Temperature dependence of the resistance of metallic nanowires of diameter $⩾15\mathrm{nm}$: Applicability of Bloch-Grüneisen theorem

We have measured the resistances (and resistivities) of Ag and Cu nanowires of diameters ranging from $15\text{to}200\mathrm{nm}$ in the temperature range $4.2–300\mathrm{K}$ with the specific aim of assessing the applicability of the Bloch-Grüneisen 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 $T=4.2\mathrm{K}$ and the temperature dependence of resistance can be fitted to the Bloch-Grüneisen formula in the entire temperature range with a well-defined transport Debye temperature $\left({\Theta }_R\right)$. The values of the Debye temperature obtained from the fits lie within 8% of the bulk value for Ag wires of diameter $15\mathrm{nm}$ 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 ${\alpha }_{\mathit{el}\text{−}\mathit{ph}}$ or ${\alpha }_R$) 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).

Figure 1

(Color online) (a) A schematic of the setup used for making the nanowires. (b) XRD of $100\mathrm{nm}$ Ag wires. The peaks have been indexed to fcc Ag.

Figure 2

SEM image of (a) $100\mathrm{nm}$ Ag wires taken after dissolving off the template in dichloromethane and (b) a single $200\mathrm{nm}$ Ag wire.

Figure 3

HRTEM image of a $15\mathrm{nm}$ Ag wire. The arrow points out the twin boundary present in the sample. The inset shows the selective area diffraction pattern. The growth direction of the wires as seen from TEM is [100].

Figure 4

(Color online) Resistance of arrays of Ag wires of diameter (a) 15, (b) 30, (c) 50, and (d) $200\mathrm{nm}$. The inset shows the resistance with the temperature axis in a logarithmic scale in order to show more clearly the low-temperature behavior of the resistance.

Figure 5

(Color online) Resistance of arrays of Cu wires of diameter (a) 15, (b) 30, (c) 50, and (d) $200\mathrm{nm}$.

Figure 6

(Color online) Fit to Bloch-Grüneisen formula [Eq. (1)] to the measured resistivity data for a $15\mathrm{nm}$ Ag wire. The inset shows the fit error (for definition see text) for $n=5$.

Figure 7

(Color online) Plot of $(R_T-R_{4.2\mathrm{K}})∕R_D$ as a function of $T∕{\Theta }_R$ [see Eq. (3)]. Here $R_D$ is the value of the measured resistance at $T={\theta }_R$.

Figure 8

(Color online) Plot of the dominant phonon wavelength ${\lambda }_d$ as a function of temperature for $n=5$. The arrow points to $15\mathrm{nm}$, the diameter of the narrowest wire measured by us.

Figure 9

(Color online) Plot of the resistivity of Ag and Cu nanowires as a function of temperature. The resistivity was obtained using the method described in the text.

Figure 10

(Color online) (a) Plot of the resistivity of Ag nanowires at $295\mathrm{K}$ and the residual resistivity as a function of the wire diameter. The line is the fit to the data using Eq. (6). The arrow is the bulk value of the resistivity of Ag, $1.629\times {10}^{-8}\Omega \mathrm{m}$. (b) Plot of $l_{\mathit{mfp}}∕d$ as a function of $d$ for Ag nanowires. The solid line is a guide to the eye.