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We study the regularity of the solution to an obstacle problem for a class of integro–differential operators. The differential part is a second order elliptic operator, whereas the nonlocal part is given by the integral fractional Laplacian. The obtained smoothness is then used to design and analyze a finite element scheme.

We propose an adaptive finite element algorithm to approximate solutions of elliptic problems whose forcing data is locally defined and is approximated by regularization (or mollification). We show that the energy error decay is quasi-optimal in two dimensional space and sub-optimal in three dimensional space. Numerical simulations are provided to confirm our findings., 28 pages, 6 Figures

Recently, the discussion about new candidate materials that can replace traditional Cu for future interconnects in integrated circuit (IC) fabrication attracts growing interests. In this paper, we have systematically investigated the breakdown current characteristic of a novel transition metal trichalcogenide (TMTCs) $\mathbf{ZrTe_{3}}$ by the finite element simulation. First, an electro-thermal coupling model was established, which the Joule heat of the $\mathbf{ZrTe_{3}}$ nanoribbon acts as the heat source, and both the solid heat transfer and the natural convection heat transfer are considered for the finite element simulation. Then, using the established model and experimental data, the breakdown temperature of the $\mathbf{ZrTe_{3}}$ nanoribbons in the air environment has been determined. Furthermore, the distribution of both temperature and current density of $\mathbf{ZrTe_{3}}$ nanoribbons during the breakdown process was investigated for failure analysis. Finally, according to the size-independent resistivity of TMTCs, the breakdown characteristic of the $\mathbf{ZrTe_{3}}$ with 10nm width was investigated by further finite element simulation. Its breakdown current density can reach 83.8 $\mathbf{MA}/\mathbf{cm}^{2}$ , which demonstrates the high potential of $\mathbf{ZrTe}_{3}$ nanoribbon as a new type of interconnects material of replacing copper interconnects.

We propose a new nonconforming finite element algorithm to approximate the solution to the elliptic problem involving the fractional Laplacian. We first derive an integral representation of the bilinear form corresponding to the variational problem. The numerical approximation of the action of the corresponding stiffness matrix consists of three steps: (1) apply a sinc quadrature scheme to approximate the integral representation by a finite sum where each term involves the solution of an elliptic partial differential equation defined on the entire space, (2) truncate each elliptic problem to a bounded domain, (3) use the finite element method for the space approximation on each truncated domain. The consistency error analysis for the three steps is discussed together with the numerical implementation of the entire algorithm. The results of computations are given illustrating the error behavior in terms of the mesh size of the physical domain, the domain truncation parameter and the quadrature spacing parameter.

We demonstrated a feasible strategy to fabricate MoTe2/Ge heterojunction by direct growth of Ge flake on a MoTe2 film substrate with a two-step chemical vapor deposition method. A thin transition layer (∼4 nm) mainly composed of polycrystalline germanium at the MoTe2/Ge interface was verified during the Ge flake growth. The MoTe2/Ge heterojunction-based photodetector exhibits both the response speed with a rise/fall time of 7/4 μs and the photoresponsivity and detectivity with 4.87 A W−1 and 5.02 × 1011 Jones under zero bias in the near-infrared regime, respectively. The characteristics of device performance imply its practical applicability as building block for potential near-infrared integrated photonics.

We demonstrated that Schottky barrier height (SBH) at the metal/CVD-grown MoTe2 interface can be significantly reduced with tunnel contact by inserting a thin Al2O3 layer regardless of the metal work function. The existence of strong Fermi level pinning (FLP) at the metal/MoTe2 interface was verified, while depinning cannot be achieved with Al2O3 insertion. Thus, the fixed charges inside the Al2O3 were proposed to be responsible for the effective SBH reduction in virtue of the eliminated SBH reduction after the post-annealing treatment. This work provides a feasible way to solve the contact issue and favors for the fabrication of high performance MoTe2-based electronic devices.

In this paper, we focus on improving MoTe 2 nanoribbon MOSFETs device performance, with solution-processed HfAlOx high-k dielectric film compared with SiO 2 . First, 2H-MoTe 2 nanoribbons with high purity and quality were synthesized by CVD process. Then, solution-processed synthesis of HfAlOx thin film was systematically investigated. HfAlOx thin film obtained with 0.3M precursor concentration, annealing at 400°C in mixed gas of N 2 (95%) + H2(5%) showed low leakage current density and high k value. Finally, improved device performance of MoTe 2 nanoribbon transistors with mobility ~9.35 cm2V−1s−1, I on /I off ratio ~1.85×l05, and the threshold voltage ~ -3.52V were demonstrated.

Approximations of the Dirac delta distribution are commonly used to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution. In this work, we show a priori rates of convergence of this approximation process in standard Sobolev norms, with minimal regularity assumptions on the approximation of the Dirac delta distribution. The application of these estimates to the numerical solution of elliptic problems with singularly supported forcing terms allows us to provide sharp $H^1$ and $L^2$ error estimates for the corresponding regularized problem. As an application, we show how finite element approximations of a regularized immersed interface method result in the same rates of convergence of its non-regularized counterpart, provided that the support of the Dirac delta approximation is set to a multiple of the mesh size, at a fraction of the implementation complexity. Numerical experiments are provided to support our theories., Comment: 24 pages, 7 figures

We report the fabrication of high performance MoTe2/Si heterojunction photodiodes by direct growth of MoTe2 patterns on a commercial Si substrate by a feasible chemical vapor deposition method. The devices exhibit an ultrafast response speed with a rise/fall time of 5/8 μs, a broadband (400–1550 nm) photoresponse, a high on/off ratio of ∼104, and self-powered photo-detection with a zero bias responsivity of 0.26 A W−1 and a detectivity of 2 × 1013 Jones at 700 nm wavelength. The devices further show high stability in air for one month. This investigation offers the feasibility to fabricate high performance MoTe2/Si photodiodes for future vital optoelectronic applications.

We study the numerical approximation of a time dependent equation involving fractional powers of an elliptic operator $L$ defined to be the unbounded operator associated with a Hermitian, coercive and bounded sesquilinear form on $H^1_0(\Omega)$. The time dependent solution $u(x,t)$ is represented as a Dunford Taylor integral along a contour in the complex plane. The contour integrals are approximated using sinc quadratures. In the case of homogeneous right-hand-sides and initial value $v$, the approximation results in a linear combination of functions $(z_qI-L)^{-1}v\in H^1_0(\Omega)$ for a finite number of quadrature points $z_q$ lying along the contour. In turn, these quantities are approximated using complex valued continuous piecewise linear finite elements. Our main result provides $L^2(\Omega)$ error estimates between the solution $u(\cdot,t)$ and its final approximation. Numerical results illustrating the behavior of the algorithms are provided., Comment: 20 pages, 4 figures

We consider the finite element approximation of fractional powers of regularly accretive operators via the Dunford-Taylor integral approach. We use a sinc quadrature scheme to approximate the Balakrishnan representation of the negative powers of the operator as well as its finite element approximation. We improve the exponentially convergent error estimates from [A. Bonito, J. E. Pasciak, IMA J. Numer. Anal. (2016) 00, 1-29] by reducing the regularity required on the data. Numerical experiments illustrating the new theory are provided., 3 figures

In this paper, we develop a numerical scheme for the space-time fractional parabolic equation, i.e., an equation involving a fractional time derivative and a fractional spatial operator. Both the initial value problem and the non-homogeneous forcing problem (with zero initial data) are considered. The solution operator $E(t)$ for the initial value problem can be written as a Dunford-Taylor integral involving the Mittag-Leffler function $e_{\alpha,1}$ and the resolvent of the underlying (non-fractional) spatial operator over an appropriate integration path in the complex plane. Here $\alpha$ denotes the order of the fractional time derivative. The solution for the non-homogeneous problem can be written as a convolution involving an operator $W(t)$ and the forcing function $F(t)$. We develop and analyze semi-discrete methods based on finite element approximation to the underlying (non-fractional) spatial operator in terms of analogous Dunford-Taylor integrals applied to the discrete operator. The space error is of optimal order up to a logarithm of $1/h$. The fully discrete method for the initial value problem is developed from the semi-discrete approximation by applying an exponentially convergent sinc quadrature technique to approximate the Dunford-Taylor integral of the discrete operator and is free of any time stepping. To approximate the convolution appearing in the semi-discrete approximation to the non-homogeneous problem, we apply a pseudo midpoint quadrature. This involves the average of $W_h(s)$, (the semi-discrete approximation to $W(s)$) over the quadrature interval. This average can also be written as a Dunford-Taylor integral. We first analyze the error between this quadrature and the semi-discrete approximation. To develop a fully discrete method, we then introduce sinc quadrature approximations to the Dunford-Taylor integrals for computing the averages., Comment: 29 pages, 5 figures

Counter-current chromatography (CCC) was applied for preparative enantioseparation of three β-blocker drugs, including propranolol, pindolol and alprenolol. The two-phase solvent system was composed of chloroform-0.05 mol L(-1) acetate buffer containing 0.10 mol L(-1) boric acid (1:1, v/v), in which 0.10 mol L(-1) di-n-hexyl L-tartrate was added in the organic phase as chiral selector. Influence factors in the enantioseparation of propranolol were investigated. The chromatographic retention mechanism based on borate coordination complex was proposed. 116 mg of racemic propranolol was completely enantioseparated using conventional high speed CCC in a single run, yielding 48 mg of (+)-propranolol with HPLC purity of 98.9% and 47 mg of (-)-propranolol with HPLC purity of 96.3%. Recovery for propranolol enantiomers from CCC fractions was in the range of 75-82%. pH-zone-refining CCC was also successfully applied in enantioseparation of propanolol and it was found that 356 mg of racemic propranolol could be completely enantioseparated. 145 mg of (+)-enantiomer with HPLC purity of 95.6% and 148 mg of (-)-enantiomer with HPLC purity of 98.2% were recovered from pH-zone-refining mode. Separation mechanism about chiral separation by pH-zone-refining CCC was discussed.