## Abstract

We studied theoretically the absorption of acoustic phonons in the hypersound regime in Fluorine modified carbon nanotube (F-CNT) Γ q F − CNT and compared it to that of undoped single walled carbon nanotube (SWCNT) Γ q SWCNT . Per the numerical analysis, the F-CNT showed less absorption to that of SWCNT, thus ∣ Γ q F − CNT ∣ < ∣ Γ q SWCNT ∣ . This is due to the fact that Fluorine is highly electronegative and weakens the walls of the SWCNT. Thus, the π -electrons associated with the Fluorine causes less free charge carriers to interact with the phonons and hence changing the metallic properties of the SWCNT to semiconductor by the doping process. From the graphs obtained, the ratio of hypersound absorption in SWCNT to F-CNT at T = 45 K is Γ SWCNT Γ F − CNT ≈ 29 while at T = 55 K , is Γ SWCNT Γ F − CNT ≈ 9 and at T = 65 K , is Γ SWCNT Γ F − CNT ≈ 2 . Clearly, the ratio decreases as the temperature increases.

### Keywords

- carbon nanotube
- fluorinated
- acoustic effects
- hypersound

## 1. Introduction

Acoustic effects in bulk and low dimensional materials have attracted lots of attention recently. This is due to the need of finding coherent acoustic phonons for scientific applications as against the use of conventional direct current [1]. Materials such as homogenous semiconductors, superlattices (SL), graphene and carbon nanotubes (CNT) are good candidates for such studies due to their novel properties such as the high scattering mechanism, the high-bias mean-free path (

Carbon nanotubes (CNTs), on the other hand, are cylindrical hollow rod of graphene sheets whose electronic structures are determined by the localized

where

This paper is organized as follows: In Section 2, the absorption coefficient for F-CNT and SWCNT are calculated. In Section 3, the final equations are analyzed numerically and presented graphically. Section 4 presents the conclusion of the study.

## 2. Theory

Fluorination plays a significant role in the doping process, as it provides a high surface concentration of functional groups, up to

The width for the F-(

where

where

Eq. (4) can be expanded as

where

where

For

After much simplification, the phonon transition rate in the presence of the electromagnetic reduces to

that is, the imaginary part of the polarization vector. In Eq. (8)

In the region of an intense laser field, i.e.,

Taking the sum over

where

Substituting Eq. (12) into Eq. (11) and also converting the summation over

The matrix element of the electron-phonon interaction is given as

where

The electron distribution function is obtained by obtained by solving the Boltzmann transport equation in the presence of external electric field

and has a solution of

and

where

where

where

where

From the conservation laws, the momentum (

By substituting

where

and

To compare the result with an undoped SWCNT, we follow the same procedure as that of F-CNT. Using the tight-binding energy dispersion of the

where

where

Using Eq. (15), the absorption in SWCNT is calculated as

where

## 3. Results and discussions

In this formulation, we consider a novel concept of monochromatic acoustic phonon amplification at the THz frequencies regime. Impulsive phonon excitation by a femtosecond optical pulse generates coherent FCNT and SWCNT phonons propagating in the forward and backward direction along the FCNT and SWCNT axis, that is setting up an stationary acoustic wave. Interaction of the propagating acoustic wave with an electrically driven intraminiband transition electron current allows for phonon absorption, connected with electron transitions between states within an electronic miniband. The intravalley or intraminiband character of the electron transport allows for much higher currents than interminiband electron or electron tunneling and thus, a much stronger phonon absorption.

The general expressions for the hypersound absorption in F-CNT (

In order to put our observations in perspective, we display Figures 4 and 5 in a three-dimensional behavior of the sound coefficient as a function of the frequency (

## 4. Conclusion

Theoretical investigation of strong absorption of coherent acoustic phonons in an FCNT and SWCNT at low temperature utilizing the Boltzmann’s transport equation is carried out in the regime