Strong confinement in terahertz intersubband lasers by intense magnetic fields

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1 PHYSICAL REVIEW B 7, 55 7 Strong confinement in terahertz intersubband lasers by intense magnetic fields Giacomo Scalari,* Christoph Walther, and Lorenzo Sirigu Institute of Physics, University of Neuchâtel, Neuchâtel, Switzerland Marcin L. Sadowski Grenoble High Magnetic Field Laboratory, Grenoble, France Harvey Beere and David Ritchie Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom Nicolas Hoyler, Marcella Giovannini, and Jérôme Faist Institute of Physics, University of Neuchâtel, Neuchâtel, Switzerland Received January 7; revised manuscript received April 7; published September 7 Terahertz intersubband lasers based on different optical transitions are investigated under an intense magnetic field applied perpendicularly to the plane of the epilayers, reaching the limit where the cyclotron energy exceeds many times the photon energy. Lasing thresholds as low as.5 A/cm, together with a strong increase of the laser efficiency demonstrate the long intersubband relaxation times achieved in the intrawell samples by the induced three dimensional confinement. The radically different behavior observed for the bound-to-continuum structures highlights the role of interface roughness scattering at low energy subband spacing. DOI:./PhysRevB.7.55 PACS number s :.55.Px, 7..Fg, 7.7.Di,.7.Ai I. INTRODUCTION In a terahertz quantum cascade laser, intersubband ISB transitions are exploited to generate coherent photons with energies h below that of the longitudinal optical LO phonon of the hosting material LO mev for GaAs. The efforts to reach both long wavelengths, i.e., low photon energies and high operating temperatures have highlighted the crucial influence of in-plane scattering on both subband lifetime and intersubband absorption. A control over the inplane degree of freedom may be an essential element to obtain population inversion and optical gain at high temperatures. A three dimensional confinement of the electron by patterning nanostructures, 5 embedding a dot in a cascade structure, or exploiting intersublevel transition in a quantum dot 7 can represent a way to obtain a high temperature, long wavelength terahertz semiconductor laser. A clean way to investigate an intersubband multiquantum well system under three dimensional confinement is to apply a strong magnetic field perpendicularly to the layers; this will produce the breaking of the in-plane parabolic energy dispersion and the formation of discrete Landau levels, allowing the creation of quasi-zero-dimensional states. 9 The possibility to strongly affect the electron dynamics and the subsequent lasing properties of a terahertz quantum cascade laser QCL by varying the relative ratio between cyclotron energy and subband energy spacing has been already demonstrated in earlier works., In the present work, we investigate the properties of different intersubband lasers, all emitting in the far-infrared region of the electromagnetic spectrum, immersed in a strong perpendicular magnetic field. We show that the nature of the wave functions that define the lasing transition is ultimately responsible for the observed behavior. II. SAMPLE DESCRIPTION We will analyze three samples all based on an intrawell transition and tuned to emit at.9 THz m, sample A9,. THz m, sample A5, and.5 THz 5 m, sample V9 and two samples based on a bound-to-continuum transition, emitting at. THz m, sample N and. THz m, sample A95. All samples are realized in the Al.5 Ga.5 As/GaAs material system. The intrawell samples do not show laser action without applied magnetic field; on the contrary, the two samples based on a bound-tocontinuum transition display quite high performance in terms of operating temperature and emitted power without any applied magnetic field. At low temperatures, the properties of the lasing system will ultimately depend on the ratio between the cyclotron energy c = eb/m * E and the energy dispersion E p characteristic of the active region architecture. For c E p, the Landau levels originated from the different electronic states of the active region will be coupled by elastic scattering phenomena as electron-electron e-e scattering, interface roughness IR scattering, and impurity scattering.,5 When the ratio c E p, Landau states are well separated in energy and only resonances corresponding to many-body processes can be observed. The trend in this scenario will be a strong quenching of the nonradiative scattering processes due to the reduced phase space available for the diffused electrons. This will heavily affect the subband electron lifetimes see Ref. 7 for the case of e-e scattering : the lasing threshold, which expression contains the electronic lifetimes as well as the intrasubband dephasing through the linewidth, is expected to reflect those two different regimes see Fig. a. The three different active region architectures are reported in Figs. b d. In the first series of structures Fig. b, 9-/7/7 / The American Physical Society

2 SCALARI et al. Energy [mev] (a) (b) elastic regime V9, A5, A9 Miniband strong confinement Ep A9 Ep A5 hω C hω LO Ep N Ep A95 Ep V9 Ep Applied bias [V] 7 5 (a) Current density[a/cm ] (b) PHYSICAL REVIEW B 7, 55 7 V9 A5 N A95 A9 (c) (d) Miniband A95 N Miniband 9 7 Ep Ep Applied bias [V] 5 7 Current density[a/cm ] FIG.. Color online a J-V characteristics for the intrawell transition samples at T= K and b J-L-V characteristics for the bound-to-continuum samples at T= K. Lsaser emission [a.u.] FIG.. Color online a Energy diagram highlighting the two different regimes of elastic resonances and strong confinement for the different samples studied. The intrawell samples present a very similar energy dispersion for the injector miniband. Lower panels: self-consistent calculation of the band structures for the investigated samples at T= K; the dashed line marks doped layers. b Sample V9 for layer sequence see Ref., and for sample A9 see Ref.. The layer sequence for sample A5, starting from the injection barrier, is in nm.5/.5/./57././././././5././.9/.5/5././.. The figures in boldface represent the Al.5 Ga.5 As barriers and the underlined layers are doped with Si,. cm in density. c Sample A95 Ref. and d sample N Ref.. the optical transition intrawell takes place between the second and first excited states of a large quantum well. The strong overlap and the localization in the same well of the wave functions representing the states i i=,, yield E p =E for this intrawell design. In these samples, both sheet carrier density n s av =5. cm and period length L p av =7 Å have been kept as close as possible to ease the comparison. The second Fig. c and third Fig. d active regions are based on a bound-to-continuum transition and the wave functions are delocalized over several quantum wells. In the bound-to-continuum structures, we assume IR as the main scattering mechanism coupling the Landau states see Ref. 5. The relevant states in the injector are only those who share at least one interface with the upper lasing level IR scattering vanishes when the two considered subbands do not share any interface 9. In the case of structure A95 n s =. cm, we get E p A95 =. mev. For structure N n s = cm, where E 7 LO, we can set E p N =E 9 = mev because states and are not coupled by IR to state 9. In Fig. are plotted the transport and laser emission characteristics for the five samples studied without applied magnetic field at a temperature T= K. III. THRESHOLD CURRENT AND POPULATION INVERSION IN STRONG MAGNETIC FIELD The samples are processed in laser ridges employing the single plasmon waveguide, already described in Ref.. For the long wavelength, intrawell devices m, a modified version of the mentioned waveguide is employed. The devices are then mounted on an insert and placed in the center of the coils of a MW dc resistive magnet. The emitted radiation is guided via an oversized cylindrical metallic lightpipe to an external He-cooled bolometer and to a Fourier transform infrared spectrometer. The whole light path is kept under vacuum to minimize losses coming from atmospheric absorption. The devices are immersed in liquid He in order to reach the long-lifetime and low waveguide 55-

3 STRONG CONFINEMENT IN TERAHERTZ INTERSUBBAND PHYSICAL REVIEW B 7, 55 7 Threshold current density [A/cm] B - V9 A9 N A95 A5 Bound-to-continuum Intra-well FIG.. Color online Threshold current density as a function of applied magnetic field for the five different samples at T=. K. Wide-dashed black line represents the inverse of the magnetic field B. loss regime attributed to a disorder-induced localization already observed on the intrawell sample A9. The lasers are driven in pulsed mode, with a micropulsemacropulse modulation scheme in order to match the bolometer s frequency response. The pulse length ranges from ns to 7 s for an equivalent maximum duty cycle of.5%. In Fig., the threshold current density for the five different devices is plotted as a function of the applied magnetic field. As expected, the J th -B plane is divided into two regions: a low magnetic field regime where elastic resonances between Landau levels dominate and a high magnetic field regime where no oscillations of the threshold current density are observed. Due to the different values of E p for the different structures see Fig. a, the strong confinement limit is placed at different values of the magnetic field. The ultralow threshold current density regime J th A/cm is accessible only to the intrawell transition samples. For this class of devices, characterized by a high value of the dipole matrix element z A9 =.9 nm, z A5 =. nm, z V9 =. nm, the relative reduction of the threshold current density is more than a factor of between the lowest value of magnetic field where they show laser action and the highest values of the applied magnetic field, above 7 T. For sample A9, the threshold reaches B=7 J T th =.5 A/cm. This extremely low value, previously unobserved for any kind of intersubband laser, is a remarkable proof of the very long intersubband lifetimes reached when the wave functions involved in the optical transition are localized in one very wide w 5 nm quantum well, interacting principally with only two interfaces. The lasing regime changes when the system enters the strong confinement region, as shown in Fig.. The threshold current density drops by a factor of 5: the successive local maxima, due to many-body resonances see Refs. and, will be analyzed in detail elsewhere but they are clearly superimposed on a decreasing background. The threshold current density shows the same trend as the inverse of the magnetic field B, plotted along the data dashed black line in Fig.. This gives a scale for the increase of the 5. B-[T - ] th th (dv/di dv/di)i>i I>Ith (dv/di dv/di)i<i I<Ith Current [ma] FIG.. Color online Population inversion deduced from the differential resistance below and above threshold for sample A5 operated in cw. Inset: surface plot of d V/dI B as a function of injected current and applied magnetic field. It is evident that the discontinuity at threshold increases as a function of the applied magnetic field. The other features correspond to modulations of the laser emission intensity. effective lifetime, since the threshold current density is ultimately proportional to / eff = up dn / up dn see Ref.. The increase of the effective lifetime eff is then proportional to the applied magnetic field. From a previous research, waveguide losses were measured and found of very reduced value cm for applied fields larger than T. Assuming that the value of the waveguide losses does not change significantly between and T and that the linewidth is essentially constant between and T, we derive from the ratio of thresholds J th /J th =.5 the same ratio for the effective lifetimes eff = eff /.5 ps. It is possible to analyze in more detail the dependence of the population inversion on the applied magnetic field by inspecting the transport of the laser in continuous wave cw operation. As demonstrated elsewhere,, the discontinuity of the I-V curve at the lasing threshold is due to the reduction of the radiative lifetime caused by stimulated emission. The ratio of the differential resistance dv di above and below threshold is then directly related to the slope efficiency dp/di and to the population inversion through the relation:, dp di eff dv/di I I th. eff + dn dv/di I Ith This quantity is reported in Fig. for sample A5 operated in cw. The improvement of the population inversion term is clear, showing an increase of about seven times between the value at T and the maximum inversion at T. The population inversion factor scales sublinearly as a function of the applied field. To give a complete overview, an intensity graph in the inset of Fig. is reported, where the second order derivative of the voltage with respect to current d V/dI is plotted as a function of injected current and applied magnetic field. This graph summarizes the threshold current behaviors through the discontinuity black line and 55-

4 SCALARI et al. the population inversion factor through the height of the discontinuity itself. The strong link between laser emission and charge transport is highlighted also by the other features present for I I th that correspond to intensity modulations of the laser emission. The devices based on a bound-to-continuum transition do not show a dramatic reduction of the threshold current density as the ones based on the intrawell transition. The lasing threshold is reduced by a factor of. in the case of sample N and. for sample A95 and displays a minimum as a function of the applied magnetic field. The minimum value of the threshold current density for the bound-to-continuum design is 5 A/cm sample A95, B=9 T. The threshold behavior is a clear signature of the strong weight of IR scattering which affects the lifetime of the bound-to-continuum transitions, where the upper state wave function spans four or more quantum wells interacting with at least eight interfaces. The comparison of the strength of the IR in the two classes of lasers is meaningful: all the five samples are grown with the same aluminum mole fraction.5, which appears in the IR scattering matrix element through the barrier height. It is important to note that the profiles of the threshold current densities for the two bound-to-continuum designs look identical if scaled for the corresponding difference in the photon energy h A95 =9. mev, h N =. mev. Also in this case, the low magnetic field region is dominated by intersubband Landau resonances related to different scattering processes., No huge decrease in the threshold is observed when entering the region c E p : moreover, the behavior results independent from the detailed architecture of the lower miniband. In this case, the strongly confined regime does not correspond to a huge enhancement of the gain. To summarize the results of this section, we believe that the localization induced by the magnetic field and interface roughness leads the system to a transition from a homogeneously broadened system to an inhomogeneously broadened ensemble of dots. 9 The gain of bound-to-continuum transition results improved less from this localization regime because the presence of many interfaces introduces a higher broadening in the ensemble of microsamples created. 5 IV. LASER EMISSION IN STRONG MAGNETIC FIELD The laser emission intensity as a function of the applied magnetic field is reported in Fig. 5 for the bound-tocontinuum samples and in Fig. for the intrawell ones and presents two radically different behaviors. For the bound-tocontinuum structures, the intensity maximum occurs at zero applied magnetic field and the dynamic range J max J th /J max is strongly reduced starting from B=T.Inthe intrawell case, the maximum of laser emission is achieved for high values of the applied magnetic field and shifts toward lower values of the applied magnetic fields as the wavelength is increased. We believe that this difference has to be ascribed to the effects of the magnetic field on the transport in the injector region and on the injection process. We assume that we can employ a density-matrix-based approach to describe the injection process into the upper state of the lasing transition, adopting the same analysis Current density [A/cm] Current density [A/cm] PHYSICAL REVIEW B 7, 55 7 λ [µm] FIG. 5. Color online Laser emission as a function of injected current density and applied magnetic field for the bound-tocontinuum samples a A95 h =9. mev and b N h =. mev at T=. K. Insets: representative spectra of the two devices. and symbols presented in Ref. for the resonant tunneling and lasing between two subbands. In the case of tunneling between Landau levels, the intersubband relaxation time up becomes the inter-landau level up,n dn,n scattering time and the intrasubband relaxation time, which is the relaxation time for the coherences of the density matrix, has to be identified with the intra-landau level dephasing time. The off-diagonal elements in the Hamiltonian still read that represents the splitting energy between the Landau states in resonance inj,n and up,n n= for a coherent process 9. The maximum current flowing in the structure at resonance as a function of magnetic field B is given by the formula e n s B J max B = + B up B. Here e is the elementary charge, n s the sheet carrier density in the injector, represents the energy splitting of the injector-upper state doublet, up B is the upper state lifetime, and B represents the intrasubband dephasing time. The magnetic field affects intersubband lifetimes involved in the laser action, up and dn, as well as the in-plane relaxation time. The full width at half maximum of the nonamplified electroluminescence can be written as in energy units 5 = + up. From a previous research, it is known that the contribution to the linewidth of a terahertz intersubband emitter is mainly λ [µm] (a) (b) 55-

5 STRONG CONFINEMENT IN TERAHERTZ INTERSUBBAND FIG.. Color online Laser emission as a function of injected current density and applied magnetic field for the vertical transition samples a V9 h =5.9 mev, b A5 h =.9 mev, and c A9 h =7.9 mev. The maximum of laser emission moves as a function of the applied magnetic field for the different samples. Insets: representative spectra of the three devices. PHYSICAL REVIEW B 7, 55 7 due to the intrasubband dephasing: the typical linewidths for an intrawell transition are mev, too wide to be dominated by the lifetime broadening up ps, = / up.7 mev. Electroluminescence measurements in magnetic field, report a strong narrowing of the terahertz emission with increasing field: the electron localization induced by the magnetic field and the disorder at the interfaces reduces the intra-landau level scattering rate. This will lead to a narrowing of the resonance described by Eq. and at the same time to a transition from a homogeneously broadened system to an inhomogeneously broadened ensemble of dots. 9 Although not directly applicable to the case of tunnel-coupled Landau states, recent investigations of time evolution of coherences in the quantum Hall regime at higher electron densities with respect to our case point toward an intralevel dephasing rate much lower with respect to the interlevel one. 7 Following Ref., the term that governs the injection process is up. When this term is much greater than unity, the injection barrier strongly couples the two states and the laser is in its optimal regime. In this regime, Eq. reduces to J max B = e n s up B and is essentially governed by up B. The magnetic field enhances this injector coupling via an increase of the term B up B. As the magnetic field is increased, both inversion lifetime ratio and injection coupling are then improved with the exception of the Landau level resonances. The bound-to-continuum structures are optimized for operation without magnetic field: the coupling strength between the lower state of injector and the upper state of the lasing transition is chosen in order to obtain up for the expected values of up = ps see Ref. and =.. ps from electroluminescence measurements. The calculated energy splittings at resonance are reported in Table I. The observed reduction of the dynamic range is a consequence of the increasing of the upper state lifetime up B, which governs J max B. The maximum current that can flow in the structure at resonance is deduced from the laser emission rollover. For sample A95 Fig. 5 a, wefindj max /J max =/5, and for sample N Fig. 5 b, wefindj max /J max =/, reflecting a similar change in up B. As already proposed in Sec. III, the delocalized nature of the bound-to-continuum wave function leads to a loss of oscillator strength because of the higher broadening with respect to the intrawell transition. This ultimately limits the threshold reduction and we finally observe a reduced dynamic range as a function of magnetic field for this class of devices. This interpretation finds support also in the results presented in Ref., where a reduced dynamic range as a function of the applied field was observed on terahertz QCL employing an optical transition based on strongly delocalized wave functions. TABLE I. Relevant parameters for the investigated samples. The energies are those calculated with the self-consistent simulation at T= K. N A95 A9 A5 V9 Design btc btc Intrawell Intrawell Intrawell h mev E p mev inj mev

6 SCALARI et al. The situation for the intrawell samples is different: to avoid parasitic coupling of the injector state with the first excited state of the large quantum well state, the injection barrier is fractioned and the coupling strength is very low. The calculated values for the three intrawell samples, reported in Table I, are in average a factor of lower with respect to the bound-to-continuum samples. The injector is then weakly coupled to the upper state of the lasing transition and in this regime J max, where the strong effect of is evident, as it appears to the square power. The increased dynamic range for the intrawell samples results from a combination of two effects: the very long intersubband lifetime up strongly reduces the threshold current density see Sec. III but hardly affects the value of J max because of the low injector coupling. In the strong confinement region B= T, the laser efficiency results a factor of higher with respect to what was observed for low magnetic field values. Since the emitted intensity is proportional to up B, 5 this constitutes another evidence of the long intersubband lifetime reached in these structures. Samples A5 and V9 see Fig. exhibit a maximum of laser emission at lower values of the magnetic field with respect to sample A9. The threshold current, on the contrary, still continues to decrease in the three samples Fig.. Our interpretation is the following: the threshold term is governed only by the lifetimes dn B, up B, and up dn B : hence, it continues to improve because of the increasing of the lifetimes due to magnetic confinement, as shown in Fig.. From Table I we deduce a higher injector coupling for samples A5 and V9 in respect to A9. The strong coupling of the injector is then reached for lower values of PHYSICAL REVIEW B 7, 55 7 the magnetic field, reducing the dynamic range. V. CONCLUSIONS In conclusion, we observed extremely low values for the threshold current density of terahertz quantum cascade lasers based on an intrawell optical transition immersed in strong perpendicular magnetic field. The possibility to achieve such low thresholds is directly related to the nature of the optical transition which affects the electron wave function localization. Structures based on the bound-to-continuum transition, where the wave function is delocalized, do not show such dramatic reduction of the threshold current highlighting the role of interface roughness as the main scattering mechanism for this low energy separation of the subbands. The results obtained on the intrawell samples indicate the possibility of obtaining extremely low threshold densities by acting on the in-plane degree of freedom of the electrons. It is clear that the main difference with a true intersublevel 7 system resides in the degeneracy of the dot states that, in the case of magnetic confinement, grows linearly with the applied field. ACKNOWLEDGMENTS The authors would like to acknowledge Romain Terazzi and Carlo Sirtori for discussions and Lassaad Ajili for processing one of the devices. The work has been partially funded by the Swiss National Science Fund and the EU project TERANOVA. The GHMFL is a Laboratoire conventionné avec l UJF et l INPG de Grenoble. *giacomo.scalari@unine.ch jerome.faist@unine.ch R. Köhler, A. Tredicucci, F. Beltram, H. Beere, E. Linfield, A. Davies, D. Ritchie, R. Iotti, and F. Rossi, Nature London 7, 5. C. Walther, G. Scalari, J. Faist, H. Beere, and D. Ritchie, Appl. Phys. Lett. 9,. G. Scalari, C. Walther, J. Faist, H. Beere, and D. Ritchie, Appl. Phys. Lett.,. B. Williams, S. Kumar, Q. Hu, and J. Reno, Opt. Lett., C. Hsu, J. O., P. Zory, and D. Botez, IEEE J. Sel. Top. Quantum Electron., 9. 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7 STRONG CONFINEMENT IN TERAHERTZ INTERSUBBAND Rev. B 9, M. Helm, in Intersubband Transitions in Quantum Wells: Physics and Device Applications I, edited by H. Liu and F. Capasso Academic, New York,, Vol., Chap., pp. 99. S. Blaser, M. Rochat, M. Beck, D. Hofstetter, and J. Faist, Appl. Phys. Lett., 7. PHYSICAL REVIEW B 7, K. M. Dani, J. Tignon, M. Breit, D. S. Chemla, E. G. Kavousanaki, and I. E. Perakis, Phys. Rev. Lett. 97, 57. We plotted the second order derivative instead of the first order one because it results in a clearer grayscale image. The trend as a function of the magnetic field is unaltered and the quantitative information is reported in the main part of the figure. 55-7