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1226 Graphene – Synthesis, Characterization, Properties anWdill-bAe-psept-lbiyc-IaN-tTiEoCnHs Despite of the good agreement obtained for sample A and for HOPG bulk samples (García et al., 2008), Eq. (2) suffers from an important limitation since it cannot describe correctly ballistic transport phenomenon in which the wave nature of the electrons plays a crucial role, i.e. in samples where λF W. As discussed before, in this case the ballistic contribution to the resistance is better described by the quantization of the transverse electron momentum in the constricted region. In this case the value of the resistance is given by the inverse of a sum of an energy-dependent and transverse wave vectors qn-dependent transmission probabilities Tn , where n = 0, ±1, ±2, . . . Nc (Stauber et al., 2007). These values correspond to the one dimensional electric sub-bands and Nc is the largest integer smaller than 2W/λF. In this case the increase in resistance is expected to show an oscillatory behavior as a function of W or λF (García & Escapa, 1989; Snyman et al., 2008) as observed experimentally in bismuth nanowires (Costa-Krämer et al., 1997) as well as in GaAs devices (van Wees et al., 1988; Wharam et al., 1988). Note that the obtained mean free path for sample A is smaller than the distance between the electrodes. The larger the sample, larger is the probability to have defective regions with larger carrier concentration and smaller mean free path within the voltage electrodes (Arndt et al., 2009). In order to increase the probability to observe the expected quantization phenomenon in multigraphene samples, it is necessary to have a mean free path larger than the sample size in order to be completely in the ballistic regime. Therefore, we repeated the experiment with sample B that shows lower resistivity and with a smaller voltage-electrode distance, see Fig. 6. Figures 7(c) and (d) show the measured resistance normalized by its value at a constriction W = 3 μm for sample B. The normalization is necessary because in this way we pay attention to the huge relative increase of R decreasing W and we need neither the absolute value of ρ nor of a to compare the data with theory. We realize that for sample B Eq. (2) does not describe accurately the experimental data even assuming the largest possible mean-free path equal to the voltage-electrode distance of ≃ 2.7 μm. The data can be reasonably well fitted dividing the ballistic term in Eq. (2) by the function trunc(2W/λF)λF/2W, which generates steps whenever the constriction width W ≃ iλF/2 with i an integer. From the fits we obtain the parameters λF = 1.0(1.5) μm and l = 2.2(2.7) ± 0.3 μm at 300(10) K. Using other values of l, for example l = 1.3 μm, see Fig. 7(d), the function does not fit the data indicating indeed that the carriers behave ballistically between the voltage electrodes, leaving actually λF the only free parameter. The ballistic analytical function we used resembles the theoretical results with similar steps obtained by Snyman et al. (2008) where the conductance vs. W was calculated numerically for a single layer graphene with an electrostatically potential landscape that resembles a constriction. An analytical average value or envelope of this stepped function is obtained replacing the truncation function by exp(−λF/2W), see Figs. 7(c) and (d). This exponential function represents the impossibility of an electron to propagate in the constriction when W < λF/2. The important result obtained for sample B is the huge increase of the resistance for W < 2 μm indicating clearly a larger l than the one obtained in sample A, see Dusari, Barzola-Quiquia, Esquinazi & García (2011) for further details. The temperature dependence of R(T,W) can be used now to obtain λF(T) and the mobility per graphene layer, this last given by μ(T) = (e/h)λF (T)l(T). Since the density of carriers per graphene layer can be calculated from n = 2π/λ2F we show in Fig. 8 the mobility vs. carrier density for the two samples and for a bulk HOPG sample and compare them with data from literature for suspended single layer graphene. From these results we clearly recognize the much larger mobility and smaller density of carrier for the graphene layers inside graphite,PDF Image | GRAPHENE SYNTHESIS CHARACTERIZATION PROPERTIES
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