A superlattice of GaAs/Ga(1 – x)Al(x)As quantum wells forms a Stark ladder under the influence of a perpendicular electric field. A two level incoherent emitter system, formed by radiative intersubband transitions between adjacent wells, is investigated as a tunable far-infrared radiation source. Intersubband transition rates are calculated at 4, 77, and 300 K for applied fields from 0 to 40 kV cm(–1). It is shown that the quantum efficiency of the radiative emission reaches a maximum at low temperatures for a field of 32 kV cm(–1). Under these conditions the emission wavelength is 38 µm with an estimated power output of 1.1 mW. © 1998 American Institute of Physics

K. Donovan

P. Harrison

R. W. Kelsall

Roberts R. G.

English

Donovan, K.

Harrison, P.

Kelsall, R.W.

White Rose Research Online

This is a repository copy of Stark ladders as tunable far-infrared emitters .White Rose Research Online URL for this paper:http://eprints.whiterose.ac.uk/1680/Article:Donovan, K., Harrison, P. and Kelsall, R.W. (1998) Stark ladders as tunable far-infrared emitters. Journal of Applied Physics, 84 (9). pp. 5175-5179. ISSN 1089-7550 https://doi.org/10.1063/1.368810eprints@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. Stark ladders as tunable far-infrared emittersK. Donovan,a) P. Harrison, and R. W. KelsallInstitute of Microwaves and Photonics, School of Electronic and Electrical Engineering,University of Leeds, Leeds LS2 9JT, United Kingdom~Received 31 March 1998; accepted for publication 15 July 1998!A superlattice of GaAs/Ga12xAl xAs quantum wells forms a Stark ladder under the influence of aperpendicular electric field. A two level incoherent emitter system, formed by radiative intersubbandtransitions between adjacent wells, is investigated as a tunable far-infrared radiation source.Intersubband transition rates are calculated at 4, 77, and 300 K for applied fields from 0 to 40kV cm21. It is shown that the quantum efficiency of the radiative emission reaches a maximum atlow temperatures for a field of 32 kV cm21. Under these conditions the emission wavelength is 38mm with an estimated power output of 1.1 mW. ©1998 American Institute of Physics.@S0021-8979~98!04620-9#I. INTRODUCTIONRecently there have been developments of semiconduc-tor emitters based on intersubband transitions between con-duction band states, in particular quantum cascade lasers op-erating in the mid-infrared.1 Additionally, the use of anelectric field to tune the output wavelength has beeninvestigated2 and there have been reports of tunable emissionin the far-infrared region~1–10 THz or 30–300mm!.3 In thisarticle a theoretical study is made of a two level emitterbased on intersubband transitions between the conductionband states of adjacent wells in a Stark ladder. The mainadvantage of this two level system is the range of tunableoutput.Intersubband transitions can take place via severalmechanisms in semiconductor systems.4 Obviously the mostimportant electron transitions in a system to be used as aterahertz emitter are those which result in the emission ofphotons—the radiative transitions. However, nonradiativetransitions may also take place via electron–phonon scatter-ing and electron–electron scattering.5,6 These nonradiativetransitions are undesirable because they compete with radia-tive emission and hence reduce the efficiency, even thoughin the design of lasers they can be used to good effect increating population inversion.7A possible system that would fulfil the criteria for a tun-able far-infrared emitter based on intersubband transitions isa Stark ladder structure. A Stark ladder is formed when anelectric field is applied to a superlattice. The field changesthe potential of adjacent wells, resulting in the formation oflocalized energy states, or subbands.8 Electron transitions be-tween subbands can result in the emission of photons. Oneimportant feature of the Stark ladder is that the subbandseparation is proportional to the applied electric field, whichmeans that the intersubband separation, and therefore thephoton energy, can be tuned.To obtain coherent emission, a population inversionmust be achieved between two subbands, and this is onlypossible in a system with three or more levels. The Starkladder structure is a two level system so all emission wouldbe incoherent. However, an advantage of the system sug-gested here is that the subband separation can be tuned from5 to 37 meV, resulting in a range of emission wavelengthstunable from 33 to 248mm.II. THEORYWhen an electric fieldF is applied to a superlattice, itintroduces an extra term to the potential,2eFz.9 Using theenvelope function and effective mass approximations theSchrödinger equation would be written asS 2\22 ]]z 1m* ]]z 1V~z !2eF~z2zo! Dc5Ec, ~1!where zo is the origin of the field at the center of thequantum well defined by the one-dimensional potentialV(z). This equation was solved using a numerical shootingtechnique.10 The material parameters of effective mass andband discontinuity were assumed to be independent of tem-perature, with the values taken from Adachi.11 In part-icular, m* (GaAs)50.067m0 , m* (Ga0.8Al0.2As)50.836m0 ,V(GaAs)50, andV(Ga0.8Al0.2As)5167 meV.This article presents a study of a five well structure of 50Å Ga0.8Al0.2As barriers and 50 Å GaAs wells in which theintersubband separation in the Stark ladder regime lies in thefar-infrared region. This is demonstrated by the graph of sub-band minima versus field shown in Fig. 1. It can be seen thatthe intersubband separation is proportional to the field forfields >7 kV cm21, enabling the output wavelength to beeasily tuned. At smaller fields the wave functions are notvery well localized and the system does not exhibit Starkladder behavior.Figure 2 displays the Stark ladder effect when an electricfield is applied to a multiple quantum well. It can be seenthat the wave functions are localized in each well and theenergy separations between them form an equally spaced‘‘ladder.’’a!Electronic mail: phykd@sun.leeds.ac.ukJOURNAL OF APPLIED PHYSICS VOLUME 84, NUMBER 9 1 NOVEMBER 199851750021-8979/98/84(9)/5175/5/$15.00 © 1998 American Institute of PhysicsDownloaded 01 Nov 2006 to 129.11.21.2. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jspThe numerical wave functions were used to calculate theradiative and nonradiative transition rates between the Starkladder states; the aim is to determine the quantum efficiencyof the system. This will determine its feasibility as a device.Bulk phonon modes were assumed for the calculation of theelectron–phonon transition rates. The electron–electron scat-tering and radiative rates were calculated using the approachof Smetet al.12 The longitudinal optical~LO! phonon energyis 36 meV in GaAs~corresponding to a wavelength of 34mm! thus implying that LO phonon emission~which repre-sents a loss process! will be suppressed for subband separa-tions less than 36 meV.4Scattering rates were calculated for all intersubbandtransitions in the five well Stark ladder from levelun11&→un&, i.e., 5→4, 4→3, 3→2, and 2→1 at 4, 77, and 300 Kfor an applied field of 40 kV cm21. These are displayed for 4and 300 K in Tables I and II, respectively. To decrease com-putational time, future calculations were performed only forthe transitions between levels 4 and 3~later labeled levels 2and 1!. This is justified because it can be seen from Tables Iand II that the scattering rates for each of the four transitionsbetween initial and final states are almost identical, thusshowing that this system can be used—to a very goodapproximation—as a model of an infinite superlattice. Theonly exceptions are the transition rates from level 2 to level1. These are slightly different from the other rates because ofend effects~the wave function from the lowest level in theStark ladder is modified by the adjacent barrier potential, asshown in Fig. 2!.13 All transition rate calculations, radiativeand nonradiative, were repeated at intervals of 5 kV cm21from 0 to 40 kV cm21 at temperatures of 4, 77, and 300 K.There are three possible mechanisms for electron–electron scattering from the initial to the final energy state:22→11, 22→21, and 21→11, where 2 and 1 are the initialand final subbands, respectively. The last two processes areAuger type transitions which only occur when an electricfield is applied to the superlattice. This is because there is anasymmetry introduced to the superlattice potential on appli-cation of a field; a condition necessary for Auger transitionsto take place.14The total electron–electron scattering rate (1/tee) for atwo level system is the sum of all three mechanisms:1tee51t222111t221111t2111, ~2!where 1/t2221, 1/t2211, and 1/t2111are the transition rates forthe electron–electron scattering mechanisms 22→ 1, 22→11, and 21→11, respectively. The internal quantum effi-ciency of the Stark ladder system is given byh51trS 1tnr11trD 21, ~3!where 1/tr and 1/tnr are the total radiative and nonradiativetransition rates, respectively. The total nonradiative rate,1/tnr , is given by1tnr51tLO11tAC11tee, ~4!where 1/tLO , 1/tAC , and 1/tee are the longitudinal optical~LO! phonon, acoustic~AC! phonon, and electron–electronscattering rates, respectively. Quantum efficiencies were cal-culated for each value of electric field at temperatures of 4,77, and 300 K. A carrier concentration of 131010 cm22 wasassumed for these calculations. Self-consistent Schro¨dinger–Poisson calculations on similar quantum well systems havedemonstrated that the band bending at these low carrier den-sities is negligible~,1 meV!.15 The radiative and nonradia-tive transition rates vary with electron density, although theradiative rates do not vary appreciably with temperature.Therefore for a field of 32 kV cm21 and a temperature of 4FIG. 2. A schematic representation of the energy levels and wave functionsin the Stark ladder regime.TABLE I. Scattering rates at 4 K with an applied field of 40 kV cm21.Transition 1/tLO 1/tAC 1/tee5→4 3.64231011 2.0813108 8.99031074→3 3.66031011 2.0643108 8.89931073→2 3.65231011 2.0573108 8.83431072→1 3.48531011 1.9903108 8.3423107TABLE II. Scattering rates at 300 K with an applied field of 40 kV cm21 .Transition 1/tLO 1/tAC 1/tee5→4 2.45731011 2.7843109 8.63631074→3 2.45231011 2.7633109 8.57331073→2 2.44731011 2.7543109 8.53431072→1 2.37231011 2.6663109 8.0963107FIG. 1. The effect of electric field on the intersubband separation of a fivewell system.5176 J. Appl. Phys., Vol. 84, No. 9, 1 November 1998 Donovan, Harrison, and KelsallDownloaded 01 Nov 2006 to 129.11.21.2. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jspK, the radiative and nonradiative transition rates were calcu-lated for different carrier concentrations~from 53109 to 131011 cm22) and the quantum efficiency was calculated foreach case.III. RESULTS AND DISCUSSIONFigures 3–6 show graphs of LO phonon, acoustic pho-non, and electron–electron scattering rates as a function ofincreasing electric field. Scattering rates are shown at differ-ent temperatures to determine the conditions under whichlosses are minimized and at which nonradiative processes aredominant. Data are shown only for fields greater than 7kV cm21. This represents the point of Stark ladder forma-tion. Below this field the system is in the ‘‘mini-bandbreakup’’ region which is not appropriate for emitter appli-cations.It can be seen that at a temperature of 4 K no LO phonontransitions take place for electric fields below 36 kV cm21because the intersubband separation for these fields is lessthan the LO phonon energy. At a field of 36 kV cm21, theintersubband separation is equal to the LO phonon energyand LO phonon scattering becomes effective. At 77 K theLO phonon scattering rate peaks at a field of 36 kV cm21,corresponding to a resonance at the LO phonon energy.However, the peak is not as large as that at 4 K because thereis a slight thermal broadening, causing some of the electronsto have energies greater than the LO phonon energy. At 300K, the thermal energy of the system iskT'26 meV and thiscauses substantial thermal broadening of the electron distri-bution function. Consequently, there are many electrons hav-ing sufficient energy to emit a LO phonon, so the LO scat-tering rate is high at low fields, even though the intersubbandseparation is much lower than 36 meV. As the field is in-creased, the states become more localized, hence the wavefunction overlap decreases. The probability of an electronscattering into a lower state is therefore reduced, resulting ina lower LO phonon scattering rate at higher fields.The AC phonon scattering rate decreases as the electricfield is increased and is not very significant at 4 and 77 K,except at low fields. The AC phonon interaction strength isproportional to the energy of the emitted phonon, so at lowfields, where the intersubband separation is small, the ACphonon transition rate is higher. At its peak rate at a tem-perature of 300 K and a field of 5 kV cm21, the AC scatter-ing rate is about 5% of the LO rate.The graph of individual electron–electron scatteringrates from level 2 to level 1 at 4 K~Fig. 5! shows that theAuger-type processes are not as significant as the transitionswhere both electrons are scattered from level 2 to level 1~2211! until a field of 10 kV cm21 is applied. The rates forAuger processes 2111 and 2221 are about 60% and 75%,respectively, of the rate of the non-Auger process 2211 at 7kV cm21. Indeed, the selection rule forbidding Auger-typeprocesses at zero field arises naturally from the results of thecalculations, as can be seen from Fig. 5. There needs to be alarge enough field to antisymmetrize the electron wave func-tions and, when this field is attained, the Auger scatteringprocesses can become dominant. All the electron–electronscattering rates decrease with increasing field, as illustratedin Fig. 6, because, as the electrons gain more energy, a largermomentum change is needed for an intersubband scatteringFIG. 4. Interwell AC phonon scattering rates for the Stark ladder as afunction of the field.FIG. 5. Individual electron–electron scattering rates at 4 K, showing thethree mechanisms by which interwell electron–electron transitions can takeplace.FIG. 6. The total interwell electron–electron scattering rates at 4, 77 and300 K.FIG. 3. Interwell LO phonon scattering rates for the Stark ladder as a func-tion of the electric field.5177J. Appl. Phys., Vol. 84, No. 9, 1 November 1998 Donovan, Harrison, and KelsallDownloaded 01 Nov 2006 to 129.11.21.2. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jspevent to take place. The probability of electron–electronscattering decreases as a larger field is applied and the elec-trons gain momentum. At low fields the electron–electronscattering rate at 4 K is about 40% greater than that at 300 Kand 20% greater than the electron–electron scattering rate at77 K. This is because the Fermi–Dirac distribution at lowtemperatures is such that there is a greater probability of anelectron occupying a state near the Fermi energy.The quantum efficiency of the system is shown in Fig. 7.It can be seen that the quantum efficiency is low (;131025) at low fields for all values of temperature. However,at 4 K the quantum efficiency peaks at a field of 32 kV cm21and has a value of 1.55531025. This is because the mostsignificant loss mechanism, LO phonon scattering, is sup-pressed at 4 K below fields of 36 kV cm21. The AC phononand electron–electron scattering rates also decrease substan-tially as the field is increased to this value. Although;231025 would appear to be a low quantum efficiency, itshould be considered that 231025 photons are emitted everytime an electron is scattered from wellm11 to well m. Fora typical superlattice having 100 repeats this would give 231023 photons per electron, which approaches an accept-able output. It can be seen that the quantum efficiency de-creases at longer wavelengths. The maximum wavelength forF57 kV cm21 is 177 mm, and the quantum efficiency herehas decreased by a factor of 10 from its peak value, whichoccurs at a field of 32 kV cm21 ~corresponding to a wave-length of 38mm!.At 77 K the peak in quantum efficiency is a factor of 8less than the peak at 4 K, and the peak at 300 K is a factor of20 less than that at 4 K. Both these peaks occur at a field of20 meV, corresponding to a wavelength of 62mm.In Fig. 8 the quantum efficiency is shown as a functionof varying carrier concentration. The quantum efficiency de-creases with increasing carrier concentration, as a direct re-sult of increased electron–electron scattering.Figure 9 shows the number of photons per well per unitarea per unit time~given by n/tr) as a function of currentdensity,J, which is given byJ5netnr. ~5!At a field of 32 kV cm21, with a carrier concentration ofn51010 cm22, the current density is 2.23103 A cm22 andthere are 2.231019 photons emitted per well per unit area persecond. For a typical device having 100 wells of area 1 cm2,the power output would be approximately 1.1 mW, whichapproaches useful levels.IV. CONCLUSIONSCalculations of intersubband transition rates by both ra-diative and nonradiative processes indicate the potential of aStark ladder based on GaAs technology for tunable far-infrared emission. The range of output wavelengths for thedesign advanced here would be tunable from 38 to 177mm,with a maximum efficiency at a temperature of 4 K in theshorter wavelength region.ACKNOWLEDGMENTSThe authors would like to thank the School of Electronicand Electrical Engineering and the University of Leeds forfinancial support.1J. Faist, F. Capasso, C. Sitori, D. I. Sivco, A. L. Hutchinson, and A. Cho,Appl. Phys. Lett.67, 3057~1995!.2G. Scarmarcio, F. Capasso, J. Faist, C. Sirtori, D. L. Sivco, and A. L.Hutchinson, Appl. Phys. Lett.70, 1796~1997!.3B. Xu, Q. Hu, and M. R. Melloch, Appl. Phys. Lett.71, 440 ~1997!.4P. Harrison, inNew Directions in Terahertz Technology, edited by J. M.Chamberlain and R. E. Miles~Kluwer, Amsterdam, 1997!, p. 130.5Carrier Scattering in Metals and Semiconductors, edited by V. F. Gant-makher and Y. B. Levinson~Elsevier, Amsterdam, 1987!.6P. Harrison and R. W. Kelsall, J. Appl. Phys.81, 7135~1997!.7P. Harrison, Semicond. Sci. Technol.12, 1487~1997!.FIG. 8. Quantum efficiency as a function of carrier concentration for anelectric field of 32 kV cm21 and a temperature of 4 K.FIG. 7. Quantum efficiency as a function of field at 4, 77, and 300 K.FIG. 9. Emitted photon density per well for an electric field of 32 kV cm21and a temperature of 4 K.5178 J. Appl. Phys., Vol. 84, No. 9, 1 November 1998 Donovan, Harrison, and KelsallDownloaded 01 Nov 2006 to 129.11.21.2. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp8R. G. Roberts, P. Harrison, T. Stirner, and W. E. Hagston, J. Phys. II3,203 ~1993!.9A. Goswami,Quantum Mechanics ~Brown, Dubuque, 1992!10P. Harrison, W. E. Hagston, and T. Stirner, Phys. Rev. B47, 16404~1993!.11S. Adachi, GaAs and Related Materials ~World Scientific, Singapore,1994!.12J. H. Smet, C. G. Fonstad, and Q. Hu, J. Appl. Phys.79, 9305~1996!.13E. L. Ivchenko and G. E. Pikus,Superlattices and Other Heterostructures~Springer, Berlin, 1997!.14P. Kinsler, P. Harrison, and R. W. Kelsall, Phys. Rev. B58, 4771~1998!.15P. Harrison~unpublished!.5179J. Appl. Phys., Vol. 84, No. 9, 1 November 1998 Donovan, Harrison, and KelsallDownloaded 01 Nov 2006 to 129.11.21.2. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

Stark ladders as tunable far-infrared emitters

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Stark ladders as tunable far-infrared emitters

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