We describe nonsymmorphic four-band tight-binding models in one dimension with Kramers degeneracy, and propose topolectric-circuit realizations of their topological phases. We begin with a representative model in the nonsymmorphic AII class with symmorphic time-reversal symmetry and nonsymmorphic charge-conjugation and chiral symmetries, resulting in a Z2 invariant. We also provide a Bogoliubov-de Gennes model in the nonsymmorphic D class with symmorphic charge-conjugation symmetry and nonsymmorphic time-reversal and chiral symmetries, which support a Z4 classification. To compute these invariants, we extend an openpath winding-number formulation previously used for non-Kramers-degenerate nonsymmorphic Z2 phases to Kramers-degenerate four-band systems. We propose topolectric-circuit implementations of nonsymmorphic systems, beginning with a comparison between the symmorphic Su-Schrieffer-Heeger and nonsymmorphic charge-density-wave models. We then extend this methodology to topolectric realizations of the AII and Z4 models, finding that the impedance response reproduces the predicted phase boundaries and associated zero-energy modes. Finally, we analyze disorder in the Z4 model and find that, although disorder breaks the nonsymmorphic symmetries, certain domain-wall zero modes in the minimal nearest-neighbor model remain pinned at zero energy because the disorder has no first-order coupling to the soliton subspace; longer-range terms satisfying relevant symmetries lift this emergent property.

Using a parent Hermitian tight-binding model on a bipartite lattice with chiral symmetry, we theoretically generate non-Hermitian models for free fermions with p orbitals per unit cell satisfying a complex generalization of chiral symmetry. The p complex energy bands in k space are given by a common k -dependent real factor, determined by the bands of the parent model, multiplied by the p th roots of unity. When the parent model is the Su-Schrieffer-Heeger (SSH) model, the single-particle energy levels are the same as those of free parafermion solutions to Baxter's non-Hermitian clock model. This construction relies on fully unidirectional hopping to create Bloch Hamiltonians with the form of generalized permutation matrices, but we also describe the effect of partial unidirectional hopping. For fully bidirectional hopping, the Bloch Hamiltonians are Hermitian and may be separated into even and odd parity blocks with respect to inversion of the orbitals within the unit cell. Partially unidirectional hopping breaks the inversion symmetry and mixes the even and odd blocks, and the real energy spectrum evolves into a complex one as the degree of unidirectionality increases, with details determined by the topology of the parent model and the number of orbitals per unit cell, p . We describe this process in detail for p = 3 and 4 with the SSH model. We also apply our approach to graphene, and we show that A A -stacked bilayer graphene evolves into a square-root Hamiltonian of monolayer graphene with the introduction of unidirectional hopping. We show that higher-order exceptional points occur at edge states and solitons in the non-Hermitian SSH model, and at the Dirac point of non-Hermitian graphene.

We model p -state Fock parafermions on a lattice in one dimension (with occupation per orbital of 0 , 1 , ... , p − 1 ). For p a composite number, they may be mapped to q m -state parafermions where q m are the prime factors of p . For a Hamiltonian with a single-particle spectrum, the parafermions decompose into q m -state parafermions. When p is a power of two, the decomposition is into fermions. We use this to construct a parafermionic Hamiltonian for p = 4 with a single-particle spectrum using a fermionic tight-binding model which is bilinear in creation and annihilation operators. The single-particle levels may be determined by diagonalizing a square matrix whose order scales linearly with system size, and they are the same as those of the fermionic model. We show that the intermediate statistics of the thermodynamic distribution function for the occupation numbers (known as Gentile statistics) are consistent with the mapping to fermions, and we provide an example calculation of the internal energy and heat capacity for a simple linear chain.

Abstract Typical thermoelectric phenomena require a junction between two dissimilar or chemically/electrically modified materials with different Seebeck coefficients, where temperature gradients generate a voltage (Seebeck effect) and applied electrical currents induce localized heating/cooling (Peltier effect). Here we show that periodic geometrical patterning alone is sufficient to generate spatially extended thermoelectric responses, such as the Seebeck coefficient, governed by a characteristic thermoelectric relaxation length, without requiring compositional modification or heterojunctions. Specifically, we found that the Seebeck coefficient, the key parameter governing thermoelectric performance, can be engineered over arbitrarily shaped, large-area regions of a uniform two-dimensional (2D) material through geometrical patterning. The modification of the Seebeck coefficient extends exponentially from the geometric discontinuity with a characteristic “decay length”, dTE ∼ 0.4 μm. By constructing nanopatterns of voids with pitch smaller than dTE, we effectively achieve a thermoelectric effect without material “junctions”, providing alternative routes to manipulate the directionality of thermoelectric response in diverse scaled 2D devices for nanoelectronics, photodetectors, sensors and energy applications.

Graphene is atomically thin, possesses excellent thermal conductivity, and is able to withstand high current densities, making it attractive for many nanoscale applications such as field-effect transistors, interconnects, and thermal management layers. Enabling integration of graphene into such devices requires nanostructuring, which can have a drastic impact on the self-heating properties, in particular at high current densities. Here, we use a combination of scanning thermal microscopy, finite element thermal analysis, and operando scanning transmission electron microscopy techniques to observe prototype graphene devices in operation and gain a deeper understanding of the role of geometry and interfaces during high current density operation. We find that Peltier effects significantly influence the operational limit due to local electrical and thermal interfacial effects, causing asymmetric temperature distribution in the device. Thus, our results indicate that a proper understanding and design of graphene devices must include consideration of the surrounding materials, interfaces, and geometry. Leveraging these aspects provides opportunities for engineered extreme operation devices.

We describe the mean-field model of a one-dimensional topological superconductor with two orbitals per unit cell. Time-reversal symmetry is absent, but a nonsymmorphic symmetry, involving a translation by a fraction of the unit cell, mimics the role of time-reversal symmetry. As a result, the topological superconductor has Z4 topological phases, two that support Majorana bound states and two that do not, in agreement with a prediction based on K-theory classification [K. Shiozaki, M. Sato, and K. Gomi, ]. As with the Kitaev chain, the presence of Majorana bound states gives rise to the 4π-periodic Josephson effect. A random matrix with nonsymmorphic time-reversal symmetry may be block diagonalized, and every individual block has time-reversal symmetry described by one of the Gaussian orthogonal, unitary, or symplectic ensembles. We show how this is manifested in the energy level statistics of a random system in the Z4 class as the spatial period of the nonsymmorphic symmetry is varied from much less than to of the order of the system size. Published by the American Physical Society 2024

We model the electronic properties of thin films of binary compounds with stacked layers where each layer is a two-dimensional honeycomb lattice with two atoms per unit cell. The two atoms per cell are assigned different on -site energies in order to consider six different stacking orders: ABC, ABA, AA, ABC ' , ABA ' , and AA ' . Using a minimal tight -binding model with nearest -neighbor hopping, we consider whether a fault in the texture of on -site energies in the vertical, stacking direction supports localized states, and we find localized states within the bulk band gap for ABC, ABA, and AA ' stacking. Depending on the stacking type, parameter values, and whether the soliton is atomically sharp or a smooth texture, there are a range of different band structures including soliton bands that are either isolated or that hybridize with other states, such as surface states, and soliton bands that are either dispersive or flat, the latter yielding narrow features in the density of states. We discuss the relevance of our results to specific materials including graphene, hexagonal boron nitride, and other binary compounds.

We classify Hermitian tight-binding models describing noninteracting electrons on a one-dimensional periodic lattice with two energy bands. To do this, we write a generalized Rice-Mele model with two orbitals per unit cell, including all possible complex-valued long-range hoppings consistent with Hermicity. We then apply different forms of time-reversal, charge-conjugation and chiral symmetry in order to constrain the parameters, resulting in an array of possible models in different symmetry classes. For each symmetry class, we define a single, canonical form of the Hamiltonian and identify models that are related to the canonical form by an off-diagonal unitary transformation in the atomic basis. The models have either symmorphic or nonsymmorphic nonspatial symmetries (time $T$, chiral and charge-conjugation). The nonsymmorphic category separates into two types of state of matter: an insulator with a $\mathbb{Z}_2$ topological index in the absence of nonsymmorphic time-reversal symmetry or, in the presence of nonsymmorphic time-reversal symmetry, a metallic state. The latter is an instance of Kramer's degeneracy with one degeneracy point in the Brillouin zone as opposed to no degeneracy points in symmorphic systems with $T^2 = 1$ and two in symmorphic systems with $T^2 = - 1$.

We model the influence of an in-plane magnetic field on the orbital motion of electrons in rhombohedral graphene multilayers. For zero field, the low-energy band structure includes a pair of flat bands near zero energy which are localized on the surface layers of a finite thin film. For finite field, we find that the zero-energy bands persist and that level bifurcations occur at energies determined by the component of the in-plane wave vector $q$ that is parallel to the external field. The occurrence of level bifurcations is explained by invoking semiclassical quantization of the zero field Fermi surface of rhombohedral graphite. We find parameter regions with a single isoenergetic contour of Berry phase zero corresponding to a conventional Landau level spectrum and regions with two isoenergetic contours, each of Berry phase $\pi$, corresponding to a Dirac-like spectrum of levels. We write down an analogous one-dimensional tight-binding model and relate the persistence of the zero-energy bands in large magnetic fields to a soliton texture supporting zero-energy states in the Su-Schreiffer-Heeger model. We show that different states contributing to the zero-energy flat bands in rhombohedral graphene multilayers in a large field, as determined by the wave vector $q$, are localized on different bulk layers of the system, not just the surfaces.

This chapter presents the electronic properties of monolayer and bilayer graphene. It begins with an overview of graphene including its history, fabrication methods, properties, and current and potential applications. We then detail the tight-binding model in order to describe the emergence of the Dirac equation near the Fermi level in monolayer graphene and its consequences including massless charge carriers with a linear electronic dispersion, pseudospin and chirality, Berry phase π, Klein tunneling, and an unusual sequence of plateaus in the integer quantum Hall effect. We explain how this is generalized in Bernal-stacked bilayer graphene as characterized by a novel Hamiltonian describing massive chiral charge carriers and in AA-stacked bilayer graphene which displays two shifted copies of the monolayer bands at low energy.

The Rice-Mele model has two topological and spatially-inversion symmetric phases, namely the Su-Schrieffer-Heeger (SSH) phase with alternating hopping only, and the charge-density-wave (CDW) phase with alternating energies only. The chiral symmetry of the SSH phase is robust in position space, so that it is preserved in the presence of the ends of a finite system and of textures in the alternating hopping. However, the chiral symmetry of the CDW wave phase is nonsymmorphic, resulting in a breaking of the bulk topology by an end or a texture in the alternating energies. We consider the presence of solitons (textures in position space separating two degenerate ground states) in finite systems with open boundary conditions. We identify the parameter range under which an atomically-sharp soliton in the CDW phase supports a localized state which lies within the band gap, and we calculate the expectation value $p_y$ of the nonsymmorphic chiral operator for this state, and the soliton electric charge. As the spatial extent of the soliton increases beyond the atomic limit, the energy level approaches zero exponentially quickly or inversely proportionally to the width, depending on microscopic details of the soliton texture. In both cases, the difference of $p_y$ from one is inversely proportional to the soliton width, while the charge is independent of the width. We investigate the robustness of the soliton level in the presence of disorder and sample-to-sample parameter variations, comparing with a single soliton level in the SSH phase with an odd number of sites.

Induced disorder in graphene enables changes in electrical and thermal transport. It has been shown previously that disorder is very important for electron cooling in graphene through disorder-assisted electron-phonon scattering, particularly via the supercollisions process. Here we study electron-momentum relaxation due to electron-phonon scattering while increasing the degree of disorder. With in-situ scanning thermal microscopy we monitor the temperature rise in the constriction region of a bowtie-shaped graphene device while increasing the disorder by means of feedback-controlled voltage ramps at high-current densities. Analysis of the combined thermal and electrical measurements in the low bias regime shows that the relative change of the momentum scattering rate vs temperature, as measured at room temperature, increases with strong local disorder. By excluding other candidate mechanisms for this phenomenon, including a change of the charge carriers density and activation of optical phonons, we conclude that the increase we observe in the temperature-dependent component of the scattering rate is likely due to new acoustic phonon scattering channels that open up as disorder increases.

We apply the mean-field Hartree Fock theory of gapped electronic states at charge neutrality in bilayer graphene to thin films of rhombohedral graphite with up to thirty layers. For the ground state, the order parameter (the separation of bands at the valley center) saturates to a constant non-zero value as the layer number increases, whereas the band gap decreases with layer number. We consider chiral symmetry breaking disorder in the form of random layer potentials and chiral preserving disorder in the form of random values of the interlayer coupling. The former reduces the magnitude of the mean band gap whereas the latter has a negligible effect, which is due to self-averaging within a film with a large number of layers. We determine the ground state in the presence of an individual stacking fault which results in two pairs of low-energy bands and we identify two separate order parameters. One of them determines the band gap at zero temperature, the other determines the critical temperature leading, overall, to a temperature dependence of the band gap that is distinct to that of pristine rhombohedral graphite. In the presence of stacking faults, each individual rhombohedral section with $m$ layers contributes a pair of low-energy flat bands producing a peak in the Berry curvature located at a characteristic $m$-dependent wave vector. The Chern number per spin-valley flavor for the filled valence bands in the ground state is equal in magnitude to the total number of layers divided by two, the same value as for pristine rhombohedral graphite.

We consider the heat equation for monolayer two-dimensional materials in the presence of heat flow into a substrate and Joule heating due to electrical current. We compare devices including a nanowire of constant width and a bow tie (or wedge) constriction of varying width, and we derive approximate one-dimensional heat equations for them; a bow tie constriction is described by the modified Bessel equation of zero order. We compare steady state analytic solutions of the approximate equations with numerical results obtained by a finite element method solution of the two-dimensional equation. Using these solutions, we describe the role of thermal conductivity, thermal boundary resistance with the substrate and device geometry. The temperature in a device at fixed potential difference will remain finite as the width shrinks, but will diverge for fixed current, logarithmically with width for the bow tie as compared to an inverse square dependence in a nanowire.

Topologically non-trivial states appear in a number of materials ranging from spin-orbit-coupling driven topological insulators to graphene. In multivalley conductors, such as mono- and bilayer graphene, despite a zero total Chern number for the entire Brillouin zone, Berry curvature with different signs concentrated in different valleys can affect the material’s transport characteristics. Here we consider thin films of rhombohedral graphite, which appear to retain truly two-dimensional properties up to tens of layers of thickness and host two-dimensional electron states with a large Berry curvature, accompanied by a giant intrinsic magnetic moment carried by electrons. The size of Berry curvature and magnetization in the vicinity of each valley can be controlled by electrostatic gating leading to a tuneable anomalous Hall effect and a peculiar structure of the two-dimensional Landau level spectrum.

With 2D materials such as graphene (GR) and hexagonal boron nitride possessing highest known thermal conductivities, one-atom thick nature of these materials makes thermal transport in them drastically dependent on the local environment. Moreover, the equally extraordinary electronic properties of GR such as relativistic carrier dynamics combined with GR highly anisotropic thermal conductance may point to unusual thermoelectric properties. In order to study thermal and thermoelectric phenomena in these nanoscale materials, we applied scanning thermal microscopy (SThM) that uses a sharp tip in contact with the probed surface that can create a controlled local sample temperature rise in the few nm acros spot, while measuring the resulting sample temperature and a heat flow. We used high vacuum environment that eliminates spurious heat dissipation channels to boost accuracy and sensitivity and to allow cryogenic measurements. We show that the thermal resistance of GR on SiO2 is increased by one order of magnitude by the addition of a top layer of MoS2, over the temperature range 150- 300 K with DFT calculations attributing this increase to the phonon transport filtering in the weak vdW coupling and vibrational mismatch between dissimilar 2D materials. By measuring the heat generated in the nanoscale constrictions in monolayer GR devices, we have discovered unconventional thermoelectric Peltier effect due to geometrical shape of 2D material and not requiring a junction of dissimilar materials, with phenomenon confirmed by measuring the Seebeck thermovoltage map due to local heating by the SThM tip. The novel nonlinear thermoelectric phenomena due to “electron wind”, and effects of GR doping and layer number are also reported.

The influence of nanostructuring and quantum confinement on the thermoelectric properties of materials has been extensively studied. While this has made possible multiple breakthroughs in the achievable figure of merit, classical confinement, and its effect on the local Seebeck coefficient has mostly been neglected, as has the Peltier effect in general due to the complexity of measuring small temperature gradients locally. Here we report that reducing the width of a graphene channel to 100 nm changes the Seebeck coefficient by orders of magnitude. Using a scanning thermal microscope allows us to probe the local temperature of electrically contacted graphene two-terminal devices or to locally heat the sample. We show that constrictions in mono- and bilayer graphene facilitate a spatially correlated gradient in the Seebeck and Peltier coefficient, as evidenced by the pronounced thermovoltage Vth and heating/cooling response ΔTPeltier, respectively. This geometry dependent effect, which has not been reported previously in 2D materials, has important implications for measurements of patterned nanostructures in graphene and points to novel solutions for effective thermal management in electronic graphene devices or concepts for single material thermocouples.

We model the magnetic ratchet effect in bilayer graphene in which a dc electric current is produced by an ac electric field of frequency $\omega$ in the presence of a steady in-plane magnetic field and inversion-symmetry breaking. In bilayer graphene, the ratchet effect is tunable by an external metallic gate which breaks inversion symmetry. For a tilted magnetic field, the perpendicular field component induces cyclotron motion with frequency $\omega_c$ and we find that the dc current displays cyclotron resonance at $\omega_c = \omega$, although this peak in the current is actually smaller than its value at $\omega_c = 0$. Second harmonic generation, however, is greatly enhanced by resonances at $\omega_c = \omega$ and $\omega_c = 2\omega$ for which the current is generally much larger than at $\omega_c = 0$.

We explore the electronic ground states of Bernal-stacked multilayer graphenes using the Hartree-Fock mean-field approximation and the full-parameter band model. We find that the electron-electron interaction tends to open a band gap in multilayer graphenes from bilayer to 8-layer, while the nature of the insulating ground state sensitively depends on the band parameter γ_2, which is responsible for the semimetallic nature of graphite. In 4-layer graphene, particularly, the ground state assumes an odd-spatial-parity staggered phase at γ_2 = 0, while an increasing, finite value of γ_2 stabilizes a different state with even parity, where the electrons are attracted to the top layer and the bottom layer. The two phases are topologically distinct insulating states with different Chern numbers, and they can be distinguished by spin or valley Hall conductivity measurements. Multilayers with more than five layers also exhibit similar ground states with potential minima at the outermost layers, although the opening of a gap in the spectrum as a whole is generally more difficult than in 4-layer because of a larger number of energy bands overlapping at the Fermi energy.