Efficient and transferable machine learning potentials for the simulation of crystal defects in bcc Fe and W

Data-driven, or machine learning (ML), approaches have become viable alternatives to semiempirical methods to construct interatomic potentials, due to their capacity to accurately interpolate and extrapolate from first-principles simulations if the training database and descriptor representation of atomic structures are carefully chosen. Here, we present highly accurate interatomic potentials suitable for the study of dislocations, point defects, and their clusters in bcc iron and tungsten, constructed using a linear or quadratic input-output mapping from descriptor space. The proposed quadratic formulation, called quadratic noise ML, differs from previous approaches, being strongly preconditioned by the linear solution. The developed potentials are compared to a wide range of existing ML and semiempirical potentials, and are shown to have sufficient accuracy to distinguish changes in the exchange-correlation functional or pseudopotential in the underlying reference data, while retaining excellent transferability. The flexibility of the underlying approach is able to target properties almost unattainable by traditional methods, such as the negative divacancy binding energy in W or the shape and the magnitude of the Peierls barrier of the $\frac{1}{2}\langle 111\rangle$ screw dislocation in both metals. We also show how the developed potentials can be used to target important observables that require large time-and-space scales unattainable with first-principles methods, though we emphasize the importance of thoughtful database design and degrees of nonlinearity of the descriptor space to achieve the appropriate passage of information to large-scale calculations. As a demonstration, we perform direct atomistic calculations of the relative stability of $\frac{1}{2}\langle 111\rangle$ dislocations loops and three-dimensional C15 clusters in Fe and find the crossover between the formation energies of the two classes of interstitial defects occurs at around 40 self-interstitial atoms. We also compute the kink-pair formation energy of the $\frac{1}{2}\langle 111\rangle$ screw dislocation in Fe and W, finding good agreement with density functional theory informed line tension models that indirectly measure those quantities. Finally, we exploit the excellent finite-temperature properties to compute vacancy formation free energies with full anharmonicity in thermal vibrations. The presented potentials thus open up many avenues for systematic investigation of free-energy landscape of defects with ab initio accuracy.

Figure 1

Histogram distribution of the DFT data for energy (in eV), force (in eV/Å), and error noise deviation of LML and QNML force fields on W database. The absolute DFT energy is presented using rescaled form $(E-\overline{E})/\sigma$ where $\overline{E}$ is the average energy of the database and $\sigma$ is the standard deviation. The error distribution for energy and force has Gaussian shape. The distribution is narrower for QNML force field.

Figure 2

Formation energies of vacancy clusters $V_n$ (from $n=1$ to 10 vacancies) in bcc W computed with LML, QNML, GAP14 [94] and GAP19 [106] potentials compared with the DFT GGA-PBE values from this work. (a) Correlation between the formation energies computed with ML potentials and with DFT. The dashed black line indicates a perfect correlation. (b) Binding energies of the vacancy clusters $E_{\text{b}}^n=E_{\text{f}}^n-n\times E_{\text{f}}^1$, where $E_{\mathrm{for}}^n$ and $E_{\text{f}}^1$ are the formation energies of a cluster with $n$ defects and of a single defect, respectively. A negative value of the binding energy implies energetic instability and dissociation of the defect cluster. DFT-SC and DFT-noSC correspond to the DFT calculations using the pseudopotentials with and without semicore states, respectively.

Figure 3

Migration barriers of trivacancy clusters $V_3$ in (a) bcc Fe and (b) bcc W. The comparison is made between the LML and QNML potentials, GAP [93, 94, 106], and some commonly used EAM potentials for Fe [81, 83, 115] and W [79, 83]. The gray dotted lines labeled with $V_{1-3}$ refer to the energy barriers for migration of monovacancies, divacancies, and trivacancies from DFT calculations in Fe [81] and W [79]. The saddle-point structure in the inset plot of (a) is detected using the distortion score of local atomic environments [14]. The atoms are colored according to the magnitude of the distortion score (yellow, high; green, medium; blue, low).

Figure 4

Peierls barriers of the $\frac{1}{2}\langle 111\rangle$ screw dislocation gliding in the {110} plane (a) in Fe and (b) in W. The comparison is made between the LML potentials, DFT calculations [86, 87], GAP [93, 94, 106], and some commonly used EAM potentials for Fe [115, 126] and W [83]. The higher Peierls barrier in bcc Fe predicted by GAP18 [93] results from different DFT calculations in the training database of the potential. The dislocation core structures in the inset plot are detected using the distortion score of local atomic environments [14]. The color of the atoms is set according to the magnitude of the distortion score (yellow, high; green, medium; blue, low).

Figure 5

Formation energies of C15 clusters and interstitial $\frac{1}{2}\langle 111\rangle$ loops in bcc Fe from LML and QNML potentials and DFT calculations. (a) The relative stability of C15 clusters and $\frac{1}{2}\langle 111\rangle$ loops. The shaded gray area between 35 and 45 SIAs represents the crossover size predicted by our ML potentials, the purple shaded area between 40 and 49 SIAs is the crossover predicted by the discrete-continuum model (DC) [71]. (b) Formation energies of C15 clusters. (c) Formation energies of $\frac{1}{2}\langle 111\rangle$ loops. In all the subplots,the $x$ axis corresponds to the number of SIAs; open circles and diamonds represent the energies of $\frac{1}{2}\langle 111\rangle$ loops and C15 clusters, respectively. The color of points is explained in the legends in (b) and (c). The DFT-RA calculations are taken from Ref. [71] and are the calculations used in the parametrization of the DC model.

Figure 6

The variation $\mathrm{\Delta }{a}_0\left(T\right)$ of the lattice parameter $a_0\left(T\right)$ in bcc Fe and W at the temperature relative to the experimental value $a_0^{\text{expt}}\left(0\right)$ at $T=0$ K [139]. The variation $\mathrm{\Delta }{a}_0\left(T\right)=a_0^{\text{ML}}\left(T\right)-a_0^{\text{expt}}\left(300\right)$. We report the interpolation of the experimental values [139] (black lines) and the corresponding values of the present LML and QNML potentials (green circles). The numerical values for ML potentials are computed using the evaluation of the minimal free energy of the bcc systems using thermodynamic integration.

Figure 7

The anharmonic free energy of monovacancy formation at zero pressure as a function of temperature computed using LML and QNML potentials. The dashed lines are the vibrational formation free energy based on the values of zero-pressure vacancy formation entropy from self-diffusion experiments [119] (see the text for more details).

Figure 8

Principal component analysis (PCA) representation of the databases for (a) bcc Fe and (b) bcc W. Each point on the plot corresponds to an atomic environment. The depicted atomic environments are outliers with respect to defect-free bcc bulk structures (see text), as provided by the distortion scores of LAEs [14]. Different colors represent different classes of structures in the database. The same color coding is used for Fe and W, with two additional classes, fcc and $\gamma$ surfaces, present in W.