Kubo Formula Research Articles

Anomalous Hall transport in tilted multi-Weyl semimetals

We study the effect of a perpendicular magnetic field B on a multinode Weyl semimetal (mWSM) of arbitrary integer monopole charge n, with the two Weyl multinodes separated in k-space. Besides type-I mWSMs, there exist type-II mWSMs which are characterized by the tilted minimal dispersion for low-energy excitations; the Weyl points in type-II mWSMs are still protected crossings but appear at the contact of the electron and hole pockets, after the Lifshitz transition. We find that the presence of a perpendicular magnetic field quantizes the occupation pockets due to the presence of Fermi tubes. In this theory, the Hilbert space is spanned by a set of n chiral degenerate ground states, and a countably infinite number of particle-hole symmetric Landau levels (LLs). We calculate the Hall conductivity for the tilt-symmetric case of type-I mWSM using the Kubo formula, in the zero-frequency (DC) limit, and recover the well-known vacuum contribution. We compute the Fermi surface corrections and show that the expression generalizes from the formula for elementary (n = 1) type-I WSMs. We derive an expression for the type-II mWSM Hall conductivity, which is bounded by a LL cutoff introduced on physical grounds. Interestingly, we find that the anomalous vacuum Hall conductivity is vanishing in the type-II phase at all temperatures. The corresponding thermal Hall and Nernst conductivities are evaluated and characterized for both phases. The qualitative and quantitative observations presented here may serve in the characterization of generic mWSMs of both types.

Even-odd-dependent optical transitions of zigzag monolayer black phosphorus nanoribbons

We analytically study the electronic structure and optical properties of zigzag-edged phosphorene nanoribbons (ZPNRs) using the tight-binding Hamiltonian and Kubo formula. By directly solving the discrete Schrodinger equation, we obtain the energy spectra and wavefunctions for N-ZPNR (where TV is the number of transverse zigzag atomic chains) and classify the eigenstates according to the lattice symmetry. Then, we obtain the optical transition selection rule of ZPNRs on the basis of symmetry analysis and analytical expressions of optical transition matrix elements. Under incident light that is linearly polarized along the ribbon, we determine that the optical transition selection rule for N-ZPNR with even- or odd-N is qualitatively different. Specifically, for even-N ZPNRs, the inter- (intra-) band selection rule is An =odd (even) because the parity of the wavefunction corresponding to the n-th subband in the conduction (valence) band is (-1)n[(-1)(n+1)] owing to the presence of C2x symmetry. However, the optical transitions between any subbands are possible owing to the absence of C2x symmetry. Our results provide a further understanding on the electronic states and optical properties of ZPNRs, which are useful for explaining the optical experiment data on ZPNR samples.

Optical properties of photonic crystals based on graphene nanocomposite within visible and IR wavelengths

In this work, we present a new type of the one dimensional photonic crystals that contain graphene based nanocomposite and dielectric layers. Here, the nanocomposite layer is designed form graphene nanoparticles into a dielectric host material. By using transfers matrix method, Maxwell–Garnett formula, and Kubo formula, the properties of photonic band gap within visible and infrared regions are discussed. Our results reflect the significant effect of graphene on the optical characteristics of the nanocomposite layer and the transmittance properties as well. Thus, the tunability of the photonic band gaps could be expected. Meanwhile, we demonstrate the effect of the volume fraction of graphene's nanoparticles, the constituent materials thicknesses, the chemical potential and the permittivities of the host dielectric material on the transmittance properties of the proposed design. This design could be of potential interest in many fields of applications such as stop band filters and switches with high reflectivity.

Microscopic approach to the macrodynamics of matter with broken symmetries

A unified set of hydrodynamic equations describing condensed phases of matter with broken continuous symmetries is derived using a generalization of the statistical-mechanical approach based on the local equilibrium distribution. The dissipativeless and dissipative parts of the current densities and the entropy production are systematically deduced in this approach by expanding in powers of the gradients of the macrofields. Green–Kubo formulas are obtained for all the linear transport coefficients. The consequences of microreversibility and spatial symmetries are investigated, leading to the prediction of cross effects resulting from Onsager–Casimir reciprocal relations. Crystalline solids and liquid crystals are potential examples of application. The approach is clarifying the links between the microscopic Hamiltonian dynamics and the thermodynamic and transport properties at the macroscale.

Viscosity of Gases in Nanochannels

The viscosity of a rarefied gas in nanochannels under normal conditions is studied. The viscosity coefficient is calculated from the Green–Kubo formula using the stochastic molecular method. The interaction of gas molecules with the wall is described by the specular, diffuse, or specular-diffuse laws. Gas temperature and channel height are varied. It is shown that the viscosity in the nanochannel is essentially anisotropic. Along the channel, it almost always coincides with its corresponding bulk viscosity. By varying the accommodation coefficient, the total viscosity of the gas can be either reduced or increased several-fold.

Spin Hall conductivity in insulators with nonconserved spin

We study the linear response of a spin current to a small electric field in a two-dimensional crystalline insulator with non-conserved spin. We adopt the spin current operator proposed in [J. Shi et al., Phys. Rev. Lett. 96, 076604 (2006)], which satisfies a continuity equation and fits the Onsager relations. We use the time-independent perturbation theory to present a formula for the spin Hall conductivity, which consists of a "Chern-like" term, reminiscent of the Kubo formula obtained for the quantum Hall systems, and a correction term that accounts for the non-conservation of spin. We illustrate our findings on the Bernevig-Hughes-Zhang model and the Kane-Mele model for time-reversal symmetric topological insulators and show that the correction term scales quadratically with the amplitude of the spin-conservation-breaking terms. In both models, the spin Hall conductivity deviates from the quantized value when spin is not conserved.

Enhanced Faraday rotation in proximitized monolayer transition metal dichalcogenides

Monolayer transition metal dichalcogenides (TMDCs) under the application of a magnetic exchange field carry the nontrivial optical Hall conductivity and thus exhibit the nonzero Faraday rotation (FR) angle. However, the tradeoff between the FR angle and transmission hinders their possible applications in magnetic-optical (MO) devices. Here, we theoretically show that a heterostructure of two photonic crystals with proximitized monolayer TMDCs enables the enhancement of the FR angle and transmission simultaneously through the combination of a four-band Hamiltonian model, Kubo formula and transfer matrix method. The MO improvement in the hybrid structure in both the FR angle and transmission is determined by the combined effects from the localized electromagnetic field at the interface between the two photonic crystals and the satisfaction of the phase match condition. Our work opens up an alternative opportunity to use TMDCs in two-dimensional MO fields in the visible frequency range.

A non-iterative method for vertex corrections of the Kubo formula for electric conductivity

When computing electric conductivity based on the Kubo formula, vertex corrections describe such effects as anisotropic scattering and quantum interference and are important in determining quantum transport properties. These vertex corrections are obtained by solving the Bethe–Salpeter equation, which can become numerically intractable when a large number of k-points and multiple bands are involved. We introduce a non-iterative approach to vertex correction based on rank factorization of the impurity vertices, which significantly alleviates the computational burden. We demonstrate that this method can be implemented along with effective Hamiltonians extracted from electronic structure calculations on perfect crystals, thereby enabling the quantitative analysis of quantum effects in electron conduction for real materials.

Theory of exciton transport in molecular crystals strongly coupled to a cavity: A temperature-dependent variational approach.

We present a semianalytical theory for the exciton transport in organic molecular crystals interacting strongly with a single cavity mode. Based on the Holstein-Tavis-Cummings model and the Kubo formula, we derive an exciton mobility expression in the framework of a temperature-dependent variational canonical transformation, which can cover a wide range of exciton-vibration coupling, exciton-cavity coupling, and temperatures. A closed-form expression for the coherent part of the total mobility is obtained in the zeroth order of the exciton-vibration coupling, which demonstrates the significance of vibrationally dressed dark excitons in the determination of the transport mechanism. By performing numerical simulations on both the H- and J-aggregates, we find that the exciton-cavity coupling has significant effects on the total mobility: (1) At low temperatures, there exists an optimal exciton-cavity coupling strength for the H-aggregate at which a maximal mobility is reached, while the mobility in the J-aggregate decreases monotonically with an increase in the exciton-cavity coupling and (2) at high temperatures, the mobility in both types of aggregates get enhanced by the cavity. We illustrate the above-mentioned low-temperature optimal mobility observed in the H-aggregate by using realistic parameters at room temperature.

Linear interband optical refraction and absorption in strained black phosphorene

Strain effects have been widely addressed in monolayer black phosphorus (MBP) due to its significant influence on the orbital hybridization of atoms. In this theoretical contribution, we use the tight-binding model, the Harrison rule and the Kubo formula to describe the optical refraction and absorption of MBP in detail. The analytical study of the band gap in strained MBP demonstrates electronic phase transitions from semiconductor-to-semimetal/metal and semiconductor-to-insulator, in which both the compressive and tensile strains act linearly on the band gap alterations. The critical strains corresponding to these phase transitions are fully characterized as well. Our calculations show that the variation of the refraction inflections and absorption peaks depends on the strained band gap, however; the band gap changes under out-of-plane strains are different than the in-plane ones. The conditions under which this discrepancy is significant and/or negligible are investigated. Moreover, the dedication of minimal/maximal optical refraction and/or absorption in MBP to both in-plane and out-of-plane strains are fully addressed. Our theoretical results clarify the strain-induced interplay between the band gap and optical properties to propose a wide range of applications in nano-optoelectronics.

Quantum Hall studies of a semi-Dirac nanoribbon

Here we comprehensively investigate Landau levels, Hofstadter butterfly and transport properties of a semi-Dirac nanoribbon in a perpendicular magnetic field using a recently developed real-space implementation of the Kubo formula based on Kernel Polynomial Method. A Dirac ribbon is considered to compare and contrast our results for a semi-Dirac system. We find that the Landau levels being non-equidistant from each other for the semi-Dirac case (true for a Dirac as well), the flatness of the energy bands vanishes in the bulk and becomes dispersive for a semi-Dirac ribbon in contrast to a Dirac system. This feature is most discernible for intermediate values of the external field. We further compute the longitudinal ($\sigma_{xx}$ and $\sigma_{yy}$) and the transverse or Hall ($\sigma_{xy}$) conductivities where the Hall conductivity shows a familiar quantization, namely, $\sigma_{xy} \propto 2n$ (the factor `2' includes the spin degeneracy) which is highly distinct from a Dirac system, such as graphene. We also observe anisotropic behavior in magneto-transport in a semi-Dirac ribbon owing to the dispersion anomalies in two different longitudinal directions. Our studies may have important ramifications for monolayer phosphorene.

Bulk viscosity of resonating fermions revisited: Kubo formula, sum rule, and the dimer and high-temperature limits

The bulk viscosity of two-component fermions with a zero-range interaction is revisited both in two and three dimensions. We first point out that the "standard" Kubo formula employed in recent studies has flaws to give rise to an unphysical divergent bulk viscosity even in a limit where it is supposed to vanish. The corrected Kubo formula as well as the sum rule is then carefully rederived so as to confirm that the bulk viscosity indeed vanishes in the free, unitarity, and dimer limits. We also discuss that the recently found discrepancy between the Kubo formalism and the kinetic theory for the bulk viscosity is attributed to the fact that the quasiparticle approximation assumed by the latter breaks down even in the high-temperature limit.

Symmetrized decomposition of the Kubo-Bastin formula

The Smrcka-Streda version of Kubo's linear response formula is widely used in the literature to compute non-equilibrium transport properties of heterostructures. It is particularly useful for the evaluation of intrinsic transport properties associated with the Berry curvature of the Bloch states, such as anomalous and spin Hall currents as well as the damping-like component of the spin-orbit torque. Here, we demonstrate in a very general way that the widely used decomposition of the Kubo-Bastin formula introduced by Smrcka and Streda contains an overlap, which has lead to widespread confusion in the literature regarding the Fermi surface and Fermi sea contributions. To remedy this pathology, we propose a new decomposition of the Kubo-Bastin formula based on the permutation properties of the correlation function and derive a new set of formulas, without an overlap, that provides direct access to the transport effects of interest. We apply these new formulas to selected cases and demonstrate that the Fermi sea and Fermi surface contributions can be uniquely addressed with our symmetrized approach.

The Invariance of the Diffusion Coefficient with Iterative Operations of the Charged Particle Transport Equation

Abstract The spatial parallel diffusion coefficient (SPDC) is one of the important quantities describing energetic charged particle transport. There are three different definitions for the SPDC: the displacement variance definition , the Fick’s law definition with , and the Taylor–Green–Kubo (TGK) formula definition . For a constant mean magnetic field, the three different definitions of the SPDC give the same result. However, for a focusing field, it is demonstrated that the results of the different definitions are not the same. In this paper, from the Fokker–Planck equation, we find that different methods, e.g., the general Fourier expansion and iteration method, can give different equations of the isotropic distribution function (EIDFs). But it is shown that one EIDF can be transformed into another by some derivative iterative operations (DIOs). If one definition of the SPDC is invariant for the DIOs, it is clear that the definition is also invariant for different EIDFs; therefore, it is an invariant quantity for the different derivation methods of the EIDF. For the focusing field, we suggest that the TGK definition is only an approximate formula, and the Fick’s law definition is not invariant to some DIOs. However, at least for the special condition, in this paper we show that the definition is an invariant quantity to the DIOs. Therefore, for a spatially varying field, the displacement variance definition , rather than the Fick’s law definition and TGK formula definition , is the most appropriate definition of the SPDCs.

Polymer rheology predictions from first principles using the slip-link model

The discrete slip-link theory is a hierarchy of strongly connected models that have great success predicting the linear and nonlinear rheology of high-molecular-weight polymers. Three of the four parameters of the most-detailed model, which can be extracted from primitive-path analysis, give quantitative agreement with experimental data for all examined chemistries (polystyrene, polyisoprene, polybutadiene, and polyethylene). Here, we attempt to extract the remaining friction parameter from atomistic simulations. In particular, an available quantum chemistry-based force field for polyethylene oxide (PEO) was used to perform molecular-dynamics simulations of a 12 kDa melt. The Kuhn friction is obtained from the mean-squared displacement of the center-of-mass of the chains (MSD of COM) in the melt. The result is also corroborated using the relaxation modulus calculated through the Green–Kubo formula. Once the four parameters are determined for any chemistry, all parameters for all members of the slip-link hierarchy are determined. Then, using a coarser member of the hierarchy, the dynamic modulus of a 256 kDa PEO melt was predicted. The predictions are compared to experimental measurements performed at the same temperature. Unfortunately, the extracted friction is about 30% larger than the one observed in the experiment. However, two fundamentally different methods, one utilizing the MSD of COM and the other the relaxation modulus, gave consistent results for the extracted Kuhn friction. Therefore, the discrepancy presumably arises from insufficient accuracy in the force field. Nonetheless, the work demonstrates that theory predictions without adjustable parameters should be possible.

Electronic properties of different configurations of double-strand DNA-Like nanowires

The band structure, density of states (DOS), and electrical conductivity (EC) of different configurations of a double-strand poly-GC-poly-AT DNA-like nanowire are investigated within the tight-binding Hamiltonian model and the Green's function formalism. Three different configurations of the double-strand DNA-like nanowire were considered: infinite, finite, and cyclic. The change of behavior in the band structure due to the introduction of dimerization effects of the longitudinal hopping terms was studied. Without dimerization, the DNA-like nanowire behaves as a conductor and with dimerization, a semiconductor behavior was observed. This behavior change was also verified through the calculation of the DOS via the Green's function formalism. The effects of the finite size of the DNA-like nanowire and also the effect of length increase on the EC of the system were studied using Kubo formula. There exists a direct relationship between EC and length of DNA-like nanowire. As length increases, so does the EC of the DNA-like nanowire. This information can be useful in both theoretical research and experimental development of DNA-based molecular electronics.

Molecular dynamics simulation of water-based nanofluids viscosity

The shear viscosity coefficients of water and water-based nanofluids with copper particles are calculated by the molecular dynamics method. Copper nanoparticles with a diameter of 2, 4 and 10 nm were used in the simulation. The volume fraction of nanoparticles was varied from 1 to 5%. The interaction of water molecules with each other was modeled using the Lennard–Jones potential. The Rudyak–Krasnolutskii and Rudyak–Krasnolutskii–Ivanov potentials were used as nanoparticle–molecule and nanoparticles interaction potentials, respectively. The viscosity coefficient was calculated using the fluctuation–dissipation theorem by the Green–Kubo formula. It is shown that the viscosity of the nanofluid significantly exceeds the viscosity of the coarse-grained suspension and increases with a decrease in the nanoparticles size at their fixed volume fraction. The correlation functions determining the viscosity coefficient of the nanofluid were analyzed in detail. The radial distribution functions of pure water and nanofluids are also presented in the paper. It is shown that the liquid near the nanoparticle is structured much more strongly than in the bulk. This greater ordering of the nanofluid is one of the main factors determining the increase in nanofluids viscosity.

Molecular Dynamics Study of Thermal Conductivity of Silver Chalcogenides

To obtain an accurate thermal conductivity of Ag2Se considering heat‐flux effects, computational requirements such as simulation times and number of atoms are investigated for two methods: the Green–Kubo (GK) formula in equilibrium molecular dynamics (EMD) simulations and perturbed MD (pMD). An empirical interatomic potential is used, which reproduces both the superionic and nonsuperionic phases of Ag2Se. It is found that the accurate thermal conductivity based on the GK formula requires at least the correlation time of 12 ps, the simulation time of 24 ns, and the number of atoms of 3000. In contrast, for the pMD, the simulation time of ≈1.2 ns is needed to obtain a comparable thermal conductivity. It is also confirmed that the heat flux must be considered in MD simulations to qualitatively reproduce the temperature dependence of thermal conductivity observed experimentally.

Numerical analysis of effective refractive index bio-sensor based on graphene-embedded slot-based dual-micro-ring resonator

We propose a novel bio-sensor structure composed of slot dual-micro-ring resonators and mono-layer graphene. Based on the electromagnetically induced transparency (EIT)-like phenomenon and the light-absorption characteristics of graphene, we present a theoretical analysis of transmission by using the coupled mode theory and Kubo formula. The results demonstrate the EIT-like spectrum with asymmetric line profile. The mode-field distributions of transmission spectrum are obtained from 3D simulations based on finite-difference time-domain (FDTD) method. Our bio-sensor exhibits theoretical sensitivity of 330 nm/RIU, a minimum detection limit of [Formula: see text] RIU, the maximum extinction ratio of 4.4 dB, the quality factor of [Formula: see text] and a compact structure of [Formula: see text]. Finally, the bio-sensor’s performance is simulated for glucose solution. Our proposed design provides a promising candidate for on-chip integration with other silicon photonic element.

Floquet-Drude conductivity

This letter presents a generalization of the Drude conductivity for systems which are exposed to periodic driving. The probe bias is treated perturbatively by using the Kubo formula, whereas the external driving is included non-perturbatively using the Floquet theory. Using a new type of four-times Green's functions disorder is approached diagrammatically, yielding a fully analytical expression for the Floquet-Drude conductivity. Furthermore, the Floquet Fermi's golden rule is generalized to $tt'$-Floquet states, connecting the Floquet-Dyson series with scattering theory for Floquet states. Our formalism allows for a direct application to numerous systems e.g. graphene or spin-orbit systems.