Electronic transport in graphene: towards high...

199

© 2014 Woodhead Publishing Limited

9 Electronic transport in graphene:

towards high mobility

K. I. B O L O T I N , Vanderbilt University , USA

DOI : 10.1533/9780857099334.3.199

Abstract : Strong carrier scattering perturbs the intrinsic response of Dirac fermions in graphene and limits potential applications of graphene-based devices. Multiple scattering mechanisms including Coulomb scattering, lattice disorder scattering and electron–phonon scattering play roles in realistic graphene devices. Moreover, different types and preparations of graphene are characterized by different dominant scattering mechanisms. In this chapter, we review the recent progress towards reduction of carrier scattering in graphene. We start by discussing different metrics – such as carrier mobility, mean free path, and scattering time – that are used to quantify the scattering strength. Then, we review the strategies to reduce scattering and to improve carrier mobility. These strategies include: lowering defect density, suspending graphene, depositing graphene onto high-quality substrates, and covering it with high- k dielectrics. Finally, we briefl y address the physical phenomena and device applications that are specifi c to ultraclean high-mobility graphene.

Key words : graphene , electronic transport , carrier mobility , carrier scattering , lattice disorder.

9.1 Introduction

Weak scattering of charge carriers in graphene is one of the most notable properties of this material. The scattering strength is usually quantifi ed by mobility, the ratio of carrier drift velocity to an applied electrical fi eld. The fi rst experiments probing electrical transport in graphene reported room temperature mobility around 10 000 cm 2 V − 1 s − 1 , an order of magnitude better than the typical mobility of the most commonly used electronic material, silicon (Novoselov et al ., 2012 ). This initial observation spawned concerted efforts to use graphene as a replacement for silicon in electronics applications. At the same time, until very recently, carrier scattering in the existing samples was strong enough to mask signatures of interactions between Dirac fermions in graphene.

Graphene samples with low carrier scattering are required both for experiments probing the intrinsic physical phenomena of graphene ’ s Dirac fermions and to realize potential applications of graphene. Only with improvements in carrier mobility does it become possible to observe a cornucopia of phenomena in graphene including ballistic transport, the

200 Graphene

symmetry-broken quantum Hall effect, the fractional quantum Hall effect (FQHE), and Klein tunneling. More anticipated phenomena including Bose–Einstein condensation, superconductivity, appearance of a mass gap, and the appearance of ν = 5/2 fractional quantum Hall state remain unobserved in part owing to the insuffi cient quality of graphene samples. Potential applications of graphene that would benefi t from high mobility include high-frequency transistors, sensors, optoelectronic modulators, and transparent conductive electrodes.

In this chapter, we review the progress towards reduction of carrier scattering in graphene. First, we review different metrics quantifying scattering strength in graphene. Second, we discuss different types and preparations of graphene, each characterized by different dominant scattering mechanisms. We then discuss various mechanisms contributing to scattering in different types of graphene, including Coulomb scattering, lattice disorder scattering, and electron–phonon scattering. Next, we discuss various approaches towards reduction of scattering in graphene: lowering defect density, suspending graphene, depositing graphene onto high-quality substrates, and covering it with high- k dielectrics. Finally, we briefl y address physical phenomena appearing in high-mobility graphene and consider potential applications of such a material.

9.2 Metrics for scattering strength

To compare the scattering strength of various graphene samples measured by different research groups, a common metric of sample quality is needed. The most commonly used metric is carrier mobility, a ratio of carrier drift velocity to an applied electrical fi eld. Despite the ubiquitous use of mobility, it has several accepted defi nitions; even for the same device, these defi nitions can yield different values. In addition, other metrics such as the mean free path, transport and quantum scattering times, and the full width at half maximum of the Dirac peak are often used. Because of the importance of well-defi ned metrics of sample quality, we briefl y review the experimental approaches of measuring scattering in graphene, and devote special attention to possible confounding artifacts. Mechanically exfoliated graphene on SiO 2 , one of the most common types of graphene devices, is used as our comparison benchmark.

Carrier mobility is typically defi ned as μ ≡ v/E = σ /en , where v is the Drude carrier drift velocity, E is applied electrical fi eld, assumed to be small, σ is conductivity, n is carrier density. It is straightforward to measure carrier mobility for a graphene fi eld effect transistor (FET) confi gured in a Hall-bar geometry using low-fi eld magnetoresistance measurements (Fig. 9.1 ). First, by applying an out-of-plane magnetic fi eld B and recording the Hall resistance R xy , one determines the carrier density n = B/eR xy and its

Electronic transport in graphene: towards high mobility 201

dependence on gate voltage V G . Longitudinal resistance R xx (which is constant for small B ) and resistivity ρ = R xx (W/L) are determined next, where W and L are the width and length of the Hall bar. Finally, Hall mobility is calculated as 1/ en ρ . The mobility value determined using this approach is termed the Hall mobility, μ H .

Alternatively, by measuring the resistivity ρ as a function of gate voltage V G in a graphene FET, the fi eld effect mobility

μ ρ σFE g GC d dV d d en= = ( )− −1 1

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202 Graphene

can be determined, where C g is a gate capacitance per unit area of graphene FET. The fi eld effect mobility can be measured for two-probe devices without employing magnetic fi elds, and it yields the same value as the Hall mobility under the assumption σ ∼ n , often a good approximation for graphene. For mechanically exfoliated graphene on SiO 2 , both μ H and μ FE typically vary between ∼ 2000 and ∼ 30 000 cm 2 V − 1 s − 1 .

The mobility values obtained using the procedures above should be taken with a grain of salt for the following reasons:

1. Although for the majority of graphene samples conductivity is propor-tional to the carrier density, this dependence is not exact. As a result, the mobility is nearly always carrier-density dependent. Moreover, because different scattering mechanisms in graphene have different dependence on n , the mobility can be limited by different scattering mechanisms at low and at high densities. Therefore, it is only prudent to compare carrier mobility obtained in different graphene devices at the same carrier density ( n ∼ 10 12 cm − 2 is commonly used).

2. The fi eld effect mobility does not capture the effect of short-range scattering, the mechanism producing carrier-density-independent contribution to σ (discussed in more detail in 9.3).

3. The gate capacitance per unit area C g entering into the defi nition of fi eld effect mobility is typically estimated using a parallel–plate capacitor approximation, C g = κ ε 0 /d, where κ is the dielectric constant of the gate material, and d is its thickness. However, this formula is only applicable when parallel-plate approximation is valid, i.e. when the lateral size of the graphene is much larger than d . For graphene devices with dimensions similar to or smaller than d , the carrier density varies across the devices and standard formulas for mobility are inapplicable. In addition, the parallel-plate-capacitor formula ignores quantum capacitance of graphene, an important contribution for devices with d of the order of nanometers (Xia et al ., 2009 ).

4. For small samples in the Hall bar geometry, the width of the ‘arms’ of the Hall bar is often similar to the width and the length of the device. In that instance, errors in mobility can come from the determination of the geometrical factor relating resistivity and measured resistance.

5. Determination of the mobility can be affected by spatial nonuniformities in the carrier density resulting from random dopant distribution. For multiprobe samples in Hall bar geometry, one can check for possible density nonuniformities by comparing inferred mobility measured using pairs of electrodes on both sides of a device.

Several additional metrics of carrier scattering strength, typically strongly correlated with carrier mobility, are discussed in the following subsections.


9.2.1 Mean free path and transport scattering time

The Drude mean time between scattering events τ and the mean free path l MFP ≡ v F τ are commonly used to characterize scattering strength and sample quality. For graphene, the scattering length (and, hence, τ ) can be related to carrier mobility via the Drude formula:

l e n vMFP F= ( )μ π�

where v F ∼ 10 6 m s − 1 is the Fermi velocity in graphene. The mean free path can be estimated for graphene either by observing the onset of ballistic transport for samples shorter than l MFP (Du et al., 2008 ), or by measuring the bend resistance (Mayorov et al ., 2011 ).

9.2.2 Onset fi eld of the Shubnikov–de Haas oscillations

At moderate magnetic fi eld, the conductivity of a 2D electron gas exhibits reproducible oscillations, which develop into the Quantum Hall plateaus at higher fi elds. These oscillations are quantum-mechanical in nature and appear when the magnetic fi eld is strong enough for a carrier to complete a cyclotron orbit without scattering. The onset fi eld of these oscillations B c can be related to the mobility via the semiclassical formula μ ~ Bc

−1 . Although measurements of the onset fi eld of Shubnikov–de Haas oscillations provide only a qualitative estimation of the mobility, these measurements are not affected by possible inaccuracies in the determination of the geometrical factor W/L and gate capacitance.

9.2.3 Temperature dependence of the minimum conductivity

A large variation in the minimum conductivity of graphene with temper-ature is only expected for k B T larger than the characteristic energy E v npuddle F= � πδ where δ n is the scale of the inhomogeneities in the carrier density of graphene (Morozov et al. , 2008 ). The parameter δ n is, in turn, related to the carrier mobility (Adam et al ., 2007 ). For graphene samples on SiO 2 , δ n ∼ 10 10 –10 11 cm − 2 , which corresponds to an energy of 10–30 meV (Martin et al ., 2008 ). This energy scale determines how close one can experimentally approach the Dirac point of graphene (Mayorov et al ., 2012 ).

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9.2.4 Width and position of the resistivity peak at the charge neutrality point

The full width at half maximum (FWHM) of the resistivity peak at the charge neutrality point is also related to the extent of charge inhomogeneity δ n and is hence correlated with the mobility. The geometrical factor does not affect determination of the FWHM. For graphene on SiO 2 , the typical FWHM is 10 11 –10 12 cm − 2 . The position of the minimum conductivity point is typically shifted from zero for low-quality samples, as several types of disorder introduce electrostatic doping in addition to carrier scattering (Adam et al ., 2007 ).

9.3 Methods of graphene synthesis

Many experimental routes towards synthesis of graphene have been discovered over the years. Different types of graphene are affected by differing scattering mechanisms and require differing approaches to scattering reduction. Below, we briefl y review the most common routes to produce graphene: mechanical exfoliation, chemical vapor deposition, epitaxial growth on silicon carbide, and the chemical route from graphene oxide. We then highlight the dominant scattering mechanisms for graphene obtained via these routes. For a more in-depth discussion of problems associated with graphene synthesis we direct the reader to several excellent recent reviews (Avouris and Dimitrakopoulos, 2012 , Novoselov et al ., 2012 ).

9.3.1 Mechanical exfoliation

Cleavage of highly oriented pyrolitic graphite crystals using the sticky tape, or ‘micromechanical exfoliation’, as it became known in the scientifi c literature, was the approach used to obtain graphene in the pioneering experiments of Geim and Novoselov (Novoselov et al ., 2012 ). Despite its simplicity and low yield, this method still produces graphene with the lowest defect density. Indeed, multiple studies reporting record mobility in graphene employ micromechanical exfoliation. The typical mobility recorded for exfoliated graphene samples on SiO 2 substrates is 5000–30 000 cm 2 V − 1 s − 1 .

9.3.2 Chemical vapor deposition on metals

Large-area graphene can be grown on nickel, copper, iridium, ruthenium and other metals via chemical vapor deposition (CVD) (Avouris and Dimitrakopoulos, 2012 ). In this approach, a feedstock gas (typically methane) decomposes onto the metal surfaces to produce either single or


multilayer graphene with a relatively low density of defects. After growth, the graphene layer can then be transferred onto an appropriate substrate for characterization. Typical mobility values for CVD-grown graphene are 1000–10 000 cm 2 V − 1 s − 1 ; the dominant scatterers are probably grain boundaries, dislocations, and other substrate-related features.

9.3.3 Graphene derived from chemical reduction of graphite oxide

The images of single sheets of graphene obtained via chemical reduction of graphene oxide date back to the early 1960s (Dreyer et al ., 2010 ). In this approach, bulk graphite is fi rst chemically oxidized and dispersed in solution via sonication. The resulting sheets of graphene oxide are then reduced via a chemical reaction with reducing agent such as hydrazine. Although large quantities of graphene can be obtained via this approach, the resulting graphene is characterized by a large degree of disorder and defects. Typical mobility reported for thin multilager fi lms of graphene obtained via chemical routes ranges from 0.1 to 1 cm 2 V − 1 s − 1 and is probably limited by hopping between individual graphene fl akes (Wang et al ., 2010 ).

9.3.4 Epitaxial graphene

Epitaxial graphene on silicon carbide (SiC) can be produced by thermal decomposition of SiC. Heating to high ( > 1000 °C) temperatures in ultrahigh vacuum causes silicon to sublimate leaving behind a carbon-rich layer, which can later undergo graphitization (Riedl et al ., 2010 ). Because bulk SiC can be obtained in an insulating state, this type of graphene can be measured without transferring it onto other substrates. The typical mobility of graphene obtained by this method is ∼ 1000 cm 2 V − 1 s − 1 . In general, epitaxial graphene is characterized by strong interactions with the underlying substrate.

9.4 Sources of scattering in graphene

All scattering mechanisms in graphene can be divided into two broad classes. Intrinsic scattering mechanisms, such as defect, grain boundary, or phonon scattering are associated with graphene itself. In contrast, the extrinsic mechanisms, such as Coulomb scattering on charged impurities on or under graphene and scattering caused by phonons in the substrate supporting graphene are associated with other materials in the vicinity of graphene. In this section, we discuss both intrinsic and extrinsic scattering mechanisms limiting carrier mobility in realistic graphene devices. For a more detailed discussion, we refer the reader to the excellent comprehensive

206 Graphene

reviews on electron transport and defects in graphene (Banhart et al ., 2011 , Das Sarma et al ., 2011 ).

9.4.1 Coulomb scattering and other long-range mechanisms

Even the fi rst experimental studies of electrical transport in single-layer graphene (Novoselov et al ., 2005 , Zhang et al ., 2005 ) reported its two notable features: roughly linear dependence of the conductivity σ on the carrier density n (and by implication nearly constant carrier mobility μ = σ /en ) and nonvanishing conductivity σ min of graphene at the charge neutrality point, Fig. 9.1 (b). The simplest scattering mechanism consistent with this type of σ (n) is Coulomb scattering of charge carriers in graphene by charged impurities residing either in the substrate under graphene (typically SiO 2 gate dielectric) or on the surface of graphene (Nomura and MacDonald, 2006 , Ando, 2006 , Adam et al ., 2007 , Hwang et al ., 2007 ). Under the assumption of a density of charged impurities in SiO 2 , n imp ∼ 50 × 10 10 cm − 2 , a theoretical model employing the Boltzmann kinetic equation with random phase approximation (Adam et al ., 2007 ) reproduces signatures observed for graphene fi eld effect transistors on SiO 2 substrates – linear σ (n) with carrier mobility μ in the range 1000–30 000 cm 2 V − 1 s − 1 , non-universal value of minimum conductivity σ 0 ranging from 2–12 e 2 /h , and the charge neutrality point shifted from zero by ∼ 10–50 V (Tan et al ., 2007 ). The effects of Coulomb scattering in graphene are rendered relatively ineffective by the Dirac character of charge carriers. First, the conservation of pseudospin in graphene in the absence of lattice disorder directly forbids backscattering of its charge carriers (McEuen et al ., 1999 ). Second, near the charge neutrality point, the charge carriers can propagate through potential barriers separating electron- and hole-doped regions via Klein tunneling (Katsnelson et al ., 2006 , Young and Kim, 2009 ). These features contribute to the relatively high mobility of graphene prepared on SiO 2 and other substrates with large density of charged impurities (Huang et al ., 2011 ).

Subsequent experiments provided strong evidence of the importance of the Coulomb scattering in graphene on common substrates. First, random fl uctuations in the local carrier density of graphene due to Coulomb fi elds of charged impurities in SiO 2 were directly imaged using a scanning probe technique (Martin et al ., 2008 ). The average magnitude of these fl uctuations δ n ∼ 10 11 cm − 2 is consistent with the expected density of the impurities. As a result of these fl uctuations, the electron gas in graphene breaks up into localized ‘puddles’ of electrons and holes near the charge neutrality point. Second, the model describing Coulomb scattering was directly tested by depositing potassium ions onto graphene (Chen et al., 2008a ). The observed changes in carrier mobility, position of the charge neutrality point, and the


value of minimum conductivity agreed with the prediction of the Boltzmann model of charged impurity scattering (Fig. 9.2 ). Third, the reduction of scattering in graphene observed after a layer of ice was deposited onto it was found to be consistent with dielectric screening of the Coulomb scattering (Jang et al ., 2008 ).

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9.2 Studying the Coulomb scattering in graphene by depositing potassium ions onto it. Reproduced with permission from Chen et al. , 2008a . (a) The conductivity ( σ ) vs. gate voltage ( V g ) curves for a graphene sample with four progressively increasing concentrations of potassium. The data are obtained at 20 K in ultra-high vacuum environment. Lines are fi ts to the self-consistent equation describing the Coulomb scattering. (b) Inverse of electron mobility 1/ μ e and hole mobility 1/ μ h versus doping time.

208 Graphene

Substrate-related Coulomb scattering became the default explanation of scattering in graphene. Several recent developments, however, introduce signifi cant complications to this seemingly simple model:

• Close values of μ measured for graphene samples deposited on different substrates (Ponomarenko et al ., 2009 ), similarity in μ between suspended and supported graphene before current annealing (Bolotin et al ., 2008b ), as well as other evidence (Hong et al ., 2009b ) all suggest that the Coulomb scattering may be associated with charged impurities on the surface of graphene or the impurities trapped between graphene and the substrate, rather than with the charge traps inside SiO 2 .

• Mechanisms such as midgap states resonant scattering (Ni et al ., 2010 , Wehling et al ., 2010 ) or frozen ripple scattering (Katsnelson and Geim, 2008 ) can produce conductivity that is roughly linear with carrier density and hence can be mistaken for the Coulomb scattering.

• Strictly linear dependence σ ∼ n is only expected for randomly distributed Coulomb scatterers. In contrast, clustered charged impurities yield nonlinear σ ( n ) (Li et al., 2011 ). Such nonlinearities have been observed after migration of mobile potassium atoms on graphene caused by thermal annealing (Yan and Fuhrer, 2011 ).

9.4.2 Lattice disorder scattering

Strong sp 2 bonds contribute to the high intrinsic strength of graphene and make lattice defects in it relatively rare. However, various types of defects, including vacancies, topological defects such as Stone–Wales defects, foreign atom substitutions, chemical bonds, grain boundaries, and even mechanical distortions of the lattice have been directly observed in graphene via atomic force microscopy, transmission electron microscopy and other techniques (Hashimoto et al ., 2004 , Gass et al ., 2008 , Meyer et al ., 2008 , Tapaszto et al ., 2008 , Ugeda et al ., 2010 ). Perhaps the easiest approach to quantify the presence of defects in graphene is through Raman spectroscopy (Dresselhaus et al ., 2010 ). Indeed, the so-called D-peak at ∼ 1350 cm − 1 , owing to zone-boundary phonons, does not appear in translationally invariant graphene and is only seen in defective samples. Multiple studies report Raman signatures of different types of defects and disorder including grain boundaries, dislocations, and sample edges. By comparing the intensities of the D-peak I D and the intensity of the G-peak ( ∼ 1580 cm − 1 ) I G , the average defect density can be estimated via an empirical formula:

n I Id D G~ .1 8 1022 4× ( )−λ

where λ is the wavelength of the excitation source (Cancado et al ., 2011 ).


The effects of different types of disorder on electrical transport in graphene are signifi cantly less understood than that of Coulomb scattering. The simplest model treats disorder in graphene as uncorrelated delta-function potentials and predicts a carrier-density-independent contribution to conductivity resulting from disorder scattering (Ando, 2006 , Nomura and MacDonald, 2006 , Hwang et al ., 2007 , Jang et al. , 2008 ). More realistic models for defects such as vacancies (Stauber et al ., 2007 ) or adsorbates (Wehling et al ., 2010 ) predict formation of localized states at or near the charge neutrality point. Such ‘midgap states’ resonantly scatter charge carriers in graphene and give rise to conductivity that is roughly linear with n, similar to the effect of charged impurities. Such conductivity behavior was recently observed for scattering on grain boundaries in graphene (Huang et al ., 2011 ) as well as for defects produced by ion irradiation and hydrogen adsorption (Ni et al ., 2010 , Wehling et al ., 2010 ).

Graphene samples produced by different techniques tend to feature different types of defects:

• In mechanically exfoliated graphene, defects are very rare (Ishigami et al ., 2007 , Meyer et al ., 2007 , Stolyarova et al ., 2007 , Gass et al ., 2008 ) and are typically created as a result of electron or ion irradiation (Gass et al ., 2008 , Tapaszto et al ., 2008 , Ugeda et al ., 2010 ). However, the appearance of the weak Raman D-peak and a density-independent component of the resistivity in high-quality exfoliated samples suggests that disorder does contribute to scattering even in exfoliated devices (Ni et al ., 2010 ).

• Covalent bonding to the underlying substrate probably plays an important role in limiting the mobility of epitaxial graphene. Dislocations with relatively low density have been observed in such graphene (Rutter et al ., 2007 , Sutter et al ., 2008 ).

• Graphene grown by chemical vapor deposition is signifi cantly more defective. Defects, typically in the form of grain boundaries, are routinely observed in such graphene (Coraux et al ., 2008 , Sutter et al ., 2008 , Park et al ., 2010 , Huang et al ., 2011 , Kim et al ., 2011 , Zhao et al ., 2011 ). Several studies have demonstrated that grain boundaries produce strong scattering limiting the carrier mobility (Huang et al ., 2011 , Jauregui et al ., 2011 , Song et al ., 2012 ).

• Graphene synthesized via a chemical oxidation–reduction approach is much more defective and features multiple heptagon–pentagon defects and substitutions (Mattevi et al ., 2009 , Gomez-Navarro et al ., 2010 ). For thin reduced graphene oxide fi lms the main contribution to scattering probably stems from hopping between individual graphene fl akes (Wang et al ., 2010 ).

210 Graphene

9.4.3 Electron–phonon scattering

The intrinsic room-temperature mobility of many electronic materials is often limited by scattering of charge carriers on phonons. For example, although the carrier mobility of the 2D electron gas in AlGaAs/GaAs heterojunctions can reach > 10 7 cm 2 V − 1 s − 1 at low temperatures, scattering on optical phonons lowers the room temperature mobility of this system to < 2000 cm 2 V − 1 s − 1 (Kawamura and Dassarma, 1992 ). In contrast, the energies of optical phonons are relatively high in graphene, ∼ 160 meV for longitudinal optical (LO) zone-boundary phonons (Yao et al ., 2000 ), and the dominant intrinsic phonon–induced mechanism around and below room temperature is relatively weak scattering owing to longitudinal acoustic (LA) phonons (Stauber et al ., 2007 , Hwang and Das Sarma, 2008 ). The transport signature of such scattering is the carrier-density-independent contribution to resistivity that is linear with temperature T in the range 10–200 K, with the coeffi cient of proportionality ∼ 0.1–0.2 Ω K − 1 (Morozov et al ., 2008 , Chen et al ., 2008b , Efetov and Kim, 2010 ). It has been argued that if all other types of scattering were eliminated, this phonon-induced scattering sets a very high limit for intrinsic room-temperature mobility in graphene μ ∼ 200 000 cm 2 V − 1 s − 1 at n ∼ 10 12 cm − 2 (Chen et al ., 2008b , Morozov et al ., 2008 ). This value signifi cantly exceeds the highest mobility recorded in a semiconductor at room temperature ( ∼ 7.7 × 10 4 cm 2 V − 1 s − 1 for InSb, Hrostowski et al ., 1955 ).

In some situations, carriers in graphene may interact with out-of-plane (fl exural) phonons. For suspended and unstrained graphene samples, scattering on fl exural phonons dominates scattering due to acoustic phonons and limits the room temperature mobility to < 10 000 cm 2 V − 1 s − 1 (Castro et al ., 2010 , Mariani and von Oppen, 2010 ). The contribution of fl exural phonons for graphene samples supported on substrates is less clear and is still debated. Some workers interpreted a rapid increase of the resistivity of graphene supported on SiO 2 at T > 100 K as evidence of carrier scattering of fl exural phonons excited inside ripples, the regions of the graphene sheet partially attached to the substrate (Morozov et al ., 2008 ).

Finally, phonons in a polar substrate material under graphene can produce fl uctuating electrical fi elds that scatter charge carriers in graphene (Moore and Ferry, 1980 ). For the case of graphene on SiO 2 , this remote interfacial phonon (RIP) scattering mechanism is probably responsible for the exponential contribution to graphene resistivity for T > 200 K (Chen et al. , 2008b , Fratini and Guinea, 2008 ). It is believed that RIP scattering limits the room temperature mobility of graphene on SiO 2 to less than ∼ 40 000 cm 2 V − 1 s − 1 at n = 10 12 cm − 2 .


9.4.4 Multiple scattering mechanisms

In realistic graphene devices, multiple scattering mechanisms contribute simultaneously. For example, for a reasonably high-quality exfoliated graphene device on SiO 2 /Si substrate at a technologically relevant carrier density n = 10 12 cm − 2 , the largest limitation of mobility μ Coulomb ∼ 10 000 cm 2 V − 1 probably results either from Coulomb scattering or resonant impurity scattering, with further leading contributions from scattering on remote interfacial phonons in SiO 2 μ RIP ∼ 40 000 cm 2 V − 1 , and scattering on graphene acoustic phonons μ phonons ∼ 200 000 cm 2 V − 1 . The overall mobility of a device affected by several different scattering mechanisms can be estimated via the Matthiessen ’ s rule :

μ μ μ μ− − − −= + +1 1 1 1Coulomb RIP phonons

It has recently become common to use a simple rule-of-thumb to separate contributions from multiple scattering mechanisms. The resistivity at high carrier density is fi tted by ( ne μ L ) − 1 + ρ s , with two fi tting parameters μ L and ρ s . The fi rst, density-dependent term is then interpreted as a contribution of long-range scattering mechanisms, such as the Coulomb scattering, resonant impurities scattering, or ripple scattering. The second, density-independent term, ρ s , is a contribution caused by short-range mechanisms, such as phonon or disorder scattering. Despite the simplicity of the approach, one must exercise caution in using it. As discussed above, long-range scattering mechanisms, such as Coulomb and resonant impurity scattering can produce nonlinear σ ( n ), whereas short-range scattering can sometimes yield density-dependent σ .

9.5 Approaches to increase carrier mobility

The goal of improving the mobility of graphene devices requires simultaneous suppression of multiple scattering mechanisms. First, to avoid defect-induced scattering, high-mobility graphene devices should not contain many defects or grain boundaries. Second, graphene should be deposited onto a fl at substrate with a low density of charge impurities to avoid scattering on corrugations in graphene and Coulomb scattering. Finally, the ideal substrate under graphene may only feature high-energy phonon modes to avoid remote interfacial phonon scattering. In this section, we consider possible approaches to reduce scattering and to increase carrier mobility in graphene.

9.5.1 Annealing

The surface of as-fabricated graphene that underwent either a transfer or lithographic processes is almost always heavily contaminated (Chapter 11).

212 Graphene

The dominant and notoriously hard to remove contaminants are polymeric residues of electron beam resists, such as polymethylmethecrylate (PMMA) or photoresists. Although common solvents, such as acetone, remove some contaminants, the surface of graphene devices is typically covered with a nonuniform nanometer-thick residue layer even after extensive chemical cleaning, Fig. 9.3 (a). Such a polymeric residue layer can contribute to scattering in graphene, for example by serving as a host for ionized impurities or by producing resonant scattering.

Unfortunately, the most common approach to remove polymeric fi lms, oxygen plasma etching, damages graphene as well. In contrast, thermal annealing has been shown to preserve the high quality of graphene while removing some – but not all (Lin et al ., 2012 ) – contaminants, and allowing atomic resolution imaging of graphene (Ishigami et al ., 2007 ), Fig. 9.3 (b and c). The majority of the methods used are variations of the method fi rst reported by Ishigami et al . ( 2007 ): annealing in hydrogen/argon atmosphere at 400 °C for longer than an hour (Fig. 9.3 ). It is also possible to anneal graphene via ohmic heating, by sending a large current between two terminals connected to graphene (Moser et al ., 2007 ). The latter approach allows in situ removal of contamination from graphene samples that are measured in high vacuum, inside a cryogenic apparatus, thereby avoiding atmospheric contamination.

Most studies report modest but consistent improvement in mobility of supported graphene after annealing. It is also notable that almost every study reporting graphene devices with high carrier mobility employed some form of annealing. For mechanically exfoliated graphene on SiO 2 , the typical mobility before annealing is 2000–30 000 cm 2 V − 1 s − 1 and

(a)

(b)

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9.3 Annealing of graphene. Reproduced with permission from Ishigami et al. , 2007 . Atomic force microscopy image of graphene (a) before and (b) after annealing. (c) Atomically resolved image of a graphene sheet obtained via scanning tunneling microscopy after annealing. Scale bars are 300 nm in (a) and (b), and 2.5 Å in (c).


10 000–30 000 cm 2 V − 1 s − 1 after (Tan et al ., 2007 , Ponomarenko et al ., 2009 ). For graphene grown by chemical vapor deposition and deposited onto boron nitride substrates, the typical mobility increases from 2000–5000 cm 2 V − 1 s − 1 to 10 000–30 000 cm 2 V − 1 s − 1 (Gannett et al ., 2011 ). In contrast, the mobility of suspended graphene devices has been shown to improve by more than order of magnitude upon current annealing (Bolotin et al ., 2008b ).

9.5.2 Substrate engineering

To minimize substrate-induced scattering, the ideal substrate material should (i) contain few or no charge impurities, (ii) be fl at, to minimize scattering caused by mechanical deformations of graphene and to avoid trapping impurities between graphene and the substrate, and (iii) have no interfacial polar phonon modes.

Although early reports suggested that the substrate material does not infl uence carrier mobility (Ponomarenko et al ., 2009 ), subsequent studies found that the transport quality can be dramatically improved via substrate engineering. For example, mobility in excess of 70 000 cm 2 V − 1 s − 1 has been measured for multilayered graphene on single-crystal Pb(Zr 0.2 Ti 0.8 )O 3 (PZT) (Hong et al ., 2009a ). Covering the SiO 2 /Si substrate with a thin hydrophobic layer of hexamethyldisilazane (HDMS) signifi cantly decreased residual doping in graphene, although it did not improve the carrier mobility (Lafkioti et al ., 2010 ). However, the fi nal breakthrough did not come until the study by Dean et al. , which reported consistent improvement in mobility up to 60 000 cm 2 /Vs by placing graphene onto crystalline hexagonal boron nitride (hBN) substrates (Dean et al ., 2010 ) (Fig. 9.4 ).

Multiple considerations make boron nitride an enticing substrate material for graphene devices (Dean et al ., 2010 ). First, strong bonding in the atomic structure of boron nitride renders it relatively inert and free of charge traps. Second, boron nitride is a dielectric with a large ( ∼ 6 eV) bandgap. Third, the lattice constant of hBN is well matched (98.3%) to that of graphene. Finally, optical phonon modes of hBN have twice the energy of the optical modes of SiO 2 .

To create graphene-on-hBN devices, Dean et al. fi rst exfoliated hBN fl akes onto SiO 2 /Si substrates using techniques developed for exfoliation of graphene. Graphene was exfoliated onto a separate polymer substrate and mechanically transferred onto hBN under accurate optical alignment. To remove the fabrication residues the devices were annealed in a Ar/H 2 atmosphere. Subsequent studies demonstrated multiple improvements of the transfer technique including the development of a completely dry transfer process and encapsulation of graphene by another layer of hBN on top to protect it from ambient contamination (Mayorov et al ., 2011 ,

214 Graphene

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9.4 High-mobility exfoliated graphene on hBN. Reproduced with permission from Dean et al. , 2010 . (a) AFM image of electrically-contacted graphene on hBN. White dashed lines indicate the edge of the graphene fl ake. Scale bar is 2 μ m. (b) Histogram of the height distribution indicates that roughness of graphene is indistinguishable from that of hBN. (c) Resistance versus applied gate voltage for monolayer graphene on hBN at different temperatures. (d) Longitudinal and Hall conductivity versus gate voltage at B = 14 T (dotted line) and 8.5 T (dashed line) for graphene on hBN.

Zomer et al ., 2011 ). Multiple measurements by multiple groups confi rm low scattering in graphene-on-hBN devices. Field effect mobilities exceeding 300 000 cm 2 /Vs have been reported at low temperatures and ∼ 100 000 cm 2 /Vs at room temperature. Even higher mobility values seem to be possible (Dean et al ., 2010 , Mayorov et al ., 2011 , Zomer et al ., 2011 ). The mean free path has been estimated to exceed 3 μ m at n ∼ 10 12 cm 2 . The magnitude of charge density fl uctuation of graphene, inferred both from transport and scanning spectroscopy measurements, is reduced to ∼ 10 9 cm − 2 (Dean et al ., 2010 , Xue et al ., 2011 ). No signature of scattering arising from


interfacial phonons in hBN has been detected in temperature-dependent measurements (Dean et al ., 2010 ).

It is worth noting that the mobility of graphene grown by chemical vapor deposition can also be dramatically improved by transferring it onto hBN substrates. Mobility values up to ∼ 60 000 cm 2 V − 1 s − 1 have been recorded in such devices (Petrone et al ., 2012 , Gannett et al ., 2011 ), Fig. 9.5 (d)–(f).

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9.5 High mobility in large-grain graphene grown via CVD. Reproduced with permission from Petrone et al. , 2012 . (a) Optical micrograph of large-grain CVD graphene transferred onto SiO 2 . (b) Large-grain CVD graphene crystal (outlined with a dashed line for clarity) transferred onto a hBN fl ake using the dry transfer method. (c) Completed Hall bar device fabricated on a hBN substrate. Plots of (d) conductivity (black line, fi t; gray line, data) and (e) fi eld effect mobility as a function of carrier density taken at 1.6 K for the graphene/hBN device shown in (c). (f) Landau fan diagram for the device.

216 Graphene

9.5.3 Elimination of defects/grain boundaries

For mechanically exfoliated graphene, the defect density is rather low and can probably be made even lower by thermal annealing (Ni et al ., 2010 ). However, the main challenge is to avoid defect formation in the process of device fabrication and measurement. Indeed, electron irradiation of graphene during electron beam lithography that is typically used to pattern graphene devices can produce signifi cant damage in graphene (Ryu et al ., 2008 , Teweldebrhan and Balandin, 2009 ). Rough edges obtained after graphene patterning can also contribute to scattering.

For graphene grown by chemical vapor deposition, disorder scattering plays a large role. Recent work demonstrated dramatic improvement in carrier mobility resulting from the elimination of crystallites boundaries. By using the growth technique yielding graphene with crystallites larger than hundreds of micrometers, Fig. 9.5 (a), the mobility up to ∼ 45 000 cm 2 V − 1 s − 1 has been achieved (Petrone et al ., 2012 ). In these devices substrate-induced scattering mechanisms were neutralized by transferring CVD-grown graphene onto hBN substrates. Similar improvements of mobility with grain size have been also observed for epitaxial graphene (Emtsev et al ., 2009 ).

Electrical transport in fi lms of reduced graphene oxide is dominated by hopping between interlocking graphene crystallites. The mobility of such fi lms can be increased to ∼ 5 cm 2 V − 1 s − 1 by using fi lms with large crystallites. Even larger mobilities ∼ 100 cm 2 V − 1 s − 1 have been reported for thicker reduced graphene oxide fi lms (Wang et al ., 2010 ).

9.5.4 Dielectric engineering

Most researchers now agree that Coulomb scattering is an important, if not dominant, scattering mechanism in graphene devices. This type of scattering can be reduced by depositing a material with high dielectric constant κ onto or under graphene. Because the lines of electric fi eld connecting carriers in graphene and charged impurities near it propagate through the dielectric surrounding graphene, by choosing a material with high dielectric constant these lines can be ‘spread apart’, thereby reducing the effective strength of Coulomb interactions (Jena and Konar, 2007 ).

Indeed, a ∼ 30% increase in the mobility has been reported for graphene covered by ice. This increase is consistent with screening of the Coulomb scattering by a thin layer of ice (Jang et al ., 2008 ). Multiple subsequent studies reported greatly improved mobility of graphene and other 2D materials on high- κ substrates, although not directly investigating the effect of varying κ (Hong et al., 2009a , Radisavljevic et al., 2011 ). For graphene fully embedded inside a dielectric material, the effects caused by screening are expected to be the strongest. Indeed, for graphene that is suspended


inside liquids with varying κ , a signifi cant effect of screening on mobility has been reported, with room temperature mobilities reaching 60 000 cm 2 V − 1 s − 1 for large κ (Fig. 9.6 ) . Finally, we note that it may be possible to ‘screen out’ the effects of the Coulomb scattering in graphene by placing another graphene layer next to it. A metal–insulator transition observed in such a system has been attributed to a reduced density of electron-hole puddles resulting from screening (Ponomarenko et al ., 2011 ).

9.5.5 Suspended devices

Perhaps the most radical approach to eliminate substrate-induced scattering employs suspended graphene devices. The fi rst electrical measurements of suspended graphene devices reported a tenfold improvement in mobility

–5

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9.6 High mobility in graphene suspended in liquids with high dielectric constant. Reproduced with permission from Newaz et al. , 2012 . (a) A diagram and (b) an AFM image of a multiterminal graphene device suspended inside a liquid with controlled composition. (c) Conductivity vs . carrier density for the same graphene device surrounded by liquids with different dielectric constants. (d) Mobility of graphene devices vs . the dielectric constant κ of graphene surrounding.

218 Graphene

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9.7 High mobility in suspended graphene. Reproduced with permission from Bolotin et al. , 2008b . (a) Suspended monolayer graphene in a Hall-bar geometry. (b) Resistivity vs . gate voltage for the device shown in (a) before and after ohmic annealing. The annealed device exhibits carrier mobility > 200 000 cm 2 V − 1 s − 1 . (c) Fractional quantum Hall state state υ = 1/3 (labeled ‘A’) observed in suspended graphene at high magnetic fi eld.

compared with graphene devices in SiO 2 (Bolotin et al ., 2008b , Du et al ., 2008 ). The mobilities of order ∼ 200 000 cm 2 V − 1 s − 1 have been reported in multiterminal suspended devices at n = 2 × 10 11 cm − 2 , mobilities of the order of millions have been estimated for two-terminal suspended graphene at low temperature (Tombros et al ., 2011b , Mayorov et al ., 2012 ).

Several approaches that have been used to create high mobility suspended graphene employed mechanically exfoliated graphene to reduce the possible contribution of disorder. In the fi rst approach, a graphene FET contacted with metal electrodes is fabricated on a SiO 2 /Si substrate, Fig. 9.7 (a). The SiO 2 sacrifi cial layer is then removed via a chemical etch, typically a diluted hydrofl uoric acid. Interestingly, the acid is very effectively wicked between graphene and the substrate and therefore only a very short etch is needed to create a suspended device. The fi nal fabrication step, sample drying, is


crucial as the force arising from the surface tension of the drying liquid may rip or collapse graphene onto the substrate. Either critical-point drying in CO 2 or drying in a solvent with low surface tension helps to avoiding graphene damage (Bolotin et al ., 2008b , Du et al ., 2008 ).

Typically, immediately after fabrication, suspended graphene is con-taminated with fabrication residues and its transport quality is not signifi cantly different from that of graphene on SiO 2 . However, the residues can be effectively removed via ohmic heating (Bolotin et al ., 2008b ). By measuring changes in the mechanical resonant frequency of suspended graphene upon annealing, it has been estimated that a residue layer that is several nanometers thick is fully removed by annealing (Chen et al ., 2009 ), Fig. 9.7 (b).

Several alternative fabrication techniques have been developed to combat fabrication residue contamination. First, graphene can be directly exfoliated onto pre-patterned electrodes (Bunch et al ., 2007 ). Second, suspended graphene can be created by depositing electrodes by metal evaporation through a shadow mask followed by a chemical etch of the underlying substrate (Bao et al ., 2010 ). In both of these techniques, graphene retains its high mobility because it is never exposed to polymeric resists.

Multiple metrics indicate very low carrier scattering in suspended graphene devices. The Hall effect mobility in suspended graphene can exceed 200 000 cm 2 V − 1 s − 1 at low temperature (Bolotin et al ., 2008b ). Field effect mobility and mobility estimated from the onset of the Shubnikov–de-Haas oscillation exceeding 10 6 cm 2 V − 1 s − 1 have been reported. Charge density inhomogeneity in the highest quality devices is as low as 10 8 cm − 2 , whereas the mean free path reaches micrometers (Tombros et al ., 2011b , Mayorov et al ., 2012 ). At room temperature, some studies report mobilities exceeding 100 000 cm 2 V − 1 s − 1 (Bolotin et al ., 2008a ), while others suggest that at least in non-strained devices the mobility is limited to < 40 000 cm 2 V − 1 s − 1 by scattering on fl exural phonons (Castro et al ., 2010 ).

The electronic transport in ultraclean suspended graphene devices at low temperatures is dramatically different from that in low-mobility graphene on SiO 2 . An insulating behavior becomes apparent in bilayer graphene, Fermi velocity get renormalized by electron–electron interactions, symmetry breaking is observed in the integer quantum Hall effect (IQHE), and the fractional quantum Hall effect emerges. We discuss these phenomena in the next section.

9.6 Physical phenomena in high-mobility graphene

Physical properties of graphene change dramatically as its mobility is increased. Below, we very briefl y review a range of exciting physical phenomena observed in clean graphene devices.

220 Graphene

Ballistic conduction starts to play a role in the electrical transport of graphene with the mean free path l MFP close to the characteristic device size. The manifestations of ballistic transport include σ ~ n dependence and the quantization of σ in the units of 2 e 2 /h (Du et al., 2008 , Tombros et al., 2011a ). Klein tunneling of relativistic Dirac fermions is observed through potential barriers that are sharp on the scale of l MFP (Katsnelson et al., 2006 , Young and Kim, 2009 ).

The Shubnikov–de-Haas effect and the integer quantum Hall effect (IQHE) dominate magnetoconductance of graphene when l MFP is larger than the magnetic length l eBB = � . The four-fold degeneracy and signatures of the Berry phase are particularly notable signatures of the graphene IQHE (Novoselov et al ., 2005 , Zhang et al ., 2005 ). It is interesting to note that recent improvements in sample quality allow observation of IQHE at magnetic fi elds lower than ∼ 100 mT.

For even cleaner graphene samples, the effects arising from electron–electron interactions become apparent (Kotov et al ., 2012 ). One such effect is breaking of the four-fold degeneracy in the Landau level spectrum of graphene indicated by the appearance of ν = 1 states and an insulating state at ν = 0 in the IQHE. In even higher quality suspended graphene samples and graphene on hBN, a rich panoply of phenomena has been observed:

• Fractional quantum Hall states characterized by large activation energies and unusual hierarchy (Bolotin et al ., 2009 , Du et al ., 2009 , Dean et al ., 2011 , Feldman et al ., 2012 , Lee et al ., 2012 ), Fig. 9.7 (c).

• Phase transition from a metallic to an insulating state in bilayer graphene probably driven by electron–electron interactions (Bao et al ., 2012 ).

• Renormalization of the Fermi velocity owing to electron-electron interactions (Li et al ., 2008 , Elias et al ., 2011 ).

• The step in conductivity at σ ∼ 0.6 × 2 e 2 /h observed in patterned graphene. This step is similar to the ‘0.7 anomaly’ seen in conventional two-dimensional electron gases (2DEGs) and may have origins in electron–electron interactions (Tombros et al ., 2011a ).

Multiple predicted phenomena rooted in electron–electron interactions that are not yet observed include Bose–Einstein condensation of Dirac fermions, superconductivity, the appearance of the non-Abelian υ = 5/2 state in FQHE, and the emergence of a mass gap in graphene (Kotov et al ., 2012 ).

Nearly all proposed applications of graphene rely on the availability of high-mobility samples. In graphene fi eld effect transistors, high carrier mobility leads to superior high-frequency operation and low heat dissipation. The sensitivity of graphene-based sensors is typically directly proportional to the mobility. For transparent conductive electrodes based on graphene, mobility is a key parameter determining whether graphene can compete


with traditional indium–tin oxide fi lms. The switching ratio of graphene-based electro-optical modulators is, in part, determined by carrier mobility. Finally, we note that recent work proposes using the effects of electron–electron interaction in ultraclean graphene for applications in electronics (Banerjee et al ., 2009 ).

9.7 Conclusion

In eight years following the fi rst measurements of electrical transport in isolated graphene, the carrier mobility of graphene was increased from ∼ 10 4 to > 10 6 cm 2 V − 1 s − 1 through various technological advancements. These high-mobility and low-scattering graphene devices provide a unique platform to investigate the rich physics of interactions between Dirac fermions (Kotov et al ., 2012 ). Device applications of high mobility graphene may also follow soon. It is interesting to note that graphene is following in the footsteps of its close relative, two-dimensional electron gases in GaAs/AlGaAs heterojunctions. In two decades, the mobility of 2D electron gases was improved by three orders of magnitude with mobility of the best devices now exceeding 3 × 10 7 cm 2 V − 1 s − 1 (Stormer, 1999 ). If it would be possible to further reduce scattering in graphene to reach these levels of mobility, anticipated phenomena including non-Abelian FQHE states and Bose–Einstein condensation of Dirac fermions may emerge (Kotov et al ., 2012 ).

It is important in this regard to answer the question of intrinsic mobility of graphene that would ultimately limit the reduction of scattering. At room temperature, the already demonstrated mobility of > 10 5 cm 2 V − 1 s − 1 (Mayorov et al ., 2011 ) is close to the intrinsic mobility limitation due to phonon scattering ∼ 2 × 10 5 cm 2 V − 1 s − 1 (Chen et al ., 2008b ). Similar values have been obtained at room temperature for graphene grown via an approach compatible with large-scale fabrication, chemical vapor deposition. At low temperatures, the situation is less clear. The values μ > 10 6 cm 2 V − 1 s − 1 at low carrier density have been reported for some graphene devices. Moreover, mobility in excess of 10 7 cm 2 V − 1 s − 1 has been measured for graphene on top of graphite (Neugebauer et al ., 2009 ) suggesting that it may be possible to realize graphene-based devices with quality exceeding the best AlGaAs/GaAs heterojunctions.

9.8 Acknowledgments

The author gratefully acknowledges support by ONR (award N000141310299), NSF (CAREER DMR-105685), and the Sloan foundation.

222 Graphene

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