Lattice Structure and Convergence of a Game of Cards

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巴黎伦敦落魄记英文版

巴黎伦敦落魄记英文版Paris and London: Tales of Triumph and DesolationIntroduction:In the cities of Paris and London, two vibrant metropolises on opposite sides of the English Channel, there lies a tale of contrasting fates. This article delves into the narratives of these two cities, exploring their rich history, cultural significance, and the challenges that have afflicted each at various points in time. Paris and London, often referred to as sister cities, share a long-standing rivalry and a unique bond shaped by their similarities and differences.1. A Tale of Triumph: Paris1.1 The Eiffel Tower and the City of LightParis, famously known as the "City of Light," stands as a beacon of triumph. At the heart of this remarkable city stands the Eiffel Tower, an iconic symbol of grace and elegance, capturing the imagination of millions of visitors each year. Its soaring iron lattice structure and breathtaking views serve as a testament to human ingenuity and architectural genius. The Tower's allure has played a significant role in establishing Paris as a global cultural capital.1.2 The Louvre Museum – A Treasure Trove of ArtWithin the enchanting streets of Paris lies an artistic marvel – the Louvre Museum. Home to some of the world's most renowned masterpieces, including the enigmatic Mona Lisa by Leonardo da Vinci, the Louvre is atestament to Paris' triumph in preserving and celebrating artistic heritage. Its vast collection spans centuries, offering visitors a mesmerizing journey through time and culture.1.3 Parisian Cuisine – A Culinary TriumphBeyond its architectural splendor and artistic treasures, Paris is renowned for its gastronomic delights. From the delicate pastries of patisseries to the rich flavors of traditional French cuisine, the city's culinary scene is a testament to its triumph in the realm of fine dining. Parisian food culture, characterized by elegance and sophistication, has captivated taste buds across the globe, cementing the city's position as a culinary capital.2. A Tale of Desolation: London2.1 The Blitz and ResilienceWhile Paris boasts triumphs, London has faced its fair share of desolation. During the Second World War, the city endured relentless bombings, known as the Blitz, causing widespread destruction and loss. Despite the devastation, Londoners displayed an indomitable spirit, epitomizing resilience in the face of hardship. The rebirth of London from the ashes showcases the city's determination to preserve its heritage and emerge stronger.2.2 The Tower of London – A Dark PastWithin the heart of London stands the Tower of London, a structure shrouded in a dark past. Built as a palace, prison, and fortress, it witnessed the rise and fall of kings, the execution of queens, and tales of betrayal and intrigue. This historic landmark reflects the desolation that has markedcertain chapters of London's history, serving as a constant reminder of both the city's resilience and the shadows it has endured.2.3 The East End – From Squalor to RenaissanceThe East End of London, once synonymous with poverty and despair, has experienced a renaissance in recent times. Long associated with the Dickensian era and the hardships faced by the working class, the area has undergone extensive urban regeneration. The transformation of the East End testifies to London's resolve to uplift marginalized communities and channel desolation into hope.Conclusion:Paris and London are cities steeped in history, each with its own tale of triumph and desolation. Paris shines as a city of light, celebrated for its architectural marvels, artistic heritage, and culinary prowess. Meanwhile, London's desolation throughout history has been a catalyst for resilience and transformation. As these sister cities continue to shape the course of history, their stories remind us of the capacity of human spirit to overcome adversity and triumph over desolation.。

半导体分离芯材料英语

半导体分离芯材料英语Semiconductor Materials for Isolation Cores.Semiconductor materials play a crucial role in modern electronics, particularly in the fabrication of isolation cores. Isolation cores are essential components in integrated circuits, ensuring that different sections of the circuit operate independently without interference. This article delves into the world of semiconductor materials suitable for isolation cores, discussing their properties, applications, and challenges.1. Introduction to Semiconductor Materials.Semiconductors are materials that have an electrical conductivity falling between that of conductors and insulators. They exhibit unique electronic properties, making them ideal for use in electronic devices. Silicon (Si) and germanium (Ge) are the most commonly used semiconductors, but others such as gallium arsenide (GaAs)and silicon carbide (SiC) are also finding applications in specific areas.2. Properties of Semiconductor Materials.Bandgap Energy: The bandgap energy is a measure of the energy required to excite an electron from the valence band to the conduction band. Materials with larger bandgap energies are better suited for high-temperature applications.Doping: Semiconductors can be doped with impurities to alter their conductivity. Dopants such as boron (B) or phosphorus (P) are introduced to create p-type or n-type semiconductors, respectively.Lattice Structure: The atomic lattice structure of semiconductors determines their physical and electrical properties. Silicon and germanium have diamond-like lattice structures, which contribute to their widespread use.3. Isolation Cores and Their Importance.Isolation cores are critical in integrated circuits, where they prevent electrical signals from leaking between different circuit sections. This isolation ensures that signals are contained within their designated paths, preventing crosstalk and noise. Isolation cores are typically made from insulating materials, but semiconductor materials can also be used to achieve isolation.4. Semiconductor Materials for Isolation Cores.Silicon-on-Insulator (SOI): SOI technology involves a thin layer of silicon sandwiched between two layers of insulating material, such as silicon dioxide or sapphire. This structure provides excellent isolation between different circuit sections. SOI wafers are widely used in high-performance microelectronics, as they offer reduced parasitic capacitance and improved thermal performance.Silicon Carbide (SiC): SiC is a wide-bandgap semiconductor material with excellent thermal conductivity and chemical stability. It is suitable for high-temperatureand high-power applications. SiC-based isolation cores can withstand extreme operating conditions, making them idealfor use in power electronics and aerospace applications.Gallium Arsenide (GaAs): GaAs has a smaller bandgap than silicon but offers higher electron mobility and saturation velocity. GaAs-based isolation cores are commonly used in high-frequency applications such as microwave and millimeter-wave circuits. GaAs also finds applications in optoelectronics due to its ability to emit and detect light.5. Challenges and Future Outlook.Despite the many advantages of semiconductor materials for isolation cores, there are still challenges to overcome. One major challenge is the scalability of these materialsfor smaller and more complex integrated circuits. Another challenge lies in the fabrication process, which requires precise control over doping levels, lattice structures, and defect densities.Future research in this area will focus on developing new semiconductor materials with improved properties and on optimizing fabrication processes for better scalability and performance. Materials such as two-dimensional semiconductors and topological insulators are beingactively explored for their potential in next-generation electronics.Conclusion.Semiconductor materials play a pivotal role in the fabrication of isolation cores, enabling the reliable operation of integrated circuits. Silicon, silicon carbide, and gallium arsenide are among the most commonly used semiconductors for this purpose, each offering its unique advantages and applications. Future research in this field will focus on addressing challenges related to scalability and fabrication processes while exploring novel materials with improved properties.。

Lecture14_Lattice_Structures

Digital Signal ProcessingLecture 14Lattice StructuresTesheng Hsiao,Associate ProfessorThe lattice structure is a modular structure consisting of cascaded stages.Digital ﬁlters implemented by the lattice structure can be transformed into direct from and vice versa.In this lecture,we are going to investigate the implementation of the lattice form and the conversion between it and the direct form.1Recursive Lattice StructureFig.(1)is a single lattice stage.It is a two-input,two-output system consisting of two multipliers,two adders,and one delayelement.Figure 1:A single lattice stageThe constant κn is referred to as reﬂection coeﬃcient .The input and output relation can be expressed in the z-domainU n (z )=U n +1(z )−κn z −1V n (z )(1)V n +1(z )=κn U n (z )+z −1V n (z )(2)Rearrange the equation and we have[U n +1(z )V n +1(z )]=[1κn z −1κn z −1][U n (z )V n (z )](3)The lattice structure of an N th order LTI digital ﬁlter is a cascade of N stages in the way shown in Fig.(2).Given the lattice structure of Fig.(2),we are going to answer the following questions:(1)how the input signal x propagates to the output signal y ,(2)what is the system function implemented by the lattice structure,and (3)how a system function can be converted to a lattice structure.Figure2:Recursive lattice structure•Lattice FilteringThe output of the lattice structure can be calculated in a recursive way.Assume that the system is at initial rest;hence v n[−1]=0,n=0,1,···,N−1.From Fig.(2),for each time step k≥0,we haveinitial conditions:v n[−1]=0,n=0,1,···,N−1for k=0,1,2,···u N[k]=x[k]for n=N−1to0u n[k]=u n+1[k]−κn v n[k−1]v n+1[k]=κn u n[k]+v n[k−1]endv0[k]=u0[k]y[k]=N∑n=0λn v n[k]end•The system function of the lattice structureLetP n(z)=U n(z)U0(z),Q n(z)=V n(z)V0(z),n=0,1,2,···,NHence,Eq.(3)can be rewritten as[P n+1(z) Q n+1(z)]=[1κn z−1κn z−1][P n(z)Q n(z)],n=0,1,2,···,N−1(4) =[1κnκn1][P n(z)z−1Q n(z)],n=0,1,2,···,N−1(5)Note thatU0(z)=V0(z),P0(z)=Q0(z)=1,X(z)U0(z)=P N(z),Y(z)U0(z)=N∑n=0λn Q n(z)Therefore the system function H (z )isH (z )=Y (z )X (z )=∑N n =0λn Q n (z )P N (z )(6)If we expand Eq.(4),we obtain [P n (z )Q n (z )]=[1κn −1z −1κn −1z −1]···[1κ0z −1κ0z −1][11],n =0,1,···,N (7)It is clear from Eq.(7)that P n (z )and Q n (z )are polynomials of z −1of order n .From Eq.(6),P N (z )is the denominator of H (z )while ∑N n =0λn Q n (z )is the numerator ofH (z ).Note that the number of parameters in the lattice structure (κn ,n =0,···N −1and λn ,n =0,···,N )is the same as the number of coeﬃcients of an N th order rational function.In summary,the system function H (z )can be determined by applying Eq.(4)recur-sively to ﬁnd P n (z )and Q n (z ),n =1,2,···,N ,given Q 0(z )=P 0(z )=1.Then use Eq.(6)to determine H (z ).•Convert the direct form to the lattice structureLet P n (z )=p n 0+p n 1z −1+···+p n n z−n and Q n (z )=q n 0+q n 1z −1+···+q n n z −n ;From Eq.(7),we have [P 1(z )Q 1(z )]=[1κ0z −1κ0z −1][11]=[1+κ0z −1κ0+z −1][P 2(z )Q 2(z )]=[1κ1z −1κ1z −1][1+κ0z −1κ0+z −1]=[1+(κ0+κ0κ1)z −1+κ1z −2κ1+(κ1κ0+κ0)z −1+z −2]...=...Hence we conclude by induction that p n n =κn −1and q n n =1for n =0,1,2,···,N .Moreover we have the following lemma.Lemma 1Q n (z )=z −n P n (z −1),n =0,1,···,NProof:This lemma can be proved by induction.The n =0case is trivialFor n =1,P 1(z )=1+κ0z −1and Q 1(z )=κ0+z −1.Thus the equality holds.Suppose that the equality holds for n =k ,i.e.Q k (z )=z −k P k (z −1).Equivalently,z −k Q k (z −1)=P k (z )For n =k +1,from Eq.(4)we haveP k +1(z )=P k (z )+κk z −1Q k (z )Q k +1(z )=κk P k (z )+z −1Q k (z )Thereforez −(k +1)P k +1(z −1)=z −k −1P k (z −1)+κk z −k Q k (z −1)=z −1Q k (z )+κk P k (z )=Q k +1(z )Thus,by mathematical induction Q n (z )=z −n P n (z −1)for n =0,1,···,NQ.E.D.Assume that κn =1for all n .If we inverse Eq.(5),we have[P n (z )z −1Q n (z )]=11−κ2n[1−κn −κn 1][P n +1(z )Q n +1(z )],n =0,1,···,N −1Hence,P n(z )=P n +1(z )−κn Q n +1(z )1−κ2n ,n =N −1,N −2,···,0(8)Let H (z )=B (z )A (z )=∑N n =0b n z −n 1−∑N n =1a n z −n ,where A (z )andB (z )are polynomials of z −1.Since p n n =κn −1for all n ,thereﬂection coeﬃcients κn ’s can be determined recur-sively by ﬁrst setting P N (z )=A (z )and Q N (z )=z −N P N (z −1).Then κN −1=p N N is determined.Applying Eq.(8)and Lemma 1recursively to ﬁnd P n (z ),κn ’s can be determined successively.To determine λn ,we observe that the coeﬃcient of z −N in the numerator must beλN since B (z )=∑Nn =0λn Q n (z )and q N N =1.Therefore λN =b N .We can removeλN Q N (z )from B (z ),resulting in a (N −1)th order polynomial,and determine λN −1by taking advantage of the property q n n =1for all n .The whole process continuous until all λn ’s are determined.In summaryP N =A (z ),S N =B (z ),λN =b Nfor n =N −1to 0κn =p n +1n +1Q n +1(z )=z −(n +1)P n +1(z −1)P n (z )=P n +1(z )−κn Q n +1(z )1−κ2nS n (z )=S n +1(z )−λn +1Q n +1(z )λn =s n nend•Stability of the Lattice Structure Form Eq.(4),we haveP 1(z )=1+κ0z −111is stable if and only if |κ0|<1.Since the lattice structure is a cascade of N similar stages,the stability of the ﬁlter can be veriﬁed easily as follows.Lemma 2The lattice structure in Fig.(2)is stable if and only if |κn |<1for all n .2All-pole SystemsAn all-pole system has no nonzero zeros,i.e.the system function is H (z )=1A (z ).In thelattice structure in Fig.(2),if λ0=1and λn =0for n >0,thenH (z )=∑N n =0λn Q n (z )P N (z )=1P N (z )Hence the all-pole system has a simpler lattice structure shown in Fig.(3).Figure 3:The lattice Structure for an all-pole systemOne interesting feature of the lattice structure in Fig.(3)is that the system function from x to v N is an all-pass system.This can be seen as follows.H all (z )=V N (z )X (z )=Q N (z )P N (z )=z −N P N (z −1)P N (z )If z 0is a pole of H all (z ),then 1/z 0must be a zero of H all (z )and vice versa.Due to the symmetry of poles and zeros,H all (z )is indeed an all-pass system.3Nonrecursive lattice structureIf H (z )=B (z ),i.e an FIR ﬁlter,the lattice structure becomes nonrecursive.We will explore its properties in this section.We would like to maintain the symmetric structure in Fig.(2)or Fig.(3)because previous results (e.q.Lemma 1)can be applied directly by doing so.In other words,Eq.(1)and Eq.(2)must hold for each stage.If H (z )is FIR,then G (z )=H −1(z )is an all-pole system.If we implement the all-pole system G (z )in the lattice form of Fig.(3),we haveG (z )=1H (z )=1P N (z )=U 0(z )U N (z )By exchanging its input and output,we get the desired FIR system.Note that in this FIR lattice structure,signals ﬂow from u 0to u N .Hence Eq.(3)should be used to compute the signal propagation from stage to stage.The corresponding lattice structure is shown in Fig.(4).Notice that the structure is nonrecursive.The system function implemented by the nonrecursive lattice structure can be constructed in the same way as the recursive lattice structure:P 0(z )=Q 0(z )=1for n =1to N [P n (z )Q n (z )]=[1κn −1z −1κn −1z −1][P n −1(z )Q n −1(z )]endH(z)=P N(z)Figure4:Nonrecursive Lattice StructureTo convert from a system function to the nonrecursive lattice structure,the algorithm is similar to that of the recursive version:P N=B(z)=1+b1z−1+···+b N z−Nfor n=N to1κn−1=p nnQ n(z)=z−n P n(z−1)P n−1(z)=P n(z)−κn−1Q n(z)1−κ2n−1endNote that from Lemma1,p n0=q nn=1for all n.Therefore the coeﬃcient of the constantterm in H(z),i.e.b0,must be1.If b0=1,an intuitive approach is to divide B(z)by b0. However,if b N=b0,as in the case of the linear phaseﬁlter,this will result inκN−1=1, and again we will run into trouble in computing the reﬂection coeﬃcients.A preferable way is to implement the FIR system H′(z)=1+(B(z)−b0)in a lattice structure and subtract 1−b0from its output.The idea is shown in Fig.(5)Figure5:Nonrecursive lattice structure for b0=1If we apply Lemma2to the nonrecursive lattice structure,we observe that B(z)is a minimum phase system if and only if|κn|<1for all n.If B(z)=P N(z)is a minimum phase system,then the system function from x=u0to v N,i.e.Q N(z),becomes a maximum phase system according to Lemma1,i.e.all its zeros are outside the unit circle.Aﬁnal remark of this lecture:According to Lemma2,each stage of the stable(or minimum phase)lattice structure is an attenuator,i.e.it does not amplify the signals.Thisproperty gives the lattice structure great computational stability and this is the primary reason that the lattice structure is implemented.However,the price for this property is the complex computation of the signalﬂow.。

半导体物理与器件英语

半导体物理与器件英语Semiconductor Physics and DevicesThe field of semiconductor physics and devices is a crucial aspect of modern technology, as it underpins the development of a wide range of electronic devices and systems that have transformed our daily lives. Semiconductors, which are materials with electrical properties that lie between those of conductors and insulators, have been the backbone of the digital revolution, enabling the creation of integrated circuits, transistors, and other essential components found in smartphones, computers, and a myriad of other electronic devices.At the heart of semiconductor physics is the study of the behavior of electrons and holes within these materials. Electrons, which are negatively charged particles, and holes, which are the absence of electrons and carry a positive charge, are the fundamental charge carriers in semiconductors. The interactions and movement of these charge carriers within the semiconductor lattice structure are governed by the principles of quantum mechanics and solid-state physics.One of the fundamental concepts in semiconductor physics is theenergy band structure. Semiconductors have a unique energy band structure, with a filled valence band and an empty conduction band separated by an energy gap. The size of this energy gap determines the semiconductor's electrical properties, with materials having a smaller energy gap being more conductive than those with a larger gap.The ability to manipulate the energy band structure and the behavior of charge carriers in semiconductors has led to the development of a wide range of electronic devices. The most prominent of these is the transistor, a fundamental building block of modern electronics. Transistors are used to amplify or switch electronic signals and power, and they are the essential components in integrated circuits, which are the heart of digital devices such as computers, smartphones, and various other electronic systems.Another important class of semiconductor devices are diodes, which are two-terminal devices that allow the flow of current in only one direction. Diodes are used in a variety of applications, including power supplies, rectifiers, and light-emitting diodes (LEDs). LEDs, in particular, have become ubiquitous in modern lighting and display technologies, offering improved energy efficiency, longer lifespan, and enhanced color quality compared to traditional incandescent and fluorescent light sources.Semiconductor devices are not limited to electronic applications; they also play a crucial role in optoelectronics, a field that deals with the interaction between light and electronic devices. Photodetectors, such as photodiodes and phototransistors, are semiconductor devices that convert light into electrical signals, enabling a wide range of applications, including imaging, optical communication, and solar energy conversion.The development of semiconductor physics and devices has been a continuous process, driven by the relentless pursuit of improved performance, efficiency, and functionality. Over the past several decades, we have witnessed remarkable advancements in semiconductor technology, with the miniaturization of devices, the introduction of new materials, and the development of innovative device architectures.One of the most significant trends in semiconductor technology has been the scaling of transistor dimensions, often referred to as Moore's Law. This observation, made by Intel co-founder Gordon Moore in 1965, predicted that the number of transistors on a microchip would double approximately every two years, leading to a dramatic increase in computing power and a corresponding decrease in device size and cost.This scaling has been achieved through a combination ofadvancements in fabrication techniques, material engineering, and device design. For example, the use of high-k dielectric materials and the implementation of FinFET transistor architectures have allowed for continued scaling of transistor dimensions while maintaining or improving device performance and power efficiency.Beyond the scaling of individual devices, the integration of multiple semiconductor components on a single integrated circuit has led to the development of increasingly complex and capable electronic systems. System-on-a-chip (SoC) designs, which incorporate various functional blocks such as processors, memory, and input/output interfaces on a single semiconductor die, have become ubiquitous in modern electronic devices, enabling greater functionality, reduced power consumption, and improved overall system performance.The future of semiconductor physics and devices holds immense promise, with researchers and engineers exploring new materials, device architectures, and application domains. The emergence of wide-bandgap semiconductors, such as silicon carbide (SiC) and gallium nitride (GaN), has opened up new possibilities in high-power, high-frequency, and high-temperature electronics, enabling advancements in areas like electric vehicles, renewable energy systems, and communication networks.Additionally, the integration of semiconductor devices with otheremerging technologies, such as quantum computing, neuromorphic computing, and flexible/wearable electronics, is paving the way for even more transformative applications. These developments have the potential to revolutionize fields ranging from healthcare and transportation to energy and communication, ultimately enhancing our quality of life and shaping the technological landscape of the future.In conclusion, the field of semiconductor physics and devices is a cornerstone of modern technology, underpinning the development of a vast array of electronic devices and systems that have become indispensable in our daily lives. The continuous advancements in this field, driven by the relentless pursuit of improved performance, efficiency, and functionality, have been instrumental in driving the digital revolution and shaping the technological landscape of the21st century. As we move forward, the future of semiconductor physics and devices promises even more remarkable innovations and transformative applications that will continue to shape our world.。

1997-JPC-(CdSe)ZnS Core-Shell Quantum Dots Synthesi_1`s and Characterization of a Size Series of

(CdSe)ZnS Core-Shell Quantum Dots:Synthesis and Characterization of a Size Series of Highly Luminescent NanocrystallitesB.O.Dabbousi,†J.Rodriguez-Viejo,‡F.V.Mikulec,†J.R.Heine,§H.Mattoussi,§R.Ober,⊥K.F.Jensen,‡,§and M.G.Bawendi*,†Departments of Chemistry,Chemical Engineering,and Materials Science and Engineering,Massachusetts Institute of Technology,77Massachusetts A V e.,Cambridge,Massachusetts02139,andLaboratoire de Physique de la Matie`re Condense´e,Colle`ge de France,11Place Marcellin Berthelot,75231Paris Cedex05,FranceRecei V ed:March27,1997;In Final Form:June26,1997XWe report a synthesis of highly luminescent(CdSe)ZnS composite quantum dots with CdSe cores ranging indiameter from23to55Å.The narrow photoluminescence(fwhm e40nm)from these composite dotsspans most of the visible spectrum from blue through red with quantum yields of30-50%at room temperature.We characterize these materials using a range of optical and structural techniques.Optical absorption andphotoluminescence spectroscopies probe the effect of ZnS passivation on the electronic structure of the dots.We use a combination of wavelength dispersive X-ray spectroscopy,X-ray photoelectron spectroscopy,smalland wide angle X-ray scattering,and transmission electron microscopy to analyze the composite dots anddetermine their chemical composition,average size,size distribution,shape,and internal ing asimple effective mass theory,we model the energy shift for the first excited state for(CdSe)ZnS and(CdSe)-CdS dots with varying shell thickness.Finally,we characterize the growth of ZnS on CdSe cores as locallyepitaxial and determine how the structure of the ZnS shell influences the photoluminescence properties.I.IntroductionSemiconductor nanocrystallites(quantum dots)whose radii are smaller than the bulk exciton Bohr radius constitute a class of materials intermediate between molecular and bulk forms of matter.1Quantum confinement of both the electron and hole in all three dimensions leads to an increase in the effective band gap of the material with decreasing crystallite size.Conse-quently,both the optical absorption and emission of quantum dots shift to the blue(higher energies)as the size of the dots gets smaller.Although nanocrystallites have not yet completed their evolution into bulk solids,structural studies indicate that they retain the bulk crystal structure and lattice parameter.2 Recent advances in the synthesis of highly monodisperse nanocrystallites3-5have paved the way for numerous spectro-scopic studies6-11assigning the quantum dot electronic states and mapping out their evolution as a function of size.Core-shell type composite quantum dots exhibit novel properties making them attractive from both an experimental and a practical point of view.12-19Overcoating nanocrystallites with higher band gap inorganic materials has been shown to improve the photoluminescence quantum yields by passivating surface nonradiative recombination sites.Particles passivated with inorganic shell structures are more robust than organically passivated dots and have greater tolerance to processing conditions necessary for incorporation into solid state structures. Some examples of core-shell quantum dot structures reported earlier include CdS on CdSe and CdSe on CdS,12ZnS grown on CdS,13ZnS on CdSe and the inverse structure,14CdS/HgS/ CdS quantum dot quantum wells,15ZnSe overcoated CdSe,16 and SiO2on Si.17,18Recently,Hines and Guyot-Sionnest reported making(CdSe)ZnS nanocrystallites whose room tem-perature fluorescence quantum yield was50%.19This paper describes the synthesis and characterization of a series of room-temperature high quantum yield(30%-50%) core-shell(CdSe)ZnS nanocrystallites with narrow band edge luminescence spanning most of the visible spectrum from470 to625nm.These particles are produced using a two-step synthesis that is a modification of the methods of Danek et al.16 and Hines et al.19ZnS overcoated dots are characterized spectroscopically and structurally using a variety of techniques. The optical absorption and photoluminescence spectra of the composite dots are measured,and the lowest energy optical transition is modeled using a simplified theoretical approach. Wavelength dispersive X-ray spectroscopy and X-ray photo-electron spectroscopy are used to determine the elemental and spatial composition of ZnS overcoated dots.Small-angle X-ray scattering in solution and in polymer films and high-resolution transmission electron microscopy measurements help to deter-mine the size,shape,and size distribution of the composite dots. Finally,the internal structure of the composite quantum dots and the lattice parameters of the core and shell are determined using wide-angle X-ray scattering.In addition to having higher efficiencies,ZnS overcoated particles are more robust than organically passivated dots and potentially more useful for optoelectronic device structures. Electroluminescent devices(LED’s)incorporating(CdSe)ZnS dots into heterostructure organic/semiconductor nanocrystallite light-emitting devices may show greater stability.20Thin films incorporating(CdSe)ZnS dots into a matrix of ZnS using electrospray organometallic chemical vapor deposition(ES-OMCVD)demonstrate more than2orders of magnitude improvement in the PL quantum yields(∼10%)relative to identical structures based on bare CdSe dots.21In addition,these structures exhibit cathodoluminescence21upon excitation with high-energy electrons and may potentially be useful in the*To whom correspondence should be addressed.†Department of Chemistry,MIT.‡Department of Chemical Engineering,MIT.§Department of Materials Science and Engineering,MIT.⊥Colle`ge de France.X Abstract published in Ad V ance ACS Abstracts,September1,1997.9463J.Phys.Chem.B1997,101,9463-9475S1089-5647(97)01091-2CCC:$14.00©1997American Chemical Societyproduction of alternating current thin film electroluminescent devices(ACTFELD).II.Experimental SectionMaterials.Trioctylphosphine oxide(TOPO,90%pure)and trioctylphosphine(TOP,95%pure)were obtained from Strem and Fluka,respectively.Dimethylcadmium(CdMe2)and di-ethylzinc(ZnEt2)were purchased from Alfa and Fluka,respec-tively,and both materials were filtered separately through a0.2µm filter in an inert atmosphere box.Trioctylphosphine selenide was prepared by dissolving0.1mol of Se shot in100mL of TOP,thus producing a1M solution of TOPSe.Hexamethyl-disilathiane((TMS)2S)was used as purchased from Aldrich. HPLC grade n-hexane,methanol,pyridine,and1-butanol were purchased from EM Sciences.Synthesis of Composite Quantum Dots.(CdSe)ZnS.Nearly monodisperse CdSe quantum dots ranging from23to55Åin diameter were synthesized via the pyrolysis of the organome-tallic precursors,dimethylcadmium and trioctylphosphine se-lenide,in a coordinating solvent,trioctylphosphine oxide (TOPO),as described previously.3The precursors were injected at temperatures ranging from340to360°C,and the initially formed small(d)23Å)dots were grown at temperatures between290and300°C.The dots were collected as powders using size-selective precipitation3with methanol and then redispersed in hexane.A flask containing5g of TOPO was heated to190°C under vacuum for several hours and then cooled to60°C after which 0.5mL of trioctylphosphine(TOP)was added.Roughly0.1-0.4µmol of CdSe dots dispersed in hexane was transferred into the reaction vessel via syringe,and the solvent was pumped off.Diethylzinc(ZnEt2)and hexamethyldisilathiane((TMS)2S) were used as the Zn and S precursors.The amounts of Zn and S precursors needed to grow a ZnS shell of desired thickness for each CdSe sample were determined as follows:First,the average radius of the CdSe dots was estimated from TEM or SAXS measurements.Next,the ratio of ZnS to CdSe necessary to form a shell of desired thickness was calculated based on the ratio of the shell volume to that of the core assuming a spherical core and shell and taking into account the bulk lattice parameters of CdSe and ZnS.For larger particles the ratio of Zn to Cd necessary to achieve the same thickness shell is less than for the smaller dots.The actual amount of ZnS that grows onto the CdSe cores was generally less than the amount added due to incomplete reaction of the precursors and to loss of some material on the walls of the flask during the addition. Equimolar amounts of the precursors were dissolved in2-4 mL of TOP inside an inert atmosphere glovebox.The precursor solution was loaded into a syringe and transferred to an addition funnel attached to the reaction flask.The reaction flask containing CdSe dots dispersed in TOPO and TOP was heated under an atmosphere of N2.The temperature at which the precursors were added ranged from140°C for23Ådiameter dots to220°C for55Ådiameter dots.22When the desired temperature was reached,the Zn and S precursors were added dropwise to the vigorously stirring reaction mixture over a period of5-10min.After the addition was complete,the mixture was cooled to 90°C and left stirring for several hours.A5mL aliquot of butanol was added to the mixture to prevent the TOPO from solidifying upon cooling to room temperature.The overcoated particles were stored in their growth solution to ensure that the surface of the dots remained passivated with TOPO.They were later recovered in powder form by precipitating with methanol and redispersed into a variety of solvents including hexane, chloroform,toluene,THF,and pyridine.(CdSe)CdS.Cadmium selenide nanocrystallites with diam-eters between33.5and35Åwere overcoated with CdS to varying thickness using the same basic procedure as that outlined for the ZnS overcoating.The CdS precursors used were Me2-Cd and(TMS)2S.The precursor solution was dripped into the reaction vessel containing the dots at a temperature of180°C and a rate of∼1mL/min.The solution became noticeably darker as the overcoat precursors were added.Absorption spectra taken just after addition of precursors showed a significant shift in the absorption peak to the red.To store these samples,it was necessary to add equal amounts of hexane and butanol since the butanol by itself appeared to flocculate the particles.Optical Characterization.UV-vis absorption spectra were acquired on an HP8452diode array spectrophotometer.Dilute solutions of dots in hexane were placed in1cm quartz cuvettes, and their absorption and corresponding fluorescence were measured.The photoluminescence spectra were taken on a SPEX Fluorolog-2spectrometer in front face collection mode. The room-temperature quantum yields were determined by comparing the integrated emission of the dots in solution to the emission of a solution of rhodamine590or rhodamine640 of identical optical density at the excitation wavelength. Wavelength Dispersive X-ray Spectroscopy.A JEOL SEM 733electron microprobe operated at15kV was used to determine the chemical composition of the composite quantum dots using wavelength dispersive X-ray(WDS)spectroscopy. One micrometer thick films of(CdSe)ZnS quantum dots were cast from concentrated pyridine solutions onto Si(100)wafers, and after the solvent had completely evaporated the films were coated with a thin layer of amorphous carbon to prevent charging.X-ray Photoelectron Spectroscopy.XPS was performed using a Physical Electronics5200C spectrometer equipped with a dual X-ray anode(Mg and Al)and a concentric hemispherical analyzer(CHA).Data were obtained with Mg K R radiation (1253.6eV)at300W(15keV,20mA).Survey scans were collected over the range0-1100eV with a179eV pass energy detection,corresponding to a resolution of2eV.Close-up scans were collected on the peaks of interest for the different elements with a71.5eV pass energy detection and a resolution of1eV.A base pressure of10-8Torr was maintained during the experiments.All samples were exchanged with pyridine and spin-cast onto Si substrates,forming a thin film several monolayers thick.Transmission Electron Microscopy.A Topcon EM002B transmission electron microscope(TEM)was operated at200 kV to obtain high-resolution images of individual quantum dots. An objective aperture was used to selectively image the(100), (002),and(101)wurtzite lattice planes.The samples were prepared by placing one drop of a dilute solution of dots in octane onto a copper grid supporting a thin film of amorphous carbon and then wicking off the remaining solvent after30s.A second thin layer of amorphous carbon was evaporated onto the samples in order to minimize charging and reduce damage to the particles caused by the electron beam.Small-Angle X-ray Scattering(SAXS)in Polymer Films. Small-angle X-ray scattering(SAXS)samples were prepared using either poly(vinyl butyral)(PVB)or a phosphine-func-tionalized diblock copolymer[methyltetracyclododecene]300-[norbornene-CH2O(CH2)5P(oct)2]20,abbreviated as(MTD300P20), as the matrix.23Approximately5mg of nanocrystallites of dispersed in1mL of toluene,added to0.5mL of a solution containing10wt%PVB in toluene,concentrated under vacuum to give a viscous solution,and then cast onto a silicon wafer. The procedure is the same for MTD300P20,except THF is used9464J.Phys.Chem.B,Vol.101,No.46,1997Dabbousi et al.as the solvent for both nanocrystallites and polymer.The resulting∼200µm thick film is clear to slightly opaque.X-ray diffraction spectra were collected on a Rigaku300Rotaflex diffractometer operating in the Bragg configuration using Cu K R radiation.The accelerating voltage was set at60kV with a300mA flux.Scatter and diffraction slits of1/6°and a0.3 mm collection slit were used.Small-Angle X-ray Scattering in Dilute Solutions.The X-ray source was a rotating copper anode operated at40kV and25mA.The apparent point source(electron beam irradiated area on the anode)was about10-2mm2.The beam was collimated onto a position sensitive detector,PSPE(ELPHYSE).A thin slit,placed before the filter,selects a beam with the dimensions of3×0.3mm2on the detector.The position sensitive linear detector has a useful length of50mm,placed at a distance D)370mm from the detector.The spatial resolution on the detector is200µm.This setup allows a continuous scan of scattering wavevectors between6×10-3 and0.40Å-1,with a resolution of about3×10-3Å-1.The samples used were quartz capillary tubes with about1 mm optical path,filled with the desired dispersion,and then flame-sealed after filling.The intensity from the reference,I ref, is collected first,and then the intensity from the sample,I s.The intensity used in the data analysis is the difference:I)I s-I ref.Wide-Angle X-ray Scattering(WAXS).The wide-angle X-ray powder diffraction patterns were measured on the same setup as the SAXS in polymer dispersions.The TOPO/TOP capped nanocrystals were precipitated with methanol and exchanged with pyridine.The samples were prepared by dropping a heavily concentrated solution of nanocrystals dispersed in pyridine onto silicon wafers.A slow evaporation of the pyridine leads to the formation of glassy thin films which were used for the diffraction experiments.III.Results and AnalysisA.Synthesis of Core-Shell Composite Quantum Dots. We use a two-step synthetic procedure similar to that of Danek et al.16and Hines et al.19to produce(CdSe)ZnS core-shell quantum dots.In the first step we synthesize nearly mono-disperse CdSe nanocrystallites ranging in size from23to55Åvia a high-temperature colloidal growth followed by size selective precipitation.3These dots are referred to as“bare”dots in the remainder of the text,although their outermost surface is passivated with organic TOPO/TOP capping groups. Next,we overcoat the CdSe particles in TOPO by adding the Zn and S precursors at intermediate temperatures.22The resulting composite particles are also passivated with TOPO/ TOP on their outermost surface.The temperature at which the dots are overcoated is very critical.At higher temperatures the CdSe seeds begin to grow via Ostwald ripening,and their size distribution deteriorates, leading to broader spectral line widths.Overcoating the particles at relatively low temperatures could lead to incomplete decom-position of the precursors or to reduced crystallinity of the ZnS shell.An ideal growth temperature is determined independently for each CdSe core size to ensure that the size distribution of the cores remains constant and that shells with a high degree of crystallinity are formed.22The concentration of the ZnS precursor solution and the rate at which it is added are also critical.Slow addition of the precursors at low concentrations ensures that most of the ZnS grows heterogeneously onto existing CdSe nuclei instead of undergoing homogeneous nucleation.This probably does not eliminate the formation of small ZnS particles completely so a final purification step in which the overcoated dots are subjected to size selective precipitation provides further assurance that mainly(CdSe)ZnS particles are present in the final powders.B.Optical Characterization.The synthesis presented above produces ZnS overcoated dots with a range of core and shell sizes.Figure1shows the absorption spectra of CdSe dots ranging from23to55Åin diameter before(dashed lines)and after(solid lines)overcoating with1-2monolayers of ZnS. The definition of a monolayer here is a shell of ZnS that measures3.1Å(the distance between consecutive planes along the[002]axis in bulk wurtzite ZnS)along the major axis of the prolate-shaped dots.We observe a small shift in the absorption spectra to the red(lower energies)after overcoating due to partial leakage of the exciton into the ZnS matrix.This red shift is more pronounced in smaller dots where the leakage of the exciton into the ZnS shell has a more dramatic effect on the confinement energies of the charge carriers.Figure2shows the room-temperature photoluminescence spectra(PL)of these Figure 1.Absorption spectra for bare(dashed lines)and1-2 monolayer ZnS overcoated(solid lines)CdSe dots with diameters measuring(a)23,(b)42,(c)48,and(d)55Å.The absorption spectra for the(CdSe)ZnS dots are broader and slightly red-shifted from their respective bare dot spectra.Figure2.Photoluminescence(PL)spectra for bare(dashed lines)and ZnS overcoated(solid lines)dots with the following core sizes:(a) 23,(b)42,(c)48,and(d)55Åin diameter.The PL spectra for the overcoated dots are much more intense owing to their higher quantum yields:(a)40,(b)50,(c)35,and(d)30.(CdSe)ZnS Core-Shell Quantum Dots J.Phys.Chem.B,Vol.101,No.46,19979465same samples before (dashed lines)and after (solid lines)overcoating with ZnS.The PL quantum yield increases from 5to 15%for bare dots to values ranging from 30to 50%for dots passivated with ZnS.In smaller CdSe dots the surface-to-volume ratio is very high,and the PL for TOPO capped dots is dominated by broad deep trap emission due to incomplete surface passivation.Overcoating with ZnS suppresses deep trap emission by passivating most of the vacancies and trap sites on the crystallite surface,resulting in PL which is dominated by band-edge recombination.Figure 3(color photograph)displays the wide spectral range of luminescence from (CdSe)ZnS composite quantum dots.The photograph shows six different samples of ZnS overcoated CdSe dots dispersed in dilute hexane solutions and placed in identical quartz cuvettes.The samples are irradiated with 365nm ultraviolet light from a UV lamp in order to observe lumines-cence from all the solutions at once.As the size of the CdSe core increases,the color of the luminescence shows a continuous progression from blue through green,yellow,orange,to red.In the smallest sizes of TOPO capped dots the color of the PL is normally dominated by broad deep trap emission and appears as faint white light.After overcoating the samples with ZnS the deep trap emission is nearly eliminated,giving rise to intense blue band-edge fluorescence.To understand the effect of ZnS passivation on the optical and structural properties of CdSe dots,we synthesized a large quantity of ∼40Ådiameter CdSe dots.We divided this sample into multiple fractions and added varying amounts of Zn and S precursors to each fraction at identical temperatures and addition times.The result was a series of samples with similar CdSe cores but with varying ZnS shell thickness.Figure 4shows the progression of the absorption spectrum for these samples with ZnS coverages of approximately 0(bare TOPO capped CdSe),0.65,1.3,2.6,and 5.3monolayers.(See beginning of this section for definition of number of monolayers.)The spectra reflect a constant area under the lowest energy 1S 3/2-1S e absorption peak (constant oscillator strength)for the samples with varying ZnS coverage.As the thickness of the ZnS shell increases,there is a shift in the 1S 3/2-1S e absorption to the red,reflecting an increased leakage of the exciton into the shell,as well as a broadening of the absorption peak,indicating a distribution of shell thickness.The left-hand side of Figure 4shows an increased absorption in the ultraviolet with increasing ZnS coverage as a result of direct absorption into the higher band gap ZnS shell.The evolution of the PL for the same ∼40Ådiameter dots with ZnS coverage is displayed in Figure 5.As the coverage of ZnS on the CdSe surface increases,we see a dramatic increase in the fluorescence quantum yield followed by a steadydeclineFigure 3.Color photograph demonstrating the wide spectral range of bright fluorescence from different size samples of (CdSe)ZnS.Their PL peaks occur at (going from left to right)470,480,520,560,594,and 620nm (quartz cuvettes courtesy of Spectrocell Inc.,photography by F.Frankel).Figure 4.Absorption spectra for a series of ZnS overcoated samples grown on identical 42Å(10%CdSe seed particles.The samples displayed have the following coverage:(a)bare TOPO capped,(b)0.65monolayers,(c)1.3monolayers,(d)2.6monolayers,and (e)5.3monolayers (see definition for monolayers in text).The right-hand side shows the long wavelength region of the absorption spectra showing the lowest energy optical transitions.The spectra demonstrate an increased red-shift with thicker ZnS shells as well as a broadening of the first peak as a result of increased polydispersity.The left-hand side highlights the ultraviolet region of the spectra showing an increased absorption at higher energies with increasing coverage due to direct absorption into the ZnS shell.9466J.Phys.Chem.B,Vol.101,No.46,1997Dabbousi et al.after∼1.3monolayers of ZnS.The spectra are red-shifted (slightly more than the shift in the absorption spectra)and showan increased broadening at higher coverage.The inset to Figure 5charts the evolution of the quantum yield for these dots as a function of the ZnS shell thickness.For this particular sample the quantum yield starts at15%for the bare TOPO capped CdSe dots and increases with the addition of ZnS,approaching a maximum value of50%at approximately∼1.3monolayer coverage.At higher coverage the quantum yield begins to decrease steadily until it reaches a value of30%at about∼5 monolayer coverage.In the following sections we explain the trends in PL quantum yield based on the structural characteriza-tion of ZnS overcoated samples.C.Structural Characterization.Wa V elength Dispersi V e X-ray Spectroscopy.We analyze the elemental composition of the ZnS overcoated samples using wavelength dispersive X-ray spectroscopy(WDS).This method provides a quantitative analysis of the elemental composition with an uncertainty of less than(5%.We focus on obtaining a Zn/Cd ratio for the ZnS overcoated samples of interest.Analysis of the series of samples with a∼40Ådiameter core and varying ZnS coverage gives the Zn/Cd ratios which appear in Table1.The WDS analysis confirms that the Zn-to-Cd ratio in the composite dots increases as more ZnS is added.We also use this technique to measure the Se/Cd ratio in the bare dots.We consistently measure a Se/Cd ratio of∼0.8-0.9/1,indicating Cd-rich nanoparticles.X-ray Photoelectron Spectroscopy.Multiple samples of ∼33and∼40Ådiameter CdSe quantum dots overcoated with variable amounts of ZnS were examined by XPS.Figure6shows the survey spectra of∼40Ådiameter bare dots and ofthe same sample overcoated with∼1.3monolayers of ZnS.Thepresence of C and O comes mainly from atmospheric contami-nation during the brief exposure of the samples to air(typicallyaround15min).The positions of both C and O lines correspondto standard values for adsorbed species,showing the absenceof significant charging.24As expected,we detect XPS linesfrom Zn and S in addition to the Cd and Se lines.Althoughthe samples were exchanged with pyridine before the XPSmeasurements,small amounts of phosphorus could be detectedon both the bare and ZnS overcoated CdSe dots,indicating thepresence of residual TOPO/TOP molecules bound to Cd or Znon the nanocrystal surfaces.25The relative concentrations ofCd and Se are calculated by dividing the area of the XPS linesby their respective sensitivity factors.24In the case of nano-crystals the sensitivity factor must be corrected by the integral∫0d e-z/λd z to account for the similarity between the size of the nanocrystals and the escape depths of the electrons.26Theintegral must be evaluated over a sphere to obtain the Se/Cdratios in CdSe dots.In the bare CdSe nanocrystals the Se/Cdratio was around0.87,corresponding to46%Se and54%Cd.This value agrees with the WDS results.We use the Auger parameter,defined as the difference inbinding energy between the photoelectron and Auger peaks,toidentify the nature of the bond in the different samples.24Thisdifference can be accurately determined because static chargecorrections cancel.The Auger parameter of Cd in the bare andTABLE1:Summary of the Results Obtained from WDS,TEM,SAXS,and WAXS Detailing the Zn/Cd Ratio,Average Size, Size Distribution,and Aspect Ratio for a Series of(CdSe)ZnS Samples with a∼40ÅDiameter CdSe Cores and Varying ZnS CoverageZnS coverage(TEM)measd TEM size measd averageaspect ratiocalcd size(SAXSin polymer)measd Zn/Cdratio(WDS)calcd Zn/Cd ratio(SAXS in polymer)calcd Zn/Cd ratio(WAXS)bare39Å(8.2% 1.1242Å(10%0.65monolayers43Å(11% 1.1646Å(13%0.460.580.71.3monolayers47Å(10% 1.1650Å(18% 1.50 1.32 1.42.6monolayers55Å(13% 1.233.60 2.9 5.3monolayers72Å(19% 1.23 6.80 6.8 Figure5.PL spectra for a series of ZnS overcoated dots with42(10%Ådiameter CdSe cores.The spectra are for(a)0,(b)0.65,(c)1.3,(d)2.6,and(e)5.3monolayers ZnS coverage.The position of themaximum in the PL spectrum shifts to the red,and the spectrumbroadens with increasing ZnS coverage.(inset)The PL quantum yieldis charted as a function of ZnS coverage.The PL intensity increaseswith the addition of ZnS reaching,50%at∼1.3monolayers,and then declines steadily at higher coverage.The line is simply a guide to the eye.Figure6.(A)Survey spectra of(a)∼40Ådiameter bare CdSe dots and(b)the same dots overcoated with ZnS showing the photoelectron and Auger transitions from the different elements present in the quantum dots.(B)Enlargement of the low-energy side of the survey spectra, emphasizing the transitions with low binding energy.(CdSe)ZnS Core-Shell Quantum Dots J.Phys.Chem.B,Vol.101,No.46,19979467overcoated samples is466.8(0.2eV and corresponds exactly to the expected value for bulk CdSe.In the case of ZnS the Auger parameter for Zn in the1.3and2.6monolayer ZnS samples is757.5eV,which is also very close to the expected value of758.0eV.The degree of passivation of the CdSe surface with ZnS is examined by exposing the nanocrystal surface to air for extended periods of time and studying the evolution of the Se peak. The oxidation of CdSe quantum dots leads to the formation of a selenium oxide peak at higher energies than the main Se peak.27Figure7shows the formation of a SeO2peak at59eV after an80h exposure to air in both the bare,TOPO capped, CdSe and0.65monolayer ZnS overcoated samples.These results indicate that in the0.65monolayer samples the ZnS shell does not completely surround the CdSe nanocrystals,and there are still Se sites at the surface that are susceptible to oxidation. In samples with an estimated coverage of∼1.3monolayers ZnS or more the oxide peak does not appear even after prolonged exposure to air,indicating that the CdSe surface is possibly protected by a continuous ZnS shell.After exposure to air for 16h,the bare CdSe nanocrystals display a selenium oxide peak which represents13%of the total Se signal,and the Se/Cd ratio decreases to0.77,corresponding to43%Se and57%Cd.The same sample after80h exposure to air had a ratio of Se/Cd of 0.37(28%Se and72%Cd),and the SeO2peak area was22% of the total Se signal.For a∼40Ådiameter sample,34%of the atoms are at the surface which means that in the sample measured most of the surface Se has been desorbed from the surface after80h.In the samples with more than 1.3 monolayers of ZnS coverage no change in the Se/Cd ratio was detected even after exposure to air for80h.Although no Cd-(O)peak appears after similar exposure to air,the Cd Auger parameter shifts from466.8eV for bare unoxidized CdSe to 467.5eV for particles exposed to air for80h.The Auger parameter for the1.3and2.6monolayer coverage samples remains the same even after prolonged exposure to air. Another method to probe the spatial location of the ZnS relative to the CdSe core is obtained by comparing the ratios of the XPS and Auger intensities of the Cd photoelectrons for bare and overcoated samples.14,28The depth dependence of the observed intensity for the Auger and XPS photoemitted electrons iswhere J0is the X-ray flux,N(z)i is the number of i atoms,σi is the absorption cross section for atoms i,Y i,n is the emission quantum yield of Auger or XPS for atoms i,F(KE)is the energy-dependent instrument response function,andλ(KE)is the energy-dependent escape depth.Taking the ratio of the intensities of the XPS and Auger lines from the same atom,Cd or Zn,it is possible to eliminate the X-ray flux,number of atoms, and absorption cross sections from the intensity equations for the Auger and the primary X-ray photoelectrons.The value of the intensity ratio I)i overcoated(Cd)/i bare(Cd),where i)i XPS-(Cd)/i Auger(Cd),is only a function of the relative escape depths of the electrons.Therefore,due to the smaller escape depths of the Cd Auger electrons in both ZnS(13.2Å)and CdSe(10Å)compared to the Cd XPS photoelectron(23.7Åin ZnS and 15Åin CdSe),the intensity I should increase with the amount of ZnS on the CdSe surface.Calculated values of1.28and 1.60for the0.65and2.6monolayer,respectively,confirm the growth of ZnS on the surface of the CdSe dots. Transmission Electron Microscopy.High-resolution TEM allows us to qualitatively probe the internal structure of the composite quantum dots and determine the average size,size distribution,and aspect ratio of overcoated particles as a function of ZnS coverage.We image the series of(CdSe)ZnS samples described earlier.Figure8shows two dots from that series, one with(A)no ZnS overcoating(bare)and one with(B)2.6 monolayers of ZnS.The particles in the micrographs show well-resolved lattice fringes with a measured lattice spacing in the bare dots similar to bulk CdSe.For the2.6monolayer sample these lattice fringes are continuous throughout the entire particle; the growth of the ZnS shell appears to be epitaxial.A well-defined interface between CdSe core and ZnS shell was not observed in any of the samples,although the“bending”of the lattice fringes in Figure8B s the lower third of this particle is slightly askew compared with the upper part s may be suggestive of some sort of strain in the material.This bending is somewhat anomalous,however,as the lattice fringes in most particles were straight.Some patchy growth is observed for the highest coverage samples,giving rise to misshapen particles,but we do not observe discrete nucleation of tethered ZnS particles on the surface of existing CdSe particles.We analyze over150 crystallites in each sample to obtain statistical values for the length of the major axis,the aspect ratio,and the distribution of lengths and aspect ratios for all the samples.Figure9shows histograms of size distributions and aspect ratio from these same samples.This figure shows the measured histograms for(A)Figure7.X-ray photoelectron spectra highlighting the Se3d core transitions from∼40Åbare and ZnS overcoated CdSe dots:(a)bare CdSe,(b)0.65monolayers,(c)1.3monolayers,and(d)2.6monolayers of ZnS.The peak at59eV indicates the formation of selenium oxide upon exposure to air when surface selenium atoms areexposed.Figure8.Transmission electron micrographs of(A)one“bare”CdSe nanocrystallite and(B)one CdSe nanocrystallite with a2.6monolayer ZnS shell.I)JN(z)iσiYi,nF(KE)e-z/λ(KE)(1)9468J.Phys.Chem.B,Vol.101,No.46,1997Dabbousi et al.。

关于我喜欢的建筑埃菲尔铁塔的英语作文

关于我喜欢的建筑埃菲尔铁塔的英语作文The Eiffel Tower is an iconic structure that has captivated people from around the world for more than a century. Its striking design and towering presence make it a symbol of both French culture and engineering marvel. From its construction to its cultural significance, the Eiffel Tower continues to fascinate and inspire visitors.埃菲尔铁塔是一个标志性建筑，已经吸引了世界各地的人们超过一个世纪。

它醒目的设计和高耸的存在使其成为法国文化和工程奇迹的象征。

从它的建造到其文化意义，埃菲尔铁塔继续着迷和激发着游客。

Designed by Gustave Eiffel, the tower was initially met with mixed reactions when it was unveiled for the 1889 Exposition Universelle in Paris. Some saw it as an eyesore, while others praised its innovation and daring design. Standing at a height of 324 meters, it held the record for the tallest man-made structure until the completion of the Chrysler Building in New York City in 1930.由居斯塔夫·埃菲尔设计，这座塔在1889年巴黎世界博览会上首次亮相时，曾引起反响不一。

对苯二甲酸锌

对苯⼆甲酸锌Hydrothermal Synthesis and Crystal Structure of a Novel 2-Fold Interpenetrated Framework Based on Tetranuclear Homometallic ClusterRong-Yi Huang ?Xue-Jun Kong ?Guang-Xiang LiuReceived:15December 2007/Accepted:11January 2008/Published online:5March 2008óSpringer Science+Business Media,LLC 2008Abstract A novel 2-fold parallel interpenetrated polymer,Zn 2(OH)(pheno)(p -BDC)1.5áH 2O (1)(pheno =phenan-threne-9,10-dione;p -BDC =1,4-benzenedicarboxylate)was prepared by hydrothermal synthesis and characterized by IRspectra,elemental analysis and single crystal X-ray /doc/c97a12ccf61fb7360b4c65f3.html plex1crystallizes in the orthorhombic space group Pbca and affords a three-dimensional (3D)six-connected a -Ponetwork.Keywords Carboxylate ligand áHomometallic complex áa -Po1IntroductionIn the last decade,the construction by design of metal-organic frameworks (MOFs)using various secondary building units (SBUs)connected through coordination bonds,supramolecular contacts (e.g.,hydrogen bonding,p áááp stacking,etc.),or their combination has been an increasingly active research area [1].The design and controlled assembly of coordination polymers based on nano-sized MO(OH)clusters and multi-functional car-boxylates have been extensively developed for their crystallographic and potential applications in catalysis,nonlinear optics,ion exchange,gas storage,magnetism and molecular recognition [2].In most cases,multinu-clear metal cluster SBUs can direct the formation of novel geometry and topology of molecular architectureand help to retain the rigidity of the networks [3].A number of carboxylate-bridged metal clusters have been utilized to build extended coordination frameworks.Among these compounds,frameworks from multinuclear zinc cluster SBUs,including dinuclear (Zn 2)[4],trinu-clear (Zn 3)[5],tetranuclear (Zn 4)[6],pentanuclear (Zn 5)[7],hexanuclear (Zn 6)[8],heptanuclear (Zn 7) [9],and octanuclear (Zn 8)[10]clusters have attracted great interest and have been investigated extensively.Addi-tionally,a series of systematic studies on this subject has demonstrated that an interpenetrated array cannot prevent porosity,but enhances the porous functionalities of the supramolecular frameworks [11].More importantly,the research upsurge in interpenetration structures was pro-moted by the fact that interpenetrated nets have been considered as potential super-hard materials [12]and possess peculiar optical and electrical properties [13].Herein we present the synthesis,structure,and spectral properties of a new coordination polymer based on tetranuclear homometallic cluster,Zn 2(OH)(pheno)(p -BDC)1.5áH 2O (1).2Experimental2.1Materials and MeasurementsAll commercially available chemicals are reagent grade and used as received without further puri?cation.Sol-vents were puri?ed by standard methods prior to use.Elemental analysis for C,H and N were carried with a Perkin-Elmer 240C Elemental Analyzer at the Analysis Center of Nanjing University.Infrared spectra were obtained with a Bruker FS66V FT IR Spectrophotometer as a KBr pellet.R.-Y.Huang áX.-J.Kong áG.-X.Liu (&)Anhui Key Laboratory of Functional Coordination Compounds,College of Chemistry and Chemical Engineering,Anqing Normal University,Anqing 246003,P.R.China e-mail:liugx@/doc/c97a12ccf61fb7360b4c65f3.htmlJ Inorg Organomet Polym (2008)18:304–308DOI 10.1007/s10904-008-9199-72.2Preparation of Zn2(OH)(pheno)(p-BDC)1.5áH2O(1)A mixture containing Zn(NO3)2á6H2O(0.20mmol), p-1,4-benzenedicarboxylic acid(H2BDC)(0.20mmol), phenanthrene-9,10-dione(pheno)(0.10mmol)and NaOH (0.20mmol)in water(10mL)was sealed in a18mL Te?on lined stainless steel container and heated at150°C for72h.The reaction product was dark yellow block crystals of1,which were washed by deionized water sev-eral times and collected by?ltration;Yield,78%. Elemental Analysis:Calcd.for C24H15N2O10Zn2:C,46.33;H,2.43;N,4.50%.Found:C,46.38;H,2.47;N,4.48%.IR (KBr pellet),cm-1(intensity):3437(br),3062(m),1587(s),1523(m),1491(w),1424(m),1391(s),1226(w),1147 (w),1103(w),1051(w),875(w),843(m),740(w),728 (m),657(w).2.3X-ray Structure DeterminationThe crystallographic data collections for complex1were carried out on a Bruker Smart Apex II CCD with graphite-monochromated Mo-K a radiation(k=0.71073A?)at 293(2)K using the x-scan technique.The data were inte-grated by using the SAINT program[14],which also did the intensities corrected for Lorentz and polarization effects.An empirical absorption correction was applied using the SADABS program[15].The structures were solved by direct methods using the SHELXS-97program; and,all non-hydrogen atoms were re?ned anisotropically on F2by the full-matrix least-squares technique using the SHELXL-97crystallographic software package[16,17]. The hydrogen atoms were generated geometrically.All calculations were performed on a personal computer with the SHELXL-97crystallographic software package[17].The details of the crystal parameters,data collection and re?nement for four compounds are summarized in Table1. Selected bond lengths and bong angles for complex1are listed in Table2.3Results and DiscussionThe X-ray diffraction study for1reveals that the material crystallizes in the orthorhombic space group Pbca and features a2-fold parallel interpenetrated3D?3D net-work motif.The asymmetric unit contains two Zn(II) atoms,one hydroxyl,one pheno ligand,one and half of p-BDC molecules and one solvent water molecule.Selected bond lengths for1are listed in Table2.As shown in Fig.1, the Zn1ion,which is in the center of a tetrahedral geom-etry,is surrounded by three carboxylic oxygen atoms (Zn–O=1.918(5)–1.964(5)A?)from three p-BDC ligands and one l3-OH oxygen atom(O9).The Zn–O distance is1.965(5)A?.Two nitrogen atoms(N1and N2)that belong to pheno,one p-BDC oxygen atom(O3A)and one hydroxyl oxygen atom(O9A)are ligated to the Zn2center in the equatorial plane with another oxygen atom(O9)that arises from the second hydroxyl group and one oxygen atom(O5)that arises from the second p-BDC molecule situated in the axial position.EachZn2lies approximately in the equatorial position with a maximum deviation (0.048A?)from the basal plane.In the structure,Zn–O and Zn–N bond distances are in the range of 2.0530(5)–2.112(5)and2.157(5)–2.184(2)A?,respectively. There exist two types of p-BDC found in1(Scheme1); namely,monobidentate bridging(l3)and bi-bidentatebridging(l4)coordination modes.The bidentate bridging p-BDC connects mixed metals,where the smallest ZnáááZn distance is3.163A?,to complete a homodinuclear cluster, which is further linked by l3-OH into a six-connected Table1Crystal data and summary of X-ray data collection for1Zn2(pheno)(OH)(BDC)1.5áH2O Empirical formula C24H15N2O10Zn2Molecular mass/g mol-1622.12Color of crystal Dark yellowCrystal fdimensions/mm0.1890.1690.12 Temperature/K293Lattice dimensionsa/A?18.777(9)b/A?13.657(6)c/A?19.983(9)a/°90b/°90c/°90Unit cell volume(A?3)5125(4)Crystal system OrthorhombicSpace group PbcaZ8l(Mo-K a)/mm-1 1.931D(cacl.)/g cm-3 1.613Radiation type Mo-K aF(000)2504Limits of data collection/° 2.04B h B25.05Total re?ections24155Unique re?ections,parameters4545,347No.with I[2r(I)2821R1indices[I[2r(I)]0.0657w R2indices0.1858Goodness of?t 1.060Min/max peak(Final diff.map)/e A?-3-0.658/2.322tetranuclear cluster that is jointly coordinated by six p-BDC molecules(Fig.2).The clusters are further extended by p-BDC into a single3D framework(Fig.3).For clarity, we used the topological method to analyze this3D framework.Thus,the six-connected SBU is viewed to be a six-connected node.Furthermore,based on consideration of the geometry of thisnode,the3D frame is classi?ed as an a-Po net with41263topology(Fig.4).Of particular interest,the most intriguing feature of complex1is that a pair of identical3D single nets is interlocked with each other,thus directly leading to the formation of a2-fold interpenetrated3D?3D architecture(Fig.4)and the two pcu(a-Po)frameworks are related by a screw axis21[18]. Recently,a complete analysis of3D coordination networks shows that more than50interpenetrated pcu(a-Po)frames have been documented in the CSD database[18],including 2-fold,3-fold[19],and4-fold[20]interpenetration.In addition,several non-interpenetration motifs with a-Po topology have been reported to date[21].ZnZnO ZnZnZnZnO Znbidentate bidentate bidentate monodentateI IIScheme1Coordination modesof the bdc ligands in the structure of1;I is bis(bidentate),II is bi/monodentateFig.1ORTEP representation of complex1(the H atoms have been omitted for the sake of clarity).The thermal ellipsoids are drawn at 30%probabilityTable2Selected bond lengths(A?)and angles(°)for1Symmetry transformations usedto generate equivalent atoms:#1x-1/2,y,-z+1/2;#2-x,-y+1,-z;#3-x+1/2,-y+1,z-1/2Zn(1)–O(1) 1.918(5)Zn(2)–O(9)#2 2.091(4)Zn(1)–O(4)#1 1.953(5)Zn(2)–O(9) 2.103(5)Zn(1)–O(6) 1.964(5)Zn(2)–O(3)#3 2.112(5)Zn(1)–O(9) 1.965(5)Zn(2)–N(1) 2.157(6)Zn(2)–O(5)#2 2.053(5)Zn(2)–N(2) 2.184(6)O(1)–Zn(1)–O(4)#197.9(2)O(9)–Zn(2)–O(3)#388.81(18)O(1)–Zn(1)–O(6)112.9(2)O(5)#2–Zn(2)–N(1)94.7(2)O(4)#1–Zn(1)–O(6)104.7(2)O(9)#2–Zn(2)–N(1)170.7(2)O(1)–Zn(1)–O(9)122.9(2)O(9)–Zn(2)–N(1)91.6(2)O(4)#1–Zn(1)–O(9)109.7(2)O(3)#3–Zn(2)–N(1)88.9(2)O(6)–Zn(1)–O(9)107.0(2)O(5)#2–Zn(2)–N(2)87.1(2)O(5)#2–Zn(2)–O(9)#291.9(2)O(9)#2–Zn(2)–N(2)98.3(2)O(5)#2–Zn(2)–O(9)173.72(19)O(9)–Zn(2)–N(2)94.5(2)O(9)#2–Zn(2)–O(9)81.82(19)O(3)#3–Zn(2)–N(2)164.3(2)O(5)#2–Zn(2)–O(3)#391.3(2)N(1)–Zn(2)–N(2)75.7(2)O(9)#2–Zn(2)–O(3)#397.42(19)Fig.2Polyhedral representation of the homotetranuclear unit as asix-connected node linked by p-BDC ligandsMoreover,rich inter and intra hydrogen-bonds between the water molecules and the carboxylate groups (Table 3)further strengthen the stacking of the supra-architecture (Fig.5).4Supplementary MaterialsCrystallographic data (excluding structure factors)for thestructures reported in this paper have been deposited with the Cambridge Crystallographic Data Center as supple-mentary publication /doc/c97a12ccf61fb7360b4c65f3.html DC-666555.Copies of the data can be obtained free of charge on application to CCDC,12Union Road,Cambridge CB21EZ,UK (Fax:+44-1223-336033;e-mail:deposit@/doc/c97a12ccf61fb7360b4c65f3.html ).Acknowledgments This work was supported by the National Nat-ural Science Foundation of China (20731004)and the Natural Science Foundation of the Education Committee of Anhui Province,China(KJ2008B004).Fig.3Polyhedral presentation of one set of the 3D network along a -axis (a )and b -axis (b )Table 3Distance (A ?)and angles (°)of hydrogen bonding for com-plex 1D–H áááADistance of D áááA (A ?)Angle of D–H–A (°)O1W–H1WB áááO2#1 2.677(9)164O9–H19áááO1W#2 2.841(9)151C13–H13áááO3#3 3.045(10)121C22–H22áááO1W#43.353(10)167Symmetry transformations used to generate equivalent atoms:#1x,y,1+z;#2-x,1-y,-1+z;#3-x+1/2,-y+1,z -1/2;#4-x,1-y,1-zFig.4Simpli?ed schematic representation of the 3D ?3D two-fold interpenetrated a -Po network in1Fig.5Projection of the structure of 1along b -axis (dotted lines represent hydrogen-bonding)References1.(a)P.J.Hagrman,D.Hagrman,J.Zubieta,Angew.Chem.Int.Ed.38,2638(1998);(b)S.Leininger,B.Olenyuk,P.J.Stang,Chem.Rev.100,853(2000);(c)A.Erxleben,Coord.Chem.Rev.246, 203(2003);(d)K.Biradha,Y.Hongo,M.Fujita,Angew.Chem. 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三角反铁磁材料Mn3Z（Z=Ga,_Ge,_Sn）的磁性和输运性质

㊀第40卷㊀第11期2021年11月中国材料进展MATERIALS CHINAVol.40㊀No.11Nov.2021收稿日期:2021-07-14㊀㊀修回日期:2021-08-31基金项目:国家自然科学基金资助项目(51671024,91427304)第一作者:张强强,男,1995年生,博士研究生通讯作者:柳祝红,女,1976年生,教授,硕士生导师,Email:zhliu@DOI :10.7502/j.issn.1674-3962.202107017三角反铁磁材料Mn 3Z (Z =Ga,Ge,Sn)的磁性和输运性质张强强1,柳祝红1,马星桥1,刘恩克2(1.北京科技大学物理系,北京100083)(2.中国科学院物理研究所北京凝聚态物理国家实验室,北京100190)摘㊀要:反铁磁材料具有零磁矩或非常小的磁矩,不易受外磁场干扰㊂相对于铁磁材料,反铁磁材料具有更低的能量损耗和更高的响应频率等优点,因在自旋电子学领域的实际应用方面具有巨大潜力而备受关注㊂作为一种兼具Kagome 晶格及三角反铁磁性的特殊自旋电子学材料,六角Mn 3Z (Z =Ga,Ge,Sn)合金展现出巨大的反常霍尔效应㊁拓扑霍尔效应㊁自旋霍尔效应以及反常能斯特(Nernst)效应等㊂这些物理效应涉及到当今凝聚态物理研究中最前沿的问题,对它们的研究不仅可以深化对凝聚态磁性物理的理解,而且也驱动了反铁磁自旋电子学的发展㊂首先介绍了Mn 3Z 合金的晶格结构及特殊的磁结构,简要分析了理论计算得到的电子结构对材料输运性能的影响㊂结合实验报道的Mn 3Z 的磁性及输运性质等对3种六角结构合金的优异性能及研究进展进行了概述,揭示了磁结构和电子结构对材料输运性质的物理机制,并对Mn 3Z 系列合金拓扑相关的输运性质进行了展望㊂关键词:Mn 3Z (Z =Ga,Ge,Sn);反铁磁材料;拓扑材料;霍尔效应;能斯特(Nernst)效应中图分类号:O469㊀㊀文献标识码:A㊀㊀文章编号:1674-3962(2021)11-0861-10Magnetic and Transport Properties of Triangular Antiferromagnetic Materials Mn 3Z (Z =Ga ,Ge ,Sn )ZHANG Qiangqiang 1,LIU Zhuhong 1,MA Xingqiao 1,LIU Enke 2(1.Department of Physics,University of Science and Technology Beijing,Beijing 100083,China)(2.Beijing National Laboratory for Condensed Matter Physics,Institute of Physics,Chinese Academyof Sciences,Beijing 100190,China)Abstract :Antiferromagnetic materials exhibit zero or rather low moment,so it would not be affected by external magneticfield.Furthermore,they have advantages of lower power consumption and higher frequency response compared with ferro-magnetic materials,which makes them have great potential applications in the field of spintronics.Hexagonal Mn 3Z (Z =Ga,Ge,Sn)alloys,with both Kagome lattice and triangular antiferromagnetism,exhibit large anomalous Hall effect,topological Hall effect,spin Hall effect and anomalous Nernst effect.These effects involve the most advanced problems in condensed matter physics.The study of them can not only deepen the understanding of condensed matter magnetic physics,but also drive the development of antiferromagnetic spintronics.In this paper,the research progress in magnetic and transport proper-ties are reviewed.The crystal structure and the special magnetic structure of Mn 3Z alloys are introduced.The influence of the electronic structure on the transport properties is briefly analyzed.An overview of the excellent properties of the Mn 3Z (Z =Ga,Ge,Sn)alloys and their research progress is given in relation to the experimentally reported magnetic and transport proper-ties.An outlook is given for the topologically relevant transport properties of the Mn 3Z alloys.Key words :Mn 3Z (Z =Ga,Ge,Sn)alloys;antiferromagnetic materials;topological materials;Hall effect;Nernst effect1㊀前㊀言当前的自旋电子器件主要基于铁磁性材料㊂反铁磁材料由于具有零磁矩或者非常小的磁矩,没有杂散场,不受外磁场干扰,故具有更高的稳定性㊂同时,反铁磁博看网 . All Rights Reserved.中国材料进展第40卷材料具有更快的响应速度(响应频率高)㊁更低的能耗以及更高的存储密度等特性,为发展下一代非易失性低功耗反铁磁存储器件提供了契机,可能对磁性随机存储器㊁人工神经网络㊁太赫兹存储器件和探测器等领域产生重大影响㊂在众多的反铁磁材料中,非共线反铁磁材料Mn3Z(Z= Ga,Ge和Sn)中出现了许多引人关注的新颖物性,如反常霍尔效应(anomalous Hall effect,AHE)㊁自旋霍尔效应(spin Hall effect,SHE)㊁拓扑霍尔效应(topological Hall effect, THE)㊁反常能斯特效应(anomalous Nernst effect,ANE)等,已经成为当前凝聚态物理研究中的前沿与热点㊂六角Mn3Z(Z=Ga,Ge和Sn)合金具有DO19型结构,如图1a所示,空间群为P63/mmc(No.194)㊂其中Mn原子占据(1/6,1/3,1/4)位置,Z原子占据(1/3,2/3, 3/4)位置㊂在六角Mn3Z结构中,两种镜面对称的Mn3Z 反铁磁平面沿着c轴方向叠加嵌套,每一层的Mn位形成一个由共享等边三角形组成的二维网格,即Kagome晶格[1]㊂在六角Mn3Z(Z=Ga,Ge和Sn)合金中,所有Mn 原子的磁矩都位于ab平面,形成一个手性自旋结构,其矢量手性与通常的120ʎ结构相反㊂Mn3Z合金已经被证明具有多种类型的非共线反铁磁结构[2-6]㊂早在1990年Brown等[7]发现Mn3Z有两种最有可能的磁结构排列,分别如图1b和1c所示[2],这两种磁结构具有相反的手性,且磁结构数据与实验测量值高度吻合㊂因此,当前对Mn3Z(Z=Ga,Ge和Sn)合金的研究既有采用图1b型磁结构的,也有采用图1c型磁结构的㊂图1㊀六角Mn3Z的晶体结构(a)[1];Mn3Z合金的两种不同的磁矩构型(b,c)[2]Fig.1㊀Lattice structure of hexagonal Mn3Z(a)[1];the two different magnetic moment configurations of Mn3Z alloy,respectively(b,c)[2]㊀㊀由于六角Mn3Z合金在基态下会展现出极小的净磁矩,表现出弱铁磁性,实际上并不算严格的反铁磁材料㊂Mn3Z中倒三角形磁矩排列具有正交对称性,每个Mn原子组成的三角形中只有一个Mn原子的磁矩平行于局域易磁化轴,因此另外两个自旋磁矩向局域易磁化轴的倾斜被认为是Mn3Z弱铁磁性的来源[4,8]㊂在凝聚态物理中,材料所展现的许多物性都与其电子结构密切相关,而电子的行为反映在能带结构中㊂Mn3Ga㊁Mn3Ge和Mn3Sn的能带结构看起来非常相似,如图2所示[1]㊂由于Ga原子的价电子数相对于Ge 和Sn原子少一个,因此Mn3Ga的费米能级(E F)相对于Mn3Ge和Mn3Sn的向下移动约0.34eV㊂在Mn3Z合金中,能带结构在E F附近具有线性交叉,产生外尔(Weyl)点㊂Weyl点是动量空间中的奇点,可以被理解为磁单极子㊂这些点成对出现,并且产生特有的表面性质,即所谓的费米弧㊂Weyl点处具有极强的贝利(Berry)曲率磁通分布,这个Berry曲率可以看作是动量空间中的赝磁场㊂这3种合金的能带中价带和导带在E F附近多次交叉,产生多对Weyl点,其中大部分为II型(II型Weyl点与I型Weyl点的区别在于其能带中Weyl锥在某个动量方向上发生倾斜)[9]㊂Weyl点的位置和手性与磁晶格的对称性一致㊂其次,在高对称点K和A处可以发现看似相似的能带简并点,如图中红色圆圈所示㊂有趣的是,部分Mn3Z合金除了可以形成DO19型六角结构之外,还可能形成DO22型四方结构或DO3型哈斯勒(Heusler)立方结构㊂例如,Mn3Ga在623K的温度下退火会形成DO22型四方结构,在883K的温度下退火会形成DO19型六角结构,在1073K的温度下退火则会形成DO3型Heusler立方结构[10]㊂DO22型Mn3Ge在大约800K 的温度下会向DO19型六角结构转变[11]㊂因此,为了确保合金可以以稳定的DO19型六角结构结晶,合适的热处理是必要的㊂2㊀六角Mn3Sn合金的输运性质2.1㊀Mn3Sn中的AHEAHE是磁性材料中比较常见的输运效应,由于其在自旋电子学器件材料方面具有潜在的应用前景,使其迅速成为材料科学等领域的研究热点之一㊂一般认为,铁磁性材料的AHE与其磁化强度成正比㊂由于反铁磁材料缺乏净剩磁矩,普遍认为反铁磁材料中不会出现AHE㊂268博看网 . All Rights Reserved.㊀第11期张强强等:三角反铁磁材料Mn 3Z (Z =Ga,Ge,Sn)的磁性和输运性质图2㊀Mn 3Ga(a)㊁Mn 3Ge(b)和Mn 3Sn(c)的能带结构[1]Fig.2㊀Electronic band structure for Mn 3Ga (a),Mn 3Ge (b)andMn 3Sn(c)[1]后来的研究表明,AHE 起源于两种不同的机制:一种是由杂质原子散射所引起的外禀散射机制,包括边跳机制和螺旋散射机制;另一种是晶体能带的Berry 曲率所驱动的内禀机制,与外部散射无关㊂Berry 曲率相当于布里渊区中的赝磁场,可以使电子获得一个额外的群速度,从而产生内禀反常霍尔电导(anomalous Hall conductivity,AHC)㊂内禀AHE 仅与材料的能带结构相关,这为在反铁磁材料中发现AHE 提供了条件㊂Mn 3Sn 在E F 附近的Weyl 点处所具有的Berry 曲率磁通分布是导致该材料出现大的反常霍尔电导的关键㊂2015年,日本研究人员首次报道了Mn 3Sn 单晶中产生的巨大的反常霍尔电导[12]㊂图3a 为室温下磁场沿(2110)方向测得的Mn 3Sn 单晶的AHE 曲线,可以看到反常霍尔电阻率在低磁场区域展现出一个相当大的跳跃㊂当磁场沿(0110)方向时,Mn 3Sn 单晶在100~400K 的温度范围内均表现出较大的AHE,如图3b 所示㊂相应地,在外加磁场沿(2110)和(0110)方向时的霍尔电导曲线也展现出较大的跃变和比较窄的滞后(图3c 和3d)㊂例如,当磁场B //(0110)轴测量时,反常霍尔电导率σH 在零场时就具有比较大的值,其中在室温下约为20Ω-1㊃cm -1,在100K 的温度下接近100Ω-1㊃cm -1,这对于反铁磁材料来说是非常大的㊂图3㊀Mn 3Sn 单晶的AHE 测量曲线[12]Fig.3㊀Magnetic field dependence of the AHE in Mn 3Sn [12]368博看网 . All Rights Reserved.中国材料进展第40卷㊀㊀Mn 3Sn 的可变磁结构会影响费米面附近的能带结构,进而影响其AHE㊂为了更好地对AHE 进行调控,北京科技大学陈骏团队对多晶Mn 3Sn 复杂的磁结构及其与AHE 的相关性进行了研究[13]㊂研究发现,Mn 3Sn 在不同的外部磁场下带场冷却(field-cooling,FC)的得到测量曲线在磁转变温度T S =190K 时存在明显的磁相变(图4a)[13]㊂早期研究表明,Mn 3Sn 在奈尔温度T N =420K 以下是三角反铁磁结构[14],并且三角反铁磁结构在T S 温度下转变为非公度自旋结构[15]㊂在自旋玻璃转变温度T g =50K 的温度以下,磁化强度随着温度的降低而升高,这主要是由于低温下的自旋玻璃态引起的[16]㊂外加磁场的大小几乎不影响FC 曲线的形状和磁转变温度的大小,并且外加磁场强度的增加只导致Mn 3Sn 磁化强度的小幅增加,表明Mn 3Sn 中的磁结构非常稳定㊂图4b 为Mn 3Sn 在不同温度下的M-H 曲线,可以看到所有温度下的M-H 曲线在6000Oe 的外场下都没有达到饱和㊂当温度高于200K 时M-H 曲线展现出明显的磁滞,表明非共线反铁磁Mn 3Sn 存在弱铁磁性㊂为了明确其磁转变所产生的不同磁结构,采用中子衍射测量之后分析发现Mn 3Sn 的磁相图可分为4个区域:①10<T <190K,②190<T <250K,③250<T <430K,④T >430K㊂其中,250<T <430K 下为反三角的反铁磁(antiferromagnetic,AFM)结构,10<T <190K 为余弦或摆线磁结构㊂宏观磁性测量结果表明,Mn 3Sn 在50K 温度以下存在自旋玻璃态;然而中子衍射测量结果显示,Mn 3Sn 在50K 温度以下并没有任何异常,因此可以认为Mn 3Sn 在50K 温度下存在自旋玻璃态与长程螺旋磁结构的共存㊂在不同温度下对Mn 3Sn 的霍尔电阻率(ρH )进行测量发现,该曲线具有明显的磁滞(图5a)㊂当T =235K 时,|ρH |为2.5μΩ㊃cm;当T =190K 时,ρH 接近于0,且|ρH |随着外加磁场磁感应强度B 的增加而线性增加㊂从ρH -B 曲线中提取了零场(B =0T)下的ρH 来揭示AHE自发分量的温度依赖性(图5b),发现|ρH |在190K 温度以下几乎保持为0,在大约235K 时增加到最大值,然后随着温度的升高而降低㊂很明显,ρH 的变化与温度导致的磁结构的变化密切相关㊂根据这一关系,可以通过改变Mn 3Sn 的磁性结构来调整其AHE㊂图4㊀Mn 3Sn 在不同外加磁场下带场冷却得到的热磁曲线(a),Mn 3Sn 在不同温度下的磁滞回线(b)[13]Fig.4㊀Magnetization as a function of the external magnetic field and temperature[field-cooling (FC)modes](a),hysteresis loops ofMn 3Sn at different temperatures (b)[13]图5㊀Mn 3Sn 不同温度下的霍尔电阻率随磁场的变化曲线(ρH -B 曲线)(a),零场下霍尔电阻率的温度依赖性(b)[13]Fig.5㊀Field dependence of Hall resistivity ρH at different temperatures(a),temperature dependence of Hall resistivity at zero field(b)[13]468博看网 . All Rights Reserved.㊀第11期张强强等:三角反铁磁材料Mn3Z(Z=Ga,Ge,Sn)的磁性和输运性质2.2㊀Mn3Sn中的THE将拓扑学的概念引入到物理学中来描述随参数连续变化而保持不变的物理量时,能够解释很多关于磁输运方面的问题和现象[17,18]㊂拓扑非平庸自旋结构的局部磁矩在几何阻挫或Dzylashinsky-Moriya相互作用(DMI)的驱动下发生空间变化,产生了一种不同类型的霍尔效应,即THE[19]㊂THE的起源可归因于非零的标量手性自旋X ijk=S i㊃(S jˑS k),其中S i㊃(S jˑS k)代表3个自旋矢量形成的立体角,打破了时间对称性,称为实空间的Berry曲率㊂由于同样具有120ʎ非共线反铁磁结构的Fe1.3Sb已经被报道具有THE[20],Nayak等对Mn3Sn合金中的THE进行了研究[21]㊂霍尔效应总的贡献可以表示为ρxy=ρN+ρM AH+ρT xy,其中ρN和ρT xy分别是正常和拓扑霍尔电阻率[17]㊂ρM AH是与Mn3Sn的磁化强度成正比的反常霍尔电阻率㊂正常霍尔电阻率与外加磁场强度成正比㊂通过从测得的霍尔数据ρxy中扣除正常和反常霍尔电阻率,可以得到拓扑霍尔电阻率ρT xy㊂图6a为在不同测试温度下得到的Mn3Sn的拓扑霍尔电阻率曲线,可以看到在低温下Mn3Sn中存在大的THE,这是由于低温下施加磁场会导致Mn3Sn中非共面的三角反铁磁转变为受拓扑保护的非平庸自旋结构(类似Skyrmions),导致实空间的Berry曲率出现[21]㊂同时,他们还发现Mn3Sn中存在3种不同的霍尔效应,包括在相对高温下的由共面三角AFM结构演化出的自发AHE(ρS xy)㊁低温下的THE(ρT xy)以及中间温区域中两种霍尔效应的共存,如图6b所示㊂2.3㊀Mn3Sn中的ANEANE是由热电流引起的自发横向电压降,与磁化强度成正比[22]㊂AHE由所有占据态能带的Berry曲率决定,而ANE是由E F处的Berry曲率决定的[23]㊂因此,能观察到大的AHE并不能保证观察到大的ANE㊂ANE的测量对于明确E F附近的Berry曲率和验证最近提出的Mn3Sn中Weyl点存在的可能性具有重要价值[9]㊂Ikhlas等对单晶Mn3.06Sn0.94和Mn3.09Sn0.91的Nernst 效应进行了研究[24]㊂结果表明,零磁场下Mn3.06Sn0.94的Nernst信号(横向热电势)-S zx在室温下为~0.35μV㊃K-1 (图7a),与室温下的FePd(0.468μV㊃K-1)㊁L10-MnGa (-0.358μV㊃K-1)等铁磁体的报道值相当[25];在200K 的温度下达到了~0.60μV㊃K-1(图7b)㊂面内的Nernst 信号表现出几乎没有各向异性的滞后现象,零场下展现的Nernst信号值与高场下的饱和Nernst信号几乎相同,表明单晶Mn3Sn中具有大的自发Nernst信号㊂但面外c 轴分量在实验精度范围内测量值为0,表明在这个方向上没有自发Nernst效应㊂通过ANE与磁化强度M的对比图6㊀Mn3Sn在不同温度下拓扑霍尔电阻率ρT xy的磁场依赖性(a),不同霍尔效应贡献的相图(b)[21]Fig.6㊀Field dependence of topological Hall resistivityρT xy at different temperatures(a),phase diagram showing contribution from dif-ferent Hall effects(b)[21]发现(图7a),低场下-S zx和M的滞后几乎相互重叠㊂另外,在大于~100G的磁场区域,ANE效应几乎保持不变,而M随着磁场的增加呈线性增加,表明正常的Nernst效应和传统的ANE的贡献可以忽略不计㊂在Mn3.09Sn0.91中也可以观察到类似的行为(图7b)㊂3㊀六角Mn3Ge合金的输运性质3.1㊀Mn3Ge单晶中的AHE与Mn3Sn相比,Mn3Ge在磁性和AHE方面有所不同,在Mn3Ge中测量得到的反常霍尔电导值比Mn3Sn中的高将近3倍㊂此外,Mn3Ge并不会展现出类似于Mn3Sn中的任何磁转变,为其AHE的稳定性提供了保障㊂Nayak等采用如图8a所示的磁结构通过第一性原理计算预测了Mn3Ge合金中的反常霍尔电导[26]㊂结果表明,Mn3Ge在xy(σz xy)和yz(σx yz)部分的反常霍尔电导接近于零,只有在xz(σy xz)部分发现了反常霍尔电导的存在(图8b)㊂其中,σk ij表示电流沿着j方向的反常霍尔电导,产生的霍尔电压沿i方向㊂Mn3Ge反铁磁结构的两个原始单元具有两个磁性层面,相互之间可以通过相对于xz平面的镜面反射加上沿c轴平移c/2转换㊂由于镜像对称,Mn2Ge合金的σk ij平行于镜面的话就会消失,从568博看网 . All Rights Reserved.中国材料进展第40卷图7㊀300K 的温度下Mn 3.06Sn 0.94的Nernst 信号-S ji 在不同测量方向下的磁场依赖性(a);200K 的温度下Mn 3.06Sn 0.94和Mn 3.09Sn 0.91的-S zx 的磁场依赖性(b)[24]Fig.7㊀Anisotropic field dependence of the Nernst signal -S ji of Mn 3.06Sn 0.94at 300K for comparison,the field dependence of the magneti-zation M (right axis)is shown(a);-S zx of Mn 3.06Sn 0.94and Mn 3.09Sn 0.91measured at 200K(b)[24]图8㊀计算中所采用的Mn 3Ge 的磁结构(a),第一布里渊区和动量依赖的反常霍尔电导(b)[26]Fig.8㊀The magnetic structure of Mn 3Ge used in the calculation(a),first Brillouin zone and momentum-dependent AHC(b)[26]而导致σz xy 和σx yz 的值为零㊂但是因为平面内残存的净磁矩作为镜面对称的扰动,导致σz xy 和σx yz 可以获得非零但是很小的值㊂相比之下,σk ij 垂直于镜面的分量(σy xz )是非零的㊂接着,他们在实验上对预测的AHE 进行了实验验证㊂当电流沿(0001)方向㊁磁场平行于(01-10)(这种测量方式称为构型1)时(图9a),ρH 在2K 的温度下达到5.1μΩ㊃cm 的大饱和值,即使在室温下也展现出了1.8μΩ㊃cm 的饱和值㊂在霍尔电导率曲线σxz -μ0H 中可以看到(图9b),反常霍尔电导在2K 的温度下具有~500Ω-1㊃cm -1的较大的值,在室温下则为50Ω-1㊃cm -1㊂为了进一步研究实验中的AHE 是否具有理论预测的各向异性,测量了电流沿(01-10)方向㊁磁场平行于(2-1-10)方向(构型2)时的霍尔电阻率,如图9c 所示㊂在这种测量方向下,ρH 在2K 的温度下约为4.8μΩ㊃cm,在室温下约为1.6μΩ㊃cm,略小于构型1得到的值㊂图9d 为构型2下的反常霍尔电导曲线,可以看到尽管在2K 的温度下构型2的σH (σyz )(约为150Ω-1㊃cm -1)要小于构型1的σH (σxz ),但在室温下具有与构型1相似的值㊂在这两种情况下,对于正(负)场,ρH 为负(正)㊂第3种测量方式为电流沿着(2-1-10)方向㊁磁场平行于(0001)方向(图9e 和9f),被称为构型3㊂在这种构型下,所有温度下的ρH 和σH 都具有比较小的值㊂此外,AHE 的符号和前两种构型的符号相反,即相对于正(负)场,ρH 为正(负)㊂虽然在正常条件下Mn 3Ge 并不会展现出类似Mn 3Sn中的磁转变,但是如果对Mn 3Ge 施加外部压力的话其磁结构会发生显著变化㊂研究表明,随着压力的增大,Mn 3Ge 的非共线三角磁结构逐渐变为均匀倾斜的非共线三角磁结构,当压力增大到5GPa 以上时变为共线铁磁结构[27]㊂由于Mn 3Sn 合金中磁结构的变化在很大程度上会影响其输运性能,因此可以通过施加不同的压力来改变Mn 3Ge 合金中的磁结构,从而进一步研究Mn 3Ge 的磁结构与AHE 的关系㊂Nicklas 等测量了静水压力与AHE 之间的关系[28],测量装置如图10a 所示,电流平行于(0001)轴,磁场平行于(2-1-10)轴㊂研究发现,随着压力的增大,霍尔电导668博看网 . All Rights Reserved.㊀第11期张强强等:三角反铁磁材料Mn3Z(Z=Ga,Ge,Sn)的磁性和输运性质图9㊀电流和磁场沿不同方向(3种构型)下的霍尔电阻率(ρH)(a,c 和e)和霍尔电导率(σH)(b,d和f)的磁场依赖性[26] Fig.9㊀Hall resistivity(ρH)(a,c and e)and Hall conductivity(σH) (b,d and f)as a function of magnetic field(H),for three differ-ent current and magnetic field configurations[26]率σyz的饱和值先降低,当压力为1.53GPa时完全消失;继续增大压力,σyz的饱和值反向并逐渐增大,如图10b所示㊂在2.85GPa的压力下,Mn3Ge合金中Mn 原子的磁矩会由图10c顶部的磁结构变化为底部的磁结构㊂可以看到压力会导致磁矩向面外倾斜,进而影响电子能带结构,从而导致Berry曲率的变化㊂除了反常霍尔电导之外,理论预测在Mn3Ge中还可以获得高达1100(ћ/e)Ω-1㊃cm-1的自旋霍尔电导率[26]㊂在对Mn3Ge薄膜样品的研究中,在Permalloy/Mn3Ge表面发现了高达90.5nm-2的自旋混合电导系数,并且Mn3Ge的自旋霍尔角是Pt的8倍左右[29]㊂3.2㊀Mn3Ge中的ANE由于Mn3Sn的磁结构在T=50K以下缺乏磁有序性,并且形成了玻璃态的磁基态,从而导致ANE消失[15]㊂而Mn3Ge的磁有序和反常输运性质通常持续到最低温度,与Mn3Sn形成鲜明对比㊂Wuttke等对Mn3Ge单晶的Nernst效应进行了测量,如图11所示[30]㊂结果表明,Nernst信号S xz不依赖于磁场,表现出反常的行为,在非常低的磁场下即表现出步进特征,并且在B>0.02T时就已经达到了饱和值㊂S yz 也表现出非常弱的场依赖性,如图11b所示㊂两种方向都显示出高达室温的特殊饱和行为,随着温度的逐渐降低,Nernst信号从0.4逐渐升高到1.5μV㊃K-1㊂S xy则显图10㊀压力元件内使用的电传输测量样品装置示意图(a),室温下施加不同压力的Mn3Ge的霍尔电导率(b),在环境压力(顶部)和压力为2.85GPa(底部)下的Mn3Ge的反三角自旋结构(c)[28]Fig.10㊀Schematic drawing of the sample device for the electrical-transport measurements used inside the pressure cell(a),field dependence Hall conductivity for Mn3Ge at room temperature for selected pressures(b),the inverse triangular magnetic structure of Mn3Ge at ambient pressure(top)and P=2.85GPa(bottom)(c)[28]768博看网 . All Rights Reserved.中国材料进展第40卷图11㊀Mn3Ge单晶的Nernst信号测量曲线[30]Fig.11㊀Nernst signal of the Mn3Ge single crystals[30]示出不同的行为,如图11c所示㊂在这种配置中,Nernst 信号非常小,阶梯状的行为只是略微可见,并且显示出非常弱的温度依赖性㊂除了单晶Mn3Ge之外,Mn3Ge薄膜在室温下也展现出0.1μV㊃K-1的反常Nernst信号,与铁磁性Fe薄膜的反常Nernst信号(0.4μV㊃K-1)相当[29]㊂4㊀六角Mn3Ga合金的输运性质4.1㊀Mn3Ga中的AHE到THE的转变在同样具有手性的非共线三角反铁磁Mn3Ga中依然存在大的AHE㊂不同的是Mn3Ga存在六角到正交的晶格畸变,原来的共面磁结构会向c轴转变,使得非共面磁结构产生,这就导致Mn3Ga中THE的出现[31]㊂Mn3Ga在100Oe磁场下的磁化强度会随温度的降低先增加(图12)[31],到140K左右出现一个磁转变,并且升降温曲线在此处展现出很明显的热滞,此处即为六角结构到正交结构的轻微畸变[32]㊂变频交流磁化率的测量表明这个转变没有频率依赖(图12插图),和结构变化相对应㊂图12㊀Mn3Ga在100Oe磁场下的热磁曲线,插图为不同频率的交流磁化率实部随温度的变化关系[31]Fig.12㊀M-T curves measured at100Oe field for Mn3Ga,the inset is temperature dependence of the real part of AC susceptibilitymeasured at different frequencies[31]图13a和13b为不同温度下多晶Mn3Ga的霍尔电阻率测量图[31]㊂在较低的磁场范围内,ρxy随着磁场的增大迅速增大,并且展现出比较小的磁滞㊂在低于100K 的温度范围内,曲线的形状及ρxy的值并不随温度明显变化(图13a)㊂当温度高于100K时,曲线的形状类似,随着磁场的增加,ρxy先迅速增大后趋于平缓㊂自发霍尔效应的符号在高于100K时发生改变,这个温度临界点对应于Mn3Ga的六角结构到正交结构的转变温度㊂随着磁场的增大,霍尔电导率σxy先迅速增加,当磁场高于0.03T之后,又逐渐减小(图13c)㊂图13d为ρxy中提取到的ρT xy,可以看到ρT xy几乎不随温度的变化而变化㊂同时,ρT xy随着磁场增加先迅速增大继而减小,表现出一个极值㊂ρT xy的极值大小也几乎与温度无关,最大值约为0.255μΩ㊃cm,比块体MnNiGa(~0.15μΩ㊃cm)和MnGe(~0.16μΩ㊃cm)的值都大[33,34]㊂THE的出现是由于在Mn3Ga中伴随着六角结构到正交结构的转变,磁矩排列由非共线向非共面转变导致的㊂4.2㊀Mn3Ga/PMN-PT中的AHE室温反铁磁自旋电子器件的主要瓶颈之一是反铁磁材料中有限的各向异性磁电阻导致的小信号读出㊂这可以通过在非共线反铁磁物质中利用Berry曲率诱导的反常霍尔电阻或者基于反铁磁自旋的有效操纵建立磁隧道结器件来克服㊂因此,刘知琪团队在300ħ的溅射温度下在(100)取向的铁电0.7PbMg1/3Nb2/3O3-0.3PbTiO3(PMN-PT)单晶衬底上生长了50nm厚的Mn3Ga薄膜,并通过压电应变调制对反常霍尔电阻进行了研究[35]㊂研究结果表明,在50~300K的温度范围内,随着温度的降低,零磁场下的霍尔电阻从~0.112Ω增加到~0.364Ω,用于切换反常霍尔电阻的矫顽场从93mT显868博看网 . All Rights Reserved.㊀第11期张强强等:三角反铁磁材料Mn 3Z (Z =Ga,Ge,Sn)的磁性和输运性质图13㊀不同温度下六角Mn 3Ga 的AHE(a~c)和THE(d)[31]Fig.13㊀Anomalous Hall effect (a~c)and topological Hall effect (d)of hexagonal Mn 3Ga at different temperatures [31]著增加到667.6mT(图14a ~14c)㊂由于静电调制机制对50nm 厚的Mn 3Ga 金属薄膜几乎不起作用,因此通过在PMN-PT 衬底上垂直施加4kV㊃cm-1的栅极电场E G ,分析了压电应变对AHE 的影响,如图14d ~14f 所示㊂可以看到E G =4kV㊃cm-1的AHE 在所有温度下都表现出巨大的变化㊂例如在50K 的温度下,零场的霍尔电阻从E G =0kV㊃cm-1时的~0.364Ω变化到了E G =4kV㊃cm-1时的~0.010Ω㊂由于非共线反铁磁体中的AHE 是其自旋结构的敏感探针,压电应变下AHE 的巨大变化表明其自旋结构在应变调控下发生了巨大的变化㊂4.3㊀Mn 3Ga 薄膜中的逆自旋霍尔效应和自旋泵浦自旋泵浦效应是产生自旋流的重要方法,进一步利用逆自旋霍尔效应(ISHE),可以将自旋流转化为可探测的电荷信号,从而实现自旋泵浦的电测量㊂因此,自旋泵浦效应结合ISHE 成为研究各种材料中自旋-电荷转换的经典手段㊂Singh 等对室温下多晶Mn 3Ga /CoFeB 异质结中的ISHE 和自旋泵浦效应进行了系统的研究[36]㊂实验中通过对ISHE 进行不同角度的测量来分解各种自旋整流效应㊂最终得到的自旋混合电导系数㊁自旋霍尔角和自旋霍尔电导率的值分别为(5.0ʃ1.8)ˑ1018m -2㊁0.31ʃ0.01和7.5ˑ105(ћ/2e)Ω-1㊃m -1㊂如此高的自旋霍尔角和自旋霍尔电导率使得Mn 3Ga 在未来的自旋电子器件中具有很好的应用前景㊂5㊀结㊀语本文对具有非共线反铁磁的DO 19型六角Mn 3Z (Z=图14㊀在300K(a)㊁200K(b)和50K(c)的温度下,E G =0kV㊃cm -1时Mn 3Ga /PMN-PT 异质结构的反常霍尔电阻;在300K (d)㊁200K(e)和50K(f)的温度下,E G =4kV㊃cm -1时Mn 3Ga /PMN-PT 异质结构的反常霍尔电阻[35]Fig.14㊀Magnetic-field-dependent anomalous Hall resistance of the Mn 3Ga/PMN-PT heterostructure at E G =0kV㊃cm -1at 300K(a),200K (b)and 50K(c);magnetic-field-dependent anomalous Hall re-sistance of the Mn 3Ga/PMN-PT heterostructure at E G =4kV㊃cm -1at 300K(d),200K(e)and 50K(f)[35]968博看网 . All Rights Reserved.中国材料进展第40卷Ga,Ge,Sn)合金的磁性和输运性质进行了综述㊂发现通过理论计算对Mn3Z合金的输运性质进行预测之后,在实验上都得到了验证,并观察到了非常优异的物理性能㊂这表明通过理论计算能带结构,调控和发现E F附近具有Weyl点的材料,从而寻找输运性能优异的材料是可行的㊂当前对Mn3Z(Z=Ga,Ge,Sn)合金的报道已经提供了明确的经验框架,为后期及进一步制作具备优良性能的非共线反铁磁材料打下了坚实的基础㊂因此,还需要大量的理论计算及实验以进一步指导六角反铁磁材料输运性能的有效调控㊂另外,通过对其他材料体系的研究发现,适当的无序掺杂会明显提高材料拓扑能带引起的Berry曲率,进而提升其输运性能,这为将来进一步提升六角反铁磁Mn3Z(Z=Ga,Ge,Sn)合金的性能提供了重要思路㊂参考文献㊀References[1]㊀ZHANG Y,SUN Y,YANG H,et al.Physical Review B[J],2017,95(7):075128.[2]㊀KÜBLER J,FELSER C.EPL(Europhysics Letters)[J],2018,120(4):47002.[3]㊀NAGAMIYA T.Journal of the Physical Society of Japan[J],1979,46(3):787-792.[4]㊀TOMIYOSHI S,YAMAGUCHI Y.Journal of the Physical Society ofJapan[J],1982,51(8):2478-2486.[5]㊀SANDRATSKII L M,KÜBLER J.Physical Review Letters[J],1996,76(26):4963.[6]㊀ZHANG D,YAN B,WU S C,et al.Journal of Physics:CondensedMatter[J],2013,25(20):206006.[7]㊀BROWN P J,NUNEZ V,TASSET F,et al.Journal of Physics:Condensed Matter[J],1990,2(47):9409.[8]㊀NYÁRI B,DEÁK A,SZUNYOGH L.Physical Review B[J],2019,100(14):144412.[9]㊀YANG H,SUN Y,ZHANG Y,et al.New Journal of Physics[J],2017,19(1):015008.[10]LIU Z H,TANG Z J,TAN J G,et al.IUCrJ[J],2018,5(6):794-800.[11]KALACHE A,KREINER G,OUARDI S,et al.APL Materials[J],2016,4(8):086113.[12]NAKATSUJI S,KIYOHARA N,HIGO T.Nature[J],2015,527(7577):212-215.[13]SONG Y,HAO Y,WANG S,et al.Physical Review B[J],2020,101(14):144422.[14]OHMORI H,TOMIYOSHI S,YAMAUCHI H,et al.Journal ofMagnetism and Magnetic Materials[J],1987,70(1-3):249-251.[15]LI X,XU L,DING L,et al.Physical Review Letters[J],2017,119(5):056601.[16]FENG W J,LI D,REN W J,et al.Physical Review B[J],2006,73(20):205105.[17]GALLAGHER J C,MENG K Y,BRANGHAM J T,et al.PhysicalReview Letters[J],2017,118(2):027201.[18]KANAZAWA N,KUBOTA M,TSUKAZAKI A,et al.Physical Re-view B[J],2015,91(4):041122.[19]BRUNO P,DUGAEV V K,TAILLEFUMIER M.Physical ReviewLetters[J],2004,93(9):096806.[20]SHIOMI Y,MOCHIZUKI M,KANEKO Y,et al.Physical ReviewLetters[J],2012,108(5):056601.[21]ROUT P K,MADDURI P V P,MANNA S K,et al.Physical Re-view B[J],2019,99(9):094430.[22]HUANG S Y,WANG W G,LEE S F,et al.Physical Review Let-ters[J],2011,107(21):216604.[23]XIAO D,YAO Y,FANG Z,et al.Physical Review Letters[J],2006,97(2):026603.[24]IKHLAS M,TOMITA T,KORETSUNE T,et al.Nature Physics[J],2017,13(11):1085-1090.[25]HASEGAWA K,MIZUGUCHI M,SAKURABA Y,et al.AppliedPhysics Letters[J],2015,106(25):252405.[26]NAYAK A K,FISCHER J E,SUN Y,et al.Science Advances[J],2016,2(4):e1501870.[27]SUKHANOV A S,SINGH S,CARON L,et al.Physical Review B[J],2018,97(21):214402.[28]DOS REIS R D,ZAVAREH M G,AJEESH M O,et al.PhysicalReview Materials[J],2020,4(5):051401.[29]HONG D,ANAND N,LIU C,et al.Physical Review Materials[J],2020,4(9):094201.[30]WUTTKE C,CAGLIERIS F,SYKORA S,et al.Physical Review B[J],2019,100(8):085111.[31]LIU Z H,ZHANG Y J,LIU G D,et al.Scientific Reports[J],2017,7(1):1-7.[32]NIIDA H,HORI T,NAKAGAWA Y.Journal of the Physical Societyof Japan[J],1983,52(5):1512-1514.[33]WANG W,ZHANG Y,XU G,et al.Advanced Materials[J],2016,28(32):6887-6893.[34]KANAZAWA N,ONOSE Y,ARIMA T,et al.Physical Review Let-ters[J],2011,106(15):156603.[35]GUO H,FENG Z,YAN H,et al.Advanced Materials[J],2020,32(26):2002300.[36]SINGH B B,ROY K,CHELVANE J A,et al.Physical Review B[J],2020,102(17):174444.(编辑㊀吴㊀锐)078博看网 . All Rights Reserved.。

艾菲尔铁塔作文英语

艾菲尔铁塔作文英语Title: The Magnificence of the Eiffel Tower。

The Eiffel Tower stands as an iconic symbol of Paris, France, and indeed, of human ingenuity and architectural prowess. Rising majestically over the Parisian skyline,this iron lattice structure has captured the imagination of millions since its completion in 1889. In this essay, we delve into the rich history, architectural significance,and enduring allure of the Eiffel Tower.First and foremost, the Eiffel Tower serves as a testament to the vision and engineering brilliance of Gustave Eiffel and his team. Constructed as the entrance arch for the 1889 World's Fair, it was initially met with skepticism and criticism. However, over time, it hasevolved into one of the most beloved landmarks in the world. Standing at a height of 330 meters (1,083 feet), it heldthe title of the world's tallest man-made structure forover four decades.From an architectural perspective, the Eiffel Tower represents a marvel of iron construction. Comprised of over 18,000 individual iron parts, meticulously designed and assembled, it showcases the beauty of structural engineering. Its distinctive lattice design not only allows for stability but also contributes to its aesthetic appeal. The interplay of light and shadow on its intricate framework creates a mesmerizing sight, especially during sunrise and sunset.Beyond its architectural significance, the Eiffel Tower holds immense cultural and historical importance. It has served as a backdrop for countless films, novels, and works of art, becoming synonymous with romance, sophistication, and the allure of Paris. Its observation decks offer panoramic views of the city, attracting millions ofvisitors annually. Moreover, it has witnessed numerous historic events, from hosting the world's first radio transmission to serving as a symbol of resilience during times of adversity.Furthermore, the Eiffel Tower serves as a beacon of sustainability and innovation. Despite being over a century old, it continues to undergo renovations and upgrades to ensure its structural integrity and safety. Efforts to reduce its environmental footprint, such as theinstallation of energy-efficient lighting and waste management systems, highlight its commitment to sustainability.In conclusion, the Eiffel Tower stands as an enduring symbol of human achievement and creativity. From its humble beginnings as a controversial construction project to its status as a global icon, it has transcended its original purpose to become a cherished cultural landmark. Asvisitors gaze upon its towering silhouette, they are reminded not only of its architectural grandeur but also of the enduring spirit of innovation that defines the human experience.。

ntopology多孔点阵结构

ntopology多孔点阵结构English Answer:What is an ntopology lattice structure with multiple porosity?An ntopology lattice structure with multiple porosity is a three-dimensional structure that is made up of a network of interconnected struts. The struts are arranged in a repeating pattern, and the size and shape of thestruts can be varied to create different properties. The porosity of a lattice structure is the amount of void space within the structure. A lattice structure with multiple porosity has multiple levels of porosity, which can be useful for a variety of applications.What are the advantages of using an ntopology lattice structure with multiple porosity?There are a number of advantages to using an ntopologylattice structure with multiple porosity. These advantages include:High strength and stiffness: Lattice structures are very strong and stiff, which makes them ideal for use in applications where weight is a concern.Low density: Lattice structures are very lightweight, which can help to reduce the overall weight of a product.High permeability: Lattice structures have a high permeability, which allows fluids to flow through them easily.Tailorable properties: The properties of a lattice structure can be tailored to meet the specific requirements of an application.What are some applications for ntopology lattice structures with multiple porosity?Ntopology lattice structures with multiple porosityhave a wide range of applications, including:Aerospace: Lattice structures are used in aerospace applications to reduce the weight of aircraft and spacecraft.Automotive: Lattice structures are used in automotive applications to reduce the weight of vehicles and improve fuel efficiency.Biomedical: Lattice structures are used in biomedical applications to create implants and scaffolds.Energy: Lattice structures are used in energy applications to create lightweight and efficient heat exchangers.中文回答：什么是具有多孔性的ntopology 晶格结构？具有多孔性的ntopology 晶格结构是一种三维结构，由相互连接的支柱网络构成。

点阵结构技术在航天飞行器的应用分析

DOI：10.16660/ki.1674-098X.2011-5640-0139点阵结构技术在航天飞行器的应用分析①钟杰华1 赵文利2 蔡昱1(1.中国运载火箭技术研究院北京 100076；2.首都航天机械有限公司北京 100076）摘要：未来航天飞行器向着隐身化、多功能化、智能化等方向发展，对飞行器结构系统提出了轻质、结构功能一体化等需求。

点阵结构是先进轻质超强韧材料之一，是实现轻质结构功能一体化的有效载体。

介绍点阵结构技术的发展与应用，对点阵结构技术在未来航天飞行器的应用前景进行了初步探讨，并指出点阵结构在航天飞行器的应用方向主要包括点阵夹层圆柱壳结构、翼舵类点阵结构、飞行器有效载荷支架点阵结构等。

关键词：点阵结构航天飞行器结构功能一体化应用中图分类号：TJ450 文献标识码：A 文章编号：1674-098X(2021)03(c)-0001-05 Application Analysis of Lattice Structure Technology in SpacecraftZHONG Jiehua1 ZHAO Wenli2 CAI Yu1( 1.China Academy of Launch Vehicle Technology, Beijing, 100076 China; 2.Capital Aerospace MachineryCo., Ltd., Beijing ,100076 China)Abstract:In the f ut ure,the spacecraf t veh icle w i l l develop towards stea lth,mut i-f unct ion and intelligence,which puts forward the requirements of light weight and integration of incorporation of structure and function for the spacecraft structure system. Lattice structure is one of the advanced lightweight super-strong materials and is an effective carrier to realize the functional integration of lightweight structures.This paper introduces the development and application of lattice structure technology,and probes into the application prospect of lattice structure technology In the future spacecraft vehicle.It is pointed out that the application direction of lattice structure in space vehicle mainly includes lattice sandwich cylindrical shell structure,wing and rudder lattice structure ,payload support lattice structure and so on.Key Words: Lattice structure; Spacecraft ;Incorporation of structure and function; Application新概念航天飞行器向着隐身化、多功能化、智能化等方向发展，对飞行器结构系统提出了轻质、结构功能一体化等需求，包括承载/隐身一体、承载/隔热一体、承载/隔振一体等多功能集成化设计需求。

北京建筑英文作文

北京建筑英文作文Beijing is a city filled with a diverse range of architectural styles. From the ancient Forbidden City to the modern skyscrapers of the Central Business District, the city's architecture tells a story of its rich history and rapid development.Walking through the streets of Beijing, one can see traditional courtyard houses known as "siheyuan" alongside towering glass and steel structures. This contrast between old and new creates a unique visual landscape that reflects the city's dynamic nature.The iconic Bird's Nest stadium, built for the 2008 Olympics, is a prime example of Beijing's modern architectural marvels. Its intricate lattice structure and innovative design have made it a symbol of the city's ambition and forward-thinking spirit.In addition to its modern landmarks, Beijing is alsohome to numerous ancient architectural wonders such as the Great Wall and the Temple of Heaven. These timeless structures stand as a testament to the city's enduring legacy and cultural significance.The hutongs, narrow alleys lined with traditional courtyard homes, offer a glimpse into Beijing's historical architecture and way of life. These charming neighborhoods provide a stark contrast to the bustling city center and serve as a reminder of Beijing's humble origins.From the majestic imperial palaces to the sleek skyscrapers, Beijing's architecture is a reflection of its past, present, and future. The city's diverse architectural landscape is a testament to its rich cultural heritage and its status as a global metropolis.。

两栋建筑的比较英语作文

Exploring the Aesthetics and Functionality of Two Iconic Buildings: A Comparative Analysis Architecture, being a reflection of society, culture, and technology, often serves as a powerful medium to communicate ideas and values. Two buildings that stand as testaments to this fact are the Eiffel Tower in Paris and the Burj Khalifa in Dubai. These two structures, although situated in different continents and representing distinct eras of architecture, share a common thread of excellencein design, engineering, and cultural significance. In this essay, we delve into the comparisons between these twoiconic buildings, exploring their unique features, aesthetics, and functionality.The Eiffel Tower, a symbol of Paris and France, stands tall as one of the most recognizable landmarks in the world. Completed in 1889 for the World's Fair, this iron lattice tower revolutionized the concept of skyscrapers,introducing a new era of vertical construction. Its elegant and intricate design, with its lattice structure and four legs supporting a central platform, not only provides stability but also offers breathtaking views of Paris fromits observation deck. The tower's design, which wasinitially criticized for its boldness and lack oftraditional aesthetic values, has now become a symbol of romance and elegance.On the other hand, the Burj Khalifa, located in the heart of Dubai, is a testament to modern engineering marvels. Completed in 2010, it holds the title of being the tallest building in the world, surpassing even the Eiffel Tower in terms of height. The Burj Khalifa's design is a blend of traditional Islamic architecture and contemporary skyscraper technology. Its distinctive shape, with its spiraling design and tapered form, not only complements the surrounding desert landscape but also maximizes wind flow and reduces energy consumption. The building's interior is home to luxury hotels, offices, and residences, offering a complete lifestyle experience.When comparing these two buildings, it's evident that they both possess unique aesthetic and functional qualities. The Eiffel Tower, with its elegant lattice design and historical significance, represents a bygone era of romance and grandeur. It's a symbol of France's industrialrevolution and a testament to the nation's technological prowess. On the other hand, the Burj Khalifa, with its modern design and state-of-the-art technology, represents the future of architecture. It's a symbol of Dubai's rapid urbanization and a testament to the city's vision of becoming a global hub for business and culture.In terms of functionality, both buildings offer unique experiences. The Eiffel Tower, with its observation deck, offers breathtaking views of Paris, attracting millions of tourists every year. It's also a popular venue for events and weddings, offering a romantic and iconic backdrop. The Burj Khalifa, on the other hand, offers a completelifestyle experience, with luxury hotels, offices, and residences. It's a vertical city within a city, providing residents and visitors with access to a range of amenities and services.In conclusion, the Eiffel Tower and the Burj Khalifa, although situated in different parts of the world and representing distinct eras of architecture, both possess unique aesthetic and functional qualities that make them iconic landmarks. The Eiffel Tower, with its elegant designand historical significance, represents a bygone era of romance and grandeur, while the Burj Khalifa, with its modern design and state-of-the-art technology, represents the future of architecture. Both buildings offer unique experiences, attracting millions of people from around the world. They are not just structures; they are symbols of their respective cities and countries, testaments to the power of architecture in shaping our world.**探索两座标志性建筑的审美与功能：比较分析**建筑，作为社会、文化和技术的反映，通常作为一种强大的媒介来传达思想和价值观。

半导体结构及其形成方法与流程

半导体结构及其形成方法与流程1.半导体结构是一种具有特定能带结构的材料。

Semiconductor structure is a material with a specificband structure.2.它通常由多层材料堆积而成。

It is usually composed of multiple layers of materials stacked together.3.常见的半导体材料包括硅、锗和氮化镓等。

Common semiconductor materials include silicon, germanium, and gallium nitride.4.半导体的形成方法有多种多样。

There are various methods for the formation of semiconductors.5.最常见的方法是化学气相沉积。

The most common method is chemical vapor deposition.6.该方法利用化学反应将气态前驱物沉积在衬底表面。

This method uses chemical reactions to deposit gaseous precursors onto a substrate surface.7.其他方法包括物理气相沉积、分子束外延和激光熔化。

Other methods include physical vapor deposition, molecular beam epitaxy, and laser melting.8.这些方法可以根据不同的要求选择和应用。

These methods can be selected and applied according to different requirements.9.形成半导体的过程需要严格控制温度、压力和气氛。

The process of forming semiconductors requires strict control of temperature, pressure, and atmosphere.10.半导体的结构对于其性能有着重要的影响。

meshes化学英文学术名词

meshes化学英文学术名词英文回答：Meshes are a fundamental concept in chemistry, serving as a way to describe the arrangement of atoms or molecules in a solid. A mesh refers to a regular arrangement of points or nodes in a three-dimensional space, forming a lattice-like structure. This lattice structure is used to represent the crystal structure of a solid, which is crucial for understanding its properties and behavior.In chemistry, meshes are commonly used in the field of crystallography. Crystallography is the study of crystal structures, and it involves determining the arrangement of atoms within a crystal lattice. By analyzing the arrangement of atoms in a crystal lattice, scientists can gain insights into the physical, chemical, and mechanical properties of the material.One example of a mesh in chemistry is the unit cell. Aunit cell is the smallest repeating unit in a crystal lattice. It is a parallelepiped with edges defined by the lattice vectors. The unit cell represents the repeating pattern of atoms or molecules in the crystal lattice. By studying the unit cell, scientists can understand the symmetry and packing of atoms within the crystal lattice.Another example of a mesh in chemistry is the computational mesh used in molecular simulations. Molecular simulations involve using computer algorithms to model the behavior of molecules. In these simulations, the molecules are represented as a collection of points or nodes connected by bonds. The computational mesh allows for the calculation of various properties, such as energy, forces, and trajectories of the molecules.中文回答：网格是化学中的一个基本概念，用于描述固体中原子或分子的排列方式。

传热系数英文

传热系数英文Thermal ConductivityHeat transfer is a fundamental concept in physics and engineering, and one of the key parameters that governs this process is thermal conductivity. Thermal conductivity, denoted by the symbol 'k', is a measure of a material's ability to conduct heat. It is a crucial property that determines how effectively a material can transfer heat from one region to another. In this essay, we will delve into the intricacies of thermal conductivity, its underlying principles, and its practical applications.At its core, thermal conductivity is a measure of the rate at which heat flows through a material. It is defined as the amount of heat that flows through a unit area of a material per unit time, per unit temperature difference. The higher the thermal conductivity of a material, the faster heat can be transferred through it. Conversely, materials with low thermal conductivity are often used as insulators, as they impede the flow of heat.The mechanism of heat transfer through a material is primarily governed by the movement of electrons and lattice vibrations,known as phonons. In metals, the dominant mode of heat transfer is through the movement of free electrons, which can quickly transport thermal energy. In non-metallic solids, such as ceramics and polymers, the heat transfer is predominantly facilitated by the vibration of the atomic lattice, or phonons.The thermal conductivity of a material can be influenced by various factors, including its composition, structure, and temperature. For example, the thermal conductivity of a pure element is generally higher than that of an alloy, as the presence of impurities and defects in the alloy can disrupt the flow of heat. Additionally, the thermal conductivity of a material can change with temperature, as the mobility of electrons and the amplitude of lattice vibrations can be affected by temperature variations.One of the most widely used equations to describe thermal conductivity is Fourier's law of heat conduction. This law states that the rate of heat transfer through a material is proportional to the temperature gradient across the material. Mathematically, Fourier's law can be expressed as:q = -k * (dT/dx)where 'q' is the heat flux (the rate of heat transfer per unit area), 'k' is the thermal conductivity of the material, and 'dT/dx' is thetemperature gradient (the change in temperature per unit distance).The applications of thermal conductivity are numerous and span a wide range of industries. In the field of electronics, for example, thermal conductivity plays a crucial role in the design and performance of electronic devices. High-performance computer chips and power electronics generate significant amounts of heat during operation, and efficient heat dissipation is essential to prevent overheating and ensure the reliability of these devices. Materials with high thermal conductivity, such as copper and aluminum, are often used as heat sinks or heat spreaders to facilitate the transfer of heat away from the heat-generating components.In the construction industry, thermal conductivity is a key factor in the design of energy-efficient buildings. The choice of building materials, such as insulation, windows, and walls, can significantly impact the overall energy efficiency of a structure. Materials with low thermal conductivity, like fiberglass or foam insulation, are used to minimize heat transfer and reduce the energy required for heating and cooling.Another important application of thermal conductivity is in the design of heat exchangers, which are devices used to transfer heat between two or more fluids or between a fluid and a solid. Heat exchangers are found in a wide range of applications, includingpower generation, refrigeration, and chemical processing. The efficiency of a heat exchanger is directly related to the thermal conductivity of the materials used in its construction.In the field of materials science, the measurement and understanding of thermal conductivity is crucial for the development of new and improved materials. For example, the search for materials with exceptionally high or low thermal conductivity has led to the discovery of novel materials, such as graphene and aerogels, which have applications in areas like thermal management and insulation.In conclusion, thermal conductivity is a fundamental property that plays a crucial role in the way heat is transferred through materials. Its understanding and application are essential in a wide range of industries, from electronics and construction to materials science and energy production. As our technological landscape continues to evolve, the importance of thermal conductivity in the design and development of new and improved products and systems will only continue to grow.。

paris英文作文

paris英文作文Paris is a city that never fails to captivate me. The moment I step foot on its streets, I am instantly enchanted by its charm and elegance. The architecture, with its ornate details and grandeur, speaks volumes about thecity's rich history and cultural heritage. Every corner seems to hold a story, and I am eager to explore and uncover the secrets that lie within.The bustling streets of Paris are a feast for the senses. The aroma of freshly baked croissants wafts through the air, mingling with the scent of coffee brewing in quaint cafes. The sound of laughter and animated conversations fills the streets, creating a vibrant atmosphere that is uniquely Parisian. As I walk along the Seine River, I am mesmerized by the sight of artists capturing the beauty of the city on their canvases. Paris truly is a city that celebrates art and creativity in all its forms.One cannot talk about Paris without mentioning itsworld-renowned landmarks. The Eiffel Tower, standing tall and majestic, is a symbol of the city and a testament to human ingenuity. Its iron lattice structure is a sight to behold, especially when it is illuminated at night, casting a golden glow over the city. The Louvre Museum, with itsvast collection of art masterpieces, is a treasure trovefor art enthusiasts. From the Mona Lisa to the Venus de Milo, the museum houses some of the most iconic works ofart in the world.Paris is also a city of fashion and style. The streets are filled with impeccably dressed individuals,effortlessly exuding an air of sophistication and elegance. The fashion boutiques that line the streets offer a glimpse into the latest trends and designs. From haute couture to street style, Paris is a melting pot of fashion inspiration.But beyond its beauty and glamour, Paris is a city that embraces diversity and inclusivity. Its multicultural neighborhoods are a testament to the city's openness and acceptance of different cultures. From the vibrantChinatown to the lively African markets, Paris is a melting pot of cultures, where people from all walks of life come together to create a vibrant and dynamic community.In conclusion, Paris is a city that captures the heart and imagination of all who visit. Its beauty, culture, and history make it a truly unique and enchanting destination. Whether you are strolling along the Seine River, exploring the art galleries, or simply enjoying a cup of coffee at a sidewalk cafe, Paris offers a multitude of experiences that will leave a lasting impression.。

关于法国巴黎的英文作文

关于法国巴黎的英文作文Paris, a city known affectionately as the "City of Light," is not only the capital of France but also a global center for art, fashion, and culture. It is a place where history and modernity intertwine, offering a rich tapestry of experiences for visitors and residents alike.Nestled along the Seine River, Paris is home to some of the world's most recognizable landmarks. The Eiffel Tower, a symbol of French engineering prowess, stands tall as a beacon of romance and elegance. Visitors often marvel at itsintricate iron lattice structure and enjoy the panoramic views of the city from its various levels.Another architectural marvel is the Notre-Dame Cathedral, a masterpiece of French Gothic architecture. Despite the recent tragic fire, it remains a testament to the city's enduring spirit and is currently undergoing restoration to regain its former glory.Art lovers will find themselves in paradise with institutions such as the Louvre Museum, which houses thousands of works of art, including the enigmatic Mona Lisa and the majestic Winged Victory of Samothrace. The Muséed'Orsay, too, is a treasure trove of Impressionist and Post-Impressionist art, with masterpieces by Monet, Van Gogh, and Renoir gracing its halls.Paris is also a city of fashion, with its designer boutiques and haute couture houses lining the streets of the Champs-Élysées and the avenues of the Golden Triangle. The city's fashion week is a major event that draws fashionistas from around the world.Culinary delights are abundant in Paris, with its world-class restaurants and traditional bistros offering a range of gastronomic experiences. The city is famous for its croissants, baguettes, and fine cheeses, as well as its decadent pastries and desserts.For those seeking a more leisurely experience, the city's numerous parks and gardens provide a serene escape from the urban bustle. The Luxembourg Gardens and the Tuileries Garden are ideal for a peaceful stroll or a relaxing picnic.Despite its urban sophistication, Paris maintains astrong sense of community and charm. Local markets, such asthe Marché d'Aligre, offer a glimpse into the daily life of Parisians and provide an opportunity to sample fresh, local produce.In conclusion, Paris is a city that captivates the hearts and minds of all who visit. Its blend of history, culture, fashion, and gastronomy make it a destination that is at once familiar and endlessly fascinating. Whether it's the allureof its iconic landmarks, the charm of its cobblestone streets, or the simple pleasure of a café au lait at a sidewalk café, Paris offers an experience that is uniquely its own.。

基于金刚石体系的固态量子计算

第４４卷第５期Ｖｏｌ．４４，Ｎｏ．５２０１４年５月ＪＯＵＲＮＡＬＯＦＵＮＩＶＥＲＳＩＴＹＯＦＳＣＩＥＮＣＥＡＮＤＴＥＣＨＮＯＬＯＧＹＯＦＣＨＩＮＡＭａｙ２０１４文章编号：０２５３－２７７８（２０１４）０５－０３６２－１２基于金刚石体系的固态量子计算王鹏飞，石发展，杜江峰（中国科学技术大学近代物理系，合肥微尺度物质科学国家实验室，量子信息与量子科技前沿协同创新中心，安徽合肥２３００２６）摘要：量子计算科学是近年来物理学领域最活跃的研究前沿之一，其开拓了与经典方式具有本质区别的全新的信息处理模式．量子计算研究的根本目标是建造基于量子力学基本原理的量子信息处理技术，能在许多复杂计算问题上大大超越经典计算性能的新型计算模式．量子计算需要一个良好的量子体系作为载体．基于自旋的量子体系由于其实用的可操作性，成为量子计算载体的优秀候选．自旋的所有量子性质表现在自旋的叠加态、自旋之间的纠缠和对自旋的量子测量上．基于系综的量子计算演示实验已经被多次实现，但是系综体系在可扩展性上有其原理上的缺陷．要实现可扩展的大规模室温固态量子信息处理和量子计算的突破，实现单量子态的寻址和读出是一个最重要特的前提．在已经提出的单自旋固态量子计算载体中，比较突出的一类是基于金刚石中的氮－空位色约心单电子自旋体系．金刚石中的氮－空位色心单电子自旋量子态可以在室温下初始化、操控与读出，评成为室温量子计算机载体的优良候选者．我们首先回顾金刚石氮－空位色心单电子自旋体系作为量述子计算机载体的重要进展；然后讨论了该体系在纳米尺度灵敏探测和成像方面的重要应用；最后，描述了此领域的前景．关键词：量子计算；金刚石；氮－空位色心；电子自旋；光探测磁共振；核磁共振中图分类号：Ｏ４７４；Ｏ４８２．５３＋３文献标识码：Ａｄｏｉ：１０．３９６９／ｊ．ｉｓｓｎ．０２５３－２７７８．２０１４．０５．００２引用格式：ＷａｎｇＰｅｎｇｆｅｉ，ＳｈｉＦａｚｈａｎ，ＤｕＪｉａｎｇｆｅｎｇ．Ｑｕａｎｔｕｍｃｏｍｐｕｔａｔｉｏｎｂａｓｅｄｏｎｎｉｔｒｏｇｅｎ－ｖａｃａｎｃｙｃｅｎｔｅｒｉｎｄｉａｍｏｎｄ［Ｊ］．ＪｏｕｒｎａｌｏｆＵｎｉｖｅｒｓｉｔｙｏｆＳｃｉｅｎｃｅａｎｄＴｅｃｈｎｏｌｏｇｙｏｆＣｈｉｎａ，２０１４，４４（５）：３６２－３７３．王鹏飞，石发展，杜江峰．基于金刚石体系的固态量子计算［Ｊ］．中国科学技术大学学报，２０１４，４４（５）：３６２－３７３．Ｑｕａｎｔｕｍｃｏｍｐｕｔａｔｉｏｎｂａｓｅｄｏｎｎｉｔｒｏｇｅｎ－ｖａｃａｎｃｙｃｅｎｔｅｒｉｎｄｉａｍｏｎｄＷＡＮＧＰｅｎｇｆｅｉ，ＳＨＩＦａｚｈａｎ，ＤＵＪｉａｎｇｆｅｎｇ（ＤｅｐａｒｔｍｅｎｔｏｆＭｏｄｅｒｎＰｈｙｓｉｃｓ，ＨｅｆｅｉＮａｔｉｏｎａｌＬａｂｏｒａｔｏｒｙｆｏｒＰｈｙｓｉｃｓＳｃｉｅｎｃｅｓａｔＭｉｃｒｏｓｃａｌｅ，ａｎｄＳｙｎｅｒｇｅｔｉｃＩｎｎｏｖａｔｉｏｎＣｅｎｔｅｒｏｆＱｕａｎｔｕｍＩｎｆｏｒｍａｔｉｏｎａｎｄＱｕａｎｔｕｍＰｈｙｓｉｃｓ，ＵｎｉｖｅｒｓｉｔｙｏｆＳｃｉｅｎｃｅａｎｄＴｅｃｈｎｏｌｏｇｙｏｆＣｈｉｎａ，Ｈｅｆｅｉ２３００２６，Ｃｈｉｎａ）Ａｂｓｔｒａｃｔ：Ｑｕａｎｔｕｍｃｏｍｐｕｔａｔｉｏｎｔｅｃｈｎｏｌｏｇｙｉｓｏｎｅｏｆｔｈｅｈｏｔｔｅｓｔｔｏｐｉｃｓｉｎｐｈｙｓｉｃｓｆｏｒｔｈｅｐａｓｔｄｅｃａｄｅｓ．Ｉｔ收稿日期：２０１３－１１－１５；修回日期：２０１３－１２－１０基金项目：国家重点基础研究发展（９７３）计划（２０１３ＣＢ９２１８００），国家自然科学基金（１１２２７９０１，９１０２１００５，１０８３４００５，１１０２８５１０），中国科学院战略性先导科技专项Ｂ类（ＸＤＢ０１０３０４００），中央高校基本科研业务费专项资金资助．ｃｎｌ：ｗｐｆ＠ｕｓｔｃ．ｅｄｕ．作者简介：王鹏飞，男，１９８６年生，博士．研究方向：基于金刚石氮－空位色心的纳米尺度磁探测和成像．Ｅ－ｍａｉ通讯作者：杜江峰，男，１９６９年生，中国科学技术大学教授，教育部长江学者特聘教授，国家杰出青年科学基金获得者，国家重大科学研究计划项目首席科学家，首批国家万人计划“中青年科技创新领军人才”入选者，“新世纪百千万人才工程”国家级人选．２０００年获中国科学技术大学原子核物理专业博士学位．长期从事量子调控和量子计算的实验研究，是国际上自旋量子相干保持、量子计算、量子模拟实验研究方面有突出贡献的学者之一，在包括Ｎａｔｕｒｅ（２篇）、Ｓｃｉｅｎｃｅ（１篇）、ＮａｔｕｒｅＰｈｙｓｉｃｓ（１篇）、ＮａｔｕｒｅＣｏｍｍｕｎｉｃａｔｉｏｎｓ（２篇）和ＰｈｙｓｉｃａｌＲｅｖｉｅｗＬｅｔｔｅｒｓ（２０篇）在内的国际学术期刊上发表论文１１０余篇，ＳＣＩ他引１５００余次．成果入选２００９年度“中国高校十大科技进展”和两院院士评选的“２００９年度中国十大科技进展新闻”．曾获得国家自然科学二等奖、中国物理学会黄昆物理奖、教育部自然科学一等奖等奖项，以及“中国科学院杰出青年”、“江淮十大杰出青ｅｄｕ．ｃｎｆ＠ｕｓｔｃ．年”、“安徽青年五四奖章”等多项荣誉称号．Ｅ－ｍａｉｌ：ｄｊ第５期基于金刚石体系的固态量子计算３６３ｏｐｅｎｓｕｐａｎｅｗｗａｙｆｏｒｃｏｍｐｕｔｉｎｇ．Ｍａｉｎｌｙ，ｑｕａｎｔｕｍｃｏｍｐｕｔａｔｉｏｎｉｓａｉｍｅｄｔｏｂｕｉｌｄｑｕａｎｔｕｍｉｎｆｏｒｍａｔｉｏｎｐｒｏｃｅｓｓｉｎｇｓｙｓｔｅｍｂａｓｅｄｏｎｑｕａｎｔｕｍｔｅｃｈｎｏｌｏｇｙ，ｗｈｉｃｈｉｓｍｕｃｈｆａｓｔｅｒｔｏｓｏｌｖｅｈａｒｄｐｒｏｂｌｅｍｓｔｈａｎｃｌａｓｓｉｃｃｏｍｐｕｔｅｒｓ．Ａｇｏｏｄｑｕａｎｔｕｍｓｙｓｔｅｍｉｓｎｅｃｅｓｓａｒｙｔｏｂｕｉｌｄａｑｕａｎｔｕｍｃｏｍｐｕｔｅｒ．Ｔｈａｎｋｓｔｏｉｔｓａｂｉｌｉｔｙｉｎｍａｎｉｐｕｌａｔｉｏｎ，ｑｕａｎｔｕｍｓｐｉｎｓｙｓｔｅｍｈａｓｂｅｃｏｍｅｏｎｅｏｆｔｈｅｂｅｓｔｃａｎｄｉｄａｔｅｓｆｏｒｑｕａｎｔｕｍｃｏｍｐｕｔｅｒ．Ｔｈｅｑｕａｎｔｕｍｓｕｐｅｒｐｏｓｉｔｉｏｎ，ｅｎｔａｎｇｌｅｍｅｎｔ，ａｎｄｍｅａｓｕｒｅｍｅｎｔｓｈｏｗｔｈｅｑｕａｎｔｕｍｎａｔｕｒｅｏｆｓｐｉｎ．Ｔｈｅｄｅｍｏｎｓｔｒａｔｉｏｎｏｆｑｕａｎｔｕｍｃｏｍｐｕｔｅｒｏｎｅｌｅｃｔｒｏｎｉｃｓｐｉｎｅｎｓｅｍｂｌｅｗａｓｒｅａｌｉｚｅｄｙｅａｒｓａｇｏ．Ｈｏｗｅｖｅｒ，ｍａｎｙｑｕａｎｔｕｍｅｆｆｅｃｔｓｈｉｄｅｉｎｔｈｅｅｎｓｅｍｂｌｅｏｂｓｅｒｖａｔｉｏｎ．Ｉｔｗａｓｏｎｌｙｉｎｒｅｃｅｎｔｙｅａｒｓｔｈａｔｓｏｍｅｑｕａｎｔｕｍｅｆｆｅｃｔｓｈａｖｅｂｅｅｎｏｂｓｅｒｖｅｄｉｎｐｕｒｅ，ｓｉｎｇｌｅａｎｄｉｎｄｉｖｉｄｕａｌｑｕａｎｔｕｍｓｙｓｔｅｍｓｓｕｃｈａｓｎｉｔｒｏｇｅｎ－ｖａｃａｎｃｙ（ＮＶ）ｃｅｎｔｅｒｓｉｎｄｉａｍｏｎｄ．ＴｈｅｅｌｅｃｔｒｏｎｉｃｓｐｉｎｓｔａｔｅｏｆａＮＶｃｅｎｔｅｒｃａｎｂｅｉｎｉｔｉａｌｉｚｅｄ，ｍａｎｉｐｕｌａｔｅｄａｎｄｒｅａｄｏｕｔａｔｒｏｏｍｔｅｍｐｅｒａｔｕｒｅ．Ｔｈｕｓｉｔｂｅｃｏｍｅｓａｂｅｓｔｃａｎｄｉｄａｔｅｆｏｒｓｃａｌａｂｌｅｑｕａｎｔｕｍｃｏｍｐｕｔｅｒ．Ｉｎｔｈｉｓｐａｐｅｒ，ａｒｅｖｉｅｗｏｆｑｕａｎｔｕｍｃｏｍｐｕｔａｔｉｏｎｂａｓｅｄｏｎＮＶｃｅｎｔｅｒｓｉｎｄｉａｍｏｎｄｗａｓｇｉｖｅｎ．Ｔｈｅｎ，ｓｏｍｅｅｘｐｅｒｉｍｅｎｔｓｉｎｎａｎｏｓｃａｌｅｄｅｔｅｃｔｉｏｎａｎｄｉｍａｇｉｎｇｗｅｒｅｒｅｖｉｅｗｅｄ．Ｆｉｎａｌｌｙ，ｉｔｓｆｕｔｕｒｅｗａｓｄｉｓｃｕｓｓｅｄ．Ｋｅｙｗｏｒｄｓ：ｑｕａｎｔｕｍｃｏｍｐｕｔａｔｉｏｎ；ｄｉａｍｏｎｄ；ｎｉｔｒｏｇｅｎ－ｖａｃａｎｃｙｃｅｎｔｅｒ；ｅｌｅｃｔｒｏｎｉｃｓｐｉｎ；ｏｐｔｉｃａｌｌｙｄｅｔｅｃｔｅｄｍａｇｎｅｔｉｃｒｅｓｏｎａｎｃｅ；ｎｕｃｌｅａｒｍａｇｎｅｔｉｃｒｅｓｏｎａｎｃｅ０引言自从１９９７年针对金刚石中氮－空位色心（以下简称ＮＶ色心）单电子自旋的光探测磁共振技术［１］实现以来，基于ＮＶ色心的量子技术产生了快速的进步．在单电子自旋体系上可以观测到由于系综的平均效应而掩盖的一些重要的物理现象，并且由于ＮＶ色心体系的特殊性，室温下具有超长的退相干时间、良好的可扩展性、微波操控、光学读出，ＮＶ色心体系成为固态量子计算的明星体系．很快，基于ＮＶ色心的量子纠缠［２－３］和ｓｉｎｇｌｅ－ｓｈｏｔ测量［４－５］也相继被实现．２０１０年，在ＮＶ色心体系上完成了第一个量子计算算法演示实验［６］．为了达到延长退相干效果，我们把动力学去耦脉冲序列引入到ＮＶ色心上来［７－１０］，并深入了解了核自旋热库的量子力学行为［８，１１－１３］．为了达到实用量子计算的标准，量子比特数至少要达到３０比特以上，计算性能才能明显地超过经典计算机，进而完成一些经典计算机所不能完成的事情［１４－１５］．量子计算的一个重要的要求是可扩展性．在ＮＶ色心体系上，人们尝试了非常多的方案，旋超长的退相干时间，ＮＶ色心在磁探测和磁共振成像领域有着非常重要的应用［２１－２５］．另外，由于ＮＶ色心发光属于电偶极发光，它与外界电偶极矩还可以用于进行近场光学成像，并突破光学衍射极限，达到数十纳米的空间分辨率［２２］．本文分以下几部分来介绍ＮＶ色心与固态量子计算相关的几个方面：①ＮＶ色心的基本特性；②量子计算算法演示实验；③量子反常退相干；④连续波动力学去耦延长退相干时间；⑤量子相位精密测量；⑥室温下体外小核自旋系综的核磁共振谱．１ＮＶ色心的基本特性金刚石中的一个取代碳原子的氮原子，外加氮原子旁边的一个空位，这样就组成了ＮＶ色心结构．ＮＶ色心的空位中的未成对电子表现出自旋为１的性质．当使用５３２ｎｍ的绿光激发ＮＶ色心时，ＮＶ色心可以发出红色荧光，荧光的零声子线在６３７ｎｍ．通过５３２ｎｍ的激光激发后，可以以大于９０％的概率将ＮＶ色心制备到ｍＳ＝０的状态．通过微波和射频可以操控其状态．ＮＶ色心荧光强度与其电子自旋状态有关：当电子自旋处于ｍＳ＝０的状态包括ＮＶ色心与核自旋的纠缠［２－３］、近距离有耦合的时，荧光较电子自旋处于ｍＳ＝±１时强．通过与电两个ＮＶ色心的纠缠［１６－１７］、远距离上两个ＮＶ色心的纠缠［１８］．进一步地，ＮＶ色心的电子云在空间上局域在一个很小的范围之内，利用这一特点，ＮＶ色心可以作为高空间分辨率的探针，进行探测和成像［１９－２０］．电子自旋有着较高的旋磁比，加之ＮＶ色心电子自子自旋处于ｍＳ＝０的状态时的荧光强度对比，可以得知电子自旋目前所处的状态．相比于量子点、超导体系，ＮＶ色心电子自旋的相干性质非常好．引起ＮＶ色心退相干的主要因素是ＮＶ色心周围数目众多的自旋．在含氮量较高的金刚石中，退相干的主要来源是ＮＶ色心周围氮原特约评述３６４中国科学技术大学学报第４４卷子的未成对电子，退相干时间在微秒量级；当将金刚石中含氮量降为１０－８（质量分数）以下时，退相干来源主要是碳－１３（１３Ｃ）核自旋，其自然丰度为１．１％，退相干时间为百微秒量级；进一步将１３Ｃ核自旋减少至０．０３％以下时，ＮＶ色心退相干时间可以提升至毫秒量级，接近自旋－晶格弛豫时间［２６］．ｓｉｎｇｌｅ－ｓｈｏｔ测量已经实现［４－５］．ＮＶ色心体系满足成为量子计算机载体的最基本的要求，成为一个优秀的量子计算机载体候选者．２量子计算算法演示实验Ｄｅｕｔｓｃｈ－Ｊｏｚｓａ算法（ＤＪ算法）是最早提出来的量子算法之一，它可以显示出量子算法比任何已知的经典算法快指数倍．对于一个Ｎ位数集：Ｘｎ＝｛ｘｎｘｎ－１…ｘ２ｘ１｜ｘｍ＝０，１｝，函数ｆ（ｘ）：Ｘｎ→｛０，１｝．如果函数ｆ（ｘ）将Ｘｎ中一半的元素变为０，另一半变为１，那么这个函数是平衡函数（ｂａｌａｎｃｅｄｆｕｎｃｔｉｏｎ）；如果函数ｆ（ｘ）将Ｘｎ中所有的元素变为０或１，那么这个函数是常函数（ｃｏｎｓｔａｎｔｆｕｎｃｔｉｏｎ）．ＤＪ算法就是区分这种函数整体性质的量子算法．图１ＮＶ色心的晶格结构和自旋能级通常，实现ｎ位ＤＪ算法需要ｎ＋１个量子比特，而特Ｆｉｇ．１ＴｈｅｃｒｙｓｔａｌｌａｔｔｉｃｅｓｔｒｕｃｔｕｒｅａｎｄｅｌｅｃｔｒｏｎｉｃｓｐｉｎｅｎｅｒｇｙｌｅｖｅｌｏｆＮＶｃｅｎｔｅｒ约评为了实现真正意义上的实用的量子计算，量子述计算机的载体必须满足由ＤｉＶｉｎｃｅｎｚｏ在１９９６年提对应的经典算法却需要２ｎ个存储空间和运行２ｎ次．最简单的ＤＪ算法是单量子位的ＤＪ算法，如图２所示．其中一个量子比特存储数集，另外一个量子出的５条判断标准，简称ＤｉＶｉｎｃｅｎｚｏ判据［２７］．比特作为辅助比特．依靠ＮＶ色心自旋为１的特性，（Ⅰ）量子计算机必须有可识别的定义明确的量子比特．ＮＶ色心体系中包含一个自旋为１的电子自旋体系和多个核自旋，这些自旋都可以作为量子比特；（Ⅱ）量子计算机必须可以进行可信的初态制备．ＮＶ色心体系中的电子自旋可以通过激光激发［２８］单个量子比特就可以完成单量子位的ＤＪ算法．更改后的ＤＪ算法的脉冲序列图如图３［６］所示．的方式制备到ｍＳ＝０的状态，其周围的核自旋可图２单量子位ＤＪ算法的量子线路图以使用动力学核极化［２９］或者极化传递［２］的方式进行高保真度的初态制备；（Ⅲ）量子计算机必须具有较弱的退相干效应，保证量子计算有着较高的精度．在氮含量低于５× １０－９（质量分数）的超纯净金刚石中，ＮＶ色心电子自旋的退相干时间长达几百微秒［１１］，核自旋的退相干时间长达几十毫秒［２］，并且经过对ＮＶ色心电子自旋周围的核自旋进行纯化后，电子自旋的退相干时间可以长达几毫秒［２６］，相对于单个量子非门操作可达ＧＨｚ的速度来说［３０］，这些足以保证量子计算的高精度．（Ⅳ）量子计算机必须可以进行精确的量子门Ｆｉｇ．２ＱｕａｎｔｕｍｃｉｒｃｕｉｔｏｆｓｉｎｇｌｅｑｕａｎｔｕｍｄｉｇｉｔＤＪａｌｇｏｒｉｔｈｍ实验采用金刚石纳米颗粒样品，其中含有少量的单个ＮＶ色心．纳米金刚石中的ＮＶ色心存在较强的Ｃ３ｖ对称破坏，所以其在零场下的连续波谱会劈裂成两条峰．当加２×１０－３Ｔ的外磁场时，由于纳米金刚石的朝向是随机的，磁场不沿［１１１］轴方向，故两条跃迁分别为２．８４５０ＧＨｚ和２．８７１５ＧＨｚ．经过测量ＮＶ色心荧光的二阶关联函数ｇ２（τ）确定它是单个的ＮＶ色心．通过测量回波的衰减，可以得知该ＮＶ色心的退相干时间为２．８８μｓ和２．９６μｓ，分别对应两个不同的叠加态．其哈密顿量如下所示：《《《《《《《《操作．目前ＮＶ色心体系的单个量子门操作的保真度已经可以达到９９％以上［３１］．（Ⅴ）量子计算机必须建立非常强的量子测量机制．目前，ＮＶ色心体系对单电子和单核自旋的Ｈ＝ｇｅβｅＳ·Ｂ－ｇｎβｎＩ·Ｂ＋Ｓ·Ａ·Ｉ＋Ｓ·Ｄ·Ｓ其中，第１项、第２项分别是ＮＶ色心电子自旋和核自旋的塞曼分裂，第３项是ＮＶ色心与周围核自旋的超精细相互作用项，最后１项是ＮＶ色心电子自第５期基于金刚石体系的固态量子计算３６５（ａ）金刚石中ＮＶ色心的结构；（ｂ）共聚焦显微镜下ＮＶ色心的荧光光点；（ｃ）荧光光子的二阶关联函数ｇ２（τ），最低点在０．５以下，表明这是一个单个的ＮＶ色心；（ｄ）ＮＶ色心电子自旋的能级，ＭＷ１和ＭＷ２分别是操控不同能级的不同频率的微波；（ｅ）２×１０－３Ｔ和０Ｔ磁场下的连续波谱；（ｆ）测得两个不同叠加态｜０〉＋｜１〉和｜０〉＋｜－１〉的自旋回波衰减图３ＮＶ色心的性质Ｆｉｇ．３ＰｒｏｐｅｒｔｉｅｓｏｆＮＶｃｅｎｔｅｒ旋本身的零场分裂项．在纳米金刚石中，由于含氮量非常高，ＮＶ色心电子自旋与周围电子自旋的相互作用非常强，导致了快速的退相干．实验在自制的共聚焦显微镜上进行，一根直径２０μｍ的铜丝被用来作为微波辐射天线．当用共振的微波驱动ＮＶ色心电子自旋时，ＮＶ色心电子自旋的状态会不断地翻转．测量荧光强度，其荧光在做周期性的变化，如图４中的拉比振荡．这种单量子比特的ＤＪ算法称为ＲｅｆｉｎｅｄＤＪ算法，简称ＲＤＪ算法．我们采用｜０〉和｜－１〉作为存储数集的量子比特，｜＋１〉作为辅助的态．脉冲序列如图５所示．为了抵抗由于外磁场的抖动和自旋环境的快速量子涨落引起的退相位，我们将整个序列放在一个回波序列中．回波序列可以消除由于外磁场抖动等原因引起的快速退相位．从图６中可以看出，常函数和平衡函数可以很清晰地被区分出来．３量子反常退相干量子对象的退相干在量子科学和量子技术中是一个非常重要的现象．通常，人们认为更强的噪音会特约评述３６６中国科学技术大学学报第４４卷（ａ）和（ｂ）色心｜０〉和｜１〉（｜－１〉）之间的拉比振荡．（ｃ）和（ｄ）对拉比振荡曲线的快速傅里叶变换表明拉比振荡频率为６．９４ＭＨｚ图４ＮＶ色心电子自旋的拉比振荡Ｆｉｇ．４ＲａｂｉｏｓｃｉｌｌａｔｉｏｎｏｆｅｌｅｃｔｒｏｎｉｃｓｐｉｎｏｆＮＶｃｅｎｔｅｒ特约评述从上到下分别对应两个常函数和两个平衡函数图５ＲＤＪ的实验脉冲序列Ｆｉｇ．５ＥｘｐｅｒｉｍｅｎｔａｌｐｕｌｓｅｓｅｑｕｅｎｃｅｏｆＲＤＪ（ａ）和（ｂ）中是正向回波表明Ｖｆ１和Ｖｆ２是常函数，反之，（ｃ）和（ｄ）中是负向回波表明Ｖｆ３和Ｖｆ４是平衡函数图６ＲＤＪ算法实验结果Ｆｉｇ．６ＥｘｐｅｒｉｍｅｎｔｒｅｓｕｌｔｏｆＲＤＪ导致更快的退相干．然而，最近我们的工作却表明，在某些特定条件下，在ＮＶ色心中，电子自旋的退相干显现出一些反常效应［８］．在含氮量小于５×１０－９（质量分数）的样品中，Ｎ原子非常稀少，几乎不引起ＮＶ色心的退相干．此时，ＮＶ色心退相干的主要来源是其周围的１３Ｃ原子核自旋，这些自旋的丰度为１．１％．与ＮＶ色心距离５ｎｍ之内的１３Ｃ与ＮＶ色心的相互作用强度在百赫兹以上，它们之间主要以超精细相互作用为主．在低磁场下，１３Ｃ核自旋的塞曼分裂在百赫兹量级，与超精细相互作用相当．这时，１３Ｃ在外场作用下和在电子自旋作用下的拉莫尔进动方式的不同，会使ＮＶ色心表现出一些反常现象．如图７所示，当电子自旋处于叠加态时，核自旋在｜０〉和｜±１〉的共同作用下做进动．对于不同的自旋状态，核自旋演化路径出现了分歧．且由于不同的１３Ｃ与电子自旋的相互作用强度不同，不同核自旋演化不同，导致了电子自旋的退相干．当磁场在１０Ｇ以上时，核自旋演化的周期性使得在用自旋回波测量电子自旋的退相干时间Ｔ２时，有相干的“崩溃”和“复生”现象．这一点也表现出了核自旋环境的量子性（图８）．（ａ）ＮＶ色心处在１３Ｃ核自旋环境中．（ｂ）在ＮＶ色心的作用下，１３Ｃ核自旋会沿着两个不同的方向进动图７ＮＶ色心周围的自旋热库Ｆｉｇ．７ＳｐｉｎｂａｔｈｏｆＮＶｃｅｎｔｅｒ实验中采用自旋回波或动力学去耦序列来观察核自旋环境对中心ＮＶ色心自旋相干的影响．自旋回波可以在一定程度上去除外界静磁场和Ｂ１场的缓慢涨落，使得退相干的核自旋环境的性质可以显现出来．在自旋回波下，Ｔ２的长度一般在０．１～０．５ｍｓ之间；当使用动力学去耦序列以后，Ｔ２可以被延长至２ｍｓ以上．由于ＮＶ色心是一个自旋为１的体系，这种特殊的体系下可以观测到核自旋环境的一些量子特性，或者称之为反常退相干效应．在经典噪声环境第５期基于金刚石体系的固态量子计算３６７中，ｍＳ＝＋１－１之间禁戒跃迁的相干项比ｍＳ＝０ ±１之间的允许跃迁相干项受到２倍的噪音，其退（ａ）允许跃迁；（ｂ）禁戒跃迁；（ｃ）允许跃迁和禁戒跃迁的前２５μｓ对比图．在１．３５×１０－３Ｔ下，禁戒跃迁比允许跃迁退相干速度快，并且观测不到相干的“回复”现象图８磁场大小为１．３５×１０－３Ｔ下的自旋回波实验Ｆｉｇ．８Ｓｐｉｎｅｃｈｏｄｅｃａｙｕｎｄｅｒ１．３５×１０－３Ｔ相干速度要更快；但是在量子环境下，如丰度为１．１％的１３Ｃ核自旋环境，禁戒跃迁却比允许跃迁有着更长的退相干时间．这个特性只有在低磁场（例如５×１０－４Ｔ）下才会有，当磁场稍高（例如１．３５×１０－３Ｔ）时，核自旋环境的这种特性便会消失．这种反常效应可以作如下理解：在单量子跃迁下，核自旋演化的两条路径之间的夹角较大，演化路径的终点相距较远，最后会产生较大的退相干；而在双量子跃迁下，两条路径近似于反平行，演化路径的终点相距较近，退相干效应较弱（图９）．当磁场加大时，双量子跃迁演化路径中反平行被破坏，这种效应不再存在（图９）．深入了解引起ＮＶ色心退相干效应的来源后，我们可以使用一些去耦方法将ＮＶ色心与其他自旋隔绝开，阻止退相干的发生．除了采用周期性和非周期性翻转的动力学去耦脉冲序列之外，使用连续波动力学去耦（ＣＷＤＤ）方法，我们不但可以延长相干时间２０倍，而且可以同时进行高保真度的门操作［１０］．加上退相位噪音后，ＮＶ色心的哈密顿量可以写成（ａ）和（ｂ）分别是ＰＤＤ－１和ＰＤＤ－５去耦下实验测得的退相干曲线．其中黑色方块点和线代表允许跃迁，红色圆圈和线代表禁戒跃迁；作为对比，根据允许跃迁退相干曲线和经典退相干噪音计算的禁戒跃迁的退相干曲线，用蓝色倒三角表示．（ｃ）和（ｄ）退相干的物理图像．核自旋在ｍＳ＝０，±１下的进动路径在Ｂｌｏｃｈ球中的表示．经过多个π脉冲翻转后，两条路径终点之间的距离代表退相干效应的强弱．其中δ＋，－＜δ０，＋表示在禁戒跃迁下的退相干效应弱于允许跃迁图９磁场为５×１０－４Ｔ时的反常退相干效应Ｆｉｇ．９Ｑｕａｎｔｕｍａｎｏｍａｌｏｕｓｄｅｃｏｈｅｒｅｎｃｅｅｆｆｅｃｔｕｎｄｅｒａｍａｇｎｅｔｉｃｆｉｅｌｄｏｆ５×１０－４Ｔ４连续波动力学去耦延长退相干时间特约评述［３６８中国科学技术大学学报第４４卷Ｈ＝Ｈ０＋γｅｂｚＳｚ其中，ｂｚ是引起退相干的随机磁场．如果将ＮＶ色心电子自旋制备在一个特殊｜±１〉叠加态上，这个叠加态便会对这个随机磁场不敏感，于是压制了退相干．采用两个偏共振的微波场可以将ＮＶ色心初始化到这个特殊的叠加态．这时其哈密顿量为ＨＮＶ＝ωｄｇ．我们选后两者作为量子比特的两个能级．缀饰态的能级和ｂ２有关，所以跟｜δω０，－１｜和｜ｂ｜的线性的相关相比，缀饰态对于ｂ是不敏感的．于是，当微波驱动场远大于ｂ时，｜ｄ〉和｜ｇ〉之间的退相位会被压制（图１０）．实验的样品是氮含量小于５×１０－９（质量分数）的金刚石单晶，磁场为１．２×１０－３Ｔ．在这种样品中，ＮＶ色心的退相位时间为０．９３μｓ，主要诱导机制为电子自旋周围的１３Ｃ核自旋的随机涨落，推算 ∑ （Δ＋γｅｂι）｜ι〉〈ι｜＋ Ω （｜０〉〈ι｜＋｜ι〉〈０｜）２］出ｂ＝０．１７ＭＨｚ．通过施加两路非共振微波制备这个哈密顿量对角化后，会产生３个缀饰态：｜ｅ〉，｜ｄ〉和｜ｇ〉．它们之间的跃迁频率分别为 ωｅｄ和ＮＶ色心到缀饰态后，退相位时间变为１８．９μｓ，增产了约２０倍．特约评述（ａ）ＮＶ色心自旋能级结构；（ｂ）能级在外磁场下的劈裂；（ｃ）能级跃迁频率与外界随机涨落磁场的线性关系；（ｄ）在双偏共振微波场下产生的缀饰态的能级和跃迁频率；（ｅ）缀饰态的能级结构与外界随机涨落磁场的关系；（ｆ）缀饰态之间的跃迁频率与外界随机涨落磁场之间的平方关系．这里取微波频率偏置 Δ ＝２ＭＨｚ，拉比振荡频率 Ω ＝４Δ图１０缀饰态下ＮＶ色心的能级Ｆｉｇ．１０ＥｎｅｒｇｙｌｅｖｅｌｏｆＮＶｃｅｎｔｅｒｉｎｄｒｅｓｓｅｄｓｔａｔｅι第５期基于金刚石体系的固态量子计算３６９（ａ）连续波谱，３条峰对应电子自旋与氮核的超精细相互作用；（ｂ）自由感应衰减（ＦＩＤ）信号；（ｃ）在微波缀饰态下的连续波谱．（ｄ）微波缀饰态之间的ＦＩＤ信号图１１ＮＶ色心的退相干Ｆｉｇ．１１ＤｅｃｏｈｅｒｅｎｃｅｏｆＮＶｃｅｎｔｅｒ进一步地，使用射频来操纵缀饰态，可以得到一个ＣＷＤＤ保护的高保真的量子操作．从对比中可以得知，在同等时间长度下，操作的保真度远远超过了普通的微波操作（图１１和图１２）．在ＣＷＤＤ的情况下，退相位被有效地压制，并且ＣＷＤＤ产生了一个缀饰态空间，在这个空间中可以进行一系列的量子逻辑门操作．金刚石ＮＶ色心的一个重要应用是作为量子干涉仪对外界电磁信号进行测量．其原理是将ＮＶ色心电子自旋或核自旋制备到叠加态，叠加态在外界电磁场下会产生相对相位，最后读出这个相对相位，便可以得知外场的数值．这其中可以使用量子相位估计算法（ＱＰＥＡ）提高读出精度．ＱＰＥＡ是量子计算中的一个非常重要的算法，也是量子精密测量中的一个重要工具，采用ＱＰＥＡ可以精确地得到当前量子比特各量子态之间的相对相位信息．在经典的测量方法中，测量的精度与测量资源的消耗是平方根反比关系，并且测量的精度受标准图１２在无和有ＣＷＤＤ的情况下，Ｒａｂｉ振荡（ａ）和Ｒａｂｉ振荡包络（ｂ）的衰减Ｆｉｇ．１２Ｒａｂｉｏｓｃｉｌｌａｔｉｏｎ（ａ）ａｎｄｅｎｖｅｌｏｐｅ（ｂ）ｄｅｃａｙｕｎｄｅｒｎｏＣＷＤＤａｎｄＣＷＤＤ量子极限限制．而量子测量却可以将测量精度与测量自旋消耗之间的关系变成线性反比关系，并且可５量子相位精密测量特约评述３７０中国科学技术大学学报第４４卷以突破标准量子极限限制，达到海森堡量子极限［３２－３４］．在量子测量中需要用到量子系统的相干．我们都知道动力学去耦可以保持量子系统的相干性，而在量子测量中，相干时间越长，信号的衰减就越小，就能达到越高的精度．如果将动力学去耦脉冲序列和量子相位估计算法序列结合，就能达到更好的效果．ＮＶ色心体系由于其超长的退相干时间和干净的退相干环境，非常适合做量子测量的探针．将ＮＶ色心电子自旋的｜０〉和｜１〉能级编码为量子干涉仪，用｜－１〉能级作为要读出的外界的信号，通过改变它可以读出由于几何演化路径不重合产生的几何相位（ＡＡ相）．在实验中，首先ＮＶ色心被制备到叠加态上，然后两个相位不同的微波 π 操作被用来产生ＡＡ相，最后通过读取叠加态上的相对相位得到ＡＡ相．当重复Ｎ次Ｕ操作时，可以将ＡＡ特相放大Ｎ倍，进一步提高精度（图１３）．约测量精度和灵敏度与ＮＶ色心的退相干时间有评关，退相干时间越长，信号越强，精度和灵敏度就越述高．于是，引入周期性的动力学去耦脉冲序列后，可以得到更高的精度．如图１４所示，每一次产生ＡＡ（ａ）标准的量子干涉仪的量子线路图；（ｂ）在Ｕ操作下，量子比特状态的几何路径；（ｃ）将量子线路图转化为脉冲序列；（ｄ）ＮＶ色心的能级结构和所用的微波频率的对应关系图１３相位测量的量子线路图Ｆｉｇ．１３Ｑｕａｎｔｕｍｃｉｒｃｕｉｔｓｆｏｒｐｈａｓｅｍｅａｓｕｒｅｍｅｎｔ相之后，加入一个用于去耦的 π脉冲，可以将退相干时间延长至百微秒以上．实验结果如图１５所示．通过这种多次通过机制，可以获得比经典上更强的测量精度．（ａ）将动力学去耦序列和量子干涉仪序列结合后的脉冲序列；（ｂ）在两个不同的相位 Ω 值下的拉比振荡；（ｃ）测得的ＡＡ相与理论值的对比图１４ＡＡ相测量的脉冲序列和实验结果Ｆｉｇ．１４ＰｕｌｓｅｓｅｑｕｅｎｃｅａｎｄｅｘｐｅｒｉｍｅｎｔｒｅｓｕｌｔｏｆＡＡｐｈａｓｅｍｅａｓｕｒｅｍｅｎｔ第５期基于金刚石体系的固态量子计算３７１从结果可以看出，采用动力学去耦与相位估计算法结合的测量结果的精确度已经超过了经典的测量方法图１５多次通过相位估计测量的脉冲序列和实验结果Ｆｉｇ．１５Ｐｕｌｓｅｓｅｑｕｅｎｃｅａｎｄｅｘｐｅｒｉｍｅｎｔｒｅｓｕｌｔｏｆｍｕｌｔｉ－ｐａｓｓｐｈａｓｅｅｓｔｉｍａｔｉｏｎ６室温下体外小核自旋系综的核磁共振谱核磁共振（ＮＭＲ）可以在不用标记的情况下进行化学结构分析，是生物学、化学、医学等学科中重要的分析手段．一旦能够进行纳米尺度的ＮＭＲ结构分析将会对这些学科产生重大的影响．之前，人们已经用磁共振力显微镜实现了（４ｎｍ）３的质子系综的探测［３５］，但这项技术非常有难度：超低温和高真空．然而，使用ＮＶ色心，可以在室温下进行核自旋的探测．目前，人们已经实现依靠ＮＶ色心与核自旋之间的超精细相互作用，探测金刚石体内的单个１３Ｃ核自旋［３６－３８］．当ＮＶ色心处在非常接近金刚石表面的位置时，它才可以被用来探测体外的自旋，如图１６所示．在ＮＶ色心电子自旋的作用下，体外核自旋在做进动，产生了一个类似于ＡＣ信号的弱磁场．将ＮＶ色心电子自旋制备到叠加态，然后进行动力学去耦操作保持相干，当动力学去耦序列的周期与核自旋进动产生的ＡＣ场的周期相对应时，ＮＶ色心叠加态上可以累加最大的相位，这时信号幅度最强．通过对得到的信号进行分析，可以得知外界核自旋的旋磁比等信息，从而知晓核自旋的种类．氢（１Ｈ）核自旋（也就是单个的质子）是非放射图１６小核自旋系综ＮＭＲ的样品示意图Ｆｉｇ．１６ＳｍａｌｌｎｕｃｌｅａｒｓｐｉｎｅｎｓｅｍｂｌｅＮＭＲ性原子核中旋磁比最高的核自旋，然而样品中的１３Ｃ核自旋也是不可去除的噪音．将１３Ｃ同位素尽可能地减少后，通过改变磁场的大小测试核自旋信号对应的旋磁比，可以将１３Ｃ的影响减到最小，然后显现出要观测的核自旋信号（图１７）．图１７在不同磁场下外界核自旋的拉莫尔进动频率对比Ｆｉｇ．１７Ｕｎｄｅｒｄｉｆｆｅｒｅｎｔｍａｇｎｅｔｉｃｆｉｅｌｄ，ｔｈｅＬａｒｍｏｒｆｒｅｑｕｅｎｃｙｏｆｎｕｃｌｅａｒｓｐｉｎｓａｒｅｄｉｆｆｅｒｅｎｔ目前，多数近表面ＮＶ色心的制备都是采用离子注入的方法．在文献［２４］中，采用２．５ｋｅＶ１５Ｎ＋离子注入，可以产生离表面大约５～１０ｎｍ的色心．通过在金刚石表面涂ＰＭＭＡ（聚甲基丙烯酸甲酯）、显微镜镜浸油，并与裸金刚石作对比，可以得到探测对象中的１Ｈ核自旋信号．如图１８所示，ＮＶ色心距离金刚石表面，也就是距离探测对象大约７ｎｍ，测量得到的全部信号对应于４＊１０５个１Ｈ核自旋，约（５ｎｍ）３的ＰＭＭＡ贡献了７０％的信号．特约评述。

古今中外建筑英语作文

古今中外建筑英语作文Title: Architectural Wonders: A Journey Through Time and Space。

Architecture, as a manifestation of human creativity and ingenuity, transcends geographical boundaries and time periods. From ancient civilizations to modern metropolises, the world is adorned with architectural marvels that narrate the story of human civilization. In this essay, we will embark on a captivating journey through the annals of architectural history, exploring significant structures from diverse cultures across the globe.Ancient Architecture:Ancient civilizations such as Egypt, Mesopotamia, Greece, and Rome have left an indelible mark on architectural history. The Great Pyramid of Giza, constructed over 4,500 years ago, stands as a testament to the advanced engineering skills of the ancient Egyptians.Its monumental presence continues to awe and inspire visitors from around the world.Moving eastward, the Forbidden City in Beijing, China, represents the pinnacle of imperial architecture. With its grandiose palaces, intricately designed courtyards, and majestic gates, it served as the political and ceremonial center of the Ming and Qing dynasties.Medieval and Renaissance Europe:The Middle Ages witnessed the rise of Gothic architecture, characterized by soaring cathedrals adorned with intricate stained glass windows and ribbed vaults. Notre-Dame Cathedral in Paris, with its iconic facade and majestic rose windows, epitomizes the Gothic style's spiritual essence and architectural brilliance.During the Renaissance, Europe experienced a revival of classical principles, leading to the construction of magnificent structures such as the Florence Cathedral and St. Peter's Basilica in Rome. These masterpieces ofRenaissance architecture reflect a harmonious blend of mathematical precision, artistic expression, and spiritual symbolism.Modern Marvels:The Industrial Revolution brought about revolutionary changes in architectural design and construction techniques. The Eiffel Tower, erected in 1889 as the centerpiece of the Paris World's Fair, symbolizes the triumph of modern engineering and innovation. Its lattice-like structure and towering height made it a pioneering feat of iron construction.In the 20th century, architects like Frank Lloyd Wright and Le Corbusier redefined the principles of modern architecture with their groundbreaking designs. Fallingwater, Wright's masterpiece nestled amidst theforests of Pennsylvania, seamlessly integrates nature with architecture, embodying his philosophy of organic architecture.Contemporary Architecture:Today, architecture continues to evolve with advancements in technology, sustainability, and cultural diversity. The Burj Khalifa in Dubai, standing as theworld's tallest building, showcases the marriage ofcutting-edge engineering and architectural design. Its sleek silhouette pierces the sky, symbolizing human ambition and achievement.Moreover, sustainable architecture has emerged as a pressing concern in response to environmental challenges. The Bosco Verticale in Milan, Italy, is a pioneering example of green architecture, featuring residential towers adorned with thousands of trees and plants, which help purify the air and mitigate urban heat island effects.Conclusion:In conclusion, the rich tapestry of architectural heritage encompasses a vast array of styles, techniques, and ideologies spanning millennia. From the timelesswonders of antiquity to the innovative skyscrapers of the modern era, architecture serves as a tangible expression of human aspiration, culture, and creativity. As we marvel at these architectural wonders, we are reminded of the enduring legacy of human ingenuity and the boundless possibilities of the built environment.。

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a rXiv:cs /0136v1[cs.D M]31O ct2Lattice structure and convergence of a Game of Cards ⋆Eric Goles 1,Michel Morvan 2,and Ha Duong Phan 31Departamento de Ingenier ´ıa Matem´a tica,Escuela de Ingenier ´ıa,Universidad de Chile,Casilla 170-Correo 3,Santiago,Chile,2LIAF A Universit´e Denis Diderot Paris 7and Institut universitaire de France -Case 7014-2,Place Jussieu-75256Paris Cedex 05-France,3LIAF A Universit´e Denis Diderot Paris 7-Case 7014-2,Place Jussieu-75256Paris Cedex 05-France,egoles@dim.uchile.cl,morvan@liafa.jussieu.fr,phan@liafa.jussieu.fr Abstract.We study the dynamics the so-called Game of Cards by us-ing tools developed in the context of discrete dynamical systems.We extend a result of [4]and of [10](this last one in the context of dis-tributed systems)who established a necessary and suﬃcient condition for the game to converge.We precisely describe the structure of the set of conﬁgurations (that we show to be very closed to a lattice structure)and we state bounds for the convergence time.Keywords:Integer composition,Order,Lattice,Convergence.1Introduction This paper is devoted to the study of the dynamics of a discrete system related to some self stabilizing protocol on a ring of processors.As explained in [4],this protocol can be seen in terms of a game of cards described as follows.“Assume a ﬁnite set of players sitting around a table.Initially,each player holds a ﬁnite number of non-distinguishable cards.The only move a player can make is passing a card to his/her right neighbor,provided that this neighbor has fewer cards than the player itself.The game terminates when no move is possible.”In the cited paper,the following theorem is proved.Theorem 1.The Game of Cards terminates if the total number of cards is amultiple of the number of players.The proof given by the authors for this theorem was simpler than the one proposed for the equivalent result in [10].Moreover,the authors werw pointing out the fact that studying some distributed protocols in terms of discrete dy-namical systems could be fruitful.In this paper,we replace the game of cardsin the broader context of the study of transition systems on compositions of a given integer,where the total number of cards which is decomposed in each conﬁguration is the sum of the number of cards of each player.The dynamics of such transition systems on compositions has been intensively studied by various authors and provides a powerful framework to derive structural and dynamical properties[2,6,9,7,8].In this paper,we are going to investigate in more details the structure of the set of all possible conﬁgurations of the game with n cards and p players. For that,let us represent a conﬁguration by a list of p integers a=(a1,...,a p) where a i is the number of cards of player i.At each step player i can give a card to player(i+1)(modulo p).Let n=kp+q with0≤q<p.Let us call G the graph deﬁbed on the set of all possible conﬁgurations of the game—that is the set of p-dimensional vectors of integers such that the sum of the components is n,and having for arcs the set of couples(a,b)of conﬁgurations such that b can be obtained from a in one step.Following[4],let us call dual those conﬁguration that do not belong to a non trivial circuit.The Game of Cards can be coded by a Chip Firing Game of which many propeties have been studied in[11].Moreover, due to its special rule,the Game of Cards has other interesting propeties that we will studying in this paper.We willﬁrst characterize dual conﬁguration and will show that if n is not a multiple of p,the unique non trivial strongly connected component of the graph G is the set of dual conﬁgurations.We will also study the subgraph of G induced by the set of conﬁgurations that can be reached from a given conﬁguration O.We will characterize the partial order naturally associated to this graph when the set of dual conﬁgurations is replaced by a unique vertex, and we will establish its lattice structure.We willﬁnish by bounding the number of steps necessary to arrive to a dual conﬁguration.In the following,we are going to discuss some lattice properties of the above dynamical systems.Let us recall that aﬁnite lattice can be described as aﬁnite partial order such that any two elements a and b admit a least upper bound(de-noted by sup(a,b))and a greatest lower bound(denoted by inf(a,b)).Sup(a,b) is the smallest element among the elements greater than both a and b.Inf(a,b) is deﬁned similarly.A useful result aboutﬁnite lattices is that a partial order is a lattice if and only if it admits a greatest element and any two elements admit a greatest lower bound.For more informations about lattice theory,see[1,3]. 2Basic structure of the Game of CardsLet usﬁrst state the following corollary of the main theorem of[4].Corollary1.If q=0there is no dual conﬁguration,which means that the game terminates;if q>0,the game does not terminates and the dual conﬁgurations are exactly the q p conﬁgurations such that each player owns either k or k+1 cards.Proof.The proof comes immediately from the proof of[4]⊓⊔See Figure 1for two examples:2,2,2,02,3,14,1,13,1,25,0,13,2,12,2,24,0,1,13,2,0,13,1,1,12,3,0,12,1,2,11,2,2,12,1,1,21,2,1,21,1,2,22,2,1,16,0,05,1,04,2,03,3,06,0,0,05,1,0,04,2,0,05,0,1,03,3,0,04,1,1,05,0,0,14,1,0,13,2,1,02,3,1,03,1,2,0Fig.1.The Game of Cards in two cases:6cards and 3players,and 6cards and 4players.a −→b signiﬁes that b can obtained from a by moving a card from a player positioned between 1and p −1.a −−>b signiﬁes that b can obtained from a by moving a card from the p −th player to the ﬁrst player.We can now state the following theorem.Theorem 2.The unique non trivial strongly connected component of G is the set of dual conﬁgurations.Proof.The result is obvious if q =0.If q >0,it is clear from the deﬁnition that a non dual conﬁguration can not belong to a strongly connected component since it does not belong to a circuit.So we just have to prove that the set of dual conﬁgurations is a strongly connected component.Let a =(a 1,...,a p )and b =(b 1,...b p )be two dual conﬁgurations.We have ∀i ∈[1,p ],the value of a i and b i is k or k +1.Let P be the dual conﬁguration deﬁned byP =k +1,...,k +1 q ,k,...,k p −q.We are going to show that there exists a path from a to P and a path from P to b ,which implies the existence of a path from a to b .Similarly,the existence of a path from b to a could be stated.The path from a to P is built by following an arbitrary maximal path start-ing from a and in which no transition is made in which player q gives a card to its right neighbor (such a path exists since in each inﬁnite path player q plays an inﬁnite number of times).The unique possible transition from the last con-ﬁguration of this path is the one in which player q plays,which proves that this conﬁguration is P .We have now to ﬁnd a path from P to b .Let i be the last index smaller than or equal to p such that b i =k +1.By consecutively applying the playing rule from conﬁguration P on players q ,q +1,...,i −1,we obtain the following conﬁguration:k +1,...,k +1 q −1,k,...,k,k +1 ith,k,...,k.Let j be the last index smaller than i such that b j =k +1.We can apply the same techniques and by iterating the process,we obtain the conﬁguration b .This achieves the proof.⊓⊔If we remark now that when q >0there is no ﬁxed point,the previous result allows to state:Corollary 2.There is a path from any conﬁguration to any dual conﬁguration.If q >0,any inﬁnite sequence of transitions starts by a ﬁnite sequence of transitions applied to a non dual conﬁguration and leading to another non dual conﬁguration and is followed by an inﬁnite sequence of transitions applied to dual conﬁgurations and leading to dual conﬁgurations.In some sense,these dual conﬁgurations represent a generalization of ﬁxed points,since when the system is in such conﬁguration it cannot reach a non dual conﬁguration but it can reach any dual one.So it is natural to consider the reduced graph R (G )obtained from G by replacing the set of dual conﬁgurations by a unique vertex ⊥with no outgoing arc and with one ingoing arc coming from each non dual conﬁguration that can lead in one step to a dual conﬁguration.(R (G )is then the quotient graph relatively to the equivalence relation “is in the same strongly connected component of”).We can now consider the partial order <gc on R (G )by considering the transitive closure of the graph:b <gc a if b is reachable from a by following a path in the graph.We are now going to focus on the relation between non dual conﬁgurations by ﬁrst showing that between two such conﬁgurations the length of all paths is the same.For that,we are going to introduce the notion of “shot vector”,following [5],which describes for each player the number of cards it gave to its neighbor.Let a and b be two non dual conﬁgurations such that b <gc a .Let C be a sequence of transitions from a to b :C :a →c 1→c 2→...→c l →b.Let s i(C)be the number of cards given by player i to its neighbor during this sequence.Let s(C)be the sequence(s1(C),s2(C),...,s p(C))and call it the shot vector of the sequence C.Let|s(C)|= i=p i=1s i(C)be the length of the sequenceC.It immediately comes that:s2(C)−s1(C)=a2−b2,s3(C)−s2(C)=a3−b3,...s1(C)−s p(C)=a1−b1,which implies thats(C)=(s p(C),...,s p(C))+(a1−b1,a1+a2−b1−b2,...,a1+...+a p−1−b1−...−b p−1,0). If we denoted(a,b)=(a1−b1,a1+a2−b1−b2,...,a1+...+a p−1−b1−...−b p−1,0),we obtains(C)=s p(C)∗(1,...,1)+d(a,b).Let us now introduce the following partial order between shot vectors:given twoshot vectors s(C)and s(D),s(C)≤s(D)if∀i s i(C)≤s i(D).Let a and b be two elements reachable from a given conﬁguration O.Let us consider two sequencesof transitions,one from O to a,the other from O to b:C:O→c1→c2→...→c u→a,D:O→d1→d2→...→d v→b.We can state the following lemma:Lemma1.Assume that there exists an index j such that s j(C)≤s j(D)and∀j′=j,s j′(C)≥s j′(D).If it is possible to apply the rule on position j of b,then it is also possible to apply this rule to a in the same position j.Proof.From the shot vector deﬁnition,we obtain:a1=O1−s1(C)+s p(C),a2=O2−s2(C)+s1(C),...a p=O p−s p(C)+s p−1(C).Since the necessary and suﬃcient condition to apply the rule on position j of bis b j−b j+1≥1,we are going to focus on the diﬀerence a j−a j+1.a j−a j+1=O j−s j(C)+s j−1(C)−(O j+1−s j+1(C)+s j(C))=O j−O j+1−2s j(C)+s j−1(C)+s j+1(C)≥O j−O j+1−2s j(D)+s j−1(D)+s j+1(D)=b j−b j+1≥1,which proves the lemma.⊓⊔We can now establish the following result,which states that the shot vector associated to a sequence of transitions only depends on the initial andﬁnal conﬁgurations.Proposition1.Let a and b be two non dual conﬁgurations such that b<gc a. Then all the sequences of transitions from a to b have the same shot vector and so have the same length.Proof.Let C and D be two sequences of transitions from a to b:C:a→c1→c2→...→c u→b,D:a→d1→d2→...→d v→b.Let us recall that:s(C)=s p(C)∗(1,...,1)+d(a,b),s(D)=s p(D)∗(1,...,1)+d(a,b).Assume that s p(D)>s p(C).We have s(D)>s(C).We are going to show that there exists a path of positive length from b to b,which is a contradiction.For that,we are going to build step by step a sequence of transitions from b to b: b→e1→...→e l→b.For i≤v let us denote by D i the following sequence:D:a→d1→d2→...→d i.There exists aﬁrst index i such that s(D i)≤s(C),which implies that there exists j such that s j(D i)>s j(C)and∀j′=j,s j′(D i)≤s j′(C).Since i is the ﬁrst index having this property,we have s(D i−1)≤s(C),so s j(D i−1)=s j(C) and s j(D i)=s j(C)+1.Since d i−1and a satisfy the conditions of Lemma1,we can apply the rule on position j to b to obtain a new conﬁguration denoted by e1.Let E1be the following sequence of transitions:a→c1→c2→...→c u→b→e1.We then have s(D i)≤s(E1)≤s(D).By iterating this process,we can deﬁne e2,e3,...Since|s(E i)|−|s(C)|=i and s(C)≤s(E i)≤s(D),after l=|s(D)|−|s(C)|steps,we will obtain e l→b,which is the contradiction.Since the case where s p(D)<s p(C)is similar,s p(D)=s p(C)and therefore s(D)=s(C),which achieves the proof.⊓⊔Using this result,we can deﬁne the shot vector s(a,b)for any couple of non dual conﬁgurations a and b such that b<gc a as being equal to the shot vector of any sequence of transitions from a to b.This shot vector will be very useful in understanding more precisely the structure and properties of the game.3Lattice structure of the Game of CardsWe dispose now of the tools we need for studying the structure of the set of all conﬁgurations that can be obtained from a given initial conﬁguration O= (O1,...,O p).Let us denote by GC(O)the set of all non dual conﬁgurations reachable from O to which we add⊥as unique minimal element if q>0(GC(O) is then the restriction of R(G)to the conﬁgurations reachable from O).We are going to study the order(GC(O),<gc)(in the following,for simplicity reasons, GC(O)will both denote the set itself and the associated partial order).Let us ﬁrst characterize this order.Theorem3.Let a and b be two non dual conﬁgurations of GC(O),thena>gc b⇔s(O,a)<s(O,b).Proof.In order to show thats(O,a)<s(O,b)⇒a>gc b,we can consider two sequences of transitions,one from O to a and the other from O to b,and then make a similar proof as the one used in Proposition1.On the other hand,let a and b be two non dual conﬁgurations of GC(O)such that b<gc a.Let E be a sequence of transitions from a to b.The sequence D built by concatenating the sequence C from O to a and the sequence E is clearly a sequence from O to b,and so we obtain:s(O,b)=s(D)=s(C)+s(E)=s(O,a)+s(a,b)>s(O,a).⊓⊔We can now establish the underlying lattice structure of the Game of Cards.Theorem4.GC(O)is a lattice.If a and b are two elements of GC(O)diﬀerent from⊥,then the conﬁguration c such that s(O,c)=((max(s i(O,a),s i(O,b)))i∈[1,p] is reachable from O;if c is not dual,then inf gc(a,b)=c,otherwise inf gc(a,b)=⊥.Proof.Let us assume that s(O,a)and s(O,b)are incomparable(otherwise a and b are comparable and the result is obvious).We areﬁrst going to prove that c is reachable from a.For that,we just have toﬁnd a conﬁguration a′such that a→a′and s(O,a′)≤s(O,c).LetO→d1→d2→...→d v→bbe a sequence of transitions from O to b and let i be theﬁrst index such that s(O,d i)≤s(O,a)and s(O,d i+1)≤s(O,a).Let us call j the position on which the transition is applied on d i.Clearly s j(O,d i)≤s j(O,a)and s j(O,d i+1)>s j(O,a).Since a and d i verify the condition of Lemma1,we can apply thetransition on position j of a to obtain a new conﬁguration a′.We have∀j′=j,s j′(O,a′)=s j′(O,a)≤s j′(O,c)and s j(O,a′)=s j(O,d l+1)≤s j(O,b)≤s j(O,c),which proves that c is reachable from a.The proof is similar for b.This implies that c is reachable from O,and by deﬁnition c is the greatest conﬁguration smaller than both a and b.If c is not dual,it is the greatest lower bound of a et b,and if c is dual,⊥is then this lower bound.Since GC(O)also has a greatest element,it is a lattice,which ends the proof.⊓⊔4Convergence timeWe are now going to focus on the time needed either to arrive to the unique stable conﬁguration or to twice through to the same conﬁguration.For that, we are going to use Proposition1which states that all the sequences between two non dual conﬁgurations have the same length.So we are going to build a particular path between a given initial conﬁguration and either theﬁxed point if q=0or a particular dual conﬁguration if q>0.Let usﬁrst study the case q=0where all the sequence converge to a unique ﬁxed point(k,...,k)denoted by P.Theorem5.If q=0,then from any initial state O,there always exists a player that never can give a card to its neighbor.Proof.Assume that O=P.Let i be such that d i(O,P)=min1≤j≤p(O,P). We areﬁrst going to show step by step that there exists a path from O to P in which player i never plays.Let m be such that O m is maximal among the(O j)1≤j≤p and such that O m+1<O m.Since O=P,such m exists and O m>k.We have d m(O,P)=d m−1(O,P)+O m−k≥d m−1(O,P)+1,so m=i and d m(O,P)≥d i(O,P)+1.Let a be the conﬁguration obtained from O by applying the rule on position m.We are going to show that a is such that d i(a,P)=min1≤j≤p d j(a,P).If m=p,then d m(a,P)=d m(O,P)=0and for all j=p,d j(a,P)= d j(O,P)+1.If m=p,then d m(a,P)=d m(O,P)−1and for all j=p,d j(a,P)= d j(O,P).In the two cases d m(O,P)≥d i(O,P)+1and so d m(a,P)≥d i(O,P), which d i(a,P)=min1≤j≤p d j(a,P).By iterating the process,we arrive to theﬁxed point P by a path with no transition in position i.Therefore s i(O,P)=0and then for all conﬁguration a between O and P,s i(O,a)=0(for a given i,s i can only increase when following a path).⊓⊔We obtain now the following corollary which directly comes from the previous proof:Corollary3.If q=0and if the initial conﬁguration is O,the game ends after t steps,witht=p∗(−min1≤i≤p d i(O,P))+i=pi=1d i(O,P).Let us ﬁnish by considering the case q >0.Here,it is more diﬃcult to give an exact formula of the time necessary to arrive on a dual conﬁguration,since all the path leading to such a conﬁguration may not have the same length.However,it is possible to give an upper bound to this time.Let us consider a particular dual conﬁguration P =k +1,...,k +1 q ,k,...,k p −q.We obtain the following result (the proof is identical to the proof of Theorem 5).Theorem 6.Assume q >0and let O be an arbitrary conﬁguration.Let i be the ﬁrst index such that d i (O,P )is a minimal component of d (O,P ).Then all the paths from O to P without circuits have the same shot vector where s i (O,P )=0,that is in which player i never plays.This shot vector is given by the following formula:s (O,P )=−(min 1≤i ≤p d i (O,P )∗(1,...,1))+d (O,P ).Moreover,the time to reach P from O is equal tot =p ∗(−min 1≤i ≤p d i (O,P ))+i =pi =1d i (O,P ).Corollary 4.If q >0and if the game starts from a given conﬁguration O ,then the conﬁguration obtained afterp ∗(−min 1≤i ≤p d i (O,P ))+i =p i =1d i (O,P )+q (p −q )+1steps has been obtained earlier.Proof.Let us consider the dominance ordering on dual conﬁgurations,in which a conﬁguration a is greater or equal to a conﬁguration b if and only if ∀j ∈[1,n ], i =j i =1a i ≥ i =j i =1b i .The greatest element of this order is clearly P and the maximal length of a chain in this order is q (p −q ).Let a and b be two dual conﬁgurations such that a covers b in this order.It is clearly possible to go from a to b by a transition,so the covering relations are a subset of the set of transitions between dual conﬁgurations.Since the dual conﬁgurations are the unique non trivial strongly connected component of G ,it is clear that the maximal length of a path between two dual conﬁgurations in G is q (p −q )+1,which proves the corollary.⊓⊔References1.Garrett Birkhoﬀ.Lattice Theory .American Mathematical Society,1967.2.T.Brylawski.The lattice of interger partitions.Discrete Mathematics ,(6):201–219,1973.3. 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