Tunable Hybrid Plasmonic Semiconductor Laser Based on Loss Perturbation

We propose a tunable plasmonic semiconductor laser that exploits loss perturbation as a tuning mechanism. A metal oxide semiconductor (MOS) capacitive structure is added on top of an edge-emitting Fabry-Perot (FP) diode laser, such that a hybrid plasmonic TM mode that overlaps partly with the MOS capacitor and the semiconductor gain region is supported as the lasing mode. We also propose the use of a layer of conductive oxide, e.g., indium tin oxide (ITO), as the semiconductor of the MOS structure, because the epsilon near zero (ENZ) condition can be attained therein under accumulation, thereby producing a very large change in the effective index of the hybrid plasmonic TM mode. The change in the imaginary part of the effective index is used to tune the lasing wavelength - exploiting loss perturbation to achieve laser tuning is paradigm-shifting. The laser proposed operates at telecom wavelengths, requiring an electrical forward bias to pump the active layer, and a gate voltage to drive the MOS tuning capacitor. Simulations yield a tuning range of over 7 nm in the O-band for a $100 \mu \text{m}$ long FP laser cavity.

Abstract-We propose a tunable plasmonic semiconductor laser that exploits loss perturbation as a tuning mechanism.A metal oxide semiconductor (MOS) capacitive structure is added on top of an edge-emitting Fabry-Perot (FP) diode laser, such that a hybrid plasmonic TM mode that overlaps partly with the MOS capacitor and the semiconductor gain region is supported as the lasing mode.We also propose the use of a layer of conductive oxide, e.g., indium tin oxide (ITO), as the semiconductor of the MOS structure, because the epsilon near zero (ENZ) condition can be attained therein under accumulation, thereby producing a very large change in the effective index of the hybrid plasmonic TM mode.The change in the imaginary part of the effective index is used to tune the lasing wavelength -exploiting loss perturbation to achieve laser tuning is paradigm-shifting.The laser proposed operates at telecom wavelengths, requiring an electrical forward bias to pump the active layer, and a gate voltage to drive the MOS tuning capacitor.Simulations yield a tuning range of over 7 nm in the O-band for a 100 µm long FP laser cavity.
Index Terms-semiconductor laser, plasmonic laser, hybrid laser, epsilon-near-zero material, ITO, MOS capacitor I. INTRODUCTION N recent years, there have been many studies on materials with a vanishing real part of permittivity at specific wavelengths [1][2][3].These materials are commonly known as epsilon-near-zero (ENZ) materials.The initial research on these materials suggested a significant increase in the electric field in the material when the real part of permittivity is zero [4,5].Examples of such materials, include degenerate semiconductors like tin-doped indium oxide (ITO) and aluminum-doped zinc oxide (AZO), in which the high level of doping causes them to show a dielectric behavior at visible wavelengths and a metallic behavior in the infrared.The two examples mentioned have their zero-permittivity region in the near-infrared range, suggesting applications in telecom systems.
By matching the ENZ wavelength with the operating wavelength, one can exploit the characteristics of such materials.ENZ materials have generated significant interest due to their substantial optical nonlinearity, including wave mixing, harmonic generation, and frequency conversion [6,7].Applications of these materials can be expanded to electro-optic components since ITO is compatible with electronics [8][9][10].
Shayan Saeidi is with the School of Electrical Engineering and Computer Science and the Centre for Research in Photonics, University of Ottawa, Ottawa, ON K1N 6N5, Canada.(e-mail: ssaeidi@uottawa.ca).
Pierre Berini is with the School of Electrical Engineering and Computer Science, the Department of Physics, and the Centre for Research in Photonics, Various options exist to achieve optical tuning in lasers, such as modifying the laser cavity length, light propagation direction, or changing the effective index of the optical mode [11].In principle, the latter can be produced by ITO as it exhibits a significant change in refractive index in the ENZ region.To achieve tunability via ITO, one should induce a change in the free carrier concentration of ITO, which overlaps with the propagating laser mode.
In this paper, we aim to exploit the properties of ITO by incorporating it in InP-based lasers to achieve broad tunability following a novel approach based on perturbing mode loss.The mechanism is simple yet unique and is explained as follow: First, the design of the laser's active region (gain medium) plays a vital role in our approach as the peak of the gain spectrum can blueshift or redshift (depending on the design) as a function of carrier density.In a Fabry-Perot (FP) laser, the overlap of the gain spectrum with the cavity modes determines the wavelength of the lasing modes.As the mode loss increases, a higher current (carrier density) must be injected into the active region to compensate for the loss.Therefore, by injecting different current levels, the lasing frequency changes with the frequency of the gain peak.So, a wide tuning range can be achieved if the gain medium compensates for the loss.Second, perturbation of mode loss can be achieved by perturbing the free carrier concentration in the ITO layer, which operates as the semiconductor layer in a plasmonic metal-oxide-semiconductor (MOS) capacitor that is placed in proximity to the gain region, in the optical path.The carrier refraction effect in the ITO layer within a MOS capacitor has been exploited previously to enable a beam steering device [12].We show that a short FP cavity (~ 100 μm) can be tuned over 7 nm (~ 1.2 THz) in the O-band.Such broad tunability from a conventional edge-emitting FP laser in III-V platform has never been reported.However, since tuning relies on loss perturbation, it is clear that this mechanism comes at the expense of high power consumption.
This paper is organized as follow.In Section II, the proposed device is described.Semiconductor modeling is discussed in Section III-A, followed by the waveguide design (Section III-B), then the design of the active region including thermal effects University of Ottawa, Ottawa, ON K1N 6N5, Canada.(e-mail: pberini@uottawa.ca).

Tunable hybrid plasmonic semiconductor laser based on loss perturbation
Shayan Saeidi, Pierre Berini, Fellow IEEE I (Section III-C).We present laser characteristics in Section III-D, followed by conclusions in Section IV.

II. PROPOSED STRUCTURE
Fig. 1 shows the proposed structure (layers not drawn to scale).The structure contains a core (active) layer bounded by p-type and n-type InP claddings, forming a p-i-n diode.The active layer consists of multiple quantum wells (MQWs) which will be discussed in detail in Section III-C.A MOS capacitor is implemented on top by using gold as the metal, hafnia as the oxide, and ITO as the semiconductor.We can exploit the carrier refraction effect in the ITO layer by operating the MOS capacitor in its accumulation regime, as will be discussed in Section III-A.The n-type ITO and the n-type InP layers provide a common ground for the MOS capacitor and the p-i-n diode.Electrical injection to pump the active layer occurs by applying a voltage Vp to the p-InP cladding layer relative to ground.A heavily doped (2×10 19 cm -3 ) p-InGaAs layer is used to form an ohmic contact with the metal.The cavity is of length L which can be tailored to determine the spacing between cavity modes, as will be discussed in Section III-D.The MOS structure is biased by the voltage Vg applied to the top contact relative to ground, as shown in Fig. 1.
To exploit the properties of ITO and of the gain medium, the propagating (lasing) mode should overlap strongly with both regions.This requires forming a hybrid plasmonic mode [13] with the MOS metal contact to strongly localize fields to this region.However, there is a trade-off between the overlap of the hybrid plasmonic mode with the metal and ITO layers to exploit ENZ, and with the active layer to compensate the attenuation.
The laser structure is described in Table I (starting from the top layer).The refractive indices are given at 1300 nm.The refractive indices of gold, hafnia, and InP are extracted from [14][15][16].The refractive index of ITO is calculated from the electrostatic simulations discussed in Section III-A using the Drude model.The refractive index of InP has negligible change with doping level of 10 17 ~10 18 [17].The positive sign of the imaginary part of the refractive index indicates loss (or gain when negative).The thicknesses are optimized to satisfy the loss/gain trade-off (discussed further in Section III-B).The refractive indices of the AlGaInAs layers are calculated using the methodology described in [18].To calculate the refractive index of heavily doped InGaAs, we need to take into account carrier absorption effects.For p-type InGaAsP, the dominant absorption mechanisms involving holes is intervalence band transitions.Using the Kramers-Kronig relation gives the index change ∆ associated with this absorption mechanisms [19]: where h is Planck's constant, c is the speed of light in vacuum, e is the electron charge, E is the photon energy in eV, p is the hole density in cm -3 , α0 = 4.252×10 -20 m 2 , b = 3.657 eV -1 , and  is Euler's constant.

III. SIMULATIONS
The simulation results presented throughout the paper are carried out by Lumerical Photonic Software [20].

A. Electrostatic modeling
The permittivity of ITO as a function of frequency () can be represented at optical wavelengths by the Drude model, where  ∞ is the high-frequency permittivity and ζ is the electron damping term.The plasma frequency   varies with carrier density N according to   = √ 2 /( 0   * ) , where e is the electron charge,  0 is the vacuum permittivity, and   * is the effective mass of electrons for conductivity.For ITO we set  ∞ = 4.2345 , ζ = 1.75 × 10 14 rad/s, and   * = 0.35  with   being the free electron mass [12].The carrier density can be tuned electrically over a thin region of ITO within a MOS capacitor, where ITO is used as the semiconductor.In this subsection, we carry out electrostatic modeling to investigate our MOS structure.Fig. 2(a) shows the 1-D cross-section of the MOS structure, where we apply a gate voltage Vg to the top contact while grounding the bottom contact.We used gold as the metal and HfO2 as the insulator.To carry out the electrostatic simulations, we assumed for ITO: DC permittivity   = 9.3 , electron mobility   = 25 cm 2 V -1 s -1 , bandgap energy   = 2.8 eV, electron affinity  = 4.8 eV, hole mobility  ℎ = 1 cm 2 V -1 s -1 , and hole effective mass  ℎ * =   (the hole mobility and effective mass in degenerately doped ITO are not well-known as they do not contribute significantly to transport processes) [21].We consider the bulk charge carrier (electron) density of ITO to be 3 × 10 20 cm -3 , such that ITO is an n-type degenerate semiconductor.We assume that the variation in doping level of ITO does not change the band-structure significantly, so the electron effective mass remains constant in our calculations.The work function of gold is taken as 5.1 eV [14].We assume a breakdown field of   = 6.4 MV/cm [22] for HfO2 which leads to a breakdown voltage of   = 3.8 V for a HfO2 thickness of 6 nm (Vg < Vbk).We use the drift-diffusion model to compute the charge distribution in ITO [23].However, the accuracy of the charge carrier density distribution in the accumulation regime could be improved by adopting the Schrödinger-Poisson model [24].
Fig. 2(b) shows that the carrier density in ITO increases near the HfO2 interface as Vg increases.When no voltage is applied, the structure is slightly depleted.The flatband voltage is ~0.32 V and as Vg is increased further, the capacitor enters the accumulation regime.The carrier density that produces ENZ in the accumulation region is shown by the dashed black line.Therefore, ENZ can be reached by applying a gate voltage of ~2.9 V.In Figs.2(c) and 2(d), we show the real and imaginary parts of the relative permittivity, εRe and εIm, calculated at 1300 nm as a function of depth into ITO, using the Drude model given above, and the charge distribution plotted in Fig. 2(b).We see that by increasing Vg, εIm increases, which signifies the metallic behavior of ITO.Figs.2(e) and 2(f) show the real and imaginary parts of the refractive index,  and , as a function of depth into ITO.For unperturbed ITO we have   = 1.718 and   = 0.045.Carrier accumulation is localized in ITO to within 1 ~ 2 nm of the interface with HfO2.We use a mesh resolution of 0.1 nm at the ITO/Hafnia interface in the forthcoming optical computations (discussed in Section III-B).

B. Waveguide modeling
It is generally known that plasmonic waveguides are subject to a trade-off between mode confinement and propagation losses due to strong absorption in metal [25].A configuration supporting hybridized plasmonic-dielectric modes [13,26,27] is appropriate here, given the overall device structure.The hybrid mode used is a transverse-magnetic (TM) mode, arising from the coupling between a dielectric mode propagating in the active region sandwiched between dielectric claddings and a surface plasmon polariton (SPP) mode on the metal surface.This mode arises naturally as we place the MOS capacitor (discussed in Section III-A) on top of the III-V laser cavity, such that the hybrid mode overlaps with the ITO layer of the MOS capacitor.The fraction of mode power overlapping with the active region and the metal can be tuned by changing the thickness of the layers separating them.Tuning these thicknesses directly changes the net modal power gain (  ) which is described by: The mode power attenuation (  ) is calculated as: where k is the vacuum wavenumber and   is the imaginary part of the effective index of the mode.Γ is the fraction of mode power overlapping with the gain region (i.e., the QWs), and   is the power gain coefficient provided by the active material (discussed in Section III-C).The modal gain of the TM polarized hybrid mode should be higher than that of the transverse-electric (TE) polarized mode for lasing to occur in the TM mode (which also overlaps with the ITO layer).
In a conventional semiconductor laser, the TE mode is the dominant lasing mode due to lower loss compared to the TM mode and the prevalence of TE gain in QWs.We address the latter point in the next subsection, and the former by utilizing a narrow waveguide (e.g., a nanowire waveguide [28]), such that the TE mode loses confinement and Γ becomes small, while the hybrid TM mode retains confinement and a large Γ -we find that a ridge width of 1 µm satisfies these requirements.Fig. 3(a) shows the electric field profile of the fundamental hybrid TM mode of the waveguide for the case where ITO is unperturbed.The overlap of the mode with the active layer is around 45%.
The effective index of the hybrid TM mode changes significantly as the MOS capacitor is driven into accumulation and the ITO is perturbed by the increased carrier density.Changes in the real part of the effective index (  ) typically contribute to the tuning, while an increase in the imaginary part of the effective index (  ) corresponds to increasing optical losses -conventionally, the former is exploited for tuning, whereas the latter is undesired.Fig. 3(b) shows the real and imaginary parts of the effective index as we vary the applied gate voltage.We note that the real part of the effective index changes by less than 0.1%, yielding only about 0.4 nm of tuning range (estimated from the Bragg equation).But the imaginary part of the effective index, and consequently the mode loss, changes by a factor of 2.5.Our proposed laser concept exploits this latter effect as the tuning mechanism.

C. Active region modeling
The active region used in our model is an InP-based laser diode designed to provide TM gain, as required for lasing in the hybrid TM mode of interest.Previous studies show that tensilestrained AlGaInAs quantum wells provide higher TM gain than TE gain [29,30].Therefore, we establish our active layer based on tensile-strained AlGaInAs MQWs.
We want the transition wavelength to correspond to a wavelength in O-band.However, the transition wavelength is a function of the well and barrier thickness and composition.The choice of these parameters is non-trivial.To find the right material composition, we start by considering the bandgap wavelength of AlxGayIn1-x-yAs for several cases of tensile and compressive strain for barriers and wells, respectively.Fig. 4 shows the bandgap wavelength as a function of x, with tensile strain (dotted lines) varying from 0.7% to 1.1% and compressive strain (solid lines) varying from 0.3% to 0.7%.It is desirable for the tensile strain in the wells to be as high as possible to achieve a high TM gain.Consequently, the required compressive strain in the barriers must increase to compensate at least partially the tension in the wells.On the other hand, our simulations show that the gain for the TE mode increases with compression in wells/barriers, which is undesirable.For the target transition wavelength to be in O-band, we need a range of compositions where the bandgap wavelength is greater than 1.3 µm for the wells and less than 1.2 µm for the barriers, corresponding to high tension wells and comparatively low compressed barriers.So, we select one of many possible solutions: Al0.04Ga0.58In0.38Aswhich has a bandgap wavelength of 1.3 µm (1.1% tensile strain) for the wells, and Al0.42In0.58Aswhich has a bandgap wavelength of 0.96 µm (0.3% compressive strain) for the barriers.Since the selected tensile strain in the wells is more than three times the compressive strain, we choose the barriers to be 3× thicker than the wells to partially compensate the strain mismatch.
The transition wavelength is also dependent on the carrier concentration in quantum well structures.The peak of the gain spectrum can blueshift or redshift as a function of the QW carrier density.The frequency of the gain peak as a function of   carrier density is determined by two effects: band filling and bandgap shrinkage.The former causes a blueshift in the gain peak with increasing carrier density because injected carriers fill the lowest energy states of the bands first, such that transitions between the conduction and valence bands occur at increasingly higher photon energies.The latter results from many-body effects.At high concentrations, the carriers interact via Coulombic repulsion causing a decrease in electron energy in the conduction band and an increase in hole energy in the valence band.Therefore, the gain peak will experience a redshift.Because of these two effects, there is a net increase or decrease in the wavelength of the gain peak as the carrier density changes.Band filling scales linearly with carrier density whereas bandgap shrinkage scales with the cube root of the carrier density [19], so at low carrier densities (less than ~5×10 18 cm -3 ) the former dominates.The carrier density eventually saturates to its value at the lasing threshold, so the band filling effect also saturates, such that bandgap shrinkage becomes dominant.Both effects are considered in our simulations.The bandgap (Δ) shrinkage at high carrier densities is modelled as Δ = − 1/3 where  is a fitting parameter.To our knowledge, there is no data available for the fitting parameter in the case of AlGaInAs in the literature and very little for InGaAsP (10 -8 [31] and 3.2×10 −8 [32] eV.cm).Thus, we set  = 10 -8 eV.cm in our simulations.Another important effect is self-heating.It is known that temperature elevation in MQWs affects key device parameters, including optical gain.In our simulations, we include heating in the proposed p-i-n diode and the effects of a temperature rise in the QWs on the material gain.We use a finite-element mesh in two dimensions to evaluate the thermal response to Joule and recombination heating under electrical drive.We implement a coupled electro-thermal simulation to estimate the thermal response to increasing bias voltage and injected current.In our setup, we assume that the ridge is surrounded by SU-8 polymer which passivates the ridge in practice.The electrical simulation parameters are discussed in Section III-D.The thermal parameters of the materials used in the simulation listed in Table II.These parameters are taken from [14,[33][34][35] with interpolation of binary constituents.
We set Neumann boundary conditions on all the boundaries except the bottom edge of the substrate.The bottom boundary is fixed to a temperature of 300 K, acting as a heat sink.The steady-state simulation results are presented in Fig. 5. Fig. 5 (a) shows the highest temperature in the diode vs. input power.The slope gives the thermal resistance of 107 °C/W, defined as ∆/  [36], where ∆ is the temperature change and   the dissipated power, taken as the input DC power calculated from the V-I characteristics of Section III-D.Fig. 5(b) plots the temperature distribution over the diode cross-section at an input power of 250 mW.The maximum temperature rise of 29 K occurs in the QWs mainly due to the recombination heat.
After many numerical calculations of the TM material gain, involving different quantum well thicknesses, we picked a well thickness of 6 nm to maximize the peak gain shift vs. carrier density.To take heating into account, we assumed a thermal bandgap reduction of 0.334 meV/K [37] for all barriers and wells.Fig. 6 shows the TM material gain of the active region vs. frequency for various average carrier densities in the MQW region, assuming a null carrier density in the barriers.The blue curves show the material gain in an isothermal situation, where Fig. 6.TM material gain spectrum for various carrier densities as labelled (×10 18 cm -3 ).Red curves take self-heating into account while blue curves do not.The dashed line follows the gain peak.heating is not considered.The red curves factor in heating effects as the carrier density increases.The dashed lines follow the gain peak for both cases.In general, as the QW carrier density increases, the blue shift due to band filling dominates, but becomes less significant as the temperature of the QWs increases.We henceforth include heating effects.
The thickness and refractive indices of the QWs and barriers designed are presented in Table I.The active region consists of eleven 6-nm-thick Al0.04Ga0.58In0.38AsQWs, each of tensile strain of 1.1%.To partially compensate the strain, the 18-nmthick Al0.42In0.58Asbarriers each produce a compressive strain of 0.3%.In our modeling, a 0.1 ps intraband relaxation time was assumed, causing spectral broadening of the gain.
For the AlGaInAs material system, the band discontinuity ratio between conduction band edge (EC) and the valence band edge (EV) of Δ/Δ = 0.7/0.3 is typically assumed [37][38][39].At the heterojunction of wells and barriers we assume in our simulations that Δ = 0.218 and Δ = 0.122, which is extracted from work functions of binary constituents of AlGaInAs.The MQW energy band diagram is shown in Fig. 7.The strong barrier in the MQW region in the conduction band will help prevent electron leakage at high temperatures.
Majority of quaternary parameters Qijkl used in the simulations, such as band deformation potentials, elastic stiffness coefficients, and Luttinger parameters are estimated via the interpolation of binary constituents using: where the corresponding binary alloy parameters Qij are extracted from [40] and [41].

D. Laser characteristics
In an FP laser, the overlap of the peak of the gain spectrum with the cavity modes determines the wavelength of a lasing mode.As the mode loss (i.e., imaginary part of effective index) increases, a higher current (carrier density) must be injected into the active region to compensate for the loss.Therefore, by injecting different current levels, the lasing frequency changes with the frequency of the gain peak.
As we saw earlier, the frequency of material gain peak varies with carrier density.Therefore, the cavity mode that overlaps best with the peak gain frequency will lase, leading to tuning by mode hopping.By properly choosing the cavity length, the lasing wavelengths will occur at desired wavelengths.In addition, the length of the laser should be kept relatively short to ensure a large spacing between modes, thereby ensuring a high side-mode suppression ratio (SMSR).We picked a cavity length of L = 100 µm to satisfy these conditions.Vg can be increased as high as 3.8 V, limited by the breakdown voltage of HfO2, to perturb the ITO's comple permittivity.
The lasing characteristics were modeled by Lumerical INTERCONNECT, which uses a traveling wave laser model to capture the time evolution of the average carrier density as the propagating light interacts with the gain material.
The dominant nonradiative recombination current is the Auger effect.In this process, the excess energy released from an electron-hole recombination is given to a third carrier.The Auger recombination rate increases at higher carrier concentrations because it is determined by energy exchange between charge carriers.We expect that the recombination rate will be proportional to n 2 p for electron-electron-hole processes and to np 2 for electron-hole-hole process, where n and p are electron and hole concentrations, respectively.So, the Auger recombination rate can be defined as RAuger = CN 3 where C is the Auger coefficient dependent on the material and N is the carrier density.After reviewing previous reports [37,38] and comparing them to interpolated data calculated via (4), we chose the Auger coefficient for Al0.04Ga0.58In0.38As to be 10 -28 cm 6 s -1 at room temperature.
Trap-assisted Shockley-Read-Hall (SRH) nonradiative recombination is less important but still considered.A SRH lifetime of 20 ns is assumed for electrons and holes in the QWs.
The radiative recombination rate is dominated by spontaneous recombination rate, which is obtained from the integral of the spontaneous emission spectrum.The surface recombination can become important due to the narrow width of the ridge.We assume a surface recombination velocity of 10 5 cm/s for InP in our simulations [42].
The mobility of carriers in wells and barriers were derived from binary materials using (4) because experimental data does not exist in the literature for our AlGaInAs compositions.
The simulated light-current-voltage (LIV) characteristics are shown in Fig. 8.The doping levels (presented in Table I) were optimized so that large currents can be injected under low applied voltage.The VI curve (dashed line) reveals a forward  bias turn-on voltage of VP ~ 1 V.The LI curves are shown for different gate voltages.The case Vg = 3.7 V corresponds to the situation where the hybrid TM mode has the highest loss.Therefore, a larger threshold current is needed in this case (~ 85 mA) for the onset of lasing.As Vg decreases to 2.2 V, the threshold current decreases to ~ 60 mA.For Vg < 2.2 V, the threshold current remains approximately the same as the imaginary effective index of the hybrid TM mode remains constant (cf.Fig. 3 (b)).We assumed facet reflectivies of 28% at both ends of the FP cavity, commensurate with air/InP interfaces.Our simulations show that the threshold current is highly dependent on the reflectivity of the facets, and can be reduced significantly by forming a high reflectance mirror on at least one end.For example, using an 80% reflectivity at one of the facets reduces the threshold current by ~15 mA.
Fig. 9 gives the distribution of the electron concentration through the MQW stack near threshold for the same Vgs in Fig. 8.The average QW electron concentration increases from ~3×10 18 cm -3 to 4.9×10 18 cm -3 with Vg, which is consistent with average carrier densities assumed in Fig. 6.This increase causes an increase in Auger recombination as well as in spontaneous and SRH recombination, contributing to an increase in threshold current with gate voltage as shown in Fig. 8.
The simulated lasing spectra for a driving current at 100 mA are shown in Fig. 10 for the same cases of gate voltage used in Fig. 8 and 9.We assumed a cavity length of L = μm in the simulations to produce a large spacing between cavity modes, leading to fewer modes overlapping with the gain spectrum and a higher SMSR.As observed from Fig. 10, the worst SMSR is 10.5 dB, corresponding to a gate voltage of Vg = 3.7 V. Simulations show that the SMSR increases as high as 20 dB as the mode loss decreases.The spacing between cavity modes is determined by [43]: Δ =  (2  ) ⁄ where  is the speed of light in vacuum, ng is the group index, and L is the physical length of the cavity.The group index is calculated in the vicinity of the lasing frequency as [43]:   =   +   / ω is the angular frequency) and taken to be locally weakly frequency dependent.The theoretical prediction of Δ is about 430 GHz.
As can be seen in Fig. 10, discrete lasing wavelength tuning of up to 1.25 THz (~ 7 nm) can be achieved.The tuning range is limited by the breakdown voltage of hafniaa greater Vg if possible, would result in larger loss perturbation.

Fig. 2 .
Fig. 2. (a) 1D cross-section of the MOS capacitor of interest.(b) Carrier density across the ITO layer for various Vg.Dashed black line shows the carrier density required for ITO to reach ENZ.(c), (d) Real (εRe) and imaginary (εIm) parts of relative permittivity across the ITO layer for various Vg.(e), (f) Real (n) and imaginary (κ) parts of refractive index across the ITO layer for various Vg.

Fig. 4 .
Fig. 4. Bandgap wavelength of AlxGayIn1-x-yAs vs. x for different strains (indicated in percentage).Solid curves denote compressive strain and dotted curves are tensile strain.

Fig. 3 .
Fig. 3. (a) Electric field amplitude of the hybrid mode at the ridge of the modelled waveguide.(b) Effective index of the hybrid mode with varying Vg.

Fig. 5 .
Fig. 5. (a) Highest temperature in the device versus the power consumption.(b) 2D temperature profile of the device at input power of ~ 250 mW.

Fig. 7 .
Fig. 7. Energy band diagram of MQW region along with InP claddings near threshold.Solid lines show conduction (Ec) and valence (Ev) band edges, and dashed lines show electron (Efn) and hole (Efp) Fermi levels.

Fig. 10 .
Fig. 10.Simulated lasing spectrum with a driving current of 100 mA for different cases of gate voltages Vg (shown in the insets).

TABLE II THERMAL
PARAMETERS OF THE MATERIALS IN OUR STRUCTURE