Spectral functions of Sr 2 IrO 4 From Cluster Dynamical Mean-Field - - PowerPoint PPT Presentation

Spectral functions of Sr2IrO4

From Cluster Dynamical Mean-Field Theory

Benjamin Lenz

IMPMC - Sorbonne Université, CNRS, MNHN, IRD benjamin.lenz@sorbonne-universite.fr

HPC online lectures on Computational Materials Physics 2020-11-17, Paris/Singapore

HPC online lectures - C-DMFT for Sr2IrO4 - Benjamin Lenz

Outline

• m cluster DMFT
• Sr2IrO4: A (not so) typical strongly correlated material
• Density functional theory and downfolding
• DMFT to the rescue:

(Cluster-) Dynamical Mean-Field Theory in a nutshell

• Spectral functions from cluster-DMFT
• Limitations of the

model

jeff = 1/2

HPC online lectures - C-DMFT for Sr2IrO4 - Benjamin Lenz

• Ir4+ (5d5) state: ⟶ extended 5d orbital
• Modest Coulomb interactions of 2eV (~bandwidth)

Sr Ir O

• C. Martins et al., PRL 107, 266404 (2012)
• R. Arita et al., PRL 108, 086403 (2012)

3 mm

110 001

• C. Martins et al.,

JPCM 29, 263001 (2017)

Sr2IrO4

HPC online lectures - C-DMFT for Sr2IrO4 - Benjamin Lenz

Ge et al. (2011) PRB 84, 100402(R)

• Ir4+ (5d5) state: ⟶ extended 5d orbital
• Modest Coulomb interactions of 2eV (~bandwidth)
• C. Martins et al., PRL 107, 266404 (2012)
• R. Arita et al., PRL 108, 086403 (2012)

Δ = 0.25 eV

• Interplay of Coulomb correlations and

spin-orbit coupling: spin-orbit Mott insulator ( )

(1 0 17) Intenstiy T (K) Magnetization (µB/Ir) 300 200 100

0.075 0.050 0.025 0.000

F. Ye et al. (2013) PRL 87, 140406(R)

• Canted antiferromagnet below TN~240K
• B. J. Kim et al. (2008)

PRL 101, 076402

• B. J. Kim et al. (2009)

Science 323, 1329

• C. Martins et al.,

JPCM 29, 263001 (2017)

Sr2IrO4

HPC online lectures - C-DMFT for Sr2IrO4 - Benjamin Lenz

/2)

• 0.4
• 0.2

0.0 E - EF (eV) x = 0.05 (π, 0) de la Torre et al. (2015) PRL 115, 176402 (0,0) (π/2,π/2) (π,0)

EB = 200 meV

x = 0 (0,0) (π/2,π/2) (π,0) x = 0.01 (0,0) (π/2,π/2) (π,0) x = 0.05

EB = 10 meV EB = EF

Angle-resolved photoemission spectroscopy

• Antinodal region shows depletion of spectral weight at Fermi level

⟶ Pseudogap phase (?)

2.0 1.5 1.0 0.5 0.0 Binding energy (eV)

(a) Γ

X M Γ

• B. J. Kim et al. (2008)

PRL 101, 076402

⟶ Fermi pockets emerge at sufficient doping

• Electron-doped (Sr1-xLax)2IrO4:

PM metal down to lowest temperatures for x≥0.04

• Sr2IrO4 isostructural to superconducting oxides of La2CuO4 family

Ge et al. (2011) PRB 84, 100402(R)

Sr2IrO4

HPC online lectures - C-DMFT for Sr2IrO4 - Benjamin Lenz

• Unit cell contains two layers in c-direction, which are shifted

by (1/2, 0) in the a-b plane

• IrO6 octahedra alternately tilted (anti)clockwise

⟶ two different configurations for the Ir atoms ⟶ symmetry lowered from I4/mmm to I4/acd

• Rotations of IrO6 octahedra cause a doubling of the unit cell

⟶ halved first BZ, redefinition of high symmetry points

• 1st Brillouin

zone 2nd Brillouin zone (1st Brillouin zone undistorted)

kz=0

a* a* b* b* M X=M

= =X

ion

Sr Ir O

11°

• Sr2IrO4 - Structure and Brillouin zone

HPC online lectures - C-DMFT for Sr2IrO4 - Benjamin Lenz

Details of DFT calculation:

• Ir4+ (5d5) state: extended d-orbitals
• Strong spin-orbit coupling: ζSO(Ir) = 0.40 eV

Energy (eV)

jeff = 1 2 , jeff = 3 2

DFT+SOC calculation, projection on states

jeff

⟶ t2g states turn into states

jeff = 1 2 jeff = 3 2 |mj| = 3 2 |mj| = 1 2

• C. Martins et al.
• J. of Phys. Cond. Mat. (2017)

Sr2IrO4 - DFT and Downfolding

HPC online lectures - C-DMFT for Sr2IrO4 - Benjamin Lenz

H = − X

i,j

tij ⇣ c†

icj + h.c.

⌘ + µ X

i

ni + U X

i

ni↑ni↓ Energy (eV)

DFT+SOC calculation, projection on states

jeff

⟶ Solve low-energy model for U=1.1eV

U

SO

jeff = 1/2 jeff = 3/2 band

LHB UHB

jeff = 1/2

• Construct effective low-energy tight-binding model for jeff=1/2 manifold
• Add electronic interactions between the jeff=1/2 states via a Hubbard U

Scheme adapted from:

• B. J. Kim et al.
• Phys. Rev. Lett. (2008)

upper Hubbard band lower Hubbard band bands

Sr2IrO4 - DFT and Downfolding

HPC online lectures - C-DMFT for Sr2IrO4 - Benjamin Lenz

Dynamical Mean-Field Theory in a nutshell

+ –

+ ¬ +R¬ +RA¬

Vn Vn e– e– Electron reservoir DMFT R A

Time

• G. Kotliar & D. Vollhardt, 


Physics Today (2004)

Σ

Σimp Σ

Σ Σ Σ

• A. Georges et al. (1996) 

• Rev. Mod. Phys. 68 6861
• Solve the local problem

impurity solver 


(continuous-time quantum Monte-Carlo, exact diagonalisation
 iterative perturbation theory, numerical renormalization group,…)

• Iterate until self-consistency:


local lattice Green’s function = impurity Green’s function

⟶

• W. Metzner & D. Vollhardt, PRL 62 324 (1989)

• A. Georges & G. Kotliar, PRB 45 6479 (1992)
• A. Georges et al., Rev. Mod. Phys. 68 6861 (1996)

• G. Kotliar et al., Rev. Mod. Phys. 78 865 (2006)
• Map many-body lattice problem to many-body local problem
• Approximate lattice self-energy by local self-energy:


Σ(k, ω) ≈ Σimp(ω)

HPC online lectures - C-DMFT for Sr2IrO4 - Benjamin Lenz

Dynamical mean-field theory and beyond

• Diagrammatic extensions of DMFT


(D A, TRILEX, QUADRILEX, 
 dual fermions, dual bosons, etc.)

• Quantum cluster extensions of DMFT


(DCA, Cellular DMFT, etc.)

Γ

How to include non-local fluctuations?

• Weakly correlated regime at small U:


Density of states resembles the band density of states

• Complex spectral function at intermediate U:


Quasiparticle bands at low energies & Hubbard bands at high energy

• Mott insulator at large U:


Two Hubbard bands, separated by a gap ~U

Local spectral function

Increase of e-e interaction strength U

Success of DMFT: Captures phase transition 
 from metal to Mott insulator

Energy LDOS

U ≫ 1 U = 0 ∼ U X.Y. Zhang et al. 
 PRL 70 6666 (1993)

• G. Rohringer et al. (2018)

• Rev. Mod. Phys. 90 025003

• T. Maier et al. (2005)

• Rev. Mod. Phys. 77 1027

HPC online lectures - C-DMFT for Sr2IrO4 - Benjamin Lenz

H = − X

i,j

tij ⇣ c†

icj + h.c.

⌘ + µ X

i

ni + U X

i

ni↑ni↓ Energy (eV)

DFT+SOC calculation, projection on states

jeff

⟶ Solve low-energy model for U=1.1eV

U

SO

jeff = 1/2 jeff = 3/2 band

LHB UHB

jeff = 1/2

• Construct effective low-energy tight-binding model for jeff=1/2 manifold
• Add electronic interactions between the jeff=1/2 states via a Hubbard U

Scheme adapted from:

• B. J. Kim et al.
• Phys. Rev. Lett. (2008)

upper Hubbard band lower Hubbard band bands

Sr2IrO4 - DFT and Downfolding

HPC online lectures - C-DMFT for Sr2IrO4 - Benjamin Lenz

H = − X

i,j

tij ⇣ c†

icj + h.c.

⌘ + µ X

i

ni + U X

i

ni↑ni↓ Energy (eV)

DFT+SOC calculation, projection on states

jeff

⟶ Solve low-energy model for U=1.1eV

U

SO

jeff = 1/2 jeff = 3/2 band

LHB UHB

jeff = 1/2

• Construct effective low-energy tight-binding model for jeff=1/2 manifold
• Add electronic interactions between the jeff=1/2 states via a Hubbard U

Scheme adapted from:

• B. J. Kim et al.
• Phys. Rev. Lett. (2008)

upper Hubbard band lower Hubbard band bands

Sr2IrO4 - DFT and Downfolding

−1 −0.5 0.5 1 Γ M X Γ Energy (eV) −1 −0.5 0.5 1 Γ M X Γ Energy (eV)

(A) (B)

HPC online lectures - C-DMFT for Sr2IrO4 - Benjamin Lenz

• Calculated spectral function
• ARPES

1st Brillouin zone

• ARPES

2nd Brillouin zone

• Good agreement between theory and experiment
• No matrix-element effects included in the calculated spectral function, but strong

effect in ARPES measurements ⟶ Fourier component of symmetry lowering potential is weak

• Identification of character of spectral density:

1st BZ: dominated by jeff=1/2 band, jeff=3/2 contribution at -1.1eV along Γ-X 2nd BZ: dominated by jeff=3/2 band (mainly mj=1/2 band) Oriented-cluster DMFT applied to the half-filled jeff=1/2 band, filled jeff=3/2 bands from LDA+DMFT

• A. Louat, PhD thesis (2018)
• C. Martins et al. (2011)

PRL 107, 266404

Undoped Sr2IrO4 - Spectral Function

HPC online lectures - C-DMFT for Sr2IrO4 - Benjamin Lenz

(A) (B)

• Energy cuts of the spectral function in the paramagnetic phase
• Good agreement between experiment and theory at both energies
• Lowest energy excitations disperse up to -0.25eV at X and never cross the Fermi level
• Spectrum similar to the antiferromagnetically ordered one
• Undoped Sr2IrO4 - Spectral Function
• C. Martins et al., Phys. Rev. Mat. 2 032001(R) (2018)

HPC online lectures - C-DMFT for Sr2IrO4 - Benjamin Lenz

−2 −1.5 −1 −0.5 Γ M X Γ Energy (eV) Nie 2015 de la Torre 2015 Liu 2015 Cao 2016

• Y. F. Nie et al.

PRL 114, 016401 (2015)

• A. de la Torre et al.

PRL 115, 176402 (2015)

• Y. Liu et al.
• Sci. Rep. 5, 13036

(2015)

• Y. Cao et al.
• Nat. Comm. 7, 11367

(2016)

Antiferromagnetic Phase

• B. Lenz et al., JPCM 31 293001 (2019)

HPC online lectures - C-DMFT for Sr2IrO4 - Benjamin Lenz

−2 −1 1 Γ M X Γ Energy (eV)

• Lower branch at X point close to Fermi level
• Good agreement with ARPES data on

(Sr1-xLax)2IrO4

• Upper and lower branch gapped at M point

⟶ no Dirac point at -0.1eV

(f)

• 0.4

0.0 0.4 k - kM (Å

• 1)

x = 0.05

de la Torre et al. PRL (2015) Brouet et al. PRB(R) (2015)

x=0.04

Calculation done for Hubbard interaction strength of Ueff=0.6eV

• 0.4
• 0.2

0.0 E - EF (eV)

Undoped system: Ueff=1.1eV ⟶ consistent with enhanced electronic screening jeff=1/2 jeff=3/2

• Upper branch of jeff=1/2 band crosses

Fermi level ⟶ Fermi pocket around M point

Electron-doped Sr2IrO4 - Spectral Function

HPC online lectures - C-DMFT for Sr2IrO4 - Benjamin Lenz

• Size and shape of lense-shaped Fermi pocket around M point in good agreement with ARPES results

de la Torre et al. (2015) PRL 115, 176402 Gretarsson et al. (2016) PRL 117, 107001

x=0.05

Electron-doped Sr2IrO4 - Spectral Function

HPC online lectures - C-DMFT for Sr2IrO4 - Benjamin Lenz

• Size and shape of lense-shaped Fermi pocket around M point in good agreement with ARPES results

de la Torre et al. (2015) PRL 115, 176402

• (Remnant) Fermi surface as used in experiment ⟶ symmetrized spectral function ⟶ extract “pseudogap“

Electron-doped Sr2IrO4 - Spectral Function

HPC online lectures - C-DMFT for Sr2IrO4 - Benjamin Lenz

• Spectral function agrees well with ARPES spectra
• Deviations at ~-1.3eV around Γ
• Fermi surface in good agreement with ARPES

Sr2Ir1-xRhxO4 (b) (a)

−2 −1.5 −1 −0.5 Γ X2 M Γ Energy (eV)

(b)

• 1
• 0.5

0.5 1

• 1
• 0.5

0.5 1 ky [π] kx [π] E = EF

η = 0.08

kΓX2

F

X2

Low High

(c)

Γ

Γ

η = 0.03

• A. Louat et al. PRB 97, 161109(R) (2018)
• A. Louat et al. PRB 100 205135 (2019)

10% hole-doped Sr2IrO4

HPC online lectures - C-DMFT for Sr2IrO4 - Benjamin Lenz

• Spectral function agrees well with ARPES spectra
• Deviations at ~-1.3eV around Γ
• Fermi surface in good agreement with ARPES

Sr2Ir1-xRhxO4 (b) (a)

−2 −1.5 −1 −0.5 Γ X2 M Γ Energy (eV)

(b)

• 1
• 0.5

0.5 1

• 1
• 0.5

0.5 1 ky [π] kx [π] E = EF

η = 0.08

kΓX2

F

X2

Low High

(c)

Γ

Γ

η = 0.03

• A. Louat et al. PRB 97, 161109(R) (2018)
• A. Louat et al. PRB 100 205135 (2019)

10% hole-doped Sr2IrO4

So far: Single-band description ( ) — is this sufficient?

jeff = 1/2

HPC online lectures - C-DMFT for Sr2IrO4 - Benjamin Lenz

• ARPES finds

band to be close to

• f

mainly character along mainly character along

• Isotropic

band on k-average

• DFT+SOC confirms k-dependent modulation of
• rbital weight

jeff = 1/2 EF dxz kx dyz ky jeff = 1/2

Hole doping: Sr2Ir1-xRhxO4 - Orbital Composition of the jeff=1/2 Band

• A. Louat et al., Phys. Rev. B 100 205135 (2019)

HPC online lectures - C-DMFT for Sr2IrO4 - Benjamin Lenz

Polarized neutron diffraction ⟶ magnetic form factor for FM moment ⟶ reconstructed magnetization density: 5dxy dominant Large spatial extent No oxygen moment Deviation from local jeff=1/2 picture

Model: k-dependence of Wannier hole state crucial

• J. Jeong et al. Phys. Rev. Lett. 125 097202 (2020)

(b) (a)

Magnetization Density - Deviation from a Local jeff=1/2 State

(a)

Collaborators

Cyril Martins

LCPQ, Toulouse

Silke Biermann Steffen Backes

CPHT, Palaiseau

Theory Funding: ARPES

Alex Louat Véronique Brouet Fabrice Bert

LPS, Orsay

Neutron Scattering

Jaehong Jeong Yvan Sidis Philippe Bourges Arsen Gukasov Xavier Fabreges Dalila Bounoua

LLB, Gif-sur-Yvette HPC online lectures - C-DMFT for Sr2IrO4 - Benjamin Lenz

Luca Perfetti

LSI, Palaiseau

François Bertran Patrick Le Fèvre Julien E. Rault

SOLEIL, Gif-sur-Yvette

Vladimir Hutanu Andrew Sazonov

RWTH Aachen

• C. Martins et al., Phys. Rev. Mat. 2 032001(R) (2018)
• B. Lenz et al., JPCM 31 293001 (2019)
• A. Louat et al., Phys. Rev. B 100 205135 (2019)
• J. Jeong et al. Phys. Rev. Lett. 125 097202 (2020)

HPC online lectures - C-DMFT for Sr2IrO4 - Benjamin Lenz

• DFT (LDA+SOC) ⟶ effective low-energy model for jeff=1/2 band
• Oriented-cluster DMFT applied to effective model

⟶ including non-local correlations

• Spectral functions of pure and doped Sr2IrO4 agree well with experiment
• PM and AF spectra of pure Sr2IrO4 do not differ much

⟶ AF fluctuations in PM phase important

• Pseudogap-like features of the spectral function obtained

as a result of the non-local fluctuations

• No pseudogap at X point of hole-doped Sr2IrO4

⟶ disorder effect in ARPES of Sr2Ir1-xRhxO4?

• Polarized neutron scattering:

Magnetization density of Sr2IrO4 deviates from local jeff=1/2 picture

Thank you for your attention!

• J. Jeong et al. Phys. Rev. Lett. 125 097202 (2020)
• A. Louat et al., Phys. Rev. B 100 205135 (2019)
• B. Lenz et al., JPCM 31 293001 (2019)
• C. Martins et al., Phys. Rev. Mat. 2 032001(R) (2018)

Summary