MEMS Accelerometer

A mass on a spring, shrunk a thousand times

If you tied a weight to a vertical spring and accelerated the whole assembly sideways, the weight would lag behind the support until the spring force matched the inertial force — a static deflection proportional to acceleration. A MEMS accelerometer is that exact experiment, scaled down by three orders of magnitude in size and twelve in mass. The "weight" is a slab of silicon a fraction of a millimetre across, weighing a few millionths of a gram. The "spring" is a set of slender silicon flexure beams. The "ruler" reading the deflection is not optical or mechanical — it is electrical, in the form of capacitance between interdigitated comb fingers on the moving mass and on a fixed anchor.

The genius of the architecture is that everything — proof mass, flexures, sense fingers, anchors, and travel stops — is etched from a single piece of silicon in one set of lithography steps. There is no assembly. A typical 3-axis accelerometer fits in a 2 × 2 × 0.8 mm package and costs the phone OEM about ten cents in volume. That economic miracle is why every smartphone, drone, vibration monitor, fitness band, AR headset, and car since roughly 2007 contains at least one.

Newton's second law, scaled to nanograms

The core physics is the simplest possible. Suspend a proof mass m on a spring of stiffness k. If the case is accelerated at a, the proof mass — inertia keeping it in place — appears to displace within the case by

x = m a / k        (static, below resonance)

For m = 10 μg = 10⁻⁸ kg and k = 10 N/m, the deflection at 1 g (a = 9.81 m/s²) is

x = (10⁻⁸ × 9.81) / 10  ≈  10⁻⁸ m  =  10 nm

Ten nanometres per g. That is a slightly humbling number: roughly the diameter of a small protein, or a few dozen silicon lattice constants. To resolve milli-g (acceleration of a slow-moving elevator floor, or your fingertip pressing on the phone glass), the readout must measure picometre-scale displacement of a mass smaller than a grain of pollen, on a chip cheap enough to give away.

The comb-finger transducer

The standard transducer is a comb-drive interdigitated capacitor. Many fingers attached to the proof mass interleave with anchored fingers; the capacitance between them depends on either the overlap area (sensitivity to in-plane motion along the finger axis) or the gap (sensitivity to motion perpendicular to the finger axis). The classical out-of-plane and gap-varying topologies are the most common for accelerometers, because ΔC/Δx scales as 1/d² and so a small gap yields very high sensitivity.

For N pairs of plates of thickness t with gap d, and a small displacement Δx that closes one set of gaps by Δx while opening the opposite set by the same amount, the differential capacitance change is

ΔC ≈ ε₀ · N · (t · Δx / d²)   per gap-direction displacement

where ε₀ = 8.854 pF/m. With N = 100 fingers, t = 20 μm, and d = 2 μm, a 1 nm displacement gives

ΔC ≈ 8.854 × 10⁻¹² × 100 × (20 × 10⁻⁶ × 10⁻⁹) / (2 × 10⁻⁶)²
   ≈ 4.4 × 10⁻¹⁹ F
   ≈ 0.4 aF

Sub-attofarad per nanometre. To detect milli-g — that is, tens of femtometres per finger pair on a 10 μg mass — the readout electronics must resolve a tiny fraction of an attofarad against a background capacitance of several picofarads. The trick is to make the measurement differential: a second comb on the opposite side of the proof mass deflects in the opposite sense, so subtracting the two capacitances both doubles the signal and cancels every common-mode drift (temperature, package stress, EMI). A switched-capacitor charge amplifier modulates a high-frequency carrier (typically 100 kHz to several MHz) onto the sense node and synchronously demodulates it; the resulting baseband voltage is the acceleration signal.

Resonance, bandwidth and damping

A mass on a spring is a harmonic oscillator with undamped resonance

ω₀ = √(k / m)

For typical numbers (m = 10 μg, k = 10 N/m): ω₀/2π ≈ 5 kHz. Below ω₀ the sensor response is flat — every g of acceleration produces the static deflection m a / k. Around ω₀ the response peaks (depending on damping) and may ring. Above ω₀ the mass cannot keep up and sensitivity rolls off as 1/ω². The resonance therefore sets the bandwidth: a sensor with ω₀/2π = 5 kHz can usefully read accelerations from DC up to perhaps a few kHz.

Damping is supplied by viscous gas flow through narrow gaps — air or nitrogen sealed in the cavity is squeezed in and out as the comb fingers move, and the squeeze-film viscous drag damps the motion. Designers target a quality factor Q ≈ 0.7 (slightly below critical) so the sensor neither rings at its resonance nor over-damps and rolls off too early. Achieving the right Q involves choosing both the cavity gas (often nitrogen) and the seal pressure (a few hundred millibar to a few bar) at wafer bonding time.

Parameter	Symbol	Typical value	Note
Proof mass	m	1–100 μg	Larger m → lower noise, more sensitivity
Spring constant	k	1–100 N/m	Set by flexure geometry (Eb·t³/L³)
Resonance	ω₀/2π	0.5–10 kHz	√(k/m)
Finger count	N	50–500	Linear gain on ΔC
Comb gap	d	1–3 μm	Limited by DRIE sidewall fidelity
Static deflection at 1 g	x	1–100 nm	m g / k
Noise floor	—	10–500 μg/√Hz	Brownian (k_B T) + electronic

Closed-loop force-feedback

In the open-loop scheme described above, the proof mass really moves; the readout reports its position. That introduces two problems at the limits. First, at high acceleration the deflection becomes large enough that the spring is no longer linear and the mass eventually slams against its travel stops — destroying linearity and risking damage. Second, the bandwidth is fundamentally tied to the mechanical resonance.

The fix is electrostatic force-feedback. The same comb-drive that senses position can be driven with a voltage to apply an electrostatic force F_e ∝ V². In a closed-loop accelerometer the readout continuously adjusts the drive voltage to hold the proof mass at the centre — true zero deflection. The mass barely moves; the applied feedback voltage is what scales with acceleration. The mechanical resonance now lies inside the loop, so the closed-loop bandwidth is set by the electronic loop gain rather than by ω₀. Linearity becomes excellent, the dynamic range expands by orders of magnitude, and the same physical chip can read both micro-g (steady gravity tilt) and many tens of g (impact) without saturation. Navigation-grade IMUs — used in missiles, guided munitions, and inertial-only undersea navigation — are almost always closed-loop MEMS or quartz accelerometers.

Where the noise comes from

The fundamental noise floor of a MEMS accelerometer has two components.

Brownian noise. The proof mass is in thermal contact with the cavity gas (and via the springs with the silicon at room temperature). Thermal fluctuations excite the mass and produce an irreducible mechanical noise. Using the fluctuation–dissipation theorem on a damped harmonic oscillator, the equivalent acceleration noise density is

a_n  =  √(4 k_B T ω₀ / (m Q))    (m²/s²/Hz)

For T = 300 K, m = 10 μg, ω₀ = 2π·5 kHz, Q = 0.7: a_n ≈ 30 μg/√Hz. Doubling the mass cuts the noise by √2; reducing Q (more damping) raises it.

Electronic noise. Charge-amplifier kT/C noise, 1/f noise of the front-end transistor, and ADC quantisation all contribute. In modern commercial parts (e.g. ADXL355, BMI088) the total noise density is in the 25–500 μg/√Hz range, often electronics-limited. Allan-variance plots are the canonical way to characterise these sensors — bias instability typically bottoms out at a few μg, and velocity random walk at a few cm/s/√hr.

Sensing principles compared

Type	Transducer	DC?	Bandwidth	Range	Typical use
Capacitive	Comb-finger ΔC	Yes	DC – 5 kHz	±2 to ±200 g	Phones, drones, IMUs
Piezoresistive	Doped Si strain gauge	Yes	DC – 20 kHz	±50 to ±10,000 g	Airbags, ballistics, shock
Piezoelectric	AlN / PZT film charge	No (AC only)	1 Hz – 50 kHz	±100 to ±100,000 g	Vibration monitoring
Thermal (convective)	Heated-gas bubble + thermopile	Yes	DC – 50 Hz	±2 to ±10 g	Toys, low-cost tilt
Resonant	Force-detuned beam, frequency output	Yes	DC – 1 kHz	±10 g	Tactical, gravimetry

Capacitive dominates the consumer market because of its combination of DC response, low power, low noise, and CMOS-compatibility for monolithic integration. Piezoresistive is preferred for high-g and harsh-environment work because the strain gauges are simple and rugged. Piezoelectric is the choice for fast vibration analysis because of its wide AC bandwidth, even though it cannot read static tilt. Thermal accelerometers — no moving solid parts at all — are immune to mechanical shock and find niches in toys and survivable instrumentation.

Fabrication: DRIE on SOI

Almost every modern capacitive MEMS accelerometer uses a silicon-on-insulator (SOI) starting wafer: a thick handle wafer, a buried oxide (BOX) layer typically 1–4 μm, and a device layer of 10–50 μm of single-crystal silicon. The flow looks like:

1. Mask & pattern. Photolithography defines the geometry of mass, flexures, fingers, anchors, and travel stops on the device layer.
2. DRIE etch. The Bosch process — alternating SF₆ etch and C₄F₈ passivation pulses — cuts vertical sidewalls through the device-layer silicon at aspect ratios of 30:1 or higher, stopping on the buried oxide. The characteristic scalloped sidewall is a visual signature.
3. Release. Vapor HF (or BHF) selectively dissolves the buried oxide under the moving parts. Anchors keep their oxide beneath them and remain tethered; the proof mass and flexures are freed.
4. Cap bonding. A second wafer with cavities is bonded on top (anodic to glass, or eutectic via Au-Si / Al-Ge). The bond is done under the target gas and pressure so the cavity is pre-filled with the right damping medium.
5. Singulate and co-package. Wafers are diced, MEMS die is wire-bonded or stacked with a CMOS ASIC inside a plastic QFN package, and the part ships.

The whole process is shared with MEMS gyroscopes (the proof mass is just driven into oscillation instead of free), MEMS microphones (a membrane instead of a mass), and MEMS pressure sensors (a diaphragm). That foundry-level commonality is why MEMS economics work: one Bosch-process line serves a billion-units-per-year market across many sensor types.

Where MEMS accelerometers are everywhere

• Smartphone screen rotation. A 3-axis ±2 g part senses the gravity vector and reports tilt; the OS rotates the UI between portrait and landscape. The same data drives step counting (Fourier peaks at 1–3 Hz), image stabilisation (high-frequency shake), and tap detection for "double-tap to wake".
• Airbag deploy. A high-g (±50 to ±250 g) accelerometer on the floor pan watches the integrated velocity change. When the crash-discrimination algorithm decides this is a real crash, a squib fires the bag in 20–40 ms. Modern cars use a network of front, side, and rollover sensors, each MEMS-based.
• Drone & UAV IMU. A 9-DOF stack (3-axis accelerometer + 3-axis gyro + 3-axis magnetometer) feeds an Extended Kalman Filter that fuses the three to a drift-corrected attitude. Without it the flight controller could not even hold level; the InvenSense MPU-6000 and Bosch BMI088 are de-facto standards.
• Gaming controllers. The Wii Remote (2006) put a 3-axis accelerometer in front of a hundred million households and proved the consumer-tilt input thesis. Today every game controller, VR headset, and AR glasses unit has at least one IMU.
• Earthquake & fall detection. The Apple Watch crash-detection feature uses a high-g (±200 g) accelerometer and a Kalman-filtered velocity-change algorithm; smartphones run free-fall and seismic-detection apps (MyShake, Earthquake Network) that crowd-source quake early warning.
• Industrial vibration monitoring. Piezoelectric and capacitive MEMS sensors on pumps, motors, and gearboxes log spectral RMS levels; an FFT trend over weeks catches bearing wear and unbalance long before failure. The "industry 4.0 condition-monitoring" market is built on these parts.
• Satellite and rocket guidance. Closed-loop tactical-grade MEMS units (e.g. Honeywell HG1900, Northrop LN-200) provide arc-minute-per-hour bias stability and now replace fibre-optic gyros and mechanical accelerometers in many guidance applications.

Real-world parts

• Analog Devices ADXL family. ADXL345 (3-axis, ±16 g, the classic Arduino sensor) and ADXL355/357 (high-precision, ±2/8 g, <25 μg/√Hz noise) define the consumer and industrial reference designs.
• STMicroelectronics LIS3DH / LIS2DH12. Ultra-low-power 3-axis, ±2/4/8/16 g, used in hundreds of millions of fitness trackers and feature phones.
• Bosch BMI series. The BMI088 (6-DOF accelerometer + gyro, drone-grade) and BMA series (consumer) ship at the highest volumes in the industry; Bosch alone ships well over a billion MEMS sensors a year.
• InvenSense / TDK MPU and ICM lines. 6-DOF and 9-DOF integrated IMUs (MPU-6050, ICM-20948) standard in drones, robotics, and consumer electronics.
• Murata / Colibrys / Honeywell tactical units. Closed-loop, navigation-grade, with bias stability in the single-μg range — used in undersea, aerospace, and defence inertial systems.

Common pitfalls

• Forgetting that 1 g of gravity is always there. The accelerometer reads "specific force" — gravity plus linear acceleration. A stationary phone lying flat reads (0, 0, 1 g), not zero. Apps that count vertical jumps must subtract the gravity vector first, usually with a low-pass filter or via IMU fusion.
• Confusing tilt with motion. A constant tilt and a constant linear acceleration look identical to a single accelerometer. Disambiguation requires a gyroscope (rate of change of orientation) or a magnetometer (absolute heading) in the fusion algorithm.
• Aliasing high-frequency vibration. If the ADC samples at 100 Hz but the mechanical resonance is at 5 kHz, narrow-band shocks alias into the baseband and look like real acceleration. Modern parts include an analog antialias low-pass before the ADC; designers must still set it appropriately for their application.
• Ignoring temperature drift. Both bias and sensitivity vary with temperature (typically tens of μg/°C and hundreds of ppm/°C). Datasheets specify these, but in tactical-grade work the user must calibrate the sensor over its operating range — temperature-modelled bias compensation can reduce error by an order of magnitude.
• Soldering-induced stress. Plastic QFN packages strain the MEMS die when the PCB warps from solder reflow. The die experiences picostrain-scale stress that produces offset shifts of tens of milli-g. Best practice: use stress-isolated dual-die packages, or characterise and store an offset trim after PCB assembly.
• Latching travel stops. Very high shocks can drive the proof mass into the travel stops with enough force to make it adhere via van der Waals stiction. Designs include anti-stiction bumps and surface treatments; some parts ship with a "shock recovery" routine in firmware.

A two-billion-unit market

The global MEMS market crossed roughly USD 18 billion in 2024, with accelerometers accounting for several billion units shipped per year — well over two billion accelerometer dice when you include integrated IMUs. Three OEMs (Bosch, ST, TDK/InvenSense) account for most of the consumer volume; Analog Devices and Honeywell dominate higher-grade instrumentation; and a long tail of fabless players (Kionix, Murata, Silicon Sensing, Colibrys) fills the rest. The marginal cost of a 3-axis consumer accelerometer is now under USD 0.20 in volume, which is why every Bluetooth toothbrush, vape pen, and ear-bud charging case can afford one. The chip that decided which way "up" is — for a $10⁻⁸ kg piece of silicon — has become one of the most economically important inventions of the 21st century.