Evidence for Klein Tunneling in Graphene $p\mathrm{\text{−}}n$ Junctions

Transport through potential barriers in graphene is investigated using a set of metallic gates capacitively coupled to graphene to modulate the potential landscape. When a gate-induced potential step is steep enough, disorder becomes less important and the resistance across the step is in quantitative agreement with predictions of Klein tunneling of Dirac fermions up to a small correction. We also perform magnetoresistance measurements at low magnetic fields and compare them to recent predictions.

Figure 1

Schematic diagram of a top-gated graphene device with a four-probe measurement setup. Graphene sheet is black, metal contacts and gates are dark gray.

Figure 2

(a) Four-probe resistance measured on device C540 (see Table ), as a function of $V_{\mathrm{bg}}$ and $V_{\mathrm{tg}}$. The color scale can be inferred from the cuts shown in  (b). The densities $n_{\mathrm{bg}}$ and $\Delta n_{\mathrm{tg}}$ are estimated using $V_{\mathrm{tg}}^0=2.42\mathrm{V}$, $V_{\mathrm{bg}}^0=18.65\mathrm{V}$, and $C_{\mathrm{tg}}=107\mathrm{nF}·{\mathrm{cm}}^{-2}$. (b) Resistance as a function of $V_{\mathrm{tg}}$ at several values of $V_{\mathrm{bg}}$. The two bold curves show a clear asymmetry with respect to the peak ($n_{\mathrm{tg}}=0$) for both $V_{\mathrm{bg}}<V_{\mathrm{bg}}^0$ (red) and $V_{\mathrm{bg}}>V_{\mathrm{bg}}^0$ (yellow).

Figure 3

(a) The series resistance $2R_{\mathrm{odd}}$ of the barrier interfaces as a function of $V_{\mathrm{tg}}$, for several values of $V_{\mathrm{bg}}$ for device A60 (corresponding densities $n_{\mathrm{bg}}$ are labeled). The measured resistance $2R_{\mathrm{odd}}$ (dots) is compared to the predicted value $2(R_{\mathrm{pn}}-R_{\mathrm{pp}})$ using either a diffusive model, Eq. (2) (dashed lines), or a ballistic model, Eq. (3), with the value $c_1=1.35$ chosen to best fit all six devices (solid lines). (b) Same as (a) for device C540.

Figure 4

Symbols: Ratio $\eta =R_{\mathrm{odd}}/(R_{\mathrm{pn}}-R_{\mathrm{pp}})$ as a function of top gate length $L$ for the devices of Table . $R_{\mathrm{pn}}$ is calculated with $c_1=1.35$. The vertical lines show the width of the histogram of $\eta$ for densities such that $|n_{\mathrm{bg}}|,|n_{\mathrm{tg}}|>{10}^{12}{\mathrm{cm}}^{-2}$. The dashed line at $\eta =1$ corresponds to perfect agreement between theory and experiment, in the case where the total resistance is the sum of the resistances of two $p\mathrm{\text{−}}n$ interfaces in series.