1. Introduction

In recent years, micro-electro-mechanical system (MEMS) devices are rapidly developed with the advances of science and technology. Among them, MEMS gyroscopes are playing an increasingly important role in inertial navigation applications owing to the advantages of small volume, low cost, low power consumption and so on^[1–3]. The MEMS gyroscope is the sensor that measures the rate of rotation of an object^[4] based on the Coriolis effect. It is a kind of active device, which requires both actuation and detection mechanisms. The most common actuation method and Coriolis response detection technique are electrostatic actuation and capacitive sensing, respectively. The stability of the resonant frequency is one of the most basic characters for gyroscopes.

However, it is an extremely challenging work to make such devices highly reliable. Some researchers have observed that the resonance frequency can drift either due to mechanical changes of the resonator^[5] or due to charging^[6]. The frequency drifts due to charging have a relation with DC voltage bias which, in general, is required for the electrostatic actuation of the MEMS devices^[7].

It has been reported that many MEMS devices such as micro switches, micro resonators and micro mirrors all suffer from dielectric charging problems^[8–10]. In this paper, we study the mechanism and variation of the parasitic charge in the MEMS gyroscopes, and analyze the effect of the parasitic charge on the output stability. Firstly, we briefly introduce the working principle of the gyroscopes in Section 2.1. Next in Section 2.2, we analyze the effect of the frequency drift due to the parasitic charge accumulation on the output of the gyroscopes in theory. Finally, we experimentally demonstrate that the parasitic charge is harmful to the output stability of MEMS gyroscopes in Section 3.

2. Statement of the problem

In this section, we introduce the working principle of vibratory gyroscopes and the effect of the frequency drift on the output of the gyroscopes.

2.1 Working principle

We take linear vibratory gyroscopes for example. The basic architecture of a vibratory gyroscope is comprised of a drive-mode oscillator that generates and maintains a constant linear momentum, coupled to a sense-mode Coriolis accelerometer that measures the sinusoidal Coriolis force induced due to the combination of the drive vibration and an angular rate input^[4]. Simply it can be equivalent to a mass-spring-damper model in both drive and sense directions. For a closed-loop drive and open-loop sense scheme, the equations of motion of the resonator are

$${m_{ x}}\frac{{{{\rm d}^2}x}}{{{\rm d}{t^2}}} + {c_{ x}}\frac{{{\rm d}x}}{{{\rm d}t}} + {k_{ x}}x = {F_0}\sin {\omega _{\rm d}}t,$$

(1)

$${m_{ y}}\frac{{{{\rm d}^2}y}}{{{\rm d}{t^2}}} + {c_{ y}}\frac{{{\rm d}y}}{{{\rm d}t}} + {k_{ y}}y = - 2{\varOmega _z}{m_{\rm c}}\frac{{{\rm d}x}}{{{\rm d}t}} - {k_{ {yx}}}x,$$

(2)

where m_x, m_y and m_c are the proof-mass of the drive mode, sense mode and the portion of the drive mode that contributes to the Coriolis force, k_x and k_y are the stiffness of the drive and sense mode, while c_x and c_y are damping factors of the drive and sense mode, respectively. F₀ is the amplitude of a harmonic force with the frequency ω_d (usually equals to the resonant frequency of drive mode ω_x). In the y direction, the displacement of the proof-mass contains the rate signal and quadrature error. The former is related to the input Ω_z and the latter is related to the coupling stiffness k_yx resulting from fabrication imperfections^[4]. They are orthogonal so that can be extracted by demodulation. The steady-state component of the response is also harmonic, of the form

$$x = {x_0}\sin \left( {{\omega _{\rm d}}t + {\varphi _{\rm d}}} \right),$$

(3)

$$y = {y_1}\cos \left( {{\omega _{\rm d}}t + {\varphi _{\rm y}}} \right) + {y_2}\sin\left( {{\omega _{\rm d}}t + {\varphi _{\rm y}}} \right),$$

(4)

where x₀ and φ_d are the amplitude and phase of the drive mode, y₁ and y₂ are the amplitude of the displacement induced by rate signal and quadrature error respectively, while φ_y is the phase of the sense mode output. When Ω_z = 0, y₁ = 0, then the output is called ZRO (zero rate output). The ZRO is

$$\begin{split} {u_{\rm {ZRO}}} = & \frac{1}{2}\frac{{{k_{\rm {yx}}}{x_0}/{m_{\rm y}}}}{{\sqrt {{{\left( {{\omega _{\rm y}}^2 - {\omega _{\rm d}}^2} \right)}^2} + {\omega _{\rm y}}^2{\omega _{\rm d}}^2/{Q_{\rm y}}^2} }} \hfill \\ & \times \sin \left[ { - \,\, {\rm {arctan}}\frac{{{\omega _{\rm y}}{\omega _{\rm d}}}}{{{Q_{\rm y}}\left( {{\omega _{\rm y}}^2 - {\omega _{\rm d}}^2} \right)}} - {\varphi _{\rm d}}} \right], \end{split}$$

(5)

where ω_y and Q_y are the resonant frequency and quality factor of the sense mode. It is shown that the ZRO is affected by ω_y which is a time-dependent variable determined by dielectric parasitic charges. We will discuss this problem in detail in the next section.

2.2 Frequency drift

The model in Fig. 1 shows a device that is actuated by the electrostatic force. The bias voltage V will soften the resonant element and generate a negative stiffness effect^[11–13].

According to the deduction in Ref. [11], the variation of stiffness caused by the bias voltage could be expressed as

$$\Delta k = - \frac{{xz}}{{{y^3}}}{\varepsilon _0}{V^2}.$$

(6)

where x, y, z are dimensions in three directions, ε₀ is the air permittivity and V is the bias voltage.

The dielectric layer in MEMS devices is created either by a natural chemical reaction like oxidation or by artificial means to achieve a certain function (such as Pyrex substrate). In the dielectric layer, the parasitic charge would accumulate due to the DC voltage required for the device operation. The physical mechanisms of the charging phenomena are not fully understood, yet it can be illustrated with a contactless or contact model^[8]. Polarization and migration will occur in both models. Besides, in the contact model, the dielectric layers and the electrodes are adhesive to each other, while ion injection is another charging mechanism^[14–18]. All of these three charge movements can interrupt the internal balance of the electric field and form an extra electrostatic force on the MEMS devices. These charge behaviors in dielectric materials are considered to be the main cause of drifting of many micro devices, such as accelerometer, micro switch, micro resonators, micro mirrors and so on.

In the capacitive MEMS device shown in Fig. 3, a dielectric layer of thickness d and of charge density ρ will induce a built-in voltage. We can use the simple model in Fig. 2 to give a quantitative description of the problem^[19]. ρ is the charge density of the dielectric on the bottom electrode, and is a time-dependent variable determined by applied voltage, dielectric material properties and environmental param-eters^[20–23]. Applying Gauss’s Law for a Gaussian surface that surrounds the bottom sheet charge,

$${\varepsilon _0}\left( {{\varepsilon _{\rm r}}{E_2} - {E_1}} \right) = {\rm{\rho }},$$

(7)

where ε_r is the relative permittivity of the dielectric, E₁ is the electric field in the gap between the top electrode and bottom dielectric, E₂ is the electric field in the dielectric between ρ and the bottom plate. Summing the electric fields according to Faraday’s law,

$$\left( {g - d} \right){E_1} + d{E_2} = V,$$

(8)

where g is the equilibrium gap, and d is the thickness of the dielectric layer. The effective potential drop V_eff across the air gap is

$${V_{\rm {eff}}} = \left( {g - d} \right){E_1}.$$

(9)

$${V_{\rm {eff}}} = \left( {V - \frac{{d{\rm{\rho }}}}{{{\varepsilon _0}{\varepsilon _{\rm r}}}}} \right)/\left( {1 + \frac{d}{g}} \right).$$

(10)

The portion of the effective voltage due to trapped surface charge is called the “built-in” voltage^[24],

$${V_{\rm{bi}}} = V - {V_{{\rm {eff}}}} \approx \frac{{d{\rm{\rho }}}}{{{\varepsilon _0}{\varepsilon _{\rm {r}}}}}{\rm{ = }}\frac{{dQ}}{{{\varepsilon _0}{\varepsilon _{\rm {r}}}A}},$$

(11)

where Q is the total charges in the dielectric layer and A is the area of the electrode.

Three models of the dielectric charge dynamics were described in Ref. [25]. However, the dielectric charges are all time-dependent variables. In the most general case, we use the exponential dynamics model to describe the dielectric charge. The dynamics of the charge resembles an exponential form

$${{Q}}\left( t \right) = {Q_{\rm {max}}}\left( {1 - {{\rm e}^{ - t/\tau }}} \right).$$

(12)

So

$${{{V}}_{\rm {bi}}} = \frac{{d\rho }}{{{\varepsilon _0}{\varepsilon _{\rm r}}}} = \frac{d}{{{\varepsilon _0}{\varepsilon _{\rm r}}A}} \times \left( {{Q_{\rm {max}}}\left( {1 - {{\rm e}^{ - t/\tau }}} \right)} \right),$$

(13)

$${{{V}}_{\rm s}} = \frac{d}{{{\varepsilon _0}{\varepsilon _{\rm r}}A}} \times {Q_{\max }},$$

(14)

where V_s is the saturation voltage eventually reached by dielectric charges and time constant τ represents the characteristic time scale of the process.

We could obtain that the effective voltage is a function of ρ, and is not the same with the applied DC voltage at most times. For the devices that are actuated by the electrostatic force, the bias voltage will soften the resonant element and generate a negative stiffness effect^{[12, 13]}. When the electrostatic stiffness co-exists with the mechanical stiffness, the resonant frequency of the sense mode can be expressed as

$${\rm{\omega }} = \sqrt {\frac{{k + \Delta k}}{m}} = \sqrt {\frac{{k - \displaystyle\frac{{xz}}{{{y^3}}}{\varepsilon _0}{{\left( {V + {V_{\rm {bi}}}} \right)}^2}}}{m}} .$$

(15)

$${\rm{\omega }}\left( t \right) = \sqrt {\frac{{k + \Delta k}}{m}} = \sqrt {\frac{{k - \displaystyle\frac{{xz}}{{{y^3}}}{\varepsilon _0}{{\left( {V + {V_{\rm s}}\left( {1 - {{\rm e}^{ - t/\tau }}} \right)} \right)}^2}}}{m}} .$$

(16)

It suggests that the resonant frequency is related to the effective voltage. So the parasitic charge in the dielectric layer would lead to the frequency drift of the gyroscopes and affect the output stability.

Take Eq. (16) into Eq. (5) and make the necessary omission in the calculation, the output of the gyroscope is

$${u_{\rm {ZRO}}}\left( t \right) = \frac{G}{{\left| {\frac{{k - \displaystyle\frac{{xz}}{{{y^3}}}{\varepsilon _0}{{\left( {V + {V_{\rm s}}\left( {1 \,-\, {{\rm e}^{ - t/\tau }}} \right)} \right)}^2}}}{m} - {\omega _{\rm d}}^2} \right|}},$$

(17)

where G is a gain constant.

3. Results and discussion

In this section, we discuss the analytical and experimental results of the MEMS gyroscopes working at different bias conditions.

The gyroscope in the experiment is based on the well proven Sensonor ButterflyGyro^TM structure^[26] and encapsulated with Pyrex glass which act as the dielectric material in this case. The asymmetric driving beams of this structure lead to an obvious mechanical properties difference between the drive and sense mode. The parameters of the gyroscope in the experiment and some constants used in the simulation are shown below in Table 1.

Fig. 3 shows both the drive and sense resonant frequency with different DC bias voltage of the MEMS gyroscopes. According to the observation data, the resonant frequency of the drive mode keeps constant. This is owing to the fact that the mechanical stiffness of the drive mode is relatively high so that the Δk is omitted. Yet the mechanical stiffness of the sense mode is several magnitudes lower than the drive mode and the Δk is unneglectable. So there is a frequency drift of the sense mode.

The drifting tendency is related to the τ and V_s; we give the simulation results in Fig. 4. The time constant τ represents the characteristic time scale of the drifting process and does not show a significant difference with different bias voltage. They both turn out to be about 1300 s. V_s is the saturation voltage eventually reached by dielectric charges which has a positive correlation with the bias voltage. The value of V_s is 0.052 V with 6 V bias voltage while it is 0.032 V with 5 V bias voltage.

Due to the frequency drift, the ZRO of the gyroscope is shown in Fig. 5. Fig. 5(a) is the output applied with 6 V DC voltage, while Fig. 5(b) is with 5 V DC voltage. We can see that the output presents an obvious tendency, and takes about 2000 s to reach a stable value. Fig. 5(c) shows the output tendency during a DC voltage changing process. Initial 7000 s shows the output with 6 V DC voltage and it corresponds with Fig. 5(a). Then the voltage reduced to 5 V during 7000–13 000 s, the output voltage changed to −1.25 V, which is the same as the stable state in Fig. 5(b). Between 13 000–18 000 s the voltage changed to 6 V again and the output turned out to be −0.86 V, which is equal to the first period. Fig. 5(c) shows the simulation and test results of ZRO with 6 and 5 V DC bias voltage. The test result matches the simulation perfectly when the bias voltage is 6 V while there is an error between them when the bias voltage is 5 V. As we can see in Figs. 3(a) and 3(c), the mismatch between drive mode and sense mode is clearer with 6 V bias voltage, so the drift of sense mode has less impact on the ZRO. However, the mismatch is less clear when the bias voltage is 5 V and the ZRO is more sensitive to the frequency drift, and what is more, G is no longer a constant anymore, which contributes to the error between the test and simulation results.

According to the previous discussion in Section 2, it is the DC voltage rather than AC voltage that makes the dielectric charge undergo a redistribution process. To verify this phenomenon and exclude the thermal effect, we use the testing scheme in Ref. [27] for reference. As in Fig. 6, we make the gyroscope only applied with AC ± 6 V or DC 6 V bias voltage but not actuated, and write down the outputs every twenty minutes. The results are shown in Fig. 7.

As we can see in Fig. 7, the output of the gyroscope with DC stressing has a similar trend with Fig. 5(a), while the output with AC stressing is basically stable, verifying that the DC voltage is the cause of the dielectric charge accumulating and frequency drift.

According to the analysis above, the drift induced by charging only occurs when both dielectric layers and DC bias exist at the same time, therefore, it can be overcome by eliminating one or both of their existences.

The dielectric layers in MEMS devices could be divided into two kinds. They are created either by natural chemical reaction or by artificial means. For the former, it is difficult to totally prevent the materials from such oxidation, so what we could do is to control the processing environment to the utmost extent. For the latter, we could improve the dielectric quality by optimizing the deposition process to reduce the defects in the dielectric layers^{[28, 29]} or by using a dielectric which has higher dielectric constants and lower leakage current density.

On the other hand, the DC bias is another cause of drifting, so a commonly adopted measure is to replace the DC bias by AC bias. However, the change of bias would lead to the failure of the origin actuation mode, so we have to redesign a feasible method. Other approaches include reducing the actuation voltage, applying the non-electrostatic actuation and so on.

4. Conclusion

Output voltage drift was observed in MEMS gyroscopes. We studied the variation of the parasitic charge and analyzed the effect of the parasitic charge on the output stability. Due to the DC voltage required for the electrostatic actuation, the charges in dielectric will undergo a process of redistribution to induce a residual voltage. The voltage is a time-dependent variable so that it affects the resonant frequency of the gyroscopes, furthermore, the output of the gyroscopes. According to the simulation and experimental results, the saturation voltage eventually reached by dielectric charges has a positive correlation with the bias voltage. The time constant τ representing the characteristic time scale of the drifting process does not show significant difference with different bias voltage. It takes about 2000 s for the output voltage to reach a stable value.