Analysis of the size effect in electroplated fine copper wires and a realistic assessment to model copper resistivity

Abstract

The size effect in electroplated copper wires has been widely studied recently. However, there is no consensus on the role of various scattering mechanisms. Therefore, an in-depth analysis to reveal the origin of the size effect is needed. In this article, we study the resistivity of fine copper wires whose feature sizes shrink in two dimensions. It is shown that the residual resistivity (at 5 K) increases with decreasing wire width or height and the temperature-dependent resistivity slightly deviates from that of bulk copper. This is mainly attributed to surface scattering rather than grain boundary scattering. In fact, the influence of grain boundary scattering in these well annealed copper wires is relatively small. In addition, for copper wires with a constant height, a linear dependence of the copper resistivity on 1/width (w) or 1/cross-sectional area (A), namely ρ=ρic+c*∕w (or ρ=ρic+c**∕A), is derived from the classic surface and grain boundary scattering models and validated experimentally. In this simple description, the contributions of different scattering mechanisms, such as surface reflectivity, p, and grain boundary reflection coefficient, R, defect and impurity density, combine together in parameters of ρic and c* (or c**). Especially, c* is a good indicator of scattering strength, from which one can quantitatively analyze the impact of nonsurface scattering contribution with a reference slope of c*=32.14.

Surface and grain boundary scattering in nanometric Cu thin films: A quantitative analysis including twin boundaries

“…4,5,9 Brongersma et al 4 reported a value of R ¼ 0.46, which also compares well with the value of R ¼ 0.43 reported by Sun et al 13 It is also worth noting that the film resistivities reported by Sun et al 13 were comparable to the lower resistivities of the electroplated Cu films rather than the higher resistivities of the self-ionized plasma-deposited Cu films reported by Brongersma et al 4 Additionally, in a later study, Barmak et al 15 showed that the values of p ¼ 0.52 and R ¼ 0.43 obtained by Sun et al 13 could be used to satisfactorily describe the resistivity of Cu wires reported in a representative set of studies from 2002 to 2010, assuming that the impurity content and the grain size/width ratio were varied. This finding was qualified by the facts that, in several of the prior studies, the impurity content was not measured and was viewed as an adjustable variable, and that the grain size, if actually measured rather than assumed equal to the sample line width, was not measured for statistically significant populations of grains.…”

“…1 Determining the relative contributions of surface and grain-boundary scattering toward this resistivity increase is an important step toward a quantitative understanding of the classical size effect. [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] A commonly used description for the surface scattering contribution to the resistivity increase is the Fuchs-Sondheimer (FS) model. 18 In this model, the resistivity increase is a result of diffuse scattering of conduction electrons at the conductor's exterior surfaces with a probability of 1 À p, where p (0 p 1) is a specular scattering coefficient that is typically inferred from experimental data.…”

Grain boundary and surface scattering in interconnect metals

“…An example of this is the increase in surface scattering (Δρ FS ) with phonon scattering observed for the FS model, which is understood as the electrons in the region of the Fermi surface having little momentum towards the conductor surfaces being scattered by phonons into Fermi surface states with greater momentum towards the surfaces [11]. Grain boundary scattering has a much weaker interaction with phonon scattering, which provides a difference in the temperature dependence of surface and grain boundary scattering resistivity contributions that has been used in attempts to separately quantify these two mechanisms [12], [13]. It is generally expected that impurity scattering, because it is isotropic, would result in electrons being scattered towards …”

Electron transport properties of sub-3-nm diameter copper nanowires

“…As a consequence, understanding the effects of sizedependent phenomena on material properties is becoming ever more critical to enable efficient device performance. Present day transistor technologies use copper as an interconnect material, however, it is unknown if at the small cross sections required for future technologies copper will be able to fulfill this function effectively due to increased resistances; measured line resistivity increases dramatically for Cu nanowires (NWs) compared to bulk materials [2,3]. It is required that the electrical conductivity of small cross-section copper nanostructures be understood to maintain acceptable power consumption in future nanoelectronics generations.…”