Current-voltage characteristics of graphene devices: interplay between Zener-Klein tunneling and defects

We report a theoretical/experimental study of current-voltage characteristics (I-V) of graphene devices near the Dirac point. The I-V can be described by a power law (I \propto V^\alpha, with 1<\alpha<= 1.5). The exponent is higher when the mobility is lower. This superlinear I-V is interpreted in terms of the interplay between Zener-Klein transport, that is tunneling between different energy bands, and defect scattering. Surprisingly, the Zener-Klein tunneling is made visible by the presence of defects.

We report a theoretical/experimental study of current-voltage characteristics (I-V ) of graphene devices near the Dirac point. The I-V can be described by a power law (I ∝ V α , with 1 < α ≤ 1.5). The exponent is higher when the mobility is lower. This superlinear I-V is interpreted in terms of the interplay between Zener-Klein transport, that is tunneling between different energy bands, and defect scattering. Surprisingly, the Zener-Klein tunneling is made visible by the presence of defects. Zener tunneling 1 is a concept known since the 30's, which, in a solid, refers to the tunneling of carriers from one band to another through the forbidden energy gap (for example from the conduction to the valence band). This tunnel process is very intriguing in graphene because the energy gap is suppressed to zero and because of the peculiar charge carriers behaving as Dirac fermions 2, 3 . In particular, some of the carriers (those with the velocity parallel to the electric field) experience Zener tunneling without being backscattered 3-6 , a behavior which is markedly different from the one in conventional semiconductors. The physics is the same as for relativistic electrons tunneling through a barrier, a phenomenon called Klein tunneling 7 and, for this reason, we will use the term Zener-Klein (ZK) tunneling.
In view of the remarkable properties of ZK tunneling in graphene, it is understandable that an intensive endeavor was made to challenge it. So far, the effort was focused on graphene p-n junctions [8][9][10][11][12][13] . In these devices, carriers tunnel through a sharp energy barrier induced with external local gate electrodes. Sophisticated nanofabrication techniques were employed to structure these local gates. For instance the insulator layer was very thin 8,9,12 , the local gate was separated from the graphene by an air gap 10,11 , or the local gate was extremely narrow 13 .
Here, we argue that Zener-Klein tunneling can be observed in graphene with the most common device layout (undoped, four-point configuration, and without any local gates) by simply measuring the I-V at room temperature. First, we provide an analytical semi-classical expression for the I-V s as a function of the doping. In graphene, the ZK current manifests itself with a superlinear current I ∝ V α , with α = 1.5. Then, we study the role of defects with the "exact" (non-perturbative) nonequilibrium Green-function approach finding the counterintuitive result that charged impurities enhance the visibility of the ZK current. Finally, we report measurements showing that the I-V s at the Dirac point is indeed described by power laws, I ∝ V α , with α ranging from 1 to 1.4. The exponent α is higher when the mobility is lower, consistently with our theoretical predictions.
In graphene ZK tunneling leads to unusual I-V s as compared to those of semiconductors/insulators. Let us consider transport through a piece of a material and apply a voltage -V between the right (R) and left (L) sides. For a semiconductor with electronic gap E g , ZK tunneling is possible only for eV > E g , where e > 0 is the electron charge (Fig. 1). On the contrary, in graphene (usually defined as a semi-metal) the gap is zero and, thus, ZK tunneling is possible for arbitrarily small V .
More specifically, the two-dimensional electronic-band dispersion of graphene is a cone: ǫ = ± v F k 2 ⊥ + k 2 , where k (k ⊥ ) is the wavevector-component parallel (perpendicular) to the current flow. During ballistic transport (in absence of scattering) k ⊥ is conserved. For a fixed k ⊥ , the bands are hyperbolae with gap ∆ = 2 v F k ⊥ (Fig. 1). For any V , there are conducting channels for which the tunneling is possible (with k ⊥ such that ∆ < V ). We will show that this results in a tunneling current I ∝ V 3/2 . By contrast, the ZK tunneling current in semiconductors vanishes exponentially at low V . In graphene, within the Landauer approach, the current per unit of lateral length, J, is where the factor 4 accounts for spin and valley degeneracy and the transmission T (ǫ, k ⊥ , V ) is the probability that an electron (with energy ǫ and perpendicular mo-mentum k ⊥ ) is transmitted through the channel. We assume a uniform drop of the electrostatic potential along the current-flow direction, with constant electric field V /l, being l the distance between the contacts.
e long-range defects: The transmission can be calculated with the non equilibrium Green function (NEGF) method. To describe the purely ballistic case, we also use a semiclassical approach, based on the Wentzel-Kramers-Brillouin (WKB) approximation. The transmission T W KB (ǫ, k ⊥ , V ) can be equal to 1, 0, or to T ZK = exp[−πl∆ 2 /(4 v F eV )] 4,6,14 (see the example in Fig. 2ab). We call non-tunneling current (Fig. 2a), the one associated with carriers that always remain in the same band π or π * (T W KB = 1, light-shadowed (yelow) area in Fig. 2a). We call "Zener-Klein" current, the one associated with carriers that tunnel from the π to the π * band (T W KB = T ZK , dark- shadowed (cyan) area in Fig. 2a). From Fig. 2b, the T W KB transmission is a good approximation to our most precise NEGF calculations 15 . In a graphene-based field-effect device, the density of the carriers n can be varied by changing the gate voltage V g . n = V g C g /e, being C g the gate-channel capacitance. Fig. 2d reports the current-voltage (I-V ) curves in the ballistic regime obtained with the semiclassical WKB approach (by letting T = T W KB in Eq. 1) and with the "exact" NEGF method 15 , for various dopings (we use C g = 1.15 × 10 −4 F/m 2 18 ). The two methods give almost identical results (for the WKB case, we report in note 19 an analytical expression for the I-V as a function of ǫ F ). For zero-doping (V g = 0 V) there is no contribution from the non-tunneling current, the current is entirely due to ZK tunneling, and the I-V curve is superlinear (I ∝ V 3/2 ) 19 . As soon as the system is doped (already for V g = 5 V) the ZK current is no more dominant (with respect to the non-tunneling current) and for small bias (V < 0.1 V) the I-V is ohmic (linear).
Do we expect the superlinear ZK current to be visible in actual devices? At first sight the answer is no for two reasons. First, in actual devices, the carrier concentration is never exactly zero. Indeed it has been observed 21 that the presence of charge impurities induces a spatial fluctuation of the Fermi level with respect to the Dirac point. As a result, it is difficult to achieve the experimental condition where ZK tunneling is observable (V g =0 in Fig. 2d). Second, the scattering of the carriers with optical phonons with energy ω=0.15 eV causes the current to saturate when increasing V to high values 18,20 . This process occurs for eV > ωl/l el (l el is the carrier elastic scattering length, due to defects) and is, thus, particularly relevant for high-quality high-mobility samples (with high l el ). This saturation of the non-tunneling current induces a sublinearity (I ∝ V α , with α < 1) which tends to cancel the superlinearity (α > 1) of the ZK current, further masking it (see 17 for further discussion).
The situation is possibly changed by the presence of defects. Actual devices are characterized by defects which scatter electrons elastically (that is conserving the energy) 18 . Elastic defects can be neutral point defects or charged Coulomb impurities 22 outside the graphene plane (usually at a distance ∼ 1−2 nm) 23 . Point defects affect the electrostatic potential seen by the carriers on a length scale smaller than the graphene unitary cell (short − range) and, thus, the carriers cannot be described in terms of electronic bands. On the other hand, charged impurities modify the potential uniformly on a length scale much longer than the unit cell (long −range) and the electronic bands are still a meaningful concept. The ZK current is expected to be more sensitive to shortrange defects than to long-range ones. Indeed, the ZK current is determined by a transition between two bands whose relative energy is not affected by long-range defects. Moreover, long-range defects are expected to diminish the non-tunneling current. Overall, one could wonder whether the presence of long-range defects can be used to suppress the non-tunneling current and, thus, to make visible the ZK one.
To verify this hypothesis, we simulate disordered graphene within NEGF by considering both long-and short-range elastic defects 15 . We remark that the NEGF approach provides an exact (non-perturbative) atomistic treatment of disorder. Defects are simulated by changing randomly the on-site potential by V d = 0.1 eV. This V d is a realistic choice since it provides a low-bias conductivity in reasonable agreement with measurements 17 .
From NEGF simulations, the presence of longrange defects diminishes the non-tunneling transmission ( Fig. 2c) but, in general does not reduce the ZK one. For V =0.1 V, long-range defects even increase the ZK transmission (Fig. 2c). We checked that short-range defects diminish, as expected, both the non-tunneling and the ZK transmission, with respect to the ballistic case (Fig. 2c). To see whether the relative increase of the ZK transmission with respect to the non-tunneling one can lead to measurable effects, in Fig. 2e we show the theoretical I-V curves in the presence of long-range defects. The superlinear behavior (the signature of the ZK current) is still visible at V g = 0 (I ∝ V α , with α = 1.4 in Fig. 2e) and is also visible at finite V g .
We now turn our attention to measurements, carried out on single-layer graphene devices 24 . Different devices were fabricated in a four-point configuration and have different mobilities µ ranging from 80 to 20000 cm 2 V −1 s −1 (low mobility corresponds to a higher density of defects) 24 . Fig. 3a shows a typical set of I-V characteristics for different V g applied on the backgate for a sample with a relatively modest mobility (µ = 1700 cm 2 V −1 s −1 ). The I-V is superlinear at the V g of the Dirac point, consistent with the above prediction of ZK tunneling. The superlinearity is better seen in a double-logarithmic scale plot (Fig. 3c) where the I-V is reasonably well described by a power law I ∝ V α with α = 1.3. Both α and the current values are in a remarkable qualitative agreement with calculations given the simplicity of the model as can be seen by comparing Fig. 2e (l=1µm) and Fig. 3a (l=1.1µm) for small V g . More elaborated models (e.g. with a more realistic description of impurities and including electronphonon scattering) are required to reach a quantitative agreement between theory and measurements. We observe that the superlinearity vanishes for devices with high µ. Fig. 3d shows α (extracted at the Dirac point) as a function of the mobility µ of 22 different devices. Indeed, as the mobility increases, α tends to 1 (corresponding to linear I-V ). In an additional experiment, we introduced defects in a high-mobility graphene device by bombarding it with 10 keV electrons. From Fig. 3b, before bombardment the mobility µ =7000 cm 2 V −1 s −1 and the I-V is linear with α = 1.0. After bombardment µ drops to 260 cm 2 V −1 s −1 and the I-V becomes superlinear (α = 1.2). These observations are consistent with the above discussion that in high-mobility samples the ZK superlinearity is masked by the non-tunneling current. Namely, the reduction of disorder increases the contribution of the non-tunneling current with respect to the ZK one and, also, favors the non-tunneling current saturation (due to scattering with optical phonons 18,20 ).
We now discuss other mechanisms that could lead to superlinear I-V s. It could be related to the physics occurring at tunnel barriers (such as the Luttinger liquidlike behavior in nanotubes or the breakdown of insulating barriers). However, measurements are done on high-quality devices in a four-point configuration, which makes the presence of tunnel barriers unlikely. Superlinear I-V s could also be attributed to quantum effects, such as weak localization or electron-electron interaction, but these effects should be negligible since the applied current is large, heating the graphene layer to several hundreds of Celsius 25 . Overall, Zener-Klein tunneling remains the most plausible mechanism to explain our measurements.
We finally stress that previous observations of Klein tunneling 10,12,13 in graphene were done using very different device setups. In 10,12,13, the carriers tunnel from conduction to valence bands in a p-n junction. In these nanostructured devices, the ZK tunneling is observed thanks to a configuration which allows to eliminate the non-tunneling current and thanks to the intense electric field at the p-n junction (∼ 10 −3 eV/Å, see suppl. info of 13 and 9). In our devices, which are not p-n junc-tions, the non-tunneling (intraband) current is present (this current can mask the ZK tunneling one) and the electric field (∼10 −5 eV/Å) is substantially weaker. Despite these unfavorable conditions, it is possible to probe the Zener-Klein effect.
Concluding, measurements and calculations show, consistently, that the I-V s of graphene devices become superlinear in the presence of disorder (in low-mobility samples). The superlinearity is attributed to Zener-Klein tunneling (tunneling between different energy bands, from π to π * ). In high-mobility (high-quality) graphene samples, the superlinearity is masked by the contribution of the non-tunneling current (due to carriers always remaining the same band). In low-mobility samples, the Zener-Klein tunneling current is visible because the higher density of defects decreases (filters) the nontunneling current.