quasi-2D Case Research Articles

Slow Many-Body Delocalization beyond One Dimension.

We study the delocalization dynamics of interacting disordered hard-core bosons for quasi-1D and 2D geometries, with system sizes and timescales comparable to state-of-the-art experiments. The results are strikingly similar to the 1D case, with slow, subdiffusive dynamics featuring power-law decay. From the freezing of this decay we infer the critical disorder W_{c}(L,d) as a function of length L and width d. In the quasi-1D case W_{c} has a finite large-L limit at fixed d, which increases strongly with d. In the 2D case W_{c}(L,L) grows with L. The results are consistent with the avalanche picture of the many-body localization transition.

Entropy and Adjoint Methods

Aerodynamic drag can be partially approximated by the entropy flux across fluid domain boundaries with a formula due to Oswatitsch. In this paper, we build the adjoint solution that corresponds to this representation of the drag and investigate its relation to the entropy variables, which are linked to the integrated residual of the entropy transport equation. For inviscid isentropic flows, the resulting adjoint variables are identical to the entropy variables, an observation originally due to Fidkowski and Roe, while for non-isentropic flows there is a significant difference that is explicitly demonstrated with analytic solutions in the shocked quasi-1D case. Both approaches are also investigated for viscous and inviscid flows in two and three dimensions, where the adjoint equations and boundary conditions are derived. The application of both approaches to mesh adaptation is investigated, with especial emphasis on inviscid flows with shocks.

Spherical porous particle drying using BEM approach

The paper covers a porous particle drying problem, which can be divided into two or three stages. The first stage is drying of surface moisture, the second stage is drying inside the particle, and the third, one which is relevant only to hygroscopic material, represents the change of particle moisture due to the change of environmental moisture. The second stage is the most relevant for the porous materials, and is, therefore, covered in more detail in this paper, with the main focus on the heat transfer inside the particle which affects the drying kinetics. The heat transfer problem inside the spherical particle has been solved using BEM, by transforming a 3D problem into a quasi 1D case by assuming uniform boundary conditions all around the particle, resulting in the solution depending only on the radial direction. The solution of the heat transfer needs to be calculated accurately as it directly affects the evaporation rate of the liquid on the interface between the dry crust and the wet core, which dictates the drying speed and affects the drying time. An in-depth analysis of space and time discretisation was performed on a typical spray drying example, where it is shown that a choice of a correct time step is cruicial for achieving good computational accuracy of the drying kinetics. The proposed numerical approach has also been tested on various drying conditions, with changing the particle size and the temperature of the drying gas, which have a largest effect on the drying kinetics. Finally, an analysis of the computed drying times is made as this is the most important parameter from the practical point of view, especially when designing the drying chambers.

Thermalization of dipole oscillations in confined systems by rare collisions

We study the relaxation of the center-of-mass, or dipole oscillations in the system of interacting fermions confined spatially. With the confinement frequency $\omega_{\perp}$ fixed the particles were considered to freely move along one (quasi-1D) or two (quasi-2D) spatial dimensions. We have focused on the regime of rare collisions, such that the inelastic collision rate, $1/\tau_{in} \ll \omega_{\perp}$. The dipole oscillations relaxation rate, $1/\tau_{\perp}$ is obtained at three different levels: by direct perturbation theory, solving the integral Bethe-Salpeter equation and applying the memory function formalism. As long as anharmonicity is weak, $1/\tau_{\perp} \ll 1/ \tau_{in}$ the three methods are shown to give identical results. In quasi-2D case $1/\tau_{\perp} \neq 0$ at zero temperature. In quasi-1D system $1/\tau_{\perp} \propto T^3$ if the Fermi energy, $E_F$ lies below the critical value, $E_F < 3 \omega_{\perp}/4$. Otherwise, unless the system is close to integrability, the rate $1/\tau_{\perp}$ has the temperature dependence similar to that in quasi-2D. In all cases the relaxation results from the excitation of particle-hole pairs propagating along unconfined directions resulting in the relationship $1/\tau_{\perp} \propto 1/\tau_{in}$, with the inelastic rate $1/\tau_{in} \neq 0$ as the phase-space opens up at finite energy of excitation, $\hbar \omega_{\perp}$. While $1/\tau_{\perp} \propto \tau_{in}$ in the hydrodynamic regime, $\omega_{\perp} \ll 1/\tau_{in}$, in the regime of rare collisions, $\omega_{\perp} \gg 1/\tau_{in}$, we obtain the opposite trend $1/\tau_{\perp} \propto 1/\tau_{in}$.

Thomas-Fermi and Thomas-Fermi-Dirac models in two-dimension – Effect of strong quantizing magnetic field

Using Thomas-Fermi (TF) and Thomas-Fermi-Dirac (TFD) models, we have investigated the properties of electron gas inside two-dimensional (2D) Wigner-Seitz (WS) cells in presence of a strong orthogonal quantizing magnetic field. The electron-electron Coulomb exchange interaction in quasi-2D case is obtained. The exact form of exchange term in 2D is derived making the width of the system tending to zero. Further, using the exchange term, the Thomas-Fermi-Dirac equation in 2D is established. It has been observed that only the ionized WS cell can have finite radius in the Thomas-Fermi model, even in presence of a strong quantizing magnetic field. On the other hand, in the Thomas-Fermi-Dirac model a neutral WS cell can have finite radius.

Fermi gases in the two-dimensional to quasi-two-dimensional crossover

We smoothly tune the dimensionality of pancake-shaped $^{6}\mathrm{Li}$ Fermi gas clouds from quasi-two-dimensional to two-dimensional (2D) to measure radio-frequency spectra and cloud profiles in both regimes. In the quasi-2D case, where ${E}_{F}/h{\ensuremath{\nu}}_{z}\ensuremath{\simeq}1$, with ${E}_{F}$ the Fermi energy and $h{\ensuremath{\nu}}_{z}$ the harmonic oscillator energy in the tightly confined direction, we confirm that the radio-frequency spectra strongly disagree with 2D mean-field theory. Then we tune to the 2D regime, ${E}_{F}&lt;&lt;h{\ensuremath{\nu}}_{z}$, where the measured radio-frequency spectra are in very good agreement with 2D mean-field theory. Nevertheless, the measured cloud profiles strongly disagree, confirming predictions that a beyond mean-field approach is required throughout the 2D to quasi-2D crossover.

Universality class of the two-dimensional polymer collapse transition.

The nature of the θ point for a polymer in two dimensions has long been debated, with a variety of candidates put forward for the critical exponents. This includes those derived by Duplantier and Saleur for an exactly solvable model. We use a representation of the problem via the CP^{N-1}σ model in the limit N→1 to determine the stability of this critical point. First we prove that the Duplantier-Saleur (DS) critical exponents are robust, so long as the polymer does not cross itself: They can arise in a generic lattice model and do not require fine-tuning. This resolves a longstanding theoretical question. We also address an apparent paradox: Two different lattice models, apparently both in the DS universality class, show different numbers of relevant perturbations, apparently leading to contradictory conclusions about the stability of the DS exponents. We explain this in terms of subtle differences between the two models, one of which is fine-tuned (and not strictly in the DS universality class). Next we allow the polymer to cross itself, as appropriate, e.g., to the quasi-two-dimensional case. This introduces an additional independent relevant perturbation, so we do not expect the DS exponents to apply. The exponents in the case with crossings will be those of the generic tricritical O(n) model at n=0 and different from the case without crossings. We also discuss interesting features of the operator content of the CP^{N-1} model. Simple geometrical arguments show that two operators in this field theory, with very different symmetry properties, have the same scaling dimension for any value of N (or, equivalently, any value of the loop fugacity). Also we argue that for any value of N the CP^{N-1} model has a marginal odd-parity operator that is related to the winding angle.

Charge density waves and phonon-electron coupling inZrTe3

Charge-density-wave (CDW) order has long been interpreted as arising from a Fermi-surface instability in the parent metallic phase. While phonon-electron coupling has been suggested to influence the formation of CDW order in quasi-two-dimensional (quasi-2D) systems, the presumed dominant importance of Fermi-surface nesting remains largely unquestioned in quasi-1D systems. Here we show that phonon-electron coupling is also important for the CDW formation in a model quasi-1D system ZrTe$_3$. Our joint experimental and computational study reveals that particular lattice vibrational patterns possess exceedingly strong coupling to the conduction electrons, and are directly linked to the lattice distortions associated with the CDW order. The dependence of the coupling matrix elements on electron momentum further dictates the opening of (partial) electronic gaps in the CDW phase. Since lattice distortions and electronic gaps are the defining signatures of CDW order, our result demonstrates that the conventional wisdom based on Fermi-surface geometry needs to be substantially supplemented by phonon-electron coupling even in the simplest quasi-1D case. As prerequisites for the CDW formation, the highly anisotropic electronic structure and strong phonon-electron coupling in ZrTe$_3$ give rise to a distinct Raman scattering effect, namely, measured phonon linewidths depend on the direction of momentum transfer in the scattering process.

Behavior of entanglement entropy in interacting quasi-one-dimensional systems and its consequences for their efficient numerical study

The density matrix renormalization group (DMRG) method allows an efficient computation of the properties of interacting 1D quantum systems. Two-dimensional (2D) systems, capable of displaying much richer quantum behavior, generally lie beyond its reach except for very small system sizes. Many of the physical properties of 2D systems carry into the quasi-1D case, for which, unfortunately, the standard 2D DMRG algorithm fares little better. By finding the form of the entanglement entropy in quasi-1D systems, we directly identify the reason for this failure. Using this understanding, we explain why a modified algorithm, capable of cleverly exploiting this behavior of the entanglement entropy, can accurately reach much larger system sizes. We demonstrate the power of this method by accurately finding quantum critical points in frustration induced magnetic transitions, which remain inaccessible using the standard DMRG or the Monte Carlo methods.

MoS2 nanostructures: Semiconductors with metallic edges

We present theoretical calculations based on Density Functional Theory for MoS2 nanoribbons. By studying nanoribbons with various widths, the energy for the (10) edge in single-layer MoS2 is obtained. We calculate their electronic structure and use linear-response theory to obtain the dielectric function and other optoelectronic properties such as static relative permittivity, refractive index and reflectivity. We compare the dielectric properties of bulk (3D), single-layer (2D) and ribbons (quasi-1D) of MoS2 to find, among other trends, that the dielectric constant is almost halved when the dimensionality is reduced by one. The static relative permittivity drops from 7.1 in 3D to 3.7 in 2D, which is consistent with experimental data, and down to 2.4 in the quasi-1D case. We discuss the edge conducting states in ribbon MoS2, an otherwise semiconducting material, as well as its dielectric properties as a function of the ribbon width.

Three-dimensional solitons in cross-combined linear and nonlinear optical lattices

The existence and stability of three-dimensional (3D) solitons, in cross-combined linear and nonlinear optical lattices, are investigated. In particular, with a starting optical lattice (OL) configuration such that it is linear in the x-direction and nonlinear in the y-direction, we consider the z-direction either unconstrained (quasi-2D OL case) or with another linear OL (full 3D case). We perform this study both analytically and numerically: analytically by a variational approach based on a Gaussian ansatz for the soliton wavefunction and numerically by relaxation methods and direct integrations of the corresponding Gross–Pitaevskii equation. We conclude that, while 3D solitons in the quasi-2D OL case are always unstable, the addition of another linear OL in the z-direction allows us to stabilize 3D solitons both for attractive and repulsive mean interactions. From our results, we suggest the possible use of spatial modulations of the nonlinearity in one of the directions as a tool for the management of stable 3D solitons.

Three-dimensional simulation of ionized metal plasma vapor deposition

A physically based model for full three-dimensional (3D) feature-scale simulation of ionized metal plasma (IMP) deposition has been implemented. It assumes a dynamical equilibrium of particle fluxes where the sum of delivering (positive) and consuming (negative) fluxes is zero. This allows to calculate the deposition rates of metal at the different positions on the feature surface. The simulator is validated by considering the quasi-2D case of a long trench and comparing the results with data from the literature. Vias are investigated by comparing the profiles to the results for long trenches. Finally, the application to 3D structures without any symmetry is shown.

Anisotropic superconductivity in systems with coexisting electrons and local pairs

The properties of systems consisting of a mixture of local electron pairs and itinerant fermions coupled via charge exchange mechanism, which mutually induces superconductivity in both subsystems, are analyzed. The case of anisotropic pairing of various symmetries is discussed in detail for a two-dimensional (2D) square lattice within the BCS-mean field approximation and the Kosterlitz-Thouless theory. The effects of electronic band dispersion as well as a quasi-2D case with interplanar hopping of electrons are also briefly analyzed. We determined the phase diagrams and superconducting characteristics as a function of the position of the local pair (LP) level and the total electron concentration. The possible types of crossovers from BCS like behavior to that of LP's are examined. In addition, the Uemura plots are obtained for extended s and ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ pairing symmetries. Some of our findings are discussed in connection with a two-component scenario of preformed pairs and unpaired electrons for the cuprate high-temperature superconductors.

Addenda to “Theoretical Description of Resistive Behavior near a Quantum Vortex–Glass Transition”

In our recent paper, the description of microscopic aspects was based on the ordinary dirty limit for twodimensional (2D) case, and the interplay between the electron–electron repulsive interaction and the microscopic disorder, particularly important in explaining the thickness (d)-dependence of observations in real quasi 2D disordered thin films, could be incorporated perturbatively. However, the extension in the ordinary dirty limit (with no repulsive interaction included) to the quasi 2D case was not described adequately there. Below, this extension will be explained because it enables one to quantitatively examine resistance data, suggesting a field-tuned superconductor-insulator quantum transition, in systems showing a superconductivity with conventional s-wave pairing. Main results in ref. 1 and the companion paper, ref. 2, remain unchanged through the modifications shown below. The quasi 2D case corresponds to assuming 3D-like behaviors on a single quasiparticle level while the bosonic modes, such as the pair-field fluctuation and diffusion propagators carrying low wavevectors, to be 2D-like. Among the ‘‘GL parameters’’, ð0Þ, , U4, and Up defined in refs. 1 and 2, only U4 and Up in O(ð 1Þ) are modified through this extension to quasi 2D case. First, ðNð0ÞÞ 1 appearing in them is replaced by the corresponding 3D expression =ðkFNð0ÞÞ. Next, the r.h.s. of the replicated action (2.3) is multiplied by an overall factor d. After the scale transformation, d ð Þ 0 ! ð Þ 0 , both U4 and Up are divided by d. Further, since the dominant diagrams contributing to Up commonly include, even without the repulsive interaction, additional diffusion propagators carrying a low wavenumber q, an extra factor d 1 appears in Up through the quasi 2D q-integral. By combining them altogether, U4 of eq. (2.28) and Up of eq. (2.33) are replaced in quasi 2D case of O(ð 1Þ) by U4=ðkFdÞ and Up=ðkFdÞ, respectively. Consequently, the key result eq. (2.34) [as well as eq. (3.13) in ref. 2], leading to the statement that GvgðB 1⁄4 B vg;TÞ in O(ð 1Þ ) is a universal constant value, remains unchanged. Further, the parameter ðEF Þ 1 appearing in the text and corresponding to U4=ðr B Þ or U p =rB is replaced by Rr=ð3RQÞ, and hence, eq. (3.5) can be rewritten as GðRÞ s ðT 1⁄4 0Þ 1⁄4 R 1 r j ð0Þj 1⁄4 Gnj ð0Þj. Therefore, by also taking account of the identification 1 1⁄4 Rr=ð8 RQÞ of the repulsive interaction constant, a material dependence of dirty s-wave thin films in T < T cr seems to be largely due to a difference in the Rr=RQ-value. In addition, there are a couple of minor errors in the text of ref. 1. First, the 2 in the expression defining x4 just below eq. (2.9) is a misprint and should be replaced by . Second, vg in the paragraph (starting with ‘‘On the other hand, ’’) prior to eq. (3.2) should be replaced by vg because the classical (or, thermal) VG fluctuation was assumed there to be in the Gaussian region. In this connection, the expression Uðx ! 1Þ ! ðxþ 1Þ z in the text below eq. (3.3) should be also replaced by Uðx ! 1Þ ! ðxþ 1Þ , while other z ’s appearing in the same paragraph remain unchanged. Finally, a minor error is also present in ! 1⁄4 0 terms of eq. (3.1) of ref. 1, and the correct expression replacing eq. (3.1) is

Heterogeneous dynamics at the glass transition in van der Waals liquids, in the bulk and in thin films

It has been shown over the last few years that the dynamics close to the glass transition is strongly heterogeneous, both by measuring the diffusion coefficient of tagged particles or by NMR studies. Recent experiments have also demonstrated that the glass transition temperature of thin polymer films can be shifted as compared to the same polymer in the bulk. We propose here first a thermodynamical model for van der Waals liquids, which accounts for experimental results regarding the bulk modulus of polymer melts and the evolution of the density with temperature. This model allows us to describe the density fluctuations in such van der Waals liquids. Then, by considering the thermally induced density fluctuations in the bulk, we propose that the 3D glass transition is controlled by the percolation of small domains of slow dynamics, which allows to explain the heterogeneous dynamics close to T g. We show then that these domains percolate at a lower temperature in the quasi-2D case of thin suspended polymer films and we calculate the corresponding glass transition temperature reduction, in quantitative agreement with experimental results of Jones and co-workers. In the case of strongly adsorbed films, we show that the strong adsorption amounts to enhance the slow domains percolation. This effect leads to 1) a broadening of the glass transition and 2) an increase of T g in quantitative agreement with experimental results. For both strongly and weakly adsorbed films, the shift in T g is given by a power law, the exponent being the inverse of that of the correlation length of 3D percolation.

Path-integral approach to the electron density of states at the interface of a single modulation-doped heterojunction

The electronic density of states (DOS) at the interface of a single modulation-doped heterojunction is calculated both in the fluctuation band tail and in the semiclassical limit using the path-integral method. Due to charge density inhomogeneities in the heavily doped barrier region, random potential fluctuations are generated in whose minima carriers are localized, resulting in a band-tail density spectrum. The screening of the long-range potential fluctuations, which are important for the problem considered, is accounted for by using the two-dimensional Thomas-Fermi model. The statistical properties of the random impurity charge distribution are taken into account using the binary correlation function of the random potential for the specific geometry of the problem in two limiting cases of the general correlation function. Analytical expressions for the dimensionless functions of the exponential and preexponential of the band-tail DOS, as a function of the energy, are obtained in the weak and strong screening limit for the quasi-2D case under consideration and are compared to the respective functions for the general d-dimensional case. The dependence of the band-tail DOS behavior on the relevant parameters of the system, namely, spacer layer thickness, doping layer thickness, 2D EG thickness, and 3D-impurity concentration is studied numerically for ${\mathrm{Al}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{A}\mathrm{s}\ensuremath{-}\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}$ modulation doped heterostructures and numerical results for the 2D-DOS are presented. The band-tail results for the 2D-DOS are compared with the Kane approximation derived by taking the limit $\stackrel{\ensuremath{\rightarrow}}{t}0$ in order to determine $\ensuremath{\rho}(E)$ in the whole energy range. We compare the results from the two computed cases of the path-integral expression for the DOS corresponding to the two limits of the general correlation function with each other. On the other hand, we compare the semiclassical limit and the white Gaussian noise limit of the above results with many other theoretical methods such as the generalized semiclassical method, multiple scattering method and the simulations resulting from the tight-binding model.

Localization in an imaginary vector potential

Eigenfunctions of 1d disordered Hamiltonian with constant imaginary vector potential are investigated. Even within the domain of complex eigenvalues the wave functions are shown to be strongly localized. However, this localization is of a very unusual kind. The logarithm of the wave function at different coordinates $x$ fluctuates strongly (just like the position of Brownian particle fluctuates in time). After approaching its maximal value the logarithm decreases like the square root of the distance $\bar{(\ln|\psi_{max}/\psi|)^2} \sim |x-x_0|$. The extension of the model to the quasi-1d case is also considered.

Hartree-Fock energy bands of YBa 2Cu 3O 7

Hartree-Fock energy bands were computed for YBa 2Cu 3O 7. For that purpose charged quasi two-dimensional and quasi one-dimensional elementary cells, modeling plane- and chain-like structural units of YBa 2Cu 3O 7, are considered. To ensure charge neutrality and to include long-range Coulomb interactions of these low-dimensional models with a three-dimensional neighborhood, partially self-consistent (quasi 2D case) and arrays of fixed point charges (quasi 1D case), respectively, were introduced. We find that the overall binding characteristics as well as the Fermi surface of YBa 2Cu 3O 7 can be roughly understood on the basis of our Hartree-Fock crystal orbital calculations, while other features (for instance density of state curves) appear to be less reasonable because of the lack of correlation.

Excitonic gain and laser emission in ZnSe-based quantum wells.

We show spectroscopically that the origin of optical gain and laser emission in (Zn,Cd)Se/ZnSe quantum wells at blue-green wavelengths is of excitonic nature. This circumstance derives from the large enhancement in the exciton binding and its oscillator strength which occurs in the quasi-2D case, so that an exciton gas is stable against ionization by optical phonons up to room temperature and that gain in the context of partial phase-space filling can develop at pair densities below the onset to an electron-hole plasma.

Specific heat of the local-pair superconductor

The temperature dependence of the specific heat C( T) of the local-pair (LP) superconductor is calculated in the mean-field-random-phase-approximation, using the mapping of the LP model at constant carrier density onto the anisotropic Heisenberg pseudo-spin system at constant magnetization. Both the 3D and quasi 2D (highly anisotropic) LP superconductors are investigated, both for a small carrier concentration n. Below T c, C( T) obeys power laws with different exponents in the 3D and quasi 2D cases, and rises steeply as T approaches T c. Above T c the model predicts a linear T-dependence for C( T) for the quasi 2D case in a substantial temperature range before C( T) reaches its classical value ≈ n. Such a contribution to C( T) above T c has been reported in some experiments. The model may also account for the small values for entropy at T c found in the experiments in the superconducting oxides. For temperatures much bigger than the LP-band width, we find C( T)∼ T −2 in both cases, in agreement with ref. [5] (R. Micnas et al., Rev. Mod. Phys. 62 (1990) 113).