An effective method to calculate RHEED rocking curves from nanoheteroepitaxial systems

Abstract

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: https://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 $U_n^{\text{substrate}}$ means the potential of the full-filled substrate layers, $U_n^{\text{layer}}$ means the potential of growing monolayer, $U_{\mathrm{add}}^{\text{layer}}$ component is responsible for diffuse scattering, and $\theta \left(n\right)$ 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 ${\tau }_n$ 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 $1∕{\tau }_{\mathrm{n<=2}}=1.0$, $1∕{\tau }_{\mathrm{n>=3}}=0.8$, 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 0.2°, which remains within the limits of experimental error.

Acknowledgments

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

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