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We theoretically reveal the unusual features of the Josephson effect in a strained zigzag graphene nanoribbon with a small length relative to the superconducting coherence length and an arbitrary width $W$. We find a step-wise variation of the critical supercurrent with the width of the nanoribbon, showing additional small width plateaus placed between the wide steps of a non-strained structure. We further demonstrate the peculiar quantization of the critical supercurrent in terms of the strain, resulted from the coupling of the pseudospin of Dirac fermions with the strain-induced gauge potential, where the height of the steps decreases with growing the strength of the fictitious gauge potential. Moreover, our results determine the potential of the proposed superconducting quantum point contact for the realization of the supercurrent switch under an applied strain. Besides, we find that the local density of states of the strained zigzag nanoribbon displays a crossover between the decaying and oscillating behavior with the distance from the edges, by tuning the width and Fermi wavelength of the nanoribbon.
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