Emergent Dirac Gullies and Gully-Symmetry-Breaking Quantum Hall States in $ABA$ Trilayer Graphene

We report on quantum capacitance measurements of high quality, graphite and hexagonal boron nitride encapsulated Bernal stacked trilayer graphene devices. At zero applied magnetic field, we observe a number of electron density- and electrical displacement-tuned features in the electronic compressibility associated with changes in Fermi surface topology. At a high displacement field and low density, strong trigonal warping gives rise to three new emergent Dirac cones in each valley, which we term “gullies.” The gullies are centered around the corners of a hexagonal Brillouin zone and related by threefold rotation symmetry. At low magnetic fields of $B=1.25\mathrm{T}$, the gullies manifest as a change in the degeneracy of the Landau levels from two to three. Weak incompressible states are also observed at integer filling within these triplet Landau levels, which a Hartree-Fock analysis indicates are associated with Coulomb-driven nematic phases that spontaneously break rotation symmetry.

Figure 1

Trilayer graphene band structure and penetration field capacitance measurements at $B=0$. (a) Lattice structure of $ABA$ trilayer graphene with hopping parameters identified. In addition to the ${\gamma }_i$, the electronic structure is determined by the interlayer potentials ${\mathrm{\Delta }}_1\propto D$ and the relative potential of the inner layer with respect to the outer layers, ${\mathrm{\Delta }}_2$. (b) Electronic band structure of trilayer graphene in the absence of an applied displacement field. The linear monolayerlike and parabolic bilayerlike bands are labeled. The momentum is relative to the $K$ point in the ${\overrightarrow{k}}_x\parallel \mathrm{\Gamma }-K$ direction. (c) Band structure evolution under applied electric field. For a wide range of electric fields, the low energy structure is described by three isolated Dirac cones slightly displaced from the $K(K^{\prime })$ points. (d) False color electron micrograph of the measured trilayer graphene device. The active region is indicated in cyan. (e) Device schematic: trilayer graphene encapsulated in $\sim 20\mathrm{nm}$ BN with few-layer graphite top and bottom gates. Independent contacts to the gates and graphene layer allow independent control of charge density $n=c_t{V}_t+c_b{V}_b$ and displacement electric field $D={\epsilon }_{\mathrm{BN}}(V_t/d_t-V_b/d_b)$, where ${\epsilon }_{\mathrm{BN}}\approx 3$ and $d_{t(b)}=18$, 20 nm are the distances to the gates. (f) Penetration field capacitance $C_P$ at $B=0\mathrm{T}$ and $T\approx 50\mathrm{mK}$ as a function of $n$ and $D$. $D$ breaks the mirror symmetry of the $ABA$-stacked trilayer graphene and induces an on-site energy difference ${\mathrm{\Delta }}_1$ between the outer layers. Main features visible in the experimental data are indicated by dashed lines and numerals. The $D<0$ region is shaded to increase the visibility of the features. Data are plotted on a saturated color scale (see Fig. S5).

Figure 2

Fermi surface topology (a) Left: Inverse electronic compressibility $\partial n/\partial \mu$ calculated from a six-band tight binding Hamiltonian as a function of interlayer asymmetry ${\mathrm{\Delta }}_1$ and electron density $n$. Right: schematic showing regions (indexed by the Roman numerals) separated by sharp changes in the compressibility. (b) Fermi contours calculated at each of the points indexed by roman numerals in (a). Color indicates the band and follows the convention of Fig. 1; note that panels (i)–(v) are Fermi surfaces of electrons while (vi)–(xi) are Fermi surfaces of holes. (c) Simulated $\partial n/\partial \mu$ and (d) measured $C_P$ at $n=-1\times {10}^{12}{\mathrm{cm}}^{-2}$. The discontinuous jump in the data at $D\approx \pm .3\mathrm{V}/\mathrm{nm}$ coincides with population of the second hole subband [(xi)–(x) transition], while the jump at $D\approx \pm .95\mathrm{V}/\mathrm{nm}$ coincides with the opening of internal electronlike Fermi surfaces within the main hole pocket [(x)–(ix) transition]. (e) Simulated $\partial n/\partial \mu$ and (f) measured $C_P$ at $n=.15\times {10}^{12}{\mathrm{cm}}^{-2}$. The sharp minimum at $D\approx \pm .5\mathrm{V}/\mathrm{nm}$ coincides with a Lifshitz transition from one multiply connected electron pocket [(iii)] to three disconnected Dirac cones [(v)]. At the discontinuity at $D\approx \pm .9\mathrm{V}/\mathrm{nm}$, the Dirac cones are joined by three additional auxiliary pockets. (g) Band structure near $K$ point for ${\mathrm{\Delta }}_1=75\mathrm{meV}$ showing the emergent Dirac gullies.

Figure 3

Triplet Landau levels and interaction induced symmetry breaking. (a) Right panel: Penetration field capacitance $C_P$ measured at $B=1.25\mathrm{T}$ as a function of $D$ and $n$. The dashed lines indicate the region of low carrier density near the valence and conduction band minima where trigonal warping has the strongest effect and leads to a formation of new Dirac points. Left panel: simulated inverse compressibility at $B=1.25\mathrm{T}$ based on band structure parameters. A phenomenological thermal broadening of 0.1 meV is assumed to generate contrast, so that only the largest gaps are visible in green. (b) $C_P$ traces for $n\in [-0.5,0.5]\times {10}^{12}{\mathrm{cm}}^{-2}$ at $D=0$ (red) and at $D=1.46\mathrm{V}/\mathrm{nm}$ (blue). The $D=0$ line trace shows strong capacitance peaks at even filling factors, in contrast to the peaks at multiples of three ($\nu =\pm 3$, 6, 9, 12) for $D=1.46\mathrm{V}/\mathrm{nm}$. (c) Evolution of LLs at $B=1.25\mathrm{T}$ as a function of interlayer potential difference ${\mathrm{\Delta }}_1$. As the electric potential increases, 12 distinct LLs at ${\mathrm{\Delta }}_1\approx 0$ intertwine into four quasidegenerate triplets, denoted $T_1,\dots ,T_4$, separated from a near- continuum of closely spaced LLs by energy gaps. (d) Expanded view of the triplet $T2$ with the average energy of the triplet subtracted. Insets show the real-space probability distribution for a coherent state formed from functions in each of the component LLs. All respect rotation symmetry. (e) Real-space probability distribution of the Hartree-Fock ground state at $1/3$ filling of the spin-polarized triplet $T2$ at ${\mathrm{\Delta }}_1=80\mathrm{meV}$, showing strongly broken threefold rotation symmetry.