We investigated the interface modes in a heterostructure consisting of a semi-infinite metallic layer and a semi-infinite Fibonacci quasiperiodic structure. Various properties of the interface modes, such as their spatial localizations, self-similarities, and multifractal properties, are studied. The interface modes decay exponentially in different ways, and the highest localized mode is found to be a mode in the lower stable gap with the largest gap width. A localization index is introduced to understand the localization properties of the interface modes. We found that the localization index of the interface modes in some of the stable gaps will converge to two slightly different constants related to the parity of the Fibonacci generation. In addition, the localization-delocalization transition is also found in the interface modes of the transient gaps.
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