Intersubband Transitions in Quantum Wells Physics and Device Applications I1
SEMICONDUCTORS AND SEMIMETALS Volume 66
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Intersubband Transitions in Quantum Wells Physics and Device Applications I1
SEMICONDUCTORS AND SEMIMETALS Volume 66
Semiconductors and Semimetals A Treatise
Edited by R. K. Willardson
Eicke R. Weber
CONSULTING PHYSICISTDEPARTMENT OF MATERIALS SCIENCE SPOKANE, WASHINGTONAND MINERALENG~VEERING UNIVERSITY OF CALIFORNIA AT BERKELEY
Intersu bband Transitions in Quanturn Wells Physics and Device Applications I1
SEMICONDUCTORS AND SEMIMETALS Volume 66 Volume Editors
H. C. LIU INSTITUTE FOR MICROSTRUCTURAL SCIENCES NATIONAL RESEARCH COUNCIL OTTOWA, ONTARIO, CANADA
FEDERICO CAPASSO BELL LABORATORIES, LUCENT TECHNOLOGIES MURRAY HILL, NEW JERSEY
A CADEMIC PRESS San Diego San Francisco London Sydney Tokyo
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Contents PREFACE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LISTOF CONTRIBUTORS . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
Chapter 1 Quantum Cascade Lasers . . . . . . . . . . . . . .
1
xi
Jerome Faist. Federico Capasso. Carlo Sirtori. Deborah L . Sivco and Alfred Y. Cho THEORETICAL FRAMEWORK . . . . . . . . . . . . . . . . . . . . . . . ENERGY BANDDIAGRAM. . . . . . . . . . . . . . . . . . . . . . . . MATERIAL ASPECTS. . . . . . . . . . . . . . . . . . . . . . . . . . . OPTICAL CONSTANTS . . . . . . . . . . . . . . . . . . . . . . . . . . STABILITY REQUIREMENTS:WHYTHEINJECTORMUSTBEDOPED . . . . . . . QC LASER WITH DIAGONAL TRANSITION AT i= 4.3 pm . . . . . . . . . . . 1. Characterization of the Active Region: Photocurrent and Absorption Spectra 2. Infiuence of the Doping Projilr . . . . . . . . . . . . . . . . . . . . . 3 . Anticrossing qfthe States in the Active Region . . . . . . . . . . . . . 4 . Band Structure at Threshold . . . . . . . . . . . . . . . . . . . . . . 5. Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Device Processing . . . . . . . . . . . . . . . . . . . . . . . . . . I . Laser Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . VIII. QC LASERS WITH VERTICALTRANSITION AND BRAGG CONFINEMENT . . . . . 1. Quantum Design . . . . . . . . . . . . . . . . . . . . . . . . . . 2. RateEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . IX. OPTIMIZATION OF THE VERTICAL TRANSITION LASERAND CONTINUOUS WAVE OPERATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Spectral Characteristics . . . . . . . . . . . . . . . . . . . . . . . x. VERTICAL TRANSITION QC LASER WITH FUNNEL INJECTOR A N D ROOMTEMPERATURE OPERATION . . . . . . . . . . . . . . . . . . . . . 1. Pulsed Operation . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . Threshold Current Density . . . . . . . . . . . . . . . . . . . . . . 3 . HighTemperature, HighPower Continuous Wave Operation . . . . . . . 4. Spectral Properties . . . . . . . . . . . . . . . . . . . . . . . . . XI . LONGWAVELENGTH ( A 2 8 pm) QUANTUM CASCADE LASERS. . . . . . . . 1. PlasmonEnhanced Waveguide . . . . . . . . . . . . . . . . . . . . . I1. 111. IV . V. VI . VII.
V
. . .
.
.
6 10 11 12 13 14 14 16 18 20 24 21 21 29 30 35 40 41
.
.
44 46 48 50 51
52 53
vi
CONTENTS 2. Quantum Design of a QC Laser with Diagonal Transition at I = 8.4 pm . . 3. Long Wavelength Quantum Cascade Laser Based on a Vertical Trunsition . 4. RoomTemperature Long Wavelength (2 = 11 pm) QC Laser . . . . . . . 5 . Semiconductor Lasers Based on Surface Plasmon Waveguides . . . . . . . 6. Distributed Feedback Quantum Cuscade Lasers . REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . .
Chapter 2 Nonlinear Optics in CoupledQuantumWell QuasiMolecules . . . . . . . . . . . . . . . . . . . . . . .
56 56 61
65 68 81
85
Federico Cupusso. Curlo Sirtori. D . L. Sivco. and A . Y. Cho I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. NONLINEAR OPTICALSUSCEPTIBILITIES IN THE DENSITY MATRIXFORMALISM . . I11. NONLINEAR OPTICALPROPERTIES OF COUPLED QUANTUM WELLS . . . . . . . IV . INTERSUBBAND ABSORPTION AND THE STARK EFFECTI N COUPLEDQUANTUM WELLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GENERATION IN COUPLED QUANTUM WELLSAND V. SECONDHARMONIC RESONANT STARK TUNINGOF x‘2’(20) . . . . . . . . . . . . . . . . . . . VI . FARINFRARED GENERATION BY RESONANT FREQUENCY MIXING . . . . . . . . VII. THIRDHARMONIC GENERATION AND TRIPLYRESONANTNONLINEAR IN COUPLED QUANTUM WELLS . . . . . . . . . . . . . . . SUSCEPTIBILITY VIII . MULTIPHOTON ELECTRON EMISSION FROM QUANTUM WELLS . . . . . . . . . Ix. RESONANT THIRDHARMONIC GENERATION VIA A CONTINUUM RESONANCE. . . X . CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 3 PhotonAssisted Tunneling in Semiconductor Quantum Structures . . . . . . . . . . . . . . . . . . . . .
85 87 90 93 102 109 112
116 121 122 123
127
Karl Unterrainer I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . TI. THEORYON PHOTONASSISTED TUNNELING . . . . . . . . . . . . . . . . . ].General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Perturbative Limit . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Nonperturbative Limit ( TienGordon Model) . . . . . . . . . . . . . . TRANSPORT IN EXTERNAL AC FIELDS . . . . . . . . . . . . . . I11. COHERENT 1. Transport in Minibands . . . . . . . . . . . . . . . . . . . . . . . 2 . The WannierStark Ladder . . . . . . . . . . . . . . . . . . . . . . 3. Classical Description of Minibund Transport in External A C Fields . . . . . 4. Quantum Mechanical Description ojsuperluttices in External A C Fields . . . 5 . Analogy to the A C Josephson Effect . . . . . . . . . . . . . . . . . . METHODS. . . . . . . . . . . . . . . . . . . . . . . . IV . EXPERIMENTAL V . EXPERIMENTS ON PHOTONASSISTED TRANSPORT IN A RESONANT TUNNELING DIODE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TUNNELING IN WEAKLY COUPLED SUPERLATTICES . . . . . . VI . PHOTONASSISTED 1. PhotonAssisted Tunneling between Ground and Excited States . . . . . . . 2. PhotonAssisted Tunneling between Ground States (Dynamic Localization. Absolute Negative Conductance. Stimulated Multiphoton . Emission) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII . TEMRTZ TRANSPORT IN SUPERLATTICE MINIBANDS . . . . . . . . . . . .
127 129 129 130 130 134 134 138 140 144 146 147 149 155 155
163 172
CoNTENTs VIII . PHOTONASSISTED TUNNELING A N D TBRAHERTZ AMPLIFICATION . . . . . . . . Ix. SUMMARY AND OUTLOOK . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii 182 183 184
Chapter 4 Optically Excited Bloch OscillationsFundamentals and Application Perspectives . . . . . . . . . . . . . . . . . . 187 P . Huring Bolivar. T. Dekorsy. and H . Kurz I. INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. HISTORICAL BACKGROUND  BLOCHOSCILLATIONS IN THE SEMICLASSICAL MODEL . . . . . . . . . . . . . . . . . . . . . . . . . 111. WANNIERSTARK DESCRIPTION OF BLOCHOSCILLATIONS . . . . . . . . . . . I v . TIMERESOLVED INVESTIGATION OF BLOCHOSCILLATIONS . . . . . . . . . . . v. BLOCHOSCILLATIONS AS MODELSYSTEM FOR COHERENT CARRIER DYNAMICS M SEMICONDUCTORS . . . . . . . . . . . . . . . . . . . . . . . . . . . VI . APPLICATIONOF BLOCHOSCILLATIONS AS A COHERENT SOURCE OF TUNABLE TERAHERTZ RADIATION. . . . . . . . . . . . . . . . . . . . . . . . . VII. SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CONTENTS OF VOLUMES IN THISSERIES. . . . . . . . . . . . . . . . . .
187 188 192 197 203
210 214 215 219 225
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Preface Research on intersubband transitions in quantum wells has led to several practical devices, such as the QWIP (quantum well infrared photodetector) and the QCL (quantum cascade laser). These are two of the success stories in using quantum wells for practical device applications. Research activities in this area have been very intense over the past ten years, resulting in many new devices that are presently being developed for the market. We therefore feel that the time is right to collect a comprehensive review of the various topics related to intersubband transitions in quantum wells. We hope that this volume will provide a good reference for researchers in this and related fields and for those individuals graduate students, scientists, and engineers who are interested in learning about this subject. The eight chapters in Volumes 62 and 66 of the Academic Press Semiconductors and Semimetals serial cover the following topics: Chapters 1 and 2 in Volume 62 discuss the basic physics and related phenomena of intersubband transitions. Chapters 3 and 4 in Volume 62 present the physics and applications of QWIP. Chapter 1 in Volume 66 reviews the development of QCL. Chapter 2 in Volume 66 studies nonlinear optical processes. Chapters 3 and 4 in Volume 66 introduce two related topics: photonassisted tunneling and optically excited Bloch oscillation. We thank all the contributors who have devoted their valuable time and energy in putting together a timely volume. We also thank Dr. Zvi Ruder of Academic Press for providing assistance and keeping us on schedule.
H. C. LIU FEDERICO CAPASSO
ix
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List of Contributors
Numbers in parentheses indicate the pages on which the authors’ contribution begins.
P. HARINGBOLIVAR(187), lnstitut fur Halbleitertechnik 11, RMTH Aachen, Germany FEDERICO CAPASSO (1, 85), Bell Laboratories, Lucent Technologies, Murray Hill, New Jersey
ALFREDY. CHO(1, 8 9 , Bell Laboratories, Lucent Technologies, Murray Hill, New Jersey T. DEKORSKY (187), lnstitut .fur Halbleitertechnik 11, RWTlrl Aachen, Germany JEROME FAIST(l), Institute of Physics, University of NeuchZltel, Neuchitef, Switzerland H. KURZ(187), Institut ,fur Halhleitertechnik 11, R WTH Aachen, Germany CARLOSIRTORI (1, 85), ThornsonCSF, Laboratoire Centrul de Recherches, Orsay, France DEBORAH L. SIVCO(1, 85), Bell Laboratories, Lucent Technologies, Murray Hill,New Jersey KARL UNTERRAINER (127), lnstitut fur Festkorperelektronik, Technische Universitat W e n , Vienna, Austria
xi
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SEMICONDUCTORS AND SEMIMETALS. VOL. 66
CHAPTER 1
Quantum Cascade Lasers Jerome Faist INSTITUTE OF PHYSICS UNIVERSITY OF N e u c H h L NEUCHATEL, SWITZERLAND
Federico Capasso BELLLABORATORIES, LUCENT TECHNOLOOES MURRAYHILL.NEWJERSEY
Carlo Sirtori THOMSONCSF LABORATOIRE CENTRAL DE RECHERCHES ORSAY, FUNCE
Deborah L. Siuco and AEfred Y. Cho BELLLABORATORIES, LUCENT TECHNOLOGIES MURRAY HILL, NEWJERSEY
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. THEORETICAL FRAMEWORK . . . . . . . . . . . . . . . . . . . . . . . 111. ENERGY BANDDIAGRAM . . . . . . . . . . . . . . . . . . . . . . . . IV. MATERIAL ASPECTS . . . . . . . . . . . . . . . . . . . . . . . . . . V. OPTICAL CONSTANTS . . . . . . . . . . . . . . . . . . . . . . . . . VI. STABILITY REQUIREMENTS: WHYTHE INJECTOR MUSTBE DOPED . . . . . . . VII. QC LASERWITH DLAGONAL TRANSITION AT I = 4.3 pm . . . . . . . . . . . 1. Characterization of the Active Region: Photocurrent and Absorption Spectra . 2. Influence of the Doping Profile . . . . . . . . . . . . . . . . . . . . 3. Anticrossing of the States in the Active Region . . . . . . . , . . . . . 4. Band Structure at Threshold . . . . . . . . . . . . . . . . . . . . . 5. Waveguide . , . . . . . , . . . . . . . . . . . . . . . , . . . . 6. Device Processing . . . . . . . . . . . . . . . . . . . . . . . . . 7 . Laser Characteristics . . . . . . . . . . . . . . . . . . . . . . . . VIII. QC LASERSWITH VERTICALTRANSITION AND BRAGG CONFINEMENT , . . . . .
2 6 10 11
12 13 14 14 16 18 20 24 21 21 29
1 Copyrigbt C 2000 by Academic Press All rights of reproduction in any form reserved. ISBN 0127521755 ISSN 00808784100$30 00
2
JEROMEFAISTET AL. 1. Quantum Design . . . . . . . . . . . . . . . . . . . . . . . . . . 2. RateEquations . . . . . . . . . . . . . . . . . . . . . . . . . . IX. OPTIMIZATION OF THE VERTICAL TRANSITION LASERAM) CONTINUOUS WAVE OPERATION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Spectral Characteristics . . . . . . . . . . . . . . . . . . . . . . . x. VERTICAL TRANSITION QC LASERWITH FUNNEL INJECTOR AND ROOMTEMPERATURE OPERATION. . . . . . . . . . . . . . . . . . . . 1. Pulsed Operation . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Threshold Current Density . . . . . . . . . . . . . . . . . . . . . . 3. HighTemperature, HighPower Continuous Wave Operation . . . . . . 4. Spectral Properties . . . . . . . . . . . . . . . . . . . . . . . . . XI. LONGWAVELENGTH ( A 3 8 pm) QUANTUM CASCADE LASERS . . . . . . . 1. PlasmonEnhanced Waveguide . . . . . . . . . . . . . . . . . . . . 2. Quantum Design of a QC Laser with Diagonal Transition at /1 = 8.4 pm . 3. Long Wavelength Quantum Cascade Laser Based on a Vertical Transition . 4. RoomTemperature Long Wavelength ( A = 11pm)QC Laser . . . . . . 5. Semiconductor Lasers Based on Surface Plasmon Waveguides . . . . . . 6. Distributed Feedback Quantum Cascade Lasers . . . . . . . . . . . .
R E F E R E N C E S .. . . . . . . . . . . . . . . . . . . . . . . . . . .
30 35 40 41
44 46 48
. .
. . . . .
50 51 52 53 56 56 61 65 68 81
I. Introduction
Most solidstate and gas lasers rely on narrow optical transitions connecting discrete energy levels between which population inversion is achieved by optical or electrical pumping (Yariv, 1988). In contrast, semiconductor diode lasers, including quantum well lasers, rely on transitions between energy bands in which conduction electrons and valence band holes, injected into the active layer through a forwardbiased pn junction, radiatively recombine across the bandgap. The bandgap essentially determines the emission wavelength. In addition, because the electron and hole populations are broadly distributed in the conduction and valence bands according to Fermi's statistics, the resulting gain spectrum is quite broad and is of the order of the thermal energy kT. The unipolar intersubband laser or quantum cascade laser that we discuss here differs in many fundamental ways from diode lasers. It relies on only one type of carrier (in our case electrons), making electronic transitions between conduction band states (subbands) arizing from size quantization in a semiconductor heterostructure. These transitions are denoted as intersubband transitions. As shown in Fig. 1, their initial and final states have the same curvature and, therefore, if one neglects nonparabolicity, the joint density of state is very sharp and typical of atomic transitions. In contrast to interband transitions, the gain linewidth now depends only indirectly on temperature through collision processes. The gain in an interband transition
3
1 QUANTUM CASCADE LASERS
l
4
.
4
gIa n ,i,
Ene~y
FIG. 1. Comparison between an intersubband transition (a) and an interband transition in a quantum well (b).
is limited by the joint density of states and saturates when the electron and hole quasiFermi levels are well within the conduction and valence band, respectively. In contrast, the intersubband gain has no such limitation, the gain being limited only by the amount of current one is able to drive in the structure to sustain the population in the upper state. Another fundamental aspect of our intersubband laser is the multistage cascaded geometry of the structure, where electrons are recycled from period to period, contributing each time to the gain and photon emission. Thus each electron injected above threshold generates N , laser photons, where N , is the number of stages, leading to a differential efficiency and therefore an optical power proportional to N,. The cascaded geometry has some significant advantages over the usual arrangement where the individual gain regions (quantum wells) are electrically pumped in parallel as in conventional diode lasers. In the latter case, a uniform gain across the active region is limited by the ratio of the effective transit time between wells, including capture, of the slower carrires (usually holes) and the recombination time; that is, the reciprocal of the total recombination rate due to stimulated, spontaneous, and nonradiative emission in the gain region. For this reason, the number of quantum wells in a multiquantum well laser is usually limited
4
JEROMEFAISTET AL.
to 5 or 10. In the cascaded scheme, the number of stages N , is only limited by the ratio between the effective width of the optical mode and the length of an individual stage. Depending on the wavelength, our structures have in general a number of stages N , between 16 and 35. Since the gain is proportional to N , , a large value of N , enables us to decrease the inversion density in each stage, strongly reducing electronelectron scattering processes and therefore the broadening of the distribution associated with hot carrier effects. There is also a strong effort to develop interband midinfrared quantum cascade lasers based on type I1 heterostructures, which resulted in the demonstration of the first device operating up to 200 K at 3.8 pm wavelength (Yang et a/., 1997). The idea of a unipolar laser based on transitions between states belonging to the same band (conduction or valence) is quite old, and can be traced to the original proposal of B. Lax (1960) of a laser based on an inversion between magnetic Landau levels in a solid (cyclotron laser). The latter was first demonstrated experimentally in the far infared using lightly pdoped germanium. Interested readers can consult a comprehensive review by Andronov (1987). The seminal work of Kazarinov and Suris (1971) represents the first proposal to use intersubband transitions in quantum wells, electrically pumped by tunneling, for light amplification. In their scheme, electrons tunnel from the ground state of a quantum well to the excited state of the neighboring well, emitting a photon in the process (photonassisted tunneling). Following nonradiative relaxation to the groundstate electrons are injected into the next stage and so forth sequentially for many stages. Population inversion in this structure is made possible by the relatively long ( 2 10 ps) nonradiative relaxation time associated with the “diagonal” transition between adjacent wells, compared with the short intrawell relaxation. During the 1970s, intense experimental work proceeded on the twodimensional electron gas formed at the SiSiO, interface in silicon metaloxide semiconductor field effect transistors (MOSFET), focusing mainly on the electrical properties (Ando et al., 1982).The first intersubband emission spectra in an electron gas, heated by a parallel current, was reported in the SiSiO, system (Gornik and Tsui, 1976). The advent of molecular beam epitaxy (MBE) (Cho, 1994) in the late 1960s for the first time enabled the fabrication of heterostructures based on 111V compounds (and now extended to almost all semiconductors) with very sharp interfaces (the size of the interface fluctuations is about a monolayer) and excellent compositional control, and created a great interest in the study of multiquantum well structures. This led to the first observation of intersubband absorption in a GaAsAlGaAs multiquantum well (West and Eglash, 1985). Early attempts were made to implement experimentally the proposal of Kazarinov and Suris (1972) in a GaAsAlGaAs superlattice. They led to the observation of intersubband luminescence pumped by resonant tunneling by M. Helm and coworkers (Helm et al., 1988).
1 QUANTUM CASCADE LASERS
5
Kazarinov’s paper spurred an intense theoretical activity in the 1980s and early 1990s, with a large number of proposals for intersubband lasers. In one variant (Capasso et al., 1986), electrons are injected by resonant tunneling into the second excited state of a quantum well, followed by a laser transition to the lower excited state and nonradiative relaxation (via phonon emission) from the latter to the ground state. Electrons are then reinjected into the next stage and so on, sequentially through many quantum wells. The use of a stack of double quantum wells in a sequential resonant tunneling structure was proposed by Liu (1988). Later, many proposals considered structures with resonant tunneling injection from the ground state of a quantum well into the first excited state of an adjacent quantum well (Kastalsky et al., 1988; Borenstein and Katz, 1989; Loehr et al., 1991; Choe et al., 1991, Belenov et al., 1988; Hu and Feng, 1991; Mii et al., 1990; Henderson et al., 1993) and laser action between the latter and the ground state, repeated sequentially through many wells. For population inversion, this scheme requires that the tunneling escape time from the ground state (i.e., the reciprocal of the resonant tunneling rate) be smaller than the intersubband scattering time between the states of the laser transition. This is difficult to implement since it necessitates very short tunneling escape times to compete against the optical phonon scattering time of about a picosecond. Optically pumped structures were also proposed (Sun and Khurgin, 1991; Berger, 1994). To establish population inversion, other proposals relied on the impossibility of emission of optical phonons for subbands spaced by less than an optical phonon. This requires operation of the device in the far infrared, which is a difficult task because waveguides in this wavelength range are very lossy and difficult to manufacture. Apart from an interest stemming from basic physics, these attempts to develop lasers based on intersubband transitions in the midinfrared and far infrared were motivated by the lack of convenient semiconductor optical sources in this wavelength region. The midinfrared is a very important spectral region for spectroscopy and gassensing since most molecules have their fundamental vibration modes in the 3 to 12pm wavelength region. The existing technology, based on interband leadsalt lasers (Tacke, 1995), has the major drawback of necessitating cryocoolers, since, in practical systems, these devices operate at temperatures between 50 and 130K. Alternative diode technologies, so far limited to the 3 to 5pm wavelength band, now include antimonybased quantum well lasers with type I (Choi et al., 1996; Lane et al., 1997) or type I1 (Chow et al., 1995; Meyer et al., 1995) transitions, and, as mentioned, quantum cascade lasers based on interband transitions (Yang et ul., 1997). So far, however, only quantum cascade (QC) lasers based on intersubband transitions have demonstrated good room temperature performance virtually across the whole midinfared region.
6
JEROMEFAISTET AL.
QC lasers have been made possible by the convergence of two techniques: molecular beam epitaxy and bandstructure engineering (Capasso, 1987; Capasso and Cho, 1994). With the latter technique, energy diagrams with nearly arbitrary shapes can be designed using building blocks such as compositionally graded alloys, quantum wells, and superlattices. In this way, entirely new materials and semiconductor devices can be designed and their properties tailored for specific applications. MBE, with its ability to grow atomically abrupt heterojunctions and precisely tailored composition and doping profiles, is the best epitaxial growth technique to fabricate QC lasers.
11. Theoretical Framework
We computed the electronic states of our multiquantum well heterostructures in the envelope function approximation (Bastard, 1990), including the nonparabolicity through an energydependant effective mass (Nelson et al., 1987): (1) where ( E  V )is the energy of the electron E measured from the conduction band edge V(z) of the material. In this approach, E,, the material's bandgap, is related to the nonparabolicity coefficient y (Nelson et al., 1987) through
(2)
y  l = 2m*E,/h2
We found that this model predicts the correct resonant energies with a typical accuracy of a few millielectron volts even for scattering states located above the barrier height (Capasso et al., 1992). However, since we solve a onedimensional Schrodinger equation, which includes the energy dependant effective mass [Eq. (l)], the wave functions that we compute with this approach are not orthogonal, which makes the definition of the dipole matrix element between the state i and j
(3)
zij = (CPiIZlPj)
somewhat problematic. The solution to this problem is to go back to the twoband model and compute the matrix element including the valence band part. The dipole matrix element now reads (Sirtori et al., 1994) 2..= "
ii 2(Ej  Ei)
(CPilpz
1 1 m*(Ei, z) k m*(Ej, z)
where the momentum operator p z is defined, as usual, p ,
(4) =
ih(a/az). To
1 QUANTUM CASCADE LASERS
account for the underlying valence band component, the wave functions and ‘pj must be normalized according to
’
+
E  V(Z) E  V ( Z )+ E,(z)
7 ‘pi
(5)
In contrast to the square well case, for which this more accurate approach leads to modification of the value of the matrix elements in the order of only about 5lo%, we found that the use of Eqs. (4)and ( 5 ) instead of Eq. (3) in our laser structures leads to much more significant differences, up to 40%. The reason is that we compute our dipole matrix elements for transitions in which the nonparabolicity effects are enhanced, such as having at least one highenergy state lying close to the top of the barrier, and those being diagonal in real space. The effect of an applied electric field on the structure is taken into account by digitizing the potential in 0.5 to 1nm steps and enclosing the whole structure in a large “box.” The latter procedure alleviates the normalization problems associated with pure scattering states. When needed, selfconsistency with Poisson’s equation is introduced assuming (i) that all the impurities are ionized and (ii) periodic boundary conditions on each stage of the structure. The main challenge in the design of a laser based on intersubband transitions is to obtain a population inversion. Unlike most other laser systems, population inversion between subbands does not come from an intrinsic physical property of the material system but from a careful design of the scattering times. The dominant scattering mechanism between subbands separated by more than an optical phonon energy (i.e., > 30 meV for InGaAs) is the emission of optical phonons. This process is always allowed and is very efficient, leading to lifetimes of the order of a picosecond. For this reason, it is essential to include the computation of the optical phononemission rate in the general design of the laser. The optical phononscattering rate between two subbands is computed for electrons emitting bulk phonons (Price, 1981; Ferreira and Bastard, 1989). Since our excited states densities are very low, the electron is always assumed to be at k,, = 0. In this case, the optical phonon scattering rate reads:
where 2m*(Eis  ha,,)
(7)
8
JEROME FAISTET
AL.
is the momentum exchanged in the transition. Absorption of optical phonons at finite temperatures is taken into account by evaluating Eq. (6) using the relevant momentum for optical phonon absorption
Although this procedure neglects the inherent complexity of the phonon spectrum of an InGaAsAlInAs superlattice, we found that it gave good agreement with our measurement of the intersubband scattering rate in a 10nm square well (Faist et al., 1993). In general, this lifetime has a minimum (0.25 ps) for a subband spacing equal to the optical phonon energy and increases monotonically with subband spacing up to a few picosecond for subbands spaced by 300 meV, when phonons with large momentum must be emitted. On the other hand, the radiative rate ( T , ~ ~ ~ ) for ' spontaneous photon emission in a single polarization mode is given by
where YE is the refractive index, c is the light velocity, E, is the vacuum permitivitty, and q is the electronic charge. The calculated radiative efficiency &d = T ~ ~ Jforz spontaneous , ~ ~ ~ emission of a square well is plotted as a function of the transition energy in Fig. 2. The strong increase above the optical phonon energy is due both to the increase of zopt with transition energy, as described in the previous paragraph and to the E,, dependence of z , contained ~ ~ ~ in Eq. (9). For transition energies below the optical phonon energy and at low temperatures, optical phonon scattering is forbidden and the lifetime, limited by emission of acoustic phonons, is much longer, of the order of hundreds of picoseconds (Faist et al., 1994a). Relatively high values of the radiative efficiency can be obtained (Helm et al., 1988). However, since the acoustic phonon energy is of the order of a millivolt, the BoseEinstein factor increases very rapidly with temperature. As shown in Fig. 2, the radiative efficiency in the far infrared drops by approximatively two order of magnitude between 0 and 100 K while remaining practically constant in the midinfrared. This makes the design of a laser structure in the far infrared very challenging since selfheating effects are expected to be strong in a laser close to threshold. The peak material gain between subbands i and j assuming a Lorenzian
1
.9 c m .D
2
QUANTUM
9
CASCADE LASERS
I0"
1o
10"
1 o4
Io
1o  ~
~
Io8 1o
~
~
1 o'
10
100
10.'
Transition Energy (mev) FIG.2. Calculated radiative efficiency of a quantum well for energies above and below the optical phonon energy. The shaded area comprises the wavelengths for which light cannot propagate (Reststrahlenband) and those at which multiphonon absorption is important.
line reads
where q is the electron charge, 1 is the emission wavelength; n is the mode refractive index; 2yij is the full width at half maximum (FWHM) of the transition in energy units determined from the luminescence spectrum (spontaneous emission); nj and n, are the sheet electron densities in subband j and i; and L, is a normalization length, which in this work is chosen as the length of a period of the active region. The peak modal gain G, is defined as G, = G,T, where r is the confinement factor of the waveguide. The latter is simply the product of the number of periods N , and the overlap factor of the mode with a single period, rp.The overall factor to the left of the population difference in Eq. (10) is called the gain cross section. We denote it g, to distinguish it from the gain coefficient g, used in our papers on QC lasers and defined as G J J , the ratio of the peak material gain to the current density J . Note that n,  n,, in steady state, is proportional to J . It is interesting to note that in contrast to the spontaneous emission, the material gain has no strong wavelength dependence and therefore the variation of threshold current as a function of design wavelength is expected to be weak and mainly determined by freecarrier absorption.
10
JEROMEFAIST ET AL.
111. Energy Band Diagram
Our lasers consist of a cascaded repetition of typically 16 to 35 of an identical stage. As shown in Fig. 3, each stage consists of an active region followed by a relaxationinjection region. The active region is the region where population inversion and gain takes place. As shown in Fig. 3, the electronic spectrum of the active region consists usually of a ladder of three states engineered so as to maintain a population inversion between the third and second states. Assuming a simplified model with no nonparabolicity and a 100% injection efficiency in the n = 3 state, the population inversion condition is simply z32 > z2 where ~3;’is the nonradiative scattering rate from level 3 to level 2 and z;’ is the total scattering rate out of level n = 2. Note, however, that since the nonradiative channel 31 is usually not negligible, especially in coupledwell structures, this condition is actually less stringent than the condition t 3 > z2 between the total lifetimes of states n = 3 and n = 2. The lifetime t2 in most QC laser structures is determined by the scattering rate 1/(z2J to a lower subband ( n = 1). The ladder of electronic states with a suitable energy spectrum and scattering properties can be obtained in a variety of multiquantum well structures. We distinguish these structures by the character of the 32 transition. The transition is said to be vertical when the two wave functions of states n = 3 and n = 2 have a strong overlap, and diagonal when this overlap is reduced.
............
..."............. 4
Active region
L
A
v v
b
Relaxation + injection
FIG.3. General philosophy of the design: each stage of the multistage active region consists of a gain (or active) region followed by a relaxationinjection region.
1 QUANTUM CASCADE LASERS
11
It is usually beneficial to leave the active region undoped since, as we have shown experimentally, the presence of dopants in the active region significantly broadens and red shifts the lasing transition by introducing a tail of impurity states (Faist et at., 1994b). The relaxationinjection region is the section of the period where the electrons cool down and are reinjected into the next period. This region must be doped to prevent the strong spacecharge buildup that would arise if the electron population was injected from the contact. In the most recent structures, doping was restricted to the center of the injector to separate electrons in the ground state of the latter from the parent donors, as in modulationdoped heterostructures. This reduces scattering thus enhancing the injection efficiency.
IV. Material Aspects In principle, a heterostructure that fulfills the preceding requirement and is based on the design philosophy displayed in Fig. 3 can be realized using almost any semiconductor material. All of our experiments were performed on In,~,,Ga,,,,AsA1,,,,In,,,,As heterojunction material latticematched to InP and grown by MBE. This material combination has the following advantages over the more conventional GaAsAlGaAs heterostructures. First, the electron masses in both well and barriers are lighter, enabling simultaneously larger oscillator strengths and smaller optical phonon scattering rates. The large conduction band discontinuity AE, = 0.52 eV enables the design of lasers in a large wavelength range from 1 = 4.3 to 17 pm. In addition, the InP substrate has very good waveguiding properties: having a lower refractive index than both AlInAs and GaInAs, it can be used as a cladding material. It is a binary material and thus provides a good thermal transport, reducing the thermal impedance of the laser, as opposed to an alloy. QC lasers based on GaAsAlGaAs require the growth of a thick lower cladding based on AlGaAs with a large A1 mole fraction. However, the GaAsbased material has advantages of its own, namely, a lattice match obtained regardless of the A1 mole fraction, enabling more flexibility in the design and somewhat relaxed growth requirements. Intersubband electroluminescence (Li et al., 1998; Strasser et al., 1997) and laser action (Sirtori et al., 1998a) were demonstrated in quantum cascade GaAsAlGaAs heterostructures. This will soon enable a comparison of both technologies. The SiSiGe heterostructure material also has advantages, including the absence of a restrahlen band, which should enable the design and operation of QC lasers in the 20 to 50pm wavelength range, and compatibility with silicon technology.
12
JEROMEFAISTET
AL.
V. Optical Constants The multistage gain region is inserted in an optical waveguide designed for the laser operating wavelength. Since the optical constants are fixed by the material used (in our case, InGaAs and AIInAs), the physical dimensions of the waveguide scale with the wavelength. This makes the waveguide design in the 4 to 10pm wavelength range a challenge since one needs a waveguide with low loss and a high confinement factor (to minimize the threshold current) while maintaining the total thickness of the layers grown by epitaxy to a minimum. For a reliable waveguide design, a good knowledge of the optical constants of InGaAs and AlInAs is needed. The values obtained by linear interpolation between the binary compounds = 3.071. To obtain (Lynch and Hunter, 1991) are nlnCaAs= 3.324 and nInAIAs more accurate values, three periods of an optical quarterwave Bragg reflector (LinCaAs = 320 nm and LinAIAs = 346 nm) were grown by MBE, latticematched to a semiinsulating InP substrate. The optical transmission, measured with a Fourier transform infrared (FTIR) spectrometer is shown in Fig. 4.A best fit of this data, also displayed in Fig. 4, leads to the value n l n ~ a= ~ 3.458 s and n l n ~ l A s= 3.198. As shown later in the text, these values gave very good agreement between the measured and calculated spacing of
0.6 c
.0 cn .v, 0.4 E cn c
E 0.2 0
FIG.4. Solid line: midinfrared transmission spectrum of a threeperiod quarterwave stack with 320nmthick InGaAs layers and 346nmthick InAlAs layers. Dashed line: computed value of the transmission assuming the refractive index displayed in the figure.
1 QUANTUM CASCADE LASERS
13
the longitudinal modes of QC lasers. The good agreement of the fit up to 1000cm' shows that no significant dispersion of the indices occurs up to 10 pm.
VI. Stability Requirements: Why the Injector Must Be Doped In t heir original paper Kazarinov and Suris (1971) proposed to use an undoped periodic superlattice as the gain medium. In this scheme, a strong applied electric field is supposed to position the ground state 1' of well i above the second state 2'" of the following well i 1 (i.e., E'; > E;+'). A photon of energy hv = E:  E? is then supposed to be emitted by photonassisted tunneling. Population inversion is obtained automatically since the upper state of the lasing transition is also the ground state of each well. However, this band alignment is inherently unstable because the current is at a minimum between the two maxima occurring for the band alignments 1'  2i+l and 1'  3"'. Indeed it is well known from experiments and theory that superlattices break into different electric field domains within which the energy levels are locked in resonance. In a laser structure, a homogenous and stable electric field distribution is an absolute necessity. Therefore, the structure (i) must be doped so that the integrated negative charge is always exactly compensated by the fixed positive donors even in the situation of strong injection to prevent spacecharge formation and (ii) the operating point must be a stable point of the currentvoltage (IV) characteristics to prevent the breakdown of the active region into different field domains. Both conditions are fulfilled in our design. Below the threshold bias, the structure has a high electrical resistance because the band diagram has an overall sawtooth shape. The bias for which the band diagram goes from sawtooth to staircase corresponds approximately to the threshold. Close to threshold, the current is controlled by resonant tunneling between the ground state of the graded gap injector and the excited state of the adjacent active region. The laser threshold occurs before the peak current (full resonance) is reached, thus preventing the formation of domains since the device is always operated in the stable region of the IV characteristic. As already mentioned, overall charge neutrality must be maintained under operating conditions. Therefore, for the injector, which acts as an electron reservoir, not to be depleted at threshold, its donor sheet density N , must be much larger than the excited state density at threshold nth. On the other hand, freecarrier losses must also be minimized and we choose as a rule of thumb N , r r ~25 x nthM 14 x 10" cm2.
'
+
14
JEROMEFAISTET AL.
VII. QC Laser with Diagonal Transition at il = 4.3 pm
1. CHARACTERIZATION OF THE ACTIVEREGION:PHOTOCURRENT AND ABSORPTION SPECTRA We start the discussion of our different QC laser designs with the original structure designed for operation at 4.3pm wavelength (Faist et al., 1994~). This design has some unique features. First, it is the one that enables the shortest wavelength to be reached for a given band discontinuity. Second, it makes use of a diagonal transition with an upper state anticrossed with another level higher in energy, a scheme that has many interes6ng features. In addition, it is the only structure for which a complete characterization was carried out, which includes, beside electroluminescence, photocurrent and absorption spectroscopy. This detailed characterization was carried out on samples with an active region similar to that of the laser samples but grown with a fewer number of period ( 5 or 10) on a semiinsulating InP substrate (Faist et al., 1994d). A semiinsulating substrate with its high transparency enables an accurate measurement of luminescence, absorption and photocurrent through a polished 45" wedge. Such a coupling is in principle not as efficient as a waveguide coupled through a cleaved facet or a twodimensional grating because the light is not completely polarized normal to the layer but has the advantage of having a wellcontrolled coupling efficiency. The band diagram of one period of the structure with and without bias applied is displayed in Figs. 5a and 5b, respectively. At zero applied bias, charge transfer between the doped injector region (short period superlattice at the righthand side of the potential profile) was taken into account in our calculations, which solved Poisson's and Schrodinger's equation selfconsistently. In the same figure, the photocurrent spectrum of the sample, grown by MBE using InGaAsAlInAs latticematched on an InP substrate, is displayed for various applied biases. The three observed photocurrent peaks, corresponding to the 13, 14, and 15 transitions in the active region are easily identified and correspond closely to the calculated transitions displayed in the band diagram. In addition, the strong Stark shift of the 13 transition is expected since level 3 and level 1 belong to two distinct wells and have not yet anticrossed with levels 2 and 4. This anticrossing is discussed later in the text. Absorption from the active region was also measured in a large mesa (800 x 800pm) by a differential absorption technique with the light incident and exiting through a 45" polished wedge. In our structure, at an electric field 50 kV/cm (applied bias U 1 V) the injected current is negligible (< 100 pA) but the active region is already depleted. As a resuIt, the differential transmission AT/T= [T(U = 1 V) 

1 QUANTUM CASCADE LASERS
r
0.8
i
I

200 (bl
15
300 400 Photon Energy (rneV)
FIG. 5. Quantum cascade emitter structure: (a) calculated conduction band diagram of a portion of the AlInAs (barriers)GaInAs (wells) structure under positive bias conditions and an electric field of lo5V/cm. The dashed lines are the effective conduction band edges of the digitally graded 18.1nmthck electron injector. The latter comprises six 3nmthick AIInAsGaInAs periods of varying AlInAs duty factor to grade the average alloy composition. Electrons are tunnelinjected through a 7.4nmthick barrier into the n = 3 subband of the active region. The latter comprises 1.1 and 3.7nm quantum wells separated by a 3.5nmthick barrier and a 3nm well sandwiched between 3nmthick barriers. The third 3.0nmthick well is sandwiched between 3nm barriers. The 1.1nmthick well and its barriers are ntype doped to 1017c ~ I  This ~ . gives rise to a small band bending. (b) Measured photocurrent spectra at various bias voltages below the onset of strong electron injection and band diagram at 0 bias. The positions of the energy levels with respect to the bottom of the flat well are E , = 175meV, E , = 230meV, E , = 410meV, E , = 525 meV, and E , = 620meV. The peaks in the spectra correspond to the optical transitions indicated by the arrows in the inset.
16
JEROME FAISTET
2.0
1
AL.
   Photocurrent  Absorption
1.0
0.5 0
L
200
250
300
350
Photon Energy (mew FIG. 6. Absorption (full line) and photocurrent (dashed line) spectrum from the active region at zero bias from the structure of Fig. 5.
T(U = O)]/T(U = 0) is a measure of the absorption from the active region at U = 0 V. Here A T / T measured at TL= 10 K for our samples is shown in Fig. 6. The same 13 and 14 transitions are again easily identified. 2. INFLUENCE OF
THE
DOPING PROFILE
A study was also carried out to identify the best doping profile (Faist et al., 1994b). To this end, two otherwise identical samples were grown with a similar sheet density per period but a different doping profile. The reference sample, similar in active and injector region design, to that of Fig. 5, was doped with Si to n = 1 x 1017cm3 across the graded injector, the smallest (0.8nm) well of the active region and its adjacent barriers. In the second sample, however, the dopants are set back from the active region and only the injector is doped to n = 1.5 x l O I 7 emp3. A comparison of the electroluminescence spectra performed at an identical injection current was carried out at low temperature with a FTIR spectrometer by a lockin and stepscan technique. The luminescence quantum efficiencies, which are directly proportional to the radiative efficiencies, are measured to be identical in the two structures. This is a strong indication that, for transitions energies above the optical phonon threshold, the impurity potential has a negligible influence on the intersubband lifetime, as expected from theoretical calculations (Ferreira and Bastard, 1989). In contrast, the spectra taken at an identical injected current ( I = 50 mA, corresponding to an injected current density of J = 1.1 kA/cm2), and compared in Fig. 7, are very different. The spectrum of the sample with setback shows a dramatic narrowing of the luminescence
17
1 QUANTUM CASCADE LASERS
150
200 250 300 350 Photon Energy (mev)
400
FIG. 7. Electroluminescence spectrum for the reference sample (left peak) and the sample with setback; that is, with an undoped active region and injection barrier (right peak). The structures and corresponding energy diagrams are similar in design to those of Fig. 5. The diagonal transition is between 0.8 and 3.5nmthick wells separated by a 3.5nm barrier. The third 2.8nmthick well is sandwiched between 3nm barriers. Electrons are injected into state 3 through a 4.5nm barrier. The drive current is I = 50mA. The spectra are fitted with a Gaussian and Lorentzian line shape (dashed curves) for the reference sample and sample with doping setback, respectively.
line down to 21 meV from its value of 50 meV for the reference sample. The sample with doping setback shows a very good agreement between the calculated (E23 = 293 meV) and measured (294meV) values of the transition energy. On the contrary, the luminescence peak of the sample with a doped barrier has a transition energy of E,, = 245 meV, 30.5 meV lower than the calculated value ( = 275.5 meV). Moreover, the electroluminescent spectra compared in Fig. 7 exhibit a very different line shape: while the spectrum of the sample with doping setback is nearly Lorentzian, the spectrum of the reference sample is well fitted by a Gaussian line shape. These observations are in qualitative agreement with a band tail picture of impurity disorder. Because of the doping level used (  1017cm3), the impurity states merge to create a tail on the lowenergy side of the twodimensional density of states. We therefore interpret the discrepancy between the calculated and measured E , , transition energy as well as the broadening of the electroluminescence in the sample doped in the barrier as a red shift induced by the dopant impurities. We stress here that for both samples, the optical transitions we observed in both absorption and electroluminescence follow the polarization selection rules for intersubband transitions. They are not transitions between discrete impurity states.
18
JEROMEFAISTET AL.
Since the peak gain [Eq. (lo)] is proportional to the inverse of the luminescence linewidth, an undoped active region is needed to reduce the threshold current density and to achieve laser action.
3. ANTICROSSING OF THE STATES IN
THE
ACTIVEREGION
As shown in Fig. 8, the line shape and the peak position of the luminescence of the sample with doping setback, for relatively low injected currents is strongly dependent on the injected current and therefore on the applied bias. As the current is increased, the peak blue shifts and narrows significantly. Measurement of the peak position and linewidth as a function
1.oo
0.75
b
0.50
g
a
0.25
220
n
240
260
280
300
320
Photon Energy (meV) I (mA) 10
s
20
50
100
I
I
I
200
300
30
p v
A
e
E v
295
I
B 25
c
.0 4
LL
v)
c
E
I
20
1.80
1.85
1.90
290
FIG.8. (a) Luminescence spectra of the sample with doping setback for different drive current: 7.5 mA (dashed line), 30mA (dotted line), and 150 mA (full line). (b) Peak position and FWHM of the luminescence spectrum as a function of applied bias.
19
1 QUANTUM CASCADE LASERS
of applied field gives insight into the narrowing mechanism. The electroluminescence spectrum shows a strong narrowing of the line from an FWHM of 32 meV at I = 7.5 mA to 20.5 meV at I = 150 mA as the peak position shifts from 290 to 297 meV. We interpret these observations in the following way. The reader is referred to the diagram of Fig. 5, which is qualitatively identical and of similar dimensions. Because they are very thin, the wells on both sides of the center well support only one bound state. The center well, being thicker has an excited state very close to the barrier edge, as shown experimentally in Figs. 5 and 6, in our photocurrent and absorption measurements. The position of the energy level differences E ,  E , = E,, and E ,  E l = E l , as a function of the applied field is displayed in Fig. 9. The ground states of the 2.8 and 3.5nm quantum wells anticross as they are brought into resonance at a field of about 60 kV/cm. Since the barrier between these two wells is only 3nm, both the minimum splitting energy is relatively large (E'$ = 20meV) and the two states remain very strongly coupled over a broad range of fields. The Stark shift of the E,, transition is clearly sublinear in the applied electric field. This sublinearity originates from the repulsion
40
2 (u
"9 :.
w (u
c
A .
0
5 200 A= a
100
30
mt
0
13
1 u 60 90
30
I 0
120
Electric Field (kV/cm) FIG. 9. Computed transition energies El, and El, of the 13 and 23 transitions and of the products of the square of the matrix elements of the latter with E,, and El,, which are proportional to the oscillator strengths, as a function of applied electric field for the sample with doping setback. These calculations are also valid for the L = 4.3 pm quantum cascade laser of Fig. 10.
20
JEROMEFASTET AL.
between level 3 in the 0.8nm well and the second excited state of the center 3.5nm well, close to the top of the barrier. Level 3 is clearly being “pushed” into the 3.5nm well. This sublinear dependence of the 23 transition is clearly observed experimentally in the electroluminescence data (Fig. 8, bottom). Due to this shift of the center of charge of the n = 3 wave function, the energy of this state is now less sensitive to the thickness fluctuations in the 0.8nm well, yielding a narrowing of the intersubband luminescence and therefore a higher peak material gain. Figure 9 also shows the field dependence of the oscillator strength for the two transitions. The matrix element of the 23 transition rapidly increases for fields >50kV/cm as a result of the anticrossing between the n = 3 and n = 4 states, as the latter is pushed into the 3.5nm well, thus enhancing the overlap between these states. This increase of the oscillator strength is at the expense of that of the 31 transition, which rapidly decreases, as required by the wellknown sum rule. Laser designs based on a diagonal transition with anticrossing between the upper level and an excited state close by in energy are attractive because they still lead to a longer lifetime for the upper state, compared to a vertical transition, which yields a good population inversion, while at the same time providing a narrower linewidth than a purely diagonal transition. This can be easily seen in a tightbinding picture. One easily shows that when states n = 3 and n = 4 are completely anticrossed (i.e., when the splitting between the bonding and antibonding states is minimum), the upper state lifetime is about twice the value of a single well for the same optical transition energy. This is because at the anticrossing point the wave functions of the n = 3 state is spread nearly equally between the two wells, decreasing the squared matrix element for emission by optical phonons by a factor of 2. As we shall see, this design has also the advantage of a better injection efficiency even at high temperatures.
4. BAND STRUCTURE AT THRESHOLD
The overall band structure of two periods of the active region with an applied bias corresponding to the approximate threshold field ( z 95 kvjcm) is displayed in Fig. 10. A transmission electron micrograph of the structure is shown in Fig. 11. The active region design is identical to that of the samples with undoped active regions used in the luminescence experiments discussed in the two preceding section. Lasing occurs between states n = 3 and n = 2. Parallel to the layers, these states have planewavelike energy dispersion. The corresponding energy subbands are nearly parallel (Fig. lob) because of the small nonparabolicities for wave number k lI corresponding
21
1 QUANTUM CASCADE LASERS
U
FIG. 10. (a) Conduction band energy diagram of a portion of the 25period section of the quantum cascade laser based on a diagonal transition and operating at I = 4.3 pm, corresponding to the energy difference between states 3 and 2. The dashed lines are the effective conduction band edges of the digitally graded electroninjecting regions. Electrons are injected through an AlInAs barrier into the n = 3 energy level of the active region. The wavy arrow indicates the laser transition. (b) Schematic representation of the n = 1, 2, and 3 states parallel to the layer; kll is the corresponding wave vector. The bottoms of these subbands correspond to the energy levels n = 1, 2, and 3 indicated in (a). The wavy arrows indicate that all radiative transitions originating from the electron population (shown as shaded) in the n = 3 state have essentially the same wavelength. The straight arrows represent the intersubband opticalphononscattering processes; note the fast (subpicosecond) relaxation processes with zero momentum transfer between the n = 1 and n = 2 subbands separated by one optical phonon.
22
JEROMEFAISTET AL.
GalnAs
AllnAs
wells
(nm) L
0.8 ___
3.5 ____. 2.8
Active
3.0 region 30 1
Injection region
FIG. 11. Transmission electron micrograph of a portion of the cleaved cross section of the quantum cascade laser based on a diagonal transition and operating at 2 = 4.3 pm (see Fig. 10). Three periods of the 25stage structure are shown. The superlattice period of the digitally graded region is 3 nm and the duty cycle of the AlInAs barrier layers varies from 40 to 77% top to bottom, creating a graded gap pseudoquaternary alloy.
to a small Fermi energy E , < 13 meV of the electrons injected in the n = 3 state. As a result, electrons making radiative transitions to a lower subband (for example, from n = 3 to n = 2) will all emit photons of essentially the same energy hv = E ,  E,. It is worth noticing that, at low temperatures, a narrow emission line is also expected even when the nonparabolicity is not negligible (i.e., for large values of Ef)because the electronelectron interaction condenses the intersubband absorption in a collective mode with a narrow spectrum. This effect was clearly demonstrated in absorption experiments in heavily doped InAs wells (Gauer et al., 1995) and subsequently theoretically explained (Nikonov, 1997). In a single electron picture, the joint density of states of these transitions is therefore similar to a delta function in the absence of broadening. If a population inversion is the created between these excited states, the gain spectrum will be correspondingly narrow, nearly symmetric, and much less sensitive to thermal broadening of the electron distribution, unlike the gain spectrum associated with interband transitions in diode lasers. Recent experiments have shown convincingly that, in the midinfrared, the broaden
1 QUANTUM CASCADE LASERS
23
ing of intersubband transitions is dominated by interface roughness (Campman, 1996) and therefore the broadening increases strongly for narrower wells. The coupled wells are engineered so as to provide a reduced spatial overlap between the initial and final states, n = 3 and n = 2 of the laser transition. This reduces tunneling out of the n = 3 level into the broad quasicontinuum of states, thus enabling a large enough electron population buildup in this state to achieve laser action. The resulting intersubband nonradiative scattering rate of the initial state ( r 3 )  ' , equal to sum of the phonon relaxation rate (T~~)’= 0.25 ps' computed using Eqs. (6)(8) = 0.17 psl, is relatively small. and the escape rate to the continuum (TJ’ The resulting lifetime of the upper state is z 3 = 2.5ps, calculated at the threshold electric field ( F = 95100kV/cm) at which laser action was observed in these structures. The reduced spatial overlap of states 3 and 2 also enhances the scattering time from state 3 to state 2. The calculated value at the bias field of Fig. 10 is 2 3 2 = 4.3 ps; this ensures population inversion between the two states ( ~ 3 2> zZ1) since the lower of the two ~. inelastic relaxation by empties with a relaxation time T~~ 0 . 6 ~ Strong means of optical phonons with near zero momentum transfer occurs between the strongly overlapped and closely spaced n = 2 and n = 1 subbands since their separation E , , is chosen by design to be equal to an optical phonon (34 meV). The calculations of Fig. 9, in fact, show that their splitting becomes equal to the optical phonon energy at an electric field ( F r 90 kV/cm) close to the value for which the energy diagram acquires a staircase shape and that these two states remain strongly coupled (i.e., anticrossed) with a separation slightly greater than 34 meV over a relatively broad range of fields (90120 kV/cm). This feature ensures population inversion in a range of operating fields that includes the threshold for laser action, making the design of the laser more robust. For a given active region and injector layer sequence, the threshold current density Jthdepends on the waveguide design (mode confinement factor and waveguide losses) and on the mirror losses, which are proportional to the inverse of thc cavity lcngth. Note also that in this range of fields, the oscillator strength is high, to enhance the gain for a fixed population inversion (Eq. 10) and weakly dependent on the electric field, along with E 2 , . Finally, the tunneling escape time out of the n = 1 state is extremely short (about 1ps), further facilitating the population inversion. Electron injection in the n = 3 state is achieved via a digitally graded injector region followed by a 4.5nmthick AlInAs tunnel barrier. The applied electric field flattens the average conduction band edge in the injector (the dashed horizontal lines in bold, Fig. lo), converting the overall sawtoothlike conduction band diagram of the structure at zero bias into an

24
JEROMEFAISTET AL.
energy staircase. This sawtoothtostaircase transition, which allows a rapid increase of the injected current once the correct band alignment has been achieved, was first introduced in the design of staircase avalanche photodiodes and solidstate photomultipliers (Capasso et al., 1983). In our first report (Faist et al., 1994c) we neglected size quantization in this 18.3nmthick graded region. A more careful subsequent analysis of these effects showed however that the bound states in the injector region do not play a significant role in the operation of this laser (Faist et al., 1998a). In further designs with more complex injector designs we will however take into account the quantization of the ground state of the injector. We conclude this section with the observation that laser designs based on a diagonal transition and anticrossing of the upper excited state are attractive because they provide a relatively long lifetime for the latter, which yields a good population inversion, but at the same time they provide a narrower linewidth than a purely diagonal transition. As we shall see, they are particularly useful at longer wavelengths since at smaller transition energies the electronoptical phonon intersubband scattering times become shorter due to the smaller momentum transfer.
5. WAVEGUIDE
The complete layer sequence of the laser, including the waveguide, is shown in Fig. 12; the corresponding refractive index profile and calculated mode profile is shown in Fig. 13. The waveguide must fulfill the requirements of low optical losses while maintaining at the same time a minimum thickness of grown material for a TM propagating mode, as required by the optical selection rules for intersubband transitions. The latter requirement is important to optimize the thermal transport across the device and to minimize the growth time and the number of defects. The waveguide comprises, on both sides of the 25periods active regioninjector region, two 300nmthick GaIn As guiding layers, which enhance the optical confinement by increasing the average refractive index difference between the core and cladding regions of the waveguide. The bottom cladding consists of a 500nmthick AlInAs layer grown on top of the InP substrate. The top cladding consists of a 2500nmthick AlInAs layer followed by a 670nmthick GaInAs cladding region. The purpose of this GaInAs layer is to decouple the high loss (a = 140cm') metai contactsemiconductor interface plasmon mode from the laser mode by enhancing the difference of the effective refractive indices of the two modes. In this case, we raised the refractive index of the plasmon mode. It is actually usually more efficient to reduce the latter, by heavily doping the top layer, depressing its refractive
25
1 QUANTUM CASCADE LASERS
GaInAs Sn doped
n = 2.0x 1020 cm3
20.0nm c
0
GaInAs
1.O X 101*
AlGaInAs Graded
1.ox 1018
AlInAs
5.0 x l O l 7
1500.0
AlInAs
1.5 x 1017
100o.o
670.0
30.0
stl s2
0
a 00 3 .= 05
:x p
Digitally graded
Active region
undoped
21.1
GaInAs
1.0 X 10"
300.0
AlGaIn As Digitally graded
i . 5 x 1017
AlInAs
1.5 X 1 Oi7
I
33.2
Q)
500.0
rrn,
3 c
0%
a0 Doped n+ InP substrate
7.0 x 10l8
’rn
9"
FIG. 12. Schematic cross section of the complete laser. The overall structure has a total of about 500 layers.
index. The latter approach was used to develop our socalled plasmonenhanced waveguides for longer wavelengths, and is discussed in more detail ~ ) of further in the text. By using lowdoped ( n = 15 x 1017~ m  instead heavily doped InP substrate (where n = 59 x 10l8cmP3) we were able, in later designs, to remove the bottom 500nmthick AlJnAs, shortening the growth and improving the heat dissipation. In all our waveguide designs, the transitions between the barrier and well materials are graded (either with
26
JEROME FAISTET AL.
1.o
7

0.8 
I
I
AllnAs
I
1
MQW
I
I
I
I

InP substrate 1
4.0
3.5 $ U K
Q)
3.0 .1 4l
0
a, 0.4
m
U
t
0
=
2.5
0.2 0
2
2
2.0 6
Distance (pm) FIG. 13. Calculated refractive index and mode profile for the waveguide of the I QC laser structure of Fig. 12.
= 4.3 pm
analog or digital graded regions) over a distance of about 30nm. This grading has the purpose of reducing the series resistance by forming a smooth band profile, preventing the formation of a barrier between the cladding layers. Another important feature of the waveguide design is to minimize the optical losses due to free carriers. This is obtained by reducing to its minimum value the doping level around the waveguide region while maintaining low resistivity. It is also important to prevent intersubband absorption at the laser wavelength, particularly in the injector regions where most of the electron density of each period resides. The optical loss is computed by solving the wave equation for a planar waveguide with a complex propagation constant, modeling each layer with its complex refractive index. The imaginary (loss) part of the refractive index was obtained through a Drude model (Jensen, 1985), where the dielectric constant E is
with a plasma frequency
1 QUANTUM CASCADE LASERS
27
where n, is the electron concentration, E , is the (highfrequency) dielectric constant, and m* is the electron’s effective mass. We used a scattering time z = 0.20.5 ps, depending on the doping level and wavelength (Jensen, 1985). This value of scattering time gives a good agreement with the measurement of freecarrier absorption in bulk GaAs or InP, as reported in the literature (Jensen, 1985) or measured by us. Note, however, that we systematically underestimate our waveguide losses. This discrepancy remains basically unexplained.
6. DEVICE PROCESSING The samples were processed into 10 to 1Cpmwide mesa waveguides by wet etching through the active region down to about l p m into the substrate. An insulating layer was then grown by chemical vapor deposition to provide an insulation between the contact pads and the doped InP substrate. For minimum losses, we chose SiO, for short wavelengths (A < 5 pm) and at long wavelengths (A = 11pm) and Si3N4 for all the intermediate ones. Windows are defined through the insulating layer by plasma etching, exposing the top of the mesa. For this process, Si3N, is preferred over SiO, whenever possible because it is easier to remove by plasma etching in a CF, gas. TiAu nonalloyed ohmic contacts were provided to the top layer and the substrate. A scanning electron microscope picture of the cleaved facet of a processed device is displayed in Fig. 14. The devices are then cleaved in 0.5 to 3mmlong bars, soldered to a copper holder, wirebonded, and mounted in the cold head of a temperature controlled He flow cryostat.
7. LASERCHARACTERISTICS A set of electroluminescence spectra of a 500pmlong and 1Cpmwide laser is displayed in Fig. 15 for various injected currents. The drive current consisted of 80nslong electrical pulses with a 80kHz repetition rate. The spectrum below a 600mA drive current is broad, indicative of spontaneous emission. Above a drive current of 850mA, corresponding to a threshold current of 15 kA/cm2, the signal increases abruptly by orders of magnitude, accompanied by a dramatic line narrowing. This is direct manifestation of laser action. A plot of the optical power versus drive current for various temperatures is displayed in Fig. 16 for a longer device (1 = 1.2mm). The optical power is measured by focusing the light with a fll.5 optics on a fast, calibrated, roomtemperature HgCdTe detector. The threshold current
28
JEROME FAISTET AL.
FIG. 14. Scanning electron micrograph of the cleaved facet of a processed device.
density was lowered to f,, = 5.4 kA/cm2 due to the lower mirror losses of this longer device. The currentvoltage characteristics at 10 and 100 K are displayed in the inset. As expected, no significant current flows for bias below 8 V , when the band diagram still has an overall sawtooth shape. Above this voltage, a current of several hundred milliamps flows across the device. These early devices already showed a fundamental property of intersubband QC lasers: a weak temperature dependence of the threshold current density. As shown in the inset of Fig. 16, the threshold current has the typical J exp( TIT,) temperature dependence. The value of To = 112 K is much larger than the value typical for interband lasers (To 2050 K). The weak temperature dependence of QC laser threshold can be ascribed to the following: (a) the material gain is insensitive to the thermal broadening of the electron distribution in the excited state since the two subbands of the laser transition are nearly parallel; (b) Auger intersubband recombination rates are negligible compared to the optical phonon scattering rates; (c) the variation of the excitedstate lifetime with temperature is small, being controlled by the BoseEinstein factor for optical phonons; and (d) the measured luminescence linewidth is weakly temperature dependent.


1 QUANTUM CASCADE LASERS
29
FIG. 15. Emission spectrum of the laser at various drive currents. The strong line narrowing and large increase of the optical power above I = 850mA demonstrate laser action. The spontaneous emission and the laser radiatron are polarized normal to the layers. The I = 4.26 prn emission wavelength is in excellent agreement with the calculated value.
VIII. QC Lasers with Vertical Transition and Bragg Confinement
In summary, population inversion in structures based on a diagonal transition is obtained through the combination of two design features. First, the laser transition proceeds by photonassisted tunneling; that is, it is diagonal in real space between states with reduced spatial overlap. This increases the lifetime of the upper state and also decreases the escape rate (zest)' of electrons into the continuum. Second, a third state, located approximately one phonon energy below the lower state of the lasing transition, is added. The resonant nature of the optical phonon emission between these two states reduces the lifetime of the lower one to about 0.6 ps. However, being less sensitive to interface roughness and impurity fluctuations, a laser structure based on a vertical transition (i.e., with the initial and final states centered in the same well) would exhibit a narrower gain spectrum and thus a lower threshold, provided that the resonant phononemission scheme is sufficient to obtain a population inversion and that electrons in the upper state can be prevented from escaping into the continuum.
30
JEROMEFASTET AL.
0
0.5
1.o
1.5
2.0
Current (A) FIG. 16. Measured peak optical power P from a single facet of the QC laser versus drive current at different heatsink temperatures. The temperature dependence of the threshold current is shown in one of the insets. The solid line is an exponential fit, A exp(T/T,). In the other inset, the drive current is shown as a function of applied bias at 10K (solid line) and 80 K (dashed line).
1. QUANTUM DESIGN The main challenge in designing a vertical transition QC laser is to suppress tunneling out of the n = 3 state. Sirtori et al. (1992) showed that electronic quarterwave stacks can be designed to confine an electronic state in the classical continuum. To enhance the confinement of the upper state in a structure based on vertical transitions, a straightforward idea would be to substitute the digitally graded region with a quarterwave stack. However, this design would suppress the escape from the state ( n = l) due to the formation of localized states in the quarterwave stack above the n = 1 level (Sirtori et al., 1992), preventing population inversion. Instead, we chose to keep the effective conduction band edge of the digitally graded superlattice flat under the applied field, as was done in the previous devices, while now requiring that each well and barrier pair accommodates a half electron de
1 QUANTUM CASCADE LASERS
31
Broglie wavelength, thus satisfying the Bragg reflection condition (Faist et al., 1995a). The barrier length 1, and well length 1, will, however, depart individually from a quarter wavelength. In mathematical terms, we require that the effective conduction band potential V ( x j ) of the injector at the position x j of the j t h period, be approximated by
where AE, is the conduction band discontinuity between the barrier and well material (= 0.52 eV). This creates a quasi electric field, which exactly cancels the applied field at threshold Fth:
We also have for each layer pair l,,j and lb,j the Bragg reflection condition:
where k,,j and kb,j are the wave numbers in the well and barrier materials. This condition ensures the constructive interference of the electronic waves reflected by all the periods. For our given upper state energy, this set of equations is solved iteratively for each consecutive layer pair 1, and 1, of the graded superlattice. This procedure yields successive values of 1, = 2.1, 2.1, 1.6, 1.7, 1.3, and 1.0nm and I , = 2.1, 1.9, 2.0, 2.3, and 2.7 nm, righttoleft in Fig. 17. In a solidstate picture, a region is created that has, under bias, an electronic spectrum similar to the one of a regular superlattice, with a miniband facing the lower states of the active region for efficient carrier escape from the ground state of the lasing transition and a minigap facing the upper state for efficient carrier confinement (Fig. 17a). This confinement is clearly apparent in Fig. 17b, where the calculated transmission of the superlattice is plotted versus electron energy at the field Fth= 85 kV/cm corresponding to the laser threshold. The transmission is very small at the energy E , corresponding to the upper state n = 3, while remaining sufficiently large (> 10 ') at the energy E l (Fig. 17b) to ensure a short escape time (, 8 pm) Quantum Cascade Lasers After our demonstration of the first quantum cascade laser operating at
1 = 4.3 pm, the first question that arose was whether it would be possible, by a modification of the design, to operate the QC laser in the second atmospheric window (813 pm). The answer to this question was very
1 QUANTUM CASCADE LASERS
53
important because it would determine whether the main claim of the QC laser was true: the possibility to design lasers with very different operating wavelengths from of the same semiconductor material. From the beginning, it was quite clear that the longer wavelength would actually bring some advantages to the quantum design. The states would be deeper in the band, with a lighter effective mass and thus a larger oscillator strength and wider wells meant a lower sensitivity to interface roughness. However, the waveguide losses would increase significantly [the freecarrier loss term grows approximately with R2, see Eqs. (11)(12)] and the lifetime tg limited by optical phonon emission would decrease, along with t32;because the momentum exchanged during the scattering would be decreased. This would increase the threshold current density fthor equivalently reduce the population inversion (J/q)z3(1  t2/tg2)for a given J . The problem of the optical losses was solved by refining the original waveguide design and using ~ ) substrates. The problem of the reduced lowdoped ( n 1017~ m  InP lifetime and population inversion can be alleviated using a diagonal transition with upper state anticrossing (Sirtori et al., 1995), as discussed in a previous section, which gives upper state lifetimes of a few picoseconds. Designing a structure with vertical transition at comparable wavelengths (1 8 pm) appears at first sight to be very difficult since in this case z3 < 1ps. This problem was solved by using a suitably optimized coupledquantumwell active region, in which the nonradiative relaxation by emission of optical phonons, is split into two paths (32 and 31) of comparable relaxation times of which only one (32) fills the lower state of the lasing transition. In addition, 721 can be made shorter than in a structure with diagonal transition by increasing the spatial overlap of states 2 and 1. For this reason, the ratio t 3 2 / ~ 2 1is still very favorable ( 10) even for a laser based on a vertical transition at long wavelengths. Indeed, a laser based on a vertical transition was demonstrated with a lasing wavelength as long as 11 pm (Sirtori et al., 1996a).


WAVEGUIDE 1. PLASMONENHANCED As already mentioned, the problem was to design a waveguide that would provide at the same time low waveguide losses and keep the thickness of the grown MBE material to a reasonable value. In particular, it was realized that growing on a lowdoped ( n 1017cmW3)InP substrate instead of the doped substrates available at that time, which had a doping level in the high 10l8cm3 would bring an enormous advantage. A highly doped substrate would necessitate the growth of a lower AlInAs cladding about 2 pm thick.

54
JEROMEFAETET AL.
This layer would increase the growth time, add to the thermal resistance, and dramatically increase the number of defects below the active regmn. However, since these low doped substrates were commercially available only by mid1995, the first samples were grown on samples that had been grown inhouse during a previous study. As for the longer wavelengths, it was imperative to decouple the guided mode from the interface plasmon propagating at the interface between the top metal contact and the semiconductor, which is also TM polarized. However, to limit the thickness of the top cladding layer, it was found that it was much more efficient to decrease the refractive index of the top layer, instead of increasing it as we did in the first QC laser. At these long wavelengths, a lowrefractiveindex layer can easily be obtained using the dispersion provided by a heavily doped semiconductor layer. The schematic of a waveguide designed using these design rules is shown in Fig. 35 (Sirtori et al., 1995). As shown in this figure, the growth finishes with a 600nmthick GaInAs layer Sidoped to n = 7 x lo1*cm3. The calculated plasma frequency for such a doping level corresponds to a wavelength A = 9.4 pm. The anomalous dielectric dispersion near the plasma frequency (Jensen, 1985) strongly depresses the refractive index of the layer at the laser wavelength of A = 8.4pm from n = 3.5 refractive index of an undoped layer, to n = 1.1, as shown in Fig. 35b, where the calculated optical mode is displayed along with the refractive index profile. This low refractive index has a twofold advantage: it completely decouples the guided mode from the interface plasmon and increases the confinement factor of the mode by “pushing” the mode away from the metal contact. The calculated waveguide loss for such a waveguide is only a, = 7.8 cm’. Waveguide loss measurements performed with the HakkiPauli technique in similar waveguide at such wavelengths show, however, much higher losses (a, = 30 cm ’) (Sirtori et al., 1996b). The origin of this discrepancy is still not known. Part of this discrepancy could arise from material and substrate quality since more recent results tend to show somewhat lower losses (a, = 24 cm ’) (Gmachl et al., 1998a). This waveguide design was kept in all subsequent QC lasers. The only significant modification was to increase the thickness of the confining InGaAs layer adjacent to the active region with a concomitant decrease of the AlInAs top cladding layer. Even if this modification does not lead to an improvement of the calculated waveguide loss, it is believed to improve the device performance since, in general, InGaAs has a much better material quality (lower defect concentration) than AlInAs and therefore can be doped much more lightly. This modification was especially important for the realization of the QC laser based on GaAs material because AlGaAs cannot be doped below 25 x loi7 cmP3 without encountering carrier freezeout problems (Sirtori et al., 1998a).

55
1 QUANTUM CASCADE LASERS n++ n
GalnAs
n = 7 x 1018 ~ r n  600 ~ nrn
AlGalnAs Graded
5 x 1017
n
AllnAs
5 x 1017
20
n
AllnAs
3x1Ol7
1200
n
AllnAs
1.2 x 1017
1200
n
AllnAs
1 x 1018
n
AlGalnAs Graded
2 x 1017
n
GalnAs
6 x 1016
n
Injector
undoped
 (a)
30 al
.’Do)
.r
> m
2.p
10 40
500
1 . 5 ~ 1 0 ~ ~ 19.6
Active region
3
m'D a0
T 2
7
0 0
27.3
n
GalnAs
6 x 1016
700
n
AlGalnAs Digitally graded
1.2 x 1017
25
i
h
cn
c
X
3 *
c 
1
FIG. 35. (a) Schematic cross section of the quantum cascade laser operating at I = 8.4pm. (b) Refractive index profile of the waveguide structure. The calculated profile of the fundamental mode is also shown.
56
JEROMEFAIST ET AL.
2. QUANTUM DESIGNOF A QC LASERWITH DIAGONAL TRANSITION AT II = 8.4 pm To a large extent, the first active region design (Sirtori et a!., 1995) followed the same design philosophy of the first QC laser at II = 4.3 pm. As shown in Fig. 36b, the structure is based on an periodic repetition of a doped injector followed by a threewell active region. At the threshold current, the structure is designed in such a way that the upper state of the lasing transition (state n = 7 in Fig. 36b) is anticrossed with the next upper state ( n = 8 in Fig. 36b). In addition, we noted that a structure based on an upper state, which is anticrossed, results in a narrower luminescence linewidth. This is confirmed by the data of Fig. 36a, which show a FWHM of 10 meV. To ensure population inversion the wave functions and energy levels are designed so that the electron intersubband scattering time from the higher ( n = 7) to the lower ( n = 5) state of the laser transition (z7, = 3.4 ps), which is dominated by optical phonon emission, is longer than the scattering time between the n = 5 and n = 4 energy levels. The latter is estimated to be d 1 ps due to the near resonance of the energy difference E,, with the typical optical phonon energies ( 30 meV) in the AlInAsGaInAs alloys. The fact that the n = 7 and n = 5 states are not centered in the same well (“diagonal” transition) enhances the relaxation time z7, and the nonradiative lifetime (z7 = 2.7 ps) of the upper state, which increases the population inversion at given injection current density. The calculated matrix element for the 75 radiative transition is 1.7 nm. As shown in Fig. 37, these first devices exhibited already a low threshold current density of 2.1 kA/cm2 at T = 10 K and operation up to T = 130 K in pulsed mode. This low value of threshold current at low temperature allowed continuous wave operation in these devices (Sirtori et al., 1998b). However, the poor substrate quality prevented the manufacture of a sufficient number of devices. Good cw operation was however realized with qc lasers based on a vertical transition (Sirtori et al., 1996b).


QUANTUM CASCADE LASERBASEDON 3. LONGWAVELENGTH VERTICAL TRANSITION
A
The improvement in performance obtained with the lasers based on a vertical transition at shorter wavelengths stimulated the development of similar structures at longer wavelength. The optimized design is shown in Fig. 38 (Sirtori et at., 1997a). Compared to an earlier vertical transition version (Sirtori et al., 1996b), it has a significant improvement: the strong
1

QUANTUM
57
CASCADE LASERS
(4 1.0
.
I
c
2
Y
5 0.5
2 
.8 c
0
50
100 150 200 Photon Energy (rnev)
250
n r
1'
FIG. 36. (a) Electroluminescence spectra of a quantum cascade structure with diagonal transition at L = 8.4 pm and five activeplusinjector regions with no cladding layers, and processed into a mesa, to avoid gain effects. The drive current is 150mA corresponding to a bias electric field across the active region of 6 x 104V/cm. The position of the (75) peak agrees with the calculated separation between the n = 7 and n = 5 energy levels ( E 7 5 = 145.4meV). The shift between the two peaks in the spectrum at 200 K is in excellent agreement with the calculated energy difference between the n = 8 and n = 7 anticrossing states ( E 8 , = 41.7meV). (b) Conduction band diagram of a portion of the Ga,~,,In,,,,AsAlo,481no,5zAsquantum cascade laser with diagonal transition at under positive bias corresponding to an electric field of 6.0 x 104V/cm. The design is identical to that of the electroluminescence sample active regions and injectors. The solid curves are the moduli in the active region and in the injector. The baselines of the squared of the wave functions /$Iz indicate the positions of the energy levels. The states of the laser transition are labeled 7 and 5. Electrons in the ground state 1' of one of the injectors tunnel into state 7 of the adjacent active region through the 4.5nmthick AlInAs barrier. The 25 active regions each comprise three GaInAs quantum wells of thicknesses 3.5, 7.5, and 5.8 nm, from right to left, separated by 2nmthick AlInAs barriers. Electrons tunneling into the next injector through a 2nmthick AlInAs barrier encounter in sequence 4.8, 4.2, 4.0, and 4.3nm GaInAs quantum wells, separated by 0.5, 0.8, and 1nm AlInAs barriers, respectively. The calculated energy separations are El, = 36.8 meV, E , , = 52.6 meV, E,, = 25.5 meV, E45 = 29.9 meV, E , , = 21.8 meV, E,, = 123.6meV, and E , , = 41.7 meV.
58
JEROME FAIST ET AL.
Current (A) FIG. 37. Measured peak optical power as a function of injected drive current for various temperatures for the QC laser of Fig. 36b operating at 1, = 8.4pm.
reduction of thickness of the barrier coupling the two quantum wells of the active region from 1.5 to 1.0 nm. This modification leads to a much stronger coupling of the lower n = 1 and n = 2 states of the active region, maximizing their spatial overlap (Fig. 38) and leading to a further reduction in zzl (=0.2ps) compared with a diagonal transition structure ( T ~ 1 ~ps) or vertical one with thicker barrier (zzl = 0.4 ps, as in the work of Sirtori et al., 1996b). It also enhances T~~ to a value approximately double ( 1 . 4 ~ s )the lifetime z3( = 0.8 ps) by splitting the nonradiative channel from the upper level of the lasing transition n = 3 into paths of comparable strength to the n = 2 and n = 1 states without changing significantly the n = 3 state lifetime. This design strategy maximizes the population inversion An for the vertical transition design although An is still smaller than that achieved in a diagonal transition structure (by a factor s 2 compared with the QC laser of Fig. 36) for the same current density. This factor, however, can be more than compensated by the enhancement of the matrix element of the vertical transition ( z =~ 2.6 ~ nm in Fig. 38 compared to z75 = 1.7 nm in Fig. 36). The net effect is that the threshold current densities [Jth= (aw + ol,)/gT), for the same total loss (waveguide and mirrors) to confinement factor (r) ratio, should be comparable in the designs of Figs. 36 and 38. This is indeed confirmed by experiments, which give 2kA/cm2 for the threshold at

1 QUANTUM CASCADE LASERS
59
F = 68kVlcm
lniector Active Region
h
r
..
..
.
FIG. 38. Schematic conduction band diagram of a portion of the Ga,,,,In,,,,As1 = 8.4pm quantum cascade laser based on a vertical transition under an A10,481no,52As applied positive bias corresponding to an electric field of 6.2 x 104V/cm. The solid curves represent the moduli squared of the relevant wave functions in the active region. In the injector, the dashed lines indicate the positions of the energy levels. The latter are expected to be broadened by impurity scattering and interface roughness. The states of the laser transition are labeled 3 and 2. The calculated energy level differences are E,, = 145 meV and E,, = 41 meV. Each of the 25 periods comprises the following layer sequence (AIInAsGaInAs), in nanometers, left to right and starting from the injection barrier: (4.5KO), (1.05.7), (2.44.4), (1.43.6), (1.23.6), (1.23.4), and (1.03.4). The structure is left undoped with the exception of the 4 layers in the center of the injector C(1.23.6), (1.23.4)] region, which are ntype doped with Si to obtain N , = 2 x lo', C I K ~ The . waveguide design is based on the plasmon enhanced confinement previously discussed. The lower cladding layer is the InP substrate.
cryogenic temperatures in the two structures (Sirtori et al., 1995; Sirtori et al., 1997a). It is worth noting that in coupledwell active regions of the type shown in Fig. 38, the state n = 2 has a significant oscillator strength only to the states n = 1 and n = 3. On the other hand, we know that in an isolated well, the oscillator strength of the 12 transition fij" is almost equal to the maximum oscillator strength available from the n = 1 state A,, = Xifli. Keeping in mind that f12 = fil, we conclude immediately than compared to the single well case, the oscillator strength of the 32 transition is increased by an amount approximately equal to the oscillator strength
60
JEROMEFAISTET AL.
between states 21 (i.e., f 2 3 2 ff’j” + f i 2 ) . The reduction of the coupling barrier thickness in a vertical transition structure will increase this oscillator strength, decreasing the threshold current density. One could describe the superlattice active region QC laser (Scamarcio et al., 1997) as an extreme case where one adds not one but many levels below the lower state of the laser transition, increasing even further the oscillator strength of the laser transition. Another modification in the design shown in Fig. 38 to previous vertical designs is the reduction of the injection barrier thickness. This increase in the coupling of the n = 3 state to the injector follows the same philosophy than already described in the paragraph dedicated to the description of the room temperature design at 2 = 5 pm. The advantages brought by the socalled “strongly coupled injector” are discussed in more detail in (Sirtori et al., 1998a). This analysis, confirmed by experiments, shows that in an injector design optimized for high peak power in the LI curve, the maximum possible resonant tunneling current density is J,,, E qn,/2z3 (i.e., the current is limited by the injector doping and the lifetime of the upper state). In this design, the tunnel splitting between the ground state of the injector and the upper state of the laser transition is large enough that they remain in resonance for a large range of drive currents. At the same time, the splitting cannot be too large, otherwise the laser transition would lose oscillator strength to the transition between the ground state of the injector and the lower state of the laser transition. The best compromise is a splitting of the order of half the luminescence width of the laser transition (i.e., E 5 meV for the design of Fig. 38). From the preceding discussion it should be clear that the vertical transition design has some key advantages over the diagonal one, for the same wavelength, losses, and cavity design. First the slope efficiency is higher 2 Second because [see Eq. (20)] of the significantly smaller ratio ~ 2 1 / ~ 3ratio. they have a higher power, for the same doping sheet density in the injector, since they can be driven to higher currents without the injector going out of resonance with the active region, since the lifetime of the upper state of the laser transition is considerably shorter. If the structure is designed correctly, vertical transition structures have a threshold current density comparable to that of diagonal transition designs. These longwavelength lasers based on a vertical transitions with an optimized design have demonstrated a very high level of performance. Shown in Fig. 39 are the optical power versus injected current of such devices at T = 300 K. The lasing threshold is below 3 A, corresponding to a threshold current density of 8 kA/cm2, the same order of magnitude than obtained in the shorter wavelength lasers. The devices operated still at T = 320 K.
1 QUANTUM CASCADE LASERS
61
Current (A) FIG. 39. Measured peak optical power from a single facet versus drive current at 300 K for two vertical transition QC lasers with different lengths: 1.8mm (a) and 2.5mm (b). Insets: highresolution spectra at 3.0 A (top) and 3.6 A (bottom). The active region design is shown in Fig. 38.
Lasers operating at longer wavelengths (A = 11 pm) were made following basically the same design as the one shown in Fig. 38 (Sirtori et al., 1996a). The larger losses at this longer wavelength, however, did allow operation only up to a maximum temperature of 220K in pulsed operation. Figure 40 shows the operation of such a laser in continuous wave operation at T = 20 K. Optical output power up to 10 mW were obtained. These results demonstrated that a laser based on a vertical transition could be operated with good performances down to wavelengths longer than 11 pm. 4. ROOMTEMPERATURE LONGWAVELENGTH (A = 11 pm) QC LASER
A few design changes were then made to increase the maximum operating temperature of QC lasers with wavelengths between 11 and 12 pm (Faist et
62
JEROME FASTET AL.
6 A
2
F E
4 
v
4.5 &

nB
Q)
3 : 0
>
2r
0’ 0
10.8
10.5
11.1
11.4
11.7
Wavelength (w) ‘
I
I
0.2
0.4
I
0.6 Current (A)
I
0.8
1
FIG. 40. Continuous optical power versus injected current for a device with vertical transition design operating at 1 = 11pm wavelength. The design used is very similar to the 1 = 8.4 pm device shown in Fig. 38.
al., 1998~).As shown in Fig. 41, we chose to use an active region consisting of three coupled quantum wells. For a longwavelength laser, this choice has two main advantages: it allows a longer lifetime of the upper level of the lasing transition (n = 3) by reducing its overlap with the lowest states n = 1 + z;i)’ = 1.3 ps and n = 2. The resulting lifetime of level 3 is z3 = ;7;( (with T~~ = 2 ps). Even if this is accompanied by a concomitant decrease in optical matrix element z 3 2 = 2.4nm (instead for z 3 2= 3.1 nm for lasers based on vertical transitions; Sirtori et al., 1996a) threshold current densities and slope efficiencies benefit from a more favorable population inversion between states 3 and 2 since the ratio of lifetimes is T 3 2 / T 2 = 5, instead of z32/z2 = 3.4 for the vertical transition design (note that t2 E zZ1 0 . 4 ~ ~ ) . In addition, this geometry will improve the injection efficiency by reducing the overlap between the ground state wave function of the injectionrelaxation region and the states 1 and 2 of the active region while maintaining a good injection efficiency into state 3, similar to the 5.2pm roomtemperature laser with a funnel injector. As in all our previous designs, the injectionrelaxation region is designed with a minigap facing level 3 to prevent escape from the latter into the injector. Moreover, the width of the lower miniband decreases toward the injection barrier to “funnel” electrons into the ground state g of the relaxationinjection region.
63
1 QUANTUM CASCADE LASERS
injection barrier
U ‘ (b)
D2286
In,&a,,As GaamAs
InP
5x10’’
1100
7 ~ 1 0 ’ ~1100
InP
8x10”
2000
6x10’’
800
6x10‘’
35 stage active In,&a,,,As
02297
10
IxlOM
Ino~szAlo,,As 1.5~10” In, &anA,As
(C)
1Nlo
800 1676
6x10’’
700
In, &a,,
A7
As
stage active +35iniector @n In,&aOA7As
1675 6x10’’
700
InP substrate
InP substrate
FIG. 41. (a) Schematic conduction band diagram of one stage (active plus injector region) of the room temperature 1 = 11 pm QC laser structure under an applied electric field of 4.6 x 104V/cm. The energy separation between states 3 and 2 corresponds to I = 11 pm and that between level 2 and 1 is slightly larger than one optical phonon. The layer sequence of one period of structure, in nanometers, left to right and starting from the injection barrier where Ino,52Alo,48As is 4.23.40.86.80.96.22.84.21.04.01.0~3.91.03.91.13.9, layers are in bold, In,,,,Ga,,,,As layers are in roman, and underlined numbers correspond to doped layer with Si to N , = 2.5 x 10” ~ m  (b) ~ ,and (c) Layer sequences of the waveguide. The layer thicknesses are in nanometers. The dashed line indicates the interface where the growth was interrupted.
To maximize the gain, the active region consisted of 35 stages. As shown in Fig. 41, two different optical waveguides were grown. They consists of the InP substrate on one side and of AlInAs (sample D2286) or InP (sample D2297) grown by solidsource MBE. As discussed earlier, the latter has a lower thermal resistance than the random alloy, enabling larger powers. Figure 42 shows the optical power versus drive current from a single facet of the InPclad sample D2297 obtained using f/O.8 optics and a calibrated,
64
JEROMEFAISTET AL.
2
0
6
4
8
10
Current (A)
10
5 7
2
1
t 0
* '
+ 0
1
D2286 L = 1.5 mm D2286 L=0.75mm D2297 L=2.25mm
100 200 Temperature (K)
300
FIG.42. (a) Collected pulsed optical power from a single facet versus injection current for heat sink temperatures of T = 275, 300, and 320K and for the InPclad sample D2197 (28 pm x 2.25 mm). The collection efficiency is estimated to be 50%. (b) Threshold current density in pulsed operation for samples D2286 (ternary cladding) and D2297 (binary cladding) as a function of temperature. The solid line is the theoretical prediction for sample D2297. Cavity lengths are also indicated.
roomtemperature HgCdTe detector. Maximum peak output powers of about 55 mW at 300 K and 35 mW at 320 K are obtained. In comparison, at T = 300 K, sample D2286 (AlInAs cladding) exhibits very similar thresholds but about onetenth of the power levels and slope efficiencies. We attribute this difference to the enhanced heat removal during the pulse by the top InP cladding as compared to the AlInAs cladding. In Fig. 42b, the
65
1 QUANTUM CASCADE LASERS
threshold current density Jthis plotted as a function of temperature for both samples D2297 and D2286. At T = 300 K, the threshold current densities are very similar for both samples and are about 810kA/cm2. Between 160 and 320K, the data can be described by the usual exponential behavior J exp(T/To) with an average To = 172 K. This value is similar to that reported ( T = 180 K) by an other group working on intersubband QC lasers (Slivken et al., 1997) at a somewhat lower temperature range T = 200280 K and is much larger than that obtained for the competing technologies (leadsalts or 111V). The gain was obtained from an HakkiPaoli analysis of the subthreshold luminescence spectrum, yielding a value of the waveguide loss of IX, = 74 cm'. This number allows a computation of the threshold current density Jth [see Eq. (19)], plotted along the experimental points in Fig. 42. The good agreement between calculated and measured hightemperature behavior shows that in these devices, our simple model, which does not include hot carrier effect (Gelmont et al., 1996), nevertheless predicts the correct thresholds. Physically, it means that the electron density is low enough for the carriercarrier scattering to be ineffective in heating the electron population of the injector. It is maybe no surprise that we observe such a good agreement in a device where the voltage drop, and therefore the energy loss per period is the smallest due to the relatively small photon energy.

5.
SEMICONDUCTOR LASERS BASEDON SURFACE PLASMON
WAVEGUIDES
So far, for all the waveguides used for the QC lasers, we tried to avoid a coupling of the guided mode with the interface plasmon propagating at the semiconductormetal interface. A radically different approach consists, on the contrary, of using this interface plasmon as the main guiding mechanism (Sirtori et al., 1998~).The main advantage of this approach is the complete suppression of the cladding layers. The losses are controlled by the correct choice of the metal. For electromagnetic surface waves at a metaldielectric interface, the modal losses are strongly dependent on the dielectric constants of the materials. The attenuation coefficient a can be written as
where n and k are the real and imaginary parts of the complex refractive index of the metal, nd is the refractive index of the dielectric, and A is the
66
JEROME FAST ET AL.
wavelength in vacuum. In the previous equation, k is assumed to be much larger than n and nd. From this expression, it is apparent that the losses at the interface are inversely proportional to /2. and can be minimized by choosing metals having a refractive index with a strong imaginary component ( k >> n, i.e., the dielectric constant is almost real and strongly negative). Around the 10pm wavelength, palladium is the most suitable material for our application since it has very large k and can be easily deposited. Its complex refractive index at the 11.3pm wavelength, corresponding to the emission frequency of one of our laser structure, is ii = 3.85 + i49.2. Introducing in Eq. (25) the n and k values for Pd and the value of the refractive index of the active region (nd = 3.38) we obtain CI = 14cm'; using the complex refractive index of gold we would obtain CI 30cm'. In fact, our waveguide structure is more complicated than a pure surface waveguide and the simple Eq. (25) offers only qualitatively results, which are nevertheless important for the choice of the metal to use at the interface. A significant difference is caused by the InP substrate, which has a lower refractive index than the laser active region. This strongly reduces the penetration of the mode into the semiconductor material and enhances the overlap factor with the active region. To obtain a more accurate prediction of the shape and the losses of the confined TM interface mode a numerical solution based on the transfer matrix method has been used. The results of the mode calculations are presented in Fig. 43. It is important to note that in these waveguides, in spite of the strong increase of the optical losses, there is an enhancement of the overlap factor (r E 70%) between the guided mode and the active material over a regular dielectric waveguide (r z 40%) with the same thickness of waveguide core. The laser structure indicated in Fig. 43a corresponds to that of sample D2295 and consists of a 700nm GaInAs buffer layer grown on a lowdoped InP substrate, followed by a 25period QC active material. The active material is identical to that presented in the preceding paragraphs (samples D2286 and D2297). The top contact is a 10nmthick GaInAs, highly doped layer (1 x lo2' crn')). The lasers are fabricated as 20pmwide, deepetched ridge waveguides. Si,N, is used for electrical insulation at the sidewalls of the ridges. For sample D2295 (Fig. 43b), at the center of the top contact, into a 10pmwide window Pd is deposited directly onto the semiconductor. The Pd does not cover the whole top surface but leaves two narrow stripes (2 pm) on the sides, which are filled with TiAu (300&3000A) during the final deposition for the nonalloyed electrical top contact. This procedure is necessary, since Pd did not prove to be a suitable contact material. It is important to note that the etching of the stripe is not a necessary step for laser fabrication, we chose to do it only to avoid spreading of the current in


67
1 QUANTUM CASCADE LASERS
(4 n = 3.31 a = 57 ern’ r = 70%
InP Substrate
I
2
2
0
4
6
Distance ( prn)
5
Substrate
*
w=1420vm FIG. 43. (a) Mode profile of the waveguide in the direction perpendicular to the layers for laser D2295. Given are the calculated values of the effective refractive index (n), total internal waveguide loss (aW),and the overlap factor (r).The complex refractive index of palladium at 1, 11.2 pm is ii = 3.85 + i49.2. (b) Schematic of the device structure. Note the palladium layer encapsulated between the gold and the top surface of the laser.
the lateral direction. In our waveguides, the optical confinement occurs where the metal is deposited, consequently the whole laser fabrication could be reduced, in principle, to a simple metal stripe deposition on an unprocessed waver. Figure 44 reports the light output (at different temperatures) and voltage versus current characteristics of a laser processed from sample D2295. The threshold current density is Jth 11 kA/cm2 at 10 K and lasing action has been observed up to 110 K. The maximum output power is 1 mW. The threshold current density is about 3.3 times higher than the lasers with the


JEROMEFAISTET
68
AL.
Wavelength (vm)
Current (A) FIG.44. Applied bias and measured peak output power from a single facet as a function of injected current for a 20pmwide and 0.8mmlong laser at three different heat sink temperatures. Inset: Highresolution laser spectrum of the device at 30 K. The maximum peak output power is estimated to be several milliwatts.
same active materials and dielectric waveguides. From this comparison and the ratio between the overlap factors, we can derive a rough estimation of 100 to 150cm'. the waveguide losses a,

6. DISTRIBUTED FEEDBACK QUANTUM CASCADE LASERS In the preceding subsection, we described quantum cascade lasers operating in pulsed mode with high optical powers at and above room temperature. Under pulsed operation, these lasers exhibited a relatively broadband (1020 cm I), multimode operation, as expected for devices based on FabryPtrot cavities. On the other hand, applications such as remote chemical sensing and pollution monitoring require a tunable source with a narrow linewidth. For most sensor applications, the linewidth must be narrower than the pressurebroadened linewidth of gases at room temperature, which is about one wave number, and the source must be tunable over a few wave numbers. By incorporating a grating in a QC laser structure, we have demonstrated a tunable distributed feedback (DFB) laser
69
1 QUANTUM CASCADE LASERS
operating at and above room temperature with a linewidth consistent with . these applications. a.
LossCoupled QCDFB Lasers
The active region of our QC lasers were optimized for hightemperature operation and included a threewell active region with vertical transition, a funnel injector, and low doping. Two samples were designed, D2189 and D2195 for operation at 5.4 and 8pm wavelengths, respectively (Faist et al., 1997). Only the results from the 5.4pm laser are discussed in this chapter; results on the A = 8 pm laser can be found in the original publication (Faist et al., 1997). The active region of sample D2189 comprises 29 periods and included a MBEgrown InP top cladding for optimal heat dissipation. The layer sequence is shown in Fig. 45. The energy band diagram of D2189 is shown in Fig. 27. In DFB lasers, the coupling constant K quantifies the amount of coupling between the forward and backward waves traveling in the cavity. In the coupledmode theory of DFB lasers (Kogelnik and Shank, 1972), K is written as
where n, is the amplitude of the periodic modulation of the real part of the effective index (neff) of the mode, induced by the grating of periodicity A. The corresponding modulation of the absorption coefficient has an amplitude cxl. The wavelength AB is determined by the Bragg reflection condition 1, = 2neffA(firstorder grating). For optimum performance in slope eficiency and threshold current, the product K L ~ ~where ,, L,,, is the cavity length, must be kept close to unity. For good roomtemperature perform23 mm so the coupling constant should ance in our QC structures, L,,, for an be designed with a value / K / 5 cm I , corresponding to y1, indexcoupled or a, 2.5 cm for a losscoupled structure. To this end, both devices incorporated a plasmonenhanced waveguide designed for an optimal coupling with a corrugated grating on the surface of the device (Fig. 43). The amplitude of the confined mode at the grating surface is about 0.51% of its maximum value in the center of the guide. The firstorder grating of periodicity A = &/(2neff)( = 850 nm) for the A = 5.4 pm and 1250 nm for the A = 8 pm devices) was exposed by contact photolithography and subsequently etched to a depth of 250nm by wet chemical etching as the first processing step. In the grating grooves, the thickness of
 '

70
JEROMEFASTET AL.
the heavily doped plasmonconfining layer is reduced and the guided mode interacts more strongly with the metal contact, therefore locally increasing the loss. We expect therefore the coupling constant to be complex (i.e., to exhibit both a real and imaginary parts). Our estimate for both samples is n, 5 x and a, N 0.52 cm '. Complexcoupled DFB lasers exhibit, in general, better singlemode yield because the loss component lifts the degeneracy between the two modes on each side of the stopband (Kogelnik and Shank, 1972). In addition, since the loss component originates from the metal and is therefore offresonant, we do not expect any saturation behavior with increasing optical intensity. A scanning electron micrograph of the cleaved facet of a processed device is shown in Fig. 45a. Control samples without grating exhibited performances comparable to the results shown in Fig. 29 with 300 K threshold current densities of 79 kA/cm2 and optical powers above 100 mW for the sample 5.4 pm. Therefore the reduction of the cladding D2189 operating at thickness did not decrease the performances in an appreciable manner. Figure 46 shows the LI characteristic two 3mmlong DFB laser from sample D2189. Laser A exhibits optical powers of about 12mW with a slope efficiency dP/dI = 10 mW/A at T = 300 K. At the same temperature, laser B exhibits much larger powers up to 60mW, with a slope efficiency dP/dI = 44 mW/A. The laser spectrum is single mode over a temperature range from 80 to 315K for laser A. As shown on Fig. 46b, in this temperature interval, the device tunes continuously from A = 5.31 to 5.38 pm. Laser B exhibits singlemode operation from 324 to about 260 K, where a second mode appears. From our data, the presence of the DFB grating seems to reduce the slope efficiency by a factor of 310 as compared to FabryPQot devices, but leaves the threshold current density essentially unchanged. We attribute this reduction in slope efficiency to an understimate of K, implying that our devices operate in the regime of S L > 1. Assuming a simple rateequation model with a constant injected current density, the (total) differential quantum efficiency for one period of the active region is N

where G,, is the total threshold modal gain, a, is the waveguide loss, and I(z) is the (average) intensity along the DFB cavity of length L, normalized such that (1/JL) [L'& f ( z ) dz = 1. Equation (27) predicts a strong reduction of slope efficiency compared to the FabryPkrot case when kL >> 1.
1 QUANTUM CASCADE LASERS
71
(4
02189 n = 7x101scm” 9OOnm
FIG,45. (a) Scanning electron micrograph of a cleaved facet end of loss coupled DFBQC laser. (b) Layer structure of the waveguide of the sample D2189. The dashed line indicates the interface where the growth was interrupted. The layer sequence of sample D2189 is described in Fig. 27.
At T = 300 K, our lasers dissipate too much power (  80 kW/cm2) to be operated in coutinuous wave. This large amount of thermal power heats the device during the electrical pulse, causing its emission wavelength to drift. For this reason, the spectra of these devices was found to be very narrow, limited by our spectrometer’s resolution (0.125 cml) for very short pulses (51011s) just above threshold and increased with pulse length up to 1 cm’ for a 100nslong pulse. To quantify this effect and check that the broadening was caused only by the thermal drift, we performed timeresolved spectra of our lasers. To this end, laser B of sample D2189 is excited with a 100nslong electrical pulse with a current I = 3.14A. The peak optical
72
JEROME FAIST ET AL.
FIG. 46. (Upper) Collected pulsed optical power from a single facet versus injection current for heat sink temperatures for two primarily DFB lasers from sample D2189. (Lower) Tuning range of the same two lasers.

power at this current is P 50mW. The laser emission is detected with a roomtemperature MCT detector with a subnanosecond time constant. The signal is sampled by a boxcar integrator, the output of which is fed back into the FTIR spectrometer. The resulting spectra taken at a 10ns interval with a 3nslong gate are shown in Fig. 47. From these spectra, it is clear that the laser keeps a narrow emission line, which drifts with time a t a rate of 0.03 cm'/ns. From this and the value of the tuning rate shown in Fig. 47, we extrapolate a temperature increase of 20K during a 100nslong pulse.
73
1 QUANTUM CASCADE LASERS
Wavelength(pm)
5.370
1862
5.365
1864
5.360
1866
5.355
1868
Wavenumber (cm I) FIG. 47. Timeresolved spectra of laser B from sample D2189 at T = 300 K. The length of the electrical pulse is 100 ns. The spectra are taken 10 ns apart with a 3nslong gate. The peak optical power is 50 mW.
The rate of wavelength shift is proportional to the power dissipated in the active region and therefore will decrease with operating point and threshold current density.
b. IndexCoupled DFB Lasers Losscoupled DFB lasers with an etched grating on the surface have some very advantageous features: because they are losscoupled, they are much more likely to be monomode regardless of the position of the cleaved facet. Most importantly, their manufacture is very simple and requires only one growth step at the beginning. The grating periodicity can be adjusted u posteriori by trial and error on the grown layer. Although this grating strongly influences the QC through its plasmonenhanced waveguide, it is located away (>2 ym) from the active region and the region of maximum intensity of the laser mode. Strong coupling is attainable only with a
74
JEROMEFAIST ET AL.
reduced uppercladdinglayer thickness. This increases the waveguide loss through the absorbing top metal layer and in turn decreases the performance of the DFB laser. A still higher level of performance can be reached using a grating etched directly inside the active region because it allows strong coupling with negligible additional loss. The scanning electron microscope (SEM) cross section of the structure is shown in Fig. 48 (Gmachl et al., 1997). In the first MBE step, the QC active regions section is grown embedded between two InGaAs layers. The upper InGaAs layer serves as the host region for the firstorder grating. The latter is transferred by contact lithography and wet chemical etching. In the second MBE step, InP is epitaxially grown directly on top of the grating. The grating strength is controlled by the grating depth and duty cycle during grating fabrication (etching) and the reflow of material in the regrowth process. It has an approximately trapezoidal shape with a duty cycle of 3050%. The tuning curve of two such devices with an active region designed for the 8.5mm wavelength is displayed in Fig. 49 (Gamchl et al., 1998b). Continuously tunable singlemode operation is achieved from 8.38 to 8.61 pm (AAem = 230 nm) in the same wafer using two different grating periods: A, = 1.35pm (8.38 to 8.49 pm) and A, = 1.375 pm (8.47 to 8.61 pm). The tuning is achieved by changing the heatsink temperature
FIG.48. Scanning electron micrograph of a cross section of an indexcoupled grating QCDFB laser designed for a wavelength near 8 pm. The grating (period A = 1.350pm) is clearly visible just above the active region.
75
1 QUANTUM CASCADE LASERS
F Y
0
8.5
8.6
WAVELENGTH
8.7
m]
TEMPERATURE [Kj FIG. 49. Singlemode tuning characteristics of two lasers with two different grating periods A and operating at I. 8.5 pm. The squares and circles are data obtained under pulsed operation, the triangles under cw operation. The laser with A = 1350 nm displays lasing on FabryPerot modes below a temperature of 80 K. This is due to the detuning of the peak gain with respect to the Bragg resonance. This allows FabryPerot modes, which are lasing at the peak gain, to reach laser threshold under high pumping conditions. The data shown for A = 1350nm are from a shallow etched device (see inset of Fig. 50). Inset: Singlemode spectrum of a laser operating in pulsed mode at room temperature. A sidemode suppression ratio of better than 30 dB is obtained.

between 20 and 320 K mainly through the temperature dependence of the material refractive indices (thermal red shift). Above ~ 2 0 K0 (average) linear tuning coefficient of +0.58 nm/K is obtained. The thermal tuning coefficient of the peak gain has been measured from the subthreshold luminescence spectra as z 1.2 nm/K. With regard to this relative detuning of the peak gain wavelength and the Bragg resonance, the latter has been designed (by choice of the grating period) such that their overlap improves in the hightemperature operating range. We attribute the fact that the device can be operated with such a wide temperature range while maintaining a dynamical singlemode operation with a sidemode suppression ratio larger than 30dB to the stronger coupling provided by the index grating. The index modulation is estimated as the difference between the modal refractive indices of the “undisturbed” waveguides at the location of the grating grooves (InP) and plateaus (InGaAs). The deviation of the grating
76
JEROME FAIST ET AL.
shape from a sinusoidal shape as well as a duty cycle other than 50% reduces the modulation amplitude of the grating. A correction factor of 0.8 (estimated for a trapezoidal grating with 50% duty cycle) finally results in An = 1.79 x From this value, and using Eq. (26), we obtain a coupling coefficient of K~~ % 33 cm A clear Bragg stopband with width ALBragg= A n . A % 24 nm follows from the strong indexcoupling of the QCDFB laser. The two Bragg resonances on either side of the stopbandlocated on the slope of the narrow gain spectrumexperience a strong discrimination with respect to each other due to the large value of AABraggleading to singlemode operation. Tunable singlemode lasing has also been achieved in cw operation. The lasing resonance has been tuned from 8.47 to 8.54 pm by changing the heat sink temperature from 20 to 120 K. Finally we discuss the LI characteristics of the QCDFB devices at various heat sink temperatures. Devices cleaved to a length of 1.5 mm are compared. In pulsed operation (pulse duration: 50 ns; frequency: 5 kHz) the deep etched ridge waveguide devices (see the inset in Fig. 50 for a schematic of the device cross section) display a low threshold current density of 2.2 kA cm2 (at 20 K) and 8.8 kA cm2 (at 300 K). A collected peak output power of 13 mW (slope efficiency 23 mW/A) is obtained for a 11ymwide ridge at 300 K and 23 mW (30 mW/A) for a 17ymwide ridge. Figure 50 shows the continuous wave (cw) L1 curves of the same devices operated up to 125K with z l 0 m W of cw output power at 120K. The maximum power measured at 80 K is 50 mW. From the data taken from deep etched QCDFB devices, it: appears that wider stripes result in an higher peak output power without degrading the singlemode yield. Therefore shallow etched ridge waveguide lasers have been fabricated. Owing to the subsequent current spreading higher threshold currents are obtained. Assuming the same threshold current densities as for deep etched devices, an effective lasing stripe width of ~ 6 pm 0 can be assumed. Figure 51 shows the pulsed LI characteristics of a 1.5mmlong device with shallow etched ridge around room temperature, resulting in 60mW (singlemode) peak output power at 300 K and 44 mW at 320 K. It should be noted that the indexcoupled 8pmwavelength QCDFB lasers described in this section had a threewell vertical transition design with a funnel injector, which further improves the performance of the vertical transition design discussed in a previous section for devices of the same wavelength. In fact, further optimization of this structure led to the recordings of cw optical powers (200mW at 80K) and pulsed room temperature powers (325 mW) (Gmachl et al., 1998a) for a 30stage device. / T (O.l), ~ ~a short lifetime 't3 = 1.65 ps, a This structure has a small T ~ ~ ratio large matrix element of the laser transition z 3 2 = 2nm, and a further
'.

1
QUANTUM CASCADE
77
LASERS
T=20K
0
0.5
1 .o

1.5
FIG. 50. Light outputtocurrent characteristics at various heat sink temperatures of a deep etched ridge waveguide QCDFB driven in continuous wave (A = 1.375 pm; length: 1.5 mm; width: 1 1 pm). The light is collected from one facet with near unity collection efficiency. The kink at high power levels in the temperature range 1 is not fulfilled for the whole voltage range. The authors extracted a scattering time of 100 fs from the DC IV and from the dynamical behavior at small bias. For the situation w.7 < 1, only classical rectification is expected, which does not show quantum effects such as photonassisted tunneling peaks. The preceding results of the Regensburg group have shown that THz radiation directly interacts with the electrons in a superlattice and influences the miniband transport. The response is fast and not due to heating, which makes it attractive for detector applications.
3 TUNNELING IN SEMICONDUCTOR QUANTUM STRUCTURES
177
UIU, FIG. 29. Measured (dashed line) and calculated (solid line) currentvoltage characteristics and fieldinduced change: (a) without radiation, (b) for 90 GHz radation (calculation for WT = 0.07), and (c) for 3.9THz radiation at different power levels (calculation for WT = 3) (after Winner1 et al., 1997).
In another work by the UCSB group it was shown that DCcurrentdriven Bloch oscillation couples to external radiation either by emission or absorption of THz photons. They explored this phenomenon by investigating the inverse Bloch oscillator efect, which senses changes in the DC conductivity under the influence of an external THz field (Unterrainer et al., 1996b).
178
KARL UNTERRAINER
The samples used in this study are GaAsAl,Ga, ,As (x = 0.3) superlattices grown on a semiinsulating GaAs substrate by molecular beam epitaxy. The superlattice structure consists of 40 periods of 80Awide GaAs wells and 20 Athick AlGaAs barriers. The superlattice is homogeneously Si doped with a concentration of n = 3 x 10'5cm3. The superlattice is embedded in 3000 Athick GaAs layers with a carrier concentration of n = 2 x 1018cm3, which serve as contact regions. The highly doped contact layers are separated from the superlattice by 80A GaAs lightly doped setback layers. A bandstructure calculation in the envelope function approximation results in a width for the lowest miniband A = 22 meV (Bastard, 1981). This means that electrons moving in the lowest miniband are not scattered by optical phonons at low temperatures. The effective mass of electrons at the bottom of the miniband is mSL= 0.07 m,. The second miniband is separated from the first one by about 100 meV. Thus, for low applied Bias (< 200 mV) we do not have to consider tunneling to the second miniband. Superlattice mesas with an area of 8pm2 were integrated with bowtie antennas as described in Section 111. The experiments were performed at 10 K in a temperaturecontrolled flow cryostat with Zcut quartz windows. The conductance of the superlattice was measured during the microsecond long pulses of THz radiation provided by the UCSB free electron lasers. The current voltage characteristics of the superlattice device without THz radiation (DC IV) can be seen in Fig. 30 (curve at the top). The current is linear for bias voltages below 20 mV. The negative differential conductivity (NDC) region begins at a bias of 20mV, which corresponds to a critical electric field of 500 V/cm. Assuming that the onset of the NDC region is due to EsakiTsu type localization when uB.t> 1, we find a scattering time z = 1.3 ps. The maximum current density is about 100A/cmz. The IV curve shows a small asymmetry: For positive bias (injection from the top contact) we find a more pronounced NDC than for negative bias where the bottom contact is the emitter. This asymmetry is present in all devices and we think is due to the different geometry of top and bottom contacts or due to inhomogeneity in the doping. From the preceding scattering time of z = 1.3ps, we would expect to see coherent effects for a frequency of a few 100 GHz. The irradiated IVs in Fig. 30 for 0.24THz show only classical rectification effects and no quantum effects can be observed. The curves are shown for increasing AC field strength (the curves are displaced downward with increasing intensity for clarity). The conductivity decreases with increasing intensity and the main peak shifts to higher voltages. This is an indication that the scattering time extracted from the maximum of the D C IV is not correct.
3 TUNNELING IN SEMICONDUCTOR QUANTUM STRUCTURES
179
1 v=0.24THz
0
0.05
0.1
DC Bias (V) FIG. 30. Currentvoltage characteristics of the 8 nm2 nm superlattice without (curve at the top) and with 0.24THz radiation. With increasing intensity, the conductivity decreases and the main peak shifts to higher voltages. No quantum effects are observed at this frequency.
Figure 31a, shows the influence of an external THz electric field at a frequency of 0.6 THz on the superlattice current. At low intensities, an additional peak emerges in the NDC region. We attribute the first additional peak to a resonance of the external laser field with the Bloch oscillation wB= a.When the intensity is increased further, the first peak starts to decrease and a second peak at about twice the voltage of the first peak is observed and assigned to a twophoton resonance. At the highest intensities, we observe a fourphoton resonance. The initial current peak of the DC IV decreases with increasing intensity indicating the onset of dynamic localization. At very high intensities, a small bump at the original position of the peak recovers. The position of the peaks do not change with intensity of the FEL. We observe quantum effects for frequencies larger than 480 GHz, which corresponds to a scattering time of z = 0.33 ps, approximately 34 times shorter than the previous estimate. This value is consistent with earlier cyclotron resonance measurements (magnetic field perpendicular to the growth direction) (Duffield et al., 1986). Figure 31b shows the results for a laser frequency of 1.5 THz. The peaks are shifted to higher voltages and are much more pronounced. Only the fundamental and the second harmonic are observed, since, for a given Em, eE,d/ho is smaller at higher frequencies. In addition, we observe a sup
180
KARLUNTER RAINER 1
0
,
0.05 0.1 0.15 DC Bias (V)
1
0.2 0
0.05
0.1
0.15
0.2
DC Bias (V)
FIG. 31. DC currentvoltage curve for increasing FEL intensity (the curves are shifted downward for increasing laser intensity). The FEL frequency was fixed to 0.6 THz (a) and to 1.5THz (b). In the NDC region, additional features occur attributed to resonances a t the Bloch frequency and its subharmonics.
pression of the current value in between the peaks. The peaks show a clear asymmetry with a steeper slope on the highvoltage side. This asymmetry is different from the shape of the peak of original DC IV, which shows a steeper slope at the low voltage side. Figure 32 shows the peak positions as a function of FEL frequency. The relationship is linear and the slopes of the Nth harmonic are N times the slope of the onephoton resonance. The magnitude of the slope is larger than expected from a voltage drop across the whole superlattice. The most reasonable explanation is that a high electric field domain is formed that extends over approximately onethird of the superlattice. For this stable situation, the electric field in the lowfield domain is below the critical field for localization and puts this part of the superlattice in the high conductive miniband transport regime. The formation of a highfield domain could also explain the discrepancies between the values for the scattering times deduced from the DC IV and from the THz measurements. If the onset of the NDC is more likely caused by domain formation, localization over a fraction of the superlattice, and not by the onset of localization over the entire superlattice, the value for the scattering time that we derived from the assumption of uniform localization is incorrect. Figure 33 shows the intensity dependence of the current at the different resonances at 0.6THz. In addition, the predicted current from Eq. (32) is
3 TUNNELING IN SEMICONDUCTOR QUANTUM STRUCTURES
181
FIG. 32. DC bias positions of the induced current peaks versus FEL frequency (after Unterrainer et al., 1996).
0
2
4 6 edE Ihv
8
10
FIG. 33. Data points show the intensity dependence of the current for constant DC bias at the different resonance positions at 0.6 THz. The upper curve shows the behavior at a bias of 45 mV, where the onephoton resonance occurs; the other curves show the behavior at the bias positions of the higher resonances. The lines show the predicted current at these resonances as a function of AC field strength (after Unterrainer et al., 1996).
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KARLUNTERRAINER
plotted. The absolute value of the electric field is obtained from a fit to the maximum of the current of the onephoton resonances. The coincidence of the first maxima for the higher photon resonances is very good. Thus, we can use the fit to the maxima to calibrate our electric field in the sample. A discrepancy exists for the predicted smaller oscillations at higher intensities, which do not show up in the experimental data. Multiphoton resonances with Bloch oscillation in a superlattice in a DC electric field have been observed. These results show clearly that the external radiation couples to Bloch oscillations, contrary to theoretical suggestions that THz radiation would not couple to a uniform WannierStark ladder. This result is intimately related to dissipation and line broadening of the otherwise identical states in the ladder: absorption appears above the WannierStark splitting (CD, < w ) and gain below (w, > w). The effect is an analogy to Shapiro steps in SIS junctions that support the AC Josephson effect.
VIII. PhotonAssisted Tunneling and Terahertz Amplification We can discuss the THz amplification of resonant tunneling diodes from the results of Drexler (1995). The current of the peak at the higher voltage side of the main tunneling peak is due to stimulated emission of photons. From the value of the current density of the onephotonstimulated emission peak at the lowest intensity we can estimate the power emitted from the sample. The current density of the main tunneling peak is about 70 Ajcm' and that of the stimulated photon emission peak for a frequency of 1 THz has about the same current density. The current of this photon assisted peak is only mediated by the emission of photons. Thus, the stimulated photon emission rate can be estimated to be j / e = 70j1.6 x = 4.3 x 1O2's' cm'. The emitted power density for a photon energy of 4.1 meV is thus about 0.3 W/cm2. This is not a very high value compared to the estimated input power density of for the bowtie device. 50kW/cm2, which results in a low gain of Since the current density of the measured RTD is quite low compared to that of RTDs designed for high current densities, there is room for further improvement. However, higher current densities should be realized only when considering that the tunneling peaks do not broaden significantly. In addition, the device geometry must be changed to ensure a larger interaction volume for the radiation. In the experiments with miniband superlattices, we attribute a positive photocurrent in the NDC to stimulated emission of photons (electrons move downward in the WannierStark ladder). From the value of the photocur
3 TUNNELING INSEMICONDUCTOR QUANTUM STRUCTURES
183
rent we can estimate the power transfer from the DC electric field into the photon field to be about 50 nW (0.56 W/cm2 in the mesa). The intensity of the AC field inside the superlattice at the maximum of the onephoton resonance is 42 kW/cmz, which leads to a total THz amplification coefficient of our superlattice mesa of 1.3 x 10W5.Assuming a 500 impedance of the bowtie structure we estimate a negative THz conductance of 1.3 x lop4 (Qcrn)’, which compares to the theoretical value of 5 x lop3(Qcm)’. The experimentally determined THz conductance is more than one order of magnitude smaller than the theoretical prediction. The values for the amplification are quite small; however, these results show that the theoretical predictions for the nonlinear behavior of semiconductor superlattices are correct. The observation of this parametric gain of a THzdriven superlattice proves that the assumptions we had to make in the calculations of the nonlinear behavior are valid. In consequence, the prediction of negative AC conductivity in a superlattice without a driving AC field seems to be quite realistic. However, the direct experimental proof of small signal gain in a biased superlattices remains missing,
IX. Summary and Outlook The experiments on photonassisted tunneling in semiconductor quantum structures have produced new results for the nonlinear interaction with external AC fields. Some of these effects had already been predicted in the 1970s and their relevance for realistic semiconductor quantum structures has been quite disputed. The results for the resonant tunneling diodes and for the sequential resonant tunneling superlattices show in a very impressive way by the observation of photonassisted side peaks and by the occurrence of dynamic localization that these models are correct. The observation of absolute negative conductance was not predicted for such systems. More refined models show that absolute negative conductance is a consequence of dynamic localization. Furthermore, the results from the resonant tunneling diode showed that the occupation of the wells influences the photonassisted tunneling process. This together with a better theory for photonassisted tunneling could lead to quantitative determination of the occupation in the wells from photonassisted tunneling measurements. For sequential resonant tunneling structures, it was shown that photonassisted tunneling experiments reveal the “instantaneous” currentvoltage characteristics, which is very important in these structures where, under DC bias, domain formation dominates the currentvoltage characteristics. The experiments on miniband superlattices have also proven the theoreti
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KARLUNTERFMNER
cal predictions for the nonlinear interaction with external AC fields. Furthermore, the successful experiments in superlattices showed that in a superlattice supplied with a constant current, Bloch oscillation couples to an external AC field. With the observation of photonassisted peaks in the negative differential conductivity region, the analogy to Shapiro steps in AC Josephson junctions is established. The suppression of the lowfield conductance by an external AC field through the dynamic localization is explained by the semiclassical model and by the full quantum mechanical approach. This effect is also important for detector applications for THz electronics. The dynamic localization is very fast since it is an instantaneous effect and does not involve carrier recombination. Resonant tunneling diodes also have the potential to be used in detector applications using the photonabsorptionassisted tunneling channel. The observed nonlinear characteristics of superlattices are important for modulators and frequency converters (mixers, harmonic generators). These applications could be equally important as the most discussed possible application of a superlattice as a tunable THz source.
ACKNOWLEDGMENTS The author would like to thank the staff at the Center for FreeElectron Laser studies: J. R. Allen, D. Enyeart, J. P. Kaminski, G. Ramian and D. White, S. J. Allen, B. J. Keay, M. C . Wanke, H. Drexler, S. Zeuner, and E. Schomburg for their collaboration, support, and for supplying their material for this review, and M. Helm for discussions and for reading the manuscript. Most of the samples were grown by K. L. Campman, K. D. Maranowski, A. C. Gossard, D. Leonard, and G. MedeirosRibeiro.
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S E M I C O N D U C I O R S A N D SEMIMETALS, VOL 66
CHAPTER 4
Optically Excited Bloch Oscillations Fundamentals and Application Perspectives P. Haring Bolivar, T Dekorsy, and H. Kurz INSTlTUT
HALBLEI~ERTECHNIK 11
RWTH AACHEN,GERMANY
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. HISTORICAL BACKGROUND BLWH OSCILLATIONS IN THE SEMICLASSICAL MODEL . . . . . . . . . . . . . . . . . . . . . . . . 111. WANNIERSTARK DESCRIPTION OF BLOCHOSCILLATIONS . . . . . . . . . . IV. TIMERESOLVED INVESTIGATION OF BLOCH OSCILLATIONS . . . . . . . . . . V. BLOCHOSCILLATIONS AS A MODELSYSTEM FOR COHERENT CARRIER DYNAMICS IN SEMICONDUCTORS . . . . . . . . . . . . . . . . . . . . . VI. APPLICATIONOF BLOCHOSCILLATIONS AS A COHERENT SOURCE OF TUNABLE TERAHERTZ RADIATION . . . . . . . . . . . . . . . . . . . . . . . . VII. SUMMARY.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
187
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
188 192 197 203
210 214 215
I. Introduction One of the most intriguing phenomena in solid state physics is the behavior of an electronic charge in the periodic potential of a crystal lattice under the influence of a constant electric field. Counterintuitively and opposing our everyday experience from the wellknown Ohmic law, in an ideal scattering free system a charge carrier will not follow uniformly the electric field and give rise to a constant electrical current, but will perform a periodic oscillatory motion in real spaceit will perform Bloch osciflations.
Despite that the theoretical concept of Bloch oscillations was postulated more than 70 years ago by F. Bloch (1928) and C. Zener (1934), only recent experiments exploiting the high time resolution achievable with femtosecond lasers and the high quality of modern molecularbeamepitaxy grown heterostructures have enabled the observation of this fundamental phenomenon (Feldmann et al., 1992a; Leo et al., 1992). This chapter contains a 187 Copyright ;( 2000 by Academic Press All rights of reproduction in any form reserved ISBN 012.7521755 ISSN 00808784’00 $3000
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general overview of fundamental aspects of Bloch oscillations and presents various experimental work performed to investigate these oscillations and derive their characteristic dependencies. The study of Bloch oscillations has not only opened the path for prospective applications especially as tunable sources of electromagnetic radiation in the terahertz range, but has also enabled us to gain deep insight into fundamental properties of ultrafast coherent charge carrier dynamics in semiconductors. The outline of this chapter is as follows: the first section contains an overview of the historical background of Bloch oscillations and their theoretical description in the semiclassical model. Section I11 introduces the WannierStark description of Bloch oscillations and discusses difficulties for the correct theoretical modeling of impulsively, optically excited coherent charge carrier dynamics in semiconductor superlattices. The third section describes different experimental approaches that have enabled the timeresolved observation of this fundamental phenomenon. The first two sections focus on the investigation of Bloch oscillations by terahertz emission experiments: both fundamental aspects (Section V) and application perspectives (Section VI) are discussed.
11. Historical Background Bloch Oscillations in the Semiclassical Model
In 1928 Felix Bloch (1928) analyzed theoretically the behavior of an electronic charge e in a periodic potential under the influence of a static electric field F. Two equivalent models can be adopted to describe this problem: It can either be solved by the solution of Schrodinger’s equation including the electric field explicitly, which corresponds to a description in a WunnierStarkBasis in real space and which is described later. A common alternative ansatz, the semiclassical model, however, enables us to gain a more illustrative picture of Bloch oscillations in a kspace formulation. The basic idea behind this semiclassical description is a formal transformation of the Schrodinger equation (Haug, 1964; Ashcroft and Mermin, 1976; Krieger and Iafrate, 1985): hk
=
eF
where h is Planck’s constant, k is the momentum, e is the elementary charge, and F is the electric field. This mathematical transformation enables the elimination of the electric field in the time dependent Schrodinger equation describing the system. The problem is thus reduced to a description of Bloch electrons (ie., to a description of the system with the wave functions of the
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189
fieldfree case) by introducing a timedependent kvector. Besides this entirely mathematical point of view, Eq. (1) can also be interpreted as an equation of motion for the momentum k. It is therefore also known as the uccelerution theorem, as any charge carrier, or more precisely, any wave packet with a narrow kspace distribution (a Houston state; Houston, 1940), will move in kspace with a constant velocity k given by Eq. (1). Since the potential of the crystal E(k) in which the wave packet moves is periodic (period 2n/d), where d is the realspace lattice constant, the wave packet will return to its original state after a finite time. In other words, due to the periodicity of E(k), the wave packet cannot gain an arbitrary high kvector, but will be Braggreflected at the end of the Brillouin zone at k = n/d. In the reduced Brillouin zone, it will thus continue its motion at the left end of the Brillouin zone at k =  n / d (see, e.g., Fig. 1). Formally speaking, the momentum is only a good quantum number modulo the reciprocal wave vector of the periodic potential defined by 2 4 d . In the absence of scattering it will thus return to its original initial state with a frequency VBloch (period TBloch = l/VBl,,ch) given by
VBloch
=
~
k  eFd __ 2n/d  h
Equation (2) already demonstrates that any electronic wave packet in any periodic potential will perform a periodic motion in kspace with a characteristic frequency vBloch that is linearly dependent on the static electric field F. Note that the Bloch frequency is independent on the specific bandstructure of the material; that is, as it is independent of the electronic mass, it behaves utterly differently in comparison to classic behavior. This frequency
FIG. 1. Semiclassical picture of Bloch oscillations: (a) dispersion relation E ( k ) and realspace group velocity u and (b) associated realspace oscillation.
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dependence [Eq. (2)] is hence one of the primary characteristics of Bloch oscillations. To calculate the motion in real space of electrons performing Bloch oscillations, one can derive the realspace group velocity u of a wave packet from the dispersion relation E(k) since 0
i a h ak
=  E(k)
(3)
As an example for the realspace dynamics of Bloch oscillations, one can take an ideal onedimensional KronigPenney model for the crystal as depicted in Fig. 1 (spatial coordinate z, field F in the  z direction), which delivers E(k) = A/2(1  cos(kd)), where A represents the width of the considered electronic band. Assuming an ideally narrow electronic wave packet with an initial wave vector k = k , at position z = z , for t = 0 one can easily calculate from Eqs. (1) and (3) the velocity v(t), which yields after time integration the position z(t) of the wave packet. For this simplified ideal system one gets
which demonstrates that the periodic motion in kspace can directly be associated with an oscillation in real space. The phase of the oscillation will then depend on the initial condition k,. An interesting property of this oscillation is that the oscillation amplitude
LZ. A
2eF
is inverse proportional to the applied field. This amplitude L will later
reappear as a localization length in the quantum mechanical description. From Eq. (4)one can also derive directly the velocity of the wave packet. Interestingly, despite the absence of scattering in this model, the maximum velocity a carrier can reach vmax = Ad/2h is independent of the field. This characteristic behavior is in pronounced contrast to what one would expect for the acceleration of a free carrier in an electric field and thus underlines the Bloch character of the motion of charge carriers in a periodic potential. As already mentioned, the phase of a particle performing Bloch oscillations depends critically on the initial conditions. Hence, when not only one but an ensemble of carriers is excited coherently with an ultrashort laser
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pulse, the initial distribution in kspace can strongly modify the motion of the total charge distribution. Interference effects can then influence the total dynamics drastically. Wellknown special cases are: (i) the excitation of carriers in the center of the band at E(k) = A/2 (at k , = +_n/2d)of a onedimensional system, which leads to the characteristic breathing mode, where the center of mass of the ensemble remains constant, but the wave packet expands and contracts with the Bloch frequency, or (ii) the homogeneous excitation of carriers over the entire Brillouin zone, which leads to no net carrier motion at all. Nevertheless, for the excitation of an ensemble of carriers close to the bandedge (at k, z 0), the behavior of the ensemble will closely follow the oscillation of a single particle as described by Eq. (4). It is important to note that the semiclassical picture of Bloch oscillations is a mathematically exact transformation which allows to employ the wellknown fieldfree band structure and wave functions for the description of the dynamics of wave packets in the system. The problem is that all time dependencies are transformed by the acceleration theorem, which makes the correct description of initial conditions particularly difficult. Although the preceding calculation of the realspace dynamics [Eq. (4)] delivers the correct general dependencies for Bloch oscillations, the adopted energetically and temporally sharp initial conditions do not fulfill the uncertainty principle. A true quantum mechanical calculation must be performed to take into account initial conditions. This can then include effects such as, for example that a photoexcited carrier distribution will already be accelerated and thus redistributed in kspace during the finite temporal duration of the exciting laser pulse. Zener tunneling (Zener, 1934) that is, the coupling of the carriers to other bands is also frequently neglected in the semiclassical approach to simplify the modeling. In the sample structures described later, however, Zener tunneling is found to be negligible. The entire concept of the Bloch oscillation of an electron in an electronic band under the influence of an electric field seems to contradict our everyday experience and intuition, and in fact although crystalline materials with a high degree of perfection like semiconductors exist, Bloch oscillations are extremely difficult to observe. The reason is that in most material systems, scattering of the carrier momentum occurs on an ultrashort timescale in typical semiconductors scattering times are in the subpicosecond range. To observe at least one Bloch oscillation before scattering has affected the motion of the carrier, one has thus to tune the Bloch frequency to above several terahertz (THz). For typical semiconductors with lattice constants of the order of a few angstrom this would require [see Eq. (2)] fields beyond or very close to typical breakdown fields. This is one of the main factors that has prevented the observation of Bloch oscillations. In 1970 L. Esaki and R. Tsu made a proposal that allowed circumvention ~
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of this problem. The idea is to grow by molecular beam epitaxy (MBE)the growth technique pioneered by Cho and Arthur (1975) artificial crystal structures with a much higher lattice period than in natural crystalline semiconductors by growing alternating layered sequences of semiconductors with different bandgaps. These superlattices now enable us to design artificial periodic potentials with tunable miniband widths and larger spatial periods in the tens of nanometers range. With these parameters Bloch oscillation periods below typical scattering times can be attained already for fields on the order of several kilovolts per centimeter, which can easily be applied in pin diode structures. Superlattices have therefore opened the possibility for observing Bloch oscillations experimentally for the first time, and, in fact, the great majority of experiments described later were performed in these heterostructure systems.
III. WannierStark Description of Bloch Oscillations
As elucidated previously, Bloch oscillations can be described theoretically also in the WannierStark picture, which we use now to derive the dynamics of Bloch oscillations in superlattices after photoexcitation with femtosecond laser pulses. By using laser pulses with a duration shorter than the Bloch oscillation phase, one can easily guarantee that photoexcited carriers performing Bloch oscillations have a narrow initial distribution in kspace, which is essential to avoid destructive interference. In the WannierStark picture the basis for the description of the dynamics of the system are the eigenstates of the Hamiltonian including the electrical field. A complete quantum mechanical calculation is necessary to derive these states correctly at each field position. This demands a greater effort, but in contrast to the semiclassical approach, this description lacks the difficulty for the description of initial conditions, which makes the modeling of timeresolved optically excited experiments often simpler. To demonstrate the dynamics of Bloch oscillations in a superlattice in the WannierStark picture, a tightbinding approach enables us to derive the essential characteristics. This approach is analogous to assuming a cosineshaped E(k) dispersion, as discussed previously in the semiclassical approach, and constitutes a very good model for a semiconductor superlattice with square well potentials. In this approach, the WannierStark (WS) states 'Fnare determined by (Wannier, 1960)
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if nearestneighbor coupling is taken into account and coupling to other bands (Zener tunneling) is neglected (Bleuse et al., 1988). Here @(z) is the wave function of an isolated well and .Ii is the Bessel function of order i. The argument of the Bessel function corresponds to a localization length L = A/2eF and enables us to distinguish three different field regimes, as depicted in Fig. 2. Note that L is exactly also the amplitude of the Bloch oscillations in the semiclassical picture [see Eq. ( 5 ) ] . For low fields ( F s 0), the Bessel functions are delocalized over the superlattice and give rise to the miniband states. For very high fields (eFd > A) the localization length is smaller than the superlattice period, and hence the system behaves only like a group of uncoupled single quantum wells. The interesting regime is the intermediate regime, where wave functions extend over several lattice periods. It is easily derived, from the translational invariance against simultaneous shift of energy by eFd and space by d, that the energy levels and consequently also the optical transitions in the miniband will split up into a series

valence band
FIG. 2. Scheme of the eigenstates in a superlattice in diverse field regimes: (a) smallfield, (b) WannierStark ladder (WSL), and (c) highfield regime.
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of discrete levels, the WSL E
=E,
+ neFd
n
= 0,f 1,
f 2 , . ..
(7)
where eFd corresponds to the potential drop over one superlattice period due to the field, and E , is the energy of the transition in the isolated wells. A calculated typical field dependency of the linear absorption of the WSL in an onedimensional A = 18 meV Al,Ga, ,AsGaAs superlattice is plotted in Fig. 3a as an example. Here E = 0 is set at the energy of the n = 0 transition. The thickness of the lines indicates the oscillator strength of the transition. In this intermediate field range, the wave functions of neighboring wells overlap, as indicated by the finite oscillator strength of the n # 0 transitions. Hence quantum mechanical superpositions (wave packets) of several WS wave functions can be constructed with femtosecond laser pulses exciting transitions from hole to several electron states. The wave packets will oscillate with the energy difference of the constituting WS wave functions. Hence, in analogy to the semiclassical description, a coherent superposition of neighboring WS states (e.g., Yo and Y  will perform Bloch oscillations in real space, as depicted in Fig. 4. Clearly also in this case of a WS description, the characteristic frequency dependence VBloch = eFd/h can directly be derived from Eq. (7). Other typical situations can easily be constructed by a superposition of WS states, such as, for example, a breathing mode by a superposition of the WS Yo and Y t: states (Sudzius et al., 1998).
$'.'""P
20 10
0
10 20 sFd [me4
30
0
5
10 15 eFd [mew
20
FIG.3 Numerical calculation of a fan chart of the WSL as a function of the electric field (a) single particle calculation and (b) calculation including the electronhole Coulomb interaction (courtesy G. Bartels, 1998).
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U
10 nm i =OfS
t = 300 fs
t = 600 fs FIG. 4. Numerically calculated electronic wave functions in a semiconductor superlattice with 97A Al,,,Ga,,,As wells and 17A GaAs barriers. The upper part of the figure illustrates the Yo and Y  WSL wave functions in the periodic potential of the superlattice at 6 kV/cm. The lower part of the figure depicts the dynamics of the electronic probability density as a function of time. The dark grey line depicts the location of the electronic center of mass as a function of time.
This ideal case for the WSL contains the basic ingredients for the description of Bloch oscillations in semiconductor superlattices. A first important modification is nevertheless introduced by the Coulomb interaction between the electronic and hole WS states (Dignam and Sipe, 1990, 1991). As an example, Fig. 4b contains numerical calculations for a onedimensional superlattice including this interaction. In comparison to the singleparticle calculation that clearly exhibits the equally spaced WS translations described by Eq. (7), the inclusion of Coulomb effects modifies the ladder. At low fields, the modification can be quite substantial and a characteristic series of anticrossings between the WSL states become visible. At high fields, the influence of the Coulomb interaction is less and the splitting between the WS states becomes again nearly proportional to the applied field. The influence of the Coulomb interaction on Bloch oscillations is discussed later in Section V. Another difficulty arises from the three
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dimensional character of real superlattice samples, which leads to a continuum of states in the k , and k, directions (i.e., the momentum perpendicular to the growth direction). As a consequence, a large number of states must be taken into account to describe Bloch oscillations in semiconductor superlattices and to account for interference effects at higher excitation energies. One should be aware that the correct theoretical description of the coherent dynamics of Bloch oscillations after excitation by the electromagnetic field is even far more demanding than described here. The problem of this elementary formulation is that the dynamics of the system are calculated within a fixed set of wave functions calculated as stationary solutions of the system. Nevertheless, the timescale of the dynamics that are analyzed with ultrashort laser pulses reaches far into a time regime, where Heisenberg’s uncertainty relation yields energy uncertainties ( h = 6.5 meV. 100 fs) of the order of the typical energetic spacings in the experiments. Static solutions are therefore not valid solutions for describing the dynamics of the system on this ultrashort timescale. The calculation of the dynamical wave functions is therefore necessary, which is extraordinarily difficult for real semiconductors. In simple words, it is necessary to attain the timedependent wave functions for the interacting N electron system that determine the macroscopic observables. As N varies with time, the whole basis of wave functions of the system must be recalculated for each time step. A less demanding formulation, in the second quantization (Heisenberg picture) is therefore more appropriate and the dynamics of the system are therefore frequency expressed in terms of electronic creation (2f) and annihilation (2) operators in a density matrix formalism. In semiconductors, it is reasonable to introduce also operators for the holes (it and 2). Another difficulty arises due to the Coulomb interaction of the Nparticle system, which gives rise to an infinite hierarchy of different orders of correlation functions (including higherorder correlations) describing the interaction between all N carriers. This represents an insurmountable difficulty and approximations are necessary to deal with the resulting infinite set of coupled equations of motion. A very successful and widespread approach is based on a reduction of all higher order correlations via a HartreeFock decoupling scheme into products of twopoint correlations (e.g., (2!j]jkt,)+( a j c i ) * (jk2,))and solving the equations of motion at a twopoint level. This ansatz was originally developed in a realspace representation (Huhn and Stahl, 1984; Stahl and Balslev, 1987), but is more widely employed in its kspace formulation (SchmittRink and Chemla, 1986; Haug and Koch, 1993). The approach is known as the semiconductor bloch equations (SBE). The SBE have extensively and very successfully been applied to a multitude of problems, specifically also for diverse observations of the coherent dynamics of Bloch oscillations (for a review see, e.g., Rossi,
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1998; and Meier et al., 1998) and calculations now include even microscopic descriptions of scattering mechanisms, avoiding simplifications related to the introduction of phenomenological relaxation constants. Despite its wide acceptance and success, the SBE approach is nevertheless based on the HartreeFock decoupling (random phase approximation) of higherorder correlations, which is rigorously only valid for a Gaussian ensemble in equilibrium, and can thus be problematic for ultrafast dynamical processes which are normally performed far from equilibrium. An alternative approach was therefore derived, the dynamics controlled truncation (DCT) scheme (Axt and Stahl, 1994). This approach incorporates systematic truncation concepts to define a finite set of density matrices (including higherorder correlations) that must be taken into account as independent dynamic variables to describe experimental observables exactly within a prescribed order of the electromagnetic field. Under many experimental situations, the inclusion of higher correlations beyond the HartreeFock approximation yields only minor corrections, but shown in Section V, severe deviations are possible when the experimental observables are dominated by densitylike quantities that are strongly influenced by Coulomb interaction. There are many more approaches for the description of ultrafast processes in semiconductors. A very popular formalism is based on the nonequilibrium Green functions (Miiller et al., 1987). This overview is intended only to emphasize that the theoretical description of coherent dynamics in semiconductors is still a subject of intense research and that ultrafast experiments, as the observation of Bloch oscillations dynamics described later, are important aids for the further development of appropriate models.
IV. TimeResolved Investigation of Bloch Oscillations One of the ingredients that opened the path for the experimental observation of Bloch oscillations was the impressive advance in MBE growth techniques for semiconductor heterostructures. These welldeveloped techniques enabled the growth of highquality Al,Ga, ,AsGaAs superlattices exhibiting sharp and homogeneous optical transitions. The initial experimental step relevant for the observation of Bloch oscillations was the demonstration of a WSL in cw experiments. The first observation was made in 1988 by photocurrent spectroscopy at low temperatures (Mendez et al., 1988; Voisin et al., 1988) and later even at room temperature (AgulloRueda et al., 1989; Mendez et al., 1990; Kawashima et al., 1991). Closely related to the experimental demonstration of Bloch oscillations was the observation of
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negative differential conductance in superlattices by Sibille (Sibille et al., 1990, 1992), which is closely related to the concept of a localization length decreasing with the electric field. Parallel to the progress in the MBE growth techniques, a key aspect that made the observation of Bloch oscillations feasible was the development of reliable femtosecond lasers sources. With the upcoming of KerrLens modelocked titan:sapphire lasers in 1991 (Spence et al., 1991), a flexible and stable source of tunable ultrafast laser pulses with a duration of few tens of femtoseconds was developed that enables us to perform sensitive optical correlation experiments with a temporal resolution given only by the duration of the laser pulses. Hence, ultrafast electronic processes can be analyzed directly in the time regime. Bloch oscillations could then be observed experimentally for the first time in Al,Ga,,AsGaAs superlattices in 1992 (Feldmann et al., 1992; Leo et al., 1992) by transient fourwave mixing (FWM), which is an elegant experimental technique for detecting coherent interband dynamics (Yajima and Taira, 1979). A typical FWM setup is depicted in Fig. 5. These experiments followed a theoretical prediction that Bloch oscillations should be observable as a modulation of the thirdorder nonlinear optical polarization (von Plessen and Thomas, 1992). Data of these first FWM measurements (Haring Bolivar et al., 1993) are represented in Fig. 6, where the characteristic VBloch F dependency could clearly be demonstrated in an AI,,,Ga,,,AsGaAs superlattice with a combined miniband width of A = 18 meV. In this

’ I I
Z
’ I t
FIG.5. Scheme of a fourwave mixing experimental setup. The thirdorder nonlinear polarization P::!cr generated by both laser pulses in the material, leads to the diffraction of a signal which is proportional to the coherent interband dynamics in the material.
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? 3.
0
C
$
1
0
1 2 time delay (ps)
3
FIG. 6 . Data of one of the first four wave mixing measurements of Bloch oscillations for different applied biases (T = 10 K) (Haring Bolivar et al., 1993). The inset shows the extracted Bloch oscillation frequencies as a function of the applied field.
experimental technique, the optical interband coherence of the system is probed by two laser pulses via the thirdorder polarization. The coherent superposition of optical transitions to different WSL states leads then to quantum interference effects (known as quantum beats from atomic spectroscopy) manifested as a modulation of the transients. The beats observed in the experiments reflect the coherent superposition of WSL states, which corresponds to wave packets performing Bloch oscillations. The first experiments showed already a tunability of these oscillations between 500 GHz and 2.5 THz. Later, measurements with a higher time resolution quickly demonstated a higher tuning range from 300 GHz to 5 THz (Leisching et al., 1994), where the upper frequency was limited only by the employed laser pulsewidth. Further FWM experiments were performed to study the dephasing behavior of Bloch oscillations in diverse structures with different miniband widths, to investigate excitation parameter, electric field and sample temperature dependencies. Even the appearance of higher harmonics or the influence of manyparticle effects was analyzed by this experimental technique. For a further review of FWM observations, see Feldmann (1992b) and Leisching et al. (1994) and references therein. Although the coherent superposition of WSL states demonstrated by the FWM experiments corresponds directly to a Bloch oscillation in real space,
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as demonstrated mathematically in Bastard and Fereirra (1989), this first experimental observation of Bloch oscillations remained debated. The point of discussion was that as the FWM technique probes only interband contributions, no direct information regarding the realspace (intraband) dynamics of the coherent oscillation within the minibands is attainable. The discussion was nevertheless quickly resolved when THz emission spectroscopy measurements directly probing the coherent dynamics within the minibands (the intraband dynamics) proved in 1993 that Bloch oscillations take place in real space (Waschke et al., 1993). The advantage of the timeresolved THz emission spectroscopy technique (see experimental setup in Fig. 7) is that as the emitted electric field is directly proportional to the acceleration of charge carriers within the sample. By monitoring the time evolution of the emitted radiation one can thus detect the spatial motion of the charge carrier ensemble directly in the time domain. Making use of this advantage, a wide variety of THz experiments on Bloch oscillations have been performed. The next sections of this chapter focuses on a more detailed discussion of these results, but before moving on to the THz experiments, further experimental approaches are referred to in this section. Another experimental technique was later established for analyzing Bloch oscillations. This technique, depicted in Fig. 8, is based on a pumpprobe scheme where the transient polarization changes of the probe pulse according to the electrooptic effect are detected as a function of time delay (Dekorsy et al., 1994). By employing this transmittive electrooptic sampling
FIG. 7. Optical setup for timeresolved terahertz (THz) emission spectroscopy, which enables us to monitor the domain of the emitted electric field in amplitude and phase directly in time. The emitted radiation is collected with parabolic mirrors and detected with a photoconductive THz antenna gated by the second timedelayed laser pulse.
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FIG. 8. Experimental setup for transmittive electrooptic sampling. The general idea is to monitor the change in the optical polarization of the probe pulse in a differential technique. The TEOS signal is then directly proportional to the internal electric field in the sample Pi:/ra, due to the linear electrooptic or Pockels effect. The signal is nevertheless also influenced by ) resonantly probed experiments (Lovenich et al., third order correction terms (  P ~ ~ ! , ,for 1997).
(TEOS) technique with shorter laser pulses, Bloch oscillations could then be observed within a wider tuning range from 250GHz to 8THz in a A = 37 meV superlattice at 10 K. These high frequencies correspond to oscillation periods below typical scattering times at room temperature, and indeed TEOS experiments in this A = 37 meV sample enabled the first observation of Bloch oscillations at room temperature (Dekorsy et al., 1995). Figure 9 depicts the corresponding oscillatory traces and Fourier transforms of the data. The highest observed frequencies are in the vicinity of the optical phonon resonance in GaAs. The effects of resonance conditions of Bloch oscillations and optical phonons are an intriguing subject concerning the influence on the electronic dephasing time, which is under present investigation (Dekorsy et al., 1998). The dephasing time in the subpicosecond range a t room temperature stems from the strong population of optical phonons, which leads to an ultrafast randomization of the carrier momentum via LO phonon absorption. This process must be distinguished from the phonon emission process
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I
"
"
'
1
f
2