From first principles calculations to statistical physics and simulation of magnetic materials

From first principles calculations to statistical physics and simulation of magnetic materials Markus Eisenbach Oak Ridge National Laboratory This work was sponsored in parts by the Center for Defect Physics, an Energy Frontier Research Center funded by the US DOE Office of Basic Energy Sciences and by the US DOE Offices of Advanced Scientific Computing Research and Basic Energy Sciences. This research used resources of the Oak Ridge Leadership Computing Facility at ORNL, which is supported by the US DOE, Office of Science.

Technological Opportunities in Nano-Magnetism Motivation Ø Magnetic recording/memory: Blocking temperature Super-paramagnetism Ø Magnetic refrigeration: Curie temperature High entropy change Brass bound lodestone Ferrite block NdFeB magnet Ø Permanent magnets: Curie temperature Magnetic moment Large Energy product Issue Ø Understanding and controlling magnetism at the nanoscale: R Harris, A.J. Williams Materials World, Vol. 7 No. 8 pp. 478-81, August 1999. Managed by UT-Battelle for the U.S. Department of Energy

Oak Ridge National Laboratory Tennessee Spallation Neutron Source Oak Ridge Leadership Computing Facility OAK RIDGE NATIONAL LABORATORY U. S. DEPARTMENT OF ENERGY 3

Opportunities for Modeling Magnetic Nanostructures Spallation Neutron Source Center for Nanophase Materials Sciences Synthesis Characterization Fe 5 nm ~12,000 atoms ~4,000 surface atoms Managed by UT-Battelle for the U.S. Department of Energy

Opportunities for Modeling Magnetic Nanostructures Spallation Neutron Source Center for Nanophase Materials Sciences Opportunity Synthesis Fe Characterization Simulation size Real device size Need First Principles approaches capable 5 nm of simulating ~10 4 atoms Methods for dealing with complex magnetic ground states Approaches for dealing ~12,000 with finite atoms temperature statistical physics ~4,000 surface atoms Managed by UT-Battelle for the U.S. Department of Energy

Length & Time Scales in Magnetism Time Scale (s) 10-6 10-9 10-12 10-15 Loss of detailed physics Increasing computational complexity Micromagnetics Spin Models Quantum 10-10 10-8 10-6 10-4 Length Scale (m) Managed by UT-Battelle for the U.S. Department of Energy

Surface Magnetic Nanostructures Co-chains at a Pt(111) Step Edge Ø STM image & magnetic measurements Magnetic moment is canted towards step θ θ Expt =43 o Expt: P. Gambardella et al., Nature 416, 301 (2002) Energy Scales Ø Exchange and Magneto-crystalline anisotropy treated on same footing E Tot ~10 3 Ry E Coh ~10-1 Ry t=1 E Exch ~10-3 Ry E MAE ~10-6 Ry Managed by UT-Battelle for the U.S. Department of Energy

Modeling surface nanostructures Extended impurity Green s function embedding Ø Fully relativistic theory (Dirac equation) Ø Screened KKR layer approach for pristine Pt (111)-surface c - Sites in region to be replaced Ø Green s function embedding method for extended impurity Vacuum Semi-Infinite Solid Managed by UT-Battelle for the U.S. Department of Energy

Modeling surface nanostructures Extended impurity Green s function embedding Ø Fully relativistic theory (Dirac equation) Ø Screened KKR layer approach for pristine Pt (111)-surface c - Sites in region to be replaced Ø Green s function embedding method for extended impurity Vacuum Extended Impurity Semi-Infinite Solid Managed by UT-Battelle for the U.S. Department of Energy

Modeling surface nanostructures Extended impurity Green s function embedding Ø Fully relativistic theory (Dirac equation) Ø Screened KKR layer approach for pristine Pt (111)-surface c - Sites in region to be replaced Ø Green s function embedding method for extended impurity Vacuum Extended Impurity Semi-Infinite Solid Model geometry Ø 7 Co atoms embedded into a cavity of the Pt(111) surface layer Managed by UT-Battelle for the U.S. Department of Energy

Modeling surface nanostructures Extended impurity Green s function embedding Ø Fully relativistic theory (Dirac equation) Ø Screened KKR layer approach for pristine Pt (111)-surface c - Sites in region to be replaced Ø Green s function embedding method for extended impurity Vacuum Extended Impurity Semi-Infinite Solid Model geometry Ø 7 Co atoms embedded into a cavity of the Pt(111) surface layer z y θ y φ y x Managed by UT-Battelle for the U.S. Department of Energy

Managed by UT-Battelle for the U.S. Department of Energy

Co-Chain at a Pt(111) Step: SD t=1 Managed by UT-Battelle for the U.S. Department of Energy

Co-Chain at a Pt(111) Step: SD t=1 t=10 t=25 t=100 t=800 Managed by UT-Battelle for the U.S. Department of Energy

Co-Chain at a Pt(111) Step: SD Full convergence in about 800 time steps t=1 t=10 t=25 t=100 t=800 Managed by UT-Battelle for the U.S. Department of Energy

Co-Chain at a Pt(111) Step: SD Full convergence in about 800 time steps t=1 t=10 t=25 t=100 t=800 θ θ Calc =42 o θ Expt =43 o Expt: P. Gambardella et al., Nature 416, 301 (2002) Managed by UT-Battelle for the U.S. Department of Energy

LSDA and Finite Temperature Magnetism LSDA is theory of ground states Correct magnetic ground state FM AFM NM Moment (µ B ) Fe Ni Calc. LSDA 2.17 0.59 Expt. 2.08 0.57 ~0.5eV

LSDA and Finite Temperature Magnetism LSDA is theory of ground states Correct magnetic ground state FM AFM NM Moment (µ B ) Fe Ni Calc. LSDA 2.17 0.59 Expt. 2.08 0.57 LSDA magnetism at 7inite temperature T C (K) Fe Ni Stoner 20,000 10,000 Expt. 1040 640 ~0.5eV T=0K; FM

LSDA and Finite Temperature Magnetism LSDA is theory of ground states Correct magnetic ground state FM AFM NM Moment (µ B ) Fe Ni Calc. LSDA 2.17 0.59 Expt. 2.08 0.57 ~0.5eV LSDA magnetism at 7inite temperature T C (K) Fe Ni Stoner 20,000 10,000 PM Disordered Local Moment (DLM) state Expt. 1040 640 T C (K) Fe Ni DLM 1015 450 Expt. 1040 640 Missing Longitudinal Fluctuations T=0K; FM

Constrained Local Moments LSDA is theory of ground states Ø General non-collinear state is not a ground state Ø Local moment orientation New degrees of freedom e i i dr m M ( r) Ω = i dr m ( r) Ω i Ø Define general orientational state Set of orientations Selfconsistentcy of the constraint is enforced by ρ δ E i M ( r) = Tr ˆ ρ( r) ( ) = Trσρ ˆ ˆ ( ) M r e e r con " # i $ % & i OAK RIDGE NATIONAL LABORATORY U. S. DEPARTMENT OF ENERGY ρ i e $ i % & ' ( r) ; M( r) ; B ( r) $ con % & ' δ e = 0, i Orientational State e " # i $ % & 11

A parallel implementation and scaling of the LSMS method: perfectly scalable at high performance 1 2 3 j j m i k l N-2 N-1 N Need only block i of A C B D 1 = (A BD 1 C) 1 m k Calculation dominated by ZGEMM Sustained performance similar to Linpack l

13 Wang-Landau Method F. Wang and Landau, PRL 86, 2050 (2001) Conventional Monte Carlo methods calculate expectation values by sampling with a weight given by the Bolzmann distribution In the Wang-Landau Method we rewrite the partition function in terms of the density of states which is calculated by this algorithm Z( ) = { i } To derive an algorithm to estimate g(e) we note that states are randomly generated with a probability proportional to 1/g(E) each energy interval is visited with the same frequency (flat histogram) OAK RIDGE NATIONAL LABORATORY U. S. DEPARTMENT OF ENERGY e H({ i }) = g 0 g(e)e E de

Metropolis Method Wang-Landau Method Metropolis et al, JCP 21, 1087 (1953) F. Wang and Landau, PRL 86, 2050 (2001) Z = e E[x]/kBT dx Z Z = g(e)e E/kBT de Compute partition function and other averages with configurations that are weighted with a Boltzmann factor Sample configuration where Boltzmann factor is large. 1. Select configuration E i = E[x i ] 2. Modify configuration (move) E f = E[x f ] 3. Accept move with probability A i f = min{1,e β(e i E f ) } If configurations are accepted with probability 1/g all energies are visited equally (flat histogram) 1. Begin with prior estimate, eg 2. Propose move, accepted with probability 3. Increase DOS at new energy 4. Iterate 2 & 3 until histogram is flat f f = f g 0 (E) =1 A if =min{1,g 0 (E i )/g 0 (E f )} g 0 (E f ) g 0 (E f ) f f>1 5. Reduce and go back to 2

Metropolis Method Wang-Landau Method Z Z = e E[x]/k BT dx Z = g(e)e E/k BT de Sample configuration space with probability e E[x]/k BT 1/g(E[x]) Samples mainly regions around energy minima Samples all energies equally -

16 Wang-Landau vs Metropolis [same CPU time] Specific heat, Cv / N 1.0 0.8 0.6 0.4 0.2 Metropolis (10 8 trials each run) Wang-Landau 0.0 0.0 0.2 0.4 0.6 0.8 Effective Temperature, k B T / ε HH OAK RIDGE NATIONAL LABORATORY U. S. DEPARTMENT OF ENERGY HP Protein model [Ying Wai Li]

Not quite embarrassingly parallel Metropolis MC acceptance: A i f =min{1,e β(e i E f ) } random walker 1 random walker 2

Not quite embarrassingly parallel Metropolis MC acceptance: A i f =min{1,e β(e i E f ) } Wang-Landau acceptance:a if =min{1,e [g(x f ) g(x i )] } random walker 1... global update of joint DOS at every MC step random walker 2 limited by latency ~ microseconds local calculation of energy and observable ~ millisecond to minutes

Organization of the WL-LSMS code using a master-slave approach Communicate moment directions and energy Master/driver node controlling WL acceptance, DOS, and histogram LSMS running on N processors to compute energy of particular spinconfigurations

Sustained performance of WL-LSMS on Cray XT5 1.836 Petaflop/s on 223,232 cores 1.755 Petaflop/s on 223,752 cores 829 Teraflop/s on 100,008 cores

What it takes to compute a converged DOS on a Cray XT5 1000 800 16 atoms (2x2x2 unit cells) 60000 50000 250 atoms (5x5x5 unit cells) ln g(e) 600 400 ln g(e) 40000 30000 20000 200 10000 0 69.05 69.1 69.15 69.2 69.25 69.3 69.35 E [Ry] 0 1079 1079.5 1080 1080.5 1081 1081.5 1082 E [Ry] 16 atoms 250 atoms WL walkers 200 400 total cores 3,208 100,008 WL samples 23,200 590,000 CPU-core hours 12,300 4,885,720 Just 2 days

Wang-Landau Magnetic Phase Transition: Fe 5x5x5 cell: Curie temperature of Fe from first principles. T C =980K (Exp: 1043K) The internal energy jumps at the Curie temp. leading to a spike in the specific heat. Managed by UT-Battelle for the U.S. Department of Energy

22 Cementite (Fe 3 C) Important constituent of carbon steel and cast iron Orthorombic (Pnma) Unit cell: 16 atoms: 4 C 4 + 8 Fe Picture: Wikipedia Commons OAK RIDGE NATIONAL LABORATORY U. S. DEPARTMENT OF ENERGY

23 Cementite (Fe 3 C) 600 Wang-Landau DOS Specific Heat [Ry/supercell/K] 500 0.08 ln g(e) 400 300 200 Speci c Heat 0.06 0.04 0.02 100 0 556.8 557 557.2 557.4 557.6 557.8 558 E [Ry] 0 300 350 400 450 500 550 600 Temperature [K] Cementite 128 atom unit cell leads to a Tc of 425K OAK RIDGE NATIONAL LABORATORY U. S. DEPARTMENT OF ENERGY

24 Comparison with Cementite (Fe3C) exp Curie Temperature calc Curie Temperature Fe 1050K 980K Fe3C 480K 425K Tc(Fe)/ Tc(Fe3C) 2.2 2.3 J. Appl. Phys. 109, 07E138 (2011) OAK RIDGE NATIONAL LABORATORY U. S. DEPARTMENT OF ENERGY

Conclusions (I) It is now possible to compute free energies in nanoscale systems - using ab initio methods based on Density Functional Theory - fully taking into account entropy Ab initio calculation of ferromagnetic transition temperature in Fe that does not rely on mean-field approximation - LDA answer based on WL-LSMS underestimate Tc by (only) 7% Future: Improve WL sampling methods - bin-less Wang-Landau - perturbative improvements ( warping ) of model based density of states using first principles calculations. Perturbation calculation of thermodynamic density of states, G. Brown, T. C. Schulthess, D. M. Nicholson, M. Eisenbach, G. M. Stocks, Phys Rev. E 84, 061116 (2011). - Convergence improvements of the WL sampling Convergence for the Wang-Landau Density of States, G. Brown, Kh. Odbadrakh, D. M. Nicholson, M. Eisenbach, Phys. Rev. E 84, 065702(R) (2011).

26 Effect of lattice vibration on magnetic phase transition in BCC Iron Although BCC Fe is not an ideal Heisenberg magnet [1], Critical exponents Classical Heisenberg Model [2] Experiment [3] α -0.118(11) -0.12(1) β 0.3669(32) 0.363(4) γ 1.385(10) 1.33(2) [1] V. Heine etc, Europhys. Lett. 12 (1990) 545; A.V. Ruban etc, Phys. Rev. B 75.(2007 ) 054442 [2] K. Chen etc, Phys. Rev. B 48 (1993) 3249 [3] L.W. Shacklette, Phys. Rev. B 9 (1974) 3789; N. Stusser etc, 31 (1985) 5905 OAK RIDGE NATIONAL LABORATORY U. S. DEPARTMENT OF ENERGY

Model MD generated configuration with displacement at T = 900K Sample magnetic configurations using WL-LSMS method [1] Fit J values from J ij = 1 4 (Eij + Eī ȷ Eīj E i ȷ ) [1] M. Eisenbach etc, J. Appl. Phys. 109 (2011) 07E138

Model H = U({r n }) X J ij ({r n })ŝ i ŝ j i6=j J(r ) is a cubic function of separation distance. J(r, v, d) is a function of both separation distance and atomic volume v and volume distortion d. d i = X k2nn(i) r k r i r 0 k + r 0 i d is the displacement of nearest neighbors from the ideal crystal position

29 Specific Heat OAK RIDGE NATIONAL LABORATORY U. S. DEPARTMENT OF ENERGY

30 Magnetization OAK RIDGE NATIONAL LABORATORY U. S. DEPARTMENT OF ENERGY

Magnetism at dislocations Dislocations and other defects are important for the understanding of the behavior of structural materials (e.g. Steel, Ni alloys) Magnetism might play a role in the dynamics of dislocations and its effect has not been investigated in detail yet. (Most force fields used in molecular dynamics don t include explicit magnetism) Screw dislocation OAK RIDGE NATIONAL LABORATORY U. S. DEPARTMENT OF ENERGY 31

increase in the moment is a coordination effect, not a volumetric effect. We see in Fig. 4 that the atomic volume of atoms near the core is very near the average atomic volume, 78.93 a.u. In addition to the coordination effect at the core, there is a volume or strain effect between the dislocations. The effect is either compressive or dilational, depending on the handedness of the neighboring dislocations. The moments under compressive strain are reduced and those in dilation are enhanced, as is expected from the typical rela- Screw Dislocation in bcc Fe with LSMS magnetic moments and local volumes 07E159-3 Odbadrakh et al. J. Appl. Phys. 109, 07E159 (2011) bulk tiona from cates lated caus valu core trost addr in re Fully axis held ques from ACK tural the Offic Ridg Labo Depa 1 M. S S. L 3 C. W 4 B. U 5 Y. W 6 S. Is 7 S. Pl 8 M. I 2 3. (Color) local moments are shown for the quadrapole of screw OAKFIG. RIDGE NThe ATIONAL LABORATORY dislocations. U. S. DEPARTMENT OF ENERGY FIG. 5. (Color online) Distribution of local moments from the center of the simulation supercell. FIG. 4. (Color) The Voronii volumes of each site are shown for the quadrapole of screw dislocations. 32

Local Magnetization N = 9240 T = 965 K M bulk = 0.189336

Combined MD- SD Simula1ons of BCC Iron

Equilibra1on via la:ce thermostat Combined MD- SD Simula1ons of BCC Iron

Conclusion (II) l Distribution of local magnetic moment is not uniform but affected by atomic environment parameters. l In BCC Fe, the effect of atomic distance and local volume cancel each other, and an uniform interaction coupling already works well. l Possible applications to dislocation systems.

Team Oak Ridge National Laboratory Markus Eisenbach Ying Wai Li University of North Carolina at Asheville Donald M. Nicholson University of Tennessee Khorgolkhuu Odbadrakh Junqi Yin Florida State University Gregory Brown University of Georgia David Landau Dilina Perera