Dynamics of particle deposition on a disordered substrate. I. Near-equilibrium behavior

A growth model that describes the deposition of particles (or the growth of a rigid crystal) on a disordered substrate is investigated. The dynamic renormalization group is applied to the stochastic growth equation using the Martin, Siggia, and Rose formalism [Phys. Rev. A 8, 423 (1973)]. The periodic potential and the quenched disorder, upon averaging, are combined into a single term in the generating functional. Changing the temperature (or the inherent noise of the deposition process) two different regimes with a transition between them at ${\mathit{T}}_{\mathit{s}\mathit{r}}$ are found: for T>${\mathit{T}}_{\mathit{s}\mathit{r}}$ this term is irrelevant and the surface has the scaling properties of a surface growing on a flat substrate in the rough phase. The height-height correlations behave as C(L,τ)∼ln[Lf(τ/${\mathit{L}}^2$)]. While the linear response mobility is finite in this phase it does vanish as (T-${\mathit{T}}_{\mathit{s}\mathit{r}}$ ${)}^{1.78}$ when T→${\mathit{T}}_{\mathit{s}\mathit{r}}^{+}$. For T<${\mathit{T}}_{\mathit{s}\mathit{r}}$ there is a line of fixed points for the coupling constant. The surface is super-rough: the equilibrium correlation functions behave as (lnL${)}^2$ while their short time dependence is (lnτ${)}^2$ with a temperature-dependent dynamic exponent z=2[1+1.78(1-T/${\mathit{T}}_{\mathit{s}\mathit{r}}$)]. While the linear response mobility vanishes on large length scales, its scale dependence leads to a nonlinear response. For a small applied force F the average velocity of the surface v behaves as v∼${\mathit{F}}^{1+\mathrm{\zeta }}$. To first order ζ=1.78(1-T/${\mathit{T}}_{\mathit{s}\mathit{r}}$). At the transition, v∼F/(1+C‖ln(F)‖${)}^{1.78}$ and the crossover to the behavior to T<${\mathit{T}}_{\mathit{s}\mathit{r}}$ is analyzed. These results also apply to two-dimensional vortex glasses with a parallel magnetic field.