Intersubband Landau Level Couplings Induced by In-Plane Magnetic Fields in Trilayer Graphene

We observed broken-symmetry quantum Hall effects and level crossings between spin- and valley- resolved Landau levels (LLs) in Bernal stacked trilayer graphene. When the magnetic field was tilted with respect to the sample normal from 0° to 66°, the LL crossings formed at intersections of zeroth and second LLs from monolayer-graphene-like and bilayer-graphene-like subbands, respectively, exhibited a sequence of transitions. The results indicate the LLs from different subbands are coupled by in-plane magnetic fields ($B_{\parallel }$), which was explained by developing the tight-binding model Hamiltonian of trilayer graphene under $B_{\parallel }$.

Figure 1

(a) $R_{xx}$ vs $V_g$ measured at 2 K. (b) Color plot of $R_{xx}$ as a function of $B_{\perp }$ and $\nu$ measured at 100 mK. (inset) Optical microscopy image of the device studied. The scale bar corresponds to $2\mu \mathrm{m}$. (c) Calculated LL spectrum as a function of $B_{\perp }$. The red (black) curves correspond to the $K^{+}$ $(K^{-})$ valleys. Solid (dashed) curves indicate up (down) spins.

Figure 2

(a) Color plot of $R_{xx}$ as a function of $B_{\perp }$ and $\nu$. Red solid circles indicate the LL crossings. (b) Calculated LL spectrum at LL crossings. The red (black) curves correspond to the $K^{+}$ $(K^{-})$ valleys. Solid (dashed) curves indicate up (down) spins. The numbers indicate $\nu$ at corresponding LL gaps.

Figure 3

(a) Color plots of $R_{xx}$ as a function of $B_{\perp }$ and $\nu$ measured at varying $\theta =0°$, 20°, 40°, and 66° (from left to right). (b) Line cuts of (a) between $\nu =5$ and 7. Each curve was offset vertically. The bottom (top) curves were measured at $B_{\perp }=4.0\mathrm{T}$ ($B_{\perp }=5.7\mathrm{T}$). The color bars overlaid on these plots indicate the ranges of $B_{\perp }$ where QHS emerged.

Figure 4

(a) LL spectrum as a function of $B_{\perp }$ for $\theta =0°$, 20°, 40°, and 66° (from left to right) calculated using $H^{\prime }({\mathrm{\Delta }}_1,B_{\parallel })$. The red (black) curves indicate $K^{+}$ $(K^{-})$ valleys. Solid (dashed) curves indicate up (down) spins. The figures indicate $\nu$ at corresponding LL gaps. At $\theta =66°$ LL gaps at $\nu =5$, 6, and 7 are highlighted by red, blue, and green arrows, respectively. (b) The values of LL gaps $\mathrm{\Delta }E$ as a function of $B_{\perp }$ at $\nu =$ (purple) 5, (blue) 6, and (green) 7 for $\theta =0°$, 20°, 40°, and 66° (from left to right), calculated by $H^{\prime }({\mathrm{\Delta }}_1,B_{\parallel }=0)$ (dotted curves) and $H^{\prime }({\mathrm{\Delta }}_1,B_{\parallel })$ (solid curves), respectively.