Blaha krakow 2004
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This document provides an overview of the WIEN2k software package for calculating crystal properties using the augmented plane wave plus local orbital (APW+lo) method. It summarizes the history and development of APW-based methods, describes the program structure and usage of WIEN2k, and highlights some applications including calculations of phase transitions, crystal structures, total energies, forces and structural relaxations. The document also discusses international collaborations and the use of WIEN2k as a benchmark for electronic structure calculations.
![Linearization of energy dependence
LAPW suggested by
( rˆ)k n l m
l m
[ A ( k ) u ( E , r ) B ( k ) u& ( E , r )] Y
l m n l l l m n l l
Atomic sphere
PW
O.K.Andersen,
Phys.Rev. B 12, 3060
(1975)
join PWs in value and slope
ul expanded at El („energy parameter“)
€additional constraint requires more
PWs than APW
basis flexible enough for single
diagonalization
ul /u&l
LAPW
APW](https://image.slidesharecdn.com/blahakrakow2004-170112163525/85/Blaha-krakow-2004-6-320.jpg)
![Extending the basis: Local orbitals (LO)
■ LO
■ is confined to an atomic sphere
■ has zero value and slope at R
■ can treat two principal QN n for
each azimuthal QN l (3p and 4p)
■ corresponding states are strictly
orthogonal (no “ghostbands”)
■ tail of semi-core states can be
represented by plane waves
■ only slight increase of basis set
(matrix size)
D.J.Singh,
Phys.Rev. B 43 6388 (1991)
1
(rˆ)E
LO l ml m ll m l l m l
& C uE2
]Y [A uE1
B u
4p 3p](https://image.slidesharecdn.com/blahakrakow2004-170112163525/85/Blaha-krakow-2004-11-320.jpg)
![New ideas from Uppsala and Washington
E.Sjöstedt, L.Nordström, D.J.Singh,
An alternative way of linearizing the augmented plane wave method,
Solid State Commun. 114, 15 (2000)
•Use APW, but at fixed El (superior PW convergence)
•Linearize with additional lo (add a few basis functions)
E
1
E
1 l m ll m l
& [A u B u ]Y (rˆ)
l m
lokn
Al m(kn)ul (El ,r)Yl m(rˆ)
lm
optimal solution: mixed basis (K.Schwarz, P.Blaha, G.K.H.Madsen,
Comp.Phys.Commun.147, 71-76 (2002))
•use APW+lo for states which are difficult to converge:
(f or d- states, atoms with small spheres)
•use LAPW+LO for all other atoms and angular momenta](https://image.slidesharecdn.com/blahakrakow2004-170112163525/85/Blaha-krakow-2004-12-320.jpg)
![Total energies and atomic forces
(Yu et al.; Kohler et al.)
2
1
2
1
3
Z V (r)(r)
xcxc
effi i i
es
(r
r
)
(r
r
)
r r
rr3r r
E [] d3
r
r
(r
r
)
d r (r )Vn T[]
d r (r )VU[]
es
■ Total Energy:
■ Electrostatic energy
■ Kinetic energy
■ XC-energy
■ Force on atom
■ Hellmann-Feynman-force
■ Pulay corrections
■ Core
■ Valence
■ expensive, contains a summation
of matrix elements over all
occupied states (done only in last
scf-iteration)
dR
F F
F
F
HF core val
dE
r tot
r
K Ki
i iieff val
val
effcorecore
es
HF
VF
F
r
F
(r) dS i(K K) K H i K
(K ) (r)
n c (K )c (K )(r) dr (r)
r Y (rˆ)V (r )
Z lim
*2
*
k,i K,K
1m
1
m1
1m
r 0
r
(r)V (r) dr
r](https://image.slidesharecdn.com/blahakrakow2004-170112163525/85/Blaha-krakow-2004-27-320.jpg)
![The Aurivillius Compounds
[Bi2O2]-[An-1BnO3n+1]
n-sized perovskite slab
cation deficient
perovskites:
[Bi,A]B1-xO3
A21am
Bi2O2
B2ab
A21am
n=2
n=3 n=4
orthorhombic
Dipl. Thesis -S. Borg 2001Ps](https://image.slidesharecdn.com/blahakrakow2004-170112163525/85/Blaha-krakow-2004-37-320.jpg)
More Related Content
Blaha krakow 2004
- 1. Electronic structure, atomic forces and structural relaxations by WIEN2k Peter Blaha Institute of Materials Chemistry TU Vienna, Austria
- 2. Outline: ■ APW-based methods (history and state-of-the-art) ■ WIEN2k ■ program structure + usage ■ forces, structure relaxation ■ Applications ■ Phasetransitions in Aurivillius phases ■ Structure of Pyrochlore Y2Nb2O7
- 3. APW based schemes ■ APW (J.C.Slater 1937) ■ Non-linear eigenvalue problem ■ Computationally very demanding ■ LAPW (O.K.Anderssen 1975) ■ Generalized eigenvalue problem ■ Full-potential total-energies (A.Freeman etal.) ■ Local orbitals (D.J.Singh 1991) ■ treatment of semi-core states (avoids ghostbands) ■ APW+lo (E.Sjöstedt, L.Nordstörm, D.J.Singh 2000) ■ Efficience of APW + convenience of LAPW ■ Basis for K.Schwarz, P.Blaha, G.K.H.Madsen, Comp.Phys.Commun.147, 71-76 (2002)
- 4. APW Augmented Plane Wave method The unit cell is partitioned into: atomic spheres Interstitial region Bloch wave function: atomic partial waves Plane Waves (PWs) Rmt unit cell PW:e rr r i(k K).r Atomic partial waves l m K al m Ylm (rˆ) u (r, ) l join matching coefficient, radial function, spherical harmonics
- 5. Slater‘s APW (1937) Atomic partial waves Energy dependent Radial basis functions lead to K l m l m l lm ˆ(r )a u (r ,)Y Non-linear eigenvalue problemH Hamiltonian S Overlap matrix Computationally very demanding One had to numerically search for the energy, for which the det|H-ES| vanishes.
- 6. Linearization of energy dependence LAPW suggested by ( rˆ)k n l m l m [ A ( k ) u ( E , r ) B ( k ) u& ( E , r )] Y l m n l l l m n l l Atomic sphere PW O.K.Andersen, Phys.Rev. B 12, 3060 (1975) join PWs in value and slope ul expanded at El („energy parameter“) €additional constraint requires more PWs than APW basis flexible enough for single diagonalization ul /u&l LAPW APW
- 7. Full-potential total-energy (A.Freeman etal.) ■ The potential (and charge density) can be of general form (no shape approximation)SrTiO3 Full potential Muffin tin approximation ■ Inside each atomic sphere a local coordinate system is used (defining LM) LM VLM (r)YLM (rˆ) r R K VKeiK.rr rI TiO2 rutile Ti O V (r ) {
- 8. Core, semi-core and valence states ■ Valences states ■ High in energy ■ Delocalized wavefunctions ■ Semi-core states ■ Medium energy ■ Principal QN one less than valence (e.g.in Ti 3p and 4p) ■ not completely confined inside sphere ■ Core states ■ Low in energy ■ Reside inside sphere For example: Ti
- 9. Problems of the LAPW method: EFG Calculation for Rutile TiO2 as a function of the Ti-p linearization energy Ep exp. EFG „ghostband“ region P. Blaha, D.J. Singh, P.I. Sorantin and K. Schwarz, Phys. Rev. B 46, 1321 (1992).
- 10. Problems of the LAPW method Problems with semi-core states
- 11. Extending the basis: Local orbitals (LO) ■ LO ■ is confined to an atomic sphere ■ has zero value and slope at R ■ can treat two principal QN n for each azimuthal QN l (3p and 4p) ■ corresponding states are strictly orthogonal (no “ghostbands”) ■ tail of semi-core states can be represented by plane waves ■ only slight increase of basis set (matrix size) D.J.Singh, Phys.Rev. B 43 6388 (1991) 1 (rˆ)E LO l ml m ll m l l m l & C uE2 ]Y [A uE1 B u 4p 3p
- 12. New ideas from Uppsala and Washington E.Sjöstedt, L.Nordström, D.J.Singh, An alternative way of linearizing the augmented plane wave method, Solid State Commun. 114, 15 (2000) •Use APW, but at fixed El (superior PW convergence) •Linearize with additional lo (add a few basis functions) E 1 E 1 l m ll m l & [A u B u ]Y (rˆ) l m lokn Al m(kn)ul (El ,r)Yl m(rˆ) lm optimal solution: mixed basis (K.Schwarz, P.Blaha, G.K.H.Madsen, Comp.Phys.Commun.147, 71-76 (2002)) •use APW+lo for states which are difficult to converge: (f or d- states, atoms with small spheres) •use LAPW+LO for all other atoms and angular momenta
- 13. Convergence of the APW+LO Method E. Sjostedt, L. Nordstrom and D.J. Singh, Solid State Commun. 114, 15 (2000). x 100 Ce (number of PWs)
- 14. Improved convergence of APW+lo ■ changes sign and converges slowly in LAPW ■ better convergence in APW+lo K.Schwarz, P.Blaha, G.K.H.Madsen, Comp.Phys.Commun.147, 71-76 (2002) Force (Fy) on oxygen in SES (sodium electro sodalite) vs. # plane waves
- 15. Quantum mechanics at work
- 16. WIEN2k software package WIEN97: ~500 users WIEN2k: ~730 users 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 http://www.wien2k.at
- 17. Development of WIEN2k ■ Authors of WIEN2k P. Blaha, K. Schwarz, D. Kvasnicka, G. Madsen and J. Luitz ■ Other contributions to WIEN2k ■ C. Ambrosch-Draxl (Univ. Graz, Austria), optics ■ U. Birkenheuer (Dresden), wave function plotting ■ T. Charpin (Paris), elastic constants ■ R. Dohmen und J. Pichlmeier (RZG, Garching), parallelization ■ R. Laskowski (Vienna), non-collinear magnetism ■ P. Novák and J. Kunes (Prague), LDA+U, SO ■ B. Olejnik (Vienna), non-linear optics ■ C. Persson (Uppsala), irreducible representations ■ M. Scheffler (Fritz Haber Inst., Berlin), forces ■ D.J.Singh (NRL, Washington D.C.), local oribtals (LO), APW+lo ■ E. Sjöstedt and L Nordström (Uppsala, Sweden), APW+lo ■ J. Sofo and J. Fuhr (Barriloche), Bader analysis ■ B. Yanchitsky and A. Timoshevskii (Kiev), spacegroup ■ and many others ….
- 18. International co-operations ■ More than 700 user groups worldwide ■ 40 industries (Canon, Eastman, Exxon, Fuji, A.D.Little, Mitsubishi, Motorola, NEC, Norsk Hydro, Osram, Panasonic, Samsung, Sony, Sumitomo). ■ Europe: (EHT Zürich, MPI Stuttgart, Dresden, FHI Berlin, DESY, TH Aachen, ESRF, Prague, Paris, Chalmers, Cambridge, Oxford) ■ America: ARG, BZ, CDN, MX, USA (MIT, NIST, Berkeley, Princeton, Harvard, Argonne NL, Los Alamos Nat.Lab., Penn State, Georgia Tech, Lehigh, Chicago, SUNY, UC St.Barbara, Toronto) ■ far east: AUS, China, India, JPN, Korea, Pakistan, Singapore,Taiwan (Beijing, Tokyo, Osaka, Sendai, Tsukuba, Hong Kong) ■ Registration at www.wien2k.at ■ 400/4000 Euro for Universites/Industries ■ code download via www (with password), updates, bug fixes, news ■ usersguide, faq-page, mailing-list with help-requests
- 19. WIEN code as benchmark
- 20. WIEN2k- hardware/software ■ WIEN2k runs on any Unix/Linux platform from PCs, workstations, clusters to supercomputers ■ Fortran90 (dynamical allocation) ■ many individual modules, linked together with C-shell or perl-scripts ■ f90 compiler, BLAS-library (mkl, Atlas), perl5, ghostview, gnuplot, Tcl/Tk (Xcrysden), www-browser •web-based GUI – w2web •real/complex version (inversion) •10 atom cells on 128Mb PC •100 atom cells require 1-2 Gb RAM •k-point parallel on clusters with common NFS (slow network) •MPI/Scalapack parallelization for big cases (>50 atoms) and fast network •installation support for many platforms
- 21. W2web – A web-interface ■ Based on www ■ WIEN2k can be managed remotely via w2web ■ Secure (Password) ■ Important steps: ■ start w2web on all your hosts ■ use your favorite browser and connect to the master host:port ■ define a session on the desired host
- 22. GUI (Graphical user interface) ■ Structure generator ■ cif-file converter ■ spacegroup selection ■ 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
- 23. Program structure of WIEN2k ■ init_lapw ■ initialization ■ symmetry detection (F, I, C- centering, inversion) ■ input generation with recommended defaults ■ quality (and computing time) depends on k-mesh and R.Kmax (determines #PW) ■ run_lapw ■ scf-cycle ■ optional with SO and/or LDA+U ■ different convergence criteria (energy, charge, forces)
- 24. Task for electron density plot ■ A task consists of ■ a series of steps ■ that must be executed ■ to generate a plot ■ For electron density plot ■ select states (e.g. valence e-) ■ select plane for plot ■ generate 3D or contour plot with gnuplot or Xcrysden
- 25. Properties with WIEN2k - I ■ Energy bands ■ classification of irreducible representations ■ ´character-plot´ (emphasize a certain band-character) ■ Density of states ■ including partial DOS with l and m- character (eg. px , py , pz ) ■ Electron density, potential ■ total-, valence-, difference-, spin-densities, of selected states ■ 1-D, 2D- and 3D-plots (Xcrysden) ■ X-ray structure factors ■ Bader´s atom-in-molecule analysis, critical-points, atomic basins and charges ( .n 0 ) ■ spin+orbital magnetic moments (spin-orbit / LDA+U) ■ Hyperfine parameters ■ hyperfine fields (contact + dipolar + orbital contribution) ■ Isomer shift ■ Electric field gradients
- 26. Properties with WIEN2k - II ■ Spectroscopy ■ core levels (with core holes) ■ X-ray emission, absorption, electron-energy-loss ■ (core - valence/conduction-band transitions including matrix elements and angular dep.) ■ EELS inclusion of possible non-dipol transititons (momentum transfer) ■ optical properties (dielectric function, JDOS including momentum matrix elements and Kramers-Kronig) ■ fermi surface (2D, 3D) ■ Total energy and forces ■ optimization of internal coordinates, (MD, BROYDEN) ■ cell parameter only via Etot (no stress tensor) ■ elastic constants for cubic cells ■ Phonons via supercells ■ interface to PHONON (K.Parlinski) – bands, DOS, thermodynamics, neutrons
- 27. Total energies and atomic forces (Yu et al.; Kohler et al.) 2 1 2 1 3 Z V (r)(r) xcxc effi i i es (r r ) (r r ) r r rr3r r E [] d3 r r (r r ) d r (r )Vn T[] d r (r )VU[] es ■ Total Energy: ■ Electrostatic energy ■ Kinetic energy ■ XC-energy ■ Force on atom ■ Hellmann-Feynman-force ■ Pulay corrections ■ Core ■ Valence ■ expensive, contains a summation of matrix elements over all occupied states (done only in last scf-iteration) dR F F F F HF core val dE r tot r K Ki i iieff val val effcorecore es HF VF F r F (r) dS i(K K) K H i K (K ) (r) n c (K )c (K )(r) dr (r) r Y (rˆ)V (r ) Z lim *2 * k,i K,K 1m 1 m1 1m r 0 r (r)V (r) dr r
- 28. Optimization of atomic posistions (E-minimization via forces) • damped Newton scheme (not user friendly, input adjustment by hand) • new quite efficient Broyden scheme W impurity in Bi (2x2x2 cell: Bi15W) 0 2 4 10 12 14 -40 -20 20 40 60 for01 for04x for04z for06x for06z 0 forces(mRy/a) 6 8 timestep 0 2 4 10 12 14 -679412.54 -679412.52 -679412.48 0 -679412.50 -679412.46 -679412.44 Energy(Ry) 6 8 time step 0 2 4 6 8 10 12 14 -0.04 -0.02 0.00 0.02 0.04 pos01 pos04x pos04z pos06 position 0 2 4 6 8 10 12 14 -4 -2 0 2 4 6 8 EFG(1021 V/m2 ) Energy Forces Positions EFG
- 29. PHONON-I ■ PHONON ■ by K.Parlinski (Crakow) ■ runs under MS-windows ■ uses a „direct“ method to calculate Force- constants with the help of an ab initio program ■ with these Force- constants phonons at arbitrary k-points can be obtained ■ Define your spacegroup ■ Define all atoms at previously optimized positions
- 30. PHONON-II ■ Define an interaction range (supercell) ■ create displacement file ■ transfer case.d45 to Unix ■ Calculate forces for all required displacements ■ init_phonon_lapw ■ for each displacement a case_XX.struct file is generated in an extra directory ■ runs nn and lets you define RMT values like: ■ 1.85 1-16 • init_lapw: either without symmetry (and then copies this setup to all case_XX) or with symmetry (must run init_lapw for all case_XX) (Do NOT use SGROUP) • run_phonon: run_lapw –fc 0.1 –i 40 for each case_XX
- 31. PHONON-III ■ analyze_phonon_lapw ■ reads the forces of the scf runs ■ generates „Hellman-Feynman“ file case.dat and a „symmetrized HF- file case.dsy (when you have displacements in both directions) ■ check quality of forces: ■ sum Fx should be small (0) ■ abs(Fx) should be similar for +/- displacements ■ transfer case.dat (dsy) to Windows ■ Import HF files to PHONON ■ Calculate phonons
- 32. Ferroelectricity in Aurivillius phases ■ *Bilbao: J. Manuel Perez-Mato, M. Aroyo ■ Universidad del País Vasco ■ *Vienna: P. Blaha, K. Schwarz, ■ J. Schweifer ■ *Cracow: K. Parlinski
- 33. Perovskites: PbTiO3 Classical Example of a Ferroelectric PbTiO3 (perovskite distorted structure) Features: • spontaneous polarization- frozen polar mode • phase transition: paraelectric -------- ferroelectric Tc
- 34. Perovskites: PbTiO3 PbTiO3 • Symmetry break at Tc: cubic ---- tetragonal Pm-3m ----- P4mm • A single (degenerate) normal mode responsible for the transition • Structure in Ferroelectric Phase: high-symmetry structure + frozen polar mode • Strain: spontaneous strain 10-2
- 35. Ab-initio “prediction” of the ferroelectric inestability PbTiO3 King-Smith&Vanderbilt 1994Waghmare&Rabe 1997
- 36. Ab-initio Phonon Branches Ghosez et al. 1999 Polar unstable mode
- 37. The Aurivillius Compounds [Bi2O2]-[An-1BnO3n+1] n-sized perovskite slab cation deficient perovskites: [Bi,A]B1-xO3 A21am Bi2O2 B2ab A21am n=2 n=3 n=4 orthorhombic Dipl. Thesis -S. Borg 2001Ps
- 38. Bi2SrTa2O9 - SBT Bi O3 O2 O1 O4 Sr O4 Spontaneous Polarization a b c polar mode Eu (deg. 2)
- 39. Previous ab-initio calculations in SBT Stachiotti et al. 2000 (110) (100) • Unstable Eu polar mode • Strong contribution of Bi displacement : Bi-O(2) hybridization
- 40. SBT – An Usual Ferroelectric? NO ! I4/mmm F2mm Eu A21am Amam Abam X3 - X2 + high and low T structures From symmetry analysis (group/ subgroup relations) the Eu mode alone cannot explain the complete phase transition Ferroelectric Phase = mode Eu + mode X3- + mode X2+
- 41. Frozen distortions in SBT - mode X3 + mode X2 Hervoches et al. 2002Shimakawa et al. 1999 Ferroelectric Phase = mode Eu + mode X3- + mode X2+ eEu eX3 eX2 QEu = 0.82 QX3= -1.20 QX2= -0.35 Eigenvectors: Mode Amplitudes (bohrs) :
- 42. All Aurivillius: “Atypical” Ferroelectrics: Prototype tetragonal paraelectric phase Ferroelectric orthorhombic phase Not possible through the freezing of a polar mode! •A single order parameter can not produce this transition. According to Landau theory: •Either discontinuous transition or intermediate phases
- 43. Phonons in SBT X3 - Eu X2 + Eu X3 - SBT CALCULATED “PHONON” BRANCHES
- 44. Comparing calculated and exp. soft-modes Eigenvectors: Ferroelectric Phase = mode Eu + mode X3- + mode X2+ In general, good agreement with experimental frozen modes Except !…. The experimental frozen mode X2 +
- 45. + Comparing exp. and calculated X2 Modes a b c “Part 1” “Part 2” a b c 2 coincides approx. with calculated soft-mode X + Two independent normal- modes coincides approx. with a 2second (hard) mode X +
- 46. Energy Landscape around the tetragonal configuration 2 2 2Energy = Eo + ½ Eu QEu + ½ X3 QX3 + ½ X2 QX2 + …… QEu QX3 QX2 QEu ground state Tc1 Tc2 Tc1 Tc2 QX3 Eu < 0, X3 <0 Transition Sequence: Thermal Free Energy = Fo + ½ Eu(T)Q 2 + ½ (T)Q 2 + ½ Q 2 + …… Eu X3 X3 X2 X2
- 47. ENERGY MAP – SBT Main axes QX3 (bohr) QEu(bohr) E(mRy) E(mRy) 33 mRy 26 mRy QEu QX3 Space GroupAmam EX3 = -40.2 QX3 2 + 10.9 QX3 4 + 0.87QX3 6 QEu QX3 Space Group F2mm EEu = -19.9 QEu 2+ 2.6 QEu 4+ 0.52 QEu 6
- 48. ENERGY MAP – SBT: Coupling Eu –X3 - QEu QX3 QEu QX31.25 EEu = -19.9 QEu 2+ 2.6 QEu 4+ 0.51 QEu 6 EEu = +3.1 QEu 2+ 1.6 QEu 4+ 0.53 QEu 6 +14.7 QEu 2 QX3 2 - 0.63 QEu 4QX3 2 Strong biquadratic coupling Eu –X3 -
- 49. SBT – Energy Map: Coupling Eu –X3 - Predicted Ground State: Only mode X3 - frozen ! Space group Amam -1.5 -1 -0.5 0 0.5 1 1.5 -2 -1 0 1 2 QX3 QEu Essential reason: Strong biquadratic coupling penalizes “mixed” states Something missing? •coupling with X2 +QX2=0
- 50. Coupling with X2 + • Coupling with soft X2 + mode relatively weak ! • Strong renormalization of hard (second) X2 + mode
- 51. Energy landscape in SBT including only soft X2 + mode with additional hard X2 + mode only with the additional hard X2 + mode the experimentally observed combination of 3 different modes can be established
- 52. Phase diagram for Bi2SrTa2O9 - SBT •Finite temperature renormalization of the T=0 energy map •Landau theory suggests linear T-variation of the quadratic stiffness coefficients Eu and - X3 The topolgy of this phase diagram predicts two second order phase transitions and implies the existence of an intermediate phase of Amam symmetry. A single first order PT is impossible.
- 53. Pyrochlore Y2Nb2O7 Insulating and non-magnetic 4d TM-oxide P.Blaha, D.Singh, K.Schwarz, PRL in print (2004) Metal Sublattice: Corner-shared tetrahedral network
- 54. Pyrochlore structure of Y2Nb2O7: ■ Experimental structure from powder diffraction (Fukazawa,Maeno): ■ Pyrochlore structure: Fd-3m, a0=10.33 Å ■ 22 atoms (2 FU/cell) ■ One free parameter for O2: (0.344,1/8,1/8) ■ Corner shared Nb - O octahedra ■ Corner shared O1 - Y tetrahedra ■ Y surrounded by 8 oxygens (~ cube) ■ Metal sublattices ■ Nb-tetrahedra (empty) ■ Y-tetrahedra (around O) ■ Insulator ■ Nonmagnetic Nb Nb Y O1O2 Y metal sublattice Nb Y.Maeno et al, Nature 372, 532 (1994) H.Fukazawa, Y.Maeno,Phys.Rev. B 67, 054410 (2003)
- 55. First theoretical results: O-2p Nb-t2g Nb-eg 3+ 4+ 2- ■ Ionic model: Y2 Nb2 O7 Nb4+: 4d1 ■ € metallic or localized system with spin ½ (neither one observed in exp.) ■ LDA gives nonmagnetic metallic ground-state with conventional t2g-eg splitting due to the octahedral crystal field of the oxygen atoms. “degenerate” t2g states are only partly filled. EF EF
- 56. How could one obtain a non-magnetic insulator ? ■ Antiferromagnetic s=1/2 solution ■ (on geometrically frustrated lattice !?) ■ Localization, strong e--e- correlation: ■ 4d (not 3d !) electrons, ■ thus correlation should be small (Hubbard-U ~ 2-3 eV) ■ LDA+U with U=6 eV gives insulator (FM ground state, no AFM) ■ bandwidth of t2g bands: 2.5 eV (similar to U) ■ structural distortion, which breaks the dominant octahedral crystal field ■ € Search for phonon-instabilities ■ 88 atom supercell, 46 symmetry adapted selected distortions from PHONON; resulting forces € back into PHONON
- 57. Phonon-bandstructure of Y2Nb2O7 (100) (110) (111) ■ strong Phonon-instabilities, lowest at X, K, L ■ select a certain (unstable) phonon, freeze it with certain amplitude into the structure and perform full structural optimization
- 58. , X and K-point phonons: ■ energy lower than in ideal pyrochlore structure, but still not insulating EF
- 59. L-point (111) phonon: ■ Relaxed structure is an Insulator ■ energy gain of 0.5 eV/FU gap metal EF
- 60. Relaxed structure: ■ Primitive supercell with 88 atoms ■ all atoms inequivalent due to numerical optimization of the positions in P1 ■ Symmetrization using KPLOT (R.Hundt, J.C.Schön, A.Hannemann, M.Jansen: Determination of symmetries and idealized cell parameters for simulated structures, J.Appl.Cryst. 32, 413-416 (1999)) ■ Tests possible symmetries with increasing tolerance ■ Space group € P-43m, 88 atoms/cell, ■ Inequivalent atoms: ■ 2 Y ■ 2 Nb ■ 3 O1 ■ 5 O2
- 61. Relaxed structure Nb2 Nb1 Experimental (ideal) relaxed: Nb2 moves „off-center“, O octahedra changes little, Nb2-Nb2 bonds ?
- 62. Main change in structural relaxation Original pyrochlore Relaxed structure Nb1NbNb equal large 3.65 Å Nb2 small 2.91 Å 3.90 Å Nb1 - Nb2 3.89 Å Nb1–triangle: 3.40Å
- 63. Chemical bonding: original structure peak A: Nb-Nb bonds within the Nb tetrahedra by d-z2 orbitals O-2p (Nb-4d) (Y-4d) Nb t2g Nb Y O peak A B
- 64. Chemical bonding: original structure Nb t2g O-2p (Nb-4d) (Y-4d) peak A B peak B: mixture of all 3 t2g orbitals Nb Y O
- 65. DOS of relaxed structure O-2p (Nb-4d) (Y-4d) Nb-4d (O-2p), Y-4d Nb1 B Nb2 A EF gap Nb2 C
- 66. Peak A (Nb2) 4-center bond (d-z2) O-2p (Nb-4d) (Y-4d) Nb-4d (O-2p), Y-4d Nb1 B Nb2 A Nb2 C EF gap Nb1 Nb1 Nb1 Nb2 Nb2 Nb2 Nb2
- 67. Peak B (Nb1) 3-center bond (new !) O-2p (Nb-4d) (Y-4d) Nb-4d (O-2p), Y-4d Nb1 B Nb2 A Nb2 C EF gap Nb1 Nb1 Nb1 Nb1 Nb2
- 68. Peak C (Nb2) 3-center bonds an all 4 faces O-2p (Nb-4d) (Y-4d) Nb-4d (O-2p), Y-4d Nb1 B Nb2 A Nb2 C EF gap Nb1 Nb1Nb1 NbNb2 Nb2 Nb2 Nb2
- 69. Results for Y2Nb2O7 ■ Orbital ordering of Nb-4d orbitals and structural distortion ! ■ Can be described with LDA !! ■ M-M interaction (second NN): 2 types of bonds: ■ strong Nb-Nb 4-center bonds (only Nb2 dz 2) ■ strong Nb-Nb 3-center bonds (both, on Nb1 and Nb2) ■ Relaxation ■ Two types of Nb ■ Nb1 (12i) highly elongated tetradedron along (111) ■ Nb2 (4e) perfect tetrahedron with short distances ■ Metal/insulator ■ Ideal structure metallic, relaxed structure insulating P.Blaha, D.J.Singh, K.Schwarz, Charge singlets in pyrochlore Y2Nb2O7 Phys.Rev.Lett. (in print) (2004)
- 70. Advantage/disadvantage of WIEN2k + robust all-electron full-potential method + unbiased basisset, one convergence parameter (LDA-limit) + all elements of periodic table (equal expensive), metals + LDA, GGA, meta-GGA, LDA+U, spin-orbit + many properties + w2web (for novice users) - ? speed (+ memory requirements) + very efficient basis for large spheres (2 bohr) (Fe: 12Ry, O: 9Ry) - less efficient for small spheres (1 bohr) (O: 25 Ry) - large cells (n3, iterative diagonalization not perfect) - no stress tensor - no linear response
- 71. Thank you for your attention !