33.2 Temperature Compensation of Silicon ... - Semantic Scholar
Temperature Compensation of Silicon Micromechanical Resonators via Degenerate Doping Ashwin K. Samarao and Farrokh Ayazi School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA, 30332-0250 [email protected]; [email protected] Abstract We report on the degenerate doping of a silicon resonator as a new method for reducing its temperature coefficient of frequency (TCF). This is the first TCF reduction technique reported till date that takes advantage of free charge carrier effects on the elastic constants of silicon. The TCF of silicon bulk acoustic resonators (SiBAR) are reduced from -29 ppm/˚C to -1.5 ppm/˚C on 5 µm thick devices using degenerate boron doping and to -2.72 ppm/˚C on 20 µm thick devices using boron-assisted aluminum doping while maintaining a high quality factor (Q) of 28000 in vacuum.
electrode, the electrostatic force applied to the corresponding face of the resonator induces an acoustic wave that propagates through the bar, resulting in a width-extensional resonance mode whose frequency is primarily defined by W. Small changes in the size of the capacitive gap on the other side of the device induce a voltage on the sense electrode whose amplitude peaks at the mechanical resonant frequency.
Introduction With an increasing demand for higher levels of integration in existing and emerging microsystem applications, integrated alternatives to discrete frequency-selective devices are necessary. Silicon micromechanical resonators can enable a cost-effective integrated platform for timing, wireless connectivity and multi-band spectral processing. However, successful insertion of these devices into cost and power sensitive consumer applications require effective tuning, trimming and compensation techniques. The uncompensated temperature coefficient of frequency (TCF) for a native silicon resonator is on the order of -30 ppm/˚C [1], which is far greater than that of quartz resonators. To achieve stable low-phase-noise frequency references, the TCF needs to be compensated without compromising on the quality factor (Q) of the resonator. Circuit-based compensation techniques add to the power budget whereas depositing compensating materials with positive TCF on silicon [2] is impractical on certain resonator geometries and may reduce the Q of the resonator. It is well known that the number of free charge carries in a silicon material can be significantly increased by doping. The effect of these free charge carriers on the elastic constants of silicon have long been studied [3] but are being used for the first time in this work to achieve temperature compensation in silicon micromechanical resonators. TCF compensation is demonstrated using Silicon Bulk Acoustic Resonators (or SiBARs) [1] as test vehicle. As shown in Figure 1, a SiBAR consists of a rectangular bar element placed between two electrodes, and supported symmetrically by two thin tethers on the sides. A DC polarization voltage (Vp) applied between the resonator and the electrodes generates an electrostatic field in the narrow capacitive gaps. When an AC voltage is applied to the drive
97-4244-5640-6/09/$26.00 ©2009 IEEE
Fig. 1: (a) Structure, (b) SEM view, and (c) simulated width-extensional mode shape of the SiBAR device in ANSYS (t=10 µm).
Resonance modes like the width-extensional mode of the SiBAR involve the propagation of longitudinal waves through the bulk of the resonating microstructure. The Young’s modulus (E) that determines the elastic resonance frequency possesses a negative temperature coefficient. This results in a material softening which causes the stiffness and thereby the frequency of the silicon resonators to decrease with increasing temperature. Though the linear thermal expansion coefficient (α) of silicon also contributes to the TCF, its contribution is negligible compared to the temperature coefficient of Young’s modulus (TCE). Temperature compensation techniques that have been reported so far combat the effect of TCE [2] whereas this work utilizes the understanding on the cause of TCE [3-5] as explained in the following section, to achieve compensation. Principle of TCF Compensation via Degenerate Doping An acoustic wave propagating through a semiconducting resonator like the SiBAR induces a flow of free charge carriers by distorting the valence and conduction bands [3]. For example, the valence band in silicon consists of three
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bands of energy in k-space of which two of the energy surfaces are degenerate and are energetically favorable to contain almost all the holes. The strain produced by the propagation of acoustic waves through such a material momentarily splits the degenerate bands leading to a flow of holes from lower to higher energy levels [4]. The amount of band splitting and the resulting flow of holes increases with temperature. The principle of conservation of energy requires that such a temperature dependant change in the electronic energy of the system manifest itself as a corresponding temperature dependant change in the elastic energy of the system, which causes a negative TCE in silicon. To minimize the effect of the momentary strain on the energy bands produced by the acoustic waves, a relatively large permanent strain can be created in the resonating microstructure by doping. For example, a boron dopant has a smaller radius than silicon and is bonded strongly to only three of the four adjacent silicon atoms when diffused in the crystal lattice. Such an atomic arrangement produces a very strong shear strain in the silicon lattice, which leads to a large permanent separation of the degenerate valence bands forcing more of the holes to occupy the higher energy band [5]. The additional band-splitting contributed by the propagation of acoustic waves or the rise in temperature as discussed before are now minimal in comparison, which serves to compensate the TCF of silicon resonators. At degenerate levels of boron doping, the acoustic wave can potentially be shielded entirely from the k-space contours of the valence band which in turn should completely compensate the TCE component of the TCF. In this work, TCF compensation is demonstrated in p-type silicon resonators using degenerate boron doping and boron-assisted aluminum doping. TCF Compensation via Degenerate Boron Doping SiBARs were fabricated with 100 nm capacitive airgaps using the HARPSS process [6] on a 10 µm thick boron-doped silicon wafer, with a starting resistivity of ~10-2 Ω-cm and a TCF of -29 ppm/˚C. The boron dopant density in silicon could be increased to ~ 7 × 1019 atoms/cm3 (i.e., ~10-3 Ω-cm) by repeated doping using conventional solid boron sources. But achieving degenerate levels of doping that requires a boron density of ~ 2 × 1020 atoms/cm3 (i.e., < 10-4 Ω-cm) [7] needed repeated doping using liquid spin-on boron dopant sources [8]. Throughout our experiment, the resistivity of the wafer was accurately monitored using four-point probe measurement. Table I illustrates the boron doping recipes using solid and liquid boron sources. In both cases, the borosilicate glass (BSG) layer that forms after every dope/anneal cycle was removed using hydrofluoric acid. The silicon wafer with the starting resistivity of ~10-2 Ω-cm was processed through five repetitions of the solid boron dope/anneal recipe to reduce the resistivity to ~10-3 Ω-cm at which a corresponding TCF of -18.9 ppm/˚C is measured (Figure 2). As seen from the square of the correlation coefficient of linearity (R2) reported in Figure 2, the linearity
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of the TCF curve is compromised to some extent due to the heavy boron doping. Table I: Boron doping recipe using solid and liquid boron sources Boron Dopant Dope Anneal Type Recipe Recipe Solid Boron Disks Liquid Boron Spin-on-Dopants
3 hours in furnace at 1050 ˚C Spin : 800 rpm for 40s / Bake : 200 ˚C for 3 min
5 hours at 1100 ˚C in N2 8 hours at 1100 ˚C in N2
Fig. 2: TCF measurements before (10-2 Ω-cm) and after (10-3 Ω-cm) boron doping (using solid boron sources) on a 10 µm thick SiBAR.
For further reduction of silicon resistivity, spin-on-dopant (SOD) boron sources were used [Futurrex Inc., BDC1-2000]. After six repetitions of liquid boron dope/anneal recipe on a silicon wafer with a starting resistivity of ~10-3 Ω-cm, the resistivity was further reduced by an order of magnitude to 10-4 Ω-cm (Figure 3). A reduction in TCF from -16.7 ppm/˚C at 10-3 Ω-cm to -10.5 ppm/˚C at 10-4 Ω-cm was measured (Figure 4). The increase in resonance frequency after doping is due to the increase in the Young’s modulus of silicon as a result of degenerative doping [9]. The maximum thickness of the heavily boron doped layer that can be achieved using SOD is limited to 7~8 µm [8] which leads to a non-uniform doping profile in SiBARs thicker than 8 µm. Hence, a 5 µm thick SiBAR with a resistivity of ~10-4 Ω-cm measured a much lower TCF of -3.56 ppm/˚C (Figure 5) indicating a more uniform doping profile along its thickness.
Fig. 3: Reduction in resistivity from 10-3 Ω-cm to 10-4 Ω-cm on a 10 µm thick device layer of an SOI.
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Fig .4: TCF Measurements before (~10-3 Ω-cm) and after (~10-4 Ω-cm) boron doping (using liquid boron sources) on a 10 µm thick SiBAR.
Fig. 5: TCF measurements of a 5 µm thick SiBAR after six and ten repetitions of spin-on-dope/anneal processes
Further repetition of the liquid boron dope/anneal cycles on the 5 µm thick SiBAR reduces the TCF to -1.5 ppm/˚C and not any lower (Figures 5 & 6). At such very high levels of doping, the resistivity values could not be accurately measured and hence are not reported.
doping. Hence, it is inferred that degenerate boron doping compensates for almost all of the frequency variation contributed by the temperature coefficient of Young’s modulus and the measured residual TCF of -1.5 ppm/˚C stems from the linear thermal expansion coefficient of the resonating heavily doped silicon material. Thus, an overall reduction in TCF from -29 ppm/˚C to -1.5 ppm/˚C can be achieved in silicon micromechanical resonators using degenerate boron doping. However, the longer hours of annealing required for achieving degenerate boron doping and the limited thickness of such achievable heavily doped boron layers are some of the drawbacks of this technique. TCF Compensation via Boron-assisted Aluminum Doping Boron diffuses as an interstitial dopant [11], which demands for longer hours of annealing to become electrically active. On the other hand, aluminum (also a p-type dopant) becomes readily electrically active by diffusion via self-interstitial mechanism [12]. Aluminum can be thermomigrated against a temperature gradient into hundreds of microns thick silicon within few tens of minutes [13]. Additionally, the uniformity and speed of aluminum thermomigration is enhanced by the presence of boron atoms in silicon [12]. Thus, boron-assisted aluminum thermomigration is a faster alternative to degenerate boron doping for TCF reduction. This was investigated by evaporating a thin layer of aluminum onto the SiBAR (Figure 7(b)) and joule-heating it by passing a current through the SiBAR resonator element via the narrow support elements [14] (Figure 7(c)). Aluminum on top of the relatively cold SiBAR diffuses through the silicon towards the hot support elements thereby doping it (Figure 7(d)).
Fig. 6: Saturation of TCF at ~-1.5 ppm/˚C of the 5 µm thick SiBAR after seven repetitions of spin-on-dope/anneal processes
It is known that the amount of frequency variation with temperature contributed by the linear thermal expansion coefficient (α) is ~1.5 ppm/˚C to the total TCF value [10]. This can also be verified by setting the TCE value of the orthotropic silicon model in ANSYS to zero while simulating the TCF of the SiBAR. Such an ANSYS simulation predicts a TCF value of -1.2 ppm/˚C which is in close agreement with the lowest measured TCF values via degenerate boron
Fig. 7: Schematic of aluminum thermomigration into SiBAR for TCF reduction
500 Å of aluminum was evaporated onto a boron-doped 20 µm thick SiBAR with a starting resistivity of ~10-3 Ω-cm. Thermomigration was performed by passing a current of 120 mA through the SiBAR for 10 minutes after which the TCF reduces from -22.13 ppm/˚C to -7.93 ppm/˚C (Figure 8).
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Fig. 8: TCF of a 20 µm thick [~10-3 Ω-cm; boron-doped] SiBAR before and after thermomigration with 500Å of aluminum @ 120 mA for 10 minutes.
A similar thermomigration performed on the degenerate -4 boron-doped (~ 10 Ω-cm) 20 µm thick SiBAR yields a TCF as low as -2.72 ppm/˚C (the “After” curve in Figure 9). The “Before” curve in Figure 9 shows as interesting anomalous behavior due to the non-uniform boron doping profile along the 20 µm thickness of the SiBAR.
Fig. 11: SEM images of the SiBAR reported in Figure 9 (a) before and (b) after thermomigration with 500Å thick aluminum at 120 mA for 10 minutes. The charging up of the SiBAR under the SEM after aluminum thermomigration indicates that the aluminum has diffused completely from the surface of the SiBAR into the bulk. Wirebond traces are visible in (b) since the same device was used for SEM images.
Conclusions Temperature compensation of silicon micromechanical resonators has been achieved using degenerate boron-doping and boron-assisted aluminum doping. Very low-TCF values are obtained without compromising on the quality factor (Q) of the resonator. A starting TCF of -29 ppm/˚C is reduced to -2.72 ppm/˚C by the faster boron-assisted aluminum thermomigration and to -1.5 ppm/˚C by the relatively slower degenerate boron-doping process, while maintaining a Q of at least 28000 in vacuum. References
Fig. 9: TCF of a 20 µm thick [~10-4 Ω-cm; boron-doped] SiBAR before and after thermomigration with 500Å of aluminum at 120 mA for 10 minutes
Such low-TCF is obtained via boron-assisted aluminum thermomigration without compromising on the quality factor (Q) of the resonator which is measured to be 28000 in vacuum (Figure 10). The SEM images of this SiBAR before and after thermomigration are shown in Figure 11.
[1]
S. Pourkamali, G. K. Ho and F. Ayazi, “Low-Impedance VHF and UHF Capacitive SiBARs – Part I: Concept and Fabrication,” IEEE Transaction on Electron Devices, vol.54, no.8, pp. 2017-2023, Aug. 2007.
[2]
F. Schoen, et al., “Temperature Compensation in Silicon-Based MicroElectromechanical Resonators,” Proc. IEEE MEMS, pp. 884-887, Jan. 2009.
[3]
P. Csavinszky, et al., “Effect of Doping on Elastic Constants of Silicon,” Physical Review Letters, vol. 132, no. 6, pp 2434-2440, Dec. 1963.
[4]
W. P. Wason, et al., “Ultrasonic attenuation and velocity changes in doped n-type germanium and p-type silicon and their use in determining an intrinsic electron and hole scattering time,” Phys. Rev. Letters, vol. 10, no. 5, pp. 151-154, Mar. 1963.
[5]
P. Csavinszky, et al., “Effect of holes on the elastic constant C’ of degenerate ptype Si,” Journal of Applied Physics, vol.36, no.12, pp 3723-3727, Dec. 1965.
[6]
S. Pourkamali, G.K. Ho and F. Ayazi, “”Low-Impedance VHF and UHF Capacitive SiBARs – Part II: Measurement and Characterization,” IEEE Transaction on Electron Devices, vol.54, no.8, pp. 2024-2030, Aug. 2007.
[7]
W.R.Thurber, et al., “Resistivity-dopant density relationship for boron-doped silicon,” Journal of Electrochemical Society, vol. 127, no. 10, pp. 2291, Oct 1980.
[8]
C. Iliescu, et al., “Analysis of highly doping with boron from spin-on diffusing source,” Surface & Coatings Technology, vol. 198, no. 1-3, pp 309-313, Aug. 2005.
[9]
N. Ono, et al., “Measurement of Young’s modulus of silicon single crystal at high temperature and its dependency on boron concentration using flexural vibration method,” Japanese Journal of Applied Physics, vol. 39, pp 368-371, Feb. 2000.
[10] Z. Hao, S. Pourkamali, and F. Ayazi, “VHF Single Crystal Silicon Elliptic BulkMode Capacitive Disk Resonators; Part I: Design and Modeling,” IEEE Journal of Microelectromechanical Systems, vol. 13, no. 6, pp. 1043-1053, Dec.2004. [11] B. Sadigh, et al., “Mechanism of boron diffusion in silicon: an ab initio and kinetic Monte Carlo study,” Phys. Rev. Letters, vol. 83, no. 21, pp 4341-4344, Nov. 1999. [12] O. Krause, et al., “Determination of aluminum diffusion parameters in silicon,” Journal of Applied Physics, vol. 91, no. 9, pp 5645-5649, May 2002. [13] C. C. Chung and M. G. Allen, “Thermomigration-based junction isolation of bulk silicon MEMS devices,” Journal of Microelectromechanical Systems, vol. 15, no. 5, pp 1131-1138, Oct. 2006.
Fig. 10: Measured response of the aluminum thermomigrated SiBAR with TCF = -2.72 ppm/°C showing a quality factor (Q) of 28000 in vacuum
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[14] A. K. Samarao and F. Ayazi, “Post-fabrication electrical trimming of silicon bulk acoustic resonators using joule heating,” Proc. IEEE MEMS, pp 892-895, Jan. 2009.
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electrode, the electrostatic force applied to the corresponding face of the resonator induces an acoustic wave that propagates through the bar, resulting in a width-extensional resonance mode whose frequency is primarily defined by W. Small changes in the size of the capacitive gap on the other side of the device induce a voltage on the sense electrode whose amplitude peaks at the mechanical resonant frequency.
Introduction With an increasing demand for higher levels of integration in existing and emerging microsystem applications, integrated alternatives to discrete frequency-selective devices are necessary. Silicon micromechanical resonators can enable a cost-effective integrated platform for timing, wireless connectivity and multi-band spectral processing. However, successful insertion of these devices into cost and power sensitive consumer applications require effective tuning, trimming and compensation techniques. The uncompensated temperature coefficient of frequency (TCF) for a native silicon resonator is on the order of -30 ppm/˚C [1], which is far greater than that of quartz resonators. To achieve stable low-phase-noise frequency references, the TCF needs to be compensated without compromising on the quality factor (Q) of the resonator. Circuit-based compensation techniques add to the power budget whereas depositing compensating materials with positive TCF on silicon [2] is impractical on certain resonator geometries and may reduce the Q of the resonator. It is well known that the number of free charge carries in a silicon material can be significantly increased by doping. The effect of these free charge carriers on the elastic constants of silicon have long been studied [3] but are being used for the first time in this work to achieve temperature compensation in silicon micromechanical resonators. TCF compensation is demonstrated using Silicon Bulk Acoustic Resonators (or SiBARs) [1] as test vehicle. As shown in Figure 1, a SiBAR consists of a rectangular bar element placed between two electrodes, and supported symmetrically by two thin tethers on the sides. A DC polarization voltage (Vp) applied between the resonator and the electrodes generates an electrostatic field in the narrow capacitive gaps. When an AC voltage is applied to the drive
97-4244-5640-6/09/$26.00 ©2009 IEEE
Fig. 1: (a) Structure, (b) SEM view, and (c) simulated width-extensional mode shape of the SiBAR device in ANSYS (t=10 µm).
Resonance modes like the width-extensional mode of the SiBAR involve the propagation of longitudinal waves through the bulk of the resonating microstructure. The Young’s modulus (E) that determines the elastic resonance frequency possesses a negative temperature coefficient. This results in a material softening which causes the stiffness and thereby the frequency of the silicon resonators to decrease with increasing temperature. Though the linear thermal expansion coefficient (α) of silicon also contributes to the TCF, its contribution is negligible compared to the temperature coefficient of Young’s modulus (TCE). Temperature compensation techniques that have been reported so far combat the effect of TCE [2] whereas this work utilizes the understanding on the cause of TCE [3-5] as explained in the following section, to achieve compensation. Principle of TCF Compensation via Degenerate Doping An acoustic wave propagating through a semiconducting resonator like the SiBAR induces a flow of free charge carriers by distorting the valence and conduction bands [3]. For example, the valence band in silicon consists of three
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bands of energy in k-space of which two of the energy surfaces are degenerate and are energetically favorable to contain almost all the holes. The strain produced by the propagation of acoustic waves through such a material momentarily splits the degenerate bands leading to a flow of holes from lower to higher energy levels [4]. The amount of band splitting and the resulting flow of holes increases with temperature. The principle of conservation of energy requires that such a temperature dependant change in the electronic energy of the system manifest itself as a corresponding temperature dependant change in the elastic energy of the system, which causes a negative TCE in silicon. To minimize the effect of the momentary strain on the energy bands produced by the acoustic waves, a relatively large permanent strain can be created in the resonating microstructure by doping. For example, a boron dopant has a smaller radius than silicon and is bonded strongly to only three of the four adjacent silicon atoms when diffused in the crystal lattice. Such an atomic arrangement produces a very strong shear strain in the silicon lattice, which leads to a large permanent separation of the degenerate valence bands forcing more of the holes to occupy the higher energy band [5]. The additional band-splitting contributed by the propagation of acoustic waves or the rise in temperature as discussed before are now minimal in comparison, which serves to compensate the TCF of silicon resonators. At degenerate levels of boron doping, the acoustic wave can potentially be shielded entirely from the k-space contours of the valence band which in turn should completely compensate the TCE component of the TCF. In this work, TCF compensation is demonstrated in p-type silicon resonators using degenerate boron doping and boron-assisted aluminum doping. TCF Compensation via Degenerate Boron Doping SiBARs were fabricated with 100 nm capacitive airgaps using the HARPSS process [6] on a 10 µm thick boron-doped silicon wafer, with a starting resistivity of ~10-2 Ω-cm and a TCF of -29 ppm/˚C. The boron dopant density in silicon could be increased to ~ 7 × 1019 atoms/cm3 (i.e., ~10-3 Ω-cm) by repeated doping using conventional solid boron sources. But achieving degenerate levels of doping that requires a boron density of ~ 2 × 1020 atoms/cm3 (i.e., < 10-4 Ω-cm) [7] needed repeated doping using liquid spin-on boron dopant sources [8]. Throughout our experiment, the resistivity of the wafer was accurately monitored using four-point probe measurement. Table I illustrates the boron doping recipes using solid and liquid boron sources. In both cases, the borosilicate glass (BSG) layer that forms after every dope/anneal cycle was removed using hydrofluoric acid. The silicon wafer with the starting resistivity of ~10-2 Ω-cm was processed through five repetitions of the solid boron dope/anneal recipe to reduce the resistivity to ~10-3 Ω-cm at which a corresponding TCF of -18.9 ppm/˚C is measured (Figure 2). As seen from the square of the correlation coefficient of linearity (R2) reported in Figure 2, the linearity
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of the TCF curve is compromised to some extent due to the heavy boron doping. Table I: Boron doping recipe using solid and liquid boron sources Boron Dopant Dope Anneal Type Recipe Recipe Solid Boron Disks Liquid Boron Spin-on-Dopants
3 hours in furnace at 1050 ˚C Spin : 800 rpm for 40s / Bake : 200 ˚C for 3 min
5 hours at 1100 ˚C in N2 8 hours at 1100 ˚C in N2
Fig. 2: TCF measurements before (10-2 Ω-cm) and after (10-3 Ω-cm) boron doping (using solid boron sources) on a 10 µm thick SiBAR.
For further reduction of silicon resistivity, spin-on-dopant (SOD) boron sources were used [Futurrex Inc., BDC1-2000]. After six repetitions of liquid boron dope/anneal recipe on a silicon wafer with a starting resistivity of ~10-3 Ω-cm, the resistivity was further reduced by an order of magnitude to 10-4 Ω-cm (Figure 3). A reduction in TCF from -16.7 ppm/˚C at 10-3 Ω-cm to -10.5 ppm/˚C at 10-4 Ω-cm was measured (Figure 4). The increase in resonance frequency after doping is due to the increase in the Young’s modulus of silicon as a result of degenerative doping [9]. The maximum thickness of the heavily boron doped layer that can be achieved using SOD is limited to 7~8 µm [8] which leads to a non-uniform doping profile in SiBARs thicker than 8 µm. Hence, a 5 µm thick SiBAR with a resistivity of ~10-4 Ω-cm measured a much lower TCF of -3.56 ppm/˚C (Figure 5) indicating a more uniform doping profile along its thickness.
Fig. 3: Reduction in resistivity from 10-3 Ω-cm to 10-4 Ω-cm on a 10 µm thick device layer of an SOI.
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Fig .4: TCF Measurements before (~10-3 Ω-cm) and after (~10-4 Ω-cm) boron doping (using liquid boron sources) on a 10 µm thick SiBAR.
Fig. 5: TCF measurements of a 5 µm thick SiBAR after six and ten repetitions of spin-on-dope/anneal processes
Further repetition of the liquid boron dope/anneal cycles on the 5 µm thick SiBAR reduces the TCF to -1.5 ppm/˚C and not any lower (Figures 5 & 6). At such very high levels of doping, the resistivity values could not be accurately measured and hence are not reported.
doping. Hence, it is inferred that degenerate boron doping compensates for almost all of the frequency variation contributed by the temperature coefficient of Young’s modulus and the measured residual TCF of -1.5 ppm/˚C stems from the linear thermal expansion coefficient of the resonating heavily doped silicon material. Thus, an overall reduction in TCF from -29 ppm/˚C to -1.5 ppm/˚C can be achieved in silicon micromechanical resonators using degenerate boron doping. However, the longer hours of annealing required for achieving degenerate boron doping and the limited thickness of such achievable heavily doped boron layers are some of the drawbacks of this technique. TCF Compensation via Boron-assisted Aluminum Doping Boron diffuses as an interstitial dopant [11], which demands for longer hours of annealing to become electrically active. On the other hand, aluminum (also a p-type dopant) becomes readily electrically active by diffusion via self-interstitial mechanism [12]. Aluminum can be thermomigrated against a temperature gradient into hundreds of microns thick silicon within few tens of minutes [13]. Additionally, the uniformity and speed of aluminum thermomigration is enhanced by the presence of boron atoms in silicon [12]. Thus, boron-assisted aluminum thermomigration is a faster alternative to degenerate boron doping for TCF reduction. This was investigated by evaporating a thin layer of aluminum onto the SiBAR (Figure 7(b)) and joule-heating it by passing a current through the SiBAR resonator element via the narrow support elements [14] (Figure 7(c)). Aluminum on top of the relatively cold SiBAR diffuses through the silicon towards the hot support elements thereby doping it (Figure 7(d)).
Fig. 6: Saturation of TCF at ~-1.5 ppm/˚C of the 5 µm thick SiBAR after seven repetitions of spin-on-dope/anneal processes
It is known that the amount of frequency variation with temperature contributed by the linear thermal expansion coefficient (α) is ~1.5 ppm/˚C to the total TCF value [10]. This can also be verified by setting the TCE value of the orthotropic silicon model in ANSYS to zero while simulating the TCF of the SiBAR. Such an ANSYS simulation predicts a TCF value of -1.2 ppm/˚C which is in close agreement with the lowest measured TCF values via degenerate boron
Fig. 7: Schematic of aluminum thermomigration into SiBAR for TCF reduction
500 Å of aluminum was evaporated onto a boron-doped 20 µm thick SiBAR with a starting resistivity of ~10-3 Ω-cm. Thermomigration was performed by passing a current of 120 mA through the SiBAR for 10 minutes after which the TCF reduces from -22.13 ppm/˚C to -7.93 ppm/˚C (Figure 8).
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Fig. 8: TCF of a 20 µm thick [~10-3 Ω-cm; boron-doped] SiBAR before and after thermomigration with 500Å of aluminum @ 120 mA for 10 minutes.
A similar thermomigration performed on the degenerate -4 boron-doped (~ 10 Ω-cm) 20 µm thick SiBAR yields a TCF as low as -2.72 ppm/˚C (the “After” curve in Figure 9). The “Before” curve in Figure 9 shows as interesting anomalous behavior due to the non-uniform boron doping profile along the 20 µm thickness of the SiBAR.
Fig. 11: SEM images of the SiBAR reported in Figure 9 (a) before and (b) after thermomigration with 500Å thick aluminum at 120 mA for 10 minutes. The charging up of the SiBAR under the SEM after aluminum thermomigration indicates that the aluminum has diffused completely from the surface of the SiBAR into the bulk. Wirebond traces are visible in (b) since the same device was used for SEM images.
Conclusions Temperature compensation of silicon micromechanical resonators has been achieved using degenerate boron-doping and boron-assisted aluminum doping. Very low-TCF values are obtained without compromising on the quality factor (Q) of the resonator. A starting TCF of -29 ppm/˚C is reduced to -2.72 ppm/˚C by the faster boron-assisted aluminum thermomigration and to -1.5 ppm/˚C by the relatively slower degenerate boron-doping process, while maintaining a Q of at least 28000 in vacuum. References
Fig. 9: TCF of a 20 µm thick [~10-4 Ω-cm; boron-doped] SiBAR before and after thermomigration with 500Å of aluminum at 120 mA for 10 minutes
Such low-TCF is obtained via boron-assisted aluminum thermomigration without compromising on the quality factor (Q) of the resonator which is measured to be 28000 in vacuum (Figure 10). The SEM images of this SiBAR before and after thermomigration are shown in Figure 11.
[1]
S. Pourkamali, G. K. Ho and F. Ayazi, “Low-Impedance VHF and UHF Capacitive SiBARs – Part I: Concept and Fabrication,” IEEE Transaction on Electron Devices, vol.54, no.8, pp. 2017-2023, Aug. 2007.
[2]
F. Schoen, et al., “Temperature Compensation in Silicon-Based MicroElectromechanical Resonators,” Proc. IEEE MEMS, pp. 884-887, Jan. 2009.
[3]
P. Csavinszky, et al., “Effect of Doping on Elastic Constants of Silicon,” Physical Review Letters, vol. 132, no. 6, pp 2434-2440, Dec. 1963.
[4]
W. P. Wason, et al., “Ultrasonic attenuation and velocity changes in doped n-type germanium and p-type silicon and their use in determining an intrinsic electron and hole scattering time,” Phys. Rev. Letters, vol. 10, no. 5, pp. 151-154, Mar. 1963.
[5]
P. Csavinszky, et al., “Effect of holes on the elastic constant C’ of degenerate ptype Si,” Journal of Applied Physics, vol.36, no.12, pp 3723-3727, Dec. 1965.
[6]
S. Pourkamali, G.K. Ho and F. Ayazi, “”Low-Impedance VHF and UHF Capacitive SiBARs – Part II: Measurement and Characterization,” IEEE Transaction on Electron Devices, vol.54, no.8, pp. 2024-2030, Aug. 2007.
[7]
W.R.Thurber, et al., “Resistivity-dopant density relationship for boron-doped silicon,” Journal of Electrochemical Society, vol. 127, no. 10, pp. 2291, Oct 1980.
[8]
C. Iliescu, et al., “Analysis of highly doping with boron from spin-on diffusing source,” Surface & Coatings Technology, vol. 198, no. 1-3, pp 309-313, Aug. 2005.
[9]
N. Ono, et al., “Measurement of Young’s modulus of silicon single crystal at high temperature and its dependency on boron concentration using flexural vibration method,” Japanese Journal of Applied Physics, vol. 39, pp 368-371, Feb. 2000.
[10] Z. Hao, S. Pourkamali, and F. Ayazi, “VHF Single Crystal Silicon Elliptic BulkMode Capacitive Disk Resonators; Part I: Design and Modeling,” IEEE Journal of Microelectromechanical Systems, vol. 13, no. 6, pp. 1043-1053, Dec.2004. [11] B. Sadigh, et al., “Mechanism of boron diffusion in silicon: an ab initio and kinetic Monte Carlo study,” Phys. Rev. Letters, vol. 83, no. 21, pp 4341-4344, Nov. 1999. [12] O. Krause, et al., “Determination of aluminum diffusion parameters in silicon,” Journal of Applied Physics, vol. 91, no. 9, pp 5645-5649, May 2002. [13] C. C. Chung and M. G. Allen, “Thermomigration-based junction isolation of bulk silicon MEMS devices,” Journal of Microelectromechanical Systems, vol. 15, no. 5, pp 1131-1138, Oct. 2006.
Fig. 10: Measured response of the aluminum thermomigrated SiBAR with TCF = -2.72 ppm/°C showing a quality factor (Q) of 28000 in vacuum
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[14] A. K. Samarao and F. Ayazi, “Post-fabrication electrical trimming of silicon bulk acoustic resonators using joule heating,” Proc. IEEE MEMS, pp 892-895, Jan. 2009.
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