An efficient finite-difference scheme for computation of electron states in free-standing and core-shell quantum wires

The electron states in axially symmetric quantum wires are computed by means of the effective-mass Schroedinger equation, which is written in cylindrical coordinates phi, rho, and z. We show that a direct discretization of the Schroedinger equation by central finite differences leads to a non-symmetric Hamiltonian matrix. Because diagonalization of such matrices is more complex it is advantageous to transform it in a symmetric form. This can be done by the Liouville-like transformation proposed by Rizea et al. (Comp. Phys. Comm. 179 (2008) 466-478), which replaces the wave function psi(rho) with the function F(rho)=psi(rho)sqrt(rho) and transforms the Hamiltonian accordingly. Even though a symmetric Hamiltonian matrix is produced by this procedure, the computed wave functions are found to be inaccurate near the origin, and the accuracy of the energy levels is not very high. In order to improve on this, we devised a finite-difference scheme which discretizes the Schroedinger equation in the first step, and then applies the Liouville-like transformation to the difference equation. Such a procedure gives a symmetric Hamiltonian matrix, resulting in an accuracy comparable to the one obtained with the finite element method. The superior efficiency of the new finite-difference (FDM) scheme is demonstrated for a few rho-dependent one-dimensional potentials which are usually employed to model the electron states in free-standing and core-shell quantum wires. The new scheme is compared with the other FDM schemes for solving the effective-mass Schroedinger equation, and is found to deliver energy levels with much smaller numerical error for all the analyzed potentials. Moreover, the PT symmetry is invoked to explain similarities and differences between the considered FDM schemes

On Improving Accuracy of Finite-Element Solutions of the Effective-Mass Schrodinger Equation for Interdiffused Quantum Wells and Quantum Wires

We use the Galerkin approach and the finite-element method to numerically solve the effective-mass Schrodinger equation. The accuracy of the solution is explored as it varies with the range of the numerical domain. The model potentials are those of interdiffused semiconductor quantum wells and axially symmetric quantum wires. Also, the model of a linear harmonic oscillator is considered for comparison reasons. It is demonstrated that the absolute error of the electron ground state energy level exhibits a minimum at a certain domain range, which is thus considered to be optimal. This range is found to depend on the number of mesh nodes N approximately as alpha(0) log(e)(alpha 1) (alpha N-2), where the values of the constants alpha(0), alpha(1), and alpha(2) are determined by fitting the numerical data. And the optimal range is found to be a weak function of the diffusion length. Moreover, it was demonstrated that a domain range adaptation to the optimal value leads to substantial improvement of accuracy of the solution of the Schrodinger equation

Dynamic fracture toughness of ultra‐high‐performance fiber‐reinforced concrete under impact tensile loading

The fracture toughness and fracture energy of ultra-high-performance fiber-reinforced concrete (UHPFRC) at both static and impact rates (43-92 s(-1)) were investigated using double-edge-notched tensile specimens. Two types of steel fiber, smooth and twisted fiber, were used in producing UHPFRC with different volume ratios of 0.5%, 1.0%, 1.5%, and 2%. The results indicated that UHPFRCs produced very high fracture resistance at impact rates, with first stress intensity factor (K-IC) up to 3.995 MPa root m, critical stress intensity factor (KIC*) up to 7.778 MPa root m, and fracture energy (G(F)) up to 86.867 KJ/m(2), which were 2.5, 5.0, and 16.9 times higher than those of ultra-high-performance concrete, respectively. The KIC* was clearly sensitive to the applied loading rate, whereas the K-IC and G(F) were not. Smooth fiber specimens exhibited not only higher KIC* and G(F) at impact rates but also higher dynamic increase factor than twisted fiber specimens. A minimum fiber volume content of 1% should be used in UHPFRC to provide a significant enhancement in crack resistance. The maximum value of UHPFRC crack velocity at impact rates was found to be 527 m/s by using a dynamic fracture mechanic model.11Nsciescopu