Observation of Anomalous Resistance Behavior in Bilayer Graphene
Yanping Liu 1,2, Wen Siang Lew 2,*and Zongwen Liu 3,*
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Abstract
Our measurement results have shown that bilayer graphene exhibits an unexpected sharp transition of the resistance value in the temperature region 200~250K. We argue that this behavior originates from the interlayer ripple scattering effect between the top and bottom ripple graphene layer. The inter-scattering can mimic the Coulomb scattering, but is strongly dependent on temperature. The observed behavior is consistent with the theoretical prediction that charged impurities are the dominant scatters in bilayer graphene. The resistance increase with increasing perpendicular magnetic field strongly supports the postulate that magnetic field induces an excitonic gap in bilayer graphene. Our results reveal that the relative change of resistance induced by magnetic field in the bilayer graphene shows an anomalous thermally activated property.
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Introduction:
The electronic properties of monolayer graphene have been extensively studied due to its intriguing energy band structure with linear dispersion around the Dirac point and chirality exhibiting Berry phase of [1]. There is a zero-energy Landau level (LL) with four-fold degeneracy due to interactions between electron spins and valleys in the magnetic field [2-4]. Recently, bilayer graphene became a subject of intense research due to the low energy Hamiltonian of chiral quasiparticles and a Berry phase of [5-8]. It has a double-degeneracy zero-energy Landau level that incorporates two different orbital states with the same energy under an external magnetic field. The bilayer graphene with a Bernal (A-B) configuration loses some features of monolayer graphene and has a unique band structure where the conduction and valence bands are in contact with a nearly quadratic dispersion [5]. In bilayer graphene, a parabolic band structure ( ) with an effective mass m*=0.037, has been calculated by using the interlayer coupling model [9-14]. What makes bilayer graphene an interesting material for study is that the interlayer potential asymmetry can be controlled by an electric field, thus opening an energy gap between the conduction and valence bands [16-18]. Various applications for bilayer graphene are possible due to the fact that its band gap can be modulated by using an external out-of-plane electric field and chemical doping. There is intensive research on bilayer graphene under the application of a perpendicular electric field, however, experimental reports on magnetic transport properties of bilayer graphene are not as well-studied. Recent theoretical work reports on excitonic condensation and quantum Hall ferromagnetism in bilayer graphene [22]. There are interesting features in bilayer graphene due to its extra twofold orbital degeneracy in the LL spectrum, which results in an eightfold -degenerate LL at zero energy. The scattering mechanism of graphene is currently a subject of intense research and debate. The problem of magneto-transport properties in the presence of Coulomb impurities is still an open research problem. Our understanding of the nature of the disorder and how the mesoscopic ripple effect affects the transport properties still need improvement; hence, a better understanding on the general electric and magnetic transport properties of bilayer graphene is necessary.
In this paper, we have systematically investigated the charge transport properties in bilayer graphene as a function of temperature, magnetic field, and electric field. Our measurement results have shown that bilayer graphene exhibits a semi-metallic R-T property and an unexpected sharp transition of the resistance value in the temperature region 200~250K. The longitudinal resistance decreases with increasing temperature and electric field, a behavior that is markedly different from the experimental reports of monolayer graphene. Our results reveal that the energy gap in the bilayer graphene shows an anomalous thermally activated property and increases with. We have shown that this phenomenon originates from a tuneable band structure behavior that can be controlled by a magnetic field, a property that had never previously been observed in bilayer graphene.
It has been shown that Raman spectroscopy is a reliable, non-destructive tool for identifying the number of graphene layers and it can be done through the 2D-band deconvolution procedure [23-25]. The Raman spectra of our graphene structure were measured at room temperature using a WITEC CRM200 instrument at 532 nm excitation wavelength in the backscattering configuration [26-28]. Fig.1a shows the characteristic Raman spectrum with a clearly distinguishable G peak and 2D band. The two most intense features are the G peak and the 2D band which is sensitive to the number of layers of graphene. The position of the G peak and the shape of the 2D band confirm the number of layers of graphene. Additionally, the number of layers of graphene can be easily distinguished from the full width half maximum of the 2D band, as its mode changes from a narrow and symmetric feature for monolayer graphene to an asymmetric distribution on the high-energy side for bilayer graphene [27]. The 2D band inset in Fig.1a shows that the Raman spectrum of bilayer graphene is red-shifted and broadened with respect to that of the monolayer graphene. Fig. 1b shows the four terminal resistance as a function of carrier-density n, and the sample shows a pronounced peak at density . Note that the sharp peak in resistance at low n is enhanced by the opening of the small energy gap owing to disorder-induced differences in carrier density between the top and the bottom layers of the flake.
We have characterized the current (I)-voltage (V) characteristics of the bilayer graphene via four-terminal measurement, at different temperatures and magnetic fields. Shown in Fig. 2a are the I-V curves for bilayer graphene under the application of various magnetic fields at three different temperatures: 2 K, 200 K and 340 K. The magnetic field is applied in the perpendicular direction to the plane of the graphene. For all the temperatures and magnetic field strengths, the bilayer graphene exhibits a linear I-V curve. This implies that the graphene layer is ohmic in nature. We observed that for a fixed magnetic field, the I-V curve displays a larger gradient at higher temperature than at lower temperature. Interestingly, the gradient of the I-V curve decreases with increasing magnetic field. In our structure, the gradient of the curve corresponds to the conductivity of the graphene layer. Such temperature and magnetic field dependent behaviour of conductivity is characteristic of an intrinsic semiconductor. The decrease in the conductivity of the bilayer graphene with increasing magnetic field is attributed to the excitonic energy gap induced by the magnetic field. This conductivity dependence on the magnetic field suggests that the resistance () of graphene is a qualitative fingerprint of its band gap.
In the absence of a magnetic field, the band structure of the bilayer graphene at the Dirac valley has a parabolic dispersion relation. When a magnetic field is present, the band structure is changed to a split Landau level structure [19-21]. Fig. 2(b) is an illustration of the bilayer bandgap and Landau level splitting under the influence of a magnetic field. Inset shows an optical image of the bilayer graphene with the metal contact electrodes. In Fig.2(c) we plot the resistance of the bilayer graphene, as extracted from the I-V curve, as a function of magnetic field for three different temperatures. As the magnetic field was increased in a step of 4T, the resistance increase for each step was different, resulting in a non-linear relationship between the resistance and magnetic field. Interestingly, the observed non-line relationship is markedly different from Zeeman spin-splitting theoretical model with the line relationship, where gap with a free-electron g-factor g=2, where is the Bohr magneton. This potentially indicates sublattice symmetry breaking and gap formation due to many-body correction in this LL [32-34]. This is further confirmation that magnetic field opens an excitonic gap in the bilayer graphene.
The temperature dependence of monolayer graphene resistance is mainly attributed to the different scattering mechanisms: Coulomb scattering [35-36], short range scattering [37], and phonon scattering [38-39]. However, the temperature dependence of bilayer graphene resistance has not been established yet. Shown in Fig.3a are the temperature dependence of the resistance of the bilayer graphene under the application of a magnetic field 0T and 12T, respectively. The results show that the resistance of the bilayer graphene drops following non-metallic behaviour as temperature increases from 2K to 340 K. This implies that the bilayer graphene resistors have intrinsic semiconductor properties as mentioned earlier. This can be explained by the decrease in Coulomb scattering with temperature for bilayer graphene due to its parabolic band structure. For B=12T, a similar trend as B=0T is obtained in Fig 3a, where the resistance decreases with increasing temperature. However, the resistance for the entire temperature range is much larger than for B=0T. This indicates that the magnetic field opens an excitonic gap in the bilayer graphene that is thermally activated due to the Coulomb interaction ion-driven electronic instabilities [20, 31].
Ripples are a common feature of cleaved graphene because it is never atomically flat, as it is placed on a substrate such as SiO2 in the term of nanometre-scale deformations or ripples [40-42]. Despite the magnitude of the ripples being quite small, it is still believed to be responsible for the unusual transport behaviour of graphene, also susceptible to adsorbed impurities, defects and the roughness of the underlying substrate [40-43]. On the other hand, it has been shown that suspended graphene films are corrugated on a mesoscopic scale, with out-of-plane deformations up to 1 nm [44-45]. The deformation is a typically smaller than the Fermi wavelengthand these ripples induce predominantly short-range scattering. The observed height variation shows that the surface roughness beyond the atomic-level is intrinsically present in bilayer graphene. Hence, one of the interesting features of corrugation of graphene is that it offers a new experimental opportunity to study how the corrugation-induced scattering impacts the transport properties of graphene. It is important to mention that there is a strange sharp threshold like decrease in resistance observed above 200K. The strong temperature dependence is inconsistent with scattering by acoustic phonons. One possible explanation is that the flexural phonons confined within ripples between the top and bottom layer causes the scattering. The presence of the ripple effect exhibits local out-of-plane ripples [44]. Theoretical calculations[41,46] show that the scattering rates for interripple flexural phonons with respect to two-phonon scattering process as, where is the flexural-phonon frequency, the derivative of the nearest-neighbour hopping integral with respect to deformation, a the lattice constant, , and the mass of carbon atom [46]. For low temperatures T () , few flexural modes can be excited inside ripples (). The conductivity of the surface roughness model at the limit at low temperature is[45-46]. As the temperature increases and typical wavelengths become shorter, short-range scattering excites the flexural phonons. For the high temperature limit, based on the above expression, we can estimate that , which yields ~100 to 1000 at T=300K. The model of quenched-ripple disorder [46] suggested that the electron scattering of the static ripples quenched from the flexural phonon disorder can mimic Coulomb scattering when at room temperature. One should also note that the model predicts stronger temperature dependence (above a certain quenching temperature of about 100K) which is close to our experimental result at about 200K. However, the ripple effect normally leads to a rapid increase in the R-T curve rather than the sudden decrease in R-T as observed for our bilayer graphene. In the absence of a theory to explain the stronger temperature dependence behaviour, we propose that the behaviour is consistent with the ripple effect interlayer scattering instead of interlayer scattering. Fig. 3b shows the schematically illustration of scattering mechanisms in bilayer graphene. For a bilayer graphene, the interlayer scattering between the top and bottom ripple graphene layer is similar to coulomb scattering with stronger dependence on temperature. The rapid decrease in R-T above 200K can be attributed to the transition between the low- and high-T limits in the interlayer ripple effect scattering.
On the other hand, it was suggested that the observed strong T dependence could be explained by thermally excited surface polar phonons of the SiO2 substrate [35-38]. The SiO2 optical phonons at the substrate-graphene interface induce an electric field which couples to the carriers in graphene due to it modulating the polarizability [38-39]. However, Coulomb scattering is dominant for bilayer graphene and the substrate surface polar phonon induced field is to some extent screened by the additional graphene layers [39]. Recently it has been shown that the substrate dielectric constant plays an important role in scattering in graphene. Theoretical predictions show that for dielectric constant , Coulomb scattering dominates, while for dielectric constant , short-range scattering dominates, as Coulomb scattering is more strongly screened for materials with a larger dielectric constant. In fact, our observed behaviour is consistent with the theory suggesting that scattering from charged impurities is dominant in graphene.
We introduce a relaxation-time approximation and treat the unscreened Coulomb potential as [1,5] where Q is the charge of impurities. Based on the Boltzmann transport theory, we can obtain the bilayer graphene resistivity with massless Dirac-fermions (MDF) at low energies as. For high temperature , , we can obtain the bilayer graphene resistivity as[47], where is the density of impurities per unit volume, is the permittivity of the semiconductor, and is the charge state of the impurity. This shows that the resistance of bilayer graphene limited by Coulomb scattering increases as increases and decreases with increasing temperature. Considering the above analysis, we deduce that the temperature dependence of resistance in bilayer graphene is mainly determined by Coulomb scattering. The short-range scattering is independent of temperature for bilayer graphene, as the density-of-states, the matrix element and the screening function are all energy independent. As a result of the parabolic band structure of bilayer graphene, the energy averaging of the Coulomb scattering time can give rise to the resistivity decreasing proportionality to temperature : .
Based on the above discussion, we fit the measured resistance in Fig.3a by using the following model for bilayer graphene:, where and are the resistance due to the Coulomb and short-range scatterings, respectively. Fig.3b shows the relative resistance change under the biased and unbiased magnetic field as a function of temperature, and the dotted line is the fit following the equation , where is the energy gap. The opening of the energy gap due to a potential difference between the two layers and Coulomb interactions could be a cause for this. These considerations explain qualitatively why the resistance of bilayer graphene decreases with increasing temperature. Note that the relative resistance change is a strong function of temperature. At temperatures of 2Kiˆ180K and 220Kiˆ250K, the relative resistance strongly increases as temperature increases, indicating that an energy gap forms due to many-body correction in Landau Level. When the temperature increases to T >250K, the relative resistance is roughly independent of the increasing temperature; this indicates that the energy gap is mostly stable at high temperatures. On the other hand, with the temperature increase from 180K to 220K, the relative resistance dependence of temperature shows a sharp decrease, which indicates that the energy gap shows an anomalous thermally activated behaviour as a function of temperature.
For zero gate voltage (i.e., neutrality point), we measured changes in longitudinal resistance as a function of applied perpendicular field B. Fig. 4a shows the four-terminal longitudinal resistance of bilayer graphene as a function of magnetic field at T= 2K at the charge-neutrality point. We have plotted the resistance per square, because it is independent of a size effect of the sample. As seen from Fig. 4a, the resistance increases nonlinearly with the magnetic field strength followed by a plateau-like phase. One should note that the plateau-like phase in Fig. 4b disappears at higher temperatures. One possible explanation is the augmented sublattice spin-splitting due to the high surface-impurity concentration of the graphene layer [18]. The origin of the nonlinear magnetoresistance increment behaviour is the splitting of Landau level that gives rise to a bandgap opening at the zero energy level [32-34]. In our measurements, we fit our results to an analytical approximation for the non-linear resistance , where is the Boltzmann constant. We found that our results are in good agreement with this equation. These considerations explain qualitatively why the nonlinear resistance increases with the magnetic field.
Fig. 5 shows the resistance of bilayer graphene as a function of electric field (E) under different magnetic fields. The dependent characteristics are symmetric due to the chirality of graphene electrons when an applied electric field changes from E to –E. The normalized resistance curve describes the response under the applied magnetic field in the range of B=0T to B=12T and the temperatures of 2-340K. The results demonstrate that when the magnetic field increases from 0T to 12T at low temperatures (2iˆ200K) and low electric field (E<0.001 ), the resistance of bilayer graphene drops significantly. The larger slump in the resistance at lower temperature T=2K and low electric field as the increasing of electric field are due to Coulomb scattering by impurities, which is a strong function of temperature. On the other hand, at high temperatures (T >200K) and electric fields (E>0.01 ), the resistance of bilayer graphene show a linear decrease. This can be explained by the scattering from thermally excited surface polar phonons of the substrate being screened by the additional top graphene layers [39]. This further confirms that at high temperatures, the scattering induced by the electric field on the substrate surface polar phonons is significantly screened between top and bottom layers in bilayer graphene.
In our experiment, temperature and magnetic field dependence of resistance of bilayer graphene was investigated. Intrinsic semiconductor behaviour at the range of temperature is 2K-340K was observed. The strange sharp threshold-like decrease in resistance around 200K is unexpected, and we attribute it to the presence of mesoscopic ripples between the top and bottom layer. Our results reveal that the energy gap in the bilayer graphene is thermally dependent. This potentially indicates the sublattice symmetry breaking and an energy gap formation due to Landau Level splits. The obtained results are important for the better understanding of magnetic field induced high resistance and provide indications of a theoretically predicted magnetic field induced excitonic gap.
Acknowledgements
Y. L would like to thank Prof. Wang and Prof. Yao for his useful discussions. This work was supported in part by the NRF-CRP program (Multifunctional Spintronic Materials and Devices) and the Agency for Science, Technology and Research (A*STAR) SERC grant (082 101 0015). The authors thank Sun Li and Li Yuanqing for their assistance in experimental measurements.
Experimental section
The bilayer graphene samples for this study were prepared using mechanical exfoliation techniques [2] from the bulk highly oriented pyrolitic graphite (grade ZYA, SPI Supplies) and transferred onto the surface of a lightly doped silicon substrate covered with a 300-nm thick layer of thermally grown , The doped silicon substrate and were used as back-gate and gate dielectric, respectively. Graphene electrical electrodes were patterned using photolithography techniques. A pair of ohmic Cr/Au (5nm/100nm) contacts were deposited via thermal evaporation at a background pressure of 10iˆ7 mbar and subsequently lifted off in warm acetone. Electronic transport measurements have been carried out on multiple samples, using PPMS (Quantum Design) with a fixed excitation current of 10 . Electrical measurements were performed in the temperature range 2K ~340K and a magnetic field up to 12T was applied. In order to enhance electrical transport, the sample was cleaned in situ by the magnetic and electric field. Four-terminal electrical measurements were used for transport characterization.
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