Neural Network Interatomic Potential for Laser-Excited Materials

11 The design of new materials and the prediction of their properties by 12 means of computational techniques has reached an unprecedent level 13 of accuracy thanks to the development and use of machine-learning 14 (ML) approaches. For instance, interatomic potentials based on ML algo-15 rithms have been increasingly used to perform large-scale, ﬁrst-principles 16 quality simulations of materials in the electronic ground state. How-17 ever, they are unable to describe situations in which the electrons are 18

of laser-excited systems by implementing a T e -dependency in the form of an Fig. 1 Modeling laser-excited materials with a Te-dependent neural network interatomic potential. a A femtosecond laser pulse excites the electrons of an atomic structure to high temperatures Te ≫ 300 K. For a few picoseconds, the ions remain cold until the energy starts to transfer from the electrons to the lattice due to electron-phononcoupling. b The Cartesian coordinates of the atoms are described by symmetry functions ⃗ G i , which are passed to the neural network potential together with the electronic temperature Te. The network is evaluated individually for every atom in the structure and the results are summed to obtain the total cohesive energy Φ coh . c The atomic neural network in the case of two hidden layers. The input values are the N sf symmetry function values for atom i and the electronic temperature Te. The network function f NN is also evaluated for ⃗ G i = ⃗ 0 and subtracted from the output value to obtain the atomic cohesive energy Φ i . additional input node to the neural network, which is depicted schematically  In order to determine the models generalization capability and to prevent over-

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In order to test if the T e -NNP is also able to reproduce electronic properties, (2) In Fig. 3b, these two properties are plotted against the electronic tempera-  We perform large-scale MD simulations using our derived T e -dependent inter-248 atomic potential to demonstrate its practical applicability for systems that can 249 not be treated with DFT anymore. We consider a simulation cell containing 250 a Si thin film with more than 160,000 atoms, which is more than 500 times 251 the number of atoms contained in the supercell used for the training dataset.
Thereby, ⃗ G i is a set of atom-centered symmetry functions describing the envi- Here, N hid is the number of nodes in the p-th layer, g (p) is the non-linear acti-323 vation function applied in the p-th layer, w (pq) is the weight matrix connecting 324 layer p with layer q and N sf is the number of symmetry functions. Our neural 325 network model for Φ i is depicted in Fig. 1.

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In order to calculate the total potential energy of a system using Eq. 1 and 327 to correctly reproduce the derivative of Φ tot with respect to T e , one also has 328 to calculate the energy Φ 0 (T e ) of the isolated atoms. In this work, we use an where R ij is the distance from atom i to its j-th neighboring atom and 341 f c (R ij , R c ) is the cutoff function controlling the atomic interaction range, Here, different values of η rad with η rad ∈ R, 0 < η rad ≤ 1 are used, which 343 means that the function is evaluated for different cutoff radii R c = η rad R rad c up 344 to a fixed maximum cutoff radius R rad c . Furthermore, we also take three-body 345 interactions into account by using angular symmetry functions defined as where ζ ∈ N and λ = ±1. Moreover, Θ ijk = acos( ⃗ R ij · ⃗ R ik /R ij ·R ik ) is the angle

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The parameters η rad , η ang , ζ and λ in Eq. (7) and (9)   In this work, we use an atomic neural network with two hidden layers and in the cohesive energy curves at small lattice parameters.

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The cost function for the gradient-based training process is based on the error 380 function used in Ref.
[18] and is defined by Here, E is the reference ab-initio total Helmholtz free cohesive energy and F i   for the cohesive energy as well as the three force components for each atom. 435 We decided to use a comparatively small percentage of the total available 436 data for training to emphasize the data efficiency of our model.

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The thin film used in this study has a thickness of 5.3 nm and consists of  We also showed that the T e -dependent neural network interatomic potential 493 is correctly reproducing the melting temperature of Si given by the underly-