## 4 Multiphysics Analysis for SPDR with Semiconductor Samples
However, if we consider a higher order mode, which has electric field in the form of \(\mathbf{E_{ext}}=\mathbf{a_{r}}E_{r}(r,z)+\mathbf{a_{z}}E_{z}(r,z)\), it is expected to observe effects due to local conductivity changes that occur when the mode is present. In order to be able to couple electromagnetics solver with highly nonlinear semiconductor physics, we have employed time-domain (TD) FEM, as it offers flexible capabilities for multiphysics modelling. We have first developed a TD-FEM electromagnetic solver in which a point source and a few probing locations are defined. After exciting with a Gaussian burst time-domain signal centered at 5 GHz, and taking the Fourier Transform of the time-domain signals observed at different probing locations, we were able to distinguish the modes supported by the structure. The resonance frequencies have been determined by analyzing field spectrum at the probe locations, and in all the cases, these peaks in the spectrum were clearly distinguishable and the maxima points correspond to the resonance frequencies. Additionally, the spectrum obtained from the probe locations show clear frequency shifts with respect to the source spectrum, providing information regarding the resonant behavior of the structure. The first higher mode (named as TM016) with \(\phi\) independent \(E_{r}\) and \(E_{z}\) components, without any sample is observed at 5.33 GHz, and the FDTD solver also verifies the existence of the same mode at 5.31 GHz. Verification of our coupled electromagnetic (EM) and Poisson-Drift-Diffusion (PDD) algorithm and solver was performed with the aid of TD-FEM electromagnetic analysis performed for material samples characterized also with bulk conductivity values. Table 2 gathers the values of TM01\(\delta\) frequencies obtained from simulations conducted for different material sample parameters and solvers' configurations. Traditional semiconductor simulation tools employ the Poisson equation together with the transport equations for the modelling of semiconductor devices. For the high frequency excitation where the wavelength becomes as small as device dimensions, the Poisson equation is replaced with the Maxwell's equations, as in [18], to be computed with the transport equations. Replacing [PERSON]'s equation also means that the traditional potential-based boundary conditions would have to be changed, making it challenging to apply classical semiconductor boundary conditions, i.e., ohmic, Schottky, etc. In [15], another method is developed to simulate the semiconductor devices where the Poisson equation is also solved in addition to the wave equation. This method provides flexible semiconductor modeling, making it possible to use terminal potentials for the semiconductor boundary conditions (AC or DC excitation) together with external electromagnetic illumination. Additionally, this method ensures the full coupling of the divergence-free electric field that is the outcome of the wave equation and curl-free electric field component that is coming from the Poisson equation for the charges as in the Equations (5) and (6). Unlike the popular commercial tools where the drift diffusion equations only include the curl-free part of the electric field, the proposed method contribution of both electric field components (divergence-free and curl-free) are taken into account for the calculation of the semiconductor current equations. Consequently, the position dependent electrical conductivity