Temperature dependence of the resistance of metallic nanowires of diameter<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mo>⩾</mml:mo><mml:mn>15</mml:mn><mml:mspace width="0.3em"/><mml:mi>nm</mml:mi></mml:mrow></mml:math>: Applicability of Bloch-Grüneisen theorem

We have measured the resistances (and resistivities) of Ag and Cu nanowires of diameters ranging from $15\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}200\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ in the temperature range $4.2--300\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ with the specific aim of assessing the applicability of the Bloch-Gr\"uneisen formula for electron-phonon resistivity in these nanowires. The wires were grown within polymeric templates by electrodeposition. We find that in all the samples the resistance reaches a residual value at $T=4.2\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ and the temperature dependence of resistance can be fitted to the Bloch-Gr\"uneisen formula in the entire temperature range with a well-defined transport Debye temperature $({\ensuremath{\Theta}}_{R})$. The values of the Debye temperature obtained from the fits lie within 8% of the bulk value for Ag wires of diameter $15\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ while for Cu nanowires of the same diameter the Debye temperature is significantly less than the bulk value. The electron-phonon coupling constants (measured by ${\ensuremath{\alpha}}_{\mathit{el}\text{\ensuremath{-}}\mathit{ph}}$ or ${\ensuremath{\alpha}}_{R}$) in the nanowires were found to have the same value as in the bulk. The resistivities of the wires were seen to increase as the wire diameter was decreased. This increase in the resistivity of the wires may be attributed to surface scattering of conduction electrons. The specularity $p$ was estimated to be about 0.5. The observed results allow us to obtain the resistivities exactly from the resistance and give us a method of obtaining the exact numbers of wires within the measured array (grown within the template).