Twisting the stacking of layered materials leads to rich new physics. A three-dimensional topological insulator film hosts two-dimensional gapless Dirac electrons on top and bottom surfaces, which, when the film is be...Twisting the stacking of layered materials leads to rich new physics. A three-dimensional topological insulator film hosts two-dimensional gapless Dirac electrons on top and bottom surfaces, which, when the film is below some critical thickness, will hybridize and open a gap in the surface state structure. The hybridization gap can be tuned by various parameters such as film thickness and inversion symmetry, according to the literature. The three-dimensional strong topological insulator Bi(Sb)Se(Te) family has layered structures composed of quintuple layers(QLs) stacked together by van der Waals interaction. Here we successfully grow twistedly stacked Sb_2Te_3 QLs and investigate the effect of twist angels on the hybridization gaps below the thickness limit. It is found that the hybridization gap can be tuned for films of three QLs, which may lead to quantum spin Hall states.Signatures of gap-closing are found in 3-QL films. The successful in situ application of this approach opens a new route to search for exotic physics in topological insulators.展开更多
Let SFd and ∏φ,n,d ={∑j^n=1bjφ(wj.x+θj):bj,θj∈R,wj∈R^d} be the set of periodic and Lebesgue's square-integrable functions and the set of feedforward neural network (FNN) functions, respectively. Denote ...Let SFd and ∏φ,n,d ={∑j^n=1bjφ(wj.x+θj):bj,θj∈R,wj∈R^d} be the set of periodic and Lebesgue's square-integrable functions and the set of feedforward neural network (FNN) functions, respectively. Denote by dist (SFd , ∏φ,n,d) the deviation of the set SFd from the set ∏φ,n,d. A main purpose of this paper is to estimate the deviation. In particular, based on the Fourier transforms and the theory of approximation, a lower estimation for dist (SFd and ∏φ,n,d) is proved. That is, dist(SFd and ∏φ,n,d) ≥C/(nlog2n)1/2. The obtained estimation depends only on the number of neuron in the hidden layer, and is independent of the approximated target functions and dimensional number of input. This estimation also reveals the relationship between the approximation rate of FNNs and the topology structure of hidden layer.展开更多
基金Supported by the National Natural Science Foundation of China (Grant Nos.61804056 and 92065102)。
文摘Twisting the stacking of layered materials leads to rich new physics. A three-dimensional topological insulator film hosts two-dimensional gapless Dirac electrons on top and bottom surfaces, which, when the film is below some critical thickness, will hybridize and open a gap in the surface state structure. The hybridization gap can be tuned by various parameters such as film thickness and inversion symmetry, according to the literature. The three-dimensional strong topological insulator Bi(Sb)Se(Te) family has layered structures composed of quintuple layers(QLs) stacked together by van der Waals interaction. Here we successfully grow twistedly stacked Sb_2Te_3 QLs and investigate the effect of twist angels on the hybridization gaps below the thickness limit. It is found that the hybridization gap can be tuned for films of three QLs, which may lead to quantum spin Hall states.Signatures of gap-closing are found in 3-QL films. The successful in situ application of this approach opens a new route to search for exotic physics in topological insulators.
基金Supported by the National Natural Science Foundation of China (Grant No. 60873206)the National Basic Research Program of China(Grant No. 2007CB311002)
文摘Let SFd and ∏φ,n,d ={∑j^n=1bjφ(wj.x+θj):bj,θj∈R,wj∈R^d} be the set of periodic and Lebesgue's square-integrable functions and the set of feedforward neural network (FNN) functions, respectively. Denote by dist (SFd , ∏φ,n,d) the deviation of the set SFd from the set ∏φ,n,d. A main purpose of this paper is to estimate the deviation. In particular, based on the Fourier transforms and the theory of approximation, a lower estimation for dist (SFd and ∏φ,n,d) is proved. That is, dist(SFd and ∏φ,n,d) ≥C/(nlog2n)1/2. The obtained estimation depends only on the number of neuron in the hidden layer, and is independent of the approximated target functions and dimensional number of input. This estimation also reveals the relationship between the approximation rate of FNNs and the topology structure of hidden layer.