Heavy Fermion Superconductivity
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a r X i v :c o n d -m a t /0311118v 1 [c o n d -m a t .s u p r -c o n ] 5 N o v 2003Heavy fermion d wave superconductivity:a X-boson approachLizardo H.C.M.Nunes a M.S.Figueira a E.V.L.de Melllo aa Institutode F´ısica da Universidade Federal FluminenseAv.Litorˆa nea s/n,24210-340Niter´o i,Rio de Janeiro,Brasil.C.P.100.0931.IntroductionRecently,we have used the X-boson method [1]to study the superconducting phase diagram of the PAM in the U →∞,where U is the on-site Coulomb repulsion,in the presence of an attractive interaction between the localized f -electrons,H = k ,σǫk ,σc †k ,σc k ,σ+ k ,σE f X k ,σσ+k ,σV (X †k ,0σc k ,σ+c †k ,σX k ,0σ)+k ,k ′W k ,k ′b †k b k ′,(1)where we only considered site independent localenergies E f,j,σ=E f,σand a constant hybridiza-tion V =V j,σ,k .The operator b †k =X †k ,0σX †−k ,0⋆We acknowledge the financial support from the Rio de Janeiro State Research Foundation (F APERJ)and Na-tional Research Council (CNPq).higher values of T c ,what suggests that the Kondo behavior of the system favors superconductivity in the X-boson approach.Nevertheless,as charge carriers are added to the system the superconduc-tivity was found both for configurations where the system presented intermediate valence (IV)and heavy fermion (HF)behavior and we have recov-ered the three characteristic regimes of the PAM:Kondo,IV and magnetic.This behavior which can-not be found for the same model by the slave-boson treatment [3],since it breaks down when the f -occupation number N f →1.Despite the fact that the model was primar-ily designed to the study of the heavy fermion compounds,theoretical descriptions of the su-perconducting phases based on two-band models have application for a large variety of systems and the model could fit into the description of the high temperature superconductors compounds (HTSC)[4]or into the new two-band supercon-ductor MgB 2.Meanwhile,unconventional super-conductors may exhibit different pair symmetries.For instance,HTSC cuprates appears to present d x 2−y 2superconductivity for some compounds and s -wave for others while for the HF compounds the crystallographic anisotropies and strong spin-orbit coupling make the task of determining the pairPreprint submitted to Elsevier Science2February 2008symmetry very difficult,athough d-wave model is still one of the leading candidates to describe superconductivity in UPt3[5].In this paper we employ the X-boson approach to compare the results for the superconducting phase diagram of an isotropic s-wave superconduc-tor with the results provided by a d x2−y2super-conductor.2.The MethodThe X-boson approach consists of introducing the variational parameter R=1− σ<Xσσ>, which modifies the Green’s functions(GF)so that it minimizes an adequate thermodynamic poten-tial while being forced to satisfy the“complete-ness”relation n0+nσ+n[6] A.C.Hewson,J.Phys.C:Solid State Phys.10,4973(1977).3。
1Superconductivity at 1 K in Cd 2Re 2O 7M. Hanawa, Y. Muraoka, T. Tayama, T. Sakakibara, J. Yamaura, and Z. HiroiInstitute for Solid State Physics, University of Tokyo, Kashiwanoha, Kashiwa, Chiba277-8581, Japan (Received 4 May 2001)We report the first pyrochlore oxide superconductor Cd 2Re 2O 7. Resistivity, magnetic susceptibility, and specific heat measurements on single crystals evidence a bulk superconductivity at 1 K. Another phase transition found at 200 K suggests that a peculiar electronic structure lies behind the superconductivity.Interplay between localized and itinerant electrons has been one of the most exciting subjects in solid state physics. In particular, various transition-metal (TM)oxides which exist in the vicinity of a metal-insulator (MI)transition have been studied extensively, and many intriguing phenomena such as high-temperature superconductivity in cuprates and charge/orbital ordering in manganites or others have been found. They would further attract many physicists in the future because of huge family of compounds already known and still hidden under the iceberg.Superconducting TM oxides known before the discovery of cupric oxide superconductors are rather limited. Only a few Ti, Nb, and W oxides which crystallize in the NaCl, spinel, or perovskite structure have been reported for 3d , 4d , and 5d series, respectively. The highest superconducting transition temperature T c among those non-cuprates was attained in a spinel compound LiTi 2O 4(T c = 13.7 K) [1]. Recently, Sr 2RuO 4 is studied well as representing an unusual p -wave superconductivity at 1 K [2].There is another class of compounds forming a large family of TM oxides which crystallize in the pyrochlore structure with a chemical formula A 2B 2O 7 where the B represents TMs [3]. However, no superconductivity has been observed there so far in spite of many metallic compounds present in the family. Looking in the general trend of electronic properties for pyrochlore compounds,most 3d and 4d TM pyrochlores are insulators owing to large electron correlations as well as relatively small electron transfers along the bent B-O-B bonds (110º ~140º). Molybdenum pyrochlores exist near the metal-insulator boundary, where ferromagnetic metals appear as the ionic radius of the counter cations is increased [4].On one hand, metallic pyrochlores can be found, when additional electrons are supplied from A cations like typically in Mn and Ru pyrochlores such as Tl 2Mn 2O 7 and Bi 2Ru 2O 7 [5, 6]. In contrast, 5d TM pyrochlores are mostly metallic because of relatively spreading 5d orbitals [3].A rare exception reported previously is found in Os 5+pyrochlores like Cd 2Os 2O 7 [7, 8] and Ca 2Os 2O 7 [9], wherea MI transition occurs with temperature. An Os 5+ion hasa 5d 3electron configuration and thus the t 2g orbital is half-filled, suggesting a possible Mott-Hubbard type MItransition. Note that most of Os 4+pyrochlores are metals.These examples illustrate that the effect of electron correlations is still important even for the 5d electron systems in the pyrochlore structure. We have searched for novel phenomena in pyrochlore compounds near Cd 2Os 2O 7 and reached to Cd 2Re 2O 7 which becomes the first superconductor in the pyrochlore family.Another important feature on the pyrochlore oxide is magnetic frustration on the three-dimensional tetrahedral network sharing vertices[10]. When d electrons are localized on the B site ions with interacting antiferromagnetically, classical Néel order is suppressed by the frustration and a quantum spin liquid state can be stabilized instead [11]. Even for ferromagnetic Ising spin systems on the A sublattice like Dy 2Ti 2O 7, it is shown that the effect of frustration leads to a highly degenerate ground state called ‘spin ice’ [12]. More interestingly, an exotic ground state is to be realized when d electrons keep an itinerant character near a MI transition. It is clearly illustrated with a mixed-valent compound LiV 2O 4 which has the spinel structure comprising a similar frustrated lattice and exhibits an unusual “heavy-Fermion” behavior [13, 14]. Superconductivity seen in another spinel compound LiTi 2O 4 may be interesting in this context.However, it has not been studied in detail because of difficulty in preparing single crystals.There are a few studies on Cd 2Re 2O 7. Donohue et al .prepared a single crystal, determined the crystal structure,and reported the resistivity which was metallic above 4 K [15]. Blacklock and White measured the specific heat above 1.8 K using a polycrystalline sample and found theSommerfeld coefficient γ to be 13.3 mJ/K 2mol Re [16].Since the Cd 2+ and Re 5+ have 4d 10 and 4f 145d 2 outer electronconfigurations, respectively, only the Re 5+is expected to underlie the electronic and magnetic properties of Cd 2Re 2O 7. It is known that Re 5+ is not present as a stable state in any binary or ternary oxide system [17].Subramanian et al . suggested that more stable Re 6+and Re 4+ are the entities instead of Re 5+ in Cd 2Re 2O 7 [3]. This suggests an inherent charge fluctuation present in this2compound. We prepared single crystals of Cd 2Re 2O 7 and measured resistivity, magnetic susceptibility, and specific heat down to ~ 0.4 K. Surprisingly observed at 1-2 K are a sharp drop in resistivity, a large diamagnetic signal in magnetization, and a well-defined λ-type anomaly in specific heat. These experimental facts give a strong evidence for the occurrence of superconductivity.Moreover, the resistivity and susceptibility show peculiar behaviors at high temperatures, implying interesting normal-state properties lying behind the superconductivity.Single crystals of Cd 2Re 2O 7 were prepared by assuming the chemical reaction, 2CdO + 5/3ReO 3 + 1/3Re →Cd 2Re 2O 7. Stoichiometric amounts of these starting powders were mixed in an agate mortar and pressed into a pellet. The reaction was carried out in an evaluated quartz ampoule at 800-900ºC for 70 h. The product appeared as purple octahedral crystals of a few mm on an edge which adhered to the walls of the ampoule. The crystals have grown probably by a vapor transport mechanism, as discussed in a previous study [15], because both CdO (Cd)and ReO 3 (Re 2O 7) are volatile. The chemical composition of the crystals examined by an electron-probe microanalyser (EPMA) was Cd/Re = 1.00 ± 0.02. The oxygen content was not determined in this study. The previous study using crystals prepared in similar conditions suggested that the oxygen nonstoichiometry was negligible [15]. A powder X-ray diffraction (XRD) pattern was taken at room temperature and indexed on the basis of a face-centered cubic unit cell with a = 1.0226(2) nm, which is slightly larger than the previously reported value of 1.0219nm.Resistivity and specific heat measurements were carried out on many crystals in a Quantum Design PPMS system equipped with a 3He refrigerator down to 0.4 K.The former was measured by the standard four probe method using crystals of typical dimensions 2 mm × 0.5mm × 0.1 mm. The current density for the measurementswas about 104 A/m 2. Specific heat was measured by the heat-relaxation method using crystals of 10-20 mg in weight. To perform dc magnetization measurements at very low temperature below 2 K, we used a Faraday-force capacitive magnetometer. A field gradient of 500 Oe/cm was applied to a crystal in addition to homogeneous external field. The detail of this technique was reported previously [18]. Magnetic susceptibility above 1.7 K was measured in a Quantum Design MPMS system.When the crystals were cooled below 2 K, we found a very sharp drop in resistivity probably due to superconductivity, as typically shown for two crystals in Fig. 1(a). The resistivity below the critical temperature T c was nearly zero within our experimental resolution of ~10 nV for the voltage detection. The onset temperature is 1.45 (2.15) K and a zero-resistivity is attained below 1.30(1.85) K for crystal A (B). We have carried out measurements on more than 10 crystals and found similar sharp drops for all of them. However, the T c values were scattered between 1 K and 2 K. When the magnetic fieldwas applied, the transition curves shifted to lower temperatures systematically.The temperature dependence of the resistivity at high temperatures are quite unusual as shown in Fig. 1(b). It is almost temperature independent around room temperature,while, with cooling down, it suddenly starts to decrease at about 200 K. Another small anomaly is seen around 120K, where the curvature is slightly changed. The decrease in the resistivity tends to saturate at low temperature, where the temperature dependence is approximately proportionalto T 3. The resistivity at 300 K and just above the T c are 640 (360) µΩcm and 21 (11) µΩcm for crystal A (B),respectively. The ratio is about 30. These resistivity values varied slightly from crystal to crystal, but the ratio was always nearly the same.Associated with the observation of the zero-resistive transition, a large diamagnetic signal due to the Meissner effect was observed below 1 K as shown in Fig. 2. The hysteresis loop measured at 0.45 K is characteristic of a type-II superconductor with H c1 less than 0.002 T and H c2of about 0.17 T, respectively. The magnetization of the peak top at 0.002 T is -0.19 emu/g, which corresponds to M /H = -0.0095 emu/g. This value is close to a value of -0.009 emu/g expected for perfect exclusion of vortices inFIG. 1.Temperature dependence of resistivity for two Cd 2Re 2O 7 single crystals A and B. The measurements were carried out on cooling for crystal A and on heating for crystal B.R e s i s t i v i t y (µΩc m )(a)3Cd 2Re 2O 7. Therefore, the superconducting volume fraction in our crystal is large enough to conclude a bulk property. The inset to Fig. 2 shows the magnetization measured with increasing and decreasing temperature. The applied field may be close to H c1, but not exactly known.The T c is determined as 0.98 K, which is considerably lower than decided in the resistivity measurements.The superconducting transition was also detected by specific heat C measurements. As shown in Fig. 3, a distinct λ-type anomaly is seen at zero field below 1 K,and it shifted to lower temperatures with increasing magnetic fields applied. The observed large jump at the transition evidences the bulk nature of superconductivity in the present compound. By fitting the data between 1 Kand 10 K to the form C = γT + αT 3 + βT 5, we obtained γ =15.1 mJ/K 2 mol Re, α = 0.111 mJ/K 4 mol Re, and β =1.35 × 10-6 mJ/K 6mol Re. This γ value is slightly larger than that reported previously [16]. It is rather small compared with the value of 37.5 mJ/K 2 mol reported for Sr 2RuO 4 [2]. The Debye temperature deduced from α is 458 K. The electronic specific heat C e was obtained by subtracting the lattice contribution estimated above, and C e /T is plotted in Fig.3. The T c determined from the midpoint of the jump in C e /T is 0.97 K, and the transition width between 25% and 75% height of the jump is 30mK, which is much smaller than that reported for Sr 2RuO 4[2]. The jump in C e at T c is 16.9 mJ/K mol Re, and thus ∆C e /γT c is 1.15, which is considerably smaller than that expected for superconductivity with an isotropic BCS gap,1.43. Specific heat data measured on other several crystals are essentially the same, giving nearly the same T c , ∆T c ,and jump at T c . Since the data is limited above 0.4 K, it is difficult to discuss how the specific heat reaches zero as T → 0. From the entropy balance, however, assuming thatFIG. 2.Magnetization versus magnetic field curve measured at 0.45 K showing a hysteresis loop characteristic of a type-II superconductor. The temperature dependence shown in the inset exhibits T c = 0.98 K.the normal-state specific heat is simply γT , it is considered that the specific heat must decrease with temperature rather quickly as in an exponential form than in a power law as observed in Sr 2RuO 4 [2]. This suggests that the superconducting ground state of Cd 2Re 2O 7 is possibly nodeless unlike Sr 2RuO 4. Nevertheless, the gap can be anisotropic, because there is a significant difference between the observed and calculated specific heat data assuming an isotropic gap, as compared in Fig. 3. From the field dependence of T c , the upper critical field H c2was roughly estimated to be 0.21 T, which corresponds to a coherence length of 40 nm.The T c value is consistent between the magnetization and specific heat measurements, while the resistivity measurements gave much higher values of 1-2 K. This might be because the surface of crystals has been modified in some way to raise T c . The former measurements probe the bulk superconductivity inside crystals, while the latter can be determined by the surface. We checked the surface of crystals by EPMA, but did observe neither deviation of the metal ratio nor segregation of other phases. One possible explanation would be a small change in oxygen content or metal ratio occurring at the crystal surface.Annealing experiments in various oxygen atmosphere are now in progress.Magnetic susceptibility χ at high temperature is shown in Fig. 4, where a kink is seen at 200 K which coincides with the anomaly observed in the resistivity. The χ seems to show a broad, rounded maximum around room temperature and decreases rapidly below 200 K. The χvalue at 5 K is reduced by 40 % from the maximum value at high temperature. There was little anisotropy in χbetween two measurements where a magnetic field wase Cd 2Re 2O 7 single crystal showing a λ-type anomaly associated with a superconducting transition at 0.97 K without magnetic fields. The solid line shows calculated C e /T assuming a superconducting transition with an isotropic BCS gap.4The Wilson ratio calculated from the χ value at 5 K and the γ is 0.72, much smaller than unity expected for free electron gas. Below 5 K, there is a strange downturn in the χ curve as shown in the inset to Fig. 4. No thermal hysteresis was detected between heating and cooling for this crystal. The resistivity did not show any corresponding anomalies. The magnitude of this downturn was sample dependent.Concerning the anomaly observed at 200 K,preliminary XRD, specific heat, and NMR experiments all suggested that there is a second-order phase transition without magnetic order. In particular, the XRD measurements indicated that there occurs a small structural change from cubic F d3m to another cubic F 43m , implying a slight deformation for the oxygen octahedra as well as the tetrahedral network of Re ions. The detail will be reported elsewhere. Anyway, there is a distinct phase transition at 200 K in Cd 2Re 2O 7 which dramatically affects resistivity, magnetic susceptibility, and also crystal structure, and thus must induce a significant change in the electronic structure . It would be crucial for understanding the superconductivity in Cd 2Re 2O 7 to elucidate the essential feature of this phase transition at high temperature.A Re 5+ion in Cd 2Re 2O 7 has two 5d electrons which should occupy the t 2g orbital. In the pyrochlore structure of space group F d3m a certain deformation of a BO 6octahedron is allowed. In most compounds an octahedron is compressed along the <111> direction, while it is slightly elongated in Cd 2Re 2O 7. Then, it is considered that in Cd 2Re 2O 7 the threefold degenerate t 2g level splits into a 1g and e g levels, the latter having a lower energy. Therefore,a kind of multi-band character arising from these orbitals is expected near the Fermi level in the metallic state, asFIG. 4.Temperature dependence of magnetic susceptibility in a wide temperature range. The measurement was performed on heating in a magnetic field of 1 T applied nearly parallel to the <111> axis of the crystal. The inset shows an enlargement of the data at low temperature.seen in Sr 2RuO 4. Preliminary band structure calculations by Harima suggested an interesting flat-band character just below the Fermi level [19]. The structural change at 200K must influence this basic electronic structure.In conclusion we found superconductivity in the pyrochlore compound Cd 2Re 2O 7. Although the nature of superconductivity is not clear at present, we expect a novel physics involved in this compound on the basis of electron correlations near the MI transition as well as frustration on the pyrochlore structure. In addition, the phase transition at high temperature is intriguing, which might be related to charge, spin, and orbital degrees of freedom.It is also important to investigate the other pyrochlore compounds which show metallic conductivity, in order to test whether a superconducting ground state is general in the pyrochlore oxides or Cd 2Re 2O 7 presents a special case.We would like to thank T. Yamauchi, M. Takigawa,M. Imada, K. Ueda and Y. Ueda for helpful discussions during the course of this study, and Y. Ueda for the use of the SQUID magnetometer and F. Sakai for chemical analysis. This research was supported by a Grant-in-Aid for Scientific Research on Priority Areas (A) given by The Ministry of Education, Culture, Sports, Science and Technology, Japan.[1] D. C. Johnston, J. Low Temp. Phys. 25, 145 (1976).[2] Y. Maeno, Physica B 281-282, 865 (2000).[3] M. A. Subramanian, G. Aravamudan, and G. V. S. Rao, Prog.Solid St. Chem. 15, 55 (1983).[4] J. E. Greedan, M. Sato, N. Ali, and W. R. Datars, J. Solid State Chem. 68, 300 (1987).[5] R. J. Bouchard and J. L. Gillson, Mat. Res. Bull. 6, 669 (1971).[6] Y. Shimakawa, Y. Kubo, N. Hamada, J. D. Jorgensen, Z.Hu, S. Short, M. Nohara, and H. Takagi, Phys. Rev. B 59, 1249(1999).[7] A. W. Sleight, J. L. Gilson, J. F. Weiher, and W. Bindloss,Solid State Commun. 14, 357 (1974).[8] D. Mandrus, J. R. Thompson, R. Gaal, L. Forro, J. C. Bryan,B. C. Chakoumakos, L. M. Woods, B. C. Sales, R. S. Fishman,and V. Keppens, Phys. Rev. B 63, 195104 (2001).[9] B. L. Chamberland, Mat. Res. Bull. 13, 1273 (1978).[10] A. P. Ramirez, Annu. Rev. Mater. Sci. 24, 453 (1994).[11] B. Canals and C. Lacroix, Phys. Rev. B 61, 1149 (2000).[12] A. P. Ramirez, A. Hayashi, R. J. Cava, R. Siddharthan, and B. S. Shastry, Nature 399, 333 (1999).[13] S. Kondo, D. C. Johnston, and J. D. Jorgensen, Phys. Rev.Lett. 78, 3729 (1997).[14] C. Urano, M. Nohara, S. Kondo, F. Sakai, H. Takagi, T.Shiraki, and T. Okubo, Phys. Rev. Lett. 85, 1052 (2000).[15] P. C. Donohue, J. M. Longo, R. D. Rosenstein, and L. Katz,Inrog. Chem. 4, 1152 (1965).[16] K. Blacklock and H. W. White, J. Chem. Phys. 71, 5287(1979).[17] C. N. R. Rao and G. V. S. Rao, Transition Metal Oxides:Crystal Chemistry, Phase Transitions, and Related Aspects (NBS, Wash., 1974).[18] T. Sakakibara, H. Mitamura, T. Tayama, and H. Amitsuka,Jpn. J. Appl. Phys. 33, 5067 (1994).[19] H. Harima, .private communication.。
当两个高能核发生碰撞时,相互作用区域会发射许多粒子。
由于全同粒子的交换对称性,发射出的全同粒子具有玻色-爱因斯坦关联,又称HBT关联。
利用全同粒子携带的信息可以测量碰撞区域时空信息和相干性。
正反粒子的背对背关联(Back-to-Back Correlations),简称BB 关联,与一对动量相反的粒子有关,它的出现是由于高密度发射源内的粒子质量位移。
本文用量子力学的波函数法推导了两粒子关联函数,并利用量子场论的知识研究了含质量位移效应的背对背关联和玻色-爱因斯坦关联函数。
由于实际的粒子发射源并非静态,而应该是随时间膨胀,考虑有限发射时间的影响,本文引入源的衰变随时间变化的分布,对含质量位移的HBT关联函数进行了修正。
这正是本文的创新点。
1. 高能重离子碰撞物理学1.1 高能重离子碰撞物理学简介在高能重离子碰撞以又称为高能核-核碰撞,通过高能重离子碰撞来产生极端高温度、高密度的核物质,研究产物的性质以寻找、探测可能存在的新物质相。
美国布鲁克海文实国家验室的相对论重离子对撞机RHIC和欧洲核子中心的大型强子对撞机LHC 都在做当前能量最高的相对论重离子碰撞实验。
1.2 相对论重离子碰撞的演化过程两核以较高能量碰撞时中心区域能量密度很高,靶核和入射核被高度激发后都会发生碎裂而产生了大量新粒子。
对高能核-核碰撞过程从时间上划分为四个阶段:初始阶段、压缩阶段、膨胀阶段、实验观察阶段。
1.3 夸克-胶子等离子体(QGP)自然界存在QGP的地方可能有两个,一是大爆炸后10μs左右的温度极高的初期宇宙;另一个则是重子数密度极高的中子星内部。
夸克被囚禁在强子内故不存在单个自由夸克。
QCD理论预测极高温度或极高密度下可能打破夸克禁闭形成“夸克—胶子等离子体”。
当前物理学存在两个谜题:夸克禁闭和破却的对称性,都有望在QGP 中得到解答。
1.4 强度干涉学强度干涉学最早是利用光子的强度干涉来测量星体的角径。
HBT关联与同时测量两个时空点上光子强度有关,关联程度依赖于发射源的角径。
δ-Pu态密度的动力学平均场理论研究刘以良;肖培【摘要】利用局域密度近似结合动力学平均场理论研究了δ-Pu的态密度. 借助赫伯德模型下的多体哈密顿量, 并使用赫伯德 I方法进行杂质求解. 计算结果很好的呈现出了强关联电子体系的低赫伯德带以及高赫伯德带, 并可以呈现出费米能级处的准粒子Kondo共振峰.【期刊名称】《西南民族大学学报(自然科学版)》【年(卷),期】2010(036)006【总页数】4页(P1014-1017)【关键词】态密度;赫伯德模型;赫伯德;I近似【作者】刘以良;肖培【作者单位】西南民族大学电气信息工程学院,四川成都,610041;西南民族大学电气信息工程学院,四川成都,610041【正文语种】中文【中图分类】O56钚是一种剧毒的、人造的强放射性锕系元素, 在能源、军事、航天等领域有极为广泛的应用.其在固态下与元素周期表中其他元素相比, 显示出复杂而反常的属性:钚有6种晶体结构, 低温下(273-373 K)以α相的形式存在, α相晶胞中有16个原子, 随着温度的升高, 会经历一系列的相过渡, 最终形成结构相对简单的面心立方δ相(573 –723 K)和体心立方ε相(~773 – 923 K), 温度再高就会融化; 晶体中钚原子的体积与温度的关系反常, 在从α 向δ相的过渡中原子体积扩展明显, 大约增加了25%, 而在δ相内部呈现一热膨胀的负增长, 并且从α相向更高温度的ε相过渡中原子体积大约减少了5%; 反常的电磁行为, 与其他的重费米体系类似, 钚呈现反常的电阻行为, 但是它的6个相都是没有磁性的, 磁化系数非常的小, 并且相对而言与温度无关; 反常的光谱特征, 从钚的光子发射谱可以看到在费米能级处有一个较强且窄的类Kondo峰存在, 这个现象与钚具有较大的线性比热系数一致[1-2].这些奇特的性质主要是由Pu的5f电子引起的, 5f电子同时具有定域和离域的特征.这种情况下, 电子处于强关联状态, 尽管传统的密度泛函理论(DFT)基础上的第一性原理可以成功计算很多实际材料的电子结构, 而对于强关联电子体系,例如存在未满壳层的d电子或者f电子的体系, 电子运动受到的限制明显, 轨道较窄, 电子之间的库仑相互作用较大与其带宽数值相差不大, 甚至超过电子的动能.DFT/LDA得出的结果将不再可靠, 这是因为局域密度近似(LDA)建立在弱关联电子模型基础上, 对于强关联电子体系, 电子空间密度变化剧烈, 电子局域密度是一个常数的假设不再成立, 并且电子之间相互作用明显, 单电子不能再被看成在一个静态的平均场中运动, 因此DFT通常只能处理非定域的弱关联的电子体系.例如DFT会将绝缘体CoO和La2CuO错误的计算成为导体.鉴于强关联电子体系的重要性, 其处理仍是是当今凝聚态物理发展的一个重要方向, 找到一个就像DFT解决弱关联体系一样成功的解决强关联电子体系的第一性原理方法就显的尤为重要.典型的强关联电子体系第一性原理解决方法有LDA+U、LDA++以及Local GW等的方法[3-7], 然而这些方法相对比较粗糙.作为20世纪90年代发展起来的一种非微扰多体技术, 动力学平均场理论DMFT[8-9]及其团簇扩展CDMFT[10-11]可以同时考虑电子的能带特性和类原子特性, 为研究强关联电子体系提供了新的有效途径.尤其是将DFT/LDA与DMFT结合起来,用DFT/LDA 处理模型哈密顿量的若关联部分, 用DMFT处理体系的强关联部分的LDA+DMFT方法正逐渐被越来越多的科研人员认同并发展起来[12-13].LDA+DMFT方法已经被成功的用来解决一些过渡金属氧化物, 磁过渡金属以及Ce和Pu等稀土金属的光谱、传输以及热力学等的性质[14-17].本文首先简要介绍了DMFT方法的本质、物理思想、基本条件以及对应的哈密顿量的形式.然后在653K下,利用赫伯德模型和LDA+DMFT方法计算了δ-Pu的态密度(DOS), 其中使用了赫伯德 I近似解决杂质问题, 最后将计算结果与其他的理论以及实验结果进行了比较.DMFT的本质就是用一个单位量子杂质模型替代原来的晶格模型, 这个单位量子杂质模型镶嵌在一个自洽的有效媒介中, 并且自洽条件满足平移不变性和连贯性效应, 其根本的物理思想就是对于某个确定的晶格位,其动力学可以看成是位上的自由度与一个外部“浴室”的相互作用, 这个“浴室”是由给定晶格位周围其它位所有自由度产生的.具体而言, 就是除了某个特定晶格位外, 其他的晶格位上的库仑相互作用都用自能来代替, 而这个特定位上的电子之间有相互作用并可以在整个晶格中运动, 但是电子在其他的位上传播是通过自能产生的媒介而不是通过电子间的相互作用, DMFT引入的“单位问题”与安德森杂质问题是等价的, 而且安德森杂质问题需要通过自能与格林函数的关系(Dyson方程)自洽求解.杂质模型为量子多体问题的局域动力学提供了直观的图像, 鉴于杂质问题迄今已经历四十几年的发展并得到一系列可用的解法[5][8][12][17-20], 杂质模型问题成为 DMFT方法中非常重要的一环.空间的纬数越大, DMFT方法就越精确, 近似说来, 也就是晶格的配位数越高计算结果就越精确,达到无限纬极限时最精确[8-9][13], 所以DMFT方法迄今难以进行表面电子态的计算.对于具有强关联f电子的体系, DMFT计算是将哈密顿量进行如下的分解[17]:多带周期安德森模型的非相互作用部分描述库仑相互作用的局域贡献, 用一个关联项来补充上面的公式如下:为了避免双重计算, 需要将 LDA 哈密顿量中的库仑关联减去.LDA 哈密顿量中的库仑关联可以近似用库仑相互作用能的平均值给出.新的非相互作用哈密顿量如下:右边第一项可以用LDA方法计算, 右边第二项可以用DMFT方法计算.如上所示, 关联电子的哈密顿量虽然可以精确写出, 但是非常复杂无法精确求解, 甚至高于 10个晶格位时已经不能够得出数值解, 因此需要用到近似的方法, 这里DMFT就是一种强有力的考虑了电子关联的近似方法, 在DMFT中, 库仑关联的效应由局域近似的自能算符来表示, 自能Σ(iωn)和格林函数G(iωn)在虚 Matsubara频率iωn=iπ(2n+1)/ β 下得到.如果要计算光谱性质就需要实轴上的格林函数.为了进行LDA+DMFT计算, 我们用密度泛函理论的局域密度近似结合量子杂质求解模型完成一个完整的LDA+DMFT循环.首先DFT Code 用以产生δ-Pu在k空间哈密顿量矩阵, δ-Pu为面心立方结构, 在653K下的实验晶格参数为8.759 a.u.[21], 使用LDA作为交换关联势, 电子的波函数基矢使用线性响应muffin-tin 轨道(LMTO)和原子球近似(ASA), 鉴于δ-Pu处于顺磁性态, 实际自旋的数目为1, 并且在未考虑S-O耦合的情况下, 在6×6×6四面体网格分割的k空间, 哈密顿量在spdf的LMTO下写成16×16的矩阵.并得到满足7s6p6d5f上的总电子数为14的化学势为8.875 eV.其次, 使用了赫伯德-I近似来解决杂质问题, 赫伯德-I方法相对比较近似, 但是使用的计算时间比较短, 更加趋于反应Pu的原子特性, 因此其得到的结果可以为其他的杂质求解器提供参考, 取库仑相互作用的平均值U=F0=4.0 eV, 杂质能级为9.025 eV, 温度为653 K(0.0562 eV), 考虑自旋量子数, f 轨道的兼并度取14, 结合解析四面体方法求解格林函数, 并最终使用Pade近似将Matsubara频率下的格林函数和自能转化到实频下.得出δ-Pu的DOS,如图1所示.计算的结果与文献[16]吻合的很好, 并且呈现出强关联电子体系的一个典型的特征,即低赫伯德带以及高赫伯德带的存在, 将费米能级处态密度与实验光电谱进行比较, 如图 2所示.计算结果与实验测得的光电谱[22]有一定的吻合, 可以看到在费米能级处一个较强的准粒子Kondo共振峰的存在.然而由于赫伯德 I方法比较适宜计算顺磁Mott绝缘子的光电谱, 因此计算中并未得到费米能级处的连贯峰.由于DMFT方法的相对不成熟性, 以及其不是完整意义上的ab initio计算方法,需要半经验的参数, 使得各种计算结果以及与实验的吻合度有所差异[2][14-16][22].利用局域密度近似结合动力学平均场理论研究了653K下δ-Pu的态密度, 使用赫伯德模型下的多体哈密顿量,以及使用赫伯德 I近似进行杂质模型求解.计算结果很好的呈现出了强关联电子体系的低赫伯德带以及高赫伯德带, 并可以看到在费米能级处较强的准粒子 Kondo共振峰的存在.然而由于赫伯德 I方法不够精确, 并且更适宜计算顺磁Mott绝缘子的光电谱, 所以要想得到更加符合实验结果的光电谱就需要量子Monte Carlo(QMC)方法、精确对角化方法(ED)等相对精确的杂质解决器, 而且需要进一步考虑 f轨道的电子占有数、经过 DMFT循环后电荷密度改变对哈密顿量的影响、平均库仑相互作用、能级的兼并度、自旋轨道耦合、强关联指数等因素对计算结果的影响.【相关文献】[1] SAVRASOV S Y.Spectral density functionals for electronic structure calculations[J].Phys Rev B, 2004, 69:245101-245105.[2] ZHU J X, MCMAHAN A K, JONES M D, et al.Spectral properties of δ-plutonium: Sensitivity to 5f occupancy[J].Phys Rev B, 2007,76:145118-145123.[3] SAVRASOV S Y, KOTLIAR G.Ground State Theory of δ-Pu[J].Phys Rev L, 2000, 84: 3670-3675.[4] BOUCHET J, SIBERCHICOT B, JOLLET F, et al.Equilibrium properties of δ-Pu: LDA + U calculations (LDA ≡ local density approximation)[J].J Phys: Condens Matter, 2000, 12: 1723-1727.[5] ANISIMOVY V I, ARYASETIAWANZ F, LICHTENSTEIN A I.First-principles calculations ofthe electronic structure and spectra of strongly correlated systems: the LDA +Umethod[J].J Phys Condens Matter, 1997,9: 767-772.[6] LICHTENSTEINA I, KATSNELSON M I.Ab initio calculations of quasiparticle band structure in correlated systems: LDA++approach[J].Phys Rev B, 1998, 57: 6884-6889. 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a r X i v :c o n d -m a t /9503139v 1 27 M a r 1995Singularity of the density of states in the two-dimensional Hubbard model from finitesize scaling of Yang-Lee zerosE.Abraham 1,I.M.Barbour 2,P.H.Cullen 1,E.G.Klepfish 3,E.R.Pike 3and Sarben Sarkar 31Department of Physics,Heriot-Watt University,Edinburgh EH144AS,UK 2Department of Physics,University of Glasgow,Glasgow G128QQ,UK 3Department of Physics,King’s College London,London WC2R 2LS,UK(February 6,2008)A finite size scaling is applied to the Yang-Lee zeros of the grand canonical partition function for the 2-D Hubbard model in the complex chemical potential plane.The logarithmic scaling of the imaginary part of the zeros with the system size indicates a singular dependence of the carrier density on the chemical potential.Our analysis points to a second-order phase transition with critical exponent 12±1transition controlled by the chemical potential.As in order-disorder transitions,one would expect a symmetry breaking signalled by an order parameter.In this model,the particle-hole symmetry is broken by introducing an “external field”which causes the particle density to be-come non-zero.Furthermore,the possibility of the free energy having a singularity at some finite value of the chemical potential is not excluded:in fact it can be a transition indicated by a divergence of the correlation length.A singularity of the free energy at finite “exter-nal field”was found in finite-temperature lattice QCD by using theYang-Leeanalysisforthechiral phase tran-sition [14].A possible scenario for such a transition at finite chemical potential,is one in which the particle den-sity consists of two components derived from the regular and singular parts of the free energy.Since we are dealing with a grand canonical ensemble,the particle number can be calculated for a given chem-ical potential as opposed to constraining the chemical potential by a fixed particle number.Hence the chem-ical potential can be thought of as an external field for exploring the behaviour of the free energy.From the mi-croscopic point of view,the critical values of the chemical potential are associated with singularities of the density of states.Transitions related to the singularity of the density of states are known as Lifshitz transitions [15].In metals these transitions only take place at zero tem-perature,while at finite temperatures the singularities are rounded.However,for a small ratio of temperature to the deviation from the critical values of the chemical potential,the singularity can be traced even at finite tem-perature.Lifshitz transitions may result from topological changes of the Fermi surface,and may occur inside the Brillouin zone as well as on its boundaries [16].In the case of strongly correlated electron systems the shape of the Fermi surface is indeed affected,which in turn may lead to an extension of the Lifshitz-type singularities into the finite-temperature regime.In relating the macroscopic quantity of the carrier den-sity to the density of quasiparticle states,we assumed the validity of a single particle excitation picture.Whether strong correlations completely distort this description is beyond the scope of the current study.However,the iden-tification of the criticality using the Yang-Lee analysis,remains valid even if collective excitations prevail.The paper is organised as follows.In Section 2we out-line the essentials of the computational technique used to simulate the grand canonical partition function and present its expansion as a polynomial in the fugacity vari-able.In Section 3we present the Yang-Lee zeros of the partition function calculated on 62–102lattices and high-light their qualitative differences from the 42lattice.In Section 4we analyse the finite size scaling of the Yang-Lee zeros and compare it to the real-space renormaliza-tion group prediction for a second-order phase transition.Finally,in Section 5we present a summary of our resultsand an outlook for future work.II.SIMULATION ALGORITHM AND FUGACITY EXPANSION OF THE GRAND CANONICALPARTITION FUNCTIONThe model we are studying in this work is a two-dimensional single-band Hubbard HamiltonianˆH=−t <i,j>,σc †i,σc j,σ+U i n i +−12 −µi(n i ++n i −)(1)where the i,j denote the nearest neighbour spatial lat-tice sites,σis the spin degree of freedom and n iσis theelectron number operator c †iσc iσ.The constants t and U correspond to the hopping parameter and the on-site Coulomb repulsion respectively.The chemical potential µis introduced such that µ=0corresponds to half-filling,i.e.the actual chemical potential is shifted from µto µ−U412.(5)This transformation enables one to integrate out the fermionic degrees of freedom and the resulting partition function is written as an ensemble average of a product of two determinantsZ ={s i,l =±1}˜z = {s i,l =±1}det(M +)det(M −)(6)such thatM ±=I +P ± =I +n τ l =1B ±l(7)where the matrices B ±l are defined asB ±l =e −(±dtV )e −dtK e dtµ(8)with V ij =δij s i,l and K ij =1if i,j are nearestneigh-boursand Kij=0otherwise.The matrices in (7)and (8)are of size (n x n y )×(n x n y ),corresponding to the spatial size of the lattice.The expectation value of a physical observable at chemical potential µ,<O >µ,is given by<O >µ=O ˜z (µ){s i,l =±1}˜z (µ,{s i,l })(9)where the sum over the configurations of Ising fields isdenoted by an integral.Since ˜z (µ)is not positive definite for Re(µ)=0we weight the ensemble of configurations by the absolute value of ˜z (µ)at some µ=µ0.Thus<O >µ= O ˜z (µ)˜z (µ)|˜z (µ0)|µ0|˜z (µ0)|µ0(10)The partition function Z (µ)is given byZ (µ)∝˜z (µ)N c˜z (µ0)|˜z (µ0)|×e µβ+e −µβ−e µ0β−e −µ0βn (16)When the average sign is near unity,it is safe to as-sume that the lattice configurations reflect accurately thequantum degrees of freedom.Following Blankenbecler et al.[1]the diagonal matrix elements of the equal-time Green’s operator G ±=(I +P ±)−1accurately describe the fermion density on a given configuration.In this regime the adiabatic approximation,which is the basis of the finite-temperature algorithm,is valid.The situa-tion differs strongly when the average sign becomes small.We are in this case sampling positive and negative ˜z (µ0)configurations with almost equal probability since the ac-ceptance criterion depends only on the absolute value of ˜z (µ0).In the simulations of the HSfields the situation is dif-ferent from the case of fermions interacting with dynam-ical bosonfields presented in Ref.[1].The auxilary HS fields do not have a kinetic energy term in the bosonic action which would suppress their rapidfluctuations and hence recover the adiabaticity.From the previous sim-ulations on a42lattice[3]we know that avoiding the sign problem,by updating at half-filling,results in high uncontrolledfluctuations of the expansion coefficients for the statistical weight,thus severely limiting the range of validity of the expansion.It is therefore important to obtain the partition function for the widest range ofµ0 and observe the persistence of the hierarchy of the ex-pansion coefficients of Z.An error analysis is required to establish the Gaussian distribution of the simulated observables.We present in the following section results of the bootstrap analysis[17]performed on our data for several values ofµ0.III.TEMPERATURE AND LATTICE-SIZEDEPENDENCE OF THE YANG-LEE ZEROS The simulations were performed in the intermediate on-site repulsion regime U=4t forβ=5,6,7.5on lat-tices42,62,82and forβ=5,6on a102lattice.The ex-pansion coefficients given by eqn.(14)are obtained with relatively small errors and exhibit clear Gaussian distri-bution over the ensemble.This behaviour was recorded for a wide range ofµ0which makes our simulations reli-able in spite of the sign problem.In Fig.1(a-c)we present typical distributions of thefirst coefficients correspond-ing to n=1−7in eqn.(14)(normalized with respect to the zeroth power coefficient)forβ=5−7.5for differ-entµ0.The coefficients are obtained using the bootstrap method on over10000configurations forβ=5increasing to over30000forβ=7.5.In spite of different values of the average sign in these simulations,the coefficients of the expansion(16)indicate good correspondence between coefficients obtained with different values of the update chemical potentialµ0:the normalized coefficients taken from differentµ0values and equal power of the expansion variable correspond within the statistical error estimated using the bootstrap analysis.(To compare these coeffi-cients we had to shift the expansion by2coshµ0β.)We also performed a bootstrap analysis of the zeros in theµplane which shows clear Gaussian distribution of their real and imaginary parts(see Fig.2).In addition, we observe overlapping results(i.e.same zeros)obtained with different values ofµ0.The distribution of Yang-Lee zeros in the complexµ-plane is presented in Fig.3(a-c)for the zeros nearest to the real axis.We observe a gradual decrease of the imaginary part as the lattice size increases.The quantitative analysis of this behaviour is discussed in the next section.The critical domain can be identified by the behaviour of the density of Yang-Lee zeros’in the positive half-plane of the fugacity.We expect tofind that this density is tem-perature and volume dependent as the system approaches the phase transition.If the temperature is much higher than the critical temperature,the zeros stay far from the positive real axis as it happens in the high-temperature limit of the one-dimensional Ising model(T c=0)in which,forβ=0,the points of singularity of the free energy lie at fugacity value−1.As the temperature de-creases we expect the zeros to migrate to the positive half-plane with their density,in this region,increasing with the system’s volume.Figures4(a-c)show the number N(θ)of zeros in the sector(0,θ)as a function of the angleθ.The zeros shown in thesefigures are those presented in Fig.3(a-c)in the chemical potential plane with other zeros lying further from the positive real half-axis added in.We included only the zeros having absolute value less than one which we are able to do because if y i is a zero in the fugacity plane,so is1/y i.The errors are shown where they were estimated using the bootstrap analysis(see Fig.2).Forβ=5,even for the largest simulated lattice102, all the zeros are in the negative half-plane.We notice a gradual movement of the pattern of the zeros towards the smallerθvalues with an increasing density of the zeros nearθ=πIV.FINITE SIZE SCALING AND THESINGULARITY OF THE DENSITY OF STATESAs a starting point for thefinite size analysis of theYang-Lee singularities we recall the scaling hypothesis forthe partition function singularities in the critical domain[11].Following this hypothesis,for a change of scale ofthe linear dimension LLL→−1),˜µ=(1−µT cδ(23)Following the real-space renormalization group treatmentof Ref.[11]and assuming that the change of scaleλisa continuous parameter,the exponentαθis related tothe critical exponentνof the correlation length asαθ=1ξ(θλ)=ξ(θ)αθwe obtain ξ∼|θ|−1|θ|ναµ)(26)where θλhas been scaled to ±1and ˜µλexpressed in terms of ˜µand θ.Differentiating this equation with respect to ˜µyields:<n >sing =(−θ)ν(d −αµ)∂F sing (X,Y )ν(d −αµ)singinto the ar-gument Y =˜µαµ(28)which defines the critical exponent 1αµin terms of the scaling exponent αµof the Yang-Lee zeros.Fig.5presents the scaling of the imaginary part of the µzeros for different values of the temperature.The linear regression slope of the logarithm of the imaginary part of the zeros plotted against the logarithm of the inverse lin-ear dimension of the simulation volume,increases when the temperature decreases from β=5to β=6.The re-sults of β=7.5correspond to αµ=1.3within the errors of the zeros as the simulation volume increases from 62to 82.As it is seen from Fig.3,we can trace zeros with similar real part (Re (µ1)≈0.7which is also consistentwith the critical value of the chemical potential given in Ref.[22])as the lattice size increases,which allows us to examine only the scaling of the imaginary part.Table 1presents the values of αµand 1αµδ0.5±0.0560.5±0.21.3±0.3∂µ,as a function ofthe chemical potential on an 82lattice.The location of the peaks of the susceptibility,rounded by the finite size effects,is in good agreement with the distribution of the real part of the Yang-Lee zeros in the complex µ-plane (see Fig.3)which is particularly evident in the β=7.5simulations (Fig.4(c)).The contribution of each zero to the susceptibility can be singled out by expressing the free energy as:F =2n x n yi =1(y −y i )(29)where y is the fugacity variable and y i is the correspond-ing zero of the partition function.The dotted lines on these plots correspond to the contribution of the nearby zeros while the full polynomial contribution is given by the solid lines.We see that the developing singularities are indeed governed by the zeros closest to the real axis.The sharpening of the singularity as the temperature de-creases is also in accordance with the dependence of the distribution of the zeros on the temperature.The singularities of the free energy and its derivative with respect to the chemical potential,can be related to the quasiparticle density of states.To do this we assume that single particle excitations accurately represent the spectrum of the system.The relationship between the average particle density and the density of states ρ(ω)is given by<n >=∞dω1dµ=ρsing (µ)∝1δ−1(32)and hence the rate of divergence of the density of states.As in the case of Lifshitz transitions the singularity of the particle number is rounded at finite temperature.However,for sufficiently low temperatures,the singular-ity of the density of states remains manifest in the free energy,the average particle density,and particle suscep-tibility [15].The regular part of the density of states does not contribute to the criticality,so we can concentrate on the singular part only.Consider a behaviour of the typedensity of states diverging as the−1ρsing(ω)∝(ω−µc)1δ.(33)with the valueδfor the particle number governed by thedivergence of the density of states(at low temperatures)in spite of thefinite-temperature rounding of the singu-larity itself.This rounding of the singularity is indeedreflected in the difference between the values ofαµatβ=5andβ=6.V.DISCUSSION AND OUTLOOKWe note that in ourfinite size scaling analysis we donot include logarithmic corrections.In particular,thesecorrections may prove significant when taking into ac-count the fact that we are dealing with a two-dimensionalsystem in which the pattern of the phase transition islikely to be of Kosterlitz-Thouless type[23].The loga-rithmic corrections to the scaling laws have been provenessential in a recent work of Kenna and Irving[24].In-clusion of these corrections would allow us to obtain thecritical exponents with higher accuracy.However,suchanalysis would require simulations on even larger lattices.The linearfits for the logarithmic scaling and the criti-cal exponents obtained,are to be viewed as approximatevalues reflecting the general behaviour of the Yang-Leezeros as the temperature and lattice size are varied.Al-though the bootstrap analysis provided us with accurateestimates of the statistical error on the values of the ex-pansion coefficients and the Yang-Lee zeros,the smallnumber of zeros obtained with sufficient accuracy doesnot allow us to claim higher precision for the critical ex-ponents on the basis of more elaboratefittings of the scal-ing behaviour.Thefinite-size effects may still be signifi-cant,especially as the simulation temperature decreases,thus affecting the scaling of the Yang-Lee zeros with thesystem rger lattice simulations will therefore berequired for an accurate evaluation of the critical expo-nent for the particle density and the density of states.Nevertheless,the onset of a singularity atfinite temper-ature,and its persistence as the lattice size increases,areevident.The estimate of the critical exponent for the diver-gence rate of the density of states of the quasiparticleexcitation spectrum is particularly relevant to the highT c superconductivity scenario based on the van Hove sin-gularities[25],[26],[27].It is emphasized in Ref.[25]thatthe logarithmic singularity of a two-dimensional electrongas can,due to electronic correlations,turn into a power-law divergence resulting in an extended saddle point atthe lattice momenta(π,0)and(0,π).In the case of the14.I.M.Barbour,A.J.Bell and E.G.Klepfish,Nucl.Phys.B389,285(1993).15.I.M.Lifshitz,JETP38,1569(1960).16.A.A.Abrikosov,Fundamentals of the Theory ofMetals North-Holland(1988).17.P.Hall,The Bootstrap and Edgeworth expansion,Springer(1992).18.S.R.White et al.,Phys.Rev.B40,506(1989).19.J.E.Hirsch,Phys.Rev.B28,4059(1983).20.M.Suzuki,Prog.Theor.Phys.56,1454(1976).21.A.Moreo, D.Scalapino and E.Dagotto,Phys.Rev.B43,11442(1991).22.N.Furukawa and M.Imada,J.Phys.Soc.Japan61,3331(1992).23.J.Kosterlitz and D.Thouless,J.Phys.C6,1181(1973);J.Kosterlitz,J.Phys.C7,1046(1974).24.R.Kenna and A.C.Irving,unpublished.25.K.Gofron et al.,Phys.Rev.Lett.73,3302(1994).26.D.M.Newns,P.C.Pattnaik and C.C.Tsuei,Phys.Rev.B43,3075(1991);D.M.Newns et al.,Phys.Rev.Lett.24,1264(1992);D.M.Newns et al.,Phys.Rev.Lett.73,1264(1994).27.E.Dagotto,A.Nazarenko and A.Moreo,Phys.Rev.Lett.74,310(1995).28.A.A.Abrikosov,J.C.Campuzano and K.Gofron,Physica(Amsterdam)214C,73(1993).29.D.S.Dessau et al.,Phys.Rev.Lett.71,2781(1993);D.M.King et al.,Phys.Rev.Lett.73,3298(1994);P.Aebi et al.,Phys.Rev.Lett.72,2757(1994).30.E.Dagotto, A.Nazarenko and M.Boninsegni,Phys.Rev.Lett.73,728(1994).31.N.Bulut,D.J.Scalapino and S.R.White,Phys.Rev.Lett.73,748(1994).32.S.R.White,Phys.Rev.B44,4670(1991);M.Veki´c and S.R.White,Phys.Rev.B47,1160 (1993).33.C.E.Creffield,E.G.Klepfish,E.R.Pike and SarbenSarkar,unpublished.Figure CaptionsFigure1Bootstrap distribution of normalized coefficients for ex-pansion(14)at different update chemical potentialµ0for an82lattice.The corresponding power of expansion is indicated in the topfigure.(a)β=5,(b)β=6,(c)β=7.5.Figure2Bootstrap distributions for the Yang-Lee zeros in the complexµplane closest to the real axis.(a)102lat-tice atβ=5,(b)102lattice atβ=6,(c)82lattice at β=7.5.Figure3Yang-Lee zeros in the complexµplane closest to the real axis.(a)β=5,(b)β=6,(c)β=7.5.The correspond-ing lattice size is shown in the top right-hand corner. Figure4Angular distribution of the Yang-Lee zeros in the com-plex fugacity plane Error bars are drawn where esti-mated.(a)β=5,(b)β=6,(c)β=7.5.Figure5Scaling of the imaginary part ofµ1(Re(µ1)≈=0.7)as a function of lattice size.αm u indicates the thefit of the logarithmic scaling.Figure6Electronic susceptibility as a function of chemical poten-tial for an82lattice.The solid line represents the con-tribution of all the2n x n y zeros and the dotted line the contribution of the six zeros nearest to the real-µaxis.(a)β=5,(b)β=6,(c)β=7.5.。
Superconductivity at 53.5 K in GdFeAsO1-δJie Yang , Zheng-Cai Li, Wei Lu, Wei Yi, Xiao-Li Shen,Zhi-An Ren*, Guang-Can Che, Xiao-Li Dong, Li-Ling Sun, Fang Zhou, Zhong-Xian Zhao*National Laboratory for Superconductivity, Institute of Physics and Beijing National Laboratory for Condensed Matter Physics, Chinese Academy of Sciences, P. O. Box 603, Beijing 100190, P. R. ChinaPACS numbers: 74.10.+v; 74.70.-b;Key Words: New superconductor; Arsenic-oxide superconductor; high-T c; Oxygen-deficient;Abstract:Here we report the fabrication and superconductivity of the iron-based arsenic-oxide GdFeAsO1-δcompound with oxygen-deficiency, which has an onset resistivity transition temperature at 53.5 K. This material has a same crystal structure as the newly discovered high-T c ReFeAsO1-d family (Re = rare earth metal) and a further reduced crystal lattice, while the T c starts to decrease compared with the Sm FeAsO1-δ system.Since the discovery of layered copper oxide superconductors [1], extensive researches have been performed to understand the mechanism of high temperature superconductivity and to explore higher T c materials. It is believed that strong electron correlation and layered structure may play an important role. However, the T c of the non-copper-based superconductors is still lower than 40 K by the prediction of BCS theory and the experimental facts. Recently, superconductivity in iron- and nickel-based layered quaternary compounds has been reported: LaOFeP (T c ~ 4K) [2], LaONiP (T c ~ 3K) [3], then LaOFeAs at 26 K when substitution of As for P and F-doping at oxygen site [4]. These discoveries have attracted much interest in further experiments and theoretical studies. Subsequently, the T c rapidly increases to above 50 K in the ReFeAsO1-x F x family [5-9], with the replacement of La by other light rare-earth elements like Pr, Nd and Sm etc. Instead of F-doping, we have succeeded in synthesizing the ReFeAsO1-d superconductors recently, which have better superconducting properties [10]. These quaternary superconductors crystallize with the tetragonal layered ZrCuSiAs structure, in the space group of P4/nmm, which has a structure of alternating Fe-As layer and Re-O layer, similar to that of cuprates. As for the Re = Gd system, we have discovered its superconductivity previously in some multi-phased samples, here we report the superconductivity in GdFeAsO1-δand GdFeAsO1-x F x systems.A series of superconductors with nominal compositions of oxygen-deficient GdFeAsO1-δ and F-doped GdFeAsO1-x F x were prepared by the high pressure (HP) synthesis method. The starting chemicals Gd chips, As, Fe, Fe2O3, FeF2 powders are all with the purity better than 99.99%. At the first step, GdAs powder was obtained by reacting Gd pieces and As powders at 650 o C for 12 hours and then 1150 o C for 12 hours. The starting materials were mixed together according to the nominal ratio, then ground thoroughly and pressed into small pellets. The pellets were sealed in boron nitride crucibles and sintered in the high pressure synthesis apparatus under a pressure of 6 GPa and temperature of 1350 o C for 2 hours. The HP samples are hard and can be polished to shiny mirror-surface.The phase purity and structural identification were characterized by powder X-ray diffraction (XRD) analysis on an MXP18A-HF type diffractometer with Cu-K a radiation from 20o to 80o witha step of 0.01o. The XRD results indicate that the main phase of all F-doped and oxygen-deficient samples adopts the same ReFeAsO structure with slight impurity phases. The impurity phases have been identified to be some by-products and do not superconduct at the measuring temperature. Fig.1 shows the comparison of a typical XRD pattern for a nominal GdFeAsO0.85 sample synthesized by HP method and an ambient-pressure (AP) synthesized undoped GdFeAsO sample. The lattice parameters are a = 3.890(4) Å, c = 8.383(2) Åfor the HP sample of GdFeAsO0.85, and a = 3.903(7) Å, c = 8.453(1) Å for AP sample of GdFeAsO, which indicates a clear shrinkage for the oxygen-deficient sample compared with the undoped sample. All samples with either F-doping or oxygen-deficiency have shrunken crystal lattices. The lattice parameters of Gd-system is smaller than that of Sm-system reported in our previous paper [10], which shows a further enhanced chemical pressure on the Fe-As plane in the Gd-system.The resistivity of all samples was measured by the standard four-probe method from 4 K to 300 K. Slight F-doping and oxygen-deficiency both lead to the occurrence of superconductivity in this system, while samples with oxygen-deficiency were found to have higher T c comparing with F-doped ones, and all samples have metallic behavior up to 300 k. The sample with a nominal composition of GdFeAsO0.85 was found to have the highest T c in this Gd-system, while for the F-doping system, the nominal GdFeAsO0.8F0.2 was found to have the highest T c, and the corresponding resistivity curves are shown in Fig. 2. As it can be seen in the inset that a clear superconducting onset transition (T c(onset)) occurred at 53.5 K and a zero resistivity transition (T c(zero)) at 52.3 K for the GdFeAsO0.85; while for the GdFeAsO0.8F0.2, the T c(onset) and T c(zero) are at 51.2 K and 45.5 K. Comparing with SmFeAsO1-δ superconductor, the T c starts to decrease, this may indicate that the T c has reached the maximum in the Sm-based system by the inner chemical pressure effect for the ReFeAsO1-δ family.The DC magnetization was measured using a Quantum Design MPMS XL-1 system. For an experiment cycle the sample was cooled to 1.8 K in zero field cooling (ZFC) and data were gathered when warming in an applied field, then cooled again under an applied field (FC) and measured when warming up. The DC-susceptibility data (measured under a magnetic field of 1 Oe) of the GdFeAsO0.85 are shown in Fig. 3, with the differential ZFC curve at the right panel. The sharp magnetic transitions on DC curves indicate the good quality of this superconductingcomponent. The onset diamagnetic superconducting transition temperature determined from the differential ZFC curve is 53.5 K, same as the onset transition point on the corresponding resistivity curve.In conclusion, we have succeeded in preparing the GdFeAsO superconductors by both of F-doping and oxygen-deficiency. The oxygen-deficient samples were found to have better superconducting properties in this system and the highest T c is at 53.5 K for the nominal GdFeAsO0.85 composition.Acknowledgements:We thank Mrs. Shun-Lian Jia for her kind helps in resistivity measurements. This work is supported by Natural Science Foundation of China (NSFC, No. 50571111 & 10734120) and 973 program of China (No. 2006CB601001 & 2007CB925002). We also acknowledge the support from EC under the project COMEPHS TTC.Corresponding Author:Zhi-An Ren: renzhian@Zhong-Xian Zhao: zhxzhao@[1] Bednorz J G and Muller K A 1986 Z. Phys. B. 64 189[2] Kamihara Y, Hiramatsu H, Hirano M, Kawamura R, Yanagi H, Kamiya T, and Hosono H 2006 J. Am. Chem. Soc. 128 10012[3] Watanabe T, Yanagi H, Kamiya T, Kamihara Y, Hiramatsu H, Hirano M, and Hosono H 2007 Inorg. Chem. 46 7719[4] Kamihara Y, Watanabe T, Hirano M and Hosono H 2008 J. Am. Chem. Soc. 130 3296[5] Chen X H, Wu T, Wu G, Liu R H, Chen H and Fang D F 2008 Cond-mat:arXiv, 0803.3603[6] Chen G F, Li Z, Wu D, Li G, Hu W Z, Dong J, Zheng P, Luo J L and Wang N L 2008 Cond-mat:arXiv,0803.3790[7] Ren Z A, Yang J, Lu W, Yi W, Che G C, Dong X L, Sun L L and Zhao Z X 2008 Cond-mat:arXiv, 0803.4283[8] Ren Z A, Yang J, Lu W, Yi W, Shen X L, Li Z C, Che G C, Dong X L, Sun L L, Zhou F and Zhao Z X 2008 EPL 82 57002[9] Ren Z A, Lu W, Yang J, Yi W, Shen X L, Li Z C, Che G C, Dong X L, Sun L L, Zhou F and Zhao Z X 2008 Chin. Phys. Lett. 25 2215[10] Ren Z A, Che G C, Dong X L, Yang J, Lu W, Yi W, Shen X L, Li Z C, Sun L L, Zhou F and Zhao Z X 2008 Cond-mat:arxiv. 0804.2582Figure 1: The typical XRD patterns for the nominal GdFeAsO0.85sample synthesized by HP method (upper line) and GdFeAsO by AP method (lower line); the vertical bars indicate the calculated diffraction peaks for the GdFeAsO.Figure 2: The temperature dependence of resistivity for the nominal composition of GdFeAsO0.85 and GdFeAsO0.8F0.2 synthesized by HP method.Figure 3: The temperature dependence of the DC-susceptibility, and differential ZFC curve for the nominal GdFeAsO0.85 sample synthesized by HP method..。
龚昌德教授简历1953年毕业于复旦大学物理系,1953年9月至1955年1月在华东水利学院任教。
1955年1月以后至今在南京大学物理系工作,1978 – 1981 年任副教授, 1981年任南京大学物理系教授,同年获国务院学位委员会通过我国首批博士生导师,1986-1993 年任南京大学物理系系主任,1994年至今任南京大学理学院院长,理论物理研究中心主任。
2005年当选为中国科学院数理学部院士。
从事的研究领域主要有:强关联电子系,超导物理,低维物理,光与低维固体相互作用,介观物理等。
主持项目和获奖情况:1978年因“超导物理研究”获“全国科学大会奖”;1982年,因“超导体临界温度”的研究成果获得国家自然科学奖;1984年获得首批“国家级有突出贡献中青年专家”;1985年主持国家基金项目“低维系统相变及元激发研究”;1987年主持国家重点基金项目“量子超细微粒的物理研究”1988年所著“热力学与统计物理学”获得国家教委高等学校优秀教材一等奖;1990年因“光与低维固体相互作用”的研究成果获得国家教委科技进步二等奖;1990年享受国务院政府特殊津贴;1992年因“低维系统中的相变元激发”的研究成果获得国家教委科技进步二等奖;1992年任理论物理攀登项目“九十年代理论物理重大前沿课题”专家组成员,强关联电子系统子课题组组长;1993年主持国家“863”超导项目中基础研究部分;1996年获得江苏省高等学校“红杉树”园丁奖金奖;1997年因“凝聚态物理学高层次人才培养与实践”获得国家级教学成果一等奖;1997年获得“宝钢优秀教师奖”。
1998年合作项目“介观环的持续电流及其电子输运性质”被广东省科技委列为1998年广东省重大科技研究成果。
曾任学术职务:1982年被推选为全国凝聚态理论及统计物理专业委员会领导小组成员;1985年起历任二、三、四届国务院学位委员会学科组成员1986年受聘为李政道中国高科技中心首批特别成员;1986年任江苏省物理学会理事长;1987年受聘为意大利国际理论物理中心(ICTP)协联教授;1990年任国家教委首届“物理学教学指导委员会”委员;1992年江苏省自然科学基金委员会首届委员;1992年受聘为国际核心期刊“J. Low. Temp. Phys.” 编委;1993年任国家普通高校优秀教学成果评审委员会委员;1995-1999中国物理学会常务理事1995年任江苏省青年科技奖专家评审委员会副主任;1995年任国家教委第二届“高等学校理科物理学与天文学教学指导委员会”副主任委员;1997年度香港中文大学杨振宁访问教授位置。
老师看不出是抄的我的奇思妙想英语作文全文共3篇示例,供读者参考篇1My Wildest Dreams and FantasiesYou know that feeling when your mind starts to wander and suddenly the most peculiar thoughts take over? It's like being transported to another dimension where reality is bent and anything is possible. Well, let me take you on a journey through the untamed wilderness of my imagination. Buckle up, because this ride is about to get weird!Imagine a world where dogs could talk. Not just bark or whine, but engage in full-blown conversations. I'm picturing myself strolling through the park, casually chatting with a pack of Golden Retrievers about the latest bone-chilling thriller they've been digging into (pun very much intended). We'd debate the finer points of storytelling, critique the character development, and maybe even exchange a few doggy puns for good measure. Who knows, I might even get some writing tips from thesefour-legged literary critics!Now, let's take a sharp left turn into the realm of the utterly absurd. What if clouds were made of cotton candy? Imagine gazing up at the sky, watching those fluffy, sugary formations drifting lazily overhead. I can almost taste the sweetness on my tongue as I reach up to pluck a handful of cloud-candy, savoring the delightful flavor bursting in my mouth. Of course, this could lead to some sticky situations – quite literally. Rainclouds would shower us with sugary syrup, and thunderstorms would be an absolute mess. But hey, at least we'd never run out of dessert!Speaking of food, have you ever wondered what it would be like if vegetables could scream? Imagine the horror of biting into a juicy tomato, only to be met with a blood-curdling shriek emanating from its fleshy interior. Chopping an onion would become a form of cruel and unusual punishment, unleashing a torrent of anguished wails. Cooking would be a traumatic experience, filled with the agonized cries of defenseless veggies begging for mercy. On second thought, maybe we should stick to a strict candy diet in this fantastical world.Now, let's drift into the realm of the truly outlandish. What if the moon was made of cheese? Not just any cheese, mind you, but the gooiest, most mouth-watering cheddar you can imagine. Imagine gazing up at that golden, celestial disc, its craters filledwith pools of melted deliciousness. Astronauts would be replaced by cheese connoisseurs, embarking on expeditions to mine the lunar surface for the finest, most aged varieties. The only downside? Lunar eclipses would be a real letdown, blotting out that tantalizing sight with the dull hunk of rock we call Earth.But wait, there's more! How about a world where gravity was reversed? Instead of being pulled down, we'd find ourselves constantly floating upwards, desperately clinging to anything anchored to the ground. Imagine the chaos of trying to have a casual conversation, both parties drifting lazily towards the ceiling as they exchange pleasantries. Eating would be a messy affair, with food and utensils constantly escaping our grasp and disappearing into the stratosphere. And let's not even talk about the challenges of using the restroom in such an environment. On second thought, maybe we should just stick to the ground for that particular activity.Of course, no journey through the bizarre would be complete without exploring the idea of animals walking on their hind legs. Picture yourself strolling down the street, only to encounter a group of upright dogs engaged in a lively discussion about the latest trends in chew toys. Or perhaps you'd stumble upon a gathering of bipedal cats, sipping daintily from tinyteacups as they gossip about the neighborhood's most eligible toms. It would be a surreal sight, to say the least, but also strangely endearing in its own peculiar way.As our adventure through the whimsical reaches of my mind comes to a close, I can't help but feel a sense of wonder and excitement. The ability to imagine the impossible, to dream up scenarios that defy logic and reason, is what makes the human experience so rich and fascinating. So, the next time your thoughts start to wander down an unconventional path, don't shy away from the absurdity. Embrace the weird, the wacky, and the downright bizarre. Who knows? You might just stumble upon your own personal wonderland, filled with talking dogs, cotton candy clouds, and a moon made of cheese.篇2My Wildest DreamsAs I lay in bed last night, staring at the glow-in-the-dark stars stuck to my ceiling, my mind started to wander. I found myself pondering the vast mysteries of the universe and imagining what incredibilities could exist out in the great cosmic unknown. The more I thought about it, the more my imagination ran wild with fanciful possibilities.What if there was another Earth-like planet out there with intelligent life forms vastly different from humans? Maybe there's a planet made entirely of candy with lollipop trees and cookie crumble mountains. Or a world with floating cities held aloft by anti-gravity technology. Anything seems possible when you really open your mind to the mysteries of existence.I started imagining myself on a journey through the cosmos, exploring strange new worlds. On the candy planet, I'd be able to bite into a chocolate mansion or drink from a river of root beer without any consequences. The floating cities would be glorious metropolises suspended in perpetual motion among the clouds and stars.Maybe there's even a planet made up of pure energy where life takes the form of concentrated beams of light and heat. How wild would it be to see cities built of plasma and lightning? I could soar effortlessly by bending radiation around me like a surfboard on a wave.As my mind continued racing with these fanciful ideas, I found myself hoping that somewhere out there, such extraordinary places truly exist. If nothing else, it's fun to imagine the possibilities of what could be.Of course, I'm not naive enough to think there's a real candy planet or cities of pure energy. But who's to say there aren't other planets with forms of life we can scarcely comprehend from our limited human perspective? With how little we know about the larger cosmos, anything seems possible if you allow your imagination to run free.I started pondering how life could evolve in ways entirely different from what we understand. On Earth, carbon-based life emerged from the primordial soup, slowly developing into complex multi-cellular organisms over billions of years. But what if a planet existed where life arose from pure energy or something entirely different and unexpected?There could be civilizations made up of sentient machines that have evolved their artificial intelligence to become a new form of technological life. Or what about a world where life emerged from complex neutron flows, forming itself into something akin to what we'd perceive as neutron stars that can somehow think and reason? The possibilities are endless when you shed your anthropocentric biases.My mind then drifted to what kind of unfathomable technologies could exist across the vastness of space and time. We humans have built some incredible devices like computers,rockets, and particle colliders. But those surely pale in comparison to what could be out there across the galaxies.I imagined a civilization that had mastered teleportation and could effortlessly travel between star systems in the blink of an eye. Or a race of beings that had unlocked the secrets of wormhole physics to create shortcuts across the cosmos. Maybe there are advanced species that long ago transcended physical forms altogether to become purely energetic beings capable of riding the very fabric of space-time itself.As a sci-fi fan, I'm quite familiar with concepts like hyperdrives, warp drives, and other fanciful means offaster-than-light travel that skirt around the cosmic speed limit of light speed. But who's to say more intelligent and ancient alien species haven't already cracked the code to achieve something like that or far crazier things we can't yet conceive?I then found myself wondering just how old some alien civilizations could truly be across this mind-bogglingly vast universe of ours. Our own human history only stretches back a few thousand years from an evolutionary standpoint. But there could be races of xenomorphs out there that are billion, if not trillions of years more ancient than we are.What unfathomable wisdom and technologies could they have amassed over such cosmically vast timescales? There's a chance some could be old enough to have originated during the earliest epochs after the Big Bang itself when the universe was an entirely different place. That's enough to make your head spin if you think about it too much.Of course, these are all wild thoughts and fanciful ideas with no grounding in scientific facts or evidence. I'm just an awestruck teenager allowing my imagination to run rampant across the celestial expanses of my mind's eye. But that's precisely what drives the wonderment of science and human curiosity.While the concepts I'm dreaming up are firmly rooted in the realms of science fiction and fantasy for now, the truth is we simply don't know what actual marvels could exist across this mind-bendingly massive universe of ours. It was once thought the sun revolved around the Earth and that our planet was the center of all creation. Just a few centuries ago, atoms and electromagnetic waves were still unknown to humankind.As our scientific understanding of the cosmos continues advancing by leaps and bounds, who's to say what shocking revelations tomorrow may bring? What we consider to be the wildest fiction today could turn out to be tame reality in thedecades and centuries to come. Keeping an open mind to the seemingly impossible is what continually expands the frontiers of knowledge and discovery.While my restless mind conjured up elaborate alien worlds and unfathomable cosmic technologies in my latest reverie, I realize how little we truly understand about this vast universe surrounding us. Our knowledge is but a drop in the ocean compared to the mysteries still left to be unraveled. But that ultimate uncertainty and sense of wonderment is what keeps driving humanity to explore, question, and push the boundaries of our perception.As my eyelids grew heavy from an all-nighter of vivid cosmic daydreaming, the last whimsical thought I remember before succumbing to sleep's embrace was: What if there's another universe out there entirely parallel to our own, filled with duplicates of every person and civilization? A vast multiverse of infinite realities coexisting across a baffling higher-dimensional expanse?Perhaps in another universe, a different version of me is writing down his own celestial reveries and wildest dreams. Or maybe the latest realms of my subconscious imagination have revealed glimpses of something startling yet to be discoveredabout this endlessly fascinating cosmos of ours. Either way, I can't wait to find out more and to keep exploring the depths of reality and fantasy alike.篇3The Cosmic DreamweaverI can still vividly recall the night everything changed – the night I met the Cosmic Dreamweaver. It was a sultry summer evening, the kind where the thick air seems to wrap around you like a stifling blanket. I had retreated to the solitude of my backyard, seeking refuge from the oppressive heat trapped inside the house. As I flopped onto the weathered wooden bench, I gazed up at the inky black canvas of the night sky, mesmerized by the twinkling diamonds strewn across its expanse.Suddenly, something caught my eye – a shooting star streaked across the heavens, burning brighter than any I'd ever witnessed before. But this was no ordinary meteor. As I watched, transfixed, the blazing streak shifted course, looping and pirouetting through the stars in mesmerizing patterns. My eyes must have been playing tricks on me, I reasoned, rubbing themfuriously. When I opened them again, the fiery trail had ceased its celestial dance and was hurtling directly towards me!I threw my arms up in a feeble attempt at self-defense, bracing for catastrophic impact. But instead of a searing demise, I felt an odd tingling spreading from the top of my head down to the soles of my feet. The tingling intensified until my entire being seemed to dissolve into a blinding kaleidoscope of color and sound. I slipped into a state of transfixed delirium, my consciousness unraveling and reweaving itself into strange new patterns.When my senses sluggishly rejoined, I found myself no longer in the confines of my backyard, but adrift in a realm of breathtaking alien splendor. Colossal spirals of multihued energy undulated all around me, emanating waves of pure harmonic resonance that reverberated through my very soul. Gargantuan celestial structures, their surfaces sculpted into fractal cities of hypnotic complexity, materialized and dematerialized in the cosmic expanse. I felt utterly minuscule, a solitary speck of near-nothingness amid the grandeur of this ineffable wonderland.As I spun about, gawking in open-mouthed awe, something drifted into my field of perception – an entity that defiedcomprehension, yet filled me with a strange sense of tranquility. It was an ever-shifting kaleidoscope of light and form, its visage cycling through a mesmerizing array of avatars: one moment an ascended astral being, brimming with arcane power; the next, a humble woodlands creature; a living fractal; a swirling nebula birth new worlds..."Do not be afraid, Earthchild," a warm voice reverberated through my psyche. "I am the Cosmic Dreamweaver. I have summoned you here to share with you the ultimate gift.""G-gift?" I stammered, utterly confounded. "What gift? Where... where is 'here?'"The entity's avatar shimmered and contorted, resolving into a wizened humanoid form that nonetheless thrummed with latent cosmic power."This place exists beyond the boundaries of your physical universe, in a realm of pure potentiality woven into the fabric of existence itself. And my gift to you is the ability to shape and reshape reality to your wildest imaginings."With an outstretched hand, the Cosmic Dreamweaver released an incandescent torrent of pure creative energy that engulfed my being. I felt my mind expanding immeasurably,blazing new neural pathways flaring open to accept the incomprehensible secrets of this strange place. Images, insights, and alien epiphanies flooded my consciousness in a deluge of pure inspiration."You now possess the power to reach into the dreamscapes of potential and pluck forth entire worlds from your thoughts," intoned the Dreamweaver. "Shape them through force of will and concentration alone into any form you can envisage. Let your imagination be your guide, and the only boundaries will be those you conceive for yourself."The being's avatar contracted into a shimmering sphere that hovered before me, dreamlike whispers echoing fromall sides: "Embrace your innermost fancies and yearnings... All is possible in the realm of dreamweavers..."With those parting words, the strange sphere imploded in a silent blossoming of spectral light. When the brilliance cleared, I found myself once again beneath the familiar night sky, on the worn bench where I'd been stargazing. Had it all been an fever dream hallucination brought on by the heat? I started to convince myself of that logical conclusion when a tinyalpha-numericticker tape began unfurlingitself from my mind'seye. It crystallized into a coherent stream of code – new neural programming granting me impossible abilities.I let the alien algorithms and fractal ciphers wash over me, integrating the incredible power into the core of my being. Taking a deep, centering breath, I focused my newly opened mindand willed my consciousness to expand outward, perceiving and interacting with the Potentiality Matrix underlying all of existence.Suddenly, the backyard scene distorted and fragmented into coruscating wings of light and shadow. A kaleidoscopic vortex spiraled into being, drawing me inward onto a vast formless expanse awash in the primal waters of creation. Here, in the nexus of conceptual potentials, my abilities as a Dreamweaver blossomed into full actualization.I imagined a twisting spire of glimmering crystal, and it effloresced before me. I envisioned floating isles crafted from woven condensed sunlight, and they shimmered into ethereal reality. Great migrating beasts with translucent bodies and metallic carapaces shimmered into existence, buoyed by levitating electrocycles that supercharged the prism-clouds that served as their propulsion...Over what seemed an eternity compressed into a singularity of subjective time, I unleashed my unbridled musings into evermore baroque expressions of pure, distilled dream-fancy. Cities crafted from living infiniglass, their axiro-latticed ziggurats straddling interdimensional continua. Vast sparkling rivers of thought that carried philosophies and mindstreams between shimmershrine arcologies. Fractal-generated megafauna forming symbiological relationships with hyper-sentient botanical canopies and roiling biomechondrial seas...At long last, my manic creative fervor found itself sated...at least for the moment. With an oblique sense of closure, I initiated a return vector to my own humble plane of existence, where the trappings of terrestrial familiarity gradually coalesced around me once again. But everything now seemed just the slightest bit...skewed, as if the molecules of reality itself had been subtly rearranged by the force of my re-imaginings.As I surveyed my surroundings anew through dreamweaver's eyes, I was struck by the boundless possibilities now laid before me. No longer was I bound by the strictures of mundane physics or constraints of preconceived possibility. Every wish, whimsy and unbridled imagining was now within my grasp to actualize through sheer creative force of will. The entirecosmos was my lumenoietic canvas to rearrange and recompose in whatever ever-shifting patterns I could envision.With my first foray into the Metaversal Dreamrealms now complete, I felt a curious synergy of potential and uncertainty well up within me. Where would my burgeoning abilities as a Dreamweaver lead? What breathtaking or terrifying new emanations would I call forth into substantive existence from the morphaevic aethers? Perhaps it was best I didn't know just yet what audacious, improbable, fantastic, and wondrous feats of reconstructive meta-genesis awaited...。
高温超导体中的一个正电子陷阱湮没模型(英文) IntroductionHigh-temperature superconductors (HTS) have been a topic of intense research for several decades. One of the challenges in understanding the behavior of HTS is the presence of positive charge carriers, such as holes and positive ions, which can act as traps for electrons. In this article, we will discuss a model for the annihilation of a positive charge carrier trap in HTS.BackgroundHTS are materials that exhibit superconductivity at temperatures above the boiling point of liquid nitrogen (-196°C). The discovery of HTS in 1986 revolutionized the field of superconductivity, as it opened up the possibility of practical applications of superconductivity at relatively high temperatures. However, the mechanism of superconductivity in HTS is still not fully understood.One of the challenges in understanding HTS is the presence of positive charge carriers, which can act as traps for electrons. These traps can reduce the number of available electrons for superconductivity, leading to a decrease in the critical temperature of the material.ModelA recent study proposed a model for the annihilation of a positive charge carrier trap in HTS. The model involves the interaction between a positive charge carrier and an electron, which leads to the formation of a bound state known as a polaron. The polaron is a quasi-particle that consists of an electron and a positive charge carrier, which are bound together by the Coulomb interaction.The polaron can move through the material, but it can also become trapped by defects in the crystal lattice. When a polaron becomes trapped, it can act as a trap for other electrons, reducing the number of available electrons for superconductivity.The proposed model involves the annihilation of the polaron by a process known as Auger recombination. Auger recombination is a process in which the energy of an excited electron is transferred to another electron, which is then ejected from the material. In the case of the polaron, the energy of the excited electron is transferred to the positive charge carrier, which is then ejected from the material.The annihilation of the polaron by Auger recombination leads to the release of an electron, which can then participate in superconductivity. This process can increase the number of available electrons for superconductivity, leading to an increase in the critical temperature of the material.ConclusionThe proposed model for the annihilation of a positive charge carrier trap in HTS provides a new understanding of the behavior of these materials. The model involves the interaction between a positive charge carrier and an electron, which leads to the formation of a polaron. The polaron can become trapped by defects in the crystal lattice, reducing the number of available electrons for superconductivity. The annihilation of the polaron by Auger recombination leads to the release of an electron, which can then participate in superconductivity. This process can increase the critical temperature of the material, providing a new avenue for the development of practical applications of HTS.。
常用物理英语词汇(全)力学(Mechanics)1. Force(力)2. Acceleration(加速度)3. Momentum(动量)4. Kinetic Energy(动能)5. Potential Energy(势能)6. Gravity(重力)7. Friction(摩擦力)8. Torque(扭矩)9. Angular Momentum(角动量)10. Work(功)热学(Thermodynamics)11. Temperature(温度)12. Heat(热量)13. Internal Energy(内能)14. Entropy(熵)15. Boyle's Law(波义耳定律)16. Charles's Law(查理定律)17. GayLussac's Law(盖吕萨克定律)18. Ideal Gas Law(理想气体定律)19. First Law of Thermodynamics(热力学第一定律)20. Second Law of Thermodynamics(热力学第二定律)电磁学(Electromagnetism)21. Electric Charge(电荷)22. Electric Field(电场)23. Magnetic Field(磁场)24. Current(电流)25. Voltage(电压)26. Resistance(电阻)27. Capacitance(电容)28. Inductance(电感)29. Ohm's Law(欧姆定律)30. Ampère's Law(安培定律)光学(Optics)31. Light(光)32. Refraction(折射)33. Reflection(反射)34. Diffraction(衍射)35. Interference(干涉)36. Polarization(偏振)37. Lens(透镜)38. Prism(棱镜)39. Spectrum(光谱)40. Wave Optics(波动光学)现代物理(Modern Physics)41. Quantum Mechanics(量子力学)42. Relativity(相对论)43. Photon(光子)44. Electron(电子)45. Proton(质子)46. Neutron(中子)47. Quark(夸克)48. Black Hole(黑洞)49. Higgs Boson(希格斯玻色子)50. String Theory(弦理论)这些词汇仅为物理学中常用术语的一小部分。
a r X i v :c o n d -m a t /0308083v 2 [c o n d -m a t .s t r -e l ] 11 D e c 2003Heavy fermion superconductivity and magnetic order in non-centrosymmetric CePt 3SiE.Bauer a ,G.Hilscher a ,H.Michor a ,Ch.Paul a ,E.W.Scheidt b ,A.Gribanov c ,Yu.Seropegin c ,H.No¨e l d ,M.Sigrist e and P.Rogl faInstitut f¨u r Festk¨o rperphysik,Technische Universit¨a t Wien,A-1040Wien,AustriabChemische Physik und Materialwissenschaften,Universit¨a t Augsburg,D -86159Augsburg,GermanycDept.of Chemistry,Moscow State University,Moscow,RussiadLaboratoire de Chimie du Solide et Inorganique Mol´e culaire,Universit´e de Rennes I,F-35042Rennes,FranceeTheoretische Physik,ETH-H¨o nggerberg,8093Z¨u rich,Switzerland fInstitut f¨u r Physikalische Chemie,Universit¨a t Wien,A-1090Wien,Austria(February 2,2008)CePt 3Si is a novel heavy fermion superconductor,crystallising in the CePt 3B structure as a tetrag-onally distorted low symmetry variant of the AuCu 3structure type.CePt 3Si exhibits antiferromag-netic order at T N ≈2.2K and enters into a heavy fermion superconducting state at T c ≈0.75K.Large values of H ′c 2≈−8.5T/K and H c 2(0)≈5T refer to heavy quasiparticles forming Cooper pairs.Hitherto,CePt 3Si is the first heavy fermion superconductor without a center of symmetry.PACS numbers:74.70.Tx,71.27.+a,75.30.MbCorrelation effects among electrons belong to the key causes for the occurrence of extraordinary properties of solids at low temperatures.The most exciting phe-nomenon in this respect is superconductivity.Both high temperature -and heavy fermion superconductors at-tracted great interest throughout the past two decades.Most interesting aspects to solve are the specific mecha-nisms of pairing and the symmetry of the superconduct-ing condensate [1].While in conventional superconduc-tors the binding of electrons into Cooper pairs is normally mediated by phonons,the origin of pairing in some heavy fermion superconductors is believed to be connected with spin fluctuations,giving rise to unconventional supercon-ducting phases (see e.g.Ref.[2]).Pronounced electron correlations are generally found in systems exhibiting the Kondo effect and thus Ce,Yb and U based compounds are certainly candidates for the oc-currence of superconductivity where renormalized quasi-particles form the Cooper pairs.While heavy fermion superconductors are yet missing in Yb based compounds,such phases were identified for both Ce and U systems.Ce-based heavy fermion superconductors,however,are still few in numbers.Prototypic CeCu 2Si 2exhibits su-perconductivity below T c =0.7K [3].The application of pressure in the 20to 30kbar range to members of this structure family such as CeCu 2Ge 2[4],CePd 2Si 2[5]and CeRh 2Si 2[6]is sufficient to trigger superconductiv-ity as well.Very recently,a new class of compounds,Ce M In 5,was added where at ambient conditions heavy fermion superconductivity occurs for M =Co and Ir at T c =2.3and 0.4K,respectively [7,8].Again,pres-sure initiates superconductivity,e.g.,in CeRhIn 5below T max c =2.1K [9].The crystal structure of latter com-pounds can be considered quasi-two-dimensional variants of CeIn 3(AuCu 3-type).Cubic CeIn 3becomes supercon-ducting at ∼25kbar [2].The aim of the present paper is to report on the dis-covery of both,heavy fermion superconductivity and longrange magnetic order in the compound CePt 3Si,to evalu-ate parameters characterising the superconducting state and to discuss possible pairing scenarios.CePt Si -P4mm;CePt B-type33FIG.1.Crystal structure of CePt 3Si.The bonds indicate the pyramidal coordination [Pt 5]Si around the Si atom.Ori-gin shifted by (0.5,0.5,0.8532)for convenient comparison with parent AuCu 3structure.CePt 3Si was prepared by argon arc melting and sub-sequent heat treatment under high vacuum at 870◦C for three weeks.Crystal structure was determined from Kappa-CCD single crystal X-ray data and found to be tetragonal,space group P 4mm (No.99),isotypic with the ternary boride CePt 3B [10,11](see Fig.1).Crystal-1lographic data(standardized)are:a=0.4072(1)nm and c=0.5442(1)nm;Ce in site1(b)at0.5,0.5,0.1468;Pt(1) in2(c)at0.5,0,0.6504,Pt(2)in1(a)at0,0,0(fixed)and Si in site1(a)at0,0,0.4118.CePt3Si derives from hypothetical CePt3with cubic AuCu3structure byfilling the void with Si,which in addition causes a tetragonal distortion of the unit cell with c/a=1.336.Filling of voids in structures with large cages may dramatically influence physical properties as evidenced in detail for skutterudites RETM4X12,(RE, rare earths;TM,transition metals;X,pnicogen)[12].Of particular importance with respect to superconductivity is the lack of a center of inversion in the crystal structure of CePt3Si.Electron microprobe analyses and Rietveld refinements revealed phase purity of the polycrystalline material used for bulk property measurements[13].The char-acterisation of the paramagnetic state of CePt3Si from susceptibility measurements(not shown here)reveals a Curie Weiss behaviour with an effective Ce moment µeff=2.54µB and a paramagnetic Curie temperature θp=−46K(data for thefit are taken from the inter-val100≤T≤300K).While the former quantity in-dicates a rather stable3+state of Ce at high tempera-tures,the large negativeθp value refers to pronounced antiferromagnetic interactions.In terms of Kondo type interactions,neglecting crystalfield effects,the Kondo temperature would follow from T K≈|θp|/4≈11K[14].ρof CePt3Si and LaPt3Si plotted on a logarithmic temperature scale.The magnetic contributionρmag(T)(dashed line)refers to the right axis.(b):Low temperature details ofρ(T)of CePt3Si.The solid line is a least squaresfit(see text)and the dashed line shows dρ(T)/d T.Evidence of superconductivity in CePt3Si is found from resistivity measurements,ρ(T),displayed in Fig. 2together with data of isostructural non-magnetic LaPt3Si.ρ(T)of CePt3Si drops to zero,resulting inT mid c =0.75K.At high temperatures,ρ(T)is char-acterised by a negative logarithmic term,followed by pronounced curvatures at about75K and15K,which may reflect crystal electricfield effects in the presence ofKondo type interactions.Further evaluation ofρ(T)re-quires knowledge of the phonon contribution which may be taken from homologous and isotypic LaPt3Si(a=0.4115(1)nm,c=0.5438(2)nm).LaPt3Si is metallic inthe temperature range measured and is simply accountedfor in terms of the Bloch Gr¨u neisen model with a DebyetemperatureθD≈160K[solid line,Fig.2(a)].The magnetic contribution to the resistivity,ρmag(T),is ob-tained by subtracting from the total measured effect ofCePt3Si,both the phonon part(taken from LaPt3Si)as well as the respective residual resistivities.ρmag(T)ex-hibits a distinct logarithmic contribution for T>100K;the maximum around80K may indicate the overall crys-talfield splitting of the j=5/2Ce4f1state[dashed line,Fig.2(a)].Fig.2(b)illustrates low temperature features of theelectrical resistivity of CePt3Si.Besides the onset of su-perconductivity,there is a distinct change of the slope in ρ(T)around2K,which becomes more evident from a dρ/d T plot[right axis,Fig.2(b)].In the context of spe-cific heat data(see below),this anomaly is interpreted asa signature of an initiation of long range magnetic order.A least squaresfit according toρ=ρ0+AT2,revealsthe residual resistivityρ0=5.2µΩcm and a materialdependent constant A=2.35µΩcm/K2.p of CePt3Si and LaPt3Si plotted as C p/T vs.ln T.The solid and the dashed lines are least squaresfits(see text).Right axis: Temperature dependent magnetic entropy S mag of CePt3Si (dashed-dotted line).Corroboration of bulk superconductivity can be read offfrom specific heat measurements,C p(T).Results are shown in Fig.3as C p/T vs.ln T.Data of LaPt3Si are added for comparison.Standard analysis for the latter yieldsγ=9mJ/molK2,andθLT D=255K.Two different features are obvious for CePt3Si(from higher to lower temperatures):i)a distinctλ-like anomaly at about2K and ii)a jump of the specific heat around0.7K.Theλ-like anomaly of the specific heat at2.2K sig-nifies the onset of long range magnetic order.A T3-dependence of C p(T)well below the transition may char-acterise antiferromagnetic ordering(solid line,Fig.3).2Antiferromagnetic order also follows from afield driven shift of theλ-like anomaly towards lower temperaturesand is confirmed by the absence of spontaneous magneti-sation.Above T N,C p/T of CePt3Si exhibits an extended–almost logarithmic–tail,indicative of pronounced spin fluctuations in the paramagnetic state.The anisotropy of the crystal structure(c/a≃1.34)is supposed to be anadditional cause for such enhanced short range magnetic correlations.The magnetic entropy S(T)was derived from a com-parison of C p(T)of both CePt3Si and LaPt3Si.Results are shown as dashed-dotted line in Fig.3.The entropygain of about0.22×R ln2at T=T N is well below that as-sociated with the lifting of the degeneracy of the ground state doublet and suggests ordering with substantiallyreduced magnetic moments.Involving Kondo type inter-actions as possible mechanism for the above observation would reflect a characteristic temperature T K of about10to15K.R ln2is reached around25K only.The most interesting feature,the anomaly around0.7 K,indicates the transition into a superconducting phase,in agreement with the aboveρ(T)data.In order to accu-rately determine T c,the standard procedure with an ide-alised jump at T c is applied,yielding T c=0.75K.Some estimation of the normal state Sommerfeld valueγn may be obtained from an extrapolation of the T3-dependence(dashed line,Fig.3),arriving atγn≈0.39J/molK2. This extrapolation primely satisfies the basic require-ment of entropy balance between the superconductingand the normal state.The jump of the specific heat ∆C p/T(T=T c)≈0.1J/molK2,allows calculation of the parameter∆C p/(γn T c)≈0.25,which is well below the figure expected from BCS theory(∆C p/(γT c)≈1.43). Assessing the electronic specific heat coefficient in the superconducting state,γs≈0.18(1)J/molK2,from the difference between data derived from the T3extrapola-tion and those estimated at low temperature and zero field yields∆C p/(γs T c)≈0.55,which is still under the BCS value.It should be noted that,e.g.,∆C p/(γT c) of the spin triplet superconductor Sr2RuO4is similarily downsized[15].The specific heat of CePt3Si for various externalfieldsis plotted in Fig.4(a).The application of magnetic fields reduces T c,giving rise to a rather large change of dH c2/dT≡H′c2≈−8.5T/K,in excellent agreement with the respective data from electrical resistivity[see Fig.4(b)].Extrapolation of thefield dependent tran-sition temperatures towards zero yields H c2(0)≈5T. Furthermore,an estimation of the Sommerfeld coefficient for highfields results in0.36J/molK2,in fair agreement with the value gained from an extrapolation of normal state,zerofield data(see Fig.3).The upturn of C p/T at lowest temperatures,reenforcing in magneticfields, is primarily associated with the nuclear contribution of 195Pt.By analogy to commonly employed practice concerning heavy fermion superconductors,we estimate in the fol-lowing a number of parameters from an analysis of super-conducting and normal state properties in terms of the BCS theory[16,17]assuming a spherical Fermi surface and incorporating clean and dirty limit terms.Starting parameters areγs=0.18(2)J/molK2(as a lower limit), H′c2=−8.5T/K andρ0=5.2µΩcm.Although devi-ations from a spherical Fermi surface are expected for tetragonal CePt3Si,reasonable physical parameters can be anticipated(compare,e.g.Ref.[17,18]).p/T of CePt3Si for various values of appliedfields;vertical lines indicate the SC transition.(b):Temperature dependence of the upper criticalfield H c2deduced from specific heat and resistivity measurements.The solid straight line yields H′c2≈−8.5T/K and the dashed line is a guide to the eyes.The effective Fermi surface S s is computed(Eqn.2 in Ref.[17])as S cl s≃3.7×1020m−2within the clean -and S dl s≃3.5×1020m−2for the dirty limit.Con-sidering the dirty limit only,H′c2(calc)=−0.77T/K,a value rather low with respect to the experimentally de-rived slope H′c2=−8.5T/K.This indicates that CePt3Si is not a typical dirty limit superconductor.Irrespective of all the shortcomings present,further calculations are carried out using clean limit parameters.Combining the Fermi surface withγs gives the Fermi velocity v F≃5300m/s and in the context of the residual resistivity,ρ0=5.2µΩcm,a mean free path l tr≃8×10−8m can be derived.The coherence length ξ0for T→0was obtained from two independent re-lations.One follows from the BCS equation,ξ0= 0.18¯h v F/(k B T c)≃9.7×10−9m.A second expression stems from the well known formulaµ0H c2=Φ0/(2πξ20), yieldingξ0≃8.1×10−9m,in reasonable agreement with the former.Evaluation of the Ginzburg Landau param-eterκGL=λ/ξrequires the knowledge of the thermody-namic criticalfieldµ0H c(0)≃26(2)mT,which is calcu-lated from the free energy difference between the super-conducting and the normal state:∆F(T)=F n−F s=µ0H2c(T)/2= T T c T′T c(C s−C n)3.With H c2(0)≈5T one derives a value forκGL=H c2(0)/(√2∆/µB∼3k B T c/µB≈2T<H c2(0),imply-ing per definition an effective electron g-factor of2.If this rough estimate holds for CePt3Si,the high H c2(0) would then be incompatible with spin-singlet pairing and rather signals spin-triplet superconductivity.The appar-ent conflict may be resolved by noticing that the lack of inversion symmetry is not excluding spin triplet pairing completely.In our case the space group P4mm involvesthe absence of the mirror plane z→−z which yields a Rashba-type of spin orbit coupling and leaves the triplet(equal-spin)pairing state d(k)=ˆx k y−ˆy k x(irreducible representation A2u of D4h)as a possible pairing state[22,23].In summary,CePt3Si is a heavy fermion compound un-dergoing both a magnetic transition at T N=2.2K and a superconducting transition at T c=0.75K.The ratio l tr/ξ0≃8is indicative of a clean limit superconductor. Thermodynamic data derived for CePt3Si suggest that the Cooper pairs are formed of heavy quasiparticles.The pairing is likely affected by the absence of an inversion center.Relatively large H c2might be a hint for the pres-ence of spin-triplet pairing.While still speculative,the prospect of spin triplet pairing in a system without in-version symmetry is an exciting issue for future studies. We are grateful to D.F.Agterberg for helpful discus-sions.This work was supported by the Austrian FWF P16370,15066,by INTAS project234,by the Deutsche Forschungsgemeinschaft(DFG)SFB484(Augsburg),by the ESF project FERLIN and the Swiss Nationalfonds.。
在刚刚过去的2001年,全世界的科技工作者为量子论诞生100周年和诺贝尔奖颁发100周年举行了一系列庆祝活动。
在《科学美国人》杂志的一篇纪念文章中,作者列举了33项展示量子论神奇力量的重大发现,这其中有8项与超流—凝聚有关。
它们是:1911年超导电性的发现;1924年玻色—爱因斯坦凝聚(BEC)的预言;1938年超流的发现;1957年BCS超导理论的提出;1973年核磁共振成象用于医疗诊断;1980和1982年整数和分数量子霍尔效应的发现;1986年铜氧化物高温超导体的发现和1995年BEC的实现。
此外,依本文作者看, 值得补入的重大发现还有:1956年朗道关于液氦理论的提出;1962年约瑟夫森效应的预言;1972年3He超流相的发现,等等。
传统超导体库珀对的总动量和总自旋均为零,即组成库珀对的二个电子应分别具有数值相等方向相反的动量和自旋。
施加外磁场,洛伦兹力将强迫传导电子在垂直于外场的平面内作回旋运动。
此时,组成库珀对的二个电子所受洛伦兹力的方向相反,库珀对遭到破坏。
如前所述,在s波和d波二类超导体中,磁场可以通过二种途径拆散库珀对:一是洛仑兹力所产生的轨道效应;二是强迫自旋平行取向的顺磁效应。
另一方面,理论预言:存在第3种库珀对,其中的二个电子自旋原本就是平行取向的。
这种情况被称为p波超导,库珀对的总自旋不再是零而是1,库珀对的总动量也不再是零而表现为轨道三重态。
于是,外磁场不再能通过顺磁效应拆散库珀对,只能通过轨道效应破坏超导电性。
更快、更高、更强——2001年的超导研究Faster, higher and stronger——Research into superconductivity in 2001戴闻中国科学院理化技术研究所北京 100080在刚刚过去的2001年,全世界的科技工作者为量子论诞生100周年和诺贝尔奖颁发100周年举行了一系列庆祝活动。
在《科学美国人》杂志的一篇纪念文章中,作者列举了33项展示量子论神奇力量的重大发现,这其中有8项与超流—凝聚有关。
维修工程中国医学装备2024年4月第21卷第4期 China Medical Equipment 2024 April V ol.21 No.4Structural function and failure case analysis of superconducting magnetic resonance helium compressor/Zhang Falun 1, Wang Jixi 2, Zhou Yalin 2, Shi Zhan 31Department of Medical Equipment T eaching and Research, Jiangsu Health V ocation College, Nanjing 211800, China; 2Department of Medical Engineering, Jiangsu Province Official Hospital, Nanjing 210024, China; 3Siemens Medical Systems Ltd., Shanghai 201318, ChinaCorresponding author: [Abstract] Helium compressor is the core component of the refrigeration system of superconducting magnetic resonance imaging (MRI) equipment, and timely resolution of helium compressor failures is crucial to ensure the stability of the superconducting magnet cryogenic system. T aking the Sumitomo F-70 helium compressor manufactured by Sumitomo Heavy Industries as an example, the structural composition and functional principle of the helium compressor for superconducting MRI were analyzed, and the maintenance ideas and solutions for common helium compressor faults were proposed. By developing standardized fault maintenance strategies, reference was provided for failures of other types of helium compressor, so as to improve the quality of clinical technical support.[Key words] Superconducting magnetic resonance imaging; Helium compressor; Structure and function; Fault diagnosisFund program: 2022 Faculty Research Projects of Jiangsu Health V ocation College (JKC2022006)[摘要] 氦压缩机是超导磁共振成像(MRI)设备制冷系统核心部件,及时解决氦压缩机故障对保障超导磁体低温系统的稳定性至关重要。
a rXiv:c ond-ma t/969116v112Se p1996Heat conduction in the heavy fermion superconductor UPd 2Al 3May Chiao,Benoit Lussier,Brett Ellman and Louis Taillefer Department of Physics,McGill University,3600University Street,Montr´e al,Qu´e bec,Canada H3A 2T8(February 1,2008)Abstract We present preliminary measurements of the thermal conductivity of the heavy fermion superconductor UPd 2Al 3for the normal and superconducting states at low temperatures.As T →0,the superconducting state is charac-terized by a finite linear term,about 10%of the normal state value,which suggests a residual density of low energy quasiparticle states.This agrees qualitatively with resonant impurity scattering theories applied to exotic su-perconductors with nodes in the gap parisons are made with theory and with other U-based superconductors,such as URu 2Si 2and UPt 3.PACS numbers:74.70.Tx,74.25.FyTypeset using REVT E XHeavy fermion superconductors(HFS)have drawn much attention since their discovery in the mid-80’s,yet the nature of the order parameter remains a focus of intense research. Discovered in1991,UPd2Al3exhibits co-existing antiferromagnetism and superconductiv-ity[1],with the highest T c(2K)and magnetic moment(0.85µB/U)of all the HFS.NMR relaxation rates[2]show a power law T dependence suggestive of a gap with a line of zeroes on the Fermi surface.However,the T dependence of the specific heat[1]and the penetra-tion depth[3]are quite close to that expected for an s-wave gap.In this paper,we make a preliminary study of the gap structure using thermal conductivity.The polycrystalline sample was grown and annealed using RF induction heating in ul-trahigh vacuum.We measured the thermal conductivityκusing the steady-state technique, with one heater and two thermometers.The normal state thermal conductivityκN was obtained by applying afield H>H c2(0).This sample has a T c of1.86K and an RRR of36 (ρ0=3.8µΩcm).The positive magnetoresistance is roughly20%in4.5T.To useκas a probe of gap structure,it is necessary to separate the contributions from all the heat carriers.Let usfirst consider the normal state(Fig.1).In afield of4.5T,the Lorentz number L=κρ/T at100mK is2.32×10−8WΩK−2,which is0.95L0(L0=2.44×10−8WΩK−2),in agreement with the Wiedemann-Franz law.In the absence of inelastic scattering,whose effect onκis unlikely to exceed10%below1K,such behaviour is expected from electrons.As a result,we take the electronic part ofκN/T to be L0/ρ0from0to1K. From inelastic neutron scattering[4],the antiferromagnetic magnons have a short life-time of about3×10−13s.With the magnon velocity,we get a mean free path of13˚A.Since the spin wave excitation spectrum is gapless and linear in q,the specific heat c mag goes as T3.Then the magnonκcan be estimated byκmag=c mag v mag l mag/3,where c mag=25.4Jm−3K−4×T3 [5],v mag=4.5×103ms−1[4]and l mag=13˚A.At1K,κmag=5.0×10−4mWcm−1K−1,which is negligible compared to the data.Furthermore,spin-waves are not found to be affected by the superconducting transition[4].For these reasons,we attribute the measuredκentirely to phonons and electrons.Therefore,the roughly linear increase inκN/T seen in Fig.1 must be due to phonons.Then,usingκph=c ph v phΛph/3,where the specific heat c ph=8.0×T3Jm−3K−4[5]and the average sound velocity v ph=5.73×103ms−1(from elastic constants[6]), we estimate the phonon mean free pathΛph to be50µm at0.6K.By175mK it has increased to160µm,roughly1/4the sample size.These are surprisingly long phonon mean free paths for a metal.Assuming the dominant scattering of phonons to come from grain boundaries and electrons,we can estimate the boundary scattering rate B and the e-ph coupling strength E from:κph=k4B T3B+ExT(1)Afit toκN(T)-κN(0)using Eq.(1)yields B=1.5×106s−1and E=5.9×106K−1s−1,which means that electron scattering dominates at all but the lowest temperatures.This value of E is somewhat less than in URu2Si2(1.5×107)[7],and much less than in Nb(2×109)[8] and V(3×109).This anomalously weak e-ph coupling is the most unusual feature of the normal stateκin UPd2Al3.Let us now turn to the superconducting state data,shown as the lower curve in Fig.1.The two main observations are:1)the existence of a substantial intercept inκ/T,2) the similarity in the slopes ofκ/T andκN/T at low T.A smooth extrapolation of the data to T=0gives a limitingκ/T=0.6-0.8mWK−2cm−1,about10%of the normal state value L0/ρ0,likely due to residual quasiparticle excitations.Now,as seen in BCS superconductors such as Nb[8],the large reduction in the number of quasiparticles available for scattering phonons at low T should cause a major increase inΛph.If the rise inκN/T is due to phonons scattering offelectrons,one would expect the slope ofκ/T in the superconducting state to increase dramatically.It does not.In fact,it seems that the non-electronic contribution to κN/T is unaffected by superconductivity.Whether other scattering mechanisms play a role requires further study.At this stage,we adopt the following simple procedure:the electronic thermal conductivityκe/T in the superconducting state is obtained by subtracting from the H=0data in Fig.1the slope ofκN/T.The result is shown in Fig.2,normalized by L0/ρ0.Afinite limiting value forκe/T is one of the major predictions of current theories of transport in unconventional superconducting states(see[9,10],and references therein).Amissing U-atom acts as a Kondo impurity in a compensated lattice,causing multiple scat-tering and large phase shiftsδ0=π/2.Within such a resonant impurity scattering model,a line of nodes in the gap can give afinite intercept inκ/T vs T[9].Such a linear term arises from node smearing due to impurities,which leaves part of the Fermi surface gapless,with a residual density of low energy quasiparticle states.The simplest candidate gap structures are based on a spherical Fermi surface[9,11].A polar gap has a line of nodes along the equator,an axial gap point nodes at the poles and the hybrid gaps both line and point nodes with the gap approaching zero at the poles either with a linear(type I)or quadratic (type II)k-dependence.Calculations such as those performed for UPt3[9]give the curves shown in Fig.2[11].The unitary limit is assumed,inelastic scattering neglected and the impurity scattering rate taken to beΓ0≡1debate in HF superconductors,the possibilities are being constrained.In this work on UPd2Al3,we see a clear gapless behaviour which suggests the presence of a line of nodes, consistent with NMR results and calculations based on resonant impurity scattering.We are grateful to M.R.Norman for his calculations.This work was funded by NSERC of Canada and FCAR of Qu´e bec.We acknowledge the support of the Alexander McFee Foundation(M.C.),the Canadian Institute for Advanced Research(L.T.)and the A.P. Sloan Foundation(L.T.).REFERENCES[1]C.Geibel et al.,Physica B186-188(1993)188.[2]M.Kyogaku et al.,J.Phys.Soc.Jpn62(1996)4016.[3]A.Amato et al.,Z.Phys.B86(1992)159.[4]T.Petersen et al.,Physica B199&200(1994)151.[5]R.Caspary et al.,Phys.Rev.Lett.71(1993)2146.[6]R.Modler et al.,Physica B186-188(1993)294.[7]K.Behnia et al.,Physica C196(1992)57.[8]P.H.Kes et al.,J.Low Temp.Phys.17(1974)341.[9]M.R.Norman and P.J.Hirschfeld,Phys.Rev.B53(1996)5706.[10]M.J.Graf et al.,J.Low Temp.Phys.102(1996)367.[11]M.R.Norman,private communication.[12]Y.Inada et al.,Physica B119&200(1994)119.[13]B.Lussier et al.,Phys.Rev.B53(1996)5145.FIGURESFIG.1.Thermal conductivity of UPd2Al3,divided by temperature,for H=0(solid circles) and H=4.5T>H c2(0)(open circles).FIG.2.Normalized electronic thermal conductivity(see text).The data(points)are compared with resonant impurity scattering calculations([9])using spherical harmonics for3gaps with a line node:polar(solid line),hybrid I(dashed line)and hybrid II(dotted line)[11].01234567800.20.40.60.81κ/Τ [m W /c m /K 2]T [K]H=0H=4.5 T00.10.20.30.40.50.600.10.20.30.40.5n o r m a l i z e d κe /T T /T c T c =1.86 K Γ0=0.3T c I p o l a rII。
Superconductivity in the PbO-type Structure α–FeSeFong-Chi Hsu1,2, Jiu-Yong Luo2, Kuo-Wei Yeh2, Ta-Kun Chen2, Tzu-Wen Huang2, Phillip M. Wu3, Yong-Chi Lee2, Yi-Lin Huang2, Yan-Yi Chu1,2, Der-Chung Yan2, and Maw-Kuen Wu21Department of Materials Science and Engineering, National Tsing Hua University, Hsinchu, Taiwan2Institute of Physics, Academia Sinica, Nankang, Taipei, Taiwan3Department of Physics, Duke University, Durham, North Carolina, USAThe recent discovery of superconductivity with relatively high transition temperature T c in the layered iron-based quaternary oxypnictides1 La[ O1-x F x] FeAs was a real surprise. The excitement generated can be seen by the number of subsequent works2-8 published within a very short period of time. Although there exists superconductivity in alloy9 that contains Fe element, LaOM Pn (with M= Fe, Ni; and Pn=P and As) is the first system where Fe-element plays the key role to the occurrence of superconductivity. LaOM Pn has a layered crystal structure with an Fe-based plane. It is quite natural to ask whether there exists other Fe based planar compounds that exhibit superconductivity. Here we report the observation of superconductivity with zero resistance transition temperature at 8K in the PbO-type α–FeSe compound. Although FeSe has been studied quite extensively10,11, a key observation is that the clean superconducting phase exists only in those samples prepared with intentional Se deficiency. What is truly striking, is that this compound has the same, perhaps simpler, planar crystal sublattice as the layered oxypnictides. Furthermore, FeSe is, compared with LaOFeAs, much easier to handle and fabricate. In view of the abundance of compounds with PbO typestructure, this result opens a new route to the search for unconventional superconductors.FeSe comes in several phases: 1) a tetragonal phase α-FeSe with PbO-structure 2) a NiAs-type β phase with a wide range of homogeneity showing a transformation from hexagonal to monoclinic symmetry 3) an FeSe2 phase which has the orthorhombic marcasite structure. The most studied of these compounds are the hexagonal Fe7Se8, which is a ferrimagnet with Curie temperature at ~125K, and monoclinic Fe3Se4.Unlike the high temperature (high-Tc) superconductors12 discovered more than 20 years ago, which has a CuO2 plane that is essential for the observed superconductivity, the tetragonal phase α-FeSe with PbO-structure has an Fe based planar sublattice equivalent to the layered iron-based quaternary oxypnictides, which has a layered crystal structure belonging to the P4/nmm space group1. The crystal of α-FeSe is composed of a stack of edge-sharing FeSe4-tetrahedra layer by layer, as shown schematically in Fig. 1. Polycrystalline samples with nominal concentration FeSe1-x (x=0.03 and 0.18) were synthesized and studied. X-ray diffraction analysis of the samples in Fig. 2 shows that α-FeSe is dominant and β-FeSe phases exist in trace amounts.The calculated lattice constants are a = 0.37693(1) nm and c = 0.54861(2) nm for FeSe0.82, and a = 0.37676(2) nm and c = 0. 54847(1)nm for FeSe0.88. The lattice constant slightly expands in the a-axis and shrinks in the c-axis for both samples as compared to those of α-FeSe in the JCPDS Card13 (a = 0.3765 nm and c = 0.5518 nm). This lattice distortion should be most probably due to the deficiency of Se, and this gives a quick estimate of the difference in selenium concentration between FeSe0.82 and FeSe0.88.Figure 1 | Schematic crystal structure of α-FeSe . Four unit cells are shown to reveal the layered structure. Sample preparation was done in the following way: high purity (99%) powder of appropriate selenium and iron stoichiometry (FeSe 1-x with x ~ 0.03 – 0.18) nominal compositions were mixed and grounded. The grounded powder was cold-pressed into discs with 400 kg/cm 2 uniaxial force, then the discs were sealed in an evacuated quartz tube and heat treated at 700 ℃ for 24hrs. The sample was reground and sintered again at 700 ℃ for 24hrs, then annealed at 400 ℃ for 36hrs. All the samples were kept in vacuum desiccators prior to measurement.10203040506010k100k?∗∗∗∗∗∗ββFeSe 0.88∗FeSe 0.82I n t e n s i t y (a r b .u n i t s )2q (degr ees)Figure 2 | Powder X-ray Diffraction patterns of FeSe0.82 and FeSe0.88. Phase identification of the sample was carried out with X-ray (Cu, Kα=1.5418Å) radiation in a Philip PW3040/60 diffractometer. The patterns show that the resulting sample with starting composition of Fe(53%)Se(47%) composes of primarily PbO-type tetragonal FeSe1-x (P4/nmm), the α phase, and partly ofNiAs-type hexagonal FeSe (P63/mmc), the β phase. The sample with higher initial iron content, Fe(55%)Se(45%), shows no β phase but trace amounts of possible impurity phases including elemental selenium, iron oxide and iron silicide (marked as *). Question marks in the figure represent unknown phases. This is reasonable since in the Fe-Se binary alloy system, the α phase is considered as a slightly Se-deficient phase (45 to 49.4 at.% Se) and the βphase in contrast persists in a wide range of compositions from slightly Fe-richto Se-rich (49.5 to 58 at.% Se)13. In FeSe0.82, the possible iron oxide impurity phases could come from either starting materials (99.9% Fe) or surface oxidation during sintering, and the silicides might be the product of reactions between the sample and silica ampoules. Nevertheless, the samples contained only trace amounts of these impurity phases (note that the y-axis is in log scale).Figure 3 displays the temperature dependence of electrical resistivity (ρ) ofFeSe0.88. The resistivity shows a broad bump at about 250K and exhibits metallic characteristics before the onset of the superconducting transition. The room temperature to residual resistance ratio is ~ 6. Interestingly, an anomalous downturn in resistivity at about 100K is observed. As the temperature is further lowered, the zero resistance transition occurs at about 8 K.0.00.51.01.52.0ρ (m Ω−c m )T (K)Figure 3 | Temperature dependence of electrical resistivity (ρ) of FeSe 0.88. The sample resistance was measured using the standard 4-probe method using silver paste for contact. Measurements were carried out with a Quantum Design Physical Property Measurement System - model 6000 (PPMS). Superconducting critical transition temperature (Tc) in variety of applied magnetic field up to 9T was determined by taking the temperature where the resistance drops to 50% of that at the onset (50%Rn). The transition width is rather broad suggesting the inhomogeneous nature of the sample. The left inset shows the resistivemeasurement in magnetic fields (H) of 0T, 1T, 3T, 5T, 7T and 9T below 12 K. Tc decreases linearly with increasing magnetic field. The right inset displays the temperature dependence of upper critical field (H c2), and the experimental result was fit to the relationship, ()222)0(/1)0(/C C C T T H H −=, shown in blue line. Theestimated upper critical field H c2 (0) is about 16.3 Tesla; that gives a coherence length ξ0 ~ 4.5 nm.The magnetic susceptibility as a function of temperature was measured at 30 Gauss field strength, as shown in Fig. 4a. The zero field cool (ZFC) susceptibility is essentiallytemperature independent, indicating the sample is a Pauli paramagnet before the onset of superconductivity. A sharp drop indicating the magnetic onset of superconductivity appears at ~8K, which is the same as the zero resistance temperature. The susceptibility shows a relatively large Pauli susceptibility in the normal state. This relatively large positive background above Tc can possibly be attributed to the existence of trace of Fe impurity. Further confirmation of superconductivity is shown in the inset of Figure 4a, which displays the typical magnetic hysteresis curve for a superconductor. A relatively small magnetic anomaly occurs at about 105K, which is more pronounced in field-cool measurement. This magnetic anomaly occurs at the same temperature as that observed in the resistive measurement. Figure 4b shows the specific heat of the FeSe0.88 sample. In the normal state, the derived electronic coefficient of specific heat γ=9.17 mJ/mol-K2. The superconducting transition at Tc is relatively broad and the estimated ΔCp/γT c is ~ 0.61. In the superconducting state, the best fit of data over the temperature range 1 < Tc/T < 1.8 leads to C es /γT c = 7.69 exp(−1.58 Tc/T). But it does not follow the BCS relation over the full temperature range below Tc, as shown in the inset of figure 4b. This result may originate from the presence of impurity phases in the final product. However, it may also imply that the FeSe1-x is possibly an unconventional superconductor.-2-11χ (10-2e m u /c m 3-G )T (K)Figure 4a | Temperature dependence of magnetic susceptibility measured in 30G magnetic field. DC magnetic susceptibility measurements were performed in a Quantum Design superconducting quantum interference device vibrating sample magnetometer (SQUID-VSM). FeSe 0.88 powder sample is measured in two ways: 1) the sample is cooled without an initial external magnetic field applied (ZFC, open squares); and 2) then cooled in an initial external magnetic field (FC, solid squares). After initializing, a 30G magnetic field is applied and the susceptibility is measured as a function of temperature. Inset shows the magnetic hysteresis of the sample measured at 2K. It confirms the superconducting characteristic of the sample. A small magnetic anomaly is observed at about 105K, which is more pronounced in the FC measurements.C p /T (m J /m o l -K 2)Τ2(K 2)Figure 4b Ι Low temperature specific heat of FeSe 0.88. The measurementwas carried out with thermal relaxation method in zero magnetic field. The red dotted line is the curve fitting of phonon and electronic contribution to the specific heat. The intercept at zero temperature gives γ = 9.17 mJ/mole-K 2. A specific jump appears at ~ 8K, which coincides with the zero resistancetemperature that confirms the superconducting transition. The inset shows the semi-logarithmic Ces /γTc vs. Tc /T in the superconducting state. The measured plot displays deviation from linear curve of fully gapped superconductor (solid blue straight line).Recently, it was reported 14,15 that there exists a pseudogap in La(O-F)FeAs, similar to that of the high Tc cuprates, based on the temperature dependent laser photoemission spectroscopy (PES). On the other hand, Mossbauer experiments 16 suggested that the suppression of magnetic and structural transition by F-doping in the LaOFeAs system may play the key role to the observation of superconductivity. It is possible that the anomaly observed in both the magnetic and resistive measurements in the FeSe 1-xsystem may have similar origin. In fact, high resolution X-ray diffraction experiments at low temperature indeed show the presence of a structural transformation at ~ 105K. Figure 5 shows the diffraction patterns of the sample at different temperatures. Clear splitting of the diffraction peaks at ~105K is observed. Detailed refinement of thediffraction data suggests that the crystal structure changes from the tetragonal (P4/nmm)symmetry to the triclinic (P-1) symmetry. Whether this observed structural change at low temperature is related to the appearance of superconductivity is currently under intensive investigation.4.7 4.8 4.90.10.22468I n t e n s i t y (a r b . u n i t s )M o m e n tu m tra n s fe r (10/n m )Figure 5 | Temperature dependence of the high resolution X-ray powder diffractions for FeSe0.88 by synchrotron source at BL12b2 in SPring 8 with incident beam wavelength of 0.995 Å. The moment transfer equals toM=4πsin(θ)/λ. (a) Observed (open black circle) and calculated (red solid line) powder diffraction intensities of FeSe0.88 at 300K using space groupP4/nmm. The inset in (a) shows a single peak of the (2, 2, 0), (2, 0, 3), (2, 2, 1) reflection at room temperature. But double peaks show up for (2, 2, 0) and (2, 2, 1) at low temperature, as seen in (b). The double peak structure begins to show up at ~105K. The lattice parameters at 6K change to a = 0.3773 nm, b = 0.3777nm, c = 0.5503 nm; both angles α and β remain to be 90 degree but γ angle increases from 90 degree to ~ 90.3 degree. Inset in (b) shows the temperature dependence of γangle fit with P-1 symmetry.1. Kamihara, Y., Watanabe, T., Hirano, M. and Hosono, H. Iron-based layered superconductor La[ O1-x F x] FeAs (x = 0.05–0.12) with T c = 26 K. J. Am. Chem. Soc. 130, 3296-3297 (2008).2. Takahashi, H. Superconductivity at 43 K in an iron-based layered compound LaO1-F x FeAs. Nature, 453, 376–378 (2008).x3. Chen, G. F. et al. Superconductivity at 41 K and its competition with spin-density-wave instability in layered CeO1−x F x FeAs. Preprinted at http://arXiv:0803.3790v3 (2008).4. Chen, X.H. et al. Superconductivity at 43 K in samarium-arsenide oxides SmFeAsO1−x F x. Preprinted at <http://arXiv: 0803.3603v2> (2008).5. Cheng, P. et al. Superconductivity at 36 K in gadolinium-arsenide oxidesGdO1−x F x FeAs. Preprinted at <http://arXiv: 0804.0835v2> (2008).6. Wen, H. H. et.al. Superconductivity at 25K in hole-doped (La1-x Sr x)OFeAs. Europhys. Lett.82, 17009 (2008).7. Yang, J. et al. Superconductivity at 53.5 K in GdFeAsO1-δ. Preprinted at<http://arXiv:0804.3727v3> (2008).8. Ren, Z.A. et al. Superconductivity at 55 K in iron-based F-doped layered quaternary compound Sm[O1-x F x]FeAs. Preprinted at <http://arXiv: 0804.2053v2> (2008).9. Unruh, K.M. and Chien, C.L., Phys. Rev. B. 30, 4968-4974 (1984).10. Terzieff, P. & Komarek, K. L. The paramagnetic properties of iron selenides with NiAs-type structure. Monatsheftef für Chemie109, 651-659 (1978).11. Schuster, W. MiMer, H. and Komarek, K L. Transition metal--chalcogen systems, VII.: the iron-selenium phase diagram. Monatshefte für Chemie110, 1153-1170 (1979).12. Wu, M. K. et al. Superconductivity at 93K in a new mixed phase Y-Ba-Cu-O compound system at ambient pressure. Phys. Rev. Lett.58, 908-910 (1987).13. Okamoto, H. The Fe-Se (iron-selenium) system. J. Phase Equilibria 12(3), 383-389 (1991).14. Ishida, Y. et al. Evidence for pseudogap evolutions in high-Tc iron oxypnictides Preprinted at <http://arXiv:0805.2647v1> (2008).15. Sato, T. et al. Superconducting gap and pseudogap in iron-based layered superconductor La(O1−x F x)FeAs.Preprinted at <http://arXiv:0805.3001v> (2008).16. Kitao, S. et al. Spin ordering in LaOFeAs and its suppression in superconductor LaO0.89F0.11FeAs probed by Mössbauer spectroscopy. J. Phys. Soc. Jpn (submitted) (2008).Acknowledgements The authors like to acknowledge fruitful discussions with Prof. T.K. Lee and Prof. Sungkit Yip. We also thank the National Science Council of Taiwan and the US AFOSR through its Tokyo Office AOARD for their generous financial support.。
c o nd -m a t /9506043 15 J u n 1995HEAVY FERMION SUPERCONDUCTIVITYbyRobert H. HeffnerMaterials Science and Technology Division, MS K764Los Alamos National LaboratoryLos Alamos , N. M. 87545andMichael R. NormanMaterials Science DivisionArgonne National LaboratoryArgonne, IL 60439ContentsI. Introduction (3)II. Normal-State Properties of Heavy Fermions (3)III. Superconducting Order Parameter (6)IV. Superconducting Properties of heavy fermionsA. General Thermodynamic Properties (7)B. Experimental Evidence for Unconventional Superconductivity (8)C. Interplay of Magnetism and Superconductivity (12)C1. Competition (13)C2. Coexistence (14)C3. Coupling (15)V. Magnetism in Heavy Fermion Superconductors (16)A. Magnetism Originating Below TcB. Magnetism Originating Above T (18)cVI. Phenomenological Models for the Superconducting Order Parameter (19)VII. Pairing Mechanisms (28)I. IntroductionThe study of heavy fermion (HF) superconductivity began about 16 years ago with the discovery of superconductivity in CeCu2Si2[1]. At that time the major mystery was how superconductivity could be supported in a system with strong local moments on the Ce ions. Since that time the field of HF superconductivity has expanded to include five (possibly six) new uranium-based superconductors at ambient pressure and one new rare-earth-based superconductor at elevated pressures. Collectively, these materials display a rich variety of unexpected properties, which have enhanced our knowledge, not only of superconductivity, but also of the general behavior of highly correlated electrons in metals. It is interesting to note that the original question posed above for the discovery of CeCu2Si2, namely the coexistence of magnetism and superconductivity in HF materials, remains a central challenge for the understanding of these materials today. This paper addresses the question of how HF superconductors are different from conventional superconductors like Al and Pb and how these differences have enhanced our overall understanding of condensed matter physics. The topic is addressed within the context of four general areas: (1) normal-state properties, (2) the superconducting order parameter, (3) the interplay of magnetism and superconductivity, and (4) the superconducting pairing mechanism. The scope and depth of this article is intended to provide an easy, but stimulating experience for readers familiar with the basics of superconductivity. More comprehensive reviews of HFs has been given by Grewe and Steglich, Ott and Fisk, and Hess, Riseborough and Smith[1]. Normal state properties were reviewed in an earlier article in this series by Lawrence and Mills [2].II. Normal-State PropertiesThe starting point for understanding superconductivity in a material must be a study of the normal ground state from which the superconducting state emerges. Here we emphasize those normal-state properties (given in Table I) which are most useful in establishing a picture of the superconducting state.The high-temperature ground state in HFs is reasonably well understood on the basis of extensions of single-ion Kondo theory [2]. How the variety of low temperature ground states (paramagnetic, magnetic, semiconducting and superconducting) evolve from the high-temperaturenormal state is only partially understood, however, although the following considerations are generally agreed upon. There is a crossover from local-moment behavior at high temperatures to a reduced-moment regime at low temperatures, where the f-electron moments are reduced to a fraction of their high temperature values. This compensation of the high-temperature f-moments occurs through an antiferromagnetic exchange interaction which produces a virtual bound state between the f-moments and the conduction electrons. In dilute metals this is the well-known single-ion Kondo interaction. Although this crossover from full- to reduced-moment behavior does not involve a phase transition, it can still be characterized by a coherence temperature T coh≈ 1-100K. Whether or not T coh is the only relevant energy scale at low temperatures in HFs is a matter of current investigation [3].The strong exchange coupling between the conduction electrons and the f-electrons also leads to a large resistivity in HF materials, typically several hundred µΩ-cm. at room temperature. The resistivity increases as the temperature is decreased below room temperature, reaching a maximum at T max and then dropping drastically below T max, indicating the loss of inelastic scattering and the onset of the HF state. The normal-state resistivity at temperatures just above the superconducting transition temperature T c is typically only a few µΩ-cm. At T = T coh << T max, if no magnetic ordering occurs, the system enters a Fermi liquid-like state with strongly renormalized quasiparticles of large effective mass m* ≈ 100 m e. This state is characterized by a large linear coefficient of specific heat γ, indicating a high density of states at the Fermi surface, and a resistivity which varies as T2.The dc susceptibility in HFs is dominated by the f-electrons and also exhibits a maximum in temperature, below which it reaches a constant value if the f-moments are fully compensated. The zero-temperature values of χ and γ for the family of HF materials appear to scale linearly with one another. It is interesting that among the HF materials the superconductors come closest to approaching the free-electron value (=1) for the Wilson scaling ratio R given byR = [χ(0) π2k B2] / [γ(0)J(J+1)g J2µB2].Magnetic ordering in HF materials has almost always been found to be antiferromagnetic and occurs with moments reduced considerably over the free ion value (Table I), as expected from the screening by the conduction electrons. There appear to be four categories of magnetism. The first class(U2Zn17, UCu5, UPd2Al3, UNi2Al3, NpSn3) has moments ranging from 0.2 to 1.5 µB and appears to be composed of typical magnets with a standard dipolar order parameter. The second class (URu2Si2, UPt3, UPd3) has small moments (0.01 to 0.03µB ). This class is poorly understood at this time because, despite the possession of small moments, these materials still have Neel temperatures comparable to those in the first class. The most popular current model to explain these compounds is that the primary order parameter is quadrupolar (or a more exotic variant), with a weak induced dipole moment. The problem with this is that, so far, neutron scattering experiments have only detected the dipolar part. The third class of HF magnets contains those materials with extremely weak magnetism (0.001µB ) which is connected with a. The most likely second superconducting phase transition, as seen in UPt3 and Th-doped UBe13scenario to explain these small moments is a time-reversal-breaking superconducting order parameter, but other possibilities exist as discussed below. Finally, possible “spin glass,” or random magnetic order, has been observed in CeCu2Si2 and CeAl3, constituting a fourth class of HF magnetic materials.Whether HFs exhibit an antiferromagnetic or paramagnetic ground state depends upon a competition between the local on-site exchange interaction, which compensates the local f-moment, and the non-local long-range f-f interaction, which gives rise to magnetic order through the RKKY interaction. The intra- and inter-site interactions are related through a common conduction-electron local-moment exchange coupling. Pethick and Pines [4] have shown that the competition between these two interactions may give rise to the coherence energy scale T coh. In general, if T coh is very small one has local-moment behavior; by contrast, a large T coh favors a paramagnetic ground state. Even in the paramagnetic state, however, neutron scattering experiments [5] have demonstrated that HFs exhibit strong antiferromagnetic (AFM) spin correlations between f-moments.It is evident that the normal-state properties of HF materials are very different from conventional intermetallic compounds. The principal differences discussed here probably result from a delicate balance between competing interactions, giving rise to a highly-correlated ground state which at low temperatures exhibits a small resistivity, a large effective mass, strong AFM spin correlations and highly-reduced ordered moments. As discussed below, these properties have profound implications for the superconducting state. The reason that only a relatively small family of materials exhibit these unusual characteristics resultsfrom a combination of the strong exchange interaction and an optimum f-ligand and/or f-f atom spacing which tunes the competition between the inter- and intra-site couplings.III. Superconducting Order ParameterA superconductor like Al is very well described by the well-known BCS model of superconductivity, originally constructed for a weak pairing interaction between electrons in a relative zero angular momentum, spin-singlet state. This implies a spherically symmetric order parameter ∆(k), which is the pair amplitude for electrons of momentum k. In general, because of crystalline inversion symmetry,∆(k) can be written separately for an even or odd parity pairing state [6]:∆ (k) = ψ(k) i σy(1a)∆ (k) = (σ•d(k)) i σy,(1b)respectively. Here ψ(k) and d(k) are even scalar and odd vector functions of the momentum k and σ is the Pauli spin matrix. (In the presence of strong spin-orbit coupling spin is replaced by pseudo-spin, as discussed in Section VII below.) The BCS state in Al is described by Equation (1a). The symmetry operations of the Hamiltonian include translation symmetry, inversion symmetry (parity), point group symmetry, time-reversal symmetry and gauge symmetry. When a superconducting transition occurs only gauge symmetry, which sets the overall phase of the order parameter, is necessarily broken. It is customary to refer to an "unconventional" superconducting state as one possessing broken point group symmetry, having odd parity or violating time-reversal symmetry. The latter occurs when the superconducting state possesses either orbital or spin moments. In addition, if the Cooper pairs have finite center-of-mass momentum, an additional translational symmetry will be broken. Finally, if one has a triplet superconductor (Eqn. 1b) and no spin-orbit coupling, then another symmetry connected with the rotation group of the spins can be broken. This leads to a large number of possibilities. These points are discussed in greater detail in Section VI below.When ∆(k) vanishes for certain k-vectors one measures specific power-law temperature dependencies for all experiments involving the thermal excitation of quasiparticles near the nodes of ∆ (k) at T << T c. This is illustrated in Table II for polar and axial gap structures on a spherical Fermi surface. A gap which is non-vanishing for all k will yield an exponential temperature dependence. As discussed below, the observation of power-law temperature dependencies below T c is not conclusive evidence for an unconventional gap structure, however, because gapless superconductors can also exhibit power-law behaviors. More stringent tests for unconventional superconductivity involve the detection of more than one superconducting phase, an inherently magnetic superconducting state (violating time reversal symmetry) or anisotropy in the critical fields, transverse ultrasound, thermal conductivity or penetration depth, for example [7].IV. Superconducting Properties of Heavy FermionsA. General Thermodynamic PropertiesThe six known HF materials which are superconducting at ambient pressure are shown in Table III, together with some of their relevant properties. All are type II superconductors. Also included for comparison are two elemental superconductors, Al and Nb, and an A-15 compound V3Si. A seventh HF superconductor, CeCu2Ge2, has been reported to be superconducting above 70 kbar pressure by Jaccard et al.[8]. In addition, U2PtC2 is probably a HF superconductor [9].Several trends are apparent from Table III. All of the HF superconductors have small transition temperatures (< 2K). In addition, the HFs have magnetic field penetration depths λ in excess of several thousand Å and coherence lengths ξ of the order of 100-200 Å. This is in contrast to the conventional elemental superconductors which have significantly lower values of λ and higher values of ξ. These differences are a direct consequence of the very large effective masses for the HFs (Table I), as can be seen by examining the expressions for λ and ξ (taken from conventional theories of superconductivity): λ∝ (m*/n s)1/2 and ξ∝n1/3/(m*T c), where n is the conduction electron density and n s is the superconducting pair density. This leads to anomalously small values of H c1∝ 1/λ2 and large values of H c2∝ 1/ξ2 when compared to the thermodynamic critical fields H c of the elemental type Isuperconductors. Note that if one calculates H c = (H c1H c2)1/2 for the HF superconductors, however, the values are not anomalously large compared to the elemental superconductors. Also, the H c2 values for the HFs are rather small compared to the A-15 superconductors like V3Si (H c2 = 19T) or to the copper-oxide superconductors, where H c2 > 100T. The small coherence length for V3Si derives from a moderately large effective mass and a large T c. (A discussion of the unusual properties of the A-15 materials is beyond the scope of this article. The example of V3Si is included here only for comparison.) Finally, the HF superconductors exhibit rather conventional specific heat jumps ∆C at T c compared to Al, where the BCS value for β = ∆C/γ(T c)T c is 1.43. (An exception is UBe13, where β can be up to 2.6 for single crystals.) The parameter β is a measure of the pair coupling strength. Thus the uniqueness of HF superconductors cannot be established from the magnitudes of their thermodynamic parameters alone.B. Experimental Evidence for Unconventional SuperconductivityThe earliest evidence for an unconventional gap structure in HF superconductors comes from the measurement of power-law temperature dependencies for the resistivity, specific heat, sound attenuation and NMR spin-lattice-relaxation rate below T c [1]. In no HF superconductors did these early measurements yield the exponential behavior expected for a gap without nodes on the Fermi surface. The problem is that these experimental data, while suggesting an anisotropic gap structure, do not uniqely determine the gap symmetry. One reason for this is that specific power laws (Table III) are valid only far below T c where impurities can obscure the intrinsic behavior. Another is that certain measurements may be dominated by one feature of the gap structure, such as a line of nodes. This could explain why the NMR spin-lattice-relaxation rates show a T3 behavior over a wide temperature range below T c for all of the HF superconductors. If point nodes were also to exist (as postulated for UPt3), for example, they may not affect the NMR rates because of a smaller phase space and because the experiments are generally performed on powder samples where directional information can be lost.Another indication of unconventional superconductivity in HFs occurs when dilute concentrations of substitutional impurities are introduced. In conventional s-wave superconductors, magnetic impurities with stable local moments (such as Gd) cause exchange scattering which breaks the Cooper pairing andyields a universal curve for T c vs. the impurity concentration or for the specific heat jump ∆C/∆C o vs T c/T co. This effect is well described by the theory of Abrikosov and Gor'kov [10]. By contrast, dilute concentrations of non-magnetic impurities in s-wave superconductors have little effect on the superconducting parameters. This is not the case for HF superconductors, however. In these systems non-magnetic impurities (such as Y, La or Th) are nearly as effective at breaking pairs as are magnetic impurities like Gd. This has been interpreted as evidence for non s-wave pairing, because non-magnetic impurity scattering suppresses an anisotropic Cooper pair wave function. When an impurity such as Gd is introduced into an anisotropic superconductor, magnetic impurity scattering produces the additional pair breaking described by Abrikosov and Gor’kov.Although the power-law behaviors and sensitivity to non-magnetic impurities are indicators of an anisotropic pairing state in HFs, the first definitive evidence for unconventional superconductivity in undoped HF systems came only recently with the discovery of two superconducting transitions in the specific heat of UPt3 by Fisher et al.[11], as shown in Fig. 1. The earliest specific heat measurements in UPt3 missed this small splitting (∆T ≈ 0.05K), apparently because of sample quality and less refined annealing procedures. This splitting of the specific heat jump in UPt3 was first discovered in zero applied field. Subsequent studies, most notably changes in ultrasonic velocity in applied fields of several Tesla or more by Bruls et al. and Adenwalla et al. [12], have revealed a rich field-temperature phase diagram for UPt3, displaying at least three different superconducting phases and a tetracritical point at ambient pressures. Surprisingly, it was found that the phase diagram is not very sensitive to field orientation. In particular, the same tetracritical point exists for fields along the c-axis, where no kink is observed in the upper critical field. Recently, Boukhny et al. [13] have extended the sound velocity measurements to pressures of several kbars, showing how these phases evolve under pressure. The phase diagram for UPtis shown in Figs. 2 and 3, and constitutes prima facia evidence for an unconventional 3superconducting state. The low-field transition from the A to the B phase in UPt3 at atmospheric pressure has recently been shown with muon-spin-relaxation (µSR) measurements by Luke et al. [14] to be accompanied by the onset of weak magnetic (µ≈ 0.001 µB) correlations. In addition, Keller et al. [15] have found a six-fold anisotropy in the basal-plane magnetoresistance of UPt3, and Broholm et al. [16] usedµSR to measure a different temperature dependence for the magnetic field penetration depth λ, depending upon the orientation of the applied field. These penetration depth measurements are consistent with previous transverse ultrasound studies by Shivaram et al. [17], and indicate line nodes perpendicular to the c-axis. Finally, Kleiman et al. [18] have observed the hexagonal vortex lattice in UPt3 using neutron diffraction. The implications of many of these experiments for the symmetry of ∆(k) will be discussed in Section VI.Pure UBe13 apparently exhibits a single superconducting phase transition, although there have been reports of structure in H c2(T) below T c by Rauchschwalbe et al. [19], possibly indicating a second phase. No clear evidence for two phases has been seen in the specific heat, however, which sas shown by Ott and collaborators to have a low-temperature behavior proportional to T3 [20]. Einzel et al. [21] have found that the magnetic field penetration depth varies at T2 at low temperatures. These two results taken alone imply an axial state with point nodes, as seen from Table II. MacLaughlin’s NMR relaxation rates are proportional to T3 at low temperatures, however, which taken by itself implies a polar state with a line of nodes [22]. Even though the NMR data were taken in 1.5T field (and therefore may not be comparable to the zero-field specific heat data), this comparison illustrates the problem of using power-law temperature dependencies to determine the gap symmetries. The fact that NMR has yielded a T3 behavior in all HF superconductors, as discussed above, further compounds the problem., T c When either magnetic (Gd) or non-magnetic (La) impurity atoms are substituted for U in UBe13is depressed as expected for a superconductor with an anisotropic order parameter. However, when doped with Th, Ott et al. found that the T c for U1-x Th x Be13 becomes a non-monotonic function of Th concentration x and for 0.019 ≤ x < 0.043 two specific heat transitions occur [23], as seen in Fig. 4. Early ultrasonic attenuation measurements by Batlogg et al. gave an indication for antiferromagnetism at the lower transition [24]. The existence of magnetic correlations was later confirmed in zero-field µSR experiments by Heffner et al. [25], revealing a mean-field like magnetic transition below T c2, with an, the onset of superconductivity, as approximate moment ≤ 0.001 µB. No such signature was found at Tcseen in Fig. 5. In addition, H c1 (T) was found to be proportional to (1-α(T/T c)2) above and below T c2, but with a larger slope α below T c2 than above it [25], implying an increase in the superfluid density asexpected for a second superconducting transition. Finally, Jin et al. [26] showed that the addition of Th to UBe13 can produce a relatively large zero-temperature specific heat coefficient γr (0) of order 0.5 - 1.0 J/mol-K2 , and can change the temperature dependence of the low-temperature specific heat.An important question arises: Is the lower transition in (U, Th)Be13 associated with a transition to a different superconducting state (possibly violating time-reversal symmetry) or is it simply associated with a small-moment AFM transition? Part of the answer lies in the large specific-heat jump at T c2. Simply put, a small local-moment AFM transition is unlikely to produce such a large jump in specific heat because most of the entropy associated with the exchange-coupled conduction-electron/local-moment system would already have been removed in reducing the value of the moments to 0.001µB. If the magnetic order occurs through a SDW transition (a Fermi surface instability, rather than local moment formation), the large specific heat jump would again be unlikely because the Fermi surface would already have been largely consumed by the superconducting transition at TFor a SDW transition, in other words, a large jumpc.would require an exceptional enhancement of the density of states near the zeros of the superconducting energy gap. We thus conclude that the large change in specific heat at T c2 is due primarily to a superconducting, not a magnetic, phase transition. Nevertheless, it is still true that magnetic correlations are also observed to set in at T c2. The possible nature of the magnetism and the implications for the nature of the superconductivity in UBe13 are discussed below in Sections V and VI.Further confirmation for multiple superconducting states in (U,Th)Be13 comes from Lambert’s early measurements of the pressure dependence of T c for various Th concentrations [27]. The functional dT c/dP changes from about -16 mK/kbar for 0.0 < x < 0.017 to about -50 mK/kbar for 0.019 < x < 0.026, indicating different superconducting phases for different range of Th concentrations. It was also found that higher pressures for 0.019 < x < 0.026 seemed to restore the phase seen for x < 0.017.The combined specific heat, critical field and µSR studies led to the phase diagram [25] shown in Fig. 6 for (U, Th)Be13. The vertical dotted lines were not actually been observed, however, but only postulated from measurements for various Th concentrations, differing by several tenths of a per cent Th. As discussed below, Th tends to act a as “negative pressure” when substituted in UBe13. Recently, Zieve et al. [28] measured the specific heat as a function of pressure for x = 0.022 and confirmed the existenceof the nearly vertical phase boundary near x = 0.019 in the T vs. x phase diagram. Here the pressure was tuned to achieve an equivalent resolution in Th concentration of 0.0002. They found that the relative specific heat jump ∆C/C decreases by about 10% in going across the A → C phase boundary at constant temperature (Figure 6). This is to be contrasted with an increase in ∆C/C of about 50% in decreasing the temperature at constant pressure and traversing the B → C phase boundary.Thus, both UPt3 and (U,Th)Be13 exhibit multiple superconducting transitions, providing unambiguous evidence for unconventional superconductivity in these HF systems. The origins of the magnetism below T c and the symmetry of the superconducting states in these systems are discussed in Sections V and VI below.C. Interplay of Magnetism and SuperconductivityIt is evident from the above discussion of the multiple superconducting phases of UPt3 and (U,Th)Be13 that magnetism and superconductivity seem to be intimately related in HF systems. Indeed, where magnetic or superconducting phase transitions are found in HFs, it is clear from the entropy balance above and below the transition that the heavy quasiparticles themselves (ie., the hybridized f-electron/conduction-electron states) form the ordered magnetic or superconducting state. This fact is in contradistinction to a previous class of "magnetic superconductors," the moly-sulfides and rare-earth rhodium borides, where the magnetism is carried by the local rare-earth moments but the superconductivity occurs through separate conduction-electron bands which interact only weakly with the local moments. It is also worth noting at this point that there is growing evidence, discussed below, that magnetic spin fluctuations (paramagnons) may provide the dominant pairing mechanism in HF superconductors.This interplay between magnetism and superconductivity in HF materials has shown considerable variety when viewed form the point of view of the competition, coexistence and/or coupling of the magnetic and superconducting order parameters [29]. As seen from Table I magnetism can occur both above and below the superconducting transition temperature.C1. CompetitionIn CeCu2Si2, the only rare-earth-based HF superconductor at ambient pressure, spontaneous magnetism occurs slightly above T c≅ 0.7K and exists along with superconductivity in zero-applied field. In non-zero fields the magnetism persists above H c2 for temperatures below 0.7K, as seen in the phase diagrams formulated by Nakamura, Wolf and collaborators [30]. Only one superconducting phase has been established. Experiments using µSR, specific heat and microstructural analysis by Luke et al. and[31]. Feyerherm et al. provide insight into the interplay of magnetism and superconductivity in CeCu2Si2The zero-field µSR relaxation function in CeCu2.05Si2 exhibits a two-component structure below 1.3K, where magnetism sets in, one component of which can be attributed to paramagnetic domains and the other to magnetic domains. The magnetic component exhibits a Gaussian form, characteristic of either a spin-glass-like state or an incommensurate SDW. This segregation occurs in an otherwise homogeneous sample. It is therefore possible to track the temperature dependence of the volume fractions of the magnetic and non-magnetic portions of the sample, as shown in Figure 7. The magnetic fraction increases below 1.3K, reaching a maximum of about 3/4 at T c. Below T c the magnetic fraction decreases to about 1/3 at T < 0.05K, while the superconducting (paramagnetic) volume fraction increases to about 2/3. Corroborating evidence for this effect is found in Nakamura’s NMR measurements [32] on CeCu2.02Si2, where the NMR intensity decreases strongly below about 0.9K due to the onset of magnetic correlations which produce a broad distribution of hyperfine fields. This decrease in intensity stops at T c, where the µSR measurements show that the magnetic volume fraction ceases to increase. Also, the height of the specific heat jump at T c correlates well with the fraction of the sample which is superconducting, as obtained from the µSR measurements. This effect has been studied in several different samples with varying Cu stoichiometries, where Cu excess tends to favor superconductivity. In each case the magnetic relaxation rate is unaffected by the onset of superconductivity and is relatively independent of the sample, indicating a common origin from one sample to the next.Thus in CeCu2Si2 there appears to be a competition between magnetism and superconductivity, each existing in its own domains. This effect is unique among HF superconductors (as discussed below) and may be related to the fact that the existence of magnetism and/or superconductivity in CeCu2Si2 is。