NONLINEAR MECHANICS OF GRAPHENE AND MASS-LOADING INDUCED DEPHASING IN NANORESONATORS
Doctoral thesis, 2012
I summarize the results of my research in the subjects of nonlinear mechanics of graphene
resonators, mass-loading induced dephasing in nanomechanical resonators and spintronicsbased
mesoscopic heat engines for cooling the fundamental flexural mode of a CNT resonator.
This thesis consists of three parts. In the first part I present a mechanical description for
monolayer graphene membranes. The equations of motion are derived in the long wavelength
limit starting from an atomistic model, which accounts for the energy cost to change the
length of the sp2 covalent bonds and also the angle between neighboring bonds in graphene.
I also propose to use nonlinear dynamics of square graphene resonators to measure the mass
and position of a single adsorbed particle using only narrow-band frequency sensors.
In the second part, I consider the effects of random mass loading in nanomechanical resonators.
Random mass loading leads to random modulation of the resonance frequency (dephasing
process) of the vibrational eigenmodes of the resonator. I consider first the situation
where the dephasing process is not affected by the motion of the resonator (no backaction).
Here, the random mass loading is caused by adsorption, desorption and diffusion of small
particles along the resonator. I discuss the method of interfering partial susceptibilities to calculate
the susceptibility of underdamped vibrational eigenmodes. I find that the final shape of
the eigenmode absorption spectrum line depends on the intensity and correlation time of the
frequency noise. In the presence of dephasing, the eigenmode energy relaxation rate cannot
be measured from the width of the absorption line. I also discuss a method to characterize the
dephasing process. Then, I consider the case of backaction in the dephasing process for the
case of particles diffusing along the resonator. The backaction is induced by an inertial force,
which drives the particles towards the antinode(s) of the excited eigenmode. I show that dephasing
subject to backaction can lead to bistability and rare interstate switching between
small and large amplitude vibrational states (with the particles delocalized and localized at
the antinode(s), respectively) if the particles diffuse comparatively fast. The diffusion induced
bistability in driven resonators has a different origin from the conventional bistability and interstate
switching, which occurs in driven nonlinear oscillators subject to a weak source of
Finally, the third part deals with a proposal for a physical realization of a mesoscopic heat
engine which consists of two spin polarized leads held at different temperatures (heat reservoirs)
linked by a CNT resonator in the presence of a nonuniform magnetic field. The latter
induces spin-mechanical coupling between the electronic subsystem (two-level system inside
the CNT) and the mechanical subsystem (fundamental flexural mode). One would expect that
the effective temperature of the mechanical subsystem should be between the temperatures
of the leads. However, I show that if the leads have spin polarization >50% and the coupling
between the mechanical subsystem and other baths is weaker than the spin-mechanical
coupling, then it is possible for the effective temperature of the mechanical subsystem to be
smaller than the temperature of the leads. In this regime, I discuss the conditions required to
achieve mechanical ground state cooling.
Dephasing in Nanomechanical Resonators