1. Introduction

The THz spectral range (λ = 30-300 μm) is interesting for spectroscopy and imaging with applications in explosive and drug detection, security screening, astronomy, and medical imaging. However, the scope of these applications is limited to a large extent by the availability of THz sources which typically either are very bulky in size or require cryogenic cooling. For THz technology to reach its potential, compact THz lasers operating at room temperature are a necessary component with the performance commensurate of the semiconductor diode lasers. Unfortunately, none of the semiconductors in nature has such a narrow bandgap that gives band-to-band transitions in the THz range, and even if it did, the diode lasers would not have operated at room temperature because of the severe Auger recombination. Quantum cascade lasers (QCLs) relying on intersubband transitions in semiconductor quantum wells (QWs) offer an important alternative where the lasing wavelength can be tuned from mid- to far-IR by engineering subband energy separations. Indeed, QCLs have become the mainstream commercially available sources in mid-IR (2.75µm< λ<12 µm) range. Meanwhile, the prospect of QCLs as THz sources that are compact and capable of operating at room temperature with high output power (milliwatts) has stimulated significant research effort. Since the first demonstration of a THz QCL in 2002 at 50K [1], a decade of research [2–5] has managed to raise the maximum operating temperature to about 200K in 2012 [6]. There is a reason, however, to think that further increase to room temperature is unlikely with the ubiquitous GaAs material system currently employed in the THz QCLs. One detrimental factor is that the relatively small longitudinal optical (LO) phonon energy in GaAs (36 meV) that is comparable to the room temperature thermal energy $k_{B}T\approx 26$ meV allows for a large percentage of the “hot” electrons with high in-plane kinetic energy in the upper laser subband to quickly relax to the lower laser state by emitting LO phonons. Another is the fact that the best performance that has been obtained to date in terms of operating temperature relied upon near-optical-phonon-resonance for depopulation of the lower laser state subband to the ground-state subband, i.e. the energy separation between them is roughly equal to that of a LO phonon, not much more than the thermal energy, further hindering the population inversion. We have previously proposed to use the GaN/AlGaN material system to realize room temperature THz QCLs [7], taking advantage of its large LO-phonon energy (~90meV). The advantages are threefold. First, the large LO-phonon energy can also increase the lifetime of the upper laser state by reducing the relaxation of electrons with higher in-plane kinetic energy via emission of a LO-phonon. Second, the large LO-phonon energy in GaN-based system can reduce the thermal population of the lower laser state. Third, ultrafast LO-phonon scattering in GaN/AlGaN QWs can be used for the rapid depopulation of the lower laser state [8,9].

Unfortunately, the nitride technology has not yet reached the same maturity level as the GaAs or InP in terms of material quality and epitaxial growth. The degree of complexity required for stacking a large number (~100) of periods of active region consisting of hundreds or even more than a thousand of layers that are precisely controlled and repeated in the cascading periods of a typical QCL presents an insurmountable challenge at this point for it to be implemented within the GaN system. For this reason, a QCL structure containing as few periods as possible would be a desirable scenario for its implementation in nitrides. In addition, a relatively low loss waveguide capable of tight confinement of THz modes is needed for such a QCL with a deep subwavelength active region. While previous THz QCLs based on the GaAs system have used either metal or heavily doped semiconductor cladding layers to confine the active regions of a few microns in thickness, such structures are simply too lossy when the active region thickness is reduced by a factor of ten or so and the electromagnetic field penetration into the cladding becomes significant. Here we propose to use the spoof surface plasmon (SSP) waveguide for the GaN/AlGaN QCL.

The concept of spoof plasmons was first introduced [10] to explain the observation of optical transmission through the perforated films [11]. Different from a conventional surface plasmon (SP) waveguide that confines optical modes in a dielectric active layer sandwiched between two metal layers such as gold, silver, or transparent conducting oxide [12,13], the SSP waveguide is constructed with corrugated interfaces between the metal and dielectric [Fig. 1(a)] whose optical behavior depends on the geometry of the subwavelength features of the corrugations. As a result, the strong plasmonic resonances of the SP waveguides accompanied with large propagating loss can be brought down from visible and near IR to the THz range with much lower loss. We shall analyze such a waveguide and show that it can indeed support a GaN/AlGaN QCL with as few as 5 cascading periods in the active region consisting totally of only 15 QWs using an injector-less design.

 

Fig. 1 (a) Illustration of the SSP waveguide confined THz GaN/Al_0.2Ga_0.8N QCL and (b) band structure and envelope wavefunctions of the active region of the proposed THzzTT showing two periods with each period consisting of 3 GaN QWs and 3 Al_0.2Ga_0.8N barriers with layer thicknesses (Å) starting from the tunneling barrier: 50/45/25/25/55/30 (wells in bold and barrier in plain) under an electric bias of 45 kV/cm.

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2. GaN/AlGaN QCL active region design

Nitrides are known to exist in both wurtzite and zinc-blende (ZB) structures [14]. In its wurtzite structure, nitrides possess a large spontaneous polarization and piezoelectric constants which lead to two-dimensional charges build up at nitride heterointerfaces where the polarization discontinuities occur, causing relatively strong built-in electric fields [15]. While in principle such build-in fields present no difficulty in the QCL design, we choose to work with ZB nitride structure absent of polarizations. Figure 1(b) shows the active region of the proposed GaN/Al_0.2Ga_0.8N QCL consisting of multiple cascading periods under an external bias field of F = 45kV/cm. Each period consists of 3 GaN QWs separated by Al_0.2Ga_0.8N barriers and their thicknesses are given in the caption of Fig. 1(b). The conduction band offset between GaN and Al_0.2Ga_0.8N is estimated to 250 meV [16]. The origin of each of the three subbands formed in a period can be traced to one of the QWs, where its associated envelope function is mostly localized. The three subbands form a three-level laser system with subbands 3 and 2 being the upper and lower laser states, respectively. The laser operates as follows: electrons are selectively injected into the upper laser state 3; undergo a lasing transition to lower laser state 2; and then depopulate to ground state subband 1 which is in resonance with level 3′ of next period so electrons can cascade into the next lasing stage through resonant tunneling - thus the lasing cycle repeats itself. The photon energy generated from the lasing transition is ΔE₃₂ = 13.7meV corresponding to 3.3THz radiation (90μm). The energy separation between states 2 and 1 is designed to be ΔE₂₁ = 88.6 meV, just above the LO-phonon energy for ZB GaN (~87.3meV), such that depopulation of lower laser subband 2 can be accelerated through the fast near-resonant optical phonon emission process yielding very short lifetime τ₂ for the lower laser state 2. The large spatial separation between states 3 and 1, on the other hand, ensures longer lifetime τ₃ for the upper laser state 3. The tunneling barrier (50Ǻ) has been optimized to prevent the escape of electrons from the upper laser state directly into the downstream period and to allow for efficient tunnel injection of carriers from state 1 to 3′. The periodical three-QW QCL structure [17] relying on resonant tunneling for electron transport from the ground state 1 to the upper state 3′ instead of employing the far more elaborate injector region consisting of a chirped superlattice to transport electrons to the next period is designed in order to obtain the close packing of the active regions with the benefit of large optical confinement factor, and more importantly, to minimize the number of QWs in the GaN/Al_0.2Ga_0.8N QCL.

In order to establish population inversion between the two laser states, the QCL structural parameters have been optimized to achieve favorable subband lifetimes – short lifetime for lower laser state 2, but long lifetime for upper state 3. We have calculated all the scattering rates between any two subbands following a previously developed procedure [18] and using ZB GaN material parameters [19]. Three key lifetimes that ultimately determine the population distribution between the two laser states are shown in Fig. 2 in the temperature of 100-300K. Because ${\tau }_2$completely dominated by the 2→1 scattering accompanied by the LO-phonon emission which is nearly temperature independent, ${\tau }_2\approx 0.1$ remains almost constant over temperature and is significantly shorter than τ₃, and is much shorter than in similar GaAs-based structures in good agreement with experimental observation [8]. The lifetime ${\tau }_3$, on the other hand, is determined by both 3→1 and 3→2 scattering. Although the 3→1 process allows for LO-phonon emission, the wavefunctions of these two involved subbands 1 and 3 have little overlap, and therefore this scattering is not as dominant in comparison with the 3→2 process which is a combination of LO-phonon absorption and acoustic phonon scattering, both of which are temperature dependent. Clearly the lifetime condition for population inversion, ${\tau }_{32}>{\tau }_2$, is easily satisfied throughout the temperature range.

 

Fig. 2 Temperature dependence of lifetimes of upper (${\tau }_3$) and lower laser states (${\tau }_2$), as well as the scattering between them (${\tau }_{32}$).

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With these lifetimes, we can solve the following set of rate equations in the absence of lasing for the subband populations$N_i\left(i=1,2,3\right)$,

3. Spoof surface plasmon waveguide

As mentioned above, because of the material challenge, it is preferable to have as few periods as possible in a GaN/AlGaN QCL. This presents a challenge for the waveguide design. Dielectric waveguides are of little use because the thickness of any THz QCL is always much less than its lasing wavelength. For this reason, SP waveguides made of metal or heavily-doped semiconductor cladding layers and capable of provide optical confinement for subwavelength structures are used for THz QCLs based on the GaAs system with active regions of at least a few µm thick. But for the thinner active layers (<0.5µm), such waveguides become inadequate because their losses go up rather significantly as more of the optical mode leaks into the metal claddings where the loss is more pronounced. This impairment, however, can be overcome by taking advantage of the SSP structure that was originally introduced and studied as a way to concentrate THz waves [21,22].

Let us now consider two periodically corrugated metal (Au) surfaces separated by a dielectric medium (GaN/Al_0.2Ga_0.8N) of thickness d [Fig. 1(a)]. For as long as the feature period t is substantially subwavelength, we can use an effective medium to describe the anisotropic dielectric function of the corrugated region of height h as a geometrical combination of the dielectric properties of metal ${\epsilon }_m$and the dielectric medium${\epsilon }_d$ [21],

Such a structure supports TM SSP mode propagating along the z-direction with the electric field,

We have plotted the SSP dispersion in the waveguide of a = 1.0µm, t = 2.0µm, h = 0.5µm d = 0.2µm using the normalized SSP wave vector ${\beta }_{SSP}=k_s/n{k}_0=\beta \text{'}+j\beta "$in Fig. 3(a) where the SSP resonance occurs at slightly over 60THz ($\hslash \omega =$ 250meV) which corresponds to the situation where the corrugates play the role of quarter-wavelength antennas (h = λ/4). In comparison with the dispersion of the surface plasmon polariton (SPP) shown in Fig. 3(b) that is propagating at the interface between Au and GaN with its resonance at much higher frequency in UV ($f\approx$ 860THz, $\hslash \omega =$ 3.571eV), the SSP waveguide brings the resonance down to the THz regime. Both dispersions exhibit back-bending of both real $\beta \text{'}$and imaginary part $\beta "$of the wave vector near their resonance because of the imaginary part of the metal dielectric function, but the SSP experiences less loss $2n{k}_0\beta "$because of the much smaller $k_0$at the SSP resonance compared to that of the SPP resonance. Since the lasing wavelength of 90μm corresponds to 3.3THz that is far below the SSP waveguide resonance (~60 THz), the loss is considerably smaller. This is better illustrated in Fig. 4(a) where the losses in both SSP and SP waveguides for the lasing wavelength of ${\lambda }_0\approx$90 µm are shown for a range of waveguide thickness d, the loss in SSP waveguide is consistently lower than that in the SP waveguide without the corrugated interface.

 

Fig. 3 Dispersions of (a) the Au/GaN SSP waveguide of a=1.0µm, t=2.0µm, h=0.5µm d=0.2µm, and (b) the SPP propagating at the Au/GaN interface.

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Fig. 4 (a) Loss comparison between Au/GaN SSP (a=1.0µm, t=2.0µm, h=0.5µm) and SP waveguides for ${\lambda }_0\approx$90 µm vs. the waveguide thickness d, and (b) temperature dependence of the threshold current density for 5 periods of the THz GaN/Al_0.2Ga_0.8N QCL in the SSP waveguide of a=1.0µm, t=2.0µm, h=0.5µm d=0.2µm with its mode profile ${\left|E(x,y)\right|}^2$ shown as inset in (b).

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We subsequently picked d = 0.2µm for the SSP waveguide with a loss of ~70/cm to confine 5 periods of GaN/Al_0.2Ga_0.8N QCL (occupies $d_{QCL}=$ 115nm) that is placed between two AlGaN layers (85nm), one works as injection layer and other as extraction layer for the QCL [Fig. 1(a)]. The optical confinement defined as $\Gamma ={\displaystyle {\int }_{-d/2}^{d/2}{\left|E\right|}^2}dx/{\displaystyle {\int }_{-\infty }^{\infty }{\left|E\right|}^2}dx$ of this waveguide has been calculated $\Gamma \approx 1$. Using a FDTD software from Lumerical, we have simulated the TM mode propagating in the SSP waveguide, and its mode profile ${\left|E\right|}^2$[inset in Fig. 4(b)] confirms that the tightly confined mode hardly spills into the corrugated region. If we consider other losses such as mirrors of cavity and free carrier absorption add to another 20/cm [24] so that the total loss that needs to be compensated for is $\alpha =90$/cm. The threshold current density $J_{th}$ for the QCL operating in the temperature range of 100 to 300K is shown in Fig. 4(b) that yields the model gain ${\Gamma }_{eff}g=\alpha$ where the effective confinement factor ${\Gamma }_{eff}=(d_{QCL}/d)\Gamma \approx 0.575$. The threshold current density has been found to be in the range of 2.5 to 6 kA/cm² with a characteristic temperature of $T_0=125$K. This current density range is slightly higher than what has been reported in the best GaAs/AlGaAs QCL which has operated up to about 200K,⁶ and can be easily accommodated in the GaN material system.

The implementation of such a thin SSP waveguide faces technical challenges. THz QCL wafer bonding processing has proven successful in fabricating a double-metal waveguide for confining a thick GaAs/Al_0.15Ga_0.85As active region where the high output power was sought after [25], such a technique needs to be developed further for the ultrathin active region such as the one proposed in this work, and eventually has to migrate to the GaN/AlGaN system in order for this design to be realized.

4. Summary

THz QCLs based on the nitride material system offer the potential to operate at room temperature because of its large optical phonon energy (~90meV) and ultrafast optical phonon scattering which can be utilized to increase the lifetime of the upper laser state and to reduce the lifetime of lower laser state. These promises, however, is faced with the reality that the nitride is a less mature material system that prohibits the growth of hundreds of layers that are precisely controlled in the conventional QCLs. We propose to use a SSP waveguide with subwavelength corrugated structures at the metal-dielectric interfaces to provide the optical confinement for only 5 periods of a GaN/AlGaN QCL consisting of 15 QWs in total. Our analysis shows that the waveguide loss is sufficiently low and lasing threshold can be reached at the current density of ~6 kA/cm² at room temperature. The result suggests a practical path to achieving room-temperature THz QCLs based on the GaN/AlGaN system and SSP waveguides.