Roughening transition temperature

When the bulk transition is of first order, the above mentioned arguments based on dimensionality do not apply and the would be roughening transition temperature T j may be larger than the bulk transition temperature T, in which case there is simply no roughening transition. The situation is further complicated by the wetting phenomena. When we approach T from below, the disordered phase becomes metastable and may wet the interface a large layer of disordered phase develops in between the two ordered domains. 

The relaxation of isolated, pairs of and ensembles of steps on crystal surfaces towards equilibrium is reviewed, for systems both above and below the roughening transition temperature. Results of Monte Carlo simulations are discussed, together with analytic theories and experimental findings. Elementary dynaniical processes are, below roughening, step fluctuations, step-step repulsion and annihilation of steps. Evaporation kinetics arid surface diffusion are considered. 

Figure 1. Monte Carlo configuration of an isolated step below the roughening transition temperature TR 1.24J ofthe standard SOS model, at t= 2000 MCS, using evaporation kinetics.

So far, we tacitly assumed that the upper and lower terraces next to the step are below their roughening transition temperature. By fixing the boundary heights of the terraces, away from the step, at, say, level 0 for the lower and level 1 (in units of the lattice spacing) for the upper terrace, one can study the time evolution of the step width w, defined, for instance, as the second moment of the gradient of the step profile also above roughening. Then one obtains s =1/4 for terrace diffusion and 1/2 for evaporation kinetics, as predicted by the continuum description of Mullins and confirmed by our Monte Carlo simulations. 

Figure 5. Profiles z(x, f) of a grating below the roughening transition temperature, at increasing time, of the standard SOS model with 80 x 320 sites and a miscut of a few lattice spacings, using evaportion kinetics in Monte Carlo simulations (full symbols). For comparison, a sinusoid is shown (open symbols). The initial amplitude of the grating is five lattice spacings.

Special attention has been paid to the profiles shapes, the asymptotic decay laws of the amplitude, and related scaling behavior. The roughening transition temperature of the relevant crystal surface plays a crucial role for these properties, for a given type of transport mechanism. 

Defects lead to a roughening of crystal surfaces with increasing temperature as already predicted by Burton et al. [335], However, calculations for low-index surfaces yield roughening transition temperatures well above the melting temperature. The reason is the high forma-... 

Rig. 7. Snapshot pictures of a Monte Carlo simulation of the crystal-vacuum interface in the framework of a solid-on-solid (SOS) model, where bubbles and overhangs are forbidden. Each lattice site i is characterized by a height variable h, and the Hamiltonian then is 7i = - hf - hj[. Three temperatures are shown kT/4> — 0.545 (a), 0.600 (b) and 0.667 (c). The roughening transition temperature 7r roughly coincides with case (b). From Weeks et al. (1973). 

This surface tension /int(T, H) is singular at the roughening transition temperature 7r, as well as at the bulk critical temperature Tc of the Ising model. 

Fig. 36. Schematic temperature variation of intcrfacial stiffness kn I K and interfacial free energy, for an interface oriented perpendicularly to a lattice direction of a square a) or simple cubic (b) lattice, respectively. While for tl — 2 the interface is rough for all non zero temperatures, in d — 3 il is rough only for temperatures T exceeding the roughening transition temperature 7r (see sect. 3.3). For T < 7U there exists a non-zero free energy tigT.v of surface steps, which vanishes at T = 7 r with an essential singularity. While k is infinite throughout the noil-rough phase, k Tic reaches a universal value as T - T . Note that k and fml to leading order in their critical behavior become identical as T - T. ...

Show that the equilibrium roughness of a one-dimensional interface is qualitatively larger than that of a two-dimensional interface. Show that the one-dimensional interface is always rough i.e., the roughening transition temperature is zero). 

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