FEM analysis of piezoelectric film as IDT on the diamond substrate to enhance the quality factor of SAW devices

Journal Pre-proof FEM analysis of piezoelectric film as IDT on the diamond substrate to enhance the quality factor of SAW devices

Bowei Xie, Fazhu Ding, Zebin Dong, Hongjing Shang, Daxing Huang, Hongwei Gu PII:

S0925-9635(19)30394-2

DOI:

https://doi.org/10.1016/j.diamond.2019.107659

Reference:

DIAMAT 107659

To appear in:

Diamond & Related Materials

Received date:

12 June 2019

Revised date:

6 December 2019

Accepted date:

8 December 2019

Please cite this article as: B. Xie, F. Ding, Z. Dong, et al., FEM analysis of piezoelectric film as IDT on the diamond substrate to enhance the quality factor of SAW devices, Diamond & Related Materials (2019), https://doi.org/10.1016/j.diamond.2019.107659

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© 2019 Published by Elsevier.

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FEM analysis of piezoelectric film as IDT on the diamond substrate to enhance the quality factor of SAW devices Bowei Xiea,b,c, Fazhu Ding a,b,c *, Zebin Donga,b,c, Hongjing Shanga,b,c, Daxing Huanga,b,c, Hongwei Gu a,b,c * a

Key Laboratory of Applied Superconductivity, Chinese Academy of Sciences, Beijing 100190, China

b

Institute of Electrical Engineering, Chinese Academy of Sciences, Beijing 100190, China

c

University of Chinese Academy of Sciences, Beijing 100049, China

* Corresponding author:

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E-mail address: [email protected] ( Hongwei Gu), [email protected] (Fazhu Ding).

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Keywords: Surface Acoustic Wave, AlN, Diamond, FEM

Abstract

This paper describes an innovative electro-acoustic coupling structure with a Patterned Piezoelectric film as

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Inter-Digital Transducer (PP-IDT) on the diamond substrate for Surface Acoustic Wave (SAW) devices of better quality factor and frequency resolution. Theoretical analysis and Finite Element Method (FEM) are employed to

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investigate the dependence of microwave impedance characters on the thickness of piezoelectric and metal electrode layers. The systematic studies show that the PP-IDT structure can excite the mechanical vibration on the

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top surface of diamond substrate and derive the SAW. The quality factor increases remarkably in excess to104 since the SAW is propagating below the diamond surface with little interface loss and transmission loss.

1. Introduction

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Surface Acoustic Wave (SAW) devices have been widely used in microwave communication, internet of things (IoT), environmental and biological sensor [1-3]. Despite massively reported device categories, there are mainly three kinds of chip materials for the electro-mechanical energy conversion and vice versa: piezoelectric single-crystals, ceramics and films. Among them, piezoelectric films attract extensive attention due to their compatibility with integrated circuits (IC) fabrication processes and easiness for system-on-chip (SoC) integration. Inter-Digital Transducer (IDT) structures of topologically patterned metal electrode lines are commonly employed to excite the SAW of corresponding frequency on the top surface of piezoelectric film wherein SAW propagates in lateral direction. Whereas, it’s hard to shrink the metal line width following the Moore’s law to increase the working frequency up to 2GHz.The difficulties lie in high electrode resistance caused high devices loss, low quality factor and power durability. AlN material is a very promising candidate to overcome above difficulties due to its high acoustic phase velocity, less velocity dispersion, especially in combination with diamond substrate to achieve the highest phase velocity and working frequency [4-10]. Also, the breakthrough in film deposition technology has enabled the application of AlN [3,11-14] and diamond materials [15-17] in SAW devices. In a diamond based SAW structure, a diamond thick layer（> 4λ, λ stands for the wavelength of SAW and is determined by the metal IDT pitch p） below AlN thin

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layer （< 0.5λ） is introduced to promote the SAW close to the highest acoustic phase velocity of diamond material. Due to the acoustic scattering loss at AlN-diamond interface and partial SAW propagating in the piezoelectric layer, the device insert loss is normally larger than 20dB [10, 18-19]. To reduce insert loss and enhance the frequency resolution is continuous to be a challenging problem of great importance and potential. Endeavors on this topic were made, including lamb mode research of membrane type SAW devices [6, 20], waveguide layer acoustic wave (WLAW) energy capturing structure [21-23], devices properties research at low-temperature [24], grooved [25] and suspending IDT metal electrode [26], etc. However, all of them are complex in structure, complicated in process and high in cost. The diamond based SAW structure provides a structural possibility for etching AlN layer. The diamond substrate can act as the etching barrier layer, which SAW propagates through and supports IDTs above. Given to the patterned piezoelectric film as Inter-Digital Transducer (PP-IDT) structure, both the metal electrode and piezoelectric layer are etched simultaneously with the same graph in the active area. Patterned piezoelectric and electrode layers together serve the purpose of microwave-electro-mechanical signal conversion as an energy transducer to the underlying diamond layer where SAW of a corresponding frequency is excited. The lateral SAW propagates in the diamond layer, with little transmission loss and interface loss. Consequently, a unit pair of acoustic transducers based on the PP-IDT structure will have small acoustic loss and high Q value, which will be beneficial of subsequent devices. This structure is of particular interest for the design of SAW devices such as high-frequency filters and resonators with low insert loss and narrow band, and environmental or biological sensor with high accuracy. Furthermore, the design is totally compatible with the general SAW technology and process as well. To investigate SAW characters of the PP-IDT structure, the Finite Element Method (FEM) has been adopted to calculate the electrical potential and particle displacement distribution under the resonance condition, and analyze frequency response. Then, a systematic evaluation has been conducted on how the thicknesses of piezoelectric layer and electrode to affect the frequency response. Ideal results of the quality factor over 104 at working frequency about 5GHz are obtained.

2. Theory and Design 2.1. Models design In order to simplify the calculation, the 2-dimensional (2-D) numerical dispersion relation for the alternating current（AC）steady state analysis is employed. Consequently, no variation is permitted in the direction perpendicular to the plane and all the derivatives are equal to zero. The general partial differential equations of quasi-static equation for the modeling of piezoelectric devices are Newton's law, Gauss's law, Maxwell equations and the constitutive relations. They are hard to be solved mathematically, seen in below equations (1) and (2), where C is the elasticity matrix (N/m2), u is the particle displacement, e is the piezoelectric matrix (C/m2), ε

Journal Pre-proof is the permittivity matrix (F/m), φ is the electric field vector (V/m). More details can be referred to [27-29]. The FEM (software Comsol Multiphysics®) provides an effective numerical solution by defining appropriate boundary conditions, material parameters and geometric models in advance. ∑

∑

(1)

∑

∑

( )

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Unit model of PP-IDT structure is illustrated in fig. 1, which is constituted of a pair of Al/AlN IDTs, diamond substrate and diamond absorption layer. Metallization ratio η=50% and IDT pitch p=1μm are constant in this research, resulting in the constant IDT width a and gap b, a=b=0.5μm. AlN thickness and Al electrode thickness are varying as hAlN and he. Diamond layer thickness is 3μm, khdiamond= 9.42 (k=2л/λ, λ=2p, hdiamond is the thickness of diamond layer). As per reference [7], the acoustic wave leakage into bulk which causes the wave reflection from backside of substrate can be neglected if the normalized substrate thickness kh is more than 4. Also, according to reference [9], additional diamond layer of 1μm thickness is attached at the bottom as the absorption layer, and Comsol software offers this function with perfect matching layer (PML) to simulate the infinitely extending materials. Owing to these two design methods, the influences of the substrate thickness, backside reflection are neglected in this study.

Fig. 1. 2D model of PP-IDT structure unit.

2.2. Simulation setting The mesh is divided as finer level using the default setting of FEM software. Non-piezoelectric diamond and aluminum materials are assumed as linear materials. The acoustic displacements, mechanical stresses and electric potential, electric

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displacement are continuous at the interfaces. Top surface is mechanically free, while the bottom is fixed. Mechanical and electrical periodic conditions are applied on both side of every model unit structure to simulate infinitely repeating units laterally. This function is offered by Comsol software. Since in 2-D simulation a line can be assigned as the electrical terminal, metal electrode thickness evaluation could be studied by varying the top electrode thickness, and also in order to simplify the model structure and computation, ground layer is idealized as a line at the bottom of AlN layer. Even though this simplification, ground terminal is important to confine the electrical field in piezoelectric film to excite the bulk acoustic wave (BAW) effectively. It also helps having a high electromechanical coupling coefficient K2 [6]. Following these criteria, the detail boundary condition settings of PP-IDT structure are shown in table1.

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Table1. Mechanical and electrical boundary conditions. Boundary condition

Setting

Bottom of PML

Mechanical free

Top surface

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Mechanical fixed

Mechanical Periodic

Bottom of the right Al electrode

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Terminal1(+1V)

Side of Diamond layer

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condition

Bottom of the left

Terminal2(-1V)

Al electrode Bottom of AlN layer

Electrical Periodic condition

Side of Diamond layer

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Ground

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The software default material parameters used in the simulation are listed in table 2.

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Table2. Material parameters in simulation.

Item

Unit

AlN

Diamond

Al

Density

kg/m³

3300

3515

2700

/

/

/

/

9

5.1

/

/

105

7

/

0.1

0.35

Elasticity matrix

(ordering: xx, yy,

1011Pa

zz, yz, xz, xy)

{4.1, 1.49, 4.1, 0.99, 0.99, 3.89, 0, 0, 0, 1.25, 0, 0, 0, 0, 1.25, 0, 0, 0, 0, 0, 1.305} {0, 0, -0.58, 0, 0, -0.58, 0, 0,

Coupling matrix

C/m²

1.55, 0, -0.48, 0, -0.48, 0, 0, 0, 0, 0}

Relative permittivity Young's modulus Poisson's ratio

/ 1010Pa

/

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3. Results and Discussions

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3.1. Basic theory Similar with the classic SAW theory, for a single PP-IDT structure unit, microwave signal of a same frequency but opposite phase, is applied on each IDT pairs. It is converted via the piezoelectric effect to mechanical BAW vibration which propagates longitudinally along the thickness direction. IDT pairs can be idealized as line-vibration transducers that vibrate at the same frequency but opposite phase position (solid and dash arrow in patterned AlN illustrated in fig. 2). Refer to Hashimoto’s description in [30], the underlying diamond is excited to have the SV and L type BAW, whose components excited close to the surface couple with each other through the boundary condition to generate SAW (SV, L type BAW in diamond and excited SAW at surface shown in fig.2). As to the multiple IDT pairs, according to the wave interference principle, only when the IDT pitch (p) equals to integral multiples of λ/2, SAW shows in-phase superimposition and produces the strongest amplitude.

Fig. 2. A schematic of PP-IDT structure.

Particularly for the PP-IDT structure, it's necessary to maintain opposite vibration phase at the bottom of each IDT of an Al/AlN pair. From this reason, a certain resonance condition must be satisfied inside the composite piezoelectric and electrode layers, namely the thickness of piezoelectric and electrode layers must be designed carefully. Therefore, the acoustic characteristics of PP-IDT structure, such as phase velocity, resonating frequency, transmission loss, and electromechanical coupling coefficient K2, are related to the normalized piezoelectric layer thickness khAlN and electrode thickness he, k and λ is the same as defined in section 2.1. Because of relatively low resonant frequency of fundamental Rayleigh mode, this research mainly focuses on the first-order resonance of Sezawa mode [9, 20]. Phase velocity vp is calculated by vp=f*λ (f stands for the resonating or anti-resonating frequency), and electromechanical coupling coefficient K2 is calculated by K2=2*(var-vr)/vr (var and vr stand for the phase velocity at anti-resonating frequency and resonating frequency respectively).

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Sezawa mode

Rayleigh mode

Diamond BAW

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3.2. Modeling results of typical structure FEM of the representative PP-IDT structure model with hAlN=0.25μm and he =0.25μm generates a microwave response curve shown in fig. 3 (a) and (b). The resonance frequency of Sezawa mode is at 5290.2MHz, corresponding to a phase velocity of 10580.4m/s (close to the lateral BAW velocity of diamond material). Electromechanical coupling coefficient K2 is 0.2%. Quality factor shows an evident peak between the resonant and anti-resonant frequencies in exceed to 104, while is small for other resonating modes of Rayleigh SAW and diamond BAW. Fig. 3 (c) and (d) show the response curves of conventional diamond based layered AlN SAW structure with same material parameters, boundary conditions and structure dimensions, except AlN layer is not patterned. It is apparent that the quality factor is much smaller than the one of PP-IDT structure. All other model couples (AlN layer patterned or not) with different piezoelectric and electrode thickness show the same relation, which are not shown one by one.

(a)

(b)

Sezawa mode

Rayleigh mode

(c)

Diamond BAW

(d)

Fig. 3. Microwave response of hAlN-0.25μm / he-0.25μm (a, PP-IDT admittance vs freq.; b, PP-IDT quality factor vs freq.; c, layered AlN admittance vs freq. ; d, layered AlN quality factor vs freq.).

Journal Pre-proof Compared to the FEM result of conventional diamond based layered AlN SAW structure reported in [31], quality factor is significantly enhanced for PP-IDT structure, but the electromechanical coupling coefficient K2 is decreased. As expected, high quality factor results from SAW propagating in diamond which incurs less acoustic energy loss than that of AlN and little interface transmission loss as well. The low electromechanical coupling coefficient K2 could be explained by the IDT equivalent circuit equation: ( )

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, where Nt represents the IDT numbers. As shown in fig. 1 and table 1, mechanical and electrical periodic conditions are applied on both sides of every model unit structure to simulate infinitely units laterally, so Nt could be considered as an infinite constant. K2 is inversely proportional to quality factor. Quality factor and coupling coefficient have to be balanced during devices design. As plotted in fig. 4, particle displacement at eigen frequency of Sezawa mode in (b) exhibits a stratified profile. Electrode layer, AlN layer and Diamond substrate show different deforming tendencies and vibration phases. Whereas the one of Rayleigh mode in (a) is a typical SAW displacement in a semi-infinite solid, different layers showing same deforming tendencies and vibration phases. BAW resonance in (c) occurring inside of diamond substrate is located around 5.8GHz, which is observed around this frequency in all other models with different thickness of AlN or electrode. Particle displacement legend is in (d).

(a)

(b)

(c)

(d)

Fig. 4. Mechanical displacement at resonating frequency (a, Rayleigh fundamental mode; b, Sezawa mode; c, diamond BAW mode; d, particle displacement legend).

As defined in table 1, positive signal +1V (red color) is applied on the right electrode terminal of unit model, and -1V (blue color) is on the left side. Fig. 5 exhibits the electrical potential distribution inside the piezoelectric layer around the eigen frequency of Sezawa mode. The electrical potential between Sezawa resonance

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Pr

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and anti-resonance frequencies exhibits the inversed phase (red color on the left side and blue color on the right side) to the applied electric field in (b), which shows the pure inductance characteristics. The ones below Sezawa resonance frequency in (a) and over anti-resonance frequency in (c) present the same potential phase (red color on the right side and blue color on the left side) to applied electric field, showing the pure capacitance characteristics. Electrical potential legend is in (d). This result is in consistent with the BAW theory described in [32]. Piezoelectric bulk shows pure inductive character with the impedance phase of π/2 in the resonance state, and pure capacitive character with the impedance phase of -π/2 in the dis-resonance state. So the patterned Al/AlN pair can be considered as the BAW vibration source to the underlying diamond substrate.

(a)

(b)

(c)

(d)

Fig. 5. Electrical potential between and out of resonating frequencies

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(a, potential distribution below resonating frequency; b, potential distribution between resonating frequencies; c,

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potential distribution over resonating frequency; d, electrical potential legend).

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3.3. Modeling results with the piezoelectric layer thickness fixed (002) oriented AlN film is required for SAW devices to obtain high electromechanical coupling coefficient K2. In order to get higher working frequency and quality factor, piezoelectric film thickness should be as thin as possible to take the most advantage of the AlN/Diamond structure to enhance the penetration of the SAW into the diamond substrate with high propagation velocity and low propagation loss. But on the other hand, ultra-thin AlN film will hinder the (002) oriented film growth and crystal quality, showing much high full width at half maximum (FWHM) of rocking curve corresponding to the (0002)-peak [5]. This is why the quality factor of conventional diamond-based layered AlN SAW devices can’t be improved by reducing the AlN layer to ultra-thin film. Also considering thinner film makes it easier to etch the AlN pattern profile for nanolithography process with small height to width aspect ratio, so both piezoelectric and electrode layers should not be designed too thin or thick. A systematic modeling has been performed to investigate the dependence of microwave impedance characteristics on the films thickness of piezoelectric and electrode layers varying

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from 0.05 to 0.75μm and 0.2 to 0.5μm respectively. In addition, the quality factor peaks of Sezawa mode out of this thickness range are hardly observed as well. For unit models with 0.25μm AlN layer thickness fixed, the resonant frequency and corresponding phase velocity of Sezawa mode drop with increasing electrode thickness (plotted in fig. 6). It is similar with the conventional diamond based layered AlN SAW devices that metal electrode thickness has impacts on phase velocity [31]. On one way, it could be interpreted and calculated using the coupling of modes (COM) theory described in [33], which is complicated taking the IDT second order effects like reemission, electrode resistivity, velocity dispersion, reflection, propagation loss into account. On the other way, from the PP-IDT structure perspective, according to section 3.1, the patterned piezoelectric and electrode layers must resonate at the bottom. So the electrode thickness definitely has significant impact on BAW vibrations longitudinally. It can be explained as composite piezoelectric layers by classic BAW theory: the electrode layer also exists as an acoustic path to extend the equivalent acoustic path and reduce resonant frequency. The electromechanical coupling coefficient K2 significantly decreases with increasing electrode thickness as well. 0.45

vp

Pr

11200 11000 10800

2

K

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vp(m/s)

9800

0.20 0.15 0.10

9600

0.05

9400

0.00

0.20

0.25

2

10000

0.25

K (%)

10200

0.35 0.30

10600 10400

0.40

0.30

0.35

0.40

0.45

he(m)

Fig. 6. vp and K2 dependence on electrode thickness he (hAlN is 0.25μm).

It's interesting that, a maximum extreme point of quality factor is observed at the electrode thickness of 0.25-0.35μm (plotted in fig. 7), and this point won’t shift greatly if the piezoelectric layer thickness changes. This is consistent with subsequent theoretical calculation result of the electrode resonance thickness based on the BAW theory.

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20000

hAlN=0.15m

18000

hAlN=0.25m hAlN=0.35m

16000

quality factor

14000 12000 10000 8000 6000 4000

0.20

0.25

0.30

0.35

he(m)

0.40

0.45

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0.15

f

2000 0.50

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Fig. 7. Quality factor dependence on electrode thickness he.

(

[1

(

) )

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The BAW impedance equation of composite piezoelectric layers is shown below: (

)

]

( )

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, detailed information of variables definition and impedance calculating method can be found in [32]. As long as material parameters and structural dimensions are determined, an impedance curve can be obtained to reveal the relation with electrode thickness, as shown in fig. 8. At the working frequency of 5GHz, such composite structure resonates when the electrode thickness is 0.3μm. The similar results are observed at other piezoelectric layer thicknesses of 0.15μm and 0.35μm as well.

(a)

(b)

Fig.8. BAW simulation of admittance and Quality factor at 5GHz (AlN thickness 0.25μm): a, impedance amplitude vs electrode thickness; b, quality factor vs electrode thickness.

Above results demonstrate that BAW vibration in side of PP-IDT transducer plays a key role for the model unit running. At the BAW resonance frequency, the acoustic energy loss is the smallest and quality factor reaches the maximum. Hence, the microwave performance of PP-IDT devices shows velocity dispersion. The resonant frequency, phase velocity, quality factor and electromechanical coupling coefficient K2 vary with composite layer thickness. 3.4. Modeling results with the electrode layer thickness fixed

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When the electrode layer thickness is fixed with he=0.2μm, the resonant frequency and corresponding phase velocity of Sezawa mode decrease as the piezoelectric layer thickness increases (plotted in fig.9). It is similar with the conventional diamond based layered AlN SAW devices that piezoelectric layer thickness has impacts on phase velocity [31], showing frequency dispersion. On one way, it could be interpreted and calculated by the effective permittivity method described in [34]. While for PP-IDT structure, it can also be explained by BAW theory: the increasing piezoelectric layer thickness extends the equivalent acoustic path and reduces resonant frequency. The electromechanical coupling coefficient K2 increases with increase of piezoelectric layer thickness, which is consistent with reported results [6]. The alike results are observed at other electrode layer thicknesses of 0.25μm and 0.3μm as well. The quality factor decreases with thicker piezoelectric layer, especially for khAlN less than 1 (plotted in fig. 10), and presents different variation ranges due to the electrode impacts discussed in 3.3.

1.4

Pr

1.2 1.0

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0.8 0.6

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0.4 0.2 0.0

0

1

2

khAlN

Fig.9. vp and K2 dependence on khAlN.

3

2

11000

10400

2

K (%)

vp(m/s)

11200

10600

1.6

K

11400

10800

vp

e-

11600

38000 36000 34000 32000 30000 28000 26000 24000 22000 20000 18000 16000 14000 12000 10000 8000 6000 4000 2000

he=0.2m he=0.25m

0.0

0.5

1.0

1.5

khAlN

f

he=0.3m

2.0

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quality factor

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2.5

pr

Fig. 10. Quality factor dependence on khAlN.

4. Conclusions

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So during the devices design of PP-IDT structure, in order to get high quality factor and appropriate electromechanical coupling coefficient K2, it is recommended to determinie the electrode resonance thickness first. Taking the devices configuration, process capabilities into accounted, the piezoelectric layer thickness should be as thick as possible to get a higher electromechanical coupling coefficient K2. The quality factor and K2 are always two contradictory parameters in SAW devices to design and trade-off carefully.

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The thickness of piezoelectric layer and the electrode is optimized based on the systematic analysis of diamond based innovative PP-IDT structure with patterned piezoelectric layer, yielding a quality factor large than 104. High quality factor of this structure is contributed by little interface and transmission loss of SAW propagation in diamond, rather than by reducing piezoelectric layer thickness, which hinders the form of highly c-orientated film. This work supplies a solution for commercial applications of piezoelectric film SAW devices, especially the high-frequency filter and resonator of low insert loss and narrow-band, and high-accuracy sensor. Experimental work is still ongoing with the aid of one-step etching technique on AlN/Al layer. A relatively low electro-mechanical coupling coefficient can be improved by doping the AlN piezoelectric film with Sc [2], adding a metal seeds layer [35], etc. Despite the high quality factor of the PP-IDT structure, a balance is necessary among wave velocity, quality factor, electromechanical coupling coefficient and technological tolerance for different devices.

Funding This work was supported by the National Nature Science Foundation of China (Grant no. 51577180, U1832131 and 51721005); the Youth Innovation Promotion Association of the Chinese Academy of Sciences (Grant no. 2016128).

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[30] K.Y. Hashimoto, Surface Acoustic Wave Devices in Telecommunications Modeling and Simulation, page 17-18, section 1.2.1, Springer-Verlag: Berlin Heidelberg, 2000, ISNB: 7-118-02823-1. [31] S. Maouhoub, Y. Aoura, A. Mir, FEM simulation of AlN thin layers on diamond substrates for high frequency SAW devices, Diamond & Related Materials 62 (2016) 7-13. [32] Y.F. Zhang, D. Chen, Multilayer Integrated Film Bulk Acoustic Resonators, page 31-38, chapter 3, Shanghai Jiao Tong University Press: Shanghai, 2012, ISBN 978-7-313-08736-2. [33] K.Y. Hashimoto, Surface Acoustic Wave Devices in Telecommunications Modeling and Simulation, page 200-214, section 7.2, Springer-Verlag: Berlin Heidelberg, 2000, ISNB: 7-118-02823-1. [34] K.Y. Hashimoto, Surface Acoustic Wave Devices in Telecommunications Modeling and Simulation, page 167-174, section 6.2, Springer-Verlag: Berlin Heidelberg, 2000, ISNB: 7-118-02823-1. [35] Ch.J. Zhou, Y. Yang, H. Jin, B. Feng, Sh.R. Dong, J.K. Luo, T.L. Ren, M.S. Chan, C.Y. Yang, Surface acoustic wave resonators based on (002) AlN/Pt/diamond/silicon layered structure, Thin Solid Films 548 (2013) 425-428.

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Author statement Bowei Xie: Conceptualization, Methodology, Writing - Original Draft, Formal analysis Fazhu Ding: Writing - Review & Editing, Supervision, Project administration Zebin Dong: Validation

Daxing Huang: Investigation, Data Curation

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Hongwei Gu: Resources, Funding acquisition

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Hongjing Shang: Software, Visualization

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Graphical abstract

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he is modified from l; hAlN is modified from h; SV type BAW was modified to add the two components

at the bottom.

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Highlights: Diamond based piezoelectric layer is patterned as inter-digital transducer;

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Quality factor of surface acoustic wave devices can be improved greatly;

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Thickness of piezoelectric and electrode layer effects the performance;

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Maximum quality factor is observed at optimal electrode thickness;

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