Surface plasmon polariton amplification in a single ... - OSA Publishing

Vol. 25, No. 22 | 30 Oct 2017 | OPTICS EXPRESS 27165

Surface plasmon polariton amplification in a single-walled carbon nanotube A. S. KADOCHKIN,1,2,3,* S. G. MOISEEV,1,4 Y. S. DADOENKOVA,1,5,6 V. V. SVETUKHIN,1,2 AND I. O. ZOLOTOVSKII1,2 1

Ulyanovsk State University, Ulyanovsk, 432017, Russia Institute of Nanotechnologies of Microelectronics of the Russian Academy of Sciences, Moscow, 119991, Russia 3 ITMO University, St. Petersburg, 197101, Russia 4 Kotelnikov Institute of Radio Engineering and Electronics of the Russian Academy of Sciences, Ulyanovsk Branch, Ulyanovsk, 432011, Russia 5 Novgorod State University, Veliky Novgorod, 173003, Russia 6 Donetsk Institute for Physics & Technology, Donetsk, 83114, Ukraine * [email protected] 2

Abstract: The interaction of a surface plasmon polariton wave of the far-infrared regime propagating in a single-walled carbon nanotube with a drift current is theoretically investigated. It is shown that under the synchronism condition a surface plasmon polariton amplification mechanism is implemented due to the transfer of electromagnetic energy from a drift current wave into a terahertz surface wave propagating along the surface of a singlewalled carbon nanotube. Numerical calculations show that for a typical carbon nanotube surface plasmon polariton amplification coefficient reaches huge values of the order of 106 сm−1, which makes it possible to create a carbon-nanotube-based spaser. © 2017 Optical Society of America OCIS codes: (240.6680) Surface plasmons; (140.2600) Free-electron lasers (FELs); (230.7020) Traveling-wave devices.

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#297840 Journal © 2017

https://doi.org/10.1364/OE.25.027146 Received 9 Jun 2017; revised 19 Sep 2017; accepted 21 Sep 2017; published 20 Oct 2017


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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 ⋅107 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 Eϕ 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 Ez of the SPP wave.

In a cylindrical CNT, all the electromagnetic field components are proportional to exp ( i β z + imϕ − iωt ) , where β is the longitudinal component of SPP wavevector (or SPP propagation constant), m = 0, ± 1, ± 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: Ez ( ρ < a ) = E0 z Ez ( ρ > a ) = E0 z

I0 ( qρ ) I 0 ( qa )

, H z ( ρ < a) = H0z

K0 ( qρ ) K 0 ( qa )

, H z ( ρ > a ) = H0z

I0 ( qρ )

dI 0 ( x ) / dx x = qa K0 ( q ρ )

,

∂K 0 ( x ) / ∂x x = qa

(1) ;,


c) all the other components of E and H fields at any point inside and outside the CNT are determined by the following equations:

β β dEz β β dH z , Hr = − . (2) H = −i 2 E = −i 2 q dr q dr ωε 0 ϕ ωμ0 ϕ In Eqs. (1) and (2), E0 z is the z-component of electric field at the CNT surface, Er =

q = β 2 − k02 is the transverse component of SPP wavevector in CNT, ε 0 and μ0 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 Ez 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]:

iε 0ω − σ zz0 = 0, aq I 0 ( qa ) K 0 ( qa )

(3)

2

where

σ zz0 =

ins e 2 ω me ω (ω + iγ ) − η q 2

(4)

is the component of conductivity tensor, e and me are the element charge and the electron effective mass, η = π ns  me2 is the square of the propagation speed of density disturbances in a uniform 2D homogeneous Fermi electron fluid [11], ns 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 zz0 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 ns = 1012 cm−2 [18]. From the presented dependences one can see that in the far-infrared regime the propagation constant takes values above 107 cm −1 , i. e., β >> k0 ( k0 = ω с < 104 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 ph = 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 . 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]: dEz ω 1 ω Ez = −  +i 2  V ph dz V ph

2

  BI d , 

where I d is drift current along the CNT (the z-axis), B = E0 z

2

(5) 2β 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 0Vg  E dS , with E being the modulus of the total field amplitude, and 2

Vg = ∂ω ∂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≈

I0 ( β a ) K0 ( β a ) 2πε 0Vg

(6)

.

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