Full-wave modeling of broadband near field scanning microwave microscopy

A three-dimensional finite element numerical modeling for the scanning microwave microscopy (SMM) setup is applied to study the full-wave quantification of the local material properties of samples. The modeling takes into account the radiation and scattering losses of the nano-sized probe neglected in previous models based on low-frequency assumptions. The scanning techniques of approach curves and constant height are implemented. In addition, we conclude that the SMM has the potential for use as a broadband dielectric spectroscopy operating at higher frequencies up to THz. The results demonstrate the accuracy of previous models. We draw conclusions in light of the experimental results.


S1. Low-frequency breakdown problem
Using the full-wave model, the wave equation derived from Maxwell-equations is where r = r − j r and µ r = µ r − jµ r are the relative permeability and permittivity respectively, and J is the source current. Following the standard FEM procedure, the discretized equation becomes to a matrix equation and the matrix A is the summation of where S is the stiffness matrix, and T is the mass matrices and R is the conductivity -and boundary condition-related matrix. They are assembled by the elemental contributions and the right-hand side of (2) are assembled by where N is the normalized vector basis function for electric field and V e is the element volume in the simulation domain. Suppose the length of an element is l, the norm of ∇ × N is proportional to 1/l, and the norm of S e ij is the order of l and T e ij is the order of 10 −17 l 3 .
For a nanosized probe, to represent the mesh accurately, the minimum mesh size l can be as small as 1nm. Therefore, at microwave frequency band, ω 2 T e is about 10 15 times smaller than S e . The contribution of ω 2 T e is treated as zero in A e because of the round-off error in computing. As a result, the matrix A(ω) becomes singular, and the full wave FEM solution breaks down. Generally, for the smaller tip, the low-frequency breakdown problem is more pronounced, because the minimum mesh size would be smaller.
To overcome this low-frequency breakdown problem due to the finite machine precision, we find the inverse of the nearly ill-conditioned component of matrix A by transforming it from a frequency dependent problem to a frequency independent generalized eigenvalue problem. To be more specific, we first divide the FEM matrix A into where A ss represent the ill-conditioned submatrix, and it is associated with the region near the probe tip apex, A rr is the regular component in A. Correspondingly, the unknowns x in (2) are also divided into two categories: x S and x R . Here, A sr = A T rs represent the coupling between these two sets of unknowns.
According to the Schur-complement lemma, the inverse of A can be written as here we omit the ω for simplicity. For SMMs operating in non-contact mode, the tip apex is surrounded by air which is lossless, the frequency dependency of A ss can be written as In FEM, matrices S ss and T ss are real and symmetric. Solving the following generalized eigenvalue problem where λ is the eigenvalue and ν is the associated eigenvector, we have inverse of A ss (ω) In the right-hand side of (11), V 0 is the set of eigenvectors associated with zero eigenvalues, and represents the direct current (DC) modes near the tip apex area, and V h is the set of eigenvectors associates with non-zero eigenvalues Λ h . For SMM operating in contact mode where tip touches the sample surface or surrounded by lossy media, the inverse of A ss can be found in a similar way. Matrix B rr usually is non-singular, and its inverse can be found normally or using eigenvalue decomposition again for a frequency dependent inversion . The electrical field in the simulation domain then can be calculated using the solved unknowns E and basis functions N.

S2 Validation of the line port
To validate the correctness of the line port we used in this paper, we compare the complex impedance of the tip-sample interaction calculated by formula and their definitions (formula (1) and (2) in the paper). The value of dimensional parameters of the probe tip and simulation domain shown in Fig.1A of main text are given in Table.1 . The complex impedance of the probe tip 50nm above on bare substrate calculated by these two approaches agrees very well as shown in the Table.2. Here we use two kinds of basis functions in full-wave FEM: the linear basis functions and quadratic basis functions.
The quadratic basis functions (with 20 DOFs for a tetrahedron) result in a much larger number of DOFs in the matrix equation, yet it has a higher accuracy comparing to the linear basis functions (with 6 DOFs for a tetrahedron). Thus we recommend using quadratic basis functions in the full wave FEM for SMM simulation.   (38) is shown in Figure.1A . The absolute capacitances of the tip-sample interaction with and without the dispersive sample are shown in Figure.  The dimension of the silicon sample is 8µm × 8µm × 1.6µm, and this sample is placed 50nm under the probe tip apex as shown in Figure.2A. Other parameters including the geometry of probe tip and simulation domain size are given in Table.3. The real part of the relative permittivity of the high-purity silicon is almost a constant( r = 11.6) in the simulated frequency band, the conductivity of the sample is shown in Figure.2B. The total conductance of the SMM tip-sample interaction in this study includes three parts: the radiation, the sample dissipation loss and the probe dissipation loss. The impedance of these three parts are The radiation impedance, sample dissipation loss impedance and probe dissipation loss impedance from 1GHz to 100GHz are given Figure.3A-C. The inner field inside the PEC probe is zero, thus R probe is zero for PEC boundary simulation. The radiation and sample dissipation loss are almost the same for different conductive probes, while the probe dissipation losses are different because of the skin effect. The metallic probe is highly conductive, and wave impedance of metal and of the surrounding air are so different that the reflection coefficient is just less than unity. Therefore, most of the incident energy is reflected by the metallic probe, only a small fraction of it is absorbed. The conductivity of PEC is infinitely large and all incident energy is reflected. At 100GHz, the absorbed energy of titanium probe is only 0.8% (0.24/2.74) of the total energy loss, thus the field distribution near the metallic probe are almost identical to that near the PEC probe. In this study, the high-purity silicon is semi-insulator, the sample dissipation loss is minimum comparing to the radiation and probe dissipation loss as shown in this figure. This implies that it might be difficult to accurately characterize the local dissipation factor of high-purity silicon sample in SMM without considering the skin effects.