As the only non-carbon elemental layered allotrope, few-layer black phosphorus or phosphorene has emerged as a novel two-dimensional (2D) semiconductor with both high bulk mobility and a band gap. Here we report fabrication and transport measurements of phosphorene-hexagonal BN (hBN) heterostructures with one-dimensional (1D) edge contacts. These transistors are stable in ambient conditions for >300 hours, and display ambipolar behavior, a gate-dependent metalinsulator transition, and mobility up to 4000 cm 2 /Vs. At low temperatures, we observe gatetunable Shubnikov de Haas (SdH) magneto-oscillations and Zeeman splitting in magnetic field with an estimated g-factor ~2. The cyclotron mass of few-layer phosphorene holes is determined to increase from 0.25 to 0.31 m e as the Fermi level moves towards the valence band edge. Our results underscore the potential of few-layer phosphorene (FLP) as both a platform for novel 2D physics and an electronic material for semiconductor applications. *

Real-space imaging reveals rich microscopic details of the quantum spin Hall edge conduction in monolayer WTe2.

Understanding and tuning correlated states is of great interest and significance to modern condensed matter physics. The recent discovery of unconventional superconductivity and Mott-like insulating states in magic-angle twisted bilayer graphene (tBLG) presents a unique platform to study correlation phenomena, in which the Coulomb energy dominates over the quenched kinetic energy as a result of hybridized flat bands. Extending this approach to the case of twisted multilayer graphene would allow even higher control over the band structure because of the reduced symmetry of the system. Here, we study electronic transport properties in twisted trilayer graphene (tTLG, bilayer on top of monolayer graphene heterostructure). We observed the formation of van Hove singularities which are highly tunable by twist angle and displacement field and can cause strong correlation effects under optimum conditions, including superconducting states. We provide basic theoretical interpretation of the observed electronic structure.Van der Waals heterostructures technology provides a variety of tuning knobs, including twist angle, displacement field, and stacking order, for band engineering by precise stacking of one atomically thin crystal onto another 1 . The lattice constant mismatch and relative twist angle give rise to a moiré superlattice, where, under some conditions, interlayer hybridization leads to the formation of an isolated low energy flat band, which quenches the kinetic energy of electronic system. Such low-energy subbands have been realised in several structures and emergent phenomena have been reported, including Mott-like insulators 2 , unconventional superconductivity [3][4][5] and ferromagnetism 6,7 in twisted bilayer graphene (tBLG) and twisted double bilayer graphene (tDBLG) [8][9][10][11] . Similar correlated states have also been reported in ABC-trilayer graphene (TLG) superlattice on hexagonal boron nitride (hBN) and rhombohedral stacked graphite films [12][13][14] .In this work, we study small-angle twisted trilayer graphene (tTLG) van der Waals heterostructures, where a monolayer graphene (MLG) and bilayer graphene (BLG) are stacked and rotated by a small angle with respect to each other. Compared to tBLG, more tuning knobs are expected in tTLG, since the band structures in multi-layer graphene are more tunable than that of the monolayer counterpart [15][16][17][18] . In particular, there naturally exists two stacking orders in trilayer graphene, Bernal (ABA)-stacking with mirror symmetry and rhombohedral (ABC)-stacking with inversion symmetry. The former is semimetallic, while the latter is known to be semiconducting with

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We derive a novel model-independent result for the pion susceptibility in QCD via the isovectorpseudoscalar vacuum polarisation. In the neighbourhood of the chiral-limit, the pion susceptibility can be expressed as a sum of two independent terms. The first expresses the pion-pole contribution. The second is identical to the vacuum chiral susceptibility, which describes the response of QCD's ground-state to a fluctuation in the current-quark mass. In this result one finds a straightforward explanation of a mismatch between extant estimates of the pion susceptibility.PACS numbers: 11.30. Rd, 12.38.Aw, 12.38.Lg, 24.85.+p Colour-singlet current-current correlators or, equivalently, the associated vacuum polarisations, play an important role in QCD because they are directly related to observables. The vector vacuum polarisation, e.g., couples to real and virtual photons. It is thus basic to the analysis and understanding of the process e + e − → hadrons [1,2]. In addition, analysis of the large Euclidean-time behaviour of a carefully chosen correlator can yield a hadron's mass [3,4]; and correlators are also amenable to analysis via the operator product expansion and are therefore fundamental in the application of QCD sum rules [5].In the latter connection, the vacuum pseudoscalar susceptibility (also called the pion susceptibility) plays a role in the sum-rules estimate of numerous meson-hadron couplings; e.g., the strong and parity-violating pion-nucleon couplings, g πN N and f πN N , respectively [6,7,8]. Furthermore, as will become plain herein, the pion susceptibility is as intimate a probe of QCD's vacuum structure as the scalar susceptibility [9] but its veracious analysis is more subtle, with conflicts and misconceptions being common [7,8,10,11,12].We approach the vacuum pseudoscalar susceptibility via the isovector-pseudoscalar vacuum polarization, which can be writtenwhere the trace is over flavour and spinor indices; ζ is the renormalisation scale; Z 4 (ζ, Λ) is the Lagrangian massterm renormalisation constant, which depends implicitly 1 In our Euclidean metric: {γµ, γν } = 2δµν ; γ † µ = γµ; γ 5 = γ 4 γ 1 γ 2 γ 3 ; a · b = P 4 i=1 a i b i ; and Pµ timelike ⇒ P 2 < 0.on the gauge parameter; 2 andrepresents a symmetry-preserving regularisation of the integral, with Λ the regularisation mass-scale which is taken to infinity as the last step in a complete calculation.Herein we will subsequently assume isospin symmetry; viz., equal u-and d-quark current-masses, in considering the isovector-channel. An extension to three flavours and the flavour-singlet channel can be pursued following the methods of Ref. [13].In Eq.(1), S is the dressed-quark propagator and Γ 5 is the fully-dressed pseudoscalar vertex, both of which depend on the renormalisation point. The propagator is obtained from QCD's gap equation; namely,where D µν (k) is the dressed-gluon propagator, Γ ν (q, p) is the dressed-quark-gluon vertex, and m bm is the Λ-dependent u-and d-quark current-quark bare mass. The quark-gluon-vertex and quark wave-f...

As the Fermi level and band structure of two-dimensional materials are readily tunable, they constitute an ideal platform for exploring the Lifshitz transition, a change in the topology of a material's Fermi surface. Using tetralayer graphene that host two intersecting massive Dirac bands, we demonstrate multiple Lifshitz transitions and multiband transport, which manifest as a nonmonotonic dependence of conductivity on the charge density n and out-of-plane electric field D, anomalous quantum Hall sequences and Landau level crossings that evolve with n, D, and B.

In graphite crystals, layers of graphene reside in three equivalent, but distinct, stacking positions typically referred to as A, B, and C projections. The order in which the layers are stacked defines the electronic structure of the crystal, providing an exciting degree of freedom which can be exploited for designing graphitic materials with unusual properties including predicted high-temperature superconductivity and ferromagnetism. However, the lack of control of the stacking sequence limits most research to the stable ABA form of graphite. Here we demonstrate a strategy to control the stacking order using van der Waals technology. To this end, we first visualise the distribution of stacking domains in graphite films and then perform directional encapsulation of ABC-rich graphite crystallites with hexagonal boron nitride (hBN). We found that hBN-encapsulation which is introduced parallel to the graphite zigzag edges preserves ABC stacking, while encapsulation along the armchair edges transforms the stacking to ABA. The technique presented here should facilitate new research on the important properties of ABC graphite.The possible atomic stacking arrangements for the basal planes in N-layer graphite films encompasses 2 N-2 different stacking sequences 1 . The two limiting cases are Bernal stacking (ABA) and the rhombohedral stacking (ABC), with very distinct electronic band structures 2-5 . The ABC-stacked graphite has attracted significant attention because of the promise for fascinating electronic properties. For instance, ABC trilayer, which was the main focus of research on ABC-stacked allotrope, was shown to have a semiconducting behaviour with a tunable band gap 6, 7 , insulating quantum Hall states 8, 9 , the Lifshitz transition induced by a trigonal warping 10, 11 , and the presence of chiral quasiparticles with cubic dispersion 8, 10 . The band structure of bulk rhombohedral graphite hosts 3D Dirac cones, which are gapped out in finite-N ABC graphite films 12 , thus uncovering topologically protected surface states with nearly flat band dispersions 12, 13 . These nearly flat bands at the surfaces are conducive to strongly correlated phenomena, giving rise to states with spontaneously broken symmetries, such as magnetic ordered states 14, 15 and surface superconducting states 16, 17 . Despite the promise of interesting physics, there are no experimental works on electronic transport in ABC-stacked graphite films thicker than tri-or tetralayer. This is due to the difficulty of producing the ABC allotrope on demand and of high quality. Here, we show that stacking order can be manipulated during the micromechanical assembly of graphite-hBN heterostructures, allowing the production of high-quality ABCstacked graphite films.Using bulk natural graphite crystals as a starting material, we exfoliated graphite films onto oxidised silicon substrates and used Raman spectroscopy to identify the presence of ABC stacking 18 (for optical transparency we limited the thickness of flakes to 10 nm). The optical micro...

The chiral susceptibility is given by the scalar vacuum polarisation at zero total momentum. This follows directly from the expression for the vacuum quark condensate so long as a nonperturbative symmetry preserving truncation scheme is employed. For QCD in-vacuum the susceptibility can rigorously be defined via a Pauli-Villars regularisation procedure. Owing to the scalar Ward identity, irrespective of the form or Ansatz for the kernel of the gap equation, the consistent scalar vertex at zero total momentum can automatically be obtained and hence the consistent susceptibility. This enables calculation of the chiral susceptibility for markedly different vertex Ansaetze. For the two cases considered, the results were consistent and the minor quantitative differences easily understood. The susceptibility can be used to demarcate the domain of coupling strength within a theory upon which chiral symmetry is dynamically broken. Degenerate massless scalar and pseudoscalar bound-states appear at the critical coupling for dynamical chiral symmetry breaking.Comment: 9 pages, 5 figures, 1 tabl