Enhanced intervalley scattering in artificially stacked double-layer graphene

We fabricated artificially stacked double-layer graphene by sequentially transferring graphene grown by chemical vapor deposition. The double-layer graphene was characterized by Raman spectroscopy and transport measurements. A weak localization effect was observed for different charge carrier densities and temperatures. The obtained intervalley scattering rate was unusually high compared to normal Bernal-stacked bilayer or single-layer graphene. The sharp point defects, local deformation, or bending of graphene plane required for intervalley scattering from one Dirac cone to another seemed to be enhanced by the artificially stacked graphene layers.


Introduction
Graphene, a two-dimensional (2D) sp 2 -hybridized network of carbon atoms, has received remarkable attention because of its linear dispersion relation together with unique electronic properties such as ambipolar transport, Dirac particle quantum Hall effect including anomalous corroborated by comparison with magnetotransport measurements of single-layer graphene (SLG).

Experimental details
Graphene film was grown on 25 μm thick copper foils from Alfa Aesar (99.8% pure) via thermal CVD. A mechanically polished and electropolished copper foil was inserted into the CVD furnace. The furnace was evacuated to ∼10 −4 Torr, and the temperature rose to 1010°C with H 2 gas flow (∼10 −2 Torr). After the temperature stabilized at 1010°C, CH 4 and H 2 (20 and 5 standard cubic centimeters per minute, respectively) were injected into the furnace to synthesize the graphene for 8 min, after which the sample was cooled at a rate of 50°C min −1 to room temperature [12]. The graphene film grown on Cu foil was transferred to a Si substrate by the wet transfer method. The Cu foil was spin-coated (850 rpm for 10 s, 2500 rpm for 30 s) with a thin layer of polymethylmethacrylate (PMMA). Then, the bottom Cu foil was removed by etching in a 1 M solution of ammonium persulfate (APS, (NH 4 ) 2 S 2 O 8 ), and the PMMA membrane was washed with deionized water. Next, the graphene film with the PMMA membrane was transferred to the p-doped Si substrate having a top 300 nm thick layer of SiO 2 . The graphene layers transferred onto the Si/SiO 2 substrate were heated at 80°C for 10 min to dry and then put in acetone for one day to completely dissolve the PMMA layer. An artificial double-layer graphene was formed by subsequent transfer of another layer onto the first layer of graphene. Half of the first layer of graphene in the Hall bar was removed with a combination of electron-beam lithography and oxygen plasma treatment. Therefore, we were also able to examine the characteristics of SLG. Raman spectra were measured with a Renishaw microspectrometer over a wavenumber range from 1100 to 3200 cm −1 , with a laser wavelength of 514.5 nm. The spot size was 1 μm and the power was kept at 1.0 mW to avoid local heating. Atomic force microscope (AFM) is used to analyze the surface morphology of single-and double-layer graphene. The typical Hall bar patterns were fabricated by photolithography and Cr/Au (5/30 nm) contacts were coated by using a thermal evaporation system. The magnetotransport measurements were performed by using the standard lock-in technique at low temperature (down to 0.35 K) in a cryostat.

Results and discussion
The Raman spectrum of the artificially stacked double-layer graphene grown by the CVD method is shown in figure 1(a). The G peak in the Raman spectrum corresponds to the in-plane bond stretching of C atoms with E 2g symmetry optical phonon at the Brillouin zone center [27]. The 2D peak is the second order of the D peak, which originates from a process where momentum conservation is satisfied by two phonons with opposite wave vectors. The 2D/G peak intensity ratio (I 2D /I G ) value is ∼2.7 in figure 1(a). The D peak is attributed to A 1g phonons near the K-zone boundary [27]. The low value of the D/G peak intensity ratio (I D /I G < 0.14) is indicative of the existence of a high quality CVD-grown graphene. To check that the quality of graphene was uniform, we performed Raman spectroscopy at different locations on the artificially stacked double-layer graphene device fabricated in this experiment. The I 2D /I G ratio at different points in the graphene device is shown in figure 1(b) and positions G and 2D peaks at different points are shown in figure 1(c). The I 2D /I G ratio of ∼2.7 is distinct from that of the Bernal-stacked bilayer graphene, whereas it is comparable to that of SLG. However, the I 2D /I G ratio of twisted bilayer graphene was reported to be closely correlated with twisted angle. When twisted angle increased from 5.4°to 27.2°, the Raman intensity ratio of I 2D /I G increased from ∼1 to ∼3 [28]. Since the I 2D /I G ratios show uniform values around 2.7, artificially stacked double-layer in this experiment has a uniform twisted angle rather than random rotational disorder. There were intriguing reports that stacking disorder in multilayer graphene can induce a low-energy linear dispersion as in SLG. Magnetotransport and far-infrared magnetotransmission investigations revealed the Dirac-type SLG-like characteristics of electronic states for multilayer epitaxial graphene grown on SiC substrate [29,30]. The electronic structure of SLG preserved even in multilayer epitaxial graphene was attributed to a high degree of rotational disorder in the multilayer epitaxial graphene.
The consistent characteristics in the Raman spectroscopy indicate that the layer-by-layer transfer of the CVD-grown graphene made a uniform quality of graphene with relatively low defects for the large area of the device in this experiment. Figure 1  uniformity and wrinkles were reported to affect the characteristic lengths of transport properties in the epitaxial graphene grown on SiC substrate [25,31,32].
The artificially stacked double-layer graphene was further characterized by electrical transport measurements. The resistivity of graphene devices was measured as a function of the back-gate voltage (V g ). The longitudinal resistivity showed a clear Dirac point (V Dirac ) at +10 V at 0.35 K and 300 K (see figure 2(a)). The field effect hole mobility values of the double-layer graphene device were found to be around 2150 and 2410 cm 2 V s at temperature of 300 K and 0.35 K, respectively. The charge carrier density (n) as a function of V g is shown in figure 2(b). The n controlled by V g was obtained from the Hall measurement. The black solid line shown in figure 2(b) represents a linear fit of the relation n = C g (V g − V Dirac )/e, where C g is the capacitance of the 300 nm-thick SiO 2 layer. The estimated magnitude of C g was 124 aF μm −2 , which is consistent with previous reports [33].
The WL magnetoresistance properties were measured by sweeping a magnetic field (B) perpendicular to the graphene plane. The electron scattering resulting from disorder can form a closed trajectory that interferes with the time-reversed path. If the phase coherence length is longer than the elastic scattering length, the quantum corrections to the resistance because of constructive interference of the time-reversed path lead to an enhanced resistance at zero magnetic field. For a detailed analysis of this phenomenon, the measured relative change in magnetoresistivity (Δρ/ρ) versus the B field are fitted by the following equation [17]: Here ψ(x) is the digamma function, τ ϕ is the dephasing rate because of inelastic scattering, τ i is the elastic intervalley scattering time, and τ * is the relaxation time, which is a combination of the three other main scattering terms: τ z , the elastic intravalley chirality breaking scattering term, which originates from defects or dislocation; τ w , the elastic intravalley trigonal warping scattering term, and τ i . The resultant rate, τ * . The elastic intervalley scattering time τ i comes from atomically sharp scatterers and scattering from the edges of the device. The relative change in magnetoresistivity (Δρ/ρ) versus B for different V g values was shown in figure 3(a) and the solid lines (red color) represent fits to equation (1). All solid lines fitted well with the experimental data, whereas τ ϕ , τ i, and τ * were obtained as fitting parameters. These curves show the same WL trend at various V g values in figure 3(a). However, the WL effect near the Dirac point (V g = +10 V) region deserved special consideration, where we observe that the resistivity started increasing significantly when the magnitude of magnetic field was higher than ±0.25 T (see figure 3(b)). The increase in resistance with magnetic field indicates a signature of the WAL effect in this regime. This prominent effect may be caused by enhancement of inelastic scattering near the Dirac point region caused by electron-hole puddles. The observed WAL effect may be associated with the low-energy quasiparticles of double-layer graphene which exhibit a similar chirality as that of single layer graphene. Fittings to the magnetoresistivity curves with carrier densities from −5.41 × 10 12 cm −2 (V g = −60 V) to 3.77 × 10 12 cm −2 (V g = +60 V) (see figure 3) allowed us to extract the scattering times with best-fit values for the parameters using equation (1) and these were later converted into scattering lengths. The ratio of τ τ ϕ * / as a function of τ ϕ /τ i is shown in figure 4(a), and indicates the existence of favorable conditions for the observation of both the WL and WAL phenomena. The large values of these ratios correspond to a regime where the WL effect is dominant, whereas near the Dirac point region these ratios have lower values, which is evidence for the existence of the WAL effect [19].
The most striking result of this paper is the large difference between the intervalley scattering rates for stacked layers compared to the single layers. This central result permits a more detailed discussion using theoretical study of Kechedzhi et al [34]. The scattering rates are . The rate τ − zz 1 arises from the scattering potential u zz , which describes an asymmetry in the potential felt at the A and B site in graphene. The other potentials like ⊥ u z involve some form of bond stretching. This can be a local deformation of the hexagon, or a bending of the graphene plane. An interesting relation is found between scattering rates 1/τ * and 1/τ i in figure 4(a).  is twice that of the momentum relaxation time in graphene because of chiral transport properties. The characteristics of WL usually become prominent when both the intra-and the intervalley scatterings are large. The observed phase coherence length is greater than the intervalley scattering length, i.e. L ϕ » L i , whereas the difference between L i and L * is relatively small, as shown in figure 4(b). This is in contrast to the previous studies on a mechanically exfoliated bilayer graphene by Gorbachev et al and Liao et al, where L ϕ , L i , and L * are all of comparable magnitudes [7,21].
The temperature dependence of the WL effect at V g = 0 V in the range from 0.35 to 10 K is shown in figure 5(a). The red lines represent the fitting of equation (1) for different temperatures. Figure 5(b) shows the temperature dependence of the phase coherence scattering time (τ ϕ ), which is obtained from the fits of the WL corrections and follows the ∼T −2/3 law above 4.2 K. As electron-electron interaction is enhanced with increasing temperature, τ ϕ decreases at high temperature [35]. The inelastic interactions due to electron-electron scattering were found to be the dominant mechanism in the case of graphene, limiting the coherence of quasiparticles at low temperature [35][36][37]. The extracted scattering lengths from the fitting parameters are plotted in figure 5(c). The value of L ϕ increased at lower temperature and saturated below 2 K, the red dashed line is a guide to the eye. On the other hand L i and L * are independent of temperature in the range of this experiment. The L i and L * are replotted as a function of temperature in the figure 5(d) for clarity. Since the variations of L i and L * with temperature are within the error bar, one can see that L i and L * are independent of temperature. The value of L i for the artificially stacked double-layer graphene was much smaller than for the Bernal-stacked bilayer graphene. As our sample was rather wide (∼10 μm), the edge scatterings were apparently negligible. The elastic intervalley scattering is thought to be enhanced by the presence of C atoms in the top graphene layer that work as sharp point defects, or local deformation and bending by the top graphene layer.
We have intentionally left one half of the device uncovered in the etching process to investigate the WL effect on single-layer CVD-grown graphene. In inset of figure 2(a) the upper part corresponds to artificially stacked double-layer graphene, whereas the lower part corresponds to SLG. This helped in the comparison with the artificially stacked double-layer graphene. The same magnetotransport measurements were performed for the single-layer CVDgrown graphene. The longitudinal resistivity of the graphene device as a function of back-gate voltage showed V Dirac = +2 V at the temperature of 4.2 K ( figure 6(a)). The hole mobility of the SLG region was around 2400 cm 2 Vs −1 . Shown in figure 6(b) is the relative change in magnetoresistivity Δρ/ρ versus B for different V g and the solid lines (red color) represent fits to equation (1). From the fitted curves we obtained the scattering length parameters L ϕ , L i , and L * , shown in figure 6(c) for different charge carrier density. For clarity, L i and L * are redrawn in figure 6(d) as a function of charge carrier density. We found that L ϕ , L i and L * are in line with previous reports for SLG [14,15]. The value of L i (=150 ∼ 210 nm) of the SLG is much larger than that of the artificially stacked double-layer graphene. As shown in figure 4(b), L i is much smaller (⩽50 nm) for the case of artificially stacked double-layer graphene. These results support the finding that the top graphene layer in the artificially stacked double-layer graphene device enhances the intervalley scattering. The temperature dependence of the WL effect at V g = 0 V ranging from 0.35 to 10 K for SLG is given in figure 7(a). The red lines represent the fitting of equation (1) for different temperatures. The temperature dependence of the phase coherence time is obtained from the fits of the WL corrections, which follows the ∼T −1 law  above 4.2 K. This is inline with previous reported results for single layer graphene [32]. The extracted phase coherence length from the fitting parameters are plotted in figure 7(c). The value of L ϕ increased as the temperature was lowered and saturated below 4.2 K. The red dashed line is a guide to the eye. The L i and L * are plotted as a function of temperature in the figure 7(d). However, L i and L * is independent of temperature in the range of this experiment. The L i and L * are plotted as a function of temperature with error bars in the figure 7(d).

Conclusion
We fabricated artificially stacked double-layer graphene by sequentially transferring CVDgrown graphene. Raman spectroscopy showed that the two transferred layers of graphene were of uniform quality and longitudinal resistivity measurements as function of V g showed considerably high mobility. The WL effect was observed at different charge carrier densities and temperatures, whereas the clear signature of the WAL effect was manifested near the Dirac point region. The large values of τ τ ϕ * / and τ ϕ /τ i ratios also demonstrated the existence of favorable conditions for the WL effect, whereas the small values corresponded to a dominant WAL-effect regime. The intervalley scattering rate of the artificially stacked double-layer graphene was much higher than that of Bernal-stacked bilayer graphene or that of SLG. The large momentum transfer required for intervalley scattering from one Dirac cone to another seemed to be enhanced and the sharp point defects, local deformation, or a bending of graphene plane are provided by the artificially stacked graphene layers. The observed phenomena can provide an idea on how to manipulate intervalley scattering in some attractive future applications such as valleytronics devices.