Non-linear evolution of step meander during growth of a vicinal surface with no desorption

Step meandering due to a deterministic morphological instability on vicinal surfaces during growth is studied. We investigate nonlinear dynamics of a step model with asymmetric step kinetics, terrace and line diffusion, by means of a multiscale analysis. We give the detailed derivation of the highly nonlinear evolution equation on which a brief account has been given [6]. Decomposing the model into driving and relaxational contributions, we give a profound explanation to the origin of the unusual divergent scaling of step meander (where F is the incoming atom flux). A careful numerical analysis indicates that a cellular structure arises where plateaus form, as opposed to spike-like structures reported erroneously in reference [6]. As a robust feature, the amplitude of these cells scales as t ^1/2, regardless of the strength of the Ehrlich-Schwoebel effect, or the presence of line diffusion. A simple ansatz allows to describe analytically the asymptotic regime quantitatively. We show also how sub-dominant terms from multiscale analysis account for the loss of up-down symmetry of the cellular structure. Received 4 May 2000 and Received in final form 8 September 2000     

A thin interface asymptotic for a phase-field model of epitaxial growth with different adatom diffusivities

Here we extend a phase-field model for epitaxial step-flow growth originally derived by Liu and Metiu to capture the case of different adatom diffusivities at neighboring terraces as well as an arbitrary Ehrlich-Schwoebel (ES) barrier. Our extended model approach bridges the atomic to continuum scale in the sense that it takes into account atomic attachment kinetics in full detail and likewise allows to simulate long range transport processes above the surface efficiently. To verify the model we present a matched asymptotic analysis of the derived model equations, which shows that in a special limit the presented model can be related to the Burton-Cabrera-Frank (BCF) model with different kinds of attachment coefficients at either side of a step edge. We demonstrate the capability of our approach by presenting numerical simulations with an Ehrlich-Schwoebel (ES) barrier, which reproduce the well-known step meandering instability. Thereby we show how mathematical analysis helps to specify and validate a phase-field model and thus contributes to the further development of this modeling approach at the nano- to microscale.     

Effects of the Ehrlich-Schwoebel potential barrier on the Wolf-Villain model simulations for thin film growth

Wolf-Villain (WV) model is a simple model used to study ideal molecular beam epitaxy (MBE) growth by using computer simulations. In this model, an adatom diffuses instantaneously within a finite diffusion length to maximize its coordination number. We study statistical properties of thin films grown by this model. The morphology of the WV model is found to be kinetically rough with a downhill particle diffusion current. In real MBE growth, however, there are additional factors such as the existence of a potential barrier that is known as the Ehrlich-Schwoebel (ES) barrier. The ES barrier is an additional barrier for an adatom that diffuses over a step edge from the upper to a lower terrace which is known to induce an uphill particle current. We found that with the addition of the ES barrier, the WV-ES model morphology is rough with mound formation on the surface when the barrier is strong enough. To confirm these results, the correlation function is also studied. We find no oscillation in the correlation function in the WV model. For the WV-ES model, the correlation function oscillates. These results confirm that a strong enough ES barrier can cause mound formation on the WV surface in our study.     

Ehrlich-Schwoebel barrier controlled slope selection in epitaxial growth

We examine the step dynamics in a 1+1-dimensional model of epitaxial growth based on the BCF-theory. The model takes analytically into account the diffusion of adatoms, an incorporation mechanism and an Ehrlich-Schwoebel barrier at step edges. We find that the formation of mounds with a stable slope is closely related to the presence of an incorporation mechanism. We confirm this finding using a solid-on-solid model in 2+1 dimensions. In the case of an infinite step edge barrier we are able to calculate the saturation profile analytically. Without incorporation but with inclusion of desorption and detachment we find a critical flux for instable growth but no slope selection. In particular, we show that the temperature dependence of the selected slope is solely determined by the Ehrlich-Schwoebel barrier which opens a new possibility in order to measure this fundamental barrier in experiments. Received 11 May 1999 and Received in final form 6 November 1999     

力和扩散机理下外延形貌的演化分析

本文提出了一个新的基于扩散界面的相场模型来描述外延生长中岛的形核、生长及熟化过程. 该模型同时考虑了弹性场、表面能、沉积、扩散、解吸和能量势垒等热力学及动力学过程对表面纳米形貌的影响. 采用经典的BCF模型来描述生长中的扩散形核过程, 而采用一个新的包含弹性应变能的自由能函数, 通过变分得到一个描述多层岛生长的相场方程, 该方法可以有效地描述外延生长中复杂的外延形貌. 采用有限差分格式对非线性耦合方程组进行求解. 数值结果显示, 该模型可以真实地再现外延生长中多层岛结构(即山丘状形貌)的演化过程, 模拟结果与已有实验结果一致. 同时模拟了生长过程中随外延形貌演化而形成的复杂生长应力, 研究表明, 在生长过程中, 岛中存在着复杂的应力分布, 且在岛边界处应力达到局部最大, 这与实验结果定性一致. 此外, 本文的重要发现是, 外延生长中的应力演化明显地影响原子的扩散过程, 当应力存在时, 外延结构变化较无弹性场时变快. 该项研究对理解外延生长中各物理机理的协同作用有重要的指导意义.     

Mixing length scales: step meandering and island nucleation on vicinal surfaces

Simultaneous island nucleation and step flow growth on vicinal surfaces are studied by Monte Carlo simulations. Step edges experience meandering instability under growth conditions if there is a kink barrier suppressing adatom jumps around kink sites. This instability has a characteristic length scale, with different scaling properties from the island separation scale. We show that there is a coupling between island nucleation and step edge instability. The length scale associated with nucleation begins to couple with the wavelength of the step edge patterns when islands and steps coalesce. Only in the submonolayer regime step meandering is independent of island formation. In this regime the island separation has a cross-over scaling behaviour as terrace width is varied.Received: 9 April 2003, Published online: 19 November 2003PACS: 81.15.Aa Theory and models of film growth - 68.55.Ac Nucleation and growth: microscopic aspects - 81.16.Rf Nanoscale pattern formationM. Rusanen: Present address: Institut Français du Pétrole, Groupe de Modélisation Moléculaire, BP 311, 92852 Rueil-Malmaison, FranceJ. Kallunki: Present address: Laboratory of Physics, PO Box 1100, FIN-02015 HUT, Espoo, Finland     

Numerical investigation of the submonolayer growth and relaxation of stepped Ag (110) and Au (110) surfaces

The terrace width distribution (TWD) as well as the adatom and island densities for some stepped fcc?(110) surfaces with parallel and equidistant steps of equal stiffness are studied by means of kinetic Monte Carlo simulations. Reliable energy barriers for surface diffusion are used. They have been calculated by means of many-body potentials derived within the second moment approximation to the tight-binding model. The effects of the temperature, atom deposition flux and surface diffusion on quantities of interest in the early stage of the surface evolution process have been singled out. In most cases, linear, logarithmic and power-law behaviors are recovered for the evolution of the step width and terrace defects. The TWD is well described by the Gaussian distribution (GD) in the domain of temperature investigated but deviates from the Generalized Wigner Distribution (GWD).     

Reduction of surface coverage of finite systems due to geometrical steps

The coverage of vicinal, stepped surfaces with molecules is simulated with the help of a two-dimensional Ising model including local distortions and an Ehrlich-Schwoebel barrier term at the steps. An effective two-spin model is capable to describe the main properties of this distorted Ising model. It is employed to analyze the behavior of the system close to the critical points. Within a well-defined regime of bonding strengths and Ehrlich-Schwoebel barriers we find a reduction of coverage (magnetization) at low temperatures due to the presence of the surface step. This results in a second, low-temperature transition besides the standard Ising order-disorder transition. The additional transition is characterized by a divergence of the susceptibility as a finite-size effect. Due to the surface step the mean-field specific heat diverges with a power law.     

Atomic diffusion inside a STM junction: simulations by kinetic Monte Carlo coupled to tunneling current calculations

The influence of a static scanning tunneling microscope (STM) tip on the diffusion of xenon atoms adsorbed on a Cu(1 1 0) stepped surface is studied. Semi-empirical potentials for the Xe-surface interaction and a N-body energy based method for the Xe-tip contribution are used to calculate the adsorption energy of adsorbates in the STM junction. First, we analyse the variation of this energy when the adatom is placed near a step edge and for different tip positions. When the tip is situated in the neighbourhood of the step edge, the Ehrlich-Schwoebel barrier experienced by the adatom is lowered. This opens a specific diffusion channel, allowing a possible crossing of the step edge. Second, through a kinetic Monte Carlo approach coupled to the elastic scattering quantum chemistry method, the noisy tunneling current created by the random motion of diffusing atoms in the vicinity of the tip can be analyzed. We show that, by counting the number of diffusion events, we can determine effective barriers related to the most dominant processes contributing to the diffusion at a particular temperature. We also demonstrate that the interaction mode of the tip (attractive or imaging) greatly modifies the diffusion processes.     

Evidence for diffusion-limited kinetics during electromigration-induced step bunching on Si(1 1 1)

We have measured how the initial terrace width l₀ on vicinal Si(1 1 1) surfaces influences the rate of step bunching and the minimum terrace width within a bunch when direct-current heated at 940-1290 °C. A comparison of this data with analytic solutions and numerical simulations of the conventional “sharp-step” model give strong evidence that the kinetic length d is relatively small (d < ∼20 nm) in both temperature regime I (∼850-950 °C) and regime III (∼1200-1300 °C), in which step-down current is required for step bunching. This indicates that surface mass transport is diffusion-limited in both regimes I and III when l₀ > 20 nm, and hence that the adatom attachment- and terrace diffusion-hopping rates are of comparable magnitude. We also observe similar scaling with initial terrace width in temperature regime II (∼1040-1190 °C), in which step-up current is required for bunching, suggesting a similar step bunching mechanism in all three temperature regimes.     

邻晶面外延生长机制的动力学Monte Carlo模拟

用动力学Monte Carlo模拟方法研究了GaAs(001)邻晶面的外延生长机制.Ehrlich-Schwoebel势垒对邻晶面外延机制有重要的影响.模拟结果显示，低温下Ehrlich-Schwoebel势垒几乎能完全阻止原子向下一台阶面的迁移，高温下原子已能有效地克服势垒的影响并向下一台阶面迁移.在外延生长初期，原子几乎在台阶面上均匀分布.当表面覆盖度达到一定数量后，台阶成核开始.而由于Ehrlich-Schwoebel势垒的存在，在台阶的上侧台阶面上开始有原子的积累，而如果没有Ehrlich-Schwoebel势垒，台阶上侧台阶面上的原子也能被有效地耗尽.Ehrlich-Schwoebel势垒对邻晶面上的外延生长模式有显著的影响，将明显提高达到台阶生长模式的温度. 关键词： 外延生长模式 动力学Monte Carlo Ehrlich-Schwoebel势垒     

六方小面相螺旋在各向异性、表面吸附、界面动力学作用下生长的相场

利用定量相场模型研究了强各向异性、表面吸附率以及界面动力学作用条件下六方Ga N螺旋结构的表面形貌与生长机理.通过引入小面相各向异性的相场修正方程,研究了不同各向异性的稳态螺旋形貌,发现各向异性通过改变台阶尖端的曲率作用影响螺旋生长.弱各向异性下稳态螺距及界面动力学特征相对稳定,各向异性较强时尖端的过饱和度随着各向异性的增强而增大,并使得界面平衡态向着有利于螺旋台阶推进的方向移动.研究了表面吸附率对小面相螺旋生长的作用机理,发现吸附率的增加导致了稳态螺旋间距的降低,通过分析螺旋间距随台阶宽度的变化趋势,发现增强的表面吸附和各向异性强度降低了螺旋间距的收敛性,并且具体分析了收敛性误差;通过探讨界面动力学作用条件下螺旋形貌特征以及螺旋间距变化趋势,发现界面动力学系数通过改变稳态螺旋间距与特征指数因子调控螺旋生长的动力学机理,与各向同性相比小面相螺旋生长表现出较低的界面动力学系数依赖性.     

异质扩散过程中ES势垒的计算

利用分子动力学中的静态结构计算方法对Pd，Ag及Cu原子在面心立方铜的台阶表面扩散过程中的Ehrlich-Schwoebel(ES)势垒进行了模拟计算，研究了各种台阶表面情况下增原子扩散过程中的ES势垒；讨论了与衬底互溶的金属和与衬底不互溶的金属增原子扩散的ES势垒的异同，并将模拟结果与同质情况的研究结果进行了对比. 结果表明： 1)在同质和异质扩散过程中ES势垒随着台阶高度的变化关系是相似的，即随着台阶高度的增加，ES势垒逐渐增加；当台阶高度达到某一高度时ES势垒将趋于定值. 2)在跳跃机理下，与Cu互溶的金属(Pd)在Cu表面台阶上扩散的ES势垒最大，其次是Cu，最小的是与Cu不互溶的金属 (Ag)；而在交换机理下，与Cu不互溶的金属(Ag)在Cu表面台阶上扩散的ES势垒最大，其次是Cu，最小的是与Cu互溶的金属(Pd). 3)对大多数台阶的情况，交换机理支配着原子在台阶边缘的扩散行为；且表面台阶高度对交换扩散过程影响较大.     

Dendritic islands in metal-on-metal epitaxy I. Shape transitions and diffusion at island edges

The development of dendritic island shape instabilities observed during metal-on-metal epitaxy is investigated via a lattice-gas model for the low coverage regime. The key assumption is that island structure is controlled by the competition between shape equilibration due to adatom edge diffusion, and Mullins-Sekerka-type shape instability due to diffusion-limited aggregation of adatoms with islands. From comparison with scanning tunneling microscopy data (for the island density and average width of dendritic arms), we advance estimates of the energy barrier for edge diffusion in several systems.     

扫描隧道显微镜针尖调制的薄膜表面的原子扩散

采用原子尺度的模拟方法,探讨了在零偏压下扫描隧道显微镜(STM)针尖调制的金属表面岛上原子运动及岛边的层间质量输运. 研究结果显示STM的移动对岛上及岛边的原子扩散有重要的影响. 针尖与吸附原子的交互作用及岛和基体中强的形状变化影响了岛上吸附原子的跳跃扩散及岛边的跳下扩散和交换扩散过程. 研究发现,通过调节针尖与基体的垂直距离及针尖与吸附原子的水平距离,可以降低岛上吸附原子的跳跃扩散能垒及岛边的跳下扩散和交换扩散能垒,从而实现薄膜由三维生长模式向二维生长模式的转变. 关键词： 扫描隧道显微镜 原子运动 质量输运     

Step instability of the In_0.2Ga_0.8As （001） surface during annealing

Anisotropic evolution of the step edges on the compressive-strained In0.2Ga0.8 As/GaAs(001) surface has been investigated by scanning tunneling microscopy (STM). The experiments suggest that step edges are indeed sinuous and protrude somewhere a little way along the [110] direction, which is different from the classical waviness predicted by the theoretical model. We consider that the monatomic step edges undergo a morphological instability induced by the anisotropic diffusion of adatoms on the terrace during annealing, and we improve a kinetic model of step edge based on the classical Burton–Cabrera–Frank (BCF) model in order to determine the normal velocity of step enlargement. The results show that the normal velocity is proportional to the arc length of the peninsula, which is consistent with the first result of our kinetic model. Additionally, a significant phenomenon is an excess elongation along the [110] direction at the top of the peninsula with a higher aspect ratio, which is attributed to the restriction of diffusion lengths.     

Step instability of the In0.2Ga0.8As （001） surface during annealing

Anisotropic evolution of the step edges on the compressive-strained In0.2Ga0.8 As/GaAs(001) surface has been investigated by scanning tunneling microscopy (STM). The experiments suggest that step edges are indeed sinuous and protrude somewhere a little way along the [110] direction, which is different from the classical waviness predicted by the theoretical model. We consider that the monatomic step edges undergo a morphological instability induced by the anisotropic diffusion of adatoms on the terrace during annealing, and we improve a kinetic model of step edge based on the classical Burton–Cabrera–Frank (BCF) model in order to determine the normal velocity of step enlargement. The results show that the normal velocity is proportional to the arc length of the peninsula, which is consistent with the first result of our kinetic model. Additionally, a significant phenomenon is an excess elongation along the [110] direction at the top of the peninsula with a higher aspect ratio, which is attributed to the restriction of diffusion lengths.     

Adsorption of tungsten on stepped tungsten surfaces studied by work function measurement

Work function measurements have been performed during the deposition of W on the (110)W plane and several stepped W surfaces with (110) terraces and different terrace width. For each sample the work function decreases with growing coverage. The total work function drop diminishes strongly with decreasing terrace width. The results are interpreted in terms of a reduced nucleation process on stepped surfaces as compared to the flat (110) plane. The step edges act as sinks for the deposited adatonis and cause in their proximity a “dead” zone for nuclei formation. Details of the work function change with coverage are discussed in terms of an edge roughening effect on stepped surfaces.     

Determination of interlayer diffusion parameters for Ag/Ag(111)

It is well known that the Ag/Ag(111) epitaxial system grows three dimensionally because of the existence of a relatively high excess diffusion barrier, DeltaE(s) = 0.13 eV, at the step edges. Several experimental methods have been developed to measure the step edge barrier in this system over a wide coverage range. The probability for an atom to move from a higher to a lower layer depends on both the barrier and the prefactor, so it is important to test whether the prefactors for hopping over a step, nu(s), and for hopping on a terrace, nu(t), are different. We present the results from several experiments on Ag/Ag(111) to conclude that nu(s)/nu(t) = 10(2.0+/-0. 3).     

Instability and wavelength selection during step flow growth of metal surfaces vicinal to fcc(001)

We study the onset and development of ledge instabilities during growth of vicinal metal surfaces using kinetic Monte Carlo simulations. We observe the formation of periodic patterns at [110] close packed step edges on surfaces vicinal to fcc(001) under realistic molecular beam epitaxy conditions. The corresponding wavelength and its temperature dependence are studied in detail. Simulations suggest that the ledge instability on fcc(1,1,m) vicinal surfaces is controlled by the strong kink Ehrlich-Schwoebel barrier, with the wavelength determined by dimer nucleation at the step edge. Our results are in agreement with recent continuum theoretical predictions, and experiments on Cu(1,1,17) vicinal surfaces.