PLMCN10, Cuernavaca (Mexico), 12-16 April 2010 Coherent Magneto-Optical Polarisation Dynamics in a...

PLMCN10, Cuernavaca (Mexico), 12-16 April 2010

Coherent Magneto-Optical Coherent Magneto-Optical Polarisation Dynamics in a Single Polarisation Dynamics in a Single

Chiral Carbon NanotubeChiral Carbon Nanotube

Coherent Magneto-Optical Coherent Magneto-Optical Polarisation Dynamics in a Single Polarisation Dynamics in a Single

Chiral Carbon NanotubeChiral 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


MotivationMotivation

● Fundamental point of viewFormulation 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 gyrotropyCNTs: promising candidates in the visible range


OutlineOutline

●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


SWCNT with chiral symmetrySWCNT with chiral symmetry

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:

Primary classification of nanotubes:

achiral (superimposable mirror image):

zig-zag and armchair

● chiral (non-superimposable)

Chiral vector: 1 2 , ,hC na ma n m nm 0

Chiral angle: hCa ,1 300

AL (5,4) AR (4,5)


Electronic band structure of SWCNTElectronic band structure of SWCNT

Graphene dispersion2

1

0

0

,...2,1,0,2

Kat

KatL

k

-1

-1

- 2

- 2

L=|Ch| - tube circumference

- quasiangular momentum quantum number


Optical dipole transitions for circularly polarised light Optical dipole transitions for circularly polarised light 1D electronic density of states at the K-point > 0

Linear polarisation

E||=Ez

E=Ex

Dipole selection rules: m=0 m=±1 both -1 and +1

symmetry allowed m=±1 only one transition -1 or +1

Circular polarisation:

Geometry of optical experiments on isolated SWCNTs

Depolarisation: Ex suppressed


Energy-level structure at the point K (KEnergy-level structure at the point K (K′′) of the lowest subbands) of the lowest subbands

AR-handed SWCNT

Non-superimposable energy-level diagrams

AL-handed SWCNT

Samsonidze et al., Phys. Rev. B 69, 205402 (2004)

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


Energy dispersion and 1D DOS of a AL-(5,4) SWCNTEnergy dispersion and 1D DOS of a AL-(5,4) SWCNT

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)

Nanotube diameter 0.611 nmChiral angle 26.330

Length of unit cell T = 3.3272 nmNumber of hexagons (unit cell) 122Boundary of Brillouin Zone (kzmax) (m-1) 9.4422e+08Bandgap Magnitude E, (eV) 1.321, =939 nm

E,±1 =1.982 eV, =626.5 nm


Dielectric response function and optical dipole matrix elementDielectric response function and optical dipole matrix element

Lü et al., Phys. Rev. B 63, 033401 (2000), Henrard and Lambin. J. Phys. B 29, 5127 (1996)

● 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

2Rz

yx

2R2rz

yx

z

yx

2R2r

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 CmIvchenko and Spivak, Phys. Rev. B 66, 155404

(2002)*Wang et al., Phys. Rev. Lett. 92, 177401 (2000)


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)*:

m

Theoretical formalismTheoretical formalism

]ˆ,ˆ[ˆ

H

ti

kjijki Sft

S

1,...,1,, 2 Nkji

1

2

3

i

N-1

N

1 (E=0)

23

iN-1N

*Hioe and Eberly PRL, 47, 838,1981

1N 2

Using Gell-Mann’s

-generators

of the SU(N) Lie

algebra:

jj tTrtS ˆ)(ˆ

ljklkj fi ˆ2]ˆ,ˆ[

jj tHTrt

)(ˆ1

torque vector


=60 fs -pulse

Optical excitation of Optical excitation of ± ± 1 transition by 1 transition by +(-)+(-)-polarised pulse-polarised pulse

0

0

2

100

2

100

002

1

002

10

ˆ

yx

yx

yx

yx

i

i

i

i

H

yy

xx

E

E

N=4, SU(4) Lie group

0

00 3

2,

3

2,,,0,0,0,0,,,0,0,0,0, yyxxγ

torque vector

Rabi frequencies System Hamiltonian

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=6.098108 Vm-1 ;

Resonant wavelength: 0=626.5 nm, E,±1=1.9815 eV

Density of resonant absorbers Na=6.8111024 m-3


Master equation for resonant excitation by Master equation for resonant excitation by + (-)+ (-) pulse pulse

Slavcheva, Phys. Rev. B 77, 115347 (2008)

4,...,1,ˆˆˆ iTrdiag i

ˆˆˆˆ,ˆ

ˆtH

i

t

1 1ˆˆ , 1,2,...,122

1 ˆˆ , 13,14,152

jkl k l j j jej j

jkl k l j

f S Tr S S jS T

tf S Tr j

t

tzP

z

tzH

t

tzEt

tzP

z

tzH

t

tzE

z

tzE

t

tzH

z

tzE

t

tzH

yxy

xyx

xy

yx

,1,1,

,1,1,

,1,

,1,

127

61

SSNP

SSNP

ay

ax

Medium polarisation

Pseudospin equationsMaxwell curl equations

2 20

2 20

( ) /

( ) /

0

0, cos( )

0, sin ( )

decay

decay

t t t

x o o

t t t

y o

E z t E e t

E z t E e t

2 20

2 20

( ) /

( ) /

0

0, cos( )

0, sin ( )

decay

decay

t t t

x o o

t t t

y o

E z t E e t

E z t E e t

Source optical field Finite-Difference Time-Domain (FDTD) solution: time-stepping algorithm with predictor-corrector iterative

scheme


Spatially resolved temporal dynamics for Spatially resolved temporal dynamics for - - and and ++ excitations excitations

dz500 nm50 nm 50 nm50 nm50 nm

z1 z2 z3 z4z=0 z=L

(a) (b) (c) (d)

(a) (b) (c) (d)

- - - -

+ + + +

z=z1 z=z2 z=z3 z=z4

- + Chirality determination from the ultrafast

nonlinear response using ultrashort pulses with both

helicities


Time evolution of a linearly polarised pulseTime evolution of a linearly polarised pulse

Source optical field

2 20( ) /

0, cos( )

0, 0

decayt t t

x o o

y

E z t E e t

E z t

z=z1

z=z4

Rotation of the polarisation plane during the pulse propagation

Transmission spectra of Ex,Ey at the output facet vs


Spatially resolved gain coefficient spectra for Spatially resolved gain coefficient spectra for -- and and ++- pulse- pulse

1

)()( 083.0~ mGGA LyxRyxyx 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)


Spatially resolved phase shift spectra for Spatially resolved phase shift spectra for -- and and ++- pulse- pulse

Specific rotatory power : mm

nn RL /24.2962~0

Circular birefringence:

0103.0 RL nnn

Comparison with rotatory power of birefringent materials:

NaBrO3 = 2.24 /mmquartz 21.7 /mm, |nL-nR|=7.1 10-5

Cinnabar (HgS) 32.5/mmAgGaS2 522 /mm

Liquid substances:Turpentine -0.37 /mmCorn syrup 1.18 /mm

Cholesteric liquid crystals ~1000 /mm,Artificial photonic metamaterials: ~ 2500 /mmSculptured thin films ~ 6000 /mm


Single chiral CNT in an axial magnetic fieldSingle 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)


Single chiral CNT in an axial magnetic fieldSingle 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

B=8 T: E Z~ 0.46 meV, z~7.031011 rad/s

1/2/13

203

,2/1/03

103

00

00

G

G

G

E

E

E

2~;2

1, ezeBz gBgE

*2 eB m

e

e

h0

H. Ajiki and T. Ando, J. Phys. Soc. Jap. 62, 1255 (1993)Jiang et al., PRB 62, 13209 (2000);

Energy gap shift (band gap reduction): B= 8T, EAB ~ 3.37 meV, AB~ 5.121012 rad/sBand gap renormalisation (K-point) : 0 0 -AB

Type I tube:

n-m=3q, q integer

Type II tube:

n-m=3q±1, q integer


Original (B=0) and reduced energy-level systems in an axial magnetic Original (B=0) and reduced energy-level systems in an axial magnetic fieldfield

z

3

4

1

l=0

l=1

l=1

l=0

Jz=+1/2

Jz= -1/2

Jz=+3/2

Jz=+1/2

Jz=+1/2

Jz= -1/2

Jz=+3/2

Jz=+1/2

1

1

B

2B

3B

3B

Eg2

31B

1B

BB

B

1’

4”

3”4’

3’

1”2’

”

E=0

0

0


Theoretical FormalismTheoretical Formalism

Jz=+1/2

Jz=+1/2

Jz= -1/2

Jz=+3/2

2B

3B

1B

BB

1’

2’

3’

4’ Jz= -1/2

Jz=+1/2

Jz=+1/2

B

2B

3B1B

B4”

3”

1”

2”

z

z

zyx

yxz

i

i

H

000

000

002

1

002

1

ˆ0

zyx

yxz

z

z

i

iH

0

0

0

2

100

2

100

000

000

ˆ

6

42,

3

22,2,0,0,0,0,0,,0,0,0,0,0, 00

0zz

zyx

γ

6

42,

3

2,2,,0,0,0,0,0,,0,0,0,0,0 00 zzzyx

γ

System Hamiltonian

Torque vector

Polarisation vector components

7

1

SNP

SNP

ay

ax

12

6

SNP

SNP

ay

ax

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=6.098108 Vm-1, Eres=(0-AB-2z) , B= 8T:*coup=3.6205410-29 Cm ,Na=6.8111024 m-3

Ivchenko and Spivak, Phys. Rev. B 66, 155404 (2002)

3-level -system


Simulation results for Faraday rotationSimulation results for Faraday rotation

Spatially resolved absorption/gain coefficient spectra for - and +- pulse at B=8 T

1

)()( 706.0~ mAGA LyxRyxyx Magnetic circular dichroism

Absorption dip at resonancefor + excitation


Spatially resolved phase shift spectra for Spatially resolved phase shift spectra for -- and and ++- pulse at B=8 T- pulse at B=8 T

Double-peakedphase shift curveat resonancefor + excitation

mm

nn RL /32580~0

Specific rotatory power : Magneto-chiral effect


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

SummarySummary


Acknowledgements

G. Bastard

R. Ferreira

C. Flytzanis

C. Voisin, LPA, ENS, Paris

Thank you for your attention!

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