## 1 Introduction  
During the last decade, there has been a continuous effort toward the development of plasmonic devices. One reason is surface plasmon polaritons (SPPs)--originating from the coupling of an electromagnetic wave to the free electrons of a metal--can strongly confine electromagnetic fields to overcome the optical diffraction limit of dielectric optics, and be very sensitive to the surrounding medium [1, 2]. Plasmonic devices are promising to build dense on-chip integrated circuits for next-generation information technology and high-performance nanoscale sensors [3, 4, 5, 6]. Among plasmonic devices, SPP waveguides are one of the key elements. Different configurations have been proposed and verified, and their degree of applicability analysed. For example, a triangular groove milled on a metal surface can work as the channel plasmon subwavelength waveguide [7, 8]. Dielectric-loaded SPP waveguides made of a rectangular dielectric rib deposited on a metal film or strip have recently emerged as a potential plasmonic method that can be integrated seamlessly with current silicon-on-insulator photonic circuits, and can sustain the transfer of information at high data rates [9; 10; 11]. These configurations require top-down nanofabrication. As an alternative, silver nanowires (Ag NWs) synthesized using wet chemistry approaches have proved useful for optical confinement applications below the diffraction limit and for the guiding of light on the nanometre scale, due to their properties of single crystallinity and atomic surface smoothness [12; 13; 14; 15; 16; 17; 18]. There are numerous results on SPP waveguides made of Ag NWs. For example, Ag NW SPP waveguides can also serve as highly directional broadband optical antennas, they can be used to create interferometric logic gates, nanoscale routers, and multiplexers, light modulators, and a complete set of Boolean logic functions [19; 20; 21; 22; 23; 24; 25]. Ag NW plasmonic waveguides have been integrated with planar polymer optical waveguides for the nanoscale confinement and guiding of light on a chip [26]. The photonic and plasmonic interactions between an individual Ag NW and single-layer MoS\({}_{2}\) show pronounced nanoscale light-matter interaction between plasmons and atomically thin material that can be exploited for nanophotonic integrated circuits [27]. However, to the best of our knowledge, limited effort has been given to controlling the propagation constants of the plasmonic mode supported by the Ag NW, which is beneficial for active plasmonic devices. The propagating surface plasmons on supported Ag NWs have been tuned with dielectric (Al\({}_{2}\)O\({}_{3}\)) layers of different thicknesses. This tunability provides a simple and robust way for the precise routing of optical signals in circuitries and new types of ultrasensitive on-chip optical sensors [28]. The current deposition technique can only coat the Ag NW and its substrate (such as glass slide) simultaneously, and thus the Ag NW will be fixed and cannot be moved to be integrated with other components such as the silicon waveguides. It is not clear that the dielectric layer can be coated uniformly around all surfaces of the NW. The coating of a dielectric layer may destroy the atomically smooth surface of the single-crystal Ag NW, and increase the scattering loss. In this letter, we show that the both the wavenumber (effective refractive index) and propagation distance or cut off wavelength of the SPP waveguide with the Ag NW can be tuned by placement on a dielectric multilayer as the substrate.