Surprising Effects of Electronic Correlations in Solids
Transcript of 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
Temporal/spatial correlations in everyday life
Time/space average inappropriate
Temporal/spatial correlations in everyday life
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
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
GGA+DMFT:
bcc-fcc structural transition at Tstruct ≈ 1.3 TC
Pressure, GPa
GGA+DMFT total energy Etot - Ebcc
fccbcc
bcc-fcc structural transition in paramagnetic Fe
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
GGA+DMFT total energy Etot - Ebcc
fccbcc
3.6 TC
• kinetic energy favors fcc structure• correlation energy indifferent as in GGA fcc structure stable}
bcc-fcc structural transition in paramagnetic Fe
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}
GGA+DMFT total energy Etot - Ebcc
fcc
bcc-fcc structural transition in paramagnetic Fe
Contributions to the GGA+DMFT total energy:
Etot=Ekin+Eint
1.2 TC
bcc
• kinetic energy favors fcc structure• correlation energy increases bcc structure becomes stable}
GGA+DMFT total energy Etot - Ebcc
fcc
bcc-fcc structural transition in paramagnetic Fe
Contributions to the GGA+DMFT total energy:
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}
GGA+DMFT total energy Etot - Ebcc
fcc
bcc-fcc structural transition in paramagnetic Fe
GGA+DMFT:
bcc-fcc structural transitionat Tstruct ≈ 1.3 TC
Contributions to the GGA+DMFT total energy:
Etot=Ekin+Eint
Electronic correlations responsible for Tstruct > TC
Pressure, GPa
Fe
Tstruct
bcc-fcc structural transition in paramagnetic Fe
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
bcc-fcc structural transition in paramagnetic Fe
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