Quantum Hall Effect and Semimetallic Behavior of Dual-Gated ABA-Stacked Trilayer Graphene

The electronic structure of multilayer graphenes depends strongly on the number of layers as well as the stacking order. Here we explore the electronic transport of purely ABA-stacked trilayer graphenes in a dual-gated field-effect device configuration. We find both that the zero-magnetic-field transport and the quantum Hall effect at high magnetic fields are distinctly different from the monolayer and bilayer graphenes, and that they show electron-hole asymmetries that are strongly suggestive of a semimetallic band overlap. When the ABA trilayers are subjected to an electric field perpendicular to the sheet, Landau-level splittings due to a lifting of the valley degeneracy are clearly observed.

Popular Summary

The discovery of monolayer graphene, a one-atom-thick flake of carbon atoms and a building block of graphite, won the 2010 Nobel Prize in physics for leading to the discoveries and understanding of many “exceptional properties that originate from the remarkable world of quantum physics.” Pseudorelativistic charge carriers, electronic conduction rivaling that of copper, and a new manifestation of the quantum Hall effect are among such properties. However, physicists have not stopped at monolayer graphene and have expanded the realm of investigations to include the bilayer and trilayer forms. Bilayer graphene is now known to be a semiconductor with its own peculiar quantum Hall effect. How does a trilayer graphene behave compared to monolayer and bilayer graphene at one end and graphite at the other? In this experimental paper, we take steps toward addressing this fundamental question by exploring the electronic transport of trilayer graphene.

It turns out that the properties of a trilayer graphene depend on how the three component monolayers are stacked. There are two distinct stacking orders, called ABA and ABC. We have prepared high-quality samples of trilayer graphene, unambiguously identified as being ABA-stacked. We have then fabricated a trilayer sheet into a dual-gated field-effect-transistor device and investigated its electronic properties at low temperatures and in both zero and high magnetic fields. The dual-gate configuration allows us to independently tune both the sample carrier density and the electric displacement field perpendicular to the sheet—a flexibility key to identifying phenomena that are dependent on either one or the other parameter.

A wealth of information about the electronic structure of this material has emerged from our study. The sequence of quantized Hall plateaus clearly shows that three Landau levels lie close to zero energy, an observation that can be traced to the chirality of the charge carriers in the three monolayers. At high electric-displacement fields, a new sequence of quantized Hall plateaus emerges due to the underlying mirror symmetry of the ABA lattice—a phenomenon seen for the first time. Finally, a peculiar electron-hole asymmetry is observed and is understood to be linked to the underlying semimetallic band structure. An interesting scenario may arise from this asymmetry: The lowest Landau levels cross at the sample edge, giving rise to a set of counterpropagating edge states.

Our work provides a solid basis for further explorations of the electronic properties of trilayer and multilayer graphene. One may explore the consequences of a tunable semimetallic band structure for device applications, or the impact of electron-electron interactions which are expected to be strong in multilayer graphenes.

Figure 1

(a) Crystal structure of ABA (Bernal)-stacked trilayer graphene. Arrows indicate the tight-binding hopping parameters of the Slonczewski-Weiss-McClure model appropriate for trilayer graphene [8, 9]. (b) Optical image of a $3000\mathrm{\text{−}}\mu {\mathrm{m}}^2$ dual-gated ABA-stacked trilayer field-effect device. (c) Infrared reflectance, $R_{\mathrm{tri}}$, of the trilayer flake shown in (b), normalized to the substrate reflectance, $R_{\mathrm{sub}}$. The dip near $4000{\mathrm{cm}}^{-1}$ is due to transitions to or from the split-off bands, as shown schematically in the inset to (c) for the case of light electron doping. The larger dip at $6000{\mathrm{cm}}^{-1}$ is due to interference in the thin ${\mathrm{SiO}}_2$ layer [23].

Figure 2

(a) ABA-stacked trilayer sheet resistivity, $\rho$, in units of $\mathrm{k}\Omega$ per square, vs the top and back gate voltages, $V_{\mathrm{t}}$ and $V_{\mathrm{b}}$. The arrows at the saddle point indicate the axes of increasing carrier density, $n$, and the electric displacement field, $D$. (b) Profile cuts along the $n$ and $D$ axes are plotted as the conductivity, $\sigma$, vs $n$ or $D$. The solid lines are from (a), while the dotted lines are from a second sample in which the saddle point is offset from $V_{\mathrm{b}}=V_{\mathrm{t}}=0$ due to extrinsic doping, so that a larger range of $D$ is accessible.

Figure 3

(a) The ABA-stacked trilayer quantum Hall effect, plotted as the inverse Hall resistance, $R_{xy}^{-1}$, vs $V_{\mathrm{b}}$ and $V_{\mathrm{t}}$ at $B=14\mathrm{T}$. Contour lines are drawn at filling factors $\nu =\dots -1.5$, $-0.5$, $+0.5$, $+1.5\dots$. The quantum numbers of several plateaus are indicated in white. Several profile cuts through the data to highlight the various quantized plateaus are shown in (b), (c), and (d). The cut in (b) is taken along $D=0$ following the dashed line in (a). The cuts for (c), for constant top gate voltages, are taken along the top and bottom edges of (a). The cut for (d) is taken along the right edge of (a).

Figure 4

(a) The band structure of the $1+2$ model of ABA-stacked trilayer graphene (left) and the associated lowest LLs (right). QH plateaus occur when the Fermi level lies at the dashed lines, with quantum number $+6$ ($-6$) when all levels are full (empty). (b) Band structure calculated to higher order in the Slonczewski-Weiss-McClure model (see text). Two bands overlap at zero energy, creating a semimetal. To the right, the corresponding lowest LLs allow for additional QH plateaus at $-2$, $+2$, and $+4$. Level energies are plotted against position, diverging upward (downward) for electron (hole) states near the sample boundary. All LLs are twofold degenerate for the electron spin. Closely spaced pairs mark only the number of levels, not additional splittings.

Figure 5

(a) The measured low-field Hall coefficient, $R_{\mathrm{H}}$ (red circles), and sheet resistivity, $\rho$ (black curve), of ABA-stacked trilayer graphene as a function of $n$, for $D=0$. Uncertainty in the measurements is smaller than the symbol size. (b) $R_{\mathrm{H}}$ and $\rho$ of monolayer graphene. The dotted line is $R_{\mathrm{H}}=-1/ne$, averaged over disorder (see text). (c) Simplified ABA band structure. (d) $R_{\mathrm{H}}$ vs $n$ calculated for the band structure in (c), for mobility ratios ${\mu }_{\mathrm{l}}/{\mu }_{\mathrm{p}}=3.3$ (solid blue), 1 (dashed red), and 0.3 (dotted green) (see text).