Simulations of thin film sputter deposition require the separation of the plasma and material transport in the gas phase from the growth/sputtering processes at the bounding surfaces (e.g., substrate and target). Interface models based on analytic expressions or look-up tables inherently restrict this complex interaction to a bare minimum. A machine learning model has recently been shown to overcome this remedy for Ar ions bombarding a Ti-Al composite target. However, the chosen network structure (i.e., a multilayer perceptron, MLP) provides approximately 4×106 degrees of freedom, which bears the risk of overfitting the relevant dynamics and complicating the model to an unreliable extent. This work proposes a conceptually more sophisticated but parameterwise simplified regression artificial neural network for an extended scenario, considering a variable instead of a single fixed Ti-Al stoichiometry. A convolutional β-variational autoencoder is trained to reduce the high-dimensional energy-angular distribution of sputtered particles to a low-dimensional latent representation with only two components. In addition to a primary decoder that is trained to reconstruct the input energy-angular distribution, a secondary decoder is employed to reconstruct the mean energy of incident Ar ions as well as the present Ti-Al composition. The mutual latent space is hence conditioned on these quantities. The trained primary decoder of the variational autoencoder network is subsequently transferred to a regression network, for which only the mapping to the particular low-dimensional space has to be learned. While obtaining a competitive performance, the number of degrees of freedom is drastically reduced to 15 111 (0.378% of the MLP) and 486 (0.012% of the MLP) parameters for the primary decoder and the remaining regression network, respectively. The underlying methodology is very general and can easily be extended to more complex physical descriptions (e.g., taking into account dynamical surface properties) with a minimal amount of data required.

1.
F.
Krüger
,
T.
Gergs
, and
J.
Trieschmann
,
Plasma Sources Sci. Technol.
28
,
035002
(
2019
).
2.
G. A.
Bird
,
Molecular Gas Dynamics and the Direct Simulation of Gas Flows
(
Oxford University
,
New York
,
1994
).
3.
M. A.
Lieberman
and
A. J.
Lichtenberg
,
Principles of Plasma Discharges and Materials Processing
, 2nd ed. (
Wiley
,
Hoboken
,
2005
).
4.
W. D. J.
Callister
and
D. G.
Rethwisch
,
Materials Science and Engineering: An Introduction
, 9th ed. (
Wiley
,
Hoboken
,
2013
).
5.
J. P.
Biersack
and
L. G.
Haggmark
,
Nucl. Instrum. Methods
174
,
257
(
1980
).
6.
W.
Eckstein
and
J.
Biersack
,
Nucl. Instrum. Methods Phys. Res., Sect. B
2
,
550
(
1984
).
7.
W.
Möller
and
W.
Eckstein
,
Nucl. Instrum. Methods Phys. Res., Sect. B
2
,
814
(
1984
).
8.
A. F.
Voter
, “Introduction to the Kinetic Monte Carlo method,” in Radiation Effects in Solids, NATO Science Series II, Mathematics, Physics and Chemistry No. 235 (Springer, Dordrecht, 2007).
9.
D. B.
Graves
and
P.
Brault
,
J. Phys. D: Appl. Phys.
42
,
194011
(
2009
).
10.
E. C.
Neyts
and
P.
Brault
,
Plasma Process. Polym.
14
,
1600145
(
2017
).
11.
C. K.
Birdsall
and
A. B.
Langdon
,
Plasma Physics via Computer Simulations
(
IOP Publishing
,
Bristol
,
1991
).
12.
J. V.
Dijk
,
G. M. W.
Kroesen
, and
A.
Bogaerts
,
J. Phys. D: Appl. Phys.
42
,
190301
(
2009
).
13.
V.
Serikov
,
S.
Kawamoto
, and
K.
Nanbu
,
IEEE Trans. Plasma Sci.
27
,
1389
(
1999
).
14.
R. E.
Somekh
,
J. Vac. Sci. Technol. A
2
,
1285
(
1984
).
15.
G. M.
Turner
,
I. S.
Falconer
,
B. W.
James
, and
D. R.
McKenzie
,
J. Appl. Phys.
65
,
3671
(
1989
).
16.
J.
Trieschmann
and
T.
Mussenbrock
,
J. Appl. Phys.
118
,
033302
(
2015
).
17.
E. C.
Neyts
,
Y.
Shibuta
,
A. C. T.
van Duin
, and
A.
Bogaerts
,
ACS Nano
4
,
6665
(
2010
).
18.
E. C.
Neyts
and
A.
Bogaerts
,
Theor. Chem. Acc.
132
,
2141
(
2013
).
19.
R.
Tonneau
,
P.
Moskovkin
,
A.
Pflug
, and
S.
Lucas
,
J. Phys. D: Appl. Phys.
51
,
195202
(
2018
).
20.
M. W.
Thompson
,
Philos. Mag.
18
,
377
(
2016
).
21.
22.
23.
S.
Berg
and
T.
Nyberg
,
Thin Solid Films
476
,
215
(
2005
).
24.
Reactive Sputter Deposition, Springer Series in Materials Science Vol. 109, edited by D. Depla, S. Mahieu, R. Hull, R. M. Osgood, J. Parisi, and H. Warlimont (Springer, Berlin, 2008).
25.
A.
Diaw
et al.,
Phys. Rev. E
102
,
023310
(
2020
).
26.
Z. W.
Ulissi
,
A. J.
Medford
,
T.
Bligaard
, and
J. K.
Nørskov
,
Nat. Commun.
8
,
14621
(
2017
).
27.
H.
Kino
,
K.
Ikuse
,
H.-C.
Dam
, and
S.
Hamaguchi
,
Phys. Plasmas
28
,
013504
(
2021
).
28.
D. P.
Kingma
and
M.
Welling
, “Auto-encoding variational bayes,” arXiv:1312.6114 [stat.ML] (2013).
29.
D. J.
Rezende
,
S.
Mohamed
, and
D.
Wierstra
, “Stochastic backpropagation and approximate inference in deep generative models,” in Proceedings of the 31st International Conference on Machine Learning, Beijing, China (PMLR, 2014), Vol. 32, pp. 1278–1286.
30.
I.
Higgins
,
L.
Matthey
,
A.
Pal
,
C.
Burgess
,
X.
Glorot
,
M.
Botvinick
,
S.
Mohamed
, and
A.
Lerchner
, “beta-VAE: Learning basic visual concepts with a constrained variational framework,” in Proceedings of the 5th International Conference on Learning Representations, Toulon, France (2016); available at https://openreview.net/group?id=ICLR.cc/2017/conference
31.
C. P.
Burgess
,
I.
Higgins
,
A.
Pal
,
L.
Matthey
,
N.
Watters
,
G.
Desjardins
, and
A.
Lerchner
, “Understanding disentangling in β-VAE,” in Proceedings of the 31st Conference on Neural Information Processing Systems, Long Beach, CA, USA (2017); available at https://dblp.org/rec/journals/corr/abs-1804-03599.html
32.
C.
Doersch
, “Tutorial on variational autoencoders,” arXiv:1606.05908 [cs, stat] (2021).
33.
R.
Behrisch
and
W.
Eckstein
, Sputtering by Particle Bombardment, Topics in Applied Physics Vol. 110 (Springer, Berlin, 2007).
34.
H.
Hofsäss
,
K.
Zhang
, and
A.
Mutzke
,
Appl. Surf. Sci.
310
,
134
(
2014
).
35.
D. P.
Kingma
and
J.
Ba
, “Adam: A method for stochastic optimization,” in Proceedings of the 3rd International Conference on Learning Representations, San Diego, CA, USA (dblp, 2015); available at https://dblp.org/rec/journals/corr/KingmaB14.html
36.
G. E.
Hinton
and
R. R.
Salakhutdinov
,
Science
313
,
504
(
2006
).
37.
M.
Abadi
et al., “TensorFlow: An open source machine learning framework for everyone,” see https://tensorflow.org/ (2016).
38.
F.
Chollet
et al., “Keras: The python deep learning library,” see https://keras.io/ (2015).
39.
K.
Sohn
,
H.
Lee
, and
X.
Yan
, “Learning structured output representation using deep conditional generative models,” in Proceedings of the 29th Conference on Neural Information Processing Systems, Montréal, Canada (2015), Vol. 28; available at https://proceedings.neurips.cc/paper/2015/hash/8d55a249e6baa5c06772297520da2051-Abstract.html
You do not currently have access to this content.