dynamical mean-field theory for correlated electron materials...v ~ 161.5/158.5 au 3 in bcc/fcc...

Dynamical Mean-Field Theory for Correlated Electron Materials

Dieter Vollhardt

XXIII Latin American Symposium on Solid State Physics Bariloche, Argentina; April 12, 2018

Center for Electronic Correlations and MagnetismUniversity of Augsburg

Outline:

1. Electronic correlations

2. Dynamical mean-field theory (DMFT)

3. DFT+DMFT: Ab initio approach to correlatedelectron materials

4. Applications:

- Fe: Correlation induced lattice stability

- FeSe: Phase stability and Lifshitz transitions

Correlations

correlatedmany-particle

system

Temporal/spatial correlations in every-day life

Correlation [lat.] (con + relatio): mutual relation, interdependence

correlatedmany-particle

system

factorization !?

Correlation [lat.] (con + relatio): mutual relation, interdependence

Temporal/spatial correlations in every-day life

Electronic correlations

Narrow d,f-bands electronic correlations important

Electronic Correlations in Solids

- Hartree-Fock: Particle interacts with a self-consistent potential- LDA/GGA of density functional theory (DFT)

Many-body effects (“correlations”): Particle interacts with all otherparticles at (ri,t)

single-particle excitations in general unknown

Single-particlestates (k-states)?

yes,

no (in general)

Landau quasiparticles: yes, <τ ∞

lifetime τ = ∞

But

Definition of electronic correlations (I)

quantifies correlations

ˆ ˆ ˆ ˆi j i jn n n n−

Correlations = effects beyond Hartree –Fock / single-particle (product) wave functions

kx

ky

kz

Fermi gas: Excited state

Switch on repulsive interaction unsolvable many-body problem

k-eigenstates: infinite life time

Particle

Hole

Fermi sphere

Fermi surface

Fermi liquid

kx

ky

kz

Microscopic derivation by diagrammatic perturbation theory to all orders

(Quasi-) Particle

(Quasi-) Hole

Landau (1956/58)

= elementary excitation

“Standard model of condensed matter physics“

Fermi sphere

Fermi surface

1-1 correspondence betweensingle-particle states (k,σ)

Abrikosov, Khalatnikov (1959)

Unusual properties of correlated electron materials

• sensors, switches• spintronics• thermoelectrics• high-Tc superconductor devices

With potential for technological applications:

• functional materials:oxide heterostructures, …

• huge resistivity changes• gigantic volume anomalies• colossal magnetoresistance• high-Tc superconductivity• metallic behavior at interfaces of insulators, …

†

, ,σ σ

σ↑ ↓= +− ∑ ∑H c c n nt Ui j i i

i j i

time

Theoretical challenge of many-fermion problems:Construct comprehensive, controlled, non-perturbative approximation schemes

n n n n↑ ↓ ↑ ↓≠i i i i

Theoretical approaches Gutzwiller, 1963Hubbard, 1963Kanamori, 1963Hubbard model

Þstrongest

simplification

U

Correlated material

How to combine?

Held (2004)

time

, GGA

Non-perturbative approximation schemes for real materials

U

Dynamical mean-field theory (DMFT)for correlated electrons

Metzner, DV (1989)

,d Z→∞→

Z=12

time

dynamicalmean field

Hubbard model

Face-centered cubic lattice (d=3)

Georges, Kotliar (1992)

Self-consistent single-impurity Anderson model

†

, ,σ σ

σ↑ ↓= +− ∑ ∑H c c n nt Ui j i i

i j i

Limit for lattice electrons or d Z →∞

Strong simplifications in many-body theory

U

Exact treatment of many-body dynamics

Dynamical mean-field theory (DMFT) of correlated electrons

( , )ωΣ k ( )ωΣU

Kotliar, DV (2004)

• Microscopic derivation by diagrammaticperturbation theory to all orders

• Φ-derivable, thermodynamically consistent, conserving


( , )ωΣ k ( )ωΣU

Kotliar, DV (2004)

U

Itinerant fermions

EEd+U/2Ed-U/2

U

Ed

Spec

tral

fun

ctio

n


( , )ωΣ k ( )ωΣU

Kotliar, DV (2004)

U

Spec

tral

fun

ctio

n

Single-particle excitations (Landau quasiparticles),

not “electrons“

Itinerant fermions


( , )ωΣ k ( )ωΣU

U

Spec

tral

fun

ctio

n

Kotliar, DV (2004)

Experimentallydetectable (ARPES, …)

Definition of electronic correlations (II):• transfer of spectral weight [due to Re ∑(ω)]• finite lifetime of excitations [due to Im ∑(ω)]

Itinerant fermions

Material specific electronic structure(Density functional theory: LDA, GGA, ...)

Computational scheme for correlated electron materials:

+Local electronic correlations

(Many-body theory: DMFT)

Anisimov et al. (1997)Lichtenstein, Katsnelson (1998)Held et al. (2003)Kotliar et al. (2006)

LDA+DMFT

=

Material specific electronic structure(Density functional theory: LDA, GGA, ...) or GW

Computational scheme for correlated electron materials:

+Local electronic correlations

(Many-body theory: DMFT)

=X+DMFT X= DFT (LDA, GGA); GW, …

Σ(ω)

G( )ω

i

(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., V2O3: Mott transition with lattice distortion Leonov, Anisomov, DV (2015)

Effective multi-orbital Anderson impurity modelwith self-consistency condition

Spectral functions in DMFT

k-integrated spectral function (“interacting DOS”) PES

k-resolved spectral function ARPES

0 1( )( , ) [ ( )]k kLDAω ω ω −= − −G HΣ

1( , ) I ( , )k kmA Trω ωπ

= − G

1( ) Im ( )A ω ωπ

= − G

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

Until ~ 10 years ago:

Theoretical investigations of electronic correlations in materials forgiven lattice structure

How do electronic correlationsinfluence the stability of the ionic lattice ?

Pnma primitive cell of paramagnetic LaMnO3

La

Mn eg electrons

O

Fe

• Most abundant element by mass on Earth

• Ferromagnetism with very high TC

• Most widely used metal in modern day industry (“iron age”)

Application of GGA+DMFT to

Fe: Electronic configuration [Ar] 3d64s2



DMFT: Ferromagnetism in the one-band Hubbard model

Ulmke (1998)

Generalized fcc lattice ( )Z →∞

LDA+DMFT

Lichtenstein, Katsnelson, Kotliar (2001)

Ferromagnetic order ofitinerant local moments

Microscopic origin of exchange interactions in Fe Eriksson collaboration (2016)

TC

Curie-Weiss

Structural stability of Fe

Ivan Leonov (Augsburg, Ekaterinburg)Alexander Poteryaev (Ekaterinburg)Vladimir Anisimov (Ekaterinburg)

Collaborators:

α-γ structural phase transition in paramagnetic iron:Leonov et al., Phys. Rev. Lett. 106, 106405 (2011)

Phonons:Leonov et al., Phys. Rev. B 85, 020401(R) (2012)

Phase stability up to melting:Leonov et al., Sci. Rep. 4, 5585 (2014)

Fe

ferrite

austenite

hexaferrum

Exceptional: • Abundance of allotropes: α, γ, δ, ε, … phases• Very high Curie temperature (TC = 1043 K)• bcc-phase stable for P,T0

©Lawrence Livermore National Laboratory

Abrikosov collaboration (2007)

Fe

ferrite

austenite

hexaferrum

Fe

ferrite

austenite

hexaferrum

DFT(GGA): finds paramagnetic bcc phase to be unstable

- What stabilizes paramagnetic ferrite?- What causes the bcc-fcc structural phase transition?

Tstruct

Pressure, GPa

Coulomb interaction between Fe 3d electrons: U=1.8 eV, J=0.9 eV

bcc-fcc structural transition in paramagnetic Fe

Construct Wannier functions for partially filled Fe sd orbitals

First-principles multi-band Hamiltonianincluding local interactions

Goal: Understand the structural stability ofparamagnetic bcc phase

Pressure, GPa

GGA+DMFT total energy Etot - Ebcc

fccbcc


Construct Wannier functions for partially filled Fe sd orbitals

Goal: Understand the structural stability ofparamagnetic bcc phase

Bain transformation path

Leonov, Poteryaev, Anisimov, DV (2011)

Paramagnetic α-phase stabilizedby electronic correlations

Conclusion:

GGA+DMFT:

bcc-fcc structural transitionat Tstruct ≈ 1.3 TC

GGA:Only paramagnetic fcc structure stable

• V ~ 161.5/158.5 au3 in bcc/fcc phaseincreased by electronic repulsion

Non-magnetic GGA:

• V ~ 141/138 au3 in bcc/fcc phasetoo small density too high

GGA+DMFT:

Agrees well with exp. data: Vexp ~ 165 / 158 au3

GGA+DMFT

bcc Fe

fcc Fe∆V ~ -2%

Equi

libri

um v

olum

e (a

u3) Pressure, GPa


Equilibrium volume V

bcc-fcc structural transition caused by electronic correlations

Conclusion:


Lattice dynamics and phonon spectra of Fe

Pressure, GPa

Lattice dynamics of paramagnetic bcc iron

Non-magnetic GGA: Phonon dispersion

Exp.: Neuhaus, Petry, Krimmel (1997)

1. Brillouin zone

Dynamically unstable +elastically unstable (C11, C‘ < 0)

What stabilizes the phonons?

Pressure, GPa

GGA+DMFT: Phonon dispersion at 1.2 TC

• phonon frequencies calculated with frozen-phonon method

• harmonic approximation

Stokes, Hatch, Campbell (2007)

Calculated:• equilibrium lattice constant

a~2.883 Å (aexp~2.897 Å)• Debye temperature Θ~458 K

Lattice dynamics of paramagnetic bcc iron

Exp.: Neuhaus, Petry, Krimmel (1997)


Phonons in bcc iron stabilizedby electronic correlations

Conclusion:

FeSe

Ivan Leonov (Augsburg, Ekaterinburg)Sergey Skornyakov (Ekaterinburg)Vladimir Anisimov (Ekaterinburg)

Collaborators:

Application of GGA+DMFT to

FeSe (expansion):Leonov et al., Phys. Rev. Lett. 115, 106402 (2015)Skornyakov et al., Phys. Rev. B 96, 035137 (2017)

FeSe (compression):Skornyakov et al., Phys. Rev. B 97, 115165 (2018)

Fe-based superconductors

‘11’ FeSe ‘111’ LiFeAs ‘1111’ LaFeAsO ‘122’ CaFe2As2Layered structure, butno separating layers

Superconductivity induced by doping or pressure

Kamihara et al. (2008)Si, Yu, Abrahams (2016)

FeSe: Crystal structure and properties

FeSe FeSe1-xTex , x<0.7 FeTe

no static magnetic order,short-range fluctuations with Qm=(π, π)

TN ~ 70 KQm=(π, 0)

Tc ~ 8 K Tc ~ 14 K -

Why does the electronic and magnetic structure of Fe(Se,Te) change with lattice expansion or compression ?

• Tetragonal (PbO-structure): α-FeSe• Tc = 8 K ( 37 K under hydrostatic pressure)• Isovalent substitution Se → Te (lattice expansion) Tc increases up to 14 K

Increase of Tc under expansion and compression !

Fully charge-self-consistent calculation (csc)

Non-charge-self-consistent calculation (ncsc)

GGA vs. GGA+DMFT total energy

• Isostructural transformation upon ~ 13%lattice expansion (pc = − 7.6 GPa)

csc results:lattice constant a (a.u.), c/a fixed

lattice anomaly

equilibriumstructure:

a = 7.05 a.u.

expandedstructure:

a = 7.6 a.u.

Skornyakov et al. (2017)

Leonov et al. (2015)

FeSe: Phase stability under expansion Partially filled Fe 3d, Se 4p orbitals;Coulomb interaction Fe 3d:

: 3.5 , 0.85mmU U eV J eVσσ ′′ = =


Non-charge-self-consistent calculation (ncsc)


Observed in FeTe:Suppression of magnetism under 4-6 GPa

Exp: Zhang et al. (2009)

csc results:

Equilib. volume (a = 7.05 a.u.) Expanded volume (a = 7.6 a.u.)

B = 79 GPa𝜇𝜇𝑧𝑧2 = 1.9 µB

49 GPa2.9 µB

lattice constant a (a.u.), c/a fixed



FeSe: Phase stability and local moment under expansion

• Isostructural transformation upon ~ 13%lattice expansion (pc = − 7.6 GPa)

• Strong increase of the fluctuating local magnetic moment + compressibility

FeSe upon expansion• Spectral function• Electronic structure• Fermi surface

FeSe: Spectral function

Exp

ansi

on

Van Hove singularityshifts to EF due to correlations

Suppression of spectral weight

a = 7.05 a.u.

a = 7.6 a.u.

equilibrium volume

expanded volume



Similar results: Watson, Backes, Haghighirad, Hoesch, Kim, Coldea, Valenti (2017)

FeSe: Spectral function

Exp

ansi

on

Van Hove singularityshifts to EF due to correlations

Exp.: Yokoya et al. (2012)

Spectral weight suppressed with Tesubstitution ( = lattice expansion)

a = 7.05 a.u.

a = 7.6 a.u.

equilibrium volume

expanded volume



non-magnetic GGA

Exp

ansi

onEquilibrium volume:

• Quasiparticle bands renormalized + orbital-selective shift

• Van Hove singularity at M point pushed towards EF

Expanded volume:

• Complete change of electronic structure• Correlation-induced shift of Van Hove

singularity above EF

GGA+DMFT

FeSe: Electronic structureSkornyakov et al. (2017)


non-magnetic GGA

FeSe: Fermi surface

Equilibrium volume:

• Correlations do not change FS• In-plane nesting with Qm=(π,π)

GGA+DMFT

Exp

ansi

on

Expanded volume:

• Correlations change FS topology(Lifshitz transition)

• Electron pocket at M point vanishes• In-plane nesting with Qm=(π,0)

Change of magnetic structure upon lattice expansion caused by correlations

In accord with ARPES experiments on FeSe1-xTex Chen et al. (2010), Zhang et al. (2010), Jiang et al. (2013), Liu et al. (2013,2015), Nakayama et al. (2010,2014)



Conclusion:

Dependence of Fermi surface on pressure, T=290 K

ambient pressure 10 GPa

kz=0 plane

kx=ky plane

kx kx

weak kz dependence sizable kz dependence

DFT+DMFT with structural optimization (V, c/a, zSe)

kx=ky kx=ky

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

Skornyakov et al. (2018) FeSe: Behavior under compression

Dependence of Fermi surface on pressure, T=290 K

ambient pressure 10 GPa

kz=0 plane

kx=ky plane

kx kx

weak kz dependence sizable kz dependence

DFT+DMFT with structural optimization (V, c/a, zSe)

kx=ky kx=ky

Skornyakov et al. (2018) FeSe: Behavior under compression

Conjecture:- Compression-induced increase of Tc due to phonons- Expansion-induced increase of Tc due to spin fluctuations

Leonov et al. (unpublished, 2015)

Experiment GGA or GGA+DMFT

FeSe: Size of Fermi surface pockets ?

Watson et al. (2017)

Size of pockets smaller than in GGA or GGA+DMFT due to- non-local correlations ?- frustrated magnetism ?- spin-orbit coupling ?

Conclusion

1) DFT+DMFT: powerful tool to study/predict properties of correlated materials

3) DFT+DMFT codes publically available

VASP-DMFT code probably available later in 2018

2) Electronic correlations strongly influence the

electronic properties + phase stability

of materials

dynamical mean-field theory for correlated electron materials...v ~ 161.5/158.5 au 3 in bcc/fcc...

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