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We theoretically investigate the tunneling transport through a localized region of uniaxial strain in a graphene nanoribbon. When a tunneling current flows along the nanoribbon, the gauge field generated by local strain induces a finite tunneling valley Hall response. A top gate located in the strained region can be used for selective switching of Klein tunneling in dependence of the momentum of the tunneling carriers. This produces a large valley Hall response. Due to time reversal symmetry, the transverse valley signal does not carry net charge. However, the presence of proximity-induced magnetization in the graphene nanoribbon breaks the time reversal symmetry, resulting in the emergence of a charge Hall conductance, even when the magnetization lies in the plane of the nanoribbon. Unlike the recently proposed tunneling planar Hall effect in topological insulators, the strain-induced tunneling Hall effect in graphene nanoribbons does not rely on spin-momentum locking. We discuss the various functionalities of the magnetically proximitized, strained graphene nanoribbon and the possibility of using both the tunneling longitudinal and Hall response for applications in spin straintronic devices.
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