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In this paper, we present a method to reconstruct the spatially varying conductivity tensor in isotropic and orthotropic materials, involved in a two-dimensional transient anisotropic model with Ro...

Imaging of the electrical conductivity distribution inside the human body has been investigated for numerous clinical applications. The conductivity tensors of biological tissue have been obtained from water diffusion tensors by applying several models, which may not cover the entire phenomenon. Recently, a new conductivity tensor imaging (CTI) method was developed through a combination of B1 mapping, and multi-b diffusion weighted imaging. In this study, we compared the most recent CTI method with the four existing models of conductivity tensors reconstruction. Two conductivity phantoms were designed to evaluate the accuracy of the models. Applied to five human brains, the conductivity tensors using the four existing models and CTI were imaged and compared with the values from the literature. The conductivity image of the phantoms by the CTI method showed relative errors between 1.10% and 5.26%. The images by the four models using DTI could not measure the effects of different ion concentrations subsequently due to prior information of the mean conductivity values. The conductivity tensor images obtained from five human brains through the CTI method were comparable to previously reported literature values. The images by the four methods using DTI were highly correlated with the diffusion tensor images, showing a coefficient of determination (R2) value of 0.65 to 1.00. However, the images by the CTI method were less correlated with the diffusion tensor images and exhibited an averaged R2 value of 0.51. The CTI method could handle the effects of different ion concentrations as well as mobilities and extracellular volume fractions by collecting and processing additional B1 map data. It is necessary to select an application-specific model taking into account the pros and cons of each model. Future studies are essential to confirm the usefulness of these conductivity tensor imaging methods in clinical applications, such as tumor characterization, EEG source imaging, and treatment planning for electrical stimulation.

Magnetic resonance conductivity tensor imaging (MRCTI) reconstructs high-resolution anisotropic conductivity images, which are proved to have critical importance in radio-oncological imaging as well as source localization fields. In the MRCTI technique, linearly independent current injections are applied to the region to be imaged and resulting magnetic flux densities are measured using magnetic resonance imaging techniques. In this study, a novel iterative reconstruction algorithm based on a sensitivity matrix approach is proposed and tested using both simulated and experimental measurements. Obtained results show that the proposed technique can reconstruct anisotropic conductivity images with high and position-independent spatial resolution in addition to decreased number of current injection strategies.

Aims Electric conduction in the atria is direction-dependent, being faster in fibre direction, and possibly heterogeneous due to structural remodelling. Intracardiac recordings of atrial activation may convey such information, but only with high-quality data. The aim of this study was to apply a patient-specific approach to enable such assessment even when data are scarce, noisy, and incomplete. Methods and results Contact intracardiac recordings in the left atrium from nine patients who underwent ablation therapy were collected before pulmonary veins isolation and retrospectively included in the study. The Personalized Inverse Eikonal Model from cardiac Electro-Anatomical Maps (PIEMAP), previously developed, has been used to reconstruct the conductivity tensor from sparse recordings of the activation. Regional fibre direction and conduction velocity were estimated from the fitted conductivity tensor and extensively cross-validated by clustered and sparse data removal. Electrical conductivity was successfully reconstructed in all patients. Cross-validation with respect to the measurements was excellent in seven patients (Pearson correlation r > 0.93) and modest in two patients (r = 0.62 and r = 0.74). Bland–Altman analysis showed a neglectable bias with respect to the measurements and the limit-of-agreement at –22.2 and 23.0 ms. Conduction velocity in the fibre direction was 82 ± 25 cm/s, whereas cross-fibre velocity was 46 ± 7 cm/s. Anisotropic ratio was 1.91±0.16. No significant inter-patient variability was observed. Personalized Inverse Eikonal model from cardiac Electro-Anatomical Maps correctly predicted activation times in late regions in all patients (r = 0.88) and was robust to a sparser dataset (r = 0.95). Conclusion Personalized Inverse Eikonal model from cardiac Electro-Anatomical Maps offers a novel approach to extrapolate the activation in unmapped regions and to assess conduction properties of the atria. It could be seamlessly integrated into existing electro-anatomic mapping systems. Personalized Inverse Eikonal model from cardiac Electro-Anatomical Maps also enables personalization of cardiac electrophysiology models.

We propose an efficient analytical upscaling method to compute the equivalent conductivity tensor for elliptic equations in three-dimensional space. Our approach uses perturbation expansion and Fourier analysis, and considers heterogeneity, anisotropy and geometry of coarse gridblocks. Through low-order approximation, the derived analytical solution accurately approximates the central-difference numerical solution with periodic boundary conditions. Numerical tests are performed to demonstrate the capability and efficiency of this analytical approach in upscaling fluid flow in heterogeneous formations. We test the method in synthetic examples and benchmark cases with both Gaussian random fields and channelized non-Gaussian fields. In addition, we examine the impact of each parameter on the upscaled conductivity, and investigate the sensitivity of the variance and correlation lengths to the coefficients. We also indicate how to extend this approach to multiphase flow problems.

This paper addresses the problem of the electrical conductivity tensor calculation for a transversely isotropic material that contains inhomogeneities of arbitrary orientation. For this goal, we first construct the electrical conductivity contribution tensor for an arbitrarily oriented isolated ellipsoidal anisotropic inhomogeneity embedded in a transversely isotropic matrix. The general case of an orthotropic ellipsoidal inhomogeneity unaligned in an anisotropic matrix with different classes of symmetry can be considered. This solution is used as the basic building block of various homogenization techniques: the Mori–Tanaka–Benveniste scheme, Maxwell scheme, and differential scheme. The approach is illustrated by an application to a transversely isotropic mudstone rock, composed of a clay matrix containing inhomogeneities of calcite and quartz. We analyse the origins of the extent of anisotropy of the effective conductivity tensor, distinguishing among the shape, orientation distribution, and anisotropy of the inhomogeneities on the one hand and the anisotropy of the matrix on the other hand. Numerical results show that the orientation distribution of the inhomogeneities significantly affects the overall anisotropy in the case of inhomogeneities with low aspect ratio(s). Limiting cases of aligned and randomly oriented inhomogeneities provide bounds of the extent of anisotropy for the overall conductivity tensor.

The frequently used Maxwell’s equations that contain E, B, D, and H fields are only substantiated in the framework of linear material equations and for isotropic media alone. We have shown that accounting for the deviation of magnetic permittivity μ (ω) from unity in the usually employed dispersion equation implies a false precision. Therefore, if spatial dispersion is disregarded, transverse waves only exist in the energy region where ε(ω) > 0 and have a positive group velocity.

We compared the conductivity tensor becoming important parameter of ionospheric plasma using the real geometry of Earth’s magnetic field in the Northern hemisphere for both cold and warm ionospheric plasma for equinox days. It could be that the conductivity tensor certainly depends on the vector of wave propagation (k) and the adiabatic sound speed (Ue) in warm ionospheric plasma and it is possible to say that the adiabatic sound speed for electron generally decreases the magnitudes of conductivity tensor components with respect to the cold ionosphere plasma except for $$\sigma_{{ 2 3 {\text{R}}}}^{\prime }$$ (Ue ≠ 0) = $$\sigma_{{ 2 3 {\text{R}}}}$$ (Ue = 0), $$\sigma_{{ 3 3 {\text{R}}}}^{\prime }$$ (Ue ≠ 0) = $$\sigma_{{ 3 3 {\text{R}}}}$$ (Ue = 0), $$\sigma_{{ 1 3 {\text{S}}}}^{\prime }$$ (Ue ≠ 0) = $$\sigma_{{ 1 3 {\text{S}}}}$$ (Ue = 0) and $$\sigma_{{ 3 3 {\text{S}}}}^{\prime }$$ (Ue ≠ 0) = $$\sigma_{{ 3 3 {\text{S}}}}$$ (Ue = 0). In this sense, the resistivity and reactance increase in ionospheric plasma. In this context, according to the accepted conditions, the change of conductivity with local time is similar to change of electron density with local time for both (cold and warm) conditions in ionospheric plasma as trend.

This work calculates conductivity tensors in cylindrical quantum wires with parabolic potential, with the presence of an electromagnetic wave field, a laser field, and is only considered for the case of electron-optical phonon scattering. By using the quantum kinetic equation for the electron in the quantum wire with the parabolic potential, combined with the presence of two external fields, the expression of conductivity tensor is computed for the case of optical electron-phonon scattering. This conductivity tensor expression depends on the electromagnetic wave frequency, the laser field frequency, the laser field amplitude, and the quantities that characterize the cylindrical quantum wire with parabolic potential. The figure of the dependence of the conductivity tensor to the external field will be plotted, examined, and analyzed for the quantum wire GaAs / GaAsAl.

This paper studies the influence of electromagnetic waves and laser fields on conductivity tensor with the case of electron - optical phonon scattering in quantum wells. Because electrons are confined in the quantum well in one direction, the conductivity tensor in the quantum well is different from the conductivity tensor in the bulk semiconductor. The carrier system in a quantum well is located in an electromagnetic wave field and a laser field. Using quantum kinetic equation for electrons in quantum wells in the presence of an electromagnetic wave and a laser field, from the quantum kinetic equation it is possible to calculate the conductivity tensor. The conductivity tensor expression is a function of the electromagnetic wave frequency, laser field frequency and quantum well characteristic parameters. Survey and plot the dependence of conductivity tensor on electromagnetic wave frequency, laser field frequency, laser field amplitude with GaAs / GaAsAl quantum well case.

Conductivity tensor characterizes the conductivity of the material, this work calculates the conductivity tensor in a cylindrical quantum wire with infinite potential in the presence of two external fields, the laser field and linearly polarized electromagnetic wave field. The problem is also only considered for the case of electron-optical phonon scattering. By using a quantum kinetic equation for electrons in a quantum wire in the presence of two external fields, the conductivity tensor is calculated. The expression for conductivity tensor shows its dependence on the frequency and amplitude of the external field as well as other parameters specific to the system The figure of the dependence of the conductivity tensor to the external field will be investigated and plotted for the quantum wire GaAs/GaAsAl.

In quantum well studies, the conductivity tensor problem is one of the fundamental problems. In this work the topic covered here is the effect of electromagnetic waves on the conductivity tensor with the presence of a laser field in a quantum well with a parabolic potential considering the case ofelectrons-acoustic phonon scattering. By using quantum kinetic equations for electrons in quantum wells with parabolic potential, the author has calculated conductivity tensor with electrons–acoustic phonon scattering in the presence of an electromagnetic wave field and a laser field. The expression tensor for conductivity shows its dependence on the frequency of the electromagnetic wave field, the frequency of the laser field, and other parameters specific to the system. From the conductivity tensor expression plotted the effect of the electromagnetic wave field on the conductivity tensor in the presence of the laser field. The quantum well is discussed and plotted here is the GaAs/GaAsAl quantum well.

Currently strong laser radiation can affect the conductivity and other dynamic effects in semiconductors, not only changing the carrier concentration or temperature, but also changing the probability of electron scattering. It has also been shown that light radiation can change the magnitude of these effects and also expand their range of existence. Electron scattering in the presence of strong radiation can be complemented by the absorption and radiation of photons. This paper presents the influence of electromagnetic waves on conductivity tensor with the presence of a laser field in semiconductor. The conductivity tensor was calculated in the presence of an electromagnetic wave, a laser field from which to plot the conductor tensor dependence on the electromagnetic wave frequency and the laser field frequency.

A program code was developed to calculate block hydraulic conductivity of the 2-D DFN(discrete fracture network) system based on equivalent pipe network, and implemented to examine the effect of joint orientation distribution on the hydraulic characteristics of fractured rock masses through numerical experiments. A rock block of size was used to generate the DFN systems using two joint sets with fixed input parameters of joint frequency and gamma distributed joint size, and various normal distributed joint trend. DFN blocks of size were selected from center of the blocks to avoid boundary effect. Twelve fluid flow directions were chosen every starting at . The directional block conductivity including the theoretical block conductivity, principal conductivity tensor and average block conductivity were estimated for generated 180 2-D DFN blocks. The effect of joint orientation distribution on block hydraulic conductivity and chance for the equivalent continuum behavior of the 2-D DFN system were found to increase with the decrease of mean intersection angle of the two joint sets. The effect of variability of joint orientation on block hydraulic conductivity could not be ignored for the DFN having low intersection angle between two joint sets.

Nakon što je uspješno sintetiziran grafen, prvi dvodimenzionalni sustav debljine jednog atoma, eksperimentalno i teorijsko istraživanje dvodimenzionalnih sustava značajno se inteziviralo. Posljedica je to velikih tehnoloških mogućnosti dvodimenzionalnih materijala koji su po svojim elektronskim karakteristikama vodiči (grafen) ili izolatori (heksagonalni borov nitrid i različiti dihalkogenidi s prijelaznim metalima). Stoga je vrlo važno razviti teorijski formalizam pomoću kojeg ćemo moći opisati najvažnija elektrodinamička svojstva ovih sustava. Na primjer, jednočestična pobuđenja koja imaju centralnu ulogu u objašnjenju električnog transporta, ili kolektivna pobuđenja, na primjer, plazmone i ekscitone. Sustavna formulacija teorijskih metoda uključuje nekoliko faza. Prvo će biti potrebno odrediti elektronske disperzije koje čine temelj analize elektrodinamičkih svojstava. U ovom razmatranju ograničit ćemo se na nekoliko vrpci u blizini Fermijevog nivoa. Vrpce određujemo pomoću aproksimacije čvrste veze te ćemo vidjeti da je za opis grafena i hBN-a dovoljno uzeti dvije orbitale u bazi dok je broj orbitala potrebnih za opis 2DMoS2 znatno veći. Zatim ćemo promotriti međudjelovanje elektrona i različitih bozonskih stupnjeva slobode. Prvo ćemo odrediti konstante elektron-fonon vezanja promatrajući promjenu matričnih elementa preskoka elektrona na prve susjede i promjenu energija atomskih orbitala uzrokovanih titranjem rešetke. Izračunat ćemo konstante elektron-fonon vezanja za akustične i optičke fonone i odrediti njihovu formu u dugovalnoj granici. Vezanje elektrona i vanjskih elektromagnetskih polja opisati ćemo pomoću Peierlsove supstitucije u aproksimaciji čvrste veze u režimu linearnog odziva. Izvest ćemo standardne oblike interakcije nabojnih i strujnih gustoća s vanjskim skalarnim i vektorskim potencijalima i objasniti ulogu nabojnih, strujnih i dipolnih vršnih funkcija. Transportna svojstva vodljivih elektrona studirati ćemo u režimu linearnog odziva razmatrajući semiklasične i kvantne transportne jednadžbe. U oba slučaja centralnu ulogu ima neravnotežna funkcija raspodjele koja se u semiklasičnim jednadžbama definira kao neravnotežna funkcija raspodjele za elektron impulsa ħk na položaju r, dok je u kvantnim transportnim jednadžbama povezana s propagatorom elektron-šupljina para čiji su valni vektori k+q i k. Dinamiku elektron šupljina para unutar kvantnih transportnih jednadžbi odrediti ćemo pomoću Heisenbergove jednadžbe, te ćemo pokazati da sva raspršenja elektrona, na drugim elektronima, na nečistoćama, i na fononima, možemo opisati preko tri doprinosa relaksacijskoj funkciji koju ćemo zvati memorijska funkcija. Alternativa formalizmu jednadžbi gibanja za elektron-šupljina propagator su standardne perturbativne metode računa odzivnih funkcija. Na taj način obično defniramo različite korelacijske funkcije, koje nazivamo Kubo formule; na primjer, Kubo formula za naboj-naboj korelacijsku funkciju, za struja-struja korelacijsku funkciju, itd. Pomoću jednadžbe kontinuiteta i zahtjeva baždarne invarijatnosti pokazat ćemo veze između korelacijskih funkcija te pokazati da je tenzor vodljivosti zapravo struja-dipol korelacijska funkcija. Izračunati ćemo istosmjernu i dinamičku vodljivost grafena i dopiranog 2DMoS2, analizirati strukturu momentum distribucijske funkcije i njenu ulogu u računu istosmjerne vodljivosti u nedopiranom grafenu. Zatim razmatramo unutarvpčanu dinamičku vodljivost dopiranog grafena u slučaju kada u memorijskoj funkciji zadržavamo samo raspršenja na nečistoćama i fononima. Rezultati dobiveni na taj način su uspoređeni s rezultatima aproksimacije relaksacijskog vremena. Na kraju promatramo međuvrpčanu vodljivost dopiranog grafena. Ponovno promatramo dvije aproksimacije: aproksimaciju relaksacijskog vremena i aproksimaciju međuvrpčane memorijske funkcije. Konačno, analizu kolektivnih pobuđenja vršimo razmatrajući dinamiku onog istog elektron-šupljina propagatora kojeg smo razmatrali u transportnim jednadžbama, no sada dugodosežnu kulonsku interakciju uključujemo eksplicitno u Heisenbergovu jednadžbu. U jednadžbama gibanja za elektron-šupljina propagator za generalni problem s više vrpci jasno ćemo razlikovati unutarvrpčana od međuvrpčanih kolektivnih pobuđenja. Pritom ćemo pokazati da se ladder i Fockov doprinos u unutarvpčanom kanalu međusobno dokidaju u vodećem članu. Preostaje RPA doprinos koji ponovno ulazi u definiciju makroskopskog električnog polja te Fockovi doprinosi višeg reda koji ulaze u doprinos memorijskoj funkciji koji je porijeklom od direktnih elektron-elektron interakcija, te ćemo odrediti disperziju plazmona. U slučaju izolatora izvest ćemo jednadžbu gibanja za međuvrpčani elektron-šupljina propagator i pronaći energije kolektivnih modova, ekscitona, koje imaju slične karakteristike kao energije dvodimenzionalnog pozitronija. Tako dobivene energije slabo se slažu s eksperimentalnim podacima. Stoga će biti potrebno poopćiti model i uključiti elektron-šupljina korelacijske efekte koji dolaze od međuvrpčanih prijelaza. Te efekte uključujemo u jednadžbe preko zasjenjene kulonske interakcije. CHAPTER 1 Experimental and theoretical research of two-dimensional materials is highly intensifying. This is due to their fascinating physical properties which are of technological importance. We shall give a simple overview of the history and physical properties of the three most important members of the two-dimensional materials, graphene, hexagonal boron nitride and molybdenum disulfide. CHAPTER 2 Every thorough analysis of the electromagnetic properties of crystals begins with the calculation of electronic dispersions, which generally is not an easy task. Some approximations are necessary in order to find simple analytical expressions for electronic dispersions. The most important one is to focus our attention only to a few bands nearest to the Fermi level. We calculate these dispersions by using the tight binding approximation (TBA). We start from atomic orbitals or their linear combinations (hybrids), for which the hopping probability is highest. The choice of atomic orbitals is determined by crystal symmetry, which will be different for all three systems considered in this thesis: graphene, hexagonal boron nitride and molybdenum disulfide. We shall briefly introduce the hexagonal lattice with a basis, display the electronic dispersions of the valence bands in aforementioned systems for typical TBA parameters, and estimate the values of these parameters by comparison with ab initio calculations. CHAPTER 3 In this chapter we study the interaction of conduction electrons with various bosonic modes. First, we determine the electron-phonon coupling constants by looking at how lattice vibrations change the hopping matrix elements between neighbouring sites and atomic energies on those sites. We calculate these constants for acoustic and optical phonons and find their long-wavelength forms. The electron coupling to external electromagnetic fields is described by the Peierls substitution in the TBA, for a linear response regime. We shall derive the standard form of the interaction of the charge and current densities with external scalar and vector potentials and explain the role of charge, current and dipole vertex functions. These coupling constants and vertex functions are naturally defined in the direct-space representation, or in the delocalized orbital representation. Therefore, we shall use the Fourier transformations to determine the corresponding contributions to the total Hamiltonian in the Bloch representation. CHAPTER 4 Transport properties of conducting systems are usually studied in the linear response regime. This can be done using semi-classical transport equations or quantum transport equations. In both cases, the non-equilibrium distribution function has a central role. The semiclassical Boltzmann equations define it as a non-equilibrium distribution function for an electron of momentum ħk at a position r, whereas in the quantum equations and their semiclassical regimes it is related to the propagator of an electron-hole pair with momenta k + q and k. The Landau transport equations are a generalization of Boltzmann equations, where electron-electron interactions are treated in a self-consistent way and the scattering from the static disorder and from phonons is included phenomenologically. We can reach the Landau equations following the usual Landau’s procedure, or from the Heisenberg equations of motion for the electron-hole propagator. The advantage of the latter approach is that all electron scattering, from electrons, from disorder and from phonons, can be described in the same way, by studying the three contributions to the relaxation function, which will be called here the memory function. In this chapter we shall introduce the non-equilibrium distribution function n (k, r, t), derive the semiclassical equations of motion, determine the structure of the memory function for the scattering by phonons and describe the role of its real and imaginary parts in the dynamical conductivity tensor. CHAPTER 5 An alternative to the equations of motion approach from the previous chapter are the usual perturbative methods of calculating response functions. In this way various correlation functions are usually defined, named the Kubo formulae; for example, the Kubo formula for the charge-charge or current-current correlation functions, etc. There are two basic formalisms; the first one at T = 0 and the Matsubara finite-temperature formalism. The conductivity tensor, introduced in the previous chapter, represents the current-dipole correlation function, linking the current (response of the system) induced in the system by a macroscopic field (external perturbation). The external field couples to the dipole moment operator. In a similar manner, other response functions of interest can be defined. In this chapter, we shall use the T =0 formalism where response functions represent the retarded correlation functions (the response follows the external perturbation). We shall define all correlation functions related to the conductivity tensor, derive their general relations, and make sure that these relations are satisfied in the lowest order, usually called the one fermion loop approximation. We shall study the effects of phenomenologically introduced relaxation processes. We shall show that in general considerations, the continuity equation and Kramers-Kronig relations play an important role. CHAPTER 6 In this chapter, we study the DC and dynamic conductivity of graphene and doped molybdenum disulfide, using the general expressions derived in chapters 4 and 5. First, we analyze the structure of the momentum distribution function and explain its role in the calculation of the DC conductivity of undoped graphene. Afterwards, we study the intraband dynamical conductivity of doped graphene, keeping in the memory function only the scattering by phonons by disorder. These results are compared with the relaxation-time approximation results. In the end, we consider the interband conductivity of doped graphene, once again in two approximations: the relaxation-time and interband memory function approach. CHAPTER 7 In this chapter, we analyze elementary excitations in the electron subsystem. Unlike the transport equations studied in the third chapter, where long-range Coulomb interactions were taken into account implicitly through the macroscopic electric field, here we will include various electron-electron interaction contributions step by step. The primary interest here is to study the collective excitations in the electronic subsystem. In the electron-hole equations of motion for a general problem with multiple bands, we shall clearly distinguish the intraband and interband collective modes. We shall show that the "ladder" and Fock contributions cancel out in the intraband channel. What remains is the RPA contribution (being part of the macroscopic field definition) and higher order Fock contributions which lead to the electron-electron contribution to the memory functions. For the insulating case, we shall derive the equation of motion for the interband electron-hole propagator and find the energies of collective modes, excitons, which have energies similar to the 2D positronium. Energies calculated in this way do not agree well with experimental data, so it will be necessary to include electron-hole correlation effects. These effects are included in our equations by using screened Coulomb interaction instead of the bare one.

This paper describes a method to invert marine CSEM data from anisotropic layered media. The method uses two types of constraints to generate stable solutions that improve the positioning of interfaces in anisotropic layers: applying the L1 norm equality constraints of the Total Variation method to parameters in adjacent layers, and imposing L2 norm equality constraints between the different components of the conductivity tensor in each layer. The solutions are compared favorably with those from previously published smoothing methods. The results show that the simultaneous application of the two constraints is able to improve the delineation of the anisotropic layers, including the resistive target commonly sought in the inversion of marine CSEM data. As an example, the method was applied to real data from an offshore Brazilian basin. Keywords: anisotropic layered media, inversion, CSEM. RESUMO Este artigo descreve um metodo para inverter dados do metodo marinho de fonte controlada CSEM de meios anisotropicos estratificados. O metodo usa dois tipos de vinculos para gerar solucoes estaveis que melhoram o posicionamento de interfaces entre camadas anisotropicas: aplicando os vinculos de norma L1 do metodo de Variacao Total a parâmetros de camadas adjacentes, e impondo vinculos de igualdade de norma L2 entre as componentes do tensor de impedância em cada camada. As solucoes obtidas mostraram-se melhores do que as solucoes encontradas na literatura com apenas o metodo de suavidade. Os resultados indicam que a aplicacao simultânea dos dois vinculos e capaz de melhorar o delineamento de camadas anisotropicas, incluindo as camadas-alvo resistivas normalmente buscadas na inversao de dados CSEM. O metodo foi aplicado a um conjunto de dados reais de uma bacia marinha brasileira. Palavras-chave: meios estratificados anisotropicos, inversao, CSEM.

We address the electromagnetic properties of two-dimensional electron gas confined by a dielectric environment in the presence of both Rashba and Dresselhaus spin-orbit interactions. It is demonstrated that off-diagonal components of the conductivity tensor resulting from a delicate interplay between Rashba and Dresselhaus couplings lead to the hybridization of transverse electric and transverse magnetic surface electromagnetic modes localized at the interface. We show that the characteristics of these hybrid surface waves can be controlled by additional intense external off-resonant coherent pumping., Comment: 6 pages, 2 figures

In the first part of the Chapter, we show that the non-commutative Kubo-formula for the transport coefficients is covered by the theory developed so far. In the second part, we present computer assisted calculations of the finite-temperature transport coefficients, for model Hamiltonians relevant for the integer quantum Hall effect and Chern insulators. The focus will be on the physical signature of the Anderson transition occurring at the boundary between the topological phases.

In the present study we derive some uniqueness criteria for solutions of the Cauchy problem for the standard equations of dynamical linear thermoelasticity backward in time. We use Lagrange-Brun identities combined with some differential inequalities in order to show that the final boundary value problem associated with the linear thermoelasticity backward in time has at most one solution in appropriate classes of displacement-temperature fields. The uniqueness results are obtained under the assumptions that the density mass and the specific heat are strictly positive and the conductivity tensor is positive definite.

The effective conductivity tensor of a polycrystalline medium with a texture has been calculated based on the method of self-consistent solution and the theory of rotation group representations. The medium consists of a single-type of biaxial spherical crystallites, oriented in space according to some probabilistic law, which implies the existence of uniaxial texture. An analytical solution is obtained for two cases: (i) weakly anisotropic crystallites and (ii) a small spread in the orientations of one of the crystallite axes with respect to the texture axis.

The conductivity of a weakly inhomogeneous anisotropic medium is considered. The effective conductivity tensor \( \hat \sigma _e \) is determined in the approximation quadratic in deviation of local conductivity \( \hat \sigma \) (r) from mean value 〈\( \hat \sigma \)〉 for an arbitrary anisotropy of the composite.

A canonical relativistic quantization of the electromagnetic field is introduced in the presence of an anisotropic conductor magneto-dielectric medium in a standard way in the Gupta–Bleuler framework. The medium is modeled by a continuum collection of the vector fields and a continuum collection of the antisymmetric tensor fields of the second rank in Minkowski space–time. The collection of vector fields describes the conductivity property of the medium and the collection of antisymmetric tensor fields describes the polarization and the magnetization properties of the medium. The conservation law of the total electric charges, induced in the anisotropic conductor magneto-dielectric medium, is deduced using the antisymmetry conditions imposed on the coupling tensors that couple the electromagnetic field to the medium. Two relativistic covariant constitutive relations for the anisotropic conductor magneto-dielectric medium are obtained. The constitutive relations relate the antisymmetric electric–magnetic polarization tensor field of the medium and the free electric current density four-vector, induced in the medium, to the strength tensor of the electromagnetic field, separately. It is shown that for a homogeneous anisotropic medium the susceptibility tensor of the medium satisfies the Kramers–Kronig relations. Also it is shown that for a homogeneous anisotropic medium the real and imaginary parts of the conductivity tensor of the medium satisfy the Kramers–Kronig relations and a relation other than the Kramers–Kronig relations.

For certain non-magnetic solids with low symmetry the occurrence of spin-polarized longitudinal currents is predicted. These arise due to an interplay of spin-orbit interaction and the particular crystal symmetry. This result is derived using a group-theoretical scheme that allows investigating the symmetry properties of any linear response tensor relevant to the field of spintronics. For the spin conductivity tensor it is shown that only the magnetic Laue group has to be considered in this context. Within the introduced general scheme also the spin Hall- and additional related transverse effects emerge without making reference to the two-current model. Numerical studies confirm these findings and demonstrate for (Au$_{1-x}$Pt$_{\rm x}$)$_4$Sc that the longitudinal spin conductivity may be in the same order of magnitude as the conventional transverse one. The presented formalism only relies on the magnetic space group and therefore is universally applicable to any type of magnetic order., 5 pages, 1 table, 2 figures (3 & 2 subfigures)

The frequency and wave-vector-dependent memory function in the longitudinal conductivity tensor of weakly interacting electronic systems is calculated by using the approach based on the quantum transport equations. It is shown, for the first time consistently, that there is a close relation between the single-electron self-energy, the electron-hole pair self-energy, and the memory function. It is also shown in which way singular long-range Coulomb interactions, together with other q approx 0 scattering processes, drop out of both the memory function and the related transport equations. The theory is illustrated on heavily doped graphene, which is the prototype of weakly interacting single-band electron-phonon systems. A steplike increase of the width of the quasiparticle peak in angle-resolved photoemission spectra at frequencies of the order of the frequency of in-plain optical phonons is shown to be consistent with similar behavior of the intraband plasmon peak in energy loss spectroscopy spectra. Both anomalies can be understood as a direct consequence of weak electron scattering from in-plane optical phonons.

Consider a two-dimensional two-component periodic composite made from a collection of non-overlapping, identical, circular disks, embedded in a matrix. The effective conductivity tensor can be written in the form of expansion on ‘basic elements' which depend only on locations of the disks. These elements are expressed in terms of the Eisenstein series. The representative cell of a composite is defined as the minimal size periodicity cell corresponding to the set of basic elements calculated for the composite. An algorithm to determine the representative cell for a given composite is constructed. †Dedicated to Professor Guochun Wen on the occasion of his 75th birthday.

We study the structure of the general solution of the R-linear conjugation problem with constant coefficients in a class of doubly periodic functions, i.e., the R-linear problem on torus. The results are applied to determine the effective conductivity tensor of composites.

Electrical conductivity plays a central role in many areas of space science. However, texts and research articles sometimes misquote the conductivity tensor. This highlights the need for authors and researchers to verify the accuracy of this tensor for their chosen coordinate system. This paper presents a new graphical method to analyze the correctness of the conductivity tensor, applicable to any coordinate system. This method also illuminates the physical meaning of the terms in the conductivity tensor, which is often obscured by standard mathematical derivations.

The paper considers the contribution of conductivity changes undergone in an anisotropical medium to measured resistance. This was achieved by extending the relationship proposed by Geselowitz to anisotropical materials described, therefore, by a conductivity tensor. It was found that each element of a conductivity change tensor contributed to the measured resistance only if a corresponding component of the electrical field was nonzero. Numerical calculations were performed for blood-flow-associated conductivity changes. A special experiment stand was developed which allowed experiments to be performed proving the theoretical results. It was found that the absolute value of resistance change measured in the direction perpendicular to the vessel axis was much smaller than that measured along the vessel axis. The results obtained may explain the fact that the actual change of measured resistance created by changes of conductivity induced by aortic blood flow is lower than expected from simplified models.

Reflectivity and magneto-optical Kerr spectra are measured in epitaxial FePt films prepared under reduced substrate temperatures. From these spectra off-diagonal elements of conductivity tensor was calculated. By comparison with those prepared under conventional preparation conditions, the Ll0-ordered FePt films prepared at reduced temperatures proved to have similar electronic properties to those prepared at elevated temperatures. It is also elucidated that disordered FePt films prepared at 300°C show intermediate electronic states between ordered and disordered states.

In the framework of the perturbation theory an expression suitable for calculation of the effective conductivity of 3-D inhomogeneous metals in uniform magnetic field $H$ is derived. For polycrystals of metals with closed Fermi surfaces in high magnetic fields the perturbation series defining the longitudinal and the hall elements of the perturbation series can be summed allowing us to obtain the exact expression for the leading terms of all these elements of the effective conductivity tensor.

We describe an efficient numerical approach to calculate the longitudinal and transverse Kubo conductivities of large systems using Bastin's formulation. We expand the Green's functions in terms of Chebyshev polynomials and compute the conductivity tensor for any temperature and chemical potential in a single step. To illustrate the power and generality of the approach, we calculate the conductivity tensor for the quantum Hall effect in disordered graphene and analyze the effect of the disorder in a Chern insulator in Haldane's model on a honeycomb lattice., 5 pages, 3 figures and a supplementary material (3 pages)

In the Bidomain model ( 3.42), the transmembrane potential v during the excitation phase of the heartbeat exhibits a steep propagating layer spreading throughout the myocardium with a thickness of about 0.5 mm. At every point of the cardiac domain, this upstroke phase lasts about 1 ms.

The general expression for the time-dependent mean velocity of large, heavy (l.h.) ions in a gas in any regime in time-varying (or static) electric fields and/or in static magnetic fields is obtained as solution of the appropriate equation of motion for the average l.h. ion. It is shown that such an equation follows directly from the Newton's law for a single ion once appropriate averages are carried out. The procedure puts in evidence that an effective friction coefficient can be defined whose expression tends, in the hydrodynamic regime, to the usual expression following from the Stokes' law, and, in the molecular regime, to the expression relevant to the heavy particles of a Rayleigh gas. In the case of l.h. ions in static or alternate electric fields and static magnetic fields, the conductivity tensor is also obtained. Moreover, the limits of validity of the theory are briefly discussed. Finally it is shown that, when the l.h. ions are supposed to move in a hard-sphere gas, all the obtained results can explicitly be expressed in terms of the Knudsen number.

In 1985 Mortola and Steffe conjectured a formula for the effective conductivity tensor of a checkerboard structure where the unit cell of periodicity is square and subdivided into four equal squares each having a different conductivity. In this article their conjecture is proven. The key idea is to superimpose suitably reflected potentials to obtain the solution to the dual problem. This is then related back to the original problem using a well known theorem of Keller, thereby proving the conjecture. The analysis also yields formulas relating the potentials in the four squares. Independently, Craster and Obnosov have obtained a completely different proof of the conjecture.

For the first time, an integrated piezoresistive microsensor, used for monitoring ball bonding, is rigorously simulated. The sensor is based on a CMOS n+-diffusion and located in situ below a standard bonding pad. It measures a value related to the ultrasound stress which is dissipated during ball bonding. We report on a two-dimensional (2D) simulation tool based on the implementation of the governing equation using a finite difference scheme and taking into account the stress-induced local variations of the conductivity tensor. This tool calculates sensor’s sensitivity and electrical potential distribution from piezoresistive coefficients and previously simulated stress field. Simulation result is verified with measured data. Using our custom program, the geometry parameters of the sensor are optimized, predicting 4.5 times more sensitivity than that of the prototype sensor. The analysis for the sensor was rounded off with an analysis of the influence of bond wire misalignment on the sensor’s sensitivity.

The results of studying temperature behavior of the microwave surface impedance Z s (T) and conductivity tensor $$\hat \sigma (T)$$ (T) of high-T c superconducting (HTSC) single crystals are analyzed. The emphasis is on the experimental facts that are inconsistent with the known electrodynamic concepts of conductivity mechanisms in these materials. Possible reasons for the inconsistency are discussed in the context of structural features of the HTSC crystals, and the outlook for future investigations is outlined.

We report in this paper results of experimental and theoretical studies of transitions between different integer quantum Hall phases, as well as transition between the insulating phase and quantum Hall phases at high magnetic fields. We focus mainly on universal properties of the transitions. We demonstrate that properly defined conductivity tensor is universal at the transitions. We also present numerical results of a non-interacting electron model, which suggest that the Thouless conductance is universal at integer quantum Hall transitions, just like the conductivity tensor. Finite temperature and system size effects near the transition point are also studied.

The behaviour of excess (1/f noise) in two-dimensional polycrystalline media is investigated. On the base of current trap model, it is shown that there exists a certain anisotropy value of conductivity tensor for polycrystalline media when the amplitude of 1/f noise becomes giant.

Commensurability peaks in square antidot arrays with various array periods are studied. We calculate the conductivity tensor under classical mechanics varying the period in a systematic manner. The structure and its magnetic-field position of conductivities change significantly depending on the period and peak indexes. These characteristics are analyzed by a model where the chaotic motion is regarded as hopping between antidots. It is shown that the characteristics can be explained by the magnetic-field lowering of the peak position of forward scattering components and broken symmetry between left-turning and right-turning components.

Within the framework of the noise-driven Ginzburg-Landau theory, we examine how the conductivity tensor of a superconductor scales in the charged superfluid regime, where the magnetic fluctuation is relevant. In particular, we show that an effect arises in which the divergence of the conductivity tensor is suppressed by the diamagnetic susceptibilities, resulting in an effective exponent for the conductivity tensor.

The effective conductivity tensor of a particle composite with transversely isotropic phases is evaluated accurately. The geometric model of a composite material is a continuous matrix containing a...

For 2D bounded composite material geometrically composed by a disk of variable radiusrand an outer ring it is determined in an analytic form thex-component of the effective conductivity tensor. Namely, it is shown that thex-component is a sum of geometrical progression with respect to powers ofr2for all sufficiently smallr.

Magnetic resonance conductivity tensor imaging (MRCTI) is an emerging modality which reconstructs images of anisotropic conductivity distribution within a volume conductor. Images are reconstructed based on magnetic flux density distribution induced by an externally applied probing current, together with a resultant surface potential value. The induced magnetic flux density distribution is measured using magnetic resonance current density imaging techniques. In this study, MRCTI data acquisition is experimentally implemented and anisotropic conductivity images of test phantoms are reconstructed using recently proposed MRCTI reconstruction algorithms.

In Chapter 3 we expand our theory to the solids containing an infinite array of inclusions. This generalization is not straightforward however. The well-known Rayleigh solution employing this idea was questioned for a long time before an alternate, triple Fourier series solution with the same resulting formulas has been derived. In fact, the convergence issue is the only theoretical problem arising in the application of the multipole expansion approach to the models involving an infinite set of inclusions. In what follows, we illustrate this problem for the models involving a doubly periodic (2P) array of spherical inclusions. They are regarded as composite band and composite layer models.

The densities and mobilities of the individual heavy‐ and light‐hole carriers have been simultaneously determined at various temperatures (40 K to 130 K) in three p‐type, single‐crystal Si samples. The separation of the two‐hole components is achieved by multicarrier analyses of magnetic‐field‐dependent Hall and resistivity measurements within the two‐carrier approximation of the reduced‐conductivity‐tensor scheme. The explicit experimental values for the densities and mobilities of the two‐hole components obtained in this work should be considered as a valuable addition to the existing database for silicon material parameters. They should also be useful to silicon device physics and modeling.

The anisotropy of the electrical conductivity and the electrical piezoeffect due to this anisotropy is found in a uniaxially deformed thin polycrystalline Bi film. The symmetrical and antisymmetrical parts of the nondiagonal elements of the conductivity tensor for various directions of the deformation in the presence of an external magnetic field are investigated. It is shown that all features of the anisotropy can be qualitatively explained in terms of electron repopulation between the L-valleys in the randomly orientated and deformed Bi microcrystallites. Possible ways of obtaining higher anisotropy in deformed films are discussed. [Russian Text Ignored].

In this paper, we present a solution of the flow problem in heterogeneous, non-stationary media, where the non-stationarity is manifested as a linear trend in the mean log-conductivity. The flow problem is posed in a stochastic framework, and our goal is to define an average flow equation and to derive the relationship between the mean gradient and the mean flux. For a stationary medium, such an approach would amount to the definition of the effective conductivity tensor, but in the present case, since due to the specific boundary condition the coefficient proportionality between the mean gradient and the mean flux depends on the angle between the trend and the mean gradient, we refer to it as the tensor of equivalent conductivity. We derive this tensor for one-, two- and three-dimensional flow equations.