Surface plasmon polariton amplification in a single-walled carbon nanotube

A. S. Kadochkin; S. G. Moiseev; Y. S. Dadoenkova; V. V. Svetukhin; I. O. Zolotovskii

doi:10.1364/OE.25.027165

1. Introduction

The spaser is a nanoscale quantum generator and ultrafast nanoamplifier of coherent localized optical fields. The main idea of spaser is compensation of optical losses of surface plasmon polariton (SPP) by gain in the active medium overlapping with the SPP. In the past few years, several schemes of SPP amplification have been proposed [1–9]. The most common way to compensate for losses is using the optical pump [1–5]. However, these techniques require an external high-power pump laser, and the use of a nanoscale spaser is of no practical sense. From this point of view, the use of electric pump [6–9] seems to be more promising for fabrication of nanoscale lasers. The loss compensation approach used in [1–8] is based on the idea of replenishing energy of SPPs by pumping the active component of plasmonic structure. There is also an alternative approach based on mechanism of direct energy transfer from plasma oscillations, which are sustained by direct current, to SSPs, which have electromagnetic nature [9-10]. The amplification in this case occurs due to the drift current created in the conductive component of the structure by an external current source. This scheme of surface wave amplification is almost complete analog of traveling-wave tube used in microwave technology.

One of the promising objects for fabrication of nanoscale spasers with electric pump can be a carbon nanotube (CNT), because, on the one hand, SPPs can exist in it, and on the other hand, it conducts an electric current. At present, a considerable amount of works is devoted to the investigation of SPPs in single-walled CNTs [11–17] using both the classical method based on the solution of Maxwell's equations [11, 12, 17] and the quantum approach based on the consideration of a CNT as a one-dimensional structure in which an electron gas has essential quantum properties [13–15]. The plasmons arising in CNTs have been reported to be observed experimentally [16].

A considerable amount of publications has also been devoted to the study of the CNTs conductivity. It is assumed that electrons in a single-walled CNT, like in the graphene [18], move in ballistic regime without scattering and the maximal velocity of such a motion is the Fermi velocity [19]. For semiconducting CNTs, quantum-mechanical calculation in the tight-binding approximation gives the maximum value of the electron drift velocity $5 \cdot 10^{7}$ cm/s [20]. For effective interaction of the electron beam and the plasmon wave, the SPP wave velocity in the nanotube should be comparable in order of magnitude with velocity of the charge carriers. As is known, in a waveguide with a characteristic transverse dimension a an electromagnetic wave can propagate with a propagation constant of the order of 1/a [21]. From this statement it follows that at a frequency of the order of 100 THz, corresponding to the far infrared range, the propagation velocity of a wave in a CNT with a diameter of 10 nm is of the order of 10⁷cm/s, which is close to the above mentioned electron drift velocity. Thus, our estimates show that for CNTs there exists a potentiality of implementing the synchronism condition and, consequently, fabricating a travelling-wave-tube-like nanoscale spaser.

2. Amplification coefficient of SPP under the synchronism condition

In this section we determine the conditions of SPP amplification in a CNT due to the potential difference applied to its ends. We consider the CNT as an infinitely thin cylindrical shell of the length L and radius a ( $L > > a$ ), as shown in Fig. 1(a), so that electrons in a nanotube can be considered as a two-dimensional electron gas. The application of the hydrodynamic approximation in combination with Maxwell's equations and standard boundary conditions at cylindrical shell with radius a (continuity of the electric field components $E_{z}$ and $E_{φ}$ and discontinuity of the magnetic field components $H_{z}$ and $H_{φ}$ due to the conductivity tensor associated with the CNT) makes it possible to find the field distribution and the dispersion relations of the TE- and TM- modes of the nanotube [11, 17].

Fig. 1 (a) Single-walled CNT of the length L. The direction of the electric current is shown with red arrow, and $(ρ, φ, z)$ are cylindrical coordinates. (b) Spatial distribution of the electric field component E_z of the SPP wave.

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In a cylindrical CNT, all the electromagnetic field components are proportional to $\exp (i β z + i m φ - i ω t),$ where $β$ is the longitudinal component of SPP wavevector (or SPP propagation constant), $m = 0, \pm 1, \pm 2...$ is the mode number, and $ω$ is the angular frequency of SPP. We consider the fundamental (m = 0) non-radiative TM-mode only, which is characterized by the following properties [11]:

а) no angular dependence of the fields;
b) inside $(ρ < a)$ and outside $(ρ > a)$ the CNT the components of the electric E and magnetic H fields depend on the coordinate ρ as follows: $\begin{array}{l} E_{z} (ρ < a) = E_{0 z} \frac{I_{0} (q ρ)}{I_{0} (q a)}, H_{z} (ρ < a) = H_{0 z} \frac{I_{0} (q ρ)}{{d I_{0} (x) / d x |}_{x = q a}}, \\ E_{z} (ρ > a) = E_{0 z} \frac{K_{0} (q ρ)}{K_{0} (q a)}, H_{z} (ρ > a) = H_{0 z} \frac{K_{0} (q ρ)}{{\partial K_{0} (x) / \partial x |}_{x = q a}};, \end{array}$
c) all the other components of E and H fields at any point inside and outside the CNT are determined by the following equations: $E_{r} = \frac{β}{ω ε_{0}} H_{φ} = - i \frac{β}{q^{2}} \frac{d E_{z}}{d r}, H_{r} = - \frac{β}{ω μ_{0}} E_{φ} = - i \frac{β}{q^{2}} \frac{d H_{z}}{d r} .$

In Eqs. (1) and (2), $E_{0 z}$ is the z-component of electric field at the CNT surface, $q = \sqrt{β^{2} - k_{0}^{2}}$ is the transverse component of SPP wavevector in CNT, $ε_{0}$ and $μ_{0}$ are dielectric permittivity and magnetic permeability of vacuum, and $I_{0}$ and K₀ are the modified zero-order Bessel functions of the first and second kind, respectively. The distribution of E_z over the reduced coordinate ρ/a is shown in Fig. 1(b). The presence of a longitudinal component of the electric field at the CNT surface (ρ/a = 1) is necessary to ensure the interaction of SPP wave with an electric pump current flowing along the CNT.

The dispersion relation of the fundamental TM-mode of the CNT suspended in vacuum has the following form [11, 17]:

\frac{i ε_{0} ω}{a q^{2} I_{0} (q a) K_{0} (q a)} - σ_{z z}^{0} = 0,

where

σ_{z z}^{0} = \frac{i n_{s} e^{2}}{m_{e}} \frac{ω}{ω (ω + i γ) - η q^{2}}

is the component of conductivity tensor, e and m_e are the element charge and the electron effective mass,

η = π n_{s} ℏ / m_{e}^{2}

is the square of the propagation speed of density disturbances in a uniform 2D homogeneous Fermi electron fluid [11],

n_{s}

is the surface concentration of free charge carriers, and

γ

describes the damping due to scattering of electrons by positively charged ions. Since a ballistic electron motion regime is realized in a single-walled CNT, electron scattering by positively charged ions can be neglected and in Eq. (4) we assume

γ = 0.

In this case

σ_{z z}^{0}

is pure imaginary, and Eq. (3) is satisfied by real values of β

The dispersion relation β(ω) obtained from the Eqs. (3) and (4) is illustrated in Fig. 2(a) for the CNT radius a = 5 nm, and n_s = 10¹² cm⁻² [18]. From the presented dependences one can see that in the far-infrared regime the propagation constant takes values above $10^{7} {cm}^{- 1}$ , i. e., $β > > k_{0}$ ( $k_{0} = ω / с < 10^{4} {cm}^{- 1}$ is wavenumber in vacuum, с is speed of light in vacuum). From Fig. 2(a) one can see that the phase velocity $V_{p h} = ω / β$ of a longitudinal electromagnetic wave in a CNT can be reduced to values of about 10⁷ cm/s (solid blue line), which indeed makes it possible to satisfy the synchronism condition between the SPP wave and current and, consequently, to create in principle a nanoscale CNT spaser .

Fig. 2 Dispersion of the SPP and drift current parameters: (a) SPP propagation constant β (solid red line), SPP phase velocity $V_{p h}$ (solid blue line) and charge-drift velocity V₀ = 5·10⁷ cm/s (dashed blue line); (b) SPP amplification coefficient a.

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The equation describing the interaction of the longitudinal electromagnetic wave in the waveguide (travelling-wave tube) and the electron flux is well-known in microwave technology [22, 23]:

\frac{d E_{z}}{d z} + i \frac{ω}{V_{p h}} E_{z} = - \frac{1}{2} {(\frac{ω}{V_{p h}})}^{2} B I_{d},

where

I_{d}

is drift current along the CNT (the z-axis),

B = {| E_{0 z} |}^{2} / 2 β^{2} P

describes the coupling efficiency between the current (which is localized at the CNT surface) and SPP wave, and P is power carried by an electromagnetic wave. Taking into account that CNT is placed in a nondispersive medium with a relative permittivity equal to 1, one can write

P = (1 / 2) ε_{0} V_{g} \int {| E |}^{2} d S

, with

| E |

being the modulus of the total field amplitude, and

V_{g} = \partial ω / \partial β

being the group velocity of the SPP wave. Taking Eqs. (1) and (2) into account, the expression for the coupling coefficient can be rewritten as:

B \approx \frac{I_{0} (β a) K_{0} (β a)}{2 π ε_{0} V_{g}} .

To describe the interaction of current and the SPP wave in the case of strong coupling, when under the influence of the wave field the amplitude of the current becomes modulated along the CNT, the Eq. (5) should be supplemented with an equation “electric current – electromagnetic field” linearized with respect to small perturbations J of the current amplitude ( $J (z) = I_{d} (z) - I_{d 0}$ , $J < < I_{d 0}$ , with $I_{d 0}$ being an unperturbed current) which has the form [22, 23]:

\frac{d^{2} J}{d z^{2}} + 2 i \frac{ω}{V_{0}} \frac{d J}{d z} - \frac{1}{V_{0}^{2}} (ω^{2} - ω_{q}^{2}) J = i \frac{ω}{V_{0}} \frac{I_{d 0}}{2 U_{0}} E_{z},

where

U_{0}

is an applied potential difference that induces a direct current

I_{d 0}

,

V_{0}

is charge carriers velocity in CNT, and

ω_{q} = \sqrt{n_{s} e^{2} / 2 π a ε_{0} m_{e}}

is reduced plasma frequency which takes into account the waveguide geometrical parameters.

Let’s assume that J and $E_{z}$ change along the CNT proportionally to $\exp (- i G z)$ , where G is the wavenumber of the harmonic perturbation. The compatibility condition for the equation set Eqs. (5) and (7) leads to the following dispersion equation:

(ω - G V_{p h}) [{(ω - G V_{0})}^{2} - ω_{q}^{2}] = C^{3} ω^{3},

where the parameter C can be written as:

C = {(\frac{B I_{d}}{4 U_{0}} \frac{V_{0}}{V_{p h}})}^{1 / 3} = {(\frac{π a^{2} ω_{q}^{2}}{V_{g} V_{p h}} I_{0} (β a) K_{0} (β a))}^{1 / 3} .

The Eq. (8) has one real (and thus not interesting for us) root and two complex conjugated roots. The value of the SPP amplification coefficient can be expressed by

α = Im (G)

.

3. Results of numerical calculations of the SPP amplification coefficient

The Eq. (8) was solved numerically, and the results of the calculations are presented in Fig. 2(b). The reduced plasma frequency for the considered CNT is $ω_{q}$ = 10¹⁴ rad/s, which corresponds to far-infrared regime. It was assumed that electrons in semiconducting CNT move in the ballistic regime, and the maximum electron drift velocity is V₀ = 5·10⁷ cm/s [20] (dashed blue line in Fig. 2(a)).

The synchronism condition is fulfilled at frequency $ω_{0}$ = 3·10¹³ rad/s (see the intersection of the solid blue and dashed blue lines in Fig. 2(a). The amplification coefficient $α$ (Fig. 2(b)) reaches huge values (of the order of 10⁶ cm⁻¹) over a wide frequency range in the infrared region with the amplification line width $Δ ω$ ≈8·10¹³ rad/s. The maximal amplification coefficient in the system under consideration is about 2·10⁶ cm⁻¹.

It should be noted that amplification in the system occurs at frequencies higher than $ω_{0},$ i.e. when the SPP phase velocity (solid blue line) is less than the drift velocity (dashed blue line). This regime corresponds to Cherenkov radiation in a nanotube [24,25]. The corresponding amplification mode of the electromagnetic waves is typical, in general, for microwave amplifiers and generators (traveling and backward wave tubes) [22, 23]. For these devices, the largest amplification also occurs under the conditions when the drift velocity of the current exceeds the value of the phase velocity of the amplified wave.

The SPP loss coefficient in the CNT can be estimated as $β^{″} = {1 / l}_{p}$ , where $l_{p} ~ V_{g} τ_{p}$ is the mean free path of SPP, and the SPP lifetime $τ_{p} = 3 \cdot 10^{- 12}$ s [15, 26]. In our case we can assume $V_{g} ~ V_{p h} \approx V_{0}$ , and $l_{p} \approx V_{0} τ_{p} \approx 1.5$ μm is much larger than the CNT length (L < 1 μm). Thus, the effective amplification coefficient α ~10⁶ cm⁻¹ is much larger than the loss coefficient $β^{″} ~ 10^{4}$ cm⁻¹, i. e. despite the SPP slowing-down leads to loss coefficient increase, this circumstance does not play an important role in the presence of a huge amplification.

4. Conclusions

We have investigated the interaction of a slow plasmon polariton wave of the far-infrared regime propagating in a single-walled carbon nanotube with a drift current. It is shown that in such a structure it is possible to achieve a significant amplification of the surface plasmon polaritons with amplification coefficient of the order of 10⁶ cm⁻¹. The predicted amplification coefficient of surface plasmon polariton waves significantly exceeds the values obtained by any other known technique. Thus, the considered properties of carbon nanotubes as structures with huge effective amplification of surface plasmon polaritons make them extremely attractive for the development of spasers. However, it should be noted that the results presented in this paper are obtained in the approximation of inexhaustible pump (in our case, by the drift current). In real structures, an amplification coefficient larger than 1 μm⁻¹ can be obtained only over a very small scale. For instance, even at the scale of nanotube length of about 1 μm it is necessary to take into account the pump degeneracy, when the amplification coefficient rapidly decreases.

In this paper we do not consider the excitation methods of surface plasmon polaritons in carbon nanotubes, however, we would like to make some comments on this matter. The excitation of ultra-slow surface plasmon polaritons involves natural problems. surface plasmon polariton cannot be introduced into a nanotube using standard techniques (for example, using prisms or diffraction gratings). The most suitable way to excite such waves is the direct generation of surface plasmon polaritons due to the feedback realized somehow in a carbon nanotube. For example, feedback can be obtained either by reflection from the ends of the nanotube or by distributed feedback. The direct exitation of surface plasmon polariton in the generation mode looks especially promising because in this case carbon nanotube is a sufficiently high-Q resonator (such as a Fabry-Perot interferometer) with an unavoidably strong reflection of the surface wave from the ends of the tube. In the case of feedback, a circuit similar to the backward-wave tube [22, 23] and a DFB laser can be implemented. This problem will be further studied and published elsewhere.

Funding

Ministry of Education and Science of the Russia (Project 14.Z50.31.0015, State Contracts 3.7614.2017/P220, 3.5698.2017/P220, 3.3889.2017 and 16.2773.2017/4.6); and the Russian Foundation for Basic Research (Project No. 17-02-01382).

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Surface plasmon polariton amplification in a single-walled carbon nanotube

Abstract

1. Introduction

2. Amplification coefficient of SPP under the synchronism condition

3. Results of numerical calculations of the SPP amplification coefficient

4. Conclusions

Funding

References and links

Cited By

Figures (2)

Equations (9)

Optics Express