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Valley currents and non-local resistances of graphene nanostructures withbroken inversion symmetry are considered theoretically in the linear responseregime. Scattering state wave functions of electrons entering the nanostructurefrom the contacts represented by groups of ideal leads are calculated bysolving the Lippmann- Schwinger equation and are projected onto the valleystate subspaces to obtain the valley velocity fields and total valley currentsin the nanostructures. In the tunneling regime when the Fermi energy is in thespectral gap around the Dirac point energy, inversion symmetry breaking isfound to result in strong enhancement of the nonlocal 4 terminalButtiker-Landauer resistance and in valley currents several times stronger thanthe conventional electric current. These strong valley currents are the directresult of the injection of electrons from a contact into the graphene in thetunneling regime. They are chiral and occur near contacts from which electronsare injected into the nanostructure whether or not a net electric current flowsthrough the contact. It is also pointed out that enhanced non-local resistancesin the linear response regime are not a signature of valley currents arisingfrom the combined effect of the electric field and Berry curvature on thevelocities of electrons.

Valley currents and non-local resistances of graphene nanostructures with broken inversion symmetry from the perspective of scattering theory