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In the present work, the design and the environmental conditions of a micromachined thermal accelerometer, based on convection effect, are discussed and studied in order to understand the behavior of the frequency response evolution of the sensor. It has been theoretically and experimentally studied with different detector widths, pressure and gas nature. Although this type of sensor has already been intensively examined, little information concerning the frequency response modeling is currently available and very few experimental results about the frequency response are reported in the literature. In some particular conditions, our measurements show a cut-off frequency at -3 dB greater than 200 Hz. By using simple cylindrical and planar models of the thermal accelerometer and an equivalent electrical circuit, a good agreement with the experimental results has been demonstrated.

Thermal micromachined accelerometers have recently attracted attention and been intensively studied because of their high shock reliability due to the absence of seismic mass and their small size inducing a low fabrication cost. These last years, many kinds of accelerometers have been developed for measuring vibration, shock, inertial motion and studied because they are widely used in the field of aviation, automation in machine condition monitoring and more especially in automotive industry, such as air bag crash detection. The most widely used principles of operation are based on piezoelectric or piezoresistive effects or capacitive changes [

In this document, the thermal accelerometer response is theoretically evaluated and experimentally measured. Specific parameters are taken into account to design the sensor, like the size of the detectors and simple models are used for a better understanding of the thermal behavior and the frequency response evolution. Indeed, we give the influence of several parameters such as detector width, gas nature and pressure on the sensor response, as well as its evolution depending on temperature variation. For the different prototypes we experimentally used, the overall system is enclosed in a gas chamber and the detectors are usually placed at 300 μm from the heater. Different measurements are described depending on the experimental conditions.

_{x} [_{x} and platinum layers are 5,000 and 3,000 Å respectively, and a Cr adhesion-promoting layer is used. Platinum is electron-beam evaporated at 400 °C and vacuum-annealed at 500 °C. Its electrical resistivity is about 15 μΩ.cm with a temperature coefficient of resistance (TCR) of 3.1 × 10^{−3} °C^{−1}. The platinum resistors and the SiN_{x} are successively patterned by a Corial 200 IL Reactive Ion Etching (RIE) device. To obtain suspended resistors on SiN_{x} bridges, the silicon is anisotropically etched out using a KOH solution at 85 °C. For all experimental studies carried out here, the silicon cell size is the same whereas the detector width and the gas nature inside the cavity will be different: N_{2}, CO_{2} and He.

The electronic control of the LDS (Ling Dynamic Systems) vibrating pot is equipped with an automatic control mechanism in amplitude and acceleration frequency, LDS DSC 4 (Digital Sine Controller). Indeed, the acceleration is compared with a reference accelerometer linked to the conditioning Nexus system. The whole system is limited to a maximum acceleration of 40 g. All samples are typically submitted to a sinusoidal acceleration of ±2 g covering a frequency domain from 10 Hz up to 1,000 Hz.

In this section, thermal behavior of thermal resistor is modeled by using equivalent electrical components: Equivalent thermal capacitance initially and then equivalent thermal resistance. Geometric models used to establish these components are developed. The two values will allow us to estimate the detector bandwidth in the next section and to compare it to the sensor cut-off frequency.

A thermal capacitance _{th}_{P,mat}

A detector is composed of the superposition of three thin layers: A layer of SiN_{x} covered by a Pt one, with a Cr-adhesion layer. Each of them creates a thermal capacitance, connected in parallel considering their superposition. As we cannot measure thermo-physical parameters of the three different thin layers, bulk parameters will be considered for each one.

Two models are proposed to evaluate the thermal resistance: A cylindrical model on one side and a planar model on the other.

The following assumption is made: The evaluated thermal resistance is the one between the detector surface and its surrounding gas. Indeed, the heat exchange we focus on is an interface phenomenon and it can be located in the thermal boundary layer, where thermal convection is assimilated to thermal conduction. Thus thermal parameters linked to convection heat transfer are calculated through conduction phenomenon.

Firstly, we consider a two-heated-concentric-circular-cylinder model, with a length _{1} with a radius r_{1} and the outer being the substrate with a temperature T_{2} < T_{1} and a radius r_{2}.

With this geometry, the thermal heat flux exchange can be calculated by:

By definition, the thermal resistance between the gas and the detector is given by:

The thermal resistance between the surrounding gas and the surface of the detector can also be calculated using a second model: A planar one as described in _{1} and T_{2}, the silicon cap temperature and the element under consideration respectively.

With this 2D complex geometry, a simple expression of a thermal flux, as in a cylindrical case, cannot be extracted. The problem is simplified by using a shape factor

Therefore, the thermal resistance between gas and detector surface is given by:

For this study, the cavity dimension is constant and respectively equal to 1,000 μm × 2,000 μm × 800 μm and the cavity is filled with a different gas, N_{2}, CO_{2} and He, for which the Prandtl numbers are close to 0.7 [

A first assumption for the sensor response time is to consider the time response of the detectors negligible compared to the bubble gas one. In this way the system is equivalent to a first order RC circuit and its response time is set by the product of the thermal resistance with the heat capacitance of the bubble:
_{P,gas}

The corresponding cut-off frequency at −3 dB is consequently proportional to:

Frequency response analysis includes both the evolution of thermal sensitivity in the bandwidth and the evolution of cut-off frequency at −3 dB with detector width.

Even if the main objective is not to demonstrate the evolution of thermal sensitivity with detector width, we point out in

We can expect that the presence of the detector within the gas will influence the value of the frequency bandwidth of the sensor. Thus, in order to take this effect into account, we consider two first order systems, one after the other, to model the frequency response of the bubble and the detectors respectively. They have been implemented as two electrical RC circuits and simulated in order to obtain a value of frequency bandwidth.

The transfer function in dB is reiterated here:
_{c,b} the cut-off frequency of the gas bubble and f_{c,d} the detector one.

The bubble frequency response taken into account for f_{c,b} has already been simulated and is equal to 175 Hz [

In _{c,b} = 175 Hz, for a detector width equal to 2 μm. We can conclude that the experimental answer is in concordance with the simulated sensor frequency bandwidth at -3 dB.

As a conclusion, the frequency response of the sensor is an important characteristic and is a function of at least two phenomena. The first corresponds to the frequency response of gas under the acceleration inducing a temperature difference on the sensitive axis. The second corresponds to the frequency response of detector to a temperature variation in the gas. Therefore, the frequency response of the detector mainly depends on the detectors thermal resistance and its thermal capacitance.

We can extract a theoretical value of the detector cut-off frequency from the two models studied in Section 3, assuming that this value is inversely proportional to the product of the thermal resistance between the gas and the detector R_{th} with its thermal capacitance C_{th}, as done in (7) for cut-off frequency of gas.

Using the three models, we finally obtain the frequency bandwidth behaviors given in

We have seen that if the detector width becomes less than 10 μm for this cavity size, the system heads toward a first order circuit response. In such a case, it will only be the gas nature (see Section 4) and the gas pressure which will condition the bandwidth of the thermal accelerometer. In the next section, the influence of this last parameter on the global sensor response will be demonstrated.

Concerning the bandwidth measurements as a function of the gas pressure, these have been done with an accelerometer which has the same size as given in Section 4 and a detector width equal to 10 μm.

With model (8) and thermal diffusivity expression equal to _{P,gas}

We noticed that all experimental results present a similar behavior law with a curve slope n equal to minus one in a log-log scale when the pressure of nitrogen, carbon dioxide or helium gas increases, for a detector width of 10 μm. Saturation at lower and higher pressure can be explained by the response time of the detector not becoming negligible compared to the gas bubble. Moreover we assume that the heater temperature rise and the propagation of the gradient shape are both modified at a higher pressure.

The following measurements have been made with a typical-dimension sensor as defined in Section 6 in an air gas environment.

As mentioned in (8), cut-off frequency is a function of gas diffusivity ^{1.62} does for air [

Inducing a variation of external temperatures, the gas temperature was modified, as were detectors and the heater. In each case, bandwidth was estimated and results are reported in ^{1.58}, value close to air thermal diffusivity behavior, T^{1.62}.

These studies essentially investigate the bandwidth of a thermal accelerometer based on thermal exchange. The sensor was micromachined by using micro-electronics techniques. The response of the sensor has been modeled and experimentally investigated as a function of detector width and gas nature and pressure. The cylindrical and planar model results completed with an equivalent electrical circuit simulation have shown a good accordance with experimental results. This theoretical model has shown the variation of gas frequency response with a good accordance to the experimental results and shed light on the influence of the detectors on the global frequency response of the sensor. Finally, a frequency bandwidth at −3 dB of 320 Hz has been experimentally measured for a given cavity dimension for Helium gas at 2.15 bar.

Temperature profile with and without acceleration.

SEM images of the sensor: Global view and view of different detector widths, on top. Cavity scheme, underneath.

Cylindrical model shape.

Planar model shape.

Sensor bandwidth at −3 dB

Experimental thermal sensitivity

Experimental sensor bandwidth, experimental detector bandwidth (confirmed by the electrical model) and theoretical frequency response of gas

Theoretical electrical model and the experimental response of the thermal accelerometer for a detector width of 2 μm.

Detector bandwidth at −3 dB obtained with cylindrical, planar and electrical models

Experimental results of the sensor bandwidth at −3 dB

Sensor bandwidth

The authors would like to sincerely acknowledge the work of MM J. Lyonnet, J.M. Peiris and F. Pichot for their technical assistance.