Main

TLG has two natural stable allotropes: (1) ABA or Bernal stacking, where the atoms of the topmost layer lie exactly on top of those of the bottom layer; and (2) ABC or rhombohedral stacking, where one sublattice of the top layer lies above the centre of the hexagons in the bottom layer (Fig. 1a,b insets). This subtle distinction in stacking order results in a dramatic difference in band structure. The dispersion of B-TLG is a combination of the linear dispersion of single layer graphene (SLG) and the quadratic relation of bilayer graphene (BLG; Fig. 1a), whereas the dispersion of r-TLG is approximately cubic, with its conductance and valence bands touching at a point close to the highly symmetric K- and K′-points (Fig. 1b; refs 9, 10, 12, 13). These distinctive band structures are expected to give rise to different transport properties. For instance, owing to the cubic dispersion relation, r-TLG is expected to host stronger electronic interactions than B-TLG. This is because the interaction strength rs is approximately the ratio of the inter-electron Coulomb energy to the Fermi energy. In graphene, rs∝n−(p−1)/2, where n is charge density and pis the power of the dispersion relation; p=1, 2, 3 for SLG, BLG and r-TLG, respectively5. Consequently, at low n, the interaction strength in r-TLG is significantly higher than that in SLG, BLG and B-TLG (the last can be considered as a combination of SLG and BLG; ref. 18). Hence, r-TLG potentially allows the observation of interaction-driven phenomena, for example spontaneous gap formation, that are not easily accessible in BLG or B-TLG. Thus we seek to experimentally explore the transport properties of TLG with different stacking orders.

Figure 1: Characteristics of suspended B-TLG (ABA) and r-TLG (ABC) devices and their band structures.
figure 1

a,b, Band structures (main panel) and schematics (inset) of B- and r-stacked TLG, respectively. c, G(Vg)for two different suspended TLG devices at T=1.5 K. Upper inset: R(Vg) in log-linear scales for the same devices. Lower inset: SEM image of a suspended graphene device. Scale bar, 2 μm.

Full size image

Figure 1c shows the two-terminal conductance G of two suspended TLG devices with different stacking orders as a function of back gate voltage Vg at T=1.5 K. Both curves are ‘V’-shaped,characteristic of high-mobility samples. Surprisingly, the two devices exhibit drastically different minimum conductance Gmin at the charge neutrality point (CNP)— Gmin for B-TLG is ∼50 μS, but <∼1 μS for r-TLG. The strikingly large difference in minimum conductivity σmin, as well as the very low σmin in certain high-mobility samples, is unexpected and unique to TLG.

To systematically examine the effect of stacking order on σmin, we investigated 21 substrate-supported and 22 suspended devices. After electrical measurements, the stacking order of the devices are identified using Raman spectroscopy 19. In particular, the 2D peak of r-TLG is more asymmetric, with a pronounced shoulder, than that of B-TLG (Fig. 2a). Our findings are summarized in Fig. 2b, which plots σmin at T=4 Kversus the field effect mobility μ, revealing several interesting observations. For instance, for all B-TLG devices, σmin decreases with increasing sample mobility but remains finite, presumably because the same scattering mechanisms that yield low mobility also give rise to electron and hole puddles20, hence smearing Dirac points and leading to higher σmin. Amazingly, σmin for r-TLG devices is significantly smaller than for B-TLG. The difference is at least a factor of two or three for substrate-supported devices, and becomes dramatic for suspended devices— σmin,B-TLGremains almost constant at ∼100 μS for μ>5×104 cm2 V−1s−1, whereas σmin,r-TLG∼0, suggesting the presence of metallic and insulating states, respectively.

Figure 2: Different Raman and transport characteristics of B- and r-TLG.
figure 2

a, Raman spectroscopy of TLG with different stacking orders. b, Minimum conductivity σminversus field effect mobility μ at 4 K for suspended and non-suspended graphene devices.

Full size image

The insulating state in neutral r-TLG is not anticipated from non-interacting electron pictures. To elucidate its nature and compare transport in TLG with different stacking orders, we investigate the temperature dependence of the devices. Figure 3a,b plots the G(Vg) curve for B- and r-TLG devices, respectively, at T between 1 K and 120 K. In both data sets, G at small n declines quickly with temperature, but stays almost constant or increases modestly for high n. The opposite G(T) dependence in these two density regimes is similar to that observed in SLG (refs 21, 22), where the weak T-dependence at large n is attributed to electron–phonon interactions 22.

Figure 3: Transport data from B-TLG and r-TLG devices.
figure 3

a,b, G(Vg) for B-TLG and r-TLG devices, respectively, taken at different temperatures. Inset in b, Zoom-in plot of G(Vg) curves at T=0.6, 0.8, 5.2, 7.7 and 10 K (bottom to top). The curves at 0.6 and 0.8 K are indistinguishable. c,d, Gmin versus 1/T. The blue lines are best fits to equation (1), with EA/kB=25 K and 32 K for B-stacked and r-stacked TLG, respectively. Insets: Gmin(T) for the same data sets shown in the main panels. e, dI/dV (Vg,V) for an r-TLG at B=0 and T=300 mK. f, Line trace of e at Vg=0.

Full size image

At the CNP, Gmin,B-TLG exhibits a moderate T-dependence, typically decreasing by a factor of 2–8 when T is reduced from 200 to 1.4 K (Fig. 3c). Data from another device with a small T-dependence is shown in the Supplementary Information. Variable range hopping, which has an stretched exponential T-dependence,cannot adequately describe the data. We thus compare data in Fig. 3c to a model of thermally activated transport

where EA is the activation energy, kBthe Boltzmann constant, and G0 and A are fitting parameters. An adequate fit to equation (1) can be obtained by using EA/kB=25 K, although the fit is not entirely satisfactory.

In contrast, Gmin of r-TLG exhibits an exceedingly strong temperature dependence—it decreases exponentially with 1/Tby 2–3 orders of magnitude for 5<T<105 K, crossing over to a constant value at lower temperatures (Fig. 3d). Using EA/kB=32.0 K,we obtain excellent agreement between the experimental data and equation (1), demonstrating that transport in r-TLG at the CNP occurs by thermal activation through an energy gap of 2EA∼5.5 meV. The constant G0 is sample-dependent, and decreases from 10 to 0.1 μS with improved mobility, indicating that it arises from scattering from residual impurities on the suspended membranes.

To gain further insight into the insulating state of r-TLG, we measure its differential conductance dI/dV at T=300 mK versus Vg and source–drain bias V (Fig. 3e). The resulting stability diagram reveals a diamond-like structure centred at CNP, where dI/dV∼0 at V=0. For V>0.7 mV, dI/dV increases almost linearly with bias up to 15 mV, with a width of ∼7.5 mV, consistent with that determined from the activation energy. As V increases further to ∼21 mV, dI/dV rises sharply to ∼400 μS within 2 mV (Fig. 3f). Such an abrupt jump in dI/dV strongly resembles the density of states for gapped phases such as superconductors or charge density waves, suggesting the presence of an intrinsic insulating state at the CNP with spontaneous symmetry breaking.

To sum up our experimental findings: at B=0, we find that B-TLG remains metallic at the CNP, whereas r-TLG becomes insulating at low temperatures. Gminof the latter is thermally activated for T>5 K, with a gap-like feature in its dI/dV curve. Taken together, these results strongly suggest the presence of an intrinsic band gap in r-TLG. Such a gap is not anticipated from tight-binding calculations, and probably arises from electronic interactions, as expected from the large interaction parameter rs of r-TLG. For instance, a band gap may occur if spatial inversion symmetry is broken by strain or an external electric field, or if electronic interactions cause spontaneous symmetry breaking such as those predicted23,24,25 or reported26 for BLG.

Last, we focus on the transport characteristics of TLG devices in the quantum Hall (QH) regime. From tight-binding calculations that include only nearest-layer coupling, the Landau level (LL) spectrum for B-TLG is a superposition of those for SLG and BLG (refs 9, 27, 28, 29):

For r-TLG, the LL energies are given by 9,30

In these expressions, Nis an integer denoting the LL index, vF∼106 m s−1 is the Fermi velocity, e is the electron charge, ℏ is Planck’s constant and is the nearest-layer coupling energy. For both types of stacking order, the LL at zero energy is 12-fold degenerate, giving rise to quantized conductance plateaux with integer values …−10, −6, 6, 10, 14… of e2/h. When other interlayer and intralayer hopping terms are included, certain degeneracies could be broken11,31, although the LL are expected to retain 4-fold degeneracy for B-TLG and 2-fold degeneracy for r-TLG.

Experimentally, in contrast to theoretical predictions above, all r-TLG and most B-TLG devices develop an insulating state at the Dirac point in finite B. This insulating state, with filling factor ν=0, is often the first QH plateau that is energetically resolved. Figure 4a plots the resistance at the Dirac point Rmaxas a function of magnetic field B for a r-TLG device at different temperatures. Rmax increases exponentially (with some localization-induced fluctuations) with increasing B and decreasing T, spanning more than three orders of magnitude and reaching ∼109 Ω, which is the de facto limit of our measurement circuit.

Figure 4: Magnetotransport data of a r-TLG device.
figure 4

a, Rmax(B)at different temperatures. b, dG/dB(Vg,B). The numbers indicate the filling factors of the features. c,d, G(Vg) and G(ν) at T=1.5 K and B=0.5 (blue), 0.6 (dark green), 0.8 (green), 1 (yellow), 1.2 (orange), 1.5 (red), and 1.7 T (pink). 1.25 kΩ has been subtracted from the device resistance to account for the contact resistance and line resistance of the cryostat. e, Calculated density of states for r-TLG versus B and n.

Full size image

For QH states at finite energies, Shubnikov–de Haas (SdH) oscillations start to emerge at B as small as 0.2 T. QH plateaux with filling factors 0≤ν≤4 can be identified32. Figure 4b plots the differentiated conductance dG/dB(B,Vg) of a r-TLG device, which allows the oscillations to be clearly discerned. The QH states appear as features radiating outwards from the CNP at B=0. As the device has a large aspect ratio (length/width ∼3), we use conductance peaks to identify the filling factors of the QH plateaux33, which are determined from their slopes in the Vg−B plane: ν=n h/B e=α Vgh/B e, where α, the gate coupling efficiency, is estimated to be 2.5×1010 cm−2V−1 from geometrical considerations as well as the periods of SdH oscillations32. Using this relation, the features in Fig. 4b are determined to correspond to ν=−30±1.2, −18±1, −9.3±0.5, 0, 9±0.5, 18±1, 30±1.2 and 42±2, respectively, as indicated on the figure. Figure 4c plots the device conductance G in units of e2/h as functions of Vgtaken at different values of B; when plotted against ν, the seven curves almost collapse into one, with plateaux at ∼±9 and −18, respectively (Fig. 4d). We note that the conductance values approximately agree with the filling factors.

The emergence of filling factors at 9 and 18 are unexpected from equation (2). A close examination of Fig. 4b reveals yet another surprising feature—some plateaux that appear at low fields unexpectedly disappear at higher values of B. For instance, the ν=−18 feature is visible at B=0.25 T and develops into a well-quantized plateau for 0.5<B<0.7 T, yet it disappears for B>0.8 T. Similarly, the ν=−9 state is a well-developed plateau at 0.5 T, but vanishes for B>1.5 T. Instead, each of the ν∼±9,±18 and −30 QH features splits into three branches at Vg∼13–16 V and B∼0.6–1.3 T. The splittings at ν∼9and−18 are indicated by the dotted circles in Fig. 4b. Such apparent 3-fold degeneracy of QH plateaux is highly surprising, and has not been observed in BLG or B-TLG devices with comparable mobility.

Such splittings are signatures of the Lifshitz transition, a topological change in the Fermi surface as a function of electron doping or other parameters such as strain. For multilayer graphene, it may be induced by trigonal warping28,30,34: at very low n, the Fermi surface in r-TLG breaks up into 3-legged pockets, thus leading to triply degenerate LLs (ref. 30); these LLs should split in higher B or n, corresponding to the merging of the pockets at the Lifshitz transition. Indeed, the observed splittings occur at |Vg|∼15 V and B∼1 T, within 60% of the theoretically predicted values. The overall device behaviour is in semi-quantitative agreement with theoretical simulations of the density of states for r-TLG (Fig. 4e), which is satisfactory, considering that the simulation uses bulk graphite parameter values that are probably different for sheets of atomic thickness. We note that the biggest discrepancy between the data and simulation lies in the filling factor of the first non-zero plateau. Theoretically, one expects the ν=6 plateau to be the most energetically stable; however, ν∼9 was observed instead, suggesting the presence of large valley and spin splitting.

Our comprehensive study of Bernal- and rhombohedrally-stacked trilayer devices reveal a number of fascinating phenomena, including spontaneous gap opening in undoped r-TLG, an insulating ν=0 QH state, and the Lifshitz transition induced by trigonal warping. This opens the door to exploring a number of interesting questions, such as the nature of the insulating state in r-TLG, the unexpected ν=0 and ν=9 QH states, and transport across Bernal–rhombohedral stacking domains, and could enable new graphene electronics based on band gap and stacking-order engineering.

Methods

Graphene devices are fabricated by shadow-mask evaporation of electrodes onto graphene sheets that are either supported on substrates or suspended across pre-defined trenches in Si/SiO2 substrates35. These devices have no contaminants introduced by lithographical processes, with field effect mobility μ ranging from 210 to 1,900 cm2 V−1s−1 for non-suspended devices, and 5,000 to 280,000 cm2 V−1s−1 for suspended samples, which are significantly higher than those fabricated by lithography. We measure their electrical properties using standard lock-in techniques in a He3 or pumped He4 cryostat.