An electrically-driven Carbon nanotube-based plasmonic laser on Silicon

Photonic signal processing requires efficient on-chip light sources with higher modulation bandwidths. Todays conventional fastest semiconductor diode lasers exhibit modulation speeds only on the order of a few tens of GHz due to gain compression effects and parasitic electrical capacitances. Here we theoretically show an electrically-driven Carbon nanotube (CNT)-based laser utilizing strong light-matter-interaction via monolithic integration into Silicon photonic crystal nanobeam (PCNB) cavities. The laser is formed by single-walled CNTs inside a combo-cavity consisting of both a plasmonic metal-oxide-semiconductor hybrid mode embedded in the one dimensional PCNB cavity. The emission originates from interband recombinations of electrostatically-doped nanotubes depending on the tubes chirality towards matching the C-band. Our simulation results show that the laser operates at telecom frequencies resulting in a power output>3 (100) uW and>100 (1000)GHz modulation speed at 1x (10x) threshold. Such monolithic integration schemes provide an alternative promising approach for light source in future photonics integrated circuits.


Introduction
Semiconducting single-walled Carbon nanotubes (CNTs) are being recently explored for photonic integrated circuits due to their unique electronic and optical properties [1,2]. Light amplification in Carbon nanotubes was experimentally demonstrated in the near-infrared wavelength range at cryo [3] and room temperatures [4], as a single photon emitter through dimensionality modification [5], by tuning the direct band-gap, controlling excitonic recombinations, and enabling exciton radiatively-decaying. Device examples of light emission from CNTs have previously demonstrated a p-n diode [6,7], tube to waveguide-coupling [8,9], flat plane-emission panels [10], and flexible light-emitting sources [11]. However, CNTsbased laser devices operating at a telecom wavelength, which are desired for on-chip optical interconnects, are not reported to date. Carbon nanotubes sorting (i.e. semiconducting, diameter, or single chirality) and placement (i.e. position precisely at a predefined location and orientation) are two of the key challengers in the development of CNT-based optoelectronic devices [Tule14]. For the sorting, surfactant-based separation solutions are utilized counting on CNTs post-growth processing through electronic type and diameter, such as density gradient ultracentrifugation technique [33,34], showing a semiconducting purity of >99%, column chromatography method [35] due to metallic and semiconducting CNTs' moving at different rates for separation. Other types of polymer extractions techniques are also effective in sorting CNTs, for instance, large (1.2~1.5 nm) and small (0.6 ~ 1.0 nm) diameter CNTs from solution can be successfully extracted by the addition of water-soluble polymers [36]. In terms of the CNTs replacement, up to date two different placement strategies are classified depending on CNTs' growth, purification, and placement accomplished either in one step or in three completely separated process [32]. The aim is to enable the sorted CNTs to transfer on a complementary metaloxide-semiconductor (MOS) compatible substrate. Among these methods, directed assembly using dielectrophoresis including alternating current [37], and radio frequency [38] exhibits a promising method for alignment of CNTs between metal contacts, where a large scale assembly of individual CNTs can bridge each electrode pair.
Compared to conventional bulk Silicon MOS field-effect transistors (FET), CNTFET exhibits superior performance due to its transconductance and drive currents by a factor of a few per unit width, making it an attractive alternative to Silicon [40]. CNT field-effect transistors were first demonstrated as early as 1998 [41,42]. With applying proper bias scheme these CNTFET create p-n junction and behave as diode device, operating more closely as rectifiers with a forward bias and limited current flow with the reverse direction. However, here we focus on electrically-induced light emission (i.e. electroluminescence) with a gain option from carbon nanotubes for laser applications. Different aspects of light emission mechanisms depend on CNT device structures, such as using various gate configurations (e.g. bottom gate, and top split-gate). Optical emission, which originates from radiative recombination of electrons and holes simultaneously injected into the undoped nanotube, was first observed from a three-terminal ambipolar type CNTFET having with a forward-biased pn junction [43]. However, two-terminal CNT-based light emitting diodes are usually the basic building block in modern optoelectronic circuits due to their significant advantages (e.g. lower power consumption and cost, relative simpler drive circuitry) as light sources [7], which is used in this work. Basically light emission from CNT devices involves radiative combination of electron and holes, either as free carriers or bound in the form of excitons.
A laser is constructed from three principal parts including gain medium, optical cavity, and pump source (either optical or electrical). The observation of optical gain in semiconducting single-walled CNTs is of great importance to the proper design of laser devices. Fortunately, the significant optical gain in (8, 7) single-walled CNTs embedded in host polymer thin film was experimentally demonstrated at a wavelength of 1.3 µm at room temperature [4], showing that carbon nanotubes are able to amplify light. Therefore, here a laser can be potentially obtained by inserting single-walled CNTs material (i.e. gain medium) into the optical cavity (e.g. photonic crystal nanobeam cavity for our case). Lasing effect may be achieved as the optical gain exceeds a threshold value determined by the cavity loss mechanism resulting from stimulated absorption and intrinsic loss.
With the aim to design a CNT-based laser, a significant challenge is the inherently small overlap factor between the tube's gain material with the optical mode, requiring light-matter interaction (LMI) enhancement techniques. Next we briefly outline some LMI options to be considered including one-dimensional (1-D) interference grating (i.e. distributed Bragg reflectors), photonic crystal, metal-clad, and plasmonic [12][13][14][15]. Regarding the latter, the metal-oxide-semiconductor (MOS) configuration can support a hybrid plasmon-polariton (HPP) waveguide mode, where the peak of the electric field intensity is mainly concentrated in the thin oxide gap, which can be collocated with the CNT gain material (i.e. placing the CNT inside the oxide gap) [16]. This mode provides synergies relating to photonic integration and active optoelectronics [17], such as enhanced LMIs via deep sub-diffraction limited modes, seamless access to semiconductors and integration with the Silicon-on-insulator (SOI)-platform for low loss routing. A 1-D photonic crystal nanobeam (PCNB) cavity operates as a Fabry-Perot-like resonator, offering optical confinement between Bragg mirrors consisting of a periodic array of air holes along the waveguide direction. For instance, an electrically driven, room-temperature 1-D PCNB laser with 0.35 ( ⁄ ) 3 mode volume was demonstrated at a lasing wavelength of 1578 nm [18].In this work we aim to deploy strongly enhanced LMIs by using both the 1-D PCNB and the plasmonic MOS mode simultaneously towards realizing a high gain material-mode overlap for a CNT-based integrated nanolaser. We recently show a 44 times enhanced interaction strength for a square plasmon resonator with III-V materials embedded in a Silicon-based PCNB cavity, due to the highly compressed mode volume compared to the inline plasmon resonator without the cavity [19].
Towards enhancing the LMI between the CNTs and a cavity, we combine the MOS structure with the PCNB cavity and placing single-walled CNTs inside this combo-cavity. We theoretically show this approach for CNTs-based lasers to be seamlessly integrated into onchip Silicon waveguides delivering potential high modulation bandwidth for planar chip architectures. Investigations of the cavity quality ( ) factor and Purcell factor, result in laser performance as derived by the light-matter interaction modified rate equations that outperforms classical laser devices. Here the cavity length of =260 nm is optimized in our previous work [19], and the cavity height of =220 nm is held constant for the compatibility of commercially available SOI wafers. A photonic ridge waveguide on SOI with the cross-section of height ( ) 220 nm and width ( ) 400 nm supporting a transverse-electric (TE) mode is deployed as a core building block for the laser design. The photonic crystal mirror and the taper parameters, including the hole period ( ), hole radius ( ), minimum hole spacing in the taper section ( ), and number of taper and mirror pairs ( , ), are optimized for a highest factor by sweeping , , and . This design is performed by using a commercial software package FDTD Solutions distributed by Lumerical. The input of complex refractive indices (i.e. and ) of Gold, SiO2, and Si are taken from the solver's built-in material database. A reasonable high cavity of ~2 × 10 4 is found as =380 nm, =350 nm, =0.2 , =8, and =10 for a cavity resonant wavelength of ~1550 nm. Next we embed the plasmonic MOS mode into the PCNB cavity towards enhancing the LMI. Inside this high electric field of the combo-cavity we inserted 10 single-walled CNTs (the chiral number of (9,2), the diameter of ~1.0 nm, and the bandgap of ~0.85 eV) at the collocated with a thin oxide layer (Fig. 1a, b). Excitation of semiconducting CNTs can be done either optically or electrically. However, electrically pumping is more preferred for our structure since the metal pad forming the plasmonic mode can be conveniently used as a gate electrode to electrostatically dope the CNTs. Here the excitation of CNTs is considered driven electrically via a p-n junction at the nanotubes. Light created by spontaneous emission through electrons and holes recombination has a fixed polarization state along with long-axis of carbon nanotube [43,44]. Hence we treat the generated light classically using electromagnetic point dipole source. The CNT emitters were first modeled by a dipole source with all 3 spatial positions and 9 polarization orientations (Fig. 1c). Among them, we found the -polarized dipole source excitation is preferred for a PCNB cavity that is typically compatible with TE polarized light supported in the photonic SOI ridge waveguide. In addition, here single-walled CNTs are parallel-aligned along the coordinate axis inside the oxide layer, which physically meet the requirement of polarized electroluminescence emission in single-walled CNTs along the axis of the nanotube. The resulting electroluminescence of the nanotubes is generated in the thin oxide layer forming a hybrid HPP waveguide mode, which contributes to a PCNB cavity lasing mode. The transmission (reflection) spectrum was recorded at the output (input) port, respectively. At the resonant frequency of ~197 THz, showing ~60% transmitted light, we thus conclude that the lasing power can be ~60% efficiently coupled out along the photonic rib waveguides (Fig. 1d, e). Our device requires 10 single-walled CNTs with pitch variation less than ~5 nm in the oxide layer. Experimentally we prefer to choose the separation method of CNT placement from solution due to the advantage of intending to select highly purified semiconducting single-walled CNTs and placing them onto a substrate with a specific pitch and orientation. Towards addressing the feasibility of placement of single chirality CNTs, here we can deploy the dielectrophoretic assembly method combining with polymer-mediated chirality sorting [39], showing an example of seven electrode pairs successfully bridged by an array of single chirality (9, 7) single-walled CNTs among the 10 electrode pairs. Note, the unbridged parts are caused by the nanotubes' length in the solution shorter than the electrode gap. The further experiment can be improved by narrowing the length distribution of CNTs in solution [39], such as by density gradient ultracentrifugation separating single-walled CNTs ranging in average length from <50 nm to ∼2 μm [34].  3 ], is the effective mode volume, is the resonant free space wavelength of the cavity, and ( ) is the effective cavity index. A rather classical approach is to enhance the cavity factor [24]. However, this is somewhat unpractical due to the required increased wafer space and the lower modulation speed for lasers with high-cavity (i.e. long photon lifetimes). The second possible approach is to decrease the . Since is ultimately limited in practice by these factors of bandwidth, material absorption, and fabrication tolerance, here we show that minimizing for a given is a preferred solution. The internal dynamics leading towards the laser threshold are more efficiently utilized as the optical mode volume is smaller (i.e. higher , and spontaneous emission coupling factor, ), and the smaller mode volume translates into a low pump power requirement to reach threshold [17,19]. Here we find a relatively large for a reasonable by scanning the oxide thickness and the cavity length, respectively (Fig. 2). Since depends on the polarization of dipole source excitation (i.e. orientation is preferred) and the position of dipole source, we purposely place a dipole source at the peak of the electric field in the cavity (e.g. the position i or iii in Fig. 1c as we refer to Fig. 3c). A high Purcell factor of ~300, which is similar for a two-dimensional photonic crystal slab cavity [25], can be achieved due to the combo-cavity effect [19]. However the latter relies on a high which introduces the aforementioned photon lifetime, footprint, and potential wavelength stabilization restrictions. Note, the maximum value of the LMI are observed at =5 nm and =260 nm, respectively, owing to the corresponding smallest cavity mode volumes (i.e. ~0.8 ( 2 ⁄ ) 3 ) observed (dashed line Fig. 2a, b). Using this configuration, the plasmonic cavity exhibits a lasing peak wavelength of ~1522 nm (Fig. 1d). We conclude that a high Purcell factor can be achieved at a modest cavity , leading to a broader bandwidth and thus enabling broadband light sources with a high spontaneous emission rate [26], due to the relatively high coupling of CNTs emitter to the cavity.

Carbon nanotube laser performance
The Purcell effect enables the CNTs-based PCNB laser to significantly improve its performance via increasing the LMI, and hence the photon built-up efficiency (i.e. -factor) inside the laser cavity. Here, we are particularly interested in the power output and the modulation speed characteristics of the Carbon-gain material driven laser. The steady state rate equations are utilized under continuous pumping without considering non-radiative recombination rate (Eq. 1, 2) [27], and the power output, , is related to the photon number derived from the rate equations, yet associated with the other parameters from the previous optical simulation results (Eq. 3) [28].
where is the injection current, and = , is the pump rate, is the current injection efficiency, and we use the electroluminescence efficiency, =1.0×10 -4 of a CNT [6], is the active gain volume, here it is the volume of single-walled CNTs, is the photon number of a single lasing mode, is the carrier density, is the spontaneous emission rate and is enhanced by the Purcell effect via = , where is the natural spontaneous emission rate of the material, and = 1 0 ⁄ , 0 is the spontaneous emission lifetime. Key for a fast gain re-modulation are the spontaneous emission lifetime, here of CNT, which is in the range of 20~200 ps [6], and the short photon lifetime of the plasmonic cavity (  Q), and here 0 =100 ps. is the spontaneous emission coupling factor, Γ quantifies the overlap between the spatial distribution of Carbon nanotube relative to a lasing mode, and Γ = 5% is estimated from the ratio between the area of 10 pieces of Carbon nanotube placed side by side and a ~200 nm 2 cross-section of a hybrid plasmonic mode. is the total cavity loss rate per unit volume, 0 is the carrier density at transparency, and 0 4.9×10 -13 /cm 3 is used for chiral (9,2) Carbon nanotubes [29]. is the waveguide transmission efficiency of the PCNB cavity, is the mirror loss, is the intrinsic loss of the cavity, is the photon life time, and is proportional to the cavity (i.e. = /(2 ), is the cavity resonant frequency), ℎ is the planck constant, is the light speed in vacuum, is the lasing wavelength, is the effective optical mode volume, and ℎ is the photon density. Here we introduce a penetration length, , into the PCNB cavity due to the undefined cavity length between the two Bragg mirror sections. The effective cavity length is = + 2 , where can be written by where is the group index of the cavity mode cycling in the resonator, the derivative of phase delay ( / ) is related to the factor (Eq. 5), is the modal reflectivity. Using both Eq. 4 and 5, can be calculated and the effective cavity volume is thus evaluated by = . The photon density may be further estimated via ℎ = / . For the CNT laser we obtain the output power of about 3 (100) µW at a 1.0 (10) of the threshold pump rate (Fig. 4a). This is remarkable given the small gain volume, but can be understood by the high photon density (e.g. ~2×10 17 /cm 3 at the threshold) inside the oxide layer of the laser cavity [19]. Below the threshold current (i.e. ~970 µA calculated for our case) the cavity laser behaves as an amplified spontaneous emission light source, showing the power output less than 3 µW with the injection current in the range of 0~1000 µA (inset Fig.  4a).
Higher modulation frequencies of directly-modulated semiconductor lasers allowing larger data rates are desired in relatively short-distance data transmissions. However, conventional semiconductor laser sources have their bandwidths limited to around 40 GHz due to gain compression effects and parasitic electrical capacitances. The 3-dB role off modulation bandwidth ( 3 , defined as the frequency at which the response function decays to half of its zero-frequency value) is estimated through the small signal response (direct modulation) of the CNT laser by observing the spectral response function [31], where is the optical cavity angular frequency, = + Γ (1 − 0 + 0 ), and 2 = Γ [ (1 + 0 ) − (1 − )Γ 0 ] , 0 and 0 are the steady-state photon number and population inversion number, respectively, and = 1/ , Γ is the transition rate of excited state population, which is equal to the spontaneous emission rate, . The frequency response of the device below lasing threshold is also calculated using spontaneous emission lifetime, which is equal to carrier lifetime but neglecting non-radiative recombination lifetime. Here we theoretically show the frequency response of the device with up to 2 times of the threshold pump rate, delivering a 3-dB bandwidth of more than 100 (400) GHz at a 1.0 (2.0) of the threshold pump rate (Fig. 4b). The modulation bandwidth increases with higher injection current, which can be understood as an interplay between photonic and electronic rates of both the cavity and the external pump (i.e. driving current). If the internal laser cavity is fast enough, the higher pump rate drives the gain medium faster into population inversion. Given the lossy plasmonics cavity, this inversion is rapidly depleted and hence can be re-excited more promptly compared to larger diffraction limited devices.

Sensitivity to fabrication imperfections
The deviation in hole size can lead to various effects on the optical properties of the photonic crystal laser. Firstly, it can cause shifts in the position of the photonic bandgap, altering the range of wavelengths that the laser can efficiently emit or reflect. This shift can have implications for the laser's output characteristics and operating parameters. Furthermore, the broadening of the bandgap due to hole size deviations can limit the laser's ability to effectively block specific wavelengths, affecting its performance in applications requiring precise control over the emission spectrum. Additionally, altered dispersion characteristics and group velocity can impact the laser's propagation properties, leading to dispersioninduced broadening and changes in pulse characteristics.
To mitigate the effects of hole size deviations, it is crucial to implement tighter fabrication tolerances during the manufacturing process. This requires advancements in fabrication techniques, such as lithography and etching, to achieve greater precision in hole dimensions.
Design optimization that considers anticipated hole size deviations can also help in minimizing their impact. Post-fabrication characterization plays a vital role in assessing the extent of hole size deviations and their influence on the laser's performance. Understanding and addressing hole size deviations are critical steps in achieving the desired optical properties and optimal performance in 1D photonic crystal lasers. By refining fabrication techniques, optimizing design parameters, and conducting thorough characterization, we can enhance the reliability and functionality of photonic crystal lasers for various applications. Fig. 5. Illustrates the effect of random imperfections in nanoholes on the performance of an electrically driven nanobeam laser. Despite a significant variation of 20 nm in hole size from the ideal design, lasing at room temperature is expected, highlighting the robustness of the laser against these imperfections.

Conclusion
In conclusion, our theoretical investigation focuses on plasmonic photonic crystal hybrid lasers utilizing Carbon nanotubes (CNTs) as the gain material. These hybrid lasers offer significant potential as high-performance on-chip light sources for telecom applications. Simulation results demonstrate that the hybrid HPP (Hybrid Plasmon-Photon) waveguide mode, generated from the emission of CNTs, exhibits strong coupling efficiency of approximately 60% with the 1-D photonic crystal cavity lasing mode. This coupling enables efficient energy transfer and enhanced light emission within the device. Compared to gain compression-limited devices, our proposed light source enables faster modulation due to two key factors: the strong Purcell effect, resulting in an enhancement factor of approximately 300 (Fp), and the short spontaneous emission lifetime of CNTs. These combined characteristics allow for modulation speeds of hundreds of GHz with a 3dB roll-off, along with tens of microwatts of optical power above the laser threshold. The integration of CNT internal processes with the plasmonic cavity architecture presents an alternative approach for the realization of active components in next-generation photonic circuits. The monolithic integration schemes proposed in this study offer a promising path for the development of advanced photonic devices, enabling improved performance, faster modulation speeds, and higher optical power output.