Electron-Phonon Interaction and Charge Instabilities in Strongly Correlated Electron Systems

Transcription

1 Electron-Phonon Interaction and Charge Instabilities in Strongly Correlated Electron Systems p. 1/21 Electron-Phonon Interaction and Charge Instabilities in Strongly Correlated Electron Systems Rome, October 21, 2008 Ph-D Candidate - Andrea Di Ciolo andrea.diciolo@roma1.infn.it Physics Department, University of Rome Sapienza Piazzale A. Moro, Rome, Italy

2 Electron-Phonon Interaction and Charge Instabilities in Strongly Correlated Electron Systems p. 2/21 PH-D ADVISORS Dr. J. G. Lorenzana University of Rome Sapienza, SMC-INFM and ISC-CNR, Italy Prof. M. Grilli University of Rome Sapienza and SMC-INFM, Italy COLLABORATOR Prof. G. Seibold Institut für Physik, BTU Cottbus, Germany

3 Electron-Phonon Interaction and Charge Instabilities in Strongly Correlated Electron Systems p. 3/21 Contents Charge Density Waves + Anomalous Phonon Softening in the Cuprates Hubbard-Holstein Model + Gutzwiller-RPA Method Homogeneous Metal = Charge Inhomogeneities Stripes = Anomalous Phonon Softening Conclusions

4 Electron-Phonon Interaction and Charge Instabilities in Strongly Correlated Electron Systems p. 4/21 Charge Density Wave (CDW) Materials K 0.30 MoO 3 La 2 x Sr x CuO 4 Y Ba 2 Cu 3 O 7 x CLASSICAL CDW s 1D-blue bronzes (A 0.30 MoO 3 with A = K, Rb); 2D-dichalcogenides (MC 2 with M=Ta,Ti,Nb,Mo and C=S,Se)... STRONGLY CORRELATED CDW s cuprates, manganites, nickelates, cobaltites...

5 Electron-Phonon Interaction and Charge Instabilities in Strongly Correlated Electron Systems p. 5/21 Charge Density Wave (CDW) Materials Rb 0.3 MoO 3 [Brun05] Ca 2 x Na x CuO 2 Cl 2 [Hanaguri04] Dy-BSCCO[Kohsaka07] CDW s in STM (Scansion Tunnelling Microscope) experiments

6 Electron-Phonon Interaction and Charge Instabilities in Strongly Correlated Electron Systems p. 6/21 Anomalous Phonon Softening in the Cuprates Stripe Phase[Tranquada96,Bosch01] Bond-Stretching Phonon Branch in LBCO[Reznik06] OUR IDEA: STRIPES Nearly 1D-metallic structures = [Kohn Anomaly] ANOMALOUS SOFTENING OF THE BOND-STRETCHING PHONON BRANCH

7 Electron-Phonon Interaction and Charge Instabilities in Strongly Correlated Electron Systems p. 7/21 Hubbard-Holstein Model + Gutzwiller-RPA Method ijσ t ijc iσ c jσ i Uˆn i ˆn i i = ELECTRONIC KINETIC TERM = E-E INTERACTIONS i βx i(ˆn i n) = E-PH COUPLING ( 1 2M P i 2 + ) 1 2 Kx2 i = PHONONIC TERM ADIABATIC LIMIT (M ) κ eph q = λ = χ 0 0 β2 /K e-ph coupling κ q 1 λκ q /χ 0 0 exact relation χ 0 0 density of states of the non-interacting system GUTZWILLER APPROXIMATION (GA) = electronic GROUND STATE + low-energy excitations (quasiparticles) RANDOM PHASE APPROXIMATION (RPA) = FLUCTUATIONS dressed ELECTRONIC SUSCEPTIBILITY: κ q (without e-ph coupling) κ eph q (with e-ph coupling)

8 Electron-Phonon Interaction and Charge Instabilities in Strongly Correlated Electron Systems p. 8/21 Homogeneous Metal = Charge Inhomogeneities n=1, U=U c Metal-Insulator Transition κ eph q = κ q 1 λκ q /χ 0 0 U=0 = κ q=2kf maximum = strong phonon anomaly (KOHN ANOMALY) SMALL U = PEIERLS CDW LARGE U = PHASE SEPARATION (PS)

9 Electron-Phonon Interaction and Charge Instabilities in Strongly Correlated Electron Systems p. 9/21 Homogeneous Metal = Charge Inhomogeneities GA is accurate U WEAKENS THE EFFECTIVE E-P H COUPLING g q = z0 2 κ q κ 0 q z 0 GA renormalized hopping t tz 2 0

10 Electron-Phonon Interaction and Charge Instabilities in Strongly Correlated Electron Systems p. 10/21 Homogeneous Metal = Charge Inhomogeneities 2D - PHASE DIAGRAM OF THE PARAMAGNETIC METAL

11 Electron-Phonon Interaction and Charge Instabilities in Strongly Correlated Electron Systems p. 11/21 Kohn Anomaly in 1D and 2D ω 0 bare phonon frequency Results for n = 0.8 and λ = 0.5. U SUPPRESSES THE KOHN ANOMALY.

12 Electron-Phonon Interaction and Charge Instabilities in Strongly Correlated Electron Systems p. 12/21 Stripes = Anomalous Phonon Softening REALISTIC PARAMETERS= n = 0.875, U/t = 8, next-nearest hopping t /t = 0.2

13 Electron-Phonon Interaction and Charge Instabilities in Strongly Correlated Electron Systems p. 13/21 Stripes = Anomalous Phonon Softening OUR OPTICAL PHONON BRANCHES VS EXPERIMENTAL BRANCHES STRONG ASYMMETRY WELL REPRODUCED NOT TRIVIAL RESULT

14 Electron-Phonon Interaction and Charge Instabilities in Strongly Correlated Electron Systems p. 14/21 Conclusions e-ph coupling suppressed by e-e interaction Homogeneous Metal λ Charge Inhomogeneities Peierls CDW = U = PS Optical Phonons + Stripes U Anomalous Phonon Softening

15 Electron-Phonon Interaction and Charge Instabilities in Strongly Correlated Electron Systems p. 15/21 Anomalous Phonon Softening in the Cuprates Half-Breathing Phonon Mode Longitudinal Optical Branch[Pintschovius99]

16 Electron-Phonon Interaction and Charge Instabilities in Strongly Correlated Electron Systems p. 16/21 Hubbard-Holstein Model + Gutzwiller-RPA Method The susceptibility κ q has all the information on how e-e interaction renormalizes e-ph interaction: κ eph q = κ q 1 λκ q /χ 0 0 = κ q 1 λ q, λq = λ κ q χ 0 0 In the absence of other (first-order) instabilities, for λ = λ c = χ 0 0/κ qc the system undergoes a transition to a CDW state with a typical q c. Increasing U κ q reduced with respect to κ 0 q λ q reduced = U WEAKENS the e-ph coupling. The problem is then reduced to the study of the electronic susceptibility for λ = 0.

17 Electron-Phonon Interaction and Charge Instabilities in Strongly Correlated Electron Systems p. 17/21 Hubbard-Holstein Model + Gutzwiller-RPA Method The Hubbard model: H e = ijσ t ij(c iσ c jσ + c jσ c iσ) + i Uˆn i ˆn i GZW TRIAL STATE: ψ G = ˆP g Sd = Sd SLATER DETERMINANT ˆP g = i [1 (1 g)ˆn i ˆn i ] = i [1 (1 g) ˆD i ] For g < 1, ˆPg suppresses double occupancies D i. For g = 0, all the configurations with D i 0 are ruled out. E e [ρ, D] = ψ G H e ψ G ijσ t ijz iσ z jσ ρ jiσ + i UD i z iσ [ρ iiσ, ρ ii, σ, D i ] GZW-renormalized hopping factors ρ jiσ = Sd c iσ c jσ Sd uncorrelated single fermion DENSITY MATRIX

18 Electron-Phonon Interaction and Charge Instabilities in Strongly Correlated Electron Systems p. 18/21 Hubbard-Holstein Model + Gutzwiller-RPA Method Computations with the GA+RPA method [Seibold-Lorenzana0103], generalization of the HF+RPA technique [Ring80, Blaizot86] An external field induces small amplitude deviations δd and δρ around the GA saddle point = E e [ρ, D] = E e0 + Tr[h 0 δρ] δρ L 0 δρ + δds 0 δρ δdt K 0 δd with h = δe e /δρ. E = 0 eliminates the δd deviations = δd 2nd order energy deviation: δe e = 1 2 δρ Wδρ The interaction kernel W mediates local and intersite charge deviations.

19 Homogeneous Metal = Charge Inhomogeneities Electron-Phonon Interaction and Charge Instabilities in Strongly Correlated Electron Systems p. 19/21

20 Electron-Phonon Interaction and Charge Instabilities in Strongly Correlated Electron Systems p. 20/21 Homogeneous Metal = Charge Inhomogeneities K q 3 2 η q =1 η q =0.5 η q =0 η q =-0.5 η q =-1 K q n=0.99 K q η q =1 η q =0.5 η q =0 η q =-0.5 η q =-1 K q n= U/U c U/U c U/U c U/U c D = η q = 1 d d ν=1 cosq ν

21 Electron-Phonon Interaction and Charge Instabilities in Strongly Correlated Electron Systems p. 21/21 Stripes = Anomalous Phonon Softening Π Π Π 2 0 Π 2 Π Π Π 2 Π 2 qy 0 0 Π 2 Π 2 Π Π Π 2 0 Π 2 q x Π Π 4 2 UNIT CELL = FIRST REDUCED BRILLOUIN ZONE