Introduction to Electrostatic (Capacitive) Sensors

Introduction to MEMS Transduction Principles

Micro-Electromechanical Systems (MEMS) are fundamentally defined by their ability to perform transduction—the conversion of energy between the physical and electrical domains. For engineers, this necessitates a rigorous understanding of how external physical stimuli are converted into measurable electrical signals (sensors) and how electrical inputs are leveraged to generate controlled mechanical motion (actuators).

In modern micro-scale engineering, 4 primary physical principles dominate:

• Piezoelectric: Stress-induced polarization in specific crystal structures.
• Thermal: Exploiting differences in thermal expansion coefficients to produce displacement.
• Magnetic: Utilizing Lorentz forces or magnetic attraction, often requiring complex integration of coils or permanent magnets.
• Capacitive (Electrostatic): Leveraging the interaction between stored charge and electrode geometry.

However, these principles can also be used to transduce external input into electrical output $\to$ sensors!

Why it Matters: The Versatility of the Capacitive Method The capacitive method is arguably the most ubiquitous in commercial MEMS. Unlike piezoelectric or thermal methods, it requires no "smart" or exotic materials; it can be implemented using standard structural materials like polysilicon or aluminum. Furthermore, capacitive devices exhibit exceptionally low power consumption (being voltage-driven rather than current-driven) and provide a high-fidelity mechanical-to-electrical response. While the operational distance is limited, the scaling benefits at the micro-level make it the gold standard for high-speed, high-precision sensing and actuation.

Ferroelectrics and the Piezoelectric Effect

Piezoelectricity is a property of non-centrosymmetric materials where mechanical stress induces electric polarization. To master these materials, one must understand the hierarchy of properties:

All ferroelectrics (铁电材料) are pyroelectric (热电) and therefore piezoelectric.

• Examples of materials that are piezoelectric but not ferroelectric: quartz (${\mathrm{SiO}}_2$), ZnO, AlN.
• Examples of Ferroelectric materials (and thus piezoelectric): ${\mathrm{BaTiO}}_3$, PZT, PVDF (in certain phases).

This nested relationship is critical for material selection in sensor design.

Material Classification

• Piezoelectric (Non-Ferroelectric): Materials such as Quartz (${\mathrm{SiO}}_2$), Zinc Oxide (ZnO), and Aluminum Nitride (AlN). These lack switchable domains but are highly stable for frequency control.
• Ferroelectric (and Piezoelectric): Materials like Barium Titanate (${\mathrm{BaTiO}}_3$), Lead Zirconate Titanate (PZT), and Polyvinylidene fluoride (PVDF). These possess a spontaneous polarization that can be reversed by an external field.

The Direct and Converse Effects: The Direct Effect involves generating an electric displacement ($D$) from mechanical stress ($T$): $D=dT+\epsilon E$ Where $d$ is the piezoelectric coefficient and $\epsilon$ is the permittivity. Conversely, the Converse Effect describes the generation of mechanical strain when an electric field is applied.

Mathematical Modeling of Voltage Generation In a sensing application, the voltage ($V$) generated across a piezoelectric structure of thickness $L$ and cross-section area $A$ when subjected to a force ($F$) is:

$$V=\frac{d\cdot F\cdot L}{{\epsilon }_0\epsilon A}$$

(Conversely, if a voltage is applied across a structure made of a piezoelectric material, a displacement will occur on the structure due to $F$ generate across the material.)

Technical Note: Dimensionally, the thickness $L$ is critical; a larger thickness, higher stress ($T$), and a higher piezoelectric coefficient lead to a significantly larger generated voltage.

Practical Application: Quartz resonators remain the most common application of this principle, with approximately 10 million units produced daily. These single-crystal devices are vacuum-packaged and valued for their high Q-factor in timing circuits.

Fundamentals of Electrostatic (Capacitive) Sensing and Actuation

Electrostatic devices operate as variable capacitors. The sensing mechanism relies on the fact that capacitance ($C$) changes when physical parameters are altered by distance (gap), position (overlap area), or the dielectric media (e.g., air vs. liquid).

Actuation: electrostatic force (attraction) between moving and fixed plates as a voltage is applied between them.

2 major Configurations:

1. Parallel Plate capacitor: Typically used for out-of-plane (vertical) motion.
2. Interdigitated Fingers / IDT: Commonly called "comb drives," these are designed for in-plane motion.

Relative Merits

• Pros: Universal sensing/actuation without special materials; ultra-low power; high speed (limited only by mechanical response and charging time).
• Cons: Inverses scaling (force $\propto 1/x^2$), vulnerability to particles in small gaps, and susceptibility to stiction (sticking due to molecular forces).

Parallel Plate Capacitor

Mathematical Analysis

For a parallel plate with area $A$ and an initial gap $d$, the capacitance is:

$$C=Q/V=Q/(Ed),E=Q/\epsilon A⟹C=\frac{Q}{\frac{Q}{\epsilon A}d}=\frac{\epsilon A}{d}.$$

Note:

• Equations without considering fringe electric field.

  • The fringe field is frequently ignored in first-order analysis. It is nonetheless important. Its effect can be captured accurately in finite element simulation tools.

• Stored energy:

  $$U=\frac{1}{2}C{V}^2=\frac{1}{2}\frac{Q^2}{C}$$

• Force is derivative of energy w.r.t. pertinent dimensional variable

  $$F=\frac{\partial U}{\partial d}=\frac{1}{2}\frac{\partial C}{\partial d}{V}^2$$

• Plug in the expression for capacitor: $C=\frac{Q}{\frac{Q}{\epsilon A}d}=\frac{\epsilon A}{d}.$

• Arrive at the expression for force:

  $$F=\frac{\partial U}{\partial d}=-\frac{1}{2}\frac{\epsilon A}{d^2}{V}^2=-\frac{1}{2}\frac{C{V}^2}{d}.$$

Relative Merits

Relative Merits of Capacitor Actuators: Pros:

• Nearly universal sensing and actuation; no need for special materials.
• Low power. Actuation driven by voltage, not current.
• High speed. Use charging and discharging, therefore realizing full mechanical response speed.

Cons:

• Force and distance inversely scaled - to obtain larger force, the distance must be small.
• In some applications, vulnerable to particles as the spacing is small - needs packaging.
• Vulnerable to sticking phenomenon due to molecular forces.

Analysis of Electrostatic Actuator

What happens to a parallel plate capacitor when the applied voltage is gradually increased?

The Electromechanical Model

An electrostatic actuator is a competitive system between non-linear electrical forces and linear mechanical restoring forces.

Figure 1: An Equivalent Electromechanical Model.

Equilibrium and Hooke’s Law: The suspension acts as a spring with a mechanical spring constant $K_m$. Equilibrium is reached when: $|F_{\text{elec}}|=|F_{\text{mech}}|⟹-\frac{1}{2}\frac{\epsilon A{V}^2}{d^2}=K_m|x|$

Force Balance Equation at Given Applied Voltage $V$:

$$-K_{m}x=\frac{\epsilon A{V}^2}{2(x+x_0{)}^2}$$

Figure 2: Curves for force balance equation. This graph shows the intersection of the linear mechanical restoring force and the family of non-linear electrical force curves for V_1, V_2, V_3. As the voltage reaches the Pull-In Voltage (V_PI), the curves no longer intersect.

Design Implications: The "First Root" Rule The displacement can be found by solving the third-order equation: $K_{m}x(x_0+x{)}^2+\frac{1}{2}\epsilon A{V}^2=0$ When solving this equation, three roots are generated. From a physical perspective, we accept only the first root (the smallest displacement). The other roots represent unstable equilibrium points beyond the pull-in threshold.

Determining Equilibrium Position Graphically:

• At each specific applied voltage, the equilibrium position can be determined by the intersection of the linear line and the curved line.
• For certain cases, two equilibrium positions are possible. However, as the plate moves from top to bottom, the first equilibrium position is typically assumed.
• Note that one curve intersects the linear line only at one point.
• As voltage increases, the curve would have no equilibrium position.
• This transition voltage is called pull-in voltage.
• The fact that at certain voltage, no equilibrium position can be found, is called pull-in effect.

The "Pull-In" Effect

Definition: pull-in / snap in effect

As the voltage bias increases from $0$ across a pair of parallel plates, the distance between such plates would decrease until they reach $2/3$ of the original spacing, at which point the 2 plates would be suddenly snapped into contact.

Figure 3: A threshold point.

Mathematical Determination of Pull-in Voltage: Defining Electrical Force Constant Let’s define the tangent of the electric force term. It is called electrical force constant, $K_e$:

$$K_e=\frac{\partial F}{\partial x}=\frac{C{V}^2}{d^2}$$

When voltage is below the pull-in voltage, the magnitude of $K_e$ and $K_m$ are not equal at equilibrium.

Key Takeaway:

• Reliable tuning is limited to only $1/3$ of the total gap. Once the plates reach $2/3$ of the original spacing (a displacement of $-x_0/3$), positive feedback causes the plates to "snap" into contact.
• Electrostatic micro mirrors: reduced range of reliable position tuning
• Electrostatic tunable capacitor
  • Reduced range of tuning and reduced tuning range
  • Tuning distance less than $1/3$, tuning capacitance less than $50\mathrm{\%}$.

Example problem

A parallel plate capacitor suspended by 2 fixed-fixed cantilever beams, each with length, width and thickness denoted $l,w$ and $t$, respectively. The material is made of polysilicon, with a Young’s modulus of $120\mathrm{GPa}$.

• $L=400\mu \mathrm{m},w=10\mu \mathrm{m},andt=1\mu \mathrm{m}$.
• The gap $x_0$ between 2 plates is $2\mu \mathrm{m}$.
• The plate area is $400\mu \mathrm{m}$ by $400\mu \mathrm{m}$.

Calculate the amount of vertical displacement when a voltage of $0.4$ volts is applied.

Figure 4: Example image.

1. Find mechanical force constants

   • Calculate force constant of one beam first: Under force $F$, the max deflection is $d=F{l}^3/(12EI)$. The force constant is therefore
   $$K_m=\frac{F}{d}=\frac{12EI}{l^3}=\frac{Ew{t}^3}{l^3}=0.01875\mathrm{N}/\mathrm{m}.$$
   • Total force constant encountered by the parallel plate is $K_m=0.0375\mathrm{N}/\mathrm{m}$.

2. Find out the Pull-in Voltage: Find out pull-in voltage and compare with the applied voltage.

   1. Find the static capacitance value $C_0$:
   $$C_0=\frac{8.85\times {10}^{-12}(\mathrm{F}/\mathrm{m})\times (400\times {10}^{-6}{)}^2}{2\times {10}^{-6}}=7.083\times {10}^{-13}\mathrm{F}.$$

   Find the pull-in voltage value
   

   $$V_p=\frac{2x_0}{3}\sqrt{\frac{k_m}{1.5C_0}}=0.25(\mathrm{volts}).$$

Hence, when the applied voltage is $0.4\mathrm{volt}$, the beam has been pulled-in. The displacement is therefore $2\mu \mathrm{m}$.

What if the applied voltage is $0.2\mathrm{V}$?

• Not sufficient to pull-in

• Deformation can be solved by solving the following equation

  $$V^2=\frac{-2k_{m}x(x+x_0{)}^2}{\epsilon A}=\frac{-2k_{m}x(x+x_0)}{C}.$$

  or

  $$x^3+2x_0{x}^2+x_0^{2}x+\frac{v^2\epsilon A}{2k_m}=0$$

Solving Third Order Equation To solve $x^3+a{x}^2+bx+c=0$: Apply $y=x+a/3,p=\frac{-a^2}{3}+b,q=2{\left(\frac{a}{3}\right)}^3-\frac{ab}{3}+c$, then

$$y=A+B,\text{where }A=\sqrt[3]{\frac{-q}{2}+\sqrt{Q}},B=\sqrt[3]{\frac{-q}{2}-\sqrt{Q}}$$

and $x=A+B-a/3$.

others

Mechanical Spring:

• Cantilever beams with various boundary conditions
• Torsional bars with various boundary conditions

Example of MEMS system: Optical Micro Switches (MEMS Scanning Mirrors)

• Torsional parallel plate capacitor support
• 2 stable positions (+/- 10 degrees w.r.t. rest)
• All aluminum structure
• No process steps entails temperature above $300-{350}^{\circ }\mathrm{C}$

Interdigitated Comb Drives and Accelerometers

Lateral vs. Transverse Drives

• Lateral Comb Drives: Preferred for actuators because force is independent of displacement (as long as overlap is maintained). However, force is a function of finger thickness; thicker fingers yield higher force.

  Total capacitance is proportional to the overlap length and depth of the fingers, and inversely proportional to the distance.

  • Pros:
    • Frequently used in actuators for its relatively long achievable driving distance.
  • Cons
    • force output is a function of finger thickness. The thicker the fingers, the large force it will be.
    • Relatively large footprint

• Transverse Comb Drives: Preferred for sensing due to extreme sensitivity to small displacements, though they have limited travel. Direction of finger movement is orthogonal to the direction of fingers.

  • Pros: Frequently used for sensing for the sensitivity and ease of fabrication
  • Cons: not used as actuator because of the physical limit of distance.

Sensing Devices Based on Transverse Comb Drive: Analog Devices ADXL Accelerometer The ADXL series utilizes a polysilicon proof mass supported by cantilever beams.

Summary

Most common MEMS actuators operate with one of the following physical principles:

• Piezoelectric, Thermal, Magnetic, or Capacitive (Electrostatic)
  • However, these principles can also be used to transduce external input into electrical output è SENSORS!
• Capacitive Sensing Principle is one of the most widely used method for commercial MEMS sensors
  • Low current (i.e., low power consumption)
  • Very good mechanical input to electrical output response
  • However, measurable input range is relatively small (due to small operation distance of capacitive plates.)