, if | | ph ; ep ( ) = (electron self-energy, weak coupl.) 0 - - PowerPoint PPT Presentation

Electron-phonon coupling in cuprates

Olle Gunnarsson

• Property-dependent apparent electron-phonon coupling.
• Vertex corrections.
• Undoped cuprates: Polaronic behavior.

Collaboration: Oliver R¨

• sch

Max-Planck Institut Stuttgart, Germany

1

Important effects of electron-phonon coupling

• Kink in nodal direction of photoemission spectrum.

(Lanzara et al., Nature 412, 510 (2001)).

• Anomalous softening, broadening of half-breathing phonon.

(Pintschovius phys. stat. sol. 242, 30 (2005)).

= (0.5,0,0) q

2−

Cu 2+ O

Interplay with Coulomb interaction important

• LDA underestimates width of half-breathing phonon by one order of magnitude.

(Bohnen, Heid and Krauss, Europhys. Lett. 64, 104 (2003)).

Here use t-J model or Hubbard model.

t-J model

t-J model with phonons can be derived from the three-band model.

• Electron-phonon coupling mainly due to modulation of tpd.
• One-site term order of magnitude larger than off-site term (neglected).

(R¨

• sch and Gunnarsson, PRL 92, 146403 (2004)).

MPI-FKF Stuttgart

2

Apparent el-ph coupling

Electron-phonon coupling usually studied for non-interacting electrons.

• Σep(ω) =

−λω,

if |ω| ≪ ωph;

0,

if |ω| ≫ ωph (electron self-energy, weak coupl.)

• 2 Im Π(ωph) = 2πω2

phN(0)λ

(phonon self-energy, weak coupl.) If λ determined from one experiment (e.g., phonon width), another experiment (e.g., PES) can be predicted. Similar relations are often implicitly assumed for strongly correlated systems. If this is not true, different experiments may appear contradictory.

MPI-FKF Stuttgart

3

Sum rule. Phonon self-energy. t-J model Π(q, ω) =

(g2

q/N)χ(q,ω)

1+(g2

q/N)χ(q,ω)D0(q,ω).

Π phonon self-energy, χ charge response function, D0 noninteracting phonon Green’s function.

Sum-rule: (δ doping).

1 πN

• q=0

∞

−∞ |Imχ(q, ω)|dω = 2δ(1 − δ)N∼ δN. (Khaliullin and Horsch)

As δ → 0, χ → 0

1 πN

P

q=0 1 g2

q

R ∞

−∞ |ImΠ(q, ω)|dω ≈ 2δ(1 − δ)∼ δ.

As δ → 0, Π → 0. Natural: For zero doping all states filled. No response possible. Is the electron self-energy reduced in a similar way?

MPI-FKF Stuttgart

4

Sum rule. Electron self-energy

Define Green’s function

G(k, z) =

a(k) z−ε(k)−Σ(k,z);

Σ(k, z)

→ |z|→∞ b(k) z

.

ωnk = ∞

−∞ ωnA(k, ω)dω/a(k)

A(k, ω) = 1

πImG(k, ω − i0+). 1 π

∞

−∞ ImΣ(k, ω −i0+)dω = b(k) = ω2k −(ωk)2.

Undoped t-J model with on-site coupling (Σep = Σ − Σ(g = 0)):

1 π −∞ ImΣep(k, ω − i0+)dω = 1 N

• q |gq|2 ≡ ¯

g2.

Identical to the lowest order result for noninteracting electrons (but valid for any ¯

g). k-independent! Nontrivial. Not valid for off-site coupling.

In contrast to phonon self-energy, correlation does not suppress Σep.

• O. R¨
• sch and O. Gunnarsson, PRL 93, 237001 (2004).

5

Phonon self-energy: System responds to phonons by transferring singlets to sites with lower singlet energies. But there are only fraction δ singlets available. Electron self-energy Singlet created in PES. Is easily scattered by pho- nons to other states as only fraction δ blocked by

• ther singlets.

Strong asymmetry between carriers and phonons.

k q k−q

Phonon induced carrier-carrier interaction: One singlet emits a phonon, which is absorbed by another singlet. Both scattering processes allowed in limit δ → 0. No suppression of carrier-carrier interaction.

q k’ k k+q k’−q

Strong interaction without soft phonons possible.

• O. R¨
• sch and O. Gunnarsson, PRL 93, 237001 (2004).

6

Relation between Hubbard and t-J models Consider large U limit, half-filling and symmetric parameters.

AH(k, ω) = At-J(k, ω + U

2 ) + At-J(−k, U 2 − ω).

Define self-energies

GH(k, z) =

• dω AH(k,ω)

z−ω

≡

1 z−ΣH(k,z).

Gt−J(k, z) =

• dω At−J(k,ω)

z−ω

≡

0.5 z−Σt−J(k,z).

This requires

ΣH(k, z) ≈ 2Σt-J(k, z + U

2 ) + U 2 4z .

Then Im ΣH(k, z) = 2ImΣt-J(k, z + U

2 ), which gives the sum rule 1 π

−U/4

−∞

ImΣep

H (k, ω − i0+)dω = 2¯

g2.

Integration only over the photoemission part. Twice the value for the t-J model.

MPI-FKF Stuttgart

7

Sum rules. Hubbard model. El.-ph. part of electron self-energy

−U/4 U/4 t−J ω

Nonint

Hubbard

No vert g2

2

g g2 2g2 g2 2g2 /2 g2/2 /2 /2 −3g

2

Sum rule for Im Σ over photoem. and inverse photoem. parts, each 2¯

g2.

Total sum rule just ¯

g2.

Im Σ(k, ω) has large positive contribution U 2/4 at ω = 0. The electron-phonon interaction slightly reduces this by −3¯

g2.

Calculate lowest order diagram in electron-phonon in- teraction, neglecting vertex corrections (Γ = 1). Sum rule over photoemission spect. violated by factor 4. Γ Γ k+q q Vertex corrections important for electron-phonon part of electron self-energy.

MPI-FKF Stuttgart

8

Vertex corrections phonon self-energy. Large U

In limit of weak electron-phonon coupling, phonon self- energy given by density-density correlation function. Ne- glect vertex corrections (Γ = 1).

Γ k+q k

Assume that doping δ results in a weight ∼ δ on the in- verse photoemission side close to Fermi energy. Assume that this weight and photoemission spectrum within energy range 2∆. Exact sum rule fulfilled

1 πN 2

• q

2∆

−2∆ |ImχH,No vert.(q, ω)|dω = 2δ(1 − δ)

Sufficient to use dressed electron Green’s functions. Neglect of vertex corrections violates sum rule for electron- phonon contribution to electron but not phonon self-energy.

ε N(ε) δ 1−δ ( )/2 2∆

MPI-FKF Stuttgart

9

q-dependence

1 πN 2

• q=0

2∆

−2∆ |ImχH(q, ω)|dω = 2δ(1 − δ). 1 π

2∆

−2∆ |ImχH,No vert.(q, ω)|dω =2 k[wP(k)wIP(k+q)+wP(k+q)wIP(k)] 1 πN 2

• q

2∆

−2∆ |ImχH,No vert.(q, ω)|dω = 2δ(1 − δ).

Calculate wP(k) and wIP(k) for

√ 18 × √ 18 t-J model with two holes.

1 πN

2∆

−2∆ |ImχH(q, ω)|dω

q/ π

3

(0, 0) (1, 1) (2, 0) (2, 2) (3, 1) (3, 3)

No vert. 0.1660 0.1848 0.1927 0.2103 0.2025 0.2285 Exact 28.44 0.2100 0.1961 0.2191 0.2085 0.2212 Ratio 0.8804 0.9825 0.9597 0.9714 1.0330 Even q-dependence rather well described.

10

Two-site model

Consider a two-site model:

H = t(n+ − n−) + U 2

i=1 ni↑ni↓ + ωphb†b + g(c† +c− + c† −c+)(b + b†).

Huang, Hanke, Arrigoni, Scalapino (PRB 68, 220507 (2003)):

Γ(k, q) =

G2(k,q) G(k+q)G(k).

G2(k, q) = β

0 dτei(ωn+ωm) β 0 dτ

′e−iωmτ ′

×

pqσ′ Tτc† p+qσ′ (τ

′)cpσ′ck+qσ(τ)c†

pσ(0).

Γ k+q k q Γ(ω

′, +, ω, −) = ω ′(ω+ω ′)+ωt+(U/2)2

(ω′+t)(ω+ω′−t)

. (ω

′

=electron; ω=phonon).

ω

′ ≈ ± U

2 ⇒ Γ(ω

′, +, ω, −) = 2.

Fixes up electron self-energy sum rule in photoemission and inverse photoemission ranges.

MPI-FKF Stuttgart

11

Small q and ω

Huang, Hanke, Arrigoni, Scalapino (PRB 68, 220507 (2003)): Koch, Zeyher (PRB 70, 094510 (2004)):

• Z(p)Z(p + q)Γ(p, q) reduced in static case (phonon frequency zero).

Fermi liquid arguments (Grilli, Castellani PRB 50, 16880 (1994)): Small |q| and ω:

• Z(p)Z(p + q)Γ(p, q) reduced for ω = 0 but not for |q| = 0.

Here integration over all q and ω and study of Γ(p, q). Vertex corr. give enhancement of electron but not phonon self-energy sum rule.

MPI-FKF Stuttgart

12

Polaronic behavior

Undoped CaCuO2Cl2. K.M. Shen et al., PRL 93, 267002 (2004). Spectrum very broad, even at top of band (insulator!). Shape Gaussian, not like a quasi-particle. Chemical potential always well above broad peak A, although expected to be anywhere in the gap depending on sample preparation. Polaronic behavior. Broad boson side band. Quasi-particle (≈ 0 weight, small dispersion) at ε ≈ 0. Strong coupling to bosons. Phonons, spin fluctuations?

13

Electron-phonon coupling. Undoped system. Shell model

Find the electron-phonon coupling strength to a Zhang-Rice singlet. Zhang-Rice singlet is an additional hole in a linear combination of four O holes. Use a shell model to describe phonons. Phonon eigenvectors ⇒ Potential on a singlet due to a phonon. Screening by the ”shells”, but otherwise no screening. Add coupling due to modulation of tpd and εd − εp as in the treatment of the half-breathing mode.

MPI-FKF Stuttgart

14

Electron-phonon coupling strength. La2CuO4 Hep =

1 √ N

• qνi Mqνi(1 − ni)(bqν + b†

−qν)

Dimensionless coupling λ = 2 1

8t

• qν

|Mqν|2 ωqν

. We find λ = 1.2. Due to (half-)breathing modes (80 meV), Oz mo- des (60-70 meV) and La (Cu) modes (20 meV).

40 80 120 160 20 40 60 80 100 γ(ω) ω [meV]

t-J Holstein model:

Polarons for λ > 0.4. Mishchenko and Nagaosa Phonons sufficient to put undoped cuprates well onto the polaronic side.

MPI-FKF Stuttgart

15

Boson side-band. Dispersion

Undoped CaCuO2Cl2. K.M. Shen et al., PRL 93, 267002 (2004). Boson side-band disperses as quasi-particle peak in t-J model without bosons. But quasi-particle now at smaller binding energy. Small dispersion. Numerical calculation for t-J model with phonons: Phonon side-band follows quasi-particle peak in t-J model without phonons.

(A. S. Mishchenko and N. Nagaosa Phys. Rev. Lett. 93, 036402 (2004))

MPI-FKF Stuttgart

16

Adiabatic approximation g(k, z) = E0|c†

k 1 z−H+E0 ck|E0

=

• dQdQ

′E0|QQ|c†

k 1 z−H+E0 ck|Q

′Q ′|E0

H = Hel + Hep(Q) + 1

2κQ2 + 1 2M ˆ

P 2 ≡ H(Q) +

1 2M ˆ

P 2.

• I. Neglect

1 2M ˆ

P 2.

Phonon satellites smeared out.

g(k, z) =

• dQdQ

′E0|QQ|c†

k 1 z−H(Q)+E0 ck|Q

′Q ′|E0

Then there is only a contribution for Q = Q

′

.

g(k, z) =

• dQE0|QQ|c†

k 1 z−H(Q)+E0 ck|QQ|E0

• II. Ground state: Q|E0 ≈ φ0(Q)|Φ0(Q). (Adiabatic approximation).

g(k, z) ≈

• dQφ2

0(Q)G(k, z, Q).

G(k, z, Q) = Φ0(Q)|c†

k 1 z−H(Q)+E0 ck|Φ0(Q).

G(k, z, Q) calculated without (dynamic) electron-phonon coupling but for distorted

lattice (Q treated as c-number). Weighted by φ2

0(Q).

• O. R¨
• sch and O. Gunnarsson, (Eur. Phys. J. B 43, 11 (2005)).

17

Accuracy of approximations

0.2 0.4 0.6 0.8 1

• 7
• 6
• 5
• 4
• 3
• 2
• 1

1 ω adiabatic approximation exact diagonalization ARPES (k=(3, 1) π

5 ). Undoped

10-site Holstein-t-J model

t=1, J=0.4, ωph=0.1, g= √ 0.05

Quite accurate. Phonon satellites smeared out.

MPI-FKF Stuttgart

18

Undoped system g(k, z) ≈

• dQφ2

0(Q)G(k, z, Q).

In general: φ0(Q) very complicated. Potential energy surface has several minima for Q0 = 0. φ0(Q) large around Q0. Knowledge of G(k, z, Q = 0) not sufficient. Undoped system: No electron-phonon coupling in initial state (for t-J model).

φ2

0(Q) centered around Q = 0. Spectrum essentially the spectrum without el-ph.

coupling (G(k, z, Q = 0)) but broadened due to contributions from (small) Q = 0. Explains why t-J model with phonons puts phonon sideband at quasi-particle in t-J model without phonons for undoped system.

MPI-FKF Stuttgart

19

10-site t-J model with one phonon mode (π,π). Exact diagonalization. Coupl.

g > 0 g = 0

1 2

• 2

2 4 6

• 2

ω

Undoped

g > 0 broadened version of g = 0

spectrum

0.5 1

• 5

5 10 15 20

• 1

ω

Doped

g > 0 can be quite different

from g = 0 spectrum

MPI-FKF Stuttgart

20

Method for calculating spectra for undoped system

When phonons included, Hilbert space very large. In particular if coupling strong.

g(k, z) ≈

• dQφ2

0(Q)G(k, z, Q).

For the undoped system φ0(Q) is known and simple. Sample Q using Monte-Carlo. Calculate G(k, z, Q) for each sampled Q value.

G(k, z, Q) is the Green’s function for a distorted lattice but without (dynamical)

electron-phonon coupling. Hilbert space has the same size as for a system without electron-phonon coupling. In this approximation the introduction of electron-phonon coupling is not a drastic

• complication. Possible to consider all 21 modes in La2CuO4 with q-dependent

coupling constants.

MPI-FKF Stuttgart

21

• ARPES. LaCuO4

EDC Intensity (A.U.)

• 0.8
• 0.4

0.0

• 0.8
• 0.4

0.0

a b

(0.32,0.32) (0.76,0.76)

E - EF (eV)

Background subtracted Very broad phonon satellite. No visible quasiparticle peak.

R¨

• sch, Gunnarsson, Zhou,Yoshida,Sasagawa, Fujimori, Hussain, Shen and Uchida, cond-mat/0504660

MPI-FKF Stuttgart

22

Width of phonon side-band

• X. Zhou et al.: Width (π/2, π/2) (2×HWFM)∼ 0.48 eV (La2CuO4).

K.M. Shen et al.: Width ∼ binding energy (Ca2CuO2Cl2) (PRL 93, 267002 (2004)).

0.2 0.4 0.6 0.8 1 1.2

• 1.4
• 1.2
• 1
• 0.8
• 0.6
• 0.4
• 0.2

A(ω) ω (π/2,π/2) (0,π/2) (π/2,π) (0,0)

Peak heights aligned.

0.2 0.4 0.6 0.8 1 1.2

• 1.4
• 1.2
• 1
• 0.8
• 0.6
• 0.4
• 0.2

A(ω) ω (π/2,π/2) (0,π/2) (π/2,π) (0,0)

Heights and positions aligned.

• La2CuO4. 21 phonon modes. q-dependent coupling constants. 4 × 4 cluster.

50 000 samples. Width (π/2, π/2) (2×HWFM) ∼ 0.5 eV. Increases with binding energy.

R¨

• sch, Gunnarsson, Zhou,Yoshida,Sasagawa, Fujimori, Hussain, Shen and Uchida, cond-mat/0504660

23

Tails of phonon side-bands

0.2 0.4 0.6 0.8 1 1.2

• 1.4
• 1.2
• 1
• 0.8
• 0.6
• 0.4
• 0.2

A(ω) ω (π/2,π/2) (0,π/2) (π/2,π) (0,0)

Coupling for La2CuO4.

q-independent coupling. 4 × 4 lattice. ≈ 5000 samples.

1 πImG(k, ω) = n |n, N − 1|ck|0, N|2δ(ω − E0(N) + En(N − 1))

Ordered system without phonons: Momentum conservation ⇒ k-dependent lower limit for En(N − 1) coupling to ck|0, N, upper limit for spectrum. Disordered system (phonons approximately included): No momentum conservation

⇒ Coupling to lowest state independent of k.

Peak extends up to the same threshold (in this approximation).

24

Summary

• Apparent el.-ph. coupling in strongly correlated materials

depends strongly on property studied.

• Sum rule for phonon but not electron self-energy satisfied,

even when vertex corrections neglected.

• El.-ph. coupling in undoped system strong enough

to give polaronic behavior.

• Adiabatic approximation explains why phonon sideband follows

quasiparticle dispersion in absence of el.-ph. coupl. for undoped system.

• Side-band width reasonable. Increases with binding energy.

MPI-FKF Stuttgart

25