Colloquium: The transport properties of graphene: An introduction

An introduction to the transport properties of graphene combining experimental results and theoretical analysis is presented. In the theoretical description simple intuitive models are used to illustrate important points on the transport properties of graphene. The concept of chirality, stemming from the massless Dirac nature of the low-energy physics of the material, is shown to be instrumental in understanding its transport properties: the conductivity minimum, the electronic mobility, the effect of strain, the weak (anti)localization, and the optical conductivity.

Figure 1

(Color online) An optical image of a graphene flake, obtained form the exfoliation of graphite, with an area of $\lesssim 1{\mathrm{mm}}^2$, on top of a silicon oxide wafer (courtesy of P. Blake).

Figure 2

(Color online) Real-space lattice and Brillouin zone of graphene. Left: The hexagonal lattice of graphene, with the nearest neighbor ${\mathbf{\delta }}_i$ and the primitive ${\mathit{a}}_i$ vectors depicted. The area of the primitive cell is $A_c=3\sqrt{3}{a}_0^2∕2\simeq 5.1{\mathrm{\AA }}^2$ and $a_0\simeq 1.4\mathrm{\AA }$. Right: The Brillouin zone of graphene, with the Dirac points $\mathit{K}$ and ${\mathit{K}}^{\prime }$ indicated. Close to these points, the dispersion of graphene is conical and the density of states is proportional to the absolute value of the energy.

Figure 3

(Color online) At the $\mathit{K}$ valley, electrons have positive helicity, $h=1$, whereas at the ${\mathit{K}}^{\prime }$ one, the helicity is negative ($\mathit{Q}={\mathit{K}}^{\prime }-\mathit{K}$ represents the transferred momentum when a scattering event between the valleys takes place). In a head-on collision of the electron on a potential barrier, the backscattered electron has to change its momentum from $\mathit{q}$ to $-\mathit{q}$. For such a head-on collision (taken here along the $x$ direction), $\widehat{h}$ is a constant of motion, with eigenvalue $+1$, but backscattering would imply a modification of this eigenvalue to $-1$. This, however, cannot be because $\widehat{h}$ is a conserved quantity, then the transmission probability through the barrier, for such type of collision, has to be one. Thus, backscattering is suppressed for intravalley scattering events. On the other hand, electrons in the ${\mathit{K}}^{\prime }$ and $\mathit{K}$ valleys have opposite chirality, thus intervalley backscattering can take place (if the potential is short range) since in this case the eigenvalue of $\widehat{h}$ does not change sign. This discussion will be of importance for Sec. 5B.

Figure 4

(Color online) Numerically exact density of states (solid lines), in the limit $v_0\to \infty$, and the corresponding SCBA calculation (dashed lines), for the same impurity concentrations $n_i$. The maximum of the SCBA density of states, at $E=0$, follows the rule $\rho \left(E\right)\simeq 0.2\sqrt{n_i}{\mathrm{eV}}^{-1}$. Inset: The function $K\left(0\right)$ is plotted for a range of impurity concentrations spanning three orders of magnitude. (The numerically exact calculations are courtesy of Vitor M. Pereira.)

Figure 5

(Color online) The conductivity of graphene at the neutrality point. Left: Conductivity minimum from the Manchesters group [data from 60]. Right: Conductivity minimum from the Columbias group [data from 206] as a function of the mobility of the devices. In both panels, several devices with different mobilities $\mu$ were measured. The mobility is defined as $\mu =\sigma \left({ϵ}_F\right)∕en$, where $n$ is the electron density, and is a measure of the amount of disorder in the system. The constant $g_0$ stands for twice the quantum of conductance, $2e^2∕h\simeq 0.078{\left(\mathrm{k}\Omega \right)}^{-1}$, and is represented by the dashed line. The dash-dotted line represents the value $\sigma ={\sigma }_{\min }$, obtained in Sec. 3B.

Figure 6

(Color online) Dependence of the absolute value of electron density on the gate voltage $V_g$. In Sec. 4, we show that the electron density $n$ as a function of the gate voltage follows $n=7.2\times {10}^{10}{V}_g{\mathrm{cm}}^{-2}$. Using this, the electron density for $V_g=3\mathrm{V}$ should be $n\simeq 2.2\times {10}^{11}{\mathrm{cm}}^{-2}$, whereas for $V_g=2\mathrm{eV}$ we should have $n\simeq 1.4\times {10}^{11}{\mathrm{cm}}^{-2}$. It is clear that for $V_g=3$ the electronic density is the predicted one; however, $n$ never equals its predicted value for $V_g\lesssim 2\mathrm{V}$. Note that the vertical scale is logarithmic. [Data from 133, courtesy of Monteverde.]

Figure 7

(Color online) The conductivity of suspended and nonsuspended graphene as function of the gate voltage. Top panels: Conductivity of suspended graphene from two different devices, after current annealing $(\ell \sim 1\mu \mathrm{m})$. The top left panel corresponds to a device with $\mu \sim 170000{\mathrm{cm}}^2{\mathrm{V}}^{-1}{\mathrm{s}}^{-1}$, at electronic density of $n=2\times {10}^{11}{\mathrm{cm}}^{-2}$ (data from 21, courtesy of Bolotin). The top right panel corresponds to a device with $\mu \sim 200000{\mathrm{cm}}^2{\mathrm{V}}^{-1}{\mathrm{s}}^{-1}$, at electronic density of $n=2\times {10}^{11}{\mathrm{cm}}^{-2}$ [data from 22, courtesy of Bolotin]. Bottom panels: Conductivity of graphene on top of silicon oxide, corresponding to devices with $\mu \sim 10000{\mathrm{cm}}^2{\mathrm{V}}^{-1}{\mathrm{s}}^{-1}$ [data from 206, courtesy of Kim]. In all panels the fits use the model developed in Sec. 4B, with $n_i$ as the only fitting parameter; $R$ has to be of the order of $a_0$ (219). In the legends, the concentration of impurities is per unit cell.

Figure 8

(Color online) Resistivity and conductivity of graphene on top of silicon oxide. Top left panel: Raw data of a measurement of the resistivity ${\rho }_{\text{measured}}$ of an exfoliated graphene sheet [data from 134]. Right panel: Fit of the conductivity using Eq. (20) for the case where ${\sigma }_{\mathrm{sub}}=1∕{\rho }_{\text{measured}}$ and using Eq. (25) for the case where ${\sigma }_{\mathrm{lin}}=1∕({\rho }_{\text{measured}}-{\rho }_S)$, with ${\rho }_S=100\Omega$ [the value of ${\rho }_S=100\Omega$ is that used by 134]. Bottom left panel: Data ${\sigma }_{\mathrm{lin}}=1∕({\rho }_{\text{measured}}-{\rho }_S)$ fitted with the conductivity formula given by Eq. (26) (squares). For comparison, we give the theoretical conductivity curve (triangles) considering that the experiment had been done using $\mathrm{Hf}{\mathrm{O}}_2$ as a dielectric. This allows one to compare the modification of the numerical values of $\sigma$ due to a substrate change. [Data from 134, courtesy of Geim.]

Figure 9

(Color online) The conductivity data ${\sigma }_{\mathrm{sub}}=1∕{\rho }_{\text{measured}}$, also plotted in Fig. 8, but fitted using Eq. (27), which combines the effect of resonant and charged scatterers. In the legend, $n_{\text{short}}$ and $n_{\text{Coloumb}}$ refer to the concentration of resonant and charged scatterers, respectively. The concentration of impurities is the only fitting parameter used in the theory. [Data from 134, courtesy of Geim.]

Figure 10

(Color online) Scheme of a device made of strained graphene. The central region is the strained part. In the assumed model, the effect of strain is to modify the hopping connecting a given carbon atom to its three neighbors in such a way that two of the hoppings are equal. Shown is the momentum of the electrons in the leads and in the strained region.

Figure 11

(Color online) Resonances, bound states, and surface states. Top: the parameters are $L=100$ and $\delta =0.1$. In the resonances region, $ϵ>k_y$, the energy curves correspond to the momenta $q_x=n\pi ∕L$. The horizontal axis refers to the transverse momentum $k_y$, and the vertical one to the dimensionless energy $ϵ$. Bottom: representation in the plane energy versus $k_y$ of the five types of states appearing in this problem. The two Dirac cones, that of the contacts and that of the strained region, are represented for $k_x=0$ and $q_x=0$, respectively. The offset of the apex of the two cones is $\delta$. All states with $k_x\ne 0$ and $q_x\ne 0$ lie above the two respective cones.

Figure 12

(Color online) Pictorial representation for two types of scattering processes, whose physical interpretation is given in the text. The presence a finite magnetic field is represented by the flux $\Phi$ piercing the area defined by the two time reversed trajectories.

Figure 13

(Color online) The quantity $\delta \sigma \left(B\right)$ is defined as in Eq. (38). Left: Weak antilocalization behavior (the gate voltage used was $V_g\lesssim 1\mathrm{V}$, corresponding to an electron density $n\lesssim 7\times {10}^{10}{\mathrm{cm}}^{-2}$). Center: Weak localization behavior (the gate voltage used was $V_g=11\mathrm{V}$, corresponding to an electron density $n\simeq 8\times {10}^{11}{\mathrm{cm}}^{-2}$). Right: Dependence of $\delta \sigma \left(B\right)$ on the electronic density (it grows from I to III) at $T=27\mathrm{K}$. A fit of the data must use the rigorous formula derived in the literature (6; 123). [Data from 209, 210.]

Figure 14

(Color online) Pictorial description of the optical excitation of electrons in graphene. The absorption of a photon can only induce vertical interband transitions. From left to right we have graphene doped with holes, neutral, and doped with electrons.

Figure 15

(Color online) An optical image of an aperture partially covered with graphene and its bilayer (from left to right: air/graphene/bilayer), taken in a light transmission experiment (courtesy of A. K. Geim).

Figure 16

(Color online) Transmittance of graphene. For photon energies below $1.2\mathrm{eV}$, we plot data from reflectance measurements taken by 118; different curves correspond to different graphene devices (courtesy of Mak). For photon energies larger than $1.2\mathrm{eV}$, we plot data from two different types of measurements (142): narrow bandpass filters, with a full band width at half maximum of $10\mathrm{nm}$, and standard spectroscopy measurements. The dashed line is the result of Eq. (47). The solid line, to the right of $1.2\mathrm{eV}$, is a plot of $T$ obtained from a calculation of the conductivity, including trigonal warping corrections (203).

Figure 17

(Color online) Optical conductivity of gated graphene on top of silicon oxide, in units of ${\sigma }_0$. The data are given by the curve. The dashed curve is a calculation taking disorder into account, and the dot-dashed curve is a calculation including disorder and electron-electron interactions. The experimental curves refer to a gate voltage of $28\mathrm{V}$, which corresponds a Fermi energy of $\mu \simeq 0.18\mathrm{eV}$. The calculation took an effective temperature of $120\mathrm{K}$. In the left (right) panel we have the real (imaginary) part of the conductivity. In the inset, the measured dc conductivity of the device (solid curve) depicted together with a fit of $\sigma \left({\epsilon }_F\right)$ using Eqs. (20, 25). The fit of the dc conductivity fixes the concentration of impurities to the values of $n_i=2\times {10}^{11}{\mathrm{cm}}^{-2}$, for resonant scatterers, and $n_i=1\times {10}^{11}{\mathrm{cm}}^{-2}$, for charged ones (dashed curve). [Data from 112, courtesy of Li.]