下载文档原格式
第一性原理发展简史（2）——霍恩贝格-科恩定理与里兹变分法•关于第一性原理这个词语在1900前哲学与数学中使用的问题很多人追问文中一段话出处，其实严格意义上逐字逐句的表述应该算作者原创，并不是直接引用，但是这样论述并非是毫无根据的。
这段内容最初是作者学生时代一门课程老师所述，写文章时已经默认这是本领域中学界的一个共识。
今天的文字资料中第一性原理(first principle)一般已经默认就是指计算材料学中的第一性原理计算，个别国家特别是德国、奥地利等德语区为了学术严谨起见则更愿意使用ab-initia（又译为从头算），美国、澳大利亚、英国等英语区国家则是二者并用，中文环境下由于历史原因主要是源自外文翻译与编辑，因此主流说法认为汉语环境“第一性原理”仅指代计算材料学。
上期之所以说1900年前第一性原理主要用于哲学、数学、理论物理，根据与逻辑如下：亚里士多德时代已经诞生了第一性原理(firstprinciple)的定义，而计算材料学的源头——量子力学诞生于1900年之后。
1900年以前的哲学、数学著作中时常可以见到first principle 这一术语的使用，当然这些著作今天流通的修订本或者是再编版已不再使用第一性原理的表述：哲学中往往用priori-principle替代之前的第一性原理表述；数学中今天已经统一使用规范术语“公理”（axioms）表述，因此今天再说第一性原理涵盖哲学、数学已经有些不合时宜。
另外第一性原理这个词语本身的使用一定程度上也体现了欧洲16世纪以来，人文主义的兴起初期理论、知识是以人为本、以人为核心的，它的出发点是希望不依赖（那时认为是上帝或神创造的）物质实验、测量建立起一个完全由人的意识引出的（与神学足以抗衡的）理论体系。
这一点上与中华神话《夸父逐日》的精髓类似，反映出人类在探索自然时不屈、奋进的精神。
•关于分子动力学是否属于第一性原理此处的分子动力学特指molecule dynamics，简称MD，中文环境中由于各种原因“分子动力学”可一词可能涵盖其他意义，如作者接触过的物理化学中的分子马达领域将相关的内容称为分子动力学。
第50卷第3期2021年5月内蒙古师范大学学报(自然科学版)Journal of Inner Mongolia Normal University(Natural Science Edition)Vol.50No.3May2021六硼化钇纳米粒子超导及光吸收性能研究王军】，包黎红】，潮洛蒙2（1.内蒙古师范大学物理与电子信息学院，内蒙古呼和浩特010022；2.内蒙古科技大学理学院，内蒙古包头014010）摘要：采用固相烧结法成功制备出了六硼化钇（YB&）纳米粒子，首次系统研究了该纳米粉末超导及光吸收性能.结果表明，当超导转变温度T=2.75K时由正常态转变为超导态，其临界磁场为H c2=0.18T.为进一步研究YB6纳米粒子电-声子相互作用机理，采用拉曼光谱对声子振动频率进行了测量，结合McMillan公式计算出YB6纳米粒子电-声子作用常数为A=0.63,该值远小于单晶块体YB6的1.01.为进一步解释其原因，采用高分辨透射电镜对晶体缺陷进行了详细表征.结果发现，晶体缺陷导致其声子振动频率的改变，从而降低了纳米YB6电-声子相互作用常数.光吸收结果表明YB6纳米粒子吸收谷波长为785nm,对可见光具有很强的穿透性.关键词：超导性；光吸收；YB6中图分类号：O511+.3文献标志码：A文章编号：1001—8735（2021）03—0204—06doi：10.3969/j.issn.1001—8735.2021.03.003众所周知，材料的宏观物理化学性能与微观结构密切相关[12].特别是当材料晶粒尺寸减小到纳米尺度后，纳米晶材料不仅具有亚稳态的特点，而且相比于粗晶材料展现出许多新奇的物理化学性能.与此同时,材料的纳米化会改变材料电子态密度及电-声子相互等物理量，从而会对超导及光学性能有很大的影响[34].因此，如何将纳米材料微观结构与宏观性能之间进行有效关联，将对材料新性能的发现和研究具有重要作用.在众多金属硼化物中，由于六硼化钇YB6具有第二高超导转变温度T c=&4K,故其超导性能受到广泛关注.目前关于这方面的研究主要集中于单晶YB6块体材料上[8],而对于YB6纳米离子超导性能研究未见报道.研究者们为了解释单晶块体超导机理及提高临界转变温度，系统研究了压强对YB6单晶块体晶体结构和声子振动的影响[912].结果发现，当压强从0增加至40GPa时，电子-声子相互作用常数从1.44减小到0.44.与此同时，相对应的超导转变温度也从&9K减小到1.4K,表明压强对电-声子作用具有很大影响.这项研究工作的一个重要提示是,YB6的纳米化是否会改变费米能级周围的电子态密度及电子-声子相互作用，从而展现出一些新的超导性能，这是本文的重要研究内容之一.此外，虽然YB6与LaB6具有相同立方晶体结构[13],但它是否同样具有对可见光的高穿透特性，是本论文另一个重要研究内容.目前国内外对纳米YB6超导性能及光吸收实验方面的系统研究未见报道.本文首次系统研究了YB6纳米粒子超导及光吸收性能.为进一步揭示材料微观结构与宏观性能之间的内在关联，采用高分辨透射电镜和拉曼光谱等测量手段对微观结构进行了有效表征，并对超导及光吸收机理进行探讨.收稿日期：2020-11-30基金项目：国家自然科学基金资助项目（51662034）；内蒙古自治区自然科学基金资助项目（2019LH05001）；内蒙古自治区留学人员创新创业启动基金资助项目.作者简介：王军（1992-），男，内蒙古阿拉善左旗人，在读硕士研究生，主要从事纳米稀土六硼化物光吸收及热发射性能研究.通讯作者：包黎红（1983—）,男，内蒙古兴安盟人，教授，博士，主要从事纳米金属硼化物物理性能研究,E-mail:baolihong@.第3期王军等：六硼化钇纳米粒子超导及光吸收性能研究-205-1材料与方法将无水氯化钇(YCl,纯度99.95%)和硼氢化钠(NaBH4,纯度98%)粉末在空气中按摩尔比为1：&8混合研磨10〜15min.将混合均匀的粉末放入压机中，在压强为10GPa下预压成块，将其装入石英管中进行真空烧结.反应温度为1100C,保温2h.由于固相反应后产物中有YBO s的杂相，故对烧结后产物分别使用稀盐酸，蒸馏水，无水乙醇等溶液进行多次清洗.采用场发射扫描电子显微镜(日立SU-8010)和X射线衍射仪(飞利浦PW1830,CuKa)对YB6纳米粒子的物相及形貌进行表征.采用PPMS测量仪对纳米YB6交流磁化率和临界磁场进行了测量，最低温度为1.8K.采用透射电子显微镜(FEITecnai F20S-Twin200kv)观察微观结构.拉曼散射由拉曼光谱仪(LabRamHR,波长：514.5nm,激光源:Ar+)进行测量.采用分光光度计(UH4150)在光源波长350〜2500nm范围内测量其光吸收。
量子力学第一性原理：仅需五个物理基本常数——电子质量、电子电量、普郎克常数、光速和玻耳兹曼常数，通过求薛定谔方程得到材料的电子结构，而不依赖于任何经验常数即可以预测微观体系的状态和性质，预测材料的组分、结构、性能之间的关系，进一步设计具有特定性能的新材料。
作为评价事物的依据，第一性原理和经验参数是两个极端。
第一性原理是某些硬性规定或推演得出的结论，而经验参数则是通过大量实例得出的规律性的数据，这些数据可以来自第一性原理(称为理论统计数据)，也可以来自实验(称为实验统计数据)。
如果某些原理或数据来源于第一性原理，但推演过程中加入了一些假设(这些假设当然是很有说服力的)，那么这些原理或数据就称为“半经验的”。
量子化学的第一性原理是指多电子体系的Schrödinger方程，但是光有这个方程是无法解决任何问题的，量子力学能够准确的解决的问题很少很少，绝大多数都是有各种各样的近似，为此计算量子力学提出一个称为“从头计算”的原理作为第一性原理，除了Schrödinger方程外还允许使用下列参数和原理：(1) 物理常数，包括光速c、Planck常数h、电子电量e、电子质量me以及原子的各种同位素的质量，尽管这些常数也是通过实验获得的。
(在国际单位值中，光速是定义值，Planck 常数是测量值，在原子单位制中则相反。
)(2) 各种数学和物理的近似，最基本的近似是“非相对论近似”(Schrödinger方程本来就是非相对论的原理)、“绝热近似”(由于原子核质量比电子大得多，而把原子核当成静止的点处理)和“轨道近似”(用一个独立函数来描述一个独立电子的运动)。
量子化学的从头计算方法就是在各种近似上作的研究。
如果只考虑一个电子，而把其他电子对它的作用近似的处理成某种形式的势场，这样就可以把多电子问题简化成单电子问题，这种近似称为单电子近似，也称为平均场近似，例如最基本的从头计算方法哈特里－富克（Hartree-Fock）方法，是平均场近似的一种，它把所有讨论的电子视为在离子势场和其他电子的平均势场中的运动。
第一性原理根据原子核和电子互相作用的原理及其基本运动规律，运用量子力学原理，从具体要求出发，经过一些近似处理后直接求解薛定谔方程的算法，习惯上称为第一原理第一性原理通常是跟计算联系在一起的，是指在进行计算的时候除了告诉程序你所使用的原子和他们的位置外，没有其他的实验的，经验的或者半经验的参量，且具有很好的移植性。
作为评价事物的依据，第一性原理和经验参数是两个极端。
第一性原理是某些硬性规定或推演得出的结论，而经验参数则是通过大量实例得出的规律性的数据，这些数据可以来自第一性原理(称为理论统计数据)，也可以来自实验(称为实验统计数据)。
但是就某个特定的问题，第一性原理和经验参数没有明显的界限，必须特别界定。
如果某些原理或数据来源于第一性原理，但推演过程中加入了一些假设(这些假设当然是很有说服力的)，那么这些原理或数据就称为“半经验的”。
第一性原理，英文First Principle，是一个计算物理或计算化学专业名词，广义的第一性原理计算指的是一切基于量子力学原理的计算。
我们知道物质由分子组成，分子由原子组成，原子由原子核和电子组成。
量子力学计算就是根据原子核和电子的相互作用原理去计算分子结构和分子能量（或离子），然后就能计算物质的各种性质。
从头算（ab initio）是狭义的第一性原理计算，它是指不使用经验参数，只用电子质量，光速，质子中子质量等少数实验数据去做量子计算。
但是这个计算很慢，所以就加入一些经验参数，可以大大加快计算速度，当然也会不可避免的牺牲计算结果精度。
那为什么使用“第一性原理”这个字眼呢？据说这是来源于“第一推动力”这个宗教词汇。
第一推动力是牛顿创立的，因为牛顿第一定律说明了物质在不受外力的作用下保持静止或匀速直线运动。
如果宇宙诞生之初万事万物应该是静止的，后来却都在运动，是怎么动起来的呢？牛顿相信这是由于上帝推了一把，并且牛顿晚年致力于神学研究。
现代科学认为宇宙起源于大爆炸，那么大爆炸也是有原因的吧。
Nb2N3的力学和电子性质的第一性原理祝颖;徐丹;李全军;陈洪斌【摘要】The crystal structure, dynamics, lattice dynamics, and electronic properties of Nb2 N3 were studied by means of first-principles method of the density functional theory. The results show that Nb2N3 has orthogonal 7;-Ta2N3 structure at normal pressure, the elastic constants of this structure meet the Bonn-Huangkun criterion, and its lattice is dynamics stable. Nb2N3 has large bulk modulus (304 Gpa) and hardness (19. 3 Gpa). Because of the hybridization of 4d orbital of Nb and 2p orbital of N forming three-dimensional Nb-N covalent bonds, Nb2N3 is ionic semiconductor materials.%采用基于密度泛函理论的第一性原理方法研究Nb2N3的晶体结构、力学、晶格动力学和电子性质.结果表明:Nb2N3常压下具有正交η-Ta2N3结构,其弹性常数满足波恩-黄昆判据,且晶格动力学稳定；Nb2N3具有较大的体弹性模量(304 GPa)和硬度(19.3 GPa),由于Nb 4d轨道与N2p轨道杂化形成三维Nb-N共价键,因此Nb2N3为离子性较强的半导体材料.【期刊名称】《吉林大学学报（理学版）》【年(卷),期】2012(050)001【总页数】4页(P118-121)【关键词】氮化铌;硬质材料;高压物理【作者】祝颖;徐丹;李全军;陈洪斌【作者单位】吉林医药学院公共卫生院,吉林吉林132013;吉林大学超硬材料国家重点实验室,长春130012;吉林大学超硬材料国家重点实验室,长春130012;吉林医药学院公共卫生院,吉林吉林132013【正文语种】中文【中图分类】O521.2过渡族金属氮化物应用广泛[1-3], 其中TiN,VN,NbN等主要应用于硬质涂层材料[4], NbN的超导转变温度高达17.3 K[5]. Zerr等[2]在高温(>2 500 K)、高压(>16 GPa)条件下合成了Zr3N4和Hf3N4, Zr3N4薄膜的耐磨性比TiN提高约一个数量级[6]. Gregoryanz等[7]在高温(>2 000 K)、高压(>45 GPa)条件下合成了第一个过渡金属二氮化物PtN2, 理论预测其维氏硬度大于46 GPa[8]. Young等[9]合成了IrN2和OsN2, 其中IrN2的体弹性模量为428 GPa, 略低于金刚石的体弹性模量(440 GPa).Zerr等[10]在高温(>1 700 K)、高压(>11 GPa)条件下制备了含有少量O杂质且具有正交对称性的η-Ta2N3, η-Ta2N3是第一个氮离子与金属离子化学计量比为3∶2的过渡金属氮化物, 晶体的针状外形表明其具有高断裂韧度. 第一性原理研究表明, 常压下η-Ta2N3的弹性常数不满足波恩-黄昆判据, O杂质对η-Ta2N3的弹性稳定具有重要作用[11]. 本文采用第一性原理方法研究Nb2N3的晶体结构、力学、晶格动力学和电子性质.1 理论方法本文所有结构优化和电子性质计算均在密度泛函理论框架下采用“维也纳从头算模拟包”(简称VASP)[12]. 原子核与电子间的相互作用采用投影缀加波“冷冻核”全势[13]; 电子间的交换关联作用采用广义梯度近似[14]; Nb和N的价电子组态分别采用4p65s24d3和2s22p3. 平面波截断能为600 eV, 当布里渊区K点MP采样间距小于0.025 时, 计算中体系的焓(H=E+pV)收敛到每原子1 meV量级; 当局域优化时, 作用在原子上的力收敛到每纳米10 meV量级, 总的残余应力收敛到0.01 GPa量级. 弹性常数计算采用“应变-应力”方法[15]: 先对晶胞实施大小分别为±0.7%和±1.3%的应变, 再固定晶格基矢优化原子位置求得应力, 并根据胡克定律由奇异值分解方法求得晶体弹性常数的最小二乘解.2 结果与分析图1 Nb2N3的物态方程Fig.1 Equations of states of Nb2N3实验表明, η-Ta2N3具有正交结构, 空间群为Pnma, 当压力低于7.7 GPa时, 其相变到空间群为P-4m2的四方结构(以下简称为t-Ta2N3)[11], 常压和高压下t-Ta2N3的弹性常数均满足波恩-黄昆判据, 表明其力学稳定. 为考察Nb2N3是否存在类似相变, 本文将Nb2N3在-8～20 GPa压力下以4 GPa为间隔进行定压局域优化, 所得物态方程如图1所示. Pnma相的平衡晶胞参数分别为a=0.813 46 nm, b=0.301 72 nm, c=0.819 86 nm, 平衡体积为每个分子式0.051 39 nm3, 晶格中所有原子均占据Wyckoff的4c(x,1/4,z)位置, 各原子的晶体坐标列于表1. P-4m2相的平衡晶胞参数分别为a=0.300 12 nm, c=0.586 68 nm, 平衡体积为每个分子式0.052 84 nm3. 晶格Nb占据Wyckoff的2g(0,1/2,0.243 8)位置; N分别占据Wyckoff的2g(0,1/2,0.859 4)和1c(1/2,1/2,1/2)位置. 与Ta2N3不同, Nb2N3为Pnma结构(以下简称为η-Nb2N3), 其常压下的能量和平衡体积均明显低于P-4m2结构(以下简称为t-Nb2N3), 表明η-Nb2N3在常压和高压(<20 GPa)下均为Nb2N3能量稳定的结构, 而t-Nb2N3为能量亚稳相.表1 η-Nb2N3中Nb和N原子的晶体坐标Table 1 Crystal coordinates of Nb and N atoms of η-Nb2N3原子x轴y轴z轴Nb1 0.694 70.250.4958Nb20.022 20.250.686 9N10.778 80.250.798 2N20.121 30.250.4519N30.954 20.250.125 9本文拟合了η-Nb2N3和t-Nb2N3的三阶Birch-Murnaghan物态方程, 并将它们在压力作用下的体积变化趋势与η-Ta2N3,t-Ta2N3,NbN进行对比, 结果如图1所示. 由图1可见, Nb2N3和Ta2N3在高压下的四方相抵抗外力压缩性能明显优于正交相. 与t-Ta2N3体弹性模量(B0=331 GPa)略高于η-Ta2N3体弹性模量(B0=323 GPa)的结果相符[11]. 此外, Nb2N3比Ta2N3和NbN更易于压缩,Ta2N3在压力作用下的体积变化趋势更接近于NbN. 由三阶Birch-Murnaghan 物态方程得到η-Nb2N3和t-Nb2N3的体弹性模量分别为307 GPa和311 GPa, 小于NbN,η-Ta2N3,t-Ta2N3的体弹性模量.本文采用基于密度泛函理论的第一性原理计算应力. 在笛卡尔坐标系中, 材料的晶格基矢可表示为其中a1, b1, c1分别为晶格基矢a, b,c的第一个笛卡尔坐标, 其余类推. 先对R施加一组应变s=(s1 s2 s3 s4 s5 s6), 其中: s1,s2,s3为常规应变;s4,s5,s6为剪切应变. 所得畸变晶格可表示为每个畸变晶格可由第一性原理求得一个应力t=(t1 t2 t3 t4 t5 t6), 与s满足胡克定律: t=sC, 其中C是6×6维的弹性常数矩阵, 矩阵元cij服从沃伊特标记. 若存在n个应变S, 则可求得n个应力T, 进而获得n×6维的非满秩弹性常数矩阵C, 满足C=S-1T, 其中“-1”表示求逆. 采用奇异值分解方法求解该方程可得弹性常数的最小二乘解. 根据晶体不同的对称性, 计算弹性常数所需最小线性不相关应变数分别为: 立方晶系2个; 六方和三方晶系3个; 四方晶系4个; 正交、单斜和三斜晶系6个.采用应变-应力方法求得η-Nb2N3的弹性常数cij为由于η-Nb2N3的c66为正数, 且η-Nb2N3的所有弹性常数均满足波恩-黄昆弹性稳定性判据, 因此常压下η-Nb2N3可稳定存在. 由弹性常数可得正交结构的剪切各向异性因子分别为A1=4c44/(c11+c33-2c13)=1.02, A2=4c55/(c22+c33-2c23)=0.86, A3=4c66/(c11+c22-2c12)=0.81. A1,A2,A3分别对应(100),(010),(001)剪切平面, 因此(010)和(001)剪切平面存在较强的各向异性. Voigt-Reuss-Hill平均形式的多晶体弹性模量(B=304 GPa)[16], 略低于含有少量O杂质的η-Ta2N3(323 GPa), 但大于单质Nb[3]的弹性模量(172 GPa), 接近于NbC的弹性模量(301 GPa)和NbN的弹性模量(309 GPa), 表明η-Nb2N3具有较强的抗压缩能力[4]; 由η-Nb2N3的剪切模量G、杨氏模量Y、泊松比υ分别为146 GPa, 377 GPa, 0.29可知, η-Nb2N3具有较强的抗剪切能力. 由B/G≈2.1可推测η-Nb2N3为脆性材料[17]. 晶体材料微观硬度理论可预测材料的硬度[18-22]. 本文利用文献[21]的理论模型求得η-Nb2N3和η-Ta2N3的维氏硬度分别为19.3 GPa和19.7 GPa. Zerr等[10]预测致密的η-Ta2N3硬度为30 GPa, 因此可推测η-Nb2N3的硬度较大. η-Nb2N3原子轨道角动量投影的电子能态密度(分立态密度)如图2所示, 其中竖线表示Fermi能级. 由图2可见, η-Nb2N3是带隙约为0.4 eV的半导体材料. 在-14.5～0 eV能量范围内, Nb的4d轨道和N的2p轨道间存在较强的杂化, 表明Nb-N键具有较强的共价性. η-Nb2N3在(001)平面上的电荷密度如图3所示. 由图3可见, Nb-N键的电子局域行为表明了其共价性, 与η-Nb2N3的半导体特性相符. 此外, 基于AIM理论的Bader分析[23]表明, η-Nb2N3中平均每个Nb离子向N离子转移2.17个电荷, 即Nb和N之间的化学键具有强离子性.图2 η-Nb2N3的分立态密度Fig.2 Discrete density of sta te of η-Nb2N3 图3 η-Nb2N3在(001)平面上的电荷密度Fig.3 Charge density of η-Nb2N3 on the (001) plane本文以Nb和Ta体心立方相及N的α相为参照相, 计算了η-Nb2N3和η-Ta2N3的形成焓: Δ H=H(η-M2N3)-2H(M)-(3/2)H(N2), 其中M表示Nb或Ta 原子. 由式(1)可知, 零压下η-Nb2N3和η-Ta2N3的形成焓分别为-4.2 eV和-5.2eV. 表明两个相在常压下能量稳定, 未分解为金属单质和氮气. 对于η-Ta2N3, 本文的理论结果与实验相符[10]. 由于η-Nb2N3与η-Ta2N3形成焓相近, 因此η-Nb2N3可在与η-Ta2N3相似的压力(11 GPa)和温度(>1 700 K)条件下合成.综上, 本文采用基于密度泛函理论的第一性原理方法研究了Nb2N3的性质. 结果表明: η-Nb2N3为具有较强离子性的半导体材料, 其带隙约为0.4 eV; η-Nb2N3在常压下的弹性稳定, 具有较强的抗压缩性、剪切各向异性和脆性, 其硬度与η-Ta2N3的硬度相似; η-Nb2N3在常压下能量稳定, 可在高温、高压条件下合成. 参考文献【相关文献】[1] Jhi S H, Louie S G, Cohen M L, et al. Vacancy Hardening and Softening in Transition Metal Carbides and Nitrides [J]. Phys Rev Lett, 2001, 86(15): 3348-3351.[2] Zerr A, Miehe G, Riedel R. Synthesis of Cubic Zirconium and Hafnium Nitride Having Th3P4 Structure [J]. Nat Mater, 2003, 2(3): 185-189.[3] Isaev E I, Simak S I, Abrikosov I A, et al. Phonon Related Properties of Transition Metals, Their Carbides, and Nitrides: A First-Principles Study [J]. J Appl Phys, 2007, 101(12): 123519.[4] Brazhkin V V, Lyapin A G, Hemley R J. Harder Than Diamond: Dreams and Reality [J]. Philos Mag A, 2002, 82(2): 231-253.[5] Alén P, Ritala M, Arstila K, et al. The Growth and Diffusion Barrier Properties of Atomic Layer Deposited NbNx Thin Films [J]. Thin Solid Films, 2005, 491(1/2): 235-241.[6] Chhowalla M, Unalan H E. Thin Films of Hard Cubic Zr3N4 Stabilized by Stress [J]. Nat Mater, 2005, 4: 317-322.[7] Gregoryanz E, Sanloup C, Somayazulu M, et al. Synthesis and Characterization of a Binary Noble Metal Nitride [J]. Nat Mater, 2004, 3: 294-297.[8] GUO Xiao-ju, LI Lei, LIU Zhong-yuan, et al. Hardness of Covalent Compounds: Roles of Metallic Component and d Valence Electrons [J]. J Appl Phys, 2008, 104(2): 023503. [9] Young A F, Sanloup C, Gregoryanz E, et al. Synthesis of Novel Transition Metal Nitrides IrN2 and OsN2 [J]. Phys Rev Lett, 2006, 96(15): 155501.[10] Zerr A, Miehe G, LI Jing-wang, et al. High-Pressure Synthesis of Tantalum Nitride Having Orthorhombic U2S3 Structure [J]. Adv Funct Mater, 2009, 19(14): 2282-2288. [11] JIANG Chao, LIN Zhi-jun, ZHAO Yu-sheng. Thermodynamic and Mechanical Stabilities of Tantalum Nitride [J]. Phys Rev Lett, 2009, 103(18): 185501.[12] Kresse G, Furthmüller J. Efficient Iterative Schemes for ab initio Total-Energy Calculations Using a Plane-Wave Basis Set [J]. Phys Rev B, 1996, 54(16): 11169-11186. [13] Blöchl P E. Projector Augmented-Wave Method [J]. Phys Rev B, 1994, 50(24): 17953-17979.[14] Perdew J P, Burke K, Ernzerhof M. Generalized Gradient Approximation Made Simple [J]. Phys Rev Lett, 1996, 77(18): 3865-3868.[15] Shang S, Wang Y, Liu Z K. First-Principles Elastic Constants of α- and θ-Al2O3[J]. Appl Phys Lett, 2007, 90: 101909.[16] Hill R. The Elastic Behaviour of a Crystalline Aggregate [J]. Proc Phys Soc A, 1952,65(5): 349-356.[17] Ravindran P, Fast L, Korzhavyi P A, et al. Density Functional Theory for Calculation of Elastic Properties of Orthorhombic Crystals: Application to TiSi2 [J]. J Appl Phys, 1998,84(9): 4891.[18] GAO Fa-ming, HE Ju-long, WU Er-dong, et al. Hardness of Covalent Crystals [J]. Phys Rev Lett, 2003, 91(1): 015502.[19] Simunek A, Vackar J. Hardness of Covalent and Ionic Crystals: First-Principle Calculations [J]. Phys Rev Lett, 2006, 96(8): 085501.[20] GAO Fa-ming. Theoretical Model of Intrinsic Hardness [J]. Phys Rev B, 2006, 73(13): 132104.[21] Simunek A. How to Estimate Hardness of Crystals on a Pocket Calculator [J]. Phys Rev B, 2007, 75(17): 172108.[22] LI Ke-yan, WANG Xing-tao, ZHANG Fang-fang, et al. Electronegativity Identification of Novel Superhard Materials [J]. Phys Rev Lett, 2008, 100(23): 235504.[23] Henkelman G, Arnaldsson A, Jónsson H. A Fast and Robust Algorithm for Bader Decomposition of Charge Density [J]. Comput Mater Sci, 2006, 36(3): 354-360.。
a rXiv:mtrl -th/9522v14Fe b1995First-principle study of excitonic self-trapping in diamond Francesco Mauri ∗and Roberto Car Institut Romand de Recherche Num´e rique en Physique des Mat´e riaux (IRRMA)IN-Ecublens 1015Lausanne,Switzerland Abstract We present a first-principles study of excitonic self-trapping in diamond.Our calculation provides evidence for self-trapping of the 1s core exciton and gives a coherent interpretation of recent experimental X-ray absorption and emission data.Self-trapping does not occur in the case of a single valence exciton.We predict,however,that self-trapping should occur in the case of a valence biexciton.This process is accompanied by a large local relaxation of the lattice which could be observed experimentally.PACS numbers:61.80.−x,71.38.+i,71.35+z,71.55.−iTypeset using REVT E XDiamond presents an unusually favorable combination of characteristics that,in connection with the recent development of techniques for the deposition of thin diamondfilms,make this material a good candidate for many technological applications.Particularly appealing is the use of diamond in electronic or in opto-electronic devices,as e.g.UV-light emitting devices.Moreover,diamond is an ideal material for the construction of windows that operate under high power laser radiation or/and in adverse environments.It is therefore interesting to study radiation induced defects with deep electronic levels in the gap,since these can have important implications in many of these applications.Excitonic self-trapping is a possible mechanism for the formation of deep levels in the gap.The study of such processes in a purely covalent material,like diamond,is interesting also from a fundamental point of view.Indeed,excitonic self-trapping has been studied so far mostly in the context of ionic compounds,where it is always associated with,and often driven by,charge transfer effects.In a covalent material the driving mechanism for self-trapping is instead related to the difference in the bonding character between the valence and the conduction band states.Both experimental data and theoretical arguments suggest the occurrence of self-trapping processes in diamond.In particular,a nitrogen(N)substitutional impurity induces a strong local deformation of the lattice[1–3]that can be interpreted as a self-trapping of the donor electron.The structure of a1s core exciton is more controversial[4–9].Indeed the similarity between an excited core of carbon and a ground-state core of nitrogen suggests that the core exciton should behave like a N impurity.However,the position of the core exciton peak in the diamond K-edge absorption spectra is only0.2eV lower than the conduction band minimum[4,7,8],while a N impurity originates a deep level1.7eV below the conduction band edge[10].On the other hand,emission spectra[8]suggest that a1s core exciton should self-trap like a N impurity.Finally,we consider valence excitations.In this case experimental evidence indicates that a single valence exciton is of the Wannier type,i.e.there is no self-trapping.To our knowledge,neither experimental nor theoretical investigations on the behavior of a valence biexciton in diamond have been performed,although simple scalingarguments suggest that the tendency to self-trap should be stronger for biexcitons than for single excitons.In this letter,we present a detailed theoretical study of excitonic self-trapping effects in diamond.In particular,we have investigated the Born-Oppenheimer(BO)potential energy surfaces corresponding to a core exciton,a valence exciton and a valence biexciton in the context of density functional theory(DFT),within the local density approximation(LDA) for exchange and correlation.Our calculation indicates that the1s core exciton is on a different BO surface in absorption and in emission experiments.Indeed X-ray absorption creates excitons in a p-like state as required by dipole selection rules.Subsequently the system makes a transition to an s-like state associated to a self-trapping distortion of the atomic lattice,similar to that found in the N impurity case.These results provide a coherent interpretation of the experimental data.In addition,our calculation suggests that self-trapping should also occur for a valence biexciton.This is a prediction that could be verified experimentally.Let us start by discussing a simple model[11,12].In diamond,the occupied valence and the lower conduction band states derive from superpositions of atomic sp3hybrids having bonding and antibonding character,respectively.Thus,when an electron,or a hole,or an electron-hole pair is added to the system,this can gain in deformation energy by relaxing the atomic lattice.Scaling arguments suggest that the deformation energy gain E def∝−1/N b, where N b is the number of bonds over which the perturbation is localized.This localization,due to quantum confinement.The in turn,has a kinetic energy cost E kin∝+1/N2/3bbehavior of the system is then governed by the value of N b that minimizes the total energy E sum=E def+E kin.Since the only stationary point of E sum is a maximum,E sum attains its minimum value at either one of the two extrema N b=1or N b=∞.If the minimum occurs for N b=1,the perturbation is self-trapped on a single bond which is therefore stretched.If the minimum occurs for N b=∞,there is no self-trapping and the perturbation is delocalized.When N p particles(quasi-particles)are added to the system,one can showthat,for a given N b,E def scales as N2p,while E kin scales as N p.As a consequence,the probability of self-trapping is enhanced when N p is larger.This suggests that biexcitons should have a stronger tendency to self-trap than single excitons[12,13].In order to get a more quantitative understanding of self-trapping phenomena in dia-mond,we performed self-consistent electronic structure calculations,using norm-conserving pseudopotentials[14]to describe core-valence interactions.The wave-functions and the electronic density were expanded in plane-waves with a cutoffof35and of140Ry,respec-tively.We used a periodically repeated simple cubic supercell containing64atoms at the experimental equilibrium lattice constant.Only the wave-functions at theΓpoint were con-sidered.Since the self-trapped states are almost completely localized on one bond,they are only weakly affected by the boundary conditions in a64atom supercell.The effect of the k-point sampling was analysed in Ref.[3]where similar calculations for a N impurity were performed using the same supercell.It was found that a more accurate k-point sampling does not change the qualitative physics of the distortion but only increases the self-trapping energy by20%compared to calculations based on theΓ-point only[3].In order to describe a core exciton we adopted the method of Ref.[15],i.e.we generated a norm conserving pseudopotential for an excited carbon atom with one electron in the1s core level andfive electrons in the valence2s-2p levels.In our calculations for a valence exciton or biexciton we promoted one or two electrons,respectively,from the highest valence band state to the lowest conduction band state.Clearly,our single-particle approach cannot account for the(small)binding energy of delocalized Wannier excitons.However our approach should account for the most important contribution to the binding energy in the case of localized excitations.Structural relaxation studies were based on the Car-Parrinello(CP) approach[16].We used a standard CP scheme for both the core and the valence exciton, while a modified CP dynamics,in which the electrons are forced to stay in an arbitrary excited eigenstate[12,17],was necessary to study the BO surfaces corresponding to a valence biexciton.All the calculations were made more efficient by the acceleration methods of Ref.[18].Wefirst computed the electronic structure of the core exciton with the atoms in the ideal lattice positions.In this case the excited-core atom induces two defect states in the gap:a non-degenerate level belonging to the A1representation of the T d point group,0.4eV below the conduction band edge,and a3-fold degenerate level with T2character,0.2eV below the conduction band edge.By letting the atomic coordinates free to relax,we found that the absolute minimum of the A1potential energy surface correponds to an asymmetric self-trapping distortion of the lattice similar to that found for the N impurity[3].In particular, the excited-core atom and its nearest-neighbor,labeled a and b,respectively,in Fig.1, move away from each other on the(111)direction.The corresponding displacements from the ideal sites are equal to10.4%and to11.5%of the bond length,respectively,so that the (a,b)-bond is stretched by21.9%.The other atoms move very little:for instance the nearest-neighbor atoms labeled c move by2.4%of the bond length only.This strong localization of the distortion is consistent with the simple scaling arguments discussed above.As a consequence of the atomic relaxation,the non-degenerate level ends up in the gap at1.5eV below the conduction band edge,while the corresponding wavefunction localizes on the stretched bond.The3-fold degenerate level remains close to the conduction band edge,but since the distortion lowers the symmetry from T d to C3v,the3-fold degenerate level splits into a2-fold degenerate E level and a non-degenerate A1level.In Fig.2we report the behavior of the potential energy surfaces corresponding to the ground-state,the A1and the T2core exciton states as a function of the self-trapping dis-tortion.Notice that the distortion gives a total energy gain of0.43eV on the A1potential energy surface.The same distortion causes an increase of the ground-state energy of1.29 eV.Our calculation indicates that the core-exciton behaves like the N impurity[3],support-ing,at least qualitatively,the validity of the equivalent core approximation.The similar behavior of the A1level in the core exciton and in the N impurity case was also pointed out recently in the context of semi-empirical CNDO calculations[9].The differences between the core exciton and the impurity[3]are only quantitative:in particular,the relaxationenergy and especially the distance of the A1level from the conduction band edge are smaller for the core exciton than for the N impurity.Our results suggest the following interpretation of the experimental data of Refs.[4,8]: (i)During X-ray absorption the atoms are in the ideal lattice positions.Dipole transitions from a1s core level to a A1valence level are forbidden,but transitions to the T2level are allowed.In our calculation the T2level is0.2eV lower than the conduction band edge,in good agreement with the core exciton peak observed in X-ray absorption spectra[4,8].(ii) On the T2BO potential energy surface the lattice undergoes a Jahn-Teller distortion which lowers its energy(see Fig.2).(iii)Since the LO phonon energy in diamond(0.16eV)is comparable to the energy spacing between the A1and the T2surfaces,which is less than 0.2eV after the Jahn-Teller distortion,the probability of a non-adiabatic transition from the T2to the A1surface is large.(iv)On the A1level the system undergoes a strong lattice relaxation resulting in a localization of the exciton on a single bond.(v)The self-trapping distortion induces a Stokes shift in the emitted photon energy.If the atomic relaxation were complete the Stokes shift would be equal to1.9eV,which correponds(see Fig.2) to the energy dissipated in the T2-A1transition(0.2eV),plus the energy gained by self trapping on the A1surface(0.43eV),plus the energy cost of the self-trapping distortion on the ground-state energy surface(1.29eV).The data reported in Ref.[8]show a shift of about1eV in the positions of the peaks associated to the1s core exciton in X-ray absorption and emission spectra.The emission peak is very broad,with a large sideband that corresponds to Stokes shifts of up to5eV.As pointed out in Ref.[8],this large sideband is likely to be the effect of incomplete relaxation. This is to be expected since the core exciton lifetime should be comparable to the phonon period[8].As a consequence,the atomic lattice would be able to perform only a few damped oscillations around the distorted minimum structure during the lifetime of the core exciton.We now present our results for the valence excitations.While in the case of a single exciton the energy is minimum for the undistorted crystalline lattice,in the case of a biex-citon wefind that the energy is minimized in correspondence of a localized distortion of theatomic lattice.This is characterized by a large outward symmetric displacement along the (111)direction of the atoms a and b in Fig.1.As a result the(a,b)-bond is broken since the distance between the atoms a and b is increased by51.2%compared to the crystalline bondlength.This distortion can be viewed as a kind of local graphitization in which the atoms a and b change from fourfold to threefold coordination and the corresponding hy-bridized orbitals change from sp3to sp2character.Again,in agreement with the model based on simple scaling arguments,the distortion is strongly localized on a single bond.As a matter of fact and with reference to the Fig.1,the atoms c and d move by1.2%of the bondlength,the atoms e and f move by2.3%,and the atoms not shown in thefigure by less than0.9%.The self-trapping distortion of the biexciton gives rise to two deep levels in the gap: a doubly occupied antibonding level,at1.7eV below the conduction band edge,and an empty bonding level,at1.6eV above the valence band edge.Both states are localized on the broken bond.In Fig.3we show how different BO potential energy surfaces behave as a function of the self-trapping distortion of the valence biexciton.In particular,from thisfigure we see that,while for the biexciton there is an energy gain of1.74eV in correspondence with the self-trapping distortion,the same distortion has an energy cost of1.49eV for the single exciton,and of4.85eV for the unexcited crystal.We notice that,while DFT-LDA predicts self-trapping for the valence biexciton,it does not do so for the single exciton,in agreement with experiment.Similarly to the case of the core exciton the major experimental consequence of the self-trapping of the valence biexciton is a large Stokes shift in the stimulated-absorption spontaneous-emission cycle between the exciton and the biexciton BO surfaces.As it can be seen from Fig.3,this Stokes shift should be equal to3.23eV,i.e.to the sum of the energy gain of the biexciton(1.74eV)and of the energy cost of the exciton(1.49eV) for the self-trapping relaxation.The fundamental gap of diamond is indirect.Thus the spontaneous decay of a Wannier exciton in an ideal diamond crystal is phonon assistedand the radiative lifetime of the exciton is much longer than in direct gap semiconductors. However,after self-trapping of the biexciton,the translational symmetry is broken and direct spontaneous emission becomes allowed.As a consequence the radiative life time of the self-trapped biexciton is much smaller than that of the Wannier ing the DFT-LDA wavefunctions,we obtained a value of∼7ns for the radiative lifetime of the biexciton within the dipole approximation.This is several orders of magnitude larger than the typical phonon period.Therefore the self-trapping relaxation of the valence biexciton should be completed before the radiative decay.A self-trapped biexciton is a bound state of two excitons strongly localized on a single bond.Thus the formation of self-trapped biexcitons requires a high excitonic density.To realize this condition it is possible either to excite directly bound states of Wannier excitons, or to create a high density electron-hole plasma,e.g.by strong laser irradiation.In the second case many self-trapped biexcitons could be produced.This raises some interesting implications.If many self-trapped biexcitons are created,they could cluster producing a macroscopic graphitization.Moreover,since the process of self-trapping is associated with a relevant energy transfer from the electronic to the ionic degrees of freedom,in a high density electron hole plasma biexcitonic self-trapping could heat the crystal up to the melting point in fractions of a ps,i.e in the characteristic time of ionic relaxation.Interestingly,melting ofa GaAs crystal under high laser irradiation has been observed to occur in fractions of a ps[19].In Ref.[19]this phenomenon has been ascribed to the change in the binding properties due to the electronic excitations.Our study on diamond leads one to speculate that in a sub-picosecond melting experiment self-trapping phenomena could play an important role.In conclusion,we have studied excited-state BO potential energy surfaces of crystalline diamond within DFT-LDA.Our calculation predicts self-trapping of the core exciton and provides a coherent description of the X-ray absorption and emission processes,which com-pares well with the experimental data.Moreover,we also predict self-trapping of the valence biexciton,a process characterized by a large local lattice relaxation.This implies a strong Stokes shift in the stimulated absorption-spontaneous emission cycle of about3eV,whichcould be observed experimentally.It is a pleasure to thank F.Tassone for many useful discussions.We acknowledge support from the Swiss National Science Foundation under grant No.20-39528.93REFERENCES∗Present address:Departement of Physics,University of California,Berkeley CA94720, USA.[1]C.A.J.Ammerlaan,Inst.Phys.Conf.Ser.59,81(1981).[2]R.J.Cook and D.H.Whiffen,Proc.Roy.Soc.London A295,99(1966).[3]S.A.Kajihara et al,Phys.Rev.Lett.66,2010(1991).[4]J.F.Morar et al,Phys.Rev.Lett.54,1960(1985).[5]K.A.Jackson and M.R.Pederson,Phys.Rev.Lett.67,2521(1991).[6]J.Nithianandam,Phys.Rev.Lett.69,3108(1992).[7]P.E.Batson,Phys.Rev.Lett.70,1822(1993).[8]Y.Ma et al,Phys.Rev.Lett.71,3725(1993).[9]A.Mainwood and A.M.Stoneham,J.Phys.:Condens.Matter6,4917(1994).[10]R.G.Farrer,Solid State Commun.7,685(1969).[11]W.Hayes and A.M.Stoneham,Defects and defect processes in nonmetallic solids,(Wiley&Sons,New York,1985)pags.29-38.[12]F.Mauri,R.Car,(to be published).[13]The number of equal particles that can be accommodated on one bond of the crystal inthe same quantum state is limited by the Pauli principle.Thus no more than two holes or/and two electrons with opposite spins can be localized on one bond of a sp3bonded semiconductor.[14]G.Bachelet,D.Hamann,and M.Schl¨u ter,Phys.Rev.B26,4199(1982).[15]E.Pehlke and M.Scheffler,Phys.Rev.B47,3588(1993).[16]R.Car and M.Parrinello,Phys.Rev.Lett.55,2471(1985).[17]F.Mauri,R.Car and E.Tosatti,Europhys.Lett.24,431(1993).[18]F.Tassone,F.Mauri,and R.Car,Phys.Rev.B50,10561(1994).[19]orkov,I.L.Shumay,W.Rudolph,and T.Schroder,Opt.Lett.16,1013(1991);P.Saeta,J.-K.Wang,Y.Siegal,N.Bloembergen,and E.Mazur,Phys.Rev.Lett.67, 1023(1991);K.Sokolowski-Tinten,H.Schulz,J.Bialkowski,and D.von der Linde, Applied Phys.A53,227(1991).FIGURESFIG.1.Atoms and bonds in the ideal diamond crystal(left panel).Atoms and bonds after the self-trapping distortion associated with the valence biexciton(right panel).In this case the distance between the atoms a and b increases by51.2%.A similar but smaller distortion is associated with the core exciton:in this case the(a,b)distance is increased by21.9%.FIG.2.Total energy vs self-trapping distortion of the core-exciton.Thefigure displays the BO potential energy surfaces correponding to the ground-state,the A1,and the T2core exciton states.FIG.3.Total energy as a function of the self-trapping distortion of the biexciton.The BO energy surfaces correponding to the ground state,the valence exciton,and the valence biexciton are shown in thefigure.a b ce df(111)ground stateA 1−core excitonT 2−core excitonconduction ideal lattice distorted latticeground statebi−excitonexcitondistorted lattice ideal lattice。
CASTEP Scientific References - 20121.Jian Sun, Miguel Martinez-Canales, Dennis D. Klug, Chris J. Pickard, and Richard J.Needs,Persistence and Eventual Demise of Oxygen Molecules at Terapascal Pressures,Physical Review Letters108 (2012) 045503 ( abstract )2.Miguel Martinez-Canales, Chris J. Pickard, Richard J. Needs,Thermodynamically Stable Phases of Carbon at Multiterapascal Pressures,Physical Review Letters108 (2012) 045704 ( abstract )3. A. Chroneos and A. Dimoulas,Defect configurations of high-k cations in germanium,Journal of Applied Physics111 (2012) 023714 ( abstract )4.Roland Gillen and John Robertson,Hybrid functional calculations of the Al impurity in α quartz: Hole localization andelectron paramagnetic resonance parameters,Physical Review B85 (2012) 014117 ( abstract )5. A. Bhattacharya, S. Bhattacharya, and G. P. Das,Band gap engineering by functionalization of BN sheet,Physical Review B85 (2012) 035415 ( abstract )6.L. Fishwick, M. Walker, M. K. Bradley, D. P. Woodruff, and C. F. McConville,Surface structure of GaP(110): Ion scattering and density functional theory study,Physical Review B85 (2012) 045322 ( abstract )7.Takatsugu Endo, Scarlett Widgeon, Ping Yu, Sabyasachi Sen, and Keiko Nishikawa,Cation and anion dynamics in supercooled and glassy states of the ionic liquid1-butyl-3-methylimidazolium hexafluorophosphate: Results from 13C, 31P, and 19F NMRspectroscopy,Physical Review B85 (2012) 054307 ( abstract )8.James D. Aldous et al.,Cubic MnSb: Epitaxial growth of a predicted room temperature half-metal,Physical Review B85 (2012) 060403 ( abstract )9.N. D. Drummond, V. Zolyomi, and V. I. Fal'ko,Electrically tunable band gap in silicene,Physical Review B85 (2012) 075423 ( abstract )10.Philip W. Avraam, Nicholas D. M. Hine, Paul Tangney, and Peter D. Haynes,Fermi-level pinning can determine polarity in semiconductor nanorods,Physical Review B85 (2012) 115404 ( abstract )11.Naoki Imamura, Hiroshi Mizoguchi, and Hideo Hosono,Superconductivity in LaT M BN and La3T M2B2N3 (T M = Transition Metal) Synthesizedunder High Pressure,J. Am. Chem. Soc.134 (2012) 2516–2519 ( abstract )12.Jakub S. Jirkovsky, Itai Panas, Simon Romani, Elisabet Ahlberg, and David J. Schiffrin,Potential-Dependent Structural Memory Effects in Au-Pd Nanoalloys,J. Phys. Chem. Lett.3 (2012) 315–321 ( abstract )13.Martin Dracinsky, Milos Budesinsky, Beata Warzajtis, and Urszula Rychlewska,Solution and Solid-State Effects on NMR Chemical Shifts in Sesquiterpene Lactones: NMR, X-ray, and Theoretical Methods,J. Phys. Chem. A116 (2012) 680–688 ( abstract )14.Luke A. O'Dell, Christopher I. Ratcliffe, Xianqi Kong, and Gang Wu,Multinuclear Solid-State Nuclear Magnetic Resonance and Density Functional Theory Characterization of Interaction Tensors in Taurine,J. Phys. Chem. A116 (2012) 1008–1014 ( abstract )15.Yutaka Natsume et al.,Chemical-State Analysis of Organic Semiconductors Using Soft X-ray AbsorptionSpectroscopy Combined with First-Principles Calculation,J. Phys. Chem. A116 (2012) 1527–1531 ( abstract )16.Tingting Lin, Xiang-Yang Liu, and Chaobin He,Calculation of Infrared/Raman Spectra and Dielectric Properties of Various Crystalline Poly(lactic acid)s by Density Functional Perturbation Theory (DFPT) Method,J. Phys. Chem. B116 (2012) 1524–1535 ( abstract )17.Marco Delgado et al.,Evolution of Structure and of Grafting Properties of γ-Alumina with PretreatmentTemperature,J. Phys. Chem. C116 (2012) 834–843 ( abstract )ardo Cuervo Reyes, Adam Slabon-Turski, Christian Mensing, and Reinhard Nesper,Spin-Glass Behavior and Electronic Structure of LiEu2Si3,J. Phys. Chem. C116 (2012) 1158–1164 ( abstract )19.Daejin Kim et al.,Pillared Covalent Organic Frameworks with Balanced V olumetric and GravimetricHydrogen Uptake,J. Phys. Chem. C116 (2012) 1479–1484 ( abstract )ardo Cuervo Reyes and Reinhard Nesper,Electronic Structure and Properties of the Alkaline Earth Monosilicides,J. Phys. Chem. C116 (2012) 2536–2542 ( abstract )21.Jung-Il Hong et al.,Magnetism in Dopant-Free ZnO Nanoplates,Nano Letters12 (2012) 576–581 ( abstract )22.Nan Gao, Wei Tao Zheng and Qing Jiang,Density functional theory calculations for two-dimensional silicene with halogenfunctionalization,Physical Chemistry Chemical Physics14 (2012) 257–261 ( abstract )23.Marco Sacchi, Martin C. E. Galbraith and Stephen J. Jenkins,The interaction of iron pyrite with oxygen, nitrogen and nitrogen oxides: a first-principles study,Physical Chemistry Chemical Physics14 (2012) 3627–3633 ( abstract )24.Zhi Yang et al.,Density functional theory studies of Nb-benzene and Nb-borazine sandwich clusters andmolecular wires,Journal of Physics B45 (2012) 025102 ( abstract )25.P Pluengphon, T Bovornratanaraks, S V annarat and U Pinsook,The effects of Na on high pressure phases of CuIn0.5Ga0.5Se2 from ab initio calculation, Journal of Physics: Condensed Matter24 (2012) 095802 ( abstract )26.Ran He, Z S Lin, Tao Zheng, He Huang and C T Chen,Energy band gap engineering in borate ultraviolet nonlinear optical crystals: ab initiostudies,Journal of Physics: Condensed Matter24 (2012) 145503 ( abstract )27.M. G. Brik,First-principles calculations of electronic, optical and elastic properties of Ba2MgWO6 double perovskite,Journal of Physics and Chemistry of Solids73 (2012) 252–256 ( abstract )28.Y. H. Liu et al.,Study on the transformation from NaCl-type Na2TiO3 to layered titanate,Journal of Physics and Chemistry of Solids73 (2012) 402–406 ( abstract )29.Elena Bichoutskaia et al.,High-precision imaging of an encapsulated Lindqvist ion and correlation of its structure and symmetry with quantum chemical calculations,Nanoscale4 (2012) 1190–1199 ( abstract )30.Pan Li et al.,Effects of surface chemistry on the morphology transformation of ZnWO4 nanocrystals: investigated from experiment and theoretical calculations,CrystEngComm14 (2012) 920–928 ( abstract )31.Zhonghua Li et al.,Single crystal titanate-zirconate nanoleaf: Synthesis, growth mechanism and enhanced photocatalytic hydrogen evolution properties,CrystEngComm14 (2012) 1874–1880 ( abstract )32.Yonggang Wang et al.,Hydrothermal growths, optical features and first-principles calculations of sillenite-type crystals comprising discrete MO4 tetrahedra,CrystEngComm14 (2012) 1063–1068 ( abstract )33.Dajiang Mei et al.,LiGaGe2Se6: A New IR Nonlinear Optical Material with Low Melting Point,Inorganic Chemistry51 (2012) 1035–1040 ( abstract )34.Kai Feng et al.,NaGe3P3: a new ternary germanium phosphide featuring an unusual [Ge3P7] ring,Dalton Transactions41 (2012) 484–489 ( abstract )35.Su-Yun Zhang, Chun-Li Hu and Jiang-Gao Mao,New mixed metal selenites and tellurites containing Pd2+ ions in a square planar geometry, Dalton Transactions41 (2012) 2011–2017 ( abstract )36.Caixia Xu, Qian Li, Yunqing Liu, Jinping Wang, and Haoran Geng,Hierarchical Nanoporous PtFe Alloy with Multimodal Size Distributions and Its CatalyticPerformance toward Methanol Electrooxidation,Langmuir28 (2012) 1886–1892 ( abstract )37.Linjuan Zhang et al.,Lattice distortion and its role in the magnetic behavior of the Mn-doped ZnO system, New Journal of Physics14 (2012) 013033 ( abstract )38.Itziar Goikoetxea et al.,Non-adiabatic effects during the dissociative adsorption of O2 at Ag(111)? Afirst-principles divide and conquer study,New Journal of Physics14 (2012) 013050 ( abstract )39.Yongjun Zhou et al.,First-principles study on the catalytic role of Ag in the oxygen adsorption of LaMnO3(0 01) surface,Applied Surface Science258 (2012) 2602–2606 ( abstract )40.Jun Zhou et al.,Density functional theory study on oxygen adsorption in LaSrCoO4: An extended cathode material for solid oxide fuel cells,Applied Surface Science258 (2012) 3133–3138 ( abstract )41.Hui Zhao, Na Qin,The stability boundary of group-III transition metal diboride ScB2 (0 0 0 1) surfaces,Applied Surface Science258 (2012) 3328–3330 ( abstract )42.Qi-Jun Liu et al.,Structural and electronic properties of cubic SrHfO3 surface: First-principles calculations, Applied Surface Science258 (2012) 3455–3461 ( abstract )43.Baojun Wang, Luzhi Song, Riguang Zhang,The dehydrogenation of CH4 on Rh(1 1 1), Rh(1 1 0) and Rh(1 0 0) surfaces: A density functional theory study,Applied Surface Science258 (2012) 3714–3722 ( abstract )44.Li-Ming Yang, Ponniah Ravindran, Ponniah V ajeeston and Mats Tilset,Ab initio investigations on the crystal structure, formation enthalpy, electronic structure, chemical bonding, and optical properties of experimentally synthesized isoreticularmetal-organic framework-10 and its analogues: M-IRMOF-10 (M = Zn, Cd, Be, Mg, Ca, Sr and Ba),RSC Advances2 (2012) 1618–1631 ( abstract )45.Jiri Czernek, Tomasz Pawlak, Marek J. Potrzebowski,Benchmarks for the 13C NMR chemical shielding tensors in peptides in the solid state, Chemical Physics Letters527 (2012) 31–35 ( abstract )46.Min Li, Junying Zhang, Yue Zhang,First-principles calculation of compensated (2N, W) codoping impacts on band gapengineering in anatase TiO2,Chemical Physics Letters528 (2012) 63–66 ( abstract )47.X. P. Yang et al.,Preparation and XRD analyses of Na-doped ZnO nanorod arrays based on experiment and theory,Chemical Physics Letters528 (2012) 16–20 ( abstract )48.Feng Yuan et al.,Structure and hydrogen storage properties of the first rare-earth metal borohydrideammoniate: Y(BH4)3.4NH3,Journal of Materials Chemistry22 (2012) 1061–1068 ( abstract )49.Jiang Xu, Daohui Lai, ZongHan Xie, Paul Munroe and Zhong-Tao Jiang,A critical role for Al in regulating the corrosion resistance of nanocrystallineMo(Si1-x Al x)2 films,Journal of Materials Chemistry22 (2012) 2596–2606 ( abstract )50.Tae Hoon Noh et al.,Photophysical and Photocatalytic Properties of Zn3M2O8 (M = Nb, Ta),Journal of the American Ceramic Society95 (2012) 227–231 ( abstract )51.Y. H. Duan et al.,Elastic constants of AlB2-type compounds from first-principles calculations,Computational Materials Science51 (2012) 112–116 ( abstract )52.Bengt E. Tegner, Graeme J. Ackland,Pseudopotential errors in titanium,Computational Materials Science52 (2012) 2 6 ( abstract )53.D. Sen, R. Thapa, K. Bhattacharjee, K.K. Chattopadhyay,Site dependent metal adsorption on (3 ×3) h-BN monolayer: Stability, magnetic and optical properties,Computational Materials Science51 (2012) 165–171 ( abstract )54.M. G. Brik, C.-G. Ma,First-principles studies of the electronic and elastic properties of metal nitrides XN (X = Sc, Ti, V, Cr, Zr, Nb),Computational Materials Science51 (2012) 380–388 ( abstract )55.Mingjun Pang et al.,First-principles calculations on the crystal, electronic structures and elastic properties of Ag-rich γ' phase approximates in Al-Ag alloys,Computational Materials Science51 (2012) 415–421 ( abstract )56.Souraya Goumri-Said, Nawel Kanoun-Bouayed, Ali H. Reshak, Mohammed BenaliKanoun,On the electronic nature of silicon and germanium based oxynitrides and their related mechanical, optical and vibrational properties as obtained from DFT and DFPT,Computational Materials Science53 (2012) 158–168 ( abstract )57.K. Haddadi, A. Bouhemadou, S. Bin-Omran,Structural, elastic, electronic, chemical bonding and thermodynamic properties ofCaMg2N2 and SrMg2N2: First-principles calculations,Computational Materials Science53 (2012) 204–213 ( abstract )58.Huiyang Gou et al.,Origin of the rigidity in tetragonal MB (M = Cr, Mo and W) and softening of defective WB: First-principles investigations,Computational Materials Science53 (2012) 460–463 ( abstract )59.Haizhou Wang, Yongzhong Zhan, Mingjun Pang,The structure, elastic, electronic properties and Debye temperature of M2AlC (M=V, Nband Ta) under pressure from first-principles,Computational Materials Science54 (2012) 16–22 ( abstract )60.Hai-You Huang, Ming Wu,First principles calculations of hydrogen-induced decrease in the cohesive strength of α-Al2O3 single crystals,Computational Materials Science54 (2012) 81–83 ( abstract )61.Xinrui Cao, Yunsong Li, Xuan Cheng, Ying Zhang,First-principles studies of structural, electronic and optical properties of AB2 (A = Si, Ge and B = O, S) nanotubes,Computational Materials Science54 (2012) 84–90 ( abstract )62.Wei Zhou, Lijuan Liu, Mengying Yuan, Qinggong Song, Ping Wu,Electronic and optical properties of W-doped SnO2 from first-principles calculations,Computational Materials Science54 (2012) 109–114 ( abstract )63.A. Bouhemadou et al.,Theory study of structural parameters, elastic stiffness, electronic structures and lattice dynamics of RBRh3 (R = Sc, Y, La and Lu),Computational Materials Science54 (2012) 336–344 ( abstract )64.M. Romero, R. Escamilla,First-principles calculations of structural, elastic and electronic properties of Nb2SnC under pressure,Computational Materials Science55 (2012) 142–146 ( abstract )65.Benhua Luo, Xueye Wang, Y u Zhang, Yong Xia,First principle calculations for pressure induced structural phase transitions of Fe doped in SrMnO3,Computational Materials Science55 (2012) 199–204 ( abstract )66.Fanjie Kong, Yanhua Liu, Baolin Wang, Yanzong Wang, Lili Wang,Lattice dynamics of PbTe polymorphs from first principles,Computational Materials Science56 (2012) 18–24 ( abstract )67.Yong-Hui Zhang et al.,Tuning the magnetic and transport property of graphene with Ti atom and cluster,Computational Materials Science56 (2012) 95–99 ( abstract )68.Ch. Bheema Lingam, K. Ramesh Babu, Surya P. Tewari, G. Vaitheeswaran,Density functional study of electronic,bonding, and vibrational properties ofCa(NH2BH3)2,Journal of Computational Chemistry33 (2012) 987–997 ( abstract )69.Zhiwei Cui, Feng Gao, Zhihua Cui and Jianmin Qu,Developing a second nearest-neighbor modified embedded atom method interatomicpotential for lithium,Modelling and Simulation in Materials Science and Engineering20 (2012) 015014( abstract )70.Alexei Bosak et al.,New insights into the lattice dynamics of α-quartz,Zeitschrift fur Kristallographie227 (2012) 84–91 ( abstract )71.Zuocai Huang, Jing Feng, Wei Pan,Theoretical investigations of the physical properties of zircon-type YVO4,Journal of Solid State Chemistry185 (2012) 42–48 ( abstract )72.Dajiang Mei et al.,Synthesis, structure, and electronic structure of CsAgGa2Se4,Journal of Solid State Chemistry186 (2012) 54–57 ( abstract )73.Haimin Ding, Jinfeng Wang, Chunyan Li, Jinfeng Nie, Xiangfa Liu,Study of the surface segregation of carbon vacancies in TiC x,Solid State Communications152 (2012) 185–188 ( abstract )74.Mei Xia Xiao, Yong Fu Zhu, Qing Jiang,Improved electromigration reliability of Cu films by doping and interface engineering, Solid State Communications152 (2012) 210–214 ( abstract )75.R. Escamilla, M. Romero, F. Morales,Elastic properties, Debye temperature, density of states and electron-phonon coupling of ZrB12 under pressure,Solid State Communications152 (2012) 249–252 ( abstract )76.Yan Wang, Xi Zhang, Yanguang Nie, Chang Q. Sun,Under-coordinated atoms induced local strain, quantum trap depression and valencecharge polarization at W stepped surfaces,Physica B: Condensed Matter407 (2012) 49–53 ( abstract )77.Yingce Yan et al.,First-principle calculations of dilute nitride GaP1-x N x alloy in zinc-blende structures, Physica B: Condensed Matter407 (2012) 112–115 ( abstract )78.Jie Liu et al.,Influence of Ni and N on generalized stacking-fault energies in Fe-Cr-Ni alloy: A first principle study,Physica B: Condensed Matter407 (2012) 891–895 ( abstract )79.T. Chihi, M. Fatmi, B. Ghebouli,First-principles prediction of metastable niobium and tantalium nitrides M4N5 and M5N6 stoichiometry,Solid State Sciences14 (2012) 80–83 ( abstract )80.Lizhao Liu et al.,Amorphous structural models for graphene oxides,Carbon50 (2012) 1690–1698 ( abstract )81.Jingying Lu, Shang-Peng Gao, Jun Yuan,ELNES for boron, carbon, and nitrogen K-edges with different chemical environments in layered materials studied by density functional theory,ultramicroscopy112 (2012) 61–68 ( abstract )82.Tian Meng-kui, Shangguan Wen-feng,Photocatalytical water decomposition on visible light-driven solid-solution compounds K4Ce2Ta10-x Nb x O30 (x = 0-10),Materials Chemistry and Physics131 (2012) 569–574 ( abstract )83.M.G. Brik, A.M. Srivastava,First-principles calculations of structural, electronic, and elastic properties of MgZrSi2O7, Materials Chemistry and Physics132 (2012) 6–9 ( abstract )84.Maochang Liu, Yuanchang Du, Lijng Ma, Dengwei Jing, Liejin Guo,Manganese doped cadmium sulfide nanocrystal for hydrogen production from waterunder visible light,International Journal of Hydrogen Energy37 (2012) 730–736 ( abstract )85.Ying Lv, Zhian Zhang, Yanqing Lai, and Yexiang Liu,Electrodeposition of Porous Mg(OH)2 Thin Films Composed of Single-CrystalNanosheets,Journal of The Electrochemical Society159 (2012) D187–D189 ( abstract )86.Lu Peng-Fei et al.,Electronic Structure and Optical Properties of Antimony-Doped SnO2 from First-Principle Study,Communications in Theoretical Physics57 (2012) 145–150 ( abstract )87.Chuan-Hui Zhang, Qiong Ran, Jiang Shen,Structural stability of silicene-like nanotubes,Computer Physics Communications183 (2012) 30–33 ( abstract )88.A. Gray et al.,Mapping application performance to HPC architecture,Computer Physics Communications183 (2012) 520–529 ( abstract )89.Wei Li Cheah and Michael W. Finnis,Structure of multilayer ZrO2/SrTiO3,Journal of Materials Science47 (2012) 1631–1640 ( abstract )90.Qi-Jun Liu, Zheng-Tang Liu, Qian-Qian Gao, Li-Ping Feng and Hao Tian,The doping effect of N substituting for different atoms in orthorhombic SrHfO3,Journal of Materials Science47 (2012) 3046–3051 ( abstract )91.A. Yangthaisong,Electronic and Lattice Vibrational Properties of Cubic SrHfO3 from First-PrinciplesCalculations,Journal of Electronic Materials41 (2012) 535–539 ( abstract )92.M. J. Phasha, P. E. Ngoepe,An alternative DFT-based model for calculating structural and elastic properties ofrandom binary HCP, FCC and BCC alloys: Mg-Li system as test case,Intermetallics21 (2012) 88–96 ( abstract )93.Teruyasu Mizoguchi, Katsuyuki Matsunaga, Eita Tochigi, Yuichi Ikuhara,First principles pseudopotential calculation of electron energy loss near edge structures of lattice imperfections,Micron43 (2012) 37–42 ( abstract )94.D.G. McCulloch, D.W.M. Lau, R.J. Nicholls, J.M. Perkins,The near edge structure of cubic boron nitride,Micron43 (2012) 43–48 ( abstract )95.Masataka Hakamada, Fumi Hirashima, Kota Kajikawa and Mamoru Mabuchi,Magnetism of fcc/fcc, hcp/hcp twin and fcc/hcp twin-like boundaries in cobalt,Applied Physics A106 (2012) 237–244 ( abstract )96.Xing Ming et al.,First-principles study of pressure-induced magnetic transition in siderite FeCO3,Journal of Alloys and Compounds510 (2012) L1–L4 ( abstract )97.Tingting Tan, Zhengtang Liu, Yanyan Li,First-principles calculations of electronic and optical properties of Ti-doped monoclinic HfO2,Journal of Alloys and Compounds510 (2012) 78–82 ( abstract )98.Yongzhong Zhan, Mingjun Pang, Haizhou Wang, Yong Du,The structural, electronic, elastic and optical properties of AlCu(Se1-x Te x)2 compounds from first-principle calculations,Current Applied Physics12 (2012) 373–379 ( abstract )99.Mingjun Pang, Yongzhong Zhan, Haizhou Wang,Ab initio investigation into the structural, electronic and elastic properties of AlCu2TM (TM = Ti, Zr and Hf) ternary compounds,Current Applied Physics12 (2012) 957–962 ( abstract )100.N. Mazumder, A. Bharati, S. Saha, D. Sen, K.K. Chattopadhyay,Effect of Mg doping on the electrical properties of SnO2 nanoparticles,Current Applied Physics12 (2012) 975–982 ( abstract )101.Hongchen Du, Guixiang Wang, Xuedong Gong, Heming Xiao,Theoretical study on the adduct of chlorine trifluoride oxide and borontrifluoride-molecular and crystal structures, vibrational spectrum, and thermodynamic properties,International Journal of Quantum Chemistry112 (2012) 1291–1298 ( abstract )102.Wenlin Zhang, Xiaoqing Wang, Guangqiu Shen, Dezhong Shen,Top-seeded growth, optical properties and theoretical studies of noncentrosymmetricTe2V2O9,Crystal Research and Technology47 (2012) 163–168 ( abstract )103.Baodan Liu et al.,Microstructure and cathodoluminescence study of GaN nanowires without/with P-doping, Crystal Research and Technology47 (2012) 207–212 ( abstract )104.Peter A. Tanner, Guohua Jia, Bing-Ming Cheng, Mikhail G. Brik,Analysis of spectra of neat and lanthanide ion-doped KPb2Cl5 excited by synchrotron radiation,physica status solidi (b)249 (2012) 581–5872 ( abstract )105.A. Trejo, A. Miranda, L. Nino de Rivera, A. Daaz-Mendez, M. Cruz-Irisson, Phonon optical modes and electronic properties in diamond nanowires,Microelectronic Engineering90 (2012) 92–95 ( abstract )106.A. Trejo, J.L. Cuevas, R. Vazquez-Medina, M. Cruz-Irisson,Phonon band structure of porous Ge from ab initio supercell calculation,Microelectronic Engineering90 (2012) 141–144 ( abstract )107.Minoru Osada and Takayoshi Sasaki,A- and B-Site Modified Perovskite Nanosheets and Their Integrations into High-kDielectric Thin Films,International Journal of Applied Ceramic Technology9 (2012) 29–36 ( abstract ) 108.D.L. Geatches, S.J. Clark, H.C. Greenwell,Iron reduction in nontronite-type clay minerals: Modelling a complex system,Geochimica et Cosmochimica Acta81 (2012) 13–27 ( abstract )109.Ting-Feng Yi et al.,Stabilities and electronic properties of lithium titanium oxide anode material for lithium ion battery,Journal of Power Sources198 (2012) 318–321 ( abstract )110.Wenjing Tang, Dechun Li, Shengzhi Zhao, Guiqiu Li and Kejian Yang, First principle study of the elastic properties of InGaAs with different dopingconcentrations of indium,Molecular Simulation38 (2012) 84–89 ( abstract )111.C. L. Zhang et al.,First principles study on mechanical properties of Mg-MgCd interface micro-zone,Molecular Simulation38 (2012) 200–203 ( abstract )112.Mengkui Tian, Wenfeng Shangguan, Wenliang Tao,The photocatalytical activities for water decomposition of K4R2M10O30 (R = Y, La, Ce, Nd, Sm; M = Ta, Nb) and their photophysical properties based on the first principlecalculation,Journal of Molecular Catalysis A: Chemical352 (2012) 95–101 ( abstract )113.Manuel Ramos, Gilles Berhault, Domingo A. Ferrer, Brenda Torres and Russell R.Chianelli,HRTEM and molecular modeling of the MoS2-Co9S8 interface: understanding thepromotion effect in bulk HDS catalysts,Catalysis Science & Technology2 (2012) 164–178 ( abstract )114.Wei Huang, Lulu Sun, Peide Han, Jinzhen Zhao,CH4 dissociation on Co(0001): A density functional theory study,Journal of Natural Gas Chemistry21 (2012) 98–103 ( abstract )115.Lu Yong-Fang, Shi Li-Qun, Ding Wei and Long Xing-Gui,First-Principles Study of Hydrogen Impact on the Formation and Migration of Helium Interstitial Defects in hcp Titanium,Chinese Physics Letters29 (2012) 013102 ( abstract )116.Sun Hong-Guo, Zhou Zhong-Xiang, Yuan Cheng-Xun, Yang Wen-Long and Wang He, Structural, Electronic and Optical Properties of KTa0.5Nb0.5O3 Surface: A First-Principles Study,Chinese Physics Letters29 (2012) 017303 ( abstract )117.Yang Ping et al.,Uniaxial stress influence on lattice, band gap and optical properties of n-type ZnO:first-principles calculations,Chinese Physics B21 (2012) 016803 ( abstract )118.Niu Wen-Xia and Zhang Hong,Ar adsorptions on Al (111) and Ir (111) surfaces: a first-principles study,Chinese Physics B21 (2012) 026802 ( abstract )119.Ye-Yu Li et al.,Synthesis, Crystal and Electronic Structures of Ba3ZnSb2O9 with the 6H-perovskite-type Structure,Chinese Journal of Structural Chemistry31 (2012) 73–78 ( abstract )120.Wang Li, Fang Li-hong, Gong Jian-hong,First-principles study of TiC(110) surface,Transactions of Nonferrous Metals Society of China22 (2012) 170–174 ( abstract )。
你用得着的五个功能强大的（论文写作以及专业英语）翻译网站新年第一帖，我把自己搜集的几个功能非常强大的学术翻译网站分享给大家：1）句库网址：/翻译的非常好，基本上都能找到自己所需的，同时也可以发音，这有利于提高自己的专业英语听力！举例：翻译f irst principles/s ... st+principles在第4个清楚的给出了”第一原理“的解释！2）词博网址：/翻译的很杂，可以供多种选择；而且可以发音，这一点感觉很好！举例：翻译f irst principles/sea ... st+principles没有找到确切的”第一原理“的翻译，但是其他词汇的翻译还好！3）句译网址：/v iewPage.php翻译的很好，基本你都可以在他的翻译中找到合适的！举例：翻译f irst principles/v iewPage.php（这个要自己输入f irt principle)翻译结果非常好！它给出10个例句，结果又8个符合我们的要求，而且给出的例句非常有用，完全是文献里摘出来的！！！我特意放在下面：~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1. 第一过渡金属酞菁分子的电子结构的第一性原理计算A f irst-principles study on the electronic structure of the first transition metal phthalocy anines2. 首要原则First Principles3. 首要教学原理First principles of Instruction4. NiTi合金的第一性原理研究A f irst principles inv estigation on NiTi alloy5. KTa_(0.5)Nb_(0.5)O_3电子结构的第一性原理研究First Principles on KTa_0.5Nb_0.506. NiTi合金的第一原理研究First-Principles Inv estigation of NiTi Alloy7. In-N共掺杂ZnO第一性原理计算First principles study of In-N codoped ZnO8. 碳纳米管的第一性原理研究First-Principles Study of Carbon Nanotubes9. Cu掺杂ZnS的第一性原理计算First-principles Calculation of Cu-doped ZnS10. 分子电子器件第一性原理设计First Principles Design of Molecular Electric Dev ices~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~4)词都/查词也还算不错，并且给出很多来自文献的例句，非常不错！举例：翻译f irst principles给出了“第一原理”的翻译，而且给出了很多直接来自文献的例句，有些都是长句。
赝势对计算石墨烯声子谱线的第一性原理研究郭富强;王艳丽;尹国盛【摘要】石墨烯是理论与实验方面研究的热点,而探究其声子谱线结构又为研究力学、热力学等提供基础.本文采用基于密度泛函理论的第一性原理,运用不同的交换关联和赝势方法,计算了石墨烯以及石墨的声子谱线.对比研究发现:在声子谱低频率阶段,不同的赝势计算的结果差别很小;而在声子谱的高频率阶段,不同赝势计算的结果差别显著.相对于GGA交换关联,LDA交换关联计算的高频光学支有所软化,计算结果与实验值更加接近.相对于US赝势方法,PAW赝势方法计算的结果与实验值更加接近.综合比较,PAW-LDA赝势的计算结果与实验值最为接近.【期刊名称】《物理与工程》【年(卷),期】2016(026)005【总页数】5页(P66-70)【关键词】声子谱;石墨烯;赝势【作者】郭富强;王艳丽;尹国盛【作者单位】郑州工业应用技术学院,河南郑州 451151;河南建筑职业技术学院,河南郑州 450007;郑州工业应用技术学院,河南郑州 451151【正文语种】中文碳原子不同的排列能形成金刚石、石墨、C60以及碳纳米管等不同的晶体结构，从而体现出不同的物理、化学性质.近来人们通过物理及化学的方法从石墨中分离出了单层的石墨片，称之为石墨烯.由于石墨烯具有非常优良的力学、热学、电学等特性，使得它从一出现就在理论与实验方面成了研究的热点，并取得了丰硕的成果.在实验方面，人们已经能够通过不同的方法制取石墨烯，这使其在生产生活中的应用成为了可能[1].声子谱线结构的研究是其他诸如力学、热力学性质研究的基础.因此研究石墨烯声子谱线结构对于石墨烯的应用具有非常重要的意义.实验方面对声子谱线的研究主要有下面几种方法：中子散射方法是经常使用的一种方法，但它不能得到高频支声子的频率[2，3]；高分辨率电子谱镜虽能得到高频声子支的频率，但其结果与理论计算相差很大[4]；拉曼谱方法虽很精确，但仅能对Γ点进行测量；非弹性X射线衍射方法仅在高频光学支的测量方面最为准确[5].由于以上方法各有缺陷，因此实验上仍需要综合以上几种方法对晶体声子谱线进行确定.理论计算方面，目前主要有力常量方法和第一性原理方法.力常量方法通过经验势函数，计算出原子之间的力矩阵，从力矩阵得出声子谱线.除了势函数的影响，力常量方法的计算精度还受所取原子作用力半径的影响 [6].相对而言，第一性原理的计算完全独立于经验参数，因此结果更值得信赖[7].但是第一性原理的计算精度受所选赝势的影响[8].基于此，本文比较全面地研究了赝势方法以及交换关联对于石墨烯以及石墨的声子谱线的影响.第一性原理对于声子结构的计算，主要采用冷冻声子方法(frozen-phonon)[9]和微扰密度泛函方法(Density Function Perturbation Theory，简称为DFPT方法)[10].冷冻声子方法是在严格分析晶体对称性的基础上引入一些微小的位移，这些位移使原子之间存在赫尔曼-费曼(Hellmann-Feynman)力，计算出原子间的赫尔曼-费曼力，进而通过动力学矩阵即可得到声子色散曲线.该方法的优点是，对于简单晶体结构计算简单准确，但是对于复杂结构，需要很大的超胞才能计算精确，因此对计算条件要求比较高.相对于直接方法，DFPT方法通过系统对外界能量的响应求解声子谱线，它克服了直接方法的缺点，能适用于复杂的体系.由于石墨以及石墨烯的结构比较简单，因此本文对于石墨及石墨烯声子谱线的计算选用了较为简单的冷冻声子方法，计算软件为vasp软件包和frophon软件.计算采用以密度泛函理论[11，12]平面波赝势法为基础的vasp软件包[13]，由于研究的是不同赝势的对比，我们选取了几种vasp中常用的交换关联和赝势方法.电子-电子之间的交换关联作用分别采用了广义梯度近似(GGA)和局域梯度近似(LDA)两种方法；离子实与价电子间的作用分别选取了缀加平面波方法(PAW)和超软赝势(USPP).计算出赝势方法和交换关联不同组合时的声子谱线，进而研究赝势对于石墨烯及石墨声子谱线的影响.对于不同的赝势，选用不同的截断能，其中GGA取为300eV，LDA取400eV.对石墨烯声子谱线的计算，超原包采用2×2×1，K点网格由Monkhost-Pack方法[14]产生.K点网格,采用(9×9×1)，单层石墨烯的真空层取为2nm.对于石墨声子谱线的计算，超胞为2×2×2，计算K点取为9×9×9.考虑到声子对力的依赖关系，计算选取了较高的收敛标准，为0.1eV/nm.石墨是层状晶体，其原子结构及布里渊区如图1所示.石墨层内为六角结构，层间为A-B-A堆栈.石墨烯是单层石墨构成的二维晶体，计算中一层石墨外加足够的真空层即可模拟石墨烯.石墨中碳原子为SP2杂化，层内原子间为共价键，所以层内碳原子作用较强.而层间碳原子之间为很弱的范德瓦尔斯力作用，因此石墨通常层间自然解理.对于第一性原理计算，现有的GGA以及LDA赝势均不能描述范德瓦尔斯力，所以第一性原理对于石墨层间性质的计算有很大的局限性.虽然如此，文献表明LDA赝势能够给出与实验较为一致的晶格常数及层间作用[16].我们运用不同的赝势方法得到的晶格常数如表1所示.由表1可以看出：对于层内晶格常数a，LDA赝势与实验一致，而GGA方法得到的结果较实验值大0.002nm；而对于层间参数c, LDA方法基本能够描述，仅仅较实验值大了0.002nm，但GGA赝势计算不到石墨间的层间距.对于赝势方法，US-LDA赝势计算的a值为2.44nm，而PAW-LDA赝势为0.245nm，实验值为0.244nm，因此从晶格优化的角度而言，US-LDA赝势是描述石墨晶体的最好的方法.本文的计算中，计算了所有4种赝势组合的石墨烯声子谱线，同时计算了LDA所对应的两种赝势组合的石墨的声子结构.首先研究一下石墨烯声子谱的特点，不同赝势的声子谱线如图2和图3所示.石墨烯原胞中含有两个原子，所以共有3支声学支(A)和3支光学支(O).在这6支声子支中，包括垂直于平面的模式(Z)以及平行于平面的模式.平行于平面模式又分成了纵模式(L)和横模式(T).为了直观表达，在图中对这些模式进行了标注.其中的高对称点的见图1的布里渊区图.由图2和图3可知，在布里渊区的高对称点，一些声子支是高度简并的.在布里渊区中心Γ点，TA和LA呈现出线性散射关系，而ZA模式呈现q2的散射关系，所以平行于平面以及垂直于平面的模式是不同的，这与别的文献研究结果一致,声子谱线的另外一个特点就是ZA与ZO在K点相交，LA与LO在K点也发生了相交[17].接下来对比几种赝势得到的声子谱线的差别.为了对比，将不同赝势方法与交换关联组合计算，得到4种组合赝势的声子谱线，按照赝势方法和交换关联分成了两组，第一组考虑赝势方法的不同造成的影响，如图2所示；第二组考虑芯电子之间交换关联的不同所造成的影响，如图3所示.首先考虑赝势方法对声子谱线的影响.由图2可知，价电子和离子实之间相互作用的不同对声子谱有一致的影响.无论对于LDA和GGA，在低频的声学支频带，两种方法得到的声子谱几乎完全重合，即在低频带，声子谱线对赝势方法的选择不敏感.但是可以明显看到，在高频带，无论对于LDA和GGA，相对于US赝势，PAW方法计算的结果总是有所软化，与实验值更接近.然后考虑赝势方法相同时，交换关联作用对声子谱线的影响.由图3可知，在低频带，不同交换关联作用得到的声子谱线几乎完全重合，即不同的交换关联对低频声子支影响不明显.但是在高频区域，不同交换关联作用得到的声子谱线分离开来，GGA方法得到的声子频率较小，LDA方法得出的频率比较大，即GGA软化了高频带的声子谱线.为了从量上区别出来,我们给出了几个高对称K点的声子频率值，如表2所示.在表2中列出了各种赝势计算的高对称点的声子谱线，同时列出了文献计算值以及各种实验测值.为了直观比较各种赝势方法得到的频率值的优越，我们做出了各种赝势下的计算与实验值之间的相对误差比值，其计算方法如公式(1)所示各种赝势计算的相对误差如图4所示，由图4可知，总体系上各种方法得到的结果与实验值的差别均不太大，在-4%～10%以内，与其他文献的计算结果也很吻合.从交换关联的角度考虑，US-GGA赝势产生的误差大于US-LDA，而PAW-GGA赝势的误差在某些点大于PAW-LDA，在其他点则小于PAW-LDA.总体上GGA要比LDA赝势误差大.从赝势方法的角度考虑，US-GGA的误差要大于PAW-GGA，而US-LDA的误差总体上小于PAW-LDA.总的比较，在所有方法中US-LDA赝势计算石墨烯声子谱最为准确.上面研究了赝势对于石墨烯声子谱线的影响，同时我们想把石墨烯的结果推广到石墨，研究赝势对于石墨声子谱线的影响.由前面的分析可以知道，由于GGA赝势不能正确计算范德瓦尔斯力作用，而石墨层间主要靠范德瓦尔斯力作用结合，因此该赝势不能用于块体石墨的计算.所以对于石墨烯的声子谱线的计算，仅考虑US-LDA和PAW-LDA两种赝势的情况.研究离子实与价电子之间的作用对石墨声子谱的影响，其结果如图5所示.由图5可见，石墨声子谱线较石墨烯声子谱线多A-Γ一段，这是由于考虑了石墨法向周期性的原因.其他各段，石墨与石墨烯的声子谱线结构几乎完全相同.这与别的计算值以及实验值相吻合[4,5].对于US-LDA与PAW-LDA两种赝势方法，对比石墨烯的结果，PAW-LDA方法计算在高频阶段要低于US-LDA方法，即 PAW-LDA方法在高频阶段也对声子有所软化，综合石墨烯的结果，可以知道US-LDA 与实验值也较为接近.这与石墨烯的研究相一致.基于密度泛函理论的第一性原理方法，应用不同赝势计算了石墨烯以及石墨的声子谱线，本文的计算结果与别的计算、理论以及实验比较吻合.对不同赝势计算结果的比较得出：对于低频声子支，赝势对声子谱影响不显著；在高频段, 对于相同的赝势方法，LDA交换关联较GGA交换关联计算的声子谱线更加精确；计算表明：在应用第一性原理方法计算石墨以及石墨烯声子谱线中，US-LDA赝势最为精确.【相关文献】[1] Geim A K. Graphene: Status and prospects[J]. Science, 2009, 1530: 324.[2] Nicklow R, Wakabayashi N, Smith H G. Lattice dynamics of pyrolytic graphite[J]. Phys. Rev. B, 1972, 5: 4951.[3] Dolling G, Brockhouse B N. Lattice vibrations in pyrolitic graphite[J]. Phys. Rev., 1962, 128: 1120.[4] Oshima C, Aizawa T, Souda R, et al. Surface phonon dispersion curves of graphite (0001) over the entire energy region[J]. Solid State Commun., 1988, 65: 1601.[5] Maultzsch J, Reich S, Thomsen C, et al. Phonon dispersion in graphite[J]. Phys. Rev. Lett., 2004, 92: 075501.[6] Jishi R A, Venkataraman L, Dresselhaus M S, et al. Phonon modes in carbon nanotubules[J]. Chem. Phys. Lett., 1993, 209: 77.[7] Wirtz L, Rubio A. The phonon dispersion of graphite revisited[J]. Solid State Commun., 2004, 131: 141.[8] Favot F, Corso A D. Phonon dispersions: Performance of the generalized gradient approximation[J]. Phys. Rev. B, 1999, 60: 11427.[9] Fanidisa C, Van Dycka D, Van Landuyt J. Inelastic scattering of high-energy electrons ina crystal in thermal equilibrium with the environment I. Theoretical framework[J]. Ultramicroscopy, 1992, 41: 55.[10] Gonze X. First-principles responses of solids to atomic displacements and homogeneous electric fields: Implementation of a conjugate-gradient algorithm[J]. Phys. Rev. B, 1997, 55: 10337.[11] Kohn W, Sham L J. Quantum density oscillations in an inhomogeneous electron gas[J].Phys. Rev., 1965, 137: A1697.[12] Kohn W, Sham L J. Self-consistent equations including exchange and correlation effects[J]. Phys. Rev., 1965, 140: A1133.[13] Kresse G, Furthmüller J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set[J]. Phys. Rev. B, 1996, 54: 11169.[14] Monkhorst H J, Pack J D. Special points for Brillouin-zone integrations[J]. Phys. Rev. B，1976, 13: 5188.[15] Ooi N, Rairkar A, Adams J B. Density functional study of graphite bulk and surface properties[J]. Carbon, 2006, 44: 231.[16] McKie D, McKie C. Essentials of crystallography[M]. Oxford: Oxford Press, 1986, 6-8.[17] Saito R, Dresselhaus G, Dresselhaus M S. Physical properties of carbon nanotubes, London: Imperial College Press, 1998.[18] Dubay O, Kresse G.Accurate density functional calculations for the phonon dispersion relations of graphite layer and carbon nanotubes[J]. Phys. Rev. B, 2003, 67: 035401. [19] Touinstra F, Koenig J L. Raman spectrum of graphite[J]. J. Chem. Phys. 1970, 53: 1126.。
First-principles prediction of the glass-forming ability in Zr e Ni binary metallic glassesC.Y.Yu a,X.J.Liu b,C.T.Liu a,*a Centre for Advanced Structural Materials,Department of Mechanical and Biomedical Engineering,College of Science and Engineering,City University of Hong Kong,Kowloon,Hong Kong,Chinab State Key Laboratory for Advanced Metals and Materials,University of Science and Technology Beijing,Beijing100083,PR Chinaa r t i c l e i n f oArticle history:Received16March2014 Received in revised form27April2014Accepted29April2014Available online28May2014Keywords:A.Metallic glassesB.Glass forming abilityE.Ab-initio calculationsE.Molecular dynamics simulation a b s t r a c tIn the present study,the atomistic approach,developed previously by hybridizing both internal energies and atomic-scale defect structures,is extended to predict the trend of glass forming ability(GFA)from the Zr e Cu binary alloy system with good GFA to the Zr e Ni binary system with relatively poor GFA.The predicted composition dependence of the GFA in the Zr e Ni alloy system is consistent with those ob-tained from experimental results,indicating the validity of the proposed atomistic approach in both of the alloy systems.The different GFAs in the Zr e Cu and Zr e Ni systems have been puzzled by the metallic glass community for quite a long time,and our study provides the physical reasons for the different glass forming behaviors of the two alloy systems in terms of both atomic configurations and relative stability of their intermetallic phases.Ó2014Elsevier Ltd.All rights reserved.1.IntroductionIn the Periodic Table,Ni is a neighboring atom of Cu,and they possess many similar physical and metallurgical properties. Nevertheless,several bulk metallic glasses(BMGs)were success-fully made in the Zr e Cu binary system but none in the Zr e Ni system[1e7].Thus,it is of physical interest for us to understand the difference between these two alloy systems.Also,despite the fact that many efforts have been devoted to the study of the atomic structures,electronic properties and dynamic properties of exper-imentally made Zr e Ni amorphous ribbons[8e11],seldom atten-tions were paid on the glass formation and glass forming ability (GFA)of this system.Thus,in this work,we study the glass forming behavior and predict the composition dependence of the GFA in the Zr x Ni100Àx binary system by the atomistic approach we developed previously,and discuss the physical reasons for the different glass forming behaviors between the Zr e Ni and Zr e Cu alloy systems.From a physical metallurgy point of view,the glass formation can always happen in alloys if a high enough cooling rate exceeding the critical one is applied,and the GFA for an alloy system can be evaluated by measuring its corresponding critical cooling rate. However,the critical cooling rate is difficult to measure experimentally,thus various criteria based on those easily measurable parameters have been proposed to validate the GFA. Among them,the most commonly used criteria are those with characteristic temperatures[12e16];nevertheless,these methods are insufficient in predicting the GFA from compositions alone since the characteristic temperatures can be determined only after the alloys have been made into the glass state experimentally.There-fore,the development of a theoretical approach to predict GFA just from compositions alone is vitally important.The glass formation is generally considered as a competition between the amorphous phase and crystalline counterparts,and the metallic glass-forming liquid with a good GFA exhibits a high thermodynamic stability and the satisfactory ability to suppress crystallization kinetically.Many efforts have been devoted to searching for good GFA from single or a hybrid of the thermo-dynamic and kinetic effects[17e22].From a thermodynamic consideration,the formation of the metastable amorphous state should include two aspects:the relative stability of the glass phase and its resistance against crystallization.In2006,based on the Miedema’s model,Xia et al.proposed a parameter g*¼GFA fÀD H amor=ðD H amorÀD H interÞbased on the consider-ation of these two factors[20],whereÀD H amor represents theformer factor and D H amorÀD H inter represents the latter one.This approach was applied to assess the GFA in specific binary systems. However,it did not specially concern the kinetic factor,which is also believed to have a significant effect on glass formation[21].*Corresponding author.Tel.:þ852********.E-mail address:chainliu@.hk(C.T.Liu).Contents lists available at ScienceDirect Intermetallicsjou rn al homepage:/locate/intermet/10.1016/j.intermet.2014.04.0200966-9795/Ó2014Elsevier Ltd.All rights reserved.Intermetallics53(2014)177e182In comparison to the thermodynamic considerations,the kinetic effect is much difficult to follow;nevertheless,some useful thoughts have been suggested recently.Among them,the intrinsic structural properties play an important role in atom diffusion[21,22],and they are considered to be closely related to the kinetic effect.The supercooled liquid contains both close-packed(solid-like)and loosely-packed(liquid-like)areas,and some excessive atomic spaces are expected to exist in the loosely packed areas as the path for atom diffusion.A lower fraction of excessive atomic spaces in the supercooled liquid means a denser atomic packing and slower atomic mobility,which makes crys-tallization processes more difficult to take place during cooling.Based on thefirst-principles(FP)calculations and thefirst-principles molecular dynamics(FPMD)simulations,we recently proposed a theoretical expressiong FP¼g*.p Sea¼D H amor.D H interÀD H amor.p Sea;(1)which combined both of the thermodynamic and kinetic effects to predict GFA from the alloy composition alone.In this expression, the term of D H amor=ðD H interÀD H amorÞrepresents the thermody-namic effect,and p Searepresents a kinetic effect defined as the proportion of the total excessive atomic spaces(S ea)to the total atomic volumes(V tot)in the supercooled liquid.The quantity of D H amor and D H inter in the Zr x Ni100Àx system can be readily calcu-lated byD H¼EÀxE Zr=100Àð100ÀxÞE Ni=100;(2)where E is the internal energy per atom for the amorphous or crystalline phase of Zr x Ni100Àx alloy,and E Zr and E Ni are the internal energies per atom for pure hexagonal close packed(hcp)Zr and face centered cubic(fcc)Ni determined from thefirst-principles calcu-lations,respectively.The total S ea in the system is derived from a benchmark configuration to consider the contribution of the“defect structure”[23].First,an ideal polyhedron with certain close-packed coordi-nation numbers(named as N ic)around a central atom is regarded as a benchmark configuration without any S ea.Then,those atoms with a close-packed coordination numbers n i exceeding N ic are expected to have no excessive atomic spaces because of their dense packing, and those with n i less than N ic are considered to have defected bonds with a certain degree of excessive space associated with atoms.As an improvement of our previous definition[23],here n i is defined as the number of close-packed coordinate atoms of the central atom within a cutoff distance r ij of1.10r1(r1is thefirst peak position in pair distribution function(PDF)curve).This cutoff dis-tance is similar with that proposed by Fan et al.[24],and chosen by taking the thermal vibration contribution into account based on Lindemann’s melting criterion[25].The total excessive atomic spaces in the system are the sum of that in the Ni and Zr centered clusters,respectively,where the ideal close-packed coordination numbers(N ic)is taken as the nearest integral number to the cor-responding average CN for Ni(or Zr)centered polyhedrons in each composition obtained from Voronoi tessellation analysis[26]. Except for the different definition of n i,the formulation of S ea and p Seais similar with that in the reference[23],and the validity of this approach has been demonstrated successfully in the Zr e Cu binary alloy system.2.MethodIn this work,thefirst-principles calculation was performed to determine the accurate internal energies for different states of the alloys.In order to increase the speed of the calculation prior to the first-principles calculation,we carried out the classical MD simu-lation instead of FPMD simulation initially to obtain the amorphous state for each alloy.10different quenching and energy calculations from independent,random initial configurations were conducted. Meanwhile,another MD simulation with8000atoms was applied to acquire more reliable structural features and the kinetic prop-erties for each alloy with the same cooling rate.The MD simulation was performed based on the embedded atom method(EAM)potential[27]by using the LAMMPS code[28], in the NPT(constant particles number e pressure e temperature) ensemble.A cubic supercell containing200atoms was adopted;Ni or Zr atoms were distributed randomly in the cubic supercell,with the initial density obtained by average atomic mass/volume.Peri-odic boundary conditions were always imposed on the supercell during the calculations.For each alloy,the system wasfirstly equilibrated for100,000steps at2300K to obtain a fully liquid state,with a time step of1fs.Then the system was cooled with the quenching rate of2*1012K/s from2300K down to300K,in order to get thefinal amorphous state ready for the subsequentfirst-principles static calculation to determine the accurate internal energy.Furthermore,extra MD simulations based on the EAM po-tential[29]were performed on some Zr e Cu alloys to acquire the comparable properties.Both the EAM potentials used for the Zr e Ni and Zr e Cu alloy systems give reasonable values for the heats of formation of the intermetallic phases[27,29].Thefirst-principles calculations were implemented by applying Vienna ab initio Simulation Package(VASP)[30].The coulomb interaction of ion cores with the valence electrons was described by projector augmented wave(PAW)potential[31,32],while the electronic exchange was described by the generalized gradient approximation(GGA)refined by Perdew,Burke and Ernzerhof (PBE)[33].The accurate internal energies for the amorphous and intermetallic phases were determined with the initial configuration of the former phase obtained by the MD simulation with200atoms and that of the latter phase set based on the crystal lattice arrangement.Firstly,the initial configurations of the two phases for each alloy were fully relaxed to approach minimum energy of the system by using the conjugate gradient algorithm,and then RMM-DIIS(Residual minimization scheme,direct inversion in the itera-tive subspace)algorithm was used for calculating ultimately the accurate internal energy.A plane-wave energy cutoff was set at 400eV to all the structures for consistency.Brillouin zone in-tegrations were performed using the G centered Monkhorst e Pack grids[34],the k-point meshes were set as1Â1Â1for the supercell of glassy phases,9Â9Â9for the intermetallic compounds, 21Â21Â21for pure fcc-Ni,and21Â21Â12for pure hcp-Zr.For all of the structures,equilibrium cell volumes and all internal atomic positions of the supercells were fully relaxed until convergence with the total energy tolerance of1*10À4eV.And for each composition,the energy was averaged by all calculations from ten different initial amorphous structures obtained from the MD simulation.3.Results and discussionsThe evolution of the potential energy(volume)as a function of temperature during the cooling process is plotted in Fig.1.A glass transition event,instead of an abrupt energy jump symbolizing a liquid e crystal transition,is observed for each alloy,as shown in Fig.1for the Ni50Zr50alloy.Byfitting the data with different slopes, we readily get the glass transition temperature,T g,from the inter-section of the twofitting lines for each alloy and the results so obtained from Zr x Ni100Àx alloys are listed in Table1.During solidification,the glassy phase in a metastable state has to compete with all crystalline phases of solid solutions andC.Y.Yu et al./Intermetallics53(2014)177e182 178intermetallic compounds.However,in view of the fact that theformation enthalpy of the assumed solid solution(D H ss)is much higher than that of the amorphous phase in the Zr x Ni100Àx alloyswithin a wide composition range4<x<90at.%[35],the solidsolution phase is not considered as the competing one here.Theground state formation enthalpies for glasses(D H amor)and inter-metallic compounds(D H inter),in the Zr x Ni100Àx alloys can be readily calculated based on Eq.(2)from thefirst-principles calculations,and the formation enthalpy of a composition locating between twoadjacent intermetallic compounds was calculated using the leverrule.The calculated lattice parameters and formation enthalpies ofthe relevant intermetallic compounds show a good agreement withavailable experimental results[36e43]as listed in Table2.Thecalculated results of D H amor and D H inter are shown in Fig.2,it can be noticed that both of the D H amor and D H inter decreasefirstly and then increase with the Zr concentration.Relatively higher valuesofÀD H amor are observed at33,38and40at.%Zr,leaving a mini-mum at38at.%Zr.This result indicates the higher driving forces ofglass formation for these alloys,especially for the alloy Zr38Ni62.In the meantime,as indicated above,both of the relative sta-bility of the glass phase and its resistance against crystallizationcontribute to the formation of the metastable amorphous state.Therefore,the term D H amorÀD H inter representing the crystallization resistance should be also counted in,to consider about the ther-modynamic effects.We calculated the parameter g*ðg*¼ÀD H amor=ðD H amorÀD H interÞÞto consider these two aspects. The g*shown in Fig.3exhibits a similar trend as that ofÀD H amor, showing a high value at33,38and40at.%Zr with a more distinct peak at38at.%Zr.This demonstrates that besides the higher value ofÀD H amor(see Fig.2),the values of D H amorÀD H inter of these alloys are relatively lower than those of the other alloys.Therefore,the three alloys,especially the alloy Zr38Ni62,are expected to be the promising ones with a higher GFA in the studied compositions calculated from the thermodynamic consideration alone.As indicated above,the intrinsic structural features related to the kinetic effect play an important role in the glass formation. Thus,the proportions of excessive atomic spaces,p Sea,in the Zr x Ni100Àx supercooled liquids are calculated,and the results as a function of Zr concentration are shown in Fig.4.It can be seen that there are four local minima of p Seaat the compositions of10,33,62, and76at.%Zr,respectively.A lower fraction of excessive atomic spaces in the supercooled liquid means a denser atomic packing and lower atomic mobility,which frustrates the formation of crystal phases and can be treated as a kinetic factor affecting GFA. Therefore,these compositions with the local minima of p Seaare expected to have a higher GFA than others based on the kinetic consideration.It is interesting tofind that the predicted composition trends with a good GFA from thermodynamic and kinetic consider-ations are different,even conflicting at some compositions. Therefore,a synergic consideration of both effects is necessary.Table1Glass transition temperature(T g)determined from MD simulations for Zr x Ni100Àx glass-forming alloys.x(Zr at.%)101533384050626776T g/K881986888844834758717722719Table2Calculated and experimental lattice parameters( A),and formation enthalpy D H(kJ/ mol)of Zr e Ni intermetallic compounds from thefirst-principles calculation.Phase SpacegroupLattice parameters(A)Formation enthalpy/atom(kJ/mol)Calc.Expt.Diff(%)ThisworkExpt.Zr P63/mmc a 3.229 3.232[37]À0.09c 5.175 5.148[37]0.52Ni5Zr F m a 6.712 6.7100.03À36.50À34.98Æ6.7[43]Ni7Zr2C12/m1a 4.669 4.698[38]À0.62À46.42À45.94Æ3.48[43]b8.2368.235[38]0.01c12.13512.193[38]À0.48Ni21Zr8P a 6.456 6.476[39]À0.30À46.79b8.0658.064[39]0.01c8.5708.594[39]À0.29Ni10Zr7Aba2a9.1879.211[40]À0.26À47.18À52.17Æ5[43]b9.1979.156[40]0.45c12.43012.386[40]0.36NiZr Cmcm a 3.307 3.271[41] 1.08À46.89À48.53Æ2.47[43]b9.9789.931[41]0.47c 4.069 4.107[41]À0.94NiZr2I4/mcm a 6.511 6.483[42]0.43À34.38À37.24Æ6.11[43]c 5.258 5.267[42]À0.17Fig.2.Formation enthalpies of amorphous and intermetallic phases as a function of Zr concentration in Zr x Ni100Àx alloys.Fig.1.Temperature dependence of the potential energy of Zr50Ni50alloy duringcooling process at the cooling rate of2*1012K/s.C.Y.Yu et al./Intermetallics53(2014)177e182179Thecalculated results from the proposed expression g FP¼g *=p S ea ¼ðD H amor =ðD H inter ÀD H amor ÞÞ=p S ea as a function of Zr concentration are shown in Fig.5.One distinct peak of g FP at the composition with 33at.%Zr,as well as a small peak at 62at.%Zr are observed.It implies these two compositions,especially the one with 33at.%Zr,are both thermodynamically and kinetically favorable for glass formation.Since there are no experimental data available for a direct comparison of the crit-ical thickness for glassy ribbons in this system,we have thus calculated the g parameter (T x /(T g þT l ))based on the available experimental data [8,44](where T x is the onset crystallization temperature,T g is the glass transition temperature and T l is the liquidus temperature),and the results so obtained are also shown in Fig.5.The value of g is the highest at the composition of 33at.%Zr among the compositions,which implies the highest GFA of this composition as compared with others in the Zr e Ni alloy system.Meanwhile,the compositions with 62and 64at.%Zr also show a relatively higher value of g than their adjacent compositions in Refs.[8]and [44],respectively,re flecting therelatively higher GFA around the composition with 62at.%Zr.The good agreement between the FP calculations and the g pa-rameters determined from the experimental data just demon-strates the effectiveness of our FP predictions in the alloy system even with a relatively poor GFA.As mentioned above,Ni and Cu are neighboring elements in the Periodical Table,and they exhibit some similar properties;thus it is reasonable to assume that the Zr e Ni and Zr e Cu binary alloy sys-tems perform analogously in the formation of metallic glasses.However,it is well known that BMGs with a diameter of amorphous rods !1mm can be readily formed experimentally in the Zr e Cu binary alloy system,but none in the Zr e Ni system [1e 7].Never-theless,it is noted that the absolute values of g FP in the Zr e Ni system are relatively larger than those in the Zr e Cu system,sug-gesting the possibility of higher GFA of the Zr e Ni system than that of the Zr e Cu system.Therefore,the g FP parameter developed herein has its limitation that it is incapable of making a direct comparison of the GFAs obtained from different alloy systems.This is understandable because the glass formation is a very complex process involving the thermodynamic driving force,individual atom mobility,and various crystalline-phase formations upon cooling from the molten state.Thus,there are no simple ways to take care of all these parameters effectively during theoretical calculations.The atomistic approach based on the first-principles calculations we developed in this study is effective in pinpointing the promising alloy compositions in a given alloy system.We can understand the GFA difference between these two alloy systems by investigating two characteristic compositions:Zr 50Cu 50and Zr 50Ni 50.Zr 50Cu 50alloy is the best glass former in the Zr e Cu system which can be made into fully amorphous rods with a diameter of !2mm experimentally.Nevertheless,the analogous equiatomic alloy Zr 50Ni 50has the almost lowest GFA in the Zr e Ni alloy system from our prediction as shown in Fig.5,where a distinctly low value of g FP is seen at this composition.This is also consistent with the experimental g results,where the g (¼T x /(T g þT l ))value for the amorphous ribbons of Zr 50Ni 50and its neighbor alloy Zr 48Ni 52both have the lowest one among the compositions studied in Refs.[44]and [8],implying their lower GFAs as compared with other alloys in this alloy system.In order to rationalize this difference,we have made a com-parison of the physical properties and atomic structures of these two alloys.First,the melting point of the intermetallic compound NiZr is much higher than that of its adjacent alloys,while the CuZr phase shows a melting point only a few degrees higher thanitsFig.3.Dependence of g *on Zr concentration in Zr x Ni 100Àxalloys.Fig.4.Dependence of the proportion of excessive atomic spaces on Zr concentration in Zr x Ni 100Àx glass-forming supercooledliquids.Fig.5.Dependence of the parameter g FP ()and g (,,taken from Refs.[8,44],respectively)on Zr concentration in Zr x Ni 100Àx alloys.C.Y.Yu et al./Intermetallics 53(2014)177e 182180adjacent ones.This suggests that the phase stability of the NiZr intermetallic phase is much higher than that of the CuZr compound in these two alloy systems,re flecting the dif ficulty of glass forma-tion in the Zr 50Ni 50alloy.Moreover,the different glass formation behavior in the two alloys can also be understood from the different types of polyhedrons formed around Cu and Ni atoms from the Voronoi tessellation analyses.In particular,we focus on the fraction of Cu-and Ni-centered full icosahedrons,the building block in metallic glasses.The results show that there are 7.1%and 5.3%Cu-and Ni-centered full icosahedrons in the Zr 50Cu 50and the Zr 50Ni 50metallic glasses,respectively,suggesting a higher atomic packing density in the Zr 50Cu 50glass.Furthermore,this comparison is consistent with the values of p S ea in the two supercooled liquids,with 0.8%for the Zr 50Cu 50and 1.7%for the Zr 50Ni 50.The lower p S ea implies a denser atomic packing and a higher GFA.Based on all these considerations,it is understandable that the Zr 50Cu 50alloy has a signi ficantly higher GFA than that of the Zr 50Ni 50alloy.Furthermore,the different glass forming behaviors in these two alloy systems can be directly illustrated by the relationship be-tween the phase transition and the cooling rate.With the current cooling rate of 2*1012K/s used in the calculation,the glass forma-tion behavior can be noticed in each alloy of the Zr e Ni system as the example shown in Fig.1.However,by the additional calcula-tions performed at a cooling rate 2*1010K/s in both Zr e Ni and Zr e Cu systems with the same compositions,the crystallization process is detected in some Zr e Ni alloys but not in any Zr e Cu alloys.As an example to interpret this difference,the potential energy e tem-perature curves at the cooling rate 2*1010K/s for Zr 10Ni 90and Zr 10Cu 90alloys are shown in Fig.6(a)and (b)for comparison.It is clearly to see an abrupt energy step in the curve due to crystalli-zation in Zr 10Ni 90while only a glass transition revealed as a change of the slope of the cooling curve is detected in Zr 10Cu 90.This comparison of these two curves obtained at the same speci fic cooling rate indicates the better GFA of the Zr e Cu alloy system as compared with that of the Zr e Ni alloy system.4.ConclusionsIn summary,the proposed expression g FP ¼ðD H amor =ðD H inter ÀD H amor ÞÞ=p S ea is successfully applied to predict the GFA in the Zr e Ni binary system with relatively poor GFA.This atomistic approach was developed from the consideration of both the thermodynamic effect based on internal energies obtained from the FP calculations,and the kinetic effect from intrinsic structural defects at atomic level from MD simulations.The predicted good glass former from g FP is consistent with that from the calculation of the GFA based onthe g (T x /(T g þT l ))parameters calculated from experimental data reported previously.This result demonstrates that our proposed atomistic approach can be successfully extended from the Zr e Cu alloy system with a good GFA to the Zr e Ni alloy system with a relatively poor GFA.Furthermore,the reasons for the different glass forming behaviors between the two binary alloy systems are dis-cussed in terms of both atomic con figurations and relative stability of their intermetallic phases.AcknowledgmentsThis work was mainly financially supported by the Hong Kong RGC grant No.522110.X.J.L.was supported in part by the National Natural Science Foundation of China (Nos.50901006,51271212),the Beijing Nova Program of China (No.2010B017),the Program of Introducing Talents of Discipline to Universities (B07003),and the Fundamental Research Funds for the Central Universities (FRF-TP-11-005A).References[1]Wang D,Li Y,Sun BB,Sui ML,Lu K,Ma E.Appl Phys Lett 2004;84:4029e 31.[2]Xu DH,Lohwongwatana B,Duan G,Johnson WL,Garland C.Acta Mater2004;52:2621e 4.[3]Inoue A,Zhang W.Mater Trans 2004;45:584e 7.[4]Li Y,Guo Q,Kalb JA,Thompson CV.Science 2008;322:1816e 9.[5]Bendert JC,Gangopadhyay AK,Mauro NA,Kelton KF.Phys Rev Lett 2012;109:185901.[6]Tang MB,Zhao DQ,Pan MX,Wang WH.Chin Phys Lett 2004;21:901e 3.[7]Duan G,Xu DH,Zhang Q,Zhang GY,Cagin T,Johnson WL,et al.Phys Rev B2005;71:224208.[8]Buschow KHJ.J Phys F Met Phys 1984;14:593e 607.[9]Fukunaga T,Itoh K,Otomo T,Mori K,Sugiyama M,Kato H,et al.Intermetallics2006;14:893.[10]Risti c R,Stubi c ar M,Babi c E.Philos Mag 2007;87:5629.[11]Kaban I,Jóvári P,Kokotin V,Shuleshova O,Beuneu B,Saksl K,et al.Acta Mater2013;61:2509.[12]Turnbull D.Contemp Phys 1969;10:473.[13]Inoue A,Zhang T,Masumoto T.Mater Trans JIM 1989;30:965e 72.[14]Lu ZP,Liu CT.Acta Mater 2002;50:3501.[15]Du XH,Huang JC,Liu CT,Lu ZP.J Appl Phys 2007;101:086108.[16]Guo S,Liu CT.Intermetallics 2010;18:2065.[17]Busch R,Schroers J,Wang WH.MRS Bull 2007;32:620e 3.[18]Gorsse S,Orveillon G,Senkov ON,Miracle DB.Phys Rev B 2006;73:224202.[19]Cui YY,Wang TL,Li JH,Dai Y,Liu BX.Phys Chem Chem Phys 2011;13:4103e 8.[20]Xia L,Fang SS,Wang Q,Dong YD,Liu CT.Appl Phys Lett 2006;88:171905.[21]Mukherjee S,Schroers J,Johnson WL,Rhim WK.Phys Rev Lett 2005;94:245501.[22]Peng HL,Li MZ,Wang WH,Wang CZ,Ho KM.Appl Phys Let 2010;96:021901.[23]Yu CY,Liu XJ,Lu J,Zheng GP,Liu CT.Sci Rep 2013;3:2124.[24]Fan C,Ren Y,Liu CT,Liaw PK,Yan HG,Egami T.Phys Rev B 2011;83:195207.[25]Lindemann F.Z Phys 1910;11:609.[26]Finney JL.Proc R Soc Lond A 1970;319:479.Fig.6.Temperature dependence of the potential energy of (a)Zr 10Ni 90alloy and (b)Zr 10Cu 90alloy during cooling process at the cooling rate of 2*1010K/s .C.Y.Yu et al./Intermetallics 53(2014)177e 182181[27]Mendelev MI,Kramer MJ,Hao SG,Ho KM,Wang CZ.Philos Mag2012;92:4454e69.[28]Plimpton S.J Comput Phys1995;117:1..[29]Mendelev MI,Kramer MJ,Ott RT,Sordelet DJ,Yagodin D,Popel P.Philos Mag2009;89:967e87.[30]Kresse G,Furthmüller J.Phys Rev B1996;54:11169e86.[31]Blöchl PE.Phys Rev B1994;50:17953e79.[32]Kresse G,Joubert D.Phys Rev B1999;59:1758e75.[33]Perdew JP,Burke K,Ernzerhof M.Phys Rev Lett1996;77:3865e8.[34]Monkhorst HJ,Pack JD.Phys Rev B1976;13:5188.[35]Basu J,Murty BS,Ranganathan S.J Alloys Compd2008;465:163.[36]Jette ER,Foote F.J Chem Phys1935;3:605e16.[37]Russel RB.Trans Metall Soc AIME1954;200:1045e52.[38]Eshelman FR,Smith JF.Acta Crystallogr Sect B1972;28:1594e600.[39]Joubert JM,Cerny R,Yvon K,Latroche M,Percheron-Guegan A.Z Kristallog1998;213:227e8.[40]Kirkpatrick ME,Smith JF,Larsen WL.Acta Crystallogr1972;15:894e903.[41]Bououdina M,Lambert-Andron B,Ouladdiaf B,Pairis S,Fruchart D.J AlloysCompd2003;356:54e8.[42]Havinga EE,Damsma H,Hokkeling P.J Less Common Met1972;27:169e86.[43]Henaff MP,Colinet C,Pasturel A,Buschow KHJ.J Appl Phys1984;56:307.[44]Liu XJ,Hui XD,Chen GL,Liu T.Phys Lett A2009;373:2488e93.C.Y.Yu et al./Intermetallics53(2014)177e182 182。
a rX iv:c ond-ma t/974231v1[c ond-m at.str-el]28A pr1997First-principles calculations of the electronic structure and spectra of strongly correlated systems:dynamical mean-field theory V.I.Anisimov,A.I.Poteryaev,M.A.Korotin,A.O.Anokhin Institute of Metal Physics,Ekaterinburg,GSP-170,Russia G.Kotliar Serin Physics Laboratory,Rutgers University,Piscataway,New Jersey 08854,USA Abstract A recently developed dynamical mean-field theory in the iterated per-turbation theory approximation was used as a basis for construction of the ”first principles”calculation scheme for investigating electronic struc-ture of strongly correlated electron systems.This scheme is based on Local Density Approximation (LDA)in the framework of the Linearized Muffin-Tin-Orbitals (LMTO)method.The classical example of the doped Mott-insulator La 1−x Sr x TiO 3was studied by the new method and the results showed qualitative improvement in agreement with experimental photoemission spectra.1Introduction The accurate calculation of the electronic structure of materials starting from first principles is a challenging problem in condensed matter science since un-fortunately,except for small molecules,it is impossible to solve many-electron problem without severe approximations.For materials where the kinetic energy of the electrons is more important than the Coulomb interactions,the most successful first principles method is the Density Functional theory (DFT)within the Local (Spin-)Density Ap-proximation (L(S)DA)[1],where the many-body problem is mapped into a non-interacting system with a one-electron exchange-correlation potential approxi-mated by that of the homogeneous electron gas.It is by now,generally accepted that the spin density functional theory in the local approximation is a reliable starting point for first principle calculations1of material properties of weakly correlated solids(For a review see[2]).The situation is very different when we consider more strongly correlated materials, (systems containing f and d electrons).In a very simplified view LDA can be regarded as a Hartree-Fock approximation with orbital-independent(averaged) one-electron potential.This approximation is very crude for strongly correlated systems,where the on-cite Coulomb interaction between d-(or f-)electrons of transition metal(or rare-earth metal)ions(Coulomb parameter U)is strong enough to overcome kinetic energy which is of the order of band width W.In the result LDA gives qualitatively wrong answer even for such simple systems as Mott insulators with integer number of electrons per cite(so-called”undoped Mott insulators”).For example insulators CoO and La2CuO4are predicted to be metallic by LDA.The search for a”first principle”computational scheme of physical proper-ties of strongly correlated electron systems which is as successful as the LDA in weakly correlated systems,is highly desirable given the considerable impor-tance of this class of materials and is a subject of intensive current research. Notable examples offirst principle schemes that have been applied to srongly correlated electron systems are the LDA+U method[3]which is akin to orbital-spin-unrestricted Hartree-Fock method using a basis of LDA wave functions,ab initio unrestricted Hartree Fock calculations[4]and the use of constrained LDA to derive model parameters of model hamiltonians which are then treated by exact diagonalization of small clusters or other approximations[5].Many interesting effects,such as orbital and charge ordering in transition metal compounds were successfully described by LDA+U method[6].However for strongly correlated metals Hartree-Fock approximation is too crude and more sophisticated approaches are needed.Recently the dynamical mean-field theory was developed[7]which is based on the mapping of lattice models onto quantum impurity models subject to a self-consistency condition.The resulting impurity model can be solved by var-ious approaches(Quantum Monte Carlo,exact diagonalization)but the most promising for the possible use in”realistic”calculation scheme is Iterated Per-turbation Theory(IPT)approximation,which was proved to give results in a good agreement with more rigorous methods.This paper is thefirst in a series where we plan to integrate recent devel-ompements of the dynamical meanfield approach with state of the art band structure calculation techniques to generate an”ab initio”scheme for the cal-culation of the electronic structure of correlated solids.For a review of the historical development of the dynamical meanfield approach in its various im-plementations see ref[7].In this paper we implement the dynamical mean-field theory in the iterated perturbation theory approximation,and carry out the band structure calculations using a LMTO basis.The calculational scheme is described in section2.We present results obtained applying this method to La1−x Sr x TiO3which is a classical example of strongly correlated metal.22The calculation schemeIn order to be able to implement the achievements of Hubbard model theory to LDA one needs the method which could be mapped on tight-binding model.The Linearized Muffin-Tin Orbitals(LMTO)method in orthogonal approximation[8]can be naturally presented as tight-binding calculation scheme (in real space representation):H LMT O= ilm,jl′m′,σ(δilm,jl′m′ǫil n ilmσ+t ilm,jl′m′ c†ilmσ c jl′m′σ)(1)(i-site index,lm-orbital indexes).As we have mentioned above,LDA one-electron potential is orbital-inde-pendent and Coulomb interaction between d-electrons is taken into account in this potential in an averaged way.In order to generalize this Hamiltonian by including Coulomb correlations,one must add interaction term:1H int=Un d(n d−1)(3)2(n d= mσn mσtotal number of d-electrons).3In LDA-Hamiltonianǫd has a meaning of the LDA-one-electron eigenvalue for d-orbitals.It is known that in LDA eigenvalue is the derivative of the total energy over the occupancy of the orbital:ǫd=ddn d (E LDA−E Coul)=ǫd−U(n d−12)(7)(q is an index of the atom in the elementary unit cell).In the dynamical mean-field theory the effect of Coulomb correlation is de-scribed by self-energy operator in local approximation.The Green function is:G qlm,q′l′m′(iω)=1The chemical potential of the effective medium µis varied to satisfy Luttinger theorem condition:1d(iωn)Σ(iωn)=0(11)In iterated perturbation theory approximation the anzatz for the self-energy is based on the second order perturbation theory term calculated with”bath”Green function G0:Σ0(iωs)=−(N−1)U21kT,Matsubara frequenciesωs=(2s+1)πβ;s,n integer numbers.The termΣ0is renormalized to insure correct atomic limit:Σ(iω)=Un(N−1)+AΣ0(iω)β iωn e iωn0+G(iωn)),B=U[1−(N−1)n]−µ+ µn0(1−n0)(15)n0=1iω+µ−∆(iω)+δµ+n(N−1)β iωn e iωn0+G CP A(iωn)(18) D[n]=n iωn e iωn0+1energy to time variables and back:G0(τ)=1V Bd k[z−H(k,z)]−1(24)After diagonalization,H(k,z)matrix can be expressed through diagonal matrix of its eigenvalues D(k,z)and eigenvectors matrix U(k,z):H(k,z)=U(k,z)D(k,z)U−1(k,z)(25) and Green function:G(z)=1V Bd k U in(k,z)U−1nj(k,z)V Bvd kU in(k,z)U−1nj(k,z)V B(28)6v is tetrahedron volumer n i=(z−D n(k i,z))2k(=j)(D n(k k,z)−D n(k j,z))ln[(z−D n(k j,z))/(z−D n(k i,z)]1+a2(z−z2)1(30)where the coefficients a i are to be determined so that:C M(z i)=u i,i=1,...,M(31) The coefficients a i are then given by the recursion:a i=g i(z i),g1(z i)=u i,i=1,...,M(32)g p(z)=g p−1(z p−1)−g p−1(z)3ResultsWe have applied the above described calculation scheme to the doped Mott insulator La1−x Sr x TiO3is a Pauli-paramagnetic metal at room tem-perature and below T N=125K antiferromagnetic insulator with a very small gap value(0.2eV).Doping by a very small value of Sr(few percent)leads to the transition to paramagnetic metal with a large effective mass.As photoemission spectra of this system also show strong deviation from the noninteracting elec-trons picture,La1−x Sr x TiO3is regarded as an example of strongly correlated metal.The crystal structure of LaTiO3is slightly distorted cubic perovskite.The Ti ions have octahedral coordination of oxygen ions and t2g-e g crystalfield splitting of d-shell is strong enough to survive in solid.On Fig.1the total and partial DOS of paramagnetic LaTiO3are presented as obtained in LDA calculations (LMTO method).On3eV above O2p-band there is Ti-3d-band splitted on t2g and e g subbands which are well separated from each other.Ti4+-ions have d1 configuration and t2g band is1/6filled.As only t2g band is partiallyfilled and e g band is completely empty,it is reasonable to consider Coulomb correlations between t2g−electrons only and degeneracy factor N in Eq.(12)is equal6.The value of Coulomb parameter U was calculated by the supercell procedure[9]regarding only t2g−electrons as localized ones and allowing e g−electrons participate in the screening.This cal-culation resulted in a value3eV.As the localization must lead to the energy gap between electrons with the same spin,the effective Coulomb interaction will be reduced by the value of exchange parameter J=1eV.So we have used effective Coulomb parameter U eff=2eV.The results of the calculation for x=0.06(dop-ing by Sr was immitated by the decreasing on x the total number of electrons) are presented in the form of the t2g-DOS on Fig.2.Its general form is the same as for model calculations:strong quasiparticle peak on the Fermi energy and incoherent subbands below and above it corresponding to the lower and upper Hubbard bands.The appearance of the incoherent lower Hubbard band in our DOS leads to qualitatively better agreement with photoemission spectra.On Fig.3the exper-imental XPS for La1−x Sr x TiO3(x=0.06)[12]is presented with non-interacting (LDA)and interacting(IPT)occupied DOS broadened to imitate experimental resolution.The main correlation effect:simultaneous presence of coherent and incoherent band in XPS is successfully reproduced in IPT calculation.However, as one can see,IPT overestimates the strength of the coherent subband.4ConclusionsIn this publication we described how one can interface methods for realistic band structure calculations with the recently developed dynamical meanfield8technique to obtain a fully”ab initio”method for calculating the electronic spectra of solids.With respect to earlier calculations,this work introduces several method-ological advances:the dynamical meanfield equations are incorporated into a realistic electronic structure calculation scheme,with parameters obtained from afirst principle calculation and with the realistic orbital degeneracy of the compound.To check our method we applied to doped titanates for which a large body of model calculation studies using dynamical meanfield theory exists.There results are very encouraging considering the experimental uncertainties of the analysis of the photoemission spectra of these compounds.We have used two relative accurate(but still approximate)methods for the solution of the band structure aspect and the correlation aspects of this problem:the LMTO in the ASA approximation and the IPT approximation. In principle,one can use other techniques for handling these two aspects of the problem and further application to more complicated materials are necessary to determine the degree of quantitative accuracy of the method.9References[1]Hohenberg P.and Kohn W.,Phys.Rev.B136,864(1964);Kohn W.andSham L.J.,ibid.140,A1133(1965)[2]R.O.Jones,O.Gunnarsson,Reviews of Modern Physics,v61,689(1989)[3]Anisimov V.I.,Zaanen J.and Andersen O.K.,Phys.Rev.B44,943(1991)[4]S.Massida,M.Posternak, A.Baldareschi,Phys.Rev.B46,11705(1992);M.D.Towler,N.L.Allan,N.M.Harrison,V.R.Sunders,W.C.Mackrodt,E.Apra,Phys.Rev.B50,5041(1994);[5]M.S.Hybertsen,M.Schlueter,N.Christensen,Phys.Rev.B39,9028(1989);[6]Anisimov V.I.,Aryasetiawan F.and Lichtenstein A.I.,J.Phys.:Condens.Matter9,767(1997)[7]Georges A.,Kotliar G.,Krauth W.and Rozenberg M.J.,Reviews of ModernPhysics,v68,n.1,13(1996)[8]O.K.Andersen,Phys.Rev.B12,3060(1975);Gunnarsson O.,Jepsen O.andAndersen O.K.,Phys.Rev.B27,7144(1983)[9]Anisimov V.I.and Gunnarsson O.,Phys.Rev.B43,7570(1991)[10]Lambin Ph.and Vigneron J.P.,Phys.Rev.B29,3430(1984)[11]Vidberg H.J.and Serene J.W.,Journal of Low Temperature Physics,v29,179(1977)[12]A.Fujimori,I.Hase,H.Namatame,Y.Fujishima,Y.Tokura,H.Eisaki,S.Uchida,K.Takegahara,F.M.F de Groot,Phys.Rev.Lett.69,1796(1992).(Actually in this article the chemical formula of the sample was LaTiO3.03, but the excess of oxygen produce6%holes which is equivalent to doping of 6%Sr).105Figure captionsFig.1.Noninteracting(U=0)total and partial density of states(DOS)for LaTiO3.Fig.2.Partial(t2g)DOS obtained in IPT calculations in comparison with noninteracting DOS.Fig.3.Experimental and theoretical photoemission spectra of La1−x Sr x TiO3 (x=0.06).11)LJ 7L G H J '26 V W D W H H 9 D W R P (QHUJ\ H97L G W J7RWDO /D7L2 '26 V W D W H H 9 F H O O3HUWXUEDWHG)LJ'26 V W D W H V H 9 (QHUJ\ H98QSHUWXUEDWHG,Q W H Q V L W \ H 9(QHUJ\ H9。
First-principles calculations of structural,electronic,andoptical absorption properties of CaCO 3VateriteS.K.Medeiros a ,E.L.Albuquerque a ,F.F.Maia Jr.b ,E.W.S.Caetano b ,V.N.Freireb,*a Departamento de Fı´sica Teo ´rica e Experimental,Universidade Federal do Rio Grande do Norte,59072-970Natal,Rio Grande do Norte,Brazil bDepartamento de Fı´sica,Universidade Federal do Ceara ´,Centro de Cie ˆncias,Caixa Postal 6030,Campus do Pici,60455-900Fortaleza,Ceara ´,Brazil Received 29July 2006;in final form 13December 2006Available online 20December 2006AbstractFirst-principles calculations are performed within the density functional theory (DFT)for the CaCO 3vaterite polymorph considering exchange-correlation potentials in the local density approximation (LDA)and generalized gradient approximation (GGA).The crystal structure of vaterite is considered to be orthorhombic,although there is considerable debate in the literature about a possible hexagonal cell with disordered carbon and oxygen atoms.After convergence,the calculated orthorhombic lattice parameters are demonstrated to be in good agreement with experimental results.The direct energy gap for vaterite is preview to be wide and estimated in LDA and GGA tobe E LDA G ðC !C Þ¼4:68eV and E GGAG ðC !C Þ¼5:07eV,but should underestimate the actual value yet to be measured.The optical absorption inten-sity is previewed to be relevant for energies larger than 6.2eV.Ó2007Elsevier B.V.All rights reserved.1.IntroductionCalcium carbonate (CaCO 3)has three polymorphs:cal-cite,aragonite,and vaterite.Calcite is thermodynamically the most stable form at room temperature and pressure,with promising industrial (plastic,rubbers,papers,paints etc.),implants,and bio-optoelectronics applications.Ara-gonite is metastable,transforming to calcite at 300–400°C in biogenic samples,and at 660–751°C as fast transformation in natural (abiogenic)crystals.Together with a low amount (1–5%by weight)of a organic matrix formed by proteins,glycoproteins and chitin,aragonite constitutes the nacreous inner portion of mollusk shells,having an important role in geochemistry and biomineral-ization.Vaterite is the least stable CaCO 3form,and conse-quently was not studied thoroughly.It was argued that vaterite is rare and undergoes recrystallization owing to its instability [1].However,it was demonstrated by Malkaj and Dalas [2]that the presence of aspartic acid in supersat-urated CaCO 3solutions stabilizes the growth of vateritepolymorph crystals.Vaterite thin films were also prepared by Xu et al.[3]and Ajikumar et al.[4]by different methods.Theoretical calculations on the structural and electronic properties of the CaCO 3calcite were performed by Skinner et al.[5]and Medeiros (a)et al.[6],and of the CaCO 3ara-gonite by Medeiros (b)et al.[7].No first-principles calcula-tions were undertaken for the CaCO 3vaterite polymorph yet.Calcite and aragonite occurs in a rhombohedral and orthorhombic structure,respectively (see Refs.in [5–7]).However,in the case of vaterite there are two crystalline structure proposed in the literature.Meyer [8]was the first to determine the vaterite crystalline structure by X-ray,reporting an orthorhombic cell with the dimensionsa =4.13A˚,b =7.15A ˚,c =8.48A ˚,space group Pnma,and the following structure:Ca in (4a),with x =0.0,y =0.0,z =0.0;C in (4c)with x =0.67,y =1/4,z =0.157;O(1)in (4c)with x =0.67,y =1/4,z =0.471;O(2)in (8d)with x =0.67,y =0.118,z =0.0.In 1960,Meyer [9]proposed an orthorhombic cell for vaterite,but with space group Pbnm due to the considerable disorder of the carbonate molecules,which leads to partial occupan-cies of different carbonate sites.However,Kamhi [10]0009-2614/$-see front matter Ó2007Elsevier B.V.All rights reserved.doi:10.1016/j.cplett.2006.12.051*Corresponding author.Fax:+558533669450.E-mail address:valder@fisica.ufc.br (V.N.Freire)./locate/cplettChemical Physics Letters 435(2007)59–64found a hexagonal cell with dimensions a=4.13A˚, c=8.49A˚,space group P63/mmc,and the following struc-ture:Ca in(2a)with x=0.0,y=0.0,z=0.0;C in(6h) with x=0.29,y=2x,z=1/4–1/3;O(1)in(6h)with x=0.12,y=2x,z=1/4–1/3;O(2)in(12k)with x=0.38,y=2x,z=0.12–1/3.In this case,two carbon atoms are randomly distributed over six positions and six oxygen atoms over eighteen positions.Meyer[11]and Lippmann[12]also proposed a hexagonal unit cell for vate-rite,with carbonate groups disposed along planes parallel to the c-axis(in calcite and aragonite,on the other hand, the carbonate planes are perpendicular to the c-axis),but with distinct space group and carbonate site symmetries.Meyer’s1969structure[11]is fairly consistent with Kamhi’s1963proposal[10]in its unit-cell dimensions, but not in its atomic positions,especially in the site symme-try of the carbonate ion.The average Meyer’s1969struc-ture[11]agrees in general with Kamhi’s1963structure [10],but the CO3group has to be centered around(j)of the space group P63/mmc,with an occupancy of one sixth only.Details of this hexagonal structure(a=7.148A˚, c=16.949A˚)are:Ca with x=0.0,y=0.0,z=0.0;C with x=0.3134,y=0.3806,z=1/8;O(1)with x=0.2911, y=0.5255,z=1/8;O(2)with x=0.3245,y=3081, z=0.0548.Note,however,that the oxygen atoms form distorted cubes around the calcium atoms,and for a more exact treatment one has to consider the deformations of the primitive hexagonal partial lattice array of calcium atoms also.It is evident that such structure(specially the partial occupancy)is difficult to simulate,and may justify the use of an orthorhombic model for the vaterite structure. Further,there is considerable difference in the arrangementof CO2À3ions in the lattices since not conclusive experi-ments suggest that in vaterite they are aligned perpendicu-lar to the ab plane,while in calcite and aragonite they are aligned parallel[9,10,13].Together with the different typesof the lattice structures,the arrangement of CO2À3ions cangive rise to singular electronic characteristics for each cal-cium carbonate polymorph.De Leeuw[14]reported results of atomistic simulations to investigate the effect of molecular adsorption of water on the low-index surfaces of calcite,aragonite,and vaterite. These simulations were performed using the orthorhombic cell(space group Pbnm)proposed by Meyer[9]in1960due to the considerable disorder of the carbonate molecules in vaterite which leads to partial occupancies of different car-bonate sites.However,Gabrielli et al.[15]determined the Raman spectrum of vaterite prepared by electrochemical scaling in carbonically pure water.Further analysis using group theory revealed that only one of the previous crystal-lographic determinations outlined above,the one proposed by Meyer[11],with carbonate anions in a C s site of the D46h (P63/mmc)group,is consistent with the Raman measure-ments.Recently,high resolution synchrotron radiation dif-fraction(HRSRD)was employed to analyze calcium carbonate samples grown in the presence of mono-L-glu-tamic(GLU)and mono-L-aspartic(ASP)acid[16].The main phases of CaCO3produced in this way were calcite and vaterite,with some structural variability in both phases leading to varying unit cell parameters.The HRSRD mea-surements[16]showed that the structure of vaterite is dif-ferent from hexagonal structures reported in the literature,but very close to the orthorhombic structure ini-tially proposed by Meyer[9],with a remarkable presence of stacking faults.In this work we present structural,electronic,and opti-cal absorption properties for the calcium carbonate vaterite polymorph,by thefirst time to the knowledge of the authors.The quantum chemicalfirst-principle calculations were performed within the density functional theory(DFT) framework,with the exchange-correlation potential consid-ered in both the local density and generalized gradient approximations,LDA and GGA,respectively,the former for the sake of comparison.The LDA exchange-correlation energy is obtained by adding the Perdew–Zunger[17] exchange term to the Ceperley–Alder[18]correlation energy;in the GGA approach,the Perdew–Burke–Ernzer-hof[19]exchange and correlation terms were taken into account.Despite the controversial aspects outlined before in the introduction on the actual structure of CaCO3vate-rite,we consider here an orthorhombic model for its struc-ture,without considering the random distribution of carbon and oxygen atoms over all the potential sites.2.First-principles structural resultsThe computer calculations were performed with the CASTEP code,[20]using a plane-wave basis and ultrasoftVanderbilt pseudopotentials[21].For the pseudopoten-tials,the valence electron configurations were Ca-3s23p64s2,C-2s22p2,and O-2s22p4.The Monkhorst-Pack scheme for integration in the Brillouin zone with a 6·4·3grid was adopted[22,23].We have used the ortho-rhombic cell as given by Meyer[9]for the initialization, without considering the random distribution of carbon and oxygen atoms over all the potential sites.Geometry optimization was carried out using the Broyden–Fletcher–Goldfarb–Shannon algorithm[24]enabling to obtain the unit cell parameters(lattice parameters,unit cell angles,internal atomic coordinates).Table1presents the starting and thefinal(both LDA and GGA)optimized internal coordinates for the orthorhombic CaCO3vaterite. We adopted a500eV energy cutofffor the plane-wave basis set,and a tolerance of5·10À7eV/atom for the self-consis-tentfield calculations.In order tofind the lowest energy structure of vaterite,we observed the following conver-gence thresholds for geometry optimization:total energy variation smaller than5.0·10À6eV/atom in three succes-sive self-consistent steps,0.01eV/A˚for maximum force, 0.02GPa for maximum pressure,and5·10À4A˚for max-imum displacement.After the optimization,the electronic and optical properties were calculated.The density of states(DOS)was computed by means of a scheme devel-oped by Ackland[25],while the partial density of states60S.K.Medeiros et al./Chemical Physics Letters435(2007)59–64(PDOS)calculation was based on a Mulliken population analysis with the relative contribution of each atom [26,27].To obtain the carriers effective masses,parabolicfittings to the E ðk !Þcurves along the main high-symmetry directions were performed,and the effective mass m for each band was found by using the relation E = h 2k 2/2m .After reaching the geometry convergence,the final average pressure obtained was 0.0043GPa,and the symmetrized stress tensor components (also in GPa)were xx =0.009661, yy =À0.002981, zz =À0.019662(hydrostatic stress).The primitive vaterite orthorhombic unit cell contains 20atoms (four CaCO 3molecules)[10,13],space group Pbmn [9,28],consisting of alternating (111)planes of Ca atoms and carbonate groups.Experimental results [9,10]indicate that the plane defined by the carbonate ions are perpendicular to the ab plane (differently from the parallel alignment in calcite),as shown at the left of Fig.1.How-ever,in the optimized unit cell obtained from the calcula-tions,the plane defined by carbonate ions is inclined by67°with respect to the ac plane,as presented in the right of Fig.1.The same result was obtained when we replaced the pseudopotentials by norm-conserved pseudopotentials [29,30].Varying the basis set energy cutofffrom 380eV to 500eV (ultrasoft pseudopotentials)and from 600eV to 660eV (norm-conserved pseudopotentials)did not change this picture.As we have seen above,the crystalline structure of vate-rite is debated [9,10,13].This was clearly highlighted by Behrens et al.[13]in his work about the Raman-spectra of vateritic calcium carbonate.Consequently,our result indicates that improved X-ray diffraction data for highly pure vaterite is still necessary to allow for a definitive con-clusion whether the plane defined by carbonate ions is par-allel or inclined with respect to the ac plane.The structural parameters of vaterite we obtained are presented in Table 1,together with the experimental values of Meyer [9].A good agreement with the experimental data is obtained in general,with LDA and GGA approaches overestimating the lattice parameter a ,probably due to the inclination of the plane defined by carbonate ions in respect to the ac plane.The lattice parameters b and c are underestimated by LDA as usual in this approximation.The GGA results,which are often expected to be larger than the experimental ones,are smaller in our calculations.Table 2shows that the LDA and GGA calculated lattice parameters of the vate-rite unit cell after geometry optimization are in good agree-ment with experimental data.3.Electronic and optical propertiesThe band structure and partial density of states for CaCO 3vaterite is depicted in Fig.2,covering a wide range of energy levels,from À4.0eV to 8.0eV.Levels below 0eV correspond to valence electronic states,while the levels above are associated to conduction states.The valence bands have mainly O-2p state contributions.Above 6.0eV,the conduction bands are mainly due to Ca-3d and C-2p orbitals contributions.Both LDA and GGA sug-gest that vaterite is a wide direct gap insulator,with the conduction band minimum and valence band maximum occurring at the C point,with E LDA G ðC !C Þ¼4:68eV and E GGAG ðC !C Þ¼5:07eV.However,both the LDA and GGA val-ues we have calculated must be recognized as estimates,being about 20–30%smaller (we guess)than the actual vaterite energy gap,to be measured yet.We hope that our work will stimulate new experiments in this direction.Table 1The starting [9]and the final (both LDA and GGA)optimized internal coordinates for the orthorhombic CaCO 3vaterite x /a y /a z /a Occupancy LDA 0.0000.0000.000Ca 0.0280.6310.250C 0.2330.5320.250O(1)À0.0980.6800.117O(2)GGA 0.0000.0000.000Ca 0.0220.6360.250C 0.2730.5450.250O(1)À0.1020.6820.117O(2)Experimental 0.0000.0000.000Ca 0.1570.6700.250C 0.4710.6700.250O(1)0.0000.6700.118O(2)Fig. 1.The primitive vaterite orthorhombic unit cell showing the perpendicular arrangement of the carbonate ions to the ab plane according experimental data (left),and the primitive vaterite orthorhombic unit cell after geometry optimization by DFT-GGA (right).Table 2Lattice parameters of the vaterite unit cell after geometry optimization within the DFT-LDA and DFT-GGA approachesa (A ˚)b (A ˚)c (A ˚)LDA4.341 6.4328.424GGA4.531 6.6408.477Experimental4.1307.1508.480The experimental data is from Meyer [9].S.K.Medeiros et al./Chemical Physics Letters 435(2007)59–6461A better estimate of the energy gap can be obtained through an improved description of electronic correlation effects beyond GGA,which is however very numerically demanding and beyond our present computational resources.For the sake of comparison with other CaCO3crystal-line structures,we mention here a theoretical study on the structure and bonding of calcite performed by Skinner et al.[5].They have obtained an indirect energy gap E G(D!Z)=4.4±0.35eV for calcite,which is well below the experimental value6.0±0.35eV as measured by reflec-tion electron energy-loss spectroscopy[31].Recently,the present authors performed ab initio calculations within the density functional theory(DFT)for the CaCO3calcite polymorph[6],taking into account the exchange correla-tion potential in the generalized gradient approximation(GGA),obtaining an indirect energy gap E GGAGðF!ZÞ¼5:059eV in considerably better agreement with the experi-mental value than the obtained by Skinner et al.[5].We have also performed LDA and GGA calculations for the CaCO3aragonite polymorph,[7]and both have predictedvery close indirect½E LDAGðX!CÞ¼4:00eV and E GGAGðX!CÞ¼4:29eV and direct½E LDAGðC!CÞ¼4:01eV and E GGAGðC!CÞ¼4:27eV energy gaps,preventing a definitive conclusion on the aragonite band gap character.A comparison between the main energy band gaps of all three crystals, calcite,aragonite and vaterite,is shown in Table3.For both calcite and aragonite,the conduction and valence bands were shown to be veryflat and anisotropic at high symmetry points,leading to heavy carriers effective masses with a significant directional dependence.While it was not possible to conclude whether aragonite is a direct or indirect gap oxide by the LDA and GGA results,this is not the case for vaterite since the possibleindirect gap½E LDAGðC!SÞ¼5:00eV and E GGAGðC!SÞ¼5:19eV isconsiderably higher than the direct gap in the C point.Elec-tron and hole effective masses in CaCO3vaterite were obtained by selecting,in the valence band,the lines formed from the C point:C!Z,C!R.Other directions were not taken into account due to theflatness of the corresponding curves.At the conduction band,the chosen lines for effec-tive mass calculations start from the C and S points: C!Z,C!R,S!X,S!Y.In Table4we see the effec-tive mass values given in terms of the free electron mass, m0.Electron and hole effective mass reveal very little anisotropy for the selected directions,except for m h(C!R). Considering the band curves neglected due to excessive bandflatness,we must conclude that vaterite is anisotropic for carrier transport.The largest value of m h(C!R)is prob-ably due to the strong ionic bonding of electrons along this direction.We note also that the LDA and GGA estimates of the carriers effective masses are very close.Considering the lack of experimental results on the carriers effective mass of vaterite,a direct evaluation of our theoretical esti-mates is not possible,but they can stimulate experimental investigation.The optical absorption of CaCO3vaterite for polarized incident radiation and a polycrystalline sample was also calculated.Polarization vectors are denoted using the crys-tal axis directions.A100polarization direction,for instance,indicates that the incident light has its polariza-tion aligned to the crystal axis a.Five polarization planes were taken into account,as well as the absorption for a polycrystalline sample.In Fig.3the results obtained using the GGA exchange-correlation functional are depicted. The absorption intensity is very weak for energies smaller than6.0eV(but above the direct band gap).This is due to the selection rule that forbids p!pfirst order dipole transitions.For energies larger than 6.0eV,we witness the onset of a new absorption regime ruled by p!d tran-sitions.It is possible then,to distinguish two absorption regimes in thisfigure.First,for energies below6.0eV,the onset of absorption starting from5.0eV is dominated by Table3Energy band gaps for the three CaCO3crystalline structuresCrystal Energy gapsCalcite E G(D!Z)=4.4±0.35eV[5]Experimental=6.0±0.35eV[31]E G(F!Z)=5.06eV[6]Aragonite[7]E LDAGðX!CÞ¼4eV E LDAGðC!CÞ¼4:01eVE GGAGðX!CÞ¼4:29eV E GGAGðC!CÞ¼4:27eV Vaterite(thiswork)E LDAGðC!SÞ¼5eV E LDAGðC!CÞ¼4:68eVE GGAGðC!SÞ¼5:19eV E GGAGðC!CÞ¼5:07eVTable4Electron and hole effective mass of vateritem e(C!Z)m e(C!R)m e(S!X)m e(S!Y)m h(C!Z)m h(C!R) LDA 1.20 1.68 3.28 4.35 1.11 4.33 GGA 1.08 1.77 1.70 1.95 1.35 4.54 They are given in term of the free electron mass m0. 62S.K.Medeiros et al./Chemical Physics Letters435(2007)59–64transitions between mainly O-2p valence bands and mainly C-2p conduction bands.Above6.0eV,transitions between p-like valence states and Ca-3d related states prevail,with more pronounced peaks around6.5eV,6.66eV,6.98eV, 7.18eV,7.48eV,and7.85eV for all polarizations.Absorp-tion is stronger along the101and110polarization axis due to the strongest coupling of the electricfield to the electricdipoles formed by Ca2+and CO2À3ions in these directions.The polycrystalline sample exhibits an absorption pattern more similar to the observed for100and110polarizations.4.Concluding remarksIn conclusion,ab initio DFT calculations of the struc-tural,electronic(band structure and carriers effective masses),and optical properties of CaCO3vaterite were per-formed using the LDA and GGA exchange-correlation potentials.Although the calculated lattice parameters are in good agreement with experimental results,there is a dif-ference between experiment and calculations with respect to the spatial orientation of the plane defined by carbonate ions.Our results show that these carbonate ions are inclined by67°with respect to the ac plane.We hope this discrepancy will stimulate experimental investigations to obtain more accurate diffraction data to characterize vate-rite crystals.The electronic structure of vaterite reveals that the highest valence band and lowest conduction bands are formed mainly from contributions of O-2p and Ca-3d,C-2p states,respectively.The quantum chemicalfirst-princi-ples calculations allow us to suggest that CaCO3vaterite is a wide direct energy gap material,withE LDAGðC!CÞ¼4:68eV and E GGAGðC!CÞ¼5:07eV.Note that boththe LDA and GGA values we have calculated must be rec-ognized as estimates,being about20–30%smaller(we guess)than the actual vaterite energy gap,to be measured yet.We hope that our work will stimulate experimental work in this direction.Electron and hole effective masses were also obtained,but no confirmation of their values was possible due to the lack of experimental data.The intensity of optical absorption was predicted to be more pronounced only for energies larger than6.0eV.We high-light the usefulness of the results presented here pointing to two applications:(i)the investigation of the optical proper-ties of porous CaCO3vaterite[32,33];and(ii)the study of confinement features of excitons in Si@CaCO3(vaterite) and CaCO3(vaterite)@SiO2spherical quantum dots (QDs)[34],and hollow calcium carbonate vaterite nano-particles[35].Finally,a drawback of this Letter is that we have considered that vaterite is orthorhombic,while its crystalline state is today a matter of debate.However, work is in development by the present authors to obtain by quantum chemical calculations the structural,elec-tronic,and optical properties of vaterite considering a hex-agonal structure.AcknowledgementsThe authors acknowledge the comments of the anony-mous referees that have contributed strongly for the improvement of the Letter.V.N.F.and E.L.A.are senior researchers of the Brazilian National Research Council (CNPq),and would like to acknowledge thefinancial sup-port received from CNPq during the development of this work through project CT-ENERG504801/2004-0and project CNPq-Rede NanoBioestruturas555183/2005-0.E.W.S.C.was a visiting scientist at the Universidade Fed-eral do Ceara´.S.K.M.was sponsored by a graduate fel-lowship from CNPq at the Physics Department of the Universidade Federal do Rio Grande do Norte,andF.F.M.J.by a graduate fellowship from CAPES at the Physics Department of the Universidade Federal do Ceara´.References[1]K.M.Wilbur, A.S.M.Saleuddin,in:K.M.Wilbur, C.M.Yonge(Eds.),The Mollusca,vol.4,Academic Press Inc.,New York,1983,p.235,and references therein.[2]P.Malkaj,E.Dalas,Crystal Growth Des.4(2004)721.[3]X.Xu,J.T.Han,K.Cho,Chem.Mater.16(2004)1740.[4]P.K.Ajikumar,kshminarayanan,S.Valiyaveettil,CrystalGrowth Des.4(2004)331.[5]A.J.Skinner,J.P.FaFemina,H.J.F.Jansen,Am.Mineral.79(1994)205.[6]S.K.Medeiros,E.L.Albuquerque,F.F.Maia Jr.,E.W.S.Caetano,V.N.Freire,Chem.Phys.Lett.,submitted for publication.[7]S.K.Medeiros,E.L.Albuquerque,F.F.Maia Jr.,E.W.S.Caetano,V.N.Freire,Chem.Phys.Lett.430(2006)293.[8]H.J.Meyer,Angew.Chem.71(1959)678.[9]H.J.Meyer,Fortsch.Mineral.38(1960)186.[10]S.R.Kamhi,Acta Cryst.16(1963)770.[11]H.J.Meyer,Z.Kristallogr.128(1969)183.[12]F.Lippmann,Sedimentary Carbonate Materials,Springer,NewYork,1973.[13]G.Behrens,L.T.Kuhn,R.Ubic,A.H.Heuer,Spectrosc.Lett.28(1995)2953.[14]N.H.de Leeuw,S.C.Parker,J.Phys.Chem.B102(1998)2914. S.K.Medeiros et al./Chemical Physics Letters435(2007)59–6463[15]C.Gabrielli,R.Jaouhari,S.Joiret,G.Maurin,J.Raman Spectrosc.31(2000)497.[16]K.Winter,The influence of experimental parameters on the structuralvariation of calcium carbonate grown in the presence of mono-L-glutamic and mono-L-aspartic acid,Presented in The Australian X-ray Analytical Association Inc.Technical Meeting,24th November 2005.[17]J.P.Perdew,A.Zunger,Phys.Rev.B23(1981)5048.[18]D.M.Ceperley,B.J.Alder,Phys.Rev.Lett.45(1980)566.[19]J.P.Perdew,K.Burke,M.Ernzerhof,Phys.Rev.Lett.77(1996)3865.[20]M.D.Segall,P.L.D.Lindan,M.J.Probert,C.J.Pickard,P.J.Hasnip,S.J.Clark,M.C.Payne,J.Phys.:Cond.Matter.14(2002)2717.[21]J.Vanderbilt,Phys.Rev.B41(1990)7892.[22]H.J.Monkhorst,J.D.Pack,Phys.Rev.B13(1976)5188.[23]J.D.Pack,H.J.Monkhorst,Phys.Rev.B16(1977)1748.[24]B.G.Pfrommer,M.Cote,S.G.Louie,M.L.Cohen,put.Phys.131(1997)133.[25]G.J.Ackland,Phys.Rev.Lett.80(1998)2233.[26]M.D.Segall,C.J.Pickard,R.Shah,M.C.Payne,Mol.Phys.89(1996)571.[27]M.D.Segall,R.Shah,C.J.Pickard,M.C.Payne,Phys.Rev.B54(1996)16317.[28]R.G.WychoffCrystal Structures,vol.2,Interscience,New York,1954.[29]J.S.Lin,A.Qteish,M.C.Payne,V.Heine,Phys.Rev.B47(1993)4174.[30]S.G.Louie,S.Froyen,M.L.Cohen,Phys.Rev.B26(1982)1738.[31]D.R.Baer,D.L.Blanchard,Appl.Surf.Sci.72(1993)295.[32]S.-F.Chen,S.-H.Yu,J.Jiang,F.Li,Y.Liu,Chem.Mater.18(2006)115.[33]H.Co¨lfen,Curr.Opin.Coll.Interface Sci.8(2003)23.[34]S.Zhang,X.Li,Powder Technol.141(2004)75.[35]G.Hadiko,Y.S.Han,M.Fuji,M.Takahashi,Matter.Lett.59(2005)2519.64S.K.Medeiros et al./Chemical Physics Letters435(2007)59–64。
a r X i v :c o n d -m a t /0402497v 1 [c o n d -m a t .m t r l -s c i ] 19 F eb 2004First principles lattice dynamics of NaCoO 2Zhenyu Li,Jinlong Yang,∗J.G.Hou,and Qingshi ZhuLaboratory of Bond Selective Chemistry and Structure Research Laboratory,University of Science and Technology of China,Hefei,Anhui 230026,P.R.China (Dated:February 2,2008)Abstract We report first principles linear response calculations on NaCoO 2.Phonon frequencies and eigenvectors are obtained throughout the Brillouin zone for two geometries with different Na site occupancies.While most of the phonon modes are found to be unsensitive to the Na site occupancy,there are two modes dominated by out-of-plane vibrations of Na giving very different frequencies for different geometries.One of these two modes,the A 2u mode,is infrared-active,and can be used as a suitable sensor of Na distribution/ordering.The longitudinal-transverse splitting of the zone-center optical-mode frequencies,Born effective charges and the dielectric constants are also reported,showing considerable anisotropy.The calculated frequencies of Raman-active modes generally agree with the experimental values of corresponding Na de-intercalated and/or hydrated compounds,while it requires better experimental data to clarify the infrared-active mode frequencies.PACS numbers:63.20.Dj,71.20.Ps,74.25.KcI.INTRODUCTIONSince the discovery of the superconductivity in Na x CoO2·yH2O,1significant interest has been focused on this novel material,because there are some intriguing similarities between this cobalt oxide superconductor and copper oxide superconductors.2A num-ber of evidences in experiments suggest that the superconductivity in this material is unconventional.3,4,5Theoretical models such as resonating valence bond(RVB)6,7,8,9and spin-triplet superconductivity10,11,12,13with strong magnetic quantumfluctuations are also proposed.Although now it is a consensus that the cobalt oxide superconductor is not simply a BCS superconductor,some experiments still suggest the possible strong lattice coupling to the electronic density of states in this compound.14,15It is thus important to fully char-acterize the lattice dynamical properties of Na x CoO2·yH2O and related materials.While there are some Raman,16,17infrared,14,18and neutron scattering19experiments involving the lattice dynamics of the cobalt oxide compound,there is nofirst principles theoretical report found in the literature.In this article,we report a density functional perturbation theory(DFPT)20,21calculation on NaCoO2,the intrinsic(without hydration or de-intercalation of Na)insulating phase of the cobalt oxide superconductor.As shown in Figures1a and1b,NaCoO2has a hexagonal structure(space group#194,P63/mmc) consisting of CoO2layers of edge sharing CoO6octahedra and Na layers with two partly occupied Na Wickoffsites.To accommodate for the partial occupation of the two Na sites, we consider two geometries of NaCoO2with one site fully occupied and the other site empty, namely geometry A(Figure1a)with the2b site occupied and geometry B(Figure1b)with the2d site occupied.PUTATIONAL METHODThe present results have been obtained through the use of the ABINIT code,22a common project of the Universit Catholique de Louvain,Corning Incorporated,and other contribu-tors(URL ),based on pseudopotentials and plane waves.It relies on an efficient fast Fourier transform algorithm for the conversion of wave functions between real and reciprocal spaces,on the adaptation to afixed potential of the band-by-band con-jugate gradient method23,and on a potential-based conjugate-gradient algorithm for thedetermination of the self-consistent potential.24Technical details on the computation of re-sponses to atomic displacements and homogeneous electricfields can be found in Ref.21, and the subsequent computation of dynamical matrices,Born effective charges,dielectric permittivity tensors,and interatomic force constants was described in Ref.25.Troullier-Martins norm conserving pseudopotentials26are used,with Teter parametrization27of the Ceperley-Alder exchange-correlation potential.The kinetic energy cutoffof the plane wave bases is50hartree.A uniform6×6×2k-point mesh is used in Brillouin-zone integrations,which gives well converged total energies and phonon frequencies.III.RESULTS AND DISCUSSIONA.Geometries and Electronic StructuresAs afirst step,we optimize the geometry and calculate the corresponding electronic structure.During the geometry optimization,the cell parameters arefixed to the experi-mental values of Na0.74CoO2(a=2.84˚A,c=10.811˚A).28We notice that neutron scattering experiments28,29,30for samples with different Na concentrations give similar cell parameters. Considering the space group symmetry,the only freedom to be relaxed is the Co-O layer space d,which turns out to be1.897and1.899˚A for geometries A and B respectively.The electronic band structure and density of states(DOS)based on the optimized structure of geometry A are shown in Figure1c,a gap of0.94eV(from0.60to1.54eV)between the valence band and the conduction band can be clearly identified.Little difference on the electronic structure of the two geometries is found.This similarity strongly indicates that Na contributes to the electronic structure only by doping electrons to the CoO layers.B.Phonon DispersionsTo map the phonon dispersion curves throughout the Brillouin zone,the dynamical ma-trices are obtained on a uniform6×6×2grid of q points,and real-space force constants are then found by Fourier transform of the dynamical matrices.The dynamical matrix at an arbitrary wave vector q can then be computed by an inverse Fourier transform.The acoustic sum rule is applied to force the three acoustic phonon frequencies atΓequal to zerostrictly as being implied by the translation symmetry.The calculated phonon dispersions and corresponding DOS for geometries A and B are shown in Figure2.We seefirst that both structures are stable.Second,we notice that the phonon modes are separated to two groups in frequency,namely the soft group with frequencies below than about400cm−1 and the hard group with frequencies between450to650cm−1.As a whole,the phonon dispersions for the two geometries with different Na site occupancies are similar,but there are also some minor differences exist for the two geometries in the phonon band structure, unlike the nearly identical electronic band structures.First,as we will discussed in detail below,there are some soft modes giving significant different frequencies near zone center for the two geometries.Secondly,we notice that the dispersion alongΓ−A for geometry B is larger than that for geometry A.C.zone-center phonon modesThe zone-center phonon modes are of special importance,since they can be obtained by various of experimental methods.In Table I,we list the frequencies,symmetries and vi-bration modes of the zone-center phonons.The triply-degenerated acoustic mode with zero frequency is not listed.The frequency differences between the two geometries are generally small,especially for the Raman-active phonon modes.However there are an infrared-active A2u mode and a silent B1g mode giving very different frequencies for geometry A and ge-ometry B,and both of these two modes are mainly contributed by the Na out-of-plane vibrations.The other A2u and B1g modes(modes7and14)also involve the out-of-plane vibrations of Na,but we notice that the vibrations of Co and/or O in these two modes are much stronger than or at least comparable to the Na vibrations.Therefore they give relatively small frequency differences for different Na site occupancies.Wefind that the frequency difference of Raman-active A1g mode of O out-of-plane vibrations is rather small, in contrast to the conclusion of the shell model calculations by Lemmens et al..17They found a pronounced frequency dependence of the A1g mode,and argued that it would be used as a very susceptible sensor of Na distribution/ordering.However,in ourfirst principles cal-culations,the infrared-active A2u mode(node6)in the soft group may be more suitable to act as the sensor mode.Another difference betweenΓphonon modes of the two geometries is the frequency order of thefirst E2g and B2u modes(modes1and10).Since the systems we studied here are polar materials,we also get the longitudinal-transverse optical mode(LO-TO)splittings by considering response to the macroscopic electricfield.As shown in Table I,the LO-TO splittings are found only in the infrared-active2E1u+2A2u modes.The biggest LO-TO splitting is found in the hard E1u mode, with∆ω=ωLO−ωT O equal to52.4and48.6cm−1for geometries A and B respectively. Contrastingly,the soft E1u mode is split by only5.1and5.9cm−1respectively.Both these two E1u modes are”layer sliding”modes,and the hard one is mainly contributed by the Co layers sliding against the O layers.While for the soft E1u mode,the sliding of the Na layers is about an order stronger than the relative sliding of Co and O layers.Therefore our results follow the trend that displacements modulating the”covalent”bonding produce the largest LO-TO splitting,as being pointed out by Lee et al.31.We also get the Born(dynamical)effective charge tensor.In hexagonal symmetry,it is diagonal and reduces to two values Z xx=Z yy=Z//and Z zz=Z⊥.As listed in Table II,these charges show considerable anisotropy in NaCoO2.We notice that Z⊥is much larger than Z//for Na,which is consistent with the fact that the LO-TO splitting of the soft in-plane E1u mode is much smaller than that of the soft out-of-plane A2u mode,with both modes being dominated by the vibrations of Na.Despite the anisotropy,we define thefor Na,Co,and O, average effective charge¯Z as12respectively.The dynamical charges of Co and O are both smaller than their nominal ionic charges.In addition,the static electronic dielectric constants are calculated to beǫ//∞=9.72, =4.34for geometry A andǫ//∞=9.78,ǫ⊥∞=4.27for geometry B respectively.ǫ⊥∞There are some experimental data on the zone-center phonon frequencies of Na x CoO2 or Na x CoO2·yH2O.In principle,these frequencies can not be directly compared with our results,since the systems we studied here is NaCoO2.But as we have seen,the Na site occupancy may not affect the frequencies much,especially for the high-frequency in-plane vibration modes.So we also list the experimental Raman and infrared frequencies in Table I for reference.For NaCoO2with only one Na site occupied,symmetry analysis leads to the Raman-active modesΓRaman=A1g+E1g+2E2g and the infrared-active optical phonon modesΓIR=2A2u+2E1u.The experimental Raman frequencies listed in Table I generally agree with our calculated frequencies well.Among the four infrared-active modes,only the frequency of the hard E1u mode is reported in experiments.The frequency of Na0.57CoO2 reported by Lupi et al.14is near our calculated TO frequency of E1u mode.It is very strangethat infrared spectra by Wang et al.18gave four different infrared frequencies for doubly-degenerated in-plane E1u mode in metallic Na0.7CoO2,where no LO-TO splitting exists. There should be only two frequencies even considering the partial Na occupation of the2b and2d site at the same time.We argue that part of these frequencies may be contributed by defects or grain surface,since the IR reflectivity data are very sensitive to surface treatment. In a word,the IR modes remain to be verified and understood.The peak of optical phonon at161.3cm−1(20meV)in neutron inelastic scattering experiment19is near the frequency of the soft E2g mode(mode1)here.IV.CONCLUSIONIn conclusion,we have calculated the lattice dynamics of NaCoO2.Most of the phonon modes are only little affected by the Na site occupancy,and the calculated zone-center phonon frequencies generally agree with those by recent Raman and neutron scattering spectra on corresponding Na de-intercalated and/or hydrated compounds well.Therefore our calculations may also be helpful for understanding the lattice dynamics of the cobalt oxide superconductor.However,to clarify the phonon contribution in the superconductivity requires much further work on the lattice dynamics and electron-phonon interaction of the superconductor itself.Experimental data on infrared-active modes are difficult to compare with our calculated results now.In addition to phonon frequencies,other useful data such as Born effective charges and dielectric constants of NaCoO2are also presented.AcknowledgmentsThis work is partially supported by the National Project for the Development of Key Fundamental Sciences in China(G1999075305,G2001CB3095),by the National Natural Science Foundation of China(50121202,20025309,10074058),by the Foundation of Ministry of Education of China,and by ICTS,CAS.∗Corresponding author.E-mail:jlyang@1K.Takada,H.Sakural,E.Takayama-Muromachi,F.Izumi,R.A.Dilanian,and T.Sasaki, Nature422,53(2003)2J.V.Badding,Nature Materials2,208(2003)3H.Sakurai,K.Takada,S.Yoshii et al.,Phys.Rev.B68,312507(2003)4K.Ishida,Y.Ihara,Y.Maeno et al.,arXiv:cond-mat/0308506.5W.Higemoto,K.Ohishi,A.Koda et al.,arXiv:cond-mat/0310324.6G.Baskaran,Phys.Rev.Lett.91,97003(2003)7 B.Kumar and B.S.Shastry,Phys.Rev.B68,104508(2003)8 C.Honerkamp,Phys.Rev.B68,104510(2003)9Q.-H.Wang,D.-H.Lee,and P.A.Lee,arXiv:cond-mat/030437710 A.Tanaka and X.Hu,Phys.Rev.Lett.91,257006(2003)11 D.J.Singh,Phys.Rev.B68,20503(2003)12H.Ikeda,Y.Nisikawa and K.Yamada,arXiv:cond-mat/030847213Y.Tanaka,Y.Yanase,and M.Ogata,arXiv:cond-mat/031126614S.Lupi,M.Ortolani,and P.Calvani,arXiv:cond-mat/031251215 ne, D.N.Argyriou, A.Chemseddine,N.Aliouane,J.Veira,and D.Alber,arXiv:cond-mat/0401273.16M.N.Iliev,A.P.Litvinchuk,R.L.Meng et al.,Phys.C402,239(2004)17P.Lemmens,V.Gnezdilov,N.N.Kovaleva et al.,arXiv:cond-mat/030918618N.L.Wang,P.Zheng,D.Wu et al.,arXiv:cond-mat/031263019 A.T.Boothroyd,R.Coldea,D.A.Tennant,arXiv:cond-mat/031258920S.Baroni,S.de Gironcoli,A.Dal Corso,and P.Gianozzi,Rev.Mod.Phys.73,515(2001)21X.Gonze,Phys.Rev.B55,10337(1997)22X.Gonze,J.-M.Beuken,R.Caracas et al.,Comput.Mater.Sci.25,478(2002)23M.C.Payne,M.P.Teter,D.C.Allan et al.,Rev.Mod.Phys.64,1045(1992)24X.Gonze,Phys.Rev.B54,4383(1996)25X.Gonze and C.Lee,Phys.Rev.B55,10355(1997)26N.Troullier and J.L.Martins,Phys.Rev.B43,1993(1991)27See the Appendix of S.Goedecker,M.Teter,and J.Hutter,Phys.Rev.B54,1703(1996)28R.J.Balsys and R.L.Davis,Solid State Ionics93,279(1996)29J.D.Jorgensen,M.Avdeev,D.G.Hinks et al.,arXiv:cond-mat/030762730J.W.Lynn,Q.Huang,C.M.Broun et al.,arXiv:cond-mat/0307263 31K.-W.Lee and W.E.Pickett,Phys.Rev.B68,85308(2003)TABLE I:Zone center optical phonon modes of NaCoO2.The three groups separated by horizontal lines are Raman-active,infrared-active and silent modes respectively.The involved atoms in each mode and their vibration directions(//for in-plane,⊥for out-of-plane)are listed.The atomic displacements in parentheses are obtained by dividing the normalized eigenvector components by the square root of the atomic mass,and with the unit of10−2for convenience of reading.These displacements are only refer to geometry A,since they are very similar to those of geometry B.f A and f B stand for calculated frequencies for geometries A and B respectively,and f EXP stands for experimental frequencies.For the four infrared-active modes,both the LO and TO frequencies are listed.Mode Symmetry Vibrations f A(cm−1)f B(cm−1)f EXP(cm−1)5E1u Na//(0.31)Co//(0.09)O//(0.05)201.2216.5206.3220.46A2u Na⊥(0.31)Co⊥(0.09)O⊥(0.05)397.5337.0418.1361.67A2u Na⊥(0.03)Co⊥(0.12)O⊥(0.24)569.8566.5618.7615.18E1u Na//(0.03)Co//(0.12)O//(0.24)586.7590.0570e,(505,530,560,575)f638.5638.6TABLE II:The Born effective charges of NaCoO2atoms.See text for the definition of Z//,Z⊥and¯Z.Superscripts A and B refer to data for geometries A and B.Z//Z⊥¯ZZ B(Na)0.85 1.40 1.04Z B(Co) 2.410.78 1.86Z B(O)-1.63-1.09-1.45FIG.1:(color online)(a)Structure model of geometry A.The smallest balls represent cobalt atoms,and the balls in red bonding to Co are oxygen atoms.Between CoO layers,there exist sodium(big purple ball)layers.(b)Structure model of geometry B.(c)Electronic band structure and density of states of NaCoO2.11FIG.2:Phonon band structure and density of states of NaCoO2for(a)geometry A and(b) geometry B respectively.(a)(b)200400600Frequency(cm-1)ΓΓM K A DOS200400600Frequency(cm-1)MΓKΓA DOS12。
