Microcavity resonators and schemes for dynamical control of terahertz quantum cascade lasers

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Università di Pisa

Dipartimento di Fisica "E. Fermi"

XXXII PhD cycle

PhD thesis

Microcavity resonators and schemes

for dynamical control of

terahertz quantum cascade lasers

Candidate:

Supervisors:

Andrea Ottomaniello

Prof. Alessandro Tredicucci

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Acknowledgements

At the end of this journey, the person who I would give my sincere gratitude is Prof. Alessandro Tredicucci. He inspired me to the study of the fascinating field of photonics since his first lectures, and he started me to the scientific research, guiding me from my master de-gree to the end of my university education. I thank him for his never missed help, his brilliant ideas and acute observations, but I mostly thank him for let me the freedom to choice and follow my research interests, together with the responsibility to carry on independently my research activity. This led me to grow a lot in these years from a professional and a human point of view.

The other point of reference in this path was Dr. Alessandro Pitanti, from whose collaboration I probably learned the most in these years, starting from my master thesis and more intensively during the PhD. He was always present for productive discussions, to clarify my doubts and to give me precious suggestions. I have really appreciated his con-structive criticism, which has always improved the quality of my work and his concrete support without which I would not have had the re-sults I achieved. I am really grateful for the time he dedicated me. I had also the fortune to have other precious collaborations in my PhD experience. A special mention goes to the two master thesis students, Veronica Leccese and Gloria Conte, with who I shared some parts of my work. They accompanied me during some fabrications and mea-surements, and they constituted a great help in the moments when only two hands were not enough.

I was just as lucky to collaborate also with the colleagues of the Uni-versity of Leeds, Dr. Paul Dean, Dr. James Keeley and Dr. Pierluigi Rubino, together with I organized and spent intense and exciting days of experiments in their department. Even if that experience was rela-tively short, I have learned a lot thanks to their great expertise and kind hospitality. I really hope to have the possibility to collaborate again

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with them in the future. Thanks are also due to Prof. Giles Davies and Prof. Edmund Linfield for their collaboration and for giving me the opportunity to exploit their labs. Another mention must also go to Dr. Riccardo Degl’Innocenti from the University of Lancaster and Dr. Mario Pagano from CREA in Florence for their fruitful collaboration and kind availability.

I would like to thank some special colleagues from the NEST lab: Dr. Luca Masini, who started me to the art of quantum cascade laser fab-rication and who taught me the skills necessary to manage with this challenging practice; Dr. Simone Zanotto for his useful suggestions and inspiring conversations about physics and more; and Dr. Federica Bianco for her helpful suggestions and expertise, and for representing always a friendly voice at NEST, together with Dr. Ji-Hua Xu who never refuses to offer me his smiling help when I need it in the lab. Finally, the biggest thanks is for my family, which has never stopped to believe in me and which has always encouraged me in my studies and passions. An incomparable thanks is for Alessia, your patience, trust and support constitute my extra gear which recharges me every day with new energy. I could not have asked for anything better.

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Abstract

Terahertz radiation is the object of a wide range of technological efforts, in fields as diverse as solid-state fundamental physics, biomedicine and astrophysics. Terahertz light is particularly suitable for sensing, imaging, spectroscopy and communication. In order to unlock the full potential of these applications on a large scale, compact and versa-tile terahertz sources are needed. A promising candidate is the quan-tum cascade laser, a compact laser device based on electrically injected semiconductor heterostructures. Quantum cascade lasers operating in the terahertz have already shown high emitted powers (several hun-dred milliwatts in both pulsed and continuous wave) and spectral cov-erage throughout the 1-5 THz range with single mode and broadband devices. However, the research community strives to achieve the per-formances (such as room temperature operation, large frequency tun-ability and low power consumption) necessary for the large-scale ex-ploitation of quantum cascade lasers in terahertz technology.

Here, we investigated the possibility to add novel functionalities to terahertz quantum cascade lasers allowing to dynamically control their operation and enlarge their versatility.

One part of the work is dedicated to the photonic engineering of their cavity. We first considered a particular microcavity resonator consist-ing of two coupled whisperconsist-ing gallery resonators, which shows low-threshold, high efficiency, vertical collimated single mode emission in continuous wave operation. We thus developed this microresonator design allowing the tuning of the laser emission by exploiting dif-ferent integrated effects. Directly embedding a suspended mechani-cal resonator, the proposed microcavity concept presents an optome-chanical interaction between the confined electromagnetic field and the resonator mechanical motion, which can affect the laser emission fre-quency. In an optimized design, where self-oscillation in the system is possible, the device can show a dynamical frequency sweep at the

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me-chanical resonance frequency. A second functionality originates from the engineering of the device injection architecture. In a slightly modi-fied version of the same microcavity, a large continuous frequency tun-ing and an unconventional far-field modification from radial to vertical emission can be obtained by spatially controlling the pumping strength within the device. The system operation for both microcavity designs is shown through finite-element simulations, the corresponding fabri-cation is described and a preliminary characterization is performed. Exploiting a different concept for the terahertz quantum cascade laser cavity, relying on line defects in a photonic crystal structure, several devices are designed by simulation, fabricated and characterized to fi-nally show features such as in plane directional emission and mode se-lectivity. Slow-light effects were shown to produce laser current thresh-old reduction with respect to a reference Fabry-Pérot laser with the same active region. Thanks the defect-line ability to waveguide light in the structure, an example of an active platform integrating multiple line-defects to provide single mode emission in different directions is experimentally shown as proof-of-concept of the potential of the line-defect architecture.

Finally, we investigated the modification of quantum cascade laser pa-rameters, such as the emission frequency and laser voltage, in a self-mixing configuration where the intracavity electromagnetic field inter-feres with the emitted light, reinjected as optical feedback inside the laser cavity, after reflecting on an external element. Specifically, the ex-ternal element was a mechanical resonator constituted by a suspended silicon-nitride membrane. An optomechanical response with nanome-ter resolution to membrane mechanical vibrations was observed in the self-mixing signal. The proposed laser feedback interferometry can be viewed as a starting point for more complex schemes for the dynami-cal control of terahertz quantum cascade laser operation. A promising perspective is represented by the realization of a self-mixing configu-ration at terahertz frequencies where different lasers could be coupled through the motion of a mechanical resonator driven by radiation pres-sure.

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Publications

• A. Ottomaniello, J. Keeley, P. Rubino, L. Li, M. Cecchini, E. H. Linfield, A. Giles Davies, P. Dean, A. Pitanti, “Optomechanical re-sponse with nanometer resolution in the self-mixing signal of a terahertz quantum cascade laser”, Optics Letters 2019 44 (23), 5663-5666. • M. Pagano, L Baldacci, A. Ottomaniello, G. de Dato, F. Chianucci,

L. Masini, G. Carelli, A. Toncelli, P. Storchi, A. Tredicucci and P. Corona, “THz Water Transmittance and Leaf Surface Area: An Ef-fective Nondestructive Method for Determining Leaf Water Content”, Sensors 2019 19 (22), 4838.

• A. Klimont, A. Ottomaniello, R. Degl’Innocenti, L. Masini, F. Bianco, Y. Wu, Y. D. Shah, Y. Ren, D. S. Jessop, A. Tredicucci, H. E. Beere, D. A. Ritchie, “Line-defect photonic crystal terahertz quantum cascade laser”, Journal of Applied Physics 2019 126 (15), 153104.

• A. Ottomaniello, S. Zanotto, L. Baldacci, A. Pitanti, F. Bianco, A. Tredicucci, “Symmetry enhanced non-reciprocal polarization rotation in a terahertz metal-graphene metasurface”, Optics Express 2018 26 (3), 3328-3340.

Conference contributions

• 2019 Infrared Terahertz Quantum Workshop (ITQW), September 15-20, Ojai (USA). Oral presentation: “Self-mixing optomechanics with nanometer resolution in a THz QCL”.

• 8th International Quantum Cascade Laser School and Workshop (IQCLSW), 2nd-7th September 2018, Cassis (France). Oral

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pre-sentation: “Manipulation of THz QCLs exploiting a dipole-antenna microresonator”.

• International workshop: frontiers of photonics, plasmonics and electronics with 2D nanosystems, July 14-20 2018, Erice (Italy). Oral presentation: “Symmetry enhanced non-reciprocal polarization rotation in a terahertz metal-graphene metasurface”.

• Son et Lumière 2017: Combining Light and Sound at the Nanoscale. April 17-28, Les Houches, France. Poster presentation: “Towards optomechanics with terahertz quantum cascade lasers”.

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Introduction

1.1 Terahertz radiation

The range of the electromagnetic (EM) spectrum which spans frequen-cies from 300 GHz to 10 THz (corresponding to the wavelength in-terval from 30 to 1000 µm) is defined as Terahertz (THz) radiation. This frequency region represents the crossover from microwaves to the infrared and thus it bridges the world of electronics with that of pho-tonics, as highlighted in Figure 1.1. Owing to its particular positioning,

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1 Introduction

The terahertz (THz) frequency range of the electromagnetic spectrum2,3_{(roughly defined}

as the frequencies between 300 GHz to 10 THz) sits in the region between areas covered by electronic techniques (radio and microwave) and photonic techniques (infrared and visible). Due to sitting at this crossing point, it has been a region rarely utilised with few high power sources available. It remains today one of the least developed regions of the electromagnetic spectrum, mostly due to the lack of compact, efficient sources, but recent interest in the range has led to a rise in development of the region.

109 ₁₀10 ₁₀11 ₁₀12 ₁₀13 ₁₀14 ₁₀15 ₁₀16 30 cm 3 cm 3 mm 300 μm 30 μm 3 μm 300 nm 30 nm f (Hz) λ UV Near and Mid-infrared Microwaves Radio Waves X-Rays

Electronics THz Gap Photonics

Terahertz

Figure 1.1: The electromagnetic spectrum. Adapted from Williams.4

1.1 Terahertz radiation

These frequencies are of great interest to both the scientific and industrial communities due to unique properties of the THz region and the abundance of proposed applications

including imaging,5 _astronomy6 _{and spectroscopy}7_{. The ability to penetrate clothing}

and packaging materials make it ideal for security purposes8_{such as airport and postal}

scanning where it can be used to detect concealed weapons.9 _{The non-ionising,}10

non-destructive properties of THz radiation make it an attractive alternative to X-rays in regard to airport imaging in particular, as a way of reducing exposure to damaging ra-diation. Another area of application of THz where this is a highly inviting feature is

medical imaging11 _{where THz imaging systems have been demonstrated for such uses as}

the detection of cancerous tissue8,12_{and imaging of an extracted human tooth.}13_{Due to}

many chemical species presenting strong rotational and vibrational absorption lines in the

Figure 1.1: Scheme of the electromagnetic spectrum. Adapted from [1].

THz radiation almost completely lacked technological applications due to the absence of any strong and compact sources until few decades ago. In fact, this region represents an upper limit for the operation frequency of electronic devices, such as Gunn oscillators and Shot-tky diode multipliers, which have difficulty working beyond 0.1-1 THz due to capacitance and transit-time limitations causing high frequency roll-offs [2]. Similarly, the higher end of the THz range has been the

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watershed for the operation of coherent photonic sources due to the material bandgap that dictates the emission frequency in interband devices. For these reasons, the THz region became known as the "Ter-ahertz gap" [3], mainly used for long time by a restricted part of the scientific community made by astronomers and spectroscopists. Nev-ertheless, this scenario has drastically changed since the development of improved THz sources and detection schemes like the laser-based THz time-domain spectroscopy (TDS) in the 1980s and 1990s [4,5], and the realization of the first semiconductor heterostructure laser at THz frequencies in 2002 [6]. These have allowed the field of THz research to largely and rapidly expand, at the point that it now covers many areas of science from fundamental research to "real world" applica-tions [7]. The range of applicability of THz radiation can be catego-rized into four main groups, which are sensing, imaging, spectroscopy and communication [8]. THz light is in fact particularly suitable to characterize and investigate the dynamics of an almost countless num-ber of existing materials, both organic and inorganic, which have an optical response in this spectral range. For example, condensed matter physics is home to many important low-energy elementary excitations

− including plasmons, phonons, magnons, and correlation-induced energy gaps−which can be addressed by the new emergent THz mi-croscopy techniques able to get nanoscale spatial resolution [9, 10]. A huge number of molecules have vibration and rotation resonances in this range which can be spectroscopically analyzed not only to obtain their chemical fingerprints, but also to stimulate certain intermolecular excitations allowing investigation of the way molecules interact with their immediate environment. This is particularly interesting for bio-logical systems, where the additional THz strong absorbance by water and the low possibility of tissue damage by the non-ionizing energy carried by THz photons, make the use of THz radiation even more ap-pealing [8]. The sensitivity and specificity of THz spectroscopy can en-able investigation of the crystalline state of drugs, i. e. polymorphism, and in connection with the use of pulsed THz imaging in proteomics and drug discovery it can determine protein 3D structure, folding and characterization. In the same way the hybridized and denatured state of DNA can be distinguished, so that research can progress towards the development of label-free DNA chips. There have been several studies on the imaging-based medical diagnosis of cancers (especially on in vivo human skin cancer detection, but not limited to) arising from the differences in absorption coming from both water content and

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tis-sue structure. Apart from oncology, also in dermatology and dentistry THz imaging showed a lot of promise for non-invasive healthcare di-agnosis for the same reasons expressed before [8, 11]. Thanks to other peculiar properties of THz radiation − being able to penetrate plas-tic, paper, clothing, packaging materials, semiconductors, polymers, ceramics, just to name a few, and being totally reflected by metals−

THz technology has wide potential and societal value in a plethora of other areas. In security, it is promising both for the stand-off detection of hidden objects (weapons, or drugs, or explosive materials) and the promising realization of new-generation THz body scanners. Quality-check and detection of cracks and defects in a broad range of materials and devices (solar cells, plastics, nanocomposites, polymeric and di-electric films, IC packages, pharmaceutical products, etc) are relevant for broad and large-scale industrial applications [8]. The monitoring of crop and plant hydration levels is supplying useful tool for agricul-ture activities [12], while the non-contact and non-destructive imaging of paintings, manuscripts and artifacts can play an important role in the preservation of cultural heritage. THz light is acquiring a strate-gic role in environmental monitoring of the concentration of chemical species, probing at the same time the atmospheric content of pollution and extending the possibilities of the continuous global weather mon-itoring service. In the field of space science, THz is a major spectral window for the observation of various astrophysical events (interest-ingly, 90% of the total signal coming from space falls in the THz region) and it represents the best evaluation method now available to inspect the foam insulation used for space shuttles. Moreover, information and communication technology is continuously expanding interest in THz radiation, in order to satisfy the strong requirement of even larger bandwidth imposed by the exponentially increasing wireless data traf-fic.

Even if the above description represents only a limited picture of the field, it gives an idea of the enormous potential and versatility of THz based-technology. However, only a very small percentage of the possi-ble THz applications have now an effective and concrete role out of the laboratories where the first THz devices originated. In order to cross more easily the threshold of frontier research and to get access to a widespread use of THz technology in everyday life, the strong effort made by the global scientific community to improve the performances and to lower the cost of THz technology has to be continued.

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1.2 Terahertz Quantum Cascade Lasers

The development of THz photonic sources [13], in parallel with that of THz detectors [14], has been the main lever in the process of filling the THz gap with technological applications. Being the operation of solid-state electronic devices available up to 1 THz, and being conventional photonic approaches to terahertz generation from interband transitions limited by the lack of appropriate materials with small bandgaps1_{, a}

wide variety of alternative techniques have been developed as sources of THz radiation. Down-conversion from the visible regime by us-ing nonlinear or photoconductive effects, multiplication up from the millimeter-wave regime, direct generation through optically pumped molecular gas lasers or free electron lasers are some examples of par-ticular significance. However, each of these have their own drawbacks, such as being inherently limited to pulsed operation, characterized by low output power or requiring large spaces, cost and complexity. Consequently, a compact, coherent, continuous wave (c.w.) solid-state source, analogous to the semiconductor laser diode in the visible and infrared regimes, was for a long time greatly desired for many applica-tions. The first proposal of radiation amplification exploiting electronic transitions between quantized subbands in two-dimensional quantum wells arising from semiconductor heterostructures was first proposed by Kazarinov and Suris in 1971 [15]. Their idea inspired to artificially engineer materials to obtain the desired energy transitions for laser op-eration. Moreover, the later development of growth techniques such as molecular beam epitaxy (MBE) or metallorganic chemical vapor phase deposition (MOCVD), which provided unprecedented control of layer thickness down to the atomic level, allowed to achieve the control of the electronic localization and transport as well as the optical processes in semiconductor devices at the sub-nanometer scale. Despite the first experimental observation of intersubband emission in 1988 at THz fre-quencies [16], the first laser based on a series of consecutive intersub-band transitions, the first quantum cascade laser (QCL), was demon-strated at a much shorter wavelength (4 µm, 75 THz) by Faist et al. at Bell Laboratories in 1994 [17]. In the years following the demon-stration of the first QCL operating in the mid-infrared, a lot of effort was made to realize a QCL working below the Reststrahlen band, but only electroluminesce, and not lasing, was observed in a variety of

1_{The longest achievable wavelength with lead-salt laser diodes do not extend below} 15 THz.

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structures [16]. The first realization came in 2002, when Köhler et al. demonstrated a THz QCL emitting (at 4.4 THz, equivalent to a wave-length of 67 µm) below the GaAs longitudinal optical (LO) phonon energy by combining a QC structure with high gain and an appropri-ate low-loss waveguide [6].

1.2.1 Principles of operation

QCLs are semiconductor devices which exploit optical transitions be-tween quantized electronic states in the conduction (valence) band, named conduction (valence) subbands, arising from the spatial con-finement of carriers in multi-quantum wells which define the active region of the laser. This series of quantum wells and barriers, repre-senting a square-like potential for electrons (holes) in the conduction (valence) band along the growth direction, is created by stacking two semiconductor materials with different bandgaps. The extended het-erostructure which comes out from this repetition can be considered as an artificial crystal with a hypermodulation of the lattice, called super-lattice. In particular, quantum wells and barriers arise from the material with smaller and higher bandgap, respectively. Although superlattices can be made both in conduction and valence band, for the sake of discussion, in the next we refer only to electrons in the conduction band, which are also the most exploited in QCLs. In the conduction band of a QC heterostructure electrons have quantized energy levels in the growth direction while they are free to move in the perpendicu-lar plane. In principle, QCLs may be realized using any semiconduc-tor material system, but, to date, the best performances are obtained by using III-V semiconductor material in four combinations: suitable for mid-IR such as GaInAs/AlInAs grown on InP substrate [18], AlS-b/InAs grown on InAs [19], InGaAs/AlAsSb and InGaAs/AlInGaAs grown on InP [20,21], or for the THz such as GaAs/AlGaAs grown on GaAs [22] and InGaAs/AlInAs on InP [23] (also InGaAs/GaAsSb on InP can be an alternative [24]).

Intersubband transitions

The choice of the materials to use, their alloy composition and the thickness of wells and barriers provide several degrees of freedom to engineer the properties of the device by design. The peculiar feature of QCLs is in fact the possibility to tune the emission frequency over a large bandwidth while using the same semiconductor system. In

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k

||

E

k

||

0-D

2 -D or

3 -D

(a)

(b)

3 -D or 2-D

0-D

DOS DOS Distance En erg y Distance En erg y

2-D

0-D

Figure 1.2: Comparison between interband and intersubband transitions. (a) Electronic transitions arising between states with different Bloch functions (arising from different bands) and same envelope function (interband transi-tions). The states have parabolic dispersion (energy E vs. in-plane wavevector k_k) with opposite sign (positive and negative for conduction and valence band, respectively) giving rise to a step-like joint density of states (DOS) for a two-dimensional system. (b) Electronic transition arising between states with same Bloch functions (same band) and different envelope functions (intersubband transitions). The two states have parabolic dispersion with same sign giving rise to a δ-like joint density of states peaked at the resonance frequency, typical of atomic or 0D-systems. Here, a representative lineshape taking into account the linewidth broadening is shown.

stark contrast to the intrinsically bipolar interband lasers, where the emission wavelength uniquely depends on the bandgap between the conduction and valence bands, the energy of the intersubband transi-tions, i.e. the energy difference between the two subbands involved in the lasing of a QCL, can be modified by changing the width of the quantum wells. As a result, the emission frequency will tend towards zero as the well width is increased. Moreover, intersubband transitions occur between subband states with parabolic in-plane dispersions par-allel to each other. Electrons can thus always contribute to the gain at the same frequency, independently of their in-plane momentum. It consequently implies that the joint density of states is δ-like,

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mean-ing that the system has an optical behavior very similar to an atomic one. Gain and absorption then result in spectral lines centered around the transition energy broadened by both homogeneous (due to the up-per state lifetime, dephasing from scattering processes) and inhomo-geneous component (such as non-parabolicity, size fluctuations, etc). Importantly, elastic or inelastic transitions allow the scattering of elec-trons from the upper to the lower subband, providing that the neces-sary momentum to reach the final state is exchanged in the process. This results in a very short transition lifetime, typically of the order of the picosecond, dominated by non-radiative processes like LO-phonon emission even in ideal samples. In Fig. 1.2 a summary of the charac-teristic properties of intersubband transitions in comparison with that of interband transitions is shown. Another fundamental implication arising from intersubband transitions concerns the selection rules. In the envelope function approximation, particularly useful to reduce the determination of the electronic wavefunctions in heterostructures to an effective one-dimensional problem, the wavefunction can be written as a Bloch function for the conduction band modulated by a slowly vary-ing envelope function with the periodicity of the superlattice. Takvary-ing into account the in-plane translational symmetry of the heterostruc-ture, the envelope function, which characterizes each subband state, is given by [25]:

Fn(r_k, z) = √1

Se

ik_kr_k_χ

n(z) (1.1)

Here, k_kand r_kare the wavevector and radial coordinate of electrons in the plane perpendicular to the growth direction z, respectively, χn(z)

(n being the subband index) is a function taking into account the po-tential modulation, and S is a normalization constant. The dipole ma-trix element between an initial i and a final subband state f is given byDFi|ˆe·p|Ff

E

where ˆe is the light polarization unit vector and p is the electron momentum; clearly, only the matrix element involving pz

is different from zero, because of the orthogonality between different envelope functions. As a consequence, only the z-component of the electric field couples to intersubband transitions. In other words, only transverse magnetic (TM) modes are allowed in the system, implying that QCLs are intrinsically in-plane (perpendicular to the heterostruc-ture growth direction) emitting sources.

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Cascade scheme

Engineering the thickness of wells and barriers together with the alloy composition is crucial in order to face the picosecond lifetime of elec-trons. This is done by controlling the transport inside the heterostruc-ture, which can be done by tuning the radiative and non-radiative tran-sition rates and the resonant tunneling. At the same time, the net gain requirement inside the active region makes population inversion nec-essary for the laser operation. To this end, a particular architecture for the whole heterostructure is chosen. It consists of a periodic arrange-ment of unit cells, each one consisting in a complex potential made by a certain number of quantum wells and barriers. In a simplified picture, each cell can be described by the presence of two fundamental parts; the active region and the injection/extraction region. While the active region allows to create and maintain a population inversion between the two subbands of the laser transition, the injection/relaxation region is the structure which behaves both as the "electron reservoir" to feed the next period with carriers, and as an efficient collector of carriers from the previous period. Since the injection is electrical, the applied electric field unavoidably implies the bending of the heterostructure potential. In order to compensate the voltage bias, and to be able to transfer electrons from the lower to the upper state of the laser transi-tion, the injector is designed as a particular superlattice with a chang-ing duty cycle between barriers and wells, called "digitally graded al-loy". Together with the presence of doping to prevent the formation of space-charge domains, this allows to have charge neutrality in every single period and the necessary current transport for the laser opera-tion. Moreover, the superlattice structure of the injection stage implies the presence of multiple levels very closely spaced in energy, which in turn results in the formation of the so-called minibands and related minigaps. Minibands and minigaps are crucial for both injection and extraction processes. In fact, they are exploited to ease electron injec-tion in the upper state while at the same time preventing injecinjec-tion in the lower state, or unwanted transition from the upper state to the next injection stage2_{. Although a non-trivial and convenient balance of the}

thicknesses and doping of the layers is required, this scheme allows to cascade the active-region periods in an almost straightforward and natural way. A quantum cascade laser in fact consists of a large num-ber Npof repeated periods, typically some tens to a few hundreds. In 2_{For example, this is done by locating a minigap at the energly level of the upper} state.

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e -3 2 1 active region injector active region injector e -e -e -Distance En erg y

Figure 1.3: Schematic diagram of a quantum cascade laser. The conduction band potential along the growth direction of two periods of a representative QC heterostructure under an applied electric field is represented by a series of wells and barriers. The miniband and minigap of the superlattice injection stage are described by green and light blue colors, respectively. The dashed line crossing the injector indicates its minimum energy level, while the black curve solid line represents the wavefunction of the injector state aligned with the upper state of the transition (level 3). The red solid lines instead represent the two states of the laser transition. The blue solid line indicates the state of injector (level 1) which extracts electrons from the active region by depopulat-ing the lower state of the optical transition (level 2) through optical phonon emission. Adapted from [26].

this manner, electrons are "recycled" from period to period as they cas-cade through the structure. Increasing Np implies both the reduction

of the threshold current density of the device, thanks to the improved overlap with the waveguide, and the enhancement of the slope effi-ciency (defined as the derivative of the optical power P with respect to the injected current I, i.e dP/dI), due to potential emission of a pho-ton in each period by each electron (thus implying: dP/dI ∝ Np). A

simplified representative scheme showing two consecutive periods of a quantum cascade structure under the presence of an applied electric bias is shown in Fig. 1.3.

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Population inversion and threshold condition

3

2 1

⌧

32

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21

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injector

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esc

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⌧

inj

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Figure 1.4: Population inversion scheme in the QC active region. A four levels system is considered. The injector provides the electron pumping rate J/e in the upper state of the transition (level 3) and it represents also the depopulating final state for the lower radiative state (level 2) with lifetime τ−1

2 =τ21−1+τinj−1.

The lifetime of the transitions (blue arrows) from level i to level j are indicated as τ_ij. τescis the escape time for electrons in level 3 to continuum of states. The

red arrow represents the stimulated emission process given by the product between the number of photons S and the population difference between level 3 and level 2.

The population inversion can be described by a four levels scheme similarly to what is done is solide-state lasers working on atomic tran-sitions. Fig. 1.4 shows a schematic diagram of the process considering only one period of the active region. The injector is assumed to be a single level aligned with the upper level of the next period, defined as level 3, where electrons are injected at a rate equal to J/e, with J the current density and e the electric charge. From level 3 electrons can scatter to level 2, the lower state of the laser transition, and to level 1 with rates τ−1

32 and τ31−1, respectively. The lifetime of level 3 is

thus determined by the sum of the rates of the last two mechanisms plus the escape rate 1/τesc of electrons to the continuum of the

fol-lowing injector region, i.e. τ−1

3 =τ32−1+τ31−1+τesc−1. Level 2 is instead

assumed to be populated only by the transition from level 3. Its life-time τ2is determined by all the processes connecting it to level 1 with

rate τ21and to the following aligned injection stage with rate τinj, thus

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en-ergy difference between the chemical potential of the injector and level 2 is designed to be larger than the thermal energy. Considering the be-havior below threshold, where no stimulated emission is present and spontaneous emission τsp is negligible (τsp τ3), the rate equations

for populations N2of level 2 and N3of level 3 can be written as [25]:

dN3 dt = J e − N3 τ3 (1.2a) dN2 dt = N3 τ32 − N2 τ2 (1.2b)

Looking for stationary conditions by setting the time derivatives to zero, the requirement for population inversion ∆N can be found [25]:

∆N=N3−N2= _eJτeff>0 (1.3)

where τeff is the effective lifetime defined as τeff = τ3

1− τ2

τ32

. This condition is only satisfied when the lifetime of the lower state of the transition is smaller than the time necessary to reach the lower energy level from the upper one, i.e. τ2< τ32. Counter-intuitively, there exist

some QCLs which work in the case of τ2 > τ3, but which

simultane-ously satisfy the inversion population condition. Nevertheless, being ∆N ∝ n3, it is convenient to have τ3as long as possible. The injection

efficiency, which quantifies the fraction of electrons injected in the up-per state, has to be optimized by design. The result for ∆N allows to calculate the threshold current density Jth for the laser operation, i.e.

the current density at which the total gain G inside the system equals the total losses αtot, and its dependence on the system parameters. The

modal gain can be calculated as the product between the population inversion ∆N and the peak-gain cross-section gc, i.e. the maximum

gain at the transition frequency per unit population of the upper-state. This can be obtained using Fermi’s golden rule for the scattering rate and it is found to be [25]: gc = 4πe 2 e0neffλ z2 32 2γ32Lp (1.4)

where e0is the vacuum dielectric constant, neffthe effective refractive

index of the active region with length period Lp, and, λ, 2γ32 and

z2

32 are the wavelength, the full-width at half-maximum (FWHM) and

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threshold current density can be written as [25]: Jth=e_gαtot cτeff = e0 4πe neffLpλ(2γ32)αtot Γz2 32τeff (1.5)

where αtotis the sum of mirror αmand waveguide αw losses and Γ is

the confinement factor which quantifies the overlap between the EM field and the active region (AR), defined as:

Γ= R AR|Ez|2dV R space|E|2dV (1.6) In order to minimize the threshold current density, apart from the pre-dictable but crucial requirement to minimize all kind of system losses and to maximize the peak-gain cross section, Eq. 1.5 also shows that the device must be engineered considering a multi-dimensional pa-rameter space, where the papa-rameters are not decoupled. For example, the dipole element z23 and the lifetimes of the electronic states

(enter-ing in τeff) are connected due to the dependence of z23on the

wavefunc-tion overlap. The luminescence linewidth does not only depend on de-phasing processes, but it is also strongly affected by the kind of optical transition, such as how much "diagonal" (or vertical) is the transition in real space. Depending on the distance between the quantum wells where the upper and lower states are mostly localized, electrons can move more or less far away in space after the optical transition. This consequently affects several parameters. In fact, while vertical transi-tions are preferred because of a higher dipole element and a narrower linewidth with respect to more "diagonal" transitions, the latter are advantageous to weaken the depletion mechanism of the upper state thereby benefiting of a longer lifetime with respect to vertical transi-tions. A good compromise between the two types of transition should be found. Moreover, the period length has a complex dependence on all the before-mentioned parameters and it is also connected to the amount of waveguide losses in the system due to the large amount of detrimental absorption by free carriers (i.e. doping). The radiative and waveguide losses in the system are then non-trivially related with the confinement factor, which quantifies the goodness of waveguide con-finement, but which often implies a consequent increase of the overall losses.

As a result, the engineering of the whole heterostructure−the ac-tive region to maximize the total amount of gain, the injection stage

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to minimize the parasitic channels in the carrier flow and the waveg-uide to optimize the optical feedback for laser operation keeping losses under an acceptable value− is a mandatory requirement to obtain a low-threshold, highly efficient quantum cascade laser. In the next Sec-tion, the typical active region designs and adopted waveguides for the operation of quantum cascade lasers will be presented, with particular focus on THz QCLs.

1.2.2 Active-region designs

In order to simultaneously fulfill all the before-mentioned conflicting requirements, a large variety of QCL designs have been developed. De-spite the different concepts on which these designs are based, they are all united by the goal to accomplish the necessary condition for pop-ulation inversion, which is making the ratio τ2/τ32 (much) lower than

1. Focusing at THz frequencies, the active region designs had to solve the issues which were preventing QCLs to operate at such long wave-lengths; namely, the lack of an efficient mechanism to achieve popula-tion inversion (to face the very short non-radiative lifetime of∼0.5 ps), the huge free carrier absorption (∝ λ2_{) and the difficulty to control the}

electron transport. The design which first allowed QCL to work below the LO phonon energy (i.e at THz frequencies) is the so-called chirped superlattice design [27] shown in Fig. 1.5a. In this design, the breaking of the superlattice translational symmetry allows charge neutrality in each period without the need of additional charges. Changing both the width and the period of the superlattice along the growth direc-tion, the minibands are aligned whenever an appropriate value of the electric field is applied to the heterostructure. Importantly, only for the states at the minigap, where the population inversion occurs, a perfect translational symmetry is maintained, and the lower miniband is com-pletely merged with the injector optimizing the injection efficiency. The bound-to-continuum design [29, 30] (reported in Fig. 1.5b) con-stitutes an improvement to the chirped superlattice. It continues to exploit the chirped superlattice for the active region, but it is char-acterized by the presence of a defect producing a bound state which originates from the upper minband. The latter allows to strongly in-crease the injection efficiency because the upper state of the transition is no longer a miniband, but a single isolated level. Furthermore, al-though the oscillator strength slightly drops, the radiative transition becomes more "diagonal" in real space implying a longer upper state

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nature photonics | VOL 1 | SEPTEMBER 2007 | www.nature.com/naturephotonics 519

RESONANT-PHONON

The other major active region type is the RP scheme (shown in Fig. 2c, refs 25,54,55). As is common for most mid-infrared QC lasers, collector and injector states are designed to be below the lower radiative state ‘1’ by approximately ELO = 36 meV, so that

electrons in the lower state will scatter very quickly into the injector states by emitting an LO-phonon. Many early terahertz QC emitter designs suffered from a fundamental difficulty: because of the close subband energy spacing, it was difficult to use LO-phonon scattering to depopulate the lower radiative state without also depopulating the upper state38,39_{. The key development of the RP scheme was bringing}

the lower radiative state into a broad tunnelling resonance with the excited state in the adjacent quantum wells, so that its wavefunction is spread over several quantum wells. As a result, the lower radiative state maintains a strong spatial overlap with the injector states and experiences subpicosecond LO-phonon scattering. The upper state ‘2’, however, remains localized and has very little overlap with the injector states, which suppresses scattering to the injector states and preserves a lifetime of several picoseconds. The lack of a miniband means that RP designs tend to have a smaller oscillator strength (f21 ≈ 0.5−1) than

the BTC designs, but this is partially compensated by the fact that the length Lmod of an RP module is typically half that of a BTC module,

which results in a higher density of gain producing transitions (that is, g L–1

mod, ref. 47). In addition, hybrid structures have been developed

in which phonon-assisted depopulation has been incorporated with a BTC optical transition (Fig. 2d). These are often known as ‘interlaced’ structures, owing to alternating photon- and phonon-emission events. Their impact has been limited, but they are particularly notable for achieving very long wavelength operation.

WAVEGUIDES

Because of the strong absorption owing to free carriers at long wavelengths, unique waveguides for terahertz QC lasers have been developed so as to minimize the overlap of the mode with any doped cladding layers. There are two types of waveguides used at present for terahertz QC lasers: the semi-insulating surface-plasmon (SI-SP), and the metal–metal (MM) waveguide, as shown in Fig. 3. It is useful to characterize a laser waveguide by its loss coefficient αw,

which accounts for scattering and absorption inside the waveguide, its confinement factor Γ, which describes the overlap of the mode with the active region, and the mirror loss coefficient αm, which

accounts for losses due to optical coupling, usually owing to finite facet reflectivities. These factors determine the required gain gthto

reach the lasing threshold, where the modal gain must equal the total losses: Γgth = αw + αm.

The SI-SP waveguide involves the growth of a thin (0.2−0.8 μm thick) heavily doped layer underneath the 10-μm-thick active region, but on top of a semi-insulating GaAs substrate23,24,56_._The

result is a compound surface-plasmon mode bound to the top metal contact and the lower plasma layer. Although the mode extends substantially into the substrate, the overlap with any doped semiconductor is small, so that the free-carrier loss is minimized. The confinement factor typically lies in the range Γ = 0.1−0.5, and in general, modes are somewhat loosely confined, which enables there to be relatively wide ridges without supporting multiple lateral modes. The downside is that ridges narrower than about 100 μm tend to squeeze the mode into the substrate, which limits the minimum device area57_.

Chirped superlattice Bound to continuum

Resonant-phonon Hybrid/interlaced 2 2 2 1 1 1 2 2 1 1 2 1 2 2 1 1 inj inj inj inj Module ~105 nm Module ~110 nm Module ~130 nm Module ~55 nm hLO hLO hLO hLO

Figure 2 Conduction-band diagrams for major terahertz QC design schemes. Examples are shown for: a, CSL, b, BTC, c, RP and d, hybrid/interlaced designs. Two identical

modules of each are shown here, although typically 100–200 cascaded modules are grown to form active regions 10–15-μm thick. The squared magnitude of the

wavefunctions for the various subband states are plotted, with the upper- and lower-radiative state shown in red and blue respectively and the injector states specifically labelled.The grey shaded regions correspond to minibands of states.

(a) (b) (c) (d) Chirped superlattice Resonant-phonon Bound-to-continuum Hybrid/Interlaced

Figure 1.5: Conduction-band diagrams of the four most exploited QC active region designs: (a) chirped superlattice, (b) bound-to-continuum, (c) resonant-phonon and (d) hybrid/interlaced scheme. Two consecutive identical modules of the cascade are shown. For each design the length of the period is reported. The blue and red solid lines indicate the electron probability density (squared modulus of the wavefunction) of the upper- and lower-radiative state, respec-tively. The black arrow represents the optical transition connecting them. The injector states are also specifically labelled, while the minibands of states are described by the grey shaded regions. Adapted from [28].

lifetime. As a consequence, this design displayed improved tempera-ture and power performance compared with the chirped superlattice design.

The other major active region type is the resonant-phonon scheme [31, 32] (Fig. 1.5c). Despite being very efficient and of easy realiza-tion in mid-infrared QCLs, in THz QCLs it was difficult to implement due to the close subband energy spacing which does not allow to use the LO phonon to depopulate the lower state without also depopu-lating the upper one. The solution consisted in spatially separating the region where the optical transition takes place from that where the resonant-phonon scattering arises. This was done by bringing the lower radiative state into a broad tunneling resonance in the adjacent quantum wells, so that its wavefunction is spread over a series of

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quan-tum wells, at the end of which electrons can finally resonantly emit an optical phonon. Although the lack of a miniband implies a smaller oscillator strength with respect to the bound-to-continuum design, the resonant-phonon scheme gives rise to a highly efficient depopulation of the lower state thanks to sub-picosecond LO-phonon scattering. Another class of designs instead aims at combining the advantages of both bound-to-continuum and resonant-phonon schemes. This hybrid structures are also known as "interlaced" designs, owing to their pecu-liar characteristic of alternating photon- and phonon-emission events [32,33]. This kind of scheme, shown in Fig. 1.5d is of particular signif-icance for having led recently to the route in the reduction of the num-ber of states per period, and consequently of the numnum-ber of quantum wells. Although the room temperature operation of THz QCLs remains an open challenge, due to the combination of both back-filling and temperature activated phonon processes, the engineering of QC active regions with the minimum possible number of states has made big steps forward in this direction. First, the resonant-phonon 3-quantum wells and, then, the 2-quantum wells design, based on a series of "di-agonal" radiative and resonant-phonon transitions without the need of an injector, pushed laser operation to temperatures larger than that achievable with a thermoelectric cooler [34, 35].

1.2.3 Waveguides

Apart from the active material, the most convenient waveguide to con-fine the EM field in the vertical direction strictly depends on the op-erating wavelength. This is particularly true for QCLs, for which the waveguide design strongly changes passing from short wavelengths to mid-infrared and finally to THz. While for QCLs based on In-GaAs/AlInAs/InP material system (operating mostly at λ < 10 µm) the material combination is very suitable for dielectric waveguides thanks to the good refractive index contrast of the substrate, at longer wavelengths the situation starts to be different. As the cladding thick-ness of dielectric waveguides scales linearly with wavelength, the max-imum growth thickness via MBE3_{, in addition to the λ}2_{-dependence of}

Drude-like absorption by the doping, limits the exploitation of this type of waveguide for wavelengths longer than 10-15 µm. Neverthe-less, the dielectric refractive index contrast approach remains valid for

3_{The relatively low epitaxial growth rate of about 0.5-1 µm per hour and the} de-crease of layer accuracy for growth larger than 10-15 µm limit the maximum achievable thickness of the QC heterostructure.