TMDC-Based Topological Nanospaser: Single and Double Threshold Behavior

TMDC-based topological nanospaser: single and double threshold behavior

Rupesh Ghimire,∗ Jhih-Sheng Wu,† Fatemeh Nematollahi,‡ Vadym Apalkov,§ and Mark I. Stockman¶ Center for Nano-Optics (CeNO) and Department of Physics and Astronomy, Georgia State University, Atlanta, Georgia 30303 (Dated: January 19, 2021)

We theoretically study a topological nanospaser, which consists of a silver nanospheroid and MoS2 monolayer flake of a circular shape. The metal nanospheroid acts as a plasmonic nanoresonator that supports two rotating modes, which are coupled to the corresponding valleys of MoS2. We apply external circularly polarized light that selectively pumps only one of the valleys of MoS2. The generated spaser dynamics strongly depends on the size (radius) of the MoS2 nanoflake. For small radius, the system has only one spasing regime when only chirally-matched plasmon mode is generated, while at larger size of MoS2, depending on the pump intensity, there are two regimes. In one regime, only the chirally-matched plasmon mode is generated, while in the other regime both chirally-matched and chirally-mismatched modes exist. Different regimes of spaser operation have also opposite handedness of the far-field radiated of the spaser system. Such topological nanospaser has potential applications in different areas of infrared spectroscopy, sensing, probing, and biomedical treatment.

I. INTRODUCTION element[27, 44]. The size of such spaser can be as small as tens of nanometers. The advent of spaser (surface plasmon amplificaiton The other type of spaser that consists of a nanorod by stimulated emission of radiation) stems from a work (gain medium) placed near a plasmonic metal has been published in 2003 [3], an idea, which initiated a distinct proposed in Ref. [30]. Such spaser is generally a mi- new approach for generation and amplification of nanolo- crometer in size and can be used as a good source of calized fields. Spaser itself has seen tremendous develop- the far-field radiation. The operation of such plasmonic ment due to its boundless possibilities. Such metal res- nanolasers is based on the same principles as the ones of onator system allows one to scale down the spatial size of the nanoshell spaser but with one key difference. Namely, a laser and, at the same time, to satisfy all the ingredients their plasmonic modes are not the localized modes as of a perfect lasing system, i.e., optical confinement, feed- in the case of the nanoshell spaser but the propagating back, and thermal management. After the first experi- modes, which are called the surface plasmon polaritions mental observation of a spaser [29, 30], many theoritical (SSPs) [23–25, 40, 47, 49]. developments have set in to the scene [2, 12, 20, 36, 50], A spaser design, which is based on a periodic array of which ushered to plethora of new designs and applica- nano-resonators that have coherent extended collective tions. The operational frequencies of spasers cover a wide modes, has been proposed in Ref.[16, 31, 50]. Such spaser range of spectrum from infrared to near-untraviolet and consists of either an array of nanocavities embedded in this is one of the key factors of its rapid emergence in the a metal film [39] or a periodic set of metal nanoparticles optical research [7, 13, 19, 22, 25, 26, 38, 39, 49]. suspended in a gain medium [51]. For a special arrange- Spaser, apart from fundamental research, has proved ment of metal nanoparticles, the system has topologically itself viable for different potential applications as an ele- nontrivial collective plasmonic excitations. The corre- ment of opto-electrionic systems [8–10, 18, 28, 33–35], as sponding spaser, which is called a topological spaser of a sensor of chemical and biological agents [6, 41, 42, 52], type I, has been proposed in Ref. [45]. In such spaser, and as a biological probe [11, 14] for therapeutics and two types of metal nanoshells form the honeycomb crys- diagnostics of different diseases, e.g., cancer. tal structure with the broken inversion symmetry. The arXiv:2101.07094v1 [physics.optics] 18 Jan 2021 The first experimentally realized spaser is a nanoshell collective plasmonic excitations of such system have two spaser[29]. It consists of a metal nanocore sorrounded types of topologically nontrivial chiral modes with op- by a dielectric gain medium. The metal core supports posite chiralities, i.e., opposite topological charges. The the plasmonic modes, which optically interact with the modes are located at the K and K0 valleys in the corre- gain medium. The gain medium usually consists of sponding reciprocal space[45]. In such topological spaser, dye molecules [14, 29], however, different materials such the nontrivial topology is introduced into the plasmonic as perovskites and 2D systems can be used as a gain excitations of the system. There is also another type of topological spaser, called the topological spaser of type II[15], in which the plas- ∗ [email protected] monic system is topologically trivial but the gain medium † [email protected] has nontrivial topology. The example of such gain ‡ [email protected] medium is a nanopatch of a transition-metal dichalco- § [email protected] genide (TMDC) monolayer. In the reciprocal space, ¶ [email protected] TMDC monolayer has two valleys, K and K0, which have 2 chiral electron states and are characterized by nonzero II. MODEL AND MAIN EQUATIONS and opposite topological charges. Plasmonic modes in such nanospaser are generated in a metal nanoparticle The topological nanospaser of type II consists of a that is placed atop of TMDC nanopatch. Thus the topol- metal nanospheroid placed on the top of a TMDC ogy in the system is introduced only through the gain nanopatch - see Fig. 1(a). We assume that the medium. We have shown in Ref. [15] that the plasmonic nanospheroid has an oblate shape with the radius of 12 modes of such nanospaser are topologically protected and nm and the height of 1.2 nm. The TMDC nanopatch determined by the topological charge of the excited val- has a circular shape with the radius that is considered ley of TMDC nanopatch. The topological spaser of type as a parameter below. The whole system is placed in II has been studied in Ref. [15] for small size of TMDC dielectric medium with dielectric constant = 2. nanopatch, which is comparable to the size of the metal The TMDC monolayer has two valleys, K and K0, nanoparticle. In the present paper we extend this the- which have opposite topological charges (chiralities). oretical analysis to large sizes of TMDC monolayer and When TMDC monolayer is illuminated with circularly show that the topological nanospaser under this condi- polarized light, only one of the valleys with the chiral- tion has very rich dynamics with different types of plas- ity that corresponds to the handedness of the circularly monic modes that are generated. pilarized light is excited [5, 48]. For concreteness, we assume that circularly polarized light excites only the K valley [see Fig. 1(b)]. We characterize such continuous excitation by the corresponding rate gK. The metal nanospheroid has asimuthal symmetry and the corresponding plasmonic modes are characterized by azimuthal quantum number m. Because of the chiral nature of TMDC valleys, the K valley is predominantly coupled to the m = 1 plasmon mode, while the K0 valley is mainly coupled to the m = −1 mode of the nanospheroid. In Fig. 1(b), these couplings are marked as the direct couplings. The selection rules corresponding to the direct couplings are exact at the center of TMDC nanopatch (~r = 0) and approximate at any other points within TMDC. As a result there are also cross couplings of the plasmonic modes and TMDC valleys, which are shown schematically in Fig. 1(b). Thus in the topological nanospaser of type II, the circularly polarized light excites one of the valleys (say K) of TMDC monolayer (gain medium), which then excites the m = 1 plasmonic mode through the direct coupling. Such mode is also coupled to the other valley (K0) through the cross coupling and excites electrons in the K0 valley, which finally excites the m = −1 plasmonic mode through the direct coupling. Thus all modes of the system are coupled and the final kinetics of the system strongly depends on its parameters, e.g., the radius of TMDC nanopatch, which controls the magnitudes of both direct and cross couplings. Within the quasistatic approximation, the plasmonic FIG. 1. Schematics of the topological spaser. (a) Spaser con- eigenmodes of nanospheroid, φm(r), satisfy the following sists of a silver nanospheroid placed on the top of TMDC equation[37] nanoflake of a circular shape. The silver nanospheroid has 2 oblate shape with radius 12 nm and height 1.2 nm. (b) ∂ ∂ ∂ θ (r) ϕn (r) − sn ϕn (r) = 0, (1) Schematic of spaser operation. A circular-polarized light ex- ∂r ∂r ∂r2 Z cites the valley with the chirality that match the light helic- 2 3 ity. The metal nanospheroid supports two plasmon modes θ(r)|∇φm(r)| d r All Space with azimuthal quantum numbers m = −1 and m = 1. The sn = Z , (2) stimulated CB→VB transitions at the corresponding K or |∇φ (r)|2d3r 0 m K points couple to these plasmon modes through direct and All Space cross couplings. where sn is the eigenvalue of the nth mode, θ is a charac- terstic function which is 1 inside and 0 outside the metal. For the oblite nanospheroid, which has azimuthal symme- 3 try, the eigenmodes are characterized by two spheroidal the SP frequency ωsp is equal to the transition frequency quantum numbers[43]: multipolarity l = 1, 2,... and az- ω21. imuthal or magnetic quantum number m = 0, ±1,... . In The spaser dynamics is described within the semi- our case the relevant modes are the dipole modes with classically approach. Within this approach the plasmonic l = 1 and m = ±1. system is treated classically, i.e., the creation and annihi- † ∗ The Hamiltonian of our system has the following form lation operators (âm = am anda ˆm = am) are considered as complex numbers, and, at the same time, the electron H = HSP + Hgain + Hint, (3) system in TMDC (gain medium) is described quantum mechanically using the density matrixρ ˆ (r, t). In the where H is the Hamiltonian of surface plasmons (SP) K SP rotating wave approximation (RWA) [1, 4] the density (nanospheroid), H is the Hamiltonian of TMDC gain matrix has the following form nanopatch (gain medium), and Hint describes the interaction between the plasmonic system and the gain (c) iωt ! ρK (r, t) ρK(r, t)e medium. ρˆK(r, t) = . (9) ρ∗ (r, t)e−iωt ρ(v)(r, t) The Hamiltonian of SP, HSP , in the quantum mechan- K K ical description, is given by the following expression Then the corresponding semiclassical equations can be X ∗ derived from Hamiltonian (3), HSP = ~ωsp aˆmaˆm, (4) m=±1 a˙ m = [i(ω − ωsp) − γsp]am+ † Z X where ωsp is the SP frequency,a ˆm anda ˆm are creation iν d2r ρ∗ (r)Ω˜ ∗ (r) , (10) and annihilation operators of the corresponding plasmon K K m,K S K with quantum number m = ±1. Then the electric field operator is [3, 36] X h ˜ i n˙ K(r) = − 4 Im ρK(r)Ωm,K(r)am + X −iωt ∗ ∗ iωt Fm(r, t) = − Asp(∇φm(r)âme + ∇φm(r)âme ), m=1,−1 m=1,−1 g [1 − n (r)] − γ (r) [1 + n (r)] , (11) (5) K K 2K K s 4π s dRe[s(ω)] A = ~ n and s0 ≡ . (6) ρ˙K(r) = [−i(ω − ω21)−Γ12]ρK(r)+ sp 0 n dsn dω ω=ωsp X ˜ ∗ ∗ inK(r) Ωm,Kam , (12) The electric field depends on the position within TMDC m=1,−1 monolayer through the position dependent plasmonic where S is the area of TMDC nanoflake, Γ12 is the po- mode, φm(r). The interaction Hamiltonian describes the larization relaxation rate, gK is the pumping rate of the dipole coupling of the plasmonic excitations and the elec- K valley , and Ω˜ m,K(r) is the Rabi frequency, tron system in TMDC monolayer and is given by the following expression ˜ 1 ∗ Ωm,K(r) = − Asp∇φm(r)dK . (13) Z ~ X 2 X ˆ Hint = −νK d r Fm(r)dK(r) , (7) Here the population inversion, nK, is defined as K=K,K0 m=±1 (c) (v) 0 nK ≡ ρK − ρK , (14) where K is the valley index, K or K , νK is the density of electron states at the K valley, which can be found from and the spontaneous emission rate, γ (r), of the SPs is 12 −2 2K the experimental data[32, 46]: νK = 7.0 × 10 cm . [36] Here the interband dipole matrix element, dK, between 2 the VB and CB states of TMDC, is proportional to the 2(γsp + Γ12) X γ (r) = Ω˜ (r) non-Abelian Berry connection 2K (ω − ω )2 + (γ + Γ )2 m,K sp 21 sp 12 m=1,−1 (cv) dK = eA (k) , (15) (cv) ∂ A (k) = i uck uvk , (8) ∂k k=K III. RESULTS AND DISCUSSIONS where uck and uvk are the lattice Bloch functions. The optical transitions in TMDC nanopatch, which are the transitions between the valence band (VB) and the A. Parameters of topological spaser conduction band (CB), are characterized by the transi- 0 ton frequency ω21, which is the same for both K and K The system consists of three components: a silver valleys due to time reversal symmetry of the system. Be- nanospheroid, a TMDC monolayer nanoflake, and a sur- low we choose the parameters of nanospheroid for which rounding dielectric medium. A silver nanospheroid has 4

FIG. 2. The real part (a) and imaginary part (b) of the Rabi frequency. The Rabi frequency determines the coupling of the plasmon mode m and the K or K0 valleys of TMDC. The radius of the metal spheroid is a = 12 nm. an oblate shape with the radius of 12 nm and the height of B. The dynamics of topological spaser 1.2 nm. These are the semi-major (a) and semi-minor (c) axes of an oblate system. We choose a MoS2 monolayer We solve the system of Eqs. (10)-(49) numerically with as a TMDC material. The MoS nanoflake has a circu- 2 2 the given initial conditions, which are Nm=1 = |a1| = 9 lar shape and is placed on the top of the nanospheroid. (a1 = 3), i.e., there are nine m = 1 plasmons, and the Below we change the radius of MoS2 nanoflake. Both conduction band populations of both K and K0 valley are the nanospheroid and MoS2 monolayer are suspended in zero. We obtain the spaser dynamics for different values dielectric medium with the permittivity of d = 2. of the gain, g, and different radii of TMDC nanoflakes. The height of nanospheroid, c, is chosen in a such way Here we assume that only one valley, i.e., the K valley, that the SP frequency ωsp matches the dipole transi- is pumped by the circularly polarized light. tion frequency ω21 = ∆g/~ of MoS2. Here ∆g is the The dynamics of the spaser is completely determined bandgap of MoS2 monolayer. The band structure of by the coupling of plasmonic excitations and the TMDC MoS2 was calculated within the three-band tight bind- system. Such coupling is characterized by the Rabi fre- ing model[21]. From this model we obtained both the quency, Ω˜ m,K(r), which is a function of the radius vec- bandgap and the dipole matrix elements between the tor, r, within TMDC nanoflake and also depends on the 0 conduction and valence bands at the K and K points: type of the plasmon, m = 1 or m = −1, and the val- 0 ∆g = 1.66 eV; dK = 17√.7 e+ D, and dK0 = 17.7 e− D, ley of TMDC, K or K . The real and imaginary parts of where e± = (ex ± iey) / 2 are the chiral unit vectors. Ω˜ m,K(r) are shown in Fig. 2 for different combinations of The other parameters, which have been used in our cal- m and TMDC valley. The radius of metal nanospheroid, −1 culations, are the coherent relaxaton rate Γ12 = 15.1 ps which is 12 nm in the x-y plane, determines two different for MoS2 monolayer and the plasmon relaxation rate for dependences of Ω˜ m,K(r). If r < 12 nm, then the Rabi −1 silver nanoparticle, γsp =13.4 ps . We also assume that frequency, both the real and imaginary parts, is isotropic only the K valley is pumped by external circularly po- function of radius. It mainly follows the ”angular mo- larized light, i.e., gK = g and gK0 = 0. With the above mentum” selection rule, i.e., m = 1 is coupled to the K parameters, from the nonlinear system of Eqs. (10)-(49), valley, while m = −1 is coupled to the K0 valley. This we obtain both the the stationary solution and the kinet- selection is exact at r = 0, but for r > 0 it is a good ics of topological spaser for given initial conditions. approximation. 5

For r > 12 nm, i.e., a point is outside the metal To illustrate different regimes of spaser dynamics, we nanospheroid in the x-y plane, the plasmonic electric field show in Fig. 4 the distributions of the population inver- 0 has a dipole nature. As a result, Ω˜ m,K(r) behaves com- sions at the K and K valleys for the radius of TMDC pletely differently. Namely, both Ω˜ m=1,K0 and Ω˜ m=−1,K flake of 16 nm and different values of gain. If the gain is less than the first threshold, g < g , then no plas- are large, while Ω˜ m=−1,K0 and Ω˜ m=1,K are small. Also, th,1 because of the dipole nature of the plasmonic electric mons are generated and the population inversion of the field, the Rabi frequency acquires strong angular depen- K valley is close to one, while the population inversion 0 dence of type exp(2iϕ) - see Fig. 2, where ϕ is the polar of the K valley is exactly -1, i.e., the valence band is in-plane angle. completely occupied and the conduction band is empty. Thus, from the properties of the Rabi frequency, we This case is shown in Fig. 4(a)-(b). can conclude that if the radius of TMDC nanoflake is In Fig. 4(c)-(d) the gain is greater than gth,1 but less less than 12 nm, then m = 1 plasmon mode is mainly than gth,2. In this case only one plasmon mode, m = 1, coupled to the TMDC valley of the same chirality, i.e, is generated. The distribution of population inversion is the K valley, while m = −1 plasmon mode - to the K0 isotropic and it is close to zero for r < 12 nm for the 0 valley. But if the radius of TMDC is greater than 12 nm, K valley and for r > 12 nm for the K valley, which then m = 1 and m = −1 plasmon modes are coupled illustrates strong coupling of these spatial regions to the to both K and K0 valleys. The larger the radius of the m = 1 mode -see also Fig. 2. TMDC flake, the stronger the coupling of the plasmonic Another possibility, when the gain is greater than gth,2, mode to the TMDC valley of opposite chirality. is shown in Fig. 4(e)-(f). Under this condition, both 2 The number of plasmons, Nm = |am| , as a function m = 1 and m = −1 plasmon modes are generated. They gain, g, is shown in Fig. 3 (a)-(c) for three different radii are coupled to both valleys at all spatial regions (r < 12 of TMDC nanoflake. The solid and dashed lines corre- nm and r > 12 nm), as a result, the population inver- spond to m = 1 and m = −1 plasmons, respectively. sions for both K and K0 valleys are close to zero. Because If r = 12 nm then only co-rotating (m = 1) plasmon of coexistence of two plasmon modes, the resulting elec- mode, i.e., the mode that is strongly coupled to the ex- tric field show interference features and the correspond- cited K valley, is generated. There is a characteristic ing population inversion distribution is anisotropic - see −1 spaser threshold, gth ≈ 20 ps , when the plasmon mode Fig. 4(c)-(d). starts generating. The large radius of TMDC nanoflake, r = 18 nm, is For a larger radius, r = 16 nm, see Fig. 3(b), the illustrated in Fig. 5. In this case there is only one thresh- system show different behavior. Now, the K valley is old. If the gain is less than the threshold, see Fig. 5(a)- coupled to both co-rotating m = 1 and counter-rotating (b), then no plasmons are generated and the conduction m = −1 modes (although the coupling to the counter- band of the K valley is highly populated, while the popu- rotating mode is still relatively weak). There are two lation inversion of the K0 valley is -1 at all spatial points. thresholds, gth,1 and gth,2. At lower threshold, gth,1 ≈ 49 If the gain is greater than the threshold, see Fig. 5(c)- ps−1, only one mode, m = 1, is generated, while at larger (d), then two plasmon modes, m = −1 and m = 1, are −1 threshold, gth,2 ≈ 70 ps , both plasmon modes, m = 1 generated. The population inversion is close to zero for and m = −1, cogenerated. At the second threshold the both K and K0 valleys. The population inversion distri- energy is transferred from the m = 1 mode to m = −1 bution is also anisotropic, which is due to interference of mode so the number of m = 1 plasmons decreases. An- the plasmonic fields of two modes. other unique feature of the second regime, g > gtr,2, is The temporal dynamics of the topological spaser is that there are more counter-rotating plasmons than the shown in Fig. 6. The initial number of plasmons is nine co-rotating ones, N−1 > N1. for both modes, m = 1 and m = −1. In Fig. 6(a), the When the radius of the TMDC nanoflake increases gain is fixed, g = 82 ps−1, and the results are shown for even more, r = 18 nm, two thresholds, gth,1 and gth,2, different radii of TMDC nanoflake. For all parameters, −1 merge into a single one, gth,1 = gth,2 ≈ 52 ps , at which the number of plasmons, N1 and N−1, show similar initial two plasmon modes are generated simultaneously - see dynamics (at t . 0.15 ps). Namely, first, both N1 and Fig. 3(c). Similar to a smaller radius, the number of N−1 sharply decrease to almost zero values, then show counter-rotating plasmons is more than the number of small oscillations and finally monotonically increase to co-rotating ones. the stationary values. At the stationary stage, for small The dependence of two thresholds, gth,1 and gth,2, on radius of TMDC nanoflake, r = 14 nm, only m = 1 is the TMDC radius, r, is shown in Fig. 3(d). At r . 15 generated, while for large radius, r = 16 nm or r = 18 nm, there is only one spaser regime when only co-rotating nm, both modes, m = 1 and m = −1, are generated - see mode is generated. At 15nm . r . 17nm there are Fig. 6(a). two thresholds and the system can generate either one This property is also illustrated in Fig. 6(b), in which plasmon mode, m = 1, or two plasmons modes, m = 1 the radius is fixed, r = 16 nm, and the gain is varied. For and m = −1, depending on the gain, g. At r > 17 nm, g = 50 ps−1 and g = 60 ps−1, which are less then the two thresholds merge into a single one and there is only second threshold, only m = 1 plasmons are generated in one regime with two generated plasmon modes. the stationary regime. For large gain, g = 70 ps−1, mode 6 m = −1 coexists with m = 1 plasmonic mode.

FIG. 3. (a)-(c) Number of plasmons, Nm, as a function of gain, g. The solid and dashed lines correspond to the plasmons with m = 1 and m = −1, respectively. The radius of TMDC nanoflake is (a) 12 nm, (b) 16 nm, and (c) 18 nm. (d) The topological spaser thresholds as a function of radius of TMDC nanoflake. If gth,2 > g > gth,1 then only m = 1 plasmon mode exists in the stationary regime, while if g > gth,2 0 then both modes m = 1 and m = −1 are generated. FIG. 4. Inversion population of K and K valleys of MoS2 nanoflake with the radius of 16 nm. The gain is (a),(b) 46 ps−1, (c),(d) 49 ps−1, and (e),(f) 70 ps−1. The panels (a), (c), and (e) correspond to the K valley, while the panels (b), (d), and (f) describe the K0 valley. C. Far-Field Radiation

Although it is not its primary role, the topological nanolaser can be used as a miniature source of far-field Here the integral is calculated over the volume of the radiation, which is due to the oscillating electric dipole of nanospheroid. The electric field, Fm(r, t), inside the the spaser system. The induced electric dipole moment metal depends on the number of m = 1 and m = −1 of the system is the sum of two contributions: the dipole plasmons. moment of TMDC nanoflake, dtmdc, and the dipole mo- The x and y components of the total dipole moment, ment of the metal nanospheroid, dmetal, which is the sum of Eqs. (48) and (43), can be expressed in the following forms dtotal = dtmdc + dmetal. (16) The dipole moment of TMDC nanoflake can be ex- iωt ˜ −iωt pressed in terms of the non-diagonal part of the density dtotal,x = B1e + B1e , (19) iωt −iωt matrix of TMDC system dtotal,y = C1e + C˜1e , (20) X iωt ∗ ∗ −iωt dtmdc = (ρKdKe + ρKdKe ) K=K,K0 where, iωt ∗ ∗ = (ρKdK + ρK0 dK0 )e + (ρKdK ∗ ∗ ∗ ∗ −iωt B = −κA E V(â +a ˆ ) + f d + f 0 d , (21) + ρK0 dK0 )e . (17) 1 sp 0 1 −1 K 0 K 0 ∗ ∗ C = iκA E V(â − aˆ ) + if d − if 0 d , (22) The dipole moment of the metal nanospheroid can be 1 sp 0 1 −1 K 0 K 0 found from the known electric field [see Eq. (5)] inside the metal, Z Re[metal − d] Re[metal − d] dmetal = Fm(r, t) dr. (18) κ = (23) V 4π 4π 7

FIG. 7. Polarization ellipse of the far field radiation of topological spaser for its two regimes of continuous wave operation. (a) Radius of TMDC nanoflake is 16 nm and the gain is 49 ps−1. Only m = 1 plasmon mode is generated. The far field radiation is left circularly polarized. (b) Radius of TMDC nanoflake is 16 nm and the gain is 70 ps−1. Two plasmon modes, m = 1 and m = −1 are generated. The far field radiation is right elliptically polarized. The electric field is shown in arbitrary units.

0 FIG. 5. Inversion population of K and K valleys of MoS2 nanoflake with the radius of 18 nm. The gain is (a),(b) 49 ps−1 and (c),(d) 61 ps−1. The panels (a), (c) correspond to the K valley, while the panels (b), (d) describe the K0 valley. which the corresponding dipole moment is almost two order of magnitude larger than the dipole moment of TMDC nanoflake. The components of the dipole radiated field are proportional to the total dipole moment, Ex ∝ dtotal,x, Ey ∝ dtotal,y. The corresponding polarization ellipse is shown in Fig. 7 for two regimes of operation of the topological spaser: (a) only one m = 1 plasmon mode is generated and (b) two modes, m = 1 and m = −1, are cogenerated. For case (a), the far field radiation is left circularly polarized, which is the same polarization as the one of the pump light. This is consistent with the condition that only one plasmon mode is generated in

FIG. 6. The number of surface plasmons, Nm, as a function this case. of time t for topological spaser with MoS2 nanoflake as a gain If two plasmon modes are generated [Fig. 7(b)], then medium. The solid and dashed lines correspond to m = 1 the corresponding polarization ellipse describes the right and m = −1 plasmons, respectively. The initial number of elliptically polarized radiation. Note, that the pump light −1 plasmons is N1 = N−1 = 9. (a) The gain is g = 82 ps is left circularly polarized. The change of the handedness and the radii of TMDC nanoflake are 14 nm, 16 nm, and 18 of polarization from left to right is due to the fact that the nm. The corresponding lines are shown by different colors as number of m = −1 plasmons is greater than the number marked in the panel. (b) The radius of TMDC nanoflake is 16 nm and the gain is 50 ps−1, 60 ps−1, and 70 ps−1. The of m = 1 plasmons in the continuous wave regime of the corresponding lines are shown by different colors as marked spaser. in the panel. Thus, for a given radius of TMDC nanoflake, by changing the gain, i.e., the intensity of the circularly polarized light, we can switch the handedness of the far field radiation from left to right and vica versa. For example, if P ∗ ∗ inK(r) Ω˜ a X m=1,−1 m,K m gth1 < g < gth2 then the far field radiation is left circu- fK = −ν , (24) −(ω − ∆g) + iΓ12 larly polarized, but if gth2 < g then it is right elliptically S polarized. P ˜ ∗ ∗ X inK0 (r) m=1,−1 Ωm,K0 am fK0 = −ν , (25) −(ω − ∆g) + iΓ12 S IV. CONCLUSION dK = d0e+ & dK0 = d0e−. (26)

The derivation of Eqs. (21)- (25) is given in the Sup- A topological nanospaser of type II consists of two porting Materials. The main contribution to the total main components: a metal nanospheroid and a TMDC, dipole moment comes from the metal nanospheroid, for e.g., MoS2, monolayer flake of a circular shape. The 8 nanospheroid functions as a plasmonic nanoresonator VI. SUPPORTING INFORMATION with two relevant plasmonic modes, which rotate in the opposite directions and are characterized by azimuthal A. Metallic Oblate Spheroid: Geometry and quantum numbers m = ±1. The MoS2 monolayer is a Modes gain medium with nontrivial topology. It is placed atop 0 of a nanospheroid and has two chiral valleys, K and K . We consider an oblate spheroid, which in the Cartesian The system is pumped by a circular polarized light, which coordinate system is described by the following equation populates the conduction band states of only one valley, say the K valley. x2 + y2 z2 + = 1 , (27) Such topological spaser has been theoretically pro- a2 c2 posed in Ref. [15]. In the present paper we show that it q has very rich dynamics, which strongly depends on the where a and c are semi-axes, ε = 1 − c2 is the eccen- radius of the gain medium (TMDC nanoflake). If the ra- a2 dius of TMDC is small, then the K and K0 valleys are tricity of the spheroid. mainly coupled to the co-rotating plasmonic modes, e.g., It is convenient to introduce the spheroidal coordi- the K valley is coupled to the m = 1 mode. In this nates, ξ, η and ϕ, which are related to the Cartesian coordinates, x, y and z through the following expressio[43]: case the nanospaser has one threshold, gth, so that if the gain is larger than gth then the co-rotating plamon mode p 2 p 2 is generated. For larger radius of nanoflake, the valleys x = f ξ + 1 1 − η cos(ϕ), (28) p p of TMDC become also strongly coupled to the counter- y = f ξ2 + 1 1 − η2 sin(ϕ), (29) rotating modes, and the nanospaser has two thresholds, z = fξη, (30) gth,1 and gth,2, so that if gth,2 > g > gth,1 then only the co-rotating mode is generated, while if g > gth,2 then where 0 ≤ ξ < ∞, −1 ≤ η ≤ 1, 0 ≤ ϕ < 2π and both co-rotating and counter-rotating modes are gener- f = εa. ated. For even larger radius of TMDC, the two thresholds Then the surface plasmon eigenmodes of the metal merge into one and the nanospaser has only one regime spheroid are described by the quasistatic equation [37] when two modes, m = 1 and m = −1, are cogenerated. In this case the number of counter-rotating plasmons is 2 ∇ [θ(r)∇φm] = ssp∇ φm, (31) larger than the number of co-rotating ones. Because of that property the far-field radiation of nanospaser where ssp is the eigenvalue of the corresponding mode shows interesting behavior. Namely, by changing the gain φm. Here θ(r) is the characteristic function that is 1 strength, one can change the handedness of the far-field inside the metal and 0 elsewhere. For oblate spheroid, radiation from left to right and vice versa. the eigenmodes are characterized by multipole quantum All these unique properties of topological nanospaser number l and magnetic quantum number m. For the make it an extremely viable option for several nanoscopic relevant modes of topological nanospaser, the multipole applications. Main areas are near-field spectroscopy and quantum number is 1, l = 1. Then the corresponding sensing where a plasmon frequency of a nanospaser can eigenmodes are described by the following expressions be tuned to work at the required condition. However, the m topological nanospaser can also be used in optical inter- ( P1 (iξ) m , 0 < ξ < ξ0, m imφ P (iξ0) connects and probing. Another key area is the biomedical φm = CNP (η)e 1m (32) 1 Q1 (iξ) m , ξ0 < ξ, one where similar systems have been previously adopted Q1 (iξ0) [11, 14] for the diagnosis and therupatics of cancer. With where P m(x) and Qm(x) are the Legendre functions of added topoligical chiral benefits, this nanolaser can be l l √ 1−ε2 more effective in such detection. In addition to all these, the first and second kind, respectively, and ξ0 = ε . the topological nanospaser has also a potential as an ex- The constant CN is determined by normalization condi- cellent far-field raditation source. tion, Z 2 3 |∇φ(r)m| d r = 1. (33) V. ACKNOWLEDGMENT All Space Due to axial symmetry of the nanospheroid, the cor- Major funding was provided by Grant No. DE- responding eigenvalues, ssp, do not depend on m. They SC0007043 from the Materials Sciences and Engineering can be also found from the following expression[3, 36] Division of the Office of the Basic Energy Sciences, Office of Science, U.S. Department of Energy. Numerical simu- Z 2 3 lations have been performed using support by Grant No. θ(r)|∇φm(r)| d r All Space DE-FG02-01ER15213 from the Chemical Sciences, Bio- ssp = Z . (34) 2 3 sciences and Geosciences Division, Office of Basic Energy |∇φm(r)| d r Sciences, Office of Science, US Department of Energy. All Space 9

Semi-principal axis Dipole elements (D) Band gap TMDC c (nm) dK dK0 (eV)

MoS2 1.20 17.68e+ 17.68e− 1.66

MoSe2 1.45 19.23e+ 19.23e− 1.79

WSe2 0.85 18.38e+ 18.38e− 1.43

MoTe2 1 20.08e+ 20.08e− 1.53

TABLE I. Parameters employed in the calculations: Semi- principal axes of the spheroids, and the dipole matrix elements and band gaps of the TMDCs.

C. Far-ﬁeld radiation

The total dipole moment of the spaser can be expressed FIG. 8. Absolute value of the left-rotating chiral dipole com- in the following form ponent, d− = e+d, in MoS2 dtotal = dmetal + dtmdc, (39)

where dmetal is the dipole moment of the metal Using explicit expression (32) for φ , we derive the ﬁnal m nanospheroid and dtmdc is the dipole moment of the equation for the eigenvalue TMDC nanoﬂake.

m dP1 (x) s = dx . (35) sp dP m(x) P m(x) dQm(x) 1 1 1 dx − Qm(x) dx 1 x=iξ0 1. Dipole moment of the metal nanospheroid

To find the plasmon frequency, ωsp, and the plasmon relaxation rate, γsp, we use the following relations [3, 36]: The electric field inside the metal, which is produced ssp = Re[s(ωsp)], (36) by generated plasmon modes, both m = 1 and m = −1, Im[s(ω )] dRe[s(ω)] is uniform and is given by the following expression r = sp , s0 ≡ , (37) sp 0 sp ssp dω ω=ωsp X −iωt ∗ ∗ iωt Fm(r, t) = − Asp(∇φmaˆme + ∇φmaˆme ), where the Bergman spectral parameter is defined as m=1,−1

d (40) s(ω) = . (38) where d − m(ω) s Here is the dielectric constant of surrounding medium, 4π s(ω) d A = ~ (41) and (ω) is the dielectric function of the metal (silver). sp 0 m ds (ω) In our computations, for silver, we use the dielectric function from Ref. [17]. and s(ω) = d , (42) B. TMDC Parameters d − m(ω) Then the dipole moment of the metal nanospheroid can In our calculations the TMDC (MoS2) monolayer is be found from the following expression characterized by its bandgap and the dipole matrix elements between the conduction and valence bands at the Z 0 dmetal = µFm(r, t) dv (43) K and K points. To find these parameters we have used V a three-band tight binding model [21]. The calculated values of the transition dipole matrix elements and the where bandgap are given in Table I. The dipole matrix elements Re[metal − d] at the K and K0 points are purely chiral. They are pro- µ = . (44) −1/2 4π portional to e± = 2 (ex ± iey), where ex and ey are the Cartesian unit vectors. The plot of the absolute value Taking into account that the electric field inside the ∗ of the chiral dipole, |d±|, where d± = e±d, is shown in metal is a constant, E0 = |∇φm|, we derive the fol- Fig. 8. lowing expressions for the dipole moment of the metal 10 nanospheroid We substitute Eq. (52) into Eq. (48) and obtain the following expression for the dipole moment of TMDC −iωt ∗ iωt iωt iωt dmetal,x = −µAspE0V (â1e +a ˆ1e )+ dtmdc = fKdKe + fK0 dK0 e + h.c., (53) where the following notations were introduced −iωt ∗ iωt P ˜ ∗ ∗ (â−1e +a ˆ−1e ) (45) X inK(r) m=1,−1 Ωm,Kam fK = −ν (54) −(ω − ∆g) + iΓ12 S −iωt ∗ iωt dmetal,y = −µAspE0V i(â1e − aˆ1e )− P ˜ ∗ ∗ X inK0 (r) m=1,−1 Ωm,K0 am fK0 = −ν . (55) −(ω − ∆g) + iΓ12 −iωt ∗ iωt S i(â−1e − aˆ−1e ) (46) Taking into account that dK = d0(1, i) and dK0 = d0(1, −i) we obtain the x and y components of the dipole moment iωt iωt dtmdc,x = fKd0e + fK0 d0e + h.c. (56) 2. Dipole moment of TMDC monolayer iωt iωt dtmdc,y = ifKd0e − ifK0 d0e + h.c. (57)

The density matrix of TMDC nanoﬂake has the following structure 3. Far ﬁeld dipole radiation

(c) iωt ! ρK (r, t) ρK(r, t)e ρˆK(r, t) = . (47) The total dipole moment of the spaser system is the ρ∗ (r, t)e−iωt ρ(v)(r, t) K K sum of the dipole moment of the metal nanospheroid and TMDC nanoflake. Its x and y components can be ex- 0 where K is the valley index, K or K . The off-diagonal pressed as elements, i.e., coherences, determine the dipole moment −iωt −iωt of TMDC system dtotal,x = −µAspE0V aˆ1e +a ˆ−1e + h.c. iωt iωt + fKd0e + fK0 d0e + h.c. X X iωt ∗ ∗ −iωt dtmdc = (ρK(r)dKe +ρK(r)dKe )+h.c., (58) S K=K,K0 d = −µA E V (iaˆ e−iωt − iaˆ e−iωt + h.c. (48) total,y sp 0 1 −1 iωt iωt P + if d e − if 0 d e + h.c. where S is the sum (integral) over all points r of TMDC K 0 K 0 nanoflake. (59) These expressions have the following structure The coherences satisfy the following stationary equa- iωt tion (see Eq. (12) of the main text) dtotal,x = 2Re Bxe , (60) iωt dtotal,y = 2Re Bye , (61) X ∗ ∗ [−i(ω − ω21) − Γ12]ρK(r) + inK(r) Ω˜ (r)a = 0, m,K m where, m=1,−1 ∗ ∗ (49) Bx = −µAspE0V(â1 +a ˆ−1) + fKd0 + fK0 d0, (62) ∗ ∗ By = iµAspE0V(â1 − aˆ−1) + ifKd0 − ifK0 d0. (63) where Γ12 is the polarization relaxation rate, nK is the population inversion defined as The total dipole moment of the system determines the far-field radiation of the spaser. The polarization of ra- n ≡ ρ(c) − ρ(v) , (50) K K K diation is characterized by the x and y components of the far electric field, which are proportional to the corre- and sponding components of the dipole moment, i.e., dtotal,x and dtotal,y, while the total radiation power is given by ˜ 1 Ωm,K(r) = − Asp∇φm(r)dK . (51) the following expression ~ 4 ω 3 ( )1/2 From Eq. (49) we can find the stationary coherences of I = d h|d |2i 3 c total TMDC monolayer 0 ~ 3 1/2 8 ω (d) 2 2 P ˜ ∗ ∗ = (|Bx| + |By| ) (64) inK(r) m=1,−1 Ωm,K(r)am 3 c0 ~ ρK(r) = − . (52) −(ω − ω21) + iΓ12 where h...i means the time average. 11

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