摘要
元素单质是构成物质世界的最基本物质,其高压研究一直备受关注。高压能够显著地改变物质的晶体结构及物理性质,如使某些元素单质相变到复杂的非公度结构,使绝缘体转变为金属,甚至超导体。对元素单质高压性质的研究能丰富人们对物质世界的认识,还可以揭示新的物理现象和物理规律。本论文利用最新发展的基于粒子群优化算法的晶体结构预测技术,结合第一性原理研究,对典型元素单质镁和氯的高压新结构进行了系统的研究,深入地讨论了这些高压新相的奇特物理行为,获得如下创新性成果:
1.发现了镁的两个高压新相:面心立方fcc和简单六角sh相,提出fcc→sh相变的相变路径为fcc结构的α和γ角同时由60°增加到90°。电子局域函数计算发现镁在高压下转化为“Electride”材料,此时价电子高度局域在晶格间隙区域,但镁并没有像钠一样转变为绝缘体,仍然呈现弱的金属性。
2.预言了固体氯单质中存在非公度调制结构。获得了固体氯在高压下的相变序列:分子相Ⅰ→非公度调制相Ⅴ→原子相Ⅱ→fcc相,其中非公度调制相Ⅴ是分子相Ⅰ和原子相Ⅱ之间的过渡相,其平均结构为fco,波矢接近于2/7。
3.固体氯的分子相Ⅰ在130 GPa发生了能带交叠导致的金属化,在157 GPa压力以上发生分子解离,相变为原子相Ⅱ。
4.电子和声子耦合计算表明,固体氯在高压下转化为超导体;其中原子fcc相在380 GPa的超导转变温度可以达到13.0 K,明显高于碘和溴的超导温度,这主要是因为固体氯fcc相的费米面附近电子能带具有“平带-陡带”的特征,有利于形成高温超导体。
The elements are the most fundamental materials and their high-pressure behavior has always attracted the attention of many researchers. At ambient conditions, the majority of the elements adopt very simple high symmetry structures. High pressure can reduce the distance between atoms, so that the crystal structure of the materials will be rearranged and the phase transition will be occurred and the new phases will be formed. For elements, there may be structural transitions to phases with lower symmetry and less close packing than found at ambient pressure. For example, iodine and bromine form incommensuration modulated structures under pressure. Changes of structure and electronic energy levels can also give rise to dramatic changes in physical phenomena and behaviors. Elements that are insulators at ambient pressure exhibit structural phase transitions accompanied by metallization and the onset of superconductivity. Indeed, the number of known elemental superconductors has been increased by over 70% by high-pressure studies. Applying pressure can also induce transitions from molecular to non-molecular structural forms. The more intriguing is metals Na will transform under pressure into insulating states. And the conversion of gaseous molecular elements to a monatomic metallic superconductor phase is, for many, the‘‘holygrail’’of high-pressure physics. Combined with modern computational methods, much is now understood about the behavior of elements at high pressure. However, there are still a number of outstanding questions and further studies are needed. Of particular interest would be the discovery of further new incommensurate structures.
Most recently, with the help of improved theory and computational capability, the first principle method base on the density functional theory has been used widely in the high pressure structures and physics behaviors. The newly developed crystal structure prediction method extended the research field for theorist once again. CALYPSO (Crystal structure AnaLYsis by Particle Swarm Optimization) is a package for crystal structure prediction through particle swarm optimization algorithm. CALYPSO requires only chemical compositions for a giver compound to predict stable or metastable structures. It is proved to be powerful with high efficiency and a high success rate, such as the high pressure structure research of Li.
Within this thesis, we studied the high pressure structures and properties of two typical elements, Mg and Cl, by using first principle method. We first determined their high pressure crystal structures based on the crystal structure prediction CALYPSO algorithm; then we studied their physical properties under high pressure.
The first part of our work is to find high pressure structures of Mg, through the CALYPSO code. Then, we studied their electronic properties, chemical bonding features and the phase diagrams. We predicted correctly the hcp and bcc structures at low pressures. The predicted transition pressures, EOS, and temperature-dependent phase diagrams of the hcp and bcc structures are in excellent agreement with the experiments. A high-pressure face-centred cubic (fcc) phase and simple hexagonal (sh) phase has been uncovered to be stable 456-756 GPa and above 756 GPa. The sh structure can be derived from the fcc lattice by distortion of the ? andγangles from 60°to 90°. The fcc and sh structures are both metallic, but the metallicity is weaken than the low-pressure hcp structure. We also calculated the electronic DOS for Cl at different pressures. We found that the weakening of metallicity is attributed to the drop of 3d bands in energy relative to the 3p bands and the increased p-d hybridization upon compression. The ELF indicated that the valence electrons of fcc and sh are mostly localized in the interstitial sites, similar to alkali and other alkaline-earth elements, such as Na, Li, K, and Ca. The fcc and sh structures can be described as electrides formed by Mg ions cores and localized interstitial electrons. The valence electron localization of sh-Mg is not as strong as hp4-Na, and the core-valence overlap is not as forceful as Na. This could be the reason why hp4-Na is an insulator and sh-Mg is a metal. We have carefully calculated the complete phase transition diagrams of Mg at high pressures and temperatures within the quasi-harmonic approximation. Our results suggest that the temperature contribution does not affect the phase transition order, but slightly changes the transition pressures. The calculated phase boundaries for both bcc to fcc and fcc to sh transition are found to have positive Chaperon slopes.
The second part of our work is to study the crystal structure and properties of Cl under high pressure. It divided into three parts. In part one our predicted high pressures structures, especially the incommensuration modulated structures through CAYLPSO are shown. The structural stability and transition mechanism are discussion. In part two, we discussions the pressure-induced metallization and molecular dissociation of Cl. Finally, in part three we focus our study on the superconductivity of Cl under pressure.
First of all, our CALYPSO simulation predicted the Cmca structures to be stable, in agreement with experiment. Our simulation predicted that the most stable structure at 157-372 GPa is the bco structure (phaseⅡ), which is isomorphic to solid bromine and iodine. The low enthalpy structure found above 372 GPa is the fcc structure. No imaginary phonon frequencies are found in the pressure ranges of 0-142 and 157-372 GPa for the Cmca and bco structure repeatedly and above 372 GPa for the fcc structure in the whole Brillouin zone, suggesting that the three structures are dynamically stable. In addition, at 150 GPa we also predicted a series of similar structures with different of space group and similar low-enthalpy. They all have similar“modulation wave”feature as incommensurate modulated structure. The x-ray diffraction patterns of our structure are very similar with the simulated incommensurate phase. The main peaks are the same with fco. The distances between neighboring atoms are not the same value, but there is a change of interval. Accordingly, we believe that chlorine also exists similar incommensurate structures with iodine, we call this phase as phaseⅤ. Through optimization, we found that modulated phase oF28 which wave vector is 2/7 has the lowest enthalpy in the pressure range from 142 GPa to 157 GPa. For the analysis of mechanical nature of chlorine, we found that the elastic constants C_(66) < 0 at 150 GPa, where the stability criterion is violated, implying that the structure is unstable at a-b plane. The modulated structure (such as oF28) is most stability structure. When approaching 160 GPa, the elastic constants of Cmca structure all taken place obvious mutations, particularly the C 45and C 55 drop to zero at about 170 GPa, implying that the phase transition happens. We assume that incommensurate structure coexists with phaseⅡ over a pressure range of 157-170 GPa until fully transformation the bco above 170 GPa.
Later on, we calculated the electronic band structure, the partial electronic density of states (DOS), and the valence electron localization function and the scaled volume versus pressure. It can be clearly seen that chlorine elements transformed from atmospheric pressure insulator structure to high pressure metal structures at 130 GPa, and this change is caused by the gap closure under pressure. The crystal structure remains the same as that of ambient pressure diatomic molecular phase. By the structural scaling rule for Cl, its semi-metallization will take place at 130 GPa. The ELF clearly shows the electrons localize condition of molecular phase, modulated phase and monatomic phases. Above 130 GPa, the electrons which located at the sides of atoms transfer to the intermolecular space, resulting in metallization. At 157 GPa, the molecular dissociation takes place and the structure transforms to a monatomic bco structure. On the basis of a comparison of interatomic distances, elastic stiffness coefficients and Raman patterns under pressures, we found that Ag mode and shear elastic stiffness coefficient C 55 soften induces the metallization. This will lead to the emergence of incommensurate modulated structure ultimately.
Finally, we studied the superconductivity of chlorine bco and fcc metal phases under high pressure. We found that element chlorine become superconducting at high pressures. The superconducting temperatures of bco and fcc phases are 3.73 and 13.03 K at 160 and 380 GPa, respectively. The superconducting temperature of bco decreased slowly under pressure. The superconducting temperature of fcc decreased under pressure and then increased. The superconducting temperature is 11.2 K at 420 GPa and 13.21 K at 500 GPa, respectively. Although the trends in the calculated superconducting properties are similar with iodine and bromine, the superconducting transition temperature of fcc phase is higher in chlorine than in iodine and bromine, even within the same structural phase. This is probably because that the electronic crossing Fermi level features apparently satisfy the“flatband-steep band”scenario, which has been suggested to be a favorable condition for the occurrence of superconductivity.
引文
[1]池元斌.《高压物理》(参考资料)[M].吉林大学超硬材料国家重点实验室.
[2] KENICHI T, KYOKO S, HIROSHI F, et al. Modulated structure of solid iodine during its molecular dissociation under high pressure[J]. Nature, 2003, 423(6943):971-974.
[3] REICHLIN R, BRISTER K E, MCMAHAN A K, et al. Evidence for the insulator-metal transition in xenon from optical, X-ray, and band-structure studies to 170 GPa[J]. Physical Review Letters, 1989, 62(6):669.
[4] ASHCROFT N W. Hydrogen dominant metallic alloys: high temperature superconductors?[J]. Physical Review Letters, 2004, 92(18):187002.
[5] SHIMIZU K, ISHIKAWA H, TAKAO D, et al. Superconductivity in compressed lithium at 20 K[J]. Nature, 2002, 419(6907):597-599.
[6] STRUZHKIN V V, EREMETS M I, GAN W, et al. Superconductivity in dense lithium[J]. Science, 2002, 298(5596):1213-1215.
[7] YABUUCHI T, MATSUOKA T, NAKAMOTO Y, et al. Superconductivity of Ca exceeding 25 K at megabar pressures[J]. Journal of the Physical Society of Japan, 75(8):083703.
[8] DEBESSAI M, HAMLIN J J,SCHILLING J S. Comparison of the pressure dependences of TC in the trivalent d -electron superconductors Sc, Y, La, and Lu up to megabar pressures[J]. Physical Review B, 2008, 78(6):064519.
[9] MA Y, EREMETS M, OGANOV A R, et al. Transparent dense sodium[J]. Nature (London), 2009, 458(7235):182-185.
[10]张流.地震地质论文集[M].天津科学技术出版社. 1986.
[11]出内智子等.高压科学与加压食品[M].日本:三英出版社. 1991.
[12]徐如人,庞文琴,霍启.无机合成与制备化学[M].北京:高等教育出版社. 2009.
[13] BRIDGMAN P W. Collected experimental papers[M]. Cambridge, MA, USA: Harvard University Press. 1964. 4721.
[14] MCMAHON M I,NELMES R J. High-pressure structures and phase transformations in elemental metals[J]. Chemical Society Reviews, 2006, 35(10):943-963.
[15] PETRICEK V, MALY K, COPPENS P, et al. The description and analysis ofcomposite crystals[J]. Acta Crystallographica Section A, 1991, 47(3):210-216.
[16] NELMES R J, ALLAN D R, MCMAHON M I, et al. Self-hosting incommensurate structure of barium IV[J]. Physical Review Letters, 1999, 83(20):4081.
[17] MCMAHON M I, BOVORNRATANARAKS T, ALLAN D R, et al. Observation of the incommensurate barium-IV structure in strontium phase V[J]. Physical Review B, 2000, 61(5):3135.
[18] MCMAHON M I, REKHI S,NELMES R J. Pressure dependent incommensuration in Rb-IV[J]. Physical Review Letters, 2001, 87(5):055501.
[19] MCMAHON M I, DEGTYAREVA O,NELMES R J. Ba-IV-Type incommensurate crystal structure in group-V metals[J]. Physical Review Letters, 2000, 85(23):4896.
[20] DEGTYAREVA O, MCMAHON M I,NELMES R J. Pressure-induced incommensurate-to-incommensurate phase transition in antimony[J]. Physical Review B, 2004, 70(18):184119.
[21] MCMAHON M I, LUNDEGAARD L F, HEJNY C, et al. Different incommensurate composite crystal structure for Sc-II[J]. Physical Review B, 2006, 73(13):134102.
[22] MCMAHON M I, NELMES R J, SCHWARZ U, et al. Composite incommensurate K-III and a commensurate form: Study of a high-pressure phase of potassium[J]. Physical Review B, 2006, 74(14):140102.
[23] LUNDEGAARD L F, GREGORYANZ E, MCMAHON M I, et al. Single-crystal studies of incommensurate Na to 1.5 Mbar[J]. Physical Review B, 2009, 79(6):064105.
[24] SMITH H G,LANDER G H. Neutron scattering investigations of alpha -uranium in the charge-density-wave state[J]. Physical Review B, 1984, 30(10):5407.
[25] HEJNY C,MCMAHON M I. Large Structural Modulations in Incommensurate Te-III and Se-IV[J]. Physical Review Letters, 2003, 91(21):215502.
[26] MCMAHON M I, HEJNY C, LOVEDAY J S, et al. Confirmation of the incommensurate nature of Se-IV at pressures below 70GPa[J]. Physical Review B, 2004, 70(5):054101.
[27] HEJNY C, LUNDEGAARD L F, FALCONI S, et al. Incommensurate sulfur above 100GPa[J]. Physical Review B, 2005, 71(2):020101.
[28] KUME T, HIRAOKA T, OHYA Y, et al. High pressure raman study of bromineand iodine: soft phonon in the incommensurate phase[J]. Physical Review Letters, 2005, 94(6):065506.
[29] MCMAHON M I, NELMES R J,REKHI S. Complex crystal structure of cesium-III[J]. Physical Review Letters, 2001, 87(25):255502.
[30] NELMES R J, MCMAHON M I, LOVEDAY J S, et al. Structure of Rb-III: novel modulated stacking structures in alkali metals[J]. Physical Review Letters, 2002, 88(15):155503.
[31] DEGTYAREVA O, MCMAHON M I, ALLAN D R, et al. Structural complexity in gallium under high pressure: relation to alkali elements[J]. Physical Review Letters, 2004, 93(20):205502.
[32] HANFLAND M, SYASSEN K, CHRISTENSEN N E, et al. New high-pressure phases of lithium[J]. Nature, 2000, 408(6809):174-178.
[33] REED S K,ACKLAND G J. Theoretical and computational study of high-pressure structures in barium[J]. Physical Review Letters, 2000, 84(24):5580.
[34] H USSERMANN U, S DERBERG K,NORRESTAM R. Comparative study of the high-pressure behavior of As, Sb, and Bi[J]. Journal of the American Chemical Society, 2002, 124(51):15359-15367.
[35] KATZKE H, TOL, EACUTE, et al. Structural mechanisms and order-parameter symmetries for the high-pressure phase transitions in alkali metals[J]. Physical Review B, 2005, 71(18):184101.
[36] TSE J S. Crystallography of selected high pressure elemental solids[J]. Zeitschrift für Kristallographie, 2005, 220(5-6-2005):521-530.
[37] DONOHUE J.The structures of the elements[M]. Michigan:Wiley. 1974.
[38] KAZUTAKA N,ET AL. X-ray diffraction study of Be to megabar pressure[J]. Journal of Physics: Condensed Matter, 2002, 14(44):10569.
[39] OLIJNYK H,HOLZAPFEL W B. High-pressure structural phase transition in Mg[J]. Physical Review B, 1985, 31(7):4682.
[40] OLIJNYK H,HOLZAPFEL W B. Phase transitions in alkaline earth metals under pressure [J]. Physics Letters, 1984, 100A:191.
[41] BOVORNRATANARAKS T, ALLAN D R, BELMONTE S A, et al. Complex monoclinic superstructure in Sr-IV[J]. Physical Review B, 2006, 73(14):144112.
[42] KENICHI T. High-pressure structural study of barium to 90 GPa[J]. PhysicalReview B, 1994, 50(22):16238.
[43] PAULING L, KEAVENY I,ROBINSON A B. The crystal structure of [alpha]-fluorine[J]. Journal of Solid State Chemistry, 1970, 2(2):225-227.
[44] SCHIFERL D, KINKEAD S, HANSON R C, et al. Raman spectra and phase diagram of fluorine at pressures up to 6 GPa and temperatures between 10 and 320 K[J]. Journal of Chemical Physics, 1987, 87(5):3016-3021.
[45] POWELL, B M, HEAL, et al.The temperature dependence of the crystal structures of the solid halogens, bromine and chlorine[M]. London, ROYAUME-UNI:Taylor & Francis. 1984.
[46] JOHANNSEN P G,HOLZAPFEL W B. Effect of pressure on raman spectra of solid chlorine[J]. Journal of Physics C: Solid State Physics, 1983, 16:L1177.
[47] VONNEGUT B,WARREN B E. The structure of crystalline bromine[J]. Journal of the American Chemical Society, 1936, 58(12):2459-2461.
[48] TAKEMURA K, MINOMURA S, SHIMOMURA O, et al. Structural aspects of solid iodine associated with metallization and molecular dissociation under high pressure[J]. Physical Review B, 1982, 26(2):998.
[49] FUJII Y, HASE K, OHISHI Y, et al. Pressure-induced monatomic tetragonal phase of metallic iodine[J]. Solid State Communications, 1986, 59(2):85-89.
[50] FUJII Y, HASE K, HAMAYA N, et al. Pressure-induced face-centered-cubic phase of monatomic metallic iodine[J]. Physical Review Letters, 1987, 58(8):796.
[51] REICHLIN R, MCMAHAN A K, ROSS M, et al. Optical, x-ray, and band-structure studies of iodine at pressures of several megabars[J]. Physical Review B, 1994, 49(6):3725.
[52] PASTERNAK M, FARRELL J N,TAYLOR R D. Metallization and structural transformation of iodine under pressure: A microscopic view[J]. Physical Review Letters, 1987, 58(6):575.
[53] OLIJNYK H, LI W,WOKAUN A. High-pressure studies of solid iodine by Raman spectroscopy[J]. Physical Review B, 1994, 50(2):712.
[54] ZENG Q, HE Z, SAN X, et al. A new phase of solid iodine with different molecular covalent bonds[J]. Proceedings of the National Academy of Sciences, 2008, 105(13):4999-5001.
[55] FUJIHISA H, FUJII Y, TAKEMURA K, et al. Structural aspects of dense solid halogens under high pressure studied by x-ray diffraction--Moleculardissociation and metallization[J]. Journal of Physics and Chemistry of Solids, 1995, 56(10):1439-1444.
[56] FUJII Y, HASE K, OHISHI Y, et al. Evidence for molecular dissociation in bromine near 80 GPa[J]. Physical Review Letters, 1989, 63(5):536.
[57] SAN-MIGUEL A, LIBOTTE H, GAUTHIER M, et al. New phase transition of solid bromine under high pressure[J]. Physical Review Letters, 2007, 99(1):015501.
[58] SAN MIGUEL A, LIBOTTE H, GASPARD J P, et al. Bromine metallization studied by X-ray absorption spectroscopy[J]. The European Physical Journal B - Condensed Matter and Complex Systems, 2000, 17(2):227-233.
[59] MUKOSE K, FUKANO R, MIYAGI H, et al. First-principles studies of solid halogens under pressure: scaling rules for properties among I 2 , Br 2 and Cl 2[J]. Journal of Physics: Condensed Matter, 2002, 14(44):10441.
[60] MIAO M S, VAN DOREN V E, MARTINS J, et al. Density-functional studies of high-pressure properties and molecular dissociations of halogen molecular crystals[J]. Physical Review B, 2003, 68(9):094106.
[61] WIGNER E,HUNTINGTON H B. On the possibility of a metallic modification of hydrogen[J]. Journal of Chemical Physics, 1935, 3:764-770.
[62] NARAYANA C, LUO H, ORLOFF J, et al. Solid hydrogen at 342 GPa: no evidence for an alkali metal[J]. Nature, 1998, 393(6680):46-49.
[63] LOUBEYRE P, OCCELLI F,LETOULLEC R. Optical studies of solid hydrogen to 320 GPa and evidence for black hydrogen[J]. Nature, 2002, 416(6881):613-617.
[64] GUOYING G, ARTEM R O, YANMING M, et al. Dissociation of methane under high pressure[J]. Journal of Chemical Physics, 2010, 133(14):144508.
[65] LI Y, GAO G, XIE Y, et al. Superconductivity at
[68] SZABO A,OSTLUND N S.Modern quantum chemistry[M].McGraw-Hill New York. 1989.
[69] KOHN W, BECKE A D,PARR R G. Density functional theory of electronic structure[J]. Journal of Physics Chemical 1996, 100(12):974.
[70] DREIZLER R M,GROSS E K U.Density functional theory[M].Springer. 1990.
[71] PARR R G,YANG W.Density-functional theory of atoms and molecules[M].Oxford University Press, USA. 1989.
[72] KOHN W. Nobel lecture: electronic structure of matter-wave functions and density functionals[J]. Reviews of Modern Physics, 1999, 71(5):1253-1266.
[73] KOHN W,SHAM L J. Self-consistent equations including exchange and correlation effects[J]. Physical Review, 1965, 140(4A):A1133.
[74]张立军.高压下几种材料中声子及电子—声子耦合的第一性原理研究[D].长春:吉林大学,2008.
[75] KOHN W,SHAM L J. Self-consistent equations including exchange and correlation effects[J]. Physical Review, 1965, 140:A1133.
[76]阎守胜.现代固体物理学导论[M].北京:北京大学出版社. 2008.
[77] PERDEW J P. Density-functional approximation for the correlation energy of the inhomogeneous electron gas[J]. Physical Review B, 1986, 33(12):8822-8824.
[78] PERDEW J P, BURKE K,ERNZERHOF M. Generalized Gradient Approximation Made Simple[J]. Physical Review Letters, 1996, 77(18):3865-3868.
[79] PERDEW J P,WANG Y. Pair-distribution function and its coupling-constant average for the spin-polarized electron gas[J]. Physical Review B, 1992, 46(20):12947-12954.
[80] WIGNER E P. Effects of the electron interaction on the energy levels of electrons in metals[J]. Transactions of the Faraday Society 1938, 34:678.
[81] JONES R O,GUNNARSSON O. The density functional formalism, its applications and prospects[J]. Reviews of Modern Physics, 1989, 61(3):689-746.
[82] HAFNER J. Atomic-scale computational materials science[J]. Acta Materialia, 2000, 48(1):71-92.
[83] PERDEW J P,ZUNGER A. Self-interaction correction to density-functional approximations for many-electron systems[J]. Physical Review B, 1981,23(10):5048.
[84] PERDEW J,WANG Y. Pair-distribution function and its coupling-constant average for the spin-polarized electron gas[J]. Physical Review B, 1992, 46(20):12947-12954.
[85] PERDEW J, BURKE K,ERNZERHOF M. Generalized gradient approximation made simple[J]. Physical Review Letters, 1996, 77(18):3865-3868.
[86] TERSOFF J. Empirical interatomic potential for carbon, with applications to amorphous carbon[J]. Physical Review Letters, 1988, 61(25):2879-2882.
[87] COWLEY E R. Lattice dynamics of silicon with empirical many-body potentials[J]. Physical Review Letters, 1988, 60(23):2379-2381.
[88] BARONI S, DE GIRONCOLI S, DAL CORSO A, et al. Phonons and related crystal properties from density-functional perturbation theory[J]. Reviews of Modern Physics, 2001, 73(2):515-562.
[89] GONZE X, ALLAN D C,TETER M P. Dielectric tensor, effective charges, and phonons in a-quartz by variational density-functional perturbation theory[J]. Physical Review Letters, 1992, 68(24):3603-3606.
[90] PARLINSKI K.PHONON manual 4.28[CP/DK]. 2007.
[91] BORN M,HUANG K.Dynamical theory of crystal lattices[M].Oxford University Press. 1998 (有中译本,葛惟锟,贾惟义译,江丕桓校,北京大学出版社,1989).
[92] WENDEL H,MARTIN R M. Theory of structural properties of covalent semiconductors[J]. Physical Review B, 1979, 19(10):5251-5264.
[93] YIN M T,COHEN M L. Theory of static structural properties, crystal stability, and phase transformations: Application to Si and Ge[J]. Physical Review B, 1982, 26(10):5668-5687.
[94] KUNC K,MARTIN R M. Ab initio force constants of GaAs: a new approach to calculation of phonons and dielectric properties[J]. Physical Review Letters, 1982, 48(6):406-409.
[95] FRANK W, ELSASSER C,FAHNLE M. Ab initio force-constant method for phonon dispersions in alkali metals[J]. Physical Review Letters, 1995, 74(10):1791-1794.
[96] PARLINSKI K, LI Z Q,KAWAZOE Y. First-Principles Determination of the Soft Mode in Cubic ZrO_ {2}[J]. Physical Review Letters, 1997,78(21):4063-4066.
[97] QUONG A A,KLEIN B M. Self-consistent-screening calculation of interatomic force constants and phonon dispersion curves from first principles[J]. Physical Review B, 1992, 46(17):10734-10737.
[98] KOHANOFF J. Phonon spectra from short non-thermally equilibrated molecular dynamics simulations[J]. Computational Materials Science, 1994, 2:221?232.
[99] MARUYAMA S. A molecular dynamics simulation of heat conduction in finite length SWNTs[J]. Physica B: Physics of Condensed Matter, 2002, 323(1-4):193-195.
[100] MADDOX J. Crystals from first principles[J]. Nature, 1988, 335(6187):201-201.
[101] GAVEZZOTTI A. Are crystal structures predictable?[J]. Accounts of Chemical Research, 1994, 27(10):309-314.
[102] MARTONCARON, AACUTE, K R, et al. Predicting crystal structures: the parrinello-rahman method revisited[J]. Physical Review Letters, 2003, 90(7):075503.
[103] PANNETIER J, BASSAS-ALSINA J, RODRIGUEZ-CARVAJAL J, et al. Prediction of crystal structures from crystal chemistry rules by simulated annealing[J]. Nature, 1990, 346(6282):343-345.
[104] WALES D J,DOYE J P K. Global optimization by basin-hopping and the lowest energy structures of lennard-jones clusters containing up to 110 atoms[J]. The Journal of Physical Chemistry A, 1997, 101(28):5111-5116.
[105] STEFAN G. Minima hopping: An efficient search method for the global minimum of the potential energy surface of complex molecular systems[J]. Journal of Chemical Physics, 2004, 120(21):9911-9917.
[106] GLASS C W, OGANOV A R,HANSEN N. USPEX--Evolutionary crystal structure prediction[J]. Computer Physics Communications, 2006, 175(11-12):713-720.
[107] OGANOV A R, MA Y, XU Y, et al. Exotic behavior and crystal structures of calcium under pressure[J]. Proceedings of the National Academy of Sciences, 2010, 107(17):7646-7651.
[108] YOSHIDA H, KAWATA K, FUKUYAMA Y, et al. A particle swarm optimization for reactive power and voltage control considering voltage security assessment[J]. Power Systems, IEEE Transactions on, 2000, 15(4):1232-1239.
[109] PARSOPOULOS K E,VRAHATIS M N. On the computation of all global minimizers through particle swarm optimization[J]. Evolutionary Computation, IEEE Transactions on, 2004, 8(3):211-224.
[110] WANG Y, LV J, ZHU L, et al. Crystal structure prediction via particle swarm optimization [J]. 2010,
[111] LV J, WANG Y, ZHU L, et al. Predicted novel high-pressure phases of lithium[J]. Physical Review Letters, 106(1):015503.
[112]黄昆.固体物理学[M].北京:高等教育出版社. 1998.
[113] BARDEEN J, COOPER L N,SCHRIEFFER J R. Theory of Superconductivity[J]. Physical Review, 1957, 108(5):1175-1204.
[114] BARDEEN J, COOPER L N,SCHRIEFFER J R. Microscopic Theory of Superconductivity[J]. Physical Review, 1957, 106(1):162-164.
[115] G. M. Eliashberg, Zh. Eksperim. i Teor. Fiz. 38, 966 (1960); 39, 1437 (1960); [English transls.: Soviet Phys.—JETP 11, 696 (1960); 12, 1000 (1961)][M].
[116] NAMBU Y. Quasi-Particles and Gauge Invariance in the Theory of Superconductivity[J]. Physical Review, 1960, 117(3):648-663.
[117] MCMILLAN W L. Transition Temperature of Strong-Coupled Superconductors[J]. Physical Review, 1968, 167(2):331-344.
[118] ALLEN P B,DYNES R C. Transition temperature of strong-coupled superconductors reanalyzed[J]. Physical Review B, 1975, 12(3):905-922.
[119] PARLINSKI K.Software PHONON[CP/DK]. Crocow 2006.
[120] CHOI H J, ROUNDY D, SUN H, et al. The origin of the anomalous superconducting properties of MgB2[J]. Nature, 2002, 418(6899):758-760.
[121] NAGAMATSU J, NAKAGAWA N, MURANAKA T, et al. Superconductivity at 39 K in magnesium diboride[J]. Nature, 2001, 410(6824):63-64.
[122] NYE J F.Physical properties of crystals: their representation by tensors and matrices[M].Clarendon Press. 1985.
[123] YOUNG D A.Phase diagram of the elements[M]. Berkeley:University of California Press. 1990.
[124] MCMAHON M I, BOVORNRATANARAKS T, ALLAN D R, et al. Observation of the incommensurate barium-IV structure in strontium phase V[J]. Phys. Rev. B, 2000, 61(5):3135.
[125] MARQU S M, ACKLAND G J, LUNDEGAARD L F, et al. Potassium under pressure: a pseudobinary ionic compound[J]. Physical Review Letters, 2009, 103(11):115501.
[126] MA Y, OGANOV A R,XIE Y. High-pressure structures of lithium, potassium, and rubidium predicted by an ab initio evolutionary algorithm[J]. Physical Review B, 2008, 78(1):014102.
[127] NEATON J B,ASHCROFT N W. Pairing in dense lithium[J]. Nature (London), 1999, 400(6740):141-144.
[128] ERRANDONEA D, MENG Y, HUSERMANN D, et al. Study of the phase transformations and equation of state of magnesium by synchrotron x-ray diffraction[J]. Journal of Physics: Condensed Matter, 2003, 15:1277.
[129] MCMAHAN A K,MORIARTY J A. Structural phase stability in third-period simple metals[J]. Physical Review B, 1983, 27(6):3235.
[130] AHUJA R, ERIKSSON O, WILLS J M, et al. Theoretical confirmation of the high pressure simple cubic phase in calcium[J]. Physical Review Letters, 1995, 75(19):3473.
[131] KENNEDY J,EBERHART R. Particle swarm optimization[C]. in Neural Networks, 1995. Proceedings., IEEE International Conference on, 1995: 1942-1948
[132] EBERHART R,KENNEDY J. A new optimizer using particle swarm theory[C]. in Micro Machine and Human Science, 1995. MHS '95., Proceedings of the Sixth International Symposium on, 1995: 39-43.
[133] KRESSE G,FURTHMULLER J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set[J]. Physical Review B, 1996, 54(16):11169.
[134] PERDEW J P, BURKE K,ERNZERHOF M. Generalized Gradient Approximation Made Simple[J]. Physical Review Letters, 1996, 77(18):3865.
[135] BLOCHL P E. Projector augmented-wave method[J]. Physical Review B, 1994, 50(24):17953.
[136] BLAHA P, SCHWARZ K, MADSEN G K H, et al.WIEN2k, an augmented plane wave + local orbitals program for calculating crystal properties[CP/DK]. Karlheinz Schwarz, Techn. Universit?t Wien, Austria 2001.
[137] CLENDENEN G L,DRICKAMER H G. Effect of pressure on the volume and lattice parameters of magnesium[J]. Physical Review, 1964, 135(6A):A1643.
[138] XU Y, ZHANG L, CUI T, et al. First-principles study of the lattice dynamics, thermodynamic properties and electron-phonon coupling of Y B6[J]. Physical Review B, 2007, 76(21):214103.
[139] DMITRIEV V P, GUFAN Y M,TOL DANO P. Theory of the phase diagram ofiron and thallium: The Burgers and Bain deformation mechanisms revised[J]. Physical Review B, 1991, 44(14):7248.
[140] AKAHAMA Y, KOBAYASHI M,KAWAMURA H. Simple-cubic-simple—hexagonal transition in phosphorus under pressure[J]. Physical Review B, 1999, 59(13):8520.
[141] HU J Z,SPAIN I L. Phases of silicon at high pressure[J]. Solid State Communications, 1984, 51:263.
[142] HU J Z, MERKLE L D, MENONI C S, et al. Crystal data for high-pressure phases of silicon[J]. Physical Review B, 1986, 34(7):4679.
[143] OLIJNYK H, SIKKA S K,HOLZAPFEL W B. Structural phase transitions in Si and Ge under pressures up to 50 GPa[J]. Physics Letters, 1984, 103A:137.
[144] AKAHAMA Y, NISHIMURA M, KINOSHITA K, et al. Evidence of a fcc-hcp transition in aluminum at multimegabar pressure[J]. Physical Review Letters, 2006, 96(4):045505.
[145] TAMBE M J, BONINI N,MARZARI N. Bulk aluminum at high pressure: A first-principles study[J]. Physical Review B, 2008, 77(17):172102.
[146] BECKE A D,EDGECOMBE K E. A simple measure of electron localization in atomic and molecular systems[J]. The Journal of Chemical Physics, 1990, 92(9):5397-5403.
[147] JAMESON J C. Crystal structures of titanium, zirconium, and hafnium at high pressures[J]. Science, 1963, 140(3562):72-73.
[148] SIKKA S K, VOHRA Y K,CHIDAMBARAM R. Omega phase in materials[J]. Progress in Materials Science, 1982, 27(3-4):245-310.
[149] MEHTA S, PRICE G D,ALFE D. Ab initio thermodynamics and phase diagram of solid magnesium: A comparison of the LDA and GGA[J]. Journal of Chemical Physics, 2006, 125(19):194507.
[150] MORIARTY J A,ALTHOFF J D. First-principles temperature-pressure phase diagram of magnesium[J]. Physical Review B, 1995, 51(9):5609.
[151] COLLIN R. The crystal structure of solid chlorine[J]. Acta Crystallographica, 1956, 9(6):537.
[152] DONOHUE J,GOODMAN S H. Interatomic distances in solid chlorine[J]. Acta Crystallographica, 1965, 18(3):568-569.
[153] MONKHORST H J,PACK J D. Special points for brillouin-zone integrations[J]. Physical Review B, 1976, 13(12):5188-5192.
[154] ACKLAND G J,FOX H. Total energy calculation for high pressure selenium:the origin of incommensurate modulations in Se IV and the instability of proposed Se II[J]. Journal of Physics: Condensed Matter, 2005, 17(12):1851.
[155] ISHIKAWA T, NAGARA H, KUSAKABE K, et al. Determining the structure of phosphorus in phase IV[J]. Physical Review Letters, 2006, 96(9):095502.
[156] LOA I, MCMAHON M I,BOSAK A. Origin of the incommensurate modulation in Te-III and fermi-surface nesting in a simple metal[J]. Physical Review Letters, 2009, 102(3):035501.
[157] SAN X,ET AL. Theoretical calculations of phase transitions and optical properties of solid iodine under high pressures[J]. Journal of Physics: Condensed Matter, 2008, 20(17):175225.
[158] ORITA N, NIIZEKI K, SHINDO K, et al. The band structure of solid iodine under pressure and the mechanism of the pressure-induced insulator-to-metal transition[J]. Journal of the Physical Society of Japan, 1992, 61:4502.
[159] D SING E-F, GROSSHANS W A,HOLZAPFEL W B. Equation of state of solid chlorine and bromine[J]. Supplément au Journal de Physique Colloques, 1984, 45(C8):C8-203-C208-206.
[160] RUDIN S P, LIU A Y, FREERICKS J K, et al. Comparison of structural transformations and superconductivity in compressed sulfur and selenium[J]. Physical Review B, 2001, 63(22):224107.
[161] TSE J S, LI Z, UEHARA K, et al. Electron-phonon coupling in high-pressure Nb[J]. Physical Review B, 2004, 69(13):132101.
[162] SHIMIZU K, YAMAUCHI T, TAMITANI N, et al. The pressure-induced superconductivity of iodine[J]. Journal of Superconductivity, 1994, 7(6):921-924.
[163] SHIMIZU K. Review of high-pressure induced superconductivity in single elements[J]. The Review of High Pressure Science and Technology, 2010, 20(2):133-139.
[164] DUAN D, JIN X, MA Y, et al. Effect of nonhydrostatic pressure on superconductivity of monatomic iodine: An ab initio study[J]. Physical Review B, 2009, 79(6):064518.
[165] SIMON A. Superconductivity and chemistry[J]. Angewandte Chemie International Edition in English, 1997, 36(17):1788-1806.
