Abstract

We propose and study the feasibility of a THz GaN/AlGaN quantum cascade laser (QCL) consisting of only five periods with confinement provided by a spoof surface plasmon (SSP) waveguide for room temperature operation. The QCL design takes advantages of the large optical phonon energy and the ultrafast phonon scattering in GaN that allow for engineering favorable laser state lifetimes. Our analysis has shown that the waveguide loss is sufficiently low for the QCL to reach its threshold at the injection current density around 6 kA/cm2 at room temperature.

© 2013 Optical Society of America

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 [25] 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 kBT26 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/Al0.2Ga0.8N 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 Al0.2Ga0.8N 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/Al0.2Ga0.8N QCL consisting of multiple cascading periods under an external bias field of F = 45kV/cm. Each period consists of 3 GaN QWs separated by Al0.2Ga0.8N barriers and their thicknesses are given in the caption of Fig. 1(b). The conduction band offset between GaN and Al0.2Ga0.8N 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/Al0.2Ga0.8N 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 τ2completely dominated by the 2→1 scattering accompanied by the LO-phonon emission which is nearly temperature independent, τ20.1 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 τ3, 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, τ32>τ2, is easily satisfied throughout the temperature range.

 

Fig. 2 Temperature dependence of lifetimes of upper (τ3) and lower laser states (τ2), as well as the scattering between them (τ32).

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With these lifetimes, we can solve the following set of rate equations in the absence of lasing for the subband populationsNi(i=1,2,3),

dN3dt=Je+N1τ13+N2τ23N3τ3=JeN3N1eΔ31/kBTτ31N3N2eΔ32/kBTτ32dN2dt=N1τ12N2τ2+N3τ32=N3N2eΔ32/kBTτ32N2N1eΔ21/kBTτ21N=N1+N2+N3.
Here J is the injection current density, Nis the total area doping density per period, and we have related the scattering rate from subband i toj to that ofj to i with τij1=τji1eΔji/kBT where Δji=EjEito show explicitly the effect of thermal excitation [20]. Solving Eq. (1) at steady state for N2and N3, we arrive at the optical gain,
g=2e2|z23|2nεo(Γ23)2πλoN3N2L
where e is the free electron charge, εois the vacuum permittivity, n=εd=2.3 is the index of refraction of GaN, Γ23=3meV is the FWHM for the lasing line, λo = 90 µm is the lasing wavelength, L=205Å is the length of a QCL period, and z23 = 10Å is the dipole matrix element between the two laser states.

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/Al0.2Ga0.8N) 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 εmand the dielectric mediumεd [21],

εx=εy=(ta)εm+aεdt,εz=t(ta)/εm+a/εd.

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

Ε=Aej(kszωt){kscos[kh(|x|hd/2)]εxcos(khh)e^xjx|x|khsin[kh(|x|hd/2)]εzcos(khh)e^z,d2<|x|<h+d2ksεdcosh(kdx)cosh(kdd/2)e^x+jkdεdsinh(kdx)cosh(kdd/2)e^z,|x|<d2
and zero outside of this region (|x|>h+d/2), which is a good approximation considering Re(εm)>>1 in Au [23]. Here the wave vectors are related by ks2=εdk02+kd2 and kdtanh(kdd/2)=(εd/εz)khtan(khh) wherek0=2π/λ0is the free space wave vector and kh is the wave vector that describes the wave propagating along the height of the corrugate, and at THz εd<<|εm| and khnk0εm/(εm+εd)nk0. Since |kdd/2|<<1 is always satisfied for the waveguide with the narrow gap d, the SSP wave vector can be analytically evaluated,

ks2=εdk02+8εdkhtan(khh)4εzεdkhdtan(khh)1d.

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 βSSP=ks/nk0=β'+jβ"in Fig. 3(a) where the SSP resonance occurs at slightly over 60THz (ω= 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 860THz, ω= 3.571eV), the SSP waveguide brings the resonance down to the THz regime. Both dispersions exhibit back-bending of both real β'and imaginary part β"of the wave vector near their resonance because of the imaginary part of the metal dielectric function, but the SSP experiences less loss 2nk0β"because of the much smaller k0at 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 λ090 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 λ090 µm vs. the waveguide thickness d, and (b) temperature dependence of the threshold current density for 5 periods of the THz GaN/Al0.2Ga0.8N QCL in the SSP waveguide of a=1.0µm, t=2.0µm, h=0.5µm d=0.2µm with its mode profile |E(x,y)|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/Al0.2Ga0.8N QCL (occupies dQCL= 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 Γ=d/2d/2|E|2dx/|E|2dx of this waveguide has been calculated Γ1. Using a FDTD software from Lumerical, we have simulated the TM mode propagating in the SSP waveguide, and its mode profile |E|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 α=90/cm. The threshold current density Jth for the QCL operating in the temperature range of 100 to 300K is shown in Fig. 4(b) that yields the model gain Γeffg=α where the effective confinement factor Γeff=(dQCL/d)Γ0.575. The threshold current density has been found to be in the range of 2.5 to 6 kA/cm2 with a characteristic temperature of T0=125K. 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/Al0.15Ga0.85As 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.

Acknowledgments

This work was supported by the Air Force Office of Scientific Research (FA9550-10-1-0417), Mid-InfraRed Technologies for Health and the Environment Research Center (NSF MIRTHE NSF-ERC; EEC0540832), and Taiwan NSC (102-2745-M-002-005-ASP and 100-2923-M-002-007-MY3).

References and links

1. R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature 417(6885), 156–159 (2002). [CrossRef]   [PubMed]  

2. B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Operation of terahertz quantum-cascade lasers at 164 K in pulsed mode and at 117 K in continuous-wave mode,” Opt. Express 13(9), 3331–3339 (2005). [CrossRef]   [PubMed]  

3. M. A. Belkin, J. A. Fan, S. Hormoz, F. Capasso, S. P. Khanna, M. Lachab, A. G. Davies, and E. H. Linfield, “Terahertz quantum cascade lasers with copper metal-metal waveguides operating up to 178 K,” Opt. Express 16(5), 3242–3248 (2008). [CrossRef]   [PubMed]  

4. M. A. Belkin, Q. J. Wang, C. Pflugl, A. Belyanin, S. P. Khanna, A. G. Davies, E. H. Linfield, and F. Capasso, “High-temperature operation of terahertz quantum cascade laser sources,” IEEE Sel. Top. Quantum Electron. 15(3), 952–967 (2009). [CrossRef]  

5. S. Kumar, C. W. I. Chan, Q. Hu, and J. L. Reno, “A 1.8-THz quantum cascade laser operating significantly above the temperature of ω/kB,” Nat. Phys. 7(2), 166–171 (2011). [CrossRef]  

6. S. Fathololoumi, E. Dupont, C. W. Chan, Z. R. Wasilewski, S. R. Laframboise, D. Ban, A. Mátyás, C. Jirauschek, Q. Hu, and H. C. Liu, “Terahertz quantum cascade lasers operating up to ~200 K with optimized oscillator strength and improved injection tunneling,” Opt. Express 20(4), 3866–3876 (2012). [CrossRef]   [PubMed]  

7. G. Sun, R. A. Soref, and J. B. Khurgin, “Active region design of a terahertz GaN/ Al0.15Ga0.85N quantum cascade laser,” Superlattices Microstruct. 37(2), 107–113 (2005). [CrossRef]  

8. N. Iizuka, K. Kaneko, N. Suzuki, T. Asano, S. Noda, and O. Wada, “Ultrafast intersubband relaxation (150 fs) in AlGaN/GaN multiple quantum wells,” Appl. Phys. Lett. 77(5), 648–650 (2000). [CrossRef]  

9. J. D. Heber, C. Gmachl, H. M. Ng, and A. Y. Cho, “Comparative study of ultrafast intersubband electron scattering times at ~1.55 μm wavelength in GaN/ALGaN heterostructures,” Appl. Phys. Lett. 81(7), 1237–1239 (2002). [CrossRef]  

10. J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004). [CrossRef]   [PubMed]  

11. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998). [CrossRef]  

12. W. A. Murray and W. L. Barnes, “Plasmonic materials,” Adv. Mater. 19(22), 3771–3782 (2007). [CrossRef]  

13. C. Rhodes, S. Franzen, J. P. Maria, M. Losego, D. N. Leonard, B. Laughlin, G. Duscher, and S. Weibel, “Surface plasmon resonance in conducting metal oxides,” J. Appl. Phys. 100(5), 054905 (2006). [CrossRef]  

14. T. Lei, M. Fanciulli, R. J. Molnar, T. D. Moustakas, R. J. Graham, and J. Scanlon, “Epitaxial growth of zinc blende and wurtzitic gallium nitride thin films on (001) silicon,” Appl. Phys. Lett. 59(8), 944–946 (1991). [CrossRef]  

15. V. Fiorentini, F. Bernardini, and O. Ambacher, “Evidence for nonlinear macroscopic polarization in III-V nitride alloy heterostructures,” Appl. Phys. Lett. 80(7), 1204–1206 (2002). [CrossRef]  

16. H. Ünlü and A. Asenov, “Band offsets in III-nitride heterostructures,” J. Phys. D Appl. Phys. 35(7), 591–594 (2002). [CrossRef]  

17. H. Luo, S. R. Laframboise, Z. R. Wasilewski, G. C. Aers, H. C. Liu, and J. C. Cao, “Terahertz quantum-cascade lasers based on a three-well active module,” Appl. Phys. Lett. 90(4), 041112 (2007). [CrossRef]  

18. G. Sun and J. B. Khurgin, “Optically pumped four-level infrared laser based on intersubband transitions in multiple quantum wells: feasibility study,” IEEE J. Quantum Electron. 29(4), 1104–1111 (1993). [CrossRef]  

19. W. J. Fan, M. F. Li, T. C. Chong, and J. B. Xia, “Electronic properties of zinc-blende GaN, AlN, and their alloys Ga1-xAlxN,” J. Appl. Phys. 79(1), 188–194 (1996). [CrossRef]  

20. J. B. Khurgin and Y. Dikmelik, “Transport and gain in a quantum cascade laser: model and equivalent circuit,” Opt. Eng. 49(11), 111110 (2010). [CrossRef]  

21. A. Rusina, M. Durach, and M. I. Stockman, “Theory of spoof plasmons in real metals,” J. Appl. Phys. A 100(2), 375–378 (2010). [CrossRef]  

22. S. A. Maier, S. R. Andrews, L. Martín-Moreno, and F. J. García-Vidal, “Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires,” Phys. Rev. Lett. 97(17), 176805 (2006). [CrossRef]   [PubMed]  

23. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]  

24. B. S. Williams, S. Kumar, H. Callebaut, Q. Hu, and J. L. Reno, “Terahertz quantum-cascade laser operating up to 137 K,” Appl. Phys. Lett. 83(25), 5142–5144 (2003). [CrossRef]  

25. M. Brandstetter, C. Deutsch, A. Benz, G. D. Cole, H. Detz, A. M. Andrews, W. Schrenk, G. Strasser, and K. Unterrainer, “THz quantum cascade lasers with wafer bonded active regions,” Opt. Express 20(21), 23832–23837 (2012). [CrossRef]   [PubMed]  

References

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  1. R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
    [CrossRef] [PubMed]
  2. B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Operation of terahertz quantum-cascade lasers at 164 K in pulsed mode and at 117 K in continuous-wave mode,” Opt. Express13(9), 3331–3339 (2005).
    [CrossRef] [PubMed]
  3. M. A. Belkin, J. A. Fan, S. Hormoz, F. Capasso, S. P. Khanna, M. Lachab, A. G. Davies, and E. H. Linfield, “Terahertz quantum cascade lasers with copper metal-metal waveguides operating up to 178 K,” Opt. Express16(5), 3242–3248 (2008).
    [CrossRef] [PubMed]
  4. M. A. Belkin, Q. J. Wang, C. Pflugl, A. Belyanin, S. P. Khanna, A. G. Davies, E. H. Linfield, and F. Capasso, “High-temperature operation of terahertz quantum cascade laser sources,” IEEE Sel. Top. Quantum Electron.15(3), 952–967 (2009).
    [CrossRef]
  5. S. Kumar, C. W. I. Chan, Q. Hu, and J. L. Reno, “A 1.8-THz quantum cascade laser operating significantly above the temperature of ω/kB,” Nat. Phys.7(2), 166–171 (2011).
    [CrossRef]
  6. S. Fathololoumi, E. Dupont, C. W. Chan, Z. R. Wasilewski, S. R. Laframboise, D. Ban, A. Mátyás, C. Jirauschek, Q. Hu, and H. C. Liu, “Terahertz quantum cascade lasers operating up to ~200 K with optimized oscillator strength and improved injection tunneling,” Opt. Express20(4), 3866–3876 (2012).
    [CrossRef] [PubMed]
  7. G. Sun, R. A. Soref, and J. B. Khurgin, “Active region design of a terahertz GaN/ Al0.15Ga0.85N quantum cascade laser,” Superlattices Microstruct.37(2), 107–113 (2005).
    [CrossRef]
  8. N. Iizuka, K. Kaneko, N. Suzuki, T. Asano, S. Noda, and O. Wada, “Ultrafast intersubband relaxation (150 fs) in AlGaN/GaN multiple quantum wells,” Appl. Phys. Lett.77(5), 648–650 (2000).
    [CrossRef]
  9. J. D. Heber, C. Gmachl, H. M. Ng, and A. Y. Cho, “Comparative study of ultrafast intersubband electron scattering times at ~1.55 μm wavelength in GaN/ALGaN heterostructures,” Appl. Phys. Lett.81(7), 1237–1239 (2002).
    [CrossRef]
  10. J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science305(5685), 847–848 (2004).
    [CrossRef] [PubMed]
  11. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998).
    [CrossRef]
  12. W. A. Murray and W. L. Barnes, “Plasmonic materials,” Adv. Mater.19(22), 3771–3782 (2007).
    [CrossRef]
  13. C. Rhodes, S. Franzen, J. P. Maria, M. Losego, D. N. Leonard, B. Laughlin, G. Duscher, and S. Weibel, “Surface plasmon resonance in conducting metal oxides,” J. Appl. Phys.100(5), 054905 (2006).
    [CrossRef]
  14. T. Lei, M. Fanciulli, R. J. Molnar, T. D. Moustakas, R. J. Graham, and J. Scanlon, “Epitaxial growth of zinc blende and wurtzitic gallium nitride thin films on (001) silicon,” Appl. Phys. Lett.59(8), 944–946 (1991).
    [CrossRef]
  15. V. Fiorentini, F. Bernardini, and O. Ambacher, “Evidence for nonlinear macroscopic polarization in III-V nitride alloy heterostructures,” Appl. Phys. Lett.80(7), 1204–1206 (2002).
    [CrossRef]
  16. H. Ünlü and A. Asenov, “Band offsets in III-nitride heterostructures,” J. Phys. D Appl. Phys.35(7), 591–594 (2002).
    [CrossRef]
  17. H. Luo, S. R. Laframboise, Z. R. Wasilewski, G. C. Aers, H. C. Liu, and J. C. Cao, “Terahertz quantum-cascade lasers based on a three-well active module,” Appl. Phys. Lett.90(4), 041112 (2007).
    [CrossRef]
  18. G. Sun and J. B. Khurgin, “Optically pumped four-level infrared laser based on intersubband transitions in multiple quantum wells: feasibility study,” IEEE J. Quantum Electron.29(4), 1104–1111 (1993).
    [CrossRef]
  19. W. J. Fan, M. F. Li, T. C. Chong, and J. B. Xia, “Electronic properties of zinc-blende GaN, AlN, and their alloys Ga1-xAlxN,” J. Appl. Phys.79(1), 188–194 (1996).
    [CrossRef]
  20. J. B. Khurgin and Y. Dikmelik, “Transport and gain in a quantum cascade laser: model and equivalent circuit,” Opt. Eng.49(11), 111110 (2010).
    [CrossRef]
  21. A. Rusina, M. Durach, and M. I. Stockman, “Theory of spoof plasmons in real metals,” J. Appl. Phys. A100(2), 375–378 (2010).
    [CrossRef]
  22. S. A. Maier, S. R. Andrews, L. Martín-Moreno, and F. J. García-Vidal, “Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires,” Phys. Rev. Lett.97(17), 176805 (2006).
    [CrossRef] [PubMed]
  23. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
    [CrossRef]
  24. B. S. Williams, S. Kumar, H. Callebaut, Q. Hu, and J. L. Reno, “Terahertz quantum-cascade laser operating up to 137 K,” Appl. Phys. Lett.83(25), 5142–5144 (2003).
    [CrossRef]
  25. M. Brandstetter, C. Deutsch, A. Benz, G. D. Cole, H. Detz, A. M. Andrews, W. Schrenk, G. Strasser, and K. Unterrainer, “THz quantum cascade lasers with wafer bonded active regions,” Opt. Express20(21), 23832–23837 (2012).
    [CrossRef] [PubMed]

2012

2011

S. Kumar, C. W. I. Chan, Q. Hu, and J. L. Reno, “A 1.8-THz quantum cascade laser operating significantly above the temperature of ω/kB,” Nat. Phys.7(2), 166–171 (2011).
[CrossRef]

2010

J. B. Khurgin and Y. Dikmelik, “Transport and gain in a quantum cascade laser: model and equivalent circuit,” Opt. Eng.49(11), 111110 (2010).
[CrossRef]

A. Rusina, M. Durach, and M. I. Stockman, “Theory of spoof plasmons in real metals,” J. Appl. Phys. A100(2), 375–378 (2010).
[CrossRef]

2009

M. A. Belkin, Q. J. Wang, C. Pflugl, A. Belyanin, S. P. Khanna, A. G. Davies, E. H. Linfield, and F. Capasso, “High-temperature operation of terahertz quantum cascade laser sources,” IEEE Sel. Top. Quantum Electron.15(3), 952–967 (2009).
[CrossRef]

2008

2007

H. Luo, S. R. Laframboise, Z. R. Wasilewski, G. C. Aers, H. C. Liu, and J. C. Cao, “Terahertz quantum-cascade lasers based on a three-well active module,” Appl. Phys. Lett.90(4), 041112 (2007).
[CrossRef]

W. A. Murray and W. L. Barnes, “Plasmonic materials,” Adv. Mater.19(22), 3771–3782 (2007).
[CrossRef]

2006

C. Rhodes, S. Franzen, J. P. Maria, M. Losego, D. N. Leonard, B. Laughlin, G. Duscher, and S. Weibel, “Surface plasmon resonance in conducting metal oxides,” J. Appl. Phys.100(5), 054905 (2006).
[CrossRef]

S. A. Maier, S. R. Andrews, L. Martín-Moreno, and F. J. García-Vidal, “Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires,” Phys. Rev. Lett.97(17), 176805 (2006).
[CrossRef] [PubMed]

2005

G. Sun, R. A. Soref, and J. B. Khurgin, “Active region design of a terahertz GaN/ Al0.15Ga0.85N quantum cascade laser,” Superlattices Microstruct.37(2), 107–113 (2005).
[CrossRef]

B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Operation of terahertz quantum-cascade lasers at 164 K in pulsed mode and at 117 K in continuous-wave mode,” Opt. Express13(9), 3331–3339 (2005).
[CrossRef] [PubMed]

2004

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science305(5685), 847–848 (2004).
[CrossRef] [PubMed]

2003

B. S. Williams, S. Kumar, H. Callebaut, Q. Hu, and J. L. Reno, “Terahertz quantum-cascade laser operating up to 137 K,” Appl. Phys. Lett.83(25), 5142–5144 (2003).
[CrossRef]

2002

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
[CrossRef] [PubMed]

V. Fiorentini, F. Bernardini, and O. Ambacher, “Evidence for nonlinear macroscopic polarization in III-V nitride alloy heterostructures,” Appl. Phys. Lett.80(7), 1204–1206 (2002).
[CrossRef]

H. Ünlü and A. Asenov, “Band offsets in III-nitride heterostructures,” J. Phys. D Appl. Phys.35(7), 591–594 (2002).
[CrossRef]

J. D. Heber, C. Gmachl, H. M. Ng, and A. Y. Cho, “Comparative study of ultrafast intersubband electron scattering times at ~1.55 μm wavelength in GaN/ALGaN heterostructures,” Appl. Phys. Lett.81(7), 1237–1239 (2002).
[CrossRef]

2000

N. Iizuka, K. Kaneko, N. Suzuki, T. Asano, S. Noda, and O. Wada, “Ultrafast intersubband relaxation (150 fs) in AlGaN/GaN multiple quantum wells,” Appl. Phys. Lett.77(5), 648–650 (2000).
[CrossRef]

1998

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998).
[CrossRef]

1996

W. J. Fan, M. F. Li, T. C. Chong, and J. B. Xia, “Electronic properties of zinc-blende GaN, AlN, and their alloys Ga1-xAlxN,” J. Appl. Phys.79(1), 188–194 (1996).
[CrossRef]

1993

G. Sun and J. B. Khurgin, “Optically pumped four-level infrared laser based on intersubband transitions in multiple quantum wells: feasibility study,” IEEE J. Quantum Electron.29(4), 1104–1111 (1993).
[CrossRef]

1991

T. Lei, M. Fanciulli, R. J. Molnar, T. D. Moustakas, R. J. Graham, and J. Scanlon, “Epitaxial growth of zinc blende and wurtzitic gallium nitride thin films on (001) silicon,” Appl. Phys. Lett.59(8), 944–946 (1991).
[CrossRef]

1972

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

Aers, G. C.

H. Luo, S. R. Laframboise, Z. R. Wasilewski, G. C. Aers, H. C. Liu, and J. C. Cao, “Terahertz quantum-cascade lasers based on a three-well active module,” Appl. Phys. Lett.90(4), 041112 (2007).
[CrossRef]

Ambacher, O.

V. Fiorentini, F. Bernardini, and O. Ambacher, “Evidence for nonlinear macroscopic polarization in III-V nitride alloy heterostructures,” Appl. Phys. Lett.80(7), 1204–1206 (2002).
[CrossRef]

Andrews, A. M.

Andrews, S. R.

S. A. Maier, S. R. Andrews, L. Martín-Moreno, and F. J. García-Vidal, “Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires,” Phys. Rev. Lett.97(17), 176805 (2006).
[CrossRef] [PubMed]

Asano, T.

N. Iizuka, K. Kaneko, N. Suzuki, T. Asano, S. Noda, and O. Wada, “Ultrafast intersubband relaxation (150 fs) in AlGaN/GaN multiple quantum wells,” Appl. Phys. Lett.77(5), 648–650 (2000).
[CrossRef]

Asenov, A.

H. Ünlü and A. Asenov, “Band offsets in III-nitride heterostructures,” J. Phys. D Appl. Phys.35(7), 591–594 (2002).
[CrossRef]

Ban, D.

Barnes, W. L.

W. A. Murray and W. L. Barnes, “Plasmonic materials,” Adv. Mater.19(22), 3771–3782 (2007).
[CrossRef]

Beere, H. E.

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
[CrossRef] [PubMed]

Belkin, M. A.

M. A. Belkin, Q. J. Wang, C. Pflugl, A. Belyanin, S. P. Khanna, A. G. Davies, E. H. Linfield, and F. Capasso, “High-temperature operation of terahertz quantum cascade laser sources,” IEEE Sel. Top. Quantum Electron.15(3), 952–967 (2009).
[CrossRef]

M. A. Belkin, J. A. Fan, S. Hormoz, F. Capasso, S. P. Khanna, M. Lachab, A. G. Davies, and E. H. Linfield, “Terahertz quantum cascade lasers with copper metal-metal waveguides operating up to 178 K,” Opt. Express16(5), 3242–3248 (2008).
[CrossRef] [PubMed]

Beltram, F.

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
[CrossRef] [PubMed]

Belyanin, A.

M. A. Belkin, Q. J. Wang, C. Pflugl, A. Belyanin, S. P. Khanna, A. G. Davies, E. H. Linfield, and F. Capasso, “High-temperature operation of terahertz quantum cascade laser sources,” IEEE Sel. Top. Quantum Electron.15(3), 952–967 (2009).
[CrossRef]

Benz, A.

Bernardini, F.

V. Fiorentini, F. Bernardini, and O. Ambacher, “Evidence for nonlinear macroscopic polarization in III-V nitride alloy heterostructures,” Appl. Phys. Lett.80(7), 1204–1206 (2002).
[CrossRef]

Brandstetter, M.

Callebaut, H.

B. S. Williams, S. Kumar, H. Callebaut, Q. Hu, and J. L. Reno, “Terahertz quantum-cascade laser operating up to 137 K,” Appl. Phys. Lett.83(25), 5142–5144 (2003).
[CrossRef]

Cao, J. C.

H. Luo, S. R. Laframboise, Z. R. Wasilewski, G. C. Aers, H. C. Liu, and J. C. Cao, “Terahertz quantum-cascade lasers based on a three-well active module,” Appl. Phys. Lett.90(4), 041112 (2007).
[CrossRef]

Capasso, F.

M. A. Belkin, Q. J. Wang, C. Pflugl, A. Belyanin, S. P. Khanna, A. G. Davies, E. H. Linfield, and F. Capasso, “High-temperature operation of terahertz quantum cascade laser sources,” IEEE Sel. Top. Quantum Electron.15(3), 952–967 (2009).
[CrossRef]

M. A. Belkin, J. A. Fan, S. Hormoz, F. Capasso, S. P. Khanna, M. Lachab, A. G. Davies, and E. H. Linfield, “Terahertz quantum cascade lasers with copper metal-metal waveguides operating up to 178 K,” Opt. Express16(5), 3242–3248 (2008).
[CrossRef] [PubMed]

Chan, C. W.

Chan, C. W. I.

S. Kumar, C. W. I. Chan, Q. Hu, and J. L. Reno, “A 1.8-THz quantum cascade laser operating significantly above the temperature of ω/kB,” Nat. Phys.7(2), 166–171 (2011).
[CrossRef]

Cho, A. Y.

J. D. Heber, C. Gmachl, H. M. Ng, and A. Y. Cho, “Comparative study of ultrafast intersubband electron scattering times at ~1.55 μm wavelength in GaN/ALGaN heterostructures,” Appl. Phys. Lett.81(7), 1237–1239 (2002).
[CrossRef]

Chong, T. C.

W. J. Fan, M. F. Li, T. C. Chong, and J. B. Xia, “Electronic properties of zinc-blende GaN, AlN, and their alloys Ga1-xAlxN,” J. Appl. Phys.79(1), 188–194 (1996).
[CrossRef]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

Cole, G. D.

Davies, A. G.

M. A. Belkin, Q. J. Wang, C. Pflugl, A. Belyanin, S. P. Khanna, A. G. Davies, E. H. Linfield, and F. Capasso, “High-temperature operation of terahertz quantum cascade laser sources,” IEEE Sel. Top. Quantum Electron.15(3), 952–967 (2009).
[CrossRef]

M. A. Belkin, J. A. Fan, S. Hormoz, F. Capasso, S. P. Khanna, M. Lachab, A. G. Davies, and E. H. Linfield, “Terahertz quantum cascade lasers with copper metal-metal waveguides operating up to 178 K,” Opt. Express16(5), 3242–3248 (2008).
[CrossRef] [PubMed]

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
[CrossRef] [PubMed]

Detz, H.

Deutsch, C.

Dikmelik, Y.

J. B. Khurgin and Y. Dikmelik, “Transport and gain in a quantum cascade laser: model and equivalent circuit,” Opt. Eng.49(11), 111110 (2010).
[CrossRef]

Dupont, E.

Durach, M.

A. Rusina, M. Durach, and M. I. Stockman, “Theory of spoof plasmons in real metals,” J. Appl. Phys. A100(2), 375–378 (2010).
[CrossRef]

Duscher, G.

C. Rhodes, S. Franzen, J. P. Maria, M. Losego, D. N. Leonard, B. Laughlin, G. Duscher, and S. Weibel, “Surface plasmon resonance in conducting metal oxides,” J. Appl. Phys.100(5), 054905 (2006).
[CrossRef]

Ebbesen, T. W.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998).
[CrossRef]

Fan, J. A.

Fan, W. J.

W. J. Fan, M. F. Li, T. C. Chong, and J. B. Xia, “Electronic properties of zinc-blende GaN, AlN, and their alloys Ga1-xAlxN,” J. Appl. Phys.79(1), 188–194 (1996).
[CrossRef]

Fanciulli, M.

T. Lei, M. Fanciulli, R. J. Molnar, T. D. Moustakas, R. J. Graham, and J. Scanlon, “Epitaxial growth of zinc blende and wurtzitic gallium nitride thin films on (001) silicon,” Appl. Phys. Lett.59(8), 944–946 (1991).
[CrossRef]

Fathololoumi, S.

Fiorentini, V.

V. Fiorentini, F. Bernardini, and O. Ambacher, “Evidence for nonlinear macroscopic polarization in III-V nitride alloy heterostructures,” Appl. Phys. Lett.80(7), 1204–1206 (2002).
[CrossRef]

Franzen, S.

C. Rhodes, S. Franzen, J. P. Maria, M. Losego, D. N. Leonard, B. Laughlin, G. Duscher, and S. Weibel, “Surface plasmon resonance in conducting metal oxides,” J. Appl. Phys.100(5), 054905 (2006).
[CrossRef]

Garcia-Vidal, F. J.

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science305(5685), 847–848 (2004).
[CrossRef] [PubMed]

García-Vidal, F. J.

S. A. Maier, S. R. Andrews, L. Martín-Moreno, and F. J. García-Vidal, “Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires,” Phys. Rev. Lett.97(17), 176805 (2006).
[CrossRef] [PubMed]

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998).
[CrossRef]

Gmachl, C.

J. D. Heber, C. Gmachl, H. M. Ng, and A. Y. Cho, “Comparative study of ultrafast intersubband electron scattering times at ~1.55 μm wavelength in GaN/ALGaN heterostructures,” Appl. Phys. Lett.81(7), 1237–1239 (2002).
[CrossRef]

Graham, R. J.

T. Lei, M. Fanciulli, R. J. Molnar, T. D. Moustakas, R. J. Graham, and J. Scanlon, “Epitaxial growth of zinc blende and wurtzitic gallium nitride thin films on (001) silicon,” Appl. Phys. Lett.59(8), 944–946 (1991).
[CrossRef]

Heber, J. D.

J. D. Heber, C. Gmachl, H. M. Ng, and A. Y. Cho, “Comparative study of ultrafast intersubband electron scattering times at ~1.55 μm wavelength in GaN/ALGaN heterostructures,” Appl. Phys. Lett.81(7), 1237–1239 (2002).
[CrossRef]

Hormoz, S.

Hu, Q.

Iizuka, N.

N. Iizuka, K. Kaneko, N. Suzuki, T. Asano, S. Noda, and O. Wada, “Ultrafast intersubband relaxation (150 fs) in AlGaN/GaN multiple quantum wells,” Appl. Phys. Lett.77(5), 648–650 (2000).
[CrossRef]

Iotti, R. C.

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
[CrossRef] [PubMed]

Jirauschek, C.

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

Kaneko, K.

N. Iizuka, K. Kaneko, N. Suzuki, T. Asano, S. Noda, and O. Wada, “Ultrafast intersubband relaxation (150 fs) in AlGaN/GaN multiple quantum wells,” Appl. Phys. Lett.77(5), 648–650 (2000).
[CrossRef]

Khanna, S. P.

M. A. Belkin, Q. J. Wang, C. Pflugl, A. Belyanin, S. P. Khanna, A. G. Davies, E. H. Linfield, and F. Capasso, “High-temperature operation of terahertz quantum cascade laser sources,” IEEE Sel. Top. Quantum Electron.15(3), 952–967 (2009).
[CrossRef]

M. A. Belkin, J. A. Fan, S. Hormoz, F. Capasso, S. P. Khanna, M. Lachab, A. G. Davies, and E. H. Linfield, “Terahertz quantum cascade lasers with copper metal-metal waveguides operating up to 178 K,” Opt. Express16(5), 3242–3248 (2008).
[CrossRef] [PubMed]

Khurgin, J. B.

J. B. Khurgin and Y. Dikmelik, “Transport and gain in a quantum cascade laser: model and equivalent circuit,” Opt. Eng.49(11), 111110 (2010).
[CrossRef]

G. Sun, R. A. Soref, and J. B. Khurgin, “Active region design of a terahertz GaN/ Al0.15Ga0.85N quantum cascade laser,” Superlattices Microstruct.37(2), 107–113 (2005).
[CrossRef]

G. Sun and J. B. Khurgin, “Optically pumped four-level infrared laser based on intersubband transitions in multiple quantum wells: feasibility study,” IEEE J. Quantum Electron.29(4), 1104–1111 (1993).
[CrossRef]

Köhler, R.

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
[CrossRef] [PubMed]

Kumar, S.

S. Kumar, C. W. I. Chan, Q. Hu, and J. L. Reno, “A 1.8-THz quantum cascade laser operating significantly above the temperature of ω/kB,” Nat. Phys.7(2), 166–171 (2011).
[CrossRef]

B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Operation of terahertz quantum-cascade lasers at 164 K in pulsed mode and at 117 K in continuous-wave mode,” Opt. Express13(9), 3331–3339 (2005).
[CrossRef] [PubMed]

B. S. Williams, S. Kumar, H. Callebaut, Q. Hu, and J. L. Reno, “Terahertz quantum-cascade laser operating up to 137 K,” Appl. Phys. Lett.83(25), 5142–5144 (2003).
[CrossRef]

Lachab, M.

Laframboise, S. R.

Laughlin, B.

C. Rhodes, S. Franzen, J. P. Maria, M. Losego, D. N. Leonard, B. Laughlin, G. Duscher, and S. Weibel, “Surface plasmon resonance in conducting metal oxides,” J. Appl. Phys.100(5), 054905 (2006).
[CrossRef]

Lei, T.

T. Lei, M. Fanciulli, R. J. Molnar, T. D. Moustakas, R. J. Graham, and J. Scanlon, “Epitaxial growth of zinc blende and wurtzitic gallium nitride thin films on (001) silicon,” Appl. Phys. Lett.59(8), 944–946 (1991).
[CrossRef]

Leonard, D. N.

C. Rhodes, S. Franzen, J. P. Maria, M. Losego, D. N. Leonard, B. Laughlin, G. Duscher, and S. Weibel, “Surface plasmon resonance in conducting metal oxides,” J. Appl. Phys.100(5), 054905 (2006).
[CrossRef]

Lezec, H. J.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998).
[CrossRef]

Li, M. F.

W. J. Fan, M. F. Li, T. C. Chong, and J. B. Xia, “Electronic properties of zinc-blende GaN, AlN, and their alloys Ga1-xAlxN,” J. Appl. Phys.79(1), 188–194 (1996).
[CrossRef]

Linfield, E. H.

M. A. Belkin, Q. J. Wang, C. Pflugl, A. Belyanin, S. P. Khanna, A. G. Davies, E. H. Linfield, and F. Capasso, “High-temperature operation of terahertz quantum cascade laser sources,” IEEE Sel. Top. Quantum Electron.15(3), 952–967 (2009).
[CrossRef]

M. A. Belkin, J. A. Fan, S. Hormoz, F. Capasso, S. P. Khanna, M. Lachab, A. G. Davies, and E. H. Linfield, “Terahertz quantum cascade lasers with copper metal-metal waveguides operating up to 178 K,” Opt. Express16(5), 3242–3248 (2008).
[CrossRef] [PubMed]

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
[CrossRef] [PubMed]

Liu, H. C.

Losego, M.

C. Rhodes, S. Franzen, J. P. Maria, M. Losego, D. N. Leonard, B. Laughlin, G. Duscher, and S. Weibel, “Surface plasmon resonance in conducting metal oxides,” J. Appl. Phys.100(5), 054905 (2006).
[CrossRef]

Luo, H.

H. Luo, S. R. Laframboise, Z. R. Wasilewski, G. C. Aers, H. C. Liu, and J. C. Cao, “Terahertz quantum-cascade lasers based on a three-well active module,” Appl. Phys. Lett.90(4), 041112 (2007).
[CrossRef]

Maier, S. A.

S. A. Maier, S. R. Andrews, L. Martín-Moreno, and F. J. García-Vidal, “Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires,” Phys. Rev. Lett.97(17), 176805 (2006).
[CrossRef] [PubMed]

Maria, J. P.

C. Rhodes, S. Franzen, J. P. Maria, M. Losego, D. N. Leonard, B. Laughlin, G. Duscher, and S. Weibel, “Surface plasmon resonance in conducting metal oxides,” J. Appl. Phys.100(5), 054905 (2006).
[CrossRef]

Martín-Moreno, L.

S. A. Maier, S. R. Andrews, L. Martín-Moreno, and F. J. García-Vidal, “Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires,” Phys. Rev. Lett.97(17), 176805 (2006).
[CrossRef] [PubMed]

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science305(5685), 847–848 (2004).
[CrossRef] [PubMed]

Mátyás, A.

Molnar, R. J.

T. Lei, M. Fanciulli, R. J. Molnar, T. D. Moustakas, R. J. Graham, and J. Scanlon, “Epitaxial growth of zinc blende and wurtzitic gallium nitride thin films on (001) silicon,” Appl. Phys. Lett.59(8), 944–946 (1991).
[CrossRef]

Moustakas, T. D.

T. Lei, M. Fanciulli, R. J. Molnar, T. D. Moustakas, R. J. Graham, and J. Scanlon, “Epitaxial growth of zinc blende and wurtzitic gallium nitride thin films on (001) silicon,” Appl. Phys. Lett.59(8), 944–946 (1991).
[CrossRef]

Murray, W. A.

W. A. Murray and W. L. Barnes, “Plasmonic materials,” Adv. Mater.19(22), 3771–3782 (2007).
[CrossRef]

Ng, H. M.

J. D. Heber, C. Gmachl, H. M. Ng, and A. Y. Cho, “Comparative study of ultrafast intersubband electron scattering times at ~1.55 μm wavelength in GaN/ALGaN heterostructures,” Appl. Phys. Lett.81(7), 1237–1239 (2002).
[CrossRef]

Noda, S.

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M. A. Belkin, Q. J. Wang, C. Pflugl, A. Belyanin, S. P. Khanna, A. G. Davies, E. H. Linfield, and F. Capasso, “High-temperature operation of terahertz quantum cascade laser sources,” IEEE Sel. Top. Quantum Electron.15(3), 952–967 (2009).
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S. Kumar, C. W. I. Chan, Q. Hu, and J. L. Reno, “A 1.8-THz quantum cascade laser operating significantly above the temperature of ω/kB,” Nat. Phys.7(2), 166–171 (2011).
[CrossRef]

B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Operation of terahertz quantum-cascade lasers at 164 K in pulsed mode and at 117 K in continuous-wave mode,” Opt. Express13(9), 3331–3339 (2005).
[CrossRef] [PubMed]

B. S. Williams, S. Kumar, H. Callebaut, Q. Hu, and J. L. Reno, “Terahertz quantum-cascade laser operating up to 137 K,” Appl. Phys. Lett.83(25), 5142–5144 (2003).
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C. Rhodes, S. Franzen, J. P. Maria, M. Losego, D. N. Leonard, B. Laughlin, G. Duscher, and S. Weibel, “Surface plasmon resonance in conducting metal oxides,” J. Appl. Phys.100(5), 054905 (2006).
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R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
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Soref, R. A.

G. Sun, R. A. Soref, and J. B. Khurgin, “Active region design of a terahertz GaN/ Al0.15Ga0.85N quantum cascade laser,” Superlattices Microstruct.37(2), 107–113 (2005).
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A. Rusina, M. Durach, and M. I. Stockman, “Theory of spoof plasmons in real metals,” J. Appl. Phys. A100(2), 375–378 (2010).
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Sun, G.

G. Sun, R. A. Soref, and J. B. Khurgin, “Active region design of a terahertz GaN/ Al0.15Ga0.85N quantum cascade laser,” Superlattices Microstruct.37(2), 107–113 (2005).
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N. Iizuka, K. Kaneko, N. Suzuki, T. Asano, S. Noda, and O. Wada, “Ultrafast intersubband relaxation (150 fs) in AlGaN/GaN multiple quantum wells,” Appl. Phys. Lett.77(5), 648–650 (2000).
[CrossRef]

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T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998).
[CrossRef]

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R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
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[CrossRef]

Wang, Q. J.

M. A. Belkin, Q. J. Wang, C. Pflugl, A. Belyanin, S. P. Khanna, A. G. Davies, E. H. Linfield, and F. Capasso, “High-temperature operation of terahertz quantum cascade laser sources,” IEEE Sel. Top. Quantum Electron.15(3), 952–967 (2009).
[CrossRef]

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Weibel, S.

C. Rhodes, S. Franzen, J. P. Maria, M. Losego, D. N. Leonard, B. Laughlin, G. Duscher, and S. Weibel, “Surface plasmon resonance in conducting metal oxides,” J. Appl. Phys.100(5), 054905 (2006).
[CrossRef]

Williams, B. S.

B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Operation of terahertz quantum-cascade lasers at 164 K in pulsed mode and at 117 K in continuous-wave mode,” Opt. Express13(9), 3331–3339 (2005).
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B. S. Williams, S. Kumar, H. Callebaut, Q. Hu, and J. L. Reno, “Terahertz quantum-cascade laser operating up to 137 K,” Appl. Phys. Lett.83(25), 5142–5144 (2003).
[CrossRef]

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T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998).
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V. Fiorentini, F. Bernardini, and O. Ambacher, “Evidence for nonlinear macroscopic polarization in III-V nitride alloy heterostructures,” Appl. Phys. Lett.80(7), 1204–1206 (2002).
[CrossRef]

N. Iizuka, K. Kaneko, N. Suzuki, T. Asano, S. Noda, and O. Wada, “Ultrafast intersubband relaxation (150 fs) in AlGaN/GaN multiple quantum wells,” Appl. Phys. Lett.77(5), 648–650 (2000).
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IEEE J. Quantum Electron.

G. Sun and J. B. Khurgin, “Optically pumped four-level infrared laser based on intersubband transitions in multiple quantum wells: feasibility study,” IEEE J. Quantum Electron.29(4), 1104–1111 (1993).
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IEEE Sel. Top. Quantum Electron.

M. A. Belkin, Q. J. Wang, C. Pflugl, A. Belyanin, S. P. Khanna, A. G. Davies, E. H. Linfield, and F. Capasso, “High-temperature operation of terahertz quantum cascade laser sources,” IEEE Sel. Top. Quantum Electron.15(3), 952–967 (2009).
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J. Appl. Phys.

W. J. Fan, M. F. Li, T. C. Chong, and J. B. Xia, “Electronic properties of zinc-blende GaN, AlN, and their alloys Ga1-xAlxN,” J. Appl. Phys.79(1), 188–194 (1996).
[CrossRef]

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J. Appl. Phys. A

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[CrossRef]

J. Phys. D Appl. Phys.

H. Ünlü and A. Asenov, “Band offsets in III-nitride heterostructures,” J. Phys. D Appl. Phys.35(7), 591–594 (2002).
[CrossRef]

Nat. Phys.

S. Kumar, C. W. I. Chan, Q. Hu, and J. L. Reno, “A 1.8-THz quantum cascade laser operating significantly above the temperature of ω/kB,” Nat. Phys.7(2), 166–171 (2011).
[CrossRef]

Nature

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998).
[CrossRef]

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
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J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science305(5685), 847–848 (2004).
[CrossRef] [PubMed]

Superlattices Microstruct.

G. Sun, R. A. Soref, and J. B. Khurgin, “Active region design of a terahertz GaN/ Al0.15Ga0.85N quantum cascade laser,” Superlattices Microstruct.37(2), 107–113 (2005).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

(a) Illustration of the SSP waveguide confined THz GaN/Al0.2Ga0.8N 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 Al0.2Ga0.8N 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.

Fig. 2
Fig. 2

Temperature dependence of lifetimes of upper ( τ 3 ) and lower laser states ( τ 2 ), as well as the scattering between them ( τ 32 ).

Fig. 3
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.

Fig. 4
Fig. 4

(a) Loss comparison between Au/GaN SSP (a=1.0µm, t=2.0µm, h=0.5µm) and SP waveguides for λ 0 90 µm vs. the waveguide thickness d, and (b) temperature dependence of the threshold current density for 5 periods of the THz GaN/Al0.2Ga0.8N QCL in the SSP waveguide of a=1.0µm, t=2.0µm, h=0.5µm d=0.2µm with its mode profile | E(x,y) | 2 shown as inset in (b).

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

d N 3 dt = J e + N 1 τ 13 + N 2 τ 23 N 3 τ 3 = J e N 3 N 1 e Δ 31 / k B T τ 31 N 3 N 2 e Δ 32 / k B T τ 32 d N 2 dt = N 1 τ 12 N 2 τ 2 + N 3 τ 32 = N 3 N 2 e Δ 32 / k B T τ 32 N 2 N 1 e Δ 21 / k B T τ 21 N= N 1 + N 2 + N 3 .
g= 2 e 2 | z 23 | 2 n ε o ( Γ 23 ) 2π λ o N 3 N 2 L
ε x = ε y = (ta) ε m +a ε d t , ε z = t (ta)/ ε m +a/ ε d .
Ε=A e j( k s zωt) { k s cos[ k h ( | x |hd/2 ) ] ε x cos( k h h ) e ^ x j x | x | k h sin[ k h ( | x |hd/2 ) ] ε z cos( k h h ) e ^ z , d 2 <| x |<h+ d 2 k s ε d cosh( k d x) cosh( k d d/2) e ^ x +j k d ε d sinh( k d x) cosh( k d d/2) e ^ z , |x|< d 2
k s 2 = ε d k 0 2 + 8 ε d k h tan( k h h ) 4 ε z ε d k h dtan( k h h ) 1 d .

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