We report a simulation program which facilitates the calculation of changes in the intensity of specular reflection of electron beams in RHEED experiments for thin epitaxial films deposited on crystalline surfaces. It has been shown that the amplitude of the RHEED intensity oscillations greatly depends on the glancing angle of the incident electron beam, the coverages of the growing layers and the model of the scattering potential. The usefulness of the program has been tested on a well-known system of Ag grown on a Si(111) surface. The obtained experimental and computational results correspond closely. The presented algorithm, together with properly modified input data, can be applied to other systems of crystalline ultrathin layer and substrate. It also enables the implementation and tests of different combinations of the scattering potentials of the crystal, and can be applied to interpret experimental RHEED rocking curves.

New version program summary

Program title: RHEED_DIFFv2

CPC Library link to program files: http://dx.doi.org/10.17632/7c6y233rys.1

Code Ocean capsule: https://codeocean.com/capsule/0078674

Licensing provisions: GNU General Public License 3

Programming language: C++

Journal reference of previous version: Computer Physics Communications 185 (2014) 3001–3009

Does the new version supersede the previous version?: Yes

Reasons for the new version: Responding to user’s feedback we improved functionality of the program. Moreover, we added new capabilities which make the input data process easier and more efficient than the previous one.

Nature of problem: The measurement of incident-angle dependence of RHEED intensity oscillations (RHEED rocking curve) during the growth of thin layers prepared by molecular beam epitaxy is a popular technique for quantitative and qualitative investigations of epitaxial structure perfection. Rocking curves recorded from heteroepitaxial layers are used for the non-destructive characterization of atoms near the surface [1,2]. Rocking curves are also used to determine the level of strain and its relaxation mechanism in lattice-mismatched systems. In most cases the interpretation of experimental results is based on the use of dynamical diffraction approaches. Such approaches are known to be useful in qualitative and quantitative analyses of RHEED experimental data, and especially in the interpretation of changes in intensity of reflected specular electron beam.

Solution method: RHEED intensities are calculated within the general framework described in Ref. [3] with the model of the scattering potential for heterostructures: (1)$U(\theta ,z)=\sum _i\sum _n{U}_n^{\text{substrate}}(\theta \left(n\right)=1,z_i)+\sum _i\sum _n(U_n^{\text{layer}}(\theta \left(n\right),z_i)+U_{\mathrm{add}}^{\text{layer}}(\theta \left(n\right),z_i)),$where means the potential of the full-filled substrate layers, means the potential of growing monolayer, component is responsible for diffuse scattering, and is the value of nth layer coverage in the vicinity of a growth front as a function of deposited layers [4]. The description of atom deposition at growing surfaces was obtained by solving the set of nonlinear differential equations developed for the distributed growth model [4]. (2)$\frac{d{\theta }_n\left(t\right)}{\mathrm{dt}}=\frac{1}{{\tau }_n}[{\theta }_{n-1}-{\theta }_n+{\alpha }_n({\theta }_n-{\theta }_{n+1})-{\alpha }_{n-1}({\theta }_{n-1}-{\theta }_n)],$where (3)${\alpha }_n=A_n\frac{d_n\left({\theta }_n\right)}{d_n\left({\theta }_n\right)+d_{n+1}\left({\theta }_{n+1}\right)},$and (4)$d_n\left({\theta }_n\right)=\theta \sqrt{(1-{\theta }_n)}.$In these equations, ${\mathit{A}}_n$ is the parameter that measures the net rate transfer of atoms from one layer to the next, and is the deposition time of nth monolayer. Detailed explanations for Eqs. (2-4) are presented in Ref. [4]. The current distribution of the RHEED_DIFFv2 program includes an example input file CoverageProfiles.dat. It is one of the program output files [4] (but it can also be prepared independently of the program [4]). The CoverageProfiles.dat file contains coverage profiles for numerical solutions of Eqs. (2-4) with parameters gR${}_{\mathrm{n<=2}}=1.0$, gR ${}_{\mathrm{n>=3}}=0.8$, and $A_1=0.999$, $A_2=0.6$, $A_3=0.9$, $A_4=0.9$, $A_5=0.8$, $A_6=0.7$, $A_{\mathrm{n>=7}}=0.6$. The values of these parameters correspond to the Ag/Si(111) growth model proposed by Zhang and co-workers [5].

Summary of revisions: In the previous version of the program for calculating the amplitude of the reflected electron beam from growing layers, only the growth mode for the topmost surface monolayer has been taken into account [3]. In the current version it is possible to include in the calculations all growing monolayers. For this purpose, we have implemented an original algorithm for dynamical calculations of changes in the RHEED rocking curve for selected growth model of thin epitaxial films [4]. In the current version of the program, the scattering potentials of successive growing monolayers are modified by the coverage values of the monolayer near the growth front. Therefore, the program needs CoverageProfiles.dat file [4] with input data to work properly. This file stores the values of layer coverages in the vicinity of a growth front as a function of deposited layers. The data in this file are saved in two columns – the first column stores the ordinal number of deposited layers, while the second contains the values of the growth front for each of these layers.

Fig. 1(a) shows experimentally measured rocking curves for Ag layers (2.83 nm-thick) for the azimuth of incidence [11-2] and [11-2]+7°. For the [11-2] direction, and turned away by 7° from [11-2], the three dimensional crystal lattice of the Si(111) substrates and Ag(111) growing layers can be regarded as a one-dimensional array of lattice planes parallel to the surface. This azimuthal direction corresponds to the one-beam condition [3]. Figs. 1(b-c) show dynamically calculated one-beam rocking curves for Ag(111)/Si(111) layers. In the numerical calculations of the rocking curves of specular beam intensity for 2.83 nm-thick Ag layers, we used the following parameters: the electron energy of 18.8 keV, the glancing angle in the range from 0.2° to 3.5°, and the values $\alpha =0.2$, $\beta =0.0$ and 0.5 for the model of the scattering potential described by Eq. (1). Assuming that results of one-beam calculations reproduce actual experimental situations, we can conclude that for a fixed, real surface, the measured and calculated positions of Bragg reflections tally very well. The difference between them does not exceed 0.2⁰, which remains within the limits of experimental error.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work has been in part supported by the National Science Centre under Grant No. 2016/21/B/ST3/01294.

References

Y. Horio, J. Yuhara, and Y. Takakuwa, Jap. J. Appl. Phys. 58 (2019) SIIA14

T. Kawamura, K. Fukutani, Surf. Sci. 688 (2019) 7-13

A. Daniluk, Comput. Phys. Commun. 185 (2014) 3001-3009.

A. Daniluk, Comput. Phys. Commun. 170 (2005) 265–286.

Z. H. Zhang, S. Hasegawa, and S. Ino, Phys. Rev. B 55 (1997) 9983.

中文翻译：

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从纳米异质外延系统计算RHEED摇摆曲线的有效方法

我们报告了一个模拟程序，该程序有助于在RHEED实验中针对沉积在晶体表面上的薄外延膜计算电子束的镜面反射强度的变化。已经表明，RHEED强度振荡的幅度在很大程度上取决于入射电子束的掠射角，生长层的覆盖范围和散射势的模型。该程序的有效性已在众所周知的生长在Si（111）表面的Ag系统上进行了测试。获得的实验和计算结果非常接近。提出的算法，连同适当修改的输入数据，可以应用于晶体超薄层和衬底的其他系统。

新版本程序摘要

程式名称： RHEED_DIFFv2

CPC库链接到程序文件： http : //dx.doi.org/10.17632/7c6y233rys.1

代码海洋胶囊： https : //codeocean.com/capsule/0078674

许可条款： GNU通用公共许可证3

编程语言： C ++

先前版本的期刊参考： Computer Physics Communications 185（2014）3001-3009

新版本会取代旧版本吗？：是

新版本的原因：响应用户的反馈，我们改进了程序的功能。此外，我们添加了新功能，这些功能使输入数据处理比上一个更加轻松和高效。

问题性质：在分子束外延制备的薄层生长过程中，RHEED强度振荡的入射角依赖性（RHEED摇摆曲线）的测量是一种用于定量和定性研究外延结构完善性的流行技术。从异质外延层记录的摇摆曲线用于表面附近原子的非破坏性表征[1,2]。摇摆曲线还用于确定晶格不匹配系统中的应变水平及其松弛机制。在大多数情况下，对实验结果的解释是基于动态衍射方法的使用。已知这些方法可用于RHEED实验数据的定性和定量分析，尤其是在解释反射镜面电子束强度变化方面。

求解方法： RHEED强度是在参考文献1中所述的一般框架内计算的。[3]异质结构的散射势模型：（1）$ü（\theta ，ž）=\sum _{\mathrm{一世}}\sum _{ñ}{ü}_{ñ}^{\text{基质}}（\theta （ñ）=\mathrm{1个}，{ž}_{\mathrm{一世}}）+\sum _{\mathrm{一世}}\sum _{ñ}（{ü}_{ñ}^{\text{层}}（\theta （ñ），{ž}_{\mathrm{一世}}）+{ü}_{\mathrm{加}}^{\text{层}}（\theta （ñ），{ž}_{\mathrm{一世}}）），$其中，表示完全填充的基材层的电势，表示生长的单分子层的电势，是造成组分散射的原因，是生长前沿附近的第n层覆盖率随沉积层而变化的值[4] ]。通过求解为分布式生长模型开发的非线性微分方程组，获得了生长表面原子沉积的描述[4]。（2）$\frac{d{\theta }_{ñ}（Ť）}{\mathrm{dt}}=\frac{\mathrm{1个}}{{\tau }_{ñ}}[{\theta }_{ñ-\mathrm{1个}}-{\theta }_{ñ}+{\alpha }_{ñ}（{\theta }_{ñ}-{\theta }_{ñ+\mathrm{1个}}）-{\alpha }_{ñ-\mathrm{1个}}（{\theta }_{ñ-\mathrm{1个}}-{\theta }_{ñ}）]，$哪里（3）${\alpha }_{ñ}={\mathrm{一种}}_{ñ}\frac{d_{ñ}（{\theta }_{ñ}）}{d_{ñ}（{\theta }_{ñ}）+d_{ñ+\mathrm{1个}}（{\theta }_{ñ+\mathrm{1个}}）}，$和（4）$d_{ñ}（{\theta }_{ñ}）=\theta \sqrt{（\mathrm{1个}-{\theta }_{ñ}）}。$在这些方程式中 ${\mathit{一种}}_{ñ}$是测量原子从一层到另一层的净速率转移的参数，并且是第n个单层的沉积时间。公式的详细说明。（2-4）在参考资料中列出。[4]。RHEED_DIFFv2程序的当前发行版包括示例输入文件CoverageProfiles.dat。它是程序输出文件[4]之一（但也可以独立于程序[4]准备）。该CoverageProfiles.dat文件包含公式的数值解覆盖型材。（2-4）参数gR${}_{\mathrm{n\; <=\; 2}}=\mathrm{1个}。0$， ${}_{\mathrm{n>\; =\; 3}}=0。8$和 ${\mathrm{一种}}_{\mathrm{1个}}=0。999$， ${\mathrm{一种}}_2=0。6$， ${\mathrm{一种}}_3=0。9$， ${\mathrm{一种}}_4=0。9$， ${\mathrm{一种}}_5=0。8$， ${\mathrm{一种}}_6=0。7$， ${\mathrm{一种}}_{\mathrm{n>\; =\; 7}}=0。6$。这些参数的值对应于张和同事提出的Ag / Si（111）生长模型[5]。

修订摘要：在用于计算来自生长层的反射电子束幅度的程序的先前版本中，仅考虑了最顶层表面单层的生长模式[3]。在当前版本中，可以在计算中包括所有增长的单层。为此，我们为薄外延薄膜的选定生长模型实现了一种动态计算RHEED摇摆曲线变化的原始算法[4]。在该程序的当前版本中，连续生长的单分子层的散射势由生长前沿附近的单分子层的覆盖率值修改。因此，该程序需要CoverageProfiles.dat文件[4]和输入数据正常工作。该文件将生长前沿附近的层覆盖率值存储为沉积层的函数。该文件中的数据保存在两列中-第一列存储已沉积层的序数，而第二列包含每个层的生长前沿值。

图1（a）显示了入射角[11-2]和[11-2] + 7°的Ag层（厚度为2.83 nm）的实验测量摇摆曲线。对于[11-2]方向，并与[11-2]偏离7°，可以将Si（111）衬底和Ag（111）生长层的三维晶格视为一维阵列平行于表面的晶格平面。该方位角方向对应于一个光束条件[3]。无花果 图1（bc）显示了动态计算的Ag（111）/ Si（111）层的单光束摇摆曲线。在对2.83 nm厚的Ag层的镜面光束强度的摇摆曲线的数值计算中，我们使用以下参数：18.8 keV的电子能量，0.2°至3.5°范围内的掠射角以及$\alpha =0。2$， $\beta =0。0$方程0.5描述的散射势模型为0.5。（1）。假设单光束计算的结果重现了实际的实验情况，我们可以得出结论，对于固定的真实表面，布拉格反射的测量位置和计算位置非常吻合。它们之间的差异不超过0.2 ⁰，这仍在实验误差的范围内。

竞争利益声明

作者声明，他们没有已知的竞争财务利益或个人关系，这些关系或个人关系似乎可以影响本文报道的工作。

致谢

这项工作得到了美国国家科学中心的部分资助， 资助号为2016/21 / B / ST3 / 01294。

参考文献

Y. Horio，J。Yuhara和Y. Takakuwa，日本。J.应用 物理 58（2019）SIIA14

川村T.福谷K.冲浪 科学 688（2019）7-13

A. Daniluk，计算机 物理 社区 185（2014）3001-3009。

A. Daniluk，计算机 物理 社区 170（2005）265-286。

ZH Zhang，长谷川S和Ino S. Phys。版本B 55（1997）9983。