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In this work we compare two fundamentally different approaches to theelectronic transport in deformed graphene: a) the condensed matter approach inwhich current flow paths are obtained by applying the non-equilibrium Green'sfunction (NEGF) method to the tight-binding model with local strain, b) thegeneral relativistic approach in which classical trajectories of relativisticpoint particles moving in a curved surface with a pseudo-magnetic field arecalculated. The connection between the two is established in the long-wavelimit via an effective Dirac Hamiltonian in curved space. Geometrical opticsapproximation, applied to focused current beams, allows us to directly comparethe wave and the particle pictures. We obtain very good numerical agreementbetween the quantum and the classical approaches for a fairly wide set ofparameters, improving with the increasing size of the system. The presentedmethod offers an enormous reduction of complexity from irregular tight-bindingHamiltonians defined on large lattices to geometric language for curvedcontinuous surfaces. It facilitates a comfortable and efficient tool forpredicting electronic transport properties in graphene nanostructures withcomplicated geometries. Combination of the curvature and the pseudo-magneticfield paves the way to new interesting transport phenomena such as bending orfocusing (lensing) of currents depending on the shape of the deformation. Itcan be applied in designing ultrasensitive sensors or in nanoelectronics.

Current flow paths in deformed graphene: from quantum transport to classical trajectories in curved space