Graphene plasmonic gratings for few-electron strong coupling experiments

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DEPARTMENT OF PHYSICS

Master’s degree in Physics

GRAPHENE PLASMONIC GRATINGS FOR

FEW-

ELECTRON STRONG COUPLING

EXPERIMENTS

Candidate:

Giuseppe Lanza

Supervisors:

Prof. Alessandro Tredicucci Prof.ssa Alessandra Toncelli

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Introduction

Thanks to its extraordinary properties and multifunctionality, graphene has emerged in recent years as one of the most promising nanomaterials for optoelectronic applica-tions. Owing to its high carrier mobility, gapless spectrum and the possibility to deeply alter its doping level through electrical gating, it represents one of the best platforms to control cavity polaritons [1].

One of the most interesting applications of graphene is the possibility to generate

sur-face plasmon polaritons, which are quantum collective oscillations of the graphene

surface charge density coupled to electromagnetic fields localized at the surface. The generation of surface plasmons in graphene cannot occur by directly illuminating the graphene with an incident light beam, but the presence of a periodic structure that can impart an additional momentum to the incident light is necessary. In the mid-infrared spectral range, the electromagnetic field of surface plasmons shows subwave-length confinement over a distance down to tens of nanometres.

Owing to this strong optical confinement of graphene surface plasmon polaritons (GSP), graphene gratings appear a promising platform for the realization of arrays of extremely small nanocavities, where the presence of a quantum well can produce strong

light-matter coupling between surface plasmon polaritons and intersubband excitations. The strong light-matter coupling regime is a phenomenon that occurs when a light emitter exchanges energy with a single cavity mode at a rate larger than the loss rates. In this situation, the interaction gives rise to a new eigenmode of the system with an associated quasi-particle called polariton, which is superposition of light and material excitation. This phenomenon has been studied in a large number of situations and has been employed for many relevant applications in optoelectronics and fundamen-tal physics [2].

In this context, interesting phenomena occur when the number of material excitations, strongly interacting with the cavity mode field, is reduced toward unity. If the size of these nanocavities is reduced below the diffraction limit, the strong light-matter in-teraction could take place also at the level of few electrons per cavity: when this sit-uation occurs, the absorption frequency of the system is strongly dependent on the number of photons within the cavity, thus exhibiting strong non-linearities at the sin-gle photon level and an effective photon-photon repulsion, as a consequence of the anharmonicity of the Jaynes-Cummings ladder states. In the extreme limit of only one "atom" within the cavity, the presence of a photon can modify the resonance frequency of the cavity mode, thus denying the entrance of a second photon into the cavity: for this reason this situation is referred to as photon blockade regime.

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2 A periodic array of GSP-based nanocavities may represent an original solution for the realization of coupled cavity arrays, which are important so as to produce systems of strongly interacting photons. The applications of those systems range from the realiza-tion of superfluid states of light [3], to the implementation of quantum simulators [6]. Moreover, single-photon nonlinear systems have applications in the realization of all-optics devices, such as single-photon switches or transistors [4,5].

Most of the current methods employed to implement those nonlinear systems cannot be reproduced in large scale for the realization of coupled cavity arrays (e.g. quantum dots in photonic crystal cavities) or have problems related to the electromagnetic cou-pling between the cavities (e.g.silicon microdisks cavities coupled by optical fibers) [6]. This thesis work is devoted to investigating graphene gratings suitable for the strong interaction with an intersubband excitation in a single quantum well, paving the way for the realization of few-electron strong coupling experiments.

The most common periodic arrangement employed for the generation of graphene surface plasmon polaritons is the graphene nanoribbons array, which we study in depth in the presence of a GaAs substrate. Besides this configuration, in this work we present a new promising graphene periodic structure for the generation of surface plasmon polaritons: a GaAs square-wave grating covered by a flat continuous sheet of graphene. Throughout this thesis we study the frequency-domain response of both plasmonic grating configurations using finite element simulations, also analysing how they cou-ple with the intersubband plasmons of a single GaAs/AlGaAs quantum well. Moreover, both graphene grating configurations are implemented and characterized experimen-tally, so as to obtain important information which can be useful for the future observa-tion of intersubband polaritons in our graphene-based systems.

The thesis is organized as follows:

• In Chapter 1 we report the fundamentals of the physics of intersubband po-laritons, which are described by two different regimes: the Jaynes-Cummings model, which considers the interaction of a cavity mode with a single two-level system, and the Hopfield model, which describes the interaction of a cavity mode electromagnetic field and a collective electronic excitation.

• In Chapter 2 we describe the theoretical basis of graphene surface plasmon po-laritons by focusing on the properties of graphene grating systems.

• Chapter 3 is fully dedicated to the finite element simulations of different graphene plasmonic gratings able to generate surface plasmons. These plasmonic gratings are also studied in the presence of a single GaAs/AlGaAs quantum well close to the graphene layer, so as to analyse the interaction of surface plasmon polari-tons and quantum well intersubband plasmons. In this part we also investigate the effects on the strength of the light-matter interaction caused by a variation of the system parameters.

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3 • In Chapter 4 we describe the engineering and fabrication procedure adopted for the implementation of the simulated graphene plasmonic gratings. In addition, we show how we optimized the fabrication process to improve the quality of the gratings.

• In Chapter 5 we characterize the graphene gratings with Raman spectroscopy, FTIR spectroscopy, microscopy imaging techniques and field-effect measure-ments. The results arising from the measurements are shown and discussed.

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1

Quantum theory of

intersubband polaritons

The strong light-matter coupling regime is a phenomenon that occurs when a material excitation reversibly exchanges its energy with an optical mode of a microcavity. As we will point out in this chapter, light and material excitations can couple strongly if the microcavity volume is small enough to ensure the interaction of excitations with a single cavity mode. It is also necessary that loss mechanisms act with a rate that is lower than the Rabi frequency of the system, which is the rate at which the reversible energy exchange happens.

In this chapter, we will start by illustrating the basic principles of photon and electron confinement in microcavities and quantum well respectively. Afterwards, the physical problem of the interaction between cavity modes and intersubband excitations in a quantum well will be examined in two different limits:

• interaction of a two-level system with a single mode electromagnetic field, de-scribed as a single quantum harmonic oscillator (Jaynes-Cummings model); • interaction of a collective material polarization with an electromagnetic field

(Hopfield model);

In the limit of few electrons, the latter model reduces to the former. Since the Jaynes-Cummings model involves energy eigenstates that grow anharmonically with an in-creasing number of photons, it leads to the emergence of nonlinearity at the single-photon level.

1.1 Optical microcavities

In the context of quantum optics, a cavity resonator is a hollow device that is reso-nant only to certain frequencies of electromagnetic radiation. The light confined in an optical cavity goes back and forth multiple times through different mechanisms, for instance by reflection on a metallic surface or by total internal reflection at the bound-ary between two dielectrics, and, due to interference, only certain frequencies of the radiation will be sustained by the resonator, with the others being suppressed destruc-tively.

Under appropriate conditions, the energy of the optical beam can be “trapped" inside the resonator, and, in this case, the beam becomes a mode of the resonator. The elec-tromagnetic field of the cavity modes is the superposition of all the reflected waves,

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1.1. Optical microcavities 6 acquiring a specific shape when they vibrate at the resonance frequencies. This is due to the fact that only the propagating patterns that are reproduced at every round-trip of the light through the cavity are stable, and are known as the eigenmodes of the res-onator.

When the dimension of the resonator is close to or below the wavelength of light it is referred to as microcavity [7]. The simplest cavity resonator configuration we can arrange is the so called Fabry-Perot resonator, which is a cavity consisting of two high-reflectivity mirrors with a dielectric medium between them separated by a certain dis-tance L, so that only a few wavelengths of light can fit in between them. If the curvature of the mirrors corresponds to a periodically stable focusing system, and their transver-sal dimensions are large enough to ensure that we can neglect diffraction effects at the beam boundaries, the light will be trapped inside the cavity and the resonator eigen-modes will be those having a frequency

νm= m

c

2nL, (1.1)

These frequencies are integer multiples of a constant frequency c/2nL known as free

spectral rangeof the Fabry-Perot, and they realize constructive interference inside the cavity.

Besides mode spectral separation, an optical microcavity is characterised by other im-portant properties that we briefly analyse below.

Figure 1.1: Sketch of a Fabry-Perot cavity. Figure from Ref. [8].

Q-factor and photon lifetime The quality factor of a cavity is a dimensionless

pa-rameter that describes the rate at which optical energy decays from inside the cavity due to absorption, scattering or leakage through imperfect mirrors. It is defined as the ratio between the cavity resonance frequencyωc and the linewidth (FWHM) of the

modeδωc: Q = ωc δωc (1.2) or equivalently Q = 2π energy stored

energy dissipated per cycle= ω

energy stored

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1.2. Intersubband transitions in quantum wells 7 The quantity Q−1 _{represents the fraction of energy lost in a single round-trip around}

the resonator. The number of photons inside the cavity is affected by an exponential decay due to the cited energy loss mechanisms, leading to the definition of a charac-teristic time called photon lifetime of the cavity:

τc =

1 δωc

The Q-factor can be therefore expressed as a function of the photon lifetime and the round-trip period of the oscillations:

Q =τc T

Finesse The finesse of a cavity is defined as the ratio of free spectral range (frequency

separation between successive longitudinal cavity modes) to the linewidth of a cavity mode:

F =∆ωc δωc

(1.4) By this definition we can conclude that finesse and Q-factor are strongly connected and describe the quality of the cavity in terms of energy storage.

1.1.1 Photon confinement in waveguides

A waveguide is a spatially inhomogeneous structure that confines the electromagnetic waves inside a path, allowing a guided propagation of the electromagnetic energy, as opposed to "ordinary" propagation in free space. Usually, a waveguide contains a re-gion of increased refractive index (core) and a surrounding medium with lower refrac-tive index (cladding); however, guidance is also possible by the use of reflections at metallic interfaces or by exploiting plasmonic effects at metal/dielectric interfaces. When an electromagnetic wave travels inside a homogeneous and transparent medium, its wavevector increases by a factor equal to the refractive index of the medium: the wavevector is n times higher than it would be in vacuum. By contrast, in the pres-ence of a sequpres-ence of dielectric and metallic layers, i.e. a spatial dependpres-ence of the permittivity functionε(x) in one or two dimensions, the electromagnetic fields will be confined in these dimensions and propagate in the other. The guided waves propagate with a wavevector called in-plane wavevector β, and its value is forced by Maxwell’s equation to be one of the eigenmodes of the waveguides [9]. The in-plane wavevector of a certain eigenmode will be increased with respect to its value in vacuum k0= ω/c

by a factor that, this time, is not the refractive index of one of the layers where the field is confined, but it is a kind of average of those refractive indices: this quantity is called

effective refractive index

β = neffk0 (1.5)

which will be useful in the next chapters for the electromagnetic analysis of surface plasmon polariton modes in graphene.

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1.2. Intersubband transitions in quantum wells 8

Figure 1.2: Sketch of a GaAs/AlxGa1−xAs quantum well. Figure from Ref. [8].

1.2 Intersubband transitions in quantum wells

In this thesis work, quantum wells play a central role for the realization of strong light-matter coupling. A semiconductor quantum well is formed by embedding a thin layer of narrow band-gap material between two layers of larger band-gap semiconductor (for instance, a GaAs layer between AlGaAs layers); this structure ensures electron con-finement in the growth direction, producing a 2D electron gas that will be fundamental for few-electron strong coupling.

Quantum wells have energy levels gathered in 2D subbands, giving us some important advantages:

• tunability of the energy transition by varying the well thickness;

• density of states constant in energy, i.e. delta-like intersubband absorption [12]. Depending on the relative band offsets of the two semiconductor materials, both elec-trons and holes can be confined in one direction in the conduction and valence band respectively, and we obtain allowed energy levels that are quantized along the growth direction. These energy levels can be tuned by the quantum well depth and thick-ness [11].

In order to study the optical properties of quantum wells we must analyse the quantum behaviour of electrons when they are confined inside these special solid-state materi-als called heterostructures, of which quantum wells represent an example. First of all we must specify that heterojunctions are the interfaces between two layers of different solid-state materials, typically semiconductors, usually having different energy band gaps and doping. The combination of multiple heterojunctions together constitute a heterostructure: layers of two or more materials with different composition are grown one on top of the other with various epitaxial growth techniques, so the layers will be aligned in the direction of growth.

We will consider semiconductors for which:

• the energy minimum and maximum of the bands is at k = 0 (direct gap); • the energy bands can be approximated by parabolic bands.

A typical example of such semiconductors is GaAs, with an energy gap of 1.42eV [55]. In the general framework of a semiconductor possessing a periodic crystal lattice po-tential Vc(r ), and a slowly varying potential V(r ), we can treat the problem with the

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1.2. Intersubband transitions in quantum wells 9 perturbation theory formalism. In the case of V(r ) = 0 (bulk semiconductor) we find eigenstates that are the well-known Bloch wavefunctions:

ψn,k(r ) = ei krun,k(r ), (1.6)

with un,kfunctions with the same periodicity of the lattice, n labelling the band. On the

other hand, in a heterostructure as in Fig.1.2we must take into account that the energy minimum of the conduction band has a dependence on the spatial coordinate of the growth direction z, and so does the effective electron mass. The fact that V(r ) varies very slowly implies that its Fourier transform is null except for very small q near the center of the Brillouin zone; therefore, with a formalism that is directly inherited from the k · p perturbation theory [12], we can express the electronic wavefunction in each material of the heterostructure as a linear combination of the basis un,k(r ) ' un,0(r ) = un(r ): ψW,B(r ) = X n,k c_nW,B_,kei krun(r ) = X n χW,Bn (r )un(r ) (1.7)

For a quantum well, W and B indicate the well and the barrier material. The coefficients χnare known as envelope functions. As it is shown in Ref. [12], the sum over n runs over

the bands included in the analysis, depending on the desired precision. In our case we are interested in intersubband transitions, i.e. transitions within a single band (e.g. the conduction band), therefore we can consider only the conduction band term in the summation and write the wavefunction as

ψ(r ) ' χ(r )uc(r ) (1.8)

where uc(r ) is the Bloch function of the conduction band. With this approximation,

the envelope function obeys to the Schrödinger’s equation: h −∇∇∇ ~ 2 2m∗_{(r )}∇∇∇ + V(r ) i χ(r ) = Eχ(r ) (1.9) For a quantum well V(r ) depends only on the z coordinate. Therefore, Eq.1.9for a quantum well reads

h − ~ 2 2m∗_(z) ³ ∂2 ∂x2+ ∂2 ∂y2 ´ −~ 2 2 ∂ ∂z 1 m∗_(z) ∂ ∂z + V(z) i χ(r ) = Eχ(r ) (1.10) with V(z) = 0 inside the well and equal to ∆Ecoutside. The band offset∆Ecis defined as

the energy difference between the conduction band minima (or valence band maxima) of the heterojunction material. For the GaAs/AlxGa1−xAs heterojunction with x < 0.4,

the offset is about proportional to the fraction of aluminium x (∆Ec' x × 836 meV [8]),

so it can be tuned by simply varying the concentration of Al in the semiconductor. Since the Hamiltonian is separated in the sum of a free particle Hamiltonian in x and y and the Hamiltonian of a particle in a step potential in z, we can factorize the envelope function as

χ(r ) = ei k_q·rqχ(z), (1.11)

where k_q= (kx,ky). The final result of this discussion is that the energy eigenstates of

the system will be in the form

E_λ,k_q= ελ+~ 2_k2 q 2m∗ w (1.12)

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1.2. Intersubband transitions in quantum wells 10 with m∗

wthe effective mass in the well material (e.g. GaAs for a GaAs/AlGaAs quantum

well) andελthe energy levels of a one dimensional quantum well with depth∆Ec. This

result shows that in the reciprocal lattice the energy bands have a parabolic dispersion in kxand ky, while the kzdispersion is replaced by the presence of a number of discrete

levels: this new 2D bands are called subbands.

Figure 1.3: Vertical optical transition mechanisms in QW, both in reciprocal and in real space. Figure

from Ref. [8].

1.2.1 Selection rules for intersubband transitions

We know from quantum mechanics that when a two level system is subjected to an external electromagnetic field, the first contribution to the transition probability be-tween two eigenstates is given by the electric dipole contribution. If we concentrate our attention on the transition probability between an initial state |i〉 and a final state | f 〉 in different subbands, we will find that the rate of transition Wi f is given by Fermi’s

golden rule: Wi f = 2π ~ | 〈i | Hi nt| f 〉 | 2_δ(E f− Ei−~ω), (1.13)

with Hi nt the interaction Hamiltonian between the electric field, that we consider as

monochromatic with a frequency ω, and the two-level system. Taking the electric dipole as the dominant term of the interaction, we can find out which selection rule the electric field must satisfy in order to allow optical transitions between the two states. If the light is polarized along the ˆedirection, the matrix element the we must calculate

will be 〈i| ê· p |f 〉 (with p the electric dipole operator): 〈i | ê · p |f 〉 = Z e−i kiqrχ∗ i(z)ui∗(r ) ê · p e i kf qrχ_f(z)u_f(r )dr = = Z ei(kqf−k i q)rχ∗ i(z)χf(z)u∗i(r ) ˆe · p uf(r )dr + + Z e−i kqiru∗ i(r )uf(r )χ∗i(z) ˆe · p e i kf qrχ_f(z)dr .

Since the envelope function varies slowly in comparison with the lattice spacing, we can assume it constant within the unit cell and rewrite the first integral as

〈i | ˆe · p |f 〉 =X R ei(k f q−k i q)rχ∗ i(z)χf(z) Z Ωu ∗ i(r ) ˆe · p uf(r ) dr (1.14)

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1.3. The Jaynes-Cummings model 11 The same can be done with the second one. These terms are non zero only if k_qf = k_qi, thus if we replace the discrete sum over the unit cells with an integral, divided by the cell volumeΩ, we will find:

〈i | ˆe · p |f 〉 = 〈χi| |χf〉 〈ui| ˆe · p |uf〉 + 〈ui| |uf〉 〈χi| ˆe · p |χf〉 . (1.15)

For intersubband transitions, the overlap between the envelope functions is 〈χi| χf〉 = 0

and 〈ui| uf〉 = 1, then the matrix element reduces to

〈i | ˆe · p |f 〉 = 〈χi(z)| ˆe· p |χf(z)〉. (1.16)

Since the envelope functions depend only on the z coordinate, the matrix elements of

px and py are zero. Therefore, this equation gives us the important selection rule for

intersubband transitions: in order to have intersubband absorption the electric field

component along the z direction must be non-zero.

1.2.2 Quantum well effective thickness

Focusing on transitions between the first two subbands of a quantum well, we want to introduce an important quantity that will be relevant later. We define the effective

thickness Leffof a quantum well for the transition 1 → 2 as [14]

Leff= 2m ∗_ω

12

~R j(z)2d z (1.17)

where m∗ _{is the electron effective mass and j (z) is the intersubband current density}

associated with the transition 1 → 2 and calculated from the subbands wavefunctions:

j(z) = χ2(z)∂χ1(z)

∂z − χ1(z) ∂χ2(z)

∂z (1.18)

Intuitively, the effective thickness of a quantum well describes the spatial extension of the microscopic current density of the intersubband transition 1 → 2.

1.3 The Jaynes-Cummings model

Since the prediction of the existence of spontaneous emission of radiation by Einstein, physicists have believed that an excited atom decays to its ground state with a rate de-termined by its nature. However, spontaneous emission is not an immutable property of the atom since it is affected by the environment coupled to it.

In this section we want to give an idea of the cavity quantum electrodynamics behind intersubband polaritons and strong light-matter interaction in general, by considering two different situations. The first is that of the Jaynes-Cummings model, which is able to describe the interaction of a single electronic excitation between two subbands (de-picted as a two-level system) and a single photon modelled as a quantum harmonic oscillator (see Fig.1.4).

The second situation, on the other hand, must be considered when the number of electrons is very large. In this case, the electronic excitations should be depicted as a

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1.3. The Jaynes-Cummings model 12 collective material excitation. Moreover, one has to take into account Coulomb inter-actions between electrons, that give rise to an important effect called depolarization

shift.

Following [13], we start by considering a two-level system with a ground state |g 〉 and an excited state |e〉 with an energy separation~ω12placed inside a microcavity. We

as-sume that the cavity is small enough to ensure that only one mode of frequencyωccan

be confined, so as to have only one mode resonant to the transition frequency. We can imagine this microcavity as a Fabry-Perot cavity: when the separation of the mirrors become very small, the modes will be increasingly distant in energy and there will be only one mode resonant to the transition.

Figure 1.4: Interaction between a two-level system and a cavity photon with coupling constantΩ. Ex-citations are lost via non radiative processes (or spontaneous emission in free space modes) at a rateγ and via cavity decay at a rateκ. Figure adapted from Ref. [6].

The full Hamiltonian of the system is the sum of electronic, photonic and interaction Hamiltonians:

H = He+ Hp+ Hi nt. (1.19)

In the framework of second quantization, we define the transition operators • ˆσ+= |e〉 〈g |

• ˆσ−= |g 〉 〈e|

• ˆσ3=¡ |e〉〈e| − |g 〉〈g |¢/2,

respectively the raising operator (from |e〉 to |g 〉), lowering operator (from |g 〉 to |e〉) and inversion operator. They satisfy the commutation relations

[ ˆσ+, ˆσ−] = ˆσ3 [ ˆσ3, ˆσ±] = ±2 ˆσ∓. (1.20)

By the use of these operators we can write the first two terms of the Hamiltonian Eq.1.19

after setting the ground state |g 〉 at zero energy:    He=~ω12σˆ+σˆ− Hp=~ωc(a†a), (1.21) with a† and a respectively the creation and annihilation operator of a photon in the cavity mode. In order to evaluate the interaction Hamiltonian Hi nt we must introduce

the electric dipole operator between the eigenstates |g 〉 and |e〉:

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1.3. The Jaynes-Cummings model 13 with deg = e 〈e| ˆr |g 〉, thus proportional to the matrix element of the position operator.

Since we are interested in intersubband transitions, if we assume that z is the growth direction, the electric field can be taken with a polarization along the z axis and only consider the element d = e〈e| z |g 〉:

ˆE = s ~ωc 2ε0Vsin(kcx)(a † + a)ez (1.23)

Therefore, the interaction Hamiltonian will be:

Hi nt= −ˆd · ˆE = −~(Ω0σˆ++ Ω∗0σˆ−)(a†+ a) (1.24)

with~Ω0= d · E defined as the single-photon Rabi frequency: this frequency is the coupling constant of the theory and describes the interaction strength between the

cavity mode and the material excitation. Notice from the definition Eq.1.23that the smaller is the cavity volume, the stronger will be the light-matter interaction.

Rotating wave approximation When the cavity mode frequency is close to the

reso-nance, we can use an important approximation in order to neglect the anti-resonant terms of the Hamiltonian. Indeed, in the interaction picture (or Dirac picture) the Hamiltonian Hi nt has terms with a phase factor:

• e±i (ω12−ωc)t, called resonant terms,

• e±i (ω12+ωc)t, called anti-resonant terms.

Since (ω12− ωc) ¿ (ω12+ ωc), the time evolution of the anti-resonant terms is much

faster then that of the resonant terms, and we can neglect the former from the interac-tion Hamiltonian: this is the rotating wave approximainterac-tion (RWA). Its applicainterac-tion coin-cides with neglecting the factors a†σˆ+(creation of a photon and a material excitation)

and a ˆσ−(annihilation of a photon and a material excitation). In the rest of this work

we will refer to the difference∆ = (ω12− ωc) as the detuning of the frequencies.

The eigenstates of the uncoupled light-matter system correspond to the solutions of the Schrödinger’s equation with Hi nt = 0. If |n〉 is the eigenstate of Hp containing n

cavity photons with energy_~ωc, we can express these eigenstates through the

eigen-value equations

(He+ Hp)|e,n〉 =~(ω12+ nωc)|e,n〉 (1.25)

(He+ Hp)|g ,n + 1〉 = (n + 1)~ωc|g , n + 1〉 , (1.26)

Notice that the energy difference between these eigenstates is equal to the detuning

~∆, so they are degenerate at resonance. On the other hand, coupling light and matter

through the dipole interaction brings to an Hamiltonian that can be diagonalized sep-arately in each subspace {|n + 1, g 〉,|n,e〉}, giving a new set of eigenstates called dressed

states:

|+, n〉 = sin θn|e, n〉 + cos θn|g , n + 1〉 (1.27)

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1.3. The Jaynes-Cummings model 14 where sinθn= q Ωn+∆ 2Ωn , cosθn= q Ωn−∆ 2Ωn and Ωn= q Ω2 0n + ∆2. (1.29)

At resonance, the eigenenergies of the dressed states are E_±,n= (n +1

2)~ωc± 1

2~Ωn (1.30)

The eigenstates of the system are a superposition of photonic and excitation states, and are referred to as polaritons. Eq.1.30shows an important consequence of the light-matter coupling: this interaction causes a degeneracy breaking when the detun-ing is∆ = 0, with an energy splitting known as Rabi splitting.

The Rabi splitting is larger as the number of cavity photons increases, and it is the sig-nature of the interaction strength between light and material excitations (see Fig.1.5). From a dynamic point of view, if the system starts at time t = 0 from the state |g ,n + 1〉, it will oscillate back and forth between |g ,n + 1〉 and |e,n〉 at the Rabi frequency Ωn, in

a series of reversible absorptions and emissions called Rabi oscillations. It is important to notice that these oscillations occurs even in the absence of cavity photons.

Figure 1.5: Energy diagram of the two-level manifolds, known as Jaynes-Cummings ladder.

1.3.1 Interaction with the vacuum reservoir

The Jaynes-Cummings Hamiltonian does not include any coupling to an environment leading to decay. In order to include loss mechanisms, one has to consider at least non-radiative decay from |e〉 (rate γ) and decay of the field mode (rate κ) into the empty modes of the radiation field. We can take into account this interaction by considering a decay of the density matrix elements of the two level system with a constant rate; this method leads to a master equation for the density operator, as shown in Ref. [13]. Anyway, the interesting concept we want to point out is that the effect of those loss mechanisms depends on the ratioΩ2_/(γκ):

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1.4. The Hopfield model for intersubband polaritons. 15

Figure 1.6: Energy diagram and dispersion relation for the first two subbands of a quantum well.

• whenΩ ¿ γ,κ the system is affected by very large losses that lead to strong decay, thus no coherent evolution is expected, This case is known as weak coupling

regime (WCR).

• whenΩ À γ,κ, the atom-photon coupling dominates the loss rates of the system and a coherent evolution occurs for a very long time, until dephasing caused by losses destroys it. This case is referred to as strong coupling regime (SCR). In the strong coupling limit, the atom-photon interaction is faster than the irreversible processes. This mechanism makes the emission of the photon a reversible process in which the photon can be re-absorbed by the material system before it is lost from the cavity. In the weak coupling limit, by contrast, the emission of the photon by the atom is an irreversible process, as in normal free-space spontaneous emission, but the emission rate is affected by the presence of the cavity.

1.4 The Hopfield model for intersubband polaritons.

When we put a weakly doped quantum well inside a microcavity, the situation is dra-matically different. Focusing our attention on intersubband transitions, we notice that there is a large number of electronic states in each subband, of the order of the num-ber of states of the fundamental subband with |k| less then the Fermi wavevector kF,

so we must take into account that all the electrons in these states can be excited by the cavity radiation. Furthermore, the resonant mode has a particular spatial distribution

fk(z) and a dispersion with the in-plane wavevector1k = (kx,ky): as a consequence,

the analytical derivation becomes much more complex by a formal point of view. Here we will summarize the results obtained by the work of Y. Todorov [14], who pro-vided a theoretical description of the coupling between electromagnetic field and in-tersubband excitations of a multiple quantum wells system in the electrical dipole gauge. Here we are only interested in the interaction with the transitions of a single quantum well electrons, and we will neglect losses deriving from other kind of absorp-tions by the material constituting the quantum well itself, i.e. the permittivity of the quantum well materials will be taken as real.

In the most general picture we must take into account that the number of involved de-grees of freedom for electrons is very large, corresponding to the number of states k in the quantum well subbands.

1_{Notice that the interaction between the 2DEG of a quantum well and an electromagnetic mode}

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1.4. The Hopfield model for intersubband polaritons. 16 For sufficiently low doping level, the quantum well is populated up to the Fermi wavevec-tor, hence if we are interested in mid-infrared intersubband transitions we can con-sider only k wavevectors in the first subband, since ~ω12∼ 100 meV ' 4kBT. On the

other hand, if we are interested in terahertz transitions, then_~ω12∼ 10 meV and we

cannot neglect the contribute of the upper subbands.

An accurate analysis of the system requires the inclusion of the dynamic electron-electron Coulomb interactions, which in an isolated quantum well leads to a blue-shift of the resonance frequency with respect to the value that one would calculate by con-sidering only the static Coulomb interaction. This effect is also known as

depolariza-tion shift. The depolarized resonance frequency becomes

ˆ ω12=

q ω2

12+ ω2P, (1.31)

where the frequencyωP corresponds to the plasma frequency of a two dimensional

electron gas distributed in a quantum well of effective thickness Leff(defined in Eq.1.17)

and permittivityεw:

ωP=

s

e2∆n

ε0εwm∗Leff. (1.32)

Here m∗_{is the effective mass of the well’s electrons and}_{∆n is the population difference}

between the first and second subband of the quantum well.

With the appropriate transformations, the electronic Hamiltonian He can be

formu-lated by using a new set of ladder operators, so that the complete Hamiltonian reads ˆ H =X k ~ωˆ12b†_kbk+ X k ~ωc(k)(a_k†ak+ 1 2) + i X k ~Ωk(a_k†− a−k)(b†_−k+ bk). (1.33) The operators b_k† and bk are, respectively, the creation and annihilation operator of collective electronic excitations that we can call intersubband plasmons, each of them carrying an energy quantum equal to ~ωˆ12. Hence, the dynamics of the low-energy

excitations of the real Fermi gas can be described as the dynamics of an ensemble of bosons.

As we can notice from the interaction part of the Hamiltonian, the coupling coefficient of the interaction is the frequencyΩk, which depends on the in-plane wavevector and is given by Ωk= ωP 2 s ωc(k) ˆ ω12 f0fw. (1.34)

The coupling coefficient is proportional toωP/p ˆω12, thus it has a non-linear

depen-dence onωP because of the formula of the depolarization shift Eq.1.31. The constant f0is the oscillator strength of the transition 1 → 2, proportional to the matrix element

of the electric dipole along z:

f0=2m ∗_ω

12

~ 〈ψ1| z |ψ2〉 (1.35)

The quantity fwis one of the most important parameters of the entire system, because

it is related to the field profile and it is crucial in determining the strength of the light-matter interaction [15]: fw= Leff Lb+ Lw R QWεQW|Ez|2 R ε(r ,z)|E|2 (1.36)

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1.4. The Hopfield model for intersubband polaritons. 17 the quantitiesεQW, Lband Lwwill be defined in the following. The fwparameter is

de-fined as the overlap factor between the quantum well and the cavity mode. Its physical meaning is quite simple: the overlap factor can be seen as a sort of confinement fac-tor for the radiation within the quantum well: it suggests us that if we want to have the strongest interaction with the cavity field, our quantum well must be placed in a region in which the intensity profile of the electric field is high, with the largest Ezcomponent

possible (see an example in Fig.1.7).

Figure 1.7: Field profile fk(z) of a surface plasmon mode launched by a metallic grating. Since the

highest intensity takes place at the surface, the quantum well should be placed close to the surface.

The Hamiltonian in Eq.1.33can be diagonalized by the Hopfield-Bogoljubov transfor-mation

pk= xkak+ yka†_−k+ zkbk+ tkb_−k† , (1.37) with the normalization condition |xk|2−|yk|2+|zk|2−|tk|2= 1. The operator pkand its hermitian adjoint p†_kare the annihilation and creation operators of a quantum of oscil-lation that we define as a new quasi-particle called intersubband polariton. From the dependency of pk on the photon and ISB plasmon ladder operators, we can conclude that the intersubband polaritons are a mixed state between photons and intersubband plasmons.

The diagonalization of the Hamiltonian Eq.1.33brings to a biquadratic secular equa-tion, the intersubband polaritons dispersion relaequa-tion, which can be solved analytically:

(ω2_{− ˆ}_ω2

12)(ω2− ω2c(k)) = f0fwω2Pω2c(k) (1.38)

The two real and positive solutions ω = ωUP(k) and ω = ωLP(k) are the frequencies

of the upper and lower polariton modes. As we can see from the dispersion rela-tion represented in Fig.1.8, there is an anticrossing between the coupled microcavity mode dispersion and the depolarization shifted resonance frequency: the result is the appearance of the characteristic polariton branches, upper-polaritonωUP and

lower-polaritonωLP, with two horizontal asymptotes at ˆω12 and ˇω12=

q ω2

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1.4. The Hopfield model for intersubband polaritons. 18 and an oblique asymptote forω = ωc(k). Since ˇω12< ˆω12, the polariton dispersion

fea-tures a gap with the horizontal asymptotes frequencies as edges; therefore the prop-agation of light at gap energies is now forbidden. This fenomenon is a consequence of the destructive interference between the microcavity electromagnetic field and the local field created by the collective electronic oscillations [14].

Figure 1.8: Dispersion relation of intersubband polaritons normalized at the bare intersubband

transi-tionω12. Figure adapted from Ref. [14].

The minimum splitting between the polariton branches occurs whenω12= ωc, i.e. at

the resonance. and is defined as the Rabi frequency of the system: 2ΩR= ωP

q

f0fw (1.39)

The coupling between light and material excitation is stronger as the ratioΩR/ω12

be-comes higher, with the consequence that the energy gap between the polariton branches increases. Sinceω12 is fixed, we can act only on the Rabi frequency by varying either

the plasma frequencyωP or the overlap factor fw.

• ωP→ the quantity that can be experimentally controlled is the population

differ-ence∆n. Hence, ωP can be varied either through the temperature of the system

or by applying a gate voltage.

• fw → the overlap factor between the photonic mode and the intersubband

cur-rent can be varied setting the quantum well position along z, so as to tune the field amplitude felt by the 2DEG.

For very large interaction strength, the effects of the quadratic and antiresonant terms in the interaction Hamiltonian of Eq.1.33become important, and the polariton frequenciesωUPandωLP become nonlinear as a function ofΩR(and thenωP). Fig.1.9

shows this nonlinear behaviour forΩR/ω12 that approaches 1: this limit is known as

the Ultrastrong coupling regime.

In order to maximise the light-matter coupling we must enhance the overlap of the cavity mode with the quantum well. Hence, we should find a cavity mode with very high field amplitude confined in a small thickness∆z of the order of the quantum well

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1.4. The Hopfield model for intersubband polaritons. 19

Figure 1.9: Normalized polariton frequency forωc(k) = ω12as a function of the normalized Rabi

fre-quency, showing nonlinear behaviour when the interaction increases. Figure from Ref. [14].

effective thickness, i.e. an electromagnetic mode with the highest confinement possi-ble. An elegant solution is to rely on Surface Plasmon Polariton modes in graphene: as we will see in the next chapter, graphene is a 2D material that allows radiation confine-ment via the generation of electromagnetic waves that “live" at the graphene surface, in a region of the order of tens of nanometres.

1.4.1 Effective dielectric function

For the understanding of many physical aspects, like the presence of virtual photons in the polariton ground state [16], we should treat the coupling between light and in-tersubband transitions from a quantum point of view, giving a complete description of how the states couple to the external world. In order to perform this description one should adopt the master equation formalism as seen in the previous section. But if we want to analyse the polaritonic dispersion and lineshapes in the first instance, we can adopt a classical linear response model in order to describe the interaction of the quantum well medium with the classical electromagnetic field.

The optical response of a quantum well is governed by intrasubband and intersub-band transitions, and it behaves like an anisotropic dispersive dielectric medium with a Lorentz-like permittivity for intersubband absorption and a Drude-like permittivity for the intrasubband one. Taking into account that only the z component of the elec-tric field can excite intersubband transitions, we can write the quantum well dielecelec-tric function as              εx,y(ω) = εQW h 1 − ω 2 P ω(ω + iγq) i εz(ω) = εQW h 1 − f0ε 2 ε2 w ω2 P ˆ ω2 12− ω2− 2i ωγ12 i (1.40)

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1.5. Few-electron strong coupling and photon blockade 20 withεQW= (Lbεb+Lwεw)/(Lb+Lw) an average permittivity of the quantum well

mate-rials of thickness Lb,wand permittivityεb,w2,γqandγ12the intrasubband and

intersub-band scattering rates, f0andωP defined in Eq.1.35and Eq.1.32. εz has an imaginary

part that is almost a Lorentzian with FWHM equal to 2γ12, and peak amplitude

pro-portional to∆n/γ12.

The definition of the quantum well effective permittivity will be essential when we will perform numerical simulations of the systems briefly mentioned in the Introduc-tion. The result is fully compatible with the quantum treatment given in Section 1.4, as shown in Ref. [14].

1.5 Few-electron strong coupling and photon blockade

The last question we want to answer is: what happens if we decrease the number of electrons inside the microcavity from NeÀ 1 to Ne= 1?

We have seen that in the presence of a large number of electrons, Coulomb interac-tions produce a blue-shift of the resonance frequency called depolarization shift and the many-body electronic states are equispaced in energy by the amount _~ωˆ12: the

electronic ensemble behaves like a collective plasmon mode with bosonic character and a harmonic spectrum.

As shown in [17], when we reduce the number of electrons toward unity, e.g. by shrink-ing the cavity volume, the energy separation between the electronic many-body ex-cited states is reduced, thus creating a strong anharmonicity in the spectrum (Fig.1.10). Moreover, the absorption frequency of the system is strongly dependent on the num-ber of excitations. In this limit the system can no longer be described by bosons but rather as a collection of two-level systems coupled through interparticle interactions, leading to the Jaynes-Cummings model for Ne= 1.

In the few-electron regime, also defined as fermionic regime, the antiresonant terms of the Hamiltonian induce coupling of the ground state to the higher-order dressed states (e.g. with two and three photons), leading to the emergence of new peaks in the transmission spectra.

In conventional optical materials, the nonlinearity at light powers corresponding to single photons is negligibly weak. On the contrary, when the number of electrons in a microcavity is reduced to some units, strong nonlinearity takes place at the level of individual photons. Indeed, if we take the expression of the Jaynes-Cummings Rabi frequencies Eq.1.29we will find a nonlinear trend when the number of photons N is increased one-by-one:

ΩN= Ω0

p

N (1.41)

Therefore, light-matter interaction in the few-electron regime can produce strong non-linear photon-photon interactions mediated by a single intersubband excitation. The most extreme situation occurs when there is only one electron in the first sub-band which couples to radiation. Since the strong light-matter coupling leads to a 2p2ΩR splitting of the two-photon manifold of the system, if we send two photons

of frequencyω12to the cavity, they will be out of resonance with the polaritonic state

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1.5. Few-electron strong coupling and photon blockade 21

Figure 1.10: (a) Frequencies of the first four many-body states of a 2DEG (label j ), above the ground

stateωG, as a function of the total number of electrons Ne. (b) Absorption spectrum of a resonator

strongly coupled to a single quantum well electron. Hereω21= 1 THz. Figure from Ref. [17].

|2, −〉 (see Fig.1.11). On the other hand, the energy of a single photon will be quasi-resonant with the polaritonic states |1,−〉 and |1,+〉: as a result, only one photon can enter the microcavity and pass through it, while other photons will be “blocked" by the strongly coupled system. This phenomenon, in which the presence of a photon inside the cavity blocks the transmission of a second photon, is referred to as photon

blockade. As a result of this effect, the radiation that comes out from the system shows

a sub-Poissonian statistics with second-order correlation g(2)(0) < 1, resulting in an overall photon antibunching [18].

The nonlinear interactions among photons could be exploited in coupled cavities ar-rays: if we consider an array of cavities in the photon blockade regime, each containing a single two-level system resonant to the cavity mode, the effective repulsion between photons results in a strong interacting system of photons in the mesoscopic or macro-scopic scale, that could be useful for a large number of applications [6].

The observation of the photon blockade mechanism was first made by Birnbaum et al. [18] in the light transmitted by an optical cavity strongly coupled with one cesium atom’s transition. Other systems have been implemented, including quantum dot in a photonic crystal defect cavity by Faraon et al. [19] and by Reinhard et al. [20].

A nanocavity can be created in a photonic crystal by introducing a localised defect in the structure periodicity, so that the light cannot propagate outside the defect area. Photonic crystal nanocavities can reach extremely small volumes leading to large cou-pling constants when coupled to a quantum dot, but have Q factors that limit the strong coupling interaction. Quantum dots are the best solution for the realisation of a two-level system because of their delta-like density of states, but it is difficult to control the properties of individual quantum dots and to place them at desired locations. Cur-rent experiments pick a quantum dot from an ensemble and then fabricate a photonic crystal tailored to this dot around it. Such an approach cannot be used obviously for producing a whole array of sufficient quality.

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1.6. Conclusions 22

Figure 1.11: Jaynes-Cummings ladder diagram showing the photon blockade mechanism. Figure

adapted from Ref. [18].

In this thesis work we will show that a promising solution can be represented by a graphene surface plasmon mode coupled to a single GaAs/AlGaAs quantum well: the extremely tight confinement of light produced by graphene nano-gratings can be ex-ploited to couple the surface plasmon mode with a single intersubband transition; fur-thermore, graphene gratings can be tailored to produce large area arrays with cavity mode frequencies fully controlled by the graphene properties.

We will study the interaction of a single quantum well and a cavity mode by using the Hopfield model, which provides a simple tool (the effective dielectric function of Eq.1.40) to simulate the behaviour of the system with classical electromagnetism. In this way we are making an error in the definition of the transition frequency of the or-der ofω12/ ˆω12, and we are neglecting the higher-order dressed states arising from the

Jaynes-Cummings anharmonic ladder. The few-electron limit could be obtained at a later stage by introducing a complete quantum description of the system.

1.6 Conclusions

In summary, we have given a brief description of the mechanisms used to confine pho-tons and electrons by exploiting optical microcavities and quantum wells. In particu-lar, photons are confined in electromagnetic modes labelled with a certain in-plane wavevector and a certain frequency, while the confinement of electrons in a 2D geom-etry gives rise to further quantization of the electronic energies, with the generation of 2D bands called “subbands". Furthermore we have shown an important result: inter-subband transitions, which are transitions between two different quantum well sub-bands, can be excited only with TM electromagnetic modes.

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1.6. Conclusions 23 We have provided a theoretical description of the light-matter interaction with the for-malism of the Jaynes-Cummings model for a two-level system coupled to an electro-magnetic mode, and adapted this approach to show the noteworthy results for a cavity mode coupled to intersubband transitions in a quantum well. The coupling produces a mix of photonic and electronic states, called “polaritons", which have a particular dispersion relation characterised by the presence of two distinct branches (result of the level anticrossing) separated by an energy gap.

We have shown that the minimum splitting between the branches is the Rabi frequency ΩR, that is the discriminant of how strong the coupling is: in order to increase the

cou-pling strength an increment of the overlap factor fwis necessary, leading to the request

of an electromagnetic mode highly confined in the quantum well region.

Finally, we have put in evidence the main state-of-the-art experiments for the realiza-tion of strongly coupled light-matter systems in the photon blockade regime.

In the next chapter we will show that graphene gratings support surface plasmon modes in the mid-infrared and THz spectral range, giving the theoretical basis for the study of strongly coupled systems with quantum wells. Since different configurations can be adopted to produce a surface plasmon mode with the desired resonance frequency, we also demonstrate the high optical confinement of graphene surface plasmons, that could make possible to achieve light-matter coupling in the few-electron regime.

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2

Graphene Surface Plasmon

Polaritons

Graphene is a two-dimensional allotrope of carbon, with carbon atoms arranged in a honeycomb lattice constituted by hexagons. Despite its widespread presence in nature as a single sheet of graphite, isolated mono-layer graphene was discovered only in 2004 by Andre Geim and Konstantin Novoselov by mechanical exfoliation of bulk graphite. Since then, graphene has attracted tremendous interest due to its remarkable proper-ties, such as two-dimensional geometry, high carrier mobility, large thermal conduc-tivity and high transparency in the visible range. These properties make this material very attractive for applications in optoelectronics [21].

2.1 Optical properties of graphene

Most of the features of graphene are essentially governed by the special linear energy-momentum dispersion relation at the corners of the Brillouin zone.

The electronic configuration of carbon is 1s2_2s2_2p2_{. Within the framework of the}

tight-binding method, the band wavefunctions originate from the s, px and pyorbitals, are

even under reflection along the z axis and give rise to three bonding σ-bands and thee antibondingσ-bands, separated by an energy gap of ≈ 5.6eV (continuous lines in Fig.2.1); electrons in the filledσ-bands do not participate to the charge transport. On the other hand, the pzorbitals are odd under reflection along z and their

hybridiza-tion produces twoπ-bands that are degenerate at the corners of the first Brillouin zone. Of relevant importance for the physics of graphene are the two vertices K and K’ of the first Brillouin zone, called Dirac points. The energy dispersion of electrons close to the Dirac points is linear with momentum and does not depend on their mass.

The relevant physics of graphene takes place in the vicinity of the Dirac points, hence we can represent the energy dispersion of graphene as two opposite energy cones (see Fig.2.1). The cone with energy larger than that of the Dirac point is the electron con-duction band, while the other is the hole valence band. In the special case of charge neutrality, the graphene Fermi energy lies exactly at the Dirac point. In this situation, even zero-energy excitations can promote electrons to the totally empty conduction band.

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2.1. Optical properties of graphene 26

(a)

(b)

Figure 2.1: (a) Graphene band structure diagram. (b) Scheme of the conical band structure of graphene

with zero-energy band gap at the Dirac points. Figures from Ref. [22] and [25].

2.1.1 Optical conductivity of graphene

In the context of graphene optics and plasmonics, the optical conductivity of graphene plays a central role since it contains all the physics that governs the electromagnetic in-teraction with an external radiation. An analytical expression of the graphene dynam-ical conductivity can be derived within the framework of the linear response theory using Kubo’s formula [23].

Assuming no external magnetic field, the local conductivity is a diagonal tensor isotropic on the plane. The analytical expression of graphene conductivity can be split into the sum of two distinct contributions, one describing the intraband transitions of electrons within the conduction or valence band, and one describing the interband transitions between the valence and the conduction band:

σ2D(ω) = σi nt r a(ω) + σi nt er(ω) (2.1)

Depending on the spectral range under consideration, one can dominate on the other. For highly doped graphene the chemical potential is |µ| À kBT, approximately equal to

the Fermi energy Ef. Hence, in the following description we will make no distinction

between the Fermi energy and the chemical potential in graphene.

At temperature T, both contributions of the optical conductivity can be written in the form [24]: σi nt r a(ω,Ef,T) = i e2kBT π~2(ω + 2iΓ) h E_f kBT+ 2 ln(e −_kBTE f + 1) i σi nt er(ω,Ef,T) = i e2 4π~2ln h2|E_f| −~(ω + 2iΓ) 2|Ef| +~(ω + 2iΓ) i (2.2)

In the above expressions kBis the Boltzmann constant. Besides the Fermi energy, the

other important parameter for graphene plasmons physics is Γ, which is the carrier

scattering rate (or relaxation rate). The expression of the graphene Fermi energy is

determined by the sheet carrier density n of the 2D graphene layer [25]: Ef =~vf

p

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2.1. Optical properties of graphene 27

Figure 2.2: Inhibition of interband transitions in graphene due to Pauli blocking when the Fermi energy

is shifted (center and right panel) with respect to the undoped level (left panel).

where vF ≈ 106m/s is the Fermi velocity. Typical values of Ef range from 0.05eV to

0.6eV, with carrier densities from ∼ 1011cm−2_{to ∼ 10}14_cm−2_[₂₆_].

The scattering rateΓ takes into account every mechanism that could make electrons relax to the ground state, such as scattering with impurity or phonons and electron-electron interaction. Its calculation from first principles is highly complex. However, it can be regarded as a phenomenological parameter. The scattering rate is related to the relaxation timeτ of electrons by 2Γ = τ−1_{, with}

τ =µmEf ev2

F

(2.4) whereµm is the graphene carrier mobility.

Tuning the carriers density in graphene, by chemical doping or with a gate bias, shifts the graphene Fermi energy and changes the relative weight of the intraband and in-terband contributions. Indeed, when~ω < 2|Ef| the contribution of interband

transi-tions to the dynamic conductivity is negligible because of Pauli blocking; in this regime the intraband term of Eq.2.2dominates in the THz and mid-infrared spectral region (see Fig.2.2). It is important to notice that in the limit kbT ¿ |Ef| this term assumes a

Drude-like expression.

On the other hand, when_~ω > 2|Ef| the interband transitions can occur: here the

in-traband transitions become negligible, and the optical conductivity assumes a con-stant valueσ0called the “universal" conductance of graphene:

σ0= e 2

4_~ (2.5)

Since σ0= 6.09 · 10−5S, this conductivity corresponds to an absorbance A = 4_cπσ0 '

2.29%., hence graphene is practically transparent to visible light.

The conductivity expression of graphene in Eq.2.2is plotted in Fig.2.3from the THz to the near infrared spectral region. It shows the real and imaginary part of the 2D graphene conductivity normalized with respect toσ0for different values of the

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2.2. Surface plasmon polaritons 28 the optical conductivity in the mid-infrared range decreases. Furthermore, we can no-tice that for frequencies higher than 2Ef it reaches the limitσ0and, as the scattering

time increases, the function in the vicinity of 2Ef becomes even sharper, approaching

the Heaviside function behaviour in the limitΓ → 0.

Figure 2.3: Real part (left) and imaginary part (right) of the infrared optical conductivity of graphene

(normalized with respect toσ0) for different values of the scattering time. The Fermi energy is set to

Ef = 0.2 eV' 48.4 THz.

2.2 Surface plasmon polaritons

In the first chapter we have shown that collective excitations of electrons in a quan-tum well can be treated as a system of bosons called intersubband plasmons. Follow-ing the same principle, surface plasmons are the quantum collective oscillations of the surface charge density of a metal. Similar to cavity polaritons, these surface oscil-lations are coupled to electromagnetic waves producing a mixed electronic-photonic state that is called surface plasmon polariton (SPP).

From an electromagnetic point of view, SPPs are solutions of Maxwell’s equations lo-calized at the surface between two layered media, i.e. solutions with electromagnetic field that propagates along the surface directions and exponentially decays to zero into both half-spaces as the distance from the interface increases (see Fig.2.4). By match-ing the fields at the interface usmatch-ing the appropriate boundary conditions, we can find that SPP modes are generated if the permittivity of each media fulfills the conditions:

ε1(ω) · ε2(ω) < 0 (2.6)

ε1(ω) + ε2(ω) < 0 (2.7)

These conditions mean that one of the dielectric functions must be negative with an absolute value that exceeds that of the other. The dielectric function of noble metals is large and negative, thus the dielectric-metal system is commonly used for the con-finement of light in SPP modes. For a detailed analysis of surface plasmon polaritons at a dielectric-metal interface we will refer to Ref. [10]. Here we only want to show that the SPP dispersion relation, i.e. the relation between the in-plane wavevector and the frequencyω, reads

k2= ε1ε2 ε1+ ε2

ω2

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2.3. Graphene Surface Plasmon Polaritons 29 If we assume a Drude-like expression for one of the dielectric functions, Eq.2.8 indi-cates that the SPP wavevector is much larger than the incident light one: we cannot excite a SPP mode simply by illuminating a metal, since the momentum would not be conserved. Several methods are used to compensate the momentum mismatch be-tween an incident light and SPP, but the most common one is to exploit a metal grating, which provides an effective momentum due to the periodic grating structure [10]. The following section is devoted to describe a semi-classical model of surface plas-mons in a graphene sheet surrounded by two different dielectric media. We will also calculate the dispersion relation for SPPs in graphene, which will be a crucial tool for our purpose because it will guide us on how to engineer the system so as to have the desired resonance.

Figure 2.4: Schematic representation of a surface plasmon mode propagating at the interface between

a dielectric and a metal. Figure from Ref. [26]

2.3 Graphene Surface Plasmon Polaritons

Graphene surface plasmon polaritons, also abbreviated in GSP, are SPP modes that ex-ist in a region very close to a graphene layer. Owing to the two dimensional nature of the collective excitation, surface plasmons excited in graphene are much more con-fined than in systems with noble metals, giving to this material an important role in plasmonics.

The main purpose of this section is to describe the physics that governs GSPs in sim-ple systems. In particular we will obtain the dispersion relation of GSP in monolayer graphene embedded in dielectric media, following the formalism adopted in [26]. Let us consider a system constituted by a single graphene sheet at z = 0 embedded be-tween two semi-infinite dielectric media having real dielectric constantsε1 for z > 0

andε2for z < 0, as shown in Fig.2.5. We look for homogeneous solutions of Maxwell’s

equations that are localized at the interface, i.e. solutions having electromagnetic fields that exponentially decay with increasing distance from the graphene sheet in both half-spaces. Here we will assume that the electromagnetic field is p-polarized in both half-spaces (TM mode) because this condition is necessary for SPPs existence.

From a mathematical point of view, we have to solve the Helmholtz equation ∇2E +ω

2

c2ε(r ,ω)E(r ,ω) = 0 (2.9)

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2.3. Graphene Surface Plasmon Polaritons 30

Figure 2.5: Illustration of a single sheet of graphene embedded between two different dielectric media.

Figure from Ref. [26].

in the form

Ej= (Ej,xˆx + Ej,zˆz)ei kj·re−i ωt= (Ej,xˆx + Ej,zˆz)ei kxe−qj|z|e−i ωt (2.10)

Bj = Bj,yei kxe−qj|z|e−i ωtˆy (2.11)

where the index j = 1,2 refers to the dielectric medium with dielectric constant εj. In

the above expression we have defined qj,= ikj,z. Moreover, the parallel component k

of the wavevector (in-plane wavevector) must be conserved, giving us the important relation

k2+ q2_j = εjk2₀ (2.12)

with k0 = ω/c the vacuum wavevector. Since both half-spaces do not contain any

source, the displacement fields must satisfy the equation ∇∇∇ · D = 0, or equivalently

qj· E = 0:

kEj,x+ qjEj,z= 0 (2.13)

Now it is time to explain when the graphene layer starts to be relevant. When we im-pose the boundary conditions at z = 0 to match the fields at the interface, we must consider that the graphene conductivity will produce a surface current density propor-tional toσ, as expected by Ohm’s law. With this assumption we can write the boundary conditions as

E1,x= E2,x (2.14)

B1,y− B2,y= µ0Jx(x) = µ0σxxE2,x, (2.15)

with fields evaluated at z = 0. It is important to notice that the graphene conductivity enters only in the boundary condition: it contains all the electromagnetic properties of graphene. Substituting the fields into the boundary conditions 2.15 we will finally find the GSP dispersion relation

ε1 q1(k,ω)+ ε2 q2(k,ω)= −i σ(ω) ωε0 (2.16)

The solution of this equation will give us the dispersion relationω(k) of graphene TM surface plasmon polaritons. Real solutions can be found only when ℑm{σ(ω)} > 0, i.e.

when~ω < 2Ef. Unfortunately, Eq.2.16does not have an analytical solution and must

be solved numerically. However, it can be possible to obtain an approximate closed-form expression after certain simplifications.

Graphene plasmonic gratings for few-electron strong coupling experiments

DEPARTMENT OF PHYSICS

Master’s degree in Physics