Collective Excitations and Optical Response of Ultrathin Carbon Nanotube Films

For an individual SWCN, absorption of a photon by a valence-band electron excites it to the conduction band to create an exciton. This makes a longitudinal polarization effect with an induced dipole moment directed predominantly along the nanotube axis due to a strongly suppressed SWCN polarizability in the perpendicular direction [82, 81], also known as the transverse depolarization [28, 83]. For a densely packed periodically aligned array of SWCNs, the induced inter-tubule dipole-dipole coupling should then result in the anisotropic collective polarization of the array (predominantly along the CN alignment direction), leading to the anisotropic dielectric response — the focus of our study here.

The second quantized Hamiltonian of the free exciton (see, e.g., Refs. [77, 84]) on the SWCN surface can be written in the form

$$\hat{H}_{ex}\!=\!\sum_{s,q}E_{s}(q)B^{\dagger}_{s,q}B_{s,q},$$

(5)

where the operators $B^{\dagger}_{s,q}$ and $B_{s,q}$ create and annihilate, respectively, the exciton with the longitudinal quasimomentum $q$ ($y$-direction in Fig. 1) in the $s$-subband, which for the $(m,n)$-type SWCN is associated with the quantized circumferential momentum component $\hbar s/R$, $s\!=\!1,2,...,m\,(\geq\!n)$ [34, 37]. These operators satisfy the standard bosonic commutation relations

$$\big{[}B_{s,q},B^{\dagger}_{s^{\prime}\!\!,q^{\prime}}\big{]}=\delta_{ss^{\prime}}\delta_{qq^{\prime}},~{}~{}\big{[}B_{s,q},B_{s^{\prime}\!\!,q^{\prime}}\big{]}=0$$

and can be converted into their coordinate counterparts using Eq. (2), to obtain

$$B^{\dagger}_{s,y_{n}}\!\!=\frac{1}{\sqrt{N}}\sum_{q}B^{\dagger}_{s,q}e^{iqy_{n}},~{}~{}B_{s,y_{n}}\!\!=\big{(}B^{\dagger}_{s,y_{n}}\big{)}^{\dagger}$$

(6)

for the $n$-th SWCN of the array. The exciton energy is

$$E_{s}(q)=E_{exc}^{(f)}(s)+\frac{\hbar^{2}q^{2}}{2M_{ex}(s)}\,,$$

(7)

where the first term is the excitation energy of the $f$-internal-state exciton, given by $E_{exc}^{(f)}(s)\!=\!E_{g}(s)+E_{b}^{(f)}(s)$ with $E_{b}^{(f)}(s)$ being its (negative) binding energy and $E_{g}$ representing the band gap, and the second term is the translational kinetic energy of the exciton with the total effective mass $M_{ex}$ (the sum of the subband-dependent electron and hole effective masses). The exciton on the $n$-th SWCN of the array is produced by the absorption of a photon of the external EM radiation. This initiates the interband dipole electronic transitions $\langle s,f|\hat{\textbf{d}}(\bm{\rho}_{n})|0\rangle$ to create the dipole polarization, an observable quantity proportional to $|\!\sum_{f,\bm{\rho}_{n}}\!\langle s,f|\hat{\textbf{d}}(\bm{\rho}_{n})|0\rangle|^{2}$, mainly along the CN axis. Only the ground-internal-state (longest-lived) excitons are considered in what follows and so the $f$-index will be omitted for brevity.

The total Hamiltonian of our model system in Fig. 1 can now be written as

$$\hat{H}=\hat{H}_{ex}\!+\hat{H}_{int}\,,$$

(8)

where the second term is responsible for the collective excitations of the SWCN array originally created by the external EM radiation on an individual constituent SWCN. To obtain this part, we perform the series expansion of the electron interaction potential (1) about the equilibrium positions of the electron density distribution. Introducing $\bm{\rho}_{n}\!+\bm{\beta}_{n}$ for the electron ring position displaced by a vector $\bm{\beta}_{n}\!=\!\bm{\beta}(\bm{\rho}_{n})$ from the point $\bm{\rho}_{n}$ of the $n$-th SWCN electron density distribution (and same for ${\bm{\rho}}_{l}^{\prime}$; see Fig. 1), the exponential factor in Eq. (1) is expanded to the third nonvanishing term assuming that $|{\bm{\rho}}_{n}\!-{\bm{\rho}}_{l}^{\prime}|\!>\!|\bm{\beta}_{n}\!-\bm{\beta}_{l}^{\prime}|$, to give

$$e^{i\textbf{k}\cdot\bm{\rho}_{nl}}\big{[}1+i\textbf{k}\cdot\bm{\beta}_{nl}-\frac{1}{2}\big{(}\textbf{k}\cdot\bm{\beta}_{nl}\big{)}^{2}\big{]}.$$

Here, $\bm{\rho}_{nl}\!=\!\bm{\rho}_{n}-\bm{\rho}^{\prime}_{l}$ and $\bm{\beta}_{nl}\!=\bm{\beta}_{n}-\bm{\beta}^{\prime}_{l}$ with the primes still to indicate that the self-interaction in the $n\!=l$ case is to be excluded. For achiral $(m,0)$ and $(m,m)$ type SWCNs (zigzag and armchair type, respectively) we choose to focus on here, the second term in square brackets vanishes due to the presence of the inversion symmetry. This is generally not true for chiral SWCNs, where it is nonzero and turns out to be the most important one. This case will be addressed elsewhere separately.

The extra electron potential energy that comes out of Eq. (1) due to the relative electron ring displacement $\bm{\beta}_{nl}$, is given by $\sum_{\bm{\rho}_{n\!},\bm{\rho}_{l}^{\prime}}\!\!\big{[}V(\bm{\rho}_{nl}+\bm{\beta}_{nl})-V(\bm{\rho}_{nl})\big{]}$ where the summation runs over all tubules of the array and over their individual unit cells with the self-interaction excluded. With the series expansion above and the self-interaction terms dropped, this becomes

$$\frac{m^{\ast}}{N_{\rm 2D}LL_{\perp}}\!\!\!\!\!{\sum_{~{}~{}~{}\textbf{k},\bm{\rho}_{n\!},\bm{\rho}_{l}^{\prime}}}\!\!\!\!\!e^{i\textbf{k}\cdot\bm{\rho}_{nl}}\Big{[}\frac{\omega_{p}(k)}{k}\Big{]}^{2}\big{(}\textbf{k}\cdot\bm{\beta}_{n}\big{)}\big{(}\textbf{k}\cdot\bm{\beta}_{l}^{\prime}\big{)},$$

and can further be rewritten as follows

	$\displaystyle\frac{1}{v_{0}}\hskip-17.07182pt\sum_{~{}~{}~{}~{}~{}\textbf{k},\bm{\rho}_{n\!},\bm{\rho}_{l}^{\prime},\mu,\nu}\hskip-19.91684pte^{i\textbf{k}\cdot\bm{\rho}_{n}}\sqrt{\frac{m^{\ast\,}d}{N_{\rm 2D}N_{\perp}N}}\;\omega_{p}(k)\Big{[}\beta_{n\mu}T_{\mu\nu}(\textbf{k})\beta_{l\nu}^{\prime}\hskip 14.22636pt$		(9)
	$\displaystyle+\,\frac{1}{2}\,\beta_{n\mu}\beta_{l\nu}^{\prime}\delta_{\mu\nu}\Big{]}\sqrt{\frac{m^{\ast\,}d}{N_{\rm 2D}N_{\perp}N}}\;\omega_{p}(k)\,e^{-i\textbf{k}\cdot\bm{\rho}_{l}^{\prime}}.\hskip 25.6073pt$

Here, $v_{0}\!=a\Delta d$ is the elementary cell volume of the array and the second-rank tensor

$$T_{\mu\nu}(\textbf{k})=\frac{k_{\mu}k_{\nu}}{k^{2}}-\frac{\delta_{\mu\nu}}{2}\,,~{}~{}~{}\mu,\nu=x,y$$

(10)

represents the Fourier transform of the pair dipole-dipole interaction between the SWCNs in the array [77]. The second term in the square brackets equals zero. This can be proven by first summing it up over $\bm{\rho}_{l}^{\prime}$ using Eq. (3), followed by summation over k, which leads to $\omega_{p}^{2}(0)\!=\!0$.

The dipole-dipole interaction energy in Eq. (9) can be used to obtain the $\hat{H}_{int}$ part of the total Hamiltonian (8). Consider the real vector quantity

$$\textbf{d}(\bm{\rho}_{n})\!=\!\sqrt{\!\frac{m^{\ast\,}d}{4\pi N_{\rm 2D}}}\,\omega_{p}(k)\bm{\beta}(\bm{\rho}_{n})\!=\!\sqrt{\!\frac{m^{\ast\,}d}{4\pi N_{\rm 2D}N}}\,\omega_{p}(k)\beta\textbf{e}(\bm{\rho}_{n})$$

(11)

with the unit vector $\textbf{e}(\bm{\rho}_{n})\!=\!\bm{\beta}(\bm{\rho}_{n})/\beta(\bm{\rho}_{n})$ to define the direction of the electron ring displacement $\bm{\beta}(\bm{\rho}_{n})$ and

$$\beta=\sqrt{|\!\sum_{\bm{\rho}_{n}}\bm{\beta}(\bm{\rho}_{n})|^{2}}\approx\sqrt{\sum_{\bm{\rho}_{n}}|\bm{\beta}(\bm{\rho}_{n})|^{2}}=\sqrt{N}\beta(\bm{\rho}_{n})$$

to represent the net electron density displacement (consistent with the random phase approximation and equivalency of rings) for the entire $n$-th SWCN. Let Eq. (11) be a matrix element $\langle s,\bm{\rho}_{n}|\hat{\textbf{d}}(\bm{\rho}_{n})|0\rangle$ of the (Hermitian) induced dipole moment operator defined in the second quantization form as

	$\displaystyle\hat{\textbf{d}}(\bm{\rho}_{n})\!=\!\sqrt{\!\frac{m^{\ast\,}d}{4\pi N_{\rm 2D}N}}\,\omega_{p}(k)\!\sum_{s}\langle s|\hat{\bm{\beta}}|0\rangle\big{(}B_{s,\bm{\rho}_{n}}\!\!+\!B^{\dagger}_{s,\bm{\rho}_{n}}\big{)},\hskip 14.22636pt$		(12)
	$\displaystyle\langle s|\hat{\bm{\beta}}|0\rangle^{\dagger}\!=\langle 0|\hat{\bm{\beta}}|s\rangle^{\ast}\!=\langle 0|\hat{\bm{\beta}}|s\rangle\!=\langle s|\hat{\bm{\beta}}|0\rangle,\hskip 39.83368pt$

where $\hat{\bm{\beta}}$ is the quantum displacement operator to represent the classical displacement $\beta\textbf{e}(\bm{\rho}_{n})$. Then the Fourier transform

$$\hat{\textbf{d}}(\textbf{k})=\frac{1}{\sqrt{N_{\perp}N}}\sum_{\bm{\rho}_{n}}\hat{\textbf{d}}(\bm{\rho}_{n})e^{-i\textbf{k}\cdot\bm{\rho}_{n}},~{}~{}\hat{\textbf{d}}^{\dagger}(\textbf{k})\!=\hat{\textbf{d}}(-\textbf{k}),$$

obtained using Eqs. (2) and (3), takes the form

	$\displaystyle\hat{\textbf{d}}(\textbf{k})\!=\sqrt{\!\frac{m^{\ast\,}d}{4\pi N_{\rm 2D}N}}\,\omega_{p}(k)\!\sum_{s}\langle s|\hat{\bm{\beta}}|0\rangle\big{(}B_{s,\textbf{k}}\!+B^{\dagger}_{s,-\textbf{k}}\big{)},\hskip 14.22636pt$		(13)
	$\displaystyle B_{s,\textbf{k}}=\frac{1}{\sqrt{N_{\perp}N}}\sum_{\bm{\rho}_{n}}B_{s,\bm{\rho}_{n}}e^{-i\textbf{k}\cdot\bm{\rho}_{n}}\hskip 42.67912pt$

with the creation and annihilation operators of the same meaning as those in Eq. (5), and Eq. (9) suggests that

$$\hat{H}_{int}=\frac{4\pi}{v_{0}}{\sum_{\textbf{k},\mu,\nu}}\hat{d}_{\mu}(\textbf{k})T_{\mu\nu}(\textbf{k})\hat{d}_{\nu}(-\textbf{k}).$$

(14)

More explicitly

	$\displaystyle\hat{H}_{int}=\frac{1}{2}\!\sum_{\textbf{k},s,s^{\prime}}\!\!V_{ss^{\prime}}(\textbf{k})\big{(}B_{s,\textbf{k}}\!+\!B^{\dagger}_{s,-\textbf{k}}\big{)}\big{(}B_{s^{\prime}\!,-\textbf{k}}\!+\!B^{\dagger}_{s^{\prime}\!,\textbf{k}}\big{)},\hskip 14.22636pt$		(15)
	$\displaystyle V_{ss^{\prime}}(\textbf{k})=\frac{2m^{\ast}\omega_{p}^{2}(k)d}{N_{\rm 2D}Nv_{0}}\sum_{\mu,\nu}\langle 0|\hat{\beta}_{\mu}|s\rangle T_{\mu\nu}(\textbf{k})\langle s^{\prime}|\hat{\beta}_{\nu}|0\rangle.\hskip 21.33955pt$

Here, different spatial component matrix elements of the displacement operator $\hat{\bm{\beta}}$ in $V_{ss^{\prime}}$ are different, in general.

In the case where the (dominant) longitudinal induced SWCN polarization is only taken into consideration and the transverse one is neglected, one has $\beta_{n\mu}\!=\beta_{ny}\delta_{y\mu}$ and $\beta_{l\nu}^{\prime}\!=\beta_{ly}^{\prime}\delta_{y\nu}$ in Eq. (9). This, after summing up over $x_{n}$ and $x_{l}^{\prime}$ with Eq. (3), projects it on the $y$-direction to give

	$\displaystyle\frac{1}{v_{0}}\!\sum_{q,y_{n\!},y_{l}^{\prime}}\!e^{iq\cdot y_{n}}\sqrt{\frac{m^{\ast\,}d}{N_{\rm 2D}N}}\,\omega_{p}(q)\,\beta_{ny}\hskip 71.13188pt$
	$\displaystyle\hskip 71.13188pt\times\,T_{yy}(q)\,\beta_{ly}^{\prime}\sqrt{\frac{m^{\ast\,}d}{N_{\rm 2D}N}}\,\omega_{p}(q)\,e^{-iq\cdot y_{l}^{\prime}},$

where $T_{yy}(q)\!=T_{yy}(q,k_{\perp}\!\!=\!0)=1/2$ according to Eq. (10) and $\omega_{p}(q)\!=\omega_{p}(q,k_{\perp}\!\!=\!0)$, thus yielding

	$\displaystyle\hat{H}_{int}=\frac{2\pi}{v_{0}}{\sum_{q}}\,\hat{d}_{y}(q)\hat{d}_{y}(-q)=\frac{2\pi}{v_{0}}{\sum_{q}}\,\hat{d}_{y}(q)\hat{d}_{y}^{\,{\dagger}}(q),\hskip 25.6073pt$		(16)
	$\displaystyle\hat{d}_{y}(q)\!=\sqrt{\!\frac{m^{\ast}d}{4\pi N_{\rm 2D}N}}\,\omega_{p}(q)\!\sum_{s}\,\langle s|\hat{\beta}_{y}|0\rangle\big{(}B_{s,q}\!+B^{\dagger}_{s,-q}\big{)}\hskip 14.22636pt$

for Eq. (14) and

	$\displaystyle\hat{H}_{int}=\frac{1}{2}\sum_{s,q}V_{ss}(q)\big{(}B_{s,q}\!+B^{\dagger}_{s,-q}\big{)}\big{(}B_{s,-q}\!+B^{\dagger}_{s,q}\big{)},\hskip 14.22636pt$		(17)
	$\displaystyle V_{ss}(q)=\frac{m^{\ast}\omega_{p}^{2}(q)d}{N_{\rm 2D}Nv_{0}}\,|\langle s|\hat{\beta}_{y}|0\rangle|^{2}.\hskip 49.79231pt$

for Eq. (15), respectively. In this latter case, $V_{ss^{\prime}}$ takes the diagonal form $V_{ss}\delta_{ss^{\prime}}$ to only include the collinear-anticollinear dipole moment coupling between the nanotubes of the array. This is the case of practical importance we focus on in what follows.

With the interaction term of Eq. (17), the total Hamiltonian (8) of the SWCN array is of the form

	$\displaystyle\hat{H}=\!\sum_{s,q}\Big{[}E_{s}(q)B^{\dagger}_{s,q}B_{s,q}\hskip 34.14322pt$		(18)
	$\displaystyle+\,\frac{1}{2}\,V_{ss}(q)\big{(}B_{s,q}\!+B^{\dagger}_{s,-q}\big{)}\big{(}B_{s,-q}\!+B^{\dagger}_{s,q}\big{)}\Big{]},$

which can be diagonalized exactly by means of the canonical transformation technique [85]. More specifically, the Hamiltonian such as this can be brought to the form

$$\hat{H}=\!\sum_{s,q}\hbar\omega_{s}(q)\xi_{s}^{\dagger}(q)\xi_{s}(q)+E_{0}$$

(19)

by redefining the original exciton operators as follows

	$\displaystyle B_{s,q}\!=u_{s}(q)\,\xi_{s}(q)+v_{s}(q)\,\xi^{\dagger}_{s}(-q),$		(20)
	$\displaystyle|u_{s}(q)|^{2}-|v_{s}(q)|^{2}=1,\hskip 28.45274pt$		(21)

in terms of the linear superposition of the collective boson excitation operators $\xi_{s}(q)$ representing the eigen states of the entire system, the array. The reciprocal of this is

	$\displaystyle\xi_{s}(q)=u^{\ast}_{s}(q)B_{s,q}-v_{s}(q)B^{\dagger}_{s,-q}$		(22)
	$\displaystyle\big{[}\xi_{s}(q),\xi^{\dagger}_{s^{\prime}}(q^{\prime})\big{]}=\delta_{ss^{\prime}}\delta_{qq^{\prime}}.\hskip 14.22636pt$		(23)

The unknown transformation coefficients $u_{s}(q)$ and $v_{s}(q)$, and the eigen state energies $\hbar\omega_{s}(q)$ can be found from the operator identity

$$\hbar\omega_{s}(q)\xi_{s}(q)=\big{[}\xi_{s}(q),\,\hat{H}\big{]},$$

(24)

followed by finding the vacuum (no excitations present) energy $E_{0}$ from the comparison of Eqs. (18) and (19) after the eigen state energies and the transformation coefficients have been obtained.

Putting Eqs. (22) and (18) into Eq. (24) and equating the coefficients in front of the exciton creation and annihilation operators on the left and right side, one obtains the set of two simultaneous algebraic equations as follows

	$\displaystyle\big{[}\hbar\omega_{s}(q)-E_{s}(q)-V_{ss}(q)\big{]}u_{s}(q)-V_{ss}(q)v_{s}(q)=0,\hskip 14.22636pt$
			(25)
	$\displaystyle V_{ss}(q)u_{s}(q)+\big{[}\hbar\omega_{s}(q)+E_{s}(q)+V_{ss}(q)\big{]}v_{s}(q)=0.\hskip 15.6491pt$

Being supplemented by the extra constraint (21), these simultaneous equations define the transformation coefficients $u_{s}(q)$ and $v_{s}(q)$ uniquely. The condition of the nontrivial solution existence requires that the determinant of Eq. (25) be equal to zero. This gives the collective excitation eigen energies in the form

$$\hbar\omega_{s}(q)=\sqrt{E_{s}^{2}(q)+2E_{s}(q)V_{ss}(q)}$$

(26)

with $E_{s}(q)$ given by Eq. (7), $V_{ss}(q)$ defined in Eq. (17), and $s\!=\!1,2,...,m\,(\geq\!n)$ representing the excitation subbands for the periodic array composed of the $(m,n)$ type SWCNs. Using Eq. (26), the transformation coefficients come out of Eq. (25) in the compact form as follows

$$u_{s}(q)\!=\!\frac{E_{s}(q)\!+\!\hbar\omega_{s}(q)}{2\sqrt{E_{s}(q)\hbar\omega_{s}(q)}},~{}v_{s}(q)\!=\!\frac{E_{s}(q)\!-\!\hbar\omega_{s}(q)}{2\sqrt{E_{s}(q)\hbar\omega_{s}(q)}}\,.$$

(27)

Finally, substituting Eq. (22) in Eq. (19), using Eqs. (26) and (27) for simplifications, and comparing the end result with Eq. (18), one obtains the vacuum state energy of the array in the form

$$E_{0}=\frac{1}{2}\sum_{s,q}\big{[}\hbar\omega_{s}(q)-E_{s}(q)\big{]}.$$

(28)

Equations (19), (22), (23), and (26)-(28) solve the dispersion relation problem for the intrinsic collective excitations in the ultrathin periodic SWCN array. One can now see that the excited eigen states of the SWCN array involve not only the single-tube excitons of energy $E_{s}(q)$ as one might expect naturally, but also the array collective plasma oscillations of energy $\hbar\omega_{p}(q)$ given by Eq. (4) are involved, which in the TD regime can be effectively controlled by varying the array thickness [53].

We proceed with the polarizability tensor calculation for the SWCN array within the Matsubara formalism. This is given by the correlator of the SWCN induced dipole moment operator in Eq. (16) with itself, and this only includes one CN related component, $P_{yy}(q,i\omega)$, since the minor polarization perpendicular to the CN axis is agreed to be neglected. Specifically,

$$P_{yy}(q,i\omega)=-\frac{1}{\hbar}\int_{0}^{\hbar\beta}\!\!\!\!\!\!\!d\tau\,e^{i\omega\tau}\langle T_{\tau}\hat{d}_{y}(q,\tau)\hat{d}_{y}(-q,0)\rangle,$$

(29)

where the bracket $\langle\,\cdots\rangle$ notates the thermodynamical averaging, $\tau\,(=\!it)$ is the complex time, and $T_{\tau}$ is the ordering operator to place the (Heisenberg picture) dipole moment operators with the earliest $\tau$ to the right.

A straightforward way to calculate Eq. (29) is by using Eq. (20) to write the single-tube induced dipole operators in terms of their collective excitation counterparts, followed by the thermodynamical averaging with the total Hamiltonian (19) in the collective excitation Hilbert space. However, for practical use purposes it is instructive to find the connection between the single-tube excitations and the collective excitations of the entire SWCN array. To this end, we first do the averaging in Eq. (29) with the unperturbed Hamiltonian (5) to obtain the noninteracting SWCN array polarizability (see Appendix A)

$$P_{yy}^{(0)}(q,i\omega)=\frac{m^{\ast}\omega^{2}_{p}(q)d}{4\pi N_{\rm 2D}N}\sum_{s}\frac{2E_{s}|\langle s|\hat{\beta}_{y}|0\rangle|^{2}}{(i\hbar\omega)^{2}-E_{s}^{2}}\,.$$

(30)

Introducing the CN dynamical polarizability function

$$\alpha_{yy}(q,i\omega)=-\frac{e^{2}}{L}\sum_{s}\frac{2E_{s}|\langle s|\hat{\beta}_{y}|0\rangle|^{2}}{(i\hbar\omega)^{2}-E_{s}^{2}}$$

(31)

links Eq. (30) to the optical response of an isolated CN (longitudinal polarizability per unit length) to give

$$P_{yy}^{(0)}(q,i\omega)=-\frac{m^{\ast}\omega^{2}_{p}(q)v_{0}}{4\pi e^{2}N_{\rm 2D}\Delta}\,\alpha_{yy}(q,i\omega).$$

(32)

Alternatively, this can be expressed in an equivalent form in terms of the isolated CN longitudinal conductivity [28]

$$\sigma_{yy}(q,i\omega)=\frac{\hbar e^{2}}{2\pi RL}\sum_{s}\frac{-2\hbar\omega|\langle s|\hat{v}_{y}|0\rangle|^{2}}{E_{s}\big{[}(i\hbar\omega)^{2}-E_{s}^{2}\big{]}}\,.$$

(33)

Using the relations between the velocity and coordinate displacement matrix elements

$$\langle s|\hat{v}_{y}|0\rangle\!=\!\langle s|\frac{d\hat{\beta}_{y}}{d\tau}|0\rangle\!=\!-\frac{1}{\hbar}\langle s|[\hat{\beta}_{y},\hat{H}_{ex}]|0\rangle\!=\!\frac{E_{s}}{\hbar}\,\langle s|\hat{\beta}_{y}|0\rangle$$

(34)

and the fact that [75, 76]

$$\alpha_{yy}(q,i\omega)=\frac{2\pi R}{\omega}\,\sigma_{yy}(q,i\omega),$$

(35)

one then obtains

$$P_{yy}^{(0)}(q,i\omega)=-\frac{m^{\ast}\omega^{2}_{p}(q)v_{0}}{4\pi e^{2}N_{\rm 2D}\Delta}\frac{2\pi R}{\omega}\,\sigma_{yy}(q,i\omega).$$

(36)

Next, being performed with the total Hamiltonian (8), the thermodynamical averaging in Eq. (29) gives the collective polarizability of the dipole-dipole coupled SWCN array. This can be evaluated by a diagrammatic expansion in which the first term, given by Eq. (5), is treated as unperturbed and the second, given by Eqs. (14)-(17), is the perturbation. For the former, the noninteracting SWCN array polarizability is already known and given by Eqs. (32) and (36). An important feature of the latter is its separability, as can be seen from Eqs. (14) and (16), so that the perturbation factors into components summed up individually. The self-energy term for the perturbation occurs in the first order with the self-energy merely equal to $2\pi/v_{0}$, as per Eq. (16). The higher-order terms of the diagrammatic expansion just produce multiples of this term. The overall result takes the form of the Dyson equation as follows (see Appendix B)

$$P_{yy}(q,i\omega)=P_{yy}^{(0)}(q,i\omega)+\frac{2\pi}{v_{0}}P_{yy}^{(0)}(q,i\omega)P_{yy}(q,i\omega),$$

which can be easily solved to give

$$P_{yy}(q,i\omega)=\frac{P_{yy}^{(0)}(q,i\omega)}{1-2\pi P_{yy}^{(0)}(q,i\omega)/v_{0}}\,.$$

(37)

This is the collective array polarization associated with the interband electronic transitions to create excitons on individual SWCNs embedded in a finite-thickness dielectric layer. There is also the low-energy polarization part contributed by the intraband transitions of harmonically bound charges on the CN surface [39, 53]. The total collective polarization of the CN array is to include both intra- and interband term on equal footing since both of them are associated with an induced oscillating dipole moment — in the lower- and higher-frequency spectral range, respectively, — initially created by external EM radiation on an individual CN and spread out over the array thereafter. This can be formally done by redefining

$\displaystyle\sigma_{yy}(q,i\omega)\longrightarrow\sigma_{yy}^{intra}(q,i\omega)+\sigma_{yy}^{inter}(q,i\omega)$

(38)

in Eq. (36), where

$$\sigma_{yy}^{intra}(q,i\omega)=\frac{e^{2}N_{\rm 2D}}{m^{\ast}}\frac{\omega}{\omega^{2}+(v_{F}q)^{2}}$$

(39)

generally represents the single-tube intraband conductivity term with $v_{F}$ being the Fermi velocity [87], and $\sigma_{yy}^{inter}$ is the interband term still given by Eq. (33). The SWCN polarizability in Eq. (32) has to be redefined per Eqs. (35) and (38) as well.

Using Eq. (37) with $P_{yy}^{(0)}$ of Eq. (36) in the general EM response expression (Gaussian units)

$$\frac{\varepsilon_{yy}(q,i\omega)}{\epsilon}=1+\frac{4\pi}{v_{0}}\,P_{yy}(q,i\omega),$$

(40)

after simplifications one obtains the SWCN array dynamical dielectric response in the form

$$\frac{\varepsilon_{yy}(q,i\omega)}{\epsilon}=1-\!\frac{2f_{\rm CN\,}\sigma_{yy}(q,i\omega)}{f_{\rm CN\,}\sigma_{yy}(q,i\omega)+\omega e^{2}N_{\rm 2D}R/m^{\ast}\omega^{2}_{p}(q)d}\,,$$

where $f_{\rm CN}\!=\!N_{\perp}V_{\rm CN}/V\!=\!\pi R^{2}/\Delta d\,$ is the CN volumetric fraction in the planar dielectric film as sketched in Fig. 1. The analytic continuation into the real frequency domain of our interest here can now be obtained by changing $i\omega$ to $\omega+i\delta$ in the above, to finally result in

$$\frac{\varepsilon_{yy}(q,\omega)}{\epsilon}\!=\!1\!-\!\frac{2f_{\rm CN\,}\sigma_{yy}(q,\omega)}{f_{\rm CN\,}\sigma_{yy}(q,\omega)+i\omega e^{2}N_{\rm 2D}R/m^{\ast}\omega^{2}_{p}(q)d}$$

(41)

with the longitudinal conductivity of an isolated SWCN to include both intra- and interband contributions, taking per Eqs. (38), (39) and (33) the form

	$\displaystyle\sigma_{yy}(q,\omega)\!=\!\sigma_{yy}^{intra}\!+\sigma_{yy}^{inter}\!=i\frac{e^{2}N_{\rm 2D}}{m^{\ast}}\frac{\omega+i\delta}{(\omega+i\delta)^{2}-(v_{F}q)^{2}}$
	$\displaystyle+\frac{\hbar e^{2}}{2\pi RL}\sum_{s}\frac{-2i\hbar\omega|\langle s|\hat{v}_{y}|0\rangle|^{2}}{E_{s}\big{[}E_{s}^{2}-(\hbar\omega)^{2}-2i\hbar^{2}\omega\delta\big{]}}\,.\hskip 42.67912pt$		(42)

Here, $\delta$ takes the meaning of a (phenomenological) inverse energy relaxation time, $\delta\!=\!1/\tau_{r}$, associated with the inelastic (phonon, defect) scattering [30, 88] and can generally be different for intra- and interband processes.

Under continuous low-intensity light illumination, the dielectric film with the SWCN array embedded in it can still be treated as being at the thermal equilibrium. At not too low temperatures the distribution of the $s$-subband quasiparticle excitations over the $q$-momentum space is then given by the normalized Boltzmann distribution function

$$f_{s}(q,T)=\frac{1}{Q_{s}}\,e^{-\beta\hbar\omega_{s}(q)}$$

(43)

with $\hbar\omega_{s}(q)$ representing the energy of the excited quasiparticle eigen states of the SWCN array given by Eq. (26) and the normalization factor is

$$Q_{s}=\sum_{q}e^{-\beta\hbar\omega_{s}(q)}=\frac{L}{\pi\hbar}\sqrt{\!\frac{M_{s}}{2}}\!\int_{\!E_{exc}}^{\infty}\!\!\frac{dE\,e^{-\beta\hbar\omega_{s}(q)}}{\sqrt{E-E_{exc}(s)}}$$

with $q=\!\sqrt{2M_{s}\,[E-\!E_{exc}(s)]}/\hbar$ to be used in the integral. The thermal averaging of Eq. (41) with this distribution function gives the temperature dependence of the EM response of the SWCN array in the form

	$\displaystyle\varepsilon_{yy}(T,\omega)=\sum_{q}f_{s}(q,T)\,\varepsilon_{yy}(q,\omega)\hskip 8.5359pt$		(44)
	$\displaystyle=\frac{L}{\pi\hbar}\sqrt{\!\frac{M_{s}}{2}}\!\int_{\!E_{exc}}^{\infty}\!\!\!\frac{dE\,f_{s}(q,T)\,\varepsilon_{yy}(q,\omega)}{\sqrt{E-E_{exc}(s)}}\,.$

Equations (41)-(44) and (26) provide the complete description of the spatially anisotropic linear EM response of the periodically aligned SWCN array in the TD regime. This is generally a two-component spatially dispersive (nonlocal) tensor of the form

$$\hat{\varepsilon}(q,\omega)=\left[\begin{array}[]{cc}\varepsilon_{xx}&0\\ 0&\varepsilon_{yy}\end{array}\right]=\left[\begin{array}[]{cc}\epsilon&0\\ 0&\varepsilon_{yy}(q,\omega)\end{array}\right],$$

(45)

which should be averaged per Eq. (44) to represent the thickness-dependent anisotropic dynamical response to continuous low-intensity light illumination. In practice, the CN-array-embedding dielectric layer may still have inclusions of non-identical parallel aligned CNs. Assuming their (quasi-)periodic in-plane distribution, these inhomogeneities in an otherwise homogeneous SWCN array can be accounted for by means of the Maxwell-Garnett (MG) mixing method [78], whereby under continuous illumination the $yy$-component of Eq. (45) takes the form

$$\left<\varepsilon_{yy}(T,\omega)\right>=\sum_{\alpha=1}^{n}w_{\alpha}\varepsilon_{yy}^{(\alpha)}(T,\omega)\,.$$

(46)

Here, $\varepsilon_{yy}^{(\alpha)}(T,\omega)$ refers to a thermally averaged homogeneous component of weight

$$w_{\alpha}=\frac{f^{(\alpha)}_{\rm CN}}{\sum_{i=1}^{n}f^{(i)}_{\rm CN}}=\left[\sum_{i=1}^{n}\!\left(\frac{R_{i}}{R_{\alpha}}\right)^{\!\!2}\!\frac{\Delta_{\alpha}}{\Delta_{i}}\right]^{-1}$$

(47)

in the generally $n$-component inhomogeneous mixture of periodic (or quasiperiodic) SWCN arrays with radii $R_{\alpha}$ and intertube separation distances $\Delta_{\alpha}$, which is embedded in the finite-thickness dielectric layer. The weights are defined to give $\sum_{\alpha=1}^{n}w_{\alpha}\!=\!1$, accordingly.