Coherent Magneto-Optical Polarisation Dynamics in a Single Chiral Carbon Nanotube

1. Coherent Magneto-Optical Polarisation Dynamics in a Single Chiral Carbon Nanotube Gaby Slavcheva1 and Philippe Roussignol2 1 The Blackett Laboratory, Imperial College London, United Kingdom 2 Laboratoire Pierre Aigrain, Ecole Normale Supérieure, Paris, France

2. Motivation • Fundamental point of view • Formulation of a theory and model of the magneto-optical activity in chiral molecules (SWCNTs) in the nonlinear coherent regime How the chirality affects the ultrafast nonlinear optical and magneto-optical response? • Novel class of ultrafast polarisation-sensitive integrated optoelectronic devices, based on SWCNTs • Time-resolved magnetic circular dichroism (MCD) and magneto-optical rotatory dispersion (MORD) techniques provide spectroscopic information, different or impossible to obtain by other means: e.g.TR- Faraday rotation for spin dynamics • Chiral materials exhibit negative refractive index: • artificial chiral negative refractive index metamaterials: exhibit giant gyrotropy • CNTs: promising candidates in the visible range

3. Outline • Relationship between chiral symmetry and optical activity • Theoretical framework for description of the natural optical activity in a chiral SWCNT in the nonlinear coherent regime • Simulation results for the ultrafast nonlinear dynamics of the natural optical activity in chiral SWCNTs • time-resolved circular dichroism • time-resolved circular birefringence and rotatory power • Model of the Faraday effect in SWCNTs in an axial B • Zeeman splitting • Aharonov-Bohm flux • Simulation results for nonlinear Faraday rotation • Summary and conclusions

4. SWCNT with chiral symmetry • Primary classification of nanotubes: • achiral (superimposable mirror image): • zig-zag and armchair • chiral (non-superimposable) Chiral vector: Chiral angle: AL-handed or AR-handed SWCNT: depending on the rotation of 2 of the 3 armchair (A) chains of C-atoms to the L or R when looking against z: AL (5,4) AR (4,5)

5. Electronic band structure of SWCNT L=|Ch| - tube circumference  - quasiangular momentum quantum number Graphene dispersion  - 2  -1    -1  - 2

6. Optical dipole transitions for circularly polarised light Geometry of optical experiments on isolated SWCNTs 1D electronic density of states at the K-point  > 0 Linear polarisation E||=Ez E=Ex Depolarisation: Ex suppressed Dipole selection rules: m=0  m=±1 both  -1and  +1 symmetry allowed m=±1 only one transition -1or +1 Circular polarisation:

7. Energy-level structure at the point K (K′) of the lowest subbands AR-handed SWCNT AL-handed SWCNT Non-superimposable energy-level diagrams Absorption of σ+ light induces  -1 ( +1 )transition in AL-handed SWCNT Absorption of σ+ light induces  -1 transition in AR-handed SWCNT Difference in dipole selection rules for L and R circularly polarised light gives rise to optical activity Samsonidze et al., Phys. Rev. B 69, 205402 (2004)

8. Energy dispersion and 1D DOS of a AL-(5,4) SWCNT Nanotube diameter 0.611 nm Chiral angle 26.330 Length of unit cell T = 3.3272 nm Number of hexagons (unit cell) 122 Boundary of Brillouin Zone (kzmax) (m-1) 9.4422e+08 Bandgap Magnitude E, (eV) 1.321, =939 nm E,±1 =1.982 eV, =626.5 nm pulse duration  = 60 fs, excitation fluence S=20 mJ/m2 J.-S. Lauret, C. Voisin, G. Cassabois, C. Delalande, Ph. Roussignol, O. Jost, and L. Capes, Phys. Rev. Lett. 90, 057404 (2003)

9. Dielectric response function and optical dipole matrix element y y y x x x 2r 2R 2R z z 2r 2R z • Effective medium theory: Dipolar polarisability of a SWCNT m per unit length in a quasistatic approximation (m=1) • Introduce equivalent isotropic dielectric function  of a solid cylinder with radius R Lü et al., Phys. Rev. B 63, 033401 (2000), Henrard and Lambin. J. Phys. B 29, 5127 (1996) • ordinary and extraordinary ray in graphite no=2.64, ne=2.03, nSWCNT= =2.3 • Estimate of the dipole matrix element for optical transitions excited by circularly polarised light • Upper and lower bounds from the extension of the effective mass method applied to chirality effects in CNTs(coupling between the orbital momentum k() and kz) ~10 -31- 10-29 Cm • Estimate from radiative lifetime of an e-h pair spont ~10 ns *:~ 3.579×10-29 Cm Ivchenko and Spivak, Phys. Rev. B 66, 155404 (2002) *Wang et al., Phys. Rev. Lett. 92, 177401 (2000)

10. Theoretical formalism Dynamical evolution of an N-level quantum system Liouville equation (Schrödinger picture): Pseudospin equation for the real state coherence vector S =(S1, S2, ... ,S ) (Heisenberg picture)*:  N N Using Gell-Mann’s -generators of the SU(N) Lie algebra: N-1 N-1 i i 3 3 2 2 1 1 (E=0) torque vector *Hioe and Eberly PRL, 47, 838,1981

11. Optical excitation of   ± 1 transition by  +(-)-polarised pulse N=4, SU(4) Lie group =60 fs -pulse System Hamiltonian Rabi frequencies torque vector Relaxation times estimated from spontaneous emission rate 1= 2.91 ns-1, 2= 9.81 ns-1, 3= 1.23ns-1,  = 130 fs-1, = 0.8 ps-1, -1= 1.6 ps-1 Gaussian -pulse with =60 fs: E0=6.098108 Vm-1 ; Resonant wavelength: 0=626.5 nm, E,±1=1.9815 eV Density of resonant absorbers Na=6.8111024m-3

12. Master equation for resonant excitation by  + (-) pulse Pseudospin equations Maxwell curl equations Medium polarisation Finite-Difference Time-Domain (FDTD) solution: time-stepping algorithm with predictor-corrector iterative scheme Source optical field Slavcheva, Phys. Rev. B 77, 115347 (2008)

13. Spatially resolved temporal dynamics for  - and  + excitations z 50 nm 50 nm 50 nm 500 nm 50 nm z=0 z1 z2 z3 z4 z=L d  - (a)  - (b)  - (c)  - (d) z=z1 z=z2 z=z4 z=z3  + (a)  + (b)  + (c)  + (d)  -  + Chirality determination from the ultrafast nonlinear response using ultrashort pulses with both helicities

14. Time evolution of a linearly polarised pulse Rotation of the polarisation plane during the pulse propagation Source optical field z=z1 Transmission spectra of Ex,Ey at the output facetvs  z=z4

15. Spatially resolved gain coefficient spectra for  - and  +- pulse Natural circular dichroism Theor. Value* ~ 1.03 m-1 ; Experiment (artificial helicoidal bilayer)**:1.15-2.07 m-1 *Ivchenko and Spivak, Phys. Rev. B 66, 155404 (2002); **Rogacheva et al., Phys Rev. Lett. 97, 177401 (2006)

16. Spatially resolved phase shift spectra for  - and  +- pulse Comparison with rotatory power of birefringent materials: NaBrO3  =2.24 /mm quartz 21.7 /mm, |nL-nR|=7.1 10-5 Cinnabar (HgS) 32.5/mm AgGaS2 522 /mm Liquid substances: Turpentine -0.37 /mm Corn syrup 1.18 /mm Cholesteric liquid crystals ~1000 /mm, Artificial photonic metamaterials: ~ 2500 /mm Sculptured thin films ~ 6000 /mm Circular birefringence: Specific rotatory power :

17. Single chiral CNT in an axial magnetic field Magnetic energy bands H. Ajiki and T. Ando, J. Phys. Soc. Jap. 62, 1255 (1993) Jiang et al., PRB 62, 13209 (2000); Minot et al., Nature 428, 536 (2004)

18. Single chiral CNT in an axial magnetic field Energy-level structure significantly modified: • Zeeman splitting • Orbital effects - Aharonov-Bohm phase due to the flux through the tube  uniform shift in the energy levels • Energy band gap oscillates with B (or magnetic flux  ),  /  0=0.00057 Type I tube: n-m=3q, q integer B=8 T: EZ~ 0.46 meV, z~7.031011 rad/s Type II tube: n-m=3q±1, q integer Energy gap shift (band gap reduction): B= 8T, EAB ~ 3.37 meV, AB~ 5.121012 rad/s Band gap renormalisation (K-point) : 0  0 -AB H. Ajiki and T. Ando, J. Phys. Soc. Jap. 62, 1255 (1993) Jiang et al., PRB 62, 13209 (2000);

19. Original (B=0) and reduced energy-level systems in an axial magnetic field Jz=+1/2 4 Jz=+3/2 l=1 4”  B  B Jz= -1/2 0 ”    l=0 1” 2’ Jz=+1/2 1 3B 2 1B Eg 2B Jz=+1/2 0 3” 4’ l=0 3B   1 Jz= -1/2 3 1B 3’ B 3  B   Jz=+3/2 1’ l=1 z E=0 1 Jz=+1/2

20. Theoretical Formalism 2’ Jz=+1/2 4” B 2B  B Jz= -1/2 Jz=+1/2 2” 4’  3B  1B Jz= -1/2 Jz=+1/2 1B 3B 3’ 1” B B  2B Jz=+3/2 Jz=+1/2 1’ 3” 3-level -system System Hamiltonian Torque vector Polarisation vector components 1= 2.91 ns-1, 2= 9.79 ns-1, 3= 9.77 ns; = 130 fs-1, = 0.8 ps-1, -1= 1.6 ps-1 E0=6.098108 Vm-1, Eres=(0-AB-2z) , B= 8T:*coup=3.6205410-29 Cm ,Na=6.8111024m-3 Ivchenko and Spivak, Phys. Rev. B 66, 155404 (2002)

21. Simulation results for Faraday rotation Spatially resolved absorption/gain coefficient spectra for  - and  +- pulse at B=8 T Absorption dip at resonance for  + excitation Magnetic circular dichroism

22. Spatially resolved phase shift spectra for  - and  +- pulse at B=8 T Double-peaked phase shift curve at resonance for  + excitation Specific rotatory power : Magneto-chiral effect

23. Summary • Dynamical model proposed of the optical activity and the Faraday effect of a SWCNT in the nonlinear coherent regime • Provided an estimate for the dielectric response function and dipole matrix element for circularly polarised light in a single CNT • SWCNT handedness determined by optical spectroscopy using circularly and linearly polarised light • Giant natural gyrotropy demonstrated (~ 3000/mm) in a (5,4) SWCNT • Model of nonlinear Faraday rotation in a single chiral CNT • Enhancement of magneto-chiral circular dichroism and rotatory power in an external B • Method valid for an arbitrary nanotube chirality and pulse polarisation;Valid for ultrashort optical pulses and arbitrary pulse shape (including cw) • Outlook: study of the rotation angle dependence on chirality with possibility of engineering rotatory power; study of the B-field dependence of the specific rotation angle

24. G. Bastard R. Ferreira C. Flytzanis C. Voisin, LPA, ENS, Paris Acknowledgements Thank you for your attention!