Quantum Ion-Acoustic Oscillations in Single-Walled Carbon Nanotubes

1 Introduction

Carbon nanotubes (CNTs) are cylindrically wrapped graphene sheets, one of the allotropes of carbon first synthesised by Iijima in 1991 [1]. Single-walled (SW) carbon nanotubes have shown remarkable properties with the vital role foreseen in futuristic technologies [2]. Their dynamic character depends strongly on their geometrical structure and physical and chemical properties. One of the most fascinating aspects of SW CNTs is their collective electronic excitation which is helpful in understanding the interactions of electrons in tubules, their electronic structure, as well as existence and propagation of waves in various settings [3, 4]. The exact dynamics of an N-particle system cannot be solved realistically except for very small systems [5]. Then, it is necessary to develop appropriate reduced models which, although discarding some effects, yet retain the principal physical features of the system under study.

Self-consistent schemes of particles and fields have been popular in theoretical descriptions of oscillations and wave dynamics in CNTs. Quantum hydrodynamics (QH) is a macroscopic model on mean-field level [6, 7], various versions of which have been proposed in the past since the early days of quantum theory [8]. In recent years, QH description of nanoscale sized systems is seen increasing significantly due to its simplicity, numerical efficiency, and suitability for relatively larger wavelength waves. This can be noticed in dynamics of quantum electron gas [6], resonant tunneling devices [9, 10], semiconductor nanostructues and nanotubes [11, 12], and so on. However, the validity of QH equations rests in the assumption of long wavelengths λ≫λ_Fe (Thomas–Fermi length) or large resolvable length scales (i.e. l>several d̅, where d¯ ∼ n0−1/3 is the mean inter-particle separation). It is important to note that in order to obtain true quantum effects, particularly in the regimes of superfluidity and other quantum phenomena in trapped ultracold atomic systems, it is necessary to consider from the very beginning the velocity and density as true quantum operators [13, 14]. Then, the solutions to small oscillations and spectrum of such oscillatory modes can be found. However, a reduced model such as QH equations obtained via the moments of Wigner function [6, 15] is rather simpler to allow a more straightforward investigation of collective dynamics on scale lengths larger than λ_Fe.

In the common methods of formation of SW tubules such as arc discharge, laser pulse-driven vaporisation, chemical vapor deposition method, and others, it is intriguing that various types of metastable charged and neutral heavy ions and ion clusters are also formed. It is also possible to introduce such heavier charged species in nanostuctures through nanoscale engineering techniques [16, 17]. This also happens in the process of plasma treatment of metallic or semiconducting nanotubes and the deposition of metallic nanoparticles as a covering on SW CNTs [18]. As a result, a multispecies plasma system is formed where collective oscillations at different time scales are possible. The presence of such charged nanoclusters gives rise to low-frequency dust wave modes [19]. In such cases, the system is considered as a combination of SW tubules surrounded by charged nanoparticles. thus constituting a three species (electron-ion-dust) plasma.

It is worth mentioning here that the main features of some processes studied via the quantum models such as density functional theory (DFT) are also accessible by mean-field QH theory. DFTs started with the work of Thomas [20] and Fermi [21] in 1927 and successfully explain the electron dynamics with quick estimates of energies and global insight. Inspired by the procedure adopted in DFT, for the first time in [22], Manfredi and co-workers have included exchange and correlation effects in QH theory phenomenologically. The authors have used an additional exchange-correlation functional in momentum equation that gives rise to an additional force on electrons. They have also performed comparisons with DFT simulations for electrons in condensed matter and observed reasonable agreement. Using the QH approach of Crouseilles et al. [22], Shukla and Eliasson [23] have recently considered the problem of effective potential of a proton embedded in a dense quantum plasma and predicted the existence of a so-called novel attractive force between ions at atomic scales due to collective interactions of degenerate electrons. The potential is attributed mainly by the consideration of quantum forces, for example Fermi degeneracy, quantum diffraction (electron tunneling), and electron exchange-correlation effects. The model has further been extended to study the plasmon modes in CNTs [24] and bounded nanowaveguides [25].

Nanoscale encapsulation techniques have made it possible encapsulate atoms, molecules or ions in nanotubes, for instance, carbon (C₆₀)-filled nanotubes [26]. These ion-filled nanotubes have enhanced semiconducting properties. Encapsulation of transition metals, such as La, Y, and Sc inside the fullerene cage is also well known [27]. Fullerenes constitute an efficient positive-negative multi-ion plasma systems such as alkali-metal-fullerene plasma (K+, e−, C60−), as well as electron-free pure pair-ion plasma consisting of mass-symmetric positive-negative (C60+, C60−), ion species [28].

Similarly, presence of charged heavy species, for example, nanostructures or ion clusters in the background of SW tubules is inevitable in many cases. Their influence on the dynamics of electrons and ions of the tubule in turn affects the transport and dispersion properties of nanotube due to change in the density balance and wave potential [17]. Such situations can also occur due to encapsulation of heavier ions in nanotubes, their (laser) irradiation-mediated engineering, or self organisation [29] which makes the tailoring of CNTs possible in a controllable way. The heavier positive or negative ion clusters surrounding the charged nanotubes can creates space charge effect which can be dealt with in a self-consistent way. Then, dynamics of heavy species is not relevant on the time scale of the constituents of CNT.

It can also be seen that remarkable field emission properties of CNTs, metallic, or semiconducting – depending on their radii and the geometric angles – have led to extensive studies of the attachments of functional groups of other nanostructures to the surface of CNTs. Due to electric field emission, the plasma assisted CNTs can be considered as charged rods [30, 31] with the possibility of low-frequency electrostatic oscillations involving the contribution of charged CNTs on density fluctuations and wave potential. Similarly, there are few other situations when CNTs can be surrounded by charged heavier species (also known as charged ‘dust’ in plasma physics community) when their presence can affect the collective wave dynamics on ion time scale. This include processes of nanotube coating and attachment of nanoparticles [32, 33], selective heterogeneous deposition of size-controlled nanoparticles on exterior of CNTs [34, 35], anchoring, and surface treatment of nanoclusters on CNTs [36, 37], self-organisation of molecules and nanoparticles in nanoscale engineering [29], and so on. Then, it is legitimate to consider the quantum ion-acoustic oscillations in SW CNTs in the presence of charged ion clusters which behave as stationary on ion time scale. Here, our aim is to investigate the quantum ion-acoustic oscillations of SW CNTs (or quantum ion-acoustic wave) on the basis of quantum hydrodynamics and predict the influence of quantum effects associated with electrons and concentration and polarity of stationary ion clusters. Such a change in density balance influences the wave potential and modifies the dispersion characteristics of low frequency oscillations on ion time scale.

2 Governing Model

We consider the SW carbon nanotube as being a cylindrical shell structure, infinitely long and infinitesimally thin, where the electrons and ions are uniformly distributed over the cylindrical surface. When surrounded by positive or negative ion clusters, it forms a three-species plasma where electrons obey the Fermi–Dirac distribution. The self-consistent QH model describing low frequency dynamics of such system consists of the electron and ion continuity equations

and momentum balance equations

where n_j, v_j, and m_j are density, velocity, and mass of jth species, where j=e(i) for electron (ion). Further, ℏ is a reduced Planck constant, ∇ = ∇|| = e^z∂∂z + e^ϕR−1∂∂ϕ and q_j=−e, Z_je for electron, ions, respectively with Z_j being the charge state. In all practical situations, the ions follow classical dynamics, and hence the quantum and correlations effect of ions are not important. In the case of electrons, the second term on r.h.s. of (3) is the Fermi pressure Pe = πℏ2mene2, and the third term is the representative of quantum diffraction effects–the so called quantum pressure. The last term represents the gradient of the electron electron-exchange and electron-correlation potential, V_xce. Due to very high density of the system, this term plays a role on electron time scales [22, 23]. For ion dynamics, the length scale is so large that the effect of this term is not significant as compared to electrons with Fermi pressure. That is why this term is not taken into account in the study. For the sake of completeness, we have given the order parameters and a typical example in the next section to justify the assumption. The electrostatic field coupling electrons and ions is defined by E=−∇φ, where φ is self-consistent potential

with d²x′=Rdφ′dz′ being the differential area element on nanotube surface. In cylindrical geometry, Coulomb potential (5) can be expanded in the well-known form

where G(R, k, m)=4πI_m(kR)K_m(kR), I_m and K_m are the cylindrical Bessel functions of order m, α=+1(−1) for positive (negative) ion clusters, and subscript h refers to heavy species. When plasma density fluctuates, a quasi neutrality condition gives Z_in_i~n_e, and the density balance equation leads to equilibrium value ±Z_hn_h=±Z_hn_h0. Thus, the unperturbed state of the system can be described by

with ‘0’ denoting the unperturbed state.

We assume the electrons as inertialess which in turn allows neglect of the terms proportional to m_e/m_i in electron momentum equation. Equation (3) on integrating once with boundary conditions φ=0 and n_e=n_e0 as z→∞ yields

where P_e0=(πℏ²/m_e)n_e0 corresponds to the equilibrium.

3 Linearised Wave Analysis

The set of equations constitutes a nonlinear system. In order to find the oscillation spectrum on the ion time scale (long wavelength limit), the parameters v_i, n_e, n_i, and φ need to be linearised by substituting v_i=v_i1, n_e=n_e0+n_e1, n_i=n_i0+n_i1, and φ=φ₁. Then, the first order quantities give rise to the set of equations

and

where m stands for azimuthal quantum number, k is for longitudinal wave number, σ=n_e0/Z_in_i0 and ℵ=Z_hn_h0/n_e0. Applying the Fourier transform of the form

we obtain the perturbed electron and ion density and potential as

where k_Fe is Fermi wave number given by k_Fe=(2πn_e0)^1/2 and k_m=k²+m²/R². On solving (14–16), one obtains the dispersion equation in rescaled form given by

where ω is the wave frequency normalised by ω_s=c_s/a_B, a_B=ℏ²/e²m_e is the Bohr’s radius, and c_s=(E_Fe/m_i)^1/2 is the quantum ion-acoustic speed and E_Fe is the electron Fermi energy.

Let us examine (17) for various limiting cases. In the long wavelength limit (kR→0), we use the properties of Bessel functions I_m(x) and K_m(x) vis a vis when x→0, the behavior of I_m(x) and K_m(x) is observed as I_m(x)→a_mx^m, K₀(x)→ln(1.123/x), and K_m(x)→b_mx^−m with assumption of a_m=2^−m/Γ(m+1) and b_m=2^m−1Γ(m). It is seen that the r.h.s of (17) vanishes for m=0 in long wavelength limit leading to ω=0. So, no wave exists for this case.

For m≠0, we obtain

which shows that the dispersion relation depends on radius of nanotube, concentration of heavier ions, as well as the azimuthal quantum number. The characteristics of the CNTs alters for long wavelength as evident from the dispersion relation (18).

Now using the asymptotic solutions of the Bessel functions (x→∞), I_m(x)=e^x/(2πx)^1/2 and K_m(x)=(π/2x)^1/2e^−x, dispersion (17) reduces to

When the radius of the nanotube approaches infinity (R→∞), we get the following form of the dispersion equation

which is the case consistent to the case of planar geometry. In what follows, we numerically analyse the foregoing results qualitatively.

4 Numerical Results and Discussion

First, we discuss the dispersive role of the terms contained in electron equation of motion. Dynamics of an electron-ion quantum plasma on electron time scale with background of static ions exhibit two distinct dispersion regimes, namely the long wavelength regime with v_Fe≫ℏk/2m_e and short wavelength regime with v_Fe≤ℏk/2m_e [10]. Both the regimes are separated by the critical wave number

and the time scale is much shorter as compared to ωpi−1 where ω_pi is the ion oscillation frequency. The significance of exchange-correlations in fluid equations can be found when the wave phase velocity ω/k≫v_Fe [22, 23]. Conversely, in the present case, the situation changes for ion plasma oscillations, because for ion dynamics ωpe−1 ≪ ωpi−1 and v_Fi≪v_Fe with electron Fermi pressure being the most dominant parameter. This is also evident from the quantum ion-acoustic wave phase velocity ordering v_Fi≪ω/k≪v_Fe. In this case, the ion mass provides the inertia, and restoring force comes via the electron Fermi pressure which exhibits the dominance of the electron Fermi degeneracy effects over electron exchange-correlation effects for λ≫λ_Fe, and the contribution of the last term in (3) is negligible. This can be interpreted in another way for a typical case with asymptotic solution for R→∞, k_m→k including the exchange-correlation potential [22, 23]

where a_B=ϵℏ²/e2m_e* is the effective Bohr radius, ϵ is the relative dielectric constant of the system and me∗ is the effective mass of electron. This leads to the wave frequency

where vex = (0.328e2ne01/3/me)1/2 × [1 + 0.624/(1 + 18.37aBne01/3)],v_Fe=(ℏ/m_e)(2πn_e0)^1/2 and we assume m_e*=m_e and ϵ=1 for the sake of simplicity. It is well known that the diameter of the SW carbon nanotube varies from 1 to several nanometers, and the atomic density of typical graphite sheet is of the order 38 nm⁻² [38]. If we consider the surface electron density of SW nanotube 4×38 nm⁻², the last term in parenthesis in the denominator on the r.h.s. of (23) representing the strength of exchange-correlations amounts to 0.0004, showing no effect on wave dispersion.

From an experimental standpoint, techniques such as electron energy loss spectroscopy (EELS) provide reliable information on quantitative analysis of collective electron oscillations in SW CNTs [39, 40] which has also been pointed out qualitatively in various studies [24, 41]. Certain properties of phonons can also be studied by other well-established methods, for instance, the force constant model [42, 43], valence force-field model [44, 45], or the DFT [46, 47]. QH modeling actually provides a decent starting point to investigate the qualitative behavior of electrostatic oscillations in SW CNTs for the regimes where the resolvable length scale is much larger than λ_Fe. For quantitative analysis at scales of the order of λ_Fe, it is desirable to employ more accurate methods.

In (17), it can be seen that the wave dispersion depends on various parameters which are primarily affected by density balance where the electron Fermi energy satisfies the relation ℏ2kFe2/2me. We measure the wave number in the units of inverse of Bohr radius (aB−1) and wave frequency in ω_s. Similarly, it is assumed that the nanotube radius varies in units of a_B. Then, the units along the reference axes in the illustrations represent appropriate nondimensional quantities and the curves do not reveal direct dependence on physical parameters. We have seen that the normalised frequency increases for concentration of negatively charged ion clusters due to the factor (1−αℵ) becomes greater than unity on substituting the value of α and ℵ. Conversely, the factor relevant to positively charged clusters vis a vis (1−αℵ) becomes smaller than unity indicating the decrease in the frequency of the wave. This shows a strong influence of density variation.

In Figure 1 the dependence of wave dispersion on cluster concentration is shown with azimuthal quantum number m=0 and the radius of the nanotube R=13a_B. It is seen that the density of negatively charged ion clusters increases the normalised frequency of waves (Fig. 1a). In the opposite case, the change in normalised frequency with normalised wave number for positive ion clusters shows the reverse (Fig. 1b) which shows the inverse relationship of the positive cluster concentration and the normalised wave frequency. Figure 2 shows the variation in wave frequency for a nonzero azimuthal index m and fixed radius of tubule R. Typical values of tubule radius are found around 13a_B [39, 48]. Taking R=13a_B, it is seen that increase in concentration of negatively charged clusters for fixed azimuthal quantum number m=2 increases the wave frequency as shown in Figure 2a. However, with increase in the positively charged cluster concentration, the frequency decreases as illustrated in Figure 2b. It is also noted that the curve separation for various discrete values of m reduces as k increases tending to the same limit for large k. Figure 3 gives the variation in normalised frequency with a radius of nanotube for azimuthal quantum number m=2. We have shown the dispersion curves for various radii of tubes starting from very small values, vis a vis, 2a_B to larger values up to 16a_B. The curve separation is significant for smaller radii, and a saturation limit is reached for larger k as R increases for both types of clusters. The curves shift down as concentration increases. But the values of ω for negatively charged clusters are higher than those for positively charged ones. The influence of azimuthal index m is shown in Figure 4. It is seen that the normalised frequency increases with azimuthal quantum number. The frequency assumes larger values for negatively charged clusters as compared to the corresponding positively charged ones. This relationship is also due to the factors α and ℵ which provides shift to the wave frequency.

Figure 1:

The normalised wave frequency ω is plotted vs. normalised wave number for azimuthal quantum number m=0 with variation of cluster concentration. Typical radius nanotube R=13a_B is used which is relevant to laboratory systems for (a) negative and (b) positive stationary ion clusters where various concentration levels are shown by distinct lines.

Figure 2:

The dispersion equation of the quantum ion-acoustic wave is shown for nonzero azimuthal quantum number (m=2) and various concentrations of charged heavy species with same radius of tubule as in Figure 1 for (a) negative and (b) positive ion clusters.

Figure 3:

The dispersion relation of quantum ion-acoustic wave is plotted for various nanotube radii with m=2 and ℵ=0.2 for (a) α=−1 and (b) α=1. Starting from smaller radii, the upper blue (thick) curve represents R=2a_B. The radius for each lower curve is increased in units of a_B such that the typical value of R=13a_B is shown by thin curve and the lowest dotted curve represents R=16a_B. The curve separation is larger for smaller R which goes on decreasing as R increases and a saturation is seen leading to same limit for large k.

Figure 4:

Normalised dispersion relation is plotted vs. normalised wave number for various azimuthal indices shown by distinct lines. The number density is 0.2n_e0 and tube radius R=13a_B for (a) α=−1 and (b) α=1.

5 Conclusions

To conclude, we have implemented the QH model in order to investigate the quantum ion-acoustic wave in SW CNTs. The presence of stationary ion clusters, either positive or negative, influences the wave potential due to change of density balance. However, charge neutrality condition is ensured at equilibrium. We have obtained a dispersion relation by Fourier transformation of model equations which shows that the dynamics of a low-frequency ion wave is strongly influenced by the potential contributed by the heavier stationary species and their number density, in addition to quantum effects such as Fermi degeneracy. The wave also depends on nanotube’s geometry and reduces to a planar solution in the limit of large (infinite) radius of the tube. However, due to dominance of electron Fermi pressure, the role of electron exchange-correlations is not significant in the present case. We have analysed the results numerically and parametric dependence of waves on concentrations of positive and negative ion clusters, nanotube radius, and azimuthal quantum number m is described for typical systems. The study was motivated by the significant variation of physical properties of CNTs with the presence of static species. The existence of low-frequency electrostatic ion excitations results in a propagating wave under the influence of electronic quantum effects, especially Fermi degeneracy.