Size-dependent resistivity of nanometric copper wires

Abstract

Higher electrical resistivity is observed in metals when dimensions approach the mean free path of the electrons. The effects of electron scattering at surfaces and at grain boundaries are then becoming substantial. This issue has been extensively studied on thin films but rarely on wires, where both small dimensions ͑width and height͒ influence the resistivity increase. In this study, copper wires having variable width and height down to 100 nm are investigated. An alternative approach is suggested in which the resistivity of such wires at different temperatures is compared to that of films having thickness that is equal to the height of the wires. The main outcome is a reliable model that overcomes the well-known difficulty of separating the contribution of surfaces to the resistivity from that of grain boundaries. It is shown that when both width and height of the wire are larger than one third of the mean free path, its resistivity exhibits a filmlike behavior with a separate contribution to the resistivity of each small dimension. The scattering of electrons at the surfaces of the investigated wires was best described by a zero specularity parameter, indicating the importance of this effect for the resistivity increase in small wires.

Atomic size effects studied by transport in single silicide nanowires

“…Fig. 9), i.e., the values match almost the description of Fuchs [60], which can describe our data only qualitatively, in agreement with many other studies [72][73][74][75][76][77].…”

“…Thereby, the worst case with p = 0 was already assumed. Moreover, also the consideration of realistic geometries, i.e., wires instead of films [28,29,60,62,71], cannot explain our discrepancy. For instance, the blue dashed line is the result based on a theory by Moraga et al [62] for wires with a rectangular cross section and a maximum width-to-height ratio of μ max ≈ 3, discussed in the context of Fig.…”

“…Nanosized materials often exhibit larger electrical resistivities compared to their macroscopic counterparts due to the additional scattering of the conduction electrons at the material surfaces and/or grain boundaries [25,26,28,29,[60][61][62][63]. This effect is dominant if the electron mean free path becomes comparable to the extension of the nanoobject.…”

“…introducing a specular parameter p, i.e., the fraction (1 − p) of electrons scatters diffusively at the surfaces and interfaces, thus reducing the mean free path and, consequently, increasing the resistivity. Analytically, the ratio of the resistivities between bulk and film of thickness D is expressed as [60] …”

Electron scattering at surfaces and grain boundaries in Cu thin films and wires

“…Most studies find p ≈ 0 at all Cu interfaces, 8,21,25 corresponding to completely diffuse surface scattering, and 0.2 < R < 0.5. 3,26,27 However, the interdependence and the uncertainty in the quantification of grain size, thickness, and roughness of the Cu layers complicates the determination of p and R from any set of polycrystalline samples. For example, the resistivity of 230 nm high and 40-800 nm wide Cu wires with grain sizes assumed to be equal to the smallest dimension of the wires can be fit equally well with parameter pairs of (p = 1, R = 0.53), (p = 0.6, R = 0.5) and (p = 0, R = 0.43).…”