**Optical activity of a single chiral CNT in an axial magnetic field**
In the presence of a static axial magnetic field, threading the nanotube (Fig. 46):
*Fig. 46.** **Perspective view of a (20,10) nanotube, threaded by an axial magnetic field*
the electronic band structure of a single carbon nanotube, and the electronic states near the band gap edge in particular, significantly change, owing to the combined action of two effects: the spin-B interaction resulting in Zeeman splitting of the energy levels and the appearance of the Aharonov-Bohm phase in the wave function resulting in an additional energy level shift.
At B_{||}=0 the two symmetric sub-bands at K (K ') are degenerate (Fig. 47 left panel). An applied magnetic field along the nanotube axis lifts this degeneracy and shifts the energy levels. As a result the energy gap of one of the sub-bands K ' becomes larger, whereas the energy gap of the other sub-band (K) becomes smaller (see Fig. 47 right panel).
**Fig. 47.** Dispersion relations E(k||) and E '( k||), at K (K') shown in blue and red, respectively. The sub-bands are degenerate at B=0 (left panel) and the magnetic field lifts this degeneracy (right panel).
The normalised (to zero magnetic field) energy gap oscillates with magnetic flux j as shown in Fig. 48.
**Fig. 48.** Normalised band gap energy of (a) type I and (b) type II carbon nanotubes in an axial magnetic field. Solid (dashed) lines correspond to spin-B interaction absent (present)
The tube considered is a semiconducting (5,4) type II carbon nanotube. The Zeeman splitting of the energy levels near the band edge results in a energy band gap shift (reduction at K-point). The orbital effects, due to the magnetic field flux trough the tube lead to a uniform shift in the energy levels. The modified energy-level scheme under a magnetic field is shown in Fig. 49.
**Fig. 49. **Zeeman splitting (*h**w*_{z}*) of the energy levels near the band gap of a (5,4) nanotube in an external axial magnetic field. The original set of levels *(*|1**ñ** -|4**ñ*)* is labelled by the (quasi-angular) orbital momentum quantum number l=0,1; the resulting energy levels are labelled by the projection of the total angular momentum J=l+s along the nanotube axis z. Two reduced sets of levels, that energetically are nearest to the band edge, can be identified (shown in Fig. 50).*
**Fig. 50. **Two antisymmetric reduced four-level systems nearest to the Fermi level in an axial magnetic field; the allowed and forbidden dipole optical transitions are designated by dashed arrows.
Spatially resolved absorption gain coefficient spectra for a *s** *^{- }and * **s** *^{+} excitation with a pulse area p are shown in Fig. 51. The phenomenological relaxation times are calculated for resonant energy gap reduced by the Zeeman and Aharonov-Bohm shifts at B=8 T.
**Fig. 51. **Spatially resolved calculated gain coefficient per *m**m vs wavelength for E*_{x}(E_{y}) electric field vector component of a *s** *^{+}* (red line) and **s** *^{- }*(green line) circularly polarised ultrashort optical excitation and the theoretical gain coefficient of a homogeneously broadened resonant two-level system (blue line) at B=8 T for a pair of points (z*_{i},z_{j}) separated by one dielectric resonant wavelength within the carbon nanotube, shifted from the left boundary of the structure. *G*_{1}^{B}_{ }*= 2.91 ns*^{-1}, *G*_{2}^{B}*= 9.79 ns*^{-1}, *G*_{3}^{B}*= 9.77 ns;**g** *^{B}= 130 fs^{-1}, *G*_{m}*= 0.8 ps*^{-1}, *G*_{m}_{-1}*= 1.6 ps*^{-1, }*E*_{0}=6.098*´**10*^{8}* Vm*^{-1}, E_{res}=*h**(**w*_{0}*-**w*_{AB}*-**w*_{z}*):**g*^{*}_{coup}*=**3.62054**´**10*^{-29}*C, N*_{a}=6.811*´**10*^{24 }m^{-3}^{ }*.*
The calculated average magnetic circular dichroism is ~0.706 mm^{-1 }is an order of magnitude larger than the natural circular dichroism. Absorption dip at resonance for a
*s** * ^{+ }excitation is clearly discerned. The spatially resolved phase shift spectra for a *s** * ^{- }and *s** * ^{+ } p-pulse under a magnetic field B=8 T are shown in Fig. 52.
*Fig. 52. Spatially resolved calculated phase shift vs wavelength at B=8 T for E*_{x} (red line) and E_{y} (green dot) components under *s** *^{+}*excitation and for Ex (magenta line) and Ey (cyan dot) under **s** *^{- }*circularly polarised ultrashort optical excitation and the theoretical phase shift of a homogeneously broadened resonant two-level system (blue line)*
The average specific rotator power is ~ 32580 °/mm, an order of magnitude larger than the zero field value. Note that this is an estimate of the combined magneto-chiral effect. The different behaviour of the calculated gain and absorption spectra under *s* ^{- }(*s* ^{+}) polarised optical pulse excitation is a direct consequence of the discrete energy level configuration describing the two cases. While the energy-level system for a *s* ^{-} excitation is a four-level system, the one corresponding to a *s* ^{+ }excitation is a three-level L-system. The calculated spectra in the latter case are reminiscent of electromagnetically-induced transparency (EIT) and coherent population trapping effects in a three-level system. In fact, the absorption at resonance is close to zero and the shape of the spectrum is similar to the absorption dip, observed in EIT. The predicted destructive interference in an external axial magnetic field after the passage of the ultrashort pulse is a direct consequence of the specific timescales of the processes involved in the relaxation dynamics. This behaviour is confirmed by Fig. 52 : whereas the phase shift spectrum for a *s* ^{- }pulse excitation is of the type of a two-level atomic phase shift, the shape of the phase shift curve is double-peaked which is characteristic for the real part of the susceptibility in a three-level system, exhibiting EIT. |