Study of a sub-wavelength scale plasmonic traveling wave amplifier with electrically pumped multiple quantum wells

The MQW simulation is based on the following equations [25]. These symbols' explanation and simulation parameters are shown in tables A1 and A2 in the appendix respectively.

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot \left({\varepsilon }_{0}{\varepsilon }_{dc}{\rm{\nabla }}\phi \right)=-q\left(p-n+{N}_{d}^{+}-{N}_{A}^{-}\right)\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial n}{\partial t}={\rm{\nabla }}\cdot \left[-{\mu }_{n}n{\rm{\nabla }}\phi +{D}_{n}{\rm{\nabla }}n\right]-{R}_{SRH}-{R}_{Aug}-{R}_{sp}-{R}_{st}\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial p}{\partial t}={\rm{\nabla }}\cdot \left[-{\mu }_{p}p{\rm{\nabla }}\phi +{D}_{p}{\rm{\nabla }}p\right]-{R}_{SRH}-{R}_{Aug}-{R}_{sp}-{R}_{st}\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {N}_{p}}{\partial t}=\displaystyle \iint {R}_{st}dxdy-\displaystyle \frac{{N}_{p}}{{\tau }_{p}}+\beta \displaystyle \iint {R}_{sp}dxdy\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{R}_{st}={v}_{g}{N}_{p}g\left(x,y\right){\left|\psi \left(x,y\right)\right|}^{2}\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&\left({{\rm{\nabla }}}_{xy}^{2}+{k}_{0}^{2}{\varepsilon }_{opt}\right)\psi \left(x,y\right)={k}_{0}^{2}{n}_{eff}^{2}\psi \left(x,y\right)\end{eqnarray} \tag{ 8 }$$

The Poisson equation (3) and the carrier continuity equations (4) and (5) are used to get the electric potential $\phi ,$ electron density n and hole density p. The optical rate equations (6) and (7) are utilized to calculate the photon density ${N}_{p}$ and ${R}_{st}$ when the $\psi \left(x,y\right)$ and $g\left(x,y\right)$ are known. The $\psi \left(x,y\right)$ can be got by solving the Helmhertz equation (8). And the $g\left(x,y\right)$ can be gotten from equation (2) using the n and p under the Lorentzian broadening condition in which the time constant is used and shown in table A2.

All of these equations are self-consistent and solved by circular iteration as the MQW bias voltage increasing. The software Crosslight PIC3D is used in this simulation.

In the above solution process the Auger recombination, spontaneous radiation recombination and SRH recombination must be considered. In other words, ${R}_{Aug},$ ${R}_{sp},$ ${R}_{SRH}$ need to be calculated using formulas from the published works as follows.

As for Auger recombination rate ${R}_{Aug},$ the formula ${R}_{Aug}=\left({C}_{n}n+{C}_{p}p\right)\left(np-{n}_{i}^{2}\right)$ [23] is used, where $n,p,{n}_{i}$ are the electron, hole and intrinsic electron density respectively. And the Auger coefficients $C={C}_{n}={C}_{p}$ are assumed to be represented by the temperature dependent function $C\left(T\right)={C}_{0}\left({T}_{0}\right)exp\left[\displaystyle \frac{{E}_{A}}{k}\left(\displaystyle \frac{1}{{T}_{0}}-\displaystyle \frac{1}{T}\right)\right]$ [26] where k is the Boltzmann constant, ${C}_{0}=5\times {10}^{-29}\,c{m}^{6}\,{s}^{-1}$ at ${T}_{0}=298\,K$ and ${E}_{A}=100\,mev.$ Utilizing these formulae the Auger coefficients in the QW, barrier and SCH regions at $T=300\,K$ are gotten and shown in table A2.

The spontaneous radiation recombination rate ${R}_{sp}$ is calculated according to ${R}_{sp}=B{n}^{2}$ [27] where n is the electron density and B is the spontaneous radiation recombination coefficient also shown in table A2.

As to ${R}_{SRH},$ there are two cases. One is the bulk recombination and the other is the surface recombination.

The bulk recombination in the QW, barrier, SCH, n-InP and p-InP area is described by the electrons and holes life time (${\tau }_{n}$ and ${\tau }_{p}$ shown in the table A2) and it is assumed that a uniform distribution of donor mid-gap traps density is ${10}^{4}\,c{m}^{-3}.$ In the semi-insulation (SI) Fe-doped InP substrate and cladding, the bulk recombination is presented by the trap density ${N}_{t},$ electron and hole capture cross-section ${\sigma }_{n},{\sigma }_{p}$ and the deep trap ${E}_{t}$ level below the conduction band edge which are taken from [28] and also listed in the table A2.

The surface recombination mainly originates from the poor quality of the vertical interfaces between the MQW region and the n-InP, p-InP area. This recombination can be described by the surface recombination velocity on these two vertical interfaces [29], of which parameter is in table A2, too.

And the intervalence-band absorption (IVBA) loss ${\alpha }_{p}$ is calculated from this formula ${\alpha }_{p}={k}_{p}\cdot p$ in these quantum wells, barriers, SCH and bulk layer regions, where ${k}_{p}$ is the coefficient and p is the local hole density in table A2.

Apart from the above parameters, these default material parameters such as material refractive index, band gap, lattice constant, effective masses and carrier mobility are defined in the Crosslight PIC3D macrofiles for different regions.

By simulation two main results are discovered. The first one is that the MQW local gain and electron concentration rise up with the bias voltage ${v}_{b}$ increasing. This conclusion is shown in figures 4(a)–(d). The 500nm-width-MQW local gains and electron concentrations at bias voltage ${v}_{b}=4v$ are shown in figures 4(a) and (b) separately, in which along the vertical middle white dash line we get the different local gains and electron concentrations at ${v}_{b}=2v,3v,4v$ shown in figures 4(c) and (d).

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The second one is that at the fixed bias voltage the narrower the MQW width ${w}_{g},$ the higher the MQW local gain and electron concentration illustrated in the figures 4(e) and (f). Along the horizontal white dash line in figures 4(a) and (b) the local gains and electron concentrations are shown in figures 4(e) and (f) at different ${w}_{g}$ at ${v}_{b}=4v.$

To describe the MQW electrical properties, we also explore the relationship between the bias voltage ${v}_{b},$ injection current density and local gain as follows. Specifically in this plasmonic TWA the electrode width is $5\,\mu {\rm{m}}.$

In figure 5(a) it shows the relationship curves between MQW injection current density and the bias voltage ${v}_{b}$ at different ${w}_{g}.$ From this figure two conclusions are gotten. The first one is that these injection current densities thicken with bias voltage ${v}_{b}$ increasing when ${v}_{b}$ is larger than the threshold voltage. Second, these MQWs with four different ${w}_{g}$ almost have the equal injection current densities at the same bias voltage ${v}_{b}.$

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In figure 5(b) there are the curves about the total injection current density and the local gain at the center of the 7th QW which is the core of the MQW. Two points are discovered.

First, the local gains lift up with the injection current density rising. Specifically speaking, the local gains rise up dramatically with injection current density increasing as the injection current density is lower than $1.0\,kA\,c{m}^{-2}$ approximately. With the injection current density increasing further, the local gains reach saturation gradually, which is the well-known gain saturation in the optical amplifier [30].

Second is the local gain enlarges with ${w}_{g}$ narrowing even as the injection current density is fixed. This phenomenon comes from the following reasons. When ${w}_{g}$ decreases, not only the internal quantum efficiency increases [31] but also the electron concentration thickens with ${w}_{g}$ dropping shown in figure 4(f). Therefore the local gain rises with ${w}_{g}$ narrowing, which is in agreement with that in figure 4(e).

From the above analysis, we can know that it is feasible to lift MQW local gain by increasing bias voltage ${v}_{b}$ and narrowing MQW width ${w}_{g}.$

We first analyze the plasmonic TWA optical characteristics when the MQW has no applied voltage i.e. MQW has no local gain. And in the next step, we explore the MQW gain threshold in which the guided wave propagation loss is completely compensated. At last, by putting the MQW local gain into plasmonic TWA, we obtained the plasmonic TWA mode gains at different gap sizes.

Proven in section 3, at 1317 nm wavelength the MQW local gain is independent polarization, which is necessary to the plasmonic TWA. Therefore we simulate and analyze the plasmonic TWA optical properties at this wavelength.

In the following simulation the dielectric constants of these media in the plasmonic TWA are gotten as follows. The $I{n}_{1-x-y}G{a}_{x}A{l}_{y}As$ dielectric constants ${\varepsilon }_{InGaAlAs}$ are calculated from equation (9) [17].

$$\begin{eqnarray}&&{\varepsilon }_{InGaAlAs}=\left(1-x-y\right){\varepsilon }_{InAs}+x{\varepsilon }_{GaAs}+y{\varepsilon }_{AlAs}\end{eqnarray} \tag{ 9 }$$

where ${\varepsilon }_{InAs},{\varepsilon }_{GaAs},{\varepsilon }_{AlAs}$ are the InAs, GaAs and AlAs dielectric constants from the data in the [32]. All of these dielectric constants including the metal Ag, gap SiO₂ and InP are calculated and shown in table A3 in the appendix.

Under the condition of MQW local gain being zero, we get the relationship curves between the gap size and plasmonic wave characteristics including the mode effective refractive index ${n}_{eff},$ mode attenuation coefficient and mode width. And the simulation is carried out by the software COMSOL Multiphysics

The definitions of the mode attenuation coefficient and mode width are explained as follows. If after propagating distance z (cm) the mode power is weakened by ${e}^{\alpha z},$ the mode attenuation coefficient is defined as $\alpha \left({\rm{c}}{{\rm{m}}}^{-1}\right)$ [30]. Also it is proposed the mode width describes the mode field lateral distribution which is important in the high integration density PIC. The mode width is defined as ${x}_{0}$ in the x direction, if

$$\begin{eqnarray}&&\displaystyle \frac{\displaystyle {\iint }_{\begin{array}{l}-\displaystyle \frac{{x}_{0}}{2}\lt x\lt \displaystyle \frac{{x}_{0}}{2}\\ -\infty \lt y\lt +\infty \end{array}}\mathop{A}\limits^{ \rightharpoonup }\cdot d\mathop{s}\limits^{ \rightharpoonup }}{\displaystyle {\iint }_{\begin{array}{l}-\infty \lt x\lt +\infty \\ -\infty \lt y\lt +\infty \end{array}}\mathop{A}\limits^{ \rightharpoonup }\cdot d\mathop{s}\limits^{ \rightharpoonup }}=\eta \end{eqnarray} \tag{ 10 }$$

where the x and y represent the horizontal and vertical axis, the plasmonic TWA cross-section is y-axial symmetric. $\mathop{A}\limits^{\longrightarrow}$ is the plasmonic wave Poynting vector and the ratio of the mode power is $\eta$ assumed to be 50% in this paper.

By simulation several valuable results are explicit in figure 6. First, illustrated in figure 6(a) the guided mode has cut-off width of ${w}_{g}.$ If ${w}_{g}$ is less than the cut-off width, the guided mode will not available. In detail, the guided mode effective index ${n}_{eff}$ falls down with ${w}_{g}$ narrowed at fixed ${h}_{g}.$ Until ${n}_{eff}$ falls to 3.20 which is the lateral p-InP, n-InP and substrate InP refractive index, the guided mode will disappear.

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Second the mode propagation loss decreases with the gap height ${h}_{g}$ increasing, which is illustrated by the mode attenuation coefficient in figure 6(b). Therefore the mode propagation attenuation can be cut down by increasing the gap height ${h}_{g}.$

Third, the trade-off between the mode field size and mode propagation loss still exists. Indicated in figures 6(b), (c), as the ${h}_{g}$ is thickened, the mode attenuation decreases and the mode width broadens. Therefore, the reduction of the mode propagation loss is accompanied by the mode field size extending when the ${h}_{g}$ is heightened.

Fourth, the guided mode is a hybrid one from the dielectric waveguide guided wave and plasmonic one. Depicted in figures 6(b), (c), at ${h}_{g}=2\,nm$ both the mode attenuation coefficient and mode width have a minimum with ${w}_{g}$ variation from the following illustrations. In fact, the guide mode is affected by two actions. One comes from the plasmonic wave. The other one originates from the MQW-InP dielectric waveguide. If the plasmonic one is dominant, the mode width narrows and mode attenuation increases with ${w}_{g}$ diminishing. On the other hand, if the MQW dielectric waveguide action is superior, the mode width widens and mode attenuation lowers with ${w}_{g}$ falling down. Therefore the minimums mode width and maximum mode attenuation coefficient appear at the junction point of these two actions. As ${h}_{g}=2\,{\rm{nm}},$ the minimum mode width appears at ${w}_{g}=90\,{\rm{n}}{\rm{m}}$ and the corresponding mode field is made clear in figure 6(f).

In the above simulation, it is assumed that MQW has no local gain. In the next part, we will explore the plasmonic TWA properties as MQW has local gain.

For getting the plasmonic TAW gain characteristics, we explore gain threshold and the mode gain. The gain threshold is the MQW local gain at which the plasmonic guided wave propagation loss is completely compensated in the plasmonic TWA. And the mode gain is the plasmonic TWA mode gain when the MQW is applied bias voltage. Here the applied bias voltage is 4 v.

By simulation we get the gain threshold shown in figure 6(d). Some results can be obtained from figure 6(d). First, the gain threshold rises up with ${h}_{g}$ decreasing, which comes from the mode attenuation increasing with ${h}_{g}$ lowering in figure 6(b). Second, as ${h}_{g}=2\,{\rm{n}}{\rm{m}},$ the gain threshold goes up with ${w}_{g}$ narrowing for two reasons. One is the amplification area, i.e. the MQW area, shrinks with ${w}_{g}$ lowering. The other one is that the overlap between the mode field and MQW decreases as ${w}_{g}$ falling down.

The plasmonic TWA mode gain at 4v-bias-voltage is shown in figure 6(b) inset. This simulation result suggests several conclusions in the following. First, the plasmonic TWA mode gain can be risen up by thickening the ${h}_{g}$ because the mode attenuation decreases with ${h}_{g}$ heightened. Second, it suggests that as $\left[{h}_{g},{w}_{g}\right]=\left[10,240\right]{\rm{n}}{\rm{m}}$ and 4v-bias-voltage, the injection current density is $9.39\,\mathrm{kA}\,{\rm{c}}{{\rm{m}}}^{-2},$ the mode gain is $3.595\,{\rm{c}}{{\rm{m}}}^{-1}$ and the mode width is 161 nm.

As $\left[{h}_{g},{w}_{g}\right]=\left[10,240\right]{\rm{n}}{\rm{m}},$ the MQW-InP dielectric waveguide action is dominant in the plasmonic TWA described in figure 6(f). Therefore it is reasonable that in the MQW area the plasmonic TWA mode field is equal to the one in the MQW-InP dielectric waveguide approximately. And this result is utilized to explain this simulation results' reliability at the beginning of the section 4.

The thermal property is very important for semiconductor device. Therefore we explore the plasmonic TWA thermal characteristics with the same method as the [33].

Conservation of energy requires that the temperature distribution satisfies the equation (11).

$$\begin{eqnarray}&&\rho {C}_{p}\displaystyle \frac{\partial T}{\partial t}+{\rm{\nabla }}\cdot \left(-k{\rm{\nabla }}T\right)=Q\end{eqnarray} \tag{ 11 }$$

where $\rho$ is the density (kg m⁻³); ${C}_{p}$ is the specific heat capacity at constant pressure $\left(J/\left(kg\cdot K\right)\right);$ $T$ is the absolute temperature $\left(K\right);$ $k$ is the thermal conductivity $\left(W/\left(m\cdot K\right)\right);$ Q is the heat sources density (W m⁻³).

For solving the equation (11), it is necessary to find out all the heat sources. In our plasmonic TWA the heat source can be separated into contributions from Joule heat, plasmonic wave decay heat and Auger recombination heat in the following.

The Joule heat is generated when the low frequency electrical current flows in the lossy material including the electrode, n-InP, p-InP and MQW. And the Joule heat power density can be calculated as $\sigma {\left|\mathop{E}\limits^{ \rightharpoonup }\right|}^{2},$ where $\sigma$ is the electrical conductivity and $\mathop{E}\limits^{ \rightharpoonup }$ is the electric field.

The plasmonic wave decay heat comes from the attenuation while plasmonic wave is passing along the metal-semiconductor-interface. Since the SiO₂ and InP are almost lossless in 1310 nm wavelength window, only the heat from the metal Ag is calculated. This heat source density is $\omega \varepsilon ^{\prime\prime} {\left|\mathop{E}\limits^{ \rightharpoonup }\right|}^{2}$ where $\omega$ is the optical frequency; $\varepsilon ^{\prime\prime}$ is the imaginary part of the Ag permittivity. $\mathop{E}\limits^{ \rightharpoonup }$ is the plasmonic wave electric field.

Auger recombination heat is defined as the energy heat emitted from the electron-hole pairs non-radiative recombination. The Auger recombination heat is given by ${E}_{g}^{QW}\left({C}_{p}{p}^{2}n+{C}_{n}{n}^{2}p\right),$ where ${E}_{g}^{QW}$ is the band gap energy of QW, ${C}_{p}$ and ${C}_{n}$ are the Auger recombination coefficients, n and p are the electron and hole densities in QW respectively.

In our thermal simulation, there are some preconditions which are the bias voltage ${v}_{b}=4v,$ the $\left[{h}_{g},{w}_{g}\right]=\left[10,240\right]{\rm{n}}{\rm{m}},$ the $100\,\mu {\rm{m}}$-height-substrate-InP, $100\,\mu {\rm{m}}$-thickness-cladding-air and the $200\,\mu {\rm{m}}$-width-simulation-zone. And the aluminum electrodes temperature is 300k since electrodes can be connected with heat sink. And the initial temperature of the system was 300 K.

The heat flux (or Neumman) condition is applied to the up and down boundaries with the external bulk temperature 300 K, while the Dirichlet condition is applied in the left and right boundaries also with temperature 300 K.

The simulations use literature values for the density, thermal conductivity, heat capacity of Ag, electrode Al, SiO₂, InP, InAs, GaAs, AlAs [34, 35]. And those of the $I{n}_{1-x-y}G{a}_{x}A{l}_{y}As$ are obtained by interpolation. The thermal simulation is also supported by these two software Crosslight PIC3D and COMSOL Multiphysics.

The plasmonic TWA thermal characteristics are deployed in figure 7. The whole thermal temperature of our plasmonic TWA is shown in figure 7(a) at ${v}_{b}=4v$ and $\left[{h}_{g},{w}_{g}\right]=\left[10,240\right]{\rm{n}}{\rm{m}}.$ For more clear illustration, the central enlarged part is described in figure 7(b), which shows that the metal Ag and MQW are the highest temperature core. And we draw the temperature curves along the middle line of the device as guided wave power ${P}_{gw}=100\,nw,1\,\mu w,5\,\mu w$ in figure 7(c). Also from figure 7(c), it is known that the plasmonic TWA core temperature rises up with the guided wave power ${P}_{gw}$ increasing. Since the $I{n}_{1-x-y}G{a}_{x}A{l}_{y}As$ MQW has excellent performance from $20\,^\circ C$ to $80\,^\circ C$ [36] and our plasmonic TWA temperature peak is less than 346 K $\left(73\,^\circ C\right)$ as ${P}_{gw}\leqslant 5\mu w$ shown in figure 7(c), this device can work well when ${P}_{gw}\leqslant 5\,\mu w.$

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