Limit of Thermal-Vibration Noise in Magnetic Field Detection with Magnetoelectric-Composite Cantilevers

Abstract

The fundamental thermal limit of magnetic field sensing using the resonance-enhanced magnetoelectric (ME) effect in strain-coupled ferroic composite cantilevers is investigated theoretically with results applied to $({\mathrm{Fe}}_{90}{\mathrm{Co}}_{10}{)}_{78}{\mathrm{Si}}_{12}{\mathrm{B}}_{10}/\mathrm{Si}/\mathrm{Al}\mathrm{N}$ using experimental material parameters. An analytic theory of the ME response and noise from thermal vibrations, Johnson-Nyquist noise from dielectric losses in $\mathrm{Al}\mathrm{N}$, and amplifier electronics is given to yield the magnetic field detection limit for the first bending mode. A systematic investigation of scaling behavior of ME response, sensitivity, noise components, and detection limit with cantilever size, layer thicknesses, quality factor ${Q}_{f}$, and resonance frequency $f$ is presented emphasizing the low-frequency small-size regime essential for biomagnetic sensing applications. Thermal vibrations are found to dominate all investigated noise sources for practical composite sizes except at low quality factors. Detection limit scaling inversely with functional layer thickness, as $1{/\mathrm{size}}^{2}$ for constant $f$ and ${Q}_{f}$, and as ${Q}_{f}^{\ensuremath{-}1/2}$ (constant size and $f$) is found. Achievable detection limits below $100\phantom{\rule{0.2em}{0ex}}\mathrm{fT}/\mathrm{H}{\mathrm{z}}^{1/2}$ are predicted for $({\mathrm{Fe}}_{90}{\mathrm{Co}}_{10}{)}_{78}{\mathrm{Si}}_{12}{\mathrm{B}}_{10}/\mathrm{Si}/\mathrm{Al}\mathrm{N}$ composite cantilevers at sizes below $10\phantom{\rule{0.2em}{0ex}}\mathrm{mm}$ (and even lower without size constraint) with $f=1\phantom{\rule{0.2em}{0ex}}\mathrm{kHz}$, ${Q}_{f}=1000$, and $({\mathrm{Fe}}_{90}{\mathrm{Co}}_{10}{)}_{78}{\mathrm{Si}}_{12}{\mathrm{B}}_{10}$ layers of $50\phantom{\rule{0.2em}{0ex}}\ensuremath{\mu}\mathrm{m}$. The scaling behavior implies the absence of an absolute detection limit for thick ferroic layers within the range of validity of the experimental material parameters observed at $\ensuremath{\sim}2\dots{}4\phantom{\rule{0.1em}{0ex}}\ensuremath{\mu}\mathrm{m}$.