Evidence of the role of contacts on the observed electron-hole asymmetry in graphene

We perform electrical transport measurements in graphene with several sample geometries. In particular, we design “invasive” probes crossing the whole graphene sheet as well as “external” probes connected through graphene side arms. The four-probe conductance measured between external probes varies linearly with charge density and is symmetric between electron and hole types of carriers. In contrast measurements with invasive probes give a strong electron-hole asymmetry and a sublinear conductance as a function of density. By comparing various geometries and types of contact metal, we show that these two observations are due to transport properties of the metal/graphene interface. The asymmetry originates from the pinning of the charge density below the metal, which thereby forms a $p\text{−}n$ or $p\text{−}p$ junction, depending on the polarity of the carriers in the bulk graphene sheet. Our results also explain part of the sublinearity observed in conductance as a function of density in a large number of experiments on graphene, which has generally been attributed to short-range scattering only.

Figure 1

(Color online) (a) Four-probe resistance calculated from voltages measured between invasive probes as a function of gate voltage, while a steady oscillating current of 100 nA runs along the whole graphene sheet. Inset: Scanning Electron Microscope image of the graphene sample TiAu1 connected to Hall probes (A–D) and invasive probes (a–e). For clarity, graphene has been colorized according to topography measured by atomic force microscopy. (b) Given the charge neutrality gate voltage $V_g^0$ identified from quantum Hall-effect measurements (see text), we plot here the asymmetry between electrons and holes by showing the odd part of the resistance defined in Eq. (1).

Figure 2

(Color online) (a) Odd part of resistance normalized by the extent $w$ of the metal/graphene interface for four pairs of invasive probes shown in the inset of Fig. 1 (same colors/grayscale). The fluctuating region at densities smaller than $1.2\times {10}^{12}{\text{cm}}^{-2}$ has been grayed. The charge density $n$ is measured using the classical Hall voltage between external probes, implying a capacitance of $13.6{\text{nF}\text{cm}}^{-2}$, consistent with the measured oxide thickness. (b) Same odd part of the resistance scaled by the ratio of the length $w$ to the distance $L$ between contacts.

Figure 3

(Color online) Conductivity as a function of gate voltage. Each color corresponds to a pair of probes identified by two letters in Fig. 1. The slope of these curves corresponds to a mobility of $\mu \approx 4600{\text{cm}}^2{\text{V}}^{-1}{\text{s}}^{-1}$. Inset: given the charge neutrality gate voltage $V_g^0$ identified from quantum Hall-effect measurements (see text), we plot here the ratio between resistivities for electrons and holes as a function of carrier density. In contrast to the case of invasive probes, the average asymmetry is invisible to the precision of our measurement (note: the observed fluctuations are reproducible). The gray line corresponds to the ratio 1.20 as observed in Ref. 7 in the presence of chemical dopants.

Figure 4

(Color online) Odd part $R_{\text{odd}}$ of the resistance scaled by the inverse width $w^{-1}$ for various samples and metals described in Table .

Figure 5

(Color online) (a) From the resistance curves plotted in Fig. 1, we show the conductance scaled by the ratio $w/L$. The noninvasive measurement between probes $B$ and $C$ from Fig. 3 is plotted as a thin line for reference. (b) Subtracting $\lambda =0.135\text{k}\Omega \mu \text{m}$ divided by the length $w$ of the metal/graphene interface, each curve from (a) is linearized for the $p$-type carriers $\left(V_g<V_g^0\right)$. Main panel: for each four-probe measurement on the samples from Table , we plot here the specific resistance $\lambda$, which best linearizes the conductance as a function of gate voltage (see text). The best fit is obtained at the dot and the vertical size of the corresponding ellipse represents the uncertainty on $\lambda$.