Quantum efficiency

Quantum efficiency (QE, $\eta$) is the fraction of incident photons that produce a collected electron in a photodetector, or the fraction of injected electrons that produce an emitted photon in an LED or laser. For a photodetector it follows directly from the responsivity $\mathcal{R}$:

$$\eta =\frac{\mathcal{R}hc}{e\lambda }\approx \frac{1.24\mathcal{R}}{\lambda [\mu \text{m}]},$$

with $\mathcal{R}$ in A/W. A photodiode with 1.0 A/W at 1550 nm has $\eta =0.80$; one with 0.55 A/W at 850 nm has the same $\eta =0.80$, because each photon at 850 nm carries more energy, so a watt holds fewer of them. Unity quantum efficiency gives the largest possible responsivity, ${\mathcal{R}}_{\text{max}}=e\lambda \mathrm{/}hc\approx \lambda [\mu \text{m}]\mathrm{/}1.24$ A/W: 0.69 A/W at 850 nm and 1.25 A/W at 1550 nm. The Photodiode Noise Calculator accepts either the quantum efficiency or the responsivity and reports the other.

Photodetectors: external and internal quantum efficiency

External quantum efficiency (EQE) counts electrons collected per photon incident on the detector. It is what a datasheet responsivity implies and what a measurement against a calibrated power meter gives.

Internal quantum efficiency (IQE) counts electrons collected per photon absorbed. It excludes the light reflected at the surface and the light that passes through the absorbing layer without being absorbed.

The two are linked by the losses on the way in:

$${\eta }_{\text{ext}}=(1-R_s)(1-e^{-\alpha d}){\eta }_{\text{int}},$$

where $R_s$ is the reflectance of the front surface, $\alpha$ the absorption coefficient and $d$ the thickness of the absorbing layer.

Reflection. At normal incidence a bare semiconductor surface reflects $R_s=[(n-1)\mathrm{/}(n+1){]}^2$. Silicon at 850 nm ($n\approx 3.6$) reflects 32% and an InP surface at 1550 nm ($n=3.17$) reflects 27%, so an uncoated detector loses roughly a third or a quarter of the light before any is absorbed. A single-layer antireflection coating a quarter-wave thick, with index near $\sqrt{n}$, brings the reflectance close to zero at its design wavelength, and it is the main reason real detectors approach their internal efficiency.

Absorption depth. The fraction absorbed, $1-e^{-\alpha d}$, is 63% for $\alpha d=1$ and 95% for $\alpha d=3$. Near the band edge $\alpha$ falls steeply, so a detector's QE collapses as the wavelength approaches its cutoff: about 1.1 µm for silicon and about 1.7 µm for standard InGaAs.

Collection. Carriers generated too far from the junction, or at a damaged surface, can recombine before they are collected. Good detectors have ${\eta }_{\text{int}}$ close to 1 across most of their band; the external value is limited by reflection and absorption.

The photodiode rows in the table below are the typical values for each class given by Agrawal (chapter 4), the same ranges the responsivity entry converts to A/W. Devices with good antireflection coatings sit toward the top of each range.

LEDs and lasers

For light sources the conversion runs the other way, electrons in and photons out, and the efficiency splits into stages.

• Internal quantum efficiency ${\eta }_i$: the fraction of injected electron-hole pairs that recombine radiatively, as opposed to non-radiative recombination.
• Extraction efficiency ${\eta }_{\text{extr}}$ (LEDs): the fraction of internally generated photons that leave the chip. A flat surface of a high-index semiconductor passes only the light inside its escape cone, ${\theta }_c=\arcsin (1\mathrm{/}n)$, which captures a fraction $(1-\cos {\theta }_c)\mathrm{/}2$ of isotropic emission: 2.1% for $n=3.5$ and 4.2% for GaN ($n=2.5$). Textured surfaces, shaped chips and encapsulation exist to raise it.
• External quantum efficiency of an LED: photons emitted into air per injected electron, ${\eta }_{\text{EQE}}={\eta }_i{\eta }_{\text{extr}}$.

For a laser above threshold the useful figure is the differential quantum efficiency, the extra photons out per extra electron in,

$${\eta }_d={\eta }_i\frac{{\alpha }_m}{{\alpha }_i+{\alpha }_m},$$

where ${\alpha }_m$ is the mirror loss and ${\alpha }_i$ the internal loss, so ${\alpha }_m\mathrm{/}({\alpha }_i+{\alpha }_m)$ is the fraction of cavity photons that leave through the mirrors. It is measured from the slope efficiency $S$ of the light-current curve, ${\eta }_d=S\lambda [\mu \text{m}]\mathrm{/}1.24$, taken over both facets or stated per facet: a slope of 0.30 W/A at 1550 nm is ${\eta }_d=0.375$.

Optically pumped lasers are usually quoted by an optical slope efficiency in W/W, which is the quantum slope efficiency multiplied by the ratio of pump to signal wavelength. For a Yb fiber laser pumped at 976 nm and emitting at 1070 nm that ratio is 0.912, so a slope efficiency of 0.80 W/W corresponds to 0.88 signal photons per pump photon.

Quantum efficiency and power efficiency. Quantum efficiency counts photons; power efficiency, such as wall-plug efficiency, counts joules. The two differ by the quantum defect for optically pumped lasers (photon energy out over pump photon energy in) and by the ratio of photon energy to the drive voltage times $e$ for LEDs and diode lasers. For real devices power efficiency is lower than quantum efficiency because of series resistance (Joule heating), the voltage drop in excess of the photon energy, thermal effects and non-radiative recombination.

Say which one. "Quantum efficiency" means different ratios in different fields: detector EQE or IQE, LED EQE, laser IQE, or laser differential QE. Stating which one is meant, and whether reflection and coupling losses are included, avoids the most common confusion in quoted figures.

Common questions

How do you calculate quantum efficiency from responsivity?

Multiply the responsivity in A/W by 1.24 and divide by the wavelength in micrometers: $\eta =1.24\mathcal{R}\mathrm{/}\lambda$. At 1310 nm, 0.80 A/W gives $\eta =0.76$.

What is the difference between external and internal quantum efficiency?

External QE counts collected electrons per incident photon; internal QE counts them per absorbed photon. The gap between them is surface reflection and incomplete absorption, which an antireflection coating and a thick enough absorber remove.

Can quantum efficiency exceed 100%?

Not in an ordinary photodiode at visible and infrared wavelengths, where each absorbed photon creates at most one electron-hole pair. An avalanche photodiode delivers many electrons per photon, but that is gain, counted separately: its quantum efficiency is the unity-gain value. At deep ultraviolet wavelengths a single photon carries enough energy to create more than one pair in silicon, and the quantum yield there does exceed one.

References: B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); E. F. Schubert, Light-Emitting Diodes, 2nd ed. (Cambridge University Press, 2006); L. A. Coldren, S. W. Corzine and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012); G. P. Agrawal, Fiber-Optic Communication Systems, 4th ed. (Wiley, 2010), chapter 4.