Calculations of Hyperfine parameters in solids based on DFT and using WIEN2k Peter Blaha Institute of Materials Chemistry TU Wien Hyperfine parameters: Sample Reference Isomer shift: = ( 0 – 0 ); it is proportional to the electron density at the nucleus the constant is proportional to the change of nuclear radii during the transition (we use =-.291 au3mm s-1) Magnetic Hyperfine fields: Btot = Bcontact + Borb + Bdip these fields are proportional to the spin-density at the nucleus and the orbital moment of the probed atom as well as the spin moment distribution in the crystal Quadrupole splitting: e Q Vzz given by the product of the nuclear quadrupole moment Q times the electric field gradient Vzz. The EFG is proportional to an integral over the non-spherical charge density (weighted by 1/r3) Schrödinger equation From the previous slide it is obvious, that we need an accurate knowledge of the electron (and magnetization) density, which in principle can be obtained from the solution of the many- body Schrödinger equation for the corresponding solid. H = E 23 However, a many-body problem with ~10 particles is not solvable at all and we must create models for the real material and rely on an approximate solution of the Schrödinger equation. (This will be briefly discussed in the next slides and my preferred options are marked in red.) Concepts when solving Schrödingers-equation 1 2 k k k V (r ) i i i 2 • Representation of the solid: • cluster model: approximate solid by a finite cluster • periodic model: approximate “real” solid by infinite ideal solid (supercells !) BaSnO3 If calculations disagree with experiment: the structural model could be wrong Concepts when solving Schrödingers-equation 1 2 k k k V (r ) i i i 2 • Exchange and correlation: • Hartree-Fock (exact exchange, no correlation) • correlation: MP2, CC, ... • Density functional theory: approximate exchange + correlation • LDA: local density approximation, “free electron gas” • GGA: generalized gradient approximation, various functionals • hybrid-DFT: mixing of HF + GGA, various functionals • LDA+U, DMFT: explicit (heuristic) inclusion of correlations If calculations disagree with experiment: the DFT approximation could be too crude Concepts when solving Schrödingers-equation 1 2 k k k V (r ) i i i 2 • basis set for wavefunctions: • “quantum chemistry”: LCAO methods • Gauss functions (large “experience”, wrong asymptotics, ... ) • Slater orbitals (correct r~0 and r~ asymptotics, expensive) • numerical atomic orbitals • “physics”: plane wave based methods • plane waves (+ pseudopotential approximation) • augmented plane wave methods (APW) • combination of PW (unbiased+flexible in interstitial regions) • + numerical basis functions (accurate in atomic regions, cusp) Concepts when solving Schrödingers-equation 1 2 k k k V (r ) i i i 2 • Computational approximations: • relativistic treatment: •non-, scalar-, fully-relativistic treatment • treatment of spin, magnetic order • approximations to the form of the potential • shape approximations (ASA) • pseudopotential (nodeless valence orbitals) • “full potential” Concepts when solving Schrödingers-equation • in many cases, the experimental knowledge about a certain system is very limited and also the exact atomic positions may not be known accurately (powder samples, impurities, surfaces, ...) • Thus we need a theoretical method which can not only calculate HFF- parameters, but can also model the system: • total energies + forces on the atoms: • perform structure optimization for “real” systems • calculate phonons (+ partial phonon-DOS) • investigate various magnetic structures, exchange interactions • electronic structure: • bandstructure + DOS • compare with ARPES, XANES, XES, EELS, ... • hyperfine parameters • isomer shifts, hyperfine fields, electric field gradients WIEN2k software package An Augmented Plane Wave Plus Local Orbital Program for Calculating Crystal Properties Peter Blaha Karlheinz Schwarz Georg Madsen Dieter Kvasnicka Joachim Luitz November 2001 Vienna, AUSTRIA Vienna University of Technology WIEN2k: ~1700 groups mailinglist: 1800 users http://www.wien2k.at APW Augmented Plane Wave method The unit cell is partitioned into: atomic spheres Interstitial region unit cell R mtr I Basisset: ul(r, ) are the numerical solutions PW: i ( k K ). r of the radial Schrödinger equation e in a given spherical potential for a join particular energy Atomic partial waves K Alm coefficients for matching the PW K Am u (r , )Ym (rˆ ) m “Exact” solution for given (spherical) potential! w2web GUI (graphical user interface) Structure generator spacegroup selection import cif file step by step initialization symmetry detection automatic input generation SCF calculations Magnetism (spin-polarization) Spin-orbit coupling Forces (automatic geometry optimization) Guided Tasks Energy band structure DOS Electron density X-ray spectra Optics theoretical EFG calculations • The coulomb potential Vc is a central quantity in any theoretical calculation (part of the Hamiltonian) and is obtained from all charges (electronic + nuclear) in the system. (r ) V c (r ) dr V LM (r )Y LM (rˆ) r r LM • The EFG is a tensor of second derivatives of VC at the nucleus: 2V (0) (r )Y V ; V 20 dr ij zz 3 xi x j r • Since we use an “all-electron” method, we have the full charge distribution of all electrons+nuclei and can obtain the EFG without further approximations. • The spherical harmonics Y20 projects out the non-spherical (and non-cubic) part of . The EFG is proportional to the differences in orbital occupations (eg. pz vs. px,py) • We do not need any “Sternheimer factors” (these shielding effects are included in the self-consistent charge density) theoretical EFG calculations (r )Y20 V dr The charge density in the integral zz 3 can be decomposed in various ways for analysis: r EF according to energy (into various valence or semi-core contributions) according to angular momentum l and m (orbitals) spatial decomposition into “atomic spheres” and the “rest” (interstital) 3 Due to the 1/r factor, contributions near the nucleus dominate. theoretical EFG calculations We write the charge density and the potential inside the atomic spheres in an lattice-harmonics expansion (r ) LM (r )YLM (rˆ) V (r ) v LM (r )YLM (rˆ) LM LM spatial decomposit ion : (r )Y Y (r )Y LM LM 20 (r )Y V 20 d 3 r LM d 3 r 20 d 3 r zz 3 3 3 r sphere r int erstital r 20 (r ) V dr interstiti al zz 3 sphere r orbital decomposit ion : nk * nk 20 (r ) lm l m Y20 drˆ p p; d d ; (s d ) contr . k ,n.l ,l ,m ,m pp dd V zz V zz V zz ..... interstiti al theoretical EFG calculations pp dd V zz V zz V zz ..... interstitial 1 V pp 1 ( p p ) p zz 3 2 x y z r p 1 V dd d d 1 (d d ) d zz 3 xy x 2 y 2 2 xz yz z 2 r d • EFG is proportial to differences of orbital occupations , e.g. between px,py and pz. • if these occupancies are the same by symmetry (cubic): Vzz=0 • with “axial” (hexagonal, tetragonal) symmetry (px=py): =0 In the following various examples will be presented. Nuclear Quadrupole moment Q of 56Fe Compare theoretical and experimental EFGs Q 1 ( previous value) Q eQV zz FeF 2 2 FeSi FeZr3 Fe2O3 FeNi YFe 2 Fe P FeS2 2 Fe4N FeCl2 FeBr2 (exp.) F - EFGs in fluoroaluminates 10 different phases of known structures from CaF2-AlF3, BaF2-AlF3 binary systems and CaF2-BaF2-AlF3 ternary system Isolated octahedra Rings formed by four Isolated chains of octahedra sharing octahedra linked by corners corners -BaAlF5 -BaCaAlF7 -CaAlF5, -CaAlF5, Ca2AlF7, Ba3AlF9-Ib, Ba3Al2F12 -BaAlF5, -BaAlF5 -Ba3AlF9 Q and Q calculations using XRD data Q,exp = 0,803 Q,cal AlF 2 3 R = 0,38 -C aAlF Attributions= 4,712.10 performed-16 |V | with with R2 = respect 0,77 to the proportionality between5 |V | Q zz -C aAlF 5 zz 1,0 1,8e+6 Ca2AlF 7 AlF and for the multi-site compounds 3 Q -BaAlF 5 -C aAlF 5 -BaAlF 5 1,6e+6 -C aAlF 5 -BaAlF 5 Ca2AlF 7 Ba Al F 0,8 3 2 12 -BaAlF 1,4e+6 5 Ba3AlF 9-Ib -BaAlF 5 -Ba 3AlF 9 -BaAlF 5 -BaC aAlF 1,2e+6 7 Ba3Al2F 12 R egression Q Ba3AlF 9-Ib 0,6 Q ,m es. = Q , cal. (H z) Q 1,0e+6 -Ba 3AlF 9 -BaC aAlF 7 R égression 8,0e+5 0,4 Experimental Experimental 6,0e+5 4,0e+5 0,2 2,0e+5 0,0 0,0 0,0 1,0e+21 2,0e+21 3,0e+21 0,0 0,2 0,4 0,6 0,8 1,0 Calculated V (V .m -2) zz Calculated Q Important discrepancies when structures are used which were determined from X-ray powder diffraction data Q and Q after structure optimization 1,8e+6 1,0 -16 Q = 5,85.10 Vzz Q, exp = 0,972 Q,cal 1,6e+6 R2 = 0,993 R2 = 0,983 0,8 1,4e+6 1,2e+6 Q (H z) 0,6 Q AlF 1,0e+6 3 AlF 3 -C aAlF 5 -C aAlF 5 -C aAlF 5 8,0e+5 -C aAlF 5 Ca2AlF 7 Ca AlF 0,4 2 7 Experimental -BaAlF 5 Experimental -BaAlF 6,0e+5 5 -BaAlF 5 -BaAlF 5 -BaAlF 5 -BaAlF 5 Ba3Al2F 12 4,0e+5 Ba Al F Ba AlF -Ib 3 2 12 0,2 3 9 Ba AlF -Ib 3 9 -Ba 3AlF 9 -Ba 3AlF 9 2,0e+5 -BaC aAlF 7 -BaC aAlF 7 R egression R egression Q ,exp. = Q , cal. 0,0 0,0 0,0 5,0e+20 1,0e+21 1,5e+21 2,0e+21 2,5e+21 3,0e+21 0,0 0,2 0,4 0,6 0,8 1,0 -2 Calculated V zz (V .m ) Calculated Q Very fine agreement between experimental and calculated values M.Body, et al., J.Phys.Chem.