# Asymptotic exchange coupling of quasi-one-dimensional excitons in carbon nanotubes

## Abstract

An analytical expression is obtained for the biexciton binding energy as a function of the inter-exciton distance and binding energy of constituent quasi-one-dimensional excitons in carbon nanotubes. This allows one to trace biexciton energy variation and relevant non-linear absorption under external conditions whereby the exciton binding energy varies. The non-linear absorption lineshapes calculated exhibit characteristic asymmetric (Rabi) splitting as the exciton energy is tuned to the nearest interband plasmon resonance. These results are useful for tunable optoelectronic device applications of optically excited semiconducting carbon nanotubes, including the strong excitation regime with optical non-linearities.

###### pacs:

78.40.Ri, 73.22.-f, 73.63.Fg, 78.67.ChSingle-walled carbon nanotubes (CNs) — graphene sheets rolled-up into cylinders of nm in diameter and m up to cm in length (1); (2) — are shown to be very useful as miniaturized electromechanical and chemical devices (3), scanning probe devices (4), and nanomaterials for macroscopic composites (5). The area of their potential applications was recently expanded to nanophotonics (6); (7) after the demonstration of controllable single-atom incapsulation into single-walled CNs (8), and even to quantum cryptography since the experimental evidence was reported for quantum correlations in the photoluminescence spectra of individual nanotubes (9).

The true potential of CN-based optoelectronic device applications lies in the ability to tune their properties in a precisely controllable way. In particular, optical properties of semiconducting CNs originate from excitons, and may be tuned by either electrostatic doping (10), or via the quantum confined Stark effect (QCSE) by means of an electrostatic field applied perpendicular to the CN axis (11). In both cases the exciton properties are mediated by collective plasmon excitations in CNs (12). In the case of the perpendicularly applied electrostatic field, in particular, we have shown recently (11) that the QCSE allows one to control the exciton-interband-plasmon coupling in individual undoped CNs and their (linear) optical absorption properties, accordingly.

Here, I extend our studies to the strong (non-linear) excitation regime whereby photogenerated biexcitonic states may be formed in CNs (observed recently in single-walled CNs by the femtosecond transient absorption spectroscopy technique (13)). An analytical (universal) expression is obtained for the biexciton binding energy as a function of the inter-exciton distance and the binding energy of constituent excitons. The formula is consistent with the numerical results reported earlier (14); (15), and is advantageous in that it allows one to trace biexciton energy variation and relevant non-linear absorption, accordingly, as the exciton energy is tuned to the nearest interband plasmon resonance by means of the QCSE. The non-linear absorption lineshapes are calculated close to the first interband plasmon resonance for the semiconducting (11,0) CN (chosen as an example) under resonant pumping conditions (16). They exhibit the characteristic asymmetric splitting behavior similar to that reported for the linear absorption regime (11). This effect could help identify the presence and study the properties of biexcitons in individual single-walled CNs, which is not an easy task under non-linear excitation because of the strong competing exciton-exciton annihilation process (17); (18); (19).

The binding energy of the biexciton in a small-diameter
(nm) CN can be evaluated by the method pioneered by
Landau (20), Gor’kov and Pitaevski (21),
Holstein and Herring (22) — from the analysis of the
asymptotic exchange coupling by perturbation on the configuration
space wave function of the two ground-state one-dimensional (1D)
excitons. Using the cylindrical coordinate system with the
*z*-axis along the CN axis and separating out circumferential
and longitudinal degrees of freedom of each of the excitons by
transforming their longitudinal motion into their respective
center-of-mass coordinates (11); (23), one arrives at
the biexciton Hamiltonian of the form [see Fig. 1 (a)]

(1) | |||

Here, are the relative electron-hole motion coordinates of two 1D excitons separated by the center-of-mass-to-center-of-mass distance , is the cut-off parameter of the effective (cusp-type) longitudinal electron-hole Coulomb potential. Equal electron and hole effective masses are assumed (24) and ”atomic units” are used (20); (21); (22), whereby distance and energy are measured in units of the exciton Bohr radius and Rydberg energy, and , respectively, is the exciton reduced mass. First two lines in Eq. (1) represent two non-interacting 1D excitons with their individual potentials symmetrized to account for the presence of the neighbor a distance away, as seen from the - and -coordinate systems treated independently [Fig. 1 (a)]. Last two lines are the inter-exciton exchange Coulomb interactions — electron-hole and hole-hole + electron-electron, respectively (25). Biexciton binding energy is , where is the lowest eigenvalue of Eq. (1), is the single exciton binding energy with being the lowest-bound-state quantum number of the 1D exciton (23). Negative indicates that the biexciton is stable with respect to dissociation into two isolated excitons.

The Hamiltonian (1) is effectively two dimensional in
the configuration space of the two *independent* relative
motion coordinates, and . Figure 1 (b),
bottom, shows schematically the potential energy surface of the
two closely spaced non-interacting 1D excitons [second line of Eq. (1)]
in the space. The surface has four symmetrical minima [representing
isolated two-exciton states shown in Fig. 1 (b), top], separated by the potential barriers
responsible for the tunnel exchange coupling between the
two-exciton states in the configuration space. The coordinate
transformation ,
places the origin of the new coordinate
system into the intersection of the two tunnel channels between
the respective potential minima [Fig. 1 (b)], whereby the
exchange splitting formula of
Refs. (20); (21); (22) takes the form

(2) |

where are the ground-state and excited-state energies [eigenvalues of Eq. (1)] of the two coupled excitons as functions of their center-of-mass-to-center-of-mass separation, and

(3) |

is the tunnel exchange coupling integral, where is the solution to the Schrödinger equation with the Hamiltonian (1) transformed to the coordinates. The factor comes from the fact that there are two equivalent tunnel channels in the problem, mixing three equivalent indistinguishable two-exciton states in the configuration space [one state is given by the two minima on the -axis, and two more are represented by each of the minima on the -axis — compare Fig. 1 (a) and (b)].

The function in Eq. (3) is sought in the form

(4) |

where is the product of two single-exciton wave functions (26) representing the isolated two-exciton state centered at the minimum (or , ) of the configuration space potential [Fig. 1 (b)], and is a slowly varying function to take into account the deviation of from due to the tunnel exchange coupling to another equivalent isolated two-exciton state centered at , (or , ). Substituting Eq. (4) into the Schrödinger equation with the Hamiltonian (1) pre-transformed to the coordinates, one obtains in the region of interest ( dropped (26))

up to negligible terms of the order of the inter-exciton van der Waals energy and up to second derivatives of . This equation is to be solved with the boundary condition originating from the natural requirement , to result in

(5) |

After plugging Eqs. (5) and (4) into Eq. (3), and retaining only the leading term of the integral series expansion in powers of subject to , one obtains

(6) |

The ground state energy of two coupled 1D excitons in Eq. (2) is now seen to go through the negative minimum (biexcitonic state) as increases. The minimum occurs at , whereby the biexciton binding energy is . In absolute units, expressing in terms of , one has

(7) |

The energy can be affected by the QCSE as decreases quadratically with the perpendicular electrostatic field applied (11). The field dependence in Eq. (7) mainly comes from the pre-exponential factor. So, will be decreasing quadratically with the field, as well, for not too strong perpendicular fields. At the same time, the equilibrium inter-exciton separation in the biexciton, (atomic units), will be slowly increasing with the field, consistently with the lowering of . In the zero field, assuming ( is the dimensionless CN radius) as reported earlier from variational calculations (27), one has as well, which is weaker than the dependence of Ref. (14), but agrees qualitatively with the recent advanced Monte-Carlo simulations (15). Interestingly, as goes down, the ratio in Eq. (7) slowly grows up approaching the 1D limit . This tendency can also be traced in the Monte-Carlo data of Ref. (15). Finally, goes down with decreasing , thus explaining experimental evidence for enhanced exciton-exciton annihilation in small diameter CNs (17); (18); (19).

Figure 2 shows the difference calculated from Eqs. (2) and (6) for a specific example of the coupled pair of the first bright excitons in the semiconducting (11,0) CN exposed to different perpendicular electrostatic fields. The inset shows the field dependences of [as given by Eq. (7)] and of . All the curves are calculated using eV, eV, and the field dependence of reported earlier in Ref. (11). They exhibit typical behaviors discussed above.

Now consider the exciton absorption lineshape under controlled (e.g., by the QCSE) variable exciton-interband -plasmon coupling (11). In the linear (longitudinal) excitation regime, one has for the exciton with the energy close to a plasmon resonance the lineshape of the form

(8) |

where , is the spontaneous decay rate into plasmons, , is the half-width-at-half-maximum of the plasmon resonance with the energy , and is an additional exciton energy broadening (normally attributed to the exciton-phonon scattering with the relaxation time ). All quantities in Eq. (8) are dimensionless, i.e. normalized to , where eV is the C-C overlap integral, and the condition is assumed to hold.

The non-linear optical susceptibility is proportional to the linear optical response function under resonant pumping conditions (16). This allows one to use Eq. (8) to study the non-linear excitation regime with the photoinduced biexciton formation as the exciton energy is tuned to the nearest interband plasmon resonance. The third-order longitudinal CN susceptibility is then of the form (16); (14)

(9) |

where is the dimensionless binding energy of the biexciton composed of two (ground-internal-state) excitons, and is the frequency-independent constant. The first and second terms in the brackets represent bleaching due to the depopulation of the ground state and photoinduced absorption due to exciton-to-biexciton transitions, respectively.

Figure 3 compares the linear response lineshape (8) with the imaginary part of Eq. (9) representing the non-linear optical response function under resonant pumping, both calculated for the first bright (ground-internal-state) exciton in the (11,0) CN, as its energy is tuned to the nearest interband plasmon resonance (vertical dashed line in Fig. 3) (11). In this calculation, was taken to be eV as given by Eq. (7) in the zero field (29). (Weak field dependence of does not play an essential role here as eV regardless of the field strength.) The phonon relaxation time fs was used as reported in Ref. (28), since this is the shortest one out of possible exciton relaxation processes, including exciton-exciton annihilation ( ps (17)). Rabi splitting eV is seen both in the linear and in non-linear excitation regime, indicating the strong exciton-plasmon coupling both in the single-exciton and in biexciton states, almost unaffected by the phonon relaxation.

This effect can be used in tunable optoelectronic device applications of small-diameter semiconducting CNs in areas such as nanophotonics, nanoplasmonics, and cavity quantum electrodynamics, including the strong excitation regime with optical non-linearities. In the latter case, the experimental observation of the non-linear absorption line splitting predicted here would help identify the presence and study the properties of biexcitonic states (including biexcitons formed by excitons of different subbands (13)) in individual single-walled CNs, due to the fact that when tuned close to a plasmon resonance the exciton relaxes into plasmons at a rate much greater than , totally ruling out the role of the competing exciton-exciton annihilation process.

Support from NSF (ECCS-1045661 & HRD-0833184), NASA (NNX09AV07A), and ARO (W911NF-10-1-0105) is gratefully acknowledged.

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