LETTER pubs.acs.org/NanoLett
Conductance through Multilayer Graphene Films Marcelo A. Kuroda,*,†,‡ J. Tersoff,† Dennis M. Newns,† and Glenn J. Martyna† † ‡
IBM T. J. Watson Research Center, Yorktown Heights, NY, United States Department of Computer Science, University of Illinois at UrbanaChampaign, Urbana, IL, United States ABSTRACT: The ballistic conductance through junctions between multilayer graphene films and several different metals is studied using ab initio calculations within the local density approximation. The system consists of films of up to four graphene layers (Bernal stacking) between metallic electrodes, assuming reasonable metalgraphene epitaxial relationships. For some metals, the conductance decays exponentially with increasing number of layers, while for others the conductance saturates with film thickness. This difference in asymptotic behavior stems from the crystal momentum (mis)match between the bulk Fermi-level states in the electrode and those in the film. In contrast, for sufficiently thin films the bonding between the metal and the adjacent graphene layer dominates, giving a metal dependence for graphene similar to that seen experimentally for single-wall carbon nanotubes. Among the metals considered here, we find Pd to be the best for electrodes to films with up to 4 graphene layers. KEYWORDS: Multilayer graphene, metal contacts, thin films, quantum transport
G
raphene is a promising material for a variety of applications, including mechanical1 and electronic2 nanoscale devices. In particular, its high transparency3 and carrier mobility4 are attractive for transparent conducting electrodes in photovoltaic devices and flexible electronics,5 especially in view of recent progress toward fabrication of high-quality, large-scale sheets.6,7 Several experiments have demonstrated that the metals employed to contact graphene play a critical role in the performance of electronic devices.811 Theoretical studies of metal contacts1216 have mainly focused on the in-plane transport in single graphene layers for field-effect transistor applications. However, for multilayer graphene, out-of-plane transport from the metal into the graphene layers can be equally important. In photovoltaic applications, multilayer graphene is a promising material for transparent electrodes, where the conductivity of a monolayer is insufficient.17,18 In these devices, losses from lateral transport can be reduced by adding a grid of metal bus bars. Achieving the lowest possible metalgraphene contact resistance (a current challenge for carbonbased electronics11,19) is important in order to minimize the footprint of the bus bar grid. We therefore focus on identifying the most promising metals among those typically used to contact graphene, and the general principles for screening candidate metals. Here first principles calculations within the local density approximation (LDA) are used to study the ballistic junction conductance of graphene thin films placed between metallic electrodes. We find that for sufficiently thin films the conductance is much larger for metals that bind strongly to graphene (Pd, Ti) than for those that bind weakly (Cu, Al, Au). For thicker films, we identify two qualitatively different classes of behavior. For some metals, the junction conductance decreases exponentially with increasing thickness of the graphene film, while for other metals the conductance saturates. The exponential decrease occurs in metals (Cu, Ti) where the coupling between Fermi-level states in the electrodes and the r 2011 American Chemical Society
multilayer graphene is inhibited by the mismatch in crystal momentum (which hereafter we refer to as momentum). Therefore carriers tunnel from one electrode to the other, giving an exponential reduction of the transmission with increasing film thickness. Whether such mismatch occurs for a specific metal/graphene system can be anticipated simply from the electrode and graphene band structures, and their epitaxial relationship. Because of the distinct roles of bonding and momentum mismatch, some metals that form good contacts to single-layer graphene will not be optimal for multilayer graphene. Our analysis (on films with up to 4 layers) points to Pd, which is known to form good contacts with carbon nanotubes,20,21 as a general-purpose choice of metal contact for graphene, since it has both strong bonding and states with appropriate momentum near the Fermi energy. Such analysis should also prove useful for screening other candidate metals for use as electrical contacts. We determine the conductance from the transmission probability of states in (semi-infinite) metallic leads through the scattering region containing the thin film, following the formalism of Choi et al.22 These films consist of up to four graphene layers arranged in Bernal stacking, as occurs, for example, for graphene films grown on the Si face of SiC23 or exfoliated from graphite. The scattering region includes six atomic layers of metal on each side of the n graphene layers, as depicted in Figure 1a. The relaxed atom positions and self-consistent potential are previously obtained in a separate calculation where this region is periodically repeated. We study several metals that are commonly used to contact graphene and carbon nanotubes: Cu, Al, Au, Pd and Ti. For each metal, we consider the close-packed face Received: April 29, 2011 Revised: June 29, 2011 Published: August 11, 2011 3629
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)
)
)
∑i, j Ti, jðk , EÞ
ð1Þ
)
Tðk , EÞ
which accounts for the contribution of all bands and so may exceed unity. The two-terminal conductance32 per unit area of the junction at zero bias is σ ¼
G0 ÆTðEF Þæ A
ð2Þ
)
Here G0 = e2/h ≈ 4 105 S is the conductance quantum, A is the supercell’s cross-sectional area, EF is the Fermi energy, and Z A Tðk , EÞdk 2 ð3Þ ÆTðEÞæ ð2πÞ2 BZ )
[i.e., fcc (111) for the first four metals and hcp (0001) for Ti], which corresponds to the low energy face. The Cu is treated as commensurate with graphene with the 4% lattice mismatch accommodated in the metal. For the other metals, larger √ all √ supercells are employed, fitting a 3 3 supercell of the metal onto a 2 2 supercell of the graphene lattice.24 These supercells keep the mismatch similar to or smaller than that of Cu-graphene system.13 The relaxations allow the out-of-plane dimension to vary, while fixing the in-plane dimensions to graphene’s lattice constant (a0 ≈ 2.45 Å). The graphene and a few adjacent metal layers (three and four layers for the fcc and hcp metals, respectively) are fully relaxed; the rest of the atomic layers in the electrodes are kept fixed (Figure 1a). The ab initio calculations are performed using density functional theory (DFT) within the LDA25,26 with an energy cutoff of 400 eV. We determine the zero-bias ballistic conductance using the PWCOND implementation of the Quantum Espresso software package.27,28 The use of LDA for exchange/correlation effects can overestimate transmission. For example transport through organic molecules,29,30 for which LDA poorly describes the electronic spectrum, required calculations within the GW approximation31 to reproduce the experimental results. Nevertheless, such effects are likely to be smaller in metalgraphene systems and should only give quantitative modifications without changing the trends or conclusions reported here. We compute the probability Ti,j(k ,E) that an incident Bloch state with momentum k parallel to the interface and energy E in the ith band of one electrode emerges in the jth band of the other electrode. Inelastic processes such as phonon scattering are
omitted and hence k and E are conserved. We define )
Figure 1. (a) Side view for the supercell used for the self-consistent calculation of the scattering region corresponding to the Cu2-grapheneCu system (enclosed by rectangle). Carbon and copper atoms close to the interface (circles with solid edge lines) are fully relaxed. Copper atoms away from the interface (circles with dashed edge lines) preserve the crystalline structure of the bulk metal. (b) Density of states of Cu. (c) Average transmission (eq 3) as a function of the incident energy for Cu-graphene systems with up to 4 graphene layers.
Figure 2. Left column: Energy surface of bulk Cu viewed along the interface-normal direction for E EF = 0, 1, and 2 eV (from top to bottom). Different colors indicate distinct sheets of the energy surface. Right column: Profile of the minimum imaginary component of the transverse momenta (k^) of the complex band structure of graphite at E EF = 0, 1, and 2 eV. Red regions around the K-point correspond to the projected energy surface of bulk graphite (states with purely real k^). Hexagons mark the edge of the 2D-BZ.
is the transmission at energy E averaged over the two-dimensional Brillouin zone (2D-BZ) of the supercell. We begin by studying the transmission of the Cugraphene system, which is the simplest since it is treated as commensurate due to the small misfit. Figure 1c shows the average transmission ÆT(E)æ for different number of graphene layers between the Cu electrodes. In the case of bulk Cu, the transmission remains nearly constant around the Fermi level with a value slightly below 2. When E EF falls below 1.65 eV, the bulk transmission increases because of the contribution of states from the Cu d-bands. The presence of a graphene monolayer causes an approximately uniform decrease in the conductance over the entire energy range. This behavior is largely due to the weak binding between Cu and graphene.13 However, when more layers are placed in between the electrodes the transmission exhibits qualitatively different behaviors depending on the energy of the incident states. Over a broad energy interval around the Fermi level the transmission decays exponentially with the number of layers. In contrast, for energies below 1.65 eV it is remarkably insensitive to the thickness of the film for n g 2. 3630
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Figure 3. Conductance per unit area vs number n of graphene layers for Al, Au, Cu, Pd and Ti electrodes; n = 0 corresponds to bulk metal. A0 denotes the area of the graphene’s unit cell. Inset: ratio η1,0 = σ(1)/σ(0) vs the electrodegraphene distance d.
Figure 4. Top row: Fermi surface for Al, Au, Pd, and Ti as seen from the direction perpendicular to the metalgraphene interface. The outer line (dashed) shows the primitive BZ of the metal. The small inner hexagon shows the supercell 2D-BZ into which the Fermi surface should be folded to study the momentum (mis)match between Fermi level states. Bottom four rows: transmission profiles T(k ,EF) within the supercell 2D-BZ for n = 1 to 4 graphene layers in between the metal electrodes; note the expanded size relative to top column and the logarithmic color scale. )
)
We can understand these results by comparing the bulk electronic structure of electrode and film. The left column of Figure 2 shows the bulk Cu energy surfaces at E EF = 2, 1, and 0 eV, projected onto the plane of the interface. In the graphene film, there are no states near EF except in small regions of the BZ. We therefore plot in the right column of Figure 2 the minimum imaginary component of the wavevector in the direction perpendicular to the interface (k^) from the complex band structure of the film (i.e., bulk graphite) as a function of k at the same energy levels. At the Fermi level, bulk graphite states reside near the K-points, the regions with min[Im(k^)] = 0, where Cu has no states. Therefore any incident state incoming from the electrode couples to evanescent modes and decays exponentially in the film. This explains the thickness dependence near EF in Figure 1. In contrast, at E EF = 2 eV the projected energy surface of the metal electrode covers most of the 2D-BZ. In particular, some of these states match the transverse momentum of states in the graphitic film (located in a region somewhat expanded around the K-point). Indeed, after a few layers these states near K give the main contribution to the average transmission, because the transmission of states outside these regions decreases rapidly with thickness. This leads to the saturation observed in Figure 1 for n g 2. The conductance for several different metals is compared in Figure 3. For Au, Al, Cu, Pd, and Ti, we plot the junction conductance per unit area versus the number of graphene layers. Both the absolute conductivity and the sensitivity to film thickness depend strongly on the choice of metal. We find that Pd and Ti form a strong bond with graphene (chemisorption), while the rest of the metals exhibit a weaker interaction, consistent with previous DFT calculations for monolayer graphene on different metal substrates13,33 and related experiments on nanotubes.34 The distance between the electrodes and the graphene layers is roughly 1 Å smaller for the two metals (Pd and Ti) that are more reactive with carbon than for the other three metals35 and comparable to that found experimentally for Ni(111) substrates.36 The larger distances for the nonreactive metals Au, Al, and Cu are similar to the values measured with Ir and Pt substrates.37,38 The inset of Figure 3 shows that the ratio η1,0 between the conductance with one graphene layer and that of the bulk metal [η1,0 σ(1)/σ(0)]
depends roughly exponentially on the metalgraphene distance. This suggests that transmission through graphene monolayers is largely dominated by the strength of the metalgraphene binding. After the second layer, the conductance of Ti decreases exponentially with graphene film thickness, at a rate comparable to that of Cu. In contrast, the conductance for Au and Al appears to saturate with increasing film thickness. We verified for Al that this trend persists up to six graphene layers. By a thickness of four layers, the conductance values for Ti and Cu electrodes are less than those for Au and Al. These distinct behaviors are explained by the very different Fermi surfaces of these metals. The top row of Figure 4 shows the Fermi surfaces for Al, Au, Pd, and Ti viewed from the direction perpendicular to the interface. Because of the epitaxial relationship between the metal and the supercell, these Fermi √ primitive √ surfaces are folded into the smaller BZ of the 3 3 supercell, indicated by the inner hexagon (solid lines). Note that for these metal electrodes, the K-point of the primitive BZ does not fold onto the supercell’s K-point, whereas in graphene those two points coincide after the folding. While Au has no states at its primitive cell’s K-point, it does have states at the supercell’s K-point, so this epitaxial relationship (as reflected in the zone folding) allows Fermi-level states to propagate through the graphene film. Ti exhibits the opposite behavior with no electronic states available around the supercell’s K-point. Thus from the bulk bandstructure 3631
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Figure 5. Shift (Emin EF) of the minimum in the projected DOS vs metal work function ϕm41 for system with four graphene layers between the electrodes. (a) Shift for graphene layers adjacent to the electrode and (b) for inner graphene layers.
)
and epitaxial relationship, one can anticipate the momentum (mis)match between Fermi-level states in the electrode and the thin film. A similar mechanism has been proposed for spin-filtering using epitaxial ferromagnetic contacts to graphene.39 The transmission profiles T(k ,EF) for Al, Au, Pd, and Ti are shown in the bottom four rows of Figure 4. As the thickness of the film increases, the transmission of Fermi-level states in regions away from the graphene’s K-point (which here coincides with the supercell’s K-point) is drastically reduced. In these regions, transmission values decrease by more than an order of magnitude for each additional graphene layer, consistent with estimates from the complex band structure of graphite (Figure 2). As a result of this fast decay, after three layers the conductance in metals with Fermi-level states around the K-point is mostly due to the contribution of these states. As intrinsic graphene has zero density of states (DOS) at the Fermi level, charge transfer of either sign (electrons or holes) enhances the conductance. In Al, Au, and Cu systems, the graphene is physisorbed onto the metal electrode with little distortion of the graphene structure and bonding. As a result, the projected DOS of each graphene layer exhibits similar features to that of single layer graphene, though somewhat shifted and smoothed. We can estimate the shift and resulting charge transfer to a layer by locating the minimum (Emin) in the projected DOS for that layer. Figure 5 shows the results for Al, Au, and Cu systems having four graphene layers. This dependence on the metal’s work function is similar to that observed in graphene monolayers on metal substrates.13,33 We find a substantial electron doping for Al, negligible doping for Cu, and a small hole doping for Au. Despite having a larger metalgraphene separation (Figure 3), the Al contacts give larger conductance than Au. This stems from the larger charge transfer for Al, which moves graphene’s Fermi-level states away from the K point, giving a larger region of the BZ with high transmission values (Figure 4). Another feature that can be observed in Figure 5 is that the screening of the charge transfer from the contacts to the interior graphene layers is stronger for Al than for Au. Because of the vanishing DOS of graphene near the Fermi level, screening in these systems is highly nonlinear. As in the case of turbostratic graphene,40 a shorter screening length is found for larger screened charges. Doping by the metals electrodes is important
for the transmission through thin graphene films. As the thickness of the film increases, screening impedes significant charge transfer to the innermost layers and regions of high transmission values are progressively reduced to those of the projected Fermi surface of bulk graphite (Figure 2). The cases where graphene binds strongly to the electrode (Pd and Ti) are more complex. The projected DOS suffers a significant distortion,13 so we cannot reliably extract information about doping. Indeed this strong interaction was reported to open a small band gap for single layer graphene on Pd.42 Figure 4 clearly suggests that for Pd, the region of the BZ around the K-point showing high transmission is relatively large, either because of doping or the distortion in electronic structure due to the metal. Among the metals compared here, Pd offers the highest conductance through films with up to four graphene layers; it has strong bonding to graphene and Fermi-level states that match the momentum of propagating modes in the film. In contrast, the initially high conductance in Ti-graphene junctions decays rapidly with increasing film thickness because states in Ti only couple to evanescent modes in graphite. The conductance in Au and Al shows a fast decrease in the transmission with the first few layers due to the exponential decay of the transmission of modes far from the K-point. For n g 3, the contribution of these states to transmission is negligible and the conductance is mostly governed by states near the supercell’s K-point. Hence conductance tends to saturate with the number of layers, as shown in Figure 3. It is important to emphasize that while the transmission values for propagating modes must be obtained from calculations, the absence of propagating modes can be anticipated from the electronic structure of the electrode and film. This is done by examining the overlap between the Fermi surfaces of electrodes and film projected according to the epitaxial relationship. For example, Cu and Au have similar Fermi surfaces (Figures 2 and 4, respectively), however their conductance when used as electrodes with epitaxial graphene is considerably different. The large lattice misfit between Au and graphene, accounted for by a supercell with low strain, yields a momentum match between Fermi-level states of the metal and graphene, allowing the strong transmission which is lacking in Cu. For single and bilayer graphene, Pd and Ti show higher total transmission values than Al or Au (second row of Figure 4). This shows that for sufficiently thin films, the determining factor in the junction conductance is not the absence/presence of momentum-matched Fermi-level states, but rather the binding between the electrodes and the graphene layers. This is because there is a significant contribution to transmission from states throughout the BZ when the graphene film is sufficiently thin. The electrical conductance described here corresponds to ballistic transmission through highly oriented graphene films in between different metal electrodes with reasonable epitaxial relationships. We expect that deviations from this simple picture will ultimately cause saturation of the exponential thickness dependence for Ti and Cu as for the other metals. For example, if the real epitaxial relationship differs from that assumed here by a small rotation or residual misfit, we can think of this as a sparse array of screw or misfit dislocation that can scatter electrons, providing a parallel path in which the momentum is not conserved. Phonon scattering as well as scattering at the edges of the metal contact will have a similar effect. However, we cannot anticipate at what film thickness this saturation will occur, because the study of those large systems using first principles calculations is currently out of reach. When graphene layers are 3632
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Nano Letters randomly stacked (i.e., deviating from Bernal) we would expect transmission to decay exponentially with thickness, regardless of the metal, because of momentum mismatch between consecutive graphene layers.43 In conclusion, we studied the two-terminal conductance of thin films of Bernal-stacked graphene epitaxially accommodated in between metal electrodes using ab initio calculations within the LDA approximation. With increasing film thickness, the conductance saturates for some metals and decays exponentially for others. These distinct behaviors are due to the crystal momentum match/mismatch between of Fermi-level states in the electrode and propagating states in the film. The match/ mismatch can be determined from the bulk band structure of the electrode and the thin films. For thin films consisting of one or two graphene layers, the conductance is governed primarily by the binding between the electrodes and the graphene. Ab initio calculations such as those reported here could prove valuable for screening metal candidates for electrodes for thin graphene films.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT We gratefully acknowledge stimulating discussions with R. A. Nistor and A. A. Maarouf. ’ REFERENCES (1) Bunch, J. S.; van der Zande, A. M.; Verbridge, S. S.; Frank, I. W.; Tanenbaum, D. M.; Parpia, J. M.; Craighead, H. G.; McEuen, P. L. Science 2007, 315, 490–493. (2) Avouris, P.; Chen, Z.; Perebeinos, V. Nat. Nanotechnology 2007, 2, 605–615. (3) Wang, X.; Zhi, L.; Mullen, K. Nano Lett. 2008, 8, 323–327. (4) Geim, A. K.; Novoselov, K. S. Nat. Mater. 2007, 6, 183–191. (5) Kim, K. S.; Zhao, Y.; Jang, H.; Lee, S. Y.; Kim, J. M.; Kim, K. S.; Ahn, J.-H.; Kim, P.; Choi, J.-Y.; Hong, B. H. Nature 2009, 15, 706. (6) Li, X.; Cai, W.; An, J.; Kim, S.; Nah, J.; Yang, D.; Piner, R.; Velamakanni, A.; Jung, I.; Tutuc, E.; Banerjee, S. K.; Colombo, L.; Ruoff, R. S. Science 2009, 324, 1312–1314. (7) Bae, S.; et al. Nat. Nanotechnology 2010, 5, 574–578. (8) Lee, E. J. H.; Balasubramanian, K.; Weitz, R. T.; Burghard, M.; Kern, K. Nat. Nanotechnology 2008, 3, 486–490. (9) Huard, B.; Stander, N.; Sulpizio, J. A.; Goldhaber-Gordon, D. Phys. Rev. B 2008, 78, 121402. (10) Xia, F.; Mueller, T.; Golizadeh-Mojarad, R.; Freitag, M.; Lin, Y.-m.; Tsang, J.; Perebeinos, V.; Avouris, P. Nano Lett. 2009, 9, 1039–1044. (11) Xia, F.; Perebeinos, V.; Lin, Y.-m.; Wu, Y.; Avouris, P. Nat. Nanotechnology 2011, 6, 179. (12) Robinson, J. P.; Schomerus, H. Phys. Rev. B 2007, 76, 115430. (13) Giovannetti, G.; Khomyakov, P. A.; Brocks, G.; Karpan, V. M.; van den Brink, J.; Kelly, P. J. Phys. Rev. Lett. 2008, 101, 026803. (14) Ran, Q.; Gao, M.; Guan, X.; Wang, Y.; Yu, Z. Appl. Phys. Lett. 2009, 94, 103511. (15) Barraza-Lopez, S.; Vanevic, M.; Kindermann, M.; Chou, M. Y. Phys. Rev. Lett. 2010, 104, 076807. (16) Maassen, J.; Ji, W.; Guo, H. Appl. Phys. Lett. 2010, 97, 142105. (17) Li, X.; Zhu, Y.; Cai, W.; Borysiak, M.; Han, B.; Chen, D.; Piner, R. D.; Colombo, L.; Ruoff, R. S. Nano Lett. 2009, 9, 4359–4363PMID: 19845330. (18) Kasry, A.; Kuroda, M. A.; Martyna, G. J.; Tulevski, G. S.; Bol, A. A. ACS Nano 2010, 7, 3839–3844.
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