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# Absorption of Acoustic Phonons in Fluorinated Carbon Nanotubes with Non-Parabolic, Double Periodic Band

By Daniel Sakyi-Arthur, S. Y. Mensah, N. G. Mensah, Kwadwo A. Dompreh and R. Edziah

Submitted: February 27th 2018Reviewed: May 2nd 2018Published: December 12th 2018

DOI: 10.5772/intechopen.78231

## 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 (l) and their sizes which enable strong electron-phonon interaction to occur in them resulting in acoustic phonon scattering. Acoustic waves through these materials are characterized by a set of elementary resonance excitations and dynamic nonlinearity which normally leads to an absorption (or amplification), acoustoelectric effect (AE) [2], and acoustomagnetoelectric effect (AME) [3, 4]. The concept of acoustic wave amplification was first predicted in bulk materials [5], and later in n-Ge [6]. In SLs, Mensah et al. [7] studied hypersound absorption (amplification) and established its use as a phonon filter, and in [8], predicted the use of the SL as a hypersound generator which was confirmed in [1]. In Graphene, Nunes et al. [9] treated theoretically hypersound amplification, but Dompreh et al. [10] further proved that absorption also occurs in the material. Experimentally, Miseikis et al. [11] and Bandhu and Nash [12] have studied acoustoelectric effect in Graphene.

Carbon nanotubes (CNTs), on the other hand, are cylindrical hollow rod of graphene sheets whose electronic structures are determined by the localized π-electrons in the sp2- hybridized bonds. Absorption (Amplification) of hypersound in undoped CNT has been carried out theoretically by Dompreh et al. [13, 14] and experimentally by [15, 16]. Other forms of research such as hot-electron effect [17], thermopower in CNT [18] have been carried out. Flourine-modified CNT (F-CNT) is off-late attracting a lot of scientific interest. This is attained by doping the CNT with Fluorine thus forming double periodic band CNT changing from metallic to semiconductor. As per the studies conducted by Jeon et al. [19], absorption in F-CNT is less than that of SWCNT but no studies have been done on the absorption of F-CNT in the hypersound regime. In this paper, the study of absorption of acoustic phonons in metallic SWCNT and F-CNT are theoretically studied. Here, the acoustic wave considered has wavelength λ=2π/q, smaller than the mean-free path of the CNT and then treated as a packet of coherent phonons (monochromatic phonons) having a δ-function distribution as

Nk=2π3ωqvsΦδkqE1

where kis the phonon wavevector, is the Planck’s constant divided by 2π, and Φis the sound flux density, and ωqand vsare respectively the frequency and the group velocity of sound wave with wavevector q. It is assumed that the sound wave is propagated along the z-axis of the CNT.

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 C2Fwithout destruction of the tube’s physical structure. Doping is an easy, fast, exothermic reaction and the repulsive interactions of the Fluorine atoms on the surface debundles the nanotube, thus enhancing their electron dispersion [20]. Figure 1 shows a one dimensional SWCNT doped with Fluorine atoms [21]. Consider a Fluorine modified CNT (n,n) with the Fluorine atoms forming a one-dimensional chain. A nanotube of this nature is equivalent to a band with unit cell as shown in Figure 2, where bis the bond length (C-C) [22].

The width for the F-(n,n) tube equals nperiods (with a periodic length of 3b), and this unit cell contains N=4n2carbon atoms which is shown in Figure 3 [22]. Figure 3 shows the atomic numbering in the unit cell of the F-(n,n) nanotube. For a conjugated πsystem, in which there is alternation of single and double bonds along a linear chain, the Hückel matrix approximation is employed to determine the electronic energy band. Proceeding as in [8, 23], we employ the Hamiltonian of the electron-phonon system in the FCNT in the second quantization formalism as

H=p,νενppecAtapν+aνν+kωkbk+bk+1Np,kννnckmννkzapν+apk+ngνbk++bkE2

where ν=1,2and for a chemically modified F-CNT, where the Fluorine atoms form a one-dimensional chain, the energy dispersion can be deduced by using the Huckel matrix method where translational symmetry is accounted for in [22] as

εpz=εo+Ξnγ0cos2N1apzE3

where a=3b/2, Ξis a constant, Nis an integer, and εois the minimum energy of the πelectrons within the first Brillouin zone. For N=2, the energy dispersion for F-CNT at the Fermi surface at the edge of the Brillouin zone is

εpz=απ+8γocos3apzE4

Eq. (4) can be expanded as

εpz=εo+Δ1cos3apz+Δ2cosapzE5

where εois the electron energy in the first Brillouin zone with momentum po, i.e., π/apoπ/a, Δ1=Δ/kBT, Δ2=3Δ/kBTand Δ=2γo. By employing the coulombs gauge, the electromagnetic wave Et=Eosinωtis related to the vector potential Atis the vector potential related to the external electric field of the electromagnetic wave Et=Eosinωtby the relation E=1/cA/tand is directed along the F-CNT tubular axis. ap+and apare the creation and annihilation operators of an electron with quasi-momentum pin the νthminiband respectively, and bk+and bkare the phonon creation and annihilation operators respectively. Nis the number of FCNT periods, g=0,0,2π/dis the FCNT reciprocal vector, and mνν'is given by

mνν'kz=φν'zφνzeikzdzE6

where φνzis the wavefunction of the νthstate in one of the one-dimensional potential wells from which the FCNT potential is formed. The electromagnetic wave frequency is assumed to be large compared with the inverse of the electron mean free time 1/τand the wavelength is taken to be large compared with the FCNT period, electron mean free path and the de Broglie wavelength. This opens the way for us to use the dipole approximation as in [8]. Moreover, the plane electromagnetic wave of frequency ωsatisfies ω/ωp>1, where ωpis the plasma frequency. In the case of the phonons, we confine our considerations to those for which the wavevector q, satisfies the conditions ql1where lis the electron mean free path in FCNT. Such phonons constitute a well-defined elementary excitations of the system.

For ωτ1and ω>ωp, ensures that the electromagnetic wave penetrate the sample and the condition ql1means that the hypersound wavelength is far smaller than the electron mean free path. The phonon dispersion relation then reads as

itbqt=bqHt=ωqbqt+1NCqpmssqzaps+ap+qngstE7

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

Γq=ImΩ=2πΦωqVspz,n=J2ξ×fεnpzfεnpz+qδεnpz+qεnpzωqℓΩE8

that is, the imaginary part of the polarization vector. In Eq. (8) Jxis the Bessel function of order and argument x. It follows from Eq. (8) that if Γq>0we have hypersound attenuation, whereas if Γq<0we have hypersound amplification due to absorption Γq>0and emission Γq<0of photons from the intensified laser field.

In the region of an intense laser field, i.e., ξω, only the electron-phonon collisions with the absorption or emission of 1photons are significant. Accordingly, in the case of ξωthe argument of the Bessel function Jξis large. For large values, the Bessel function Jξis small except when the order is equal to the argument.

ξ=eEoa2ΔqΩ2E9

Taking the sum over using the approximation in Eq. (10)

=J2ξδEℓΩ12δEξ+δE+ξE10

where E=εpz+qεpzωq. Using the Fermi Golden Rule, the phonon transition rate reduces Γq=Un,nacwhere

Un,nac=2πΦωqVspz,pzn,n{|Gpzq,pz|2[f(εn(pzq))f(εn(pz))]δ(εn(pzq)εn(pz)+ωqξ)+|Gpz+q,pz|2[f(εn(pz+q))f(εn(pz))]δ(εn(pz+q)εn(pz)ωq+ξ)}E11

fpz=fεn,npzis the unperturbed distribution function, εn,npzis the energy band, nand ndenotes the quantization of the energy band, and Gpz±qpzis the matrix element of the electron-phonon interaction. Letting pz=pz±qand employing the principle of detailed balance, we assume that scattering into a state pzand out of the state pzis the same, and hence

Gp,p2=Gp,p2E12

Substituting Eq. (12) into Eq. (11) and also converting the summation over pz'into an integral, we obtain

Γq=2πΦωqvs.n,n'Gp,p2fεpzfεpz+qδεpz+qεpzωq+ξdpzE13

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

Gp,p=Λq2σωqE14

where Λis the deformation potential constant, and σis the density of F-CNT. Substituting Eq. (14) into Eq. (13), we obtain

Γq=2πΦωqvsΛq2σωq2n,nfεnpzfεnpz+q×δεnpz+qεnpzωq+ξdpzE15

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

frptt+vp.rfrpt+eEpfrpt=frptfopτE16

and has a solution of

fpz=0dtτexpt/τfopzeaEtE17

and fopzis the Fermi-Dirac distribution given as

fopz=1expεpzμ/kBT+1E18

where μis the chemical potential which ensures the conservation of electrons, kBis the Boltzmann’s constant, Tis the absolute temperature in energy units. Substituting Eqs. (17) and (18) into Eq. (15), we obtain an equation for Γqwhich contains Fermi-Dirac integral of the order 1/2as

F1/2ηf=1Γ1/20ηf1/21+expηηfE19

where EFEc/kBTηf. For nondegenerate electron gas, where the Fermi level is several kBTbelow the energy of the conduction band Ec(i.e., kBTEc), the integral in Eq. (19) approaches 2/πexpηf. Eqs. (18) and (19) then simplifies to

fopz=CexpεpzeaEτ/kBTE20

where Cis the normalization constant to be determined from the normalization condition fpdp=noas

C=3noa22IoΔ1IoΔ2expεoEFkBTE21

where nois the electron density concentration, Tis the absolute temperature in energy units and Ioxis the modified Bessel function of zero order.

From the conservation laws, the momentum (pz) can be deduced from the delta function part of Eq. (15) as

pz=q2+14aarcsinωq12γoaqE22

By substituting pzinto the distribution function in Eq. (15), and after some cumbersome calculations yields

ΓqFCNT=ΓosinhΔ1cos3pasinAsin32aq+Δ2cospasinBsina2q×coshΔ1cos3pacosAcos32aq+Δ2cospacosBcosa2q4Δ2sinpacosBsina2q+Δ1cosAsin3pasin32aq+Δ1Δ2sinpasin3pacosAcosBsina2qsin32aq×sinhΔ1cos3pacosAcos32aq+Δ2cospacosBcosa2q×coshΔ1cos3pasinAsin32aq+Δ2cospasinBsina2qE23

where χ=ωqa/vs, Θis defined to be the Heaviside step function, α=ωq/12γoaq=ωq/6Δ1aq. In the absence of an external electric field

ΓqFCNT=ΓosinhΔ1sin32aqsinA+Δ2sina2qsinB×coshΔ1cos32aqcosA+Δ2cosa2qcosBE24

and

Γo=noa2ΦΛ2qΘ1α248πIo2γoβIo6γoβωq2σvsγo1α2A=34arcsinωq12γoaqB=14arcsinωq12γoaqα=ωq12γoaqE25

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 pzorbital which is given as:

εpz=±γo1+4cosνπncospz3b2+4cos2pz3b2E26

where γo=2.6eVis the hopping integral parameter, b=0.142nmis the C-C bonding distance, and (+) and () signs are respectively the conduction and valence band. When ν=0, the conduction and valence bands cross each other near the Fermi points, pF=±2π/33bgiving the metallic nature to the armchair tube. Putting ν=0, and making the substitution, pz=pz+3po/2in Eq. (26) gives

εpz=±γo12cospz3b2E27

where po=2pF=4π/33b1.7×1010m1see [24]. Eq. (27) is equivalent to the energy dispersion in Eq. (5) when n=1, which is

εpz=εo+ΞγocosapzE28

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

ΓqSWCNT=π2Λ2q2ΦnoΘ1α24γo2ωq2vsσsinaq/2Io2γoβ1α2×sinhβωqcosh4γoβ1α2cosaq2E29

where

α=ωq4γosinaq/2E30

## 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 (ΓqFCNT) and in SWCNT (ΓqSWCNT) are presented in Eqs. (24) and (29) respectively. In both equations, the absorptions are dependent on the frequency (ωq), the acoustic wavenumber (q), and temperature (T) as well as other parameters such as the inter-atomic distances, the velocity of sound (vs) and the deformation potential (Λ). In both expressions (see Eqs. (24) and (29)) a transparency window is observed: for F-CNT is ωq12γoaq; and for SWCNT is ωqγosin12aq/. These are the consequence of conservation laws. The Eqs. (24) and (29), are analyzed numerically with the following parameters used: Λ=9eV, q=105cm1, ωq=1012s1, vs=5×103m/s, Φ=104Wb/m2, and T=45K. The results are graphically plotted (see Figures 4, 5, 6, 7). Figure 4 shows the dependence of the sound absorption coefficient on the frequency (ωq) for varying q. In both graphs, the absorption is initially high but falls off sharply and then changes slowly at high values of ωq. Increasing the values of qcorrespondingly increases the obtained graph in both doped F-CNT and undoped SWCNT though the magnitude of absorption obtained in SWCNT exceeds that of F-CNT, that is, ΓqSWCNT>ΓqFCNT. This is in accordance with the work of Jeon et al. [19]. In Figure 2, the graph increases to a maximum point then drops off. It then changes again slowly at high qfor both undoped SWCNT and doped F-CNT. By increasing the temperature, the amplitude of the graphs reduces. For T=45K, the maximum absorption in ΓqSWCNT=8.2×104whilst that of ΓqFCNT=2867which gives the ratio of the absorption ΓSWNTΓFCNT29, whilst at T=55K, ΓSWNTΓFCNT9and at T=65K, we had ΓSWNTΓFCNT2. Clearly, we noticed that the ratio decreases with an increase in temperature. The nonlinear behavior in Figure 5 is as a result of the fact that, increasing temperature increases the scattering process in the material. The majority of electrons in this case acquire a higher velocity, shorter collision time, and higher energy. This energetic electrons, which are the majority undergo inter-mini-band transition (tunneling) allowing only a handful to undergo intra-mini-band transition. This allows only the few intra-mini-band electrons to interact with the copropagating phonons leading to a decrease in absorption of the acoustic phonons. To aid a better understanding of the comparison between the absorption obtained in both SWCNT and F-CNT, a semilog plot is presented in Figure 6, which clearly shows that the undoped SWCNT absorbs more than the doped F-CNT. This can be attributed to the fact that the presence of F-CNT atoms leads to chemical activation of a passive surface CNT by adding additional electronic band structure and altering the carbon π-bonds around the Fermi level in a non-linear manner thus forming a band structure of width two periods [22]. As Flourine is highly electronegative it thus weakens the walls of the CNT as it approaches it. The π-electrons attached to the Flourine which causes less free charge carriers to interact with the phonons. Current researches have predicted sp2bonding charge change to sp3by F-functionlization [25, 26, 27]. This bonding charge change would reduce the density of free carriers, consequently leading to the magnitude reduction of the absorption [22] (Figures 4, 5, 6).

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 (ωq) and the wavevector q(Figure 7).

## 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 ql1. The absorption coefficient obtained is highly nonlinear and depends on the stimulated absorption of acoustic phonons by electrically determined electrons experiencing intraminiband transport. The study is appropriate and furthermore considers a strong absorption of energized FCNT and SWCNT phonons. The Flourine doping affects the absorption properties of F-CNT, whereas SWCNT absorbs better than the F-CNT as was observed by Jeon et al. [19]. The phonons absorbed in this study have THz frequencies with wavelengths in the nanometer run, and takes into account examinations with high spatial determination, e.g., in phonon filters, spectroscopy (phonon spectrometer), microbiology, micro-nanoelectronic gadgets, tetrahertz adjustment of light, nondestructive testing of microstructures, and acoustic examination.

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© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Daniel Sakyi-Arthur, S. Y. Mensah, N. G. Mensah, Kwadwo A. Dompreh and R. Edziah (December 12th 2018). Absorption of Acoustic Phonons in Fluorinated Carbon Nanotubes with Non-Parabolic, Double Periodic Band, Phonons in Low Dimensional Structures, Vasilios N. Stavrou, IntechOpen, DOI: 10.5772/intechopen.78231. Available from:

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