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NbN quantum dots anchored hollow carbon nanorods as efficient polysulfide immobilizer and lithium stabilizer for Li-S full batteries
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作者 Fei Ma Zhuo Chen +9 位作者 Katam Srinivas Ziheng Zhang Yu Wu Dawei Liu Hesheng Yu Yue Wang Xinsheng Li Ming-qiang Zhu Qi Wu Yuanfu Chen 《Journal of Energy Chemistry》 SCIE EI CAS CSCD 2024年第1期260-271,I0007,共13页
The shuttle effect of lithium polysulfides(LiPSs)and uncontrollable lithium dendrite growth seriously hinder the practical application of lithium-sulfur(Li-S)batteries.To simultaneously address such issues,monodispers... The shuttle effect of lithium polysulfides(LiPSs)and uncontrollable lithium dendrite growth seriously hinder the practical application of lithium-sulfur(Li-S)batteries.To simultaneously address such issues,monodispersed Nb N quantum dots anchored on nitrogen-doped hollow carbon nanorods(NbN@NHCR)are elaborately developed as efficient Li PSs immobilizer and Li stabilizer for high-performance Li-S full batteries.Density functional theory(DFT)calculations and experimental characterizations demonstrate that the sulfiphilic and lithiophilic NbN@NHCR hybrid can not only efficiently immobilize the soluble Li PSs and facilitate diffusion-conversion kinetics for alleviating the shuttling effect,but also homogenize the distribution of Li+ions and regulate uniform Li deposition for suppressing Li-dendrite growth.As a result,the assembled Li-S full batteries(NbN@NHCR-S||Nb N@NHCR-Li)deliver excellent long-term cycling stability with a low decay rate of 0.031%per cycle over 1000 cycles at high rate of 2 C.Even at a high S loading of 5.8 mg cm^(-2)and a low electrolyte/sulfur ratio of 5.2μL mg^(-1),a large areal capacity of 6.2 mA h cm^(-2)can be achieved in Li-S pouch cell at 0.1 C.This study provides a new perspective via designing a dual-functional sulfiphilic and lithiophilic hybrid to address serious issues of the shuttle effect of S cathode and dendrite growth of Li anode. 展开更多
关键词 Dual-functional host NbN quantum dots Shuttle effect Dendrite-free Li anode Li-S full batteries
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Kondo effect for electron transport through an artificial quantum dot 被引量:1
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作者 孙科伟 熊诗杰 《Chinese Physics B》 SCIE EI CAS CSCD 2006年第4期828-832,共5页
We have calculated the transport properties of electron through an artificial quantum dot by using the numerical renormalization group technique in this paper. We obtain the conductance for the system of a quantum dot... We have calculated the transport properties of electron through an artificial quantum dot by using the numerical renormalization group technique in this paper. We obtain the conductance for the system of a quantum dot which is embedded in a one-dimensional chain in zero and finite temperature cases. The external magnetic field gives rise to a negative magnetoconductance in the zero temperature case. It increases as the external magnetic field increases, We obtain the relation between the coupling coefficient and conductance. If the interaction is big enough to prevent conduction electrons from tunnelling through the dot, the dispersion effect is dominant in this case. In the Kondo temperature regime, we obtain the conductivity of a quantum dot system with Kondo correlation. 展开更多
关键词 kondo effect quantum dot numerical renormalization group
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Spin-Polarized Transport Through a Quantum Dot Coupled to Ferromagnetic Leads: Kondo Correlation Effect
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作者 MAJing DONGBing LEIXiao-Lin 《Communications in Theoretical Physics》 SCIE CAS CSCD 2005年第2期341-348,共8页
We investigate the linear and nonlinear transport through a single level quantum dot connected to two ferromagnetic leads in Kondo regime, using the slave-boson mean-field approach for finite on-site Coulomb repulsion... We investigate the linear and nonlinear transport through a single level quantum dot connected to two ferromagnetic leads in Kondo regime, using the slave-boson mean-field approach for finite on-site Coulomb repulsion. We find that for antiparallel alignment of the spin orientations in the leads, a single zero-bias Kondo peak always appears in the voltage-dependent differential conductance with peak height going down to zero as the polarization grows to P = 1.For parallel configuration, with increasing polarization from zero, the Kondo peak descends and greatly widens with the appearance of shoulders, and finally splits into two peaks on both sides of the bias voltage around P ~ 0.7 until disappearing at even larger polarization strength. At any spin orientation angle θ, the linear conductance generally drops with growing polarization strength. For a given finite polarization, the minimum linear conductance always appears at θ = π. 展开更多
关键词 quantum dot kondo correlation effect transport
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Effects of van Hove Singularities on Transport of Quantum Dot Systems in Kondo Regime
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作者 HU Zhi-Ming YANG Kai-Hua TIAN Guang-Shan 《Communications in Theoretical Physics》 SCIE CAS CSCD 2005年第3X期563-572,共10页
In the present paper, we study the effect of van Hove singularities of conduction electron on the transport of a single quantum dot system in the Kondo regime. By using both the equation-of-motion and the noncrossing ... In the present paper, we study the effect of van Hove singularities of conduction electron on the transport of a single quantum dot system in the Kondo regime. By using both the equation-of-motion and the noncrossing approximation techniques, we show that the corrections caused by these singularities are actually minor. It can be explained by observing that the singularities in the equations, which determine the electronic DOS on the dot, are integrable. Furthermore, we find that, although each line width function is divergent at van Hove singular points, the total divergence is canceled out in the final formula to calculate the current through the system. Therefore, as far as the qualitative properties of the system is concerned, these singularities can be ignored and the wide-band approximation can be safely used in calculation. 展开更多
关键词 quantum dot system van Hove singularities kondo effect
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Kondo transport through a quantum dot coupled with side quantum-dot structures
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作者 江兆潭 《Chinese Physics B》 SCIE EI CAS CSCD 2010年第7期1-7,共7页
This paper investigates Kondo transport properties in a quadruple quantum dot (QD) based on the slave-boson mean field theory and the non-equilibrium Green's function. In the quadruple QD structure one Kondo-type Q... This paper investigates Kondo transport properties in a quadruple quantum dot (QD) based on the slave-boson mean field theory and the non-equilibrium Green's function. In the quadruple QD structure one Kondo-type QD sandwiched between two leads is side coupled to two separate QD structures: a single-QD atom and a double-QD molecule. It shows that the conductance valleys and peaks always appear in pairs and by tuning the energy levels in three side QDs, the one-, two-, or three-valley conductance pattern can be obtained. Furthermore, it finds that whether the valley and the peak can appear is closely dependent on the specific values of the interdot couplings and the energy level difference between the two QDs in the molecule. More interestingly, an extra novel conductance peak can be produced by the coexistence of the two different kinds of side QD structures. 展开更多
关键词 quantum transport kondo effect quantum dot
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Fano-Kondo effect in the T-shaped double quantum dots coupled to ferromagnetic leads
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作者 侯涛 吴绍全 +3 位作者 毕爱华 羊富彬 陈嘉峰 樊梦 《Chinese Physics B》 SCIE EI CAS CSCD 2009年第2期783-789,共7页
Using an equation-of-motion technique, we theoretically study the Fano-Kondo effect in the T-shaped double quantum dots coupled to two ferromagnetic leads by the Anderson Hamiltonian. We calculate the density of state... Using an equation-of-motion technique, we theoretically study the Fano-Kondo effect in the T-shaped double quantum dots coupled to two ferromagnetic leads by the Anderson Hamiltonian. We calculate the density of states in this system with both parallel and antiparaIlel lead-polarization alignments, and our results reveal that the interdot coupling, the spin-polarized strength and the energy level of the side coupled quantum dot greatly influence the density of states of the central quantum dot. This system is a possible candidate for spin valve transistors and may have potential applications in the spintronics. 展开更多
关键词 Fano kondo effect density of states quantum dots
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Kondo-Fano Effect in T-shaped Double Quantum Dots Coupled to Ferromagnetic Leads
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作者 HUANG Rui WU Shao-Quan HOU Tao 《Communications in Theoretical Physics》 SCIE CAS CSCD 2012年第1期161-165,共5页
Using an equation-of-motion technique, we theoretically study the Kondo-Fano effect in the T-shaped double quantum dots coupled to two ferromagnetic leads by the Anderson Hamiltonian. We calculate the density of state... Using an equation-of-motion technique, we theoretically study the Kondo-Fano effect in the T-shaped double quantum dots coupled to two ferromagnetic leads by the Anderson Hamiltonian. We calculate the density of states in this system by solving Green function. Our results reveal that the density of states show some noticeable characteristics not only depending upon the interdot coupling tab, the energy level eal of the side coupled quantum dot QDb, and the relative angle θ of magnetic moment M, but also the asymmetry parameter a in ferromagnetic leads and so on. All these parameters greatly influence the density of states of the eentral quantum dot QDa. This system is a possible candidate for spin valve transistors and may have potential applications in the spintronies. 展开更多
关键词 T-shaped double quantum dots kondo-Fano effect density of states
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Anomalous Kondo-Switching Effect of a Spin-Flip Quantum Dot Embedded in an Aharonov-Bohm Ring
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作者 CHEN Xiong-Wen SHI Zhen-Gang SONG Ke-Hui 《Communications in Theoretical Physics》 SCIE CAS CSCD 2009年第11期945-948,共4页
We theoretically investigate the Kondo effect of a quantum dot embedded in a mesoscopic Aharonov-Bohm (AIR) ring in the presence of the spin flip processes by means of the one-impurity Anderson Hamiltonian. Based on... We theoretically investigate the Kondo effect of a quantum dot embedded in a mesoscopic Aharonov-Bohm (AIR) ring in the presence of the spin flip processes by means of the one-impurity Anderson Hamiltonian. Based on the slave-boson mean-field theory, we find that in this system the persistent current (PC) sensitively depends on the parity and size of the AB ring and can be tuned by the spin-flip scattering (R). In the small AB ring, the PC is suppressed due to the enhancing R weakening the Kondo resonance. On the contrary, in the large AB ring, with R increasing, the peak of PC firstly moves up to max-peak and then down. Especially, the PC phase shift of π appears suddenly with the proper value of R, implying the existence of the anomalous Kondo effect in this system. Thus this system may be a carldidate for quantum switch. 展开更多
关键词 persistent current quantum dot kondo effect spin-flip scattering effect
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Kondo Effect Versus Magnetic Coupling in Indirectly Coupled Double Quantum Dots
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作者 龙文 孙庆丰 《Communications in Theoretical Physics》 SCIE CAS CSCD 2010年第11期933-937,共5页
We theoretically investigate a device consisting of two quantum dots (QDs) side-coupled to a quantum wire which has many physical ingredients of an artificial heavy fermion system. An extra parameter, the distance L... We theoretically investigate a device consisting of two quantum dots (QDs) side-coupled to a quantum wire which has many physical ingredients of an artificial heavy fermion system. An extra parameter, the distance L between the two QDs, is introduced and it plays an important role on the competition of the Kondo temperature and magnetic coupling. Three different phases are found: antiferromagnetic phase, Kondo phase with spin S = 1/2, and Kondo phase with S = 1, depending on the distance L, the magnetic properties are qualitatively different for different phases: conductance tends to the unitary value 2e2 /h; for the S : the distance. coupling, and the Kondo temperature. Quantum transport for the S = 1 Kondo and the antiferromagnetic phases, the 1/2 Kondo phase the conductance is strongly dependent onthe distance. 展开更多
关键词 quantum dot kondo effect RKKY interaction
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The Kondo and Fano Effects in Triple Quantum Dots Coupled to Noncollinear Ferromagnetic Leads
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作者 HUANG Rui MING Shan-Wen WANG Yong 《Chinese Physics Letters》 SCIE CAS CSCD 2012年第4期174-177,共4页
We use the equation-of-motion technique and non-equilibrium Green's function theory to study the Kondo effect and the Fano effect in triple quantum dots (QDs) coupled to symmetrically ferromagnetic leads whose mag... We use the equation-of-motion technique and non-equilibrium Green's function theory to study the Kondo effect and the Fano effect in triple quantum dots (QDs) coupled to symmetrically ferromagnetic leads whose magnetic moments are noncollinear.We address the question of how the noncollinear ferromagnetic leads influence the Kondo effect and how the side-coupled QDs present Fano interference.The results show that the spin splitting of the density of state (DOS) takes place in an intermediate direction between the magnetic moments in the two leads.When interdot coupling strength ti is nonzero,Fano interference begins to play a major role in complicating the DOS of QD0. 展开更多
关键词 quantum kondo effects
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Influences of a Side-Coupled Triple Quantum Dot on Kondo Transport Through a Quantum Dot
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作者 江兆潭 杨彦楠 秦志杰 《Communications in Theoretical Physics》 SCIE CAS CSCD 2010年第11期925-932,共8页
Kondo transport properties through a Kondo-type quantum dot (QD) with a side-coupled triple-QD structure are systematically investigated by using the non-equilibrium Green's function method. We firstly derive the f... Kondo transport properties through a Kondo-type quantum dot (QD) with a side-coupled triple-QD structure are systematically investigated by using the non-equilibrium Green's function method. We firstly derive the formulae of the current, the linear conductance, the transmission coefficient, and the local density of states. Then we carry out the analytical and numerical studies and some universal conductance properties are obtained. It is shown that the number of the conductance valleys is intrinsically determined by the side-coupled QDs and at most equal to the number of the QDs included in the side-coupled structure in the asymmetric limit. In the process of forming the conductance valleys, the side-coupled QD system plays the dominant role while the couplings between the Kondo-type QD and the side-coupled structure play the subsidiary and indispensable roles. To testify the validity of the universal conductance properties, another different kinds of side-coupled triple-QD structures are considered. It should be emphasized that these universal properties axe applicable in understanding this kind of systems with arbitrary many-QD side structures. 展开更多
关键词 kondo effect quantum dots
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Polaron effect on the optical rectification in spherical quantum dots with electric field 被引量:5
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作者 冯振宇 闫祖威 《Chinese Physics B》 SCIE EI CAS CSCD 2016年第10期446-451,共6页
The polaron effect on the optical rectification in spherical quantum dots with a shallow hydrogenic impurity in the presence of electric field is theoretically investigated by taking into account the interactions of t... The polaron effect on the optical rectification in spherical quantum dots with a shallow hydrogenic impurity in the presence of electric field is theoretically investigated by taking into account the interactions of the electrons with both confined and surface optical phonons. Besides, the interaction between impurity and phonons is also considered. Numerical calculations are presented for typical Zn1-xCdxSe/ZnSe material. It is found that the polaronic effect or electric field leads to the redshifted resonant peaks of the optical rectification coefficients. It is also found that the peak values of the optical rectification coefficients with the polaronic effect are larger than without the polaronic effect, especially for smaller Cd concentrations or stronger electric field. 展开更多
关键词 polaron effect optical rectification spherical quantum dot
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Effective-mass theory for coupled quantum dots grown on (11N)-oriented substrates 被引量:2
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作者 李树深 夏建白 《Chinese Physics B》 SCIE EI CAS CSCD 2007年第1期1-5,共5页
The electronic structures of coupled quantum dots grown on (11N)-oriented substrates are studied in the framework of effective-mass envelope-function theory. The results show that the all-hole subbands have the smal... The electronic structures of coupled quantum dots grown on (11N)-oriented substrates are studied in the framework of effective-mass envelope-function theory. The results show that the all-hole subbands have the smallest widths and the optical properties are best for the (113), (114), and (115) growth directions. Our theoretical results agree with the available experimental data. Our calculated results are useful for the application of coupled quantum dots in photoelectric devices. 展开更多
关键词 coupled quantum dots effective-mass theory electronic structures
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Temperature Effects on the Self-trapping Energy of a Polaron in a GaAs Parabolic Quantum Dot 被引量:1
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作者 Kadi ZHU and Shiwei GU (Institute of Condensed Matter Physics, Shanghai Jiaotong University, Shanghai, 200030, China)(To whom correspondence should be addressed) 《Journal of Materials Science & Technology》 SCIE EI CAS CSCD 1994年第2期131-134,共4页
The temperature and the size dependences of the self-trapping energy of a polaron in a GaAs parabolic quantum dot are investigated by the second order Rayleigh-Schrodinger perturbation method using the framework of th... The temperature and the size dependences of the self-trapping energy of a polaron in a GaAs parabolic quantum dot are investigated by the second order Rayleigh-Schrodinger perturbation method using the framework of the effective mass approximation. The numerical results show that the self-trapping energies of polaron in GaAs parabolic quantum dots shrink with the enhancement of temperature and the size of the quantum dot. The results also indicate that the temperature effect becomes obvious in small quantum dots 展开更多
关键词 GAAS REV Temperature effects on the Self-trapping Energy of a Polaron in a GaAs Parabolic quantum dot
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Transport Properties of Two Coupled Quantum Dots Under Optical Pumping
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作者 葛传楠 温俊 +1 位作者 彭菊 王伯根 《Journal of Semiconductors》 EI CAS CSCD 北大核心 2006年第4期598-603,共6页
Using the Keldysh-Green function,we present a theoretical study on the electron transport properties of two coupled quantum dots under optical pumping. Plateaus in the I-V curve and resonant peaks in the transmission ... Using the Keldysh-Green function,we present a theoretical study on the electron transport properties of two coupled quantum dots under optical pumping. Plateaus in the I-V curve and resonant peaks in the transmission coefficient occur and can be explained by the local electron density of states in the quantum dots. The effects of the optical pumping frequency and intensity on the transport properties of the system are also discussed. The electron dynamical localization phenomenon occurs when the optical pumping frequency is equal to the discrete hole energy level. This result can be used to realize optical control switches. 展开更多
关键词 Keldysh-Green function optical pumping quantum dot electron transport dynamical localization
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Inorganic Halide Perovskite Quantum Dots:A Versatile Nanomaterial Platform for Electronic Applications 被引量:2
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作者 Chien‑Yu Huang Hanchen Li +7 位作者 Ye Wu Chun‑Ho Lin Xinwei Guan Long Hu Jiyun Kim Xiaoming Zhu Haibo Zeng Tom Wu 《Nano-Micro Letters》 SCIE EI CAS CSCD 2023年第2期1-31,共31页
Metal halide perovskites have generated significant attention in recent years because of their extraordinary physical properties and photovoltaic performance.Among these,inorganic perovskite quantum dots(QDs)stand out... Metal halide perovskites have generated significant attention in recent years because of their extraordinary physical properties and photovoltaic performance.Among these,inorganic perovskite quantum dots(QDs)stand out for their prominent merits,such as quantum confinement effects,high photoluminescence quantum yield,and defect-tolerant structures.Additionally,ligand engineering and an all-inorganic composition lead to a robust platform for ambient-stable QD devices.This review presents the state-of-the-art research progress on inorganic perovskite QDs,emphasizing their electronic applications.In detail,the physical properties of inorganic perovskite QDs will be introduced first,followed by a discussion of synthesis methods and growth control.Afterwards,the emerging applications of inorganic perovskite QDs in electronics,including transistors and memories,will be presented.Finally,this review will provide an outlook on potential strategies for advancing inorganic perovskite QD technologies. 展开更多
关键词 Inorganic perovskite quantum dots Electronics NANOCRYSTALS quantum confinement effects
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Dual-Functional Lithiophilic/Sulfiphilic Binary-Metal Selenide Quantum Dots Toward High-Performance Li-S Full Batteries 被引量:4
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作者 Youzhang Huang Liang Lin +6 位作者 Yinggan Zhang Lie Liu Baisheng Sa Jie Lin Laisen Wang Dong-Liang Peng Qingshui Xie 《Nano-Micro Letters》 SCIE EI CAS CSCD 2023年第5期169-186,共18页
The commercial viability of lithium-sulfur batteries is still challenged by the notorious lithium polysulfides(Li PSs)shuttle effect on the sulfur cathode and uncontrollable Li dendrites growth on the Li anode.Herein,... The commercial viability of lithium-sulfur batteries is still challenged by the notorious lithium polysulfides(Li PSs)shuttle effect on the sulfur cathode and uncontrollable Li dendrites growth on the Li anode.Herein,a bi-service host with Co-Fe binary-metal selenide quantum dots embedded in three-dimensional inverse opal structured nitrogen-doped carbon skeleton(3DIO FCSe-QDs@NC)is elaborately designed for both sulfur cathode and Li metal anode.The highly dispersed FCSe-QDs with superb adsorptive-catalytic properties can effectively immobilize the soluble Li PSs and improve diffusion-conversion kinetics to mitigate the polysulfide-shutting behaviors.Simultaneously,the 3D-ordered porous networks integrated with abundant lithophilic sites can accomplish uniform Li deposition and homogeneous Li-ion flux for suppressing the growth of dendrites.Taking advantage of these merits,the assembled Li-S full batteries with 3DIO FCSe-QDs@NC host exhibit excellent rate performance and stable cycling ability(a low decay rate of 0.014%over 2,000 cycles at 2C).Remarkably,a promising areal capacity of 8.41 mAh cm^(-2)can be achieved at the sulfur loading up to 8.50 mg cm^(-2)with an ultra-low electrolyte/sulfur ratio of 4.1μL mg^(-1).This work paves the bi-serve host design from systematic experimental and theoretical analysis,which provides a viable avenue to solve the challenges of both sulfur and Li electrodes for practical Li-S full batteries. 展开更多
关键词 Dual-functional host Fe_(2)CoSe_(4)quantum dots Shuttle effect Dendrite-free Li anode Li-S full batteries
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Magnetotransport through an Aharonov-Bohm ring with parallel double quantum dots coupled to ferromagnetic leads
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作者 吴绍全 侯涛 +1 位作者 赵国平 余万伦 《Chinese Physics B》 SCIE EI CAS CSCD 2010年第4期351-356,共6页
Using the Keldysh nonequilibrium Green function and equation-of-motion technique, this paper studies the mag- netotransport through an Aharonov-Bohm (AB) ring with parallel double quantum dots coupled to ferromagnet... Using the Keldysh nonequilibrium Green function and equation-of-motion technique, this paper studies the mag- netotransport through an Aharonov-Bohm (AB) ring with parallel double quantum dots coupled to ferromagnetic leads. It calculates the transmission probability in both the equilibrium and the nonequilibrium case, analyses the conduc- tance and the tunnel magnetoresistance for various parameters, and obtains some new results. These results show that this system is provided with an excellent spin filtering property, and that a large tunnelling magnetoresistance and a negative tunnelling magnetoresistance can arise by adjusting relative parameters; these facts indicate that this system is a possible candidate for spin valve transistors, and has important applications in spintronics. 展开更多
关键词 kondo effect double quantum dots Aharonov-Bohm ring MAGNETORESISTANCE
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Quantum phase transition and Coulomb blockade effect in triangular quantum dots with interdot capacitive and tunnel couplings
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作者 熊永臣 王为忠 +1 位作者 杨俊涛 黄海铭 《Chinese Physics B》 SCIE EI CAS CSCD 2015年第2期403-408,共6页
The quantum phase transition and the electronic transport in a triangular quantum dot system are investigated using the numerical renormalization group method.We concentrate on the interplay between the interdot capac... The quantum phase transition and the electronic transport in a triangular quantum dot system are investigated using the numerical renormalization group method.We concentrate on the interplay between the interdot capacitive coupling V and the interdot tunnel coupling t.For small t,three dots form a local spin doublet.As t increases,due to the competition between V and t,there exist two first-order transitions with phase sequence spin-doublet-magnetic frustration phase-orbital spin singlet.When t is absent,the evolutions of the total charge on the dots and the linear conductance are of the typical Coulomb-blockade features with increasing gate voltage.While for sufficient t,the antiferromagnetic spin correlation between dots is enhanced,and the conductance is strongly suppressed for the bonding state is almost doubly occupied. 展开更多
关键词 quantum phase transition Coulomb blockade effect triangular quantum dots strongly correlated system
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Effect of hydrostatic pressure and polaronic mass of the binding energy in a spherical quantum dot
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作者 A.Rejo Jeice Sr.Gerardin Jayam K.S.Joseph Wilson 《Chinese Physics B》 SCIE EI CAS CSCD 2015年第11期92-97,共6页
Simultaneous effect of hydrostatic pressure and polaronic mass on the binding energies of the ground and excited states of an on-center hydrogenic impurity confined in a GaAs/GaA1As spherical quantum dot are theoretic... Simultaneous effect of hydrostatic pressure and polaronic mass on the binding energies of the ground and excited states of an on-center hydrogenic impurity confined in a GaAs/GaA1As spherical quantum dot are theoretically investigated by the variational method within the effective mass approximation. The binding energy is calculated as a function of dot radius and pressure. Our findings proved that the hydrostatic pressure led to the decrease of confined energy and the increase of donor binding energy. Conduction band non-parabolicity and the polaron masses are effective in the donor binding energy which is significant for narrow dots not in the confined energy. The maximum donor binding energy achieved by the polaronic mass in the ground and excited states are 2%-19% for the narrow dots. The confined and donor binding energies approach zero as the dot size approaches infinity. 展开更多
关键词 spherical quantum dot square well confinement confined energies effective mass approximation
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