Create Neural Network force field of Ag-C system#

This article is based on data provided by Associate Professor Osamu Kurokawa (Graduate School of Engineering, Kyoto University).

Overview#

The behavior of carbon adsorbed on Ag surface was analyzed with a Neural Network force field. First, Neural Network force field was created for structure with multiple C atoms adsorbed on the Ag(111) surface and structure with C atoms located at sublattice points in an atomic layer slightly inside the Ag surface. Monte Carlo simulation was performed using the created Neural Network force field to analyze the distribution of C atoms near the Ag(111) surface.

①Sample Structures#

The following 22 sample structures were used to create training data.

(1) Ag Crystal with FCC structure Super cell models (32 atoms)	(2) Diamond Super cell models (64 atoms)
(3) Graphite Super cell models (72 atoms)	(4) Wurtzite-Type alloy Super cell models (72 atoms)
(5) Rock-salt-Type alloy Super cell models (64 atoms)	(6) Diamond-Type alloy Super cell models (64 atoms)
(7) Ag Crystal w/ point defects Percentage of point defects 10% (29 atoms)	(8) Diamond w/ point defects Percentage of point defects 10% (58 atoms)
(9) Graphite w/ 10% defect Percentage of point defects 10% (65 atoms)	(10) Ag Crystal w/ C@sublattice Ag : C = 4 : 1 (40 atoms)
(11) Ag Crystal w/ C@sublattice Ag : C = 2 : 1 (36 atoms)	(12) Ag(111) surface Slab Model (80 atoms)
(13) Ag(111) slab w/ C@sublattice Ag : C = 4 : 1 (100 atoms)	(14) Ag(111) slab w/ C@sublattice Ag : C = 2 : 1 (108 atoms)
(15) Ag(111) surface w/ C@top Three C atoms on Ag surface (83 atoms)	(16) Ag(111) surface w/ C@bridge Three C atoms on Ag surface (83 atoms)
(17) Ag(111) surface w/ C@hollow Three C atoms on Ag surface (83 atoms)	(18) Ag(111)/Graphite Graphite coverage 100% (186 atoms)
(19) Ag(111)/Graphite Graphite coverage 50% (153 atoms)	(20) Ag(111)/Graphite Graphite coverage 25% (136 atoms)
(21) Ag(334) surface Slab containing step surface (100 atoms)	(22) Ag(334) surface w/ C Atom @hollow Five C atoms on Ag surface (105 atoms)

②Random Structures#

Random structures were generated based on the 22 sample structures. The method and the number of generated random structures were as follows.

For all 22 sample structures, the atomic coordinates were randomly displaced (displacement width: 0.2 Å). →　4,400 structures were generated.
For sample structures consisting only of Ag atoms or C atoms, trajectories of MD calculations using EAM and ReaxFF force fields were included in the random structures. →　1,000 structures were generated.
Random structures were generated by reinforcement learning for the systems with C atoms adsorbed on the Ag slab model and for some of the systems (#13, 14, 19, 20, 22) containing C atoms as sublattices. →　2,000 structures were generated.

Out of the total of 7,400 structures, those for which SCF calculations converged successfully and the maximum force was less than 10 eV/Å were used to train the Neural Network. The number of structures used was 4,744 and the total number of atoms was 630,469.

③Training Data Generation with First-principles Calculations#

Training data was generated with Quantum ESPRESSO. The calculation conditions in SCF calculations were as follows.

#	The Calculation Conditions	set value
1	pseudo-potential	Ultrasoft-Type (Ag.pbe-d-rrkjus.UPF, C.pbe-rrkjus.UPF)
2	Exchange Correlation Functional	GGA-PBE
3	Spin polarization	None
4	Wave function cutoff	25Ry
5	Charge density cutoff	225Ry
6	k point sampling	Only Γ point
7	Smearing	Gaussian (0.01Ry)
8	Convergence threshold	10^-6 Ry
9	lattice constant	Use values obtained from Materials Project

④Neural Network Force Field Creation#

A Neural Network force field was created with the following specifications.

Chebyshev polynomial was used for the symmetry function. The cutoff function was composed of tanh³ and the cutoff radius was 6 Å. The number of symmetry functions was 60 for the radial component and 40 for the angular component.
The structure of the Neural Network was 2 layers x 50 nodes and the activation function was tanh.
A full batch algorithm was applied and optimization was performed using L-BFGS method.
After the calculation of 10000 epochs, the RMSE for energy was 0.027 eV/atom and the RMSE for force was 0.15 eV/Å.

⑤Monte-Carlo Simulation#

Monte-Carlo simulation was performed with Metropolis method using created Neural Network force field.

Analyical Model

The model under analysis was an 8 x 4√2 slab model of an Ag(111) surface (the figure below). The model contained 64 atoms per layer and consisted of 5 layers. The coordinates of Ag atoms were fixed for the bottom two layers of the slab model. In addition, "vacancy sites" were placed at all hollow and top sites on the surface, and at all sub-lattice points between layers 1 and 2 and between layers 2 and 3. These vacancy sites were interchangeable with the coordinates of C atoms in the Metropolis method.

Metropolis Method

The Metropolis method was applied to the model in question and calculations were performed until 2000 structures were generated. The simulation temperature was 300 K. For the atom transition operations, the translation of each atom and the exchange of C atoms and vacancy sites were taken into account. The width of the atomic translations was 0.2 Å. The number of C atoms was 2~64. Since there were 64 top sites, there were multiple variations in the initial position setting when the number of C atoms was less than 64. Therefore, for each number of C atoms, 16 patterns of initial positions were randomly generated and simulation was performed for each pattern. The average of 16 patterns was used as calculation results.

Simulation video

Videos of Monte-Carlo simulations are shown below. As examples, movies for cases with 16, 32, and 64 C atoms are shown. In all cases, it could be seen that the C atoms placed at the top site had entered the sublattice inside Ag. In the case of 32 and 64 C atoms, it could be confirmed that the excess C atoms were forming bonds with each other on the Ag surface.

C atoms＝16

C atoms＝32

C atoms＝64

Location of C atoms

The number of C atoms in each position relative to the total number of C atoms is shown in the graph below. The C atoms between the first and second layers of the Ag slab model are referred to as "at surface", and those between the second and third layers are referred to as "at next sub-surface". The number of these C atoms was counted in the final structure in Monte-Carlo simulation, which was the average of 16 simulation patterns with different initial structures.

The position of C atoms in Monte-Carlo simulation

C atoms existed on the surface or the next sub-surface and rarely on the sub-surface.
When the number of C atoms was smaller than 20, C atoms were preferentially in the next sub-surface.
When the number of C atoms was greater than 20, the number of C atoms in the next sub-surface tended to decrease slightly. Instead, the number of C atoms on the surface increased linearly.
The limit number of C atoms that could enter the next sub-surface was about 8, and all of excess C atoms above that existed on the surface.