From Quantum Mechanics to Materials Design

Transcription

1 From to Materials Design Volker Eyert Center for Electronic Correlations and Magnetism Institute of Physics, University of Augsburg October 14, 2010 From to Materials Design

2 Outline 1 Density Functional Theory Full-Potential ASW Method 2 From to Materials Design

3 Outline 1 Density Functional Theory Full-Potential ASW Method 2 From to Materials Design

4 Outline Density Functional Theory Full-Potential ASW Method 1 Density Functional Theory Full-Potential ASW Method 2 From to Materials Design

5 Calculated Electronic Properties Density Functional Theory Full-Potential ASW Method Moruzzi, Janak, Williams (IBM, 1978) E/Ryd E/Ryd K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga 0 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Cohesive Energies ˆ= Stability WSR/au WSR/au K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga 2 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Wigner-Seitz-Rad. ˆ= Volume Bulk modulus/kbar Bulk modulus/kbar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga 0 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Compressibility ˆ= Hardness From to Materials Design

6 Density Functional Theory Full-Potential ASW Method Energy band structures from screened HF exchange Si, AlP, AlAs, GaP, and GaAs Experimental and theoretical bandgap properties Shimazaki, Asai JCP 132, (2010) From to Materials Design

7 Key Players Density Functional Theory Full-Potential ASW Method Hamiltonian (within Born-Oppenheimer approximation) H = H el,kin + H el el + H ext ] [ 2 2m 2 i = i e 2 4πǫ 0 i,j j i 1 r i r j + i v ext (r i ) where v ext (r i ) = 1 2 i e 2 4πǫ 0 µν µ ν Z µ Z ν R µ R ν e2 4πǫ 0 µ i Z µ R µ r i µ: ions with charge Z µ, i: electrons From to Materials Design

8 Key Players Density Functional Theory Full-Potential ASW Method Electron Density Operator ˆρ(r) = N δ(r r i ) = αβ i=1 χ α(r)χ β (r)a + α a β χ α : single particle state From to Materials Design

9 Key Players Density Functional Theory Full-Potential ASW Method Electron Density Operator ˆρ(r) = N δ(r r i ) = αβ i=1 χ α(r)χ β (r)a + α a β χ α : single particle state Electron Density ρ(r) = Ψ ˆρ(r) Ψ = α χ α (r) 2 n α Ψ : many-body wave function, n α : occupation number Normalization: N = d 3 r ρ(r) From to Materials Design

10 Key Players Density Functional Theory Full-Potential ASW Method Functionals Universal Functional (independent of ionic positions!) F = Ψ H el,kin + H el el Ψ Functional due to External Potential: Ψ H ext Ψ = Ψ v ext (r)δ(r r i ) Ψ i = d 3 r v ext (r)ρ(r) From to Materials Design

11 Authors Density Functional Theory Full-Potential ASW Method Pierre C. Hohenberg Walter Kohn Lu Jeu Sham From to Materials Design

12 Density Functional Theory Full-Potential ASW Method Hohenberg and Kohn, 1964: Theorems 1st Theorem The external potential v ext (r) is determined, apart from a trivial constant, by the electronic ground state density ρ(r). 2nd Theorem The total energy functional E[ρ] has a minimum equal to the ground state energy at the ground state density. From to Materials Design

13 Density Functional Theory Full-Potential ASW Method Hohenberg and Kohn, 1964: Theorems 1st Theorem The external potential v ext (r) is determined, apart from a trivial constant, by the electronic ground state density ρ(r). 2nd Theorem The total energy functional E[ρ] has a minimum equal to the ground state energy at the ground state density. Nota bene Both theorems are formulated for the ground state! Zero temperature! No excitations! From to Materials Design

14 Density Functional Theory Full-Potential ASW Method Levy, Lieb, : Constrained Search Percus-Levy partition From to Materials Design

15 Density Functional Theory Full-Potential ASW Method Levy, Lieb, : Constrained Search Variational principle E 0 = inf Ψ Ψ H Ψ = inf el,kin + H el el + H ext Ψ Ψ [ = inf inf el,kin + H el el Ψ + ρ(r) Ψ S(ρ) [ ] =: inf F LL [ρ] + d 3 r v ext (r)ρ(r) ρ(r) = inf ρ(r) E[ρ] S(ρ): set of all wave functions leading to density ρ F LL [ρ]: Levy-Lieb functional ] d 3 r v ext (r)ρ(r) From to Materials Design

16 Density Functional Theory Full-Potential ASW Method Levy, Lieb, : Constrained Search Levy-Lieb functional F LL [ρ] = inf Ψ H el,kin + H el el Ψ Ψ S(ρ) = T[ρ] + W xc [ρ] }{{} +1 e 2 d 3 r 2 4πǫ 0 = G[ρ] + 1 e 2 d 3 r 2 4πǫ 0 Functionals Kinetic energy funct.: T[ρ] Exchange-correlation energy funct.: W xc [ρ] Hartree energy funct.: 1 2 e 2 4πǫ 0 d 3 r d 3 r ρ(r)ρ(r ) r r d 3 r ρ(r)ρ(r ) r r d 3 r ρ(r)ρ(r ) r r not known! not known! known! From to Materials Design

17 Density Functional Theory Full-Potential ASW Method Kohn and Sham, 1965: Single-Particle Equations Ansatz 1 use different splitting of the functional G[ρ] T[ρ] + W xc [ρ] = G[ρ]! = T 0 [ρ] + E xc [ρ] 2 reintroduce single-particle wave functions Imagine: non-interacting electrons with same density Density: ρ(r) = occ α χ α(r) 2 known! Kinetic energy funct.: T 0 [ρ] = occ α d 3 r χ α (r) [ 2 2m 2 ]χ α (r) known! Exchange-correlation energy funct.: E xc [ρ] not known! From to Materials Design

18 Density Functional Theory Full-Potential ASW Method Kohn and Sham, 1965: Single-Particle Equations Euler-Lagrange Equations (Kohn-Sham Equations) ] δe[ρ] δχ α (r) ε αχ α (r) = [ 2 2m 2 + v eff (r) ε α χ α (r) =! 0 Effective potential: v eff (r) := v ext (r) + v H (r) + v xc (r) Exchange-correlation potential: v xc (r) := δe xc[ρ] δρ Single-particle energies : ε α (Lagrange-parameters, orthonormalization) not known! From to Materials Design

19 Density Functional Theory Full-Potential ASW Method Kohn and Sham, 1965: Local Density Approximation Be Specific! Approximate exchange-correlation energy functional E xc [ρ] = ρ(r)ε xc (ρ(r))d 3 r Exchange-correlation energy density ε xc (ρ(r)) depends on local density only! is calculated from homogeneous, interacting electron gas Exchange-correlation potential [ ] v xc (ρ(r)) = ρ {ρε xc(ρ)} ρ=ρ(r) From to Materials Design

20 0 10 W L G X W K 0 10 W L G X W K Density Functional Theory Full-Potential ASW Method Kohn and Sham, 1965: Local Density Approximation Limitations and Beyond LDA exact for homogeneous electron gas (within QMC) Spatial variation of ρ ignored include ρ(r),... Generalized Gradient Approximation (GGA) Self-interaction cancellation in v Hartree + v x violated Si Energy (ev) Bandgaps Si, exp: 1.11 ev Si, GGA: 0.57 ev Ge, exp: 0.67 ev Ge, GGA: 0.09 ev Ge Energy (ev) From to Materials Design

21 Muffin-Tin Approximation Density Functional Theory Full-Potential ASW Method John C. Slater Full Potential { spherical symmetric near nuclei v σ (r) : flat outside the atomic cores Muffin-Tin Approximation { vσ MT spherical symmetric in spheres (r) = constant in interstitial region From to Materials Design

22 Muffin-Tin Approximation Density Functional Theory Full-Potential ASW Method Full Potential (FeS 2 ) Muffin-Tin Potential v / a.u From to Materials Design

23 Muffin-Tin Approximation Density Functional Theory Full-Potential ASW Method Wave Function 1 solve Schrödingers eq. partial waves 2 match partial waves basis functions, augmented partial waves 3 use to expand wave function Muffin-Tin Potential From to Materials Design

24 Muffin-Tin Approximation Density Functional Theory Full-Potential ASW Method Flavors Muffin-Tin Approximation: touching spheres Atomic Sphere Approximation: space-filling spheres interstitial region formally removed only numerical functions in spheres minimal basis set (s, p, d) very high computational efficiency O(ASA) speed!!! makes potential more realistic systematic error in total energy bad! From to Materials Design

25 Density Functional Theory Full-Potential ASW Method Iron Pyrite: FeS 2 Pyrite Pa 3 (T 6 h ) a = Å NaCl structure sublattices occupied by iron atoms sulfur pairs sulfur pairs 111 axes x S = rotated FeS 6 octahedra From to Materials Design

26 FeS 2 : Structure Optimization Density Functional Theory Full-Potential ASW Method ASA + Code E (Ryd) x S From to Materials Design

27 Density Functional Theory Full-Potential ASW Method Basic Principles of the Full-Potential ASW Method Steps to be Taken remove total energy error due to overlap of atomic spheres reintroduce non-overlapping muffin-tin spheres restore interstitial region From to Materials Design

28 Density Functional Theory Full-Potential ASW Method Basic Principles of the Full-Potential ASW Method Steps to be Taken remove total energy error due to overlap of atomic spheres reintroduce non-overlapping muffin-tin spheres restore interstitial region find representation of electron density and full potential inside muffin-tin spheres in the interstitial region From to Materials Design

29 Density Functional Theory Full-Potential ASW Method Basic Principles of the Full-Potential ASW Method Steps to be Taken remove total energy error due to overlap of atomic spheres reintroduce non-overlapping muffin-tin spheres restore interstitial region find representation of electron density and full potential find representation of products of the wave function inside muffin-tin spheres in the interstitial region From to Materials Design

30 Density Functional Theory Full-Potential ASW Method Basic Principles of the Full-Potential ASW Method Steps to be Taken remove total energy error due to overlap of atomic spheres reintroduce non-overlapping muffin-tin spheres restore interstitial region find representation of electron density and full potential find representation of products of the wave function find representation of products of the basis functions inside muffin-tin spheres in the interstitial region From to Materials Design

31 Density Functional Theory Full-Potential ASW Method Basic Principles of the Full-Potential ASW Method Steps to be Taken remove total energy error due to overlap of atomic spheres reintroduce non-overlapping muffin-tin spheres restore interstitial region find representation of electron density and full potential find representation of products of the wave function find representation of products of the basis functions inside muffin-tin spheres use spherical-harmonics expansions in the interstitial region From to Materials Design

32 Density Functional Theory Full-Potential ASW Method Basic Principles of the Full-Potential ASW Method Steps to be Taken remove total energy error due to overlap of atomic spheres reintroduce non-overlapping muffin-tin spheres restore interstitial region find representation of electron density and full potential find representation of products of the wave function find representation of products of the basis functions inside muffin-tin spheres use spherical-harmonics expansions in the interstitial region no exact spherical-wave representation available! From to Materials Design

33 Density Functional Theory Full-Potential ASW Method From Wave Functions to Electron Density Density inside MT-Spheres (Al) ρ v / a.u From to Materials Design

34 Density Functional Theory Full-Potential ASW Method From Wave Functions to Electron Density Products of Spherical Waves in Interstitial Region expand in spherical waves would be efficient coefficients/integrals not known analytically Methfessel, 1988: match values and slopes at MT-sphere surfaces From to Materials Design

35 Density Functional Theory Full-Potential ASW Method From Wave Functions to Electron Density Products of Spherical Waves in Interstitial Region expand in spherical waves match values and slopes at MT-sphere surfaces Density from Value/Slope Matching at MT-Radii (Al) ρ v / a.u. ρ v / a.u From to Materials Design

36 Comparison of Approaches Density Functional Theory Full-Potential ASW Method Ole K. Andersen 1975 ASA geometry used for basis functions minimal basis set ASA geometry used for density and potential error in total energy good! bad! From to Materials Design

37 Comparison of Approaches Density Functional Theory Full-Potential ASW Method Ole K. Andersen 1975 ASA geometry used for basis functions minimal basis set ASA geometry used for density and potential error in total energy good! bad! Michael S. Methfessel 1988 MT geometry used for density and potential accurate total energy MT geometry used for basis functions large basis set good! bad! From to Materials Design

38 Comparison of Approaches Density Functional Theory Full-Potential ASW Method Ole K. Andersen 1975 ASA geometry used for basis functions ASA geometry used for density and potential good! bad! Michael S. Methfessel 1988 MT geometry used for density and potential MT geometry used for basis functions good! bad! present approach 2006 ASA geometry used for basis functions minimal basis set O(ASA) speed MT geometry used for density and potential accurate total energy great! great! From to Materials Design

39 Density Functional Theory Full-Potential ASW Method Implementation: Augmented Spherical Wave Method 0th Generation ASW (Williams, Kübler, Gelatt, 1970s) 1st Generation (VE, 1990s) PRB 19, 6094 (1979) new implementation (accurate, stable, portable) VE, Int. J. Quantum Chem. 77, 1007 (2000) VE, Lect. Notes Phys. 719 (Springer, 2007) xanderson convergence acceleration scheme VE, J. Comput. Phys. 124, 271 (1996) all LDA- and GGA-parametrizations still based on atomic-sphere approximation VE, Höck, PRB 57, (1998) From to Materials Design

40 Density Functional Theory Full-Potential ASW Method Implementation: Augmented Spherical Wave Method 2nd Generation ASW (VE, 2000s) based on 1st generation code full-potential ASW method electron densities, spin densities electric field gradients elastic properties, phonon spectra optical properties based on linear-response theory direct calculation of Rσ and Iσ no Kramers-Kronig relations needed at O(ASA) speed! transport properties, thermoelectrics LDA+U method all flavors for double-counting terms (AMF, FLL, DFT) VE, Lect. Notes Phys. (2nd ed., Springer, 2011) From to Materials Design

41 Density Functional Theory Full-Potential ASW Method Iron Pyrite: FeS 2 Pyrite Pa 3 (T 6 h ) a = Å NaCl structure sublattices occupied by iron atoms sulfur pairs sulfur pairs 111 axes x S = rotated FeS 6 octahedra From to Materials Design

42 FeS 2 : Structure Optimization Density Functional Theory Full-Potential ASW Method ASA + Code E (Ryd) x S From to Materials Design

43 FeS 2 : Structure Optimization Density Functional Theory Full-Potential ASW Method ASA + Code E (Ryd) x S Full-Potential Code E (Ryd) x S at O(ASA) speed! From to Materials Design

44 Phase Stability in Silicon Density Functional Theory Full-Potential ASW Method ASA + Code E (Ryd) dia hex β-tin sc bcc fcc V (a B ) Bad β-tin structure most stable # nature (diamond structure) From to Materials Design

45 Phase Stability in Silicon Density Functional Theory Full-Potential ASW Method ASA + Code Full-Potential Code E (Ryd) dia hex β-tin sc bcc fcc V (a B ) E (Ryd) dia hex β-tin sc bcc fcc V (a B ) New! at O(ASA) speed! diamond structure most stable pressure induced phase transition to β-tin structure From to Materials Design

46 LTO(Γ)-Phonon in Silicon Density Functional Theory Full-Potential ASW Method ASA + Code E (Ryd) x Si Bad no stable Si position # nature From to Materials Design

47 LTO(Γ)-Phonon in Silicon Density Functional Theory Full-Potential ASW Method ASA + Code Full-Potential Code E (Ryd) E (Ryd) x Si x Si New! at O(ASA) speed! phonon frequency: f calc = THz (f exp = THz) From to Materials Design

48 Outline 1 Density Functional Theory Full-Potential ASW Method 2 From to Materials Design

49 Industrial Applications Computational Materials Engineering Automotive Energy & Power Generation Aerospace Steel & Metal Alloys Glass & Ceramics Electronics Display & Lighting Chemical & Petrochemical Drilling & Mining From to Materials Design

50 Delafossites: ABO 2 Delafossite Structure Building Blocks rhombohedral lattice triangular A-atom layers BO 2 sandwich layers B-atoms octahedrally coordinated linear O A O bonds From to Materials Design

51 Delafossites: ABO 2 Delafossite Structure Building Blocks rhombohedral lattice triangular A-atom layers BO 2 sandwich layers B-atoms octahedrally coordinated linear O A O bonds Issues dimensionality geometric frustration play chemistry From to Materials Design

52 Delafossites: ABO 2 Delafossite Structure Prototype Materials CuFeO 2, CuCrO 2 CuCoO 2, CuRhO 2 CuAlO 2, CuGaO 2, CuInO 2,... PdCrO 2, PdCoO 2, PdRhO 2, PtCoO 2 Properties semiconductors, AF interactions, (distorted) triangular non-magn. semicond., thermopower wide-gap semicond., p-type TCO very good metals, high anisotropy From to Materials Design

53 Delafossites: ABO 2 Delafossite Structure Prototype Materials CuFeO 2, CuCrO 2 CuCoO 2, CuRhO 2 CuAlO 2, CuGaO 2, CuInO 2,... PdCrO 2, PdCoO 2, PdRhO 2, PtCoO 2 Properties semiconductors, AF interactions, (distorted) triangular non-magn. semicond., thermopower wide-gap semicond., p-type TCO very good metals, high anisotropy From to Materials Design

54 Delafossites: ABO 2 Delafossite Structure Prototype Materials CuFeO 2, CuCrO 2 CuCoO 2, CuRhO 2 CuAlO 2, CuGaO 2, CuInO 2,... PdCrO 2, PdCoO 2, PdRhO 2, PtCoO 2 Properties semiconductors, AF interactions, (distorted) triangular non-magn. semicond., thermopower wide-gap semicond., p-type TCO very good metals, high anisotropy From to Materials Design

55 Delafossites: ABO 2 Delafossite Structure Prototype Materials CuFeO 2, CuCrO 2 CuCoO 2, CuRhO 2 CuAlO 2, CuGaO 2, CuInO 2,... PdCrO 2, PdCoO 2, PdRhO 2, PtCoO 2 Properties semiconductors, AF interactions, (distorted) triangular non-magn. semicond., thermopower wide-gap semicond., p-type TCO very good metals, high anisotropy From to Materials Design

56 PdCoO 2 and PtCoO 2 Delafossite Structure Experimental Results very low resistivity anisotropy ratio 200 PES: only Pd 4d states at E F PES/IPES: E F in shallow DOS minimum high thermopower on doping? From to Materials Design

57 PdCoO 2 and PtCoO 2 Delafossite Structure Experimental Results very low resistivity anisotropy ratio 200 PES: only Pd 4d states at E F PES/IPES: E F in shallow DOS minimum high thermopower on doping? Open Issues role of Pd 4d, Co 3d, and O 2p orbitals? From to Materials Design

58 Structure Optimization in PdCoO 2 Structural Data experiment theory a = 2.83 Å c = Å z O = a = Å c = Å z O = Total energy surface VE, R. Frésard, A. Maignan, Chem. Mat. 20, 2370 (2008) From to Materials Design

59 Electronic Properties of PdCoO 2 Partial Densities of States DOS (1/eV) Pd 4d Co 3d t 2g Co 3d e g O 2p (E - E F ) (ev) Results Co 3d-O 2p hybridization CoO 6 octahedra: Co 3d t 2g and e g Co 3d 6 (Co 3+ ) LS Pd 4d 9 (Pd 1+ ) Co 3d, O 2p: very small DOS at E F VE, R. Frésard, A. Maignan, Chem. Mat. 20, 2370 (2008) From to Materials Design

60 Electronic Properties of PdCoO 2 Partial Densities of States DOS (1/eV) Pd 4d xy,x 2 -y 2 Pd 4d xz,yz Pd 4d 3z 2 -r 2 O 2p z (E - E F ) (ev) Results broad Pd d xy,x 2 y 2 bands short in-plane Pd-Pd distance non-bonding Pd d xz,yz bands strong Pd 4d 3z 2 r 2-O 2p hybridization states at E F : Pd d xy,x 2 y 2, d 3z 2 r 2 VE, R. Frésard, A. Maignan, Chem. Mat. 20, 2370 (2008) From to Materials Design

61 Electronic Properties of PdCoO 2 Fermi Surface Results quasi-2d single band crossing E F but: bands below E F disperse along Γ-A VE, R. Frésard, A. Maignan, Chem. Mat. 20, 2370 (2008) From to Materials Design

62 CuFeO 2 Delafossite Structure Basics semiconductor AF interactions triangular lattice From to Materials Design

63 CuFeO 2 Delafossite Structure Basics semiconductor AF interactions triangular lattice Open Issues frustration vs. long-range order role of Cu 3d orbitals? role of Fe 3d and O 2p orbitals? From to Materials Design

64 CuFeO 2 Previous Neutron Data T N1 = 16 K, T N2 = 11 K Θ CW = 90 K magnetic supercells no structural distortion m Fe 3+ = 4.4 µ B From to Materials Design

65 CuFeO 2 Previous Neutron Data T N1 = 16 K, T N2 = 11 K Θ CW = 90 K magnetic supercells no structural distortion m Fe 3+ = 4.4 µ B Band Calculations rhombohedral structure m Fe = 0.9 µ B, m Fe = 3.8 µ B E g = 0 in LDA, GGA PES, XES From to Materials Design

66 CuFeO 2 Previous Neutron Data T N1 = 16 K, T N2 = 11 K Θ CW = 90 K magnetic supercells no structural distortion m Fe 3+ = 4.4 µ B Band Calculations rhombohedral structure m Fe = 0.9 µ B, m Fe = 3.8 µ B E g = 0 in LDA, GGA PES, XES New Neutron Data magnetic supercells monoclinic structure below 4 K From to Materials Design

67 CuFeO 2 Previous Neutron Data T N1 = 16 K, T N2 = 11 K Θ CW = 90 K magnetic supercells no structural distortion m Fe 3+ = 4.4 µ B New Neutron Data magnetic supercells monoclinic structure below 4 K Band Calculations rhombohedral structure m Fe = 0.9 µ B, m Fe = 3.8 µ B E g = 0 in LDA, GGA PES, XES Open Issues spin-state of Fe? influence of monoc. structure? From to Materials Design

68 Electronic Properties of CuFeO 2 Partial Densities of States DOS (1/eV) Cu 3d Fe 3d t 2g Fe 3d e g O 2p (E - E F ) (ev) Results Fe 3d-O 2p hybridization FeO 6 octahedra: Fe 3d t 2g and e g Cu 4d 10 (Cu 1+ ) Fe 3d t 2g sharp peak at E F VE, R. Frésard, A. Maignan, PRB 78, (2008) From to Materials Design

69 Electronic Properties of CuFeO 2 Fermi Surface Results strongly 3D FS PdCoO 2 VE, R. Frésard, A. Maignan, PRB 78, (2008) From to Materials Design

70 Magnetic Properties of CuFeO 2 Total Energies (mryd/f.u.), Magn. Moms. (µ B ), Band Gaps (ev) structure magn. order E m Fe m O E g rhomb. spin-deg rhomb. ferro (LS) rhomb. ferro (IS) rhomb. ferro (HS) VE, R. Frésard, A. Maignan, PRB 78, (2008) From to Materials Design

71 Magnetic Properties of CuFeO 2 Total Energies (mryd/f.u.), Magn. Moms. (µ B ), Band Gaps (ev) structure magn. order E m Fe m O E g rhomb. spin-deg rhomb. ferro (LS) rhomb. ferro (IS) rhomb. ferro (HS) monoc. spin-deg monoc. ferro (LS) monoc. ferro (IS) monoc. ferro (HS) VE, R. Frésard, A. Maignan, PRB 78, (2008) From to Materials Design

72 Magnetic Properties of CuFeO 2 Total Energies (mryd/f.u.), Magn. Moms. (µ B ), Band Gaps (ev) structure magn. order E m Fe m O E g rhomb. spin-deg rhomb. ferro (LS) rhomb. ferro (IS) rhomb. ferro (HS) monoc. spin-deg monoc. ferro (LS) monoc. ferro (IS) monoc. ferro (HS) monoc. antiferro 46.0 ±3.72 ± VE, R. Frésard, A. Maignan, PRB 78, (2008) From to Materials Design

73 CuRhO 2 Delafossite Structure Experimental Findings semiconductor high thermopower on hole doping Rh 3+ Mg 2+ up to 12% high T -independent power factor From to Materials Design

74 CuRhO 2 Delafossite Structure Experimental Findings semiconductor high thermopower on hole doping Rh 3+ Mg 2+ up to 12% high T -independent power factor Open Issues origin of high thermopower role of Cu 3d orbitals? role of Rh 4d and O 2p orbitals? From to Materials Design

75 Electronic Properties of CuRhO 2 Partial Densities of States DOS (1/eV) Cu 3d Rh 4d t 2g Rh 4d e g O 2p (E - E V ) (ev) Results Rh 4d-O 2p hybridization RhO 6 octahedra: Rh 4d t 2g and e g E g 0.75 ev Cu 4d 10 (Cu 1+ ) electronic structure: strongly 3D A. Maignan, VE et al., PRB 80, (2009) From to Materials Design

76 Thermoelectric Power of CuRh 1 x Mg x O 2 Experiment S (µv K -1 ) x= x=0.01 x= x= T (K) Theory: S xx S (µv/k) x = 0.10 x = 0.06 x = 0.04 x = 0.02 x = T (K) A. Maignan, VE et al., PRB 80, (2009) From to Materials Design

77 2D Electron Gas at LaAlO 3 -SrTiO 3 Interface From to Materials Design

78 2D Electron Gas at LaAlO 3 -SrTiO 3 Interface Insulator-Metal Transition Chen, Kolpak, Ismail-Beigi, Adv. Mater. 22, 2881 (2010) From to Materials Design

79 Slab Calculations for the LaAlO 3 -SrTiO 3 Interface Structural setup of calculations central region: 5 layers SrTiO 3, TiO 2 -terminated sandwiches: 2 to 5 layers LaAlO 3, AlO 2 surface vacuum region 20 Å inversion symmetry lattice constant of SrTiO 3 from GGA (3.944 Å) From to Materials Design

80 Slab Calculations for the LaAlO 3 -SrTiO 3 Interface Calculational method Vienna Ab Initio Simulation Package (VASP) GGA-PBE Steps: 1 optimization of SrTiO 3 lattice constant 2 slab calculations full relaxation of all atomic positions k-points Γ-centered k-mesh Methfessel-Paxton BZ-integration From to Materials Design

81 Slab Calculations for the LaAlO 3 -SrTiO 3 Interface Ideal Structural relaxation AlO 2 surface layers strong inward relaxation weak buckling LaO layers strong buckling AlO 2 subsurface layers buckling TiO 2 interface layers small outward relaxation Optimized From to Materials Design

82 Slab Calculations for the LaAlO 3 -SrTiO 3 Interface DOS (1/eV) L 3L 4L 5L (E - E F ) (ev) From to Materials Design

83 Tunneling Spectroscopy of LaAlO 3 -SrTiO 3 Interface What is the origin of the 2-DEG? Intrinsic mechanism or defect-doping? I t V s STM tip z x Ti 4 unit cells LaAlO 3 interface electron system SrTiO 3 M. Breitschaft et al., PRB 81, (2010) From to Materials Design

84 Tunneling Spectroscopy of LaAlO 3 -SrTiO 3 Interface VS (V) 4uc LaAlO 3 on SrTiO 3, tunneling data NDC (arb. units) (a) exp. 4uc LaAlO 3 on SrTiO 3, LDA calculations, DOS of interface Ti (b) LDA E - EF (ev) 4uc LaAlO 3 on SrTiO 3, LDA+U calculations, DOS of interface Ti (c) LDA+U E - EF (ev) From to Materials Design

85 Tunneling Spectroscopy of LaAlO 3 -SrTiO 3 Interface VS (V) bulk SrTiO 3, LDA calculations, conduction band DOS NDC (arb. units) (a) (b) exp. LDA DOS (1/eV) (c) E - EF (ev) LDA+U (E - E V ) (ev) E - EF (ev) From to Materials Design

86 2D Electron Liquid State at LaAlO 3 -SrTiO 3 Interface Al x Ga 1-x As GaAs CB LaAlO 3 SrTiO 3 E F VB (a) s E F (b) Ti 3d M. Breitschaft et al., PRB 81, (2010) From to Materials Design

87 Critical review of the Local Density Approximation Limitations and Beyond Self-interaction cancellation in v Hartree + v x violated Repair using exact Hartree-Fock exchange functional class of hybrid functionals PBE0 HSE03, HSE06 E PBE0 xc = 1 4 E HF x E PBE x + E PBE c E HSE xc = 1 4 E HF,sr,µ x E PBE,sr,µ x + E PBE,lr,µ x + E PBE c based on decomposition of Coulomb kernel 1 r = S µ(r) + L µ (r) = erfc(µr) + erf(µr) r r From to Materials Design

88 Critical review of the Local Density Approximation 0 10 W L G X W K W L G X W K Limitations and Beyond Self-interaction cancellation in v Hartree + v x violated Repair using exact Hartree-Fock exchange functional class of hybrid functionals GGA Energy (ev) Si bandgap exp: 1.11 ev GGA: 0.57 ev HSE: 1.15 ev HSE Energy (ev) From to Materials Design

89 Critical review of the Local Density Approximation 0 10 W L G X W K W L G X W K Limitations and Beyond Self-interaction cancellation in v Hartree + v x violated Repair using exact Hartree-Fock exchange functional class of hybrid functionals GGA Energy (ev) Ge bandgap exp: 0.67 ev GGA: 0.09 ev HSE: 0.66 ev HSE Energy (ev) From to Materials Design

90 Critical review of the Local Density Approximation Calculated vs. experimental bandgaps From to Materials Design

91 SrTiO 3 GGA 10 8 DOS (1/eV) (E - E V ) (ev) Bandgap GGA: 1.6 ev, exp.: 3.2 ev From to Materials Design

92 SrTiO 3 GGA 10 HSE DOS (1/eV) 6 4 DOS (1/eV) (E - E V ) (ev) (E - E V ) (ev) Bandgap GGA: 1.6 ev, HSE: 3.1 ev, exp.: 3.2 ev From to Materials Design

93 LaAlO 3 GGA DOS (1/eV) (E - E V ) (ev) Bandgap GGA: 3.5 ev, exp.: 5.6 ev From to Materials Design

94 LaAlO 3 GGA 25 HSE DOS (1/eV) DOS (1/eV) (E - E V ) (ev) (E - E V ) (ev) Bandgap GGA: 3.5 ev, HSE: 5.0 ev, exp.: 5.6 ev From to Materials Design

95 Metal-Insulator Transition of VO 2 Morin, PRL 1959 Metal-Insulator Transitions (MIT) VO 2 (d 1 ) 1st order, 340 K, σ 10 4 rutile M 1 (monoclinic) V 2 O 3 (d 2 ) 1st order, 170 K, σ 10 6 corundum monoclinic paramagn. AF order Origin of the MIT??? Structural Changes? Electron Correlations? From to Materials Design

96 Metal-Insulator Transition of VO 2 Octahedral Coordination V 3d e σ g V 3d t 2g E F σ π Rutile Structure simple tetragonal P4 2 /mnm (D 14 4h ) O 2p π σ V 3d-O 2p hybridization σ, σ (p-de σ g ) π, π (p-dt 2g ) From to Materials Design

97 Metal-Insulator Transition of VO 2 Octahedral Coordination e σ g Orbitals V 3d e σ g σ V 3d t 2g O 2p E F π π σ t 2g Orbitals V 3d-O 2p hybridization σ, σ (p-de σ g ) π, π (p-dt 2g ) From to Materials Design

98 Metal-Insulator Transition of VO 2 Octahedral Chains Rutile Structure simple tetragonal P4 2 /mnm (D 14 4h ) From to Materials Design

99 Metal-Insulator Transition of VO 2 Octahedral Coordination t 2g Orbitals V 3d e σ g σ V 3d t 2g E F π e π g = π a 1g = d O 2p π σ From to Materials Design

100 Metal-Insulator Transition of VO 2 Rutile Structure M 1 -Structure Structural Changes V-V dimerization c R antiferroelectric displacement c R From to Materials Design

101 Metal-Insulator Transition of VO 2 Rutile M 1 π π d d E F d Goodenough, metal-metal dimerization c R splitting into d, d antiferroelectric displacement c R upshift of π Zylbersztejn and Mott, 1975 splitting of d by electronic correlations upshift of π unscreenes d electrons From to Materials Design

102 Metal-Insulator Transition of VO 2 Other Phases Ì Ãµ Ù Ù Ä Ù ¼¼ ÄÄ Ä ÄÄ Ù Ù Ù Ù Ù Ù Ä Ì ¾¼¼ ÄÄ Å ½ Ä Ù ÄÄ Ù Ù Ù ÙÄ ½¼¼ Ù Ù Ù ÄÄ Ù Ù Ù Ù Ù Ä Ù Ù ÄÄ Ù Ä ¹ ¼º¼½ ¼º¼¾ ¼º¼ ¼º¼ ¼º¼ Ü Å ¾ Ê doping with Cr, Al, Fe, Ga uniaxial pressure 110 Cr x V 1 x O 2 Pouget, Launois, 1976 From to Materials Design

103 Electronic Structure in Detail Rutile Structure molecular-orbital picture octahedral crystal field = V 3d t 2g /e g V 3d O 2p hybridization DOS (1/eV) V 3d t 2g V 3d e g O 2p (E - E F ) (ev) Ann. Phys. (Leipzig) 11, 650 (2002) From to Materials Design

104 Electronic Structure in Detail Rutile Structure molecular-orbital picture V 3d x 2 -y 2 V 3d yz V 3d xz octahedral crystal field = V 3d t 2g /e g V 3d O 2p hybridization t 2g at E F : d x 2 y 2, d yz, d xz n(d x 2 y 2) n(d yz) n(d xz ) DOS (1/eV) (E - E F ) (ev) Ann. Phys. (Leipzig) 11, 650 (2002) From to Materials Design

105 Electronic Structure in Detail Rutile Structure M 1 Structure V 3d x 2 -y 2 V 3d yz V 3d xz V 3d x 2 -y 2 V 3d yz V 3d xz DOS (1/eV) DOS (1/eV) (E - E F ) (ev) (E - E F ) (ev) bonding-antibonding splitting of d bands energetical upshift of π bands = orbital ordering optical band gap on the verge of opening From to Materials Design

106 New Calculations: GGA vs. HSE Rutile Structure GGA Rutile Structure HSE 5 4 V 3d O 2p 5 4 V 3d O 2p DOS (1/eV) 3 2 DOS (1/eV) (E - E F ) (ev) (E - E F ) (ev) Rutile Structure: GGA = HSE broadening of O 2p and V 3d t 2g (!) bands splitting within V 3d t 2g bands From to Materials Design

107 New Calculations: GGA vs. HSE M 1 Structure GGA M 1 Structure HSE 5 4 V 3d O 1 2p O 2 2p 5 4 V 3d O 1 2p O 2 2p DOS (1/eV) 3 2 DOS (1/eV) (E - E F ) (ev) (E - E F ) (ev) M 1 Structure: GGA = HSE splitting of d bands, upshift of π bands optical bandgap of 1 ev From to Materials Design

108 New Calculations: GGA vs. HSE M 1 Structure GGA M 1 Structure HSE Energy (ev) Energy (ev) C Y G B A E Z C Y G B A E Z M 1 Structure: GGA = HSE splitting of d bands, upshift of π bands optical bandgap of 1 ev From to Materials Design

109 New Calculations: GGA vs. HSE Total and Summed Density of States (1/eV) Partial Density of States (1/eV) Total and Summed Density of States (1/eV) Partial Density of States (1/eV) M 2 Structure GGA M 2 Structure HSE Energy (ev) Energy (ev) M 2 Structure: GGA = HSE localized magentic moment of 1 µ B optical bandgap of 1.6 ev From to Materials Design

110 Unified Picture Rutile-Related Transition-Metal Dioxides VO 2 (3d 1 ), NbO 2 (4d 1 ), MoO 2 (4d 2 ) (WO 2 (5d 2 ), TcO 2 (4d 3 ), ReO 2 (5d 3 )) instability against similar local distortions metal-metal dimerization c R antiferroelectric displacement c R ( accidental ) metal-insulator transition of the d 1 -members VE et al., J. Phys.: CM 12, 4923 (2000) VE, Ann. Phys. 11, 650 (2002) VE, EPL 58, 851 (2002) J. Moosburger-Will et al., PRB 79, (2009) From to Materials Design

111 Success Stories Basics DFT (exact, ground state) LDA, GGA,... Percus-Levy partition From to Materials Design

112 Success Stories Basics DFT (exact, ground state) Implementation Muffins and beyond LDA, GGA,... Full-Potential ASW Percus-Levy partition Full-Potential Code E (Ryd) xsi From to Materials Design

113 Success Stories Delafossites 2D 3D From to Materials Design

114 Success Stories Delafossites 2D 3D LaAlO 3 /SrTiO 3 DOS (1/eV) L 3L 4L 5L (E - E F ) (ev) From to Materials Design

115 Success Stories Delafossites 2D 3D LaAlO 3 /SrTiO 3 DOS (1/eV) L 3L 4L 5L (E - E F ) (ev) Metal-Insulator Transitions in VO V 3d O 1 2p O 2 2p 5 4 V 3d O 1 2p O 2 2p DOS (1/eV) 3 2 DOS (1/eV) (E - E F ) (ev) (E - E F ) (ev) From to Materials Design

116 Visions Methods Materials From to Materials Design

117 Acknowledgments Augsburg M. Breitschaft, U. Eckern, K.-H. Höck, S. Horn, R. Horny, T. Kopp, J. Kündel, J. Mannhart, J. Moosburger-Will, N. Pavlenko, W. Scherer, D. Vollhardt TRR 80 Caen R. Frésard, S. Hébert, A. Maignan, C. Martin Darmstadt/Jülich P. C. Schmidt, M. Stephan, J. Sticht Europe/USA E. Wimmer, A. Mavromaras, P. Saxe, R. Windiks, W. Wolf From to Materials Design

118 Acknowledgments Augsburg M. Breitschaft, U. Eckern, K.-H. Höck, S. Horn, R. Horny, T. Kopp, J. Kündel, J. Mannhart, J. Moosburger-Will, N. Pavlenko, W. Scherer, D. Vollhardt TRR 80 Caen R. Frésard, S. Hébert, A. Maignan, C. Martin Darmstadt/Jülich P. C. Schmidt, M. Stephan, J. Sticht Europe/USA E. Wimmer, A. Mavromaras, P. Saxe, R. Windiks, W. Wolf Würzburg Thank You for Your Attention! From to Materials Design