## Abstract

Ultrathin epitaxial graphite was grown on single-crystal silicon carbide by vacuum graphitization. The material can be patterned using standard nanolithography methods. The transport properties, which are closely related to those of carbon nanotubes, are dominated by the single epitaxial graphene layer at the silicon carbide interface and reveal the Dirac nature of the charge carriers. Patterned structures show quantum confinement of electrons and phase coherence lengths beyond 1 micrometer at 4 kelvin, with mobilities exceeding 2.5 square meters per volt-second. All-graphene electronically coherent devices and device architectures are envisaged.

The fundamental limitations of silicon-based microelectronics have inspired searches for new processes, methods, and materials. We show here that the properties of epitaxial graphene are suitable for coherent nanoscale electronics applications (*1*). In particular, an ultrathin graphite layer grown on a commercial single-crystal silicon carbide by thermal decomposition has high structural integrity. The single graphene layer at the graphite-SiC interface has impressive two-dimensional (2D) electron gas properties, including long phase coherence lengths (even at relatively high temperatures) and elastic scattering lengths that are determined primarily by the micrometer-scale sample geometry. Magnetotransport measurements of patterned structures reveal signatures of quantum confinement, thus demonstrating that graphene ribbons act as electron waveguides. The material can be patterned, and intricate submicrometer structures can be constructed using standard microelectronics lithography methods, in contrast to the closely related carbon nanotube electronics. The transport properties show that electrons in the interfacial graphene layer dominate the transport and that they are Dirac fermions, as recently observed in mechanically exfoliated graphene layers (*2*, *3*). The properties of Dirac fermions, which are also responsible for transport in carbon nanotubes (*4*), can be conveniently explored in epitaxial graphene.

The electronic properties of sp^{2}-bonded carbon structures (which include bulk graphite, graphene ribbons, carbon nanotubes, and aromatic molecules) result from the overlap of p_{z} atomic orbitals on neighboring carbon atoms. Simple tight-binding calculations show that in graphene, π-bands are formed, with electronic energy dispersion *E*(*p*) = ±ν_{0}|p|, where the carrier momentum (*5*–*7*), ν_{0} is the velocity, and *ħ* is Planck's constant divided by 2π. Consequently, like photons, the velocity of electrons is independent of energy. The predicted velocity ν_{0} = 3*a*_{0}γ_{0}/2*ħ* ≈ 10^{6} m/s, where γ_{0} ∼ 3 eV is the interatomic overlap energy and *a*_{0} = 1.42 Å is the interatomic spacing. The Fermi surface of neutral graphene (Fermi energy *E*_{F} = 0) shrinks to a point, so that it is a zero-band-gap semiconductor (*6*). When carriers in graphene are confined, their properties depend on the confinement geometry, as is true for other 2D electron gases (*8*). However, in contrast to conventional 2D electron systems, which are electrostatically confined at a buried interface, epitaxial graphene (*1*, *9*, *10*) is a well-defined material that is robust, so that in principle it can be sculpted down to the molecular level.

We present production methods for epitaxial graphene (EG) and show that the transport properties of a representative patterned EG structure are due to carrier confinement and coherence. From magnetoresistance (MR) and Hall-effect measurements, we find that the material that dominates the transport is in fact graphene. This conclusion follows from the following measured properties: Berry's phase Φ_{B} = π, Fermi temperature *T*_{F} = 2490 K, Fermi velocity ν_{0} = 1.0 × 10^{6} m/s, mobility μ = 2.7 m^{2}/V·s, carrier diffusion constant *D* = 0.3 m^{2}/s, elastic mean free path *l*_{s} = 600 nm, and phase coherence lengths of = 1.1 μm at 4 K and 500 nm at 58 K. Furthermore, quantum-confined states are observed.

The production of EG on diced (3 mm by 4 mm) commercial SiC wafers (*11*) is illustrated in Fig. 1. In summary, the steps are (i) hydrogen etching to produce atomically flat surfaces (*12*), (ii) vacuum graphitization to produce an ultrathin epitaxial graphite layer (*9*, *10*), (iii) application of metal contacts (Pd, Au), (iv) electron-beam patterning and development, (v) oxygen plasma etch to define graphite structures, and (vi) wire bonding. To date, more than 200 SiC sample blanks have been processed, of which 22 have been patterned and measured in detail [see also (*1*)]. The structural order has been characterized by low-energy electron diffraction (LEED), Auger electron spectroscopy, x-ray diffraction (*13*), and scanning tunneling microscopy (STM) (*1*). The electronic structure has been characterized by angle-resolved photoelectron spectroscopy [ARPES (*14*)], scanning tunneling spectroscopy (STS) (*1*), and electronic transport [see below and (*1*)]. Patterned surfaces are routinely measured by atomic force microscopy (AFM) and electrostatic force microscopy (EFM) under ambient conditions. The results are summarized in Fig. 1. As evident from LEED, an ultrathin graphite layer grows epitaxially on the surface. X-ray diffraction shows that graphene grown on the face has a structural coherence of at least 90 nm (*13*). On the Si-terminated SiC(0001) surface, LEED and STM measurements reveal a interface reconstruc-STS measurements reveal the graphitic band structure, and STM and STS together suggest that graphene layers can remain continuous over steps on the SiC surface (*1*). The ARPES data (for two EG layers on 0001 SiC) suggest a Dirac electronic dispersion and a Fermi temperature of 2700 K (*14*). This relatively large energy indicates that the interface layer is charged, with a charge density σ ∼ 1 × 10^{12} electrons/cm^{2} (see below). As usual for such interfaces, the electric field caused by the surface charge compensates the work function difference between the materials. Only the interface layer is expected to be highly charged (*15*, *16*); the (few) other layers are essentially neutral. Thus, the interface layer should dominate the transport properties, which are essentially identical to those of isolated graphene (see below). The interface layer is further distinguished from any others by a weak superlattice structure imposed by the epitaxial match to SiC (*1*).

We briefly summarize some relevant properties of confined Dirac electrons in graphene. For a graphene ribbon of width *W*, the boundaries impose a constraint on the transverse motion so that (for not too small *n*) *k*_{y} is quantized: *k*_{y} = *n*π/*W*, where *n* is an integer (*17*). Hence, the energy of the *n*th electronic subband is , where Δ*E* = Δ*E*(*W*) = π*ħ*ν_{0}/*W* ∼ (2 eV·nm)/*W*, and *E*_{x} = *ħ*ν_{0}*k*_{x}. Hence, these electrons propagate like electromagnetic waves in waveguides. A more detailed analysis shows that undoped graphene (i.e., *E*_{F} = 0) can be tuned to be either a metal or a semiconductor with a band gap on the order of Δ*E*(*W*) (*18*, *19*). This is an important property that undoped graphene ribbons share with undoped carbon nanotubes.

For any 2D electron system (*20*–*22*), a perpendicular magnetic field *B* creates a discrete energy spectrum (Landau levels) due to quantization of the cyclotron orbits (radius *R*_{c} = *p*/*eB*). The energy states for Dirac electrons are given by (*2*, *3*, *23*) where *n* is the integer Landau level index [by contrast, for “normal” electrons, where *m** is the effective mass (*20*)]. Shubnikov–de Haas (SdH) MR maxima were observed at magnetic fields *B*_{n} such that *E*_{n}(*B*_{n}) = *E*_{F}. Hence, *B*_{n} = *B*_{0}/*n*, where (*24*–*26*).

Next, consider a graphene ribbon of width *W* in a magnetic field. It is intuitively clear that for low magnetic fields, when *W* < *2R*_{c} the ribbon cannot support a cyclotron orbit, so the above picture needs to be modified (*20*, *22*). For graphene, *2R*_{c} = *4n*/*k*_{F}, which is 260 nm for *n* = 20. Quantitatively, Berggren *et al*. (*27*) found that for a normal 2D electron system confined to a strip of width *W*, where *E*_{n}*W* = *n*π*ħ*/*W*^{2}*m** (*20*). Analogously, for graphene ribbons, *E*_{n}(*W*) = *n*π*ħ*ν_{0}/*W*, and an approximate expression for the energy levels can be obtained (*28*): (1) Recent numerical results agree very well with this analytical form (*29*). As for normal electrons, SdH peaks are expected when *E*_{n}(*B*_{n},*W*) = *E*_{F}. Consequently *B*_{n} = *B*_{0}/*n* for small *n*, whereas for large *n* ∼ *n*_{max} the peak spacing becomes more regular: *B*_{n} – *B*_{n+1} ∼ *B*_{0}/*n*_{max}, where *n*_{max} = *E*_{F}/Δ*E*(*W*) is the number of populated transverse modes.

To illustrate the properties of carrier confinement and coherence, we next present magnetotransport data from a representative patterned EG structure. The sample is a Hall bar (ribbon) of width *W* = 500 nm created on the graphitized face of a high-quality semiinsulating 4H-SiC substrate (Fig. 2A, lower inset). Contacts are bonded on Pd/Au deposited pads. Four-point measurements were made using standard lock-in methods, with excitation currents through the ribbon limited to 100 nA. Voltages were measured over a 6-μm length *L* of the ribbon. MR and Hall-effect data were acquired at six temperatures from 4 to 58 K and in magnetic fields from –9 to +9 T. Field sweeps were repeated to verify reproducibility.

Figure 2A shows that ρ_{xx} = *R*_{xx}*W**/*L* (where ρ_{xx} is the resistivity and *W** the effective ribbon width), and the Hall resistance *R*_{xy} = ρ_{xy} of the sample. ρ_{xx} increases approximately linearly with increasing field. At high fields, SdH oscillations are clearly seen in the ρ_{xx} curves and step-like features are observed in ρ_{xy}. Subtracting a common smooth curve from the MR data reveals a rich, reproducible, and temperature-dependent structure (Fig. 2B). Pronounced, regularly spaced SdH maxima are distinguished clearly at high fields, whereas at low fields MR peaks are visible but are less well defined. Increasing the temperature decreases the amplitude of the peaks, with the high-field peaks decreasing more slowly than the low-field peaks. At a given temperature the amplitudes are relatively constant for *B* < 2 T and increase uniformly for *B* > 2 T. Positions of 29 distinct SdH peaks *B*_{n} have been identified and are indicated in Fig. 2B. Features are identified as SdH peaks when they present a clear maximum, and they are present at all of the measured temperatures. A complication is the reproducible fine structure observed throughout the MR spectra, which can obscure the SdH peaks for small fields. These are (universal) conductance fluctuations. As discussed below, the temperature dependences of universal conductance fluctuations (UCFs) and SdH oscillations are well understood and quite distinct. Incorporating this information results in an ambiguity of less than 10% in the number of peaks assigned according to the above criteria.

A Landau plot of the peaks [i.e., versus *n* (*20*, *24*, *30*)] is shown in the inset of Fig. 3B. The low index peaks (*n* = 4 to 12) define a straight line: with *B*_{0} = 35.1 ± 0.8 T and γ = –0.05 ± 0.14. This value of γ is consistent with a Berry phase Φ_{B} = π, as expected for Dirac fermions and previously observed in graphene (*2*, *3*, *23*, *26*). It is specifically not consistent with γ = ±0.5, as would be the case for normal electrons (*23*). Note that the same result was found for various subsets of peaks in the interval *n* = 4 to 12. For a 2D electron gas in general, we have . From *B*_{0} we find *k*_{F} = 3.3 10^{8}/m, and the carrier density (where *g*_{s} = 2 and *g*_{v} = 2 are the spin and valley degeneracies) (*20*, *21*). The deviation from linearity for the larger index peaks (*n* > 14; *B* < 2.5 T; *2R*_{c} > 170 nm) indicates confinement (*20*, *27*), as explained below.

We next analyze the individual SdH peaks. The amplitudes of SdH peaks decrease with increasing temperature because higher Landau levels are thermally populated. The temperature dependence of the peak amplitudes is given by the Lifshitz-Kosevich (LK) equation (*23*, *31*): (2) where *A*_{n}(*T*) is the peak amplitude (or peak area) of the *n*th SdH peak at temperature *T*, and *u* = 2π^{2}*k*_{B}*T*/Δ*E*(*B*), where Δ*E*(*B*) = *E*_{n+1}(*B*) – *E*_{n}(*B*). Experimental values for Δ*E*(*B*) were determined by fitting SdH peaks to the LK equation for six different temperatures. In Fig. 4 the results are plotted as solid circles; values of Δ*E*(*B*) calculated from Eq. 1 are shown as open circles. A nearly linear increase for *B* > 4 T and saturation at low fields is observed in both theory and experiment. For large *B*, Δ*E*(*B*) ∼ *E*_{F}*B*/2*B*_{0} = *E*_{F}/*2n*; for small *B*, Δ*E*(*B*) = *ħ*πν_{0}/*W*. Consequently, the data in Fig. 4 can be used to find *E*_{F}/*k*_{B} = 2490 ± 80 K and . Our experimental ν_{0} agrees remarkably well with the accepted value for graphene (*6*). Furthermore, *E*_{F} is also consistent with recent ARPES measurements (*14*) on two-layer EG grown on SiC(0001), which found *E*_{F}/*k*_{B} = 2700 K. Hence, experimental evidence (measured Φ_{B}, ν_{0}, *E*_{F}) strongly supports the conclusion that the material is indeed graphene.

For small *B*, the saturation of Δ*E*_{n}(*B*)/*k*_{B} at 80 ± 10 K indicates quantum confinement. The number of confined subbands should be *n*_{max} ∼ *E*_{F}/Δ*E*(*B* → 0) = 31 ± 3, consistent with the observed *n*_{max} = 29 ± 3. However the hard wall boundary condition would predict that *n*_{max} = *Wk*_{F}/π = 52. This discrepancy suggests that the carrier confinement width of the ribbon is less than the lithographic width *W* = 500 nm (subsequent AFM and EFM studies confirmed that the physical width of the ribbon is compatible with the lithographic width; this does not preclude edge roughness effects). In fact, the best fit to the data using Eq. 1 is for *W** = 270 nm, as shown in Fig. 4 (see below for further evidence of reduced width). Similar discrepancies have been observed in several ribbons. The smaller effective width may be related to carrier scattering from steps on the substrate (Fig. 1; steps tend to run parallel to the ribbon) or to a stronger confinement potential caused by charge transfer to states at the ribbon edges (edge roughness, chemisorbed molecules, and intrinsic edge states could contribute), or it may have a more fundamental origin. In Fig. 3 the MR data are presented with the Landau index *n*(*B*) as the abscissa, obtained by inversion of Eq. 1 (for *E*_{n} = *E*_{F}), where the experimental values ν_{0} = 1.0 × 10^{6} m/s, *E*_{F} = 2490 *k*_{B}, and *W** = 270 nm were used. If Eq. 1 is correct, the measured SdH peaks should coincide with integers. The correspondences up to *n* = 22 are remarkable (note that the *n* = 11 peak is missing at low T), providing additional support for our conclusion that the SdH peaks are determined by both Landau orbital quantization and transverse quantum confinement.

The overall linear increase in the MR (Fig. 1A, inset) may result from a conducting layer on top of the graphene [i.e., a thin graphite layer, consistent with independent x-ray measurements on similar samples (*13*)]. The slope is removed completely by subtracting from the measurement the conductivity of a layer with resistivity ohms/square, and with negligible Hall coefficient. This procedure results in and values that are similar to those reported for isolated graphene. Hence, the resistivity of the graphene layer is ohms/square, versus 63 ohms/square for the sample before correction. The integrated carrier density derived from the Hall resistance is *n*_{Hall} = 6.5 × 10^{16}/m^{2}, compared with *n*_{s} = 3.4 × 10^{16}/m^{2} found for the graphene layer (see above). The difference can be attributed to the integrated carrier density of the conducting (presumably neutral graphite) layer. Hence, we find the graphene mobility μ* = (*n*_{s}*e*ρ*)^{–1} = 2.7 m^{2}/V·s, and carrier diffusion constant *D* = *E*_{F}/2*n*_{s}*e*^{2}ρ* = 0.30 m^{2}/s. From *D* = *l**ν_{0}/2, we obtain *l** = 600 nm, where *l** is the carrier mean free path in the graphene. This value is in excellent agreement with the limiting value *l** ∼ 3π*W**/4 = 635 nm for a ribbon 270 nm wide with only diffuse elastic boundary scattering (*20*), which implies that the resistance is determined primarily by the confining geometry and not by defect scattering in the material.

Alternatively, the conductance of a graphene ribbon can be estimated from the Landauer equation (*21*): *G* = (*e*^{2}/*h*)*g*_{s}*g*_{v}∑*T*_{n}, where ∑*T*_{n} is the sum over transmission coefficients of the *n*th modes (0 ≤ *T* ≤ 1). The *T*_{n} values are obtained approximately as *T*_{n} = *l*_{n}/*L*, where *l*_{n} is the mean free path of the *n*th mode (*21*). If we assume elastic scattering at the boundaries without mode mixing, then *l*_{n} is the distance along the wire between scattering events for the *n*th mode—that is, *l*_{n} = *k*_{x}/*k*_{n}*W* = *W*[(*k*_{F}*W*/*n*π)^{2} – 1]^{1/2}. Hence, *G* = (*e*^{2}/*h*)(*g*_{s}*g*_{v}*W**/*L*) Σ[(*n*_{max}/*n*)^{2} – 1]^{1/2}. With *W** = 270 nm, *L* = 6 μm, and *n*_{max} = 29, we find *R* = 1/*G* = 1430 ohms, or ρ_{g} = *RW**/*L* = 64 ohms/square, in remarkable agreement with experiment.

The sample resistance decreased uniformly from 1490 ohms at *T* = 180 K to a minimum of 1410 ohms at *T* = 30 K, below which it increased (Fig. 5, inset). From 300 to 30 K, the resistance decreased only 13%, which indicates [from Matthiessen's rule (*20*, *21*)] that the electron phonon-scattering time is ∼4 × 10^{–12} s at 300 K. The increase of resistance below 30 K is caused by the increasing phase coherence length (*20*, *22*, *32*). The resistance increase is a manifestation of constructive quantum interference between time-reversed trajectories, which enhances the probability for an electron to be localized at a scattering site (*20*, *22*, *32*). This well-understood weak localization (WL) effect is undone in a magnetic field because time reversal symmetry is lifted (*20*, *22*). The pronounced peak in the low-field MR at *B* = 0 shows this WL effect (Fig. 5). Clear universal conductance fluctuations flank the peak. The reproducible UCFs are caused by quantum interference from elastic scatterers (e.g., at the ribbon edges). For all fields, the widths of UCF features are similar to the WL peak width, in contrast to the SdH peaks, which are much wider. The WL peak saturates at low temperatures (peaks for 4, 6, and 9 K are similar), whereas the amplitude of the UCFs increases very rapidly with decreasing temperature (which also distinguishes UCFs from SdH peaks).

Weak localization and UCFs have been exhaustively investigated in 2D electron systems. Before saturation below 9 K, the observed decrease of the WL peak with increasing *T* fits a *T*^{–2/3} dependence, which indicates that electron-electron scattering is the primary dephasing mechanism (*22*). The decrease in amplitude of the UCFs with increasing *T* is caused by a reduction of combined with “thermal smearing,” which is characterized by the thermal length .

On the other hand, the WL is not sensitive to *l*_{T}. Its width Δ*B* is essentially determined by the field for which an electron trajectory of length encloses one flux quantum (for this sample » *W*, so that flux cancellation effects also need to be taken into account). Following the thorough development reviewed by Beenakker and van Houten (*20*), the field is parameterized in terms of , where , and *l*_{m} = (*ħ*/*eB*)^{1/2} is the magnetic length. The WL contribution to the conductance (for » *l** » *W*) is (3) and the root mean square amplitude of the UCFs is given by (4)

The only free parameter in these expressions is . The origin of the commonly observed low-temperature saturation of the WL is still debated intensely (*22*). Here we assume that it is caused by a physical limitation to , which appears to be the finite sample length. The effective phase coherence length saturates at *L*_{sat}/2π (*22*). The fits shown in Fig. 5 correspond to *L*_{sat} = 7 μm and = (7 μm)*T*^{–2/3} [at 4 K, , and ; at 58 K, , and = 6 × 10^{–13} s].

This analysis clearly shows that the resistance of the graphene ribbon is determined primarily by boundary scattering, with a coherence length far larger than the effective ribbon width, even at 58 K. Therefore, as also shown in our analysis of SdH oscillations, the low-field resistance depends directly on the eigenmodes of laterally confined carriers in the graphene ribbon. Other samples indicate that the width dependence of resistance can persist up to room temperature. Because electron-phonon scattering is weak even at room temperature (see above), electron-electron scattering should continue to be the dominant dephasing mechanism under ambient conditions (such scattering affects the resistance only through phase-breaking, i.e., by limiting ). Accordingly, on the basis of the measurements presented here, we expect to see quantum interference effects over distances exceeding 100 nm at room temperature (and beyond 400 nm at liquid nitrogen temperature). Moreover, quantum confinement should be observable at room temperature in ribbons as wide as 50 nm. In this context, we also note that a ribbon 100 nm wide has been observed to sustain a current of >100 μA at room temperature, and that, like nanotubes (*33*), the conductance of graphene ribbons increases approximately linearly with bias voltage at high bias.

These results raise new possibilities for coherent EG electronics on an attractive size scale and at relatively high temperatures. The demonstrated material and transport properties of EG could allow electronic devices and interconnects to be designed that rely on the wave properties of electrons and holes, so that interference-based electronic switches can be envisioned. Nanotubes, on the other hand, require metallic interconnects that destroy phase coherence. Furthermore, graphene is an extremely robust material that has the potential to be patterned with atomic precision down to the molecular level. Such precision might be achieved through a combination of standard lithographic and chemical methods, enabling a wide variety of coherently connected molecular structures. Finally, we note that it has been previously determined that the carrier density can be controlled electrostatically (*1*, *16*, *34*) and that chemical doping of the edges also is feasible (*1*, *35*). Consequently, epitaxial graphene provides a platform for the science and technology of coherent graphene molecular electronics.