Response Tensor Research Articles

Injection current in ferroelectric group-IV monochalcogenide monolayers

We study the injection current response tensor (also known as circular photogalvanic effect or ballistic current) in ferrolectric monolayer GeS, GeSe, SnS, and SnSe. We find that the injection current is perpendicular to the spontaneous in-plane polarization and could reach peak (bulk) values of the order of ${10}^{10}\phantom{\rule{4pt}{0ex}}\mathrm{A}\phantom{\rule{0.16em}{0ex}}{\mathrm{V}}^{\ensuremath{-}2}\phantom{\rule{0.16em}{0ex}}{\mathrm{s}}^{\ensuremath{-}1}$ in the visible spectrum. The magnitude of the injection current is the largest reported in the literature to date for a two dimensional material. To rationalize the large injection current, we correlate the injection current spectrum with the joint density of states, electric polarization, strain, etc. We find that various factors such as anisotropy, in-plane polarization and wave function delocalization are important in determining the injection current tensor in these materials. We also find that compression along the polar axis can increase the injection current (or change its sign), and hence strain can be an effective control knob for their nonlinear optical response. Conversely, the injection current can be a sensitive probe of the crystal structure.

Photo-Hall, photorefractive and photomagnetoelectric effects in tungsten bronzes and related tetragonal ferroelectrics

We present, at the phenomenological point of view, the responses of tetragonal ferroelectrics under linearly polarized light illumination, using a new theory based on the intertwining of the symmetries of both the material and the incident wave. Photopolarization, photomagnetic, photovoltaic, and phototoroidal vector responses are worked out as functions of the wave vector and wave polarization directions. Second-order response tensors associated with photoelastic, photomagnetoelectric, photorefractive, photoconductivity, and photo-Hall effects are also described in detail. We focus attention on the tetragonal and orthorhombic phases in crystals of the family of tungsten bronzes, and we discuss related tetragonal materials.

Upper limit on shift current generation in extended systems

The response functions of a material characterize its behavior under external stimuli, such as electromagnetic radiation. Such responses may grow linearly with the amplitude of the incident radiation, as is the case of absorption, or may be nonlinear. The latter category includes a diverse set of phenomena such as second harmonic generation (SHG), shift current, sum frequency generation, and excited state absorption, among others. Despite decades of research into nonlinear response theory, and the occasional discovery of materials with large nonlinear responses, there has been no systematic investigation into the maximum amount of nonlinear optical response attainable in solid-state materials. In this work, we present an upper bound on the second-order response functions of materials, which controls the SHG and shift current responses. We show that this bound depends on the band gap, band width, and geometrical properties of the material in question. We find that Kuzyk's bound for the maximum SHG of isolated molecules can be exceeded by conjugation or condensation of molecules to form molecular solids, and that strongly coupled systems generally have larger responses than weakly coupled or isolated ones. As a proof of principle, we perform first-principles calculations of the response tensors of a wide variety of materials, finding that the materials in our database do not yet saturate the upper bound. This suggests that new large SHG and shift current materials will likely be discovered by future materials research guided by the factors mentioned in this work.

Convex regularization for high-dimensional multiresponse tensor regression

In this paper, we present a general convex optimization approach for solving high-dimensional multiple response tensor regression problems under low-dimensional structural assumptions. We consider using convex and weakly decomposable regularizers assuming that the underlying tensor lies in an unknown low-dimensional subspace. Within our framework, we derive general risk bounds of the resulting estimate under fairly general dependence structure among covariates. Our framework leads to upper bounds in terms of two very simple quantities, the Gaussian width of a convex set in tensor space and the intrinsic dimension of the low-dimensional tensor subspace. To the best of our knowledge, this is the first general framework that applies to multiple response problems. These general bounds provide useful upper bounds on rates of convergence for a number of fundamental statistical models of interest including multiresponse regression, vector autoregressive models, low-rank tensor models and pairwise interaction models. Moreover, in many of these settings we prove that the resulting estimates are minimax optimal. We also provide a numerical study that both validates our theoretical guarantees and demonstrates the breadth of our framework.

Frequency-dependent current density tensors as density functions of dynamic polarizabilities.

Relationships accounting for contributions to the first-order charge density ρ(1) and current density J(1), induced in the electrons of a molecule by a monochromatic plane wave, have been obtained via time-dependent quantum mechanical perturbation theory. Their gauge invariance and invariance in passive translations of the coordinate system have been demonstrated (i) within the (long wavelengths) electric dipole approximation, in which only the electric field contributions to these densities are required, and (ii) within the electric quadrupole approximation, in which nonseparable terms provided by the magnetic field and by the electric field gradient, assumed uniform over the molecular domain, are needed. It is shown that the physical meaning of current density tensors depending on the frequency ω of the monochromatic wave shone on the molecule, and corresponding to derivatives of the current density J(1)(r, ω) with respect to components of the perturbing fields, is that of property density. Therefore, frequency-dependent current density tensors can be interpreted as the integrand function in 3d-space integrals defining dynamic molecular response tensors, e.g., electric dipole, mixed electric dipole-magnetic dipole, and electric dipole-quadrupole polarizabilites. Plots of current density tensors are expected to provide important information on the molecular domains giving dominant contributions to these properties.

Alternative stacking sequences in hexagonal boron nitride

The relative orientation of successive sheets, i.e. the stacking sequence, in layered 2D materials is central to the electronic, thermal, and mechanical properties of the material. Often different stacking sequences have comparable cohesive energy, leading to alternative stable crystal structures. Here we theoretically and experimentally explore different stacking sequences in the van der Waals bonded material hexagonal boron nitride (h-BN). We examine the total energy, electronic bandgap, and dielectric response tensor for five distinct high symmetry stacking sequences for both bulk and bilayer forms of h-BN. Two sequences, the generally assumed AA′ sequence and the relatively unknown (for h-BN) AB (Bernal) sequence, are predicted to have comparably low energy. We present a scalable modified chemical vapor deposition method that produces large flakes of virtually pure AB stacked h-BN; this new material complements the generally available AA′ stacked h-BN.

Atomic-Scale Magnetic Toroidal Dipole under Odd-Parity Hybridization

Magnetic toroidal dipole (MTD) is one of a fundamental constituent to induce magneto-electric effects in the absence of both spatial inversion and time-reversal symmetries. We report on a microscopic investigation of the atomic-scale MTD in solids by taking into account the orbital degree of freedom with a different parity. We construct an effective two-orbital $d$-$f$ tight-binding model on a polar tetragonal system for describing the atomic-scale MTD, which are obtained by incorporating the atomic spin-orbit coupling and odd-parity hybridization. The effective model exhibits two types of the MTDs: in-plane $x, y$ components activated through spontaneous ferromagnetic ordering or external magnetic field and an out-of-plane $z$ component by a spontaneous odd-parity hybridization without spin moments. We show that the intra-orbital (inter-orbital) Coulomb interaction in multi-orbital systems plays an important role in stabilizing the in-plane (out-of-plane) MTD orderings. We also examine the magneto-electric effect under each MTD ordering by calculating a linear response tensor. We show that the odd-parity hybridization enhances the magneto-electric effect for the in-plane MTDs, while it suppresses that for the out-of-plane MTD.

Charge-spin response and collective excitations in Weyl semimetals

Weyl semimetals are characterized by unconventional electromagnetic response. We present analytical expressions for all components of the frequency- and wave-vector-dependent charge-spin linear-response tensor of Weyl fermions. The spin-momentum locking of the Weyl Hamiltonian leads to a coupling between charge and longitudinal spin fluctuations, while transverse spin fluctuations remain decoupled from the charge. A real Weyl semimetal with multiple Weyl nodes can show this charge-spin coupling in equilibrium if its crystal symmetry is sufficiently low. All Weyl semimetals are expected to show this coupling if they are driven into a non-equilibrium stationary state with different occupations of Weyl nodes, for example by exploiting the chiral anomaly. Based on the response tensor, we investigate the low-energy collective excitations of interacting Weyl fermions. For a local Hubbard interaction, the charge-spin coupling leads to a dramatic change of the zero-sound dispersion: its velocity becomes independent of the interaction strength and the chemical potential and is given solely by the Fermi velocity. In the presence of long-range Coulomb interactions, the coupling transforms the plasmon modes into spin plasmons. For real Weyl semimetals with multiple Weyl nodes, the collective modes are strongly affected by the presence of parallel static electric and magnetic fields, due to the chiral anomaly. In particular, the zero-sound frequency at fixed momentum and the spin content of the spin plasmons go through cusp singularities as the chemical potential of one of the Weyl cones is tuned through the Weyl node. We discuss possible experiments that could provide smoking-gun evidence for Weyl physics.

A UNIFIED INELASTICITY AND DAMAGE MECHANICS FOR RUBBER-LIKE MATERIALS

The nonlinearities observed in the behavior of rubber and polymeric materials are influenced not only by the far-field stress conditions but also by the morphology, microstructures, and changes at the meso-level such as kinks, crosslinking, and micro tearing. Considering constants temperature applications, the occurrence of microtearing has a significant contribution on the performance of polymers, soft tissues and other rubber-like materials. In this paper, a unified damage mechanics and nonlinear elasticity approach is presented to model nonlinear behavior of rubber-like materials under constant temperatures. The theory is cast within the general framework of the internal variable theory of thermodynamics with large deformations where the Clausius-Duhem inequality is provoked to develop general damage potential. The strain energy density function is formulated in terms of an effective Lagrangian strain tensor that evolves with cumulative damage as cracking and micro-tearing take places. Piola-Kirchhoff (PK) stress tensor is presented and a new form of the damage response tensor is proposed. The model prediction is illustrated against experimental results with good agreement.

Nonlinear optical response of doped monolayer and bilayer graphene: Length gauge tight-binding model

We compute the nonlinear optical response of doped mono- and bilayer graphene using the full dispersion based on tight-binding models. The response is derived with the density matrix formalism using the length gauge and is valid for any periodic system, with arbitrary doping. By collecting terms that define effective nonlinear response tensors, we identify all nonlinear Drude-like terms (up to third-order) and show that all additional spurious divergences present in the induced current vanish. The nonlinear response of graphene comprises a large Drude-like divergence and three resonances that are tightly connected with transitions occurring in the vicinity of the Fermi level. The analytic solution derived using the Dirac approximation captures accurately the first- and third-order responses in graphene, even at very high doping levels. The quadratic response of gapped graphene is also strongly enhanced by doping, even for systems with small gaps such as commensurate structures of graphene on SiC. The nonlinear response of bilayer graphene is significantly richer, combining the resonances that stem from doping with its intrinsic strong low-energy resonances.

General form of the full electromagnetic Green function in materials physics

In this article, we present the general form of the full electromagnetic Green function which is suitable for the application in bulk materials physics. In particular, we show how the seven adjustable parameter functions of the free Green function translate into seven corresponding parameter functions of the full Green function. Furthermore, for both the fundamental response tensor and the electromagnetic Green function, we discuss the reduction of the Dyson equation on the four-dimensional Minkowski space to an equivalent, three-dimensional Cartesian Dyson equation.

Classification of atomic-scale multipoles under crystallographic point groups and application to linear response tensors

Four types of atomic-scale multipoles, electric, magnetic, magnetic toroidal, and electric toroidal multipoles, give a complete set to describe arbitrary degrees of freedom for coupled charge, spin, and orbital of electrons. We here present a systematic classification of these multipole degrees of freedom towards the application in condensed matter physics. Starting from the multipole description under the rotation group in real space, we generalize the concept of multipoles in momentum space with the spin degree of freedom. We show how multipoles affect the electronic band structures and linear responses, such as the magneto-electric effect, magneto-current (magneto-gyrotropic) effect, spin conductivity, Piezo-electric effect, and so on. Moreover, we exhibit a complete table to represent the active multipoles under 32 crystallographic point groups. Our comprehensive and systematic analyses will give a foundation to identify enigmatic electronic order parameters and a guide to evaluate peculiar cross-correlated phenomena in condensed matter physics from microscopic point of view.

Microscopic theory of refractive index applied to metamaterials: effective current response tensor corresponding to standard relation n2 = εeffμeff

In this article, we first derive the wavevector- and frequency-dependent, microscopic current response tensor which corresponds to the "macroscopic" ansatz $\vec D = \varepsilon_0 \varepsilon_{\mathrm{eff}} \vec E$ and $\vec B = \mu_0 \mu_{\mathrm{eff}} \vec H$ with wavevector- and frequency-independent, "effective" material constants $\varepsilon_{\mathrm{eff}}$ and $\mu_{\mathrm{eff}}$. We then deduce the electromagnetic and optical properties of this effective material model by employing exact, microscopic response relations. In particular, we argue that for recovering the standard relation $n^2 = \varepsilon_{\mathrm{eff}} \mu_{\mathrm{eff}}$ between the refractive index and the effective material constants, it is imperative to start from the microscopic wave equation in terms of the transverse dielectric function, $\varepsilon_{\mathrm{T}}(\vec k, \omega) = 0$. On the phenomenological side, our result is especially relevant for metamaterials research, which draws directly on the standard relation for the refractive index in terms of effective material constants. Since for a wide class of materials the current response tensor can be calculated from first principles and compared to the model expression derived here, this work also paves the way for a systematic search for new metamaterials.

Abelian Chern-Simons theory for the fractional quantum Hall effect in graphene

We develop a theory for the pseudorelativistic fractional quantum Hall effect in graphene, which is based on a multicomponent abelian Chern-Simons theory in the fermionic functional integral approach. Calculations are performed in the Keldysh formalism, directly giving access to real-time correlation functions at finite temperature. We obtain an exact effective action for the Chern-Simons gauge fields, which is expanded to second order in the gauge field fluctations around the mean-field solution. The one-loop fermionic polarization tensor as well as the electromagnetic response tensor in random phase approximation are derived, from which we obtain the Hall conductivities for various FQH states, lying symmetrically around charge neutrality.

Translational and rotational averaging of nonlocal response tensors for nano-shaped light

Orientational averaging of response tensors is an essential operation in calculating optical signals in randomly oriented liquid or gas phase molecular ensembles. When expanded in multipoles, rotational averaging can be performed simply through contraction with isotropic tensors. Alternatively, the multipolar expansion can be avoided and all multipoles may be naturally and implicitly incorporated via nonlocal response tensors which are fields and may not be rotationally averaged in a simple manner. The nonlocal response is useful when the light beam variation across the system is non negligible. Examples are nano-sculpted or short wavelength x-ray field. We derive exact expressions which recast optical signals as a convolution of a nonlocal response function of matter with a nonlocal intrinsic property of the electromagnetic field. The approach involves a gauge invariant calculation based on the minimal coupling Hamiltonian.

Strong second harmonic generation in two-dimensional ferroelectric IV-monochalcogenides

The two-dimensional ferroelectrics GeS, GeSe, SnS and SnSe are expected to have large spontaneous in-plane electric polarization and enhanced shift-current response. Using density functional methods, we show that these materials also exhibit the largest effective second harmonic generation reported so far. It can reach magnitudes up to which is about an order of magnitude larger than that of prototypical GaAs. To rationalize this result we model the optical response with a simple one-dimensional two-band model along the spontaneous polarization direction. Within this model the second-harmonic generation tensor is proportional to the shift-current response tensor. The large shift current and second harmonic responses of GeS, GeSe, SnS and SnSe make them promising non-linear materials for optoelectronic applications.

Linear electromagnetic wave equations in materials

After a short review of microscopic electrodynamics in materials, we investigate the relation of the microscopic dielectric tensor to the current response tensor and to the full electromagnetic Green function. Subsequently, we give a systematic overview of microscopic electromagnetic wave equations in materials, which can be formulated in terms of the microscopic dielectric tensor.

The abstract [formula omitted] and [formula omitted] groups of discrete [formula omitted] and T symmetries

Essential symmetry properties of physical quantities of classical mechanics and classical electromagnetism can be rationalized via the Abelian symmetry group GPT, with four operations (identity E, space inversion P, time reversal T, and combined PT) isomorphic to the spatial C2v point group. To account for charge conjugation C, a larger discrete group, GCPT, with eight operations (E,P,T,C, and their products, CP,CT,PT, and CPT), isomorphic to the spatial D2h point group, has been considered. Some features of these groups are discussed by a few examples, showing in particular that they provide group-theoretical implications for the existence of magnetic monopoles, magnetic scalar potential, magnetic charge density and magnetic current density, and magnetic-field induced electronic anapoles. A set of linearly independent vectors belonging to a representation space is constituted by eight fermion bilinears of quantum field theory. The GCPT group can be used to determine the discrete symmetry properties of molecular response tensors and provides interesting elucidations of established notions in a different, group-theoretical light, e.g., new understanding of duality transformations, which leave the Maxwell equations invariant, and a geometrical reinterpretation of Barron’s concept of true enantiomers.

Microscopic theory of the refractive index

We examine the refractive index from the viewpoint of modern first-principles materials physics. We first argue that the standard formula, n2=ɛrμr, is generally in conflict with fundamental principles on the microscopic level. Instead, it turns out that an allegedly approximate relation, n2=ɛr, which is already being used for most practical purposes, can be justified theoretically at optical wavelengths. More generally, starting from the fundamental, Lorentz-covariant electromagnetic wave equation in materials as used in plasma physics, we rederive a well-known, three-dimensional form of the wave equation in materials and thereby clarify the connection between the covariant fundamental response tensor and the various Cartesian tensors used to describe optical properties. Finally, we prove a general theorem by which the fundamental, covariant wave equation can be reformulated concisely in terms of the microscopic dielectric tensor.

Modeling RF waves in spatially dispersive inhomogeneus plasma using an iterative wavelet spectral method

The wave equation for a spatially dispersive inhomogeneous magnetized plasma is given by an integro-differential equation. The effects caused by spatial dispersion in the directions perpendicular and parallel to the magnetic field are quite different. In this study, we show how to solve the wave equation using a newly developed iterative wavelet spectral method for two cases. In the first case, the method is applied to a propagating kinetic Alfven wave in the perpendicular direction and solved to all orders in FLR. To conserve the kinetic energy flux, first order corrections in equilibrium gradients are used in the dielectric response tensor. In the second case, we verify the method for a fast wave minority heating scenario and study the up- and downshift in the parallel wave number.