Commensurability Oscillations in One-Dimensional Graphene Superlattices

We report the experimental observation of commensurability oscillations (COs) in 1D graphene superlattices. The widely tunable periodic potential modulation in hBN-encapsulated graphene is generated via the interplay of nanopatterned few-layer graphene acting as a local bottom gate and a global Si back gate. The longitudinal magnetoresistance shows pronounced COs when the sample is tuned into the unipolar transport regime. We observe up to six CO minima, providing evidence for a long mean free path despite the potential modulation. Comparison to existing theories shows that small-angle scattering is dominant in hBN/graphene/hBN heterostructures. We observe robust COs persisting to temperatures exceeding $T=150\mathrm{K}$. At high temperatures, we find deviations from the predicted $T$ dependence, which we ascribe to electron-electron scattering.

Figure 1

Sample geometry and characteristics. (a) AFM image of the PBG of sample $B$. (b) hBN/graphene/hBN heterostructure on top of a few-layer graphene patterned bottom gate after mesa etching, before contact deposition (sample $B$, PBG outlined in black, lower hBN outlined in white). Labels 1, 2, and 3 denote the modulated, the unmodulated, and the reference area, respectively (see text). (c) Schematic longitudinal section of the sample geometry, showing the influence of the two independent gates on the graphene charge-carrier density in the case of a unipolar modulation. (d) Resistance map of sample $A$ as a function of gate voltages, $V_p$ (PBG) and $V_g$ (back gate) at $B=0$. The highly regular Fabry-Pérot pattern in the bipolar regions confirms the presence of identical barriers, forming a superlattice. The white dashed line in the $n{n}^{\prime }$ quadrant represents the $n=n^{\prime }$ configuration of the two independent gates. The configurations, marked by stars, will be addressed in the text and Fig. 2.

Figure 2

Commensurability oscillations in graphene. (a)–(c) Magnetoresistance of sample $A$ ($a_A=200\mathrm{nm}$) at $T=1.5\mathrm{K}$ for fixed $V_p=0.9\mathrm{V}$ and $V_g=10$, 15, 20 V, respectively. The densities $n$ were extracted from SdHOs at higher fields. Vertical blue lines: Calculated flat band position [Eq. (1)]. The COs appear for weak modulation (a),(c) and disappear in the demodulated situation (b). The inset in (b) shows the calculated charge-carrier density profiles for blue $V_g=5\dots 25\mathrm{V}$, where situations (a)–(c) are represented by colors. (d),(e) Magnetoresistance of sample $B$ ($a_B=80\mathrm{nm}$) at $T=40\mathrm{K}$ and fixed $V_g=-25\mathrm{V}$, with (d) $V_p=0\dots -0.6\mathrm{V}$ and (e) $V_p=-0.6\dots -1.2\mathrm{V}$. (f) $T=1.5\mathrm{K}$, where minima up to $\lambda =6$ are resolvable. Inset in (e): 1D charge-carrier density distribution (units of $10^{12}{\mathrm{cm}}^{-2}$) for $V_p=-0.3\dots -1.2\mathrm{V}$, where gate configurations in (d)–(f) are represented by colors.

Figure 3

(a) Comparison of different theoretical expressions with experiment at $V_g=-25\mathrm{V}$, $V_p=-0.8\mathrm{V}$, and $T=40\mathrm{K}$ (black curve). The blue curve represents the theory for graphene [28] with isotropic scattering and $\eta =0.08$. The red curve includes small-angle impurity scattering [49], using $\eta =0.2$, and matches the experiment well. The experimental curve does not show a pronounced minimum around 6 T due to strong SdHOs setting in. (b) Longitudinal magnetoresistance of sample A at $V_g=25\mathrm{V}$ and $V_p=0.9\mathrm{V}$ at different temperatures, from 1.5 to 155 K.

Figure 4

(a) $T$ dependence of the CO peaks, marked by the red triangles in Fig. 3. Black triangles: Similar data for sample $B$ at density $p=2.82\times {10}^{12}{\mathrm{cm}}^{-2}$. Solid lines: Expected $T$ dependence $x/\sinh (x)$, where $x=T/T_c$, with $T_c$ from Eq. (2). (b) Scattering times extracted for both samples, together with ${\tau }_p$ from $T=1.5\mathrm{K}$ and predictions for ${\tau }_{e-e}$ [52] and ${\tau }_{e-ph}$ [53].