Landau level spectroscopy of graphene (Raman scattering and far-infrared absorption) Electron-phonon and electron-electron interactions Marek Potemski Laboratoire National des Champs Magnétiques Intenses Grenoble High Magnetic Field Laboratory CNRS/UJF/UPS/INSA MOMB
"The ZOO of magneto-phonon resonances in graphene" D.M. Basko, P. Leszczynski, C. Faugeras… et al., to be published PRL 114 , 126804, (2015)
Why ? Graphene: a truly two-dimensional crystal of sp 2 –bonded carbon Graphene-Based Revolutions in ICT And Beyond This talk: fundamental properties studied with magnetic fields (spectroscopy)
Dispersion relations and corresponding Landau level ladders Electronic states, generic (quasi) 2D structure of sp 2 carbon (Bernal stacking) ~ graphene + (effective) bilayers = = E E ( B ) E E ( k ) n i ! > = B 0 B 0
Dispersion relations and corresponding Landau level ladders Electronic states, generic (quasi) 2D structure of sp 2 carbon (Bernal stacking) ~ graphene + (effective) bilayers = = E E ( B ) E E ( k ) n i ?! > = B 0 B 0
Landau level spectroscopy L → L Probing inter Landau level excitations : i j E L j B L j
Absorption/transmission Selection rules ν h = 4 + + σ n ∆ n = ± σ − 1 , E + = 3 n B + ∆ = + σ + = 2 n 1 : n + = 1 n ∆ = − σ − n 1 : = ν T T ( h , B ) and = 0 n ∆ n = 2 , 4 , 5 , 7 , 8 resonant when B ν = h E if trigonal warping exc = 1 − n = 2 − n − = 3 n V.P. Gusynin & S.G. Sharapov, PRB, 2006 M. Koshino & T. Ando, PRB, 2008 − = 4 n M. Mucha-Kruczynski et al., J. Phys., 2009 M.L. Sadowski et al., SSC, 2007
Raman scattering Selection rules ν ν h ' h + σ + σ σ − σ − / / = 4 + n ∆ n = 0 E + = 3 n B strong + = 2 n σ + σ + σ − σ − / , / = ν − ν E exc h ' h = 1 + n optical fibers ∆ n = ± 2 = n 0 weaker σ + σ − σ − σ + B / , / excitation collection ∆ n = ± 1 = 1 − n if trigonal warping − = 2 or coupled to phonon B 30 T n T 1 K = 3 − n x,y,z-stage − = 4 n miniaturized optical bench Faugeras, Kossacki, Breslavetz, … O. Kashuba & V.I. Falko PRB, 2009; M. Mucha-Kruczynski et al., PRB, 2010
What can be learned from magneto-optics ? √ Band structure Scattering: efficiency ( mechanism ) ?
Scattering ? Classical condition for observation of cyclotron resonance (Landau quantization) τ > T scattering cyclotron e τ > ω 1 / × S C B µ > 1 B / rough estimate of carrier mobility min More general : Γ ← τ 1 / Spectral broadening scat ← Γ = Γ ( B , E ) Scattering mechanisms
What can be learned ? √ Band structure √ Scattering: efficiency ( and mechanism ) Interactions (?) : electron-phonon electron-electron
Interactions ? tuning the excitations in resonance E E 0 L → L n m B
Interactions ? resonant electron-phonon coupling ? E 3 E ph = E 1,3 2 1 E δ phonon n = 0 - 1 L → - 2 L n m - 3 B + more than this !! ← δ strength of interaction other magneto-phonon" resonance
Electron-electron interactions and inter Landau level transitions Parabolic dispersions E equidistant LLs n = 2 π 2 1 e = E exch ε 2 2 l B n = 1 ω E opt = E single particle = E single particle C n = 0 2 k ~ l e-h /l B Restoring single electron spectrum of excitations at k ~ 0 (Kohn theorem) k opt ~ 1/ λ << k coll ~1/l B Optics is useless to study the many-body effects !? e.g., C. Kalin, B.I. Halperin, PRB , 1984, Bychkov, Eliashberg, Iordanskii, (JETP Letters, 1981)
Electron-electron interactions and inter Landau level transitions E Linear dispersions non-equidistant spacing ≠ E E n = 2 C exch n = 1 ∆ E opt = E single particle + E single particle corr ∆ = δ ÷ ∞ ? corr n = 0 2 k ~ l e-h /l B n = -1 A. Iyengar, et al., PRB, 2008 Yu.A. Bychkov, G. Martinez, PRB, 2008 J. Sari, C. Toke, PRB, 2013 R. Roldan et al., PRB, 2010 n = -2 YU.E. Lozovik, A.A. Sokolik, Nanoscale Research Lett., 2012 Expectations: ∆ γ ~ B ? Rather large deviations from corr nm effective single electron model ?
Graphene: Electron-electron interactions at B=0
OUTLINE Band structure mono to pentalayer graphene Scattering efficiency graphene on graphite: the best ever seen graphene Electron-phonon interaction the ZOO of magneto-phonon resonances Electron-electron interaction Conclusions
What can be learned from magneto-optics ? Band structure !
What can be learned from magneto-optics ? Band structure !
What can be learned from magneto-optics ? Band structure !
What can be learned from magneto-optics ? Band structure !
What can be learned from magneto-optics ? Scattering: efficiency ( mechanism ) !
Graphene on graphite: best ever seen graphene !! G. Li et al., PPRL, 2008
Cyclotron resonance absorption : high temperature but well resolved LLs LL spacing > kT LL broadening < LL spacing E . non-equidistant spacing . . + L E 3 F + L 2 + L 1 multimode cyclotron resonance absorption L 0 − L 1 − L 2
Graphene on graphite (Very) low field cyclotron resonance absorption 1 mT perfect Dirac states : m = ⋅ 6 v F 1 . 0 10 s Γ ≈ µ LL broadening : 35 eV ( 0 . 4 K ) P. Neugebauer et. al., PRL, 2009
How perfect can graphene be ≅ ≅ ⋅ − ∗ = ≅ ⋅ − 9 2 2 3 E 6 . 5 meV , n 3 10 cm , m E / v 1 . 3 10 m F F F e 2 1 cm Landau level quantization µ > = 7 10 ⋅ down to B 0 = 1 mT 1 mT V s 2 e cm γ = µ τ ≈ µ = τ ≈ ⋅ ≈ µ 7 35 eV ( 0 . 4 K ) 20 ps , 3 10 , l 20 m ∗ ⋅ F m V s Also at 50 K ! = µ γ = 1 B T E 1 ≈ µ ≈ = > γ = B Earth 50 T E 0 . 25 meV 3 K 0 . 4 K 1 Pronounced Landau quantization in the magnetic field of the Earth P. Neugebauer et. al., PRL, 2009
Graphene on graphite: magneto Raman scattering response phonons + search for a characteristic electronic response e.g., L -1 → L 1 inter Landau level excitation E B L -1 → L 1 2D band 150 150 a) b) 125 125 G band B=10T 2D band 100 100 1500 Distance ( µ m) Distance ( µ m) λ exc. =514.53nm 75 75 Temp.=4K L -1,1 50 50 Intensity (counts) L -1,2 /L -2,1 25 25 1000 0 0 0 25 50 75 100 125 150 0 25 50 75 100 125 150 150 150 d) c) 125 125 500 100 100 Distance ( µ m) Distance ( µ m) 75 75 50 50 0 25 25 1400 1600 1800 2000 2200 2400 2600 2800 0 0 -1 ) Raman Shift (cm 0 25 50 75 100 125 150 0 25 50 75 100 125 150 Distance ( µ m) Distance ( µ m)
Graphene on graphite: magneto-Raman scattering response: an overview 1200 1000 B=5.98T 800 Intensity (counts) B=4.98T 600 B=4.38T 400 Intensity (counts) B=3.88T 200 B=0T 0 1000 1200 1400 1600 1800 2000 2200 2400 2600 -1 ) Raman shift (cm B= 6T G 2D’’ 2D B=0T 2D’ 1500 2000 2500 3000 3500 -1 ) Raman shift (cm
Graphene on graphite: magneto-Raman scattering response: an overview E 2g phonon + electonic excitations = 4 + n E + = 3 n = 2 + n = 1 + n = n 0 B − = 1 n = 2 − n = 3 − n = 4 − n C. Faugeras et al., PRL, 2011; M. Kühne et al., PRB, 2012, P. Leszczynski et. al , to be published
Graphene on graphite: magneto-Raman scattering response: an overview focus on E 2g phonon C. Faugeras et al., PRL, 2011; M. Kühne et al., PRB, 2012, D. Basko et. al , to be published
Interactions ? resonant electron-phonon coupling ! E E δ phonon L → L n m B ← δ strength of interaction
In magnetic fields Resonant coupling of E 2g phonon ("optical") with Δ n=±1 inter Landau level excitations Theoretical predictions : T. Ando, JPSJ, 2007 M.O. Goerbig, et al., PRL , 2007 δ − σ + σ + σ − σ / / δ λ ⋅ ⋅ − λ ⋅ ⋅ − E B f f B f f ~ 2 ( ) ( 1 ) ~ ( 1 ) 1 res f i res f i
Magneto-phonon resonance: graphene on graphite Raman shift (cm -1 ) 1600 1500 10 5 Magnetic field (T) Graphene on graphite: an electronic system of unprecedented quality ! J. Yan et al., PRL , 2010 C. Faugeras, et al., PRL, 2011; M. Kühne el al., PRB 2012
Experiment: magneto-phonon resonance in epitaxial graphene 1630 E T 4 T 1 T 3 T 2 1620 Raman shift (cm-1) 1610 E 1 T 3 T 2 E F 1600 B 1590 1580 1570 1 2 3 4 5 B 1/2 (T 1/2 ) Neutral graphene λ = ⋅ − 3 4 . 5 10 C. Faugeras, et al., PRL, 2009
Magneto-phonon resonance in doped graphene δ λ ⋅ ⋅ − ~ B ( 1 f ) f res f i Graphene flake on Si/SiO 2 P. Kossacki et al., Phys. Rev. B, 2012
Magneto-phonon resonances: graphene on h-BN ~ neutral and better electronic quality Experiment in qualitative agreement with simulations 30 30 25 25 20 20 B (T) 15 B (T) 15 10 10 5 5 0 0 1500 1600 1700 1500 1600 1700 Raman shift (cm-1) Raman shift (cm-1) P. Leszczynski, A. Nicolet, C. Faugeras et al., to be published
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