Valley Symmetry Research Articles

Emergent Inhomogeneity and Nonlocality in a Graphene Field-Effect Transistor on a Near-Parallel Moiré Superlattice of Transition Metal Dichalcogenides.

At near-parallel orientation, twisted bilayers of transition metal dichalcogenides exhibit interlayer charge transfer-driven out-of-plane ferroelectricity. Here, we report detailed electrical transport in a dual-gated graphene field-effect transistor placed on a 2.1° twisted bilayer WSe2. We observe hysteretic transfer characteristics and an emergent charge inhomogeneity with multiple local Dirac points evolving with an increasing electric displacement field (D). Concomitantly, we also observe a strong nonlocal voltage signal at D ∼ 0 V/nm that decreases rapidly with increasing D. A linear scaling of the nonlocal signal with longitudinal resistance suggests edge mode transport, which we attribute to the breaking of valley symmetry of graphene due to the spatially fluctuating electric field from the underlying polarized moiré domains. A quantitative analysis suggests the emergence of finite-size domains in graphene that modulate the charge and the valley currents simultaneously. This work underlines the impact of interfacial ferroelectricity that can trigger a new generation of devices.

Effective model for superconductivity in magic-angle graphene

We carry out large-scale quantum Monte Carlo simulations of a candidate field theory for the onset of superconductivity in magic-angle twisted bilayer graphene. The correlated insulating state at charge neutrality spontaneously breaks U(1) Moiré valley symmetry. Owing to the topological nature of the bands, skyrmion defects of the order parameter carry charge 2e and condense upon doping. In our calculations we encode the U(1) symmetry by an internal degree of freedom such that it is not broken upon lattice regularization. Furthermore, the skyrmion carries the same charge. The nature of the doping-induced phase transitions depends on the strength of the easy-plane anisotropy that reduces the SU(2) valley symmetry to U(1)×Z2. For large anisotropy, we observe two distinct transitions separated by phase coexistence. While the insulator to superconducting transition is of mean-field character, the U(1) transition is consistent with three-dimensional XY criticality. Hence, the coupling between the gapless charge excitations of the superconducting phase and the XY order parameter is irrelevant. At small anisotropy, we observe a first-order transition characterized by phase separation. Published by the American Physical Society 2024

Ising superconductivity induced from spin-selective valley symmetry breaking in twisted trilayer graphene

We show that the e-e interaction induces a strong breakdown of valley symmetry for each spin channel in twisted trilayer graphene, leading to a ground state where the two spin projections have opposite sign of the valley symmetry breaking order parameter. This leads to a spin-valley locking in which the electrons of a Cooper pair are forced to live on different Fermi lines attached to opposite valleys. Furthermore, we find an effective intrinsic spin-orbit coupling explaining the protection of the superconductivity against in-plane magnetic fields. The effect of spin-selective valley symmetry breaking is validated as it reproduces the experimental observation of the reset of the Hall density at 2-hole doping. It also implies a breakdown of the symmetry of the bands from C6 to C3, with an enhancement of the anisotropy of the Fermi lines which is at the origin of a Kohn-Luttinger (pairing) instability. The isotropy of the bands is gradually recovered, however, when the Fermi level approaches the bottom of the second valence band, explaining why the superconductivity fades away in the doping range beyond 3 holes per moiré unit cell in twisted trilayer graphene.

Multilayer graphenes as a platform for interaction-driven physics and topological superconductivity

Motivated by the recent discoveries of superconductivity in bilayer and trilayer graphene, we theoretically investigate superconductivity and other interaction-driven phases in multilayer graphene stacks. To this end, we study the density of states of multilayer graphene with up to four layers at the single-particle band structure level in the presence of a transverse electric field. Among the considered structures, tetralayer graphene with rhombohedral (ABCA) stacking reaches the highest density of states. We study the phases that can arise in ABCA graphene by tuning the carrier density and transverse electric field. For a broad region of the tuning parameters, the presence of strong Coulomb repulsion leads to a spontaneous spin and valley symmetry breaking via Stoner transitions. Using a model that incorporates the spontaneous spin and valley polarization, we explore the Kohn-Luttinger mechanism for superconductivity driven by repulsive Coulomb interactions. We find that the strongest superconducting instability is in the $p-$wave channel, and occurs in proximity to the onset of Stoner transitions. Interestingly, we find a range of densities and transverse electric fields where superconductivity develops out of a strongly corrugated, singly-connected Fermi surface in each valley, leading to a topologically non-trivial chiral $p+ip$ superconducting state with an even number of co-propagating chiral Majorana edge modes. Our work establishes ABCA stacked tetralayer graphene as a promising platform for observing strongly correlated physics and topological superconductivity.

Valley transport in strained silicene heterojunctions with zigzag and armchair interfaces

We study the carrier transport through a strained silicene heterojunction based on the tight-binding Hamiltonian, focusing on the effect of the interfaces, i.e., zigzag or armchair, on the valley polarization. We find that inter-valley transmission is prevented in the zigzag configuration owing to the conservation of transverse momentum, but this restriction is lifted for the armchair configuration. More importantly, the preservation of certain valley symmetries of the Fermi surfaces in the zigzag configuration even under strain distortion results in complete cancellation of the valley polarization after summing over all incident angles. In contrast, such symmetries are absent in the forward-propagating states for the strained armchair configuration, resulting in a sizable net valley polarization even after summation over all incident angles. Our findings provide insight toward the key role of interfaces in strained silicene-based valleytronics.

Spontaneous time-reversal symmetry breaking in twisted double bilayer graphene

Twisted double bilayer graphene (tDBG) comprises two Bernal-stacked bilayer graphene sheets with a twist between them. Gate voltages applied to top and back gates of a tDBG device tune both the flatness and topology of the electronic bands, enabling an unusual level of experimental control. Metallic states with broken spin and valley symmetries have been observed in tDBG devices with twist angles in the range 1.2–1.3°, but the topologies and order parameters of these states have remained unclear. We report the observation of an anomalous Hall effect in the correlated metal state of tDBG, with hysteresis loops spanning hundreds of mT in out-of-plane magnetic field (B⊥) that demonstrate spontaneously broken time-reversal symmetry. The B⊥ hysteresis persists for in-plane fields up to several Tesla, suggesting valley (orbital) ferromagnetism. At the same time, the resistivity is strongly affected by even mT-scale values of in-plane magnetic field, pointing to spin-valley coupling or to a direct orbital coupling between in-plane field and the valley degree of freedom.

Plasmonic transverse dipole moment in chiral fermion nanowires

Plasmons are elementary quantum excitations of conducting materials with Fermi surfaces. In two dimensions they may carry a static dipole moment that is transverse to their momentum which is quantum geometric in nature, the quantum geometric dipole (QGD). We show that this property is also realized for such materials confined in nanowire geometries. Focusing on the gapless, intrasubband plasmon excitations, we compute the transverse dipole moment ${\mathcal{D}}_{x}$ of the modes for a variety of situations. We find that single chiral fermions generically host nonvanishing ${\mathcal{D}}_{x}$, even when there is no intrinsic gap in the two-dimensional spectrum, for which the corresponding two-dimensional QGD vanishes. In the limit of very wide wires, the transverse dipole moment of the highest velocity plasmon mode matches onto the two-dimensional QGD. Plasmons of multivalley systems that are time-reversal symmetric have a vanishing transverse dipole moment but can be made to carry nonvanishing values by breaking the valley symmetry, for example, via a magnetic field. The presence of a nonvanishing transverse dipole moment for nanowire plasmons in principle offers the possibility of continuously controlling their energies and velocities by the application of a static transverse electric field.

Far-field thermal radiation properties of graphene under uniaxial strain

Electronic band structure and optical conductivity of single-layer graphene could be altered by applied uniaxial strain. Valley and space inversion symmetries are broken. Dirac cones are deformed. We investigate the effect of uniaxial strain on the radiative properties of graphene from the perspective of direction and modulus. Optical conductivity exhibits wealthy phenomenon due to the degeneracy of the energy band broken by strain. The total energy radiation exhibits a novel behavior of periodicity in , in accordance with the symmetry of the hexagonal honeycomb lattice.

Spin and valley ordering of fractional quantum Hall states in monolayer graphene

We study spin and valley ordering in the quantum Hall fractions in monolayer graphene at Landau level filling factors ${\ensuremath{\nu}}_{G}=\ensuremath{-}2+n/3\phantom{\rule{4pt}{0ex}}(n=2,4,5)$. We use exact diagonalizations on the spherical as well as toroidal geometry by taking into account the effect of realistic anisotropies that break the spin/valley symmetry of the pure Coulomb interaction. We also use a variational method based on eigenstates of the fully $\text{SU}(4)$ symmetric limit. For all the fractions we study, there are two-component states for which the competing phases are generalizations of those occurring at neutrality ${\ensuremath{\nu}}_{G}=0$. They are ferromagnetic, antiferromagnetic, charge-density wave, and K\'ekul\'e phases, depending on the values of Ising or $XY$ anisotropies in valley space. The varying spin-valley content of the states leads to ground-state quantum numbers that are different from the ${\ensuremath{\nu}}_{G}=0$ case. For filling factor ${\ensuremath{\nu}}_{G}=\ensuremath{-}2+5/3$, there is a parent state in the $\text{SU}(4)$ limit that has a flavor content $(1,1/3,1/3,0)$ where the two components that are one-third filled form a two-component singlet. The addition of anisotropies leads to the formation of new states that have no counterpart at ${\ensuremath{\nu}}_{G}=0$. While some of them are predicted by the variational approach, we find notably that negative Ising-like valley anisotropy leads to the formation of a state that is a singlet in both spin and valley space and lies beyond the reach of the variational method. Also fully spin polarized two-component states at $\ensuremath{\nu}=\ensuremath{-}2+4/3$ and $\ensuremath{-}2+5/3$ display an emergent $\text{SU}(2)$ valley symmetry because they do not feel point-contact anisotropies. We discuss implications for current experiments concerning possible spin transitions.

Visualizing broken symmetry and topological defects in a quantum Hall ferromagnet.

The interaction between electrons in graphene under high magnetic fields drives the formation of a rich set of quantum Hall ferromagnetic (QHFM) phases with broken spin or valley symmetry. Visualizing atomic-scale electronic wave functions with scanning tunneling spectroscopy (STS), we resolved microscopic signatures of valley ordering in QHFM phases and spectral features of fractional quantum Hall phases of graphene. At charge neutrality, we observed a field-tuned continuous quantum phase transition from a valley-polarized state to an intervalley coherent state, with a Kekulé distortion of its electronic density. Mapping the valley texture extracted from STS measurements of the Kekulé phase, we could visualize valley skyrmion excitations localized near charged defects. Our techniques can be applied to examine valley-ordered phases and their topological excitations in a wide range of materials.

Site-controlled telecom-wavelength single-photon emitters in atomically-thin MoTe2

Quantum emitters (QEs) in two-dimensional transition metal dichalcogenides (2D TMDCs) have advanced to the forefront of quantum communication and transduction research. To date, QEs capable of operating in O-C telecommunication bands have not been demonstrated in TMDCs. Here we report site-controlled creation of telecom QEs emitting over the 1080 to 1550 nm telecommunication wavelength range via coupling of 2D molybdenum ditelluride (MoTe2) to strain inducing nano-pillar arrays. Hanbury Brown and Twiss experiments conducted at 10 K reveal clear photon antibunching with 90% single-photon purity. The photon antibunching can be observed up to liquid nitrogen temperature (77 K). Polarization analysis further reveals that while some QEs display cross-linearly polarized doublets with ~1 meV splitting resulting from the strain induced anisotropic exchange interaction, valley degeneracy is preserved in other QEs. Valley Zeeman splitting as well as restoring of valley symmetry in cross-polarized doublets are observed under 8 T magnetic field.

Magnetic phases from competing Hubbard and extended Coulomb interactions in twisted bilayer graphene

We implement a self-consistent Hartree-Fock approximation based on a microscopic model in real space, which allows us to consider the interplay between the Hubbard and the extended Coulomb interaction in twisted bilayer graphene at the magic angle. These two interactions tend to favor different symmetry breaking patterns, having therefore complementary roles in the regimes where one or the other dominates. We show that, for sufficiently large values of the on-site Hubbard repulsion, magic angle graphene has an antiferromagnetic ground state at the charge neutrality point, while at half-filling of the lowest valence band the state becomes fully spin-polarized. In general, a suitable screening of the extended Coulomb interaction is required to observe the magnetic state in either case, as otherwise the instabilities take place in the charge sector, preferentially in the form of time-reversal, chiral or valley symmetry breaking.

Vanishing Thermal Equilibration for Hole-Conjugate Fractional Quantum Hall States in Graphene

Transport through edge channels is responsible for conduction in quantum Hall (QH) phases. Robust quantized values of charge and thermal conductances dictated by bulk topology appear when equilibration processes become dominant. We report on measurements of electrical and thermal conductances of integer and fractional QH phases, realized in hexagonal boron nitride encapsulated graphite-gated bilayer graphene devices for both electron and hole doped sides with different valley and orbital symmetries. Remarkably, for complex edges at filling factors ν=5/3 and 8/3, closely related to the paradigmatic hole-conjugate ν=2/3 phase, we find quantized thermal conductance whose values (3κ_{0}T and 4κ_{0}T, respectively where κ_{0}T is the thermal conductance quantum) are markedly inconsistent with the values dictated by topology (1κ_{0}T and 2κ_{0}T, respectively). The measured thermal conductance values remain insensitive to different symmetries, suggesting its universal nature. Our findings are supported by a theoretical analysis, which indicates that, whereas electrical equilibration at the edge is established over a finite length scale, the thermal equilibration length diverges for strong electrostatic interaction. Our results elucidate the subtle nature of crossover from coherent, mesoscopic to topology-dominated transport.

Optical properties of massive anisotropic tilted Dirac systems

We explore the effect of valley-contrasting gaps in the optical response of two-dimensional anisotropic tilted Dirac systems. We study the spectrum of intraband and interband transitions through the joint density of states (JDOS), the optical conductivity tensor, and the Drude spectral weight. The energy bands present an indirect gap in each valley. Thus a new possibility opens for the position of the Fermi level (an ``indirect zone''), and for the momentum space for allowed transitions. The JDOS near each gap displays a set of three van Hove singularities which are in contrast to the case of gapped graphene (an absorption edge only) or 8-$Pmmn$ borophene (two interband critical points due to the tilt). For the Fermi level lying in the gap, the JDOS shows the usual linear dependence on frequency, while when above an indirect zone it looks similar to the borophene case. These spectral characteristics determine the prominent structure of the optical conductivity. The longitudinal conductivity illustrates the strong anisotropy of the optical response. Similarly, the Drude weight is anisotropic and shows regions of nonlinear dependence on the Fermi level. The breaking of valley symmetry leads to a finite Hall response and associated optical properties. The anomalous and valley Hall conductivities present graphene-like behavior with characteristic modifications due to the indirect zones. Almost perfect circular dichroism can be achieved by tuning the exciting frequency with an appropriate Fermi level position. We also calculate the spectra of optical opacity and polarization rotation, which can reach magnitudes of tenths of radians in some cases. The spectral features of the calculated response properties are signatures of the simultaneous presence of tilt and mass and suggest optical ways to determine the formation of different gaps in such class of Dirac systems.

Anomalous Josephson effect and quantum anomaly in inversion asymmetric Weyl semimetals

We study a Josephson junction involving an inversion-asymmetric Weyl semimetal in presence of time-reversal symmetric (TRS) or time-reversal symmetry broken tilt in the Weyl spectra. We reveal that both types of tilts in the Weyl nodes lead to a Josephson $0$-$\pi$ transition and a zero bias valley/chiral supercurrent. Strikingly, the TRS tilt gives rise to a pure valley Josephson current (VJC) and TRS broken tilt induces a pure chirality Josephson current (CJC) in this system. The VJC and CJC are the manifestation of valley symmetry broken and $\mathbb{Z}_2$ symmetry broken by the respective tilt. We obtain the reversal of a pure VJC and pure CJC even in the zero bias condition controllable by the junction length. Our analysis of controllability of valley and chirality dependent transport in an inversion asymmetric Weyl semimetal junction could allow applications in valleytronics and chiralitytronics, respectively. The tilt induced Josephson effect provides an alternative route for supercurrent $0$-$\pi$ transition, different from the conventional ferromagnetism Josephson junctions where the spin polarization is essential. In the long junction and zero temperature limit, VJC and CJC are associated with a quantum anomaly which is manifested through a discontinuous jump in the current in absence of TRS and TRS breaking tilts, respectively.

Skyrmion zoo in graphene at charge neutrality in a strong magnetic field

As a consequence of the approximate spin-valley symmetry in graphene, the ground state of electrons in graphene at charge neutrality is a particular SU(4) quantum-Hall ferromagnet to minimize their exchange energy. If only the Coulomb interaction is taken into account, this ferromagnet can appeal either to the spin degree of freedom or equivalently to the valley pseudo-spin degree of freedom. This freedom in choice is then limited by subleading energy scales that explicitly break the SU(4) symmetry, the simplest of which is given by the Zeeman effect that orients the spin in the direction of the magnetic field. In addition, there are also valley symmetry breaking terms that can arise from short-range interactions or electron-phonon couplings. Here, we build upon the phase diagram, which has been obtained by Kharitonov [Phys. Rev. B \textbf{85}, 155439 (2012)], in order to identify the different skyrmions that are compatible with these types of quantum-Hall ferromagnets. Similarly to the ferromagnets, the skyrmions at charge neutrality are described by the $\text{Gr}(2,4)$ Grassmannian at the center, which allows us to construct the skyrmion spinors. The different skyrmion types are then obtained by minimizing their energy within a variational approach, with respect to the remaining free parameters that are not fixed by the requirement that the skyrmion at large distances from their center must be compatible with the ferromagnetic background. We show that the different skyrmion types have a clear signature in the local, sublattice-resolved, spin magnetization, which is in principle accessible in scanning-tunneling microscopy and spectroscopy.

Broken Symmetries and Kohn’s Theorem in Graphene Cyclotron Resonance

The magnetoplasmon spectrum of Landau level transitions in hexagonal boron nitride-encapsulated graphene is explored via infrared transmission magnetospectroscopy, as a function of the filling factor at fixed magnetic field. As the lowest Landau level occupancy is increased from half-filling, a non-monotonic progression of multiple cyclotron resonance peaks is observed, with a single peak evolving into four peaks and back to two, all with linewidths of order 0.5 meV. This provides a novel window on the interplay of electron interactions with broken spin and valley symmetries in the quantum Hall regime. Analysis of the peak energies shows an indirect enhancement of spin gaps below the Fermi energy, a Dirac mass at half-filling that is nearly 50\% larger than when the lowest Landau level is completely full, and a small but clear particle-hole asymmetry. We suggest a key role is played by the boron nitride in enabling interaction-enhanced broken symmetries to be observed in graphene cyclotron resonance.

Time-reversal symmetry breaking versus chiral symmetry breaking in twisted bilayer graphene

By applying a self-consistent Hartree-Fock approximation, we show that the mechanism of dynamical symmetry breaking can account for the insulating phase that develops about the charge neutrality point of twisted bilayer graphene around the magic angle. ($i$) If the Coulomb interaction is screened by metallic gates, the opening of a gap between the lowest-energy valence and conduction bands proceeds through the breakdown of chiral symmetry at strong coupling. Increasing the dielectric screening, however, we find a critical coupling at which chiral symmetry breaking is suppressed, triggering a very strong signal for time-reversal symmetry breaking with Haldane mass. ($ii$) If the long-range tail of the Coulomb interaction is not screened, we see the appearance of yet a different dominant pattern at strong coupling, which is characterized by breaking the time-reversal invariance but with opposite flux in the two sublattices of each carbon layer, with the consequent valley symmetry breaking. In this case a gap is also opened in the Dirac cones, but superposed to the splitting of the degeneracy of the low-energy bands at the $K$ points of the moir\'e Brillouin zone.

Charge order and Mott insulating ground states in small-angle twisted bilayer graphene

In this work, we determine states of electronic order of small-angle twisted bilayer graphene. Ground states are determined for weak and strong couplings which are representatives for varying distances of the twist-angle from its magic value. In the weak-coupling regime, charge density waves emerge which break translational and C 3-rotational symmetry. In the strong coupling-regime, we find rotational and translational symmetry breaking Mott insulating states for all commensurate moiré band fillings. Depending on the local occupation of superlattice sites hosting up to four electrons, global spin-(ferromagnetic) and valley symmetries are also broken which may give rise to a reduced Landau level degeneracy as observed in experiments for commensurate band fillings. The formation of those particular electron orders is traced back to the important role of characteristic non-local interactions which connect all localized states belonging to one hexagon formed by the AB- and BA-stacked regions of the superlattice.

Valleytronics in Two-dimensional Materials

A new type of degree of freedom in terms of valley symmetry has recently emerged, allowing an additional control, in addition to the traditional controls of the charge and the spin degrees of freedom, which are widely used in transistors and in spintronic devices, respectively. Valleytronics is a new type of electronics having great potential for faster and more efficient information processing and for high-density data storage in next-generation devices. Two-dimensional materials are considered to be ideal systems for investigating valleytronics due to many systems having two distinguishable valleys of opposite spin textures. In this article, we demonstrate the fundamental properties related to the valley degree of freedom in two-dimensional materials and its potential applications for valleytronic devices.