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 $\ensuremath{\sim}2\dots{}4\phantom{\rule{0.1em}{0ex}}\ensuremath{\mu}\mathrm{m}$.
Highlights
Significant research interest has recently emerged in magnetoelectric (ME) composites based on observed giant magnetoelectric effects, rich physics, and potential for real-world applications, such as ultrasensitive roomtemperature magnetic field sensors for biomedical fields, actuators, energy harvesters, and nonlinear frequency mixing devices [1,2,3,4,5,6,7]
For systematic analysis of the effects of cantilever dimensions and layer thicknesses on resonance frequency, ME response, sensitivity, different noise components, and the magnetic field detection limit, we proceed by focusing on scaling behavior, moving from many to a few constrained parameters
We derived a theory of thermal-vibration noise in magnetoelectric-composite cantilevers and calculate the resonant magnetoelectric response and thermal noise behavior of (Fe90Co10)78Si12B10/Si/AlN composites to determine the magnetic field detection limit for the first bending mode
Summary
Significant research interest has recently emerged in magnetoelectric (ME) composites based on observed giant magnetoelectric effects, rich physics, and potential for real-world applications, such as ultrasensitive roomtemperature magnetic field sensors for biomedical fields, actuators, energy harvesters, and nonlinear frequency mixing devices [1,2,3,4,5,6,7]. The layer sequence with the Si substrate in the center is motivated by greater ME response [37,39] and processing advantages for the functional layers, i.e., (Fe90Co10)78Si12B10 and AlN, due to low surface roughness on both sides of the Si substrate Using this theory, we systematically investigate a wide range of ME cantilever sizes and layer thicknesses, calculating magnetic-field-induced ME-response signals and individual and sums of uncorrelated noise signals at the output of matched charge amplifiers including thermal vibration, Johnson-Nyquist, amplifier, and cable-associated noise and filter effects, to yield the detection limit behavior for given resonance frequencies and quality factors. Further investigation addresses the behavior with functional layer thickness and detection limit scaling with cantilever length, resonance frequency, and quality factor
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