Response of the Shockley surface state to an external electrical field: A density-functional theory study of Cu(111)

K. Berland, T. L. Einstein, and P. Hyldgaard

Phys. Rev. B 85, 035427 – Published 18 January 2012

Abstract

The response of the Cu(111) Shockley surface state to an external electrical field is characterized by combining a density-functional theory calculation for a slab geometry with an analysis of the Kohn-Sham wave functions. Our analysis is facilitated by a decoupling of the Kohn-Sham states via a rotation in Hilbert space. We find that the surface state displays isotropic dispersion, quadratic until the Fermi wave vector but with a significant quartic contribution beyond. We calculate the shift in energetic position and effective mass of the surface state for an electrical field perpendicular to the Cu(111) surface; the response is linear over a broad range of field strengths. We find that charge transfer occurs beyond the outermost copper atoms and that accumulation of electrons is responsible for a quarter of the screening of the electrical field. This allows us to provide well converged determinations of the field-induced changes in the surface state for a moderate number of layers in the slab geometry.

Received 10 September 2011

DOI:https://doi.org/10.1103/PhysRevB.85.035427

Authors & Affiliations

K. Berland^1,*, T. L. Einstein^2,†, and P. Hyldgaard^1,‡

¹Department of Microtechnology and Nanoscience, MC2, Chalmers University of Technology, SE-41296 Göteborg, Sweden
²Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA

^*berland@chalmers.se
^†einstein@umd.edu
^‡hyldgaar@chalmers.se

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Vol. 85, Iss. 3 — 15 January 2012

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Images

Figure 1
Contours of the $\bar{Γ}$ -point surface-state density. Yellow (red, blue) surfaces correspond to high (medium, low) density. The density is largest outside the surface, and lowest in the planes formed by the atoms. Graphics generated with vesta.[26]Reuse & Permissions
Figure 2
Issues with surface-state hybridization that arise because of the finite slab geometry. The upper panel shows the sum-projected surface-state densities obtained for a six-layer-wide copper slab. The full lines shows the Kohn-Sham (KS) states with a surface-state character. The dashed lines show the surface-localized (SL) states for which we here present a simple but robust determination (since the SL states provide a more accurate description of genuine surface-state behavior). The lower panel shows the electric field variation of the surface state for a 15-layer slab calculation using the plane-wave code DACAPO. The full line shows the least-squares fit to decoupled surface-state energies as obtained for the SL states, cf. Sec. IV B.Reuse & Permissions
Figure 3
Dispersion of the surface state plotted as a function of $| k |$ for $k$ sampled evenly on a 2D grid. The full (dashed) curve gives the best-fit second-(fourth-) order polynomial. The dotted line indicates the Fermi surface. The filled (green) circles, which form a thick line, indicate the calculated values, while the [red] crosses indicate those for $k_{y} = 0$ . The insert shows the energy contours as function of $k$ , with ticks having the same spacing as in the main figure.Reuse & Permissions
Figure 4
The average potential profile and KS and SL surface-state densities at the $\bar{Γ}$ point for a Cu(111) surface in an external electrical field $E$ pointing to the left, i.e., inwards to the surface at the top (at $z \approx 0$ ) of the slab and outwards from the surface at the bottom (at $z \approx - 30 Å$ ). Thus, the force on an electron is away from the surface at the top of the slab and the minimum energy $ε (0)$ of this top-surface SL state is lowered, increasing $ε_{F} - ε (0)$ . The upper panel shows the KS surface-state densities (full and dashed) for three different strengths of E. At zero electrical field they are almost symmetric, while for larger field they almost decouple. The inserts illustrate the corresponding hybridization parameters. The lower panel shows the atom positions, filled (yellow) circles, and the decoupled SL surface states, evanescent oscillatory curves. The bottom panel also shows, by surfaces of semitransparent grey shading, the potential profile for the three different strengths of electrical fields. Field-induced changes in the potential profile are observable only outside the outermost Cu atoms.Reuse & Permissions
Figure 5
Calculated energy at the $\bar{Γ}$ -point (upper panel) and mass (middle panel) shift for different external electrical fields. Here, we take a positive value of the field strength $E$ to imply that the field $E$ (the force) points away from (towards) the surface. In terms of the underlying slab calculations, Fig. 4, the positive- $E$ (negative- $E$ ) results characterize the behavior of the SL state at the left-most (right-most) slab surface. The filled (red) circles [(white) diamond] gives the calculated values for a 15 (12) layer slab. The black line gives the least-square fit to the calculated data. In the lower panel, the relative shift in wavelength (wave vector) is given by the full (dashed) curve.Reuse & Permissions
Figure 6
Charge transfer induced by external electrical field averaged over the in-plane directions. The upper light (cyan) [dark (black), dashed (orange)] is for $E = - 0.36 [- 0.12, 0.24] V / Å$ .Reuse & Permissions

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