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. 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 青木研究室M2 森本高裕

2. K’ K sxy K’ K K’ K 10 μm rxx Graphene quantum Hall effect In the effective-mass picture the quasiparticle is described by massless Dirac eqn. (courtesy of Geim) Landau level: Cyclotron energy: sxy = 2(n+1/2)(-e2/h) (Novoselov et al, Nature 2005; Zhang et al, Nature 2005)

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

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

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

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

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

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

9. Competing process : phonon emission q 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. Effect of phonon ^ 2DEG  same order as photoemission in conventional QHE (Chaubet et al. PRB 1998) 2DEG Phonon ^ 2DEG Wavefunction with a finite thickness Graphene is only one atom thick  phonon does not compete with photoemission. However, atomic phonon modes ^ graphene will have to be examined

10. ρxy ρxx B 2DEG (Paalanen et al, 1982) 2D electron gas

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

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

13. sxy(w)in GaAs ●3D plot of sxy(eF, w) against Fermi energy and frequency sxy(w) Hall step still remains in ac regime w=0.4wC eF sxy(w) sxy(w) w w Resonance at cyclotron frequency eF

14. sxy (w)in graphene ● sxy(eF, w) of graphene 電子正孔対称 w=0 eF sxy(w) Resonance at cyclotron frequency sxy(w) w eF w Reflecting massless Dirac DOS structure Hall step remains

15. Consideration with Kubo formula ●Why does Hall step remain in ac region? ●How robust is it? Clean ordinary QHE system Hall step structure in clean system (not disturbed so much by impurity) THz Hall 効果 (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