Terahertz plasmonic instabilities in graphene:A hydrodynamical description

Pedro Afonso Cosme e Silva

Thesis to obtain the Master of Science Degree in

Engineering Physics

Supervisor: Prof. Hugo Fernando Santos Terças

Examination Committee

Chairperson: Prof. João Pedro Saraiva BizarroSupervisor: Prof. Hugo Fernando Santos Terças

Members of the Committee: Prof. Mário Gonçalo Mestre Veríssimo SilveirinhaProf. Nuno Miguel Machado Reis Peres

June 2019

ii

Dedicated to my parents and siblings.

iii

iv

Acknowledgments

Firstly and foremostly I would like to express my gratitude to my supervisor Prof. Hugo Tercas for all the

support, guidance and availability through the course of this thesis. It has been a great pleasure to work

with such a gifted physicist.

I would also like to thank Prof. J. Tito Mendonca, and all the members of Laboratory for Quantum

Plasmas at IPFN, for all the invaluable discussions and suggestions.

v

vi

Resumo1

As fontes de radiacao terahertz (THz) estimuladas electricamente sao extremamente interessantes

dada a sua versatilidade e potencialidade no que concerne a miniaturizacao, abrindo o caminho para

tecnologia THz de circuito integrado. Neste trabalho e demonstrado que explorando a plasmonica em

grafeno e possıvel gerar radiacao THz na gama 0.5 THz < ω/2π < 10 THz, podendo, em particular,

gerar quer radiacao coerente, quer frequency comb. A configuracao estudada consiste num transıstor

de efeito de campo de grafeno, sujeito a condicoes fronteira assimetricas, onde a radiacao se origina

devido a uma instabilidade plasmonica que pode ser controlada pela injeccao de corrente contınua.

Partindo desta configuracao varios esquemas sao tambem apresentados, de entre os quais um mecan-

ismo para a amplificacao da instabilidade no caso de substratos com permitividade electrica variada, o

que permite ultrapassar eventuais limitacoes associadas a implementacao experimental.

Os tratamentos analıticos e numericos da hidrodinamica do plasma em grafeno sao explanados,

mostrando que a instabilidade pode ser controlada experimentalmente pela tensao aplicada na gate,

bem como pela corrente injectada. Os calculos efectuados, assim como as simulacoes numericas,

indicam ainda que a radiacao emitida exibe graus de coerencia, temporal e espacial, (g(1)(τ) & 0.6

g(r1, r2) & 0.8) e irradiancia (107 Wm−2) apreciaveis. Este facto leva a que estes modelos sejam can-

didatos importantes para uma futura fonte de laser THz. Ademais uma configuracao electromecanica,

cujo objectivo visa acoplar a instabilidade plasmonica as oscilacoes flexurais (modos fora do plano), e

tambem analisada, deixando antever um outro mecanismo de instabilidade a estudar no porvir.

Palavras-chave: plasmonica em grafeno; instabilidade plasmonica; radiacao THz; transistor

de grafeno1Este texto foi intencionalmente escrito de acordo com a ortografia anterior ao AO90.

vii

viii

Abstract

Electrically injected terahertz (THz) radiation sources are extremely appealing given their versatility and

miniaturisation potential, opening the venue for integrated-circuit THz technology. In this work, it is

shown that with the exploitation of graphene plasmonics is possible to generate THz radiation in the

range 0.5 THz < ω/2π < 10 THz, being able, in particular, to generate both coherent radiation and

frequency combs. The setup studied consists of a graphene field-effect transistor subject to asymmetric

boundary conditions, with the radiation originating from a plasmonic instability that can be controlled by

direct current injection.

Furthermore, several additional variation designs are also brought forth, among which a mechanism

for the instability amplification is advanced for the case of substrates with varying electric permittivity,

which allows to overcome eventual limitations associated with the experimental implementation. A com-

bined analytic and numerical analysis of the graphene plasma hydrodynamics is put forward, showing

that the instability can be experimentally controlled by the applied gate voltage and the injected current.

The performed calculations and numerical simulations indicate that the emitted THz radiation exhibits

appreciable temporal and spatial coherence (g(1)(τ) & 0.6 g(r1, r2) & 0.8) and an output radiant emit-

tance (107 Wm−2). This makes these schemes appealing candidates for a future graphene-base THz

laser source. Moreover an electro-mechanical setup, aiming to couple the plasmonic instability with flex-

ural oscillations (out-of-plane modes), is also analysed hinting to yet other instability mechanism to be

studied in future works.

Keywords: graphene plasmonics; plasmonic instability; THz radiation; graphene transistor

ix

x

Contents

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii

1 Introduction 1

1.1 Graphene theoretical overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Graphene structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Band theory and density of states . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.3 Graphene bulk electrical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 State-of-the-art of Graphene THz emission . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.1 The TeraHertz problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.2 Graphene transistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Graphene Hydrodynamic Model 11

2.1 Graphene Fermi liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Density and Fermi level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.2 Fermi Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.3 Effective mass of the carriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.4 Gated graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Electronic fluid model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Adimensionalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 System analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4.1 Hyperbolicity and nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4.2 Free dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

xi

3 Plasmonic instability in gated graphene 19

3.1 Frequency and instability growth rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1.1 Limit Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Rankine-Hugoniot conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3 The impact of sound speed variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3.1 Shoaling effect simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4 Pulsed stimulation & frequency combs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4.1 Frequency comb simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.5 Cloosed-loop system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Radiation Emission 29

4.1 Reconstructed Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.1.1 Reaction to radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2 Far-field radiated power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2.1 Dissipated power by Joule effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3 Antenna attributes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3.1 Radiation pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3.2 Radiation efficiency and quality factor . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.4 Coherence properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.5 Simulated power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5 Numerical Methods 37

5.1 Hydrodynamic simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.1.1 Courant–Friedrichs–Lewy condition . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.1.2 Numerical oscillation suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.2 EM fields and antenna simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

6 Suspended Graphene 43

6.1 Kirchhoff-Love membrane coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6.2 Membrane driven excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6.3 Plasmon-flexuron hybridisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6.4 System parameters – Elasticity vs. stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.5 Kirchhoff–Love membrane simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

7 Conclusions 51

7.1 Achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Bibliography 52

A Derivation of Euler equations 63

xii

B Code flowcharts 65

C Specimina of fluid simulation results 69

D Submitted paper 73

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xiv

List of Tables

1.1 Number of publications with the keyword “2D materials” over the past decade (until May

2019). From Web of Science database . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

3.1 Increment of growth rate in the presence of a negative gradient of local sound speed. . . 23

6.1 Typical mechanical values for single layer graphene. . . . . . . . . . . . . . . . . . . . . . 47

6.2 Estimated values of ∆/v0 for polymeric membranes considering a thickness of H =

150nm, length L = 1µm and drift velocity v0 = 0.3vF . The data of Young’s modulus,

density and Poisson ratio were retrieved from [111]. . . . . . . . . . . . . . . . . . . . . . 47

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xvi

List of Figures

1.1 Graphene structure and Brillouin zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Monolayer graphene band structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Density of states of monolayer graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1 Phase diagram of carriers fluid in graphene . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Typical values of S/v0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Shallow waters dispersion relation in graphene . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1 Graphene field-effect transistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Numerical results on DS frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3 Limit cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4 Influence of gradient in S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.5 Shoaling effect on spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.6 Pulsed excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.7 THz Frequency comb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.8 Closed-loop realisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.9 Closed Loop density evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.10 Closed Loop limit cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.1 Radiation pattern of the graphene layer emitter. . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2 Degrees of coherence of emitted radiation. . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.3 Poynting vector magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.4 Radiated and Joule power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.1 Ricthmyer method stencil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.2 CFL criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.3 Moving average smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6.1 Dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6.2 Hopfield coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

B.1 Flowchart for fluid simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

B.2 Flowchart for radiation simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

xvii

B.3 Flowchart of the analysis routines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

C.1 Density evolution at drain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

C.2 Velocity evolution at drain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

C.3 Current evolution at drain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

C.4 Velocity evolution at source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

C.5 Density distribution on channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

C.6 Velocity distribution on channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

C.7 Current distribution on channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

xviii

Nomenclature

Greek symbols

γ Imaginary part of frequency; strain force.

ε Electric permittivity.

θ Polar angle.

Θ( ) Heaviside theta step function.

µ Chemical potential; carrier mobility.

µ0 Magnetic permeability of free space.

ρ Mass density.

σ Conductivity.

σ0 Graphene universal conductivity.

φ Azimuthal angle.

ω Frequency.

Ω Drive frequency.

ωp Plasma frequency.

Modifiers

〈a〉 Time average; ensemble average.

a Arithmetic mean.

a Operator; Fourier transform of a.

a Time derivative of a.

Roman symbols

a Graphene inter-carbon distance.

a0 Graphene lattice constant.

xix

B Magnetic field vector.

c Speed of light in vacuum.

Cg Gate capacitance per area.

Cq Quantum capacitance per area.

D( ) Density of states

D Bending stiffness.

E Electric field vector.

e Electron charge

EF Fermi level.

F Force.

Fj Complete Fermi-Dirac integral.

F [ ] Fourier transform.

g(1)(τ) First degree of temporal coherence.

g(r1, r2) Degree of spatial coherence between points r1 and r2.

gs Spin degeneracy.

gv Valley degeneracy.

~ Reduced Planck constant

H Hamiltonian.

H[ ] Hilbert transform.

Im( ) Imaginary part.

j Current density.

k Wave number.

k,q Wave vector.

kB Boltzmann constant.

K( ) Complete elliptic integral of the first kind.

L Lagrangian density.

L Length of graphene layer.

m? Effective mass

xx

me Electron mass.

n Numeric density.

O( ) Asymptotic order of error.

p Momentum; electric dipole moment.

P Pressure.

Prad Radiated power.

PΩ Dissipated Joule power

Q Quality factor.

Qi Ideal quality factor.

R Reflection coefficient.

Re( ) Real part.

S Poynting vector.

∆t Time discretisation.

T Absolute temperature.

t Graphene hopping integral; time.

U Electric potential difference.

r Position vector.

v Velocity vector.

v Scalar velocity.

vF Fermi velocity

vF Phase velocity

W Width of graphene layer.

∆x Space discretisation.

Z Grand canonical partition function.

Subscripts

i, j Computational indices for spatial step.

x, y, z Cartesian coordinates indices.

Superscripts

xxi

′ Derivative.

∗ Complex conjugate; adimensional quantity

† Conjugate transpose (Hermitian conjugate).

T Transpose.

k Computational index for time step.

xxii

Glossary

2DEG Two dimensional electron gas.

BZ1 First Brillouin zone.

CFL Courant–Friedrichs–Lewy.

DOS Density of states.

DS Dyakonov–Shur.

EM Electromagnetic.

FEL Free electron laser.

FET Field effect transistor.

FWHM Full width at half maximum.

GFET Graphene channel field effect transistor.

H.c. Hermitian conjugate.

HEMT High electron mobility transistor.

HPF High-pass filter.

mid-IR mid-infrared.

pGe p-type Germanium.

Q-factor Quality factor.

QCL Quantum cascade laser.

THL Terahertz laser.

THz Terahertz.

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xxiv

Chapter 1

Introduction

The study of bidimensional materials is one of the most emergent fields in both theoretical and exper-

imental physics, and engineering. Despite being apparently doomed to failure by thermal instability,

as predicted by Landau [1] and Mermin [2], 2D crystals are now not only proven to be stable at room

temperatures but are also invaluable to modern nanotechnology. Its usage has been opening new and

exciting possibilities in electronics, opto-electronics, photonics and nanomechanics [3–5]. Indeed, nowa-

days a plethora of bidimensional materials are studied and produced, as can be inferred from Tab.1.1 ,

ranging from graphene (with all its varieties) to hexagonal boron nitride and even more exotic materials

such as transition metal dichalcogenide monolayers or even MXenes.

Table 1.1: Number of publications with the keyword “2D materials” over the pastdecade (until May 2019). From Web of Science database

Year 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019

No. Publ. 3176 3011 3163 3833 4324 4923 5896 6929 8711 9073 2707

In the area of 2D materials no other has been so prominent in applications and theoretical investiga-

tion as graphene. The first theoretical work describing what would be know as graphene dates back to

1947 by Wallace [6], as a model of graphite in the limit of a single layer. Despite some previous incipient

observations of thin graphite flakes [7, 8], graphene was experimentally obtained quite recently, in 2004,

by Geim and Novoselov resorting to mechanical exfoliation of graphite [9]. This work warranted the 2010

Nobel Prize in Physics. Since then graphene has been used in a multitude of applications, among which

graphene channel transistors being one of the most significant and flexible, being used to exploit both

the tunable electrical and optical properties.

Regarding the interaction of graphene with electromagnetic radiation, one of the attributes that dis-

tinguishes it from metals and traditional semiconductors is that the spectral range of plasmons lie in

the TeraHertz (THz) and mid-infrared (mid-IR) regions, making it a candidate for generation, detection

and manipulation of such radiation. Given that THz radiation is in high demand, but its generation has

several drawbacks, such potential is a key feature to explore.

The present thesis will, therefore, focus on the plasmonic excitations on a graphene channel tran-

1

sistor, considering a hydrodynamic model for the description and simulation of the electronic transport.

Moreover, we exploit the possibility of resorting to such device for the generation of coherent THz fre-

quency combs, arising from the aforesaid plasmonic instability. The latter can be excited via the injection

of an electric current, thus forgoing the necessity of optical pumping. This opens the possibility to the

development of an all-electric, low-consumption stimulated emitters, capable of operating at room tem-

perature. Suggesting this scheme to be a competitive solution towards THz laser light with integrated-

circuit technology. In order to achieve such goal let we start by describing the properties and theoretical

background of graphene.

1.1 Graphene theoretical overview

1.1.1 Graphene structure

Graphene is a planar allotrope of carbon, where the sp2 hybridisation of the carbon atoms leads to an

hexagonal honeycomb lattice. Each carbon atom shares a covalent bound with three other in plane, as

shown in Fig.1.1, with the distance between atoms a0 ≈ 0.142 nm, and hence, leaving a free π orbital

out of plane. Since the honeycomb lattice is not a Bravais lattice, the primitive cell possesses two points

and, therefore, the lattice can be decomposed in two sub-lattices, A and B, where the first neighbours

of any given point belong to the other sub-lattice. The primitive vectors for such structure can be chosen

as

a1 = a(1, 0)

and a2 = a(− 1/2,

√3/2)

(1.1)

where a = a0

√3 ≈ 0.246 nm is the graphene lattice constant. The vectors connecting first neighbours

are given by

δ1 = a

(0,

√3

3

), δ2 = a

(1

2,−√

3

6

)and δ3 = a

(−1

2,−√

3

6

)(1.2)

and then the reciprocal lattice vectors defined by ai · bj = 2πδi,j are given by

b1 =2π

a

(1, 1/√

3)

and b2 =2π

a

(0, 2/√

3)

(1.3)

in this way the reciprocal lattice is also a hexagonal one. Hence, the constructions of the first Brillouin

zone yields the symmetry points

Γ = (0, 0), M =2π

a(0, 1/

√3), K =

2π

a(1/3, 1/

√3), K ′ =

2π

a(2/3, 0) (1.4)

for the centre of the Brillouin zone, centre of the edge, and vertices, respectively, as illustrated in Fig.1.1.

1.1.2 Band theory and density of states

The band structure of graphene can be obtained analytically by the tight binding model, as the π orbitals

decay rapidly and do not overlap in a significant way. In that context the Hamiltonian of the system can

2

a1

a2

δ1

δ2δ3

ky

kx

b2

b1

Γ

M K

K ′

Figure 1.1: Left panel – Graphene hexagonal structure, unit cell containing twolattice points and Bravais primitive vectors a1 and a2. Each lattice point is connectedto its first neighbours by one of the δ vectors. Right panel – Reciprocal latticevectors b1 and b2 with first Brillouin (shaded) zone with the symmetry points: Γ –Centre; M – Edge midpoint; K – Vertex

be approximated by the interaction between first neighbours HTB =∑R,R′ |R〉 〈R| H |R′〉 〈R′| ,with |R〉

representing the Wannier states of the π orbital at positions R. For each point of one of the sub-lattices,

for instance A, the nearest neighbours are in the second sub-lattice, B, as mentioned, and the sites are

connected by the vectors δ` (cf. Fig.1.1). Therefore the Hamiltonian can be written in the form

HTB =∑

R, `

|A,R〉 〈A,R| H |B,R+ δ`〉︸ ︷︷ ︸t

〈B,R+ δ`| (1.5)

with the hopping integral determined to be t ≈ −2.7 eV [10, 11]. Therefore, in the tight-binding approxi-

mation to first neighbours the system Hamiltonian is, in the Wannier basis,

HTB = t∑

R

3∑

`=1

|A,R〉 〈B,R+ δ`|+ H. c. (1.6)

with H.c. standing for Hermitian-conjugate. Using the fact that the Wannier states can be written in the

Bloch representation

|A,R〉 ≡ 1√N

∑

k∈BZ1

e−ik·R |A,k〉 (1.7)

the Hamiltonian in (1.6) is then further simplified to

HTB = t∑

k∈BZ1

|A,k〉 〈B,k|φ(k) + H. c. (1.8)

with φ(k) =∑3`=1 e

ik·δl = eikya0 + eikxa0√

3/2e−ikya0/2 + e−ikxa0√

3/2e−ikya0/2. Introducing the vector

Φk =(|A,k〉 , |B,k〉

)T the Hamiltonian in Eq. (1.8) can be condensed in HTB = Φ†kHBlochΦk with

HBloch =

0 tφ(k)

tφ(k)∗ 0

(1.9)

3

having the energy eigenvalues given by:

Ek = ±t|φ(k)| = ±

√√√√3 + 2 cos(√

3kxa0

)+ 4 cos

(√3

2kxa0

)cos

(3

2kya0

)(1.10)

which defines the first conduction and valence bands. The first important remark is that these bands

are symmetric, evidently if the tight-binding approximation had included more than the first neighbours,

the bands would be slightly different and lose such symmetry. However, for low energy systems such

correction would not bring considerable modifications to the bands obtained in the first neighbours ap-

proximation.

At the vertices of the Brillouin zone, K and K ′, the energy vanishes, EK = 0, leading to a gapless

band transition. Moreover, in the case of undoped graphene each carbon atom contributes with one

electron and within this model each atom has only one orbital available with two possible states, spin up

or down, therefore in the absence of doping graphene is at half filling: the lowest band is filled and the

upper one unoccupied. The Fermi surface is then restricted to the vertices of the Brillouin zone as the

Fermi energy is exactly at the crossing of the bands – the Dirac point. For this reasons the low energy

regimes will be determined by the behaviour around points K and K ′. In the vicinity of such points

φ(k) = φ(K + q) =∑δ e

i(K+q)·δ which expanded in first order q is φ(q) ≈ 32a0(−qx + iqy) so the Bloch

hamiltonian around K and K ′ is approximated by

HBloch =

0 tφ(k)

tφ(k)∗ 0

≈ 3

2a0t

0 −qx + iqy

−qx − iqy 0

(1.11)

having the linear spectrum

E± = ±3

2a0t|q| = ±vF~|q|. (1.12)

Where the last term defines the Fermi velocity vF = 3a0t/2~ ≈ c/300.

−3

−2

−1

0

1

2

3

Γ M K Γ

0

K

E/t

π∗π

−2π/a 0 2π/akx

−2π/a

0

2π/a

ky

0

0.5

1

1.5

2

2.5

3

E/tΓ

M K

Figure 1.2: Left panel – Monolayer graphene lower (π) and upper (π∗) bands alongthe Γ−M −K−Γ path in the Brillouin zone in the tight binding approximation to firstneighbours. Notice the linear approximation around Dirac point K (dashed lines onthe inset) and the stationary point at M . Right panel – Density plot of upper bandin the k-space.

The presence of a linear energy momentum relation, as well as the fact that the hamiltonian from

4

(1.11) can be written, in terms of the momentum p = ~k and the Pauli matrices σ, in the form

H = −vFσ · p (1.13)

as a massless Dirac hamiltonian, shows that near the K points the electrons behave as having zero rest

mass and obey the Dirac equation, with the Fermi velocity instead of the usual light speed, hence the

denomination Dirac points to the vertices K and K ′. The eigenstates of (1.13) associated to the energy

eigenvalues (1.12) are ψ±(r) = χ±(q)eiq·r with the spinor

χ±(q) =1√2

e−iθ(q)/2

±eiθ(q)/2

(1.14)

around point K, and where θ(q) = arctan(qx/qy) is the angle in the momentum space. Note that the

helicity operator

h =1

2σ · p|p| (1.15)

commutes with the Hamiltonian (and thus shares the same eigenstates) having two eigenstates hψ±(r) =

± 12ψ±(r) i.e. the pseudo-spin is either along or against the momentum and therefore near the Dirac

points electrons have well-defined chirality. This fact restricts the exchange of momentum in scattering

processes, and consequently the conductivity of graphene is, as will be seen, extremely high.

From the structure of the bands (1.10) it is possible to obtain the density of states [12, 13] in the form

D(E) =gsgv|E|π2t2

1√Z0

K(√

Z1

Z0

), (1.16)

where gs = 2 and gv = 2 are the spin and valley degeneracy, respectively, and K is the complete elliptic

integral of the first kind, while Z0 and Z1 are given by

Z0 =

(1 +

∣∣Et

∣∣)2 − [(E/t)2−1]2

4 , 0 ≤ |E| ≤ t

4∣∣Et

∣∣ , t < |E| ≤ 3t

(1.17)

and

Z1 =

4∣∣Et

∣∣ , 0 ≤ |E| ≤ t(1 +

∣∣Et

∣∣)2 − [(E/t)2−1]2

4 , t < |E| ≤ 3t

(1.18)

the profile of this expression can te seen in Fig. 1.3. Such function is cumbersome to handle analytically,

however, around the Dirac points, i.e. in the linear regime of the energy bands, the density of states can

be approximated by

D(E) =gsgv|E|2π~2v2

F

. (1.19)

this approximation breaks down, nonetheless, due to the stationary point in M that creates two Van

Hove singularities at E = ±t averting the validity of linear models for such energies. Thus, the models

and simulations in this thesis will refer to energies well below such values and restrain EF 3 eV.

5

0

0.5

1

1.5

2

2.5

−3t −2t −1t 0t 1t 2t 3t

D(E

)

E

Figure 1.3: Density of states from (1.16) showing the Van Hove singularities forE = ±t and the approximately linear region around the Dirac point (E = 0).

1.1.3 Graphene bulk electrical properties

The fact that the density of states is zero at the Dirac point leads to the impression that undoped mono-

layer graphene conductivity should be null. On the contrary, the experimental results show a minimum

non zero conductivity at the neutrality point, denominated universal conductivity [14–17], given by

σ0 =e2

4~. (1.20)

Beyond the neutrality point (EF 6= 0) optical conductivity in graphene can be decomposed in terms of

intra/inter band conductivity as σ = σintra+σinter, where each term can be written, at the low temperature

limit as functions of the frequency ω and relaxation γ, as [10]

σintra(ω) =σ0

π

4EF~γ − i~ω (1.21)

σinter(ω) = σ0

(Θ(~ω − 2EF ) +

i

πlog

∣∣∣∣~ω − 2EF~ω + 2EF

∣∣∣∣). (1.22)

For frequencies in the optical to infrared range the inter-band processes dominate the total conductivity.

Since useful applications of graphene should bound the Fermi level well below van Hoove singularities

and at the same time surpass thermal excitations 0.03 eV E 3 eV a crude estimation for the typical

frequencies as f ≈ E/h points to a frequency range in the TeraHertz regime 7.25 THz . f . 725 THz.

Furthermore, the two dimensional plasma frequency [18]

ω2p =

2πe2vFn

ε0~(1.23)

for a typical carrier density of n = 1012 cm−2 yields fp ≈ 200 THz and corroborates such spectral range

of interest in the infrared are of the spectrum.

6

1.2 State-of-the-art of Graphene THz emission

1.2.1 The TeraHertz problem

Terahertz radiation has numerous applications ranging from sensing and imaging to metrology and spec-

troscopy [19, 20]; in particular terahertz laser (THL) and THz frequency combs play a prominent role

within such technology [21, 22].

In the field of spectral analysis THz radiation can assay low frequency movements such as rotational

and vibrational motion of molecules, particularly in gas phase. As well as identify unknown specimen by

spectral signature.

Concerning its relevance to imaging and sensing technology, the interaction of THz radiation with

matter can be sorted in three distinct cases [23]: interaction with water, metals and dielectrics. As the

water molecule is highly polar THz radiation is strongly attenuated within it. Quantitative analysis of such

attenuation can provide information for medical imaging while being non-ionising and biologically safe,

in contrast to other medical imaging techniques. In the case of metals as a consequence of their high

conductivity, and consequently plasma frequency, THz radiation is reflected almost completely. While on

the other hand nonmetallic and nonpolar materials, such as plastics, paper and fibbers, are transparent

to THz contrary to what happens in the visible range where such materials tend to be opaque. In this

way THz radiation is well suited for non destructive sensing in security applications.

Despite the aforementioned relevance of TeraHertz radiation and THz laser their generation still

face significant difficulties given the low energy of the photons involved that excludes most of atomic

transitions. Present production of such radiation is restricted to four main technologies [24, 23]: gas

lasers [25, 26]; free electron lasers [27–29]; quantum cascade lasers [30, 31] and p-type Germanium

lasers [23], each presenting some caveats.

Gas lasers, usually operating below 10 THz, consist in a long waveguide filled with a gas at low

pressure of molecules with permanent dipole moment, viz. CH3COH , NH3, CH3F or CH2F2, that allows

them to couple with EM radiation by dipolar interaction. In such scheme the rotational transitions of the

gas molecules are optically pumped by another laser, frequently a CO2 laser, and such process displays

a low conversion efficiency.

In the case of free electron lasers (FEL) the lasing medium is a beam of relativistic electrons forced

to oscillate in the optical cavity by the presence of an array of alternating magnets – the undulator. As it

is well known in these conditions the electrons emit broad synchrotron radiation, which wavelength can

be tailored by the magnetic field strength, periodicity of the magnets and beam velocity. Furthermore,

as the emitted radiation is trapped in the optical cavity it interacts back with the electronic beam by

ponderomotive force which further accelerates or decelerates some of the electrons. This feedback

leads to the bunching of the electrons in groups that behave collectively, a vital characteristic for the

coherent properties of such radiation. Although the high power output of FEL they are bulky, scaling

to some meters, and require a high degree of maintenance and technology, thus not being suitable for

low-power applications.

Regarding quantum cascade lasers (QCL), they are composed by a periodic train of thin semicon-

7

ductor layers that creates a series of electronic sub-bands. Thus, when decaying, the electrons undergo

a first intersub-band transition followed by intrasub-band transition enabled by quantum tunneling to the

next sub-band, then this process is repeated successively. Originating several photons throughout this

cascading sequence. However QCL suffer from the inherent problem of the low energy THz photon as

thermal excitations are strong enough to undermine the electron configuration and the population inver-

sion. Therefore low temperature, tipically bellow 180K, is a necessary requirement for QCL operation.

Finally, p-type Germanium (pGe) rely in transitions of holes between Landau levels in Germanium

crystals. Heavy-holes are accelerated by a electric field to an excited state where, at cryogenic tempera-

tures, decay spontaneously by phonon emission to a excited Landau level whose down transition emit in

the THz range. Notwithstanding the important fact that pGe lasers operate on completely electrical base

the necessary temperatures below 40K restricts the usability of such technology in integrated circuit

devices.

Given the above limitations is clear that no satisfactory solutions for benchtop production, nor de-

tection, of THz exist. However the recent progress in graphene based transistors paves the way to the

possibility for all electrical miniaturised devices for low power THz radiation emission and detection.

1.2.2 Graphene transistors

The effect of an external field on graphene layers was one of the first main questions addressed right

after the experimental discovery of graphene [9] since then the concept of graphene transistors and

graphene electronics have proposed as possible substitute for modern silicon based electronics [32].

A graphene field effect transistor1 (GFET) is composed by a layer of graphene acting as a channel

placed between two metallic contacts, the source and the drain, in addition, parallel to the graphene one

metallic gate, or more in some configurations, is responsible for the imposition of a transverse electric

field. The graphene is typically laid over a dielectric base or even encapsulated in a dielectric [37, 38],

although suspended graphene transistors are also possible [39–41] and present some advantages as

will be addressed later. Mono-layer graphene renders rather unusual transistors in the sense that, given

the absence of a band gap, the conductance between source and drain can not be truly switched off,

making such apparatus unsuitable for usage as transistor for digital applications. Yet analog systems can

still exploit the high conductance properties and the usual current control by the gate voltage[32, 37, 35].

Therefore, the research on GEFT devices and its applications has grown in the last years.

Beyond the traditional use of GFET as analog transistors the plasmonic properties of graphene have

drawn attention to its use as both detectors and emitters of THz radiation [42–48], in an effort to unravel

the issue of room temperature THz. Several techniques have been brought forward, some of them

relying in optical pumping [27–29] and so, demanding an auxiliary excitation laser what counters the

purpose of an independent and totally electrical solution. More recently THz emission from dual gate

graphene FET has been reported [49, 50] due to electron/hole recombination in a p-i-n junction.

1The following discussion will be focused on large area graphene transistors of single layer, although other technologies existwith bilayer graphene or nano-ribbons among others [33–36].

8

1.3 Motivation

More than twenty years ago, well before the advent of graphene, least of all GFET, M. Dyakonov and

M. Shur predicted the occurrence of a plasmonic instability in two dimensional electron gas, [51–53],

in the conditions that the electronic flow is subjec to an electrical potential from a parallel gate and

driven by a continuous current. Such instability would be tunnable in the THz range and electronic

requirements relatively easy to implement in a circuit board. However such instability faces the obstacle

of resistivity in traditional transistors that do not allow the establishment of the instability waves, owing

to the inherently low growth rate compared to the scattering processes, as will be discussed in detail

further ahead. This fact led to the necessity of resorting to high-electron-mobility transistors (HEMT) in

order to experimentally observe the Dyakonov-Shur instability [54–57].

Since the graphene displays such a high mobility in the order of 3×103 cm2V−1s−1 [58, 59] and

even more in the case of suspended mono layer graphene, up to 104 cm2V−1s−1 at room temperature or

even 5×105 cm2V−1s−1 at low temperature [60, 61], in striking contrast with other materials for instance

silicon which has mobility below 1.4×103 cm2V−1s−1. It seems natural to focus effort in the research

and experimentation of plasmonic instabilities, such as the Dyakonov-Shur instability, in the context of

graphene and GFET, where the low mobility problem can be avoided.

1.4 Objectives

In the light of the antecedent motivation the main goal of the work developed in the course of the present

thesis is to study and characterise the existence, as well as the behaviour, of plasmonic instabilities in a

graphene channel field-effect transistor, resorting to a hydrodynamic model for the electronic conduction

in mono-layer graphene. For that purpose a a numerical code capable of solving the fluid equations was

developed.

In particular, it was made an effort to find and study instabilities similar to the Dyakonov–Shur in-

stability, and that as it dwell in the THz range. Given the technological potential of such region of the

spectrum.

Moreover, it is also the aim of this thesis to consider the radiation emitted by the aforementioned

plasmonic instabilities, and asses the feasibility of a system akin to a graphene field-effect transistor as

the core emitter of a lasing device.

1.5 Thesis Outline

After the antecedent concise introduction the following chapters of this thesis are organised in the fol-

lowing manner.

Chapter two, with title “Graphene Hydrodynamic Model” details the semi-classical model used de-

scribe the electronic properties of graphene, its regions of validity and assumptions, §2.1§2.2, as well

as first implications, §2.4.

9

The third chapter, “Plasmonic instability in 2DEG”, deals with the instabilities in graphene in the light

of the fluid model presented before, analysing its properties, §3.1, §3.2 and proposing several methods

of exploiting such instabilities, §3.3§3.4§3.5.

In chapter four, named “Radiation Emission”, the means of studying the emitted electromagnetic

radiation from the GFET with the necessary simplifications are discussed, §4.1§4.2, and the discovered

properties, such as energy flux or directivity, of such radiation are unveiled, §4.3.

After this, chapter five, denominated “Numerical Simulations”, brings the discussion, as the name

implies, to the numerical methods employed and developed to fulfil this thesis objectives. This chapter

is further split in two sections regarding the electronic fluid simulations, §5.1, and those concerning

radiation, §5.2.

Chapter six, “Suspended Graphene”, discusses the mechanical properties of suspended monolayer

graphene and the effects of out of plane oscillations in the theoretical model, §6.1, as well as the pos-

sibility of the emergence of new instabilities and behaviours due to hybridisation of mechanical and

electronic modes, §6.3.

Lastly, the final conclusions of the work that conduced to the elaboration of this thesis are patent in

the seventh, and endsome, chapter.

10

Chapter 2

Graphene Hydrodynamic Model

2.1 Graphene Fermi liquid

In order to describe, and simulate, the collective behaviour of conduction electrons, or holes, on a large1

graphene sheet a fluid model can be used. The validity of such models derive from the weak interaction

of carriers and, subsequently the large mean free path of carriers scattering. In the present chapter

a hydrodynamic model for conduction electrons in mono-layer graphene for the gated case will be put

forward from which the simulations and main results of this thesis will be drawn out limiting the analysis

of carriers in graphene to those whose momentum and energy lie in the vicinity of the Dirac points.

As fermions, even though massless, both carriers (either electrons or holes) are subject to Fermi-

Dirac distribution, where the two limit cases of degenerate µ kBT and non degenerate µ kBT play

a crucial role in the properties of the system. In graphene such distinction is particularly critical as near

the neutrality point the system is in a Dirac fluid phase [62, 63]. The Dirac fluid and Fermi liquid phases

at a given density are bounded by a critical temperature2 [63]

T ∗(n) =~vFkB

√π|n|

[1 +

e2

8εvF~log

2

A0|n0|

], (2.1)

that can be seen in Fig. 2.1.

This thesis will focus on the Fermi liquid regime, since it is more relevant for technological applica-

tions given that at room temperature and for reasonable charge densities the system is well under the

conditions of such regime.

2.1.1 Density and Fermi level

As stated before, the undoped graphene has the lowest band completely full, the conduction electrons

of doped graphene would then fill the upper band and the new Fermi surfaces are, in first approximation,

circumferences around the Dirac point. The dependency of carrier density with the Fermi level can be

1Large in the sense that it’s dimensions are much larger than the size of the graphene hexagonal cell a = 0.246 nm.2Where A0 is the area of the graphene cell

11

0

200

400

600

800

1000

1200

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

T[K

]

n [1012m−2]

Dirac fluid

Electron F. liquidHole F. liquid

Figure 2.1: Phase diagram of carriers fluid in graphene separated by the criticaltemperature T ∗(n). At room temperature, around 300K, and usual densities, circa1016m−2,the system is unquestionably in the Fermi liquid regime. Based on [63].

found, as usual, from the Fermi-Dirac distribution f(E) = 1/(e(E−µ)/kBT + 1) and the density of states

from the linearised DOS (1.19) as n(EF ) =∫∞

0D(E)f(E)dE. In the case of electron carriers, it reads:

n(E) =gsgv

2π~2v2F

∫ ∞

0

E

e(E−µ)/kBTdE =

gsgv2π~2v2

F

(kBT )2 F1

(µ

kBT

). (2.2)

where F1(·) is the complete Fermi–Dirac integral [64, 1]. The Sommerfeld expansion for the chemical

potential [65] in graphene, using the linearised DOS from (1.19), gives

µ(T ) = EF

(1− π2

6

(kBT

EF

)2

+ · · ·). (2.3)

At room temperature it is easy to guarantee the condition kBT ≈ 0.03 eV EF < t ≈ 3 eV, that is,

to set the Fermi level well bellow the Van Hove singularities while being in the degenerate limit, thus

the chemical potential and the Fermi energy are essentially equal. Taking it in account Eq.(2.2) is well

approximated by

n(EF ) =E2F

π~2v2F

. (2.4)

2.1.2 Fermi Pressure

Besides the description of density, the dependency of pressure with the energy, and density itself, is

key to the development of a consistent hydrodynamic model of electrons as they are subject to high

degeneracy pressure.

The pressure in the 2D Fermi-Dirac system can be obtained from the grand partition function Z =

1 + e(EF−E)/kBT with

P = gsgvkBT

∫d2p

(2π~)2Z (2.5)

which leads to

P =gsgv(kBT )3

2π~2v2F

F2

(EFkBT

), (2.6)

where the dependency of the chemical potential with the temperature was, once again discarded as

12

T TF . Furthermore, such relation can be expanded for the Fermi liquid regime EF kBT

P (EF ) =E3F

3π(~vF )2

[1 + π2

(kBT

EF

)2

+ . . .

](2.7)

and recurring to EF = ~vF√πn from (2.4) the leading term in (2.7) simplifies, in accordance with [66] ,

to

P (n) =~vF3π

(πn) 3

2 (2.8)

2.1.3 Effective mass of the carriers

The fact that the electrons in graphene behave as massless fermions pose the major difficulty for the

development of hydrodynamic models with explicit dependency on the mass. In fact electrons in mono-

layer graphene not only have zero rest mass but also the usual definition on solid state physics to the

effective inertial mass tensor [67]

m−1ij =

1

~2

∂2E

∂ki∂kj(2.9)

diverges since the bands are conical around the Dirac points. It is then imperious to use yet another

definition for the mass. A naive approach would dictate to simply define mass as nominal Drude mass

[66, 68], the quotient between momentum and velocity,

m? =~kFvF

=~√πn0

vF. (2.10)

In fact, such definition coincides with the cyclotron mass [13, 67], resorting to the area A in k-space

enclosed by the orbit at a given energy

m? =~2

2π

∂A

∂E

∣∣∣∣E=EF

=~kFvF

. (2.11)

The latter definition will be used throughout this work, even though it ignores the dependency with

density/energy variations, and in future developments such dependency is indeed a factor to have in

consideration as a second order correction. With such definition the effective mass is restrained to be

2.7 keV/c2 m? 270 keV/c2, fairly bellow the free electron mass with me = 511 keV/c2.

2.1.4 Gated graphene

Considering a monolayer graphene sheet in a field effect transistor (FET) structure, i.e. placed between

two metallic contacts, source and drain, and subject to a gate (cf. Fig. 3.1), the electric force in the

electrons is dominated by the imposed potential that screens the Coulomb interaction between them.

Therefore, the acceleration of electrons is solely due to the imposed potential. The applied bias potential

U has the contribution of both the geometric capacitance from the gate Cg and the quantum capacitance

Cq [58, 69–71] as

U = en

(1

Cg+

1

Cq

). (2.12)

13

The quantum capacitance Cq reflects change in potential with the occupancy of band structure of the

material and is defined as

Cq = e2D(E) = 2e2√πn/π~vF . (2.13)

As for the geometric capacitance in the approximation of parallel plates approximation for the gate and

graphene layer the geometric capacitance is give by Cg = ε/d0, where d0 is the separation and ε the

medium permittivity. However, for carrier density n & 1012 cm−2 quantum capacity dominates Cq Cg

and the potential can be approximated as U = end0/ε. Thus, the acceleration on the electron fluid is

g = −e |∇U |m?

= − e2d

m?ε

∂n

∂r(2.14)

2.2 Electronic fluid model

The Euler fluid equations can be derived3 from the Boltzmann equation [66, 72, 73]

∂f

∂t+ v ·∇f + g · ∂f

∂v=

(∂f

∂t

)

coll

, (2.15)

where f(r,v, t) is the distribution function defined over the phase space, g ≡ ∂v∂t the acceleration and(

∂f∂t

)coll

the collisional term. Considering the absence of collisions, or equivalently assuming that the

mean free path is much larger that the dimensions of the systems which is easily complied given the high

mobility of graphene carriers, and defining the number density by∫f(r,v) dv ≡ n(r, t) and the mean,

or bulk, velocity as∫vif(r,v) dv ≡ 〈vi〉n(r, t) the zero moment of (2.15) produces the usual continuity

equation: ∫∂f

∂t+ v · ∂f

∂r+ g · ∂f

∂vdv = 0 ⇐⇒ ∂n

∂t+

∂

∂r· [n〈v〉] = 0 (2.16)

and the first moment yields the momentum equation:

∫ [∂f

∂t+ vj

∂f

∂rj+ gj

∂f

∂vj

]v dv = 0 ⇐⇒ ∂〈v〉

∂t+ 〈v〉 · ∂〈v〉

∂r= g − 1

m?n

∂P

∂r(2.17)

where P is the fluid pressure. The present model will consider the one-dimensional laminar flow, dis-

regarding the transverse motion of the fluid and the finite-size effects due to the channel width, i.e.,

assuming that the boundary layer due to viscosity along the edges of graphene is much smaller than the

width of the graphene layer. Thus, for the sake of simplicity the bulk velocity and position vector will be

written as 〈v〉 ≡ v and r ≡ x respectively and the motion is governed by the Euler equations [66, 74, 68]

in the form:∂n

∂t+

∂

∂xnv = 0 and (2.18a)

∂v

∂t+ v

∂v

∂x= g − 1

m?n

∂P

∂x. (2.18b)

3See appendix A for a detailed derivation

14

The pressure term in (2.18) can be simplified with the explicit form of the pressure (2.8) and Drude

mass (2.10)1

m?n

∂P

∂x=

v2F

2√nn0

∂n

∂x=v2F

n0

∂

∂x

√n ≈ v2

F

2n0· ∂n∂x. (2.19)

The linearisation in the last step simplifies the problem analysis and numerical implementation. More-

over, the performed simulations shown no significant variation with or without such simplification, giving

support to its usage.

By combining Eqs.(2.18), (2.19) and (2.14), we finally obtain the physical model describing the elec-

tron flow in a graphene FET

∂n

∂t+

∂

∂x(nv) = 0

∂v

∂t+ v

∂v

∂x+

(e2dvFε~√πn0

+v2F

2n0

)∂n

∂x= 0

. (2.20)

In the following section a detailed analysis of such mathematical model is performed from which some

relevant properties are drawn.

2.3 Adimensionalisation

The equations (2.20) can be adimensionalised redefining the variables in terms of the channel length L

and characteristic velocity and density v0 and n0, that can be taken as the equilibrium or steady state

values, writing

x∗ ≡ x/L t∗ ≡ tv0/L v∗ ≡ v/v0 n∗ ≡ n/n0 (2.21)

so the system becomes

∂n∗

∂t∗+

∂

∂x(n∗v∗) = 0

∂v∗

∂t∗+ v∗

∂v∗

∂x∗+S′2

v20

∂n∗

∂x∗+v2F

2v20

∂n∗

∂x∗= 0

⇐⇒

∂n∗

∂t∗+

∂

∂x(n∗v∗) = 0

∂v∗

∂t∗+ v∗

∂v∗

∂x∗+S2

v20

∂n∗

∂x∗

(2.22)

where S′ ≡√

e2d0n0

m?ε and S2 = S′2 + v2F /2. Henceforward the superscripts indicating adimensional

quantities will be dropped for simplicity.

The parameter S has units of a velocity and can be interpreted as a sound speed of the electron

fluid4. Moreover, the ratio S/v0 will play a crucial role determining the properties of the hydrodynamic

system, similarly to the Froude number in fluid dynamics. For typical parameters of a graphene FET,

S/v0 scales up to a few tens as can be seen in Fig.2.2, a fact that, as will be seen, will be essential to

counteract Landau damping.

4Not to be confused with the intrinsic sound speed of phonons in graphene

15

12080

40

20

10

0.01 0.1 1 10n0 (×1012 cm−2)

0.1

1

10

d0

(×10−

6cm

)20

40

60

80

100

120

140

160

180

S/v 0

Figure 2.2: Values of S/v0 (considering v0 = 0.1vF = 105 ms−1) for typical values ofthe gated graphene, electronic density n0 and distance between gate and graphened0.

2.4 System analysis

2.4.1 Hyperbolicity and nonlinearity

The system of equations (2.20) can be written in the quasilinear matricial form

∂u

∂t+ A(u)

∂u

∂x= 0 ⇐⇒ ∂

∂t

nv

+

v n

S2/n0 v

∂

∂x

nv

= 0, (2.23)

Then, flux jacobian matrix A(u) has two distinct real eigenvalues given by

λ1 = v − S√n/n0 and λ2 = v + S

√n/n0 (2.24)

with the associated linearly independent right eigenvectors

R1 =[−√nn0, S

]T and R2 =[√nn0, S

]T. (2.25)

Therefore, the system is said to be strictly hyperbolic for any S 6= 0, once it has two real and distinct

eigenvalues. Moreover, as ∇λ · R = 32S 6= 0 it is also said to be genuinely nonlinear [75]. Such

properties made difficult for such system to be simulated, or for that matter analytically solved, but guar-

antees, nonetheless, the saturation of the instabilities, a key aspect to take advantage of self stimulated

instabilities maintaining them contained and blocking any potential hazardous outcome to the underlying

physical system.

From the eigenvalues (2.24) the characteristic curves are given by

dx

dt= v ± S

√n/n0 (2.26)

along which the Riemann invariants are r± = v± 2S√n/n0. Even taking only in account the equilibrium

16

term in n the characteristic curves from (2.26) cross each other. Leading, in consequence, to either

shock or rarefaction waves, this fact will be of preponderant importance and will define not only dynamic

of the electronic fluid but have implications in the radiation properties, as will be seen later on.

2.4.2 Free dispersion relation

Expanding the fields, n and v, around their equilibrium values, in the form a = a0 + a(x, t), one can

derive from (2.20) the linearised system in Fourier space

∂n

∂t+ n0

∂v

∂x+ v0

∂n

∂x= 0

∂v

∂t+ v0

∂v

∂x+S2

n0

∂n

∂x= 0

F−→

(ω − kv0)n− kn0v = 0

− S2

n0n+ (ω − kv0)v = 0

(2.27)

which leads to the dispersion relation

ω = k(v0 ± S) (2.28)

akin to a shallow–waters linear dispersion with a Doppler shift and from which the parameter S is ev-

idently interpreted as the sound velocity of the electron fluid. Remarkably such dispersion relation is

linear, as a result of the potential from the gate screening the direct interaction of electrons, whereas

a 2DEG dispersion relation would exhibit ω ∝√k [76, 77]. Furthermore, in order to the plasmons to

persevere their velocity must surpass the Fermi velocity and abides ω < 2EF /~− vF k in order to avoid

the Landau damping in the regions of electron-hole inter-band and intra-band excitations [10] (regions

I and II, respectively, in Fig.2.3). Fortunately, those constrains are easily complied, as is evident by the

values of S/v0 from Fig. 2.2 and setting v0 ∼ vF /10.

0

0.5

1

1.5

2

0 0.5 1 1.5 2

~ω/E

F

k/kF

I

II

Figure 2.3: The dispersion relation (2.28) with v0 = 0.1vF and S = 2vF . In regionsI and II any plasmon will be critically attenuated due to Landau damping. For com-parison a typical dispersion for surface plasmons in ungated monolayer grapheneis also displayed [78, 10](dashed blue).

17

18

Chapter 3

Plasmonic instability in gated

graphene

The hydrodynamic model (2.22) describes an instability – the Dyakonov-Shur instability [51, 53] – under

the boundary conditions of fixed density at source and fixed current density at the drain:

n(x = 0) = n0, n(x = L)v(x = L) = n0v0. (3.1)

This instability was first studied in the context of regular 2DEG in high mobility field effect transistors but

can be easily implemented in GFET with the characteristics of Fig.3.1.

GFET

ID

U

y

z

d0

x

L

W

gate

drain

source

graphene

Figure 3.1: Left panel – Circuit implementation of a GFET under the conditionsof DS instability. Right panel – Schematic diagram for gated graphene transistor.Suspended graphene monolayer placed between two metallic contacts, source anddrain, and subject to an electrostatic potential U = end0/ε imposed by a gate,placed at a distance d0 from the sheet. For the DS instability to occur a fixedcurrent ID is injected at the drain while maintaining the electronic density of thesource constant.

The DS instability rises from the multiple reflection of the plasma waves at the boundaries whilst

being amplified by the driven current at the drain with a reflection coefficient

R =

∣∣∣∣S + v0

S − v0

∣∣∣∣ . (3.2)

Such reflection of the incoming density waves is induced by the condition at source, while the imposed

19

drain current guarantees the necessary Doppler shift for the upstream current to interfere positively with

the downstream current.

3.1 Frequency and instability growth rate

To obtain the frequency of the DS instability let the density be defined by the sum of two travelling waves,

with momenta k± = ω/(v0 ± S), as given by (2.28) and arbitrary amplitudes A±, plus the steady state

average n0,

n(x) = n0 + n1(x) = n0 +A+eik+x +A−e

ik−x (3.3)

besides, recurring to the current density j = nv the continuity equation (2.18a) in Fourier space forces

j1(x) =ω

kn1(x) =

ω

k+A+e

ik+x +ω

k−A−e

ik−x; (3.4)

from the boundary condition at source n(0) = 0 the amplitudes for the density have the simple relation

A+ +A− = 0, (3.5)

while the imposition of constant current at drain, j(L) = j0, leads to

ω

k+A+e

ik+L +ω

k−A−e

ik−L = 0 (3.6)

which combined with the previous relations yields

k+

k−= ei(k+−k−)L ⇐⇒ ω = i

S2 − v20

2SLlog

v0 + S

v0 − S(3.7)

sorting out this expression in the real and imaginary parts of the complex frequency ω = ωr + iγ returns:

[51, 79, 54]

ωr =|S2 − v2

0 |2LS

πl (3.8a)

γ =S2 − v2

0

2LSlog

∣∣∣∣S + v0

S − v0

∣∣∣∣ , (3.8b)

where l is an integer number. One can observe that the instability occurs for the subsonic regime where

S > v0 as the imaginary part of the frequency, γ, becomes positive. In fact, in the supersonic regime

the Doppler shift would prevent the opposite travelling waves to interact and there would be no place for

amplification along the channel.

3.1.1 Limit Cycle

Actually, the evolution of the system under the Dyakonov-Shur conditions undergoes a global bifurcation

as the parameter S surpasses v0. The critical point (n0, v0) ceases to be a stable attraction point and

20

converts itself to a stable limit cycle in the velocity–density plane, as will be self-evident with Fig.3.3.

Although the complete analysis of such behaviour is complex in the infinite-dimension dynamical system

some qualitative properties can be extracted from the jump conditions of fluid theory.

3.1.2 Numerical results

As expected, under the boundary conditions (3.1), prescribed by Dyakov and Shur [51], the simulated

fluid spontaneously evolve to a cycle of shock fronts and rarefaction, cf. Appendix C, reflected at the

simulation domain endpoints, leading to the temporal oscillation at each point. Regarding such time

evolution, similar profiles of those found on literature [79, 80] were obtained and showing significantly

less oscillation as the discontinuities than other works, for instance that from Satou and Narahara [81],

the time evolution for the velocity and density can also be observed at Appendix C.

The simulated hydrodynamic instability shows a great accordance with its theoretical properties, in

particular frequency of the first mode and growth rate for a wide range of S/v0 values as can be observed

in Fig. 3.2 where can also be pointed that the growth rate slightly exceeds what was expected.

02468

10

5 501 10 100

ωr/2π(T

Hz)

S/v0

v0/L=0.2THz

v0/L=0.4THz

v0/L=0.6THz

v0/L=0.8THz

0

0.2

0.4

0.6

5 501 10 100

γ(1012s−

1)

S/v0

1/〈τ〉

Figure 3.2: Left panel – Frequency of first mode parameter of DS instability ingraphene. Right panel – Increment. Solid and dashed lines indicating the theoret-ical curves from (3.8), circular points for the simulation results for v0/L = 0.4THz.Dotted black line indicating the intrinsic decay rate for suspended graphene.

3.2 Rankine-Hugoniot conditions

A conservation law ∂tu + ∂xf(u) = 0, like model (2.18), whose solution comprises a discontinuity, and

hence ill defined partial derivatives, can be recast in the weak formulation as

d

dt

∫ x2

x1

u dx+ [f(u)]x2x1

= 0, (3.9)

integrating across the discontinuity. Assuming, accordingly, that a shock front occurs somewhere be-

tween the integration limits x1 < xs < x2 the previous integral can be split in two regions, before and

after the shock, leading to

d

dt

[∫ xs

x1

u dx+

∫ x2

x1

u dx

]= −[f(u)]x2

xs ⇐⇒ u(xs)dxsdt− u(x1)

dx1

dt+

∫ xs

x1

∂u

∂tdx+

+ u(x2)dx2

dt− u(xs)

dxsdt

+

∫ x2

xs

∂u

∂tdx = −[f(u)]x2

xs . (3.10)

21

In this expression the derivatives of the boundary points x1 and x2 are evidently null as such points

do not move with the fluid. Furthermore, taking the limits x1 −→ x−s and x2 −→ x+s , constricting the

integration limits to the discontinuity, the integrals vanish and the Rankine–Hugoniot condition, or jump

condition, is obtained as

cshock =f(u−)− f(u+)

u− − u+, (3.11)

where the propagation speed of the shock front is rewritten as cshock ≡ dxsdt and the superscripts dis-

tinguish quantities before (u−) and after (u+) the shock. Replacing the relation (3.11) in the continuity

equation yields

cshock =n−v− − n+v+

n− − n+. (3.12)

Relating the properties of the solution near discontinuity, provided that the Lax entropy condition

λ(n−, v−) ≤ cshock ≤ λ(n+, v+), (3.13)

with λ the characteristic velocities given by (2.24), is also respected.

Assuming that the values on the sides of the discontinuity are well approximated by the values at

the extrema, and thus specified by the boundary conditions1, which is equivalent to say that the shock

front is well approximated by a step function, with a given a shock speed, Eq.(3.11) provides a relation

between the density at source with the velocity at drain,

v(0)

v0= 1 +

cshock

v0

[1− n(L)

n0

]. (3.14)

The only thing that remains is to determine the speed of the shock itself. When the shock front is in the

imminence of the source, either approaching or leaving it, its speed is well approximated by the relative

velocity between the plasma waves and imposed current as cshock = ±(S− v0). While in proximity of the

drain appears to move with the phase velocity cshock = vp = |S2−v20 |/2S. Such relations bound the phase

space of the system to the domain(n(L); v(0)

)∈ [0; 2n0]×[2v0−S;S]; however, such considerations lose

validity as S/v0 increases, where the amplitudes of the density oscillations start to decay.

3.3 The impact of sound speed variation

Hitherto, is has been assumed that the instability growth rate is high enough to counteract the relax-

ation mechanisms, which is easily obtained by a clever choice of parameters in the case of suspended

graphene. In the case of monolayer graphene over a dielectric substrate the mobilities are lower [58]

indicating relaxation rates 1/〈τ〉 & 1012 s−1 and, therefore, it is desirable to have larger growth rates,

therefore diminishing the characteristic time until instability saturation. Indeed, the power output relies

in the maximisation of the amplitude of the travelling waves in the channel. To circumvent this issue, we

imposed a negative gradient α on S along the channel length, that is, S = S0(1 − αx), either by ma-

1That is to say that, for the examples of a shock propagating towards the source (from x = L to x = 0), the values at thediscontinuity are n− ≈ n(0) ≡ n0, n+ ≈ n(L), v− ≈ v(0) and v+ ≈ v(L) ≡ n0v0/n(L)

22

−6

−4

−2

0

2

4

6

0 0.5 1 1.5 2

v(0

)

n(L)

1− SS − 1(S2 − 1)/2S

Figure 3.3: Phase space limit cycle v(0) vs. n(L), with the relations from Rankine-Hugoniot for the various shock speeds. Simulation performed with S/v0 = 7

nipulating the permittivity ε or the gate distance d0 greatly enhances the instability growth rate, without

significant impact on the spectrum besides a small shift in the main frequency.

Such modification on the velocity along the FET channel introduces a positive feedback in the current

instability, which then leads to a faster saturation determined by a higher growth rate γ significantly larger

than γ from (3.8b) as stated in Table 3.1. This mechanism can be seen as analogous to wave shoaling

effect on shallow-waters systems near the coast and the shock wave amplitude is likewise amplified in

the presence of a velocity gradient. In fact, as the discussed model for the electronic fluid is so analogous

to the shallow waters equations, is plausible to investigate conditions for instability occurring in the latter,

assuming that they will also be present in the former in a similar way.

Table 3.1: Normalised increment of growth rate γ/γ in the presence of a negativegradient of local sound speed along the channel S = S0(1 − αx), as obtained bynumerical solution of Eq.(2.20).

S0/v0

α 20 40 60 80

0.025 1.9±0.5 3.8±0.3 6.9±0.2 10.9±0.40.05 4.5±0.4 12.5±0.5 26±1 42.3±0.90.075 7.3±0.4 21.1±0.3 37±2 68±20.1 8.5±0.3 26±2 53±1 93±3

3.3.1 Shoaling effect simulation

As seen in §3.3 the imposition of a local variation of S/v0 has a strong effect in the system, not only

leading to a faster onset and saturation of the instability but also increasing the saturation level of the

density oscillation and therefore total output power emitted as THz radiation. These effects can be

clearly observed in Fig.3.4. It deserves to be noted that the previous constrains in the density and

velocity, imposed by the Rankine–Hugoniot conditions for the original system, are now violated and the

density saturation occurs at values well over 2n0.

In addition to the simple gradient along the transistor FET, periodic configurations as sawtooth wave

were also attempted to be simulated. However, for such configuration the employed criteria for stability

23

0

0.5

1

1.5

2

0 2 4 6 8 10 12

n(L

)[1

012cm−

2]

t (ps)

1

1.5

2

2.5

3

3.5

4

0 10 20 30 40 50 60 70 80 90 100

maxn

(L)

[101

2cm−

2]

S0/v0

α = 0α = 0.05/Lα = 0.10/Lα = 0.15/L

Figure 3.4: Left panel – Evolution of electronic density at drain. Comparison ofgrowth rate and amplitude in the case of constant S = 40 (red line) vs. the presenceof linear gradient. S/v0 = 40(1 − 0.05x) (blue line). Right panel – Effect of velocitygradient on the saturation level for the electronic density

proven themselves not sufficient for most of the simulations with that condition given the fast growth

of the instability dynamic. Likewise, for gradients steeper than 20% the numerical method begins to

diverge, seemingly once the CFL condition applied in the algorithm is no longer capable to maintain the

stability of the method, as will be detailed in Chapter 5.

00.20.40.60.8

11.21.4

0 2 4 6 8 10

|P(ω

)|a.

u.

ω/2π (THz)

α = 0.15/Lα = 0.10/Lα = 0.05/L

α = 0

Figure 3.5: Spectrum of DS instability in the case of constant or varying S, notethat the besides the increment in power the slope does not entail much variation inthe spectrum – Total radiated power FFT vs. frequency. Numerical simulation withL = 0.75µm and v0 = 0.3vF , for constant S/v0 = 20.

3.4 Pulsed stimulation & frequency combs

On the previous configuration the excitation of the plasmons is solely achieved by a continuous (DC)

injection of current at the drain contact. As shown, it generates an instability that saturates itself after a

transient time, due to the nonlinear effects, leading to a continuous wave. Such wave is composed by

several modes as given by (3.8a) hence each frequency produced in the cavity of the channel is widely

spaced from the previous one.

Although continuous wave is relevant for a number of applications, wide bandwidth frequency combs

are quite sought after, for example, for THz spectroscopy. The creation of a frequency comb, which in

Fourier space is a group of close, equally spaced, frequency peaks ordinarily involves the generation of

a wave packet train in time domain.

Thereby, a scheme was idealised to generate such wave packets. If one considers that a periodic

inversion of polarity is imposed across the channel, in such a way that the time on direct polarisation

24

is enough for the saturation to occur, and then suddenly inverted, it is easy to see that the plasma

instability would be generated – saturate – and then compelled to decay in time. In this way a wave

packet is created with a envelope that can be approximated by:

A(t) =

eγt − 1 if 0 ≤ t < ts

eγts − 1 if ts ≤ t < to

e−γ(t−tp) − 1 if to ≤ t < tp

(3.15)

where ts is the characteristic time for saturation, to the duration of the excitation and tp = to+ ts the total

duration of the pulse, and that can be seen in Fig.3.6. The amplitude envelope Fourier transforms into

A(ω) = γ

√2

π

etsγω cos[ω2 (to − ts)

]− ω cos

[ω2 tp]

+ etsγγ sin[ω2 (to − ts)

]− γ sin

[ω2 tp]

ω (γ2 + ω2), (3.16)

which defines the bandwidth, and general shape, of the frequency comb. From that expression one can

deduce that, as expected, larger values of ts and to, that is to say wider envelope in time-domain, leads

to thinner envelope in Fourier space. The computed full-width at half maximum (FWHM) as function of

the pulse duration, to, is shown in 3.6, from where is also evident that the saturation time ts does not

play a significant role in the value of the FWHM.

0

0.5

1

1.5

2

0 50 100 150 200 250 300

0

1

2

ts to tp

n/n

0

t · v0/L

1/frep

1/frep

0

0.5

1

1.5

2

2.5

3

3.5

0 5 10 15 20 25 30

FWH

M

to

ts = 2ts = 3ts = 4

Figure 3.6: Top panel – Example of pulsed excitation of DS instability, using a period100t∗ with 10% duty cycle. Inset showing the amplitude envelope with the charac-teristics times. Bottom panel – Dependence of FWHM of the spectral envelope fromequation (3.16) with duration of excitation for different saturation times.

Such process can be then repeated in order to form a train of short pulses. This can, in the future,

25

be implemented with a pulse generator controlling the injected current at drain with a repetition rate frep.

Originating, consequently, a THz frequency comb around the main frequency with frequencies given by

ω = ωr + j2πfrep with j ∈ Z. Given that the typical range of frequencies available on electronic devices,

which limits the possible frep, lies in the GHz range so, the the separation between consequent peaks

in the frequency comb.

3.4.1 Frequency comb simulation

One of the more promising results presented in this thesis is, undoubtedly, the the possibility of frequency

combs generation combining the DS instability in a GFET with current GHz technology. The performed

simulations showed that the electronic fluid responds as expected to the polarisation inversion originating

the pulses that form the frequency comb. In Fig. 3.7 a paradigmatic example of such frequency combs

can be seen, with the central frequency very close to the theoretical value of (3.8a) and the shape of the

spectral envelope in accordance with (3.16), in particular denoting its the non-monotonic silhouette.

0

0.2

0.4

0.6

0.8

1

1.7 1.8 1.9 2 2.1 2.2

|P(ω

)|(a.u.)

ω/2π (THz)

Figure 3.7: Frequency comb – Total radiated power FFT vs. frequency. Numericalsimulation with L = 0.75µm and v0 = 0.3vF , for constant S/v0 = 20, repetition rateof pulses of 5 GHz with a 10% duty cycle. These Fourier spectra were calculatedwith FFTW3 library [82].

3.5 Cloosed-loop system

Taking inspiration from the previous design where the drain is actively stimulated by an external low

frequency generator, compared to the characteristic frequency of the system, a closed loop feedback

scheme can also be designed, where in addition to the steady current injection the drain receives as

input a portion of the source current. The implementation would be similar to the layout of Fig.3.8 where

the high-pass filter at the loop ensures that no DC component is further added at the drain and also to

prevent current to bypass the graphene FET.

With this configuration the system will undergo a significant shift from the typical DS instability, ex-

citing higher frequencies but diminishing the wave amplitude. The propagating shock fronts are also

smeared out leading to a somewhat rounder limit cycle (cf. Fig.3.10).

26

GFET

ID

εIS

HPFU

0 400 800 12000

5

10

15

20

25

Im(ω

)

Re(ω)

ε = 0.1

ε = 1.5

Figure 3.8: Left panel – Circuit schematics for closed loop realisation. The sourcecurrent is injected again at drain, after passing a high pass filter (HPF) and withan attenuation/gain factor ε. Right panel – Numerical results of the frequenciessolution of (3.17) in adimensional units for S = 20, sweeping ε from 0.1 to 1.5

Applying the same strategy as for the DS instability in §3.1. The boundary condition at drain is known

as j(L) = j0 + εj(0) where 0 ≤ ε ≤ 1 is some reduction factor, leading to

k+

k−=eik+L − εeik−L − ε . (3.17)

This expression, contrary to (3.7) in which all modes have the same growth rate, exhibits non analytic

solutions where not only the characteristic frequencies are different, but also each mode exhibits a

distinct growth rate, allowing the establishment of higher frequency modes as patent in Fig.3.8. It is

important to notice that for ε ≥ 1 a mode with no frequency real part, but with nonzero imaginary part,

starts to appear and as a consequence the instability ceases to exist and the system evolves to a steady

solution.

Feedback simulation

The manipulation of the system with a positive feedback as described in §3.5, brings two major con-

sequences, to wit, frequency raise and amplitude contraction. Indeed, the already foreseen presence

of higher frequencies is evident in the time evolution of the density presented in Fig. 3.9. Besides the

shorter period, also the time until saturation is lessened. However, this occurs much at the expense of

the reduction of the amplitude maximum as patent in Fig. 3.10. Hence, this design is less appropriate

to radiation emission as its debited power would be less than that of open loop situation. Furthermore,

for feedback signals not attenuated, i.e. with ε ≥ 1, the instabilities are extinguished leaving but the

constant values of density and current across the GFET.

Nevertheless, the analysis of this feedback situation is important to consider as a potential way to

achieve synchronisation between several GFET in the case of an array of emitters, where instead of a

pure feedback, the output signal of a given GFET is used to stimulate a second one, or even a set of

neighbouring emitters that ought to be synchronous.

27

0

0.5

1

1.5

2

0 5 10 15 20 25 30 35 400

0.5

1

1.5

2

0 5 10 15 20 25

0

0.5

1

1.5

2

0 2 4 6 8 10 12 140

0.5

1

1.5

2

0 2 4 6 8 10

n(L

)/n

0

t [ps]

ε = 0.1

n(L

)/n

0

t [ps]

ε = 0.25

n(L

)/n

0

t [ps]

ε = 0.5

n(L

)/n

0

t [ps]

ε = 1.0

Figure 3.9: Evolution of carrier density at drain with S/v0 = 20 and with a positivefeedback as described in §3.5. The quantity ε refers to the attenuation factor of theloop.

−15

−10

−5

0

5

10

15

0 0.5 1 1.5 2−15

−10

−5

0

5

10

15

0 0.5 1 1.5 2

−15

−10

−5

0

5

10

15

0 0.5 1 1.5 2−15

−10

−5

0

5

10

15

0 0.5 1 1.5 2

v(0

)/v 0

n(L)/n0

ε = 0.1

v(0

)/v 0

n(L)/n0

ε = 0.25

v(0

)/v 0

n(L)/n0

ε = 0.5

v(0

)/v 0

n(L)/n0

ε = 0.75

Figure 3.10: Limit cycle in the phase space of carrier density at drain and velocityat source with S/v0 = 20 and with a positive feedback as described in §3.5 (bluelines), as reference is also displayed in foreground the limit cycle in the open loopcase (grey lines) . The quantity ε refers to the attenuation factor of the loop.

28

Chapter 4

Radiation Emission

The ulterior motive for the study of plasmonic instabilities in graphene is the radiation emission, or

conversely the reception, of electromagnetic radiation, with the work in thesis being devoted to the

former. In fact, as mentioned, even before the advent of graphene plasmonic instabilities in electron

fluids were seen as a reasonable source of radiation. Along this chapter the origin and properties of

such emission are discussed in their peculiarities once it is indeed a rather unusual form of producing

radiation. One of the main features of the emission of radiation, and in particular THz radiation, by a

graphene FET is that the wavelength – in the millimetre and sub-millimetre range – is typically much

larger than the emitter itself, typically with sub-micron size, contrary to regular EM antennae that exploit

the matching of its dimensions with some portion of the wavelength. This, together with the fact that

most applications, like imaging or sensing, would require propagation along some centimetres, imposes

that the most relevant situation is that of far field regime, which will allow a more simplified approach in

some cases.

Back to the subject of the emitter size relative to the wavelength, the question of dealing with electri-

cally small antennae has some hindrances. Notably the limitation on Q-factor1, low radiative efficiency

and wide radiation pattern, in a quasi isotropic fashion. The latter two can be overcome if, instead of

considering a single GFET as the sole emitter, a two dimensional antenna array is taken in account.

While the former can be improved, if wanted, changing the total impedance of the system.

4.1 Reconstructed Fields

Having predetermined the evolution of current density and charge density on the graphene layer with

the numerical simulations the electromagnetic field can be directly obtained from Jefimenko’s integral

equations given, for a bidimensional source, by [83]

E(r, t) =en0

4πε0

∫d2r′

(n(r′, t′)|R|3 +

∂tn(r′, t′)|R|2c

)R− ∂tn(r′, t′)v(r′, t′)

|R|c2 and (4.1a)

1Defined as the quotient between reactive and active power or, similarly, between stored energy and dissipated energy.

29

B(r, t) =eµ0n0v0

4π

∫d2r′

(n(r′, t′)v(r′, t′)

|R|3 +∂tn(r′, t′)v(r′, t′)

|R|2c

)×R. (4.1b)

Here R = r − r′ is the usual displacement vector and c the speed of light, while the quantities n and v

are calculated at retarded time t′ = t −R/c. These expressions are greatly simplified by noticing that,

given the geometry of the model the velocity occurs only in x direction, v(r′, t′) = v(x′, t′)x, and the

density is also restricted to vary along the same axis, n(x′, y′, t′) = n(x′, t′). Once the fields at each

point are obtained the Poynting vector can then be calculated as

S =1

µ0E×B. (4.2)

The direct integration of (4.1), being an exact equation, allows the reconstruction of the emitted fields

both in far and near field situations. It is not able, however, to obtain the contact electrostatic field in the

thin film as 1/R→∞ ,as usual, and the integral diverges.

4.1.1 Reaction to radiation

To reckon with the effect of the radiation into the electrons themselves one has to consider both the

effect of Abraham-Lorentz force and ponderomotive force, the latter from the interaction of charges with

the gradient of electric field and the former arising from the energy loss of the charges themselves while

emitting radiation. Nevertheless, the magnitude of such effects in the studied system are small enough

to be safely neglected in first approximation, which is also supported by Fig.4.4 .

Strictly speaking the emission of radiation by the moving electronic fluid entails a loss of momentum

that can be characterised by the Abraham-Lorentz force, that in terms of adimensional velocity v and

position n, is given by

Frad =e2

6πε0L2

(v0

c

)3 ∂2v

∂t2. (4.3)

However, as c v0 this contribution is diminutive compared to the force Fg exerted in the electrons by

the electrostatic potential from the gate, in fact Fg/Frad ∼ 1013 for regular conditions of the GFET.

In addition the electrons will also be subject to the ponderomotive force from the oscillation on the

electric longitudinal electric field in the form Fpond = −e2∇E2/4meω2, resorting to the Drude model

expression v = µE, such force can then be written as

Fpond = − e2v20

4meω2µ2L

∂v2

∂x. (4.4)

Once again this provides but a small correction as in usual contexts Fg/Fpond ∼ 104, however, this can

be a relevant correction to take in account in future work while considering systems with higher velocity

or shorter lengths, for which the velocity gradients are higher.

30

4.2 Far-field radiated power

Wanting to avoid the laborious simulation of the EM fields to recover the energy flux one can resort to

an approximation, valid for far-field, based on the dipolar moment of the charge distribution. In fact, the

far-field Poynting vector S emitted by an arbitrary charge distribution [84, 85] is given by

S ∼= µ0

16π2c||p||2 f(θ, φ)

r2r (4.5)

where p is the electric dipole moment of such distribution and f(θ, φ) some angular distribution, in the

case under study it was found that f(θ, φ) = | cos θ| as will be addressed in §4.3. The time derivative of

the dipole moment can be obtained from the current density with the volume integral

p =

∫

V

j dx. (4.6)

Since the current is confined to the x direction, (4.6) is simply

p = W

∫ L

0

j dx = −eW∫ L

0

nv dx, (4.7)

where W is the graphene layer width, in the numerical implementation this integral was approximated

using Simpson’s rule [86], afterwards, in order to obtain its second order time derivative and perform

some smoothing a gaussian convolution was employed as

p =d

dt

[p ∗G

]= p ∗ d

dtG, (4.8)

with the kernel G(t) = e−t2/2ζ2

√2πζ

as will be discussed in §5.2. This method shows good agreement with

the direct method mentioned before, therefore, will be used to estimate the emitted power and efficiency

of the antenna, comparing it with the power enrolled in the conductance.

4.2.1 Dissipated power by Joule effect

Although the mobility in graphene is extremely high, even compared to metals and good conductors,

there will be some dissipation by Joule effect during the conductivity of the electronic fluid. This dissi-

pated power is given, in terms of current density and conductivity, by

PΩ =

∫j2

σdA, , (4.9)

where σ is the conductivity and dA is the surface element. Since the conductivity and mobility are related

as σ = enµ and given that j = nv the expression (4.9) becomes

PΩ =e

µ

∫nv2 dA =

en0v20

µWL

∫ 1

0

nv2 dx, (4.10)

31

A first estimate, with fairly conservative values of n0 = 1012cm−2, v0 = 105ms−1 and µ = 5×104cm2V−1s−1,

returns PΩ/A ∼ 3 × 106Wm−2, an extremely low value, as it becomes evident when compared to the

same estimate for silver, the best conductor amongst metals, which returns PΩ/V ∼ 1016Wm−2nm−1.

The contact resistance hindrance

However and in contrast with the aforementioned, the greatest limitation to the efficiency of the GFET

is by far the contact resistance [87], at the source and drain metallic contacts, that curtails the perfor-

mance of the GFET, since it lowers the effective conductivity at each end of the channel, and leads to

Joule heating in such regions [88]. This fact leads to the increment of the power dissipated in the an-

tenna effectively lowering its efficiency, thus additional ways to minimise such losses are of the utmost

importance. Although this low efficiency can be desirable for having a wider bandwidth.

4.3 Antenna attributes

4.3.1 Radiation pattern

The radiation pattern of the graphene FET in the DS configuration was calculated from the reconstructed

fields from §4.1 without reckoning radiation reflection on the metallic gate or absorption by the substrates.

This is possible since the radiation wavelength in the vacuum λ ' 4Lc/S is much larger than the typical

FET dimensions as the ratio c/v0 ∼ 1000, renders the GFET an electrically small antenna.

The pattern itself shows a wide omnidirectional profile with half power beam width of 120, as patent

in Fig.4.1. The wider profile comes from the fact that the average Poynting vector obtained follows a

〈S〉 ∝ | cos θ| law unlike a typical dipolar emitter, 〈S〉 ∝ cos2 θ [84, 85, 89] such wide lobe profile can be

explained by the fact that the collective movement of charges occurs on the entire area of the graphene

FET, similarly to patch antennae [90].

-3-10 -5 0

030

60

90

120

150180

150

120

90

60

30

dB

Figure 4.1: Radiation pattern of the graphene layer emitter at a distanceR = 1000L.Average Poynting vector normalised to its maximum value vs. polar angle θ in thevertical xz-plane. Points from numeric simulation follow 〈S(θ)〉 ∝ | cos θ| closely (redsolid line) whereas the radiation from a dipole antenna is narrower with 〈S(θ)〉 ∝cos2 θ (blue solid line). Showing a wide omnidirectional antenna behaviour in the farfield.

32

4.3.2 Radiation efficiency and quality factor

Being in the realm of electrically antennae has the known obstacles of low efficiency and high Q-factor,

which limits the bandwidth [91]. However, the Ohmic losses themselves diminish the theoretical Q-factor

allowing a larger bandwidth, desirable for the case of a frequency comb emission, but, on the other hand

for monochromatic laser applications the high Q-factor regime guarantees a very sharp frequency peak.

The efficiency of the emission is determined by the ratio between the power effectively radiated and

the total ceded, that is, adding that lost by joule effect [92],

η =Prad

Prad + PΩ=

1

1 + δ, (4.11)

where δ = PΩ/Prad is the dissipation factor. The performed simulations pointed to a high dissipation

factor leading to and efficiency η ∼ 10%, this can, we believe, be improved, in future, as no effort was

made to plan the emitter in a effective design, and the present work merely wishes to deliver a proof-

of-concept. Nonetheless such extremely low radiation efficiency has the merit of lowering the Q-factor

which opens the possibility to broad band emission or reception.

Chu–Harrington limit

For a lossless antenna there is a fundamental limit [89] stating that the ideal, or lossless, Q-factor Qi is

bounded by

Qi ≥1

k3a3, (4.12)

where k is the wave number and a the diameter of the smallest sphere that encloses the antenna,

so, considering a square graphene layer a =√

2L. Moreover as k = ω/c and the frequency is given

approximately by ω ' S/2Las seen in (3.8a) the limit can be recast as

Qi & 2√

2c3

S3(4.13)

which clearly states the extremely high Q-factor Qi ∼ 107, since c S as S ∼ 2 × 106 ms−1, and

subsequent reduced bandwidth, since Q = ω/∆ω a high Q-factor implies a small bandwidth ∆ω.

This limit, however, refers to the ideal case, taken only in account the ratio between the electromag-

netic energy stored in the antenna and that dissipated by radiation [90, 93]

Qi =2ωmaxWe,Wm

Prad, (4.14)

where We and Wm are the average electric and magnetic energies. Withal, as the actual Q-factor is

likewise impacted by the radiation efficiency as,

η =Q

Qi(4.15)

its value drops substantially allowing wider bandwidths as required frequency comb spectroscopy or

33

communications, on the other hand for lasing applications the high Q regime would be more desirable,

originating very pristine spectral lines shapes.

As a matter of fact, to balance both efficiency and bandwidth is a complex task that requires a

profound study and careful design [90]. With several works being devoted to this issue in the last

decades in the context of metallic antennae and wave-guides in distinct regions of the spectrum, in

particular to mobile phone technology. It would be most beneficial if the next years bring also such

amount of analysis to field of graphene THz emitters and sensors.

4.4 Coherence properties

Chapters 2 and 3 explored the hydrodynamic model for the electrons in graphene and stumble in the

presence of well defined shock waves forming in the FET channel. One of the most significant implica-

tions of such shock waves is that they compel the electrons to not only act collectively but, more impor-

tantly, to bunch themselves as the shock profile forces the electronic fluid to concentrate upstream. In

that sense the electrons undergo a process with some likelihood to the one in free electron lasers where

they are also automatically bunched. Then, it is no wonder that the radiation provided by such means

exhibits coherence to some extent, and that such technology could pave the way to THz lasing, beyond

a simple antenna emission.

To assess the coherence properties of the emitted radiation both the temporal and spatial degrees

of coherence [94]

g(1)(τ) =〈E∗(r, t)E(r, t+ τ)〉

〈E2(r, t)〉 and (4.16)

g(r1, r2) =〈E∗(r1, t)E(r2, t)〉√〈E2(r1, t)〉〈E2(r2, t)〉

, (4.17)

respectively, were calculated. For that purpose the analytic continuation of the electric field was obtained

resorting to the Hilbert transform2 of the real simulated values Er,

E(t) = Er(t) + iH[Er](t). (4.18)

As expected, a high coherence degree, both in far field and intermediate regimes, was obtained.

Moreover the coherences are enhanced by increasing the S as patent in Fig. 4.2. It is also observed

that the temporal degree of coherence is periodic in time and with the same period as the radiation

itself, such is due to the fact that the radiation is not truly monochromatic with the higher modes inducing

such oscillation, this factor can potentially be used to designing an appropriate optical cavity for mode

selection.

Evidently this preliminary study of the coherence has not taken in account stochastic effects on

the current during the operation of the GFET that will certainly be present in real experimental setups.

Moreover, in the case of an array of GFET emitters care needs to be taken so that all emitters are

synchronised in phase in order to preserve coherence, for that purpose one can conjecture that placing

2Defined by H[f ](t) = 1πp.v.

∫∞−∞

f(τ)t−τ dτ . Such calculation was performed resorting to the R numerical libraries.

34

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

g(1

)(τ

)τ 2π/ω0

S/v0 = 10

S/v0 = 20

S/v0 = 40

S/v0 = 60

0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500 600 700

g(r

1,r

2)

|r1 − r2|/L

S/v0 = 10

S/v0 = 20

S/v0 = 40

S/v0 = 60

Figure 4.2: Top panel – Temporal first order degree of coherence. Bottom panel– Spatial degree of coherence, where the displacement is transverse to main lobepropagation lobe.

the array of emitting GFET in an optical cavity would prompt the interaction between the radiation and

the electronic fluid.

4.5 Simulated power

Regarding the reconstruction of the emitted EM radiation the low power, characteristic of a small an-

tenna, emission was obtained by either of the discussed methods, in (4.2) and (4.5), that show good

accordance with each other as can be seen in Fig. 4.3. Moreover, such temporal profile corroborates

the presence of a multitude of frequency harmonics as already patent in the obtained spectra. The

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

18 18.5 19 19.5 20

|S|[1

03W

m−

2]

t [ps]

r →∞r = 200L

Figure 4.3: Time evolution of Poynting vector magnitude, using both the completeintegration of the fields and then resorting to (4.2) (blue line) and using the far fieldapproximation (4.5) (red line). (S/v0 = 20 with v0/L = 0.3THz)

aforesaid radiated power must be confronted with the Joule power that is lost, albeit much lesser than

in a typical metal or semiconductor, throughout the electronic conduction. Such comparison is made in

35

Fig.4.4 where is made clear that the radiated power is, de facto, quite low. However this circumstances

strengthens the assumptions made earlier, disregarding the influence of the loss of energy by radiation

to the dynamics of the electronic Fermi liquid.

0

1

2

3

4

5

6

18 18.5 19 19.5 20

Pow

er[1

0−

4W

]

t [ps]

PΩ

Prad × 10

Figure 4.4: Radiated power in far field approximation times ten (blue) and Joulepower dissipated in the electronic flow (red), the corresponding average values arealso marked by the dashed lines. (S/v0 = 20 with v0/L = 0.3THz)

36

Chapter 5

Numerical Methods

To further investigate the properties and solution of the model (2.18) whenever further analytic results

are cumbersome, or simply not possible, a program to solve it numerically was developed1, this allowed

also to perform the numerical experiments that validate the proposed setups hereinbefore presented

at §3.3, §3.4 and §3.5. Besides the fluid simulation the radiation emission was likewise numerically

solved. In this chapter the numerical methods and strategies employed are discussed in detail, while in

Appendix B the flowcharts describing the algorithms are presented.

5.1 Hydrodynamic simulation

Once the system in question is a hyperbolic set of nonlinear equation the choose of an adequate numer-

ical solver is a delicate questions on which a great number of studies were devoted over time, in fact,

contrary to parabolic partial differential equations, the hyperbolic case does not have unconditionally sta-

ble methods. In addition the solutions with physical meaning are oftentimes weak solutions comprising

a discontinuity which adds to the hardship of the numerical method operation.

To ease the numerical computation of the hydrodynamic model (2.22) the hyperbolic system of equa-

tions for density and velocity fields is written in a conservation form:

∂

∂t

nv

+

∂

∂x

nv

v2

2 + S2n

= 0 ⇐⇒ ∂

∂tu +

∂

∂xF(u) = 0, (5.1)

upon which it was chosen to apply a finite volume Lax-Wendroff type method [95, 96]. The two step

Richtmyer scheme for nonlinear systems [96] was then elected to implement, given it is computationally

light, as it is not required to explicitly evaluate the system Jacobian at each step, yet capable of returning

accurate simulations. It consists in two distinct steps, a predictor where the mid nodes are calculated

and then a corrector step that updates the simulated quantities at the central nodes (cf. Fig. 5.1) and

1The core code was developed in C++ resorting to FFTW library to the Fourier transform routines and occasionally to R for largedata handling.

37

was implemented for n an v according to

uk+1/2i+1/2 =

1

2

(uki+1 + uki

)− ∆t

2∆x

[F(uki+1

)− F

(uki)]

(5.2a)

uk+1i = uki −

∆t

∆x

[F(uk+1/2i+1/2

)− F

(uk+1/2i−1/2

)](5.2b)

using u and F(u) as defined by equation (5.1) and space (indicated at subscript indices) and time

(indicated at superscript indices) discretisation where t = k∆t and t = i∆x. Since such scheme is

second order both in time and space it will not introduce spurious diffusion in the solution as a first

order scheme would. It will, however, introduce artificial oscillations, contiguous to discontinuities, which

correction will be discussed further on.

ii− 1 i+ 1

ii− 1 i+ 1

i+ 12i− 1

2

k

k + 12

k + 1

First step

Secon

d step

Figure 5.1: Schematic representation of Ricthmyer method stencil. Every nodedepends not only of its previous state but is corrected at each iteration by the valuesof the adjacent mid nodes.

5.1.1 Courant–Friedrichs–Lewy condition

In a physical system no process can unfold faster than the propagation of information in order to uphold

causality. In a similar manner, given a time and space discretisation of a numerical scheme the prop-

agation of any feature ought not to surpass the propagation of the information itself. This statement is

encoded in the Courant–Friedrichs–Lewy condition for finite difference methods of PDE solving, which

imposes that the propagation speed, s, of the numerical solution and the discretisation parameters are

bounded by some upper limit, the maximum Courant number as s∆t∆x ≤ Cmax. Comply with this con-

dition is essential not only for the causality issue and to do not obtain specious solutions but is also a

necessary condition for the stability of these numerical methods.

The employed Richtmyer method has a CFL stability condition [97] given by

∆t

∆x≤ 1

|λmax|(5.3)

where λmax is the largest eigenvalue of the Jacobian ∂F/∂u from (5.1).

Since the equations are nonlinear the mentioned eigenvalues λ = v ± S√n are not constant in

the course of the simulation. Therefore, in order to guarantee that the CFL condition is satisfied the

38

maximum eigenvalue is overestimated in ad hoc approach. Previously studying the behaviour of a test

λmax, calculated at midpoint of the simulation domain for several S values, as seen in Fig.5.2, it was

concluded that the criterion

∆t =∆x

2S + 20(5.4)

for a given space discretisation, is enough to ensure the correct operation of the algorithm. Therefore

the first stage of the algorithm consist in determine if time step for a preselected spatial step given the S

value required for the simulation.

0

10

20

30

40

50

60

70

0 5 10 15 20 25

λm

ax

S

|vmax + S√nmax|

2S + 20

Figure 5.2: Simulated maximum jacobian eigenvalue (joined points) and imposedCFL criterion (dashed line)

5.1.2 Numerical oscillation suppression

The employed method displays a well known phenomenon of numerical oscillations near discontinuities

[96, 95, 98]. To mitigate this effect, as the model in question is expected to develop such solutions, a

moving average filter, with a window size m+ 1,

ui =

i+m/2∑

j=i−m/2uj , (5.5)

is performed along the spatial domain, between each temporal iteration, effectively reducing the oscil-

lation amplitude while preserving the discontinuity profile. It was found that a 5 points window already

perform an adequate processing, without much distortion of the shock. An example of such results can

be observed in Fig.5.3. This method was found to be not only faster but also more effective than the clas-

sical methods of artificial viscosity presented in [99] that tend to introduce diffusion on the discontinuity

edge and consequently smear it out.

5.2 EM fields and antenna simulation

Since the velocity and density profiles on which the Jefimenko’s equations (4.1) rely to reconstruct

the electromagnetic fields around the graphene layer are calculated beforehand, their time derivatives

39

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

0 0.2 0.4 0.6 0.8 1

n(x

)

x

unfiltered

5 points

7 points

Figure 5.3: Spatial moving average smoothing, comparing the unfiltered case(dashed line) with a 5 points moving average (solid blue) and 7 points moving aver-age (solid red)

can be extracted afterwards in an intermediate process resorting to a central differences second order

scheme.

Hence, obtaining the fields is then only a matter of solving the surface integrals. This numerical

integration was performed with a second order method using the approximation:

∫ b

a

∫ d

c

f(x, y)dydx =∆x∆y

4

[f(a, c) + f(b, c) + f(a, d) + f(b, d) + 2

m−1∑

i=1

f(xi, c) + f(xi, d)+

+ 2

n−1∑

j=1

f(a, yj) + f(b, yj) + 4

n−1∑

j=1

m−1∑

i=1

f(xi, yj)

+O(∆x2) +O(∆y2), (5.6)

for an integral over a rectangular region [a; b] × [c; d] with the discretisation points ximi=0 and yjnj=0

making x0 = a, xm = b, y0 = c and yn = d the endpoints of the region to integrate. As for the EM fields

reconstruction the integrand functions must be evaluated at retarded time such quantity is calculated

at each step and the computed values for the nearest simulated time are used. The integration is

calculated, with the specific integrand functions, for each of the six components of E and B fields.

For that reason the presented algorithm is very well suited to parallelization. Furthermore, a previous

estimation of the average retarded time is performed and the reconstruction proceeds only when that

yields a positive retarded time, i.e. after the EM wave effectively have reached the observation point.

The general description of the algorithm is, as mentioned before, presented at Appendix B.

As stated in §4.2 the energy flux emitted by the graphene was also computed, in the far field approx-

imation, from the second derivative in time of the electric dipole moment. While the first derivative is

obtained indirectly resorting to the integral relation of (4.7) implemented with a Simpson quadrature rule

as:

pk =∆x

3

N/2∑

i=1

nk2i−2vk2i−2 + 4nk2i−1v

k2i−1 + nk2iv

k2i, (5.7)

afterwards the second derivative is obtained by convolving this result with a gaussian kernel,

pk =1

∆t

w∑

`=−wpk−` · −`e

−`2/2ζ√

2πζ3/2, (5.8)

40

with an appropriate standard deviation ζ over a chosen window with width, 2w. Such method is superior

to the more usual finite difference methods as it handles the natural discontinuities and unwanted oscil-

lations without increasing the overall error. With the performed benchmarking was concluded that the

values of w = 50 and ζ = 65 provide accurate results.

41

42

Chapter 6

Suspended Graphene

Until this point the topography of the graphene layer has been absent in both the simulations and the

theoretical considerations. In fact, however, the corrugation, oscillations and non planar behaviour in

general are of the utmost importance to suspended graphene [100]. Not only granting it stability but also

impacting the electrical properties as the electrons scatter with out-of-plane phonons. Moreover, in the

setup that has been discussed so far, the vibration, and subsequent displacement, of the graphene layer

would have direct implications in the electric field between graphene and gate [101]. In this chapter such

interactions are analysed and a new, yet modest, hybridisation mechanism is brought forth.

Nonetheless, the performed numerical simulations are still in their inchoative stages, not having

delivered, up to the present time, conclusive results regarding the influence of the flexural movement in

the carriers dynamic. A detailed study of such interactions and simulations in a more fundamental level,

as in [102, 103], would require future work which falls out of the scope of the present dissertation.

6.1 Kirchhoff-Love membrane coupling

In the case of a suspended graphene layer, free to oscillate, the distance between the gate and graphene

is perturbed by a displacement η such that d = d0 + η(x, t), thus changing the gate electric potential

locally to U = en(d0 + η)/ε. Furthermore the Lagrangian density of the flexural phonons [104] is added

to the electromagnetic Lagrangian density in the absence of magnetic fields

L = Lflex + Lem =1

2

[ρη2 − γ(∇η)2 −D(∇2η)2

]+e2n2(d0 + η)

ε+e2n2

2ε(6.1)

where ρ is the surface mass density, D the bending stiffness and γ the clamping tension applied. There-

fore the Euler–Lagrange equation for the 1D+1 graphene layer oscillation is

ρ∂2η

∂t2− γ ∂

2η

∂x2+D

∂4η

∂x4= −e

2n2

ε, (6.2)

43

Combining the previous equation (6.2) with the hydrodynamic model (2.20), the coupled system obtained

is given by

∂n

∂t+

∂

∂x(nv) = 0

∂v

∂t+ v

∂v

∂x+e2d0

εm?

∂

∂x

[n(1 + η/d0)

]= 0

∂2η

∂t2− γ

ρ

∂2η

∂x2+D

ρ

∂4η

∂x4= −e

2n2

ρε

(6.3)

those equations are adimensionalised with the same variables defined in (2.21) so the coupled system

becomes

∂n∗

∂t∗+

∂

∂x(n∗v∗) = 0

∂v∗

∂t∗+ v∗

∂v∗

∂x∗+S2

v20

∂

∂x∗[n∗(1 + η∗)] = 0

∂2η∗

∂t∗2− Γ2

v20

∂2η∗

∂x∗2+

∆2

v20

∂4η∗

∂x∗4= −C

2

v20

n∗2

(6.4)

where the following quantities were employed:

S2 =e2d0n0

m?ε+v2F

2, Γ2 =

γ

ρ, ∆2 =

D

ρL2and C2 =

e2n02L2

ρεd0. (6.5)

6.2 Membrane driven excitation

Leaving, for a moment, the influence of the gate over the graphene aside, and regarding it purely as

a flexible membrane this section will consider the excitations arising from a purely mechanical driving

mechanism.

In the context of a GFET is reasonable to admit that the graphene layer is fixed at the metallic

contacts of source and drain while is left free standing at other two sides. Once the graphene membrane

is supposed to be clamped at its edges it is reasonable to consider the case of where it can be excited by

the action of piezoelectric actuators, varying the mechanical tension with the suitable driver frequency

Ω.

Considering the homogeneous part of (6.2) and performing a separation of variables η(x, t) =

H(x)T (t) the equation becomes:

T ′′(t)− γ(t)

ρT (t)

H ′′(x)

H(x)+D

ρT (t)

H(4)(x)

H(x)= 0 ⇐⇒ T ′′(t) +

γ(t)

ρT (t)k2

0 +D

ρT (t)k4

0 = 0, (6.6)

assuming a harmonic variation of tension γ(t) = γ0 + γ cos Ωt and re-scaling the time variable τ ≡ tΩ in

(6.6) one gets

T ′′(τ) +

[ω2

0

Ω2+k2

0 γ

ρΩ2cos τ

]T (τ) = 0 (6.7)

where ω0 is the membrane natural frequency ω20 = D

ρ k40 + γ0

ρ k20. This equation has, therefore, the form

of the Mathieu’s equation [105] u′′(τ) + [δ + ε cos τ ]u(τ) = 0 matching the parameters as

δ =ω2

0

Ω2and ε =

k20 γ

ρΩ2, (6.8)

44

the Mathieu’s equation unstable region is well know to be given by 14 − ε

2 < δ < 14 + ε

2 which leads to the

condition √4ω2

0 − 2Γ2k20 < Ω <

√4ω2

0 + 2Γ2k20, (6.9)

for the drive frequency of the membrane, where Γ2 = γ/ρ in analogy with the definition given in (6.5).

Therefore, based solely in the membrane response an instability is expected to occur when the drive

frequency is in a narrow window of values around double the natural frequency. Obviously the effect of

coupling the membrane equation with the fluid dynamics of the electrons would lead to more accurate

results for the aimed design, nevertheless this simple excitation could be used to provide feedback to

the electronic fluid similarly to the setups discussed in §3.3 and §3.5.

6.3 Plasmon-flexuron hybridisation

Attacking now the interaction between the oscillation of the graphene layer and the Fermi liquid of its

electrons, the set of equations (6.3) were linearised, while imposing the equilibrium displacement η0 = 0,

which yields the coupled system of equations in Fourier space

(ω − kv0)n− kn0v = 0

−k e2d0

m?εn+ (ω − kv0)v − k e

2n0

m?εη = 0

2e2n0

ρεn+

(γ

ρk2 +

D

ρk4 − ω2

)η = −e

2n20

ρε

, (6.10)

this leads to the secular equation

(Γ2k2 + ∆2L2k4 − ω2

) ((ω − kv0)2 − S2k2

)+

2S2C2

L2k2 = 0. (6.11)

Notice that, without surprise, it combines the flexural mode with dispersion relation ω2f = Γ2k2 + ∆2L2k4

with the Dyakonov–Shur mode of ωe = (v0 ± S)k, adding an extra term of coupling 2S2C2k2/L2. Hence

the two modes hybridise, as can be observed in Fig. 6.1, avoiding the crossing that would occur between

the two bare modes at k± =

√(v0±S)2−Γ2

∆L . Furthermore, with such dispersion relation the real part of

the frequency vanishes for a critical value, kc, of the wave number and therefore the system exhibits an

instability for 0 < |k| < kc, such critical wave number is given by

k2c =

Γ2 +√

Γ4 − 8∆2S2C2

S2−v202L2∆2

∼=2Γ2 − 4∆2C2S2

(S2−v20)Γ2

2L2∆2. (6.12)

The qualitative behaviour of this dispersion relation can be observed Fig.6.1, from which can be

inferred that for small k the upper mode is essentially the linear mode from DS and the lower mode

the flexural one, this is confirmed by the calculation of the Hopfield coefficients [106] for this coupling,

45

ωL ωU

0-kc kck- k+k

ω

Figure 6.1: Real part of dispersion relation from (6.11). Arbitrary units to pointout the general characteristics of critical wave number kc as well as the avoidedcrossing at k− and k+.

defined by

uk =ωUωe − ωLωf√

|ωUωe − ωLωf |2 + |ωUωf − ωLωe|2(6.13a)

vk =ωUωf − ωLωe√

|ωUωe − ωLωf |2 + |ωUωf − ωLωe|2(6.13b)

subject to the normalisation |uk|2 + |vk|2 = 1 and where ωU and ωL refers to the upper and lower

mode, respectively, while ωf and ωe, in its turn, indicate the flexural and plasmonic bare modes. These

coefficients quantify the relative proportion of DS or flexural behaviour in each mode. From Fig.6.2 the

fast transition between modes near the crossing points is evident, as well as the mixing in the unstable

region.

|uk2 |vk

2

0-kc kck- k+k

1

2

1

Figure 6.2: Hopfield coefficients for the membrane–electrons hybridisation.

46

6.4 System parameters – Elasticity vs. stiffness

In order to correctly estimate values for the system parameters (6.5) of the model the typical properties

of a single layer graphene sheet were taken in account, such usual values are presented in Table. 6.1

Table 6.1: Typical mechanical values for single layer graphene.

vF [ms−1] ρ [Kgm−2] n0 [cm−2] γ [Nm−1] D [eV]106 7.6× 10−7 1010 ∼ 1013 < 42 1.5

Choosing v0 = 0.3vF , n0 = 1011cm−2, d0 = 0.01µm and L = 1µm as reasonable values for a

graphene transistor, within the bounds of Table 6.1 and originating IDS ≈ 0.05mA and VG ≈ 0.2V ,

the values obtained for the simulation parameters are S = 7.38v0, C = 0.002v0 ∆ = 1.87×10−6v0 and

Γ . 0.025v0.

Since Γ ∆ ∼ 0 the graphene can be considered, in this conditions, an elastic membrane dropping

the biharmonic term. This, however, introduces a problem with the preceding analysis as the flexural

modes exhibit also a linear dispersion relation, crossing the Dyakonov–Shur modes only at the origin

and the phenomenon of avoided crossing is absent. Even so, the real part of the frequency still vanishes

near k ∼ 0 keeping the potential exploitation of instability in that region opened.

Wishing, nonetheless, to study the effect of the biharmonic term in (6.3) a scenario of deposited

graphene on a polymeric thin film membrane, as those described in [107–110], can prospected and

different values of bending stiffness can be designed, as seen in Table 6.2.

Table 6.2: Estimated values of ∆/v0 for polymeric membranes considering a thick-ness of H = 150nm, length L = 1µm and drift velocity v0 = 0.3vF . The data ofYoung’s modulus, density and Poisson ratio were retrieved from [111].

Polymer Density Young modulus Poisson ratioρ [gcm−3] E [GPa] ν ∆/v0

Polyimide (PI), Kapton® 1.42 4.0 0.34 2.6× 10−4

Polyvinyl chloride (PVC) 1.40 2.74 0.40 2.2× 10−4

Polyethylene (PE) 0.95 1.0 0.45 1.6× 10−4

Polyethylene terephthalate (PET) 1.55 6.79 0.34 4.1× 10−4

Polydimethylsiloxane (PMDS) 0.97 0.62 0.50 1.3× 10−4

Polymethyl methacrylate (PMMA) 1.20 2.45 0.35 2.2× 10−4

6.5 Kirchhoff–Love membrane simulation

In order to simulate the coupled system (6.4), as the dynamics of the membrane is necessarily slower

that the one of the electrons, it was chosen to update the membrane displacement between each itera-

tion step of the electron fluid simulation and then use such results in the next step, i.e. admitting that the

membrane is quasi static in comparison to the electrons.

47

The 1-dimensional Kirchhoff-Love equation,

∂2η

∂t2− Γ2 ∂

2η

∂x2+ ∆2 ∂

4η

∂x4= −C2n2, (6.14)

was solved implicitly with a discretisation of second order central differences in space and second order

forward in time, which takes the form [112]:

2ηk+1i − 5ηki + 4ηk−1

i − ηk−2i

∆t2− Γ2 η

k+1i−1 − 2ηk+1

i + ηk+1i+1

∆x2+

∆2 ηk+1i−2 − 4ηk+1

i−1 + 6ηk+1i − 4ηk+1

i+1 + ηk+1i+2

∆x4= −C2(nki )2. (6.15)

Limiting the case to the study of clamped membranes, the natural boundary conditions of constant

values and first derivatives at the borders can be stated as:

η(0) = H0; η(L) = HL;∂η

∂x

∣∣∣x=0

= T0 and∂η

∂x

∣∣∣x=L

= TL. (6.16)

The Dirichlet conditions impose that, at the endpoints of the discretisation the displacement is held at a

fixed value,

η0 = H0 and ηN = HL, (6.17)

while evaluating the finite difference ∂η∂x = ηi+1−ηi−1

2∆x +O(∆x2) at i = 0 and i = N yields

η−1 = η1 − 2∆xT0 and ηN+1 = ηN−1 + 2∆xTL, (6.18)

thus relating the Neumann conditions and the values on the grid, with the resort of the definition of

two ghost points,η−1 and ηN+1, outside the discretisation domain. Hence, the vector of unknowns to

be determined at each iteration is [η1 · · · ηN−1]T the finite difference linear system (6.15) can then be

written, using the relations (6.17) and (6.18), in the form

2 + 2g + 7d −g − 4d d 0

−g − 4d 2 + 2g + 6d

d d

2 + 2g + 6d g − 4d

0 d −g − 4d 2 + 2g + 7d

ηk+11

ηk+1N−1

=

=

5ηk1 − 4ηk−11 + ηk−2

1 + F k1 + gH0 + d(H0 + 2∆xT0)

5ηk2 − 4ηk−12 + ηk−2

2 + F k2 − dH0

...

5ηkN−2 − 4ηk−1N−2 + ηk−2

N−2 + F k2 − dHL

5ηkN−1 − 4ηk−1N−1 + ηk−2

N−1 + F kN−1 + gH0 + d(H0 − 2∆xT0)

(6.19)

48

where g ≡ Γ2∆t2

∆x2 d ≡ ∆2∆t2

∆x4 while F ki ≡ −∆t2C2nki corresponds to the forcing term. To solve such

gargantuan system the Gnu Scientific Library routines were employed [113].

The performed simulations showed [101] that the coupling of the Dyakonov-Shur model with a

Kirchhoff-Love vibrating membrane is able to self drive the membrane oscillations and it does not seem

to mitigate the plasma oscillations when in its presence. The inverse processes, of the membrane

exciting the electronic properties, is expected to be usable in the drive and control of future terahertz

oscillators in which the applied mechanical strain on the membrane could modify the plasmon behaviour.

However, a more in depth study of the herein presented model is required, being out of the scope of this

thesis.

49

50

Chapter 7

Conclusions

As final remarks on the hereby presented work ought to be noted that it has shown that hydrodynamic

model for carrier transport in mono-layer graphene admits the excitability of Dyakonov-Shur plasmonic

instability, solely by steady state electric conditions. This fact paves the way to the future development of

all-electrical miniature devices from which the advantageous byproducts arising from the instability can

be harvested, namely the emitted radiation.

It was also shown, both analytically and with the support of numerical simulations, that the referred

electrically excited instabilities lead to the emission of THz radiation, in the so called “THz gap”, for which

state of the art technology still faces significant difficulties for table-top production, notwithstanding being

extremely sought after. Conferring, consequently, great pertinence to the line of research herebefore

presented.

The presented technology offers a simple method for the development of THz small antennae, tun-

able by the applied gate voltage. Appropriate to applications in medical imaging or security screenings,

while the generation frequency combs has direct application in spectroscopy. Furthermore, numerical

simulations suggest that the emitted THz radiation is extremely coherent, thus being an appealing can-

didate for a THz laser source, dismissing the usage of external light sources as other THz solutions

require.

This puts graphene plasmonics and, in particular, graphene field-effect transistors in the run for com-

petitive, low-consumption THz devices based on integrated-circuit technology, allowing the fabrication of

patch arrays designed to enhance the total radiated power.

7.1 Achievements

Over the course of the work that conduced to the production of this thesis, several important objec-

tives were attained. First and foremost, the creation and development of a simulation code capable

of simulating the hydrodynamic model described, as well as reconstructing the emitted electromag-

netic field around a graphene field effect transistor, and also perform some numerical analysis over the

computed results. The performed simulations attested the Dyakonov-Shur instability transpiring in the

51

expected range of frequency, along with the theorised growth rate, featuring the saturation of amplitude

that induces the emergence of the limit cycle in the (n, v) phase space. Concerning the emitted fields

simulation the revelation of the high coherence of the radiation was a milestone of grave importance

suggesting lasing applications.

Resorting to the developed numerical tools other results followed, which were also corroborated

by analytical study. In particular new methods for the amplification and growth rate enhancement of

the plasmonic instabilities, akin to the Dyakonov-Shur one, were obtained with fairly simple electronic

setups. In first place the amplification by shoaling effect is an original development in the fields of DS

instability, that will certainly play a relevant role in the real experiments. Secondly the possibility of pulsed

excitation of a GFET recurring to usual GHz technology that delivers THz frequency combs electrically

produced and tuned. Lastly the closed loop design shown how to obtain even higher frequencies besides

hinting the effects of interconnection in net of GFET.

7.2 Future Work

The current study is but an incipient endeavour to understand the phenomena of THz plasmonic in

graphene and more work still need to be dedicated to this issue. Regarding the physical processes oc-

curring in mono-layer graphene additional effects can be taken into account in the ensuing work namely:

phonon coupling; viscous flow simulation; transverse motion of carriers; interaction with radiation and

magnetic fields.

The coupling and scattering with phonons, either in-plane or out-of-plane, which play a significant

role in the electron transport [100], should be considered in future to further test the capability of the

plasmonic instabilities to overcome the inherent losses.

Likewise the inclusion of viscous effects, arising either from the aforementioned scattering or from

finite-size effects that may arise in thin graphene ribbons [43] should also be taken in consideration,

noting, however that it would have a profound implication in the simulations as it would mutate the fluid

equations from hyperbolic to parabolic, rendering the algorithms no longer suitable.

Dealing with viscosity and subsequently with a boundary layer for the electronic plasma at either

sides of the graphene layer will require a full bidimensional description that would prove valuable to

study also other effect such as wakes in the fluid [66].

Moreover, and arguably more importantly, the feedback reaction of the electronic fluid to radiation is

also a significant aspect to add to the simulation in order to be able to assess not only the direct effect

in the electron fluid but, with greater relevance, for the analysis of the reverse case, when the graphene

layer receives THz radiation, acting as a detector rather than an emitter.

Finally, the instability amplification by resonances taking place in magnetised graphene plasmas may

also conduct to interesting solutions [114].

52

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62

Appendix A

Derivation of Euler equations

The Boltzmann equation describes the evolution of the distribution function f(r,v, t) in the phase space

of velocity v and position r subject to an acceleration field g

∂f

∂t+ v · ∇f + g · ∂f

∂v=

(∂f

∂t

)

coll

(A.1)

such distribution is subject to the normalisation∫f(r,v) dv ≡ n(r) and the average values are obtained

as usual as∫vif(r,v) dv/n(r) ≡ 〈vi〉. Therefore the zero moment of the collisioness form of equation

(A.1) provides the customary continuity equation.

∫∂f

∂t+ v · ∂f

∂r+ g · ∂f

∂vdv = 0 ⇐⇒ ∂

∂t

∫f dv +

∂

∂r·∫

vf dv = 0 ⇐⇒

⇐⇒ ∂n

∂t+

∂

∂r· [n〈v〉] = 0 (A.2)

Moreover the first moment of (A.1) can be written in the form

∫ [∂f

∂t+ vj

∂f

∂rj+ gj

∂f

∂vj

]v dv = 0 ⇐⇒ ∂

∂t

[n〈vi〉

]+

∂

∂rj

∫vivjf dv +

∫gjvi

∂f

∂vjdv = 0 (A.3)

for i, j ∈ 1, 2, 3 and with implied summation over repeated indices, the last term of the left hand side

can be simplified integrating by parts

∫vi∂f

∂vjdv =

[vif]+∞−∞ −

∫∂vi

∂vjf dv = −δijn(r) (A.4)

and noting that the distribution f must vanish at infinity. Introducing the new variable vi = vi − 〈vi〉 in

(A.3) it becomes

∂

∂t

[n〈vi〉

]+

∂

∂rj

∫〈vi〉〈vj〉f + vivjf dv − gjδijn = 0 ⇐⇒

⇐⇒ ∂

∂t

[n〈vi〉

]+

∂

∂rj

[〈vi〉〈vj〉n

]+

∂

∂rj

∫vivjf dv − gin = 0 (A.5)

63

where the term∫vivjf dv ≡ Pij/m∗ corresponds to the pressure tensor, expanding the temporal deriva-

tive and using (A.2) the second Euler equation is obtained

n∂〈vi〉∂t

+ 〈vi〉∂n

∂t+

∂

∂rj

[〈vi〉〈vj〉n

]= − ∂

∂rj

Pijm∗

+ gin, ⇐⇒

⇐⇒ n∂〈vi〉∂t−〈vi〉

∂

∂rj[n〈vj〉] +

〈vi〉∂

∂rj

[〈vj〉n

]+ n〈vj〉

∂〈vi〉∂rj

= − ∂

∂rj

Pijm∗

+ gin ⇐⇒

⇐⇒ ∂〈v〉∂t

+ 〈v〉 · ∂〈v〉∂r

= g − 1

m∗n∂P

∂r. (A.6)

64

Appendix B

Code flowcharts

Start fluid dynamics algorithm

Input S

CFL condtion

Random initalization

t < Tmax

i < N

Stop

Predictor step uk+1/2i+1/2

Corrector step uk+1i

i← i+ 1

Boundary conditions

Average filter uki =∑l ukl /m

Dipole integration

Record data

files out:

n(x, t) v(x, t) j(x, t)t← t+ dt

1 2

yes

no

yes

no

Figure B.1: Abridged description of the overall al-gorithm used to solve the fluid model. The CFLcondition was applied as described in (5.4) whilethe dipole integration according to (4.7). Thegenerated data files are afterwards imported tothe the other routines, namely the radiation emit-ter simulation and an elementary analysis rou-tine.

65

Start antenna simulation

Input S

1

Time derivatives files in

Input position r

t < Tmax

Retarded time

estimation

tr = t − ∆r/c

tr > 0 Field integration

Record datat← t+ dt

files out

E(r, t) B(r, t) S(r, t)

Stop

yes

no

yes

no

Figure B.2: Simplified flowchart for the simulationof the radiated fields and Poynting vector, resort-ing to the data, simulated previously, of density,velocity and current in the channel. This routinecan be run to calculate the fields at a provided po-sition r or, evidently, sweeping through a series ofpredefined point in order to obtain a spatial re-construction. At each time step a fast estimationof the retarded time from a central point in theGFET is performed to ensure that the EM waveshad reached the desired point. The field recon-struction is obtained solving equations (4.1).

66

Start analysis algorithm

Input S

Calculate ∆t and

find array size M

files in:

n(L, t) v(0, t)

p(t)Ek(t)

2

k < M

Gauss. con-

volution

pk=∑j<|w|

pk−jG′k

∆t

k ← k + 1

Estimate T = ∆t 2πω

and set t0 = 0

Find extrema for

t ∈ [t0, t0 + T ]

Shift t0 ← t0 + T

Record data

files out:

S(t) p(t) and

extrema of

n(L, t) v(0, t)

Stop

yes

no

Figure B.3: In addition to the main simulationroutines a concise analysis routine was also de-signed, aiming to obtain the far field approxima-tion of the Poyting vector from the second deriva-tive of the electric dipole moment (4.5), and to de-termine the maxima and minima of the time seriesat drain and source in order to study the satura-tion of amplitude as well as the dependency of itsvalue with the system characteristic.

67

68

Appendix C

Specimina of fluid simulation results

The developed code allowed the performance of several numerical experiments, the ones concerning

the fluid behaviour were of particular significance. In the ensuing pages some of the results of those

simulations are displayed, either the time evolution of the various quantities at source (x = L) and drain

(x = 0) as well as the space propagation along the channel length.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 2 4 6 8 10 12 14 16 18 20

0.5

1

1.5

15 15.5 16 16.5 17 17.5 18

n(L

)[1

012cm−

2]

t [ps]

Figure C.1: Time evolution of numerical density at drain for S/v0 = 20 with v0/L =0.3 THz

69

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14 16 18 20

0

0.5

1

1.5

2

2.5

15 15.5 16 16.5 17 17.5 18

v(L

)[1

05m

s−1]

t [ps]

Figure C.2: Time evolution of velocity at drain for S/v0 = 20 with v0/L = 0.3 THz

1− 10−6

1

1 + 10−6

0 2 4 6 8 10 12 14 16 18 20

1− 6× 10−7

1

1 + 6× 10−7

15 15.02

j(L

)[1.6

Acm−

1]

t [ps]

Figure C.3: Time evolution of current density at drain for S/v0 = 20 with v0/L =0.3 THz. Note that the boundary conditions impose j(L, t) = j0, and therefore,this plot shows the committed numerical error at this point, having average value〈j(L)〉 = 1.0000 and standard deviation σ[j(L)] = 2.12792× 10−7.

70

−15

−10

−5

0

5

10

15

0 2 4 6 8 10 12 14 16 18 20−24

−20

−16

−12

−8

−4

0

4

8

12

16

20

24

−15−10−5

05

1015

15 15.5 16 16.5 17 17.5 18−24−16−8081624

v(0

)[1

05m

s−1]

j(0)

[Acm−

1]

t [ps]

Figure C.4: Time evolution of velocity and current density at source for S/v0 = 20with v0/L = 0.3 THz

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

n(L

)[1

012cm−

2]

x [µm]

25.6ps 26.2ps 26.8ps 27.4ps

n(L

)[1

012cm−

2]

x [µm]

27.9ps 28.8ps 29.4ps 30.1ps

n(L

)[1

012cm−

2]

x [µm]

30.3ps 30.6ps 30.9ps 3.12ps

n(L

)[1

012cm−

2]

x [µm]

31.2ps 31.5ps 31.8ps 32.1ps

Figure C.5: Propagation of the density shock and rarefaction waves during a com-plete period. For S/v0 = 7 with v0/L = 0.2 THz

71

−4

−2

0

2

4

6

0 0.2 0.4 0.6 0.8 1

−4

−2

0

2

4

6

0 0.2 0.4 0.6 0.8 1

−4

−2

0

2

4

6

0 0.2 0.4 0.6 0.8 1

−4

−2

0

2

4

6

0 0.2 0.4 0.6 0.8 1

v(0

)[1

05m

s−1]

x [µm]

25.6ps 26.2ps 26.8ps 27.4ps

v(0

)[1

05m

s−1]

x [µm]

27.9ps 28.8ps 29.4ps 30.1ps

v(0

)[1

05m

s−1]

x [µm]

30.3ps 30.6ps 30.9ps 31.2ps

v(0

)[1

05m

s−1]

x [µm]

31.2ps 31.5ps 31.8ps 32.1ps

Figure C.6: Propagation of the velocity shock and rarefaction waves during a com-plete period. For S/v0 = 7 with v0/L = 0.2 THz

−4

−2

0

2

4

6

0 0.2 0.4 0.6 0.8 1

−4

−2

0

2

4

6

0 0.2 0.4 0.6 0.8 1

−4

−2

0

2

4

6

0 0.2 0.4 0.6 0.8 1

−4

−2

0

2

4

6

0 0.2 0.4 0.6 0.8 1

j(L

)[1.6

Acm−

1]

x [µm]

25.6ps 26.2ps 26.8ps 27.4ps

j(L

)[1.6

Acm−

1]

x [µm]

27.9ps 28.8ps 29.4ps 30.1ps

j(L

)[1.6

Acm−

1]

x [µm]

30.3ps 30.6ps 30.9ps 31.2ps

j(L

)[1.6

Acm−

1]

x [µm]

31.2ps 31.5ps 31.8ps 32.1ps

Figure C.7: Propagation of the current shock and rarefaction waves during a com-plete period. For S/v0 = 7 with v0/L = 0.2 THz

72

Appendix D

Submitted paper

The work developed over the course of this thesis culminated in the production of the following paper,

submitted to Physical Review B, where the main aspects and new results, discussed in this dissertation

are presented.

73

Terahertz Laser Combs in Graphene Field-Effect Transistors

Pedro Cosme1, 2, ∗ and Hugo Tercas1, 2, †

1Instituto de Plasmas e Fusao Nuclear, Lisboa, Portugal2Instituto Superior Tecnico, Lisboa, Portugal

Electrically injected terahertz (THz) radiation sources are extremely appealing given their ver-satility and miniaturization potential, opening the venue for integrated-circuit THz technology. Inthis work, we show that coherent THz frequency combs in the range 0.5 THz < ω/2π < 10 THzcan be generated making use of graphene plasmonics. Our setup consists of a graphene field-effecttransistor with asymmetric boundary conditions, with the radiation originating from a plasmonicinstability that can be controlled by direct current injection. We put forward a combined analyticaland numerical analysis of the graphene plasma hydrodynamics, showing that the instability can beexperimentally controlled by the applied gate voltage and the injected current. Our calculations

indicate that the emitted THz comb exhibits appreciable temporal coherence (g(1)(τ) > 0.6) andradiant emittance (107 Wm−2). This makes our scheme an appealing candidate for a graphene-base THz laser source. Moreover, a mechanism for the instability amplification is advanced for thecase of substrates with varying electric permitivitty, which allows to overcome eventual limitationsassociated with the experimental implementation.

PACS numbers:

Introduction.−Terahertz (THz) radiation consists ofelectromagnetic (EM) waves within the frequency rangefrom 0.1 to 10 THz, filling the gap between microwaveand infrared light. The technology for its production iscurrently a very active field of research [1, 2], since THzradiation has numerous applications, comprising sensing,imaging, metrology and spectroscopy [3, 4]. A major rea-son for the hype around THz radiation is that fact that itis able to penetrate several materials that are opaque tovisible and IR radiation, while the short wavelength pro-vides high image resolution [5]. On the other hand theattenuation in water can provide information for medicalimaging while being non-ionising and biologically safe.

Among all forms of THz radiation, THz laser (THL)combs play a prominent role within such technology[6, 7]. However, THL generation still faces significantdifficulties, being restricted to gas lasers [8, 9], with lowefficiency, quantum cascade lasers [1, 10], which requireextremely low temperatures, and free electron lasers [11],practically impossible to miniaturize. With the advent ofgraphene plasmonics, new techniques relying in opticalpumping have been put forward [12–14]. The progressin graphene based transistors [15] paved the way tothe possibility for all-electrical miniaturized devices forlow power radiation emission and detection. Yet, suchintegrated-circuit THz technology based on graphene isat its infancy, notwithstanding some experimental studiesin visible and mid-infrared light emission [16–18]. Morerecently, THz emission from dual gate graphene field-effect transistor (FET) due to electron/hole recombina-tion in a p-i-n junction has been made possible [19, 20].

∗Electronic address: pedro.cosme.e.silva@tecnico.ulisboa.pt†Electronic address: hugo.tercas@tecnico.ulisboa.pt

Practical solutions towards inexpensive, compact andeasy-to-operate lasing devices are, therefore, desirable.

In this Letter, we exploit a scheme for the generationof coherent THz frequency combs in graphene field-effecttransistors, arising from the Dyakonov-Shur (DS) plas-monic instability [21, 22]. The latter can be excited viathe injection of an electric current, thus forgoing the ne-cessity of optical pumping. This opens the possibility tothe development of an all-electric, low-consumption stim-ulated THL, capable of operating at room temperature.Our calculations are based on the hydrodynamic formu-lation of the plasma in monolayer graphene, from whichwe analytically obtain the instability criteria and numer-ically extract the radiation spectrum, intensity and cor-relation. Our findings reveal that the emitted THL combexhibits appreciably large values of both spatial and tem-poral coherences, suggesting our scheme to be a compet-itive solution towards THz laser light with integrated-circuit technology. Finally, a new passive mechanism forthe amplification of the DS instability in gated grapheneis introduced.Graphene plasma instability.−The dynamics of the

electronic flow in a graphene FET can be described withthe help of a hydrodynamic model [23–25]. Assumingthe transport to be restricted to one (say x) direction,the latter reads

∂n

∂t+

∂

∂xnv = 0,

∂v

∂t+ v

∂v

∂x=

F

m∗− 1

m∗n∂P

∂x,

(1)

where n is the 2D electronic density, v is the flow ve-locity, F is the total force exerted in the electrons,P = ~vF

√πn3/3 is the pressure (with vF ∼ 106 ms−1

denoting the Fermi speed), and m∗ the electron effective

2

x = 0 x = L

G

S Dgraphene

→

∓U

IDS

(a)

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 2 4 6 8 10 12 14

n(L

)(1

012cm−

2)

t (ps)

(b)

FIG. 1: Left panel: Schematic diagram for gated graphenetransistor. For the DS instability to occur a fixed currentIDS is injected at the drain while mantaining the electronicdensity of the source constant. Right panel: Example of theinstability growth of electronic density at the drain.

mass. The validity of the hydrodynamic model in Eq.(1) is granted thanks to the large value of the mean freepath of electron scattering with phonons and impuritiesl at room temperature (l > 0.2µm), thus assuring thatballistic transport holds. We also consider graphene tobe in the degenerate Fermi liquid regime, provided thatthe Fermi level remains bellow the Van Hove singular-ities. At room temperature, the latter holds for Fermienergies in the range 0.025 eV EF 3 eV.

The fact that electrons in graphene behave as masslessfermions poses a difficulty to the development of hydro-dynamic models with explicit dependency on the mass.Here, the Drude mass m∗ = ~

√πn0/vF , with n0 denot-

ing the equilibrium carrier density, is used as an effectivemass [23, 25, 26]. In the field-effect transistor (FET) con-figuration comprising a drain, a source and a gate (seeFig. 1a for a schematic representation), the electric forceexerted on the electrons is dominated by the externalpotential that screens Coulomb interaction. The appliedbias potential U has the contribution of both the parallelplate capacitance Cg = ε/d0 (with d0 denoting the dis-tance between the gate and the graphene sheet) and thequantum capacitance [27–30] Cq = 2e2

√πn/π~vF , as

U = en

(1

Cg+

1

Cq

). (2)

For carrier densities in the range n & 1012 cm−2, quan-tum capacity dominates, and the potential can be ap-proximated as U ' end0/ε. Keeping the pressure termup to first order in the density, the fluid model in (1) canbe recast in a dimensionless form as

∂n

∂t+

∂

∂x(nv) = 0,

∂v

∂t+ v

∂v

∂x+S2

v20

∂n

∂x= 0, (3)

where v0 is the electron mean drift velocity along thegraphene channel and S2 ≡ e2d0n0/(m

∗ε) + v2F /2 canbe interpreted as sound velocity of the carriers fluid, asthe dispersion relation for the electron fluctuations, ω =(v0±S)k, is similar to that of a shallow water. For typicalvalues, the ratio S/v0 scales up to a few tens.

0

2

4

6

8

10

5 501 10 100

ωr/2π

(TH

z)

S/v0

v0/L=0.2THz

v0/L=0.4THz

v0/L=0.6THz

v0/L=0.8THz

0

0.2

0.4

0.6

5 501 10 100

γ(1

012s−

1)

S/v0

1/〈τ〉

FIG. 2: Top panel: Frequency of first mode vs. S/v0, theparameter controlling the DS instability in graphene. Bot-tom panel: Instability growth rate vs. S/v0. The solid anddashed lines depict the theoretical curves from (4), while theopen dots the simulation results for v0/L = 0.4 THz. Thehorizontal solid line line indicates the instability thresholdγτ = 1, as determined by typical experimental conditions.

The hydrodynamic model in Eq. (3) contains an in-stability under the boundary conditions of fixed densityat source n(x = 0) = n0 and fixed current density at thedrain n(x = L)v(x = L) = n0v0, dubbed in the literatureas the Dyakonov-Shur (DS) instability [21, 31]. The laterarises from the multiple reflections of the plasma wavesat the boundaries, which provides a positive feedback forthe incoming waves driven by the current at the drain.Combining Eq. (3) with the asymmetric boundary con-ditions described above, the dispersion relation becomescomplex, ω = ωr + iγ, where ωr is the electron oscillationfrequency and γ is the instability growth rate [21, 32, 33]

ωr =|S2 − v20 |

2LSπ,

γ =S2 − v20

2LSlog

∣∣∣∣S + v0S − v0

∣∣∣∣ .(4)

Therefore, given the dependence of S with gate voltage,and as v0n0 = IDS/We, with IDS representing the source-to-drain current and W the transverse width of the sheet,the frequency can be tuned by the gate voltage and in-jected drain current, not being solely restricted to thegeometric factors of the FET.

After an initial transient time, the DS instability sat-urates due to the nonlinearities and the system goes in acycle of shock and rarefaction waves that sustain the col-lective motion of the electrons (see Fig. 1b). Evidently,such collective oscillation radiates in the same main fre-quency ωr that lies in the THz range for typical valuesof parameters, as depicted in Fig. 2. Moreover, as theelectrons are transported along density shock waves, theybunch together and behave as a macroscopic dipole. As a

3

consequence, the emitted radiation is highly coherent, afact that we will demonstrate below and that we believeto be at the basis of a THL source.

In experimental conditions, the ideal situation de-scribed above must be analysed with care. Indeed, forthe DS instability to take place, the growth rate γ hasto be larger than the total relaxation rate, 1/τ , dueto scattering with impurities and phonons. Recent ex-periments performed at room temperature point to anelectron mobility of monolayer suspended graphene ofµ ' 105 cm2V−1s−1 [27, 34], that results in an averagerelaxation rate 1/τ ≈ 1011 s−1. Fortunately, for a suit-able choice of parameters, we can safely operate in theregime γτ > 1, as shown in Fig. 2.

Numerical simulation of the THz frequency comb.−Thehyperbolic set of fluid equations in (3) have been inte-grated using a second-order (time and space) Richtmyertwo-step Lax-Wendroff scheme [35]. The suppression ofnumerical oscillations at the shock front has been imple-mented by means of a moving average filter on the spa-tial domain, from which the n(x, t) and v(x, t) profilesare obtained as well as the integrated current and ten-sion drop across the FET. From the output current anddensity, the electromagnetic field can then be calculatedfrom Jefimenko’s integral equations [36]

E(r, t) =en04πε0

∫d2r′

[n

|R|3 +∂tn

|R|2c

]R− ∂t(nv)

|R|c2 ,

B(r, t) =eµ0n0v0

4π

∫d2r′

[nv

|R|3 +∂t(nv)

|R|2c

]×R.

(5)with R = r − r′ being the displacement vector and cthe speed of light. The direct integration of Eq. (5)allows the reconstruction of the emitted fields both innear field and far field regimes. For the latter, our cal-culations provide a radiant emittance of the order of107 Wm−2. The reconstructed radiation pattern of thegraphene layer (i.e. without the reckoning radiation re-flection/absorption effects due to the metallic gate, anapproximation that holds as the ratio c/v0 ∼ 1000, im-plying the radiation wavelength λ ' 4Lc/S to be muchlarger that the typical FET dimensions), shows a wideomnidirectional profile with a half-power beam width of120, as patent in Fig. 3. In fact, the angular Poyntingvector profile is 〈S〉 ∝ | cos θ|, unlike the typical dipo-lar emitter 〈S〉dip ∝ cos2 θ. Such a wide lobe profile canbe explained by the fact that the collective movement ofcharges occurs on the entire area of the graphene FET,similarly to patch antennae [37]. The radiated spectrumconsists of a frequency comb in the THz range, with fre-quencies ω = (j + 1)ωr with j ∈ N, as can be seen in theinset of Fig. 4. Hereafter, a suitable design of a reso-nance cavity would allow the selection and amplificationof the desired mode. Furthermore, if a periodic inversion

-3-10 -5 0

0

30

60

90

120

150

180150

120

90

60

30

dB

FIG. 3: Far-field radiation pattern of the graphene FETemitter at a distance R = 1000L. Average Poynting vec-tor normalized to its maximum value vs. polar angle θ inthe vertical xz-plane. Points from numeric simulation follow〈S(θ)〉 ∝ | cos θ| closely (red solid line), whereas the radiationfrom a dipole antenna is narrower with 〈S(θ)〉 ∝ cos2 θ (bluesolid line).

0

0.2

0.4

0.6

0.8

1

1.7 1.8 1.9 2 2.1 2.2

0

0.5

1

0 2 4 6 8 10

|P(ω

)|(a.u.)

ω/2π (THz)

a.u

.

FIG. 4: Frequency comb obtained for L = 0.75µm, v0 =0.3vF and S/v0 = 20. A pulse repetition rate of 5 GHz with a10% duty cycle has been used. In the inset, the spectrum forthe dc case. Both Fourier spectra have been calculated withthe FFTW3 library [38].

of polarity is imposed across the channel, in such a waythat after the saturation time the perturbation due to DSinstability decays, a train of short pulses can be formed.This can be implemented with a pulse generator control-ling the injected current at drain with a repetition ratefrep. In Fourier space, such pulses form a THz frequencycomb around the main frequency with frequencies givenby ω = ωr + 2πjfrep with j ∈ Z as seen in Fig. 4.

THz laser coherences.−In order to determine wetheror not the emitted radiation can be understood as a THzlaser, we proceed to the calculation of both the temporal

4

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

g(1

)(τ

)

τ 2π/ω0

S/v0 = 10

S/v0 = 20

S/v0 = 40

S/v0 = 60

0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500 600 700

g(r

1,r

2)

|r1 − r2|/L

S/v0 = 10

S/v0 = 20

S/v0 = 40

S/v0 = 60

FIG. 5: Top panel: Temporal first order degree of coherencevs. delay normalized to the period. Bottom panel: Spatialdegree of coherence vs. displacement transverse to main lobepropagation.

and spatial coherences [39],

g(1)(τ) =〈E∗(r, t)E(r, t+ τ)〉

〈E2(r, t)〉 ,

g(r1, r2) =〈E∗(r1, t)E(r2, t)〉√〈E2(r1, t)〉〈E2(r2, t)〉

.

(6)

As expected, the radiated field exhibits appreciably largecoherences, both in far field and near field (not shown)regimes, that increase with S/v0, as presented in Fig. 5.As mentioned above, such an appreciable coherence de-gree is attributed to the collective motion of the electronsbunching together in the shock and rarefaction waves cy-cle. The observed oscillation in the temporal coherenceis due to the frequency mixing in the frequency comb, afeature that could be suppressed with the help of a THzcavity, allowing for mode selection in the THz comb.

Improving the DS instability.−As mentioned above,one possible limitation of our process may arise whenwe approach the threshold region γτ = 1. To circum-vent this issue without changing the setup, we found thatby varying the speed of the the plasma waves along thechannel length, i.e. by taking S = S0(1 − αx) (eitherby manipulating the permittivity ε or the gate distanced0), a remarkable enhancement of the instability growthrate can be achieved, while no significant impact on thespectrum is observable. Such modification on the veloc-ity along the FET channel introduces a positive feedbackin the current instability, which leads to a larger valueof the growth rate and, consequently, to a faster satura-

0

0.5

1

1.5

2

0 2 4 6 8 10 12

n(L

)(1

012cm−

2)

t (ps)

FIG. 6: Temporal evolution of electronic density at the drain.Comparison of the growth rate in the case of constant S = 40(red line) vs. the presence of linear gradient, S/v0 = 40(1 −0.05x/L) (blue line). An increase of ∼ 33% in the electronicdensity at saturation is attained.

tion of the DS instability. This features are illustratedin Fig. 6. As such, the electron density at saturationis higher, resulting in a ∼ 33% increase in the emittedpower for α = 0.05/L. The numerically extracted val-ued of the growth rate in the speed-gradient scheme, γ,are significantly larger than γ in Eq. (4), as it can bestated in Table I. This mechanism can be seen as anal-ogous to wave shoaling effect on shallow waters systemsand the shock wave amplitude is likewise amplified inthe presence of the velocity gradient [40]. Although outof the scope of the present work, the investigation of theDS mechanism in combination with other positive feed-back configurations, such as subtract patterning [41] andcounter flows [42], will certainly deserve our attention inthe near future.

Conclusion.−We make use of a hydrodynamic modelto describe a plasmonic instability (Dyakonov-Shur insta-bility) taking place in a graphene field-effect transistorat room temperature. Our scheme, based on the con-trol of the electron current at the transistor drain, re-sults in the emission of a THz frequency comb. Numer-ical simulations suggest that the emitted THz radiationis extremely coherent, thus be an appealing candidatefor a THz laser source. Our findings point towards amethod for the development of tuneable, all-electricalTHz antennae, dismissing the usage of external lightsources such as THz solutions based on quantum cascade

TABLE I: Normalized growth rate γ/γ in the presence of anegative gradient of the local sound speed in the FET channel,S = S0(1− αx).

S0/v0α/L 20 40 60 800.025 1.9±0.5 3.8±0.3 6.9±0.2 10.9±0.40.05 4.5±0.4 12.5±0.5 26±1 42.3±0.90.075 7.3±0.4 21.1±0.3 37±2 68±20.1 8.5±0.3 26±2 53±1 93±3

5

lasers. This puts graphene plasmonics and, in particular,graphene field-effect transistors in the run for competi-tive, low-consumption THz devices based on integrated-circuit technology, allowing the fabrication of patch ar-rays designed to enhance the total radiated power.

Additional effects can be taken into account, such asthe electron-phonon coupling. This can particularly im-portant in the case of suspended graphene, as the out-of-plane vibrations (flexural phonons) play a significant rolein the electron transport [43]. Moreover, important effectrelated to electron viscosity may arise in thin grapheneribbons [44]. The later, more relevant for very small de-vices, may hinder the plasmonic instability and, for thatreason, it is desirable to combine the Dyakonov-Shur con-figuration with other positive-feedback schemes. Finally,the instability amplification by resonances taking placein magnetized graphene plasmas may also conduct to in-teresting solutions [45].

One of the authors (H.T.) acknowledges Fundacao daCiencia e Tecnologia (FCT-Portugal) through the grantnumber IF/00433/2015.

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