## 4 Multiphysics Analysis for SPDR with Semiconductor Samples  
SPDR have already been used for semiconductor (SC) characterization as in [9]; however, modelling of such measurement scenarios were always limited to bulk semiconductor parameters. In this section, we analyze such a scenario by taking semiconductor physics into account as well, so that the local conductivity changes due to charge movements are also considered. The charge transport in semiconductors can be modelled with the well-known drift-diffusion equations. These transport equations are usually solved together with Poisson equation to give a complete picture of the charges in semiconductor regions [17]:  
\[\
abla^{2}\varphi=-\frac{q}{\varepsilon}\Big{(}p-n+N_{d}^{+}-N_{a}^{-}\Big{)}, \tag{1}\]  
\[\
abla\cdot\mathbf{J_{p}}=-q\bigg{(}R+\frac{\partial p}{\partial t}\bigg{)}, \tag{2}\]  
\[\
abla\cdot\mathbf{J_{n}}=q\bigg{(}R+\frac{\partial n}{\partial t}\bigg{)}, \tag{3}\]  
\[\mathbf{J_{p}}=qp\mu_{p}(-\
abla\varphi)-qD_{p}\
abla p, \tag{4}\]  
\[\mathbf{J_{n}}=qn\mu_{n}(-\
abla\varphi)+qD_{n}\
abla n, \tag{5}\]  
where \(\varphi\) is the electric potential, \(n\) (\(p\)) is the electron (hole) concentration, \(R\) is the net generation-recombination rate of carriers, \(\mathbf{J_{n}}\) (\(\mathbf{J_{p}}\)) is the electron (hole) current density, \(N_{d}^{+}\) and \(N_{a}^{-}\) are the ionized donor and acceptor charge densities due to doping, \(\mu_{n}\) (\(\mu_{p}\)) is the electron (hole) mobility (1450 cm\({}^{2}\)(V\(\cdot\)s)\({}^{-1}\) and 500 cm\({}^{2}\)(V\(\cdot\)s)\({}^{-1}\), respectively), \(D_{n}\Big{(}D_{p}\Big{)}\) is the electron (hole) diffusion coefficient (37.49 cm\({}^{2}\)s\({}^{-1}\) and 12.93 cm\({}^{2}\)s\({}^{-1}\) respectively), \(\varepsilon\) is the electric permittivity of semiconductor material respectively, and \(q\) (= 1.602\(\cdot\)10\({}^{-19}\) C) is the elementary charge. Equation (1) is the Poisson equation for electric potential; (2) and (3) are the continuity equations for the charge carriers. This set of equations is supplemented by carrier transport Equations (4) and (5), which take both drift and diffusion of charges into account. For the net generation-recombination rate of the charges, the widely used Read-Hall-Shockley recombination mechanism (\(R=\frac{np}{\tau_{n}(p+n_{i})+\tau_{p}(n+n_{i})}\) ), where average life times for the carriers are taken as \(\tau_{n}=\tau_{p}=10\) us, and \(n_{i}=9\cdot\)10\({}^{9}\) cm\({}^{-3}\) is the intrinsic carrier concentration), is assumed. This set of Equations (1)-(5) is valid when the semiconductors are modelled themselves without external E-field (\(\mathbf{E_{ext}}\)) illumination, if they are also illuminated through external field, the transport equations take the following form:  
\[\mathbf{J_{n}}=qn\mu_{n}\mathbf{E_{ext}}+qn\mu_{n}(-\
abla\varphi)+qD_{n}\
abla n, \tag{6}\]  
\[\mathbf{J_{p}}=qp\mu_{p}\mathbf{E_{ext}}+qp\mu_{p}(-\
abla\varphi)-qD_{p}\
abla p. \tag{7}\]  
Hence, the transport is not only affected due to curl-free internal built-in potential, but also divergence free wave part of the E-field. In addition, the movement of the carriers (depletion and accumulation) creates position dependent conductivity as \(\sigma=qp\mu_{p}+qn\mu_{n}\), which will be taken into account by the electromagnetic solver, affecting the resonance. We initially considered the scenario where a homogeneously doped silicon wafer is inserted into our SPDR. For the fundamental mode that has been used to characterize materials in the previous Section, it is known that the external E field will have the form \(\mathbf{E_{ext}}=\mathbf{a_{\phi}}E_{\phi}(r,z)\), which will cause azimuthal closed loop current flow on the silicon sample. Therefore, there is not any local carrier distribution in the r-z plane of the axisymmetric structure, and no change in resonance frequency will be observed.