aa r X i v : . [ c ond - m a t . o t h e r] J u l Extraction-Controlled Quantum Cascade Lasers
Andreas Wacker ∗ Mathematical Physics, Lund University, Box 118, 221 00 Lund, Sweden (Dated: July 29, 2010)A simple two-well design for terahertz quantum cascade lasers is proposed which is based onscattering injection and the efficient extraction of electrons from the lower laser level by resonanttunneling. In contrast to existing designs this extraction also controls the positive differential con-ductivity. The device is analyzed by calculations based on nonequilibrium Green functions, whichpredict lasing operation well above 200 K at a frequency of 2.8 THz.
Since their first realization[1], quantum cascade lasers(QCLs) have turned into versatile devices. While the firstlasers operated in the infrared region above the opticalphonon frequency (around 9 THz in the most commonIII/V semiconductor materials used), THz-QCLs, oper-ating below this frequency, could be achieved later. [2].However the operation of these THz-QCLs has only beenestablished for temperatures up to 186 K yet[3]. Theachievement of higher operation temperatures is of highsignificance for many technical applications, as one couldapply simpler cooling techniques.Based on a single concept QCLs operate over a rangeof almost two orders of magnitude in frequency from 1.2THz (250 µ m) [4] to 114 THz (2.63 µ m) [5] except fora gap around the optical phonon frequency. The basicideas, some of them essentially going back to Kazari-nov and Suris[6] are: (i) The use of electronic subbands in semiconductor heterostructures as upper (subscript u )and lower (subscript l ) laser level allowing for the widevariation of the transition energy by properly chosen het-erostructures. (ii) Electric pumping by a bias along thegrowth direction of the heterostructure. Here the flow ofcurrent through the structure feeds electrons into the up-per laser level coming from the injector level (subscript i ).While the further propagation from the upper laser levelat energy E u is essentially blocked by a gap in the en-ergy spectrum of the heterostructure, efficient pathwaysare provided for the emptying of the lower level into an extraction level (subscript e ). (iii) While a single inter-subband transition is typically not sufficient to overcomethe losses in the waveguide, a cascade of identical struc-tures (called period) is realized, all contributing to thegain of the optical mode in the waveguide.As an example the Wannier states for a recent THzQCL with high-temperature operation [3] have been dis-played in Fig. 1(a). Here the current flows via resonanttunneling from the injector into the upper laser level andis extracted from the lower laser level by a further reso-nant tunneling processes, while this extraction level isemptied by optical phonon scattering. This resonantphonon extraction design [7], has been proven to be veryeffective for THz-QCLs as it selectively empties the lower ∗ [email protected] -20 0 20 40 60z (nm)-50050100 E ne r g y ( m e V ) ilue one periodinjector upperlowerextraction(a) 0 20 40z (nm)0100200 i=eule=i one periodinjectorupper lower extraction(b) FIG. 1: (Color online) (a) absolute square of the Wan-nier states of the QCL from Ref. [3] together with the het-erostructure potential. (b) Same data for the device proposedhere. The layer sequence is 17 . / . / . / nm with effectivemasses of 0.067 in the wells and 0.0919 in the barriers (boldsymbols). The conduction band offset is 0.27 eV, which re-lates to a GaAs/Al . Ga . As structure. The underlined bar-rier is doped with 3 × / cm . (Note that Wannier statesare not the commonly plotted eigenstates of the Hamiltonianbut are better localized within the period [8].) laser level while the upper laser level is less affected dueto the resonance condition.An important issue for the operation of quantum cas-cade lasers is the achievement of positive differential con-ductivity (PDC) for each period. Otherwise, for negativedifferential conductivity (NDC), the electric field distri-bution becomes inhomogeneous over the structure [9] andprevents from achieving the designed resonances in eachperiod of the cascade. This phenomenon of domain for-mation is well known from the study of superlattices (see[10] and references cited therein) where it constitutes themain obstacle to observe the gain properties. Tunnelresonances exhibit commonly a current peak when thelevels align. This provides PDC/NDC if the left level islocated below/above the right level, respectively [11, 12].In particular this relates to the gain transition in THzstructures, where the energy separation is only slightlylarger than the broadening and thus constitutes a sourceof NDC. This has to be compensated by PDC contribu-tions in different parts of the current flow through eachperiod which is typically achieved by a second tunnel-ing transition just below resonance. In most cases this isthe transition between the injector and the upper laserlevel, see also Fig. 1(a). Albeit establishing record tem-perature operation such a tunneling injection design hastwo shortcomings. (i) As the tunneling resonance shouldbe an effective source of PDC the injector level must ex-hibit an occupation at least comparable to the upper laserlevel. This restricts the possible inversion for a given to-tal carrier density. (ii) Tunneling from the injector to thelower laser level constitutes a second resonance which is afurther source of NDC and thus of particular concern forlow lasing frequencies, when it mixes with the tunnelingresonance into the upper laser level. A possible solutionof this problem is the development of scattering injection designs [13, 14]. Such a structure was recently shown toexhibit improved temperature performance in the THzregion[15]. In these structures a tunneling resonance isincluded in the current flow before the electrons reachthe injector state in order to guarantee PDC.Here a new structure based on scattering injection isproposed where this tunneling resonance is skipped, seeFig. 1(b). Instead, the tunneling resonance from thelower laser level to the extraction level controls the cur-rent. The idea is that for biases below the designed opera-tion point the carriers are essentially located in the lowerlaser level. At the design bias the extraction level re-moves these carriers effectively. Simultaneously, this levelserves as the injection level for the upper laser level of thenext period via phonon scattering. Thus this structureis a simplified combination between the resonant phononextraction and the scattering injection scheme, whereboth features are provided by the same levels. This al-lows a design with only two wells per period and therebyincreases the number of possible periods in the waveg-uide. (Two-well designs have been already establishedfor the conventional tunneling injection design [16, 17].)The design of the structure was optimized by calcu-lations within the nonequilibrium Green function modeldescribed in [8, 18, 19] using an improved treatment ofacoustic phonon scattering and including alloy scatter-ing, see [20] for details. This model allows for a consis-tent treatment of coherent evolution and scattering in-cluding level broadening, and has been recently used byother groups as well [21, 22]. Here the following issueswere found to be of relevance for the final design: (i) Atthe operating bias, the extraction level is in resonancewith the lower laser level and located about one opticalphonon energy above the upper laser level of the subse-quent period. (ii) The higher levels do not provide furtherlevel spacings comparable to the lasing transitions in or-der to avoid reabsorption at higher temperatures, whenthey are partially filled. (iii) Increasing doping enhancesthe number of carriers in the gain transition but alsostrengthens impurity scattering associated with a largerlinewidth and a shorter lifetime of the upper laser state.The chosen doping was found to provide the strongestgain. (iv) Compensation effects [18] reduce the width ofthe gain spectrum if the same doping atoms affect bothlaser levels. Thus the placing of the doping in the barrierbetween the lasing states is advantageous.Fig. 2(a) shows the calculated current-voltage char-acteristics for different temperatures for the optimizedstructure. The currents are of the same magnitude as C u rr en t den s i t y ( A / c m )
100 K200 K300 K(a) 0.01 0.015 0.02photon energy h ω (meV)050100150200250 m a t e r i a l ga i n ( / c m )
100 K200 K300 K(b)
FIG. 2: (Color online) Simulation results for the proposedstructure of Fig. 1(b) at different lattice temperatures. (a)Current versus bias per period. (b) Gain spectrum at a biasof 48 mV per periodtemperature (K) 100 200 300upper n u (10 / cm ) 22.3 16.6 13.5lower n l (10 / cm ) 5.5 8.7 9.9 n u e − ¯ hω o pt /k B T (10 / cm ) 0.3 2.1 3.4FWHM of gain spectrum (meV) 2.2 2.9 3.9Peak gain (1/cm) 233 76 21TABLE I: Calculated quantities for the gain transition at48 mV per period. If the upper laser level were in ther-mal equilibrium with its injector level, one would expect n i = n u e − ¯ hω o pt /k B T , which is a lower bound for the popu-lation of the lower laser level, n l , being in resonance with theinjector level. in the design of [3] and thus the same thermal manage-ment should work. The gain spectrum, see Fig. 2(b),shows a peak around 12 meV (2.8 THz). At 200 K thegain maximum is 76/cm which is almost twice as largeas the calculated value for the structure of [3] (42/cm at17 meV for 200 K). Given the fact that the latter sam-ple exhibited laser operation until 186 K, laser operationwell above 200 K can be expected for the new design. At300 K the peak gain is reduced to 22/cm, which is mostlikely not sufficient to overcome the waveguide losses.In order to understand the operation, the energeti-cally and spatially resolved carrier density [19] is shownin Fig. 3. At 37 mV per period, the extraction levelis aligned with the upper laser level, which causes a pro-nounced current peak, while a significant part of the elec-trons is trapped in the lower laser level. Increasing thebias, the lower laser level is emptied at 48 mV due toits alignment with the extraction level. For low temper-atures (e.g. 100 K) the scattering lifetime of the upperlaser level is long and thus almost all electrons are col-lected in this level, see the data given in Table I. Thusthe current is determined by the tunneling from the up-per laser level into the extraction level which causes NDCat the bias of 48 mV per period. With increasing tem-perature, scattering becomes stronger leading to a higheroccupation of the lower laser level, so that the contribu-tion of the tunneling resonance between lower laser leveland the extraction level is of larger importance. This pro-vides PDC for each period as observed at 200 and 300 FIG. 3: (Color online) Electron density at 200 K for the firstcurrent peak at 37 mV per period (upper panel) and the lasingoperation point at 48 mV per period (lower panel). In addi-tion the electronic eigenstates are shown, which clearly showthe mixing of the Wannier states at the respective resonances.
K and required for stable operation. Thus the scatteringfrom the upper to the lower laser level, which reduces theinversion (see [23] for a detailed discussion), is actuallyrequired for the device operation in this design. A keyfeature is the efficient extraction, which maintains the in-version. Thus the design can be considered as extraction-dominated.The reduction of gain with temperature can be at-tributed to a combination of increased broadening of thegain profile and reduced inversion, see Table I. The dataindicates that thermal backfilling contributes to the in-crease of n l with temperature, but can only explain halfthe magnitude. The depopulation kinetics is of equal rel-evance. A principle advantage of the new design is thatthermal backfilling can not entirely destroy the inversionas the upper laser level plays the role of the reservoir,having the highest occupation at all temperatures. Thisis a common feature of properly designed scattering in-jection lasers.In conclusion, a new design for THz-QCLs has beenproposed based on the efficient depopulation of the lowerlaser level, which also ensures the positive differentialconductivity. Lasing operation above 200 K is predicted. Acknowledgments
I want to thank S.-C. Lee and R. Nelander for theircontributions to the development of the computer code.Discussions with E. Dupont, S. Fathololoumi, S. Kumar,and G. Scalari clarified several issues addressed here. Fi-nancial support by the Swedish Research Council (VR)is acknowledged. [1] J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L.Hutchinson, and A. Y. Cho, Science , 553 (1994).[2] R. K¨ohler, A. Tredicucci, F. Beltram, H. E. Beere, E. H.Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, andF. Rossi, Nature , 156 (2002).[3] S. Kumar, Q. Hu, and J. L. Reno, Applied Physics Let-ters , 131105 (2009).[4] C. Walther, M. Fischer, G. Scalari, R. Terazzi, N. Hoyler,and J. Faist, Appl. Phys. Lett. , 131122 (2007).[5] O. Cathabard, R. Teissier, J. Devenson, J. C. Moreno,and A. N. Baranov, Appl. Phys. Lett. , 141110 (2010).[6] R. F. Kazarinov and R. A. Suris, Sov. Phys. Semicond. , 707 (1971).[7] B. S. Williams, S. Kumar, H. Callebaut, Q. Hu, and J. L.Reno, Appl. Phys. Lett. , 5142 (2003).[8] S.-C. Lee and A. Wacker, Phys. Rev. B , 245314(2002).[9] S. L. Lu, L. Schrottke, S. W. Teitsworth, R. Hey, andH. T. Grahn, Phys. Rev. B , 033311 (2006).[10] A. Wacker, Phys. Rep. , 1 (2002).[11] F. Capasso, K. Mohammed, and A. Y. Cho,Appl. Phys. Lett. , 478 (1986).[12] A. Kristensen, P. E. Lindelof, C. B. Sørensen, and A. Wacker, Semicond. Sci. Technol. , 910 (1998).[13] M. Yamanishi, K. Fujita, T. Edamura, and H. Kan, Opt.Express , 20748 (2008).[14] H. Yasuda, T. Kubis, P. Vogl, N. Sekine, I. Hosako, andK. Hirakawa, Appl. Phys. Lett. , 151109 (2009).[15] S. Kumar, C. W. I. Chan, Q. Hu, and J. L. Reno, in Proceedings of CLEO/QELS (2010).[16] S. Kumar, C. W. I. Chan, Q. Hu, and J. L. Reno, AppliedPhysics Letters , 141110 (2009).[17] G. Scalari, M. I. Amanti, C. Walther, R. Terazzi,M. Beck, and J. Faist, Optics Express , 8043 (2010).[18] F. Banit, S.-C. Lee, A. Knorr, and A. Wacker,Appl. Phys. Lett. , 41108 (2005).[19] S.-C. Lee, F. Banit, M. Woerner, and A. Wacker,Phys. Rev. B , 245320 (2006).[20] R. Nelander, Ph.D. thesis, Lund University (2009).[21] T. Kubis, C. Yeh, P. Vogl, A. Benz, G. Fasching, andC. Deutsch, Phys. Rev. B , 195323 (2009).[22] T. Schmielau and M. Pereira, Appl. Phys. Lett. ,231111 (2009).[23] C. Jirauschek, G. Scarpa, P. Lugli, M. S. Vitiello, andG. Scamarcio, J. Appl. Phys.101