Fig. 2: SET-based nonlinearity.
From: Nonlinear nanomechanical resonators approaching the quantum ground state
a, Schematic of the nanotube vibrating at ωm. A quantum dot (highlighted in red) is formed along the suspended nanotube; the total electron tunnelling rate to the two leads is Γe. b, Origin of the SET-based nonlinearity. The two linear force–displacement curves (shown in black) correspond to the dot filled with either N or N + 1 electrons; the slope is given by the spring constant \(m{{\omega }_{{{{\rm{m}}}}}^{{{{\rm{o}}}}}}^{2}\) and the two curves are separated by \({{\Delta }}x=2(g/{\omega }_{{{{\rm{m}}}}}^{{{{\rm{o}}}}}){x}_{{{{\rm{zp}}}}}\) caused by the force created by one electron tunnelling onto the quantum dot. The force felt by the vibrations is an average of the two black forces weighted by the Fermi–Dirac distribution when \({{{\varGamma }}}_{{{{\rm{e}}}}} > {\omega }_{{{{\rm{m}}}}}^{{{{\rm{o}}}}}\). The resulting force (red) is nonlinear for vibration displacements smaller than \(\sim \frac{{k}_{{{{\rm{B}}}}}T}{\hslash g}{x}_{{{{\rm{zp}}}}}\); the reduced slope at zero vibration displacement indicates the decrease in ωm. c, Gate voltage dependence of conductance G of device I at T = 6 K.
