Photonic signal processing requires efficient on-chip light sources with higher modulation bandwidths. Today’s 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) µW and > 100 (1000)’s GHz modulation speed at 1 × (10 × ) threshold. Such monolithic integration schemes provide an alternative promising approach for light source in future photonic integrated circuits.
© 2015 Optical Society of America
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  and room temperatures , as a single photon emitter through dimensionality modification , 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 , and flexible light-emitting sources . However, CNTs-based 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 challenges in the development of CNT-based optoelectronic devices . 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 [13, 14], showing a semiconducting purity of >99%, column chromatography method  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 . In terms of CNT 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 . The aim is to enable the sorted CNTs to transfer onto complementary metal-oxide-semiconductor (MOS) compatible substrates. Among these methods, directed assembly using dielectrophoresis including alternating current , and radio frequency  exhibits a promising method for alignment of CNTs between metal contacts, where a large-scale assembly of individual CNT can bridge each electrode pair.
Compared to conventional bulk Silicon MOS field-effect transistors (FET), CNTFETs exhibit superior performance due to its high transconductance and drive currents, making it an interesting alternative to Silicon . CNTFETs were first demonstrated as early as 1998 [20, 21]. With applying proper bias scheme these CNTFETs 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 p-n junction . However, two-terminal CNT-based light emitting diodes are usually the basic building block in modern optoelectronic circuits due to their lower power consumption and cost, relative simpler drive circuitry as light sources . Thus, 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 a 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 , showing that carbon nanotubes are able to amplify light. Therefore, here a laser can potentially be realized by inserting single-walled CNTs material (i.e. gain medium) into the optical cavity (e.g. photonic crystal 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 [23–26]. Regarding the latter, the metal-oxide-semiconductor 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) . This mode provides synergies relating to photonic integration and active optoelectronics , 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 mode volume was demonstrated at a lasing wavelength of 1578 nm .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 .
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 on-chip Silicon waveguides delivering potential high modulation bandwidth for planar chip architectures. Investigations of the cavity quality (Q) factor and Purcell factor, result in laser performance as derived by the light-matter interaction modified rate equations that outperforms classical laser devices.
2. Laser and cavity design
A high Q 1-D PCNB cavity without the MOS structure is first designed at a target resonant wavelength of ~1550 nm. The design process of a 1-D PCNB cavity usually consists of engineering three elements [31, 32]: (1) the photonic crystal mirror, (2) the taper section, and (3) the cavity length. Here the cavity length of L = 260 nm is optimized in our previous work , and the cavity height of H = 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 (H) 220 nm and width (W) 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 (a), hole radius (r), minimum hole spacing in the taper section (amin), and number of taper and mirror pairs (n, m), are optimized for a highest Q factor by sweeping a, r, and amin . This design is performed by using a commercial software package FDTD Solutions distributed by Lumerical. The input of complex refractive indices (i.e. n and ) of Gold, SiO2, and Si are taken from the solver’s built-in material database. A reasonable high Q cavity of ~ is found as a = 380 nm, amin = 350 nm, r = 0.2 a, n = 8, and m = 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. 1(a), 1(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 [22, 33]. 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. 1(c)). Among them, we found the y-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, single-walled CNTs are parallel-aligned along the y coordinate axis inside the oxide layer, which physically meet the requirement of polarized electroluminescence emission in the single-walled CNTs along the axis of the nanotubes. 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. 1(d), 1(e)).
Our device assumes 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 envision to deploy the dielectrophoretic assembly method combining with polymer-mediated chirality sorting , 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. Further experiment can narrow the length distribution of CNTs , such as by density gradient ultracentrifugation separating single-walled CNTs ranging in average length from <50 nm to ∼2 μm .
The Purcell factor indicates the interaction strength between photons in the cavity and the laser gain medium by quantifying the spontaneous emission rate enhancement of an emitter inside a cavity. There are two methods to increase the Purcell factor , , according to the widely used formula of , where is the diffraction-limited mode volume in a cubic half-wavelength in material, i.e. , 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 . 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 [28, 30]. 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 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. 1(c) as we refer to Fig. 3(c)). A high Purcell factor of ~300, which is similar for a two-dimensional photonic crystal slab cavity , can be achieved due to the combo-cavity effect . 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 ) observed (dashed line Fig. 2(a), 2(b)). Using this configuration, the plasmonic cavity exhibits a lasing peak wavelength of ~1522 nm (Fig. 1(d)). 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 , due to the relatively high coupling of CNTs emitter to the cavity.
3. 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)) , 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) .6], is the active gain volume, here it is the volume of single-walled CNTs, S 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 , 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 , and the short photon lifetime of the plasmonic cavity ( ∝ Q), and here = 100 ps. is the spontaneous emission coupling factor, quantifies the overlap between the spatial distribution of carbon nanotube relative to a lasing mode, and = 5% is estimated from the ratio between the area of 10 pieces of carbon nanotube placed side by side and a ~200 nm2 cross-section of a hybrid plasmonic mode. is the total cavity loss rate per unit volume, is the carrier density at transparency, and ≈4.9 × 10−13/cm3 is used for chiral (9,2) carbon nanotubes . 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., is the cavity resonant frequency), is the planck constant, c is the light speed in vacuum, is the lasing wavelength, is the effective optical mode volume, and 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 , where can be written by ,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 .
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. 4(a)). This is remarkable given the small gain volume, but can be understood by the high photon density (e.g. ~2 × 1017/cm3 at the threshold) inside the oxide layer of the laser cavity . 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. 4(a)).
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 (, 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 ,Fig. 4(b)). 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.
In conclusion, we have theoretically investigated plasmonic photonic crystal hybrid lasers using carbon nanotubes as a gain material, which offers opportunities as high-performance on-chip light sources for telecom applications. Our simulation results show that the hybrid HPP waveguide mode originating from CNTs light emission can contribute to the 1-D photonic crystal cavity lasing mode with ~60% coupling efficiency. This light source are able to provide faster modulation than gain compression-limited devices due to both the strong Purcell effect (i.e. ~300 of ) and the short spontaneous emission lifetime of CNTs. The nanotube internal processes along with the plasmonic cavity therefore allow for hundreds of GHz-fast 3dB role-off modulation speeds, and tens of microwatts optical power above the threshold. Such monolithic integration schemes provide an alternative approach for active components of next-generation photonic circuits.
We acknowledge support from the Air Force Office of Scientific Research (AFOSR) under the award numbers FA9559-14-1-0215 and FA9559-14-1-0378, the National Natural Science Foundation of China under grant 61377059, the Beijing Municipal Natural Science Foundation under grant 4142004, and the Importation and Development of High-Caliber Talents Project of Beijing Municipal Institutions.
References and links
1. P. Avouris, M. Freitag, and V. Perebeinos, “Carbon-nanotube photonics and optoelectronics,” Nat. Photonics 2(6), 341–350 (2008). [CrossRef]
3. X. Wang, L. Zhang, Y. Lu, H. Dai, Y. K. Kato, and E. Pop, “Electrically driven light emission from hot single-walled carbon nanotubes at various temperatures and ambient pressures,” Appl. Phys. Lett. 91(26), 261102 (2007). [CrossRef]
4. E. Gaufrès, N. Izard, X. Le Roux, D. Marris-Morini, S. Kazaoui, E. Cassan, and L. Vivien, “Optical gain in carbon nanotubes,” Appl. Phys. Lett. 96(23), 231105 (2010). [CrossRef]
5. Y. Miyauchi, M. Iwamura, S. Mouri, T. Kawazoe, M. Ohtsu, and K. Matsuda, “Brightening of excitons in carbon nanotubes on dimensionality modification,” Nat. Photonics 7(9), 715–719 (2013). [CrossRef]
6. T. Mueller, M. Kinoshita, M. Steiner, V. Perebeinos, A. A. Bol, D. B. Farmer, and P. Avouris, “Efficient narrow-band light emission from a single carbon nanotube p-n diode,” Nat. Nanotechnol. 5(1), 27–31 (2010). [CrossRef] [PubMed]
7. S. Wang, Q. Zeng, L. Yang, Z. Zhang, Z. Wang, T. Pei, L. Ding, X. Liang, M. Gao, Y. Li, and L. M. Peng, “High-performance carbon nanotube light-emitting diodes with asymmetric contacts,” Nano Lett. 11(1), 23–29 (2011). [CrossRef] [PubMed]
10. S. Bahena-Garrido, N. Shimoi, D. Abe, T. Hojo, Y. Tanaka, and K. Tohji, “Plannar light source using a phosphor screen with single-walled carbon nanotubes as field emitters,” Rev. Sci. Instrum. 85(10), 104704 (2014). [CrossRef] [PubMed]
11. D. Yu, H. Liu, L. M. Peng, and S. Wang, “Flexible light-emitting devices based on chirality-sorted semiconducting carbon nanotube films,” ACS Appl. Mater. Interfaces 7(6), 3462–3467 (2015). [PubMed]
12. G. S. Tulevski, A. D. Franklin, D. Frank, J. M. Lobez, Q. Cao, H. Park, A. Afzali, S. J. Han, J. B. Hannon, and W. Haensch, “Toward high-performance digital logic technology with carbon nanotubes,” ACS Nano 8(9), 8730–8745 (2014). [CrossRef] [PubMed]
13. M. S. Arnold, A. A. Green, J. F. Hulvat, S. I. Stupp, and M. C. Hersam, “Sorting carbon nanotubes by electronic structure using density differentiation,” Nat. Nanotechnol. 1(1), 60–65 (2006). [CrossRef] [PubMed]
14. J. A. Fagan, M. L. Becker, J. Chun, P. Nie, B. J. Bauer, J. R. Simpson, A. Hight-Walker, and E. K. Hobbie, “Centrifugal length separation of carbon nanotubes,” Langmuir 24(24), 13880–13889 (2008). [CrossRef] [PubMed]
15. G. S. Tulevski, A. D. Franklin, and A. Afzali, “High purity isolation and quantification of semiconducting carbon nanotubes via column chromatography,” ACS Nano 7(4), 2971–2976 (2013). [CrossRef] [PubMed]
18. A. Vijayaraghavan, S. Blatt, D. Weissenberger, M. Oron-Carl, F. Hennrich, D. Gerthsen, H. Hahn, and R. Krupke, “Ultra-large-scale directed assembly of single-walled carbon nanotube devices,” Nano Lett. 7(6), 1556–1560 (2007). [CrossRef] [PubMed]
19. Y. Che, H. Chen, H. Gui, J. Liu, B. Liu, and C. Zhou, “Review of carbon nanotube nanoelectronics and macroelectronics,” Semicond. Sci. Technol. 29(7), 073001 (2014). [CrossRef]
20. S. J. Tans, A. R. M. Verschueren, and C. Dekker, “Room-temperature transistor based on a single carbon nanotube,” Nature 393(6680), 49–52 (1998). [CrossRef]
21. R. Martel, T. Schmidt, H. R. Shea, T. Hertel, and P. Avouris, “Single- and multi-wall carbon nanotube field-effect transistors,” Appl. Phys. Lett. 73(17), 2447–2449 (1998). [CrossRef]
22. J. A. Misewich, R. Martel, P. Avouris, J. C. Tsang, S. Heinze, and J. Tersoff, “Electrically induced optical emission from a carbon nanotube FET,” Science 300(5620), 783–786 (2003). [CrossRef] [PubMed]
23. G. H. Duan, C. Jany, A. L. Liepvre, A. Accard, M. Lamponi, D. Make, P. Kaspar, G. Levaufre, N. Girard, F. Lelarge, J. M. Fedeli, A. Descos, B. B. Bakir, S. Messaoudene, D. Bordel, S. Menezo, G. D. Valicourt, S. Keyvaninia, G. Roelkens, D. V. Thourhout, D. J. Thomson, F. Y. Gardes, and G. T. Reed, “Hybrid III-V on Silicon lasers for photonic integrated circuits on Silicon,” IEEE J. Sel. Top. Quantum Electron. 20(4), 6100213 (2014).
24. S. Wu, S. Buckley, J. R. Schaibley, L. Feng, J. Yan, D. G. Mandrus, F. Hatami, W. Yao, J. Vučković, A. Majumdar, and X. Xu, “Monolayer semiconductor nanocavity lasers with ultralow thresholds,” Nature 520(7545), 69–72 (2015). [CrossRef] [PubMed]
25. K. Ding, M. T. Hill, Z. C. Liu, L. J. Yin, P. J. van Veldhoven, and C. Z. Ning, “Record performance of electrical injection sub-wavelength metallic-cavity semiconductor lasers at room temperature,” Opt. Express 21(4), 4728–4733 (2013). [CrossRef] [PubMed]
27. V. J. Sorger, N. Pholchai, E. Cubukcu, R. F. Oulton, P. Kolchin, C. Borschel, M. Gnauck, C. Ronning, and X. Zhang, “Strongly enhanced molecular fluorescence inside a nanoscale waveguide gap,” Nano Lett. 11(11), 4907–4911 (2011). [CrossRef] [PubMed]
28. K. Liu, C. R. Ye, S. Khan, and V. J. Sorger, “Review and perspective on ultra-fast and wavelength-size electro-optic modulators,” Laser Photonics Rev. 9(2), 172–194 (2015). [CrossRef]
29. K. Y. Jeong, Y. S. No, Y. Hwang, K. S. Kim, M. K. Seo, H. G. Park, and Y. H. Lee, “Electrically driven nanobeam laser,” Nat. Commun. 4, 2822 (2013). [CrossRef]
30. K. Liu and V. J. Sorger, “Enhanced interaction strength for a square plasmon resonator embedded in a photonic crystal cavity,” J. Nanophotonics 9(1), 093790 (2015). [CrossRef]
31. A. R. M. Zain, N. P. Johnson, M. Sorel, and R. M. De La Rue, “Ultra high quality factor one dimensional photonic crystal/photonic wire micro-cavities in silicon-on-insulator (SOI),” Opt. Express 16(16), 12084–12089 (2008). [CrossRef] [PubMed]
33. P. Rai, N. Hartmann, J. Berthelot, J. Arocas, G. Colas des Francs, A. Hartschuh, and A. Bouhelier, “Electrical excitation of surface plasmons by an Individual carbon nanotube transistor,” Phys. Rev. Lett. 111(2), 026804 (2013). [CrossRef] [PubMed]
34. A. Vijayaraghavan, F. Hennrich, N. Stürzl, M. Engel, M. Ganzhorn, M. Oron-Carl, C. W. Marquardt, S. Dehm, S. Lebedkin, M. M. Kappes, and R. Krupke, “Toward single-chirality carbon nanotube device arrays,” ACS Nano 4(5), 2748–2754 (2010). [CrossRef] [PubMed]
36. E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69(1–2), 681 (1946).
38. T. Yoshie, J. Vučković, A. Scherer, H. Chen, and D. Deppe, “High quality two-dimensional photonic crystal slab cavities,” Appl. Phys. Lett. 79(26), 4289 (2001). [CrossRef]
39. E. J. R. Vesseur, F. J. García de Abajo, and A. Polman, “Broadband Purcell enhancement in plasmonic ring cavities,” Phys. Rev. B 82(16), 165419 (2010). [CrossRef]
40. R. M. Ma, R. F. Oulton, V. J. Sorger, and X. Zhang, “Plasmon lasers: coherent light source at molecular scales,” Laser Photonics Rev. 7(1), 1–21 (2013). [CrossRef]
41. C. Y. Lu, C. Y. Ni, M. Zhang, S. L. Chuang, and D. H. Bimberg, “Metal-cavity surface-emitting microlasers with size reduction: theory and experiment,” IEEE J. Sel. Top. Quantum Electron. 19(5), 1701809 (2013).
42. J. M. Marulanda and A. Srivastava, “Carrier density and effective mass calculations in carbon nanotubes,” Phys. Status Solidi 245(11), 2558–2562 (2008). [CrossRef]
43. P. Lalanne, C. Sauvan, and J. P. Hugonin, “Photon confinement in photonic crystal nanocavities,” Laser Photonics Rev. 2(6), 514–526 (2008). [CrossRef]
44. D. A. Genov, R. F. Oulton, G. Bartal, and X. Zhang, “Anomalous spectral scaling of light emission rates in low-dimensional metallic nanostructures,” Phys. Rev. B 83(24), 245312 (2011). [CrossRef]