## 2 Theory and Methods
### Theoretical Framework for MOSFET Detectors
The Geminate model has successfully explained the experimental results for electrons, and X-rays, and while further refinements have been made [77; 78],[PERSON]'s solution is sufficient to describe the concepts presented in this work, where we consider e-h pair separations in only the longitudinal and transverse planes. Figure 1 describes the single e-h pair considered by the Geminate model. To confirm the applicability of this model to 100 MeV electrons, first, the longitudinal spacing of e-h pairs along the track must be larger than 2\(r\), \(\sim\)16 nm. Dividing the e-h pair creation energy by the collisional stopping power of 100 MeV electrons in SiO\({}_{2}\) allows estimation of the spacing between e-hole pairs. Using a stopping power of 1.923 Mev cm\({}^{2}\)/g from the NIST E-Star database [79; 80; 81] and an e-h pair generation energy of 17 eV [71], results in a spacing of 38.94 nm. As 38.94 nm is more than double the 16 nm requirement, the e-h pairs can be considered isolated along the track, and the Geminate model is, therefore, a suitable representation of the conditions within which initial recombination will occur. In the case of the current clinical radiotherapy dose rates, the DPP and, hence, the fluence, is low enough such that the separation of e-h pairs in the transverse plane will also be much greater than twice the average thermalization radius. Hence, the Geminate model should remain applicable, and the total dose will, therefore, be proportional to the number of tracks passing through the gate oxide. Delivering ultra-high DPP, however, requires a fluence that is sufficiently high such that the probability of overlap of the ejected electrons between neighbouring tracks could increase. If this behaviour, illustrated in Figure 1 were to occur, the spatial separation of an electron and its corresponding hole could be impacted upon by e-h pairs of another track, causing a decrease in the probability that an electron escapes initial recombination within the gate oxide. A reduction in the escape probability would decrease the number of holes trapped at the Si/SiO\({}_{2}\) interface, resulting in a saturation limit upon which the measurement of \(\Delta\)V is no longer directly proportional to the absorbed dose and the response will plateau with increasing fluence. Figure 1: The applicability of the Geminate recombination model rests upon a critical assumption of large separation distances between electron–hole pairs. **Top**: When the centre-to-centre spacing is greater than twice the length of \(r\), initial recombination can be considered restricted to a single pair. **Bottom**: The limiting case, whereby increased initial recombination will occur due to decreased separation distances between neighbouring electron–hole pairs. While this is a simplification of an incredibly complex system and track structure, the effect of such overlap behaviour on the charge yield of MOS devices has been previously studied [82; 83], with custom Monte Carlo codes developed to model this phenomenon for other radiation types [84], though this is the first discussion of Geminate sphere overlap in the context of DPP independence for FLASH radiotherapy. Such a situation would lead to a DPP dependence of the MOS_kin_; hence, we consider the dose at which the distance between neighbouring e-h pairs produced by different electron tracks decreases to a point at which the probability of initial recombination could be impacted upon. To estimate the limiting case, the spheres of the Geminate model can be arranged using a square lattice distribution. At an instant in time, where the dose rate is infinite as \(t\to 0\), the fluence is distributed across the transverse plane. The dose is delivered to a medium by electrons according to \(D=\Phi\frac{1}{\rho}\frac{dE}{dx}\), where \(\Phi\) is the fluence, and \(\frac{1}{\rho}\frac{dE}{dx}\) is the mass collision stopping power. This relation can be used to calculate the required fluence for any dose.