Transition from Ohmic to ballistic transport in oriented graphite: Measurements and numerical simulations

In this work we show that the spreading Ohmic resistance of a quasi-two-dimensional system of size $\Omega$, thickness $t$, and with a constriction of size $W$ connecting two half-parts of resistivity $\rho$ goes as $\left(2\rho /\pi t\right)\text{ln}\left(\Omega /W\right)$, diverging logarithmically with the size. Measurements in highly oriented pyrolytic graphite (HOPG) as well as numerical simulations confirm this relation. Furthermore, we present an experimental method that allows us to obtain the carriers’ mean-free path $\ell \left(T\right)$, the Fermi wavelength $\lambda \left(T\right)$, and the mobility $\mu \left(T\right)$ directly from experiments without adjustable parameters. Measuring the electrical resistance through microfabricated constrictions in HOPG and observing the transition from Ohmic to ballistic regime, we obtain that $0.2\mu \text{m}\lesssim \ell \lesssim 10\mu \text{m}$, $0.1\mu \text{m}\lesssim \lambda \lesssim 2\mu \text{m}$, and a mobility $5\times {10}^4{\text{cm}}^2/\text{V}\text{s}\lesssim \mu \lesssim 4\times {10}^7{\text{cm}}^2/\text{V}\text{s}$ when the temperature $T$ decreases from 270 to 3 K. A comparison of these results with those from literature indicates that conventional, multiband Boltzmann-Drude approaches are inadequate for oriented graphite. The upper value obtained for the mobility is much larger than that for the mobility in graphene samples of micrometer size can have.

Figure 1

(Color online) (a) Sketch of the sample geometry with the electrode positions at the edges. The length and the width of the constriction are shown as $L$ and $W$. (b) Scanning electron microscope picture of the $2\mu \text{m}$ constriction done on the HOPG sample. (c) Electron backscattering image shows the in-plane orientation of the crystallites of the HOPG sample in a scan size of $200\times 110\mu {\text{m}}^2$; the crystallites observed here are in the $\sim 5,\dots ,20\mu \text{m}$ range of elongation.

Figure 2

(Color online) (a) Two-dimensional plot of the current-density distribution through a constriction of $4\mu \text{m}$ width and $4\mu \text{m}$ length. The thickness of the sample is 100 nm. The color scale represents the current density and the stream lines indicate the current flow with the density proportional to the current density too. (b) Two-dimensional plot of the electric potential distribution and the current flow through a constriction of $100\mu \text{m}$ width and $4\mu \text{m}$ length. The color scale represents the electric potential drop from 1 to 0 V and the stream lines indicate the current flow inserted with arrows which sizes are proportional to the local current density. These numerical simulations are done for the case of the resistivity of Cu at room temperature, $\rho =1.66\mu \Omega \text{cm}$, and 100 nm thickness. The result scales for any other material and thickness. (c) Numerical results for the voltage across the sample length $\left(L_s=\Omega =1000\mu \text{m}\right)$ for different constriction widths $W=1$, 4, 10, 20, 50, 100, 500, and $1000\mu \text{m}$ starting at the black continuous curve and constriction length $L=1\mu \text{m}$.

Figure 3

(Color online) Resistance versus constriction width $W$. Left $y$ axis: the symbols $\left(+,\square \right)$ correspond to the numerical values of the of a Cu plate of 100 nm thickness with 1 mm length and width and for two constrictions with $L=1\mu \text{m}(+)$ and $4\mu \text{m}(\square )$. The continuous (red) and dash-dotted lines correspond to Eq. (4) for the two constriction lengths. The parameters used are $\rho /t=0.16\times {10}^{-3}\Omega$, which corresponds to Cu, $a=1$ for a linear electrode arrangement, and mean-free path $\ell ⪡W$. Right $y$ axis: $(\ast )$ Experimental values of the measured resistance vs constriction width at $T=250\text{K}$ for the HOPG sample. The continuous line follows Eq. (4) with $\rho /t=22.8\text{m}\Omega$, $a=0.42$, and $L=W$ (note that the fit takes care only of the points where this relation holds).

Figure 4

(Color online) Resistance as a function of temperature for the HOPG sample without $(+)$ and with three constrictions of different widths $W$. The plotted resistance for the sample without constriction is obtained as the average value of $R_1$ and $R_2$, interchanging the current and voltage electrodes, following Ref. 10.

Figure 5

Ratio between the resistance $R$ of the HOPG sample with constriction and the resistivity $\rho$ divided by the thickness $t$. The three curves correspond to the measurements for the sample with the three different constriction widths $W$.

Figure 6

Mean free path $\ell (◼)$ and the Fermi wavelength $\lambda \left(o\right)$ in microns, obtained from Eqs. (5, 3), respectively, using the curve shown in Fig. 5 for $W=L=2\mu \text{m}$, the constant $a=0.42$, and the measured ratio $\Omega /W=300$. The error bars are calculated assuming an error of 10% in the ratio $\Omega /W$. The continuous line is a fit to the data using Eq. (7) and a $T^{-2}$ dependence for the mean-free path ${\ell }_e\left(T\right)$. The dashed line is calculated using Eq. (7) and the scattering rate obtained by fitting with a three-band model the magnetotransport data of HOPG as described in Ref. 13 assuming a $T$-independent Fermi velocity (see text).

Figure 7

Electron density $n$ [$(◼)$, left $y$ axis] and mobility $\mu$ [$\left(o\right)$, right $y$ axis] vs temperature for HOPG. The close stars are the values of $n$ from Ref. 13 and the open stars are the mobility values from Ref. 14, both parameters obtained using multiband models and the Boltzmann-Drude approach.

Figure 8

Mobility vs carrier density obtained for HOPG [our data, $(◼)$]. Note that the carrier density in our case changes with temperature that is why two temperature regions are depicted in the figure as a guide. The shadowed region called “ballistic” is obtained from the ballistic model prediction at 100 K from Ref. 19. The suspended graphene data are obtained from Ref. 19 (shadowed area and dashed line) for temperatures between 20 and 300 K and from Ref. 18 (vertical error bars) at 5 K.