Quantum Efficiency

Prerequisites

Understanding QE gives the intuition; this page gives the mathematical model.

Scope of this page

This page is the formal QE reference for COMPASS. It covers definitions, absorbed-power calculations, channel aggregation, crosstalk, and energy balance. For an intuitive photon-budget story, use Understanding QE.

Quantum Efficiency (QE) is the primary figure of merit computed by COMPASS. It measures how effectively a pixel converts incident photons into electrical signal.

Definition

The external quantum efficiency at wavelength $\lambda$ is:

$$\text{QE}(\lambda )=\frac{\text{Number of electron-hole pairs collected}}{\text{Number of incident photons}}$$

Equivalently, in the optical-only approximation used by most COMPASS examples:

$$\text{QE}(\lambda )=\frac{P_{\text{absorbed in PD}}(\lambda )}{P_{\text{incident}}(\lambda )}$$

where $P_{\text{absorbed in PD}}$ is the power absorbed within the photodiode volume and $P_{\text{incident}}$ is the total incident power. QE is dimensionless and ranges from 0 to 1 (0% to 100%).

This approximation is useful for optical design, but a full sensor characterization separates optical absorption from electrical charge collection.

OE, IQE, and EQE

For a production-style characterization, it is better to keep three quantities separate:

Quantity	Meaning	What it answers
OE	Optical efficiency: incident photons that generate carriers in the active silicon volume	Did the optical stack deliver light into useful silicon?
IQE	Internal quantum efficiency: generated carriers that are collected by the target node	Did the electrical pixel collect the generated charge?
EQE	External quantum efficiency: incident photons that become collected charge	What signal does the camera pixel produce?

Let $G(\mathbf{r},\lambda ,\theta ,\varphi )$ be the carrier-generation density in silicon from the optical solver, normalized to the incident photon flux. Let $W_i(\mathbf{r})$ be the electrical collection weighting function for pixel $i$: the probability that a carrier generated at $\mathbf{r}$ is collected by that pixel. Then:

$${\text{EQE}}_i(\lambda ,\theta ,\varphi )={\int }_{V_{\text{Si}}}G(\mathbf{r},\lambda ,\theta ,\varphi )W_i(\mathbf{r})dV$$

and:

$$\text{OE}(\lambda ,\theta ,\varphi )={\int }_{V_{\text{Si}}}G(\mathbf{r},\lambda ,\theta ,\varphi )dV$$

If $\text{OE}>0$, a corresponding internal efficiency can be reported as:

$${\text{IQE}}_i=\frac{{\text{EQE}}_i}{\text{OE}}$$

The common COMPASS photodiode-absorption model is the special case where $W_i(\mathbf{r})$ is treated as 1 inside the photodiode integration volume and 0 elsewhere. That is a practical optical proxy, not a replacement for a charge-transport solve when electrical crosstalk matters.

Practical rule

Use optical QE for stack tuning, BARL tuning, microlens focusing, and solver convergence. Use EQE with an electrical weighting function when comparing final sensor performance, color crosstalk, or pixel-to-pixel charge collection.

Factors affecting QE

The total QE is the product of several loss factors:

$$\text{QE}=(1-R)\times T_{\text{optics}}\times {\eta }_{\text{abs}}\times {\text{FF}}_{\text{PD}}$$

Factor	Symbol	Description
Surface reflection	$R$	Light reflected by the pixel stack (BARL reduces this)
Optical transmission	$T_{\text{optics}}$	Fraction transmitted through color filter, planarization, etc.
Silicon absorption	${\eta }_{\text{abs}}$	Fraction of light absorbed within the silicon thickness
Photodiode fill factor	${\text{FF}}_{\text{PD}}$	Fraction of absorbed photons within the PD collection volume

In COMPASS, the full-wave simulation captures all these effects simultaneously. The QE is not computed from this factored form but rather from the total electromagnetic solution.

Computing QE in COMPASS

COMPASS computes QE by two methods:

Method 1: Absorption integral

The absorbed power density at position $\mathbf{r}$ and wavelength $\lambda$ is:

$$p_{\text{abs}}(\mathbf{r})=\frac{1}{2}\omega {\epsilon }_0\text{Im}({\epsilon }_r)|\mathbf{E}(\mathbf{r}){|}^2$$

The QE for a specific pixel (photodiode region $V_{\text{PD}}$) is:

$$\text{QE}=\frac{{\int }_{V_{\text{PD}}}{p}_{\text{abs}}dV}{P_{\text{incident}}}$$

With an electrical collection model, replace the hard photodiode volume by a weighting function:

$${\text{EQE}}_i=\frac{{\int }_{V_{\text{Si}}}{p}_{\text{abs}}(\mathbf{r})W_i(\mathbf{r})dV}{P_{\text{incident}}}$$

The weighting function can be imported from a device simulator or approximated from measured collection maps. It should cover the target pixel and enough neighboring pixels to capture electrical crosstalk.

Method 2: Poynting flux difference

Alternatively, the power absorbed in a region can be found from the net Poynting flux entering and leaving the region:

$$P_{\text{absorbed}}={\oint }_S⟨\mathbf{S}⟩\cdot \hat{n}dA=S_{z,\text{top}}-S_{z,\text{bottom}}$$

where $S_{z,\text{top}}$ and $S_{z,\text{bottom}}$ are the z-components of the Poynting vector at the top and bottom of the photodiode region.

Both methods are implemented in the QECalculator class.

Per-pixel QE and color channels

A 2x2 Bayer unit cell has four pixels, each with its own photodiode:

  +--------+--------+
  | R_0_0  | G_0_1  |
  +--------+--------+
  | G_1_0  | B_1_1  |
  +--------+--------+

COMPASS computes QE independently for each photodiode. The naming convention is {Color}_{row}_{col}.

The spectral_response function averages QE across pixels of the same color to produce per-channel QE curves:

from compass.analysis.qe_calculator import QECalculator

# result.qe_per_pixel = {"R_0_0": array, "G_0_1": array, "G_1_0": array, "B_1_1": array}
channel_qe = QECalculator.spectral_response(result.qe_per_pixel, result.wavelengths)
# channel_qe = {"R": (wavelengths, qe_R), "G": (wavelengths, qe_G_avg), "B": (wavelengths, qe_B)}

Crosstalk

Optical crosstalk occurs when light intended for one pixel is absorbed by a neighboring pixel. COMPASS quantifies this with a crosstalk matrix:

$${\text{CT}}_{ij}(\lambda )=\frac{{\text{QE}}_j(\lambda ,\text{illuminating pixel }i)}{\sum _k{\text{QE}}_k(\lambda ,\text{illuminating pixel }i)}$$

The diagonal elements represent correctly detected signal; off-diagonal elements represent crosstalk. Lower crosstalk means better color separation.

The QECalculator.compute_crosstalk method computes this matrix from the per-pixel QE data.

Interactive Pixel Crosstalk Heatmap

Explore how wavelength and pixel pitch affect optical crosstalk between neighboring pixels. The center pixel is illuminated; surrounding pixels show crosstalk intensity.

Wavelength: 550 nm

Pixel pitch: 1.00 um

Show Bayer overlay

Absorption Depth (Si):1.56 um

Nearest-neighbor Crosstalk:3.8%

Total Crosstalk:55.2%

Energy balance

A fundamental physical constraint is energy conservation:

$$R(\lambda )+T(\lambda )+A(\lambda )=1$$

where $R$ is total reflection, $T$ is total transmission (through the bottom), and $A$ is total absorption (in all materials). If this balance is violated by more than 1-2%, the simulation may have numerical issues.

The total QE across all pixels is bounded by the total absorption in silicon:

$$\sum _{\text{pixels}}{\text{QE}}_i\le A_{\text{Si}}$$

The inequality is strict because some light is absorbed in the color filter, metal grid, and other non-photodiode regions.

Typical QE spectra

Interactive QE Spectrum Chart

Explore how silicon thickness, BARL quality, and metal grid width affect the quantum efficiency spectrum of Red, Green, and Blue channels.

Silicon thickness: 3.0 um

BARL quality: 70%

Metal grid width: 50 nm

Blue peak QE:46.7%

Green peak QE:54.5%

Red peak QE:42.8%

Average QE:48.0%

For a well-designed 1 um pitch BSI pixel:

Channel	Peak QE	Peak wavelength
Blue	50-70%	450-470 nm
Green	60-80%	530-560 nm
Red	50-70%	600-630 nm

The QE spectrum typically shows:

• A sharp rise at the blue edge due to increasing silicon absorption
• A peak in the passband of each color filter
• A gradual decline at the red edge due to decreasing silicon absorption (absorption depth exceeds pixel thickness)
• Spectral ripples from thin-film interference in the BARL stack

The blackbody spectrum below shows how the solar spectral irradiance relates to the wavelength range where QE is most critical for image sensor performance:

Interactive Blackbody Spectrum Viewer

Adjust the color temperature to see how the blackbody spectrum changes. Enable standard illuminant overlays for comparison.

Color Temperature: 5500 K

CIE D65 CIE A CIE F11 LED White

CCT:5500 K

λ_max:527 nm

Visible Power:71.1%

Approx. Color

INFO

QE above 80% for a single channel is rare because color filter absorption, reflection losses, and photodiode fill factor all reduce the total efficiency.

From Theory to Simulation

Use Pixel Optical Effects to decide which design variable is likely limiting the QE curve, then run a focused sweep:

Symptom in QE or crosstalk	Likely design variable	Practical workflow
Low blue QE with strong edge leakage	Microlens shape, metal grid aperture, diffraction	Microlens & CRA
Spectral ripple or broad reflection loss	BARL thickness/material stack	BARL Design
Red/NIR QE falls too early	Silicon thickness, photodiode depth, DTI depth	Si Absorption Depth
Color channels bleed into neighbors	Metal grid, CFA absorption, DTI geometry	DTI Crosstalk
Corner pixels underperform	CRA, microlens offset, angular response	Angular Response