NEMS Multimode Mass Sensing: Locating a Single Landed Particle

Beam Parameters

Axial tension T 1.00 µN

Beam length L 2000 nm

Particle mass δm 4.0 fg

Mode Frequencies (live)

Mode 1, f₁

—

Mode 2, f₂

—

Δf₁/f₁

0.00%

Δf₂/f₂

0.00%

Inverted vs. True

Estimated x₀

—

True x₀

—

Estimated δm

—

True δm

—

Position error

—

Mass error

—

One mode's frequency shift alone can't tell you whether a heavy particle landed near a node or a light one landed at an antinode — the two-mode ratio breaks that degeneracy.

How it works

A doubly-clamped nanobeam under strong axial tension T behaves like a vibrating string: its flexural mode shapes are φ_n(x) = sin(nπx/L) with resonant frequencies

f_n = (n / 2L) · sqrt(T / μ)

where μ = M/L is the mass per unit length and M is the beam's own (unloaded) mass. When a point mass δm « M lands at position x₀, first-order perturbation theory gives a mode-dependent fractional frequency shift:

Δf_n / f_n ≈ -(2δm / M) · sin²(nπx₀/L)

A single mode cannot separate δm from x₀ — a large mass near a node looks like a small mass near an antinode. But the ratio of two modes' shifts cancels δm entirely:

R = (Δf₂/f₂) / (Δf₁/f₁) = sin²(2πx₀/L) / sin²(πx₀/L) = 4cos²(πx₀/L)

x₀ = (L/π) · arccos( sqrt(R/4) )

Once x₀ is known, either equation gives δm directly. This mode-shape-ratio inversion is the working principle behind real multimode NEMS "inertial imaging" mass spectrometers, which use three or more modes to resolve mass, position and the shape of adsorbed nanoparticles from vibration alone — no optical or fluorescent label needed.

Tension — sets how stiff (string-like) the beam is; higher tension raises both f₁ and f₂ without changing their 2:1 ratio.
Length — a longer beam has a larger, more massive sensing area but lower resonant frequencies.
Particle mass — the true δm used the next time you drop a particle.
Drop particle — picks a random landing point x₀ along the beam, applies the exact perturbation to f₁ and f₂, then runs the closed-form inversion above to recover x₀ and δm from the shifted frequencies alone.

f₁ / f₂ shift

Doubly-clamped nanobeam Landed particle (true x₀)

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