THz Photodetection in Graphene Field Effect Transistors

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Università di Pisa

Facoltà di Scienze Matematiche Fisiche e Naturali

Corso di Laurea Magistrale in Fisica della Materia

A. A. 2010/2011

Tesi di Laurea Magistrale

Terahertz Photodetection

in Graphene Field Eect Transistors

Candidato: Relatori:

Leonardo Vicarelli

Prof. Vittorio Pellegrini

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Abstract

This Master thesis (Tesi di Laurea Magistrale) targets the design and test of a graphene-based Terahertz (THz) photodetector. We exploit the high carrier mobility of graphene to fabricate a top-gated graphene eld eect transistor (GFET) that is able to detect THz radiation at room temperature. The eciency of this type of detectors is directly linked to the carrier mobility, which can be quite low in traditional semiconductor FETs at room temperature. On the contrary, graphene shows extremely high-mobilities mostly independent of temperature, which makes it a promising material for a wide range of applications and for THz detec-tion. The photodetection theory of a two-dimensional electron plasma was originally developed by M. Dyakonov and M. S. Shur in 1993 [1] and it is based on the nonlinear transport properties of semiconduc-tor FETs, which lead to the rectication of an ac current induced by the incoming radiation. In our approach the modulation of the car-rier density in the graphene channel was achieved through a metallic top gate, isolated from graphene by a thin layer of Hafnium Oxide that we deposited using Atomic Layer Deposition (ALD). In addition, we ex-ploited integrated log-periodic antennas in order to enhance the coupling between the incident THz wave and the graphene transistor. We suc-cessfully detected radiation at 0.3 THz at room temperature, both with monolayer and bilayer graphene devices. We achieved a maximum re-sponsivity of 0.09 V/W and a minimum Noise Equivalent Power (NEP) of 7 × 10−8_W/√_Hz_{. To our knowledge, our work demonstrates the rst}

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1 Introduction

1.1 The Terahertz Band

THz radiation comprises electromagnetic waves with frequencies belonging to the THz band which is situated between the upper frequency limit of the radio-frequency part of the electromagnetic spectrum (electronic domain) and the lower frequency limit of the infrared part (optical domain). This is illustrated in Fig. 1.1.

As all electromagnetic waves, THz radiation can be described by Maxwell's equations. Due to its specic wavelengths, THz radiation possesses some unique properties. It is non-ionizing (unlike for instance x-rays) due to the low photon energies of less than 0.05 eV. It is therefore considered to be safe for biological samples and operators, also because radiation intensities are generally small compared to those in other frequency domains. Similar to microwaves, THz waves are able to penetrate many non-conducting and thin materials such as plastics, ceramics, cardboard, or clothing that are opaque to infrared or visible light. Compared to microwaves they allow a higher spa-tial resolution in imaging applications and smaller antenna apertures due to shorter wavelengths. They are less aected by Mie scattering than infra-red or visible light in many situations (e.g. fog; small particles in solutions etc.) due to their longer wavelengths. Water is a strong absorber of THz waves and metals reect them completely.

Figure 1.1: Electromagnetic spectrum with THz band position.

Terahertz detectors

THz detectors are commonly classied into coherent and incoherent detectors. Coherent detectors (also called indirect or phase sensitive detectors) are able

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Detector f NEP Cool- Resp. [THz] [pW/√Hz] ing Time Golay Cell 0.2-30 150 no 30 ms Pyro Detector 0.1-30 400 no 10 ms Schottky Diode 0.8 20 no < 1ns Bolometer, Nb 0.1-30 0.03 4 K 0.5 ms Bolometer, Si 2-4 ∼ 0.5 no 2 ms Si-MOSFETs 0.3-1 10 no 10 ps

Table 1.1: Overview of characteristics and performances of the most common incoherent THz detectors. Data is extracted from Ref. [2]

to measure the amplitude and the phase of a THz wave, while the incoher-ent detectors (also called indirect or energy detectors) measure only intensity. Coherent detectors provide therefore richer information and allow the mea-surement of refractive indices and the reconstruction of 3D images based on phase information; however, they require a higher system complexity.

We focus on some of the most common incoherent detectors and compare their characteristics and performances in Table 1.1, in terms of sensitivity (Noise Equivalent Power or NEP) and speed (response time).

The Golay cells are pneumatic detectors in which a gas in a hermetic cham-ber is heated up by the incident radiation and the volume expansion is mea-sured, for example, with a membrane whose curvature deects a laser beam; they have a at response over a wide frequency range, but they are also very slow and fragile.

Pyroelectric detectors consist of a particular crystal that exhibits sponta-neous electric polarization that varies strongly with temperature; the radiation heats up the crystal and the change of dielectric polarization is measured as a voltage via an external circuit; they are sensitive and robust but they are quite slow in response time.

In bolometers, the THz radiation heats up an absorber whose tempera-ture rise is measured by the thermo-resistive eect; cryo-cooled, supercon-ducting bolometers are currently the most sensitive detectors for THz radi-ation with a NEP that can be as low as 0.03 pW/√Hz; on the other hand, room-temperature microbolometers are still extensively developed and show promising performances.

Schottky barrier diodes exploit the non-linearity in the I-V characteristic to rectify the THz signal; these detectors are well performing but have relatively high fabrication costs.

Finally, FET-based THz detectors exploit the theoretical predictions of Dyakonov and Shur [1, 3] (see Section 4.1). The rst experimental demon-strations of THz detection with eld eect transistors (FETs) were done with

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dierent GaAs/AlGaAs and GaInAs/GaAs high electron mobility transistors (HEMTs) starting in 1998 [4]. At room temperature, the devices act as broad-band detectors [3], while at cryogenic temperatures and high frequency ( & 2 THz) the detector response shows resonant peaks due to resonant plasmon modes [5]. The rst demonstration of sub-THz and THz detection by FETs in silicon and at room temperature was made by Knap et al. [6].

Practical Terahertz applications

Possible applications for commercial terahertz systems include medical imaging and security applications.

Because of their low photon energies and non-ionizing character THz waves are promising candidates for applications in the medical eld. They are not expected to damage DNA or tissue molecules as x-rays can do. Even though THz waves cannot penetrate the entire body, many applications are possible. For instance the detection of skin cancers (in-vivo), breast cancer (ex-vivo), skin burns, or osteoarthritis in knee joints has been demonstrated with reec-tion imaging techniques [7]. The unhealthy tissue contrast is mostly due to a higher or lower water content. The high water absorption inside the body can be overcome in a multitude of cases with recently developed miniature THz endoscopes [7]. In dental care, small scanning devices could take 3D pictures of the teeth due to their low water content to detect early stages of caries or monitoring fabrication of dental prostheses. Figure 1.2-left shows the THz image of a leaf: water content is higher in the veins.

In the security sector, two main applications can be distinguished. One is the screening of people at airports or border check points in order to detect hid-den weapons and contraband goods. With clothing being transparent to THz waves, concealed objects can be visualized. The rst of these systems has al-ready been commercialized. The image in Figure 1.2-right is from the company QinetiQ and shows a man hiding a knife in a newspaper. QinetiQ does not disclose details of the used imaging system, but it is likely that Schottky diode based detectors operating at nearly microwave frequencies (f ∼ 100 GHz) have been applied. A second application is the detection of forbidden substances (drugs, explosives) on people or in mail envelopes. THz spectroscopes with a sucient spectral resolution can clearly distinguish harmful substances from non-oensive ones.

1.2 Graphene

Graphene is a one-atom thick layer of carbon atoms tightly packed into a two-dimensional (2D) honeycomb lattice [8]. Graphene has been studied the-oretically as the building block of graphite for over 60 years [9] but, before its isolation from graphite in 2004 by Novoselov and Geim [10], it was believed

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Figure 1.2: Left: THz image of a freshly cut leaf and the same leaf after 48 hours; attenuation of THz radiation through the leaf is largely due to water content (imagThzwaves). Right: A man hiding a knife in a newspaper imaged with THz radiation (QuinetiQ)

to be unstable with respect to the formation of curved structures (i.e. soot, nanotubes, fullerene). Since its discovery, graphene has been the focus of in-tense research activities due to its unusual electrical and magnetic properties. Its band structure is characterized by two special points in the Brillouin zone (called Dirac points) where the energy depends linearly on quasi-particle mo-mentum. This peculiar energy dispersion leads to the remarkable fact that charge carriers in graphene behave like relativistic particles described by a 2D massless Dirac equation. Graphene is gapless and shows ambipolar electric eld eect, which means that charge carriers can be tuned continuously between electrons and holes. Moreover it displays extremely high carrier mobilities, mostly independent of temperature.

If we combine two layers of graphene we obtain bilayer graphene (which is why, to avoid misunderstanding, it is preferable to speak of monolayer graphene instead of simply graphene). Bilayer graphene displays a quadratic energy-momentum dispersion and it possesses a tunable band-gap, that can be opened by applying a potential dierence between the two layers (see section 2.6). A band-gap can be opened also in monolayer graphene, through physical connement in a quasi-one-dimensional geometry (graphene nanoribbons [11]) and through the interaction with specic substrates.

Graphene based electronic, photonic and optoelectronic devices The unique electronic properties of graphene have been already exploited to create ultra-high speed FETs. The rst top-gated graphene FET (GFET) was realized in 2007 by Lemme et al. [12] and, after only three years, IBM realized a GFET with a cuto frequency of 100 GHz [13].

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photoelec-tric eect: absorbed photons excite carriers from the valence to the conduction band, outputting an electric current. So far, graphene-based photodetectors have been reported to reveal radiation in the near infrared region of the elec-tromagnetic spectrum (λ ≈ 0.5 ÷ 2.5 µm) modulated at very high frequencies (40 GHz), which makes them promising candidates for high-speed optical com-munications [14].

Graphene has also been used as a saturable absorber in a mode-locked laser and as a replacement for ITO (indium tin oxide) for transparent conduc-tive lms in touchscreens, organic photovoltaic cells and organic light-emitting diodes [15].

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2 Graphene

2.1 Band structure of graphene

Graphene is a single layer of carbon atoms arranged in a hexagonal lattice such as that shown in Fig. 2.1-left.

The unit cell, highlighted in gray in Fig. 2.1-left, is composed of two carbon atoms, labeled A and B, and the primitive vectors are:

a1 = a 2(3, √ 3) a2 = a 2(3, − √ 3) (2.1)

where a = 1, 42 Å is the carbon-carbon distance.

The reciprocal lattice is also hexagonal, shown in Fig. 2.1-right, with the high symmetry points Γ, K, K0_{and M. The two points K, K}0 _{are named Dirac}

points and they are very important for the physics of graphene for reasons that will be clear below.

Figure 2.1: Left: graphene honeycomb lattice showing the two triangular sub-lattices in dierent colors. Right: graphene Brillouin zone in momentum space. Adapted from [16].

Using a tight-binding approach, limited to the second-nearest neighbor, it is possible to derive the essential features of the band structure of graphene [9, 17].

There are three nearest neighbors, described in real space by the vectors δ1 = a 2(1, √ 3) δ2 = a₂(1, − √ 3) δ3 = −a(1, 0) (2.2)

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and six second-nearest neighbors δ0₁ = ±a1 δ 0 2 = ±a2 δ 0 3 = ±(a2− a1) (2.3)

The tight-binding Hamiltonian for electrons in graphene has the form (with ~ = 1): H = −t X hi,ji,σ (a†_σ,ib†_σ,j+ h.c.) − t0 X hi,ji,σ (a†_σ,ia†_σ,j+ b†_σ,ib†_σ,j+ h.c.) (2.4) where a†

σ,iand aσ,i are the creation and annihilation operators for electrons

with spin σ (σ =↑, ↓) on site Ri on sublattice A, while b †

σ,j and bσ,j are referred

to sublattice B.

The hopping energy for the nearest neighbor is t ≈ 2.7 eV and for the second nearest neighbor is t0 _{≈ −0.2 t}_.

The eigenvalues of this Hamiltonian have the form: E±(k) = ±t p 3 + f (k) − t0f (k), (2.5) f (k) = 2 cos(√3kya) + 4 cos( √ 3_/₂_k ya) cos(3/2kxa), (2.6)

where the plus sign refers to the upper band (π∗) and the minus sign to the lower band (π). The full band structure is shown in Fig. 2.2(a): close to the K and K' points the energy dispersion is linear (see Fig.2.2(b)).

Since each carbon atom (electronic conguration 1s2_2s2_2p2_{) hybridizes with}

its three nearest neighbors (sp2 _{hybridization), there is one electron left in the}

pz orbital. Therefore the system is half lled and the physics is governed by

the spectrum close to the K and K' points. We can expand the dispersion relation close to the K (or K' ) point and, neglecting the hopping energy t0

and restoring ~, we obtain:

E±(q) ≈ ±vF~ |q| . (2.7)

q = k − K is the momentum measured from the K point and vF = 3ta/2

is the Fermi velocity. Substituting the values for t = 2.7 eV and a = 1.42 Å we nd that vF ' 1 × 106m/s, which is roughly 1/300 the speed of light in

vacuum.

Equation 2.7 says that, close to the K and K0_{points, the energy-momentum}

relationship is linear and that the conduction and the valence band intersect at q = 0, with no energy gap. Graphene is thus a semimetal with a linear, rather than parabolic, energy dispersion for both electrons and holes in the conduction and valence bands.

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Figure 2.2: (a) Graphene band structure for t = 2.7 and t0 _{= −0.2t}_{. (b)}

Graphene band structure close to the K and K0 _{points showing the Dirac}

cones. (c) Density of states of graphene close to the Dirac point. The inset shows the density of states over the full electron bandwidth. Adapted from [17].

The linear relationship described in Eq. 2.7 is valid up to energies Ec ∼ 1

eV, above which the hopping energy t0 _{cannot be neglected any more and}

non-linear eects appear, commonly referred as trigonal-warping in the literature. We will restrict to the linear regime from now on.

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t0 = 0, close to the Dirac point: D(E) = 2Ac π~2 |E| v2 F , (2.8)

where Ac is the unit cell area given by Ac = 3

√

3a2_/2. Figure 2.2(c) shows

the density of states as function of energy (in units of t). The density of states vanishes linearly for E = 0.

2.2 Massless Dirac fermions

It can be easily shown [17] that at low energies and for k ≈ K, the Hamiltonian in Eq. 2.4 (with t0 _{= 0}_{) can be written as:}

HK = vF 0 kx− iky kx+ iky 0 = vFσ · k (2.9)

and similarly HK0 = v_Fσ∗· kfor k ≈ K0, where σ = (σ_x, σ_y)is the 2D vector

of the Pauli matrices (∗ denotes the complex conjugate).

The wave equation for electrons in graphene near the K point then reads: −ivFσ · Oψ(r) = Eψ(r), (2.10)

where ψ(r) is a two component wavefunction. Equation (2.10) is formally equivalent to the Dirac equation for massless fermions; its eigenvalues are the same we found in Eq. (2.7).

The eigenfunctions, in momentum space for k near the K and K0 _point,

have the form:

ψ±,K(k) = 1 √ 2 e−iθk/2 ±eiθk/2 (2.11) ψ±,K0(k) = 1 √ 2 eiθk/2 ±e−iθk/2 (2.12) where the ± index refers to the sign of the associated eigenvalue and θk =

arctan(kx/ky) is the angle in momentum space.

We can calculate the projections of the eigenfunctions along the pseudo spin direction. To this end, we introduce the helicity operator:

ˆ h = 1

2σ · k

|k|. (2.13)

The states ψK(r) and ψK0(r)are eigenstates of the operator ˆh:

ˆ hψK(r) = ± 1 2ψK(r) , ˆhψK0(r) = ∓ 1 2ψK0(r), (2.14)

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From Eq. (2.14) we see that, in the K valley, electrons (holes) have positive (negative) helicity. We also observe that the eigenvalues of σ can be either parallel, or anti-parallel to the momentum k. This means that the states of the system close to the Dirac points have well dened chirality or helicity (we can say that chirality is a good quantum number). We stress that this chiralityhas nothing to do with the real spin of the electrons but it is related to a pseudospin variable linked with the two components of the spinor.

One of the most striking results of this chiral symmetry is the prediction of Klein tunneling in graphene, as we will discuss in the next section.

Klein tunneling

One of the rst problems encountered in a course of quantum mechanics is the tunneling of an electron through a potential barrier. Given the square potential barrier shown in Fig. 2.3(a):

V (x) = (

V0 0 < x < L

0 otherwise (2.15) the transmission coecient for an electron with energy E < V0 going through

it, is an exponentially decreasing function of the barrier thickness L.

On the contrary, the transmission probability for Dirac particles orthogo-nally incident on a potential barrier is weakly dependent on the barrier height, approaching unity with increasing barrier height. This striking phenomenon is know as Klein tunneling and it can be understood writing the wave functions in the three regions (I, II and III in Fig. 2.3(b)) and matching them at the boundaries. In the limit |V0| |E| , the transmission coecient for carriers

in graphene incident at an angle φ on the potential barrier is [18]: T (φ) ≈ cos 2_φ 1 − cos2_(Lq x) sin2φ , (2.16) with qx = q

(V0 − E)2/v2F − ky2. The barrier is completely transparent for

orthogonal incidence (φ = 0).

Klein tunneling was observed through electrostatic barriers, which were created by gate voltages [19].

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Figure 2.3: (a) Schematic of an electron impinging on a square potential bar-rier. (b) Denition of the angle φ in the scattering formalism in regions I, II and III. Adapted from [17] (c) Quantum Hall eect in graphene as a function of charge-carrier density at T=4 K and B=14 T. The peak at n = 0 shows that, in high magnetic elds, a Landau level appears at zero energy. The eld draws electronic states for this level from both conduction and valence bands. Adapted from [20].

Quantum Hall Eect

Another consequence of the massless nature of the charge carriers in graphene is the anomalous Quantum Hall Eect (QHE). In a 2D system with a constant magnetic eld B perpendicular to the system plane, the energy spectrum is discrete (Landau quantization). For the case of massless Dirac fermions the energy spectrum takes the form [17]:

Eνσ = ±

q

2 |e| ~Bv2

F(ν + 1/2 ± 1/2) (2.17)

where ν = 0, 1, 2, ... is the Landau level quantum number.

The ±1/2 term is related to the chirality of the quasiparticles and ensures the existence of two energy levels (one electron-like and one hole-like) at exactly zero energy. Experimentally such a ladder of Landau levels manifests itself in the appearance of a half-integer quantum Hall eect, that was observed at low temperature (T∼ 2 K) in 2005 by the Novoselov/Geim group [20] and by Kim et al. [21]; it was also observed at room temperature by Novoselov and Geim in 2007 [22].

Figure 2.3(c) shows the experimental quantum Hall conductivity and the longitudinal resistivity as a function of the carrier concentration in a single graphene layer.

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2.3 Transport in monolayer graphene

Graphene obtained by micromechanical cleavage of graphite or other growth procedures contains defects and charged impurities, interacts with the sub-strate, has edges and ripples and it is aected by phonons [23]. All these per-turbations modify the properties of a perfect graphene sheet in several ways: rst, they introduce spatial inhomogeneities in the carrier density and, second, they act as scattering sources, reducing the electron mean free path [24]. Spa-tial inhomogeneities aect the minimum conductivity of graphene when the Fermi level is close to the Dirac point. Scattering sources (specied in section 2.4) reduce the carrier mobility away from the Dirac point.

We can distinguish two transport regimes, depending on the mean free path length l and the graphene length L: when l > L transport is ballistic since carriers can travel at Fermi velocity vF from one electrode to the other without

scattering. In this case transport is described by the Landauer formalism and the conductivity is expressed as [23]:

σball= L W 4e2 h ∞ X m=1 Tm, (2.18)

where W is the width of the graphene sheet and the sum is over all avail-able transport modes of transmission probability Tm. The factor of 4 at the

numerator stands for the spin degeneracy (gs = 2) and the K, K0 valley

de-generacy (gv = 2). This theory predicts a universal minimum conductivity for

W Lat the Dirac point:

σmin_ball = 4e

2

πh. (2.19)

Experiments performed on very small samples (area < 0.2 µm2_{) show that}

σmin

ball is a decreasing function of the aspect ratio W/L, approaching indeed the

limit of Eq.(2.19) for W L [25]. This result is compatible with the fact that the mean free path for carriers in graphene on a SiO2 substrate is around

100 nm.

The opposite regime appears when l < L: carriers undergo elastic and inelastic collisions and transport becomes diusive. In this case, transport is described by Boltzmann transport theory and the conductivity, away from the Dirac point and in the limit of zero temperature, can be expressed in terms of the total relaxation time τ(kF) and of the density of states D(kF):

σboltz =

e2_v2 F

2 D(kF)τ (kF), (2.20) where kF is the Fermi wave vector with reference to the K point in the Brillouin

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linear dispersion of graphene, we nd: σboltz = e2_v F ~ kF π τ (kF) = e2_v Fτ (kF) ~ r n π, (2.21) where n is the carrier density in graphene and we used the relationship kF =

p4πn/gsgv.

This equation describes the diusive motion of carriers, independently of the scattering source.

The behaviour at the Dirac point in the diusive regime has to be treated with particular attention: there is still a minimum conductivity, but the value diers from the one found in Eq 2.19. Experiments performed on large-area samples (> 3 µm2) report that the minimum conductivity is independent of the

aspect ratio W/L and it approaches the limit σmin = 4e2/h, more than three

times higher than the conductivity in the ballistic case [25]. The explanation for this behaviour is believed to originate from the formation of electron and hole puddles when the carrier density is very low and electron screening is suppressed. This inhomogeneity in the charge distribution is mostly due to charged impurities deposited on graphene or trapped between the graphene and the substrate. These puddles have been observed experimentally with an STM (Scanning Tunneling Microscopy) study [26], revealing that the average lateral dimension of the puddles is of the order of ∼ 20 nm for graphene deposited on a SiO2 substrate. From a theoretical point of view, graphene

in the regime where puddles form can be thought of as a random resistor network: this approximation is valid as long as the transport is incoherent at scales larger than the puddle sizes. In normal 2D semiconductors the formation of puddles eventually leads to an insulating phase, when percolation between puddles is no longer possible; however graphene is gapless and the insulating state is impossible to achieve, resulting in a nite minimum conductivity. The universality of the minimum conductivity in the diusive regime is still subject to debate: a recent experiment reports that σmin depends on the

charged-impurities density [27].

2.4 Scattering mechanisms

In the diusive regime, transport away from the Dirac point is ruled by the Boltzmann conductivity described by Eq.2.21. We briey discuss how the scattering time τ depends on the type of scattering source and we compare the theory with experimental data from literature.

Phonon scattering

The phonon dispersion relation of graphene comprises six branches. There are two modes corresponding to the out-of-plane vibrations, ZA (acoustic)

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and ZO (optical), and other four modes related to the in-plane vibrations: TA (transverse acoustic), TO (transverse optical), LA (longitudinal acoustic), LO (longitudinal optical). Longitudinal acoustic (LA) phonons are known to have a higher electron-phonon scattering cross-section than those in the other branches.

There are two regimes of scattering, separated by a characteristic temper-ature TBG (Bloch − Gr¨uneisen temperature [28]) dened as:

kBTBG = 2kFvph, (2.22)

where kB is the Boltzmann constant, vph is the sound velocity in graphene.

For currently available semiconductors TBG ∼ 4 K; optical phonon scattering

becomes exponentially more important for T & 100 K and dominates carrier transport at room temperature [28]. In graphene this temperature is much higher (n ∼ 1012_cm−2 _{⇒ T}

BG ∼ 54 K) and, at room temperature, phonon

scattering gives a minor contribution compared to impurity scattering. In the case T TBG, the Bose-Einstein distribution function for phonons is

N (ωq) ≈ kBT /~ωq, which leads to a linear dependence of the scattering rate

(and resistivity) on temperature. For T = 300 K and normal carrier den-sities n ∼ 1012_cm−2_{, the theoretical limit imposed by phonon scattering to}

mobility is µ = σph/en ∼ 105cm2/Vs, which is the highest value known for

intrinsic room-temperature mobility for a single graphene sheet up to date. Experiments performed on suspended graphene have conrmed the linear de-pendence on temperature, showing that phonon scattering is relevant for those devices [29]. However, graphene deposited on a substrate shows much lower mobilities (103 _{÷ 10}4_cm2_/Vs_{), implying that other forms of scattering come}

into play (see next sections). Short-range scattering

Short-range scattering is related to local defects, such as vacancies and cracks in graphene akes. Vacancies can be modeled as a deep circular potential well of radius R; the scattering time in this case is equal to [30]:

τd= kF ndvF ln2(kFR) π2 = √ n ndvF ln2(√πnR) π3/2 , (2.23)

where nd is the defect density. Inserting this result in Eq.2.21 we get the

conductivity: σd= 2e2 πh n nd ln2(√πnR). (2.24) Equation 2.24 has been tested experimentally by irradiating a graphene ake with 500 eV He and Ne ions in ultra high vacuum (UHV) [31]; the result-ing conductivity was approximately linear with carrier density and inversely

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proportional to the ion (or defect) dose nd. The defect radius was estimated

to be R ≈ 2.9 Å, compatible with a single carbon atom vacancy. Coulomb scattering

Coulomb scattering is related to the presence of charged impurities close to the graphene sheet; they may be chemical species deposited on top of graphene or trapped between the substrate and graphene. Assuming a random distribution of impurities with density nimp, the scattering time τ at T = 0 is given by [30]:

τcoul= CkF 2πvFnimp = CkF 2vFnimp r n π, (2.25)

where C is a dimensionless parameter related to the scattering strength (C ∼ 20 is calculated within the random phase approximation and the dielectric screening of SiO2 [27]). The associated conductivity is linear with the carrier

density n: σcoul = Ce2 h n nimp . (2.26)

The conductivity of graphene akes was measured in a UHV environment and the density of charged impurities was increased by depositing potassium atoms onto its surface; the n−1

imp dependance was observed [32].

2.5 Plasmons in graphene

Plasmons are self-sustained normal mode oscillations of a carrier system, aris-ing from the long range nature of the Coulomb interaction. The plasmon modes are dened by the zeros of the corresponding dynamical dielectric func-tion ε(q, ω) = 0.

Using the Random Phase Approximation (RPA), the plasmon dispersion relation for graphene in the long wavelength limit (q → 0) reads [33]:

ωp(q) = g_sgve2 2~2 0r EFq 1/2 = e 2_v Fq ~0r √ πngsgv 1/2 (2.27) where 0 is the vacuum permittivity and r is the eective dielectric

con-stant, obtained as a mean of the dielectric constants of the materials sur-rounding graphene. We can see that ωp ∝ q1/2 as in a standard 2DEG system.

However the density dependence is ωp ∝ n1/4, while in classical 2D plasmon it

is ωp ∝ n1/2.

Plasmon resonances have been observed in engineered graphene micro-ribbon arrays [34]. In this experiment, an array of graphene micro-ribbons a few mi-crons wide was probed using Fourier transform infrared spectroscopy (FTIR).

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Figure 2.4: Acoustic-plasmon dispersion (in units of EF) in a graphene

mono-layer for a xed carrier density n = 5 × 1013_cm−2 _{and air dielectric constant}

r= 1. Filled circles and squares refer to dierent values of the graphene-metal

gate distance d. Filled triangles refer to the standard plasmon (∝ q1/2) in the

absence of a metal gate. The long dashed line (ω = vFq) represents the upper

bound of the intraband electron-hole continuum. The intersection between the acoustic plasmon and the thin short-dashed line (ω = 2EF− vFq, lower bound

of the interband electron-hole continuum) gives the critical wave number at which Landau damping starts. Adapted from [35].

The absorption was measured as a function of frequency and resonances were observed in the THz range, only for light polarized perpendicular to the ribbons (as expected). With the aid of a local gate, the experiment also showed the n1/4 dependence of the plasmon resonance frequencies.

Recent theoretical studies [35] have suggested that the presence of a metal gate close to graphene implies the formation of an acoustic plasmon, with a linear dispersion as a function of momentum ωac(q) = csq. We dene the

dimensionless parameter x = de2_k

F/r0~vF, where d is the gate-graphene

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approximation: cs = vF s [1 + (1/2gsgvx)]2 [1 + (1/2gsgvx)]2− 1 ≈ vF √ 4x (2.28) where the last approximation is taken from a linear t in the x > 1 region. For d = 40 nm and r = 13, we have x = 1 ÷ 10 for charge carrier density

n = 7 × 1011_{÷ 7 × 10}13_cm−2_{, which implies that the plasma wave velocity is}

higher than the Fermi velocity. Figure 2.4 shows the dispersion ωac(q) of the

acoustic plasmon as found from the numerical solution of ε(q, ω) = 0 and the long-wavelength analytical result given in Eq.(2.28).

Another theoretical paper [36] recovers from classical considerations the same linear dispersion for plasma waves in a gated channel; however, they predict a dierent wave velocity:

cs ≈ vF

√

2x. (2.29)

Both theories recover the q1/2 dependence in the limit d → ∞.

2.6 Bilayer Graphene

Figure 2.5: (a) Lattice structure of graphene bilayer. (b)-(c) Band structure near the K point for V = 0 and V 6= 0, respectively. From [17].

The case of bilayer graphene is interesting as well. The bilayer structure, with the AB stacking of graphite, is shown in Fig. 2.5(a). Using a tight-binding approach, it can be shown that the energy spectrum for k ' K (with ~ = 1) is [17, 37]: E±(k) = V2+ v_F2k2+ t2_⊥/2 ± q 4V2_v2 Fk2 + t2vF2k2+ t4⊥/4 1/2 , (2.30) where t⊥(≈ 0.4 eV) is the eective interlayer hopping energy, while t and

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term V breaks the equivalence of the two layers and accounts for the possibility of a real shift (e.g. by an applied external electric eld perpendicular to the layers) in the electrochemical potential.

Expanding Eq.2.30 to the leading order in momentum, and assuming V t, we nd

E±(k) ≈ ±V − 2V vF2k2/t⊥2 + v4Fk4/(2t2⊥V ) . (2.31)

We can see that for V = 0 bilayer graphene is a gapless semiconductor with a parabolic dispersion relation with eective mass m = t⊥/(2v_F2) ≈ 0.04 me(we

can speak of massive chiral fermions, see Fig. 2.5(b)). For V 6= 0 a band-gap appears M= 2V − 4V3_/t2

⊥ at k =

√

2V /vF (see Fig. 2.5(c)). Experiments

report a band-gap of 0, 2 eV when an electric eld ∼ 1 V/nm is applied [38]. Another dierence from monolayer graphene concerns the Klein tunneling. It can be shown [17] that carriers in bilayer graphene behave oppositely to the monolayer case, as they are always perfectly reected for angles close to orthogonal incidence on a barrier. At the same time, there is always a magic angle at which the transmission probability is equal to one.

Plasmons in bilayer graphene behave as normal 2DEG plasmons, follow-ing the relation [37]:

ωp(q → 0) =

2πe2_nq

m0

1/2

∝ n1/2_q1/2 _(2.32)

Presumably then, also bilayer graphene shows acoustic plasmons in the pres-ence of a metallic gate, although, to our knowledge, no dedicated theory has been developed yet: we expect ωac ∝ n1/2q.

It is worth to cite also the case of a twisted bilayer (or double monolayer) in which the two graphene layers are not AB stacked but they are rotated with some angle. This twist breaks the coupling between the planes and restores the gapless, linear dispersion of monolayer graphene. Twisted bilayer has been predicted to show acoustic plasmons, even in the absence of a metallic gate: the electrons in the two layers oscillate out of phase with a frequency ωtwist

ac ∝ n1/4q

[39].

2.7 Graphene characterization

Several dierent techniques can be used to identify and characterize graphene. We discuss here two of the most important ones, with a particular emphasis on the methods that allow to identify monolayer and bilayer graphene on a substrate.

Optical microscopy

The quickest method to identify graphene akes mechanically exfoliated on a oxidized Si substrate is using an optical microscope.

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It is well known that the thickness of a SiO2 lm grown on top of a Si

sub-strate can be quantied with some precision simply by evaluating its apparent color. The latter is in fact due to the interference between the reection paths that originate from the two air-SiO2 and SiO2-Si interfaces. Depending on

their distance, interfering paths will experience relative phase shifts: thickness variations of a fraction of the wavelength lead to color shifts that can easily be appreciated by eye.

Graphene akes are suciently transparent to add a measurable optical path, which changes the interference color with respect to an empty wafer. Using the graphene thickness (d = 0.34 nm) and the refractive index of bulk graphite, it is possible to calculate the optical contrast between graphene and the substrate. Figure 2.7(a) illustrates the contrast of graphene akes de-posited on a substrate with a 300 nm thick SiO2 layer, illuminated at dierent

wavelengths: increasing the number of graphene layers, the contrast increases almost linearly and saturates at around 5 layers. Using red light (λ = 590 nm), the optical contrast reaches 12% for monolayer graphene and 24% for bilayer graphene [40, 41].

Raman spectroscopy

Raman spectroscopy is a powerful tool that enables to identify monolayer, bilayer and many-layer graphene. The three most prominent peaks in the Raman spectrum of graphene and other graphitic materials are the G band at ∼ 1580 cm−1_{, the 2D band at ∼ 2680 cm}−1 _{and the disorder-induced D band at}

∼ 1350 cm−1 _{(see Fig. 2.6(a)) . The position of the peaks can shift depending}

on the wavelength used to probe the sample (typically 633nm He-Ne laser or 514nm Argon laser).

The G band is present in all graphitic materials and it is linked with the in-plane vibrations of the atoms of the hexagonal ring; this resonance corresponds to the in-plane optical phonons at the Γ point. The 2D band comes from a two phonon resonance process, involving phonons near the K point; its shape and intensity relative to the G peak are used to count the number of layers (monolayer, bilayer, etc.). The D band corresponds to the in-plane optical phonons near the K point and appears only when graphene contains defects, i.e. missing atoms in the lattice. The intensity ratio of the G and D peak can be used to characterize the number of defects in a graphene sample.

Monolayer graphene is characterized by a very sharp, Lorentzian 2D peak (FWHM ≈ 28 cm−1_{) with an intensity greater than the G peak (typically 2÷10}

times more intense than the G peak). As the number of layers increases, the 2D peak becomes broader, less symmetric and decreases in intensity (see Fig. 2.6(b)). For example, the 2D peak of bilayer graphene can be tted by a convolution of four Lorentzians with a total FWHM≈ 60 cm−1_{, as shown in}

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A recent review [42] explores the inuence of doping on the Raman n-gerprint of graphene. The results show a correlation between the position of the 2D peak and the doping level, with hole-doping resulting in a blue shift ( ∼ 10 cm−1 _{for heavy p-doping), and the opposite for high electron doping.}

Doping also aects the 2D/G peak intensity ratio, showing the highest ratio for intrinsic graphene and a 1:1 ratio for highly doped graphene.

More details on Raman characterization of graphene can be found in Refs. [43, 44].

Figure 2.6: (a) Comparison of Raman spectra at 514 nm for bulk graphite and graphene. (b) Evolution of the 2D peak at 514 nm with the number of layers. (c) The four components of the 2D peak in bilayer graphene at 514 and 633 nm. From [43].

Other methods

Atomic Force Microscopy (AFM) can also be used to measure the thickness of graphene layers, since the 3.4 Å step height for each successive layer is well within the detection limits of modern AFM systems (see Fig. 2.7(c)). Scanning tunneling microscopy (STM) is also used on graphene: this technique is able to resolve the honeycomb structure of monolayer graphene, producing beau-tiful images like the one in Fig. 2.7(b). Scanning electron microscopy (SEM) clearly shows the shape of graphene akes, but it is not straightforward to precisely count the number of layers due to charging eects that modify the contrast of the akes (see Fig. 2.7(d)). Moreover, SEM imaging often leads to amorphous carbon deposits on the samples under observation, which makes it an unsuitable technique to use on graphene.

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Figure 2.7: (a) Optical image of graphene akes on 300 nm SiO2 at various

wavelengths. The optical contrast increases with the number of layers. From [40] (b) STM topographic image of monolayer graphene. From [45]. (c) AFM image of graphene akes. (d) SEM image of the same ake. From [41].

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3 Sample production and experimental

charac-terization

In this chapter we present the experimental procedures to produce and charac-terize a graphene eld eect transistor (or GFET). We explain how to make a graphene layer from graphite, how to nd the graphene akes on a substrate, make ohmic contacts on them and measure the eld-eect mobility using a back-gate. Then we show how to realize a top-gate on graphene devices. We nally report the resistivity measurement as a function of back and top gates, both at atmospheric pressure and in high-vacuum.

3.1 Graphene production and identication

The rst step is the production of graphene single layers. Although many tech-niques have been developed to grow graphene on metal [46] and on SiC [47], the samples with the highest quality are still produced with the micromechani-cal cleavage or scotch-tape method. This method was invented by Novoselov and Geim in 2004 [10] and it was used to fabricate the rst graphene top-gated eld-eect device in 2007 [12]: it consists of placing a ake of graphite (for example HOPG - highly oriented pyrolytic graphite, or Kish graphite) on normal scotch tape and repeatedly peel it o until it becomes almost trans-parent. Then the tape is placed on a Si substrate with thermally grown SiO2

and peeled o again. Some of the thin graphite akes adhere to the sub-strate thanks to Van der Walls forces and only a few of those, when peeled o, become monolayer or bilayer graphene, readily identied with an optical microscope as described in Section 2.7. It is very important that the substrate is as clean as possible, otherwise the adhesion can be compromised: standard methods to clean the substrate include piranha etching (mixture of sulfuric acid (H2SO4) and hydrogen peroxide (H2O2)) and oxygen plasma cleaning (we

used the latter, see Appendix A.1).

We deposited graphene obtained from Kish graphite on two dierent types of substrates: the rst one is a heavily p-doped Si substrate with 300 nm SiO2

on top, used to produce a few test devices, while the second one is a highly intrinsic (ρ = 10 kΩ cm) Si substrate with 300 nm SiO2. This latter substrate

was used for the photodetector device presented in Chapter 4. With this SiO2

thickness, graphene akes are most visible when illuminated with red light, obtained with a suitable lter in the optical microscope [41]. Once a good graphene ake is identied, a contrast analysis is performed with an image processing software to decide whether it is monolayer, bilayer or composed by a few layers. We have established through many observations that monolayer and bilayer graphene display 10 ÷ 13% and 22 ÷ 25% contrast, respectively; this variance is mainly due to the non-uniformity of the SiO2 thickness and

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partly due to the noise in the digital image.

The next step is the removal of glue residues from the substrate. We found that acetone does not fully dissolve glue, leaving large halos on the substrate. Instead, the substrate was heated for 30 minutes at 300◦_C _{in nitrogen}

atmo-sphere (100 mbar pressure), resulting in the complete evaporation of the glue. However, heating in vacuum or in nitrogen atmosphere causes heavy p-doping in graphene, as already reported in ref.[48]. According to ref.[48], the reason behind this doping is to be found in water and oxygen adsorption on graphene. Finally, Raman spectroscopy is performed to conrm the number of graphene layers. The absence of the defect-induced D peak in most of the samples con-rms the high quality of graphene, even after annealing. An example of Raman spectra from a graphene bilayer will be shown in Fig. 3.5.

3.2 Fabrication of ohmic contacts on graphene

Figure 3.1: (a) Optical image of a monolayer graphene ake deposited with the microcleavage method on the p-doped Si substrate (300 nm SiO2 on top). The

image shows only the red component (λ ≈ 590 nm) in order to maximize the contrast of the graphene ake. (b) The same ake with Cr/Au ohmic contacts deposited on it. The thin traces are connected with larger pads (see bottom left and top right portion of the gure) that are linked to the bonding pads (see Fig. 3.6 for a view of the entire device)

In order to probe the transport properties of graphene, ohmic contacts must be realized to inject and collect carriers. The standard procedure consists in creating a mask with Electron Beam Lithography (EBL) or with optical lithog-raphy, evaporating the metal and then performing the lift-o (see Appendix A.2 for a detailed explanation). Cr/Au contacts realized with this method

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dis-play ohmic behaviour, which is certainly not a trivial result. A graphene-metal junction, in fact, involves carrier transport from a three-dimensional region to the two-dimensional graphene sheet, which has very dierent density of states and work function. Several dierent metals have been tested in the literature (Cr, Ti, Pd, Ni, Au, Pt are the most commonly used). It was shown that the current preferentially enters graphene at the edge of the metal contact [49]. This implies that contact resistance cannot be lowered increasing the contact area. Saturation occurs when the contact is wider than ≈ 1 µm.

Another fact that needs to be considered is that graphene under the metal is doped by charge transfer: the sign and amount of doping can be deduced from the dierence between the metal and graphene work functions. For example, Titanium has a lower work function than graphene (ΦTi = 4, 3 eV < ΦGraph =

4, 5 eV [50],[51]), implying that Ti contacts result in n-type doping of graphene. This may lead to the formation of a junction between two regions of graphene (one under the metal and the other outside) with dierent doping, yielding a larger contact resistivity. Chromium, on the other hand, has almost the same work function as graphene (ΦCr = 4, 5 eV [51]) and it should not induce any

doping. These considerations do not take into account the metal-graphene interaction, which may modify the behaviour of the junction. More theoretical considerations on the metal graphene interaction can be found in [52] and [53]. In light of the above discussion, we decided to fabricate our devices with Cr/Au contacts (typically 5 nm Cr and 80 nm Au) and to keep the contact width xed at 500 nm for all samples, in order to minimize the formation of junctions at the metal-graphene interface and to obtain reproducible contact resistivity measurements. We have observed contact resistivities of the order of 10−5

Ω cm2, measured with a 2-probe setup (see next section). This value is supported by studies on contact resistivity found in literature [51, 54, 49, 55]. As a nal remark, we note that the contact resistivity depends on the pressure of the chamber in which the metal deposition takes place. This may be due to the desorption of impurities from the graphene surface prior to the deposition. We deposited the metal on all of our samples in a low-pressure environment (∼ 5 × 10−7_mbar_{). A few test samples were processed at a higher pressure}

(∼ 1 × 10−5_mbar_{) and showed higher contact resistivity (order of 10}−4_{Ω cm}2_).

Figure 3.1 shows a graphene ake before (a) and after (b) the contact deposition. The metal traces are terminated with large bonding pads (150 × 150 µm, see Fig. 3.6).

3.3 Experimental characterization of back-gated graphene

FETs

In order to modulate the carrier density in the graphene channel, we used the highly doped Si wafer as a back-gate, isolated from graphene by the 300 nm

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thick layer of SiO2. Given the dielectric constant (εSiO2 = 3.9), the capacitance

per unit area of the back-gate is easily found to be CBG = 11.5 nF cm−2.

To perform the measurement, the sample was mounted in a dual in-line package (DIL) and wire bonding was performed to connect the pads; contact with the back of the Si wafer was achieved with a conductive silver paste. All measurements were carried out at room-temperature and atmospheric pres-sure, unless otherwise specied.

First, we measured the current as a function of the source-drain voltage to test the ohmic contacts. The current-voltage curve is linear up to very high current densities (∼ 100 µA/µm, see Fig. 3.2), above which Joule heating starts damaging the metallic contacts and eventually burns the device. Then we xed the source-drain voltage to a certain value (from ∼ 0.1 mV to ∼ 1 mV depending on the device resistance) and we measured the current as a function of the voltage applied to the back-gate, both in DC and AC using a standard lock-in technique (17 Hz excitation frequency). We set the source-drain voltage at a value leading to a current of ∼ 100 nA owing in the device with the channel fully open.

Figure 3.2: Current as a function of the source-drain voltage. For this sample L = 4 µmand W = 6.5 µm. The linearity is lost above ∼ 120 µA/µm.

The experimental 2-probe resistance as a function of the back-gate voltage for a monolayer graphene device is shown in Fig. 3.3 (squares). The curve has a characteristic bell shape: starting from negative gate-voltage, where the majority carriers are holes, the resistance increases and reaches a maximum, corresponding to the condition of minimum carrier density in the channel; this

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condition is also know as charge neutrality point (CNP). Since graphene is gapless, the device cannot be switched o like a normal transistor. Moving to positive gate voltage, the majority carriers switch from holes to electrons and the resistance decreases again. The CNP in graphene is reached when the Fermi energy is exactly zero, at the point where the valence and the conduction bands touch. We stress that exfoliated graphene always shows some nite doping at zero gate voltage. The gate voltage that is required to shift the Fermi energy back to zero is know as the Dirac voltage. The graphene sheet is p-doped if the Dirac voltage is positive (n-doped if negative): for example, the sample in Fig 3.3 is p-doped.

We now assume that the diusive transport regime applies to our graphene devices (as it will be shown later) and that charged impurities are the dominant source of scattering. We do not consider short range scattering, related to defects in the lattice, because of the abscence of the D peak in the Raman spectra of our devices (see Fig. 3.5). In this regime, the conductivity is linear on the carrier density n, as shown by the Drude-like Eq. 2.26.

Given all of the above information, we can t the resistance as a function of the gate voltage with the following equation (assuming that the curve is symmetric with respect to the Dirac voltage):

R = Rc+ W LeµF E q n2 0+ n2 −1 , (3.1) n = CBG e (VBG− V 0 BG). (3.2)

Here we have dened the contact resistance Rc, the eld-eect mobility

µF E, the length L and the width W of the graphene channel, the residual

carrier density n0, the Dirac voltage VBG0 and the voltage applied on the

back-gate VBG. The residual carrier density n0 is linked to the charged impurities

density nimp(n0 ∝ nimp, see Ref. [24]). The lower n0, the sharper the resistance

peak.

In general the curve is not symmetric with respect to the Dirac voltage and two separate ts are required, one on the left and one on the right side of the Dirac voltage. The asymmetry arises mainly from two factors: one is the dierent scattering cross-section for electrons and holes, which leads to dierent mobilities µelect and µhole. The other one is the dependence of the

contact resistance Rc on the gate voltage.

The plot R-vs-VBG shown in Fig. 3.3 is well tted by Eq. 3.1 (red line),

leading to µhole ≈ 6500 cm2V−1s−1. The scattering length associated with this

mobility is found to be l = vFτel = vFµmc/e ≈ 75 nm, where the electron

scattering time τel has been obtained from the graphene cyclotron mass mc=

EF/vF2 at a xed carrier density n = 1×1012cm

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of our devices are of the order of a few microns, the assumption of diusive transport is reasonable.

Figure 3.3: Two-probe resistance of a monolayer graphene FET as a func-tion of the back-gate voltage (squares), tted with Eq.(3.1) for holes only (red line). The device has L = 3.5 µm, W = 4 µm, V0

BG = 23.15 V. The

parameters extracted from the t are Rc ≈ 1 kΩ, n0 ≈ 3 × 1011cm−2 and

µhole ≈ 6500 cm2V−1s−1.

3.4 Fabrication of the top-gate

As it will be explained later, the operation of the photodetector device requires the use of a top gate. In order to fabricate this top-gate on our devices, we need to deposit a dielectric insulator on top of the graphene sheet. We chose Hafnium Oxide (HfO2) deposited with Atomic Layer Deposition (ALD); HfO2

is a high-κ oxide (∼ 25 for crystalline lms) with a wide bandgap (5.8 eV) and it was already used on graphene in several works [56, 57, 58, 59]. After the deposition of graphene akes on the substrate and the fabrication of ohmic contacts, we deposited the HfO2 insulating layer on top of the whole device and

then realized a metallic top-gate on top of it. In the following paragraph we briey explain how ALD works and we give the details of our HfO2 deposition

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Atomic Layer Deposition (ALD)

Atomic Layer Deposition (ALD) is a process by which metals and multi-component oxides can be grown layer by layer. In particular, high quality, high-κ materials like Al2O3 and HfO2 can be grown via ALD. The idea

be-hind ALD is relatively simple. First, a substrate is placed in an evacuated chamber where the ALD growth will occur. Then a xed amount of precursor gas 1 is pulsed into the ALD chamber. This precursor chemically reacts with the substrate surface and binds to the surface. Specic sites (indicated by the triangular shapes in the substrate in Fig. 3.4(a)) on the surface must be catalytically suitable for this reaction to occur: if the precursor cannot nd these sites, the reaction cannot start and the precursor is pumped out of the chamber. Once all the catalytic sites are occupied, further reactions are not permitted making this process self-limited to adsorption of a single layer of the precursor, see Fig. 3.4(b) . After enough precursor 1 is introduced into the chamber to ll all the sites, a second precursor is pulsed into the chamber. Here precursor 1 provides the catalytic sites for precursor 2 to chemically ad-sorb on the surface, shown in Fig. 3.4(c) . Again, this process is self-limiting, terminating when all the sites of the precursor 1 layer are occupied, see Fig. 3.4(d). By repeating this process, oxides can grow one single atomic layer at a time with a high degree of uniformity and oxide quality.

Figure 3.4: Schematic representation of one ALD cycle: (a) precursor 1 reacts with the substrate. (b) The adsorption of precursor 1 is self-limited and stops when there are no more free slots. (c) Precursor 2 reacts with precursor 1 until saturation occurs in (d).

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ALD of Hafnium Oxide (HfO2) on Graphene

One the most common choices to deposit an insulating layer on top of graphene is ALD. However, due to the hydrophobic nature of graphene, a functionaliza-tion layer needs to be deposited on top of graphene before the ALD process. The functionalization layer (FL) sticks to graphene and provides the catalytic sites required to start the ALD deposition. Many dierent combinations of functionalization layer and oxides are present in literature, leading to an im-pressive amount of cooking recipes, well summarized in Ref. [60].

As already anticipated, we chose HfO2 as a gate dielectric: precursor 1 is

TEMAH (tetrakis(ethylmethylamido)hafnium(IV)), and precursor 2 is simply water vapor. To deposit the HfO2 lm, we started from the recipes described

in [61] and [56], we adapted them to our ALD machine and then we made some changes to improve the lm quality.

First we cleaned the graphene sheet from resist residues through annealing at 300◦_C _{in Ar atmosphere (150 mTorr) for 1 hour (performed directly in}

the ALD chamber). Then we deposited the FL, which is realized through the direct deposition on graphene of 35 ALD cycles of HfO2 (∼ 15 nm thickness)

at very low temperature (110◦ _{C). The graphene is completely covered by the}

oxide after this rst step. In the second step, the temperature was increased to 200◦ _{C and 100 ALD cycles of HfO}

2 were deposited (∼ 20 nm thickness).

The growth rate is lower at higher temperatures and comparable with results obtained in literature [62, 63]. More details about the recipe can be found in Appendix A.3.

The Raman spectrum of a representative graphene bilayer is shown in Fig. 3.5(a), before and after the HfO2 deposition: the disorder induced D peak is

not present, meaning that the oxide deposition does not damage the graphene sheet; the 2D peak, shown in Fig. 3.5(b), shifts from 2713 cm−1 _{to 2709 cm}−1_,

implying that the HfO2 deposition induces n-type doping in graphene (see

section 2.7).

Once the oxide was deposited, a metallic (Cr/Au) top-gate was patterned in the middle of the graphene channel with standard EBL and lift-o technique. Figure 3.6 shows a complete test GFET, covered with HfO2 and with the

Cr/Au top-gate. The top-gated region of the same device is shown in detail in Fig. 3.7: the metallic top-gate crosses the graphene ribbon from side to side but it does not cover the whole surface; the deposited HfO2 covers graphene

uniformly, although the roughness is higher on graphene than on the SiO2

substrate.

Considering that HfO2 deposited at a temperature T ≤ 200◦ C is

amor-phous and has a lower dielectric constant than the crystalline form ( κ ∼ 13 ÷ 19 [63]), we estimate the capacitance per unit area of our top-gate to be Ctop∼ 300 ÷ 500 nF cm−2.

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Figure 3.5: (a) Raman spectrum of a graphene bilayer, before and after the HfO2 deposition. (b) Close-up of the 2D peak

Figure 3.6: Optical image of a complete GFET testing device. It shows the same graphene ake as in Fig. 3.1, after HfO2 deposition and Cr/Au top-gate

fabrication. The inset shows a close-up of the graphene ake covered by the HfO2 and the top gate; colors have been altered to maximize the contrast of

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Figure 3.7: SEM image of the same GFET shown in Fig. 3.6. The dimensions of the graphene ake are L = 8.8 µm and W = 3.1 µm. The top-gate crosses the whole width of the ake but it is shorter in the other direction (LT G= 3.1 µm).

The inset shows an enlargement of the graphene ake covered by HfO2; the

coverage of graphene is uniform and pinhole-free.

3.5 Experimental characterization of dual-gated graphene

FETs

The carrier density in the device can now be tuned both with the top and the back-gate. Under the top-gated region we have that:

n = 1

e CBG(VBG− V

0

BG) + CT G(VT G− VT G0 ) , (3.3)

where VT Gand VT G0 are the top-gate voltage and the Dirac top-gate voltage,

respectively.

We report representative measurements performed on a top-gated test GFET. In this device the HfO2 was deposited with an earlier version of the

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grown at 110◦ _{C (∼ 30 nm thickness). The linearity of the current vs}

source-drain voltage characteristics was checked again after the HfO2 deposition.

Fig-ure 3.8(a) shows the 2D color plot of the graphene resistance as a function of top and back-gate voltages. Representative traces at dierent values of the back-gate voltage are shown in Fig. 3.8(b). The slope of the black dashed line indicates the ratio between the top and back-gate capacitance. From a linear t we obtain that CT G/CBG ∼ 50 ± 20.

3.6 Vacuum treatment for graphene FETs

We also measured the R-vs-VT G curve of the GFET shown in Fig. 3.7 after

placing it in vacuum. Figure 3.9(a) shows the evolution R-vs-VT G curve with

increasing time spent in vacuum. The two bottom curves of Fig. 3.9(a) were measured 30 minutes after the vacuum pump had started. The curves are asymmetric and hysteresis is present. After 16 hours under vacuum (two lines in the middle of Fig. 3.9(a)), the Dirac voltage shifts signicantly, the curve is broader and the hysteresis is reduced. This trend continues slowly until, after 64 hours in vacuum (two lines at the top of Fig. 3.9(a)), the hysteresis is almost suppressed. However, when the GFET is restored back to atmospheric pressure, the hysteresis comes back together with the asymmetric shape (see Fig. 3.9(b)).

This behaviour was previously reported in literature for graphene [64] and for carbon nanotubes [65] and it is most likely due to the adsorption/desorption of water and other molecules (i.e. oxygen, hydrogen) trapped in the oxide layer. We t the bottom curve in Fig. 3.9(a) (30 minutes in vacuum) with Eq. 3.1, as a representative measurement for our top-gated devices. Given the ra-tio LT G/W ∼ 1, if we suppose a top-gate capacitance (per unit area) CT G =

500 nF/cm2 _{and we average between the values obtained in the forward and}

backward sweep, we obtain a mobility µelec ≈ µhole ≈ 5000 cm2V−1s−1, a

con-tact resistance Rc∼ 5800 Ωand a residual carrier density n0 ∼ 3 × 1011cm−2.

The contact resistance is very high because it includes also the contribution of the graphene sheet covered by the HfO2 but not covered by the metallic

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Figure 3.8: (a) 2D color plot of the graphene resistance as a function of top and back-gate. The dimensions of this device are L = 4 µm, LT G = 300 nm

and W = 11 µm. (b) Resistance as a function of the top-gate voltage VT G

for dierent back-gate voltages VBG extracted from (a). The black dashed

lines indicate the positions of the charge neutrality point under the top-gated region.

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Figure 3.9: (a) Resistances as a function of the top-gate voltage of a GFET measured in vacuum, after 30 minutes (bottom lines), 16 hours (middle lines) and after 64 hours (top lines). The hysteresis tends to disappear while the curves broaden. (b) The same device is restored to atmospheric pressure. The curve after 64 hours in vacuum in shown as reference (lines at the top). After 30 minutes in atmosphere, the resistance has decreased back to the original value (before the vacuum treatment). After 8 hours in atmosphere, the hysteresis appears again, together with the asymmetric shape of the curve around the Dirac voltage. The arrows indicate the sweep direction. The gate voltage VT G

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4 THz photodetection with a graphene FET

We present in this chapter the main result of this thesis: the detection of THz radiation by a graphene FET. We begin explaining the theory underlying our photodetection and the photodetector design, then we show the experimental results.

4.1 Theory of Photodetection: the Dyakonov and Shur

model

We present here the theory developed by Dyakonov and Shur (Refs. [1] and [3]) . This theory allows for the use of FETs to detect THz frequencies exploiting plasma eects that take place in the gated region. Several experiments on Si MOSFET [6], GaAs/AlGaAs FET [4], GaAs HEMT [3] and more recently on InAs nanowires [66] have conrmed so far the validity of this theory.

The basic idea is to couple the THz radiation to the FET with a two lobed antenna: one lobe is connected with the gate and the other one with the source. In this way, both the carrier density and the carrier drift velocity are modulated simultaneously by the THz ac signal. The THz signal is rectied and leads to a dc photovoltage 4U between source and drain proportional to the received power. We now proceed to explain the theory in more details.

We recall the results obtained in section 2.5 regarding the plasma wave dispersion in the channel of a graphene FET and we proceed in the following with further simplications. We assume that an acoustic plasmon exists in the gated region, with linear dispersion law ω(k) = csk, where csis the plasma

wave velocity. We also assume that the graphene sheet is not doped (V0 T G = 0)

and we consider only the modulation induced by the top-gate, so that we simply write VG and CG instead of VT G and CT G, respectively. We rewrite the

carrier sheet density: ns = s n2 0+ CGVG e 2 = CG e q V2 0 + VG2, (4.1)

where n0 is the residual carrier density that we already dened in section

3.3 and V0 = en0/CG.

Recalling Eqs. 2.28 and 2.29, we express the plasma wave velocity cs as a

function of the gate capacitance per unit area CG, the carrier sheet density ns

and the effective mass m:

cs = s α nse 2 mCG , (4.2)

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where α = 4π or α = 2π according to Eq. 2.28 or Eq. 2.29,respectively. In a 2DEG system, m is the eective electron mass while in graphene m is taken to be equal to the cyclotron mass[17]:

m = EF v2 F = ~ vF √ πns. (4.3)

Since we are modulating the carrier density and drift velocity in the FET channel with the THz radiation, it is useful to separate the constant part of the top-gate voltage VG, from the local, time dependent part VG(x, t)(x is the

distance from the source):

U (x, t) = VG(x, t) + VG. (4.4)

The plasma wave theory is based on three fundamental equations: the dependence of the carrier sheet density ns on the local gate-source voltage

swing U(x, t) (see Eq. 4.1), the Euler equation of hydrodynamic movement, and the continuity equation:

∂v ∂t + v ∂v ∂x + e m ∂U ∂x + v τpl = 0 Euler equation, (4.5) ∂U ∂t + ∂(U v) ∂x = 0 Continuity Equation, (4.6) where ∂U/∂x is the longitudinal electric eld, v(x, t) the local electron drift velocity and τpl the plasma damping time constant.

As a rst estimate for the plasmon damping we can consider that τpl ≈ τel,

where τelcorresponds to the momentum relaxation time of electrons. The

scat-tering time τel is related to the mobility through the relation τel = µ~kF/evF

(assuming that charged-impurities are the dominant source of scattering). With a mobility µ = 2000 cm2_V−1_s−1 _{and a density n}

s = 1 × 1012cm−2, we

nd that τel ≈ 25 fs.

We will now give a condensed derivation of the low frequency detection regime following Refs. [67] and [68]. The response at high frequencies will be showed and commented afterward.

Low Frequency Regime ωτpl 1

This is the regime that we expect for our graphene FETs since at 0.3 THz: ωτpl ∼ ωτel |f =0.3 THz= 2π · 0.3 THz · 25 fs = 0.05 (4.7)

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When ωτpl 1, the damping is so high that plasma waves are overdamped

and thus decay before even one oscillation period. The THz radiation/ exci-tation at the source side of the channel causes a carrier density perturbation that decays exponentially with the distance from the source. The characteristic decay length lc can be written as (proof will be given later):

lc = cs

q

2τpl/ω (4.8)

If ωτpl 1, the inertial term ∂v/∂t in Eq. 4.5 can be neglected. If we

further neglect the nonlinear term v∂v/∂x we can rewrite Eq. 4.5 and 4.6 as a function of the d.c. conductivity σ = enµ and the current density j = env (this operation is useful for an easier interpretation of the experimental data). After some easy math we obtain:

j(x, t) = −σ∂U (x, t) ∂x Ohm 0 s law, (4.9) CG ∂U (x, t) ∂t + ∂j(x, t) ∂x = 0 continuity equation. (4.10) Combining these equations together we get:

CG ∂U (x, t) ∂t − ∂ ∂x σ∂U (x, t) ∂x = 0. (4.11) This equation must be solved together with the boundary conditions:

U (0, t) = VG+ Uacos(ωt) at the source, (4.12)

U (∞, t) = const at the drain. (4.13) In Eq. 4.12, the term Ua is the amplitude of the radiation induced

modu-lation of the source-gate voltage and ω is the angular frequency of the incident THz radiation.

The symbol ∞ in Eq. 4.13 means that the condition is valid only if the channel is very long (L lc). We look for a solution of Eq. 4.11 as an

expansion in powers of Ua:

U (x, t) = U1(x, t) + U2(x) + VG, (4.14)

where U1(x, t) is the a.c. component proportional to Ua and U2(x) is the d.c

component proportional to U2

a (the second harmonic term is not considered).

Inserting the Ansatz (4.14) in Eq. 4.11 and retaining only terms in the rst order in Ua, we obtain: CG ∂U1(x, t) ∂t − σ(VG) ∂2U1(x, t) ∂x2 = 0. (4.15)

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Here σ(VG) is the d.c. conductivity evaluated at the equilibrium carrier

density ns. The boundary conditions in Eq. 4.12 and 4.13 now become

U1(0, t) = Uacos(ωt) and U1(∞, t) = 0. The solution of equation 4.15 then

reads:

U1(x, t) = Uaexp(−x/lc) cos(ωt − x/lc), (4.16)

where l2

c = 2σ(VG)/ωCG = 2τplc2s/ω.

We need to nd also an equation for U2(x). Again, we insert the Ansatz

(4.14) in Eq. 4.11, retain only terms in the second order in Ua and then

integrate over the period T = 2π/ω: ∂ ∂x " σ(VG) ∂U2(x) ∂x + ∂σ ∂U U =VG × U1(x, t) ∂U1(x, t) ∂x # = 0. (4.17) The quantity between the brackets h...i can be readily calculated:

1 T ˆ T 0 dt U1(x, t) ∂U1(x, t) ∂x = − 1 2lc U_a2exp(−2x/lc). (4.18)

Integrating Eq. 4.17 with the boundary condition U2(0) = 0, we get:

U2(x) = U2 a 4 1 σ ∂σ ∂U U =VG (1 − exp(−2x/lc)). (4.19)

Finally, the photoresponse 4U = U2(∞) − U2(0) is:

4U = U 2 a 4 1 σ ∂σ ∂U U =VG = U 2 a 4 ∂(ln σ) ∂VG . (4.20)

In the case L ≈ lc, the photoresponse becomes:

4U = U 2 a 4 · ∂(ln σ) ∂VG · (1 − exp(−2L/lc)) . (4.21)

If we estimate lc using τpl = 25 fs (as found before), ω/2π = 0.3 THz and

CG = 500 nF/cm2, we get lc ≈ 250 nm. Since in our devices L = 200 ÷ 300 nm,

we expect that the photoresponse will experience a little reduction following Eq. 4.21.

To conclude this paragraph, we write the photoresponse in Eq. 4.20 as a function of the eective mass m (given in Eq. 4.3) and the plasma wave velocity cs (Eq. 4.2): 4U = U 2 a 4 1 σ ∂σ ∂U U =VG = U 2 a 4 CG nse + 1 µ ∂µ ∂U U =VG ! . (4.22)