# Understanding the Light Detection in an Integrated Novel Electrophotonic Wavesensor Photodetector
## 4 Electrical Behavior and Photodetection
### Energy Bands under Illumination Condition  
If the drain-source junction is biased beyond Punch-Through condition, and light with an optical power \(P_{in}\) is absorbed by the silicon below the gate area, electron-hole pairs are produced in the space charge region between drain and source. This mechanism is depicted in Figure 6.  
Figure 4: Energy band diagrams at thermal equilibrium for a WS. Figure 5: Energy band diagrams for a WS on Punch-Through condition. Absorbed photons with energy \(E_{ph}\) greater than the band gap \(E_{g}\) produce electron-hole pairs, the electric field inside the depletion region separates the pair, and a current \(I_{ph}\) is established. If it is assumed that each absorbed photon generates one electron-hole pair, then the current corresponding to photons \(I_{ph}\) under ideal conditions can be calculated with (6). \[I_{ph}=\frac{qP_{in}\lambda}{hc} \tag{6}\]  
where \(h\), is the Planck constant, \(c\) the speed of light in vacuum, \(\lambda\) the wavelength, and \(P_{in}\) the incident optical power. This is the maximum current that can be generated by the absorbed light in a photodiode or any photodetector. However, the transduction of light in silicon present losses due to recombination processes, and only a portion of the generated electron-hole pairs end up contributing to the detectable current \(I_{opt}\), which always is lower than \(I_{ph}\), depending on the quantum efficiency \(\eta\) as indicated in (7):  
\[I_{opt}=\eta I_{ph}=\eta\frac{qP_{in}\lambda}{hc} \tag{7}\]  
In addition, a Fermi quasi-level variation is produced by the optical carrier excess, and then the potential barrier is additionally reduced, promoting the injection of extra carriers from source to drain. Then, a current of electrons from the source \(j_{n}\) is added to the photogenerated current. Assuming the WS is biased at low voltage, and a low injection regime, \(j_{n}\) current is expressed as (8):  
\[j_{n}=\frac{qD_{n}n_{i}^{2}}{2\sqrt{\pi}L_{\mathrm{D}}N_{A}}\exp\left(\frac{qV _{Feff}}{kT}\right) \tag{8}\]  
Figure 6: Energy diagram for WS on Punch-Through under stimulation of light with absorbed photons with energy \(E_{ph}\). Red dots represent the electrons and blue dots the holes. Here, \(V_{Feff}\) is known as the effective induced source voltage, and is composed by a potential \(V_{F}\) induced by the drain-source bias, plus the optical potential induced by photogenerated carrier excess \(\phi_{opt}\). \[V_{Feff}=V_{F}+\phi_{opt},\ (V_{DS}>V_{PT}) \tag{9}\]  
The total electron drain current density \(j_{dudal}\) can be expressed as (10):  
\[j_{dtotal}=j_{n}+j_{opt}=\frac{qD_{n}{n_{i}}^{2}}{2\sqrt{\pi}L_{D}N_{A}}\exp \biggl{(}\frac{qV_{Feff}}{kT}\biggr{)}+j_{opt} \tag{10}\]  
Using (7), we can obtain \(j_{opt}\), which is the photocurrent density generated in the substrate within the space charge region. This current can be simplified into (11):  
\[j_{opt}=\frac{I_{opt}}{A}=\eta\frac{qP_{in}\lambda}{Ahc} \tag{11}\]  
where \(A\), is the cross-section area of the space charge region. In the presented configuration of the device, an optical current of holes is established through the substrate junction. This can be computed using (12) [39; 40]:  
Figure 4:0  
Using this current, and considering that hole and electron currents are of the same magnitude, it is possible to estimate the optical potential as:  
Figure 4:1  
Combining (9), (10) and (13), the total equivalent drain current density can be expressed as:  
Figure 4:2  
in which the second and third terms account for to the presence of light. If the dark current is subtracted, the optical contribution to current \(j_{dopt}\) can be identified:  
Figure 4:3  
Remember \(j_{opt}\) represents the photocurrent density, taking into account quantum efficiency. Then, the ratio of \(j_{dopt}\) to \(j_{opt}\) delivers a value which, in case of being larger than 1, indicates an optical gain. This is \(\beta\) expressed in (16):  
Figure 4:4  
This is the gain of the WS.