  Strain engineering is a promising approach for suppressing the OFF-state
conductance in graphene-based devices that arises from Klein tunnelling. In
this work, we derive a comprehensive tight-binding Hamiltonian for strained
graphene that incorporates strain-induced effects that have been neglected
hitherto, such as the distortion of the unit cell under strain, the effect of
strain on the next-nearest neighbor coupling, and the second-order
contributions of the strain tensor. We derive the corresponding low-energy
effective Hamiltonian about the Dirac points and reformulate the boundary
conditions at the interfaces between strained and unstrained graphene in light
of additional terms in the Hamiltonian. By applying these boundary conditions,
we evaluate the transmission across a strained graphene heterostructure
consisting of a central segment sandwiched between two unstrained leads.
Modulation of the transmitted current can be effected by varying the magnitude
and direction of the applied strain, as well as by the applying a gate voltage.
Based on realistic parameter values, we predict that high ON-OFF ratios of up
to $10^{12}$ as well as high current valley polarization can be achieved in the
strain-modulated device.
