1. グラフェン量子ホール系の発光 量子ホール系の光学ホール伝導度 1 青木研究室 M2 森本高裕 青木研究室 M2 森本高裕

2. K K K K’K’ K’K’ K’K’ Graphene quantum Hall effect Landau level: Cyclotron energy: 10 μm (courtesy of Geim)  xx  xy (Novoselov et al, Nature 2005; Zhang et al, Nature 2005)  xy = 2(n+1/2) (-e 2 /h) In the effective-mass picture the quasiparticle is described by massless Dirac eqn. 2

3. Landau-level spectroscopy in graphene (Sadowski et al, PRL 2006) Uneven Landau level spacings 0101 -1  2 -2  3 1212 Peculiar selection rule |n|  |n|+1 (usually, n  n+1) 3

4. Basic idea Population inversion  cyclotron emission  Possibility of graphene “Landau level laser” Uneven Landau levels ∝ √n + |n|  |n|+1  Population inversion Ladder of excitations Tunable wavelength -n  n+1 excitation (Aoki, APL 1986) Ordinary QHE systemsGraphene Landau levels 4

5. (Ando, Zheng & Ando, PRB 2002) Optical conductivity  (  ): method Green’ s f  SCBA  Level broadening by impurity is considered through Born approximation with self-consistent Green’s function. Solve self-consistently by numerical method ()() ()() Optical conductivity is calculated from Kubo formula : current matrix elements Singular DOS makes the calculation difficult. short range Impurity potential Cf. Gusynin et al. (PRB 2006)  no self-consistent treatment of impurity scattering 5

6. Optical conductivity : result -1  2 0101 1212 higher T (Sadowski et al, 2006) higher T 6

7. Uneven Landau levels ∝ n=0 Landau level stands alone, while others form continuum spectra Population inversion is expected between n=0 and continuum. excitation Cyclotron radiation rapid decay Population inversion Density of states suitable for radiation Impurity broadening  photoemission vs other relaxation processes (phonon) 7

8. Orders of magnitude more efficient photoemission in graphene Relaxation process : photon emission Spontaneous photon emission rate is calculated from Fermi’s golden rule. Singular B dependence of Dirac quasiparticle in graphene Magnetic field:1T 8

9. Competing process : phonon emission Ordinary QHE system  Chaubet et al., PRB 1995,1998 discussed phonon emission is the main relaxation channel. Graphene  Also obtained from golden rule and factor with and, phonon emission is exponentially small in graphene as well. 2DEG Wavefunction with a finite thickness Phonon  2DEG Effect of phonon  2DEG  same order as photoemission in conventional QHE (Chaubet et al. PRB 1998) Graphene is only one atom thick  phonon does not compete with photoemission. However, atomic phonon modes  graphene will have to be examined q 9

10. 2D electron gas 2DE G ρ xy ρ xx B 10 (Paalanen et al, 1982)

11. THz spectroscopy of 2DEG 11 Faraday rotation (Sumikura et al, JJAP, 2007) Ellipticity Resonance structure at cyclotron energy

12. Motivation ●conventional results - Hall conductivity quantization at  =0 - Faraday rotation measurement in finite  12 ● How peculiar can optical Hall conductivity  xy (  F,  ) be? ● Is ac QHE possible? Only Drude form treatment Calculating  xy (  F,  ) from … ● Kubo formula ● Self-consistent Born approximation (O'Connell et al, PRB 1982)

13.  xy (  ) in GaAs ●3D plot of  xy (  F,  ) against Fermi energy and frequency Hall step still remains in ac regime 13  =0.4  C  xy (  ) FF  Resonance at cyclotron frequency FF   xy (  )

14.  xy (  ) in graphene ●  xy (  F,  ) of graphene 14 w=0 Reflecting massless Dirac DOS structure Hall step remains Resonance at cyclotron frequency  xy (  ) FF  FF  電子正孔対称

15. Consideration with Kubo formula ●Why does Hall step remain in ac region? ●How robust is it? 15 THz Hall 効果 Hall step structure in clean system (not disturbed so much by impurity) Clean ordinary QHE system (Peng et al, PRB 1991) では ac の取り扱いが不十分 □ Future problem Effect of long-range impurity Localization and delocalization in ac field Relation to topological arguement