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High-Power Bipolar and Unipolar Quantum Cascade Lasers


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1 HIGH POWER BIPOLAR AND UNIPOLAR QUANTUM CASCADE LASERS By ARKADIY LYAKH A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007

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2 2007 Arkadiy Lyakh

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3 To the memory of my dear mother in law.

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4 ACKNOWLEDGMENTS First, I would like to thank my advisor, Prof. Peter Zory, for his guidance and encouragement throughout my research. I learned from him very diverse techniques on semiconductor laser design, processing and characterization, and I am convinced that this kn owledge will be very help ful to me in the future. In addition, in his lab he has created an atmosphere of trust and understanding. Therefore, the years that I spent working with him will always remain a pleasant memory for me. It was a privilege to have P rof. Zory as my supervisor. I would also like to thank Prof. Rakov, Prof. Xie and Prof. Holloway for being the members of my supervisory committee. Finally, I would like to thank my dear wife, mother, father and sister for their support through the four ye ars of my study at the University of Florida. Their encouragement and love was very important to my success.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ ............... 4 LIST OF TABLES ................................ ................................ ................................ ........................... 7 LIST OF FIGURES ................................ ................................ ................................ ......................... 8 ABSTRACT ................................ ................................ ................................ ................................ ... 11 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .................. 13 2 BIPOLAR CASCADE LASERS ................................ ................................ ............................ 21 2.1 Ligh t Polarization Rules for Bipolar and Unipolar QCLs ................................ ................ 21 2.2 Basic Operational Principles of Bipolar QCLs ................................ ................................ 24 2.3 Review of Previous Work on Bipolar QCLs ................................ ................................ .... 25 2.4 Experimental Results on Fabricated Bipolar QCLs ................................ .......................... 27 3 SIMULATION OF CURRENT SPREADING IN BIPOLAR QCLs ................................ .... 40 3.1 Influence of the Tunnel Junction Resistivity on the Current Spreading in Bipolar QCLs ................................ ................................ ................................ ................................ ... 40 3.2 Previous Work on Current Spreading Simula tion in Diode Lasers ................................ .. 43 3.3 Simulation of Current Spreading in SSL ................................ ................................ .......... 44 3.4 Simulation of Current Spreading in DSL ................................ ................................ ......... 48 4 UNIPOLAR QUANTUM CASCADE LASERS ................................ ................................ ... 55 4.1 Basic Operational Principles of Unipolar QCLs ................................ .............................. 55 4.2 Review of Previous Wor k of Unipolar QCLs ................................ ................................ ... 57 4.3 Low Ridge Configuration Concept ................................ ................................ .................. 59 4.4 Waveguide Structure for the Low Ridge QCLs ................................ ............................... 60 4.5 Fabrication of the Low Ridge QCLs ................................ ................................ ................ 63 4.6 Experimental Results for Fabricated Unipolar Low Ridge QCLs ................................ ... 64 5 CURRENT SPREADING MODEL FOR UNIPOLAR QCLs ................................ .............. 79 5.1 Previous Work on Current Spreading Simulation in Unipolar QCLs .............................. 79 5.2 Two Dimensional Current Spreading Model ................................ ................................ ... 80 5.2.1 Active R egion T ransverse C onductivity ................................ ................................ 80 5.2.2 Two D imensional Finite Difference M ethod ................................ ......................... 82 5.3 Model Results ................................ ................................ ................................ ................... 85

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6 6 SUMMARY AND FUTURE WORK ................................ ................................ .................... 93 6.1 Bipolar Quantum Cascade Lasers ................................ ................................ ..................... 93 6.2 Unipolar Quantum Cascade Lasers ................................ ................................ .................. 94 APPENDIX A CALCULATION OF ELECTRON ENERGY LEVELS AND WAVEFUNCTIONS IN LAYERED STRUCTURES ................................ ................................ ................................ ... 96 LIST OF REFERENCES ................................ ................................ ................................ ............. 101 BIOGRAPHICAL SKETCH ................................ ................................ ................................ ....... 106

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7 LIST OF TABLES Table page 2 1 Dependence of tunneling probability exponential factor D on indium c omposition ......... 39 3 1 List of parameters used for simulation of current spreading in bipolar QCLs. ................. 54 3 2 Current spreading simu lation results for bipolar QCLs. ................................ .................... 54 4 1 List of parameters used to calculate transverse NF distribution in the low ridge QCL under the ridge. ................................ ................................ ................................ .................. 78 4 2 List of parameters used to calculate transverse NF distribution in the low ridge QCL under the channels. ................................ ................................ ................................ ............. 78

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8 LIST OF FIGURES Figure page 1 1 Semiconductor laser and its typical output power P 0 versus current I characteristic. At threshold current I th laser action is initiated. ................................ ................................ 16 1 2 Interband mechanism ................................ ................................ ................................ ......... 17 1 3 Intersubband mechanism ................................ ................................ ................................ ... 18 1 4 Illustration showing how an electron loses energy in a bipolar QCL ................................ 19 1 5 llustration showing how an electron loses energy in a unipolar QCL. .............................. 20 2 1 Energy band diagram for SSL and DSL ................................ ................................ ............ 31 2 2 llustration of current spreading in bipolar QCLs. ................................ .............................. 32 2 3 Schematic of TJ and QWTJ. ................................ ................................ .............................. 32 2 4 Bipolar cascade lasers with different waveguide configuration. A) Separate waveguide for each stage. B) Single waveguide for all stages. ................................ ......... 33 2 5 V ertical cavity surface emitting bipolar QCL.. ................................ ................................ .. 34 2 6 Schematic of TJ. ................................ ................................ ................................ ................. 34 2 7 Quantum well tunnel junction ................................ ................................ ............................ 35 2 8 Single stage and double stage structures ................................ ................................ ........... 35 2 9 Near field pattern of the double stage laser. ................................ ................................ ...... 36 2 10 Far field pattern of the double stage taken at 2A ................................ .............................. 36 2 11 Voltage vs. current characteristics for the double stage and single stage lasers. ........... 37 2 12 Power vs. current characteristics for the double stage and single stage lasers near threshold and at high power levels. ................................ ................................ .................... 38 3 1 Structure (comprising tunnel junction) used for illustration of influence of TJ resistivity on current spreading. ................................ ................................ ......................... 52 3 2 Illustration of the current flow in stripe geometry SSL. ................................ .................... 52 3 3 Voltage vs. current characteristics for the new double stage and single stage lasers. ...... 53 3 4 Near field pattern of a new double stage laser measured at 1A. ................................ ....... 53

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9 4 1 Schematic of electron transitions in unipolar QCLs ................................ .......................... 67 4 2 Schematics of intersubband and interband transitions. ................................ ..................... 67 4 3 Transverse near field distribution for the surface plasmon waveguide ............................. 68 4 4 High ridge configuration for unipolar QCL. ................................ ................................ ..... 68 4 5 Schema tic of the low ridge laser (dimensions are given in microns). ............................... 69 4 6 Typical IV curve for a high ridge unipolar QCL ................................ ............................... 69 4 7 Energy band diagram of the active region used in the low ridge structure. ...................... 70 4 8 Transverse NF distribution in the low ridge QCL under the ridge ................................ .... 71 4 9 Far field distribution corresponding to NF under the ridge. ................................ .............. 71 4 10 Transverse NF distribution in the low ridge QCL under the channel. .............................. 72 4 11 Far field distribution corresponding to NF under the channels. ................................ ........ 72 4 12 SEM pictures of the fabricated low ridge unipolar QCLs. ................................ ................ 73 4 13 Pulsed anodization etching setup. ................................ ................................ ...................... 74 4 14 Etching rate of InP with GWA (8:4:1) mixed with BOE in ration 600 to 7. ..................... 74 4 15 Power vs. current characteristics for the realized low ridge QCL. ................................ .... 75 4 16 Laser spectra measured at 4A and 20A taken at 80K. ................................ ....................... 76 4 17 Voltage vs. current character istics and corresponding power vs. current curves measured at multiple temperatures (laser is without HR coating). ................................ .... 76 4 18 Dependence of the threshold current on temperature. ................................ ....................... 77 4 19 Far field intensity distribution for a low ridge QCL. ................................ ........................ 77 5 1 Schematic of the low ridge QCL. ................................ ................................ ...................... 86 5 2 A) High ridge IV curve. B) the corresponding dependence of the transverse active region conductivity on the voltage across the active region. ................................ ............. 87 5 3 Rectangular mesh used for 2D Finite Difference Method ................................ ................. 88 5 4 A) Rectangular mesh used for the low ridge QCL. B) Basic cell of the mesh. ................. 88 5 5 Schematic of the calculation procedure. ................................ ................................ ............ 89

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10 5 6 List of parameters used for simula tion. ................................ ................................ .............. 89 5 7 Current spreading model results. ................................ ................................ ....................... 90 A 1 Model results for a GaAs based layered structure. ................................ ............................ 99 A 2 Step sequence for the calculation procedure. ................................ ................................ ... 100

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11 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy HIGH POWER BIPOLAR AND UNIPOLAR QUANTUM CASCADE LASERS By Arkadiy Lyakh May 2007 Chair: Peter Zory Major: Electrical and Computer Engineering High power bipolar and unipolar quantum cascade lasers were designed, fabricated and characterized. The importance of lateral current spreading is emphasized since it plays an importan t role in operation of these devices. Edge emitting, gallium arsenide (GaAs) based bipolar cascade l asers were fabricated from metalorganic chemical vapor depositio n grown material containing two diode laser structures separated by a quantum well tunnel j unction (QWTJ). The QWTJ was comprised of a thin, high indium content indium gallium arsenide layer sandwiched between relatively low doped, p type and n type GaAs layers. Comparison of near field data with predictions from a one dimensional current spread ing model shows that this type of reverse biased QWTJ has a low effective resistivity. As a consequence, current spreading perpendicular to the laser length in the plane of the layers (lateral direction) is reduced leading to a relativel y low threshold cur rent for the second stage. In addition, the differential quantum efficiency ~150% of these double stage lasers is nearly twice that of single stage lasers Low ridge unipolar quantum cascade lasers operating at 5.3m were fabricated from InP based MOCVD grown material. Record high maximum output pulsed optical power of 12W at 14A was measured from a low ridge chip with a high reflectivity coated back facet at 80K. Also,

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12 Far Field measurements demonstrated current beam steering for this device. Modeling sh ows that the lateral variation of transverse conductivity is essential for an accurate description of current spreading in these devices.

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13 CHAPTER 1 INTRODUCTION Since their first demonstration in 1962, semiconductor lasers have become the enabling components in many applications. Emission wavelengths for these devices now span the range from the ultra violet (0.4m for GaN based interband lasers) to the far infra red (hundreds of microns for intersubband lasers). Optical powers from single chips are now in the 100 watt range in pulsed operation and the several watt range in continuous operation. Conversion efficiencies from electrical to optical power can be in t he 50% range for red and near infrared lasers with world record numbers now exceeding 70%. Figure 1 1 is a diagram showing a typical semiconductor laser chip and its most important characteristic, laser light output power P 0 versus current I Electrons associated with current are temporarily trapped in the active region where they lose energy either by photon emission or some non radiative process. Photons can be generated by either an electron hole recombination process between conduction and valence b ands (interband mechanism, Figure 1 2) or a simple electron transition between energy levels in the conduction band (intersubband mechanism, Figure 1 3). Stimulated photons are confined and directed by a built in waveguide and at a sufficiently high curre nt (threshold current I th ), laser action is initiated (Figure 1 1). As current increases above I th P 0 continues to increase linearly to some high level that depends on the type of laser chip used. Typical chip size as defined by L in (Figure 1 1) is abo ut one millimeter. Due to their small size, low cost and ability to be directly pumped by electrical current, low power semiconductor lasers are now widely used in applications such as printing, optical memories and fiber optic communications. For applic ations such as infra red countermeasures, free space optical communications, machining and range/ranging measurements, high power lasers are required. In order to achieve very high output powers, semiconductor lasers are

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14 operated in the pulsed mode using high current pulses, tens of nanoseconds wide. In order to get even higher powers without increasing power supply current requirements, one can employ designs using two or more optical gain (active) regions in the laser material. Since each active region (AR) is sandwiched between a number of layers, this combination being called a stage, electrons can produce photons as they move from stage to stage through the multistage device. As a consequence, it's possible in principle to obtain laser power proport ional to the number of stages in the material without increasing drive current. Semiconductor lasers with two or more stages, usually called quantum cascade lasers (QCLs), fall into two major categories: bipolar (using interband transitions) [1] (Figure 1 4) and unipolar (using intersubband transitions) [2] (Figure 1 5). In bipolar QCLs, light emission occurs due to recombination of electron hole pairs in the ARs. After the recombination process in the first stage, electrons tunnel from the valence band of the first stage into conduction band of the second stage. The tunneling process between stages takes place through a reverse biased, heavily doped p n junction. The tunnel junction is a crucial element in bipolar QCLs since it allows electron recycli ng. The number of stages in a bipolar QCL typically ranges from two to five. For unipolar QCLs, electrons emit photons by making transitions between conduction band states (subbands) arising from layer thickness quantization in the AR. After a radiative (or non radiative) transition is made, the electron "cools down" in a relaxation injection region and then tunnels to the upper laser level of the next stage. Usually, the number of stages in a unipolar QCL ranges from 20 to 35. In both cases, tunneling to the next stage provides the electron with the opportunity to generate another photon. This electron recycling process in the multistage device leads to

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15 increased output power relative to that obtained from a single stage device operating at the same cu rrent. In this work both types of QCLs are discussed. A large portion of the discussion is dedicated to the analysis of current spreading in the plane of the layered structure (lateral current spreading) since all devices studied were fabricated in either a stripe geometry as shown in Figure 1 1 or in a low ridge configuration (Figure 4 5). Since the conductivity of the layers above the active region is usually quite high, lateral current spreading is substantial and has a strong influence on laser opera tion. In Chapters 2, 3, 4 and 5, various aspects of the high power QCLs that have been designed, fabricated and characterized during this work are discussed. In Chapter 2, the relationship between the design of the reverse biased tunnel junction used in bipolar QCLs and lateral current spreading is discussed. Experimental results for realized bipolar QCLs are also presented in this chapter. Output power versus current curves show that the performance of double stage lasers made using this design are clo se to the best possible. The lateral current spreading model developed in Chapter 3 shows that resistivity of the quantum well tunnel junction used in our devices is not high enough to cause any additional current spreading. In Chapter 4, the peculiarit ies of lateral current spreading in low ridge unipolar QCLs and their influence on the operation of these devices are qualitatively described. Output power versus current curves show that this type of unipolar QCL has the unique feature of low threshold c urrent combined with very high peak output power. Results of the current spreading model for low ridge unipolar QCLs developed in Chapter 5 support the qualitative description of the spreading mechanism described in Chapter 4. In Chapter 6, the thesis wo rk is summarized and suggestions made for future research directions.

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16 Figure 1 1. Semiconductor laser and its typical output power P 0 versus current I characteristic. At threshold current I th laser action is initiated.

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17 Figure 1 2. I nterban d mechanism whereby conduction band (CB) electrons trapped in the quantum well active la yer recombine with valence band (VB) holes to produce photons with energy hv.

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18 Figure 1 3. Intersubband mechanism whereby conduction band electrons in a high energy state in the active region quantum well make a transition to a lower energy state by emitting a photon with energy hv.

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19 Figure 1 4. Illustration showing how an electron loses energy in a bipolar QCL

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20 Figure 1 5. Illustration showing how an electron loses energy in a unipolar QCL

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21 CHAPTER 2 BIPOLAR CASCADE LASE RS 2.1 Light P olarization Rules for B ipolar and U nipolar QCLs As discussed in Chapter 1, the radiative mechanism is different for bipolar and unipolar QCLs. Using quantum mechanical approach, in this section of Chapter 2 we demonstrate that this distinction leads to different light polarization for these devices. In particular, for bipolar QCLs emitted light is mostly TE polarized, while for unipolar QCLs TM polarization is dominant. Using Fermi golden rule it can be shown that the absorption (emission) coefficient in the both cases is proportional to: (2 1) where i and f are electron wavefunctions corresponding to initial and final states involved in the transition: (2 2) with f i,f envelope functions and u periodic Bloch functions. The interaction Hamiltonian H is given by (2 3) where A is the vector potential, p is the momentum operator and m* is the effective mass. Vector potential A in its turn can be expressed as a polarization unit vector e multiplied by a scalar function that slowly changes with the spatial coordinate r (its spatial variation is negligibly small within each unit cell). Therefore, the matrix element in Eq. 2 1 is proportional to (2 4)

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22 where are the band and subband indices of the initial and final states. The first right hand side term of the equation corresponds to interband transitions, while the second one corresponds to inter subband transitions. mostly TM polarized. In this case =1 for all intersubband transition, since Bloch functions are approximately the same for electrons within the same band. Using the fact that there is no size quantization in the x y plane (plane of semiconductor layers) the envelope function can be expressed as (2 5) where n (z) reflects the envelope func tion dependence along the z axis and lies in the x y plane. As a consequence, dipole matrix element in Eq. 2 4 has the following form ~ (2 6) Terms proportional to e x and e y are nonzero only in case when and In other words these components are always zero for photon absorption and emission. This is a consequence of the fact that the conservation of energy and wavevector for a transition within the same subband requires an electron interaction with the lattice (phonons). Therefore, absorption (emission) coefficient for intersubband case is nonzero only when light has the z component and it reaches the highest value when the electric field is full y polarized along the z direction (perpendicular to semiconductor layers). This explains why light emitted by intersubband lasers is always TM polarized. The first term on the right hand side of Eq. 2 4 can be analyzed in a similar way. Overlap integral be tween the envelope wavefunctions determines selection rule for interband transitions,

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23 which states that transitions are allowed only between states with the same quantum number in contrast to intersubband transitions: if ( ) (2 7) Analysis of the dipole matrix element for the Bloch functions is more complicated than for envelope functions (intersubband case). This analysis is based on the fact that hole Bloch functions can be considered as linear combinations of so called valence band basis functions. The basis functions in their turn have symmetry of atomic orbitals of an isolated atom. Using this fact and the fact that interband absorption in quantum well lasers is mostly determined by conduction band to heavy hole band transitions, analysis of Bloch functions dipole matrix element leads to the conclusion that light emitted by bipolar lasers is mostly TE polarized (electric field lies in the plane of semiconductor layers). Due to different polar ization, near field (NF) intensity distribution for intersubband lasers has some peculiarities relative to interband lasers. For example, calculated NF shows presence of intensity discontinues at boundaries between semiconductor layers in accordance with t he boundary condition 1 E 1n = 2 E 2n while for bipolar lasers it is continuous at all interfaces. In addition, since dielectric functions for a metal and semiconductor layers have different signs (dielectric function is negative for the metal since imaginar y part of the refractive index for the metal is bigger than its real part) there is possibility of a surface plasma mode propagating at the metal/semiconductor interface of unipolar lasers. This effect can be used to make unipolar lasers based on so called surface plasmon waveguide (see Section 4 1). In the rest of the text we will always assume that the radiation corresponding to bipolar lasers is TE polarized, while it is TM polarized for unipolar lasers.

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24 2.2 Basic O perational P rinciples of B ipolar QCLs The bipolar cascade laser project was dedicated to development and realization of a double stage laser (DSL) comprising a standard single stage laser (SSL) as a recurrent stage. Schematics of SSL and DSL are shown in Figure 2 1. After radiative recomb ination with holes, electrons tunnel from valence band of the first stage to conduction b and of the second stage through the potential barrie r in the tunnel j unction (TJ) Therefore, in ideal case each electron can give rise to two photons. As a consequence, DSL in principle can give twice as much power as SSL at the same current Usually, TJs for bipolar QCLs are composed of two n and p heavily doped layers. If doping of t hese layers is not high enough, potential barrier width increases. As a consequence, the electron tunneling probability reduces and the effective resistivity of the TJ increases. High resistive layers can be the reason for a strong lateral current spreadin g in the laser (spreading perpendicular to the laser axis in the plane of the layers) and, as a consequence, lower DQE and higher threshold currents. The h ighest spreading in this case is expected in the N layers ( with high mobility ) above highly resistive TJ ( Figure 2 2) The most efficient way to get low TJ resistivity is to dope both TJ layers above 19 cm 3 In some material systems, the high carrier concentrations required in one or both of the layers and/or dopant atom stability during the growth o f additional laser stages cannot be achieved. This problem can be reduced by sandwiching an appropriate quantum well between the two tunnel junction layers (Figure 2 3) since the tunneling probability exponentially depends on the barrier height. This possi bility is explored in this work. Esaki tunnel junctions are widely used in semiconductor device design beyond bipolar QCLs. For example, TJs are employed to cascade solar cells. In these devices each active region of the solar cell is optimized for light a bsorption in a particular wavelength range. This increases conversion efficiency of the solar cell.

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25 Bipolar QCLs can be designed to have either separate waveguides for each stage (Figure 2 4a) or a single waveguide shared by all stages (Figure 2 4b). Due t o reduced number of layers the latter configuration benefits from suppressed current spreading in the structure and lower strain. Also, such structures can be processed into distributed feedback lasers. However, overlap between the optical mode and highly doped TJs increases free carrier absorption and as a consequence decreases DQE for these devices. In this work we were mostly interested in getting the highest possible DQE. Therefore, we employed the configuration with separate waveguides. The bipolar cas cade mechanism is also used for making Vertical Cavity Surface Emitting (VCSE) QCLs (Figure 2 5). VCSEL emits light vertically (parallel to the growth direction) rather than horizontally, which makes it easy to use them for 2D laser arrays. One key technic al advantage of VCSEL is its ability to produce a circular, low divergence output beam. The active regions of these quantum cascade lasers are placed in antinodes of the standing wave pattern to increase the overlap with the optical mode, whereas TJs are l ocated in the vicinity of a field null to reduce the free carrier absorption. The cascade configuration is used to increase roundtrip gain, which is extremely low for these devices. Higher gain leads to lower threshold current and as a consequence to highe r optical power at the same current. Due to the low roundtrip gain DQE for bipolar cascade VCSELs is also relatively low (~0.9 for triple stage devices). 2.3 Review of P revious W ork on B ipolar QCLs The first realization of an edge emitting bipolar QCL was reported in 1982 [1]. In this work MBE grown, GaAs based, triple stage cascade lasers were fabricated and tested. Each active region of the structure had its own waveguide and cladding layers. Using Be and Sn as p and n dopant atoms respectively the author s achieved doping densities in TJs above 10 19 cm 3 This allowed them to triple DQE for the cascade lasers relative to SSLs (0.8 vs. 0.27). However, due to immature technology overall performance for these devices was quite low (nowadays

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26 DQE close to 0.7 is routinely measured for SSLs). Edge emitting bipolar QCLs with separate waveguides for each active region reported later had substantially improved characteristics [3]. For example, DQE ~1.4 was reported for a double stage 808nm lasers in this work. Bipola r cascade lasers where all the stages share the same waveguide and cladding layers were reported in [4, 5]. Due to increased Gamma factor these lasers have threshold currents lower than SSLs. As discussed above, mode overlap with highly doped TJs reduces D QE. For example, in [4] DQE for triple stage lasers was reported to be 125%. Realization of Vertical Cavity Surface Emitting bipolar QCLs was reported in [6 8]. Since effective length of the gain region is increased, threshold current density for cascade V CSELs is lower than for their single stage counterparts. For example, in [7] comparison between triple stage and a single stage VCSELs showed that threshold current density for the triple stage device was only 800A/cm 2 compared to 1.4kA/cm 2 for single sta ge laser, while DQE was 60% compared to 20%. Relatively low DQE in the both cases is explained by low roundtrip gain for the vertical cavity configuration. The importance of high doping of the layers composing TJ was emphasized in [9]. It was also mentione d there that the very high doping of the TJ layers is not always achievable. For example when Si is used as the n dopant, it becomes amphoteric as the doping concentration 18 cm 3 The idea on the reduction of TJ resistivity using QWTJ layers was suggested in [10]. In this work it was shown that resistivity of GaAs TJ can be decreased by one order through sandwiching 120 thick In 0.15 Ga 0.85 As layer between the heavily doped layers. However, in this work QWTJ were not used in the laser de sign due to expected high modal losses. The first MOCVD grown InP based bipolar laser with QWTJ was reported in [11].

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27 Employment of QWTJ allowed the authors substantially reduce voltage drop across TJ and improve laser performance. 2.4 Experimental R esults on F abricated B ipolar QCLs p n junction is given by: (2 8) where N a and N d are acceptor and donor densities and V 0 is the contact potential. Since high doping is used for the layers composing the TJ, the degenerate approach should be used to calculate V 0 (Figure 2 6): (2 9) Depletion region extends mostly in the n layer since doping concentration for the p layer is higher by approximately one order of magnitude. Also, even though N a > N d it is usually valid that fn > fp since density of states for conduction band is substantially lower than for valence band. fn can be found from th e following equation: (2 10) where is the conduction band density of states. Similar equation should be applied to find fp Using Eq. 2 9 and Eq. 2 10 it can be shown that V 0 slowly changes with carrier density concentration. Therefore, from Eq. 2 8 it can be concluded that the depletion width decreases as the doping increases. As a consequence, potential barrier width (and electron tunneling probability) seen by electrons inc reases.

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28 As discussed in Section 2 2, very high doping for TJ layers is not always achievable. In our case the highest possible doping level for GaAs 18 cm 3 (Si doping) for the n 19 cm 3 (C doping) for the p layer. Relatively low n doping is the reason that a TJ composed of these layers would have high effective resistivity. Therefore, other methods should be used to reduce TJ resistivity. QWTJ concept can be employed in this case. Schematic showing how QW transforms potential barrier seen by electrons is represented in Figure 2 7. It was assumed in this figure that QW lowers potential barrier height in the vicinity of the interface between the TJ layers. In quantum mechanics the probability of a particle tunneling through a pot ential barrier is proportional to the following factor: (2 11) where U(x) is potential barrier and E is particle energy. Therefore, we should expect lower effective resistivity for QWTJ due to lower barrier height in the v icinity of x = 0. Rough estimation of influence of QW on TJ resistivity can be done using Eqs. 2 1, 2 2, 2 3 and 2 4 and Figure 2 6, 2 7. The following approximations are used: potential barrier for electrons in depletion region has triangular shape with a deep around x = 0, electron effective mass is constant through the structure and equal to (GaAs based TJ), barrier width of the TJ is not influenced by dimensions of QW, doping for the n side GaAs is and for the p side it is equal to quantum well used is In x Ga1 xAs. Results on D dependence on In composition are represented in Table 2 1 assuming thick QW (critical thickness equal to corresponds to In composition around 40% [12]). As it mentioned above, introduction of QW in TJ with In composition 15% in [10] reduced TJR by one order of magnitude which is roughly consistent with results presented in Table 2 1:

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29 D increases approximately by factor of ten when In % increases from 0 to 15. Also, it can be seen that further increase in In composition leads to further substantial increase of the tunneling probability. For example, D is more than 2 orders higher for 40% than for 1 5%. In this work [13] we explore the use of a QWTJ between two standard GaAs based diode lasers grown by MOCVD. The quantum well used is a 10nm thick, highly strained InGaAs layer with a 25% indium content. The SSL and DSL structures are shown in Figure 2 8 respectively. Doping of the n AlGaAs cladding layer in the first stage of the DSL is reduced relative to the 17 cm 3 compared to 10 18 cm 3 ) in order to decrease current spreading. The QWTJ connecting the two stages is composed of two relatively low doped GaAs 18 cm 3 Si for n 19 cm 3 C for p doping) with a 10nm thick In 0.25 Ga 0.75 As layer sandwiched between them. The active regions in both structures have two typical quantum wells sandwiched between standard barrier layers. The DSL structure is designed such that the optical mode loss due to overlap with the QWTJ is negligible. The SSL and DSL lasers used in the experiments were 750 long with 75 wide contact stripes. The output facet reflectivity was about 5% and the rear facet reflectivity was about 95%. All measurements were performed in pulsed mode at room temperature using 500ns wide current pulses at a repetition rate of 1kHz. The near field (NF) intensity distribution for the DSL measured at 1A is shown in Figure 2 9. Since the distance between the ARs is approximately 5 we used this scale to estimate that the NF width at the second AR is around 100 25 larger than the contact stripe width. This demonstrates that the current spreading in the structure is non negligible.

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30 Far field pattern for DSL measured at 2A is shown in Figure 2 10. Double lobed lateral pro file predicted for some wide stripe diode lasers due to V shaped phase front [14] was observed for DSLs. IV curves measured in cw mode at low current ( Figure 2 11) show that the turn on voltage for the DSL is close to double that of the SSL. The additiona l voltage drop above turn on is attributed to the finite effective resistivity of the QWTJ. The power vs. current (PI) characteristics near threshold and at high power levels for a typica l SSL and DSL are shown in Figure 2 12 The slope efficiency of the DSL (2W/A) (corresponding DQE ~150%) is nearly twice that of the SSL (1.1W/A) Maximum measured optical power is determined by generator maximum current (~7A). Ratio of the threshold currents for DSL for the second AR (490mA) and the first AR (340mA) is approximately 1.5. This is another indication of the current spreading in the structure. Experimental results described above show that the goal to double slope efficiency for DSL was achieved. However, as shown by NF measurements lateral current spreading is still present in the structure. A possible reason for this effect is the finite resistivity of TJ demonstrated in Figure 2 10. To determine the degree of influence of QWTJ resistivity on current spreading, a current spreading model for DSL based on 1D spreading model for SSL was developed. Its details and results are discussed in Chapter 3.

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31 Figure 2 1. Energy band diagram for SSL and DSL

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32 Figure 2 2. Illustration of current spreading in bipolar QCLs. Dashed line shows FWHM of carrier density distribution. Highest spreading is expected in the N layers with mobility above highly resistive TJ Figure 2 3. Schematic of TJ and QWTJ. Tunneling probab ility exponentially depends on the potential barrier height. It is expected to be lower for QWTJ since in this case height of the central part of the barrier is lower then for TJ.

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33 Figure 2 4. Bipolar cascade lasers with different wave guide configuration. A ) S epara te waveguide for each stage. B) S ingle waveguide for all stages These figures demonstrate transverse intensity distribution through the layers. a) b )

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34 Figure 2 5. VCSE bipolar QCL. The quantum wells in these devices were placed in antinodes of the standing wave pattern, whereas TJs were located in the vicinity of a field null to reduce free carrier absorption. Figure 2 6 Schematic of TJ. Triangular potential barrier seen by electrons is confined by thick dark lines V 0 fn fp E g 0 x n0 x p0

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35 Figure 2 7 Quantum well tunnel junction Figure 2 8. Single stage and double stage structures x = 0

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36 Figure 2 9. Near field pattern of the double stage laser. Distance between the active regions is 5m measured at 1A. Figure 2 10. Far field pattern of the double stage taken at 2A

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37 Figure 2 11. Voltage vs. current characteristics for the double stage and single stage lasers. Turn on voltage for the double stage laser is approximately twice that of the single stage laser. The additional voltage drop above turn on is attributed to the finite effective resistivity of the QWTJ

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38 Figure 2 12. Power vs. current characteristics for the double stage and single stage lasers near threshold and at high power levels

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39 Table 2 1 Dependence of tunneling probability exponential factor D on indium composition In% D 0 1.08E 18 15 1.22E 17 20 3.07E 17 25 8.36E 17 30 2.48E 16 35 8.16E 16 40 3.07E 15

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40 CHAPTER 3 SIMULATION OF CURREN T SPREADING IN BIPOL AR QCLS 3.1 Influence of the T unnel J unction R esistivity on the C urrent S preading in B ipolar QCLs As discussed in Chapter 2, NF measurements clearly demonstrate that there is a substantial current spreading between the 1 st and the 2 nd stages in realized DSLs. In this section using simple 1D model [15] it will be shown that tunnel junction resistivity can be the reason for this spreading. For the demonstration of shown in Figure 3 1. The structure is comprised of a resistive TJ and a P layer above it. The TJ has the thickness h and the effective resistivity cm 2 ], where is the resistivity o f the TJ and h is its effective thickness ( instead of is usually used to characterize TJ since depletion width and are usually unknown for a TJ, while can be directly measured [10].). Thickness for the p layer is d and resistivity is s Stripe w idth and length are taken to be W and L layers above the tunnel junction is negligibly small and that current through TJ under the stripe ( I e ) is constant. Leakage current ( 0 ) is de is given by (3 1) For current density flowing through the TJ we have (3 2) This expression can be modi fied as (3 3)

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41 where V voltage across the TJ and dI y is the current across the TJ between y and y+dy For lateral current in the P layer we have: (3 4) where is defined as and I y is the current flowing in the y direction. From Eq. 3 3 and Eq. 3 4 we get (3 5) Solution to Eq. 3 5 is (3 6) I y (0)=I 0 we get (3 7) Current across the TJ between y and y+dy is obtained from Eq. 3 7 (3 8) As a consequence, I e is given by (3 9) Therefore, total current through the device is (3 10) From Eq. 3 10 for I 0 we get

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42 (3 11) Using Eq. 3 7 and Eq. 3 11 we get for current density flowing through the TJ (3 12) From (3 12) (3 13) where W eff is the effective output stripe width (FWHM of the lateral current density distribution right under the active region) Typical values of for TJs are in the range from to [10]. Substituting I t =200mA, and we get from Eq. 3 11 and Eq. 3 13: and Eq. 3 11 and Eq. 3 13 can not be directly applied to our structure. However, they demonstrate that a TJ with a typical resistivity in vertical direction can influence current spreading and its contribution can be reduced by lowering resistivity. The question we are trying to answer in this chapter is stated as follows: Does TJ resistivity contribute to current spreading between the 1 st and the 2 nd stages of the DSL observed in the NF measurements? To answer this question a current spreading model for DSL that assumes negligible TJ r esistivity will be presented. Validity of this assumption will be verified by comparing model results with experiment.

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43 3.2 Previous W ork on C urrent S preading S imulation in D iode L asers A simple 1D current spreading model for diode lasers was presented in [15]. Transverse voltage drop across the p layers above the active region was considered to be negligibly small and current density under the wide contact stripe to be constant. In this work the diffusion In [16, 17] this model was extended to include the diffusion component. It was shown in [16] that Ohmic current in the layers above the active region and the diffusion current in t he consistency of the problem. In [17] the following physical description of the lateral current in the active region was suggested: holes are transported in the active reg ion under the combined effect of drift and diffusion, but the field causing diffusion is such that their motion is identical to that of pure diffusion with an effective diffusion coefficient. Electrons on the other hand, are stationary in the active region Instead of moving there, the electrons are supplied from or to the N layer at just such a position dependent rate that they maintain charge neutrality in the active region. The model developed in [16, 17] is applicable only below threshold since it does into account the stimulated emission term. Since the stimulated term is also involved in the scalar wave equation, coupling between the optical mode and diffusion and Ohmic currents should be considered in this case. A numerical model that takes i nto account the stimulated emission term was reported in [18]. In addition, it comprises Poisson equation and photon rate equation. This model was used to predict an optimal ridge width for a diode laser. In this work we adopt the subthreshold model devel oped in [16, 17]. The model (Section 3 3) and its employment for simulation of the current spreading in DSL (Section 3 4) are discussed in the next two sections.

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44 3.3 Simulation of C urrent S preading in SSL The exact solution to the lateral current flow pro blem in a laser requires the solving of the 2D current continuity equations. A major simplification used here is the reduction of what is naturally a 2D problem to a problem in one dimension. As discussed in the Section 3 2, there are two major components to the lateral carrier flow in a diode laser: a lateral current spreading in the layers away from the active region ( Ohmic current) and a lateral diffusion current in the active layer. It was also mentioned in Section 3 2 that these two currents are couple across the P layers above the active region is negligibly small, Ohmic current spreading is characterize d by normalized conductance (Figure 3 2): (3 14) where N is the number of P layers and is normalized resistance. When is big ( is small), lateral current spreading is big. In this work we assume that the lateral voltage gradient is small at the interface between active layer and N cladding layer compared to the lateral voltage gradient at interface betwee n P cladding layer and active layer. This condition is usually satisfied for single stage semiconductor lasers since the resistance of the layers below the active region is much smaller than the resistance for the ones above it. E ffective current width is defined here as FWHM of spatial current distribution. Following [16] we have for Ohmic sheet current density (lateral Ohmic current per unit of stripe length (A/cm) in layers above active region) (Figure 3 2)

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45 (3 15) (3 16) where is voltage drop across the active region at location is active region injected current density and (see Eq. 3 14) normalized resistance of the layers above active region. Eq. 3 15 and Eq. 3 16 take into account that the current density injected in the active region originates from the decrease in the lateral Ohmic current flowi ng above the active region. It is assumed that the evolution of the electron concentration in the active layer can be described by a diffusion equation: (3 17) where ( see Section 3 2 ) is the effective diffusion coefficient [17] and is a hole diffusion current density ( ). The diffusion current has as its source the junction current density and, as its sink, the concentration dependent recombination rate (3 18) where active layer thickness and recombination rate with and taken from [19]. Eqs. 3 15 through Eq. 3 18 have to be solved self consistently taking into account that the voltage across the active region is a function of the concentr ation n. For this dependence we use the following formula (see [16] and references therein) (3 19)

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46 where at , effective density of states in the conduction band. From Eq. 3 15 and Eq. 3 16 we can get (3 20) Solving Eq. 3 17 and Eq. 3 18 we have (3 21) First lets conside r the case when Equations Eq. 3 20 and Eq. 3 21 are coupled through Elimination of J in these equations and some mathematical manipulations give (see [16]) (3 22) where Since it is assumed that current density under the stripe J 0 is constant, the first integral of Eq. 3 21 has the form (3 23) where is the carrier concentration in the active region below the center of the stripe, uniform current density under the stripe and represents stripe width. Eq. 3 22 (or Eq. 3 23) can then be integrated to yield as the solution for the carrier concentratio n profile (3 24) where at

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47 Calculation of current density distribution in the active region is done using the following procedure. Carrier concentration at the stripe edge n e is used as an input parameter. Using Eq. 3 22 and Eq. 3 23 f(n) is found for and respectively. Also, f(n) under the stripe at this step is a function of two unknown parameters: n 0 and J 0 Relation between J 0 and n 0 (J 0 (n 0 )) is found using the fact that Eq. 3 22 and Eq. 3 23 should be equal under the stripe edge. Using J 0 (n 0 ) Eq. 3 23 and (3 25) n 0 (and as a consequence J 0 ) is found as a function of n e Lateral carrier concentration n(x) is found using Eq. 3 24. Using n(x) injected electron current density from the n side of the p n junction can be found from the fact that it is equal to local recombination current density [17] (3 25) where A is the coefficient corresponding to non radiative recombination (for example through interface states), B bimolecular radiative recombination coefficient and C the coefficient corresponding to Auger non radiative recombination. According to definition output current effective width which is used as an effective stripe width for the next stage is taken to be equal to: (3 26) where is determined by: (3 27) In addition, for diffusion current leakage we have:

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48 (3 28) while for Ohmic sheet current den sity we get from Eq. 3 2 and Eq. 3 4: (3 29) Leakage current is sum of these two currents (multiplied by a factor of two) evaluated under the stripe edge. Therefore total leakage current is given by (3 30) where (3 31) is the relative importance of the Ohmic and diffusion current. Total current corresponding to carrier density in the active region at the edge of the contact stripe n e (which is used as an input parameter) is found integrating Eq. 3 25. 3.4 Simulation of C urrent S preading in DSL In order to determine the degree of current spreading at the two DSL active regions (ARs) a model was developed based on the 1D discussed in t he previous section. This model estimated the effective current width (FWHM of the current density distribution) at each AR. It was assumed that the QWTJ contribution to current spreading was negligibly small due to its low effective resistivity. Comparis on between the model and experiment was supposed to show wither this approximation is valid. It can be seen from Eq. 3 1 that spreading should be especially important for the bipolar cascade lasers where several active regions are connected via tunnel junc tions (TJs). Total thickness of such structure is substantially bigger than for a common semiconductor laser and

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49 therefore current leakage is expected to be enhanced in this case. Also, the resistivity of TJs (TJR) in the current flow direction should cont ribute to total spreading since the current tends to spread before entering a less conducting material. In the discussion below by stage we mean active region and all the layers above it but up to the active region of the previous stage. For example, accor ding to this definition, second stage comprises 2 nd active region and all the layers between 1 st and 2 nd active regions. For simplicity, current spreading in each stage is considered independently. This approach is accurate when we can assume that the lat eral voltage gradient is small at the interface between active layer and N cladding layer compared to the lateral voltage gradient at interface between P cladding layer and active layer. This condition is usually satisfied for single stage semiconductor la sers since the resistance of the layers below the active region is much smaller than the of effective current width for the first stage (its value at the a ctive region) is considered as an effective width of the stripe for the second stage. This approximation substantially simplifies calculations. Each layer above active layer for a stage under consideration is characterized by its thickness, doping and mob ility. Dependence of mobility on concentration and doping is taken from [20] and [21] respectively. Parameters used for the calculations below are listed in Table 3 1. The following parameters were fixed in the program: , , and Also, was adjusted before each calculation in such way that to tal current was around 200mA (below threshold for the structure).

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50 Simulation results are presented in Table 3 2. As expected, current spreading in the layers of the first stage is low. However, current width at the second active region was calculated to b e 97m, 22 m wider than the stripe width. Near field for the DSL measured at 1A is shown in Figure 2 7. FWHM of the intensity distribution at the second active region was estimated to be 100 m. It was also observed that NF to the threshold of the second active region. Therefore, assuming that NF width is close to the width of the current density distribution, simulation and experimental results are in good agreement. As mentioned above, the current spreading model did not include the effective resistivity of the QWTJ under the assumption that it should be small. This assumption is validated by th e good agreement between the model used and the measured NF of the second AR. Therefore, we showed that even though TJR is finite (Figure 2 cause any additional current spreading in the bipolar cascade laser. This model conclusi on was consistent with the following experimental results. The DSL structure was grown again using the design described above. However, QWTJ used in the new structure was different: GaAs (n doped with Te above 10 19 cm 3 )/ In 0.15 Ga 0.85 As (10nm thick quantum well)/ GaAs (p 19 cm 3 ). Measured voltage vs. current characteristics and near field intensity distribution for a laser fabricated from this structure are shown in Figure 3 3 and Figure 3 4 respectively. The IV curve shows that QWTJ e ffective resistivity is very low (due to higher n doping of the GaAs QWTJ). However, near field measurements demonstrate that despite this fact there is still a considerable lateral current spreading between the first and the second active regions.

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51 In conclusion, in this work it has been demonstrated that the employment of a deep QW inserted between TJ layers with relatively low doping densities can be used to fabricate DSLs with slope efficiencies and DQEs close to twice that of SSLs. It was also d emonstrated that a 1D model can be used to accurately calculate the current spreading in DSLs provided that a QWTJ with low effective resistivity is used. It is expected that this type of QWTJ should be of use in any device requiring monolithically stacke d diodes where material growth limitations require that the doping level densities in the TJ layers be kept relatively small. Future work in this area could be related to understanding how width of QW in the TJ influences its effective resistivity. Such in formation would be very useful for future device design.

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52 Figure 3 1. Structure (comprising tunnel junction) used for illustration of influence of TJ resistivity on current spreading. Figure 3 2. Illustration of the current flow in stripe geometry SSL.

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53 Figure 3 3. Voltage vs. current characteristics for the new double stage and single stage lasers Figure 3 4 Near field pattern of a new double stage laser measured at 1A

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54 Table 3 1. List of parameters used for simulation of current spreading in bipolar QCLs. 1 st stage Al composition Thickness, nm Doping, Mobility, P+ 0 200 200 70 P 0.3 50 10 67 P 0.6 1300 10 25 P 0.3 700 4 78 P 0.1 20 0.05 280 2 nd stage N 0.1 20 0.05 5900 N 0.3 300 4 950 N 0.6 20 4 70 N 0.3 2500 10 630 N 0.2 50 10 1100 n 0 50 50 1200 p 0 50 1000 40 p 0.3 50 10 67 p 0.6 1300 10 25 p 0.3 700 4 78 p 0.1 20 0.05 280 Table 3 2 Current spreading simulation results for bipolar QCLs Effective stripe width, Output effective width, Leakage current, mA Ohms 1 st stage 75 80 35 2 37 185 2 nd stage 80 97 73 1 74 38

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55 CHAPTER 4 UNIPOLAR QUANTUM CAS CADE LASERS 4.1 Basic O perational P rinciples of U nipolar QCLs Operation of unipolar (intersubband) QCLs rel ies only on one type of carrier in contrast to bipolar QCLs where light emission occurs due to radiative recombination between holes and electrons. The best performance for intersubband QCLs was demonstrated for n doped devices. In these devices light is generated due to radiative elect ron transitions between energy levels localized in conduction band (Figure 4 1). Electrons tunnel through the injector barrier from the injection region to the upper laser level. The radiative transitions occur between the 3 rd and the 2 nd energy levels. Ca lculation procedure of electron energy levels and wavefunctions in layered structures is discussed in Appendix. Initial and final states of the intersubband transitions have approximately the same curvature of the energy vs. wavevector dependence (Figure 4 2). As a consequence, joint density of states corresponding to these transitions and gain spectrum are substantially narrower than for interband transitions. In addition to radiative transitions between the 3 rd and the 2 nd levels, there are parallel non r adiative transitions between these levels through emission of longitudinal optical phonons. These transitions are very fast (~5ps) and as a consequence strongly increase laser threshold current density. To create population inversion between the upper and lower laser levels, energy separation 21 between the 2 nd and 1 st energy levels is usually designed to be equal to the energy of the longitudinal optical phonon (~34meV). 21 2 1 32 However, 21 is smaller than the energy of the longitudinal optical phonon, this fast process is prohibited and transition between the 2 nd and the 1 st levels occurs through emission of acoustical phonons which is a much slower process (~100ps). As a consequence, lasing can be unachievable in this case. Electrons get recycled through tunneling from the 1 st energy level to

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56 the miniband (multitude of closely spaced (in energy) levels) of the injection region. A reverse, undesirable effect (so calle levels occurs when the quasi Fermi level of this miniband is located close to the lower laser level. As discussed in Section 1 1 light emitted by unipolar QCLs has TM polarization in cont rast to TE polarization typical for interband transitions (diode lasers). Therefore, to avoid high losses usually cladding layers in unipolar QCLs are designed to decouple the guided mode from the plasmon mode propagating at the metal/semiconductor interfa ce. The opposite design approach is to employ surface plasmon waveguide. In this configuration there is no need to use cladding layers which can be helpful to improve heat dissipation in these devices. To illustrate corresponding mode the following 3 layer structure is used. One micron thick active region characterized by refractive index equal to 3.2 is sandwiched between a metal with and substrate with n=2.8 Calculated transverse intensity distribution for this structure is sho wn in Figure 4 3. Gamma factor (active region (including injector layers) confinement factor) was found to be 93%, mu ch higher than typical values ~6 0 70 % (including injector layers) Calculated effective refractive index and intensity loss were 3.13 and ~ 100cm 1 respectively. Therefore, disadvantage of this configuration is high loss and as a consequence high threshold current density. Figure 4 3 shows discontinuity at the interface between the active region a nd the substrate, the consequence of the boundary conditions particular to TM polariz ation Unipolar QCLs are usually fabricated in the high ridge configuration since surface recombination is not present in unipolar devices. This helps to reduce threshold current densities for these devices. Typi cal length for unipolar QCLs is in the range of several millimeters.

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57 U nipolar QCLs are used in atmospheric sensing, medical breath analysis, process monitoring and food production. Future possible applications for high power devices include infra red count ermeasures and free space optical communications. 4.2 Review of P revious W ork of U nipolar QCLs Concept of the intersubband cascade configuration for light amplification was suggested in 1971 [22]. However, the first quantum cascade laser was demonstrated more than twenty years later at Bell Labs in 1994 [2]. It became possible due to high growth precision of molecular beam epitaxy (MBE) and development of band structure engineering. InP based structures were used for fabrication of the first QCLs. This ch oice of material allows employment of heterojunctions based on In 0.53 Ga 0.47 As Al 0.48 In 0.52 As layers lattice matched to InP. High conduction band discontinuity (~0.5eV) of this composition makes it possible to fabricate QCLs emitting at relatively low wavel ength (below 5m). In addition, InP has low refractive index and as a consequence can be effectively used as a cladding layer. Strain compensated InP based QCL s were reported in [23]. In these structures barrier height can be increased relative to unstrain ed In 0.53 Ga 0.47 As Al 0.48 In 0.52 As composition. However, In and Al percentage in the barrier and quantum well layers should be changed simultaneously to avoid strain build up in the structure. InP based QCLs were demonstrated to operate at room temperature i n continuous mode [24] with hundreds mW of output optical power [25]. GaAs based QCLs were realized for the first time in 1998 [26]. Since AlGaAs layers are almost lattice matched to the GaAs layer independent of Al composition, this structure allows more design flexibility compared to InP based material. However, it should be taken into account that Al x Ga 1 x As structure becomes indirect when x>0.45. In this case, scattering to X valleys can be harmful for laser performance [27]. GaAs based QCLs performance at 80K was demonstrated to be as good, if not better, as for InP based lasers and it is steadily improving [28].

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58 It was also shown that employment of active region with deep quantum wells (In 0.3 Ga 0.7 As) can be used to substantially reduce carrier leakage from the injector region to continuum and as a consequence increase tunneling injection efficiency to the upper laser level [29]. Gas spectroscopy applications for unipolar QCLs require laser linewidth to be below 1cm 1 substantially less than typical li n ewidth for edge emitting QCLs (>10 cm 1 ). Distributed feedback (DFB) configuration, where a grating is introduced in the structure, proved to be very efficient for reduction of the linewidth for diode lasers. First DFB unipolar QCL was demonstrated in [30] Linewidth for DFB QCLs was reported to be below resolution of FTIR spectrometer (0.125cm 1 ). Wavelength in this case can be adjusted with temperature and current variation since refractive indices of the layers composing the structure depend on these par ameters. Typical adjustment rates are 0.5nm/K and 20nm/A respectively and wavelength adjustment usually lies in the range 30nm 100nm. Further increase of the scanning range can be achieved through employment of the external cavity configuration and bound to continuum active region design [31]. In these devices emission wavelength can be controlled with position and angle of the external grating. Wavelength for external cavity QCLs can be varied by ~ 100 400nm. Output power in cw mode in this case can be as high as several hundreds of mW [32, 33]. Second order DFB (surface emitting) configuration can be used to reduce strong beam divergence typical for edge emitting QCLs from ~60x15 to ~ 1x15 [34]. Maximum output optical power for these devices workin g in cw mode on a Peltier cooler is in the range of tens of mW [35]. New type of intersubband QCLs emitting in far infrared range (~100m and above), terahertz QCLs, was reported in 2002 [36]. In this work lasing was based on radiative transitions between minibands. Also, guided mode was confined by two metallic claddings, which

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59 decreased mode losses and increased confinement factor for the active region. Terahertz QCLs are still limited to low power, low temperature operation. 4.3 Low R idge C onfiguration C oncept High peak power, pulsed operated quantum cascade lasers (QCLs) operating in the first and the second atmospheric windows are being developed for use in application areas such as infra red countermeasures, free space optical communication and laser d etection and ranging (LADAR). Previous work on such devices [37, 38] employed optimized structural designs in a narrow width, standard high ridge configurations (Figure 4 4). A nother approach of getting high power QCLs is to increase width of the high rid ge. In this work an alternat ive approach for achieving high peak pulsed power QCLs is described, that uses a narrow width, low ridg e configuration (Figure 4 5). Figure 4 5 includes low ridge laser dimensions used in this work To understand operation of low physical mechanisms responsible for the shape of IVs for high ridge QCLs. Typical IV curve for a high rid ge unipolar QCL is shown in Figure 4 6. At low bias the injector and the upper laser levels (Figure 4 1) are misaligned. In this range the active region is in the high differential resistance mode (low effective conductivity). As the applied voltage increases (8V 12V) these levels line up and differential resistance substantially decreases. At voltages above 12V this alignment breaks again and the active region becomes resistive. In this work it is suggested that this mechanism can be used to design high power, low ridge unipolar QCLs with rela tively low threshold currents. The basic mechanism that allows narrow width, low ridge QCLs provide high output power with relatively low threshold currents is lateral current spreading (spreading perpendicular to the laser axis in the plane of the layers ) As shown in [39], lateral current spreading mainl y

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60 occurs in layers above the active region and decreases when the active region transverse conductivity increases. At low bias, the active region conductivity is low and the current density distribution is wide. At higher bias, the conductivity of the cent ral part of the active region begins to increase (as the injector and upper laser levels align). As a consequence, the width of the current density distribution (characterized by its full width at half maximum, FWHM) decreases. This narrowing of the curren t density distribution is the reason for the relatively low threshold current of these devices. When the applied voltage exceeds the voltage value that causes misalignment between energy levels, the conductivity of the central part of the active region dim inishes causing current to spread laterally and the current density width to increase. This additional lateral current spreading effect, not taken into account in a previous current spreading model of low ridge QCLs [39], allows higher peak powers than exp ected to be achieved. The previous model [39] and the modifications required to include the additional lateral current spreading effect are described below. Low ridge unipolar QCLs were reported in several previous works. In [40, 41] it was demonstrat ed t hat low ridge QCLs can to give substantially higher output optical power than high ridge lasers fabricated from the same wafer (for the same ridge width and length). However, low ridge lasers in this case were treated just as broad area devices. In [42] lo w ridge configuration helped to improve heat dissipation, which substantially increased characteristic temperature of the laser. In this work proton implantation through the active region was used to suppress the lateral current spreading. 4.4 Waveguide S tructure for the L ow R idge QCLs Low ridge quantum cascade structure realized by our group comprised the active region design reported in [43] (Figure 4 7 ) embedded in the waveguide discussed below. In [44] using this active region design it was demonstrate d that MOCVD grown InP based QCLs are capable

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61 to demonstrate as good performance as MBE grown devices. The sequence of layers for the low ridge QCL was the following: InP low doped substrate (S, 1 17 cm 3 ), 2 m InP cladding layer (Si, 10 17 cm 3 ), 300 In GaAsP graded layer (Si, 10 17 cm 3 ), 3000 InGaAs waveguide layer (Si, 16 cm 3 ), 1.5 m active region comprising 30 stage AlInAs InGaAs QC strain balanced structure [43, 44], 3000 16 cm 3 ), 300 InGaAsP graded layer (Si, 10 1 7 cm 3 ), 2 m InP cladding layer (Si, 10 17 cm 3 ), 0.2 m InP contact layer (Si, 10 17 cm 3 ), 100 InGaAs top layer (Si+). Input parameters required for transverse waveguide calculations include imaginary and real parts of the refractive index for each layer. At low photon energy limit (comparable with thermal electron energy k 0 T) these parameters can be obtained using classical Drude theory. Theory results are (4 1) (4 2) where and are real and imaginary parts of the complex dielectric constant n and k are real and imaginary parts of the complex refractive index N. (4 3) (4 4) where is the high frequency dielectric constant, is the electron scattering time (4 5) (4 6) (4 7)

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62 For the real and imaginary part s of the refractive index we get (4 8) (4 9) where Quantum mechanical extension of this theory described in [45, 46] gives (4 10) where (4 11) Also, relaxat 4 was found to be a function of both photon energy and layer InP and GaAs at 300K can be found in [46]. Near field (NF) transverse intensity distribution calculated based on the parameters listed in Table 4 1 (calculated based on Eq. 4 1 Eq. 4 11) is shown in Figure 4 8 Gamma factor for the active region (includi ng injector layers) was found to be ~78% with real and imaginary part of the refractive index equal to 3.35 and 1.4E 5 respectively. Losses corresponding to the imaginary part of the refractive index can be calculated using (4 12) In our case mode loss is below 1cm 1 It is important to mention that calculated losses are usually significantly smaller than losses obtained from experiment. The reason for this effect is still not completely understood [48]. Full width at hal f maximum (FWHM) of the corresponding Far Field (FF) transverse intensity dependence on the emission angle (Figure 4 9 ) was calculated to

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63 be approximately 68. Calculated Gamma F actor for the active region (including injector layers) was found to be ~80% w ith real and imaginary parts of the refractive index equal to 3.35 and 1.46E 5 respectively. Also, FWHM of the FF was found to be ~70 0 Therefore, transverse NFs (and corresponding FFs) are almost the same under the ridge and under the channels despite su bstantially thinner cladding layer thickness in the latter case (0.8 m vs. 2.0m). The reason for this effect is the low refractive index of the Si 3 N 4 used as an insulator for the low ridge QCL. Basically this layer acts as a cladding layer separating mode from lossy gold contact above it. 4.5 Fabrication of the L ow R idge QCLs The QC wafer was grown by low pressure MOCVD at a slow rate (0.1nm/sec) in the same reactor as in [44] and under essentially the same conditions (except for growth uncertainties). 1.4 m high, 25 m wide ridges were etched in the wafer using pulsed anodization etching (PAE) [47]. The channel width on the both sides of the ridge was defined to be 50 m. The surface was then passivated by 300nm of Si 3 N 4 deposited by plasma enhanced CVD at 300 o C. Metal contact windows 12 m wide were opened on top of the ridges by photo lithography. The Si 3 N 4 in the openings was etched by RIE. The substrate was then thinned to approximately 120 m by mechanical lapping. Non alloyed contact metals of Ge(12 nm)/Au(27nm)/Ag(50nm)/Au(100nm) were deposited on the substrate side of the wafer and Ti(10nm)/Au(400nm) were deposited on the top by metal evaporation. Finally, the wafer was scribed into chips of dimensions 2.5mm by 500 m. SEM picture s of the fabricated low ridge QCLs are shown in Figure 4 12 Both insulator and contact layers are smooth and etching quality is good. As discussed above etching was done using so called pulsed anodization etching technique. This is a fast, inexpensive and safe procedure. PAE setup is shown in Figure 4 13

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64 Solution is composed of glycol (40): water (20): 85% phosphoric acid (1) (GWA). Generator drives 50V, 100Hz and 0.7ms wide pulses through the solution. When the pulse is on OH ions are attracted to the positive polarity applied to the sample. As a consequence, due to chemical reaction native oxide is growing on the sample surface. When pulse is off GWA solution mildly etches the native oxide. In result, the oxide slowly propagates through the structure. When desired etch ing depth is achieved, native oxide can be removed with BOE or KOH solutions. Etching rate can be increased by adding a small percentage o f BOE (GWA (8:4:1) 650ml: BOE (7:1) 7ml) (Figure 4 14 ). 4.6 Experimental R esults for Fabricated Unipolar Low Ridge QCL s Operating parameters for all testing procedures were 60ns pulse width at 5kHz repetition rate. The lasi ng wavelength measured with Fourier Transform Infrared Spectrometer (FTIR) equipped with Mercury Cadmium T elluride ( MCT ) detector cooled to 80K was found to be 5.3 m. Figure 4 1 5 shows the power vs. current (P I) curves for the low ridge QCLs. The laser was placed in cryostat and light was focused using two Ge lenses on a room temperature MCT detector. At 80K threshold curr ent, maximum optical power per facet and slope efficiency per facet were measured to be 2A, 6.7W and 730mW/A respectively. The same chip with high reflectivity coated back facet demonstrated at 80K a threshold current of about 1.3A and a peak output pow er of about 12W. This is the record high power reported at this temperature. At 300K these characteristics were 4A and 2.2W respectively. spectral cha nges with current increase (Figure 4 16 ). I V characteristics and corresponding P I curves measured up to 5A at multipl e temperatures are shown in Figure 4 1 7 At higher currents precision of the I V measurements

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65 decreased due to impedance mismatch between the laser and a transformer used to extend the driving range of the current generator. Measurements of threshold current temperature de pendence based on P I data (Figure 4 1 8 ) showed that the characteristic temperature T 0 was 172K Far Field intensity dependence on later al angle is shown in Figure 4 19. Electroluminescence below threshold current is symmetrical. However, it becomes asymmetrical above threshold. The angle corresponding to intensity maximum slowly increases as current increases. Also, spectrum broadens with current increase. As discussed in [49] the beam steering in high power QCLs can be explained using the concept of interference between the two lowest order lateral modes. A small difference in the effective refractive index between these modes causes beat ing along the stripe length. As a consequence, the angle corresponding to the maximum of the FF lateral intensity distribution shifts from one lateral side to another depending on the phase shift between the modes at the output facet (depending on the posi tion within beating period at the output facet). This phase shift is influenced by the current since it changes the effective refractive indices of the modes. As a consequence, the lateral angle corresponding to maximum of the FF distribution shifts with c urrent change. Maximum steering angle in [49] was reported ~10 o approximately the same as measured for our low ridge lasers. For low ridge QCLs however there is a possibility of existence of several lateral modes at high current since effective stripe wid th (current width) is substantially bigger in this case (>70m instead of 13m in [49]). The experimental data presented above demonstrate that low ridge QCLs are capable of giving very high peak pulsed optical powers with relatively low threshold curre nts. Also, the qualitative mechanism on current spreading responsible for operation of these devices was

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66 suggested. Far field measurements demonstrated beam steering. This indicates presence of several lateral modes. Spectrally resolved near field measurem ents would be very useful for better understanding of the FF behavior. High ridge lasers from the same wafer were made with the goal of comparing their performance with the low ridge lasers. However, due to processing issues, reliable lasing was not achi eved for the high ridge lasers. Since we couldn't repeat the experiment due to lack of material, we modeled current spreading in the low ridge lasers based on high ridge data presented in [44]. In the next section a 2 D current spreading model quantitative ly supporting the current spreading mechanism suggested in this work will be developed.

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67 Figure 4 1 Schematic of electron transitions in unipolar QCLs Figure 4 2 Schematics of intersubband and interband transitions. For intersubband transitions joint density of states is substantially narrower

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68 Figure 4 3 Transverse near field distribution for the surface plasmon waveguide Fi gure 4 4. High ridge configuration for unipolar QCL.

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69 Figure 4 5. Schematic of the low ridge laser (dimensions are given in microns). Figure 4 6. Typical IV curve for a high ridge unipolar QCL

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70 Figure 4 7. Energy band diagram of the active region used in the low ridge structure.

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71 Figure 4 8. Trans verse NF distribution in the low ridge QCL under the ridge Figure 4 9 Far field distribution corresponding to NF under the ridge

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72 Figure 4 10 Transverse NF distribution in the low ridge QCL under the channel. Figure 4 11 Far field distribution corresponding to NF under the channels.

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73 Figure 4 12. SEM pictures of the fabricated low ridge unipolar QCLs.

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74 Figure 4 13 Pulsed anodization etching setup. Figure 4 14 Etching rate of InP with GWA (8:4 :1) mixed with BOE in ra tion 600 to 7

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75 Figure 4 15 Power vs. current characteristics for the realized low ridge QCL.

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76 Figure 4 16 Laser spectra measured at 4A and 20A taken at 80K. Figure 4 17 Voltage vs. current characteristics and corresponding power vs. current curves measured at multiple temperatures (laser is without HR coating).

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77 Figure 4 18 Dependence of the threshol d current on temperature. Figure 4 19 Far field intensity distribution for a low ridge QCL.

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78 Table 4 1 List of parameters used to calculate transverse NF distribution in the low ridge QCL under the ridge Layer Thickness, um n Gold inf 1.83 42.3 InP, 1E19 0.2 2.63 0.005 InP, 1E17 2 3.1 0.0000362 InGaAsP, 1E17 0.03 3.18 0.0000471 InGaAs, 3E16 0.3 3.26 0.000008058 AR 1.53 3.49 0.000013 InGaAs, 3E16 0.3 3.26 0.00000806 InGaAsP, 1E17 0.03 3.18 0.0000471 InP, 1E17 2 3.1 0.0000362 InP, 2E17 inf 3 0.0000697 Table 4 2 List of parameters used to calculate transverse NF distribution in the low ridge QCL under the channels Layer Thickness, um N Gold Inf 1.83 42.3 Si 3 N 4 0.3 2 0 InP, 1E17 0.8 3.1 0.0000362 InGaAsP, 1E17 0.03 3.18 0.0000471 InGaAs, 3E16 0.3 3.26 0.000008058 AR 1.53 3.49 0.000013 InGaAs, 3E16 0.3 3.26 0.00000806 InGaAsP, 1E17 0.03 3.18 0.0000471 InP, 1E17 2 3.1 0.0000362 InP, 2E17 Inf 3 0.0000697

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79 CHAPTER 5 CURRENT SPREADING MODEL FOR UNIPOLAR QCLS 5.1 Previous W ork on C urrent S preading S imulation in U nipolar QCLs Current spreading model for unipolar, stripe contact QCLs was developed in [31].In this work it was suggested that this configuration can be useful for optimal heat dissipation and single spatial optical mode operation. It was also explained that current spreading in these devices is substantially different from current spreading in diode lasers. In particular, since carrier concentration in each stage of the active region is fixed by injector doping, there is no diffusion spreading component in the active region. As a consequence, there is only Ohmic current spreading component given by (5 1) where (5 2) Also, the effective charge separation in each stage is negligible compared to typical scale of the current spreading, it can be assumed that there is no space charge in the active region. As a consequence, using charge conservation law we get (5 3) Substitution of Eq. 5 1 into Eq. 5 3 gives for each layer (5 4) where (5 5)

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80 Solution to Eq. 5 4 was obtained in [31] extending solution given in [49]. For each layer of the QC st ructure (excluding the active region ), the conductivities and were taken to be isotropic ( ) and equal to the bulk conductivity of the layer material. Lateral conductivity for the active region was calculated as a weighted average of the bulk conductivities of al l of the layers composing the active regi on Transverse conductivity for the active region was assumed to be constant and was found by fitting simulated IV curve to the measured one. Using this approach, it was found in [39] that current spreads mostly in the layers above the active region and current width decreases as cu rrent increases. In our work we extended this model by taking into account lateral variation of the active region effective transverse conductivity. Also, we extended simulation above voltage corresponding to roll over point of the PI characteristic. 5. 2 Two Dimensional C urrent S preading M odel 5.2.1 Active R egion T ransverse C onductivity Schematic diagram of the low ridge QCL is shown in Figure 5 1. X axis was chosen to be along the lateral direction and y axis along the transverse direction and pointing downwards. These axes intersected at the center of the top of the contact stripe. It was assumed that voltage applied to the ridge was V 0 and voltage applied to the substrate was zero. Also, since there was no electrical current flowing through the rest of the boundaries, corresponding spatial voltage derivatives or taken at these boundaries were also assumed to be zero (see Figure 5 1). As mentioned above, each layer of the structure is characterized by its transverse and lateral conductivities and thickness.

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81 Eq. 5 4 is valid in a layer with constant transverse and lateral conductivities. In the general case, both of these quantities can have spatial dependence. Therefore, Eq. 5 4 should be modified to (5 6) In this form it is applicable to the whole layered structure. Following [39 ], for the AR was calculated as a weighted average of the bulk conductivities of all of the layers composing the AR. However, for the active region depends on the active region desi gn and the voltage across the active region : (5 7 ) The dependence of the active region transverse conductivity on the voltage across the active region (Eq. 5 7) can be calculated from the active region IV for a high ridge laser fabricated from the same wafer as the low ridge laser The wafer discussed in [44, Fig.2a] was grown in the same MOCVD reactor as our structure under similar conditions and has the same active region design. Since cladding layers, waveguide layers and substrate have high condu ctivities we believe that the I V measured for the h igh ridge laser reported in [44] is a good appro ximation to its active region I V (For example, voltage a cross the high ridge QCL in Figure 2a at 0.75A is equal to 12V. Using typical conductivity values for waveguid e layers, cladding layers and substrate it can be shown that total voltage drop across these layers at 0.75A is less than 0.3V. Therefore, if voltage drop across the contacts is small the laser I V is basically determined by its active region IV). As a con sequence, we believe it is appropriate to use the IV reported in [44, Figure 2a] for simulation of current spreading mechanism in realized low ridge QCLs. This IV and corresponding dependence of active region transverse conductivity on the

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82 voltage drop acr oss active region is shown in Figure 5 2a, b. Part of the IV curve in the range from 13V to 18V was obtained by the extrapolation (dashed line) to emphasize the effect of current spreading when transverse conductivity of central part of AR gets low. Figure 5 2b was obtained assuming a uniform current density flowing through the active region in the high ridge lasers reported in [44, Figure 2a]. Using shown in Figure 2b, Eq. 5 6 and Eq. 5 7 were solved self consistently for a low ri dge QCL using the 2D Finite Difference Method. Calculation procedure is described in the next section. 5.2.2 Two D imensional Finite Difference M ethod The two dimensional Finite Difference method is a numerical type of solution to a differential equation ( 5 8) using (5 9) (5 10) (5 11) (5 12) Usually the calculation procedure is done as follows. The structure is divided by a rectangular mesh (Figure 5 3) into cells. Boundary u values corresponding to the physical problem are set at u values are set at the all inner points.

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83 Boundary conditions are left unchanged through the calculation procedure. U values at each point P are found from the u values of the four neighbor points of the mesh using Eq. 5 8 through Eq. 5 12. This procedure ( calculation of u values at all the points) is repeated iteratively u values at the inner points). Using found values at each point of the mesh all related quantities such as can be found. In our case we had to solve Eq. 5 6. Using 2D Finite Difference Method this equation can be modified as (5 13) or (5 14) Making the transformations and then gives (5 15) Using Eq. 5 15 we can find V values at each point P(i, j) using V values at neighbor points (i+1,j; i 1,j; i,j+1; i,j 1) Figure 5 4. Also, conductivity values should be set at points C (i+1/2,j; i 1/2,j; i,j+1/2; i,j 1/2). Figure 5 5a, b illustrate the rectangular mesh used for the numerical calculation. Increments along x and y directions, and we chosen to be 1 m and 0.2 m

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84 respectively As discussed above, voltage on the top of the ridge and at the bottom of the substrate we fixed at V 0 and 0 respectively. Since there is no current through all the other boundaries we can assume that voltage for the neighbor points at each boundary should be the same. For example, since there is no current flow through the left vertical edge of the laser, at this edge the following equation should hold (5 16) Employing the discrete approach used here we get (5 17) Therefore, at the left vertical boundary the following condition is true (5 18) Calculation procedure is shown in Figure 5 5. As mentioned above, initial voltage values important. Therefore, for simpl icity initial voltage values at the all inner points were taken to be equal to zero. Initial value for is taken as 1 Each cycle of the solution of Eq. 5 6 (using Eq. 5 15) involves calculation of the voltage value at each point of the structure using voltage and conductivity values at the neighbor points (Figure 5 4b). This cycle is repeated until the convergence condition is satisfied. After that the active region transverse conductivity values are up dated using Eq. 5 7 that reflects the dependence given in Figure 5 2b. If new values are close enough to the old ones, current density distribution can be calculated using obtained voltage distribution. If not, Eq. 5 6 should be solved again using the new active region transverse conductivity values.

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85 5. 3 Model R esults Parameters used for calculation are listed in Figure 5 6. Since substrate conductivity is high it is appropriate to neglect by voltage drop across the substrate. Therefore, its thickness was taken to be ~7m instead of 150m defined by the wafer processing Active region lateral sensitive to the choice of this parameter. A convenient way to interpret the model results is to examine the evolution of the lateral dis tribution of the active region as a function of bias change (Figure 2a), and by doing so, determine its influence on cu rrent density distribution (Figure 2b), current width and the low ridge IV curve (Figure 2c). When voltage acr oss the low ridge device is below approximately 6V, transverse conductivity of the AR is low and almost constant. Due to the low conductivity, current density distribution at the AR for these voltages is broad and has large tails. When the device bias incr eases above 6V, the conductivity of the central part of the AR increases rapidly with increasing voltage, as can be seen in Figure 2b. The high transverse conductivity of the AR central part and the peak shape of the transverse conductivity lateral distrib ution lead to a strong narrowing of the current density profile. Further voltage increase leads to decrease of the transverse conductivity of the central part of the AR. Due to the decrease in conductivity, current starts to spread in the lateral direction and the effec tive current width goes up (Figure 2c), extending the laser dynamic range. Spreading is expected to be stronger at lower temperatures since mobility increases rapidly as the device temperature decreases. The IV curve for the low ridge laser does not show a strong increase in differential resistance at high current (Fig ure 2c), the effect shown in Fig.1a for the high ridge laser. This

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86 difference is a consequence of lateral current spreading in the low ridge structure when the transverse conduc tivity of the central part of the AR goes down. The simulation results show that low ridge QCLs have relatively low threshold current due to current narrowing when the central part of the active region enters high conductivity mode of 2 3 higher than threshold for high ridge lasers processed from the same wafer). It is also demonstrated that high powers achievable by these lasers may be due to lateral current spreading at higher voltages. The work discussed here on the low ridge, uni polar QCLs is summarized in [51]. Figure 5 1 Schematic of the low ridge QCL.

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87 Figure 5 2 A ) H igh ridge IV curve B) the corresponding dependence of the transverse active region conductivity on the voltage across the active region.

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88 Figure 5 3. Rectangular mesh used for 2D Finite Difference Method Figure 5 4 A ) R ectangular me sh used for the low ridge QCL. B) B asic cell of the mesh

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89 Figure 5 5 Schematic of the calculation procedure. Figure 5 6 List of parameters used for simulation.

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90 Figure 5 7. Current spreading model results. A) Lateral dependence of the active region transverse conductivity at different current values. B) Corresponding lateral distribution of the current density at the active region. C) Low ridge IV curve and current width dependence on the current calculated based on (B).

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91 Figure 5 7. Continued

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92 Figure 5 7. Continued

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93 CHAPTER 6 SUMMARY AND FUTURE WORK 6.1 Bipolar Quantum Cascade Lasers It was demonstrated in Chapters 2 and 3 that deep quantum wells can be effectively used for designing tunnel junctions with relatively low effective resistivity. Employment of the quantum well tunnel junctions is esp ecially important when very high doping density is not possible for the layers composing the tunnel junction. In this work this approach allowed us to achieve double stage laser slope 2.0 W/A twice that of the single stage laser even though maximum doping level for the n 18 cm 3 ). Results of the developed current spreading model for the double stage laser showed that the substantial current spreading between the first and the second active regions in our str ucture (and corresponding relatively high threshold current for the second active region) was not a consequence of the quantum well tunnel junction resistivity but rather caused by presence of the n layers with high mobility and relatively large total thic kness of the layers between the active regions. Further experimental and theoretical study of the influence of quantum well parameters (such as quantum well thickness and depth) on the tunnel junction resistivity at different doping levels of the layers co mposing the tunnel junction would be very helpful. This information would make the future design of semiconductor devices incorporating quantum well tunnel junctions (bipolar QCLs, multiple junction collar cells, etc.) much more flexible since it would sub stantially relax the requirement of the very high doping in the tunnel junction layers. Also, wall plug efficiency could be reduced due to lower voltage drop across the tunnel junction at the same current. Since the tunnel junction comprises the highly str ained InGaAs layer, reliability

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94 testing is necessary to verify wither the strain in the structure reduces the lifetime of the double stage laser. 6.2 Unipola r Quantum Cascade Lasers In Chapters 4 and 5 we demonstrated that low ridge unipolar quantum cascad e lasers can deliver very high peak pulsed optical powers (12W at 80K for the laser with HR coated back facet) with relatively low threshold current (~1.5A at 80K for the same laser). Developed 2D numerical current spreading model takes into account latera l dependence of transverse active region effective conductivity, in contrast to the original model. Also, the new model was extended to the high differential resistance range at high bias caused by the coupling breaking between the injector and upper laser electron energy levels. It was shown that the low threshold current was a consequence of the current density distribution narrowing at voltages corresponding to a good alignment between the lowest injector energy level and the upper laser level. It was al so suggested that the high optical power achieved by the low ridge laser was due to the additional current spreading in the structure caused by the increase in the differential resistance of the central part of the active region at voltages when this align ment breaks. Far field measurement results demonstrated current beam steering for the low ridge lasers. Possible reason for this effect is the co existence of several lateral modes with slightly different effective refractive indices. Far field intensity d istribution is determined by the interference between these modes and, as a consequence, depends on the phase shift between them at the output laser facet. The current beam steering occurs as a result of the effective refractive indices dependence on the c urrent since the phase shift also changes in this case. Further study of this effect can include Near Field measurements at different currents. In particular, spectrally resolved Near Field observation would allow thorough analysis of the Far Field behavio r. Also, numerical simulation of the Near Field and Far Field intensity distributions

PAGE 95

95 would be very useful for understanding the current steering. This simulation should take into account the lateral variation of the current density distribution and coupli ng between the current density and the optical modes. Finally, since the low ridge devices operate at high voltages (up to 40V), reliability testing has to be done to verify whether these lasers can be used in practical high power mid infrared applications If performance degradation with time is observed, structural changes improving device reliability should be made. For example, increase of the contact window width on the top of the ridge to reduce current density flowing through this region could be hel pful.

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96 APPENDIX A CALCULATION OF ELECT RON ENERGY LEVELS AN D WAVEFUNCTIONS IN LAYERED STRUCTURES Calculation procedure described here is to a large degree based on the approach discussed in [52]. In general case it is required to find electron energy levels and wavefunctions in a layered structure with doped layers taking into account band non parabolicity. Also, the electron effective mass can change at the interfaces between the layers. The governing equations in this case are coupled Schrdinger and Poisson equations: (A 1) (A 2) where m* is the electron effective mass, V(z) is the potential profile and is the spatial charge density. 1 and then extend the solution to general case by taking into account its coupling with Eq. A 2. Eq. A 1 can be modified using the expansion given in Eq. 5 10 (taking int o account effective mass discontinuity) (A 3) or (A 4) which can be modified as

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9 7 (A 5) Therefore, (z+ z) can be found using (z), (z z) and m (z+ z/2), m (z z/2). Solution to Eq. A 1 is found as follows. The (left) boundary condition used for Eq. A 5 is (0) = 0 and ( z ) = 1 Using it, (z) at all the other points is found using Eq. A 5. When the energy E is not a solution of Eq. A 1, (z) diverges as z approaches the right boundary (accumulation of the error). The sign of this divergence changes when the energy passes its solution value. During the calculation process the energy is varied in a particular range (usually from the energy corresponding to t he bottom of the quantum well layers up to the energy corresponding to the top of the barrier layers). When the sign of the divergence changes an iterative procedure is used to find the solution. When external electric field E E F (z) is applied to the struc ture, V(z) should be taken as (A 6) Figure A 1 shows results of the calculation procedure described above for a GaAs based structure with AlGaAs barriers and InGaAs quantum wells. Applied electric field was assumed to be taken into account to create the potential profile for this structure (transition layer thickness was assumed to be equal to two monolayers). Inclusion of Poisson equa tion can be done using found electron energy levels and wavefunctions. The approximation that is typically used in this case is that all donor atoms are fully ionized and electrons from these atoms are distributed among the found electron energy levels. So lution of the rate equations is necessary to find the electron population on each of these

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98 levels. Potential created by positive ionized donor atoms and electrons distributed among the energy levels can be found using the following formulas. Assuming t hat the layered structure is infinite in the x y plane, electric field ( EF(z) ) created by electrons and ionized dopant atoms (with doping density d(z) ) can be found using (A 7) (A 8) (A 9) where is the areal charge density given by Eq. A 8 and N i is the electron population on the i electron energy level (found using the rate equations). The latter equation reflects charge neutrality in the structure. Using found EF(z) corresponding potential can be calculated using (A 10) Therefore, the full calcul ation procedure for solution of the coupled Eqs. A 1 and A 2 can be described by the scheme presented in Figure A 2. Eq. A 1 is solved using the expansion given by Eq. A 5. Found electron energies and wavefunctions are used in the rate equations to find el ectron population on each energy level. Using Eqs. A 7 through A 10 potential profile is updates and compared to the previous profile. If convergence is achieved, all desired quantities can be calculated using obtained energy values and corresponding wavef unctions. If no, the new potential profile should be used in the next iteration. Example of an employment of this procedure for doping optimization in quantum cascade lasers can be found in [53].

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99 Figure A 1 Model results for a GaAs based layered structure.

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100 Figure A 2. Step sequence f or the calculation procedure Solution of Eq.(A1) using Eq.(A5) (1 st iteration is done neglecting by spatial charge) Using found i and E i solve rate equations, which gives N i Using found i N i and d(z) find V (z) V(z)[new] = V(z)[w/o spatial charge]+ V (z) V(z)[new] =? V(z)[old] NO YES Output parameters V(z) = V(z)[new]

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101 LIST OF REFERENCES [1] J. van der Ziel and W. T junctions, Appl. Phys. Lett. vol. 41, pp. 499 501 1982. [2] J. Faist F. Capasso, D. Sivco C. Sitori A. Hu tchinson Science vol. 264, pp. 553 556, 1994. [3] power AlInGaAs GaAs LEOS 1999 12th Annu. Meeting vol. 1, pp. 80 81, 1999. [4] J. Kim E. Hall, O. Sjolund, and L. stacked multiple active region Appl. Phys. Lett. vol. 74, pp. 3251 3253, 1999. [5] F. Dross, F. van Dijk, O. Parillaud, B. transverse mode InGaAsP InP edge IEEE J. Quantum Electron. vol. 41 pp. 1356 1360, 2005. [6] junction connected distrib uted feedback vertical cavity surface Appl. Phys. Lett. vol. 73, pp. 1475 1477, 1998. [7] J. room temperature continuous wave operation of multiple active region 1.55m vertical cavity su rface emitting lasers wit h Appl. Phys. Lett. vol. 77, pp. 3137 3139, 2000. [8] bipolar cascade vertical cavity surface emitting lasers: IEEE J. of Select. Topics in Quantum Electron. vol. 9, pp. 1406 1414 2003. [9] consumption in GaA s Electron. Lett. vol. 38, pp. 1259 1261, 2002. [10] J. Garcia, E. Rosencher, P. Collot, N. Laurent, J. Guyaux, B. Vinter, and J. Nagle Appl. Phys. Lett. vol. 71, pp. 3752 3754, 1997 [11] Appl. Phys. Lett. vol. 94, pp. 7370 7372, 2003. [12] Quantum Well Lasers P. Zory (Ed.). San Diego, CA: Academic, 1993. [13] Arsenide based bipolar cascade lasers with deep quantum Photon. Technol. Lett. vol. 18, pp. 2656 2658, 2006.

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102 [14] R. Lang, A. Larsson, and G. IEEE J. Quantum Electron. vol. 27, pp. 312 320, 1991. [15] AlxGa1 xAs Double Heterostru Japanese J. Appl. Phys. vol. 12, pp. 1585 1592, 1973. [16] crowded carrier confinement in double J. Appl. Phys. vol. 51, pp. 2394 2401, 1980. [17] in double J. Appl. Phys., vol. 53, pp. 7235 7239, 1982. [18] Z consistent two dimensional model of quantum well semiconductor lasers: optimization of a GRIN SCH SQW las er IEEE J. Quantum Electron. vol. 28, pp. 792 802, 1992. [19] L. Coldren and S. Corzine, Diode Lasers and Photonic Integrated Circuits New York, NY: Willey, 1995, p. 199. [20] Y. Goldberg, Handbook Series on Semiconductor Parameters vol. 2. London, UK: World Scientific, 1999. [21] B. Streetman and S. Banerjee, Solid State Electronic Devices Upper Saddle River, NJ: Prentice Hall, 2000, p. 99. [22] c waves in a semiconductor Sov. Phys. Semicond. vol. 5, pp. 707 709, 1971. [23] 3.4m) quantum cascade lasers based on strained compensated I Appl. Phys. Lett. vol. 72, pp. 680 682, 1998. [24] M. Beck, D. Hofstetter, T. Aellen, J. Faist, E. Gini, H. Melchior, U. Oesterle, and M. infrared semiconductor laser at room Scie nce vol. 295, pp. 301 305, 2002. [25] temperature, high power, continuos wave operation of buried heterostructure quantum Appl. Phys. Lett. vol. 84, pp. 314 316, 2004. [26] C. Sirtori, P. Kruck, S. Barberi, P. Collot, J. Nagle, M. Beck, J. Faist, and U. Oesterle, Appl. Phys. Lett. vol. 73, pp. 3486 3488, 1998.

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106 BIOGRAPHICAL SKETCH Physics and Technology in 2001 and 2003 respectively. From 2001 until 2003, he also had a part time job as a researcher at Lebedev Physical Institute, Moscow, Russia. During that time, he studied semiconductor laser gain spectrum and its dependence on driving current and temperature. Since 2003, he has been working toward his PhD degree at the University of s research in this group has been primarily focused on quantum cascade laser physics and technology. In February 2007, he accepted a job offer for a Senior Scientist position at Pranalytica Inc. based in Santa Monica, California. His job responsibilities w ill include design and realization of high performance unipolar quantum cascade lasers.


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Copyright Date: 2008

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Title: High-Power Bipolar and Unipolar Quantum Cascade Lasers
Physical Description: Mixed Material
Copyright Date: 2008

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HIGH-POWER BIPOLAR AND UNIPOLAR QUANTUM CASCADE LASERS


By

ARKADIY LYAKH















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007































O 2007 Arkadiy Lyakh



































To the memory of my dear mother-in-law.









ACKNOWLEDGMENTS

First, I would like to thank my advisor, Prof. Peter Zory, for his guidance and

encouragement throughout my research. I learned from him very diverse techniques on

semiconductor laser design, processing and characterization, and I am convinced that this

knowledge will be very helpful to me in the future. In addition, in his lab he has created an

atmosphere of trust and understanding. Therefore, the years that I spent working with him will

always remain a pleasant memory for me. It was a privilege to have Prof. Zory as my

supervisor. I would also like to thank Prof. Rakov, Prof. Xie and Prof. Holloway for being the

members of my supervisory committee.

Finally, I would like to thank my dear wife, mother, father and sister for their support

through the four years of my study at the University of Florida. Their encouragement and love

was very important to my success.











TABLE OF CONTENTS


page

ACKNOWLEDGMENT S .............. ...............4.....

LI ST OF T ABLE S ................. ...............7....____......

LIST OF FIGURES .............. ...............8.....

AB S TRAC T ........._. ............ ..............._ 1 1...

CHAPTER

1 INTRODUCTION ................. ...............13......... .....

2 BIPOLAR CASCADE LASERS ............ ..... ._ ...............21...


2. 1 Light Polarization Rules for Bipolar and Unipolar QCLs ......____ ...... ...__ ...........21
2.2 Basic Operational Principles of Bipolar QCLs ................. ...............24........... ..
2.3 Review of Previous Work on Bipolar QCLs ................. ...............25..............
2.4 Experimental Results on Fabricated Bipolar QCLs ................. ................ ......... .27

3 SIMULATION OF CURRENT SPREADING IN BIPOLAR QCLs .............. ..................40

3.1 Influence of the Tunnel Junction Resistivity on the Current Spreading in Bipolar
Q C Ls ................ ....... ... .. .. ... .. ... ................4
3.2 Previous Work on Current Spreading Simulation in Diode Lasers .............. ..............43
3.3 Simulation of Current Spreading in SSL ......__....._.__._ ......._._. ..........4
3.4 Simulation of Current Spreading in DSL .............. ...............48....

4 UNIPOLAR QUANTUM CASCADE LASERS .............. ...............55....

4.1 Basic Operational Principles of Unipolar QCLs .............. ...............55....
4.2 Review of Previous Work of Unipolar QCLs ....._.__._ ..... ... .__. ........_........5
4.3 Low-Ridge Configuration Concept ................. ...............59...
4.4 Waveguide Structure for the Low-Ridge QCLs .............. ...............60....
4.5 Fabrication of the Low-Ridge QCLs .............. .... ........_.._ .............. ...........6
4.6 Experimental Results for Fabricated Unipolar Low-Ridge QCLs .............. ..................64

5 CURRENT SPREADING MODEL FOR UNIPOLAR QCLs .............. .....................7

5.1 Previous Work on Current Spreading Simulation in Unipolar QCLs ............... ...............79
5.2 Two-Dimensional Current Spreading Model .............. ...............80....
5.2.1 Active Region Transverse Conductivity .............. ...............80....
5.2.2 Two-Dimensional Finite-Difference Method.........._..._.._ .........................._82
5.3 M odel Results .............. ...............85....











6 SUMMARY AND FUTURE WORK .............. ...............93....


6. 1 Bipolar Quantum Cascade Lasers ........._._ ........... ...............93...
6.2 Unipolar Quantum Cascade Lasers .............. ...............94....

APPENDIX

A CALCULATION OF ELECTRON ENERGY LEVELS AND WAVEFUNCTIONS IN
LAYERED STRUCTURES .............. ...............96....

LIST OF REFERENCES ................ ...............101................

BIOGRAPHICAL SKETCH ................. ...............106......... ......










LIST OF TABLES


Table page

2-1 Dependence of tunneling probability exponential factor D on indium composition........ .39

3-1 List of parameters used for simulation of current spreading in bipolar QCLs. .................54

3-2 Current spreading simulation results for bipolar QCLs. .................. ................5

4-1 List of parameters used to calculate transverse NF distribution in the low-ridge QCL
under the rid ge. ............. ...............78.....

4-2 List of parameters used to calculate transverse NF distribution in the low-ridge QCL
under the channels s ................. ...............78........... ....











LIST OF FIGURES


Figure page

1-1 Semiconductor laser and its typical output power Po versus current I characteristic.
At threshold current Ith, laser action is initiated ................. ...............16..............

1-2 Interband mechanism ................. ...............17................

1-3 Intersubband mechanism .............. ...............18....

1-4 Illustration showing how an electron loses energy in a bipolar QCL ........._..._... ..............19

1-5 illustration showing how an electron loses energy in a unipolar QCL. .............. ...............20

2-1 Energy band diagram for SSL and DSL ................ ...._.._ ...............31. .

2-2 Ilustration of current spreading in bipolar QCLs. ............. ...............32.....

2-3 Schematic of TJ and QWT J. ............. ...............32.....

2-4 Bipolar cascade lasers with different waveguide configuration. A) Separate
waveguide for each stage. B) Single waveguide for all stages. ............. ....................33

2-5 Vertical cavity surface emitting bipolar QCL. ...._.._.._ .... .._._. ...._.._...........3

2-6 Schematic of TJ. ........._. ............ ...............34...

2-7 Quantum well tunnel junction ................. ...............35...............

2-8 Single-stage and double-stage structures ................ ...............35........... ...

2-9 Near-field pattern of the double-stage laser ................. ...............36........... ..

2-10 Far-field pattern of the double-stage taken at 2A ................. ...............36........... .

2-11 Voltage vs. current characteristics for the double-stage and single-stage lasers. ...........37

2-12 Power vs. current characteristics for the double-stage and single-stage lasers near
threshold and at high power levels............... ...............38.

3-1 Structure (comprising tunnel junction) used for illustration of influence of TJ
resistivity on current spreading. .............. ...............52....

3-2 Illustration of the current flow in stripe geometry SSL. ....._____ ... ....___ .............52

3-3 Voltage vs. current characteristics for the new double-stage and single-stage lasers. ......53

3-4 Near-field pattern of a new double-stage laser measured at 1A ................... ...............53











4-1 Schematic of electron transitions in unipolar QCLs ....._._._ ... ....... ........_.......67

4-2 Schematics of intersubband and interband transitions ................. .......... ...............67

4-3 Transverse near field distribution for the surface plasmon waveguide .............................68

4-4 High-ridge configuration for unipolar QCL. ............. ...............68.....

4-5 Schematic of the low-ridge laser (dimensions are given in microns). .............. .............69

4-6 Typical IV curve for a high-ridge unipolar QCL ................. ...............69........... .

4-7 Energy band diagram of the active region used in the low-ridge structure. ................... ...70

4-8 Transverse NF distribution in the low-ridge QCL under the ridge ................. ........._....71

4-9 Far Hield distribution corresponding to NF under the ridge. ............. .....................7

4-10 Transverse NF distribution in the low-ridge QCL under the channel. ............. ................72

4-11 Far Hield distribution corresponding to NF under the channels. ............. ....................72

4-12 SEM pictures of the fabricated low-ridge unipolar QCLs. ................ .......................73

4-13 Pulsed anodization etching setup. ............. ...............74.....

4-14 Etching rate of InP with GWA (8:4: 1) mixed with BOE in ration 600 to 7. .....................74

4-15 Power vs. current characteristics for the realized low-ridge QCL ................. ................75

4-16 Laser spectra measured at 4A and 20A taken at 80K. ............. ...............76.....

4-17 Voltage vs. current characteristics and corresponding power vs. current curves
measured at multiple temperatures (laser is without HR coating) .................. ...............76

4-18 Dependence of the threshold current on temperature. ...........__......_ ..............77

4-19 Far-Hield intensity distribution for a low-ridge QCL. ............. ...............77.....

5-1 Schematic of the low-ridge QCL. ........._. ...... .__ ...............86..

5-2 A) High-ridge IV curve. B) the corresponding dependence of the transverse active
region conductivity on the voltage across the active region. ............. .....................8

5-3 Rectangular mesh used for 2D Finite-Difference Method ........._.. ....... ._. ............88

5-4 A) Rectangular mesh used for the low-ridge QCL. B) Basic cell of the mesh. .................88

5-5 Schematic of the calculation procedure. ................ ...............89...............











5-6 List of parameters used for simulation. ................ ........................ ..............89

5-7 Current spreading model results. ............. ...............90.....

A-1 Model results for a GaAs-based layered structure. .............. ...............99....

A-2 Step sequence for the calculation procedure. ...._.._................. .. ......_..........10









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

HIGH-POWER BIPOLAR AND UNIPOLAR QUANTUM CASCADE LASERS

By

Arkadiy Lyakh

May 2007

Chair: Peter Zory
Major: Electrical and Computer Engineering

High-power bipolar and unipolar quantum cascade lasers were designed, fabricated and

characterized. The importance of lateral current spreading is emphasized since it plays an

important role in operation of these devices.

Edge-emitting, gallium arsenide (GaAs) based bipolar cascade lasers were fabricated from

metalorganic chemical vapor deposition-grown material containing two diode laser structures

separated by a quantum-well tunnel junction (QWTJ). The QWTJ was comprised of a thin, high

indium content indium gallium arsenide layer sandwiched between relatively low-doped, p-type

and n-type GaAs layers. Comparison of near field data with predictions from a one dimensional

current spreading model shows that this type of reverse-biased QWTJ has a low effective

resistivity. As a consequence, current spreading perpendicular to the laser length in the plane of

the layers (lateral direction) is reduced leading to a relatively low threshold current for the

second stage. In addition, the differential quantum efficiency ~150% of these double stage lasers

is nearly twice that of single stage lasers.

Low-ridge unipolar quantum cascade lasers operating at 5.3Clm were fabricated from InP-

based MOCVD-grown material. Record-high maximum output pulsed optical power of 12W at

14A was measured from a low-ridge chip with a high reflectivity coated back facet at 80K. Also,









Far-Field measurements demonstrated current beam steering for this device. Modeling shows

that the lateral variation of transverse conductivity is essential for an accurate description of

current spreading in these devices.









CHAPTER 1
INTRODUCTION

Since their first demonstration in 1962, semiconductor lasers have become the enabling

components in many applications. Emission wavelengths for these devices now span the range

from the ultra-violet (0.4Clm for GaN-based interband lasers) to the far infrared (hundreds of

microns for intersubband lasers). Optical powers from single chips are now in the 100-watt

range in pulsed operation and the several-watt range in continuous operation. Conversion

efficiencies from electrical to optical power can be in the 50% range for red and near infrared

lasers with world record numbers now exceeding 70%.

Figure 1-1 is a diagram showing a typical semiconductor laser chip and its most important

characteristic, laser light output power Po versus current I. Electrons associated with current are

temporarily trapped in the active region where they lose energy either by photon emission or

some non-radiative process. Photons can be generated by either an electron-hole recombination

process between conduction and valence bands (interband mechanism, Figure 1-2) or a simple

electron transition between energy levels in the conduction band (intersubband mechanism,

Figure 1-3). Stimulated photons are confined and directed by a built-in waveguide and at a

sufficiently high current (threshold current Ith), laser action is initiated (Figure 1-1). As current

increases above Ith, Po continues to increase linearly to some high level that depends on the type

of laser chip used. Typical chip size as defined by L in (Figure 1-1) is about one millimeter.

Due to their small size, low cost and ability to be directly pumped by electrical current,

low power semiconductor lasers are now widely used in applications such as printing, optical

memories and fiber-optic communications. For applications such as infra-red countermeasures,

free space optical communications, machining and range/ranging measurements, high power

lasers are required. In order to achieve very high output powers, semiconductor lasers are










operated in the pulsed mode using high current pulses, tens of nanoseconds wide. In order to get

even higher powers without increasing power supply current requirements, one can employ

designs using two or more optical gain (active) regions in the laser material. Since each active

region (AR) is sandwiched between a number of layers, this combination being called a stage,

electrons can produce photons as they move from stage to stage through the multistage device.

As a consequence, it's possible in principle to obtain laser power proportional to the number of

stages in the material without increasing drive current.

Semiconductor lasers with two or more stages, usually called quantum cascade lasers

(QCLs), fall into two major categories: bipolar (using interband transitions) [1] (Figure 1-4) and

unipolar (using intersubband transitions) [2] (Figure 1-5). In bipolar QCLs, light emission

occurs due to recombination of electron-hole pairs in the ARs. After the recombination process

in the first stage, electrons tunnel from the valence band of the first stage into conduction band of

the second stage. The tunneling process between stages takes place through a reverse-biased,

heavily doped p-n junction. The tunnel junction is a crucial element in bipolar QCLs since it

allows electron recycling. The number of stages in a bipolar QCL typically ranges from two to

five.

For unipolar QCLs, electrons emit photons by making transitions between conduction band

states (subbands) arising from layer thickness quantization in the AR. After a radiative (or non-

radiative) transition is made, the electron "cools down" in a relaxation-inj section region and then

tunnels to the upper laser level of the next stage. Usually, the number of stages in a unipolar

QCL ranges from 20 to 35.

In both cases, tunneling to the next stage provides the electron with the opportunity to

generate another photon. This electron recycling process in the multistage device leads to










increased output power relative to that obtained from a single stage device operating at the same

current.

In this work both types of QCLs are discussed. A large portion of the discussion is

dedicated to the analysis of current spreading in the plane of the layered structure (lateral current

spreading) since all devices studied were fabricated in either a stripe geometry as shown in

Figure 1-1 or in a low-ridge configuration (Figure 4-5). Since the conductivity of the layers

above the active region is usually quite high, lateral current spreading is substantial and has a

strong influence on laser operation.

In Chapters 2, 3, 4 and 5, various aspects of the high power QCLs that have been designed,

fabricated and characterized during this work are discussed. In Chapter 2, the relationship

between the design of the reverse-biased tunnel junction used in bipolar QCLs and lateral current

spreading is discussed. Experimental results for realized bipolar QCLs are also presented in this

chapter. Output power versus current curves show that the performance of double stage lasers

made using this design are close to the best possible. The lateral current spreading model

developed in Chapter 3 shows that resistivity of the quantum-well tunnel junction used in our

devices is not high enough to cause any additional current spreading. In Chapter 4, the

peculiarities of lateral current spreading in low-ridge unipolar QCLs and their influence on the

operation of these devices are qualitatively described. Output power versus current curves show

that this type of unipolar QCL has the unique feature of low threshold current combined with

very high peak output power. Results of the current spreading model for low-ridge unipolar

QCLs developed in Chapter 5 support the qualitative description of the spreading mechanism

described in Chapter 4. In Chapter 6, the thesis work is summarized and suggestions made for

future research directions.



































P~








Figure 1-1. Semiconductor laser and its typical output power Po versus current I characteristic.
At threshold current Ith, laser action is initiated.



















: Guide Layer


e


Guide Layer


1,


CIB


EC/







EH2


VB


I


hC


Figure 1-2. Interband mechanism whereby conduction band (CB) electrons trapped in the
quantum well active layer recombine with valence band (VB) holes to produce
photons with energy hy.





























eL Qw


Energy

r e-1
1


Figure 1-3. Intersubband mechanism whereby conduction band electrons in a high energy state
in the active region quantum well make a transition to a lower energy state by
emitting a photon with energy hy.


r :~~J. I
;--,-s















































Figure 1-4. Illustration showing how an electron loses energy in a bipolar QCL.

























mnjector








injector













Figure 1-5. Illustration showing how an electron loses energy in a unipolar QCL.









CHAPTER 2
BIPOLAR CASCADE LASERS

2.1 Light Polarization Rules for Bipolar and Unipolar QCLs

As discussed in Chapter 1, the radiative mechanism is different for bipolar and unipolar

QCLs. Using quantum-mechanical approach, in this section of Chapter 2 we demonstrate that

this distinction leads to different light polarization for these devices. In particular, for bipolar

QCLs emitted light is mostly TE-polarized, while for unipolar QCLs TM-polarization is

dominant.

Using Fermi golden rule it can be shown that the absorption (emission) coefficient in the

both cases is proportional to:


(7, | H'| 7,l (2-1)

where ~, and cpyare electron wavefunctions corresponding to initial and final states involved in

the transition:


7:f ,, r; (2-2)

withJ&- envelope functions and uv,v- periodic Bloch functions. The interaction Hamiltonian

H' is given by

H' = ~ )(A p +p ) (2-3)


where A is the vector potential, p is the momentum operator and m*" is the effective mass. Vector

potential A in its turn can be expressed as a polarization unit vector e multiplied by a scalar

function that slowly changes with the spatial coordinate r (its spatial variation is negligibly small

within each unit cell). Therefore, the matrix element in Eq. 2-1 is proportional to









where v v and n, n are the band and subband indices of the initial and final states. The first

right-hand side term of the equation corresponds to interband transitions, while the second one

corresponds to intersubband transitions.

Using the intersubband term it' s easy to show that light emitted by unipolar lasers is

mostly TM-polarized. In this case (21 u ,)=1 for all intersubband transition, since Bloch

functions are approximately the same for electrons within the same band. Using the fact that

there is no size quantization in the x-y plane (plane of semiconductor layers) the envelope

function can be expressed as

fnk, =7 iZ)8-zkir (2-5)


where
plane. As a consequence, dipole matrix element in Eq. 2-4 has the following form

~f~ e- pln fi'k' ~ d're-'*~zkirp (Z) exI~x ey~ p zkire" *,, (z) (2-6)


Terms proportional to ex and e, are nonzero only in case when n=n and k, = k's In other words

these components are always zero for photon absorption and emission. This is a consequence of

the fact that the conservation of energy and wavevector for a transition within the same subband

requires an electron interaction with the lattice (phonons). Therefore, absorption (emission)

coefficient for intersubband case is nonzero only when light has the z-component and it reaches

the highest value when the electric field is fully polarized along the z-direction (perpendicular to

semiconductor layers). This explains why light emitted by intersubband lasers is always TM-

polarized.

The first term on the right-hand side of Eq. 2-4 can be analyzed in a similar way. Overlap

integral between the envelope wavefunctions determines selection rule for interband transitions,









which states that transitions are allowed only between states with the same quantum number in

contrast to intersubband transitions:


(n If = 0 if (ns n') (2-7)

Analysis of the dipole matrix element for the Bloch functions is more complicated than for

envelope functions (intersubband case). This analysis is based on the fact that hole Bloch

functions can be considered as linear combinations of so-called valence band basis functions.

The basis functions in their turn have symmetry of atomic orbitals of an isolated atom. Using this

fact and the fact that interband absorption in quantum well lasers is mostly determined by

conduction band to heavy hole band transitions, analysis of Bloch functions dipole matrix

element leads to the conclusion that light emitted by bipolar lasers is mostly TE-polarized

(electric field lies in the plane of semiconductor layers).

Due to different polarization, near field (NF) intensity distribution for intersubband lasers

has some peculiarities relative to interband lasers. For example, calculated NF shows presence of

intensity discontinues at boundaries between semiconductor layers in accordance with the

boundary condition eEEn= E2En,, while for bipolar lasers it is continuous at all interfaces. In

addition, since dielectric functions for a metal and semiconductor layers have different signs

dielectricc function is negative for the metal since imaginary part of the refractive index for the

metal is bigger than its real part) there is possibility of a surface plasma mode propagating at the

metal/semiconductor interface of unipolar lasers. This effect can be used to make unipolar lasers

based on so-called surface plasmon waveguide (see Section 4-1).

In the rest of the text we will always assume that the radiation corresponding to bipolar

lasers is TE-polarized, while it is TM-polarized for unipolar lasers.









2.2 Basic Operational Principles of Bipolar QCLs

The bipolar cascade laser project was dedicated to development and realization of a

double-stage laser (DSL) comprising a standard single-stage laser (SSL) as a recurrent stage.

Schematics of SSL and DSL are shown in Figure 2-1. After radiative recombination with holes,

electrons tunnel from valence band of the first stage to conduction band of the second stage

through the potential barrier in the tunnel junction (TJ). Therefore, in ideal case each electron

can give rise to two photons. As a consequence, DSL in principle can give twice as much power

as SSL at the same current.

Usually, TJs for bipolar QCLs are composed of two n and p heavily doped layers. If

doping of these layers is not high enough, potential barrier width increases. As a consequence,

the electron tunneling probability reduces and the effective resistivity of the TJ increases. High-

resistive layers can be the reason for a strong lateral current spreading in the laser (spreading

perpendicular to the laser axis in the plane of the layers) and, as a consequence, lower DQE and

higher threshold currents. The highest spreading in this case is expected in the N-layers (with

high mobility) above highly-resistive TJ (Figure 2-2). The most efficient way to get low TJ

resistivity is to dope both TJ layers above 5-1019 cm-3. In SOme material systems, the high carrier

concentrations required in one or both of the layers and/or dopant atom stability during the

growth of additional laser stages cannot be achieved. This problem can be reduced by

sandwiching an appropriate quantum well between the two tunnel junction layers (Figure 2-3)

since the tunneling probability exponentially depends on the barrier height. This possibility is

explored in this work. Esaki tunnel junctions are widely used in semiconductor device design

beyond bipolar QCLs. For example, TJs are employed to cascade solar cells. In these devices

each active region of the solar cell is optimized for light absorption in a particular wavelength

range. This increases conversion efficiency of the solar cell.










Bipolar QCLs can be designed to have either separate waveguides for each stage (Figure 2-

4a) or a single waveguide shared by all stages (Figure 2-4b). Due to reduced number of layers

the latter configuration benefits from suppressed current spreading in the structure and lower

strain. Also, such structures can be processed into distributed feedback lasers. However, overlap

between the optical mode and highly-doped TJs increases free-carrier absorption and as a

consequence decreases DQE for these devices. In this work we were mostly interested in getting

the highest possible DQE. Therefore, we employed the configuration with separate waveguides.

The bipolar cascade mechanism is also used for making Vertical-Cavity Surface-Emitting

(VCSE) QCLs (Figure 2-5). VCSEL emits light vertically (parallel to the growth direction)

rather than horizontally, which makes it easy to use them for 2D laser arrays. One key technical

advantage of VCSEL is its ability to produce a circular, low-divergence output beam. The active

regions of these quantum cascade lasers are placed in antinodes of the standing wave pattern to

increase the overlap with the optical mode, whereas TJs are located in the vicinity of a field null

to reduce the free-carrier absorption. The cascade configuration is used to increase roundtrip

gain, which is extremely low for these devices. Higher gain leads to lower threshold current and

as a consequence to higher optical power at the same current. Due to the low roundtrip gain DQE

for bipolar cascade VCSELs is also relatively low (~0.9 for triple-stage devices).

2.3 Review of Previous Work on Bipolar QCLs

The first realization of an edge-emitting bipolar QCL was reported in 1982 [1]. In this

work MBE-grown, GaAs-based, triple-stage cascade lasers were fabricated and tested. Each

active region of the structure had its own waveguide and cladding layers. Using Be and Sn as p

and n dopant atoms respectively the authors achieved doping densities in TJs above 1019cm-3

This allowed them to triple DQE for the cascade lasers relative to SSLs (0.8 vs. 0.27). However,

due to immature technology overall performance for these devices was quite low (nowadays









DQE close to 0.7 is routinely measured for SSLs). Edge-emitting bipolar QCLs with separate

waveguides for each active region reported later had substantially improved characteristics [3].

For example, DQE ~1.4 was reported for a double-stage 808nm lasers in this work.

Bipolar cascade lasers where all the stages share the same waveguide and cladding layers

were reported in [4, 5]. Due to increased Gamma factor these lasers have threshold currents

lower than SSLs. As discussed above, mode overlap with highly doped TJs reduces DQE. For

example, in [4] DQE for triple-stage lasers was reported to be 125%.

Realization of Vertical-Cavity Surface-Emitting bipolar QCLs was reported in [6-8]. Since

effective length of the gain region is increased, threshold current density for cascade VCSELs is

lower than for their single-stage counterparts. For example, in [7] comparison between triple-

stage and a single-stage VCSELs showed that threshold current density for the triple-stage

device was only 800A/cm2 COmpared to 1.4kA/cm2 foT Single-stage laser, while DQE was 60%

compared to 20%. Relatively low DQE in the both cases is explained by low roundtrip gain for

the vertical-cavity configuration.

The importance of high-doping of the layers composing TJ was emphasized in [9]. It was

also mentioned there that the very high doping of the TJ layers is not always achievable. For

example when Si is used as the n-dopant, it becomes amphoteric as the doping concentration

increases beyond 5-10 scm-3. The idea on the reduction of TJ resistivity using QWTJ layers was

suggested in [10]. In this work it was shown that resistivity of GaAs TJ can be decreased by one

order through sandwiching 120-thick Ino.15Gao.ssAs layer between the heavily doped layers.

However, in this work QWTJ were not used in the laser design due to expected high modal

losses. The first MOCVD-grown InP-based bipolar laser with QWTJ was reported in [1l].









Employment of QWTJ allowed the authors substantially reduce voltage drop across TJ and

improve laser performance.

2.4 Experimental Results on Fabricated Bipolar QCLs

It' s well known that depletion width for a p-n junction is given by:


2ck 1 1
q N, N,f 28

where N, and Nd are acceptor and donor densities and Yo is the contact potential. Since high

doping is used for the layers composing the TJ, the degenerate approach should be used to

calculate Yo (Figure 2-6):

Vo = Eg + AE,, + AE, (2-9)

Depletion region extends mostly in the n-layer since doping concentration for the p-layer is

higher by approximately one order of magnitude. Also, even though N,>Nd~ it is usually valid

that AEf>AEf, since density of states for conduction band is substantially lower than for valence

band. AEf, can be found from the following equation:


Nrl=2 Pe~(E-Ec)cdE (2-10)


where P, is the conduction band density of states. Similar equation should be applied to find

AEfp.

Using Eq. 2-9 and Eq. 2-10 it can be shown that Yo slowly changes with carrier density

concentration. Therefore, from Eq. 2-8 it can be concluded that the depletion width decreases as

the doping increases. As a consequence, potential barrier width (and electron tunneling

probability) seen by electrons increases.









As discussed in Section 2-2, very high doping for TJ layers is not always achievable. In

our case the highest possible doping level for GaAs-based TJ was 5-10lscm-3 (Si doping) for the

n-layer and 5-1019cm-3 (C doping) for the p-layer. Relatively low n-doping is the reason that a TJ

composed of these layers would have high effective resistivity. Therefore, other methods should

be used to reduce TJ resistivity. QWTJ concept can be employed in this case. Schematic showing

how QW transforms potential barrier seen by electrons is represented in Figure 2-7. It was

assumed in this figure that QW lowers potential barrier height in the vicinity of the interface

between the TJ layers. In quantum mechanics the probability of a particle tunneling through a

potential barrier is proportional to the following factor:


D = exp ~- 2m [Tx) E~d (2-11)

where U(x) is potential barrier and E is particle energy. Therefore, we should expect lower

effective resistivity for QWTJ due to lower barrier height in the vicinity of x = 0.

Rough estimation of influence of QW on TJ resistivity can be done using Eqs. 2-1, 2-2, 2-3

and 2-4 and Figure 2-6, 2-7. The following approximations are used: potential barrier for

electrons in depletion region has triangular shape with a deep around x = 0, electron effective

mass is constant through the structure and equal to 0.067mo (GaAs-based TJ), barrier width of

the TJ is not influenced by dimensions of QW, doping for the n-side GaAs is 5*"10' m and for

the p-side it is equal to 5*"1019 -m3, quantum well used is InxGal -xAs.

Results on D dependence on In composition are represented in Table 2-1 assuming 100A -

thick QW (critical thickness equal to 100A corresponds to In composition around 40% [12]).

As it mentioned above, introduction of QW in TJ with In composition 15% in [10] reduced

TJR by one order of magnitude which is roughly consistent with results presented in Table 2-1:









D increases approximately by factor of ten when In % increases from 0 to 15. Also, it can be

seen that further increase in In composition leads to further substantial increase of the tunneling

probability. For example, D is more than 2 orders higher for 40% than for 15%.

In this work [13] we explore the use of a QWTJ between two standard GaAs-based diode

lasers grown by MOCVD. The quantum well used is a 10nm thick, highly strained InGaAs layer

with a 25% indium content. The SSL and DSL structures are shown in Figure 2-8 respectively.

Doping of the n-AlGaAs cladding layer in the first stage of the DSL is reduced relative to the

corresponding region in the SSL (4-101 cm-13 COmpared to 101s cm-3) in order to decrease current

spreading. The QWTJ connecting the two stages is composed of two relatively low-doped GaAs

layers (5-101s cm-3 Si for n-doping and 5-1019 cm-3 C for p-doping) with a 10nm thick

In0.25Gao.75As layer sandwiched between them. The active regions in both structures have two

typical quantum wells sandwiched between standard barrier layers. The DSL structure is

designed such that the optical mode loss due to overlap with the QWTJ is negligible. The SSL

and DSL lasers used in the experiments were 750' long with 75' wide contact stripes. The

output facet reflectivity was about 5% and the rear facet reflectivity was about 95%. All

measurements were performed in pulsed mode at room temperature using 500ns wide current

pulses at a repetition rate of 1kHz.

The near field (NF) intensity distribution for the DSL measured at 1A is shown in Figure

2-9. Since the distance between the ARs is approximately 5' we used this scale to estimate

that the NF width at the second AR is around 100' 25' larger than the contact stripe width.

This demonstrates that the current spreading in the structure is non-negligible.









Far-field pattern for DSL measured at 2A is shown in Figure 2-10. Double-lobed lateral

profile predicted for some wide-stripe diode lasers due to V-shaped phase front [14] was

observed for DSLs.

IV curves measured in cw-mode at low current (Figure 2-11) show that the turn-on voltage

for the DSL is close to double that of the SSL. The additional voltage drop above turn-on is

attributed to the finite effective resistivity of the QWTJ.

The power vs. current (PI) characteristics near threshold and at high power levels for a

typical SSL and DSL are shown in Figure 2-12. The slope efficiency of the DSL (2W/A)

(corresponding DQE ~150%) is nearly twice that of the SSL (1.1W/A).

Maximum measured optical power is determined by generator maximum current (~7A).

Ratio of the threshold currents for DSL for the second AR (490mA) and the first AR (340mA) is

approximately 1.5. This is another indication of the current spreading in the structure.

Experimental results described above show that the goal to double slope efficiency for

DSL was achieved. However, as shown by NF measurements lateral current spreading is still

present in the structure. A possible reason for this effect is the finite resistivity of TJ

demonstrated in Figure 2-10. To determine the degree of influence of QWTJ resistivity on

current spreading, a current spreading model for DSL based on 1D spreading model for SSL was

developed. Its details and results are discussed in Chapter 3.




















contact gold -
stripe"\


Ilchi~
cmlrrln,


75pmr


10m




750pmi


I~
:i":""IJ


300pm ,










A co


300am /


Figure 2-1. Energy band diagram for SSL and DSL


75pm


100am ri





750um












p-layer

AR

n-layer
TJ

p-layer /
AR

n-substrate


Figure 2-2. Illustration of current spreading in bipolar QCLs. Dashed line shows FWHM of
carrier density distribution. Highest spreading is expected in the N-layers with
mobility above highly-resistive TJ.


Figure 2-3. Schematic of TJ and QWTJ. Tunneling probability exponentially depends on the
potential barrier height. It is expected to be lower for QWTJ since in this case height
of the central part of the barrier is lower then for TJ.










Cladding layer
Guiding layer
QW
Guiding layer
Cladding layer
TJ
Cladding layer
Guiding layer
QW
Guiding layer
Cladding layer


Cladding layer
Guiding layer
QW
TJ
QW
Guiding layer
Cladding layer




Figure 2-4. Bipolar cascade lasers with different waveguide configuration. A) Separate
waveguide for each stage. B) Single waveguide for all stages. These figures
demonstrate transverse intensity distribution through the layers.


















-r I --1


cc~


~-~C~


DBR


SQuantum
wells


SIntensity
distribution


junc tons


DBR

Substrate


Figure 2-5. VCSE bipolar QCL. The quantum wells in these devices were placed in antinodes of
the standing wave pattern, whereas TJs were located in the vicinity of a field null to
reduce free-carrier absorption.






Vo














Figure 2-6. Schematic of TJ. Triangular potential barrier seen by electrons is confined by thick
dark lines.


Emitted light











































p-GaaAs contact, 21019
p-AlGaAs cladding, l10
p-AlGaAs waveguide, 4 101
ACTIVE REGION
n-AlGaAs wnaveguide, 4 100
n-AlGaAs cladding, 101X

n-GaAs substrate, 2*10'8


x=0


Figure 2-7. Quantum well tunnel junction


1" model
stage




2nd model
stage


st" Stage




/ tg


Figure 2-8. Single-stage and double-stage structures



































Figure 2-9. Near-field pattern of the double-stage laser. Distance between the active regions is 5pm measured at 1A.


Figure 2-10. Far-field pattern of the double-stage taken at 2A.




















-SSL
---- SSL voltage is doubled

(at the same current)
----- DSL .C...----

-


4.0

3.5-

3.0
- .-






1.0

o -



0.0


I


I
C
C


I


..
5


..


..
15


Current (mA)




Figure 2-11. Voltage vs. current characteristics for the double-stage and single-stage lasers.
Turn-on voltage for the double-stage laser is approximately twice that of the
single-stage laser. The additional voltage drop above turn-on is attributed to the
finite effective resistivity of the QWTJ.











































I I I I II III I I I I
0 1 2 3 4 5 6 7 8


2.0

1.5






0.0
0.0


DSL
I =340mA


16
1-


14
1-

8-

6-


0.4 0.8 1.2
Current (A)


* DSL with ~
* SSL with ~


2WV/A
1.1WV/A


3
L
a>
r
o


Current (A)




Figure 2-12. Power vs. current characteristics for the double-stage and single-stage lasers near
threshold and at high power levels.










Table 2-1. Dependence of tunneling probability exponential factor D on indium composition

In% D

0 1.08E-18

15 1.22E-17

20 3.07E-17

25 8.36E-17

30 2.48E-16

35 8.16E-16

40 3.07E-15









CHAPTER 3
SIMULATION OF CURRENT SPREADING IN BIPOLAR QCLS

3.1 Influence of the Tunnel Junction Resistivity on the Current Spreading in Bipolar QCLs

As discussed in Chapter 2, NF measurements clearly demonstrate that there is a substantial

current spreading between the 1st and the 2nd stages in realized DSLs. In this section using simple

1D model [15] it will be shown that tunnel junction resistivity can be the reason for this

spreading.

For the demonstration of current spreading induced by TJ resistivity we'll use a structure

shown in Figure 3-1. The structure is comprised of a resistive TJ and a P-layer above it. The TJ

has the thickness h and the effective resistivity p '=ph [G2-cm2], where p is the resistivity of the TJ

and h is its effective thickness (p' instead of p is usually used to characterize TJ since depletion

width and p are usually unknown for a TJ, while p 'can be directly measured [10].). Thickness

for the p-layer is d and resistivity is p,. Stripe width and length are taken to be W and L

respectively. In this model it's assumed that voltage drop across the P-layers above the tunnel

junction is negligibly small and that current through TJ under the stripe (Ic) is constant. Leakage

current (2-lo) is defined as the current that doesn't flow under the stripe. Therefore, total current

is given by

frot = Ie + 210 (3-1)

For current density flowing through the TJ we have

E= pJ (3-2)

This expression can be modified as


V = pJh = p' J= Ly p' (3-3)









where V voltage across the TJ and -dI,- is the current across the TJ between y and y+dy. For

lateral current in the P-layer we have:

-dPy = p,ISdy (3 -4)

where p, is defined as p, /(Ld) and I,- is the current flowing in the y-direction. From Eq. 3-3

and Eq. 3-4 we get


--;L/ (p= L I (3-5)

Solution to Eq. 3-5 is


I, ,ex -, Ly+ C, expP y (3-6)


Using the fact that current can't grow exponentially and I,-(0)=Io we get


Iv = IO expl Iy(37


Current across the TJ between y and y+dy is obtained from Eq. 3-7


-dl= j~, (0 = I (3-8)


As a consequence, le is given by


I = WI, (3-9)


Therefore, total current through the device is


I,pp = : I,W + 21, (3-10)


From Eq. 3-10 for Io we get










I, (3-11)



Using Eq. 3-7 and Eq. 3-11 we get for current density flowing through the TJ


J(x) = 7toa exp -x rP x >0 (3-12)
2+ W I-


From (3-12)


Weg (outptut) = -2 n (0.5) I+ W (3-13)


where Wegis the effective output stripe width (FWHM of the lateral current density distribution

right under the active region)

Typical values of p' for TJs are in the range from 10-3 to 10-' Ohms -cm2 [10].

Substituting p' = 10' Ohms cm2 t=200mA, W = 80pum and p, /d = 300hms we get from Eq.

3-11 and Eq. 3-13: I1otle = 27mA and Wfs(output) = 88um .

Eq. 3-11 and Eq. 3-13 can not be directly applied to our structure. However, they

demonstrate that a TJ with a typical resistivity in vertical direction can influence current

spreading and its contribution can be reduced by lowering resistivity. The question we are trying

to answer in this chapter is stated as follows: Does TJ resistivity contribute to current spreading

between the 1st and the 2nd stages of the DSL observed in the NF measurements? To answer this

question a current spreading model for DSL that assumes negligible TJ resistivity will be

presented. Validity of this assumption will be verified by comparing model results with

experiment.










3.2 Previous Work on Current Spreading Simulation in Diode Lasers

A simple 1D current spreading model for diode lasers was presented in [15]. Transverse

voltage drop across the p-layers above the active region was considered to be negligibly small

and current density under the wide contact stripe to be constant. In this work the diffusion

current in the active region wasn't taken into account.

In [16, 17] this model was extended to include the diffusion component. It was shown in

[16] that Ohmic current in the layers above the active region and the diffusion current in the

active region are coupled and can't be considered independently without losing self-consistency

of the problem. In [17] the following physical description of the lateral current in the active

region was suggested: holes are transported in the active region under the combined effect of

drift and diffusion, but the field causing diffusion is such that their motion is identical to that of

pure diffusion with an effective diffusion coefficient. Electrons on the other hand, are stationary

in the active region. Instead of moving there, the electrons are supplied from or to the N-layer at

just such a position-dependent rate that they maintain charge neutrality in the active region.

The model developed in [16, 17] is applicable only below threshold since it doesn't take

into account the stimulated emission term. Since the stimulated term is also involved in the scalar

wave equation, coupling between the optical mode and diffusion and Ohmic currents should be

considered in this case. A numerical model that takes into account the stimulated emission term

was reported in [18]. In addition, it comprises Poisson equation and photon rate equation. This

model was used to predict an optimal ridge width for a diode laser.

In this work we adopt the subthreshold model developed in [16, 17]. The model (Section 3-

3) and its employment for simulation of the current spreading in DSL (Section 3 -4) are discussed

in the next two sections.









3.3 Simulation of Current Spreading in SSL

The exact solution to the lateral current flow problem in a laser requires the solving of the

2D current continuity equations. A maj or simplification used here is the reduction of what is

naturally a 2D problem to a problem in one dimension.

As discussed in the Section 3 -2, there are two maj or components to the lateral carrier flow

in a diode laser: a lateral current spreading in the layers away from the active region (Ohmic

current) and a lateral diffusion current in the active layer. It was also mentioned in Section 3-2

that these two currents are coupled and can't be evaluated separately without losing self-

consistency of the problem [16]. Since it' s usually safe to assume that transverse voltage drop

across the P-layers above the active region is negligibly small, Ohmic current spreading is

characterized by normalized conductance E (Figure 3-2):


F = f(qny)h, = 0 (3-14)


where N is the number of P-layers and 0 is normalized resistance. When r is big ( 0 is small),

lateral current spreading is big.

In this work we assume that the lateral voltage gradient is small at the interface between

active layer and N-cladding layer compared to the lateral voltage gradient at interface between P-

cladding layer and active layer. This condition is usually satisfied for single stage semiconductor

lasers since the resistance of the layers below the active region is much smaller than the

resistance for the ones above it.

Effective current width is defined here as FWHM of spatial current distribution.

Following [16] we have for Ohmic sheet-current density C(x) (lateral Ohmic current per

unit of stripe length (A/cm) in layers above active region) (Figure 3 -2)









dv
= -szC(x) (3-15)
dx

dC
= -J (3-16)
dx

where v(x) is voltage drop across the active region at location x J is active region injected-

current density and 0Z = T1 (see Eq. 3-14) normalized resistance of the layers above active

region. Eq. 3-15 and Eq. 3-16 take into account that the current density inj ected in the active

region originates from the decrease in the lateral Ohmic current flowing above the active region.

It is assumed that the evolution of the electron concentration n in the active layer can be

described by a diffusion equation:

dn
qDeg d = J d (3-17)

where Deff (see Section 3-2 ) is the effective diffusion coefficient [17] and JdIfs is a hole-diffusion

current density (A /cm2 ). The diffusion current has as its source the junction-current density J

and, as its sink, the concentration dependent recombination rate R(n)

dJJ
df qR(n) (3-18)
dx d

where d -active layer thickness and R(n) = An + Bn2 Cn3 TOCOmbination rate with A = 0 ,

B = 0. 8*10-tocm3 / S and C = 3.5*"10-30 C6 / S taken from [19]. Eqs. 3-15 through Eq. 3 -18 have

to be solved self-consistently taking into account that the voltage across the active region is a

function of the concentration n. For this dependence we use the following formula (see [16] and

references therein)

kT n n
v(n) = -421n-n +A At (3-19)









where n = n, at x = x,, ns n,, A, 0.35 N,-effective density of states in the conduction

band. From Eq. 3-15 and Eq. 3-16 we can get

d~v
= 11 (3-20)


Solving Eq. 3-17 and Eq. 3-18 we have

d2n
D + R(n) (3 -21)
e dx 2 qd

First lets consider the case when x 2 W/2 Equations Eq. 3-20 and Eq. 3-21 are coupled through

J. Elimination of Jin these equations and some mathematical manipulations give (see [16])


dn dy(R 2 dy 1
-- R~) d = f (n) x >- W (3-22)
dlx dnl D, dn 2

where y = n + v(n) / qD~d Since it is assumed that current density under the stripe Jo is constant,

the first integral of Eq. 3-21 has the form




dn 2 J

where nO is the carrier concentration in the active region below the center of the stripe,

uniform current density under the stripe and W represents stripe width. Eq. 3-22 (or Eq. 3-23)

can then be integrated to yield as the solution x(n) for the carrier-concentration profile


1 "rdn
x =- W (3-24)
2 f(n)

where ne = n at x = xe W .









Calculation of current density distribution in the active region is done using the following

procedure.

Carrier concentration at the stripe edge n, is used as an input parameter.
1 1
Using Eq. 3-22 and Eq. 3-23 f(n) is found for x 2 W and 0 I x < W
2 2
respectively. Also, f(n) under the stripe at this step is a function of two unknown parameters: no

and Jo.

Relation between Jo and no (Jo(no)) is found using the fact that Eq. 3-22 and Eq. 3-23 should be
equal under the stripe edge.

Using Jo(no), Eq. 3-23 and

1 "" dn
-W = (3-25)
2 f(n)

no (and as a consequence Jo) is found as a function of ne.

Lateral carrier concentration n(x) is found using Eq. 3-24.
Using n(x) inj ected electron current density from the n-side of the p-n junction can be found from
the fact that it is equal to local recombination current density [17]
Jcx) = qd An (x) +Bn2 (tn3 (x)) (3-25)

where A is the coefficient corresponding to non-radiative recombination (for example through

interface states), B bimolecular radiative recombination coefficient and C the coefficient

corresponding to Auger non-radiative recombination.

According to definition output current effective width which is used as an effective stripe

width for the next stage is taken to be equal to:

W, = 2xJO 2 (3 -26)

where x,, is determined by:


J~y )_1,xJo/ > 0 (3 -27)
J(0) 2

In addition, for diffusion current leakage we have:









dn
Jdyr = -qD = qDf (n) (3 -28)

while for Ohmic sheet current density we get from Eq. 3-2 and Eq. 3-4:

1 dv dn dv /dn
C(x) Jdyd (3 -29)
0Z dn dx qD~d

Leakage current is sum of these two currents (multiplied by a factor of two) evaluated under the

stripe edge. Therefore, total leakage current is given by

ITotalLeakge = 2C(xe )L +2Jdy eX,)Ld = 2Jdyf ex)Ld [1+ z] (3-30)

where


z 2kT/nq2D~d (3-31)


is the relative importance of the Ohmic and diffusion current.

Total current corresponding to carrier density in the active region at the edge of the contact

stripe ne (which is used as an input parameter) is found integrating Eq. 3-25.

3.4 Simulation of Current Spreading in DSL

In order to determine the degree of current spreading at the two DSL active regions (ARs),

a model was developed based on the 1D discussed in the previous section. This model estimated

the effective current width (FWHM of the current density distribution) at each AR. It was

assumed that the QWTJ contribution to current spreading was negligibly small due to its low

effective resistivity. Comparison between the model and experiment was supposed to show

wither this approximation is valid.

It can be seen from Eq. 3-1 that spreading should be especially important for the bipolar

cascade lasers where several active regions are connected via tunnel junctions (TJs). Total

thickness of such structure is substantially bigger than for a common semiconductor laser and









therefore current leakage is expected to be enhanced in this case. Also, the resistivity of TJs

(TJR) in the current flow direction should contribute to total spreading since the current tends to

spread before entering a less conducting material.

In the discussion below by stage we mean active region and all the layers above it but up to

the active region of the previous stage. For example, according to this definition, second stage

comprises 2nd active region and all the layers between 1st and 2nd active regions.

For simplicity, current spreading in each stage is considered independently. This approach

is accurate when we can assume that the lateral voltage gradient is small at the interface between

active layer and N-cladding layer compared to the lateral voltage gradient at interface between P-

cladding layer and active layer. This condition is usually satisfied for single stage semiconductor

lasers since the resistance of the layers below the active region is much smaller than the

resistance for the ones above it. In our case we believe it's true for the both stages. Output value

of effective current width for the first stage (its value at the active region) is considered as an

effective width of the stripe for the second stage. This approximation substantially simplifies

calculations.

Each layer above active layer for a stage under consideration is characterized by its

thickness, doping and mobility. Dependence of mobility on concentration and doping is taken

from [20] and [21] respectively. Parameters used for the calculations below are listed in Table 3-



The following parameters were fixed in the program: L = 750pum, T = 300K,


D, =19.35cm2/Vs, A=0, B=0.8*10-tocm3/iS, C.=3.5*10-30cm 6Sand d=140A. Also, n,

was adjusted before each calculation in such way that total current was around 200mA (below

threshold for the structure).









Simulation results are presented in Table 3-2. As expected, current spreading in the layers

of the first stage is low. However, current width at the second active region was calculated to be

97Clm, 22 Clm wider than the stripe width.

Near field for the DSL measured at 1A is shown in Figure 2-7. FWHM of the intensity

distribution at the second active region was estimated to be 100 lm. It was also observed that NF

width didn't substantially change with current and it was approximately the same at current close

to the threshold of the second active region. Therefore, assuming that NF width is close to the

width of the current density distribution, simulation and experimental results are in good

agreement. As mentioned above, the current spreading model did not include the effective

resistivity of the QWTJ under the assumption that it should be small. This assumption is

validated by the good agreement between the model used and the measured NF of the second

AR. Therefore, we showed that even though TJR is finite (Figure 2-8) it's not high enough to

cause any additional current spreading in the bipolar cascade laser.

This model conclusion was consistent with the following experimental results. The DSL

structure was grown again using the design described above. However, QWTJ used in the new

structure was different: GaAs (n-doped with Te above 1019cm-3) 0.15lGao.ssAs (10nm thick

quantum well)/ GaAs (p-doped with C above 5-1019cm-3). Measured voltage vs. current

characteristics and near field intensity distribution for a laser fabricated from this structure are

shown in Figure 3-3 and Figure 3-4 respectively. The IV curve shows that QWTJ effective

resistivity is very low (due to higher n-doping of the GaAs QWTJ). However, near field

measurements demonstrate that despite this fact there is still a considerable lateral current

spreading between the first and the second active regions.









In conclusion, in this work it has been demonstrated that the employment of a deep QW

inserted between TJ layers with relatively low doping densities can be used to fabricate DSLs

with slope efficiencies and DQEs close to twice that of SSLs. It was also demonstrated that a 1D

model can be used to accurately calculate the current spreading in DSLs provided that a QWTJ

with low effective resistivity is used. It is expected that this type of QWTJ should be of use in

any device requiring monolithically stacked diodes where material growth limitations require

that the doping level densities in the TJ layers be kept relatively small. Future work in this area

could be related to understanding how width of QW in the TJ influences its effective resistivity.

Such information would be very useful for future device design.
























7 vi
I
I
h p'= p-h V TJ
I
I
I


* "


Figure 3-2. Illustration of the current flow in stripe geometry SSL.


x= 0


| 10 I


Figure 3-1. Structure (comprising tunnel junction) used for illustration of influence of TJ
resistivity on current spreading.




xo x,


i=1


P- layers


Voltage gradient is negligibly small


N-layers


Sle 1


Pp= ps/(Ld)













4.0 SSL

3.5 j~voltage twice that for SSL


~


0.0 0.5 1.0 1.5 2.0

I (mA)


2.5 3.0 3.5 4.0


Figure 3-3. Voltage vs. current characteristics for the new double-stage and single-stage lasers.


Figure 3-4. Near-field pattern of a new double-stage laser measured at 1A.











Table 3-1. List of parameters used for simulation of current spreading in bipolar QCLs.
1s stage

Al composition Thickness, nm Doping, 1017cm-3 Mobility, cm2 Vs
P+ 0 200 200 70
P 0.3 50 10 67
P 0.6 1300 10 25
P 0.3 700 4 78
P 0.1 20 0.05 280

2nd Stage


N 0.1 20 0.05 5900
N 0.3 300 4 950
N 0.6 20 4 70
N 0.3 2500 10 630
N 0.2 50 10 1100
n 0 50 50 1200
p 0 50 1000 40
p 0.3 50 10 67
p 0.6 1300 10 25
p 0.3 700 4 78
p 0.1 20 0.05 280


Table 3-2. Current spreading simulation results for bipolar QCLs.
Effective Output Leakage current, mA 0, Ohms
stripe effective
w width, p~m IOhmic if total

1st stage 75 80 35 2 37 185

2nd Stage 80 97 73 1 74 38









CHAPTER 4
UNIPOLAR QUANTUM CASCADE LASERS

4.1 Basic Operational Principles of Unipolar QCLs

Operation of unipolar (intersubband) QCLs relies only on one type of carrier in contrast to

bipolar QCLs where light emission occurs due to radiative recombination between holes and

electrons. The best performance for intersubband QCLs was demonstrated for n-doped devices.

In these devices light is generated due to radiative electron transitions between energy levels

localized in conduction band (Figure 4-1). Electrons tunnel through the injector barrier from the

inj section region to the upper laser level. The radiative transitions occur between the 3rd and the

2nd energy levels. Calculation procedure of electron energy levels and wavefunctions in layered

structures is discussed in Appendix. Initial and final states of the intersubband transitions have

approximately the same curvature of the energy vs. wavevector dependence (Figure 4-2). As a

consequence, joint density of states corresponding to these transitions and gain spectrum are

substantially narrower than for interband transitions. In addition to radiative transitions between

the 3rd and the 2nd levels, there are parallel non-radiative transitions between these levels through

emission of longitudinal optical phonons. These transitions are very fast (~5ps) and as a

consequence strongly increase laser threshold current density. To create population inversion

between the upper and lower laser levels, energy separation AE21 between the 2nd and 1st energy

levels is usually designed to be equal to the energy of the longitudinal optical phonon (~34meV).

In this case transition time 221 between these levels is very short (~0.5ps) and T21< 232. However,

if AE21 1S Smaller than the energy of the longitudinal optical phonon, this fast process is

prohibited and transition between the 2nd and the 1st levels occurs through emission of acoustical

phonons which is a much slower process (~100ps). As a consequence, lasing can be

unachievable in this case. Electrons get recycled through tunneling from the 1st energy level to









the miniband (multitude of closely spaced (in energy) levels) of the inj section region. A reverse,

undesirable effect (so-called backfillingg') that reduces population inversion between the laser

levels occurs when the quasi Fermi level of this miniband is located close to the lower laser

level .

As discussed in Section 1-1 light emitted by unipolar QCLs has TM polarization in

contrast to TE polarization typical for interband transitions (diode lasers). Therefore, to avoid

high losses usually cladding layers in unipolar QCLs are designed to decouple the guided mode

from the plasmon mode propagating at the metal/semiconductor interface. The opposite design

approach is to employ surface plasmon waveguide. In this configuration there is no need to use

cladding layers which can be helpful to improve heat dissipation in these devices. To illustrate

corresponding mode the following 3-layer structure is used. One micron thick active region


characterized by refractive index equal to 3.2 is sandwiched between a metal with n = 2.0 + 32i

and substrate with n=2.8. Calculated transverse intensity distribution for this structure is shown

in Figure 4-3. Gamma factor (active region (including inj ector layers) confinement factor) was

found to be 93%, much higher than typical values ~60-70% (including inj ector layers).

Calculated effective refractive index and intensity loss were 3.13 and~-100cml respectively.

Therefore, disadvantage of this configuration is high loss and as a consequence high threshold

current density. Figure 4-3 shows discontinuity at the interface between the active region and the

substrate, the consequence of the boundary conditions particular to TM-polarization.

Unipolar QCLs are usually fabricated in the high-ridge configuration since surface

recombination is not present in unipolar devices. This helps to reduce threshold current densities

for these devices. Typical length for unipolar QCLs is in the range of several millimeters.










Unipolar QCLs are used in atmospheric sensing, medical breath analysis, process

monitoring and food production. Future possible applications for high-power devices include

infra-red countermeasures and free space optical communications.

4.2 Review of Previous Work of Unipolar QCLs

Concept of the intersubband cascade configuration for light amplification was suggested in

1971 [22]. However, the first quantum cascade laser was demonstrated more than twenty years

later at Bell Labs in 1994 [2]. It became possible due to high growth precision of molecular beam

epitaxy (MBE) and development of band structure engineering.

InP-based structures were used for fabrication of the first QCLs. This choice of material

allows employment of heterojunctions based on In0.53Ga0.47As-Al0.48In0.52As layers lattice-

matched to InP. High conduction band discontinuity (~0.5eV) of this composition makes it

possible to fabricate QCLs emitting at relatively low wavelength (below 5 Cm). In addition, InP

has low refractive index and as a consequence can be effectively used as a cladding layer. Strain-

compensated InP-based QCLs were reported in [23]. In these structures barrier height can be

increased relative to unstrained In0.53Ga0.47As-Al0.48In0.52As composition. However, In and Al

percentage in the barrier and quantum-well layers should be changed simultaneously to avoid

strain build-up in the structure. InP-based QCLs were demonstrated to operate at room

temperature in continuous mode [24] with hundreds mW of output optical power [25].

GaAs-based QCLs were realized for the first time in 1998 [26]. Since AlGaAs-layers are

almost lattice-matched to the GaAs-layer independent of Al composition, this structure allows

more design flexibility compared to InP-based material. However, it should be taken into

account that AlxGal-xAs-structure becomes indirect when x>0.45. In this case, scattering to X-

valleys can be harmful for laser performance [27]. GaAs-based QCLs performance at 80K was

demonstrated to be as good, if not better, as for InP-based lasers and it is steadily improving [28].









It was also shown that employment of active region with deep quantum wells (In0.3Gao.7As) can

be used to substantially reduce carrier leakage from the inj ector region to continuum and as a

consequence increase tunneling inj section efficiency to the upper laser level [29].

Gas spectroscopy applications for unipolar QCLs require laser linewidth to be below Icm~

,substantially less than typical linewidth for edge-emitting QCLs (>10cm '). Distributed

feedback (DFB) configuration, where a grating is introduced in the structure, proved to be very

efficient for reduction of the linewidth for diode lasers. First DFB unipolar QCL was

demonstrated in [30]. Linewidth for DFB QCLs was reported to be below resolution of FTIR

spectrometer (0.125cm- ). Wavelength in this case can be adjusted with temperature and current

variation since refractive indices of the layers composing the structure depend on these

parameters. Typical adjustment rates are 0.5nm/K and 20nm/A respectively and wavelength

adjustment usually lies in the range 30nm 100nm. Further increase of the scanning range can

be achieved through employment of the external cavity configuration and bound-to-continuum

active region design [31]. In these devices emission wavelength can be controlled with position

and angle of the external grating. Wavelength for external cavity QCLs can be varied by ~ 100 -

400nm. Output power in cw-mode in this case can be as high as several hundreds of mW [32,

33]. Second order DFB (surface-emitting) configuration can be used to reduce strong beam

divergence typical for edge-emitting QCLs from ~60ox150 to ~ lox150 [34]. Maximum output

optical power for these devices working in cw-mode on a Peltier cooler is in the range of tens of

mW [35].

New type of intersubband QCLs emitting in far-infrared range (~100Clm and above),

terahertz QCLs, was reported in 2002 [36]. In this work lasing was based on radiative transitions

between minibands. Also, guided mode was confined by two metallic claddings, which









decreased mode losses and increased confinement factor for the active region. Terahertz QCLs

are still limited to low-power, low-temperature operation.

4.3 Low-Ridge Configuration Concept

High-peak-power, pulsed-operated quantum cascade lasers (QCLs) operating in the first

and the second atmospheric windows are being developed for use in application areas such as

infra-red countermeasures, free space optical communication and laser detection and ranging

(LADAR). Previous work on such devices [37, 38] employed optimized structural designs in a

narrow width, standard high-ridge configurations (Figure 4-4). Another approach of getting high-

power QCLs is to increase width of the high-ridge.

In this work an alternative approach for achieving high-peak-pulsed power QCLs is

described, that uses a narrow width, low-ridge configuration (Figure 4-5). Figure 4-5 includes

low-ridge laser dimensions used in this work

To understand operation of low-ridge unipolar QCLs it' s first necessary to understand

physical mechanisms responsible for the shape of IVs for high-ridge QCLs. Typical IV curve for

a high-ridge unipolar QCL is shown in Figure 4-6. At low bias the inj ector and the upper laser

levels (Figure 4-1) are misaligned. In this range the active region is in the high differential

resistance mode (low effective conductivity). As the applied voltage increases (8V 12V) these

levels line-up and differential resistance substantially decreases. At voltages above 12V this

alignment breaks again and the active region becomes resistive. In this work it is suggested that

this mechanism can be used to design high power, low-ridge unipolar QCLs with relatively low

threshold currents.

The basic mechanism that allows narrow width, low-ridge QCLs provide high output

power with relatively low threshold currents is lateral current spreading (spreading perpendicular

to the laser axis in the plane of the layers). As shown in [39], lateral current spreading mainly









occurs in layers above the active region and decreases when the active region transverse

conductivity increases. At low bias, the active region conductivity is low and the current density

distribution is wide. At higher bias, the conductivity of the central part of the active region

begins to increase (as the inj ector and upper laser levels align). As a consequence, the width of

the current density distribution (characterized by its full width at half maximum, FWHM)

decreases. This narrowing of the current density distribution is the reason for the relatively low

threshold current of these devices. When the applied voltage exceeds the voltage value that

causes misalignment between energy levels, the conductivity of the central part of the active

region diminishes causing current to spread laterally and the current density width to increase.

This additional lateral current spreading effect, not taken into account in a previous current

spreading model of low-ridge QCLs [39], allows higher peak powers than expected to be

achieved. The previous model [39] and the modifications required to include the additional

lateral current spreading effect are described below.

Low-ridge unipolar QCLs were reported in several previous works. In [40, 41] it was

demonstrated that low-ridge QCLs can to give substantially higher output optical power than

high-ridge lasers fabricated from the same wafer (for the same ridge width and length). However,

low-ridge lasers in this case were treated just as broad-area devices. In [42] low-ridge

configuration helped to improve heat dissipation, which substantially increased characteristic

temperature of the laser. In this work proton implantation through the active region was used to

suppress the lateral current spreading.

4.4 Waveguide Structure for the Low-Ridge QCLs

Low-ridge quantum cascade structure realized by our group comprised the active region

design reported in [43] (Figure 4-7) embedded in the waveguide discussed below. In [44] using

this active region design it was demonstrated that MOCVD-grown InP based QCLs are capable









to demonstrate as good performance as IVBE-grown devices. The sequence of layers for the low-

ridge QCL was the following: InP low-doped substrate (S, 1-2-107cm-3), 2Clm InP cladding layer

(Si, 1017cm-3), 300A InGaAsP graded layer (Si, 1017cm-3), 3000A InGaAs waveguide layer (Si,

3-1016cm-3), 1.5Clm active region comprising 30-stage AllnAs-InGaAs QC strain-balanced

structure [43, 44], 3000A InGaAs waveguide layer (Si, 3-1016cm-3), 300A InGaAsP graded layer

(Si, 107cm-3), 2Clm InP cladding layer (Si, 1017cm-3), 0.2Clm InP contact layer (Si, 1017cm-3,

100A InGaAs top layer (Si+).

Input parameters required for transverse waveguide calculations include imaginary and real

parts of the refractive index for each layer. At low photon energy limit (comparable with thermal

electron energy koT) these parameters can be obtained using classical Drude theory. Theory

results are

e = 8,-ie, = N (4-1)

N =n -ik (4-2)

where El and E, are real and imaginary parts of the complex dielectric constant E n and k are

real and imaginary parts of the complex refractive index N.

e &(-25 09 (4-3)


82 __ (,/2 29)1/W (4-4)

where E, is the high frequency dielectric constant, r is the electron scattering time

nr = 47tn e2 ,, (4-5)
-g=m2 2F
m /e (4-6)


q=_1+ 1/02 2) (4-7)









For the real and imaginary parts of the refractive index we get


n= (e+e,)/2 1/2 (4-8)


k= 62 /2nZ= (E-e//2 1/2 (4-9)

where

e= e2 1/2,

Quantum-mechanical extension of this theory described in [45, 46] gives

-[e_/(1-X)] 1-8 u02o) (4-10)

where

X = Aco/E, (4- 11)

Also, relaxation time z in Eq. 4-4 was found to be a function of both photon energy and layer

doping concentration. Plots of z for InP and GaAs at 300K can be found in [46].

Near field (NF) transverse intensity distribution calculated based on the parameters listed

in Table 4-1 (calculated based on Eq. 4-1 Eq. 4-11) is shown in Figure 4-8. Gamma factor for

the active region (including inj ector layers) was found to be ~78% with real and imaginary part

of the refractive index equal to 3.3 5 and 1.4E-5 respectively. Losses corresponding to the

imaginary part of the refractive index can be calculated using

47r Im n7*
a = (4-12)

In our case mode loss is below Icm l. It is important to mention that calculated losses are usually

significantly smaller than losses obtained from experiment. The reason for this effect is still not

completely understood [48]. Full width at half maximum (FWHM) of the corresponding Far-

Field (FF) transverse intensity dependence on the emission angle (Figure 4-9) was calculated to









be approximately 68o. Calculated Gamma Factor for the active region (including inj ector layers)

was found to be ~80% with real and imaginary parts of the refractive index equal to 3.3 5 and

1.46E-5 respectively. Also, FWHM of the FF was found to be ~700. Therefore, transverse NFs

(and corresponding FFs) are almost the same under the ridge and under the channels despite

substantially thinner cladding layer thickness in the latter case (0.8Clm vs. 2.0plm). The reason for

this effect is the low refractive index of the Si3N4 USed as an insulator for the low-ridge QCL.

Basically this layer acts as a cladding layer separating mode from lossy gold contact above it.

4.5 Fabrication of the Low-Ridge QCLs

The QC wafer was grown by low pressure MOCVD at a slow rate (0.1nm/sec) in the same

reactor as in [44] and under essentially the same conditions (except for growth uncertainties).

1.4Clm-high, 25Clm-wide ridges were etched in the wafer using pulsed anodization etching (PAE)

[47]. The channel width on the both sides of the ridge was defined to be 50Clm. The surface was

then passivated by 300nm of Si3N4, deposited by plasma-enhanced CVD at 3000 C. Metal

contact-windows 12Clm wide were opened on top of the ridges by photo-lithography. The Si3N4

in the openings was etched by RIE. The substrate was then thinned to approximately 120Clm by

mechanical lapping. Non-alloyed contact metals of Ge(1 2nm)/Au(27nm)/Ag(50nm)/Au( 100nm)

were deposited on the substrate side of the wafer and Ti(10nm)/Au(400nm) were deposited on

the top by metal evaporation. Finally, the wafer was scribed into chips of dimensions 2.5mm by

500Cim.

SEM pictures of the fabricated low-ridge QCLs are shown in Figure 4-12. Both insulator

and contact layers are smooth and etching quality is good.

As discussed above etching was done using so-called pulsed anodization etching

technique. This is a fast, inexpensive and safe procedure. PAE setup is shown in Figure 4-13.









Solution is composed of glycol (40): water (20): 85% phosphoric acid (1) (GWA). Generator

drives 50V, 100Hz and 0.7ms-wide pulses through the solution. When the pulse is on, OH- ions

are attracted to the positive polarity applied to the sample. As a consequence, due to chemical

reaction native oxide is growing on the sample surface. When pulse is off GWA solution mildly

etches the native oxide. In result, the oxide slowly propagates through the structure. When

desired etching depth is achieved, native oxide can be removed with BOE or KOH solutions.

Etching rate can be increased by adding a small percentage of BOE (GWA (8:4: 1) 650ml: BOE

(7:1) 7ml) (Figure 4-14).

4.6 Experimental Results for Fabricated Unipolar Low-Ridge QCLs

Operating parameters for all testing procedures were 60ns pulse width at 5kHz repetition

rate. The lasing wavelength measured with Fourier Transform Infrared Spectrometer (FTIR)

equipped with Mercury-Cadmium-Telluride (MCT) detector cooled to 80K was found to be

5.3Clm. Figure 4-15 shows the power vs. current (P-I) curves for the low-ridge QCLs. The laser

was placed in cryostat and light was focused using two Ge lenses on a room-temperature MCT

detector. At 80K threshold current, maximum optical power per facet and slope efficiency per

facet were measured to be 2A, 6.7W and 730mW/A respectively.

The same chip with high-reflectivity coated back facet demonstrated at 80K a threshold

current of about 1.3A and a peak output power of about 12W. This is the record high power

reported at this temperature. At 300K these characteristics were 4A and 2.2W respectively.

Spectral measurements performed at 80K at currents 4A and 20A didn't reveal substantial

spectral changes with current increase (Figure 4-16).

I-V characteristics and corresponding P-I curves measured up to 5A at multiple

temperatures are shown in Figure 4-17. At higher currents precision of the I-V measurements









decreased due to impedance mismatch between the laser and a transformer used to extend the

driving range of the current generator. Measurements of threshold current temperature

dependence based on P-I data (Figure 4-18) showed that the characteristic temperature To was

172K.

Far-Field intensity dependence on lateral angle is shown in Figure 4-19.

Electroluminescence below threshold current is symmetrical. However, it becomes asymmetrical

above threshold. The angle corresponding to intensity maximum slowly increases as current

increases. Also, spectrum broadens with current increase. As discussed in [49] the beam steering

in high-power QCLs can be explained using the concept of interference between the two lowest

order lateral modes. A small difference in the effective refractive index between these modes

causes beating along the stripe length. As a consequence, the angle corresponding to the

maximum of the FF lateral intensity distribution shifts from one lateral side to another depending

on the phase shift between the modes at the output facet (depending on the position within

beating period at the output facet). This phase shift is influenced by the current since it changes

the effective refractive indices of the modes. As a consequence, the lateral angle corresponding

to maximum of the FF distribution shifts with current change. Maximum steering angle in [49]

was reported ~100, approximately the same as measured for our low-ridge lasers. For low-ridge

QCLs however there is a possibility of existence of several lateral modes at high current since

effective stripe width (current width) is substantially bigger in this case (>70Clm instead of 13Clm

in [49]).

The experimental data presented above demonstrate that low-ridge QCLs are capable of

giving very high peak-pulsed optical powers with relatively low threshold currents. Also, the

qualitative mechanism on current spreading responsible for operation of these devices was










suggested. Far-field measurements demonstrated beam steering. This indicates presence of

several lateral modes. Spectrally resolved near-field measurements would be very useful for

better understanding of the FF behavior.

High-ridge lasers from the same wafer were made with the goal of comparing their

performance with the low-ridge lasers. However, due to processing issues, reliable lasing was not

achieved for the high-ridge lasers. Since we couldn't repeat the experiment due to lack of

material, we modeled current spreading in the low-ridge lasers based on high-ridge data

presented in [44]. In the next section a 2-D current spreading model quantitatively supporting the

current spreading mechanism suggested in this work will be developed.











Electric field across the active layers is in the range of 10Ot of kV/cm


mnJector


232>22


Figure 4-1. Schematic of electron transitions in unipolar QCLs


mmmmmmmmm


Figure 4-2. Schematics of intersubband and interband transitions. For intersubband transitions
joint density of states is substantially narrower.


mnjector





0.0 0.5 1.0 1.5 2.0


2.5 3.0


Transverse direction (clm)




Figure 4-3. Transverse near field distribution for the surface plasmon waveguide.


Figure 4-4. High-ridge configuration for unipolar QCL.



68












25
,--- 1 .0

Active









Figure 4-5. Schematic of the low-ridge laser (dimensions are given in microns).


1 l


i
f
r




i


0.0


0


0.2


0.6


Current (A)

Figure 4-6. Typical IV curve for a high-ridge unipolar QCL.













































1,


S~i 1 ;111


Figure 4-7. Energy band diagram of the active region used in the low-ridge structure.















4.0




3.5



-h
3.0 m








2.0




1.5


Trasnverse distance (pm)


Figure 4-8. Transverse NF distribution in the low-ridge QCL under the ridge.




1.0-

FWHMI ~68o







*- 0.4




aJ 0.2



Ll..





0.0 g g g g g g g .
-80 -60 -40 -20 0 20 40 60 80

Angle





Figure 4-9. Far field distribution corresponding to NF under the ridge.



















j 08 lk =1.46E-5 3.




'r 0.6~ I 3.0



C 0.4- J1 2.5



S0.2 -1 -1 2.0




0.0 I 1.5
0 2 4 6

Transverse direction (pLm)




Figure 4-10. Transverse NF distribution in the low-ridge QCL under the channel.








I FWHM ~700





C~ 0.6






CI.

S0.0 6 ~ ~ ~ g g g g g



Angl



Fiur 41. arfel dsriuto crrsonin o F nerth hanes





Figure 4-12. SEM pictures of the fabricated low-ridge unipolar QCLs.


i:


































'i'cRan f9ver J i


--
GltlA
Efectmfyl:
It




Figure 4-13. Pulsed anodization etching setup.


2.5-



2.0-

E
1.5-



c- 1.0-



0.5-



0.0-


0 2 4 6 8 10

Time (min)




Figure 4-14. Etching rate of InP with GWA (8:4:1) mixed with BOE in ration 600 to 7.


Etching rate through InP cladding
layer ~ 0.3pLmmimn (linear region)

















110 -80K, HR Coated

0 8-
80K, No HR Coating



40 /
Sm 300K, HR Coated



0 5 10 15 20
Current (A)

Figure 4-15. Power vs. current characteristics for the realized low-ridge QCL.











2,0-








1.0-




4,5


*apr
r
3

E
k,
O
C

m


5.0 5.5
Wa~velength (pmf)


6.0


Figure 4-16. Laser spectra measured at 4A and 20A taken at 80K.


10




15









0,01


0.0 1.0 2.0 3,0 4.0
Current (A)


Figure 4-17. Voltage vs. current characteristics and corresponding power vs. current curves
measured at multiple temperatures (laser is without HR coating).



































_


4.0-



3.5



3.0



2.5



2.0



1.5


......


240


Figure 4-18. Dependence of the threshold current on temperature.


- 0.95 x Ith (x10)
- Ith
- 1.52xlth
- 2.05 x Ith
- 3.1 x Ih
- 4.43 x Ith


40000

35000

30000

25000 -

20000

15000

10000

5000 -

0-


/-7
--
-


-30~ ~ -20 -1 0 0 0
Far Field Angle (~deg.)


30 40


Figure 4-19. Far-field intensity distribution for a low-ridge QCL.


To = 172K


120 180

Temperature (K)






































calculate transverse NF distribution in the low-ridge QCL


Thickness,
Layer um n 2*nr*k/A

Gold inf 1.83 42.3

InP, 1E19 0.2 2.63 0.005

InP, 1E17 2 3.1 0.0000362

InGaAsP, 1E17 0.03 3. 18 0.0000471
InGaAs, 3E16 0.3 3.26 0.000008058

AR 1.53 3.49 0.000013

InGaAs, 3E16 0.3 3.26 0.00000806

InGaAsP, 1E17 0.03 3. 18 0.0000471

InP, 1E17 2 3.1 0.0000362

InP, 2E17 inf 3 0.0000697


Layer Thickness, um N 2*nr*k/A

Gold Inf 1.83 42.3

Si3N4 0.3 2 0

InP, 1E17 0.8 3.1 0.0000362

InGaAsP, 1E17 0.03 3. 18 0.0000471

InGaAs, 3E16 0.3 3.26 0.000008058

AR 1.53 3.49 0.000013

InGaAs, 3E16 0.3 3.26 0.00000806

InGaAsP, 1E17 0.03 3. 18 0.0000471

InP, 1E17 2 3.1 0.0000362

InP, 2E17 Inf 3 0.0000697


Table 4-1.


List of parameters used to calculate transverse NF distribution in the low-ridge QCL
under the ridge


Table 4-2. List of parameters used to
under the channels.









CHAPTER 5
CURRENT SPREADING MODEL FOR UNIPOLAR QCLS

5.1 Previous Work on Current Spreading Simulation in Unipolar QCLs

Current spreading model for unipolar, stripe-contact QCLs was developed in [31].In this

work it was suggested that this configuration can be useful for optimal heat dissipation and

single spatial optical mode operation. It was also explained that current spreading in these

devices is substantially different from current spreading in diode lasers. In particular, since

carrier concentration in each stage of the active region is fixed by inj ector doping, there is no

diffusion spreading component in the active region. As a consequence, there is only Ohmic

current spreading component given by

J(x, y) = e V V(x, y) (5-1)

where

a= (x,,"y) (5-2)

Also, the effective charge separation in each stage is negligible compared to typical scale of the

current spreading, it can be assumed that there is no space charge in the active region. As a

consequence, using charge conservation law we get

V-J=0 (5-3)

Substitution of Eq. 5-1 into Eq. 5-3 gives for each layer

82V(x, y) 1 82V(x, y)
+ = 0 (5-4)
8x2 2 ~2

where


a2 (5-5)









Solution to Eq. 5-4 was obtained in [31] extending solution given in [49]. For each layer of

the QC structure (excluding the active region), the conductivities ax and oy were taken to be

isotropic ( ox = o, ) and equal to the bulk conductivity of the layer material. Lateral conductivity


x,, for the active region was calculated as a weighted average of the bulk conductivities of all of

the layers composing the active region. Transverse conductivity oy for the active region was

assumed to be constant and was found by fitting simulated IV curve to the measured one. Using

this approach, it was found in [39] that current spreads mostly in the layers above the active

region and current width decreases as current increases.

In our work we extended this model by taking into account lateral variation of the active

region effective transverse conductivity. Also, we extended simulation above voltage

corresponding to roll-over point of the PI characteristic.

5.2 Two-Dimensional Current Spreading Model

5.2.1 Active Region Transverse Conductivity

Schematic diagram of the low-ridge QCL is shown in Figure 5-1. X-axis was chosen to be

along the lateral direction and y-axis along the transverse direction and pointing downwards.

These axes intersected at the center of the top of the contact stripe. It was assumed that voltage

applied to the ridge was Vo and voltage applied to the substrate was zero. Also, since there was

no electrical current flowing through the rest of the boundaries, corresponding spatial voltage

aV dV
derivatives -or taken at these boundaries were also assumed to be zero (see Figure 5-1).
ax Sy

As mentioned above, each layer of the structure is characterized by its transverse and lateral

conductivities and thickness.









Eq. 5-4 is valid in a layer with constant transverse and lateral conductivities. In the general

case, both of these quantities can have spatial dependence. Therefore, Eq. 5-4 should be modified




e + a = 0 1 ; (5-6)


In this form it is applicable to the whole layered structure. Following [39], ax for the AR was

calculated as a weighted average of the bulk conductivities of all of the layers composing the

AR. However, a, for the active region depends on the active region design and the voltage

across the active region:

o, =f (V,) (5-7)

The dependence of the active region transverse conductivity on the voltage across the active

region o,, (V,) (Eql. 5-7) can be calculated from the active region IV for a high-ridge laser

fabricated from the same wafer as the low-ridge laser. The wafer discussed in [44, Fig.2a] was

grown in the same MOCVD reactor as our structure under similar conditions and has the same

active region design. Since cladding layers, waveguide layers and substrate have high

conductivities we believe that the IV measured for the high-ridge laser reported in [44] is a good

approximation to its active region IV (For example, voltage across the high-ridge QCL in Figure

2a at 0.75A is equal to 12V. Using typical conductivity values for waveguide layers, cladding

layers and substrate it can be shown that total voltage drop across these layers at 0.75A is less

than 0.3V. Therefore, if voltage drop across the contacts is small, the laser IV is basically

determined by its active region IV). As a consequence, we believe it is appropriate to use the IV

reported in [44, Figure 2a] for simulation of current spreading mechanism in realized low-ridge

QCLs. This IV and corresponding dependence of active region transverse conductivity on the









voltage drop across active region is shown in Figure 5-2a, b. Part of the IV curve in the range

from 13V to 18V was obtained by the extrapolation (dashed line) to emphasize the effect of

current spreading when transverse conductivity of central part of AR gets low. Figure 5-2b was

obtained assuming a uniform current density flowing through the active region in the high-ridge

lasers reported in [44, Figure 2a]. Using o- (91) shown in Figure 2b, Eq. 5-6 and Eql. 5-7 were

solved self-consistently for a low-ridge QCL using the 2D Finite-Difference Method. Calculation

procedure is described in the next section.

5.2.2 Two-Dimensional Finite-Difference Method

The two dimensional Finite-Difference method is a numerical type of solution to a

differential equation

82 u2 8 u du
a C2+bZ+c-d+d -+ eu + f = 0 (5-8)


usmng


~1 (5-9)
dUx2 ;,idUx2 hy~+


d221 d221 21 +1 2uI~ +ulg+
= -(5-10)



I-I I- I -1~(5-11)
\ x I,\ x I, 2h


-I= -I, 2 (5-12)


Usually the calculation procedure is done as follows. The structure is divided by a rectangular

mesh (Figure 5-3) into cells. Boundary u-values corresponding to the physical problem are set at

the boundary points of the mesh and initial 'best guess' u-values are set at the all inner points.









Boundary conditions are left unchanged through the calculation procedure. U-values at each

point P are found from the u-values of the four neighbor points of the mesh using Eq. 5-8

through Eq. 5-12. This procedure (calculation of u-values at all the points) is repeated iteratively

until the convergence condition is achieved (convergence rate strongly depends on the 'initial

guess' of the u-values at the inner points). Using found values at each point of the mesh all

au\
related quantities such as can befound.
\ x P,

In our case we had to solve Eq. 5-6. Using 2D Finite-Difference Method this equation can

be modified as




+- = 0 (5-13)
23x 23y



(e)' 28 ex' 8x0 26y -1 2 y =0 (5-14)


26x 26y



Making the transformations 26x 3 x and 23y S y then gives




6x Sy

Using Eq. 5-15 we can find V-values at each point P(i, j) using V-values at neighbor points (i+1,j;

i-lj; i3j+1; i3j-1) Figure 5-4. Also, conductivity values should be set at points C (i+1/2,j; i-1/2,j;



Figure 5-5a, b illustrate the rectangular mesh used for the numerical calculation.

Increments along x and y-directions, Sx and 6y we chosen to be 1 Clm and 0.2 Clm









respectively. As discussed above, voltage on the top of the ridge and at the bottom of the

substrate we fixed at Vo and 0 respectively. Since there is no current through all the other

boundaries we can assume that voltage for the neighbor points at each boundary should be the

same. For example, since there is no current flow through the left vertical edge of the laser, at

this edge the following equation should hold

dV
= 0 (5-16)


Employing the discrete approach used here we get


0, "= (5-17)
6x

Therefore, at the left vertical boundary the following condition is true

Vo~ = V~~ (5-18)

Calculation procedure is shown in Figure 5-5. As mentioned above, initial voltage values

can strongly influence convergence rate. However, in our case convergence rate wasn't very

important. Therefore, for simplicity initial voltage values at the all inner points were taken to be

equal to zero. Initial value for o~ is taken as 1R Omv Each cycle of the solution of Eq. 5-6

(using Eq. 5-15) involves calculation of the voltage value at each point of the structure using

voltage and conductivity values at the neighbor points (Figure 5-4b). This cycle is repeated until

the convergence condition is satisfied. After that the active region transverse conductivity values

are updated using Eq. 5-7 that reflects the dependence given in Figure 5-2b. If new values are

close enough to the old ones, current density distribution can be calculated using obtained

voltage distribution. If not, Eq. 5-6 should be solved again using the new active region transverse

conductivity values.









5.3 Model Results

Parameters used for calculation are listed in Figure 5-6. Since substrate conductivity is

high it is appropriate to neglect by voltage drop across the substrate. Therefore, its thickness was

taken to be ~7Clm instead of 150Clm defined by the wafer processing. Active region lateral

conductivity was borrowed from [31]. In this work it was shown that model results aren't very

sensitive to the choice of this parameter.

A convenient way to interpret the model results is to examine the evolution of the lateral

distribution of the active region o, as a function of bias change (Figure 2a), and by doing so,

determine its influence on current density distribution (Figure 2b), current width and the low-

ridge IV curve (Figure 2c). When voltage across the low-ridge device is below approximately

6V, transverse conductivity of the AR is low and almost constant. Due to the low conductivity,

current density distribution at the AR for these voltages is broad and has large tails. When the

device bias increases above 6V, the conductivity of the central part of the AR increases rapidly

with increasing voltage, as can be seen in Figure 2b. The high transverse conductivity of the AR

central part and the peak shape of the transverse conductivity lateral distribution lead to a strong

narrowing of the current density profile. Further voltage increase leads to decrease of the

transverse conductivity of the central part of the AR. Due to the decrease in conductivity, current

starts to spread in the lateral direction and the effective current width goes up (Figure 2c),

extending the laser dynamic range. Spreading is expected to be stronger at lower temperatures

since mobility increases rapidly as the device temperature decreases.

The IV curve for the low-ridge laser does not show a strong increase in differential

resistance at high current (Figure 2c), the effect shown in Fig.1la for the high-ridge laser. This

































Active Region


1


difference is a consequence of lateral current spreading in the low-ridge structure when the

transverse conductivity of the central part of the AR goes down.

The simulation results show that low-ridge QCLs have relatively low threshold current due

to current narrowing when the central part of the active region enters high-conductivity mode

(it' s usually by factor of 2-3 higher than threshold for high-ridge lasers processed from the same

wafer). It is also demonstrated that high powers achievable by these lasers may be due to lateral

current spreading at higher voltages. The work discussed here on the low-ridge, unipolar QCLs is

summarized in [51].


Vo


x-axis



~=0
ex


y-axis


Figure 5-1. Schematic of the low-ridge QCL.


aV _













16-( r


^12-









0.0 0.2 0.4 0.6 0.8
Current (A)



3.5 b
"i 3.0-

S' 2.5.

oU 2.0-
o n5
ar 1.5-


> 0 10.5-


0.0 ,
U 4 8 12 16 20
Voltage (V)

Figure 5-2. A) High-ridge IV curve. B) the corresponding dependence of the transverse active
region conductivity on the voltage across the active region.


















1~ l -1 1 +1



0 h 1, -


Figure 5-3. Rectangular mesh used for 2D Finite-Difference Method.


a) ... x

-- --I 1----r----I -r -







b):-----i -1)








(i-lJ j)P(i3j) (i1)

(i(j+1/2J)


i(ij+1)



Figure 5-4. A) Rectangular mesh used for the low-ridge QCL. B) Basic cell of the mesh.



88














Solve Eq. (5-6)


Find new values for oy in the
active region using Eq.(5-7)


Are new values for a, equal to
the old ones?

1Yes
Calculate current density
distribution


InGaAs 100OA
InP 1E19 0.2um
InP 1E17 2.0um
InGaAsP grade 300OA
InGaAs 3E16 0.3um
AR 1.5um
InGaAs 3E16 0.3um
InGaAsP grade 300OA
InP 1E17 2.0um
InP 1E17


Update
AR cr


No


Figure 5-5. Schematic of the calculation procedure.


Structure


Parameters for the model


2.2um, 9600 0 nzm


0.3um, 6400 0 Z'm
1.5um, lateral 400 0 Z'm


0.3um, 6400 0 nzm-


7.0um, 9600 0 Z'm


Figure 5-6. List of parameters used for simulation.


Set initial values for ox, ay, V and the
boundary conditions at each point of the mesh
















2.8-1 I 1 8V
o 1I4V
C E 2.4
c: 12V
0 -2.0-

oa

:.o 0 .


03O 0.4 6V


0.0
U 100 200 300 400 500
Lateral distance (pm)



Figure 5-7. Current spreading model results. A) Lateral dependence of the active region
transverse conductivity at different current values. B) Corresponding lateral
distribution of the current density at the active region. C) Low-ridge IV curve and
current width dependence on the current calculated based on (B).















3.0 ] .

2.5- 14V

2.0- 1 2V






0.5-

S0.0
SO 10200 300 400 500
Lateral distance (pFm)

Figure 5-7. Continued
















20 150
C)
1 8 140 a.
16-
130 "*

-12 0 -o -

e 10- -o 7 1~10~
-8 -100
O
6- -90



-70

U 2 4 6 8 10 12
Current (A)


Figure 5-7. Continued









CHAPTER 6
SUMMARY AND FUTURE WORK

6.1 Bipolar Quantum Cascade Lasers

It was demonstrated in Chapters 2 and 3 that deep quantum wells can be effectively used

for designing tunnel junctions with relatively low effective resistivity. Employment of the

quantum well tunnel junctions is especially important when very high doping density is not

possible for the layers composing the tunnel junction. In this work this approach allowed us to

achieve double stage laser slope 2.0 W/A twice that of the single stage laser even though

maximum doping level for the n-GaAs tunnel junction layer was relatively low (5-10lscm-3.

Results of the developed current spreading model for the double stage laser showed that the

substantial current spreading between the first and the second active regions in our structure (and

corresponding relatively high threshold current for the second active region) was not a

consequence of the quantum well tunnel junction resistivity but rather caused by presence of the

n-layers with high mobility and relatively large total thickness of the layers between the active

regions.

Further experimental and theoretical study of the influence of quantum well parameters

(such as quantum well thickness and depth) on the tunnel junction resistivity at different doping

levels of the layers composing the tunnel junction would be very helpful. This information would

make the future design of semiconductor devices incorporating quantum well tunnel junctions

(bipolar QCLs, multiple junction collar cells, etc.) much more flexible since it would

substantially relax the requirement of the very high doping in the tunnel junction layers. Also,

wall-plug efficiency could be reduced due to lower voltage drop across the tunnel junction at the

same current. Since the tunnel junction comprises the highly strained InGaAs layer, reliability









testing is necessary to verify wither the strain in the structure reduces the lifetime of the double

stage laser.

6.2 Unipolar Quantum Cascade Lasers

In Chapters 4 and 5 we demonstrated that low-ridge unipolar quantum cascade lasers can

deliver very high peak pulsed optical powers (12W at 80K for the laser with HR coated back

facet) with relatively low threshold current (~1.5A at 80K for the same laser). Developed 2D

numerical current spreading model takes into account lateral dependence of transverse active

region effective conductivity, in contrast to the original model. Also, the new model was

extended to the high differential resistance range at high bias caused by the coupling breaking

between the inj ector and upper laser electron energy levels. It was shown that the low threshold

current was a consequence of the current density distribution narrowing at voltages

corresponding to a good alignment between the lowest inj ector energy level and the upper laser

level. It was also suggested that the high optical power achieved by the low-ridge laser was due

to the additional current spreading in the structure caused by the increase in the differential

resistance of the central part of the active region at voltages when this alignment breaks. Far-

field measurement results demonstrated current beam steering for the low-ridge lasers. Possible

reason for this effect is the co-existence of several lateral modes with slightly different effective

refractive indices. Far-field intensity distribution is determined by the interference between these

modes and, as a consequence, depends on the phase shift between them at the output laser facet.

The current beam steering occurs as a result of the effective refractive indices dependence on the

current since the phase shift also changes in this case.

Further study of this effect can include Near-Field measurements at different currents. In

particular, spectrally resolved Near-Field observation would allow thorough analysis of the Far-

Field behavior. Also, numerical simulation of the Near-Field and Far-Field intensity distributions









would be very useful for understanding the current steering. This simulation should take into

account the lateral variation of the current density distribution and coupling between the current

density and the optical modes. Finally, since the low-ridge devices operate at high voltages (up

to 40V), reliability testing has to be done to verify whether these lasers can be used in

practical high-power mid-infrared applications. If performance degradation with time is

observed, structural changes improving device reliability should be made. For example, increase

of the contact window width on the top of the ridge to reduce current density flowing through

this region could be helpful.









APPENDIX A
CALCULATION OF ELECTRON ENERGY LEVELS AND WAVEFUNCTIONS IN
LAYERED STRUCTURES

Calculation procedure described here is to a large degree based on the approach discussed

in [52]. In general case it is required to find electron energy levels and wavefunctions in a

layered structure with doped layers taking into account band non-parabolicity. Also, the electron

effective mass can change at the interfaces between the layers.

The governing equations in this case are coupled Schroidinger and Poisson equations:


,,sIy(z)+ V(z)-E yWz)= (A-1)


V2V = _P(A-2)


where nz* is the electron effective mass, V(z) is the potential profile and p is the spatial charge

density.

First let' s consider calculation procedure for Eq. A-1 and then extend the solution to

general case by taking into account its coupling with Eq. A-2. Eq. A-1 can be modified using the

expansion given in Eq. 5-10 (taking into account effective mass discontinuity)


m ( z1 87(z)d I ( z1 87(z)d
=+"= A2" [V(z)-E y(z) (A-3)




1( + z y(z + 23z) ry(z) 1 "( z y(z) ry(z 23z)
(A-4)
2(23z)~[()CV

which can be modified as










ry(z + z) 23
[V(z)-E]+ + yz
nz(z3:2)A2 n*(z + :/ 2) n *(z :/ 2)


n *(z :/ 2)

Therefore, p(z+5z) can be found using p(z), p(z-5z) and m(z+5z/2), m(z-6z/2).

Solution to Eq. A-1 is found as follows. The (left) boundary condition used for Eq. A-5 is

@(0) = 0 and y(5z) = 1. Using it, p(z) at all the other points is found using Eq. A-5. When the

energy E is not a solution of Eq. A-1, p(z) diverges as : approaches the right boundary

(accumulation of the error). The sign of this divergence changes when the energy passes its

solution value. During the calculation process the energy is varied in a particular range (usually

from the energy corresponding to the bottom of the quantum well layers up to the energy

corresponding to the top of the barrier layers). When the sign of the divergence changes an

iterative procedure is used to find the solution.

When external electric field EEF(z) is applied to the structure, V(z) should be taken as

V(z) 4 V(z) + q EEF(z) (A-6)

Figure A-1 shows results of the calculation procedure described above for a GaAs -based

structure with AlGaAs barriers and InGaAs quantum wells. Applied electric field was assumed

to be 80kV/cm. Interface grading (composition doesn't change abruptly at the interfaces) was

taken into account to create the potential profile for this structure (transition layer thickness was

assumed to be equal to two monolayers).

Inclusion of Poisson equation can be done using found electron energy levels and

wavefunctions. The approximation that is typically used in this case is that all donor atoms are

fully ionized and electrons from these atoms are distributed among the found electron energy

levels. Solution of the rate equations is necessary to find the electron population on each of these









levels. Potential created by positive ionized donor atoms and electrons distributed among the

energy levels can be found using the following formulas.

Assuming that the layered structure is infinite in the x-y plane, electric field (EF(z))

created by electrons and ionized dopant atoms (with doping density d(z)) can be found using


EFz)= sign(z -z ') (A-7)






CN = N= d(z)dz (A-9)

where o-(z) is the areal charge density given by Eq. A-8 and N, is the electron population on the i

electron energy level (found using the rate equations). The latter equation reflects charge

neutrality in the structure. Using found EF(z) corresponding potential can be calculated using


V (z)=- EF(z')dz' (A-10


Therefore, the full calculation procedure for solution of the coupled Eqs. A-1 and A-2 can

be described by the scheme presented in Figure A-2. Eq. A-1 is solved using the expansion given

by Eq. A-5. Found electron energies and wavefunctions are used in the rate equations to find

electron population on each energy level. Using Eqs. A-7 through A-10 potential profile is

updates and compared to the previous profile. If convergence is achieved, all desired quantities

can be calculated using obtained energy values and corresponding wavefunctions. If no, the new

potential profile should be used in the next iteration. Example of an employment of this

procedure for doping optimization in quantum cascade lasers can be found in [53].














80 kVlcm
Graded structure transitionn layer thickness is 2ML)


Distance, Al2


Figure A-1. Model results for a GaAs-based layered structure.

















I I


Using found Wi and Ei solve rate equations,
which gives Ni


Using found 9;i, Ni and d(z) find V,(z)


V(z)[new] = V(z)w/o spatial charge]+ V,(z)


SYES
Output parameters


a


Solution of Eq.(A1I) using Eq.(A5) (1st iteration
is done neglecting by spatial charge)


I


V(z) = V(z)[new]





NO


V (z) [new ] = ? V(z)[ol d]


Figure A-2. Step sequence for the calculation procedure.