# Multi-Frequency Resonance Behaviour of a Si Fractal NEMS Resonator

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Device and Fabrication

_{2}layer. The gap between the hanged structure and the insulator is produced when the 2 μm thick top Si-layer has been etched. The middle SiO

_{2}-layer has thickness of 2 μm too. The structure is suspended over a 10 μm trench and has width of 12 μm. The nanobeams that form the device are 0.12 μm wide and 0.04 μm thick. The thickness of 0.04 μm is defined by the fabrication process and prescribes Young’s modulus of 76 GPa which is significantly lower than the value of the bulk material.

^{2}chips from a silicon on insulator wafer with 2 μm thick device layer oriented in the 100 crystallographic direction. Before the fabrication, each dice have been cleaned in an organic solvent (acetone) to remove potential organic contamination and rinsed in isopropanol. The samples have been directly patterned by focused ion beam (FIB) implantation of Ga+ at 30 keV, beam current of 10 pA and dose of $1\times {10}^{16}$ per cm

^{2}in a Cross-Beam system (1560xB from Zeiss, Oberkochen, Germany). The ion beam has been controlled with the nanolithography kit Elphy Quantum (from Raith, Dortmund, Germany) to enable the definition of different geometries [11]. It is remarkable to mention that this resist free lithographic step permits to modify the crystallinity of the irradiated silicon creating an etching mask that can be functional from the electrical point of view with the appropriate treatment [35]. According to SRIM (The Stopping and Range of Ions in Matter) simulations, the implanted and damaged volume in silicon goes up to 40nm in depth when ions goes perpendicular to the surface. This has been concluded from the TEM measurements after the ion implantation [35] and the SEM images after the silicon etching [36]. Working at the above mentioned conditions the thickness of the devices is fixed at 40 nm. Supporting pillars have been defined by ion milling and lateral ion implantation to sustain the fractal resonator, in a similar way as in [37] where a suspended lateral electrode was developed for the gatebility of suspended single charged transistors. Tetramethylammonium hydroxide at 80 degree Celsius and 25% concentration has been prepared to selectively etch the non-implanted volume of silicon. This approach permits to fabricate free suspended mechanical structures in only two steps [38]. The fractal pattern has been generated by an equation based home-written Python code which is able to produce wide variety of polygonal fractals while ensuring that the substructures of the resulted fractal shape do not overlap; see [28]. We decided for a structure that consist of $\left(\genfrac{}{}{0pt}{}{6}{2}\right)$-polygons where the self-similarity is up to second iteration [39] in the sense of IFS. Finally, in order to improve the electrical conductivity, we put through the devices under an annealing process at high temperature (up to 1000 °C) with boron in a nitrogen reach atmotmosphere to promote the recrystallization and doping (p-type) of the device [35,38]. This final step serves two important purposes for the electrical excitation of the devices to be achieved. It promotes the crystallisation of the amorphous material and it introduces electrical carriers through their doping.

## 3. Experimental Set-Up and Characterisation

#### 3.1. The Peaks at 2.37 MHz and 5.1 MHz

#### 3.2. The Peaks at 6.1 MHz and 7.1 MHz

#### 3.3. The Peaks at 9.8 MHz and 10.1 MHz

#### 3.4. The Peaks at 12.3 MHz and 13.05 MHz

#### 3.5. The Peaks at 13.54 MHz and 14.3 MHz

## 4. FEM Simulations

_{2}layer. The insulation layer has fixed boundary conditions as well. Also, we define the Electrostatic Interface where the excitation pad and the ground(the lower boundary of the SiO

_{2}layer) are specified. The next step is to mesh the domains. As the height of our device is relatively small in comparison with its length and width, we first use triangle mesh for the upper boundary of the structure which has been swiped down through the domains. When the mesh was evaluated, we need to define the different solvers. For the eigenfrequency study of the relaxed structure shown in the following subsection we use only the eigenfrequency solver. For the investigation of the natural frequencies of the compressed structure that computes the results of Section 4.2, a previous study that defines the bending must be run. The bending is computed by a Stationary study where the boundary conditions that define the side electrodes are displaced towards each other. Furthermore, DC voltage is applied and then gradually decreased to 0V so the structure is firstly attracted towards the bottom Si-layer and then left at its bent equilibrium. The resulted bent shape has its specific stresses that resemble the postfabrication stress, hence this numerical output is plugged again in the Eigenfrequency study where we compute the eigenfrequencies of the compressed structure. For both, Eigenfrequency and Stationary studies we use the standard solver settings of COMSOL. The eigenvale solver uses MUMPS direct solver while the stationary study uses combination of MUMPS and segregated solver where the mechanics and electrostatics are decoupled

#### 4.1. Symmetry Axes and Mode Shapes

#### 4.2. Mode Alternation Due to Compression Stress

^{3}are also kept the same as in the successful nanobeam model [40]. Finally, an eigenvalue study was performed where the temperature was set to 296 K. In Table 2 we compare the frequency measurements of the first five modes with the frequencies resulted from the numerical study and the first three of them fit very well. When the natural frequencies of the device before and after compression get compared (see Table 1 and Table 2), we detected that the standard order of appearance of the elliptical membrane modes is altered due to the stress of the device. Without compression we have even-(0,1), odd-(1,1), even-(1,1), even-(2,1) and odd-(2,1), and after compression we have even-(1,1), even-(0,1), odd-(1,1), odd-(2,1) and even-(2,1) modes.

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Extended Measurements

#### Appendix A.1. Resonance Peaks between 16 MHz and 24 MHz

**Figure A1.**Experimental frequency swipes centred at about 16.6 MHz for V${}_{pp}$ = 18 V. In panel (

**a**) the blue, cyan and green curves denote V${}_{off}$ = 0 V, V${}_{off}$ = 3 V and V${}_{off}$ = 5 V, respectively. In panel (

**b**) the blue curve denote swipe down for V${}_{pp}$ = 18 V, V${}_{off}$ = 0 V.

**Figure A2.**Experimental frequency swipes-up centred at about 17.4 MHz for V${}_{pp}$ = 18 V, where the blue, cyan and green curves denote V${}_{off}$ = 1 V, V${}_{off}$ = 3 V and V${}_{off}$ = 5 V, respectively.

**Figure A3.**Experimental frequency swipe-up centred at about 18.6 MHz for V${}_{pp}$ = 18 V, where the blue and cyan curves denote V${}_{off}$ = 1 V and V${}_{off}$ = 3 V, respectively.

**Figure A4.**Experimental frequency swipe up centred at about 19.6 MHz for V${}_{pp}$ = 18 V, the blue and cyan curves denote V${}_{off}$ = 0 V and V${}_{off}$ = 5 V, respectively.

**Figure A5.**Experimental frequency swipe-up centred at about 20.5 MHz for V${}_{pp}$ = 18 V, where the blue and cyan curves denote V${}_{off}$ = 0 V and V${}_{off}$ = 5 V, respectively.

**Figure A6.**Experimental frequency swipe-up centred at about 22.2 MHz for V${}_{pp}$ = 10 V, where the blue, cyan and green curves denote V${}_{off}$ = 1 V, V${}_{off}$ = 3 V and V${}_{off}$ = 5 V, respectively.

**Figure A7.**Experimental frequency swipe-up centred at about 23.6 MHz for V${}_{pp}$ = 18 V, where the blue, cyan and green curves denote V${}_{off}$ = 0 V, V${}_{off}$ = 3 V and V${}_{off}$ = 5 V, respectively.

#### Appendix A.2. Measurements in Air and Spectra of Other Devices with Defects

**Figure A8.**In the background (yellow) is the experimental frequency response spectrum as in Figure 2 where the frequency range was swiped-up in vacuum. The frequency response in air is added on the top, where the blue, cyan and green curves denote V${}_{DC}$ = 20 V, V${}_{DC}$ = 30 V and V${}_{DC}$ = 40 V, respectively. V${}_{pp}$ = 16 V between 3 MHz and 8 MHz and V${}_{pp}$ = 18 V between 9 MHz and 15 MHz. Resonant peaks were not detected between 8 MHz and 9 MHz.

#### Appendix A.3. Q Factors

**Table A1.**Q-factors computed using MATLAB ‘gaussian’ smoothed peaks detected in the range of 2 to 24 MHz.

Frequency | Q-Factor | Excitation V${}_{\mathit{pp}}$ & V${}_{\mathit{off}}$ Applied | MATLAB Smoothdata ‘Gaussian’ |
---|---|---|---|

2.378 MHz | 805.9849 | 18 V${}_{pp}$ 0 V${}_{off}$ | 30 |

5.154 MHz | 912.2355 | 20 V${}_{pp}$ 0 V${}_{off}$ | 5 |

5.154 MHz | 896.3706 | 10 V${}_{pp}$ 3 V${}_{off}$ | 10 |

5.154 MHz | 851.8230 | 10 V${}_{pp}$ 1 V${}_{off}$ | 5 |

5.154 MHz | 937.1147 | 10 V${}_{pp}$ 0 V${}_{off}$ | 5 |

6.03 MHz | 517.5697 | 4 V${}_{pp}$ 0 V${}_{off}$ | 40 |

7.121 MHz | 543.5703 | 10 V${}_{pp}$ 0 V${}_{off}$ | 20 |

9.247 MHz | 451.0302 | 10 V${}_{pp}$ 0 V${}_{off}$ | 20 |

10.1 MHz | 594.0296 | 10 V${}_{pp}$ 0 V${}_{off}$ | 30 |

11.2 MHz | 395.8694 | 2 V${}_{pp}$ 0 V${}_{off}$ | 40 |

11.21 MHz | 253.2587 | 4 V${}_{pp}$ 0 V${}_{off}$ | 40 |

12.3 MHz | 289.5149 | 10 V${}_{pp}$ 0 V${}_{off}$ | 40 |

12.3 MHz | 752.5982 | 18 V${}_{pp}$ 0 V${}_{off}$ | 20 |

13.07 MHz | 654.8430 | 10 V${}_{pp}$ 0 V${}_{off}$ | 30 |

13.07 MHz | 811.7344 | 14 V${}_{pp}$ 0 V${}_{off}$ | 20 |

13.07 MHz | 950.7295 | 18 V${}_{pp}$ 0 V${}_{off}$ | 20 |

13.08 MHz | 961.4365 | 20 V${}_{pp}$ 0 V${}_{off}$ | 20 |

13.45 MHz | 739.0293 | 10 V${}_{pp}$ 0 V${}_{off}$ | 30 |

14.29 MHz | 1226.6 | 12 V${}_{pp}$ 0 V${}_{off}$ | 30 |

16.68 MHz | 582.0690 | 10 V${}_{pp}$ 0 V${}_{off}$ | 20 |

16.68 MHz | 809.6446 | 18 V${}_{pp}$ 0 V${}_{off}$ | 20 |

17.54 MHz | 308.8481 | 10 V${}_{pp}$ 0 V${}_{off}$ | 30 |

18.63 MHz | 423.9597 | 18 V${}_{pp}$ 0 V${}_{off}$ | 20 |

18.63 MHz | 388.8721 | 18 V${}_{pp}$ 3 V${}_{off}$ | 20 |

20.37 MHz | 368.0044 | 18 V${}_{pp}$ 3 V${}_{off}$ | 20 |

20.37 MHz | 437.1467 | 18 V${}_{pp}$ 5 V${}_{off}$ | 20 |

**Figure A10.**Matlab approximation by ’smoothdata’ function with Gaussian filter followed by Q-factor derivation that result in 806 and 912 for the 2.378 MHz and 5.154 MHz peaks respectively.

**Figure A11.**Matlab approximation by ’smoothdata’ function with Gaussian filter followed by Q-factor derivation that result in 518 and 544 for the 6.03 MHz and 7.121 MHz peaks respectively.

## Appendix B. Extended Simulations

#### Appendix B.1. Poisson Ratio of the Structure

**Figure A12.**FEM simulation of the displacement of the structure when it is stretched along x-axis. The colours denote the y-component of the displacement where red is positive and blue negative displacement in micrometers, see the legend. Note that the center of the structure is at $x=0,y=0$.

#### Appendix B.2. Piezoresistance Simulations

**E**, and the current,

**J**, within a piezoresistor is: $\mathbf{E}=\rho \mathbf{J}+\Delta \rho \mathbf{J}$. where $\rho $ is the resistivity and $\Delta \rho $ is the induced change in the resistivity, and both are rank 2 tensors. The change in resistance is related to the stress, $\sigma $: $\Delta \rho =\Pi \sigma $ where $\Pi $ is the piezoresistance tensor (SI units: Ω m/Pa) [49]. Silicon has cubic symmetry, and as a result the $\Pi $ matrix can be described in the following manner:

**Figure A13.**The first three resonant modes of the device that has no initial compression (see Table 1, column 3) and the corresponding resistance peaks. The resistance changes are represented by the expression $R-\Delta R$ where R = 420,219.19 Ω is the resistance of the structure at rest and $\Delta $ R is the stress dependent resistance.

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**Figure 1.**SEM images of the fractal nano structure: (

**a**) 45 degree tilted SEM image of the suspended fractal resonator. (

**b**) Top-view SEM image of the suspended structure.

**Figure 2.**Experimentally measured frequency response spectrum produced by using electrostatic actuation and optical interferometric detection.

**Figure 3.**Experimental frequency swipes centred at about 2.37 MHz and 5.15 MHz. In panel (

**a**) the blue curve denotes swipe-up and the red curve denotes swipe-down. The excitation AC voltage is 18 V peak-to-peak (V${}_{pp}$ = 18 V) and 3 V offset (V${}_{off}$ = 3 V). In panel (

**b**) V${}_{pp}$ = 10 V where the blue, cyan and green curves denote V${}_{off}$ = 0 V, V${}_{off}$ = 1 V and V${}_{off}$ = 3 V, respectively.

**Figure 4.**Experimental frequency swipes centred at about 6.1 MHz and 7.1 MHz. In panel (

**a**), V${}_{pp}$ = 18 V, V${}_{off}$ = 0 V where, the blue curve denotes swipe-up and the red curve denotes swipe-down. In panel (

**b**) the blue and cyan curves denote V${}_{pp}$ = 18 V, V${}_{off}$ = 0 V and V${}_{pp}$ = 10 V, V${}_{off}$ = 3 V, respectively.

**Figure 5.**Experimental frequency swipes centred at about 9.8 MHz in panel (

**a**) and 10.1 MHz in panel (

**b**). In panel (

**a**) V${}_{off}$ = 0, where the blue curve denote swipe up for V${}_{pp}$ = 10 V and the cyan and red curves denote swipe up and down for V${}_{pp}$ = 18 V. In panel (

**b**) V${}_{off}$ = 0 V and the blue and cyan curves denote swipe-up for V${}_{pp}$ = 10 V and V${}_{pp}$ = 20 V, respectively. The peak at about 10.1 MHz appears at twice the frequency of the peak shown in Figure 3b.

**Figure 6.**Experimental frequency swipe up centred at about 12.3 MHz and 13.05 MHz. In panel (

**a**) V${}_{pp}$ = 18 V. The blue and cyan curves denote V${}_{off}$ = 0 V and V${}_{off}$ = 3 V, respectively. In panel (

**b**) V${}_{off}$ = 0 V. The the blue, cyan, green and yellow curves denote: V${}_{pp}$ = 10 V, V${}_{pp}$ = 14 V, V${}_{pp}$ = 18 V, V${}_{pp}$ = 20 V. The peak at about 12.3 MHz appears at twice the frequency of the peak shown in Figure 4a.

**Figure 7.**Experimental frequency swipes centred at about 13.45 MHz and 14.3 MHz. In panel (

**a**) V${}_{off}$ = 0 V. The blue curve denote V${}_{pp}$ = 10 V while the cyan together with the red curve are the swipe-up and -down for V${}_{pp}$ = 18 V. In panel (

**b**) the blue curve denotes swipe up for V${}_{off}$ = 3 V and V${}_{pp}$ = 10 V. Cyan and red curves denote swipe-up and -down for V${}_{off}$ = 5 V and V${}_{pp}$ = 10 V. The peak at about 14.3 MHz appears at twice the frequency of the peak shown in Figure 4b.

**Figure 8.**The first five resonant modes of vibration and their corresponding frequencies for the compressed fractal structure. Red colour denotes maximum displacements while + and − signs denote the positive and negative amplitudes along z-axis (orthogonal to the xy-plane). In Table 1 and Table 2 the frequencies at which the modes appear for the uncompressed and compressed structures are shown.

**Table 1.**The first five resonant frequencies for the cases where AD = BE = CF amd AD = BE ≠ CF. The different mode shapes can be seen in Figure 8.

Sections | AD = BE = CF | AD = BE ≠ CF |

Compression | 0 nm | 0 nm |

Maximum Bending Displacement | 0 μm | 0 μm |

even-(0,1) | 3.176 MHz | 3.837 MHz |

odd-(1,1) | 6.577 MHz | 7.428 MHz |

even-(1,1) | 6.577 MHz | 8.422 MHz |

even-(2,1) | 10.654 MHz | 12.106 MHz |

odd-(2,1) | 10.654 MHz | 13.004 MHz |

**Table 2.**Table of the first five resonant modes together with their frequencies for compressed beam, Columns 2 and 3, and the compressed fractal structure, Columns 5 and 6. The last row shows the fitted material properties of the suspended structures.

Nanobeam | Fractal resonator | ||||

Compression | 8.6 nm | Compression | 4 nm | ||

Maximum Bending Displacement | 114.6 nm | Maximum Bending Displacement | 82 nm | ||

Temperature | 298 K | Temperature | 296 K | ||

Mode | Experimental | Numerical | Mode | Experimental | Numerical |

second | 26.2 MHz | 25.97 MHz | even-(1,1) | 2.4 MHz | 2.783 MHz |

first | 30.9 MHz | 31.255 MHz | even-(0,1) | 5.15 MHz | 5.107 MHz |

torsional | 71.8 MHz | 71.826 MHz | odd-(1,1) | 6.1 MHz | 6.06 MHz |

third | 76.9 MHz | 76.322 MHz | odd-(2,1) | 7.15 MHz | 8.043 MHz |

fourth | 96 MHz | 106.48 MHz | even-(2,1) | 9.22 MHz | 8.5151 MHz |

Material Properties | Young modulus = 76 GPa; Poisson ratio = 0.3; $\rho $ = 2328 kg/m^{3} |

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**MDPI and ACS Style**

Tzanov, V.; Llobet, J.; Torres, F.; Perez-Murano, F.; Barniol, N.
Multi-Frequency Resonance Behaviour of a Si Fractal NEMS Resonator. *Nanomaterials* **2020**, *10*, 811.
https://doi.org/10.3390/nano10040811

**AMA Style**

Tzanov V, Llobet J, Torres F, Perez-Murano F, Barniol N.
Multi-Frequency Resonance Behaviour of a Si Fractal NEMS Resonator. *Nanomaterials*. 2020; 10(4):811.
https://doi.org/10.3390/nano10040811

**Chicago/Turabian Style**

Tzanov, Vassil, Jordi Llobet, Francesc Torres, Francesc Perez-Murano, and Nuria Barniol.
2020. "Multi-Frequency Resonance Behaviour of a Si Fractal NEMS Resonator" *Nanomaterials* 10, no. 4: 811.
https://doi.org/10.3390/nano10040811