Spaser Based on Graphene Tube

Scientia Iranica D (2020) 27(6), 3084{3095

Sharif University of Technology Scientia Iranica Transactions D: Computer Science & Engineering and Electrical Engineering http://scientiairanica.sharif.edu

Spaser based on graphene tube

S. Behjati Ardakani and R. Faez

Department of Electrical Engineering, Sharif University of Technology, Azadi Avenue, Tehran, Iran.

Received 23 January 2018; received in revised form 11 December 2018; accepted 4 February 2019

KEYWORDS Abstract. This study proposes a structure for graphene spaser (surface plasmon ampli cation by stimulated emission of radiation) and develops an electrostatic model for Plasmonics; quantizing plasmonic modes. Using this model, one can analyze any spaser consisting of Graphene; graphene in the electrostatic regime. The proposed structure is investigated analytically Spaser; and the spasing condition is derived. We show that spasing can occur at some frequencies Quantum optics; where the quality factor of plasmonic modes is higher than some particular minimum Quantum wire. values. Finally, an algorithmic design procedure is proposed by which one can design a viable structure at a given frequency. As an example, a spaser with plasmon energy of 0.1 eV is designed.

1. Introduction smaller than their wavelengths. In 2003, Stockman and Bergman published the rst paper about the idea and Spaser, as its name suggests, is a counterpart of laser introduced the word spaser to the literature [1]. Since in sub-wavelength dimensions. The di erence between then, many people and groups have focused on ana- spaser and laser is that the latter emits photons, while lyzing and realizing the spaser. In 2009, Noginov et al. the former emits intense coherent Surface Plasmons demonstrated an experimental spaser using an aqueous (SPs). This idea was emerged after trying to overcome solution of gold nanoparticles, each surrounded by dye- the main shortcoming of laser. Emission of photons doped silica shell as a gain medium [2]. In 2010, in laser restricts its use in small dimensions due to Stockman proposed a plasmon ampli er using spaser the di raction limit of light. The electromagnetic eld and analyzed its equation of motion using optical of photons cannot be concentrated in spots that are Bloch equations. The author claimed that the spaser smaller than half their wavelength, qualitatively. This could not be analyzed classically [3]. Zhong and Li is a fundamental theoretical limit and thus, cannot tried to analyze the spaser semi-classically in 2013 [4]. be circumvented. So, spaser inventors, Bergman and Dorfman et al. focused on the full quantum mechanical Stockman, suggested using another particle, instead description of spaser in 2013 [5]. In 2014, Apalkov and of a photon, which does not have this theoretical Stockman proposed a graphene-based spaser [6]. Until constraint [1]. Their idea was to utilize the extra con- now, many papers have been published covering many ned nature of SPs. SPs can con ne in regions much aspects of spaser [7{31]. Spaser, similar to laser, consists of two main parts: a medium for supporting SP modes and an active *. Corresponding author. E-mail addresses: [email protected] (S. Behjati or gain medium. SPs can propagate along interface Ardakani); [email protected] (R. Faez) between two materials, one of which has negative dielectric constant. Metals have negative permittivity doi: 10.24200/sci.2019.50322.1635 below their plasma frequencies and thus, a majority S. Behjati Ardakani and R. Faez/Scientia Iranica, Transactions D: Computer Science & ... 27 (2020) 3084{3095 3085 of papers focus on them as a medium for supporting and propagating plasmon modes. However, metals are not the ideal ones. Metal losses avoid plasmons to propagate along long distances. In this paper, we will use graphene instead of metal. Graphene is a material that forms by a 2D arrangement of carbon atoms in a honeycomb lattice bonding by strong sp2 hybridized covalent bonds [32]. The pz electrons of carbons, lying in orbitals, give the graphene some extraordinary electronic properties, making it an interesting potential candidate in many applications [33{35]. The graphene electrons, near Dirac points, have a linear dispersion; thus, they behave like massless Dirac fermions. Plasmons can propagate along and con ne close to graphene about Figure 1. The proposed structure. The materials which an order of magnitude stronger than metals [36]. are included in the structure are distinguished by di erent The active medium provides the energy required colors. The shown structure is embedded in a matrix of for initiating and maintaining the spasing process. The InP. main factor in choosing the active medium is that which pumping mechanism we wish to use. Similar Table 1. Physical parameters of materials which are used to laser, pumping method can be optical, chemical, in this paper. All the alloys are chosen such that to be electrical, and so forth. In our research, we are going to lattice matched with InP at 295 K. Dielectric constants of use electrical pumping method by utilizing a Quantum ternary alloys are calculated using interpolation Wire (QW) as the gain medium. method [37,38]. In this table, r, m , and m0 represent In this paper, the full quantum mechanical ap- dielectric constant, electron's e ective mass, and electron proach is used for analyzing the structure. The mass, respectively. most important quantity in quantum mechanics is the Material r m =m0 system's Hamiltonian. The Hamiltonian of the entire InP 12.56 0.077 system is H = Hsp +Ham +Hint, where Hsp, Ham, and Al0:48In0:52As 12.46 0.075 H are SP, active medium, and interaction Hamilto- int Ga In As 13.60 0.041 nians, respectively. The individual Hamiltonian parts 0:47 0:53 are quantized in the subsequent sections. The paper is organized as follows: Section 2 inner rod only senses the weak tail of SPs' eld. For introduces our proposed structure which will be used numerical calculations, the speci c material system, throughout the paper. Section 3 is devoted to the Al0:48In0:52As/Ga0:47In0:53As, is used. All the materi- Hamiltonian of SP eld and its quantization. Sec- als are chosen such that they are lattice matched to the tion 4 concentrates on active medium, and Section 5 is matrix, InP, at room temperature 295 K. The required dedicated to the interaction mechanism and derivation physical parameters are listed in Table 1. of the spasing condition. Moreover, in this section, a The graphene tube will support plasmonic modes. procedure for designing the structure is suggested. The cylindrical symmetry of the con guration makes it possible to derive plasmonic modes, analytically. In 2. The main structure addition, choosing the cylindrical structure has the ad- vantage of dealing with fewer geometrical parameters, Our proposed structure consists of a graphene-coated i.e., tube's radius, to design. tube made of layered semiconductor heterostructure, The QW is used as an active medium to provide as shown in Figure 1. The average dielectric constant energy for plasmons. By applying electric potential of materials inside the tube is 1. The heterostructure di erence between graphene and QW, the electrons makes up of two layers of semiconductors with di erent in QW excite. Depending on the degree of coupling energy gaps. The energy gap of inner rod is lower strength between QW and SPs, the energy can in- than that of outer shell so that the heterostructure terchange among electrons and SPs. The oscillation forms a QW system. The inner rod plays the role of of energy exchange can continue steadily under some QW and the outer shell is its barrier. The graphene- conditions. The next sections deal with nding this coated tube is embedded in a matrix with dielectric condition. constant 2. Regarding the extra-con ne nature of Throughout the paper, the tube is assumed to SPs, it can be assumed that 1 is equal to the outer be in nitely long such that the edge e ects can be shell dielectric constant because, roughly speaking, the neglected, and also its radius to be large enough 3086 S. Behjati Ardakani and R. Faez/Scientia Iranica, Transactions D: Computer Science & ... 27 (2020) 3084{3095 such that size quantization e ects do not in uence its is calculation of modes. The quasi-electrostatic ap- conductivity, signi cantly. proximation is utilized throughout the paper. It can be shown that this approximation describes plasmon 3. SP Hamiltonian modes quite well [40]. We will discuss the approximations in Appendix A in detail. For quantizing the SP Hamiltonian, the orthogonal Considering the cylindrical symmetry of the struc- potential modes of the structure should be derived. ture, the following ansatz can be used for the electric Therefore, this section is divided into two subsections. potential: The rst subsection deals with extracting the potential ( modes and the second one is about writing SP Hamil- AmIm(k) exp i(kz+m'!k;mt) a tonian in the quantized form. (r; t)= BmKm(k) exp i(kz+m'!k;mt) >a (5) 3.1. Graphene tube's plasmonic modes Before dealing with derivation of modes, the surface where Am and Bm are dependent arbitrary coecients conductivity of graphene is introduced. This quantity to be determined; k, m, !k;m,Im, and Km are mode can be derived using the well-known linear response indices, the corresponding frequencies, and m'th order theory, Kubo formula, and Random Phase Approxi- modi ed Bessel functions of rst and second kind, mation (RPA). In the low momentum regime, which respectively. In this paper, , , and z symbols means de nitely not for the ultra-low doping [39] are reserved for radial and angular coordinates in cases, the rst order or local response approximation cylindrical coordinate system. The unit vector along of conductivity leads to (!) = intra(!) + inter(!), any direction is denoted by adding a hat symbol above where: the vector associated with that direction; moreover, 2 the hat symbol is reused for representing operators, 2e kBT i EF = ln 2 cosh ; (1) without adding any ambiguity. Furthermore, without intra ~2 ! + i 1 2k T B any loss of generality, it is assumed that graphene tube and: is oriented along the z direction.

e2 For the ansatz to be a valid solution, it must ful ll = H(!=2) boundary conditions. After application of potential inter 4~ continuity across the boundary, = a, the following Z ! is obtained: 4i(! + i 1) 1 [H() H(!=2)] d + ; (! + i 1)2 42 Am Km(ka) 0 (2) = : (6) Bm Im(ka) with the following de nition: By using Ohm's law, Js = 2DEt (where Js, 2D, and sinh(~=k T ) H() = B : (3) Et are surface current density, surface conductivity, cosh(EF=kBT ) + cosh(~=kBT ) and tangential electric eld, respectively) and exploiting current continuity equation, the following relation In the above relations, e, kB, ~, T , EF, and ' 0:4 ps [6] are elementary charge, Boltzmann, and for the surface charge is derived: reduced Planck constants, temperature, Fermi en- (! ) m2 ergy, and electron relaxation time, respectively. In = 2D k;m A I (ka) + k2 s m m 2 i!k;m a circumstances where ~! < 2EF and ~! < ~!oph (where ~! ' 0:2 eV is optical phonon energy in oph exp i(kz + m' ! t) + c.c.; (7) graphene), Drude-like pro le is a good approximation k;m for graphene's conductivity [36]: where c.c. stands for the complex conjugate of previous e2E i terms. By using Eq. (7) and substituting it in per- = F : (4) ~2 ! + i 1 pendicular electric eld boundary condition, another relation for coecients is derived: Out of the speci ed range of frequencies, two main 2 damping channels are opened. For ~! > 2EF the kI0 (ka) 2D(!k;m) m + k2 I (ka) B 0 1 m i! a2 m most remarkable mechanism is due to vertical Landau m = k;m : A kK0 (ka) damping, and for ~! > ~!oph, optical phonon damping m 0 2 m (8) manifests itself. Just for simplicity of analyses, we stick to the range where these damping mechanisms are not In the above relation, 0 is vacuum permittivity and a trouble. primes denote derivation with respect to the argument. After this short introduction to graphene's con- Combining Eq. (6) with Eq. (8) leads to the dispersion ductivity, we turn back to the main target, which relation of plasmons: S. Behjati Ardakani and R. Faez/Scientia Iranica, Transactions D: Computer Science & ... 27 (2020) 3084{3095 3087

0 0 Im(ka) Km(ka) (! ) 1 2 2D k;m = Im(ka) Km(ka) ka: (9) 2 2 i!k;m0a [m + (ka) ] Until now, no assumption has been used for surface conductivity pro le. Thus, the derived dispersion relation is generalizable to any arbitrary pro le of surface conductivity and is not restricted to graphene. For further simpli cation, we assume Drude approximation of graphene conductivity, Eq. (4), in the lossless regime, i.e., ! 1: s e2E 1 ! (k) = F p ; (10) m 2 ~ 0a m(ka) where:

0 0 Im(x) Km(x) 1 2 (x) x Im(x) Km(x) : (11) m (m2 + x2)

If we further assume 1 = 2 , then the result is more simpli ed: e2E !2 = F g (x); (12) k;m 2 m ~ a0 where: 2 2 gm(x) = (m + x )Im(x)Km(x): (13) In the derivation of the above relation, the Wron- skian property of the modi ed Bessel functions, 0 0 Figure 2. Plasmons dispersion for di erent Fermi Im(x)Km(x)Im(x)Km(x) = 1=x, is utilized [41]. This result is exactly the same as the one derived from [42] energies. In deriving these curves, The Drude approximation is assumed: (a) Normalized plasmon by a completely di erent method using zeros of RPA energy as a function of normalized wavenumber and (b) dielectric constant. Figure 2 shows dispersion curves, normalized plasmon energy versus dimensionless ka. Eq. (12), for some lower order modes. In drawing this These gures show that for large enough wavenumber, all gure, it is assumed that a = 100 nm and EF = the modes become degenerate. 0:4 eV. Figure 2(a) depicts ~! normalized to Fermi energy versus k normalized to Fermi wavenumber, kF = small argument approximation of the modi ed Bessel 6 EF=~vF , where vF = 10 m/s is Fermi velocity, and functions is used, I0(x) ! 1 and K0(x) ! ln x, Figure 2(b) draws normalized plasmon energy versus then it turns out that low momentum plasmons in ka. From this gure, it can be seen that there are graphene tube behave almost like 1D plasmons, i.e. two main regions that can be discussed. For large p !0(k) ' !pka ln ka, where plasma frequency is k=kF's (especially in this case k & 0:2kF or equivalently 2 1=2 de ned as !p = (e n0=me) , as in 3D case except for ~! & 0:2EF) all the modes, regardless of the value of m, me which must be replaced by our speci c de nition become degenerate and have the square root feature of made later in Subsection 3.2. In case of m 6= 0, conventional 2D plasmons. This result was predictable the situation is di erent. It would be interesting to a priori, because for large enough wavenumbers, SPs nd a relation for !m(0). By exploiting the small are mostly con ned to the graphene and do not sense argument approximation of modi ed Bessel functions, tube's radius, practically. The asymptotic form of it can be shownp that gm(0) = m=2 and therefore dispersion relation, assuming Drude-like conductivity, ! (0) = ! m=2, roughly speaking, like the rst- is: m p s order approximation of 3D plasmons. 2 p It is worth mentioning that these features of e EF !(k) = k: (14) dispersion are mostly due to the 2D nature of graphene 2~2 0 rather than its peculiar band-structure [40], and the This result resembles that of the extended graphene. results are almost the same for other 2D materials. Let's look at the gure more precisely for small values It must be noted that for the Drude approxima- of k=kF. There are two di erent categories of behaviors tion to be applicable, EF must lie in one of the following for m = 0 and m 6= 0. In case of m = 0, if the two regions: 3088 S. Behjati Ardakani and R. Faez/Scientia Iranica, Transactions D: Computer Science & ... 27 (2020) 3084{3095

0:080 di erent values of E using Eq. (18). If Drude 0:1 < EF < ; (15) F ek conductivity is inserted into Eq. (18), the following ek linear approximation can be made: < EF < 0:1: (16) 80 !(k) Q ' : (19) 1 These intervals are obtained if we consider that ~! < 2 2EF and ~! < ~!oph ' 0:2 eV. The rst interval is The linear behavior is apparent from Figure 4(b) in the obtained if 2EF > ~!oph and the second when 2EF < frequency range where Drude model is more accurate, ~!oph. At the above intervals, EF is in eV unit. i.e., ~!=EF 2. Finally, the normalized potential can be written in the following form: 3.2. Quantization of SP Hamiltonian In the electrostatic regime, the SP Hamiltonian is 8 Im(k) composed of kinetic and potential parts, H = H + <> exp i(kz+m'!k;mt) a sp kin Im(ka) H , where for the potential part: (r; t)= pot > Z : Km(k) (17) K (ka) exp i(kz+m'!k;mt) > a 1 2 m Hpot = sd r; (20) 2 S The normalized potential pro les of rst four lower g order modes are shown in Figure 3. such that s is the surface charge density of graphene The other important parameter, which will be due to the existence of plasmons and is total electric encountered later in this work, is the Quality factor potential. The integration runs over the graphene's of modes. If we write (k) = !(k) i (k) and surface, Sg. The kinetic part is [43]: 0 00 2D = +i , and replace !(k) by (k) in Eq. (9), Z 2D 2D 1 and then equating the real and imaginary parts of both H = n m jv j2 d2r; (21) kin 2 s0 e e sides, assuming ! (which is valid for the frequency Sg range which will be used), the Quality factor of SP where n , m and v are the surface density of modes is derived by using the de nition Q = !=2 : s0 e e electrons in equilibrium, a suggested plasmonic electron 00 (!(k)) e ective mass (not equal to common electron e ective Q(!(k)) = 2D ; (18) 20 (!(k)) mass), and average drift velocity of electrons, respec- 2D tively. The above kinetic Hamiltonian resembles that Figure 4 shows the Quality factor of SP modes as of 3D electron gas one. Indeed, we suggest using the a function of plasmon wavenumber and energy for same formulation for graphene, but with a modi ed

Figure 3. Cross-section view of the rst four normalized potential modes so that their maximums become unity. The notation Mm is used for modes, where m is the mode index de ned in the paper. S. Behjati Ardakani and R. Faez/Scientia Iranica, Transactions D: Computer Science & ... 27 (2020) 3084{3095 3089

+ k;m() = k;m() + k;m(); (24)

Im(k) k;m() = ( + a) ; (25) Im(ka)

+ Km(k) k;m() = ( a) : (26) Km(ka)

Ck;m's are expansion coecients and stands for Heaviside step function. Using this potential, surface charge density can be derived exploiting perpendicular electric eld boundary condition, X (! ) m2 = C 2D k;m + k2 s k;m 2 i!k;m a k;m

exp i(kz + m' !k;mt) + c.c. (27) The only remaining quantity is drift velocity. The drift velocity can be derived using Newton's second law, eE = medve=dt, where E = r is electric eld. After some algebra, the following result is obtained: e X hm i 1 v (r ; t) = '^ + kz^ C e k k;m m a !k;m e k;m

exp i(kz + m' !k;mt) + c.c.; (28)

where rk is the in-plane position vector. By substituting Eq. (27) and Eq. (28) into Hamiltonians, Eq. (20) and Eq. (21), and after some cumbersome algebra, the following results are found: Figure 4. The quality factor of modes as a function of A X (! ) m2 (a) wavenumber and (b) frequency for di erent Fermi H = g 2D k;m + k2 pot 2 2 i!k;m a energies. In Figure (b) one can see the range which linear k;m approximation is valid.

[Ck;mCk;m exp i(!k;m electron e ective mass. We propose nding e ective mass by equating the Drude conductivity of graphene +!k;m)t + Ck;mCk;m + c.c.; (29) to the 3D electron gas one [44]: A e2n X m2 H = g s0 + k2 2 kin 2 i0! 2m a (!) = p ; (22) e k;m ! + i C C 2 2 k;m k;m where plasma frequency is de ned by !p = e n0=0me exp i(!k;m + !k;m)t !k;m!k;m and n0 is electron number density. By equating # Eq. (22) to graphene conductivity, Eq. (4), one can 2 Ck;mCk;m nd me = ns0~ =EF. + 2 + c:c: (30) For the purpose of quantizing the plasmon eld, !k;m we write all the eld variables in the Hamiltonian, Throughout the paper, we assume that A and L are Eq. (20) and Eq. (21), as a linear combination of g the hypothetical area and length of graphene tube, plasmon modes. Therefore, the electric potential can respectively, and k and m run over all possible index be written in the following form: values. In deriving the above relations, the following X orthogonality properties are exploited: (r; t)= Ck;mk;m() exp i(kz+m'!k;mt)+c.c.; Z k;m (23) L=2 i(kk0)z e dz = Lk;k0 ; (31) where: L=2 3090 S. Behjati Ardakani and R. Faez/Scientia Iranica, Transactions D: Computer Science & ... 27 (2020) 3084{3095

Z 2 i(mm0)' 4. Active medium Hamiltonian e d' = 2m;m0 ; (32) 0 In the present work, we propose utilizing a QW of where represents Kronecker delta function. By radius b and volume VQW as the gain medium. For combining Eq. (29) with Eq. (30) and assuming that further investigation of the structure, energy levels and !k;m = !k;m the SP Hamiltonian is obtained: wavefunctions should be derived using the well-known A X (! ) m2 Schrodingerequation. For the sake of simplicity and H = g 2D k;m + k2 sp 2 obtaining a rule of thumb, appropriate for design pur- 2 i!k;m a k;m poses, the in nite wall boundary condition is applied. The wavefunctions of such a structure are: C C + C C : (33) k;m k;m k;m k;m 8 exp i(k z+n) >p z xnl The above Hamiltonian is analogous to harmonic os- > Jn b b < VQWJn+1(xnl) cillator's one such that by the following substitution (; ; z) = (43) nl > and assuming negligible damping, Hsp recasts to the :> operator form: 0 > b C ! (! )^a ; (34) k;m k;m k;m k;m where nl, xnl, and kz are nl'th eigenfunction, the l'th zero of n'th order Bessel function of the rst y Ck;m ! k;m(!k;m)^ak;m; (35) kind, and wavenumber along the longitudinal direction, where k;m is de ned as follows: respectively. The rst four lower order modes are !1=2 sketched in Figure 5. The eigenenergy associated with ~!2 ~2k2 k;m 2 2 2 z (! ) = (m =a + k ) : nl'th mode is E = Enl + Ec, where Ec = 2m and k;m k;m A j00 j (! ) w g 2D k;m (36) energy levels Enl are:

Using these relations, the SP Hamiltonian is simpli ed 2 2 ~ xnl in the following operator form: Enl = ; (44) 2m b2 X w ^ ~!k;m y y Hsp = a^k;ma^k;m +a ^k;ma^k;m 2 where mw is the electron's e ective mass in the wire. k;m Maximum coupling between the QW and graphene is achieved when both are in resonance with X 1 = ~! a^y a^ + ; (37) each other, i.e., ~! = E E , where ! , E , and k;m k;m k;m 2 sp e g sp e k;m Eg are SP's angular frequency, and excited and ground states energy, respectively. Thus, for design purposes, b wherea ^ anda ^y are annihilation and creation km km maybe chosen such that the energy di erence between operators of an SP in the mode k, m, respectively, and excited and ground states coincides with plasmon obey bosonic algebra [45]: h i energy. After some substitution and rearranging, the y following result is found for QW's radius: a^k;m; a^k0;m0 = k;k0 m;m0 ; (38) s [â ; a^ 0 0 ] = 0; (39) ~ k;m k ;m b = x2 x2 : (45) nele ng lg h i 2mw!sp y y a^k;m; a^k0;m0 = 0: (40) Another important quantity with a vital role in the By substituting Eq. (36) in Eq. (23), the electric eld next section is the dipole moment. The value of dipole operator is obtained: moment represents how much the coupling strength is. X ^ It can be shown that for our structure, dipole moments E (r; t) = k;m(!k;m) k;m only have nonzero values between states which have the same quantum number n. In addition, it is simple to h i M (r)â (t) + M (r)ây (t) ; show that dipole moment has only radial component. k;m k;m k;m k;m 0 (41) Dipole moment between nl and nl states is: where: dnlnl0 = 2ebfnll0 ;^ (46) im M (r) = 0 ()^ + ()' ^ + ik ()z^: k;m k;m k;m k;m where: (42) Z1 In Eq. (41), we switched to the Heisenberg picture, 1 2 where the time dependence is completely transferred fnll0 = Jn(xnl)Jn(xnl0 ) d; Jn+1(xnl)Jn+1(xnl0 ) to the operators. 0 (47) S. Behjati Ardakani and R. Faez/Scientia Iranica, Transactions D: Computer Science & ... 27 (2020) 3084{3095 3091

Figure 5. Cross-section view of the absolute moduli squared of the rst four lower order eigenfunctions. The notation

Mnl shows the mode with quantum number nl discussed in the paper. is a dimensionless number, which depends on n, l, As a rule of thumb, the more the distance between QW 0 and l and it is independent of b. If 01 and 02 are and graphene, the more accurate the results will be. By considered as the states which are in resonance with using this assumption, the interaction Hamiltonian can ^ ^ ^ ^ ^ a speci c plasmon mode, then this number is equal to be written as Hint = d E, where d and E are dipole 0:09722. moment and electric eld operators, respectively. d^ can According to the spectral decomposition theo- be written down in the following form [45]: rem [46], the active medium Hamiltonian, in the basis ^ which diagonalizes itself, can be written in the following d = dqp (^+ + ^) ; (52) form: X where d is a dipole matrix element, associated with ^ qp Ham = Eiîi; (48) p ! q transition, and ^+ = jqihpj and ^ = jpihqj i are rising and lowering ladder operators, respectively. where i is a representative of all the discrete and After using these relations and considering energy con- continuous quantum numbers and runs over all the servation, the interaction Hamiltonian can be written possible states and îi = jiihij. If we assume that the as follows: only transition, strongly coupled to the plasmon eld, ^ y is p ! q, and further, setting the zero level of energy to Hint = ~ kmqp(r0)^+a^ + kmqp(r0)â ^ ; (53) the halfway between these two states, then the active medium Hamiltonian can be written in the simple form: where r0 is position vector of dipole and Rabi fre- ^ ^ quency, kmqp = d E=~ [45], is written as: ^ ~!qp Ham = ^z; (49) 2 (! ) (r ) = k;m k;m d M (r ): (54) where the following de nitions are used: kmqp 0 ~ qp k;m 0 ^ = jqihqj jpihpj ; (50) z Taking into account that dqp has only a component along the radial direction and substituting Eq. (42) into ~!qp = Eq Ep: (51) Eq. (54), the following result is obtained: 8 (! ) I0 (k ) 5. Interaction Hamiltonian and spasing k;m k;m m 0 > dqpk 0 < a < ~ Im(ka) (r ) = (55) We assume that the active medium can be approxi- kmqp 0 > : (! ) K0 (k ) mated as a dipole. Accuracy of this approximation k;m k;m m 0 ~ dqpk K (ka) 0 > a depends on how large the multipole terms are, relative m to dipole term, in the potential multipole expansion. where 0 is QW radial position. 3092 S. Behjati Ardakani and R. Faez/Scientia Iranica, Transactions D: Computer Science & ... 27 (2020) 3084{3095

The spasing condition can be written as follows [3,6]:

0 2 X ( km + qp) 2 0 j kmqpj qp; 0 2 2 km ( +qp) +(!qp !k;m) km kz (56)

0 where qp, and km are the damping rate of polar- ization and plasmon mode k, m, respectively, and kz runs over all possible transverse wavenumbers. By substituting the rabi frequency, Eq. (55), into spasing condition, Eq. (56), and assuming near resonance region, after some manipulations, we nd that for spasing to be able to occur, the quality factor of SP modes should be higher than Qmin:

22a~ Z2 (ka) Figure 6. The solid, dashed, dotted, and dash-dotted Q = 0 qp m min 2 02 lines show the quality factor of modes versus plasmon jdqpj kFkZ (k0) m energy for di erent values of Fermi level as a parameter. 0 0 Circle-marked line represents minimum required value of Im(ka) Km(ka) 1 2 ; (57) quality factor. Im(ka) Km(ka) or: After this long discussion, we have arrived at 2 2 the point where we can design a graphene tube-based ae vF qpZm(ka) Qmin = 0 spaser by using derived formulas. We intend to propose jd j2 !2 Z 2(k ) qp k;m m 0 a design procedure: I0 (ka) K0 (ka) 1. For a given ! , calculate b from Eq. (45); m m ; (58) sp 1 I (ka) 2 K (ka) m m 2. Calculate a using Eq. (62) so that !sp > !th or where: equivalently: ( 2 2 Im(x) 0 < a b fnll0 3 Zm(x) = (59) a < !sp: Km(x) 0 > a: vF qp

For large values of k, which is the case for SPs, the 3. Determine EFk using Eq. (14); condition for minimum quality factor reduces to: 4. Derive EF and k separately so that the validation 2 ranges, Eqs. (15) or (16), are satis ed. 4 a~0qp Qmin ' 2 ; (60) jdqpj kFk There exists a freedom for assigning EF and k values, separately, as long as the validation range is ful lled. or as a function of angular frequency: Increasing k con nes SPs more and more; thus, EF can 2 2ae vF qp avF qp be utilized for changing spot size. Qmin ' = : (61) 2 2 2 2 2 As an example, we design a spaser for ~! = jdqpj ! (k) 2fnll0 b ! (k) sp 0:1 eV. The calculated b for this speci c frequency is Figure 6 illustrates, graphically, which frequency re- 15.1 nm, the maximum value for a is 1.73 m and gions are allowable for spasing. The gure shows 7 EFk = 4:33 10 . The maximum value for EFk is that for a xed E , there can exist an intersection 7 F 17:39 10 provided that we choose EF > 0:1 eV, point between Q and Q curves, which from now min Eq. (15). As long as EF > 0:1 eV, we can change EF to on we call it threshold frequency and denote it by focus SPs beam. The smallest spot is obtained where ! . Spasing can occur for frequencies higher than th EF = 0:1 eV. In drawing Figure 6, we use these values ! . This dependency can be driven out analytically th for a and b and further assume that qp = 3:6 meV [6]. by combining Eq. (61) with Eq. (4): av 6. Conclusion ! = F qp : (62) th b2f 2 nll0 In summary, this paper suggested a structure for It can be seen that threshold frequency depends on the spasing. The structure was analyzed theoretically ratio of the tube's radius and cross-section area of the using full quantum mechanical approach, which treats QW. both the eld and matter quantum mechanically. In S. Behjati Ardakani and R. Faez/Scientia Iranica, Transactions D: Computer Science & ... 27 (2020) 3084{3095 3093 quantizing the Surface Plasmon (SP) eld and also in 13. Totero Gongora, J.S., Miroshnichenko, A.E., Kivshar, writing the kinetic energy of electrons inside graphene, Y.S., and Fratalocchi, A. \Energy equipartition and a special e ective mass was de ned. The spasing unidirectional emission in a spaser nanolaser", Laser condition for the structure was derived by quantizing & Photonics Reviews, 10(3), pp. 432{440 (2016). the Hamiltonian of the system. Finally, a design 14. Richter, M., Gegg, M., Theuerholz, T.S., and Knorr, procedure was proposed and a spaser for plasmon A. \Numerically exact solution of the many emitter- energy of ~!sp = 0:1 eV was designed. Throughout cavity laser problem: Application to the fully quan- the paper, the electrostatic approximation was used. tized spaser emission", Physical Review B, 91(3), p. 035306 (2015). References 15. 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Rupasinghe, C., Rukhlenko, I.D., and Premaratne, M. modeling of mesoscopic CQED systems", In Ultra- \Spaser made of graphene and carbon nanotubes", fast Phenomena and Nanophotonics XIX, 9361, p. ACS Nano, 8(3), pp. 2431{2438 (2014). 93610Y. International Society for Optics and Photon- ics (2015). 12. Jayasekara, C., Premaratne, M., Stockman, M.I., and Gunapala, S.D. \Multimode analysis of highly tunable, 25. Petrosyan, L. and Shahbazyan, T. \Spaser quenching quantum cascade powered, circular graphene spaser", by o resonant plasmon modes", Physical Review B, Journal of Applied Physics, 118(17), p. 173101 (2015). 96(7), p. 075423 (2017). 3094 S. Behjati Ardakani and R. Faez/Scientia Iranica, Transactions D: Computer Science & ... 27 (2020) 3084{3095

26. Warnakula, T., Stockman, M.I., and Premaratne, M. 40. Christensen, T., From Classical to Quantum Plasmon- \Improved scheme for modeling a spaser made of ics in Three and Two Dimensions, Springer (2017). identical gain elements", JOSA B, 35(6), pp. 1397{ 41. Abramowitz, M. and Stegun, I.A., Handbook of Math- 1407 (2018). ematical Functions: With Formulas, Graphs, and 27. Zheng, C., Jia, T., Zhao, H., Zhang, S., Feng, D., and Mathematical Tables, 55, Courier Corporation (1964). Sun, Z. \Low threshold tunable spaser based on mul- 42. Longe, P. and Bose, S. \Collective excitations in tipolar fano resonances in disk-ring plasmonic nanos- metallic graphene tubules", Physical Review B, 48(24), tructures", Journal of Physics D: Applied Physics, p. 18239 (1993). 49(1), p. 015101 (2015). 43. Arista, N.R. and Fuentes, M.A. \Interaction of charged 28. Wan, M., Gu, P., Liu, W., Chen, Z., and Wang, Z. particles with surface plasmons in cylindrical channels \Low threshold spaser based on deep-subwavelength in solids", Physical Review B, 63(16), p. 165401 (2001). spherical hyperbolic metamaterial cavities", Applied Physics Letters, 110(3), p. 031103 (2017). 44. Ashcroft, N. and Mermin, N., Solid State Physics, HRW International Editions, Holt, Rinehart and Win- 29. Song, P., Wang, J.H., Zhang, M., Yang, F., Lu, H.J., ston (1976). URL https://books.google.com/books? Kang, B., Xu, J.J., and Chen, H.Y. \Three-level spaser id=1C9HAQAAIAAJ. for next-generation luminescent nanoprobe", Science Advances, 4(8), p. eaat0292 (2018). 45. Scully, M. and Zubairy, M., Quantum Optics, Cambridge University Press (1997). URL 30. Ardakani, S.B. and Faez, R. \Doped silicon quan- https://books.google.com/books?id=9lkgAwAAQBAJ. tum dots as sources of coherent surface plasmons", Journal of Optics, 20(12), p. 125001 (2018). URL 46. Hassani, S., Mathematical Physics: A Modern In- http://stacks.iop.org/2040-8986/20/i=12/a=125001 troduction to Its Foundations, Springer Interna- tional Publishing (2013), URL https://books.google. 31. Ardakani, S.B. and Faez, R., A Tunable Spherical com/books?id=uRa4BAAAQBAJ. Graphene Spaser, arXiv preprint arXiv:1712.01322 (2017). 47. Mikhailov, S. and Ziegler, K. \New electromagnetic 32. Novoselov, K.S., Geim, A.K., Morozov, S., Jiang, D., mode in graphene", Physical Review Letters, 99(1), p. Katsnelson, M., Grigorieva, I., Dubonos, S. and Firsov, 016803 (2007). A. \Two-dimensional gas of massless dirac fermions in 48. Gao, Y., Ren, G., Zhu, B., Liu, H., Lian, Y., and Jian, graphene", Nature, 438(7065), pp. 197{200 (2005). S. \Analytical model for plasmon modes in graphene- 33. Majdi, M. and Fathi, D. \Graphene-based nano bio- coated nanowire", Optics Express, 22(20), pp. 24322{ sensor: Sensitivity improvement", Scientia Iranica, 24331 (2014). 24(6), pp. 3531{3535 (2017). 49. Jalali-Mola, Z. and Jafari, S. \Electromagnetic modes 34. Faramarzi, V., Ahmadi, V., Ghane Golmohamadi, F., from stoner enhancement: Graphene as a case study", and Fotouhi, B. \A biosensor based on plasmonic wave Journal of Magnetism and Magnetic Materials, 471, excitation with di ractive grating structure", Scientia pp. 220{235 (2019). Iranica, 24(6), pp. 3441{3447 (2017). 35. Derakhshi, M. and Fathi, D. \Terahertz plasmonic Appendix A. Beyond exploited approximations switch based on periodic array of graphene/silicon", Scientia Iranica, 24(6), pp. 3452{3457 (2017). As we mentioned in the text, electrostatic treatment can describe most features of plasmons. But, for a 36. Jablan, M., Buljan, H., and Soljacic,M. \Plasmonics while, let's examine what happens if full electrodynam- in graphene at infrared frequencies", Physical Review ics is considered, qualitatively. The rst and most in- B, 80(24), p. 245435 (2009). teractive feature is the appearance of TE mode, which 37. Chuang, S., Physics of Photonic Devices, Wiley Series is not predictable by electrostatics [47,48]. However, it in Pure and Applied Optics, John Wiley & Sons (2009). is worthwhile to mention several important notes about URL https://books.google.com/books?id=x5Cd TE mode in graphene tube: PDf1kC. 38. Vurgaftman, I., Meyer, J., and Ram-Mohan, L. \Band 1. While TE and TM modes are completely dis- parameters for iii{v compound semiconductors and tinguished in extended graphene, it is not so in their alloys", Journal of Applied Physics, 89(11), pp. graphene tube. Actually, there are no pure TE 5815{5875 (2001). and TM modes in graphene tube, except for the 39. Elias, D., Gorbachev, R., Mayorov, A., Morozov, S., case of total azimuthal symmetry, i.e., m = 0. All Zhukov, A., Blake, P., Ponomarenko, L., Grigorieva, I., the other m 6= 0 modes are essentially hybrid in Novoselov, K., Guinea, F., et al. \Dirac cones reshaped character. m = 0 requires some further attention. by interaction e ects in suspended graphene", Nature By solving Maxwell equations, it can be shown that Physics, 7(9), p. 701 (2011). for m = 0, dispersion relation reduces to [48]: S. Behjati Ardakani and R. Faez/Scientia Iranica, Transactions D: Computer Science & ... 27 (2020) 3084{3095 3095

only complicates the analysis while achieving noth- ing more than what has been achieved by neglecting it; 3. It can be proven that TE mode exists for the cases where the absolute value of the di erence between two dielectric constants surrounding graphene is quite small, which is practically hard to achieve [40]. Even if we could nd two materials with very close dielectric constants, there is no guarantee that small perturbations in dielectric constants, which are common in reality, keep the criterion satis ed; 4. In case of TE mode, we have to solve full vectorial Maxwell equations. One may propose considering the TE mode by inserting it to the derived electrostatic relations, manually. Although it seems a good idea, it is absolutely wrong. Because the Figure A.1. Quality factor of TE mode as a function of eigen-solutions of Laplace equation build an orthog- normalized frequency in the valid range. The curves onal basis and thus, inserting the TE mode makes belong to di erent Fermi energies speci ed on the legend. the basis over-complete which is unacceptable and may lead to unphysical results. I ( a) K ( a) 1 1 1 + 2 1 2 1 I0(1a) 2 K0(2a) j!0 Apart from TE mode which is a consequence of electromagnetic consideration, there exists a new hybrid mode I ( a) K ( a) for the undoped graphene that is absent normally. 0 1 + 0 2 j! = 0; 1 I ( a) 2 K ( a) 0 This is due to the addition of spin- ip corrections to 1 1 1 2 (A.1) the response function [49]. It can be shown that for the doped case, spin- ip excitations only slightly shift where is vacuum permeability and = (k2 0 i plasmons energy towards generically lower values [49]. !2 )1=2. 0 i 0 However, fortunately, it is not a concern of our work Apparently, TE and TM modes are decoupled. because we decided not to use undoped graphene. The rst parenthesis on LHS led to the TM disper- Finally, it can be shown that in graphene tubes, sion relation (for k !2 this equation matches 0 0 the condition k a 1 should be satis ed for the Eq. (9) for m = 0) and likewise, the second one to F derived dispersion relation to be valid [42]. It can that of TE mode. So, the TE dispersion relation is: be checked in the nal step of design procedure. For I ( a) K ( a) instance, considering EF = 0:1 eV, the condition means 0 1 + 0 2 = j! : (A.2) that a 6:9 nm, far below our designed radius, which 1 I ( a) 2 K ( a) 0 1 1 1 2 is in the order of micrometer. It is clear that TE mode exists only if = < 0, which corresponds to ~! & 1:6671EF; Biographies 2. The Quality factor of TE mode can be found Sadreddin Behjati Ardakani received his BSc and straightforwardly, using Eq. (A.2), the same as the MSc degrees both in Electrical and Electronics En- procedure that has already been done in deriving gineering from Amirkabir University of Technology, Eq. (18). Doing some algebra leads to the following Tehran, Iran in 2009 and 2011, respectively, and result: his PhD degree in Micro- and Nano-Electronics at 00 Sharif University of Technology, Tehran, Iran, in 2019. Q = (00 < 0): (A.3) His current research interests include nanooptics and TE 20 optoelectronics. Figure A.1 shows the Quality factor of TE mode as a function of frequency in the speci ed range, Rahim Faez received his BS degree from Sharif 1:6671 < ~!=EF < 2 where TE mode exists and for University of Technology in 1977 and the MS and PhD di erent values of Fermi Energies. By comparing degrees from UCLA in 1979 and 1985, respectively. this gure with Figure 4(b), it is obvious that Then, he joined Sharif University of Technology and Quality factor of TE mode is an order of magnitude currently he is an Associate Professor. His research less than those of other TM and hybrid modes. interests include design and simulation of advanced Therefore, including the TE mode in calculations semiconductor nano and quantum devices.