Optics and magneto-optics of graphene. Vladimir Falko

Transcription

1 Optics and magneto-optics of gaphene Vadimi Fako

2 Intoduction: symmeties and notations. Optics and magneto-optics of gaphene: absoption. Magneto-phonon esonance and fiing facto dependent fine stuctue of the G-ine in the Raman spectum of phonons. Eectonic ecitations in the Raman spectum of gaphene. Magneto-optics of biaye gaphene. Abege VF - PR (7) Mucha-ucynski McCann VF - SSC 49 (9) Abege VF - PR (7) Abege Russe VF - APL 9 65 (7) Goebig Fuchs echedhi VF - PRL (7) ashuba VF unpubished (9) ashuba VF aiv:9655 (9)

3 Eectons in gaphene as obseved in ARPES ψ ϕa ϕ IARPES ~ ϕ A + ϕ ~ sin k + R A ϑ ε vp k G ± + p Mucha-ucynski Tsypyatyev Gishin McCann VF oswick Rotenbeg - PR (8) ARPES of heaviy doped gaphene synthesied on siicon cabide ostwick et a - Natue Physics 6 (7)

4 vp ε Eectons in gaphene obseved using ARPES p G k + ± ARPES of heaviy doped gaphene synthesied on siicon cabide ostwick et a - Natue Physics 6 (7) A ϕ ϕ ψ ) ( π π μ π π ςv H y y ip p ip p + + π π vaey tigona waping tems ' ' + ς ς A A p

5 ; ; ; ' ' A A 4-dimensiona epesentation of the symmety goup of the honeycomb attice Tansation A A T π π π π i i i i e e e e Geneating eements: A A S C T π C π Rotation π π π π i i i i e e e e ' A Mio efection S A { 6 T} C G v ψ

6 Σ Σ I t t invet sign symmetic Σ Σ y y y Σ ] [ s s s s s s i Σ Σ Σ ε y i i ] [ i ε [ ] Σ s asis of 44 matices: 6 geneatos of U 4 ' ' + ς ς A A SU Lie ageba with subattice matices vaey matices SU Lie ageba with

7 fou D-epesentations Σ Σ I T s C π A A one 4D-epesentation Σ (y) (y) π C s T fou D-epesentations E E Ieducibe mati epesentation of G{ TC 6v } XU U X U X ˆ ˆ ˆ ˆ ] [ ˆ + echedhi McCann VF Suuua Ando Atshue - Eu. Phys. J. Specia Topics (7)

8 Σ Σ I t t invet sign symmetic Σ y y y / / / Σ ] [ s s s s s s i Σ Σ Σ ε SU Lie ageba with subattice matices ζ y i i vaey matices ] [ i ε SU Lie ageba with: [ ] Σ s asis of 44 matices: 6 geneatos of U 4 intevaey scatteing asymmety between vaeys asymmety between subattices hopping between subattices ' ' + ς ς A A

9 Hˆ Σ / y Diac tem waping tem vσ p the same in both vaeys / y / y v 6 γ ζ Σ ( Σ p) Σ ( ) Σ p Σ asymmetic between vaeys captues p ' p symmety ξ t t Σ invet sign Σ symmetic 6γ McCann echedhi VF Suuua Ando Atshue - PRL (6)

10 Diac eecton inteaction with photons positive paity (vaeysymmetic): same in both vaeys nagative paity (opposite signs in diffeent vaeys): vaey-antisymmetic Γ-point optica phonons ( G-ine ) Hˆ Hˆ vσ ( p vσ p e c A ) ζ Σ ( u u) g u A u ζ Mω Feai et a PRL (6)

11 Intoduction: symmeties and notations. Optics and magneto-optics of gaphene: absoption. Magneto-phonon esonance and fiing facto dependent fine stuctue of the G-ine in the Raman spectum of phonons. Eectonic ecitations in the Raman spectum of gaphene. Magneto-optics of biaye gaphene. Abege VF - PR (7) Mucha-ucynski McCann VF - SSC 49 (9) Abege VF - PR (7) Abege Russe VF - APL 9 65 (7) Goebig Fuchs echedhi VF - PRL (7) ashuba VF unpubished (9) ashuba VF aiv:9655 (9)

12 Absoption coefficient πe hc.% Ω Ω Nai ake Gigoenko Novoseov ooth Staube Pees Geim - Science (8) γ μ Ω (in units of )

13 Gaphene fakes ae bette visibe in efection when the oide aye in SiO /Si wafe acts as a ceaing optica fim if λ ε s N + sin α s s nm δα o s μm δα o Abege Russe VF - App. Phys. Lett (7) ake Hi Casto Neto Novoseov Jiang Yang ooth Geim - App. Phys. Lett (7)

14 Infaed absoptions due to inte-ll tansitions A A Landau eve ( n ) M ω + + n n A ± n McCue Phys. Rev (956) ± ε ± nhv / λ hv + ω + ( n + n + ) ( n+ ) ( n+ ) n λ Sadowski et a - PRL (6) n M (n + ) + + A Jiang Heniksen Tung Wang Schwat Han im Stome - PRL (7)

15 Intoduction: symmeties and notations. Optics and magneto-optics of gaphene: absoption. Magneto-phonon esonance and fiing facto dependent fine stuctue of the G-ine in the Raman spectum of phonons. Eectonic ecitations in the Raman spectum of gaphene. Magneto-optics of biaye gaphene. Abege VF - PR (7) Mucha-ucynski McCann VF - SSC 49 (9) Abege VF - PR (7) Abege Russe VF - APL 9 65 (7) Goebig Fuchs echedhi VF - PRL (7) ashuba VF unpubished (9) ashuba VF aiv:9655 (9)

16 Hˆ e vσ ( p A c Ω ) Absoption of a photon geneates vaey-symmetic ecitations (positive paity) [ Φ + Φ ' eh eh ] ε ' Φeh ' Φ eh [ Φ ' eh eh Absoption of a phonon geneates vaey-antisymmetic ecitations (negative paity) Hˆ vσ p Φ ζ Σ ( ] u) g Mω

17 Couping of optica phonon with vaey anti-symmetic eectonic ecitations shifts the phonon enegy. ~ω ω Continuous spectum of ~ ω g ω + γ Π [ Φ Φ ' eh eh Casto Neto Guinea PR (7) Ando J. Phys. Soc. Jpn (7) ] ω ω ω ω g γ Π

18 Couping of optica phonon with vaey anti-symmetic eectonic ecitations ω ~ ω ~ g ω ω + Π( μ) γ Continuous spectum of [ Φ Φ ' eh eh Casto Neto Guinea PR (7) Ando J. Phys. Soc. Jpn (7) ] Yan Zhang im Pincuk - PRL (7) S. Pisana et a Natue Mat (7)

19 M + A ~ + i A ~ i y y M M + M Couping of modes in a magnetic fied Opticay active (couped to IR ight) ' [ Φ + Φ ] ' Φ Φ ' [ Φ Φ ' Opticay inactive and couped to the phonon u ~ + i y u ] ~ i y Φ Φ

20 Couping of optica phonon with vaey antisymmetic magneto-ecitons in undoped gaphene M M + ω ~ discete spectum of ω ' [ Φ Φ ] empty haf-fied Ando - J. Phys. Soc. Jpn (7) Goebig Fuchs echedhi VF - PRL (7) fu Φ Φ

21 Couping of optica phonon with magneto-ecitons in doped gaphene ω ~ [ Φ Φ ' ] M M + ω ~ ω g g ( n) ( n) ga λ ga λ π π [ Φ ( + δ ( + δ Φ ' n n ] )[ ν )[ ν n ( n+ ) ν ω ( n+ ) ν n + + ] ] empty fu fu ω Φ Φ

22 Couping of optica phonon with magneto-ecitons in doped gaphene ω ~ [ Φ Φ ' ] M M + ω ~ ω [ Φ Φ ' ] ω ν empty fu ω fu ~ ω ~ ω ~ a λ g ω ( v / λ v / λ ) ω Φ Φ

23 Magneto-phonon esonance in the Raman spectum g g ( n) ( n) ga λ ga λ π π ( + δ ( + δ n n )[ ν )[ ν n ( n+ ) ν ( n+ ) ν n + + ] ] G-band ω ω o Ω n hv ( n + n + ) λ Goebig Fuchs echedhi VF - PRL (7)

24 Intoduction: symmeties and notations. Optics and magneto-optics of gaphene: absoption. Magneto-phonon esonance and a fiing facto dependent fine stuctue of the G-ine in the Raman spectum of phonons. Eectonic ecitations in the Raman spectum of gaphene. Magneto-optics of biaye gaphene. Abege VF - PR (7) Mucha-ucynski McCann VF - SSC 49 (9) Abege VF - PR (7) Abege Russe VF - APL 9 65 (7) Goebig Fuchs echedhi VF - PRL (7) ashuba VF unpubished (9) ashuba VF aiv:9655 (9)

25 Eectonic ecitations in the Raman spectum: eecton-hoe pai eft afte scatteing a photon + ω ε ( p) ε ( p) H p H p i p j dominant if Ω dominant if Ω sma due to the e-h symmety Patmann and Woff Waves and inteactions in soid state pasmas

26 Hˆ vσ p v 6 γ ζ Σ ( Σ p) Σ ( ) Σ p Σ Ω vaey-symmetic vσ vσ Ω vσ vσ vaey-antisymmetic contact inteaction v ζ Σ 6γ α

27 ) ( 4 ) ( * μ ω θ ω π ω Ω c v v e I out in h vaey-symmetic e-h ecitations geneated by scatteing of ight ashuba VF aiv:9655 (9) 5 ~

28 ± in Δn M vaey-symmetic (positive paity) decouped fom the phonon ± out Δn vσ vσ ω n n + vσ vσ hv λ ecitations ae weak in Raman due to the canceation between diagams vaey-antisymmetic (negative paity) couped to the phonon v ζ Σ 6γ hv ω ( n + n + ) λ α ± in Δn ΔM m m out ± tansfeed to the attice ashuba VF aiv:9655 (9)

29 Signatue of the eectonic ecitations in the Raman spectum I * in out n ± in n + ± out v e / λ I ~ ~ fo ~ T Ω ~ ev c πω ashuba VF aiv:9655 (9)

30 Concusions Magneto-phonon esonance and a fiing facto dependent fine stuctue of the G-ine in the Raman spectum of phonons. Goebig Fuchs echedhi VF - PRL (7) ashuba VF unpubished (9) Eectonic ecitations in the Raman spectum of gaphene. ashuba VF aiv:9655 (9) Magneto-optics of biaye gaphene. Abege VF - PR (7) Mucha-ucynski McCann VF - SSC 49 (9)

31 m ~. 5 m e Monoaye J McCue Phys. Rev (956) ε ± ± hv n λ ψ εψ m ~. 5m e iaye J ± ε ± hω c n( n ) with 8-fod degeneate ε Landau eve McCann VF - Phys. Rev. Lett (6)

32 Infaed absoptions in mono/bi-aye gaphene - epeiment Jiang Heniksen Tung Wang Schwat Han im Stome (7) Heniksen Jiang Tung Schwat Takita Wang im Stome (8)

33 Inteaye asymmety gap Δ Δ U U McCann VF - PRL (6) McCann - PR (6) Mucha-ucynski Tsypyatyev Gishin McCann VF oswick Rotenbeg Phys. Rev (8) T. Ohta et a Science 95 ( 6) (Rotenbeg s goup at ekeey NL) SiC-based highy doped biaye gaphene

34 U U U w + e d ε n( ν ) gate n e n + n Mucha-ucynski McCann VF - SSC 49 (9)

35 Concusions Magneto-phonon esonance and fiing facto dependent fine stuctue of the G-ine in the Raman spectum of phonons. Goebig Fuchs echedhi VF - PRL (7) ashuba VF unpubished (9) Eectonic ecitations in the Raman spectum of gaphene. ashuba VF aiv:9655 (9) Magneto-optics of biaye gaphene. Abege VF - PR (7) Mucha-ucynski McCann VF - SSC 49 (9)