Flat-band States Research Articles

Investigation of the static electric field effect of strontium silicate layers on silicon substrates

We studied the effective net charge density (Qeff) of strontium silicate (SrxSiOx+2, x = 1, 2, 3) films grown on silicon (Si) (100) substrates. The SrxSiOx+2 layers were deposited from a Sr2SiO4 polycrystalline target by pulsed laser deposition, and then annealed at 400–600 °C in an oxygen atmosphere with a tube furnace. The Qeff values of the SrxSiOx+2/Si (100) samples were obtained from the shift in the voltage of the flat band state in their capacitance–voltage curves. The SrxSiOx+2/Si (100) samples with a thickness of 15 nm annealed at 400 °C showed the maximum Qeff/q value of 1.03 × 1013 cm−2, where q is the elementary charge. With increasing annealing temperature, the SrxSiOx+2 layer penetrated into the Si (100) substrate. This penetration may degrade the interfacial properties and decrease the Qeff value of the layers. For all the samples, the charges concentrated near the SrxSiOx+2/Si (100) interface. Our results suggest that anion and cation migration plays an important role in charge generation at the SrxSiOx+2/Si (100) interface, consistent with the findings of a previously reported molecular dynamics calculation.

Observation of robust flat-band localization in driven photonic rhombic lattices.

We demonstrate that a flat-band state in a quasi-one-dimensional rhombic lattice is robust in the presence of external drivings along the lattice axis. The lattice was formed by periodic arrays of evanescently coupled optical waveguides, and the external drivings were realized by modulating the paths of the waveguides. We excited a superposition of flat-band eigenmodes at the input and observed that this state does not diffract in the presence of static, as well as high-frequency sinusoidal drivings. This robust localization is due to destructive interference of the analogous wavefunction and is associated with the symmetry in the lattice geometry. We then excited the dispersive bands and observed Bloch oscillations and coherent destruction of tunneling.

Topological properties of nearly flat bands in two-dimensional photonic crystals

Based on exact numerical calculations and physical analyses, we have demonstrated that there are two types of flat band in two-dimensional (2D) magnetic photonic crystals (PhCs). One has trivial topology with a zero Chern number, and the other has a nontrivial topology with a nonzero Chern number. The former originates from resonant scatterings of single scatters or cavity modes encircled by scatters in PhCs with complex lattices, while the latter comes from strong coupling interactions of fields among neighboring unit cells. Two types of flat bands exhibit very different topological properties. When a point source with the frequency corresponding to the trivial flat band is placed inside the PhCs, its radiation is easily cloaked by metal obstacles. In contrast, a topological flat band state can bypass the obstacles.

Quantum collision theory in flat bands

We consider quantum scattering of particles in media exhibiting strong dispersion degeneracy. In particular, we study flat-banded lattices and linearly dispersed energy bands. The former constitute a prime example of single-particle frustration while the latter show degeneracy at the few- and many-particle level. We investigate both impurity and two-body scattering and show that, quite generally, scattering does not occur, which we relate to the fact that transition matrices vanish on the energy shell. We prove that scattering is instead replaced by projections onto band-projected eigenstates of the interaction potential. We then use the general results to obtain localized flat band states that are eigenstates of impurity potentials with vanishing eigenvalues in one-dimensional flat bands and study the particular case of a sawtooth lattice. We also uncover the relation between certain solutions of one-dimensional systems that have been categorized as ‘strange’, and the scattering states in linearly dispersed continuum systems.

Localization of weakly disordered flat band states

Certain tight binding lattices host macroscopically degenerate flat spectral bands. Their origin is rooted in local symmetries of the lattice, with destructive interference leading to the existence of compact localized eigenstates. We study the robustness of this localization to disorder in different classes of flat band lattices in one and two dimensions. Depending on the flat band class, the flat band states can either be robust, preserving their strong localization for weak disorder W, or they are destroyed and acquire large localization lengths $\xi$ that diverge with a variety of unconventional exponents $\nu$, $\xi \sim 1/W^{\nu}$.

Edge states and quantum phase transition in graphene under in-plane effective exchange fields

We investigated the edge states and quantum phase transition in graphene under an in-plane effective exchange field. The result shows that the combined effects of the in-plane effective exchange field and a staggered sublattice potential can induce zero-energy flat bands of edge states. Such flat-band edge states can evolve into helical-like ones in the presence of intrinsic spin–orbit coupling, with a unique spin texture. We also find that the bulk energy gap induced by the spin–orbit coupling and staggered sublattice potential can be closed and reopened with the in-plane effective exchange field, and the reopened bulk gap can be even larger than that induced by only the spin–orbit coupling and staggered sublattice potential, which is different from the case of an out-of-plane effective exchange field. The calculated spin-dependent Chern numbers suggest that the bulk gap closing and reopening is accompanied by a quantum phase transition from a trivial insulator phase across a metal phase into a spin-dependent quantum Hall phase.

Correlation between chemical-bonding states and fixed-charge states of Sr-silicate film on Si(100) substrate

The authors studied the correlation between the chemical bonding (CB) states and fixed charge (FC) states of Sr-silicate films grown on Si(100) substrates [Sr-silicate/Si(100)]. The Sr-silicate/Si(100) samples were synthesized by silicate reaction of SrO layers on the Si substrates through the diffusion of Si atoms from the substrates by thermal annealing in oxygen atmosphere. The CB states and the FC states of the Sr-silicate/Si(100) samples were obtained from their O 1s core-level x-ray photoemission spectra and the shift in voltage from the flat band state in their capacitance–voltage curves, respectively. Peak fittings of the O 1s core-level spectra for each sample were carried out with the three components of Si–O, Si–O–Sr, and Sr–O bonds to determine the CB state. The thin Sr-silicate layers were mainly constructed of the Si-O-Sr component. With increasing thickness, the amount of Si-O-Sr component decreased while that of Sr-O increased. The thickness dependency of the FC density showed a good agreement with that of the Si-O-Sr component, revealing a clear correlation between FC and Sr–O–Si bonding. Our results suggest that silicate bonding plays an important role in FC generation in Sr-silicate systems.

Adjustable localized states in perfect and single C-chain doped zigzag AlN nanoribbons

The localized feature of electronic states is often related to some interesting electronic properties in graphite-like materials. We use the first principles calculations to investigate two important localized states, that is, flat-band states and border states, in zigzag AlN nanoribbons (zAlNNRs). Our results indicate that both localized states can exist in zAlNNRs and the border states have a close relationship with electrical conductivity. It is found that the flat-band states of perfect zAlNNRs result from edge N-pz orbitals, while the border states of doped zAlNNRs are due to π/π* orbitals of C and Al/N atoms. These findings enrich our understanding on localized states in zAlNNRs, showing a potential application in functional nanodevices.

Edge instabilities of topological superconductors

Nodal topological superconductors display zero-energy Majorana flat bands at generic edges. The flatness of these edge bands, which is protected by time-reversal and translation symmetry, gives rise to an extensive ground-state degeneracy. Therefore, even arbitrarily weak interactions lead to an instability of the flat-band edge states towards time-reversal and translation-symmetry-broken phases, which lift the ground-state degeneracy. We examine the instabilities of the flat-band edge states of d_{xy}-wave superconductors by performing a mean-field analysis in the Majorana basis of the edge states. The leading instabilities are Majorana mass terms, which correspond to coherent superpositions of particle-particle and particle-hole channels in the fermionic language. We find that attractive interactions induce three different mass terms. One is a coherent superposition of imaginary s-wave pairing and current order, and another combines a charge-density-wave and finite-momentum singlet pairing. Repulsive interactions, on the other hand, lead to ferromagnetism together with spin-triplet pairing at the edge. Our quantum Monte Carlo simulations confirm these findings and demonstrate that these instabilities occur even in the presence of strong quantum fluctuations. We discuss the implications of our results for experiments on cuprate high-temperature superconductors.

Completely Flat Band in a Crystal of Finite Thickness

Conditions for the existence of a completely flat band in a crystal of finite thickness are clarified. Furthermore, the condition for the localization of the flat band states near the surfaces is also discussed. It is also found that a completely flat band can appear in a crystal where a lattice point has multiple orbital states. In addition to the known results for honeycomb and diamond lattices, a localized completely flat band is found in a crystal of the wurtzite structure of finite thickness. A completely flat band is also found in many other crystals, but it is extended in the direction perpendicular to the surface.

Observation of localized flat-band states in Kagome photonic lattices.

We report the first experimental demonstration of localized flat-band states in optically induced Kagome photonic lattices. Such lattices exhibit a unique band structure with the lowest band being completely flat (diffractionless) in the tight-binding approximation. By taking the advantage of linear superposition of the flat-band eigenmodes of the Kagome lattices, we demonstrate a high-fidelity transmission of complex patterns in such two-dimensional pyrochlore-like photonic structures. Our numerical simulations find good agreement with experimental observations, upholding the belief that flat-band lattices can support distortion-free image transmission.

Demonstration of flat-band image transmission in optically induced Lieb photonic lattices.

We present a simple, yet effective, approach for optical induction of Lieb photonic lattices, which typically rely on the femtosecond laser writing technique. Such lattices are established by judiciously overlapping two sublattices (an "egg-crate" lattice and a square lattice) with different periodicities through a self-defocusing photorefractive medium. Furthermore, taking advantage of the superposition of localized flat-band states inherent in the Lieb lattices, we demonstrate distortion-free image transmission in such two-dimensional perovskite-like photonic structures. Our experimental observations find good agreement with numerical simulations.

Engineering flat electronic bands in quasiperiodic and fractal loop geometries

Exact construction of one electron eigenstates with flat, non-dispersive bands, and localized over clusters of various sizes is reported for a class of quasi-one-dimensional looped networks. Quasiperiodic Fibonacci and Berker fractal geometries are embedded in the arms of the loop threaded by a uniform magnetic flux. We work out an analytical scheme to unravel the localized single particle states pinned at various atomic sites or over clusters of them. The magnetic field is varied to control, in a subtle way, the extent of localization and the location of the flat band states in energy space. In addition to this we show that an appropriate tuning of the field can lead to a re-entrant behavior of the effective mass of the electron in a band, with a periodic flip in its sign.

Quantitative Characterization of Near-Interface Oxide Traps in 4H-SiC MOS Capacitors by Transient Capacitance Measurements

1．Introduction The characteristics of SiC MOS devices are often suffered from the effects of near-interface oxide traps, however, the quantitative analysis method of those traps has not been established. In this study we propose a method to estimate their density by transient capacitance measurement. We assume the oxide trap model [1], in which carriers in semiconductors interact with the traps in oxide by tunneling, with a widely-spread response time of the traps, due to the variation of the tunneling distance between the interface and the traps. 2. Characterization method Transient response of capacitance (C-t) gives the information of trapping behaviors of the carriers in MOS capacitors [2-3], since the change of amount of trapped charges shifts the C-V curve. In this study we employed the MOS capacitors fabricated by thermal oxidation of n-type 4H-SiC. The gate bias on a capacitor was kept in accumulation state (Vg=Vtrap) for 100 sec to fill the traps, before a sudden shift to flatband state (Vg=Vmeas) to observe the C-t characteristics. Taking account of widely-spread response time of the traps due to their depth distribution in oxide, the time-dependent change of capacitance would be described by extended-Debye relaxation model [4] instead of a conventional relaxation with a single time constant, as |C(t)-Ceq|≈|DC0|exp (-(t/τeff)β), where Ceq is the capacitance in thermal equilibrium, β is the stretched exponential factor (0<β<1) of response time, τeff is the effective relaxation time of the group of traps with distributed response time, and DC0 is the total change of capacitance. C-t characteristics at 200K and 300K were fitted by the above equation to determine DC0, from which the amount of trapped charges in near-interface traps per area (Qox ) was estimated by assuming that the distance of the traps from the interface is significantly shorter than the film thickness. Note that low temperature slows down the de-trapping process by a few orders because of the change of the Fermi-Dirac distribution. Even though our measurement system detects only relatively slow response (≥0.1 s), low temperature measurement enables us to evaluate the traps locating very shallow region(~ a few ongstroms from the interface). 3. Results and Discussions The observed C-t characteristics were well-fitted by the extended-Debye relaxation model as desribed above. The lower temperature measurements resulted in larger DC0 which is indicating more traps exist in the region closer to the interface. By changing the Vtrap while unchanging the Vmeas, we can vary the width of energy region of the traps contributing to the C-t characteristics. From the SiC surface potential dependence of Qox, we can deduce the effective density of states (Dox ) of the near-interface traps. We employed two kinds of MOS capacitors with different interface state density (Dit ): ~1012 cm-2eV-1 and ~1011 cm-2eV-1 at 0.2 eV below conduction band edge. For the near-interface oxide traps in shallow region (measured at 200K), we could detect Dox of 3-4×1012 cm-2eV-1 for the higher Dit sample, but only ~5×1011 cm-2eV-1 for the lower Dit sample. These traps are estimated to locate at a few angstroms distance from the interface, taking account of the tunneling mass of electrons in SiO2. On the other hand, both of the two samples showed relatively less density of traps for the deep region (measured at 300K), in the order of 1011 cm-2eV-1. The location of the traps detected at 300K is estimated to be ~ 1 nm depth from the interface. These results indicates that the oxide trap density in shallow region is significantly sensitive to the preparation condition of SiC/SiO2 interface. 4. Conclusions The method to determine the near-interface oxide trap density in MOS capacitors from transient capacitance characteristics was proposed and demonstrated for 4H-SiC MOS capacitors, by taking account of the wide distribution of distance between the traps and the interface. The effective state density of the near-interface traps was estimated to be 1011 ~1012 cm-2eV-1 for shallow location from the interface but ~1011 cm-2eV-1 for deep location for our MOS capacitors. 5. Acknowledgement This work was partly supported by Council for Science, Technology and Innovation (CSTI), Cross-ministerial Strategic Innovation Promotion Program (SIP), “Next-generation power electronics” (funding agency: NEDO), and by JSPS KAKENHI.

Observation of a Localized Flat-Band State in a Photonic Lieb Lattice.

We demonstrate the first experimental realization of a dispersionless state, in a photonic Lieb lattice formed by an array of optical waveguides. This engineered lattice supports three energy bands, including a perfectly flat middle band with an infinite effective mass. We analyze, both experimentally and theoretically, the evolution of well-prepared flat-band states, and show their remarkable robustness, even in the presence of disorder. The realization of flat-band states in photonic lattices opens an exciting door towards quantum simulation of flat-band models in a highly controllable environment.

Strongly correlated flat-band systems: The route from Heisenberg spins to Hubbard electrons

On a large class of lattices (such as the sawtooth chain, the kagome and the pyrochlore lattices), the quantum Heisenberg and the repulsive Hubbard models may host a completely dispersionless (flat) energy band in the single-particle spectrum. The flat-band states can be viewed as completely localized within a finite volume (trap) of the lattice and allow for construction of many-particle states, roughly speaking, by occupying the traps with particles. If the flat-band happens to be the lowest-energy one, the manifold of such many-body states will often determine the ground-state and low-temperature physics of the models at hand even in the presence of strong interactions. The localized nature of these many-body states makes possible the mapping of this subset of eigenstates onto a corresponding classical hard-core system. As a result, the ground-state and low-temperature properties of the strongly correlated flat-band systems can be analyzed in detail using concepts and tools of classical statistical mechanics (e.g., classical lattice-gas approach or percolation approach), in contrast to more challenging quantum many-body techniques usually necessary to examine strongly correlated quantum systems. In this review, we recapitulate the basic features of the flat-band spin systems and briefly summarize earlier studies in the field. The main emphasis is made on recent developments which include results for both spin and electron flat-band models. In particular, for flat-band spin systems, we highlight field-driven phase transitions for frustrated quantum Heisenberg antiferromagnets at low temperatures, chiral flat-band states, as well as the effect of a slight dispersion of a previously strictly flat-band due to nonideal lattice geometry. For electronic systems, we discuss the universal low-temperature behavior of several flat-band Hubbard models, the emergence of ground-state ferromagnetism in the square-lattice Tasaki–Hubbard model and the related Pauli-correlated percolation problem, as well as the dispersion-driven ground-state ferromagnetism in flat-band Hubbard systems. Closely related studies and possible experimental realizations of the flat-band physics are also described briefly.

Superconductivity and the periodic table: from elements to materials.

Based on the normal-state electronic band structure, the necessary condition for a metal to become a superconductor is the simultaneous occurrence of flat and steep bands at the Fermi level. The sufficient condition at least for conventional superconductors is a strong enough coupling of the flat band states to the lattice, e.g. via phonons. Selected elements (Te) and compounds of the rare earth metals (RE(2)C(3), REC(2), RE(2)X(2)C(2) with X=halogen) and MgB(2) serve as examples to illustrate the idea.

Flat band analogues and flux driven extended electronic states in a class of geometrically frustrated fractal networks

We demonstrate, by explicit construction, that a single band tight binding Hamiltonian defined on a class of deterministic fractals of the b = 3N Sierpinski type can give rise to an infinity of dispersionless, flat-band like states which can be worked out analytically using the scale invariance of the underlying lattice. The states are localized over clusters of increasing sizes, displaying the existence of a multitude of localization areas. The onset of localization can, in principle, be ‘delayed’ in space by an appropriate choice of the energy of the electron. A uniform magnetic field threading the elementary plaquettes of the network is shown to destroy this staggered localization and generate absolutely continuous sub-bands in the energy spectrum of these non-translationally invariant networks.

Novel Electronic Structures of Ru-pnictides RuPn (Pn = P, As, Sb)

Density-functional-theory-based electronic structure calculations are made to consider the novel electronic states of Ru-pnictides RuP and RuAs where the intriguing phase transitions and superconductivity under doping of Rh have been reported. We find that there appear nearly degenerate flat bands just at the Fermi level in the high-temperature metallic phase of RuP and RuAs; the flat-band states come mainly from the 4dxy orbitals of Ru ions and the Rh doping shifts the Fermi level just above the flat bands. The splitting of the flat bands caused by their electronic instability may then be responsible for the observed phase transition to the nonmagnetic insulating phase at low temperatures. We also find that the band structure calculated for RuSb resembles that of the doped RuP and RuAs, which is consistent with experiment where superconductivity occurs in RuSb without Rh doping.

Majorana vortex-bound states in three-dimensional nodal noncentrosymmetric superconductors

Noncentrosymmetric superconductors (NCSs), characterized by antisymmetric spin-orbit coupling and a mixture of spin-singlet and spin-triplet pairing components, are promising candidate materials for topological superconductivity. An important hallmark of topological superconductors is the existence of protected zero-energy states at surfaces or in vortex cores. Here we investigate Majorana vortex-bound states in three-dimensional nodal and fully gapped NCSs by combining analytical solutions of Bogoliubov--de Gennes (BdG) equations in the continuum with exact diagonalization of BdG Hamiltonians. We show that depending on the crystal point-group symmetries and the topological properties of the bulk Bogoliubov-quasiparticle wave functions, different types of zero-energy Majorana modes can appear inside the vortex core. We find that for nodal NCSs with tetragonal point group ${C}_{4v}$ the vortex states are dispersionless along the vortex line, forming one-dimensional Majorana flat bands, while for NCSs with ${D}_{4}$ point-group symmetry the vortex modes are helical Majorana states with a linear dispersion along the vortex line. NCSs with monoclinic point group ${C}_{2}$, on the other hand, do not exhibit any zero-energy vortex-bound states. We show that in the case of the ${C}_{4v}$ $({D}_{4})$ point group the stability of these Majorana zero modes is guaranteed by a combination of reflection $(\ensuremath{\pi}$ rotation), time-reversal, and particle-hole symmetry. Considering continuous deformations of the quasiparticle spectrum in the presence of vortices, we show that the flat-band vortex-bound states of ${C}_{4v}$ point-group NCSs can be adiabatically connected to the dispersionless vortex-bound states of time-reversal symmetric Weyl superconductors. Experimental implications of our results for thermal transport and tunneling measurements are discussed.