Nonharmonic oscillations of nanosized cantilevers due to quantum-size effects

Using a one-dimensional jellium model and standard beam theory we calculate the spring constant of a vibrating nanowire cantilever. By using the asymptotic energy eigenvalues of the standing electron waves over the nanometer-sized cross-section area, the change in the grand canonical potential is calculated and hence the force and the spring constant. As the wire is bent more electron states fits in its cross section. This has an impact on the spring “constant” which oscillates slightly with the bending of the wire. In this way we obtain an amplitude-dependent resonance frequency of the oscillations that should be detectable.

Figure 1

A cross section $S$ of the cantilever before and after a perpendicular force is applied on the wires end. Inset: side view and top view of the cantilever nanowire. The expansions and contractions of the cross sections are larger nearer the fixed end if the cross section can contract and expand freely.

Figure 2

Example of spring constant $k$ variation as a function of bending $Z^{\prime }=\sqrt{\frac{9}{8}}\frac{\nu d_{0}Z}{L^2}$ obtained from an equation having the same type of dependency on $n$ and $Z$ as Eq. (32) has, i.e., Eq. (38). The three curves show the forms of the curve obtained from Eq. (38) for $\frac{E_F}{E_0}$ below, at and above an integer. We see that Eq. (38) is independent of the sign of $Z$ as is to be expected by symmetry. The $k$ axis always intersects the curve at a local maxima or minima. However, this maxima or minima may not be very wide depending on how close we are that a new electron state will fit when bending the wire slightly. We see in the figure that the width is small when $\frac{E_F}{E_0}$ is close to an integer (middle curve) and broader farther from an integer.

Figure 3

The amplitude response of a driven weakly nonlinear harmonic oscillator. $\beta =0$ corresponds to the harmonic case where the resonance frequency is ${\omega }_0$. When the amplitude is increased (or $\beta$ is increased), the resonance frequency is shifted toward higher values for positive $\beta$ and toward lower values for negative $\beta$, from ${\omega }_0$ to $\sqrt{{\omega }_0^2+\frac{3}{4}\beta A^2}$.

Figure 4

(Color online) $\beta /\left(C{\omega }_0^2\right)$ as a function of $N=\frac{E_F}{E_0}$ obtained by dividing the second term in Eq. (39) with the first term and identifying the spring constant in Eq. (35). $C=\frac{9{\nu }^2{d}_0^2}{8L^4}$. We note the singularities at integer numbers, corresponding to when we have $k=k_0+\text{const}\times \left|Z\right|$ instead of $k=k_0+\text{const}\times Z^2$, which is the normal behavior near $Z=0$.

Figure 5

(Color online) Example of spring constant as a function of different bending $Z^{\prime }=\sqrt{\frac{9}{8}}\frac{\nu d_{0}Z}{L^2}$ for four different temperatures using Eq. (43). Curve a: $k_{B}T=0.0001\text{eV}$, curve b: $k_{B}T=0.01\text{eV}$, curve c: $k_{B}T=0.025\text{eV}$ (room temperature), and curve d: $k_{B}T=0.05\text{eV}$. The plot is made using $\mu =2.9\text{eV}$, $E_0=0.47\text{eV}$ (i.e., $d_0=1\text{nm}$), and the upper energy limit of integration is taken to be $E_{cut}=5.0\text{eV}$. Increasing $E_{cut}$ to 7.0 yields no visible change in the curves. For $E_{cut}>\mu$ the contribution to the expression decreases rapidly due to the Fermi-Dirac function.