The anisotropic size effect of the electrical resistivity of metal thin films: Tungsten

Abstract

The resistivity of nanoscale metallic conductors is orientation dependent, even if the bulk resistivity is isotropic and electron scattering cross-sections are independent of momentum, surface orientation, and transport direction. This is demonstrated using a combination of electron transport measurements on epitaxial tungsten layers in combination with transport simulations based on the ab initio predicted electronic structure, showing that the primary reason for the anisotropic size effect is the non-spherical Fermi surface. Electron surface scattering causes the resistivity of epitaxial W(110) and W(001) layers measured at 295 and 77 K to increase as the layer thickness decreases from 320 to 4.5 nm. However, the resistivity is larger for W(001) than W(110) which, if describing the data with the classical Fuchs-Sondheimer model, yields an effective electron mean free path k* for bulk electron-phonon scattering that is nearly a factor of two smaller for the 110 vs the 001-oriented layers, with k à ð011Þ ¼ 18.8 6 0.3 nm vs k à ð001Þ ¼ 33 6 0.4 nm at 295 K. Boltzmann transport simulations are done by integration over real and reciprocal space of the thin film and the Brillouin zone, respectively, describing electron-phonon scattering by momentum-independent constant relaxation-time or mean-free-path approximations, and electron-surface scattering as a boundary condition which is independent of electron momentum and surface orientation. The simulations quantify the resistivity increase at the reduced film thickness and predict a smaller resistivity for W(110) than W(001) layers with a simulated ratio k à ð011Þ /k à ð001Þ ¼ 0.59 6 0.01, in excellent agreement with 0.57 6 0.01 from the experiment. This agreement suggests that the resistivity anisotropy in thin films of metals with isotropic bulk electron transport is fully explained by the non-spherical Fermi surface and velocity distribution, while electron scattering at phonons and surfaces can be kept isotropic and independent of the surface orientation. The simulations correctly predict the anisotropy of the resistivity size effect, but underestimate its absolute magnitude. Quantitative analyses suggest that this may be due to (i) a twofold increase in the electron-phonon scattering crosssection as the layer thickness is reduced to 5 nm or (ii) a variable wave-vector dependent relaxation time for electron-phonon scattering. Published by AIP Publishing.

The electrical resistivity of rough thin films: A model based on electron reflection at discrete step edges

“…This value is close to the reported first-principles predictions for the ballistic conductance for W along the [100] and [110] transport directions of 9.5 Â 10 14 and 8.7 Â 10 14 X À1 m À2 , respectively. 63 We emphasize here again that the primary conclusion from the experimental measurements is the linear relationship in Fig. 4 that confirms our new model and particularly the resistivity prediction of Eq.…”

“…(14) for the thin film resistivity as a function of the root mean square surface roughness x and the lateral correlation length n of the surface morphology. We have chosen as an experimental model system epitaxial W(001) films, primarily because the high melting point facilitates epitaxial (single-crystal) growth of thin continuous layers on insulating substrates down to thicknesses of 4 nm, 62,63 and we have previously developed in situ annealing procedures that allow variations in the surface roughness with negligible changes in the crystalline quality. 39 4.5-52 nm thick W(001) films were deposited on MgO(001) substrates in a three chamber ultrahigh vacuum DC magnetron sputter deposition system with a base pressure <10 À9 Torr following the procedure in Ref.…”

“…where k ¼ 15.5 nm is the reported bulk mean free path at room temperature, 69 p is the specularity parameter of the two surfaces which is set to zero (completely diffuse scattering) according to Ref. 63, and the bulk resistivity q bulk ¼ 5.6 and 1.08 lX cm at 295 and 77 K, respectively, is obtained from the measured resistivity of thick samples (320 and 390 nm) which is corrected for by the relatively small 1.8% and 8.3% resistivity size effect in these samples at 295 and 77 K. The mean free path at 77 K, k ¼ 80.4 nm, is determined from the room temperature mean free path and the fact that the product k  q bulk is temperature independent. The flat film resistivity determined with Eq.…”

Surface roughness dependence of the electrical resistivity of W(001) layers

“…We start in this discussion with the latter approach, primarily because the mean free path for tungsten is not well established, yet, with reported values varying considerably 42,44,60,61 and some reports suggesting that it may even be orientation dependent. 45,62 One convenient approach is to assume completely diffuse surface scattering, that is p ¼ 0. This assumption is correct for various systems including, for example, Cu exposed to air 18,63 or coated with Ta 64 or Ti, 34 and also provides a method to determine the lower bound for k, since less diffuse scattering would lead to a larger value for p which, in turn, results in a larger value for k. 60 Correspondingly, we first fit the measured room temperature resistivity data with the FS model using a fixed p ¼ 0, but allow different effective mean free paths for the two sets of samples.…”

“…Tungsten is considered a potential material for <10nm-wide lines in future integrated circuits including middle-of-line local interconnects 40 and 3D through silicon structures, 41 because of a possibly lower effective resistivity associated with the smaller mean free path than Cu, [42][43][44][45] superior electromigration resistance, 46,47 and good process compatibility with CMOS devices. 48 Therefore, measurements on the effect of the surface roughness of W layers have not only impact on the fundamental understanding of the resistivity size effect, but also direct value to assess the potential benefits of W as a possible barrier-free interconnect metal.…”