Electron-phonon coupling in cuprates Olle Gunnarsson • Property-dependent apparent electron-phonon coupling. • Vertex corrections. • Undoped cuprates: Polaronic behavior. Collaboration: Oliver R¨ osch 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)). Cu 2+ 2− O • Anomalous softening, broadening of half-breathing phonon. (Pintschovius phys. stat. sol. 242 , 30 (2005)). q = (0.5,0,0) 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 t pd . • One-site term order of magnitude larger than off-site term (neglected). (R¨ osch and Gunnarsson, PRL 92 , 146403 (2004)). MPI-FKF Stuttgart 2
Apparent el-ph coupling Electron-phonon coupling usually studied for non-interacting electrons. � − λω, if | ω | ≪ ω ph ; • Σ ep ( ω ) = (electron self-energy, weak coupl.) 0 , if | ω | ≫ ω ph • 2 Im Π( ω ph ) = 2 πω 2 ph N (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 ( g 2 q /N ) χ ( q ,ω ) Π( q , ω ) = q /N ) χ ( q ,ω ) D 0 ( q ,ω ) . 1+( g 2 Π phonon self-energy, χ charge response function, D 0 noninteracting phonon Green’s function. Sum-rule: ( δ doping). � ∞ 1 � −∞ | Im χ ( q , ω ) | dω = 2 δ (1 − δ ) N ∼ δN. (Khaliullin and Horsch) q � =0 πN As δ → 0 , χ → 0 R ∞ 1 1 P −∞ | ImΠ( q , ω ) | dω ≈ 2 δ (1 − δ ) ∼ δ. g 2 πN q � =0 q 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 a ( k ) → b ( k ) G ( k , z ) = z − ε ( k ) − Σ( k ,z ) ; Σ( k , z ) . | z |→∞ z � ∞ � ω n � k = −∞ ω n A ( k , ω ) dω/a ( k ) A ( k , ω ) = 1 π Im G ( k , ω − i 0 + ) . � ∞ 1 −∞ ImΣ( k , ω − i 0 + ) dω = b ( k ) = � ω 2 � k − ( � ω � k ) 2 . π Undoped t - J model with on-site coupling ( Σ ep = Σ − Σ( g = 0) ): � 0 q | g q | 2 ≡ ¯ 1 −∞ ImΣ ep ( k , ω − i 0 + ) dω = 1 g 2 . � π N 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 Σ ep . O. R¨ osch 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- q k nons to other states as only fraction δ blocked by other singlets. k−q Strong asymmetry between carriers and phonons. Phonon induced carrier-carrier interaction: One singlet emits a phonon, which is absorbed by k k+q another singlet. Both scattering processes allowed q in limit δ → 0 . No suppression of carrier-carrier k’−q k’ interaction. Strong interaction without soft phonons possible. O. R¨ osch and O. Gunnarsson, PRL 93 , 237001 (2004). 6
Relation between Hubbard and t - J models Consider large U limit, half-filling and symmetric parameters. A H ( k , ω ) = A t - J ( k , ω + U 2 ) + A t - J ( − k , U 2 − ω ) . Define self-energies dω A H ( k ,ω ) 1 � G H ( k , z ) = ≡ z − Σ H ( k ,z ) . z − ω dω A t − J ( k ,ω ) 0 . 5 � G t − J ( k , z ) = ≡ z − Σ t − J ( k ,z ) . z − ω This requires 2 ) + U 2 Σ H ( k , z ) ≈ 2Σ t - J ( k , z + U 4 z . Then Im Σ H ( k , z ) = 2ImΣ t - J ( k , z + U 2 ) , which gives the sum rule � − U/ 4 ImΣ ep 1 H ( k , ω − i 0 + ) dω = 2¯ g 2 . π −∞ 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 2g 2 2 2g 2 −3g Hubbard g 2 g 2 /2 No vert /2 Sum rule for Im Σ over photoem. and g 2 . g 2 inverse photoem. parts, each 2 ¯ t−J g 2 . Total sum rule just ¯ 2 g 2 g Nonint /2 /2 ω −U/4 0 U/4 Im Σ( k , ω ) has large positive contribution U 2 / 4 at ω = 0 . g 2 . The electron-phonon interaction slightly reduces this by − 3¯ q Calculate lowest order diagram in electron-phonon in- teraction, neglecting vertex corrections ( Γ = 1 ). Γ Γ Sum rule over photoemission spect. violated by factor 4. k+q Vertex corrections important for electron-phonon part of electron self-energy. MPI-FKF Stuttgart 8
Vertex corrections phonon self-energy. Large U k+q In limit of weak electron-phonon coupling, phonon self- Γ energy given by density-density correlation function. Ne- glect vertex corrections ( Γ = 1 ). k Assume that doping δ results in a weight ∼ δ on the in- ε verse photoemission side close to Fermi energy. Assume that this weight and photoemission spectrum within energy range 2∆ . Exact sum rule fulfilled N( ε) � 2∆ 1 � − 2∆ | Im χ H , No vert . ( q , ω ) | dω = 2 δ (1 − δ ) πN 2 q δ Sufficient to use dressed electron Green’s functions. 2∆ 1−δ ( )/2 Neglect of vertex corrections violates sum rule for electron- phonon contribution to electron but not phonon self-energy. MPI-FKF Stuttgart 9
q -dependence � 2∆ 1 � − 2∆ | Im χ H ( q , ω ) | dω = 2 δ (1 − δ ) . πN 2 q � = 0 � 2∆ 1 − 2∆ | Im χ H , No vert . ( q , ω ) | dω = 2 � k [ w P ( k ) w IP ( k + q ) + w P ( k + q ) w IP ( k )] π � 2∆ 1 � − 2∆ | Im χ H , No vert . ( q , ω ) | dω = 2 δ (1 − δ ) . πN 2 q √ √ Calculate w P ( k ) and w IP ( k ) for 18 × 18 t - J model with two holes. � 2∆ 1 − 2∆ | Im χ H ( q , ω ) | dω πN q / π (0 , 0) (1 , 1) (2 , 0) (2 , 2) (3 , 1) (3 , 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 n i ↑ n i ↓ + ω ph b † b + g ( c † + c − + c † − c + )( b + b † ) . Huang, Hanke, Arrigoni, Scalapino (PRB 68 , 220507 (2003)): G 2 ( k,q ) Γ( k, q ) = G ( k + q ) G ( k ) . q � β 0 dτe i ( ω n + ω m ) � β ′ ′ e − iω m τ G 2 ( k, q ) = 0 dτ Γ k k+q pqσ ′ � T τ c † ′ ) c pσ ′ c k + qσ ( τ ) c † × � p + qσ ′ ( τ pσ (0) � . ′ ( ω + ω ′ )+ ωt +( U/ 2) 2 ′ , + , ω, − ) = ω ′ Γ( ω ( ω =electron; ω =phonon). . ( ω ′ + t )( ω + ω ′ − t ) ′ ≈ ± U ′ , + , ω, − ) = 2 . ω 2 ⇒ Γ( ω 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 CaCuO 2 Cl 2 . 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 t pd and ε d − ε p as in the treatment of the half-breathing mode. MPI-FKF Stuttgart 14
Electron-phonon coupling strength. La 2 CuO 4 160 q νi M q νi (1 − n i )( b q ν + b † 1 H ep = � − q ν ) √ N 120 | M q ν | 2 Dimensionless coupling λ = 2 1 � . q ν 8 t ω q ν γ ( ω ) 80 We find λ = 1 . 2 . 40 Due to (half-)breathing modes (80 meV), O z mo- des (60-70 meV) and La (Cu) modes (20 meV). 0 0 20 40 60 80 100 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
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