PERSPECTIVE pubs.acs.org/NanoLett
Graphene: Electronic and Photonic Properties and Devices Phaedon Avouris* IBM T. J. Watson Research Center, Yorktown Heights, New York 10598, United States ABSTRACT Graphene is in many respects a nanomaterial with unique properties. Here I discuss the electronic structure, transport and optical properties of graphene, and how these are utilized in exploratory electronic and optoelectronic devices. Some suggestions for needed advances are made. KEYWORDS Graphene, electronic structure, electrical transport, optical properties, high-frequency transistors, photodetectors
G
raphite, the source of graphene, has been used since ancient times, although its exact nature was not known (it was thought to be a kind of lead ore, “plumbago”) until 1777, when Scheele showed that it is pure carbon. There were early reports of graphene isolation by exfoliation of oxidized graphite followed by reduction,1 formation of graphene by thermal decomposition of SiC,2 and the generation of graphene layers on transition metals.3 However, although discussions of the electronic structure of graphene were common in textbooks, and graphene had been used as the basic structure to describe the electronic states of carbon nanotubes and fullerenes,4 direct experimental work on graphene itself was lacking. In 2004, Novoselov, Geim, and co-workers at the University of Manchester used a simple mechanical exfoliation technique to obtain supported single layer graphene and study its electrical properties.5,6 This study has generated enormous interest and intense activity on graphene research.7-12 Electronic Structure and Carrier Transport. There is good reason for this interest in graphene. The π electrons in graphene provide an ideal 2D system: single atom thickness with the π and π* states noninteracting. Unlike conventional 2D systems formed at buried semiconductor interfaces, this structure is directly amenable to physical and chemical modifications and measurements. The hexagonal honeycomb lattice of graphene, with two carbon atoms per unit cell (Figure 1a), leads to a rather unique bandstructure that was first calculated by Walace in 194713 (Figure 1b). The π-states form the valence band and the π* states the conduction band. These two bands touch at six points, the so-called Dirac or neutrality points. Symmetry allows these six points to be reduced to a pair, K and K′, which are independent of one another. If we limit ourselves to low energies, which are the most relevant in electron transport, the bands have a linear
dispersion and the bandstucture can be viewed as two cones touching at EDirac (Figure 1c,d). This is because the orthogonal π and π* states do not interact, so their crossing is allowed. The fact that these bands touch at EDirac indicates that graphene has zero band gap, and it is therefore usually described as a zero-gap semiconductor. Since the bandstructure is symmetric about the Dirac point, electrons and holes in pure, free-standing graphene should have the same properties. The linear dispersion is reminiscent of the dispersion of light
E ) cpk where c is the light velocity. Moreover, the fact that there are two sublattices, A and B, in the structure of graphene (Figure 1a) allows the Hamiltonian describing it to be written in the form of a relativistic Dirac Hamiltonian
H ) vFσ · pk where σ is a spinor-like wave function, vF is the Fermi velocity of graphene, and k the wavevector of the electron. However, the spinor character of the graphene wave function arises not from spin but from the fact that there are two atoms in the unit cell. We can define a pseudospin corresponding to the hopping of the electron between the A and B sites. Thus, electrons in graphene can be described in the same way as relativistic particles. The energy of a relativistic particle is given by
E ) √m2c4 + p2c2 where m is the rest mass of the particle, p is its momentum, and c its velocity. However, electrons in graphene have linear dispersion, i.e.,EG ) vFp, which implies that electrons in graphene behave as zero rest-mass, relativistic Dirac Fermions. Unlike the constant density of states (DOS) of 2D systems with parabolic dispersion, the graphene DOS increases linearly with energy. Graphene exhibits outstanding transport properties.14,15 In the ballistic transport regime, carriers move with a Fermi velocity of vF ≈ 106 ms-1. As in the case of metallic nanotubes, backscattering through long-
* E-mail
[email protected]. Published on Web: 09/29/2010
© 2010 American Chemical Society
4285
DOI: 10.1021/nl102824h | Nano Lett. 2010, 10, 4285–4294
PERSPECTIVE
FIGURE 1. (a) Hexagonal honenycomb lattice of graphene with two atoms (A and B) per unit cell. (b) The 3D bandstructure of graphene. (c) Dispersion of the states of graphene. (d) Approximation of the low energy bandstructure as two cones touching at the Dirac point. The position of the Fermi level determines the nature of the doping and the transport carrier.
depending on the nature of the dominant scatterers.21,22 In suspended, exfoliated graphene, where interactions with the substrate are eliminated, mobilities of about 200000 cm2 V-1 s-1 have been observed14,15 (×10 higher than that of an InP HEMPT), and even higher values are currently being obtained. On insulators, such as amorphous SiO2, the mobility is significantly lower: a few thousand to tens of thousands cm2 V-1 s-1, depending on the nature and purity of the insulator. On single crystal substrates with high surface phonon frequencies, such as boron nitride, higher mobilites have been reported.31 Currently, epitaxial and chemical vapor deposition (CVD) grown graphene typically shows lower mobilities, on the order of a few thousand cm2 V-1 s-1.32 It should be noted that conventional transistors are typically operated at the so-called velocity saturation regime. This is the limiting velocity which carriers acquire when they have enough kinetic energy to excite the optical phonons of the material, i.e., EKin g pωop. The optical phonons of graphene and carbon nanotubes have much higher frequency (∼1600 cm-1) than those of inorganic semiconductors such as GaAs (∼300 cm-1) or Si (∼500 cm-1). Similarly, the intrinsic saturation velocities of graphene and carbon nanotubes are higher, ∼4 × 107cm
range interactions, such as charged impurities or phonons, is forbidden, and the elastic mean-free paths in clean samples are typically of the order of hundred nanometers. Scattering by acoustic phonons is rather weak and the optical phonons have very high frequency (∼1,600 cm-1) so that scattering by them becomes important only at high applied electric fields. In long graphene channels the carriers undergo elastic and inelastic collisions and transport becomes diffusive.9,16,17 The elastic scattering mechanisms discussed in the literature involve Coulomb scattering by charged impurities (primarily trapped charges in the underlying insulating substrate),18-22 short-range scatterers (defects, adsorbates) in or on graphene,9 and surface roughness or ripples of the graphene structure.23 Inelastic scattering results from the phonons of graphene;24-26 it can also involve the surface phonons of a polar insulating substrate.27-30 These thermally excited surface phonons generate an electric field that extends away from the substrate surface, which couples to the graphene carriers and leads to scattering. An important factor affecting the efficiency of the carrier mobility in graphene is the carrier density. Increasing this density in monolayer graphene generally decreases the mobility, with the exact behavior © 2010 American Chemical Society
4286
DOI: 10.1021/nl102824h | Nano Lett. 2010, 10, 4285-–4294
PERSPECTIVE s-1, than those in GaAs (∼2 × 107 cm s-1), Si (∼1 × 107 cm s-1), or InP (∼0.5 × 107 cm s-1).40 However, there is evidence that the graphene substrate through the coupling of its surface phonons to the carriers can influence the value of saturation velocity.30,44 This is still an active area of research. Finally, comparing the transport properties of graphene and carbon nanotubes, we find that they are, in many respects, similar. For example, in both cases the carriers can have very high mobilities and ambipolar transport. However, there are also differencessthe most important, from the device point of view, being the difference in dimensionality (1D vs 2D) and the resulting changes in the Landauer (contact) resistance and, of course, the presence of a band gap in semiconducting nanotubes. From the practical point of view, nanotubes, with their many different chiralities, pose a problem in producing a well-defined starting material for technology. Graphene, on the other hand, is well-defined and its planar geometry allows the use of the highly advanced techniques already developed in the semiconductor industry. While graphene itself has excellent electrical properties, carriers have to be injected into it and then collected through metal contacts. These contacts generate potential energy barriers that have to be circumvented and can profoundly affect the performance of graphene devices. Graphene and metal typically have different work functions, which causes charge transfer between them. The resulting dipole layer leads to the doping of graphene underneath the metal and a bending of the graphene bands that extends some distance into the graphene channel. In contacts involving more reactive metals, like Ni or Pt, there is also significant rehybridization of graphene. A carrier injected into graphene must pass both the dipole barrier and the doped-undoped graphene (p-n junctionlike) barrier.33-36 The height of the latter depends on the extent and direction of the charge transfer and the initial doping of the graphene. This height is greater for p-n junctions than for p-p′ or n-n′ junctions. This barrier, which is usually treated as a case of Klein tunneling (see below), also introduces an asymmetry between electrons and holes.33,35 The contact resistance, Rc, is usually measured by fabricating variable channel length devices and extrapolating to zero channel length. It was found that Rc is gate bias dependent, varying from about 100 Ω µm to a few kΩ µm, with a maximum near the Dirac point.37 This behavior can be understood on the basis of electrical screening of the doping barrier and the change of the number of transport channels in graphene by the gate field. Once the carriers have been injected into the graphene channel, their transport is controlled by the gate. Negative gate bias raises the Fermi level, while positive lowers it. Thus, for Vg < VDirac transport in graphene involves holes, while for Vg>VDirac electrons are transported (Figure 1d). © 2010 American Chemical Society
As the Fermi energy is changed by the gate, the DOS and corresponding carrier density are also changed. This is the basis of “switching” in graphene. However, unlike transistors made of conventional semiconductors with a band gap, a graphene transistor does not turn off completely, even though DOS ) 0 at EDirac. The relativistic, massless Dirac Fermion behavior of quasi-particles (electron and holes) in graphene combined with the absence of a band gap has important implications for the electrostatic confinement of these quasi-particles. Back in 1929 Klein38 discussed the problem of an electron encountering a potential barrier. He found that, while nonrelativistic electrons tunnel through the barrier with an exponential damping, relativistic electrons with energies higher than 2mc2 (the Dirac energy gap) travel through the barrier as if it was nearly transparent. This transparency can be understood in terms of particle (electron)-antiparticle (positron) formation that requires an energy of ∼1 MeV, or electric fields of the order of 1016 V/cm. However, since there is no band gap in graphene, this type of tunneling (Klein tunneling) can occur in graphene even when the energy barrier and fields induced by the gate are small (∼100 meV and ∼105 V/cm, respectively). In monolayer graphene, electrons with wavevectors perpendicular to the barrier are transmitted, while other, oblique incident angles are reflected.39 In bilayer graphene, where the electrons behave as “massive” Dirac Fermions, the opposite is true.39 Graphene Transistors. Both conventional semiconductors with band gaps >0.5 eV and semiconducting nanotubes which have band gaps that scale inversely with their diameter allow for excellent gate switching with current on/off ratios of the order of 104-107, appropriate for digital switches. It is this ability of Si CMOS devices that leads to their extremely low static power dissipation and has made them the dominant digital technology. Graphene, on the other hand, does not have a band gap and therefore does not have that capability. Switching in graphene is controlled by the variation of the DOS with EF and the carrier scattering time τ. The conductivity σ in the diffusive transport regime varies as σ ∝ DOSτ.19 Experimentally, most studies of graphene FETs (GFETs) obtain Ion/Ioff ratios e10, although in very pure samples (long τ) higher values can be obtained. In Figure 2a we show an example of the change of the conductance of a graphene field-effect transistor device with top-gate bias for several backgate biases, which tune the position of the Dirac point. So, if graphene devices are not appropriate for digital switches, why is there such a strong interest in graphene electronics? The key attractions of graphene are its outstanding carrier mobility, the good transconductance of graphene devices, and the ultimate thinness and stability of the material. These characteristics suggest that graphene may be an ideal material for radio frequency (rf) analog electronics. In analog rf operation complete 4287
DOI: 10.1021/nl102824h | Nano Lett. 2010, 10, 4285-–4294
PERSPECTIVE
FIGURE 2. (a) Conductance of a single layer graphene used as the channel of a field effect transistor, as a function of the top-gate bias voltage at different back-gate voltages. Inset: The structure of the device which uses a gate stack composed of a thin NFC (polystyrene) polymer layer followed by 10 nm of HfO2. (b) Electron and hole mobilities of exfoliated single layer graphene incorporated as the channel of a transistor on 300 nm SiO2, as a function of carrier density and temperature based on Hall effect measurements. From refs 57 and 22, respectively.
of graphene on Cu,53 and thermal segregation of graphene from transition metals,54,55 have been used to grow large scale graphene. Most of the rf GFETs reported have been fabricated on SiC-derived graphene, and the mobilities in this material has been lower than those in exfoliated graphene (typically 130 meV opens up in the graphene bilayer. From ref 69.
their difference, Db - Dt, determines the position of the Fermi level. At the neutrality point, Db ) Dt. The current on/off ratio achieved from the dual-gated FET at room temperature is 100 (Figure 4b), while an otherwise identical monolayer GFET exhibits an on/off ratio of 5. The induced band gap of this device is field-tunable with a maximum value of 130 meV.69 Graphene Photonics. Graphene’s optical properties are also an area of strong interest, from both fundamental science and technological points of view. Assuming that only vertical (k-conserving) transitions are allowed (Figure 5a), light transmittance (T) through free-standing graphene can be derived using the Fresnel equations for a thin film with a universal optical conductance of G0 ) e2/ 4p to be71,72
via electron-electron intraband scattering and phonon emission within ∼100 fs to reach a new Fermi-Dirac distribution, followed by interband carrier relaxation and hot phonon emission on the picosecond time scale.78 Resonant fluorescence is not observed in graphene due to the fast relaxation. The high conductivity and large light transmission of graphene suggest that it will make an excellent conductive electrode for a number of applications including solar cells, flat panel displays, touch screens, and OLEDs. Indium tin oxide (ITO) is currently used in most of these applications. ITO has a sheet resistance
