Tension-induced non-linearities of flexural modes in nanomechanical resonators

We consider the tension-induced non-linearities 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 $u(x)\gg h$ compared to the thickness $h$ of the resonator and show that in this case the third-order coupling can become non-zero due to the induced dc deformation and offers the possibility to realize 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.