Cooling Benefit Calculator
Dark current reduction and SNR improvement from thermoelectric (TEC) or cryogenic cooling.
Si E_g: 1.125 eV at 25 °C, 1.140 eV at -40 °C
InGaAs E_g: 0.738 eV at 25 °C, 0.760 eV at -40 °C
Si Dark Reduction
×9.5e+2
1.05e-4 e⁻/s at -40 °C
InGaAs Dark Reduction
×1.9e+4
5.26e-2 e⁻/s at -40 °C; pure diffusion, an upper bound
Si SNR (25 °C → cooled)
31.5 → 31.5
InGaAs SNR (25 °C → cooled)
22.3 → 31.5
Si (depletion generation): I ∝ n_i ∝ T^(3/2) · exp(−E_g/2k_BT)
InGaAs (diffusion-limited): I ∝ n_i² ∝ T³ · exp(−E_g/k_BT)
SNR = S / √(S + I_dark·t + σ_read²)
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
Si dark current scales as depletion generation (∝ n_i), InGaAs as diffusion (∝ n_i²), with a fixed band gap or E_g(T): Varshni for Si, Vurgaftman's In₀.₅₃Ga₀.₄₇As gap for InGaAs. The temperature-dependent gap raises the 25 → −40 °C reduction from ×630 to ×950 for Si and from ×7158 (fixed 0.75 eV) to ×1.90e4 for InGaAs; E_g(T) is the default, and a fixed gap fits it best at the 0 K value (≈ 1.20 eV for Si). One mechanism per material: a generation part of the InGaAs current would fall only as n_i, so its reduction is an upper bound. Surface leakage and tunnelling are left out.
- Sze S. M., Ng K. K. (2007). Physics of Semiconductor Devices, 3rd ed. Wiley. n_i(T), generation (∝ n_i) and diffusion (∝ n_i²) currents. doi:10.1002/0470068329
- Varshni Y. P. (1967). Temperature dependence of the energy gap in semiconductors. Physica 34, 149–154. Si coefficients E_g(0) = 1.17 eV, α = 4.73e-4 eV/K, β = 636 K as in Sze, 2nd ed. (1981). doi:10.1016/0031-8914(67)90062-6
- Vurgaftman I., Meyer J. R., Ram-Mohan L. R. (2001). Band parameters for III–V compound semiconductors and their alloys. J. Appl. Phys. 89, 5815–5875. Varshni sets (Tables I–X) and the Ga₁₋ₓInₓAs bowing C = 0.477 eV (Table XIII). doi:10.1063/1.1368156