Structural Calculations phase stability, surfaces, interfaces etc Keith Refson STFC Rutherford Appleton Laboratory September 19, 2007 Phase Equilibrium 2 Energy-Volume curves.................................................................. 3 Model equations of State................................................................ 4 Birch-Murnaghan Equation of State........................................................ 5 First-order phase transitions.............................................................. 6 More degrees of freedom................................................................ 7 Technical aspects, convergence............................................................ 8 Zero-point energy..................................................................... 9 Zero-point energy and the quasi-harmonic approximation......................................... 10 Elastic constants.................................................................... 11 Elastic properties of CaO............................................................... 12 CASTEPs use of symmetry............................................................. 13 Surfaces 14 Surface modelling with slab geometry...................................................... 15 Slab models I....................................................................... 16 Surface Energy calculations............................................................. 17 Polarization in slabs.................................................................. 18 Defect calculations 19 Defect models...................................................................... 20 1
Phase Equilibrium 2 / 20 Energy-Volume curves Consider simple structure such as rocksalt (B1) one of simplest calculations is to compute E as function of V B1 rocksalt every atom is on a crystallographic high-symmetry site. No geometry optimisation needed! Equation of state (Really PV curve, which is commonly measured experimental quantity.) V(A 3 ) 40 30 20 10 CaO Equation of State 0 0 50 100 150 P (GPa) Energy (ev) -1428-1428.5-1429 -1429.5-1430 -1430.5-1431 CaO E vs V 80 100 120 140 160 V (A 3 ) CASTEP Workshop 2007: York 3 / 20 Model equations of State Real EOS are PVT relations, but this lecture restricted to easy case, T=0. Model EOS for fluids, eg Van der Waals, Redlich-Kwong There are several common models of EOS for solids based on elasticity theory Birch Murnaghan (Phys. Rev 71 809 (1947)) is most commonly used EOS E(V ) = E 0 + 9V 0B 0 16 j h` V0 V 2 i 3 3 1 B 0 + h` V0 V 2 3 1i 2 h 6 4 ` V 0 V 2 3 i ff Two definitions of bulk modulus B 0; B 0 = V 0 d 2 P dv 2 and as a fit to B-M or other EOS. These are not equivalent. Compression/EOS experiments use a B-M fit whereas ultrasound methods measure 2nd derivative CASTEP Workshop 2007: York 4 / 20 2
Birch-Murnaghan Equation of State Birch-Murnaghan fits to CaO EOS Effect of fitting range -1428-1428.5 Castep calculation V 0 =110.2, B 0 =28.4 GPa, B 0 =4.16 V 0 =110.3, B 0 =27.9 GPa, B 0 =4.23 range of data changes fit result. Only V <= V 0 accessible to experiment. V > V 0 also accessible to calculation. B 0 should be compared using fit over same or similar range to expt. Need to understand how B 0 was measured to make valid comparison. B 0 from ultrasonic expt. should be compared to V 0 d 2 P Energy (ev) -1429-1429.5-1430 -1430.5-1431 80 100 120 140 160 dv 2 V (A 3 ) CASTEP Workshop 2007: York 5 / 20 First-order phase transitions Phase equilibrium occurs when 2 criteria met simultaneously Pressures of 2 phases are equal, ie P = de 1 = de 2 dv Enthalpies are equal E 1 + PV 1 = E 2 + PV 2 E 2 E 1 = P (V 2 V 1). dv. Then P eq is gradient of common tangent CaO B1/B2 transition 4 40 CaO Equation of State 3 30 Energy (ev) 2 1 V(A 3 ) 20 0 10-1 15 20 25 30 35 V (A 3 ) 0 0 50 100 150 P (GPa) Alternatively use enthalpy-pressure plot, where most stable phase is that of lowest enthalpy. Need accurate calculation of pressures for this approach to work, so higher degree of PW convergence needed. CASTEP Workshop 2007: York 6 / 20 3
More degrees of freedom Eg TiO2 - Tetragonal rutile phase has 3 parameters a, c and internal coord u. Geometry optimization at range of P cheaper than exploring 3d energy surface. Complex phases with many internal degrees of freedom can be handled the same way, eg high-pressure Mg 2SiO 4 polymorphs (right). Olivine-Wadsleyite and Wadsleyite-Ringwoodite transitions detectable as seismic discontinuities in Earth s Mantle due to different elastic properties. CASTEP geometry optimiser has (unique?) capability of optimising cell at fixed volume. Lattice Energy (ev) -416-417 -418-419 -420-421 Forsterite P6 3 mc P - 3m1 Ringwoodite Wadsleyite P-3m1-422 110 120 130 140 150 160 170 Volume (A 3 ) CASTEP Workshop 2007: York 7 / 20 Technical aspects, convergence K-point integration accuracy varies with volume (not usually serious for insulators) K-point sampling - No error cancellation between different phases need absolute convergence with k-point. Gradient of common tangent sensitive to small absolute errors. Need high-accuracy calculations. Enthalpy minimum calculations equally sensitive, but pressure harder to calculate accurately than energy, even with FSBC. Does not account for quantum nature of nuclear motion. Need to include zero-point energy correction even at 0K. Care needed using symmetry - constraint restricts search space If using cell optimisation fixed-basis calculation gives systematic basis-set error with volume. Use fixed-cutoff option. No prescription for determining which phases to try. Crystal structure prediction is very hard problem. (But recent work is making progress). CASTEP Workshop 2007: York 8 / 20 4
Zero-point energy Zero-point energy contribution P β ω i can be significant and is in general different for different phases 2 Can calculate if vibrational phonon spectrum or DOS known. Use lattice dynamics calculation or (less satisfactory) experimental data. Harmonic Free R energy is ` F = E + 1 β g(ω)log `2sinh β ω 2 dω where g(ω) is the phonon density of states (DOS). To calculate DOS need phonon frequencies at all q in BZ. MgH2 phonon DOS g(ω 0.6 0.4 0.2 0 0 200 400 600 800 1000 1200 1400 ω (cm -1 ) CASTEP Workshop 2007: York 9 / 20 Zero-point energy and the quasi-harmonic approximation -111-111.2 Energy (ev) -111.4 E (athermal) E(0K) E(20K) E(300K) Quasi-harmonic approximation assumes that phonon frequencies depend only on the cell parameters. Ignores intrinsic anharmonic thermal effects on DOS. Works for relatively harmonic systems Valid for T 0.5 T m. QHA is cheapest way of extending ab-initio to TT > 0 In Zero Static Internal Stress Approximation geometry optimise coordinates at each volume (or strain) Quasi-Harmonic Free energy geometry optimization possible with empirical force fields but not yet ab-initio. -111.6-111.8-4 -2 0 2 4 Pressure (GPa) CASTEP Workshop 2007: York 10 / 20 5
Elastic constants Elastic strain theory gives E(ǫ) = E 0 + V/2 6X C ijǫ iǫ j + O(3) Programmed strains ǫ may be used to extract individual elastic constants, e.g. if 2 3 ǫ = 1 0 δ δ 4δ 0 δ5 then 2 δ δ 0 ij E = E 0 + 3 2 V C44δ2 + O(3) for a cubic crystal. See book by Nye on elastic theory for symmetry-adapted strains. For low symmetry crystals more efficient to compute stress and use σ αβ = C αβγδ ǫ γǫ δ. But need very well converged stress. No automated calculation built into CASTEP; available in Materials Studio CASTEP Workshop 2007: York 11 / 20 Elastic properties of CaO Theory Expt(1) Expt (2) a 0 4.784 ± 0.001 4.8105 4.8105 K 0 111.9 ± 1.5 117 112.5 ± 0.6 K 0 4.3 ± 0.3 5.9 ± 2 4.8 ± 0.1 C s 83.1 ± 0.9 81.9 ± 3 82.0 ± 0.3 C 11 223.1 ± 1.9 226.2 ± 0.9 221.9 ± 0.6 C 12 56.9 ± 1.6 62.4 ± 0.9 57.8 ± 0.7 C 44 83.9 ± 0.4 80.6 ± 0.3 80.3 ± 0.1 1. Dragoo and Spain, J. Phys. Chem. Solids (1977) 38, 705 2. Chang and Graham J. Phys. Chem. Solids (1977) 38, 1355 CASTEP Workshop 2007: York 12 / 20 6
CASTEPs use of symmetry CASTEP uses symmetry to optimize BZ integration. K-point grid is reduced to Irreducible BZ using a weighted sum. Geometry optimization preserves initial symmetry of atomic co-ordinates and cell vectors. Warning 1. The result of a symmetry-constrained SPE or Geom. Opt. calculation may be mechanically unstable to a symmetry-breaking perturbation. System sits on saddle point of energy hypersurface. Forces are still zero at saddle point. A phonon calculation at q = 0 will give an imaginary frequency at a saddle point diagnostic test. Alternative test is to break symmetry and re-optimize. Warning 2. The size of unit cell may still be a constraint. e.g. cell-doubling may arise from imaginary gamma-point phonon at BZ boundary. CASTEP Workshop 2007: York 13 / 20 7
Surfaces 14 / 20 Surface modelling with slab geometry Surfaces can be modelled as a slab cleaved from bulk crystal. Can calculate Surface energy or free energy Energies of steps Adsorption energies and structures of adsorbates Surface chemical reaction energies. CASTEP Workshop 2007: York 15 / 20 8
Slab models I Choice of slab: usually need to make 2 surfaces identical. Surfaces related by symmetry operation are more easily geometry optimised. Simulation cell should not be optimised. (use fix all cell =T). In-plane cell parameters usually fixed at values from relaxed bulk crystal. CASTEP Workshop 2007: York 16 / 20 9
Surface Energy calculations Surface free energy defined as E surf = (E slab E bulk )/A A is total area of both surfaces. E surf is sensitive quantity requiring well-converged total energies. Can sometimes gain some k-point error cancellation between slab and bulk calcs by using non-primitive bulk cell with same in-plane vectors as slab. (not CaCO 3 10 14 shown). Need only 1 k-point in direction perpendicular to slab. Any dispersion in bands is error due to insufficient vacuum gap - no point in calculating accurately! Need to test convergence with both slab thickness and vacuum gap CASTEP Workshop 2007: York 17 / 20 10
Polarization in slabs Electric dipoles perpendicular to surface raise theoretical difficulties Energy does not converge with slab thickness. P.W. Tasker (Surf. Sci 87, 315 (1979) described 3 classes of surface. In classical charge model, Type III unstable and must always reconstruct. In ab initio calculation, surfaces can instead become metallic. CASTEP Workshop 2007: York 18 / 20 Defect calculations 19 / 20 Defect models Point- and extended- defects may be modelled using supercell approach. Several tricky convergence issues with supercells, to converge defect-defect interaction to zero. See M. I. J. Probert and M. C. Payne Phys. Rev. B67 075204 (2003). Only need to converge energy to a few mev, but still need accuracy in forces to correctly describe strain relaxation. Strain relaxation. Local strain around defect decreases as 1/R. Can model long-range strain relaxation using classical models if suitable potential exists. Charged defects can be modelled using periodic interaction correction terms (M. Leslie and M. Gillan, J. Phys. Cond. Mat. 18, 973 (1985)) Corrections for higher multipoles also available G. Makov and M. C. Payne Phys. Rev. B51, 4104 (1995), L. N. Kantorovich Phys. Rev. B60, 15476 (1999)). CASTEP Workshop 2007: York 20 / 20 11