### Mathematical approach

In Figure

1, the fluorochrome molecules bound to the Aβ peptides are excited by blue laser irradiation (λ

_{ex}) at a high energy level and emit visible fluorescent light (λ

_{em}) at a low energy level. Through this process, the filter surface unoccupied by the cells reflects the unwanted range of blue laser irradiation and only transmits the fluorescently emitted light through the filter. When the fluorescent light (dotted red line) emitted by the excitation (dotted blue line) is transmitted through the optical filter and approaches the p-FET sensing area (i.e., α-Si surface), excess charge carriers are generated by the absorbed photons within the α-Si sensing layer to give rise to a measurable photocurrent, which can be theoretically expressed as follows:

${I}_{\mathsf{\text{p}}}=\left(\frac{e}{hv}\right)\left(1-R\right)\eta \left(1-{e}^{-\alpha vd}\right){P}_{\mathsf{\text{FET}}}=\left(\frac{e}{hv}\right){C}_{\mathsf{\text{FET}}}{P}_{\mathsf{\text{FET}}},$

(1)

where

*I*_{p} is the photocurrent generated by p-FET,

*e* is the electron charge,

*h* is the Planck constant, and

*ν* is the photon frequency of fluorescence emission. The remaining parts of Equation

1 represent the optical properties related to the p-FET device which can be designated as a single constant,

*C*_{FET},

*R* is the reflectance,

*η* is the quantum efficiency,

*α* is the absorption coefficient of the photon, and

*d* is the p-FET sensing layer thickness. Therefore,

*I*_{p} is directly proportionate to

*P*_{FET} which represents the intensity of the incident fluorescent light transmitted through the filter. Also, the intensity of the fluorescent transmittance onto the p-FET is linearly proportional to the multiplication of the intensity of emitted fluorescence (

*P*_{em}) from fluorochrome before transmission and the transmittance coefficient (

*T*) of the optical filter for the emitted wavelength:

${P}_{\mathsf{\text{FET}}}=T\phantom{\rule{0.5em}{0ex}}{P}_{\mathsf{\text{em}}}.$

(2)

Meanwhile,

*P*_{em} definitely relies on the number of the emitted photons from the fluorochrome excited by the blue laser. By the definition of the quantum yield as follows:

$\mathsf{\text{\Phi}}\equiv \frac{\mathsf{\text{\#ofemittedphotons}}}{\mathsf{\text{\#ofabsorbedphotons}}},$

(3)

the number of emitted photons indicates the multiplication of the quantum yield and the number of photons absorbed within the fluorochrome. Subsequently,

*P*_{em} can be mathematically expressed as follows:

${P}_{\mathsf{\text{em}}}=\mathsf{\text{\Phi}}\times \left(\mathsf{\text{\#ofabsorbedphotons}}\right)=\mathsf{\text{\Phi}}\left(1-{e}^{-{A}_{\lambda}}\right),$

(4)

where

*A*_{λ} is the absorbance of fluorochrome for the specified wavelength which is defined as the logarithmic value of the ratio of the intensity of light passed through fluorochrome to the intensity of incident light. Therefore, Equation

1 can be completely rewritten as follows:

${I}_{\mathsf{\text{p}}}=\left(\frac{e}{h\nu}\right){C}_{\mathsf{\text{FET}}}T\mathsf{\text{\Phi}}\left(1-{e}^{-{A}_{\lambda}}\right)=C\left(1-{e}^{-{A}_{\lambda}}\right),$

(5)

where *C* represents an arbitrary constant. Since the absorbance, *A*_{λ}, is proportionate to the volume (or mass, or thickness) and the concentration of the absorbing species, Equation 5 meaningfully indicates that high electrical current may be generated by the p-FET device for a large amount of the Aβ-conjugated fluorochrome and vice versa. Eventually, since the quantity of Aβ peptides is directly proportionate to that of the tagged fluorochrome, the photocurrents generated by p-FET are a function of the amount of Aβ peptides specifically conjugated with fluorochrome.

Even though the exact relationship would not be available in a form of an equation in this study, the photocurrent generated by the p-FET device would be expressed as a linear function of the amount of the Aβ-conjugated fluorochrome and potentially provide the quantified Aβ concentration.