Responsivity and NEP from first principles

The number

A photodiode datasheet gives the responsivity in A/W at a stated wavelength and the noise-equivalent power (NEP) in W/√Hz. NEP is the optical power whose signal equals the noise current in a 1 Hz bandwidth: the smallest resolvable power, referred to 1 Hz.

A diode with a quantum efficiency of 0.90 at 900 nm, 1 nA of dark current and 1 GΩ of shunt resistance at 25 °C has a responsivity of 0.653 A/W and an NEP of 2.81 × 10⁻¹⁴ W/√Hz.

What it means

Each photon of energy $hc/\lambda$ that yields a collected carrier gives a charge $q$, so with quantum efficiency $\eta$ the responsivity is

The dark current carries shot noise (Schottky 1918) and the shunt resistance carries Johnson noise (Johnson 1928; Nyquist 1928). The two are uncorrelated and add in quadrature:

For the example the shot term is 1.79 × 10⁻¹⁴ A/√Hz and the Johnson term 4.06 × 10⁻¹⁵ A/√Hz, so dark current sets the noise. $D^*$ normalizes NEP to the active area $A$: 7.96 × 10¹² cm·√Hz/W for 5 mm². The constants are the exact SI values of $h$, $c$, $q$ and $k$ (BIPM 2019).

Compute it

Where it misleads

NEP referred to 1 Hz is not the noise of a working receiver. The noise in a bandwidth $B$ is $i_n\sqrt{B}$, and the load or feedback resistor adds its own current noise $\sqrt{4kT/R_L}$. With a 100 kΩ feedback resistor that term is 4.06 × 10⁻¹³ A/√Hz, 22 times the diode's own noise, so the front end, not the diode, sets the NEP of the receiver.

The conditions matter as much as the value. Responsivity changes with wavelength and falls near the band edge, where it also depends on temperature. Generation dark current scales with the intrinsic carrier density, which in silicon roughly doubles every 10 K near room temperature, so an NEP quoted at 25 °C is optimistic at 70 °C. Reverse bias raises the dark current and lowers the shunt resistance.

The live photodiode datasheet computes these values against bias and temperature for a fictional device. The glossary defines responsivity, NEP, shot noise and Johnson noise.

What to specify

• Responsivity at the working wavelength, with the bias and temperature of the measurement.
• Dark current and shunt resistance at the working bias and at the highest operating temperature.
• NEP with its wavelength, bias, temperature and measurement bandwidth, or the noise spectrum when 1/f noise matters at low frequency.
• The front-end noise budget: feedback resistor, amplifier voltage and current noise, and the signal bandwidth.

References

• Sze, S. M. and Ng, K. K. (2007). Physics of Semiconductor Devices, 3rd ed. Wiley, Hoboken. Photodiode sections.
• Saleh, B. E. A. and Teich, M. C. (2019). Fundamentals of Photonics, 3rd ed. Wiley, Hoboken.
• Schottky, W. (1918). Über spontane Stromschwankungen in verschiedenen Elektrizitätsleitern. Annalen der Physik 57, 541–567.
• Johnson, J. B. (1928). Thermal agitation of electricity in conductors. Phys. Rev. 32, 97–109. doi:10.1103/PhysRev.32.97 (opens in a new tab)
• Nyquist, H. (1928). Thermal agitation of electric charge in conductors. Phys. Rev. 32, 110–113. doi:10.1103/PhysRev.32.110 (opens in a new tab)
• BIPM (2019). The International System of Units (SI), 9th ed. Bureau International des Poids et Mesures, Sèvres.