A technique for calculating the temperature of the nonequilibrium electron distribution functions in general quantum well intersubband devices is presented. Two recent GaAs/Ga(1–x)Al(x)As quantum cascade laser designs are considered as illustrative examples of the kinetic energy balance method. It is shown that at low current densities the electron temperature recovers the expected physical limit of the lattice temperature, and that it is also a function of current density and the quantised energy level structure of the device. The results of the calculations show that the electron temperature T(e) can be approximated as a linear function of the lattice temperature T(l) and current density J, of the form T(e) = T(l) + a(e–l)J, where a(e–l) is a coupling constant (~6–7 K/kA cm(–2) for the devices studied here) which is fixed for a particular device. © 2002 American Institute of Physics

Harrison, P.

Indjin, D.

Kelsall, R.W.

English

White Rose Research Online

This is a repository copy of Electron temperature and mechanisms of hot carrier generation in quantum cascade lasers .White Rose Research Online URL for this paper:http://eprints.whiterose.ac.uk/1689/Article:Harrison, P., Indjin, D. and Kelsall, R.W. (2002) Electron temperature and mechanisms of hot carrier generation in quantum cascade lasers. Journal of Applied Physics, 92 (11). pp. 6921-6923. ISSN 1089-7550 https://doi.org/10.1063/1.1517747eprints@whiterose.ac.ukhttps://eprints.whiterose.ac.uk/Reuse See Attached Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. Electron temperature and mechanisms of hot carrier generation inquantum cascade lasersP. Harrison,a) D. Indjin, and R. W. KelsallSchool of Electronic and Electrical Engineering, University of Leeds, Leeds LS2 9JT, United Kingdom~Received 22 May 2002; accepted 4 September 2002!A technique for calculating the temperature of the nonequilibrium electron distribution functions ingeneral quantum well intersubband devices is presented. Two recent GaAs/Ga12xAl xAs quantumcascade laser designs are considered as illustrative examples of the kinetic energy balance method.It is shown that at low current densities the electron temperature recovers the expected physical limitof the lattice temperature, and that it is also a function of current density and the quantised energylevel structure of the device. The results of the calculations show that the electron temperatureTecan be approximated as a linear function of the lattice temperatureT l and current densityJ, of theform Te5T l1ae2lJ, whereae2l is a coupling constant~;6–7 K/kA cm22 for the devices studiedhere! which is fixed for a particular device. ©2002 American Institute of Physics.@DOI: 10.1063/1.1517747#The rapid development of the quantum cascade laser asan application of the mid- and far-infrared radiative intersub-band transitions available in semiconductor heterostructureshas been impressive and continues to be prominent in theapplied physics literature. However, in addition, quantumcascade lasers have also given new insight into more physi-cal aspects of the quantum mechanics of electron transport,see for example, Ref. 1. In particular, an earlier work byHarrison2 deduced that:~i! the electron distributions in quan-tum cascade lasers are thermalized, i.e., the energy distribu-tion of electrons in each subband can be described by aFermi–Dirac distribution function and~ii ! the electron tem-perature for each subband is the same. Both of these conclu-sions were substantiated by experimental measurements ofthe electroluminescence spectra of superlattice cascadelasers3 and Monte Carlo simulations.4 While Monte Carlosimulations can be valuable for investigating the physics ofintersubband devices on a microscopic scale, they are diffi-cult to implement, and are very demanding computationally.In this work an alternative more accessible technique forcalculating theaverage electron temperature throughout thedevice is developed, which can be applied to all intersubbandelectron- or hole-based quantum devices.The approach adopted is based on energy balance. Giventhat any intersubband device will always reach an equilib-rium under continuous operating conditions, then the rate atwhich the electron~or hole! distributions gainkinetic energy~relative to the particular subband minimum! through scatter-ing will balance with the rate at which they lose kinetic en-ergy to the lattice. In the steady state, the lattice temperaturewill also reach an equilibrium. The lattice temperature itselfwill always be controlled by the device design, packaging,and cooling systems—in developmental laboratories manymeasurements are made under cryogenic conditions whichgive the lattice temperature more specifically.It is acknowledged that, in areas of high current densityand intense phonon production, the phonon distribution maynot be in equilibrium with the lattice, however, these effectsare likely to be small. The aim of this work is to understandand demonstrate a technique to calculate global effects—theelectron temperature in the device as a single number.Consider the two instances of intersubband scattering asin Figs. 1~a! and 1~b!. In Fig. 1~a! the subband separationE i –E f between the initial electron stateui& and the final stateuf& is greater than the dominant phonon energy@in III–V’s thepolar longitudinal optical~LO! phonon#, hence any phononemission event produces a carrier in the lower subbandwithan increased kinetic energy. This energy adds to the totalkinetic energy of the electron distributions, and through thesubpicosecond intrasubband carrier–carrier scatteringevents,2 is quickly absorbed by the distributions as they re-thermalize. Bi-intrasubband carrier–carrier scattering~of theform u j& ug&→u j& ug&) serves to further redistribute this en-ergy throughout the set of subbands within the quantum de-vice and to equalize the temperatures of the distributions.2 IFig. 1~b! the subband separationE i –E f is less than the pho-non energy, hence a scattering event from the upper to thelower subband reduces the total kinetic energy of the elec-trons.In both cases, the change in the total kinetic energy ofthe electron distributions can be written as:DE5E i2E f2ELO , whereDE.0 should be interpreted as an increaseand DE,0 a decrease in this total energy. If there aren icarriers in the initial state and the LO phonon transitionshave associated scattering rates of 1/t i fem. and 1/t i fabs.for emis-sion and absorption processes, respectively, then the net ki-netic energy generation rate fromintersubband scattering is(f Þi(iF n it i fem.~E i2E f2ELO!1n it i fabs.~E i2E f1ELO!G ,where the indices on the summations imply over all initialand final states in the quantum system.a!Author to whom correspondence should be addressed; electronic mail:p.harrison@physics.orgJOURNAL OF APPLIED PHYSICS VOLUME 92, NUMBER 11 1 DECEMBER 200269210021-8979/2002/92(11)/6921/3/$19.00 © 2002 American Institute of PhysicsDownloaded 02 Nov 2006 to 129.11.21.2. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jspOne of the main features of quantum cascade lasers isthe bridging ~injector! regions between the active layers.These regions serve to remove the electrons from one activeregion and supply them, at the correct energy, to the nextactive region. There is, however, another important role thatthey provide—the opportunity for hot~high kinetic energy!carriers to cool. The injector–collector regions can be opti-mized to do this by designing the separation of the subbandsto be less than the LO phonon energy. Hence, they encouragethe carriers to scatter as in Fig. 1~b!, thus losing kinetic en-ergy and cooling the distribution. Another mechanism thatcontributes to this cooling is illustrated in Fig. 1~c!—intrasubband phonon emission. Such transitions lead to ade-crease in the energy by an amountELO , again if this has ascattering rate of 1/t iiem., then the corresponding kinetic en-ergy loss rate fromintrasubband scattering is(iF n it iiem.ELO1n it iiabs.~2ELO!G ,where 1/t iiabs.accounts for intrasubband phonon reabsorption,which reduces the above energy loss rate by2ELO .Carrier–carrier~electron–electron or hole–hole! scatter-ing events are described as elastic, which means that the totalenergy of the particles before the event is the same as thatafter. However, intersubband electron–electron transitions doconvert potential energy into kinetic energy~or vice versa!,which from the viewpoint of this work would lead to anincrease~decrease! in the total kinetic energy of a subbandpopulation. Note, the potential energy as defined here in-cludes the quantized component of the kinetic energy.It has been argued that at equilibrium the intersubbandgeneration rate is equal to the intrasubband loss rate. Thiscan be written succinctly asD5 (em.,abs.,c –c(f(in it i f~E i2E f1dE !50, ~1!where the change in energydE is equal to2ELO for phononemission~em.!, 1ELO for absorption~abs.!, and zero forcarrier–carrier (c –c) scattering.Equation~1! is a kinetic energy balance equation. Nowthe scattering rates in Eq.~1! are functions of the subbandpopulationsn i and the electron temperatures@if the electrontemperature is too low, the number of intrasubband scatteringevents, as in Fig. 1~c! is too small and the kinetic energyequation cannot be balanced#.The scattering rates 1/t and subband carrier densitiesn iin multilevel quantum cascade structures, can be calculatedusing the self-consistent scattering theory approach of Dono-vanet al.5 and later developed~and validated by comparisonwith current-voltage curves and laser output characteristics,such as threshold current, gain and temperature dependence!by Indjin et al.6 The procedure now is to vary the electrontemperature~which, as argued above, is assumed to be thesame for all subbands! until the kinetic energy balance equa-tion is satisfied self consistently.For the purpose of this illustration two significant GaAs/AlGaAs quantum cascade laser designs were employed. Oneof the lasers was the first reported GaAs/Ga12xAl xAs laser7~with x50.33) which has been extensively studied1,2,4 andadapted still further, see for example Refs. 8 and 9. Thesecond design is more recent10 and was the firstGaAs/Ga12xAl xAs (x50.45) device to lase at room~lattice!temperature, again this device has been the attention of fur-ther studies.11 For specific details of the layer structures anddoping profiles, see the original reports.7,10Figure 2 illustrates the method for the calculation of theelectron temperature. Following our standard procedure fordeducing the carrier densities and lifetimes across all levelsof the device~the model includes all states across an injector/active region/collector, i.e., 1 1/2 periods, which involves 15levels in total!, the self-consistent iteration is extended toinclude the effect of the electron temperature. The data inFig. 2 show the variation of the net kinetic energy generationrateD with the electron temperatureTe for the x50.33 cas-cade laser. The solutions for the electron temperatures occurwhenD50. Detailed study of the contributions to the elec-tron temperature highlight two interesting points: first thatphonon absorption mechanisms are important even at low~77 K! lattice temperatures, and second that, for lattice tem-peratures above 50 K, electron–electron scattering is not animportant electron heating mechanism~accounting for lessthan 1% of the total heat generation rate! in these mid-infrared lasers.Figure 3 summarizes the electron temperatureTe ~givenby the circles! as a function of the current densityJ throughboth the quantum cascade lasers. It can be seen that at verylow current densities~low input powers! the model recoversthe expected physical limit of the electron temperature beingequal to the lattice temperature, but as the current density isincreased, the electron temperature increases too—in agree-FIG. 1. Schematic diagram illustrating inelastic~polar optic phonon! inter-subband scattering processes for subband separations~a! greater than and~b! less than the LO phonon energy.~c! illustrates an inelastic intrasubbandscattering process.FIG. 2. The calculated value of the kinetic energy generation rateD as afunction of the electron temperature and for several different lattice tempera-tures for thex50.33 device at the operating field of 48 kV cm21.6922 J. Appl. Phys., Vol. 92, No. 11, 1 December 2002 Harrison, Indjin, and KelsallDownloaded 02 Nov 2006 to 129.11.21.2. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jspment with experiment.3 For both devices at a lattice tempera-ture of 77 K, the relationship betweenTe andJ turned out tobe linearTe5T l1ae2lJ, ~2!where the electron–lattice (e2l) coupling constantae2lwas deduced from the straight line fits to the data~ s on Fig.3! and had values of 7.6 and 6.1 K/~kA cm22! for the x50.33 andx50.45 devices, respectively. At the higher lat-tice temperature of 300 K, the calculations showed that thereis some deviation from this linear relationship, however, Eq.~2! ~with the same coupling constants! i still a reasonableapproximation, as can be seen by the straight lines on Fig. 3.It is expected that the technique developed in this workwill now allow the important aspect of the energy within theelectron distributions of intersubband devices to be consid-ered in the design and optimization stages of development.Such knowledge will be crucial in solving the problems oftemperature dependence of the lasing characteristics of thenew generations of very long wavelength~Terahertz! quan-tum cascade lasers reported recently.12,13 Application of thetechniques presented here to the Terahertz laser of Ko¨hleret al.12 has shown that the coupling constantae2l is muchlarger ~;47 K/kA cm22! than in the mid-infrared devicesabove. This is a consequence of the increased role ofelectron–electron scattering in this device operating at low~5–50 K! lattice temperatures and demonstrates the greatersensitivity of the electron distribution functions to the injec-tion current—a fact that will have to be controlled, eitherthrough reduced threshold currents or improved injector de-sign, if high temperature operation of Terahertz lasers is to beachieved.In summary, a general technique has been presentedwhich allows the temperature of the nonequilibrium electrondistributions in quantum intersubband devices to be calcu-lated as a function of all the device operating parameters,such as the electric field, the current density, and the latticetemperature. In the illustrative examples of quantum cascadelasers it was found that the electron temperature has a nearlinear dependence on the input current density, although thecoupling constant~slope! is different for different device de-signs.The authors would like to thank EPSRC~U.K.! for fi-nancial support and Z. Ikonic´ for useful discussions.1R. C. Iotti and F. Rossi, Phys. Rev. Lett.87, 146603~2001!.2P. Harrison, Appl. Phys. Lett.75, 2800~1999!.3M. Troccoli, G. Scamarcio, V. Spagnolo, A. Trediccuci, C. Gmachl, F.Capasso, D. L. Sivco, and M. Striccoli, Appl. Phys. Lett.77, 1088~2000!.4R. C. Iotti and F. Rossi, Appl. Phys. Lett.78, 2902~2001!.5K. Donovan, P. Harrison, and R. W. Kelsall, J. Appl. Phys.89, 3084~2001!.6D. Indjin, P. Harrison, R. W. Kelsall, and Z. Ikonic, J. Appl. Phys.91,9019 ~2002!.7C. Sirtori, P. Kruck, S. Barbieri, P. Collot, J. Nagle, M. Beck, J. Faist, andU. Oesterle, Appl. Phys. Lett.73, 3486~1998!.8L. R. Wilson, J. W. Cockburn, M. J. Steer, D. A. Cavder, and M. S.Skolnick, Appl. Phys. Lett.78, 413 ~2001!.9S. Barbieri, C. Sirtori, H. Page, M. Stellmacher, and J. Nagle, Appl. Phys.Lett. 78, 282 ~2001!.10H. Page, C. Backer, A. Roberston, G. Glastre, V. Ortiz, and C. Sirtori,Appl. Phys. Lett.78, 3529~2001!.11D. Indjin, P. Harrison, R. W. Kelsall, and Z. Ikonic, Appl. Phys. Lett.81,400 ~2002!.12R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G.Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, Nature~London! 417, 156~2002!.13M. Rochat, L. Ajili, H. Willenberg, J. Faist, H. Beere, G. Davies, E.Linfield, and D. Ritchie, Appl. Phys. Lett.81, 1381~2002!.FIG. 3. The calculated electron temperature as a function of current densityfor the two different quantum cascade laser structures considered and atlattice temperatures of 77 and 300 K.6923J. Appl. Phys., Vol. 92, No. 11, 1 December 2002 Harrison, Indjin, and KelsallDownloaded 02 Nov 2006 to 129.11.21.2. 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Electron temperature and mechanisms of hot carrier generation in quantum cascade lasers

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