# Hybrid Electro-Optical Pumping of Active Plasmonic Nanostructures

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Results and Discussion

_{A}= 2.3 × 10

^{18}cm

^{−3}) and pump the structure electrically [26]. To avoid the complexity of the structure and investigate the fundamental properties of the hybrid pumping, we assume the top contact in Figure 1b to be an ideal ohmic contact, which might be realized by using a transparent electrode [36]. However, we note that for practical applications, the usage of a heterojunction [24,32] is more reasonable. The structure can also be pumped optically. The operating free-space light wavelength is chosen to be equal to λ

_{0}= 3.26 μm (ћω

_{0}= 0.38 eV), which roughly corresponds to the wavelength at which the maximum optical gain in the InAs can be achieved in the regime of full loss compensation, and is dictated by the bandgap energy of InAs [26]. If there is no loss or gain in the InAs layer, the SPP propagation length is equal to L

_{SPP}= 52 μm (ε

_{Au}= −561 + 30i [24], ε

_{InAs}= 12.8 [37] at λ

_{0}= 3.26 μm). InAs can also be pumped optically at a free-space light wavelength of about 2 μm (ћω

_{p}= 0.6 eV). Such a short pump wavelength allows the generation of electron-hole pairs even at a high degree of population inversion in InAs. The high density of electron-hole pairs created in InAs by electrical pumping and optical pumping gives the possibility to compensate for the SPP propagation losses and even amplify the SPP.

_{n}and j

_{p}are the electron and hole electric current densities, D

_{n}and D

_{p}are the diffusion coefficients, μ

_{n}and μ

_{p}are the electron and hole mobilities. Finally, R

_{spont}, R

_{stim}, R

_{nr}, R

_{opt}are the rates of spontaneous emission, stimulated emission, nonradiative Auger recombination, and recombination associated with optical pumping, respectively.

_{spont}via the Einstein relation between the spontaneous and stimulated emission processes:

_{sp}, given by

_{e}and F

_{h}are the quasi-Fermi levels for electrons and holes, respectively, and k

_{B}T is the thermal energy. The spontaneous emission rate R

_{spont}accounts for the enhanced spontaneous emission due to the existence of the guided plasmonic mode strongly-confined to the metal-semiconductor interface. The Purcell factor is evaluated as [38]

_{free}(ω,z) is the Purcell factor associated with emission into free space modes, β is the wavenumber of the SPP, n

_{g}is its group index, n

_{InAs}is the refractive index of InAs, w

_{E}and w

_{M}are the electric and magnetic energy densities of the SPP,

**E**(z) is the complex amplitude of the SPP electric field. In our simulations, we assumed that P

_{free}(z) ≈ 1. In the studied waveguide geometry, the Purcell factor is as large as 2 near the Au/InAs interface and decreases as the distance from the interface increases. The stimulated emission recombination rate R

_{stim}is a function of the SPP power density per unit waveguide width P

_{SPP}and the spatial distribution of the material gain in InAs:

_{InAs}(ω

_{p}) = 3.45 is the refractive index of InAs at the pump wavelength, α(ω

_{p}, n(z), p(z)) = −g(ω

_{p}, n(z), p(z)) is the material loss of InAs at the pump wavelength, and ${w}_{E}^{\mathrm{pump}}$ is the electric energy density of the pump wave, which is very non-uniform in the InAs layer due to the reflection from the InAs/Au interface and formation of a standing wave (Figure 1b). The material gain used in Equations (2), (5), and (6) is a function of the photon energy and the electron and hole densities and is calculated in every point of the InAs layer at every simulation step using the approach presented in [39].

_{pump}show the same typical diode-like behavior with a turn-on voltage of about 0.35 V (Figure 1c). However, it should be noted that as P

_{pump}increases, the absolute value of the current density at V < 0.35 V rapidly increases and can be as high as 30 kA/cm

^{2}. The optical pumping generates free carriers in InAs, and the electrons flow towards the bottom gold contact (negative electrode), and the holes flow towards the top ohmic contact (positive electrode), which results in a negative current. Such unusual directions of the carrier flow are determined by the band bending of InAs at voltages below about 0.4 eV (Figure 1d). Since the carrier recombination rate is much lower than the carrier generation rate, the current almost does not depend on the bias voltage at V < 0.35 V. However, as the bias voltage increases, the current increases and eventually becomes positive (Figure 1c) at voltages above ~0.4 V, which is due to the decrease of the strength of the “photodetection” effect due to flattening of InAs bands (Figure 1e) and increase in the electron and hole forward currents with the increase in the bias voltage.

^{18}cm

^{−3}(Figure 1f), while in the bulk of InAs, the electron density is more than ten orders of magnitude lower in equilibrium. In the absence of optical pumping, under high forward bias (V ≳ 0.35 eV), electrons are efficiently injected into the bulk of p-type InAs (Figure 1f), which allows to create population inversion in InAs required for the amplification of SPPs [26,32]. If the structure is also pumped optically, the spatial distribution of the electron density is significantly different. The inversion layer near the Au/InAs interface also exists; however, the electron density in the bulk of InAs is determined by the optical pumping and is as high as 2 × 10

^{15}cm

^{−3}at P

_{pump}= 0.5 mW/μm

^{2}and bias voltages below V ≈ 0.35 V. Only at higher bias voltages, the strong electron injection from gold gives the possibility to increase the electron density. The hole density, which is much higher than the electron density in the p-type-doped InAs layer, is not sensitive to optical pumping (Figure 1g), which explains the same diode-like behavior for the current-voltage characteristics under different optical pumping levels. The spatial distribution of holes is determined only by the change in the thickness of the depletion region near the Au/InAs interface, which decreases as the bias voltage increases (Figure 1g).

_{0}, n, p) is mostly determined by the electron density distribution since the hole density is almost the same across the InAs layer at any bias voltage and optical pump power density. At a very low electron density, g(ω

_{0}) is about −180 cm

^{−1}(Figure 2a), and it increases as n increases. At n = 2.2 × 10

^{15}cm

^{−3}, InAs becomes transparent, and accordingly, if one creates a higher electron density, it is possible to compensate for the SPP propagation losses. Figure 2b shows the spatial distribution of the material gain at two bias voltages in the presence and absence of optical pumping. At V < 0.3 V and P

_{pump}= 0, the material gain is equal to −180 cm

^{−1}over the whole InAs layer due to the absence of pumping. The optical pumping at a power density of P

_{pump}= 0.5 mW/μm

^{2}creates gain in InAs (Figure 2b) and make it almost transparent for the SPP propagating along the Au/InAs interface, since the modal gain G of the SPP mode is given by [43]:

_{SPP}= 1/L

_{SPP}= 190 cm

^{−1}is the SPP modal loss at an optically transparent InAs. Equation (7) shows that the SPP modal gain is determined by the spatial distribution of material gain in InAs and its overlap with the spatial distribution of the electric field of the SPP mode. At a very high forward bias voltage, the material gain of InAs is primarily determined by electrical pumping (Figure 2b), and the contribution of optical pumping to the net SPP modal gain is lower than at V = 0 due to the nonlinear dependence of the material gain and non-radiative recombination on the electron density, which reduces population inversion faster at a higher free carrier density. Thus, the hybrid electro-optical pumping cannot be considered as a linear superposition of the electrical pumping and optical pumping.

^{2}. Under such strong optical pumping, the current density at zero voltage (the short-circuit condition) is as high as −50 kA/cm

^{2}, which significantly reduces population inversion in InAs. Therefore, it is more efficient to operate under the open-circuit condition (at j = 0 and non-zero bias voltage) (Figure 2c), i.e., under pure optical pumping, since a much higher SPP modal gain can be achieved. However, a significantly higher gain can be achieved at j > 0, which is one of the advantages of hybrid pumping. Under hybrid electro-optical pumping, the SPP modal gain at high forward currents (high forward bias voltages) can be several times higher than at j = 0. Thus, electrical pumping can enhance optical pumping. Similarly, optical pumping can be used to control and enhance electrical pumping, as shown in Figure 2d. From the practical point of view, it is easier to control the bias voltage rather than the injection current. Since the turn-on voltage of the considered diode is about 0.35 V, the voltage of at least 0.36 V should be applied (Figure 2d).

_{stim}(see Equations (1) and (5)) is negligibly small. However, in practical applications, the SPP power cannot be equal to zero. Moreover, the SPP power can be quite high. For example, in on-chip communication, each optical pulse must carry a non-zero amount of energy to be efficiently detected at a photodetector and not to be significantly distorted during propagation. Assuming this energy to be of about 1–5 fJ [45,46], the bit rate to be of the order of 100 Gbit/s, and the waveguide width to be of about 300 nm [24,39], we obtain an average SPP power per unit waveguide width of as high as 150–750 μW/μm. Thus, in a practical system, the SPP power density is comparable with the power density of optical pumping, which can strongly affect the possibility of compensating for the SPP propagation losses. Figure 3 shows the SPP modal gain at an SPP power per unit waveguide width of 500 μW/μm. It can be seen that at zero optical pump power and zero bias voltage, the current is negative, which is due to the “photodetection” effect. At V = 0 and P

_{pump}= 0, the material gain in InAs is negative, and therefore, it absorbs the SPP electromagnetic field. Hence, the curves at a high SPP power are left shifted compared with curves at nearly zero SPP power. The slopes of the curves are lower than at P

_{SPP}= 1 nW/μm, since it is more difficult to create gain in InAs at a higher SPP power due to the stronger depopulation of electron and hole densities in InAs through the stimulated emission (see Equations (1) and (5)). However, the high SPP power does not prevent us from achieving full compensation for the SPP propagation losses.

## 3. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Jiang, N.; Zhuo, X.; Wang, J. Active Plasmonics: Principles, Structures, and Applications. Chem. Rev.
**2018**, 118, 3054–3099. [Google Scholar] [CrossRef] [PubMed] - Stockman, M.I.; Kneipp, K.; Bozhevolnyi, S.I.; Saha, S.; Dutta, A.; Ndukaife, J.; Kinsey, N.; Reddy, H.; Guler, U.; Shalaev, V.M.; et al. Roadmap on plasmonics. J. Opt.
**2018**, 20, 043001. [Google Scholar] [CrossRef][Green Version] - Oulton, R.F.; Bartal, G.; Pile, D.F.P.; Zhang, X. Confinement and propagation characteristics of subwavelength plasmonic modes. New J. Phys.
**2008**, 10, 105018. [Google Scholar] [CrossRef] - Fang, Y.; Sun, M. Nanoplasmonic waveguides: Towards applications in integrated nanophotonic circuits. Light Sci. Appl.
**2015**, 4, e294. [Google Scholar] [CrossRef][Green Version] - Krasavin, A.V.; Zayats, A.V. Active Nanophotonic Circuitry Based on Dielectric-loaded Plasmonic Waveguides. Adv. Opt. Mater.
**2015**, 3, 1662–1690. [Google Scholar] [CrossRef][Green Version] - McPeak, K.M.; Jayanti, S.V.; Kress, S.J.P.; Meyer, S.; Iotti, S.; Rossinelli, A.; Norris, D.J. Plasmonic Films Can Easily Be Better: Rules and Recipes. ACS Photonics
**2015**, 2, 326–333. [Google Scholar] [CrossRef] - Fedyanin, D.Y.; Yakubovsky, D.I.; Kirtaev, R.V.; Volkov, V.S. Ultralow-Loss CMOS Copper Plasmonic Waveguides. Nano Lett.
**2016**, 16, 362–366. [Google Scholar] [CrossRef] - Yakubovsky, D.I.; Arsenin, A.V.; Stebunov, Y.V.; Fedyanin, D.Y.; Volkov, V.S. Optical constants and structural properties of thin gold films. Opt. Express
**2017**, 25, 25574–25587. [Google Scholar] [CrossRef][Green Version] - Khurgin, J.B.; Sun, G. In search of the elusive lossless metal. Appl. Phys. Lett.
**2010**, 96, 181102. [Google Scholar] [CrossRef] - Yakubovsky, D.I.; Fedyanin, D.Y.; Arsenin, A.V.; Volkov, V.S. Optical constant of thin gold films: Structural morphology determined optical response. AIP Conf. Proc.
**2017**, 1874, 040057. [Google Scholar] - Oulton, R.F.; Sorger, V.J.; Genov, D.A.; Pile, D.F.P.; Zhang, X. A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation. Nat. Photonics
**2008**, 2, 496–500. [Google Scholar] [CrossRef][Green Version] - Zenin, V.A.; Choudhury, S.; Saha, S.; Shalaev, V.M.; Boltasseva, A.; Bozhevolnyi, S.I. Hybrid plasmonic waveguides formed by metal coating of dielectric ridges. Opt. Express
**2017**, 25, 12295–12302. [Google Scholar] [CrossRef] - Sorger, V.J.; Ye, Z.; Oulton, R.F.; Wang, Y.; Bartal, G.; Yin, X.; Zhang, X. Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales. Nat. Commun.
**2011**, 2, 4674. [Google Scholar] [CrossRef][Green Version] - Nezhad, M.; Tetz, K.; Fainman, Y. Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides. Opt. Express
**2004**, 12, 4072–4079. [Google Scholar] [CrossRef] [PubMed] - Leosson, K. Optical amplification of surface plasmon polaritons: Review. J. Nanophotonics
**2012**, 6, 061801. [Google Scholar] [CrossRef][Green Version] - Sudarkin, A.N.; Demkovic, P.A. Excitation of surface electromagnetic waves on the boundary of a metal with an amplifying medium. Sov. Phys. Tech. Phys.
**1989**, 34, 764. [Google Scholar] - Leon, I.D.; De Leon, I.; Berini, P. Amplification of long-range surface plasmons by a dipolar gain medium. Nat. Photonics
**2010**, 4, 382–387. [Google Scholar] [CrossRef] - Gather, M.C.; Meerholz, K.; Danz, N.; Leosson, K. Net optical gain in a plasmonic waveguide embedded in a fluorescent polymer. Nat. Photonics
**2010**, 4, 457–461. [Google Scholar] [CrossRef] - Oulton, R.F.; Sorger, V.J.; Zentgraf, T.; Ma, R.-M.; Gladden, C.; Dai, L.; Bartal, G.; Zhang, X. Plasmon lasers at deep subwavelength scale. Nature
**2009**, 461, 629–632. [Google Scholar] [CrossRef][Green Version] - Kéna-Cohen, S.; Stavrinou, P.N.; Bradley, D.D.C.; Maier, S.A. Confined surface plasmon-polariton amplifiers. Nano Lett.
**2013**, 13, 1323–1329. [Google Scholar] [CrossRef][Green Version] - Fedyanin, D.Y.; Arsenin, A.V. Semiconductor Surface Plasmon Amplifier Based on a Schottky Barrier Diode. AIP Conf. Proc.
**2010**, 1291, 112–114. [Google Scholar] - Fedyanin, D.Y.; Arsenin, A.V. Surface plasmon polariton amplification in metal-semiconductor structures. Opt. Express
**2011**, 19, 12524–12531. [Google Scholar] [CrossRef] [PubMed] - Hill, M.T.; Marell, M.; Leong, E.S.P.; Smalbrugge, B.; Zhu, Y.; Sun, M.; van Veldhoven, P.J.; Geluk, E.J.; Karouta, F.; Oei, Y.-S.; et al. Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides. Opt. Express
**2009**, 17, 11107–11112. [Google Scholar] [CrossRef] [PubMed][Green Version] - Fedyanin, D.Y.; Krasavin, A.V.; Arsenin, A.V.; Zayats, A.V. Surface plasmon polariton amplification upon electrical injection in highly integrated plasmonic circuits. Nano Lett.
**2012**, 12, 2459–2463. [Google Scholar] [CrossRef] [PubMed] - Bergman, D.J.; Stockman, M.I. Surface plasmon amplification by stimulated emission of radiation: Quantum generation of coherent surface plasmons in nanosystems. Phys. Rev. Lett.
**2003**, 90, 027402. [Google Scholar] [CrossRef][Green Version] - Fedyanin, D.Y. Toward an electrically pumped spaser. Opt. Lett.
**2012**, 37, 404–406. [Google Scholar] [CrossRef] - Gwo, S.; Shih, C.-K. Semiconductor plasmonic nanolasers: Current status and perspectives. Rep. Prog. Phys.
**2016**, 79, 086501. [Google Scholar] [CrossRef] - Ning, C.-Z. Semiconductor nanolasers and the size-energy-efficiency challenge: A review. Adv. Photonics
**2019**, 1, 1. [Google Scholar] [CrossRef] - Lee, C.J.; Yeh, H.; Cheng, F.; Su, P.-H.; Her, T.-H.; Chen, Y.-C.; Wang, C.-Y.; Gwo, S.; Bank, S.R.; Shih, C.-K.; et al. Low-Threshold Plasmonic Lasers on a Single-Crystalline Epitaxial Silver Platform at Telecom Wavelength. ACS Photonics
**2017**, 4, 1431–1439. [Google Scholar] [CrossRef] - Chou, Y.-H.; Wu, Y.-M.; Hong, K.-B.; Chou, B.-T.; Shih, J.-H.; Chung, Y.-C.; Chen, P.-Y.; Lin, T.-R.; Lin, C.-C.; Lin, S.-D.; et al. High-Operation-Temperature Plasmonic Nanolasers on Single-Crystalline Aluminum. Nano Lett.
**2016**, 16, 3179–3186. [Google Scholar] [CrossRef] - Zhang, Q.; Li, G.; Liu, X.; Qian, F.; Li, Y.; Sum, T.C.; Lieber, C.M.; Xiong, Q. A room temperature low-threshold ultraviolet plasmonic nanolaser. Nat. Commun.
**2014**, 5, 4953. [Google Scholar] [CrossRef] [PubMed][Green Version] - Vyshnevyy, A.A.; Fedyanin, D.Y. Self-Heating and Cooling of Active Plasmonic Waveguides. ACS Photonics
**2016**, 3, 51–57. [Google Scholar] [CrossRef] - Fedyanin, D.Y. Electrically pumped double-heterostructure surface plasmon polariton amplifier. AIP Conf. Proc.
**2012**, 1475, 56. [Google Scholar] - Tiwari, S.; Frank, D.J. Empirical fit to band discontinuities and barrier heights in III–V alloy systems. Appl. Phys. Lett.
**1992**, 60, 630–632. [Google Scholar] [CrossRef] - Costantini, D.; Bousseksou, A.; Fevrier, M.; Dagens, B.; Colombelli, R. Loss and Gain Measurements of Tensile-Strained Quantum Well Diode Lasers for Plasmonic Devices at Telecom Wavelengths. IEEE J. Quantum Electron.
**2012**, 48, 73–78. [Google Scholar] [CrossRef] - Widmann, M.; Niethammer, M.; Fedyanin, D.Y.; Khramtsov, I.A.; Rendler, T.; Booker, I.D.; Hassan, J.U.; Morioka, N.; Chen, Y.-C.; Ivanov, I.G.; et al. Electrical Charge State Manipulation of Single Silicon Vacancies in a Silicon Carbide Quantum Optoelectronic Device. Nano Lett.
**2019**, 19, 7173–7180. [Google Scholar] [CrossRef][Green Version] - Adachi, S. Optical dispersion relations for GaP, GaAs, GaSb, InP, InAs, InSb, AlxGa1−xAs, and In1−xGaxAsyP1−y. J. Appl. Phys.
**1989**, 66, 6030. [Google Scholar] [CrossRef] - Vyshnevyy, A.A.; Fedyanin, D.Y. Spontaneous Emission and Fundamental Limitations on the Signal-to-Noise Ratio in Deep-Subwavelength Plasmonic Waveguide Structures with Gain. Phys. Rev. Appl.
**2016**, 6, 064024. [Google Scholar] [CrossRef] - Svintsov, D.A.; Arsenin, A.V.; Fedyanin, D.Y. Full loss compensation in hybrid plasmonic waveguides under electrical pumping. Opt. Express
**2015**, 23, 19358–19375. [Google Scholar] [CrossRef][Green Version] - Aydaraliev, M.; Zotova, N.V.; Karandashov, S.A.; Matveev, B.A.; Stus’, N.M.; Talalakin, G.N. Low-threshold long-wave lasers ( lambda =3.0-3.6 mu m) based on III-V alloys. Semicond. Sci. Technol.
**1993**, 8, 1575–1580. [Google Scholar] [CrossRef] - Melngailis, I.; Rediker, R.H. Properties of InAs Lasers. J. Appl. Phys.
**1966**, 37, 899–911. [Google Scholar] [CrossRef] - Bhargava, S.; Blank, H.-R.; Narayanamurti, V.; Kroemer, H. Fermi-level pinning position at the Au–InAs interface determined using ballistic electron emission microscopy. Appl. Phys. Lett.
**1997**, 70, 759–761. [Google Scholar] [CrossRef] - Visser, T.D.; Blok, H.; Demeulenaere, B.; Lenstra, D. Confinement factors and gain in optical amplifiers. IEEE J. Quantum Electron.
**1997**, 33, 1763–1766. [Google Scholar] [CrossRef][Green Version] - Casey, H.C., Jr.; Stern, F. Concentration-dependent absorption and spontaneous emission of heavily doped GaAs. J. Appl. Phys.
**1976**, 47, 631–643. [Google Scholar] [CrossRef] - Miller, D.A.B. Rationale and challenges for optical interconnects to electronic chips. Proc. IEEE
**2000**, 88, 728–749. [Google Scholar] [CrossRef][Green Version] - Manipatruni, S.; Lipson, M.; Young, I.A. Device Scaling Considerations for Nanophotonic CMOS Global Interconnects. IEEE J. Sel. Top. Quantum Electron.
**2013**, 19, 8200109. [Google Scholar] [CrossRef][Green Version] - Vyshnevyy, A.A.; Fedyanin, D.Y. Noise reduction in plasmonic amplifiers. Appl. Phys. Express
**2018**, 11, 062002. [Google Scholar] [CrossRef][Green Version] - Humar, M.; Yun, S.H. Intracellular microlasers. Nat. Photonics
**2015**, 9, 572–576. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**(

**a**) Schematic illustration of surface plasmon polariton amplification in an electro-optically pumped active planar plasmonic waveguide, β

_{0}is the SPP wavevector and λ

_{0}is the free-space wavelength at which the SPP is excited. (

**b**) Spatial distributions of the electric energy density of the SPP mode and the optical pump wave. The SPP is excited at a free space wavelength of 3.26 μm (ε

_{Au}= −561 + 30i [24], ε

_{InAs}= 12.8 [37]), while the wavelength of the pump wave is 2.07 μm (ε

_{Au}= −225 + 7.8i [24], ε

_{InAs}= 11.9 [37]). (

**c**) Simulated current-voltage characteristics of the device shown in panel (

**a**) at different optical pump power densities. (

**d**,

**e**) Energy band diagrams of the device at bias voltages of 0.2 V (panel (

**d**)) and 0.4 V (panel (

**e**)) in the vicinity of the Au/InAs interface in the absence and presence (P

_{pump}= 0.5 mW/μm

^{2}) of optical pumping. E

_{c}and E

_{v}denote the band edges of the conduction and valence bands, respectively. (

**f**,

**g**) Spatial distribution of the electron (panel (

**f**)) and hole (panel (

**g**)) densities in the vicinity of the Au/InAs interface at three bias voltages in the absence and presence (P

_{pump}= 0.5 mW/μm

^{2}) of optical pumping. Abbreviations: SPP, surface plasmon polariton.

**Figure 2.**(

**a**) Material gain of p-type-doped InAs at a free-space light wavelength of 3.26 μm as a function of free electron density calculated employing Stern’s model for interband transitions in heavily-doped semiconductors [39,44], which accounts for the band tails in the density of states arising due to a chaotic screened potential of charged dopants. (

**b**) Spatial distribution of the material gain across the InAs layer at bias voltages of 0 and 0.4 V in the presence and absence of optical pumping (0.5 mW/μm

^{2}). To estimate the overlap of the gain and mode profiles, the spatial distribution of the electric energy density of the SPP field is also shown. (

**c**) SPP modal gain as a function of the current density at different levels of optical pumping. The dots on the curves correspond to V = 0 V, while the arrows show the direction of the bias voltage increase. (

**d**) SPP modal gain as a function of the optical pump power density at five different bias voltages. In panels (

**c**) and (

**d**), the region that corresponds to the net SPP amplification is shown in red.

**Figure 3.**SPP modal gain in the hybridly pumped active plasmonic waveguide as a function of the current density at different levels of optical pumping for the SPP powers of 500 μW/μm (solid lines) and 1 nW/μm (dashed lines). The dots on the curves correspond to V = 0 V, while the arrows show the direction of the bias voltage increase.

**Figure 4.**Heatmap of the SPP modal gain as a function of the pump current and optical pump power for a low-power signal (P

_{SPP}= 1 nW/μm) (panel (

**a**)) and a high-power signal (P

_{SPP}= 500 μW/μm) (panel (

**b**)). In both panels, the white line corresponds to the regime of lossless SPP propagation, i.e., the net SPP modal gain is equal to zero, and the propagation losses are fully compensated by gain in InAs. The white area in the left bottom corner corresponds to negative bias voltages, which are out of practical interest for hybrid pumping.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Vyshnevyy, A.A.; Fedyanin, D.Y. Hybrid Electro-Optical Pumping of Active Plasmonic Nanostructures. *Nanomaterials* **2020**, *10*, 856.
https://doi.org/10.3390/nano10050856

**AMA Style**

Vyshnevyy AA, Fedyanin DY. Hybrid Electro-Optical Pumping of Active Plasmonic Nanostructures. *Nanomaterials*. 2020; 10(5):856.
https://doi.org/10.3390/nano10050856

**Chicago/Turabian Style**

Vyshnevyy, Andrey A., and Dmitry Yu. Fedyanin. 2020. "Hybrid Electro-Optical Pumping of Active Plasmonic Nanostructures" *Nanomaterials* 10, no. 5: 856.
https://doi.org/10.3390/nano10050856