Valley Symmetry Research Articles

Valley Jahn-Teller Effect in Twisted Bilayer Graphene

The surprising insulating and superconducting states of narrow-band graphene twisted bilayers have been mostly discussed so far in terms of strong electron correlation, with little or no attention to phonons and electron-phonon effects. We found that, among the 33492 phonons of a fully relaxed $\theta=1.08^\circ$ twisted bilayer, there are few special, hard and nearly dispersionless modes that resemble global vibrations of the moir\'e supercell, as if it were a single, ultralarge molecule. One of them, doubly degenerate at $\Gamma$ with symmetry $A_1+B_1$, couples very strongly with the valley degrees of freedom, also doubly degenerate, realizing a so-called $\text{E}\otimes\text{e}$ Jahn-Teller (JT) coupling. The JT coupling lifts very efficiently all degeneracies which arise from the valley symmetry, and may lead, for an average atomic displacement as small as $0.5~$mA, to an insulating state at charge neutrality. This insulator possesses a non-trivial topology testified by the odd winding of the Wilson loop. In addition, freezing the same phonon at a zone boundary point brings about insulating states at most integer occupancies of the four ultra-flat electronic bands. Following that line, we further study the properties of the superconducting state that might be stabilized by these modes. Since the JT coupling modulates the hopping between AB and BA stacked regions, pairing occurs in the spin-singlet Cooper channel at the inter-(AB-BA) scale, which may condense a superconducting order parameter in the extended $s$-wave and/or $d\pm id$-wave symmetry.

Quantum Hall skyrmions at ν=0,±1 in monolayer graphene

Monolayer graphene under a strong perpendicular field exhibit quantum Hall ferromagnetism with spontaneously broken spin and valley symmetry. The approximate SU(4) spin/valley symmetry is broken by small lattice scale effects in the central Landau level corresponding to filling factors $\nu=0,\pm 1$. Notably the ground state at $\nu=0$ is believed to be a canted antiferromagnetic (AF) or a ferromagnetic (F) state depending on the component of the magnetic field parallel to the layer and the strength of small anisotropies. We study the skyrmions for the filling factors $\nu=\pm 1,0$ by using exact diagonalizations on the spherical geometry. If we neglect anisotropies we confirm the validity of the standard skyrmion picture generalized to four degrees of freedom. For filling factor $\nu=- 1$ the hole skyrmion is an infinite-size valley skyrmion with full spin polarization because it does not feel the anisotropies. The electron skyrmion is also always of infinite size. In the F phase it is always fully polarized while in the AF phase it undergoes continuous magnetization under increasing Zeeman energy. In the case of $\nu=0$ the skyrmion is always maximally localized in space both in F and AF phase. In the F phase it is fully polarized while in the AF it has also progressive magnetization with Zeeman energy. The magnetization process is unrelated to the spatial profile of the skyrmions contrary to the SU(2) case. In all cases the skyrmion physics is dominated by the competition between anisotropies and Zeeman effect but not directly by the Coulomb interactions, breaking universal scaling with the ratio Zeeman to Coulomb energy.

Development of soil spectral allocation models considering the effect of soil moisture

Soil spectral allocation or classification is usually conducted on air-dried soils. However, the field soils are not all air-dried, and the change of soil moisture will affect soil reflectance. We introduce a soil allocation model that considers the effect of soil moisture for the purpose of eliminating the effect of soil moisture. The topsoil spectral curves of four typical soils from the Songnen Plain in Northeast China were re-sampled to 10-nm intervals and converted to first-derivative spectral curves and continuum removal curves. The spectral feature parameters were extracted from continuum removal curves in the visible-near infrared (VNIR) range (350–2500 nm), and the range of 430–2400 nm was used to build soil allocation models for reducing the effect of noise. Samples with different soil moisture were mixed into air-dried soils and we calculated the coefficient of variation (CV) of different inputs to assess the effect of soil moisture and to find allocation indices that were not affected by soil moisture. We used allocation indices of Zhang et al. (2018) because of the high accuracy of their DT (Decision Tree) model to allocate mixed-soil samples. We also used allocation indices that were not affected by soil moisture to allocate mixed-soil samples with decision tree (DT), multinomial logistic regression (MLR) and multi-layer perception neural network (MLPNN), and compared the results of the two methods. The results show the following: 1) As SFPs were built with shorter bands, SFP was less sensitive to soil moisture than PCR and PCFD and thus SFP is more suitable to build soil allocation models that consider the effect of soil moisture as input than PCR and PCFD. 2) Differences in soil moisture had little effect on absorption valley shoulders, symmetry and absorption positions, moderate effect on absorption area and depth, and a major effect on the slope of different bands. 3) The effect of soil moisture on continuum removal curves of different soil classes was variable. There was little effect on Arenosols, a moderate effect on Chernozems and Cambisols, and a large effect on Phaeozems. 4) The accuracy of the DT model using allocation indices that were not affected by soil moisture was 91.892% with a Kappa coefficient of 0.888. Our results suggest that it is feasible to build soil spectral allocation models that are not affected by soil moisture, and this improves the universality of soil spectral allocation, especially to field soils, which can be of considerable help in soil classification.

Interplay between destructive quantum interference and symmetry-breaking phenomena in graphene quantum junctions

We study the role of electronic spin and valley symmetry in the quantum interference (QI) patterns of the transmission function in graphene quantum junctions. In particular, we link it to the position of the destructive QI antiresonances. When the spin or valley symmetry is preserved, electrons with opposite spin or valley display the same interference pattern. On the other hand, when a symmetry is lifted, the antiresonances are split, with a consequent dramatic differentiation of the transport properties in the respective channel. We demonstrate rigorously this link in terms of the analytical structure of the electronic Green function, which follows from the symmetries of the microscopic model, and we confirm the result with numerical calculations for graphene nanoflakes. We argue that this is a generic and robust feature that can be exploited in different ways for the realization of nanoelectronic QI devices, generalizing the recent proposal of a QI-assisted spin-filtering effect [A. Valli et al., Nano Lett. 18, 2158 (2018)].

Superconducting states and Majorana modes in transition-metal dichalcogenides under inhomogeneous strain

We study the effect of inhomogeneous strain on transition-metal dichalcogenides with a large intrinsic gap in their spectrum. It is found that, by tuning the chemical potential, superconductivity can preserve within the strain-induced discrete pseudo Landau levels (LLs), which introduce interesting topological properties to these systems. As we show, the superconductivity for integer fillings is quantum critical, and the quantum critical coupling strength is determined by the spacing between the two LLs closest to the Fermi level. For partial fillings, the superconducting gap is scaled linearly with the coupling strength, and decreases rapidly when the chemical potential shifts away from the middle of each LL. In the presence of a Zeeman field, a pair of Majorana modes emerge simultaneously in the two valleys of strained dichalcogenides. When valley symmetry is further destroyed, a single Majorana mode can be expected to emerge at the edges of the strained monolayer dichalcogenides.

Origin of Mott Insulating Behavior and Superconductivity in Twisted Bilayer Graphene

A remarkable recent experiment has observed Mott insulator and proximate superconductor phases in twisted bilayer graphene when electrons partly fill a nearly flat mini-band that arises a `magic' twist angle. However, the nature of the Mott insulator, origin of superconductivity and an effective low energy model remain to be determined. We propose a Mott insulator with intervalley coherence that spontaneously breaks U(1) valley symmetry, and describe a mechanism that selects this order over the competing magnetically ordered states favored by the Hunds coupling. We also identify symmetry related features of the nearly flat band that are key to understanding the strong correlation physics and constrain any tight binding description. First, although the charge density is concentrated on the triangular lattice sites of the moir$\text{\'e }$ pattern, the Wannier states of the tight-binding model must be centered on different sites which form a honeycomb lattice. Next, spatially localizing electrons derived from the nearly flat band necessarily breaks valley and other symmetries within any mean-field treatment, which is suggestive of a valley-ordered Mott state, and also dictates that additional symmetry breaking is present to remove symmetry-enforced band contacts. Tight-binding models describing the nearly flat mini-band are derived, which highlight the importance of further neighbor hopping and interactions. We discuss consequences of this picture for superconducting states obtained on doping the valley ordered Mott insulator. We show how important features of the experimental phenomenology may be explained and suggest a number of further experiments for the future. We also describe a model for correlated states in trilayer graphene heterostructures and contrast it with the bilayer case.

Graphene quantum dot with a Coulomb impurity: Subcritical and supercritical regime

We study the influence of confinement on the atomic collapse due to a Coulomb impurity placed at the center of a graphene quantum dot of radius $R$. We apply the zigzag or infinite-mass boundary condition and consider both a point-size and a finite-size impurity. As a function of the impurity strength $Z\ensuremath{\alpha}$, the energy spectra are discrete. In the case of the zigzag boundary condition, the degenerate (with respect to the angular momentum $m$) zero-energy levels are pulled down in energy as $Z\ensuremath{\alpha}$ increases, and they remain below $\ensuremath{\epsilon}=\ensuremath{-}Z\ensuremath{\alpha}$. Our results show that the energy levels exhibit a $1/R$ dependence in the subcritical regime $[Z\ensuremath{\alpha}l|km+1/2|$, $k=1\phantom{\rule{4pt}{0ex}}(\ensuremath{-}1)$ for the $K$ (${K}^{\ensuremath{'}}$) valley]. In the supercritical regime ($Z\ensuremath{\alpha}g|km+1/2|$) we find a qualitatively very different behavior where the levels decrease as a function of $R$ in a nonmonotonic manner. While the valley symmetry is preserved in the presence of the impurity, we find that the impurity breaks electron-hole symmetry. We further study the energy spectrum of zigzag quantum dots in gapped graphene. Our results show that as the gap increases, the lowest electron states are pushed into the gap by the impurity.

Two-dimensional topological superconducting phases emerged from d-wave superconductors in proximity to antiferromagnets

Motivated by the recent observations of nodeless superconductivity in the monolayer CuO2 grown on the Bi2Sr2CaCu2O8+δ substrates, we study the two-dimensional superconducting (SC) phases described by the two-dimensional model in proximity to an antiferromagnetic (AF) insulator. We found that i) the nodal d-wave SC state can be driven via a continuous transition into a nodeless d-wave pairing state by the proximity-induced AF field. ii) The energetically favorable pairing states in the strong field regime have extended s-wave symmetry and can be nodal or nodeless. iii) Between the pure d-wave and s-wave paired phases, there emerge two topologically distinct SC phases with symmetry, i.e., the weak and strong pairing phases, and the weak pairing phase is found to be a Z2 topological superconductor protected by valley symmetry, exhibiting robust gapless nonchiral edge modes. These findings strongly suggest that the high-Tc superconductors in proximity to antiferromagnets can realize fully gapped symmetry-protected topological SC.

Broken-Symmetry Quantum Hall States in Twisted Bilayer Graphene.

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 dependence on the induced displacement field are opposite between even and odd filling factor states. These results strongly suggest that the QH states with broken valley and spin symmetries for individual layer become hybridized via interlayer tunnelling, and the hierarchy of the QH states is sensitive to both magnetic field and displacement field due to charge imbalance between layers.

Zero-energy modes and valley asymmetry in the Hofstadter spectrum of bilayer graphene van der Waals heterostructures with hBN

We investigate the magnetic minibands of a heterostructure consisting of bilayer graphene (BLG) and hexagonal boron nitride (hBN) by numerically diagonalizing a two-band Hamiltonian that describes electrons in BLG in the presence of a moir\'e potential. Due to inversion-symmetry breaking characteristic for the moir\'e potential, the valley symmetry of the spectrum is broken, but despite this, the zero-energy Landau level in BLG survives, albeit with reduced degeneracy. In addition, we derive effective models for the low-energy features in the magnetic minibands and demonstrate the appearance of secondary Dirac points in the valence band, which we confirm by numerical simulations. Then, we analyze how single-particle gaps in the fractal energy spectrum produce a sequence of incompressible states observable under a variation of carrier density and magnetic field.

Valley-symmetry-preserved transport in ballistic graphene with gate-defined carrier guiding

Two distinct valleys in the electronic band structure of graphene provide an additional degree of freedom that could be exploited for devices. Conservation of this valley symmetry can now be seen in the quantized conductance of graphene nanoribbons.

Atomistic study of band structure and transport in extremely thin channel InP MOSFETs

Abstractauthoren III–V channel materials have emerged as one of the major contenders to replace silicon as the channel material in sub-10 nm transistors. Motivated by this, we study the feasibility of using InP as a channel material in extremely scaled MOSFETs. In this work, we have performed a comprehensive analysis of the band structure of extremely thin InP channels with different surface orientations and transport directions using first-principle density functional theory calculations. We show that the effective masses in the valley and the bandgap increase monotonically as the thickness decreases for each orientation. Valley symmetry is found to be orientation dependent. Further, the performance of extremely thin InP channel double-gate MOSFET is analyzed via semi-classical as well as full quantum ballistic transport simulations using the non-equilibrium Green's function (NEGF) approach.

Energy Gaps and Layer Polarization of Integer and Fractional Quantum Hall States in Bilayer Graphene.

Owing to the spin, valley, and orbital symmetries, the lowest Landau level in bilayer graphene exhibits multicomponent quantum Hall ferromagnetism. Using transport spectroscopy, we investigate the energy gaps of integer and fractional quantum Hall (QH) states in bilayer graphene with controlled layer polarization. The state at filling factor ν=1 has two distinct phases: a layer polarized state that has a larger energy gap and is stabilized by high electric field, and a hitherto unobserved interlayer coherent state with a smaller gap that is stabilized by large magnetic field. In contrast, the ν=2/3 quantum Hall state and a feature at ν=1/2 are only resolved at finite electric field and large magnetic field. These results underscore the importance of controlling layer polarization in understanding the competing symmetries in the unusual QH system of BLG.

Dirac cones beyond the honeycomb lattice: A symmetry-based approach

Recently, several new materials exhibiting massless Dirac fermions have been proposed. However, many of these do not have the typical graphene honeycomb lattice, which is often associated with Dirac cones. Here, we present a classification of these different two-dimensional Dirac systems based on the space groups, and discuss our findings within the context of a minimal two-band model. In particular, we show that the emergence of massless Dirac fermions can be attributed to the mirror symmetries of the materials. Moreover, we uncover several novel Dirac systems that have up to twelve inequivalent Dirac cones, and show that these can be realized in (twisted) bilayers. Hereby, we obtain systems with an emergent SU(2N) valley symmetry with N=1,2,4,6,8,12. Our results pave the way to engineer different Dirac systems, besides providing a simple and unified description of materials ranging from square- and $\beta$-graphynes, to Pmmn-Boron, TiB$_2$, phosphorene, and anisotropic graphene.

Chiral symmetry breaking by a magnetic field in graphene

We report a novel many-body mechanism for the chiral symmetry breaking in the lowest Landau levels (LLs) of graphene, which is responsible for the new quantum Hall states near the charge neutral Dirac points corresponding to ν=±1 plateaus observed in magnetotransport measurements (Zhang et al., 2006 [31], Jiang et al., 2007 [32]). We show that, when a uniform magnetic field is switched on, valley symmetry breaking emerges in graphene due the interaction of electrons with zone-boundary phonons having A1g-symmetry. The induced valley gap is about 14.68 KkBB(T) whose magnitude and field dependence are compatible with the observed quantum Hall plateaus in graphene.

Valley-polarized interlayer conduction of anisotropic Dirac fermions in SrMnBi2.

We report the valley-selective interlayer conduction of SrMnBi(2) under in-plane magnetic fields. The c-axis resistivity of SrMnBi(2) shows clear angular magnetoresistance oscillations indicating coherent interlayer conduction. Strong fourfold variation of the coherent peak in the c-axis resistivity reveals that the contribution of each Dirac valley is significantly modulated by the in-plane field orientation. This originates from anisotropic Dirac Fermi surfaces with strong disparity in the momentum-dependent interlayer coupling. Furthermore, we found a signature of broken valley symmetry at high magnetic fields. These findings demonstrate that a quasi-two-dimensional anisotropic Dirac system can host a valley-polarized interlayer current through magnetic valley control.

Erratum: Spin-Orbit Coupling, Quantum Dots, and Qubits in Monolayer Transition Metal Dichalcogenides [Phys. Rev. X 4, 011034 (2014)

where ħω c 1⁄4 ejBzj=m eff . Accordingly, the second sentence below Eq. (2) should read as follows: “The next term ~ Hvl 1⁄4 τ 2 ~ gvlμBBz breaks the valley symmetry of Landau levels.” The ~ gvl values shown in the first row of Table II of the manuscript need to be corrected because they are actually half of the correct values. In addition, a minus-sign change comes from the above-mentioned redefinition of ~ gvl. Furthermore, since gvl in Eq. (4) is defined through ~ gvl, both its definition and its values (shown in the third row of Table II) are also modified. After redefinition, gvl is given by gvl 1⁄4 ð2me=meffÞ þ ~ gvl. The new Table II of the manuscript, with the corrected values of the valley g factors ~ gvl and gvl, is shown below. Note that the spin g factors g⊥so and g⊥sp do not need to be changed. Some of the gvl values also appear in the captions of Figs. 5 and 6, which therefore also need to be corrected. The caption of Fig. 5 should read as follows. “Spectrum of a 40-nmWS2 quantum dot as a function of the perpendicular magnetic field Bz > 0. Black (red) lines show the spin ↑ (↓) states from the valley K (K0). The values ofm eff can be found in Table I, whereas gvl 1⁄4 −3.36 and g⊥sp 1⁄4 1.99 (see Table II).”

Fractional and integer quantum Hall effects in the zeroth Landau level in graphene

Experiments on the fractional quantized Hall effect in the zeroth Landau level of graphene have revealed some striking differences between filling factors in the ranges $0<|\ensuremath{\nu}|<1$ and $1<|\ensuremath{\nu}|<2$. We argue that these differences can be largely understood as a consequence of the effects of terms in the Hamiltonian which break SU(2) valley symmetry, which we find to be important for $|\ensuremath{\nu}|<1$ but negligible for $|\ensuremath{\nu}|>1$. The effective absence of valley anisotropy for $|\ensuremath{\nu}|>1$ means that states with an odd numerator, such as $|\ensuremath{\nu}|=5/3$ or 7/5, can accommodate charged excitations in the form of large-radius valley skyrmions, which should have a low energy cost and may be easily induced by coupling to impurities. The absence of observed quantum Hall states at these fractions is likely due to the effects of valley skyrmions. For $|\ensuremath{\nu}|<1$, the anisotropy terms favor phases in which electrons occupy states with opposite spins, similar to the antiferromagnetic state previously hypothesized to be the ground state at $\ensuremath{\nu}=0$. The anisotropy and Zeeman energies suppress large-area skyrmions, so that quantized Hall states can be observable at a set of fractions similar to those in GaAs two-dimensional electron systems. In a perpendicular magnetic field $B$, the competition between the Coulomb energy, which varies as ${B}^{1/2}$, and the Zeeman energy, which varies as $B$, can explain the observation of apparent phase transitions as a function of $B$ for fixed $\ensuremath{\nu}$, as transitions between states with different degrees of spin polarization. In addition to an analysis of various fractional states from this point of view and an examination of the effects of disorder on valley skyrmions, we present new experimental data for the energy gaps at integer fillings $\ensuremath{\nu}=0$ and $\ensuremath{\nu}=\ensuremath{-}1$, as a function of magnetic field, and we examine the possibility that valley skyrmions may account for the smaller energy gaps observed at $\ensuremath{\nu}=\ensuremath{-}1$.

Microscopic theory of a quantum Hall Ising nematic: Domain walls and disorder

We study the the interplay between spontaneously broken valley symmetry and spatial disorder in multivalley semiconductors in the quantum Hall regime. In cases where valleys have anisotropic electron dispersion a previous long-wavelength analysis [D. A. Abanin, S. A. Parameswaran, S. A. Kivelson, and S. L. Sondhi, Phys. Rev. B 82, 035428 (2010)] identified two new phases exhibiting the QHE. The first is the quantum Hall Ising nematic (QHIN), a phase with long-range orientational order manifested in macroscopic transport anisotropies. The second is the quantum Hall random-field paramagnet (QHRFPM) that emerges when the Ising ordering is disrupted by quenched disorder, characterized by a domain structure with a distinctive response to a valley symmetry-breaking strain field. Here we provide a more detailed microscopic analysis of the QHIN, which allows us to (i) estimate its Ising ordering temperature, (ii) study its domain-wall excitations, which play a central role in determining its properties, and (iii) analyze its response to quenched disorder from impurity scattering, which gives an estimate for domain size in the descendant QHRFPM. Our results are directly applicable to AlAs heterostructures, although their qualitative aspects inform other ferromagnetic QH systems, such as Si(111) heterostructures and trilayer graphene with trigonal warping.

Insulating Behavior at the Neutrality Point in Single-Layer Graphene

The fate of the low-temperature conductance at the charge-neutrality (Dirac) point in a single sheet of graphene on boron nitride is investigated down to 20mK. As the temperature is lowered, the peak resistivity diverges with a power-law behavior and becomes as high as several megohms per square at the lowest temperature, in contrast with the commonly observed saturation of the conductivity. As a perpendicular magnetic field is applied, our device remains insulating and directly transitions to the broken-valley-symmetry, ν=0 quantum Hall state, indicating that the insulating behavior we observe at zero magnetic field is a result of broken valley symmetry. Finally we discuss the possible origins of this effect.