Atomistic behavior of metal surfaces under high electric fields

Combining classical electrodynamics and density functional theory (DFT) calculations, we develop a general and rigorous theoretical framework that describes the energetics of metal surfaces under high electric fields. We show that the behavior of a surface atom in the presence of an electric field can be described by the polarization characteristics of the permanent and field-induced charges in its vicinity. We use DFT calculations for the case of a W adatom on a W{110} surface to confirm the predictions of our theory and quantify its system-specific parameters. Our quantitative predictions for the diffusion of W-on-W{110} under field are in good agreement with experimental measurements. This work is a crucial step towards developing atomistic computational models of such systems for long-term simulations.

Figure 1

Charge redistribution induced by (a) the presence of an adatom, (b) a positive 1 GV/m applied field (anode) on a system with adatom (atoms fixed at their original zero-field positions for illustration purposes). The open surface of the slab is {110} oriented. Cyan and magenta colored areas correspond to increased and decreased electron densities, respectively, that exceed 1% of the maximum electron density of the reference system for (a) and 0.1% for (b).

Figure 2

Top view of the slab models for the W{110}: (a) a slab with the W adatom at the lattice site, (b) with the W adatom at the bridge (saddle), and (c) a flat surface.

Figure 3

Total energy of the four W systems simulated by DFT, vs the applied field. Black dots and green diamonds in (a) correspond to the system with an adatom at the saddle point and at the lattice site, respectively. Black dots in (b) and (c) correspond to the flat reference and isolated atom systems, respectively. The corresponding solid-line curves indicated by arrows are obtained by Eq. (3) with parameters that are fitted to the same DFT data.

Figure 4

Migration barrier $E_m$ [(a), black, left axis], binding energy $E_b$ [(a), green, right axis] and removal work $\mathrm{\Lambda }$ [(b), green, right axis] of a W adatom on a W{110} surface vs the uniform applied electric field as calculated by Eq. (9) and (5), respectively. The gray shadow area around the curves gives the error margin calculated by the error propagation rule (EPR) applied on the uncertainties of the parameters given in Table 2. In the insets we zoom in the low field region and plot with markers (dots for $E_m, \mathrm{\Lambda }$, and squares for $E_b$) the corresponding direct DFT data. The marker error bars are calculated by applying the EPR to the 1 meV error estimation for the DFT ground-state energy (see Sec. 3).

Figure 5

Migration barrier vs the relative field increment $(F_s-F_l)/F_l$ for various applied fields $F_l$. Solid lines correspond to the anode ($F_l>0$) and dashed ones to the cathode ($F_l<0$), as calculated by Eq. (10). Markers correspond to values directly calculated by DFT according to Eq. (9). The error bars are obtained as in Fig. 4.