10⁻⁶ = 54.8 µA.
3. Vo = −I_p·Rf = −54.8 × 10⁻⁶ × 10⁴ = −0.548 V.
4. Silicon: λc = 1.24/1.12 = 1.11 µm < 1.3 µm, so no — use Ge or InGaAs.

Answer: R = 0.548 A/W, I_p = 54.8 µA, Vo = −0.548 V; 1.3 µm is beyond silicon's cut-off.

Example 2 — LDR divider and phototransistor (GATE level). (a) An LDR is 10 kΩ at 100 lux with γ = 0.7. It is the upper arm of a divider with a 10 kΩ lower resistor on 5 V; the output is taken across the 10 kΩ. Find the LDR resistance and output at 500 lux. (b) A phototransistor's collector–base junction produces 2 µA of photocurrent and β = 100. Find I_C.

1. R = 10 × (500/100)^(−0.7) = 10 × 5^(−0.7) = 10 × 0.3241 = 3.24 kΩ.
2. Vo = 5 × 10/(10 + 3.24) = 3.78 V (it was 2.50 V at 100 lux).
3. I_C = (β + 1)·I_p = 101 × 2 µA = 202 µA ≈ 0.2 mA.

Answer: (a) 3.24 kΩ, 3.78 V; (b) about 0.20 mA.

Common mistakes

• Forgetting the electron charge: I = η·q·P/(hf), not η·P/(hf) (that is photons per second).
• Using frequency and wavelength inconsistently; convert nm to m.
• Assuming an LDR is linear or exactly inversely proportional to light; use the γ law from the datasheet.
• Claiming phototransistors are faster than photodiodes — gain costs bandwidth.
• Forward-biasing a photodiode; it must be reverse or zero biased.
• Ignoring the cut-off wavelength when choosing a detector material.

For GATE IN

NAT questions: photon energy, photocurrent from power and quantum efficiency, responsivity, cut-off wavelength from band gap, transimpedance output, phototransistor collector current, LDR divider outputs. MCQs: compare photoemissive/photoconductive/photovoltaic devices, modes of photodiode operation, speed and linearity rankings. Keep hc/q = 1.24 µm·eV in mind.

Quick check

1. What is the cut-off wavelength of a material with Eg = 1.42 eV?
2. Responsivity 0.5 A/W, power 20 µW, Rf = 100 kΩ. Output magnitude?
3. Rank photodiode, phototransistor and LDR by speed.
4. Which photodiode mode gives no dark current? Answers: 1. 1.24/1.42 = 0.873 µm; 2. 10 µA × 100 kΩ = 1 V; 3. photodiode fastest, then phototransistor, then LDR; 4. photovoltaic (zero-bias) mode.