# Advanced Modelling Techniques for Resonator Based Dielectric and Semiconductor Materials Characterization

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## Abstract

**:**

## Featured Application

**Non-destructive dielectric and semiconductor material characterization using dielectric resonator.**

## Abstract

## 1. Introduction

## 2. Eigenvalue Analysis of Dielectric Resonator

## 3. Results for SPDR with Dielectric Samples

## 4. Multiphysics Analysis for SPDR with Semiconductor Samples

^{15}cm

^{−3}and 10

^{16}cm

^{−3}, is presented. As the doping of the semiconductor increases, its electrical conductivity also increases, and the sample starts to show behavior that is more metallic. For the highly conductive samples, it is observed that the EM-PDD coupled solver predicts resonant frequencies very close to the ones that are obtained by the EM solver with the bulk parameters. For shallow doping, however, the nonlinear effects of the semiconductor domain might shift the resonance slightly. This change can also be observed in Figure 4b for n-type doping of 10

^{15}cm

^{−}

^{3}.

## 5. Discussion of the Modelling of SPDR Loaded with Semiconductor Samples

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Lee, J.H.; Hyun, S.; Char, K. Quantitative analysis of scanning microwave microscopy on dielectric thin film by finite element calculation. Rev. Sci. Instrum.
**2001**, 72, 1425–1434. [Google Scholar] [CrossRef] - Lai, K.; Kundhikanjana, W.; Kelly, M.; Shen, Z.X. Modelling and characterization of a cantilever-based near-field scanning microwave impedance microscope. Rev. Sci. Instrum.
**2008**, 79, 063703. [Google Scholar] [CrossRef] [PubMed][Green Version] - Wei, T.; Xiang, X.D.; Wallace-Freedman, W.G.; Schultz, P.G. Scanning tip microwave near-field microscope. Appl. Phys. Lett.
**1996**, 68, 3506–3508. [Google Scholar] [CrossRef] - Hoffmann, J.; Gramse, G.; Niegemann, J.; Zeier, M.; Kienberger, F. Measuring low loss dielectric substrates with scanning probe microscopes. Appl. Phys. Lett.
**2014**, 105, 013102. [Google Scholar] [CrossRef] - Gungor, A.C.; Celuch, M.; Smajic, J.; Olszewska-Placha, M.; Leuthold, J. Flexible electromagnetic modelling of SMM setups with FE and FDTD methods. In Proceedings of the IEEE MTT-S International Conference on Numerical Electromagnetic and Multiphysics Modelling and Optimization (NEMO), Boston, MA, USA, 29–31 May 2019; pp. 1–4. [Google Scholar]
- Moertelmaier, M.; Huber, H.P.; Rankl, C.; Kienberger, F. Continuous capacitance–voltage spectroscopy mapping for scanning microwave microscopy. Ultramicroscopy
**2014**, 136, 67–72. [Google Scholar] [CrossRef] [PubMed] - Krupka, J. Frequency domain complex permittivity measurements at microwave frequencies. Meas. Sci. Technol.
**2006**, 17, R55–R70. [Google Scholar] [CrossRef] - Krupka, J. Contactless methods of conductivity and sheet resistance measurement for semiconductors, conductors and superconductors. Meas. Sci. Technol.
**2013**, 24, 062001. [Google Scholar] [CrossRef] - Krupka, J.; Mazierska, J. Contactless measurements of resistivity of semiconductor wafers employing single-post and split-post dielectric-resonator techniques. IEEE Trans. Instrum. Meas.
**2007**, 56, 1839–1844. [Google Scholar] [CrossRef] - Blackburn, J. Solving the double eigenvalue problem: A study of mode matching in arbitrary-layer dielectric resonators. IEE Proc. Microw.
**2006**, 153, 447–455. [Google Scholar] [CrossRef] - H2020 MMAMA Project Reports. Available online: www.mmama.eu (accessed on 28 September 2020).
- Celuch, M.; Gwarek, W.; Wieckowski, A. Enhanced-resolution material imaging with dielectric resonators: A new implicit space-domain technique. In Proceedings of the IEEE MTT-S International Microwave Symposium (IMS), Boston, MA, USA, 2–7 June 2019; pp. 55–58. [Google Scholar]
- QWED Split Post Dielectric Resonators (SPDR). Available online: Qwed.eu/resonators_spdr.html (accessed on 28 September 2020).
- European Standard: IEC 61189-2-721:2015. Available online: https://webstore.iec.ch/publication/22343 (accessed on 28 November 2020).
- Gungor, A.; Celuch, M.; Smajic, J.; Olszewska-Placha, M.; Leuthold, J. Electromagnetic and semiconductor modelling of scanning microwave microscopy setups. IEEE J. Multiscale Multiphys. Comput. Tech.
**2020**, 5, 209–216. [Google Scholar] [CrossRef] - Jin, J.M. The Finite Element Method in Electromagnetics, 3rd ed.; John Wiley & Sons: Hoboken, NJ, USA, 2015. [Google Scholar]
- Sze, S.M.; Ng, K.K. Physics of Semiconductor Devices; John Wiley & Sons: Hoboken, NJ, USA, 2006. [Google Scholar]
- Grondin, R.O.; El-Ghazaly, S.M.; Goodnick, S. A review of global modeling of charge transport in semiconductors and full-wave electromagnetics. IEEE Trans. Microw. Theory Tech.
**1999**, 47, 817–829. [Google Scholar] [CrossRef]

**Figure 1.**Visualization of the considered SPDR: (

**a**) in 3D; (

**b**) materials used and geometry with details; (

**c**) 2D representation using axial symmetry.

**Figure 2.**Normalized ${E}_{\varphi}$ component of the fundamental mode of the considered SPDR without any sample (air) is obtained from FD-FEM solver.

**Figure 3.**Resonant frequency shifts with respect to the case without inserting a sample for FEM and FDTD simulations for the samples (

**a**) with 0.5 mm thickness, (

**b**) with 1.0 mm thickness, and (

**c**) with 1.5 mm thickness. (

**d**) Resonant frequency shifts for the quartz sample; (

**e**): resonant frequency shifts for the sapphire sample.

**Figure 4.**(

**a**) Source and probe locations shown on the axisymmetric representation of the resonator; (

**b**) spectrum obtained through EM solver and EM-PDD coupled solvers for a Si sample with n-type doping 10

^{15}cm

^{−3}; (

**c**) spectrum obtained through EM solver and EM-PDD coupled solvers for a Si sample with n-type doping of 10

^{16}cm

^{−3.}

**Table 1.**Resonant frequency of the fundamental mode of SPDR loaded with dielectric samples varying in thickness and permittivity: simulations and measurements performed by the authors.

$\mathbf{Thickness}{\mathit{t}}_{\mathit{s}\mathit{a}\mathit{m}\mathit{p}\mathit{l}\mathit{e}}\left[\mathbf{mm}\right]$ | $\mathbf{Permittivity}{\mathit{\epsilon}}_{\mathit{s}\mathit{a}\mathit{m}\mathit{p}\mathit{l}\mathit{e}}$ | Fundamental Mode Frequency [GHz] | |||
---|---|---|---|---|---|

FD-FEM | FDTD | Measurement 1 | Measurement 2 | ||

No sample | 1.0 | 4.9502 | 4.9466 | 5.1588 | 5.1117 |

0.5 | 2.0 | 4.9384 | 4.9347 | X | X |

0.5 | 5.0 | 4.9027 | 4.8990 | X | X |

0.5 | 10.0 | 4.8434 | 4.8396 | X | X |

1.0 | 2.0 | 4.9268 | 4.9232 | X | X |

1.0 | 5.0 | 4.8567 | 4.8529 | X | X |

1.0 | 10.0 | 4.7411 | 4.7371 | X | X |

1.5 | 2.0 | 4.9151 | 4.9114 | X | X |

1.5 | 5.0 | 4.8108 | 4.8069 | X | X |

1.5 | 10.0 | 4.6415 | 4.6373 | X | X |

0.542-deformed | 3.82 | 4.9148 | 4.9111 | X | X |

0.542 | 3.82 | 4.9140 | 4.9103 | 5.1223 | 5.0770 |

0.475 | 9.40 | 4.8554 | 4.8516 | 5.0629 | 5.0200 |

**Table 2.**Resonant frequencies of SPDR for TM01$\delta $ mode for different sample types and simulated by different solver configurations.

$\mathbf{Permittivity}{\mathit{\epsilon}}_{\mathit{s}\mathit{a}\mathit{m}\mathit{p}\mathit{l}\mathit{e}}$ | $\mathbf{Thickness}{\mathit{t}}_{\mathit{s}\mathit{a}\mathit{m}\mathit{p}\mathit{l}\mathit{e}}\left[\mathbf{mm}\right]$ | Bulk Conductivity [S/m] | n-Type Doping [cm^{−3}] | Used TD-FEM Solver | $\mathbf{TM}01\mathit{\delta}$ Frequency [GHz] |
---|---|---|---|---|---|

1.0 | 0.5 | 0.0 | 0.0 | EM | 5.33 |

11.7 | 0.5 | 0.0 | 0.0 | EM | 5.10 |

11.7 | 0.5 | 23.2316 | (corresponds to 10^{15}) | EM | 5.05 |

11.7 | 0.5 | 232.316 | (corresponds to 10^{16}) | EM | 5.03 |

11.7 | 0.5 | bulk value not used | 10^{15} | EM-PDD | 5.06 |

11.7 | 0.5 | bulk value not used | 10^{16} | EM-PDD | 5.03 |

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**MDPI and ACS Style**

Gungor, A.C.; Olszewska-Placha, M.; Celuch, M.; Smajic, J.; Leuthold, J. Advanced Modelling Techniques for Resonator Based Dielectric and Semiconductor Materials Characterization. *Appl. Sci.* **2020**, *10*, 8533.
https://doi.org/10.3390/app10238533

**AMA Style**

Gungor AC, Olszewska-Placha M, Celuch M, Smajic J, Leuthold J. Advanced Modelling Techniques for Resonator Based Dielectric and Semiconductor Materials Characterization. *Applied Sciences*. 2020; 10(23):8533.
https://doi.org/10.3390/app10238533

**Chicago/Turabian Style**

Gungor, Arif Can, Marzena Olszewska-Placha, Malgorzata Celuch, Jasmin Smajic, and Juerg Leuthold. 2020. "Advanced Modelling Techniques for Resonator Based Dielectric and Semiconductor Materials Characterization" *Applied Sciences* 10, no. 23: 8533.
https://doi.org/10.3390/app10238533