Lattice Green Function Research Articles

Resistance calculation of the face-centered cubic lattice: Theory and experiment

The effective resistance between two arbitrary lattice points in an infinite, face-centered cubic lattice network of identical resistors is calculated using the lattice Green's function method. Theoretical results have been verified experimentally by constructing actual finite networks of resistors. This problem could be useful in undergraduate courses (e.g., advanced mathematical methods course) and would provide a good example for introducing the concept of Green's function.

Y-Shaped Spin Filter in Graphene with Rashba Spin–Orbit Coupling

We propose a graphene Y-shaped spin filter with three zigzag-edged terminals in the presence of the Rashba spin–orbit interaction (RSOI). The three terminal spin-dependent conductances are calculated numerically using the lattice Green function. The results show that the spin polarization is nearly 100% for strong spin–orbit coupling and a suitable Fermi energy. The polarization direction can be tuned by controlling the structure parameters. It is interesting that the conductance of the first channel is heavily suppressed by the edge states localized at the edges of the zigzag ribbon terminals and can be significantly enhanced by Rashba coupling at the junction.

ELECTRICAL NETWORKS WITH INTERSTITIAL SINGLE CAPACITOR

We investigate the equivalent capacitance between two arbitrary nodes in a perturbed network (i.e. an interstitial capacitor is introduced between two arbitrary points in the perfect lattice) based on the lattice Green's function approach. An explicit formula for the capacitance of the perturbed lattice is derived in terms of the capacitances of the perfect lattice by solving Dyson's equation exactly. Numerical results are presented for the infinite perturbed square network. Finally, the asymptotic behavior of the effective capacitance has been studied.

Ruderman-Kittel-Kasuya-Yosida interaction in biased bilayer graphene

We study the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between two contact magnetic impurities placed on bilayer graphene (BLG). We compute the interaction mediated by the carriers of the pristine and biased BLG as well as the conduction electrons of the doped system. The results are obtained from the linear-response expression for the susceptibility written in terms of the integral over lattice Green's functions. For the unbiased system, we obtain some analytical expressions in terms of the Meijer G-functions, which consist of the product of two oscillatory terms, one coming from the interference between the two Dirac points and the second coming from the Fermi momentum. In particular, for the undoped BLG, the system exhibits the RKKY interaction commensurate with its bipartite nature as expected from the particle-hole symmetry of the system. Furthermore, we explore a beating pattern of oscillations of the RKKY interaction in a highly doped BLG system within the four-band continuum model. Besides, we discuss the discrepancy between the short-range RKKY interaction calculated from the two-band model and that obtained from the four-band continuum model. The final results for the applied gate voltage are obtained numerically and are fitted with the functional forms based on the results for the unbiased case. In this case, we show that the long-range behavior is scaled with a momentum that depends on Fermi energy and gate voltage, allowing the possibility of tuning of the RKKY interaction by gate voltage.

Ising n-fold integrals as diagonals of rational functions and integrality of series expansions

We show that the n-fold integrals χ(n) of the magnetic susceptibility of the Ising model, as well as various other n-fold integrals of the ‘Ising class’, or n-fold integrals from enumerative combinatorics, like lattice Green functions, correspond to a distinguished class of functions generalizing algebraic functions: they are actually diagonals of rational functions. As a consequence, the power series expansions of the, analytic at x = 0, solutions of these linear differential equations ‘derived from geometry’ are globally bounded, which means that after just one rescaling of the expansion variable, they can be cast into series expansions with integer coefficients. We also give several results showing that the unique analytical solution of Calabi–Yau ODEs and, more generally, Picard–Fuchs linear ODEs with solutions of maximal weights are always diagonals of rational functions. Besides, in a more enumerative combinatorics context, generating functions whose coefficients are expressed in terms of nested sums of products of binomial terms can also be shown to be diagonals of rational functions. We finally address the question of the relations between the notion of integrality (series with integer coefficients, or, more generally, globally bounded series) and the modularity of ODEs.

Gap-modulated doping effects on indirect exchange interaction between magnetic impurities in graphene

A dilute distribution of magnetic impurities is assumed to be present in doped graphene. We calculate the interaction energy between two magnetic impurities which are coupled via the indirect-exchange or Ruderman-Kittel-Kasuva-Yosida (RKKY) interaction by the doped conduction electrons. The current model is a half-filled AB-lattice structure. Our calculations are based on the retarded lattice Green's function formalism in momentum-energy space which is employed in linear response theory to determine the magnetic susceptibility in coordinate space. Analytic results are obtained for gapped graphene when the magnetic impurities are placed on the A and B sublattice sites of the structure. This interaction, which is important in determining spin ordering, has been found to be significantly different for AA and BB exchange energies in doped graphene due to the existence of an energy gap and is attributed to a consequence of the local fields not being equal on the A and B sublattices. For doped graphene, the oscillations of all three RKKY interactions from ferromagnetic to antiferromagnetic with increasing Fermi energy is significantly modified by the energy gap both in magnitude and phase. Additionally, the AB exchange energy may be modified by the presence of a gap for undoped graphene but not for doped graphene due to the dominance of doped conduction electrons.

Lattice Green functions of the higher-dimensional face-centered cubic lattices

We study the face-centered cubic (fcc) lattice in up to six dimensions. In particular, we are concerned with lattice Green functions (LGFs) and return probabilities. Computer algebra techniques, such as the method of creative telescoping, are used for deriving an ODE for a given LGF. For the four- and five-dimensional fcc lattices, we give rigorous proofs of the ODEs that were conjectured by Guttmann and Broadhurst. Additionally, we find the ODE of the LGF of the six-dimensional fcc lattice, a result that was not believed to be achievable with current computer hardware.

Lattice Green's functions for kagome, diced, and hyperkagome lattices

Lattice Green's functions (LGF) and density of states (DOS) for non-interacting models on 3 related lattices are presented. The DOS and LGF at the origin for the kagome and diced lattices are rederived. Furthermore, from the form obtained for the DOS of the hyperkagome lattice, it seems highly suggestive that a closed form expression can be obtained if a particular complicated yet highly symmetrical integral can be solved. The results have been checked numerically. The continuum of energy states in the hyperkagome lattice and the appearance of a zero in the kagome lattice may be explicitly seen from the formulae obtained.

Infinite 3D Cubic Lattices of Identical Resistors

We expressed the resistance between the origin and any lattice point in an infinite perfect Simple Cubic (i.e., SC) network rationally in terms of the known value of the Lattice Green's Function at the origin (i.e., ), and . On the other hand, we investigated the asymptotic behavior of the resistance. Finally, some numerical results has been calculated.

Localization for a line defect in an infinite square lattice

Localized defect modes generated by a finite line defect composed of several masses, embedded in an infinite square cell lattice, are analysed using the linear superposition of Green's function for a single mass defect. Several representations of the lattice Green's function are presented and discussed. The problem is reduced to an eigenvalue system and the properties of the corresponding matrix are examined in detail to yield information regarding the number of symmetric and skew-symmetric modes. Asymptotic expansions in the far field, associated with long wavelength homogenization, are presented. Asymptotic expressions for Green's function in the vicinity of the band edge are also discussed. Several examples are presented where eigenfrequencies linked to this system and the corresponding eigenmodes are computed for various defects and compared with the asymptotic expansions. The case of an infinite defect is also considered and an explicit dispersion relation is obtained. For the case when the number of masses within the line defect is large, it is shown that the range of the eigenfrequencies can be predicted using the dispersion diagram for the infinite chain.

Spanning tree generating functions and Mahler measures

We define the notion of a spanning tree generating function (STGF) ∑anzn, which gives the spanning tree constant when evaluated at z = 1, and gives the lattice Green function (LGF) when differentiated. By making use of known results for logarithmic Mahler measures of certain Laurent polynomials, and proving new results, we express the STGFs as hypergeometric functions for all regular two and three dimensional lattices (and one higher-dimensional lattice). This gives closed form expressions for the spanning tree constants for all such lattices, which were previously largely unknown in all but one three-dimensional case. We show for all lattices that these can also be represented as Dirichlet L-series. Making the connection between STGFs and LGFs produces integral identities and hypergeometric connections, some of which appear to be new.This article is part of ‘Lattice models and integrability’, a special issue of Journal of Physics A: Mathematical and Theoretical in honour of F Y Wu's 80th birthday.

Robust signatures in the current–voltage characteristics of DNA molecules oriented between two graphene nanoribbon electrodes

In this work, we numerically calculate the electric current through three kinds of DNA sequences (telomeric, λ-DNA and p53-DNA) described by different heuristic models. A bias voltage is applied between two zigzag edged graphene contacts attached to the DNA segments, while a gate terminal modulates the conductance of the molecule. Calculation of the current is performed by integrating the transmission function (calculated using the lattice Green's function) over the range of energies allowed by the chemical potentials. We show that a telomeric DNA sequence, when treated as a quantum wire in the fully coherent low-temperature regime, works as an excellent semiconductor. Clear steps are apparent in the current–voltage curves of telomeric sequences and are present independent of length and sequence initialization at the contacts. We also find that the molecule–electrode coupling can drastically influence the magnitude of the current. The difference between telomeric DNA and other DNAs, such as λ-DNA and DNA for the tumour suppressor p53, is particularly visible in the length dependence of the current.

Spin-dependent electron transport in a Rashba quantum wire with rough edges

We investigate theoretically the spin-dependent electron transport in a Rashba quantum wire with rough edges. The charge and spin conductances are calculated as function of the electron energy or the wire length by adopting the spinresolved lattice Green function method. For a single disordered Rashba wire, it is found that the charge conductance quantization is destroyed by the edge disorder. However, a nonzero spin conductance can be generated and its amplitude can be manipulated by the wire length, which is attributed to the broken structure symmetries and the spin-dependent quantum interference induced by the rough boundaries. For a large ensemble of disordered Rashba wires, the average charge conductance decreases monotonically, however, the average spin conductance increases to a maximum value and then decreases, with increasing wire length. Further study shows that the influence of the rough edges on the charge and spin conductances can be eliminated by applying a perpendicular magnetic field to the wire. In addition, a very large magnitude of the spin conductance can be achieved when the electron energy lies between the two thresholds of each pair of subbands. These findings may not only benefit to further apprehend the transport properties of the Rashba low-dimensional systems but also provide some theoretical instructions to the application of spintronics devices.

Recursive evaluation of space-time lattice Green's functions

Up to a multiplicative constant, the lattice Green's function (LGF) as defined in condensed matter physics and lattice statistical mechanics is equivalent to the Z-domain counterpart of the finite-difference time-domain Green's function (GF) on a lattice. Expansion of a well-known integral representation for the LGF on a ν-dimensional hyper-cubic lattice in powers of Z−1 and application of the Chu–Vandermonde identity results in ν − 1 nested finite-sum representations for discrete space-time GFs. Due to severe numerical cancellations, these nested finite sums are of little practical use. For ν = 2, the finite sum may be evaluated in closed form in terms of a generalized hypergeometric function. For special lattice points, that representation simplifies considerably, while on the other hand the finite-difference stencil may be used to derive single-lattice-point second-order recurrence schemes for generating 2D discrete space-time GF time sequences on the fly. For arbitrary symbolic lattice points, Zeilberger's algorithm produces a third-order recurrence operator with polynomial coefficients of the sixth degree. The corresponding recurrence scheme constitutes the most efficient numerical method for the majority of lattice points, in spite of the fact that for explicit numeric lattice points the associated third-order recurrence operator is not the minimum recurrence operator. As regards the asymptotic bounds for the possible solutions to the recurrence scheme, Perron's theorem precludes factorial or exponential growth. Along horizontal lattices directions, rapid initial growth does occur, but poses no problems in augmented dynamic-range fixed precision arithmetic. By analysing long-distance wave propagation along a horizontal lattice direction, we have concluded that the chirp-up oscillations of the discrete space-time GF are the root cause of grid dispersion anisotropy. With each factor of ten increase in the lattice distance, one would have to roughly double the pulse width of the source signature to keep pulse distortion at bay. The GF time sequences can also be used for an efficient computation of discrete space-frequency LGFs, especially if one employs Aitken's δ2 process for the acceleration of the convergence of the consecutive partial sums.

Simulation of solid-friction dependence on number of surface atoms and theoretical approach for infinite number of atoms

We investigate the dependence of solid friction on the number of surface atoms in a unit cell under periodic boundary conditions. A two-dimensional friction model between a single asperity and an elastic solid surface is examined. The solid surface is modeled using a coupled-oscillator lattice that consists of an infinite number of inner solid atoms; the dynamics are simulated by the recently proposed semi-infinite dynamic lattice Green's function method. A significant dependence of the friction on the number of surface atoms is observed. The dependence is attributed to the requirement for a large number of surface atoms for the excitation of the energy-dissipative surface phonons with nonzero wave numbers. In order to eliminate the problematic dependence, a correction method combined with the continuum contact theory is developed to evaluate friction in the limit of the infinite surface atoms. In addition, we found a relationship between the friction and the power spectrum of the temporal fluctuation in the force; the latter quantity does not significantly depend on the number of surface atoms.

The nondissipative spin Hall current in a nonuniform Rashba effect system

The spin Hall current in a two-dimensional electron system with nonuniform Rashba spin–orbit interaction (SOI) is investigated by means of the lattice Green's function method. Large electric and spin Hall currents are produced by this nonuniform Rashba SOI, while the electric Hall current vanishes in the uniform Rashba SOI system. A nondissipative spin Hall current is also produced, without any longitudinal voltage bias, any external magnetic field and any special class of band insulators.

Spin Filtering in a Rashba Electron Waveguide Induced by Edge Disorder

We theoretically study the spin-dependent electron transport in a Rashba electron waveguide with rough edges, attached to ideal leads without spin-orbit interaction. The influence of the edge disorder on the charge and spin conductances is clarified by using the spin-resolved lattice Green function method. It is found that a spin-polarized current can be generated in the output lead and its spin polarization can be manipulated by varying the waveguide length. The underlying physics is attributed to the broken longitudinal symmetry and the spin-dependent quantum interference induced by the rough boundaries. Our results may provide a new method to design a spin filter without using magnetic materials or applying a magnetic field.

Application of Mahler measure theory to the face-centred cubic lattice Green function at the origin and its associated logarithmic integral

The mathematical properties of the face-centred cubic lattice Green functionand the associated logarithmic integralare investigated, where c(θ) ≡ cos (θ) and w = w1 + iw2 is a complex variable in the w plane. In particular, the theory of Mahler measures is used to obtain a closed-form expression for S(w) in terms of 5F4 generalized hypergeometric functions. The method of analytic continuation is then applied to this result in order to prove thatNext the relation dS/dw = G(w), where w ∈ (3, +∞), is used to derive the simple formulawhere w lies in a restricted region of the cut w plane. The limit functionis also evaluated in the intervals w1 ∈ ( − 1, 0] and w1 ∈ (0, 3]. It is shown that GR(w1) and GI(w1) can be expressed in terms of 2F1[z(w1)] hypergeometric functions, where the independent variable z(w1) is a real-valued rational function of w1. Finally, new formulae are derived for the number of random walks rn on the face-centred cubic lattice which return to their starting point (not necessarily for the first time) after n nearest-neighbour steps.

Effective elastic moduli of triangular lattice material with defects

This paper presents an attempt to extend homogenization analysis for the effective elastic moduli of triangular lattice materials with microstructural defects. The proposed homogenization method adopts a process based on homogeneous strain boundary conditions, the micro-scale constitutive law and the micro-to-macro static operator to establish the relationship between the macroscopic properties of a given lattice material to its micro-discrete behaviors and structures. Further, the idea behind Eshelby’s equivalent eigenstrain principle is introduced to replace a defect distribution by an imagining displacement field (eigendisplacement) with the equivalent mechanical effect, and the triangular lattice Green's function technique is developed to solve the eigendisplacement field. The proposed method therefore allows handling of different types of microstructural defects as well as its arbitrary spatial distribution within a general and compact framework. Analytical closed-form estimations are derived, in the case of the dilute limit, for all the effective elastic moduli of stretch-dominated triangular lattices containing fractured cell walls and missing cells, respectively. Comparison with numerical results, the Hashin–Shtrikman upper bounds and uniform strain upper bounds are also presented to illustrate the predictive capability of the proposed method for lattice materials. Based on this work, we propose that not only the effective Young’s and shear moduli but also the effective Poisson’s ratio of triangular lattice materials depend on the number density of fractured cell walls and their spatial arrangements.

Direct calculation of the lattice Green function with arbitrary interactions for general crystals

Efficient computation of lattice defect geometries such as point defects, dislocations, disconnections, grain boundaries, interfaces, and free surfaces requires accurate coupling of displacements near the defect to the long-range elastic strain. Flexible boundary condition methods embed a defect in infinite harmonic bulk through the lattice Green function. We demonstrate an efficient and accurate calculation of the lattice Green function from the force-constant matrix for general crystals with an arbitrary basis by extending a method for Bravais lattices. New terms appear due to the presence of optical modes and the possible loss of inversion symmetry. By separately treating poles and discontinuities in reciprocal space, numerical accuracy is controlled at all distances. We compute the lattice Green function for a two-dimensional model with broken symmetry to elucidate the role of different coupling terms. The algorithm is generally applicable in two and three dimensions to crystals with arbitrary number of atoms in the unit cell, symmetry, and interactions.