All-electron first-principles investigations of the energetics of vicinal Cu surfaces

Abstract

Using first-principles calculations we studied the energetics ͑surface energy, step energy, stability with respect to faceting͒ of the low-and high-Miller-index ͑vicinal͒ Cu surfaces, namely, the ͑111͒, ͑100͒, ͑110͒, ͑311͒, ͑331͒, ͑210͒, ͑211͒, ͑511͒, ͑221͒, ͑711͒, ͑320͒, ͑553͒, ͑410͒, ͑911͒, and ͑332͒ surfaces. Our calculations are based on density-functional theory employing the all-electron full-potential linearized augmented planewave ͑FLAPW͒ method. We found that the unrelaxed vicinal Cu surfaces between ͑100͒ and ͑111͒ are unstable relative to faceting at 0 K, while fully relaxed vicinal surfaces between ͑100͒ and ͑111͒ are stable relative to faceting, which is in agreement with the observed stability of vicinal Cu surfaces at room temperature. Thus atomic relaxations play an important role in the stability of the vicinal Cu surfaces. Using the surface energies of Cu͑111͒, Cu͑100͒, and Cu͑110͒ and employing the effective pair-potential model, which takes into account only the changes in the coordination of the surface atoms, the surface energies of the vicinal Cu surfaces can be calculated with errors smaller than 1.0% compared with the calculated FLAPW surface energies. This result is due to the almost perfect linear scaling of the surface energies of the Cu͑hkl͒ surfaces as a function of the total number of broken nearest-neighbor bonds. Furthermore, we calculate step-step interactions as a function of terrace widths and step energies of isolated steps.

First-Principles Study of the Stability and Interfacial Bonding of Tilt and Twist Grain Boundaries in Al and Cu

“…6,7,30) Recent ab initio calculations also indicate that the surface energy of Cu (110) is 0.901 eV per surface atom in contrast to 0.708 eV (78%) for Al(110). 31,32) Thus the present results of the smaller boundary energies for Al than for Cu are consistent with other types of defect energies.…”

The three-dimensional equilibrium crystal shape of Pb: Recent results of theory and experiment

“…An analogous situation exists for step energies of vicinal Cu(111) surfaces [17]. We evaluated the Cu first principles surface energies in the 110 zone according to (1) and obtained excellent straight lines whose slopes correspond to the formation energies of A and B steps of Cu(111).…”

“…In the general context of the broken bond model mentioned above it has been suggested that surface energies of vicinal orientations may not have to be calculated but instead be estimated by assuming a strict linear relationship between values calculated for low-index surfaces and the number of broken bonds for any vicinal surface [17]. However, this degree of simplification is most likely leading to unreliable results when it comes to a description of properties of vicinal surfaces [9,63].…”

“…More recently, full potential first principles calculations have produced surface energies of many crystalline materials [7][8][9][10][11][12][13][14][15][16][17][18] mostly for low-index orientations. Although there can be considerable scatter in theoretical values for a single material and orientation [16,19], it has been suggested by u Fax: +49 2461 61 2950, E-mail: h.bonzel@fz-juelich.de some authors that the lack of experimental surface energy data for well-defined orientations may be replaced by ab initio calculations [15,17,20].…”

“…More recently, full potential first principles calculations have produced surface energies of many crystalline materials [7][8][9][10][11][12][13][14][15][16][17][18] mostly for low-index orientations. Although there can be considerable scatter in theoretical values for a single material and orientation [16,19], it has been suggested by u Fax: +49 2461 61 2950, E-mail: h.bonzel@fz-juelich.de some authors that the lack of experimental surface energy data for well-defined orientations may be replaced by ab initio calculations [15,17,20].Theoretical attempts of describing the more or less complete anisotropy of the surface free energy of crystals by invoking attractive pairwise atomic interaction potentials have been numerous, beginning in 1931 with the work by Stranski et al [21][22][23][24][25][26] followed by others [27][28][29][30]. A different phenomenological description of the anisotropy and to some extent of the temperature dependence of the surface free energy was based on the terrace-step-kink model of a crystalline surface [31] assuming the existence of steps of monoatomic height, their thermal formation, interaction and roughening [32,33].…”