Accuracy and transferability of GAP models for tungsten

Transcription

1 Accuracy and transferability of GAP models for tungsten Wojciech J. Szlachta Albert P. Bartók Gábor Csányi Engineering Laboratory University of Cambridge 5 November 214

2 Motivation Number of atoms ms Finite element 1 µs Qualitative accuracy Coarse -grained Time scale 1ns 1ps TB Empirical DFT.1 ev accuracy 1fs QMC 1 Å 1 nm 1 nm 1 nm 1 µm Length scale Calculations from first principles lead to predictive capabilities that allow discovering novel materials with desired properties. Need for quantum mechanical accuracy, but with no electrons!

3 Empirical Potentials Typical interatomic potentials: fixed functional form with a set number of adjustable parameters (i.e. empirical, analytical formula) parameters fit to reproduce arbitrarily chosen target properties E = NX NX NX V 1 (x (i) )+ V 2 (x (i), x (j) ) i i j j<i {z } bonds NX NX NX + V 3 (x (i), x (j), x (k) ) +... i j k j<i k<j {z } angles

4 Empirical Potentials E = N P i i = N P j i A r 12 B r 6 Energy, Repulsive + A r 12 Attractive B r 6 r Atomic distance, r Lennard-Jones potential: choose target properties fit parameters A and B to reproduce target properties

5 Gaussian Approximation Potential Apply Bayesian probability / machine learning to infer underlying function Bayes theorem posterior = likelihood prior marginal likelihood Start with prior distribution over functions Set of target values is observed Calculate posterior distribution for any new prediction with mean i and variance 2 i

6 Gaussian Approximation Potential Interpolation: Gaussian process regression A. P. Bartók, M. C. Payne, R. Kondor, and G. Csányi, Phys. Rev. Lett. 14, (21) QM is inherently many dimensional large function space, overfitting Representation: Smooth overlap of atomic positions A. P. Bartók, R. Kondor, and G. Csányi, Phys. Rev. B 87, (213) faithful representation of atomic environments rotational & permutational invariance, smoothness Database: Tungsten interatomic potential W. J. Szlachta, A. P. Bartók, and G. Csányi, Phys. Rev. B 9, 1418 (214) domain specificity predictive power

7 Gaussian Process Regression Optimal way of interpolating many-dimensional functions No fixed functional form Number of free parameters not fixed i (q i )= X j covariance function z } { j k(q j, q i ) 2 i (q i )=k(q i, q i ) X j jk(q j, q i ) Covariance function k(q j, q i ) Prior distribution over functions is determined by choice of the covariance function.

8 Gaussian Process Regression Optimal way of interpolating many-dimensional functions No fixed functional form Number of free parameters not fixed i (q i )= X j covariance function z } { j k(q j, q i ) 2 i (q i )=k(q i, q i ) X j jk(q j, q i ) Training data { j, q j } j Fit quality determined by both training data and choice of the covariance function.

9 Gaussian Process Regression Optimal way of interpolating many-dimensional functions No fixed functional form Number of free parameters not fixed i (q i )= X j covariance function z } { j k(q j, q i ) 2 i (q i )=k(q i, q i ) X j jk(q j, q i ) Prediction mean i (q i ) and variance 2 i (q i ) Resulting potential is only as good as the training data!

10 Gaussian Process Regression (Notes) i (q i )= X j j k(q j, q i ) Calculation of (and ) requires inversion of the covariance matrix K: K ij = k(q i, q j ) =(K + I) 1 y Linear regression can be quickly recovered with dot product covariance: K ij = w 2 q i q j i (q i )= q i

11 Gaussian Process Regression

12 Smooth Overlap of Atomic Positions Need rotationally and permutationally invariant description of atomic environment! ({x j x i } N j )! ( i ) i (r) = X j e r r ij 2 /2 2 f cut ( r ij )

13 Smooth Overlap of Atomic Positions ( i )! (q i ) i (r) = X nlm c i nlm g n( r )Y lm (ˆr ) We can now define: ( ) X q i = (c i nlm) c i n lm m nn l ˆq i = q i/ q i And we can demonstrate that: K ij = k(q i, q j) = 2 w ˆq i ˆq j! Z d ˆR Z dr i(r) j( ˆRr) 2 = k( i, j)

14 Gaussian Approximation Potential (Caveats) Atomic energies cannot be directly computed from QM data! Training from total energies, forces and stresses Inferring function (q i ) from linear combination of its values / partial derivatives (and atomic positions) possible. NX total energies: E = i i X N atomic forces: {f (i) = r (i) j } N i j NX stress virials: = x (i) NX j j Pseudo training points to deal with large data sets e ciently (which optimally represent the underlaying teaching data).

15 Tungsten Interatomic Potential Non-magnetic BCC transition metal with the highest melting temperature (368 K). Often thought of as a prototype for this class of elements - good toy model before attempting iron!

16 GAP Databases Database: Computational cost [ms/atom] Elastic constants [GPa] Phonon spectrum [THz] Vacancy formation [ev] Surface energy [ev/å 2 ] Dislocation structure [Å 1 ] Dislocation-vacancy binding energy [ev] Peierls barrier [ev/b] GAP 1 : 2 primitive unit cell with varying lattice vectors GAP 2 : GAP atom unit cell GAP 3 : GAP 2 + GAP 4 : GAP 3 + GAP 5 : GAP 4 + GAP 6 : GAP 5 + vacancy in: 4 53-atom unit cell, atom unit cell (1), (11), (111), (112) surfaces atom unit cell (11), (112) gamma surfaces atom unit cell vacancy in: (11), (112) gamma surface atom unit cell 1 2h111i dislocation quadrupole atom unit cell

17 Elastic Constants & Lattice Defects 15% 5% GAP BOP MEAM FS C11 C12 C44 Elastic const. Vacancy energy (1) (11) (111) (112) Surface energy

18 Elastic Constants & Lattice Defects 15% 5% GAP BOP MEAM FS C11 C12 C44 Elastic const. Vacancy energy (1) (11) (111) (112) Surface energy

19 Phonon Spectrum 8 a) 6 4 Frequency [THz] 2 6 b) 4 2 GAP FS DFT H N P HP N

20 Stress-Strain Curves 8 4 a) longitudinal, xx vs. xx -4-8 Stress [GPa] 2-2 b) transverse, yy vs. xx -4 1 c) shear, yz vs. yz GAP -1 FS DFT Strain

21 Towards Description of Plasticity Plasticity behaviour largely attributed to lattice crystallography Dominant dislocation type in bcc metals is 1 2h111i screw h11i dislocations observed, but believed to be product of the dominant 1 2h111i screw dislocations b k disloc. line b? disloc. line Screw dislocation Edge dislocation

22 Screw Dislocation Core Structure Screw dislocation core structure determined by the properties of the interatomic potential and the boundary conditions Dislocation mobility is a direct consequence of the core structure energetics 8 Å 4 Å -4 Å Non-symmetric core Symmetric core -8 Å

23 Screw Dislocation Peierls Barrier.5 GAP, r= r=.25 r=.5 r=.75 r=1. Nye tensor [Å 1 ] (screw comp.) FS, r= r=.25 r=.5 r=.75 r=1..3 DFT, r= GAP, quadrupole(135at.) GAP, quadrupole(459at.) GAP, quadrupole(1215at.) GAP, cylinder(33633at.) FS, cylinder(33633at.) DFT, quadrupole(135at.).2 Energy [ev/b] Reaction coordinate, r

24 Dislocation-Vacancy Binding Energy GAP FS (112) (112) (11)+ (11)+ (11) (11) Binding energy [ev] GAP FS (11)+ -.5 (11) -.5 (112) Å -1 Å -5 Å 5 Å 1 Å 15 Å

25 Further Work Screw dislocation kink in tungsten Other defects/phases are also on the way Automatic generation of GAP databases for other elements Code and data available at:

26 Thank you for your attention! More details in: W. J. Szlachta, A. P. Bartók, and G. Csányi, Phys. Rev. B 9, 1418 (214)

27

28 Appendix Compute many-body atomic energy function i = ({x j x i } N j )= (q i ) Fit to arbitrary precision QM data Need rotationally and permutationally invariant description of atomic environment! Decriptor vs. covariance function symmetries k(q j, q i ) +! k ({x x k j } k, {x k x i } k ) {x k x i } k! q i

29 Appendix Bispectrum expand atomic density using 4d spherical harmonics basis Smooth Overlap of Atomic Positions expand atomic density using 3d spherical harmonics and radial basis atoms represented by atoms represented by Dirac function Gaussian function square exponential covariance dot product covariance

30 Appendix Both bispectrum and SOAP are based on expansion in spherical harmonics that needs to be truncated: Fourier / bispectrum space Real space Mapping from {x j x i } j to q i should be an invertible function such that both the function and its inverse are smooth!

31 Appendix Calculation of (and matrix K: ) requires inversion of the covariance K ij = k(q i, q j ) =[K MM + K MN L 1 L T K NM ] 1 K MN L 1 y Where: K DD = L T K NN L = 2 I

32 Appendix Database: Total GAP GAP GAP GAP GAP GAP

33 Appendix Slip can occur along nearest neighbour direction 1 2 h111i (also the shortest Burgers vector) The most densely packed planes of the h111i zone are the {11} planes (111) plane {11} (11) plane {112} h112i 1 2 h111i h111i

34 Appendix Slip plane separation distance h111i{11}! 1 p 2 a h111i{112}! 1 p 6 a h111i{123}! 1 p 14 a h111i{134}! 1 p 26 a

35 Appendix.5 8 Å Screw component (Nye tensor [Å 1 ]) 4 Å -4 Å.1-8 Å Edge component (Nye tensor [Å 1 ]) Nye tensor Displacement map

36 Appendix Dislocation Quadrupole Quadrupolar arrangement b 2 Dipole simulation cell b 1 Active region (atomic positions relaxed) Inactive region (atomic positions fixed) Vacuum Isolated Dislocation

37 Appendix hard lattice site Path A Path B Path C soft lattice site

38 Appendix Energy [ev/å 2 ] I-S-GAP FS/b-GAP FS DFT 1 8 [111] 1 4 [111] 3 8 [111] 1 2 [111]

39 Appendix 5 4 I-S-GAP FS/b-GAP FS DFT Energy [ev/å] [111] 1 2 [111] 1 4 [111] 1 4 [111] 1 2 [111] 3 4 [111]

40 Appendix Log1 core local energy error [log(ev)] at at at. 537 at at.1893 at. Dislocations annihilate (separation distance too small) FS quadrupole FS cylinder Average dislocation separation distance [Å]

41 Appendix Log1 core local energy error [log(ev)] at at at. 537 at at.1893 at. S-GAP quadrupole S-GAP cylinder Average dislocation separation distance [Å]

42 Appendix at at at. S-GAP cylinder Binding energy [ev] Cylinder depth