Tension-induced nonlinearities of flexural modes in nanomechanical resonators

We consider the tension-induced nonlinearities of mechanical resonators and derive the Hamiltonian of the flexural modes up to the fourth order in the position operators. This tension can be controlled by a nearby gate voltage. We focus on systems which allow large deformations, $u\left(x\right)\gg h$, compared to the thickness $h$ of the resonator and show that in this case, the third-order coupling can become nonzero due to the induced dc deformation and offers the possibility to realize radiation-pressure-type equations of motion encountered in optomechanics. The fourth-order coupling is relevant especially for relatively low voltages. It can be detected by accessing the Duffing regime and by measuring frequency shifts due to mode-mode coupling.

Figure 1

Schematic picture of the studied setup: a metallic beam whose deformation is controlled by a gate voltage $V_g$ coupled to the beam via capacitance $C[d-u(y,t\left)\right]$.

Figure 2

Duffing constant and frequency response function (inset) of the first mode with ${\tau }_0=0$. The Duffing constant is plotted from a numerical solution of the full Euler-Bernoulli equation obtained from (2). The response function is plotted with ${\gamma }^2=0.04{\omega }_s^2$ and $\frac{f^2}{{\omega }_1^2}=0.002m^2{\omega }_s^2{h}^2$. Here $E_{\mathrm{stress}}=ES{h}^4/\left(8L^3\right)$.

Figure 3

Duffing constant for the first mode with different initial tensions ${\tau }_0$. From left to right, the dimensionless initial tension ${\tau }_0$ starts from 0 and increases with the step of 10. The dashed lines are our analytic expressions and the full lines are numerical solutions of the full Euler-Bernoulli equations obtained from Eq. (2). The deviation between the two sets of curves at low $V_g$ is due to our scheme of approximating mode functions by harmonic functions. Here, $E_{\mathrm{stress}}=ES{h}^4/\left(8L^3\right)$.

Figure 4

Duffing constant for the third mode with different initial tensions ${\tau }_0$. From left to right, the dimensionless initial tension ${\tau }_0$ starts from 0 and increases with the step of 10. The dashed lines are our analytic expressions and the full lines are numerical solutions of the full Euler-Bernoulli equations obtained from Eq. (2). Note that now the $V_g$ induced effect in the analytics is very small. The approximation with harmonic mode functions underestimates ${\Theta }_{mp}^{no}$, which is the reason for the discrepancy between the full numerical solutions and our analytic approximations. Here, $E_{\mathrm{stress}}=ES{h}^4/\left(8L^3\right)$.

Figure 5

Crossover voltage $V_c$ for the sign change of the Duffing constant with respect to the initial tension ${\tau }_0$. Here, $E_{\mathrm{stress}}=ES{h}^4/\left(8L^3\right)$.

Figure 6

Effective frequency and damping (inset) of mode $n=1$ when driving mode $m=3$ with initial tension ${\tau }_0=0$ (no symbols) and $\tau =10$ (circles) in the case of red detuning (red lines, lower) with $\Delta ={\omega }_1$ and in the case of blue detuning (blue lines, upper) with $\Delta =-{\omega }_1$. The full lines are numerical results obtained by solving the full Euler-Bernoulli equation obtained by requiring $u(x,t)$ to minimize the energy in Eq. (2), and dashed lines follow Eqs. (5), (14), and (A8). Here, $\chi =4m{\omega }_s^2{\left|{\alpha }_m\right|}^2{x}_{z{p}_m}^2\frac{x_{z{p}_n}^2}{h^2}$ and $E_{\mathrm{stress}}=ES{h}^4/\left(8L^3\right)$.

Figure 7

Effective frequency and damping (inset) of mode $n=1$ when driving mode $m=5$ and when initial tension ${\tau }_0=0$ in the case of red detuning (red lines, lower) with $\Delta ={\omega }_1$ and in the case of blue detuning (blue lines, upper) with $\Delta =-{\omega }_1$. The full lines are numerical results obtained by solving the full Euler-Bernoulli equation and the dashed lines are analytical results derived in the text.

Figure 8

Effective frequency and damping (inset) of mode $n=3$ when driving mode $m=5$ and when initial tension ${\tau }_0=0$ in the case of red detuning (red lines, lower) with $\Delta ={\omega }_3$ and in the case of blue detuning (blue lines, upper) with $\Delta =-{\omega }_3$. The full lines are numerical results obtained by solving the full Euler-Bernoulli equation and dashed lines are analytical results derived in the text.