Dynamical Mean-Field Theory for Correlated Electron Materials Dieter Vollhardt

Transcription

1 Center for Electronic Correlations and Magnetism University of Augsburg Dynamical Mean-Field Theory for Correlated Electron Materials Dieter Vollhardt XXIII Latin American Symposium on Solid State Physics Bariloche, Argentina; April 12, 2018

2 Outline: 1. Electronic correlations 2. Dynamical mean-field theory (DMFT) 3. DFT+DMFT: Ab initio approach to correlated electron materials 4. Applications: - Fe: Correlation induced lattice stability - FeSe: Phase stability and Lifshitz transitions

3 Correlations

4 Correlation [lat.] (con + relatio): mutual relation, interdependence Temporal/spatial correlations in every-day life correlated many-particle system

5 Correlation [lat.] (con + relatio): mutual relation, interdependence Temporal/spatial correlations in every-day life correlated many-particle system factorization!?

6 Electronic correlations

7 Electronic Correlations in Solids Narrow d,f-bands electronic correlations important

8 Definition of electronic correlations (I) Correlations = effects beyond Hartree Fock / single-particle (product) wave functions nn ˆˆ ˆ ˆ i j ni nj quantifies correlations - Hartree-Fock: Particle interacts with a self-consistent potential - LDA/GGA of density functional theory (DFT) Many-body effects ( correlations ): Particle interacts with all other particles at (r i,t) single-particle excitations in general unknown But Single-particle states (k-states)? yes, no (in general) Landau lifetime τ = quasiparticles: yes, τ <

9 Fermi gas: Excited state k z Particle Fermi surface Hole k y k x Fermi sphere k-eigenstates: infinite life time Switch on repulsive interaction unsolvable many-body problem

10 Fermi liquid Standard model of condensed matter physics Fermi surface k z (Quasi-) Hole Landau (1956/58) 1-1 correspondence between single-particle states (k,σ) (Quasi-) Particle = elementary excitation k y k x Fermi sphere Microscopic derivation by diagrammatic perturbation theory to all orders Abrikosov, Khalatnikov (1959)

11 Unusual properties of correlated electron materials huge resistivity changes gigantic volume anomalies colossal magnetoresistance high-t c superconductivity metallic behavior at interfaces of insulators, With potential for technological applications: sensors, switches spintronics thermoelectrics high-t c superconductor devices functional materials: oxide heterostructures,

12 Theoretical approaches Gutzwiller, 1963 Correlated material strongest simplification Þ Hubbard model Hubbard, 1963 Kanamori, 1963 time U H = t c c + U n n ij,, σ iσ jσ i i i n n n n i i i i Theoretical challenge of many-fermion problems: Construct comprehensive, controlled, non-perturbative approximation schemes

13 Non-perturbative approximation schemes for real materials, GGA How to combine? time U Held (2004)

14 Dynamical mean-field theory (DMFT) for correlated electrons

15 Limit d or Z H = t c c + U n n ij,, σ for lattice electrons iσ jσ i i i Hubbard model Face-centered cubic lattice (d=3) U time Metzner, DV (1989) dz, Strong simplifications in many-body theory dynamical mean field Z=12 Self-consistent single-impurity Anderson model Georges, Kotliar (1992)

16 Dynamical mean-field theory (DMFT) of correlated electrons Σ( k, ω) Σ( ω) U Kotliar, DV (2004) Exact treatment of many-body dynamics Microscopic derivation by diagrammatic perturbation theory to all orders Φ-derivable, thermodynamically consistent, conserving

17 Dynamical mean-field theory (DMFT) of correlated electrons Σ( k, ω) Σ( ω) U Kotliar, DV (2004) Itinerant fermions Spectral function E d -U/2 E d U E d +U/2 E U

18 Dynamical mean-field theory (DMFT) of correlated electrons Σ( k, ω) Σ( ω) U Kotliar, DV (2004) Itinerant fermions Spectral function Single-particle excitations (Landau quasiparticles), not electrons U

19 Dynamical mean-field theory (DMFT) of correlated electrons Σ( k, ω) Σ( ω) U Kotliar, DV (2004) Itinerant fermions Spectral function U Definition of electronic correlations (II): transfer of spectral weight [due to Re (ω)] finite lifetime of excitations [due to Im (ω)] Experimentally detectable (ARPES, )

20 Computational scheme for correlated electron materials: Material specific electronic structure (Density functional theory: LDA, GGA,...) + Local electronic correlations (Many-body theory: DMFT) = LDA+DMFT Anisimov et al. (1997) Lichtenstein, Katsnelson (1998) Held et al. (2003) Kotliar et al. (2006)

21 Computational scheme for correlated electron materials: Material specific electronic structure (Density functional theory: LDA, GGA,...) or GW + Local electronic correlations (Many-body theory: DMFT) = X+DMFT X= DFT (LDA, GGA); GW,

22 Effective multi-orbital Anderson impurity model with self-consistency condition Σ(ω) i G( ω) (i) Effective single impurity problem (ii) k-integrated Dyson equ. Solve self-consistently with an impurity solver, e.g., QMC, CT-QMC, NRG, DMRG, ED If correlations strongly redistribute charge density need also charge self-consistency e.g., V 2 O 3 : Mott transition with lattice distortion Leonov, Anisomov, DV (2015)

23 Spectral functions in DMFT k-integrated spectral function ( interacting DOS ) PES 1 A( ω) = Im G( ω) π k-resolved spectral function ARPES G( k, ω) = [ ω Σ( ω) H ( k)] 0 1 LDA 1 A( k, ω) = I mtrg( k, ω) π

24 Held, Nekrasov, Keller, Eyert, Blümer, McMahan, Scalettar, Pruschke, Anisimov, DV (Psi-k, 2003)

25 Until ~ 10 years ago: Theoretical investigations of electronic correlations in materials for given lattice structure La O Mn e g electrons Pnma primitive cell of paramagnetic LaMnO 3 How do electronic correlations influence the stability of the ionic lattice?

26 Application of GGA+DMFT to Fe Most abundant element by mass on Earth Ferromagnetism with very high T C Most widely used metal in modern day industry ( iron age )

27 Electronic Correlations in Solids Narrow d,f-bands electronic correlations important Fe: Electronic configuration [Ar] 3d 6 4s 2

28 DMFT: Ferromagnetism in the one-band Hubbard model Generalized fcc lattice ( Z ) Ulmke (1998) Ferromagnetic order of itinerant local moments Curie-Weiss T C LDA+DMFT Lichtenstein, Katsnelson, Kotliar (2001) Microscopic origin of exchange interactions in Fe Eriksson collaboration (2016)

29 Structural stability of Fe Collaborators: Ivan Leonov (Augsburg, Ekaterinburg) Alexander Poteryaev (Ekaterinburg) Vladimir Anisimov (Ekaterinburg) α-γ structural phase transition in paramagnetic iron: Leonov et al., Phys. Rev. Lett. 106, (2011) Phonons: Leonov et al., Phys. Rev. B 85, (R) (2012) Phase stability up to melting: Leonov et al., Sci. Rep. 4, 5585 (2014)

30 Fe austenite ferrite hexaferrum Exceptional: Abundance of allotropes: α, γ, δ, ε, phases Very high Curie temperature (T C = 1043 K) bcc-phase stable for P,T 0

31 Fe austenite ferrite hexaferrum Abrikosov collaboration (2007) Lawrence Livermore National Laboratory

32 Fe T struct austenite ferrite hexaferrum DFT(GGA): finds paramagnetic bcc phase to be unstable - What stabilizes paramagnetic ferrite? - What causes the bcc-fcc structural phase transition?

33 bcc-fcc structural transition in paramagnetic Fe Goal: Understand the structural stability of paramagnetic bcc phase Construct Wannier functions for partially filled Fe sd orbitals Pressure, GPa First-principles multi-band Hamiltonian including local interactions Coulomb interaction between Fe 3d electrons: U=1.8 ev, J=0.9 ev

34 bcc-fcc structural transition in paramagnetic Fe Goal: Understand the structural stability of paramagnetic bcc phase Construct Wannier functions for partially filled Fe sd orbitals Pressure, GPa GGA+DMFT total energy E tot - E bcc GGA: Only paramagnetic fcc structure stable GGA+DMFT: bcc-fcc structural transition at T struct 1.3 T C Conclusion: Paramagnetic α-phase stabilized by electronic correlations bcc fcc Bain transformation path Leonov, Poteryaev, Anisimov, DV (2011)

35 bcc-fcc structural transition in paramagnetic Fe Equilibrium volume V Equilibrium volume (au 3 ) GGA+DMFT bcc Fe V ~ -2% fcc Fe Non-magnetic GGA: V ~ 141/138 au 3 in bcc/fcc phase too small density too high GGA+DMFT: Pressure, GPa V ~ 161.5/158.5 au 3 in bcc/fcc phase increased by electronic repulsion Leonov, Poteryaev, Anisimov, DV (2011) Agrees well with exp. data: V exp ~ 165 / 158 au 3 Conclusion: bcc-fcc structural transition caused by electronic correlations

36 Lattice dynamics and phonon spectra of Fe

37 Lattice dynamics of paramagnetic bcc iron Non-magnetic GGA: Phonon dispersion Pressure, GPa 1. Brillouin zone Dynamically unstable + elastically unstable (C 11, C < 0) Exp.: Neuhaus, Petry, Krimmel (1997) What stabilizes the phonons?

38 Lattice dynamics of paramagnetic bcc iron phonon frequencies calculated with frozen-phonon method harmonic approximation Stokes, Hatch, Campbell (2007) Pressure, GPa GGA+DMFT: Phonon dispersion at 1.2 T C Calculated: equilibrium lattice constant a~2.883 Å (a exp ~2.897 Å) Debye temperature Θ~458 K Leonov, Poteryaev, Anisimov, DV (2012) Exp.: Neuhaus, Petry, Krimmel (1997) Conclusion: Phonons in bcc iron stabilized by electronic correlations

39 Application of GGA+DMFT to FeSe Collaborators: Ivan Leonov (Augsburg, Ekaterinburg) Sergey Skornyakov (Ekaterinburg) Vladimir Anisimov (Ekaterinburg) FeSe (expansion): Leonov et al., Phys. Rev. Lett. 115, (2015) Skornyakov et al., Phys. Rev. B 96, (2017) FeSe (compression): Skornyakov et al., Phys. Rev. B 97, (2018)

40 Electronic Correlations in Solids Narrow d,f-bands electronic correlations important

41 Fe-based superconductors Kamihara et al. (2008) Si, Yu, Abrahams (2016) 11 FeSe 111 LiFeAs 1111 LaFeAsO 122 CaFe 2 As 2 Layered structure, but no separating layers Superconductivity induced by doping or pressure

42 FeSe: Crystal structure and properties Tetragonal (PbO-structure): α-fese T c = 8 K ( 37 K under hydrostatic pressure) Isovalent substitution Se Te (lattice expansion) T c increases up to 14 K Increase of T c under expansion and compression! FeSe FeSe 1-x Te x, x<0.7 FeTe no static magnetic order, short-range fluctuations with Q m =(π, π) T N ~ 70 K Q m =(π, 0) T c ~ 8 K T c ~ 14 K - Why does the electronic and magnetic structure of Fe(Se,Te) change with lattice expansion or compression?

43 FeSe: Phase stability under expansion Non-charge-self-consistent calculation (ncsc) Fully charge-self-consistent calculation (csc) Leonov et al. (2015) Skornyakov et al. (2017) Partially filled Fe 3d, Se 4p orbitals; Coulomb interaction Fe 3d: σσ U : U = 3.5 ev, J = 0.85eV mm lattice constant a (a.u.), c/a fixed csc results: Isostructural transformation upon ~ 13% lattice expansion (p c = 7.6 GPa) GGA vs. GGA+DMFT total energy lattice anomaly equilibrium structure: a = 7.05 a.u. expanded structure: a = 7.6 a.u.

44 FeSe: Phase stability and local moment under expansion Non-charge-self-consistent calculation (ncsc) Fully charge-self-consistent calculation (csc) Leonov et al. (2015) Skornyakov et al. (2017) lattice constant a (a.u.), c/a fixed csc results: Isostructural transformation upon ~ 13% lattice expansion (p c = 7.6 GPa) Strong increase of the fluctuating local magnetic moment + compressibility Equilib. volume (a = 7.05 a.u.) Expanded volume (a = 7.6 a.u.) B = 79 GPa 49 GPa μμ 2 zz = 1.9 µ B 2.9 µ B Observed in FeTe: Suppression of magnetism under 4-6 GPa Exp: Zhang et al. (2009)

45 FeSe upon expansion Spectral function Electronic structure Fermi surface

46 FeSe: Spectral function Leonov et al. (2015) Skornyakov et al. (2017) Van Hove singularity shifts to E F due to correlations equilibrium volume a = 7.05 a.u. Expansion Suppression of spectral weight expanded volume a = 7.6 a.u. Similar results: Watson, Backes, Haghighirad, Hoesch, Kim, Coldea, Valenti (2017)

47 FeSe: Spectral function Leonov et al. (2015) Skornyakov et al. (2017) Van Hove singularity shifts to E F due to correlations equilibrium volume a = 7.05 a.u. Expansion expanded volume a = 7.6 a.u. Exp.: Yokoya et al. (2012) Spectral weight suppressed with Te substitution ( = lattice expansion)

48 FeSe: Electronic structure Leonov et al. (2015) Skornyakov et al. (2017) non-magnetic GGA GGA+DMFT Expansion Equilibrium volume: Quasiparticle bands renormalized + orbital-selective shift Van Hove singularity at M point pushed towards E F Expanded volume: Complete change of electronic structure Correlation-induced shift of Van Hove singularity above E F

49 FeSe: Fermi surface Leonov et al. (2015) Skornyakov et al. (2017) non-magnetic GGA GGA+DMFT Expansion Equilibrium volume: Correlations do not change FS In-plane nesting with Q m =(π,π) Expanded volume: Correlations change FS topology (Lifshitz transition) Electron pocket at M point vanishes In-plane nesting with Q m =(π,0) Conclusion: Change of magnetic structure upon lattice expansion caused by correlations In accord with ARPES experiments on FeSe 1-x Te x Chen et al. (2010), Zhang et al. (2010), Jiang et al. (2013), Liu et al. (2013,2015), Nakayama et al. (2010,2014)

50 FeSe: Behavior under compression Skornyakov et al. (2018) DFT+DMFT with structural optimization (V, c/a, z Se ) Dependence of Fermi surface on pressure, T=290 K ambient pressure 10 GPa k z =0 plane k x =k y plane k x k=k x y weak k z dependence k x =k y sizable k z dependence Upon compression: (i) Band degeneracy lifted (orbital dependence) (ii) 2d-3d transition (Lifshitz transition) (iii) χ(q): Reduction of spin fluctuations and (π,π) nesting

51 FeSe: Behavior under compression Skornyakov et al. (2018) DFT+DMFT with structural optimization (V, c/a, z Se ) Dependence of Fermi surface on pressure, T=290 K ambient pressure 10 GPa k z =0 plane k x =k y plane k x k=k x y weak k z dependence k x =k y sizable k z dependence Conjecture: - Compression-induced increase of T c due to phonons - Expansion-induced increase of T c due to spin fluctuations

52 FeSe: Size of Fermi surface pockets? Experiment GGA or GGA+DMFT Watson et al. (2017) Size of pockets smaller than in GGA or GGA+DMFT due to - non-local correlations? - frustrated magnetism? - spin-orbit coupling? Leonov et al. (unpublished, 2015)

53 Conclusion 1) DFT+DMFT: powerful tool to study/predict properties of correlated materials 2) Electronic correlations strongly influence the electronic properties + phase stability of materials 3) DFT+DMFT codes publically available VASP-DMFT code probably available later in 2018