  Parametric amplification is widely used in nanoelectro-mechanical systems to
enhance the transduced mechanical signals. Although parametric amplification
has been studied in different mechanical resonator systems, the nonlinear
dynamics involved receives less attention. Taking advantage of the excellent
electrical and mechanical properties of graphene, we demonstrate electrical
tunable parametric amplification using a doubly clamped graphene nanomechanical
resonator. By applying external microwave pumping with twice the resonant
frequency, we investigate parametric amplification in the nonlinear regime. We
experimentally show that the extracted coefficient of the nonlinear Duffing
force {\alpha} and the nonlinear damping coefficient {\eta} vary as a function
of external pumping power, indicating the influence of higher-order
nonlinearity beyond the Duffing (~x^3) and van der Pol (~x^2 dx/dt) types in
our device. Even when the higher-order nonlinearity is involved, parametric
amplification still can be achieved in the nonlinear regime. The parametric
gain increases and shows a tendency of saturation with increasing external
pumping power. Further, the parametric gain can be electrically tuned by the
gate voltage with a maximum gain of 10.2 dB achieved at the gate voltage of 19
V. Our results will benefit studies on nonlinear dynamics, especially nonlinear
damping in graphene nanomechanical resonators that has been debated in the
community over past decade.
