An unconditionally energy stable finite difference scheme for a stochastic Cahn-Hilliard equation

In this work, the MMC-TDGL equation, a stochastic Cahn-Hilliard equation is solved numerically by using the finite difference method in combination with a convex splitting technique of the energy functional. For the non-stochastic case, we develop an unconditionally energy stable difference scheme which is proved to be uniquely solvable. For the stochastic case, by adopting the same splitting of the energy functional, we construct a similar and uniquely solvable difference scheme with the discretized stochastic term. The resulted schemes are nonlinear and solved by Newton iteration. For the long time simulation, an adaptive time stepping strategy is developed based on both first- and second-order derivatives of the energy. Numerical experiments are carried out to verify the energy stability, the efficiency of the adaptive time stepping and the effect of the stochastic term.Comment: This paper has been accepted for publication in SCIENCE CHINA Mathematic

Microscopic theory of quantum anomalous Hall effect in graphene

We present a microscopic theory to give a physical picture of the formation of quantum anomalous Hall (QAH) effect in graphene due to a joint effect of Rashba spin-orbit coupling $\lambda_R$ and exchange field $M$. Based on a continuum model at valley $K$ or $K'$, we show that there exist two distinct physical origins of QAH effect at two different limits. For $M/\lambda_R\gg1$, the quantization of Hall conductance in the absence of Landau-level quantization can be regarded as a summation of the topological charges carried by Skyrmions from real spin textures and Merons from \emph{AB} sublattice pseudo-spin textures; while for $\lambda_R/M\gg1$, the four-band low-energy model Hamiltonian is reduced to a two-band extended Haldane's model, giving rise to a nonzero Chern number $\mathcal{C}=1$ at either $K$ or $K'$. In the presence of staggered \emph{AB} sublattice potential $U$, a topological phase transition occurs at $U=M$ from a QAH phase to a quantum valley-Hall phase. We further find that the band gap responses at $K$ and $K'$ are different when $\lambda_R$, $M$, and $U$ are simultaneously considered. We also show that the QAH phase is robust against weak intrinsic spin-orbit coupling $\lambda_{SO}$, and it transitions a trivial phase when $\lambda_{SO}>(\sqrt{M^2+\lambda^2_R}+M)/2$. Moreover, we use a tight-binding model to reproduce the ab-initio method obtained band structures through doping magnetic atoms on $3\times3$ and $4\times4$ supercells of graphene, and explain the physical mechanisms of opening a nontrivial bulk gap to realize the QAH effect in different supercells of graphene.Comment: 10pages, ten figure

Unbalanced edge modes and topological phase transition in gated trilayer graphene

Gapless edge modes hosted by chirally-stacked trilayer graphene display unique features when a bulk gap is opened by applying an interlayer potential difference. We show that trilayer graphene with half-integer valley Hall conductivity leads to unbalanced edge modes at opposite zigzag boundaries, resulting in a natural valley current polarizer. This unusual characteristic is preserved in the presence of Rashba spin-orbit coupling that turns a gated trilayer graphene into a ${Z}_2$ topological insulator with an odd number of helical edge mode pairs.Comment: 5 pages, 4 figure

Determining layer number of two dimensional flakes of transition-metal dichalcogenides by the Raman intensity from substrate

Transition-metal dichalcogenide (TMD) semiconductors have been widely studied due to their distinctive electronic and optical properties. The property of TMD flakes is a function of its thickness, or layer number (N). How to determine N of ultrathin TMDs materials is of primary importance for fundamental study and practical applications. Raman mode intensity from substrates has been used to identify N of intrinsic and defective multilayer graphenes up to N=100. However, such analysis is not applicable for ultrathin TMD flakes due to the lack of a unified complex refractive index ($\tilde{n}$) from monolayer to bulk TMDs. Here, we discuss the N identification of TMD flakes on the SiO$_2$/Si substrate by the intensity ratio between the Si peak from 100-nm (or 89-nm) SiO$_2$/Si substrates underneath TMD flakes and that from bare SiO$_2$/Si substrates. We assume the real part of $\tilde{n}$ of TMD flakes as that of monolayer TMD and treat the imaginary part of $\tilde{n}$ as a fitting parameter to fit the experimental intensity ratio. An empirical $\tilde{n}$, namely, $\tilde{n}_{eff}$, of ultrathin MoS$_{2}$, WS$_{2}$ and WSe$_{2}$ flakes from monolayer to multilayer is obtained for typical laser excitations (2.54 eV, 2.34 eV, or 2.09 eV). The fitted $\tilde{n}_{eff}$ of MoS$_{2}$ has been used to identify N of MoS$_{2}$ flakes deposited on 302-nm SiO$_2$/Si substrate, which agrees well with that determined from their shear and layer-breathing modes. This technique by measuring Raman intensity from the substrate can be extended to identify N of ultrathin 2D flakes with N-dependent $\tilde{n}$ . For the application purpose, the intensity ratio excited by specific laser excitations has been provided for MoS$_{2}$, WS$_{2}$ and WSe$_{2}$ flakes and multilayer graphene flakes deposited on Si substrates covered by 80-110 nm or 280-310 nm SiO$_2$ layer.Comment: 10 pages, 4 figures. Accepted by Nanotechnolog

An efficient projection-type method for monotone variational inequalities in Hilbert spaces

We consider the monotone variational inequality problem in a Hilbert space and describe a projection-type method with inertial terms under the following properties: (a) The method generates a strongly convergent iteration sequence; (b) The method requires, at each iteration, only one projection onto the feasible set and two evaluations of the operator; (c) The method is designed for variational inequality for which the underline operator is monotone and uniformly continuous; (d) The method includes an inertial term. The latter is also shown to speed up the convergence in our numerical results. A comparison with some related methods is given and indicates that the new method is promising

Topological phases in gated bilayer graphene: Effects of Rashba spin-orbit coupling and exchange field

We present a systematic study on the influence of Rashba spin-orbit coupling, interlayer potential difference and exchange field on the topological properties of bilayer graphene. In the presence of only Rashba spin-orbit coupling and interlayer potential difference, the band gap opening due to broken out-of-plane inversion symmetry offers new possibilities of realizing tunable topological phase transitions by varying an external gate voltage. We find a two-dimensional $Z_2$ topological insulator phase and a quantum valley Hall phase in $AB$-stacked bilayer graphene and obtain their effective low-energy Hamiltonians near the Dirac points. For $AA$ stacking, we do not find any topological insulator phase in the presence of large Rashba spin-orbit coupling. When the exchange field is also turned on, the bilayer system exhibits a rich variety of topological phases including a quantum anomalous Hall phase, and we obtain the phase diagram as a function of the Rashba spin-orbit coupling, interlayer potential difference, and exchange field.Comment: 15 pages, 17figures, and 1 tabl