Workshop on Quantum Fielt Theory aspects of Condensed Matter Physics, LNF, Frascati, 7 September 2011 Infrared phonon activity and quantum Fano interference in multilayer graphenes Emmanuele Cappelluti Instituto de Ciencia de Materiales de Madrid (ICMM) , CSIC, Madrid, Spain Institute of Complex Systems (ISC), CNR, Rome, Italy, Lara Benfatto ISC, CNR, Rome, Italy Alexey B. Kuzmenko Dept. Physics Uni. Geneve, Switzerland and: Z.Q. Li, C.H. Lui, T. Heinz (Columbia, NY, USA) Outline motivations (limits of Raman spectroscopy) experimental measurements (intensity and Fano asymmetry modulation) theoretical approach unified theory for phonon intensity (charged phonon) and Fano asymmetry tunable phonon switching effect comparison with experiments conclusions Probing interactions (and characterization) in graphenes electronic states ARPES - dispersion anomalies - renormalization - linewidth A Bostwick et al., NJP 9, 385 (2007) DC Elias et al., Nat Phys 7, 701 (2011) Probing interactions (and characterization) in graphenes electronic states optical conductivity ZQ Li et al., Nat. Phys. 4, 532 (2008) - doping dependence - electronic interband features - possible to extract bandgap KF Mak et al, PRL 102, 256405 (2009) Probing interactions (and characterization) in graphenes lattice dynamics optical transitions single layer in-plane in-plane E2g (G) out-of-plane bilayer Eg Raman Eu IR Raman spectroscopy phonon intensity C Casiraghi, PRB 80, 233407 (2009) I Calizo et al, JAP 106, 043509 (2009) difficult access to absolute phonon intensity relative intensity between different peaks instead used Raman spectroscopy focus on: ph. frequency ph. linewidth J Yan et al, PRL 98, 166802 (2007) Raman spectroscopy - not only characterization, also fundamental physics doping dependence of phonon frequency and linewidth: evidence of nonadiabatic breakdown of Born-Oppenheimer S Pisana et al, Nat Mat 6, 198 (2007) Raman spectroscopy investigation tools: peak frequency peak linewidth relative (non absolute) peak intensity but no modulation of intensity no asymmetric peak lineshape J Yan et al, PRL 98, 166802 (2007) IR phonon spectroscopy suitable tool??? IR phonon spectroscopy IR phonon peak best resolved in ionic systems -Z +Z Z: dipole effective charge (related to oscillator strength S, f) ex. Na+ Cl- Z = 1 integrated area W' VG Baonza, SSC 130, 383 (2004) d '() ' BG W' Z 2 IR phonon spectroscopy bilayer graphene one allowed in-plane IR mode: antisymmetric (A) Eu homo-atomic compound first approximation: all the C atoms equal charge equally distributed q no net dipole q q no IR activity q IR phonon spectroscopy taking into account the slight difference between atomic sites small charge disproportion finite dipole Z ≈ (q1-q2) q1 q2 q2 however q 1, q 2 < n limited by the total amount of doped charge n Z ≈ 10-3 q1 (static dipole) no hope, thus..... but..... Exp. results: Geneve group tunable phonon peak intensity Zmax ~ 1.2!! huge! as large as 1 electron over N=4 (sp3) !! AB Kuzmenko et al, PRL 103, 116804 (2009) Exp. results: Geneve group tunable phonon peak intensity neutrality point (NP) n=0 also problem: negative peak area… Z not defined…? AB Kuzmenko et al, PRL 103, 116804 (2009) Negative peak: Fano effect and quantum interference arising from quantum interference (coupling) between a discrete state (phonon) with continuum spectrum (electronic) q2 -1 - 2qz A = ABG + A' 2 2 q z +1 non coupled phonon z= |q| ≈ weakly coupled q = asymmetry Fano parameter strongly coupled symmetric lineshape - 0 asymmetric lineshape |q| ≈ 1 negative peak |q| ≈ 0 Exp. results: Geneve group four independent parameter fit p2 q2 -1 - 2qz '( ) 'BG ( ) = 4 q2 z2 +1 - 0 z= p : related to intensity q : Fano asymmetry 0 : phonon frequency : phonon linewidth p2 1 W'= 1 2 8 q W= p2 8 “bare” intensity (in the absence of Fano) AB Kuzmenko et al, PRL 103, 116804 (2009) Exp. results: Geneve group phonon softening with doping: ok with LDA and TB theory Eg (S) mode Eu (A) mode T Ando, JPSJ 76, 104711 (2007) AB Kuzmenko et al, PRL 103, 116804 (2009) Exp. results: Geneve group phonon linewidth: strong increase at NP: why?? Eu (A) mode? T Ando, JPSJ 76, 104711 (2007) AB Kuzmenko et al, PRL 103, 116804 (2009) Exp. results: Geneve group linear dependence of bare intensity with doping: where from? why so huge Z? NB: tight-binding calculations AB Kuzmenko et al, PRL 103, 116804 (2009) Exp. results: Geneve group linear dependence of bare intensity with doping: where from? why so huge Z? Fano asymmetry: where from? related to el. optical background? points out finite intensity at n=0…! AB Kuzmenko et al, PRL 103, 116804 (2009) Charge-phonon effect doped insulators: organic and C60 systems KxC60 huge intensity increase of selected IR modes upon electron doping x doping SC Erwin, in Backminsterfullerenes (1993) K-J Fu et al, PRB 46, 1937 (1992) Charge-phonon effect el ( ) i ( ) : el. polarizability (interband transitions) ( ) electronical background of optical conductivity direct light-phonon coupling but these no polar materials:.... Charge-phonon effect el ( ) i ( ) : el. polarizability (interband transitions) ( ) electronical background of optical conductivity direct light-phonon coupling but these no polar materials:.... no intrinsic dipole further channels to be considered Rice (Michael) theory electronic polarizability provides finite IR intensity to phonon modes allowed but otherwise not active el ( ) i ( ) : el. polarizability (interband transitions) ( ) irreducible diagrams electronical background of optical conductivity no phonon resonance phonon mediated contribution giving rise to resonance at phonon energy Rice (Michael) theory fundamental ingredients: tot ( ) i ( ) x ( ) ( )Dph ( ) phonon resonance Rice (Michael) theory fundamental ingredients: current/ electron-phonon response function tot ( ) i ( ) x ( ) ( )Dph ( ) intensity ruled by the current/electron-phonon response function Rice theory in bilayer graphene : real function (α doping) tuning the phonon intensity Rice theory in bilayer graphene : real function (α doping) tuning the phonon intensity Rice theory: in its original application: semiconductors effective theory: interesting peculiarities of bilayer graphene: zero gap semiconductor: low energy interband transitions Fano asymmetry : complex quantity tunable charged-phonon effects controlled by external voltage biases (doping and gap) Microscopic Rice theory in bilayer graphene three different response functions: jj (el.background) AA (ph. self-energy) jA (charged-phonon effect) we can compute microscopically each of them Fano-Rice theory in bilayer graphene interband transitions at low energy: jA = RejA +iImjA DAA ( ) 'ep ( ) jA complex quantity!!! (in gapped systems: ImjA = 0) 1 A iA A 2 'jA (A ) AA 2 q 1 2zqA q2A (1 z2 ) 2 A 'jA (A ) qA "jA (A ) Fano formula! Fano and charged-phonon effects same origin! it permits a microscopical identification Peak parameters in Fano systems Fano fit 'ep ( ) 2WA q 1 2zqA A q2A (1 z2 ) 2 A A -integrated area W'A 'jA (A ) 2 WA A 'jA (A ) "jA (A ) 2 2 A |qA| ≈ 0 (RejA=0) negative peak but WA=0 not good |qA| ≈ 1 (RejA = ImjA) asymmetric peak but W’A=0 not good pA 2 'jA (A ) "jA (A ) A 2 phonon strength Phonon intensity in bilayer graphene Step by step analysis: gating induces doping but not Ez in this case low-energy transitions between 2 and 3 system like a gapped semiconductor Im = 0 4 3 2 1 no Fano effect doping depedence of -integrated area W’ perfectly reproduced what about WA? negative area? E Cappelluti et al, PRB 82, 041402 (2010) Exp. results: Berkeley group double-gated device possible tuning doping and in independent way n=0 n = 0 and 0: negative peak like us Fano effect as a function of they attribute origin of negative peak at n = 0 to Eg (S) (Raman-active) mode (S allowed by symmetry in IR when 0) T-Ta Tang et al, Nat Nanotechn 5, 32 (2010) Different phonon channels in optical conductivity gating induces z-axis asymmetry Ez >0 Eg (S) mode also IR active! two main IR channels present probes DAA ph. propagator probes DSS ph. propagator relative “intensity” ruled by pA and pS total spectra dependent on the relative dominance of one channel vs. the other one Optical channels and phonon switching in optical conductivity - phase diagram Berkeley Eu-A and Eg-S modes dominant in different regions of phase diagram: possible switching of intensity from one mode to other one Geneve E Cappelluti et al, PRB 82, 041402 (2010) Phonon switching in optical conductivity Geneve group Eu (A) Eg (S) Eu (A) E Cappelluti et al, PRB 82, 041402 (2010) experimental integrated area and Fano asymmetry interpolates and switches from A to S mode AB Kuzmenko et al, PRL 103, 116804 (2009) Trilayer graphenes and stacking order ABA and ABC deeply different stacking revealed phonon intensity and phonon frequency strongly doping dependent in ABC but not in ABA good agreement with theory CH Lui et al, submitted to PRL (2011) Trilayer graphenes and stacking order fundamental ingredient: electronic band structure reminder: phonon activity is triggered by electronic particle-hole excitations upon doping, el. transitions at ω = √2 γ1 ≈ 0.55 eV in ABA, at ω ≤ γ1 ≈ 0.39 eV in ABC ABC closer to ω0 ≈ 0.2 eV CH Lui et al, submitted to PRL (2011) phonon activity amplified Raman spectroscopy in bilayer graphene remarkable features: |q| ≈ no Fano asymmetry !!! (in IR S mode had q ≈ 0) intensity does not depend on doping !!! J Yan et al, PRL 98, 166802 (2007) C Casiraghi, PRB 80, 233407 (2009) unlike IR probes! why? Fano-Rice theory for Raman spectroscopy effective mass approximation dHˆ k ˆ xy dkx dky Raman vertex electronic Raman background ( ) T ( ) Rice theory Raman active S mode tot irr irr S irr S () () ()DSS () () Fano-Rice theory for Raman spectroscopy IR Raman EC RejA ~ const. ReS ~ EC ImjA ~ const. ImS ~ const. ReS scaling with UV dispersion cut-off Ec ReS >> ImS W’S ≈ WS Ec2 qS ReS (A ) ImS (A ) weakly dependent on band-structure details (doping, ) no Fano profile Conclusions source of microscopic IR phonon intensity unified theory of IR intensity and Fano profile more information encoded in phonon intensity and Fano factor phonon mode switching predicted (and observed) differences between IR and Raman spectroscopy accounted for alternative and powerful tool to characterize ML graphenes Additional slides Raman spectroscopy in bilayer graphene focus on Eg symmetric mode Raman active present also in single-layer graphene J Yan et al, PRL 101, 136804 (2008) T Ando, JPSJ 76, 104711 (2007) frequency and linewidth OK with theoretical calculations Fano-Rice theory for Raman spectroscopy ex.: isotropic Raman scattering two main quantities: S, A EC scaling with UV dispersion cut-off Ec ReS ~ EC, ImS ~ const. ReA ~ const., ImA ~ const. irr ep ( ) irr S ( )DSS ( ) S ( ) pS » pA qS dominant DSS channel ReS (A ) ImS (A ) W’S ≈ WS Ec2 irr irr A ( )DAA ( ) A ( ) irr + irr ( )D ( ) S SA A ( ) h.c. no Fano profile weakly dependent on band-structure details (doping, ) Fano-Rice theory for Raman spectroscopy effective mass approximation dHˆ k ˆ xy dkx dky Raman vertex electronic Raman background ( ) T ( ) Rice theory =0 only S mode coupled tot irr irr S irr S () () ()DSS () () Fano-Rice theory for Raman spectroscopy effective mass approximation dHˆ k ˆ xy dkx dky Raman vertex electronic Raman background ( ) T ( ) Rice theory ep irr S irr S irr A irr A ( ) ( )DSS ( ) ( ) ( )DAA ( ) ( ) irr + irr ( )D ( ) S SA A ( ) h.c. 0 phonon switching possible (in principle) Fano-Rice theory for Raman spectroscopy ex.: isotropic Raman scattering two main quantities: S, A EC scaling with UV dispersion cut-off Ec ReS ~ EC, ImS ~ const. ReA ~ const., ImA ~ const. irr ep ( ) irr S ( )DSS ( ) S ( ) pS » pA qS dominant DSS channel ReS (A ) ImS (A ) W’S ≈ WS Ec2 irr irr A ( )DAA ( ) A ( ) irr + irr ( )D ( ) S SA A ( ) h.c. no Fano profile weakly dependent on band-structure details (doping, ) Probing electronic spectrum: optical conductivity bilayer (BL) AB Kuzmenko et al, PRB 80, 165406 (2009) KF Mak et al, PRL 102, 256405 (2009) possible to extract gap and doping n vs. gate voltage Vg Effective charge in IR spectroscopy integrated area effective charge W' d '() ' W’ BG 2VW ' M C Z CNe 2 V: volume unit cell, MC: carbon mass, C constant, N: # atoms/cell Z: effective charge put on ion positions to produce exp. dipole upon lattice distortion as an the same ionic crystal -Z +Z (related to oscillator strength S, f) ex. Na+ Cl- Z = 1 VG Baonza, SSC 130, 383 (2004) Phonon intensities in Fano systems?? q2 -1 - 2qz = BG + A' 2 2 q z +1 two main popular choices: A' p2 /4 for q 0, A’ 0 -1/(z2+1) negative peak, but p = 0 no good parameter for q W and W’ coincide phonon intensity well defined W = d () - BG integrated spectral area however W' p2 /8 p: phonon oscillator strength however for q 1, W 0 -2z/(z2+1) negative and positive areas cancel out no good parameter Peak parameters in Fano systems Fano fit 'ep ( ) 2WA q 1 2zqA A q2A (1 z2 ) 2 A A -integrated area W'A 'jA (A ) 2 WA A 'jA (A ) "jA (A ) 2 2 A |qA| ≈ 0 (RejA=0) negative peak but WA=0 not good |qA| ≈ 1 (RejA = ImjA) asymmetric peak but W’A=0 not good pA 2 'jA (A ) "jA (A ) A 2 phonon strength Rice theory in bilayer graphene multiband structure /2 v(kx ik y ) v(k x iky ) /2 Hˆ k /2 v(kx ik y ) v(k x iky ) /2 ˆ d H k ˆj x e dkx jj ( ) T j( ) j jj ( ) ( ) i electronic background EJ Nicol & JP Carbotte, PRB 77, 155409 (2008) Microscopic Rice theory in bilayer graphene el-ph interaction =0 Eu (antisymmetric) mode 0 i ˆ VA ig 0 0 Hep kVˆAk A k el-ph contribution to () irr irr irr tot ( ) ( ) ( )D ( ) jj jj jA AA Aj () i 0 0 0 0 0 0 0 i 0 i 0 Doping dependence of phonon intensity in bilayer graphene 4 = 0 : analytical calculations 3 jA () () () () () 12 jA 13 jA 24 jA 34 jA nm nm mn jA () jA () jA () () ge vNsNv k nm jA 1 4 ( vk) 2 2 f (kn ) f (km ) kn km i damping: disorder/impurities/inhomogeneities = 0 particle-hole symmetry jA=0 pA = 0 : no phonon intensity 2 Raman spectroscopy in bilayer graphene A A S A S S >0 S A =0 S LM Malard et al, PRL 101, 257401 (2008) double peaks A-S evolve upon S mode intensity constant why no phonon switching? why no Fano asymmetry in both of them? Fano-Rice theory for Raman spectroscopy no Fano effect only direct coupling to S-channel in Raman response not two channels making phonon switching possible irr tot () irr() irr ( )D ( ) S SS S () double-peak? encoded in DSS, Optical channels and phonon mixing in optical conductivity >0 mode mixing in phonon propagators AA() but also: current directly coupled to Eg S mode!!! jS() 0 irr irr irr irr ep ( ) ( )D ( ) ( ) ( )D ( ) jj jA AA Aj jS SS Sj ( ) peak at A irr + irr ( )D ( ) jA AS Sj ( ) h.c. peak at S Phonon hybridization self-energy A and S lattice vibrations eigenmodes only for = 0 0 mode mixing through coupling to electronic excitations DAA DSA DAS D0AA DSS 1 1 AA 0 DSS SA AS SS DAA double peaked: contains a second (weaker) pole at S DSS double peaked: contains a second (weaker) pole at A LM Malard et al (2008); T Ando M Koshino (2009); P Gava et al (2009) double peak only at very large origin of double peak deeply different from phonon switching it could never produce a dominant S peak in IR neither a dominat A peak in Raman ~ Double peaks in Raman spectroscopy Raman spectroscopy only probed the direct S-channel but for > 0: mode mixing in phonon propagators double paks in DSS Z Z DSS ( ) conditions to resolve the double-peak structure: Z- ≈ Z+ (triggered by ) |+ - +| ph) Double peaks in Raman spectroscopy Raman spectroscopy in bilayer graphene one problem: difficult to obtain absolute intensities (estimated indirectly by looking at some reference phonon peak) at =0 only S Eg mode Raman active J Yan et al, PRL 101, 136804 (2008) T Ando, JPSJ 76, 104711 (2007) frequency and linewidth OK with theoretical calculations |q| ≈ : while no Fano asymmetry? (in IR S mode had q ≈ 0)