Superconductivity in a two-dimensional Electron Gas
Quantum spin liquid emerging in 2D correlated Dirac fermions
Superconducting qubits II Decoherence
The transition from quantum to classical physics, now known as decoherence, has intrigued physicists since the formulation of quantum mechanics (Giulini et al., 1996; Leggett, 2002; Peres, 1993; Feynman and Vernon, 1963; Zurek, 1993). It has been put into the poignant Schr¨ odinger cat paradox (Schr¨ odinger, 1935) and was considered an open fundamental question for a long time.
and compare it to the corresponding classical mixture leading to the same expectation value of σz 1 1 0 ρmix = (2) 2 0 1 we can see that the von-Neumann entropy ρ = −kB Tr [ρ log ρ] rises from Spure = 0 to Smix = kB ln 2. Hence, decoherence taking ρpure to ρmix creates entropy and is irreversible. Quantum mechanics, on the other hand, is always reversible. It can be shown, that any isolated quantum system is described by the Liouville von-Neumann equation i¯ hρ ˙ = [H, ρ] (3)
双层硼烯的电输运和热输运性质的研究
摘要随着2004年石墨烯的问世，人们不断地设计和制备出新型的二维材料，但只有极少数是单质二维材料，而且大多数结构都存在离面屈曲，目前只有硼烯存在与石墨烯一样的纯平面结构。
随着硼烯的成功制备，人们通过研究发现该材料在力学、电学、光学、和热输运等方面具有优异的性能。
硼烯在力学性能方面是集坚、柔、韧于一体的二维材料；硼烯目前是质量最轻、厚度最薄的金属；在硼烯的β12结构结构中费米能级附近发现狄拉克锥。
由于其强的电声耦合，让其具有声子调节和χ3的高温超导性，高的能量存储容量以及近可见光波段的等离子激发，硼烯可广泛应用于柔性电子器件、光控器件、锂离子和钠离子电池、储氢以及复合材料设计等方面。
本文基于第一性原理和非平衡格林函数方法，首先得到了双层硼烯的新结构，分析了该结构的热输运和电输运，并基于半导体性的硼烯的无结晶体管，分析电输运性质。
主要内容具体如下：（1）得到了双层硼烯的新结构。
我们以实验制备出的硼烯β12单层结构为基础，进行双层结构研究，发现靠范德瓦尔斯力结合的双层硼烯不稳定，色散谱中有虚频；从而缩小双层距离发现了靠B-B共价键结合的稳定的双层硼烯，具有AA/AB 两种堆垛方式，从能量角度分析AB结构比AA结构更稳定。
另外，我们构建了双层AA和AB结构靠范德瓦尔斯力结合的体结构，计算后发现它们也是稳定的。
（2）研究了双层硼烯的电输运和热输运性质。
电输运性质研究发现β12构成的AB和AA结构的I-V曲线具有欧姆特性，表现出很好的金属性，而且具有很强的各向异性，另外由于AB结构的电子分布更局域，从而当偏压相同时电流相对要小一些。
热输运性质研究发现AB结构和AA结构的热导值相当，虽然AB和AA 结构具有很好的导热性，但比单层β12的热导要小很多，主要是由于低频段的声子谱降低很多。
（3）基于半导体硼烯的无结晶体管的电输运性质。
我们以半导体硼烯为研究对象，通过掺杂形成N型和P型无结晶体管，通过计算不同方向的I-V曲线，结果表明P型无结晶体管沿着α方向传输时性能最好，开关比达到1.2*104，而且可通过加大栅极电压的长度来改善其性能。
Superconductivity at 1 K in Cd2Re2O7
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.。
凝聚态物理相关诺贝尔奖Word版
凝聚态物理相关诺贝尔化学奖1970-凝聚态物理相关诺贝尔物理学奖1970-约翰·巴丁美国美国约翰·罗伯特·施里弗美国日本“发现"for their experimental discoveriesregarding tunneling phenomena insemiconductors and superconductors,respectively"挪威英国“他理论上预测出通过隧道势垒的超电流的性质，特别是那些通常被称为象”"for his theoretical predictions of the properties of a supercurrent through a tunnel barrier, in particular those phenomena which are generally known as the Josephson effect"美国英国美国年彼得·列昂尼多维奇·卡皮苏联美国“对与相转变有关的临界现象理论的贡献”"for his theory for critical phenomena inconnection with phase transitions"克劳斯·冯·克利青德国“发现"for the discovery of thequantized Hall effect"德国德国瑞士约翰内斯·贝德诺尔茨德国卡尔·米勒瑞士法国美国美国美国若雷斯·阿尔费罗夫俄罗斯“发展了用于高速电子学和半导体异质结构"for developing semiconductorheterostructures used in high-speed-and optoelectronics"德国美国“在发明"for his part in the invention of theintegrated circuit"美国“在碱性原子稀薄气体的聚态质的早期基础性研究”"for the achievement of Bose-Einsteincondensation in dilute gases of alkaliatoms,of美国德国美国俄罗斯“对性贡献”"for pioneering contributions to thesuperfluids"俄罗斯英国美国法国德国英国美国[110]美国美国荷兰俄罗斯“在二维"for groundbreaking experiments regarding the two-dimensional material graphene"康斯坦丁·诺沃肖洛夫英国俄罗斯（注：可编辑下载，若有不当之处，请指正，谢谢!）。
准一维超导体K2Mo3As3中超导电性的再现
Reemergence of superconductivityin pressurized quasi-one-dimensional superconductor K2Mo3As3Cheng Huang1,2*, Jing Guo1,4*, Kang Zhao1,2*, Fan Cui1,2, Shengshan Qin1,2, Qingge Mu1, Yazhou Zhou1, Shu Cai1,2, Chongli Yang1, Sijin Long1,2, Ke Yang3, Aiguo Li3,Qi Wu1, Zhian Ren1,2, Jiangping Hu1,2 and Liling Sun1,2,4†1Institute of Physics and Beijing National Laboratory for Condensed Matter Physics,Chinese Academy ofSciences, Beijing 100190, China2University of Chinese Academy of Sciences, Beijing 100190, China3Shanghai Synchrotron Radiation Facilities, Shanghai Institute of Applied Physics, Chinese Academy of Sciences,Shanghai 201204, China4Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China Here we report a pressure-induced reemergence of superconductivity in recently discovered superconductor K2Mo3As3, which is the first experimental case observed in quasi-one-dimensional superconductors. We find that, after full suppression of the ambient-pressure superconducting (SC-I) state at 8.7 GPa, an intermediary non-superconducting state sets in and prevails to the pressure up to 18.2 GPa, however, above this pressure a new superconducting (SC-II) state appears unexpectedly. High pressure x-ray diffraction measurements demonstrate that the pressure-induced dramatic change of the lattice parameter c contributes mainly to the emergence of the SC-II state. Combined with the theioretical calculations on band strcture, our results suggest that the reemergemce of superconductivity is associated with the change of the complicated interplay among different orbital electrons, driven by the pressure-induced unisotropic change of the lattice.The discoveries of the copper oxide and iron-based high-T c unconventional superconductors have generated considerable interest over the past 30 years due to their unusal superconducting mechanism and great potential for application. A common feature of these superconductors is that they contain d orbital electrons in their quasi-two-dimentional lattice that results in intriguing superconductivity and other exotic properties [1-6]. Recently, a family of CrAs-based superconductors with superconducting transition temperatures (T c) of 2.2 - 8.6 K were found in X2Cr3As3 (X=Na, K, Rb, Cs) compounds, which possess the features of containing d orbital electrons in a quasi-one-dimensional lattice [7-10] and many unconventional properties [11]. Soon after, a new family of quasi-one-dimmensional MoAs-based superconductors Y2Mo3As3(Y= K, Rb, Cs) with T c s from 10.4 K to 11.5 K was discovered [12-14]. The properties of the upper critical field and specific-heat coefficient of the Y2Mo3As3 superconductors are similar to those of X2Cr3As3.The crystal structure of the Y2Mo3As3 superconductors is same as that of X2Cr3As3, crystallized in a hegxagonal unit cell without an inversion symmetry [12]. The alkali metallic atoms serve as the charge reservoir and the Mo-As chains are responsible for their superconductivity [12]. Since these superconductors host non-centrosymmetric structure that usually connects to the unconventional pairings and exotic physics, they have received considerable attentions in the field of superconductivity researches [15-19].Pressure tuning is a clean way to provide significant information on evolution among superconductivity, electronic state, and crystal structure without changing the chemical composition, and eventually benefits for a deeper understanding of the underlying physics of the puzzling state emerged from ambient-pressure materials[2,20-24]. Earlier high pressure studies on X2Cr3As3(X=K and Rb) below ~4 GPa found that its superconductivity can be dramatically suppressed by external pressure [25,26], however, high pressure studies on the K2Mo3As3 superconductor is still lacking. In this study, we performed the in-situ high pressure transport measurements on the K2Mo3As3 sample up to 50 GPa to know what is new for this compound under high pressure.The polycrystalline (wire-like) samples, as shown in the inset of Fig. 1a, were synthesized using a conventional solid-state reaction method, as described in Ref. [12]. High pressure was generated by a diamond anvil cell made from a Be-Cu alloy with two opposing anvils. Diamond anvils with 300 culets (flat area of the diamond anvil) were used for the experiments. In the resistance measurements, we used platinum foil as electrodes, rhenium plate as gasket and cubic boron nitride as insulating material. The four-probe method was employed to determined pressure dependence of superconducting transition temperature. In the high-pressure ac susceptibility measurements, the sample was surrounded by a secondary coil (pickup coil), above which a field-generating primary coil was wounded [21, 27]. High-pressure angle dispersive x-ray diffraction (XRD) measurements were carried out at beamline 15U at the Shanghai Synchrotron Radiation Facility. A monochromatic x-ray beam with a wavelength of 0.6199 Å was adopted. The pressure was determined by ruby fluorescence method [28]. Given that the sample reacts with the pressure transmitting mediums under pressure, no pressure medium was adopted in all high-pressure measurements.Figure1a shows the temperature dependence of the electrical resistance of the ambient-pressure sample with an onset T c of about 10.4 K, in good agreement with the previous report [12]. Figure 1b-1d show the high pressure results. It is seen that theresistance at 1.2 GPa shows a remarkable drop starting at ~9.2 K and reaches zero at ~ 4.5 K (inset of Fig. 1b), indicating that the applied pressure decreases the onset T c from 10.4 K to 9.2 K. Upon further increasing pressure to 1.5 GPa, the sample loses its zero resistance and its T c shifts to lower temperature, reflecting that the superconductivity of this polycrystalline sample is highly sensitive to the applied pressure. Then, T c decreases continuously with a rate of d T c/d P= -1.19 K/GPa until cannot be detected at 8.7 GPa down to 1.6 K (Fig. 1b). The non-superconducting state persists to the pressure of 18.2 GPa, at which unexpectedly another remarkable resistance drop is found at ~ 4.6 K (Fig. 1c). This drop becomes more pronounced with further compression (Fig. 1d), and plunges about 92% at 47.4 GPa. Moreover, we find that the onset temperature of the new resistance drop shifts to high temperature with increasing pressure initially (the inset of Fig. 1d), reaches a maximum (~8.1 K) at ~ 30 GPa and saturats up to ~38 GPa. By applying higher pressure to 47.4 GPa, the temperature of the drop displays a slow decline (inset of Fig. 1d).To confirm whether the new resistance drop observed in K2Mo3As3 is related to a superconducting transition, we applied magnetic field to the sample subjected to 19.6 GPa and 41.8 GPa, respectively (Fig. 2a and 2b). It can be seen that this new resistance drop shifts to lower temperature with increasing magnetic field and almost suppressed under the magnetic field of 3.5 T and 4.0 T for the compressed sample at 19.6 GPa and 41.8 GPa, respectively. These results indicate that the new resistance drop should be resulted from a superconducting transition. The alternating-current (ac) susceptibility measurements were also performed for the sample subjected to 20 GPa - 44 GPa down to 1.5 K, the lowest temperature of our instrument, but the diamagnetism is not detected. By our analysis, the failure of the measurements on the diamagnetism may be related to the low volume fraction of the pressure-induced superconducting phase.We extract the field (H) dependence of T c for K2Mo3As3 at 19.6 GPa and 41.8 GPa (Fig. 2a and 2b) and plot the H(T c) in Fig. 2c. The experimental data is fitted by using Ginzburg-Landau (GL) formula, which allows us to estimate the values of the upper critical magnetic field (H C2) at zero temperature: 4.0 T at 19.6 GPa and 4.8 T at 41.8 GPa (Fig. 2c). Note that the upper critical fields obtained at 19.6 GPa and 41.8GPa are lower than their corresponding Pauli paramagnetic limits (10.3 T and 14.8T, respectively), suggesting that the nature of the pressure-induced superconducting state may differ from that of the initial superconducting state.To investigate whether the observed reemergence of superconductivity in pressurized K2Mo3As3 is associated with the pressure-induced crystal structure phase transition, we performed in-situ high pressure XRD measurements. The XRD patterns collected at different pressures are shown in Fig. 3. No structure phase transition is observed under pressure up to 51.6 GPa. And all peaks shift to higher angle due to the shrinkage of the lattice, except for the (002) peak. It is found that the (002) peak, which is realted to the parameter c, shifts toward to the lower angle starting at 8.8 GPa. Upon further compression, it becomes more pronounced. We propose that the left shift of the (002) peak may be a consequence of the pressure-induced elongation of the polycrystalline wire-like samples (the direction of the wire length is the c-axis of the K2Mo3As3 crystal lattice) due to their preferred orientation under pressure (see the right panel of Fig. 3). At higher pressure, the wire-like samples aline themselves perpendicular to the pressure direction applied.We summarise our results in Fig. 4a.The pressure-T c phase diagram clearly reveals three distinct superconducting regions: the initial superconducting state (SC-I), intermediary non-superconducting state (NSC) and the pressure-induced superconducting state (SC-II). In the SC-I region between 1 bar and 8.7 GPa, T c issuppressed with applied pressure, and not detectable above 8.7 GPa. In the SC-II region, T c increases with pressure and reaches the maximum (8.1 K) at 30 GPa. Upon further compression, T c shows slow a slight decline. This is the first observation of reemerging superconductivity in one-dimensional superconductors, to the best of our knowledge.We extract the lattice parameters and volume as a function of pressure, and summarise these results in Fig. 4b. It is found that the lattice constant a displays a decrease monotonously with pressure, while the lattice constant c shows a complicated relation with pressure applied. In the SC-I region, the parameter c shrinks upon increasing pressure, but it unusually expands in the NSC region. At the pressure of 18.2 GPa, the superconductivity reemerges and the lattice constant a and c decrease simultaneously with pressure again as what is seen in the SC-I region, implying that the pressure-induced elongation effect on the wire-like sample is satuated. The pressure dependence of the volume for K2Mo3As3 is shown in the inset of the Fig. 4b, displaying that the sample volume remains almost unchanged due to the increase of parameter c and the decrease of parameter a concurrently with elevating pressure in the NSC region. The strong correlation between the lattice parameters and the superconductivity in pressurized K2Mo3As3 suggests that the remarkable change of parameter c may paly an important role for the development of the SC-II state.To understand the underlying correlation between T c and the electronic state in K2Mo3As3 further, we performed the first-principles calculations on its electronic structure, based on our XRD results, by using the projector-augmented wave (PAW) method (see the Supplemental Material [29]). We find that the percentage of the density of states (P-DOS) at Fermi level for K2Mo3As3 is dominated by the electrons from the d xy and d x2-y2 orbitals in the pressure range investigated, with a secondary contribution from the d z2,p x and p y orbitals, and the P-DOSs of the p z, d xz, d yz and sorbitals are relatively small (see the Supplemental Material [29]). We note that, in the NSC and SC-II regions, the P-DOSs of the d xy, and d x2-y2orbitals decrease continuously with elevating pressure over the experimental range investigated, however, the change trend of the P-DOSs contributed by the d z2 and the p x as well as the p y orbitals displays differently. In the NSC region, the P-DOS of the d z2 orbital exibits a remarkable increase, while that of the p orbitals shows a slow decline (Fig. 4c). As the P-DOSs of the d z2 orbital and p orbitals reach a maximum and minimum respectively, SC-II state appears. Note that the T c value of the SC-II state increases upon compression in the pressure range of 18.2 GPa – 27 GPa, just where the P-DOS of the d z2 orbital displays a decrease again. Meanwhile, the P-DOSs of the p x and p y orbitals appear an increase in the same pressure range. Further compression from 27 GPa to 40 GPa, T c of the SC-II state stays almost constant, and the corresponding P-DOSs of the d z2 and p orbitals show a small change (Fig. 4c). These results suggest that the emergence of the SC-II state in K2Mo3As3 and its T c change with pressure are the consequence of the interplay among the different orbital electrons.In conclusion, the pressure-induced reemergence of the superconductivity is observed for the first time in the qausi-one-demensional superconductor K2Mo3As3. An intimate correlation between T c’s of the ambient-pressure and high-pressure superconducting states, lattice parameters and the density of state contributed by d z2, p x and p y orbitals have been revealed. We find that the initial superconducting state (SC-I) is suppressed by pressure at 8.7 GPa, and then an intermediary non-superconducting (NSC) state sets in and stabilizes up to 18 GPa. Subsequently, a new superconduting state (SC-II) emerges and prevails up to 47.4 GPa. Our synchrotron x-ray diffraction results indicats that the reemergence of superconductivity is not associated with anycrystal structure phase transition. In combination of theoretical calculations on the band strcture, our results suggest that the appearance of the SC-II state found in this material is a consenquence of the dramatic interplay among different orbital electrons due to the pressure-induced lattice change. We hope that the results found in this study will shed new light on understanding the correlation among superconductivity, electronic and lattice structures in unconventional quasi-one dimentional superconductors.AcknowledgementsWe thank Prof. V. A. Sidorov for useful discussions. The work was supported by the National Key Research and Development Program of China (Grant No. 2017YFA0302900, 2016YFA0300300 and 2017YFA0303103), the NSF of China (Grants No. U2032214, 12004419 and 12074414) and the Strategic Priority Research Program (B) of the Chinese Academy of Sciences (Grant No. XDB25000000). J. G. is grateful for support from the Youth Innovation Promotion Association of the CAS (2019008).*contributed equally to this work.†To whom correspondence may be addressed. Email: llsun@Reference[1].J. G. Bednorz and K. A. Müller, Possible high T c superconductivity in the Ba-La-Cu-O system, Z. Phys. B 64, 189 (1986).[2].M. K. Wu, J. R. Ashburn, C. J. Torng, P. H. Hor, R. L. Meng, L. Gao, Z. J. 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[29].See the supplementary material for electronic structure calculations; also see Refs.[30-36][30].P. E. Blöchl, Phys. Rev. B 50, 17953 (1994).[31].G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).[32].G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993).[33].G. Kresse and J. Furthmüller, Comput. Mater. Sci. 6, 15 (1996).[34].G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).[35].J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).[36].H. J. Monkhorst and J. Pack, Phys. Rev. B 13, 5188 (1976).Fig. 1 (a) Resistance as a function of temperature for quausi-one-dimensional superconductor K2Mo3As3 at ambient pressure and its polycrystallined-sample’s image taken by a scanning electron microscope (inset). (b)-(d) Temperature dependence of the resistance in the pressure range of 1.2 GPa -8.7 GPa, 8.7-18.2 GPa, and 18.2-47.4 GPa, respectively. The insets of figure (b) and (d) display enlarged views of the the resistance in the low temperature regime.Fig. 2 Temperature dependence of resistance under different magnetic fields for K2Mo3As3measured at 19.6 GPa (a) and 41.8 GPa (b), respectively. (c) Plot of superconducting transition temperature T c versus critical field (H C2) for K2Mo3As3 at 19.6 GPa and 41.8 GPa, respectively. The dished lines represent the Ginzburg-Landau (GL) fits to the data of H C2.Fig. 3 X-ray diffraction patterns collected in the pressure range of 3.8 – 51.6 GPa for K2Mo3As3. The right panel schematically shows the evolution of the preferred orientation of the wire-like samples under pressures. SC-I and SC-II stand for the ambient-pressure superconducting state and the reemergence supercnducting states, respectively. NSC represents non-superconducting state.Fig 4 Superconductivity, crystal and electronic structure information for the qausi-one-dementional superconductor K2Mo3As3. (a) Pressure-Temperature phase diagram. (b) Pressure dependence of lattice parameters and volume (see inset). (c) Plots ofpercentage of the density of state (P-DOS) contrinuted by the d z2, p x and p y orbitals versus pressure.。
第一性原理的基本概念
Condensed matter physicsFrom Wikipedia, the free encyclopediaCondensed matter physics is the field of physics that deals with the macroscopic physical properties of matter. In particular, it is concerned with the "condensed" phases that appear whenever the number of constituents in a system is extremely large and the interactions between the constituents are strong. The most familiar examples of condensed phases are solids and liquids, which arise from the bonding and electromagnetic force between atoms. More exotic condensed phases include the superfluid and the Bose-Einstein condensate found in certain atomic systems at very low temperatures, the superconducting phase exhibited by conduction electrons in certain materials, and the ferromagnetic and antiferromagnetic phases of spins on atomic lattices.Condensed matter physics is by far the largest field of contemporary physics. Much progress has also been made in theoretical condensed matter physics. By one estimate, one third of all American physicists identify themselves as condensed matter physicists. Historically, condensed matter physics grew out of solid-state physics, which is now considered one of its main subfields. The term "condensed matter physics" was apparently coined by Philip Anderson and Volker Heine when they renamed their research group at Cavendish Laboratory - previously "solid-state theory" - in 1967. In 1978, the Division of Solid State Physics at the American Physical Society was renamed as the Division of Condensed Matter Physics. Condensed matter physics has a large overlap with chemistry, materials science, nanotechnology and engineering.One of the reasons for calling the field "condensed matter physics" is that many of the concepts and techniques developed for studying solids actually apply to fluid systems. For instance, the conduction electrons in an electrical conductor form a type of quantum fluid with essentially the same properties as fluids made up of atoms. In fact, the phenomenon of superconductivity, in which the electrons condense into a new fluid phase in which they can flow without dissipation, is very closely analogous to the superfluid phase found in helium 3 at low temperatures.Fermi energyThe Fermi energy is a concept in quantum mechanics usually referring to the energy of the highest occupied quantum state in a system of fermions at absolute zero temperature. This article requires a basic knowledge of quantum mechanics.IntroductionIn quantum mechanics, a group of particles known as fermions (for example, electrons, protons and neutrons are fermions) obey the Pauli exclusion principle. This principle states that two identical fermions can not be in the same quantum state. The states are labeled by a set of quantum numbers. In a system containing many fermions (like electrons in a metal) each fermion will have a different set of quantum numbers. To determine the lowest energy a system of fermions can have, we first group the states in sets with equal energy and order these sets by increasing energy. Starting with an empty system, we then add particles one at a time, consecutively filling up the unoccupied quantum states with lowest-energy. When all the particles have been put in, the Fermi energy is the energy of the highest occupied state. What this means is that even if we have extracted all possible energy from a metal by cooling it down to near absolute zero temperature (0 kelvins), the electrons in the metal are still moving around, the fastest ones would be moving at a velocity that corresponds to a kinetic energy equal to the Fermi energy. This is the Fermi velocity. The Fermi energy is one of the important concepts of condensed matter physics. It is used, for example, to describe metals, insulators, and semiconductors. It is a very important quantity in the physics of superconductors, in the physics of quantum liquids like low temperature helium (both normal 3He and superfluid 4He), and it is quite important to nuclear physics and to understand the stability of white dwarf stars against gravitational collapse.The Fermi energy (E F) of a system of non-interacting fermions is the increase in the ground state energy when exactly one particle is added to the system. It can also be interpreted as the maximum energy of an individual fermion in this ground state. The chemical potential at zero temperature is equal to the Fermi energy.Illustration of the concept for a one dimensional square wellThe one dimensional infinite square well is a model for a one dimensional box. It is a standard model-system in quantum mechanics for which the solution for a single particle is well known. The levels are labeled by a single quantum number n and the energies are given by.Suppose now that instead of one particle in this box we have N particles in the box and that these particles are fermions with spin 1/2. Then only two particles can havethe same energy i.e. two particles can have the energy of , or two particles can have energy E2 = 4E1 and so forth. The reason that two particles can have the same energy is that a spin-1/2 particle can have a spin of 1/2 (spin up) or a spin of -1/2 (spin down), leading to two states for each energy level. When we look at the total energy of this system, the configuration for which the total energy is lowest (the ground state), is the configuration where all the energy levels up to n=N/2 are occupied and all the higher levels are empty. The Fermi energy is therefore.The three-dimensional caseThe three-dimensional isotropic case is known as the fermi sphere.Let us now consider a three-dimensional cubical box that has a side length L (see infinite square well). This turns out to be a very good approximation for describing electrons in a metal. The states are now labeled by three quantum numbers n x, n y, and n z. The single particle energies aren x, n y, n z are positive integers.There are multiple states with the same energy, for example E100 = E010 = E001. Now let's put N non-interacting fermions of spin 1/2 into this box. To calculate the Fermi energy, we look at the case for N is large.If we introduce a vector then each quantum state corresponds to a point in 'n-space' with EnergyThe number of states with energy less than E f is equal to the number of states that liewithin a sphere of radius in the region of n-space where n x, n y, n z are positive. In the ground state this number equals the number of fermions in the system.The free fermions that occupy the lowest energy states form a sphere in momentum space. The surface of this sphere is the Fermi surface.the factor of two is once again because there are two spin states, the factor of 1/8 is because only 1/8 of the sphere lies in the region where all n are positive. We findso the Fermi energy is given byWhich results in a relationship between the fermi energy and the number of particles per volume (when we replace L2 with V2/3):The total energy of a fermi sphere of N0 fermions is given byTypical fermi energies White dwarfsStars known as White dwarfs have mass comparable to our Sun, but have a radius about 100 times smaller. The high densities means that the electrons are no longer bound to single nuclei and instead form a degenerate electron gas. The numberdensity of electrons in a White dwarf are on the order of 1036 electrons/m3. This means their fermi energy is:Another typical example is that of the particles in a nucleus of an atom. The radius of the nucleus is roughly:where A is the number of nucleons.The number density of nucleons in a nucleus is therefore:Now since the fermi energy only applies to fermions of the same type, one must divide this density in two. This is because the presence of neutrons does not affect the fermi energy of the protons in the nucleus, and vice versa.So the fermi energy of a nucleus is about:The radius of the nucleus admits deviations around the value mentioned above, so a typical value for the fermi energy usually given is 38 MeV.Fermi levelThe Fermi level is the highest occupied energy level at absolute zero, that is, all energy levels up to the Fermi level are occupied by electrons. Since fermions cannot exist in identical energy states (see the exclusion principle), at absolute zero, electrons pack into the lowest available energy states and build up a "Fermi sea" of electron energy states. [1] In this state (at 0 K), the average energy of an electron is given by:where E f is the Fermi energy.The Fermi momentum is the momentum of fermions at the Fermi surface. The Fermi momentum is given by:where m e is the mass of the electron.This concept is usually applied in the case of dispersion relations between the energy and momentum that do not depend on the direction. In more general cases, one must consider the Fermi energy.The Fermi velocity is the velocity of fermions at the Fermi surface. It is defined by:where m e is the mass of the electron.Below the Fermi temperature, a substance gradually expresses more and more quantum effects of cooling. The Fermi temperature is defined by:where k is the Boltzmann constant.Quantum mechanicsAccording to quantum mechanics, fermions -- particles with a half-integer spin, usually 1/2, such as electrons -- follow the Pauli exclusion principle, which states that multiple particles may not occupy the same quantum state. Consequently, fermions obey Fermi-Dirac statistics. The ground state of a non-interacting fermion system is constructed by starting with an empty system and adding particles one at a time, consecutively filling up the lowest-energy unoccupied quantum states. When the desired number of particles has been reached, the Fermi energy is the energy of the highest occupied molecular orbital (HOMO). Within conductive materials, this is equivalent to the lowest unoccupied molecular orbital (LUMO); however, within other materials there will be a significant gap between the HOMO and LUMO on the order of 2-3 eV.Pinning of Fermi levelWhen the energy density of surface states is very high (>1013/cm2), the position of the Fermi level is determined by the neutral level of the Surface states [2] and becomes independent of Work Function [3] variations.Free electron gasIn the free electron gas, the quantum mechanical version of an ideal gas of fermions, the quantum states can be labeled according to their momentum. Something similar can be done for periodic systems, such as electrons moving in the atomic lattice of a metal, using something called the "quasi-momentum" or "crystal momentum" (see Bloch wave). In either case, the Fermi energy states reside on a surface in momentum space known as the Fermi surface. For the free electron gas, the Fermi surface is the surface of a sphere; for periodic systems, it generally has a contorted shape (see Brillouin zones). The volume enclosed by the Fermi surface defines the number of electrons in the system, and the topology is directly related to the transport properties of metals, such as electrical conductivity. The study of the Fermi surface is sometimes called Fermiology. The Fermi surfaces of most metals are well studied both theoretically and experimentally.The Fermi energy of the free electron gas is related to the chemical potential by the equationwhere E F is the Fermi energy, k is the Boltzmann constant and T is temperature. Hence, the chemical potential is approximately equal to the Fermi energy at temperatures of much less than the characteristic Fermi temperature E F/k. The characteristic temperature is on the order of 105 K for a metal, hence at room temperature (300 K), the Fermi energy and chemical potential are essentially equivalent. This is significant since it is the chemical potential, not the Fermi energy, which appears in Fermi-Dirac statistics.ReferencesKroemer, Herbert; Kittel, Charles (1980). Thermal Physics (2nd ed.). W. H. Freeman Company. ISBN 0-7167-1088-9.Table of fermi energies, velocities, and temperatures for various elements.a discussion of fermi gases and fermi temperatures.Fermi Surface:In condensed matter physics, the Fermi surface is an abstract boundary useful for predicting the thermal, electrical, magnetic, and optical properties of metals, semimetals, and doped semiconductors. The shape of the Fermi surface is derived from the periodicity and symmetry ofthe crystalline lattice and from the occupation of electronic energy bands. The existence of a Fermi surface is a direct consequence of the Pauli exclusion principle, which allows a maximum of one electron per quantum state.TheoryFormally speaking, the Fermi surface is a surface of constant energy in -spacewhere is the wavevector of the electron. At absolute zero temperature the Fermi surface separates the unfilled electronic orbitals from the filled ones. The energy of the highest occupied orbitals is known as the Fermi energy E F which, in the zero temperature case, resides on the Fermi level. The linear response of a metal to an electric, magnetic or thermal gradient is determined by the shape of the Fermi surface, because currents are due to changes in the occupancy of states near the Fermi energy.Free-electron Fermi surfaces are spheres of radius determinedby the valence electron concentration where is the reduced Planck's constant. A material whose Fermi level falls in a gap between bands is an insulator or semiconductor depending on the size of the bandgap. When a material's Fermi level falls in a bandgap, there is no Fermi surface.A view of the graphite Fermi surface at the corner H points of the Brillouin zone showing the trigonal symmetry of the electron and hole pockets.Materials with complex crystal structures can have quite intricate Fermi surfaces. The figure illustrates the anisotropic Fermi surface of graphite, which has both electron and hole pockets in its Fermi surface due to multiple bands crossing the Fermi energyalong the direction. Often in a metal the Fermi surface radius k F is larger than the size of the first Brillouin zone which results in a portion of the Fermi surface lying in the second (or higher) zones. As with the band structure itself, the Fermi surface canbe displayed in an extended-zone scheme where is allowed to have arbitrarilylarge values or a reduced-zone scheme where wavevectors are shown modulowhere a is the lattice constant. Solids with a large density of states at the Fermi level become unstable at low temperatures and tend to form ground states where the condensation energy comes from opening a gap at the Fermi surface. Examples of such ground states are superconductors, ferromagnets, Jahn-Teller distortions and spin density waves.The state occupancy of fermions like electrons is governed by Fermi-Dirac statistics so at finite temperatures the Fermi surface is accordingly broadened. In principle all fermion energy level populations are bound by a Fermi surface although the term is not generally used outside of condensed-matter physics.Experimental determinationde Haas-van Alphen effect. Electronic Fermi surfaces have been measured through observation of the oscillation of transport properties in magnetic fields H, for example the de Haas-van Alphen effect (dHvA) and the Shubnikov-De Haas effect (SdH). The former is an oscillation in magnetic susceptibility and the latter in resistivity. The oscillations are periodic versus 1 / H and occur because of the quantization of energy levels in the plane perpendicular to a magnetic field, a phenomenon first predicted by Lev Landau. The new states are called Landau levels and are separated by an energywhere ωc = eH / m*c is called the cyclotron frequency, e is the electronic charge, m* is the electron effective mass and c is the speed of light. In a famous result, Lars Onsager proved that the period of oscillation ΔH is related to the cross-section of theFermi surface (typically given in ) perpendicular to the magnetic field directionby the equation . Thus the determination of the periods ofoscillation for various applied field directions allows mapping of the Fermi surface.Observation of the dHvA and SdH oscillations requires magnetic fields large enough that the circumference of the cyclotron orbit is smaller than a mean free path. Therefore dHvA and SdH experiments are usually performed at high-field facilities like the High Field Magnet Laboratory in Netherlands, Grenoble High Magnetic Field Laboratory in France, the Tsukuba Magnet Laboratory in Japan or the National High Magnetic Field Laboratory in the United States.Fermi surface of BSCCO measured by ARPES. The experimental data shown as an intensity plot in yellow-red-black scale. Green dashed rectagle represents the Brillouin zone of the CuO2 plane of BSCCO.Angle resolved photoemission. The most direct experimental technique to resolve the electronic structure of crystals in the momentum-energy space (see reciprocal lattice), and, consequently, the Fermi surface, is the angle resolved photoemission spectroscopy (ARPES). An example of the Fermi surface of superconducting cuprates measured by ARPES is shown in figure.Two photon positron annihilation.With positron annihilation the two photons carry the momentum of the electron away; as the momentum of a thermalized positron is negligible, in this way also information about the momentum distribution can be obtained. Because the positron can be polarized, also the momentum distribution for the two spin states in magnetized materials can be obtained. Another advantage with De Haas-Van Alphen-effect is that the technique can be applied to non-dilute alloys. In this way the first determination of a smeared Fermi surface in a 30% alloy has been obtained in 1978.ReferencesN. Ashcroft and N.D. Mermin, Solid-State Physics, ISBN 0-03-083993-9.W.A. Harrison, Electronic Structure and the Properties of Solids, ISBN0-486-66021-4.VRML Fermi Surface DatabaseBrillouin zoneFrom Wikipedia, the free encyclopediaJump to: navigation, searchBrillouin zoneIn mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell of the reciprocal lattice in the frequency domain. It is found by the same method as for the Wigner-Seitz cell in the Bravais lattice. The importance of the Brillouin zone stems from the Bloch wave description of waves in a periodic medium, in which it is found that the solutions can be completely characterized by their behavior in a single Brillouin zone.Taking the surfaces at the same distance from one element of the lattice and its neighbours, the volume included is the first Brillouin zone. Another definition is as the set of points in k-space that can be reached from the origin without crossing any Bragg plane.There are also second, third, etc., Brillouin zones, corresponding to a sequence of disjoint regions (all with the same volume) at increasing distances from the origin, but these are used more rarely. As a result, the first Brillouin zone is often called simply the Brillouin zone. (In general, the n-th Brillouin zone consist of the set of points that can be reached from the origin by crossing n− 1 Bragg planes.)A related concept is that of the irreducible Brillouin zone, which is the first Brillouin zone reduced by all of the symmetries in the point group of the lattice.The concept of a Brillouin zone was developed by Leon Brillouin (1889-1969), a French physicist.Critical pointsFirst Brillouin zone of FCC lattice showing symmetry labels for high symmetry lines and points Several points of high symmetry are of special interest – these are called critical points.[1]Symbol DescriptionΓ Center of the Brillouin zoneSimple cubeM Center of an edgeR Corner pointX Center of a faceFace-centered cubicK Middle of an edge joining two hexagonal facesL Center of a hexagonal faceU Middle of an edge joining a hexagonal and a square faceW Corner pointX Center of a square faceBody-centered cubicH Corner point joining four edgesN Center of a faceP Corner point joining three edgesHexagonalA Center of a hexagonal faceH Corner pointK Middle of an edge joining two rectangular facesL Middle of an edge joining a hexagonal and a rectangular faceM Center of a rectangular faceReferences1.^ Harald Ibach & Hans Lüth, Solid-State Physics, An Introduction to Principles of MaterialsScience, corrected second printing of the second edition, 1996, Springer-Verlag, ISBN3-540-58573-72.Charles Kittel, Introduction to Solid State Physics (Wiley: New York, 1996).3.Neil W. Ashcroft and N. David Mermin, Solid State Physics (Harcourt: Orlando, 1976).4.Léon Brillouin Les électrons dans les métaux et le classement des ondes de de Brogliecorrespondantes Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, 191, 292 (1930). (original article)Bloch waveFrom Wikipedia, the free encyclopediaA Bloch wave or Bloch state, named after Felix Bloch, is the wavefunction of a particle (usually, an electron) placed in a periodic potential. It consists of the product of a plane wave envelope function and a periodic function (periodic Bloch function)which has the same periodicity as the potential:Bloch wave in siliconThe result that the eigenfunctions can be written in this form for a periodic system is called Bloch's theorem. The corresponding energy eigenvalue is Єn(k)= Єn(k + K), periodic with periodicity K of a reciprocal lattice vector. Because the energies associated with the index n vary continuously with wavevector k we speak of an energy band with band index n. Because the eigenvalues for given n are periodic in k,all distinct values of Єn(k) occur for k-values within the first Brillouin zone of the reciprocal lattice.More generally, a Bloch-wave description applies to any wave-like phenomenon in a periodic medium. For example, a periodic dielectric in electromagnetism leads to photonic crystals, and a periodic acoustic medium leads to phononic crystals. It is generally treated in the different forms of the dynamical theory of diffraction.The plane wave wavevector (or Bloch wavevector) k (multiplied by the reduced Planck's constant, this is the particle's crystal momentum) is unique only up to a reciprocal lattice vector, so one only needs to consider the wavevectors inside the first Brillouin zone. For a given wavevector and potential, there are a number of solutions, indexed by n, to Schrödinger's equation for a Bloch electron. These solutions, called bands, are separated in energy by a finite spacing at each k; if there is a separation that extends over all wavevectors, it is called a (complete) band gap. The band structure is the collection of energy eigenstates within the first Brillouin zone. All the properties of electrons in a periodic potential can be calculated from this band structure and the associated wavefunctions, at least within the independent electron approximation.A corollary of this result is that the Bloch wavevector k is a conserved quantity in a crystalline system (modulo addition of reciprocal lattice vectors), and hence the group velocity of the wave is conserved. This means that electrons can propagate without scattering through a crystalline material, almost like free particles, and that electrical resistance in a crystalline conductor only results from things like imperfections that break the periodicity.It can be shown that the eigenfunctions of a particle in a periodic potential can always be chosen in this form by proving that translation operators (by lattice vectors) commute with the Hamiltonian. More generally, the consequences of symmetry on the eigenfunctions are described by representation theory.The concept of the Bloch state was developed by Felix Bloch in 1928, to describe the conduction of electrons in crystalline solids. The same underlying mathematics, however, was also discovered independently several times: by George William Hill (1877), Gaston Floquet (1883), and Alexander Lyapunov (1892). As a result, a variety of nomenclatures are common: applied to ordinary differential equations, it is called Floquet theory (or occasionally the Lyapunov-Floquet theorem). Variousone-dimensional periodic potential equations have special names, for example, Hill's equation:[1],where the θ's are constants. Hill's equation is very general, as the θ-related terms may viewed as a Fourier series expansion of a periodic potential. Other much studied periodic one-dimensional equations are the Kronig-Penney model and Mathieu's equation.References1.^ W Magnus and S Winkler (2004). Hill's Equation. Courier Dover, p. 11. ISBN0-0486495655.Further reading∙Charles Kittel, Introduction to Solid State Physics (Wiley: New York, 1996).∙Neil W. Ashcroft and N. David Mermin, Solid State Physics (Harcourt: Orlando, 1976).∙Felix Bloch, "Über die Quantenmechanik der Elektronen in Kristallgittern," Z. Physik52, 555-600 (1928).∙George William Hill, "On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon," Acta. Math.8, 1-36 (1886). (This work wasinitially published and distributed privately in 1877.)∙Gaston Floquet, "Sur les équations différentielles linéaires à coefficients périodiques,"Ann. École Norm. Sup.12, 47-88 (1883).∙Alexander Mihailovich Lyapunov, The General Problem of the Stability of Motion (London: Taylor and Francis, 1992). Translated by A. T. Fuller from Edouard Davaux'sFrench translation (1907) of the original Russian dissertation (1892).Retrieved from "/wiki/Bloch_wave"Density of statesFrom Wikipedia, the free encyclopediaIn statistical and condensed matter physics, the density of states (DOS) of a system describes the number of states at each energy level that are available to be occupied.A high DOS at a specific energy level means that there are many states available for occupation. A DOS of zero means that no states can be occupied at that energy level.ExplanationWaves, or wave-like particles, can only exist within quantum mechanical (QM) systems if the properties of the system allow the wave to exist. In some systems, the interatomic spacing and the atomic charge of the material allows only electrons of certain wavelengths to exist. In other systems, the crystalline structure of the material allows waves to propagate in one direction, while suppressing wave propagation in another direction. Waves in a QM system have specific wavelengths and can propagate in specific directions, and each wave occupies a different mode, or state. Because many of these states have the same wavelength, and therefore share the same energy, there may be many states available at certain energy levels, while no states are available at other energy levels. For example, the density of states for electrons in a semiconductor is shown in red in Fig. 2. For electrons at the conduction band edge, very few states are available for the electron to occupy. As the electron increases in energy, the electron density of states increases and more states become available for occupation. However, because there are no states available for electrons to occupy within the bandgap, electrons at the conduction band edge must lose at least E g of energy in order to transition to another available mode. The density of states can be calculated for electron, photon, or phonon in QM systems. The DOS is usually represented by one of the symbols g, ρ, D, n, or N, and can be given as a function of either energy or wavevector k. To convert between energy and wavevector, the specific relation between E and k must be known. For example, the formula forelectrons isAnd for photons, the formula isDerivationThe density of states is dependent upon the dimensional limits of the object itself. The role dimensions play is evident from the units of DOS (Energy-1Volume-1). In the limit that the system is 2 dimensional a volume becomes an area and in the limit of 1 dimension it becomes a length. It is important to note that the volume being referenced is the volume of k-space,the space enclosed by the constant energy surface of the system derived through a dispersion relation that relates E to k. An example of a specific k-space is given in Fig. 1. It can be seen that the dimensionality of the system itself will confine the momentum of particles inside the system.。
龚昌德教授简历 - 海峡两岸统计物理与凝聚态理论研究中心
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。
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a r X i v :c o n d -m a t /9709168v 3 [c o n d -m a t .s t r -e l ] 21 S e p 1998Superconductivity in a two-dimensional Electron GasPhilip Phillips,Yi Wan,Ivar Martin,Sergey Knysh,and Denis DalidovichLoomis Laboratory of PhysicsUniversity of Illinois at Urbana-Champaign 1100W.Green St.,Urbana,IL,61801-3080In a series of recent experiments,Kravchenko and colleagues 1,2observed unexpectedly that a two-dimensional electron gas in zero magnetic field can be a conductor.The two-dimensionality was imposed by confining the electron gas to move laterally at the interface between two semi-conductors.The observation of a conductor in two dimensions (2D)is surprising as the conven-tional theory of metals precludes the presence of a metallic state at zero temperature in 2D 3.Nonetheless,there are now several experiments confirming the existence of the new conducting phase in a dilute two-dimensional electron gas in zero magnetic field 4–7.Here we argue based on an analysis of the experiments and general theoreti-cal grounds that this phase is a zero-temperature superconductor with an inhomogeneous charge density.While the specific details differ as the semiconduc-tor that confines the 2D electron gas is changed,sev-eral key similarities 1,2,4–7exist among the reported ob-servations of the transition to the conducting state:1)the existence of a critical electron or hole density,n c ,above which the conducting phase appears,2)a char-acteristic temperature,typically on the order of half the Fermi temperature,T F ,below which the resistivity on the conducting side decreases,3)critical scaling,in-dicative of a quantum critical point,in the vicinity of the insulator-conducting phase transition,4)non-linear current-voltage (I-V)curves that exhibit a symmetry around the I-V curve at criticality,and 5)suppression of the conducting phase by a magnetic field.In Si metal-oxide-field-effect-transistors (MOSFET’s)samples 1,tem-perature (T)and electric field (E)scaling have made it possible to extract the dynamical and correlation length exponents,z and ν,respectively.By fitting the resis-tivity measurements to functions of the form,ρ(T,n )=f 1(|δ|/T b )and ρ(E,n )=f 2(|δ|/E a ),with n,the electron density,b =1/zν,a =1/[(z +1)ν],and δ=(n −n c )/n c ,Kravchenko and colleagues 1found that ν=1.5and z =0.8.Analogous scaling occurs in GaAs 5but only in the most disordered samples.This indicates that the underlying transition in the clean system is first order,whereas in the presence of disorder,it becomes second order and critical scaling applies.Three postulates anchor the conventional theory of metals 3:1)Fermi liquid theory accurately describes the low-temperature physics of conventional clean metals,2)the classical and quantum corrections to the conductivityare additive 8as the system size increases,and 3)the log-arithmic derivative (β)of the dimensionless conductance with respect to the system size is a continuous mono-tonic single-valued function as the strength of the disor-der is increased from weak to strong.From (2)and (3)it follows 3,8that βis always negative in two dimensions.Hence,as the system size increases,the conductance de-creases and insulating behavior necessarily obtains in two dimensions.The general applicability of the scaling anal-ysis rests firmly on the extensive numerical and experi-mental work that have confirmed its central predictions and assumptions 9.Strictly speaking,however,Fermi liquid behaviour ob-tains only for T ≪T F .Hence,the onset of a conducting phase at T ≈T F does not necessarily pose a paradox pro-vided that the resistivity turns upwards at sufficiently low temperatures.However,such behaviour is inconsistent with the experimental observations for two primary rea-sons.First,there is no experimental indication that the resistivity turns upwards at low temperatures.Secondly,the critical scaling observed 1signifies that the conductor-insulator transition is a bona-fide quantum phase transi-tion.The conventional theory of metals forces us then to come to grips with the new conducting state in 2D by abandoning one or all of the postulates of the stan-dard view.In the context of the experiments 1,2,4–7,the Coulomb interaction V ee exceeds the Fermi energy,ǫF ,by at least an order of magnitude.For long-range interac-tions,non-Fermi liquid behaviour is likely to obtain when V ee ≫ǫF .It is for this reason and the localization con-straint placed on Fermi liquids in 2D by the conventional theory that the experiments force us to consider non-Fermi liquid scenarios only.In this vein,Chakravarty and co-workers 10proposed a Luttinger liquid state to explain the experimental observations.However,such a state has yet to be proven to exist except in 1-dimension.In fact in 2D,a T =0superconductor is the only conduct-ing non-Fermi liquid state proven to exist in the pres-ence of disorder 11and zero magnetic field.Hence,rather than speculate as to the existence of some yet-unproven non-Fermi liquid state in 2D,we take the conservative approach and propose that superconducting fluctuations mediate the new conducting state in 2D.We support this proposal with three subsidiary ar-guments:1)the generic features 12of the conduct-ing transition resemble those of known 13insulator-superconductor phase transitions,2)the magnetoresis-tance measurements 1,5provide evidence for the existence1of a critical magneticfield above which the conduct-ing phase is destroyed,and3)the conducting transition lies in close proximity to an incipient electron crystal state in which strong charge retardation effects can lead to Cooper pair formation.Further,we draw a paral-lel between the2D electron gas experiments and those on inhomogeneous superconductors in which an appar-ent saturation of the resistivity has been observed at low temperatures14,15.We argue that the enhanced role of classicalfluctuations of the superconducting phase16in 2D can significantly suppress the temperature at which the resistivity vanishes.In support of the superconducting scenario are numer-ous transport measurements on the electron and hole sys-tems.The value of the dynamical exponent13,z=1,the non-linearity in the I-V curves as well as the observed temperature and electricfield scaling of the resistivity are all consistent with what is observed in known insulator-superconductor transitions(IST’s).We now analyze the experimental data on the magneto-resistance and show that they offer evidence for a critical parallel magnetic field,consistent with singlet pairing.Although we limit our discussion to a parallelfield(H||),Kravchenko and colleagues2showed that the response of the conducting phase to a magneticfield is independent of the direction of thefield,indicating that it is a spin effect that de-stroys the conducting phase.Considerfirst the magneto-resistance measurements in GaAs5.Measurements of the conductivity as a function of the hole density in GaAs in the temperature interval[1.4K,0.3K]reveal that even in the presence of a magneticfield,the conductivity curves cross at a unique value of the hole density.The single crossing point signifies that the conductivity is temper-ature independent at a particular density.Hence,this density demarcates the transition between the conduct-ing and insulating phases.If the conducting phase were destroyed by an arbitrarily small magneticfield,such a crossing point would not occur atfinitefield.However, the experiments show that even forfields as high as1T,a unique crossing point exists.By H||=3.0T,the unique crossing point vanishes,indicating that there is a thresh-old parallel magneticfield above which the conducting phase is extinguished.The criticalfield is,however,den-sity dependent.From the crossing point infinitefield,we conclude that forδ=0.02andδ=0.03,the criticalfieldsare H c||=0.5T and H c||=1.0T,respectively.For GaAs,the Zeeman energies at suchfield strengths correspond to an energy at least an order of magnitude smaller than the Fermi temperature.Hence,the conducting phase in GaAs is characterized by an internal energy scale distinct fromǫF.What about Si MOSFET’s?Based on the measure-ments ofρ(H,E),Kravchenko,et.al.1concluded that the conducting phase is destroyed for an arbitrarily small parallel magneticfield,indicating a vanishing of the crit-icalfield.This conclusion poses a distinct problem if the conducting transition in Si MOSFET’s and GaAs is assumed to be driven by the same physics.That is,ei-ther both or neither should display a criticalfield,H c.We point out,however,that the observation of a criti-calfield in GaAs was based on an analysis ofρ(H,T),notρ(H,E).If this conclusion is correct,ρ(H,T)should also confirm this result.Hence,we investigate preciselywhat is contained inρ(H,T)for Si MOSFET’s.The raw data for Si MOSFET’s indicates that for H||≈9kOe the slope ofρ(T)changes sign2.Hence,from the raw data,there is no indication at least within the temper-ature regime studied that an arbitrarily small magnetic field suppresses the conducting phase.As a consequence, we analyzedρ(H,T)for a Si MOSFET(D.Simonian,S. Kravchenko,and M.Sarachik,personal communication) with a scaling function of the form,f(|H||−H c|/T1/α).In field-tuned2D IST’s13,17,the resistivity scales on either side of H c as a universal function17of|H−H c|/T1/z BνB, where z B andνB are the dynamical and correlation length exponents,respectively in a magneticfield.If a criticalfield exists,then above and below H c the experi-mental values for the resistivity should collapse onto two distinct branches.Such collapse is shown in Fig.(1) above and below a criticalfield of H c=9.5kOe with α=0.6±0.1.Here again,the criticalfield corresponds to an energy scale that is an order of magnitude smaller than the Fermi energy.It is interesting to note that ex-periments on the IST in bismuthfilms(N.Markovic, Christiansen,and A.M.Goldman,personal communi-cation)show that z BνB=0.7±0.2,which is remarkably close to the value,α=0.6±0.1,obtained in the scaling plot in Fig.(1).The stiffness in the conducting phase to a parallel mag-neticfield signifies that the ground state is a singlet. While spin glass and antiferromagnetic order are also consistent with a singlet ground state,such phases are insulating in2D.If we entertain the possibility of other non-Fermi liquid states,it is unclear how such a state will differ from a superconducting one,as the experiments dictate that such a state must also have a singlet energy gap and conduct in the presence of disorder.In terms of the dimensionless measure of the density, r s=1/√phase exhibiting quasi-long range order still forms.The melting density for such a phase is shifted to higher val-ues,typically r s≈10(though the amount and type of disorder might change this value)because disorder sta-bilizes the solid relative to the liquid phase.As disorder is undoubtedly present in the experimental systems and the conducting transition occurs for r s≈10,it makes sense to think about the transition to the conducting state as a transition from an insulating incipient Wigner crystal phase.Experimentally,the density dependence of the threshold electricfield24required to initiate trans-port on the insulating side,E t∝δ1.5,is consistent with the prediction25for quantum tunneling in an incipient Wigner crystal.Hence,experimental evidence also cor-roborates the proximity of the conducting transition to an insulating phase with quasi-long range order.The proximity of the conducting phase to a quasi-crystalline dilute electron phase makes this problem quite analogous to high temperature superconductivity(high T c).While this phenomenon is currently unexplained, experimentally it is clear that superconductivity in the copper oxides obtains as doped holes disrupt the per-fect antiferromagnetic order in the Mott insulating phase. The correspondence with high T c is even more striking as such systems are quasi-2D with r s∼10for the holes. In both systems,destruction of the long-range or quasi-long range spin or charge order in the insulating phase is accompanied by a charge or spin retardation effect which attempts to preserve the memory of the correlations in the insulating phase.As an incipient Wigner crystal has at best frustrated antiferromagnetic21spin order,the charge retardation effect is expected to dominate26.We illustrate its role as follows.Consider an electron crystalline or quasi-crystalline phase in which the electrons are locked into‘home’posi-tions by the electron interaction.Such a crystalline phase is stable when the zero-point energy¯hω0much exceeds the kinetic energy,ǫF,of an electron in each unit cell(or Wigner-Seitz cell in the context of a Wigner crystal)of edge length r s.A distinct feature of an electron crystal or quasi-crystalline phase is the dominant role played by the correlations.The correlations manifest themselves in the form of a correlation hole which is centered at the aver-age position of each electron in a unit cell.Increasing the electron density leads to an increase in the electron ki-netic energy and an eventual melting of a Wigner crystal. In the melted phase,correlation holes still form around electrons.However,because they are massive,relative to a single electron,their response is delayed.The re-tardation timescale is set by the inverse of the plasma frequency.Within this timescale,a correlation hole lags behind its associated electron and hence appears posi-tively charged to another electron.As a consequence, the partially-vacated correlation hole can attract another electron.In so doing,the correlation hole mediates a dy-namic attraction between electrons and the subsequent onset of Cooper pair formation.Because plasmons are ungapped in2D,the retardation effect is strongest forthe low frequency plasmons.We anticipate that it is from such plasmons that the dominant electron attrac-tion arises.For a3D electron gas,Takada26,28has made the anal-ogous observation regarding the proximity of supercon-ductivity to the melting of a Wigner crystal.The work of Kelly and Hanke27and Ren and Zhang28also relates to this proximity.More recently,Belitz and Kirkpatrick29 have proposed a similar charge polarization mechanism in which the long-time tail of the charge density correla-tion function,which appears in the presence of disorder, assists pairing.Also,in the Monte Carlo simulations21ona clean Wigner crystal,a precipitous drop of the spin sus-ceptibility is observed in the melted phase,a pre-requisite for singlet superconductivity.In addition,capacitance charging experiments on GaAs quantum dots30,in which a2D electron gas is constructed one electron at a time from the veryfirst electron,have reported the occur-rence of pair electron charging events in the density range2.8<r s<9.6.As pair-tunneling events correspondto two electrons charging the same quantum state on the dot,such states form only if an electronic attraction screens the Coulomb repulsion between the two electrons.In so far as the electrons in a quantum dot realistically model a dilute(r s>3)2D electron gas,it is certainly reasonable to suspect that the electron attraction persists in Si MOSFET’s as well.In2D superconductors,the resistivity vanishes not at the mean-field temperature,T c,at which the pair ampli-tude is established but at a lower temperature,Tθ,where global phase coherence obtains.That is,rather than defining the superconducting transition temperature,T c determines the characteristic temperature at which the resistivity initially decreases.Both disorder31and low superfluid densities16can significantly lower Tθrelative to T c in2D systems.In fact,Emery and Kivelson16have argued that a key feature which suppresses Tθin high T c materials is their notoriously low superfluid density.In the electron systems of interest,disorder is present (leading to inhomogeneous electron density)and the su-perfluid density is expected to be small as the electron gas is dilute.As a consequence,we anticipate that in Si MOSFETS’s and GaAs,Tθ,will be greatly suppressed relative to the mean-field T c.In the case of GaAs,6,5ex-ponential drop of the resistivity on the conducting side is accompanied by a plateau below some characteristic tem-perature.This latter experimental trend is identical to the behaviour observed by Goldman and collaborators14 and Imry and Strongin15in inhomogeneous2D super-conductors.The strong similarity between the inhomo-geneous systems and the GaAs data suggests that the phase-locked zero-resistance state occurs at a temper-ature significantly lower than that probed experimen-tally.Nonetheless,the non-linear I-V characteristics1 observed in the Si MOSFET’s is consistent with a2D superconducting state on the brink of global phase co-herence.However,as long as T>Tθ,pairfluctuations are quenched in a magneticfield only when the Zeeman 3and pairing energies are comparable.In the dense limit,correlation and retardation effects are negligible and consequently,Cooper pair formation ceases.The anticipated termination of the electron at-traction at small r s suggests the schematic phase diagram depicted in Fig.(2).Beyond the upper density at which superconducting pairfluctuations cease,the electron liq-uid is most likely insulating.Hence,a direct transition from an insulator to a metal in2D by changing the den-sity seems unlikely.In GaAs samples,a re-entrant(M. Y.Simmons,personal communication)insulating phase at r s≈8was observed consistent with the analysis here. Further experiments are needed on the MOSFET’s to see the re-entrant insulating phase.In closing,because Fermi iquids are localized in2D, the new conducting state must be some type of non-Fermi liquid.As T=0superconductivity is the only conducting,disordered non-Fermi liquid proven to exist in2D,this must be viewed as the leading candidate to explain the experimental observations.The existence of a critical parallel magneticfield is a clear indicator of a singlet energy gap in the conducting phase in2D.How-ever,further experiments are needed to probe the low temperature physics as well as the magneto-resistance in Si MOSFET’s.In addition to disorder and a suppressed superfluid density,the inhomogeneities introduced in a dilute electron gas(r s>3)as a result of the nega-tive compressibility32(an instability relative to a uni-form charge density)are expected to enhance phasefluc-tuations as well.Consequently,we anticipate that the transition to the phase-locked state will occur at a tem-perature significantly below the mean-field T c.29Belitz,D.&Kirkpatrick,T.Are Si MOSFET’s Triplet Su-perconductors,cond-matt/9705023.30Zhitenev,N.B.,Ashoori,R.C.,Pfeiffer,L.N.,West,&K.W.,Periodic and aperiodic bunching in the addition spec-tra of quantum dots,Phys.Rev.Lett.77,1833-6(1996).31Beasley,M.R.,Mooij,J.E.,&,Orlando,T.P.Possibility of vortex-antivortex pair dissociation in two-dimensional superconductors,Phys.Rev.Lett.42,1165-1168(1979).32Moroni,S.,Ceperley,D.&Senatore,G.Static response of 2D electrons from quantum Monte Carlo,in Proceedings of the Rochester Symposium on Strongly Coupled Plasmas eds.H.van Horn and S.Ichimaru,U of Rochester Press, Rochester(1993)p.445-452.FIG.1.Scaling curve for the magnetoresistance obtained from the experimental data of Simonian,Kravchenko,and Sarachik illustrating clearly the existence of a criticalfield at 9.5kOe withα=0.6.FIG.2.Schematic mean-field phase diagram for a2D dis-ordered electron system.DWC=Disordered Wigner crystal and SC=superconductor.The black circle at n c is determined from the experimental data in Ref.1ACKNOWLEDGMENTSWe thank Bob Laughlin for his most generous,copi-ous,and characteristically level-headed remarks regard-ing the presentation.We also acknowledge useful con-versations with E.Fradkin,N.Markovic,A.Goldman, A.Yazdani,A.Castro-Neto,Tony Leggett,D.Ceper-ley,T.Giamarchi,P.Parris,and S.Wan.We also the ACS Petroleum Research Fund and the NSF grant No. DMR94-96134.5c。