Retinomorphic sensor

While the equation above is valid for any form of ${\displaystyle C(t)}$, it cannot be solved analytically unless the input form of the optical stimulus is known. The simplest form of optical stimulus would be a step function going from zero to some finite optical power density ${\displaystyle P}$ at a time ${\displaystyle t=t_{0}}$. While real-world applications of retinomorphic sensors are unlikely to be accurately described by such events, it is a useful way to understand and benchmark the performance of retinomorphic sensors. In particular, we are primarily concerned with the maximum height of the ${\displaystyle V_{out}}$ immediately after the light has been turned on.

In this case the capacitance could be described by:

${\displaystyle C(t)={\begin{cases}C_{dark},&{\text{for }}t<t_{0}\\C_{light},&{\text{for }}t\geq t_{0}\end{cases}}}$

The capacitance under illumination will depend on ${\displaystyle P}$. Semiconductors are known[19] to have a conductance, ${\displaystyle G}$, which increases with a power-law dependence on incident optical power density: ${\displaystyle G\propto P^{\gamma }}$, where ${\displaystyle \gamma }$ is a dimensionless exponent. Since ${\displaystyle G}$ is linearly proportional to charge density, and capacitance is linearly proportional to charges on the plates for a given voltage, the capacitance of a retinomorphic sensor also has a power-law dependence on ${\displaystyle P}$. The capacitance as a function of time in response to a step function, can therefore be written as:

${\displaystyle C(t)={\begin{cases}C_{dark},&{\text{for }}t<t_{0}\\C_{dark}+\alpha _{0}P^{\gamma },&{\text{for }}t\geq t_{0}\end{cases}}}$

where ${\displaystyle \alpha _{0}}$ is the capacitance prefactor. For a step function we can re-write our differential equation for ${\displaystyle V_{in}}$ as a difference equation:

${\displaystyle V_{in}=V_{C}+R{\bigg [}C{\frac {\Delta V_{C}}{\Delta t}}+V_{C}{\frac {\Delta C}{\Delta t}}{\bigg ]}}$

where ${\displaystyle \Delta V_{C}}$ is the change in voltage dropped across the capacitor as a result of turning on the light, ${\displaystyle \Delta C}$ is the change in capacitance as a result of turning on the light, and ${\displaystyle \Delta t}$ is the time taken for the light to turn on. The variables ${\displaystyle V_{C}}$ and ${\displaystyle C}$ are defined as the voltage dropped across the capacitor and the capacitance, respectively, immediately after the light has been turned on. I.e. ${\displaystyle V_{C}}$ is henceforth shorthand for ${\displaystyle V_{C}(t_{0})}$, and ${\displaystyle C}$ is henceforth shorthand for ${\displaystyle C(t_{0})}$. Assuming the sensor has been held in the dark for sufficiently long before the light is turned on, the change in ${\displaystyle V_{C}}$ can hence be written as:

${\displaystyle \Delta V_{C}=V_{C}-V_{in}}$

Similarly, the change in ${\displaystyle C}$ can be written as

${\displaystyle \Delta C=\alpha _{0}P^{\gamma }}$

Putting these into the difference equation for ${\displaystyle V_{in}}$:

${\displaystyle V_{in}=V_{C}+R{\bigg [}C{\frac {V_{C}-V_{in}}{\Delta t}}+V_{C}{\frac {\alpha _{0}P^{\gamma }}{\Delta t}}{\bigg ]}}$

Top: Illustration of transient response (output voltage ${\displaystyle V_{out}}$ as a function of ${\displaystyle t}$) of retinomorphic sensor in response to a step change in light intensity from 0 to ${\displaystyle P}$, for increasing values of ${\displaystyle P}$. Bottom: Illustration of how one can extract the retinomorphic figure of merit, ${\displaystyle \Lambda }$, from experimental retinomorphic sensor data. By plotting the ratio of ${\displaystyle V_{in}/V_{max}}$ on the y-axis against ${\displaystyle 1/{\sqrt {P}}}$ on the x-axis, then fitting with a straight line, the gradient will yield ${\displaystyle 1/\Lambda }$.

Multiplying this out:

${\displaystyle V_{in}\Delta t=V_{C}\Delta t+R{\bigg [}C(V_{C}-V_{in})+V_{C}\alpha _{0}P^{\gamma }{\bigg ]}}$

Since we are assuming the light turns on very quickly we can approximate ${\displaystyle \Delta t\approx 0}$. This leads to the following:

${\displaystyle V_{C}=V_{in}{\frac {C_{dark}+\alpha _{0}P^{\gamma }}{C_{dark}+2\alpha _{0}P^{\gamma }}}}$

Using the relationship ${\displaystyle V_{in}=V_{C}+V_{out}}$, this can then be written in terms of the output voltage:

${\displaystyle V_{max}=V_{in}{\frac {\alpha _{0}P^{\gamma }}{C_{dark}+2\alpha _{0}P^{\gamma }}}}$

Where we have defined the peak height as ${\displaystyle V_{max}=V_{out}(t_{0})}$, since he peak occurs immediately after the light has been turned on.

The retinomorphic figure of merit, ${\displaystyle \Lambda }$, is defined as the ratio of the capacitance prefactor and the capacitance of the retinomorphic sensor in the dark:[18]

${\displaystyle \Lambda ={\frac {\alpha _{0}}{C_{dark}}}}$

With this parameter, the inverse ratio of peak height to input voltage can be written as follows:

${\displaystyle {\frac {V_{in}}{V_{max}}}=2+{\frac {1}{\Lambda P^{\gamma }}}}$

The value of ${\displaystyle \gamma }$ will depend on the nature of recombination in the semiconductor,[20] but if band-to-band recombination dominates and the charge density of electrons and holes are equal, ${\displaystyle \gamma =0.5}$. For systems where this is approximately true[21] the following simplification to the above equation an be made:

${\displaystyle {\frac {V_{in}}{V_{max}}}=2+{\frac {1}{\Lambda P^{1/2}}}}$

This equation provides a simple method for evaluating the retinomorphic figure of merit from experimental data. This can be carried out by measuring the peak height, ${\displaystyle V_{max}}$, of a retinomorphic sensor in response to a step change in light intensity from 0 to ${\displaystyle P}$, for a range of values ${\displaystyle P}$. Plotting ${\displaystyle V_{in}/V_{max}}$ as a function of ${\displaystyle P^{-1/2}}$ should yield a straight line with a gradient of ${\displaystyle 1/\Lambda }$. This approach assumes that ${\displaystyle V_{max}}$ is linearly proportional to ${\displaystyle V_{in}}$.