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Experimental Study of the Intrinsic and Exxpetriminenstaicl StTudryaonf tshepInotrintsiPc arnodpExetrintsieic sTraonsfpoGrt rParopehrtietseofaGnradphiMte aundltMigurltiagrpaphenne SeamSpalesmples 12151 (Wexler, 1966). In Eq. (1) the first and the second terms of the right-hand side (rhs) correspond with the Knudsen-Sharvin and Ohmic resistances, respectively. The spreading Ohmic resistance in three dimensions can be estimated within a factor 2/π off from the exact Maxwell’s solution assuming a hemisphere in which the electric field E(r) = J3D(r)ρ(T). The radius r is taken at the constriction middle point and J3D(r) is the current density equal to the total current I divided by half of a sphere, i.e. J3D(r) = I/(2πr2), assuming that due to symmetry the current is radial far away from the constriction. From a similar calculation but in the two dimensional case, appropriate for graphite due to the weak coupling between the graphene planes, we have (García et al., 2008): R2D(T) = aρ(T)l(T)+a2ρ(T)γ(κ)ln(Ω/W)|W<Ω + ρ(T)L . (2) 4Wt πt Wt The first term at the right-hand side (rhs) of Eq. (2) corresponds to the ballistic Knudsen-Sharvin resistance; the second, logarithmic term to the ohmic, spreading resistance in two-dimensions; here Ω is the total sample width and t its thickness. The logarithmic dependence on the constriction width of this ohmic, diffusive contribution is due to the quasi-two dimensionality of the transport in graphite and supports the assumption of weakly interacting graphene layers inside the sample. The last term is due to the ohmic resistance of the constriction tube itself. From previous works it was clear that the position and shape of the voltage electrodes affects the outcome of the experiment in mesoscopic devices (McLennan et al., 1991). Therefore the constant a was introduced, which takes care of the influence of the sample shape, the topology, and the location of the electrodes in the sample. For the usual electrode positions through the whole sample width as shown in Fig. 6, a = 1. The validity of Eq. (2), especially the logarithmic dependence of the ohmic part, for HOPG as well as for multigraphene samples has been verified by García et al. (2008) and Dusari, Barzola-Quiquia, Esquinazi & García (2011). In the following, we review some of our experimental results for two multigraphene samples, A and B, with different geometry and resistivity. The sample details as well as their preparation and fabrication were described by Barzola-Quiquia et al. (2008); Dusari, Barzola-Quiquia, Esquinazi & García (2011). The constrictions in the middle of the samples, see Fig. 6, were prepared with the focused-ion beam of a dual-beam microscope. It is important to note that we avoided the modification of the crystalline structure of the samples due to the ion beam spread by protecting them with a ∼ 300 nm thick negative-electron beam resist (AR-N 7500), a method successfully tested by Barzola-Quiquia et al. (2010). To obtain the mean free path without further adjustable parameter we measured the resistance R as a function of the constriction width W and use Eq. (2). Figure 7 shows the results for samples A and B at two temperatures. The results show that for W < 1 μm the ballistic contribution (dashed lines in Figs. 7(a) and (b)) overwhelms the ohmic ones indicating that the mean-free path should be of this order. Having only l as free parameter Eq. (2) can be used to fit the behavior of R vs. W for sample A. From the theoretical fits one obtains l = 1.2 μm and 0.8 μm at 60 K and 250 K, respectively. The Fermi wavelength per graphene layer can be calculated using (García et al., 2008): 1 2πρ(T)l(T)e2 λF = a0 h , (3) where a0 = 0.335 nm is the distance between graphene planes in the graphite Bernal stacking configuration. For sample A we obtain then λF = 0.5(0.8) ± 0.25 μm at 250 K (60 K).PDF Image | GRAPHENE SYNTHESIS CHARACTERIZATION PROPERTIES
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