Towards intersubband polariton lasing

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Towards Intersubband Polariton

Lasing

Francesco Pisani

Dipartimento di Fisica ”Enrico Fermi”

Universit`

a di Pisa

A thesis presented for the degree of

Doctor of Philosophy

Supervised by: Alessandro Tredicucci

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Chapter 1 Introduction

Since the first laser was built in 1960 [1], the field of Photonics developed countless devices with applications ranging from fundamental research to the PC display the reader is probably watching this thesis in. A variety of light sources, from semicon-ductor lasers to LEDs and quantum cascade lasers [2, 3] were developed, covering different frequency ranges and/or power outputs. Alongside, fabrication techniques continue to improve. The photonic response of a material can be tailored, with nanometer scale features, to achieve properties not available in nature or to improve a specific characteristic of a material: photonic crystal waveguides [4], perfect lenses [5], cloaking devices [6] are just few examples of the modern technology possibilities. Usually light and matter are treated as separate entities but it is known that the strength of interaction between light and a material emitter (like an electron in a quantum well) can be enhanced or suppressed by engineering its electromagnetic environment [7]. It is here that this thesis comes into play. We will be studying a particular regime of light-matter coupling which gives rise to intersubband po-laritons. We will be working with a variety of instruments to design, fabricate and characterize a novel type of light source: the intersubband polariton laser. ISB polariton lasers can efficiently fill in the so-called THz gap in modern laser sources and work even at room-temperature; the emission can be tuned by changing few fabrication parameters making them very versatile. Apart from its practical appli-cations, the ISB polariton laser can also become a powerful instrument to study the ultra-strong light-matter coupling regime [8]. In this regime the usual rotating wave approximation in the Hopfield model breaks down [8, 9], allowing the observation of interesting phenomena like the dynamic Casimir effect [10], superradiant phase transitions [11] or strong diamagnetic effects [12].

This thesis work will be a theoretical and experimental investigation of the prop-erties of intersubband polaritons, quasi-particles arising from the strong interaction between light and a material excitation constituted by the intersubband transition in a quantum well. Light will be coupled to the device through the generation of a surface plasmon polariton: another quasi-particle generated by the interaction be-tween light and free surface electrons in a metal. After a theoretical introduction of some general aspects of the ISB polariton generation, we will describe an experimen-tal campaign with the aim of verifying the intersubband polariton emission (chapter

2). The emission of light from a polaritonic state lacks of a clear experimental veri-fication; therefore this experiment would be the first observation of polariton lasing emission. We will see that, despite very promising results, it is still not evident if

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lasing was achieved or if we measured spontaneous emission. The sample we used to perform those measurements represents the most promising candidate to study the polariton emission as it is easily reproducible from a fabrication point of view. It is also quite robust against mechanical and electromagnetic stress. Nonetheless in chapter3 we will describe some improvements that can be envisaged in order to increase the efficiency of the polariton generation. In particular we will describe how to increase the quality factor of the resonance by means of simulations. The possi-bility to use graphene to design a better optical resonance will be largely discussed. We will also present the results of a preliminary experimental campaign aimed at the fabrication of a device in which the strength of the light-matter interaction can be tuned with a gate potential. Such a device would be the perfect benchmark to investigate different coupling regimes.

In the two last chapters of this thesis we will report two side projects developed during this PhD. The first project (section 4) regards the generation of ultrashort surface plasmons with a rotating wavefront laser pulse. Surface plasmons lasting only few cycles are powerful instruments that can find very interesting applications, such as ultra-fast surface-enhanced Raman spectroscopy [13] or to study phenomena with fast (femptoseconds) time scales, e.g. the effects of the vibration of a lattice on the electric and thermal transport [14]. In the second side project we will il-lustrate a concept to manipulate light with 3D printed free-form optics [15]. The technique allows to redistribute the light intensity to achieve a desired luminosity distribution profile, allowing a plethora of applications, from microscopy [16] to se-curity controls. The final objective of this project would be the design of optics able to correct aberrations from sub-wavelength sources (in particular Terahertz sources).

We will now start from the theoretical description of intersubband polaritons. We will first need to introduce the two main ingredients needed to generate the polaritons: the intersubband transition in a quantum well and the electromagnetic environment in which it will be placed, a double metal cavity tailored to couple light in the active region through the generation of surface plasmons.

1.1 Intersubband Transitions

An intersubband transition (IST) is an electronic transition between two different subbands in a Quantum Well [17]. A Quantum Well (QW) is constituted by a material with a low band-gap embedded between a material with a large band-gap (e.g. gallium arsenide and aluminum gallium arsenide, GaAs-AlGaAs). Such a structure can be fabricated by epitaxial growth.

The electrons (and holes) are trapped by the potential drop, in the direction perpendicular to the interface, of the QW (fig.1.1). This potential creates quantized states with an energy related to the width and depth of the well. In the other two directions the electrons (and holes) move “freely” in the crystal, thus creating subbands in the valence and conduction bands as we will explain in detail in this section. An electron can recombine with a hole in the valence band with an interband transition or can make a transition in the same band but to another subband with an IST. The latter are the most interesting ones for our purposes. Indeed while the interband transition energy is set by the chosen material, the subbands are separated by an energy related mostly to the thickness of the well. The transition energy can

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Figure 1.1: Sketch of an heterostructure showing the interband and intersubband electronic transitions. The simulation on the right shows the transmittance of an intersubband transition in an AlGaAs/GaAs QW happening at 32 THz, for different angles of incidence. Due to the selection rules the absorption increases for higher angles of incidence, i.e. for higher values of the electric field component parallel to the growth direction.

thus be easily tuned by increasing or decreasing the thickness during the growth. The first experimental evidence of these quantized states was achieved in 1974 [18] where the interband absorption of an AlGaAs-GaAs QW was measured. The first measurements of an intersubband absorption was instead in 1985 [19], again with an AlGaAs-GaAs QW in the mid-IR frequency range. From those pioneering works a new research field arose and with it a whole plethora of applications, mainly concerning detectors and lasers [20, 21]. The key feature that led to the wide diffusion of IST concepts in the physics community is indeed the wide tunability of the transition energy that can span, depending on the material, from tens of meV to around one eV, ranging thus from the THz to the visible range. In this section we will briefly report the main theoretical results we are going to need in this thesis, starting from the quantum derivation of the subband wavefunctions and energies. We will then give an effective medium approach for the absorption that will be extremely useful for the design of the more complex devices we will employ and in the end we will talk about some important effects that must be taken into account when designing our devices.

1.1.1 Selection rule

The wavefunction of an electron in a QW can be written as a product of two func-tions: the Block function ub(r), with the lattice periodicity, and the envelope

func-tion f (r), where b represent the band (valence or conducfunc-tion). If the growth direcfunc-tion is z then the QW potential will be V = V (z) and the envelope function is again the product of the in-plane (x-y, perpendicular to the growth direction) free particle and the out of plane (z) wavefunctions:

Ψ(r) = ub(r)f (r) = ub(r) 1 √ Ae ik⊥r⊥_ϕ n(z) = ub(r)fn,k⊥(r), (1.1)

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where n is the quantum number of the subband and k⊥ is the in-plane momentum.

Solving the Schr¨oedinger equation leads to the energy of each subband:

En,k⊥ = En+

¯ h2k_⊥2

2m∗ , (1.2)

where En depends on the confining potential V (z) and m∗ is the electron effective

mass, a parameter related to the chosen material1. The electrons can interact with an external electromagnetic field so that the transition rate between an initial state i and a final state f is:

Wf i = |hψf| HEM|ψii|2 ∝ D uf_bff e · p ui_bfi 2 , (1.3)

where e is the polarization of the impinging wave and p the dipole operator. If we consider only the transition between two consecutive subbands in the conduction band, the final envelope function has a different symmetry in the z axis with respect to the initial function. Thus, the space integral is different from zero only if the polarization has a z component: pz = ez. In other words the selection rule for

the transition between two consecutive subbands in a QW is that the electric field must have a component in the growth direction. The simulation in fig.1.1shows the transmittance of a QW as a function of the incidence angle. One can clearly see how the absorption of the QW increases for higher incidence angles, that is indeed due to the increase of the electric field component in the growth direction. This result states that an electromagnetic wave impinging at normal incidence on a device with a QW will not excite any IST; therefore one needs to find other solutions to couple the external field to the QW as we will see in section1.2.

It should be noted that the envelope function Hamiltonian is a 4×4 matrix which includes the contribution of the electrons in the conduction band as well as the holes in the heavy-hole band, light-hole band and split-off band [22]; indeed the envelope function is a four components vector. It has been shown though that, while the heavy-hole contribution is always negligible for the conduction band, the light-hole and split-off band can be replaced with an “effective” valence band, reducing the Hamiltonian to a 2 × 2 matrix [23]. Moreover the problem can be recast into an one-band model by including the effective band contribution in the effective mass:

m(E, z) = me

E − Ev(z)

EP

, (1.4)

where Ev(z) and EP are parameters that can be measured empirically [24]. All

these contributions are usually small for the first subband in the conduction band, especially when the confinement energy ∆Ecis much smaller than the band gap Eg,

which is the case of an AlGaAs-GaAs QW (∆Ec ∼ 0.2 − 0.3 eV and Eg ∼ 1.4 eV).

Therefore the one band approximation given in this text holds.

1.1.2 Absorption and effective medium approximation

The absorption coefficient of a QW is defined by the ratio between the energy absorbed per unit volume and time ¯hωWf i/V and the incident power flux I =

1_{In this case we assume m}∗ _{to be constant. This is generally a good approximation, but more}

complex derivations also include a z dependency. Different subbands may also have different effective masses, leading to a broadening of the transition for high k⊥ as we will see later.

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1 2ε0cnE

2 _[₂₀_{], where n is the refractive index of the material. At zero temperature,}

the absorption coefficient of the transition between the first and second subband takes the form of a lorentzian with a half width at half maximum (HWHM) Γ2_:

α12 = nee2¯h 2m∗_cnε 0 f12 Γ (E2− E1− ¯hω)2+ Γ2 (1.5)

where ne is the two dimensional electron density inside the QW, n is the refractive

index, f12 is the oscillator strength, defined as 2m∗_(E 2−E1) ¯ h2 |hϕ2| z |ϕ1i| 2 and E1,2 are

the energies of the first two levels.

It is useful to notice that we can describe the absorption of the QW with the dielectric function and simulate its interaction with the environment by numerically solving Maxwell’s equations. The core of this thesis, as anticipated, will be the interaction of the IST with a photonic resonance giving birth to the Intersubband Polariton. Despite the fully quantum description we will give in 1.3, the possibility to simulate the process classically is fundamental for the design and optimization of the device. The dielectric function can be written as:

ε(ω) = ε∞ 1 + iσ ε0ω = ε∞ 1 + iσ2D ε0ωLef f (1.6)

Due to the bi-dimensional nature of the QW the conductivity σ is replaced with σ2D/Lef f, where Lef f is the effective length in which the ISTs occur [25]. The

absorption of a material is given by:

α = Re(σ2D) ε0c Re(n)Lef f

= α2D Lef f

(1.7)

where n = √ε is the refractive index of the well. Comparing α2D with eq.(1.5) we

see that we obtain the same result employing a Drude-Lorentz oscillator:

σ2D = nee2 m∗ f12 −iω ω2 12− ω2− 2iγ12ω (1.8)

with ω12 = E2 − E1/¯h and γ12 the HWHM of the lorentzian. From this we can

find the out of plane component of the dielectric function describing the lorentzian absorption of the QW: εz(ω) = ε∞ h 1 − ω 2 P ω2 12− ω2− 2iωγ12 i−1 , (1.9)

were ε∞ is the high frequency dielectric constant of the well. In the plane the

electrons move freely, thus the dielectric function will be that of a Drude metal with a plasma frequency ωP =pnee2/ε0ε∞m∗Lef f, εxy(ω) = ε∞ h 1 − ω 2 P ω(ω + iγ12) i (1.10)

1.1.3 Depolarization shift and multiple quantum wells

When the QW is highly doped, the interaction between electrons becomes important and gives rise to other terms in the potential. The one that influences the most3

2_{A complete derivation of the absorption coefficient can be found in many textbook of quantum}

physics [20] and it is not reported for the sake of concision.

3_{Other terms, namely the Hartree self consistent potential, the exchange-correlation energy}

and the exciton shift [20] are usually an order of magnitude less than the depolarization shift and partially compensate each other, thus we will not talk about them.

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Figure 1.2: Transmittance simulations for a single QW with different dopings (left), or for a different number of QWs at a fixed doping, 6 · 1011 _cm−2 _{(right). The}

reso-nance blueshifts for higher dopings and becomes more intense for a higher number of QWs.

the IST energy is the depolarization shift [20]. The electrons partially screen the oscillating field inducing the transition, leading to a blue-shift of the resonance. It can be shown that the modified transition frequency becomes:

˜

ω2₁₂= ω2₁₂+ f12ωP2 (1.11)

This phenomenon is physically the same as the Debye screening length [26] in a plasma, where the restoring Coulomb force on the electron screens the low frequency external electromagnetic oscillations. To describe a highly doped QW we can simply substitute ω12 in equation (1.9) with ˜ω12 . As a result the transition frequency

blue-shifts as the QW doping increases and the transmittance decreases since there are more electrons to absorb the light (fig.1.2, left).

In this thesis we will also study devices with multiple QWs. Two cases must be distinguished when considering this kind of devices: if the barriers between the wells are very thin, the electron wavefunction extends to the nearest wells. The electrons will thus feel a periodic potential in the growth direction acting like a lattice (called super-lattice). In this case the dispersion E(ω) is characterized by the presence of minibands of allowed states. This will not be the case of our interest as we will use only barriers thick enough, so that the electron wavefunction is localized in each well. Each state is N times degenerate as each of the N QWs has the same levels. It is very useful from a computational point of view to describe all the QWs with a single effective medium. Indeed when the thickness of all the QWs is much less than the wavelength of the impinging light, the behaviour of the multiple QWs is described in a very simple way [27]:

εz(ω) = ¯ε h 1 − ω˜ 2 P ω2 12− ω2− 2iωγ12 i−1 εxy(ω) = ¯ε h 1 − ω˜ 2 P ω(ω + iγ12) i (1.12) ¯

ε is the average between the barrier and well high frequency dielectric constant: ¯

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plasma frequency is calculated as if the electrons were distributed in the barrier and well layers: ωP = pnee2/ε0ε∞m∗(Lb+ Lw). This effective medium approach

is of incredible help in simulations as it allows to use only one “active” layer to simulate all the wells, instead of having to compute the fields for every well and barrier separately4_{. The absorption of the whole system increases as the number of}

QWs increases without blue-shifting the resonance (fig.1.2,right).

4_{The codes commonly used to simulate the dielectric response of a device use a grid in which}

the fields are calculated. A very rough simulation of a barrier/well may use one grid point at the interface and one in the middle of each layer. Thus if the barriers and wells have a thickness of 20 nm the simulation requires 1 point every 10 nm. In comparison, to solve the behaviour of the effective medium is usually enough to take a point every λ/5. In the Mid-IR this means a point every ∼ 500 nm, so 50 times less for each dimension of the simulation.

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1.2 Microcavities

Figure 1.3: Sketch of the three main types of microcavities. a) Fabry-Perot cavity with metallic mirrors. b) Whispering gallery modes in a ring resonator. c) Photonic crystal (metallic grating) with the surface plasmon polariton field in red.

An electromagnetic cavity is a region of space in which the electromagnetic field is confined due to the boundary conditions; a very general definition from which a large variety of cavity designs and applications sparkled, from the first laser [1] which involved stimulated emission in a ruby crystal to the state of the art Quan-tum Cascade Lasers (QCLs). In this thesis we are interested in microcavities, in which the electromagnetic field is confined in a very small region of space, typically even smaller than the wavelength. There are mainly three types of microcavities: Fabry-Perot, whispering gallery and photonic crystals [28, 29]. A Fabry-Perot cav-ity (Fig.1.3,a) employs two mirrors (being metallic or distributed Bragg reflector) to confine the light in the region between. This is the most common configuration for a solid state laser, in which an active material is placed inside the cavity to stimulate the emission. A whispering gallery cavity (Fig.1.3,b) instead is based on the total internal reflection in a circular (or spherical) medium. Usually it takes the form of a disk or a ring in which the light is coupled and trapped due to the inter-nal reflection. Photonic Crystals (PhCs) are the most complex: they are composed by a periodic repetition of different materials (or even one material and vacuum) arranged in way that resembles a crystal. The photons behave like the electrons do in a crystal, occupying energy bands with photonic gaps. These cavities can be fabricated to be one, two or three dimensional. We are interested in a particular type of PhC: a mono-dimensional metallic grating (Fig.1.3,c). This type of structure may allow the excitation of Surface Plasmon Polaritons (SPPs), mixed light-matter quasi-particles that will be the main topic of this thesis. Trough the excitation of SPPs the external electromagnetic field will be coupled to basically all sample de-sign used in this work, allowing to study the strong light-matter coupling regime. Indeed the SPP electromagnetic field may interact with another material excita-tion, the intersubband transiexcita-tion, creating another quasi-particle: the Intersubband Polariton.

Before we find the condition under which a metallic grating can sustain a SPP, it is useful to introduce a more general concept: the quality factor [30]. The Q-factor can be defined in many ways and it represents the ratio between the energy stored in a cavity and the energy lost per cycle5. It can be seen [29] that this definition

5_{A cycle is defined by the time the light needs to complete a loop inside the cavity: e.g. 2L/c}

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leads to the more practical equation:

Q = ω0/∆ω, (1.13)

where ω0 is the cavity resonant frequency and ∆ω is the Full Width Half Maximum

(FWHM) of the resonance. The sharper the resonance is the higher the quality factor of the cavity. Usual Q-factors in micro-rings and PhCs can easily exceed 103

with a record value of ∼ 1010 reached with a silica micro-sphere [31]. The type of gratings we will use usually reach Q-factors around 50. Despite having such a low Q-factor the grating has many advantages with respect to other geometries. The most important benefit is that it can be fabricated directly on top of a sample covering extremely large area6, thus maximizing the interaction volume. Moreover we will see that when coupled together with the intersubband transition in QWs, the resulting polariton Q-factor will be the square root average of the grating and QW Q-factor. The QWs we will use usually have a quality factor around 10-20 [32], thus they will mostly determine the final polaritonic Q-factor.

1.2.1 Surface Plasmon Polaritons

Figure 1.4: The sketch on the left represents the system used: the two mediums are described by the dielectric constant ε1,2. The SPP is propagating at the interface in

the x direction and it is evanescent in the z direction. The graph reports the SPP dispersion at the interface between vacuum and metal together with the light line propagating parallel to the interface (dotted line) or at an angle θ.

SPPs are electromagnetic excitations propagating at an interface between two media with dielectric permittivity of opposite sign, e.g. a metal and vacuum [33]. In the direction perpendicular to the surface they are confined, i.e. the field is evanescent on both sides of the interface; the skin depth plays a fundamental role in the generation of polaritons as we will explain later. SPPs can be excited by a light source impinging on the surface, provided the matching conditions are satisfied,

6_{The grating is fabricated trough standard lithography and metal deposition techniques that}

can be performed directly on the already grown wafer. The grating may cover all the wafer surface creating a very large area in which the PhC field may interact with the sample. This will be fundamental for absorption measurement and for the pump and probe experiment we will describe later.

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meaning that both ωL and kL of the light source must be equal to ωSP and kSP of

the SPP. We will see that this is possible if the dispersion relation of the SPP is folded in the first Brillouin zone by means of a periodic medium: a grating.

In order to obtain the resonance condition we need to find the dispersion relation of the SPP. We start from Maxwell’s equations:

         ∇ · E = 4πρ ∇ × E = −1 c∂tB ∇ · B = 0 ∇ × B = 1 c 4πJ + ∂tE (1.14)

Assuming a harmonic dependence for the electric and magnetic fields we can find the wave equation:

∇2_{E +} ω 2

c2εE = 0 (1.15)

We assumed that in both media B · k = 0 (TM modes7_{) and ∇ · E = 0, i.e. the}

volume charge density ρ is zero.

Since we are interested in finding the SPP dispersion relation, we will find so-lutions of eq.1.15 in a medium with a discontinuity in the dielectric constant: we assume the z > 0 region to be filled by a medium with dielectric constant ε2 and

z < 0 by a medium with dielectric constant ε1 (fig.1.4). In this configuration, the

SPP propagates on the interface between the two media. Without loss of generality we can take x as propagation direction and we can write E(r, t) = E(z)eikxx_e−iωt_,

with kx the component of the wave vector in the x direction. In z the field must

de-cay exponentially since it is confined on the interface. To find the dispersion relation we need to solve the equations in both regions.

z > 0                By(z) = A2eikxx−k2z Ex(z) = −iA2 c 2 ωε2k2e ikxx−k2z Ez(z) = −A2kxc 2 ωε2e ikxx−k2z z < 0                By(z) = A1eikxx+k1z Ex(z) = −iA1 c 2 ωε1k1e ikxx+k1z Ez(z) = −A1kxc 2 ωε1e ikxx+k1z Where k_1,22 = k2_x− ω 2

c2ε1,2 is the component of the wave vector perpendicular to the

interface, representing the inverse of the decay length. Continuity of By, Ex and

εEz at the interface requires that A1 = A2 and k2/k1 = −ε2/ε1. We see from these

conditions that ε2 and ε1 must have opposite signs in order to obtain evanescent

waves in the z direction.

From the equations for kj we finally get the dispersion relation for the SP:

k2_x = k_{SP P}2 = ω 2 c2 ε1ε2 ε1+ ε2 (1.16)

The condition for the dielectric constants becomes ε1ε2/(ε1 + ε2) > 0 in order to

obtain a real wave vector. Since we have ε1ε2 < 0, we must chose ε1 and ε2 such that

7_{The choice of the TM modes will lead to the SPP dispersion relation, while it can be shown}

that TE modes lead to longitudinal waves propagating inside the medium called “plasma waves” [33]. These solutions exist for a frequency higher than the plasma frequency, thus well above the region of interest for this work.

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ε1+ε2 < 0. A possible choice is ε1 = 1 and ε2 = 1− ω2 p ω2_{− iγω}, with 2− ω2 p ω2_{− iγω} < 0,

which represent a metal-vacuum interface. Fig.1.4,b) shows the dispersion relation of a SPP at the interface between those two media.

Now we will discuss how to excite a SPP with an electromagnetic wave imping-ing from the vacuum region. The excitation occurs when the matchimping-ing between the component of the beam’s wave vector parallel to the surface k|| = k sin θ and the

SP’s wave vector is achieved: kSP = k||, where θ is the incidence angle. We can see

(Fig.1.4,b) that the light cone ω = k||c (dotted lines) always lies on the left of the

dispersion relation of the SPP. Since there is no matching between the wave vectors of the SPPs and light, it is not possible to excite SPP with a laser beam at a flat metal-vacuum interface.

1.2.2 Grating coupling

Figure 1.5: a) Sketch of the grating coupler with the impinging electromagnetic wave on the left and the SPP propagating at the interface. b) SPP theoretical dispersion relation with matching points highlighted by red dots at which a light source excites the SPP. The straight lines are the light lines of the source impinging on the grating with an angle θ. c) Simulated reflectance of a gold grating on a GaAs substrate as a function of the frequency and angle of the x component of the wavevector, kx. The

yellow lines represent the dispersion relation.

Various methods have been developed to solve the matching problem and excite SPPs. A widely used one is a grating coupler. A metal surface with a periodic structure, a grating (Fig.1.5,a), is used instead of the flat interface. With the periodic medium, the wave equation has periodic coefficients and its solutions are given by the Floquet-Bloch theorem: the dispersion relation is folded in the first Brillouin zone (Fig.1.5,b) and the matching condition between the light wave vector and the SPP’s wave vector becomes:

ω

c sin θ = kSP ± n 2π

d (1.17)

where d is the grating period. For any given angle θ a set of discrete solutions for the wave vector kSP P of the SPP always exists in this case. We can see in fig.1.5,b) the

possible solutions, i.e. the intersections between the light line and the SPP dispersion relation. Here we assumed that the dispersion relation of the SPP is the same as in

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the flat interface case. This assumption is an approximation: the dispersion relation changes as the grating grooves become deeper, affecting the matching condition [34]. Even band gaps can appear at the center and at the edges of the Brillouin zone as can be seen from the simulation in fig.1.5,c) where we computed the reflectance of a gold grating on top of a gallium arsenide (GaAs) substrate8_{. Usually shallow grooves are}

used9 _{and the perturbation to the dispersion relation remains small. In fact, it can}

be shown that the contribution of the surface roughness in the dispersion relation is only of the second order in the groove depth h [36]. Despite this, the simulations we performed to design each sample naturally take into account these effects, as they simply solve Maxwell’s equations without any assumption.

1.2.3 Metal-Insulator-Metal cavity and Critical Coupling

Figure 1.6: a) Sketch of the Metal-Insulator-Metal cavity. b) Simulated z compo-nent of the electric field in the cavity; the width of the ribbon was ∼ 2 µm and the thickness h = 0.1 µm. Two orders of resonance (N=1,3) are reported. The imping-ing light creates an electron displacement in the metal ribbon generatimping-ing localized surface plasmons.

Surface Plasmons may be generated with many different geometries [37], the one that maximizes the active area is the grating. The SPP field penetrates inside the dielectric structure below the metallic grating with a decay length called “skin depth” [3] δd= λ 2πRe s −(εd+ εm) ε2 d ! (1.18)

where εd,m are the dielectric functions of the dielectric and the metal10. The

distri-bution of the fields inside the sample is thus non-uniform as it exponentially decays with the distance from the metal. This can be detrimental when the light coupled by the SPP interacts with another resonance inside the sample, e.g. a system com-prising multiple QWs: the QWs closer to the grating experience a field intensity higher than the ones farther from it. In this work we are interested in studying the light matter coupling and the non-uniformity of the field may compromise the results: we will see in the next section that the light-matter coupling strength is

8_{The simulations where performed with a rigorous coupled wave analysis based on the scattering}

matrix algorithm PPML[35]. The code calculates the scattering matrix elements, so that the reflectance or the transmittance of the system can be obtained as a function of the incidence angle.

9_{In this thesis the worst case for this approximation will be described in chapter}₂ _{where the}

grooves are 100 nm deep and the pitch is 4 µm; thus the shallow grooves approximation still holds.

10_{The skin depth can be easily calculated substituting eq.(}_1.16_{) in the decay wavevector k} 1,2

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proportional to the electric field intensity, thus it is important to have it as con-stant as possible in all the active region, rather than exponentially decaying from the grating layer. A widely used solution to this problem is the Metal-Insulator-Metal (MIM) cavity. The insulator (or the active region) is sandwiched between two metallic layers (Fig.1.6,a). The top one is patterned in order to allow coupling of the light from an external source, while the other is continuous. In this configuration the SPP is generated at both metal-insulator interfaces and if the thickness of the insulator is less than the decay length, the field distribution in the growth direction is approximately constant (Fig.1.6,b).

Two different regimes can be identified depending on the thickness h: h << λ [38] and h ∼ λ [39]. In the first case the system behaves as a Fabry-Perot cavity with length w, the metal ribbon width. The resonant frequencies are given by:

νN =

N 2nE

c

w (1.19)

Where N is the mode index and nE is the effective modal index: the index of

refrac-tion of the dielectric medium plus a contriburefrac-tion due to the impedance mismatch between the metal-metal region and the single metal region (where the top metal is removed to fabricate the grating) [40]. The oscillation of the electrons inside each metal ribbon generates localized SPP and allows the coupling of light with the de-vice11_{. In figure} _1.6_{,b) we reported a simulation of two order of resonance (N=1,3)}

occurring at ∼ 13 and 36 THz. When the thickness is comparable to the wavelength we instead recover the propagating SPP dispersion of the previous section. In chap-ter 2we will measure the polariton emission of both types of sample focusing on the advantage of each structure12.

An interesting phenomenon arising from these types of structure is the Critical Coupling (CC) [35,41], a condition in which all the impinging light is absorbed in the device. Let us consider a dielectric slab with complex refractive index ¯nd = nd+ikdof

thickness h placed over a perfectly reflective substrate (Fig.1.7,a). If light impinges on the slab from the air side the reflection amplitude follows [42]:

r = r12− e

2iβ

1 − r12e2iβ

(1.20)

where r12 = (1 − ¯nd)/(1 + ¯nd) and β = k0n¯dh are the Fresnel reflection coefficient

for the air-dielectric interface and the phase acquired propagating in the medium. One can design a system to achieve the desired value of reflectivity by varying the thickness h. This principle can be applied to a obtain perfect reflector or vice versa an anti-reflection coating. When the dielectric has low losses (kd << nd) the

condition for zero reflectance is h = λ/4n. In general the CC condition is achieved when the radiative losses equal the internal (non-radiative) losses[35]:

R =γr− γnr γr+ γnr

2

(1.21)

11_{The impinging electric field must have an x component 6= 0 in order to create the electron}

displacement in the metal ribbon.

12_{We will see in the next section that confining the field in a smaller volume leads to higher}

coupling constant. Despite this effect, the dispersive behavior of the thicker configuration allows to have more versatility in choosing the resonant frequencies and/or the angle of incidence/emission.

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Figure 1.7: a) Sketch of a dielectric slab over a reflective substrate. Light under-goes multiple reflections at each interface. The sum of all reflections may give a constructive or destructive interference. b) Simulated reflectance of a MIM cavity. The losses of the dielectric were artificially increased to show the perfect absorption obtained when the radiative and non-radiative losses are equal.

In figure1.7,b) the non-radiative losses where artificially increased in a simulation to better present the phenomenon, but in a real experiment the losses can be controlled with various parameters: the angle of incidence, the grating thickness, pitch and filling fraction influence the radiative losses while the material chosen as dielectric control the non-radiative ones. The doping inside of a QW or the number of QWs in the layer are examples of how to change these losses. Exploiting this principle we can design a MIM cavity with perfect absorption, a fundamental condition to exper-imentally reveal the strong light-matter coupling regime and achieve the polariton lasing. Indeed the increased contrast eases up the spectroscopic characterization of the resonance while the increased absorption allows a more efficient coupling of the light, decreasing the pumping power needed to reach the lasing threshold.

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1.3 Intersubband Polaritons

Figure 1.8: a) Two identical pendulums coupled by a spring. The oscillation ampli-tude shows the two eigenstates of the whole system separated by a factor propor-tional to the strength of the spring. b) Sketch of a heterostructure with a metallic grating on top and multiple QWs below it. The red area depicts the distribution of the electric field that acts as the “spring” which couples the SP and QW resonances. c) The reflectance of the structure shows two peaks spaced by the Rabi frequency. The SP and QW resonances are plotted to show the two isolated resonances in comparison to the coupled system.

Polaritons are quasi-particles generated by the strong interaction between two resonances: a photonic and a material resonance [43–45]. The photonic resonance may exploit the mode of a MIM cavity or the SPP resonance in a grating to confine the light. In this thesis we will focus on the combination of these two: SPP gen-eration in a MIM cavity. The material resonance is instead related to the resonant response of the dielectric function, for example the transitions between the valence and conduction band in a semiconductor material or the intersubband transition in a QW. When the cavity mode resonance occurs at the same frequency of the electronic transition, the system is degenerate. But if the two resonances interact with each other, the degenerate state splits in two different states: the upper and lower polariton13 _(Fig._1.8_{, b, c). A classical picture is given by two identical}

pen-dulums coupled by a spring (Fig.1.8,a): the whole system is then described by two eigenstates with eigenfrequencies separated by a factor related to the strength of the spring. In our case this occurs due to the strong interaction between a doped QW and a SPP [46, 47] and the role of the “spring” is played by the electric field, responsible for both the intersubband transition and the SPP generation.

1.3.1 Quantum description of the Intersubband Polaritons

The strong interaction between light and matter was firstly described by the Hopfield model [48] and later applied to intersubband transitions [8,49]. We will summarize the dissertations in these articles and give a fully quantum description of the inter-subband polaritons. In particular we will show that the complete Hamiltonian of

13_{Note that the reflectance reported in fig.}_1.8_{,c shows a large splitting, fingerprint of the strong}

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the system14 can be diagonalized through a set of Bogoliubov and Hopfield trans-formations and written as a harmonic oscillator whose operators create and destroy polaritons. The full Hamiltonian of the system sketched in figure 1.8,b) can be divided in three terms:

H = He+ Hp+ Hint (1.22)

where He and Hp are the electronic and photonic Hamiltonian that can be written

in a second quantization fashion as15_:

He = X n,k ¯ hωn,kc † n,kcn,k (1.23) Hp = X q ¯ hωcq(a†qaq+ 1/2) (1.24)

Where c†_n,kis the creation operator of an electron in the subband n with a momentum k and ¯hωn,k is its energy (see section 1.1); a† is the creation operator of a photon

in the cavity with energy ¯hωcq and momentum q, ωcq = ωc(q) describes the cavity

dispersion (see section 1.2). The interaction Hamiltonian in the dipole gauge reads as: Hint= Z ₁ ε0ε(z) h − D(r) · P + 1 2P 2_(r)i_d3_r _(1.25)

Where D is the displacement field and P the polarization density; ε(z) is the di-electric function of the system layer by layer, from the metallic top and back to the wells and barriers, and defines the guided mode of the cavity. In a double metal configuration, due to the boundary conditions, the only non-zero component of the displacement field is Dz and is written in second quantization as [49]:

Dz = i X q r εε0¯hωcq 2SLc eiqrk_(a q− a † −q) (1.26)

Where S is the area of the system and Lc the cavity thickness16. The polarization

density can be found from the microscopic current:

dP/dt = 1

i¯h[P, H] = j. (1.27) In our system the only contribution to the current density is due the electrons oscillating at the intersubband transitions frequencies, so that the only component is: Pz = ¯ he 2Sm∗ X n>m,q ξnm(z) ωnm eiqrk_[B nm,−q+ Bnm,q† ] (1.28)

14_{We will use the dipole gauge, thanks to which we will find the depolarization shift due to the}

QW doping in a natural way: writing the microscopic current of the intersubband transition and diagonalizing the plasmon Hamiltonian.

15_{We dropped the hat to indicate the operators and the bold for the vectors to simplify the}

notation.

16_{The length of the system is given by the effective length of the electromagnetic mode: L} q =

R+∞

−∞ f 2

q(z)dz; where f (q) is the mode profile function which describes the lateral profile of the

guided mode [49]. In a double metal configuration the field is approximately constant in the whole system, thus Lq' Lc.

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Figure 1.9: a) Single electronic transition and b) “bright” transition. c) Anisotropic distribution of the displacement field in a cavity. The coupling strength with the QWs is depicted with the number of arrows.

where

ξnm(z) = φn(z)∂zφm(z) − φm(z)∂zφn(z) (1.29)

B_nm,q† =X

k

c†_n,k+qcm,k (1.30)

are the microscopic current density (φ is the envelope wave function of the bound state) and the operator that describes all the possible transitions from the m-th subband to the n-th with a momentum transfer of q given by the cavity. ¯hωnm =

¯

h(ωn− ωm) is the energy difference between the two subbands.

Before writing the full Hamiltonian substituting D and P in the interaction term, it is useful to simplify the discussion by assuming that only one intersubband transition may occur, i.e 1 → 2. That is the case of the devices we will study in this thesis: the only transition resonant with the cavity is the one occurring between the first two subbands. The device will be cooled down to 80 K, so that only the first subband will be populated by electrons and all other possible transitions occur at much higher energies. Under this assumption the Hamiltonian is written as:

H =X q ¯ hω21b†qbq+ X q ¯ hωcq(a†qaq+ 1/2) +X q ¯ hωP 2 r ωcq ω21 f21f21w(a † q− a−q)(b†−q+ bq) +X q ¯ hω2_P 4ω21 (b†_q+ b−q)(b†−q+ bq). (1.31)

The second and third lines are the linear and quadratic terms in the interaction Hamiltonian, while in the first line one can recognize the photonic and electronic Hamiltonians. The oscillator strength and the plasma frequency f21 and ωP were

already defined in sect.1.1, while fw

21 is the overlap factor between the cavity mode

and the intersubband current distribution [11]: fw

21 = Lef f/Lcav. It can be

intu-itively seen as the fraction of electric field shared by the two resonances: taking as reference fig.1.9,c), only a portion of the SPP field that penetrates in the substrate interacts with the QW, the higher is that portion the stronger will be the coupling. The overlap factor may change from a QW to another, as is the case of fig.1.9,c), but in a MIM cavity it can be treated as constant since the displacement field is. The operator b†_q = B_q†/√N1− N2 is the bosonic creation operator of the “bright”

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state, found in the limit of weakly excited states, which distributes the transition symmetrically among all the electrons inside of a subband (Fig.1.9,b); It can be shown that the bright state holds the whole oscillator strength of the system, while all the other Ne− 1 state are “dark”: they do not couple with light since they do

not have a dipole moment [49] (Ne is the number of electrons)17.

The choice of the dipole gauge will become now clear as the combination of the electronic Hamiltonian and the non linear part of the interaction Hamiltonian can be diagonalized with a Bogoliubov transformation. We can define the “intersubband plasmon” operator: pq = ˜ ω21+ ω21 2√ω˜21ω21 bq+ ˜ ω21− ω21 2√ω˜21ω21 b†−q; (1.32) ˜ ω21 = q ω2 P + ω212 . (1.33)

With this definition

He+ HN L = X q h ¯ hω21b†qbq+ ¯ hω2 P 4ω21 (b†_q+ b−q)(b † −q+ bq) i =X q ¯ h˜ω21p†qpq. (1.34)

The quadratic term P2 of the Hamiltonian in the dipole gauge describes the inter-action between the electrons, thus by calculating it for the intersubband transition we recovered the depolarization shift in doped quantum wells. One can recognize in eq.1.33the blue-shifted intersubband transition frequency mentioned in section 1.1. The complete Hamiltonian is written with the intersubband plasmon operators as: H =X q h ¯ h˜ω21p†qpq+ ¯hωcq(a†qa−q+ 1/2) + i¯hΩq(a†q− a−q)(p†−q+ pq) i , (1.35)

where we finally introduced the light-matter coupling constant:

Ωq = ωP 2 r ωcq ˜ ω21 f21f21w. (1.36)

The Hamiltonian can be exactly diagonalized with a Bogoliubov-Hopfield transfor-mation. Defining the intersubband polariton operator:

Πq = xqaq+ yqa †

−q+ zqpq+ tqp †

−q, (1.37)

the Hamiltonian of the whole system can be cast in the form:

H = EG+ X q h ¯ hωLP,qΠ † LP,qΠLP,q+ ¯hωU P,qΠ † U P,qΠU P,q i , (1.38)

where EG is the new ground state energy [8] and ωU P,q, ωLP,q are the upper and

lower polariton frequencies that can be found from the secular equation arising from the diagonalization process:

(ω2_{{U P,LP },q}− ˜ω₂₁2 )(ω_{{U P,LP }q}2 − ω2 cq) = f21f21wω 2 Pω 2 cq. (1.39)

17_{They do not couple with light, but the existence of these dark states does compromise the}

efficient electronic injection into polariton states [50]. This is why it is preferable to excite polaritons optically.

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In figure 1.10 we reported the reflectance of a MIM cavity containing multiple QWs. The light is coupled through a grating coupler, which excites the SPP. When the cavity resonant frequency matches the QWs intersubband frequency, the disper-sion splits in the upper and lower intersubband polaritons. The two asymptotes are at ˜ω21 and ¯ω21 = pω212 + ω2P(1 − f21f212 ). The minimum splitting between the two

branches, called Rabi splitting, is given by:

2ΩR = ωP p f21f21w = s f21e2ne εε0m∗Lcav , (1.40)

where ne is the electron density inside the QWs.

Figure 1.10: Reflectance of a device showing features of the strong coupling: the plasmonic resonance in a MIM cavity, coupled with the intersubband transition in multiple QWs, splits in two branches, the upper (ωU P) and lower (ωLP) polariton.

1.3.2 Intersubband Polariton Lasing

The bosonic nature of the intersubband polaritons makes this type of device partic-ularly interesting for building a laser based on Bose-Einstein condensation [51]. The higher the number of polaritons in the condensate the stronger is the stimulated scattering from another state to the condensate itself. This type of lasers would not require a population inversion, instead they rely on the possibility to generate the condensate with an optical pump or electrical injection. The polaritons then relax by emitting a photon that bears the properties of the emitting polariton [52], thus giving rise to coherent light. To achieve the lasing effect the polariton disper-sion must have a minimum where the condensate generates and a way to scatter polaritons from another state to the condensate itself is required. In figure 1.11

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Figure 1.11: Sketch of the polariton condensate generation. A pump excites po-laritons in the upper polariton branch. The portion of popo-laritons that decay with the phonon-polariton scattering builds the polariton condensate in the minimum (k = 0) of the lower polariton branch. The process occurs when the energy differ-ence between the two states equals that of the longitudinal optical (LO) phonon.

from the upper branch scatters into it. There are two mechanisms for which a po-lariton in the upper branch may relax to the lower branch: popo-lariton-popo-lariton and phonon-polariton scattering [53,54]. We will be focusing on the latter as it will be the process involved in the experimental campaign described in chapter 2.

In order to achieve stimulated scattering the number of polaritons scattering toward the condensate must compensate the number of polaritons leaving the con-densate. We can write a rate equation to evaluate the pump threshold necessary to achieve the lasing emission:

dn1 dt = n2(n1+ 1)Γ n2,n1 LO − n1ΓLP, dn2 dt = κIP umpS ¯ hωU P − n2ΓU P, (1.41)

where n1,2 is the polariton population in the lower and upper branch, Γn_LO1,n2 is the

phonon-polariton scattering rate, IP ump is the pump intensity, κ the absorption

co-efficient of the device, usually around 0.4 for the upper polariton (see the reflectance value in the upper polariton branch in Fig.1.10) and S is the area of the device. We dropped the wavevector index q for clarity. ΓLP (U P ) is the loss rate of the lower

(up-per) branch containing both the radiative, Γr, and non-radiative, Γnr, losses. It is

related to the cavity and intersubband transition losses by the Hopfield coefficients18:

ΓLP = |xLP|2Γcav + |zLP|2Γ12 (1.42)

Both Hopfield coefficients have a value around 0.5 close to resonance [8] while the cavity and intersubband transition losses can be directly evaluated from the quality factors of the uncoupled resonance (see for example figure1.8,c). The only missing piece to evaluate the pump threshold is the phonon-polariton scattering rate. The complete derivation involves the evaluation of the matrix element between the upper

18_{We are neglecting the anti-resonant term in the polariton operator. This approximation is}

satisfied for small value of the coupling strength ΩR/ω12but holds even for values of ΩR/ω12= 0.5

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and lower polariton states mediated by the Fr¨ohlich Hamiltonian and goes beyond the scope of this thesis. Here we report the result adapted from [54]:

n2(n1+ 1)ΓnLO1,n2 = n2(n1+ 1) S |zLP| 2 |zU P| 2ωLO ΓLO 4e2LQWfσ ¯ hεp (1.43)

where fσ is the overlap factor between the QW subband and the phonon

wavefunc-tion; εp = [ε−1∞ − ε −1 0 ]

−1 _{' 0.03 [}₅₅_{]. The term n}

1 + 1 represents the simulated

scattering toward the condensate.

Solving for a steady state solution in eq.(1.41) leads to Γn2,n1

LO = n1ΓLossLP from

which we can find the threshold power:

In2,thr

P ump =

ΓLoss_{U P} ¯hωU P

κS n2,thr, (1.44) where we assumed n1 << n2 and we neglected the deviation from ideal bosonic

behaviour [54].

1.3.3 Coupled Mode Theory

Figure 1.12: Coupled mode theory model for a MIM cavity interacting with a QW. d, k and Ω are the coupling constant of the impinging light with the cavity, of the cavity with outgoing light and of the cavity with the QW intersubband transition. A direct reflection channel is included through c.

In the context of coupled resonances, the coupled mode theory (CMT)[56] is a very general model to describe the interaction between an external input19 _{(like a}

laser beam) and coupled resonators. The CMT can be applied to many different systems that can be summarized as light interacting with a resonator, being it a single photonic resonator or a system of coupled resonances as the intersubband polaritons in MIM cavities. The theory provides a classical model to describe the optical properties of such systems, like the reflectance or the absorbance. Even if this model does not take into account the microscopical quantum nature of the resonance, it can give an insight on many fundamental parameters and can be used in our case to determine quantities like the Rabi frequency or the losses of the cavity and of the intersubband transition.

19_{The theory can be generalized for multiple input and output channels [}₅₇_{] but we will focus}

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The model is depicted in figure 1.12: the incoming light, s+, either couples to

the cavity with the coupling constant d or is reflected with the c coefficient. The cavity resonance, ωc, can scatter to the outgoing channel, s−, with amplitude k.

This represents the radiative losses of the cavity, while the non-radiative ones are described by the term γnr. The electromagnetic energy in the cavity can also couple

to the intersubband transition, ω12, with coupling constant Ω. γ12 represents the

losses of the intersubband transition. We can write a set of rate equations:

da dt = (iωc− γc)a + iΩb + ks+ db dt = (iω12− γ12)b + iΩa s−= cs++ ad (1.45)

Where a, b are complex time-dependent variable representing respectively the elec-tromagnetic energy stored in the cavity and in the matter resonator. γc= γnr+ γr is

the cavity damping rate which contributes to the outgoing light with the radiative term. Following [58], from this set of equations, it is possible to find the reflectance of the whole system:

R = 1 − 4γr

(ω − ω0)2+ γ122 γnr+ γ12Ω2

(ω − ω0)4+ (ω − ω0)2(γc2+ γ122 − 2Ω2) + (Ω2+ γcγ12)2

(1.46)

where we are assuming that the cavity and the intersubband transition have the same resonant frequency: ω0 ≡ ω12= ωc.

The reflectance is fully described by five parameters: ω0, Ω, γr, γnr and γ12. This

expression holds for both the weak coupling, in which the spectrum of the device presents a single peak at ω0, and the strong coupling in which the two strongly

interacting resonances give birth to the polariton spectrum characterized by the two absorption peaks spaced by 2ΩR. This model is thus perfect to fit the data from

experiment or simulation to find the parameters describing each resonance and the coupling strength; it will be used exactly in this way in chapter3 to investigate the strong and ultra-strong coupling regime between the surface plasmon resonance in a graphene grating and the intersubband transition of a QW.

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1.4 Graphene Plasmonics

Figure 1.13: a) Graphene hexagonal lattice; the elementary cell is represented in red. b) Electronic band structure in the reciprocal space. The conduction and valence bands touch in the Dirac point (red square). c) Dirac cone filled by electrons in the conduction band. The red lines highlight the possible intra-band or interband transitions.

Graphene is a mono-atomic layer of carbon atoms arranged in a honeycomb lattice (Fig.1.13,a). First achieved by Geim and Novoselov in 2004 [59], graphene immediately attracted the interest of the scientific community. The in-plane σ-bonds are responsible for the incredible resistance to mechanical stress [60] while the out of plane pz orbitals create the weak Van der Waals interaction that binds layers of

graphene into the bulk graphite. When dealing with a single layer, the electrons in this orbital are responsible for the optical and electrical properties [61]: graphene shows a very high optical absorption (2.3%) in a wide region of the spectrum and an electron conductivity higher than silver. Moreover graphene possesses a high thermal conductivity associated to a long phonon lifetime [62]. All these properties made graphene the perfect platform for many different applications especially in electronics and optics where it could be employed in the fabrication of sensors, transistors, batteries and more [63–65]. In this work we focus on the optical properties of graphene connected to the generation of graphene surface plasmons (GSPs); we will thus start from the band structure of graphene to find its electric conductivity and then find the dispersion relation of GSP as done in section 1.2 for their metallic counterpart. We will describe two particular geometries to launch GSPs, that will be used in the experimental campaign (see chapter 3), focusing in particular on the possibility to tune the GSP resonance with the Fermi energy.

Most of graphene peculiar properties arise from the band structure at low ener-gies (Fig.1.13,b): while electrons in a crystal have a quadratic dependence on the wavevector k, in graphene the dispersion is linear around the Dirac points: the va-lence and conduction bands form two symmetric cones that touch each other at zero energy. Intrinsic graphene, having Fermi energy equals to zero, is thus defined as a semi-metal due to the absence of a band gap and the zero density of states at the Dirac point. The conductivity in graphene can be described as the sum of two

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Figure 1.14: Dispersion relation of graphene surface plasmons for three different values of the Fermi energy. The light line is visible on the left as a straight black line. The inset shows a zoom at small q values. The light line never intersects GSPs dispersion.

contributions: the interband and the intraband, σGra= σintra+ σinter [66, 67]:

σintra(ω) = 2ie 2_k BT π¯h2(ω + i/τ )ln2 cosh EF/2kBT; (1.47) σinter(ω) = e 2 4¯h ( 1 2 + 1 πarctan ¯ h(ω + i/τ ) − 2EF 2kBT − i 2πln [¯h(ω + i/τ ) + 2EF]2 [¯h(ω + i/τ ) − 2EF]2 + (2kBT )2 ) ; (1.48)

where the Fermi energy EF = ¯hvF

√

πne, τ = µEF/evF2 is the scattering time, vF is

the Fermi velocity (∼ c/300) and µ is the graphene mobility. The two conductiv-ities describe the absorption of a photon which promotes the electron to a higher empty energy states. Note that scattering may also take part in the interaction compensating the momentum mismatch for the intraband transitions (Fig.1.13,c). The dielectric function can be obtained from:

εGra(ω) = 1 +

iσGra(ω)

ωε0δ

(1.49)

where δ ∼ 0.34 nm is the graphene thickness.

1.4.1 Surface Plasmons in Graphene

In order to find the GSP dispersion relation, we need to find a solution for a p-polarized wave with the form:

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as previously done for the SPP (see section 1.2). The index l = 1, 2 represents the dielectric layer above or below graphene. In this case graphene is treated as a boundary condition so that the continuity of the magnetic field must include the current in it: B1,y − B2,y = µ0Jx = µ0σxEx. Solving Maxwell’s equations leads to

the implicit form of the dispersion relation [68]: ε1 k1 + ε2 k2 + i σx ε0ω = 0 (1.51)

The equation can be solved analytically under some assumptions: the two dielec-tric constant are similar ε1 ∼ ε2 = ε; we can use the electrostatic regime, q >> k0,

with k0 the wavevector of the exciting light, and the interband absorption can be

neglected20_{. Within these approximations the dispersion reduce to:}

qGSP = ¯ hε 2αcEF (ω2+ iω τ) (1.52)

with α = e2/4πε0¯hc the fine structure constant and qGSP the graphene surface

plasmon wavevector. The dispersion relation is reported in figure 1.14 for three values of the Fermi energy. One can see that the resonant frequency is proportional to the square root of the Fermi energy: ωGSP ∝

√

EF. It is useful to define the

effective index of the graphene SP mode as:

nGra = Re{q} k0 = ε1+ ε2 4α ¯ hω EF (1.53)

Substituting the typical values in our experiments: ε ∼ 10, EF ∼ 0.1 eV and

¯

hω ∼ 0.1 eV, one gets an effective index (or a confinement factor) of around 500. This means that a wavelength of 10 µm will excite a GSP with a wavelength of 20 nm, confining the electromagnetic energy in an extremely small volume. This property will be fundamental for the light-matter strong coupling as we previously mentioned in section 1.3.

Figure 1.15: a) Graphene grating. The horizontal cut shows the simulated z com-ponent of the electric field. The graphene ribbon acts as a Fabry-Perot cavity, res-onating only when an integer number of wavelengths propagate within the ribbon width. b) Gold nano-antenna deposited on a graphene sheet. The dipole moment induced by the antenna launches the GSP shown by the simulation.

As one can see from the inset of figure 1.14, since the GSP dispersion never intersects the light line21_{, a coupling method is necessary to excite these SPPs.}

20_{This is indeed our case since the interband absorption is suppressed for ¯}_{hω < 2E}

F. Our

experiments occur in the terahertz region where ¯hω ∼ 0.05 eV while the Fermi energy usually is around 0.1 eV.

21_{Due to the strong confinement of graphene, the momentum mismatch for GSPs is even larger}

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In this thesis we will be using two types of coupling: the grating and the nano-antennas (Fig.1.15). These two methods rely on different principles to overcome the momentum mismatch: the grating as explained before relies on the periodic structure to fold the GSP in the Brillouin zone. A fundamental difference with respect to SPP in metals arises due to the extreme confinement factor: remembering eq.(1.17) kGSP = k0sin θ + n 2π d ∼ n 2π d . (1.54)

The approximation stems from the extremely small pitch d of the grating, that needs to be of the same order of magnitude of the GSP wavelength [69,70]. This condition implies that there is no angular dispersion in the reflectance of a graphene grating, so that the resonant frequency may be changed only by varying the grating pitch. For this reason these SPPs are also referred to as localized surface plasmons.

Another interesting consequence of this property is that the graphene ribbons in the grating act essentially as a Fabry-Perot cavity for the light (Fig.1.15,a) with a refractive index nGra and length equals to the ribbon width w [71]:

ωRes = n

πc nGraw

(1.55)

The nano-antennas instead rely on the dipole moment that is created by the charges moving inside the metal. In a similar fashion to figure 1.6,b) an impinging EM wave displaces the electrons inside the metal and creates a dipole field. By taking the spatial Fourier transform of the field one can see that it contains higher momentum with respect to the impinging wavevector k0 [72]. It can be shown that

the momentum distribution transferred by a metallic sphere with radius r follows (see supplementary of [73]):

ρ(k) = 1 12π5_/r4k

2

e−πk/r (1.56) The momentum transferred will thus contain a whole spectra of values with a peak centered around π/r. In other words the metallic sphere will transfer higher mo-mentum the smaller it is. This phenomenon can be exploited to overcome the large momentum mismatch between the GSP and the impinging light and in particular we will use a metallic nano-antenna. The antenna shown in figure 1.15,b) launches GSPs thanks to the strong dipole field edges (where the electric field is maximum). The antenna can be roughly approximated to a sphere with diameter equals to its width to estimate the momentum distribution transferred through equation (1.56).

1.4.2 Tuning the Fermi Energy

One of the most interesting properties of graphene and GSP is the possibility to tune the Fermi energy level EF with a gate potential. The geometry is basically the one of

the field effect transistor: the graphene sheet is placed between two metallic contacts over a dielectric spacing that separates it from another gold contact (Fig.1.16,a). The bottom metal acts as a back gate (BG) and the two top ones are the source (S) and drain (D). By applying a voltage between the BG and S one can induce a p or n doping in the graphene layer [74, 75]:

n(VBG) = r n2 0+ ε0εD h (VBG− VDirac) 2 _(1.57)

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Figure 1.16: a) Scheme of a graphene field effect transistor device. The Fermi energy in graphene can be tuned with the back-gate voltage (VBG). A

source-drain voltage (VSD) can be employed to test the current in the graphene layer and

extract information on the mobility and Fermi energy of the sheet. A dielectric (εD) is required to isolate the graphene sheet from the metallic back. b) Simulated

reflectance of a graphene grating placed over 500 nm of gallium arsenide and a metallic bottom gate as a function of the Fermi energy. The reflectance shows how the GSP resonance shifts with the Fermi energy.

Where n0 is a residual charge and VDirac is the potential of the Dirac point. Both

parameters arise from the interaction with the substrate and otherwise would be theoretically zero. h is the thickness of the dielectric layer and εD is its dielectric

function. ε0εD/h is the BG capacitance.

The Fermi energy is related to the electron density by the equation:

EF = ¯hvF

√

πn (1.58)

thus remembering eq.(1.52), it is possible to change the resonance frequency of the GSP simply by changing the BG potential and therefore the Fermi energy: ¯

hωGSP ∝ V 1/4

BG. In figure 1.16 we reported the reflectance of such a device in which

the dielectric was 500 nm of gallium arsenide. One can see how the resonance shifts to higher frequency for higher Fermi energy.

The system has been demonstrated to work both in the BG configuration [59] and with a Top Gate, also employing ionic gel [76] and shows (especially in the latter case) the possibility to tune the Fermi energy by even more than 1 eV. We will employ the BG configuration in chapter3both experimentally and in simulation to study how the strength of the coupling between the GSP and the IST transition can be modulated with a gate. We will see that this type of device can show features of the weak, strong and even ultra-strong coupling, depending on the values of the Fermi energy.

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Chapter 2 Intersubband Polariton Emission

Figure 2.1: a) Sketch of the sample with the laser pump impinging at the excitation angle. The light from the low polariton branch is emitted at kk = 0, i.e.

perpendic-ular to the sample face. The sample is constituted of 35 QWs sandwiched between a gold grating with pitch 4.4 µm and filling fraction 0.8 and a gold substrate. b) The simulated reflectance of this sample is plotted as a function of the frequency and angle of incidence. The circle indicates the pumping point, while the arrows indicate the polariton-phonon scattering and the polariton emission.

The main objective of this thesis was to demonstrate the intersubband polariton lasing with optical pumping, in particular employing the phonon-polariton scattering to build up the population in the polariton condensate: the pump is absorbed in the upper polariton branch which scatters towards the lower polariton branch due to the interaction with the phonons inside the material. In this chapter we will show the results of the experimental campaign we performed: we will firstly describe the sample we used, its composition and its dispersion relation. Then we will present two different setups we assembled to measure the intersubband polariton emission: the first involves the generation of the polariton condensate due to the resonant phonon-polariton scattering, and the characterization of the spectrum emitted by the sample; the second will be a pump and probe measurement in which two lasers will be focused on the sample. One will act as a pump, generating polaritons in the lower branch again due to the phonon-polariton scattering, while the other will be used to monitor the reflectance of the sample with the pump on or off. The

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probe is also used to generate polaritons in the final state and increase the phonon-polariton scattering. Indeed since the latter is proportional to population of the final state, we expect to measure a higher reflected intensity when the pump is operating. This effect is a fingerprint of the emission of polaritons generated by the stimulated scattering toward the final state. The probe laser will be impinging on the sample at an angle different from zero, thus no condensate will be generated in this case. The aim in this case is indeed measuring the pumping efficiency rather than achieving the polariton lasing. We will see a clear emission from the lower polariton in both cases. The measured power was a factor of ten higher than previous experiments, but the result seems to indicate that the lasing threshold was not reached yet.

The device is built so that the polariton dispersion has a minimum at k = 0 were the condensate may generate. In order to have a resonant scattering the pump and the condensate must have an energy difference equals to that of the phonon. For this reason it is fundamental to have access to the dispersion of the device, which can be observed by measuring the reflectance and by simulation. As an example we report in figure 2.1 the simulated reflectance of the sketched device. The design process of this sample requires to optimize the pump absorption and the resonant scattering and it is usually done through simulations. The code we used, PPML [35], allows one to calculate the reflection of this device as a function of the frequency and angle of incidence. The geometry of the grating: pitch, filling fraction and thickness can be varied to obtain the best configuration possible (see also chapter 3for more information on the code).

The fabrication was not performed by us: this thesis was performed within the MIR-Bose project collaboration, and the growth of this device was carried out in the university of Leeds: the sample is composed by 35 repetitions of the same AlGaAs-GaAs QWs, grown with standard molecular beam epitaxy techniques. On top of the sample a gold grating has been lithographically manufactured. The pitch of the grating was 4.4 µm with a filling fraction of 0.8. Below the QWs, a continuous layer of gold was attached with a wafer-bonding technique. The MIM geometry ensure an approximately constant field distribution inside the sample (see section 1.2), that is important for polariton generation as the coupling strength is proportional to the field intensity and the superposition of the photonic mode and material excitation. Once fabricated, the dispersion of the sample has been characterized first with a Fourier-Transform Infrared spectrometer (FTIR) in Paris. The sample was mounted on a motorized reflection unit and the reflectance was measured as a function of the angle of incidence1_{. From the spectra, the optimal pumping point was estimated}

to be 1078 cm−1 (or 32.3 THz) at a ∼ 40°. This point lies in the upper polariton branch at an energy distance of 293 cm−1 (8.8 THz) from the lower polariton branch at kk = 0 (Fig.2.1). That is exactly the frequency of the longitudinal optical (LO)

phonon inside gallium arsenide [77]. To have a better understanding of the polariton dispersion, one can simulate the device reflectance. The QW doping and quality factor are adjusted so that the simulation fits the data acquired experimentally (that is the case of figure 2.1,b).

1_{The measurements will be reported later in this chapter as we will be comparing them with}

Towards intersubband polariton lasing