Ultrasensitive Magnetometers Based on Carbon-Nanotube Mechanical Resonators

We describe herein how a nanoelectromechanical system based on a carbon nanotube used as a force sensor can enable one to assess the magnetic properties of a single and very small nano-object grafted onto the nanotube. Numerical simulations performed within the framework of the Euler-Bernoulli theory of beams predict that the magnetic field dependence of the nanotube mechanical resonance frequency is a direct probe for the nano-object magnetic properties and that a sensitivity around a few (few hundreds) Bohr magnetic moments at low temperature (room temperature) can be expected.

Figure 1

(a) Schematic view of a CNTNEMS. A single wall carbon nanotube is suspended between two metallic electrodes over a trench etched in a ${\mathrm{SiO}}_2/\mathrm{Si}$ wafer. The potential $V_{\mathrm{DC}}+V_{\mathrm{AC}}{e}^{i\omega t}$ actuates the nanotube. $r(z)$ is the CNTNEMS motion in the basis $(x,y,z)$. (b) Schematic view of a magnetic NO (represented in orange) grafted onto the CNTNEMS. ${\theta }_c$ is the angle between the NO anisotropy axis $c$ and the CNTNEMS. ${\theta }_0$ is the angle made by the magnetic field with regard to the $c$ axis at zero magnetic field, $\theta$ is the angle between $m$ and $c$, and $\alpha$ is the bending angle of the nanotube under magnetic field.

Figure 2

Magnetic field dependence of the CNTNEMS mechanical properties and of $\theta$. The magnetic field is in the $(x,z)$ plane. (a) Magnetic field dependence of $\theta$ for ${\theta }_0=\pi /4$. (b),(c) Shape of the CNTNEMS under magnetic field. The curve in the absence of a magnetic field shows a tiny negative bending due to the electrostatic force. The black dashed (red solid, purple dotted) curves correspond to the black dashed (red solid, purple dotted) curve in (a). (d),(e) Magnetic hysteresis loop of the NO and magnetic field dependence of the CNTNEMS mechanical resonance frequency for different ${\theta }_0$. For more clarity, the frequency curves are shifted by 75 kHz. The black arrows point to the field at which $m$ jumps.

Figure 3

Effect of the NO position on the CNTNEMS magnetic sensitivity. (a) Shape of the CNTNEMS as a function of $z$ for different NO positions $0.1$, $0.2$, $0.4$, and $0.5\mu \mathrm{m}$. $d_{\mathrm{NO}}=1\mathrm{nm}$, $B=0.85\mathrm{T}$, and ${\theta }_0=\pi /2$. (b) Corresponding frequency shift. The maximum frequency shift is achieved near $L/4$. (c) Magnetic loop hysteresis of a NO with $d=0.5\mathrm{nm}$. (d) CNTNEMS magnetic frequency shift dependency for different ${\theta }_0$ with the NO grafted at $L/4$. The curves are shifted by 4 kHz for more clarity. The magnetic frequency shift ranges between 1 and 4 kHz.