## 2 Theory and Methods  
### Theoretical Framework for MOSFET Detectors  
We first present a brief summary of the physics behind MOSFET response to ionizing radiation and the effects of total absorbed dose upon a device. As a common electrical component, MOSFETs can also be used to detect ionizing radiation. Interested readers are referred to [71; 72; 73] for further information on the general operation and background of MOSFET devices. The MOS_kin_ utilizes a p-channel MOSFET, whereby electron-hole (e-h) pairs are generated within the gate oxide when exposed to ionizing radiation. As an active dosimeter, a bias is applied to the gate during irradiation causing the holes to drift towards the Si/SiO\({}_{2}\) interface, where they become trapped within interface defect centres. This causes a shift in the gate threshold voltage (\(\Delta\)V) required to pass a constant current between the source and drain terminals of the MOSFET. This shift in \(\Delta\)V is proportional to the absorbed dose; the sensitivity of the device is expressed as \(\Delta\)V/1 cGy. During prolonged exposure, the accumulation of holes at the interface may lead to a decrease in sensitivity, proportional to the total accumulated dose. This decrease in sensitivity, although minimal, can be corrected if exhibited by increasing the positive bias on the gate, or by the introduction of a factor calculated from a linear fit to the decreasing sensitivity within a given beam quality [74]. The response of MOSFET detectors to ionising radiation can be modelled in two ways. The first is the columnar model for high linear energy transfer (LET) particles. For low LET particles such as megavoltage electrons, the Geminate recombination model is used and describes the probability of an electron escaping initial recombination [73]. In fact, for such particles, the initial recombination is considered to be restricted to electrons and holes created from the same atom [75], occurring over timescales as low as a few picoseconds [73]. To understand the independence of the MOS_kin_ to dose rate and DPP for UHDR VHEE fields, the model should be considered under these conditions. First solved by [PERSON] [71; 72; 73], the work of [PERSON] [76] further showed that at low electric fields, the probability that an electron escapes initial recombination within the gate oxide is proportional to the two terms shown in Equation (1):  
\[P(r,E)=\exp\left(\frac{-r_{c}}{r}\right)(1+\beta r_{c}(1+\cos\theta)) \tag{1}\]  
where \(r\) is the initial separation of an electron and hole following ionisation (also known as the thermalisation distance), \(r_{c}=e^{2}/\epsilon kT\), \(\beta=e\vec{E}/2 kT\), and \(\theta\) is the angle between the electric field and the line joining an e-h pair. Further, \(r_{c}\) is known as the Onsager critical radius, determining a sphere with a hole at its centre. The spherical surface defined by \(r_{c}\) represents the separation distance at which an isolated e-h pair are considered likely to escape initial recombination. The average value of the thermalisation distance, \(r\), has been determined to be approximately 8 nm in SiO\({}_{2}\) by curve fitting to experimental data [71; 73]. As an incident particle travels along its track, more e-h pairs are created; although not true in reality, the model assumes a separation distance between stochastically generated consecutive e-h pairs large enough that neighbouring pairs do not interfere with each other. As a two-term model, with no representation of dose rate, the probability that an isolated e-h pair escape recombination depends upon the value of \(r\), and the strength of the electric field. This solution was originally presented for a range of low electric fields, whereas the MOS_kin_ is operated with an electric field higher than the linear region within which Equation (1) can be used for calculations [76]. However, it highlights a geometric perspective of MOS_kin_ detector response, as the electric field is fixed and only the electron beam fluence is varied in this investigation.