ith exponents $3/8$ and $1/2$, respectively, in both main panels and insets. In (b), the slopes of the dashed and solid lines are $1/4$ and $1/2$, respectively.](w_wvdt_1d.pdf "fig:"){width="0.8\linewidth"}\
![\[width\]Time evolution of the interface width $w$ for WV (main panels) and DT (insets) models grown on (a) one- and (b) two-dimensional substrates. Both simulations with the kinetic barrier (using $N_s$ values indicated in the legend) and the original version are shown. In (a), dashed and solid lines are power-laws with exponents $3/8$ and $1/2$, respectively, in both main panels and insets. In (b), the slopes of the dashed and solid lines are $1/4$ and $1/2$, respectively.](w_wvdt_2d.pdf "fig:"){width="0.8\linewidth"}

Figure \[width\] shows the time evolution of the interface width for both models in one and two dimensions. The main panels and insets present the results for the WV and DT models, respectively, including or not the kinetic barrier. The interface width is expected to scale as $w\sim t^\beta$ where $\beta$ is the growth exponent [@barabasi]. The short time dynamics of both WV and DT models is well described by the linear version of the MBE equation [@Villain; @LSarma] $$\frac{\partial h}{\partial t} = -\nu \nabla^{4}h + \lambda \nabla^{2} (\nabla h)^{2} + \eta,
\label{Eq1}$$ with $\lambda=0$ where $\eta$ is a non-conservative Gaussian noise [@Villain; @LSarma; @DasSarma1992]. This result is confirmed in Fig. \[width\] where the short time behavior is consistent with the growth exponents $\beta=3/8$ in $d=1$ and $\beta=1/4$ in $d=2$ expected for the linear MBE universality class [@barabasi]. It is worth to mention that these models may undergo crossovers to different universality classes in the asymptotic, depending on the dimension and model [@wvbogo; @Vvedensky; @Xun; @Chen2017; @AaraoReis2004b; @Punyindu]. The curves in Fig. \[width\] are consistent with crossovers to different universality classes at long times. One expects that DT is asymptotically consistent with the non-linear 