An efficient algorithm for shadowing effect simulation under hexagonal lattice-based model

doi:10.1088/1674-1056/ae82e5

Abstract This paper introduces an efficient algorithm for simulating the shadowing effect in hexagonal-lattice networks, addressing a critical gap in existing methods that are limited to cubic-lattice models. The shadowing effect significantly influences roughness development in processes such as film growth and plasma etching but has previously been impractical for hexagonal lattices due to its computational complexity. The proposed algorithm dramatically reduces computational load by strategically reducing the calculation dimensionality from three-dimensional (3D) to two-dimensional (2D) and handling the misaligned odd and even layers separately. This enables simulations on personal computers, with the typical simulation time decreasing from several days to less than one hour (achieving $>100\times$ acceleration). Validated simulations show consistent results with cubic-lattice models in both growth (roughening) and etching (smoothing) scenarios, while demonstrating superior symmetry in roughness formation. The work facilitates broader applications of shadowing simulations and integration with other hexagonal-lattice-based mechanisms.

Keywords: thin film structure and morphology Monte Carlo method lattice theory and statistics algorithm

Received: 08 May 2026
Revised: 08 June 2026
Accepted manuscript online: 26 June 2026

PACS:	68.55.-a	(Thin film structure and morphology)
	02.50.Ng	(Distribution theory and Monte Carlo studies)
	05.50.+q	(Lattice theory and statistics)
	02.60.Gf	(Algorithms for functional approximation)

Fund: Project supported by the National Natural Science Foundation of China (Grant No. 12275254) and the Laser Fusion Research Center Funds for Young Talents (Grant No. RCFPD7-2024-7).

Corresponding Authors: Xiaolong Jiang
E-mail: xljiang@caep.cn

Cite this article:

Xiaolong Jiang(蒋晓龙), Xin Xiang(向欣), Xiaoyu Luan(栾晓雨), Haijun Wang(王海军), Baoshen Jia(贾宝申), Qinghua Deng(邓青华), Chuanchao Zhang(张传超), Wei Liao(廖威), Wei Ni(倪卫), and Qihua Zhu(朱启华) An efficient algorithm for shadowing effect simulation under hexagonal lattice-based model 2026 Chin. Phys. B 35 086801

[1] Karunasiri R P U, Bruinsma R and Rudnick J 1989 Phys. Rev. Lett. 62 788
[2] Drotar J T, Zhao Y P, Lu T M and Wang G C 2000 Phys. Rev. B 62 2118
[3] Pelliccione M, Karabacak T and Lu T M 2006 Phys. Rev. Lett. 96 146105
[4] Pelliccione M, Karabacak T, Gaire C, Wang G C and Lu T M 2006 Phys. Rev. B 74 125420
[5] Wang S Q, Zhou C H, Ru H Y and Zhang Y Y 2005 Appl. Opt. 44 4429
[6] Zhou J, Chen Y, ZhouWL, Miao X S, Yang Z, Yu N, Liu H, Lan T and Yan J 2013 Appl. Surf. Sci. 280 862
[7] Qiu R F, Lu H, Chen D J, Zhang R, Zhe Y, Yu N N, Liu H, Lan T and Yan J B 2011 Appl. Surf. Sci. 257 2700
[8] Kokkoris G, Constantoudis V, Angelikopoulos P, Boulousis G and Gogolides E 2007 Phys. Rev. E 76 193405
[9] Jiang X L, Liu Y, Liu Z K, Qiu K Q, Xu X D, Hong Y L and Fu S J 2015 Appl. Surf. Sci. 355 1180
[10] Li L, Abe T and Esashi M 2003 J. Vac. Sci. Technol. B 21 2545
[11] Choi D Y, Lee J H, Kim D S and Jung S T 2004 J. Appl. Phys. 95 8400
[12] Goyal A, Hood V and Tadigadapa S 2006 J. Non Cryst. Solids 352 657
[13] Hsu C H, Lo H C, Chen C F, Wu C T, Hwang J S, Das D, Tsai J, Chen L C and Chen K H 2004 Nano Lett. 4 471
[14] Zheng J, Yang R, Xie L, Qu J L, Liu Y and Li X G 2010 Adv. Mater. 22 1451
[15] Guo Y R, Wang T, Chen J, Zheng J, Li X G and Ostrikov K 2018 Adv. Energy Mater. 8 1800085
[16] Lee S H and Lu J Q 2010 Nanotechnology 21 155303
[17] Hein E, Fox D and Fouckhardt H 2011 J. Nanophotonics 5 051703
[18] Mundo R D, Benedictis V D, Palumbo F and Agostino R 2009 Appl. Surf. Sci. 255 5461
[19] Jiang X L, Zhang C C, Luan X Y, Tang H P, Deng Q H, Jia B S, Wang H J, Liao W, Ni W and Zhu Q H 2026 Appl. Surf. Sci. 715 164617
[20] Kokkoris G and Gogolides E 2012 J. Phys. D: Appl. Phys. 45 165204
[21] Bruschi P, Cagnoni P and Nannini A 1997 Phys. Rev. B 55 7955
[22] Angqvist M, Magnus Rahm J, Gharaee L and Erhart P 2019 Phys. Rev. B 83 073605
[23] Jiang X L, Wu L X, Yang K, Liu T X, Liao W, Zhang C C, Zhang L J, Liu Y and Jiang X D 2021 Appl. Surf. Sci. 543 148862

[1]	Reviews of algorithm-driven terahertz metamaterials: From intelligent design to multidisciplinary applications Wenyue Cao(曹文钺), Yuying Jiang(蒋玉英), Hongyi Ge(葛宏义), and Juncheng Cao(曹俊诚). Chin. Phys. B, 2026, 35(8): 080702.
[2]	Enhanced multiscale quantum approximate optimization algorithm in multibody combinatorial optimization problems Lei-Lei Chen(陈蕾蕾), Ping Zou(邹平), and Ya-Fei Yu(於亚飞). Chin. Phys. B, 2026, 35(7): 070305.
[3]	Three-party semi-quantum dialogue enhanced with Grover's algorithm based encoding and hypergraph access control Rui Tao(陶瑞), Jin-Zhe Jiang(蒋晋喆), and Zhi-Hua Zhang(章志华). Chin. Phys. B, 2026, 35(7): 070306.
[4]	Coherence and decoherence in generalized Shor’s algorithm Linlin Ye(叶琳琳), Zhaoqi Wu(吴照奇), and Nanrun Zhou(周南润). Chin. Phys. B, 2026, 35(6): 060304.
[5]	A visually meaningful medical image encryption scheme based on image steganography and memristive Hopfield neural networks Wei Yao(姚卫), Xiangyun Huang(黄翔云), Jianhua Xiao(肖捡花), Fei Yu(余飞), and Yichuang Sun(孙义闯). Chin. Phys. B, 2026, 35(6): 068702.
[6]	Intelligent identification for discrete memristive neuron map: An adaptive chaos game optimization algorithm studied from the perspectives of different sample sizes and objective functions Yuexi Peng(彭越兮), Xinyi Luo(罗馨怡), Zhijun Li(李志军), Mengjiao Wang(王梦蛟), and Minglin Ma(马铭磷). Chin. Phys. B, 2026, 35(6): 060507.
[7]	Auxiliary-qubit-free quantum approximate optimization algorithm for the minimum dominating set problem Guanghui Li(李广辉), Xiaohui Ni(倪晓慧), Junjian Su(苏俊健), Sujuan Qin(秦素娟), Fenzhuo Guo(郭奋卓), Bingjie Xu(徐兵杰), Wei Huang(黄伟), and Fei Gao(高飞). Chin. Phys. B, 2026, 35(5): 050304.
[8]	Advanced magneto-acousto-electrical tomography based on circular arrays of omnidirectional transducers and orthogonally oriented electrodes Peixia Li(李培霞), Wei Chen(陈伟), Hao Qian(钱昊), Yulong Zhou(周瑜龙), Gepu Guo(郭各朴), Juan Tu(屠娟), Dong Zhang(章东), and Qingyu Ma(马青玉). Chin. Phys. B, 2026, 35(5): 054302.
[9]	Hierarchical QAOA circuit design framework for distributed quantum computing Ting-Yu Luo(骆挺宇) and Yu-Xin Deng(邓玉欣). Chin. Phys. B, 2026, 35(4): 040309.
[10]	Distributed Kuperberg’s algorithm Peng-Yu Yang(杨鹏宇), Xin Zhang(张新), and Song Lin(林崧). Chin. Phys. B, 2026, 35(3): 030303.
[11]	Preparation of digital-encoded and analog-encoded quantum states corresponding to matrix operations Kaitian Gao(高凯天), Youlong Yang(杨有龙), and Zhenye Du(杜振叶). Chin. Phys. B, 2026, 35(1): 010202.
[12]	Effect of measurement reduction on synthetic aperture x-ray ghost imaging Haipeng Zhang(张海鹏), Jie Tang(汤杰), Nixi Zhao(赵尼西), Changzhe Zhao(赵昌哲), Jianwen Wu(吴建文), Zhongliang Li(李中亮), and Tiqiao Xiao(肖体乔). Chin. Phys. B, 2026, 35(1): 014204.
[13]	Color Fourier single-pixel imaging with random color filter array Jialiang Chen(陈佳亮), Wei Zhu(朱维), Le Wang(王乐), and Shengmei Zhao(赵生妹). Chin. Phys. B, 2025, 34(9): 094210.
[14]	Energy focusing of flexural waves via algorithmically optimized coding metasurface lenses Zi-Rui Wang(王子睿), Di-Chao Chen(陈帝超), Rui Hong(洪瑞), and Da-Jian Wu(吴大建). Chin. Phys. B, 2025, 34(9): 094302.
[15]	Progressive quantum algorithm for maximum independent set with quantum alternating operator ansatz Xiao-Hui Ni(倪晓慧), Ling-Xiao Li(李凌霄), Yan-Qi Song(宋燕琪), Zheng-Ping Jin(金正平), Su-Juan Qin(秦素娟), and Fei Gao(高飞). Chin. Phys. B, 2025, 34(7): 070304.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

An efficient algorithm for shadowing effect simulation under hexagonal lattice-based model

Cite this article:

Online attention