Multi-level multi-thermal-electron FDTD simulation of plasmonic interaction with semiconducting gain media: applications to plasmonic amplifiers and nano-lasers

X. Chen; B. Bhola; Y. Huang; S. T. Ho

doi:10.1364/OE.18.017220

1. Introduction

Optical radiation can be confined in volumes much smaller than the diffraction limit through its interaction with surface plasmons, which are free electron oscillations at the metal-dielectric interface [1]. Recently, there has been considerable interest in the field of surface plasmons due to their potential applications in the field of nanophotonics from the visible to the infrared wavelength range. Propagating surface plasmon wave confined at a planar interface between a metal and a dielectric has been studied for various applications, including waveguiding [1], fluorescence enhancement [2], and second harmonic generation (SHG) [3], to name a few. Localized surface plasmon resonance (LSPR) in metallic nano-particles has attracted much attention especially in bio-sensing partly due to the applications of surface enhanced Raman scattering (SERS), which has led to highly sensitive bio-sensors with capability to detect close to a single molecule [4]. Other interesting applications of LSPR include optical nanoantennas [5], waveguides [6], enhanced Rayleigh scattering [7], nano-lasers [8], SPASER (Surface Plasmonic Amplification with Stimulated Emission of Radiation) [9], enhanced Kerr effect [10], near field lithography [11], enhanced photochemistry [12], increased responsivity detectors [13], negative index materials [14], solar cells [15] and high resolution imaging [16], which have either been theorized or experimentally demonstrated.

The surface wave property of propagating plasmons makes it suitable to confine light in very small volumes leading to the demonstration of light confinement in sub-diffraction limited waveguiding structures [17–21]. Plasmonic waveguiding can be achieved in two planar configurations: metal-dielectric-metal (MDM) [17–20] and dielectric-metal-dielectric (DMD) [21]. In both configurations, there is a small gap between two surface plasmon or plasmon-polariton supporting metal-dielectric interfaces. The surface plasmons (plasmon-polaritons) on each interface can be coupled across the two interfaces provided the gap between them is small enough and guided optical modes can be achieved for the waveguide. The DMD configuration has been widely researched for waveguiding purposes and is used to realize long range surface plasmon (LRSP) (or plasmon-polariton) waveguides [21]. LRSP waveguides can have low propagation loss (a few dB per cm); however, the optical mode is a few microns in size at 1550nm. MDM waveguides on the other hand, can guide light in very small volumes much below the diffraction limit [17]. However, the propagation loss is very high (few dB’s per micron) due to the increased interaction of the plasmonic (or plasmon-polariton) field with the highly absorptive metal. Also, the loss increases exponentially as the gap between the metal layers is decreased to increase the optical mode confinement. Thus, there is a fundamental trade-off between optical confinement and propagation loss in a plasmonic (plasmon-polariton) waveguide.

In order to take advantage of plasmonic confinement in developing optical nano-devices such as nano-lasers, LEDs, waveguides and amplifiers, a suitable optical gain medium isrequired to compensate the plasmonic propagation loss. Many researchers have explored the possibility of incorporating a suitable gain medium along with plasmonic propagation or localization in order to demonstrate lasers, SPASERs or amplifiers [22–26].The SPASER has been explored through theoretical formulation [8,9] and recently been experimentally demonstrated by utilizing specially prepared dye-coated metal nanoparticles which act as the gain medium [27].

However, dye is not a very conducive gain medium for photonic or plasmonic device integration. In the paper, we focus on using direct-gap semiconductor as the optical gain medium for realizing active plasmonic devices, including nanolasers and lossless plasmonic waveguide with loss compensated by gain that will be conducive for integration on a semiconductor chip. In particular, we focus on the theoretical study of MDM based waveguiding structure with semiconductor as the gain medium and analyze the conditions under which a net optical gain can be realistically achieved. Furthermore, we study a novel plasmonic semiconductor nano-ring laser (PSNRL) structure and analyze under what conditions lasing can be realistically achieved. The work here will focus on devices that operate around the optical communications wavelength of 1550nm. The operating wavelength will be at off plasmon resonance but still take advantage of the higher propagation constant of the guided mode in a MSM plasmon-polariton waveguide resulting in small cavity size, which is a manifestation of the dispersion relation of plasmon-polaritons. Devices at near the regime of plasmon resonance will be presented in future works.

In order to perform the study, we will use the Finite Difference Time Domain (FDTD) method to model the interactions between the electromagnetic field and the plasmonic waveguide embedded with semiconductor gain medium. The simulation of semiconductor medium using FDTD is performed using a powerful Dynamical Thermal-Electron Quantum-Medium FDTD (DTEQM-FDTD) model developed by Huang et al. recently [28] involving multiple levels and multiple electrons that is capable of modeling the carrier dynamics of a direct-gap semiconductor at finite (e.g. room) temperature realistically. Below, we will simply refer to it as Multi-Level Multi-Thermal-Electron FDTD (MLMTE-FDTD) model. The MLMTE-FDTD simulation utilizes the FDTD approach for wave propagation coupled with a quantum mechanically derived set of rate equations for the optically active gain medium. In the MLMTE-FDTD medium model, the carrier dynamics are governed by Pauli’s exclusion principleand dynamical Fermi-Dirac thermalization process, which enable one to effectively model the complicated carrier band filling effect at finite temperature. The use of multiple energy levels enables one to effectively model or “fit” the conduction and valence energy band structures for the actual semiconductor material simulated. The use of multiple electrons with Pauli exclusion and Fermi-Dirac thermalization process enables one to effectively model the electron and hole movements in the conduction and valence bands at finite temperature, as well as the optical transitions between the bands. Furthermore, the carrier-dependant nonlinear optical refractive index and nonlinear optical gain or absorption in semiconductor are automatically accounted for in the MLMTE-FDTD model [28] as they are a result of the band-filling effect and carrier dynamics.

Section 2 outlines the simulation program for simulating the plasmon-polariton interaction with the semiconducting gain medium using the MLMTE-FDTD formulation. Section 3 explores the application of the simulation program to a simple semiconductor gain assisted plasmon-polariton waveguide. Section 4 explores the possibility of incorporating low refractive index intermediate layers between the metal and semiconductor to reduce propagation losses, and illustrates its problem. Section 5 discusses the operation of the plasmonic nano-ring laser and discusses its threshold conditions and experimental feasibility. Section 6 discusses potential improvement and experimental implementation. Section 7 concludes the paper.

2. The multi level multi thermal electronfinite difference time domain (MLMTE-FDTD) incorporating metallic response

We employ themulti-level multi-thermal-electron FDTD (MLMTE-FDTD) model which is capable of modeling the sophisticated semiconductor electron-hole dynamics in the conduction and valence bands, including carrier band filling, electron-hole relaxation dynamics, and carrier thermalization in a computationally efficient manner. The theory is based on a multi-level multi-electron model where the electronic band structure of the gain medium is divided into discrete levels as shown in Fig. 1 . As mentioned in [28], each conduction-valence band energy level pair is capable of representing a broad frequency bandwidth of about 50nm in wavelength span due to the large dipole dephasing rate of ~100fsecs in the semiconductor. Hence, a few energy-level pairs are enough to span a broad wavelength bandwidth. This makes the model computationally efficient and yet is capable of capturing the essential pumping dynamics of the electron and hole carriers. As shown in [28], Pauli exclusion and dynamical thermal up-transition terms are incorporated into the quantized energy-level rate equations, enabling simulation of electron and hole dynamics at finite temperature and the simulation model was referred to as dynamical thermal electron quantum-medium FDTD model or simply as MLMTE-FDTD model here.

Fig. 1 Energy band diagram for the MLMTE model showing the intraband and interband transitions used in the FDTD simulation.

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As discussed in [28], the wavelength spacing between two energy-level pairs should be close enough to obtain a relatively smooth absorption (or gain) curve but should be large enough to minimize computational time. In the work here we choose a wavelength spacing of 30nm that gives just a few percent ripples to the absorption curve (see for example Fig. 5) to obtain a good tradeoff between the simulation time and simulated energy bandwidth for the band structure. In this work, a model consisting of 7electrons and 7 energy-level pairs (7 valence and 7 conduction band levels) as shown in Fig. 1 effectively emulates a direct bandgap semiconducting medium [29]. Each energy level is characterized by several gain parameters as shown in the figure. To simulate external injection current, electrons are constantly excited from level 7_v to level 7_c (lowest valence band to the highest conduction band)at a rate Rpump(pumping rate), which is also illustrated in Fig. 1.The intra-band and inter-band electron-hole dynamics in the gain material are incorporated through their respective relaxation times, τ(i,i + 1)-c / (i,i + 1)-v, and τi(see [28] for the definition of the symbol labels) whereasΔωa1toΔωa7 correspond to the dephasing rates of the transition dipoles between each energy pairs. The parameter N0 is the medium’s energy-state number density which is the total volume density of states in the conduction and valence bands assuming a parabolic band like structure.

The metal response is incorporated through the Drude model (Eq. (1) which is coupled into Maxwell’s equations as a frequency dependent complex permittivity [30,31].

ε (ω) = ε_{\infty} - \frac{ω_{p}^{2}}{ω^{2} - j ω τ},

where ε∞ is the background permittivity due to bound electrons and ωp and τ are the volume plasmon frequency and the electron relaxation time for the free electron oscillations. All the simulations are done in 2D-FDTD using effective refractive index approximation. For the purpose of calculation in this paper, we take the effective refractive index of the vertical semiconductor structure to be n = 3.4 [32,33].

3. Gain assisted plasmon-polariton waveguide

Before we simulate the plasmonic nanoring laser, we first apply the model to simulate a straight plasmon-polariton waveguide with gain consisting of a semiconducting gain medium sandwiched between two metal layers. Light can propagate in this structure through the coupling of plasmon-polaritons at the two metal-semiconductor interfaces [34]. The MSM waveguide comprises of a semiconductor InGaAs core having a refractive index of 3.63 sandwiched between two gold coated surfaces (refractive index of 0.38 + 10.75i at 1550nm [35]). The vertical field confinement in the waveguide core is via the refractive index difference due to the InGaAs core surrounded vertically by lower refractive index InP cladding as shown in Fig. 2 .

Fig. 2 Schematic of the MSM plasmonic waveguiding structure. The InGaAs semiconducting core is sandwiched laterally by gold and vertically by lower refractive index InP cladding.

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The propagation losses of the straight waveguide are calculated using a passive 2D-FDTD simulation by exciting a plasmon-polariton waveguide mode at one end and monitoring its power transmission.

In order to compare with the FDTD numerical results, the relation between the complex propagation constant of the 2D MSM plasmon-polariton waveguide and width of the core is also solved analytically by solving the Eigen-value equation for a two dimensional MSM waveguide [36]. Applying Maxwell’s equations with appropriate boundary conditions, the characteristic equation for a symmetric Metal-Semiconductor-Metal (MSM) waveguide can be written as Eq. (2).

- \frac{k_{m}}{k_{d}} = \frac{ε_{m} (ω)}{ε_{d}} \tanh (\frac{d}{2} \cdot k_{d}),

where,

k_{m} = \sqrt{β^{2} - k^{2} \cdot ε_{m} (ω)}

and

k_{d} = \sqrt{β^{2} - k^{2} \cdot ε_{d}} .

Here, β is the complex propagation constant for the guided mode, k is the propagation constant in free-space, d is the width of the waveguide, and εm and εd are the complex permittivity for the metal and dielectric respectively. Only the lowest-order symmetric mode of the MSM waveguide is considered. However, a suitable approximation for Eq. (2) can be utilized [37] where an analytical expression can be written for the complex propagation constant of the propagating plasmon-polariton wave as shown in Eqs. (3), and (4).

β = k \sqrt{ε_{d} + 0.5 \cdot {(\frac{β_{0}}{k})}^{2} + \sqrt{[ε_{d} - ε_{m} (ω) + 0.25 \cdot {(\frac{β_{0}}{k})}^{2}] \cdot {(\frac{β_{0}}{k})}^{2}}},

β_{0} = - \frac{2 \cdot ε_{d}}{d \cdot ε_{m} (ω)},

where, β0 is the propagation constant of the plasmon-polariton wave in the limited→ 0. The real part and imaginary part of β gives the absorption coefficient (α) and the propagating refractive index (nz) as shown in Eqs. (5) – (8).

n_{z} = {[\frac{{[K^{2} + L^{2}]}^{0.5} + K}{2}]}^{0.5}, α = - j \cdot 2 k_{0} {[\frac{{[K^{2} + L^{2}]}^{0.5} - K}{2}]}^{0.5},

where

K = \frac{G \cdot A + H \cdot B}{k_{0}} + ε_{d} + \frac{0.5}{k_{0}^{2}} \cdot [\frac{4 \cdot ε_{d}^{2} \cdot (ε_{m r}^{2} - ε_{m i}^{2})}{d^{2} \cdot {(ε_{m r}^{2} + ε_{m i}^{2})}^{2}}], L = \frac{G \cdot B - H \cdot A}{k_{0}} - \frac{0.5}{k_{0}^{2}} \cdot [\frac{8 \cdot ε_{d}^{2} \cdot ε_{m r} \cdot ε_{m i}}{d^{2} \cdot {(ε_{m r}^{2} + ε_{m i}^{2})}^{2}}],

A = \frac{- 2 \cdot ε_{d} \cdot ε_{m r}}{d \cdot (ε_{m r}^{2} + ε_{m i}^{2})}, B = \frac{2 \cdot ε_{d} \cdot ε_{m i}}{d \cdot (ε_{m r}^{2} + ε_{m i}^{2})}, G = \sqrt{\frac{\sqrt{E^{2} + F^{2}} + E}{2}}, H = \sqrt{\frac{\sqrt{E^{2} + F^{2}} - E}{2}},

E = ε_{d} - ε_{m r} + \frac{ε_{d}^{2} \cdot (ε_{m r}^{2} - ε_{m i}^{2})}{k_{0}^{2} \cdot d^{2} \cdot {(ε_{m r}^{2} + ε_{m i}^{2})}^{2}}, F = ε_{m i} + \frac{2 \cdot ε_{d}^{2} \cdot ε_{m r} \cdot ε_{m i}}{k_{0}^{2} \cdot d^{2} \cdot {(ε_{m r}^{2} + ε_{m i}^{2})}^{2}},

with εmr and εmi being the real and imaginary parts of the dielectric function of the metal. FDTD simulations are carried out for different widths of the waveguiding structure at a wavelength of 1550nm. Figure 3 shows the variation in the propagation loss and propagating refractive index of the lowest order symmetric mode of the MSM plasmonic waveguide and compares the FDTD results to the approximate analytical results given by Eq. (5). The propagation loss as well as the propagating refractive index increases exponentially as the width of the waveguide core is reduced [34] and matches reasonably well with the approximate expression of Eq. (5). This verifies the proper working of the numerical FDTD program. The propagation loss is largely due to the absorption coefficient of the surrounding metal at the wavelength of interest. However, the refractive index of the dielectric core also plays an important part in determining the magnitude of the plasmon-polariton waveguide propagation loss.

Fig. 3 Propagation loss (a) and propagating index (b) for a straight waveguide as a function of width of the semiconductor core. The refractive index of the core and cladding of the MSM waveguide are 3.4 and 0.38 + 10.75i respectively at 1550nm.

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Using the parameters for gold at 1550nm in Eq. (3), propagation loss as a function of the refractive index of the core is plotted in Fig. 3 for a w = 50nm wide MDM plasmon-polariton waveguide. As the refractive index of the core increases, the propagation loss for the plasmon-polariton wave also increases as shown in Fig. 4 . This is of concern when designing active plasmonic devices based on semiconductors due to their relatively high dielectric permittivities [38].

Fig. 4 Plasmonic propagation loss as a function of refractive index of the dielectric or semiconductor.

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For simulating semiconductor based plasmon-polariton MSM waveguiding structures with gain, the simulation parameters for InGaAs are summarized in Table 1 . The physical meaning of the various simulation parameters have been explained in Sec 2. The bulk InGaAs medium is assumed to have a parabolic band structure. A total of 7 pairs of energy levels are chosen by taking into account the trade-off between simulation accuracy and computational time. Pair 1 - 6 each represents an energy broadened electron transition dipole with discrete optical transition wavelengths that cover the range from 1585nm to 1435nm and are equally spaced by 30nm in our simulation. Each transition dipole is homogeneously broadened with a spectral line width of 50nm. The pumping level (level 7) is intentionally spaced 135nm away from level 6 to account for those higher energy levels not described by level 1-6. The volume density of states of each level is respectively determined from Eq. (9) below (Eq. (22) in [28]) and normalized to the result of level 1.Please refer to the section 7 in [28] for definitions of parameters and the detailed derivation of this equation. The interband decay time, τ interband, is set as 1 ns. Downward intraband transition times, τ intraband, are set as200fs and 100fs for conduction band and valence band, respectively. The corresponding upward transition times calculated from Eq. (26) in [28] are based on the Fermi-Dirac thermalization effect which depends on the temperature, the energy difference, and the respective volume density of states as explained in [28]. The temperature for the thermalization effect is taken to be room temperature at T = 300K.

Table 1. Simulation parameters utilized for simulating InGaAs semiconducting medium in the MLMTE-FDTD program.

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N_{C, V i}^{0} (\bar{r}) = \int_{Δ E_{C, V (i - 1)}}^{Δ E_{C, V i}} ρ (Δ E) \cdot d E = \frac{16 \sqrt{2} \cdot m_{C}^{3 / 2} m_{V}^{3 / 2} \cdot [{(E_{i +}^{B} - E_{_{G}})}^{3 / 2} - {(E_{i -}^{B} - E_{_{G}})}^{3 / 2}]}{3 π^{2} ℏ^{3} {(m_{C} + m_{V})}^{3 / 2}} .

By propagating an optical pulse through a bulk InGaAs medium (un-pumped, absorbing) and calculating the ratio of the output power to the input power, the bulk absorption coefficient is calculated. The total valence band volume density of states N0, representing the sum of density of states of all levels in the valence band, is considered as a fitting parameter and is adjusted so that the absorption spectrum matches the typical spectrum obtained for InGaAs [39]. The absorption spectrum obtained from the simulation with N0 = 8.16 × 1016cm-3is plotted for bulk InGaAs in Fig. 5, which matches reasonably well with that of InGaAs semiconductor [39].It is possible, in theory, to achieve a gain of ~0.5μm-1if we pump the semiconducting medium hard enough reaching the condition of complete inversion where all the electrons are residing in the conduction band. The condition of complete inversion is generally not preferred in practice as it requires injecting electrons in the gain material at very high pumping rates which generates a considerable amount of heat. Thus, to keep the calculations as realistic as possible, we limit the pumping rate such that the carrier distribution reaches about 50% inversion corresponding to a bulk medium gain of 0.25μm-1at 1550nm. This is particularly important for simulating Plasmon-polariton nano-lasers.

Fig. 5 Bulk absorption coefficient of InGaAs simulated using the MLMTE-FDTD model with total valence band volume density of states N0 = 8.16 x 1016 cm-3.

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Next we estimate the plasmon-polariton waveguide width at which the propagation loss matches the semiconductor gain, i.e. the lossless propagation condition for a bulk gain of 0.25μm-1.A phenomenon that is of considerable interest is the propagation refractive index of the guided mode in a MSM plasmon-polariton waveguide being higher than the bulk refractive index of the core. This is a manifestation of the dispersion relation of plasmon-polaritons and is very useful in developing semiconductor based plasmon-polariton nano-waveguides. The higher propagating group refractive index leads to slowing down of light inside the waveguide due to the lower group velocity of the plasmon-polariton wave, effectively increasing the interaction between the plasmon-polaritons and the atoms of the gain medium. A longer interaction time leads to an increased gain per unit length that can be achieved from a plasmon-polariton waveguide as compared to bulk gain medium [40].Fig. 6 shows our simulated small signal gain coefficient obtained for both bulk and plasmon-polariton MSM waveguide plotted as a function of bulk small signal gain coefficient obtained from the semiconducting gain medium. When compared to bulk semiconductor medium, the gain per unit length increases as the width of the plasmon-polariton waveguide is reduced. However, this gain is accompanied by increased propagation loss with reduced width which requires careful consideration while choosing the optimum width of the waveguide core.

Fig. 6 Effective propagating gain coefficient as a function of bulk gain coefficient of the semiconducting gain medium (4 um). This enhancement effect can be attributed to a reduction in group velocity of the surface plasmonic wave.

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The gain enhancement effect can only be seen in plasmon-polariton waveguiding structures as in a dielectric waveguide, the propagation refractive index of the electromagnetic wave is usually smaller than that in a bulk medium. The propagating gain coefficient (gain per unit length) in a plasmon-polariton MSM waveguide can then be written as Eq. (10) below.

g_{z} = g_{b u l k} \cdot \frac{n_{z}}{n_{b u l k}} \cdot Γ,

where, gbulk is the gain coefficient of the bulk semiconducting medium, nzis the propagating group refractive index of the MSM waveguide mode and nbulk is the group refractive index of the bulk semiconducting medium. Γ is the overlap integral, which is very close to unity for MSM plasmon-polariton waveguides. This gain enhancement effect can be clearly observed in Fig. 6, in which we show simulated propagating gain coefficients corresponding to plasmon-polariton waveguides of different widths.

Finally, a plot of the straight waveguide loss along with the transmission from an MSM waveguide with gain is plotted in Fig. 7 for different widths of the waveguide core. The bulk gain in the MSM waveguide is set at 0.25μm-1 for reasons discussed earlier. The point at which the two curves intersect determines the width of the plasmon-polariton waveguide that will achieve lossless propagation. This occurs approximately at a waveguide core width of 75nm.

Fig. 7 Straight waveguide loss and enhanced gain from an MSM waveguide with a gain coefficient of 0.25μm-1. The total number density is set at 8.16 x 1016 cm-3. The optimum plasmon-polariton waveguide width for loss-less propagation is 75nm.

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In summary, what we have learned in these simulations is that a lossless Plasmonic MSM waveguide for 1550nm applications can be achieved with gold cladding and InGaAs semiconductor core provided the width of the semiconductor core is around 75nm or wider (assuming a pumping current to reach a bulk material gain of 0.25μm-1for the InGaAs material).

4. Gain-assisted plasmon-polariton waveguide with intermediate low index buffer layers

The plasmon-polariton waveguide propagation loss is dependent on the permittivity of the dielectric medium as discussed in the previous section (Fig. 3). In order to reduce these losses, a thin layer of a low refractive index material such as SiO2 can be introduced between the metal and gain medium in the MSM waveguiding structure. The low refractive index layer induces plasmon-polariton wave at the metal-SiO2 interface which couples through the semiconducting gain medium to the other metal-SiO2interface [41]. This configuration according to Eq. (2) above should reduce propagation losses due to a lower propagating refractive index.

The plasmon-polariton waveguide amplifying structure is illustrated in Fig. 8 (inset) along with normalized electric field profiles for waveguides with (Fig. 8b) and without (Fig. 8a) the intermediate SiO2 layers. The eigen modes were calculated using MODE solutions simulation software from Lumerical [42]. The total width of the waveguiding structure (SiO2 buffer layer + semiconductor gain medium) is kept constant at w = 50nm to keep the total mode volume identical.

Fig. 8 Field profiles for the MSM waveguides a) without any SiO2 buffer b) with 5nm SiO2buffer layers on each side between metal and semiconductor.

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Two cases of w = 50nm and w = 100nm of total waveguide core width are explored for a TM polarized (with E-field vector perpendicular to the metal-dielectric interface) wave with varying widths of the intermediate low refractive index SiO2 layers. These two structures are then simulated using the MLMTE-FDTD program with InGaAs as the active semiconducting medium having the same parameters as described in Table 1. However, in this case, the small signal gain coefficient is set at 0.346μm-1 at 1550nm in order to achieve the lossless propagation condition for a w = 50nm wide MSM Plasmon-polariton waveguide having an propagating refractive index of 4.78 (see Fig. 2, Eq. (4) and Fig. 5) (note: gz = 0.346*4.78/3.4 = 0.49μm-1, is approximately equal to the loss of the w = 50nm wide MSM waveguide).

The simulation results for the w = 50nm and w = 100nm wide plasmon-polariton waveguide are summarized in Table 2 . The total power transmission is represented as gain (positive) or loss (negative) in μm-1. The results show that although propagation loss is significantly reduced by the addition of low refractive index SiO2 layer, the total gain is reduced as well. In fact, the gain is reduced faster. This is because of the concentration of the field energy in the SiO2 layer (see Fig. 8b) due to the higher electric field strength in the intermediate SiO2 layers compared with the semiconductor core layer arising from the dielectric boundary conditions. Note that the field strength scales up as (nInGaAs/nSiO2)2 across from SiO2 layer to InGaAs layer, where nInGaAs~3.4 is the refractive index of InGaAs and nSiO2~1.5 is the refractive index of SiO2. The energy density given by εE2 also scales up as (nInGaAs/nSiO2)2.As a result, there is a faster reduction in the overlap integral between the plasmon-polariton mode and the active semiconducting region and hence a faster reduction in the gain coefficient compared to the reduction in the loss coefficient due to the SiO2layer. The last two rows of Table 2 compares the transmission achieved through MLMTE-FDTD simulations with that calculated from Eq. (4). There is agreement between the two results to within ± 5%. To quantify these two effects, Fig. 9 (a) and (b) plots the overlap integral and the propagating refractive index of the guided plasmon-polariton mode as a function of SiO2 thickness for a total waveguide width of w = 50 and w = 100nm. As expected, both the propagating refractive index and the overlap integral decrease monotonically as the width of the intermediate SiO2 layer is increased. However, the overlap integral decreases faster than the propagating refractive index as shown in Fig. 9(c) and 9(d).

Table 2. Simulation Results for w = 50 nm and w = 100 nm Surface Plasmonic Waveguides with InGaAs Gain Medium for SiO2 Thicknesses of 5 nm and 10 nm for w = 50 nm, and SiO2 Thicknesses of 10 nm and 20 nm for w = 100 nm^a

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Fig. 9 The overlap integral, (a), and propagating refractive index, (b), of the surface plasmonic waveguide as a function of width of the intermediate SiO2 layer for a total core width of 50nm (red) and 100nm (black).(c) and (d) show the variation in the total propagating gain or loss coefficient (μm-1) as a function of SiO2 intermediate layer width for the plasmon-polariton waveguide total width of w = 50nm and 100nm respectively.

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In summary, what we have learned in this section is that the faster decrease in the gain overlapping factor compared with the decrease in the propagating refractive index actually overwhelms the lower loss advantage with the SiO2 layer and results in lower net gain. The faster decrease in gain is due to the enhanced field energy density in the SiO2 layer compared to the semiconductor layer by (nInGaAs/nSiO2)2.

5. The Plasmon-polariton nano-ring laser

Conventional dielectric lasers based on photonic crystals [43], Distributed Bragg Reflectors (DBR) [44], microdisks [45] and micro-loop mirrors [46] have an inherent limitation on the geometrical volume of the cavity in which electromagnetic radiation can be confined and amplified for sustained oscillations. This is due to the propagating wavelength limitation that dictates the length of the optical cavity to a value greater than or equal to around λ/(2nmed) and the mode width to a similar value(λ is the radiation wavelength in free space and nmed is the dielectric medium’s refractive index), resulting in a smallest possible laser cavity volume of larger than(λ/2nmed)3. The lateral mode size limitation is attributed to diffraction limitation imposed on optical radiation (the mode propagating angle is equivalent to “diffraction angle” from the finite mode size. This angle is limited to be smaller than 90 degree at which the smallest λ/(2nmed) mode size is achieved). Thus, (λ/2nmed) 3 is typically referred to as diffraction limited volume. Photonic crystals have been used to demonstrate semiconductor based lasing cavities with a volume around 5 times of (λ/2nmed) 3 [49]. Recently, there are reports that provide [50–52] a way to reduce the effective volume of a cavity by introducing a low refractive index slot in a photonic crystal cavity. Cavity volumes as low as 0.1*(λ/2nmed)3 have been reported but such small volumes are only possible in laser cavities where the active medium is inside the slot such as gas lasers or erbium doped silica lasers [50]. For semiconductor lasers, this scheme is very difficult to implement as the spatial overlap between the confined optical radiation and the semiconducting gain medium is very small resulting in very long cavities to achieve net gain. Thus, the inability to further shrink integrated semiconductor light sources could create potential bottleneck in future ultra-dense, integrated photonic chips. Another advantage of the plasmonic nanoring geometry is that unlike the dielectric ring cavity, radiation loss is basically absent due to the metallic field confinement. Photonic crystal also limits radiation loss but has an effective field penetration length into the crystal structure, which results in a larger mode size.

In this section, we apply the method and results obtained above to explore a plasmon-polariton nano-ring cavity with a semiconductor sandwiched between gold to achieve a smaller cavity volume than the diffraction limited volume of (λ/2nmed)3. The configuration for the plasmonic nano-laser cavity considered in this paper is illustrated schematically in Fig. 10 below.If the semiconducting gain medium can provide enough gain to overcome metal absorption loss in one round-trip, self-sustained oscillations can be generated inside this plasmonic nano-cavity.

Fig. 10 3D schematic of the plasmonic nano-laser considered for MLTME-FDTD simulations.

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MLMTE-FDTD simulations were performed for a w = 80nm wide, dID = 138nm inner diameter and dOD = 298nm outer diameter nano-ring laser. The total valence-band volume density of states is set at 8.16 x 1016 cm-3 as before. An outer diameter of dOD = 298nm is specifically chosen so that the passive cavity resonance is at 1550nm.The quality factor (Q) is first estimated by launching a TM (in-plane) polarized pulse in the nano-cavity and observing the spectrum of the circulating light. The spectrum is shown in Fig. 11 and the calculated Qvalue is 27. The round trip loss in the nano-cavity can be calculated by using Eqs. (11) and (12) [53].

Q = \frac{2 \cdot π^{2} \cdot R \cdot n_{z} \cdot \sqrt{a}}{(1 - a) \cdot λ_{0}},

a = \exp (- α \cdot π \cdot R),

where R is the radius of the cavity, nz is the propagating index, λ0 is the resonant wavelength of the cavity, a is the amplitude loss factor which represents the decay in amplitude of the electric field after one round trip inside the cavity, and α is the power loss coefficient. Solving the above expression for a by using the values of Q = 27, R = 109nm, nz = 4.23, and λ0 = 1550nm, we obtain α = 0.322μm-1. This agrees well with the straight waveguide loss of 0.32μm-1 and also suggests that bending loss can be neglected as compared to absorption loss in a plasmonic cavity.

Fig. 11 Power spectrum plot of lasing cavity

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The simulation parameters for the active simulation are the same as before (Table 1). The steady state magnetic field (Hz) of the lasing mode is shown in Fig. 12 along with a diametric cross-section for the field profile along with the lasing spectrum in Fig. 13 .. The magnetic field profile in the cavity shows the characteristic feature of a bound (i.e. guided)plasmon-polariton field having its maxima at the metal-semiconductor interface. The lasing simulation is done by first pumping the medium computationally with a constant pump rate that is equivalent to a certain external injection current (see [28]) and then the lasing is initiated with a short external pulse to replicate the effect of lasing initiation by spontaneous emission (the MLMTE-FDTD model does not include spontaneous emission noise though it includes spontaneous carrier population decay).

Fig. 12 Simulation results: (a) Steady state magnetic field distribution in the lasing cavity; (b)The cross-sectional plot of the Hz magnetic field strength across a diameter of the laser.

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Fig. 13 Lasing spectrum for the plasmonic nano-ring laser. The lasing peak is at 1548nm.

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In the plasmon-polariton nano-cavity here, there are no pure dielectric modes that can be excited, which is not the case for the metal coated nano-lasers reported in [48] where the lasing mode is primarily a HE11dielectric waveguide mode. The modes supported by the metal coated nano-resonator in [48] contains both dielectric and plasmon-polariton modes. However, the mode with lowest round-trip loss is usually the dielectric mode as demonstrated experimentally in [48]. By increasing the shield thickness in between the metal and the semiconductor gain medium in the nano-resonator, the mode increasingly becomes dielectric in nature and loses its plasmonic character. This limits the size of the cavity to a few hundred nanometers in diameter in accordance with the diffraction limit (310 ± 25nm total diameter of Si3N4coated InGaAs pillar structure). However, the addition of a dielectric buffer layer in between the metal and the semiconductor can help reduce surface recombination loss of excited electrons.

To illustrate the onset of lasing threshold, the total energy density in the cavity which is determined by is plotted as a function of the current density in Fig. 14 [54]. The threshold current density for the laser is approximately 1kA/cm2. This value is close to the current densities used in conventional semiconductor lasers and hence is a realistic value. The total current density is adequately predicted by the MLMTE-FDTD model due to the inclusion of band-filling effect. The gain spectrum for the semiconductor medium is also plotted for different injection current densities as shown in Fig. 15 . The simulation is carried out for varying pumping rates and the steady state behavior of the nano-laser is determined based on the knowledge of electron population of the various electronic energy levels in the semiconducting medium. Gain pinning can be observed at 1550nm with the gain saturating at a value of 0.258μm-1 for the bulk medium. The propagating refractive index for an 80nm wide plasmon-polariton MSM waveguide is 4.23 from Fig. 2. This gain, according to Eq. (4) corresponds to a propagating gain of 0.32μm-1 which is very close to the lossless condition for an 80nm-wide straight MSM plasmon-polariton waveguide. Also, this threshold gain matches well with the loss coefficient α calculated using Q value.

Fig. 14 2D Energy density inside of lasing cavity as a function of different current density of the semiconductor medium.

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Fig. 15 Gain spectrum at lasing steady state for three cases corresponding to 1.3, 5.2, and 10.4kA/cm2.

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The effective mode volume of the plasmon-polariton lasing cavity (80nm radius) is calculated using the expression, Eq. (13) [46], for a 2D structure assuming the mode size in the third dimension is the geometrical size of the core.

V_{e f f} = \frac{\int_{V} μ (\vec{r}) \cdot {| H (\vec{r}) |}^{2} d^{2} r}{\max [μ (\vec{r}) \cdot {| H (\vec{r}) |}^{2}]} \cdot h .

Here, h is the height of the structure and μ is the magnetic permeability. The effective mode volume is 0.0071μm-3 which is about an order of magnitude smaller than the smallest photonic crystal cavity demonstrated [49]. The plasmon-polariton mode volume can also be written as 0.59* (λ/2n) 3 which is considerably smaller the diffraction-limited volume for dielectric cavities. This volume is for a metal semiconductor plasmon-polariton cavity at room temperature and further reduction is possible by lowering the temperature as described in the following section.

In summary, from the results of this section, we have learnt that the waveguide width of the plasmonic semiconductor nanoring laser (PSNRL) can be as small as about 80nm and the outer diameter of the PSNRL can be around 300nm. The lasing mode has only two optical cycles in the ring cavity as shown in Fig. 12(a). To achieve lasing, a bulk InGaAs materials can be used as the gain material at room temperature, which will give a lasing threshold current density of about 1kA/cm2. This corresponds to a threshold current of only about 550nA given the area of lasing cavity is 5.48 × 10-10 cm2 [55].

6. Potential improvement and experimental implementation

In practical implementations, it is also possible to reduce the metal absorption loss by cooling. Lowering the temperature increases the mean free path of electrons which corresponds to larger electron relaxation times and hence, lowers absorption loss. This effect can be described in the Drude model for modeling metals. The real and imaginary parts of the dielectric function of a metal as a function of frequency can be written according to the Drude model as follows (Eq. (14) [47]:

ε^{'} (ω) = 1 - \frac{ω_{p}^{2}}{ω^{2} + τ^{2}}, ε^{''} (ω) = \frac{ω_{p}^{2} \cdot τ}{ω^{3} + ω \cdot τ^{2}} .

The real part of the dielectric function does not change significantly with temperature. However, the imaginary part has a strong dependence on temperature due to the large variation in the electronic relaxation time, τ [47]. Also, at lower temperatures, semiconducting medium are known to exhibit higher gain [39]. The net effect is a lowering of the optical gain requirement on the semiconducting medium.

For a practical device, a schematic of the cross-section of the waveguiding geometry along with the waveguiding eigen-mode is shown in Fig. 16 . The eigen-mode is calculated by using the COMSOL RF module software package. In the lateral direction the confinement is due to the plasmon-polariton interaction while in the transverse direction it is due to dielectric confinement.

Fig. 16 a) The cross-sectional schematic of the plasmonic nano-waveguide; b) The fundamental mode profile of the w = 50nm wide plasmonic waveguide.

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7. Conclusion

To conclude, electromagnetic interaction of a plasmon-polariton wave with a semiconducting gain medium is studied for two very specific cases of a gain assisted MSM waveguide and a plasmon-polariton nano-ring laser. The study was performed using a powerful Multi-Level Multi-Thermal-Electron FDTD method to realistically simulate electromagnetic wave interaction with the plasmonic waveguide and cavity embedded with semiconductor gain medium. It was found that even though the addition of a low refractive index buffer layer between the metal side wall and the semiconductor reduced the plasmonic propagation loss for the straight waveguide amplifier considerably, the reduction in the gain overlap integral results in reduction in the net gain and hence adding a buffer layer brings no advantage in theory. However, in practice a thin buffer layer may help to reduce surface recombination loss of the excited carriers. Plasmon-polariton waveguide based nano-ring lasers are simulated and their threshold parameters investigated for experimental feasibility. The mode volume for an 80nm radius plasmon-polariton nano-ring with 298nm outer ring diameter is calculated as 0.0071μm3 which is about an order of magnitude smaller than the smallest photonic crystal based laser and approximately one third of the diffraction limited volume. At 298nm diameter, the cavity has a length of two optical cycles. The threshold current density for the plasmon-polariton nano-laser is of the order of 1kA/cm2 which is very similar to the current densities used in conventional semiconductor lasers and hence can be realistically achieved. The dimensions used in this paper and the laser parameters obtained lead us to believe that experimental realization of a plasmon-polariton semiconductor nano-laser is indeed possible at room temperature.

The optical gain requirement on the semiconducting medium is on the higher side partly due to the high absorption losses in gold and partly due to the wavelength of operation. Low temperature operation can reduce the gain requirement on the semiconducting material considerably due to the reduction in the propagation loss in metals and increased gain from semiconductors.

Acknowledgements

The authors thank Reuben M. Bakker and Koustuban Ravi for help in certain numerical calculations and paper editing.

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54. Energy density at the steady state in the cavity is determined by: $η = 0.5 \cdot \int_{S} (ε (\bar{r}) \cdot {| E (\bar{r}) |}^{2} + μ (\bar{r}) \cdot {| H (\bar{r}) |}^{2}) \cdot d \bar{r}$ (S is the area of the lasing cavity).

55. The area of lasing cavity is calculated by: $S = π \cdot {(d_{OD} / 2)}^{2} - {(d_{ID} / 2)}^{2}]$ = 5.48 × 10–10 cm2.

MLMTE-FDTD simulation parameters	Value
Spatial resolution, dx, dy [nm]	2
Total valence band volume density of states [cm-3]	8.16 × 1016
Bandgap wavelength (λbandgap[μm]	1.600
Transition dipole wavelengths λ 1 –λ7[μm]	1.585, 1.555, 1.525, 1.495, 1.465, 1.435, 1.300
Volume density of states for each energy level (normalized to level 1) (i1 – i7)	1, 1.901, 2.582, 3.212, 3.831, 4.461, 7.700
ngold	0.38 + 10.75 i
tinterband [ns]	1
tintraband(conduction band)[fs]	200
tintraband(valence band)[fs]	100
ωa1 toωa7[rad/s] (dipole broadening)	3.92 x 1013
Temperature (K)	300

Waveguide total width, w = 50nm
SiO2 Thickness (nm)	0	5	10
Loss (μm-1) w/o gain	−0.54	−0.348	−0.277
Propagating index (nz)	4.78	3.47	2.95
Overlap (%), Γ	~100	43.7	22.8
Transmission(μm-1) (Loss (μm-1) w/gain)	−0.04	−0.19	−0.21
gz = $g_{b u l k} \cdot \frac{n_{z}}{n_{b u l k}} \cdot Γ$	0.495	0.154	−0.068

Waveguide total width, w = 100nm
SiO2 Thickness (nm)	0	10	20
Loss (μm-1) w/o gain	−0.34	−0.188	−0.15
Propagating index (nz)	4.17	3.03	2.49
Overlap (%), Γ	~100	44.5	23.2
Transmission (μm-1) (Loss (μm-1) w/gain)	0.084	−0.051	−0.089
gz = $g_{b u l k} \cdot \frac{n_{z}}{n_{b u l k}} \cdot Γ$	0.424	0.137	0.061

MLMTE-FDTD simulation parameters	Value
Spatial resolution, dx, dy [nm]	2
Total valence band volume density of states [cm-3]	8.16 × 1016
Bandgap wavelength (λbandgap[μm]	1.600
Transition dipole wavelengths λ 1 –λ7[μm]	1.585, 1.555, 1.525, 1.495, 1.465, 1.435, 1.300
Volume density of states for each energy level (normalized to level 1) (i1 – i7)	1, 1.901, 2.582, 3.212, 3.831, 4.461, 7.700
ngold	0.38 + 10.75 i
tinterband [ns]	1
tintraband(conduction band)[fs]	200
tintraband(valence band)[fs]	100
ωa1 toωa7[rad/s] (dipole broadening)	3.92 x 1013
Temperature (K)	300

Waveguide total width, w = 50nm
SiO2 Thickness (nm)	0	5	10
Loss (μm-1) w/o gain	−0.54	−0.348	−0.277
Propagating index (nz)	4.78	3.47	2.95
Overlap (%), Γ	~100	43.7	22.8
Transmission(μm-1) (Loss (μm-1) w/gain)	−0.04	−0.19	−0.21
gz = $g_{b u l k} \cdot \frac{n_{z}}{n_{b u l k}} \cdot Γ$	0.495	0.154	−0.068

Multi-level multi-thermal-electron FDTD simulation of plasmonic interaction with semiconducting gain media: applications to plasmonic amplifiers and nano-lasers

Abstract

1. Introduction

2. The multi level multi thermal electronfinite difference time domain (MLMTE-FDTD) incorporating metallic response

3. Gain assisted plasmon-polariton waveguide

4. Gain-assisted plasmon-polariton waveguide with intermediate low index buffer layers

5. The Plasmon-polariton nano-ring laser

6. Potential improvement and experimental implementation

7. Conclusion

Acknowledgements

References and links

Cited By

Figures (16)

Tables (2)

Equations (15)

Optics Express