Miniaturization of Terahertz Quantum Cascade Lasers exploiting a graphene-based heterostructure

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DIPARTIMENTO DI FISICA ‘E. FERMI”

Corso di Laurea Magistrale in Fisica

Miniaturization of Terahertz Quantum Cascade Lasers

exploiting a

graphene-based heterostructure

Candidato:

Veronica Leccese

Relatori:

Prof. Alessandro Tredicucci

Dr. Camilla Coletti

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Introduction

The Terahertz (THz) spectral region (ν ∼ 0.5−10 Thz, λ ∼ 600−30µm, ¯hω ∼ 2−40 meV) has remained underdeveloped for years in large part due to the lack of suitable techniques to generate coherent high-power ra-diation. The interest in THz radiation comes from its numerous possible applications in many fields such as medical diagnostics, safety, quality con-trol and observational astrophysics [1].

Among the THz sources (e.g. free electron lasers, Guun diode and high-frequency transistors [2]), Quantum Cascade Lasers (QCLs) have emerged as the most promising mainly thanks to the small device footprint and good optical powers.

QCLs are unipolar devices exploiting optical transitions between electronic conduction subbands created by spatial confinement in semiconductor multi-quantum-wells. The latter constitute the laser active region and can be en-gineered on a nanometer scale to have the desired wavefunctions .

QCLs were first demonstrated in the mid-infrared [3], while the first THz QCL was realized in 2002 [4]. THz QCLs exhibit excellent coherence prop-erties and good optical powers in the frequency range between 1.2 and 5.4 THz [3, 4,5] opening the way to a variety of THz photonic applications. Despite their remarkable properties, some challenges still remain open. A strong research effort is dedicated to increase their maximum operating temperature, now fixed at 199,5 K in pulse mode [6], as well as to decrease their power consumption and to improve frequency tuning and outcou-pling efficiency.

At THz frequencies, two types of waveguides are commonly used: semi-insulating surface-plasmon and double metal waveguides, which exploit metal Surface Plasmon Polaritons (SPPs). Between the two, double metal waveguides are the most suitable since they guarantee near unity confine-ment factor (i.e. the overlap of the guided mode with the active region) and low lasing threshold current [7]. Moreover, thanks to the lack of a cut-off frequency, they enable to confine a transverse magnetic mode (that couples to the intersubband electronic transition of the active region) inside the cavity independently from its transverse dimension. This property has been useful for device miniaturization since it allows to focus on the

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reduc-with double metal waveguides; in particular, microdisk resonators, operat-ing on whisperoperat-ing-gallery modes (WGMs), are known to yield lasers with very low threshold currents thanks to the combination of a small-mode size and high mode quality factors [8]. However, being the used metals not perfect conductors, there is a relevant light absorption that increases as the wavelength exceeds the waveguide thickness. Thus, ohmic losses in the metal are the main factor that limits the lowest possible achievable threshold current [5].

There is a growing need, therefore, to identify new and better plasmonic materials. One of the most promising candidates for THz applications is graphene [9]. Graphene SPPs are characterized by a wavelength of the or-der of λ0/100, where λ0 is the free-space wavelength, which is one order

smaller than in metal SPPs wavelength. Thus, the radiation is confined much more strongly than for conventional metals, leading to the possibil-ity of a further miniaturization of the devices.

The propagation of SPPs as well as the graphene mobility are affected by the scattering from charged surface states, impurities and substrate surface roughness [10]. An increase in mobility can be obtained using materi-als as graphene encapsulants, that minimize extrinsic sources of scattering coming from both the interfaces with the substrate and air. The ultimate graphene encapsulant is h-BN, that is an insulator isomorph of graphite with atomically flat layers nearly free of charge trapping [11]. It has been demonstrated that graphene embedded in two h-BN flakes leads to electron mobilities comparable with the ones reached in free-standing graphene (∼ 200000 cm2/Vs in suspended samples [12] and ∼ 140000 cm2/Vs in encapsulated graphene [13]).

Hence, the idea of this thesis is to exploit a new innovative low-loss waveg-uide based on graphene SPPs, which would allow also a further miniatur-ization of THz QCLs. This novel structure is similar to the double metal waveguide but the top metal layer is replaced by graphene encapsulated in two layers of hexagonal-Boron Nitride (h-BN).

In particular, in this work, a microdisk resonator based on h-BN/graphene/ h-BN heterostructure is presented. It was studied via Finite Element Method (FEM) simulations and compared with a microdisk exploiting double metal waveguides.

Since graphene SPPs confine the electromagnetic energy inside the ac-tive region in deeply subwavelength dimensions, a further reduction of the mode volume and, thus, a strong enhancement of the Purcell factor should be expected. The high Purcell factor could result in ultra-small las-ing threshold.

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CONTENTS

the carrier density via doping or electrostatic gating, leads to tunable SPPs. This could be exploited for laser emission control, in particular for tuning the laser frequency and the far-field profiles of the emitting modes.

Besides the study via simulation, all the steps for the final fabrication of the device were experimentally implemented, using different nano-fabrication techniques, such as Electron Beam Lithography (EBL), Reactive Ion Etching (RIE), thermal metal deposition and Inductively Coupled Plasma-Reactive Ion Etching (ICP-RIE). Both graphene and h-BN were mechanically exfoli-ated and assembled in h-BN/graphene/h-BN heterostructures. The latter were characterized, firstly, by Atomic Force Microscopy and Raman spec-troscopy and, then, by electrical measurements, carried out on a Field Effect Transistor (FET) based on h-BN/graphene/h-BN hetrostructure.

The thesis outline is presented below.

Chapter 1reports the fundamental physics of the THz QCLs. Specific focus is given to suitable THz QCLs waveguides and microcavities.

Chapter 2 discusses the remarkable properties of surface plasmon polari-tons in h-BN/graphene/h-BN heterostructures. The main structural and electronic properties of the latter are also described.

Chapter 3 is fully dedicated to the finite element simulations of the mi-croresonators. The first studied system is a microresonator exploiting a double metal waveguide, the results are then compared with the ones obtained from the study of the microcavity exploiting a graphene-based waveguide. Once the interesting properties of the new waveguide design has been pointed out, the system is studied by varying the h-BN/graphene/h-BN properties. Finally vertical emission was achieved thanks to the cou-pling between two identical subwavelength microdisk resonators in close proximity.

Chapter 4describes the engineering and fabrication procedure adopted for the implementation of the simulated THz QCL with a h-BN/graphene/h-BN heterostructure. All the procedures and the attempts carried out for the optimization of each fabrication step are described in detail.

Chapter 5 is centred on the characterization of the h-BN/graphene/h-BN heterostructure through Raman spectroscopy, allowing to estimate the graphene carrier density, and electrical measurements, carried out on a FET.

Finally, in Conclusions and perspectives the major findings of this work are summarized and a series of perspectives are proposed.

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Chapter 1 Overview of operational

principles of THz Quantum

Cascade Lasers

Terahertz Quantum Cascade Lasers (THz QCLs) promise to provide power-ful new sources of long-wavelength radiation that may impact applications in spectroscopy and imaging.

In this chapter the fundamental theory behind QCLs and, in particular THz QCLs, is reviewed starting with intersubband transitions, that are the ba-sis of their operation and continuing with the main lasing principles and with the suitable waveguide for THz QCLs. Particular attention is given to double metal waveguide which allow to confine radiation in deeply sub-wavelength microcavities, very appealing for the device miniaturization. The main properties of microcavities are reported in the last section.

1.1 Intersubband transition

In traditional bipolar semiconductor lasers, the emitted wavelength is mainly determined by the material bandgap.

The extension of the bipolar laser technology to the terahertz region is not possible, as materials with bandgaps less than 20 meV would be required. A solution can be found in radiative transitions, denoted as intersubband transitions, between two quantized states of the conduction band. The quantized states arise from heterostructure quantum wells. Indeed, when thin (on the order of a De Broglie wavelength (<hundreds of Angstroms)) semiconductor layers of differing composition are sequentially grown, dis-continuities are introduced in the band edges, which leads to the quantum confinement of carriers in the growth direction. For example, if a semi-conductor B is grown between two layers of semisemi-conductor A, which has a

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higher bandgap, A-B semiconductors can form a heterojunction of the first type, where the extrema of the conduction and valence bands produce a potential well corresponding to the change in materials (Fig.1.1). As a

re-Figure 1.1: Pictorial view of the quantum well model heterostructure. EgA and EgB are

the bandgaps of A and B semiconductors.

sult of the discontinuities, the electronic bands break up into ”subbands,” where the energy is quantized in the growth direction and parabolic free carrier dispersion is retained in the in-plane directions.

The wavefunctions that describe the states in the heterostructures, using the effective mass approximation and a simplified single band expansion, can be written as:

Ψ(r) = F(r)u_n,0(r) (1.1) where un,0(r) is the Bloch state wavefunction at the band minimum (0

in-dicates k = 0, that is the so called Γ-point) and F(r) = χ(z)eikk·r is the

envelope function, with kk indicating the in-plane wavevector and χ(z)

de-scribing the extension of the electron states in the direction perpendicular to the heterostructure layers.

The varying material composition is represented by a spatial variation of the effective mass m∗(z)and the potential Ec(z), which represents the

con-duction band edge profile, including any externally applied field and local variations due to space charge. Thus, the effective mass equation for the envelope function can be written as:

" −¯h2∆k2 2m∗(z) − ¯h2 2 ∂ ∂z 1 m∗₍_z₎ ∂ ∂z +Ec(z) # F(r) =EF(r) (1.2) 2

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1.2 Lasing principles

where ∆k is the in-plane differential operator [14]. The solution of Eq. 1.2

for the envelope function, neglecting the coupling between the in-plane and z directions introduced by the spacially dependent mass, is given by:

F(r) = q1 Sk eikk·rk χn(z) (1.3) where χn(z)satisfies " −¯h 2 2 d dz 1 m∗(z) d dz +Ec(z) # χn(z) = Enχn(z) (1.4)

n and Sk are the subband index and the normalization area of the layers,

respectively.

Thus, the total energy is given by

En(kk) = En+

¯h2kk

2m∗ (1.5)

where m∗ is the well material effective mass. It is clear that in the recipro-cal lattice the energy bands have a parabolic dispersion in kx and ky, while

the kz dispersion is replaced by the presence of a number of discrete levels,

thereby forming a series of subbands.

Transitions between two subbands have two main advantages. The first one is that the joint density of states is delta-like, similar to the case of atomic systems. The broadening of gain and absorption lines is due mostly to scattering and inhomogeneities ((Fig.1.2 (a)). On the contrary, in the case of interband transitions, the broadening of the gain spectrum is due to elec-tron and hole distributions within the band (Fig.1.2 (b)).

The second advantage is that the photon energy resulting from the transi-tions can be chosen by tailoring the thickness of the coupled wells and barriers, which makes such structures ideal for the generation of long-wavelength radiation.

1.2 Lasing principles

1.2.1 Cascading scheme and population inversion

A periodic scheme made up of wells and barriers, with the addition of an appropriate voltage, gives rise to a cascading scheme, which is the basis of the QCL operation.

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Figure 1.2: (a) The intersubband transition connects two envelope functions of the same band, giving a narrow atom-like joint density of states. (b) The interband transition be-tween different Bloch functions and the gain as a function of energy.

The periodic repetition of wells and barriers is called superlattice, where the single well states couple together to form the so called minibands, which are separated by minigaps. The bias voltage applied to the superlattice makes it possible to have resonant states in each period of the structure. Thus, due to the tunnelling, once an electron has undergone an intersubband transition and emitted a photon in one period of the superlattice, it can be recycled from period to period, contributing each time to the gain and the photon emission. So, each electron can generate, in principle, Np laser

photons, where Np is the number of periods of the superlattice. This leads

to a cascading emission of photons.

The cascading scheme makes quantum efficiency (i.e. the percentage of electrons which contribute to the amplification of radiation) greater than unity, leading to higher output powers than semiconductor laser diodes of the same wavelength.

A typical cascading scheme of QCLs is a periodic repetition of active re-gions and relaxation/injection rere-gions (see Fig. 1.3 (a)). The active region is the region where population inversion and optical gain takes place and, in this example, it is based on a three-quantum-well scheme as reported for the first time by Faist et al. in 1994 [3]. One period of the active zone can be approximated to a three-level system (see Fig. 1.3 (b)). To achieve population inversion, the lifetime of the transition between levels 3 and 2 has to be longer that the lifetime of level 2.

Non-radiative relaxation brings electrons to the ground-state (level 1) and they are then injected into the next stage thanks to tunnelling. Thus, the role of the injector region is to transport electrons from level 1 to level 3 of the next period. Moreover, part of the injection region is doped in

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der to provide the electron charge needed for the transport. The transition between levels 3 and 2 can be vertical or diagonal. It is said to be vertical when the two wave functions of states n = 3 and n = 2 have a strong overlap and diagonal when this overlap is reduced.

Once the structure of a QCL is defined, it is possible to calculate the popula-tion inversion, approximating one period of the active zone to a three-level system, as above. The following treatment is based on that in Paiella [15]. First of all, the injection efficiency has be to introduced. The injection effi-ciency is the ratio of the current injected into level 3 to the total current:

ηi =

J3

J (1.6)

In the ideal case it should be 1. η < 1 means that electrons have been injected in the lower states (1 and 2) or there was a thermal activation from the injector to the continuum. Both of these problems can be avoided by using a narrow quantum well after the injection barrier and a higher energy barrier, respectively. The population density of the level 3 (in the steady-state case) can be written as

n3 =

ηiJ

e τ3 (1.7)

where (τ3) is the total lifetime of electrons in level 3. Assuming that level 2

is populated only by electrons from level 3, the electron density of level 2 is given by

n2 =n3 τ2

τ32

(1.8) where (τ2) is the lifetime of electrons in level 2 and (τ32) is the scattering

time from level 3 to 2. Backfilling, i.e. thermal population of n2 originating

from carriers thermally excited to the lower state, is neglected. Thus, the population inversion can be written as

n3−n2= ηiJ e τ3 1− τ2 τ32 (1.9)

In the worst-case scenario, if τ32 =τ3, Eq. 1.9 becomes

n3−n2 = ηiJ

e (τ3−τ2) (1.10) This means that if τ3 > τ2, population inversion and optical gain are

achieved for any current density. As a consequence, the laser threshold current density is directly proportional to the optical cavity losses, which can be reduced as much as possible through low-loss waveguides.

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Figure 1.3: (a) Schematic conduction band diagram of a quantum cascade laser. Each period of the structures is made up of an active region and a relaxation/injection region. Each electron can emittes one photon per period and it can be recycled from period to period thanks to tunneling. (b) The active region scheme, i.e. three-level system. The lifetime of the 3→2 transition (τ32) has to be longer than the lifetime of level 2 (τ2) to

obtain population inversion [16].

In THz spectral range the development of intersubband lasers proved to be much more difficult than for mid-infrared QCLs for two main reasons: the difficulty to achieve a population inversion for such a small subband separation and the challenge of obtaining a low-loss waveguide for such long wavelengths. The latter one will be dealt with in the next section. In mid-infrared QCLs, the population inversion is obtained with typically electron-longitudinal-optical-phonon (LO-phonon) resonant scattering that helps the depletion of the lower level (level 2). In THz QCLs the radia-tive state separation is less than the LO-phonon energy ELO (see Fig. 1.4).

The higher LO-phonon energy with respect to the radiative state separa-tion makes hard the achievement of the populasepara-tion inversion because the phonons contribute to the depletion of both levels 2 and 3. Thus, the op-eration of THz QCLs is limited at low temperature (<200 K [6]), which limit both the phonon scattering and the back-filling. Indeed, at low tem-perature, LO-phonon scattering is nominally forbidden and only the high energy tail of the thermal electron distribution can emit LO-phonons. As a result the lifetime is determined by a combination of electron-electron scattering, electron-impurity scattering, interface roughness scattering and thermally activated LO-phonon scattering [7]. Both electron-electron and electron-impurity scattering are highly dependent on the electron

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Figure 1.4: Schematic of in-plane momentum space (kk) with typical allowed relaxation

paths for the 3 →2 transition. The non radiative transitions 3 → 1 and 2 →1 are also depicted [17].

bution, which causes τ32 to drop with increasing injection and limits the

maximum upper state population. Moreover, for small energy separations E32, it becomes difficult to selectively inject electrons into level 3 and not

level 2.

Possible active region designs to solve these problems are the chirped superlattice concept, bound-to-continuum and spatially separate resonant phonon extraction, which are depicted in Fig.1.5.

In chirped superlattice the wells and barriers have gradually changing size along the growth direction to allow a properly enginereed alignment of the levels (see Fig. 1.5 (a)). The radiative transition involves the lowest state of the upper miniband 2 and the highest state of the lower miniband 1. Population inversion is guaranteed by the fact that the electron scattering is stronger between the tightly coupled states of the same miniband (intra-miniband scattering: τ ∼1 ps) than between states of different minibands (inter-miniband scattering: τ ∼10 ps). Thus, electrons relax to the bottom of each miniband and tend to populate 2 and deplete 1.

The bound-to-continuum design [19] consists of a chirped superlattice, where the active region spans the whole period. The lower miniband is tilted and its width is maximum in the center and decreases on both sides close to the injection barriers (see Fig.1.5(b)). The upper state is in the first minigap, thanks to the presence of a small well adjacent to the injection barrier and it is well-separated from the higher-lying states of the superlat-tice. The large energy separation prevents injection of electrons into higher energy states.

Another active region design for THz QCLs is the resonant phonon

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Figure 1.5: Schematic of THz QCL active regions [5]: (a) chirped superlattice [18], (b) bound-to-continuum [19] and (c) renonant-phonon [20]. The red and blue lines are the upper and lower radiative subbands respectively. Grey shaded regions indicate minibands of states.

sign [20], which is different from the other two designs described above. It uses a combination of resonant tunneling and direct electron-LO-phonon scattering (see Fig.1.5 (c)). A collector/injector state is engineered to lie at LO-phonon energy (¯hωLO) below the lower radiative state 1 involved in the

THz transition: electrons undergo fast sub-picosecond relaxation from 1 by emission of a LO-phonon. This selective phonon-assisted depopulation is obtained by creating a huge overlap between 1 and the collector/injector states of the adjacent wells. On the contrary, the upper state 2 is kept localised in the active region suppressing scattering towards the collector states, so that its lifetime is few ps.

1.2.2 Intersubband radiative transitions and gain derivation

In order to calculate the gain of QCLs, the derivation of intersubband tran-sition probabilities, induced by an incident electromagnetic wave, is neces-sary. The following description is based on that in Paiella [15].

Considering a linearly polarized electromagnetic wave with an electric field

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E = E0εcos(ωt−q·r), where ε is the polarization, ω the pulsation and q

the propagation vector, the Hamiltonian describing the interaction between the incident electromagnetic wave and the heterostructure, with a refractive index n, can be written as

H = − e

m∗A·p=V

eiωt−e−iωt (1.11) where e in the electron charge, m∗ the effective mass in the quantum well material, A the vector potential associated with the incident electromag-netic wave

A= −E0

2ω ε

ei(ωt−q·r)₋_e−i(ωt−q·r) _(1.12)

Using the Fermi golden rule, the probability for an electron to make a transition from an initial state|iito a final one|fi is

Wi f(¯hω) = 2π ¯h |hf|V|ii| 2 δ(Ef −Ei±¯hω) =2π ¯h e2E2₀ 4m∗2_ω2 |hf|ε·p|ii| 2 δ(Ef −Ei±¯hω) (1.13)

The term -¯hω corresponds to the absorption of an incident photon while +¯hω corresponds to the stimulated emission of a photon.

The initial state|ii and a final one|fican be replaced by the wave function of heterostructure states having the form of Eq.1.1. Since for intersubband transitions the initial and final states originate from the same band, u_i,0 =

uf ,0, the matrix element of Eq.1.13 becomes hf|ε·p|ii =hψf|ψii 1 S Z dxdy e−kk f·rk εxpx+εypy ekk f·rk +δ(kkf −kki) εzhψf|pz|ψii| (1.14)

The first term hψf|ψii vanishes if i6=f due to the orthogonality of the

en-volpe functions. Thus, rewriting the matrix elements by making use of the commutation relation for the unperturbed Hamiltonian H0

i

¯h [H0, z] = pz

m∗ (1.15)

the transition rate, expressed in r representation, is given by

Wi f(¯hω) = 2π ¯h e2E₀2 4 ε 2 z|hf|z|ii|2×δ(kkf −kki)δ(Ef −Ei±¯hω) (1.16)

Eq. 1.16 contains the well known intersubband selection rule: only

transi-tions with the E field polarized along the growth axis z are permitted.

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Since the transitions have a certain finite width due to dephasing mecha-nisms, the Dirac delta-function has to be replaced with a Lorentzian func-tion, assuming a homogeneously broadened transition. Thus Eq.1.16 can be rewritten as Wi f(¯hω) = πe2E₀2|hzi fi|2 2¯h γ/π (Ei f −¯hω)2+γ2 (1.17)

where γ is the half width at half maximum of the Lorentzian function. Eq. 1.17has a maximum when ¯hω =Ei f, so

W_{i f}max(¯hω) = e 2_E2

0|hzi fi|2

2γ¯h (1.18)

Using the expressions for the transition rate, the gain in the laser medium can be derived. In order to do this, a heterostructure of width w and thick-ness Lp (in the x and z directions, respectively) containing one quantum

cascade active period with three levels and an electromagnetic plane wave propagating in the y direction can be considered. The number of photons of energy ¯hω that cross the heterostructure per unit time is given by multi-plying the ratio between the power density carried by the plane wave and the energy of the single photon with Lpw:

Φ = 1

2

ε0ncE02

¯hω Lpw (1.19) Considering ¯hω =E32, the variation of the photon flux, over a distance dy,

can be written as

dΦ =W₃₂max n3w dy−W32max n2w dy (1.20)

The first term corresponds to the stimulated emission of photons due to the presence of electrons in level 3 (the number of electrons in level 3 is given by n3w dy), while the second corresponds to the absorption of photons due

to the presence of electrons in level 2 (the number of electrons in level 2 is n2w dy). Finally, it is possible to define the material gain as

G= dΦ/dy

Φ (1.21)

By using Eqs. 1.20, 1.19and 1.18 it obtains G= 2e 2_|h_z 32i|2ω ε0nc2γLp (n3−n2) (1.22) 10

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1.3 Waveguides

By replacing (n3−n2) with Eq.1.9 and defining the gain coefficient g as

g= 4πe|hz32i| 2 ε0nλ2γLp ηiτ3 1− τ2 τ32 (1.23)

where λ is the wavelength of the incident light (λ =2πc/ω), the material gain can be rewritten as

G=gJ (1.24)

Actually, in QCLs, as described in detail in next section, the optical mode is guided and it extends in the z direction outside of the active region. Thus, the gain is reduced by a factorΓ called confinement/overlap factor, that is the overlap of the guided mode with the active region, defined as

Γ = R AR|E(z)|2dz R∞ −∞|E(z)|2dz (1.25)

where AR stands for Active Region. Thus, the gain can be rewritten as

G =ΓgJ (1.26)

This is the gain for each period, if the active regions has Np periods, Γ has

to be replaced by ΓpNp, so the gain will be proportional to the number of

periods because by increasing the number of periods, the overlap of the guided mode with the active region increases.

Now, it is possible to write the threshold current density of the laser, that is directly proportional to the cavity losses, which include waveguide losses

αw and mirror losses αm:

ΓgJth=αw+αm → Jth= αw +αm

Γg (1.27)

Increasing the number of periods allows to reduce threshold; however, it implies also higher applied voltage and, so, an increase of the total dis-sipated power injected into the device. Hence, the only way to achieve low threshold is finding a waveguide that minimizes the optical losses and maximizes the overlap factor.

1.3 Waveguides

Besides population inversion, the second challenge of THz QCLs is design-ing an appropriate low-loss waveguide for mode confinement. For most mid-infrared QCLs, as in traditional semiconductor diode lasers, confine-ment is provided by dielectric cladding with a lower refractive index than

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the core. Because of the long wavelength at terahertz frequencies, such cladding would need to be prohibitively thick (about 100 µm [7]), indeed the necessary cladding thickness has to be on the order of λ/2 . Further-more, the optical mode penetrates into the doped claddings causing high optical losses (that grow as λ2) owing to the strong absorption by free car-riers.

A solution to these problems is found in Surface-Plasmon-based waveg-uides, that are divided into: semi-insulating surface-plasmon waveguide and double metal waveguide [7] (see Fig.1.6).

Figure 1.6: Drawing of the two waveguides: (a) semi-insulating surface plasmon waveg-uide and (b) double metal wavegwaveg-uide [17].

Before describing the two kinds of waveguides, a brief description of Sur-face Plasmons (SPs) is given. SPs are solutions of Maxwell’s equations at the interface between two materials with dielectric constants of oppo-site signs (i.e. between a metal and a dielectric) [21]. Basically they are light waves trapped on the surface because of their interaction with the free electrons of the metal. Due to this interaction, the free electrons begin to oscillate in resonance with the light wave. SPs arise from the resonant interaction between the surface charge oscillation and the electromagnetic field of the light [22]. Actually, for their nature SPs should be called Surface Plasmon Polaritons (SPPs).

By requiring continuity of electric and magnetic field at the interface, it can be derived that only TM polarized waves can interact with surface charges. Thus, SPs only exist for TM polarization [21].

The dispersion relation of SPs propagating at the interface between two materials is given by:

kSP =k0

r

εdεm

εd+εm (1.28)

where εd (> 0) is the permittivity of the dielectric material, while εm (<0)

is the metal permittivity.

From the dispersion relation (Eq. 1.28) it results that, once the frequency is fixed, the momentum of the SP mode (¯hkSP) is greater than the one of a

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1.3 Waveguides

free-space photon (¯hk0) (see Fig. 1.7 (a)). This mismatch between the two

momenta must be overcome in order to couple light to SP modes (for ex-ample through special phase matching techniques such as grating or prism coupling [21]).

Another important consequence of the interaction between the surface charges and the electromagnetic field is that the field perpendicular to the surface is evanescent, i.e. it decays exponentially with the distance from the surface (see Fig. 1.7 (b)). The penetration depths in the dielectric and the metal δd

and δm are given by [23]

δd = 1 k0 εd+εm −ε2_d 1/2 (1.29) and δm = 1 k0 εd+εm −ε2_m 1/2 (1.30) Once light has been converted into an SP mode, it will propagate, but

(a) (b)

Figure 1.7: (a) Dispersion curve of a SP wave; kSP and k0 are the SP and free-space

wavevectors, respectively. It is evident that the momentum (¯hkSP) of the SP wave is larger

than that of the light in free space (¯hk0) for the same frequency ω. (b) The electric field

component perpendicular to the surface is enhanced near the surface and decay exponen-tially with distance from the surface itself [22].

it will gradually attenuate owing to losses arising from absorption in the metal. The propagation length δSPof the SP is thereby limited by the

imagi-nary part of the complex SP wavevector kSP (k00SP) due to the internal

damp-ing and it can be written as

δSP = 1 2k00_SP = c ω ε0m +εd ε0_mεd 3/2 (ε0m)2 ε00_m (1.31) 13

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where εm and εm are the real and imaginary parts of the metal dielectric

function and ω=ck0.

Thus, for their properties, SPs are suitable for concentrating and chan-nelling light in subwavelength structures.

1.3.1 Semi-insulating surface plasmon waveguide

By contrast to dielectric waveguides, where losses grow as λ2, the losses of a single plasmon waveguide [7] can be written as

αw =2k00SP =

4πnmn3_d

k3 mλ

(1.32)

where nm and km are the real and imaginary part of the refractive index of

the metal, respectively; nd is the refractive index of the dielectric material.

Eq. 1.32 can be obtained using Eq. 1.28, reminding that εm = (nm+km)2

and εd=n2_d.

Eq. 1.32 suggests that the waveguide losses decrease with the increase of the wavelength λ and refractive indices nm and nd. In this kind of

waveg-uide, the strong coupling between the metal and the optical mode is used to manipulate the confinement of the mode in the plane of layers.

A limitation of single surface plasmon waveguide in the THz region is that, although the penetration of light into the top metal layer is low at such long wavelenghts, only a very small overlap of the guided mode with the active region can be achieved.

A solution could be found in a thin (0.1-1 µm) heavily doped contact layer grown directly under the active region and above the semi-insulating GaAs substrate. The doped layer behaves as a low density metal with a dielectric function εsc < 0, but it is thinner than its own skin depth, so, the mode

extends substantially into the substrate. Since the mode overlap with the doped contact layer is small, the free carrier loss is minimized. At the same time confinement can be maximized by suitable choice of the charge den-sity.

Fig.1.8shows the mode profile along the growth direction of the first THz QCL exploiting a semi-insulating surface-plasmon waveguide [4].

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1.3 Waveguides

Figure 1.8:Mode profile in the growth direction of the first THz QCL in a semi-insulating surface-plasmon waveguide. The active region is ”sandwiched” between a metal layer (on top) and a thin doped GaAs layer, acting also as bottom electric contact. GaAs layer is on top of a semi-insulating GaAs substrate. This structure allows losses limitation (αw =13 cm−1). The confinement factorΓ is 0.47.

1.3.2 Double metal waveguide

In double metal waveguides (also called metal-metal (MM) waveguides), the doped semiconductor contact is replaced by a layer of metal. This leads to a full confinement of the mode (Γ ∼ 1) between the two metals at the expense of a more difficult fabrication (including wafer bonding, that will be described in chapter 4).

Futhermore, the vertical and lateral dimensions of the device can be made much smaller than λ. In this way, thermal dissipation and threshold cur-rent are reduced, leading to a good high-temperature performance. How-ever, due to the sub-wavelength nature of the double metal waveguide, huge diffractive effects affect the beam quality and low output power re-sults from the high end-facet reflectivity [5]. Moreover, there are non-negligible losses by the metal producing somewhat larger absorption coef-ficients.

Fig. 1.9 shows the typical mode profile in THz QCL exploiting a double metal waveguide. By comparing double metal and semi-insulating surface plasmon waveguides, it results that the former one have the best high-temperature performance, while the latter one has higher output powers and better beam patterns. MM waveguides have a higher Γ, however las-ing threshold is comparable to the semi-insulatlas-ing ones because the ratio

αw/Γ is similar in both waveguides (a little bit better in MM waveguides).

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Figure 1.9: Mode profile in the growth direction of the THz QCL reported in [24], based on double metal waveguide. The confinement factor Γ is very close to unit (0.98) and losses waveguide are αw =17.8 cm−1) .

1.4 Microcavities

Thanks to the lack of a cut-off frequency, MM waveguides enable to con-fine a TM mode inside a cavity independently from its transverse dimen-sion. Besides having an already sub-wavelength thickness, also the lateral dimensions can be decreased from usual dimensions of the order of wave-length. In this way, really sub-wavelength microcavities can be realized, very appealing for their compactness and potential massive integration. Due to the sub-wavelength dimensions, the free spectral range of this type of cavities (i.e. the frequency distance between adjacent cavity modes) can be large enough to ensure only one cavity mode provides feedback for the lasing transition. In this way, single-mode emission is guaranteed having the other cavity modes out of the gain bandwidth of the medium.

An example of deeply sub-wavelength microcavities are the resonators where the electric field can be confined and spatially separated from the magnetic field [25]. For this reason, these devices act as hybrid electronic-photonic systems, operating on circuit-like resonances, the so called inductor-capacitor (LC) resonances. Recently, a microcavity laser oscillating in a circuit-based resonator has been demonstrated to operate at THz frequen-cies with a strongly sub-wavelength effective mode volume [26]. The effec-tive mode volume Ve f f is a measure of the volume in which the radiation

is mostly confined: Ve f f = R V|E|2ε(r) dV max(|E|2_ε₍_r₎₎ (1.33) 16

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1.4 Microcavities

where E is the electric field, V is the volume of the whole system and ε(r)

the relative permittivity.

This parameter is crucial because it determines the density of final states of the microcavity [27]:

ρc = _∆ν1 cVe f f

(1.34)

where∆νc is the resonance width.

In the case of weak coupling regime, where the light-matter interaction is treated as a perturbation on the system dynamics, the enhancement or in-hibition of the spontaneous emission can arise, depending on whether a spectral overlap between the photonic cavity mode and the electronic reso-nance is present or not, respectively.

When the cavity mode frequency ideally matches the electronic transition frequency, an enhancement of the spontaneous emission can be achieved. This enhancement is quantified by the so called Purcell factor given by the ratio between the emission rate inside the cavity and the one in vac-uum [27]: FP = 3 4π λc nc 3 Q Ve f f (1.35)

where λc is cavity mode wavelength, nc represents the refractive index of

the cavity medium and Q is the so called quality factor (Q-factor) given by:

Q = νc

∆νc (1.36)

The Q-factor is a dimensionless parameter describing the rate at which optical energy decays from inside the cavity due to absorption, scattering or leakage through imperfect mirrors.

In a microcavity, the Purcell factor can reach very large values due to the Q/Ve f f proportionality. The enhancement of the spontaneous emission

corresponds to an identical increase of the stimulated emission. Indeed, in a ideal microcavity, the total emission rate can be written as [27]:

Remi =BFP(s+1) (1.37)

where s is the number of photons in the cavity, B is the Einstein coeffi-cient for the spontaneous emission, the terms BFPs and BFP are related

to the stimulated emission rate and to the spontaneous emission one, re-spectively. Considering an ideal laser, where the population inversion is achieved and non-radiative effects are neglected, the time evolution of the photons density in the cavity, s, and the one of the excited electrons, n, are given by [27]:

˙s= FP(s+1)n−γs (1.38)

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˙n= p−FP(s+1)n (1.39)

where γ is the rate of the photons escaping from the cavity and p is the pumping rate of the system. In steady state conditions, the solution of the system of Eqs.1.38, 1.39is:

s= p

γ; n=

pγ

FP(p+γ) (1.40)

So, ideally, it is possible to have no-threshold laser, where the first photon spontaneously emitted turns the laser on because spontaneous and stimu-lated emissions are no more distinguishable.

Furthermore, in the limit p → ∞, the density of electrons n → γ

Fp. Thus,

the population inversion arises immediately with negligible pumping and reaches asymptotically the threshold charge density for a conventional laser.

However, in a realistic system, only a fraction of the spontaneous emission rate in the solid angle is affected by the presence of the microcavity. This fraction is quantified by the so-called spontaneous emission coupling factor, β. This value is typically below 10−5 in bulk lasers but can be much larger in microcavities, achieving value > 10%. The laser threshold can then be written as [27]

Pth = πhν 2 c

Qβ (1.41)

In order to get large values of β and, thus, lower threshold currents, high Q-values and very small volumes are necessary, obtaining huge FP values

for almost all the spontaneous emission solid angle.

Moreover, a small optical cavity means a smaller volume that is pumped to provide gain, so a major decrease of the total energy needed to reach population inversion.

In order to measure these effects, microcavities with volumes of the order of tens µm−3 and Q-factors of 103 are used. The simplest type is consti-tuted by a microdisk resonator that exploits total internal reflection to con-fine light in the lateral dimensions, the so-called whispering gallery mode (WGM) resonators. In particular, WGM resonators that allow to confine the radiation in the vertical direction sandwiching the active material be-tween two metal layers as in the MM waveguides have been realized. After the first demonstration for interband semiconductor lasers [28], disk res-onators were exploited for mid-infrared QCLs [29, 30] and, more recently, at THz frequencies. For example, a WGM laser at THz frequencies has been shown to emit in the vertical direction with regular beam profiles and with high output power by implementing appropriate diffraction gratings

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1.4 Microcavities

along the disk circumference [31].

A very low-threshold laser emission in a collimated beam pattern has been achieved through two microdisks coupled by a suspended metallic bridge [32]. This design merges the high quality factor of WGMs with the subwavelength antenna design, which allows to achieve emissions with regular far-field profile.

Starting from these results, the idea of this thesis is to exploit a microcavity based on an innovative waveguide taking advantage of the graphene sur-face plasmon polaritons, that will be discussed in next chapter. This novel structure would allow to achieve an extreme miniaturization of THz QCLs with unprecedented low lasing threshold current.

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Surface Plasmon Polaritons in

graphene/hexagonal-Boron Nitride

heterostructures

Graphene is one of the most promising materials for THz plasmonics. Ow-ing to the two-dimensional nature of collective excitations, surface plas-mons in graphene are confined much more strongly than in conventional metals, thus representing a good alternative to metal waveguides.

To preserve its remarkable properties, graphene is usually encapsulated in peculiar dielectric insulators. In this way, carrier mobility is enhanced by minimizing extrinsic sources of scattering, coming from both interface with the substrate and air. The most suitable graphene encapsulant is hexagonal-Boron Nitride (h-BN), an insulator isomorph of graphite with atomically flat layers nearly free of charge trapping [33].

In this chapter a brief review of graphene properties is given before examin-ing graphene plasmonics. More details on h-BN and on h-BN/graphene/h-BN heterostructures are also provided.

2.1 Graphene and hexagonal Boron Nitride

struc-tures

2.1.1 Graphene properties

Crystal and electronic structure

Graphene is a two-dimensional allotropic form of carbon with a honeycomb-like lattice structure. It was obtained, for the first time, in 2004 by A. Geim and K. Novoselov [34, 35] (who were awarded the 2010 Nobel Prize in

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2.1 Graphene and hexagonal Boron Nitride structures

Physics) by mechanical exfoliation of bulk graphite. Since its discovery, graphene has received considerable scientific interest thanks to its excel-lent electrical, optical and mechanical properties, like for instance electron mobility (it can reach ∼200000 cm2V−1s−1 at room temperature in sus-pended samples [12]), thermal conductivity (∼5000 Wm−1K−1) and flat optical transmittance (∼97.7%)) [36].

The crystalline structure of graphene is reported in Fig.2.1(a). Its unit cell is rhombic and contains two non equivalent carbon atoms (A and B) with unit cell vectors given by

~a1,2 =

a 2(3,±

√

3) (2.1)

where a=2.46 ˚A is the carbon-carbon bond length. The reciprocal lattice of graphene is also hexagonal (see Fig. 2.1 (b)) and the reciprocal lattice vectors are

~_b_1,2 ₌ 2π

a (1,±

√

3) (2.2)

Each atom of the sub-lattice is linked to three nearest-neighbour atoms of the other sub-lattice by the three vectors:

~_δ₁ ₌ a 2(1, √ 3), ~δ2= a 2(1,− √ 3), ~δ3= a(−1, 0) (2.3)

The high symmetry points in the first Brillouin zone (BZ) (the hexagonal

Figure 2.1: (a)Honeycomb lattice of graphene made up two interpenetrating triangular lattices with lattice unit vector a1and a2.~δ1,~δ2and~δ3are the nearest-neighbor vectors. (b)

The corresponding Brillouin zone showing the positions of the Dirac points~K and~K0[37].

one in Fig.2.1(b)) are the~Γ point at the zone center (origin of the reciprocal space with~k = 0), the M point in the middle of the hexagonal sides, and~ the two non equivalent ~K and~K0 points at the hexagon corners, where

~ K = 2π 3a, 2π 3√3a , ~K0 = 2π 3a,− 2π 3√3a (2.4) 21

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The~K and ~K points are known as the Dirac points.

The electronic configuration of an isolated carbon atom is 1s22s22p2, thus it has four valence electrons to form chemical bonds. In graphene, three of the four valence electrons are hybridized into three sp2 orbitals and form in-plane σ-bonds, while the remaining electron of each carbon atom forms π-orbitals perpendicular to the graphene plane. The σ-bond is the strongest covalent chemical bond, which is the reason for graphene out-standing mechanical properties, while the π-orbitals determine the low-energy electronic structure of graphene and are responsible for its high electrical conductivity [38].

Using the tight-binding method that consists in expanding the crystal states in linear combinations of atomic orbitals of the composing atoms, it is pos-sible to obtain a reasonable description for the electronic band structure of graphene. The model concerned takes into account only the interac-tion between the nearest neighbour carbon atoms (A-B) and it assumes the orthonormality of local orbitals. The energy dispersion relation obtained from tight binding model calculations gives

E±(~k) = ±t q 3+f(~k) (2.5) where f(~k) = 2 cos(√3ky) +4 cos √ 3 2 kya ! cos 3 2kxa (2.6)

where~k is the quasi-momentum in the first Brillouin zone of the reciprocal space, the signs +, - indicate respectively the upper and lower π band and t∼2.8 eV [37] is the nearest neighbours hopping energy. Hence, considering the interaction only between the nearest neighbours, the two bands are symmetric around the zero energy. Close to the Dirac points, ~K and ~K0, Eq. 2.5can be written as

E±(~k) = ±¯hvF|δ~k| (2.7)

where~k = ~K+δ~k, with |δ~k| |~K|; and vF = 3/2t ≈ 106 m/s is the

Fermi velocity of electrons. Thus, close to the Dirac points, the energy of electrons does not depend on their mass and they can be described by the Dirac equation, instead of the Sch ¨odinger one. For these reasons they are called massless Dirac fermions.

Fig. 2.2 shows the energy dispersion of graphene, where the conduction and valence bands touch at the six non equivalent ~K and ~K0 points of the first BZ. Therefore, graphene is defined as a gapless semiconductor or a semimetal.

The Dirac fermions exhibit a two-component wavefunction that describes

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the sublattices A and B, leading to a different chirality in the graphene con-duction and valence bands at the ~K and ~K0. Therefore, the band structure of graphene at the different Dirac points is not equivalent.

For undoped graphene, the Fermi energy lies exactly at the Dirac points, meaning that all the states below are completely filled by electrons, so, even zero-energy excitations can promote electrons to the totally empty conduction band. The electrons behaviour near the Dirac points leads to extraordinary properties, e.g. ultra-high mobility of carriers, long mean free path and gate-tunable carrier density [37].

The linear energy dispersion is of particular interest also because the

den-Figure 2.2: Electronic dispersion of graphene. The valence band (lower red band) and conduction band (upper blue band) touch at the ~K-points of the BZ. In the vicinity of these points, the energy dispersion relation is linear [39].

sity of states (DOS) near the Dirac points is different from the one of ordi-nary two-dimensional materials with a parabolic energy dispersion. While the DOS for two-dimensional materials is given by the Heaviside step-like function, the DOS of graphene is linear and symmetric in energy (E) and it is described by [37]

ρ(E) = 2|E| π¯h2v2_F

(2.8)

from which the relation between the carrier density and Fermi Energy can be derived [40]: n= E 2 F πv2_F¯h2 (2.9)

Carrier density can be tuned from 1011cm−2 to 1013cm−2 [41] by applying an appropriate electric voltage with external gate or by chemical doping.

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Optical properties

The optical response of graphene results from both interband and intra-band processes. These can be evaluate through the conductivity σ with the optical absorption, linked to the real part of σ.

Taking into account both interband and intraband contributions, the ex-pression of σ is given by σ(ω) =σintra(ω) +σinter(ω) = = e 2 ω iπ¯h2 Z ∞ −∞dE |E| (ω+iγ)2 d f(E) dE − Z ∞ −∞dE f(−E) − f(E) (ω+iδ)2−4E2 (2.10) where f(E) = e (E−µ) kBT ₊₁ −1

is the Fermi-Dirac distribution, ω is the fre-quency, e is the electron charge, γ and δ are the damping rates taking into account of the carrier relaxation for interband and intraband transitions, respectively.

The first term of Eq. 2.10 corresponds to the intraband electron-photon scattering and its integration (with µ= EF) gives

σintra(ω) = 2ie 2_k BT π¯h2(ω+iγ) ln 2 cos µ 2kBT (2.11)

In the limit µkBT (µ ≈EF), Eq.2.11 can be rewritten as

σintra(ω) = ie

2_|_E F|

π¯h2(ω+iγ)

(2.12)

Defining the so called Drude weight as

D= e 2_E F 2¯h2 (2.13) Eq. 2.12becomes σintra(ω) = 2iD π(ω+iγ) (2.14)

Comparing this result with the one obtained in the case of massive parti-cles, it is possible to define a mass, m∗, for graphene:

m∗ = ¯h √

πn

vF

(2.15)

The mobility can be derived from the Drude model:

µD =

eτ

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where τ is the relaxation time of electrons, related to γ through 2γ =τ−1

and m is the electron mass. Substituting Eq. 2.15 and Eq. 2.9 in Eq. 2.16, we obtain:

µg = eτv 2 F

EF (2.17)

Thus, the carrier mobility in graphene sheets depends on EF and τ.

The second term of Eq.2.10 gives the intraband contribution. The integra-tion of this term gives

σinter(ω) = e

2

4¯hθ(¯hω−2|µ|) (2.18) where θ(ω) is the Heaviside function. From Eq. 2.18, it is clear that the

interband transition originates from the transition of an electron from an occupied state in the valence band to an unoccupied state in the conduction band only when the energy of the incident photon is two-times higher than the chemical potential of graphene (¯hω > 2µ). If ¯hω < 2µ, the interband transition does not occur due to Pauli blocking because the final transition states are filled and, in this case, the optical conductivity is dominated by intraband processes (free charge carrier absorption) (see Fig.2.3(a)).

Figure 2.3: (a)llustration of the various optical transition processes in graphene. If ω is smaller than the thermal energy, transitions occur via intraband processes. (b) Typical absorption spectrum of doped graphene, which shows a universal absorbance of 2.3% in the visible range; graphene becomes transparent in the mid-infrared region and shows a strong Drude absorption peak in the far-infrared and THz frequency regions [41].

When ¯hω > 2µ, the optical conductivity due to interband transitions is found to be independent on the frequency of the light ω. For this reason it is called ”universal conductivity” and it can be described by

σ0 =

e2

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The conductivity is related to the transmission T through [42] T = 1+ σ0 c2_ε 0 −2 =1+πα 2 −2 ≈1−πα≈97.7% (2.20) where α = _4πεe2 0¯h ≈ 1

137 is the fine structure constant.

Thus, the independence of conductivity from the frequency in the visible leads to a constant absorption (≈2.3% of the incident radiation). A typical absorption spectrum of doped graphene is shown in Fig.2.3 (b). Typically, the optical absorption spectrum at THz frequencies, that is the region we are interested in, is dominated by the intraband transitions. By contrast, from mid-infrared to optical frequencies the main contribution is given by interband transitions.

2.1.2 hexagonal-Boron Nitride structure and properties

Boron nitride (BN) is a III-V compound that can exist in hexagonal, cubic, or wurtzite crystalline structures, as well as in amorphous form [43]. Of particular interest for this thesis is the hexagonal boron nitride (h-BN), which is an insulator with a structure analogous to the graphene. Indeed, Boron (B) and Nitrogen (N) atoms of 2D h-BN are alternately arranged to form a honeycomb structure (see Fig.2.4 (a)).

Figure 2.4: (a)2D structure of h-BN and (a) AB stack of h-BN layers [44].

Each cell contains B atoms and N atoms and the B-N bond length is 1.45 ˚

A , which forms through sp2 hybridization. Adjacent layers of h-BN are combined with weak van der Waals forces and in each layer B atoms and N atoms are joined by covalent bonds due to a different electronegativity between B and N atoms (see Fig.2.4 (b)).

Despite the similarity with the graphene crystal structure, hBN does not possess the same properties mainly because of the slight ionicity of the B-N bond. Indeed, while graphene is a zero bandgap semiconductor, h-BB-N is

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2.2 Graphene and Graphene/hexagonal-Boron Nitride Surface Plasmon Polaritons

a wide bandgap insulator (the experimental bandgap value is 5.97 eV [45]) (see Fig.2.5). Besides its dielectric nature, h-BN has a really smooth surface

Figure 2.5:Electronic bands of h-BN calculated by LDA method [46].

(owing to the strong, in-plane, ionic bonding of the planar hexagonal lattice structure), free of trapped charges and dangling bonds that can limit, for example, the mobility of graphene (as it happens with SiO2substrates). For

these reasons it is regarded as an excellent substrate material for graphene. h-BN is also an interesting optical material as it is a natural hyperbolic material, meaning that its permittivity can be written as [48]:

ε=   εx 0 0 0 εy 0 0 0 εz   (2.21)

where εx = εy 6= εz with εz having opposite sign with respect to εx,y. This

property results in h-BN supporting tunable propagating phonon polari-tons in the bulk [49]. Combining h-BN with graphene leads to uncon-ventional plasmon-phonon hybridization, that, in turn, gives rise to new surface-phonon-plasmon-polariton (SPPP) modes.

2.2 Graphene and Graphene/hexagonal-Boron

Ni-tride Surface Plasmon Polaritons

In the last few years graphene plasmonics has attracted great interest. The main reason is that, owing to the 2D nature of collective excitations, SPs in graphene are confined much more strongly than in conventional noble metals. Moreover, controllable graphene carrier density by a gate voltage or chemical doping leads also to tunable SPs.

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by a single graphene sheet embedded between two semi-infinite dielec-tric media with dielecdielec-tric constant εr and ε0r. From Maxwell’s equations

and Green’s functions, the dispersion for TM modes (like SPs) can be de-rived [41]: εr q k2_TM− (εrω2/c2) + ε 0 r q k2_TM− (ε0_rω2/c2) +iσ(ω) ωε0 (2.22)

where kTM is the wavelength of the TM modes, σ(ω) is the conductivity

shown in Eq.2.10 and ω is the frequency of the radiation (ω =k0c).

For suspended graphene, εr =ε0r =1, thus

kTM =k0 s 1− 2 σ(ω)η0 2 (2.23) where η0 = p

µ0/ε0 ≈377Ω is the intrinsic impedance of free space.

Con-sidering highly doped graphene (i.e. EF kBT) on a substrate (εr 6=1, ε0r =

1), in the non-retarded regime, i. e. when kSP k0, the dispersion relation

Eq. 2.22becomes [50]

kSP ≈ iε0

(εr+1)ω

σ(ω) (2.24)

and, therefore, the wavelength of SPs is expressed by:

λSP ≈ λ0α 4EF εr+1

1

¯h(ω+iγ) (2.25)

where α = e2/(4πε0¯hc) ≈ 137 is the fine structure constant and λ0 = (2πc)/ω. Since, under the above conditions the intraband contribution is dominant, σ(ω) in Eq. 2.24 can be replaced by σintra(ω) (see Eq. 2.12)

leading to: kSP ≈ π¯h 2 e2_E Fε0 (εr+1)ω(ω+iγ) (2.26)

If the scattering rate is small enough to be negligible with respect to the frequency ω, the term γ in Eq. 2.12 can be ignored, leading to the most simplified form of the conductivity:

σintra(ω) = ie

2_|_E F|

π¯h2ω

(2.27)

By replacing Eq.2.27 in Eq.2.24, ωSP can be found:

ωSP = s e2_E FkSP ε0(εr+1)¯h2π (2.28)

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2.2 Graphene and Graphene/hexagonal-Boron Nitride Surface Plasmon Polaritons

Figure 2.6:Graphene SPPs dispersion.

Given that ωSP ∝ √

kSPEF and EF ∝ n1/2, we can conclude that the

plas-monic resonant frequency ωSP ∝ n1/4.

The degree of in-plane confinement provided by graphene can be evaluated through the SPs effective index, defined as

n_SPg = λ0 λSP ≈ 4α (εr+1) EF ¯hω (2.29)

where λ0 is the wavelength of the incident radiation and g stands for

graphene. We can compare the refractive index of SPs in graphene with the one in metals, which is given by

nm_SP = √

εdεm

(εd+εm) (2.30)

where εm is the real part of the metal dielectric function and εd is the

di-electric function of the didi-electric medium. Since −εm εd, the refractive

index of SPs in metal cannot be much larger than √εd (for dielectric

mate-rials εd is not more than 10). By contrast, ng_SP can achieve values between

40 and 70 [51]) due to the smaller λSP and the possibility to increase the

Fermi energy by gating or doping. This means that graphene SPs allow to achieve a much stronger confinement with a penetration depth:

δ=λSP/2π. (2.31)

while the in-plane propagation distance is given by

δSP =

1

2Im(kSP) (2.32)

The propagation distance in graphene can reach values above 100 λSP [52].

To improve even more the SPs properties in graphene, h-BN/graphene/h-BN heterostructures can be used. Beyond improving the propagation dis-tance δSP by reducing scattering, it has been proven that encapsulating

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graphene in h-BN leads to even stronger field confinement through cou-pling with phonon bands [48]. The penetration depth was found to be

∼20 nm in a frequency range from about 27 THz to 32 THz, that is com-parable to the penetration depth in metals in the visible spectrum [51]. Furthermore, the surface polaritons wavelength was demonstrated to be 70 nm, about 150 times smaller than the free space wavelength, leading to high confinement of the propagating optical field of∼107compared to the modal volume in free space.

The strong field confinement provided by SPs in h-BN/graphene/h-BN heterostructures together with their low ohmic losses offers an ideal sub-stitute of gold as the top metal layer in the double metal waveguide for THz QCLs. In next chapter the improvement of the performances of this kind of waveguide will be quantified via finite elements simulations .

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Chapter 3 Simulations of THz microcavity

lasers in graphene-based

resonators

In this chapter WGM resonators exploiting an innovative waveguide based on graphene SPPs is presented. Basically the novel structure merges the re-markable properties of a microcavity based on the double metal waveguide with the ones of graphene SPPs. In particular, the top gold layer of the dou-ble metal waveguide is replaced by h-BN/graphene/h-BN heterostructure. Thanks to the graphene THz optical properties, this kind of waveguide would allow a stronger electromagnetic (e.m.) field confinement and a dra-matic miniaturization of THz QCLs with respect to what possible with the MM waveguide.

In order to investigate the e.m. field distribution in such microcavity, we calculated its eigenmodes and relative eigenfrequencies. We were partic-ularly interested in quantifying the light confinement in the planar and transverse directions, in particular in calculating the parameters that de-scribed the performances of the system (i.e. the quality factor, the con-finement factor, the mode volume and, thus, the Purcell Factor). This was achieved through finite element method simulations that allow to recon-struct the e.m. field in the whole three dimensional space both inside and outside the cavity. In this way it was possible to retrieve the near-field of the resonant modes and their out-coupling to the far-field.

First, a mode analysis of the microcavity exploiting a double metal waveg-uide was performed; then, these results were compared with the ones ob-tained studying the resonator with the h-BN/graphene/h-BN heterostructure-based waveguide. Finally, the dependence on h-BN/graphene/h-BN het-erostructure properties was explored.

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com-mercial software COMSOL Multiphysics .

3.1 Basic principles of the finite element method

(FEM)

The finite element method (FEM), is a numerical method allowing to ob-tain approximate solutions to boundary value problems, such as problems of electromagnetic analysis, where a set of Maxwell’s equation have to be solved under given boundary conditions. To solve the problems, this method subdivides a large system into smaller, simpler parts that are called finite elements.

The discretization in a finite number of elements allows to pass from a con-tinuum of degrees of freedom to a discrete number. The result is that the geometric structure is decomposed in a mesh of finite elements connected one to each other by nodal points. The discretization of the structure has a crucial role in any finite element analysis because the manner in which the domain is discretized will affect the computer memory requirements, the computation time and the accuracy of the numerical results.

The simple equations that model the finite elements are then assembled into a larger system of equations that model the entire problem. The ele-ment equations are simple equations that locally approximate the original complex equations to be studied, where the original equations are often partial differential equations (PDE).

A typical boundary-value problem is governed by a differential equation in a domain Ω that has the the form

Lφ= f (3.1)

where L is a differential operator, f is the excitation or forcing function, and φ is the unknown test function which has to be found using boundary conditions that enclose the domain.

The idea adopted in the finite element method is to approximate the value of the function φ in each element by a linear combination of the values of

φat each node of the element:

φ≈φe=

N

∑

i=1

ue_iφ_ie (3.2)

where N is the number of nodes of the element, ue_i are the so called inter-polation functions of the element e and φe_i is the value of the variable at the node i in the element.

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3.2 Simulations of an h-BN/graphene/h-BN waveguide

The approximation introduces a discrepancy between the exact solution and the numerical one. This discrepancy must be minimized in order to make the numerical method converge. Thus, the solution can be found by solving the system of equations obtained by setting to zero the weighted residual:

Ri =

Z

ΩN(Lφ− f)dΩ=0 (3.3)

The results is a set of algebraic equations at nodes that must be solved numerically with the corresponding boundary conditions so as to find a good approximation of φ. If the problem is well posed, i.e. the system has a unique solution that continuously depends on the problem data, the discrepancy between the exact solution and the numerical one decreases with the typical element size l, so that the smaller the elements are cho-sen , the more accurate the simulation will be. However, choosing a mesh with smaller elements will increment the number of nodes and, as a con-sequence, the number of degrees of freedom. This leads to higher compu-tation time and makes the convergence of the system more difficult.

3.2 Simulations of an h-BN/graphene/h-BN

waveg-uide

3.2.1 Simulations structure

In order to perform simulations we had to define the characteristic pa-rameters of the system: first of all, the geometric papa-rameters (microdisk radius and the thickness of both active region and h-BN/graphene/h-BN heterostructure), the materials of the various components and relative elec-tronic/optical properties, the boundary conditions and the mesh.

The studied systems were all characterized by an air shell in which the microcavity is embedded. In this way, also the mode out-coupling was investigated. The space dimensions were kept infinite by positioning per-fectly matched layers (PMLs) around the air shell, that acted as a perfect absorber.

The physics of the systems is governed by Maxwell’s equation:

∇ × 1 µr ∇ ×E − ω 2 c2 εrE=0 (3.4)

where εr and µr are the complex dielectric permittivity and magnetic

per-meability tensor assigned to each medium, respectively.

For the simulations we considered a particular active region that we in-tended to use for the subsequent device realization. It consists in a GaAs/

Miniaturization of Terahertz Quantum Cascade Lasers exploiting a graphene-based heterostructure

DIPARTIMENTO DI FISICA ‘E. FERMI”

Corso di Laurea Magistrale in Fisica