Scattering and Interference in Epitaxial Graphene

Science  13 Jul 2007:
Vol. 317, Issue 5835, pp. 219-222
DOI: 10.1126/science.1142882


A single sheet of carbon, graphene, exhibits unexpected electronic properties that arise from quantum state symmetries, which restrict the scattering of its charge carriers. Understanding the role of defects in the transport properties of graphene is central to realizing future electronics based on carbon. Scanning tunneling spectroscopy was used to measure quasiparticle interference patterns in epitaxial graphene grown on SiC(0001). Energy-resolved maps of the local density of states reveal modulations on two different length scales, reflecting both intravalley and intervalley scattering. Although such scattering in graphene can be suppressed because of the symmetries of the Dirac quasiparticles, we show that, when its source is atomic-scale lattice defects, wave functions of different symmetries can mix.

Built of a honeycomb of sp2-bonded carbon atoms, graphene has a linear, neutrino-like energy spectrum near the Fermi energy, EF. This results from the intersection of electron and hole cones in the graphene band structure at the Dirac energy, ED. The linear energy dispersion and concomitant topological constraints give rise to massless Dirac quasiparticles in graphene, with energy-independent propagation speed vF ≈ 106 m/s (where vF is the Fermi velocity). Distinctive symmetries of the graphene wave functions lead to unusual quantum properties, such as an anomalous integer quantum Hall effect (1, 2) and weak antilocalization (3, 4), that have spurred an intense scientific interest in graphene (5). Bilayer graphene (57) is equally distinctive: Quasiparticle states are chiral (6) with Berry's phase 2π for the bilayer versus π for the monolayer (6). High carrier mobilities, chemical inertness, and the two-dimensional (2D) nature of graphene make it a promising candidate for future electronic-device applications (1, 2, 5, 8, 9). In particular, graphene grown epitaxially on SiC substrates and patterned via standard lithographic procedures has been proposed as a platform for carbon-based nanoelectronics and molecular electronics (8, 9).

Epitaxial graphene was grown on the silicon-terminated (0001) face of high-purity semiinsulating 4H-SiC by thermal desorption of silicon at high temperatures (8, 10). This method produces an electron-doped graphene system, with the Fermi level 200 to 400 meV above ED. The data that we present were obtained from a region identified as bilayer graphene (11). Scanning tunneling microscopy (STM) measurements were performed in a custom-built ultrahigh-vacuum, low-temperature instrument. We measured the scanning tunneling spectroscopy (STS) differential conductance dI/dV (where I is current and V is voltage) with lock-in detection by applying a small modulation to the tunnel voltage at ≈500 Hz. Differential conductance maps were obtained by recording an STS spectrum at each spatial pixel in the topographic measurement. All measurements reported here were taken at 4.3 K.

STM topographic images (Fig. 1) show the atomic structure and different types of disorder for epitaxial graphene on SiC(0001). At the atomic scale, the graphene is imaged as a triangular lattice (Fig. 1B), characteristic of imaging only one of the two graphene sublattices. Superimposed on this atomic structure is a modulation period of ≈2 nm caused by a reconstruction of the SiC interface beneath the graphene: a SiC “6 × 6” superstructure (12). Survey images reveal two categories of defects. Type A defects, such as mounds (red arrow in Fig. 1A), have an unperturbed graphene structure that is continuous, akin to a blanket. These defects are due to irregularities in the interface layer between graphene and the SiC bulk. In contrast, type B defects are atomic defects within the graphene lattice itself (Fig. 1, A, C, and D) and are accompanied by strong distortions in the local lattice images. These distortions are of electronic origin and are accompanied by large increases in the local density of states (LDOS) at the defect site (13, 14). Quasiparticle scattering from type B defects gives rise to spectacular patterns in the topographic images (Fig. 1, C and D) resulting from the symmetry of the graphene Bloch states (1517).

Fig. 1.

STM topographic images of defects in the bilayer epitaxial graphene sample. (A) Large field of view showing a variety of defects. Type A defects (red arrow) are subsurface irregularities blanketed by graphene. The defect indicated by the red arrow has a height of 2 Å. Type B defects are atomic-scale defects in the graphene lattice. Higher-magnification images from the boxed regions in (A) are shown: a defect-free region (B) and complex scattering patterns around type B defects [(C) and (D)]. Tunneling setpoint: I = 100 pA, V = 300 mV.

Detailed information on scattering from both types of defects is obtained from STS maps of dI/dV (Fig. 2), which is determined by the LDOS. By comparing the topographic and spectroscopic images, we find that type B defects in the graphene lattice are the dominant scattering centers. Over much of the energy range studied, these atomic-scale defects have a large central density of states surrounded by a strong reduction in the LDOS that appears to pin the phase of the scattering pattern nearby. For example, the type B defects labeled by blue arrows in Fig. 2 show a bright central spot encircled by a dark region and a bright ring (Fig. 2, B to E). In contrast, the dI/dV maps show that type A defects, over which the graphene is continuous (red arrows in Fig. 2), have dramatically less influence on the LDOS.

Fig. 2.

Defect scattering in bilayer epitaxial graphene. (A) STM topography and (B to E) simultaneously acquired spectroscopic dI/dV maps. Type A defects (mounds) and type B defects are labeled with red and blue arrows, respectively. Sample biases are: (B) –90 mV, (C) –60 mV, (D) –30 mV, and (E) 30 mV. I = 500 pA, V = 100 mV, ΔV = 1 mVrms where ΔV is the modulation voltage.

Over large length scales, the dI/dV maps exhibit long-wavelength fluctuations that change with sample bias voltage (Fig. 2, B to E). As the sample voltage increases from –100 to +100 mV, the dominant wavelength decreases correspondingly from 9 to 5 nm. Fluctuations of much shorter wavelength are also present in these dI/dV maps, but they are not apparent over such a large displayed area. Figure 3 shows the short-wavelength modulations in dI/dV maps taken with atomic-scale spatial resolution. The interference patterns in these maps display a local (Math) R30 ° structure (Fig. 3, B to E) with respect to the graphene lattice, with a superimposed long-wavelength modulation. Both the long-wavelength standing-wave modulations and the Math periodicity are due to quasiparticle scattering from type B defects through wave vectors determined by the electronic structure of epitaxial graphene.

Fig. 3.

Bilayer graphene topography (A) and simultaneous dI/dV maps at sample bias voltages of (B) –31 mV, (C) –13 mV, (D) 1.0 mV, and (E) 21 mV. The type B scattering centers lie outside the image region [see lower left corner of (A)]. (F to I) dI/dV (color scale) versus sample bias (horizontal axis) and distance (vertical axis) along corresponding red lines in (B) to (E). The blue-white-red color scale spans the conductance values observed in (J) to (M). (J to M) Line-averaged dI/dV spectra obtained from regions marked by red lines in (B) to (E). The spectra are averages of nine curves acquired at positions of the Formula interference maxima in the region of the red lines. Peaks in the dI/dV spectra correlate with maxima in the long-wavelength modulation of the Formula interference pattern. Blue arrows indicate the bias (energy) position of the corresponding conductance images in (B) to (E). I = 500 pA, V = 100 mV, ΔV = 0.7 mVrms.

The 2D constant-energy contours in reciprocal space (Fig. 4A) are used to understand the scattering vectors that define the interference patterns observed in the STS maps of Figs. 2 and 3 (18). For graphene, the constant-energy contours near EF cut through the electron or hole conical sheets, resulting in small circles of radius κ, centered at the wave vectors K+ locate and K, that each locate three symmetry-equivalent corners of each the 2D Brillouin zone. The scattering wave vectors q connect different points on the constant-energy contours (Fig. 4A). Two dominant families of scattering vectors, labeled q1 and q2, give rise to the patterns observed in the spectroscopic conductance maps. Wave vectors q1 connect points on a single constant-energy circle (intravalley scattering) and determine the observed long-wavelength patterns. Wave vectors q2 connect constant-energy circles at adjacent K and K points (intervalley scattering), yielding scattering wave vectors close in length to that of K±. K+ (K) is related to the reciprocal lattice vectors G by a rotation of 30° (–30°) and a length that is shorter by Math in reciprocal space. This relationship gives rise to the (Math) R30 ° real-space superstructures observed in the high-resolution maps (Fig. 3, B to E). The vectors q2 will differ from the exact K± wave vector because of the finite size of the constant-energy circles. The combination of different lengths contributing to q2 leads to the modulation of the Math scattering patterns in Fig. 3.

Fig. 4.

(A) Schematic of the 2D Brillouin zone (blue lines), constantenergy contours (green rings) at the K± points, and the two dominant classes of scattering vectors that create the interference patterns. k1 and k2 denote the wave vectors of incident and scattered carriers. Scattering wave vectors q1 (short red arrow) are seen to connect points on a single constant-energy circle, and wave vectors q2 (long red arrow) connect points on constant-energy circles between adjacent K+ and K points. Red circles indicate graphene reciprocal lattice points with origin Γ. (B) q-space map of scattering amplitudes, obtained from the Fourier transform power spectrum of the dI/dV map in Fig. 2D. q1 scattering forms the small ring at q = 0, whereas q2 events create the six circular disks at K± points. (C) Angular averages of the central q1 ring from the q-space maps, at bias voltages from –100 to –20 mV shown in 10-mV increments, arb., arbitrary units. (D) Energy dispersion as a function of κ for bilayer graphene determined from the q-space profiles in (C) and similar data. Values shown are derived from the radii of the central q1 scattering rings (red squares) and from the angle-averaged radii of the scattering disks at K+ and K (blue triangles). Dashed line shows a linear fit to the data with vF = (9.7±0.6) × 105 m/s and an energy intercept of –330 ± 20 meV. Similar results are found for a single monolayer of graphene (fig. S2D) (18). The error bars represent the typical combined statistical and systematic uncertainties estimated for each data set.

To quantify the observed interference patterns and deduce the local band structure, we obtain q-space images of the scattering vectors (Fig. 4B) from Fourier transform power spectra of the spectroscopic dI/dV maps (19, 20). In Fig. 4B, q1 scattering appears as a bright ring centered at q = 0. The ring is a consequence of the enhanced phase space for scattering near the spanning vectors of the constant-energy circle. Circular disks appear centered at the K and K points because of the distribution of q2 wave vectors. We determined ring radii for the central ring (Fig. 4C) and the K± point disks using angular averages to maximize the signal-to-noise ratio. Both features change radius as a function of bias voltage because of dispersion in the graphene electronic states, and for these extremal q values, the scattering geometry determines |q| = 2κ or |q ± K±| = 2κ. The resulting κ values vary linearly with energy (Fig. 4D), with vF = (9.7 ± 0.6) × 105 m/s (18). The κ = 0 energy intercept gives the Dirac energy, EFED = 330 ± 20 meV. This local measurement of E(κ) agrees well with photoemission studies of bilayer epitaxial graphene (7), and the parameters are close to those reported from transport studies on epitaxial graphene grown on Math (9). Similar results are found for a single monolayer of graphene (fig. S1) (18).

In addition to states localized on defect sites, sharp conductance peaks, ≈5 meV in width, are found several nanometers from the nearest type B defect (Fig. 3). The peaks are clearly associated with the q2-induced Math LDOS modulation, as can be seen in the dI/dV maps (Fig. 3, B to E) and the spectral line profiles (Fig. 3, F to I). Furthermore, the data show that these conductance peaks are spatially localized, with maximum intensity in regions of constructive interference (i.e., over broad maxima modulating the Math pattern in Fig. 3, B to E). We attribute these conductance peaks to scattering resonances, which localize quasiparticles because of constructive interference in scattering from the random arrangement of defects found within a phase coherence length (18).

In support of these conclusions, Fig. 3, F to I, displays sequences of dI/dV spectra taken along the red lines shown in Fig. 3, B to E (the red lines are in regions of maximum intensity modulation for the four different energies of the dI/dV maps in panels B through E). Each of the figures shows a very prominent q2 modulation along the vertical (distance) axis at the energy of the corresponding dI/dV map (B to E). The lower set of panels (J to M) shows dI/dV spectra obtained at positions of the Math maxima, in the general areas of constructive interference (i.e., near the red lines). Clearly, the energy-dependent standingwave patterns are associated with conductance peaks of different energies. Across the series of maps and spectra, resonances decrease in intensity as new ones acquire increased spectral strength, each corresponding to a particular spatial location of constructive interference in panels B to E. Resonances are seen at –31 mV (Fig. 3, F and J), at –13 mV (Fig. 3, G and K), straddling the Fermi energy at ±1 mV (Fig. 3, H and L), and at several energies above the Fermi level (Fig. 3, I and M). Many more spectral peaks are observed for different spatial locations in the data set in Fig. 3, with equally narrow line widths.

Of particular interest is the influence of the observed scattering centers on the transport properties of epitaxial graphene. For perfect monolayer graphene, the lattice A-B site symmetry and the K± valley symmetry give rise to wave functions with distinct values of pseudospin and chirality (3, 21, 22). Both quantities are tied directly to the group velocityofthe quasiparticle wave function, and their near-conservation in the presence of weak potentials is equivalent to a suppression of backscattering. Our measurements of both q1 and q2 scattering processes show very directly that in-plane atomic defects are a dominant source of both intravalley (pseudospin-flip) and intervalley (chirality-reversal) backscattering. This result may explain the observation of weak localization in similar samples (8, 18). The related phenomenon of weak antilocalization was recently confirmed in epitaxial graphene grown by a different method on carbon-terminated Math substrates (4), indicating a very low density of inplane atomic scattering centers in those samples. Thus, the transport properties in epitaxial graphene are critically influenced by the microscopic properties of the sample, determined (at least) by the substrate and growth conditions. For carbon-based electronics, this work highlights the need for further microscopic studies that are correlated closely with macroscopic transport measurements.

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