Since the lattices of ABA-stacked graphene multilayers with an even number of layers, as well as that of monolayer graphene, satisfy spatial-inversion symmetry, their electronic bands must be spin degenerate in the presence of time-inversion symmetry. In intrinsic monolayer and bilayer graphene, when symmetry is not broken by external fields, the only spin-orbit coupling present at low energy near the corner of the Brillouin zone is the Kane-Mele term, that opens a bulk energy gap but does not break the spin degeneracy of the energy bands [C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005)]. However, spin splitting is allowed in multilayers with an odd number of layers (greater than or equal to 3) because their lattices do not satisfy spatial inversion symmetry. We show that, in trilayer graphene, in addition to the Kane-Mele term, there is a second type of intrinsic spin-orbit coupling present at low energy near the corner of the Brillouin zone. It introduces a Zeeman-like spin splitting of the energy bands at each valley, with an opposite sign of the effective magnetic field in the two valleys. We estimate the magnitude of the effective field to be ~2T.
We study spatial symmetry in general ABA-stacked multilayer graphene to illustrate how electronic spectra at the two valleys are related in a magnetic field. We show that the lattice of multilayers with an even number of layers, as well as that of monolayer graphene, satisfy spatial inversion symmetry, which rigorously guarantees valley degeneracy in the absence of time-reversal symmetry. A multilayer with an odd number of layers (three or more) lacks inversion symmetry, but there is another transformation imposing an approximate valley degeneracy, which arises because the low-energy Hamiltoni…
We analyze the ordered phases of Bernal stacked multilayer graphene in the presence of interaction induced band gaps due to sublattice symmetry breaking potentials, whose solutions can be analyzed in terms of light-mass and heavy-mass pseudospin doublets which have the same Chern numbers but opposite charge polarization directions. The application of a perpendicular external electric field reveals an effective Hund's rule for the ordering of the sublattice pseudospin doublets in a tetralayer, while a similar but more complex phase diagram develops with increasing layer number.
Zero-modes, their topological degeneracy and relation to index theorems have attracted attention in the study of single- and bilayer graphene. For negligible scalar potentials, index theorems explain why the degeneracy of the zero-energy Landau level of a Dirac hamiltonian is not lifted by gauge field disorder, for example due to ripples, whereas other Landau levels become broadened by the inhomogenous effective magnetic field. That also the bilayer hamiltonian supports such protected bulk zero-modes was proved formally by Katsnelson and Prokhorova to hold on a compact manifold by using the At…
The dynamic polarization for kekulé-patterned graphene is studied within the Random Phase Approximation (RPA). It is shown how the breaking of the valley degeneracy by the lattice modulation is manifested through the dielectric spectrum, the plasmonic dispersion, the static screening and the optical conductivity. The valley-dependent splitting of the Fermi velocities due to the kekulé distortion leads to a similar splitting in the dielectric spectrum of graphene, introducing new characteristic frequencies, which are given in terms of the valley-coupling amplitude. The valley coupling also spli…
We analyze theoretically the electronic properties of Aharonov-Bohm rings made of graphene. We show that the combined effect of the ring confinement and applied magnetic flux offers a controllable way to lift the orbital degeneracy originating from the two valleys, even in the absence of intervalley scattering. The phenomenon has observable consequences on the persistent current circulating around the closed graphene ring, as well as on the ring conductance. We explicitly confirm this prediction analytically for a circular ring with a smooth boundary modelled by a space-dependent mass term in …
Measuring degeneracy and broken-symmetry states of a system at nanoscale requires extremely high energy and spatial resolution, which has so far eluded direct observation. Here, we realize measurement of the degeneracy and subtle broken-symmetry states of graphene at nanoscale for the first time. By using edge-free graphene quantum dots, we are able to measure valley splitting and valley-contrasting spin splitting of graphene at the single-electron level. Our experiments detect large valley splitting around atomic defects of graphene due to the coexistence of sublattice symmetry breaking and t…
We show that valley degeneracy in rotationally faulted multilayer graphene may be broken in the presence of a magnetic field and interlayer commensurations. This happens due to a simultaneous breaking of both time-reversal and inversion symmetries leading to a splitting of Landau levels linear in the field. Our theoretical work is motivated by an experiment [Y. J. Song et al., Nature 467, 185 (2010)] on epitaxially grown multilayer graphene where such linear splitting of Landau levels was observed at moderate fields. We consider both bilayer and trilayer configurations and, although a linear s…
Using transport measurements, we investigate multicomponent quantum Hall (QH) ferromagnetism in dual-gated rhombohedral trilayer graphene (r-TLG), in which the real spin, orbital pseudospin and layer pseudospins of the lowest Landau level form spontaneous ordering. We observe intermediate quantum Hall plateaus, indicating a complete lifting of the degeneracy of the zeroth Landau level (LL) in the hole-doped regime. In charge neutral r-TLG, the orbital degeneracy is broken first, and the layer degeneracy is broken last and only the in presence of an interlayer potential U. In the phase space of…
We demonstrate that charge density wave (CDW) phase transition occurs on the surface of electronically doped multilayer graphene when the Fermi level approaches the M points (also known as van Hove singularities where the density of states diverge) in the Brillouin zone of graphene band structure. The occurrence of such CDW phase transitions are supported by both the electrical transport measurement and optical measurements in electrostatically doped multilayer graphene. The CDW transition is accompanied with the sudden change of graphene channel resistance at T$_m$= 100K, as well as the split…
Suspended Bernal-stacked graphene multilayers up to an unexpectedly large thickness exhibit a broken-symmetry ground state, whose origin remains to be understood. Here we show that a finite-temperature second order phase transition occurs in multilayers whose critical temperature Tc increases from 12 K in bilayers to 100 K in heptalayers. A comparison of the data to a phenomenological model inspired by a mean field approach suggests that the transition is associated with the appearance of a self-consistent valley- and spin-dependent staggered potential changing sign from one layer to the next,…
Janus dichalcogenide multilayers provide a paradigmatic platform to engineer electronic phenomena dominated by spin-orbit coupling. Their unique spin-orbit effects stem from local mirror symmetry breaking in each layer, which induces a colossal Rashba spin-orbit effect in comparison with the conventional dichalcogenide counterparts. Here we put forward twisted dichalcogenide bilayers as a simple platform to realize spin-orbit correlated states. We demonstrate the emergence of flat bands featuring strong spin-momentum locking and the emergence of non-collinear symmetry broken states when intera…
We show that Floquet engineering with circularly polarized light (CPL) can selectively split the valley degeneracy of a twisted multilayer graphene (TMG), and thus generate a controlled valley-polarized Floquet Chern flat band with tunable large Chern number. It offers a feasible optical way to manipulate the valley degree of freedom in moiré flat bands, and hence opens new opportunities to study the valleytronics of morié flat band systems. We thus expect that many of the valley-related properties of TMG, e.g. orbital ferromagnetism, can be switched by CPL with proper doping. We reveal a Cher…
We show that opacity of a clean multilayer graphene flake depends on the helicity of the circular polarized electromagnetic radiation. The effect can be understood in terms of the pseudospin selection rules for the interband optical transitions in the presence of exchange electron-electron interactions which alter the pseudospin texture in momentum space. The interactions described within a semi-analytical Hartree--Fock approach lead to the formation of the topologically different broken--symmetry states characterized by Chern numbers and zero-field anomalous Hall conductivities.
We have studied the magnetic properties of multilayers composed of ferromagnetic metal Co and heavy metals with strong spin orbit coupling (Pt and Ir). Multilayers with symmetric (ABA stacking) and asymmetric (ABC stacking) structures are grown to study the effect of broken structural inversion symmetry. We compare the perpendicular magnetic anisotropy (PMA) energy of symmetric Pt/Co/Pt, Ir/Co/Ir multilayers and asymmetric Pt/Co/Ir, Ir/Co/Pt multilayers. First, the interface contribution to the PMA is studied using the Co layer thickness dependence of the effective PMA energy. Comparison of th…
We report on transport and capacitance measurements of graphene devices in magnetic fields up to 30 T. In both techniques, we observe the full splitting of Landau levels and we employ tilted field experiments to address the origin of the observed broken symmetry states. In the lowest energy level, the spin degeneracy is removed at filling factors $ν=\pm1$ and we observe an enhanced energy gap. In the higher levels, the valley degeneracy is removed at odd filling factors while spin polarized states are formed at even $ν$. Although the observation of odd filling factors in the higher levels poin…
Mainstream among topological insulators, GaSb/InAs quantum wells present a broken gap alignment for the energy bands which supports the quantum spin Hall insulator phase and forms an important building block in the search of exotic states of matter. Such structures allow the band-gap inversion with electrons and holes confined in adjacent layers, providing a fertile ground to tune the corresponding topological properties. Using a full 3D 8-band ${\bf k}\cdot{\bf p}$ method we investigate the inverted band structure of GaSb/InAs/GaSb and InAs/GaSb/InAs multilayers and the behavior of the helica…
We present low temperature transport measurements on dual-gated suspended trilayer graphene in the quantum Hall (QH) regime. We observe QH plateaus at filling factors ν=-8, -2, 2, 6, and 10, in agreement with the full-parameter tight binding calculations. In high magnetic fields, odd-integer plateaus are also resolved, indicating almost complete lifting of the 12-fold degeneracy of the lowest Landau levels (LL). Under an out-of-plane electric field E, we observe degeneracy breaking and transitions between QH plateaus. Interestingly, depending on its direction, E selectively breaks the LL degen…
We study the electronic structures and topological properties of $(M+N)$-layer twisted graphene systems. We consider the generic situation that $N$-layer graphene is placed on top of the other $M$-layer graphene, and is twisted with respect to each other by an angle $θ$. In such twisted multilayer graphene (TMG) systems, we find that there exists two low-energy flat bands for each valley emerging from the interface between the $M$ layers and the $N$ layers. These two low-energy bands in the TMG system possess valley Chern numbers that are dependent on both the number of layers and the stacking…
A transfer matrix approach to study ballistic charge transport in few-layer graphene with chiral-symmetric stacking configurations is developed. We demonstrate that the chiral symmetry justifies a non-Abelian gauge transformation at the spectral degeneracy point (zero energy). This transformation proves the equivalence of zero-energy transport properties of the multilayer to those of the system of uncoupled monolayers. Similar transformation can be applied in order to gauge away an arbitrary magnetic field, weak strain, and hopping disorder in the bulk of the sample. Finally, we calculate the …
We have realized encapsulated trilayer MoS$_2$ devices with gated graphene contacts. In the bulk, we observe an electron mobility as high as 7000~cm$^{2}$/(V s) at a density of 3 $\times$ 10$^{12}$~cm$^{-2}$ at a temperature of 1.9~K. Shubnikov--de Haas oscillations start at magnetic fields as low as 0.9~T. The observed 3-fold Landau level degeneracy can be understood based on the valley Zeeman effect. Negatively biased split gate electrodes allow us to form a channel that can be completely pinched off for sufficiently large gate voltages. The measured conductance displays plateau-like feature…
Layer stacking and crystal lattice symmetry play important roles in the band structure and the Landau levels of multilayer graphene. ABA-stacked trilayer graphene possesses mirror-symmetry-protected monolayer-like and bilayer-like band structures. Broken mirror symmetry by a perpendicular electric field therefore induces hybridization between these bands and various quantum Hall phases emerge. We experimentally explore the evolution of Landau levels in ABA-stacked trilayer graphene under electric field. We observe a variety of valley and orbital dependent Landau level evolutions. These evoluti…
Circular polarization resolved magneto-infrared studies of multilayer epitaxial graphene (MEG) are performed using tunable quantum cascade lasers in high magnetic fields up to 17.5 T. Landau level (LL) transitions in the monolayer and bilayer graphene inclusions of MEG are resolved, and considerable electron-hole asymmetry is observed in the extracted electronic band structure. For monolayer graphene, a four-fold splitting of the $n=0$ to $n=1$ LL transition is evidenced and attributed to the lifting of the valley and spin degeneracy of the zeroth LL and the broken electron-hole symmetry. The …
The absence of an energy gap separating valence and conduction bands makes the low-energy electronic properties of graphene and its multi-layers sensitive to electron-electron interactions. In bilayers, for instance, interactions are predicted to open a gap at charge neutrality, turning the system into an insulator, as observed experimentally. In mono and (Bernal-stacked) trilayers, interactions, although still important, do not have an equally drastic effect, and these systems remain conducting at low temperature. It may be expected that interaction effects become weaker for thicker multilaye…
We theoretically study the correlated insulator states, quantum anomalous Hall (QAH) states, and field-induced topological transitions between different correlated states in twisted multilayer graphene systems. Taking twisted bilayer-monolayer graphene and twisted double-bilayer graphene as examples, we show that both systems stay in spin polarized, $C_{3z}$-broken insulator states with zero Chern number at 1/2 filling of the flat bands under finite displacement fields. In some cases these spin polarized, nematic insulator states are in the quantum valley Hall phase by virtue of the nontrivial…
Optical Faraday rotation is one of the most direct and practically important manifestations of magnetically broken time-reversal symmetry. The rotation angle is proportional to the distance traveled by the light, and up to now sizeable effects were observed only in macroscopically thick samples and in two-dimensional electron gases with effective thicknesses of several nanometers. Here we demonstrate that a single atomic layer of carbon - graphene - turns the polarization by several degrees in modest magnetic fields. The rotation is found to be strongly enhanced by resonances originating from …
Valley Valves at Domain Walls in Symmetry-Broken Rhombohedral Graphene
Florida State University（佛罗里达州立大学）；National High Magnetic Field Laboratory（国家高磁场实验室）；Instituto de Ciencia de Materiales de Madrid (ICMM), CSIC（马德里材料科学研究所，西班牙国家研究委员会）；IMDEA Nanoscience（IMDEA纳米科学研究所）；Donostia International Physics Center（圣塞巴斯蒂安国际物理中心）；University of Pennsylvania（宾夕法尼亚大学）
Recent experiments indicate that the Bernal stacked graphene multilayer (BGM) have an interaction induced gapped (or pseudo gapped) ground state. Here, we propose that, due to the electron correlation, the BGM can be induced into a half metallic phase by applying a vertical electric field and doping. The half metallic states in even-layer and odd-layer BGMs have totally different behaviors, due to their different band structures. We systematically calculate the graphene tetralayer (4L-BGM) and trilayer (3L-BGM) as the typical examples of the even-layer and odd-layer BGMs, respectively. In 4L-B…
Recent progresses in condensed matter physics, such as graphene, topological insulator and Weyl semimetal, often origin from the specific topological symmetries of their lattice structures. Quantum states with different degrees of freedom, e.g. spin, valley, layer, etc., arise from these symmetries, and the coherent superpositions of these states form multiple energy subbands. The pseudospin, a concept analogy to the Dirac spinor matrices, is a successful description of such multi-subband systems. When the electron-electron interaction dominates, many-body quantum phases arise. They usually ha…
We present analytical expressions for the eigenstates and eigenvalues of electrons confined in a graphene monolayer in the presence of a disclination. The calculations are performed in the continuum limit approximation in the vicinity of the Dirac points, solving Dirac equation by freezing out the carrier radial motion. We include the effect of an external magnetic field and show the appearence of Aharonov-Bohm oscillation and find out the conditions of gapped and gapless states in the spectrum. We show that the gauge field due to a disclination lifts the orbital degeneracy originating from th…
The dependence of mesoscopic conductance fluctuations on parallel magnetic field is used as a probe of spin degeneracy in open GaAs quantum dots. The variance of fluctuations at high parallel field is reduced from the low-field variance (with broken time-reversal symmetry) by factors ranging from roughly two in a 1 square-micron dot at low temperature, to four or greater in 8 square-micron dots. The factor of two is expected for simple Zeeman splitting of spin degenerate channels. A possible explanation for the unexpected larger factors in terms of field-dependent spin orbit scattering is prop…
At a large displacement field, in rhomboedral and Bernal-stacked graphene a normal paramagnetic state transitions to a correlated state. Recent experiments showed that such systems have several phase transitions as a function of the carrier density. The phase adjacent to a paramagnetic state has anomalously high resistance and reduced degeneracy of the Fermi sea. We show that both phenomena can be explained through a concept of partial intervalley exciton condensation: a fraction of particles condenses into excitons, and another forms an intervalley coherent Fermi liquid. The exciton part of t…
Essential properties of multilayer graphenes are diversified by the number of layers and the stacking configurations. For an $N$-layer system, Landau levels are divided into $N$ groups, with each identified by a dominant sublattice associated with the stacking configuration. We focus on the main characteristics of Landau levels, including the degeneracy, wave functions, quantum numbers, onset energies, field-dependent energy spectra, semiconductor-metal transitions, and crossing patterns, which are reflected in the magneto-optical spectroscopy, scanning tunneling spectroscopy, and quantum tran…
Near-infrared magneto-optical spectroscopy of single-walled carbon nanotubes reveals two absorption peaks with an equal strength at high magnetic fields ($>$ 55 T). We show that the peak separation is determined by the Aharonov-Bohm phase due to the tube-threading magnetic flux, which breaks the time-reversal symmetry and lifts the valley degeneracy. This field-induced symmetry breaking thus overcomes the Coulomb-induced intervalley mixing which is predicted to make the lowest exciton state optically inactive (or ``dark'').
Bernal-stacked multilayer graphene is a versatile platform to explore quantum transport phenomena and interaction physics due to its exceptional tunability via electrostatic gating. For instance, upon applying a perpendicular electric field, its band structure exhibits several off-center Dirac points (so-called Dirac gullies) in each valley. Here, the formation of Dirac gullies and the interaction-induced breakdown of gully coherence is explored via magnetotransport measurements in high-quality Bernal-stacked (ABA) trilayer graphene. In the absence of a magnetic field, multiple Lifshitz transi…
Energy spectroscopy of strongly interacting phases requires probes which minimize screening while retaining spectral resolution and local sensitivity. Here we demonstrate that such probes can be realized using atomic sized quantum dots bound to defects in hexagonal Boron Nitride tunnel barriers, placed at nanometric distance from graphene. With dot energies capacitively tuned by a planar graphite electrode, dot-assisted tunneling becomes highly sensitive to the graphene excitation spectrum. The spectra track the onset of degeneracy lifting with magnetic field at the ground state, and at unoccu…
Twisted bilayer graphene offers a unique bilayer two-dimensional-electron system where the layer separation is only in sub-nanometer scale. Unlike Bernal-stacked bilayer, the layer degree of freedom is disentangled from spin and valley, providing eight-fold degeneracy in the low energy states. We have investigated broken-symmetry quantum Hall (QH) states and their transitions due to the interplay of the relative strength of valley, spin and layer polarizations in twisted bilayer graphene. The energy gaps of the broken-symmetry QH states show an electron-hole asymmetric behaviour, and their dep…
Degeneracy is an omnipresent phenomenon in various physical systems, which has its roots in the preservation of geometrical symmetry. In electronic and photonic crystal systems, very often this degeneracy can be broken by virtue of strong interactions between photonic modes of the same energy, where the level repulsion and the hybridization between modes causes the emergence of photonic bandgaps. However, most often this phenomenon does not lead to a complete and inverted bandgap formation over the entire Brillouin zone. Here, by systematically breaking the symmetry of a two-dimensional square…
In condensed matter physics, the study of electronic states with SU(N) symmetry has attracted considerable and growing attention in recent years, as systems with such a symmetry can often have a spontaneous symmetry-breaking effect giving rise to a novel ground state. For example, pseudospin quantum Hall ferromagnet of broken SU(2) symmetry has been realized by bringing two Landau levels close to degeneracy in a bilayer quantum Hall system. In the past several years, the exploration of collective states in other multi-component quantum Hall systems has emerged. Here we show the conventional ps…
Inherent symmetries of a system lead to multiple degeneracies of its energy spectra. Introducing individual atomic impurities can locally break these symmetries, which is expected to lift the degenerate degrees of freedom around the impurities. Although central to our understanding of the fundamental properties of solids, the broken-symmetry states induced by individual atomic impurities have so far eluded observation. Here, we report nanoscale probing of the broken-symmetry states in graphene induced by two types of individual atomic impurities, i.e., isolated nitrogen dopants and isolated hy…
We review progress in developing epitaxial graphene as a material for carbon electronics. In particular, improvements in epitaxial graphene growth, interface control and the understanding of multilayer epitaxial graphene's electronic properties are discussed. Although graphene grown on both polar faces of SiC is addressed, our discussions will focus on graphene grown on the (000-1) C-face of SiC. The unique properties of C-face multilayer epitaxial graphene have become apparent. These films behave electronically like a stack of nearly independent graphene sheets rather than a thin Bernal-stack…
The behavior of a dc SQUID, based on a dirty point contacts between a single-band and three-band superconductor with broken time-reversal symmetry is investigated. Using earlier obtained results for Josephson effects in such systems new features in characteristics of a dc SQUID are revealed. It is shown that in the case of a BTRS (broken time-reversal symmetry) three-band superconductor for the applied external magnetic flux, which is divisible by the half-integer flux, strong degeneracy of ground states of a dc SQUID is taken place. This can lead to the appearance of possible multi-hysteresis…
We use the DMRG to study the correlated electron states favored by the Coulomb interaction projected onto the narrow bands of twisted bilayer graphene within a spinless one-valley model. The Hilbert space of the narrow bands is constructed from a pair of hybrid Wannier states with opposite Chern numbers. Depending on the parameters in the BM model, the DMRG in this basis determines the ground state at one particle per unit cell to be either QAH state or a state with no Hall effect which is nearly a product state. Based on this form, we then apply the variational method to study their competiti…
We show that Majorana fermions associated with Majorana flat bands emerge as zero energy modes on the [110] edge of single layer or multilayer nodal superconductors with $d_{x^2-y^2}$-wave pairing and Rashba spin-orbit coupling. Moreover, as long as the global inversion symmetry of the single layer or multilayer superconductor is broken, and in the presence of an in-plane magnetic field parallel to the [110] edge, the Majorana fermions together with usual fermionic Andreev bound states induce a triple-peak structure in tunnelling spectroscopy experiments. Importantly, we show that the zero bia…
