Question 4 of 6: Silicon Photodiode — Measured Responsivity, Bandwidth, and Noise

Question 4: Silicon Photodiode — Measured Responsivity, Bandwidth, and Noise (equal value)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

The measured responsivity vs. wavelength curve (both 0 V and 30 V bias) is reproduced below from the exam's own figure.

Find. (a) The physical origin of the responsivity curve's shape. (b) Quantum efficiency at 850 nm at 0 V and 30 V bias. (c) Incident intensity giving $I_{photo}=I_d$ at 30 V bias. (d) Detection bandwidth and response time. (e) Quantum- and thermal-noise-limited SNR and the NEP at 10 µW incident, 30 V bias. (f) Operating mode (photovoltaic vs. photoconductive) with I-V curve and load line.

Approach. Read responsivity values directly off the given curve at 850 nm; convert to quantum efficiency via $\eta=Rh\nu/q$. Combine the RC time constant (load + amplifier capacitance) with the transit-time-limited bandwidth to get the overall detection bandwidth, then invert for response time. Compute shot- and thermal-noise-limited SNR and NEP at the stated bandwidth and optical power.

Part (a) — main features of the curve

The responsivity rises roughly linearly from a short-wavelength cutoff near 400–500 nm, peaks around 850–900 nm, then falls sharply above about 1000–1100 nm. The short-wavelength roll-off occurs because high-energy (blue/UV) photons are absorbed within a few tens of nanometres of the surface, in the heavily doped, poorly-collected $p^+$ (or $n^+$) surface layer, so many photogenerated carriers recombine before reaching the depletion region. The long-wavelength cutoff near 1100 nm is set by silicon's indirect bandgap, $E_g=1.11$ eV $\Rightarrow\lambda_g=hc/E_g=1117$ nm: photons longer than this lack the energy to create an electron-hole pair at all, and the absorption coefficient falls steeply as $\lambda\to\lambda_g$ from below (silicon is an indirect-gap semiconductor, so absorption near the edge is weak and requires phonon assistance). The 30 V reverse bias curve sits slightly above the 0 V curve across the whole peak region because the wider depletion width at higher reverse bias collects photogenerated carriers from deeper in the silicon (closer to where longer-wavelength light is actually absorbed) before they can recombine, raising the effective quantum efficiency.

Part (b) — quantum efficiency at 850 nm

1. Photon energy at 850 nm. $$h\nu=\frac{hc}{\lambda}=\frac{1239.8\ \text{eV}\cdot\text{nm}}{850\ \text{nm}}=1.459\ \text{eV}.$$
2. Quantum efficiency, $\eta=R\,(h\nu/q)$. Reading $R(0\text{V})\approx0.54$ A/W and $R(30\text{V})\approx0.57$ A/W off the curve at 850 nm, $$\eta_{0V}=\frac{(0.54)(1.459)}{1}=0.788\ (78.8\%),\qquad \eta_{30V}=\frac{(0.57)(1.459)}{1}=\boxed{0.831\ (83.1\%)}.$$ Both exceed typical commercial photodiode quantum efficiencies because $R\approx0.5$ A/W near the peak already implies $\eta$ close to unity at this wavelength — consistent with a well-designed silicon detector operating near its responsivity maximum.

Part (c) — intensity giving $I_{photo}=I_d$ at 30 V

1. Required incident power. $$P=\frac{I_d}{R_{30V}}=\frac{10\times10^{-9}}{0.57}=1.75\times10^{-8}\ \text{W}=17.5\ \text{nW}.$$
2. Intensity. $$I=\frac{P}{A}=\frac{1.75\times10^{-8}\ \text{W}}{5\times10^{-6}\ \text{m}^2}=\boxed{3.51\times10^{-3}\ \text{W/m}^2}=0.351\ \mu\text{W/cm}^2.$$

Part (d) — bandwidth and response time

1. RC-limited bandwidth. Total capacitance loading the load resistor is $C_T=C_j+C_{amp}=3+7=10$ pF: $$f_{RC}=\frac{1}{2\pi R_LC_T}=\frac{1}{2\pi(50)(10\times10^{-12})}=318\ \text{MHz}.$$
2. Transit-time-limited bandwidth. $$f_{tr}=\frac{0.44}{t_{tr}}=\frac{0.44}{0.5\times10^{-9}}=880\ \text{MHz}.$$
3. Combined 3 dB bandwidth (independent limits combine as reciprocal-squares): $$f_{3dB}=\left(\frac{1}{f_{RC}^2}+\frac{1}{f_{tr}^2}\right)^{-1/2}=\boxed{299\ \text{MHz}}.$$
4. Response time. Using the standard risetime-bandwidth product $t_r\approx0.35/f_{3dB}$, $$t_r=\frac{0.35}{299\times10^6}=\boxed{1.17\ \text{ns}}.$$

Part (e) — noise limits and NEP at 10 µW, 30 V

1. Signal photocurrent. $$I_p=R_{30V}P_{inc}=(0.57)(10\times10^{-6})=5.70\times10^{-6}\ \text{A}.$$
2. Quantum (shot-noise) limit. The fundamental shot-noise-limited SNR, using the detection bandwidth from part (d), $$\mathrm{SNR}_{quantum}=\frac{I_p^2}{2qI_pB_{det}}=\frac{I_p}{2qB_{det}} =\frac{5.70\times10^{-6}}{2(1.602\times10^{-19})(2.99\times10^{8})}=\boxed{5.94\times10^{4}\ (47.7\ \text{dB})}.$$
3. Thermal-noise limit. $$\overline{i_{th}^2}=\frac{4kTB_{det}}{R_L}=\frac{4(1.381\times10^{-23})(300)(2.99\times10^8)}{50}=9.92\times10^{-14}\ \text{A}^2,$$ $$\mathrm{SNR}_{thermal}=\frac{I_p^2}{\overline{i_{th}^2}}=\frac{(5.70\times10^{-6})^2}{9.92\times10^{-14}}=\boxed{327\ (25.1\ \text{dB})}.$$ The thermal limit is far below the quantum limit here, i.e. thermal noise (not shot noise) sets the practical noise floor for this receiver — consistent with Q2(c)'s finding for the LED link.
4. NEP (thermal-noise-referred, at $B_{det}=1$ Hz). $$\mathrm{NEP}=\frac{\sqrt{\overline{i_{th}^2}/B_{det}}}{R_{30V}} =\frac{\sqrt{4kT/R_L}}{R_{30V}}=\boxed{3.19\times10^{-11}\ \text{W}/\sqrt{\text{Hz}}}.$$

Part (f) — operating mode, I-V curve and load line

With a 30 V external reverse bias applied and a small load resistor carrying the photocurrent, the diode is operated in the photoconductive mode (as opposed to the unbiased, photovoltaic/solar-cell mode where the diode itself supplies the power). Reverse bias widens the depletion region (improving speed and, as seen in part (b), quantum efficiency) and keeps the diode current an essentially linear function of incident optical power over a wide range, which is why photoconductive operation is preferred for high-bandwidth detection.