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CN 11-5639/O4
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HighLights

SymTopo:An automatic tool for calculating topological properties of nonmagnetic crystalline materials Hot!

Yuqing He(贺雨晴), Yi Jiang(蒋毅), Tiantian Zhang(张田田), He Huang(黄荷), Chen Fang(方辰), Zhong Jin(金钟)
Chin. Phys. B, 2019, 28 (8): 087102 doi: 10.1088/1674-1056/28/8/087102
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Topological materials have novel properties both in their bulk and boundaries, thereby attracting a wide interest in the theoretical and experimental communities. The recent development of the topological quantum chemistry and symmetry-based indicator theory in this field has significantly simplified the procedure for determining the topological properties of nonmagnetic crystalline materials. Accordingly, a large number of new topological materials have been found by scanning large crystal databases. This study provides details on the algorithm used in the Catalogue of Topological Electronic Materials. Moreover, based on the algorithm, we develop an automatic package named SymTopo, which calculates the symmetry representations of any given nonmagnetic crystalline material and predicts its topological properties. This package may facilitate the discovery of more topological materials in the future.

First-principles study of the band gap tuning and doping control in CdSexTe1-x alloy for high efficiency solar cell Hot!

Jingxiu Yang(杨竞秀), Su-Huai Wei(魏苏淮)
Chin. Phys. B, 2019, 28 (8): 086106 doi: 10.1088/1674-1056/28/8/086106
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CdTe is one of the leading materials for low cost, high efficiency thin-film solar cells with a nearly ideal band gap of 1.48 eV. However, its solar to electricity power conversion efficiency (PCE) is hindered by the relatively low open circuit voltage (VOC) due to intrinsic defect related issues. Here, we propose that alloying CdTe with CdSe could possibly improve the solar cell performance by reducing the “ideal” band gap of CdTe to gain more short-circuit current from long-wavelength absorption without sacrificing much VOC. Using the hybrid functional calculation, we find that the minimum band gap of the CdTe1-xSex alloy can be reduced from 1.48 eV at x=0 to 1.39 eV at x=0.32, and most of the change come from the lowering of the conduction band minimum. We also show that the formation of the alloy can improve the p-type doping of CuCd impurity based on the reduced effective formation energy and nearly constant effective transition energy level, thus possibly enhance VOC, thus PCE.

Non-Stokes drag coefficient in single-particle electrophoresis:New insights on a classical problem Hot!

Mai-Jia Liao(廖麦嘉), Ming-Tzo Wei(魏名佐), Shi-Xin Xu(徐士鑫), H Daniel Ou-Yang(歐陽新喬), Ping Sheng(沈平)
Chin. Phys. B, 2019, 28 (8): 084701 doi: 10.1088/1674-1056/28/8/084701
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We measured the intrinsic electrophoretic drag coefficient of a single charged particle by optically trapping the particle and applying an AC electric field, and found it to be markedly different from that of the Stokes drag. The drag coefficient, along with the measured electrical force, yield a mobility-zeta potential relation that agrees with the literature. By using the measured mobility as input, numerical calculations based on the Poisson-Nernst-Planck equations, coupled to the Navier-Stokes equation, reveal an intriguing microscopic electroosmotic flow near the particle surface, with a well-defined transition between an inner flow field and an outer flow field in the vicinity of electric double layer's outer boundary. This distinctive interface delineates the surface that gives the correct drag coefficient and the effective electric charge. The consistency between experiments and theoretical predictions provides new insights into the classic electrophoresis problem, and can shed light on new applications of electrophoresis to investigate biological nanoparticles.

Evolution of incommensurate superstructure and electronic structure with Pb substitution in (Bi2-xPbx)Sr2CaCu2O8+δ superconductors Hot!

Jing Liu(刘静), Lin Zhao(赵林), Qiang Gao(高强), Ping Ai(艾平), Lu Zhang(张璐), Tao Xie(谢涛), Jian-Wei Huang(黄建伟), Ying Ding(丁颖), Cheng Hu(胡成), Hong-Tao Yan(闫洪涛), Chun-Yao Song(宋春尧), Yu Xu(徐煜), Cong Li(李聪), Yong-Qing Cai(蔡永青), Hong-Tao Rong(戎洪涛), Ding-Song Wu(吴定松), Guo-Dong Liu(刘国东), Qing-Yan Wang(王庆艳), Yuan Huang(黄元), Feng-Feng Zhang(张丰丰), Feng Yang(杨峰), Qin-Jun Peng(彭钦军), Shi-Liang Li(李世亮), Huai-Xin Yang(杨槐馨), Jian-Qi Li(李建奇), Zu-Yan Xu(许祖彦), Xing-Jiang Zhou(周兴江)
Chin. Phys. B, 2019, 28 (7): 077403 doi: 10.1088/1674-1056/28/7/077403
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High-quality Bi2-xPbxSr2CaCu2O8+δ (Bi2212) single crystals have been successfully grown by the traveling solvent floating zone technique with a wide range of Pb substitution (x=0-0.8). The samples are characterized by transmission electron microscope (TEM) and measured by high resolution laser-based angle-resolved photoemission spectroscopy (ARPES) with different photon energies. A systematic evolution of the electronic structure and superstructure with Pb substitution has been revealed for the first time. The superstructure shows a significant change with Pb substitution and the incommensurate modulation vector (Q) decreases with increasing Pb substitution. In the meantime, the superstructure intensity from ARPES measurements also decreases dramatically with increasing Pb concentration. The superstructure in Bi2212 can be effectively suppressed by Pb substitution and it nearly disappears with a Pb substitution of x=0.8. We also find that the superstructure bands in ARPES measurements depend sensitively on the photon energy of lasers used; they can become even stronger than the main band when using a laser photon energy of 10.897 eV. These results provide important information on the origin of the incommensurate superstructure and its control and suppression in bismuth-based high temperature superconductors.

A simple rule for finding Dirac cones in bilayered perovskites Hot!

Xuejiao Chen(陈雪娇), Lei Liu(刘雷), Dezhen Shen(申德振)
Chin. Phys. B, 2019, 28 (7): 077106 doi: 10.1088/1674-1056/28/7/077106
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A simple rule for finding Dirac cone electronic states in solids is proposed, which is neglecting those lattice atoms inert to particular electronic bands, and pursuing the two-dimensional (2D) graphene-like quasi-atom lattices with s- and p-bindings by considering the equivalent atom groups in the unit cell as quasi-atoms. Taking CsPbBr3 and Cs3Bi2Br9 bilayers as examples, we prove the effectiveness and generality of this rule with the density functional theory (DFT) calculations. We demonstrate that both bilayers have Dirac cones around the Fermi level and reveal that their corresponding Fermi velocities can reach as high as~0.2×106 m/s. This makes these new 2D layered materials very promising in making new ultra-fast ionic electronic devices.

Fabrication of large-scale graphene/2D-germanium heterostructure by intercalation Hot!

Hui Guo(郭辉), Xueyan Wang(王雪艳), De-Liang Bao(包德亮), Hong-Liang Lu(路红亮), Yu-Yang Zhang(张余洋), Geng Li(李更), Ye-Liang Wang(王业亮), Shi-Xuan Du(杜世萱), Hong-Jun Gao(高鸿钧)
Chin. Phys. B, 2019, 28 (7): 078103 doi: 10.1088/1674-1056/28/7/078103
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We report a large-scale, high-quality heterostructure composed of vertically-stacked graphene and two-dimensional (2D) germanium. The heterostructure is constructed by the intercalation-assisted technique. We first synthesize large-scale, single-crystalline graphene on Ir(111) surface and then intercalate germanium at the interface of graphene and Ir(111). The intercalated germanium forms a well-defined 2D layer with a 2×2 superstructure with respect to Ir(111). Theoretical calculations demonstrate that the 2D germanium has a double-layer structure. Raman characterizations show that the 2D germanium effectively weakens the interaction between graphene and Ir substrate, making graphene more like the intrinsic one. Further experiments of low-energy electron diffraction, scanning tunneling microscopy, and x-ray photoelectron spectroscopy (XPS) confirm the formation of large-scale and high-quality graphene/2D-germanium vertical heterostructure. The integration of graphene with a traditional 2D semiconductor provides a platform to explore new physical phenomena in the future.

Possible nodeless s±-wave superconductivity intwisted bilayer graphene Hot!

Zhe Liu(刘哲), Yu Li(李宇), Yi-Feng Yang(杨义峰)
Chin. Phys. B, 2019, 28 (7): 077103 doi: 10.1088/1674-1056/28/7/077103
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The recent discovery of superconductivity in the twisted bilayer graphene has stimulated numerous theoretical proposals concerning its exact gap symmetry. Among them, the d+ id or p+ ip-wave was believed to be the most plausible solution. Here, considering that the superconductivity emerges near a correlated insulating state and may be induced by antiferromagnetic spin fluctuations, we apply the strong-coupling Eliashberg theory with both inter- and intraband quantum critical pairing interactions and discuss the possible gap symmetry in an effective low-energy four-orbital model. Our calculations reveal a nodeless s±-wave as the most probable candidate for the superconducting gap symmetry in the experimentally relevant parameter range. This solution is distinctly different from previous theoretical proposals. It highlights the multi-gap nature of the superconductivity and puts the twisted bilayer graphene in the same class as the iron-pnictide, electron-doped cuprate, and some heavy fermion superconductors.

Global phase diagram of a spin-orbit-coupled Kondo lattice model on the honeycomb lattice Hot!

Xin Li(李欣), Rong Yu(俞榕), Qimiao Si
Chin. Phys. B, 2019, 28 (7): 077102 doi: 10.1088/1674-1056/28/7/077102
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Motivated by the growing interest in the novel quantum phases in materials with strong electron correlations and spin-orbit coupling, we study the interplay among the spin-orbit coupling, Kondo interaction, and magnetic frustration of a Kondo lattice model on a two-dimensional honeycomb lattice. We calculate the renormalized electronic structure and correlation functions at the saddle point based on a fermionic representation of the spin operators. We find a global phase diagram of the model at half-filling, which contains a variety of phases due to the competing interactions. In addition to a Kondo insulator, there is a topological insulator with valence bond solid correlations in the spin sector, and two antiferromagnetic phases. Due to the competition between the spin-orbit coupling and Kondo interaction, the direction of the magnetic moments in the antiferromagnetic phases can be either within or perpendicular to the lattice plane. The latter antiferromagnetic state is topologically nontrivial for moderate and strong spin-orbit couplings.

Hard carbons derived from pine nut shells as anode materials for Na-ion batteries Hot!

Hao Guo(郭浩), Kai Sun(孙凯), Yaxiang Lu(陆雅翔), Hongliang Wang(王洪亮), Xiaobai Ma(马小柏), Zhengyao Li(李正耀), Yong-Sheng Hu(胡勇胜), Dongfeng Chen(陈东风)
Chin. Phys. B, 2019, 28 (6): 068203 doi: 10.1088/1674-1056/28/6/068203
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Hard carbons as promising anode materials for Na-ion batteries (NIBs) have captured extensive attention because of their low operation voltage, easy synthesis process, and competitive specific capacity. However, there are still several disadvantages, such as high cost and low initial coulombic efficiency, which limit their large-scale commercial applications. Herein, pine nut shells (PNSs), a low-cost biomass waste, are used as precursors to prepare hard carbon materials. Via a series of washing and heat treatment procedures, a pine nut shell hard carbon (PNSHC)-1400 sample has been obtained and delivers a reversible capacity of around 300 mAh/g, a high initial coulombic efficiency of 84%, and good cycling performance. These excellent Na storage properties indicate that PNSHC is one of the most promising candidates of hard carbon anodes for NIBs.

Improved electrochemical performance of Li(Ni0.6Co0.2Mn0.2)O2 at high charging cut-off voltage with Li1.4Al0.4Ti1.6(PO4)3 surface coating Hot!

Yi Wang(王怡), Bo-Nan Liu(刘柏男), Ge Zhou(周格), Kai-Hui Nie(聂凯会), Jie-Nan Zhang(张杰男), Xi-Qian Yu(禹习谦), Hong Li(李泓)
Chin. Phys. B, 2019, 28 (6): 068202 doi: 10.1088/1674-1056/28/6/068202
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Li(Ni0.6Co0.2Mn0.2)O2 has been surface-modified by the lithium-ion conductor Li1.4Al0.4Ti1.6(PO4)3 via a facile mechanical fusion method. The annealing temperature during coating process shows a strong impact on the surface morphology and chemical composition of Li(Ni0.6Co0.2Mn0.2)O2. The 600-℃ annealed material exhibits the best cyclic stability at high charging cut-off voltage of 4.5 V (versus Li+/Li) with the capacity retention of 90.9% after 100 cycles, which is much higher than that of bare material (79%). Moreover, the rate capability and thermal stability are also improved by Li1.4Al0.4Ti1.6(PO4)3 coating. The enhanced performance can be attributed to the improved stability of interface between Li(Ni0.6Co0.2Mn0.2)O2 and electrolyte by Li1.4Al0.4Ti1.6(PO4)3 modification. The results of this work provide a possible method to design reliable cathode materials to achieve high energy density and long cycle life.

Direct deposition of graphene nanowalls on ceramic powders for the fabrication of a ceramic matrix composite Hot!

Hai-Tao Zhou(周海涛), Da-Bo Liu(刘大博), Fei Luo(罗飞), Ye Tian(田野), Dong-Sheng Chen(陈冬生), Bing-Wei Luo(罗炳威), Zhang Zhou(周璋), Cheng-Min Shen(申承民)
Chin. Phys. B, 2019, 28 (6): 068102 doi: 10.1088/1674-1056/28/6/068102
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Uniform mixing of ceramic powder and graphene is of great importance for producing ceramic matrix composite. In this study, graphene nanowalls (GNWs) are directly deposited on the surface of Al2O3 and Si3N4 powders using chemical vapor deposition system to realize the uniform mixing. The morphology and the initial stage of the growth process are investigated. It is found that the graphitic base layer is initially formed parallel to the powder surface and is followed by the growth of graphene nanowalls perpendicular to the surface. Moreover, the lateral length of the graphene sheet could be well controlled by tuning the growth temperature. GNWs/Al2O3 powder is consolidated by using sparking plasma sintering method and several physical properties are measured. Owing to the addition of GNWs, the electrical conductivity of the bulk alumina is significantly increased.

Crystal structures and sign reversal Hall resistivities in iron-based superconductors Lix(C3H10N2)0.32FeSe (0.15 < x < 0.4) Hot!

Rui-Jin Sun(孙瑞锦), Shi-Feng Jin(金士锋), Jun Deng(邓俊), Mu-Nan Hao(郝木难), Lin-Lin Zhao(赵琳琳), Xiao Fan(范晓), Xiao-Ning Sun(孙晓宁), Jian-Gang Guo(郭建刚), Lin Gu(谷林)
Chin. Phys. B, 2019, 28 (6): 067401 doi: 10.1088/1674-1056/28/6/067401
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Heavy electron-doped FeSe-derived materials have attracted attention due to their uncommon electronic structures with only ‘electron pockets’, and they are different from other iron-based superconductors. Here, we report the crystal structures, superconductivities and normal state properties of two new Li-doped FeSe-based materials, i.e., Li0.15(C3H10N2)0.32FeSe (P-4) and Lix(C3H10N2)0.32FeSe (P4/nmm, 0.25 < x < 0.4) with superconducting transition temperatures ranging from 40 K to 46 K. The determined crystal structures reveal a coupling between Li concentration and the orientation of 1,3-diaminopropane molecules within the largely expanded FeSe layers. Superconducting fluctuations appear in the resistivity of the two superconductors and are fitted in terms of the quasi two-dimensional (2D) Lawrence-Doniach model. The existence of a crossing point and scaling behavior in the T-dependence of diamagnetic response also suggests that the two superconductors belong to the quasi-2D system. Interestingly, with the increase of temperature, a sign of Hall coefficient (RH) reversing from negative to positive is observed at~185 K in both phases, suggesting that ‘hole pockets’ emerge in these electron-doped FeSe materials. First principle calculations indicate that the increase in FeSe layer distance will lift up a ‘hole band’ associated with dx2-y2 character and increase the hole carriers. Our findings suggest that the increase in two dimensionalities may lead to the sign-reversal Hall resistivity in Lix(C3H10N2)0.32FeSe at high temperature.

The universal characteristic water content of aqueous solutions Hot!

Xiao Huang(黄晓), Ze-Xian Cao(曹则贤), Qiang Wang(王强)
Chin. Phys. B, 2019, 28 (6): 065101 doi: 10.1088/1674-1056/28/6/065101
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Raman measurements at room temperature reveal a characteristic concentration for a series of aqueous solutions of electrolytes, through which O-H stretching vibration of H2O or dilute HDO obviously changes their concentration dependence. This characteristic concentration is very consistent with another, through which the solutions undergo an abrupt change in their glass-forming ability. Interestingly, the molar ratio of water to solute at these two consistent concentration points is almost solute-type independent and about twice the hydration number of solutes. We suggest that just when the concentration increases above this characteristic concentration, bulk-like free water disappears in aqueous solutions and all water molecules among closely-packed hydrated solutes exhibit the characteristics of confined water.

Epitaxial fabrication of two-dimensional TiTe2 monolayer on Au(111) substrate with Te as buffer layer Hot!

Zhipeng Song(宋志朋), Bao Lei(雷宝), Yun Cao(曹云), Jing Qi(戚竞), Hao Peng(彭浩), Qin Wang(汪琴), Li Huang(黄立), Hongliang Lu(路红亮), Xiao Lin(林晓), Ye-Liang Wang(王业亮), Shixuan Du(杜世萱), Hong-Jun Gao(高鸿钧)
Chin. Phys. B, 2019, 28 (5): 056801 doi: 10.1088/1674-1056/28/5/056801
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Two-dimensional (2D) materials provide a platform to exploit the novel physical properties of functional nanodevices. Here, we report on the formation of a new 2D layered material, a well-ordered monolayer TiTe2, on a Au(111) surface by molecular beam epitaxy (MBE). Low-energy electron diffraction (LEED) measurements of the samples indicate that the TiTe2 film forms (√3×√7) superlattice with respect to the Au(111) substrate, which has three different orientations. Scanning tunneling microscopy (STM) measurements clearly show three ordered domains consistent with the LEED patterns. Density functional theory (DFT) calculations further confirm the formation of 2H-TiTe2 monolayer on the Au(111) surface with Te as buffer layer. The fabrication of this 2D layered heterostructure expands 2D material database, which may bring new physical properties for future applications.

Variational and diffusion Monte Carlo simulations of a hydrogen molecular ion in a spherical box Hot!

Xuehui Xiao(肖学会), Kuo Bao(包括), Youchun Wang(王友春), Hui Xie(谢慧), Defang Duan(段德芳), Fubo Tian(田夫波), Tian Cui(崔田)
Chin. Phys. B, 2019, 28 (5): 056401 doi: 10.1088/1674-1056/28/5/056401
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The variational and diffusion Monte Carlo approaches are used to study the ground-state properties of a hydrogen molecular ion in a spheroidal box. In this work, we successfully treat the zero-point motion of protons in the same formalism with as of electrons and avoid the Born-Oppenheimer approximation in density function theory. The study shows that the total energy increases with the decrease in volume, and that the distance between protons decreases as the pressure increases. Considering the motion of protons, the kinetic energy of the electron is higher than that of the fixed model under the same conditions and increases by 5%. The kinetic energy of the proton is found to be small under high pressure, which is only a fraction of the kinetic energy of the electron.

Optically manipulated nanomechanics of semiconductor nanowires Hot!

Chenzhi Song(宋晨之), Shize Yang(杨是赜), Xiaomin Li(李晓敏), Xiao Li(李晓), Ji Feng(冯济), Anlian Pan(潘安练), Wenlong Wang(王文龙), Zhi Xu(许智), Xuedong Bai(白雪冬)
Chin. Phys. B, 2019, 28 (5): 054204 doi: 10.1088/1674-1056/28/5/054204
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Opto-electromechanical coupling at the nanoscale is an important topic in new scientific studies and technical applications. In this work, the optically manipulated electromechanical behaviors of individual cadmium sulfide (CdS) nanowires are investigated by a customer-built optical holder inside transmission electron microscope, wherein in situ electromechanical resonance took place in conjunction with photo excitation. It is found that the natural resonance frequency of the nanowire under illumination becomes considerably lower than that under darkness. This redshift effect is closely related to the wavelength of the applied light and the diameter of the nanowires. Density functional theory (DFT) calculation shows that the photoexcitation leads to the softening of CdS nanowires and thus the redshift of natural frequency, which is in agreement with the experimental results.

Isotope effect and Coriolis coupling effect forthe Li + H(D)Cl→LiCl + H(D) reaction Hot!

Hongsheng Zhai(翟红生), Guanglei Liang(梁广雷), Junxia Ding(丁俊霞), Yufang Liu(刘玉芳)
Chin. Phys. B, 2019, 28 (5): 053401 doi: 10.1088/1674-1056/28/5/053401
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A time-dependent quantum wave packet method is used to investigate the dynamics of the Li+ H(D)Cl reaction based on a new potential energy surface (J. Chem. Phys. 146 164305 (2017)). The reaction probabilities of the Coriolis coupled (CC) and centrifugal sudden (CS) calculations, the integral cross sections, the reaction rate constants are obtained. The rate constants of the Li+ HCl reaction are within the error bounds at low temperature. A comparison of the CC and CS results reveals that the Coriolis coupling plays an important role in the Li+ H(D)Cl reaction. The CC cross sections are larger than the CS results within the entire energy range, demonstrating that the Coriolis coupling effect can more effectively promote the Li+ DCl reaction than the Li+ HCl reaction. It is found that the isotope effect has a great influence on the title reaction.

Observing the steady-state visual evoked potentials with a compact quad-channel spin exchange relaxation-free magnetometer Hot!

Peng-Cheng Du(杜鹏程), Jian-Jun Li(李建军), Si-Jia Yang(杨思嘉), Xu-Tong Wang(王旭桐), Yan Zhuo(卓彦), Fan Wang(王帆), Ru-Quan Wang(王如泉)
Chin. Phys. B, 2019, 28 (4): 040702 doi: 10.1088/1674-1056/28/4/040702
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We observed the steady-state visually evoked potential (SSVEP) from a healthy subject using a compact quad-channel potassium spin exchange relaxation-free (SERF) optically pumped magnetometer (OPM). To this end, 30 s of data were collected, and SSVEP-related magnetic responses with signal intensity ranging from 150 fT to 300 fT were observed for all four channels. The corresponding signal to noise ratio (SNR) was in the range of 3.5-5.5. We then used different channels to operate the sensor as a gradiometer. In the specific case of detecting SSVEP, we noticed that the short channel separation distance led to a strongly diminished gradiometer signal. Although not optimal for the case of SSVEP detection, this set-up can prove to be highly useful for other magnetoencephalography (MEG) paradigms that require good noise cancellation. Considering its compactness, low cost, and good performance, the K-SERF sensor has great potential for biomagnetic field measurements and brain-computer interfaces (BCI).

Interaction of two symmetric monovacancy defects in graphene Hot!

Wen-Yan Xu(徐文焱), Li-Zhi Zhang(张礼智), Li Huang(黄立), Yan-De Que(阙炎德), Ye-Liang Wang(王业亮), Xiao Lin(林晓), Shi-Xuan Du(杜世萱)
Chin. Phys. B, 2019, 28 (4): 046801 doi: 10.1088/1674-1056/28/4/046801
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We investigate the interactions between two symmetric monovacancy defects in graphene grown on Ru (0001) after silicon intercalation by combining first-principles calculations with scanning tunneling microscopy (STM). First-principles calculations based on free-standing graphene show that the interaction is weak and no scattering pattern is observed when the two vacancies are located in the same sublattice of graphene, no matter how close they are, except that they are next to each other. For the two vacancies in different sublattices of graphene, the interaction strongly influences the scattering and new patterns' emerge, which are determined by the distance between two vacancies. Further experiments on silicon intercalated graphene epitaxially grown on Ru (0001) shows that the experiment results are consistent with the simulated STM images based on free-standing graphene, suggesting that a single layer of silicon is good enough to decouple the strong interaction between graphene and the Ru (0001) substrate.

Entangled multi-knot lattice model of anyon current Hot!

Tieyan Si(司铁岩)
Chin. Phys. B, 2019, 28 (4): 040501 doi: 10.1088/1674-1056/28/4/040501
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We proposed an entangled multi-knot lattice model to explore the exotic statistics of anyons. Long-range coupling interaction is a fundamental character of this knot lattice model. The short-range coupling models, such as the Ising model, Hamiltonian model of quantum Hall effect, fermion pairing model, Kitaev honeycomb lattice model, and so on, are the short-range coupling cases of this knot lattice model. The long-range coupling knot lattice model bears Abelian and non-Abelian anyons, and shows integral and fractional filling states like the quantum Hall system. The fusion rules of anyons are explicitly demonstrated by braiding crossing states. The eigenstates of quantum models can be represented by a multi-layer link lattice pattern whose topology is characterized by the linking number. This topological linking number offers a new quantity to explain and predict physical phenomena in conventional quantum models. For example, a convection flow loop is introduced into the well-known Bardeen-Cooper-Schrieffer fermion pairing model to form a vortex dimer state that offers an explanation of the pseudogap state of unconventional superconductors, and predicts a fractionally filled vortex dimer state. The integrally and fractionally quantized Hall conductance in the conventional quantum Hall system has an exact correspondence with the linking number in this multi-knot lattice model. The real-space knot pattern in the topological insulator model has an equivalent correspondence with the Lissajous knot in momentum space. The quantum phase transition between different quantum states of the quantum spin model is also directly quantified by the change of topological linking number, which revealed the topological character of phase transition. Circularized photons in an optical fiber network are a promising physical implementation of this multi-knot lattice, and provide a different path to topological quantum computation.

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