1. Introduction

Recent demands in technologies for optoelectronic integration have placed much emphasis on the research of low cost, low power, high speed, lightweight and compact devices with small device footprints. With the exception of a compact light source, downwards scaling has been possible for most of the other components necessary for optoelectronic integration, including waveguides, modulators and detectors [1]. Electrical plasmon sources provide an avenue to realize compact light sources [2, 3] and are essential to generate surface plasmon polaritons (SPPs) which are surface eigenmodes [4,5] that occur at the interfaces of metals and dielectrics. While plasmons can be electrically excited using generated electron hole pairs [6–11] via electrostatic interactions to form excitons [12], electrical means have been used in metal-insulator-metal (MIM) structures to generate plasmon sources, making use of roughened dielectric [13, 14] or induced change in molecular dipole moments from charge transfer in molecular tunnel junctions [15–21] to excite SPPs. The use of these rough dielectric or uniform molecular junctions makes it difficult to control the location and types of excited plasmon polaritons. It is also very difficult to couple these excited plasmons out [21]. For electrically excited MIM structures, the insulator layer is usually very thin, leading to very short SPP propagation lengths [22].

There has been much research on the creation of electrical plasmon sources using scanning tunneling microscopy (STM) [23–26]. While STM allows control over the site of excitation, it is difficult to translate to practical applications due to bulky structures and high cost. Theoretically, the STM configuration can be likened to a 2-layer insulator metal (IM) structure due to the small size of the tip as compared to the overall surface on which plasmons are generated. Assuming that the total input power goes into excitation of confined eigenwaves and non-confined leaky waves, the efficiency of SPP excitation using STM is extremely low due to high leakage radiation losses. The potential advantages for a compact electrical plasmon source for on-chip integration far outweigh difficulties in overcoming the above challenges and serve to make the best use of SPPs despite its’ inherent weakness of short propagation length.

2. Theory and modelling

In this study, we propose a protruded MIM (pMIM) structure as shown in Fig. 1(a). This structure has a protruded segment and an MIM waveguiding segment. The protruded segment of the pMIM makes use of the concept of tunneling junctions to excite propagating plasmons. Different from the conventional MIM structure with thin uniform or roughened junctions for electrical excitations, the pMIM structure has a thick insulator layer and a protrusion on the top Au cathode which can be excited by an applied external bias. The distance between top Au cathode and bottom Au anode ranges from sub-nanometers to nanometers, is the shortest at the site of the protrusion and provides a passage for the tunneling current which acts as a source of excitation of propagating eigenmodes. Combining the advantages of an equivalent STM [23–26] and MIM [13–20] excitation, the pMIM structure allows controllability over the location of the source and facilitates electrical excitation due to the presence of a short tunneling path. It also allows for highly efficient plasmon-polariton generation due to minimized leakage radiation and dissipation losses especially for a thick insulator layer. The presence of the protrusion also significantly enhances plasmon-polariton generation and allows smooth coupling onto the MIM waveguiding segment of the pMIM.

 

Fig. 1 (a) Protruded Au-SiO2-Au configuration with a tunneling path for propagating plasmon source generation along the MIM waveguiding segment. Dashed line indicates distinction between protruded segment and MIM waveguiding segment. (b) Image plots for real values of H_ф captured between 0 to 5µm away from the source at photon energies of (i) 0.6 eV, (ii) 1.1 eV and (iii) 2.0 eV.

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For this pMIM structure, the tunneling current source excites transverse magnetic (TM) waves which have a magnetic field component H_ф perpendicular to the plane of incidence and an electric field E_r parallel to the plane of incidence in the direction of the propagating waves. The image plots for magnetic field H_ф excited at the photon energies of 0.6, 1.1 and 2.0 eV for an Au-SiO₂-Au configuration are illustrated in Figs. 1 (b)(i), 1(b)(ii) and 1(b)(iii) respectively. Using this pMIM design, we will show that different plasmon polariton modes can be electrically excited for different applied biases. The presence of multiple propagating modes allows waveguide mode manipulation for parallel data transmission and wider communication bandwidth [27, 28].

In the case of the pMIM structure, the addition of a metal layer on top of the IM structure significantly lowers the excitation of leaky waves [29], thereby minimizing leakage radiation losses [30]. In an ideal situation, all the input power can be converted into confined eigenmodes with no leaky wave generation. Further, the excitation of a single long travelling eigenmode would mean that all input power would go into the excitation of that mode with optimal efficiency. A comparison between efficiency of exciting eigenmodes using IM and pMIM structures is shown in Fig. 2.

 

Fig. 2 Efficiency of eigenmode excitation in semi-infinite (a) STM configuration and (b) pMIM configuration. Arrows indicate direction of power flow. Leaky waves are almost negligible in pMIM structures.

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3. Results and discussion

To show that our proposed pMIM structure can electrically excite the intrinsic eigenmodes of the MIM waveguiding segment efficiently, we choose an MIM structure consisting of a gold (Au) cathode, 500 nm thick silicon dioxide (SiO₂) insulator and an Au anode. From our calculations, this configuration is by far the most efficient amongst other practical material options that would include Eutectic Gallium Indium (EGaIn), Silver (Ag) and Aluminum (Al) for metals, as well as Air, Silicon Nitride (Si₃N₄) and Aluminum Oxide (Al₂O₃) for insulators. The criterion for choice of material is to achieve the longest possible propagation lengths of excited plasmon-polaritons in the energy range (0.5-2.0 eV) of interest.

Here we perform an eigenmode analysis [31] for the energy range of 0.5 to 2.0 eV to extract mode indices and propagation lengths for excited plasmon-polaritons that can be supported within the Au-SiO₂-Au MIM waveguiding segment of the pMIM. The excited eigenmodes comprise both volume and surface eigenmodes that can generally be distinguished with the effective mode index. The volume and surface eigenmodes feature propagating and evanescent fields in the insulator [32, 33] respectively, with the former and latter having effective mode indices below and above the insulator refractive index in the MIM waveguiding segment.

For the Au-SiO₂-Au MIM waveguiding segment of the pMIM, we are able to identify 3 different excited eigenmodes [31, 34]. Since each of these modes is excited at different energies, we identify three operating regimes whereby each regime shows either an independent mode or the start of the rise of additional modes. The cutoff energies of the 3 regimes can be identified by comparing the real and imaginary components of the calculated mode indices. The plots of mode index and propagation length as function of energy as well as the associated regimes are shown in Figs. 3(a) and 3(b) respectively.

 

Fig. 3 Eigenmode analysis of intrinsic modes supported by the Au-SiO₂-Au MIM waveguiding segment of the pMIM structure calculated in theory. (a) Mode index vs photon energy. (b) Propagation length vs photon energy. Dashed lines indicate cutoff energies of the three regimes. (c) Theoretical mode profile of modes 1, 2 and 3. Mode 1 achieves the longest propagation length of 30 µm at 0.6 eV. Dashed lines indicate different material interfaces.

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The three regimes of the MIM waveguiding segment indicate the possibility of exciting either single or multiple plasmon-polariton modes with respect to different applied voltage bias. The amount of applied voltage bias defines the maximum energy of plasmon polaritons excited inelastically by the tunneling current. For each of these different modes, we can identify their individual mode profiles as shown in Fig. 3(c). Modes 1 and 3 are seen to be even while mode 2 is odd with a strong dip in the center.

In the above discussions, we have performed eigenmode analysis for the Au-SiO₂-Au MIM waveguiding segment of the pMIM, demonstrating the intrinsic eigenmodes. To show that the pMIM structure can indeed efficiently electrically excite eigenmodes of the MIM waveguiding segment, we study pMIM structure with the configuration shown in Fig. 1(a) using finite element method (FEM) simulation. First, we find that the pMIM structure is able to enhance the power transferred by tunneling current to excite eigenwaves as compared to MIM structures. Second, the fact that the pMIM simulation agrees well with eigenmode analysis of MIM waveguiding segment [see Appendix] provides clear evidence of good coupling from the protruded segment to the MIM waveguiding segment of the pMIM.

In our pMIM simulation, we assume ideal uniform surfaces and interfaces over a domain size of 50 µm. Perfectly matched layer (PML) boundary conditions are used to absorb waves (both propagating and evanescent) with near zero reflections. For the smallest wavelength studied, a mesh size of 10 elements per wavelength is used to ensure accuracy. The Au cathode, SiO₂ insulator and Au anode are assigned a thickness of 1 μm, 500 nm and 1 μm respectively. The protrusion has a base radius of 250 nm and a tip radius of 1 nm. It is placed 1 nm above the bottom Au anode. An external tunneling current between the protrusion and the bottom Au anode acts as the excitation source. To study the impact of the protruded segment of the pMIM, we performed another simulation by taking out the protrusion while keeping the tunneling current as an excitation source. This mimics the case of MIM structures with an external excitation source.

Figure 4 shows a comparison of the two cases. Compared to the MIM structures, the protrusion in pMIM structures allows for a huge enhancement in emission from the tunneling junction. For a fair comparison, the power emitted by the tunneling current,$\text{P}\left(\text{ω}\right)$, is measured at the site of the tunneling current as shown in Fig. 4(a) and is computed as

 

Fig. 4 Emission enhancement in excitation of pMIM over MIM structures. (a) Site (red highlighted region) in vicinity of tunnel current measures average power transferred by tunneling current to excited eigenwaves in MIM and pMIM structures. (b) Emission enhancement in pMIM is around two to three orders larger than MIM structures.

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Since the pMIM structure is seen to have stronger emission than the MIM stuctures, we show next that all the power is used efficiently to electrically excite eigenmodes of the MIM waveguiding segment of the pMIM. To do this, an energy value corresponding to each operation regime is selected as a reference of study. Hence, we select 0.6 eV, 1.1 eV and 2.0 eV for energies lying within the boundaries of single, dual and multi mode regimes respectively. From the image plots presented earlier in Figs. 1(b)(i), 1(b)(ii) and 1(b)(iii), a photon energy of 0.6 eV shows excitation of a single propagating SPP mode while energies of 1.1 eV and 2.0 eV show excitation of multiple eigenmodes. To obtain further insights into the types of excited modes and to verify that the modes observed do correlate to theory, we extract individual modes from the image plots of the real values of H_ф shown in Figs. 1(b)(i), 1(b)(ii) and 1(b)(iii). The real values of the H_ф field profile corresponding to image plots at 0.6 eV, 1.1 eV and 2.0 eV along the direction of the propagating waves are as shown in Figs. 5(a), 5(b) and 5(c), respectively. The H_ф field profile is calculated at the top interface between Au cathode and SiO₂ insulator for distances up to 50µm away from the tunneling current source.

 

Fig. 5 Field plots of real(H_ф) vs propagating distance at photon energies of (a) 0.6 eV, (b) 1.1 eV, (c) 2.0 eV; normalized absolute value of Fourier transformed (H_ф) vs mode index for (d) 0.6 eV, (e) 1.1 eV, (f) 2.0 eV; |H_ф| across vertical interface for (g) 0.6 eV, (h) 1.1 eV, (i) 2.0 eV. The simulated mode profile closely mirrors the theoretical mode profile, demonstrating the existence of excited eigenmodes in the different regimes and the obvious absence of leaky modes.

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To investigate excited fields in the pMIM structure, a Fourier analysis on the complex H_ф field profile is performed. The normalized absolute values of the Fourier transform (F.T) are plotted against mode index and shown in Figs. 5(d), 5(e) and 5(f) for energies of 0.6 eV, 1.1 eV and 2.0 eV respectively. Energy of 0.6 eV shows a single peak around mode index of 1.5 while energy of 1.1 eV shows two peaks corresponding to the mode indices of 1.1 and 1.5. The peak at mode index 1.5 is a result of mode 1 while the new peak at mode index 1.1 is a result of mode 2. At an energy of 2.0 eV, only two peaks at mode indices 1.6 and 1 are seen. The peak seen at 1.6 can be attributed to both modes 1 and 2 having very similar mode indices at energy 2.0 eV as shown in Fig. 3(a) while the peak at mode index 1.0 is due to mode 3. The obtained spectra align closely with those for omnidirectionally excited eigenwaves calculated from theory [see Appendix]. Also, this is in accordance with our study in Fig. 3(a) where we observe that only one, two and three modes are excited strongly in single, dual and multi mode regimes respectively. Similarly, the normalised Fourier images of real and imaginery values of the F.T of H_ф are in accordance with both theory and simulation [see Appendix].

To further confirm that the theoretically calculated modes have been observed, we consider H_ф mode profile across the vertical interfaces. For an accurate study, we place appropriate field monitors at suitable distances away from the tunneling current source. We consider the propagation lengths of each excited mode in order to know how far they propagate and how the extent of their influence on the mode profile varies with propagation length. In accordance with Fig. 3(b), mode 1 has a propagation length of 30 µm at 0.6 eV. At 1.1 eV, mode 1 and mode 2 has a propagation length of 20 µm and 8 µm respectively. At 2.0 eV, mode 1 and mode 2 have similar propagation lengths of 2 µm while mode 3 has a propagation length of 1 µm. This means that modes 1 and 2 diminish while mode 3 only starts to show its influence at higher energies.

Hence, in order to observe the individual contributions to the mode profile for real values of H_ф, we use monitors placed 5µm away from the source for energies 0.6 eV and 1.1 eV and monitors placed 0.5 µm away from the source for energy 2.0 eV. We can then perform superposition for the theoretically calculated individual mode profiles shown in Fig. 3(c) for particular energies. The final mode profile for absolute values of H_ф calculated in theory is shown together with simulated values in Figs. 5(g), 5(h) and 5(i) for energies of 0.6 eV, 1.1 eV and 2.0 eV. Similarly, the mode profiles for real and imaginery values of H_ф are in accordance with both theory and simulation [see Appendix]. For both real and imaginery values of H_ф, the simulated mode profile closely mirrors that of the theoretical mode profiles, further demonstrating the efficient excitation of the confined eigenwaves with absence of leakage radiation.

Having demonstrated that an externally applied bias induced tunneling current can be used to excite desired eigenmodes in different operating regimes of a pMIM structure, we provide some design guidelines to tune the cutoff energies of these regimes. The variation of cutoff energy with insulator thickness and insulator permittivity for the single and dual modes regimes are shown in Figs. 6(a) and 6(b) respectively. From the results obtained, it can be concluded that thinner insulators with lower permittivity values exhibits higher cutoff energies. This means that we can tune cutoff energies to excite desired modes with the required voltage bias. Knowledge of cutoff energies signifies control over the number of modes excited [35].

 

Fig. 6 Cutoff energies of single and dual mode regimes as a function of (a) insulator thickness and (b) insulator permittivity. Thinner insulators with lower permittivity exhibit higher cutoff energies. This means that cutoff energies can be tuned to excite desired modes with the required voltage bias.

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From the information obtained thus far, we notice that long travelling waves of more than 10 µm can be electrically excited at relatively small voltage bias and hence low energies. Taking Au-SiO₂(500 nm)-Au MIM waveguiding segment of the pMIM as an example, the operation voltage bias for single-mode regime is 0.2 to 0.7 volts. These long travelling waves operating in the single mode regime are the focus of propagating SPP excitations in the creation of a low power compact on-chip propagating plasmon source. Further analysis is imperative for this single mode regime. In particular, we study the influence of the applied voltage bias on the performance of the electrically excited pMIM plasmon source.

An increase in applied voltage bias causes an increase in tunneling current density $\text{J}\left(\text{ω}\right)$ for a constant tunneling path of 1nm. $\text{J}\left(\text{ω}\right)$is dependent on the static tunneling current density, ${\text{J}}_{\text{0}}$ and voltage bias V in accordance to the relation

Due to the nature of the tunneling current being formed via a short tunneling pass of 1 nm, ${\text{J}}_{\text{0}}$ is calculated self consistently from a solution of 1D time independent Poisson and Schrodinger’s equation with the consideration of image potential, exchange correlation effects and space charge limited effects [36]. The calculated ${\text{J}}_{\text{0}}$ which has the profile shown in Fig. 7(a) is then substituted into Eqs. (2) and (1) to obtain the power $\text{P}\left(\text{ω}\right)$. We find that an increase in applied voltage bias enhances the amount of power at the site of the tunneling current that goes into exciting the SPPs in pMIM as shown in Fig. 7(b). By increasing the applied voltage bias, we can also shift the resonance energy with peak emission to higher energies. This allows us to excite mode 1 more strongly at higher energies. For example, increasing the voltage bias from 0.5 to 0.7 volts shifts the peak emission from 0.2 eV to 0.25 eV.

 

Fig. 7 (a) Calculated static current density profile as a function of applied bias (b) Power transferred from tunneling current in pMIM into the excitation of SPPs as a function of energy for different applied biases in the single mode regime.

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4. Conclusion

In conclusion, a tunneling current can be electrically induced in our proposed pMIM structure via a small applied voltage bias, locally exciting long traveling plasmon polaritons with high efficiencies and no leakage radiation. The pMIM structures enhance the total amount of power that is transferred from the tunneling current into the excitation of eigenwaves. By controlling the applied voltage bias, we can excite desired modes in the single, dual and multi modes regimes. We can also tune the cutoff energies of eigenmodes in pMIM structures by varying insulator thickness and permittivity. In particular, we have studied excitation of SPPs in the single mode regime which is desired for its long propagation length. Utilising pMIM structures to create on-chip plasmon sources allows downwards scaling and integration into optoelectronic circuits, thereby reducing operation time and lowering power consumption.

5 Appendices

5.1 Eigenmode analysis of MIM waveguiding segment of the pMIM

We performed an eigenmode analysis of intrinsic modes supported by the MIM waveguiding segment of the pMIM both in theory and in simulation. The values of mode index and propagation length were theoretically obtained from calculation of the complex wavenumber k_x for eigenwaves propagating in the MIM waveguiding segment and compared with those derived in simulation by fitting spectra of the excited magnetic field (Fig. 8).

 

Fig. 8 Eigenmode analysis of intrinsic modes supported by Au-SiO₂-Au MIM waveguiding segment of the pMIM calculated in theory and in simulation. (a) Mode index as a function of photon energy for first three modes. (b) Propagation length vs photon energy for first three low-order modes.

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5.2 Fourier analysis of the omnidirectional excited waves

To study the mode index of the waves excited by the source, we did a Fast Fourier Transform of the complex magnetic field H_ϕ (r) obtained from FEM simulation along the MIM waveguiding segment of the pMIM at different photon energies. To extract the complex mode index of the excited plasmon-polaritons, we fitted the Fourier images with the Fourier image of omnidirectionally propagating eigenwaves in the MIM waveguiding segment of pMIM with the complex wavenumber k_x,

$$\begin{array}{l}H\left(k_r,z_0\right)=\frac{1}{\sqrt{2\pi }}\underset{\frac{1}{Re\left(k_x\right)}}{\overset{\infty }{{\displaystyle \int }}}H\left(r,z_0\right)e^{-i{k}_{r}r}dr=\frac{1}{\sqrt{2\pi }}\underset{\frac{1}{Re\left(k_x\right)}}{\overset{\infty }{{\displaystyle \int }}}{H}_0\left(z_0\right)H_1^1\left(k_{x}r\right)e^{-i{k}_{r}r}dr\\ \approx -\frac{1+i}{\sqrt{2\pi }} \frac{H_0\left(z_0\right)}{\sqrt{k_x\left(k_r-k_x\right)}}\end{array}$$

For all the regimes, we used several wave fitting that demonstrated excellent agreement for both complex magnetic field H_ϕ(r) as shown in Fig. 9 and complex wavenumber k_x as shown in Fig. 8, as well as confirmed the number of excited waves in all three regimes considered.

 

Fig. 9 Normalized Fourier images of Re(H_ϕ) as a function of mode index for photon energies of (a) 0.6 eV, (b) 1.1 eV, (c) 2 eV; normalized Fourier images of Im(H_ϕ) vs mode index for (d) 0.6 eV, (e) 1.1 eV, (f) 2 eV.

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5.3 Field analysis of the omnidirectional excited waves

To analyze the excited waves in the MIM waveguiding segment of pMIM by a compact source, we investigated the complex magnetic field H_ϕ(z) obtained from FEM simulations across the MIM waveguiding segment at different photon energies. We fitted those field profiles with the corresponding distribution of the magnetic field H_y(z) taken from the eigenvalue analysis for several low-order modes. This fitting showed excellent agreement between the eigenvalue theory and FEM simulation as shown in Fig. 10 and confirmed the excitation of one eigenwave in the single mode regime, two eigenwaves in the dual mode regime, and three eigenwaves in the multi mode regime.

 

Fig. 10 Real part of H_ϕ across the MIM structure for photon energies of (a) 0.6 eV, (b) 1.1 eV, (c) 2 eV; imaginary part of H_ϕ across vertical interface for (d) 0.6 eV, (e) 1.1 eV, (f) 2 eV.

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Acknowledgments

We acknowledge the support of the National Research Foundation Singapore under its Competitive Research Programme (Grant No. NRF-CRP 8-2011-07). We would like to thank Dr. Christian Albertus Nijhuis, Dr. Nikodem Tomczak, Dr. Wang Tao and Du Wei for the insightful discussions leading to this manuscript.

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