Scaling laws for step bunching on vicinal surfaces: Role of the dynamical and chemical effects,Physical Review E

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Scaling laws for step bunching on vicinal surfaces: Role of the dynamical and chemical effects
Physical Review E ( IF 2.2 ) Pub Date : 2021-09-28 , DOI: 10.1103/physreve.104.034802
L Benoit-Maréchal _{1,

2} , M E Jabbour _{1,

3} , N Triantafyllidis _{1,

3,

4}

Affiliation

We study the evolution of step bunches on vicinal surfaces using a thermodynamically consistent step-flow model. By accounting for the dynamics of adatom diffusion on terraces and attachment-detachment at steps (referred to collectively as the dynamical effect), this model circumvents the quasistatic approximation that prevails in the literature. Furthermore, it generalizes the expression of the step chemical potential by incorporating the necessary coupling between the diffusion fields on adjacent terraces (referred to as the chemical effect). Having previously shown that these dynamical and chemical effects can explain the onset of step bunching without recourse to the inverse Ehrlich-Schwoebel (iES) barrier or other extraneous mechanisms, we are here interested in the evolution of step bunches beyond the linear-stability regime. In particular, the numerical resolution of the step-flow free boundary problem yields a robust power-law coarsening of the surface profile, with the bunch height growing in time as

H \sim t^{1 / 2}

and the minimal interstep distance as a function of the number of steps in the bunch cell obeying

ℓ_{min} \sim N^{- 2 / 3}

. Although these exponents have previously been reported, the novelty of the present approach is that these scaling laws are obtained in the absence of an iES barrier or adatom electromigration. In order to validate our simulations, we take the continuum limit of the discrete step-flow system via Taylor expansions with respect to the terrace size, leading to a novel nonlinear evolution equation for the surface height. We investigate the existence of self-similar solutions of this equation and confirm the 1/2 coarsening exponent obtained numerically for

H

. We highlight the influence of the combined dynamical-chemical effect and show that it can be interpreted as an effective iES barrier in the setting of the standard Burton-Cabrera-Frank theory. Finally, we use a Padé approximant to derive an analytical expression for the velocity of steadily moving step bunches and compare it to numerical simulations.

中文翻译：

相邻表面阶梯聚束的缩放定律：动力学和化学效应的作用

我们使用热力学一致的阶梯流模型研究邻近表面上阶梯束的演变。通过考虑台阶上吸附原子扩散的动力学和台阶上的附着-分离（统称为动力学效应），该模型规避了文献中普遍存在的准静态近似。此外，它通过在相邻平台上的扩散场之间引入必要的耦合（称为化学效应）来概括阶跃化学势的表达）。之前已经表明，这些动力学和化学效应可以解释阶跃聚束的开始，而无需求助于逆 Ehrlich-Schwoebel (iES) 势垒或其他外来机制，我们在这里对超出线性稳定机制的阶跃聚束的演变感兴趣。特别是，阶梯流自由边界问题的数值分辨率产生了表面轮廓的稳健幂律粗化，束高随着时间的推移而增长

H ～ 吨^{1 / 2}

以及作为束单元中步数的函数的最小步距

ℓ_{分钟} ～ N^{- 2 / 3}

. 虽然这些指数以前已经被报道过，但本方法的新颖之处在于这些缩放定律是在没有 iES 屏障或吸附原子电迁移的情况下获得的。为了验证我们的模拟，我们通过关于阶地大小的泰勒展开来获取离散阶梯流系统的连续极限，从而得出表面高度的新非线性演化方程。我们研究了这个方程的自相似解的存在，并确认了数值上获得的 1/2 粗化指数