Landau level spectra and the quantum Hall effect of multilayer graphene

Koshino, Mikito; McCann, Edward

doi:10.1103/physrevb.83.165443

Cited by 92 publications

(168 citation statements)

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“…The LLs in TLG are not truly fourfold degenerate even in a single-particle picture, owing to the finite values of γ 2 , γ 5 and δ, which break valley degeneracy 23 ( Fig. 2c), in addition to the Zeeman interaction, which breaks spin degeneracy.…”

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“…2c). In addition, the fourfold-degenerate N = 0 LL of the SLG-like subband splits into two twofold-degenerate valley-polarized LLs, and the eightfold-degenerate (spin, valley and N = 0,1 LLs) zero-energy LLs of the BLG-like subband splits into two fourfold-degenerate LLs (the splitting between N = 0 and N = 1 LLs remains relatively small compared with the valley splitting) 23 . We note that the Zeeman splitting is at least an order of magnitude smaller than other types of splitting even at 9 T, which is the reason why LLs remain spin degenerate in this non-interacting model.…”

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Quantum Hall effect and Landau-level crossing of Dirac fermions in trilayer graphene

et al. 2011

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The physics of Dirac fermions in condensed-matter systems has received extraordinary attention following the discoveries of two new types of quantum Hall effect in single-layer and bilayer graphene [1][2][3] . The electronic structure of trilayer graphene (TLG) has been predicted to consist of both massless single-layer-graphene-like and massive bilayer-graphene-like Dirac subbands [4][5][6][7] , which should result in new types of mesoscopic and quantum Hall phenomena. However, the low mobility exhibited by TLG devices on conventional substrates has led to few experimental studies 8,9 . Here we investigate electronic transport in high-mobility (>100,000 cm Bernal-or ABA-stacked TLG (Fig. 1b) is an intriguing material to study Dirac physics and the quantum Hall effect (QHE) because of its unique band structure, which, in the simplest approximation, consists of massless single-layer-graphene (SLG)-like and massive bilayer graphene (BLG)-like subbands at low energy (Fig. 1c;. The energies of the Landau levels (LLs) for massless charge carriers depend on the square root of the magnetic field √ B (refs 1, 2,11-13), whereas for massive charge carriers they depend linearly on B (refs 3,11,12,14). Therefore, the LLs from these two different subbands in TLG should cross at some finite fields, resulting in accidental LL degeneracies at the crossing points. However, one of the main challenges so far to observe the QHE in TLG has been its low mobility on SiO 2 substrates 8,9 . To overcome this problem, we use hexagonal boron nitride (hBN) single crystals 15 as local substrates, which have been shown to reduce carrier scattering in graphene devices 16 (See Methods and Supplementary Information for fabrication). Substrate-supported devices also enable us to reach higher carrier density than suspended samples 17 , which is necessary for the observation of the LL crossings. Figure 1e,f shows the resistivity and conductivity of a TLG device at zero magnetic field. The resistivity at the Dirac peak exhibits a strong temperature dependence, which in SLG is a strong indication of high device quality 18,19 . In addition, we also observe a doublepeak structure at low temperatures (Fig. 1e). This double-peak structure is probably due to the band overlap that occurs in TLG when all SWMcC parameters are included in the tight-binding calculation of its band structure, as we show below. The field-effect mobility of this device reaches 110,000 cm 2 V −1 s −1 at 300 mK at densities as high as 6 × 10 11 cm −2 . This mobility value is two orders of magnitude higher than previously reported values for supported TLG (refs 8,9) and comparable to suspended samples 17,19 . The low disorder and high mobility enable us to probe LL crossings of Dirac fermions through the measurement of Shubnikov-de Haas oscillations (SdHOs). Figure 2a shows longitudinal resistivity ρ xx as a function of 1/B, for a carrier density n = −4.4 × 10 12 cm −2 . At low B (below ∼1 T), there are a number of oscillations characterized by broad minima separated by relatively n...

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Quantum Hall effect and Landau-level crossing of Dirac fermions in trilayer graphene

et al. 2011

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“…Considering only nearest-neighbor interactions, the Landau-level spectrum of all these forms of N -layer graphene can be described by a 4N -fold degenerate zero-energy level, shared equally between electrons and holes, and fourfold degenerate higher Landau levels for electrons and holes separately. [9][10][11] When taking more than only interlayer coupling into account, the situation becomes more complicated. In particular, for trilayer graphene the two possible stacking sequences ABA and ABC lead to different band structures 12 and a distinctly different Landaulevel spectrum.…”

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Fine structure of the lowest Landau level in suspended trilayer graphene

et al. 2013

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Magneto-transport experiments on ABC-stacked suspended trilayer graphene reveal a complete splitting of the twelve-fold degenerated lowest Landau level, and, in particular, the opening of an exchange-driven gap at the charge neutrality point. A quantitative analysis of distinctness of the quantum Hall plateaus as a function of field yields a hierarchy of the filling factors: \nu=6, 4, and 0 are the most pronounced, followed by \nu=3, and finally \nu=1, 2 and 5. Apart from the appearance of a \nu=4 state, which is probably caused by a layer asymmetry, this sequence is in agreement with Hund's rules for ABC-stacked trilayer graphene

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“…30 However, if the system is sufficiently doped, this will not alter our results. For the undoped case, the results may be slightly altered, but our results could definitely be used as a starting point to investigate the full parameter model in more detail.…”

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confidence: 71%

Spin and band ferromagnetism in trilayer graphene

2011

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We study the ground-state properties of an ABA-stacked trilayer graphene. The low-energy band structure can be described by a combination of both a linear and a quadratic particle-hole symmetric dispersion, reminiscent of monolayer and bilayer graphene, respectively. The multiband structure offers more channels for instability toward ferromagnetism when the Coulomb interaction is taken into account. Indeed, if one associates a subband-index 1/2 degree of freedom to the bands (parabolic and linear), it is possible to realize also a band-ferromagnetic state, where there is a shift in the energy bands since they fill up differently. By using a variational procedure, we compute the exchange energies for all possible variational ground states and identify the parameter space for the occurrence of spin-and band-ferromagnetic instabilities as a function of doping and interaction strength.

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Landau level spectra and the quantum Hall effect of multilayer graphene

Cited by 92 publications

References 92 publications

Quantum Hall effect and Landau-level crossing of Dirac fermions in trilayer graphene

Quantum Hall effect and Landau-level crossing of Dirac fermions in trilayer graphene

Fine structure of the lowest Landau level in suspended trilayer graphene

Spin and band ferromagnetism in trilayer graphene

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