AIAA-2003-1632

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AIAA-2003-1632

Piezoresistive microphone design Pareto optimization: tradeoff between sensitivity and noise floor Melih Papila, Raphael T. Haftka Toshikazu Nishida and Mark Sheplak, University of Florida, Gainesville, FL

44th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics, and Materials Conference

07-10 April 2003 / Norfolk, Virgina

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics, 1801 Alexander Bell Drive, Suite 500, Reston, Virginia 20191-4344.

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Piezoresistive microphone design Pareto optimization: tradeoff between sensitivity and noise floor

Melih Papila1* (papila@mae.ufl.edu) Raphael T. Haftka2* (haftka@ufl.edu) Toshikazu Nishida3‡ (nishida@ufl.edu) and Mark Sheplak4* (ms@mae.ufl.edu)

*Department of Mechanical and Aerospace Engineering, ‡Department of Electrical and Computer Engineering,

University of Florida Gainesville, FL 32611-6250

1 Post-Doctoral Research Associate, Member AIAA 2 Distinguished Professor, Fellow AIAA 3 Associate Professor 4 Associate Professor, Member AIAA Copyright© 2003 by Authors. Published by American Institute of Aeronautics and Astronautics, Inc. with permission.

ABSTRACT

This paper addresses tradeoffs between pressure sensitivity and electronic noise floor in optimizing the performance of a piezoresistive microphone. A design optimization problem was formulated to find optimum dimensions of the diaphragm, and the optimum piezoresistor geometry and location for two objective functions, maximum pressure sensitivity and minimum electronic noise floor. The Pareto curve of optimum designs for both objectives was generated. The minimum detectable pressure was also employed as an objective function, generating a point on the Pareto curve that may be the best compromise between the two objectives. The results also indicated that the critical constraints are the linearity and power consumption. The minimum MDP design was found to be favorable regarding the sensitivity to parameter uncertainty.

INTRODUCTION

Miniature microphones are particularly valuable for fundamental acoustics and fluid mechanics measurements because they reduce flow disturbances and diffraction effects relative to conventional sensors. The small size enables mounting in a confined space and permits close spacing of multiple sensors. For instance, small microphones can be flush mounted to a curved airfoil surface for wind tunnel studies. Microelectromechanical systems (MEMS) technology has been used to fabricate miniature micromachined pressure sensors or microphones for a variety of applications (see, for example, Scheeper et al. 1994). The reduction in cost afforded by batch-fabrication also makes it feasible to use large numbers of MEMS sensors for arrays (Arnold et al. 2003). While cost is a key consideration, ultimately overall performance metrics such as sensitivity, electronic noise, and minimum detectable signal (MDS) determine the usefulness of MEMS sensors for fundamental acoustics and fluid mechanics measurements.

The performance metrics depend on the transduction principles employed. All microphones are based on the measurement of a pressure-induced structural deflection.

Incident pressure waves result in the deflection of a diaphragm, thus translating acoustic energy to mechanical energy. The resulting structural deflection may be detected using several different transduction mechanisms including capacitive, piezoresistive, piezoelectric, and optical techniques. Performance metrics such as sensitivity, electronic noise, and MDS depend on the geometrical dimensions and material properties of the structure and electronic properties of the transducer. Theoretical studies indicate that piezoresistive-sensing schemes can transduce lower minimum detectable pressure levels than commonly used capacitive schemes for diaphragm edge-lengths of less than 0.5 mm (Spencer et. al 1988). However, whether a piezoresistive MEMS microphone achieves optimally low minimum detectable pressure (MDP) depends on the design of the diaphragm and piezoresistors and the fabrication process used to realize the design. Noise studies of poly-crystalline silicon resistors have shown high levels of low-frequency excess noise that is attributed to charge carrier interaction with the grain boundaries (de Graaf and Huybers 1983). Single crystal silicon piezoresistors were used in subsequent MEMS microphone designs (Sheplak et al. 1999, Arnold et al. 2003). A MEMS microphone with dielectrically isolated single-crystal silicon piezoresistors fabricated on top of a 210µm diameter low-stress silicon nitride membrane possessed a sensitivity of 22 PaV /µ when biased at 10V and a MDP of 92 dB sound pressure level (SPL) for 1Hz bin centered at 1kHz (Sheplak et al 1999). The sensitivity normalized to bias voltage was 2.2 PaVV //µ . This device achieved a five-fold increase in sensitivity and a two-order-of-magnitude decrease in power consumption over commercially available MEMS piezoresistive microphones, but exhibited a higher than expected noise floor that was attributed to large resistor values and high process-related silicon piezoresistor defect densities. A similar MEMS microphone geometry with larger diameter diaphragm (1mm), plasma-enhanced chemical vapor deposited (PECVD) nitride passivation, and higher piezoresistor doping (1020 boron/cm3) exhibited

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a lower normalized sensitivity, 0.6 PaVV //µ , but also a lower MDP (52 dB SPL for 1Hz bin centered at 1kHz) (Arnold et al. 2003). These two piezoresistive microphone designs illustrate tradeoffs between sensitivity and noise. Since both the sensitivity and noise performance metrics depend on the geometry of the piezoresistor, doping concentration and profile, and mechanical and electronic properties, the parameter design space is complex. In their detailed analysis of piezoresistive cantilevers, Harley and Kenny (2000) have elucidated design guidelines and graphical design charts for optimization of the displacement resolution (minimum detectable displacement) of piezoresistive cantilevers through analysis of the geometry, doping, process, and bias dependencies of the mechanical sensitivity and electronic noise (Harley and Kenny 2000).

In this paper, the sensitivity and noise tradeoffs of a piezoresistive microphone are analyzed as an optimization problem with both sensitivity and noise performance metrics as the alternating objective function, in addition to setting constraints on one of the metrics in order to display the tradeoffs on a Pareto curve (e.g., Belegundu and Chandrupatla, 1999). Solutions of the optimization problem using different levels of coupling between the sensitivity and noise offer further insight into the trade offs when a power constraint is added. The minimum detectable pressure is also employed as an objective function to introduce a compromise between the two competing metrics.

The organization of the paper is as follows. First, device models are summarized for the mechanical structure and piezoresistor relating pressure fluctuation to radial and tangential stress and relating stress in the piezoresistor to resistance modulation. Models of the dominant electronic noise sources are used to estimate the electronic noise present in the output signal. The performance metrics, sensitivity, noise, and minimum detectable pressure, are formulated in terms of the structural and electronic parameters. Next, a mathematical representation of the optimization problem is given using objective functions and constraints. Then, optimization results are given for maximizing the sensitivity, minimizing the noise, and minimizing the minimum detectable pressure. A Pareto curve is computed to elucidate the tradeoff between sensitivity and noise floor. Finally, the effect of parameter uncertainty for the optimized designs is investigated via Monte Carlo simulations.

DESCRIPTION OF THE DEVICE MODELS

Formulation of the objective function for performance optimization begins with the structural and electronic device models. The structural response of the diaphragm, specifically transverse deflection and stress distribution, directly determine the sensitivity, bandwidth, and linearity of the dynamic response. The piezoresistor design determines the overall sensitivity and the electronic noise of the device. The following sections describe the

structural and piezoresistor models employed in this study and their use in prediction of the performance parameters.

Structural modeling of the diaphragm

The model consists of a clamped circular diaphragm of radius a and thickness h. The diaphragm (Figure 1), designed to sense incident ac or time-varying pressure fluctuations, is also subjected to dc or quasi-static in-plane residual stress due to packaging or residual stress from film deposition (Senturia, 2001). For simplicity, the structural analysis assumed a homogeneous and isotropic silicon nitride diaphragm. Table 1 presents the material properties, such as Young’s modulus, Ediaph, and Poisson ratio, ν . Assuming small deflection and linearity, the radial stress, rσ , and tangential stress, tσ , that result on the top side of the diaphragm when a pressure load, pz, is applied in the presence of in-plane tension stress is given by (Sheplak and Dugundji 1998),

−

−

−+

=)(

)1(

)(13

)( *1

2*

*

1

*1

*

*

0

2*2

2

kIrk

arkIa

kIk

arkI

khap

r zr

ννσ , (1)

and

−

+

−+

=)(

)1(

)(13

)( *1

2*

*

1

*1

*

*

0

2*2

2

kIrk

arkIa

kIk

arkI

khap

r zt

νννσ ,

(2)

where, zp is the pressure, I0( ) and I1( ) are modified Bessel functions of the first kind, and k* is the tension parameter as given in Eq. (3) due to residual in-plane tension stress, 0σ , that the diaphragm sustains in addition to pressure loading.

diaphEhak 0

2* )1(12 σν−= . (3)

To avoid geometrical nonlinearity, a constraint on the maximum incident pressure, pmax, was defined for 5% departure from linear deflection at the diaphragm center (Sheplak and Dugundji 1998),

03.2)1(12

02.023

0234

max ≤−

−

−

Eha

ha

EP σν

. (4)

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Piezoresistor modeling With application of stress, a piezoresistor responds

with a change in resistance from its nominal unstressed value. The conversion of the pressure-induced resistance change into a voltage change requires the presence of a bias current through the piezoresistor. The bias current may be driven by a constant current source or is a result of a constant voltage source. Typically, four piezoresistors are arranged in a balanced Wheatstone-bridge configuration driven by constant voltage excitation. In a balanced bridge, the output of the bridge is null when the mean resistances in all four legs are equal. Ideally, common mode disturbances are attenuated while differential disturbances are linearly converted into the bridge output. To achieve a differential signal from pressure fluctuations, the piezoresistors are oriented such that the resistance modulation in each resistor of a given leg is equal in magnitude, but of opposite sign. For a polar geometry, these conditions are achieved by placing a tapered resistor opposite to an arc-shaped resistor in the bridge configuration (Figure 1c).

For simplicity, we employ a piezoresistor model that assumes constant stress through the thickness in the piezoresistors but accounts for a different modulus of elasticity in the silicon piezoresistor (Table 1) from the underlying silicon nitride diaphragm when calculating the stress field. A uniform doping concentration dopn is assumed throughout the thickness of the piezoresistors. These approximations are valid for thin silicon piezoresistors fabricated on top of a silicon nitride membrane using a wafer-bond and thin-back process as discussed in Sheplak et al. (1998). The optimization techniques described herein may be extended to non-uniform stress and doping cases.

The nominal unstressed resistance is determined for the arc and taper geometries by computing the differential resistances and integrating over the arc and radial lengths. For the arc-shaped resistor, the current is in the tangential direction with respect to resistor orientation. Therefore, the point resistances on a given arc of radius r are connected in series within Arcθ , and point resistances along the radial direction are connected in parallel between ar and br . Integration yields an expression for the mean sheet resistance of the arc piezoresistor,

( )ab

ArcSArc rr

RRlnθ

= , (5)

where RRS tR /ρ= is the sheet resistance, and Rρ is the resistivity given as a function of doping concentration by

doppR nqµ

ρ 1= . (6)

Here, pµ is the hole mobility that itself is a function of

dopn and q is the electronic charge (q =1.602x10-19 C).

The dependence of pµ on dopn is modeled using the physical mobility model of Nishida and Sah (1987). For a tapered resistor, however, the current is in the radial direction with respect to the resistor orientation. Therefore, the point resistances are connected in series between 1r and 2r . In addition, the point resistances on a given arc of radius r are connected in parallel within the angle Taperθ . Integration over the resistor area yields the following expression for the mean resistance,

( )Taper

STaperrr

RRθ

12ln2= . (7)

Note that there is a factor of two in the expression due to the fact that two identical tapered resistors are connected with a turn-around piece (shown in Figure 1b) in order to achieve radial current flow through both tapered resistor pieces.

In a piezoresistor, the resistor modulation is a linear sum of the product of the applied stress and piezoresistive coefficients that are parallel to the current flow and perpendicular to the current flow. In a circular geometry, the basis directions are the radial and tangential directions. The equations for the normalized resistance modulation,

RR∆ , for the tapered and arc-shaped resistors are given by

)()()()( θπσθπσ TtLrTaper

rrRR

+=∆ , (8)

and

)()()()( θπσθπσ TrLtArc

rrRR

+=∆ , (9)

where the subscripts L and T denote the longitudinal and transverse directions with respect to current direction at any point of the piezoresistor, and θ represents the polar orientation of any point with respect to the silicon <100> crystal axis. The values of the longitudinal and transverse piezoresistive coefficients may be computed for arbitrary θ by using piezoresistive coefficient data, 441211 ,, πππ . The data are modification of the reference low concentration and room temperature piezoresistive coefficients 441211 ,, πππ (Table 1, due to Smith (1954)) to take into account their dependence on operating temperature and doping concentration, dopn . At room temperature, the doping concentration dependence is modeled using the experimental fit obtained by Harley and Kenny (2000),

=

dopijdopij n

xn221053.1log2014.0)( ππ . (10)

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DESCRIPTION OF THE PERFORMANCE PARAMETERS

Using the structural and electronic device models, the key performance parameters, sensitivity, electronic noise, and minimum detectable pressure are defined.

Sensitivity

The sensitivity of the piezoresistive microphone is defined as the slope of the calibration curve relating the output voltage to the input pressure, pVS oEM ∂∂= / , and is given by Eq. (11) for a linear device with zero offset voltage,

maxPV

S oEM

∆= . (11)

The differential output voltage, oV∆ , on a balanced Wheatstone bridge (RArc = RTaper) with constant bias voltage, BV , is expressed as

BTaperArcArc

TaperArco V

RRRRR

V

∆+∆+

∆−∆=∆

2. (12)

For incident pressures below Pmax, the calibration curve is linear provided that the resistance modulation of the arc and tapered piezoresistors are equal in magnitude. The voltage modulation versus pressure loading relation is nonlinear if ∆RArc ≠ ∆RTaper. On the other hand, if the mean resistance is much greater than the resistance modulations, Eq. (12) is approximately linear,

BArc

TaperArco V

RRR

V

∆−∆≈∆

2. (13)

Electronic noise The key contributors to the electronic noise of the

piezoresistor are thermal noise and low frequency 1/ f noise (Harley and Kenny 2000). Physical fluctuations of the diaphragm at equilibrium at a temperature, T, can result in random motion of the diaphragm; however, the contribution of thermomechanical displacement noise to the piezoresistor output noise has been shown to be much smaller than the electronic noise sources except at mechanical resonance (Harley and Kenny 2000). In addition, Brownian motion of diaphragm due to random scattering of gas molecules is also negligible for the mass of the diaphragm under consideration (Bhardwaj et al. 2001). Thus, the electronic noise will be computed using the power spectral densities of thermal and low frequency noise.

Thermal noise

Voltage fluctuations, at the external terminals of a resistor, are produced when electrons are scattered by the thermal vibrations of the lattice (Van der Ziel, 1956). These fluctuations are present in any device that dissipates

energy due to thermal vibrations. Since higher temperatures lead to increased vibration motion, thermal noise power spectral density (PSD) is directly proportional to temperature. Moreover thermal noise PSD is independent of frequency since random thermal vibrations are not characterized by discrete time constants. The thermal noise PSD ( TSν ) is given by (Nyquist, 1928)

RTkS KBT 4=ν , (14)

where kB is the Boltzmann constant, R is the resistance, and KT is the temperature in Kelvin. In a piezoresistor, the rms noise voltage, tRE , due to thermal noise is obtained by taking the square root of the thermal noise PSD integrated over the frequency range of operation,

12 fff −=∆ (Nyquist, 1928, Van der Ziel, 1956),

fRTkdfSE KB

f

fTtR ∆== ∫ 4

2

1

ν . (15)

Low frequency 1/ f noise Piezoresistors also exhibit noise with a PSD that varies

inversely with frequency when an external dc bias is applied. Since the PSD is more prevalent at lower frequencies, it is also known as low frequency noise. Two physical mechanisms have been proposed to account for the low frequency noise, random trapping/detrapping of carriers at surface and bulk electronic traps and random mobility fluctuations. Since the dominant mechanism may depend on the surface and electronic material properties, an empirical formulation for f/1 noise given by Hooge (1976) will be used to model the f/1 noise in the piezoresistor. Hooge’s relation for f/1 noise PSD is given by

NfV

S Hf

2

/1α

ν = , (16)

where Hα , known as the Hooge’s parameter, is an empirically obtained constant which ranges from 5x10-6 to 2x10-3 (we use Hα = 1x10-3 for this study) and is sensitive to bulk crystalline silicon imperfections as well as to the interface quality, V is the applied voltage, and N is the total number of carriers. The rms noise voltage due to 1/ f noise is obtained by integrating the noise PSD over the frequency range of operation,

1

22

/1 lnff

NV

E HfR

α= . (17)

Unlike the thermal noise, f/1 noise occurs under nonequilibrium conditions and is proportional to the applied voltage. From the inverse relation to the square

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root of the number of charge carriers, we see that 1/ f noise increases in smaller volume and higher resistivity piezoresistors.

In a Wheatstone bridge utilizing two arc and two taper resistors (see Figure 1c), the rms thermal and 1/ f noise voltages at the bridge output are given by

( )12||8 ffRTkV KBT −= , (18)

and

+

=

1

222

22||/1 ln11

22

ff

RNRNV

RVTaperTaperArcArc

BHf α ,

(19)

where TaperArc

TaperArc

RRRR

R+

=|| , BV is the bias or excitation

voltage for the bridge, and ArcN and TaperN are the total number of carriers in the arc and taper piezoresistors, respectively. Assuming that the Wheatstone bridge is balanced, TaperArc RR = , the total rms noise voltage may be expressed as

( )

++−=

1

2212 ln11

814

ff

NNVffRTkV

TaperArcBHArcKBN α

. (20)

In this work, the noise voltage is computed for a 1Hz bandwidth centered at 1kHz, i.e., 5.9991 =f Hz. and

5.10002 =f Hz. Minimum detectable pressure

The minimum detectable pressure, minP , defines the lower end of the dynamic range which the piezoresistive microphone can resolve in the presence of noise. It is computed as the ratio of electronic noise to sensitivity and provides a joint performance metric of the two parameters.

max0min PV

VP N

∆= . (21)

This metric is generally reported in SPL using a reference pressure of 61020 −×=refP Pa,

=

refPP

MDP minlog20 . (22)

DESIGN OPTIMIZATION OF MEMS MICROPHONE

The optimization problem is formulated in terms of objective functions based on the performance parameters: sensitivity, noise, and minimum detectable pressure. The design variables and their constraints are first defined, followed by the mathematical formulation of the objective function.

Design variables

The design variables specify the diaphragm geometry, piezoresistor geometry and material, and operating bias. The diaphragm geometry is defined by its radius, a, and thickness, h. Each piezoresistor is described using two radii ( ba rr , ) for the arc and two radii ( 21, rr ) for the tapered piezoresistor, and one angle ( Arcθ and Taperθ ). We set the outer radii for the piezoresistors to be equal to the diaphragm radius, i.e. arrb == 2 . We also assume a balanced Wheatstone bridge, TaperArc RR = . Under this condition, one of the piezoresistor design variables may be determined from the remainder, for example by equating Eqs. (5) and (7),

( ) ( )Arc

aTaper

raraθ

θlnln

2 1= . (23)

Other design variables are the piezoresistor thickness tpiezo (same for both types), the piezoresistor doping concentration, ndop, and the bias voltage, VB. In total, the following eight design parameters are simultaneously varied in the optimization: a, h, θArc, ra, r1, tpiezo, ndop, and VB.

Limitations-constraints

The constraints may be categorized as physical bounds, manufacturing limits, and operational requirements. The specific values are adjustable for a given design and fabrication process. The constraints included in the problem formulation are as follows:

i) Lower (LB) and upper (UB) bounds for the variables. The following bounds are listed in Table 2:

LB ≤ a, h, θArc, ra, r1, tpiezo, ndop,VB ≤UB. (24)

ii) Minimum line-width, i.e. the smallest possible dimension. The minimum line-width for the resistors is set at wline=10 µm. In practice, the minimum line-width is determined by lithography and etching limitations:

.0,0

,0,0

1

1

≤−≤−≤−+≤−+

Taperline

Arcaline

line

aline

rwrw

arwarw

θθ

(25)

iii) Piezoresistor thickness. The piezoresistor thickness is constrained to be thinner than half of the diaphragm

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thickness. This is required to approximate the stress to be constant throughout the piezoresistor:

05.0 ≤− ht piezo . (26)

iv) Linearity.

0 lim

≤

−

ha

ha . (27)

Equation (4) leads to ( ) 513/ lim =ha for 800 =σ MPa:

v) Power consumption. The power dissipation in the bridge is given by Power=VB

2/RArc. If the thermal conduction and convection rates are exceeded, the piezoresistor temperature will rise due to Joule heating, causing temperature drift of material parameters and eventually failure due to accelerated electromigration and melting:

0 lim ≤− PowerPower . (28)

The power limit is set to 0.1W in the optimization analysis;

however, other power limits are also considered.

Problem formulation The mathematical representation of the optimization

problem is the minimization of an objective function denoted as )(xobjf ,

,0

0

05.0

0 0

0 0

UBxLB such that

),x( Minimize

lim

lim

1

1

obj,,,,,,x 1

≤−

≤

−

≤−

≤−≤−≤−+≤−+

≤≤

=

PowerPowerha

ha

ht

rwrw

arwarw

f

piezo

Taperline

Arcaline

line

aline

Vntrra,h BdoppiezoaArc

θθ

θ

(29)

where )(xobjf is customized for a performance parameter, sensitivity, noise, or minimum detectable pressure. The optimization problem was implemented in MATLAB (2002) using its optimization toolbox that employs sequential quadratic programming for non-linear constrained problems and calculates the gradients by finite difference method.

Pareto optimality The goal of optimizing (maximizing) sensitivity is at

odds with the goal of optimizing (minimizing) noise. For example, a larger piezoresistor volume reduces the

f/1 noise, but a larger piezoresistor volume also encompasses a lower stress region of the diaphragm, reducing the sensitivity since the stress is concentrated near the clamped boundary. Thus, optimization of engineering systems often faces the challenge of conflicting design criteria or goals. There are number of multi-objective optimization approaches as described in Belegundu and Chandrupatla, (1999). Setting one criterion as the objective function, and adding constraints ic on the others, for instance, can be employed in order to evaluate the multi-objective nature of the problems as given by

L,3,2 such that

minimize1

=≤ icf

f

iiobj

obj. (30)

By using different sets of possible ic , we obtain Pareto optimal points that are designs where one objective cannot be improved without deterioration in one of the other objectives, and construct a Pareto hypersurface (Belegundu and Chandrupatla, 1999). For the two objectives in our study, the hypersurface becomes a curve called a Pareto curve on which there is no point that provides an improvement compared to any other with respect to both sensitivity (increase) and electronic noise (decrease).

For the implementation of Pareto optimality or Pareto curve, the problem formulation given in Eq. (29) is modified by adding another constraint, 0)x(trade ≤g , to take into account of the sensitivity and noise interaction. We investigated two approaches: i) maximizing the sensitivity, but constraining the electronic noise not to exceed a specified limit and ii) minimizing the electronic noise, but constraining the sensitivity not to decrease beyond a specified limit. The two strategies can be summarized by

NEMLim

EMLimN

VfSSg

SfVVg

=−=

−=−=

)x(for )x(and

)x(for )x(

objtrade

objtrade

(31)

It is clear that a design minimizing MDP must lie on the Pareto curve. Pareto optimality provides one way of selecting the best combination of the two objectives.

OPTIMIZATION RESULTS

The design values in Arnold et al. (2003) were used as a reference and also as the initial design for performing optimization runs. The optimization results are summarized for the goals of maximizing the sensitivity, minimizing the noise, and minimizing the minimum detectable pressure. A Pareto curve is computed to

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investigate the trade off between sensitivity and electronic noise.

Maximizing sensitivity

Since the problem formulation is for minimization of an objective function, )x(objf in Eq. (29) is replaced by the negative of the sensitivity, Eq. (11), after substituting Eq. (12). For this performance metric, the electronic noise floor was not constrained at first. For maximum sensitivity one can expect an increase in aspect ratio ha / such that the linearity constraint associated with the small deformation theory is not violated ( ha / =513). Results of the optimization are presented in the 3rd column of Table 3. The diaphragm radius did not change since the reference/initial design already is at the upper bound, whereas the thickness of the diaphragm was reduced until the linearity constraint, Eq. (27) became active. The piezoresistor dimensions were mainly determined by the manufacturing limitations considered via the line-width constraint Eq. (25). The radial length reached its possible minimum 10 micron for both types of piezoresistor, because maximum stress, and as a result, maximum resistance modulations occur near the clamped boundary. The piezoresistor thickness is redundant due to the assumption of having constant stress distribution through the resistor thickness if noise is not considered. The doping concentration went to its lower limit because lower doping concentration level means higher piezoresistor coefficients due to Eq. (10). Finally, the bias voltage is at its upper bound as expected from Eq.(12).

The sensitivity was improved by more than a factor of 10 compared to the reference design, but the electronic noise floor of the design is quite high since no noise constraint was applied. As a result, the minimum detectable pressure increased substantially compared to the reference design (from 37.55dB=0.0015Pa to 79.18dB=0.182Pa). Minimizing electronic noise

The objective function, )x(objf in Eq. (29), is replaced by the rms noise voltage, Eq.(20). The sensitivity, at first, was not constrained to be higher than a given level. Results of the optimization are presented in the 4th column of Table 3. For minimum noise, Eq. (20) indicates that optimization is dominated by the piezoresistor design (resistance and number of carriers). In order to achieve low noise, a higher number of carriers is favorable, and the resistance needs to be lower. A higher number of carriers is necessary for lower 1/f noise component and is obtained by increasing the piezoresistor volume. Thus, the piezoresistor angles are set at high values. In fact, the arc angle is at its upper bound for the optimum result. The number of carriers also drives the diaphragm radius to its upper bound since the design has the outer radius of the piezoresistor to be equal to the diaphragm radius. The diaphragm thickness is also driven to its upper bound since

it allows a maximum piezoresistor thickness via the constraint relating these two thicknesses. The large piezoresistor thickness also results in low resistance. The doping concentration determines resistivity [Eq. (6)] and resistance. Since a low resistance directly results in a low thermal or Johnson noise component, the lowest resistivity (highest concentration) corresponds to the minimum noise level. Setting the bias voltage as low as possible for less 1/f noise contribution also favors the low power requirement. Power consideration is also a factor on the resistance levels. Although low thermal noise floor results from low resistance, power dissipation is increased. Therefore, the power constraint was active along with the piezoresistor and diaphragm thickness constraint.

The noise floor was reduced by about a factor of 18, but the sensitivity also dropped by about a factor of 20 compared to the reference design. As a result, the minimum detectable pressure increased compared to the reference design (from 37.55dB=0.0015Pa to 39.28dB=0.0018 Pa).

Minimizing minimum detectable pressure

With high sensitivity and low noise floor favoring simultaneously a small and large piezoresistor volume, respectively, minimization of the MDP can be expected to lead to an intermediate design. The objective function,

)(xobjf in Eq. (29) is now replaced by Eq. (22) in order to minimize the MDP. The final column of Table 3 presents the results for this optimization problem. Optimum design parameters indeed seem to be a compromise between the previous two cases. Comparing the outcome of the optimization with the reference design, although sensitivity is slightly decreased, the minimum detectable pressure is reduced to 0.0007 Pa (30.50dB) compared to 0.0015 Pa (37.55dB). For the optimum design, the line-width, geometric linearity, piezoresistor and diaphragm thickness, and power constraints became active.

Operational bias voltage parameter, BV , is crucial for the sensitivity, the noise voltage, the power consumption and as a result crucial for the MDP. Although it is used in the optimization as a design variable, it can be varied during operation of the device. Figure 2(a) shows the power consumption as a function of BV , identifying the MDP for the 0.1W power constraint. The power versus performance metrics plots are shown in Figure 2(b) for the MDP optimum design summarized above, but as a function of BV . It is clear that with the increased bias voltage both sensitivity and noise voltage increase, but their cumulative effect is in favor of the MDP metric as it decreases. Therefore, the power consumption that limits the functionality of the piezoresistor is the key factor for the optimum operational bias voltage BV .

Since the power constraint is important for optimizing the device, we also ran the MDP minimization for two additional power limits. The optimizations for power limits of 0.01W, 0.1W and 1W resulted in MDP levels of

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35.34dB, 30.50dB and 28.41dB, respectively. Figure 3 presents the noise spectrums for the two optimum MDP designs with 0.1W and 1W power consumption, for which the optimal BV is about 4 V and 13 V, respectively (as the power varies with the square of BV ). It shows that the noise floor increases due to increase in 1/f noise component, but less than a factor of 10 . The sensitivity,

however, also increases by about a factor of 10 , and the resultant MDP can be lowered by about 2 dB.

Sensitivity versus noise – trade off via Pareto optimality

The results presented so far demonstrate the fact that

neither maximizing the sensitivity nor minimizing the noise floor alone is an efficient way to improve the performance of the design since the trade off between the two metrics should be taken into account. In this section, we introduce a more complete trade off study for optimizing the device performance in order to evaluate the efficiency of the MDP as a metric for optimization.

As discussed, the two extreme cases are minimizing the electronic noise alone, denoted as d1 ( 11 , dNdEM VS ) and maximizing the sensitivity alone, denoted as d2 ( 22 , dNdEM VS ). The trade off or compromise between these two metrics can be shown by performing both optimization cases by modifying Eq. (29) via Eq. (30) for several defined limits 21 dNLimdN VVV ≤≤ and

21 dEMLimdEM SSS ≤≤ , respectively. Figure 4 presents the results in a noise versus

sensitivity Pareto curve (left axis) and noise versus MDP (right axis) plot, where two extreme cases d1 and d2 are the end points of the plot. Note that designs on the Pareto curve are compared to the reference design (Arnold et al. 2003), indicating that the reference design may be improved by up to 7 dB given the assumed material and electronic properties. The design obtained by minimizing the MDP is also shown on the Pareto curve and shows that MDP minimization is an effective method to compromise the trade off unless there is a strong noise or sensitivity requirement.

EFFECT OF UNCERTAINTY

It is quite common to achieve different performance than the theory predicts when the device is actually built and tested. One reason for this is the uncertainty in the parameters such as the material properties and geometry that were used in the theoretical calculations. Here we present an uncertainty evaluation for the optimum designs on the Pareto curve by performing Monte Carlo simulations. We first considered material properties, design variables and process parameters as normally distributed random parameters, N( paramparam σµ , ) where

paramµ denotes the average value for the parameter, and

paramσ denotes the standard deviation for the parameter. For each design on the Pareto curve, we assumed paramµ as the values used for that particular design. The standard deviation was calculated as,

paramparamparam µσ cov= (32)

where paramcov is the coefficient of variation for the parameter. Table 4 reports the random parameters and the associated paramcov for generating the statistical distributions. For each design on the Pareto curve (Figure 4), for example the ith Pareto optimal design, we performed 1000 simulations based on the random sampling of the parameters from the associated normal distributions. The sensitivity SEM, noise floor VN, and minimum detectable pressure minP for all 1000 sampled designs were calculated. Then the statistics associated with the three performance metrics were computed; the mean,

iSEMµ ,

iVNµ , and

iPminµ , and the standard deviation,

iSEMσ ,

iVNσ , and

iPminσ . Finally, the coefficients of variation

were used as a measure of the uncertainty given by,

.cov

,cov

,cov

min

min

min

iP

iP

iP

iV

iV

iV

iS

iS

iS

N

N

N

EM

EM

EM

µ

σ

µ

σ

µ

σ

=

=

=

(33)

Figure 5 shows the effect of about 10% parameter uncertainty on the three metrics, SEM, VN , and minP for the Pareto optimal designs. One can expect the least sensitivity to uncertainty for a metric to occur for a design which is the optimum for that particular metric. For instance, the noise floor, VN , is the least sensitive to uncertainty at design d1 obtained by minimizing VN (

1cov

dVN=5.2%). Also minP is the least sensitive to

uncertainty at the optimal MDP design (

MDPPmincov =13.6%). For sensitivity, SEM, the design

least sensitive to uncertainty was also the optimal MDP design. This is significant because it leads us to the conclusion that the design obtained by minimizing the MDP is also favorable in terms of sensitivity to parameter uncertainty. As a summary, for the MDP optimal design, uncertainty levels for the three metrics are:

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10 American Institute of Aeronautics and Astronautics

MDPSEMcov =12.8%,

MDPVNcov =6.0%, and

MDPPmincov =13.6%.

CONCLUDING REMARKS

The design optimization problem for piezoresistive microphone performance parameters was formulated. Optimum dimensions of the diaphragm and the optimum piezoresistor geometry and location were sought for maximum pressure sensitivity and minimum electronic noise. The formulation also included several constraints such as linearity and power dissipated by the piezoresistors. It was demonstrated that maximizing the sensitivity alone or minimizing the noise alone do not provide an effective performance optimization since there is a tradeoff between the two. The tradeoff was studied in detail by solving the optimization problem for one metric limiting the other one at different levels. This enabled us to generate a Pareto curve demonstrating the trade off between electronic noise and sensitivity. Their ratio, the minimum detectable pressure, was also employed as the objective function, and demonstrated a way of dealing with the trade off. The results also indicated that the critical constraints are the linearity and power consumption. The minimum MDP design was also found to be favorable regarding the sensitivity to parameter uncertainty.

ACKNOWLEDGEMENTS This work is supported by the Office of Naval

Research (contract #N00014-00-1-0343) monitored by Dr. Kam Ng and by NASA (contract # NAG-1-2133) monitored by Dr. William H. Humphreys, Jr.

REFERENCES

Arnold, D. P., Nishida, T., Cattafesta, L., Sheplak, M., "MEMS-Based Acoustic Array Technology," J. Acoust. Soc. Am. 113 (1), pp.289-298, 2003.

Belegundu, A. D. and Chandrupatla, T. R., Optimization Concepts and Applications in Engineering, Prentice Hall, 1999.

Bhardwaj, S., Sheplak, M, and Nishida, T., “S/N Optimization and Noise Considerations for Piezoresistive Microphones,” Proceedings of the 16th International Conference on Noise in Physical Systems and 1/f Fluctuations, (World Scientific, New Jersey, 2001), pp. 549-552, 2001.

de Graaf, H. C., Huybers, M. T. M., “1/f noise in polycrystalline silicon resistors,” J. Appl. Physics, Vol. 54, pp. 2504-2507, 1983.

Harley, J. A. and Kenny, T. W., “1/f noise considerations for the design and process optimization of piezoresistive cantilevers,” Journal of Microelectromechanical Systems, 9 (2), pp. 226-235, 2000.

Hooge, F. N., “1/f Noise,” Physica, Vol. 83B, pp. 14-23, 1976

Kanda, Y., “A Graphical Representation of the Piezoresistance Coefficients in Slilicon,” IEEE Transactions on Electron Devices, ED-29 (1), pp. 64-70, 1982.

Mathworks Inc., Matlab Version 6.5, 2002. Melvas, P., Kalvesten, E. and Stemme, G., “A

Temperature Compensated Dual Beam Pressure Sensor,” Sensors and Actuators A, Vol. 100 pp. 46-53, 2002.

Nishida, T and Sah, C.-T., “A physically based mobility model for MOSFET numerical simulation,” Trans. Electron Devices, Vol. ED-34, No. 2, pp. 310-320, 1987.

Nyquist, H., “Thermal Agitation of Electric Charge in Conductors”, Phys. Rev., 32, July 1928.

Saini R., Bhardwaj, S., Nishida, T. and Sheplak, M., “Scaling Relations for Piezoresistive Microphones,” Proceednings of IMECE 2000. Internatioanl Mechanical Engineering Congress and Exposition, Orlando, 2000

Scheeper, P. R., van der Donk, A. G. H., Olthuis, W., Bergveld, P., 1994, “A Review of Silicon Microphones,” Sensors and Actuators A, Vol. 44, pp. 1-11.

Schellin, R., Hess, G., “A silicon subminiature microphone based on piezoresistive polysilicon strain gauges,” Sensors and Actuators A, Vol. 32, pp. 555-559, 1992.

Sheplak, M. and Dugundji, J., “Large Deftlections of Clamped Circular Plates under Initial Tension and Transitions to Membrane Behavior,” Journal of Applied Mechanics, 65, pp. 107-115, 1998.

Sheplak, M., Schmidt, M.A., and Breuer, K.S., “Dielectrically-Isolated, Single-Crystal Silicon, Piezoresistive Microphone,” Technical Digest, Solid-State Sensor and Actuator Workshop, Hilton Head, SC, pp. 23-26, 1998.

Sheplak, M., Seiner, J. M., Breuer, K. S. and Schmidt, M. A., “A MEMS Microphone for Aeroacoustic Measurements,” AIAA-99-0606, 1999.

Smith, C. S., “Piezoresistance Effects in Germanium and Silicon,” Physical Review, 94, pp. 42-49, 1954.

Spencer, R. R, Fleischer, B. M., Barth, P. W., Angell, J. B., 1988, “A Theoretical study of Transducer Noise in Piezoresistive and capacitive Silicon Pressure Sensors,” IEEE Transactions on Electron Devices, Vol. 35, No. 8, pp. 1289-1297.

Van der Ziel, A. , Noise, (Prentice-Hall), 1954.

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Table 1: Material properties for silicon nitride diaphragm and p-type silicon piezoresistors.

Young’s modulus for diaphragm, Ediaph (GPa)

270

Young’s modulus for piezoresistors, Epiezo (GPa)

160

Poisson’s ratio for diaphragm, ν 0.27

Residual stress, 0σ (MPa) 80

Piezoresistive coefficient, 11π , (Pa-1) 6.6x10-11

Piezoresistive coefficient, 12π , (Pa-1) -1.1x10-11

Piezoresistive coefficient, 44π , (Pa-1) 138.1x10-11

maxP , (Pa) 2000

Table 2: Lover (LB) and upper (UB) limits for the design variables.

a ( mµ ) h ( mµ ) Arcθ ( o ) ar ( mµ )

100-500 0.01-2 1-30 100-500

1r ( mµ ) piezot ( mµ ) dopn (cm-3) BV (V)

100-500 0.01-2 1018-4x1020 1-15

Table 3: Reference and optimum designs.

Parameters

Reference

design, 2nd gen.

Maximized

sensitivity, EMS

Minimized

noise floor, NV

Minimized

MDP

2,, rra b ( mµ ) 500 500 500 500 h ( mµ ) 1.000 0.975 2.000 0.975 ar ( mµ ) 480 490 415 490 1r ( mµ ) 475 490 357 175 piezot ( mµ ) 0.500 0.006 1.000 0.488

Arcθ ( o ) 16.95 2.92 30.00 30.00

Taperθ ( o ) 0.96 1.32 14.20 5.57 dopn (cm-3) 1x1020 1x1018 4x1020 4x1020 0σ (MPa) 80 80 80 80 Hα 0.001 0.001 0.001 0.001

BV (Volt) 3 15 1 4 Power total (W) 0.012 0.001 0.100 0.100 Sensitivity, EMS (µV/Pa) 5.18 71.87 0.24 3.79 Noise floor, NV (µV) 0.0078 13.076 .0004 .0025 MDP (dB) 37.55 79.18 39.28 30.50

Table 4: Uncertainty of the parameters: paramµ , mean value for the parameter; paramparamparam µσ cov= ,

standard deviation for the parameter ( paramcov : coefficient of variation).

Parameters Uncertainty paramcov (%)

h , piezot , Arcθ , Taperθ 1.0 dopn , 0σ , Hα , Ediaph, Epiezo, ν 10.0

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+++++

++

++

++

++

++

Power (W)

S EM(µ

V/Pa

),V N

(nV)

MD

P(d

B)

0.25 0.5 0.75 1 1.250

5

10

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40SEM (µV/Pa)VN (nV)MDP (dB)

+

minimizedMDP

VB (V)

Pow

er(W

)

5 10 15

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

minimizedMDP

Figure 1 Piezoresistive microphone: a) Arc piezoresistor, b) Taper piezoresistors, c) Wheatstone bridge configuration.

(a) (b) Figure 2: Effect of bias voltage on minimum MDP design obtained for Powerlim =0.1W: a) voltage versus power consumption, b) power versus performance.

RTaper

RArc RTaper

RArc

VBOVDiaphragm

(1 mm x 1 µm)

Arc Resistor

[110]ra

rb

θArc

Taper Resistor

Turn around

r1

r2

θTaper

(a)

(b)

(c)

[110]

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13 American Institute of Aeronautics and Astronautics

Pareto optimal design

coef

ficie

ntof

vari

atio

n(%

)

4

5

6

7

8

9

10

11

12

13

14

15

16

SEMVNPmin

d1 1 2 3 4 5 6 7 8 9 10 d2MDP

f (Hz.)

V/√H

z

100 101 102 103 10410-10

10-9

10-8

10-7

10-6

1/f noiseJohnson Noise

f (Hz.)V/√H

z100 101 102 103 104

10-10

10-9

10-8

10-7

10-6

1/f noiseJohnson Noise

(a) (b) Figure 3: Noise spectrum: a) minimum MDP design for Powerlim =0.1W, b) minimum MDP design for Powerlim =1W.

Figure 4: Trade off between sensitivity (to maximize) and electronic noise (to minimize): Pareto optimal designs-Pareto curve and their resultant minimum detectable pressure.

Figure 5: Coefficient of variation (%) for the performance metrics SEM, VN and Pmin

VN (µV)

S EM(µ

V/Pa

)

MD

P(d

B)

10-4 10-3 10-2 10-1 100 101 10210-1

100

101

102

30

35

40

45

50

55

60

65

70

75

80

SEM (µV/Pa)MDP (dB)SEM (µV/Pa)MDP (dB)

minimizedMDP

12

3

4

56

78

910

d1

d2

2nd gen.