Metal-Insulator Transitions of the Vanadates. New Perspectives of an Old Mystery

Transcription

1 : New Perspectives of an Old Mystery Volker Eyert Center for Electronic Correlations and Magnetism Institute of Physics, University of Augsburg November 15, 21

2 Outline Basic Theories 1 Basic Theories Density Functional Theory Full-Potential ASW Method 2

3 Outline Basic Theories 1 Basic Theories Density Functional Theory Full-Potential ASW Method 2

4 Outline Basic Theories Density Functional Theory Full-Potential ASW Method 1 Basic Theories Density Functional Theory Full-Potential ASW Method 2

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 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 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Compressibility ˆ= Hardness

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, (21)

7 Key Players Basic Theories 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 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 µν µ ν Z µ Z ν R µ R ν e2 4πǫ µ i Z µ R µ r i µ: ions with charge Z µ, i: electrons

8 Key Players Basic Theories 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

9 Key Players Basic Theories 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)

10 Key Players Basic Theories 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)

11 Authors Basic Theories Density Functional Theory Full-Potential ASW Method Pierre C. Hohenberg Walter Kohn Lu Jeu Sham

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 eual to the ground state energy at the ground state density.

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 eual to the ground state energy at the ground state density. Nota bene Both theorems are formulated for the ground state! Zero temperature! No excitations!

14 Basic Theories Density Functional Theory Full-Potential ASW Method Levy, Lieb, : Constrained Search Percus-Levy partition

15 Density Functional Theory Full-Potential ASW Method Levy, Lieb, : Constrained Search Variational principle E = 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)

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 = G[ρ] + 1 e 2 d 3 r 2 4πǫ Functionals Kinetic energy funct.: T[ρ] Exchange-correlation energy funct.: W xc [ρ] Hartree energy funct.: 1 2 e 2 4πǫ 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!

17 Density Functional Theory Full-Potential ASW Method Kohn and Sham, 1965: Single-Particle Euations Ansatz 1 use different splitting of the functional G[ρ] T[ρ] + W xc [ρ] = G[ρ]! = T [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 [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!

18 Density Functional Theory Full-Potential ASW Method Kohn and Sham, 1965: Single-Particle Euations Euler-Lagrange Euations (Kohn-Sham Euations) ] δe[ρ] δχ α (r) ε αχ α (r) = [ 2 2m 2 + v eff (r) ε α χ α (r) =! 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!

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)

20 1 W L G X W K 1 W L G X W K Basic Theories 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:.57 ev Ge, exp:.67 ev Ge, GGA:.9 ev Ge Energy (ev)

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

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

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

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!

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

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

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

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

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

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

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

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!

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

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

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

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!

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!

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 26 ASA geometry used for basis functions minimal basis set O(ASA) speed MT geometry used for density and potential accurate total energy great! great!

39 Density Functional Theory Full-Potential ASW Method Implementation: Augmented Spherical Wave Method th Generation ASW (Williams, Kübler, Gelatt, 197s) 1st Generation (VE, 199s) PRB 19, 694 (1979) new implementation (accurate, stable, portable) VE, Int. J. Quantum Chem. 77, 17 (2) VE, Lect. Notes Phys. 719 (Springer, 27) 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)

40 Density Functional Theory Full-Potential ASW Method Implementation: Augmented Spherical Wave Method 2nd Generation ASW (VE, 2s) 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, 211)

41 ASW Method: Further Reading Density Functional Theory Full-Potential ASW Method

42 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

43 FeS 2 : Structure Optimization Density Functional Theory Full-Potential ASW Method ASA + Code E (Ryd) x S

44 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!

45 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)

46 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

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

48 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 freuency: f calc = THz (f exp = THz)

49 Outline Basic Theories 1 Basic Theories Density Functional Theory Full-Potential ASW Method 2

50 Metal-Insulator Transition of VO 2 Morin, PRL 1959 Metal-Insulator Transitions (MIT) VO 2 (d 1 ) 1st order, 34 K, σ 1 4 rutile M 1 (monoclinic) V 2 O 3 (d 2 ) 1st order, 17 K, σ 1 6 corundum monoclinic paramagn. AF order Origin of the MIT??? Structural Changes? Electron Correlations?

51 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 )

52 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 )

53 Metal-Insulator Transition of VO 2 Octahedral Chains Rutile Structure simple tetragonal P4 2 /mnm (D 14 4h )

54 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

55 Metal-Insulator Transition of VO 2 Rutile Structure M 1 -Structure Structural Changes V-V dimerization c R antiferroelectric displacement c R

56 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

57 Metal-Insulator Transition of VO 2 Other Compounds d d 1 d 2 d 3 d 4 d 5 d 6 3d TiO 2 VO 2 CrO 2 MnO 2 (S) (M S) (F M) (AF S) 4d NbO 2 MoO 2 TcO 2 RuO 2 RhO 2 (M S) (M) (M) (M) (M) 5d TaO 2 WO 2 ReO 2 OsO 2 IrO 2 PtO 2 (?) (M) (M) (M) (M) (M) deviations from rutile, M = metal, S = semiconductor F/AF = ferro-/antiferromagnet

58 Metal-Insulator Transition of VO 2 Other Compounds c/a number of d electrons 3d 4d 5d

59 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 11 Cr x V 1 x O 2 Pouget, Launois, 1976

60 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, 65 (22)

61 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, 65 (22)

62 Electronic Structure in Detail u u u u u u... d = d x 2 y 2 π = (d yz, d xz ) E ev EF k x Z k z A M R X k y E ev EF -1. Γ X R Z Γ R A Γ M A Z -1. Γ X R Z Γ R A Γ M A Z 1D-dispersion of d bands 3D-dispersion of π bands no hybridization between d and π

63 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 = embedded Peierls instability energetical upshift of π bands = orbital ordering optical band gap on the verge of opening

64 Electronic Structure in Detail u... u. u. u u u.. u.. u... Rutile Structure E ev Γ X R Z Γ R A Γ M A Z EF k x Z k z E B D A C Y k y M 1 Structure E ev Γ Y C Z Γ A E Z Γ B D Z EF bonding-antibonding splitting of d bands = embedded Peierls instability energetical upshift of π bands = orbital ordering optical band gap on the verge of opening

65 Further Investigations Cluster-DMFT Calculations Rutile-VO 2 moderately correlated metal M 1 -VO 2 correlations strong/weak on d /π optical band gap of.6 ev Phase Transition correlation-assisted Peierls transition S. Biermann, A. Poteryaev, A. I. Lichtenstein, A. Georges PRL 94, 2644 (25)

66 Further Investigations Rutile M 1 1. ρ(ω) LDA (dashed) DMFT (solid) a 1g π e g VO 2 rutile ρ(ω) LDA (dashed) cluster DMFT (solid) a 1g e g π VO 2 M ω[ev] ω [ev]

67 Transition-Metal Dioxides: Partial d t 2g DOS VO 2 (3d 1 ) NbO 2 (4d 1 ) MoO 2 (4d 2 ) V 3d x 2 -y 2 V 3d yz V 3d xz Nb 4d x 2 -y 2 Nb 4d yz Nb 4d xz 2 Mo 4d x 2 -y 2 Mo 4d yz Mo 4d xz DOS (1/eV) 1.5 DOS (1/eV).8.6 DOS (1/eV) (E - E F ) (ev) (E - E F ) (ev) (E - E F ) (ev) V 3d x 2 -y 2 V 3d yz V 3d xz Nb 4d x 2 -y 2 Nb 4d yz Nb 4d xz Mo 4d x 2 -y 2 Mo 4d yz Mo 4d xz DOS (1/eV) DOS (1/eV).8.6 DOS (1/eV) (E - E F ) (ev) (E - E F ) (ev) (E - E F ) (ev)

68 Transition-Metal Dioxides: Partial d t 2g DOS VO 2 (3d 1 ) X R Z R A M A Z E ev EF Y C Z A E Z B D Z E ev EF NbO 2 (4d 1 ) Ê Ê Å Î ¾º¼ º¼ È Æ Î ¾º¼ º¼ MoO 2 (4d 2 ) X R Z Z A M A Z E ev EF Y C Z A E Z B D Z E ev EF

69 Electronic Structure in Detail: M 2 -VO 2 dimerized chains DOS (1/eV) V 1 3d x 2 -y 2 V 1 3d yz V 1 3d xz DOS (1/eV) zigzag chains V 2 3d x 2 -y 2 V 2 3d yz V 2 3d xz (E - E F ) (ev) (E - E F ) (ev)

70 Electronic Structure in Detail: M 2 -VO 2 (AF) dimerized chains zigzag chains 2 V 1 3d x 2 -y 2 V 1 3d yz V 1 3d xz 2 V 2 3d x 2 -y 2 V 2 3d yz V 2 3d xz 1 1 DOS (1/eV) DOS (1/eV) (E - E F ) (ev) (E - E F ) (ev)

71 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 PBE HSE3, HSE6 E PBE 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

72 Critical review of the Local Density Approximation 1 W L G X W K 1 1 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:.57 ev HSE: 1.15 ev HSE Energy (ev)

73 Critical review of the Local Density Approximation 1 W L G X W K 1 1 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:.67 ev GGA:.9 ev HSE:.66 ev HSE Energy (ev)

74 Critical review of the Local Density Approximation Calculated vs. experimental bandgaps

75 SrTiO 3 GGA 1 8 DOS (1/eV) (E - E V ) (ev) Bandgap GGA: 1.6 ev, exp.: 3.2 ev

76 SrTiO 3 GGA 1 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

77 LaAlO 3 GGA 25 2 DOS (1/eV) (E - E V ) (ev) Bandgap GGA: 3.5 ev, exp.: 5.6 ev

78 LaAlO 3 GGA 25 HSE DOS (1/eV) 15 1 DOS (1/eV) (E - E V ) (ev) (E - E V ) (ev) Bandgap GGA: 3.5 ev, HSE: 5. ev, exp.: 5.6 ev

79 2D Electron Gas at LaAlO 3 -SrTiO 3 Interface

80 2D Electron Gas at LaAlO 3 -SrTiO 3 Interface Issues Role of electronic correlations? SrTiO 3, LaAlO 3 : band insulators SrTiO 3 /LaAlO 3 interface: MIT (# LaAlO 3 layers) magnetic properties of the interface superconductivity below 2 mk What is the origin of the 2-DEG? intrinsic mechanism? defect-doping?

81 2D Electron Gas at LaAlO 3 -SrTiO 3 Interface Insulator-Metal Transition Chen, Kolpak, Ismail-Beigi, Adv. Mater. 22, 2881 (21)

82 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 2 Å inversion symmetry lattice constant of SrTiO 3 from GGA (3.944 Å)

83 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

84 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

85 Slab Calculations for the LaAlO 3 -SrTiO 3 Interface DOS (1/eV) L 3L 4L 5L (E - E F ) (ev)

86 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

87 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

88 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

89 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

90 Unified Picture Basic Theories 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 (2) VE, Ann. Phys. 11, 65 (22) VE, EPL 58, 851 (22) J. Moosburger-Will et al., PRB 79, (29)

91 From VO 2 to CrO 2 : Applications half-metallic ferromagnet T C 391 K Partial DOS 4 2 Cr 3d t 2g Cr 3d e g O 2p DOS (1/eV) (E - E F ) (ev) J. Phys. I France 2, 315 (1992) J. Phys. I France 4, 1199 (1994)

92 From VO 2 to CrO 2 : Applications half-metallic ferromagnet T C 391 K Spin Density ρ v / a.u. Partial DOS DOS (1/eV) Cr 3d t 2g Cr 3d e g O 2p (E - E F ) (ev) J. Phys. I France 2, 315 (1992) J. Phys. I France 4, 1199 (1994)

93 From VO 2 to V 2 O 3 : Magnéli-Phases V n O 2n 1 = V 2 O 3 + (n 2)VO 2 n : VO 2 n = 2: V 2 O 3 variation of d-band filling metal-insulator transitions structural transformations rutile-type slabs of thickness n corundum-like shear planes

94 From VO 2 to V 2 O 3 : Magnéli-Phases V n O 2n 1 = V 2 O 3 + (n 2)VO 2 n : VO 2 n = 2: V 2 O 3 variation of d-band filling metal-insulator transitions structural transformations rutile-type slabs of thickness n corundum-like shear planes VO 2 (n ) O octahedra occupied by V b prut Magnéli phase characterized by V-chain length n c prut

95 From VO 2 to V 2 O 3 : Magnéli-Phases V 4 O 7 (n = 4) V n O 2n 1 = V 2 O 3 + (n 2)VO 2 n : VO 2 n = 2: V 2 O 3 variation of d-band filling metal-insulator transitions structural transformations rutile-type slabs of thickness n corundum-like shear planes O octahedra occupied by V b prut Magnéli phase characterized by V-chain length n c prut

96 From VO 2 to V 2 O 3 : Magnéli-Phases V n O 2n 1 = V 2 O 3 + (n 2)VO 2 n : VO 2 n = 2: V 2 O 3 variation of d-band filling metal-insulator transitions structural transformations rutile-type slabs of thickness n corundum-like shear planes V 2 O 3 (n = 2) O octahedra occupied by V b prut Magnéli phase characterized by V-chain length n c prut

97 From VO 2 to V 2 O 3 : Magnéli-Phases V n O 2n 1 = V 2 O 3 + (n 2)VO 2 n : VO 2 one-dimensional Peierls instability n = 2: V 2 O 3 localized electrons electronic correlations V n O 2n 1 interpolate between VO 2 and V 2 O 3 charge order, orbital order V n O 2n 1, Ti n O 2n 1 Europhys. Lett. 61, 361 (23) Europhys. Lett. 64, 682 (23) CPL 39, 151 (24) Ann. Phys. 13, 475 (24) J. Phys.: CM 18, 1955 (26) V 2 O 3, Ti 2 O 3 PRL 86, 5345 (21) PRL 9, (23) PRB 7, (24) Europhys. Lett. 7, 782 (25) pss (b) 243, 2599 (26)

98 Basic Theories Success Stories Basics DFT (exact, ground state) LDA, GGA,... Percus-Levy partition

99 Basic Theories 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

100 Success Stories 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)

101 Acknowledgments Augsburg U. Eckern, K.-H. Höck, S. Horn, R. Horny, T. Kopp, J. Kündel, J. Mannhart, J. Moosburger-Will Darmstadt/Jülich P. C. Schmidt, M. Stephan, J. Sticht Europe/USA M. Christensen, C. Freeman, M. Halls, A. Mavromaras, P. Saxe, E. Wimmer, R. Windiks, W. Wolf

102 Acknowledgments Augsburg U. Eckern, K.-H. Höck, S. Horn, R. Horny, T. Kopp, J. Kündel, J. Mannhart, J. Moosburger-Will Darmstadt/Jülich P. C. Schmidt, M. Stephan, J. Sticht Europe/USA M. Christensen, C. Freeman, M. Halls, A. Mavromaras, P. Saxe, E. Wimmer, R. Windiks, W. Wolf Bielefeld Thank You for Your Attention!