Surprising Effects of Electronic Correlations in Solids

Surprising Effects of Electronic Correlations in Solids

Dieter Vollhardt

Center for Electronic Correlations and MagnetismUniversity of Augsburg

Sólidos 2015, November 10, 2015; La Plata, Argentina

Outline:

• What are electronic correlations?

• Modeling of correlated materials

• Ab initio approach to correlated electron materials

• Applications:

- Lattice stability of Fe

- FeSe: Parent compound for Fe-based superconductors

Correlations

Correlation [lat.]: con + relatio ("with relation")

Correlations in mathematics, natural sciences:

Definition of Correlations (I): Effects beyond factorization approximations (e.g., Hartree-Fock)

AB A B≠

2( ) ( ') ( ) ( ')n n n n n≠ =r r r r

e.g., densities:

Temporal/spatial correlations in everyday life

Correlations?

But: External periodic potential long-range order enforced no genuine correlations


Time/space average inappropriate


Electronic Correlations in Solids

Narrow d,f-orbitals/bands electronic correlations important

Correlated electron materials have unusual properties

} How to study correlated systemstheoretically?

• sensors, switches, Mottronics• spintronics• thermoelectrics• high-Tc superconductors

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

material realistic modelÞmodeling

quantum many-particle problem

material realistic modelÞmodeling

maximal reduction: Hubbard modelÞU

time

†

, ,σ σ

σ↑ ↓= +− ∑ ∑H c c n nt Ui j i i

i j i

Hubbard model

time

Dimension of Hilbert spaceL: # lattice sites

(4 )LO

Computational time for N2 molecule: ca. 1 year with 50.000 compute nodes

n n n n↑ ↓ ↑ ↓≠i i i i

Static (Hartree-Fock-type) mean-field theories generally insufficient

Gutzwiller, 1963Hubbard, 1963Kanamori, 1963

†

, ,σ σ

σ↑ ↓= +− ∑ ∑H c c n nt Ui j i i

i j i

Theoretical challenge of many-fermion problems:Construct reliable, comprehensive

non-perturbative approximation schemes

Hubbard model

time

n n n n↑ ↓ ↑ ↓≠i i i i

Static (Hartree-Fock-type) mean-field theories generally insufficient

Purely numerical approaches (d=2,3): hopeless

Gutzwiller, 1963Hubbard, 1963Kanamori, 1963

How to combine?

Held (2004)

time

Non-perturbative approximation schemes for real materials

Dynamical Mean-Field Theory (DMFT)of Correlated Electrons

Metzner, Vollhardt (1989)

,d Z→∞→

Z=12

time

dynamicalmean-field

Theory of correlated electrons

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

Solve with an „impurity solver“, e.g., QMC, NRG, ED,...

Exact time resolved treatment of local electronic interactions

Dynamical mean-field theory (DMFT) of correlated electrons

U

Spec

tral

fun

ctio

n

( , )ωΣ k ( )ωΣ

Metzner, DV (1989)Georges, Kotliar (1992)

Kotliar, DV (2004)

U

Landau quasiparticles, not “electrons“

Exact time resolved treatment of local electronic interactions

Dynamical mean-field theory (DMFT) of correlated electrons

U

Spec

tral

fun

ctio

n

Experimentallydetectable!

Definition ofCorrelations (II):

Transfer of spectral weight

( , )ωΣ k ( )ωΣ

Metzner, DV (1989)Georges, Kotliar (1992)

Kotliar, DV (2004)

U

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

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=LDA, GGA; GW, …

1) Calculate LDA band structure: ' 'ˆ( )lml m LDAk Hε →

ˆLDAH

,

ˆd

d

i mi i

dm

n σσ

ε==

− ∆∑∑

double counting correction''mmU σσ

local Coulomb interaction

basis' ' e.g. LMTO

ˆ( )lml m LDAk Hε →

LDA+DMFT: Simplest version

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

Goal: Dynamical mean-field approach with predictive power for strongly correlated materials

FOR 1346 Research Unit

Fe

Application of LDA+DMFT

• Most abundant element by mass on Earth

• Ferromagnetism: Longest known quantum many-body phenomenon

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

Narrow d,f-orbitals/bands electronic correlations important

DMFT: Ferromagnetism in the one-band Hubbard modelUlmke (1998)

Generalized fcc lattice ( )Z →∞

ferromagnetic metal

Ferromagnetic order of itinerant local moments

Lichtenstein, Katsnelson, Kotliar (2001)

LDA+DMFT

DMFT: Ferromagnetism in the one-band Hubbard modelUlmke (1998)

Generalized fcc lattice ( )Z →∞

ferromagnetic metal

Ferromagnetic order of itinerant local moments

Lichtenstein, Katsnelson, Kotliar (2001)

LDA+DMFT

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

Fe

ferrite

austenite

hexaferrum

Fe

ferrite

austenite

hexaferrum

©Lawrence Livermore National Laboratory

Fe

ferrite

austenite

hexaferrum

©Lawrence Livermore National Laboratory

Mikhaylushkin, Simak, Dubrovinsky, Dubrovinskaia, Johansson, Abrikosov (2007)

Until recently: LDA+DMFT investigations of correlated materials for

given lattice structure

• How do electrons + ions influence each other ? • Which lattice structure is stabilized ?

Investigation of the structural stability of Fe

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

Collaborators:

Leonov, Poteryaev, Anisimov, DV; Phys. Rev. Lett. 106, 106405 (2011)

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

Bain and Dunkirk (1924)

• continuous transformation path from bcc-phase to fcc-phase• volume per atom fixed at exp. value of α-Fe

Total energies calculated along bcc-fcc Bain transformation path:

bcc-fcc structural transition in paramagnetic Fe

bcc (c/a=1) fcc (c/a = )2

Z = 8 Z = 12 c/a

21

c c

aa

Pressure, GPa

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


Construct Wannier functions for partially filled Fe sd orbitals

First-principles multi-band Hamiltonianincluding local interactions

Goal: Understand the structural stability ofparamagnetic bcc phase

GGA+DMFT:

bcc-fcc structural transition at Tstruct ≈ 1.3 TC

Pressure, GPa

GGA+DMFT total energy Etot - Ebcc

fccbcc


Construct Wannier functions for partially filled Fe sd orbitals

What determines the strong temperature dependence of the total energy?

GGA:Only paramagnetic fcc structure stable

Goal: Understand the structural stability ofparamagnetic bcc phase


fccbcc

3.6 TC

• kinetic energy favors fcc structure• correlation energy indifferent as in GGA fcc structure stable}


Contributions to the GGA+DMFT total energy:

Etot=Ekin+Eint

1.8 TC

bcc

• kinetic energy favors fcc structure• correlation energy increases fcc structure still stable}


fcc



Etot=Ekin+Eint

1.2 TC

bcc

• kinetic energy favors fcc structure• correlation energy increases bcc structure becomes stable}


fcc



Etot=Ekin+Eint

GGA+DMFT:

bcc-fcc structural transitionat Tstruct ≈ 1.3 TC

0.9 TC

bcc

• kinetic energy favors fcc structure• correlation energy increases bcc structure remains stable}


fcc


GGA+DMFT:

bcc-fcc structural transitionat Tstruct ≈ 1.3 TC


Etot=Ekin+Eint

Electronic correlations responsible for Tstruct > TC

Pressure, GPa

Fe

Tstruct


Conclusion:

Vexp ~ 165 / 158 au3

• 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:

GGA+DMFT

bcc Fe

fcc Fe∆V ~ -2%

Equi

libri

um v

olum

e (a

u3) Pressure, GPa


Equilibrium volume V

Leonov, Poteryaev, Anisimov, DV; Phys. Rev. B 85, 020401(R) (2012)

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)

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)

Lattice dynamics of paramagnetic fcc iron

Non-magnetic GGA phonon dispersion 1. Brillouin zone

Elastic constants much too large

Pressure, GPa

Exp.: Zarestky, Stassis (1987)

GGA+DMFT phonon dispersion at 1.4 TC

Exp.: Zarestky, Stassis (1987)

Calculated:• equilibrium lattice constant

a~3.605 Å (aexp~3.662 Å)• Debye temperature Θ~349 K

Pressure, GPa

Lattice dynamics of paramagnetic fcc iron

Strong anharmonic effectsT1

Dynamically and elastically unstable

Pressure, GPa

Instability of bcc phase due to soft transverse T1 acoustic mode near N-point •

Leonov, Poteryaev, Gornostyrev, Lichtenstein, Katsnelson, Anisimov, DV;Scientific Reports 4, 5585 (2014)

N [110] displacement at 1.4 TC:

What causes the bcc-fcc structural phase transition ?

GGA+DMFT phonon dispersion for bcc structure

T1 mode becomes strongly anharmonic high lattice entropy lowers free energy F = E – TS

What stabilizes the high temperature bcc (δ) phase ?

Pressure, GPa

bcc (δ) phase of iron stabilized byelectronic correlations + lattice entropy

Quantitative estimate:

Leonov, Poteryaev, Anisimov, DV; Phys. Rev. B 85, 020401(R) (2012)

Leonov, Skornyakov, Anisimov, DV; Phys. Rev. Lett. 115, 106402 (2015)

FeSe

Fe-based superconductors

‘11’ FeSe ‘111’ LiFeAs ‘1111’ LaFeAsO ‘122’ BaFe2As2

• new class of superconducting materials (Tc ~ 55 K in SmFeAsO1-xFx)• parent compounds are metals (typically do not superconduct)• superconductivity induced by doping or pressure

Y. Kamihara et al. (2008)

similarity to high-Tc cuprates

FeSe: Crystal structure and properties

• How does the electronic and magnetic structure of FeSe change with lattice expansion ?

• Why does Tc increase ?

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 -

• Simplest structure of/parent compound forFe-based superconductors

• tetragonal crystal structure• Tc ~ 8 K (~ 37 K under hydrostatic pressure)• isovalent substitution Se → Te:

- lattice expansion- Tc increases up to ~ 14 K

GGA+DMFT total energy U = 3.5 eV, J = 0.85 eV

FeSe: Phase stability and local magnetism

• isostructural transformation upon ~ 10%expansion of the lattice (pc ~ -6.4 GPa)

• strong increase of the fluctuating magnetic moment

Observed in FeTe:- tetragonal to collapsed-tetragonalphase transformation under pressure

- suppression of magnetism > 4-6 GPaExp: Zhang et al. (2009)

FeSe: Spectral function

Low-volume phase

High-volume phase

Van Hove singularityshifts to EF due to correlations

Exp

ansi

on

Exp.: Yokoya et al. (2012)

Spectral weight suppressed with Tesubstitution ( = lattice expansion)

T = 290 K

non-magnetic GGA

Exp

ansi

on

GGA+DMFT

Lifshitz transition

Low-volume phase:

• Orbital-selective shift and renormalization of quasiparticle bands

• Van Hove singularity at M point pushed towards EF

High-volume phase:

• Complete reconstruction of the electronic structure

• Correlation-induced shift of Van Hove singularity above EF

Lifshitz transition

FeSe: Electronic structure

Leonov et al., Phys. Rev. Lett. 115, 106402 (2015)

Shift of Van Hove singularity to EF goes along with increase of Tc in FeSe1-xTex

Conjecture:Superconductivity in FeSe1-xTex depends sensitively on Van Hove singularity

Correlation-induced change of magnetic structure upon expansion of lattice

In accord with experiments on FeSe1-xTex Xia et al. (2009), Tamai et al. (2010), Maletz et al. (2014), Nakayama et al. (2014)

non-magnetic GGA

Low-volume phase:

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

High-volume phase:

• Correlations induce topological change of FS Lifshitz transition

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

GGA+DMFT

Exp

ansi

on

FeSe: Fermi surface

Leonov et al., Phys. Rev. Lett. 115, 106402 (2015)

Conclusion:

The structural stability of solidsdepends strongly on

Coulomb correlations between electrons

Surprising Effects of Electronic Correlations in Solids

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