Experimental review of graphene

Experimental review of graphene

Abstract

This review examines the properties of graphene from an experimental perspective. The intent is to review the most important experimental results at a level of detail appropriate for new graduate students who are interested in a general overview of the fascinating properties of graphene. While some introductory theoretical concepts are provided, including a discussion of the electronic band structure and phonon dispersion, the main emphasis is on describing relevant experiments and important results as well as some of the novel applications of graphene. In particular, this review covers graphene synthesis and characterization, field-effect behavior, electronic transport properties, magneto-transport, integer and fractional quantum Hall effects, mechanical properties, transistors, optoelectronics, graphene-based sensors, and biosensors. This approach attempts to highlight both the means by which the current understanding of graphene has come about and some tools for future contributions.

pacs:
81.05.ue, 72.80.Vp, 63.22.Rc, 01.30.Rr
1
Contents

I Introduction

Graphene is a single two-dimensional layer of carbon atoms bound in a hexagonal lattice structure. It has been extensively studied in the last several years even though it was only isolated for the first time in 2004 Novoselov et al. (2004). Andre Geim and Konstantin Novoselov won the 2010 Nobel Prize in Physics for their groundbreaking work on graphene. The fast uptake of interest in graphene is due primarily to a number of exceptional properties that it has been found to possess.

There have been several reviews discussing the topic of graphene in recent years. Many are theoretically oriented, with Castro Neto et al.’s review of the electronic properties as a prominent example Castro Neto et al. (2009) and a more focused review of the electronic transport properties Das Sarma et al. (2011). Experimental reviews, to name only a few, include detailed discussions of synthesis Choi et al. (2010) and Raman characterization methods Ni et al. (2008a), of transport mechanisms Avouris (2010); Giannazzo et al. (2011), of relevant applications of graphene such as transistors and the related bandgap engineering Schwierz (2010), and of graphene optoelectronic technologies Bonaccorso et al. (2010). We feel, however, that the literature is lacking a comprehensive overview of all major recent experimental results related to graphene and its applications. It is with the intent to produce such a document that we wrote this review. We gathered a great number of results from what we believe to be the most relevant fields in current graphene research in order to give a starting point to readers interested in expanding their knowledge on the topic. The review should be particularly well-suited to graduate students who desire an introduction to the study of graphene that will provide them with many references for further reading.

We have attempted to deliver an up-to-date account of most topics. For example, we present recent results on the fractional quantum Hall effect and some of the newest developments and device details in optoelectronics. In addition, we include a summary of the work that has been done in the field of graphene biosensors. As a practical tool, we also give comparative analyses of graphene substrate properties, of available bandgap engineering techniques and of photovoltaic devices in order to provide the researcher with useful and summarized laboratory references.

This review is structured as follows: We start by reviewing the electronic band structure and its associated properties in Sec. II before introducing the vibrational properties, including the phonon dispersion, in Sec. III. We then detail the various synthesis methods for graphene monolayers and their corresponding characterization. The main properties are then discussed, starting with the electric field effect, which originally spurred the intense activity in graphene research. This is then followed by a review of the magneto-transport properties, which includes the quantum Hall effect and recent results on the fractional quantum Hall effect. We conclude the review of the main properties with a discussion of the mechanical properties of graphene, before turning to some important applications. The transistor applications are discussed first, with particular emphasis on band gap engineering, before reviewing optoelectronic applications. We finish the applications section with a discussion of graphene-based sensors and biosensors.

Ii Electronic Structure

Graphene has a remarkable band structure thanks to its crystal structure. Carbon atoms form a hexagonal lattice on a two-dimensional plane. Each carbon atom is about 1.42 Å from its three neighbors, with each of which it shares one bond. The fourth bond is a -bond, which is oriented in the z-direction (out of the plane). One can visualize the orbital as a pair of symmetric lobes oriented along the z-axis and centered on the nucleus. Each atom has one of these -bonds, which are then hybridized together to form what are referred to as the -band and -bands. These bands are responsible for most of the peculiar electronic properties of graphene.

Figure 1: Triangular sublattices of graphene. Each atom in one sublattice (A) has 3 nearest neighbors in sublattice (B) and vice-versa.

The hexagonal lattice of graphene can be regarded as two interleaving triangular lattices. This is illustrated in Fig. 1. This perspective was successfully used as far back as 1947 when Wallace calculated the band structure for a single graphite layer using a tight-binding approximation Wallace (1947).

Figure 2: First Brillouin zone and band structure of graphene. The vertical axis is energy, while the horizontal axes are momentum space on the graphene lattice. The first Brillouin zone of graphene is illustrated in the horizontal plane and labeled with some points of interest. K and K’ are the two non-equivalent corners of the zone, and M is the midpoint between adjacent K and K’ points. is the zone center. K and K’ are also known as the Dirac points. The Dirac points are the transition between the valence band and the conduction band.

Band structure is most often studied from a standpoint of the relationship between the energy and momentum of electrons within a given material. Since graphene constrains the motion of electrons to two dimensions, our momentum space is also constrained to two dimensions. A plot of the energy versus momentum dispersion relation for graphene can be found in Fig. 2 using the tight binding approximation discussed below.

Graphene is a zero-gap semiconductor because the conduction and valence bands meet at the Dirac points (see Fig. 2). The Dirac points are locations in momentum space, on the edge of the Brillouin zone. There are two sets of three Dirac points. Each set is not equivalent with the other set of three. The two sets are labeled K and K’. The two sets of Dirac points give graphene a valley degeneracy of . The K and K’ points are the primary points of interest when studying the electronic properties of graphene. This is noteworthy in comparison to traditional semiconductors where the primary point of interest is generally , where momentum is zero.

Following the original work Wallace (1947), the tight binding Hamiltonian, written in a basis of sublattice positions A and B, is given by:

(1)

where are the on-site energies of the carbon atoms on sites A and B, 2.7 eV is the next nearest hopping element (between A and B sites), , , are the positions of the three nearest neighbors, and the complex conjugate of the off-diagonal matrix element. The eigenvalues of this tight binding Hamiltonian are shown in Fig. 2 as a function of .

ii.1 Massless Dirac fermions

Looking closely at the region near one of the Dirac points (K or K’) in Fig. 2, the cone-like linear dispersion relation is evident. The Fermi energy for neutral (or ideal) graphene is at the Dirac energy, which is the energy of the Dirac point. In graphene devices, the Fermi energy can be significantly different from the Dirac energy.

Electrons within about 1 eV of the Dirac energy have a linear dispersion relation. The linear dispersion region is well-described by the Dirac equation for massless fermions. That is, the effective mass of the charge carriers in this region is zero. The dispersion relation near the K points is generally expressed as follows:

(2)

which corresponds to the spectrum of the Dirac-like Hamiltonian for low-energy massless Dirac fermions (again in the sublattice basis {A,B}):

(3)

This Dirac Hamiltonian is simply the tight binding Hamiltonian from Eq. (1) expanded close to K with . Close to K’ the Hamiltonian becomes , where is the 2D vector of the Pauli matrices (and denotes the complex conjugate), is the wavevector, and the Fermi velocity is m/s, or 1/300th the speed of light in vacuum. Charge carriers in graphene behave like relativistic particles with an effective speed of light given by the Fermi velocity. This behavior is one of the most intriguing aspects about graphene, and is responsible for much of the research attention that graphene has received.

ii.2 Chirality

Transport in graphene exhibits a novel chirality which we will now briefly describe. Each graphene sublattice can be regarded as being responsible for one branch of the dispersion. These dispersion branches interact very weakly with one another.

This chiral effect indicates the existence of a pseudospin quantum number for the charge carriers. This quantum number is analogous to spin but is completely independent of the ‘real’ spin. The pseudospin lets us differentiate between contributions from each of the sublattices. This independence is called chirality because of the inability to transform one type of dispersion into another Das Sarma et al. (2011). A typical example of chirality is that you cannot transform a right hand into a left hand with only translations, scalings, and rotations. The chirality of graphene can also be understood in terms of the Pauli matrix contributions in the Dirac-like Hamiltonian described in the previous section.

ii.3 Klein paradox

A peculiar property of the Dirac Hamiltonian is that charge carriers cannot be confined by electrostatic potentials. In traditional semiconductors, if an electron strikes an electrostatic barrier that has a height above the electron’s kinetic energy, the electron wavefunction will become evanescent within the barrier and exponentially decay with distance into the barrier. This means that the taller and wider a barrier is, the more the electron wavefunction will decay before reaching the other side. Thus, the taller and wider the barrier is, the lower the probability of the electron quantum tunneling through the barrier.

However, if the particles are governed by the Dirac equation, their transmission probability actually increases with increasing barrier height. A Dirac electron that hits a tall barrier will turn into a hole, and propagate through the barrier until it reaches the other side, where it will turn back into an electron. This phenomenon is called Klein tunneling.

An explanation for this phenomenon is that increasing barrier height leads to an increased degree of mode-matching between the wavefunctions of the holes within the barrier and the electrons outside of it. When the modes are perfectly matched (in the case of an infinitely tall barrier), we have perfect transmission through the barrier. In the case of graphene, the chirality discussed earlier leads to a varying transmission probability depending on the angle of incidence to the barrier Katsnelson et al. (2006).

Some experimental results have been interpreted as evidence for Klein tunneling. Klein tunneling has been observed through electrostatic barriers, which were created by gate voltages Stander et al. (2009). Similar effects have also been observed in narrow graphene resonant heterostructures Young and Kim (2009).

ii.4 Graphene vs traditional materials

Here we summarize some of the interesting properties of graphene by comparing them with more traditional materials such as 2D semiconductors.

  1. Traditional semiconductors have a finite band gap while graphene has a nominal gap of zero. Normally, the study of electron and hole motion through a semiconductor must be done with differently doped materials. However, in graphene the nature of a charge carrier changes at the Dirac point from an electron to a hole or vice-versa. On a related note, the Fermi level in graphene is always within the conduction or valence band while in traditional semiconductors the Fermi level often falls within the band gap when pinned by impurity states.

  2. Dispersion in graphene is chiral. This is related to some very distinctive material behaviors like Klein tunneling.

  3. Graphene has a linear dispersion relation while semiconductors tend to have quadratic dispersion. Many of the impressive physical and electronic properties of graphene can be considered to be consequences of this fact.

  4. Graphene is much thinner than a traditional 2D electron gas (2DEG). A traditional 2DEG in a quantum well or heterostructure tends to have an effective thickness around 5-50 nm. This is due to the constraints on construction and the fact that the confined electron wavefunctions have an evanescent tail that stretches into the barriers. Graphene on the other hand is only a single layer of carbon atoms, generally regarded to have a thickness of about 3 Å (twice the carbon-carbon bond length). Electrons conducting through graphene are constrained in the z-axis to a much greater extent than those that conduct through a traditional 2DEG.

  5. Graphene has been found to have a finite minimum conductivity, even in the case of vanishing charge carriers Novoselov et al. (2005); Tan et al. (2007). This is an issue for the construction of field-effect transistors (FETs), as we will see in more detail in section VI, since it contributes to relatively low on/off ratios for graphene-based transistors.

Readers seeking further reading about distinctly electronic properties of graphene should refer to Sec. VI. The next section will introduce and discuss the vibrational properties of graphene.

Iii Vibrational properties

While the electronic properties have attracted the lion’s share of the interest in graphene, the vibrational properties are of great importance too. They are responsible for several fascinating properties such as record thermal conductivities. Since graphene is composed of a light atom, where the in-plane bonding is very strong, graphene exhibits a very high sound velocity. This large sound velocity is responsible for the very high thermal conductivity of graphene that is useful in many applications. Moreover, vibrational properties are instrumental in understanding other graphene attributes, including optical properties via phonon-photon scattering (e.g. in Raman scattering) and electronic properties via electron-phonon scattering.

iii.1 Phonon dispersion

Most of the vibrational properties of graphene can be understood with the help of the phonon dispersion relation. Interestingly, the phonon dispersion has some similarity with the electronic band structure discussed in the previous section, which stems from the identical honeycomb structure (the out-of-plane modes are shown in Fig. 3). In order to obtain the phonon dispersion it is necessary to consider the vibrational modes of the crystal in thermal equilibrium. This is done by considering the displacement of each atom from its equilibrium position, written for the atom labeled . Each atom is effectively coupled to its neighbors by some torsional and longitudinal force constants, which only depend on the relative positions of the atoms. This allows one to write the Newtonian coupled equation of motion in frequency space as:

(4)

where in graphene the sum over is typically over the second or fourth nearest neighbors, with the corresponding coefficients , also known as the dynamical matrix.

In graphene, and similarly to the electronic structure, the two sublattices A and B have to be considered explicitly to solve for the eigenspectrum of the dynamical matrix. However, the atoms can vibrate in all three dimensions, hence the dynamical matrix has to be written in terms of both the sublattices A and B as well as the 3 spatial dimensions. This leads to a dynamical matrix in reciprocal space , which is given by a matrix when assuming . Here one applies Bloch’s theorem, where is the equilibrium position of atom in the sublattice A and B, respectively. Two of the eigenvalues correspond to the out-of-plane vibrations, ZA (acoustic) and ZO (optical), and the remaining 4 correspond to the in-plane vibrations: TA (transverse acoustic), TO (transverse optical), LA (longitudinal acoustic) and LO (longitudinal optical).

Figure 3: First Brillouin zone and out-of-plane phonon modes. The phonon dispersion relation of the ZA and ZO modes as a function of the in-plane reciprocal vector . The vertical axis is the phonon frequency, while the horizontal axes are momentum space on the graphene lattice. The dispersion relation is obtained using the second-nearest-neighbor model for graphene Falkovsky (2007). The gray surface corresponds to the ZO (optical) mode, whereas the pink surface shows the ZA (acoustic) mode. Also shown are the corresponding K, , and M points of the Brillouin zone.

The ZA and ZO modes are often assumed to be decoupled from the in-plane modes Falkovsky (2007), which leads to a simple dispersion relation very similar to the electronic band structure as shown in Fig. 3. It is interesting to note that the dispersion is quadratic at the point, which is unusual for acoustic modes. In contrast, a simple graphene phonon model based on atomic potentials containing only three parameters leads to a linear dispersion for the ZA mode Adamyan and Zavalniuk (2011). However, the experimental data seems to be more consistent with a quadratic dispersion (see Fig. 5) Popov (2004); Dresselhaus and Eklund (2000). At the K and K’ points we recover a cone structure similar to the Dirac cones in the electronic structure. However, the phonon density of states does not vanish at these points because of the presence of the in-plane modes.

Figure 4: First Brillouin zone and in-plane phonon modes. The phonon dispersion relation of the TO and LO modes in gray and the TA and LA modes in pink as a function of the in-plane reciprocal vector . The longitudinal modes are on top of the transverse modes. The vertical axis is the phonon frequency, while the horizontal axes are the momentum space on the graphene lattice.

The in-plane modes are constituted by two acoustic modes and two optical modes. These modes can be obtained from the reduced in-plane dynamical matrix, which can be described by a matrix, assuming no coupling to the out-of-plane modes. Using the parameters given by reference Falkovsky (2007), the in-plane modes are shown in Fig. 4.

Figure 5: Experimental phonon dispersion relation in graphite. All the phonon modes are shown, including the in-plane and out-of-plane modes. The data (full symbols) is shown along the special symmetry points , K, and M along with results from ab initio calculations (lines). Figure adapted from Wirtz and Rubio (2004).

The in-plane modes show the expected linear dispersion at the point. The transverse modes closely follow the longitudinal modes but with a slightly lower frequency, for both the acoustic and optical modes. For comparison with experiments, it is more instructive to show the dispersion relation following surface cuts along the lines to M, M to K and K to in the Brillouin zone. An extensive collection of experimental data for graphite is shown in Fig. 5 along with ab initio calculations. The data was obtained using several techniques, including neutron scattering, electron energy loss spectroscopy, X-ray scattering, infrared absorption and double resonant Raman scattering experiments Wirtz and Rubio (2004). There is no comparably extensive data on graphene yet, but it is expected to be very similar to graphite, since the coupling between planes is very weak.

Figure 6: Theoretical phonon dispersion of graphene and Raman spectrum of large scale graphene. The labels (G, D, 2D, etc) identify the peaks from the Raman spectrum with the corresponding phonon energies Venezuela et al. (2011); Rao et al. (2011). Only the in-plane modes are shown since they are the only ones which are Raman active. Several Raman data sets of graphene C are shown in blue taken from different spots of the same sample at a laser wavelength of 514nm. The red line corresponds to the average over these data sets. The phonon dispersion is calculated using the mass of C.

Most of the data on graphene stems from Raman scattering, which allows for the determination of the phonon spectrum close to special symmetry points. This is illustrated in Fig. 6, where the theoretical phonon dispersion with parameters from Falkovsky (2007) is shown with the labels corresponding to the various Raman peaks. The Raman peaks determine the phonon energies for some values of the momentum. The Raman data was obtained from chemical vapor deposition (CVD) C grown graphene. More conventional C graphene is very similar except for a rescaling of the Raman peaks due to the change of mass. Other aspects of the Raman spectrum are discussed in Sec. V.

An important consequence of the phonon dispersion relation in graphene is the very high value of the in-plane sound velocity, close to km/s Adamyan and Zavalniuk (2011), which leads to very high thermal conductivities.

iii.2 Thermal conductivity

From the kinetic theory of gases, the thermal conductivity due to phonons is given by , where is the specific heat per unit volume and is the phonon mean free path. This implies that since is very large in graphene, one can expect a large thermal conductivity. Indeed, experiments at near room temperature obtain 3080-5150 W/mK and a phonon mean free path of nm for a set of graphene flakes Ghosh et al. (2008); Balandin et al. (2008).

These results indicate that graphene is a good candidate for applications to electronic devices, since a high thermal conductivity facilitates the diffusion of heat to the contacts and allows for more compact circuits. Phonons also play an important role in electronic transport via electron-phonon scattering, which is discussed in Sec. VI. The mechanical properties of graphene are discussed in Sec. VIII, whereas the synthesis of graphene is treated in the next section.

Iv Synthesis

Since graphene was isolated in 2004 by Geim and Novoselov using the now famous Scotch tape method, there have been many processes developed to produce few-to-single layer graphene. One of the primary concerns in graphene synthesis is producing samples with high carrier mobility and low density of defects. To date there is no method that can match mechanical exfoliation for producing high-quality, high-mobility graphene flakes. However, mechanical exfoliation is a time consuming process limited to small scale production. There is great interest in producing large scale graphene suitable for applications in flexible transparent electronics, transistors, etc. Some concerns in producing large scale graphene are the quality and consistency between samples as well as the cost and difficulty involved in the method.

Table 1 shows a summary of some of the most important synthesis methods. The typical number of graphene layers produced as well as currently achievable dimensions are given. For comparison the mobilities listed are for graphene transferred to Si/SiO wafers, since the electron mobility of graphene is heavily substrate dependent, as discussed in Sec. VI.

Method Layers Size Mobility (cmVs)
Exfoliation 1 to 10+ 1 mm 15000
Thermal SiC 1 to 4 50 m 2000
Ni-CVD 1 to 4 1 cm 3700
Cu-CVD 1 65 cm 16000
Table 1: Comparison of graphene synthesis methods. Shows typical number of layers produced, size of graphene layers (largest dimension) and mobility on Si/SiO. a:Geim (2009), b:Novoselov et al. (2005), c:Emtsev et al. (2009), d:Kim et al. (2009), e:Bae et al. (2010), f:Li et al. (2010b).

iv.1 Mechanical exfoliation

Developed by Geim and Novoselov, the exfoliation process uses HOPG (highly oriented pyrolitic graphite) as a precursor. The HOPG was subjected to an oxygen plasma etching to create 5 m deep mesas and these mesas were then pressed into a layer of photoresist. The photoresist was baked and the HOPG was cleaved from the resist. Scotch tape was used to repeatedly peel flakes of graphite from the mesas. These thin flakes were then released in acetone and captured on the surface of a Si/SiO wafer Novoselov et al. (2004).

Figure 7: Monolayer graphene produced by mechanical exfoliation. Large sample with length of 1 mm on Si/SiO.

These few-layer graphene (FLG) flakes were identified using the contrast difference in an optical microscope and single layers using an SEM. Using this technique Geim and Novoselov were able to generate few- and single-layer graphene flakes with dimensions of up to 10 m Novoselov et al. (2004). The few-layer graphene flakes were found to have ballistic transport at room temperature and mobilities as high as 15000 cmVs on Si/SiO wafers Novoselov et al. (2005). The scotch tape method can generate flakes with sides of up to 1 mm in length Geim (2009), of excellent quality and well suited for fundamental research. However, the process is limited to small sizes and cannot be scaled for industrial production.

iv.2 Thermal decomposition of SiC

The thermal decomposition of silicon carbide is a technique that consists of heating SiC in ultra-high vacuum (UHV) to temperatures between 1000C and 1500C. This causes Si to sublimate from the material and leave behind a carbon rich surface. Low-energy electron microscopy (LEEM) studies indicate that this carbon layer is graphitic in nature, which suggests that the technique could be used to form graphene Hass et al. (2008).

Berger and De Heer produced few-layer graphene by thermal decomposition of SiC. The Si face of a 6H-SiC single crystal was first prepared by oxidation or H etching in order to improve surface quality. The sample was then heated by electron bombardment in UHV to 1000C to remove the oxide layer. Once the oxide was removed the samples were heated to 1250-1450C, resulting in the formation of thin graphitic layers. Typically between 1 and 3 layers were formed depending on the decomposition temperature. Using this method, devices were produced with mobilities of 1100 cmVs Berger et al. (2004).

This technique is capable of generating wafer-scale graphene layers and is potentially of interest to the semiconductor industry. Several issues still remain, notably controlling the number of layers produced, repeatability of large area growths and interface effects with the SiC substrate Choi et al. (2010).

Emtsev et al. found that by heating SiC in Ar at 900 mbar as opposed to UHV they were able to reduce surface roughness and produce much larger continuous graphene layers, up to 50 m in length. The graphene on SiC was characterized using atomic force microscopy (AFM) and LEEM, as shown in Fig. 8. They measured electron mobilities of up to 2000 cmVs Emtsev et al. (2009).

Figure 8: Graphene produced by thermal decomposition of SiC. (a) AFM image of graphene growth on SiC annealed at UHV. (b) LEEM image of UHV grown graphene film. (c) AFM image of graphene annealed in Ar at 900 mbar. (d) LEEM image of graphene on Ar annealed SiC substrate showing terraces up to 50 m in length Emtsev et al. (2009).

Juang et al. synthesized millimeter size few- to single-layer graphene sheets using SiC substrate coated in a thin Ni film. 200 nm of Ni was evaporated onto the surface of the SiC and the sample was heated to 750C in vacuum. Graphene was found to segregate to the surface of the Ni on cooling. This gave a continuous graphene layer over the entire nickel surface Juang et al. (2009).

Unarunotai et al. have developed a technique to transfer graphene synthesized on SiC onto arbitrary insulating substrates. Graphene was first produced using a typical thermal decomposition of SiC technique. A bilayer film of gold/polyimide was deposited onto the SiC wafer and then peeled off. The gold/polyimide film was then transferred onto a Si/SiO substrate and the gold/polyimide layers were removed using oxygen plasma reactive ion etching. This yielded single-layer graphene flakes with mm areas Unarunotai et al. (2009).

iv.3 Chemical vapor deposition

In contrast to the thermal decomposition of SiC, where carbon is already present in the substrate, in chemical vapor deposition (CVD), carbon is supplied in gas form and a metal is used as both catalyst and substrate to grow the graphene layer.

Growth on nickel

Yu et al. grew few-layer graphene sheets on polycrystalline Ni foils. The foils were first annealed in hydrogen and then exposed to a CH-Ar-H environment at atmospheric pressure for 20 mins at a temperature of 1000C. The foils were then cooled at different rates between 20C/s and 0.1C/s. The thickness of the graphene layers was found to be dependent on the cooling rate, with few-layer graphene (typically 3-4 layers) being produced with a cooling rate of 10C/s. Faster cooling rates result in thicker graphite layers, whereas slower cooling prevents carbon from segregating to the surface of the Ni foil Yu et al. (2008).

To transfer the graphene layers to an insulating substrate, the Ni foil with graphene was first coated in silicone rubber and covered with a glass slide then the Ni was etched in HNO.

Growth on copper

Li et al. used a similar process to produce large scale monolayer graphene on copper foils. 25 m thick copper foils were first heated to 1000C in a flow of 2 sccm (standard cubic centimeters per minute) hydrogen at low pressure and then exposed to methane flow of 35 sccm and pressure of 500 mTorr. Raman spectroscopy and SEM imaging confirm the graphene to be primarily monolayer independent of growth time. This indicates that the process is surface mediated and self limiting. They fabricated dual gated FETs using graphene and extracted a carrier mobility of 4050 cmVs Li et al. (2009).

Recently a roll-to-roll process was demonstrated to produce graphene layers with a diagonal of up to 30 inches as well as transfer them to transparent flexible substrates Bae et al. (2010). Graphene was grown by CVD on copper, and a polymer support layer was adhered to the graphene-copper. The copper was then removed by chemical etching and the graphene film transferred to a polyethylene terephthalate (PET) substrate. These films demonstrate excellent sheet resistances of 125 for a single layer. Using a repeated transfer process, doped 4-layer graphene sheets were produced with sheet resistances as low as 30 and optical transmittance greater than 90%. These 4-layer graphene sheets are superior to commercially available indium tin oxide (ITO) currently used in flat panel displays and touch screens in terms of sheet resistance (100 for ITO) and optical transmittance (90% for ITO).

Figure 9: Multiple CVD graphene sheets transferred to PET. A roll-to-roll process was used to produce graphene sheets with up to 30 inch diagonal Bae et al. (2010).

Li et al. have shown the dependence on the size of graphene domains synthesized by CVD with temperature, methane flow and methane pressure. Performing the growth at 1035C with methane flow of 7 sccm and pressure 160 mTorr led to the largest graphene domains with average areas of 142 m. A two-step process was used to first grow large graphene flakes and then by modifying the growth conditions to fill in the gaps in the graphene sheet. Using this technique they were able to produce samples with carrier mobility of up to 16000 cmVs Li et al. (2010b). In general, the graphene layer is slightly strained on the copper foil due to the high temperature growth Yu et al. (2011).

Figure 10: Controlling domain size in CVD graphene. Effect of temperature, methane flow and methane partial pressure on the size of graphene domains in CVD growth, scale bars are 10 m Li et al. (2010b).

Recently Lee et al. have demonstrated a technique to produce uniform bilayer graphene by chemical vapor deposition on copper using a similar process but with modified growth conditions. They determined optimal bilayer growth conditions to be: 15 minutes at 1000C with methane flow of 70 sccm and pressure of 500 mTorr. The bilayer nature of the graphene was confirmed by Raman spectroscopy, AFM, and transmission electron microscopy (TEM). Electrical transport measurements in a dual gated device indicate that a band gap is opened in CVD bilayer graphene Lee et al. (2010).

Figure 11: Bilayer CVD growth on copper. (a) 2 x 2 inch bilayer graphene on Si/SiO. (b) Raman spectrum with 514 nm laser source of 1 and 2 layers of graphene produced by exfoliation and CVD Lee et al. (2010).

iv.4 Molecular beam deposition

Zhan et al. succeeded in layer-by-layer growth of graphene using a molecular beam deposition technique. Starting with an ethylene gas source, gas was broken down at 1200C using a thermal cracker and deposited on a nickel substrate. Large area, high quality graphene layers were produced at 800C. This technique is capable of forming one layer on top of another, allowing for synthesis of one to several layers of graphene. The number of graphene layers produced was found to be independent of cooling rate, indicating that carbon was not absorbed into the bulk of the Ni as in CVD growth on nickel. Results were confirmed using Raman spectroscopy and TEM Zhan et al. (2011).

Figure 12: Molecular beam deposition produced graphene. (a) Diagram of thermal cracker setup. (b) TEM image of graphene film, scale bar 100 nm Zhan et al. (2011).

iv.5 Unzipping carbon nanotubes

Multi-walled carbon nanotubes were cut longitudinally by first suspending them in sulphuric acid and then treating them with KMnO. This produced oxidized graphene nanoribbons which were subsequently reduced chemically. The resulting graphene nanoribbons were found to be conducting, but electronically inferior to large scale graphene sheets due to the presence of oxygen defect sites Kosynkin et al. (2009).

iv.6 Sodium-ethanol pyrolysis

Graphene was produced by heating sodium and ethanol at a 1:1 molar ratio in a sealed vessel. The product of this reaction is then pyrolized to produce a material consisting of fused graphene sheets, which can then be released by sonication. This yielded graphene sheets with dimensions of up to 10 m. The individual layer, crystalline and graphitic nature of the samples was confirmed by TEM, selected area electron diffraction (SAED) and Raman spectroscopy Choucair et al. (2009).

iv.7 Other methods

There are several other ways to produce graphene such as electron beam-irradiation of PMMA nanofibres Duan et al. (2008), arc discharge of graphite Subrahmanyam et al. (2009), thermal fusion of PAHs Wang et al. (2008a), and conversion of nanodiamond Subrahmanyam et al. (2008).

iv.8 Graphene oxide

Another approach to the production of graphene is sonication and reduction of graphene oxide (GO). The polar O and OH groups formed during the oxidation process render graphite oxide hydrophilic, and it can be chemically exfoliated in several solvents, including water Zhu et al. (2010). The graphite oxide solution can then be sonicated in order to form GO nanoplatelets. The oxygen groups can then be removed in a reduction process involving one of several reducing agents. This method was used by Stankovich et al. using a hydrazine reducing agent, but the reduction process was found to be incomplete, leaving some oxygen remaining Stankovich et al. (2007).

Graphene oxide (GO) is produced as a precursor to graphene synthesis. GO is useful because its individual layers are hydrophilic, in contrast to graphite. GO is suspended in water by sonication McAllister et al. (2007); Paredes et al. (2008), then deposited onto surfaces by spin coating or filtration to make single or double layer graphene oxide. Graphene films are then made by reducing the graphene oxide either thermally or chemically Marcano et al. (2010). The exact structure of graphene oxide is still a matter of debate, although there is considerable agreement as to the general types and proportion of oxygen bonds present in the graphene lattice He et al. (1996).

Wet chemical synthesis

The chemical methods to produce GO were all developed before 1960. The most recent and most commonly employed is the Hummers procedure Hummers and Offeman (1958). This process treats graphite in an anhydrous mixture of sulfuric acid, sodium nitrate, and potassium permanganate for several hours, followed by the addition of water. The resulting material is graphite oxide hydrate, which contains approximately 23% water. Subsequent nuclear magnetic resonance and X-ray diffraction studies of the structure of GO have led to fairly detailed models based on a combination of hydroxide, carbonyl, carboxyl and epoxide groups covalently bonded to the graphene lattice. Fig. 13 shows a predicted structure for GO produced using the Hummers method He et al. (1998).

Figure 13: Structure of a monolayer of graphite oxide He et al. (1998).
Figure 14: Summary of the Hummers method and thermal reduction. Bulk graphite is oxidized then separated in water. Then it is thermally reduced to make single-layer graphene McAllister et al. (2007).

Understandably, the degree of oxidation strongly affects the in-plane electrical and thermal conductivity of graphene oxide. Increased introduction of oxygen groups into the graphene lattice interrupts the hybridization of electron orbitals. Epoxide groups can be reduced by thermal treatment, or reaction with potassium iodide (KI), leading to a similar structure, where only hydroxyl groups are present. This leads to improved electrical conductivity and unaffected hydrophilicity. In each case the procedure requires tight temperature control and long reaction times of several hours. The basic process, including thermal treatment, is shown in Fig. 14.

Plasma functionalization

Following the realization of the potential importance of graphene as a replacement for semiconductor materials and indium tin oxide (ITO), as discussed in Sec. X.2, alternative methods for graphene production have been explored. Other approaches have been sought to produce the same hydrophilicity in graphite without the time and material requirements of the Hummers method. Very recently, glow discharge treatment has been proven to introduce oxygen species into the lattices of all forms of graphitic materials (e.g. buckyballs, CNTs, graphene, carbon nanofibers, and graphite) Vandsburger et al. (2009). The resulting graphene/graphite oxides have a structure very similar to Hummers GO, and can be thermally treated to selectively reduce epoxides. Unlike the Hummers method, plasma functionalization requires no strong acids, can proceed at room temperature and can be completed very quickly, often in a matter of seconds or minutes.

Aside from its potential to replace the Hummers method, plasma treatment in itself is interesting for altering the electrical conductivity of graphene or thin graphite. This allows for bandgap engineering as well as phenomena like photoluminescence (PL).

RF plasma

Radio frequency (RF) plasma refers to a processing technique whereby a capacitive plasma is ignited in an isolated volume. RF treatment is used almost exclusively for surface treatment of graphene, because ion bombardment is significantly reduced in RF treatments, as opposed to DC discharges. In RF plasma, electrodes need not be in contact with the plasma gas, current is supplied with an alternation frequency of 13.57 MHz, and power ranges from 10 W to 50 W. RF treatment has been shown to selectively affect the outermost surface of graphene Hazra et al. (2011). Other work using only oxygen for RF treatment allows for layer-by-layer etching of a graphite surface, producing islands of GO You et al. (1993).

Photoluminescence

Single- and dual-layer graphene do not exhibit photoluminescence, due mainly to the negligible bandgap of native graphene. GO does have a photoluminescent response, but the typical oxidation methods, sonication of bulk graphite oxide, are inappropriate for use in photoluminescence applications. For that reason, RF plasma oxidation has been the subject of recent work at producing photoluminescent single layers of GO. The procedure for producing GO thin films from single layer graphene was reported in Gokus et al. (2009). Rather than oxidizing bulk graphite to produce single layers of graphene, single- or few-layer graphene is oxidized after isolation. Typically, graphene is prepared by micro-cleavage using the scotch tape or other methods, and electrical contacts or other additions are put in place. Following installation, RF plasma treatment in Ar-O mixes is applied in one to six second intervals. The plasma power reported was 10 W, the pressure 0.04 mbar, and the gas composition ratio was 2:1 Ar to O.

An important finding reported by Childres et al. (2011) is the temporal evolution of the Raman spectrum of graphene with increasing plasma treatment time. The spectra are reproduced in Fig. 15. The most noticeable change in the spectra is the gradual reduction in intensity of the 2D and 2D’ peaks, which are indicative of hybridization. This indicates the disruption of the graphene lattice by introduction of oxygen groups and demonstrates oxidation. Further changes in the region of interest involve the development of a G peak that arises from the increased presence of “disordered” carbon. Similar findings were reported in other work using pulsed RF plasma treatment rather than continuous treatments.

Figure 15: Raman spectra of graphene with increasing number of 1.5 sec plasma treatments Childres et al. (2011).

In a sample with a heterogeneous surface, it was shown that only regions of single-layer graphene oxide displayed PL, while untreated graphene or multi-layer graphene did not. The first image shown in Fig. 16 is an image of the PL produced by laser fluorescence. The bright regions correspond to low intensity regions in an elastic scattering image (b), revealing that they are single-layer graphene, labeled 1L in the chart. (c) shows both a PL curve and a contrast curve taken along the white dotted line in (a). The middle-contrast regions in the blue curve correspond exactly to high PL.

Figure 16: Photoluminescence in oxygen plasma treated GO. (a) Dark regions are pristine graphene, while bright regions are single layer graphene oxide. (b) Few-layer graphene is bright while single-layer graphene oxide is dark. (c) Shaded zone shows the correlation between contrast and PL-intensity taken along the white dotted line in (a) Gokus et al. (2009).

Photoluminescence occurs after plasma treatment as a result of the introduction of defects in the graphene lattice. Such defects disrupt the electrical properties of pristine graphene and introduce a bandgap that is absent from native graphene monolayers. A bandgap is desirable for more than PL, and other work has reported plasma oxidation of single and few layer graphene for these purposes Nourbakhsh et al. (2010). Specifically, a bandgap would allow for logic and optoelectronic applications as we shall see respectively in Sec. IX and Sec. X.

Once graphene has been produced, it is important to identify it and to characterize its structure, which is the topic of the next section.

V Characterization

A great many techniques are being used to characterize graphene. We discuss here some of the most important ones with a particular emphasis on the identification of graphene.

v.1 Raman spectroscopy

Raman spectroscopy is an important characterization tool used to probe the phonon spectrum of graphene as discussed in section Sec. III. Raman spectroscopy of graphene can be used to determine the number of graphene layers and stacking order as well as density of defects and impurities. The three most prominent peaks in the Raman spectrum of graphene and other graphitic materials are the G band at 1580 cm, the 2D band at 2680 cm and the disorder-induced D band at 1350 cm.

The G band results from in-plane vibration of carbon atoms and is the most prominent feature of most graphitic materials. This resonance corresponds to the in-plane optical phonons at the point. The 2D band arises as a result of a two phonon resonance process, involving phonons near the K point, and is very prominent in graphene as compared to bulk graphite Ni et al. (2008a).

The D band is induced by defects in the graphene lattice (corresponding to the in-plane optical phonons near the K point), and is not seen in highly ordered graphene layers. The intensity ratio of the G and D band can be used to characterize the number of defects in a graphene sample Pimenta et al. (2007).

The line shape of the 2D peak, as well as its intensity relative to the G peak, can be used to characterize the number of layers of graphene present as illustrated in Fig. 17. Single-layer graphene is characterized by a very sharp, symmetric, Lorentzian 2D peak with an intensity greater than twice the G peak. As the number of layers increases the 2D peak becomes broader, less symmetric and decreases in intensity Wang et al. (2008b).

Figure 17: Layer dependence of graphene Raman spectrum. Raman spectra of N = 1-4 layers of graphene on Si/SiO and of bulk graphite. Figure adapted from Yu (2010).

v.2 Optical microscopy

Monolayer graphene becomes visible on SiO using an optical microscope. The contrast depends on the thickness of SiO, the wavelength of light used Blake et al. (2007) and the angle of illumination Yu and Hilke (2009).

Figure 18: Optical microscope images of graphene. Multilayer graphene sheet on Si/SiO showing optical contrast at different wavelengths and thicknesses Blake et al. (2007).

This feature of graphene is useful for the quick identification of few- to single-layer graphene sheets, and is very important for mechanical exfoliation. Fig. 18 shows the optical contrast of one, two and three layers of exfoliated graphene under different wavelengths of illumination and different thicknesses of SiO.

v.3 Electron microscopy

Transmission electron microscopy has been used to image single-layer graphene suspended on a microfabricated scaffold. It was found that single-layer graphene displayed long range crystalline order despite the lack of a supporting substrate Meyer et al. (2007). Suspended graphene was found to have considerable surface roughness with out-of-plane deformations of up to 1 nm.

Figure 19: Atomic scale TEM image of suspended graphene. Few- to single-layer graphene sheet showing long range crystalline order, scale bar 1 nm Meyer et al. (2007).

Aberration-corrected annular dark-field scanning transmission electron microscopy (ADF-STEM) was used in order to image CVD grown graphene suspended on a TEM grid Huang et al. (2011). They found that along grain boundaries the hexagonal lattice structure breaks down and the grains are “stitched together” with pentagon-heptagon pairs as seen in Fig. 20.

Figure 20: ADF-STEM imaging of graphene suspended on TEM grid. (a) SEM image of graphene transferred to TEM grid, scale bar 5 m. (b) Atomic scale ADF-STEM image showing the hexagonal lattice in the interior of a graphene grain. (c) ADF-STEM image showing intersection of two grains with a relative rotation of 27. (d) Same image with pentagons, heptagons and deformed hexagons formed along grain boundary highlighted (b, c and d have scale bars of 5 Å) Huang et al. (2011).

v.4 Measuring the electronic band structure

A wide variety of experimental techniques exist for measuring the band structure of materials. Due to its particular characteristics, graphene places severe limitations on the techniques available. Most band structure measurement techniques are highly sensitive to the bulk of a material rather than the surface. Since graphene is so thin, we need techniques that are very sensitive to surface layers.

Angle-resolved photoemission spectroscopy (ARPES) is the most popular technique for measuring the band structure of graphene. Photons of sufficient energy (20-100 eV) are shot at the surface of the material being probed. Each photon is energetic enough that if it interacts with an electron, it has a significant chance of transferring enough energy to launch the electron out of the material completely. The electron must be given enough energy to overcome the work function of the material.

The electron, once free of the material, will have a chance of hitting the ARPES detector. The detector is oriented so that it can measure one specific angle of electron emission. Note that there are two degrees of freedom in angle, typically called and . Using these two together, one can specify any direction. The detector is also able to accurately measure the energy of the outgoing electron. This means that ARPES will simultaneously measure the three variables , , and . The three components (x,y,z) of the scattered electron’s momentum prior to being struck by the photon can be found using the measured quantities. In this way the experimenter can map out the correspondence between energy and momentum within the material with high resolution.

Figure 21: Band structure of graphene on top of SiC. Vertical axis is the electron’s energy, and horizontal axis is its momentum. Note the key locations in momentum space with reference to Fig. 2, and also that is zone center, corresponding to zero momentum. The black line is a theoretical prediction based on the tight binding approximation. The fainter bands are believed to be due to interactions between the substrate and the graphene. Image adapted from Bostwick et al. (2007).
Figure 22: Substrate-induced band gap in single layer graphene on top of SiC. (a) Real space and momentum space structure of graphene. (b) Band structure of graphene taken along vertical black line near the K point in panel (a). The black lines are dispersion relations estimated from energy distribution curves. Figure from Zhou et al. (2007).

ARPES is capable of scanning to within about 5 Å of the surface when using electrons of 20-100 eV. This means that most of the signal will be from the first few atomic layers of the surface in question. This property makes ARPES particularly well-suited to measuring the band structure of incredibly thin materials such as graphene. On the other hand, this introduces some experimental difficulties because it means that the sample surface must be kept under ultra-high vacuum (UHV). Creating and measuring graphene without leaving UHV is a significant experimental challenge. Another commonly used method is to anneal the graphene by running a significant electrical current through it. The annealing process does a good job of cleaning the graphene so that it can be measured by techniques like ARPES even after it has been exposed to atmosphere.

ARPES measurements have been made on graphene in a wide variety of circumstances and with many different goals. For the purposes of this review, only a few studies will be mentioned out of this vast field. Fig. 21 shows the experimental band structure of graphene grown on top of SiC Bostwick et al. (2007). The intent of this study was to better understand the dynamics of charge carriers in graphene. Fig. 22 is from a different group that was also probing the behavior of graphene grown on top of SiC Zhou et al. (2007). This second experiment observed a notable band gap in their single-layer graphene samples. Additionally, they noticed that this gap shrank as the number of layers of graphene was increased from one to four. It is believed that the existence of the observed band gap is due to interactions with the substrate that cause the symmetry of graphene’s -bonds to be broken Zhou et al. (2008).

There are a number of variations of ARPES that differ only in the wavelength of the probing photons. Angle-resolved ultraviolet photoemission spectroscopy (ARUPS) has also been used to study graphene’s band structure Gierz et al. (2008). The primary reason why this technique is employed is convenience. In ARUPS, a laboratory-based ultraviolet wave source can be used to produce the probing photons. This is a less expensive and simpler setup than ARPES, which typically uses X-rays produced from a synchrotron. It is also worth noting that information about the electronic structure of graphene can be inferred from the results of other material techniques such as optical spectroscopy Mak et al. (2010).

In the previous sections we have discussed various ways to obtain and identify graphene. We now turn our attention to its physical properties, starting with electronic transport measurements.

Vi Electronic Transport and Field Effect

Owing to its unique band structure (see Sec. II), graphene exhibits novel transport effects such as ambipolar field effect and minimum conductivity which are absent in most conventional materials Wu et al. (2010). This unusual electronic behavior leads to exceptional transport properties in comparison to common semiconductors. This can be seen on Table 2 which compares two of the main electronic properties (carrier mobility and saturated velocity) of graphene with those of common bulk semiconductors and 2DEGs. In what follows, we will first describe the experimental methods that are commonly used to measure the ambipolar field effect. We will then discuss the transport properties that can be extracted from this experimental data. The effect of different scattering mechanisms on the carrier mobility and minimum conductivity will then be discussed in detail. Finally, other electrical properties relevant to transistor technology will be reviewed.

Property Si Ge GaAs 2DEG Graphene
at 300 K 1.1 0.67 1.43 3.3 0
(eV)
1.08 0.55 0.067 0.19 0
at 300 K 1350 3900 4600 1500-2000 2
(cmVs)
1 0.6 2 3 4
( cm/s )
Table 2: Comparison between the electronic properties of graphene and common bulk semiconductors. Energy band gap (), electron effective mass (), electron mobility () and electron saturation velocity () of graphene is compared to those of conventional semiconductors and AlGaN/GaN 2DEG Giannazzo et al. (2011).

vi.1 Measurement of the ambipolar field effect

Transport properties are typically measured with a graphene device similar to those shown in Fig. 23. To fabricate these devices, graphene (exfoliated, CVD, etc) is often deposited on an oxidized silicon wafer (SiO/Si). Later on we discuss other substrates that are sometimes used Ponomarenko et al. (2009); Dean et al. (2010b). Unless otherwise specified, all measurements reported here were made using exfoliated graphene flakes. Electrical contacts, usually made of gold, are then defined using a lithographic process or a stencil mask to avoid photoresist contamination. Electrodes are generally patterned in a 4-lead (Fig. 23a) or Hall bar (Fig. 23b) configuration. Lastly, the device can be cleaned by annealing at ultrahigh vacuum or in H/Ar gas, or by applying a large current density ( A/cm) through it to remove adsorbed contamination Moser et al. (2007).

Figure 23: Schematic representation of common electronic devices. (a) 4-lead Dorgan et al. (2010) and (b) Hall bar Novoselov et al. (2004).

With this graphene device in hand, one can tune the charge carrier density between holes and electrons by applying a gate voltage () between the (doped) silicon substrate and the graphene flake. The gate voltage induces a surface charge density where is the the permittivity of SiO, is the electron charge and is the thickness of the SiO layer. This charge density change shifts accordingly the Fermi level position () in the band structure (see the insets of Fig. 24a). At the Dirac point, should theoretically vanish Geim and Novoselov (2007), but as will be explained further on, thermally generated carriers () and electrostatic spatial inhomogeneity () limit the minimum charge density Dorgan et al. (2010). Fig. 24b, which shows the calculated carrier density as a function of gate voltage, clearly illustrates the fact that charge density is well controlled by the gate away from the Dirac point. The linear relation between and was verified experimentally Novoselov et al. (2004) in that region by measuring the Hall coefficient as a function of (see Sec. VII.1.2). Typically, charge density can be tuned from to cm by applying a gate voltage that moves 10 to 400 meV away from the Dirac point Giannazzo et al. (2011).

Figure 24: Ambipolar electric field effect in graphene. The insets of (a) show the changes in the position of the Fermi level as a function of gate voltage Geim and Novoselov (2007). (b) Calculated charge density vs. gate voltage at 300 K and 500 K Dorgan et al. (2010). Solid lines include contribution from , and . Dashed line shows only the contribution from the gating ().

In 2004, the ambipolar field effect corresponding to the change in resistivity (or conductivity ) that occurs when the charge density is modified by the gate voltage was observed and analyzed Novoselov et al. (2004). Experimentally, is measured using a standard 4-probe technique Van Der Pauw (1958) and is given by where and are respectively the width and the length of graphene, is the voltage across electrodes 2 and 3 (see Fig. 23) and is the current between contact 1 and 4. Note that because of the uncertainty on the aspect ratio , the error on the absolute magnitude of is usually around 10% Chen et al. (2008b, a). As Fig. 24b shows, resistivity rapidly increases as we remove charge carriers, reaching its maximum value at the Dirac point. From this curve one can extract the carrier mobility and the minimum conductivity . Other definitions for mobility are sometimes used such as the field effect mobility (where is the gate capacitance) Dean et al. (2010b) and the Hall mobility Novoselov et al. (2004). Note that in practice, the carrier mobility is only meaningful away from the Dirac point, where is accurately tuned by the gate voltage.

vi.2 Transport and scattering mechanisms

In contrast with the ideal, theoretical graphene, experimental graphene contains defects Chen et al. (2009) and impurities Chen et al. (2008a); Zhang et al. (2009a), interacts with the substrate Chen et al. (2008b), has edges and ripples Katsnelson and Geim (2008) and is affected by phonons Bolotin et al. (2008a). These perturbations alter the electronic properties of a perfect graphene sheet first by introducing spatial inhomogeneities in the carrier density and, second, by acting as scattering sources which reduce the electron mean free path Giannazzo et al. (2011). The former effect dominates when the Fermi level is close to the Dirac point and alters the minimum conductivity of graphene whereas the latter effect prevails away from the Dirac point and affects the carrier mobility. The impact of these perturbations has been subjected to intensive and ongoing investigation, on both the theoretical and experimental side, in order to determine the mechanisms that limit the mobility and the minimum conductivity. From a theoretical point of view, two transport regimes are often considered depending on the mean free path length and the graphene length . When , transport is said to be ballistic since carriers can travel at Fermi velocity () from one electrode to the other without scattering. In this regime, transport is described by the Landauer formalism Peres (2009) and the conductivity is expressed as:

(5)

where the sum is over all available transport modes of transmission probability . For ballistic transport mediated by evanescent modes, this theory predicts that at the Dirac point the minimum conductivity is:

(6)

On the other hand, when , carriers undergo elastic and inelastic collisions and transport enters the diffusive regime. This regime prevails when the carrier density is much larger than the impurity density . In that case, transport is often described by the semiclassical Boltzmann transport theory Das Sarma et al. (2011) and at very low temperature carrier mobility can be expressed in terms of the total relaxation time as:

(7)

This equation describes the diffusive motion of carriers scattering independently off various impurities. The relaxation time depends on the scattering mechanism dominating the carrier transport or a combination thereof. The scattering mechanisms mostly discussed in the literature include Coulomb scattering by charged impurities (long range scattering), short-range scattering (defects, adsorbates) and electron-phonon scattering. In the following, we provide a brief theoretical introduction of these scattering processes and relevant transport measurements.

Phonon scattering

Phonons can be considered an intrinsic scattering source since they limit the mobility at finite temperature even when there is no extrinsic scatterer. As explained in section Sec. III, the dispersion relation of graphene comprises six branches. Longitudinal acoustic (LA) phonons are known to have a higher electron-phonon scattering cross-section than those in the other branches Hwang and Das Sarma (2008). The scattering of electrons by LA phonons can be considered quasi-elastic since the phonon energies are negligible in comparison with , the Fermi energy of electrons.

In order to determine the effect of electron-phonon scattering on resistivity, one must consider two distinct transport regimes separated by a characteristic temperature called the Bloch-Grüneissen temperature, defined as Hwang and Das Sarma (2008):

(8)

where is the Boltzmann constant, is the sound velocity and is the Fermi wave vector with reference to the K point in the BZ.

(9)

where is the electron density in the conduction band Pisana et al. (2007). If one measures in units of cm we get K.

Consider first , the equipartition limit. In this case the Bose-Einstein distribution function for the phonons is , which leads to a linear -dependence of the scattering rate and hence the resistivity . In the or degenerate regime, on the other hand, where , one obtains at very low , Hwang and Das Sarma (2008) as shown in Fig. 25.

Figure 25: Electric resistivity of graphene at ultrahigh carrier densities. Resistivity over wide range of , showing the cross-over from the low regime to the high one Efetov and Kim (2010).

Coulomb scattering

Coulomb scattering stems from long-ranged variations in the electrostatic potential caused by the presence of charged impurities close to the graphene sheet. These impurities are often thought of as trapped ions in the underlying substrate and screened by the conduction electrons of graphene. Assuming random distribution of charged impurities with density and employing a semiclassical approach, it was predicted Adam et al. (2007) that the charged-impurity scattering is proportional to . With Eq. (7), the conductivity at high carrier density () is given by:

(10)

where is a dimensionless parameter related to the scattering strength. Considering the random phase approximation and the dielectric screening from the SiO substrate, it was predicted that .

Chen et al. investigated experimentally the effect of charged impurities on the carrier mobility and conductivity by doping a graphene flake with a controlled potassium flux in UHV Chen et al. (2008a). Fig. 26a shows the conductivity as a function of gate voltage for a pristine sample and three different doping concentrations. It can be clearly seen that the gate voltage of minimum conductivity becomes more negative with increasing doping. As it was previously shown Schedin et al. (2007a), this is because K atoms dope graphene with electrons (n-doping), which in effect moves the Fermi level up with respect to the Dirac point. From Fig. 26a, one can also see that becomes more linear and mobility decreases as the doping concentration increases which is in good agreement with Eq. (10). The dashed line in Fig. 26b shows that mobility scales linearly with when transport is limited by charged-impurity scattering. In these measurements in Eq. (10) was obtained Chen et al. (2008a); Tan et al. (2007).

Figure 26: Effect of charge impurities and defects on transport properties of graphene. (a) The conductivity () vs. gate voltage for a pristine sample (black curve) and three different potassium doping concentrations at 20 K in UHV. Lines represent empirical fits Chen et al. (2008a). (b) Inverse of mobility () as a function of ion dosage for different samples and irradiated ions (Ne and He). The dashed line corresponds to the behavior for the same concentration of potassium doping Chen et al. (2009).

Short-range scattering

Finally, short-range defects such as vacancies and cracks in graphene flakes are predicted to produce midgap states in graphene Stauber et al. (2007). Vacancies can be modeled as a deep circular potential well of radius and this strong disorder gives rise to a conductivity which is roughly linear in :

(11)

where is the defect density. This equation mimics the one for charged impurities (Eq. (10)), with a slightly logarithmic dependence of the conductivity on the charge carrier density. Defect scattering was experimentally studied by irradiating a graphene flake with 500 eV He and Ne ions in UHV Chen et al. (2009). The resulting conductivity was also demonstrated to be approximately linear with charge density, with mobility inversely proportional to the ion (or defect) dose . As shown in Fig. 26b, the mobility decrease was found to be 4 times larger than the same concentration of charged impurities. From the linear fits of Fig. 26 with Eq. (11), the impurity radius was found to be Å which is a reasonable value for a single-carbon vacancy.

vi.3 Mobility

Graphene on SiO

In a graphene electronic device, all of the scattering mechanisms mentioned above come into play. From a technological point of view, determining the exact nature of the scattering that limits the mobility is essential in order to develop high-speed electronic devices. To do so, one must also take into account the effect of the underlying substrate on the electronic transport. Chen et al. performed a general study of scattering mechanisms in graphene on SiO by measuring the temperature dependence of the ambipolar effect Chen et al. (2008b). The measurements were fitted using three terms: . Each term was associated with a certain scattering mechanism. As shown in Fig. 27a, the first two terms were determined with a linear fit at low . The y-intercepts correspond to and were found to scale linearly with . According to Eq. (10) and Eq. (11), this behavior can be associated with charged impurities and defects. was determined by the slope of the linear fit in Fig. 27 and appears to be independent of the charge density. These results agree very well with predictions for longitudinal acoustic (LA) phonon scattering in the regime where . Finally, the third term , was found to have a strong temperature dependence. Such behavior was explained by considering the surface polar phonons (SPP) of the SiO substrate which produce an electrical field that couples to electrons in graphene. Theoretical expressions for SPP-limited resistivity showed good agreement with the data of Fig. 27a at high temperature.

Fig. 27b presents the temperature-dependent mobility and the theoretical limits of the three scattering mechanisms (LA phonons, SPP and impurities/defects). One can see that at room temperature, SPP and impurities/defects are by far the two dominant scattering mechanisms for graphene on SiO. At low and room temperature, mobility is mainly limited by the former type of scattering. Typically, mobility of graphene on SiO ranges from 10000 to 15000 cmVs. It was suggested that the scattering impurities are trapped charges in the underlying SiO substrate, but this remains under debate Ponomarenko et al. (2009).

Figure 27: Temperature dependence of resistivity and mobility in graphene on SiO. (a) Temperature-dependent resistivity of graphene on SiO for different gate voltages, samples and runs. Dashed lines are linear fits at low temperature. (b)Temperature-dependent mobility in graphene on SiO for two samples at cm. Mobility limits due to LA phonons (dark red solid line), substrate surface phonons (green dashed line), and impurities/defects (red and blue dashed lines) are presented. Matthiessen’s rule was used to obtain the net mobility for each sample (red and blue solid lines) Chen et al. (2008b).

Suspended graphene

Removing the substrate or using one which is free of trapped charge are two possible ways to improve the carrier mobility. The former approach was used by Du et al. (2008), in which the authors fabricated a suspended graphene device by chemically etching the underlying SiO (see the inset of Fig. 28a). From this device, they obtained mobilities as high as 200,000 cmVs for charge density below 5 cm. Fig. 28a shows the charge density dependence of mobility for suspended and non-suspended (on SiO) graphene devices at 100 K. The arrows indicate the minimum charge density that is accurately controlled by the gate voltage. From this figure, one can see that mobility in suspended graphene approaches the ballistic model prediction. Furthermore, the mobility deterioration caused by adsorbed impurities on the surface of suspended graphene has been demonstrated Bolotin et al. (2008a, b). As Fig. 28b shows, mobility and mean free path clearly increase after current-induced cleaning and high-temperature annealing of the device. At large carrier density ( cm) and temperatures above 50 K, resistivity of clean suspended graphene was found to be linear with temperature, suggesting LA phonon scattering. At low temperature ( 5 K), mobility as high as 170,000 cmVs was obtained and the mean free path reached the device dimension. In these conditions, conductivity is well described by the ballistic model as shown in Fig. 28b.

Figure 28: Conductance and mobility in suspended graphene as a function of charge density. (a) Mobility vs. charge density for suspended (red line) and non-suspended (black line) graphene at = 100 K. The blue line represents the ballistic model prediction. Inset: schematic representation of the suspended graphene device Du et al. (2008). (b) Conductance vs. gate voltage (at = 40 K) for a suspended graphene device before (blue line) and after (red line) annealing and current-induced cleaning. The red dotted line was calculated using a ballistic model. Inset: AFM image of the suspended graphene device Bolotin et al. (2008a).

Other substrates

Although suspended graphene shows impressive transport properties, this geometry imposes evident constraints on the device architecture. To overcome this problem, boron nitride (BN) was proposed as a substrate Dean et al. (2010b). Compared to SiO, BN has an atomically smooth surface, is relatively free of charged impurities, has a lattice constant similar to that of graphene and high surface phonon frequencies. All these advantages result in a mobility about three times higher than that of graphene on SiO. However, graphene/BN devices are troublesome to fabricate and are thus not ideal for industrial applications. Currently, large area graphene synthesized by CVD or thermal segregation is the most promising material for technological applications since it can be produced on a wafer scale Li et al. (2009); Kim et al. (2009). Depending on the technique, large scale graphene (on SiO) typically shows mobilities lower than 5000 cmVs Avouris (2010). The scattering mechanisms responsible for this reduced mobility are still under investigation. Preliminary studies suggested that grain boundaries Yazyev and Louie (2010), doping defects Wei et al. (2009) and ineffective gate voltage control Cao et al. (2010) might limit the mobility. In addition to mobility, a common way to characterize the quality of large scale graphene is to measure its sheet resistance (resistivity at ). This property varies with the synthesizing technique used and can be as low as 125 Bae et al. (2010). It was also shown that sheet resistance increases with decreasing temperature Park et al. (2010). This observation contrasts with the metallic behavior () of pristine graphene flakes but agrees with the insulating behavior of irradiated graphene (with defects) Chen et al. (2009).

Substrate Production Ref.
technique ( cmVs) ()
SiO/Si Exfoliation 10-15 4 a
Boron nitride Exfoliation 25-140 6 b
Suspended Exfoliation 120-200 1.7/ c
SiC Thermal-SiC 1-5 - d
SiO/Si Ni-CVD 2-5 - e
SiO/Si Cu-CVD 1-16 - f
Table 3: Mobility range () and minimum conductivity () of graphene produced by different techniques and deposited on different substrates. a: Novoselov et al. (2004), b: Dean et al. (2010b), c: Bolotin et al. (2008b), d: Emtsev et al. (2009), e: Kim et al. (2009), f: Li et al. (2010b)

vi.4 Minimum conductivity

The presence of disorder in graphene (ripples, defects, impurities, etc.) produces fluctuations in its electrostatic potential. We mentioned above that these perturbations are significant at the Dirac point where the screening of the potential fluctuations is weak due to the low charge density. These fluctuations in the charge density can be thought of as electron-hole puddles and have been observed experimentally using scanning probe methods Zhang et al. (2009a); Martin et al. (2008) on graphene/SiO samples. Experimentally, these disordered graphene samples have a minimum conductivity about times larger than that predicted for ballistic transport given in Eq. (6). This discrepancy between theory and experiments was observed in 2004 Novoselov et al. (2004) and was known as “the mystery of the missing pi.” The effect of doping on the minimum conductivity was also investigated in a study of charged-impurity scattering Chen et al. (2008a). It was found that decreases on initial doping and reaches a minimum near only for non-zero charged impurities. This suggests that charged impurities trapped in the SiO substrate, located at the graphene/SiO interface or on the graphene surface are responsible for the minimum conductivity obtained experimentally.

To verify this conclusion, the minimum conductivity of clean suspended graphene was measured as a function of temperature Du et al. (2008). As Fig. 29a shows, minimum conductivity decreases with temperature and approaches the ballistic prediction down to a factor of 1.7. However, the linear temperature dependence was not expected. Another group also investigated the near-ballistic regime by measuring the minimum conductivity on samples with different aspect ratios () and surface areas at low temperature (1.5 K) Miao et al. (2007). For large-area samples (3 m), the minimum conductivity is mostly independent of the aspect ratio (inset of Fig. 29b). On the other hand, small-area devices (0.2 m) yield results that are clearly dependent on the aspect ratio (Fig. 29b). Devices with wide electrodes and short channels (large ) approach the theoretical minimum conductivity in the ballistic regime. Table 3 compares the carrier mobility and minimum conductivity of graphene produced by different techniques and on different substrates.

Figure 29: Minimum conductivity of graphene. (a) Minimum conductivity vs. temperature for a suspended graphene device. The upper dashed line represents the limit for graphene with charged impurities. The lower dashed line corresponds to the theoretical value in the ballistic regime Du et al. (2008). (b) Minimum conductivity vs. aspect ratio for small and large (inset) graphene/SiO devices Miao et al. (2007).

vi.5 Other transport properties

Concerning electronic applications, graphene has attracted considerable attention due to its high mobility. However, other transport properties must be taken into account in order to develop graphene-based technologies successfully. Among those properties, saturation velocity is particularly relevant for field-effect transistor (FET) applications. In modern FETs, short channel lengths result in high electrical fields Schwierz (2010) of around 100 kV/cm. In such conditions, the carriers acquire enough kinetic energy to excite the optical phonon modes of graphene Avouris (2010). As a consequence, carrier velocity saturates, reducing the relevance of mobility to device performance. The optical phonons of graphene have higher energy (160 meV) than those of common semiconductors such as Si (55 meV) Dorgan et al. (2010). Consequently, the intrinsic saturation velocity is higher in graphene than in conventional semiconductors.

Numerical values of saturation velocity are presented in Table 2, and Fig. 30 shows the experimental and theoretical dependence of saturation velocity on charge density for graphene/SiO. A simple phonon emission model Meric et al. (2011) predicts that saturation velocity is proportional to . In Fig. 30, the upper and lower theoretical curves correspond to saturation velocities limited by optical phonons of graphene and SiO, respectively. The experimental results suggest that both kinds of phonons play a role in limiting but that substrate phonons of SiO are dominant for this device Dorgan et al. (2010). Some studies also point out that current never reaches a complete saturation in some graphene devices. This incomplete saturation might be due to a competition between disorder and optical phonon scattering Barreiro et al. (2009), or the formation of a “pinch-off” region Meric et al. (2011). Mechanisms limiting the saturation velocity are under active investigation.

Figure 30: Electron saturation velocity of graphene on SiO. Electron saturation velocity vs. charge density for an electric field of 2 V/m at = 80 K and 300 K. The upper and lower dashed curves correspond to theoretical saturation velocities limited by optical phonons of graphene and SiO, respectively Dorgan et al. (2010).

Graphene has excellent transport properties, but to take advantage of them, carriers must be injected and collected through metal contacts. These electrical contacts produce energy barriers that limit the charge transfer at the graphene/metal junction Avouris (2010). This limitation results in a contact resistance which can be measured and should be reduced to create high-performance graphene devices. One way to determine the contact resistance is to fabricate variable channel length devices and to extrapolate the resistance to zero channel length. It was shown that contact resistance is temperature and gate voltage dependent, and that it varies from about 100 m to a few km Xia et al. (2011). Finally, it is worth pointing out that graphene can sustain current densities greater than A/cm, which is 100 times higher than those supported by copper Moser et al. (2007). Graphene can thus be used as interconnects in integrated circuits.

Since Geim’s and Novoselov’s seminal work Novoselov et al. (2004), graphene’s unique electronic properties have attracted massive interest and created an explosion of scientific activity. Much ink has been spilled about the ambipolar field effect, the ultra-high mobility of graphene and the limiting scattering mechanisms, as well as the minimum conductivity. In this section, we reported some of the main experimental studies on these ever growing subjects. In the next section, we turn our attention to the related but more specific topics of magnetotransport and quantum Hall effect in graphene.

Vii Magnetoresistance and Quantum Hall Effect

The first demonstration, in the early 1980s, of the existence of a so-called quantum Hall effect (QHE) of the two-dimensional electron gas (2DEG) at the interfaces of semiconductor heterostructures sparked great interest in the experimental study of the properties of low-dimensional systems Klitzing et al. (1980). Thus it is perhaps no surprise that when the one atom thick, two-dimensional lattice of graphene could finally be isolated in 2004, physicists immediately tried to characterize its magnetotransport properties, which are still an important field of research today. The purpose of this section is to recapitulate the present state of experimental knowledge about magnetotransport measurements in graphene. Firstly, we will describe the general experimental procedures that are necessary to carry such measurements. Secondly, we will briefly review the fundamental physics relevant to localization phenomena and to the QHE. Finally, we will show how these techniques provide information on the electronic properties of graphene through weak (anti-)localization, the integer QHE and the fractional quantum Hall effect (FQHE).

vii.1 Experimental procedures

We first describe the basic experimental procedures required to perform low temperature magnetotransport measurements in graphene.

Setup

The setup consists of a superconducting electromagnet (typically Nb based type-II superconductor, e.g. NbTi or NbSn) immersed in a liquid helium cryostat. It is represented in Fig. 31. Thus the sample can be refrigerated to temperatures of 4 K and below. Lower temperatures are achieved by reducing the vapor pressure of helium by pumping. Furthermore, in liquid helium the fields generated in the electromagnet can reach high maximal values typically ranging between 6 and 14 T.

Figure 31: Setup for magnetotransport measurements. The cryostat is cooled to = 4 K in two steps using liquid nitrogen and liquid helium. A solenoidal superconducting magnet creates a magnetic field perpendicular to the plane of the graphene. The helium bath can be pumped on to yield even lower temperatures.

The sample is placed at the bottom of the cryostat in such a way that the field is perpendicular to the plane of the graphene. The device is usually conceived for 4-terminal measurements, as illustrated in Fig. 32. A small current is driven through the end terminals and voltage measurements are carried out between different combinations of the lateral terminals. This avoids the measurement of the contact resistance at the current injection points. The longitudinal resistivity is the ratio of the longitudinal voltage to the current normalized by the aspect ratio. The Hall resistivity and the Hall Resistance are identical and are defined as the ratio of the transverse voltage to the current. The Hall conductivity is related to the resistivity by the inverse tensor relation .

Figure 32: Magnetotransport measurements. Top: Graphene is etched into a 4-terminal Hall bar. Gold electrodes are used to perform electrical measurements. The width of the central wire is 0.2 m. Bottom: Graphene is placed on a thin SiO layer (285-300 nm), which is itself on a doped Si wafer. A gate voltage allows the carrier density to be tuned. Figure from Novoselov et al. (2005).

Carrier density tuning

In graphene, the carrier density can be tuned using the electric field effect, as discussed in Sec. VI. A graphene sheet is placed on a SiO substrate about 330 nm thick, which in turn lies on a doped Si wafer. A gate voltage is applied on the Si to inject electron carriers in the graphene or withdraw them. In the latter case, holes carry the current. Thus graphene can be studied above and below the neutral point corresponding to the Fermi energy and for which in principle .

The calibration is done using the conventional Hall effect. In a (low) applied magnetic field, the Hall resistance is related to the current by , where is the Hall constant and is the charge of the carriers. A measurement of for different gate voltages allows one to establish a linear map between the carrier density and for both holes and electrons. An example of calibration is shown on Fig. 33.

Figure 33: Measurement of the conventional Hall effect for different gate voltages gives . The Hall constant is related to the carrier density by . Figure from Novoselov et al. (2005).

Observation of the quantum Hall effect

The observation of the QHE is conditional on many factors.

  1. High quality samples must be used in order to maximize the momentum scattering time of the electrons.

  2. High magnetic fields are required in order for the cyclotron period of the carriers to be much shorter than the scattering time . Equivalently, this means that the cyclotron frequency of an electron must be much higher than the broadening of the carrier energy levels (the Landau levels, see Sec. VII.2): .

  3. Low temperatures are needed in order for the thermal energy to be much less than the spacing of the Landau levels: .

In graphene, the integer QHE can be observed at liquid helium temperature for fields ranging from about 1 T and higher. For high magnetic fields the effect can be observed for much higher temperatures Novoselov et al. (2005). As an extreme example it has been shown that for a gigantic field of = 45 T, the QHE remains detectable at room temperature Novoselov et al. (2007). The observation of the FQHE requires much more extreme conditions. The FQHE can be observed in graphene for fields above 2 T for temperatures lower than 1 K and it can subsist up to = 20 K at B = 12 T, provided that the sample is ultraclean and that the graphene is suspended to suppress scattering. This yields very low charge inhomogeneity of order cm Du et al. (2009). In such small devices the contacts of the 4-terminal measurement prevent the appearance of the QHE and one must perform a two-terminal measurement instead Du et al. (2009). More recently, FQHE measurements in graphene have been performed on graphene lying on a hexagonal boron-nitride substrate at fields of the order of B = 35 T and temperatures of the order of = 0.3 K Dean et al. (2010a). Such fields cannot be reached with the simple setup illustrated in Fig. 31. Instead, these measurements are performed in High Magnetic Field Laboratories such as the National High Magnetic Field Laboratory at Florida State University or Grenoble High Magnetic Field Laboratory in France.

vii.2 Theoretical background

Chiral electrons and pseudospin

Imagine a gas of electrons in the x-y plane confined by some potential well in the z-direction. We assume that this well is deep and narrow enough, and hence that the energy levels of the well are separated enough, so that the electrons are forbidden to access its excited states through any excitations. Then we say that the electron motion is two-dimensional.

At low carrier density , the dispersion of graphene is linear around a Dirac point (call it K) so that we cannot define an effective mass such that as in semiconductor samples. Instead it can be shown that in the continuum limit (i.e. taking the position of the electron on the lattice to be a continuous variable), the motion of the electron around a single Dirac point should be described by the Dirac-Weyl equations for massless fermions discussed in Sec. II using the Hamiltonian of Eq. (3) Castro Neto et al. (2009):

(12)

where is a spinor. It can be shown that the state acquires an extra phase (a Berry phase) of on a closed trajectory Castro Neto et al. (2009). This is key to understanding the weak localization measurements in graphene discussed below.

Since in graphene there are two inequivalent Dirac valleys K and K’ with the same dispersion, there is an equation similar to Eq. (12) for K’. This leads to an extra valley degeneracy for the state of the electron and to chiral electrons: the state in valley K has a different helicity than a state in valley K’. We call the doublet (K, K’) the valley pseudospin. We conclude that the energy levels should be 4 times degenerate: two times for the pseudospin and two times for the actual spin of the electron.

In a graphene bilayer, two sheets of graphene are stacked onto each other (in the natural Bernal stacking of graphite) Castro Neto et al. (2009). In such a system, the carriers can hop between the layers and the dispersion relation is not linear. Instead, the valence and conduction bands consist of two parabolic bands of the same curvature touching at the neutral point Castro Neto et al. (2009). The theory of Eq. (12) can then be extended to show that the carriers should be massive Dirac fermions with a Berry phase of Novoselov et al. (2006).

Landau levels

Eq. (12) can be solved in the presence of a magnetic field and the result predicts a sequence of energy levels called the Landau levels (LL) Ezawa (2008); Castro Neto et al. (2009); Gusynin and Sharapov (2005); Apalkov and Chakraborty (2006):

(13)

The energy is measured with respect to the Dirac point (i.e. the energy at the Dirac point is zero). Each such level should also have the 4-fold degeneracy discussed above. Note that in a semiconductor 2DEG, the LLs would be equally separated by an amount where is the cyclotron frequency and the cyclotron mass Datta (1997), which from Eq. (13) is clearly not the case for graphene. We sketch a figure of the splitting of the bands in LLs for graphene in Fig. 34.

Figure 34: Schematic representation of the formation of Landau levels in graphene (top) and bilayer graphene (bottom). In both cases the levels are not equally spaced and the neutral point has an associated LL. In graphene each level is 4 times degenerate because of the spin and valley pseudospin. In bilayer graphene the lowest LL is 8 times degenerate because of the extra layer degree of freedom.

For bilayer graphene, we find a different expression:

(14)

It can be shown that the extra layer degree of freedom should give rise to an extra degeneracy for the level and only for that level. Thus, the level is 8 times degenerate while the others keep the 4-fold degeneracy of graphene. The splitting of the bands in LLs for bilayer graphene is illustrated schematically in Fig. 34.

vii.3 Measurements of weak localization and antilocalization in graphene

Quantum interferences

Quantum interferences have long been known to affect transport measurements at low temperatures. In particular, two-dimensional electron systems in the presence of a (low) magnetic field show variations of conductance with respect to the value in the absence of a field. This effect is known as weak (anti-)localization.

Consider an electron hitting an impurity and going back in the direction from which it came. In a classical picture, this could happen in a number of ways. In particular, the electron could go around the impurity clockwise or counter-clockwise (see Fig. 36). However, since electrons are quantum mechanical waves, we must add each of the possible paths and make them interfere together to get the net probability of the electron to have backscattered. Electrons in conventional 2DEGs, in the absence of spin-orbit interaction and scattering by magnetic impurities, gain the same phase on both trajectories and interfere constructively, leading to an increased probability of backscattering compared to the value expected from the Drude model. We say that the electrons exhibit weak localization (WL). If a magnetic field is applied, an additional (Aharonov-Bohm) phase is added between the two paths, destructive interference occurs and the conductance increases (positive magnetoconductance) Tikhonenko et al. (2009).

In graphene, however, it can be shown that because of the additional Berry phase of of the wave function, the two paths end up in opposite phase in the absence of spin-orbit interaction and scattering within or between Dirac valleys. Therefore application of a magnetic field can only restore the constructive interference and decrease the conductance (negative magnetoconductance) Morozov et al. (2006); Tikhonenko et al. (2009). We say that the electrons exhibit weak anti-localization (WAL). Note that conventional electrons which couple strongly through spin-orbit interactions can also exhibit WAL. However, this effect is not expected in graphene because of the small mass of the carbon atom Tikhonenko et al. (2009).

Measurements

The early quantum interference measurements on graphene showed no WAL and a strongly suppressed WL. This was attributed to suppression of interference within one Dirac valley due to ripples and large defects Morozov et al. (2006). More recently, it was demonstrated that both WAL and WL could be observed in graphene under the proper conditions Tikhonenko et al. (2009). The behavior of the magnetoconductance is shown on Fig. 35.

Figure 35: Weak localization and anti-localization in graphene. (a) The sample is studied for various carrier densities (here I, II and III). The QHE shows that the sample is indeed two-dimensional (see Sec. VII.4). The sample is annealed at 400 K for 2 hours to increase its homogeneity. (b), (c) and (d) Variation of the magnetoresistance behavior as and are changed. A transition occurs between WL and WAL and negative magnetoconductance (WAL) is observed for low and high due to the competition between valley scattering and dephasing processes (see text). Figure from Tikhonenko et al. (2009).

The data shows WAL (negative magnetoconductance) for both decreasing carrier density and increasing temperature. The increasing temperature reduces the dephasing time of the phase of the electron due to thermal fluctuations, while decreasing carrier density increases the intervalley and intravalley elastic scattering times and defined in the theory of reference McCann et al. (2006). The latter are roughly temperature independent. Thus it is observed that the dephasing time is not the only parameter controlling weak localization behavior in graphene. Instead, one must consider the ratios and . When they are small, anti-localization occurs while when they are big, localization occurs. Note that there exist combinations of these parameters for which the correction to the magnetoconductance is suppressed. This is shown in Fig. 36.

Figure 36: Phase diagram for weak (anti-)localization in graphene. (a) A schematic representation of interference between electron paths. For graphene it is expected to be naturally destructive and a magnetic field is expected to partly restore constructive interference (see text). (b) A phase diagram for weak localization behavior in graphene. It is seen that the ratios of the dephasing time to the two valley scattering times determine the magnetoresistance behavior. Figure from Tikhonenko et al. (2009).

In graphene, the quantum corrections to the conductance survive at much higher temperatures than for 2DEG semiconductor structures because the electron-phonon scattering is expected to be weak in the system. Indeed, weak (anti-)localization can be observed up to much high temperatures in graphene, in particular in large scale CVD grown graphene Whiteway et al. (2010). It turns out that electron-electron scattering could be responsible for the disappearance of magnetoconductance at high temperatures Tikhonenko et al. (2009).

vii.4 Measurements of the quantum Hall effect in graphene

Here we review the principal features of the QHE in graphene and compare them to the conventional results. We first discuss the general features of the QHE.

Shubnikov-de Haas oscillations

The existence of LLs implies the appearance of the so-called Shubnikov-de Haas oscillations (SdHO) in the QH system. The SdHO consist of a strong modulation of the longitudinal resistivity as a function of the carrier density . The minima of these peaks correspond to a completely vanishing longitudinal resistivity. This occurs when the chemical potentials of both leads sit between the LLs, as illustrated in Fig. 37. Then all the carrying states of the LLs are completely filled so that the electrons cannot carry current in a LL. The only states that carry current are the edge states at the Fermi level (see Fig. 37). Furthermore, the magnetic field is such that each edge carries current in different directions. Hence, the only way for backscattering to occur is for an electron to go from one edge to another, which is not feasible. Thus scattering is suppressed and no voltage can build up, hence vanishes. As the Fermi level is raised, a LL may sit at the chemical potential of the leads, which allows for some backscattering, hence producing a maximum in resistivity.

Figure 37: Quantization of Hall conductance in graphene. (a) The density of states is peaked at the LLs and broadened by the presence of disorder. (b) Edge states topologically separated by the magnetic field conduct the current between the leads at fixed potential. In the absence of scattering, the quantum of conductance carried by each channel translates into the quantization of the Hall conductivity (see text).

Conductance quantization

Associated with the SdHO is the celebrated phenomenon of Hall conductance quantization. It is observed that at the values of for which the zeros in occur, the Hall resistivity is constant and forms a plateau that is a multiple of . The integer is said to be the filling factor of the QH state and represents the total number of levels below the Fermi energy. In fact, there are as many conducting edge channels as there are filled levels, and one can show that each such channel should contribute one quantum of conductance Datta (1997). In the absence of backscattering, one expects each edge to be at the same potential as the lead that provides its carriers: this implies that the transverse Hall voltage is the same as the potential difference between the leads. Therefore the Hall conductance should be the same as the sum of the conductance of each of these channels. Thus:

(15)

This phenomenon is known as the quantum Hall effect. The reader is referred to Datta (1997) for a thorough theoretical treatment of ballistic transport in micron scale samples.

Integer quantum Hall effect

Monolayer graphene For the plateaus of the conventional QHE in semiconductor heterostructures, is a positive integer and a finite number of charge carriers is required to occupy the lowest LL. At low magnetic field, only the even filling factors appear because the LLs can be filled with both spin up and spin down electrons. We have the sequence:

(16)

Therefore the sequence of ’s provide information on the degeneracies of the electron states or equivalently, on their symmetries. If the magnetic field is increased, the degeneracies can be completely lifted via Zeeman interaction of the spin with the magnetic field, revealing the entire sequence of filling factors.

Figure 38: Typical measurement of the integer quantum Hall effect. The Hall resistivity exhibits plateaus quantized in exact multiples of . Image created by D.R. Leadley, Warwick University (1997).

As a two-dimensional system, graphene is expected to exhibit the QHE. It was observed in 2005 by two research groups Novoselov et al. (2005); Zhang et al. (2005) and showed features that were characteristic of the linear dispersion of graphene. A measurement of the conventional QHE in graphene is shown in Fig. 39.

Figure 39: Integer quantum Hall effect in graphene. The carrier density can be tuned across the neutral point . The absence of a plateau at indicates the presence of a LL at the neutral point. The inset shows bilayer graphene, discussed in Sec. VII.4.3. Figure from Novoselov et al. (2005).

The sequence of observed plateaus is very different from that of the conventional QHE. The precise sequence is:

(17)

We notice that the carrier density (i.e. the Fermi level) of graphene can be tuned through both the valence and conduction band of graphene, resulting in negative for holes and positive for electrons. From the sequence we immediately see that each plateau contributes to the conductance, instead of . This is indicative of the expected extra valley degeneracy of the states, in addition to the spin degeneracy. Another interesting feature of this sequence is that there is no plateau at zero carrier density. Thus there is a LL at the neutral point with degenerate holes and electrons as seen from Eq. (13). The structure of electron and hole states is also completely symmetric.

Fig. 40 shows a complete lift of the degeneracy of the LL at ultrahigh magnetic fields, confirming explicitly the 4-fold degeneracy of the level Zhang et al. (2006). The Zeeman coupling in graphene is too weak to cause the lifting of the spin degeneracy and does not explain the breaking of the valley pseudospin symmetry Zhang et al. (2005, 2006). Instead, the lifting of the degeneracies is suspected to be caused by enhancement of Coulomb and exchange interactions in the sample. This is due to the fact that as the field is increased the cyclotron orbits become smaller and the carriers come closer to each other Dean et al. (2010a); Ezawa (2008). All integer filling fractions can be observed for the cleanest samples Dean et al. (2010a).

Figure 40: Complete lifting of the degeneracy of the lowest LL with fields from 9 T (circle) up to 45 T (star). The expected 4-fold degeneracy is shown explicitly by the presence of plateaus at and . The plateaus at show partial degeneracy lifting of the levels . All degeneracies can be lifted in clean samples, see Fig. 44. Figure from Zhang et al. (2006).

Just like the conductance quantization, the SdHO oscillations show unusual behavior. We still observe their minima at the positions of the plateaus, as seen in Fig. 39.

Figure 41: Determination of the dispersion of graphene through SdHO oscillations. (a) The SdHO decay more rapidly with temperature as the carrier density is increased. The data can be fitted to yield the cyclotron mass . (b) The cyclotron mass varies with carrier density as , implying the linear dispersion of graphene. Figure from Novoselov et al. (2005).

However, their amplitude decays more rapidly with temperature as the density is increased. Theoretical studies have shown that for a given , the amplitude of the SdHO should follow Castro Neto et al. (2009):

(18)

where is the cyclotron mass of the electrons at the Fermi level Novoselov et al. (2005).

A fit to the data shows that the rapid decay of the SdHO implies that , as shown in Fig. 41. Unlike in semiconductor devices, the cyclotron mass in graphene varies with density. It is possible to show that this dependence of on implies the linear dispersion of graphene and thus the form of the LLs given in Eq. (13) Novoselov et al. (2005).

Bilayer graphene The quantum Hall effect can also be observed in multilayer graphene. The inset of Fig. 39 as well as Fig. 42 shows measurements carried out on the graphene bilayer Novoselov et al. (2006). The observed sequence is as follows:

(19)
Figure 42: Quantum Hall effect in bilayer graphene. As in the case of the monolayer, the quantum Hall effect in graphene bilayer shows no plateau at , indicating the presence of a LL at zero carrier density and the relativistic behavior of the bilayer. However, contrarily to the monolayer, the level is 8-fold degenerate as the height of the step indicates. This is consistent with the theoretical prediction of a “parabolic Dirac point” at , i.e. massive Dirac fermions. Figure from Novoselov et al. (2006).

The absence of a plateau at is a common characteristic of the QHE in graphene and bilayer graphene. Once again it indicates the presence of a LL at zero energy. However, this level is eight times degenerate as can be seen from the jump in the Hall conductivity. The other levels keep the 4-fold degeneracy of graphene. This is what we expect from the additional layer degree of freedom of the electron for a given spin and valley Novoselov et al. (2006); Ezawa (2008). The degeneracy of the lowest LL can be completely lifted by many-body interactions in ultra-high magnetic fields Zhao et al. (2010), as shown in Fig. 43.

Figure 43: The degeneracy of the lowest LL of bilayer graphene is lifted by magnetic fields ranging from 9 T to 35 T. We clearly distinguish for levels above the neutral point and 4 others start to be visible below the neutral points, accounting for the expected 8-fold degeneracy of this level. Figure from Zhao et al. (2010).

Fractional quantum Hall effect

The extreme two-dimensional confinement in graphene is expected to enhance many-body interactions between electrons. Since 1982, we know that these interactions can manifest through the so called fractional quantum Hall effect (FQHE), consisting of the appearance of Hall plateaus at fractional (rational) values of the filling factor Tsui et al. (1982). The efforts to observe the FQHE in graphene were without success for a long time because of the existence of a competing insulating state induced by disorder near the neutral point . The improvements in the sample fabrication made its observation possible in recent years.

FQH states can be understood as the realization of the integer QHE for weakly interacting quasiparticles called composite fermions. In a heuristic picture, an even number of magnetic flux vortices bind with an electron to form an object with reduced effective charge that is a fraction of the elementary charge Jain (1989); Dean et al. (2010a). In graphene, the multiple components of the electron wave function are expected to give rise to new interacting ground states with various spatial arrangements of pseudospin (textures) that depend on the symmetries of the states. These special states are theoretically predicted to occur at filling factors that are multiples of , where is an odd integer Ezawa (2008). The FQHE allows for a characterization of these symmetries and puts hard constraints on theoretical models. Different symmetry scenarios are presented in Fig. 45.

Figure 44: The FQHE in graphene. Multiple plateaus at rational filling factors show that electron interactions give rise to new states. Figure from Dean et al. (2010a).
Figure 45: Schematic SdHO structures corresponding to different possible symmetries of the lowest LL in graphene. (a) Total symmetry breaking. (b) Either the spin or the valley symmetry is broken. (c) Full symmetry of the spin and valley degrees of freedom. The latter is the best fit to the observations. Figure from Dean et al. (2010a).

Fig. 44 shows a clean and recent 6-terminal measurement of the FQHE in graphene made on a hexagonal boron-nitride substrate at ultrahigh fields Dean et al. (2010a). The filling factors appear in the following sequence:

(20)

The measurements for hole carriers were not performed for . The filling factors are all multiples of , although is missing. Furthermore, a peak in the SdHO may indicate the emergence of a plateau at . The effect is much more robust under increase in temperature than its counterpart in semiconducting devices, revealing the suspected enhancement of electron-electron interactions in graphene compared to those occurring in 2DEGs Dean et al. (2010a); Du et al. (2009). The absence and the presence of is consistent with a global symmetry of the FQH state in the lowest LL Dean et al. (2010a).

Previous measurements have been performed on very small (2 m) 2-terminal ultraclean samples of suspended graphene Du et al. (2009); Bolotin et al. (2009). For a discussion of the properties of suspended graphene see Sec. VI.3.2. Their results are presented in Fig. 46. They also obtain the FQHE in the lowest LL at and . While 2-terminal measurements make the results less precise and harder to interpret (for example, the contact resistance makes it very hard to extract information from the longitudinal resistance data and may produce unusual features at ), they possess the advantage of being realizable with lower field magnets in standard physics laboratories.

Figure 46: High quality suspended graphene samples used by Bolotin et al. (left) and Du et al. (right). The contacts are less than 2 m apart in both cases, making a 4-terminal measurement impossible. However, the FQHE can be observed clearly at relatively low fields ranging from 5 T to 12 T. Figures from Bolotin et al. (2009); Du et al. (2009).

The observation of the anomalous quantum Hall effect, including its fractional daughter are one of the brightest highlights, which propelled graphene as one of the most studied materials in recent years. We now move away from magneto-transport to discuss the mechanical properties in the next section, which are very important for future applications in nanodevices.

Viii Mechanical Properties

Microelectromechanical systems (MEMS) have been deployed to perform common tasks such as opening and closing valves, regulating electric current, or turning mirrors. This is realized by employing microscopic machines such as beams, cantilevers, gears and membranes. These MEMS are found in many commercially available products from accelerometers found in air bag deployment systems, gyroscopes in car electronics for stability control and ink jet printer nozzles Craighead (2000). The nanoscopic version of MEMS, nanoelectromechanical systems (NEMS), demonstrate their own advantages in engineering and fundamental science in areas such as mass, force and charge detection Ekinci and Roukes (2005). All these electromechanical devices are functional only in response to an external applied force. The utmost limit would be a one atom thick resonator. Robustness, stiffness and stability is thus important when reaching this limit Bunch et al. (2007).

The fabrication of graphene-based mechanical resonators is still under active development. Presented here is an overview of experimental results for single and multilayer graphene sheets placed over predefined trenches. A simple drive and detection system is used to probe the mechanical properties of these graphene resonators in order to extract the fundamental resonance frequency. This allows us to characterize the quality factor, Young’s modulus and built-in tension Bunch et al. (2007). One can also extract this information from a separate experiment using an atomic force microscope (AFM) tip Frank et al. (2007). In addition to their use as mechanical resonators, graphene sheets are robust (impermeable) enough to act as a thin membrane between two dissimilar environments Bunch et al. (2008).

viii.1 Overview of the harmonic oscillator

In order to study the mechanical properties of graphene and graphite sheets, one needs to comprehend the basics of a harmonic oscillator. A classic example is a mass attached to a spring. In an ideal situation, the spring obeys Hooke’s law, but for large displacements, the spring does not follow Hooke’s law as the system is subject to damping, denoted by , which dissipates the vibrational energy of the system. In addition, a driving force with frequency is necessary to drive oscillations to the system. Therefore, the equation of motion for a damped harmonic oscillator is given by:

(21)

The general solution is given by:

(22)

where is the phase shift, the amplitude

(23)

and the quality factor

(24)

For a given peak with resonance frequency , the peak amplitude is and its full-width half maximum (FWHM) determines the Q factor, given by f/Q.

viii.2 Tuning resonance frequency by electrical actuation

The mechanical resonators are exfoliated graphene sheets that are suspended over trenches on SiO/Si substrates. These micron-sized sheets are doubly-clamped beams which are secured to the SiO surface via van der Waals forces. A gold electrode defined by photolithography is used to electrically actuate the graphene resonator, filling the role of the driving force from Eq. (21). All resonator measurements are performed in ultra-high vacuum at room temperature. The actuation is an applied alternating electric field used to drive the resonant motion of the beam. A capacitor is then formed between the bottom gate electrode and the contacted graphene, shown in Fig. 47. A voltage applied to the capacitor induces electric charges onto the beam. A small time-varying radio frequency (RF) voltage at frequency is used to electrically modulate the beam on top of a constant DC voltage Bunch et al. (2007):

(25)

The resulting electrostatic force is then given by:

(26)

where z is the distance between the graphene and the gate electrode.

Figure 47: Schematic of a suspended graphene resonator for electrical actuation Bunch et al. (2007).

The response due to a varying RF voltage is monitored by a 632 nm He-Ne laser focused on the resonator. As a result, the light creates an interference between the suspended graphene sheets and the silicon back plane. Variations of the reflected light intensity are monitored by a photodiode.

Plugging Eq. (26) into Eq. (23), one can speculate on the behavior of the resonant peak as a function of applied voltage. The data for a few-layer graphene stack is shown in Fig. 48 for electrical drive on resonance. One can see that the amplitude and frequency increase linearly with at a fixed for the higher mode (Fig. 48B) while the frequency of the fundamental mode (Fig. 48A) does not change significantly. In addition, if is increased while is fixed (Fig. 48C), both the amplitude of the fundamental mode and that of the higher mode increase linearly as expected from Eq. (23). As for the frequency (Fig. 48D), it seems that there is evidence of positive tuning for the higher mode but that the fundamental mode remains unaffected. A possible explanation for the tuning may originate from an electrostatic attraction to the gate which increases the tension from stretching the graphene.

Figure 48: Resonance spectrum taken with an electrostatic drive. Plot of the amplitude versus frequency of the (a) fundamental mode at 10 MHz and (b) the higher mode at 35 MHz with increasing while remains fixed. Inset: Plot of the resonant peak amplitude with increasing . Dependence of the (c) amplitude and (d) frequency as function of at fixed where the solid squares and triangles represent the fundamental and higher mode respectively Bunch et al. (2007).

viii.3 Resonance spectrum by optical actuation

Another technique to actuate vibrations in resonators is an optical drive. A diode laser is shined on the suspended graphene in which the graphene actuates by itself via thermal expansion and contraction. This motion is controlled by the intensity of the diode laser set to a frequency . This is then monitored by a photodiode using a He-Ne laser through the procedure given in the previous section Bunch et al. (2007).

Figure 49: Plots showing the resonance spectrum of graphene and graphite resonators. (a) Resonance spectrum showing the fundamental mode and higher modes for a 15 nm thick multilayer graphene sheet taken with optical drive. Inset: Optical image of the resonator. Scale bar = 5 m. (b) Amplitude vs. frequency for a single layer graphene. The red curve represents the Lorentzian fit of the data Bunch et al. (2007).

Fig. 49A shows the resonance spectrum of a few-layer graphene sheet suspended over a SiO trench. Several resonant peaks were measured; the first peak is the fundamental vibrational mode which will be the main focus of the discussion. The fundamental peak is the most pronounced peak, so it is easy to obtain the mechanical characteristics of the resonator. Analyzing higher modes is unnecessary as they depend on the fundamental frequency . The most appropriate fit to extract the resonance frequency is a Lorentzian fit, giving to be 42 MHz with a quality factor Q = 210. Fig. 49B shows similar results for a single layer graphene, = 70.5 MHz and Q = 78. The same measurements were repeated for 33 resonators of different geometries with thicknesses ranging from 1 atomic layer to 75 nm. They are plotted in Fig. 50. The data shows that the fundamental frequency largely varies from 1 MHz to 166 MHz with quality factors ranging between 20 and 850. Knowing the fundamental resonance , the following equation for a doubly-clamped geometry is used as a reference to extract the Young’s modulus of the sheets Weaver et al. (1990):

(27)

where is Young’s modulus, is the tension per width, is the mass density, and are the thickness and length of the suspended graphene sheet, and is a geometrical constant; = 1.03 for doubly-clamped beams and 0.162 for cantilevers. The parameter represents the built-in tension on the graphene sheet which may come from the fabrication process or from the van der Waals interaction between the substrate and graphene. Assuming that the tension is sufficiently small, Eq. (27) predicts that the fundamental frequency scales as . Fig. 50A is a plot of the resonance frequency for resonators with nm (solid squares) and nm (hollow squares) as a function of and the slope gives the Young’s modulus . Taken the density for bulk graphite as = 2200 kg/m, the Young’s modulus of the graphene sheets plotted as dashed lines correspond to values between 0.5 and 2 TPa. These values correspond well to the Young’s modulus of 1 TPa of bulk graphite Kelly (1981). This is the highest modulus resonator to date. This is in stark contrast to the 12-300 nm thick Si cantilevers Sazonova et al. (2004) which achieve values ranging between 53 - 170 GPa, and which are plotted as solid triangles in Fig. 50A. For graphene resonators with nm, the data points are not as linear as for the thick resonators. To elucidate these results, Scharfenberg et al. studied the elasticity effect of thin and thick graphene samples on a corrugated elastic substrate such as polydimethylsiloxane (PDMS). For thin graphene samples, it adheres fully to the substrate so that the graphene follows the topography of the substrate. As for thicker samples, it flattens the corrugated substrate. These observations suggest that thin graphene flakes, despite their size, are highly sensitive to tension, making them easier to deform than the thicker samples Scharfenberg et al. (2011).

Figure 50: Measurements of the Young’s modulus and Q factor for doubly-clamped beams. (a) Plot showing the frequency of the fundamental mode of all the doubly-clamped beams and Si cantilevers versus . The cantilevers are shown as solid triangles. The doubly clamped beams with nm are indicated as solid squares while those with 7 nm are shown as hollow squares. The solid line is the theoretical prediction with no tension. (b) Plot of the quality factor as a function of the thickness for all resonators Bunch et al. (2007).

The quality factor is an important parameter to consider when dealing with resonators. It gives a good indication of the resonator’s sensitivity to external perturbations. A high Q is thus essential for most applications. A plot of Q as a function of the thickness for all graphene resonators is shown in Fig. 50B. It seems that there is no clear dependence of quality factor on resonator thickness observed and the Q factor is on the order of hundreds. These Q factors are lower than diamond NEMS (Q2500-3000) Sekaric et al. (2002) and significantly lower than high tensile SiN (Q200,000) Verbridge et al. (2006). Thicker variants of graphene, like graphene oxide (Q4000) Robinson et al. (2008b) and multilayer epitaxially grown graphene (Q1000) Shivaraman et al. (2009) have shown higher quality factors. Despite the usability of graphene as a mechanical resonator, the source of dissipation is still poorly understood. Recent progress on graphene resonators suspended over circular holes on SiN membranes have shown some improvements on the Q factor. Thus, Fig. 51 demonstrates a clear dependence of quality factor on the resonator diameter. The highest Q factor is about 2400 for a resonator with a diameter of 22.5 m Barton et al. (2011).

Figure 51: Resonance frequency and Q factor of graphene membranes. (a) Fundamental frequency as a function of diameter for 29 graphene membranes. The red line is a fit of the data in which . (b) Quality factor of membranes as a function of diameter. Inset: the highest Q factor peak observed fitted with a Lorentzian indicating a value of 2400 Barton et al. (2011).

viii.4 Measurements by atomic force microscopy

In the previous subsections, the Young’s modulus is measured by considering the tension being small, neglecting the second term in Eq. (27) for a doubly-clamped beam. The built-in tension in the suspended graphene may indeed play a role in the mechanical measurements, a plausible explanation to the scattered data found in Fig. 50 for thin graphene resonators. To measure the built-in tension, an AFM tip with a calibrated spring constant is pressed onto the suspended graphene Frank et al. (2007). The spring constants on the sheet are then extracted to calculate the Young’s modulus and built-in tension. With an applied static force for a doubly-clamped beam and employing the relation and Eq. (27), the resulting effective spring constant is:

(28)

where is the width of the sheet.

Figure 52: Schematic of a deflected AFM tip while pushing on the suspended graphene. is the height that the tip is deflected by the graphene and is the height pushed by the tip Frank et al. (2007).

Fig. 52 shows the schematic of an AFM tip pushing down the suspended graphene sheet. The deflection of the tip then helps to find the effective spring constant of the suspended sheet. The chosen AFM tip has a spring constant 2 N/m which enables the graphene to be deflected by a detectable amount. The tip is pushed slowly against the sheet and the relation between tip displacement and the position of the piezo is then plotted in Fig. 53. As the AFM tip comes into contact with the suspended graphene, the cantilever is pulled down onto the surface resulting in a dip in the deflection as observed in Fig. 53a. From the tip’s spring constant, a graph of the force exerted on the tip versus the displacement of the graphene sheets can be extracted as plotted in Fig. 53b. The displacement can be written as:

(29)

where is the deflection of the tip, is the location of the piezo moving the tip, and is the deflection of the graphene sheet. From Hooke’s law, the slope yields the effective spring constant of the suspended graphene sheet.

Figure 53: Measurement of the graphene’s spring constant. (a) Curve obtained from the deflection of the AFM tip by pushing down the suspended graphene sheet. The right axis represents the force corresponding to the tip displacement. (b) Plot of the force as a function of the displacement of the graphene sheet Frank et al. (2007).
Figure 54: Plot of the spring constant in the center of the suspended region versus for eight different samples with various thicknesses. The linear fit provides information about the built-in tension and Young’s modulus of the graphene sheets Frank et al. (2007).

Fig. 54 shows the plot of the spring constant as a function of the dimensions of the suspended graphene sheet for eight samples. The spring constant varies between 1-5 N/m. A linear fit is then used to extract the built-in tension and Young’s modulus of the suspended graphene sheets. From Eq. (28), the slope suggests that a Young’s modulus of 0.5 TPa, in good agreement with suspended graphene actuated optically and bulk graphite. The offset of the linear fit gives a tension of 300 nN, suggesting that the built-in tension of all suspended graphene sheets are on the order of nN.

Other work on AFM nanoindentation was realized on graphene suspended over micron-sized circular wells to investigate the elasticity properties of graphene. Indenting defect-free graphene with an AFM tip similar to Fig. 52, one can probe the elastic stress-strain response. A non-linear elastic response model is fitted to find that the stiffness is about 340 N/m with intrinsic breaking strength of 42 N/m Lee et al. (2008).

As discussed in this section, graphene as an ultra-thin membrane with extremely high bulk modulus is very promising for potential applications in mechanical systems, if the source of dissipation can be reduced substantially, which should lead to a significant increase in Q factor. We now turn to applications of graphene as transistors.

Ix Graphene Transistors

This section provides an overview of current experimental work being pursued toward the development of graphene field effect transistors (GFET). The two most common varieties of transistor are the logic transistor and the analog transistor. The former is characterized, among other things, by a high ratio to ensure low energy consumption (of the off state) and to maintain a high logic interpretation yield. The latter is characterized primarily by its cutoff frequency, and is often used in high frequency applications as an amplifier. Both types of devices, as we shall see, could benefit from the remarkable electrical properties of graphene.

ix.1 Logic transistors

For more than forty years CMOS technology has dominated the logic transistor industry with the fabrication of MOSFETs. A general trend over the years has been to reduce the length of the transistor’s gate in order to achieve higher transistor densities but also increased performance produced by the higher electrical fields in the channel region. However, as the size of the device is reduced, more and more issues (commonly known as short channel effects) begin to appear Frank et al. (2001). These issues include but are not limited to such problems as hot electron effects, velocity saturation effects, and punchthrough effects. It has been suggested that graphene, due to its monoatomic thickness, would reduce these parasitic effects Schwierz (2010). Graphene could be included in the channel region of transistors and thus provide a high mobility channel which would help to reduce the effect of short channel effects. One problem is that graphene does not possess a bandgap since the conduction and valence bands touch at the Dirac point. This reduces the on-off ratio of the transistor by several orders of magnitude, such that a graphene FET will typically have an on-off ratio . This ratio is unacceptable if one wants to replace CMOS technologies where ratios are commonly achieved. It has been proposed by the International Technology Roadmap of Semiconductors (ITRS) that a ratio of would be required for logic applications. Such a ratio could be achieved in graphene only by opening a bandgap at the Dirac point in order to suppress the band-to-band tunneling Fiori and Iannaccone (2009). A bandgap of 0.4 eV could achieve the desired on-off ratio Schwierz (2010). Several techniques are being envisioned for engineering a graphene bandgap. The following sections provide an overview of these methods.

Bilayer graphene

The first technique that we shall investigate is the use of bilayer graphene. Theoretical calculations have predicted the possibility of a bandgap opening in this configuration McCann (2006). Such a structure allows the opening of a bandgap by breaking the symmetry of the bilayer stack (Bernal stacking) with the application of a transverse electrical field. One of the first investigations of this approach is given in reference Ohta et al. (2006). The authors used bilayer graphene on a SiC substrate. In such a structure, light doping is provided to the bottom graphene layer by the substrate and further doping can be introduced to the top graphene layer by the adsorption of potassium atoms. This setup was designed and fabricated to facilitate band structure analysis using the ARPES technique. Fig. 55 presents the experimental band structure for increasing amounts of potassium doping and a comparison to theoretical calculations. One may notice the good agreement between probed and theoretical bandstructure. A maximal bandgap of approximately 0.2 eV is observed using this technique.

Figure 55: Bandgap evolution as a function of the potassium doping. The doping level is controlled using the potassium adsorption of bilayer graphene. Shown from left to right is a diagram of the evolution of the bandgap for an increasing doping level per unit cell Ohta et al. (2006).

The results presented above show the potential of bilayer graphene for bandgap engineering. Several groups used this material for designing FET devices Oostinga et al. (2008). In these devices doping is induced using the field effect instead of using potassium atoms. Fig. 56 presents a schematic representation of such a device. The double gate configuration allows one to control both the Fermi position and the bandgap size.

Figure 56: Schematic of a GFET device. Graphene is deposited on an isolating material (e.g. SiO) after which regular photolithographic patterning can be performed Oostinga et al. (2008).

After characterization of the device, a maximal ratio of approximately 100 could be achieved at 4.2 K. Decreasing performance was observed with increasing temperature. These low on-off ratios are explained by the insufficiently large bandgap. Theoretical calculations for bilayer graphene predict a maximal achievable bandgap that would be below the 0.4 eV gap needed for an on-off ration over . These predictions are corroborated with experimental results. Therefore, bilayer graphene seems to be impractical for logic application.

Graphene nanoribbons

A second technique used for engineering a bandgap in graphene is the use of graphene nanoribbons. The idea is to provide additional charge carrier confinement in the plane of the graphene sheet, effectively making it a one-dimensional structure. This additional confinement can lead to the opening of a transport gap. There are two common techniques for producing nanoribbons. The first technique is the lithographic patterning of graphene. On one hand, this approach makes it easy to control the position of the transistor’s elements (e.g. channel, gate, electrons, etc.). On the other hand, making the width of a nanoribbon arbitrarily small is limited by the lithographic process. Furthermore, the edge quality of the patterned nanoribbons has been found to be rough, leading to increased scattering and decreased performance Han et al. (2007). Fig. 57 presents a graph of the bandgap energy () as a function of the width of the lithographically patterned nanoribbons at K. Again decreased performance is correlated with increasing temperature. One can notice a maximal bandgap of 200 meV for a nanoribbon width of 15 nm. These results are similar to the ones obtained for bilayer graphene.

Figure 57: Energy bandgap as a function of nanoribbon width at K Han et al. (2007).

A second approach for producing nanoribbons is chemical synthesis Li et al. (2008). This method, unlike lithographic patterning, produces smooth edges on the nanoribbons, while achieving widths in the sub-10 nm range. However, this method leads to a distribution of nanoribbon sizes and makes their positioning difficult. Scalability of such a technique is thus hard to achieve. Fig. 58a presents an I-V curve for a 5 nm wide nanoribbon in a GFET device. One may notice that even at room temperature ratios above are observed. This behavior is explained by the size of the bandgap, which reaches 0.4 eV, as illustrated in Fig. 58b. For such small widths, carrier confinement is appreciable leading to an asymptotic behavior of the bandgap as a function of width.

(a) I-V curve for a 5 nm wide nanoribbon
(b) Energy bandgap as a function of nanoribbon width at room temperature.
Figure 58: Chemically derived nanoribbons Li et al. (2008).

Other techniques

This section presents three other techniques for engineering a bandgap in graphene: strained graphene, nanomesh graphene and patterned hydrogen adsorption graphene.

Strained graphene Several studies suggest that a bandgap would be induced in graphene by the application of strain. By keeping track of the G and 2D bands while applying strain on a graphene sheet, one can monitor the amount of strain induced. Strain of 0.8% has been experimentally measured using this technique Ni et al. (2008b). Theoretical calculations predict that such an amount of strain would lead to a 300 meV bandgap. However, experimental evidence of this gap has not been demonstrated.

Nanomesh graphene Another novel way of inducing a bandgap in graphene is to pattern holes in a graphene sheet Bai et al. (2010). This technique led to an ratio of the order of 100 at room temperature for patterned holes of 7 nm in width. The on-off ratio tends to increase with decreasing neck width of the holes. A major advantage of this technique is its scalability. Work remains to be done in order to study whether an on-off ratio of can be achieved.

Patterned hydrogen adsorption This novel technique makes use of half-hydrogenated graphene on Ir(111) Balog et al. (2010). Theoretical DFT simulation predicts that this technique can open a bandgap of 0.43 eV in graphene, thus reaching our desired value of 0.4 eV. Graphene grown on Ir give rises to a superperiodic potential, also called a Moiré pattern. Such a pattern remains after the hydrogen adsorption and it alters the electronic properties of graphene leading to a bandgap opening. Fig. 59 presents ARPES measurements of the hydrogen adsorbed graphene. One can easily see the the opening of a large bandgap around the Fermi level. This structure is also stable at room temperature.

Figure 59: Valence band evolution. Exposition time from left to right goes from 0 sec (clean graphene) to 30 sec and finally 50 sec. A bandgap of almost 0.5 eV is observed after 50 sec exposition a result that is in good agreement with theoretical predictions Balog et al. (2010).

Techniques comparison

Presented below is a summary table that presents achieved results for the different bandgap engineering methods presented in this section.

Method Bandgap Ref.
Bilayer 0-0.25 eV 100 Ohta et al. (2006); Oostinga et al. (2008)
Nanoribbons 0-0.4 eV Han et al. (2007)          Li et al. (2008)
Strain 0.3 eV* NA Ni et al. (2008b)
Nanomesh NA Bai et al. (2010)
Hydrogen   adsorption eV NA Balog et al. (2010)
Table 4: Bandgap engineering techniques *No experimental data available

ix.2 Analog transistors

A second type of transistor is the analog transistor. These devices do not require a non-conducting off-state like logic transistors, and therefore do not require a bandgap. They are used in radio frequency applications and are characterized by their cutoff frequency . The high mobility of graphene possibly leads to very high cutoff frequencies, making graphene an attractive material for designing these transistors. One major challenge for the integration of graphene in these transistors is the preservation of high mobility while integrating graphene in a device.

Since 2008 there has been increased research interest in the GFET analog transistor Meric et al. (2008). Some of the initial work showed that a 14.7 GHz transistor was achievable using graphene with a 500 nm device length. Following these results, a 26 GHz device was fabricated using a shorter gate length of 150 nm and mechanically exfoliated graphene Lin et al. (2009). The cutoff frequency of GFET devices has proven to follow its well-known behavior for this type of transistors given by:

(30)

Shorter gate length increases the drift velocity of charge carriers by increasing the electrical field. It also leads to a reduced distance between the source and drain contacts, which also reduces the transit time across the FET. It was observed that , so reducing the gate length from 500 nm to 150 nm resulted in the cutoff frequency rising from 3 GHz to 26 GHz Lin et al. (2009). For these devices the effective mobility was estimated to be 400 cmVs. This value is well below the 10000 cmVs mobility of exfoliated graphene at room temperature that could be achieved Novoselov et al. (2004). Indeed, the effective mobility in graphene is highly sensitive to its environment and decreases as the amount of scattering elements and defects are increased (e.g. insulating substrate, gate oxide, source/drain contacts, etc.).

A similar device configuration using a higher quality graphene (graphene grown on SiC) achieved a higher effective mobility of 1000-1500 cmVs. For this device, a 100 GHz cutoff frequency been measured with a gate length of 240 nm Lin et al. (2010).

Lately, other device configurations that minimize scattering with the graphene sheet are being engineered. Fig. 60 shows a GFET device using a nanowire gate.

Figure 60: GFET schematics using a nanowire gate Liao et al. (2010).

This configuration minimizes scattering and defects in the graphene sheet and allows one to achieve cutoff frequencies in the 100-300 GHz with gate lengths in the 200 nm range. Further work is still required in order to achieve higher effective mobility. In the following section we shall see how the optical properties of graphene are of a great interest for optoelectronic devices.

X Optoelectronics

x.1 Optical properties

The properties of graphene make it an attractive choice for use in optoelectronic devices. In particular, graphene’s high transparency, low reflectance, high carrier mobility and near-ballistic transport at room temperature make it a promising choice for transparent electrodes. Optically, single-layer graphene (SLG) has an unusually high absorption (given its thickness) that can be described in terms of the fine structure constant, (a fundamental physical constant that describes the interaction of matter and electromagnetic fields). Light transmittance through free-standing graphene can be derived using the Fresnel equations for a thin film with a universal optical conductance of :

(31)

where . For the most part, each SLG will contribute an absorbance of to visible light. Because graphene sheets behave as a 2-dimensional electron gas, they are optically almost non-interacting in superposition (though the same cannot be said of their electronic interaction Luican et al. (2011)), so the absorbance of few-layer graphene (FLG) sheets is roughly proportional to the number of layers. The absorption spectrum of graphene from the ultraviolet to infrared is notably constant around 2-3% absorption, as shown in Fig. 61a, compared to some other materials. The reflectance is very low at 0.1%, though increases to  2% for 10 layers Casiraghi et al. (2007). Graphene has found its way into many optoelectronic and photonic devices, a portion of which will be covered here. Other uses, including photodetectors, touch screens, smart windows, saturable absorbers and optical limiters have recently been covered in detail elsewhere Bonaccorso et al. (2010).

Figure 61: Exploring the utility of graphene for thin conducting electrodes. Reprinted from Bonaccorso et al. (2010). Theoretical ranges are shown bound by black lines, as calculated using typical values with Eq. (32). (a) Transmittance as a function of wavelength for graphene, ITO and two other metal oxides, as well as SWNTs. (b) values as a function of thickness, illustrating the potential of graphene to achieve substantially lower for a given thickness. (c) Transmittance as a function of for the same materials. (d) Transmittance as a function of for a number of graphene fabrication methods: triangles, CVD; blue rhombuses, micromechanical cleavage (MC); red rhombuses, organic synthesis from polyaromatic hydrocarbons (PAHs); dots, liquid-phase exfoliation (LPE) of pristine graphene; and stars, reduced graphene oxide (rGO).

x.2 Transparent conducting electrodes

Transparent conducting electrodes (TCEs) are critical components in a wide variety of devices, from those found in academic and industrial settings to commercial devices that define modern technology. They function primarily by injecting or collecting charge depending on the purpose of the device, and are highly transparent across some part of the electromagnetic spectrum (most often to visible light). The most widely used material for TCEs is indium tin oxide (ITO), an n-type, wide-bandgap semiconductor ( = 3.75 eV), which typically consists of 90% indium(III) oxide doped with 10% tin(IV) oxide by weight. The tin atoms function as n-type donors. ITO has favorable electronic and optical properties for most TCE applications, but suffers from practical limitations. It is relatively inflexible and fragile, and a limited supply of indium makes it increasingly costly for large-scale production. Though it is highly transparent across most of the visible range and near-infrared, it becomes increasingly opaque at UV and reflects IR wavelengths, because of band-to-band absorption (excitation of an electron from the valence to the conduction band) and free carrier absorption (excitation within the conduction band), respectively Keshmiri et al. (2002). Despite the utility of ITO in optoelectronic devices, alternatives are sought due to the aforementioned reasons. One popular inorganic substitute is doped zinc oxide, which has comparable properties, though suffers from limited etching capability due to acid sensitivity, among other things. It is typically doped with aluminum, gallium or indium. The list of inorganic semiconductor materials is extensive, but ITO has become the standard to beat for many applications.

In contrast to inorganic materials, organic TCEs are characterized by low cost, flexibility and stability, though do not achieve the same level of charge mobility as inorganic materials. Many varieties of intrinsic conducting polymers (ICPs) have found use in optoelectronics. In particular, their electroluminescence, flexibility and mechanical strength makes them particularly suitable for OLEDs and touch screens, though processability is a known issue. PEDOT (poly(3,4-ethylenedioxythiophene)) is a conducting polymer that is widely used as a transparent conductive film in the form of PEDOT:polystyrene sulfonic acid (PSS) dispersions (due to the insolubility of PEDOT in water). Like other ICPs, the conductive properties of PEDOT arise from a chain of conjugated double bonds, and by controlling the extent of -electron cloud overlap, the PEDOT band gap can be varied from 1.4 to 2.5 eV in a manner analogous to doping (earning such polymers the nickname “organic metals”) Groenendaal et al. (2000). For organic semiconducting materials, the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) are analogous to the valence and conduction band of inorganic semiconductors.

x.3 Graphene as TCE

Two of the most vital parameters for TCE materials are low sheet resistance and high optical transmittance, . Sheet resistance (in units of ) is directly obtained using a four-terminal sensing measurement (see Sec. VI), and can be thought of as resistance per aspect ratio. Under most circumstances, graphene matches or exceeds the transmittance of competing materials, though the sheet resistance has proven somewhat less predictable, and varies depending on production method, with CVD being the closest to optimal for optoelectronics (highest and lowest ). This is illustrated graphically in Fig. 61. and in doped FLG are related fairly simply by: