Capillary fluctuations of surface steps: An atomistic simulation study for the model Cu(111) system

Molecular dynamics (MD) simulations are employed to investigate the capillary fluctuations of steps on the surface of a model metal system. The fluctuation spectrum, characterized by the wave number ($k$) dependence of the mean squared capillary-wave amplitudes and associated relaxation times, is calculated for $\langle 110\rangle$ and $\langle 112\rangle$ steps on the $\left\{111\right\}$ surface of elemental copper near the melting temperature of the classical potential model considered. Step stiffnesses are derived from the MD results, yielding values from the largest system sizes of $(37\pm 1)\text{meV}/\mathring{\mathrm{A}}$ for the different line orientations, implying that the stiffness is isotropic within the statistical precision of the calculations. The fluctuation lifetimes are found to vary by approximately four orders of magnitude over the range of wave numbers investigated, displaying a $k$ dependence consistent with kinetics governed by step-edge mediated diffusion. The values for step stiffness derived from these simulations are compared to step free energies for the same system and temperature obtained in a recent MD-based thermodynamic-integration (TI) study [Freitas, Frolov, and Asta, Phys. Rev. B 95, 155444 (2017)]. Results from the capillary-fluctuation analysis and TI calculations yield statistically significant differences that are discussed within the framework of statistical-mechanical theories for configurational contributions to step free energies.

Figure 1

System dimensions and orientation: $d$ is the step-step separation distance, $L$ is the step length, and $h$ is the bulk depth. The step average direction is along $\widehat{\mathbf{x}}$, and the step line profile is given by the curve $y\left(x\right)$. Periodic boundary conditions are applied on the $x$ and $y$ directions.

Figure 2

Step orientations on the $\left(111\right)$ surface of face-centered cubic copper. The blue (dark gray) atoms on the right are in the first layer, and the red ones (light gray) on the left are in the second layer. The orientations are (a) $\left[110\right]\mathrm{A}$, (b) $\left[110\right]\mathrm{B}$, and (c) $\left[211\right]$. The difference between the $\left[110\right]$ A and B steps are the nearest neighbors on the layer immediately below the step.

Figure 3

Illustration of the grid algorithm for step profile determination. (a) We select the atoms belonging the first surface layer by selecting all atoms with height above some threshold based on the number of atomic layers in the system. (b) Then we define a grid along the $\widehat{\mathbf{x}}$ direction and divide each of these stripes in equally spaced bins along the $\widehat{\mathbf{y}}$ direction. (c) The step height is defined as the average height between the first cell with zero density along a stripe and the cell immediately below it.

Figure 4

Illustration of the cluster algorithm for step profile determination. (a) We determine the atoms belonging the first surface layer by selecting all atoms with height above some threshold based on the number of atomic layers in the system. (b) Then we find the largest cluster of atoms in that layer (light gray atoms), where we consider two atoms to be neighbors if the distance between them is smaller than some distance $r_{\text{max}}$. (c) The system is divided in bins along the $\widehat{\mathbf{x}}$ direction, and the step height in each bin is determined as the atom with the highest value of $y\left(x\right)$ inside that bin.

Figure 5

Comparison of the power spectrum of the Fourier transform of the step line profile obtained using different algorithms to determine the step profile. The cluster algorithm has shown to result in a power spectrum which better follows the $k_n^{-2}$ behavior expected from the capillary-wave model (CFM).

Figure 6

Typical autocorrelation functions obtained for the normal modes' amplitudes (normal mode index increases in the clockwise direction or from the upper right to the bottom left). The normal modes relaxation times (${\tau }_n$) are obtained by fitting $C_n\left(t\right)$ curves to Eq. (11).

Figure 7

Relaxation time of each normal mode of the step line profile. ${\tau }_n$ is obtained by fitting $C_n\left(t\right)$ curves to Eq. (11). The horizontal line corresponds to ${\tau }_n=1\text{ns}$ and the vertical line marks the wave vector value at which the adjusted curve (black solid line) intercepts the ${\tau }_n=1\text{ns}$ relaxation time, i.e., the vertical line is at $k_n=k_{\text{min}}$.

Figure 8

Power spectrum plot of the Fourier transform of the step line profile. Long wave vectors were discarded since they deviate from the $\langle {\left|A_n\right|}^2\rangle \propto k_n^{-2}$ behavior due to the presence of atomic vibrations not accounted for the CFM, as presented in Sec. 2. Short wave vectors were discarded because their relaxation time was long enough to be comparable to the total simulation time, hence they cannot be adequately sampled in a MD simulation. The dashed lines delimit the wavelengths used for adjusting the black solid curve to Eq. (10). The inset highlights the agreement of the normal modes considered with the CFM predictions.

Figure 9

Comparison of the step free energy obtained from thermodynamic integration simulations (Ref. [29]) with the predictions of the capillary-wave model.