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Ab initio Berechungen für Datenbanken Jörg Neugebauer University of Paderborn Lehrstuhl Computational Materials Science Computational Materials Science Group CMS Group
Scaling Problem in Modeling length (m) 1 10-3 10-6 10-9 dislocations interfaces point defects chemical bonds Phenomenological approaches: FEM materials properties Phase fields crystal growth : growth mechanism microstructure formation 10-15 10-9 10-3 1 10 3 time (s)
Scaling Problem in Modeling length (m) 1 10-3 10-6 10-9 dislocations interfaces point defects chemical bonds Free energy: growth mechanism microstructure formation 2 F = ε φ + f φ, T, cα, σ,... 10-15 V 2 10-9 10-3 Phenomenological approaches: FEM materials properties Phase fields crystal growth : http://quattro.me.uiuc.edu/~provatas/ 1 2 ( ) dv 1 empirical, fitted parameters/functions 10 3 time (s)
Atomistic picture: Ab initio Description r 1,e r 2,e Rr 1, Z 1 R r 2, Z 2 Hamiltonian: H = T el + T ion I r I, i R I Z r i + 1 2 i, j i j 1 r r i j + 1 2 i, j i j 1 r r R I R J e-ion e-e Ion-Ion Solve Schrödinger equation: HΨ = EΨ r r r ( ) Ultimate information: E Ψ R, KR, r, r, K,, 1 M 1 2 r N Advantage: Free of adjustable/empirical parameters (ab initio) r r
Scaling Problem in Modeling length (m) 1 Phenomenological approaches: FEM materials properties Phase fields crystal growth : 10 Density-Functional Al (111) -3 Theory: microstructure 2 h r 2 ext r H r xc r formation r r + v ( ;{ RI, Z I }) + v [ n( )] + v [ n( )] ϕ i ( ) = ε iϕi ( ) 10-6 2me dislocations interfaces growth material point defects mechanism universal 10-9 r functional DFT chemical bonds ({ R Z }) E, tot I I Energy barrier 10-15 10-9 10 Phase II -3 Phase I 1 x int 10 3 time (s)
Scaling Problem in Modeling length (m) 1 10-3 10-6 10-9 DFT dislocations interfaces point defects ~2050 ~2020 growth mechanism microstructure formation materials properties crystal growth 10-15 10-9 10-3 1 10 3 time (s)
Scaling Problem in Modeling length (m) 1 10-3 10-6 10-9 DFT dislocations interfaces point defects Phenomenological approaches: FEM materials properties Phase fields crystal growth : microstructure formation How to growth bridge mechanism length and time scales? 10-15 10-9 10-3 1 10 3 time (s)
Ab initio based Multiscale Simulations Continuum models Thermodyn. Kinetics Strain Electr. struct. DFT: E tot ({R I }), ε i, φ i DFT code library: SFHIngX (www.sfhingx.de)
Potential Energy Surfaces Example: Crystal Structure of Aluminum reaction coordinates: angle length
Potential Energy Surfaces Example: Aluminum bcc fcc Energy [H] lattice constant [Bohr] Angle between lattice vectors
Potential Energy Surfaces Example: Aluminum Angle between lattice vectors fcc barrier bcc Angle between lattice vectors
Application: Phonon Dispersion Example: Aluminum
Ab initio based Multiscale Simulations Continuum models Thermodyn. Kinetics Strain Electr. struct. DFT: E tot ({R I }), ε i, φ i DFT code library: SFHIngX (www.sfhingx.de)
Method: Ab initio Thermodynamics Key: Calculate partition function DFT = r BO E RI Z( V, T) e r R { } I V ({ } ) V / k Experiment B T Z Phase ( V, T) = N deg e E tot r Phase { } vib R I V / kbt E { R } I e r { R } I V r Phase V / k B T F(V,T) = k B T ln G ( p, T ) = F V, T + n G(A ) µ A = n { Z(V,T) } ( ) pv T,p equil. config. D I, J E = tot u ({ r } ) Phase R I I u V vibronic config. J
Diluted Alloys Example: Mg in GaN Solubility/ Miscibility: Mg 3 N 2 Mg bulk µ Mg µ Ga Incorporation on other sites: Mg Ga Mg N Mg i
Intrinsic Solubility/Doping Efficiency Formation Energy (ev) 3 2 1 0 Formation Energies: V N Mg Ga -1 0.0 0.5 1.0 1.5 2.0 Fermi Energy (ev) Concentration (log 10 cm -3 ) Defect Concentrations: 22 20 18 16 14 12 Mg Ga V N 10 400 600 800 1000 1200 1400 Temperature (K) Strong compensation at characteristic growth temperatures! very limited doping efficiency J. Neugebauer and C. Van de Walle, Appl. Phys. Lett. 68, 1829 (1996)
Co-Doping Example: Co-doping with hydrogen Formation Energy (ev) 3 2 1 0 Formation Energies: V N H + Mg Ga -1 0.0 0.5 1.0 1.5 2.0 Fermi Energy (ev) Concentration (log 10 cm -3 ) Defect Concentrations: 22 H free case 20 H-rich conditions 18 16 14 12 Mg Ga H V N 10 400 600 800 1000 1200 1400 Temperature (K) Co-Doping: significantly increases dopant concentration significantly suppresses defect formation But: fully compensated material J. Neugebauer and C. Van de Walle, Appl. Phys. Lett. 68, 1829 (1996)
Example: Dislocations
Example: Dislocations continuum elastic theory emp. pot. DFT Main Main challenge: How How to to connect zones zones at at boundaries? New New approach: Projection method without explicit boundaries!
Comparison with Experiment Theory Experiment L. Lymperakis, J. Neugebauer, M. Albrecht, H. Strunk, PRL in print. EC-TMR Project: IPAM
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