Active plasmonic waveguiding has become a key requirement for designing and implementing nanophotonic devices. We study theoretically the performance of an Au/GaSb-based, metal–insulator–semiconductor (MIS) structure acting as a hybrid electrically pumped waveguide with gain. The surface-plasmon polariton (SPP) mode supported by this configuration is analyzed in the third telecommunication window and discussed in detail. Changes in the effective mode index, confinement factor and effective mode area are illustrated for different core widths and layer thicknesses. Electrical behavior of the MIS junction is analyzed using a self-consistent numerical technique and used to study variations in the material and model gains within the semiconducting region of the device. Our results indicate the possibility of achieving low loss SPP propagation while maintaining a strong field confinement.
© 2014 Optical Society of America
Conventional electronic circuitry that is at the heart of all modern electronic devices and systems may not be able to cope up with the ever increasing demand for higher and higher speeds, owing to inherent speed limitations of the electronic chips . The use of optical circuitry may seem a potential solution as it can support higher bandwidths; yet miniaturizing such a system into a sub-wavelength scale presents considerable challenges due to the diffraction limit of light . To overcome this limit and to produce highly integrated optical circuitry systems, the use of plasmonic structures is considered a promising solution. These sub-wavelength structures use surface-plasmon polariton (SPP) fields as the information carrier; SPPs are excited from the collective oscillation of electrons at the interface of a metal and a dielectric in the presence of an electromagnetic field [3, 4].
The main issue in realizing the plasmonic circuitry is related to intrinsic losses of SPPs, which limit the propagation length of the SPP mode. A number of proposed solutions for the waveguiding of SPPs in this context are metallic gap waveguides [5, 6], nanowires , nano-particle arrays , metallic groove/wedge channel waveguides [9,10], dielectric-loaded waveguides [11–14], and hybrid SPP-waveguide structures [15–19]. These passive SPP waveguides improve the SPP propagation length and confinement but, as far as the practical applications are concerned, there is a greater need of active waveguides.
There are a few potential solutions for active waveguiding. In all cases, a portion of the SPP mode lies inside an optical gain medium consisting of dye molecules [20, 21], quantum-dot inclusions , semiconductor heterostructures, or quantum wells  to compensate losses by the process of stimulated emission. The recent work on Schottky junction-based amplification has shown a promising path to active plasmonic systems with simple operation [14, 24–27].
In this work we consider a hybrid structure, which integrates dielectric waveguiding with plasmonics using a low-index nano-slot that couples the SPP mode between the metal and a high-index dielectric medium . The use of a nano-slot results in stronger field localization and improved propagation length. In the active waveguiding approach, semiconductor materials are commonly selected as the high-index medium . Here, we analyze how SPP propagation losses can be compensated using a metal–insulator–semiconductor (MIS) junction inside a hybrid plasmonic waveguide with an electrically active core material. We discuss in Section 2 the formation and propagation of the SPP mode in the proposed configuration. Section 3 focuses on the active medium and its electrical behavior. Section 4 illustrates a thermal analysis for the proposed device. In Sections 5 we discuss our results and their practical importance.
2. SPP mode propagation in the passive configuration
Figure 1 shows the schematic cross section (a) and three-dimensional view (b) of our proposed configuration for a simple hybrid plasmonic waveguide structure. The waveguide consists of a high-index GaSb substrate (ns = 3.8) also serving as the gain medium, a low-index dielectric cladding, a nano-slot formed by a thin layer of silica (nd = 1.44), and a metal cap made of gold (Au). The width and the height of the structure are chosen to be w = 0.6 μm and h = 0.5 μm, resulting in a cross-section area of only 0.3 μm2. The heights of metal cap and substrate are hcap = hsub = 100 nm. The nano-slot thickness is fixed to a value of 10 nm in all cases discussed later. In contrast, the width wc of the nano-slot is varied from 50 to 300 nm. By solving the wave equation for electric field (E),3,4]. Here, k0 = ω/c is the vacuum propagation constant at the frequency ω, ε is the dielectric constant of the medium, and β is the propagation constant of the SPP mode.
We use the finite-element method (FEM) for calculating β of the SPP mode supported by our waveguide at the telecommunication wavelength of λ = 1.55 μm. Figure 2 shows the transverse profiles for the Hx and Ey components of the transverse-magnetic (TM) SPP mode supported by the waveguide. Top [(a) and (b)] and bottom [(c) and (d)] panels compare two cases with wc = 50 nm and wc = 200 nm, respectively, while keeping other parameters fixed. In both cases, a strong field intensity can be seen around the nano-slot region, but the field is relatively high in the semiconductor gain region only for the wider nano-slot (wc = 200 nm).
The complex propagation constant of the SPP mode propagating along the z direction can be used to obtain the effective mode index neff = β/k0. The SPP propagation length (without gain) is related to the imaginary part of neff as Lsp = 2k0 Im(neff). Figures 3(a) and 3(b) show the real and imaginary parts of neff as a function of the core width wc for several combinations of hs and hm (see Fig. 1). Values of the real part of neff vary in the range of 1.8–2.8 for a 10 nm thick slot and are considerably lower than the 2.1–3.6 range found for t = 0 nm (absence of a nano-slot). In contrast, the imaginary parts of neff is nearly constant for all the core widths for t = 10 nm case although it decreases with hs. The value of Im(neff) is more than twice for t = 0 nm compared to that of t = 10 nm, showing relatively higher losses for the configuration without the nano-slot.11, 19], 3]
The right panel of Figs. 3(c) and 3(d) show variations of Γg and Aeff with the core width for the same device parameters used in the left panel. The presence of a nano-slot (t ≠ 0) reduces both Γg and Aeff compared to their values obtained for t = 0. However, Γg remains relatively large with values exceeding 50% for wider nano-slots. Such large values indicate that that the SPP mode should be amplified considerably if the semiconductor region is pumped electrically. Our results obtained in a passive waveguide are in good agreement with the previous work [19,23]. The advantage of having a nano-slot is that the the presence of even a 10 nm thick slot forces the field to be localized in its vicinity. As a result, penetration of the SPP mode into the metallic region is reduced considerably which, in turn, reduces ohmic losses. For the case of wc = hs = 200 nm, the propagation losses are 2270 cm−1 and 810.7 cm−1 for t=0 and t=10 nm, respectively.
3. Electrical pumping and optical gain
The hybrid waveguide shown in Fig. 1 is designed to have a standard electrical contact on the metallic side and an ohmic contact on the semiconductor side. The electrical contact is a physical connection to the voltage source and the ohmic contact is a non-rectifying contact between a metal and a semiconductor that is also connected to the same voltage source. The presence of a SiO2-based nano-slot between the metal cap and semiconductor ridge leads to the formation a MIS junction with a Schottky barrier , as shown in Fig. 4(a). Similar to a metal–semiconductor Schottky junction, the MIS junction also has the Fermi level pining at the junction attached to the interface layer. The junction can be biased by applying voltage to the terminals. The injection of minority carriers is used to pump the semiconductor gain medium , which is a simple technique compared to other available pumping techniques. In our design, GaSb is a direct band-gap semiconductor (Eg = 0.72 eV) of the III–V group and it supports radiative recombination and generation of light at λ = 1.55 μm. The stimulated-emission process can be established at high forward-biased conditions by developing an inversion region near the junction when ϕB > Eg [24, 31], where ϕB and Eg denote the Schottky barrier height and energy gap of the semiconductor. The inversion region changes the electron and hole quasi-Fermi levels (Fe and Fh) near the junction. The condition for the population inversion is Fe − Fh − Eg > 0. As shown schematically in Fig. 4(b) by the dashed line, this condition can be satisfied for our proposed device .
3.1. Electrostatic analysis of MIS junction
To obtain the gain characteristics of the semiconductor, the distribution of electrons and holes across the entire structure needs to be analyzed in a self-consistent manner. For this purpose, we solve the following Poisson’s equation ,3], 30],
In the absence of an applied voltage (Va = 0, thermal equilibrium), the potential and carrier densities can be obtained in an analytic form and are given by32, 33]:
The generation and recombination rates are calculated using Rn,p = RSRH + Rsp + Rst, where the first term accounts for nonradiative recombination [29, 34]. Its origin lies in the Shockley–Read–Hall recombination and it has the form29, 34]. The stimulated emission rate of SPP is given by Rst = gmP/h̄ω [24, 34], where P is the optical power density and h̄ω is the photon energy.
We solve Eqs. (5)–(8) numerically using the finite element method [35, 36]. The solution of this self-consistent approach provides the electron and hole densities n and p. These are used to calculate the material gain gm of the semiconductor through gm = Gst (min(n, p) − Nt) [24,27,29,34], where Gst is a material constant and Nt is the carrier density at transparency (see Table 1). Figure 5 shows the electron density, hole density, current density and material gain as a function of y (x = 0) for several values of applied voltage Va in the range of 0.8–1.05 V. All material parameters used for these calculations are listed in Table 1 [37, 38]. These results agree very well with the data reported in references [14, 24]. Comparing carrier densities in Figs. 5(a) and 5(b), we can see that p ≫ n closer to the MIS interface (y = 0.3 μm). Figures 5(c) and 5(d) indicate that both the current density and material gain also increase substantially near the MIS interface. In particular, the material gain approaches 2000 cm−1 near the junction for Va = 1.05 V, leading to a considerable high modal gain. The modal gain is generally defined as [24, 34]Fig. 6(a) [24, 29].
We have calculated the modal gain gM as a function of the core width wc using Eq. (12), and the results are shown by the red curve in Fig. 6(b) for Va = 1.05 V and hs = 200 nm. We have already calculated propagation loss in Section 2 using its definition Lsp = 2k0 Im(neff), and this loss is shown by the black curve in that figure. The net amplification of the SPP mode is governed by the net gain, defined as the difference of modal gain and propagation loss an shown by the blue curve in Fig. 6(b). As seen there, net gain is positive for all core widths in the range of 150 to 300 nm, and it can exceed 400 cm−1 for wider cores. Simultaneously, a strong mode confinement can be achieved by selecting larger semiconductor ridge heights (hs > 200 nm) and smaller metal cap heights (hm < 100 nm) for a nano-slot thickness of 10 nm.
4. The thermal analysis of the device
It is important to consider the heat generation associated with the operation of the MIS waveguide in order to properly engineer the device. The joule heating originates from three different processes; current injection, non-radiative recombination and Ohmic losses of SPPs. The temperature rise of the device associated with these heat sources can be characterized by the transient heat flux equation below:Figs. 7(a) and 7(b) for ts=0.01 μs and ts=0.3 μs. The material parameters are listed in Table 2.
Assuming a 3 ns full width at half-maximum (FWHM) pulse, the time evolution of the internal temperature is shown in Fig. 8(a) for 0.3 μs duration of pulse operation of the device. We observe around 80 K temperature rise during the 0.3 μs duration (i.e. 100 pulses). Figure 8(b) shows the temperature distribution across MIS contact in the y-direction. Very similar to what has been already used in DFB lasers and microprocessor systems [39,40], to maintain the temperature within acceptable limits, appropriate heat sink (eg. Copper, Aluminium) has to be installed as an outer cladding without interfering with the optical mode [41, 42]. The effectiveness of this method is shown in Fig. 8(c) where a heat sink is placed to maintain a constant temperature along the sides of the waveguide.
5. Discussion and conclusions
The proposed MIS structure is a novel approach for hybrid plasmonic waveguide design, providing a simple gain mechanism route towards electrical injection scheme that do not interfere with the confined mode. This study combines lasing properties and SPP propagation in the metal - semiconductor structures in one platform to incite experimental work to compensate losses in SPP waveguides [43–48]. We have presented a highly accurate theoretical analysis based on the experimental data obtained for theoretical approximations and material parameters. Refs.  and  show better agreement for the carrier and current distribution near the interface for high bias condition. The Ref.  demonstrates the minority carrier injection in the Schottky junction, in which, Fig. 2 has a similar trend for the hole distribution illustrated in Fig. 5(a). Ref.  experimentally demonstrates Schottky junctions between PbTe/ Au and InAs/Au, that have similar inversion regions near the interfaces. Ref.  illustrates high current and carrier densities near interface under high level carrier injection.
In this study we have developed a theoretical model for the MIS waveguide that shows net positive amplification of SPPs under practical operating conditions. The main advantage of our technique is that it avoids bulky optical amplification systems, while maintaining a strong field confinement. The latter property is important because not only it controls the mode size but also affects the bending angle and the resulting device length. Our proposed structure, shown in Fig. 1, contains a nano-slot formed by an ultrathin (∼10 nm thick), low-index dielectric layer sandwiched between the metal and semiconductor layers on its opposite sides. This nano-slot plays a key role in localizing the SPP field to its vicinity. Also, the permittivity and the thickness of the dielectric layer are the two key factors that affect the main electrical parameters of the MIS junction. The thickness of the nano-slot is chosen to be close to 10 nm to support the electrical conduction. The effect of interface states is neglected in our calculations because the nano-slot is ultra thin. The cross section dimensions of 0.3 μm2 for the SPP waveguide were selected avoiding the higher-order modes.
The passive MIS hybrid waveguide demonstrates greater neff which leads towards the stronger degree of confinement [see Fig. 3(a)]. For hs = wc = 200 nm, Aeff is calculated 0.01 μm2 or in terms of wavelength it is equal to λ2/220. Accounting the propagation loss 810.7 cm−1, SPP propagation distance is calculated as 12.5 μm. The experimental study discussed in  on MIS SPP waveguide has similar ridge dimensions (height-230 nm, width-174 nm, slot width-10 nm) has a mode area of λ2/157 at wavelength of 1.427 μm and Lsp = 20 μm. Ref.  described another hybrid waveguide with a cylindrical ridge [diameter (200–500) nm and t=(2–100) nm] which achieved the mode area over λ2/400 with Lsp = 40 μm. It is obvious that having a high confinement leads to low propagation length. In our device with the mentioned values, for about 90% loss compensation, we can achieve a propagation length close to 125 μm.
The parameters wc and hs are important because that determine the amount of modal gain need for full compensation of SPP losses. For instance let’s consider hs = wc = 200 nm and an effective conduction area of around 50 μm2 for a waveguide length of 100 μm and thickness of 0.5 μm. At Va=1.05 V the terminal current is set at 5.9 mA, thus the current density equals to 11.9 kA/cm2. Under this condition, the required minimum gain for the loss compensation is of about 810.7 cm−1 for operating optical power of 1 mW. Even though it seems that 11.9 kA/cm2 of current is comparatively high for a CW operating amplifier, it is acceptable for pulsed operation because it is expected that these waveguides will eventually be used to carry digital signals. The emission rates Rst ≫ Rsp under such conditions, indicating that spontaneous emission has little effect on the modal gain in the mentioned power range. The materials of the metal cap, cladding, and substrate are carefully selected to favor to the plasmonic mode. Although we used GaSb for the semiconducting material in this study, composite materials such as GaAs0.45Sb0.55, Al0.16In0.85As and In0.53Ga0.47As are also suitable for the high-index gain medium at the telecommunication wavelengths near 1550 nm. For the low-index cladding material, Al2O3 or Ga2O3 can be used in place of SiO2. Since our proposed design is a planer structure, its fabrication is relatively easy compared to curved nano structures. Indeed, conventional lithography in combination with etching, electron-beam lithography, and chemical-vapor deposition can be employed to fabricate this structure.
The SPP frequency (ω) and group velocity (vg) dispersion for MIS waveguide are illustrated in Fig. 9. The group velocity dispersion parameter defined as D = 2πcβ2/λ2 where β2 = d2β/dω2. At 0.8 eV, D has the value of 414 ps/nm.km for the MIS structure with hs = wc = 200 nm. As discussed in  this is low enough to achieve signal bandwidths over 100 Gbit/s. For the considered case, the maximum single-channel bandwidth can be calculated around 3.92 Tbit/s for waveguide length of 1 mm . There is more room to improve the mode confinement, lossy propagation length and group velocity by further making adjustment to the waveguide geometry and materials.
In summery, the SPP mode propagation and electrical behavior of the proposed MIS junction was analyzed using a self-consistent numerical approach. Our results show that, once the bias voltage exceeds 0.8 V, the minority carrier density increases drastically close to the junction. The distribution of material gain resulting from this population inversion was determined at the wavelength of 1.55 μm and used to calculate the modal gain. It was found that the modal gain can exceed modal loss, resulting in a net amplification of the SPP, over a wide range of practical device parameters. Clearly, we have demonstrated the possibility of low loss SPP propagation in our structure. Compared to other gain-assisted waveguide structures, our electrically pumped hybrid waveguide is much simpler and smaller in volume. Our analysis should facilitate the design and development of ultra-compact active plasmonic integrated systems.
The authors thank Australian Research Council for the financial support.
References and links
1. R. Kirchain and L. Kimerling, “A roadmap for nanophotonics,” Nat. Photonics 1(6), 303–305 (2007). [CrossRef]
3. M. Premaratne and G. P. Agrawal, Light Propagation in Gain Media: Optical Amplifiers (Cambridge University, 2011). [CrossRef]
4. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).
6. S. A. Maier, “Gain-assisted propagation of electromagnetic energy in subwavelength surface plasmon polariton gap waveguides,” Opt. Commun. 258, 295–299 (2006). [CrossRef]
7. D. Handapangoda, I. D. Rukhlenko, M. Premaratne, and C. Jagadish, “Optimization of gain–assisted waveguiding in metal–dielectric nanowires,” Opt. Lett. 35(24), 4190–4192 (2010). [CrossRef] [PubMed]
8. I. B. Udagedara, I. D. Rukhlenko, and M. Premaratne, “Complex–ω approach versus complex–k approach in description of gain–assisted SPP propagation along linear chains of metallic nano spheres,” Phys. Rev. B 83, 115451 (2011). [CrossRef]
10. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by sub-wavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005). [CrossRef]
12. T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B 75(24), 245405 (2007). [CrossRef]
13. A. V. Krasavin and A. V. Zayats, “Passive photonic elements based on dielectric-loaded surface plasmon polariton waveguides,” Appl. Phys. Lett. 90(21), 211101 (2007). [CrossRef]
14. D. Y. Fedyanin, A. V. Krasavin, A. V. Arsenin, and A. V. Zayats, “Surface plasmon polariton amplification upon electrical injection in highly integrated plasmonic circuits,” Nano Lett. 12(5), 2459–2463 (2012). [CrossRef] [PubMed]
15. R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for sub-wavelength confinement and long-range propagation,” Nat. Photonics 2, 496–500 (2008). [CrossRef]
17. Y. S. Bian, Z. Zheng, X. Zhao, J. S. Zhu, and T. Zhou, “Symmetric hybrid surface plasmon polariton waveguides for 3D photonic integration,” Opt. Express 17(23), 21320–21325 (2009). [CrossRef] [PubMed]
18. J. Zhang, L. Cai, W. Bai, Y. Xu, and G. Song, “Hybrid plasmonic waveguide with gain medium for lossless propagation with nanoscale confinement,” Opt. Lett. 36(12), 2312–2314 (2011). [CrossRef] [PubMed]
19. D. Dai, Y. Shi, S. He, L. Wosinski, and L. Thylen, “Gain enhancement in a hybrid plasmonic nano-waveguide with a low-index or high-index gain medium,” Opt. Express 19(14), 12925–12936 (2011). [CrossRef] [PubMed]
20. J. Seidel, S. Grafstrom, and L. Eng, “Stimulated emission of surface plasmons at the interface between a silver film and an optically pumped dye solution,” Phys. Rev. Lett. 94(17), 177401 (2005). [CrossRef] [PubMed]
21. I. D. Leon and P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nat. Photonics 4, 382–388 (2010). [CrossRef]
22. J. Grandidier, G. C. D. Francs, S. Massenot, A. Bouhelier, L. Markey, J. C. Weeber, C. Finot, and A. Dereux, “Gain-assisted propagation in a plasmonic waveguide at telecom wavelength,” Nano Lett. 9(8), 2935–2939 (2009). [CrossRef] [PubMed]
23. X. Zhang, Y. Li, T. Li, S. Y. Lee, C. Feng, L. Wang, and T. Mei, “Gain-assisted propagation of surface plasmon polaritons via electrically pumped quantum wells,” Opt. Lett. 35(18), 3075–3077 (2010). [CrossRef] [PubMed]
26. T. M. Wijesinghe and M. Premaratne, “Surface plasmon polaritons propagation through a Schottky junction: influence of the inversion layer,” IEEE Photonics J. 5(2), 4800216 (2013). [CrossRef]
28. R. F. Oulton, G. Bartal, D. F. P. Pile, and X. Zhang, “Confinement and propagation characteristics of subwavelength plasmonic modes,” New J. Phys. 10(10), 105018 (2008). [CrossRef]
29. S. L. Chuang, Physics of Photonic Devices, 2nd ed. (John Wiley, 2009).
30. S. M. Sze and K. K. Ng, Physics of Semiconductor Devices, 3rd ed. (John Wiley, 2006).
31. M. A. Green and J. Shewchun, “Minority carrier effects upon small signal and steady-state properties of schottky diodes,” Solid State Electron. , 16(10), 1141–1150 (1973). [CrossRef]
32. K. Horio and H. Yanai, “Numerical modelling of heterojunctions including the thermionic emission mechanism at the heterojunction interface,” IEEE Trans. Electron. Dev. 37, 1093–1098 (1990). [CrossRef]
33. M. S. Lundstrom and R. J. Schuelke, “Numerical analysis of heterostructure semiconductor devices,” IEEE Trans. Electron. Dev. 30, 1151–1159 (1983) [CrossRef]
34. Z. M. Li, “Two-dimensional numerical simulation of semiconductor lasers,” Prog. Electromagn. Res. 11, 301–344 (1995).
35. C. M. Snowden, Introduction to Semiconductor Device Modelling (World Scientific, 1998). [CrossRef]
36. R. Millett, J. Wheeldon, T. Hall, and H. Schriemer, “Towards modelling semiconductor heterojunctions,” in Proceedings of the COMSOL Users Conference, Boston, 2006.
37. S. Adachi, Properties of Group-IV, III-V and II-VI Semiconductors (John Wiley, 1950).
38. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]
39. K. Banerjee, S. C. Lin, and N. Srivastava, “Electrothermal engineering in the nanometer era: from devices and interconnects to circuits and systems,” in Proceedings of ASP-DAC (IEEE, 2006), pp. 223–230. [CrossRef]
40. J. Darabi and K. Ekula, “Development of a chip-integrated micro cooling device,” Microelectr. J. 34(11), 1067–1074 (2003). [CrossRef]
41. W. Nakwaski, “Dynamical thermal properties of stripe-geometry laser diodes,” IEEE Proc. Electron Dev. 131(3), 94 (1984).
42. O. J. Martin, G. L. Bona, and P Wolf, “Thermal behavior of visible AlGaInP-GaInP ridge laser diodes,” IEEE J. Quantum Electron. 28(11), 2582–2588 (1992). [CrossRef]
43. S. Kameda and W. Carr, “Analysis of proposed MIS laser structures,” IEEE J. Quantum Electron 9(2), 374–378 (1973). [CrossRef]
44. H. C. Card and B. L. Smith, “Green injection luminescence from forward-biased Au/GaP Schottky barriers,” J. Appl. Phys. 42(13), 5863–5865 (1971). [CrossRef]
45. K. W. Nill, A. R. Calawa, T. C. Harman, and J. N. Walpole, “Laser emission barriers on PbTe and PbSnTe,” Appl. Phys. Lett. 16(10), 375–377 (1970). [CrossRef]
46. C. Sirtori, C. Gmachl, F. Capasso, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho, “Long-wavelength (λ≈ 811.5μm) semiconductor lasers with waveguides based on surface plasmons,” Opt. Lett. 23(17), 1366–1368 (1998). [CrossRef]
48. C. Wang, H. J. Qu, W. X. Chen, G. Z. Ran, H. Y. Yu, B. Niu, J. Q. Pan, and W. Wang, “Polarization of the edge emission from Ag/InGaAsP Schottky plasmonic diode,” Appl. Phys. Lett. 102(6), 061112 (2013). [CrossRef]
49. J. N. Walpole and K. W. Nill, “Capacitance-voltage characteristics of metal barriers on p PbTe and p InAs: Effects of the inversion layer,” J. Appl. Phys. 42(13), 5609–5617 (1971). [CrossRef]
50. M. Alavi, D. K. Reinhard, and C. C. W. Yu, “Minority-carrier injection in PtSi Schottky-barrier diodes at high current densities,” IEEE Trans. Electron Dev. 34(5), 1134–1140 (1987). [CrossRef]
51. V. J. Sorger, Z. Ye, R. F. Oulton, Y. W. G. Bartal, X. Yin, and X. Zhang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. 2, 331 (2011). [CrossRef]
53. E. Dulkeith, F. Xia, L. Schares, W. Green, and Y. Vlasov, “Group index and group velocity dispersion in silicon-on-insulator photonic wires,” Opt. Express 14, 3853–3863 (2006). [CrossRef] [PubMed]