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Solid-gap wine-glass mode disks VB-FET resonators applied to biomass sensing
Microelectronic Engineering 145 (2015) 53–57
Contents lists available at ScienceDirect
Microelectronic Engineering journal homepage: www.elsevier.com/locate/mee
Solid-gap wine-glass mode disks VB-FET resonators applied to biomass sensing M. Maqueda López ⇑, M. Fernandéz-Bolaños Badía, W. Vitale, A.M. Ionescu École Polytechnique Fédérale de Lausanne, Lausanne 1015, Switzerland
a r t i c l e
i n f o
Article history: Received 24 October 2014 Received in revised form 8 January 2015 Accepted 4 March 2015 Available online 11 March 2015 Keywords: Wine-glass mode disk MEMS resonator Motional resistance Q-factor Mass sensitivity Solid-gap
a b s t r a c t The working principle of an electromechanical resonator for mass sensing applications is based on monitoring the characteristic resonance frequency downshift due to the particles attachment on the surface of the device. Solid-Gap Vibrating-Body Field Effect Transistor (VB-FET) resonators offer an interesting solution for extreme biomass sensing due to their high motional current at resonance, their compatibility with Complementary Metal-Oxide-Semiconductor (CMOS) circuits, high reproducibility of characteristics and the possibility to have exploitable values of the quality factor (Q-factor) in air and liquid operation. In this work a detailed micromechanical analysis of VB-FET disk resonators is carried out, considering geometrical design parameters, geometrical variability, motional resistance for air and solid-gaps, Q-factor and variable particle-resonator sensing cases, permitting quantitative discussions on their suitability for biosensing. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction The actual tendency of decreasing the size of the mechanical resonators into submicron domain is very promising for the development of ultrasensitive chemical and biological mass sensors able to reach extremely low mass resolutions. The working principle of such sensors is based on monitoring the characteristic resonance frequency downshift due to the particles attachment on the surface of the resonator. Recently, resonant mechanical nano-balances have been proven to achieve an experimental point mass sensitivity of 1.6 kHz pg1 for individual submicron airborne particles [1]. In the same way, airborne nanoparticles have been detected by vertically aligned silicon nanowire (SiNW)-based resonators revealing a point mass sensitivity of 7.1 Hz fg1 [2]. For applications where the mass is deposited as a uniform layer on the surface of the resonator the Figure of Merit (FOM) to be considered is the distributed mass sensitivity that measures the minimum mass that can be detected per square meter of sensing surface [3]. Mass sensors based on Quartz Crystal Microbalances (QCM) have yielded a distributed mass resolution of 700 pg cm2 and those based on Thin-Film Acoustic Wave Resonators (TFBAR) technologies have accomplished a distributed mass resolution of 1000 pg cm2 [4],
⇑ Corresponding author. http://dx.doi.org/10.1016/j.mee.2015.03.005 0167-9317/Ó 2015 Elsevier B.V. All rights reserved.
nevertheless they cannot be easily integrated into silicon process. Silicon resonant cantilever and mass sensors based on Carbon Nano-Tubes (CNT) have achieved subatomic distributed mass resolution, however the needed readouts have not been miniaturized and their manufacturing reproducibility has been proven to be low. In order to integrate the actuation and the readout, square bulk mode resonators with electrostatic transduction and a distributed mass sensitivity of 125 pg cm2 [5] as well as disk bulk mode resonators achieving 8.7 pg cm2 [3] have been implemented, nonetheless, due to their reduced transduction area a small motional current is extracted from the devices and is interfered by the frequency floor noise, revealing the importance of better amplification in the mass sensors readout. The high motional resistance Rx, due to the relative inefficiency of the air–gap in electrostatic transduction has boosted the fabrication of 2 lmsolid-gap Radial Contour Mode (RCM) resonators for the detection of goat immunoglobulin G (IgG) achieving a mass sensitivity of 1617 Hz nm2 and a motional resistance of 6.5 MX [4], which is still too high to be compatible with Integrated Circuits (IC). Solid-gap VB-FET resonators are vibrating structures embedded into FET transistors fabricated in Silicon-On-Insulator (SOI) that integrate both actuation and readout on-chip and perform a remarkable amplification in the readout [6] as well as Rx reduction due to the high-k dielectric into the gap. In this work we present the preliminary micromechanical analysis to fabricate a VB-FET
54
M. Maqueda López et al. / Microelectronic Engineering 145 (2015) 53–57
disk resonator, taking into account materials, geometrical design parameters, motional resistance, Q-factor and point and distributed mass sensitivities.
2. Principles of operation The device consists in a silicon disk suspended by two anchors corresponding to the drain and the source electrodes of the VB-FET (see Fig. 1). The structure is enclosed by two polysilicon capacitive gate electrodes that are fixed and spaced from the perimeter of the movable transistor body by high-k dielectric solid-gaps. The body of the resonator is p-type doped and the drain and source regions are highly n-type doped what results in the creation of two modulated channels in the areas of the disk that are controlled by the gates. The VB-FET is electrostatically driven in resonance with a twoport AC-DC actuation: a bias voltage Vp, is applied to the disk by means of the drain electrode, and two AC-DC voltages, VG1 and VG2, are applied to the gates. When the frequency of the AC electrical signals matches the Wine-Glass Mode (WGM) resonance frequency of the disk, the resulting mechanical force drives the disk into a characteristic vibration mode shape in which the disk expands and contracts modulating the gaps. The mechanical modulation of the gaps results in an electrical modulation of drain-tosource current ID existing in the channels that face the gaps. ID is collected in the source electrode and contains the contributions of the capacitive, piezoresistive and field-effect currents as it has been already proven in [6,7].
Fig. 2. Perspective-view schematic of the simulated micromechanical wine-glass mode disk resonator.
3. Design The present work is focused on the micromechanical Finite Elements Methods (FEM) analysis of disk resonators to study the benefits of the solid-gap regarding to resonance frequency, Rx, Qfactor and point and distributed mass sensitivity. Fig. 3 shows the perspective-view schematic for the resonator: a solid disk with radius R, thickness t, anchor length lanchor, anchor width wanchor and solid-gap d. The two anchors correspond with the drain (connected to Vp) and source (connected to ground), as it has been reported in Figs. 1 and 2. The gates surround the solid-gaps, (in blue) and excite the disk by means of a harmonic perturbation as it is mentioned in Section 2. Aside from the variants that are specified later, the typical dimensions considered in this work are: R = 5 lm, t = 1 lm,
Fig. 3. Motional resistance, Rx, in function of the bias voltage, Vp, from analytical simulations for bulk disks by considering different parameters.
d = 50 nm, lanchor = 1 lm and wanchor = 750 nm, and four anchors have been simulated. 4. Results 4.1. Motional resistance The equivalent LCR circuit for the wine-glass disk is governed by the total integrated kinetic energy in the resonator, its mode shape and the parameters associated with its transducer ports [8]. Using the procedure of [8–10] the motional resistance for a vibrating disk can be approximated as in Eq. (1).
Rx ¼
Fig. 1. Perspective-view schematic of the wine-glass mode disk resonator in a typical two-port bias and excitation configuration. The solid-gaps face the inversion channel regimes.
x0 mre Q g2e
ð1Þ
where x0 is the angular resonance frequency, mre is the equivalent mass of the disk vibrating at the fundamental wine-glass mode of resonance, Q is the quality factor of the disk and ge is the electromechanical coupling factor in the high-k solid-gaps. The electromechanical transduction is modeled by the integrated change in the electrode-disk capacitance per unit of displacement as it is included in Eq. (2), where C denotes the capacitance between the disk and each electrode. A is the coupling
M. Maqueda López et al. / Microelectronic Engineering 145 (2015) 53–57
55
area given by URt, U is the angular overlap between the gate electrodes and the disk, e0 is the vacuum permittivity and er is the relative permittivity of the high-k dielectric in the solid-gap.
ge ¼ V p
dC e0 er A ffi Vp 2 dr d
ð2Þ
The equivalent mass mre for a vibrating disk can be approximated as it is shown in Eq. (3) where q, E and r are density, Young modulus and Poisson’s ratio, respectively, of the disk structural material, Jn(x) is the Bessel function of first kind of order n = 1, 2, 3 and r concerns the radial polar coordinate of the disk. The rest of the used parameters are defined in Eq. (5) [1,8–12].
mre ðrÞ ¼
qpt
RR G
2 G 2B 2r J 3 ðGÞ þ AR J 2 ðGnÞ rdr 2 G G 2B J f 2r J 3 nf þ AR J 2 ðfÞ 2r 1 n
0
B ¼ 4:5236 A
J ðGÞ 2r 1
G¼
fr nR
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2qx20 ð1 þ rÞ f¼R E
ð3Þ
ð4Þ rffiffiffiffiffiffiffiffiffiffiffiffi 2 n¼ 1r
ð5Þ
Fig. 5. FEM simulations for the resonance frequency shifts considering different particle-disk contact areas with the same mass. The shape of the wine-glass mode of resonance is included on the right for a sketched disk resonator.
The final approximated expression for the motional resistance is shown in Eq. (6), where U1 and U2 are the overlap angles between the gates electrodes (solid-gaps) and the disk.
Rx ffi
x0 mre 1 Q
V 2p
4
d
U1 U2 ðe0 er RtÞ2
ð6Þ
Fig. 3 depicts the analytical simulations that relate Rx with the supply bias, Vp, and the Q-factor for silicon wine-glass mode disk resonators of 5–10 lm-radius, 1–2 lm-thickness and 50–200 nmgap, respectively. Both air and solid-gap have been simulated and a reduction in Rx about 600 has been achieved by using HfO2, (er = 25), instead of air for a 50 nm-gap, and about 200 by comparing both disks geometries for smaller values of Vp and Q = 5000. 4.2. Point mass sensing In order to obtain the point mass sensitivity, Pt particles of 1 lm3 have been deposed and simulated by means of FEM analysis onto the surface of a 2 lm-thick 10 lm-radius 200 nm-Si3N4-gap bulk disk that resonates at 189 MHz. Several deposition cases have been considered in order to analytically compare the sensitivity of
Fig. 4. Modal FEM simulations of the resonance frequency shifts and mass sensitivity for different deposited Pt particles distributions onto a disk.
Fig. 6. Modal FEM simulations of the frequency shifts considering a mass deposited onto an antinode of resonance of disks resonating at 189 and 394 MHz and their respective disk-particle mass ratio relationship.
the disk depending on the area where the particles attach (see Fig. 4): in case 1 the particle is deposited on a resonance node, (minimum displacement); in cases 2–3 the particle is attached to a resonance antinode, (maximum displacement) considering the cases of maximum compression (case 2) and expansion (case 3) of the resonance shape; in case 4 it has been proven that deposition in the node of resonance does not substantially contribute to the frequency downshift; in cases 5–9 particles have been added sequentially in order to approximate a distributed mass deposition, attaining mass sensitivities up to 796 kHz pg1. Fig. 5 shows the additional resonant frequency downshifting effect due to variations in the particle-disk contact area that have been simulated. Mass sensitivity approximately increases x2 when the contact area between the disk and a deposited Pt particle increases x16. In Fig. 6 an interesting FOM has been plotted: in green, it includes the resonant frequency shifts regarding to a Pt particle deposited onto an antinode, with a mass that varies from 1 fg to 20 pg. Frequency shifts up to x2 those of the 189 MHz are achieved by means of the smallest disk at 394 MHz.
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M. Maqueda López et al. / Microelectronic Engineering 145 (2015) 53–57
4.4. Geometry variability As one of the main applications of the VB-FET disks is mass sensing, variations in the resonance frequency due to parasitic effect have to be avoided to ensure a robust design. In order to study the resonance frequency shifts regarding to variations in the geometry, FEM simulations have been carried out and the results have been plotted in Fig. 8. While one of the dimensions has been varied, the remaining dimensions have been kept as indicated in Section 3. In the case of thickness, variations between 0.25 and 2 lm suppose a slight growing deviation of the order of few hundreds of kHz. It can be concluded that certain flexibility in the design of this dimension can be accepted. Contrary, simulating the anchors width and length into the same geometrical range entails a deviation of several MHz of the original resonance frequency for both of the cases, rising and falling respectively, therefore the anchors width and length have been indicated as critical parameters in the disk resonator design. Fig. 7. Modal FEM simulations of the frequency shifts considering uniform layers of biomass deposited onto the disk.
Fig. 8. Modal FEM simulations of the frequency shifts due to variations in the geometry of the disk: thickness, length and width of the anchors.
Nevertheless, the smallest disk stops being sensitive for particles heavier than 3 pg while the disk at 189 MHz keeps on resonance for particles up to 20 pg. This effect is a consequence of the diskparticle mass ratio: the larger the ratio, the smaller the frequency shift and the faster the disk vibrations are attenuated. 4.3. Distributed mass sensing In order to simulate biomass sensing by virtue of FEM analysis, uniform layers of 10 nm-thickness of a typical antibody such a goat IgG have been attached onto the disk resonator [4]. The resonance frequency variation after every deposited layer of IgG is represented in Fig. 7. The frequency downshift achieved by the smallest disk resonating at 394 MHz is x4 the one of the largest disk. The distributed mass sensitivity is defined in Eq. (7) [4].
Df 1 Dm!0 f 0 Dm
Sm ¼ lim
5. Conclusions In this paper, the micromechanical design for high-k solid-gap Vibrating Body FET bulk silicon resonators excited at the fundamental wine-glass mode of resonance has been simulated and has been revealed as an excellent candidate for biomass applications. Effects of mass of deposed particles, position of attached particles on the disk and contact area particle-resonator on the resonance frequency have been included achieving a point mass sensitivity of 796 kHz pg1 and a distributed mass sensitivity of 161 lm2 ng1 in the best cases. Geometrical variations of the resonating disk have been studied concluding that anchors design is the major factor that causes a deviation of several MHz from the target resonance frequency. We have proposed HfO2 as a promising high-k dielectric to fill the solid gap while lowering the motional resistance and keeping a quality factor of several thousand as it has been already proven in [7,13] for air–gaps and in [14,15] for silicon nitride gaps. As HfO2’s Young modulus is one order of magnitude smaller than the one of silicon nitride, the solid-gap would be less stiff in the former case, and so higher Q-factor than for silicon nitride gap would be expected for the same structure. Future work includes comparison of FEM simulations and theoretical approximations with experimental data for the electromechanical model of the HfO2 solid-gap disk resonator. Considering sensing applications under ambient conditions, variations in temperature would modify the mechanical material properties and the electrical transport parameters of the VB-FET and hence the resonance frequency [7]. In the same way, atmospheric pressure and humidity would increase the air damping surrounding the resonator, and Q-factor could drop. Consequently, the study of these effects is of major importance for the coming design steps of applied VB-FET biosensors. Acknowledgement This research was supported by the funding of the Swiss National Science Foundation (SNSF). References
ð7Þ
Our designs can achieve a distributed mass sensitivity of 161 lm2 ng1 for the smallest disk and 86 lm2 ng1 for the largest disk, so while disks are kept in resonance as it has been discussed above, smaller and higher frequency disks are more suitable to be implemented as mass and/or biomass sensors.
[1] [2] [3] [4] [5] [6] [7] [8]
A. Hajjam et al., IEEE Sens. J. 11 (11) (2011) 2883–2890. H.S. Wasisto et al., Micro Nano Lett. 8 (10) (2013) 554–558. A. Cagliani et al., J. Micromech. Microeng. 21 (4) (2011) 6. A. Heidari et al., Microelectron. Eng. 88 (2011) 1729–1732. J.E. -Y et al., Appl. Phys. Lett. 91 (23) (2007) 4. D. Grogg, et al., IEDM08, (2008) 1–4. D. Grogg et al., IEEE Trans. Electron. Dev. 58 (7) (2011) 2113–2121. D. Grogg, et al., ESSDERC09, (2009) 379–382.
M. Maqueda López et al. / Microelectronic Engineering 145 (2015) 53–57 [9] [10] [11] [12]
M.A. Abdelmoneum, et al., IEEE MEMS03, (2003) 698–701. F.D. Bannon et al., IEEE J. Solid-State Circuits 35 (4) (2000) 512–526. M. Onoe et al., J. Acoust. Soc. Am. 28 (6) (1956) 1158–1162. J. Basu et al., Microsyst. Technol. 17 (2011) 1557–1580.
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[13] J. Wang et al., Control 51 (12) (2004) 1607–1628. [14] Y.-W. Lin, et al., Proceedings Joint IEEE Int. Frequency Control/Precision Time & Time Interval Symposium, 2005, 128–134. [15] D. Weinstein et al., Micro Nano Lett. 10 (4) (2010) 1234–1237.
Contents lists available at ScienceDirect
Microelectronic Engineering journal homepage: www.elsevier.com/locate/mee
Solid-gap wine-glass mode disks VB-FET resonators applied to biomass sensing M. Maqueda López ⇑, M. Fernandéz-Bolaños Badía, W. Vitale, A.M. Ionescu École Polytechnique Fédérale de Lausanne, Lausanne 1015, Switzerland
a r t i c l e
i n f o
Article history: Received 24 October 2014 Received in revised form 8 January 2015 Accepted 4 March 2015 Available online 11 March 2015 Keywords: Wine-glass mode disk MEMS resonator Motional resistance Q-factor Mass sensitivity Solid-gap
a b s t r a c t The working principle of an electromechanical resonator for mass sensing applications is based on monitoring the characteristic resonance frequency downshift due to the particles attachment on the surface of the device. Solid-Gap Vibrating-Body Field Effect Transistor (VB-FET) resonators offer an interesting solution for extreme biomass sensing due to their high motional current at resonance, their compatibility with Complementary Metal-Oxide-Semiconductor (CMOS) circuits, high reproducibility of characteristics and the possibility to have exploitable values of the quality factor (Q-factor) in air and liquid operation. In this work a detailed micromechanical analysis of VB-FET disk resonators is carried out, considering geometrical design parameters, geometrical variability, motional resistance for air and solid-gaps, Q-factor and variable particle-resonator sensing cases, permitting quantitative discussions on their suitability for biosensing. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction The actual tendency of decreasing the size of the mechanical resonators into submicron domain is very promising for the development of ultrasensitive chemical and biological mass sensors able to reach extremely low mass resolutions. The working principle of such sensors is based on monitoring the characteristic resonance frequency downshift due to the particles attachment on the surface of the resonator. Recently, resonant mechanical nano-balances have been proven to achieve an experimental point mass sensitivity of 1.6 kHz pg1 for individual submicron airborne particles [1]. In the same way, airborne nanoparticles have been detected by vertically aligned silicon nanowire (SiNW)-based resonators revealing a point mass sensitivity of 7.1 Hz fg1 [2]. For applications where the mass is deposited as a uniform layer on the surface of the resonator the Figure of Merit (FOM) to be considered is the distributed mass sensitivity that measures the minimum mass that can be detected per square meter of sensing surface [3]. Mass sensors based on Quartz Crystal Microbalances (QCM) have yielded a distributed mass resolution of 700 pg cm2 and those based on Thin-Film Acoustic Wave Resonators (TFBAR) technologies have accomplished a distributed mass resolution of 1000 pg cm2 [4],
⇑ Corresponding author. http://dx.doi.org/10.1016/j.mee.2015.03.005 0167-9317/Ó 2015 Elsevier B.V. All rights reserved.
nevertheless they cannot be easily integrated into silicon process. Silicon resonant cantilever and mass sensors based on Carbon Nano-Tubes (CNT) have achieved subatomic distributed mass resolution, however the needed readouts have not been miniaturized and their manufacturing reproducibility has been proven to be low. In order to integrate the actuation and the readout, square bulk mode resonators with electrostatic transduction and a distributed mass sensitivity of 125 pg cm2 [5] as well as disk bulk mode resonators achieving 8.7 pg cm2 [3] have been implemented, nonetheless, due to their reduced transduction area a small motional current is extracted from the devices and is interfered by the frequency floor noise, revealing the importance of better amplification in the mass sensors readout. The high motional resistance Rx, due to the relative inefficiency of the air–gap in electrostatic transduction has boosted the fabrication of 2 lmsolid-gap Radial Contour Mode (RCM) resonators for the detection of goat immunoglobulin G (IgG) achieving a mass sensitivity of 1617 Hz nm2 and a motional resistance of 6.5 MX [4], which is still too high to be compatible with Integrated Circuits (IC). Solid-gap VB-FET resonators are vibrating structures embedded into FET transistors fabricated in Silicon-On-Insulator (SOI) that integrate both actuation and readout on-chip and perform a remarkable amplification in the readout [6] as well as Rx reduction due to the high-k dielectric into the gap. In this work we present the preliminary micromechanical analysis to fabricate a VB-FET
54
M. Maqueda López et al. / Microelectronic Engineering 145 (2015) 53–57
disk resonator, taking into account materials, geometrical design parameters, motional resistance, Q-factor and point and distributed mass sensitivities.
2. Principles of operation The device consists in a silicon disk suspended by two anchors corresponding to the drain and the source electrodes of the VB-FET (see Fig. 1). The structure is enclosed by two polysilicon capacitive gate electrodes that are fixed and spaced from the perimeter of the movable transistor body by high-k dielectric solid-gaps. The body of the resonator is p-type doped and the drain and source regions are highly n-type doped what results in the creation of two modulated channels in the areas of the disk that are controlled by the gates. The VB-FET is electrostatically driven in resonance with a twoport AC-DC actuation: a bias voltage Vp, is applied to the disk by means of the drain electrode, and two AC-DC voltages, VG1 and VG2, are applied to the gates. When the frequency of the AC electrical signals matches the Wine-Glass Mode (WGM) resonance frequency of the disk, the resulting mechanical force drives the disk into a characteristic vibration mode shape in which the disk expands and contracts modulating the gaps. The mechanical modulation of the gaps results in an electrical modulation of drain-tosource current ID existing in the channels that face the gaps. ID is collected in the source electrode and contains the contributions of the capacitive, piezoresistive and field-effect currents as it has been already proven in [6,7].
Fig. 2. Perspective-view schematic of the simulated micromechanical wine-glass mode disk resonator.
3. Design The present work is focused on the micromechanical Finite Elements Methods (FEM) analysis of disk resonators to study the benefits of the solid-gap regarding to resonance frequency, Rx, Qfactor and point and distributed mass sensitivity. Fig. 3 shows the perspective-view schematic for the resonator: a solid disk with radius R, thickness t, anchor length lanchor, anchor width wanchor and solid-gap d. The two anchors correspond with the drain (connected to Vp) and source (connected to ground), as it has been reported in Figs. 1 and 2. The gates surround the solid-gaps, (in blue) and excite the disk by means of a harmonic perturbation as it is mentioned in Section 2. Aside from the variants that are specified later, the typical dimensions considered in this work are: R = 5 lm, t = 1 lm,
Fig. 3. Motional resistance, Rx, in function of the bias voltage, Vp, from analytical simulations for bulk disks by considering different parameters.
d = 50 nm, lanchor = 1 lm and wanchor = 750 nm, and four anchors have been simulated. 4. Results 4.1. Motional resistance The equivalent LCR circuit for the wine-glass disk is governed by the total integrated kinetic energy in the resonator, its mode shape and the parameters associated with its transducer ports [8]. Using the procedure of [8–10] the motional resistance for a vibrating disk can be approximated as in Eq. (1).
Rx ¼
Fig. 1. Perspective-view schematic of the wine-glass mode disk resonator in a typical two-port bias and excitation configuration. The solid-gaps face the inversion channel regimes.
x0 mre Q g2e
ð1Þ
where x0 is the angular resonance frequency, mre is the equivalent mass of the disk vibrating at the fundamental wine-glass mode of resonance, Q is the quality factor of the disk and ge is the electromechanical coupling factor in the high-k solid-gaps. The electromechanical transduction is modeled by the integrated change in the electrode-disk capacitance per unit of displacement as it is included in Eq. (2), where C denotes the capacitance between the disk and each electrode. A is the coupling
M. Maqueda López et al. / Microelectronic Engineering 145 (2015) 53–57
55
area given by URt, U is the angular overlap between the gate electrodes and the disk, e0 is the vacuum permittivity and er is the relative permittivity of the high-k dielectric in the solid-gap.
ge ¼ V p
dC e0 er A ffi Vp 2 dr d
ð2Þ
The equivalent mass mre for a vibrating disk can be approximated as it is shown in Eq. (3) where q, E and r are density, Young modulus and Poisson’s ratio, respectively, of the disk structural material, Jn(x) is the Bessel function of first kind of order n = 1, 2, 3 and r concerns the radial polar coordinate of the disk. The rest of the used parameters are defined in Eq. (5) [1,8–12].
mre ðrÞ ¼
qpt
RR G
2 G 2B 2r J 3 ðGÞ þ AR J 2 ðGnÞ rdr 2 G G 2B J f 2r J 3 nf þ AR J 2 ðfÞ 2r 1 n
0
B ¼ 4:5236 A
J ðGÞ 2r 1
G¼
fr nR
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2qx20 ð1 þ rÞ f¼R E
ð3Þ
ð4Þ rffiffiffiffiffiffiffiffiffiffiffiffi 2 n¼ 1r
ð5Þ
Fig. 5. FEM simulations for the resonance frequency shifts considering different particle-disk contact areas with the same mass. The shape of the wine-glass mode of resonance is included on the right for a sketched disk resonator.
The final approximated expression for the motional resistance is shown in Eq. (6), where U1 and U2 are the overlap angles between the gates electrodes (solid-gaps) and the disk.
Rx ffi
x0 mre 1 Q
V 2p
4
d
U1 U2 ðe0 er RtÞ2
ð6Þ
Fig. 3 depicts the analytical simulations that relate Rx with the supply bias, Vp, and the Q-factor for silicon wine-glass mode disk resonators of 5–10 lm-radius, 1–2 lm-thickness and 50–200 nmgap, respectively. Both air and solid-gap have been simulated and a reduction in Rx about 600 has been achieved by using HfO2, (er = 25), instead of air for a 50 nm-gap, and about 200 by comparing both disks geometries for smaller values of Vp and Q = 5000. 4.2. Point mass sensing In order to obtain the point mass sensitivity, Pt particles of 1 lm3 have been deposed and simulated by means of FEM analysis onto the surface of a 2 lm-thick 10 lm-radius 200 nm-Si3N4-gap bulk disk that resonates at 189 MHz. Several deposition cases have been considered in order to analytically compare the sensitivity of
Fig. 4. Modal FEM simulations of the resonance frequency shifts and mass sensitivity for different deposited Pt particles distributions onto a disk.
Fig. 6. Modal FEM simulations of the frequency shifts considering a mass deposited onto an antinode of resonance of disks resonating at 189 and 394 MHz and their respective disk-particle mass ratio relationship.
the disk depending on the area where the particles attach (see Fig. 4): in case 1 the particle is deposited on a resonance node, (minimum displacement); in cases 2–3 the particle is attached to a resonance antinode, (maximum displacement) considering the cases of maximum compression (case 2) and expansion (case 3) of the resonance shape; in case 4 it has been proven that deposition in the node of resonance does not substantially contribute to the frequency downshift; in cases 5–9 particles have been added sequentially in order to approximate a distributed mass deposition, attaining mass sensitivities up to 796 kHz pg1. Fig. 5 shows the additional resonant frequency downshifting effect due to variations in the particle-disk contact area that have been simulated. Mass sensitivity approximately increases x2 when the contact area between the disk and a deposited Pt particle increases x16. In Fig. 6 an interesting FOM has been plotted: in green, it includes the resonant frequency shifts regarding to a Pt particle deposited onto an antinode, with a mass that varies from 1 fg to 20 pg. Frequency shifts up to x2 those of the 189 MHz are achieved by means of the smallest disk at 394 MHz.
56
M. Maqueda López et al. / Microelectronic Engineering 145 (2015) 53–57
4.4. Geometry variability As one of the main applications of the VB-FET disks is mass sensing, variations in the resonance frequency due to parasitic effect have to be avoided to ensure a robust design. In order to study the resonance frequency shifts regarding to variations in the geometry, FEM simulations have been carried out and the results have been plotted in Fig. 8. While one of the dimensions has been varied, the remaining dimensions have been kept as indicated in Section 3. In the case of thickness, variations between 0.25 and 2 lm suppose a slight growing deviation of the order of few hundreds of kHz. It can be concluded that certain flexibility in the design of this dimension can be accepted. Contrary, simulating the anchors width and length into the same geometrical range entails a deviation of several MHz of the original resonance frequency for both of the cases, rising and falling respectively, therefore the anchors width and length have been indicated as critical parameters in the disk resonator design. Fig. 7. Modal FEM simulations of the frequency shifts considering uniform layers of biomass deposited onto the disk.
Fig. 8. Modal FEM simulations of the frequency shifts due to variations in the geometry of the disk: thickness, length and width of the anchors.
Nevertheless, the smallest disk stops being sensitive for particles heavier than 3 pg while the disk at 189 MHz keeps on resonance for particles up to 20 pg. This effect is a consequence of the diskparticle mass ratio: the larger the ratio, the smaller the frequency shift and the faster the disk vibrations are attenuated. 4.3. Distributed mass sensing In order to simulate biomass sensing by virtue of FEM analysis, uniform layers of 10 nm-thickness of a typical antibody such a goat IgG have been attached onto the disk resonator [4]. The resonance frequency variation after every deposited layer of IgG is represented in Fig. 7. The frequency downshift achieved by the smallest disk resonating at 394 MHz is x4 the one of the largest disk. The distributed mass sensitivity is defined in Eq. (7) [4].
Df 1 Dm!0 f 0 Dm
Sm ¼ lim
5. Conclusions In this paper, the micromechanical design for high-k solid-gap Vibrating Body FET bulk silicon resonators excited at the fundamental wine-glass mode of resonance has been simulated and has been revealed as an excellent candidate for biomass applications. Effects of mass of deposed particles, position of attached particles on the disk and contact area particle-resonator on the resonance frequency have been included achieving a point mass sensitivity of 796 kHz pg1 and a distributed mass sensitivity of 161 lm2 ng1 in the best cases. Geometrical variations of the resonating disk have been studied concluding that anchors design is the major factor that causes a deviation of several MHz from the target resonance frequency. We have proposed HfO2 as a promising high-k dielectric to fill the solid gap while lowering the motional resistance and keeping a quality factor of several thousand as it has been already proven in [7,13] for air–gaps and in [14,15] for silicon nitride gaps. As HfO2’s Young modulus is one order of magnitude smaller than the one of silicon nitride, the solid-gap would be less stiff in the former case, and so higher Q-factor than for silicon nitride gap would be expected for the same structure. Future work includes comparison of FEM simulations and theoretical approximations with experimental data for the electromechanical model of the HfO2 solid-gap disk resonator. Considering sensing applications under ambient conditions, variations in temperature would modify the mechanical material properties and the electrical transport parameters of the VB-FET and hence the resonance frequency [7]. In the same way, atmospheric pressure and humidity would increase the air damping surrounding the resonator, and Q-factor could drop. Consequently, the study of these effects is of major importance for the coming design steps of applied VB-FET biosensors. Acknowledgement This research was supported by the funding of the Swiss National Science Foundation (SNSF). References
ð7Þ
Our designs can achieve a distributed mass sensitivity of 161 lm2 ng1 for the smallest disk and 86 lm2 ng1 for the largest disk, so while disks are kept in resonance as it has been discussed above, smaller and higher frequency disks are more suitable to be implemented as mass and/or biomass sensors.
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