Pixel Response: Linking Semiconductor Properties and Pixel Design

Hello. The previous module ended at the recording surface: its position had to be controlled precisely enough for a focused image to land on it. We now move inside that surface. This module follows the photographic signal from photons in silicon through sensor readout and, later, computational image formation.

Today’s question is what determines a pixel’s response before any image processing: how many incident photons become stored electrons, how many electrons the pixel can hold, and how strongly that charge changes its electrical output. Allow about 40–45 minutes, including the selected resources.

From light at the sensor to carriers in silicon

The optics deliver an irradiance to the sensor, but a pixel detects individual photons. For a narrow wavelength band centred on $\lambda$, photon energy is $hc/\lambda$. If the irradiance in that band is $E$, the illuminated pixel area is $A$, and the exposure lasts $t$, the expected number of incident photons is

The bars matter: this is an expectation, not a promise that every exposure receives precisely that count. For broadband light, the corresponding expression integrates spectral irradiance $E_{\lambda }$ over wavelength:

Read the selected pages of Silicon Photodetectors for the device physics beneath this photon budget. The absorption calculation is especially useful: it distinguishes a photon entering silicon from one being absorbed at a depth where its charge can be collected.

In ordinary silicon imaging, a photon needs energy above the band gap, about $1.12\mathrm{eV}$, to generate an electron–hole pair. This puts the long-wavelength cutoff near $1.1\mu \mathrm{m}$. A visible photon normally produces at most one useful electron; making that photon more energetic does not proportionally increase the charge it deposits.

For a uniform material and monochromatic flux $F_0$ entering at depth $x=0$, the notes derive

where $\alpha$ is the wavelength-dependent absorption coefficient and $G$ is the carrier-generation rate per unit volume. The characteristic absorption depth is $1/\alpha$. Blue light tends to be absorbed nearer the illuminated surface; longer-wavelength light penetrates farther. Neither “shallow” nor “deep” is inherently better: carriers made very near a defective surface may recombine, while photons that travel beyond an effective collection region may also fail to contribute.

Quantum efficiency and the area that gathers light

A useful pixel-level definition of external quantum efficiency, ${\mathrm{QE}}_{\mathrm{ext}}(\lambda )$, is the mean number of collected photoelectrons divided by the number of photons incident on the pixel’s specified entrance area at that wavelength. Stating that reference plane matters. Efficiency measured relative to photons already inside silicon excludes losses that an efficiency measured at the top of the pixel includes.

A simplified model makes the distinction between absorption and collection explicit:

Here $T_{\mathrm{opt}}$ represents transmission through the structures above silicon, $d$ is the relevant silicon thickness, and $P_{\mathrm{collect}}$ is the probability that a carrier generated at a given depth contributes to this pixel’s stored signal. Lateral losses and charge collected by a neighbour can be included in that probability. This is a model for reasoning, not a universal formula for every fabricated pixel.

The image below shows why optical access to the silicon requires engineering. It pairs pixel layouts with cross-sections of light passing microlenses and metal layers toward pinned photodiodes (PPDs). The shallow-trench isolation (STI) separates neighbouring structures.

A pixel’s geometrical area and its photosensitive silicon area are not identical. Wiring and transistors compete for space, while a microlens can direct photons arriving over much of the pixel pitch into a smaller unobstructed region. Consequently, a quoted fill factor is not interchangeable with QE: fill factor describes geometry under a particular definition; QE also incorporates transmission, absorption, and charge collection.

At fixed irradiance and exposure time, geometrical pixel area sets the first-order photon budget. For monochromatic light,

where ${\bar{n}}_{\mathrm{ph}}$ is the expected stored photoelectron count before saturation. Halving a square pixel’s pitch quarters its area and, if irradiance and QE remain unchanged, quarters its expected electrons. In a real redesign, QE may change too: shrinking a pixel changes its optical stack, photodiode geometry, and vulnerability to losses. A smaller pitch therefore cannot be assessed from pitch alone.

Storage is not the same as electrical gain

The generated electrons must remain somewhere until their signal is measured. To see the distinction between storing charge and measuring it as voltage, watch the short pixel explanation by Albert Theuwissen, then read the relevant portions of the research review by Eric Fossum, Nobukazu Teranishi, and Albert Theuwissen. The review connects the pinned photodiode’s physical structure to its storage capacity and to the separate readout node.

A pinned photodiode is a charge-storage region: illumination generates carriers, and electrons accumulate in its well. Its surface pinning layer helps suppress generation at interface defects, while the junction structure supports charge storage. At readout, the electrons are transferred to a floating diffusion (FD), a much smaller electrical node whose voltage is measured.

That distinction prevents two common confusions:

• Full-well capacity, $N_{\mathrm{FW}}$, is the largest useful number of photoelectrons the pixel can record without saturating. The photodiode’s storage design matters, but the readout node can impose a lower usable limit.
• Conversion gain, $g_{\mathrm{CG}}$, is the voltage change per electron at the sensing node. For a roughly constant total FD capacitance $C_{\mathrm{FD}}$,

where $q$ is the elementary charge.

If $n$ electrons reach the FD, its voltage falls by approximately $nq/C_{\mathrm{FD}}$. The signal can be represented as the positive difference between its reset voltage and its post-transfer voltage:

Thus a high conversion gain does not mean more photons were captured. It means each collected electron produces a larger voltage difference. This can make a given downstream voltage disturbance less significant when expressed in electrons. But a small FD capacitance also reaches a fixed permissible voltage swing after fewer electrons. To a first approximation, the usable count is bounded by both storage and sensing:

Actual capacitances and permissible swings depend on operating point and device design, so this is an engineering estimate rather than a universal full-well law. In particular, increasing photodiode storage capacity and decreasing FD capacitance act on different parts of the pixel. Designs that switch additional sensing capacitance can deliberately offer a lower-gain, higher-readout-capacity setting for bright scenes.

A complete pixel budget

Consider a hypothetical pixel of pitch $4\mu \mathrm{m}$, with area $16\mu {\mathrm{m}}^2$. Suppose it receives monochromatic $550\mathrm{nm}$ irradiance of $0.010W/{\mathrm{m}}^2$ for $10\mathrm{ms}$, and its external QE at that wavelength is $0.60$. Ignoring saturation for the moment, the incident photon expectation is

The expected photocharge is therefore about $0.60(4,430)=2,660$ electrons. If the FD capacitance is $4\mathrm{fF}$, its conversion gain is approximately $40\mu \mathrm{V}$ per electron, giving a voltage difference of about $0.107\mathrm{V}$.

Now suppose the storage well holds $10,000$ electrons and the FD permits a $0.40\mathrm{V}$ signal swing. That swing corresponds to about $10,000$ electrons at the stated conversion gain, so storage and readout capacities are approximately matched. At this exposure time and QE, saturation begins at roughly $0.038W/{\mathrm{m}}^2$ of the same monochromatic irradiance. Beyond that point, the incident-photon estimate continues to rise, but the pixel cannot provide a proportionally larger usable signal.

If pitch instead becomes $2\mu \mathrm{m}$ under the explicit assumption of unchanged irradiance and QE, the expected photocharge falls to about $665$ electrons because the area is one quarter as large. If storage capacity also happened to scale with area, it would fall from $10,000$ to $2,500$ electrons; the saturation irradiance could then remain similar even though the pixel collects fewer electrons at every unsaturated exposure. Neither unchanged QE nor proportional capacity scaling is guaranteed in a fabricated sensor. Doping, operating voltages, perimeter effects, and layout all matter.

Finally, this budget predicts the mean light response, not a perfectly repeatable result. Photon arrival and collection are discrete, and charge can also arise in the dark. We will separate those fluctuations and readout noise quantitatively later in this module.

Conclusion

A pixel’s response is a chain with distinct constraints. Photon energy and silicon’s absorption coefficient govern where carriers can be generated; wavelength-dependent quantum efficiency tells us what fraction of incident photons become collected electrons. Pixel area and exposure time set the incident-photon budget. Full-well and readout capacities limit the largest usable electron count, while conversion gain determines the voltage produced per electron, not the number of electrons captured.

The next lesson examines how a CCD moves accumulated charge through the sensor for readout. That architecture changes the route the charge takes, while retaining the photon-generation and capacity principles established here.