Thermal mass transport mechanism of an adatom on a crystalline surface

The diffusion of particles on surfaces subjected to a thermal gradient driven by phonons is numerically and theoretically investigated. Performing molecular dynamics (MD) simulations, we show that the thermal gradient induces a drift velocity of the adatom toward the cold regions. Beyond the bare study of the adatom trajectories for various thermal gradients, we propose to use a Massieu function as a thermodynamic potential that drives the adatom motion. We generalize the thermodynamic integration method to this out-of-equilibrium system to compute this thermodynamic potential. The thermal-gradient-induced effect is decorrelated from the stochastic diffusion from the analysis of the thermodynamic potential. Based on these results, we propose a model to evaluate the drift velocity and trajectory of an adatom that compares with a good agreement with trajectories provided by MD.

Figure 1

(a) Sketch of the simulation model. Atoms of the hot and cold regions and the diffusing adatom are respectively in red, blue, and orange. (b) Temperature profiles (and standard deviations) in the substrate for $T_{\mathrm{hot}}=0.3$ and various values of $T_{\mathrm{cold}}$. Standard deviations are so small that they are barely visible.

Figure 2

The $x$ coordinates of the adatom as a function of time (only 5 trajectories out of 50 calculated trajectories are represented); blue and red areas respectively designate cold and hot regions; (a) for $\langle \frac{\partial T}{\partial x}\rangle =0.0038$ (the inset shows a zoom on a trajectory); (b) for $\langle \frac{\partial T}{\partial x}\rangle =0.0027$.

Figure 3

(b) Thermodynamic potential and (a) its derivative as a function of the adatom position $x_0$ computed from simulations with the thermal gradient $\langle \frac{\partial T}{\partial x}\rangle =0.0038$.

Figure 4

Thermodynamic potential for several gradients as a function of the inverse of the temperature.

Figure 5

(a) ${\mathrm{\Phi }}_{\mathrm{diff}}=\mathrm{\Phi }-{\mathrm{\Phi }}_{\mathrm{TGIP}}$ as a function of the inverse of the temperature. (b) Amplitude $A_{\mathrm{diff}}$ of the oscillation of ${\mathrm{\Phi }}_{\mathrm{diff}}$ as a function of the inverse of the temperature and its affine regression (green solid line). Data are extracted from simulations with the thermal gradient $\langle \frac{\partial T}{\partial x}\rangle =0.0038$.

Figure 6

Schematic representation of the thermodynamic potential as a function of the adatom position.

Figure 7

Model trajectory (solid black line) computed from integration of Eq. (7) compared to the average MD trajectory of the adatom; (a) for $\langle \frac{\partial T}{\partial x}\rangle =0.0038$; (b) for $\langle \frac{\partial T}{\partial x}\rangle =0.0027$.

Figure 8

Sketch of the effective potential seen by the particle.