Etienne Schwyter

Wulf Glatz

Lukas Durrer

Christofer Hierold

Flexible Micro Thermoelectric Generator based on Electroplated Bi 2+x Te 3-x

We present and discuss the fabrication process and the performance of a flexible micro thermoelectric generator with electroplated Bi 2+x Te 3-x thermocouples in a SU-8 mold. Demonstrator devices generate 278µWcm -2 at ∆T meas =40K across the experimental set up. Based on model calculations, a temperature difference of ∆T G =21.4K across the generator is assumed. Due to the flexible design and the chosen generator materials, the performance stays high even for curved contact surfaces. The measurement results correlate well with the model based design optimization predictions. I.

INTRODUCTION

Power MEMS are proposed -among other applicationsto provide electrical energy for autonomous microsystems and sensors. Micro thermoelectric generators (µTEGs) based on the Seebeck effect convert thermal energy, represented by heat flow (Q

• H ,Q • C
) and driven by a temperature difference across the generator, into an electrical potential difference. For thermoelectric generators the material of choice has to have not just a high Seebeck coefficient but also a good electrical conductivity for a low internal resistance of the generator and a high thermal resistance in comparison to the thermal resistances of the generator's package, which provides the thermal interface to the environment. Bismuth telluride (Bi 2+x Te 3-x ) is the state of the art thermoelectric material for power harvesting and cooling applications near room-temperature.

II. MICRO FABRICATION PROCESS

A. µTEG Chip Processing

In our previous publication [START_REF] Glatz | Optimization and Fabrication of Thick Flexible Polymer Based Micro Thermoelectric Generator[END_REF] on the fabrication, testing and modelling of fully flexible µTEGs, a first proof of concept was presented. This fabrication concept has been adopted and extended for the use of Bi 2+x Te 3-x as thermoelectric material, replacing the nickel and copper thermolegs. The fabrication process consists of four deposition, photolithography and etching steps followed by an electrochemical deposition (ECD) of both Bi-rich and Terich Bi 2 Te 3 in 200-500µm thick SU-8 molds, respectively. The chips are mechanically polished and three subsequent deposition, photolithography and etching steps complete the fabrication of the micro thermoelectric generator. For the Bi 2+x Te 3-x µTEGs, the material of choice for the electrical interconnects and contacts is gold, since copper would have been oxidized in the electrolyte solution and copper ion diffusion into the Bi 2+x Te 3-x thermo legs would have deteriorated the generator's performance. The SU-8 mold is left in place and serves as package providing flexibility and stability to the device. The Bi 2+x Te 3-x µTEG demonstrator device is shown in Fig. 1. Fig. 2 shows the working principle of a generator device. ISBN: 978-2-35500-006-5

B. Electrochemical Deposition of Bi 2+x Te 3-x

The electrochemical deposition of Bi 2+x Te 3-x has been studied intensively during the last few years [START_REF] Takahashi | Electric and thermoelectric properties of electrodeposited bismuth telluride (Bi2Te3) films[END_REF][START_REF] Martin-Gonzalez | Insights into the Electrodeposition of Bi2Te3[END_REF]. The full control over the stoichiometric ratio between the compounds is the most challenging task for ECD of Bi 2+x Te 3-x . However, electroplating is cheap, fast and efficient, which makes this method interesting for mass production.

The electrolyte solutions are prepared by dissolving 80mmol O 2 Te and 20-60mmol Bi 2 O 3 in 2M HNO 3 (65%). Thus a 2M HNO 3 solution containing Bi 3 + and HTeO 2 + ions with a measured pH-value of 0.02 is achieved. All depositions are performed with a three electrode configuration, where a Mercury Sulfate Electrode (MSE) is used as reference electrode. A Pt grid is used as counter electrode and the structured SU-8 molds were connected to the working electrode. It is advantageous to apply a pulsed plating method to homogeneously fill molds with high aspect ratios. Primarily the ion concentration at the reaction surface does not deplete and secondly the undesired reaction products have time to dissipate. The pulse and pause duration (1-200ms and 4-5s, respectively), the pulse current density (40-120Acm -2 ) or pulse potential (0.4-0.8V), the ion concentration (see above), the electrolyte convection (mechanical stirrer) and the temperature (room temperature) is set in order to deposit both, bismuth and telluride rich thermolegs [START_REF] Glatz | Novel mixed method for the electrochemical deposition of thick layers of Bi2+xTe3-x with controlled stoichiometry[END_REF] (Fig. 3). With shorter pulses, the material becomes denser, but at the same time the material's internal stress increases. Hence, cracks appear and the stoichiometric control cannot be guaranteed over the whole deposition area. Due to the high aspect ratio of the thermopile holes, ion diffusion is dominant over convection, for which reason the pause duration has to be rather long. The deposition of Birich Bi 2 Te 3 is performed in electrolyte baths with Bi 2 O 3 concentrations of 40-60mmol. The Te-rich Bi 2 Te 3 is deposited using Bi 2 O 3 concentrations of 20-40mmol.

III. MEASUREMENT SETUP

The manufactured µTEGs are tested and characterized with the same measurement setup as used for the Cu/Ni generators [START_REF] Glatz | Optimization and Fabrication of Thick Flexible Polymer Based Micro Thermoelectric Generator[END_REF]. The generators are clamped between a Peltier cooling element mounted on a water cooling system and three thermoresistors integrated in a metal bar. The temperature difference is measured and controlled by two thermoresistive sensors close to the generator contact area. The obtained potential difference and the internal electrical resistance of the µTEGs were measured in a four probe configuration. The matched load case maximum output power was subsequently calculated. Due to the limited thickness of the device and the non perfect thermal contact to heat source and sink, the temperature difference across the generator surfaces ∆T G is significantly lower than the temperature difference measured between the sensors ∆T meas . The fraction of ∆T meas that is lost in the interface is directly depending on the thermal interface resistance K, the TC length and the ratio of insulating to thermoactive area, A V . The specific electrical contact resistance ρ C directly influences the power output by influencing the internal resistance R i of the generator.

IV. ANALYTICAL MODEL

The model based analysis [START_REF] Glatz | Optimization and Fabrication of Thick Flexible Polymer Based Micro Thermoelectric Generator[END_REF] of the performance of the µTEGs is shown in Fig. 4. The characteristics are calculated for a ∆T meas of 40K and the average K=3.9KW -1 for Ni/Cu and Bi 2 Te 3 generators for comparison. The optimal TC length for Bi 2 Te 3 generators is around 100 -300µm. This fits well to the capabilities of microsystems technology, applying SU-8 as photosensitive resist and flexible substrate material. V.

MEASUREMENT RESULTS AND DISCUSSION

A. µTEG Chip Processing

The first set of Bi 2+x Te 3-x µTEGs yielded 71.6±1.0µWcm -2 for a ∆T meas of 40.3±0.1K (see Fig. 6). After an annealing step for 18 hours at 200°C, the performance increased by a factor of 3.9 and the generator delivered 278.5±5.4µWcm -2 for a similar ∆T meas of 40.0±0.3K. In comparison with the Cu/Ni generator, the Bi 2+x Te 3-x generator is over 60 times better for the same operating conditions. The maximum output power measured so far was 344.1µWcm -2 for ∆T meas =44.4K and results in an efficiency factor ZT of 0.18µWcm -2 K -2 .

B. Electrochemical Deposition of Bi 2+x Te 3-x

As depicted in Fig. 4 and5, thermolegs with constant stoichiometry and a thickness of over 300µm are deposited at a rate of 20µm/h. All the stoichiometric analysis have been made by Energy Dispersive X-ray Detection (EDX) and the measured stoichiometric ratios covering all the fabricated generators did range from 0.8 (Bi rich) to 2.1 (Te rich). Subsequent qualitative Seebeck measurements have reconfirmed that both, Bi-rich and Te-rich thermo legs with positive and negative Seebeck coefficients are achieved.

VI. CONCLUSION

The fabrication of vertical, flexible Bi 2+x Te 3-x µTEGs could be shown, by using a novel, fast and low cost ECD approach with deposition rates exceeding 20µm/h. This novel ECD method further more allows full control of the stoichiometry over the whole deposit. The deposition of nand p-rich Bi 2+x Te 3-x with a thickness of over 300µm and an aspect ratio of at least 1.5 could be proven. According to our analytical model, the stated process enables µTEG fabrication in the ideal thermoleg thickness regime. The µTEG performance could further be improved by a factor of 3.9, if thermal annealing was performed for 18h at 200°C.

In addition to annealing, we see several routes to optimization: adjust the stoichiometry of Te-rich and Bi-rich thermolegs for higher Seebeck coefficients by optimizing the process parameters during electroplating, reduce grain boundaries by adjusting the deposition speed, and reduce the internal resistance by improving contact and interconnect resistances. Electroplating allows increasing the thermo legs' length towards 200 -300µm, which is the simulated optimum for the given material combination (see Fig. 3). Electroplating in general offers several advantages above sputtering of Bi 2+x Te 3-x as thermoelectric material, which has yielded the current state-of-the-art µTEGs with an efficiency factor of 2.4µWcm -2 K -2 [START_REF] Bottner | New thermoelectric components using microsystem technologies[END_REF]. These advantages are: high deposition speed and thicker layers, low material and energy consumption and low costs on large substrate sizes.

Fig 1

 1 Fig 1 Micro thermoelectric generator with Bi2+xTe3-x thermolegs and gold interconnects.

Fig 2

 2 Fig 2 Working principle of a µTEG connected to a heat source and heat sink.

Fig 3 Fig 4

 34 Fig 3 Electrochemically deposited Bi2Te3 thermolegs with high aspect ratios.

Fig 5 Fig 6

 56 Fig 5 SEM image with overlaid EDX line scan of an electrochemically deposited Bi2+xTe3-x thermoleg measured in its growth direction.


Mustafa Calis

Omar Laghrouche

Marc P Y Desmulliez

O Lagrouche

Modelling methodology of MEMS structures based on Cosserat theory

Modelling MEMS involves a variety of software tools that deal with the analysis of complex geometrical structures and the assessment of various interactions among different energy domains and components. Moreover, the MEMS market is growing very fast, but surprisingly, there is a paucity of modelling and simulation methodology for precise performance verification of MEMS products in the nonlinear regime. For that reason, an efficient and rapid modelling approach is proposed that meets the linear and nonlinear dynamic behaviour of MEMS systems.

I.

INTRODUCTION

At the present time, modelling MEMS is widely made using either Finite Element Analysis (FEA) [START_REF] Cao | Three Dimensional Nonlinear Dynamics of Slender Structures: Cosserat Rod Element Approach[END_REF], component level modelling [START_REF] Lorenz | Network-type modeling of micromachined sensor systems[END_REF][3], Boundary-Element Method (BEM) [START_REF] Senturia | A Computer-aided design system for microelectromechanical systems (MEMCAD)[END_REF] or lumped level modelling [START_REF] Osterberg | Selfconsistent simulation and modelling of electrostatically deformed diaphragms[END_REF]. The main disadvantage of the FEA is the dependency of stress on the number of elements used to represent a component such as a microbeam cantilever. For accurate representation, the number of equations increases significantly and the model tends to be cumbersome and complicated, preventing thereby designers from performing real-time simulation. Even with the complexity involved by the FEA approach, this technique is unsuitable for taking into account large scale deformation/motion. Component level (also call nodal) methods have been introduced in the form of MEMS design tools containing a library parameterised behavioural models. This tool describes each component as a single element enabling a considerable reduction in simulation time compared to finite element models where the component is often subdivided into many elements. A major disadvantage of this component level method is the limitation of its library containing only frequently used MEMS components. In lumped level modelling the simulation time is also low since only the behaviour of the transducer is modelled. BEM requires only surface discretization and the treatment of boundary conditions at infinity, however discretizing the boundary integral equations leads to cumbersome systems whose memory costs scale as O(n 2 ) and solution costs scale with O(n 3 ) with n being the number of discretization unknowns [START_REF] Altman | FFTSVD: A Fast Multiscale Boundary-Element Method Solver Suitbale for Bio-MEMS and Biomolecule Simulation[END_REF].

A new approach is therefore unavoidable to reduce the design process and to enable simulation of complex MEMS structures. In that respect, this paper presents a new approach for modelling linear and especially nonlinear MEMS structures based on Cosserat theory for a better representation of stress in miniaturized systems, especially in the nonlinear regime. The use of Cosserat theory leads to a reduction of the complexity of the modelling and thus increases its capability to simulate microstructures in realtime, indispensable for haptic technology. To demonstrate the feasibility of the proposed model, a cantilever microbeam undergoing loads modelled with ANSYS, SABER and Cosserat theory are compared.

II. METHODS FOR MODELLING MEMS COMPONENTS

Methods for modelling MEMS components can usually be classified into two categories as shown in Figure 1. The exact methods include the Euler-Bernoulli [START_REF] Paz | Power series expansion of the general stiffness matrix for beam elements[END_REF] and the Timoshenko techniques [START_REF] Paz | Mathematical observations in structural dynamics[END_REF] which are solved using a power series expansion. The FEA, BEM and lumped mass methods are classified under the approximate methods and are solved using superposition techniques. The FEM is also known as the matrix displacement method. The word approximate is used since it assumes that displacements can be represented by simple polynomial expressions. Our approach uses a semi-analytical method based on both power series expansions and a multimodal approximate method. MEMS devices consist of a system of inter-connected slender structures and masses which is also known as a network of slenders. The Cosserat theory can implement such modelling systems as a network of Cosserat rods where each Cosserat rod will be further discretised in order to yield a dynamic description using ordinary differential equations (ODEs) instead of partial differential equations (PDEs). This considerably facilitates the use of numerical and analytical techniques; therefore it is possible to implement it using VHDL-AMS.

III. COSSERAT THEORY

A Cosserat rod can be described by defining a set of crosssections the centroids of which are connected by a curve which is referred to as the line of centroids.

( ) ( ) ( ) 1 1
, x t y t and

( ) ( ) ( ) 2 2
,

x t y t . φ 1 (t) and φ 2 (t) denote the angles between 3 d and 1 e at node P and Q , respectively. The shape function of the rod can be constructed by approximating PDEs with ODEs and interpolating ( ) In the Cosserat theory, the accuracy will depend on the method used to model the motion/deformation of the centroid line. Unlike in [START_REF] Wang | Construction of nonlinear dynamic MEMS component modesl using Cosserat theory[END_REF] where the Newton's dynamical law and analytical method are used, our approach is based on a semi-analytical method and on the Euler-Bernoulli equation of motion.

            = = + + + +                           ( 2 )
( )

2 3 0 1 2 3 ... a a a a U aq iwt w w w e = + + + + = (3)
where

{ } { } 1 2 3 4 1 2 3 4 U iwt U U U U q q q q e = = (4)
The matrices ix a and iy a represent axial and transverse displacements, respectively. w is the circular frequency. In equation [START_REF] Lorenz | Network-type modeling of micromachined sensor systems[END_REF] x u and y u can be rewritten respectively as 0 a q a q r i w t i w t

Substituting equations ( 5) and ( 6) into equation ( 1) we get the following equation

As we are only considering geometric nonlinearity, the matrix 0 a y is determined from the boundary conditions of the cantilever microbeam.

IV. RESULTS

To validate our design approach, our model is tested on a 2D microbeam cantilever undergoing transversal loads applied at the free end. Figure 3 shows the deflections for a given load obtained from the analytical solution, ANSYS and Cosserat theory using linear stiffness matrix. From the figure above, the results obtained for the different model are identical. Secondly, our model is tested in the same conditions as in [START_REF] Neul | A Modeling approach to include mechanical microsytem Computation components into the system simulation[END_REF]. The test consists of a static behaviour of a cantilever microbeam where a force of I compares the results of the analytical calculations, static analysis of the ANSYS and SABER models [START_REF] Neul | A Modeling approach to include mechanical microsytem Computation components into the system simulation[END_REF], with our model. Afterwards, to validate the nonlinear model, we compare the analytical buckling load [START_REF] Ugural | Advanced Strength and Applied Elasticity[END_REF] against the buckling load obtained using our model. The results show clearly that model is in good agreement with the SABER, ANSYS and analytical solutions (Table I). Then the nonlinear model is used to compute the first buckling load of the microbeam. Table II shows that the results obtained for 1, 4 and 10 elements for the discretization of the microbeam are very close to the analytical solution. Currently the nonlinear model is being used to compute the deflection of a cantilever microbeam when it is subjected to a vertical load at the free-end. Such a problem is not straightforward and extra physical considerations must be introduced in order to get the simulation close to the reality.

IV. CONCLUSION

In this paper, it is demonstrated that the proposed method for modelling linear effects in MEMS is valid. The Cosserat theory has been successfully used for modelling and testing simple structures such as a cantilever and a microbridge. Nonlinear problems such as the buckling of beams have also been tackled successfully. We believe that Cosserat theory will lead to a reduction of the complexity of the modelling and thus will increase its capability for real time simulation, indispensable for haptic technologies.

Fig. 1 .

 1 Fig. 1. Taxonomy of MEMS modelling Methods

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 11 April 2008 ©EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5

Fig. 2 . 1 d

 21 Fig. 2. A Cosserat Rod Figure 2 is a schematic construction of a 2D Cosserat beam element in the x -and y -plane. The motion in space of a nonlinear Cosserat rod segment can be represented as a vector ( ) , r s t , called a Cosserat curve, which describes the position of the line of centroids of the cross-sections (Figure 2, dotted line). The deformation of the slender MEMS structure represented by the deformation of the centroid line depends upon three vectors ( ) r s ,

Figure 2

 2 represents a schematic 2D Cosserat beam element in a moving frame where P and Q are two time-dependent end points having inertial Cartesian coordinates

,

  r s t for given end point values of the beams,

  the moment of inertia of the beam of cross-section A and Young modulus E . To solve the Euler-Bernoulli equation of motion, the displacements x u and y u in the transverse and axial directions are expanded in ascending powers of w [10].

Fig. 3 .

 3 Fig. 3. Deflection of 2d cantilever microbeam

-

  was applied at the free end in x and y directions. Table

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 11 April 2008 ©EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5

TABLE I STATIC

 I ANALYSIS OF THE LINEAR CANTILEVER MICROBEAM

TABLE II BUCKLING EFFECT

 IIEFFECT


M Calis

M P Y Desmulliez

Haptic Sensing for MEMS with Application for Cantilever and Casimir Effect

Haptic sensing for MEMS with application for cantilever and Casimir effect

INTRODUCTION

Commercial MEMS are usually the result of many manufacturing, characterisation, packaging and tests iteration runs used to optimise their performance and reliability. At the design and manufacturing phases, engineers employ a variety of software tools that deal with the analysis of complex geometrical structures and the assessment of various component interactions that often belong to different domains of energy. Software packages such as CoventorWare [1], MEMSCAP [2] and ANSYS [3] are based however on classical elasticity, which limits the validity of stress results for large deflections of structures. In addition, these software packages tend to be computationally intensive when dealing with complex microsystems. Commercial software packages such as CoventorWare [1], MEMS Pro [4], SABER [START_REF]SABER from Synopsys[END_REF], IntelliSuite [START_REF]IntelliSuite from IntelliSense[END_REF] do not allow real-time simulation. An efficient and rapid modelling approach that represents accurately the linear and nonlinear dynamic behaviors of MEMS is therefore called for.

II. HAPTIC SENSING TECHNOLOGY

The term haptic originates from the Greek "haptesthai" and means the sense of touch with both tactile (cutaneous) and kinesthetic (proprioception) feedback. The PHANTOM Omni [START_REF] Zilles | Haptic Rendering with the Toolhandle Haptic Interface[END_REF], an analogue interface, from the company "SensAble Technologies" enables the interaction of the user with virtual environments based on haptic rendering through touch. This haptic device is a "pen" (also called stylus, proxy and haptic interface point) located at the end of an arm controlled by electric motors. These new interfaces differ from current computer systems and are changing the way by which digital information is perceived and manipulated. One of the most powerful ideas behind haptic technologies is to design ideas creatively and not mathematically. A haptic system consists of two loops, called haptic loop and display loop that need to be maintained with an update rate of 1 KHz and 30 Hz, respectively (Figure 1) [START_REF] Calis | The application of Cosserat theory into haptic sensing technology: a new design methodology for MEMS[END_REF]. The display loop must be updated at 30 Hz since the Human Visual System (HVS) has a flicker fusion frequency around 30-60 Hz. The update rate of the haptic loop has been set at 1 KHz to avoid force artefacts and due to the human being able to detect discrete events at less than 1 KHz. Haptic computers represent an important advantage in simulated industrial product development, virtual prototyping, in assembly/disassembly process, in final design and finally for training. Haptic computers enable also designers the possibility to scale-up to a human ergonomic scale micro-assemblies (such as MEMS) which can consist of millions of parts and then scale the product back to its original size. These properties improve the manufacturing lead times and the maintenance. In addition, in VREs, scaling down MEMS allow designers to gain more information and better understanding of their devices/components since the designers will be immersed in the VRE and any part of the system will be accessible. The different needs of MEMS modelling and simulation methodology for the precise performance verification of

III. COSSERAT THEORY

Methods for modelling MEMS components can usually be classified into two categories, as shown in Figure 2. The exact methods include the Euler-Bernoulli [START_REF] Paz | Power series expansion of the general stiffness matrix for beam elements[END_REF] and the Timoshenko techniques [START_REF] Paz | Mathematical observations in structural dynamics[END_REF] which are solved using a power series expansion. The FEA, BEM and lumped mass methods are classified under the approximate methods and are solved using superposition techniques. The FEM is also known as the matrix displacement method. The word approximate is used since it assumes that displacements can be represented by simple polynomial expressions. Our approach uses a semi-analytical method based on both power series expansions and a multimodal approximate method [START_REF] Wang | Construction of nonlinear dynamic MEMS component modesl using Cosserat theory[END_REF]. In the Cosserat theory, the accuracy will depend on the method used to model the motion/deformation of the centroid line. Unlike in [START_REF] Calis | The implementation of Cosserat theoy into haptic sensing technology for large deflection beam model[END_REF] where the Newton's dynamical law and analytical solution are used, our approach is based on a semi-analytical method and on the Euler-Bernoulli equation of motion. In the solution process, the displacements x u and y u , in the transverse and axial directions, are expanded in ascending powers of w [START_REF] Przemieniecki | Quadratic Matrix Equations for Determining Vibration Modes[END_REF].

            = = + + + +                           (2)
By rewriting (2) as ( 3 ) and using the shape function and the algebraic manipulation presented in [START_REF] Calis | A New Modelling Methodology of MEMS Structures based on Cosserat Theory[END_REF] , the equation of motion for a free end -free end horizontal nonlinear microbeam (Figure 3) can be rewritten with the first order kinetic and strain terms ( )

1 1 1 2 2 2 T 2 2 T T iNL L NL 2 2 d q d q F M q K q q K q dt dt N L       = + +           (4)
where M and L K are the mass and the stiffness matrices, respectively.

'

( ) EA L L N L - =
is the axial force applied on the microbeam and NL K is the nonlinear geometric stiffness matrix. It is demonstrated in [START_REF] Calis | A New Modelling Methodology of MEMS Structures based on Cosserat Theory[END_REF] that the proposed method for modelling linear effects in MEMS valid and nonlinear problems such as the buckling of beams have also been tackled successfully.

IV. IMPLEMENTATION OF COSSERAT THEORY INTO HAPTIC SENSING TECHNOLOGY

The algorithm structure to simulate a Cosserat microbeam in real-time in a VRE using haptic sensing technologies can be broken down into five stages. The first phase is to convert/scale up the load generated with the PHANTOM Omni to the appropriate MEMS micro-world. The second step is to measure the displacement in the reference frame. This displacement is then translated into the moving frame where different forces are calculated and summed up to obtain the total force. Afterwards, the total force is transformed back to the system matrix. A scale down/conversion is subsequently carried out; the proper force feedback is then calculated and rendered. The algorithm is implemented in Object Oriented language. OpenHaptics is used to interact and to feel the MEMS components, OpenGL for graphics and Microsoft visual C++. A front end-user interface has been designed to readily interact with selected MEMS components as shown in Figure 5.

The model of a linear cantilever microbeam has then been integrated into the haptic environment as shown in figure 5. Deflection of the microbeam is occurring in real-time for a load applied by the pen of the haptic device. Figure 6 shows the deflection of another microstrcutre, a microbridge, when a load is applied with the stylus in the y-direction. Comparisons with the software packages ANSYS and SABER were not possible as these packages are not suitable for real-time simulation.

V. MODELLING SURFACES INTERACTION

The highly intensive market for faster computing power for smaller electronic die sizes drives the development of enhanced lithographic tools for greater miniaturization. This capability is likely to be translated in the field of MEMS/NEMS. Since these devices are generally movable structures on a semiconductor and as the miniaturization is increasing, the likelihood of the moveable elements to collapse onto the substrate is augmenting concurrently (Figure 7). Therefore, for efficient manufacturing process and high performance of these devices, a thorough understanding of these surfaces interactions during growth and device operation are decisive for reliable performance. A particular category of devices falling into this category is RF-MEMS switches and relays. The effect in MEMS/NEMS is known as stiction (static friction); the adhesion of contacting surfaces which, in general results in permanent stiction. Another example of phenomenon that can cause the failure or reduce the performance of MEMS/NEMS is the Casimir force, which is of quantum nature [15]. This consists in the attraction between a pair of neutral, parallel, conductive metal plates separated by a small gap causing moveable parts to collapse into the substrate. This effect arises since the average of virtual particles is greater surrounding the plates than between them since the distance between the plates are so small that the range of fluctuations is restricted causing the generation of an attractive force (Figure 8). Casimir effect has been predicted in 1948 by the Dutch physicist Hendrik Brught Gerhard Casimir [15], but it has only been confirmed experimentally in 1997, by Steven K. Lamoreaux using a torsion balance [START_REF] Lamoreaux | Demonstration of the Casimir Force in the 0.6 to 6mm Range[END_REF]. The Casimir effect is known to be a local version of the van der Walls force between molecules and a proof for the evidence of the quantum vacuum possessing infinite energy density populated by virtual particles.

In this work, only the stiction effect which occurs when component touch the substrate is considered. The modelling of this surface interaction has been implemented in our software to guide designers during the assembly process of for training engineers quickly to operate in this field (Figure 9). With our user-end interface, the cantilever can be tested for given parameters. Moreover, the range of load that can be applied on the cantilever can be tested virtually and corrected depending on the needs at the design stage. When the cantilever is very close to the substrate, a warning messages pops-up on the screen and once it touches the substrate, the cantilever sticks to it. By pressing the "Reset Failure" on the left hand side of the window, the cantilever will be placed back in its original position. The consideration of such surfaces interaction in CAD/CAM tools can enable to build physical prototypes late in the manufacturing cycle. The advantages of using haptic sensing technologies allow the designer to touch and feel the MEMS component and at the same time to see the deflection in real-time. The parameters of the beam, such as the Young modulus, the density and number of elements composing the cantilever can be changed readily using the left hand side menu.

VI. CONCLUSION

In this paper, it is demonstrated that Cosserat theory has been successfully integrated into haptic sensing technologies for modelling and testing simple structures such as a cantilever and a microbridge. In addition, the modelling surfaces interaction such as the stiction effect has also been integrated successfully in the software.

Fig. 1 .

 1 Fig. 1. Schematic of a haptic system

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 11 April 2008 ©EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5 MEMS products are summarized as shown in table 1.

Fig. 2 .

 2 Fig. 2. Taxonomy of MEMS modelling methodsOne reason of using the Cosserat theory is its capability to

Fig. 3 .

 3 Fig. 3. A Cosserat rod: Deformed beam The deformation of the slender MEMS structure represented by the deformation of the centroid line depends upon three vectors ( ) r s , ( ) 1 d s and ( ) 2 d s . In the Cosserat theory, the accuracy will

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 11 April 2008 ©EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5

Fig. 4 .

 4 Fig. 4. Haptic interface

Fig. 5 .Fig. 6 .

 56 Fig. 5. Linear cantilever microbeam in deflection for a vertical load

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 11 April 2008 ©EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5

Fig. 7 .

 7 Fig. 7. Stiction effect: a) Stuck finger on comb drive-Stiction. Courtesy of Sandia National Labs, MEMS reliability departments b) Cantilever after release etch adhering ti substrate. Stiction caused by capillary forces/condensation [16].

Fig. 8 .

 8 Fig. 8. Schematic representation of fluctuating virtual particles

Fig. 9 .

 9 Fig. 9. Stiction effect caused by the cantilever surface touching the substrate

TABLE I NEEDS OF HAPTIC TECHNOLOGIES FOR MEMS Application Needs Benefits of haptic technologies CAD/CAM design

 I

			• Manufacturing of mould
			at micro-scales for
		• Guide designers during assembly and disassembly process • Conception • Tolerance	LIGA processes • Sense surface, shape of components, deformation • Sense and effect of forces at micro-, meso-
			and nano-meters size
			structures
	Virtual	• Replace physical
	Prototyping	prototype by virtual
		model
		• Enhance product
		development

• For combining physical and digital modeling • Can guide designers • Improve manufacturing lead times Visualization • Analyses of any parts of the system • Ergonomic analysis • Scale-up/down • Increased information flow between user and the computer • Better understanding Maintenance • Verification • Diagnosis • Quick analysis of any default, the cause and solution • Security Training • Simulation • A better understanding • Useful for application related to manipulation • Force-feedback • Sense gravity, inertia • Motion of components


U Triltsch

S Büttgenbach

Next Generation of TCAD Environments for MEMS Design

Most available MEMS design environments focus on the integrated behavioral simulation of electronic and mechanical components. The fabrication process is considered a fixed sequence of well-known steps, which can be purchased from foundries. Simulation models rely on the assumption of ideal process conditions and do not take into account tolerances or intra die variations.

Recently there has been discussion that the gap between actual process results and that of predicted models are often out of a tolerable range. This leads to the demand of increased simulation accuracy for the most crucial process steps. The deviations of the comb structures in a resonator, which result from the tolerances in a Deep-RIE process, for example, can make the overall system design fail. To avoid these kinds of errors two things are needed: the technology provider has to know the parameter range of his process and be able to provide it to the designer without loosing his intellectual property. The designer in turn must be provided with a way to use this information to design a system which can perform in the range expected.

In this paper we present the latest version of the TCAD environment BICEP³S (Braunschweigs Integrated CAD-Environment for Product Planning and Process Simulation). By using a central process database, which allows the exchange of all relevant process data it is able to overcome many of the mentioned obstacles. The database and process planning tool can be used by process developers to document changes in process settings and the influence of such changes on the process result. This information can then be used by the designers to set-up a simulation file for a detailed analysis of the impact of such parameter changes on the requested design. This will be shown by the example of silicon etching using an atomistic etch simulator.

I.

INTRODUCTION

In this paper the most recent results of the design environment BICEP³S (Braunschweigs Integrated CAD-Environment for Product Planning and Process Simulation) are presented. This TCAD tool consists of a series of modules [START_REF] Straube | Modular software system for computer aided design of microsystems[END_REF] that integrate, for example, SoftMEMS system-level tools with detailed process simulators such as the etch simulator SUZANA [START_REF] Than | Simulation of anisotropic chemical etching of crystalline silicon using a cellular automata model[END_REF] as well as process design tools with a link to process-, material-and media databases [START_REF] Hansen | Büttgenbach: T-CAD for the analysis and verification of processing sequences[END_REF]. This approach combines system-level behavioral simulation and component-level design tools to provide engineers with a complete view of the overall design process.

Most of today's MEMS-EDA / TCAD environments focus on integrated behavioral simulation of electronic and mechanical components [4,5]. Fabrication technologies are considered to be a fixed sequence of well known process steps, which cannot be varied. Therefore, the main focus of the designer is on adapting the layout needed to meet the desired performance requirements of a device. There have been many academic publications regarding behavioral modeling and system-level design approaches around the turn of the millennium [START_REF] Senturia | Microsystem Design[END_REF][START_REF] Fedder | Top-Down Design of MEMS[END_REF][START_REF] Mukherjee | Hierarchical Design and Test of Integrated Microsystems[END_REF].

Recently there has been discussion that the gap between actual process results and that of predicted models are often out of a tolerable range. This leads to the demand of increased simulation accuracy for the most crucial process steps. The deviations of the comb structures in a resonator, which result from the tolerances in a Deep-RIE process, for example, can make the overall system design fail. To avoid these kinds of errors two things are needed: the technology provider has to know the parameter range of his process and be able to provide it to the designer without loosing his intellectual property. The designer in turn must be provided with a way to use this information to design a system which can perform in the range expected.

II. DESIGN ENVIRONMENT BICEP³S

The design environment BICEP³S focuses on the definition of interfaces between single modules and acts as a central control and data management platform for different phases of microsystem design (figure 1). A central database management system makes sure that all the relevant data, which is obtained during a product design process, is stored in a consistent way and can be accessed by different designers in a simultaneous engineering environment.

Each module focuses on a specialized task in componentoriented design cycles. Special consideration was given to the planning and optimization of consistent process flows for systems, which consist of components that have been developed using different technologies and are integrated ISBN: 978-2-35500-006-5 into the same innovative microsystem. Once the system is subdivided into functional elements the designer can search the database of predefined components for previously designed parts which meet the functional criteria, as described in [START_REF] Triltsch | T-CAD Environment for Multi-Material MEMS Design[END_REF]. Already existing components can be selected and their stored fabrication data as well as the parameterized layout data is used for the definition of the new component. In other cases components might have to be designed in detail, whereby the layout and process design module will support the designer during this design state.

When a 3D-model without detailed process features is sufficient, the layout and process data can be exported to commercially available 3D modelers, e.g. SoftMEMS or Coventor Designer, where a model can be created for further computational analysis. If a more detailed view of the process result is needed for certain process steps, specialized technology simulator can be used. These process simulators are able to extract parameters from the definition of the process flow and are therefore directly linked to the process design module. If the simulation results are to be used for further computational analysis, there are converters available to ensure conversion between CAD or FEM tools and the simulator. Such is the case for etch simulation results.

In some cases the mask design needs to be optimized. In theses situations the design environment offers a powerful module for the optimization of lithographic masks. An evolutionary algorithm is used to create suitable layouts. These layouts evolve over a certain number of generations starting from a predefined set of masks and a 3D reference structure. The process simulators are used for the assessment of each generated layout and a score is calculated from the predefined weights and degree of matching. To prevent the process from running into a local optimum, mutation is used to generate new individuals [START_REF] Triltsch | Optimization of Lithographic Masks Using Genetic Algorithms[END_REF].

Finally, the process planning module acts as an interface for the collection and documentation of all relevant process results during the fabrication process. For example, process engineers can store measurement protocols during the setup of the process. Such protocols are then used during inline measurements by the operator. This ensures full control over the quality criteria for the executed fabrication steps.

The connection and communication between all single modules is controlled by a workflow system. This system leads the engineers through different steps, which are associated with the current design problem.

As the performance of a system is often dependent on the actual process result, focus will be given to the capabilities of process simulation in the next section of this paper. Wet chemical etching of silicon is still an important technology for MEMS and the built in simulator along with its model and export functionality will be described in detail. As one example of the process simulation capabilities of BICEP³S, the etch simulator SUZANA is presented. SUZANA uses a cellular automaton model to simulate a wide range of three-dimensional geometries, including etchstop layers, double side etching and even locally disordered crystal zones. The first version of this simulator was developed from 1991 to 1994 [START_REF] Than | Simulation of anisotropic chemical etching of crystalline silicon using a cellular automata model[END_REF]. In the past few years the model has been further enhanced and new features such as tolerance analysis of crucial parameters have been introduced [START_REF] Straube | Modular software system for computer aided design of microsystems[END_REF]. This has made SUZANA one of the most accurate and flexible tools in this area available today. However, as it considers only the etch rates of the most relevant crystal planes, the simulation might not be as accurate in terms of predicting surface properties on very small features. A more accurate model which takes higher indexed planes into account was presented by Gosálvez [START_REF] Gosálvez | Atomistic Modelling of Anisotropic Etching of Crystaline Silicon[END_REF] and implemented into the VisualTAPAS [START_REF] Gosálvez | VisualTAPAS: an example of density functional theory assisted understanding and simulation of anisotropic etching[END_REF] project. For most design tasks the use of SUZANA is absolutely sufficient and benefits from its fast computation times and accuracy. Figure 2 demonstrates how a comb-like compensation element is removed along the etch limiting (111) planes. The pictures on the right hand side show simulation results after etching with KOH (30%) at 80 °C. The pictures on the left hand side are SEM micrographs. This example shows the high degree of matching between simulation results and the etched structures.

Fig. 2. A comparison between simulation and experimental results

shows the high accuracy of the algorithm.

A. Simulation Model

Two different types of etch simulators have emerged during the last few years. One uses the geometric model based on the construction of Wulff-Jaccodine, the other is an atomistically modeled etch process. The delimiting problem of the Wulff-Jaccodine algorithm is the occurrence of saddle points whenever concave-convex shapes are mixed. Frühauf [START_REF] Frühauf | Shape & Functional Elements of Bulk-Silicon Microtechnique[END_REF] tries to overcome these problems by performing twodimensional calculations in a first step and calculating threedimensional sidewalls from the generated line structures. However, simulations for complex undercut geometries and processes involving etching from top and bottom sides are not possible.

SUZANA uses the atomistic approach which implements a cellular automata based model of the etch process. Cellular automata can be considered discrete structures which consist of a large number of simple cells. Such cells are ordered into a grid, interact locally and can represent a certain number of predefined states (in this case etched or not etched). The interaction of all cells is defined by rules. A rule can be regarded as a function, whose arguments are the states of the cell under consideration and of the neighboring cells. Its value is the new state of the considered cell. The application of the rules to a certain cell results in a change of its state.

One basic cell of the silicon crystal structure can be described by two face-centered cubes, which are displaced by a quarter of the silicon lattice constant in each spatial direction. This results in eight relevant atoms for the complete description of one cell. In Figure 3 these atoms are depicted as darker and linked. There are two models available. If a less detailed model is sufficient, one macro cell of the model is formed by the eight corner points of one cubic cell [START_REF] Straube | Modular software system for computer aided design of microsystems[END_REF][START_REF] Than | Simulation of anisotropic chemical etching of crystalline silicon using a cellular automata model[END_REF][START_REF] Hansen | Büttgenbach: T-CAD for the analysis and verification of processing sequences[END_REF](4)(5)[START_REF] Senturia | Microsystem Design[END_REF][START_REF] Fedder | Top-Down Design of MEMS[END_REF][START_REF] Mukherjee | Hierarchical Design and Test of Integrated Microsystems[END_REF]. In this case each of the corner atoms represents the eight atoms (5, b-h). If a more detailed model is required a full set of atoms is defined, i.e. one macro cell describes the atoms 5 and b to h. The second model ensures accurate results in multi step simulations and leads to a higher resolution in the z-direction.

As described above, the interaction of the cells is described by rules. The next section will explain how rules model the behavior of the cells in the case of wet-chemical etching of silicon.

ISBN: 978-2-35500-006-5

B. Rules for Wet-Chemical Etching

The simulation uses discrete time steps to calculate the propagation of the etch front. For each step the algorithm iterates over all cells on the surface of the model and calculates the number of neighbors of the current cell. A Depending on the number of neighbors a decision is made as to whether a cell belongs to a certain crystal plane and the etch probability is calculated accordingly. For the three main planes the rules are as follows:

(100)-plane: Atoms of this plane have two neighbors of first order (fig. 5 a) and are etched with a probability P 100 . (110)-plane: Atoms of this plane have 3 neighbors of first order and less than 9 neighbors of second order OR 9 neighbors of second order and less then 3 neighbors of third order. They are etched with a probability P 110 .

(111)-plane: Atoms of this plane have 3 neighbors of first order and a minimum of 10 neighbors of second order OR nine neighbors of second order and a minimum of 9 neighbors of third order. They are etched with a probability P 111 .

If higher indexed planes have to be taken into account, appropriate rules have to be defined and etch probabilities must be gained from experimental data. Simulation speed decreases with an increasing number of rules.

C. Parameter and Layout Definition

The simulation process can be set-up in a graphical user interface. Masks for top and bottom sides as well as etch stop areas can be imported from EDA/ CAD environments or can be created from scratch.

The process panel is used to define dimensions and resolution of the simulated area. Other specific parameters such as etch rates, substrate orientation or concentration and temperature of the etchant can be specified or imported from the central database.

D. Post Processing of Simulation Results

To gain full advantage of the simulation results a link to CAD or FEM tools has been developed. The simulation results are stored in discrete voxel file at the specified model resolution. A user defined number of etch states is saved which, if chosen, can be translated into a surface data model and displayed in a built-in 3-D viewer. Another important feature, which speeds-up the design process, is the functionality of ACIS format export. For further use in FEM analysis the detailed simulation results can be simplified. If, for example, the surface topography of sidewalls is of less relevance than the transmission between a membrane and a boss structure as depicted in figure 8, these details can be removed. The ACIS file can be imported into most solid modelers. In the example below, SolidWorks was used to import the simulation result and CosmosWorks was utilized to simulate the mechanical behavior of the simulated structure. Figure 5 gives an overview of the workflow for SUZANA to FEM export.

IV. SUMMARY AND PERSPECTIVE

A TCAD tool for MEMS, which combines process and layout design and provides links to behavioral modeling tools as well as specialized modules for the optimized design of single components was presented. This tool supports the design of new fabrication processes along with behavioral simulation and layout design. The features of the latest version of the modular software environment BICEP³S (Braunschweigs Integrated CAD-Environment for Product Planning Process Simulation) was explained. Due to the importance of detailed simulation of single process steps in fabrication oriented design cycles, an atomistic etchsimulator was presented as an example for specialized simulation modules of our design environment.

One major issue of future research will be the ISBN: 978-2-35500-006-5

enhancement of the etch simulation software. Currently the implementation of a new cellular automaton for deep reactive ion-etching is being investigated.

Fig. 5. The results can be exported any CAD or FEM tool using the ACIS-format.

Fig. 1 :

 1 Fig. 1: The single modules of BICEP³S form a closed environment for MEMS design.

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 11 April 2008 ©EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5 III. ETCH SIMULATION SUZANA

Fig. 3 .

 3 Fig. 3. The elementary cell of the silicon crystal can be modeled by 8 atoms.

Fig. 4 .

 4 Fig. 4. The number of cells in the neighborhood an atom (black) is used to determine whether an atom belongs to a (100)-plane (a), (110)-plane (b) or (111)-plane (c).

ACKNOWLEDGMENT

The Deutsche Forschungsgemeinschaft (DFG) has financially supported this work within a collaborative Research Center (Sonderforschungsbereich 516) titled, 'Design and Fabrication of Active Microsystems'.


El Mehdi Boujamaa

Yannick Soulie

Frederick Mailly

Laurent Latorre
email: latorre@lirmm.fr

Pascal Nouet

Rejection of Power Supply Noise in Wheatstone Bridges: Application to Piezoresistive MEMS

Keywords: MEMS, conditioning electronics, noise optimisation I

This paper deals with the design of MEMS using piezoresistivity as transduction principle. It is demonstrated that when the sensor topology doesn't allow a perfect matching of strain gauges, the resolution is limited by the ability of the conditioning circuit (typically a Wheatstone bridge) to reject power supply noise. As this ability is strongly reduced when an offset voltage is present at the output of the bridge, the proposed architecture implements a feedback loop to control MOS transistors inserted in the Wheatstone bridge to compensate resistor mismatches. This feedback exhibits a very good offset cancellation and therefore a better resolution is achieved.

INTRODUCTION

During the last two decades, a considerable transformation of the "Sensor world" was achieved by the emergence of MEMS technologies. Even if transducers based on capacitive sensing are the most commonly used, piezoresistive sensors that are intrinsically less sensitive have a unique advantage of being fully compatible with CMOS technologies. As an example some of the authors already proposed both inertial and magnetic field sensors on the same CMOS die [START_REF] Chaehoi | Absolute Pitch, Roll and Yaw Measurement on CMOS[END_REF]. At the price of a simple postprocess, it is thus possible to integrate on a single wafer, heterogeneous systems including one or several electromechanical sensors and their conditioning circuitry. The integration of conditioning electronic close to the sensing element, improves the overall performances. Indeed having the sensor and the conditioning circuitry on the same substrate avoid the sensor to be loaded by long cables and to prevent noise and interference pickup [START_REF] Schoeneberg | A CMOS Readout Amplifier for Instrumentation Applications[END_REF].

The principle of a piezoresistive sensor is to convert mechanical stress (or strain) into a resistance variation. The transduction is realised thanks to strain gauges designed with materials that have piezoresistive properties. Those gauges are resistances that we want to measure the value with a conditioning (biasing) circuit. In our study, we will focus on the Wheatstone bridge, which is the most common conditioning structure used for this type of sensors [START_REF] Hébrard | A Chopper Stabilized Biasing Circuit Suitable for Cascaded Wheatstone-Bridge-Like Sensors[END_REF].

In the case of a perfectly balanced Wheatstone bridge, the four resistances are identical in the absence of signal and the power supply noise is totally rejected. Assuming that the differential voltage at the output of the bridge may be amplified without degradation of the signal to noise ratio, the resolution of the sensing system is then limited by the intrinsic thermal noise generated by resistors forming the Wheatstone bridge.

In this paper, a feedback conditioning architecture is proposed to reject the power supply noise and to maximise the resolution up to the intrinsic resolution of the Wheatstone bridge (SNR=1). This study is illustrated using a magnetic field sensor developed by our research team [START_REF] Dumas | Development of a low-cost piezoresistive compass on CMOS[END_REF] but can be extended to any piezoresistive resonant sensors. In the next section, the sensor principle is introduced along with the basis of power supply noise rejection. The principle of the offset cancellation scheme is discussed and validated in section 3. Finally, the hardware implementation is presented in section 4 where the efficiency of the proposed circuit is evaluated.

II. SENSOR PRINCIPLE AND PSRR BACKGROUND

The magnetic field sensor is based on a "U-shape" mechanical structure as shown on Figure1. It is obtained after wet anisotropic etching of the silicon bulk and is composed with the various backend layers of the standard CMOS process. Among them, oxide and nitride layers are the base materials for the frame, polysilicon and aluminium are respectively used to embed strain gauges and a coplanar coil. The later is flowed by a current If that interacts with the earth magnetic field and leads to a Lorenz force that bends the frame. The mechanical strain generated by the bending, translates into a change of gauge resistances placed closely to the anchor of the suspended "U-shape". The differential output voltage (VWheat) of the Wheatstone bridge writes:

( )( ) 4 2 3 1 4 1 3 2 R R R R R R R R V V cc Wheat + + ⋅ - ⋅ ⋅ = (1)
Thus when the bridge is balanced (R2.R3=R1.R4) the output voltage is null and therefore the power supply noise is perfectly rejected. In general, the ability of an electronic circuit to reject the power supply noise is characterised by the PSRR (standing for Power Supply Rejection Ratio). This figure is defined as the ratio between the gain w.r.t. the input signal and the gain w.r.t. a change of the power supply:

            ∂ ∂ ∂ ∂ = dd out in out V V V V PSRR log 20
(2)

In our case, the frame is a second order mechanical system, with a weak damping or a high quality factor. The resonance phenomenon can be used to increase the sensor sensitivity. To make so, the coil current is set to an ac current at the cantilever resonant frequency (22 kHz in our case). The output voltage is then composed with a carrier at 22 kHz, modulated by the amplitude of the earth magnetic field. This signal is detected by an amplification circuitry composed with a Low Noise differential Amplifier (LNA), high pass filters, and differential amplifiers to reach a gain of about 25000 (figure 3). The amplification chain was designed to reach an optimal performance in terms of intrinsic noise. The equivalent input noise equals to 6.4 nV/√Hz (at 22 kHz).

This noise level is equivalent to the intrinsic noise of the sensor, i.e. the thermal noise of bridge resistances (1kΩ at 300°K) thus maximising the resolution of the sensor.

However, the resolution for the complete system degrades due to resistance mismatches between gauges and reference resistors. In our case, the power supply noise is only attenuated by 49.5dB. In the literature it can be found that the power supply noise, introduced by a digital function in a circuit, can reach several hundreds of millivolts [START_REF] Badaroglu | Digital Circuit Capacitance and Switching Analysis for Ground Bounce in ICs With a High-Ohmic Substrate[END_REF]. Therefore, making a realistic assumption that a 10µV/√Hz wide band (1MHz) noise affects the power supply, the resulting noise at the bridge output equals few dozens of nV/√Hz; far more than the intrinsic noise level. Finally, it comes that the resolution limitation is due to the Wheatstone bridge PSRR.

Improving the PSRR requires a better balanced bridge. Let's review the source of mismatches. On the one hand, the reference resistors are placed over the substrate whereas the gauges are embedded in the suspended frame. This leads to different internal stresses between reference resistances and gauges and thus to different initial value of resistances. Different orientations also induce a higher susceptibility to CMOS spreading. On the other hand, gauges are subject to self-heating. Indeed, because of the bridge biasing an increase in temperature of gauges is obtained. This is due to the fact that gauges are thermally isolated from the substrate. This phenomenon is even worse considering the cumulative effect of self-heating and power in the excitation coil [START_REF] Dumas | Analysis of offset and noise in CMOS piezoresistive sensors using a magnetometer as a case study[END_REF]. Due to the temperature coefficient of the polysilicon, this leads to a new source of mismatch. Finally, there are so many reasons for having an unbalanced bridge and having a self-balanced bridge is not achievable simply by design.

III. PSRR IMPROVEMENTS

As previously established, designing a self-balanced bridge is not an easy task. It is then necessary to implement a robust design strategy in order to ensure better bridge balance, to reduce bridge output offset and hence to reach a better rejection of the power supply noise.

The following principle has been studied: a voltagecontrolled resistor is added in each branch of the bridge to balance it. The added resistor values are directly controlled by the amplified Wheatstone bridge differential output voltage. In figure 4, this principle is represented as a block diagram. For simplicity reasons, a half bridge is represented where β, A, R DS and ∆R are respectively the feedback coefficient (Ω/V), the gain of the voltage amplifier, the value of the controlled resistor (Ω), and the mismatch of the bridge.

Vcc R A Vcc R A R V s 2 1 2 β + ∆ = (3)
Partial derivations of this equation lead to the relative variation of the output voltage w.r.t. the power supply (noise) or the input signal (resistance variation):

β β 1 2 1 2 ≈ + = ∆ ∂ ∂ Vcc R A Vcc R A R V s (4) 2 2 2 2 2 1 2 Vcc A R R Vcc R A R R A Vcc V s ⋅ ∆ ⋅ ≈       + ⋅ ∆ ⋅ = ∂ ∂ β β (5)
Considering Aβ/2R >> 1, it is obvious that the output signal and the power supply induced noise are proportional respectively to 1/β and ∆R /Aβ 2 . In the following, the criterion used to evaluate the ability of the circuit to reject power supply noise is defined as:

        ∆ =             ∆ ∂ ∂ ∂ ∂ = - ² . 2 log 20 log 20 1 CC S CC S V A R R R V V V PSRR β (6)
For a perfectly balanced bridge, where ∆R is null, a perfect power supply noise rejection is achieved. Figure 5 illustrates the evolution of PSRR -1 versus the input mismatch ∆R, for different values of the amplifier gain. From these simulations, equation ( 6) is verified by demonstrating a 20 dB PSRR increase for each decade of gain. These simulations were performed using Cadence® tools. The implemented structure is illustrated figure 6, where the feedback resistors (MOS transistors), were described using a VERILOGA description language.

IV. HARDWARE IMPLEMENTATION

A first implementation using PMOS transistors in triode region has been developed. In figure 6, a simple implementation is shown with two transistors used as voltage controlled resistances and a fully differential amplifier. Transistor sizing is trivial and won't be developed here. MOS resistance is controlled by the gate-to-source voltage. The source being tied to Vcc, power supply noise leads to a V GS modulation and thus to a R DS variation. Both transistors finally act as common gate amplifiers for the power supply noise thus degrading the PSRR. It is worth noting that, even if inefficient for the power supply noise, this architecture rejects the ground noise. Simulations are reported on figure 7 where a 1 kHz resistance variation is applied to the gauge together with a 9 kHz noise on the ground. Results clearly show that the noise is completely rejected: the 9 kHz tone is not present on the differential outputs (curves b and c).

Similarly, the use of NMOS transistors placed between the ground and the bridge resistors will eliminate power supply noise while amplifying ground noise. A second implementation is depicted in figure 8 where an RC circuit has been added between each differential outputs of the amplifier and the power supply. From amplifier outputs we obtain a low pass filter that cancels only bridge static offset. From the power supply, a high pass filter is obtained and the gate-to-source voltage modulation is cancelled in the frequency range of the signal. Considering that the noise rejection architecture has been validated, we then characterize the overall performance of the proposed solution. Assuming feedback transistors are sized with a sufficient area to reduce their intrinsic noise, we have compared the open-loop architecture (figure 3) with the previously proposed one for the same overall gain in the range of 87 dB. It comes that, even if the PSRR is increased, the overall performance is reduced due to the differential amplifier noise. We then modified the differential amplifier to include the LNA of the open-loop architecture before the differential amplifier. Obtained results (last column in table 1) demonstrate a very good compromise with very high PSSR and a very good intrinsic noise level. In this paper, we have proposed a feedback conditioning architecture to improve the power supply noise rejection ratio (PSRR) of a Wheatstone bridge. The proposed architecture is applicable to any resonant piezoresistive MEMS to improve the resolution. This architecture is of primary importance in the case of highly integrated MEMS where the power supply may be polluted by the switching of digital electronics.

Efficiency of the proposed solutions has been demonstrated using high-level simulations based on Matlab or VERILOGA languages to validate the principle and transistor level simulations for final implementation and overall performance evaluation.

The final architecture exhibits the same intrinsic noise level, the same sensitivity (gain) and a 35 dB PSRR improvement. It is then possible to calculate the resolution of the sensor using only the SNR ratio within gauges.

Fig. 1 .Fig. 2 .

 12 Fig. 1. The piezoresistive magnetic field sensor.

Fig. 3 .

 3 Fig. 3. Open loop amplification chain

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 11 April 2008 ©EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5

Fig. 4 .

 4 Fig. 4. Block diagram of the enhanced-PSRR architecture From this diagram, the output voltage writes:

Fig. 5 .

 5 Fig. 5. PSSR-1 with and without feedback loop, for different values of the amplifier gain.

Fig. 6 .

 6 Fig. 6. Bridge balancing feedback with PMOS transistors.

Fig. 7 .

 7 Fig. 7. Transient simulation in presence of both signal (@1kHz) and ground noise (@9kHz): a) Vcc, b) positive and c) negative differential outputs, d) Gnd.

  Fig. 8. Enhanced PSRR circuit for both Vcc and Gnd noise rejection.

Figure 9

 9 Figure9illustrates the power supply noise rejection versus frequency. In a 1 KHz -100 KHz frequency band, the compensated bridge exhibit a PSRR more than ten times higher than the stand alone Wheatstone bridge.

Fig. 9 .

 9 Fig. 9. PSRR simulation for a compensated (a) and stand alone (b) Wheatstone bridge.

TABLE 1 OVERALL

 1 NOISE PERFORMANCE OF THE STUDIED ARCHITECTURES

Open loop Feedback loop (no LNA) Feedback loop (with LNA)

	Gain	87dB	86.9dB	85.9dB
	PSRR	-49dB	-73dB	-84dB
	Intrinsic	160	390	150
	electronic	µV/√Hz	µV/√Hz	µV/√Hz
	noise
		V.	CONCLUSION


Pascal Nouet

Frédérick Mailly

Laurent Latorre

Laurent Latorre Study

Olivier Leman

STUDY OF FIRST-ORDER THERMAL SIGMA-DELTA ARCHITECTURE FOR CONVECTIVE ACCELEROMETERS

d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Many studies report the design of open-loop convective accelerometers [START_REF] Leung | Micromachined accelerometer based on convection heat transfer[END_REF][START_REF] Milanovic | Convection-based Accelerometer and Tilt Sensor Implemented in Standard CMOS[END_REF][START_REF] Billat | Micromachined inclinometer with high sensitivity and very good stability[END_REF][4][START_REF] Luo | An optimized micromachined convective accelerometer with no proof mass[END_REF][START_REF] Kaltsas | A thermal convective accelerometer system based on a silicon sensor -Study and packaging[END_REF][START_REF] Zhu | Micromachined gas inertial sensor based on convection heat transfer[END_REF][START_REF] Mailly | Micromachined thermal accelerometer[END_REF][START_REF] Chaehoi | Experimental and finite-element study of convective accelerometer on CMOS[END_REF][START_REF] Leman | HDL modeling of convective accelerometers for system design and optimization[END_REF] but very few of them concern the conditioning electronics. Generally, a simple instrumentation amplifier is used. For long term stability and room temperature compensation, temperature control of the heater was studied by [START_REF] Kaltsas | A thermal convective accelerometer system based on a silicon sensor -Study and packaging[END_REF] and is implemented in a commercial sensor [4]. However, to our knowledge, no study deals with a real closed-loop architecture, where the effect of the acceleration on the detectors' temperature is compensated.

Sensor working principle is quite simple. The prototype realized at LIRMM (fig. 1) using a standard CMOS technology features three polysilicon resistors embedded inside three suspended bridges. The central bridge is heated by Joule effect, and a thermal gradient is thus created in the surrounding air. The two lateral bridges are resistive temperature sensors with a Positive Temperature Coefficient (PTC). Without acceleration, both sensing bridges measure the same temperature due to symmetry reasons. Acceleration modifies the temperature profile due to free convection phenomenon along the sensitive axis which is perpendicular to the bridges. The temperature difference generated between the sensing bridges is converted into a differential output voltage by a Wheatstone bridge setup.

These sensors suffer from several weaknesses, such as: • A quite low intrinsic sensitivity in the range of 1mV/g, which implies the design of a high-gain and low-noise amplifier. • A non-linear acceleration-to-differential-temperature transfer function since the temperature difference between the sensors is limited by the temperature increase of the heater.

• A thermal inertia of the detectors which limits the bandwidth of the sensor to 50Hz. This is a technological limit. It was proven in previous studies that convective bandwidth higher than 100Hz could be reached through a reduction of the cavity, making thermal inertia of detectors the limiting factor [START_REF] Chaehoi | Experimental and finite-element study of convective accelerometer on CMOS[END_REF]. These limitations lead us to develop a closed-loop architecture in the form of an electro-physical sigma-delta modulator [START_REF] Makinwa | Modeling and simulation of thermal sigma-delta modulators[END_REF]. The sensor conditioner has a direct-digital output in the form of a bit-stream and compensates for the thermal inertia of the sensing bridges. The proposed feedback loop cancels the temperature difference between the two detectors by generating a compensation thermal power. This method uses Joule's effect, so that it can elevate the temperature of the "cold" detector, but the "hot" detector cannot be cooled this way. In order to solve this problem, the modulator slightly shifts the thermal bias point of the detectors by using a common-mode bias power which is then cancelled or increased depending on the desired action.

The first part deals with the modeling of the convective accelerometer, in order to realize behavioral simulations. In the second part we propose the electronic design of a convective accelerometer with thermal sigma-delta modulator conditioning. Finally, behavioral simulations of the proposed system are performed under the Matlab- As a foreword, we want to stress the point that we have experimentally validated the thermal sigma-delta modulator principle and the proposed behavioral model [10, 12 and 13], giving strength to this design study.

II. OPEN-LOOP SENSOR MODELING

The study and modeling of the convective accelerometer sensing cell was published in previous studies [START_REF] Leman | HDL modeling of convective accelerometers for system design and optimization[END_REF][START_REF] Leman | Modeling and system-level simulation of a CMOS convective accelerometer[END_REF]. Using FEM simulations and characterization results, the model of fig. 2 was developed.

H th A H H P R T T T × = - = ∆ (1)
Where T H and T A (K) are respectively the average heater temperature and ambient temperature, R th (K.W -1 ) is the equivalent thermal resistance of the beam and P H (W) is the joule power dissipated in the heater resistance. Section 2 describes the conduction heat transfer, which is responsible for the common-mode temperature of the detectors. This model is based on a cylindrical geometry approximation and the common-mode temperature at a distance r from the heater is ruled by expression 2:

( ) ( ) 2 ln ln 2 2 2 1 1 2 2 2 2 H H A H A H T T r r r r T T T T T T γ γ γ + +         ⋅               - + - = + (2)
Section 3 expresses convective transfer from which depends the output signal. The temperature profile established in section 2 deforms, producing a temperature difference between the detectors ∆T D . This temperature difference has been found to be proportional to the Grashof number [START_REF] Leung | Micromachined accelerometer based on convection heat transfer[END_REF] for ∆T D << (T H -T A ):

( ) p l T T a S p Gr S T A H D τ µ βρ τ + ⋅ - ⋅ = + = ∆ 1 1 1 . 2 3 2
(3) with S (in K) a fitting coefficient (extracted from FEM analysis and experimental results) which represents the system sensitivity, a (m.s -2 ) the acceleration, ρ (kg.m -3 ) the gas density, β (K -1 ) the gas coefficient of expansion, µ (kg.m -1 .s -1 ) the gas viscosity, l (m) a linear dimension related to the cavity volume and geometry and τ (s) the time constant of the fluid (τ = 0.5 … 12ms, depending on the cavity geometry and dimensions [START_REF] Chaehoi | Experimental and finite-element study of convective accelerometer on CMOS[END_REF]).

Finally, section 4 implements detectors transfer function taking into account the temperature sensitivity of polysilicon (TCR = 9.10 -4 /K) and the thermal time constant of the detecting bridges (τ D = 3.3ms):

        +       ∆ ± + = p T T TCR Rd Rd D D CM nom i i τ 1 1 2 1 1 , ( 4
)
The resistance variations of the PTC sensors are converted into a differential output voltage by means of a Wheatstone bridge with a sensitivity S wheat (V/K) :

MC Wheat T TCR TCR Vdd S . 1 4 + = ( 5
)
The Johnson noise of the detectors is finally added to the model since it has been identified as the limiting factor for the sensor resolution. In our prototype, the heating power is 42mW, heater temperature is 720K and detectors commonmode temperature is 423K. The convection transfer function can thus be linearized as

MEMS D K a T . = ∆
with sensitivity K MEMS = 1.53K/g. K MEMS is an empirical, composite parameter including the parameters of expression 3. Wheatstone bridge's Johnson noise (4k b T CM R) generated by the 50kΩ resistors is 34nV/√Hz over a 6MHz bandwidth limited by the 1pF comparator input capacitance. Accelerometer sensitivity is 1.55mV/g, and its intrinsic resolution is 98µgrms over the 1-20Hz bandwidth.

In order to study the conditioning electronics, a small signal model of the sensor was derived (fig. 3). Such a simplified model is suitable for fast behavioral simulations under the Matlab-Simulink framework.

(V) S Wheat + + 4k b T CM R p K MEMS τ + 1 a (g) ∆T D (K) p D τ + 1 1 ∆T D' (K) V out (V) (V) S Wheat + + + + 4k b T CM R p K MEMS τ + 1 a (g) ∆T D (K) p D τ + 1 1 ∆T D' (K) V out (V)
Fig. 3. Small-signal model of the sensor This model was used in a previous study of open-loop amplifier architectures to show that a Wheatstone bridge bias modulation together with a synchronous detection scheme would allow to cancel the low-frequency noise of a standard CMOS amplifier and to reach resolutions close to the intrinsic performance of the convective accelerometer cell [START_REF] Leman | A comparative study of conditioning architectures for convective accelerometers[END_REF].

III. CLOSED-LOOP SENSOR ARCHITECTURE

The convective sensing element studied here is an evolution of the previous version presented in part 2. Two temperature detectors are added in the suspended bridges in order to create a Wheatstone bridge with 4 sensitive resistors (fig. 4-a). The resulting sensor features twice the original S Wheat sensitivity and then twice the previous intrinsic signalto-noise ratio. Considering the operating conditions stated in part 2, the sensitivity reaches now 3.1mV/g; the intrinsic noise of 11µg/√Hz leads to a 49µgrms resolution over the 1-20Hz bandwidth.

In order to allow a thermal feedback of the suspended bridges, self-heating of the temperature detectors due to the biasing of the Wheatstone bridge is now taken into account in the model, resulting in a slight shift of the common mode temperature of the two suspended bridges. Moreover, we noticed that this self-heating interestingly falls in the same range as the power necessary to compensate for the convection phenomenon. The thermal feedback is then simply achieved by shorting the sensitive resistors using MOS switches (fig. 4-a). Table 1 gives the power dissipation in the two suspended bridges as a function of the MOS switches command. When both feedback commands are low, the Wheatstone bridge is regularly biased, and the power dissipation is P 0 = V 2 /(2R(T MC )) = 225µW in each suspended bridge. When feedback_1 is high, resistors in bridge 2 are shorted and resistors in bridge 1 are biased at V dd . Power dissipation is then null in bridge 2 and P max = 2V 2 /(R(T MC )) = 900µW in bridge 1. Feedback_2 activation generates a complementary power dissipation effect. These two feedback paths are used to compensate the temperature difference generated by convection (i.e. by the acceleration). This principle requires a 2-phase time scheduling and fig.

4.b. presents the corresponding chronograms:

• phase Φ1, the sensor is in reading mode. Both feedback paths are disabled, and the Wheatstone bridge under regular bias delivers the error signal V out . If V out is negative, it means that the two detectors embedded on bridge 2 are the warmer and then the comparator output is low. If it is positive, the two detectors in bridge 1 are the warmer and the comparator output is high. • phase Φ2, the sensor is in the feedback mode, thus one of the two feedback paths is enabled according to the comparator output. For example, if the comparator output is low, feedback_1 is enabled: power dissipation is then null in bridge 2 and equal to P max in bridge 1. During this phase, the Wheatstone bridge output is an invalid +/-V dd .

The longer T comp is, the higher compensation power P comp is generated, according to the P comp = α P max relationship.

Sensor full-scale is adjusted through the duty-cycle α of feedback pulses. A null acceleration entails that each bridge is heated by an average P comp /2 power which slightly shifts the thermal common-mode of the detecting bridges.

IV. CLOSED-LOOP SENSOR SIMULATION

First the small signal model of the sensor (fig. 3) is modified to include the power-feedback scheme of the sigma-delta modulator (fig. 5). This is achieved by converting the temperature variations of the bridges into thermal power using the characterized thermal resistance of the detecting bridges, R th = 10 4 K/W. Consequently, system input becomes a thermal power ∆P D resulting from acceleration:

p a R K P th MEMS D τ + = ∆ 1 (6)
In this first order thermal sigma-delta modulator, the electro-thermal response of the detecting bridges replaces the integrator. This thermal filter is a first-order low-pass with a cut-off frequency of 48Hz, so that it has a DC gain of 10 4 K/W in the DC-48Hz part of the spectrum, thus acting as a leaking integrator. In order to study the impact of this non-ideal integrator, we use a linearized model of the modulator (fig. 6) where T(z) is the z-transform of the loop filter transfer function and N(z) is the quantization noise, under the white-noise assumption. The Johnson noise is not considered in this model.

Quantization noise transfer function Y(z)/N(z) is :

( )

) ( 1 1 z T z NTF + = (7)
Considering an ideal integrator as the loop filter, the NTF equals 1-z -1 . Hence, quantization noise is shaped over the full spectrum. Now if we consider the thermal filter which transfer function in the z-domain writes:

( )

1 1 1 - - ⋅ - = z z z T α (8)
with

D Te e τ α - = close to 1, the NTF becomes 1 1 - ⋅ -z α [14]
. This NTF becomes flat in the DC-48Hz part of the spectrum, meaning that the quantization noise is unshaped in this part of the spectrum (fig. 7-b).

Transient behavioral simulations of the system were carried-out in Matlab-Simulink. We studied the performance of the architecture using a spectral analysis of the output bitstream (fig. 8).

If we compare results of figures 7 and 8, we notice that the behavior of the simulation model is close to the prediction of the analytical model in the low-frequency spectrum, the lowpass thermal loop filter leads to a flat quantization noise from 1 to 48Hz (fig. 8-a). Resolution is 1.48mg over the 1-20Hz bandwidth, whereas the same modulator with an ideal integrator (fig. 8-b) would have a 0.26mg resolution. Moreover, a slight distortion appears. This phenomenon is common for first-order modulators, it corresponds to the third harmonic and its amplitude is below 0.1% of the main peak. However, figure 8 shows that quantization noise is unshaped in the upper part of the spectrum. Above 1.6kHz, the power of the error signal falls below the Johnson noise floor of the Wheatstone bridge. This results in a random output of the 1-bit quantizer. In the end, oversampling simply leads to a quantization noise spreading. In absence of shaping from 1.6kHz to f ck /2 (e.g. about 98% of the spectrum), more than 99% of the quantization noise power is unshaped. We can have a good estimation of the quantization noise spectral density in the 1.6kHz-(f ck /2) spectrum, neglecting the 48Hz-1.6kHz first order noise shaping. Assuming that the total bit-stream power is constant, the quantization noise should be given by: Simulations of the figure 8 give 7.47mg/√Hz. This proves that the 48Hz-1.6kHz 1st order shaping has little effect on the overall quantization noise shaping.

Figure 9 presents the signal transfer functions of the two modulators. As a consequence of the finite DC gain of the thermal, the DC gain of the electro-thermal modulator is 0.3dB lower than the theoretical gain. Their -3dB cut-off frequencies are both about 1.6kHz. Above this frequency it was shown that the Johnson noise of the Wheatstone bridge disturbs the modulator operating. It's likely that this phenomenon limits also the modulator bandwidth. Finally, the same system was simulated for various clock frequencies f ck and the corresponding resolutions were computed over the bandwidth 1-20Hz. The results, plotted on figure 10, show that for clock frequencies higher than 10 kHz, the signal-to-noise ratio increases at a 3dB/octave rate. This poor improvement comes from the quantization noise spreading due to the absence of quantization noise shaping above 1.6kHz.

A 132µg resolution was simulated with a 8.4MHz clock frequency. Higher clock frequencies resulted in bitstreams too large to be analyzed with Matlab. From the -3dB/octave SNR increase, a modulator with a clock frequency of about 100MHz would reach the intrinsic resolution of the convective sensor; say 49µg over the 1-20Hz bandwidth.

This result suggests the design of a second order sigma-delta modulator architecture, expecting a lower distortion and lower frequency clocks through a better noise shaping.

CONCLUSION

This study reports an original smart-sensor architecture for thermal convective accelerometers based on the thermal sigma-delta modulator principle. We consider a first order sigma-delta architecture. Using the side effects of self heating and thermal inertia of detectors, most of the analog signal processing is performed in the thermal domain, drastically reducing the amount of analog electronics. Simulation results show that a high modulator clock frequency is required to reach the intrinsic noise of the sensor, motivating the design of a higher order modulator.

Fig. 1 .

 1 Fig. 1. Prototype micrograph

  the results are discussed.

Fig. 2 .

 2 Fig. 2. Convective sensor model

9- 11

 11 April 2008 ©EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5

Fig. 4 .

 4 Fig. 4. Modified sensor architecture (a) and chronograms (b)

Fig. 6 .

 6 Fig.5. Sigma-delta modulator simulation model

Fig. 7 .Fig. 8 .

 78 Fig. 7. Noise transfer functions for an ideal integrator (b) and for the thermal integrator (a) @ fck = 131kHz.

  where S b = 2g RMS is the bit-stream signal power, and S a = 9-11 April 2008 ©EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5 1/√2g RMS is the acceleration signal.

Fig. 9 .Fig. 10 .

 910 Fig. 9. Signal transfer functions of the modulators, fck = 131kHz, ±2g fullscale