Vacuum Photodiode
Responsivity, thermionic dark current, transit-time and RC bandwidth, space-charge limit and the shot- plus Johnson-noise SNR of a planar vacuum photodiode.
Responsivity
60.492 mA/W
Threshold 620 nm (hc/φ)
Photocurrent
6.05e-8 A
3.02e-6 V across the load
Thermionic Dark Current
2.00e-17 A
0.02 fA/cm² at 25 °C, scaled to T over an effective ψ = 1.404 eV
Space-Charge Limit
5.83e-2 A
Child's law; nonlinearity starts well below it
Transit Time
674.4 ps
Transit limit 0.820 GHz (0.553/τ)
Capacitance
0.443 pF
gap ε₀A/d 0.443 pF + stray 0.00 pF; RC limit 7.190 GHz
Bandwidth
0.815 GHz
1/√(1/f_RC² + 1/f_tr²), used as the noise bandwidth
SNR
-18.7 dB
20·log₁₀(I/σ) in B; shot 3.97e-9 A, Johnson (load at 25 °C) 5.18e-7 A rms
Key Formulas
R = η·q·λ/(h·c) for λ ≤ hc/φ; Iph = P·R
Jth = A₀·T²·exp(−ψ/kBT), A₀ = 4π·me·kB²·q/h³, ψ = kBT₂₅·ln(A₀T₂₅²/J₂₅) from the 25 °C dark emission
τ = d·√(2me/(qV)); ftr = 0.553/τ; fRC = 1/(2π·RL·(ε₀A/d + Cs))
σI² = 2q(Iph + Ith)B + 4kBTL·B/RL
Imax = (4ε₀/9)·√(2q/me)·V3/2·A/d² (Child)
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Planar diode, flat QE up to hc/φ, electrons leaving at rest, a brick-wall filter at the −3 dB bandwidth and load Johnson noise at 25 °C. Dark emission is a measured 25 °C value (Burle Table I presets), carried to other temperatures over an effective Richardson barrier (order of magnitude away from 25 °C). Stray capacitance is an input; a coax-matched tube is a transmission line, not a lumped RC.
- Dushman S. (1923). Electron emission from metals as a function of temperature. Phys. Rev. 21, 623–636. J = A₀·T²·exp(−φ/k_B·T). doi:10.1103/PhysRev.21.623
- Ramo S. (1939). Currents induced by electron motion. Proc. IRE 27, 584–585. The ramp current of an accelerating electron; f_tr = 0.553/τ. doi:10.1109/JRPROC.1939.228757
- Child C. D. (1911). Discharge from hot CaO. Phys. Rev. (Series I) 32, 492–511. Space-charge-limited current ∝ V^(3/2)/d². doi:10.1103/PhysRevSeriesI.32.492
- Burle Industries (1980). Photomultiplier Handbook (TP-136), Table I: dark emission at 25 °C of Cs₃Sb, Na₂KSb, K₂CsSb, multialkali and GaAs photocathodes; p. 20: thermionic and photoemission barriers differ in semiconductor cathodes. psec.uchicago.edu
- Hamamatsu Photonics (2017). Photomultiplier Tubes: Basics and Applications, 4th ed., §4.3.6 Eq. 4-19 (thermionic dark emission over the work function) and Eq. 5-7 (f_c = 1/(2π C_s R_L), C_s including stray and wiring capacitance). hamamatsu.com
- Bowers J. E., Burrus C. A. (1987). Ultrawide-band long-wavelength p-i-n photodetectors. J. Lightwave Technol. 5, 1339–1350. Combined RC and transit bandwidth 1/f² = 1/f_RC² + 1/f_tr². doi:10.1109/JLT.1987.1075419