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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 λ* for bulk electron-phonon scattering that is nearly a factor of two smaller for the 110 vs the 001-oriented layers, with λ(011)*= 18.8 ± 0.3 nm vs λ(001)* = 33 ± 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 λ(011)*/λ(001)* = 0.59 ± 0.01, in excellent agreement with 0.57 ± 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 two-fold increase in the electron-phonon scattering cross-section as the layer thickness is reduced to 5 nm or (ii) a variable wave-vector dependent relaxation time for electron-phonon scattering.
Zheng et al. (Tue,) studied this question.