First-principles quantum transport theory of the enhanced wind force driving electromigration on Ag(111)

Kirk H. Bevan, Hong Guo, Ellen D. Williams, and Zhenyu Zhang

Phys. Rev. B 81, 235416 – Published 10 June 2010

Abstract

Herein we examine the low-bias electromigration wind force acting on quasi-one-dimensional nanoscale features within the Landauer-Büttiker conduction picture. Ordinarily the electromigration force is calculated under the approximation that the nonequilibrium carrier distribution in the vicinity of a defect is the same as that in the bulk. However, this approximation is rooted in the assumption that atomic scale defects scatter all incident electrons weakly (just as electrons weakly and diffusely scatter in the bulk). We examine this assumption by calculating the mode-resolved transmission against Ag(111) step edges and atomic wires using density-functional theory within the single-particle Green’s function Landauer scattering picture. Furthermore we show that those modes that scatter strongly give rise to a nonequilibrium electrochemical potential drop across a defect and an increased wind force. The results quantitatively explain previously not understood experimental observations of an enhanced electron wind force against Ag(111) step edges [O. Bondarchuk et al., Phys. Rev. Lett. 99, 206801 (2007)]. In general, the results underscore the challenging nanoscale reliability problem posed by surface electromigration in nanostructures and the need for a nonequilibrium quantum transport description of the electron wind force.

Received 6 February 2010

DOI:https://doi.org/10.1103/PhysRevB.81.235416

Authors & Affiliations

Kirk H. Bevan^1,2,*, Hong Guo², Ellen D. Williams³, and Zhenyu Zhang^1,4,5

¹Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
²Centre for the Physics of Materials and Department of Physics, McGill University, Montreal, Quebec, Canada H3A 2T8
³Department of Physics, University of Maryland at College Park, College Park, Maryland 20742-4111, USA
⁴Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA
⁵ICQD, University of Science and Technology of China, Hefei, Anhui, China

^*bevankh@ornl.gov

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Vol. 81, Iss. 23 — 15 June 2010

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Figure 1
(Color online) Scattering defect electrochemical potential splitting $Δ μ_{d}$ superimposed on its electrostatic potential drop $V_{d}$ very close to the defect (much less than the electron mean-free path as compared to Fig. 5). We define $T_{d}$ to be the low-bias transmission between the left $μ_{L}$ and right $μ_{R}$ electrochemical potentials (or quasi-Fermi levels), where the electrochemical potential splitting across the defect is given by $Δ μ_{d} = μ_{L} - μ_{R}$ . The equilibrium electrochemical potential $μ_{e q}$ sits midway between the nonequilibrium left $μ_{L}$ and right $μ_{R}$ electrochemical potentials. The inset shows the shifted momentum occupation for bulk conduction far from the scattering defect.Reuse & Permissions
Figure 2
(Color online) Ag(111) 12-layer surface scattering device geometries: (a) One-dimensional atomic chain and (b) scattering step edge. Semi-infinite 12-layer left and right contacts are included to calculate electron scattering and transport characteristics (Ref. 33). In each case the device region is repeated infinitely along the transverse $x$ vector pointing out of the page to form a two-dimensional slab. In each geometry the scattering atom, for which the wind force is calculated, is highlighted in red (see Secs. 2B, 3).Reuse & Permissions
Figure 3
(Color online) Unitless real-space plots of the left-scattering electron density $Δ ρ_{L}$ at a driving current density of $8 \times 10^{- 5} A / {cm}^{2}$ as given by Eq. (4).Reuse & Permissions
Figure 4
Ag(111) atomic chain surface defect local potential contribution $V^{l}$ to the $z$ -direction bulk wind force at a driving current of $8 \times 10^{- 5} A / {cm}^{2}$ [see Figs. 2a, 3a]. Figure 4a displays the local potential derivative as the integral $\int \partial V^{l} / \partial Z_{i} d x d y$ . Figure 4b displays the force integrals $F_{w_{b}, L}^{l} (z) = - \int Δ ρ_{L} \partial V^{l} / \partial Z_{i} d x d y$ (dashed) and $F_{w_{b}, R}^{l} (z) = - \int Δ ρ_{R} \partial V^{l} / \partial Z_{i} d x d y$ (dotted).Reuse & Permissions
Figure 5
(Color online) Bulk diffuse phonon scattering is shown in red where the electrostatic potential drop follows the electrochemical potential drop. The length scale $2 L_{m}$ is much greater than that shown in Fig. 1. Local-defect scattering is shown in blue, where due to the enhanced scattering strength of the defect a resistivity dipole develops (see Sec 2). The dashed line color coded to each scenario displays the average quasi-Fermi level (Ref. 15) between left bound and right bound scattering electrons. The incident bulk current scattered by the defect is denoted as $J_{s}$ . The defect ballistic scattering region with current density $J_{d}$ and surrounding bulk scattering medium with current density $J_{s} + J_{b} \approx J_{b}$ can be approximated as two resistances conducting in parallel (Ref. 46).Reuse & Permissions
Figure 6
(Color online) The transmission and scattering properties at $μ_{e q}$ for 12-layer Ag(111) atomic chain and step-edge defects (Ref. 47). Figures 6a, 6b display the mode-resolved eigenchannel transmission $T_{d, i}$ for each mode index $i$ (the number of scattering modes varies across the transverse Brillouin zone). Figures 6c, 6e display the scattering charge density $Δ ρ_{L} (μ_{e q})$ for those eigenchannels with transmission $T < 0.5$ . Figures 6d, 6f display the scattering charge density $Δ ρ_{L} (μ_{e q})$ for those eigenchannels with transmission $T \geq 0.5$ . All eigenchannels have a transmission of 1 for a perfectly crystalline clean Ag(111) 12-layer slab.Reuse & Permissions

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