基于SOI的电容式微加速度计器件物理模型与实验研究
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摘要
基于微机械工艺的电容式微加速度计因其体积小、功耗低、灵敏度高和方便与电路集成等优点,已成为民用工业与国防航空航天惯性导航领域的关键部件和研究热点。其中,在基于SOI的微加速度计器件层应力研究、开环模式非线性模型、跨机械、热、电场领域的一体化仿真方法与模型和梳齿加工偏差对性能的作用诸领域尚有待突破与创新。
本论文针对当前电容式微加速度计有待解决的理论技术问题,展开了器件层的应力分析和实验、开环工作模式非线性研究,对集成一体化仿真予以了探索,进行了MEMS加工误差与性能的关系研究。
1.分析了SOI器件层中存在的各种应力,及这些应力对微加速度计敏感质量块结构变形的作用,进行了实验,提出了一种支撑结构,导出了该支撑结构的最大变形模型。通过对基于器件层厚50μm,氧化层厚5μm键合SOI片的该结构进行测量,结果表明该模型基本能描述器件层存在应力时的变形。
2.根据力学理论和电路读出原理,针对开环模式非线性误差大和SOI器件层存在应力梯度的特点,从非线性特性中最大可测量加速度、初始工作点和吸合点与动定电极间距d的变化规律出发,并结合器件层应力梯度导致弹性梁刚度的改变,提出了电容式微加速度计非线性模型。模型指出,随着输出电压的增加,可测量加速度近似成线性增加,在靠近最大可测量加速度amax时斜率增大,其后随输出电压的增加而减小至吸合点;amax由d和弹性梁刚度与敏感质量块质量之比λ决定,且d的作用优于λ,在计入器件层应力梯度后,可测量加速度范围将增大;d在小于初始工作点时加速度计不能工作,该点由传感头参数η决定且η最佳取值范围为1×1019~2×1019,此时准静态条件下吸合点大于整个动电极行程的95%。
3.就MEMS微加速度计涉及机械、电和热等多物理领域情形,基于力学、热学、电学理论和微加速度计工作原理,依据微加速度计中SOI器件层应力梯度、温度与气膜阻尼的关系,用电压形式描述了气体阻尼、应力梯度、温度和机械结构运动微分方程,并与偏置电压产生的静电力相联系,提出了突破中间电极小位移近似的接口电路模型,从而实现了微加速度计的一体化仿真。通过与文献实验数据和相关理论结果进行比较,表明该模型能较好地应用于微加速度计的一体化仿真。
4.针对微机械工艺加工可能引起的梳齿间距与版图设计值的偏差,从概率统计和加工误差理论出发,给出了当各梳齿间距偏差在一定范围内独立且均匀分布时,电容式微加速度计电容与静电力模型、灵敏度模型和加速度信号为冲击与阶跃时的响应三物理模型。
(1)电容与静电力模型表明,电容和静电力均为类高斯分布,它们在一定区段出现的可能性可由其准期望和准方差确定,该准期望与无加工误差情况相比有小于5%的偏移;准方差依赖于梳齿数目和梳齿加工误差,当梳齿数目由10增大到60时,电容与静电力分布准方差分别增大约2倍和1倍,而当梳齿加工误差从5%增大到20%时,则分别增大约3.5倍和2.5倍。其结果与有机结合多物理场、有限元和Monte-Carlo方法原理的仿真数据较为接近。
(2)导出了单边电容和双边电容驱动的硅微加速度计冲击与阶跃信号响应物理模型。经过有限元仿真和Monte-Carlo模拟验证,结果表明理论模型与仿真值之差小于10%。模型指出,当梳齿间距偏差由0变化到20%,加速度计在受到冲击与阶跃加速度信号作用时,其可靠工作范围将比无偏差理想情况下降10%~15%。
(3)建立了可描述灵敏度偏差概率的模型。该模型与通过另一途径统计方法的仿真验证结果相差小于10%。模型表明,当偏差约为25%时,灵敏度减小5~10%的概率系40~50%,而其增大5~10%的概率约25~30%;同时,当梳齿数目越小,灵敏度偏差概率越大,在梳齿数目小于20时,灵敏度减小10%的概率约30%,但当梳齿数目增大到50~60时,该概率减小到10%左右。
Due to the small volume, low power, high sensitivity and easily being integrated with circuits, the capacitive micro-accelerometer fabricated by micro-machined process has become key part in civil industry and defense fields as inertia navigation device and therefore, has attracted many attentions and researches recently. In the researches, the residual stress in device layer,the nonlinear model of accelerometer operating in open loop mode, the integrated simulation of the device in mechanic, electric and thermal fields, and the effect of comb fingers gaps’deviation on performance of SOI accelerometers are still in need of being developed.
To deal with the current theoretical and technology problems in SOI capacitive micro-accelerometers, an investigation on nonlinear model of open loop operating mode, integrated simulation in multi-physics fields, residual stress in device layer and the relationship between performance and MEMS process error has been carried on.
Firstly, after the analysis of various residual stresses in device layer of SOI and the contribution of these stresses to the deformation of proof mass, a support structure is proposed and a maximum deflection model of the structure is investigated. Meanwhile, an experiment with different dimensions of the proof mass on SOI wafer with device layer depth 50μm and dioxide layer depth 5μm is done, in which the maximum deflection is measured. It is indicated by the experiment that the proposed model could basically describe the maximum deflection of proof mass.
Secondly, to cope with the nonlinear error of open-loop accelerometers and stresses in SOI device layer, based on the rule of maximum measurand acceleration, initial operating and pull in point changing with the distance between electrodes d and, the variation of spring beams’stiffness due to the stresses, the nonlinear model of micro-machined accelerometer is proposed. It is indicated that, by the model, with the output increasing, the measured acceleration increases roughly linear with the output voltage until it approaches the maximum acceleration amax, then it decrease till the point where electrodes are pulled in. The amax is decided by d andλ, which is the ratio between the spring constant and the mass of the microstructure, and the amax will be increased if the stresses in SOI device layer are taken into account. The accelerometer can not work when d is less than a critical point which is determined by transducer’s parameterηand the optimum ofηis from 1×1019 to 2×1019, where the pull in point is beyond 95% of the full travel range of the moveable electrode.
Thirdly, to deal with the separated situation of MEMS accelerometer’s simulation in multi-physics fields, based on the mechanics, thermal, electrical theories and mechanism of micro-accelerometer, considering the stress gradient in SOI device layer and the relationship between air damping and temperature, the air damping, the stress gradient and the movement of mechanical structure has been transferred into the form of voltage, then, after combined with the electrostatic force and interface circuits, an integrated simulation model is proposed,which can simulate the middle electrode in the whole travel range. Through the comparison of results obtained by proposed model, classic formula and experiments, it is indicated that the model could basically be applied in the integrated simulation of the micro-accelerometers.
Finally, due to the deviation between actual comb finger gap and dimension on mask, based on probability and process error theory, the device physical models, which are capacitance and electrostatic force model, sensitivity model and the pulse and step signal response model, are proposed when the deviation is distributed in the adjacent scope of the design value with equal probability.
It is indicated that, by the capacitance and electrostatic force model, both the capacitance and electrostatic force distributions are quasi-Gaussian distribution type. Hence the probability for the capacitance and electrostatic force occurring in any interval can be estimated by means of quasi-mean and quasi-variance. The quasi-mean is a little bit different from the value of capacitance or electrostatic force without process error, less than 5%. The quasi-variance depends on the comb finger number and process error degree. When the comb finger number increases from 10 to 60, the quasi-variance of capacitance distribution increases 2 times while is about 1 time for electrostatic force distribution and, as process error degree is from 5% to 20%, the quasi-variance increases 1.5 times for capacitance distribution and is about 2.5 times for electrostatic force distribution.
Then, pulse and step acceleration signal response models of capacitive micro-accelerometer with single-sided driving mode and double-sided driving mode are derived respectively. The precision of the models has been verified by the FEA and Monte-Carlo methods with ANSYS software. The deviation between them is less than 10%. It is pointed out by the models that the reliable operation ranges of accelerometers will decrease 10%-15% when the comb gaps deviate 0-20% from ideal value. The models can be used in the estimation of the reliable operation ranges of capacitive accelerometers.
Lastly, a model to describe the sensitivity deviation probability is proposed and, is verified by a simulation in a statistical way. The difference between the model and simulation results is less than 10%. According to the model, when the process error is about 25%, the sensitivity will decrease 5-10% with 40-50% probability and increase 5-10% with 25-30% probability. Meanwhile, the smaller the comb finger’s number n is, the bigger the deviation probability is. When n is less than 20, the sensitivity decreases 10% with about 30% probability, but when n is close to 50~60, the probability deceases to about 10%. Through the model, a bridge between MEMS process error and the sensitivity is set up and,in the process, a new approach based on probability and statistical theory to study the effect of the process error on performance is presented.
本论文针对当前电容式微加速度计有待解决的理论技术问题,展开了器件层的应力分析和实验、开环工作模式非线性研究,对集成一体化仿真予以了探索,进行了MEMS加工误差与性能的关系研究。
1.分析了SOI器件层中存在的各种应力,及这些应力对微加速度计敏感质量块结构变形的作用,进行了实验,提出了一种支撑结构,导出了该支撑结构的最大变形模型。通过对基于器件层厚50μm,氧化层厚5μm键合SOI片的该结构进行测量,结果表明该模型基本能描述器件层存在应力时的变形。
2.根据力学理论和电路读出原理,针对开环模式非线性误差大和SOI器件层存在应力梯度的特点,从非线性特性中最大可测量加速度、初始工作点和吸合点与动定电极间距d的变化规律出发,并结合器件层应力梯度导致弹性梁刚度的改变,提出了电容式微加速度计非线性模型。模型指出,随着输出电压的增加,可测量加速度近似成线性增加,在靠近最大可测量加速度amax时斜率增大,其后随输出电压的增加而减小至吸合点;amax由d和弹性梁刚度与敏感质量块质量之比λ决定,且d的作用优于λ,在计入器件层应力梯度后,可测量加速度范围将增大;d在小于初始工作点时加速度计不能工作,该点由传感头参数η决定且η最佳取值范围为1×1019~2×1019,此时准静态条件下吸合点大于整个动电极行程的95%。
3.就MEMS微加速度计涉及机械、电和热等多物理领域情形,基于力学、热学、电学理论和微加速度计工作原理,依据微加速度计中SOI器件层应力梯度、温度与气膜阻尼的关系,用电压形式描述了气体阻尼、应力梯度、温度和机械结构运动微分方程,并与偏置电压产生的静电力相联系,提出了突破中间电极小位移近似的接口电路模型,从而实现了微加速度计的一体化仿真。通过与文献实验数据和相关理论结果进行比较,表明该模型能较好地应用于微加速度计的一体化仿真。
4.针对微机械工艺加工可能引起的梳齿间距与版图设计值的偏差,从概率统计和加工误差理论出发,给出了当各梳齿间距偏差在一定范围内独立且均匀分布时,电容式微加速度计电容与静电力模型、灵敏度模型和加速度信号为冲击与阶跃时的响应三物理模型。
(1)电容与静电力模型表明,电容和静电力均为类高斯分布,它们在一定区段出现的可能性可由其准期望和准方差确定,该准期望与无加工误差情况相比有小于5%的偏移;准方差依赖于梳齿数目和梳齿加工误差,当梳齿数目由10增大到60时,电容与静电力分布准方差分别增大约2倍和1倍,而当梳齿加工误差从5%增大到20%时,则分别增大约3.5倍和2.5倍。其结果与有机结合多物理场、有限元和Monte-Carlo方法原理的仿真数据较为接近。
(2)导出了单边电容和双边电容驱动的硅微加速度计冲击与阶跃信号响应物理模型。经过有限元仿真和Monte-Carlo模拟验证,结果表明理论模型与仿真值之差小于10%。模型指出,当梳齿间距偏差由0变化到20%,加速度计在受到冲击与阶跃加速度信号作用时,其可靠工作范围将比无偏差理想情况下降10%~15%。
(3)建立了可描述灵敏度偏差概率的模型。该模型与通过另一途径统计方法的仿真验证结果相差小于10%。模型表明,当偏差约为25%时,灵敏度减小5~10%的概率系40~50%,而其增大5~10%的概率约25~30%;同时,当梳齿数目越小,灵敏度偏差概率越大,在梳齿数目小于20时,灵敏度减小10%的概率约30%,但当梳齿数目增大到50~60时,该概率减小到10%左右。
Due to the small volume, low power, high sensitivity and easily being integrated with circuits, the capacitive micro-accelerometer fabricated by micro-machined process has become key part in civil industry and defense fields as inertia navigation device and therefore, has attracted many attentions and researches recently. In the researches, the residual stress in device layer,the nonlinear model of accelerometer operating in open loop mode, the integrated simulation of the device in mechanic, electric and thermal fields, and the effect of comb fingers gaps’deviation on performance of SOI accelerometers are still in need of being developed.
To deal with the current theoretical and technology problems in SOI capacitive micro-accelerometers, an investigation on nonlinear model of open loop operating mode, integrated simulation in multi-physics fields, residual stress in device layer and the relationship between performance and MEMS process error has been carried on.
Firstly, after the analysis of various residual stresses in device layer of SOI and the contribution of these stresses to the deformation of proof mass, a support structure is proposed and a maximum deflection model of the structure is investigated. Meanwhile, an experiment with different dimensions of the proof mass on SOI wafer with device layer depth 50μm and dioxide layer depth 5μm is done, in which the maximum deflection is measured. It is indicated by the experiment that the proposed model could basically describe the maximum deflection of proof mass.
Secondly, to cope with the nonlinear error of open-loop accelerometers and stresses in SOI device layer, based on the rule of maximum measurand acceleration, initial operating and pull in point changing with the distance between electrodes d and, the variation of spring beams’stiffness due to the stresses, the nonlinear model of micro-machined accelerometer is proposed. It is indicated that, by the model, with the output increasing, the measured acceleration increases roughly linear with the output voltage until it approaches the maximum acceleration amax, then it decrease till the point where electrodes are pulled in. The amax is decided by d andλ, which is the ratio between the spring constant and the mass of the microstructure, and the amax will be increased if the stresses in SOI device layer are taken into account. The accelerometer can not work when d is less than a critical point which is determined by transducer’s parameterηand the optimum ofηis from 1×1019 to 2×1019, where the pull in point is beyond 95% of the full travel range of the moveable electrode.
Thirdly, to deal with the separated situation of MEMS accelerometer’s simulation in multi-physics fields, based on the mechanics, thermal, electrical theories and mechanism of micro-accelerometer, considering the stress gradient in SOI device layer and the relationship between air damping and temperature, the air damping, the stress gradient and the movement of mechanical structure has been transferred into the form of voltage, then, after combined with the electrostatic force and interface circuits, an integrated simulation model is proposed,which can simulate the middle electrode in the whole travel range. Through the comparison of results obtained by proposed model, classic formula and experiments, it is indicated that the model could basically be applied in the integrated simulation of the micro-accelerometers.
Finally, due to the deviation between actual comb finger gap and dimension on mask, based on probability and process error theory, the device physical models, which are capacitance and electrostatic force model, sensitivity model and the pulse and step signal response model, are proposed when the deviation is distributed in the adjacent scope of the design value with equal probability.
It is indicated that, by the capacitance and electrostatic force model, both the capacitance and electrostatic force distributions are quasi-Gaussian distribution type. Hence the probability for the capacitance and electrostatic force occurring in any interval can be estimated by means of quasi-mean and quasi-variance. The quasi-mean is a little bit different from the value of capacitance or electrostatic force without process error, less than 5%. The quasi-variance depends on the comb finger number and process error degree. When the comb finger number increases from 10 to 60, the quasi-variance of capacitance distribution increases 2 times while is about 1 time for electrostatic force distribution and, as process error degree is from 5% to 20%, the quasi-variance increases 1.5 times for capacitance distribution and is about 2.5 times for electrostatic force distribution.
Then, pulse and step acceleration signal response models of capacitive micro-accelerometer with single-sided driving mode and double-sided driving mode are derived respectively. The precision of the models has been verified by the FEA and Monte-Carlo methods with ANSYS software. The deviation between them is less than 10%. It is pointed out by the models that the reliable operation ranges of accelerometers will decrease 10%-15% when the comb gaps deviate 0-20% from ideal value. The models can be used in the estimation of the reliable operation ranges of capacitive accelerometers.
Lastly, a model to describe the sensitivity deviation probability is proposed and, is verified by a simulation in a statistical way. The difference between the model and simulation results is less than 10%. According to the model, when the process error is about 25%, the sensitivity will decrease 5-10% with 40-50% probability and increase 5-10% with 25-30% probability. Meanwhile, the smaller the comb finger’s number n is, the bigger the deviation probability is. When n is less than 20, the sensitivity decreases 10% with about 30% probability, but when n is close to 50~60, the probability deceases to about 10%. Through the model, a bridge between MEMS process error and the sensitivity is set up and,in the process, a new approach based on probability and statistical theory to study the effect of the process error on performance is presented.
引文
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