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The role of wall’s curvature on the quantum tunneling within subnanometer gaps

The effect of wall’s curvature on the quantum tunneling within an air gap in gold nanodimers is investigated to realize the relation between the dimer radius or the wall’s curvature and the red-shift in the surface plasmon (SP) coupling band. To apprehend the gap properties namely gap size and curvature, in which quantum tunneling starts to play a role, instead of a full quantum mechanical approach which is only tractable in small dimers, the quantum corrected model (QCM) is employed [1] . In this approach, the quantum tunneling probability within the gap is translated into a DC conductivity and consequently a fictitious material is defined for the gap, taking into account the effect of d-band electrons in gold [2] , [3] (cf. Fig. 1a ). The conductivity variations along the gap width (red dashed line within the inset) is depicted in the Fig. 1b for 15nm and 40nm dimers with various subnanometer gap sizes, which shows that the effective gap volume including the proposed DC conductivity model tends to be larger in dimers with larger radii (cf. Fig. 1c ). To find the SP coupling band in each of the dimers, in a finite element based solver, namely COMSOL Multiphysics, the hydrodynamic model via partial differential equations is implemented in addition to the classical Maxwell equations [4] , where the quantum tunneling is introduced via the fictitious material for the gap. The simulations are done for 15nm and 40nm gold dimers with gap sizes ranging from 0.1nm up to 2nm and the results are summarized in the Fig. 1d , where all three cases of the fully classical model, the hydrodynamic model as well as the QCM are included. As shown the hydrodynamic model results converge to the classical model for larger dimers (40nm dimer) or for large gap sizes, however the QCM shows a different trend and in the case of the larger dimer, at the 0.6nm gap size starts to deviate from the hydrodynamic model, which is due to the start of the quantum tunneling. This deviation happens in smaller gap sizes namely 0.52nm gap in the 15nm dimer. The underlying reason is the more modest curvature of larger radii which enhances the effective DC conductivity volume and hence the quantum tunneling starts at larger gap sizes.