POLITECNICO DI MILANO · 2017. 5. 25. · La mobilità è un parametro essenziale per sviluppare il modello della corrente fotovol-taica. I meccanismi di fotoeccitazione e successivo

POLITECNICO DI MILANO

Scuola di Ingegneria Industriale e dell'Informazione

Master Degree in Engineering Physics

Simulation and characterization of graphene basedhigh frequency photodetectors

Supervisor:

Prof. Roman Sordan

Politecnico di Milano

Co-supervisor:

Doc. Pierre Legagneux

Thales Research&Technology

Candidate:

Pierfrancesco Aversa

ID number: 835331

Academic Year 2015/2016

To my uncles Mimmo and Carmela, and my grandmother

Pasana, always with me.

Abstract

The present work thesis about simulation and characterization of graphene based high fre-quency photodetectors was developed inside the Physics Group of Thales Research&Technology,in Palaiseau. The work poses itself in a contest of study from the physical point of view of theworking principle of graphene based photodetectors and its main sources of photocurrent whena laser is shining a spot on graphene ake. The two main mechanisms are the so-called pho-tovoltaic eect, i.e. the excited electron-hole pair separated by an external electric eld, andphotobolometric eect, induced by change in conductance due to heating provided by the laserbeam.To develop this model, some physical parameters have to be extracted.In the rst part of the thesis (Chapters 1 and 2) an introduction about the physical back-

ground needed to understand the physical contest, together with the presentation of devicesfabrication and experimental setup for measurements, is present. After, (Chapter 3) a modelbased on resistivity and resistance t on graphene based transistors, supported by DC mea-surements, is developed, with which we can extract graphene mobility. Mobility is an essentialparameter in photovoltaic eect model. The photoexcitation mechanisms and subsequent elec-trons cooling are then investigated and studied theoretically (Chapter 4), with a model based onheat equation including electron-phonon cooling mechanisms to calculate electronic and latticetemperature. Finally (Chapter 5) photodetection measurements and photocurrent simulationsare present. In particular a new photocurrent model, missing in literature, is suggested, basedon the hyphothesis that the change in conductance due to heating is a direct consequence of theincrease of electronic temperature rather then lattice temperature, like instead it was suggestedin previous works. This is supported by the extraction from experimental data of recombi-nation times for photoexcited carriers and of the parameter which expresses the conductancedependence on temperature which should assume unreasonable values if the lattice temperatureincrease is considered.

Sommario

Il presente lavoro di tesi sulla caratterizzazione e simulazione di photodetector ad alta fre-quenza a base di grafene è stato sviluppato all'interno del Gruppo di Fisica di ThalesResearch&Technology, in Palaiseau. Il lavoro si pone come obiettivo quello di capire, dal pun-to di vista sico, il principio di funzionamento dei photodetector a base di grafene e le sueprincipali sorgenti di fotocorrente quando uno spot del grafene stesso viene illuminato da unasorgente laser. I due principali meccanismi sono l'eetto fotovoltaico, ossia la separazione dellecoppie elettrone-lacuna indotta da un campo elettrico esterno, e l'eetto fotobolometrico, cioèla variazione della conduttanza dovuta al riscaldamento prodotto dal laser.Per sviluppare il modello di fotocorrente, è necessario conoscere alcuni parametri sici signi-

cativi.La prima parte della tesi (Capitoli 1 e 2) consiste nella presentazione del background sico,

per inquadrare il contesto nel quale ci muoviamo, insieme ad una introduzione sulle tecniche difabbricazione dei dispositivi testati e del setup sperimentale. Successivamente (Capitolo 3) vienesviluppato un modello basato su dei t di resistenza e resistività di transistor a base di grafene,supportato sperimentalmente da misure DC. Con tale modello possiamo estrarre la mobilità delgrafene. La mobilità è un parametro essenziale per sviluppare il modello della corrente fotovol-taica. I meccanismi di fotoeccitazione e successivo rareddamento degli elettroni sono quindistudiati teoricamente (Capitolo 4), con un modello basato sull'equazione del calore, includen-do i meccanismi di rareddamento elettrone-fonone, per calcolare la temperatura elettronica edel cristallo. Inne (Capitolo 5) vengono presentate misure di photodetection e simulazioni difotocorrente. In particolare viene suggerito un modello di fotocorrente, non ancora presentein letteratura, basato sull'ipotesi che la variazione della conduttanza dovuto al riscaldamentoprodotto dalla sorgente laser sia una conseguenza diretta dell'innalzamento della temperaturaelettronica piuttosto che della temperatura del cristallo, come invece veniva aermato in prece-denti studi sull'argomento. Questo è supportato dall'estrazione, dai dati sperimentali, dei tempidi ricombinazione delle cariche fotoeccitate e del parametro che esprime la dipendenza della con-duttanza dalla temperatura; entrambi assumerebbero valori irragionevoli se venisse consideratol'aumento di temperatura del cristallo.

Acknowledgements

The realization of this work could be possible thanks to a lot of people to whom I am verygrateful. First of all I would like to thank Politecnico di Milano (in particular prof. RomanSordan and prof. Ermanno Pinotti) and Thales Research&Technology for having given me thepossibility to have this experience abroad. Then, of course, a very big and special thanks toPierre Legagneux and Alberto Montanaro, who followed me in these months of work, who advised

me, who gave me whatever I needed. Really, a very big thanks. So:THANKS. Also,I would like to thank Bernard Plaçais for the very useful and instructive discussions.Another "of course" thanks goes to my family, my father, mother and sister. You supported

me in my choices and this is a very big gift.Finally I would like to thank all the people who shared with me important moments of my life,

so really thanks, thanks, thanks to my uncles and aunts, cousins, friends from Milano, Como,Grottaglie, Palaiseau: whoever dedicated to me real and sincere moments in their life will benot forgotten and our experiences are of great importance for me.

Contents

List of gures 8

List of tables 11

1. Graphene: physical background and applications 12

1.1. Brief history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2. Basic physical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.1. Lattice structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2.2. Bandstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2.3. DOS and particle density . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.2.4. Optical absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.3. Graphene based transistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.3.1. Electrostatics of graphene gated structures . . . . . . . . . . . . . . . . . . 171.3.2. Transfer characteristic of a GFET . . . . . . . . . . . . . . . . . . . . . . 19

1.4. Applications: photodetector and optoelectronic mixer . . . . . . . . . . . . . . . . 211.4.1. Photodetectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.4.2. Optoelectronic mixer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2. Fabrication technique and experimental setup 24

2.1. Fabrication process with protection and/or passivation layer . . . . . . . . . . . . 242.1.1. Back-gated GFET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.1.2. Coplanar waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2. Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2.1. Probe station . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2.2. DC measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.2.3. Photodetection measurements . . . . . . . . . . . . . . . . . . . . . . . . . 29

3. Characterization of graphene resistivity and extraction of mobility 35

3.1. Theoretical limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1.1. Acoustic phonon scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1.2. Remote interfacial phonons . . . . . . . . . . . . . . . . . . . . . . . . . . 363.1.3. Mobility and charged impurities . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2. Four-probe method for extraction of mobility . . . . . . . . . . . . . . . . . . . . 393.2.1. The issue of contact resistances . . . . . . . . . . . . . . . . . . . . . . . . 393.2.2. General formulation of graphene resistivity . . . . . . . . . . . . . . . . . 393.2.3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3. Two-probe method for extraction of mobility . . . . . . . . . . . . . . . . . . . . 423.3.1. Starting equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.3.2. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4. Hot carrier eects and temperature simulations 48

4.1. Photoexcited carrier dynamics and cooling . . . . . . . . . . . . . . . . . . . . . . 484.1.1. Photoexcitation cascade . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.1.2. Electron-lattice cooling and supercollisions . . . . . . . . . . . . . . . . . . 51

Contents 7

4.2. Complete heat equation model for electronic temperature . . . . . . . . . . . . . 534.2.1. Analytic solution and numerical validation for heat equation with electron-

lattice cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.2.2. Solution and validation with supercollisions . . . . . . . . . . . . . . . . . 57

5. Photocurrent measurements and simulations 60

5.1. Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.1.1. Results over the tested devices . . . . . . . . . . . . . . . . . . . . . . . . 605.1.2. Laser scan over dierent positions . . . . . . . . . . . . . . . . . . . . . . 64

5.2. Photocurrent modelization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.2.1. Photovoltaic current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.2.2. Photobolometric eect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.2.3. Model validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6. Conclusions and perspectives 76

A. Appendix - Matlab script for resistivity t 77

A.0.1. Four-probe method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77A.0.2. Two-probe method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

B. Appendix - Analytic solution of heat equation including electron-lattice cooling 82

C. Appendix - Resistance t for the graphene coplanar waveguides tested 84

Bibliography 88

List of Figures

1.1. Honeycomb lattice and its Brillouin zone. Adapted from [1]. . . . . . . . . . . . . 131.2. Bandstructure of graphene. Adapted from [1]. . . . . . . . . . . . . . . . . . . . . 151.3. Geometry used to calculate density of states in a two dimension system. . . . . . 161.4. Excitation processes responsible for absorption of light in graphene. Adapted

from [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.5. Work function and electron anity in graphene. . . . . . . . . . . . . . . . . . . . 191.6. Band diagram of gate-oxide-graphene structure at VG = 0, left, and VG > 0, right.

For simplicity here φgg = 0. Picture adapted from [3]. . . . . . . . . . . . . . . . 191.7. GFET structure with a back-gate. Adaptrd from [4]. . . . . . . . . . . . . . . . . 201.8. GFET circuital scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.9. Scheme of GFET transfer characteristic. . . . . . . . . . . . . . . . . . . . . . . . 211.10. Schematic representation of graphene based photodetector. Adapted from [5]. . . 22

2.1. Standard fabrication steps for graphene back-gated transistors. . . . . . . . . . . 242.2. Process ows of graphene devices fabrication without any protection/passivation

layers (a), with protection process (b), and with protection/passivation process(c). Adapted from [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3. Representative scheme of a coplanar waveguide integrating a graphene lm. Adaptedfrom [7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4. Photo of the probe station Cascade Microtech Summit 12000. . . . . . . . . . . . 272.5. Photo of four-probe measurement conguration. . . . . . . . . . . . . . . . . . . . 282.6. Circuital scheme of two-probe measurement. . . . . . . . . . . . . . . . . . . . . . 282.7. Circuital scheme of four-probe measurement. . . . . . . . . . . . . . . . . . . . . 292.8. Photo of typical setup for photodetection measurements. . . . . . . . . . . . . . . 292.9. Complete scheme of photodetection measurements. . . . . . . . . . . . . . . . . . 302.10. Photo of MZM and coupler used for measurements. . . . . . . . . . . . . . . . . . 312.11. Photo of the optical system used for measurements. . . . . . . . . . . . . . . . . . 322.12. Photo of the VNA used for measurements. . . . . . . . . . . . . . . . . . . . . . . 322.13. IR photo of the laser spot on the graphene channel. . . . . . . . . . . . . . . . . . 332.14. Converted image from g. 2.13, in pixels and intensity in arbitrary units. . . . . . 34

3.1. Calculated resistivity (from eq. 3.2 including the behaviour described in equations3.4 and 3.5) for carrier density nS = 1, 3, 5 · 1012 cm−2 in logarithmic (left panel)and linear (right panel) scale. Adapted from [8]. . . . . . . . . . . . . . . . . . . . 36

3.2. Resistivity of two GFET samples with mechanical expholiated graphene (a, sam-ple 1; b, sample 2) from [9] in function of temperature and for dierent gatevoltages (for 300 nm SiO2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3. Same experimental values of resistivity of g. 3.2, in logarithmic scale, tted withthe expression 3.8 (dashed lines) [9]. . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4. Gate-voltage dependent mobility limits for three samples in ref. [9] at T = 300 K. 393.5. Schematic of the graphene FET with the back gate and scheme of the band

diagram showing the charge transfer region (hatched). Adapted from [10]. . . . . 403.6. Schematic of four-probe method geometry. . . . . . . . . . . . . . . . . . . . . . . 41

List of Figures 9

3.7. Experimental (red circles) and calculated (blue line) resistivity for device 5 (a)and device 7 (b) in function of VG − Vth, Vth Dirac point voltage. . . . . . . . . . 43

3.8. Experimental (red circles) and calculated (blue line) total resistance for device 5(a) and device 7 (b) in function of VG − Vth, Vth Dirac point voltage. . . . . . . . 46

3.9. Direct comparison between the value found of mobilities (a) and residual carrierdensity (b) with four-probe method (red bars) and two-probe method (blue bars)for devices 1, 3, 5, 6, 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.1. Scheme of Auger processes: Auger Recombination (AR, left) and Impact Ioniza-tion (II, right). Adapted from [11]. . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2. Impact excitation (IE) cascade of a photoexcited carrier with initial energy E0.Adapted from [12]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.3. Energy-loss rate via impact excitation, Jel (blue curve), and optical phonon emis-sion, Jph (red curve) for a typical doping of EF = 0.2 eV. Adapted from [13]. . . 51

4.4. Kinematics of supercollisions and normal collisions at T > TBG. Adapted from [14]. 524.5. Comparison between analytical (a) and numerical (b) solution for equation 4.10. 554.6. Numerical solution of eq. 4.10 without the constraint of xed electronic thermal

conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.7. Numerical solution of the heat equation from which the solution is shown in g.

4.6 but adding the supercollision term. . . . . . . . . . . . . . . . . . . . . . . . . 564.8. Numerical solution for electronic temperature in function of space for VG = 12 V

(Dirac point voltage) and power density P = 0, 175 mW(µm)2

. . . . . . . . . . . . . . . 584.9. Experimental ((a), adapted from ref. [15]) and numerical (b) solution for elec-

tronic temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.1. Magnitude (a) and phase (b) of the electrical power relative to the modulated partof the photocurrent, together with the DC current, in function of gate voltage VGfor device RF1 at VDS = 3 V and P0 = 62,9 mW. . . . . . . . . . . . . . . . . . . 62


5.3. Magnitude (a) and phase (b) of the electrical power relative to the modulated partof the photocurrent, together with the DC current, in function of gate voltage VGfor device RF3 at VDS = 3,5 V and P0 = 62,9 mW. . . . . . . . . . . . . . . . . . 63



5.6. Magnitude (a) and phase (b) of the electrical power relative to the modulated partof the photocurrent, together with the DC current, in function of gate voltage VGfor device RF6 at VDS = 3,5 V and P0 = 62,9 mW. . . . . . . . . . . . . . . . . . 64

5.7. DC current and magnitude and DC current and phase in function of gate voltageVG for device RF2 at VDS = 4 V and P0 = 62,9 mW for three dierent positionsof the laser spot in the graphene channel. (Scheme of the graphene channel is notin scale). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.8. Numerical solution of electronic (a) and lattice (b) temperature for the sampleRF1 at VGth = 0 and Popt = 62, 9 mW . . . . . . . . . . . . . . . . . . . . . . . . . 68

List of Figures 10

5.9. Numerical solution of electronic (a) and lattice (b) temperature for the sampleRF1 at x=0, in function of VG and the incident optical power P0. . . . . . . . . . 68






5.15. (a) Total measured photocurrent (blue line), simulated photobolometric current(red line) and extracted photovoltaic current (green line) in function of gate volt-age at VDS = 3 V and P0 = 62, 9 mW for the device RF1 (b) Comparisonbetween the photovoltaic peak in (a) (circles) and root mean square of calculatedphotovoltaic current from eq. 5.4 (lines) in function of VDS at dierent P0. . . . . 71


5.17. (a) Total measured photocurrent (blue line), simulated photobolometric current(red line) and extracted photovoltaic current (green line) in function of gate volt-age at VDS = 3, 5 V and P0 = 62, 9 mW for the device RF3 (b) Comparisonbetween the photovoltaic peak in (a) (circles) and root mean square of calculatedphotovoltaic current from eq. 5.4 (lines) in function of VDS at dierent P0. . . . . 72



5.20. (a) Total measured photocurrent (blue line), simulated photobolometric current(red line) and extracted photovoltaic current (green line) in function of gate volt-age at VDS = 3, 5 V and P0 = 62, 9 mW for the device RF6 (b) Comparisonbetween the photovoltaic peak in (a) (circles) and root mean square of calculatedphotovoltaic current from eq. 5.4 (lines) in function of VDS at dierent P0. . . . . 73

C.1. Experimental (red) and tted (blue) total resistance for device RF1. . . . . . . . 84C.2. Experimental (red) and tted (blue) total resistance for device RF2. . . . . . . . 85C.3. Experimental (red) and tted (blue) total resistance for device RF3. . . . . . . . 85C.4. Experimental (red) and tted (blue) total resistance for device RF4. . . . . . . . 86C.5. Experimental (red) and tted (blue) total resistance for device RF5. . . . . . . . 86C.6. Experimental (red) and tted (blue) total resistance for device RF6. . . . . . . . 87

List of Tables

2.1. Summary of 5 dierent power coming out from the EDFA, read after the beamsplitter, and corresponding calculated incident values. . . . . . . . . . . . . . . . 33

3.1. Summary of geometrical parameters of the devices tested, extracted mobilities andn0, and maximum relative error between experimental and calculated resistivity. . 42

3.2. Mean value and standard deviation for µ and n0 from table 3.1. . . . . . . . . . . 423.3. Summary of extracted mobilities, n0,2p, RC,tot, experimental range of RC,tot and

maximum relative error between experimental and calculated resistance from thetwo-probe method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.4. Mean value and standard deviation for µ2p, n0,2p and RC,tot from table 3.3. . . . 45

4.1. List of physical parameters taken from ref. [16]. . . . . . . . . . . . . . . . . . . . 554.2. List of physical parameters taken from ref. [15]. . . . . . . . . . . . . . . . . . . . 57

5.1. Summary of devices tested with respective length and width of the graphenechannel, the sweep of VDS applied and the sweep of incident optical power. . . . 61

5.2. Extracted values of mobility, residual carrier density and total contact resistancefor the devices tested together with the reported Dirac point voltage Vth. . . . . . 62

5.3. Extracted values of dGdT and τrc for all the devices. . . . . . . . . . . . . . . . . . . 75

C.1. Extracted values of mobility, residual carrier density, total contact resistance andmaximum relative error with respect to the experimental resistance for the devicestested. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

1. Graphene: physical background and

applications

1.1. Brief history

Graphene is a 2D lattice of carbon atoms disposed in a hexagonal conguration.From theoretical studies, Landau and Peierls demonstrated that a perfect 2D lattice cannot

exist, since thermal uctuations should destroy long-range order, resulting in a melting of thelattice at nite temperature. Anyway experiments show that a freely suspended graphene canexist, since interaction between bending and stretching long-wavelenght phonons stabilizes a 2Dcrystal through its deformation in the third dimension [17].The very big breakthrough came in 2004 when K. Novoselov and A. Geim put in evidence

for the rst time the stability of some layers of mechanical exfoliated graphene and studied thetransport properties [18].From these works, new paths of studies and technological applications arose, continuing to

fascinate the scientic community.

1.2. Basic physical properties

1.2.1. Lattice structure

The atomic conguration of a carbon atom is 1s22s22p2, where the 4 valence electrons ofthe two most external orbitals allow the formation of σ and π orbitals. In graphene carbonatoms are disposed in a hexagonal conguration (honeycomb) on a 2D plane; each of them has3 nearest neighbors with which a sp2 hybridization takes place, forming a covalent σ bond. Thiskind of bond makes graphene very thin but also very resistant and then attractive for exibleelectronics.The remaining electron of the pz orbital forms with the correspective nearest neighbor an out

of plane π bond. These electrons are responsible for the transport properties of graphene.As shown in gure 1.1, left panel, we can distinguish a periodicity (Bravais lattice) if we

consider this structure as a triangular lattice with a basis of two nearest-neighbours carbonatoms, labeled A and B. By taking the origin as in the gure, the lattice vectors are [1]:

a1 =a

2(3,√

3),a2 =a

2(3,−

√3) (1.1)

where a ≈ 1.42Å is the carbon-carbon distance. By taking as origin the gamma point inreciprocal space, the reciprocal lattice vectors can be easily calculated to be:

b1 =2π

3a(1,√

3),b2 =2π

3a(1,−

√3) (1.2)

From these vectors we can construct the primitive cell in the reciprocal space which, as can beseen in gure 1.1, right panel, is a hexagon with the characteristic points K and K ′. These twopoints are called Dirac points. They are of particular importance for the physics of graphene,as will be explained later. Their position are:

1.2 Basic physical properties 13

Figure 1.1.: Honeycomb lattice and its Brillouin zone. Adapted from [1].

Left: lattice structure of graphene, that can be seen as two interpenetrating triangular lattices (a1 and a2 arethe lattice unit vectors, and δi, i=1,2,3 are the nearest-neighbor vectors). Right: corresponding Brillouin zone.

K =

(2π

3a,

2π

3√

3a

),K′ =

(2π

3a,− 2π

3√

3a

)(1.3)

The three nearest-neighbor vectors in real space are given by:

δ1 =a

2(1,√

3), δ2 =a

2(1,−

√3), δ3 = −a(1, 0) (1.4)

1.2.2. Bandstructure

To understand transport and optical properties of graphene, a brief review on the particularbandstructure is useful.The electronic structure can be described using a tight-binding Hamiltonian. Since the bond-

ing and anti-bonding σ bands are well separated in energy, they can be neglected in semi-empirical calculations, retaining only the two remaining π bands [5].As shown in the last section, we can think about graphene as two compenetrating sublattices;

so we use the pz orbitals of atoms A and B as basis vectors for our wavefunction system. Thenthe general wavefunction of an electron moving in this lattice structure is the following Blochfunction:

|Ψ(k, r)〉 =∑j

ψA(k)eik·rAj

∣∣φA(r− rAj )⟩

+∑j

ψB(k)eik·rBj

∣∣φB(r− rBj )⟩

(1.5)

where k = (kx, ky) is the vector in rst Brillouin Zone, r = (rx, ry) is the vector in real space withrespect to a general origin in the 2D plane,

∣∣φA(r− rAj )⟩and

∣∣φB(r− rBj )⟩are the displaced

pz orbitals, in ket notation, of jth A and B atoms in the lattice and ψA(k) and ψB(k) are theBloch coecients to be determined.The time-independent Schrödinger equation is:

H |Ψ(k, r)〉 =

(− ~

2m∇2 + U(r)

)|Ψ(k, r)〉 = E(k) |Ψ(k, r)〉 (1.6)

with ~ = h2π , h Planck constant, m is the electron mass and U(r) is a periodic potential with

lattice periodicity. Then:

• substituting to |Ψ(k, r)〉 the expression written in the equation 1.5


• multiplying at left equation 1.6 once by 〈φA(r− rA)| and once by 〈φB(r− rB)|, which aredisplaced pz orbitals for xed chosen A and B atoms

• applying the orthogonality conditions for eigenfunctions, i.e.:

〈φA(r− rA)|∣∣φA(r− rAj )

⟩=

1 for rA = rAj0 otherwise

〈φB(r− rB)|∣∣φB(r− rBj )

⟩=

1 for rB = rBj0 otherwise

〈φA(r− rA)|∣∣φB(r− rBj )

⟩= 〈φB(r− rB)|

∣∣φA(r− rAj )⟩

= 0

• applying the hypotesis of tight-binding approach, which are:

〈φA(r− rA)|H∣∣φA(r− rAj )

⟩ 6= 0 for rA = rAj= 0 otherwise

〈φB(r− rB)|H∣∣φB(r− rBj )

⟩ 6= 0 for rB = rBj= 0 otherwise

(these quantities are the on-site energies that we can set to 0)

we end up with an eigenequation in matrix form:(0 −γf(k)

−γf∗(k) 0

)(ψA(k)ψB(k)

)= E(k)

(ψA(k)ψB(k)

)(1.7)

whereγ = −

⟨φA(r− rAj )

∣∣H ∣∣φB(r− rBj )⟩

= −⟨φB(r− rBj )

∣∣H ∣∣φA(r− rAj )⟩

(1.8)

is the hopping integral between rst neighbor orbitals (typical values for γ are 2.9-3.1eV [5]) and

f(k) =

3∑j=1

eik·δj (1.9)

The eigenvalues are:E(k) = ±γ|f(k)| (1.10)

where

|f(k)| =

√1 + 4cos2(

√3

2akx) + 4cos(

3

2aky)cos(

√3

2akx) (1.11)

With one pz electron per atom in the π − π∗ model the (-) band (negative energy branch)in the expression 1.10 is fully occupied, while the (+) branch is totally empty. These occupiedand unoccupied bands touch at the K points. The Fermi level EF (at T = 0 K) is the zero-energy reference, and the Fermi surface is dened by K and K ′. A picture of the calculatedbandstructure is shown in gure 1.2.As we can see graphene is a zero-bandgap material (semimetal) and the main physical prop-

erties are then related to K points. Then is interesting to expand eq. 1.10 near K(K ′). Thisyields the linear π − π∗ bands for Dirac fermions:

E±(q) = ±~vF |q| (1.12)

where q = k −K and vF is the electronic group velocity, vF ≈ 106m/s. The linear dispersiongiven by eq. 1.12 is the solution to the following eective Hamiltonian at the K point:

H = ~vF (σ · p) (1.13)


Figure 1.2.: Bandstructure of graphene. Adapted from [1].

Left: energy spectrum of graphene lattice. Right: zoom in of the energy bands close to one of the Dirac points.

where p = −i∇ , and σ's are the pseudo-spin Pauli matrices [5].The most interesting aspect of the graphene energy spectrum is that its charge carriers can be

described by a Dirac spectrum for massless fermions rather than the usual Schrödinger equationfor nonrelativistic particles. This means that electrons in graphene all move at a constantvelocity (vF ) regardless of their momentum. Because of this linear dispersion, the quasiparticlesin graphene behave very dierently from other semiconductors or metals, which have in generala parabolic dispersion relation [19].Finally, the Bloch coecients are:

ψ(k) =1√2

(1

∓ f∗(k)|f(k)|

)(1.14)

which near K can be expanded and rewritten as:

ψ(k) =1√2

(1±eiθq

)(1.15)

Near K ′ the solutions are quite the same, provided that in the eective Hamiltonian we have−σ∗ instead of σ; the eigenfunctions are:

ψ(k) =1√2

(1

∓e−iθq

)(1.16)

where θq is the angle formed between q and the horizontal axis of the reciprocal space.The particular shape of these wavefunctions near Dirac points are at the basis of many physical

phenomena that manifest in graphene like Klein tunneling [20] and Half-Integer Quantum Halleect [21].

1.2.3. DOS and particle density

From now on, unless otherwise specied, we will restrict the analysis around the K point. Wewill call for semplicity of notation |q| = k and then we rewrite eq. 1.12 as:

E± = ±~vFk (1.17)


Figure 1.3.: Geometry used to calculate density of states in a two dimension system.

By nding the k−space volume in an energy interval between E and E+dE, one can nd the number of allowedstates.

To calculate the density of states per unit energy (DOS(E)), we start from the reciprocalspace taking as reference gure 1.3:The number of allowed states is the k-space area between k and k + dk divided by

( (2π)2A

),

which is the k-space volume per electron state (where A is the area of graphene ake):

no of allowed states in 2D system in the ring of thickness dk = 42πkdk( (2π)2

A

) (1.18)

where the 4 takes into account spin and valley degeneracy. To nd the number of states perunit area we have to divide again by the area A:

no of allowed states in 2D system per unit area =2kdk

π(1.19)

Then, substituting the relations between E and k from eq. 1.17, we nd:

no of allowed states in 2D system per unit area per unit energy =2|E|

π(~vF )2dE (1.20)

so, we can call

DOS(E) =2|E|

π(~vF )2(1.21)

Using the equilibrium Fermi-Dirac function fD(E − µ) = 1

exp(E−µkBT

)+1, the electron density

per unit area ne at a given chemical potential µ (usually called Fermi level EF ) for nonzerotemperature T reads:

ne =

∫ +∞

0DOS(E)fD(E − µ)dE (1.22)

Using electron-hole symmetry DOS(E) = DOS(−E) we have a similar relationship for thehole density nh:

nh =

∫ 0

−∞DOS(E)

(1− fD(E − µ)

)dE (1.23)

Full charge density per unit area or the charge imbalance reads as nS = ne − nh which witha good approximation can be calculated to be [22]:

nS ≈µ2

π(~vF )2(1.24)

1.3 Graphene based transistors 17

Figure 1.4.: Excitation processes responsible for absorption of light in graphene. Adapted from[2].

1.2.4. Optical absorption

Despite being only one atom thick, graphene is found to absorb a signicant fraction of incidentwhite light, a consequence of graphene's unique electronic structure.If we consider suspended graphene and taking into account the linear approximation for band-

structure near K points, as evidenced in the previous section, the graphene absorption can bederived by calculating the absorption of light by two-dimensional Dirac fermions with Fermi'sgolden rule.Indeed sending a perpendicular incident light wave, electrons from the valence band are excitedinto empty states in the conduction band with conserving their momentum and gaining theenergy E = ~ω, as can be seen in gure 1.4 [2].The trasmittance T is calculated to be [5]:

T = (1 + 0.5πα)−2 ≈ 1− πα ≈ 97.7% (1.25)

where α = e2

4πε0~c ≈ 1/137 is the ne structure constant (c = light speed, ε0 = vacuum permit-tivity).Then the optical absorption is:

A = 1− T ≈ 2.3% (1.26)

This value is found to be quite constant for photon energies in the range of near-infrared toviolet [2].

1.3. Graphene based transistors

Thanks to its mechanical, thermal and electronic properties, graphene aroused great interestamong the scientic community. In particular, graphene was studied as a possible candidate tosubstitute Silicon in electronic devices. For this reason, one of the most important application isthe realization of graphene based transistors, where graphene is used as the conductive channel.Moreover, DC measurements were performed on this kind of devices and will be presented inChapter 3. In this section a brief review about their working principle is presented.

1.3.1. Electrostatics of graphene gated structures

As for conventional semiconductors, it is useful to start the analysis of a graphene-basedeld-eect transistor by studying the electrostatics of a metal-oxide-graphene stack.


We start by dening some key concepts: the chemical potential, the electrochemical potentialand the electrostatic energy.The chemical potential (µ) is the energy below which the one electron levels are occupied

and above which they are unoccupied in the ground state (of a metal) and for this reason isusually referred as the Fermi level EF 1. The electrostatic potential (φ) is the amount of electricpotential energy that a unitary point electric charge would have if located at any point in space,and is equal to the work done by an external agent in carrying a unit of positive charge fromthe arbitrarily chosen reference point (usually innity) to that point without any acceleration.The electrostatic energy is simply the electrostatic potential multiplied by the electric charge,and in the case of an electron is then U = −eφ.The abstract denition of the electrochemical potential is the sum of the chemical potential

and the electrostatic energy:

ε = µ+ U (1.27)

We can now start the electrostatic analysis of the gate-insulator-graphene stack, which can beseen as a two plate capacitor. The two plates are represented by metal and graphene (see alsog. 1.6). Without loss of generality we will reference the chemical potential in graphene fromthe level of charge neutrality ENP , i.e. the points where conduction and valence band touch.Near a surface we have to include also surface eects. If we consider them, for a given metal

surface the electrochemical potential can be written as 2:

εm = −eφm −Wm (1.28)

where Wm is the work function at the surface, φm is the electrostatic potential in the vacuumin the nearby surface and -e the electron charge.Similarly to semiconductors, for graphene we can write the work function as the quantity−χg + µ (see gure 1.5), with χg graphene electron anity and µ grahene chemical potentialwith respect to ENP . Then the electrochemical potential is [3]:

εg = −χg + µ− eφg (1.29)

where φg is the graphene electrostatic potential in the nearby of the surface.Applying a gate voltage (to say, positive), we induce a dierence in the electrochemical poten-

tial of the system; with reference of grounded graphene plate we increase the chemical potentialand electrostatic potential of the graphene sheet so as they exactly compensate each other keep-ing the electrochemical potential of the graphene sample unchanged, as we can see in gure1.6.Then:

eVG = εm − εg = eφgg + µ+ e(φm − φg) (1.30)

where φgg = Wm − χg. For zero oxide charge (or, for charged oxide defects located nearly theinsulator-graphene interface) the electric eld Eox is uniform across the gate thickness (dox) andone reads [3]:

φm − φg = Eoxdox = doxenSε0εr

=enSCox

(1.31)

1see [23], chap. 28, pag. 573; for a semiconductor, the "Fermi level" doesn't refer to a unique energy, since anyenergy in the gap state separates occupied from unoccupied levels at T = 0

2see [23], chap. 18, pag. 358


Figure 1.5.: Work function and electron anity in graphene.

Taking as 0 reference the vacuum energy, the work function (i.e. the work required to extract an electron to thevacuum energy, negative by convention) is, for a given chemical potential: −Wg = −χg + µ.

Figure 1.6.: Band diagram of gate-oxide-graphene structure at VG = 0, left, and VG > 0, right.For simplicity here φgg = 0. Picture adapted from [3].

where εr is the oxide dielectric constant and Cox is the oxide capacitance per unit area expressedas:

Cox =εrε0dox

(1.32)

What is usually done is to neglect the dierence φgg and the variation of µ with respect tothe variation of electrostatic potential in eq. 1.30 [4]. Then eq. 1.31 reads:

VG ≈enSCox

(1.33)

The most important consequence is then that applying a gate voltage we can tune the chemicalpotential (EF ) of graphene, through nS (see eq. 1.24).

1.3.2. Transfer characteristic of a GFET

A FET (Field Eect Transistor) includes a gate, a channel which connects source and drainelectrodes, and an insulating material, generally an oxide, which separates the gate from thechannel. The channel conductivity and thus the current between source and drain is modulatedby the electric eld induced by a gate voltage VG applied between the gate and the sourceterminal.


Figure 1.7.: GFET structure with a back-gate. Adaptrd from [4].

In gure 1.7, we can see a section of a back-gated GFET (Graphene Field Eect Transistor). Atypical GFET is made starting from a doped Si substrate which acts as a back-gate. This gate isseparated from the graphene channel by a 300-nm SiO2 layer operating as a back-gate dielectric[4]. After transfer of the graphene lm, metal contacts of drain and source are patterned usinglithographic techniques. Details about the fabrication process can be found in the next chapter.

Figure 1.8.: GFET circuital scheme.

Label: S = source, D = drain, G = gate, VDS = VD − VS (drain-source voltage), VGS = VG − VS (gate-sourcevoltage), ID = drain current.

Let's apply a positive constant drain-source voltage VDS , in order to observe a current in thechannel (g. 1.9). For simplicity, we assume this VDS "small" enough to consider VGD = VGS .The transfer characteristic is the drain current (or conductivity) dependence on the gate voltage.Starting from the ideal situation for which the Fermi level is exactly at the neutrality point ingraphene, applying a positive voltage VGS > 0 promote an electron accumulation in the channel(n-type channel) while negative voltages VGS < 0 lead to a p-type channel [4]. This arises fromthe considerations done in the previous section and from equation 1.33. The resulting transfercharacteristics is therefore composed by two branches separated by the Dirac point voltage wherewe have the minimum of the curve, for VG = 0. This peculiar behaviour is possible thanks tothe gapless nature of graphene, and is referred as electrostatic doping.A scheme of the complete transfer characteristic is presented in g. 1.9. In this case, an initial

p-doping is present at VG = 0. This situation is more realistic with respect to the ideal casepresented above.The position of the Dirac point depends on several factors: the dierence between the work

functions of the gate material and the graphene, the type and density of the charges at theinterfaces at the top and bottom of the channel, and any doping of the graphene [4]. Anothernon-ideality is the minimum of conductivity, that is never zero, neither when the Dirac pointvoltage is applied. This is due to the precence of the so-called electron-hole puddles, i.e. densityuctuation due to charged impurities from the substrate [24].

1.4 Applications: photodetector and optoelectronic mixer 21

Figure 1.9.: Scheme of GFET transfer characteristic.

An initial p-doping is present; the minimum is reached at VGS = Vth.

From GFET characteristic, two important properties have to be noted: the ambipolarity andcarrier mobility.Ambipolarity, i.e. the capability of a single GFET device to conduct with both electrons and

holes, comes directly from the graphene bandstructure, as shown in gure 1.9.We can suppose and model as a very rst approximation the graphene channel as a tunable

resistance. Then, we can relate the mobility (here called µ) to the conductivity through theformula 3 σ = nSeµ. Huge values of mobility (≈ 105cm2V −1s−1) are obtained in high qualitysuspended graphene or on particular substrates like hBN [25] [26]. Anyway values of ≈ 103 −104cm2V −1s−1 were measured on SiO2 substrate, still high with respect to other typical FETs[4].

1.4. Applications: photodetector and optoelectronic mixer

Graphene is an interesting material for photonic and optoelectronic applications for dierentreasons:

• graphene has a wide light absorption spectrum, given its particular bandstructure. Withrespect of being a monolayer material, it has a very high absorption coecient, as seen insection 1.2.4;

• graphene has a high value of carrier mobility, as seen in the previous section, associatedwith a short (≈ ps) photocarrier lifetime; this allows to have short response times fordevices like high frequency photodetectors;

• graphene is compatible with all the standard silicon CMOS technological steps; this opensthe ability to integrate electronics and optics on a single cost-eective chip [27].

3see [23], chap. 28, pag. 563


Figure 1.10.: Schematic representation of graphene based photodetector. Adapted from [5].

A large part of the work presented in this thesis is dedicated to the physical understandingand simulation of photocurrent in graphene. This investigation is necessary for properly designand optimize graphene-based photodetectors. For this reason in this section are then presentedthe working principles of graphene photodetectors and a particular function implemented withthem, that is, optoelectronic mixing. Details about the sample characteristics and experimentalsetup can be found in the next chapter.

1.4.1. Photodetectors

Photodetectors (PDs) measure photon ux or optical power by converting the absorbed pho-ton energy into electrical current. The photoelectrical response of graphene has been widelyinvestigated, both experimentally and theoretically [5].All our experiments are performed by employing a laser with 1,55 µm wavelength, that is the

typical telecom wavelength. This value corresponds to a photon excitation energy of 0,8 eV (sowe can induce a transition for an electron in the valence band from -0,4 eV to +0,4 eV, takingas zero reference the neutrality point).In gure 1.10 a scheme of graphene based photodetector is illustrated. As can be seen, a

graphene channel is contacted by two metal contacts. Light impinging the channel can generatean electrical photoresponse, which can be detected by means of the two metallic terminals.According to ref. [16], we can expect three sources of photocurrent:

photothermoelectric (PTE), i.e. the thermoelectric eect induced by light illumination; itexploits a dierence of the Seebeck coecient in the material, reachable if we have forexample a p-n junction

photovoltaic (PV), i.e. the current arising from the photogenerated electron-hole pairs

photobolometric (PB), which manifests as a change of conductance when the material is heatedby an electromagnetic radiation.

For devices like the one showed in g. 1.10 or the coplanar waveguides tested in this thesis(for which a description can be found in the next chapter), the contribution of PTE current forbiased uniform samples is negligible since we don't have a density of states variation along thechannel necessary to have a dierence in Seeback coecient [16].Both PV and PB eects need a DC bias to detect the signal generated after interaction with

light. If we consider a graphene transistor like the one described in section 1.3.2, one can controland change the photocurrent nature by simply sweeping the gate voltage. Near the Dirac pointthe PV contribution dominates, while at high doping (far away from the Dirac point) the PBcurrent is present.If we modulate the intensity of the laser impinging the channel, at a frequency fopt, an AC

photocurrent having the same frequency fopt is produced and can be detectable, with the help


of a spectrum analyzer or a VNA (vector network analyzer). Working near the Dirac point,the photocurrent was found to be proportional to the optical power of the laser and to thedrain-source voltage [27]:

Iph = αPoptVDS (1.34)

where α is a dimensional proportionality constant. But the optical power has a continuous-wave part and a modulated part:

Popt = PCW + Pmsin(2πfoptt) (1.35)

Then, the photocurrent is:

Iph = αPCWVDS + αPmVDSsin(2πfoptt) = Iph,CW + Iph,m (1.36)

So also the photocurrent has a continuous (Iph,CW ) and a modulated (Iph,m) part, which isthe one of interest.

1.4.2. Optoelectronic mixer

Properly designed graphene photodetectors can perform the function of optoelectronic mixers(OEMs). OEMs are devices that mix an electrical signal at frequency fele and an intensity-modulated optical signal at frequency fopt. Two electrical signals are generated at frequenciesfup = fele + fopt (up-conversion) and at fdown = |fele − fopt| (down-conversion).Recently, 30 GHz optoelectronic mixing in commercially-available graphene was demonstrated

for the rst time [27]. In particular, using a 30 GHz intensity-modulated optical signal and a29.9 GHz electrical signal, in ref. [27] they showed frequency downconversion to 100 MHz.The working principle is quite simple. Eq. 1.34 always holds, but now an RF signal is applied

instead of a DC bias, so VDS = Vmsin(2πfelet + φ), where Vm is the amplitude of the RFelectrical signal, and φ is a phase shift with respect to the optical power modulation. Iph,m canthus be rewritten as [27]:

Iph = αPmsin(2πfoptt)Vmsin(2πfelet+ φ) (1.37)

giving:

Iph =αPmVm

2[cos(2π(fopt − felet− φ)− cos(2π(fopt + felet+ φ)] (1.38)

so two signals are generated at fup = fele + fopt and fdown = |fele − fopt|, explaining theoptoelectronic mixing principle.

2. Fabrication technique and experimental

setup

In this thesis, two dierent kind of devices where used to investigate dierent physical proper-ties of graphene, as will be described in details in Chapters 3 and 5. This chapter is dedicated tomethods, and is divided in two sections. The rst section is dedicated to the description of thetechnological steps that were necessary to fabricate the devices just mentioned. In the secondsection, a detailed description of the experimental setup will be carried on.

2.1. Fabrication process with protection and/or passivation layer

In the following section there will be an introduction of the technological steps for devicesfabrication. In particular we will motivate and describe the use of the passivation and protectionlayer used to stabilize some crucial graphene's properties.

2.1.1. Back-gated GFET

Electrical measurements for devices characterization were done on standard back-gated graphenebased transistor for which a brief scheme was presented in section 1.3.2 and in gure 1.7.In gure 2.1 we can see the main steps used for the fabrication.

a) The studied devices are fabricated from graphene grown by CVD (Chemical Vapour De-position) on Cu foils and then wet transferred onto the target SiO2/Si substrate usingthe standard poly(methyl methacrylate) (PMMA) technique. These steps are routinelyrealized and result in standardized, commercially available Graphene on SiO2/Si wafers,and are developed in Graphenea 1.

1Graphenea is a private European company (based in Spain) focused on the production of high quality graphenefor industrial applications.

Figure 2.1.: Standard fabrication steps for graphene back-gated transistors.

2.1 Fabrication process with protection and/or passivation layer 25

Figure 2.2.: Process ows of graphene devices fabrication without any protection/passivationlayers (a), with protection process (b), and with protection/passivation process (c).Adapted from [6].

b) The metallic alignment marks are realized by the lift-o technique.

c), d) Photoresist pads are patterned to dene the graphene channel by oxygen plasma etching.

e) Source/drain electrodes are dened using a lithography step followed by Ti/Au (20 nm/80nm) layer evaporation and a subsequent lift-o process.

Thus the graphene layer is directly exposed to fabrication processes and to air. For thisreason unintentional doping of graphene due to adsorbates at the graphene surface leads tovarious charge carrier densities. Usually a strong p-doping is present, due to a redox reactionin adsorbed water molecules, because SiO2 is an hydrophilic surface and can accumulate watermolecules on its surface [28]. As a consequence, the conductance minimum of a eld eectgraphene transistor is obtained for large gate elds.Moreover, trap states due to adsorbates at the dielectric/graphene interface and on the

graphene surface lead to transfer characteristics exhibiting a large hysteresis, i.e. a shift ofthe curve sweeping the gate voltage [29].To overcome these problems, protection and/or passivation layer were applied on graphene

transistors. In particular, in ref. [6] a statistical study was conducted in order to understandthe best combination of technological process to address these issues.In a rst set of devices, the standard process described above was performed (g. 2.2(a)).

In parallel, a second set of devices (see gure 2.2(b)) was prepared with a similar fabricationprocess ow, except that a protection layer is introduced. It is a thin (1 nm) aluminum layerdeposited on top of the transferred graphene layer by electron beam evaporation. This thicknesshas been chosen to ensure full oxidation of the Al lm in O2 atmosphere. This layer avoidsgraphene contamination during the fabrication process and thus acts as a protection layer. Then,lithographic steps are prepared to pattern graphene channels and to dene metallic source/drain

2.1 Fabrication process with protection and/or passivation layer 26

Figure 2.3.: Representative scheme of a coplanar waveguide integrating a graphene lm.Adapted from [7].

contacts. The Al2O3 layer is removed solely on contact areas prior to metal deposition in aTetramethylammonium hydroxide (TMAH) based solution. A third set of devices (see gure2.2(c)) is prepared with the same protection layer and with a postfabrication 30 nm thick Al2O3

lm deposited by ALD (Atomic Layer Deposition).The graphene quality was probed with Raman spectroscopy techniques. The bare graphene

was taken as reference. In both cases (only protection and protection/passivation processes) nostructural defects were introduced.The main results evidenced that, on 300 nm SiO2 dielectric [6]:

for the initial doping level,

• all of the devices without protection layer exhibited a strong p-doping such that theDirac point wasn't even observable within the instrumental limits of the gate voltagerange;

• 58% of the devices with only protection layer exhibited an observable Dirac pointwith a residual p-doping present (6 4 · 1012cm−2);

• 75% of the devices with protection/passivation layers exhibited a relatively low initialp or n doping (6 4 · 1012cm−2) and a good control of the minimum of conductance;

for the hysteresis

• 66% of the previous 58% of the devices with only protection layer exhibited a smallhysteresis on Dirac point voltage (∆Vth ≈ 5 V );

• 73% of the previous 75% of the devices with protection/passivation layers exhibiteda small hysteresis on Dirac point voltage (∆Vth ≈ 5 V ).

These results highlight the necessity of performing both protection and passivation layers toobtain hysteresis free graphene devices based on very low doped graphene. Importantly, thestability of these results in time was observed, thanks to the passivation process [6].

2.1.2. Coplanar waveguides

As will be detailed in Chapter 5, high frequency photodetection measurement where performedin this work. In order to probe high frequency electrical signals, graphene was embedded in acoplanar waveguide. The scheme of the device is presented in gure 2.3.As can be seen, a coplanar waveguide in a typical ground-signal-ground conguration was

interrupted in the middle of the signal line (central metalization) by a monolayer graphene

2.2 Experimental setup 27

Figure 2.4.: Photo of the probe station Cascade Microtech Summit 12000.

channel. The chosen substrate is a High-Resistivity Silicon with a 2 µm thick SiO2 layerobtained by thermal oxidation.The design used was consequent to previous simulations on coplanar waveguide without

graphene in order to have:

- a 50 Ω impedance on the line;

- a value of the reection parameter S11 less then -20 dB for all the frequencies ranging from0 to 40 GHz.

The fabrication process follows the steps described in the previous subsection and in g. 2.1(with the obvious dierences of the type of substrate and the dielectric thickness), where in thelast step Ground-Signal-Ground metal pads were dened instead of source/drain contacts. Afterthis step, the graphene layer was passivated with a 30 nm layer of Al2O3 deposited by ALD 2.

2.2. Experimental setup

2.2.1. Probe station

In g. 2.4 a photo of the probe station Cascade Microtech Summit 12000 on which all themeasurements were done is shown. It is an equipment that integrates 4 DC probes (for the four-probe measurements, described below: 2 probes for source and drain and other two to measurethe potential drop in the middle of the graphene channel) and 2 RF probes. The DC probesare constituted by tungsten tips, very exible and adaptable to the design of the devices tested.The RF probes have a tip system Ground-Signal-Ground adapted to the design of coplanarwaveguides described before. Each probe has a micro-positionator which allows to move themin the XYZ directions.All the samples on which there were the tested devices were positioned on a metallic support,

the "chuck", which can be displaced along XY directions and has also a control of the angletheta to adjust the sample position. The chuck can be polarized in order to apply a voltage tothe substrate, which acts as back gate.

2The Al2O3 layer doesn't aect photodetection measurements since it is transparent at the laser wavelength1,55 µm used.


2.2.2. DC measurements

Figure 2.5.: Photo of four-probe measurement conguration.

DC measurements in this thesis were subsequent to the work presented in section 2.1.1; inparticular all the devices tested had a protection layer with a passivation of HfO2, deposited byALD, instead of Al2O3, giving anyway comparable results to those presented for Al2O3 [7]. Thethickness of SiO2 is 300 nm. Measurements were performed on devices with a typical Hall barconguration, as can be seen in g. 2.5. The fabrication process was described in the previoussection.In particular four-probe measurements were performed. The graphene channel can be modeled

as a tunable resistance, as seen in section 1.3.2. To investigate directly the graphene resistivity,from which we can extract its mobility, as will be described in the next chapter, we impose adrain-source voltage and we measure the potential drop inside the channel.Indeed, if we only impose a drain-source voltage and we measure the current owing in the

channel, there is the inuence of contact resistances, which arise from the dierent work functionbetween metal contacts and graphene (see also section 3.2.1).The circuital scheme of a typical two-probe measurement is shown in g. 2.6, where RC are

the contact resistance and RG the resistance of the grahene channel.With the four-probe measurement we put two probes in the middle of the channel where we

can measure the potential drop due only to the graphene resistance, as can be seen in g. 2.7.All the probes were connected through coaxial cables to Source Measurement Units (SMU),

on three dierent Keithley 2636:

• one to apply a voltage on the chuck, so that the back gate voltage of the transistor can besweept;

• one to to impose a drain/source voltage and measure the current owing along all thechannel;

• one connected to the two probes in the middle of the channel, so we impose a zero currentand we measure the potential drop, obtaining then a voltmeter.

Figure 2.6.: Circuital scheme of two-probe measurement.


Figure 2.7.: Circuital scheme of four-probe measurement.

Figure 2.8.: Photo of typical setup for photodetection measurements.

2.2.3. Photodetection measurements

Photodetection measurements were performed on graphene based coplanar waveguides de-scribed in section 2.1.2.In g. 2.8 a photo of the typical setup conguration for photodetection measurements is

shown. We can see the RF tips conguration and an example of the actual coplanar waveguidestested.In g. 2.9 the complete scheme for these measurements is shown. Once the sample is put on

the chuck, the coplanar waveguide is contacted by the RF probes on the two sides. Each sideof the probe is connected to a VNA and to a Keythley channel. The VNA measures the ACsignals, while the Keythley measures the DC component. This two instruments are decoupledby means of a bias-tee directly connected to the probe. A DC source/drain voltage is appliedby imposing a voltage on the two Keythley channel on each side, and the DC current owingalong the graphene channel is measured on the Keythley; also, with another Keithley we sweepthe back-gate voltage.A 1,55 µm Distributed Feed-Back (DFB) laser was used; to obtain its intensity modulation,

we link it to a Mach-Zender Modulator (MZM), which has as an input also a DC bias, to xthe working point of the MZM itself and an RF signal (generated by port 3 of the VNA), at thefrequency at which the laser light is modulated, fopt. Then at the output of the MZM the lightintensity is: Popt = PCW + Pmsin(2πfoptt), at the basis of the working principle described insection 1.4.1.The modulated output light of the MZM is split with a coupler into two parts:

• 90 % of the light goes as an input to an Erbium Doped Fiber Amplier (EDFA). Theamplied signal is then focalized on the graphene sample, as detailed after;


Figure 2.9.: Complete scheme of photodetection measurements.


• 10 % of the light goes as an input to an Opitcal Spectrum Analyzer (OSA); here wecan observe the spectral power density of the modulated light, so we see three peaks, infrequency: flaser = 1.55µm, flaser − fopt and flaser + fopt. The last two are the side peaksgenerated by the MZM. In particular we check that the dierence in amplitude betweenthe peaks of flaser and flaser−fopt and between flaser and flaser +fopt is 3 dB: in this waywe are sure that the total power (the integral in frequency of the spectral power density)is divided exactly in two equal parts, continuous and modulated (maximal modulationdepth); so: Popt = PCW + PCW sin(2πfoptt).

The amplied beam from the EDFA is connected through an optical ber to the optical systemin which there are a lens, a beam splitter and an objective. The lens collimates the beam; thiscollimated beam arrives to the beam splitter, with which a part of the light is split down andgoes through an objective that focalizes the beam on the graphene channel; the other part goeson a power-meter, with which we could control and characterize the total incident optical poweron the sample.With a IR camera, we could check the focalization of the beam on the graphene channel.Finally, we used the VNA as a spectrum analyzer: it measures (in a frequency window centered

at fopt) the root mean square of the modulated part of the photocurrent, coming in port 2.In gures 2.10, 2.11 and 2.12 photos of the actual instrumentation described are shown.

Figure 2.10.: Photo of MZM and coupler used for measurements.


Figure 2.11.: Photo of the optical system used for measurements.

Figure 2.12.: Photo of the VNA used for measurements.


Power from EDFA [mW] Power read after beam splitter [mW] Calculated incident power [mW]300 10,7 62,9250 9,1 53,7200 7,4 43,5150 5,5 32,1100 3,7 21,8

Table 2.1.: Summary of 5 dierent power coming out from the EDFA, read after the beamsplitter, and corresponding calculated incident values.

Figure 2.13.: IR photo of the laser spot on the graphene channel.

Power characterization

As seen in sec. 1.4.1, we expect the photocurrent to be proportional, or more in general todepend on, the incident optical power on the devices tested. Then is important to characterizethe power losses through the optical system and know the values of the incident power.We measured far from the probe station the power coming out from the objective with a power-

meter and we calculated it to be proportional to the optical power incident on the power-meterafter the beam splitter. In particular, the relation between the two read powers is:

Pafter beam splitter = 0, 17Pafter objective (2.1)

Of course during photodetection measurements we couldn't read directly the optical powercoming out from the objective; but thanks to the relation written before, we could monitor andread in real time what was the incident power on the sample.In particular, we could decide the power coming out from the EDFA; values that were used

during measurements are summarized in tab. 2.1 , with the corresponding values read after thebeam splitter and the calculated values from eq. 2.1 of the incident power on the device.

Beam diameter

In g. 2.13 we can see an infrared image of the laser spot in the middle of the graphenechannel.We could convert this image with a Matlab script into an image in which we can clearly see

the gaussian spot in the middle. This image is shown in g. 2.14.The diameter of the laser beam with a gaussian prole is dened as the 13.5 % of the peak

value, from the formula:


Figure 2.14.: Converted image from g. 2.13, in pixels and intensity in arbitrary units.

gaussian spot diameter =FWHM√

2ln2(2.2)

The objective, at the end of which the beam is focalized onto the device, can be modelizedthrough an equivalent lens of focal f.The laser spot can be calculated through the formula [30]:

spot diameter [µm] =4λf

πD(2.3)

where λ is the wavelenght (in µm) and D is the diameter of the collimated beam in input tothe objective. From objective and collimating lens specics, D = 7 µm and f = 10 µm; the laserwavelenght is 1, 55 µm. Putting these numbers in eq. 2.3, we obtain spot diameter ≈ 2, 82 µm.From g. 2.14, we performed a gaussian t to the beam prole with which we could extract

σgauss = FWHM2√2ln2

. This corresponds to the half of the gaussian spot diameter. The value foundconrmed the theoretical estimation of the spot diameter.

3. Characterization of graphene resistivity

and extraction of mobility

3.1. Theoretical limits

As introduced in section 1.3.2, mobility in graphene is strongly limited by the substrate onwhich it is transferred. The main theoretical limitations are both intrinsic (due to acousticphonon of the graphene lattice) and extrinsic (due to interaction with phonons arising from thedielectric substrate surface and charged impurities). In this part of the chapter, a brief reviewabout the main limitations and scattering mechanisms that aect graphene in DC measurements,and that were studied and accepted in literature, is presented.

3.1.1. Acoustic phonon scattering

If all extrinsic scattering mechanisms, e.g. charged impurities, interface rough-ness, grapheneripples etc., can be eliminated from the system, we can study the intrinsic room temperaturelimits for mobility and conductivity.There are two types of phonons in graphene: optical and acoustic. The in-plane optical and

acoustic spectra are 1 [13]:

Optical phonons : ~ω0 ≈ 200 meV, Acoustic phonons : ~ωq = ~s|q| (3.1)

where s ≈ 106 cms−1 is the sound velocity in graphene and q dened as in eq. 1.12. At typicalroom temperatures, optical phonons have too high energy to provide an eective scattering chan-nel, so only acoustic phonons are considered to deduce the intrinsic resistivity. Also, scatteringby acoustic phonons can be approximated as quasi-elastic.From the Boltzmann transport theory, the graphene conductivity can be calculated to be [8]:

σ = e2DOS(EF )vF2< τ > (3.2)

where DOS(EF ) is the density of states evaluated at the Fermi level and < τ > is the relaxationtime averaged over energy. In particular < τ > is proportional to the phonon occupation number(Nq = 1

exp(~ωqkBT

)−1), which describes the phonon population, through the scattering transition

probability Wk,k′ for an electron to go from state with momentum k to k′.We distinguish two temperature regimes in describing the behavior of resistivity [8] [15]; these

regimes are separated by the so-called Bloch-Grüneisen Temperature TBG, which is denedstarting from the energy and momentum conservation conditions for acoustic phonon scattering:the maximum momentum that can be exchanged for an electron in a Fermi level EF withmomentum kF through an acoustic phonon is 2kF ; TBG is then dened from:

kBTBG = 2~skF (3.3)

1Out-of-plane (called exural) phonons can also occur, but since graphene devices are conventionally found onsubstrates, the exural phonon contribution to transport and energy ow characteristics are expected to besmall [13].

3.1 Theoretical limits 36

Figure 3.1.: Calculated resistivity (from eq. 3.2 including the behaviour described in equations3.4 and 3.5) for carrier density nS = 1, 3, 5 · 1012 cm−2 in logarithmic (left panel)and linear (right panel) scale. Adapted from [8].

For typical values of doping, for example EF = 0.1 eV , values of TBG are around 10 K [13] [8].Then [8]:

• for T TBG we are in the non-degenerate limit, where acoustic phonons with energy~ωq ≈ kBT can be exchanged. In this case we can approximate the phonon occupationnumber as Nq ≈ kBT/~ωq and at the end 1

<τ> ∝ T so:

ρ =1

σ∝ T (3.4)

• for T TBG we are in the degenerate limit and acoustic phonon energies are ~ωq kBT ;the full expression on Nq has to be taken and 1

<τ> ∝ T4 so:

ρ =1

σ∝ T 4 (3.5)

A picture of this behavior is presented in g. 3.1.As can be seen, they are showed the two dierent behaviors described; anyway, in linear scale

(right panel) the resistivity in function of the temperature is almost linear for any relevant tem-perature. Another important feature that can be noted is also that the calculated contributionof the acoustic phonons to resistivity is weakly dependent on the carrier density.

3.1.2. Remote interfacial phonons

In many works, such as the one of ref. [9], experimental values of the resistivity of grapheneon SiO2 dielectric substrate evidenced that at temperatures above 200 K the behavior is highlynon linear and also depends on the carrier density, as can be seen in gure 3.2.


Figure 3.2.: Resistivity of two GFET samples with mechanical expholiated graphene (a, sample1; b, sample 2) from [9] in function of temperature and for dierent gate voltages(for 300 nm SiO2).

All the values of VG are referred with respect to the Dirac point voltage. Dashed lines are t of the second termof equation 3.6.

In particular in ref. [9] they studied the behavior of resistivity of mechanically exfoliatedgraphene on 300 nm SiO2 and they proposed that other interactions arising for example fromthe dielectric substrate should be included in the model to account this discrepancy.The general expression of the resistivity can be expressed as:

ρ(VG, T ) = ρ0(VG) + ρA(T ) + ρB(VG, T ) (3.6)

where ρ0(VG) is the residual resistivity (which trivially depends only on nS and so on VG),estimated at low temperatures (few Kelvin), ρA(T ) is the contribution of the acoustic phonons,like described in the previous section, and ρB(VG, T ) takes into account the Remote InterfacialPhonons (RIP), the polar optical phonons of the silicon oxide substrate. The theory for thiscontribution was developed in ref. [31].Insulating substrates induce uctuating electric elds that can couple to, and partially screen,

external perturbations. These properties are characterized by the frequency dependence of thedielectric constant of the substrate itself. When a graphene sheet is deposited on a dielectric sub-strate, the electron charges couple to the polar modes that arise at the surface of the substrate.The frequencies of the corresponding surface modes are determined by the equation (neglectingthe dielectric response of graphene):

ε(ω) + 1 = 0 (3.7)

where 1 represents the dielectric constant of the air. For SiO2 we have two surface modes:ω(1)s = 59 meV, ω

(2)s = 155 meV .

Evaluating the coupling between electrons in graphene and these surface polar modes, thedielectric properties of the substrate play a signicant role. Also, the conductivity was calculated


to depend strongly on the lower frequency polar mode and the behavior of the resistivity infunction of the temperature including this contribution reects the phonon occupation number.Finally, even the conductivity dependence on the Fermi level is aected, in particular σ ∝ E2

F ∝nS [31].These considerations can in turn explain the tted expression of ρB(VG, T ) written in ref. [9]:

ρB(VG, T ) = BV −αG

( 1

eE0/kBT − 1

)(3.8)

where B is a dimensional proportionality constant, α = 1.04 (since nS ∝ VG, see eq. 1.33, andσ ∝ nS as it was found before, ρ ∝ V −1G , at a xed temperature T, so the found value of α is

consistent), and E0 = ~ω(1)s .

Figure 3.3.: Same experimental values of resistivity of g. 3.2, in logarithmic scale, tted withthe expression 3.8 (dashed lines) [9].

The t to experimental values of resistivity in g. 3.2 including the behavior described ineq. 3.8 is shown in g. 3.3. It is then found a good agreement between the model and theexperimental results.

3.1.3. Mobility and charged impurities

Taking into account all the contributions to the resistivity described before and following theMathiessen's rule, mobility can be calculated from µ = 1/nSeρ. A picture of the calculatedmobility for xed room temperature T = 300 K in ref. [9] is shown in g. 3.4. As can be seenthe main limitation comes from the RIP of the substrate.Anyway another important contribution for the mobility limits was not considered here: the

charged impurities from SiO2 substrate, i.e. unbalanced local charges that arise near crystalimpurities.Charged impurities were found to directly inuence the transport properties of graphene, in

particular near the charge neutrality point [32] [24]. Here they induce density uctuations anda picture of inhomogeneous puddles of conducting electrons and conducting holes is necessaryto understand the nite value of conductivity near the Dirac point. Indeed the minimum con-ductivity occurs at the added carrier density n0 at which the average impurity potential is zero[33].

3.2 Four-probe method for extraction of mobility 39

Figure 3.4.: Gate-voltage dependent mobility limits for three samples in ref. [9] at T = 300 K.

Short-dashed line: contribution from acoustic phonons, long-dashed line: SiO2 surface phonons, solid lines: bothphonon contributions

3.2. Four-probe method for extraction of mobility

Once we are conscious about the dierent scattering mechanisms that aect transport ingraphene, we would like to develop a simple and reliable method to extract and evaluate graphenemobility from DC measurements of resistance and resistivity of the graphene ake.In this thesis a t on the resistivity measurements with a four probe technique was developed.

CVD Graphene based transistors with 300 nm thickness SiO2 over a Si substrate were used,realized as described in Chapter 2.

3.2.1. The issue of contact resistances

By applying a bias voltage between source and drain, and sweeping the gate voltage, wemodulate the resistance of the graphene channel but the total current that ows depends alsoon contact resistances between metal pads and graphene: usually materials used to obtain sourceand drain contacts have a work function which is dierent with respect to graphene electronanity. There is then a charge transfer from graphene to metal (since the density of states ofgraphene is smaller then that of the metal) and this can lead to local p-n junction that we canmodelize as resistance [10].In our case we have Ti deposited on graphene; here φm − χg = −0.2 eV (χg ≈ 4.5 eV ).

This means that when we are at the Dirac point voltage Vth, a local p-doping is present nearthe contacts while the entire channel has ideally the chemical potential at the neutrality point.When we apply a gate voltage such that VG − Vth < 0, all the channel is p-doped while ifVG − Vth > 0 p-n-p regions are formed, as can be seen in gure 3.5.As can be easily understood, the main problem with a back-gate is that we can modulate also

the contact resistances.

3.2.2. General formulation of graphene resistivity

The graphene resistance can be dened as [5]:

Rg = ρL

W(3.9)


Figure 3.5.: Schematic of the graphene FET with the back gate and scheme of the band diagramshowing the charge transfer region (hatched). Adapted from [10].

The traces of the Dirac point for dierent VG (here Vth = 0 is assumed) are indicated by the broken line, thedash-dotted line, and the dash-double-dotted line. The respective resistivities are also shown as a function ofposition.

where ρ is the sheet resistivity, L the length and W the width of the graphene rectangle channel.Then the total resistance is the sum of contact resistances source/graphene and drain/-

graphene (which we suppose equal) and of the graphene channel (see also g. 2.6):

VDSIDS

= Rtot = 2RC +Rg (3.10)

where IDS is the current owing from drain to source and RC the metal/graphene contactresistance.In ref. [9] and in gure 3.4 it was shown the mobility gate-voltage dependence. Actually the

relation between resistivity and mobility from which they started didn't take into account thenite minimum conductivity of the GFET characteristic due to charged impurities, as it wasintroduced at the end of section 3.1.3. We considered ref. [9] to explain what are the physicalmechanisms that limit the mobility. But if we want to model the entire GFET characteristic,we have to include the eects of charged impurities. Thus we can write [34]:

Rtot = 2RC +(L/W )√n20 + n2Seµ

(3.11)

where

ρ =1√

n20 + n2Seµ(3.12)

is the graphene resistivity. n0 is the residual carrier density, which models the equivalent chargedensity at the Dirac point voltage due to electron-hole puddles induced by charged impurities.The strength of four probe measurement is in the direct measure of the graphene resistivity

without contact resistance eect, as explained in section 2.2.2. This allows to have a directaccess to mobility.


If we suppose to have a constant mobility, as it is usually done in literature [34], we canidentify the maximum of resistivity for nS = 0. Then we can write:

ρmax =1

n0eµ(3.13)

Then combining eq. 3.12, eq. 3.13 and eq. 1.33, we obtain:

ρ(VGth) =

√1

1ρ2max

+ C2oxV

2Gthµ

2(3.14)

where VGth = VG − Vth, gate voltage dierence with respect to the Dirac point voltage.The only unknown parameter in this equation is the mobility µ, which is the tting parameter

to be found.

3.2.3. Results

Figure 3.6.: Schematic of four-probe method geometry.

D = drain, S = source, V+ and V- = measured voltage drop in the middle of the channel, L = graphene channellength, W = graphene channel width, l = distance between the two probes V+ and V- where the potential dropdue only to graphene was measured.

In g. 3.6 is shown the geometry from which the graphene channel resistivity was measured.In particular:

Rch,4p =V+ − V−IDS

= ρl

W(3.15)

The resistance that we measure, Rch,4p, with four-probe method is the one between the twoprobes in the middle of the channel, so to extract the resistivity we have to normalize withrespect to l (and not L).Measurements were done on seven devices according to experimental methods already de-

scribed in Chapter 2.From equation 3.15, the experimental value of ρ for each VG can be extracted. So, the tting

parameter µ is obtained by minimizing the sum of the squared errors between the model inequation 3.14 and the experimental values of ρ. In order to start the minimization iterationprocess, an initial value of µ has to be set to let the run start until the end of iterations (detailsof the Matlab code can be found in Appendix A, section A.0.1). Here an initial value of 1000cm2V −1s−1 was choosen according to literature for value of mobility of CVD graphene on SiO2

substrates [35] [36].In table 3.1 the main results of the t are presented: for each device mobilities were extracted

and from eq. 3.13 the residual carrier densities n0 are calculated; then, to give an estimate ofthe goodness of t, the maximum relative error with respect to the experimental resistivity (%rel. error = 100 · |ρexp−ρcalc|ρexp

) is reported. Even if with 7 devices we can't do a proper statistic,in tab. 3.2 are reported the mean values of mobility and n0 and their standard deviation.

3.3 Two-probe method for extraction of mobility 42

Device l [µm] L [µm] W [µm] µ [ cm2

V s ] n0 [cm−2] max. rel. error(%)1 10 24 6 925 1,1·1012 19%

2 20 48 12 280 1,5·1012 17%

3 10 24 6 1170 8,2·1011 18%

4 20 48 12 90 2·1012 23%

5 10 24 6 60 3,3·1012 8%

6 10 24 6 970 1,7·1012 16%

7 10 24 6 2360 1,3·1012 29%

Table 3.1.: Summary of geometrical parameters of the devices tested, extracted mobilities andn0, and maximum relative error between experimental and calculated resistivity.

Mean value St. deviationµ [ cm

2

V s ] 840 810n0 [cm−2] 1,7·1012 0.8·1012

Table 3.2.: Mean value and standard deviation for µ and n0 from table 3.1.

As can be noted, there is a lot of dispersion between mobility values; moreover there is a non-negligible contribution of residual carrier density n0. This means at the Dirac point voltage,an equivalent carrier (electron-hole puddles) density of ≈ 1012 cm−2 is present, which is highlycomparable with the charge induced applying a gate voltage, far from Dirac point. To give anidea, when we have 300 nm of SiO2 dielectric substrate, applying VGth = 10V we induce annS ≈ 0.8 · 1012cm−2.Then, we can assert that a large concentration of charged impurities are present, which are

the main limiting factor also for mobility [34] [32] [9].In gure 3.7 there are two examples of t, in particular for device 5 (g. 3.7 (a)) and device 7

(g. 3.7 (b)). The error margin is acceptable and the t follows the shape of the experimentalcurve.

3.3. Two-probe method for extraction of mobility

The direct measurement of graphene resistivity is only possible by designing "ad-hoc" struc-tures, like Hall bars presented in the previous section, from which contact resistance is experi-mentally measured and removed from total resistance. Real devices have often just two contacts.This is the case, for example, of the coplanar waveguides measured in this thesis. In this case,the only information directly obtainable from measurement is the total resistance. In order to tmobility in such a kind of structures, we also need to t the contact resistance, starting from themodel in equation 3.11. This 2-parameter t is carried on by assuming that RC in equation 3.11is constant. As seen in section 3.2.1, this Ansatz is not exact for back gated structures: sweepingthe back gate voltage has an eect not only on the channel, but also on the metal/graphenedoping. In other words, sweeping the back gate means changing the contact resistance. Theeect of this assumption will be discussed in the analysis.To validate this two-probe method for extraction of mobility, we extract from the same devices

tested and described in the previous section the total experimental resistance and we comparethe extracted values of mobility with the previous ones.


(a)

(b)

Figure 3.7.: Experimental (red circles) and calculated (blue line) resistivity for device 5 (a) anddevice 7 (b) in function of VG − Vth, Vth Dirac point voltage.


3.3.1. Starting equations

We can always identify the maximum of the total resistance when nS = 0. Then:

Rtot,max = RC,tot +(L/W )

n0eµ(3.16)

Combining eq. 3.16, eq. 3.11, and eq. 1.33, we obtain:

Rtot(VGth) = RC,tot +(L/W )√

(L/W )2

(Rtot,max−RC,tot)2 + C2oxV

2Gthµ

2(3.17)

So here we have two unknown parameters, i.e. RC,tot = 2RC and µ. We can always do a twith respect to the experimental resistance with a method which is very similar to the one usedin four-probe method, but this time we have to choose a starting point also for the total contactresistance, according to the code implemented and described in Appendix A, section A.0.2.For mobility, according to the previous results, we set as a starting point 1000 cm2V −1s−1

for all the devices. To set an initial value of total contact resistance, we can start from theexperimental values of contact resistances that can be extracted from four-probe measurements:from eq. 3.15 we can extract ρ and through eq. 3.9 and 3.10 we have experimental values of 2RC .Given the large variability from device to device of the experimental total contact resistance (seetab. 3.3), for each device the initial value of the contact resistance was chosen to be the onemeasured in the Dirac point voltage. At this voltage, we should have a better estimate of thecontact resistance. This because the graphene channel resistance is maximal and so less aectedby noise.Actually, for device 2 and 4 negative contact resistances were found, which has not physical

meaning. This negative RC,tot is not intrinsic; when Rtot and Rch,4p (see eq. 3.15) are measuredby the four-probe measurement, the Dirac points for both cases are not consistent, suggestingthat the averaged potential minima of the whole channel area (LW) for Rtot and the local channelarea (lW) for Rch,4p are dierent. This dierence seems to be due to the charge transfer from themetal contact and/or the spatially distributed interaction with the SiO2 substrate [10]. The factthat devices 2 and 4 have a dierent geometry with respect to the other ones then aect results.In particular, a longer channel (see tab. 3.1) could lead to dishomogeneities of the channel (evenduring the technological process), that could explain this behavior. For this reason these devicesare not considered here.

3.3.2. Results

Device µ2p [ cm2

V s ] n0,2p [cm−2] RC,tot [Ω] range of exp. RC,tot [Ω] max. rel. error(%)1 920 1,1·1012 8100 5660-13200 13%

3 1060 7,7·1011 43500 41100-51130 4%

5 70 3,7·1012 36400 15000-36400 12%

6 950 1,8·1012 800 730-1840 13%

7 1290 1,7·1012 1100 2250-6650 19%

Table 3.3.: Summary of extracted mobilities, n0,2p, RC,tot, experimental range of RC,tot andmaximum relative error between experimental and calculated resistance from thetwo-probe method.


Mean value St. deviationµ2p [ cm

2

V s ] 860 460n0,2p [cm−2] 1,8·1012 1.1·1012

RC,tot [kΩ] 18 20

Table 3.4.: Mean value and standard deviation for µ2p, n0,2p and RC,tot from table 3.3.

In tab. 3.3 a summary of the extracted parameters is reported. With the pedix 2p we intendthe values extracted with two-probe method, to get a distinction with the values extracted withfour-probe technique. From the Matlab code, the couple of unknown parameters µ2p and RC,totwere extracted; then, from eq. 3.16, n0,2p is calculated. To get a comparison between theextracted and experimental total contact resistance, for each device the range of the values forwhich the experimental RC,tot varies (due to gate voltage sweeping) is reported. We can notethat only for device 7 we nd a discrepancy between the experimental and tted parameter,while for other devices the extracted contact resistances fall in the experimental range. It is alsoreported the maximum relative error with respect to the experimental resistance (here the %

rel. error is 100 · |Rtot,exp−Rtot,calc|Rtot,exp).

Even here, a little statistic is done, with the mean values and standard deviations for µ2p,n0,2p and RC,tot reported in tab. 3.4.In gure 3.8 we can nd for the same devices 5 and 7 for which the resistivity was investigated

in the previous section, the experimental and tted total resistance. Also in the case of two-probe method, the relative error is acceptable and the shape of the tted curve follows quitewell the experimental one.In g. 3.9 a direct comparison between the values extracted with four-probe and two-probe

method are shown. The discrepancy on device 7 for the total contact resistance reects alsoin the values of mobility. Indeed, seeing g. 3.9 (a), the two mobilities for device 7 are alsoquite dierent. This is actually a limitation of the code implemented: the purpose of the tis to nd the couple (µ2p, RC,tot) that better minimizes the sum of squared errors betweenthe experimental data and the function supposed to represent it. Then, even if we can choosereasonable values as starting points for the t, since we have two free parameters instead of oneof the four-probe method, the t can converge to values which are not the real ones. But we canassert that in 4 cases over 5 the two-probe method is consistent with the four-probe method,supposed to be more exact from this point of view.Of course, to have a real validation of this model, a real statistic should be done on many

devices. Anyway we can conclude that in the large part of the cases, for our situation, consistentvalues of mobility and residual carrier density (g. 3.9 (b)) are found with respect to the four-probe method. Importantly, the assumption of constant total contact resistance doesn't aectthe consistence of results.


(a)

(b)

Figure 3.8.: Experimental (red circles) and calculated (blue line) total resistance for device 5 (a)and device 7 (b) in function of VG − Vth, Vth Dirac point voltage.


(a)

(b)

Figure 3.9.: Direct comparison between the value found of mobilities (a) and residual carrierdensity (b) with four-probe method (red bars) and two-probe method (blue bars)for devices 1, 3, 5, 6, 7.

In the future, to do a real statistic, it can be thought to put devices with both Hall-barconguration and coplanar waveguide conguration in the same sample. In this way, we aresure that both kind of devices are done with the same technological process at the same time; ifthere is a sucient number of devices on which a statistic can be done with four-probe technique,then a more proper estimation of the physical parameters, that should be used for simulationsof photocurrent for coplanar waveguides, can be performed.

4. Hot carrier eects and temperature

simulations

In section 1.4.1 the main contributions to the photocurrent in graphene were introduced. Inparticular, photothermoelectric (PTE) and photobolometric (PB) currents are detectable andcome from the raise of electrons and lattice temperature.For PTE current, necessary conditions are a dierence in the Seebeck coecient along the

graphene channel and an electron temperature gradient. A dierence in Seebeck coecienttranslates in graphene as a variation in density of states. Indeed, from Mott formula:

S = −π2k2BT

3eσ

dσ

dE(4.1)

where kB is the Boltzmann constant, T is temperature, e is the electron charge, σ is the conduc-tivity, E is the chemical potential. Since σ = nSeµ, supposing a constant mobility µ, we obtainthe Seebeck coecient proportional to dnS

dE , i.e. proportional to the DOS [37](see also eq. 1.24).The photocurrent is then [16]:

IPTE =1

R

∫ +L2

−L2

S(x)dTedx

dx (4.2)

where R is the resistance, L is the length of the graphene channel, Te is the electronic tempera-ture. Then to have this contribution we should have a variation of the Seebeck coecient alongthe channel, i.e. a variation of density of states, that can be obtained for example with a p-njunction or at an interface with a dierent material. The other crucial point is the temperaturegradient. Shining with a laser on a single spot of the graphene channel we induce a temperaturegradient, as shown in the simulations developed in this chapter. PTE is negligible in the devicesstudied in this work because the channel is uniformly doped by the back gate, but can be seenfor example if we shine with the laser the edge between graphene and metal contacts.For the PB current, we induce a change in conductivity: when we illuminate the device

with a laser, we are giving energy to the electronic system and to the lattice. There is then anegative dependence of conductivity on temperature, arising from the resistivity increase withtemperature explained in sections 3.1.1 and 3.1.2. What was not explicitly assumed in literature[16] [38] is that this dependence is sensible to the electronic temperature, anticipating results ofthe next chapter.As can be seen, it is important to develop a robust simulation for electronic and lattice

temperature in order to understand the principles of these thermic eects.Also, a complete model of electron temperature based on heat equation is still missing, and

will be provided in this chapter.

4.1. Photoexcited carrier dynamics and cooling

A number of experimental techniques have been employed to track the decay dynamics ofphoto-excited carriers in graphene, unraveling a complex picture of competing relaxation path-ways [39] [40] [41]. An example of these techniques is the pump-probe experiment.

4.1 Photoexcited carrier dynamics and cooling 49

Figure 4.1.: Scheme of Auger processes: Auger Recombination (AR, left) and Impact Ionization(II, right). Adapted from [11].

In a pump-probe experiment, the sample is excited with a strong pump pulse. This inducesa change in carrier population. A weaker probe pulse, which is focused onto the same spot onthe sample, is used to monitor the change in population. Optical pumping initially leads to anonequilibrium distribution of carriers. Since the momentum of the photon is negligible on thescale of the Brillouin zone of graphene, the interband transitions can be viewed as vertical inmomentum space. The optical pumping excites valence band electrons of an energy E = ~ω/2into the conduction band, where ~ω is the photon energy (see also section 1.2.4). Common to allinvestigations is the observation of two distinct time scales in dierential transmission spectra:the nonequilibrium carrier distribution quickly thermalizes into a hot carrier distribution viacarrier-carrier interaction and subsequently cools down via electron-phonon processes.

4.1.1. Photoexcitation cascade

Ultrafast optical pump-terahertz probe [39] and angle resolved photoemission spectroscopy(ARPES) [42] experiments have indicated that electron-electron scattering events occur on afast time scale, several tens of femtoseconds, highlighting the crucial role interactions play inthe photo-excitation cascade (and optical response) of graphene. In particular, in ref. [39]they obtained detailed information on the energy absorbed by the electronic system in thecascade following a short photoexcitation pulse. These results indicate that the decay of photo-excited carriers was dominated by electron-electron scattering events rather than the emissionof phonons.We have to distinguish two dierent situations: one for typical graphene electrostatic doping

(EF ≈ 100 meV ) and one for graphene with doping level near the charge neutrality point.

Carrier multiplication near Dirac point

Near the Dirac point, a challenging feature of graphene relaxation dynamics is the expectedhigh eciency of Auger relaxation, which can be exploited to obtain carrier multiplication:multiple charge carriers are generated from the absorption of a single photon [11]. We distinguishtwo types of Auger processes: Auger recombination (AR) and impact ionization (II), see g. 4.1.AR is a process, where an electron is scattered from the conduction into the valence band,

while at the same time, the energy is transferred to another electron, which is excited to anenergetically higher state within the conduction band (g. 4.1, left). II is the opposite process(inverse Auger recombination). An electron relaxes to an energetically lower state inducing theexcitation of a valence band electron into the conduction band (g. 4.1, right). The result of II


is an increase of the carrier density (carrier multiplication). Both processes also occur for holesin an analogous way.In conventional semiconductor structures, these relaxation channels are suppressed by restric-

tions imposed by energy and momentum conservation, which are dicult to fulll at the sametime due to the bandgap and the energy dispersion. In contrast, graphene is expected to showa very ecient scattering via Auger processes. Here, it is of crucial importance to addressthe question whether impact ionization is ecient enough to give rise to a signicant carriermultiplication despite the competing processes of Auger recombination and phonon inducedscattering.In ref. [11], they showed a theoretical investigation of carrier multiplication and their results

were conrmed by subsequent pump-probe experiments [43]. In particular, they simulated agaussian pulse with a temporal width of tens of fs at about 1,5 eV. They found an importantcarrier multiplication factor. This can be explained by an asymmetry between impact ionization(II) and Auger recombination (AR) resulting in a much higher probability for II. The asymmetrycan be explained as follows: in the beginning of the relaxation dynamics the probability for anelectron to be excited into the conduction band is proportional to II ∝ fvk(1 − f ck) while theopposite process is AR ∝ f ck(1 − fvk) (where fk is the Fermi-Dirac distribution referring toa given particle momentum k, and the apexes v and c refer to valence and conduction bandrespectively). In the rst femtoseconds after the optical excitation, the probability to nd anelectron (a hole) in the conduction (valence) band close to the K point is small, that is, f ck ≈ 0(fvk ≈ 1). As a result, II ≈ 1 and AR ≈ 0. In other words, the Auger recombination is suppressedby Pauli blocking, since its nal states in the valence band are occupied. With increasing time,an equilibrium between II and AR is reached resulting in a constant carrier density.To understand the eciency of carrier multiplication, electron-phonon scattering has to be

taken into account. This is an important relaxation channel, which is in direct competition withAuger-type processes. Phonons reduce the eciency of carrier multiplication, since a part of theexcited electrons are cooled by emission of phonons resulting in a loss of energy necessary forAuger processes. However, the decrease of the charge carrier density is slow. As a result, despitethe electron-phonon scattering, there is still a carrier multiplication (CM > 1) on a picosecondtime scale.

Impact excitation scattering for doped graphene

In ref. [12], they identify impact excitation (IE) as the scattering process (see g. 4.2) thatdominates carrier relaxation dynamics in the case of doped graphene.

Figure 4.2.: Impact excitation (IE) cascade of a photoexcited carrier with initial energy E0.Adapted from [12].

Multiple secondary electron-hole (e-h) pairs produced by IE scattering involving a photoex-


Figure 4.3.: Energy-loss rate via impact excitation, Jel (blue curve), and optical phonon emis-sion, Jph (red curve) for a typical doping of EF = 0.2 eV. Adapted from [13].

(Inset) Branching ratio, Jel/Jph vs ε, photoexcited carrier energy, and EF . The branching ratio can be tunedover an order of magnitude by density dependence.

cited carrier and ambient carriers in the Fermi sea can lead to ecient hot carrier redistribution.Their analysis predicts that IE processes result in a chainlike cascade consisting of sequentialsteps with relatively small energy loss per step ∆ε ≈ EF , where EF is the Fermi level (g. 4.2).Both the number of pairs produced in the cascade and the characteristic energy for the pairsare highly sensitive to doping. As a result, the key parameters of photoexcitation cascade ingraphene are expected to be gate tunable in a wide range. Morevoer, calculating typical timedecays for this initial part of the photoexcited cascade, for typical doping (EF = 200 meV) andfor an excitation energy E0 = 1 eV, they found a ∆t ≈ 0, 12 ps, far faster then typical electroncooling times found in graphene [13] [12].This IE mechanism has a fast characteristic rate that makes this scattering process a highly

ecient relaxation pathway which dominates over phonon-mediated pathways in a wide range ofenergies. To have an evaluation, a direct comparison have to be done between energy relaxationfrom IE processes and the contribution of other potentially signicant channels. In particular,electron-phonon scattering leads to a direct transfer of energy to the lattice degrees of freedomwithout creation of secondary electron-hole excitations. In ref. [12] they focus on the contri-bution of optical phonons, which under normal circumstances is more important than that ofacoustic phonons (since optical phonons are higher in energy than acoustic ones and then leadto bigger losses of energy during scattering with photoexcited electrons). Their estimate showsthat under realistic conditions the contribution of optical phonons to the energy relaxation rateis weaker than that due to carrier-carrier scattering, as can be seen in g. 4.3 where the energyloss rates for these two mechanisms are compared.

4.1.2. Electron-lattice cooling and supercollisions

After the fast carrier-carrier scattering described before, excited electronic distribution de-cays and is centered around the initial Fermi level before excitation [13]; an eective electronictemperature is established, higher than lattice temperature. Then excited electrons cool downthrough electron-phonon scattering. A unique feature of graphene is the slow cooling betweenelectron and lattice system. Thermal decoupling of electrons from the crystal lattice in mostmaterials takes place at temperatures of order a few kelvin. In contrast, the rates for electron-


Figure 4.4.: Kinematics of supercollisions and normal collisions at T > TBG. Adapted from [14].

Phonon momenta are constrained by the Fermi surface for normal collisions (white arrows), and totally un-constrained for supercollisions (qph), with the recoil momentum (qrecoil) transferred to the lattice via disorderscattering or carried away by second phonon. The energy dissipated in supercollisions is much greater than thatdissipated in normal collisions.

lattice cooling in graphene are predicted to be very slow in a much wider temperature range,reaching also room temperatures (300 K) [14].The ineciency of the standard cooling pathways mediated by optical and acoustic phonons

stems from the material properties of graphene. The large value of the optical phonon energy, ω0

= 200 meV, quenches the optical phonon scattering channel below a few hundred kelvin [13]; asmall Fermi surface and momentum conservation severely constrain the phase space for acousticphonon scattering.Indeed, since for such processes the phonon momenta are limited by 2kF , the maximal energy

transfer cannot exceed kBTBG = 2~skF per scattering event (here s and kF are the soundvelocity and Fermi momentum, recalling eq. 3.3). The TBG values are a few kelvin for typicalcarrier densities, i.e. a small fraction of kBT .In ref. [14], they argue that an unconventional, disorder-assisted pathway dominates cooling in

a wide range of temperatures, explaining key features of cooling dynamics observed in previouspump-probe measurements [44], where the cooling times were found to grow with decreasingtemperature, from ≈ 10 ps at 300 K to ≈ 200 ps below 50 K. This is very dierent fromthe dependence expected for momentum-conserving scattering by acoustic phonons, where thecooling times are predicted to increase with temperature, reaching a nanosecond scale at roomtemperature for comparable densities [45].This disorder-assisted scattering allows for arbitrarily large phonon recoil momentum values,

see g. 4.4. In this case, the entire thermal distribution of phonons can contribute to scattering,resulting in the energy dissipated per scattering of order kBT (supercollisions). This provides adramatic boost to the cooling power.For this cooling mechanism, modeling disorder by short-range scatterers, they obtain the

energy-loss power:

J = A(T 3e − T 3

ph), A = 9.62g2DOS(EF )2k3B

~kF l(4.3)

where Te(Tph) is the electron (lattice) temperature, DOS(EF ) is the density of states atthe Fermi level, g is the electron-phonon coupling, (g = D/

√2ρds2, where D is the constant

deformation potential, D ≈ 20 eV [13], ρd is the graphene mass density and s the sound velocity,see eq. 3.1), l is the electron mean free path and kBTe(ph) EF , so it is valid for high dopinglevel, if we have electronic or lattice temperature near 300 K, as will be found in the next sectionin simulations.Other terms could in principle contribute to supercollisions, like long-range scatterers graphene

4.2 Complete heat equation model for electronic temperature 53

ripples or exural phonons (out of plane phonons) but their contribution was calculated to benegligible [13].Importantly, introducing supercollision terms, cooling times were calculated to be on the order

of ps, and they are consistent with the observed behavior with respect to temperature from whichin ref. [14] they started.Moreover, even if supercollisions in relaxation processes allow for large energy transfers, in

electronic transport they are elusive, since the average scattering probability for supercollisionsis calculated to be much smaller than the one for "normal" collisions and scattering events,described in sections 3.1.1, 3.1.2 and 3.1.3.Supercollisions are also widely accepted in literature and experimentally veried [46][15] [47]

[48].In particular in ref. [15], they studied supercollisions induced by Joule heating, since also

electrical power heating can lead to this eect. They measured the electronic temperature withnoise thermometer; also, they extracted the supercollision coecient A and they saw to beproportional to nS , coherent to eq. 4.3 (A ∝ DOS2 ∝ E2

F ∝ nS , see eq. 1.24). Anyway theyobtained a dierent expression for A with respect to the one in eq. 4.3. They concluded thatthe formula in eq. 4.3 gives the right behavior and the right order of magnitude of temperature,but still the hypothesis of disorder only due to short-range scatterers is too restrictive. Theyconcluded then that for example the eect of impurity concentration has also to be taken intoaccount.This paper will be taken as a reference later for temperature simulations, since they studied

this eect in a device similar to the conditions in which we were and comparable physicalparameters (mobility and residual carrier density). Moreover, they give an empirical formulationfor the supercollision term, extendable also near Dirac point and supported experimentally andnot only predicted in theory, which will be very useful for simulations.

4.2. Complete heat equation model for electronic temperature

To calculate the electronic temperature we start temporally after the rst femtoseconds ofphotoexcitation cascade, then when an equilibrium hot carrier distribution is reached; one canstart from the heat equation [49]:

Ce(x, t)∂Te∂t

= P (x, t) +∂

∂x(ke(x, t)

∂Te∂x

) (4.4)

where Ce(x, t) = CTe is the electron heat capacity, C = π2

3 DOS(EF )k2B [50], ke is the electronthermal conductivity, related to the electrical conductivity through the Wiedemann-Franz lawke = LσTe, L = (π/3)(kB/e)

2 is the Lorenz number, and P (x, t) is the power density (electricalor optical), that we treat as a source.This is a 1D heat equation, which does not take into account all the cooling terms described

before. In a rst step we begin to validate the heat equation both analytically and numerically,introducing only an electron-lattice cooling term, already known in literature [16] [13] [51]; afterwe introduce the supercollision term, for which there is no analytic solution, validating it withresults reported in ref. [15].

4.2.1. Analytic solution and numerical validation for heat equation withelectron-lattice cooling

We will develop the analytic solution using the Green function. To have a reliable solution weuse some approximations:


• we consider a steady state heat equation; indeed, given the usual timescales of ≈ nsand ≈ ps for electron-lattice cooling and supercollisions respectively, since the intensitymodulation on the optical power fopt that we operate is on the order of GHz, the systemreaches always a steady state temperature and follows the modulation;

• we evaluate the thermal conductivity at a xed temperature Troom which will be our initialcondition.

Moreover, since our problem is actually a 2D problem, to convert it in 1D we do an averageof the photon ux along the width of the rectangular graphene ake [51]. The spatial extent ofour gaussian beam intensity is:

P (x, y) = Gexp(− x2

2σ2gauss− y2

2σ2gauss), G =

P0

2πσ2gauss(4.5)

where σgauss = FWHM2√2ln2

, P0 is the total incident optical power and G is a coecient such thatthe total integral over all the space of P (x, y) is P0.The incident photon ux is the ratio between optical power and photoexcitation energy. Its

spatial extent can then be written as:

N(x, y) =P (x, y)

ε0, N(x, y) =

G

ε0exp(− x2

2σ2gauss− y2

2σ2gauss) (4.6)

where ε0 is the photoexcitation energy. The average on width is then:

Nave(x) =G

ε0Wexp(− x2

2σ2gauss)

∫ W

0exp(− y2

2σ2gauss)dy (4.7)

The integral can be reconduced to the well known erf function; substituting all the terms weobtain:

Nave(x) =P0

ε0Wσgauss2√

2exp(− x2

2σ2gauss)erf(W ) (4.8)

so:

Pave(x) =P0

Wσgauss2√

2exp(− x2

2σ2gauss)erf(W ) (4.9)

Finally the problem that we want to solve is then:−ke ∂

2Te∂x2

= γαPave(x)−B(Te − Tph)

Te(−L2 ), Te(+

L2 ) = Troom

(4.10)

where α ≈ 0.023 is the graphene optical absorption (see section 1.2.4), γ is the fraction ofphoton energy remaining in the electronic system (see section 4.1.1 and g. 4.3), Tph is thelattice temperature, which for now we consider equal to the constant Troom [13], and B is theelectron-lattice cooling term [13] [52], B = πg2DOS(EF )2k2F s

2kB, g dened as in eq. 4.3; theinitial condition takes into account that at the extremes of the graphene channel, where grapheneis in contact with drain/source metals that we can treat as heat sinks, the electronic temperatureis xed at Troom.To solve eq. 4.10, we dene T = Te − Tph and then solve equation in T to have homogeneous

boundary conditions:


EF 0.1 eVρ 2000 Ω

L 6 µmW 1 µmP0 0.37 mWσgauss 0.35 µmγ 0.75Troom 300 K

Table 4.1.: List of physical parameters taken from ref. [16].

(a) (b)

Figure 4.5.: Comparison between analytical (a) and numerical (b) solution for equation 4.10.

−ke ∂

2T∂x2

= γαPave(x)−B(T )

T (−L2 ), T (+L

2 ) = 0(4.11)

This is a typical 1D Sturm-Liouville problem for which the solution can be represented as [53]:

T (x) =

∫ +L/2

−L/2f(x0)G(x, x0)dx0, f(x0) = −γα

kePave(x0) (4.12)

where G(x, x0) is the Green function associated with this Sturm-Liouville problem. Detailsabout the Green function and the analytic solution calculated can be found in Appendix B.We solve the equation starting from the conditions written in ref. [16]. Here they study

the dierent mechanisms of photocurrent in graphene, introduced in section 1.4.1. To studythe bolometric eect they evaluated the electronic temperature and they started from a similarheat equation written in eq. 4.10 (then not including supercollisions yet), even if the valuesfound were adjusted to justify the experimental value of photoexcited carrier density, relevantin photovoltaic contribution for photocurrent. For our equation we choose and x a Fermi level.A summary of the physical parameters found in ref. [16] and used to solve analytically theequation can be found in table 4.1.We took the geometric specics for the graphene ake, the optical power used, the beam

radius (σgauss), resistivity (ρ), Troom. For γ: since the light wavelength used in ref. [16] is 628nm (that corresponds to ≈ 1.9 eV of photoexcitation energy), as a very rough estimate fromg. 4.3, for EF = 0.1 eV (even if the actual photoexcitation energy is not present), Jel/Jph ≈ 3and so since Jel + Jph = αPave(x), then Jel ≈ 0.75αPave(x).


Figure 4.6.: Numerical solution of eq. 4.10 without the constraint of xed electronic thermalconductivity

Figure 4.7.: Numerical solution of the heat equation from which the solution is shown in g. 4.6but adding the supercollision term.

In g. 4.5 (a) the analytic solution for electronic temperature Te is shown. As can be seen,the shape is almost triangular like suggested in ref. [16] and ref. [51] where the gaussian beamwas reasonably approximated by a Dirac delta, but the peak value is of the order of ≈ 1000 Kover Troom, in disagreement with the temperature estimated in ref. [16], where it is said to beof the order of few K (around 4 K in particular for the same doping level). We compared theanalytic solution with a nite element simulation carried on COMSOL Multiphysics, where weput the same values of table 4.1 and the same coecients and expression of eq. 4.10. The resultis shown in g. 4.5 (b). Analytic and numerical results are identical, then from the analyticpoint of view we are safe.Moreover, if we relax the approximation of electronic thermal conductivity evaluated at a

xed temperature, which was useful to have an analytic solution, we have the real heat equationto be solved. The numerical solution is shown in g. 4.6; the shape is quite dierent from whatit was found before but even here the peak value is completely dierent from the one estimatedin ref. [16].But if we just add the cooling term due to supercollisions (eq. 4.3), i.e. we add −A(T 3

e −T 3ph)1

1for kf l which enters in the expression of A, we estimated the electron scattering lenght from the Einsten


µ 2800 cm2

V s

n0 4 · 10−11cm−2

L 2,2 µmW 2,8 µmVth 12 VCox 3,5 ·10−5 F

m2

Troom 4,2 K

Table 4.2.: List of physical parameters taken from ref. [15].

to the previous solution, we can note that the peak value has a more consistent value with respectto the one estimated in ref. [16], as showed in g. 4.7. The shape is anyway completely dierentfrom the one that was considered including only the electron-lattice cooling term; in particularthe shape reects the spatial prole of the laser beam.All these results suggest that we are very sensible to the model used: if we want to get the

right, or at least consistent, result for the electronic temperature starting from an ab-initioequation, the simple heat equation with only the electron-lattice cooling is most probably toosimple and not a correct interpretation of the physics that we are treating.

4.2.2. Solution and validation with supercollisions

To validate the heat equation model for electronic temperature with supercollisions, we useas a reference the work in ref. [15], introduced in section 4.1.2, where they measured directlythe electronic temperature through noise thermometer, just giving to the electronic system anelectrical (Joule) power and giving experimental evidence of supercolission theory. The exper-iments were performed on exfoliated graphene on h-BN/SiO2/Si substrates and at cryogenictemperatures, Troom = 4, 2 K.Then the complete model is:

− ∂∂x(ke(x)∂Te∂x ) = γαPave(x)−B(Te − Tph)−A(T 3

e − T 3ph)

Te(−L2 ), Te(+

L2 ) = Troom

(4.13)

Tph was calculated to be of the order of 60 K: it is the steady state lattice temperature reachedwhen the same power of the electrons is transferred to the substrate. For what concerns thesource term P, in ref. [15] it is uniform along all the graphene channel since it is simply theJoule power density consumed by the device; also, since we don't have photoexcitation we canput α = 1 and γ = 1.Even in this case we used the same physical conditions of the reference. In particular a

summary can be found in table 4.2.To calculate the electrical conductivity (and then the electronic thermal conductivity) we have

all the ingredients: we have the mobility, the residual carrier density and the Dirac voltage Vth,and so we can evaluate the resistivity from equations 3.12 and eq. 1.33. Moreover, we use thesame gate voltage and power density values used in ref. [15]. By sweeping these two values inthe same experimental range done in ref. [15], we can see also the variation of temperature andhow the dierent cooling terms change with doping. For the supercollision constant A we usedthe empirical formulation found in this ref. [15] itself, i.e. A = 1.25 · 10−16nS

Wm2K3 .

diusion relation: the Einstein diusion coecient is DE = µkBTq

, q is the electronic charge, µ the mobility

equal to 2700 cm2

V s, taken always from ref. [16]; the diusion lenght is l = 2DE

vF[3].


Figure 4.8.: Numerical solution for electronic temperature in function of space for VG = 12 V(Dirac point voltage) and power density P = 0, 175 mW

(µm)2.

(a) (b)

Figure 4.9.: Experimental ((a), adapted from ref. [15]) and numerical (b) solution for electronictemperature.

The numerical solution of eq. 5.7 was implemented in COMSOL Multiphysics using values oftab. 4.2. In g. 4.8, the found electronic temperature in function of space at xed power densityand gate voltage is shown. Since the power density applied is uniform along the channel, theelectronic temperature reaches always a plateau, as showed in the example of g. 4.8.In g. 4.9 it is shown a direct comparison between experimental values of the electronic

temperature found in ref. [15] and numerical solution evaluated at the center of the channel.As can be noted, numerical results are in optimal agreement with the experimental ones, bothin the behavior with respect to the power density and in the behavior with respect to the gatevoltage sweep.The electronic temperature behavior in the paper was demonstrated experimentally to be

Tel ∝ P 1/3, like expected; indeed this can be proven directly from eq. 5.7, evaluating it at theplateau (so when dTe

dx = 0), and neglecting the electron-lattice cooling term, which is smallerthen the supercollision cooling term, as also demonstrated in ref. [14]. Moreover the electronictemperature is higher at the Dirac point (here Vth = 12 V ), like can be predicted theoretically


from eq. 4.3: since, as it was demonstrated, the supercollision cooling term depends on carrierdensity and is the strongest cooling term in equation 5.7, it has its lower value when nS = 0,leading to a higher temperature in the Dirac point.Our model based on heat equation including supercollisions is then validated.

5. Photocurrent measurements and

simulations

The two main contributions for photocurrent that we measure in our devices are photovoltaic(PV) and photobolometric current (PB), since as asserted in the previous chapter the photother-moelectric (PTE) current is negligible in our situation.Experimental evidence of these two eects was given in ref. [16]. In particular the two currents

are characterized by a dierent sign, evidenced sweeping the gate voltage: a change of phaseof the signal is indeed seen. At low doping a photocurrent signal with a sign coherent to theDC bias applied was found, and the only physical contribution coherent with this observationis the photovoltaic one; at high doping a signal with opposite sign was registered and this wasattributed to the PB eect, due to the negative dependence of the conductance on temperature.This change of sign can be understood also by this simple model of the total current [54]:

I = IDC + IPV + IPB =

∫ W/2

−W/2σξdx+

∫ W/2

−W/2σ∗ξdx+

∫ L/2

−L/2

dG

dT∆T (x)ξdx (5.1)

where W is the width of the graphene channel, σ is the graphene conductivity, σ∗ is the pho-toinduced change in conductivity due to the the generation of electron-hole pairs due to laserexcitation, ξ is the applied electric eld, G is the graphene conductance; the last term, whichexpress the PB contribution, takes into account the change of conductance induced by heat-ing due to incident optical power. This change was asserted to be negative in literature, sincegraphene resistivity increases with increasing temperature [16] [38]; this explains the change ofsign observed. The integral in this case is done over the channel length to take into accountthe spatial variation of ∆T (x). But the question is: this change in conductance is due to elec-tronic or lattice temperature? In many works, such as the one in ref. [16], in ref. [38], in ref.[54], it was almost implicitly assumed that the bolometric response in general depends on theincrease of lattice temperature, even if in ref. [55] they asserted that the PB eect is sensible toelectronic temperature. The unique heat properties in graphene, such as electron-phonon decaybottleneck, and abundant hot carriers eects, support this last hypothesis.In this chapter, trying to answer this important question, we also assert that this dependence

is electronic temperature sensible, since the calculated lattice temperature lead to inconsistentresults for the term dG

dT and for recombination times of photoexcited carriers.Moreover, a new model of the photocurrent is introduced, dierent from the one in ref. [16],

where it is said that the bolometric eect is null at low doping, where there is only the PVeect. Here we advance the hypothesis that instead the PB part is always present at all level ofdoping, according to the behavior of electronic temperature on which we said we are sensible;at low doping, the photovoltaic part simply overcome the photobolometric one.

5.1. Measurements

5.1.1. Results over the tested devices

Photodetection measurements were performed on six graphene based coplanar waveguides,with 2µm SiO2 dielectric subsrate over High-Resistivity Si substrate. The fabrication process

5.1 Measurements 61

Device Length [µm] Width [µm] Range VDS Step VDS Sweep incident optical power P0

RF1 22 20 2-3 V 0,5 V 62,9/53,7/43,5 mWRF2 25 20 2-4 V 0,5 V 62,9/53,7/43,5/32,1/21,8 mWRF3 22 20 2-3,5 V 0,5 V 62,9/53,7/43,5/32,1/21,8 mWRF4 22 20 1-5 V 0,5 V 62,9/53,7/43,5/32,1/21,8 mWRF5 30 20 0,5-5 V 0,5 V 62,9/53,7/43,5/32,1/21,8 mWRF6 22 20 2,5-3,5 V 0,5 V 62,9/53,7/43,5/32,1/21,8 mW

Table 5.1.: Summary of devices tested with respective length and width of the graphene channel,the sweep of VDS applied and the sweep of incident optical power.

and the experimental setup are described in Chapter 2.To properly study the photodetection and to see the PV and the PB eect, we swept the gate

voltage at dierent DC bias voltage and incident optical power, since the photocurrent signal isexpected to be proportional to both the bias voltage and the optical power [16] [27], as can bededuced also from the simple model in eq. 5.1.We xed the modulation frequency at fopt = 5 GHz; analogous results are obtained also for

other modulation frequencies up to fopt = 20 GHz, where we start to be limited by the MZMbandwidth.We measured the magnitude and phase of the electrical power signal at fopt = 5 GHz entering

in the port 2 of the VNA (see scheme in g. 2.9 and the description in section 2.2.3), as afunction of the gate voltage. The gate voltage was swept between -200 V and 200 V (limits ofthe Kheithleys used). The measurement was repeated for several VDS and optical powers.Since the VNA can be modelized as an equivalent impedance of 50 Ω, this electrical power is

the power consumed by the root mean square of the modulated part of the photocurrent on thisequivalent impedance.In table 5.1, a summary of the conditions applied is shown. For the six devices are then

specied the values of the DC bias voltage applied and the values of incident optical power. Asdescribed in section 2.2.3, in the part of power characterization, we imposed the values comingout from the EDFA, and the values reported are the corresponding ones incident on the sample.The laser spot was focalized in the middle of the channel for each device.Values for VDS were chosen reasonably to have the same work conditions on all the samples.From g. 5.1 to 5.6, for all the six devices are shown the measured magnitude and phase

of the electrical power relative to the modulated part of the photocurrent, together with thecorresponding DC current for xed values of VDS and optical power. The reference phase wastaken at high doping.The change of phase is evident in all the devices, and we can then distinguish clearly the PV

eect from the PB one.To extract values of mobility, residual carrier density and total contact resistance it was used

the two probe method described in section 3.3. In particular, as initial values for the t therewere used a mobility of 1000 cm2

V s and a RC,tot = 6000 Ω, reasonably coherent to the resultsshowed in sec. 3.3. Indeed contact resistances are measured in [Ωµm] and are proportional tothe width of the graphene channel, where it is in contact with source/drain metal pads [10].From the little statistics done in section 3.3.2, the mean value was around 18 kΩ which for 6µm width gives a value on the order of 12000 Ωµm; taking this value, since here we have a 20µm width, this translates in a RC,tot = 6000 Ω. Details and plot of the experimental and ttedresistances will be shown in Appendix C. The extracted values are reported in tab. 5.2 and willbe used for simulations.

5.1 Measurements 62

Device µ [ cm2

V s ] n0 [cm−2] RC,tot [Ω] Vth [V]RF1 870 1,25 ·1012 7000 7RF2 1150 1,1 ·1012 4000 -64RF3 1200 1,3 ·1012 3000 50RF4 1150 1 ·1012 2700 -27RF5 1050 0,9 ·1012 5800 -13RF6 1050 1 ·1012 4000 -5

Table 5.2.: Extracted values of mobility, residual carrier density and total contact resistance forthe devices tested together with the reported Dirac point voltage Vth.

(a) (b)

Figure 5.1.: Magnitude (a) and phase (b) of the electrical power relative to the modulated partof the photocurrent, together with the DC current, in function of gate voltage VGfor device RF1 at VDS = 3 V and P0 = 62,9 mW.

(a) (b)


5.1 Measurements 63

(a) (b)

Figure 5.3.: Magnitude (a) and phase (b) of the electrical power relative to the modulated partof the photocurrent, together with the DC current, in function of gate voltage VGfor device RF3 at VDS = 3,5 V and P0 = 62,9 mW.

(a) (b)


5.1 Measurements 64

(a) (b)


(a) (b)

Figure 5.6.: Magnitude (a) and phase (b) of the electrical power relative to the modulated partof the photocurrent, together with the DC current, in function of gate voltage VGfor device RF6 at VDS = 3,5 V and P0 = 62,9 mW.

5.1.2. Laser scan over dierent positions

In previous works (ref. [16], ref. [27]) it was evidenced that the photovoltaic eect, herecorresponding to the part in which the phase of the signal is 180o, should be peaked around theDirac point. What can be seen is that this is not strictly true for all the devices.This could suggest that we are sensible to the local properties of the graphene investigated,

since we are shining only a relatively small spot with the laser (remembering that the laser spotis around 2,8 µm, see sec. 2.2.3, over a channel length of the order of 20-30 µm, see tab. 5.1)and so we are inducing local eects.To verify this hypothesis we took the sample RF2 and we applied the same conditions as in

g. 5.2, but we pointed the laser spot in several dierent positions along the graphene channel.Three of them are represented in g. 5.7 and for each position there are the plots of the DCcurrent together with magnitude (top plot in each panel) and phase (bottom plot in each panel).In all the dierent points we obtained dierent behaviors: the photovoltaic part is either almost

5.1 Measurements 65

absent, g. 5.7 (a), or around the Dirac point as expected in normal conditions, g. 5.7 (b), orshifted on the left with respect to Dirac point, g. 5.7 (c).These observations show then that we are very sensible to the point on which we shine with

the laser, i.e. we are sensible to the local properties of the graphene channel.To avoid these kind of problems it can be thought to shorten the graphene channel: in this

way we have less probability to have dishomogeneities of the graphene channel itself during thetechnological process; also, with this solution, we decrease the resistance and we can incrementthe photoconductive gain, since it is inversely proportional to the transit time for a carrier togo through all the channel length [27] [30]. Moreover if we shine all the graphene channel weexpect a PV eect more close to the Dirac point because we are probing globally the device, aswe do when we individuate the Dirac point from current measurement.

(a) (b)

(c)

Figure 5.7.: DC current and magnitude and DC current and phase in function of gate voltageVG for device RF2 at VDS = 4 V and P0 = 62,9 mW for three dierent positions ofthe laser spot in the graphene channel. (Scheme of the graphene channel is not inscale).

5.2 Photocurrent modelization 66

5.2. Photocurrent modelization

5.2.1. Photovoltaic current

In photovoltaic eect photo-excited electrons and holes are accelerated in opposite directionsby the external applied electric eld. To account for the photoexcited conductivity σ∗, we canstart from the expression that calculates the local photoexcited carrier density for electrons andholes [16]:

n∗e/h =αP0τrcEoptaspot

(5.2)

where P0 is the incident optical power, Eopt is the photon energy which here is 0,8 eV, since weare using a laser with 1,55 µm wavelength, and so the rate P0/Eopt represents the photon uxarriving on the sample; α ≈ 0, 023 is the graphene optical absorption (section 1.2.4), aspot isthe area of the laser spot, where a diameter of 2,8 µm was considered (from the value found insection 2.2.3) and τrc is electron-hole recombination time, usually of the order of picoseconds [16][51] [40]. In our case this is an unknown parameter and it will be extracted as will be explainedlater.The total photexcited carrier density is then:

n∗ =2n∗e/h

gf, gf =

LW

aspot(5.3)

where gf is a geometrical factor [16], with which we are normalizing the photoexcited carrierdensity over all the channel (L and W are length and width of the channel).Then we can calculate the total PV current as:

IPV =

∫ W/2

−W/2σ∗ξdx, σ∗ = n∗eµ (5.4)

where we are supposing that the mobility of the photoexcited carriers is equal to particle mobility[16] [51]; ξ is the electric eld applied, which in the case of photodetection is uniform and equalto: ξ = VDS/L.We note that in this formula it is already included the photoconductive gain, which is the rate

between the recombination and transit time for photexcited carriers [56], G = τrcτtrans

. Indeed

τtrans = L2

µVDS. As an estimate, in our cases if we put for example τrc ≈ 1 ps, L ≈ 20µm,

VDS ≈ 4 V , µ = 1000 cm2

V s , we obtain a G of the order of 10−3 which means that we havephotodetection power losses around 60 dB with respect to the case in which G = 1.With these relations, we can calculate for each P0 and for each VDS the PV contribution which

in particular we expect proportional to both quantities, as shown in ref. [27].

5.2.2. Photobolometric eect

We simulate the PB current from the temperature simulations developed in the previouschapter. In particular we took the values extracted for mobility and residual carrier density,summarized in table 5.2. Starting from these values, since we know the Dirac point voltageand the oxide capacitance (for 2µm SiO2 dielectric substrate), we can calculate the grapheneresistivity. From graphene resistivity, we can calculate the thermal conductivity through theWiedemann-Franz law. For the supercollision term A we used the same value used in section4.2.2, which is valid for all the doping levels and is experimental.We have to take into account the spatial variation of the chemical potential, which aects

directly the electron-lattice cooling term (through the density of states) and the fraction of


photoexcitation energy harvested by the electronic system (see section 4.1.1); we write a simplelinear model in function of space, along the channel length direction, i.e. for −L

2 < x < L2 [16]

[3]. When we apply a drain-source voltage, the chemical potentials at the extremes are [16]:

βD/S = −sign(VG − VD/S) · ~vF

√1

eπCox|VG − VD/S | (5.5)

Then the spatial variation of the chemical potential is:

E(x) =βD − βS

Lx+

βD + βS2

(5.6)

We introduce a model also for the phonon (or lattice) temperature. A very simple idea isto include the thermal boundary resistance [57] between graphene and SiO2 substrate and thisgives the possibility to model the vertical heat transfer between graphene and the dielectricsubstrate.Then the complete model is a coupled equation for electronic and lattice temperature:

− ∂∂x(ke(x)∂Te∂x ) = γαPave(x)√

2−B(Te − Tph)−A(T 3

el − T 3ph)

−kph∂2Tph∂x2

= (1− γ)γαPave(x)√2

+B(Tel − Tph) +A(T 3el − T 3

ph)− 1RB

(Tph − Troom)

Te(−L2 ), Te(+

L2 ) = Troom

Tph(−L2 ), Tph(+L

2 ) = Troom

(5.7)

where kph is the lattice (in-plane) thermal conductivity 1, RB = 4 · 10−8Km2

W [57] is the thermalboundary resistance introduced before and (1−γ) is the fraction of photon energy not harvestedby the electronic system (section 4.1.1); electron-lattice and cooling term for electronic systemare then source terms in lattice temperature equation. Troom was xed at 300 K.For the source term Pave(x), the expression in eq. 4.9 was taken, where for the values of P0

values in the third column of table 5.1 were used and with σgauss the half of beam diameter. Wedivided Pave(x) by

√2 to take into account the root mean square of the optical power. Indeed

we measure on the VNA the electrical power related to the root mean square of the modulatedpart of the photocurrent; P0 is the amplitude of the modulated part of the incident optical powerand the temperatures that will be calculated will enter in the modulated part of the PB currentthat we want to simulate. So we need to calculate a temperature which is the result of the rootmean square of the incident optical power.For γ, a very rough estimation can be done from the inset of gure 4.3, considering that we

know Jel/Jph from the picture and Jel + Jph = αPave(x). At the photoexcitation energy of 0,8eV, we obtain:

γ = 0.5 + 0.78 · 10191

J· EF (x) (5.8)

where EF (x) is the Fermi level, expressed in Joule, which varies along the space. Even if this isa very simple estimation, results are consistent with ref. [39], where a more proper study on γwas done.All these parameters and equations were implemented for each device on COMSOL Multi-

physics to have a numerical solution for both electronic and lattice temperature.The PB current can be then written as:

1The graphene thermal conductivity normalized with respect to the graphene thickness is 600 WmK

[58]; here,to have consistent unity measures, we multiply by the graphene thickness d = 0.34 nm, so we obtain kph =2, 04 · 10−7W/K


(a) (b)

Figure 5.8.: Numerical solution of electronic (a) and lattice (b) temperature for the sample RF1at VGth = 0 and Popt = 62, 9 mW .

IPB =

∫ L/2

−L/2

dG

dT∆T (x)ξdx (5.9)

where the term dGdT is another unknown parameter that will be extracted and that we expect

to be on the order of -50 nS/K [16] [38]. For simplicity in our simulations it will be consideredas constant with respect to the graphene electrostatic doping. This physical quantity expressesin practice the change in conductance induced by temperature increase. It is negative becausethe resistivity increases with temperature, most likely due to a greater interaction with phononsarising from the dielectric substrate, like explained in section 3.1.2.For both electronic and lattice temperature, an example of spatial extent is shown in g. 5.8,

where the numerical solution was taken for a xed VGth and P0 for the device RF1. Electronictemperature changes with incident optical power and doping, as expected (see sections 4.1.2 and4.2.2), while lattice temperature was found to not depend on the gate voltage sweep. Anyway,in both cases, the spatial extent reects the one presented in g. 5.8 such that in each case the∆T (x) (which is either Te(x) − Troom or Tph(x) − Troom) can be approximated by a triangularfunction from -4 µm to 4 µm centered at T(0), where it is positioned the laser spot.

(a) (b)

Figure 5.9.: Numerical solution of electronic (a) and lattice (b) temperature for the sample RF1at x=0, in function of VG and the incident optical power P0.


(a) (b)

Figure 5.10.: Numerical solution of electronic (a) and lattice (b) temperature for the sampleRF2 at x=0, in function of VG and the incident optical power P0.

(a) (b)


(a) (b)



(a) (b)


(a) (b)


Since the relevant temperature for the photocurrent simulation is then the one at the centerof the graphene channel, where we put the laser, from g. 5.9 to g. 5.14 they are shown, foreach device, the electronic and lattice temperature at x=0 in function of the gate voltage andthe incident optical power. As can be noted, the lattice temperature doesn't depend on the gatevoltage (all the curves for dierent VG are superimposed to the one in the gures), while theelectronic temperature highly depends on doping and in particular is maximum at the Diracpoint voltage in each device; it was found also to not vary sensibly with the bias voltage VDS .


5.2.3. Model validation

(a) (b)

Figure 5.15.: (a) Total measured photocurrent (blue line), simulated photobolometric current(red line) and extracted photovoltaic current (green line) in function of gate voltageat VDS = 3 V and P0 = 62, 9 mW for the device RF1 (b) Comparison between thephotovoltaic peak in (a) (circles) and root mean square of calculated photovoltaiccurrent from eq. 5.4 (lines) in function of VDS at dierent P0.

(a) (b)



(a) (b)

Figure 5.17.: (a) Total measured photocurrent (blue line), simulated photobolometric current(red line) and extracted photovoltaic current (green line) in function of gate volt-age at VDS = 3, 5 V and P0 = 62, 9 mW for the device RF3 (b) Comparisonbetween the photovoltaic peak in (a) (circles) and root mean square of calculatedphotovoltaic current from eq. 5.4 (lines) in function of VDS at dierent P0.

(a) (b)



(a) (b)


(a) (b)

Figure 5.20.: (a) Total measured photocurrent (blue line), simulated photobolometric current(red line) and extracted photovoltaic current (green line) in function of gate volt-age at VDS = 3, 5 V and P0 = 62, 9 mW for the device RF6 (b) Comparisonbetween the photovoltaic peak in (a) (circles) and root mean square of calculatedphotovoltaic current from eq. 5.4 (lines) in function of VDS at dierent P0.

From g. 5.15 to g. 5.20, for all the devices, the measured photocurrent, the simulatedphotobolometric current and the extracted PV current, calculated as the dierence betweentotal measured photocurrent and PB current, are presented in the panels at left at xed VDSand P0; in the right panels a comparison between the peak of the extracted photovoltaic currentand the root mean square of the value found from eq. 5.4 is done.The total measured photocurrent for each device is extracted from the registered values of

magnitude and phase of the electrical power consumed by the root mean square of the modulatedpart of the photocurrent on the characteristic VNA impedance R = 50 Ω; the sign was chosencoherently with respect to the phase of the DC current: the part in which the phase is shown to


be of 180o has the same sign of the bias current, so it ows from drain to source for VDS > 0.The electrical power was converted in linear scale and the root mean square of the modulatedpart of the photocurrent is: Im,rms =

√(Prms/R).

We suppose, like in ref. [16], that at high doping we have the large part of the contributionfor the total photocurrent arising from the photobolometric eect. We note that, taking asan example all the curves at P0 = 62, 9 mW shown in the left panels of gures 5.15-5.20, themeasured photocurrent in the high doping case is always on the order of ≈ 10−6−10−7 A. If wetook the phonon temperatures calculated, shown in gures 5.9-5.14, where the peak values atx=0 for ∆T are ≈ 0, 8−0, 9 K corresponding to the xed P0 that we are choosing, and we wouldlike to calculate the PB current from eq. 5.9, we should put a (constant) value of dGdT ranging from-500 to -1000 nS/K for all the devices, which is not coherent to what it is found in literature [16][38]. For this reason we calculated the photobolometric contribution starting from the electronictemperature. For each VDS and P0 and for each device the shapes are similar to the ones shownin left panels of pictures 5.15-5.20 (red lines), so higher in modulus near the Dirac point voltage,where indeed we have that the electronic temperatures have their maximum (see section 5.2.2).The parameter dG

dT was chosen in order to be less or equal to the total measured photocurrent(in order to have always a positive PV current, which is indeed a physical constraint) and to becoincident as much as possible to the total PC at the extremes, always being in the hypothesisthat at high doping case the main contribution to the total photocurrent is the photobolometricone.The subtraction between the measured photocurrent and the simulated PB current gives

then the photovoltaic contribution (green lines in left panels of gures 5.15-5.20). With theseassumptions, the behavior of the PV part with respect to the gate voltage is a bell shape, i.e itdecreases at high doping, like it was asserted in ref. [16]. With eq. 5.4 we can't model such abehavior, since there is no explicit dependence on gate voltage. To do this, experiments shouldbe performed in this direction. An idea for example is to have a pump-probe experiment in whichthe gate voltage can be swept; in this way it can be directly seen if the recombination timeschange with respect to the doping level. Further studies need to be done also to understandcompletely the physics behind this expected behavior. One hypothesis is that since when wehave a photoexcited carrier at low doping case we have a carrier multiplication factor (suchthat for a single photoexcited carrier we have multiple electron-hole pairs generated), while athigh doping we have simply a redistribution and scattering between the photoexcited carrierand electrons of the Fermi sea, the electron-hole pairs generated are bigger in quantity near theDirac point voltage (see section 4.1.1). Another hypothesis is that is the recombination timefor photoexcited carriers that decreases with increasing electrostatic doping, as it happens intypical semiconductors [30]. The formula in eq. 5.4 is then interpreted to be at the Dirac pointvoltage [16]. In particular, since the measured photocurrent is the root mean square of themodulated part, we have to divide also the expression of eq. 5.4 by

√2, in order to have the

root mean square value. Then the recombination time was extracted, since it is the unknownparameter, in order to have for each VDS and P0 a good agreement between the value found withthe formula and the peak of the extracted PV contribution. This comparison is directly donefor all the devices in the right panels of gures 5.15-5.20. As can be seen, we could estimate theexact order of magnitude and moreover the dependence on the optical power and on the biasdrain/source voltage is in optimal agreement with the one calculated theoretically.Finally, in tab. 5.3 the extracted values for dG

dT and the recombination times for each deviceare reported. For both quantities we found the right order of magnitude expected in literature[16] [38] [40] [51] [39]. One important thing to be noted is that, if we consider the hypothesisin which the photobolometric current depends on the lattice temperature (through ∆T (x) =Tph(x)− Troom, see eq. 5.2.2), the simulated PB part should not depend on doping and shouldalways remain on the order of the measured total photocurrent at high doping even near the


Device dGdT [nS/K] τrc [ps]

RF1 -80 0,85RF2 -85 0,65RF3 -350 2,2RF4 -120 0,9RF5 -90 0,75RF6 -90 0,6

Table 5.3.: Extracted values of dGdT and τrc for all the devices.

Dirac point voltage, like explained before; then the peak of the extracted photovoltaic current,considering always the dierence between the measured PC and PB eect, should be 3/4 timessmaller then the value extracted and reported here, as can be easily seen also from gures5.15-5.20; this directly reects in the extraction of the recombination times, which should resultalways around ≈ 100 fs, completly unreasonable. Even lower recombination time values arefound if we consider the hypothesis in ref. [16], where they asserted that near the Dirac pointvoltage the PB eect is almost zero.All these considerations lead us to think that graphene based photodetectors are actually

sensible to electronic temperature. Morever, contrarily to what was asserted in precedent works[16], the PB eect is always present and is even more present near the Dirac point voltage thenat high doping and this comes directly from the electronic temperature model developed in theprevious chapter.A more systematic study should be done anyway to conrm this new hypothesis, as well as

a more accurate analysis about the graphene local properties that, as seen in section 5.1.2, candirectly inuence measurements.

6. Conclusions and perspectives

In conclusion, in this thesis we investigated and highlighted the main mechanisms which gen-erate photocurrent in graphene based photodetectors. Modulating the light intensity of our laserwe have a modulated part of the photocurrent, that we measure. The two main contributionsfor this photocurrent are the photovoltaic eect, i.e. the electron-hole charge separation bythe external electric eld and the photobolometric eect, i.e. the change in conductance dueto the heating induced by the laser. Heating our system with an optical power means heatingboth the electron system and the lattice, inducing then an increase of both electron and latticetemperatures. For both systems a steady state temperature is reached through electron-phononscattering mechanisms (principally disorder assisted scattering, supercollisions [14]), which werealso studied in Chapter 4.In previous works [16], the change in conductance was asserted to be sensible to the increase

of the lattice temperature, while here we demonstrate that if the lattice temperature increase isconsidered, unreasonable values of dGdT , which expresses the change in conductance with respectto temperature increase and is negative, and of the photoexcited carriers recombination timesare found. Then we assert that we are sensible to electronic temperatures. In practice we aresaying that the change in conductance detected can be used as a thermometer of the electronicsystem and this is due to the weak coupling between the electronic and phonon systems, apeculiar characteristic of graphene at room temperature [14] [13][55].To understand better and exploit these mechanisms further studies should be done in this

direction, even if from the technological point of view other kind of devices are being developed.For example p-n junction graphene on silicon photonic waveguide were produced [59], with whichthe light-graphene interaction is increased (and in particular it is increased the light absorptionin the device) and the photothermoelectric eect is exploited. Anyway also here a model forelectronic temperature is always needed in order to control and improve the device eciency.

A. Appendix - Matlab script for resistivity t

A.0.1. Four-probe method

As described in Chapter 3, section 3.2.2, in four-probe technique we want to nd a constantvalue of mobility µ that better minimizes the quantity:

S =∑i

(yi − ρ(VGthi , µ))2 (A.1)

where yi are the experimental values of resistivity in function of VGth and ρ(VGthi , µ) is thequantity that recalls eq. 3.14:

ρ(VGthi , µ) =

√1

1ρ2max

+ C2oxV

2Gthi

µ2(A.2)

Function to be minimized

We rst implement the quantity S, to be minimized:

func t i on out = sum_square (param , xdata , ydata )

% th i s func t i on that we cons t ruc t take as input a param ,% de f ined as the square o f mobi l i ty ,xdata = V_Gth , ydata = exper imenta l va lue s o f r e s i s t i v i t y

mu_quad=param ;

% SiO2 d i e l e c t r i c constanteps_ox=3.9;

% vacuum pe rm i t t i v i t y in SI un i t seps_0=8.85∗1e−12;

% t o t a l d i e l e c t r i c constanteps=eps_0∗eps_ox ;

% oxide th i c kne s s in metert_ox=300∗1e−9;

% oxide capac i tanceCox=eps /t_ox ;

% f i nd rho_max[ a , b]=max( ydata ) ;

% formula o f S

78

out = sum( ( ydata−s q r t ( ( 1 . / ( 1 . / a.^2+Cox .^2 .∗mu_quad .∗ xdata . ^ 2 ) ) ) ) . ^ 2 ) ;

Fit script

To minimize the function S we wrote before we use the Matlab function 'fminsearch', whichtakes as an input a handle function of a variable, an initial value for that variable and someoptimization parameters. In particular:

% d e f i n i t i o n o f constants , the same as func t i on Seps_ox=3.9;eps_0=8.85∗1e−12;eps=eps_0∗eps_ox ;t_ox=300∗1e−9;Cox=eps /t_ox ;% e l e c t r on chargeq=1.6∗1e−19;

% handle func t i on o f the va r i ab l e x% we cons t ruc t i t us ing the func t i on we made in the prev ious s e c t i o n% x corresponds to param in func t i on sum_squarefun = @(x ) sum_square (x ,VGth , rho ) ;% f o r each device , we put as input o f sum_square ve c t o r s% o f VGth and r e s i s t i v i t y .

% i n i t i a l va lue o f mu^2 , with mu in [m^2 V^−1 s^−1]x0 = 0 . 0 1 ;

% opt imiza t i on parametersopt ions=opt imset ( 'MaxFunEvals ' , 4000000 , ' MaxIter ' , 4000000 , ' TolFun ' , 1 e−8 , 'TolX ' , 1 e−8);

% 'MaxFunEvals ' : maximum number o f t imes the rou t in e eva lua t e s% the ob j e c t i v e funct ion ,% each time with the updated parameter e s t imate s .% 'MaxIter ' : Maximum number o f i t e r a t i o n s f o r fminsearch al lowed% 'TolFun ' : i s a lower bound on the change in the value o f the% ob j e c t i v e func t i on during a step .% I f | f ( x_i)− f (x_ i +1)| < TolFun , i t e r a t i o n s end .% 'TolX ' : i s a lower bound on the s i z e o f a step ,% the norm o f ( x_i − x_ i +1).% I f the s o l v e r attempts to take a step that i s sma l l e r than TolX ,% the i t e r a t i o n s end .% The va lue s put f o r the se opt ions were choosen reasonably ,% to not be l im i t ed during i t e r a t i o n s .

% fminsearch g i v e s the value o f x that be t t e r s u i t% the prev ious cond i t i on sbestx = fminsearch ( fun , x0 , opt ions ) ;

79

% mobi l i ty mu then i s the square root o f bestxmu=sq r t ( bestx ) ;% n0 i s found us ing formula o f rho_max in Chapter 3% al ready ca l c u l a t ed in cm^−2n0 = 1e−4∗(1./(max_rho∗mu∗q ) ) ;

% now we can c a l c u l a t e the r e s i s t i v i t y with the f i t t e d parameter mumax_rho=a ;r h o f i t =(1./( sq r t ( 1 . / a.^2+Cox .^2 .∗mu_quad .∗VGth^2 ) ) ) ;

A.0.2. Two-probe method

The two-probe technique resembles strictly the one for four-probe presented.Now we want to nd the couple (µ,RC) that minimizes the quantity:

S =∑i

(yi −R(VGthi , µ,RC,tot))2 (A.3)

where yi are the experimental values of the total resistance in function of VGth andR(VGthi , µ,RC,tot)is the quantity that recalls eq. 3.17:

R(VGthi , µ,RC,tot) = RC,tot +(L/W )√

(L/W )2

(Rtot,max−RC,tot)2 + C2oxV

2Gthi

µ2(A.4)

Function to be minimized

Here we construct a function which returns the sum S:

func t i on [ knownVariables ] =sum_square ( unknownVariables , vGate , rTotal , beta , ox ideCapac i tance )% This func t i on takes as input ve c t o r s VG, t o t a l r e s i s t an c e ,% quan t i t i e s L , W, ( here beta = L/W, see below ) and Cox and% the vec to r unknownVariables , where% unknownVariables (1 ) i s R_C, to t and% unknownVariables (2 ) i s the square o f mob i l i ty mu

% f ind R_ tot ,max[ rTotalMax , vGateDiracIndex ]= max( rTota l ) ;% de f i n e V_GthvGateBar = vGate − vGate ( vGateDiracIndex ) ;

% wr i t e formula f o r R_ tot (VGth_i , mu, R_C)f i t = unknownVariables (1 ) +beta . / sq r t ( ( beta .^2 / ( ( rTotalMax − unknownVariables ( 1 ) ) . ^2 ) +( oxideCapac i tance .^2 ∗ vGateBar .^2 ∗ unknownVariables (2 ) ) ) ) ;

% func t i on output , which i s the formula f o r SknownVariables= sum( rTotal− f i t )^2 ;

80

end

Fit script

Also here we use 'fminsearch', Matlab function. So:

% Constants d e f i n i t i o n s , in SI un i t s% same cons tant s as f o r f i t s c r i p t in four−probe techniquee = 1.602 e−19;eps_ox=3.9;eps_0=8.85∗1e−12;eps=eps_0∗eps_ox ;t_ox=300∗1e−9;Cox=eps /t_ox ;

% For each device , VG, VDS, current , R_C, to t are p r ev i ou s l y ext rac t ed% and are t r ea t ed as ve c t o r s in the s c r i p t% are a l s o known va lue s o f l enght L and width W

% ca l c u l a t e R_ tot rTota l = v_Drain_Source . / i_Drain_Source ;% f i nd R_ tot ,max and so Dirac po int p o s i t i o n and VGth[ rTotalMax , vGateDiracIndex ] = max( rTota l ) ;vGateBar = vGate − vGate ( vGateDiracIndex ) ;

beta = L/W;

% Minimizing func t i on% l i k e be fore , we use the func t i on c rea ted sum_squarefun =@( unknownVariables ) sum_square ( unknownVariables , vGate , rTotal , beta , Cox ) ;

% as i n i t i a l va lue s we use :% exper imenta l va lue s o f R_C, to t in the Dirac po int% 0.01 f o r mu^2 , with mu in [m^2 V^−1 s^−1]x0 = [R_C, to t ( vGateDiracIndex ) , 0 . 0 1 ] ;

%opt imiza t i on parameter , s e e f i t s c r i p t f o r four−probe techniqueopt ions=opt imset ( 'MaxFunEvals ' , 4000000 , ' MaxIter ' , 4000000 , ' TolFun ' , 1 e−8 , 'TolX ' , 1 e−8);

% output o f fminsearchknownVariables = fminsearch ( fun , x0 , opt ions ) ;

% r e s u l t s from minimizing func t i onrContact = knownVariables ( 1 ) ;mu = sq r t ( knownVariables ( 2 ) ) ;

81

% f i n a l c a l c u l a t i o n s f o r the f i t t e d R_ tot rTota lCa l cu la ted = rContact +beta . / sq r t ( ( beta .^2/ ( ( rTotalMax−rContact ) .^2 ) +(Cox.^2 ∗ vGateBar .^2 ∗mu. ^ 2 ) ) ) ;

% f i nd n0 from the formula wr i t t en in s e c t i o n r e l a t i v e to% two−probe techniquen0 = beta . / ( ( rTotalMax − rContact ) .∗ e .∗ mu) ;

B. Appendix - Analytic solution of heat

equation including electron-lattice

cooling

We solved analytically eq. 4.11 with the Green function. We rewrite here for simplicity theproblem:

−ke ∂2T∂x2

= γαPave(x)−B(T )

T (−L2 ), T (+L

2 ) = 0(B.1)

remembering that T (x) = Te(x)−Tph, where Tph is assumed to be constant and equal to Troom.This is a typical 1D Sturm-Liouville (S-L) problem, where the S-L problem with homogeneous

boundary conditions reads as [53]:ddx(p(x)dudx) + q(x)u(x) = f(x) in Ω

u(x) = 0 on ∂Ω(B.2)

where p(x) and q(x) are real and continuous functions and Ω is the domain, in our case thechannel length. From eq. B.1, we can set p(x) = 1 and q(x) = −β, β = B

ke= constant,

f(x) = −γαkePave(x). Then eq. B.2 in our case reads:

d2T (x)dx2

− βT (x) = f(x) in Ω

T (x) = 0 on ∂Ω(B.3)

and the general solution can be expressed as [53]:

T (x) =

∫ +L/2

−L/2f(x0)G(x, x0)dx0 (B.4)

where G(x, x0) is the Green function associated to the problem B.3, solution of:

d2

dx2G(x, x0)− βG(x, x0) = δ(x− x0) (B.5)

For x 6= x0, the general solution for G(x, x0) can be written as:

G(x, x0) = c1e√βx + c2e

−√βx (B.6)

To nd the solution for G, as it is usually done, we impose the homogeneous boundaryconditions for x < x0 and x > x0, we impose the continuity of the function at x = x0 and weimpose the discontinuity of the rst derivative at x = x0.

Homogeneous boundary conditions for the Green function

For x < x0, G(−L2 , x0) = 0 leads to:

c1 = −c2e√βL (B.7)

83

For x > x0, G(L2 , x0) = 0 leads to:

c3 = −c4e−√βL (B.8)

Then:

G(x, x0) =

−c2e

√βL2 sinh(

√βx+

√βL2 ) for x < x0

−c4e−√βL2 sinh(

√βx−

√βL2 ) for x > x0

(B.9)

Jump conditions at x = x0

In eq. B.5 at x = x0 we have a singularity. From the Green function's properties and in thecase of S-L problem this translates in imposing a continuity condition for G in x = x0 and adiscontinuity condition of the rst derivative [53]:[dG

dx

]x=x−0x=x+0

=1

p(x)(B.10)

where in our case p(x) = 1.For the continuity condition:

c2 = c4e−√βL sinh(

√βx0 −

√βL2 )

sinh(√βx0 +

√βL2 )

(B.11)

For the rst derivative, applying the condition B.10 and using eq. B.11, we obtain:

c4 =1

C,

C =√βe−

√βL2

[sinh(√βx0 −

√βL2 )

sinh(√βx0 +

√βL2 )

cosh(√βx0 +

√βL

2)− cosh(

√βx0 −

√βL

2)] (B.12)

Final solution

Since we have c4, we can calculate c2 from B.11 and we can substitute them in eq. B.9 tohave the full expression of the Green function.To represent the analytic solution of T(x) we implemented and wrote the found Green function

in Matlab and we calculated the integral from the expression B.4. Then Te = T (x) + Tph andthe solution is shown in g. 4.5 (a).

C. Appendix - Resistance t for the

graphene coplanar waveguides tested

Extraction of mobility for graphene coplanar waveguides tested, and on which photodectionmeasurements and simulations were done in Chapter 5, is needed. Of course we don't have afour-probe conguration with which we can directly investigate the graphene resistivity, so wehave to use the two-probe method explained in section 3.3 and Appendix A. In this method wehave to start from initial values for both mobility and total contact resistance; here for all thedevices they were chosen: µ = 1000 cm−2/V s and RC,tot = 6000 Ω, coherent with the valuesfound from the little statistics done on graphene based transistors and shown in Chapter 3, asexplained in section 5.1.1.Here all the plots of experimental and tted total resistance are presented. They refer to the

curves presented from g. 5.1 to g. 5.6.

Figure C.1.: Experimental (red) and tted (blue) total resistance for device RF1.

85



86



87


Device µ [ cm2

V s ] n0 [cm−2] RC,tot [Ω] Vth [V] max. rel. error [%]RF1 870 1,25 ·1012 7000 7 4%

RF2 1150 1,1 ·1012 4000 -64 8%

RF3 1200 1,3 ·1012 3000 50 5%

RF4 1150 1 ·1012 2700 -27 8%

RF5 1050 0,9 ·1012 5800 -13 5%

RF6 1050 1 ·1012 4000 -5 4%

Table C.1.: Extracted values of mobility, residual carrier density, total contact resistance andmaximum relative error with respect to the experimental resistance for the devicestested.

In table C.1, they are reported, together with the reported Dirac point voltage, the ex-tracted values of mobility, residual carrier density, total contact resistance (like in table 5.2)and maximum relative error with respect to the experimental resistance (% rel. error = 100· |Rtot,exp−Rtot,calc|Rtot,exp

) with which we have a simple estimation of the goodness of the t.We can note from g. C.1 to g. C.6 that the tted resistance follows quite well in all the

cases the experimental one and also, from table C.1 the maximum relative error committed ishighly acceptable.

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