Hypersound Absorption of Acoustic Phonons in a degenerate Carbon Nanotube

Hypersound Absorption of acoustic phonons having $ql>>1$ in a degenerate Carbon Nanotube (CNT) with linear energy dispersion near the Fermi level was theoretically studied. The general expression for the absorption coefficient ($\Gamma$) under a non-quantizing electric field ($E$) with drift velocity ($V_D$)was obtained. At $T = 10K$ and scattering angle $\theta>0$, the dependence of $\Gamma$ on acoustic wave number ($\vec{q}$), frequency ($\omega_q$), and $\gamma = 1-\frac{V_D}{V_s}$, ($V_s$ being the speed of sound) were analysed numerically at $n = 0, \pm 1, \pm 2$ (where $n$ represent the various harmonics) and presented graphically. In a $3D$ representation, when $\gamma<0$, the maximum amplification was attained at $V_D = 1.1V_s$ which occurred at $E = 51.7Vcm^{-1}$. In the second harmonics, ($n =\pm 2$), the absorption obtained was compared to experimental measurement of acoustoelectric current via the Weinreich relation. From the graphs, the observed amplification of acoustic phonons caused by intraband transition shows CNT as a promising hypersound generator (SASER).


Introduction
Carbon Nanotubes (CNT) has recently attracted lot of interest for use in many semiconductor devices due to its remarkable electrical and mechanical properties which are mainly attributed to its unusual band structures [1, 2, 3].
The π-bonding and anti-bonding (π * ) energy band of CNT crosses at the Fermi level in a linear manner [4,5]. In the linear regime, electron-phonon interactions in CNT at low temperatures leads to the emission of large number of coherent acoustic phonons [6,7,8]. Studies of the effect of phonons on thermal transport [9,10], on Raman scattering [11] and on electrical transport [12] in CNT is an active area of research. Also, the speed of electrons in the linear region is extremely high. This makes CNT a good candidate for application of high frequency electronic systems such as field effect transistors (FET's) [13], single electron memories [14] and chemical sensors [15]. Another important investigation in the linear regime is interaction of acoustic phonons with drift charges in CNT. It is well known that when acoustic phonons interact with charge carriers, it is accompanied by energy and momentum exchange which give rise to the following effects: Absorption (Amplification) of acoustic phonons [16,17]; Acoustoelectric Effect (AE) [18,19,20,21,22,23]; Acoustomagnetoelectric Effect (AME) [24,25,26,27,28]; Acoustothermal Effect [29] and Acoustomagnetothermal Effect [29]. The idea of acoustic wave amplification in bulk material was theoretically predicted by Tolpygo (1956), Uritskii [30], and Weinreich [31] and in N-Ge by Pomerantz [32]. In Superlattices, the effect of hypersound absorption/amplification was extensively studied by Mensah et. al [33,34,35,36,37],Vyazovsky et. al. [38], Bau et.
al [39], while Shmelev and Zung [40] calculated the absorption coefficient and renormalization of the short-wave sound velocity. Azizyan [41] calculated the absorption coefficient in a quantized electric field. Furthermore, Acoustic wave absorption/amplification in Graphenes [42,43,44], Cylindrical quantum wires [45] , and quantum dots [46,47] have all received attention. On the concept of Acoustoelectric effect (AE) in bulk [48] and low-dimensional materials [49], much research has been comprehensively done both theoretically and experimentally. Acoustoelectric effect in CNT's is now receiving attention with few experimental work done on it. Ebbecke et. al. [50] studied the AE current transport in a single walled CNT, whilst Reulet et. al [51] studied AE in CNT. But in all these research there is no theoretically studies of AE in CNT. In this paper, the absorption (amplification) of hypersound in CNT in the regime ql >> 1 (q is the acoustic wave number and l is the electron mean free path) is considered where the acoustic wave is considered as a flow of monochromatic phonons of frequency (ω q ).
It is worthy to note that the mechanism of absorption (amplification) is due to Cerenkov effect. For practical use of the Cerenkov acoustic-phonon emission, the material must have high drift velocities and large densities of electrons [17]. Carbon Nanotubes (CNT) has electron mobility of 10 5 cm 2 /Vs at room temperature. At low temperatures (T = 10K), CNT exhibit good AE effect, which indicates that Cerenkov emission can take place in it [52]. The paper is organised as follows: In section 2, the kinetic theory based on the linear approximation for the phonon distribution function is setup, where, the rate of growth of the phonon distribution is deduced and the absorption coefficient (Γ) is obtained. In section 3, the final equation is analysed numerically in a graphical form at various harmonics where the ab-sorption obtained are related to the acoustoelectric current via the Wienrich relation [31]. Lastly the conclusion is presented in section 4.

Theory
We will proceed following the works of [48,49] where the kinetic equation for the phonon distribution is given as where N q (t) represent the number of phonons with wave vector q at time t.
The factor N q +1 accounts for the presence of N q phonons in the system when the additional phonon is emitted. The f p (1 − f p ) represent the probability that the initial p state is occupied and the final electron state p is empty whilst the factor N q f p (1 − f p ) is that of the boson and fermion statistics.
The unperturbed electron distribution function is given by the shifted Fermi- Dirac function as where f p is the Fermi-Dirac equilibrium function, with µ being the chemical potential, p is momentum of the electron, β = 1/kT , k is the Boltzmann constant and V D is the net drift velocity relative to the ion lattice site. In a more convenient form, Eqn(1) can be written as To simplifiy Eqn.
(3), the following were utilised Given that the phonon generation rate simplifies to In Eqn. (7), f p > f p if ε p < ε p . Whenhω q −h q · V D > 0, the system would return to its equilibrium configuration when perturbed where Buthω q −h q · V D < 0 leads to the Cerenkov condition of phonon instability (amplification). The linear energy dispersion ε( p) relation for the CNT is given as [53] The ε 0 is the electron energy in the Brillouin zone at momentum p 0 , b is the lattice constant , γ 0 is the tight binding overlap integral (γ 0 = 2.54eV). The ± sign indicates that in the vicinity of the tangent point, the bands exhibit mirror symmetry with respect to each point. The phonon and the electric field are directed along the CNT axis therefore p = ( p +h q)cos(θ). Where θ is the scattering angle. At low temperature, the kT << 1, Eqn.(5) reduces to Inserting Eqn.(8 and 9) into Eqn. (7), and after some cumbersome calculations where χ = √ 3γ 0 b/2h, and

Numerical analysis
Considering the finite electron concentration, the matrix element can be modified as where ℵ (el) ( q) is the electron permitivity [48]. However, for acoustic phonons, |C q | = Λ 2h q/2ρV s , where Λ is the deformation potential constant and ρ is the density of the material. From Eq.(10), taking ε 0 = p 0 = 0, the Eqn. (10) finally reduces to where I n (x) is the modified Bessel function. The parameters used in the numerical evaluation of Eqn (12) are: |Λ| = 9eV, b = 1.42nm, q = 10 7 cm −1 , ω q = 10 12 s −1 ,V s = 4.7 × 10 5 cm s −1 , T = 10K, and θ > 0. The dependence of the absorption coefficient (Γ) on the acoustic wave number ( q), the frequency (ω q ) and (γ) at various harmonics (n = 0, ±1, ±2) are presented below. For n = 0, the graph of Γ versus q at varying frequencies and that of Γ versus ω q for various acoustic wave numbers are shown in Figure 1(a and b). In Figure   1a, an amplification curve was observed, where the minimum value increases by increasing ω q but above ω q = 1.6 × 10 12 s −1 , an absorption was obtained.
In Figure 1b, it was observed that absorption switched over to amplification when the q values were increased. For n = ±1 (first harmonics), in Figure   2a, it was observed that absorption exceed amplification and the peaks shift to the right. A further increase in ω q values caused an inversion of the graph where amplification exceeds absorption (see Figure 2b). A similar observation was seen in Figure 3 (a and b), where, the peak values shift to the right and decreases with increasing q values (see Figure 3a) but in Figure 3b, an inversion of the graph occurred for increasing values of q. Figure 4 (a and   , shows the dependence of Γ on γ by varying either ω q or q. In both graphs, when γ < 0, produce non-linear graphs which satisfy the Cerenkov condition, but at γ > 0, the graph returns to zero. The observed peaks in Figure 4a, shift to the left by increasing ω q whilst in Figure 4b, shift to the right by increasing q. For further elucidation, a 3D graph of Γ versus ω q and γ or Γ versus q and γ are presented in Figure 5 (a and b). In both graphs, when γ = −0.10, a maximum amplification was obtained. For n = ±2 (Second harmonics), the dependence of the absorption coefficient Γ on ω q is presented in 2D and 3D form as shown in Figure 6 and 7. In Figure 6, an absorption graph was obtained. The insert shown is an experimental results obtained for the Acoustoelectric current in Single walled Carbon Nanotube [50]. Figure   7 (a and b) is the 3D representation of the absorption in second harmonics.
From Weinreich relation [31], the absorption coefficient is directly related to the acoustoelectric current, therefore from Figure 6, the results obtained for the absorption coefficient qualitatively agrees with the experimental results presented (see insert). In the 3D graphs,the maximum amplification and

Conclusion
The expression for Hypersound Absorption of acoustic phonons in a degenerate Carbon Nanotube (CNT) was deduced theoretically and graphically presented. In this work, the acoustic waves were considered to be a flow of monochromatic phonons in the short wave region (ql >> 1). The general expression obtained was analysed numerically for n = 0, ±1, ±2 (where n is an integer). From the graphs, at certain values of ω q and q, an Amplification was observed to exceed Absorption or vice-versa . For γ < 0, the maximum Amplification was observed at V D = 1.1V s which gave us a field of E = 51.7V cm −1 . This field is far lower than that observed in superlattice and homogeneous semiconductors permitting the CNT to be a suitable material for hypersound generator (SASER). A similar expression can be seen in the works of Nunes and Fonseca [54].
Very interesting to our work is the qualitative agreement of the absorption graph to an experimental graph resulting from an acoustoelectric current via the Weinriech relation.